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Copyright 2011, AUTHORS.txt (http://jqueryui.com/about) + * Dual licensed under the MIT or GPL Version 2 licenses. + * http://jquery.org/license + * + * http://docs.jquery.com/UI/Theming/API + */ + +/* Layout helpers +----------------------------------*/ +.ui-helper-hidden { display: none; } +.ui-helper-hidden-accessible { position: absolute !important; clip: rect(1px 1px 1px 1px); clip: rect(1px,1px,1px,1px); } +.ui-helper-reset { margin: 0; padding: 0; border: 0; outline: 0; line-height: 1.3; text-decoration: none; font-size: 100%; list-style: none; } +.ui-helper-clearfix:after { content: "."; display: block; height: 0; clear: both; visibility: hidden; } +.ui-helper-clearfix { display: inline-block; } +/* required comment for clearfix to work in Opera \*/ +* html .ui-helper-clearfix { height:1%; } +.ui-helper-clearfix { display:block; } +/* end clearfix */ +.ui-helper-zfix { width: 100%; height: 100%; top: 0; left: 0; position: absolute; opacity: 0; filter:Alpha(Opacity=0); } + + +/* Interaction Cues +----------------------------------*/ +.ui-state-disabled { cursor: default !important; } + + +/* Icons +----------------------------------*/ + +/* states and images */ +.ui-icon { display: block; text-indent: -99999px; overflow: hidden; background-repeat: no-repeat; } + + +/* Misc visuals +----------------------------------*/ + +/* Overlays */ +.ui-widget-overlay { position: absolute; top: 0; left: 0; width: 100%; height: 100%; } + + +/* + * jQuery UI CSS Framework 1.8.16 + * + * Copyright 2011, AUTHORS.txt (http://jqueryui.com/about) + * Dual licensed under the MIT or GPL Version 2 licenses. + * http://jquery.org/license + * + * http://docs.jquery.com/UI/Theming/API + * + * To view and modify this theme, visit http://jqueryui.com/themeroller/?ffDefault=Helvetica,%20Arial,%20sans-serif&fwDefault=bold&fsDefault=1.1em&cornerRadius=2px&bgColorHeader=7fbcfd&bgTextureHeader=03_highlight_soft.png&bgImgOpacityHeader=50&borderColorHeader=dae5c9&fcHeader=031634&iconColorHeader=031634&bgColorContent=ffffff&bgTextureContent=01_flat.png&bgImgOpacityContent=75&borderColorContent=dae5c9&fcContent=031634&iconColorContent=adcc80&bgColorDefault=7fbcdf&bgTextureDefault=03_highlight_soft.png&bgImgOpacityDefault=100&borderColorDefault=dae5c9&fcDefault=7a994c&iconColorDefault=adcc80&bgColorHover=bddeff&bgTextureHover=03_highlight_soft.png&bgImgOpacityHover=25&borderColorHover=7fbcdf&fcHover=7a994c&iconColorHover=adcc80&bgColorActive=023063&bgTextureActive=02_glass.png&bgImgOpacityActive=65&borderColorActive=dae5c9&fcActive=dae5c9&iconColorActive=454545&bgColorHighlight=ffffff&bgTextureHighlight=01_flat.png&bgImgOpacityHighlight=55&borderColorHighlight=cccccc&fcHighlight=444444&iconColorHighlight=adcc80&bgColorError=ffffff&bgTextureError=01_flat.png&bgImgOpacityError=55&borderColorError=fa720a&fcError=222222&iconColorError=fa720a&bgColorOverlay=eeeeee&bgTextureOverlay=01_flat.png&bgImgOpacityOverlay=0&opacityOverlay=80&bgColorShadow=aaaaaa&bgTextureShadow=01_flat.png&bgImgOpacityShadow=0&opacityShadow=60&thicknessShadow=4px&offsetTopShadow=-4px&offsetLeftShadow=-4px&cornerRadiusShadow=0px + */ + + +/* Component containers +----------------------------------*/ +.ui-widget { font-family: Helvetica, Arial, sans-serif; font-size: 1.1em; } +.ui-widget .ui-widget { font-size: 1em; } +.ui-widget input, .ui-widget select, .ui-widget textarea, .ui-widget button { font-family: Helvetica, Arial, sans-serif; font-size: 1em; } +.ui-widget-content { border: 1px solid #dae5c9; background: #ffffff url(images/ui-bg_flat_75_ffffff_40x100.png) 50% 50% repeat-x; color: #031634; } +.ui-widget-content a { color: #031634; } +.ui-widget-header { border: 1px solid #dae5c9; background: #7fbcfd url(images/ui-bg_highlight-soft_50_7fbcfd_1x100.png) 50% 50% repeat-x; color: #031634; font-weight: bold; } +.ui-widget-header a { color: #031634; } + +/* Interaction states +----------------------------------*/ +.ui-state-default, .ui-widget-content .ui-state-default, .ui-widget-header .ui-state-default { border: 1px solid #dae5c9; background: #7fbcdf url(images/ui-bg_highlight-soft_100_7fbcdf_1x100.png) 50% 50% repeat-x; font-weight: bold; color: #7a994c; } +.ui-state-default a, .ui-state-default a:link, .ui-state-default a:visited { color: #7a994c; text-decoration: none; } +.ui-state-hover, .ui-widget-content .ui-state-hover, .ui-widget-header .ui-state-hover, .ui-state-focus, .ui-widget-content .ui-state-focus, .ui-widget-header .ui-state-focus { border: 1px solid #7fbcdf; background: #bddeff url(images/ui-bg_highlight-soft_25_bddeff_1x100.png) 50% 50% repeat-x; font-weight: bold; color: #7a994c; } +.ui-state-hover a, .ui-state-hover a:hover { color: #7a994c; text-decoration: none; } +.ui-state-active, .ui-widget-content .ui-state-active, .ui-widget-header .ui-state-active { border: 1px solid #dae5c9; background: #023063 url(images/ui-bg_glass_65_023063_1x400.png) 50% 50% repeat-x; font-weight: bold; color: #dae5c9; } +.ui-state-active a, .ui-state-active a:link, .ui-state-active a:visited { color: #dae5c9; text-decoration: none; } +.ui-widget :active { outline: none; } + +/* Interaction Cues +----------------------------------*/ +.ui-state-highlight, .ui-widget-content .ui-state-highlight, .ui-widget-header .ui-state-highlight {border: 1px solid #cccccc; background: #ffffff url(images/ui-bg_flat_55_ffffff_40x100.png) 50% 50% repeat-x; color: #444444; } +.ui-state-highlight a, .ui-widget-content .ui-state-highlight a,.ui-widget-header .ui-state-highlight a { color: #444444; } +.ui-state-error, .ui-widget-content .ui-state-error, .ui-widget-header .ui-state-error {border: 1px solid #fa720a; background: #ffffff url(images/ui-bg_flat_55_ffffff_40x100.png) 50% 50% repeat-x; color: #222222; } +.ui-state-error a, .ui-widget-content .ui-state-error a, .ui-widget-header .ui-state-error a { color: #222222; } +.ui-state-error-text, .ui-widget-content .ui-state-error-text, .ui-widget-header .ui-state-error-text { color: #222222; } +.ui-priority-primary, .ui-widget-content .ui-priority-primary, .ui-widget-header .ui-priority-primary { font-weight: bold; } +.ui-priority-secondary, .ui-widget-content .ui-priority-secondary, .ui-widget-header .ui-priority-secondary { opacity: .7; filter:Alpha(Opacity=70); font-weight: normal; } +.ui-state-disabled, .ui-widget-content .ui-state-disabled, .ui-widget-header .ui-state-disabled { opacity: .35; filter:Alpha(Opacity=35); background-image: none; } + +/* Icons +----------------------------------*/ + +/* states and images */ +.ui-icon { width: 16px; height: 16px; background-image: url(images/ui-icons_adcc80_256x240.png); } +.ui-widget-content .ui-icon {background-image: url(images/ui-icons_adcc80_256x240.png); } +.ui-widget-header .ui-icon {background-image: url(images/ui-icons_031634_256x240.png); } +.ui-state-default .ui-icon { background-image: url(images/ui-icons_adcc80_256x240.png); } +.ui-state-hover .ui-icon, .ui-state-focus .ui-icon {background-image: url(images/ui-icons_adcc80_256x240.png); } +.ui-state-active .ui-icon {background-image: url(images/ui-icons_454545_256x240.png); } +.ui-state-highlight .ui-icon {background-image: url(images/ui-icons_adcc80_256x240.png); } +.ui-state-error .ui-icon, .ui-state-error-text .ui-icon {background-image: url(images/ui-icons_fa720a_256x240.png); } + +/* positioning */ +.ui-icon-carat-1-n { background-position: 0 0; } +.ui-icon-carat-1-ne { background-position: -16px 0; } +.ui-icon-carat-1-e { background-position: -32px 0; } +.ui-icon-carat-1-se { background-position: -48px 0; } +.ui-icon-carat-1-s { background-position: -64px 0; } +.ui-icon-carat-1-sw { background-position: -80px 0; } +.ui-icon-carat-1-w { background-position: -96px 0; } +.ui-icon-carat-1-nw { background-position: -112px 0; } +.ui-icon-carat-2-n-s { background-position: -128px 0; } +.ui-icon-carat-2-e-w { background-position: -144px 0; } +.ui-icon-triangle-1-n { background-position: 0 -16px; } +.ui-icon-triangle-1-ne { background-position: -16px -16px; } +.ui-icon-triangle-1-e { background-position: -32px -16px; } +.ui-icon-triangle-1-se { background-position: -48px -16px; } +.ui-icon-triangle-1-s { background-position: -64px -16px; } +.ui-icon-triangle-1-sw { background-position: -80px -16px; } +.ui-icon-triangle-1-w { background-position: -96px -16px; } +.ui-icon-triangle-1-nw { background-position: -112px -16px; } +.ui-icon-triangle-2-n-s { background-position: -128px -16px; } +.ui-icon-triangle-2-e-w { background-position: -144px -16px; } +.ui-icon-arrow-1-n { background-position: 0 -32px; } +.ui-icon-arrow-1-ne { background-position: -16px -32px; } +.ui-icon-arrow-1-e { background-position: -32px -32px; } +.ui-icon-arrow-1-se { background-position: -48px -32px; } +.ui-icon-arrow-1-s { background-position: -64px -32px; } +.ui-icon-arrow-1-sw { background-position: -80px -32px; } +.ui-icon-arrow-1-w { background-position: -96px -32px; } +.ui-icon-arrow-1-nw { background-position: -112px -32px; } +.ui-icon-arrow-2-n-s { background-position: -128px -32px; } +.ui-icon-arrow-2-ne-sw { background-position: -144px -32px; } +.ui-icon-arrow-2-e-w { background-position: -160px -32px; } +.ui-icon-arrow-2-se-nw { background-position: -176px -32px; } +.ui-icon-arrowstop-1-n { background-position: -192px -32px; } +.ui-icon-arrowstop-1-e { background-position: -208px -32px; } +.ui-icon-arrowstop-1-s { background-position: -224px -32px; } +.ui-icon-arrowstop-1-w { background-position: -240px -32px; } +.ui-icon-arrowthick-1-n { background-position: 0 -48px; } +.ui-icon-arrowthick-1-ne { background-position: -16px -48px; } +.ui-icon-arrowthick-1-e { background-position: -32px -48px; } +.ui-icon-arrowthick-1-se { background-position: -48px -48px; } +.ui-icon-arrowthick-1-s { background-position: -64px -48px; } +.ui-icon-arrowthick-1-sw { background-position: -80px -48px; } +.ui-icon-arrowthick-1-w { background-position: -96px -48px; } +.ui-icon-arrowthick-1-nw { background-position: -112px -48px; } +.ui-icon-arrowthick-2-n-s { background-position: -128px -48px; } +.ui-icon-arrowthick-2-ne-sw { background-position: -144px -48px; } +.ui-icon-arrowthick-2-e-w { background-position: -160px -48px; } +.ui-icon-arrowthick-2-se-nw { background-position: -176px -48px; } +.ui-icon-arrowthickstop-1-n { background-position: -192px -48px; } +.ui-icon-arrowthickstop-1-e { background-position: -208px -48px; } +.ui-icon-arrowthickstop-1-s { background-position: -224px -48px; } +.ui-icon-arrowthickstop-1-w { background-position: -240px -48px; } +.ui-icon-arrowreturnthick-1-w { background-position: 0 -64px; } +.ui-icon-arrowreturnthick-1-n { background-position: -16px -64px; } +.ui-icon-arrowreturnthick-1-e { background-position: -32px -64px; } +.ui-icon-arrowreturnthick-1-s { background-position: -48px -64px; } +.ui-icon-arrowreturn-1-w { background-position: -64px -64px; } +.ui-icon-arrowreturn-1-n { background-position: -80px -64px; } +.ui-icon-arrowreturn-1-e { background-position: -96px -64px; } +.ui-icon-arrowreturn-1-s { background-position: -112px -64px; } +.ui-icon-arrowrefresh-1-w { background-position: -128px -64px; } +.ui-icon-arrowrefresh-1-n { background-position: -144px -64px; } +.ui-icon-arrowrefresh-1-e { background-position: -160px -64px; } +.ui-icon-arrowrefresh-1-s { background-position: -176px -64px; } +.ui-icon-arrow-4 { background-position: 0 -80px; } +.ui-icon-arrow-4-diag { background-position: -16px -80px; } +.ui-icon-extlink { background-position: -32px -80px; } +.ui-icon-newwin { background-position: -48px -80px; } +.ui-icon-refresh { background-position: -64px -80px; } +.ui-icon-shuffle { background-position: -80px -80px; } +.ui-icon-transfer-e-w { background-position: -96px -80px; } +.ui-icon-transferthick-e-w { background-position: -112px -80px; } +.ui-icon-folder-collapsed { background-position: 0 -96px; } +.ui-icon-folder-open { background-position: -16px -96px; } +.ui-icon-document { background-position: -32px -96px; } +.ui-icon-document-b { background-position: -48px -96px; } +.ui-icon-note { background-position: -64px -96px; } +.ui-icon-mail-closed { background-position: -80px -96px; } +.ui-icon-mail-open { background-position: -96px -96px; } +.ui-icon-suitcase { background-position: -112px -96px; } +.ui-icon-comment { background-position: -128px -96px; } +.ui-icon-person { background-position: -144px -96px; } +.ui-icon-print { background-position: -160px -96px; } +.ui-icon-trash { background-position: -176px -96px; } +.ui-icon-locked { background-position: -192px -96px; } +.ui-icon-unlocked { background-position: -208px -96px; } +.ui-icon-bookmark { background-position: -224px -96px; } +.ui-icon-tag { background-position: -240px -96px; } +.ui-icon-home { background-position: 0 -112px; } +.ui-icon-flag { background-position: -16px -112px; } +.ui-icon-calendar { background-position: -32px -112px; } +.ui-icon-cart { background-position: -48px -112px; } +.ui-icon-pencil { background-position: -64px -112px; } +.ui-icon-clock { background-position: -80px -112px; } +.ui-icon-disk { background-position: -96px -112px; } +.ui-icon-calculator { background-position: -112px -112px; } +.ui-icon-zoomin { background-position: -128px -112px; } +.ui-icon-zoomout { background-position: -144px -112px; } +.ui-icon-search { background-position: -160px -112px; } +.ui-icon-wrench { background-position: -176px -112px; } +.ui-icon-gear { background-position: -192px -112px; } +.ui-icon-heart { background-position: -208px -112px; } +.ui-icon-star { background-position: -224px -112px; } +.ui-icon-link { background-position: -240px -112px; } +.ui-icon-cancel { background-position: 0 -128px; } +.ui-icon-plus { background-position: -16px -128px; } +.ui-icon-plusthick { background-position: -32px -128px; } +.ui-icon-minus { background-position: -48px -128px; } +.ui-icon-minusthick { background-position: -64px -128px; } +.ui-icon-close { background-position: -80px -128px; } +.ui-icon-closethick { background-position: -96px -128px; } +.ui-icon-key { background-position: -112px -128px; } +.ui-icon-lightbulb { background-position: -128px -128px; } +.ui-icon-scissors { background-position: -144px -128px; } +.ui-icon-clipboard { background-position: -160px -128px; } +.ui-icon-copy { background-position: -176px -128px; } +.ui-icon-contact { background-position: -192px -128px; } +.ui-icon-image { background-position: -208px -128px; } +.ui-icon-video { background-position: -224px -128px; } +.ui-icon-script { background-position: -240px -128px; } +.ui-icon-alert { background-position: 0 -144px; } +.ui-icon-info { background-position: -16px -144px; } +.ui-icon-notice { background-position: -32px -144px; } +.ui-icon-help { background-position: -48px -144px; } +.ui-icon-check { background-position: -64px -144px; } +.ui-icon-bullet { background-position: -80px -144px; } +.ui-icon-radio-off { background-position: -96px -144px; } +.ui-icon-radio-on { background-position: -112px -144px; } +.ui-icon-pin-w { background-position: -128px -144px; } +.ui-icon-pin-s { background-position: -144px -144px; } +.ui-icon-play { background-position: 0 -160px; } +.ui-icon-pause { background-position: -16px -160px; } +.ui-icon-seek-next { background-position: -32px -160px; } +.ui-icon-seek-prev { background-position: -48px -160px; } +.ui-icon-seek-end { background-position: -64px -160px; } +.ui-icon-seek-start { background-position: -80px -160px; } +/* ui-icon-seek-first is deprecated, use ui-icon-seek-start instead */ +.ui-icon-seek-first { background-position: -80px -160px; } +.ui-icon-stop { background-position: -96px -160px; } +.ui-icon-eject { background-position: -112px -160px; } +.ui-icon-volume-off { background-position: -128px -160px; } +.ui-icon-volume-on { background-position: -144px -160px; } +.ui-icon-power { background-position: 0 -176px; } +.ui-icon-signal-diag { background-position: -16px -176px; } +.ui-icon-signal { background-position: -32px -176px; } +.ui-icon-battery-0 { background-position: -48px -176px; } +.ui-icon-battery-1 { background-position: -64px -176px; } +.ui-icon-battery-2 { background-position: -80px -176px; } +.ui-icon-battery-3 { background-position: -96px -176px; } +.ui-icon-circle-plus { background-position: 0 -192px; } +.ui-icon-circle-minus { background-position: -16px -192px; } +.ui-icon-circle-close { background-position: -32px -192px; } +.ui-icon-circle-triangle-e { background-position: -48px -192px; } +.ui-icon-circle-triangle-s { background-position: -64px -192px; } +.ui-icon-circle-triangle-w { background-position: -80px -192px; } +.ui-icon-circle-triangle-n { background-position: -96px -192px; } +.ui-icon-circle-arrow-e { background-position: -112px -192px; } +.ui-icon-circle-arrow-s { background-position: -128px -192px; } +.ui-icon-circle-arrow-w { background-position: -144px -192px; } +.ui-icon-circle-arrow-n { background-position: -160px -192px; } +.ui-icon-circle-zoomin { background-position: -176px -192px; } +.ui-icon-circle-zoomout { background-position: -192px -192px; } +.ui-icon-circle-check { background-position: -208px -192px; } +.ui-icon-circlesmall-plus { background-position: 0 -208px; } +.ui-icon-circlesmall-minus { background-position: -16px -208px; } +.ui-icon-circlesmall-close { background-position: -32px -208px; } +.ui-icon-squaresmall-plus { background-position: -48px -208px; } +.ui-icon-squaresmall-minus { background-position: -64px -208px; } +.ui-icon-squaresmall-close { background-position: -80px -208px; } +.ui-icon-grip-dotted-vertical { background-position: 0 -224px; } +.ui-icon-grip-dotted-horizontal { background-position: -16px -224px; } +.ui-icon-grip-solid-vertical { background-position: -32px -224px; } +.ui-icon-grip-solid-horizontal { background-position: -48px -224px; } +.ui-icon-gripsmall-diagonal-se { background-position: -64px -224px; } +.ui-icon-grip-diagonal-se { background-position: -80px -224px; } + + +/* Misc visuals +----------------------------------*/ + +/* Corner radius */ +.ui-corner-all, .ui-corner-top, .ui-corner-left, .ui-corner-tl { -moz-border-radius-topleft: 2px; -webkit-border-top-left-radius: 2px; -khtml-border-top-left-radius: 2px; border-top-left-radius: 2px; } +.ui-corner-all, .ui-corner-top, .ui-corner-right, .ui-corner-tr { -moz-border-radius-topright: 2px; -webkit-border-top-right-radius: 2px; -khtml-border-top-right-radius: 2px; border-top-right-radius: 2px; } +.ui-corner-all, .ui-corner-bottom, .ui-corner-left, .ui-corner-bl { -moz-border-radius-bottomleft: 2px; -webkit-border-bottom-left-radius: 2px; -khtml-border-bottom-left-radius: 2px; border-bottom-left-radius: 2px; } +.ui-corner-all, .ui-corner-bottom, .ui-corner-right, .ui-corner-br { -moz-border-radius-bottomright: 2px; -webkit-border-bottom-right-radius: 2px; -khtml-border-bottom-right-radius: 2px; border-bottom-right-radius: 2px; } + +/* Overlays */ +.ui-widget-overlay { background: #eeeeee url(images/ui-bg_flat_0_eeeeee_40x100.png) 50% 50% repeat-x; opacity: .80;filter:Alpha(Opacity=80); } +.ui-widget-shadow { margin: -4px 0 0 -4px; padding: 4px; background: #aaaaaa url(images/ui-bg_flat_0_aaaaaa_40x100.png) 50% 50% repeat-x; opacity: .60;filter:Alpha(Opacity=60); -moz-border-radius: 0px; -khtml-border-radius: 0px; -webkit-border-radius: 0px; border-radius: 0px; }/* + * jQuery UI Resizable 1.8.16 + * + * Copyright 2011, AUTHORS.txt (http://jqueryui.com/about) + * Dual licensed under the MIT or GPL Version 2 licenses. + * http://jquery.org/license + * + * http://docs.jquery.com/UI/Resizable#theming + */ +.ui-resizable { position: relative;} +.ui-resizable-handle { position: absolute;font-size: 0.1px;z-index: 99999; display: block; } +.ui-resizable-disabled .ui-resizable-handle, .ui-resizable-autohide .ui-resizable-handle { display: none; } +.ui-resizable-n { cursor: n-resize; height: 7px; width: 100%; top: -5px; left: 0; } +.ui-resizable-s { cursor: s-resize; height: 7px; width: 100%; bottom: -5px; left: 0; } +.ui-resizable-e { cursor: e-resize; width: 7px; right: -5px; top: 0; height: 100%; } +.ui-resizable-w { cursor: w-resize; width: 7px; left: -5px; top: 0; height: 100%; } +.ui-resizable-se { cursor: se-resize; width: 12px; height: 12px; right: 1px; bottom: 1px; } +.ui-resizable-sw { cursor: sw-resize; width: 9px; height: 9px; left: -5px; bottom: -5px; } +.ui-resizable-nw { cursor: nw-resize; width: 9px; height: 9px; left: -5px; top: -5px; } +.ui-resizable-ne { cursor: ne-resize; width: 9px; height: 9px; right: -5px; top: -5px;}/* + * jQuery UI Selectable 1.8.16 + * + * Copyright 2011, AUTHORS.txt (http://jqueryui.com/about) + * Dual licensed under the MIT or GPL Version 2 licenses. + * http://jquery.org/license + * + * http://docs.jquery.com/UI/Selectable#theming + */ +.ui-selectable-helper { position: absolute; z-index: 100; border:1px dotted black; } +/* + * jQuery UI Accordion 1.8.16 + * + * Copyright 2011, AUTHORS.txt (http://jqueryui.com/about) + * Dual licensed under the MIT or GPL Version 2 licenses. + * http://jquery.org/license + * + * http://docs.jquery.com/UI/Accordion#theming + */ +/* IE/Win - Fix animation bug - #4615 */ +.ui-accordion { width: 100%; } +.ui-accordion .ui-accordion-header { cursor: pointer; position: relative; margin-top: 1px; zoom: 1; } +.ui-accordion .ui-accordion-li-fix { display: inline; } +.ui-accordion .ui-accordion-header-active { border-bottom: 0 !important; } +.ui-accordion .ui-accordion-header a { display: block; font-size: 1em; padding: .5em .5em .5em .7em; } +.ui-accordion-icons .ui-accordion-header a { padding-left: 2.2em; } +.ui-accordion .ui-accordion-header .ui-icon { position: absolute; left: .5em; top: 50%; margin-top: -8px; } +.ui-accordion .ui-accordion-content { padding: 1em 2.2em; border-top: 0; margin-top: -2px; position: relative; top: 1px; margin-bottom: 2px; overflow: auto; display: none; zoom: 1; } +.ui-accordion .ui-accordion-content-active { display: block; } +/* + * jQuery UI Autocomplete 1.8.16 + * + * Copyright 2011, AUTHORS.txt (http://jqueryui.com/about) + * Dual licensed under the MIT or GPL Version 2 licenses. + * http://jquery.org/license + * + * http://docs.jquery.com/UI/Autocomplete#theming + */ +.ui-autocomplete { position: absolute; cursor: default; } + +/* workarounds */ +* html .ui-autocomplete { width:1px; } /* without this, the menu expands to 100% in IE6 */ + +/* + * jQuery UI Menu 1.8.16 + * + * Copyright 2010, AUTHORS.txt (http://jqueryui.com/about) + * Dual licensed under the MIT or GPL Version 2 licenses. + * http://jquery.org/license + * + * http://docs.jquery.com/UI/Menu#theming + */ +.ui-menu { + list-style:none; + padding: 2px; + margin: 0; + display:block; + float: left; +} +.ui-menu .ui-menu { + margin-top: -3px; +} +.ui-menu .ui-menu-item { + margin:0; + padding: 0; + zoom: 1; + float: left; + clear: left; + width: 100%; +} +.ui-menu .ui-menu-item a { + text-decoration:none; + display:block; + padding:.2em .4em; + line-height:1.5; + zoom:1; +} +.ui-menu .ui-menu-item a.ui-state-hover, +.ui-menu .ui-menu-item a.ui-state-active { + font-weight: normal; + margin: -1px; +} +/* + * jQuery UI Button 1.8.16 + * + * Copyright 2011, AUTHORS.txt (http://jqueryui.com/about) + * Dual licensed under the MIT or GPL Version 2 licenses. + * http://jquery.org/license + * + * http://docs.jquery.com/UI/Button#theming + */ +.ui-button { display: inline-block; position: relative; padding: 0; margin-right: .1em; text-decoration: none !important; cursor: pointer; text-align: center; zoom: 1; overflow: visible; } /* the overflow property removes extra width in IE */ +.ui-button-icon-only { width: 2.2em; } /* to make room for the icon, a width needs to be set here */ +button.ui-button-icon-only { width: 2.4em; } /* button elements seem to need a little more width */ +.ui-button-icons-only { width: 3.4em; } +button.ui-button-icons-only { width: 3.7em; } + +/*button text element */ +.ui-button .ui-button-text { display: block; line-height: 1.4; } +.ui-button-text-only .ui-button-text { padding: .4em 1em; } +.ui-button-icon-only .ui-button-text, .ui-button-icons-only .ui-button-text { padding: .4em; text-indent: -9999999px; } +.ui-button-text-icon-primary .ui-button-text, .ui-button-text-icons .ui-button-text { padding: .4em 1em .4em 2.1em; } +.ui-button-text-icon-secondary .ui-button-text, .ui-button-text-icons .ui-button-text { padding: .4em 2.1em .4em 1em; } +.ui-button-text-icons .ui-button-text { padding-left: 2.1em; padding-right: 2.1em; } +/* no icon support for input elements, provide padding by default */ +input.ui-button { padding: .4em 1em; } + +/*button icon element(s) */ +.ui-button-icon-only .ui-icon, .ui-button-text-icon-primary .ui-icon, .ui-button-text-icon-secondary .ui-icon, .ui-button-text-icons .ui-icon, .ui-button-icons-only .ui-icon { position: absolute; top: 50%; margin-top: -8px; } +.ui-button-icon-only .ui-icon { left: 50%; margin-left: -8px; } +.ui-button-text-icon-primary .ui-button-icon-primary, .ui-button-text-icons .ui-button-icon-primary, .ui-button-icons-only .ui-button-icon-primary { left: .5em; } +.ui-button-text-icon-secondary .ui-button-icon-secondary, .ui-button-text-icons .ui-button-icon-secondary, .ui-button-icons-only .ui-button-icon-secondary { right: .5em; } +.ui-button-text-icons .ui-button-icon-secondary, .ui-button-icons-only .ui-button-icon-secondary { right: .5em; } + +/*button sets*/ +.ui-buttonset { margin-right: 7px; } +.ui-buttonset .ui-button { margin-left: 0; margin-right: -.3em; } + +/* workarounds */ +button.ui-button::-moz-focus-inner { border: 0; padding: 0; } /* reset extra padding in Firefox */ +/* + * jQuery UI Dialog 1.8.16 + * + * Copyright 2011, AUTHORS.txt (http://jqueryui.com/about) + * Dual licensed under the MIT or GPL Version 2 licenses. + * http://jquery.org/license + * + * http://docs.jquery.com/UI/Dialog#theming + */ +.ui-dialog { position: absolute; padding: .2em; width: 300px; overflow: hidden; } +.ui-dialog .ui-dialog-titlebar { padding: .4em 1em; position: relative; } +.ui-dialog .ui-dialog-title { float: left; margin: .1em 16px .1em 0; } +.ui-dialog .ui-dialog-titlebar-close { position: absolute; right: .3em; top: 50%; width: 19px; margin: -10px 0 0 0; padding: 1px; height: 18px; } +.ui-dialog .ui-dialog-titlebar-close span { display: block; margin: 1px; } +.ui-dialog .ui-dialog-titlebar-close:hover, .ui-dialog .ui-dialog-titlebar-close:focus { padding: 0; } +.ui-dialog .ui-dialog-content { position: relative; border: 0; padding: .5em 1em; background: none; overflow: auto; zoom: 1; } +.ui-dialog .ui-dialog-buttonpane { text-align: left; border-width: 1px 0 0 0; background-image: none; margin: .5em 0 0 0; padding: .3em 1em .5em .4em; } +.ui-dialog .ui-dialog-buttonpane .ui-dialog-buttonset { float: right; } +.ui-dialog .ui-dialog-buttonpane button { margin: .5em .4em .5em 0; cursor: pointer; } +.ui-dialog .ui-resizable-se { width: 14px; height: 14px; right: 3px; bottom: 3px; } +.ui-draggable .ui-dialog-titlebar { cursor: move; } +/* + * jQuery UI Slider 1.8.16 + * + * Copyright 2011, AUTHORS.txt (http://jqueryui.com/about) + * Dual licensed under the MIT or GPL Version 2 licenses. + * http://jquery.org/license + * + * http://docs.jquery.com/UI/Slider#theming + */ +.ui-slider { position: relative; text-align: left; } +.ui-slider .ui-slider-handle { position: absolute; z-index: 2; width: 1.2em; height: 1.2em; cursor: default; } +.ui-slider .ui-slider-range { position: absolute; z-index: 1; font-size: .7em; display: block; border: 0; background-position: 0 0; } + +.ui-slider-horizontal { height: .8em; } +.ui-slider-horizontal .ui-slider-handle { top: -.3em; margin-left: -.6em; } +.ui-slider-horizontal .ui-slider-range { top: 0; height: 100%; } +.ui-slider-horizontal .ui-slider-range-min { left: 0; } +.ui-slider-horizontal .ui-slider-range-max { right: 0; } + +.ui-slider-vertical { width: .8em; height: 100px; } +.ui-slider-vertical .ui-slider-handle { left: -.3em; margin-left: 0; margin-bottom: -.6em; } +.ui-slider-vertical .ui-slider-range { left: 0; width: 100%; } +.ui-slider-vertical .ui-slider-range-min { bottom: 0; } +.ui-slider-vertical .ui-slider-range-max { top: 0; }/* + * jQuery UI Tabs 1.8.16 + * + * Copyright 2011, AUTHORS.txt (http://jqueryui.com/about) + * Dual licensed under the MIT or GPL Version 2 licenses. + * http://jquery.org/license + * + * http://docs.jquery.com/UI/Tabs#theming + */ +.ui-tabs { position: relative; padding: .2em; zoom: 1; } /* position: relative prevents IE scroll bug (element with position: relative inside container with overflow: auto appear as "fixed") */ +.ui-tabs .ui-tabs-nav { margin: 0; padding: .2em .2em 0; } +.ui-tabs .ui-tabs-nav li { list-style: none; float: left; position: relative; top: 1px; margin: 0 .2em 1px 0; border-bottom: 0 !important; padding: 0; white-space: nowrap; } +.ui-tabs .ui-tabs-nav li a { float: left; padding: .5em 1em; text-decoration: none; } +.ui-tabs .ui-tabs-nav li.ui-tabs-selected { margin-bottom: 0; padding-bottom: 1px; } +.ui-tabs .ui-tabs-nav li.ui-tabs-selected a, .ui-tabs .ui-tabs-nav li.ui-state-disabled a, .ui-tabs .ui-tabs-nav li.ui-state-processing a { cursor: text; } +.ui-tabs .ui-tabs-nav li a, .ui-tabs.ui-tabs-collapsible .ui-tabs-nav li.ui-tabs-selected a { cursor: pointer; } /* first selector in group seems obsolete, but required to overcome bug in Opera applying cursor: text overall if defined elsewhere... */ +.ui-tabs .ui-tabs-panel { display: block; border-width: 0; padding: 1em 1.4em; background: none; } +.ui-tabs .ui-tabs-hide { display: none !important; } +/* + * jQuery UI Datepicker 1.8.16 + * + * Copyright 2011, AUTHORS.txt (http://jqueryui.com/about) + * Dual licensed under the MIT or GPL Version 2 licenses. + * http://jquery.org/license + * + * http://docs.jquery.com/UI/Datepicker#theming + */ +.ui-datepicker { width: 17em; padding: .2em .2em 0; display: none; } +.ui-datepicker .ui-datepicker-header { position:relative; padding:.2em 0; } +.ui-datepicker .ui-datepicker-prev, .ui-datepicker .ui-datepicker-next { position:absolute; top: 2px; width: 1.8em; height: 1.8em; } +.ui-datepicker .ui-datepicker-prev-hover, .ui-datepicker .ui-datepicker-next-hover { top: 1px; } +.ui-datepicker .ui-datepicker-prev { left:2px; } +.ui-datepicker .ui-datepicker-next { right:2px; } +.ui-datepicker .ui-datepicker-prev-hover { left:1px; } +.ui-datepicker .ui-datepicker-next-hover { right:1px; } +.ui-datepicker .ui-datepicker-prev span, .ui-datepicker .ui-datepicker-next span { display: block; position: absolute; left: 50%; margin-left: -8px; top: 50%; margin-top: -8px; } +.ui-datepicker .ui-datepicker-title { margin: 0 2.3em; line-height: 1.8em; text-align: center; } +.ui-datepicker .ui-datepicker-title select { font-size:1em; margin:1px 0; } +.ui-datepicker select.ui-datepicker-month-year {width: 100%;} +.ui-datepicker select.ui-datepicker-month, +.ui-datepicker select.ui-datepicker-year { width: 49%;} +.ui-datepicker table {width: 100%; font-size: .9em; border-collapse: collapse; margin:0 0 .4em; } +.ui-datepicker th { padding: .7em .3em; text-align: center; font-weight: bold; border: 0; } +.ui-datepicker td { border: 0; padding: 1px; } +.ui-datepicker td span, .ui-datepicker td a { display: block; padding: .2em; text-align: right; text-decoration: none; } +.ui-datepicker .ui-datepicker-buttonpane { background-image: none; margin: .7em 0 0 0; padding:0 .2em; border-left: 0; border-right: 0; border-bottom: 0; } +.ui-datepicker .ui-datepicker-buttonpane button { float: right; margin: .5em .2em .4em; cursor: pointer; padding: .2em .6em .3em .6em; width:auto; overflow:visible; } +.ui-datepicker .ui-datepicker-buttonpane button.ui-datepicker-current { float:left; } + +/* with multiple calendars */ +.ui-datepicker.ui-datepicker-multi { width:auto; } +.ui-datepicker-multi .ui-datepicker-group { float:left; } +.ui-datepicker-multi .ui-datepicker-group table { width:95%; margin:0 auto .4em; } +.ui-datepicker-multi-2 .ui-datepicker-group { width:50%; } +.ui-datepicker-multi-3 .ui-datepicker-group { width:33.3%; } +.ui-datepicker-multi-4 .ui-datepicker-group { width:25%; } +.ui-datepicker-multi .ui-datepicker-group-last .ui-datepicker-header { border-left-width:0; } +.ui-datepicker-multi .ui-datepicker-group-middle .ui-datepicker-header { border-left-width:0; } +.ui-datepicker-multi .ui-datepicker-buttonpane { clear:left; } +.ui-datepicker-row-break { clear:both; width:100%; font-size:0em; } + +/* RTL support */ +.ui-datepicker-rtl { direction: rtl; } +.ui-datepicker-rtl .ui-datepicker-prev { right: 2px; left: auto; } +.ui-datepicker-rtl .ui-datepicker-next { left: 2px; right: auto; } +.ui-datepicker-rtl .ui-datepicker-prev:hover { right: 1px; left: auto; } +.ui-datepicker-rtl .ui-datepicker-next:hover { left: 1px; right: auto; } +.ui-datepicker-rtl .ui-datepicker-buttonpane { clear:right; } +.ui-datepicker-rtl .ui-datepicker-buttonpane button { float: left; } +.ui-datepicker-rtl .ui-datepicker-buttonpane button.ui-datepicker-current { float:right; } +.ui-datepicker-rtl .ui-datepicker-group { float:right; } +.ui-datepicker-rtl .ui-datepicker-group-last .ui-datepicker-header { border-right-width:0; border-left-width:1px; } +.ui-datepicker-rtl .ui-datepicker-group-middle .ui-datepicker-header { border-right-width:0; border-left-width:1px; } + +/* IE6 IFRAME FIX (taken from datepicker 1.5.3 */ +.ui-datepicker-cover { + display: none; /*sorry for IE5*/ + display/**/: block; /*sorry for IE5*/ + position: absolute; /*must have*/ + z-index: -1; /*must have*/ + filter: mask(); /*must have*/ + top: -4px; /*must have*/ + left: -4px; /*must have*/ + width: 200px; /*must have*/ + height: 200px; /*must have*/ +}/* + * jQuery UI Progressbar 1.8.16 + * + * Copyright 2011, AUTHORS.txt (http://jqueryui.com/about) + * Dual licensed under the MIT or GPL Version 2 licenses. + * http://jquery.org/license + * + * http://docs.jquery.com/UI/Progressbar#theming + */ +.ui-progressbar { height:2em; text-align: left; } +.ui-progressbar .ui-progressbar-value {margin: -1px; height:100%; } \ No newline at end of file diff --git a/css/local.css b/css/local.css new file mode 100644 index 0000000000..288d92cacb --- /dev/null +++ b/css/local.css @@ -0,0 +1,58 @@ +#coursenav { + background-color:#031634; + padding: 4px 8px 8px 0px; + margin: 4px 8px 8px 0px; + width: 100%; + text-align: right; +} + +#left_nav { + border: 0; + padding: 0; + margin: 0; + float: left; + width: 276; +} + +#hide_acc { + min-height:600; +} + +#mainblock { + border: 0; + padding: 0; + margin: 0; + width: 200; + min-height:600px; + margin-left: 276px; + margin-right: 0px; +} + + +#bodyContent { + width: 100%; +} + +.bordered { border: 1px solid #AAAAAA; border-style : dotted; } + +.seq_problem_visited { background-color: #ccccaa;} +.seq_video_visited { background-color: #ccaacc;} +.seq_video_visited { background-color: #cccaac;} +.seq_html_visited { background-color: #caaccc;} +.seq_tab_visited { background-color: #aacccc;} +.seq_vertical_visited { background-color: #acaccc;} +.seq_sequential_visited { background-color: #cacacc;} +.seq_problem_visited { background-color: #ccacac;} +.seq_schematic_visited { background-color: #cccaca;} + +.seq_problem_inactive { background-color: #aaaa99;} +.seq_video_inactive { background-color: #aa99aa;} +.seq_video_inactive { background-color: #aaa99a;} +.seq_html_inactive { background-color: #a99aaa;} +.seq_tab_inactive { background-color: #99aaaa;} +.seq_vertical_inactive { background-color: #9a9aab;} +.seq_sequential_inactive { background-color: #a9a9aa;} +.seq_problem_inactive { background-color: #aa9a9a;} +.seq_schematic_inactive { background-color: #aaa9a9;} + +.seq_active { background-color: red;} \ No newline at end of file diff --git a/css/theme.css b/css/theme.css new file mode 100644 index 0000000000..48d12f3e1c --- /dev/null +++ b/css/theme.css @@ -0,0 +1,363 @@ + +/* +---------------------------------------------------------------------------------------- + +Copyright 2010 - Thierry Ruiz - www.dotemplate.com - All rights reserved. + +THIS TEMPLATE IS FREE AS LONG AS YOU KEEP THE LINK TO WWW.DOTEMPLATE.COM IN THE FOOTER +TO REMOVE THE LINK, PLEASE MAKE A 10 DOLLARS DONATION at www.dotemplate.com/#donate + +pmitros donated $10 + +Colors: +Light blue: bddeff + Blue: 7fbcfd +Very dark blue: #031634 + +Dark blue: #023063; +Dark Greenish: #7a994c; +Greenish: #adcc80; +Very light greenish: #dae5c9; +Bright orange: #fa720a; + + +---------------------------------------------------------------------------------------- +*/ + + +html { + background: $bottomWrapper.bg.color; + margin:0 0 0 -1px; + padding:0; + overflow-Y: scroll; +} + + +body { + padding:0; + margin :0; + display:table; + width:100%; +} + + +body { + color:#023063; + font-family: Helvetica, Arial, sans-serif; + font-size: 0.8em; + font-style: normal; + font-weight: normal; + text-transform: normal; +} + +p { + letter-spacing: normal; + line-height: 20px; +} + +h1 { + color:#adcc80; + font-size:1.6em; + margin:25px 0 10px 0; + clear:both; +} + + +h2 { + color:#adcc80; + font-size:1.1em; + margin:20px 0 10px 0; + clear:both; +} + + +h3 { + color:#adcc80; + font-size:1em; + margin:20px 0 5px 0; + clear:both; +} + + + +a:link, a:visited { + color:#adcc80; + text-decoration:none; +} + + +a:hover { + color:#fa720a; + text-decoration:underline; +} + +#sectiondesc { + font-size:8px; +} + +#wrapper { + background: transparent url( images/css/page-vbg.jpg ) repeat-y scroll 50% 0px; + margin:0 auto ; + width:100%; +} + + body { + background-color:#FFFFFF; +} + +#wrapper { + border-top: 1px solid #ffffff; +} + + + +#topWrapper { + width:100%; + height:100px; + background-color:${topWrapper.bg.color}; + background-image:url(images/css/topWrapper-bg.jpg); + background-position:center top ; + background-repeat:repeat-x; +} + + + +#topBanner { + width:800px; + margin:0 auto; + height:100px; + background-image:url(images/css/topBanner.jpg); + background-repeat:no-repeat; +} + + + +#topnav { + background:transparent url('images/css/hmenu.jpg') repeat-x top left; +} + + + + +#topnav { + height:20px; + margin:0 auto ; + text-align:center; +} + +#topnav ul { + display:table; + margin:0 auto; + padding:0; + list-style-type:none; + position:relative; + height:20px; + text-transform:uppercase; + font-size:0.9em; + font-family:Arial,sans-serif; +} + + +#topnav ul li { + display:block; + float:left; + margin:0; + padding:0; + background:transparent url('images/css/hmenu.jpg') repeat-x top left; + +} + + +#topnav ul li a { + display:block; + float:left; + color:#031634; + text-decoration:none; + padding:0px 50px ; + line-height:20px; + font-weight:bold; +} + + +#topnav ul li a:hover, #topnav li#current a { + color:#7a994c; + background:transparent url('images/css/hmenu-sel.jpg') repeat-x top left; +} + + + +#topnav ul { + width:800px; +} + + + + +body { + background-position : center 120px; +} + + + +#bg { + margin:0 auto; + padding:0; + background:transparent ; + background-image:url(images/css/hat.jpg); + background-repeat:no-repeat; + background-position: center top ; +} + + + + #header { + margin: 0 auto; + min-height:0px; + height: 0px; + width: 800px; +} + + +#page { + background: transparent url( images/css/page-bg.jpg ) no-repeat center top; +} + + + + + +#container { + margin:0 auto; + width:800px; +} + + +#content { + width:780px; + margin:0 auto ; + text-align:left; + min-height:800px; +} + + +#content p { + text-align:justify; +} + + + + + +#footerWrapper { + margin:0 auto; + height:10px; + padding:0; + background:transparent url( images/css/footerWrapper-bg.jpg ) no-repeat scroll center top; +} + + +#footer { + margin:0 auto; + padding:0; + width:800px; + height:10px; +} + + + + + +#right { + width:200px; + padding:10px 20px 20px 20px; + float:left; +} + + +#center { + width:500px; + padding:10px 20px 20px 20px; + float:left; +} + + +#content p { + margin-top:10px; + text-align:justify; +} + + + +#sidebar ul.vmenu { + list-style: none; + text-align: left; + margin: 7px 0px 8px 0px; + padding: 0; + text-decoration: none; + border-top: 1px solid #eeeeee; +} + + +#sidebar ul.vmenu li { + list-style: none; + padding: 4px 0 4px 0px; + margin: 0 2px; + border-bottom: 1px solid #eeeeee; +} + + + +#sidebar ul.vmenu li a { + text-decoration: none; + color:#023063; +} + + +#sidebar ul.vmenu li a:hover { + color:#adcc80; +} + + +#sidebar ul.vmenu ul { margin: 0 0 0 5px; padding: 0; } +#sidebar ul.vmenu ul li { border: none; } + + + + + +#bottomWrapper { + height:140px; + background-color:#031634; + background-image:url(images/css/bottomWrapper-bg.jpg); + background-position:center top ; + background-repeat:repeat-x; + text-align:center; + color:#adcc80; + text-align:center; +} + +#bottomWrapper a:link, #footer a:visited { + color:#dae5c9; +} + + +#bottomWrapper a:hover{ + color:#adcc80; + text-decoration:none; +} + + + +a img { + border:none; +} + + +.clear { + clear: both; + display: block; + overflow: hidden; + visibility: hidden; + width: 0; + height: 0; +} diff --git a/js/schematic.js b/js/schematic.js new file mode 100644 index 0000000000..0e6aca9547 --- /dev/null +++ b/js/schematic.js @@ -0,0 +1,2016 @@ +////////////////////////////////////////////////////////////////////////////// +// +// Simple schematic capture +// +//////////////////////////////////////////////////////////////////////////////// + +// Chris Terman, Nov. 2011 + +// add schematics to a document with +// +// +// +// other attributes you can add to the input tag: +// width -- width in pixels of diagram +// height -- height in pixels of diagram +// parts -- comma-separated list of parts for parts bin (see parts_map) + +// JSON schematic representation: +// sch := [part, part, ...] +// part := [type, coords, properties, connections] +// type := string (see parts_map) +// coords := [number, ...] // (x,y,rot) or (x1,y1,x2,y2) +// properties := {name: value, ...} +// connections := [node, ...] // one per connection point in canoncial order +// node := string + +// TO DO: + +// - draggable overlay window base class (dialogs, scope, ...) +// - wire labels? +// - devices: diode, nfet, pfet, opamp, scope probe +// - icons for test equipment? (scope, sig gen, counter, ...) + +// - zoom/scroll canvas +// - freeze_diagram, freeze_properties attributes (freeze certain components/properties?) +// - rotate multiple objects around their center of mass +// - rubber band wires when moving components + +// - simulation: operating points, trans, ac analysis, sweeps? + +// set up each schematic entry widget +function update_schematics() { + // set up each schematic on the page + var schematics = document.getElementsByClassName('schematic'); + for (var i = schematics.length - 1; i >= 0; i--) + if (schematics[i].getAttribute("loaded") != "true") { + new Schematic(schematics[i]); + schematics[i].setAttribute("loaded","true"); + } +} + +// add ourselves to the tasks that get performed when window is loaded +function add_schematic_handler(other_onload) { + return function() { + // execute othe onload functions first + if (other_onload) other_onload(); + + update_schematics(); + } +} +window.onload = add_schematic_handler(window.onload); + +background_style = 'rgb(220,220,220)'; +element_style = 'rgb(255,255,255)'; +thumb_style = 'rgb(128,128,128)'; +normal_style = 'rgb(0,0,0)'; // color for unselected components +selected_style = 'rgb(64,255,64)'; // highlight color for selected components +grid_style = "rgb(128,128,128)"; + +// list of all the defined parts +parts_map = { + 'g': [Ground, 'Ground connection'], + 'v': [VSource, 'Voltage source'], + 'i': [ISource, 'Current source'], + 'r': [Resistor, 'Resistor'], + 'c': [Capacitor, 'Capacitor'], + 'l': [Inductor, 'Inductor'], +}; + +// fix cursor bug in Chrome (default behavior: change to text cursor +// whenever a drag is initiated). +document.onselectstart = function() { return false; }; + +/////////////////////////////////////////////////////////////////////////////// +// +// Schematic = diagram + parts bin + status area +// +//////////////////////////////////////////////////////////////////////////////// + +// setup a schematic by populating the
with the appropriate children +function Schematic(input) { + this.div = document.createElement('div'); + // set up div so we can position elements inside of it + this.div.style.position = 'relative'; + this.div.style.cursor = 'default'; + + this.grid = 8; + this.scale = 2; + this.origin_x = 0; + this.origin_y = 0; + this.clipboard = null; + + // start with a background element with normal positioning + this.background = document.createElement('canvas'); + this.background.style.backgroundColor = background_style; + this.background.style.borderStyle = 'solid'; + this.background.style.borderWidth = '2px'; + + this.status_div = document.createElement('div'); + //this.status_div.style.borderStyle = 'solid'; + //this.status_div.style.borderWidth = '1px'; + this.status_div.style.position = 'absolute'; + this.status_div.style.padding = '2px'; + //this.status_div.style.backgroundColor = element_style; + this.status = document.createTextNode(''); + this.status_div.appendChild(this.status); + + this.connection_points = new Array(); // location string => list of cp's + this.components = []; + + // this is where schematic is rendered + this.canvas = document.createElement('canvas'); + this.canvas.tabIndex = 1; // so we get keystrokes + this.canvas.style.borderStyle = 'solid'; + this.canvas.style.borderWidth = '1px'; + this.canvas.style.borderColor = grid_style; + this.canvas.style.position = 'absolute'; + this.canvas.style.outline = 'none'; + + this.canvas.schematic = this; + this.canvas.addEventListener('mousemove',schematic_mouse_move,false); + this.canvas.addEventListener('mouseover',schematic_mouse_enter,false); + this.canvas.addEventListener('mouseout',schematic_mouse_leave,false); + this.canvas.addEventListener('mousedown',schematic_mouse_down,false); + this.canvas.addEventListener('mouseup',schematic_mouse_up,false); + this.canvas.addEventListener('dblclick',schematic_double_click,false); + this.canvas.addEventListener('keydown',schematic_key_down,false); + this.canvas.addEventListener('keypress',schematic_key_press,false); + + // toolbar + this.tools = new Array(); + this.toolbar = []; + this.tools['cut'] = this.add_tool(cut_icon,'Cut: move selected components from diagram to the clipboard',this.cut); + this.tools['copy'] = this.add_tool(copy_icon,'Copy: copy selected components into the clipboard',this.copy); + this.tools['paste'] = this.add_tool(paste_icon,'Paste: copy clipboard into the diagram',this.paste); + + // make the canvas "clickable" by registering a dummy click handler + // this should make things work on the iPad + this.canvas.addEventListener('click',function(){},false); + + this.dragging = false; + this.drawCursor = false; + this.cursor_x = 0; + this.cursor_y = 0; + this.draw_cursor = null; + this.select_rect = null; + this.wire = null; + + // repaint simply draws this buffer and then adds selected elements on top + this.bg_image = document.createElement('canvas'); + + // use user-supplied list of parts if supplied + // else just populate parts bin with all the parts + var parts = input.getAttribute('parts'); + if (parts) parts = parts.split(','); + else { + parts = new Array(); + for (var p in parts_map) parts.push(p); + } + + // now add the parts to the parts bin + var parts_left = this.width + 3 + background_margin; + var parts_top = background_margin; + this.parts_bin = []; + for (var i = 0; i < parts.length; i++) { + var part = new Part(this); + var pm = parts_map[parts[i]]; + part.set_component(new pm[0](part,0,0,0),pm[1]); + this.parts_bin.push(part); + } + + // add all elements to the DOM + this.div.appendChild(this.background); + for (var i = 0; i < this.toolbar.length; i++) + this.div.appendChild(this.toolbar[i]); + this.div.appendChild(this.canvas); + this.div.appendChild(this.status_div); + for (var i = 0; i < this.parts_bin.length; i++) + this.div.appendChild(this.parts_bin[i].canvas); + input.parentNode.insertBefore(this.div,input.nextSibling); + + // make sure other code can find us! + input.schematic = this; + this.input = input; + + // set locations of all the elements in the editor + var w = parseInt(input.getAttribute('width')); + var h = parseInt(input.getAttribute('height')); + this.set_locations(w,h); + + // process initial contents of diagram + this.load_schematic(this.input.value); +} + + +Schematic.prototype.load_schematic = function(value) { + if (value) { + // convert string value into data structure + var json = JSON.parse(value); + + // top level is a list of components + for (var i = json.length - 1; i >= 0; --i) { + var c = json[i]; + if (c[0] == 'view') { + // special hack: view component lets us recreate view + this.origin_x = c[1]; + this.origin_y = c[2]; + this.scale = c[3]; + } else if (c[0] == 'w') { + // wire + this.add_wire(c[1][0],c[1][1],c[1][2],c[1][3]); + } else { + // ordinary component + // c := [type, coords, properties, connections] + var type = c[0]; + var coords = c[1]; + var properties = c[2]; + + // make the part + var part = new parts_map[type][0](this,coords[0],coords[1],coords[2]); + + // give it its properties + for (var name in properties) + part.properties[name] = properties[name]; + + // add component to the diagram + this.add_component(part); + } + } + + // see what we've got! + this.redraw_background(); + } +} + +background_margin = 5; +part_w = 42; // size of a parts bin compartment +part_h = 42; +status_height = 18; + +// w,h are the dimensions of the canvas, everyone else is positioned accordingly +Schematic.prototype.set_locations = function(w,h) { + // limit the shrinkage factor + w = Math.max(w,120); + h = Math.max(h,120); + + this.width = w; + this.height = h; + this.bg_image.width = w; + this.bg_image.height = h; + + this.min_x = 0; + this.min_y = 0; + this.max_x = w/this.scale; + this.max_y = h/this.scale; + + var left = 2*background_margin; // space to the left + + // start with tool bar + var top = background_margin; + if (this.toolbar.length > 0) { + tool_left = left; + for (var i = 0; i < this.toolbar.length; i++) { + var img = this.toolbar[i]; + img.style.left = tool_left + 'px'; + img.style.top = top + 'px'; + tool_left += 24; // width + 2*padding + 2*border + gap + } + top += 27; // height + 2*padding + 2*border + gap; + } + + // configure canvas + this.canvas.style.left = left + 'px'; + this.canvas.style.top = top + 'px'; + this.canvas.width = w; + this.canvas.height = h; + this.redraw_background(); // redraw diagram + + // configure status bar + this.status_div.style.left = left + 'px'; + this.status_div.style.top = this.canvas.offsetTop + this.canvas.offsetHeight + 3 + 'px'; + this.status_div.style.width = (w - 4) + 'px'; // subtract interior padding + this.status_div.style.height = status_height + 'px'; + + // configure parts bin + var total_w = this.canvas.offsetLeft + this.canvas.offsetWidth; + var parts_left = total_w + 5; + var parts_top = top; + var parts_h_limit = this.canvas.offsetTop + this.canvas.offsetHeight; + for (var i = 0; i < this.parts_bin.length; i++) { + var part = this.parts_bin[i]; + part.set_location(parts_left,parts_top); + + total_w = part.right(); + parts_top = part.bottom() + 2; + if (parts_top + part_h > parts_h_limit) { + parts_left = total_w - 1; + parts_top = top; + } + } + + // configure background + var total_h = this.status_div.offsetTop + this.status_div.offsetHeight + background_margin; + total_w += background_margin; + this.background.height = total_h; + this.background.width = total_w; + + /* enable when there's support for resizing schematic + // redraw thumb + var c = this.background.getContext('2d'); + c.clearRect(0,0,w,h); + c.strokeStyle = thumb_style; + c.lineWidth = 1; + c.beginPath(); + w = total_w - 1; + h = total_h - 1; + c.moveTo(w,h-4); c.lineTo(w-4,h); + c.moveTo(w,h-8); c.lineTo(w-8,h); + c.moveTo(w,h-12); c.lineTo(w-12,h); + c.stroke(); + */ +} + +// label all the nodes in the circuit +Schematic.prototype.label_connection_points = function() { + // start by clearing all the connection point labels + for (var i = this.components.length - 1; i >=0; --i) + this.components[i].clear_labels(); + + // components are in charge of labeling their unlabeled connections. + // labels given to connection points will propagate to coincident connection + // points and across Wires. + + // let special components like GND label their connection(s) + for (var i = this.components.length - 1; i >=0; --i) + this.components[i].add_default_labels(); + + // now have components generate labels for unlabeled connections + this.next_label = 0; + for (var i = this.components.length - 1; i >=0; --i) + this.components[i].label_connections(); +} + +// generate a new label +Schematic.prototype.get_next_label = function() { + // generate next label in sequence + this.next_label += 1; + return this.next_label.toString(); +} + +// propagate label to coincident connection points +Schematic.prototype.propagate_label = function(label,location) { + var cplist = this.connection_points[location]; + for (var i = cplist.length - 1; i >= 0; --i) + cplist[i].propagate_label(label); +} + +// update the value field of our corresponding input field with JSON +// representation of schematic +Schematic.prototype.update_value = function() { + // label connection points + this.label_connection_points(); + + // build JSON data structure, convert to string value for + // input field + this.input.value = JSON.stringify(this.json()); +} + +// produce a JSON representation of the diagram +Schematic.prototype.json = function() { + var json = []; + + // output all the components/wires in the diagram + for (var i = this.components.length - 1; i >=0; --i) + json.push(this.components[i].json()); + + // capture the current view parameters + json.push(['view',this.origin_x,this.origin_y,this.scale]); + + return json; +} + +Schematic.prototype.add_component = function(new_c) { + this.components.push(new_c); + + // create undoable edit record here +} + +Schematic.prototype.remove_component = function(c) { + var index = this.components.indexOf(c); + if (index != -1) this.components.splice(index,1); +} + +// add connection point to list of connection points at that location +Schematic.prototype.add_connection_point = function(cp) { + var cplist = this.connection_points[cp.location]; + if (cplist) cplist.push(cp); + else { + cplist = [cp]; + this.connection_points[cp.location] = cplist; + } + + // return list of conincident connection points + return cplist; +} + +// remove connection point from the list points at the old location +Schematic.prototype.remove_connection_point = function(cp,old_location) { + // remove cp from list at old location + var cplist = this.connection_points[old_location]; + if (cplist) { + var index = cplist.indexOf(cp); + if (index != -1) { + cplist.splice(index,1); + // if no more connections at this location, remove + // entry from array to keep our search time short + if (cplist.length == 0) + delete this.connection_points[old_location]; + } + } +} + +// connection point has changed location: remove, then add +Schematic.prototype.update_connection_point = function(cp,old_location) { + this.remove_connection_point(cp,old_location); + return this.add_connection_point(cp); +} + +// add a wire to the schematic +Schematic.prototype.add_wire = function(x1,y1,x2,y2) { + var new_wire = new Wire(this,x1,y1,x2,y2); + this.add_component(new_wire); + new_wire.move_end(); + return new_wire; +} + +// see if connection points of component c split any wires +Schematic.prototype.check_wires = function(c) { + for (var i = this.components.length - 1; i >=0; --i) { + var cc = this.components[i]; + if (cc != c) { // don't check a component against itself + // only wires will do return non-null from a bisect call + var cp = cc.bisect(c); + if (cp) { + // cc is a wire bisected by connection point cp + + // remove biscted wire + cc.delete(); + + // add two new wires with cp in the middle + this.add_wire(cc.x,cc.y,cp.x,cp.y); + this.add_wire(cc.x+cc.dx,cc.y+cc.dy,cp.x,cp.y); + this.redraw_background(); + break; + } + } + } +} + +Schematic.prototype.unselect_all = function(which) { + for (var i = this.components.length - 1; i >= 0; --i) + if (i != which) this.components[i].set_select(false); +} + +Schematic.prototype.drag_begin = function() { + // let components know they're about to move + for (var i = this.components.length - 1; i >= 0; --i) { + var component = this.components[i]; + if (component.selected) component.move_begin(); + } + + // remember where drag started + this.drag_x = this.cursor_x; + this.drag_y = this.cursor_y; + this.dragging = true; +} + +Schematic.prototype.drag_end = function() { + // let components know they're done moving + for (var i = this.components.length - 1; i >= 0; --i) { + var component = this.components[i]; + if (component.selected) component.move_end(); + } + this.dragging = false; +} + +Schematic.prototype.cut = function() { + // clear previous contents + this.clipboard = []; + + // look for selected components, move them to clipboard. + for (var i = this.components.length - 1; i >=0; --i) { + var c = this.components[i]; + if (c.selected) { + c.delete(); + this.clipboard.push(c); + } + } + + // update diagram view + this.redraw(); +} + +Schematic.prototype.copy = function() { + // clear previous contents + this.clipboard = []; + + // look for selected components, copy them to clipboard. + for (var i = this.components.length - 1; i >=0; --i) { + var c = this.components[i]; + if (c.selected) + this.clipboard.push(c.clone(this,c.x,c.y)); + } +} + +Schematic.prototype.paste = function() { + // compute left,top of bounding box for origins of + // components in the clipboard + var left = null; + var top = null; + for (var i = this.clipboard.length - 1; i >= 0; --i) { + var c = this.clipboard[i]; + left = left ? Math.min(left,c.x) : left; + top = top ? Math.min(top,c.y) : top; + } + + // clear current selections + this.unselect_all(-1); + this.redraw_background(); // so we see any components that got unselected + + // make clones of components on the clipboard, positioning + // them relative to the cursor + for (var i = this.clipboard.length - 1; i >= 0; --i) { + var c = this.clipboard[i]; + var new_c = c.clone(this,this.cursor_x + (c.x - left),this.cursor_y + (c.y - top)); + this.add_component(new_c); + new_c.set_select(true); + } + + // see what we've wrought + this.redraw(); +} + +/////////////////////////////////////////////////////////////////////////////// +// +// Drawing support -- deals with scaling and scrolling of diagrama +// +//////////////////////////////////////////////////////////////////////////////// + +// here to redraw background image containing static portions of the schematic. +// Also redraws dynamic portion. +Schematic.prototype.redraw_background = function() { + var c = this.bg_image.getContext('2d'); + var w = this.bg_image.width; + var h = this.bg_imageheight; + + // paint background color + c.fillStyle = element_style; + c.fillRect(0,0,this.width,this.height); + + // border + //c.strokeStyle = "rgb(0,0,0)"; + //c.strokeRect(0,0,this.width,this.height); + + // grid + c.strokeStyle = grid_style; + var first_x = this.min_x; + var last_x = this.max_x; + var first_y = this.min_y; + var last_y = this.max_y; + for (var i = first_x; i < last_x; i += this.grid) + this.draw_line(c,i,first_y,i,last_y,0.1); + for (var i = first_y; i < last_y; i += this.grid) + this.draw_line(c,first_x,i,last_x,i,0.1); + + // unselected components + for (var i = this.components.length - 1; i >= 0; --i) { + var component = this.components[i]; + if (!component.selected) component.draw(c); + } + + this.redraw(); // background changed, redraw on screen +} + +// redraw what user sees = static image + dynamic parts +Schematic.prototype.redraw = function() { + var c = this.canvas.getContext('2d'); + + // put static image in the background + c.drawImage(this.bg_image, 0, 0); + + // selected components + var selections = false; + for (var i = this.components.length - 1; i >= 0; --i) { + var component = this.components[i]; + if (component.selected) { + component.draw(c); + selections = true; + } + } + this.enable_tool('cut',selections); + this.enable_tool('copy',selections); + this.enable_tool('paste',this.clipboard); + + // connection points: draw one at each location + for (var location in this.connection_points) { + var cplist = this.connection_points[location]; + cplist[0].draw(c,cplist.length); + } + + // draw new wire + if (this.wire) { + var r = this.wire; + c.strokeStyle = selected_style; + this.draw_line(c,r[0],r[1],r[2],r[3],1); + } + + // draw selection rectangle + if (this.select_rect) { + var r = this.select_rect; + c.lineWidth = 1; + c.strokeStyle = selected_style; + c.beginPath(); + c.moveTo(r[0],r[1]); + c.lineTo(r[0],r[3]); + c.lineTo(r[2],r[3]); + c.lineTo(r[2],r[1]); + c.lineTo(r[0],r[1]); + c.stroke(); + } + + // finally overlay cursor + if (this.drawCursor && this.draw_cursor) { + //var x = this.cursor_x; + //var y = this.cursor_y; + //this.draw_text(c,'('+x+','+y+')',x+this.grid,y-this.grid,10); + this.draw_cursor(c,this.cursor_x,this.cursor_y); + } +} + +// draws a cross cursor +Schematic.prototype.cross_cursor = function(c,x,y) { + this.draw_line(c,x-this.grid,y,x+this.grid,y,1); + this.draw_line(c,x,y-this.grid,x,y+this.grid,1); +} + +Schematic.prototype.draw_line = function(c,x1,y1,x2,y2,width) { + c.lineWidth = width*this.scale; + c.beginPath(); + c.moveTo((x1 - this.origin_x) * this.scale,(y1 - this.origin_y) * this.scale); + c.lineTo((x2 - this.origin_x) * this.scale,(y2 - this.origin_y) * this.scale); + c.stroke(); +} + +Schematic.prototype.draw_arc = function(c,x,y,radius,start_radians,end_radians,anticlockwise,width,filled) { + c.lineWidth = width*this.scale; + c.beginPath(); + c.arc((x - this.origin_x)*this.scale,(y - this.origin_y)*this.scale,radius*this.scale, + start_radians,end_radians,anticlockwise); + if (filled) c.fill(); + else c.stroke(); +} + +Schematic.prototype.draw_text = function(c,text,x,y,size) { + c.font = size*this.scale+'pt sans-serif' + c.fillText(text,(x - this.origin_x) * this.scale,(y - this.origin_y) * this.scale); +} + +// add method to canvas to compute relative coords for event +HTMLCanvasElement.prototype.relMouseCoords = function(event){ + // run up the DOM tree to figure out coords for top,left of canvas + var totalOffsetX = 0; + var totalOffsetY = 0; + var canvasY = 0; + var currentElement = this; + do { + totalOffsetX += currentElement.offsetLeft; + totalOffsetY += currentElement.offsetTop; + } + while(currentElement = currentElement.offsetParent); + + // now compute relative position of click within the canvas + this.mouse_x = event.pageX - totalOffsetX; + this.mouse_y = event.pageY - totalOffsetY; +} + +/////////////////////////////////////////////////////////////////////////////// +// +// Event handling +// +//////////////////////////////////////////////////////////////////////////////// + +// process special keys here since they don't get delivered correctly on keypress +function schematic_key_down(event) { + if (!event) event = window.event; + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + var code = event.keyCode; + + if (code == 8 || code == 46) { + // delete selected components + for (var i = sch.components.length - 1; i >= 0; --i) { + var component = sch.components[i]; + if (component.selected) component.delete(1); + } + sch.redraw(); + event.preventDefault(); + return false; + } + return true; +} + +// process normal characters +function schematic_key_press(event) { + if (!event) event = window.event; + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + var code = window.event ? event.keyCode : event.charCode; + var char = String.fromCharCode(code); + + // rotate + if (!event.control && !event.altKey && (char == 'r' || char == 'R')) { + // rotate + for (var i = sch.components.length - 1; i >= 0; --i) { + var component = sch.components[i]; + if (component.selected) component.rotate(1); + } + sch.redraw(); + event.preventDefault(); + return false; + } + + // cut + if ((event.ctrlKey || event.altKey) && char == 'x') { + sch.cut(); + event.preventDefault(); + return false; + } + + // copy + if ((event.ctrlKey || event.altKey) && char == 'c') { + sch.copy(); + event.preventDefault(); + return false; + } + + // paste + if ((event.ctrlKey || event.altKey) && char == 'v') { + sch.paste(); + event.preventDefault(); + return false; + } + + + return true; +} + +function schematic_mouse_enter(event) { + if (!event) event = window.event; + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + + // see if user has selected a new part + if (sch.new_part) { + // grab incoming part, turn off selection of parts bin + var part = sch.new_part; + sch.new_part = null; + part.select(false); + + // make a clone of the component in the parts bin + part = part.component.clone(sch,sch.cursor_x,sch.cursor_y); + + // unselect everything else in the schematic, add part and select it + sch.unselect_all(-1); + sch.redraw_background(); // so we see any components that got unselected + sch.add_component(part); + part.set_select(true); + + // and start dragging it + sch.drag_begin(); + } + + sch.drawCursor = true; + sch.redraw(); + sch.canvas.focus(); // capture key strokes + return false; +} + +function schematic_mouse_leave(event) { + if (!event) event = window.event; + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + sch.drawCursor = false; + sch.redraw(); + return false; +} + +function schematic_mouse_down(event) { + if (!event) event = window.event; + else event.preventDefault(); + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + + // determine where event happened in schematic coordinates + sch.canvas.relMouseCoords(event); + var x = sch.canvas.mouse_x/sch.scale + sch.origin_x; + var y = sch.canvas.mouse_y/sch.scale + sch.origin_y; + sch.cursor_x = Math.round(x/sch.grid) * sch.grid; + sch.cursor_y = Math.round(y/sch.grid) * sch.grid; + + /* + // for debugging... triggered by clicks in upper left corner + if (sch.cursor_x < 10 && sch.cursor_y < 10) { + sch.label_connection_points(); + sch.append_message(JSON.stringify(sch.json())); + } + */ + + // is mouse over a connection point? If so, start dragging a wire + var cplist = sch.connection_points[sch.cursor_x + ',' + sch.cursor_y]; + if (cplist && !event.shiftKey) { + sch.unselect_all(-1); + sch.wire = [sch.cursor_x,sch.cursor_y,sch.cursor_x,sch.cursor_y]; + } else { + // give all components a shot at processing the selection event + var which = -1; + for (var i = sch.components.length - 1; i >= 0; --i) + if (sch.components[i].select(x,y,event.shiftKey)) { + if (sch.components[i].selected) { + sch.drag_begin(); + which = i; // keep track of component we found + } + break; + } + // did we just click on a previously selected component? + var reselect = which!=-1 && sch.components[which].was_previously_selected; + + if (!event.shiftKey) { + // if shift key isn't pressed and we didn't click on component + // that was already selected, unselect everyone except component + // we just clicked on + if (!reselect) sch.unselect_all(which); + + // if there's nothing to drag, set up a selection rectangle + if (!sch.dragging) sch.select_rect = [sch.canvas.mouse_x,sch.canvas.mouse_y, + sch.canvas.mouse_x,sch.canvas.mouse_y]; + } + } + + sch.redraw_background(); + return false; +} + +function schematic_mouse_move(event) { + if (!event) event = window.event; + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + + sch.canvas.relMouseCoords(event); + var x = sch.canvas.mouse_x/sch.scale + sch.origin_x; + var y = sch.canvas.mouse_y/sch.scale + sch.origin_y; + sch.cursor_x = Math.round(x/sch.grid) * sch.grid; + sch.cursor_y = Math.round(y/sch.grid) * sch.grid; + + if (sch.wire) { + // update new wire end point + sch.wire[2] = sch.cursor_x; + sch.wire[3] = sch.cursor_y; + } else if (sch.dragging) { + // see how far we moved + var dx = sch.cursor_x - sch.drag_x; + var dy = sch.cursor_y - sch.drag_y; + if (dx != 0 || dy != 0) { + // update position for next time + sch.drag_x = sch.cursor_x; + sch.drag_y = sch.cursor_y; + + // give all components a shot at processing the event + for (var i = sch.components.length - 1; i >= 0; --i) { + var component = sch.components[i]; + if (component.selected) component.move(dx,dy); + } + } + } else if (sch.select_rect) { + // update moving corner of selection rectangle + sch.select_rect[2] = sch.canvas.mouse_x; + sch.select_rect[3] = sch.canvas.mouse_y; + //sch.message(sch.select_rect.toString()); + } + + // just redraw dynamic components + sch.redraw(); + + return false; +} + +function schematic_mouse_up(event) { + if (!event) event = window.event; + else event.preventDefault(); + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + + // drawing a new wire + if (sch.wire) { + var r = sch.wire; + sch.wire = null; + + if (r[0]!=r[2] || r[1]!=r[3]) { + // insert wire component + sch.add_wire(r[0],r[1],r[2],r[3]); + sch.redraw_background(); + } else sch.redraw(); + } + + // dragging + if (sch.dragging) sch.drag_end(); + + // selection rectangle + if (sch.select_rect) { + var r = sch.select_rect; + + // if select_rect is a point, we've already dealt with selection + // in mouse_down handler + if (r[0]!=r[2] || r[1]!=r[3]) { + // convert to schematic coordinates + var s = [r[0]/sch.scale + sch.origin_x, r[1]/sch.scale + sch.origin_y, + r[2]/sch.scale + sch.origin_x, r[3]/sch.scale + sch.origin_y]; + canonicalize(s); + + if (!event.shiftKey) sch.unselect_all(); + + // select components that intersect selection rectangle + for (var i = sch.components.length - 1; i >= 0; --i) + sch.components[i].select_rect(s,event.shiftKey); + } + + sch.select_rect = null; + sch.redraw_background(); + } + return false; +} + +function schematic_double_click(event) { + if (!event) event = window.event; + else event.preventDefault(); + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + + // determine where event happened in schematic coordinates + sch.canvas.relMouseCoords(event); + var x = sch.canvas.mouse_x/sch.scale + sch.origin_x; + var y = sch.canvas.mouse_y/sch.scale + sch.origin_y; + sch.cursor_x = Math.round(x/sch.grid) * sch.grid; + sch.cursor_y = Math.round(y/sch.grid) * sch.grid; + + // see if we double-clicked a component. If so, edit it's properties + for (var i = sch.components.length - 1; i >= 0; --i) + if (sch.components[i].edit_properties(x,y)) break; + + return false; +} + +/////////////////////////////////////////////////////////////////////////////// +// +// Status message and dialogs +// +//////////////////////////////////////////////////////////////////////////////// + +Schematic.prototype.message = function(message) { + this.status.nodeValue = message; +} + +Schematic.prototype.append_message = function(message) { + this.status.nodeValue += ' / '+message; +} + +// set up a dialog with specified title, content and two buttons at +// the bottom: OK and Cancel. If Cancel is clicked, dialog goes away +// and we're done. If OK is clicked, dialog goes away and the +// callback function is called with the content as an argument (so +// that the values of any fields can be captured). +Schematic.prototype.dialog = function(title,content,callback) { + // create the div for the top level of the dialog, add to DOM + var dialog = document.createElement('div'); + dialog.sch = this; + dialog.content = content; + + // div to hold the title + var head = document.createElement('div'); + head.style.backgroundColor = 'black'; + head.style.color = 'white'; + head.style.textAlign = 'center'; + head.style.padding = '5px'; + head.appendChild(document.createTextNode(title)); + dialog.appendChild(head); + + // div to hold the content + var body = document.createElement('div'); + body.appendChild(content); + body.style.padding = '5px'; + dialog.appendChild(body); + + // OK button + var ok_button = document.createElement('span'); + ok_button.appendChild(document.createTextNode('OK')); + ok_button.dialog = dialog; // for the handler to use + ok_button.addEventListener('click',dialog_okay,false); + ok_button.style.border = '1px solid'; + ok_button.style.padding = '5px'; + ok_button.style.margin = '10px'; + + // cancel button + var cancel_button = document.createElement('span'); + cancel_button.appendChild(document.createTextNode('Cancel')); + cancel_button.dialog = dialog; // for the handler to use + cancel_button.addEventListener('click',dialog_cancel,false); + cancel_button.style.border = '1px solid'; + cancel_button.style.padding = '5px'; + cancel_button.style.margin = '10px'; + + // div to hold the two buttons + var buttons = document.createElement('div'); + buttons.appendChild(ok_button); + buttons.appendChild(cancel_button); + buttons.style.padding = '5px'; + buttons.style.margin = '10px'; + dialog.appendChild(buttons); + + // add to DOM + dialog.style.background = 'white'; + dialog.style.zindex = '1000'; + dialog.style.position = 'absolute'; + dialog.style.left = this.canvas.mouse_x+'px'; + dialog.style.top = this.canvas.mouse_y+'px'; + dialog.style.border = '2px solid'; + dialog.callback = callback; + this.div.appendChild(dialog); +} + +// callback when user click "Cancel" in a dialog +function dialog_cancel(event) { + if (!event) event = window.event; + var dialog = (window.event) ? event.srcElement.dialog : event.target.dialog; + + // remove the dialog from the top-level div of the schematic + dialog.parentNode.removeChild(dialog); +} + +// callback when user click "OK" in a dialog +function dialog_okay(event) { + if (!event) event = window.event; + var dialog = (window.event) ? event.srcElement.dialog : event.target.dialog; + + // remove the dialog from the top-level div of the schematic + dialog.parentNode.removeChild(dialog); + + // invoke the callback with the dialog contents as the argument + if (dialog.callback) dialog.callback(dialog.content); +} + +/////////////////////////////////////////////////////////////////////////////// +// +// Toolbar +// +//////////////////////////////////////////////////////////////////////////////// + +Schematic.prototype.add_tool = function(icon,tip,callback) { + var img = document.createElement('img'); + img.src = icon; + img.style.borderWidth = '1px'; + img.style.borderStyle = 'solid'; + img.style.borderColor = background_style; + img.style.position = 'absolute'; + img.style.padding = '2px'; + + img.addEventListener('mouseover',tool_enter,false); + img.addEventListener('mouseout',tool_leave,false); + img.addEventListener('click',tool_click,false); + + img.sch = this; + img.tip = tip; + img.callback = callback; + this.toolbar.push(img); + + img.enabled = false; + img.style.opacity = 0.2; + + return img; +} + +Schematic.prototype.enable_tool = function(tname,which) { + var img = this.tools[tname]; + img.style.opacity = which ? 1.0 : 0.2; + img.enabled = which; + + // if disabling tool, remove border and tip + if (!which) { + img.style.borderColor = background_style; + img.sch.message(''); + } +} + +// highlight tool button by turning on border, changing background +function tool_enter(event) { + if (!event) event = window.event; + var img = (window.event) ? event.srcElement : event.target; + + if (img.enabled) { + img.style.borderColor = normal_style; + img.sch.message(img.tip); + img.opacity = 1.0; + } +} + +// unhighlight tool button by turning off border, reverting to normal background +function tool_leave(event) { + if (!event) event = window.event; + var img = (window.event) ? event.srcElement : event.target; + + if (img.enabled) { + img.style.borderColor = background_style; + img.sch.message(''); + } +} + +// handle click on a tool +function tool_click(event) { + if (!event) event = window.event; + var img = (window.event) ? event.srcElement : event.target; + + if (img.enabled) img.callback.call(img.sch); +} + +cut_icon = 'data:image/gif;base64,R0lGODlhEAAQALMAAAAAAIAAAACAAICAAAAAgIAAgACAgMDAwICAgP8AAAD/AP//AAAA//8A/wD//////yH5BAEAAAcALAAAAAAQABAAAAQu8MhJqz1g5qs7lxv2gRkQfuWomarXEgDRHjJhf3YtyRav0xcfcFgR0nhB5OwTAQA7'; + +copy_icon = 'data:image/gif;base64,R0lGODlhEAAQALMAAAAAAIAAAACAAICAAAAAgIAAgACAgMDAwICAgP8AAAD/AP//AAAA//8A/wD//////yH5BAEAAAcALAAAAAAQABAAAAQ+8MhJ6wE4Wwqef9gmdV8HiKZJrCz3ecS7TikWfzExvk+M9a0a4MbTkXCgTMeoHPJgG5+yF31SLazsTMTtViIAOw=='; + +paste_icon = 'data:image/gif;base64,R0lGODlhEAAQALMAAAAAAIAAAACAAICAAAAAgIAAgACAgMDAwICAgP8AAAD/AP//AAAA//8A/wD//////yH5BAEAAAcALAAAAAAQABAAAARL8MhJqwUYWJnxWp3GDcgAgCdQIqLKXmVLhhnyHiqpr7rME8AgocVDEB5IJHD0SyofBFzxGIQGAbvB0ZkcTq1CKK6z5YorwnR0w44AADs='; + +/////////////////////////////////////////////////////////////////////////////// +// +// Parts bin +// +//////////////////////////////////////////////////////////////////////////////// + +// one instance will be created for each part in the parts bin +function Part(sch) { + this.sch = sch; + this.component = null; + this.selected = false; + + // set up canvas + this.canvas = document.createElement('canvas'); + this.canvas.style.borderStyle = 'solid'; + this.canvas.style.borderWidth = '1px'; + this.canvas.style.borderColor = background_style; + this.canvas.style.position = 'absolute'; + this.canvas.style.cursor = 'default'; + this.canvas.height = part_w; + this.canvas.width = part_h; + this.canvas.part = this; + + this.canvas.addEventListener('mouseover',part_enter,false); + this.canvas.addEventListener('mouseout',part_leave,false); + this.canvas.addEventListener('mousedown',part_mouse_down,false); + this.canvas.addEventListener('mouseup',part_mouse_up,false); + + // make the part "clickable" by registering a dummy click handler + // this should make things work on the iPad + this.canvas.addEventListener('click',function(){},false); +} + +Part.prototype.set_location = function(left,top) { + this.canvas.style.left = left + 'px'; + this.canvas.style.top = top + 'px'; +} + +Part.prototype.right = function() { + return this.canvas.offsetLeft + this.canvas.offsetWidth; +} + +Part.prototype.bottom = function() { + return this.canvas.offsetTop + this.canvas.offsetHeight; +} + +Part.prototype.set_component = function(component,tip) { + this.component = component; + this.tip = tip; + + // figure out scaling and centering of parts icon + var b = component.bounding_box; + var dx = b[2] - b[0]; + var dy = b[3] - b[1]; + this.scale = 0.8; //Math.min(part_w/(1.2*dx),part_h/(1.2*dy)); + this.origin_x = b[0] + dx/2.0 - part_w/(2.0*this.scale); + this.origin_y = b[1] + dy/2.0 - part_h/(2.0*this.scale); + + this.redraw(); +} + +Part.prototype.redraw = function(part) { + var c = this.canvas.getContext('2d'); + + // paint background color + c.fillStyle = this.selected ? selected_style : background_style; + c.fillRect(0,0,part_w,part_h); + + if (this.component) this.component.draw(c); +} + +Part.prototype.select = function(which) { + this.selected = which; + this.redraw(); +} + +Part.prototype.update_connection_point = function(cp,old_location) { + // no connection points in the parts bin +} + +Part.prototype.draw_line = function(c,x1,y1,x2,y2,width) { + c.lineWidth = width*this.scale; + c.beginPath(); + c.moveTo((x1 - this.origin_x) * this.scale,(y1 - this.origin_y) * this.scale); + c.lineTo((x2 - this.origin_x) * this.scale,(y2 - this.origin_y) * this.scale); + c.stroke(); +} + +Part.prototype.draw_arc = function(c,x,y,radius,start_radians,end_radians,anticlockwise,width,filled) { + c.lineWidth = width*this.scale; + c.beginPath(); + c.arc((x - this.origin_x)*this.scale,(y - this.origin_y)*this.scale,radius*this.scale, + start_radians,end_radians,anticlockwise); + if (filled) c.fill(); + else c.stroke(); +} + +Part.prototype.draw_text = function(c,text,x,y,size) { + // no text displayed for the parts icon +} + +function part_enter(event) { + if (!event) event = window.event; + var canvas = (window.event) ? event.srcElement : event.target; + var part = canvas.part; + + canvas.style.borderColor = normal_style; + part.sch.message(part.tip+': drag onto diagram to insert'); + return false; +} + +function part_leave(event) { + if (!event) event = window.event; + var canvas = (window.event) ? event.srcElement : event.target; + var part = canvas.part; + + canvas.style.borderColor = background_style; + part.sch.message(''); + return false; +} + +function part_mouse_down(event) { + if (!event) event = window.event; + var part = (window.event) ? event.srcElement.part : event.target.part; + + part.select(true); + part.sch.new_part = part; + return false; +} + +function part_mouse_up(event) { + if (!event) event = window.event; + var part = (window.event) ? event.srcElement.part : event.target.part; + + part.select(false); + part.sch.new_part = null; + return false; +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Rectangle helper functions +// +//////////////////////////////////////////////////////////////////////////////// + +// rect is an array of the form [left,top,right,bottom] + +// ensure left < right, top < bottom +function canonicalize(r) { + var temp; + + // canonicalize bounding box + if (r[0] > r[2]) { + temp = r[0]; + r[0] = r[2]; + r[2] = temp; + } + if (r[1] > r[3]) { + temp = r[1]; + r[1] = r[3]; + r[3] = temp; + } +} + +function between(x,x1,x2) { + return x1 <= x && x <= x2; +} + +function inside(rect,x,y) { + return between(x,rect[0],rect[2]) && between(y,rect[1],rect[3]); +} + +// only works for manhattan rectangles +function intersect(r1,r2) { + // look for non-intersection, negate result + var result = !(r2[0] > r1[2] || + r2[2] < r1[0] || + r2[1] > r1[3] || + r2[3] < r1[1]); + + // if I try to return the above expression, javascript returns undefined!!! + return result; +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Component base class +// +//////////////////////////////////////////////////////////////////////////////// + +property_size = 5; // point size for Component property text + +function Component(sch,x,y,rotation) { + this.sch = sch; + this.x = x; + this.y = y; + this.rotation = rotation; + this.selected = false; + this.properties = new Array(); + this.bounding_box = [0,0,0,0]; // in device coords [left,top,right,bottom] + this.bbox = this.bounding_box; // in absolute coords + this.connections = []; +} + +Component.prototype.json = function() { + var props = {}; + for (var p in this.properties) props[p] = this.properties[p]; + + var conns = []; + for (var i = 0; i < this.connections.length; i++) + conns.push(this.connections[i].json()); + + var json = [this.type,[this.x, this.y, this.rotation],props,conns]; + return json; +} + +Component.prototype.add_connection = function(offset_x,offset_y) { + this.connections.push(new ConnectionPoint(this,offset_x,offset_y)); +} + +Component.prototype.update_coords = function() { + var x = this.x; + var y = this.y; + + // update bbox + var b = this.bounding_box; + this.bbox[0] = this.transform_x(b[0],b[1]) + x; + this.bbox[1] = this.transform_y(b[0],b[1]) + y; + this.bbox[2] = this.transform_x(b[2],b[3]) + x; + this.bbox[3] = this.transform_y(b[2],b[3]) + y; + canonicalize(this.bbox); + + // update connections + for (var i = this.connections.length - 1; i >= 0; --i) + this.connections[i].update_location(); +} + +Component.prototype.rotate = function(amount) { + var old_rotation = this.rotation; + this.rotation = (this.rotation + amount) % 8; + this.update_coords(); + + // create an undoable edit record here + // using old_rotation +} + +Component.prototype.move_begin = function() { + // remember where we started this move + this.move_x = this.x; + this.move_y = this.y; +} + +Component.prototype.move = function(dx,dy) { + // update coordinates + this.x += dx; + this.y += dy; + this.update_coords(); +} + +Component.prototype.move_end = function() { + var dx = this.x - this.move_x; + var dy = this.y - this.move_y; + + if (dx != 0 || dy != 0) { + // create an undoable edit record here + + this.sch.check_wires(this); + } +} + +Component.prototype.delete = function() { + // remove connection points from schematic + for (var i = this.connections.length - 1; i >= 0; --i) { + var cp = this.connections[i]; + this.sch.remove_connection_point(cp,cp.location); + } + + // remove component from schematic + this.sch.remove_component(this); + + // create an undoable edit record here +} + +Component.prototype.transform_x = function(x,y) { + var rot = this.rotation; + if (rot == 0 || rot == 6) return x; + else if (rot == 1 || rot == 5) return -y; + else if (rot == 2 || rot == 4) return -x; + else return y; +} + +Component.prototype.transform_y = function(x,y) { + var rot = this.rotation; + if (rot == 1 || rot == 7) return x; + else if (rot == 2 || rot == 6) return -y; + else if (rot == 3 || rot == 5) return -x; + else return y; +} + +Component.prototype.draw_line = function(c,x1,y1,x2,y2) { + c.strokeStyle = this.selected ? selected_style : normal_style; + var nx1 = this.transform_x(x1,y1) + this.x; + var ny1 = this.transform_y(x1,y1) + this.y; + var nx2 = this.transform_x(x2,y2) + this.x; + var ny2 = this.transform_y(x2,y2) + this.y; + this.sch.draw_line(c,nx1,ny1,nx2,ny2,1); +} + +Component.prototype.draw_circle = function(c,x,y,radius,filled) { + if (filled) c.fillStyle = this.selected ? selected_style : normal_style; + else c.strokeStyle = this.selected ? selected_style : normal_style; + var nx = this.transform_x(x,y) + this.x; + var ny = this.transform_y(x,y) + this.y; + + this.sch.draw_arc(c,nx,ny,radius,0,2*Math.PI,false,1,filled); +} + +rot_angle = [ + 0.0, // NORTH (identity) + Math.PI/2, // EAST (rot270) + Math.PI, // SOUTH (rot180) + 3*Math.PI/2, // WEST (rot90) + 0.0, // RNORTH (negy) + Math.PI/2, // REAST (int-neg) + Math.PI, // RSOUTH (negx) + 3*Math.PI/2, // RWEST (int-pos) +]; + +Component.prototype.draw_arc = function(c,x,y,radius,start_radians,end_radians) { + c.strokeStyle = this.selected ? selected_style : normal_style; + var nx = this.transform_x(x,y) + this.x; + var ny = this.transform_y(x,y) + this.y; + this.sch.draw_arc(c,nx,ny,radius, + start_radians+rot_angle[this.rotation],end_radians+rot_angle[this.rotation], + false,1,false); +} + +Component.prototype.draw = function(c) { +} + +// result of rotating an alignment [rot*9 + align] +aOrient = [ + 0, 1, 2, 3, 4, 5, 6, 7, 8, // NORTH (identity) + 2, 5, 8, 1, 4, 7, 0, 3, 6, // EAST (rot270) + 8, 7, 6, 5, 4, 3, 2, 1, 0, // SOUTH (rot180) + 6, 3, 0, 7, 4, 1, 8, 5, 3, // WEST (rot90) + 2, 1, 0, 5, 4, 3, 8, 7, 6, // RNORTH (negy) + 8, 5, 2, 7, 4, 1, 6, 3, 0, // REAST (int-neg) + 6, 7, 8, 3, 4, 5, 0, 1, 2, // RSOUTH (negx) + 0, 3, 6, 1, 4, 7, 2, 5, 8 // RWEST (int-pos) +]; + +textAlign = [ + 'left', 'center', 'right', + 'left', 'center', 'right', + 'left', 'center', 'right' +]; + +textBaseline = [ + 'top', 'top', 'top', + 'middle', 'middle', 'middle', + 'bottom', 'bottom', 'bottom' +]; + +Component.prototype.draw_text = function(c,text,x,y,alignment,size) { + var a = aOrient[this.rotation*9 + alignment]; + c.textAlign = textAlign[a]; + c.textBaseline = textBaseline[a]; + c.fillStyle = this.selected ? selected_style : normal_style; + this.sch.draw_text(c,text, + this.transform_x(x,y) + this.x, + this.transform_y(x,y) + this.y, + size); +} + +Component.prototype.set_select = function(which) { + if (which != this.selected) { + this.selected = which; + // create an undoable edit record here + } +} + +Component.prototype.select = function(x,y,shiftKey) { + this.was_previously_selected = this.selected; + if (inside(this.bbox,x,y)) { + this.set_select(shiftKey ? !this.selected : true); + return true; + } else return false; +} + +Component.prototype.select_rect = function(s) { + this.was_previously_selected = this.selected; + if (intersect(this.bbox,s)) + this.set_select(true); +} + +// if connection point of component c bisects the +// wire represented by this compononent, return that +// connection point. Otherwise return null. +Component.prototype.bisect = function(c) { + return null; +} + +Component.prototype.edit_properties = function(x,y) { + if (inside(this.bbox,x,y)) { + var content = document.createElement('table'); + content.style.marginBotton = '5px'; + content.fields = []; + + // add an field for each property + for (var i in this.properties) { + var label = document.createTextNode(i + ': '); + var field = document.createElement('input'); + field.type = 'text'; + field.value = this.properties[i]; + field.size = 10; + content.fields.push([i,field]); + + var col1 = document.createElement('td'); + col1.appendChild(label); + var col2 = document.createElement('td'); + col2.appendChild(field); + var row = document.createElement('tr'); + row.appendChild(col1); + row.appendChild(col2); + row.style.verticalAlign = 'center'; + + content.appendChild(row); + } + + var component = this; // capture in closure below + this.sch.dialog('Edit Properties',content,function(content) { + var fields = content.fields; + for (var i = fields.length - 1; i >= 0; i--) + component.properties[fields[i][0]] = fields[i][1].value; + component.sch.redraw(); // component is selected, so this will redraw it + }); + return true; + } else return false; +} + +// clear the labels on all connections +Component.prototype.clear_labels = function() { + for (var i = this.connections.length - 1; i >=0; --i) { + this.connections[i].clear_label(); + } +} + +// default action: don't propagate label +Component.prototype.propagate_label = function(label) { +} + +// give components a chance to generate default labels for their connection(s) +// default action: do nothing +Component.prototype.add_default_labels = function() { +} + +// component should generate labels for all unlabeled connections +Component.prototype.label_connections = function() { + for (var i = this.connections.length - 1; i >=0; --i) { + var cp = this.connections[i]; + if (!cp.label) + cp.propagate_label(this.sch.get_next_label()); + } +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Connection point +// +//////////////////////////////////////////////////////////////////////////////// + +connection_point_radius = 2; + +function ConnectionPoint(parent,x,y) { + this.parent = parent; + this.offset_x = x; + this.offset_y = y; + this.location = ''; + this.update_location(); + this.label = null; +} + +ConnectionPoint.prototype.toString = function() { + return ''; +} + +ConnectionPoint.prototype.json = function() { + return this.label; +} + +ConnectionPoint.prototype.clear_label = function() { + this.label = null; +} + +ConnectionPoint.prototype.propagate_label = function(label) { + // should we check if existing label is the same? it should be... + + if (this.label == null) { + // label this connection point + this.label = label; + + // propagate label to coincident connection points + this.parent.sch.propagate_label(label,this.location); + + // possibly label other cp's for this device? + this.parent.propagate_label(label); + } +} + +ConnectionPoint.prototype.update_location = function() { + // update location string which we use as a key to find coincident connection points + var old_location = this.location; + var parent = this.parent; + var nx = parent.transform_x(this.offset_x,this.offset_y) + parent.x; + var ny = parent.transform_y(this.offset_x,this.offset_y) + parent.y; + this.x = nx; + this.y = ny; + this.location = nx + ',' + ny; + + // add ourselves to the connection list for the new location + parent.sch.update_connection_point(this,old_location); +} + +ConnectionPoint.prototype.coincident = function(x,y) { + return this.x==x && this.y==y; +} + +ConnectionPoint.prototype.draw = function(c,n) { + if (n != 2) + this.parent.draw_circle(c,this.offset_x,this.offset_y,connection_point_radius,n > 2); +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Wire +// +//////////////////////////////////////////////////////////////////////////////// + +near_distance = 2; // how close to wire counts as "near by" + +function Wire(sch,x1,y1,x2,y2) { + // arbitrarily call x1,y1 the origin + Component.call(this,sch,x1,y1,0); + this.dx = x2 - x1; + this.dy = y2 - y1; + this.add_connection(0,0); + this.add_connection(this.dx,this.dy); + + // compute bounding box (expanded slightly) + var r = [0,0,this.dx,this.dy]; + canonicalize(r); + r[0] -= near_distance; + r[1] -= near_distance; + r[2] += near_distance; + r[3] += near_distance; + this.bounding_box = r; + this.update_coords(); // update bbox + + // used in selection calculations + this.len = Math.sqrt(this.dx*this.dx + this.dy*this.dy); +} +Wire.prototype = new Component(); +Wire.prototype.constructor = Wire; + +Wire.prototype.toString = function() { + return ''; +} + +Wire.prototype.json = function() { + var json = ['w',[this.x, this.y, this.x+this.dx, this.y+this.dy]]; + return json; +} + +Wire.prototype.draw = function(c) { + this.draw_line(c,0,0,this.dx,this.dy); +} + +Wire.prototype.clone = function(sch,x,y) { + return new Wire(sch,x,y,x+this.dx,y+this.dy); +} + +Wire.prototype.near = function(x,y) { + // crude check: (x,y) within expanded bounding box of wire + if (inside(this.bbox,x,y)) { + // compute distance between x,y and nearst point on line + // http://www.allegro.cc/forums/thread/589720 + var D = Math.abs((x - this.x)*this.dy - (y - this.y)*this.dx)/this.len; + if (D <= near_distance) return true; + } + return false; +} + +Wire.prototype.select = function(x,y,shiftKey) { + this.was_previously_selected = this.selected; + if (this.near(x,y)) { + this.set_select(shiftKey ? !this.selected : true); + return true; + } else return false; +} + +// selection rectangle selects wire only if it includes +// one of the end points +Wire.prototype.select_rect = function(s) { + this.was_previously_selected = this.selected; + if (inside(s,this.x,this.y) || inside(s,this.x+this.dx,this.y+this.dy)) + this.set_select(true); +} + +// if connection point of component c bisects the +// wire represented by this compononent, return that +// connection point. Otherwise return null. +Wire.prototype.bisect = function(c) { + for (var i = c.connections.length - 1; i >= 0; --i) { + var cp = c.connections[i]; + var x = cp.x; + var y = cp.y; + + // crude check: (x,y) within expanded bounding box of wire + if (inside(this.bbox,x,y)) { + // compute distance between x,y and nearst point on line + // http://www.allegro.cc/forums/thread/589720 + var D = Math.abs((x - this.x)*this.dy - (y - this.y)*this.dx)/this.len; + // final check: ensure point isn't an end point of the wire + if (D < 1 && !this.connections[0].coincident(x,y) && !this.connections[1].coincident(x,y)) + return cp; + } + } + return null; +} + +Wire.prototype.move_end = function() { + this.sch.check_wires(this); +} + +// wires "conduct" their label to the other end +Wire.prototype.propagate_label = function(label) { + // don't worry about relabeling a cp, it won't recurse! + this.connections[0].propagate_label(label); + this.connections[1].propagate_label(label); +} + +// some actual component will start the labeling of electrical nodes, +// so do nothing here +Wire.prototype.label_connections = function() { +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Ground +// +//////////////////////////////////////////////////////////////////////////////// + +function Ground(sch,x,y,rotation) { + Component.call(this,sch,x,y,rotation); + this.add_connection(0,0); + this.bounding_box = [-6,0,6,8]; + this.update_coords(); + this.type = 'g'; +} +Ground.prototype = new Component(); +Ground.prototype.constructor = Ground; + +Ground.prototype.toString = function() { + return ''; +} + +Ground.prototype.draw = function(c) { + this.draw_line(c,0,0,0,8); + this.draw_line(c,-6,8,6,8); +} + +Ground.prototype.clone = function(sch,x,y) { + return new Ground(sch,x,y,this.rotation); +} + +// give components a chance to generate a label for their connection(s) +// default action: do nothing +Ground.prototype.add_default_labels = function() { + this.connections[0].propagate_label('0'); // canonical label for GND node +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Resistor +// +//////////////////////////////////////////////////////////////////////////////// + +function Resistor(sch,x,y,rotation,name,r) { + Component.call(this,sch,x,y,rotation); + this.properties['name'] = name; + this.properties['r'] = r ? r : '1'; + this.add_connection(0,0); + this.add_connection(0,48); + this.bounding_box = [-4,0,4,48]; + this.update_coords(); + this.type = 'r'; +} +Resistor.prototype = new Component(); +Resistor.prototype.constructor = Resistor; + +Resistor.prototype.toString = function() { + return ''; +} + +Resistor.prototype.draw = function(c) { + this.draw_line(c,0,0,0,12); + this.draw_line(c,0,12,4,14); + this.draw_line(c,4,14,-4,18); + this.draw_line(c,-4,18,4,22); + this.draw_line(c,4,22,-4,26); + this.draw_line(c,-4,26,4,30); + this.draw_line(c,4,30,-4,34); + this.draw_line(c,-4,34,0,36); + this.draw_line(c,0,36,0,48); + if (this.properties['r']) + this.draw_text(c,this.properties['r']+'\u03A9',5,24,3,property_size); + if (this.properties['name']) + this.draw_text(c,this.properties['name'],-5,24,5,property_size); +} + +Resistor.prototype.clone = function(sch,x,y) { + return new Resistor(sch,x,y,this.rotation,'',this.properties['r']); +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Capacitor +// +//////////////////////////////////////////////////////////////////////////////// + +function Capacitor(sch,x,y,rotation,name,c) { + Component.call(this,sch,x,y,rotation); + this.properties['name'] = name; + this.properties['c'] = c ? c : '1p'; + this.add_connection(0,0); + this.add_connection(0,48); + this.bounding_box = [-8,0,8,48]; + this.update_coords(); + this.type = 'c'; +} +Capacitor.prototype = new Component(); +Capacitor.prototype.constructor = Capacitor; + +Capacitor.prototype.toString = function() { + return ''; +} + +Capacitor.prototype.draw = function(c) { + this.draw_line(c,0,0,0,22); + this.draw_line(c,-8,22,8,22); + this.draw_line(c,-8,26,8,26); + this.draw_line(c,0,26,0,48); + if (this.properties['c']) + this.draw_text(c,this.properties['c']+'F',9,24,3,property_size); + if (this.properties['name']) + this.draw_text(c,this.properties['name'],-9,24,5,property_size); +} + +Capacitor.prototype.clone = function(sch,x,y) { + return new Capacitor(sch,x,y,this.rotation,'',this.properties['c']); +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Inductor +// +//////////////////////////////////////////////////////////////////////////////// + +function Inductor(sch,x,y,rotation,name,l) { + Component.call(this,sch,x,y,rotation); + this.properties['name'] = name; + this.properties['l'] = l ? l : '1n'; + this.add_connection(0,0); + this.add_connection(0,48); + this.bounding_box = [-4,0,5,48]; + this.update_coords(); + this.type = 'l'; +} +Inductor.prototype = new Component(); +Inductor.prototype.constructor = Inductor; + +Inductor.prototype.toString = function() { + return ''; +} + +Inductor.prototype.draw = function(c) { + this.draw_line(c,0,0,0,14); + this.draw_arc(c,0,18,4,6*Math.PI/4,3*Math.PI/4); + this.draw_arc(c,0,24,4,5*Math.PI/4,3*Math.PI/4); + this.draw_arc(c,0,30,4,5*Math.PI/4,2*Math.PI/4); + this.draw_line(c,0,34,0,48); + + if (this.properties['l']) + this.draw_text(c,this.properties['l']+'H',6,24,3,property_size); + if (this.properties['name']) + this.draw_text(c,this.properties['name'],-3,24,5,property_size); +} + +Inductor.prototype.clone = function(sch,x,y) { + return new Inductor(sch,x,y,this.rotation,'',this.properties['l']); +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Source +// +//////////////////////////////////////////////////////////////////////////////// + +function Source(sch,x,y,rotation,name,type,value) { + Component.call(this,sch,x,y,rotation); + this.type = type; + this.properties['name'] = name; + this.properties['value'] = value ? value : '1'; + this.add_connection(0,0); + this.add_connection(0,48); + this.bounding_box = [-12,0,12,48]; + this.update_coords(); +} +Source.prototype = new Component(); +Source.prototype.constructor = Source; + +Source.prototype.toString = function() { + return '<'+this.type+'source '+this.properties['params']+' ('+this.x+','+this.y+')>'; +} + +Source.prototype.draw = function(c) { + this.draw_line(c,0,0,0,12); + this.draw_circle(c,0,24,12,false); + this.draw_line(c,0,36,0,48); + + if (this.type == 'v') { // voltage source + // draw + and - + this.draw_line(c,8,5,8,11); + this.draw_line(c,5,8,11,8); + this.draw_line(c,5,40,11,40); + // draw V + this.draw_line(c,-3,20,0,28); + this.draw_line(c,3,20,0,28); + } else if (this.type == 'i') { // current source + // draw arrow: pos to neg + this.draw_line(c,0,16,0,32); + this.draw_line(c,-3,24,0,32); + this.draw_line(c,3,24,0,32); + } + + if (this.properties['name']) + this.draw_text(c,this.properties['name'],-13,24,5,property_size); + if (this.properties['value']) + this.draw_text(c,this.properties['value']+(this.type=='v'?'V':'A'),13,24,3,property_size); +} + +Source.prototype.clone = function(sch,x,y) { + return new Source(sch,x,y,this.rotation,'',this.type,this.properties['value']); +} + +function VSource(sch,x,y,rotation,name,value) { + Source.call(this,sch,x,y,rotation,name,'v',value); + this.type = 'v'; +} +VSource.prototype = new Component(); +VSource.prototype.constructor = VSource; +VSource.prototype.toString = Source.prototype.toString; +VSource.prototype.draw = Source.prototype.draw; +VSource.prototype.clone = Source.prototype.clone; + +function ISource(sch,x,y,rotation,name,value) { + Source.call(this,sch,x,y,rotation,name,'i',value); + this.type = 'i'; +} +ISource.prototype = new Component(); +ISource.prototype.constructor = ISource; +ISource.prototype.toString = Source.prototype.toString; +ISource.prototype.draw = Source.prototype.draw; +ISource.prototype.clone = Source.prototype.clone; diff --git a/js/schematic.js-old b/js/schematic.js-old new file mode 100644 index 0000000000..903a678973 --- /dev/null +++ b/js/schematic.js-old @@ -0,0 +1,1587 @@ +////////////////////////////////////////////////////////////////////////////// +// +// Simple schematic capture +// +//////////////////////////////////////////////////////////////////////////////// + +// Chris Terman, Nov. 2011 + +// add schematics to a document with +// +// +// +// other attributes you can add to the input tag: +// width -- width in pixels of diagram +// height -- height in pixels of diagram +// parts -- comma-separated list of parts for parts bin (see parts_map) + +// TO DO: + +// - read initial diagram from value of hidden input field +// - write diagram state into value of hidden input field +// - wire labels? +// - devices: diode, nfet, pfet, opamp, scope probe +// - icons for test equipment? (scope, sig gen, counter, ...) + +// - rotate multiple objects around their center of mass +// - rubber band wires when moving components +// - add help messages/tooltips +// - tool bar (zoom in/zoom out, rotate, mode: wire/select, save/restore, simulate, help) +// - scroll canvas +// - freeze_diagram, freeze_properties attributes (freeze certain components/properties?) +// - add thumb to resize work area + +// - label nodes, extract component netlist +// - simulation: operating points, trans, ac analysis, sweeps? +// - how to integrate plot display? + +// add ourselves to the tasks that get performed when window is loaded +window.onload = add_schematic_handler(window.onload); + +function update_schematics() { + // set up each schematic on the page + var schematics = document.getElementsByClassName('schematic'); + for (var i = schematics.length - 1; i >= 0; i--) + if (schematics[i].getAttribute("loaded") != "true") { + new Schematic(schematics[i]); + schematics[i].setAttribute("loaded","true"); + } +} + +function add_schematic_handler(other_onload) { + return function() { + // execute othe onload functions first + if (other_onload) other_onload(); + + update_schematics(); + } +} + +background_style = 'rgb(200,255,200)'; +element_style = 'rgb(255,255,255)'; + +// list of all the defined parts +parts_map = { + 'g': Ground, + 'v': VSource, + 'i': ISource, + 'r': Resistor, + 'c': Capacitor, + 'l': Inductor, +}; + +/////////////////////////////////////////////////////////////////////////////// +// +// Schematic = diagram + parts bin + status area +// +//////////////////////////////////////////////////////////////////////////////// + +// setup a schematic by populating the
with the appropriate children +function Schematic(input) { + var div = document.createElement('div'); + // set up div so we can position elements inside of it + div.style.position = 'relative'; + + // grab attributes from the
that created us + this.div = div; + this.grid = 8; + + this.scale = 2; + this.origin_x = 0; + this.origin_y = 0; + + // start with a background element with normal positioning + this.background = document.createElement('canvas'); + this.background.style.backgroundColor = background_style; + + this.status_div = document.createElement('div'); + this.status_div.style.borderStyle = 'solid'; + this.status_div.style.borderWidth = '1px'; + this.status_div.style.position = 'absolute'; + this.status_div.style.padding = '2px'; + this.status_div.style.backgroundColor = element_style; + this.status = document.createTextNode('Ready.'); + this.status_div.appendChild(this.status); + + this.connection_points = new Array(); // location string => list of cp's + this.components = []; + + // this is where schematic is rendered + this.canvas = document.createElement('canvas'); + this.canvas.tabIndex = 1; // so we get keystrokes + this.canvas.style.borderStyle = 'solid'; + this.canvas.style.borderWidth = '1px'; + this.canvas.style.position = 'absolute'; + + this.canvas.schematic = this; + this.canvas.addEventListener('mousemove',this.mouse_move,false); + this.canvas.addEventListener('mouseover',this.mouse_enter,false); + this.canvas.addEventListener('mouseout',this.mouse_leave,false); + this.canvas.addEventListener('mousedown',this.mouse_down,false); + this.canvas.addEventListener('mouseup',this.mouse_up,false); + this.canvas.addEventListener('dblclick',this.double_click,false); + this.canvas.addEventListener('keydown',this.key_down,false); + this.canvas.addEventListener('keypress',this.key_press,false); + + // make the canvas "clickable" by registering a dummy click handler + // this should make things work on the iPad + this.canvas.addEventListener('click',function(){},false); + + this.dragging = false; + this.drawCursor = false; + this.cursor_x = 0; + this.cursor_y = 0; + this.draw_cursor = null; + this.select_rect = null; + this.wire = null; + + // repaint simply draws this buffer and then adds selected elements on top + this.bg_image = document.createElement('canvas'); + + // use user-supplied list of parts if supplied + // else just populate parts bin with all the parts + var parts = input.getAttribute('parts'); + if (parts) parts = parts.split(','); + else { + parts = new Array(); + for (var p in parts_map) parts.push(p); + } + + // now add the parts to the parts bin + var parts_left = this.width + 3 + background_margin; + var parts_top = background_margin; + this.parts_bin = []; + for (var i = 0; i < parts.length; i++) { + var part = new Part(this); + part.set_component(new parts_map[parts[i]](part,0,0,0)); + this.parts_bin.push(part); + } + + // add all elements to the DOM + div.appendChild(this.background); + div.appendChild(this.canvas); + div.appendChild(this.status_div); + for (var i = 0; i < this.parts_bin.length; i++) + div.appendChild(this.parts_bin[i].canvas); + input.parentNode.insertBefore(div,input.nextSibling); + + // make sure other code can find us! + input.schematic = this; + this.input = input; + + // set locations of all the elements in the editor + var w = parseInt(input.getAttribute('width')); + var h = parseInt(input.getAttribute('height')); + this.set_locations(w,h); +} + +background_margin = 5; +part_w = 56; // size of a parts bin compartment +part_h = 56; +status_height = 18; +thumb_style = 'rgb(128,128,128)'; + +// w,h are the dimensions of the canvas, everyone else is positioned accordingly +Schematic.prototype.set_locations = function(w,h) { + // limit the shrinkage factor + w = Math.max(w,120); + h = Math.max(h,120); + + this.width = w; + this.height = h; + this.bg_image.width = w; + this.bg_image.height = h; + + this.min_x = 0; + this.min_y = 0; + this.max_x = w/this.scale; + this.max_y = h/this.scale; + + // configure canvas + this.canvas.style.left = background_margin + 'px'; + this.canvas.style.top = background_margin + 'px'; + this.canvas.width = w; + this.canvas.height = h; + this.redraw_background(); // redraw diagram + + // configure status bar + this.status_div.style.left = background_margin + 'px'; + this.status_div.style.top = this.canvas.offsetTop + this.canvas.offsetHeight + 3 + 'px'; + this.status_div.style.width = (w - 4) + 'px'; // subtract interior padding + this.status_div.style.height = status_height + 'px'; + + // configure parts bin + var total_w = this.canvas.offsetLeft + this.canvas.offsetWidth; + var parts_left = total_w + 3; + var parts_top = background_margin; + var parts_h_limit = this.canvas.offsetTop + this.canvas.offsetHeight; + for (var i = 0; i < this.parts_bin.length; i++) { + var part = this.parts_bin[i]; + part.set_location(parts_left,parts_top); + + total_w = part.right(); + parts_top = part.bottom()-1; + if (parts_top + part_h > parts_h_limit) { + parts_left = total_w - 1; + parts_top = background_margin; + } + } + + // configure background + var total_h = this.status_div.offsetTop + this.status_div.offsetHeight + background_margin; + total_w += background_margin; + this.background.height = total_h; + this.background.width = total_w; + + /* enable when there's support for resizing schematic + // redraw thumb + var c = this.background.getContext('2d'); + c.clearRect(0,0,w,h); + c.strokeStyle = thumb_style; + c.lineWidth = 1; + c.beginPath(); + w = total_w - 1; + h = total_h - 1; + c.moveTo(w,h-4); c.lineTo(w-4,h); + c.moveTo(w,h-8); c.lineTo(w-8,h); + c.moveTo(w,h-12); c.lineTo(w-12,h); + c.stroke(); + */ +} + +// update the value field of our corresponding input field with JSON +// representation of schematic +Schematic.prototype.update_value = function() { + // to do: fill in this.input.value +} + +Schematic.prototype.add_component = function(new_c) { + this.components.push(new_c); + + // create undoable edit record here +} + +Schematic.prototype.remove_component = function(c) { + var index = this.components.indexOf(c); + if (index != -1) this.components.splice(index,1); +} + +// add connection point to list of connection points at that location +Schematic.prototype.add_connection_point = function(cp) { + cplist = this.connection_points[cp.location] + if (cplist) cplist.push(cp); + else { + cplist = [cp]; + this.connection_points[cp.location] = cplist; + } + + // return list of conincident connection points + return cplist; +} + +// remove connection point from the list points at the old location +Schematic.prototype.remove_connection_point = function(cp,old_location) { + // remove cp from list at old location + var cplist = this.connection_points[old_location]; + if (cplist) { + var index = cplist.indexOf(cp); + if (index != -1) { + cplist.splice(index,1); + // if no more connections at this location, remove + // entry from array to keep our search time short + if (cplist.length == 0) + delete this.connection_points[old_location]; + } + } +} + +// connection point has changed location: remove, then add +Schematic.prototype.update_connection_point = function(cp,old_location) { + this.remove_connection_point(cp,old_location); + return this.add_connection_point(cp); +} + +// add a wire to the schematic +Schematic.prototype.add_wire = function(x1,y1,x2,y2) { + var new_wire = new Wire(this,x1,y1,x2,y2); + this.add_component(new_wire); + new_wire.move_end(); + return new_wire; +} + +// see if connection points of component c split any wires +Schematic.prototype.check_wires = function(c) { + for (var i = this.components.length - 1; i >=0; --i) { + var cc = this.components[i]; + if (cc != c) { // don't check a component against itself + // only wires will do return non-null from a bisect call + var cp = cc.bisect(c); + if (cp) { + // cc is a wire bisected by connection point cp + + // remove biscted wire + cc.delete(); + + // add two new wires with cp in the middle + this.add_wire(cc.x,cc.y,cp.x,cp.y); + this.add_wire(cc.x+cc.dx,cc.y+cc.dy,cp.x,cp.y); + this.redraw_background(); + break; + } + } + } +} + +/////////////////////////////////////////////////////////////////////////////// +// +// Drawing support -- deals with scaling and scrolling of diagrama +// +//////////////////////////////////////////////////////////////////////////////// + +// here to redraw background image containing static portions of the schematic. +// Also redraws dynamic portion. +Schematic.prototype.redraw_background = function() { + var c = this.bg_image.getContext('2d'); + var w = this.bg_image.width; + var h = this.bg_imageheight; + + // paint background color + c.fillStyle = element_style; + c.fillRect(0,0,this.width,this.height); + + // border + //c.strokeStyle = "rgb(0,0,0)"; + //c.strokeRect(0,0,this.width,this.height); + + // grid + c.strokeStyle = "rgb(128,128,128)"; + var first_x = this.min_x; + var last_x = this.max_x; + var first_y = this.min_y; + var last_y = this.max_y; + for (var i = first_x; i < last_x; i += this.grid) + this.draw_line(c,i,first_y,i,last_y,0.1); + for (var i = first_y; i < last_y; i += this.grid) + this.draw_line(c,first_x,i,last_x,i,0.1); + + // unselected components + for (var i = this.components.length - 1; i >= 0; --i) { + var component = this.components[i]; + if (!component.selected) component.draw(c); + } + + this.redraw(); // background changed, redraw on screen +} + +// redraw what user sees = static image + dynamic parts +Schematic.prototype.redraw = function() { + var c = this.canvas.getContext('2d'); + + // put static image in the background + c.drawImage(this.bg_image, 0, 0); + + // selected components + for (var i = this.components.length - 1; i >= 0; --i) { + var component = this.components[i]; + if (component.selected) component.draw(c); + } + + // connection points: draw one at each location + for (var location in this.connection_points) { + var cplist = this.connection_points[location]; + cplist[0].draw(c,cplist.length); + } + + // draw new wire + if (this.wire) { + var r = this.wire; + c.strokeStyle = selected_style; + this.draw_line(c,r[0],r[1],r[2],r[3],1); + } + + // draw selection rectangle + if (this.select_rect) { + var r = this.select_rect; + c.lineWidth = 1; + c.strokeStyle = selected_style; + c.beginPath(); + c.moveTo(r[0],r[1]); + c.lineTo(r[0],r[3]); + c.lineTo(r[2],r[3]); + c.lineTo(r[2],r[1]); + c.lineTo(r[0],r[1]); + c.stroke(); + } + + // finally overlay cursor + if (this.drawCursor && this.draw_cursor) { + //var x = this.cursor_x; + //var y = this.cursor_y; + //this.draw_text(c,'('+x+','+y+')',x+this.grid,y-this.grid,10); + this.draw_cursor(c,this.cursor_x,this.cursor_y); + } +} + +// draws a cross cursor +Schematic.prototype.cross_cursor = function(c,x,y) { + this.draw_line(c,x-this.grid,y,x+this.grid,y,1); + this.draw_line(c,x,y-this.grid,x,y+this.grid,1); +} + +Schematic.prototype.draw_line = function(c,x1,y1,x2,y2,width) { + c.lineWidth = width*this.scale; + c.beginPath(); + c.moveTo((x1 - this.origin_x) * this.scale,(y1 - this.origin_y) * this.scale); + c.lineTo((x2 - this.origin_x) * this.scale,(y2 - this.origin_y) * this.scale); + c.stroke(); +} + +Schematic.prototype.draw_arc = function(c,x,y,radius,start_radians,end_radians,anticlockwise,width,filled) { + c.lineWidth = width*this.scale; + c.beginPath(); + c.arc((x - this.origin_x)*this.scale,(y - this.origin_y)*this.scale,radius*this.scale, + start_radians,end_radians,anticlockwise); + if (filled) c.fill(); + else c.stroke(); +} + +Schematic.prototype.draw_text = function(c,text,x,y,size) { + c.font = size*this.scale+'pt sans-serif' + c.fillText(text,(x - this.origin_x) * this.scale,(y - this.origin_y) * this.scale); +} + +// add method to canvas to compute relative coords for event +HTMLCanvasElement.prototype.relMouseCoords = function(event){ + // run up the DOM tree to figure out coords for top,left of canvas + var totalOffsetX = 0; + var totalOffsetY = 0; + var canvasY = 0; + var currentElement = this; + do { + totalOffsetX += currentElement.offsetLeft; + totalOffsetY += currentElement.offsetTop; + } + while(currentElement = currentElement.offsetParent); + + // now compute relative position of click within the canvas + this.mouse_x = event.pageX - totalOffsetX; + this.mouse_y = event.pageY - totalOffsetY; +} + +/////////////////////////////////////////////////////////////////////////////// +// +// Event handling +// +//////////////////////////////////////////////////////////////////////////////// + +// process special keys here since they don't get delivered correctly on keypress +Schematic.prototype.key_down = function(event) { + if (!event) event = window.event; + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + var code = event.keyCode; + + if (code == 8 || code == 46) { + // delete selected components + for (var i = sch.components.length - 1; i >= 0; --i) { + var component = sch.components[i]; + if (component.selected) component.delete(1); + } + sch.redraw(); + event.preventDefault(); + return false; + } + return true; +} + +// process normal characters +Schematic.prototype.key_press = function(event) { + if (!event) event = window.event; + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + var code = window.event ? event.keyCode : event.charCode; + var char = String.fromCharCode(code); + + if (char == 'r' || char == 'R') { + // rotate + for (var i = sch.components.length - 1; i >= 0; --i) { + var component = sch.components[i]; + if (component.selected) component.rotate(1); + } + sch.redraw(); + event.preventDefault(); + return false; + } + return true; +} + +Schematic.prototype.mouse_enter = function(event) { + if (!event) event = window.event; + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + + // see if user has selected a new part + if (sch.new_part) { + // grab incoming part, turn off selection of parts bin + var part = sch.new_part; + sch.new_part = null; + part.select(false); + + // make a clone of the component in the parts bin + part = part.component.clone(sch,sch.cursor_x,sch.cursor_y); + + // unselect everything else in the schematic, add part and select it + sch.unselect_all(-1); + sch.redraw_background(); // so we see any components that got unselected + sch.add_component(part); + part.set_select(true); + + // and start dragging it + sch.drag_begin(); + } + + sch.drawCursor = true; + sch.redraw(); + sch.canvas.focus(); // capture key strokes +} + +Schematic.prototype.mouse_leave = function(event) { + if (!event) event = window.event; + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + sch.drawCursor = false; + sch.redraw(); +} + +Schematic.prototype.unselect_all = function(which) { + for (var i = this.components.length - 1; i >= 0; --i) + if (i != which) this.components[i].set_select(false); +} + +Schematic.prototype.mouse_down = function(event) { + if (!event) event = window.event; + else event.preventDefault(); + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + + // determine where event happened in schematic coordinates + sch.canvas.relMouseCoords(event); + var x = sch.canvas.mouse_x/sch.scale + sch.origin_x; + var y = sch.canvas.mouse_y/sch.scale + sch.origin_y; + sch.cursor_x = Math.round(x/sch.grid) * sch.grid; + sch.cursor_y = Math.round(y/sch.grid) * sch.grid; + + // is mouse over a connection point? If so, start dragging a wire + var cplist = sch.connection_points[sch.cursor_x + ',' + sch.cursor_y]; + if (cplist && !event.shiftKey) { + sch.unselect_all(-1); + sch.wire = [sch.cursor_x,sch.cursor_y,sch.cursor_x,sch.cursor_y]; + } else { + // give all components a shot at processing the selection event + var which = -1; + for (var i = sch.components.length - 1; i >= 0; --i) + if (sch.components[i].select(x,y,event.shiftKey)) { + if (sch.components[i].selected) { + sch.drag_begin(); + which = i; // keep track of component we found + } + break; + } + // did we just click on a previously selected component? + var reselect = which!=-1 && sch.components[which].was_previously_selected; + + if (!event.shiftKey) { + // if shift key isn't pressed and we didn't click on component + // that was already selected, unselect everyone except component + // we just clicked on + if (!reselect) sch.unselect_all(which); + + // if there's nothing to drag, set up a selection rectangle + if (!sch.dragging) sch.select_rect = [sch.canvas.mouse_x,sch.canvas.mouse_y, + sch.canvas.mouse_x,sch.canvas.mouse_y]; + } + } + + sch.redraw_background(); +} + +Schematic.prototype.mouse_up = function(event) { + if (!event) event = window.event; + else event.preventDefault(); + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + + // drawing a new wire + if (sch.wire) { + var r = sch.wire; + sch.wire = null; + + if (r[0]!=r[2] || r[1]!=r[3]) { + // insert wire component + sch.add_wire(r[0],r[1],r[2],r[3]); + sch.redraw_background(); + } else sch.redraw(); + } + + // dragging + if (sch.dragging) sch.drag_end(); + + // selection rectangle + if (sch.select_rect) { + var r = sch.select_rect; + + // if select_rect is a point, we've already dealt with selection + // in mouse_down handler + if (r[0]!=r[2] || r[1]!=r[3]) { + // convert to schematic coordinates + var s = [r[0]/sch.scale + sch.origin_x, r[1]/sch.scale + sch.origin_y, + r[2]/sch.scale + sch.origin_x, r[3]/sch.scale + sch.origin_y]; + canonicalize(s); + + if (!event.shiftKey) sch.unselect_all(); + + // select components that intersect selection rectangle + for (var i = sch.components.length - 1; i >= 0; --i) + sch.components[i].select_rect(s,event.shiftKey); + } + + sch.select_rect = null; + sch.redraw_background(); + } +} + +Schematic.prototype.double_click = function(event) { + if (!event) event = window.event; + else event.preventDefault(); + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + + // determine where event happened in schematic coordinates + sch.canvas.relMouseCoords(event); + var x = sch.canvas.mouse_x/sch.scale + sch.origin_x; + var y = sch.canvas.mouse_y/sch.scale + sch.origin_y; + sch.cursor_x = Math.round(x/sch.grid) * sch.grid; + sch.cursor_y = Math.round(y/sch.grid) * sch.grid; + + // see if we double-clicked a component. If so, edit it's properties + for (var i = sch.components.length - 1; i >= 0; --i) + if (sch.components[i].edit_properties(x,y)) break; +} + +Schematic.prototype.drag_begin = function() { + // let components know they're about to move + for (var i = this.components.length - 1; i >= 0; --i) { + var component = this.components[i]; + if (component.selected) component.move_begin(); + } + + // remember where drag started + this.drag_x = this.cursor_x; + this.drag_y = this.cursor_y; + this.dragging = true; +} + +Schematic.prototype.drag_end = function() { + // let components know they're done moving + for (var i = this.components.length - 1; i >= 0; --i) { + var component = this.components[i]; + if (component.selected) component.move_end(); + } + this.dragging = false; +} + +Schematic.prototype.mouse_move = function(event) { + if (!event) event = window.event; + var sch = (window.event) ? event.srcElement.schematic : event.target.schematic; + + sch.canvas.relMouseCoords(event); + var x = sch.canvas.mouse_x/sch.scale + sch.origin_x; + var y = sch.canvas.mouse_y/sch.scale + sch.origin_y; + sch.cursor_x = Math.round(x/sch.grid) * sch.grid; + sch.cursor_y = Math.round(y/sch.grid) * sch.grid; + + if (sch.wire) { + // update new wire end point + sch.wire[2] = sch.cursor_x; + sch.wire[3] = sch.cursor_y; + } else if (sch.dragging) { + // see how far we moved + var dx = sch.cursor_x - sch.drag_x; + var dy = sch.cursor_y - sch.drag_y; + if (dx != 0 || dy != 0) { + // update position for next time + sch.drag_x = sch.cursor_x; + sch.drag_y = sch.cursor_y; + + // give all components a shot at processing the event + for (var i = sch.components.length - 1; i >= 0; --i) { + var component = sch.components[i]; + if (component.selected) component.move(dx,dy); + } + } + } else if (sch.select_rect) { + // update moving corner of selection rectangle + sch.select_rect[2] = sch.canvas.mouse_x; + sch.select_rect[3] = sch.canvas.mouse_y; + //sch.message(sch.select_rect.toString()); + } + + // just redraw dynamic components + sch.redraw(); +} + +/////////////////////////////////////////////////////////////////////////////// +// +// Status message and dialogs +// +//////////////////////////////////////////////////////////////////////////////// + +Schematic.prototype.message = function(message) { + this.status.nodeValue = message; + this.recompute_height(); +} + +Schematic.prototype.append_message = function(message) { + this.status.nodeValue += ' / '+message; + this.recompute_height(); +} + +// set up a dialog with specified title, content and two buttons at +// the bottom: OK and Cancel. If Cancel is clicked, dialog goes away +// and we're done. If OK is clicked, dialog goes away and the +// callback function is called with the content as an argument (so +// that the values of any fields can be captured). +Schematic.prototype.dialog = function(title,content,callback) { + // create the div for the top level of the dialog, add to DOM + var dialog = document.createElement('div'); + dialog.sch = this; + dialog.content = content; + + // div to hold the title + var head = document.createElement('div'); + head.style.backgroundColor = 'black'; + head.style.color = 'white'; + head.style.textAlign = 'center'; + head.style.padding = '5px'; + head.appendChild(document.createTextNode(title)); + dialog.appendChild(head); + + // div to hold the content + var body = document.createElement('div'); + body.appendChild(content); + body.style.padding = '5px'; + dialog.appendChild(body); + + // OK button + var ok_button = document.createElement('span'); + ok_button.appendChild(document.createTextNode('OK')); + ok_button.dialog = dialog; // for the handler to use + ok_button.addEventListener('click',dialog_okay,false); + ok_button.style.border = '1px solid'; + ok_button.style.padding = '5px'; + ok_button.style.margin = '10px'; + + // cancel button + var cancel_button = document.createElement('span'); + cancel_button.appendChild(document.createTextNode('Cancel')); + cancel_button.dialog = dialog; // for the handler to use + cancel_button.addEventListener('click',dialog_cancel,false); + cancel_button.style.border = '1px solid'; + cancel_button.style.padding = '5px'; + cancel_button.style.margin = '10px'; + + // div to hold the two buttons + var buttons = document.createElement('div'); + buttons.appendChild(ok_button); + buttons.appendChild(cancel_button); + buttons.style.padding = '5px'; + buttons.style.margin = '10px'; + dialog.appendChild(buttons); + + // add to DOM + dialog.style.background = 'white'; + dialog.style.zindex = '1000'; + dialog.style.position = 'absolute'; + dialog.style.left = this.canvas.mouse_x+'px'; + dialog.style.top = this.canvas.mouse_y+'px'; + dialog.style.border = '2px solid'; + dialog.callback = callback; + this.div.appendChild(dialog); +} + +// callback when user click "Cancel" in a dialog +function dialog_cancel(event) { + if (!event) event = window.event; + var dialog = (window.event) ? event.srcElement.dialog : event.target.dialog; + + // remove the dialog from the top-level div of the schematic + dialog.parentNode.removeChild(dialog); +} + +// callback when user click "OK" in a dialog +function dialog_okay(event) { + if (!event) event = window.event; + var dialog = (window.event) ? event.srcElement.dialog : event.target.dialog; + + // remove the dialog from the top-level div of the schematic + dialog.parentNode.removeChild(dialog); + + // invoke the callback with the dialog contents as the argument + if (dialog.callback) dialog.callback(dialog.content); +} + +/////////////////////////////////////////////////////////////////////////////// +// +// Parts bin +// +//////////////////////////////////////////////////////////////////////////////// + +// one instance will be created for each part in the parts bin +function Part(sch) { + this.sch = sch; + this.component = null; + this.selected = false; + + // set up canvas + this.canvas = document.createElement('canvas'); + this.canvas.style.borderStyle = 'solid'; + this.canvas.style.borderWidth = '1px'; + this.canvas.style.position = 'absolute'; + this.canvas.height = part_w; + this.canvas.width = part_h; + this.canvas.part = this; + + this.canvas.addEventListener('mousedown',this.mouse_down,false); + this.canvas.addEventListener('mouseup',this.mouse_up,false); + + // make the part "clickable" by registering a dummy click handler + // this should make things work on the iPad + this.canvas.addEventListener('click',function(){},false); +} + +Part.prototype.set_location = function(left,top) { + this.canvas.style.left = left + 'px'; + this.canvas.style.top = top + 'px'; +} + +Part.prototype.right = function() { + return this.canvas.offsetLeft + this.canvas.offsetWidth; +} + +Part.prototype.bottom = function() { + return this.canvas.offsetTop + this.canvas.offsetHeight; +} + +Part.prototype.set_component = function(component) { + this.component = component; + + // figure out scaling and centering of parts icon + var b = component.bounding_box; + var dx = b[2] - b[0]; + var dy = b[3] - b[1]; + this.scale = 1; //Math.min(part_w/(1.2*dx),part_h/(1.2*dy)); + this.origin_x = this.scale*(b[0] + dx/2.0) - part_w/2.0; + this.origin_y = this.scale*(b[1] + dy/2.0) - part_h/2.0; + + this.redraw(); +} + +Part.prototype.redraw = function(part) { + var c = this.canvas.getContext('2d'); + + // paint background color + c.fillStyle = this.selected ? selected_style : element_style; + c.fillRect(0,0,part_w,part_h); + + if (this.component) this.component.draw(c); +} + +Part.prototype.select = function(which) { + this.selected = which; + this.redraw(); +} + +Part.prototype.update_connection_point = function(cp,old_location) { + // no connection points in the parts bin +} + +Part.prototype.draw_line = function(c,x1,y1,x2,y2,width) { + c.lineWidth = width*this.scale; + c.beginPath(); + c.moveTo((x1 - this.origin_x) * this.scale,(y1 - this.origin_y) * this.scale); + c.lineTo((x2 - this.origin_x) * this.scale,(y2 - this.origin_y) * this.scale); + c.stroke(); +} + +Part.prototype.draw_arc = function(c,x,y,radius,start_radians,end_radians,anticlockwise,width,filled) { + c.lineWidth = width*this.scale; + c.beginPath(); + c.arc((x - this.origin_x)*this.scale,(y - this.origin_y)*this.scale,radius*this.scale, + start_radians,end_radians,anticlockwise); + if (filled) c.fill(); + else c.stroke(); +} + +Part.prototype.draw_text = function(c,text,x,y,size) { + // no text displayed for the parts icon +} + +Part.prototype.mouse_down = function(event) { + if (!event) event = window.event; + var part = (window.event) ? event.srcElement.part : event.target.part; + + part.select(true); + part.sch.new_part = part; +} + +Part.prototype.mouse_up = function(event) { + if (!event) event = window.event; + var part = (window.event) ? event.srcElement.part : event.target.part; + + part.select(false); + part.sch.new_part = null; +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Rectangle helper functions +// +//////////////////////////////////////////////////////////////////////////////// + +// rect is an array of the form [left,top,right,bottom] + +// ensure left < right, top < bottom +function canonicalize(r) { + var temp; + + // canonicalize bounding box + if (r[0] > r[2]) { + temp = r[0]; + r[0] = r[2]; + r[2] = temp; + } + if (r[1] > r[3]) { + temp = r[1]; + r[1] = r[3]; + r[3] = temp; + } +} + +function between(x,x1,x2) { + return x1 <= x && x <= x2; +} + +function inside(rect,x,y) { + return between(x,rect[0],rect[2]) && between(y,rect[1],rect[3]); +} + +// only works for manhattan rectangles +function intersect(r1,r2) { + // look for non-intersection, negate result + var result = !(r2[0] > r1[2] || + r2[2] < r1[0] || + r2[1] > r1[3] || + r2[3] < r1[1]); + + // if I try to return the above expression, javascript returns undefined!!! + return result; +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Component base class +// +//////////////////////////////////////////////////////////////////////////////// + +property_size = 5; // point size for Component property text +normal_style = 'rgb(0,0,0)'; // color for unselected components +selected_style = 'rgb(64,255,64)'; // highlight color for selected components + +function Component(sch,x,y,rotation) { + this.sch = sch; + this.x = x; + this.y = y; + this.rotation = rotation; + this.selected = false; + this.properties = new Array(); + this.bounding_box = [0,0,0,0]; // in device coords [left,top,right,bottom] + this.bbox = this.bounding_box; // in absolute coords + this.connections = []; +} + +Component.prototype.add_connection = function(offset_x,offset_y) { + this.connections.push(new ConnectionPoint(this,offset_x,offset_y)); +} + +Component.prototype.update_coords = function() { + var x = this.x; + var y = this.y; + + // update bbox + var b = this.bounding_box; + this.bbox[0] = this.transform_x(b[0],b[1]) + x; + this.bbox[1] = this.transform_y(b[0],b[1]) + y; + this.bbox[2] = this.transform_x(b[2],b[3]) + x; + this.bbox[3] = this.transform_y(b[2],b[3]) + y; + canonicalize(this.bbox); + + // update connections + for (var i = this.connections.length - 1; i >= 0; --i) + this.connections[i].update_location(); +} + +Component.prototype.rotate = function(amount) { + var old_rotation = this.rotation; + this.rotation = (this.rotation + amount) % 8; + this.update_coords(); + + // create an undoable edit record here + // using old_rotation +} + +Component.prototype.move_begin = function() { + // remember where we started this move + this.move_x = this.x; + this.move_y = this.y; +} + +Component.prototype.move = function(dx,dy) { + // update coordinates + this.x += dx; + this.y += dy; + this.update_coords(); +} + +Component.prototype.move_end = function() { + var dx = this.x - this.move_x; + var dy = this.y - this.move_y; + + if (dx != 0 || dy != 0) { + // create an undoable edit record here + + this.sch.check_wires(this); + } +} + +Component.prototype.delete = function() { + // remove connection points from schematic + for (var i = this.connections.length - 1; i >= 0; --i) { + var cp = this.connections[i]; + this.sch.remove_connection_point(cp,cp.location); + } + + // remove component from schematic + this.sch.remove_component(this); + + // create an undoable edit record here +} + +Component.prototype.transform_x = function(x,y) { + var rot = this.rotation; + if (rot == 0 || rot == 6) return x; + else if (rot == 1 || rot == 5) return -y; + else if (rot == 2 || rot == 4) return -x; + else return y; +} + +Component.prototype.transform_y = function(x,y) { + var rot = this.rotation; + if (rot == 1 || rot == 7) return x; + else if (rot == 2 || rot == 6) return -y; + else if (rot == 3 || rot == 5) return -x; + else return y; +} + +Component.prototype.draw_line = function(c,x1,y1,x2,y2) { + c.strokeStyle = this.selected ? selected_style : normal_style; + var nx1 = this.transform_x(x1,y1) + this.x; + var ny1 = this.transform_y(x1,y1) + this.y; + var nx2 = this.transform_x(x2,y2) + this.x; + var ny2 = this.transform_y(x2,y2) + this.y; + this.sch.draw_line(c,nx1,ny1,nx2,ny2,1); +} + +Component.prototype.draw_circle = function(c,x,y,radius,filled) { + if (filled) c.fillStyle = this.selected ? selected_style : normal_style; + else c.strokeStyle = this.selected ? selected_style : normal_style; + var nx = this.transform_x(x,y) + this.x; + var ny = this.transform_y(x,y) + this.y; + + this.sch.draw_arc(c,nx,ny,radius,0,2*Math.PI,false,1,filled); +} + +rot_angle = [ + 0.0, // NORTH (identity) + Math.PI/2, // EAST (rot270) + Math.PI, // SOUTH (rot180) + 3*Math.PI/2, // WEST (rot90) + 0.0, // RNORTH (negy) + Math.PI/2, // REAST (int-neg) + Math.PI, // RSOUTH (negx) + 3*Math.PI/2, // RWEST (int-pos) +]; + +Component.prototype.draw_arc = function(c,x,y,radius,start_radians,end_radians) { + c.strokeStyle = this.selected ? selected_style : normal_style; + var nx = this.transform_x(x,y) + this.x; + var ny = this.transform_y(x,y) + this.y; + this.sch.draw_arc(c,nx,ny,radius, + start_radians+rot_angle[this.rotation],end_radians+rot_angle[this.rotation], + false,1,false); +} + +Component.prototype.draw = function(c) { +} + +// result of rotating an alignment [rot*9 + align] +aOrient = [ + 0, 1, 2, 3, 4, 5, 6, 7, 8, // NORTH (identity) + 2, 5, 8, 1, 4, 7, 0, 3, 6, // EAST (rot270) + 8, 7, 6, 5, 4, 3, 2, 1, 0, // SOUTH (rot180) + 6, 3, 0, 7, 4, 1, 8, 5, 3, // WEST (rot90) + 2, 1, 0, 5, 4, 3, 8, 7, 6, // RNORTH (negy) + 8, 5, 2, 7, 4, 1, 6, 3, 0, // REAST (int-neg) + 6, 7, 8, 3, 4, 5, 0, 1, 2, // RSOUTH (negx) + 0, 3, 6, 1, 4, 7, 2, 5, 8 // RWEST (int-pos) +]; + +textAlign = [ + 'left', 'center', 'right', + 'left', 'center', 'right', + 'left', 'center', 'right' +]; + +textBaseline = [ + 'top', 'top', 'top', + 'middle', 'middle', 'middle', + 'bottom', 'bottom', 'bottom' +]; + +Component.prototype.draw_text = function(c,text,x,y,alignment,size) { + var a = aOrient[this.rotation*9 + alignment]; + c.textAlign = textAlign[a]; + c.textBaseline = textBaseline[a]; + c.fillStyle = this.selected ? selected_style : normal_style; + this.sch.draw_text(c,text, + this.transform_x(x,y) + this.x, + this.transform_y(x,y) + this.y, + size); +} + +Component.prototype.set_select = function(which) { + if (which != this.selected) { + this.selected = which; + // create an undoable edit record here + } +} + +Component.prototype.select = function(x,y,shiftKey) { + this.was_previously_selected = this.selected; + if (inside(this.bbox,x,y)) { + this.set_select(shiftKey ? !this.selected : true); + return true; + } else return false; +} + +Component.prototype.select_rect = function(s) { + this.was_previously_selected = this.selected; + if (intersect(this.bbox,s)) + this.set_select(true); +} + +// if connection point of component c bisects the +// wire represented by this compononent, return that +// connection point. Otherwise return null. +Component.prototype.bisect = function(c) { + return null; +} + +Component.prototype.edit_properties = function(x,y) { + if (inside(this.bbox,x,y)) { + var content = document.createElement('table'); + content.style.marginBotton = '5px'; + content.fields = []; + + // add an field for each property + for (var i in this.properties) { + var label = document.createTextNode(i + ': '); + var field = document.createElement('input'); + field.type = 'text'; + field.value = this.properties[i]; + field.size = 10; + content.fields.push([i,field]); + + var col1 = document.createElement('td'); + col1.appendChild(label); + var col2 = document.createElement('td'); + col2.appendChild(field); + var row = document.createElement('tr'); + row.appendChild(col1); + row.appendChild(col2); + row.style.verticalAlign = 'center'; + + content.appendChild(row); + } + + var component = this; // capture in closure below + this.sch.dialog('Edit Properties',content,function(content) { + var fields = content.fields; + for (var i = fields.length - 1; i >= 0; i--) + component.properties[fields[i][0]] = fields[i][1].value; + component.sch.redraw(); // component is selected, so this will redraw it + }); + return true; + } else return false; +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Connection point +// +//////////////////////////////////////////////////////////////////////////////// + +connection_point_radius = 2; + +function ConnectionPoint(parent,x,y) { + this.parent = parent; + this.offset_x = x; + this.offset_y = y; + this.location = ''; + this.update_location(); +} + +ConnectionPoint.prototype.toString = function() { + return ''; +} + +ConnectionPoint.prototype.update_location = function() { + // update location string which we use as a key to find coincident connection points + var old_location = this.location; + var parent = this.parent; + var nx = parent.transform_x(this.offset_x,this.offset_y) + parent.x; + var ny = parent.transform_y(this.offset_x,this.offset_y) + parent.y; + this.x = nx; + this.y = ny; + this.location = nx + ',' + ny; + + // add ourselves to the connection list for the new location + parent.sch.update_connection_point(this,old_location); +} + +ConnectionPoint.prototype.coincident = function(x,y) { + return this.x==x && this.y==y; +} + +ConnectionPoint.prototype.draw = function(c,n) { + if (n != 2) + this.parent.draw_circle(c,this.offset_x,this.offset_y,connection_point_radius,n > 2); +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Wire +// +//////////////////////////////////////////////////////////////////////////////// + +near_distance = 2; // how close to wire counts as "near by" + +function Wire(sch,x1,y1,x2,y2) { + // arbitrarily call x1,y1 the origin + Component.call(this,sch,x1,y1,0); + this.dx = x2 - x1; + this.dy = y2 - y1; + this.add_connection(0,0); + this.add_connection(this.dx,this.dy); + + // compute bounding box (expanded slightly) + var r = [0,0,this.dx,this.dy]; + canonicalize(r); + r[0] -= near_distance; + r[1] -= near_distance; + r[2] += near_distance; + r[3] += near_distance; + this.bounding_box = r; + this.update_coords(); // update bbox + + // used in selection calculations + this.len = Math.sqrt(this.dx*this.dx + this.dy*this.dy); +} +Wire.prototype = new Component(); +Wire.prototype.constructor = Wire; + +Wire.prototype.toString = function() { + return ''; +} + +Wire.prototype.draw = function(c) { + this.draw_line(c,0,0,this.dx,this.dy); +} + +Wire.prototype.clone = function(sch,x,y) { + return new Wire(sch,x,y,x+this.dx,y+this.dy); +} + +Wire.prototype.near = function(x,y) { + // crude check: (x,y) within expanded bounding box of wire + if (inside(this.bbox,x,y)) { + // compute distance between x,y and nearst point on line + // http://www.allegro.cc/forums/thread/589720 + var D = Math.abs((x - this.x)*this.dy - (y - this.y)*this.dx)/this.len; + if (D <= near_distance) return true; + } + return false; +} + +Wire.prototype.select = function(x,y,shiftKey) { + this.was_previously_selected = this.selected; + if (this.near(x,y)) { + this.set_select(shiftKey ? !this.selected : true); + return true; + } else return false; +} + +// selection rectangle selects wire only if it includes +// one of the end points +Wire.prototype.select_rect = function(s) { + this.was_previously_selected = this.selected; + if (inside(s,this.x,this.y) || inside(s,this.x+this.dx,this.y+this.dy)) + this.set_select(true); +} + +// if connection point of component c bisects the +// wire represented by this compononent, return that +// connection point. Otherwise return null. +Wire.prototype.bisect = function(c) { + for (var i = c.connections.length - 1; i >= 0; --i) { + var cp = c.connections[i]; + var x = cp.x; + var y = cp.y; + + // crude check: (x,y) within expanded bounding box of wire + if (inside(this.bbox,x,y)) { + // compute distance between x,y and nearst point on line + // http://www.allegro.cc/forums/thread/589720 + var D = Math.abs((x - this.x)*this.dy - (y - this.y)*this.dx)/this.len; + // final check: ensure point isn't an end point of the wire + if (D < 1 && !this.connections[0].coincident(x,y) && !this.connections[1].coincident(x,y)) + return cp; + } + } + return null; +} + +Wire.prototype.move_end = function() { + this.sch.check_wires(this); +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Ground +// +//////////////////////////////////////////////////////////////////////////////// + +function Ground(sch,x,y,rotation) { + Component.call(this,sch,x,y,rotation); + this.add_connection(0,0); + this.bounding_box = [-6,0,6,8]; + this.update_coords(); +} +Ground.prototype = new Component(); +Ground.prototype.constructor = Ground; + +Ground.prototype.toString = function() { + return ''; +} + +Ground.prototype.draw = function(c) { + this.draw_line(c,0,0,0,8); + this.draw_line(c,-6,8,6,8); +} + +Ground.prototype.clone = function(sch,x,y) { + return new Ground(sch,x,y,this.rotation); +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Resistor +// +//////////////////////////////////////////////////////////////////////////////// + +function Resistor(sch,x,y,rotation,name,r) { + Component.call(this,sch,x,y,rotation); + this.properties['name'] = name; + this.properties['r'] = r ? r : '1'; + this.add_connection(0,0); + this.add_connection(0,48); + this.bounding_box = [-4,0,4,48]; + this.update_coords(); +} +Resistor.prototype = new Component(); +Resistor.prototype.constructor = Resistor; + +Resistor.prototype.toString = function() { + return ''; +} + +Resistor.prototype.draw = function(c) { + this.draw_line(c,0,0,0,12); + this.draw_line(c,0,12,4,14); + this.draw_line(c,4,14,-4,18); + this.draw_line(c,-4,18,4,22); + this.draw_line(c,4,22,-4,26); + this.draw_line(c,-4,26,4,30); + this.draw_line(c,4,30,-4,34); + this.draw_line(c,-4,34,0,36); + this.draw_line(c,0,36,0,48); + if (this.properties['r']) + this.draw_text(c,this.properties['r']+'\u03A9',5,24,3,property_size); + if (this.properties['name']) + this.draw_text(c,this.properties['name'],-5,24,5,property_size); +} + +Resistor.prototype.clone = function(sch,x,y) { + return new Resistor(sch,x,y,this.rotation,'',this.properties['r']); +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Capacitor +// +//////////////////////////////////////////////////////////////////////////////// + +function Capacitor(sch,x,y,rotation,name,c) { + Component.call(this,sch,x,y,rotation); + this.properties['name'] = name; + this.properties['c'] = c ? c : '1p'; + this.add_connection(0,0); + this.add_connection(0,48); + this.bounding_box = [-8,0,8,48]; + this.update_coords(); +} +Capacitor.prototype = new Component(); +Capacitor.prototype.constructor = Capacitor; + +Capacitor.prototype.toString = function() { + return ''; +} + +Capacitor.prototype.draw = function(c) { + this.draw_line(c,0,0,0,22); + this.draw_line(c,-8,22,8,22); + this.draw_line(c,-8,26,8,26); + this.draw_line(c,0,26,0,48); + if (this.properties['c']) + this.draw_text(c,this.properties['c']+'F',9,24,3,property_size); + if (this.properties['name']) + this.draw_text(c,this.properties['name'],-9,24,5,property_size); +} + +Capacitor.prototype.clone = function(sch,x,y) { + return new Capacitor(sch,x,y,this.rotation,'',this.properties['c']); +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Inductor +// +//////////////////////////////////////////////////////////////////////////////// + +function Inductor(sch,x,y,rotation,name,l) { + Component.call(this,sch,x,y,rotation); + this.properties['name'] = name; + this.properties['l'] = l ? l : '1n'; + this.add_connection(0,0); + this.add_connection(0,48); + this.bounding_box = [-4,0,5,48]; + this.update_coords(); +} +Inductor.prototype = new Component(); +Inductor.prototype.constructor = Inductor; + +Inductor.prototype.toString = function() { + return ''; +} + +Inductor.prototype.draw = function(c) { + this.draw_line(c,0,0,0,14); + this.draw_arc(c,0,18,4,6*Math.PI/4,3*Math.PI/4); + this.draw_arc(c,0,24,4,5*Math.PI/4,3*Math.PI/4); + this.draw_arc(c,0,30,4,5*Math.PI/4,2*Math.PI/4); + this.draw_line(c,0,34,0,48); + + if (this.properties['l']) + this.draw_text(c,this.properties['l']+'H',6,24,3,property_size); + if (this.properties['name']) + this.draw_text(c,this.properties['name'],-3,24,5,property_size); +} + +Inductor.prototype.clone = function(sch,x,y) { + return new Inductor(sch,x,y,this.rotation,'',this.properties['l']); +} + +//////////////////////////////////////////////////////////////////////////////// +// +// Source +// +//////////////////////////////////////////////////////////////////////////////// + +function Source(sch,x,y,rotation,name,type,value) { + Component.call(this,sch,x,y,rotation); + this.type = type; + this.properties['name'] = name; + this.properties['value'] = value ? value : '1'; + this.add_connection(0,0); + this.add_connection(0,48); + this.bounding_box = [-12,0,12,48]; + this.update_coords(); +} +Source.prototype = new Component(); +Source.prototype.constructor = Source; + +Source.prototype.toString = function() { + return '<'+this.type+'source '+this.properties['params']+' ('+this.x+','+this.y+')>'; +} + +Source.prototype.draw = function(c) { + this.draw_line(c,0,0,0,12); + this.draw_circle(c,0,24,12,false); + this.draw_line(c,0,36,0,48); + + if (this.type == 'v') { // voltage source + // draw + and - + this.draw_line(c,8,5,8,11); + this.draw_line(c,5,8,11,8); + this.draw_line(c,5,40,11,40); + // draw V + this.draw_line(c,-3,20,0,28); + this.draw_line(c,3,20,0,28); + } else if (this.type == 'i') { // current source + // draw arrow: pos to neg + this.draw_line(c,0,16,0,32); + this.draw_line(c,-3,24,0,32); + this.draw_line(c,3,24,0,32); + } + + if (this.properties['name']) + this.draw_text(c,this.properties['name'],-13,24,5,property_size); + if (this.properties['value']) + this.draw_text(c,this.properties['value']+(this.type=='v'?'V':'A'),13,24,3,property_size); +} + +Source.prototype.clone = function(sch,x,y) { + return new Source(sch,x,y,this.rotation,'',this.type,this.properties['value']); +} + +function VSource(sch,x,y,rotation,name,value) { + Source.call(this,sch,x,y,rotation,name,'v',value); + +} +VSource.prototype = new Component(); +VSource.prototype.constructor = VSource; +VSource.prototype.toString = Source.prototype.toString; +VSource.prototype.draw = Source.prototype.draw; +VSource.prototype.clone = Source.prototype.clone; + +function ISource(sch,x,y,rotation,name,value) { + Source.call(this,sch,x,y,rotation,name,'i',value); + +} +ISource.prototype = new Component(); +ISource.prototype.constructor = ISource; +ISource.prototype.toString = Source.prototype.toString; +ISource.prototype.draw = Source.prototype.draw; +ISource.prototype.clone = Source.prototype.clone; diff --git a/js/video_player.js b/js/video_player.js new file mode 100644 index 0000000000..811f69eca4 --- /dev/null +++ b/js/video_player.js @@ -0,0 +1,317 @@ +// Things to abstract out to another file + +function postJSON(url, data, callback, csrf) { + $.ajax({ + url: url, + dataType: 'json', + data: data, + success: callback + }); +} + +var global=5; + +// Video player + +var load_id = 0; + +var video_speed = 1.0; + +function change_video_speed(speed, youtube_id) { + new_position = ytplayer.getCurrentTime() * video_speed / speed; + video_speed = speed; + ytplayer.loadVideoById(youtube_id, new_position); +} + +function caption_at(index) { + if (captions==0) + return " "; + + text_array=captions.text + + if ((index>=text_array.length) || (index < 0)) + return " "; + return text_array[index]; +} + +function caption_time_at(index) { + if (captions==0) + return 0; + + time_array=captions.start; + + if (index < 0) + return 0; + if (index>=time_array.length) + return ytplayer.getDuration(); + + return time_array[index] / 1000.0 / video_speed; +} + +function caption_index(now) { + // Returns the index of the current caption, given a time + now = now * video_speed; + + if (captions==0) + return 0; + + time_array=captions.start + + // TODO: Bisection would be better, or something incremental + var i; + for(i=0;i(now*1000)) { + return i-1; + } + } + return i-1; +} + +function update_captions(t) { + var i=caption_index(t); + $("#std_n5").html(caption_at(i-5)); + $("#std_n4").html(caption_at(i-4)); + $("#std_n3").html(caption_at(i-3)); + $("#std_n2").html(caption_at(i-2)); + $("#std_n1").html(caption_at(i-1)); + $("#std_0").html(caption_at(i)); + $("#std_p1").html(caption_at(i+1)); + $("#std_p2").html(caption_at(i+2)); + $("#std_p3").html(caption_at(i+3)); + $("#std_p4").html(caption_at(i+4)); + $("#std_p5").html(caption_at(i+5)); + $("#std_p6").html(caption_at(i+6)); +} + +function title_seek(i) { + // Seek video forwards or backwards by i subtitles + current=caption_index(getCurrentTime()); + new_time=caption_time_at(current+i); + + ytplayer.seekTo(new_time, true); +} + +function updateHTML(elmId, value) { + document.getElementById(elmId).innerHTML = value; +} + +function setytplayerState(newState) { + // updateHTML("playerstate", newState); +} + +// Updates server with location in video so we can resume from the same place +// IMPORTANT TODO: Load test +// POSSIBLE FIX: Move to unload() event and similar +var ajax_video=function(){}; + +function onYouTubePlayerReady(playerId) { + ytplayer = document.getElementById("myytplayer"); + setInterval(updateytplayerInfo, 1000); + setInterval(ajax_video,1000); + ytplayer.addEventListener("onStateChange", "onytplayerStateChange"); + ytplayer.addEventListener("onError", "onPlayerError"); + if((typeof load_id != "undefined") && (load_id != 0)) { + var id=load_id; + loadNewVideo(id, 0); + } + +} + +function videoDestroy() { + load_id = 0; + // TODO/BUG: Figure out why removeEventListener doesn't work + ytplayer.removeEventListener("onStateChange", "onytplayerStateChange"); + ytplayer.removeEventListener("onError", "onPlayerError"); + ytplayer = false; +} + +function log_event(e, d) { + // CRITICAL TODO: Change to AJAX + //$("#eventlog").append("
"); + //$("#eventlog").append(JSON.stringify(e)); + + // TODO: Figure out + // XMLHttpRequest cannot load http://localhost:7000/userlog. Origin http://localhost:8000 is not allowed by Access-Control-Allow-Origin. + + /*window['console'].log(JSON.stringify(e)); + $.get("http://localhost:7000/userlog", + {'user':'pmitros', + 'key':'key', + 'event_type':'unknown', + 'data':'e'}, + function(data) { + });*/ +} + +function seek_slide(type,oe,value) { + //log_event('video', [type, value]); + if(type=='slide') { + // HACK/TODO: Youtube recommends this be false for slide and true for stop. + // Works better on my system with true/true. + // We should test both configurations on low/high bandwidth + // connections, and different browsers + // One issue is that we query the Youtube window every 250ms for position/state + // With false, it returns the old one (ignoring the new seek), and moves the + // scroll bar to the wrong spot. + ytplayer.seekTo(value, true); + } else if (type=='stop') { + ytplayer.seekTo(value, true); + log_event('video', [type, value]); + } + + update_captions(value); +} + +function get_state() { + if (ytplayer) + return [ytplayer.getPlayerState(), + ytplayer.getVideoUrl(), + ytplayer.getDuration(), ytplayer.getCurrentTime(), + ytplayer.getVideoBytesLoaded(), ytplayer.getVideoBytesTotal(), + ytplayer.getVideoStartBytes(), + ytplayer.getVolume(),ytplayer.isMuted(), + ytplayer.getPlaybackQuality(), + ytplayer.getAvailableQualityLevels()]; + return []; +} + +function onytplayerStateChange(newState) { + setytplayerState(newState); + log_event('video', ['State Change',newState, get_state()]); +} + +function onPlayerError(errorCode) { + alert("An error occured: " + errorCode); +} + +function updateytplayerInfo() { + if(ytplayer.getPlayerState()!=3) { + $("#slider").slider("option","max",ytplayer.getDuration()); + $("#slider").slider("option","value",ytplayer.getCurrentTime()); + } + if (getPlayerState() == 1){ + update_captions(getCurrentTime()); + } + + // updateHTML("videoduration", getDuration()); + // updateHTML("videotime", getCurrentTime()); + // updateHTML("startbytes", getStartBytes()); + // updateHTML("volume", getVolume()); +} + +// functions for the api calls +function loadNewVideo(id, startSeconds) { + captions={"start":[0],"end":[0],"text":["Attempting to load captions..."]}; + $.getJSON("/static/subs/"+id+".srt.sjson", function(data) { + captions=data; + }); + load_id = id; + //if ((typeof ytplayer != "undefined") && (ytplayer.type=="application/x-shockwave-flash")) { + // Try it every time. If we fail, we want the error message for now. + // TODO: Add try/catch + try { + ytplayer.loadVideoById(id, parseInt(startSeconds)); + load_id=0; + } + catch(e) { + window['console'].log(JSON.stringify(e)); + } +} + +function cueNewVideo(id, startSeconds) { + if (ytplayer) { + ytplayer.cueVideoById(id, startSeconds); + } +} + +function play() { + if (ytplayer) { + ytplayer.playVideo(); + } +} + +function pause() { + if (ytplayer) { + ytplayer.pauseVideo(); + } +} + +function stop() { + if (ytplayer) { + ytplayer.stopVideo(); + } +} + +function getPlayerState() { + if (ytplayer) { + return ytplayer.getPlayerState(); + } +} + +function seekTo(seconds) { + if (ytplayer) { + ytplayer.seekTo(seconds, true); + } +} + +function getBytesTotal() { + if (ytplayer) { + return ytplayer.getVideoBytesTotal(); + } +} + +function getCurrentTime() { + if (ytplayer) { + return ytplayer.getCurrentTime(); + } +} + +function getDuration() { + if (ytplayer) { + return ytplayer.getDuration(); + } +} + +function getStartBytes() { + if (ytplayer) { + return ytplayer.getVideoStartBytes(); + } +} + +function mute() { + if (ytplayer) { + ytplayer.mute(); + } +} + +function unMute() { + if (ytplayer) { + ytplayer.unMute(); + } +} + +function getEmbedCode() { + alert(ytplayer.getVideoEmbedCode()); +} + +function getVideoUrl() { + alert(ytplayer.getVideoUrl()); +} + +function setVolume(newVolume) { + if (ytplayer) { + ytplayer.setVolume(newVolume); + } +} + +function getVolume() { + if (ytplayer) { + return ytplayer.getVolume(); + } +} + +function clearVideo() { + if (ytplayer) { + ytplayer.clearVideo(); + } +} \ No newline at end of file diff --git a/lib b/lib new file mode 120000 index 0000000000..79deb8f3f6 --- /dev/null +++ b/lib @@ -0,0 +1 @@ +../3rdParty/static/js/ \ No newline at end of file diff --git a/pixel.jpg b/pixel.jpg new file mode 100644 index 0000000000..8fbe381c7a Binary files /dev/null and b/pixel.jpg differ diff --git a/staff/agarwal-mit-news-small.jpg 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Good morning.", + "Let's get going. In today's lecture we continue", + "with the operational amplifier, \"op amp\" for short.", + "And what we are going to do is just build up a bunch of fun", + "building blocks using the op amp.", + "As a quick review --", + "", + "To quickly review what we've seen about the op amp --", + "", + "We represented the op amp as a device that looked like this", + "where the amplifier had an incredibly high gain.", + "So, if I had a small voltage difference here --", + "I call this v plus and this v minus with respect to ground.", + "And if I had a small voltage difference then this gain here", + "would multiply the difference by a large number and thereby", + "giving me an output that was on the order of a million times", + "greater than this difference. And because of that when I use", + "the op amp in a mode like this without any negative feedback", + "the output would usually crank up to the positive rail or the", + "negative rail. We also saw that it had", + "infinite input resistance so that the current flowing in here", + "or here was zero and also had zero output resistance.", + "This is my ideal op amp where irrespective of what load I", + "connect here the op amp would supply pretty much any current.", + "Now, in practical op amps that's not the case.", + "But suffice it to say that when used as an ideal op amp the", + "output impedance, the output resistance is going", + "to be zero. The op amp is a huge workhorse", + "of the analog industry. You will see based both on what", + "you've done on Tuesday and Wednesday but also today that", + "it's very, very simple to build circuits using the op amp.", + "When you use the amplifier, you don't have to worry about", + "things like nonlinear analysis. You don't have to worry about", + "am I really meeting the criteria for saturation limits and so on?", + "To some extent you have to think about that with the op", + "amp, too, because if the output hits the positive rail or", + "negative rail it isn't going to behave like you expect it to.", + "But fundamentally with this primitive model,", + "this idea model it becomes really simple to build circuits", + "with the op amp. Therefore it has become a key", + "building block for circuits. When circuit designers build", + "analog circuits very often their primitive building blocks are", + "really an amplifier of this sort, an op amp,", + "resistors, capacitors and some of our other primitive building", + "elements. If you look at the course notes", + "the readings are -- There are a bunch of examples", + "solved in Chapter 16. And you will see that using the", + "op amp it is indeed possible to build current sources that look", + "like more or less ideal current sources.", + "It is also possible to build voltage sources and so on.", + "It is an incredibly neat building block using which you", + "can do all kinds of cool stuff.", + "", + "In this course you will see a whole bunch of example circuits", + "using the op amp. In today's lecture you will see", + "things like a subtractor. You will also see integrators", + "and a differentiator. And then in your lab,", + "lab four, you will build a really fun mixed signal circuit", + "involving both digital and analog components.", + "And you will build what is called a digital to an analog", + "converter using the op amp. And of course I can build all", + "our good-old amplifiers and circuits of that sort.", + "In a later lecture you will also see how we can build", + "filters using an op amp. This is going to be using the", + "knowledge you learn in terms of connecting resistors,", + "capacitors and inductors together and doing a frequency", + "domain analysis, well we can throw the op amp in", + "there and build filters, too.", + "This is just to give you a preview of upcoming attractions.", + "For today I am going to focus on these circuits.", + "I won't be covering any new theory or any new set of", + "foundations but pretty much take the simple properties that I", + "have explained to you about the op amp.", + "And using those simple properties very quickly build up", + "a bunch of circuits that you can use to analyze signals in a", + "variety of ways. Let's start with the following", + "circuit. With op amps I start with this", + "little guy. And what I am going to do is", + "use two voltage sources, v1, and this is a resistor,", + "not an inductor. And value R1,", + "value R2. So, I have a voltage connected", + "by a divider, voltage divider to the plus", + "input. And I am going to provide some", + "negative feedback in the following way.", + "This is going to be R2, the same as this one here,", + "a resistor R1. And then a voltage source v2", + "that I connect out here. So notice that- Oh,", + "and I take the output vOUT out here.", + "And that vOUT of course is with respect to ground,", + "and R2, v1 and v2 are also connected to ground.", + "What I am going to do is analyze the circuit it two", + "different ways, and as I analyze it describe", + "some other interesting properties to you.", + "In the last lecture the technique I used to analyze op", + "amps was one in which I replaced the op amp with its ideal model", + "involving a dependent source and so on with a large gain A and", + "showed that. I wrote the expression and then", + "I let A increase to infinity to the limits and got an expression", + "that was independent of A. And then in recitation", + "yesterday you would have covered another technique which makes it", + "much simpler to analyze op amps. Let me very quickly review that", + "method. We fondly call that technique,", + "there is no formal name for it, but we fondly call that v plus", + "more or less equal to v minus method.", + "This is also variously called the virtual ground method and so", + "on, but we shall call it the v plus more or less equal to v", + "minus method. The insight here is that", + "whenever I use the op amp in a way in which I am giving it", + "negative feedback, so I am feeding some portion of", + "the output to its negative input.", + "I am giving it negative feedback.", + "That's one property. Second property is that my", + "inputs, v1 and v2, and my resistance values are", + "chosen such that the output is not in saturation.", + "So, the op amp is not at the plus VS rail or minus VS rail.", + "Rather it's somewhere in the middle in its active region.", + "When that happens we claim that the v minus and v plus for the", + "op amp are more or less equal. And to give you some intuition", + "as to why that is so, let's say the output is 6 volts", + "and my supply is plus/minus 12. This is 6 volts and the", + "amplifier is a gain of a million, ten to the six.", + "To sustain 6 volts at the output all I need is a", + "difference of 6 microvolts here. Six divided by ten to the six", + "is the difference between v plus and v minus.", + "It's very, very, very small.", + "It's so small as to make v plus more or less equal to v minus.", + "All it takes is a very small differential voltage here to", + "give you 6 volts at the output. The key thing to observe is", + "under negative feedback, when the op amp is not in", + "saturation the property that v plus equals v minus holds.", + "And the way it works is that it's not that it's a magical", + "property. It is simply that when I apply", + "negative feedback the negative feedback is such that it will", + "force this v minus node here to be at more or less the same", + "voltage as v plus. Remember the when in doubt", + "simply go back and think about the anti lock brakes example we", + "did last time. For example if v plus increases", + "the output will increase and so will the voltage here and tend", + "to make these two equal. What we can do,", + "being rather tricky here, what we'll do is say look,", + "if we know for a fact that under negative feedback the op", + "amp is going to engineer these two node voltages to be more or", + "less equal then why don't I just use that fact to begin with and", + "analyze my circuit assuming that it's true.", + "This is just a bit of inverted logic here that says look,", + "the circuit is going to make that happen.", + "If the circuit is going to make that happen to analyze the", + "circuit in its steady state, why don't I just go ahead and", + "assume that to begin with? This again goes back to us", + "wanting to be engineers here and do whatever is simply and find", + "the simplest possible way of getting some place.", + "I want to use that method, the v plus equals v minus", + "method. Let me just first write down", + "some values that I know about. I know that v plus is simply a", + "voltage divider relation here. That's v1 times R2 divided by", + "R1 plus R2. And by the v plus equals v", + "minus method I know that this is going to be equal to v minus.", + "And this is going to be true because I am giving you negative", + "feedback here. And we are going to engineer", + "the values of R1, R2, v1 and v2 such that the op", + "amp is not in saturation. So, we know that.", + "The next thing that we know, let's say this is a current i.", + "This current i flows here. Know that there is no current", + "going in here. Op amp has an infinite input", + "resistance so there is nothing going in there.", + "There is no current going in there.", + "If there is no current going in here, what must happen to i?", + "Remember, from the foundations of the universe Maxwell's", + "equations and therefore KVL and KCL hold.", + "KVL and KCL simply come straight from nature.", + "You and I cannot mess with that.", + "Bad things happen to you if you do.", + "So, nature, Maxwell's equations, KVL,", + "KCL. It's simply nature.", + "So, KCL applies here. Current comes in here.", + "Nothing goes there. Don't argue.", + "The current has to go here, period.", + "No if, ands or buts. There is i coming in here,", + "nothing goes there, so that current must flow here.", + "It has no choice. It's from basic nature.", + "I can write down what my current i is going to look like.", + "What is i going to look like? Well, I know v2,", + "I know v minus. v minus is the same as v plus.", + "And v plus is the i expression given here.", + "So, I can write i as v2 minus v minus divided by R1.", + "Let me keep track of those two and then go ahead and compute", + "vOUT. So, my goal in life is compute", + "vOUT as a function of the two input voltages v1 and v2.", + "And just for kicks I have gone ahead and computed some of the", + "intermediate node voltages and currents.", + "How do I write vOUT? What is vOUT?", + "vOUT is simply v minus from KVL.", + "vOUT is simply v minus minus the drop across this resistor.", + "So, the drop across that resistor is simply iR2.", + "From good-old KVL from the first lecture,", + "a voltage minus the drop across the resistor is equal to vOUT.", + "Therefore it's simply v minus minus iR2.", + "One thing to be very cautious about, I will tell you right", + "now, is that the output here relates to the inversion of the", + "voltage across this resistor R2. Be very, very careful in that", + "if I have a voltage across this resistor here that impacts vOUT", + "with a minus sign attached to it.", + "Notice that iR2 is the voltage across R2 and vOUT relates to", + "the negative of that. Be very cautious.", + "That's one of the commonest silly mistakes I have seen", + "people make in solving problems like this.", + "Let's go ahead. I know v minus and I don't know", + "i. Let me substitute for i for", + "now, and that is v2 minus v minus divided by R1 times R2.", + "Let me go ahead and collect all the v minuses.", + "v minus, I get a one here, minus minus becomes a plus,", + "and so I get R2 divided by R1 out there.", + "And then I minus v2 R2 divided by R1.", + "That is vOUT. Now let me go ahead and", + "substitute for v minus. And that is simply v1 R2", + "divided by R1 plus R2. That is v minus.", + "And this character here is simplified to be R1,", + "R1 plus R2 minus v2 R2 divided by R1.", + "What do we get? I cancel these two suckers out", + "and what I end up with is v1 R2 divided by R1 minus v2 R2", + "divided by R1, which is simply R2/R1(v1-v2).", + "What is interesting here is that what I have ended up", + "building is a very primitive subtractor.", + "So, my output relates to v1 minus v2 multiplied by the", + "constant factor given by R2 divided by R1.", + "", + "Again, as I pointed out to you at the beginning of this", + "lecture, no knew foundations today, no new theories,", + "no new disciplines, no new laws.", + "We are just going to take what you have learned --", + "Three simple things, infinite gain,", + "infinite input resistance, zero output resistance,", + "plus this new thing v plus equals v minus.", + "And just being armed with those four principles we are just", + "going to charge ahead and analyze a bunch of circuits.", + "It is purely intellectual and pure applications today.", + "This is one way of doing it. There is another way of solving", + "it. We can solve the circuit.", + "Remember, whenever you see a linear circuit and you see two", + "sources or three sources, just think superposition,", + "right? You see a linear circuit and", + "two or three sources, think superposition.", + "We should be able to apply superposition to this.", + "The op amp is simply another building block.", + "It's a linear circuit. So, let's see if we get the", + "same answer. Let's try to solve the circuit", + "using superposition and see if we get the same answer.", + "To do superposition what I am going to do is build two", + "subcircuits. One subcircuit in which v1 is", + "zero, and that subcircuit looks like this.", + "If I set v1 to be zero then I get R1 parallel R2 going to", + "ground. So, if v1 is set to zero then", + "R1 goes to ground. And I get R1 parallel R2 here.", + "And of course I have v2 as before.", + "And this was R1, this was R2,", + "and let me call that vOUT1. Oh, I'm sorry.", + "Let me call it vOUT2 corresponding to that component", + "of the output that relates to v2 acting alone.", + "Remember superposition? Build two subcircuits,", + "one that depends on v2 and another one that depends on v1.", + "Let's do the second one, too.", + "Second one is v2 going to zero. Here is my little op amp.", + "And what I will do is simply flip the op amp just to see if", + "you can identify some interesting patterns.", + "Just flip the op amp around. And this is v1 as before.", + "And recall that v1 was going to the plus node through a resistor", + "R1. And then I had a R2 to ground.", + "And then let me short v2 to ground.", + "And when I short v2 to ground what happens?", + "When I short v2 to ground what happens is that the tail of R1", + "here goes to ground. And so it is as if the output", + "is connected to the node v minus through a resistor,", + "so it as if the output v R2 is connected to the minus input", + "through a resistor. We will draw it like this.", + "And the minus input goes through a resistor R1,", + "to ground. If you thought that patterns", + "were important in the earlier part of the course doing voltage", + "divider patterns and current divider patterns and amplifier", + "pattern, the source follower pattern, op amps is all about", + "patterns. You should remember two or", + "three simple patterns and be able to write down the", + "expression for those just by observation.", + "So, this is one common pattern that you have seen before in the", + "very first lecture. And I just wrote it down in", + "that manner. Let me go ahead and solve this", + "circuit. It turns out that this is also", + "a pattern. I will analyze it today but in", + "the future v2 going to this node through R1 and then R2 to the", + "output. You have probably also seen", + "this in your recitation. This one is called an inverting", + "connection and this one here is called a non-inverting", + "connection. Let's go ahead and do vOUT2.", + "vOUT2 is simply given by, notice that since this is", + "ground, no current flowing here, this voltage is zero.", + "If this voltage is zero, this voltage is zero by the v", + "plus equals v minus method. If this is zero,", + "the current that goes through here is v2 divided by R1.", + "And that same current must flow through the resistance R2 as", + "well. If the current v2 divided by R1", + "flows through this resistor, the drop across this resistor", + "is simply given by, let me hide this for a second,", + "is simply given by v2. So, v2 divided by R1 is the", + "current here. This is zero.", + "So, the drop across this resistor is v2 R1 multiplied by", + "R2. That's a drop across this", + "resistor. This voltage is simply zero", + "minus a drop across the resistor.", + "So, it's zero minus the drop across the resistor and that", + "gives me v2. Again, remember this minus sign", + "comes in when I want to convert this to get the output voltage", + "from that. This is a very common pattern.", + "It's called an inverting connection where the output is", + "some factor of the input voltage and the factor is given by R2", + "divided by R1. Let's go ahead and analyze this", + "guy now. What is vOUT1 equal to?", + "I should have called this vOUT1 because it relates to v1.", + "vOUT1. There is a v plus here.", + "From our first lecture I know that vOUT1 relates to v plus in", + "the following way. I know that it is v plus times", + "the sum of the resistances divided by R1.", + "Based on the first lecture this is true.", + "vOUT1 is simply an amplified version of v plus where the", + "amplification factor is given by R1 plus R2 divided by R1.", + "And I know v plus is simply a voltage divider action here.", + "And I can take a simple voltage divider action here because the", + "current going in is zero. Looking in here this is as if", + "it's an infinite resistance, so it is as if the element", + "simply does not exist. The voltage here is simply v1", + "divided by R1 plus R2 multiplied by R2, our voltage divider", + "pattern. So, I get v1 times R2 divided", + "by R1 plus R2 times R1 plus R2 divided by R1.", + "These two cancel out which gives me vOUT1 is simply v1 R2", + "divided by R1. To get vOUT I add up the two.", + "vOUT is vOUT1 plus vOUT2, which is my goal.", + "And that is simply v1 R2 by R1 minus v2 R2 by R1.", + "Thankfully what we have here is the same as here.", + "Again, there is really nothing new that I am going to cover", + "today. Simply apply,", + "apply, apply, four simple principles.", + "Here I have used superposition and I am showing you a circuit.", + "So, it turns out with op amps you should really remember that", + "pattern. You will see it again and again", + "and again. And each time you see it,", + "it will save you six minutes of having to solve the circuit", + "without knowing the pattern. So, remember this pattern.", + "You can pick up another three or four minutes by remembering", + "this pattern here. This pattern is simply v2 R2", + "divided by R1. Imprint those two patterns into", + "your brains. OK, so those are a couple of", + "simple circuits using the op amp.", + "We built a subtractor. The next step,", + "let's go ahead and try to build an integrator.", + "Using this little building block we can go ahead and try to", + "build a bunch of circuits. We can build filters,", + "A to D converters and so on. Let's build an integrator.", + "Abstractly I need to build this box.", + "Which when fed a vI, I want that box to integrate", + "and give me a vO which is vI integrated over time.", + "That is what I want to build. How do I go about building it?", + "What I would like to do next is give you some flavor for design.", + "How do you go about designing things with an op amp?", + "Knowing that you do not know the pattern for this yet,", + "how do you go about designing things?", + "Well, let's start with the following intuition.", + "The intuition that I begin with is that if I have a current i,", + "and remember that capacitors and inductors related to,", + "you saw differentiation and integration happening when we", + "dealt with capacitors and inductors.", + "So, I think we have to invoke a capacitor here or an inductor.", + "In this example I invoke a capacitor.", + "Notice that if I stick a capacitor in here this current", + "is i, capacitance C, then my voltage vO is given by", + "what? Voltage is simply the integral", + "of the current flowing through it or vice versa i is C dv/dt.", + "If i is C dv/dt then v is simply one by C integral.", + "If I can pass the current through a capacitor then the", + "voltage across the capacitor must be a current.", + "Notice then that vO is related to i dt.", + "I have some multiplying constants and so on,", + "but fundamentally what I have found is if I can stick a", + "current through a capacitor then the voltage across the capacitor", + "relates to the integral of the current.", + "OK, that's interesting. So, I have an integral in", + "there. But I have a current.", + "Notice my goal was to integrate a voltage.", + "What I figured out how to do was if I can turn that voltage", + "into a current -- If I can turn that voltage into", + "a proportional current and then pump that current through a", + "capacitor I will get the integration that I want.", + "How do I convert my vI to i? How do I do that?", + "Well, let's take a stab at it. Here is my vI.", + "Let's take the resistor R. And remember I need to stick", + "the capacitor here. I have some current I here.", + "I don't know what the current is yet.", + "And I stick a voltage here. And what I am trying to do is", + "trying to see if I stick a voltage and a resistance in", + "series then there is some relationship between the current", + "and this voltage. Recall that I am trying to make", + "this current be directly proportional to the voltage vI.", + "But it turns out that i here is not equal to vI divided by R.", + "If i was vI divided by R somehow, I am done.", + "If i was vI divided by R, by some magic,", + "then I have converted my voltage to a current,", + "I feed that current through my capacitor and vO is my integral", + "that I am looking for. But unfortunately i is not", + "equal to vI divided by R. You know that.", + "i relates to vI minus the capacitor voltage divided by R.", + "So, i is not simply vI divided by R for all time but i is", + "really vI minus the capacitor voltage divided by R.", + "And, in fact, when we did RC circuits you", + "wrote this equation to represent the dynamics of the circuit,", + "RC dvO by dt plus vO equals vI. We wrote down this circuit for", + "a first order RC, wrote this equation for a first", + "order RC circuit. Now, it does turn out,", + "to wrap up on this wild goose chase that we went on,", + "it does turn out that if this term here is much bigger than", + "that term. If this term is much bigger", + "than that term then I can ignore that term and write down RC dvO", + "by dt more or less equal to vI. If that were true,", + "this would be true, and then vO would be more or", + "less equal to one by RC integral of vI dt.", + "Again, if this were true. If this were true for all time", + "then vO would be integral of vI dt.", + "Again, remember this is all a wild goose chase.", + "Just write down WGC there just so you don't get confused.", + "I am on this wild goose hunt here trying to find a way to get", + "a current from a voltage which I can then feed into a capacitor.", + "This was one thing I knew, but this was not what I want.", + "But it does turn out to be what I want when vO is very,", + "very small. So, I see some glimmer of hope", + "but not quite. It turns that in R and C,", + "if I make R and C very, very big, if I have a huge time", + "constant, with a huge time constant the voltage vO looks", + "like an integral of vI, but only when I have a very", + "huge time constant. So, I give up on that track.", + "Instead I try something else.", + "", + "Another try. I would like you to notice if", + "you take your op amp, here is your op amp,", + "if you take this op amp and you stick the positive terminal to", + "ground, under reasonable feedback, under reasonable", + "negative feedback what do you notice about the current?", + "If I had a current i flowing here what did you notice?", + "Look at this picture. I had a current i flowing in", + "here, v2 divided by R1. And because this resistance was", + "infinite all the current went through the upper terminal.", + "So, this is zero volts. And by the v plus equals v", + "minus method this is also more or less equal to zero.", + "And I have a current i flowing in here, nothing goes here,", + "so then the i must flow up there.", + "So, all I am doing here is causing a reflection of the", + "current from this grounded node. My current is being reflected", + "into, or deflected if you feel like it, the upper edge here", + "after coming in through this edge.", + "That is interesting. We are just one step away from", + "the key insight.", + "", + "I have an i coming in here, an i going out there.", + "Notice that, as I said before,", + "this is zero volts. How do I get my voltage vI to", + "look like a current, to become proportional to a", + "current? It is simple now.", + "All I do is put a voltage vI and put a resistor R out there.", + "If I do that, and since this is zero,", + "the current i is given by vI divided by R.", + "I have gotten to where I want to be.", + "So, by using an op amp and using the fact that the minus", + "node here, v minus is at the same potential as v plus when", + "there is negative feedback then I can stick a resistor here.", + "And because this is zero the current here is simply vI", + "divided by R. I have gotten to the first", + "place. Now all I need to do is simply", + "pump this current through a capacitor and I get the integral", + "of the, the voltage becomes an integral of the current.", + "That is easy. I stick my capacitor here and I", + "get my answer out there as vO. Notice that when I do this,", + "let's say this is plus/minus VC.", + "This is zero. So, vO is minus VC.", + "Again, I will keep emphasizing it maybe 17 times throughout", + "this course that if this is zero then the output here is related", + "to the negative of this voltage, common, common,", + "common mistake. I will be very upset after", + "doing all this if I see this mistake happen in any of the", + "future homeworks or finals or whatever.", + "This should not happen. So, vO is a minus sign here VC.", + "And I know that if I have a current i through a capacitor", + "what is VC? If I have current i through a", + "capacitor than this is simply t i dt.", + "And i by design is -- So, I have my integrator.", + "It is a two-step process. I stuck a resistor here,", + "so the current became equal to vI divided by R.", + "Then I took that current and pumped it through a capacitor", + "through this terminal here, and the voltage across the", + "capacitor for a current i is given by this expression.", + "This is Capacitors 101. OK Capacitors 101 says that the", + "voltage across the capacitor is simply one by C integral i dt.", + "Another way of looking at it is the voltage across the capacitor", + "is C, I'm sorry, the current through a capacitor", + "is C dv/dt. This is simply the integral", + "form of that equation. And I am done with my", + "integrator. So, this is another very common", + "building block. Remember this.", + "Most of the circuits we will be seeing with op amps simply", + "involve something here and some there.", + "And the output in this inverting connection is the", + "output times, if it is a resistance it is", + "simply R2 divided by R1, if it's a capacitor I get the", + "integral form looking like this. Yes.", + "", + "Can someone tell me where the negative sign went?", + "The blackboard ate it up. Good catch.", + "", + "After all that lecture about watching the negative sign.", + "After this little bit of faux pas here, now I will be doubly", + "mad if you guys make that mistake.", + "All right. Now that we have built the", + "integrator, I could give this out as a homework problem.", + "And you should be able to design a differentiator based on", + "what you've learned here. You now have the tools to go", + "and do some design like this, but we don't have any more", + "homeworks left so I guess I will go ahead and solve this for you", + "right here and do the design for you.", + "The building block that we need looks like this,", + "d/dt here. Let me take a vI and stick a vI", + "in there. That's what I want to build.", + "And what I built here is that different integrator box.", + "And what I would like to do now is build a differentiator box.", + "How do I go about doing it? I will go really slow here so", + "you will have some time to think about it for yourselves and see", + "if you folks are crack op amp circuit designers already,", + "if you have the right instincts here.", + "Again, when you see differentiation integration", + "think capacitors or inductors, it doesn't matter.", + "In fact, as a homework exercise, you may want to go", + "back and see how you can get a similar effect using inductors.", + "Can you play with inductors and get a similar effect?", + "So, inductors are devices that are a dual of the capacitor.", + "Whatever we will do with capacitors, there must be a", + "corresponding way with inductors.", + "You can try it out in your spare time.", + "Let's go back to this one here. I will stick with the capacitor", + "way of looking at things. I need a differentiation now.", + "Remember this. If I have a vI and I stick this", + "across a capacitor, I have a current C and some", + "voltage vc across the capacitor, what does i relate to?", + "i is simply C dv/dt and vc in this case is simply C dvI/dt.", + "If I can stick a voltage across a capacitor, if my input voltage", + "is stuck across a capacitor then the resulting current relates to", + "dvI/dt. Here we have the opposite", + "problem. By doing this simple trick,", + "I can obtain a current that has the right form.", + "Now what I need to do is somehow convert that current", + "into a voltage because the abstraction that I need is a", + "voltage to voltage. The next step,", + "what I need to do is somehow convert a current to a voltage.", + "How do I go about doing that? Again, remember for the op amp,", + "if I have a current i flowing here then by the reflection", + "property i gets pushed up into this edge, provided that the", + "whole circuit is working with descent negative feedback.", + "Given this trick what I can do is say look, suppose I did this.", + "Remember, my goal here is how do I convert a current to a", + "voltage? I have a current i coming in", + "here, and I can turn that into a voltage because I know the", + "current must come out here, I know this current must come", + "out there. All I have to do is stick a", + "resistor in there. If I stick a resistor in there", + "what is vO equal to? vO is simply iR,", + "right? That's right.", + "vO, I get i here, so i pumps through here.", + "Remember, what comes in here must get reflected up because", + "the current going in here is zero.", + "All the i must come out here. So, that i must pump through", + "this resistor. The drop across this resistor", + "is iR. That's the voltage drop across", + "that resistor. And since this at a virtual", + "ground the output here is simply zero minus this drop which is", + "minus iR. So, I have gotten to where I", + "want to be. I have my current i being", + "converted to a voltage. I have taken my current,", + "and I have been able to convert that into a voltage by sticking", + "a resistor in here. As a final step,", + "I simply need to produce the current.", + "And that is pretty easy to do. Abstractly what I need to do,", + "again, this is design here so we will talk about abstract", + "stuff. If I had a voltage vI,", + "I need to produce a current which relates to C dvI/dt.", + "And I know I can do that by simply doing this.", + "By doing this I know my i is C dvI, correct?", + "If I can get this effect, I put this in quotes because", + "that's my pattern. I am looking for a pattern,", + "where a voltage vI is directly applied across a capacitor.", + "And when that happens the current relates to C dv/dt.", + "Let's go back to our op amp pattern here,", + "op amp circuit. So far I have achieved --", + "I just repeated this out there. And so somehow I need to take", + "this pattern here and learn from that pattern and apply the", + "pattern here. So, what I can do is,", + "this is a ground node, correct?", + "Now, the poor little capacitor, what does it care,", + "whether it's a ground node or a virtual ground node?", + "As long as it's a zero volt node down here what does it", + "care? What I am going to do is stick", + "this point, not here but into a virtual ground node.", + "I am going to grab that point, take it here and stick it here.", + "The poor little capacitor doesn't know the difference.", + "I have really suckered the little beast.", + "This is vI. Remember this.", + "My i through the capacitor is proportional to C dv/dt.", + "Instead what I have done is taken this guy and stuck it here", + "to get something like this. Just remember these four or", + "five little tricks. And you apply them in op amp", + "circuits again and again and again and again.", + "So, this is vI, this is my virtual ground.", + "As far as this poor little capacitor is concerned,", + "it is chugging along merrily thinking that it is connected to", + "ground. Little does it know it is only", + "a virtual ground, all right?", + "But the current i here is simply C dvI/dt.", + "And that current, the C dvI/dt,", + "that current flows through here and gives me vO as iR.", + "So, vO is simply minus R. Let me substitute for i there,", + "C dvI/dt. OK, so notice then that my vO", + "is now proportional to dvI/dt. So, vO is some RC time constant", + "times dvI/dt. Therefore, I have my", + "differentiator circuit. Remember this as a closing", + "thought. Remember this v plus more or", + "less equal to v minus trick. And to the extent possible", + "simply use that trick to analyze op amp circuits under feedback", + "and not in saturation. Just remember these two.", + "Very quickly for the demo, I have a square wave input here", + "to the op amp, that's my vI to the integrator.", + "And this is the output vO. The integral of a square wave", + "is a triangular wave, as you can see.", + "And we will do the same thing for a differentiator.", + "And for the differentiator, I input the square wave to this", + "differentiator circuit. And I get this,", + "wherever there is a sharp rise, I get this huge negative spike", + "and a positive spike because of the minus sign.", + "So, this is the differentiator circuit.", + "Then I feed this into the op amp.", + "OK. Thank you." + ] +} \ No newline at end of file diff --git a/subs/2vHGYdepKLw.srt.sjson b/subs/2vHGYdepKLw.srt.sjson new file mode 100644 index 0000000000..5a80933787 --- /dev/null +++ b/subs/2vHGYdepKLw.srt.sjson @@ -0,0 +1,1943 @@ +{ + "start": [ + 0, + 1541, + 5779, + 10691, + 14832, + 19263, + 22827, + 28317, + 34000, + 41131, + 44971, + 52651, + 58000, + 63348, + 67867, + 73400, + 75982, + 80317, + 85204, + 90000, + 95740, + 102946, + 110641, + 117603, + 123801, + 127128, + 131079, + 135237, + 138564, + 140990, + 144455, + 147851, + 150623, + 154504, + 159438, + 164415, + 169800, + 174066, + 177825, + 184123, + 189000, + 192830, + 196895, + 201039, + 204478, + 208856, + 213000, + 216016, + 220745, + 224088, + 228736, + 231589, + 235340, + 239091, + 243575, + 248437, + 252320, + 255771, + 258718, + 262529, + 266699, + 271837, + 276094, + 281297, + 286689, + 291608, + 295864, + 301256, + 307027, + 313388, + 316374, + 321236, + 325246, + 329000, + 332865, + 337676, + 340768, + 345922, 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OK.", + "Let's get started. We have one handout today.", + "That's your lecture notes. There's some copies still", + "outside for those who haven't picked one up.", + "In general, what I do is, in the lecture notes,", + "I leave out large amounts of material.", + "So, this will enable you to keep your hands busy while I'm", + "lecturing and take down some notes and so on.", + "So, don't assume that everything that I talk about is", + "on here. Please follow along.", + "OK, so as is my usual practice, let me start with a quick", + "review of what we covered so far.", + "So what we did primarily was looked at this discipline that", + "we call the lump matter discipline, which was very", + "similar, very reminiscent of the point mass simplification in", + "physics. And this discipline,", + "this set of constraints we imposed on ourselves,", + "allowed us to move from Maxwell's equations to a very,", + "very simple form of algebraic equations.", + "And specifically, the discipline took two forms.", + "One is, we said that we will deal with elements for whom the", + "rate of change of magnetic flux is zero outside of the elements,", + "and for whom the rate of change of charge I want to charge", + "inside the element was zero. So, if I took any element,", + "any element that I called a lump circuit element,", + "like a resistor or a voltage source, and I put a black box", + "around it, then what I'm saying is that the net charge inside", + "that is going to be zero. And this is not true in", + "general. We will see examples where,", + "if you choose some piece of an element for example,", + "there might be charge buildup, but net inside the,", + "if I put a box around the entire element,", + "I am going to assume that the rate of change of charge is", + "going to be zero. So, what this did was it", + "enabled us to create the lump circuit abstraction,", + "where I could take elements, some element of the sort,", + "this could be a resistor, a voltage source,", + "or whatever, and I could now ascribe a", + "voltage, some voltage across an element, and also some current,", + "\"i,\" that was going into the element.", + "And as I go forward, when I label the voltages and", + "currents across and through elements, I'm going to be", + "following a convention. OK, the convention is that I'm", + "going to label, if I label V in the following", + "manner, then I'm going to label \"i\" for that element as a", + "current flowing into the positive terminal.", + "It's just a convention. By doing this,", + "it turns out that the power consumed by the element is \"vi\"", + "is positive. OK, so by choosing I going in", + "this way into the positive terminal, the power consumed by", + "the element is going to be positive.", + "OK, so in general of even simply following this", + "convention, when I label voltages and currents,", + "I'll be labeling the current into an element entering in", + "through the plus terminal. Remember, of course,", + "if the current is going this way, let's have one amp of", + "current flowing this way, then when I compute the", + "current, \"i\" will come out to be negative.", + "OK, so by making these assumptions, the assumptions of", + "the lumped matter discipline, I said I was able to simplify", + "my life tremendously. And, in particular what it did", + "was it allowed me to take Maxwell's equations,", + "OK, and simplify them into a very simple algebraic form,", + "which has both a voltage law and a current law that I call", + "Kirchhoff's voltage law, and Kirchhoff's current law.", + "KVL simply states that if I have some circuit,", + "and if I measured the voltages in any loop in the circuit,", + "so if I look at the voltages in any loop, then the voltages in", + "the loop would sum to zero. OK, so I measure voltages in", + "the loop, and they will sum to zero.", + "Similarly, for the current, if I take a node of a circuit,", + "if I build the circuit, a node is a point in the", + "circuit where multiple edges connect.", + "If I take a node, then the current coming into", + "that node, the net current coming into a node is going to", + "be zero. OK, so if I take any node of", + "the circuit and sum up all the currents going into that node,", + "they will all net sum to zero. So, notice what I've done is by", + "this discipline, by this constraint I imposed on", + "myself, I was able to make this incredible leap from Maxwell's", + "equations to these really, really simple algebraic", + "equations, KVL and KCL. And I promise you,", + "going forward to the rest of 6.002, if this is all you know,", + "you can pretty much solve any circuit using these two very", + "simple relations. It's actually really,", + "really simple. It's all very simple algebra,", + "OK? So, just to show you an", + "example, let me do a little demonstration.", + "Let me build let me build a small circuit and measure some", + "voltages for you, and show you that the voltages,", + "indeed, add up to zero. So, here's my little circuit.", + "", + "So, I'm going to show you a simple circuit that looks like", + "this, and let's go ahead and measure some voltages and", + "currents. In terms of terminology to", + "remember, this is called a loop. So if I start from the point C", + "and I travel through the voltage source, come to the node A down", + "through R1 and all the way down through R2 back to C,", + "that's a loop. Similarly, this point A is a", + "node where resistor R1 the voltage source V0,", + "and R4 are connected. OK, just make sure your", + "terminology is correct. So, what I'll do is I'll make", + "some quick measurements for you, and show you that these KVL and", + "KCL are indeed true. So, the circuits up there,", + "could I have a volunteer? Any volunteer?", + "All you have to do is write things on the board.", + "Come on over. OK, so let me take some", + "measurements, and why don't you write down", + "what I measure on the board? What I'll do is,", + "let me borrow another piece of chalk here.", + "What I'll do is focus on this loop here, and focus on this", + "node and make some measurements.", + "", + "All right, so you see the circuit up there.", + "OK, so I get 3 volts for the voltage from C to A.", + "so why don't you write down 3 volts?", + "", + "OK, so the next one is -1.6. And so that will be,", + "I'm doing AB, V_AB.", + "OK, and then let me do the last one.", + "It is -1.37. The measurements,", + "I guess, have been this way. So, what's written is V_AC.", + "But it's OK for now. Don't worry about it.", + "So, well, thank you. I appreciate your help here.", + "OK, so within the bonds of experimental error,", + "noticed that if I add up these three voltages,", + "they nicely sum up to zero. OK, next let me focus on this", + "node here. And at this node,", + "let me go ahead and measure some currents.", + "What I'll do now is change to an AC voltage so that I can go", + "ahead and measure the current without breaking my circuit.", + "OK, this time around, you'll get to see the", + "measurements that I'm taking as well.", + "So, what I have here, I guess you can see it this", + "way. What I have here is three wires", + "that I have pulled out from D. And this is the node D,", + "OK? So, I have three wires coming", + "into the node D just to make it a little bit easier for me to", + "measure stuff. OK, so everybody keep your", + "fingers crossed so I don't look like a fool here.", + "I hope this works out. So, you roughly get,", + "what's that, 10 mV.", + "OK, so it's about 10 mV peak to peak out there,", + "and let's say that if the waveform raises on the left-hand", + "side, it's positive. So, it's positive 10 mV.", + "And another positive 10 mV, so that's 20 mV.", + "And this time, it's a negative,", + "roughly 20, I guess, -20.", + "So, I'm getting, in terms of currents,", + "I have a -10, -10, I'm sorry,", + "positive 10, positive 10,", + "and a -20 that adds up to zero. But more interestingly,", + "I can show you the same thing by holding this current", + "measuring probe directly across the node.", + "And, notice that the net current that is entering into", + "this node here is zero. OK, so that should just show", + "you that KCL does indeed hold in practice, and it is not just a", + "figment of our imaginations. So, before I go on,", + "I wanted to point one other thing out.", + "Notice that I've written down two assumptions of the lumped", + "matter discipline, OK?", + "There is a total assumption of the lump matter discipline,", + "and that assumption is, in spirit, at least,", + "shared by the point mass simplification in physics as", + "well. Can someone tell me what that", + "assumption is? A total assumption,", + "which I did not mention, which you can read in your", + "notes in section 8.2 in the appendix, what's a total", + "assumption that is shared in spirit with the point mass", + "simplification? Anybody?", + "A total assumption to be made here is that in all the signals", + "that we will study in this course, we've made the", + "assumption that the signal speeds of interest,", + "transition speeds, and so on, are much slower than", + "the speed of light. OK, that my signal transition", + "speeds of interest are much slower than the speed of light.", + "Remember, the laws of motion, the discrete laws of motion", + "break down if your objects begin moving at the speed of light.", + "OK, the same token here, our lump circuit abstraction", + "breaks down if we approach the speed of light.", + "And there are follow on courses that talk about waveguides and", + "other distributed analysis techniques that deal with", + "signals that travel close to speeds of light.", + "OK, so with that, let me go on to talking about", + "method one of circuit analysis. This is called the basic KVL", + "KCL method. So just based on those two", + "simple algebraic relations, I can analyze very interesting", + "and complicated circuits. The method goes as follows.", + "So, let's say our goal is, given a circuit like this,", + "our goal is to solve it. OK, in this course,", + "we will do two kinds of things: analysis and synthesis.", + "Analysis says, given a circuit,", + "OK, what can you tell me about the circuit?", + "OK, so we'll solve existing circuits for all the voltages", + "and currents, voltages across elements,", + "and currents through those elements.", + "Synthesis says, given a function,", + "I may ask you to go and build circuits.", + "OK, so for analysis here, we can apply this method that I", + "want to show you. And the idea here is that,", + "given a circuit like this, let us figure out all the", + "voltages and currents that are a function of the way these", + "elements are connected. So, the basic KVL and KCL", + "method has the following steps. The first step is to write down", + "the element VI relationships. OK, right down the element VI", + "relationships for all the elements.", + "The second step is write KCL for all the nodes,", + "and the third step is to write KVL for all the loops in the", + "circuit. That's it.", + "Just go ahead and write down element rules,", + "KVL, and KCL, and then go ahead and solve the", + "circuit. So, what we'll do,", + "we'll do an example, of course.", + "But, just as a refresher, we've looked at a bunch of", + "elements so far, and for the resistor,", + "the element relation says that V is pi R, where R is the", + "resistance of the element here. For a voltage source,", + "V is equal to V nought. That's the element", + "relationship. And for a current source,", + "the element is the relation is, \"i\" is simply the current", + "flowing through the element. OK, so these are some of the", + "simple element rules for the devices that the current source,", + "voltage source, and the resistor.", + "So let's go ahead and solve this simple circuit.", + "And what I'll do is go ahead and solve the circuit for you.", + "OK, if you turn to page five of your notes, I'm going to go", + "ahead and edit the circuit here. You can scribble the values on", + "your notes on page five. OK, so as a first step of my", + "KVL KCL method, I need to write down all my", + "element VI relationships. So, before I do that,", + "let me go ahead and label all the voltages and currents that", + "are unknowns in the circuit. So, let me label the voltages", + "and currents associated with the voltage source as here.", + "Notice, I continue to follow this convention where whenever I", + "label voltages and currents for an element, I will show the", + "current going into the positive terminal of the element", + "variable, OK, after element variable voltage.", + "So here, I have V nought and I nought.", + "Let me pause here for five seconds and show you a point of", + "confusion that happens sometimes.", + "Often times, people confuse between what is", + "called the variable that is associated with the element", + "versus the element value. OK, notice that here,", + "capital V nought is the voltage that this voltage source", + "provides, while this name here, v nought, is simply a variable", + "that we've used to label the voltage across that element.", + "So, similarly, I can label v1 as the voltage", + "across the resistor, and i1 is the current flowing", + "through the resistor. So this method of labeling,", + "where I follow the convention, that the current flows into the", + "positive terminal is called the associated variables discipline.", + "I was trying to use the word discipline in situations where", + "you have a choice, OK, and of a variety of", + "possible choices, you pick one as the convention.", + "OK, so here, as a convention,", + "we use the associated variables discipline, and use that method", + "to consistently label the unknown voltages and currents in", + "our circuits. OK, so let me continue the", + "labeling here, v4, i4, i3, v3 here,", + "and v2 and i2, v5, and i5.", + "I think that's it. So, I've gone ahead and labeled", + "all my unknowns. So each of these voltages and", + "currents are the voltages and currents associated with each of", + "the elements. And my goal is to solve for", + "these. OK, so in terms of our solution", + "here, let's follow the method that I outlined for you.", + "So, as the first step I am simply going to go ahead and", + "write down all the element VI relationships.", + "OK, so as a first step, I'm going to go ahead and write", + "down all the VI relationships. So, can someone yell out for me", + "the VI relationship for the voltage source?", + "OK, good. So, v0 is capital V nought,", + "that is that the variable V nought is simply equal to the", + "voltage, v0. Similarly, I can write the", + "others. v1 is i1, R1.", + "v2 is i2, R2, and so on.", + "OK, and I have one, two, three, four,", + "five, six elements. So, I will get six such", + "equations. Step two, I'm going to go ahead", + "and write KCL for the nodes in my system.", + "So, let me start with node A. So, for node A,", + "let me take as positive the currents going out of the node.", + "So, I get i nought flowing out, plus i1 flowing out,", + "plus i4 flowing out, and they must sum to zero for", + "node A. Then, I can go ahead and do the", + "other nodes, let's say, for example,", + "I do node B. For node B, I have i2 going", + "out. That's positive,", + "i3, and i1 is coming in, so I get -i1 equals zero.", + "OK, so I have one, two, three, four,", + "I have four nodes. OK, so I would get four", + "equations. It turns out that the fourth", + "equation is not independent. You can derive it from the", + "others. So, I get three independent", + "equations out of this. I can then write KVL.", + "And for KVL, I just go down my loops here.", + "And let me go through this first loop here in this manner.", + "OK, and a simple trick that I use, you have to be incredibly", + "careful when you go through this in keeping your minuses and", + "pluses correct. Otherwise you can get", + "hopelessly muddled. Once you label it,", + "you need to be sure that you get all your minuses and pluses", + "correct. So, for KVL,", + "what I'd like to do is, let's say I start at C,", + "and from C I'm going to go to A.", + "For A I go to B, and from B I'm going to come", + "back to C. OK, that's how I traverse my", + "loop. And, the trick that I'm going", + "to follow is, as my finger walks through that", + "loop, I'm going to label the voltage as the first sign that I", + "see for that voltage. OK, so I'm going to start with", + "C, and I go up. I start by punching into the", + "voltage source element, and then punch into it,", + "I hit the minus sign for the V nought.", + "OK, so I'm just going to write down minus V nought,", + "plus then I go through and as I come up to A and go down to B,", + "I punch to the plus sign of the V1.", + "So, that's plus V1. And then I punch into the plus", + "sign of the V2, and so I get plus V2,", + "and that is zero. OK, good.", + "So, that matches what you have in your notes as well.", + "So, this is the first equation. Similarly, I can go through my", + "other loops and write down equations for each of the loops.", + "OK, and the convention that I like to follow is as I go", + "through the loop, I write down as a sign for the", + "voltage the first sign that I counter for that element.", + "OK, you can do the exact opposite, if you want,", + "just to be different. But, as long as you stay", + "consistent, you'll be OK. All right, so in the same", + "manner here, there are four loops that I can have,", + "so four equations. Again, one of them turns out to", + "be dependent on the others. So I end up getting three", + "independent equations. So, I get a total of 12", + "equations. I get 12 equations.", + "There are six elements, OK, voltage source,", + "and five resistors. So, there are six unknown", + "voltages, and six unknown currents.", + "So, I have 12 equations, and 12 unknowns.", + "OK, I can take all of the equations and put them through a", + "big crank, and sit there and grind.", + "And if I was really cruel, I'd give this as a homework", + "problem, and have you grind, and grind, and grind until you", + "get your six voltages and six currents.", + "OK, it works. OK, so you get 12 equations,", + "and this method just works. However, notice that this is", + "quite a grubby method. It's quite grungy.", + "I get 12 equations, and it's quite a pain even for", + "a simple circuit like this. However, suffice it to say that", + "this fundamental method is one step away from Maxwell's", + "equations, simply works. OK?", + "So what you'll do is the rest of this lecture,", + "I'll introduce you to a couple more methods.", + "One is an intuitive method, and another one called the node", + "method is a little bit more formal, but is much more,", + "I guess, terse Than the KVL KCL method.", + "Method 2. So the relevant section to read", + "in the course notes is section 2.4.", + "One of the things that I will be stressing this semester is", + "intuition. What you'll find is that as you", + "become EECS majors, and so on, and go on,", + "or if you talk to your TAs or your professors and so on,", + "you will find that very rarely do they actually go ahead and", + "apply the formal methods of analysis.", + "OK, by and large, engineers are able to look at a", + "circuit and simply by observation write down an", + "answer. And usually in the past,", + "what we have tried to do is kind of ignore that process and", + "told our students, look, we teach you all the", + "formal methods, and you will develop your own", + "intuition and be able to do it. What we'll try to do this term", + "is try to stress the intuitive methods, and try to show you how", + "the intuitive process goes, so you can very quickly solve", + "many of these circuits simply by inspection.", + "OK, so this method that I'm going to show you here is one", + "such an intuitive method. And I'll call it element", + "combination tools. OK, for many simple circuits,", + "you can solve them very quickly by applying this method.", + "The components of this method are these.", + "I learned about how to compose a bunch of elements.", + "So, let's say, for example,", + "I have a set of resistors, R1 through RN,", + "in series. OK, you can use KVL and KCL to", + "show that this is equivalent to a single resistor whose value is", + "given by the sum of the resistances.", + "OK, so if I have resistors in series, then effectively it's", + "the same as if there was a single resistor whose value is", + "the sum of all the resistances. OK, you can look at the course", + "notes for a proof for derivation of this fact.", + "Similarly, if I have resistances in parallel,", + "so let me call them conductances.", + "A conductance is the reciprocal of a resistance.", + "If resistance is measured in ohms, conductance is measured in", + "mhos, M-H-O-S. OK, so that's the conductance", + "is G1, G2, and G3. And effectively,", + "this is the same as having a single conductance whose", + "effective value is given by the sum of the conductances.", + "OK, the conductances in parallel add,", + "and resistances in series add. Similarly, for voltage sources,", + "if I have voltage sources in series, then they are tantamount", + "to the sum of the voltages. And similarly,", + "for currents, if I have currents in parallel,", + "then they can be viewed as a single current source,", + "whose currents are the sum of the individual parallel", + "currents. So, let's do a quick example.", + "So let's do this example. So, let's say I have a circuit", + "that looks like this, and three resistances.", + "And let's say all I care about is the current,", + "I, that flows through this wire.", + "All I care about is that current.", + "Of course, you can go ahead and write KVL and KCL.", + "You will get four equations, and there are four unknowns.", + "And you can solve it. But, I can apply my element", + "combination rules, and very quickly figure out", + "what the current I is, using the following technique.", + "So, what I can do is, I can, first of all,", + "take this circuit. And, I can compose these two", + "resistances and show that the circuit is equivalent as far as", + "this current, I, is concerned to the", + "following circuit, R1.", + "And I take the sum of the two conductances,", + "OK, and that comes out to be R1, R2, R3, R2 plus R3.", + "And then, I can further simplify it, and I get a single", + "resistance, whose value is given by R1 plus R2,", + "R3, R3. OK, I'm just simplifying the", + "circuit. Now, from this circuit,", + "I can get the answer that I need.", + "I is simply the voltage, V, divided by R1 plus.", + "OK, so in situations like this where I'm looking for a single", + "current, I can very quickly get to the answer by applying some", + "of these element combination rules.", + "And, I can get rid of having to go through formal steps.", + "So, in general, whenever you encounter a", + "circuit, by and large attempt to use intuitive methods to solve", + "it. And go to a formal method only", + "if some intuitive method fails. Even in your homework,", + "by and large, the homeworks are not meant to", + "be grungy. OK, if you find a lot of grunge", + "in your homework, just remember you're probably", + "not using some intuitive method. OK, so just be cautious.", + "All right, so let me go on to the third method of circuit", + "analysis, and the third method is called the node method.", + "So, the node method is simply a specific application of the KVL", + "KCL method and results in a much, much more compact form of", + "the final equations. If there's one method that you", + "have to remember for life, then I would say just remember", + "this method. OK, the node method is a", + "workhorse of the easiest industry.", + "OK, if there's one method that you want to consistently apply,", + "then this is the one to remember.", + "So, let me quickly outline for you to method,", + "and then work out an example for you.", + "The first step of the node method will be to select a", + "reference or a ground node. This is the symbol for a ground", + "node. The ground node simply says", + "that I'm going to denote voltages at that point to be", + "zero, and measure all my other voltages with reference to that", + "point. So, I'm going to select a", + "ground node in my circuit. Second, I want to label the", + "remaining voltages with respect to the ground node.", + "So, label voltages for all the other nodes with respect to the", + "ground node. Next, write KCL for each of the", + "nodes write KCL. OK, but don't write KCL for the", + "ground node. Remember, if you have N nodes,", + "the node equations will give you N-1 independent equations.", + "So, write KCL for the nodes, but don't do so for the ground", + "node. Then, solve for the node", + "voltages. So, let's say when we label", + "voltages. I want to be labeling them as E", + "something or the other. So, solve for the unknown node", + "voltages. And then, once I know all the", + "voltages associated with the nodes, I can then back solve for", + "all the branch voltages and currents.", + "OK, once I know all the node voltages, I can then go ahead", + "and figure out all the branch voltages and the branch", + "currents. So, let's go ahead and apply", + "this method, and work out an example.", + "Again, remember, if there's one method that you", + "should remember, it's the node method.", + "OK, and when in doubt, consistently apply the node", + "method and it will work whether your circuit is linear or", + "nonlinear, if the resistors are built in the US or the USSR it", + "doesn't matter. OK, the node method will simply", + "work, linear or nonlinear, OK?", + "So, what I'm going to do is I'm going to build a circuit that's", + "my old faithful. It's our old faithful,", + "plus I'll make it a little bit more complicated by adding in", + "the current source. So, let's go have some fun.", + "Let's do this. So here's my voltage source,", + "as before. OK, what I'll do is for fun,", + "add a current source out there. And, you can convince", + "yourselves that if you go ahead and apply the KVL KCL method,", + "it'll really be a mess of equations.", + "OK, so R1, R3, R4, R2, R5.", + "OK, so let's follow our method and just plug and chug here.", + "So let's apply the first step. I select a ground node.", + "It's a reference node from which I'll measure all my other", + "voltages. OK, now without knowing", + "anything about the node method, try to use intuition as to", + "which node you should choose as a ground node.", + "Remember, you want to label the ground node with the voltage", + "zero, and measure all the other voltages with respect to that", + "node. OK, a usual trick is to pick a", + "node which has the largest number of elements connected to", + "it as the ground node. OK, and in particular,", + "you will find out later it's useful to pick a node in which", + "all your voltage sources, the maximum number of your", + "voltage sources are also connected.", + "OK, so in this instance, I'm going to choose this as my", + "ground node. OK, that's my first step.", + "I chose that as my ground node. And I'm going to label that as", + "having a voltage zero. Second step,", + "I'll label voltages of the other branches with respect to", + "the ground node. OK, so what I'll do is add this", + "node here. So I'm going to label that", + "voltage E1. These are my unknowns.", + "Remember, node method, because my node voltages are my", + "unknowns. So, I label this as E1.", + "I label this one as my unknown voltage, E2.", + "What about this one here? Is that voltage unknown?", + "No. I know what the voltage is", + "because I know that this node is at a voltage,", + "V0, higher than the ground node.", + "OK, notice that to go from here to here, I directly go through a", + "voltage source. And so, this node has voltage", + "V0. And I'll simply write down V0.", + "OK, try to simplify the number of steps that you have to go", + "through, so directly go ahead and write down the voltage,", + "V0, for that node. What I will also do,", + "is for convenience, I'm going to write down,", + "I'm going to use conductances. So I'm going to use GI in the", + "place of one by RI, and write down a bunch of node", + "equations. OK, so step one,", + "I've chosen my ground node. Step two, I've labeled my node", + "voltages, E, OK? I've done that with two of my", + "steps. Now, let me go ahead and --", + "", + "OK, so let me go ahead and apply step three.", + "And, step three says go ahead and apply KCL for each of the", + "nodes at which you have an unknown node voltage.", + "And then that will give you your equations.", + "So let me start by applying KCL at E1.", + "So, let me write KCL at E1. I do one more thing.", + "Notice, I don't have any currents there.", + "OK, so how do I write KCL? KCL simply says the sum of", + "currents into a node is zero again, remember,", + "by the lump matter discipline. So, if I don't have currents in", + "there, so the trick that I adopt is that to write KCL,", + "I use the node voltages, and implicitly substitute for", + "the node voltages, divide by the elemental the", + "resistance, for instance, so I take the node voltages,", + "and divide by the resistance, get the current.", + "OK, so I implicitly apply element relationships to get the", + "node currents. So, the example that make it", + "clear, so I take node E1 and, again, for currents going out", + "I'm going to assume to have, to be positive.", + "So, the current going up is E1 minus V nought,", + "divide by R1, so I multiplied by the G1.", + "That's the current going up. Plus, the current going down is", + "E1 minus zero where the ground node potential is zero,", + "G2, OK, plus the current that is going to resistor R3,", + "which is simply E1 minus E2, divide by R3.", + "So, E1 minus E2, divide by R3,", + "or multiplied by G3 is equal to zero.", + "OK, see how I got this? This is simply KCL,", + "but to get my currents, I simply take the differences", + "of voltages across elements, and divide by the element of", + "resistance, and I get the currents.", + "OK, so I can similarly write KCL at E2.", + "So, at KCL at E2, again, let me go outwards.", + "So, the current going up is E2 minus V nought multiplied by G4.", + "The current going left is E2 minus E1 divided by R3 or", + "multiplied by G3. The current going down is E2", + "minus zero multiplied by G5. And, the current going down is", + "-I1. OK, you've got to be careful", + "with your polarities here. So all the currents going out", + "sum to zero. And here are the currents that", + "are going out at this point. So what I do next is I can move", + "the constant terms to the left-hand side and collect my", + "unknowns. So, let me write them out here.", + "So, let's say I get E1 here, OK, and from this equation,", + "I have a V nought, G1, which comes out here.", + "So, minus V nought G1 comes over to the other side.", + "And, let me collect all the values that multiply E1.", + "So I get, G1 is one example. I have G2, and I have G3.", + "And then, for E2, I have minus G3.", + "OK, so I'll simply express this as the element voltages", + "multiplied by some terms in parentheses, and I put my", + "external sources on the right hand side.", + "Similarly, I go ahead and do the same thing here.", + "In this instance, let me move my sources to the", + "right. So, I get I1 coming out there,", + "and I get V nought G4 coming out there.", + "By the way, I just want to mention to you that if you're", + "looking to fall asleep, this is a good time to do so", + "because as soon as I write down these two equations,", + "OK, from now on it's nap time. There's nothing new that you're", + "going to learn from here on. It's just Anant Agarwal having", + "fun at the blackboard, pushing symbols around.", + "So, once you write down these two node equations,", + "the rest of it is just grubby math.", + "So, let me just have some fun. So let me just go ahead and do", + "that. So, I moved my voltages and", + "currents to the other side. And let me collect all the", + "coefficients for E1 here. So, E1 minus G3,", + "and that's it, I guess.", + "OK, and then I'll do the same for E2.", + "So, I get G4, and I get G3,", + "and I get G5. OK, so notice here that I have", + "two equations, and two unknowns.", + "OK, the two equations are on the right hand side,", + "I have some voltages and currents which are my dry", + "voltages and dry currents. OK, so actually this is getting", + "quite boring. I'm going to pause here,", + "and talk about something else. So, you can take this and you", + "can put it in matrix form, so I've done that for you on", + "page ten. It's all matrix form.", + "Yeah, I know that. You can use any technique to", + "solve it, use algebraic techniques, use linear algebraic", + "methods to solve it, use a computer,", + "whatever you want. And, computers,", + "when computers analyze circuits, they write down these", + "equations, and deal with solving matrices.", + "So, when you take the linear algebra across,", + "how many people here have taken a linear algebra class?", + "How many people here have heard of Gaussian elimination?", + "How can more people have heard of Gaussian elimination than", + "took a linear algebra class? Well anyway,", + "so now you know why you took those linear algebra classes.", + "And so, if I just collected these into matrix form --", + "", + "OK, so I just simply expressed those two equations in linear", + "algebraic form, and here's my column vector of", + "unknowns, and you can apply any of the techniques you've learned", + "in linear algebra to solve for this.", + "Gaussian elimination works. OK, and in computer,", + "people doing research in computer techniques,", + "or solving such equations simply deals with huge equations", + "like this, building computer programs that,", + "given equations like this, can go ahead and solve them.", + "OK, so let me stop here and reemphasize that what you've", + "done is made a huge leap from Maxwell's equations to using the", + "lump matter discipline to KVL and KCL, which ended up giving a", + "simple algebraic equation to solve, and not having to worry", + "about partial differential equations that were the form of", + "Maxwell's equations." + ] +} \ No newline at end of file diff --git a/subs/3GdMaDzIUeQ.srt.sjson b/subs/3GdMaDzIUeQ.srt.sjson new file mode 100644 index 0000000000..07db11ddb8 --- /dev/null +++ b/subs/3GdMaDzIUeQ.srt.sjson @@ -0,0 +1,2477 @@ +{ + "start": [ + 0, + 32000, + 33573, + 38098, + 41737, + 46655, + 51180, + 57081, + 62000, + 65917, + 69344, + 72352, + 76479, + 79836, + 83054, + 87181, + 90553, + 94719, + 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Good morning.", + "Let's get going. As always, I will start with a", + "review. And today we embark on another", + "major milestone in our analysis of lumped circuits.", + "And it is called the \"Sinusoidal Steady-state\".", + "Again, I believe this will be the second and the last lecture", + "for which I will be using view graphs.", + "And the idea here is that, just like in the last lecture,", + "there is a bunch of mathematical grunge that needs", + "to be gone through. And I want to show you a", + "sequence in a little chart today that talks about the effort", + "level in doing some problems based on time domain", + "differential equations, as in the last lecture,", + "something slightly different today and something much better", + "on Thursday. And so, in some sense,", + "Thursday's lecture and today's lecture involve talking about", + "the foundations of the behavior of certain types of circuits.", + "And it is good for you to have this foundation as background,", + "but when you actually go to solve circuits you don't quite", + "use these methods. You use much easier techniques,", + "which I will talk about next Thursday.", + "Let's start with a quick review, and then we will go into", + "sinusoidal steady state. The last lecture we talked", + "about this circuit and we did the same two lectures ago on", + "Tuesday. We had one inverter driving", + "another inverter. And we said that the wire over", + "ground had some inductance. CGS is the capacitor of the", + "gate and R is the resistance at the drain of the first inverter.", + "And if you look at this circuit, that circuit formed an", + "RLC pattern. And what we did was we said", + "let's drive this with a one to zero transition at the input.", + "And the one to zero transition at the input would cause this", + "transistor to switch off, and this node would then go", + "from a very low value to a high value.", + "So it as if a 5 volt step was applied at this input.", + "We also saw that using time domain differential equations", + "that by applying a step input here the output looked like", + "this. The output would show some", + "oscillatory behavior when we have a capacitor and inductor.", + "I also gave you some insight as to why it oscillates like this.", + "And you also heard in recitation that the reason for", + "this oscillation was because of these two storage elements.", + "Each of these storage elements tries to hold onto its state", + "variable. For example,", + "the capacitor tries to maintain its voltage while the inductor", + "tries to maintain its current. And, much like a pendulum which", + "oscillates back and forth, it swaps potential energy", + "versus kinetic energy down here and swings back and forth.", + "In the same way, in an LC circuit like this,", + "energy swaps back and forth between a potential energy and a", + "kinetic energy equivalent, which swaps back and forth", + "between energy stored in the inductor and energy stored in", + "the capacitor and sloshes back and forth.", + "And because of this resistor the energy eventually dissipates", + "and you end up getting a final value which corresponds to the 5", + "volts appearing here. And why is that?", + "That is because remember the capacitor is a long-term open", + "for DC. It is a DC voltage.", + "After a long time this capacitor looks like an open", + "circuit and the inductor looks like a complete short circuit,", + "an ideal inductor as a complete short circuit for DC.", + "And so therefore in the long-term it is as if this guy", + "is a short, this guy is an open, so 5 volts simply appears here.", + "And this is the transient behavior.", + "Then we just switch the first transistor off.", + "In the last lecture, I left off with intuitive", + "analysis. Let me quickly cover that and", + "redo the intuitive analysis for you.", + "I left it the last time by having you think about whether", + "the transient response would begin by going down or begin by", + "going up. And I will cover that today.", + "This was the circuit that we analyzed.", + "A VI input with a step and an RLC out here.", + "And we were plotting the capacitor voltage.", + "And intuitively we can plot this in the following way.", + "I have also marked for you the section number in the course", + "notes which has a discussion of this intuitive analysis,", + "(Section 13.8). Let's do the easy stuff first.", + "Notice that the capacitor wants to hold its voltage.", + "And so given that we don't have any infinite impulse here,", + "I am going to start out with the capacitor voltage being", + "where it is. And the initial conditions are", + "given to you. You are given that the", + "capacitor voltage starts out being positive at v zero and the", + "current starts out being negative at time zero.", + "So I am telling you that there is a voltage v across the", + "capacitor at time t=0 and there is a current that is flowing.", + "Since i is negative there is a current initially that is", + "flowing in the opposite direction to this arrow here.", + "The i zero is negative. In light of that,", + "I can start plotting my curve here by intuition.", + "I start by saying at time t=0 I am told that the initial voltage", + "of the capacitor is at zero. This is about as simple as it", + "gets. Completely intuitive.", + "I also know that after a long time, can someone tell me after", + "a long time what the voltage will be at the end of the", + "capacitor? You should be able to get his", + "by observation? VI.", + "And why is it VI? It is vI because this is a DC", + "value VI. And after a long time this guy", + "behaves like an open circuit to DC.", + "This guy behaves like a short circuit to DC.", + "So since this is an open circuit, VI will appear here", + "after a long time. And so therefore,", + "after a long time, the capacitor voltage is going", + "to be at VI. And I just showed you that.", + "There you go. You already have the two", + "endpoints of your curve completely by observation,", + "intuition. No DEs, no nothing.", + "Just by staring at it and understanding the fundamentals", + "of how simple primitive circuit elements work.", + "Absolutely simple stuff. So you've nailed the two ends", + "now and you cannot go wrong with the stuff in the middle.", + "Let's see. As a next step what I do is I", + "need to understand what the dynamics of the circuit looks", + "like here. What I do is I develop the", + "characteristic equation. And initially you will write", + "the differential equation and then substitute e^st and get", + "this characteristic equation. But you could also remember it", + "as a pattern. For a series RLC circuit you", + "always get an equation of this form, always.", + "If this were R, L and C.", + "And whether you are looking at L up here or C up here,", + "as long as you're looking at the capacitor voltage,", + "the capacitor voltage is going to behave the same.", + "And for this circuit the characteristic equation remains", + "the same as well for a series RLC.", + "It is exactly this. And I write the standard", + "canonic form as s squared plus two alpha s + omega nought", + "squared. And omega nought is simply one", + "by square root of LC and alpha is simply R divided by L and I", + "have two in the denominator as well.", + "And then I get omega d which is my damped frequency given by", + "omega nought squared minus alpha squared.", + "And Q is simply called the quality factor.", + "And we will learn about Q in a lot more detail in about a", + "couple of lectures from today. That is given where omega", + "nought divided by two alpha. These parameters,", + "alpha, omega nought, Q and omega d pretty much", + "characterize everything else that I need to know about the", + "circuit. First of all,", + "omega d is the frequency of oscillation.", + "And so since omega d is a frequency of oscillation then I", + "know that the time period of oscillation is two pi by omega", + "d. Omega is in radians.", + "Notice that for typical values of circuits like this when R is", + "pretty small, alpha squared is going to be", + "very small. It's a common case for", + "underdamped circuit that omega d will more or less be equal to", + "omega nought. Commonly that is going to be", + "the case. This frequency is governed by", + "LC. And if R is large it is", + "governed by this omega d here. So I have the frequency of", + "oscillation. I also know that Q is related", + "to the frequency of oscillation divided by alpha.", + "It is a ratio of the frequency divided by how badly I am being", + "damped. So it is a fight between the", + "frequency of oscillation and now heavily I damp.", + "And the ratio of that is an indication of how many cycles I", + "ring. So Q tells me that the ringing", + "stops approximately after Q cycles.", + "These four values, omega d, Q, alpha and omega", + "nought are telling me more and more now.", + "So I have got these two factors.", + "So I know now, based on omega d and Q,", + "that it is going to look something like this.", + "Some ringing here and then I stop at this point.", + "The last thing that is left to do here for me for now is to", + "figure out whether I start out going down or I start out going", + "up. I start out going down or I", + "start out going up? I don't know that yet.", + "And I stopped at this point in the last lecture and let you", + "think about how you can stare at the circuit and intuitively", + "figure out whether this goes down or that goes up.", + "So here is the insight. Let's stare at this for a", + "minute purely through intuition. The capacitor has a voltage v", + "across it, right? And that is because I am", + "telling you that it has an initial voltage v.", + "Now, I want to find out at prime t equals zero plus,", + "in which direction does a capacitor voltage go?", + "Increase or decrease? What do I do?", + "Let me take a look at the inductor current.", + "I am told that the inductor current is negative which means", + "I am told that the inductor current is going in this", + "direction initially. The inductor current is pushing", + "in this direction. Now, remember,", + "just as the capacitor is one stubborn nut trying to hold its", + "voltage, the inductor is as stubborn.", + "It is trying to hold its current.", + "It is trying to maintain its current.", + "And its initial current i(0) is in this direction.", + "Capacitor has a voltage here, that is v(0),", + "and the inductor is yanking the current in that direction.", + "So what should happen to the capacitor voltage initially?", + "If I am at v(0) and someone is yanking current out,", + "at least initially in that direction, what should the", + "initial dynamics of the capacitor voltage look like?", + "Pardon? It should drop,", + "which means that if the initial current is being pulled in that", + "direction the capacitor voltage must droop to begin with.", + "Completely through intuition. No math here.", + "This means that i(0) is negative.", + "It is as if i(0) is being pulled out in this manner,", + "so the capacitor voltage must drop to begin life.", + "Therefore, the dynamics look somewhat like this.", + "Notice that this is very reminiscent of the ringing that", + "we saw at the gate node of the second inverter.", + "Let's stop here in terms of time domain analysis of RLC,", + "and today let's take another big step forward.", + "Today marks another transition in life here.", + "This is actually a huge transition so I want to just", + "pause and take like ten seconds of a breather just to clearly", + "demarcate the fact that we have a huge transition coming up.", + "The key to this transition is that I want to look at today the", + "steady-state response of networks to a sinusoidal drive.", + "We are going to do two things differently starting today on", + "this new journey of ours. In the past,", + "we looked at time domain behavior of circuits.", + "For RLC, for example, we looked at the transient", + "response. And what happened the instant I", + "turn something on? The transient response.", + "Today we are going to look at a steady-state response.", + "Steady-state response says that if I wait long enough,", + "for whatever the circuit wants to do in the beginning of life", + "to die out. If I wait long enough,", + "how is the circuit going to behave after a long time?", + "I will tell you why that is important in a second.", + "I look at the steady-state behavior.", + "Second, I am going to look today at sinusoidal drive.", + "Two things that are different from, say for example,", + "what I covered in the past ten minutes.", + "In the past ten minutes I covered two things which were", + "different. One is that I looked at the", + "transient response and then steady-state.", + "And remember for a DC input, for a DC voltage the", + "steady-state was a DC voltage across the capacitor,", + "correct? So the steady-state was pretty", + "boring, it was steady-state DC. But what we are going to do", + "today is instead of having DC inputs or step inputs which", + "settle to a DC value after some time, we are going to drive a", + "circuit through the sinusoidal input.", + "So you may ask me, Agarwal, are you nuts?", + "Why do you want to drive something with a sinusoidal", + "input? Why not just steps in DC and so", + "on? That was painful enough.", + "Why sinusoidal? Why not do triangular or why", + "not do some other exponentially decaying stuff or something", + "really cool like a whacko music signal?", + "What is special about sinusoidal stuff?", + "The key thing to realize is that, well, let me ask you a", + "question first. How many people here know about", + "Fourier series? Good.", + "It looks like some of you have taken the prerequisites.", + "Good. Need I say more as to why this", + "is important? Just that question should give", + "you the answer, right?", + "You've learned about Fourier series.", + "And when you learned about Fourier series you were", + "wondering why on earth are we learning about Fourier series?", + "Who cares that you can represent the periodic signals", + "as a summation of a series of sine waves?", + "Why is that interesting? Why are you telling me that I", + "can take a square wave and represent that as a summation of", + "periodic square waves and represent that as a summation of", + "sines? Who cares that I can take a", + "sequence of pulses with a fixed period and represent that as a", + "sum of sines? Who cares that I can take a", + "triangular wave and represent that as a sum of sines?", + "I am not sure what answer your math professors gave you when", + "they taught you Fourier series. But in math they are purists.", + "They don't care about applications.", + "The answer could well have been because it is aesthetically", + "pleasing. I mean isn't it cool that you", + "can represent a sequence of pulses as a sum of sines?", + "That is good enough for mathematicians.", + "But I am an engineer. If it I cannot see how it helps", + "humanity in the short-term then I probably don't care too much", + "about it. Let me give you some practical", + "significance of this. So it turns out that,", + "since we know that we can represent periodic signals with", + "sums of sines. What this means is that if I", + "can figure out the behavior of networks to a sinusoidal input,", + "if I can understand how to analyze a network for a", + "sinusoidal input that means that if the circuit is linear I can", + "then compute the response of the circuit to any periodic", + "waveform. Here is the argument.", + "I can represent any periodic waveform as a sum of sines.", + "The Fourier series, remember?", + "If I just figure out the response of my network for a", + "sine wave, then if this is a linear network,", + "I can compute the response to any sequence of scaled sum of", + "sines. A some sine,", + "B sine two, omega whatever, C sine something or the other.", + "I can simply take the response of the one sine.", + "And from there I can go ahead, and knowing the response of a", + "sine wave I can compute the response to a sum of sines.", + "That is pretty cool. Therefore, doing it for", + "sinusoidal drives is really important.", + "Why steady-state now? Hopefully, I have convinced you", + "that looking at the response of circuits to sinusoidal drive is", + "important and interesting because we can long ways from", + "there. What about steady-state?", + "Well, it turns out that, and I am going to show you,", + "when you listen to music, you have an amplifier and", + "listen to music, what you are observing by and", + "large is the steady-state behavior of the amplifier.", + "You are listening to something over many seconds or many hours.", + "And the transients used for most of our common circuits,", + "the transients die out pretty quickly.", + "And so the transients are quite complicated and they die out", + "quickly. We say we are engineers.", + "Let's focus on what is practically important.", + "And we focus on the steady-state behavior as a large", + "part of our analysis and just completely ignore the transient", + "response when we care about the response to sinusoidal input.", + "The transient response will die away, and I will show that", + "mathematically to you in a few seconds.", + "And let's focus on the steady-state because that what I", + "am going to hear most of the time.", + "I am going to listen to an average over long periods of", + "time. That's the motivation behind", + "this. And let me put this in", + "perspective for you. By now this should bring", + "memories to your mind. This is the playground that we", + "are in. This is the lumped circuit", + "playground here. Remember we came from the", + "playground of nature to the playground of EECS where we made", + "the big leap from Maxwell's equations to lumped circuits,", + "that's lumped circuit abstraction.", + "And within there we spent a large part of the last couple of", + "months doing linear circuits. We also analyzed nonlinear", + "circuits. Remember the amplifier circuit", + "of the MOSFET large signal analysis was nonlinear?", + "Well, there is linear and nonlinear.", + "Within linear we also showed that if you take a digital", + "circuit, at least as we understood them,", + "and draw the subcircuit for a given set of switch settings,", + "if I set the switches in a given way what I was left with", + "was another linear circuit for a given value of all the switch", + "settings. My small signal analysis was", + "also linear. If I focused on small signals I", + "also had linear analysis. Today what we are going to do", + "is this. We are going to articulate a", + "different part of the playground.", + "This was a big linear playground.", + "We've done this. We've done this.", + "We are going to explore this territory.", + "This is that territory of the playground in which we have", + "sinusoidal inputs to circuits. Furthermore,", + "we are going to look at a subcircuit of that region which", + "is steady-state. We will look at sinusoidal", + "input and in the steady-state. So that is going to be our", + "focus for the next two or three lectures just to give you a", + "perspective of where we are. To motivate this,", + "what I would like to do is consider your amplifier.", + "This is our friend the amplifier circuit,", + "this part here. And remember,", + "even though this is an amplifier, I am using a MOSFET", + "here. And a MOSFET,", + "as you know, has this gate capacitance CGS.", + "I am explicitly drawing it out for you here.", + "And I am going to drive this with a bias voltage plus some", + "small signal vI, the usual template for", + "amplifiers. And there is some Thevenin", + "resistance attached to that source.", + "I am going to model my source as a bias voltage,", + "a small signal plus some source resistance.", + "And I want to apply a sine wave here and I am going to look at", + "what this looks like. You may think,", + "look, this is a linear amplifier.", + "And so if I apply a sine wave here I should see some response", + "here, and that should be the same amplitude if I feed the", + "same amplitude here over any frequency.", + "But let's see what happens. Keep a look at,", + "switch over to that view graph while I show you a little", + "demonstration here. What you see here are three", + "sine waves, a yellow sine wave which is the input here,", + "you see a green sine wave which is the input vC at the gate of", + "the MOSFET, and then you see the blue which is the output here.", + "For now simply focus on the yellow and the blue.", + "The yellow is the input and the blue is the output.", + "So I apply some input and I get an output that looks more or", + "less some linear function of this input here.", + "It is a small signal. What I am going to do is I am", + "going to change the frequency of the input.", + "Remember, I want to look at the response of the circuit to a", + "sinusoid. I am feeding a sinusoid here.", + "I look at the response. I am going to change the", + "frequency. That is a big shift that we are", + "making in that the curve that we drew in the last lecture had to", + "do with varying time. Now I am going to focus on", + "sinusoids and vary their frequency.", + "I am going to change the frequency.", + "Stare at the blue curve as I increase the frequency and just", + "think of what you might expect. Based on the knowledge you have", + "so far you would expect that look, as I change the frequency,", + "the frequency should change but I should see the same amplitude.", + "OK but take a look. Let me increase the frequency", + "of the input. What do you see at the output?", + "I am increasing the frequency.", + "", + "What do you see happening there?", + "If you don't see anything changing there you will need to", + "see an optometrist. What do we see here?", + "As I changed the frequency, as I increased the frequency", + "what happened to the blue? The blue kept decreasing in", + "amplitude. And you are saying whoa,", + "what is happening here? We don't have the tools to deal", + "with this. I expected that when I changed", + "my frequency, my frequency here should change", + "of course, but why is the amplitude changing?", + "What is happening here? That is weird.", + "I noticed that this amplitude became smaller because the", + "amplitude of the green became smaller.", + "And remember the green was the voltage across the capacitor.", + "So this is your RC. And here is my input.", + "My input has the amplitude, which I am holding more or less", + "constant. And notice that vC decreased in", + "value as I increased my frequency.", + "Just hold that thought. As I increased the frequency of", + "my input the amplitude of the output kept diminishing.", + "In other words, the gain of the system seemed", + "to have decreased as I increased by frequency.", + "And today we will look at why that is so and how we can", + "analyze that. The other thing that is not so", + "obvious here is there is a phase shift.", + "What I am going to do is try to see if we can observe the phase", + "shift here.", + "", + "Notice here. What we have been used to is", + "for the amplifier we get a complete inversion at the", + "output. Inversion means a phase", + "difference of 180 degrees for a sine wave, right?", + "This peak should have been here, but notice that there is a", + "shifting of the peak. In other words,", + "if the yellow was my input my output should have had its", + "minimum when the input had its maximum.", + "But notice there is a shifting of the signal such that the", + "output is a maximum, not quite at the point where", + "the input is a minimum but a little bit after that.", + "Also weird. Not only has this little", + "circuit here lost its gain somehow, but also it seems to", + "have shifted the signal in phase.", + "That is weird. And today we will look at why", + "that is so and try to understand the frequency behavior of this", + "little subcomponent here. Notice that vC is exactly 180", + "out of phase with vO. So vO is faithfully an inverted", + "amplified form of the input vC. However, vC itself should have", + "been the same as vI but it looks quite different.", + "So let's understand why that is so.", + "The subcircuit to model is the subcircuit comprising the", + "source, resistor and the capacitor.", + "And I am just showing that to you here.", + "I have a vI, a resistor and capacitor.", + "And I am going to understand how this behaves.", + "And it is a first order circuit, single capacitor.", + "My input is a vI cosine of omega t.", + "vI is a real number for t greater than zero.", + "And I am telling you that the capacitor voltage starts out", + "being zero. And my vI is a sinusoid.", + "It's not a step this time. It's a sinusoid.", + "So vI is a sinusoid and I want to find out what vC looks like.", + "The behavior here tells me, I will give you the answer,", + "that when I feed a sinusoidal input as the frequency", + "increases, vC should be decreasing somehow and also be", + "shifting in phase. Let's do the derivation for", + "that and see what happens. To give you some insight as to", + "how to go about analyzing this let me draw a little graph as to", + "the effort level of doing this. To determine vC of t on the", + "y-axis here is our effort. How hard do we have to work to", + "solve this circuit for a sinusoidal input?", + "And on this graph, down here is easy and up here", + "is pure agony. Sheer agony up here.", + "So it's the scale of effort level ranging from easy to", + "complete agony. And this is the time axis.", + "And the time axis starts out at 11 o'clock, the early part of", + "today's lecture, and ends at roughly 12,", + "that is today's lecture and this is next lecture.", + "What I am going to show you today is a method that uses a", + "usual differential equation approach, and this is going to", + "be pure agony. If you thought last Thursday", + "was agony watch today. This is going to be sheer,", + "sheer, sheer hell. So I am going to grunge through", + "that, and I think I will sort of give up halfway because it's", + "just too painful even for me here.", + "Then what I am going to do is at the end of this lecture I am", + "going to show you an approach that I give a cutesy name.", + "I call it the \"sneaky approach\".", + "We are going to sneak something in there it is going to be a lot", + "easier. And then in the next lecture I", + "am going to show you an even sneakier approach that is just", + "going to be absolute bliss. So let's start here.", + "Indulge me as I go through the agony part.", + "I am going to blast through it because that is not of how we", + "are going to do things, but you just need to know that", + "that is agony. Let's do a usual differential", + "equation approach. Steps one, two,", + "three and four. Set up differential equation,", + "find the particular solution, find the homogenous solution,", + "add up the two and solve for the unknowns.", + "It's a mantra. The four-step manta.", + "Let's do it. Step one, write the DE.", + "That's easy. We have done this before the RC", + "circuit. It's RC dvc/dt+vc=vI.", + "This is no different from what you got from what you got from", + "your RC circuit with a step input, just that my input is VI", + "cosine of omega t in this case. It is not just a DC voltage VI.", + "Stare at that. Enjoy it while the going is", + "easy. It's like traversing rapids,", + "and before you come to a class five, you have calm and raging", + "waters there, you kind of sit there saying", + "oh, the scenery around here looks really good and so on.", + "All you are doing is stalling before you have dive down to", + "class five. We will get to class five", + "rapids in a few seconds here, so just enjoy this.", + "RC dvc/dt+vC=vI. You've seen this before.", + "Nothing fancy. Good old stuff.", + "VI cosine of omega t. You want to hold onto your", + "seatbelts? OK.", + "Let's find the particular solution to the cosine input.", + "Let's use our standard method. What I will do is just so,", + "there is going to be so much crapola up there,", + "so that I draw your attention to vP, which is what we are", + "trying to get, I am just going to put a box", + "around vP in red. If you see like all sorts of", + "garbage appear, just look for the red square.", + "That is what we are trying to get at.", + "That's the equation. Let's try.", + "First try, A worked before. A constant value A worked", + "before for DC inputs. Let's try that again.", + "If it worked then it may work now.", + "If I use A, a constant value, and I substitute it here,", + "I get dA/dt goes to zero, vP is A, but on the right-hand", + "side I have VI cosine omega t. So clearly A doesn't work.", + "Sorry. I struck out.", + "Well, cosine omega t here, let's try A cosine omega T as", + "my particular solution. Things are getting a little", + "harder now, a little more painful.", + "So substitute A cosine omega t here.", + "So I do get A cosine omega T for vP, but out here I get RCA", + "sine omega t times omega times minus one.", + "So I have a sine and a cosine, and I have a cosine on the", + "right-hand side. Sorry, it doesn't work.", + "Nope, doesn't work either. Well, let's try A cosine omega", + "t plus phi. We are now embarking into the", + "rapids here. You can begin feeling the", + "pressure. Just to refresh your memories", + "of sines and cosines. A is the amplitude of the", + "cosine. Omega is the frequency.", + "Phi is the phase. Remember the signal I showed", + "you earlier? If something happens to the", + "amplitude of the sine, something happens to the phase.", + "A cosine omega t plus phi. Let me plug it in here and go", + "by standard practice. Here is what I get.", + "I plug in A cosine omega t to this equation,", + "and this is what I get. I differentiate it.", + "I get omega out minus sine, sine of negative d plus phi,", + "A cosine omega t plus phi equals VI cosine omega t.", + "That might work. Now we get into the class six", + "part of the class five. All class fives have a little", + "bit of class six rapids. Remember, the rapids go up on", + "an exponential scale so it like earthquakes.", + "What I do now is expand out sine omega t plus phi,", + "blah, blah, blah, it goes on and on.", + "I could go on and on, but this is even tiring me.", + "This can be made to work, but I am not sure I want to put", + "all of us through this trig nightmare here.", + "If I am really, really nasty I could give this", + "to you as a homework assignment or something,", + "but I am not that nasty so you won't see that.", + "But if I go down this path it will get me to the answer,", + "but I would have to soon negotiate class six,", + "class seven rapids to get to where I want.", + "So let me punt on it, let me start from scratch.", + "I am at step two, let me start from scratch.", + "Instead what I would like to do is let's get sneaky here.", + "Rather than negotiating the class five rapids,", + "what we can say is ah-ha, we can take our canoes and jump", + "onto shore and run down and then get back onto the river.", + "Let's do that. That is called the sneaky", + "approach. So that all our friends who are", + "behind us think we are negotiating the rapids,", + "but what we are going to do is get sneaky and take the shore", + "path. Let's get sneaky,", + "just walk down the shore and see what is there.", + "I want to do something completely weird here.", + "I want to look at solving this equation through the shore", + "method. S stands for shore or S stands", + "for sneaky, whatever you want. What I am going to do is rather", + "than trying to solve for VI cosine omega t.", + "I am going to say let's try a different input all together.", + "And you will understand why in a second.", + "It's like I am the captain of my canoe and I tell my", + "teammates, hey, let's not negotiate the rapids,", + "let's go and explore the shore. Maybe down the shore we can", + "find a path that gets us across to the other side more easily.", + "So here is me and my colleagues carrying our canoe and getting", + "onto shore and taking a sneaky path.", + "This is not what I set out to solve.", + "I don't know where this will lead me.", + "But let's see where the shore path leads us.", + "I want to try solving this equation Vie^st.", + "S stands for shore or sneaky or whatever you want.", + "I don't know where I am going, but let's see where this leads", + "us. Let's explore.", + "Make believe you are Columbus or something.", + "I don't know. Let's use the usual techniques", + "and see how this works out. Let's try a particular", + "solution, Vpe^st. My input is Vie^st.", + "I am trying to solve the circuit for a different input.", + "And let me try solution Vpe^st and see if that works out", + "nicely. I substitute Vpe^st into my", + "equation here, so RCVpe^st blah blah blah.", + "What I get here is Vie^st, Vpe^st stays the same,", + "but here vP comes out, s comes out and e^st stays the", + "same. That is nice property of", + "exponentials. This is what I get.", + "A really cool property of exponentials is that when I", + "differentiate it I get the exponential back.", + "Unlike a cosine I got a sine, and for a sine I got a cosine.", + "Exponentials are very plain and simple, are straightforward.", + "What you see is what you get. You differentiate it.", + "You get the same thing, you get scaling vP,", + "S and so on, and some scaling here.", + "You get S scaling here, but the basic form stays the", + "same. This is quite nice.", + "I have e^st in all three places, so I can cancel those", + "out and I get this expression. And I get this.", + "Wow. So if I go down the shore I get", + "some place fast. I don't know where I am yet,", + "but whatever I did, it was easy.", + "I am just exploring this path, down the shore path.", + "I am making progress. I don't know where I have", + "gotten yet. We will see where we got to in", + "a second, but I got some place quickly, fast.", + "I was able to solve for this input Vie^st and get this", + "solution very quickly. So what happened here?", + "I assumed the solution of the form Vpe^st, substituted it", + "back, and found that if vP were equal to Vi/(1+sRC) then Vpe^st", + "is a solution. What I have done here is that", + "Vi/(1+sRC) is a particular solution to this equation for", + "the input Vie^st. I put a box around it because", + "this is important. This was easy.", + "We went down the shore, and said let's try this other", + "input. We made rapid progress on shore", + "and I got some place. I don't know where I am yet.", + "I got this. Let me pause here and let me", + "give you the final answer. I am going to show you over the", + "next five minutes that this is our answer.", + "You are staring at the answer already.", + "I am a party, I have taken a shore path and", + "we have gotten some place. We see the river there,", + "so it turns out we are exactly where we want to be,", + "just after the rapids. All I have to do now is get my", + "colleagues into the river with myself in the canoe and we are", + "done. You don't know that yet.", + "My colleagues and I are sitting on the shore looking at the", + "river. We've gotten some place.", + "There are no rapids there. We have gotten some place.", + "We don't quite know is this just after the rapids or not.", + "We don't know yet, but I got there very quickly.", + "And I will tell you right now, that is the place we wanted to", + "go to. The next five view graphs I am", + "going to blast through. There is going to be more pain", + "and agony to show you why that is the case.", + "It's me thinking I am Columbus and proving to my colleagues", + "that this is where we want to be.", + "And pulling out my sextant, and the compasses and so on", + "that cartographers and people use to prove to my colleagues", + "that this is where I want to be. This is the answer.", + "The next five view graphs will be demonstrating that this is", + "indeed the answer, or close enough to the answer", + "that we will be satisfied. Isn't this spectacular?", + "I am going to show you in about five minutes that this gives us", + "all the information we need to know to compute the sinusoidal", + "steady-state response to this differential equation.", + "Let me write that down here just so you know.", + "", + "Just so you remember. I am going to put a marker on", + "the shore that says this is where we are right now.", + "Now let me prove to you. As I just said,", + "vPS is Vi, it's this stuff here multiplied by e^st is the", + "solution to Vie^st. This guy here is a solution for", + "Vie^st and vP is simply the coefficient that multiplies", + "e^st. Similarly, if I substitute S", + "equals j omega. I told you five view graphs of", + "more hell, but I am just going to prove to you that this is it.", + "I am substituting S equals j omega.", + "This is Columbus giving a big speech at the end convincing his", + "colleagues that we are where we want to be.", + "I substitute j omega for S and this is what I get.", + "This is a solution for e to the st, and so this is a solution", + "for e to the j omega t. And let me mark this for you as", + "something to remember. Vi/(1+j omega RC).", + "In terms of that, I am substituting j omega for", + "S. And this is a complex number.", + "It is a complex amplitude. It is a complex number because", + "of j here, and it multiplies e to the j omega t.", + "Just keep this in mind. So that was easy.", + "The steps were easy. I am still proving to you that", + "this is where we want to be. Now let me draw the connection", + "back to vP. And the first fact was that", + "finding the response to Vie^(j omega t) was easy.", + "We know that. Second was the following", + "observation that Vi cosine omega t is simply the real part of", + "this number here. So Vi cosine omega t is simply", + "the real part of Vie^(j omega t) from the Euler relation.", + "So cosine omega t is simply the real part of this guy.", + "Light bulbs beginning to go off?", + "The first fast was that finding the response to Vie^(j omega t)", + "was easy. And the response was this,", + "right? Times e to the j omega t.", + "That was easy. All right.", + "And the second part is that the real part of this is Vi cosine", + "omega t was our input. Draw the connection between two", + "steps. Finding the response to Vie^(j", + "omega t) was easy. The real part of that was the", + "input we cared about. Are light bulbs going off?", + "Let me draw you a little picture here to show you what is", + "happening. Response to vI is vP.", + "It's the particular response we are looking for.", + "Remember the red square? But we threw in a sneaky input", + "vIS and we formed the response vPS to that.", + "This step was easy. This step was hard.", + "vI to vP was hard, trig nightmare,", + "remember? But vIS to vPS was easy.", + "It was a simple Vpe^st thing. We also know that the real part", + "of vIS is vI. The real part of this is simply", + "vI. If you have a real circuit,", + "if you have a real linear circuit, for a linear circuit,", + "if the real part of this gives me vI then the real part of the", + "solution should give me vP. So computing vPS was easy.", + "If I take the real part of this, I take the corresponding", + "real part of this. This is sort of an inverse", + "superposition argument. Superposition,", + "I said, take the response for A, take the response for B,", + "add them up and you get the response for A plus B.", + "Here what we are saying is that get the response to A plus B,", + "or to A plus jB and the real part of the input will produce", + "the response given by the real part of the output.", + "So this is an inverse superposition argument.", + "If it is a linear circuit, then if vI is the real part of", + "this sneaky input then I find the response to the sneaky input", + "and take its real part I should get vP.", + "Here I am, Columbus, staring down at the entrance to", + "this part of the river. I just proved to my colleagues", + "that all we have to do is take the real part of what we have.", + "We can just jump right back into the river and get back to", + "vP. And what I am going to do next", + "is just grind through the math and just show you that.", + "I will just blast through it. It is not important,", + "but you have it in your notes. I am telling you that vP is", + "simply the real part of the sneaky output.", + "And I take the real part of vP e to the j omega t.", + "And I take the real part. And just a bunch of math here.", + "I am just taking the real part and doing a bunch of complex", + "math. Remember vP was given by this", + "quantity here. And I take the real part and I", + "end up with vP is simply this quantity multiplied by cosine", + "omega t plus phi, where phi is given by is given", + "by tan inverse of omega RC, and this is the coefficient", + "multiplying the cosine. So by taking the sneaky path", + "and then taking the real part of that output answer,", + "I was able to very quickly get to where I wanted to be.", + "So from here to here it is only math.", + "Recall, that vP, the thing in the red was what", + "we set out to find out, which was the particular", + "response to VI cosine of omega t.", + "And remember that two grunge is all of this stuff.", + "I am going to blast through two or three more view graphs that", + "just give you more insight and more math, nothing particular.", + "And remember to solve the equation we have to find a", + "homogenous solution, too.", + "But recall that the homogenous solution for an RC circuit is of", + "the form Ae^-t/RC. This means that as time becomes", + "very large this part goes to zero.", + "As time becomes large in the steady state,", + "remember I care about the steady state?", + "This goes to zero. I don't care about the", + "homogenous solution. Isn't that fantastic?", + "Most the circuits we will deal with, except for purely", + "oscillatory ones, the homogenous part dies away.", + "You have something like e to the -t whatever.", + "It just dies away. It's gone.", + "So the total solution has vH going away.", + "And what I end up with is just vP.", + "My total solution in the steady state is simply vP.", + "And A is given by this that we just calculated.", + "I just have a bunch more insight that I talk about that", + "you can look through in your notes.", + "And I just want to show you a very quick summary.", + "In summary, what we have is we computed vP.", + "It was a complex coefficient. And all these steps,", + "2 grunge, 3 and 4 were a waste of time.", + "And what I showed you was that for the input VI the coefficient", + "vP was complex. And I can take the ratio and", + "represent it in this manner as well.", + "And from vP, I can then compute the", + "multiplier for the cosine as follows.", + "I divide by vP here. Remember the cosine was", + "multiplied by, in the mathematical step that I", + "did, VI divided one plus, this stuff here,", + "so I could get the magnitude and phase of the transfer", + "function of this circuit in the following manner.", + "And to wrap up very quickly, I am going to cover this again", + "the next time and show you a magnitude plot.", + "Notice here that if I plot Vp/Vi.", + "Remember this was Vp here. That's the answer.", + "The magnitude looks like this. On a log scale Vp/Vi for small", + "frequencies omega is at one, but as omega increases Vp/Vi", + "keeps decreasing. That is the output.", + "Remember Vp was the amplitude of the output?", + "That keeps decreasing. And this is the reason why.", + "As I increase the frequency, the amplitude of my output", + "cosine kept decreasing. I could also plot the phase for", + "you. And the phase,", + "in the same manner as omega increased, my phase also kept", + "shifting from zero initially to pi/2 finally.", + "Let me stop here and start with this the next time and revisit", + "this. Unfortunately,", + "I won't have time for the demo. I will show it to you next", + "time." + ] +} \ No newline at end of file diff --git a/subs/4TCnYYpZxEc.srt.sjson b/subs/4TCnYYpZxEc.srt.sjson new file mode 100644 index 0000000000..da0f8e0a5f --- /dev/null +++ b/subs/4TCnYYpZxEc.srt.sjson @@ -0,0 +1,1967 @@ +{ + "start": [ + 0, + 9067, + 16945, + 24675, + 30770, + 36145, + 41491, + 46000, + 51870, + 57846, + 61019, + 64902, + 69407, + 73679, + 75000, + 82000, + 88237, + 93197, + 99487, + 105776, + 111959, + 116649, + 119954, + 124644, + 132000, + 135431, + 139411, + 141676, + 144627, + 147372, + 150186, + 154746, + 159349, + 162126, + 166492, + 169428, + 174031, + 178238, + 183000, + 186881, + 189681, + 192800, + 195154, + 197699, + 200563, + 204254, + 206227, + 209472, + 212590, + 216000, + 225000, + 227611, + 231000, + 242000, + 250000, + 255000, + 259230, + 261883, + 265898, + 269913, + 274000, + 279336, + 284396, + 287800, + 291204, + 295160, + 300118, + 306471, + 313285, + 320100, + 325528, + 331881, + 336811, + 340692, + 344121, 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2876215, + 2880140, + 2884778, + 2890040, + 2893875, + 2897354, + 2902527, + 2906540, + 2911000, + 2915891, + 2920072, + 2925052, + 2928610, + 2931456, + 2936000 + ], + "text": [ + "So today we are going to talk about another process of lumping", + "or another process of discretization what will lead to", + "the digital abstraction. So today's lecture is titled", + "\"Go Digital\". So let me begin with a usual", + "review. And so in the first lecture we", + "started out by looking at elements and lumping them.", + "For example, we took an element and said for", + "the purpose of analyzing electrical properties let's lump", + "this element into a single lumped value called a resistor,", + "R. And this led to the lumped", + "circuit abstraction. The lumped circuit abstraction", + "says let's take these elements, connect them with wires and", + "analyze the properties of these using a sort of analysis", + "technique.", + "", + "So a set of a methods. We've looked at the KVL,", + "KCL method. Another example of a method we", + "looked at was the node method. And of this category there is", + "one method you should remember, which you can apply to every", + "single circuit and it will simply work, is the node method.", + "For linear circuits other methods also apply,", + "and these include superposition,", + "Thevenin method, and in recitation or in your", + "course notes you would have looked at the Norton method.", + "So that's what we did so far. So this is a toolkit.", + "So now you have a utility belt with a bunch of tools in it,", + "and you can draw from those tools.", + "And, just like any good carpenter, you know,", + "the carpenter has to cut a piece of wood.", + "He could use a chisel. He could use a saw.", + "He could use an electric saw. And the reason you pay", + "carpenters $80 an hour in the Boston region is because they", + "know which tool to use for what job.", + "So what we'll learn today is, so this was one process of", + "discretization. We discretized matter.", + "This gave us the discipline here that we decided to follow,", + "lumped matter discipline, that moved us from Maxwell's", + "equations into this new playground called EECS.", + "Where all elements looked like these rinky-dinky little values", + "like resistors and voltage sources and so on.", + "What we'll do today, if that wasn't simple enough,", + "let's simplify our lives even further.", + "What we're going to do is lump some more.", + "So what else can we lump? We've lumped matter,", + "so all matter is taken care of. So what can we lump to make", + "life even easier? When in doubt,", + "if things are complicated, discretize it or lump it,", + "right? So what do you think?", + "What we will do today is lump signal values.", + "", + "So we'll just deal with lumped values.", + "And this will lead to the digital abstraction.", + "", + "And the related reading is Chapter 5 of the course notes.", + "", + "So before we do this kind of lumping, let me motivate why we", + "do this. One reason is to simplify our", + "lives, but there is no need to just go around simplifying", + "things just because we can. Let's try to see if there are", + "other reasons motivating the digital abstraction.", + "So what I would like to start with is a simple example of a", + "analog processing circuit that you should now be able to", + "analyze. So I'm going to be motivating", + "digital. So let's start with an analog", + "circuit that looks like this, two resistors,", + "R1 and R2. And what I'm going to do is", + "apply a voltage source here, V1, apply another one here,", + "V2, and make this connection. And let me call this voltage V", + "nought and call this my output. This voltage with respect to", + "ground node, rather than drawing this wire here,", + "I often times draw a ground here and simply throw ground", + "wherever I want. This symbol simply refers to", + "the fact that the other terminal is taken at the ground node.", + "So here is my V nought. Now, let's go and analyze this", + "and see what it gives us. In this example,", + "V1 and V2 may be outputs of two sensors, maybe heat sensors or", + "something like that. This is a heat sensor on that", + "side of the room and this is a heat sensor on this side of the", + "room. And I pass their signals", + "through two resistors and I look at the voltage there.", + "So by now you should be able to write the answer V nought,", + "or the value V nought almost by inspection.", + "Just to show you, let me use superposition.", + "When you see multiple sources, the first thing you should", + "think about is can I use superposition to simplify my", + "life? And let me do that.", + "V nought here is the sum of two voltages, one due to V1 acting", + "alone and one due to V2 acting alone.", + "So what's the voltage here due to V1 acting alone?", + "To find out that I short this voltage, I zero out this voltage", + "and look at the effect of V1. So the effect of V1,", + "if this were shorted out, is simply V1 x R2 / R1 + R2.", + "This is now a voltage divider, right?", + "A voltage V applied across two resistors and the output taken", + "across one resistor. So that's this value.", + "Then I could do the second part.", + "To look at the effect of V2, what I will do is short this", + "voltage and look at the effect of this.", + "Now, this voltage is across this resistor divider.", + "And so I get R1 / (R1 + R2) here.", + "So you'll notice that for something like this,", + "if I had applied KVL and KCL of the node method I would have", + "gotten a bunch of equations, but here I wrote it just by", + "inspection. You should be able to look at", + "circuit patterns like this and write the answers down very", + "quickly. Let's say if I chose R1 to be", + "equal to R2 then V nought would simply be (V1 + V2) / 2.", + "So if these two values were equal, I simply get the output,", + "the average of the two voltages.", + "So this guy is an adder circuit.", + "It adds up these two voltages. But more precisely it's an", + "averaging circuit. It takes two voltages and gives", + "me the average value. Now, if you have two sensors in", + "the room, you might think of why you want to take that average", + "value to control the temperature of the room.", + "But suffice it to say that V nought is the average of the two", + "values. So let me show you a quick demo", + "of this example and then look at what the problems are with this", + "example. So let's say,", + "as one example, I applied a square wave at V1,", + "which is the top curve, the green curve,", + "and I applied a triangular wave at V2, that's the second one.", + "As you expect, the output is going to be the", + "sum of the two voltages scaled appropriately.", + "So notice that I have a square wave with a superimposed", + "triangular wave on top. And I can play around.", + "What I could do is change the amplitude of my wave form here.", + "And, as you notice, the amplitude of the output", + "component also changes accordingly.", + "So this is one simple example of an adder circuit,", + "and the two wave forms get summed up and I get the output.", + "So I'll switch to Page 3. Let me just draw a little", + "sketch for you here. Here, what I showed you was I", + "had a triangular wave coming on one of these inputs and I had a", + "square wave on the other one, and the output looks something", + "like this.", + "", + "OK? No surprise here.", + "This is a simple analog signal processing circuit which gives", + "me the average of two wave forms.", + "Now, let me do the following. Often times I may need to look", + "at this value some distance away.", + "So let's say this person here wants to look at the value.", + "So I bring this wire here. And I also bring the ground", + "connection and I look at it. I look at this value here.", + "And when I have a long wire I can get noise added onto the", + "circuit. So let's say a bunch of noise", + "gets added into the signal there.", + "And what I end up seeing here is not something that looks like", + "this but something that looks like that.", + "That's not unusual. And the problem with this is", + "now when I look at this, if I'm looking to distinguish", + "between, say, a 3.9 and a 3.8,", + "it's really hard to do that because my noise is overwhelming", + "my signal. I have a real problem,", + "a real problem here. Noise is a fact of life.", + "So what do we do? This is so fundamental.", + "Large bodies of courses in electrical engineering are", + "devoted to how do I carefully analyze signals in the presence", + "of noise? You'll take courses in speech", + "processing that look at clever techniques to recognize speech", + "in the presence of noise and so on and so forth.", + "One technique we adopt that we'll talk about here,", + "which is fundamental to EECS, is using the digital", + "abstraction. Let me show you how it can", + "really help with the noise problem.", + "So the idea is value lumping or value discretization.", + "Much like we lumped matter, we've discretized matter into", + "discrete chunks, let's discretize value into two", + "chunks. Let's simply say that now I'm", + "going to deal with two values and I can, say,", + "call them high, low.", + "I have a bunch of choices here. I may call it 5 volts and 0", + "volts. I may call it true and false.", + "What I'm doing is I'm just restricting my universe to deal", + "with just two values, zero and one.", + "This is like dealing with a number system with only two", + "digits. And these are zero and one.", + "So what I've now done is I'm saying that rather than dealing", + "with all possible continuous values, 0.1, 3.9999 recurring", + "and so on and so forth, what I'm going to do is simply", + "deal with a high and a low. Dealing with this whole", + "continuum of numbers is really complicated.", + "Let me simplify my life and just postulate that I am going", + "to be looking at high and low. Whenever I see something I'll", + "look at it and say high or low, is it black or white,", + "period. There's no choice here,", + "just two individual values. So that sounds simple,", + "and nice and so on, but what's the point?", + "What do we get by doing that? Let's take our example.", + "Let's take what might be a digital system.", + "Let's take a digital system and let's say I have a sender.", + "Much like I sent a signal value a long distance,", + "let me have a sender, and I have a ground as well and", + "here is a receiver. This symbol simply says that", + "both of them share a ground wire.", + "So the sender and a receiver. And what I'm interested in", + "doing, the sender is interested in sending a signal to the", + "receiver. And in the digital system,", + "the way I would send a digital signal is all I can use is ones", + "and zeros, OK? So let's say the sender sends", + "something like this. The sender wants to send a", + "value. This is my time axis and this", + "is 2.5 volts, this is 0 volts and this is 5", + "volts. My sender has some agreement", + "with the receiver and says I'm just going to be sending to you", + "low values and high values. And this signal here would", + "correspond to \"0\" \"1\" \"0\". It's a symbol.", + "That's why I have input zero in quotes there.", + "We'll go into this in much more detail later,", + "but for now suffice it to say that I'm sending a set of", + "signals here \"0\" \"1\" \"0\". This simplistic scheme will not", + "work in many situations but go along with this for a few", + "seconds. So I send the signal sequence", + "\"0\" \"1\" \"0\" out here. And notice that there is a high", + "and a low. And the agreement the sender", + "and the receiver have is that, look, if you see a value that's", + "higher than 2.5 volts that's a high.", + "If you see a value below 2.5 volts in the wire that's a low.", + "And I'm going to send a 0 volt and a 5 volt from here.", + "So now at the sending site let's say I don't have any noise", + "in this system. Let's say this is my Vn,", + "some noise being added. And let's say Vn is 0.", + "Then in that case I will receive exactly what is sent \"0\"", + "\"0\" 5, 2.5, 0 volts. And this is time.", + "Nothing fancy here, right?", + "My receiver receives a \"0\" \"1\" \"0\".", + "Now, the beauty of this is that now suppose I were to impose", + "noise much like I had noise out there and Vn was not 0.", + "Rather Vn was some noise voltage, let's say 0.2 volts", + "peak to peak. Let's say that simply got", + "superposed on the signal. In which case what do I get?", + "What I end up here with is a signal that looks like this.", + "So the receiver gets that signal because a noise is added", + "into my signal and that's what I get.", + "But guess what? No problem.", + "The receiver says oh, yeah, this is a 0 because the", + "values are less than 2.5, this is a 1 and this is a 0.", + "\"0\" \"1\" \"0\". So here my receiver was able to", + "receive the signal and correctly interpret it without any", + "problems. So because I used this value", + "discretization and because I had this agreement with the", + "receiver, I had better noise immunity.", + "", + "Consequently, I had what is called a noise", + "margin. Noise margin says how much", + "noise can I tolerate? And in this situation,", + "because the sender sends 5 volts and 0 volts,", + "the 5 volts can creep all the way down to 2.5,", + "I'll still be OK. Similarly, 0 could go all the", + "way up to 2.5, I'd still be OK.", + "So in this case I have a noise margin of 2.5 volts for a 1 and", + "similarly 2.5 volts for a 0, because there are 2.5 volts", + "between a 0 volt and 2.5. So notice that I have a nice", + "little noise margin here, which simply is the English", + "meaning of the term there is a margin for noise.", + "And even though I can change the signal value by up to 2.5", + "volts, the receiver will still correctly interpret the signal.", + "So I've decided to discretize values into highs and lows.", + "And because of that, if all I wanted to do in life", + "is send highs and lows I can send them very effectively.", + "There are many complications, but if all I care about is", + "sending highs and lows I can send it with a lot of tolerance", + "to noise. So many of you are saying but", + "what about this, but what about that?", + "There are lots of buts here. And let's take a look at some", + "of them. If you look up there.", + "What I ended up doing was creating a design space that", + "looked like this. This is on Page 6.", + "What I did was I said with a range of values from 0 to 5,", + "what I'm going to do is at 2.5 I drew a line and I said as a", + "sender if you wanted to send a 0 then you would send a value", + "here. And if you wanted to send a 1", + "you would send a value here. Similarly, for a receiver.", + "And if the sender sent a value all the way up in 5 volts that", + "was the best thing, but technically the sender", + "could send any value between 2.5 and 5.", + "And if there was no noise then the receiver could correctly", + "interpret a 1 if it was above this and 0 if it was below this.", + "The problem with this approach really is that if I allow the", + "sender to send any value above 2.5 all the way to 5 then there", + "really is no noise margin in this situation.", + "", + "OK? Because if I allowed the sender", + "to send any value between 2.5 and 5 then what if I have a", + "value 2.5 for a 1? Then I may end up getting very", + "little noise margin on the other side.", + "Worse yet, what if I get a value 2.5?", + "That's a much worse situation. What if the receiver receives a", + "value of 2.5? Now what?", + "What does the receiver do? The receiver cannot tell", + "whether it's a 1 or a 0. The receiver gets hopelessly", + "confused. So to deal with that,", + "I'm going to fix this, what I'm going to do is the", + "following. Switch to Page 7.", + "What I'll do here is to prevent the receiver from getting", + "confused, if the receiver saw 2.5, what I'm going to do is", + "define what is called \"no man's land\".", + "I'm going to define the region of my voltage space called the", + "forbidden region. And what I'm going to do is,", + "say, let's say I defined it as 2 volts, 3 volts and 5 volts,", + "0, 2, 3 and 5. With my forbidden region,", + "if I have a sender then I tell the sender you can send any", + "value between 3 and 5 for a 1. And you can send any value", + "between 2 and 0 for a 0. To send the symbol 0,", + "I can send any voltage between 0 and 2, and similarly for 1.", + "At the receiving side, if I see any value between 3", + "and 5, I read that as a 0, and any value between 0 and 2 I", + "read that as 2 volts. So I may label this value VH", + "and label this threshold VL, so there's a high threshold and", + "a low threshold. So this solves one problem.", + "Now the receiver can never see a value in the forbidden region.", + "Now, I can stand her and pontificate and say,", + "oops, that's a forbidden region, thou shalt not go there.", + "But what if I get some noise and a value goes in there?", + "In real systems values may enter there.", + "But what I'm saying, so this is the beauty of using", + "a discipline. Let me use my playground", + "analogy. This is my playground.", + "We got into this playground using the discrete matter of", + "discipline, the playground of EECS, but in that playground", + "some region of that playground deals with just high and low", + "values. I further restrict the", + "playground and I say I'm only going to focus on that", + "playground in which all signal values have a forbidden region.", + "All senders and receivers adhere to a forbidden region.", + "And if there is any signal in this space, in the forbidden", + "space then my behavior is undefined.", + "I don't care. You want to go there?", + "Sure. I don't know what's going to", + "happen to you. Now, we're engineers,", + "right? So we've disciplined ourselves", + "to play in this playground. It's like I tell my 9-year-old,", + "don't go there, right?", + "And of course he wants to go there.", + "He says what will happen if I go there?", + "And the answer here will be undefined, OK?", + "Something really bad could happen to you.", + "I don't know what it is but something really bad,", + "you know, a lightening bolt or who knows what,", + "but something really bad. And you as a designer of a", + "circuit can, let's say you were Intel.", + "Intel designs its chips. And let's say Intel decides to", + "play in this playground and there is a forbidden region.", + "So Intel says oh, it's really easy for me if in", + "the forbidden region the chip simply burns up and catches", + "fire, we'll sell more chips. That's fine.", + "Whatever you want. The key here is that all I'm", + "saying is that I am going to discipline myself into playing", + "in this playground and that's where I will define my rules,", + "and you stay within the boundaries and all the rules", + "will apply. It's called a \"discipline.\"", + "You're disciplining yourselves to stay within it.", + "There's no logic to it. It's just a discipline.", + "Just do it and you'll be OK. When we look at practical", + "circuits and so on, we have to address the issue of", + "what happens when things go in there.", + "But let's postpone that discussion.", + "For now I've solved one of my problems, which is,", + "the previous problem was what does a receiver do if it saw a", + "2.5? Now it can't see a 2.5.", + "But then the receiver asks, Agarwal, but what if I see a", + "2.5? I can tell the receiver you can", + "do whatever you want to do. You can stomp it.", + "You can squish it. You can burn it.", + "You can chuck it. Whatever you want.", + "It's up to you. Do whatever you want.", + "You won't see a value. If you do, do whatever you", + "want. It's undefined.", + "That works. So you, as the receiver", + "designer can do whatever you want when you see a 2.5.", + "You can say yeah, I'll just put out a 1 if I see", + "a 2.5 or a 2.6. I'll just do something.", + "No one cares. So this is pretty good.", + "This is pretty good. We still have a problem,", + "though. Do people see the problem here?", + "This still doesn't quite work. If Intel did this,", + "instead of your laptops failing and blue-screening every hour", + "they'd be doing it every millisecond.", + "So the problem is this discipline have allowed the", + "sender to send any value between 3 and 5 as a 1.", + "And any value between 3 and 5 at the receiver is treated as a", + "1. Do you see where the problem", + "is? Yes?", + "The sender sends a 1.99 and the noise pumps it into forbidden", + "region. Exactly.", + "So the sender says it's legitimate, I'm Intel.", + "They've told me stick to 0 and 2.", + "And Intel parts will be sending to values between 0 and 2.", + "And Motorola parts, which are receivers,", + "you know they have to receive 0 and 2.", + "So Intel can send the value, 2.", + "They can because it's 1.9 out of 2.", + "It's legal. This way I can make really", + "cheap parts. But now the problem is that", + "even the smallest amount of noise will bump it into the", + "forbidden region, and so therefore this one has a", + "problem. And the problem is that this", + "one offers zero noise margin. There is no noise margin.", + "There is no margin for noise in the discipline.", + "All right, back to the drawing board, folks.", + "Switch to Page 8. Let's get rid of all this stuff", + "and go back to the drawing board.", + "", + "OK, so what do we do now? How about the following?", + "How, about as before I say, as a receiver,", + "if you see a value between 3 and 5 you treat that as a 1 and", + "a value between 0 and 2 you treat that as a 0.", + "No difference. So as a receiver same as", + "before. But now what I do is I hold the", + "sender to tougher standards. I hold the feet of the sender", + "to the fire and say you have to adhere to tougher standards.", + "So what I'm going to do is hold the sender to tougher standards,", + "maybe four walls. That is tell the sender that if", + "you want to send to 0 or a 1, for a 1 you have to send a", + "value between 4 and 5, and for a 0 a value between 0", + "and 1. Sender is now held to tougher", + "standards. This is what my chart looks", + "like. So now I do have some noise", + "margin. Can someone tell me what is the", + "noise margin here for a 1? 1 volt.", + "And the reason is that the lowest voltage a sender can send", + "is 4 volts, OK? If the 4 leaks down to 2.99", + "that's in the forbidden region, I'm in trouble.", + "2.99. This is my forbidden region", + "here. And 2.99 is in the forbidden", + "region. I'm in trouble.", + "So notice that the lowest value that the receiver can receive is", + "3 volts. So if I sent the 4 and sent", + "this over a long cable to you, the value can be beaten up by", + "noise to such an extent that you may begin receiving 3s but", + "nothing lower than a 3. So this is a noise margin,", + "1 volt. Similarly, for a 0 the noise", + "margin is also 1 volt. So let me label these.", + "There are four important thresholds here.", + "This threshold is called VOL. V output low.", + "These have special meanings. This threshold here is called", + "VOH, V output high. This threshold here is called V", + "input high and this threshold here is called V input low.", + "So VOH simply says that senders must send voltages higher than", + "VOH. Receivers must receive values", + "higher than VIH as a 1. So these four thresholds", + "together give you your threshold.", + "", + "For the sender gets 2.5, what does sender do?", + "It could do that. So, in that case,", + "you can do that. If all you want to do is have", + "one value here then what you have is an infinitesimal value", + "here for the forbidden region. That's fine.", + "It's up to you to design it that way.", + "You can. But it turns out that when you", + "design circuits, when we see some examples in", + "the next lecture it turns out to be fairly practical and easy to", + "do it this way. But, again, these are design", + "choices. If I'm Intel,", + "Intel wants all its parts to work together.", + "So parts that follow a common discipline can work together,", + "right? Because senders will send", + "values, receivers will receive these values here,", + "so it will simply work. So the noise margin for a 1", + "here is simply VOH minus VIH and the noise margin for a 0 is VIL", + "minus VOL. VIL minus VOL is the noise", + "margin for a 0. So what do we have here?", + "What we have here is a discipline that we've agreed to", + "follow where senders are held to a tough standard and receivers", + "are held to a different standard so that I allow myself some", + "margin for error. And it's up to you as a", + "designer to choose ranges for the forbidden region.", + "Now, you may say that I want to make my forbidden region as", + "small as possible. But you will see in practical", + "circuits it's very hard to achieve that.", + "Practical devices that you get, they have a natural region that", + "gets very, very hard to break apart, and that tends to", + "establish what that region looks like.", + "So to continue with an example here, I may have the following", + "voltage wave form for a sender. So I have some sender,", + "I have a sender here.", + "", + "I have VOL, VIL, VIH, VOH and some other high", + "voltage. And then, as a sender,", + "if I want to send a \"0\" \"1\" \"0\" then I send a 0.", + "I have to be within this band. And then for a 1 I have to be", + "within this band. So this is an example of,", + "say, \"0\" \"1\" \"0\" \"1\". And at the receiver --", + "", + "Let's have VOL, VIL, VIH, VOH.", + "So at the receiver, I interpret any signal below", + "VIL as a 0. So I may get some signal that", + "looks like this.", + "", + "And I'll still interpret that as a \"0\" \"1\" \"0\" \"1\".", + "So to summarize here, this discipline that forms the", + "foundations of digital systems is called \"a static discipline\".", + "", + "The static discipline says if inputs meet input thresholds --", + "So if an input to a digital system meets the input", + "thresholds then outputs will meet, or the digital system", + "should ensure that the outputs --", + "", + "Output thresholds. So this means that if I have a", + "system like this then if I give it good inputs.", + "And by giving it good inputs I mean for 1s I have signal values", + "that are greater than VIH and for 0s signal values which are", + "less than VIL. These are valid inputs.", + "So if my inputs are valid, that is below VIL for a 0 and", + "above VIH for a 1 then this digital system D will produce", + "corresponding outputs that follow output thresholds.", + "For a 1 it will produce outputs that are greater than VOH and if", + "it needs to produce a 0 it will produce outputs that are less", + "than VOL. So notice that there is this", + "tough requirement in digital systems that for the inputs,", + "I should recognize as a 1 anything higher than a VIH.", + "But if I want to produce a 1, I have to produce a tough 1", + "like a 4-volt 1. So there is a discipline that", + "all my digital systems must follow, and that discipline is", + "called a static discipline. So static discipline encodes", + "the thresholds, encodes four thresholds that", + "all digital systems must follow so that they can talk to each", + "other. So if Intel and Motorola want", + "to make parts that are compatible with,", + "say, Pentium 4 devices then they will all talk over the", + "phone or something and agree on a static discipline.", + "We will say that, all right, all my peripherals", + "will follow a static discipline with the following volted", + "thresholds. And this way parts made by", + "different manufacturers can interoperate and still provide", + "immunity to noise. Yes.", + "Question?", + "", + "Absolutely. There are many constraints on", + "how you as a designer choose the noise margin.", + "As a designer you want to make your noise margin as large as", + "possible. The larger the noise margin the", + "better you can tolerate noise which is why,", + "how many people have heard of some devices called rad hard", + "devices, radiation hard devices? Some of you have.", + "There are a bunch of devices. Different manufacturers make", + "different kinds of devices for different markets.", + "For consumer markets they use parts which may have relatively", + "poor noise margins because consumers can tolerate more", + "faults. But if you're building devices", + "for, say, the medical industry or for spaceships and so on,", + "you need to be held to a much, much tougher standard.", + "So for those devices you may end up having much,", + "much tighter bands in which you have to operate so you have a", + "tougher noise margin. So that leads us to,", + "given these sort of voltage thresholds, we now move into the", + "digital world. And in the digital world we can", + "build a bunch of digital devices.", + "The first device we will look at is called a combinational", + "gate.", + "", + "A combinational gate is a device that adheres to the", + "static discipline, Page 11, and this is a device", + "whose outputs are a function of inputs alone.", + "", + "So I can build little boxes which take some inputs,", + "produces an output where the outputs are a function of the", + "existing inputs. And this kind of a device is", + "called a combinational gate. And I can analyze such devices", + "for the kinds of things that I would like to do.", + "Before I go into the kinds of devices I'd like to build,", + "let's spend a few minutes talking about how to process", + "signals. How to process digital signals,", + "Page 10. So notice that you have two", + "values, 0 and a 1. So devices like my", + "combinational gate, for example,", + "can only deal with 0s and 1s. So I have to come up with some", + "kind of a mathematics or some kind of a set of processing that", + "can work with 0,1 values. So 0,1 map completely natural", + "to the logic true and false. So I can borrow from logic and", + "use true and false to do my processing of signals.", + "So if all I care about is processing logic values,", + "0s and 1s, trues and falses then that's all I need.", + "I can also use numbers. How do I represent a number?", + "3.9 which is 0s and 1s. It turns out that this is a", + "whole field in itself. You'll hear more about this in", + "recitation. Let me also point you to the", + "last section of the course notes, Chapter 5.6 I believe,", + "that talks about how to represent numbers.", + "The basic insight is much like you can represent arbitrary long", + "numbers with the digits 0 through 9 in the same way,", + "but concatenating digits you can represent arbitrary long", + "numbers with 0-1-1-1-0-0 and so on.", + "So you can have a whole sequence of digits and you can", + "build a binary number system. So you can read A&L Section", + "5.6, I believe. It's the last section for", + "numbers. And you will also discuss this", + "in your recitation tomorrow. Let me spend some more time", + "talking about Boolean logic, two-valued logic,", + "and how to process these systems.", + "So one way of processing it is using logic statements of the", + "following form. If X is true and Y is true then", + "Z is true, else is Z false. So this is a logic statement.", + "It says if X is true and Y is true then Z is true,", + "else Z is false. So I can process this with 0s", + "and 1s, trues and falses. And I do this all the time so I", + "have a succinct notation for this.", + "I express this as Z is X anded with Y.", + "X and Y is Z. So Z is true if X is true and Y", + "is true. A shorthand notation for this", + "is just a dot. And a circuit notation for this", + "is called an \"AND gate\". That's a little circuit.", + "I haven't told you what's inside it.", + "It's an abstract little device called an AND gate which takes", + "two inputs, produces one output Z where the output is related to", + "the inputs in the following manner.", + "That's a little device called an AND gate.", + "I could also represent logic in truth tables.", + "And truth tables simply enumerate all the values and the", + "corresponding outputs. Inputs can be 0-0-0-1-1-0 or", + "1-1. For an AND system output is 1,", + "only if both are ones, it's a 0 otherwise.", + "So that's a truth table for AND gate.", + "So from 0s and 1s we deal with logic and we create devices like", + "the AND gate to process digital signals.", + "And what we will do is look at a whole bunch of little symbols", + "like this, like the AND gate to process our input signals.", + "And these devices might look like other functions like OR", + "gates and so on. Let me show you a quick demo.", + "What I'm going to show you is a signal feeding an AND gate.", + "And one signal is going to look like this, and my signal Y is", + "going to look like this. So you expect a processed", + "output. So 1-0-1-0-1-0-1.", + "And the output is simply going to be --", + "This is my time axis going this way.", + "It is going to be an AND-ing of these two signal values like so.", + "What I'm also going to show you is I'm going to superimpose", + "noise on this wire. I'm going to superimpose noise", + "on the wire, and what I want you to observe is the output of this", + "digital gate. The output will stay exactly", + "like this, even though I impose noise.", + "The ultimate test. So stay right there.", + "Let's do this demo. Give me a couple of seconds.", + "", + "If you look at the signal up there, look at the middle wave", + "form, and I'm imposing let's have a digital system in a noisy", + "environment like a lumberyard, for example,", + "or chopping a bunch of trees in my backyard and building digital", + "systems on the side. And if I have my buddies", + "revving up chainsaws superimposing noise on my second", + "input, but look at the output. And just to show that I'm not", + "bluffing here, what I'll do is I'll pass the", + "noise through and make the noise larger.", + "And you'll notice that when the noise begins to surpass the", + "noise margins the output begins to go berserk.", + "Watch. Can you increase it gradually?", + "Notice that as I put in a lot more noise then the output", + "begins to go berserk, but as long as my input is", + "within the noise margin my output stays perfectly stable.", + "So that's the \"Intro to Digital Systems\".", + "You'll see numbers in recitation.", + "And we'll see you at lecture on Tuesday." + ] +} \ No newline at end of file diff --git a/subs/9RqFFlZgf60.srt.sjson b/subs/9RqFFlZgf60.srt.sjson new file mode 100644 index 0000000000..e0b4f402d0 --- /dev/null +++ b/subs/9RqFFlZgf60.srt.sjson @@ -0,0 +1,1607 @@ +{ + "start": [ + 0, + 7777, + 15555, + 23333, + 31111, + 38600, + 45800, + 53000, + 66000, + 74000, + 90000, + 95454, + 100908, + 106362, + 111816, + 117272, + 122000, + 126000, + 130000, + 134000, + 138000, + 142000, + 146000, + 150214, + 154642, + 159070, + 163498, + 167926, + 172355, + 176785, + 181900, + 187700, + 193500, + 199300, + 205100, + 209882, + 213646, + 217410, + 221174, + 224938, + 228702, + 232468, + 236234, + 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Good morning. I hope you guys did", + "not spend all of last night celebrating the Red Sox victory,", + "but there is one more tonight. OK. Let's see. I trust the quiz went", + "OK. What I will do today is take off from where we left", + "off on Tuesday. And continue our discussion of the", + "large signal and small signal analysis of our amplifier.", + "Today the focus will be on \"Small Signal Analysis\".", + "So let me start by reviewing some of", + "the material. And, as you know, our MOSFET amplifier", + "looks like this.", + "One of the things you will notice in circuits, as I have been mentioning", + "all along in this course, is that certain kinds of patterns", + "keep repeating time and time again. And this is one such pattern. A", + "three terminal device like the MOSFET with an input and the drain", + "to source port connected to RL and VS in series in the following manner,", + "this is a very common pattern. There are several other common", + "patterns. The voltage divider is a common pattern.", + "We keep running into that again and again and again.", + "The Thevenin form, a voltage source in series with the", + "resistor is another very common form. The Norton equivalent form,", + "which is a current source in parallel with a resistor is also", + "very common. And it behooves all of us to be very familiar with the", + "analyses of these things. Voltage dividers in particular are", + "just so common that you need to be able to look at it and boom,", + "be able to write down the expression for voltage dividers.", + "I would also encourage you to go and look at current dividers.", + "When you have two resistors in parallel and you have some current", + "flowing into the resistors to find out the current in one branch versus", + "the other very quickly. The expression is very analogous to", + "the voltage divider expression. And some of these very common", + "patterns are highlighted in the summary pages in the course notes,", + "so it is good to keep track of those and be extremely familiar with those", + "patterns to the point where if you see it you should be able to jump up", + "and shout out the answer just by looking at it without having to do", + "any math. So here was an amplifier. And then we noticed that when the", + "MOSFET was in saturation it behaved like a current source.", + "And this circuit would give us amplification while the MOSFET was", + "in saturation. So we agreed to adhere to the", + "saturation discipline which simply said that I was going to use my", + "circuit in a way that the MOSFET would always remain in saturation in", + "building things like amplifiers and so on. And by doing that throughout", + "the analysis I could make the assumption that the MOSFET was in", + "saturation. I didn't have to go through --", + "Analysis became easier. I didn't have to figure out now,", + "what region is the MOSFET in? Well, because of my discipline it is", + "always going to be in saturation. But in turn what we had to do was", + "conduct a large signal analysis.", + "Again, in follow on courses you will be given circuits like this.", + "In fact, this very circuit with a very high likelihood.", + "And you will be looking at more complicated models of the MOSFET.", + "Or you will be given the MOSFET like this and,", + "let's say in that course the designers do not adhere to the", + "saturation discipline, in which case you have to first", + "figure out is my MOSFET in its triode region or in the saturation", + "region? And depending on the region it is in you have to apply different", + "equations. So it is one step more complicated than in 002.", + "In 002 we simplified our lives by following a discipline.", + "And let me tell you that following a discipline is quite OK.", + "When it simplifies our lives and we can do good things with it,", + "it is quite OK to do that. We are not wimps or anything like that.", + "It is quite OK to have a discipline and agree that we are going to play", + "in this region of the playground and build circuits in that manner.", + "By doing so, we could assume the MOSFET was in saturation all the", + "time. And analysis simply used a current source model.", + "By the same token, what becomes important is to figure", + "out what are the boundaries of valid operation of the MOSFET in", + "saturation? To do that we conducted a large signal analysis.", + "And it had two components to it. One of course was to figure out the", + "output versus input response. And what this usually does is that", + "it does a nonlinear analysis of this circuit.", + "If it is a linear circuit it is a linear analysis.", + "And figures out what the values of the various voltages and currents", + "are in the circuit as a function of the applied inputs and chosen", + "parameters. And the second step we said was to figure out valid", + "operating ranges --", + "-- for input and corresponding ranges for the other dependent", + "parameters such as VO. You could also find out the", + "corresponding operating range for the current IDS and so on.", + "So by doing this you could first analyze the circuit,", + "find out the \"bias\" parameters, find out the values of VI and VO and", + "so on. And then you could say all right,", + "provided, as long as VI stays within these bounds my assumption that this", + "is in saturation will hold and everything will be fine.", + "The reading for this is Chapter 8. And today we will take the next", + "step and revisit small signal analysis. In the demo that I showed", + "you at the end of last lecture, I showed you an input triangular", + "wave. And the input triangular wave gave", + "rise to an output. And we noticed that we did have", + "amplification, I had a small input and a much", + "bigger output. I did have amplification when the", + "MOSFET was in saturation but it was highly nonlinear.", + "The input was a triangular wave and the output was some funny,", + "it kind of looked like a sinusoid whose extremities had been whacked", + "down and kind of flattened. And its upward going peak had been", + "shrunk. So it was a kind of weird nonlinear behavior.", + "I will show that to you again later on. And so it amplified but it was", + "nonlinear. And remember our goal of two weeks ago?", + "We set out to build a linear amplifier. So today we will walk", + "down that path and talk about building a linear amplifier.", + "So to very quickly revisit the input versus output characteristic,", + "VI versus VO, this is VT and this is VS, this is what things looked like.", + "Also to quickly review the valid ranges, until some point here the", + "amplifier was in saturation, the MOSFET was in saturation and", + "somewhere here I had VO being equal to VI minus a threshold drop.", + "At that point the MOSFET went into its triode region and I no longer", + "was following the saturation discipline. So therefore this is my", + "valid region of operation. We also know that the output was", + "given by VS minus K (VI-VT) all squared RL over 2.", + "Again assuming the MOSFET is in saturation. It is very important to", + "keep stating this because this is true only when the MOSFET is in", + "saturation, when I am following the discipline. Notice that this is a", + "nonlinear relationship. So VO depends on some funny square", + "law dependence on VI. The key here is how do we go about", + "building our amplifier? Take a look at this point here.", + "At this point here let's say I have a VI input. Corresponding output is", + "VO. Focus is this point. And left to itself this was a", + "nonlinear curve. Remember the trick that we used in", + "our nonlinear Expo Dweeb example? We used the Zen Method.", + "Remember the Zen Method? We said look, this is nonlinear,", + "but if you can focus your mind on this little piece of the curve here", + "this looks more or less linear. If I look at a small itty-bitty", + "portion of the curve and I do the Zen thing, and kind of zoom in on", + "here. This looked more or less linear. This means that if I could", + "work with very small signals and apply the signal in a way that I", + "also had a DC offset of some sort. Then I would be in a region of the", + "curve, I would be delineating a small region of the curve which", + "would be more or less linear. This was a small signal trick.", + "And what we will do here is simply revisit the small signal model.", + "Most of what I am going to do from here on will be more or less a", + "repeat of what you saw for the light emitting expo dweeb.", + "Just that here I have a three terminal device,", + "with a little bit more complication. The equation is different. I don't", + "have to resort to a Taylor series expansion. I will just do a", + "complete expansion of this expression and develop the small", + "signal values for you. Recall the small signal model.", + "It had the following steps. The first step will operate at some", + "bias point, VI, VO, and of course some corresponding", + "point IDS. This is Page 3. And then superimpose a small signal", + "VI on top of the big fat bias. Remember the \"boost\"?", + "So VI is the boost. Boom. And above VI,", + "I have small signal VI that I apply. And our claim is that response of", + "the amplifier to VI is approximately linear.", + "The key trick with this is that for", + "my small signal model here, this is Page 3 here, and Page 2.", + "The key trick here is that with the small signal model,", + "I operate my amplifier at some operating point,", + "VO, VI. I superimpose a small signal VI on top of small VI on top", + "of big VI. And then I claim that the response to VI is", + "approximately linear. And let me just embellish that curve", + "a little bit more.", + "Notice that in this situation this was my VI, which is my bias voltage,", + "this is VO, which is the output bias, and of course not shown on this", + "graph is the output operating current which is IDS.", + "One nice way of thinking about this is to redraw this and think that", + "your coordinate axes have kind of shifted in the following manner.", + "This is VI. This is also on your Page 3. This is VT.", + "Remember this was the operating point, VO and VI.", + "And notice that we were operating in this small regime of our", + "transfer curve here. And in effect what we are saying is", + "that I am going to apply small variations about VI and call those", + "variations delta VI or small VI. And the resulting variations are", + "going to look like delta VO. Also referred to as small V,", + "small O. So I will have small variations here.", + "And they give rise to corresponding small variations there.", + "One way to view this is as if we are working with a new coordinate", + "system. Another way to view this is that so the capital VI and capital", + "VO correspond to my VI and VO as the total voltages in my circuit,", + "but at this bias point I can think of another coordinate system here", + "with small VI and VO out there. And for small changes to VI,", + "I can figure out the corresponding small changes to VO.", + "Just that all the analysis I perform here is going to be linear.", + "And I will prove it to you in a couple of different ways in the next", + "few seconds. When I am doing small signal analysis I am operating here", + "in this regime at some bias point. You have also seen this before.", + "How do I get a bias? This is my amplifier RL and VS.", + "This is Page 4. VO. The way I get a bias is I apply", + "some DC voltage VI and superimpose on top of that my small signal small", + "VI. This is my DC bias that has boosted up the signal to", + "an interesting value. And because of that what I can get", + "is by varying VI as a small signal with a very small amplitude,", + "I am going to get a linear response here. And I can draw", + "that for you as well.", + "This is my bias point here. And if I vary my signal like so then", + "my output should look like this. This is point VI, this is point VO,", + "and this is my small signal VI and this is my small signal VO and this", + "is capital VO. So this small thing here is VI.", + "I would like to show you a little demo.", + "I will start with the same demo I showed you the last time.", + "I showed you the amplifier. In the demo I am going to apply a", + "triangular wave. And initially I start with a large", + "signal. And you will see that the output looks really corny,", + "is going to look something like this. That's large signal response.", + "And then I will begin playing with the input making it smaller,", + "and you can see how it looks yourselves.", + "There you go. So this is where I stopped the last time.", + "The last lecture I applied this input, time is going to the right,", + "and the purple curve in the background is the output.", + "It looks much more like a sinusoid with some flattening of its tips.", + "Nothing like an interesting triangular wave.", + "What I will do next is that let me make sure I have enough of a boost", + "here, enough of a DC voltage so that I am operating at some point here.", + "I believe I already have that. Notice that I can shift up the", + "triangular wave input, or I can shift it down.", + "So let me bias it here. I have chosen a VI that's about,", + "I forget how many volts per division it is, but I have chosen", + "some VI here. And I biased it such that this is", + "the input. You get a nonlinear response. It is amplified.", + "It is much bigger. What I will do next is make VI that I apply smaller", + "and smaller. I have already done the boosting. Boom,", + "that's a boost. So I have boosted up your VI already.", + "Next is I am going to shrink it, and hopefully you will see that if", + "all that I am saying is truthful here you will see a triangular", + "response. Let's go try it out.", + "Watch the yellow. I am going to shrink the yellow and make it", + "smaller and smaller. There you go. It is great when", + "nature works like you expect it to. I have never seen a triangular wave", + "looks so pretty in my life. It is awesome. Look at this.", + "Here is a tiny triangular wave. And the output is also a triangular", + "wave but it is much more linear. Yes. Question? What's that? The", + "question is that the output here is only as big as the input used to be", + "before. That's a good question. What I have done here is I am", + "showing you a laboratory experiment. And let's assume that this input is", + "the input I am getting from some sensor in the field.", + "Assume that this is my input, not what I had before. Assume that", + "this is my input to begin with and this is the amplified output.", + "What I can also do is I can also change the bias.", + "And we will see this at the end of the lecture, in the last ten minutes", + "of lecture. How do you select a bias point? By changing your bias", + "point you can change the properties of an amplifier to give you a", + "preview of upcoming attractions. Let me ask you,", + "what do you think should happen if I change the bias point?", + "I have not shown you the math yet, so intuitively what do you think", + "should happen? If I increase the bias what do you", + "think is going to happen? Yes. Good insight. Higher bias", + "will be more amplification. Let's see if our friend is correct.", + "Let me set a higher bias.", + "Not necessarily,", + "I guess. You're actually right, by the way. I am playing a trick on", + "everybody here.", + "As I change my input bias. Notice that under certain", + "conditions my output becomes smaller and gets more distorted.", + "Under other conditions what is going to happen to my output is that", + "it is becoming smaller and is going to get distorted again.", + "So there are a bunch of funny effects happening that reflect on", + "the bias point, but for an appropriate choice of", + "bias point as I increase the bias the amplification should increase.", + "And I will show you that in a few minutes. But it is a complicated", + "relationship. Yes.", + "This is finally getting fun. Here is the question. Professor", + "Agarwal, we love your song and dance, but if you really want to get a high", + "signal at the output and you want to amplify your big input signal", + "how do you do it? So the question is let's say I have", + "an input that is this big here, if it is this big, I have shown you", + "how I can get things that are this big, but what if my input was this", + "big? How do I get an output that is this big? Well,", + "I will use one of those learned by questioning methods and have you", + "tell me the answer. Someone tell me the answer.", + "How do I do that? Yes. Use another amplifier.", + "So the answer is I will use one amplifier to go from here to here.", + "And the suggestion is use another amplifier to go from here to here.", + "And, in fact, I believe that you may have a problem in your problem", + "set where you will do that. And so you have only yourselves to", + "blame. So how do you make this work?", + "What you have to do is this VI has to be much smaller than the bias", + "point VI on this one. I have to build a different", + "amplifier, choose a different set of parameters such that VI prime,", + "which is the VI for this guy, is much less than V capital I prime for", + "this guy. It's a design question. You need to design it in a way that", + "the signals of interest need to be much smaller than the bias voltage", + "of this amplifier. So you may have to use much higher", + "supply voltages. My amplifier, I believe,", + "has a 4 volt supply or 5 volt supply. You might have to use an amplifier", + "with a much bigger supply, different values of RL and so on.", + "And I know that the course notes also have some exercises and problem", + "sets that discuss that in more detail. Yes. This is even more fun.", + "The question is, good question. The question is why do you need this", + "guy here? Just use this guy, right? Why do you need this guy?", + "Big guys rule, right? Who needs the little guys?", + "Well, let me use the Socratic method again. Why don't you give me", + "the answer? You guys are smart. Why do you need little guys? Why", + "do you need the small guy here? Anybody with the answer?", + "Yeah. The big guy may not be as sensitive. I like that.", + "You know what? He is almost correct. I will show you why in a", + "second. Anything else? Any other reason? Yes.", + "Bingo. That is another good answer.", + "So let me address both the answers. The answer given was that look,", + "this amplifier is amplifying the signal by a certain amount,", + "by a factor of 7. And I have designed this such that this", + "amplifies a signal by a factor of maybe 10.", + "So in all I am getting an amplification of 70.", + "This would be a great design question for lab next year.", + "I give you a bunch of components and ask you to design an amplifier", + "given the constraints with the highest amount of amplification.", + "It turns out that when you design your amplifier,", + "in order to meet the saturation discipline and so on,", + "you have to choose values of RL and VS and stuff like that and be within", + "power constraints so the amplifier doesn't blow up and stuff.", + "And by the end of it all you are going to get a measly 7X gain out of", + "it. The same way here, to be able to deal with a very small", + "signal here and get some amplification,", + "another set of values and you get 10X. So they multiply.", + "It is much harder to build one amplifier with a much larger gain.", + "You know what? I just realized that we will be looking at this in", + "the last five or seven minutes of lecture. I am going to show you", + "what the amplification depends upon. It depends upon K. It depends upon", + "RL. It depends upon VI. Now the question is I have had all", + "this time to think about how to stitch in sensitive into this,", + "and I believe I can. It turns out that when you have large voltages", + "and so on and you have practical devices, it turns out that the more", + "current you pump through devices they tend to produce noise of", + "various kinds. So very powerful amplifiers are not", + "very good at dealing with really tiny signals because they have some", + "inherent noise capabilities. And so I guess that is sensitive.", + "It is sensitive to noise. Another question? Yes.", + "Ask me the question again.", + "I didn't follow.", + "Let me just explain it. It turns out that I will not be", + "able to pass this through the big amplifier to begin with because it", + "is just going to give me a gain of just a factor of 7.", + "However, if I have a signal that is this big to begin with then I may", + "just need this amplifier. I don't need the smaller guy.", + "If my signal was this big to begin with, if I had a strong sensor that", + "produced a strong signal to begin with, yeah, I can deal with", + "just a single stage. I don't need to two stages.", + "It is all a matter of design. And it is actually a fun design", + "exercise. Given a budget, dollars, right? You go to your", + "supply room and look at the parts that you have and you go to build", + "what you have to build with the parts that you have.", + "And so sometimes you need to build two amplifiers to get the gain or", + "build a signal amplifier. It's all a design thing.", + "All right. Moving on to Page 7. That brings us to the small signal", + "model.", + "Page 5. What I showed you up on the little", + "demo was that provided the signal input in this example VI was much", + "smaller than capital VI out there as I shrank my input,", + "I was able to get a more or less linear response at the output.", + "And so to repeat my notation at the input, the total input is a sum of", + "the operating point input plus a small signal input.", + "This is called the total variable. This is called the DC bias. It is", + "also called the operating point voltage. And this is called my", + "small signal input. It is also variously called", + "incremental input. This is more a mathematical term", + "relating to incremental analysis or perturbation analysis.", + "So VI, call it small signal, call it small perturbation, call it", + "increment, whatever you want. Similarly, at the output I have my", + "total variable at the output a sum of the output operating voltage and", + "the small signal voltage. I do not like using Os in symbols", + "because big O and small O is simply a function of how big you write them.", + "It is not super clear. And in terms of a graph,", + "let me plot the input and output for you. Let's say this is the total", + "input and that is the total output. I may have some bias VI.", + "And corresponding to that I may have some bias VO. Hold that thought for", + "a second while I give you a preview of something that we will be", + "covering in about three or four weeks. Notice that as I couple", + "amplifiers together, the output operating point voltage", + "of this amplifier in this connection becomes the input operating point", + "voltage of this amplifier, right? So when they connect this output to", + "this input, the output operating point voltage becomes coupled to the", + "input here so it becomes the input operating point voltage here.", + "Now I have a nightmare on my hands. As I adjust the bias of this guy,", + "the bias of this guy changes, too. The two are dependent. It is", + "a pain in the neck. And we being engineers find ways to", + "simplify our lives. And you will learn another trick in", + "about three or four weeks. And that trick will let you decouple", + "these two stages in a way that you can design this stage in isolation,", + "go have a cup of coffee and then come back to this stage and design", + "this stage in isolation. For those of you who want to run", + "ahead and think about how to do it, think about it. What trick can you", + "use to get them in isolation? Moving on.", + "What I would like to do next is address this from a mathematical", + "point of view. And much as I did for the light", + "emitting expo dweeb analyze this mathematically and show you that if", + "VI is much smaller than capital VI, I indeed get a linear response.", + "This time around I won't use Taylor series because it turns out that", + "this expression can be expanded fully.", + "So you don't have to buy into Taylor series and so on.", + "I am going to list everything down for you. We know,", + "to begin with, that VO for the amplifier is VS-RLK/2 (VI-VT)^2.", + "What I am going to do for this, much as I did for the LED,", + "what I'm going to do is derive for you the output as a function of the", + "input when the input VI is very small.", + "In other words, when I substitute for VI,", + "V capital I squared plus small VI. Much as I did for the expo dweeb, I", + "want to substitute for VI a big DC VI. So VI is much smaller than VI.", + "And show you for yourselves that the output response,", + "V small O is going to be linearly connected to VI.", + "Notice that, let me write another equation here.", + "This is a total variable. This simply says that if the input", + "is VI then the output is going to be VO, which means that the operating", + "point input voltage should satisfy this equation,", + "correct? In other words, the operating point output voltage V", + "capital O should equal VS-RLK/2 (VI-VT)^2.", + "This is at VI equals capital VI. This is very simple but may seem", + "confusing. All this is saying is that look, this equation gives me", + "the relationship between VI and VO. Therefore, if I apply capital VI as", + "the input, I'm given that my corresponding output is capital VO,", + "so they must satisfy this equation, right?", + "Those are bias point values and that must satisfy this equation.", + "Simple. I know that. So hold that thought. Stash it away in the back", + "of your minds. Now let me go through a bunch of", + "grubby math and substitute for VI in this expression here.", + "Let me go ahead and do that. VS-RLK/2((VI+vi)-VT)^2. When I do", + "something that is other than math I will wake you up.", + "I will just keep doing a bunch of steps that are pure math.", + "No cheating. No nothing. Watch my fingers. When I do", + "anything that is not obvious math I will wake you up.", + "Next I am going to simply move VT over and rewrite this as follows,", + "RLK/2((VI-VT)+vi)^2. Again, I haven't done anything interesting so", + "far. I have just substituted this. I am just juggling things around", + "just to pass away some time, I guess. All right.", + "Next what I am going to do is simply expand this out and write it this", + "way RLK/2, expand that out and treat this as one unit VS -", + "RLK/2((VI-VT)^2+2(VI-VT)vi+vi^2). Nothing fancy here.", + "This is like the honest board. Nothing fancy here. Standard stuff.", + "Only math. I will move to this blackboard here where I do some fun", + "EE stuff. Yes.", + "Good. At least one person isn't asleep here.", + "Thank you. So just math here. Nothing fancy. Plain old simple", + "math. I have not done any trickery. I still have all my ten fingers.", + "Now what I am going to do, now watch me. I am not using Taylor", + "series here because this expression lends itself to this analysis.", + "Notice VI squared here. I made the assumption that VI is", + "much smaller than capital VI, so what I can do is assuming that VT", + "is small enough that VI minus VT is still a big number compared to small", + "VI, what I can do is ignore this in comparison to the capital VI terms.", + "So I have a capital VI term here. I am going to ignore VI squared.", + "So, for example, if capital VI was 5 volts and small", + "VI was 100 millivolts 0. , so 0.1 squared is 0.01.", + "So it is comparing 0.01 to 5. So I am off by a factor of 500.", + "So now watch me. Now I begin playing some fun and games here.", + "I eliminate this, and because I eliminate that it now becomes", + "approximately equal. What I do in addition is let me", + "write down the output. The total variable is the sum of", + "the DC bias and some variation of the output. And let me simply", + "expand that term and write it down again. VS-RLK/2(VI-VT)^2-RLK/2.", + "I get a two here. And I get VI-VT. I won't forget", + "the VI this time. Again, from here to there nothing", + "fancy. This is the one step where I have used a trick.", + "I have said small VI is much smaller than capital VI,", + "and so I have simply expanded this out and written it here.", + "So do you see the obvious next trick here?", + "From star look at this guy.", + "I can cancel this out from star because I know that at the operating", + "point these two expressions are equal, and so therefore I can cancel", + "out the operating point voltage and this.", + "What I am left with is small VO is simply minus RLK(VI-VT) times vi.", + "Only one place where I did something funny.", + "Other than that it is purely math. So this is what I get.", + "Notice that this whole thing is a constant, minus RLK(VI-VT).", + "This whole thing is a constant. And so VO is equal to some constant", + "times VI. Let me just define some terms for you that you will use", + "again and again. For reasons that will be obvious", + "next lecture, I am going to call this term here GM.", + "I am going to call this term a", + "constant, K(VI - VT). It is a constant for a given bias", + "point voltage. So I am going to call that GM.", + "And then I am going to call this whole thing A.", + "And of course this is VI. There you go.", + "I have my linear amplifier. A is the gain times small VI.", + "And the gain has these terms in it. I just call this GM. You will see", + "why later. But notice that the gain relates to RL.", + "The size of the load resistor RL, how big it is, 1K, 10K, whatever. K,", + "this is a MOSFET parameter, and VI minus VT.", + "That is a constant for a given bias point voltage and small VI.", + "So VO equals small VI.", + "I won't give you a graphical interpretation,", + "but I encourage you to go and look at Figure 8.9 in the course notes.", + "And it gives you a graphical interpretation of that expression.", + "Move to Page 7. Another way of looking at this,", + "another way of mathematically analyzing it, here I went through a", + "full blown expansion and pretty much deriving the small signal response.", + "What I can also do is take a shortcut here.", + "So let me just give you the shortcut. You might find this handy.", + "VO=VS-KRL/2(VI-VT)^2. And my shortcut is as follows.", + "My small signal response is simply this relationship.", + "I find the slope at the point capital VI and multiply by the", + "increment. Slope times the increment gives me the incremental", + "change in VO as follows. d/dI (VS-KRL/2(VI-VT)^2) evaluated", + "at vI=VI times vi. This is math again.", + "I want to find out the change in VO for a small change in VI,", + "and I do that by taking the first derivative of this with respect to", + "VI substituting V capital I and multiplying by the small change", + "delta VI or small VI. So this is simply the slope of the", + "VO versus VI curve at VI. And so therefore taking the", + "derivative here of this. This is a constant so it vanishes.", + "But twice 2 to cancel out, so I get KRL(VI-VT) times small vi", + "evaluated at capital VI. So I get twice KRL,", + "VI evaluated at capital VI, so it is VI minus VT times small VI.", + "Same thing. Oh, and I have a minus sign here.", + "I get the same expression that I derived for you up there,", + "and this is just taking the slope and going with it.", + "And this, as I mentioned before, this is A. The last few minutes let", + "me kind of pull everything together and also hit upon something that", + "many of your questions are touched upon.", + "And that all relates to how to choose the bias point.", + "So here I have taken an analysis approach. When teaching we often", + "teach you are given something, you analyze it, but as you begin to", + "master it you can begin to design things where you can ask a lot of", + "questions and so on. And here what we have is an", + "analysis given a value of RLK, VI and so on.", + "How to choose the bias point becomes more of a design issue.", + "If you are designing an amplifier, you asked me the question, how do I", + "choose two small amplifiers versus one big amplifier,", + "that sort of stuff? It boils down to how do you choose", + "the bias point? How do you choose VI?", + "How do you choose RL and so on? What I would like to do is touch", + "upon some of these things. First of all, gain or the", + "amplification. One of the most important design", + "perimeters for an amplifier is what is the gain? Let's say you get a", + "job at Maxim Integrated Technologies, and they say we would like you to", + "build a linear power amplifier for cell phones. You can say I know how", + "to do that. And then they say the next stage needs a 100", + "millivolt input. While this thing coming from the", + "antenna is only a few tens or a few hundreds of a microvolt.", + "So you sit down and say oh, my gosh, I need an amplification of", + "so much, and you go design an amplifier. So gain tends to be a", + "key parameter. And notice that gain is", + "proportional to RL. It relates to VI minus VT,", + "so proportional to VI. It is also related to RL.", + "The second point is the gain point determines where I bias something.", + "If I choose my bias too high I get distortion, or if I choose my bias", + "too low I get distortion.", + "So depending on how I choose my bias point, as a signal goes up it may", + "begin clipping or begin distorting. And I will show you a demo the next", + "time on that particular example. So bias point will determine how", + "big of a signal you can send without getting too much distortion.", + "And the other thing is that, relates to how big of an input,", + "what is a valid input range? So let's say you have a signal.", + "And you want that signal to have both positive and negative", + "excursions of the same value. Then, depending on where you choose", + "a bias point, your input range may become smaller or larger.", + "And we will go through these in the context of and amplifier and look at", + "some design issues in the next lecture." + ] +} \ No newline at end of file diff --git a/subs/AfQxyVuLeCs.srt.sjson b/subs/AfQxyVuLeCs.srt.sjson new file mode 100644 index 0000000000..d29f672744 --- /dev/null +++ b/subs/AfQxyVuLeCs.srt.sjson @@ -0,0 +1,2003 @@ +{ + "start": [ + 0, + 4400, + 8922, + 15033, + 22366, + 28111, + 35195, + 38123, + 41620, + 44954, + 48858, + 52762, + 57397, + 62870, + 67407, + 71111, + 75277, + 80185, + 84259, + 90000, + 93539, + 97230, + 101447, + 104158, + 108150, + 111012, + 114025, + 117112, + 121255, + 126000, + 130561, + 134064, + 138300, + 141233, + 145876, + 150438, + 155000, + 157695, + 161739, + 165782, + 168553, + 171323, + 174243, + 178736, + 182106, + 186000, + 189977, + 193253, + 196763, + 201130, + 203158, + 207526, + 211192, + 215728, + 218321, + 220847, + 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2269842, + 2275442, + 2278655, + 2283980, + 2287997, + 2290959, + 2294403, + 2298743, + 2301498, + 2304667, + 2308524, + 2312175, + 2316750, + 2319333, + 2322166, + 2327333, + 2330250, + 2334666, + 2338000, + 2340833, + 2344465, + 2349199, + 2353032, + 2356978, + 2361599, + 2368476, + 2374000, + 2378205, + 2382773, + 2385601, + 2389226, + 2393794, + 2398000, + 2403881, + 2406416, + 2411690, + 2417369, + 2422743, + 2428726, + 2434000, + 2439055, + 2440875, + 2446032, + 2451492, + 2457458, + 2462817, + 2466963, + 2471000 + ], + "text": [ + "So, one question to ask ourselves is,", + "what is engineering? How do we define,", + "what is engineering? Well, the definition I like to", + "use is one put forth by Steve Senturia, one of our professors", + "who is now retired. He defined engineering to be", + "the purposeful use of science. All right, so what is 6.002", + "about? So, 6.002 is a first course in", + "engineering. And I like to view 6.002 as the", + "gainful employment of Maxwell's equations.", + "Many of you have seen Maxwell's equations before.", + "Most of you should have. And they are hard stuff.", + "6.002 is all about teaching you how to simplify our lives,", + "make things simple. So, if you can gainfully employ", + "Maxwell's equations, gainfully employ the facts of", + "nature to build very interesting systems.", + "So let me show you how the transition is made.", + "So, there's a world around us, nature, so we made some", + "observations in nature. We make measurements,", + "and we can write down large tables of measurements.", + "So, for example, we can take objects and measure", + "the voltage across them, and look at the resulting", + "current through the elements. So, we may end up getting a", + "bunch of values such as [CHALKBOARD].", + "So, we start out life with making measurements on what", + "exists. And we build a bunch of tables.", + "Now, we could directly take these tables,", + "and based on observations of these tables,", + "we could go ahead and build very interesting engineering", + "systems that help us out in day-to-day lives.", + "But that's incredibly hard. Imagine having to resort to a", + "set of tables to do any kind of useful work.", + "So what we do as engineers, we first layer a level of", + "abstraction. We look at all the data,", + "and somehow layer abstraction such that we can simplify or", + "much more succinctly put in a simple equation or a simple", + "statement what these numbers are telling us.", + "OK, so for example, our physics laws,", + "so laws of physics for example are simply abstractions,", + "the laws of abstractions. So, these sets of numbers can", + "be codified by Ohm's law, for example,", + "V is equal to RI, the voltage current,", + "relates to the resistance of the object.", + "So, V is equal to RI is a law that succinctly describes a set", + "of experiments, and replaces a large number of", + "tables with a very simple statement.", + "You could call this the law, or you could call it an", + "abstraction. OK so you see laws of physics,", + "call them abstractions of physics if you like.", + "Similarly, there are Maxwell's equations and so on and so", + "forth. So, this is what is.", + "This is what's out there. OK, and a law as an abstraction", + "describe the properties of nature, as we see it,", + "in some succinct form. Now, if you want to go and", + "build useful things, we could take these", + "abstractions, take Maxwell's equations,", + "and go and build things. But it's hard.", + "It's really, really hard.", + "And what you learn in, at MIT is this place is all", + "about simplifying things. Take complicated things,", + "build layers of abstraction, and simplify things so that we", + "can build useful systems. Even in 6.002 we start life by", + "making a huge leap from Maxwell's equations to a couple", + "of very, very simple laws. OK, I'm going to show you that", + "leap that we will make today. So, the first abstraction that", + "we layer is called the lump circuit abstraction.", + "OK, in the lump circuit abstraction, what we do is we", + "make a set of simplifications that allows us to view a set of", + "objects as discrete or lumped elements.", + "So, we may, I will define voltage sources.", + "We'll define resistors. We'll define capacitors,", + "and so on. OK, and I'm going to make the", + "jump, and show you how we make the jump in a few minutes.", + "So, on that sort of abstraction, we then layer yet", + "another abstract layer. And let me call that the", + "amplifier abstraction. OK, remember,", + "here we are absolutely down and dirty.", + "We are setting the probes, measuring objects,", + "and building huge tables. We abstracted things into", + "simple laws, and life got a little better.", + "OK, I'm going to show you can abstract things further out and", + "build discrete objects, and, you could build even more", + "interesting components called amplifiers and begin playing", + "around with amplifiers. OK, so when you are using", + "amplifiers, you don't really have to worry about the details", + "of Maxwell's equations. OK, I'll give you some very", + "simple abstract rules of behavior for an amplifier,", + "and you can go build very interesting systems without", + "really, really knowing how Maxwell's equations applies to", + "that because you will be working at this abstract layer.", + "However, since you're engineers, and you are good at", + "building such systems, it's very important for you to", + "understand how we make this leap from the laws of physics into", + "some of our very primitive engineering abstractions.", + "So, once we make the amplified abstraction in 6.002,", + "by the way, 6.002 starts here. We start from the laws of", + "physics and then proceed all the way out.", + "So, once we talk about amplifiers we will take two", + "pads. On the amplifier,", + "you will build the next abstraction called the digital", + "abstraction. OK, and with the digital", + "abstraction, we will build new elements such as inverters and", + "combinational gates, OK?", + "So, notice we are building bigger, and bigger things,", + "which have more and more complicated behavior inside", + "them, but which are very simple to describe, right?", + "So, following the digital abstraction, we will superimpose", + "the combinational logic abstraction on top of that,", + "and define functional blocks that look like this:", + "some inputs, some function,", + "some outputs. The next abstraction on top of", + "that will be the clock digital abstraction, where we will have", + "some notion of time introduced into the system.", + "There will be a clock, and this will be some function.", + "And there will be a clock that introduces time into the sort of", + "logic values that functions operate upon.", + "Following that, the next level of abstraction", + "that we build is called instruction set abstraction.", + "OK, now you begin to see things that consumers get to look at.", + "Can someone give me an example of, or name an instruction set,", + "or instruction set abstraction? Bingo.", + "So, x86 is one set of abstractions.", + "And in fact, in many universities,", + "education could well start just by saying, OK,", + "here's an abstraction. These are the x86 instructions,", + "OK? Some MIT gurus have designed", + "this awesome little microprocessor,", + "OK? So you just worry about,", + "you take this abstraction layer here, the assembly instructions,", + "and you go and build systems on top of that.", + "OK, so this is an abstraction layer called the x86 layer.", + "There are other abstraction layers.", + "In 6.004, you will learn about, I believe, the alpha or the", + "beta, OK, and various other abstractions at this point.", + "So, 6.002 kind of goes until here.", + "6.002 takes me from the world of physics all the way to the", + "world of interesting analog and digital systems.", + "OK, 004, the course on computation structures,", + "will show you how to build computers all the way from", + "simple digital objects all the way to big systems.", + "Following that, you learn about language", + "abstractions, Java, C, and other languages,", + "and that's in 6.002. And there are several other", + "courses that will cover that. Following this,", + "you learn about software system abstractions,", + "and software systems, you will learn about operating", + "systems. Any example of an operating", + "system abstraction that people know out there?", + "What's that? Linux.", + "What else? I'm just wondering how long", + "I'll have to go before I hear what I want to hear.", + "[LAUGHTER] OK, so we have a bunch of software", + "systems. So, if we have a bunch of", + "software systems, these are nothing but", + "abstractions. Linux simply implies a set of", + "system calls that the programs must adhere to.", + "Windows is another set of system calls.", + "That's it. And see how much money they", + "made out of it? OK, it's all about abstraction", + "layers, that all start from nature.", + "All right? Build abstraction upon", + "abstraction upon abstraction upon abstraction,", + "and someone out here are lots of dollars.", + "OK, so based on these abstractions,", + "we can then build useful things for human beings.", + "We can build very useful things, video games,", + "so we can send space shuttles up, and a whole bunch of other", + "systems. But it's based on these", + "abstraction layers. What's unique about education", + "at MIT? What's unique about 6.002 and", + "EECS? Is to my knowledge,", + "there are not many other places in the world where you will get", + "an education in everything going all the way from nature to how", + "to build very complicated analog and digital systems.", + "OK, we will show you layer upon layer upon layer upon layer,", + "peel away the onion until you are down to raw nature,", + "OK, through Maxwell's equations.", + "So, 6.002, 004, this is 033,", + "OK, 6.170, and so on. OK, the whole EECS is about", + "building abstraction layers, one on top of the other.", + "So that's one path. There's the analog path.", + "The analog path would take an amplifier, and build an", + "abstraction layer called the op-amp.", + "See how similar they all look? You know the amplifier,", + "the inverter of the digital world, and the operational", + "amplifier in the analog world, just different ways of looking", + "at the same devices. So, to build an analog system,", + "to build an operational amplifier, and then,", + "here we go end up building a whole bunch of different", + "interesting analog system components.", + "OK, and these components might look like oscillators.", + "They might look like filters. OK, they look like power", + "supplies, a whole bunch of very interesting abstract components,", + "which pulled together can then give you the next set of", + "systems. And these systems might be", + "toasters, or say for example other analog systems like the", + "various control systems for various power plants and so on", + "and so forth, and ultimately,", + "fun and dollars. OK, so 6.002 is about going", + "from physics all the way to this point.", + "We will build interesting analog systems,", + "and take you up to interesting digital system components,", + "from which 004 will take you all the way to building computer", + "architectures. So that, in a nutshell,", + "kind of gives you a feel for the space of EECS.", + "OK, this chart here is almost a vignette of what EECS at MIT is", + "all about. And this is the world according", + "to Agarwal, because he's teaching 002.", + "OK, so this is 6.002, and the rest of EECS is", + "somewhere out there. OK, so I'm going to do now is", + "throughout this course; I want you to think about which", + "part in this vignette we are in. So, right now,", + "I'm going to start here and take you here.", + "OK, and as you get closer and closer, things get simpler,", + "and simpler, and simpler.", + "Still, the final abstractions are pedal, brake,", + "steering wheel. I mean, that's the abstraction", + "to play a game, right, four or five very simple", + "interfaces, and that's all you need to know.", + "And everybody in the world can play stuff.", + "So remember, this stuff is complicated.", + "This stuff is very, very simple.", + "OK, and the more we build abstractions and come to this", + "side, things get simpler and simpler.", + "So, a large part of what I'll cover today is make the biggest", + "simplification. The biggest simplification we", + "will make his go from Maxwell's equation to some very,", + "very simple algebraic rules. OK, I did Maxwell's equations", + "myself. And I tell you,", + "they were very interesting stuff but complicated.", + "I can't imagine building efficient systems using", + "Maxwell's equations. So, let's take an example,", + "OK? So, let's say I have a battery.", + "Just switch to page three of your course notes.", + "And let's say I connect that to a bulb.", + "OK, and this is a wire. And, the battery supplies some", + "voltage, V, and I ask you a simple question.", + "What is the current through the bulb?", + "OK, so here is something that I can build using objects.", + "I can pick a round from stores and so on.", + "And I can collect them up in this way, and ask the question,", + "what is the current, I?", + "Now, if all you've done is learn about Maxwell's equations,", + "you can roll up your sleeves and say, ah-ha!", + "The first step is to write down all of Maxwell's equations,", + "and you can say, del cross E is minus del and go", + "on, and on, and on, OK, and write out all of", + "Maxwell's equations and say, now how do I get from there to", + "here? OK, it's very good.", + "You can do it. OK, you can do it,", + "but it's very complicated. OK, so instead,", + "what you're going to do is take the easy way.", + "So, what I want to remind you is that this course is actually", + "very easy. OK remember,", + "we're going to be building abstraction upon abstraction to", + "make your lives easier. If you think your lives are", + "getting more complicated, then you are not using", + "intuition enough. OK, just remember the big I", + "word. It's all about making things", + "simple. OK, so let me give you an", + "analogy. So, suppose you have an object.", + "OK, and I apply a force to the object.", + "It's an analogy, OK to get some insight into how", + "to do this. So, I say here's an object.", + "I apply a force, and I ask you the question.", + "What is the acceleration of the object when I apply a force,", + "F? So, how would you do it?", + "OK, and eighth, or ninth, or tenth grader can", + "do this. OK, they would ask me,", + "what's the mass of the object? OK, I ask you what is the", + "acceleration? You would turn around and ask", + "me, what is the mass of the object?", + "I tell you, the mass of the object is M.", + "And then you say, oh sure, A is F divided by M,", + "done. It's as simple as that.", + "OK, I could have gone into all kinds of differential equations", + "and so on to figure that out, but you asked me for the mass.", + "And you gave me the answer, A is F divided by M.", + "So, you ignored a bunch of things.", + "You ignored the shape of the object.", + "You ignored its color. You ignored its temperature.", + "OK, and you ignored the soft or hard or whatever.", + "OK, you ignored a whole bunch of things.", + "You were focused on one thing. OK, you're focused on its mass.", + "And, it turns out that the process really was developed", + "from a set of simplifications. That is called,", + "does anybody remember this? Point mass simplification.", + "OK, so, in physics, you've done this before.", + "OK, you've simplified your lives by viewing objects as", + "having a mass at a point, and force is acting at that", + "point. OK, M is that property of the", + "object that is of interest to you.", + "This process is called, in physics, point mass", + "discretization. OK, now using an analogy,", + "and I'm going to show you a similar simple process to do the", + "problem with the light bulb. OK, so take my light bulb", + "again,", + "", + "And I focus on the filament of the light bulb.", + "OK, all I care about is the current flowing through the", + "light bulb. OK, I don't care about whether", + "the filament is twisted, whether it's hot.", + "I don't care about its shape. I don't care about its color.", + "All I care about is the current.", + "OK, so to do that, what we can do here at a very", + "high level is since we just need the current and don't care about", + "a bunch of other properties, we will simply replace the bulb", + "with a discrete object called a resistor.", + "So the discrete object is a resistor, much like the point", + "mass simplification that we did earlier that replaced the bulb", + "filament with a object called a resistor, a discrete object", + "called a resistor. Or a lump object called", + "resister, and put a value next to it just like the mass for the", + "object, a resistance value, R.", + "OK, now what I can do is in the same manner, replace the battery", + "with an object called a battery object, and connect that here,", + "the voltage, V, applied to it.", + "V falls across the resistor, and I get my I simply from", + "Ohm's law as we divide by R. So, notice here,", + "to replace this complicated bulb, this really twisty,", + "weird old thing with this discreet thing called a", + "resistor, and its only property of interest was its resistance", + "value, R, direct analogy to what we did there.", + "So, since R represents the only property of interest,", + "we can simply ignore all the other things.", + "So, notice here, we've done things the simple", + "way. And remember,", + "in EE, in the electrical engineering, we do things the", + "simple way. OK, we could go the hard route", + "and do Maxwell's equations, and get PhD's in physics,", + "and so on. But out here,", + "we are looking to do useful, interesting systems in the", + "simplest way that we can. OK, we do things a simple way.", + "All right, so we just did this, and boom, I found out what the", + "current was. Now, I cheated a little bit.", + "I've cheated a little bit. R is a lumped abstraction for", + "the bulb. So, you look at this resistor", + "here. That is simply a placeholder.", + "It's a stand-in for this complicated thing called a bulb.", + "It's a discreet object. It's a lumped object,", + "and represents the bulb. Now, so most of 6.002 will take", + "off from here, OK, and that's it.", + "To very simple stuff, like V is equal to IR,", + "it's a simple high school algebra to take off in that", + "direction. But before we go there,", + "it's important to understand, why was it that we were able to", + "make the simplification? OK, we did something else.", + "Something's going on under the covers here.", + "On the one hand, I say let's use Maxwell's,", + "and then I jump out and say, hey, we can just use this", + "simple thing. I did something that allowed me", + "to go from here to here. And you need to understand why", + "I did that and how I did that. Understand it once,", + "and then you won't have to need that information again.", + "You just need to understand it. So, let's take a closer look at", + "the bulb filament, and look at what we really did.", + "So, here's my filament, A, and let's say that the", + "surface area here, I label that SA,", + "and the one down here SB, my voltage, V,", + "applied there, and this is what I call my", + "black box that I've replaced with a resistor.", + "Notice that, in order for this to work,", + "V and I need to be defined. So I needs to be defined,", + "and V needs to be defined. OK, if I give you a random", + "object, and I don't tell you anything else about the object,", + "it's not clear I can do that. OK, if it's a much more general", + "situation, I have to write down Maxwell's equations,", + "and this is what I would write down.", + "Write down J dot dS as a function of the coordinate here", + "integrated over the area minus, OK, I would have to start from", + "there from one of Maxwell's equations.", + "All right, notice that this becomes IA, and this becomes IB", + "in our simplification. But, if I don't tell you", + "anything else, you have to start from here.", + "You will have some varying current here by point.", + "You might have some other current coming out here because", + "I may have some charge buildup happening inside.", + "If charge is building up inside the filament;", + "then I would have to put del q by del t out here,", + "right, the current in minus the current out must equal charge", + "buildup. Whoa, where is this and where", + "is that? So this is reality.", + "This is really, really what I have to do.", + "But how did I get there? How did I get there?", + "The key answer is, as engineers,", + "when in doubt we simplify. Remember, we are engineers.", + "Our goal in life is to build interesting systems.", + "OK and some are motivated by money.", + "OK, so our goal is to build interesting systems and do good", + "to humanity. So, as long as we can build a", + "good light bulb, we are happy.", + "So what we can do is we can say, look, all I care about is", + "building interesting systems. So I can say,", + "hey, this stuff is too hard. Let's make the assumption that", + "all the systems that we will consider will have this thing be", + "zero. OK, in other words,", + "if I take a complete object, if I take an element like a", + "resistor or a capacitor, the box around the entire", + "element, OK, and I want to just deal with those systems in which", + "this thing is zero. You can come and beat me up and", + "say, but why? Why not?", + "Why am I doing this? And I am saying the world is", + "arbitrary. I'm an engineer;", + "I want to build good systems. By making this simplification,", + "I eliminate this squiggle thing, and so on.", + "I don't want to deal with it. I want to make my life simple.", + "So this is gone to zero because, why?", + "Because I have said that in the future I will only deal with", + "those elements for which this is true.", + "I'm going to discipline myself. I'm going to discipline myself", + "to only deal with those systems. OK, Maxwell is turning around", + "and, you know, mad at me and all that stuff,", + "but tough. So this, what I've said about", + "making a simplification here, and this is one of the", + "simplifications I'm making. And I give a name to the", + "simplification. And that's called the lumped", + "matter discipline. OK, so I'm saying I will only", + "deal with elements for which if I put a black box around it,", + "this is going to be true. And if this is going to be", + "true, then notice, there is no charge buildup.", + "Current in must equal current out.", + "Ah-ha! So this becomes IA.", + "This becomes IB. Yes.", + "OK, I can now deal with IA's and IB's.", + "And IB and IA are equal because this is zero.", + "Notice that there is a whole bunch of depth here in the jump", + "from here to here. As MIT graduates,", + "you really, really need to understand why it is that we", + "made that jump, and then go and use that,", + "and do cool things. All right, this allows us to", + "define I. We have a unique I associated", + "with an element for the current through the element.", + "We still have to worry about B, and I won't go through that in", + "detail. The course notes have some", + "discussion of that and so does the textbook.", + "So V, AB is defined when del phi B, the rate of change of", + "magnetic flux is zero. So, if I take the element and I", + "take any region outside the element, this must be true.", + "And you say, why should that be true?", + "That's not true in general. Absolutely.", + "It's not true in general. But I, because I choose to,", + "I going to deal with only those elements.", + "I will discipline myself. But these are only those", + "elements for which this is true, and this is true.", + "I'm going to limit my world. I'm going to create a play", + "field for myself. You want to play;", + "follow my rules. OK, and that's called the", + "lumped matter discipline. So once you say that I'm going", + "to adhere to the lump matter discipline, and this is true", + "inside your elements. This is true outside the", + "elements. You can define VA and VB,", + "and good things happen to you. OK, let me show you a few", + "examples of lumped elements. But remember,", + "a large part of what we're doing is based on these two", + "assumptions. And to just go through the", + "background on that, I would encourage you to go to", + "chapter 1 of your course notes and read through just as how", + "this came about, that comes about.", + "So, by doing that by adhering to a lumped matter discipline,", + "we can now lump objects. We could lump a bulb into a", + "resistor. OK, so to be clear,", + "a certain number of lumped objects, and now,", + "the universe is going to be comprised into lumped objects.", + "OK, so before this, when he went home,", + "we talked about eggs, and omelets,", + "and light bulbs, and switches,", + "but once you come to MIT, and after you've taken 6.002,", + "you begin talking about lumped elements, you know,", + "resistors, voltage sources, capacitors, little inky-dinky", + "objects that follow the lumped matter discipline.", + "OK, they stick to very simple rules, and the math that you", + "have to do to analyze them is incredibly simple.", + "What could be simpler than V is equal to IR?", + "So, let me give you an example of interesting lumped elements,", + "and then show you a couple of really nasty lumped elements.", + "OK.", + "", + "OK, so what you see out here, so we characterize lumped", + "elements by the VI characteristics.", + "OK, you apply voltage, measure the current.", + "OK, so what I can do is I can plot I here, and V here,", + "and see what it looks like. OK, I can characterize elements", + "by their VI relationship. And there are a bunch of", + "elements that I can create based on the VI relationship.", + "So let me show you a few examples.", + "So for the resistor, since V is directly", + "proportional to I, and R is a constant,", + "I get a straight line. That's the I axis,", + "the V axis, and this is the resistor.", + "What I actually have is a variable resistor,", + "so I'm going to change the resistance value,", + "R, and the curve will also change slope.", + "OK, I changed the value of R because it's a variable", + "resistor, and the changes slope because my R is different.", + "OK, next, let me go to a fixed resistor, and this guy here on", + "the screen to your left is a fixed resistor.", + "And you see that its IV characteristic is a line of a", + "given slope, 1 by R, and that's it.", + "I can't change it. Number three,", + "I have another lumped element called a Zener diode that you", + "will see in the fourth week of this class, and the", + "characteristics for the Zener diode look like this:", + "IV. If my voltage goes across the", + "Zener diode goes up slightly, the current shoots up.", + "But if the voltage becomes negative I don't have any", + "current flowing into it until the voltage passes on the", + "threshold, at which point my current begins to build up.", + "OK, so I can increase the voltage a little bit,", + "and it can show that the current starts building up", + "again. So that's another interesting", + "lumped element called a Zener diode.", + "Let's switch to the next one called a diode.", + "So a diode looks like this: IV.", + "As the voltage across the diode becomes positive,", + "around .6 volts, or thereabout,", + "the current begins to shoot up. But when the voltage is below", + "that threshold of .6, then my current is almost zero.", + "It's another lumped element called a diode.", + "And you will begin using these elements in your 002 lives to", + "build interesting systems. The next example is a", + "thermistor. A thermistor is a resistor", + "whose resistance varies with temperature.", + "OK, so this is a very expensive little hairdryer,", + "and what I'm going to do is blow some hot air at my", + "resistor, and you're going to see that its value is going to", + "change depending on how much I heat it.", + "So as it cools down, let me cool it down,", + "so you can see it's coming down.", + "I can zap it again. I could do this all day.", + "This is so much fun. OK, so that's another", + "interesting lumped element. As the temperature rises,", + "its resistance changes. The next thing is called a", + "photo resistor. It's a resistor.", + "It used to be a resistor; Lorenzo?", + "Oh OK, that's fine. So this is a photo resistor.", + "And notice that it almost behaves like an open circuit.", + "But what I'm going to do is shine some light on it.", + "When I shine light on it, it begins to conduct and", + "becomes a resistor of some value.", + "There you go. OK, so that's a photo resistor.", + "So now I'm going to show you a battery.", + "Notice we did talk about batteries before.", + "I'll show you a battery. So before you show a battery,", + "just thinking your own minds, what should the IV", + "characteristic of a battery look like?", + "IV. A battery supplies a constant", + "voltage. You know your little cell,", + "the AA battery, 1.5 volts?", + "So, think of what the IV characteristic of a battery", + "should look like for three seconds before it shows you.", + "This is the one I showed, Lorenzo?.", + "It's a straight line. This is a good battery.", + "It's a straight, vertical line,", + "but says that the voltage is 1.5 volts, or thereabouts.", + "No matter what current it supplies as an ideal voltage", + "source, it has a fixed voltage, V, and no matter what the", + "current going through is. Now, I'll show you a dud,", + "a bad battery, and this is what the bad", + "battery looks like. So, many of you have had your", + "car batteries die on you. When you go to the store,", + "they check your batteries. They use exactly this", + "principle, that dead batteries have resistance.", + "By the way, you see slopes here.", + "You're thinking of resistance. OK, they can use this property", + "to figure out that your battery is dead.", + "So that's a dead battery. And finally,", + "let me show you a bulb. We started with a bulb,", + "and so I need to end, OK, we started with a bulb,", + "so I need to end with a bulb. And what you will see is that a", + "bulb simply behaves like a resistor.", + "Its IV curve is going to look like this.", + "OK, notice this is my bulb. And guess what,", + "it behaves like a resistor. It's a very interesting kind of", + "resistor, so I won't go into details for now.", + "But notice its IV characteristic behaves like a", + "resistor. OK, so those are some pretty", + "standard lumped elements. You deal with a lot more sets", + "of lumped elements, switches, MOSFETs,", + "capacitors, inductors, a bunch of other fun stuff.", + "But before we do that, what I wanted to tell you,", + "don't go berserk on this abstraction binge.", + "Too much of anything is bad for you.", + "So what I'm going to show you is, abstractions or models are", + "only valid provided you work within a set of constraints.", + "Notice, we have already had this tacit handshake which said", + "that we follow the discipline. Even after we follow the", + "discipline, there are ranges to how well physical elements can", + "behave like ideal lumped elements.", + "OK, for example, what we will do is show you the", + "resistor. And it's going to look like a", + "resistor. And I'm going to keep", + "increasing the voltage around it.", + "OK, what's going to happen at some point?", + "I just keep doing that. If it's an ideal element,", + "if you're a theorist, you say, oh yeah,", + "the curve will keep extending until I reach infinity.", + "But this is a practical resistor, so people out here can", + "cover your eyes or something. OK, so you're abstraction can't", + "predict that. All it says is the current is", + "an amp. It can't predict the heat,", + "light, or the smell. In the laboratory,", + "even, you get the smell. You know what somebody has just", + "done. So that's one example of the", + "lumped abstraction breaking down.", + "So, if I really believe that my own BS, anything is a lumped", + "element. So here's a pickle.", + "A pickle is a lumped element. I can choose it as a lumped", + "resistor. But this is a very interesting", + "lumped resistor. Don't try this at home.", + "This is a standard pickle into which you are pumping 110 V AC.", + "I promise you, this is a standard pickle.", + "So, it has a fixed resistance, but your lumped abstraction", + "cannot predict the nice light and sound effect.", + "OK, so the last two or three minutes what I want to do,", + "so remember, don't get carried away by", + "abstractions. There are limits.", + "OK, you can't predict everything.", + "OK, that's the smell of a pickle.", + "OK, so let me give you a preview of some upcoming", + "attractions, and show you one more quick simplification in the", + "last few minutes. So what we can do,", + "once we build these lumped elements, we can connect them in", + "circuits. OK, so I can build a circuit,", + "of the sort. So here's a voltage source with", + "a bunch of resistors. I can connect them with wires", + "and build a circuit of the sort. One interesting question we can", + "ask ourselves is, under the lumped matter", + "discipline, what can we say about the voltages?", + "OK, if I go around the loop, provided my world adheres to", + "the lumped matter discipline, what can I say about the", + "voltages around this loop? Ah-ha, Maxwell again,", + "right? So, I can write Maxwell's", + "appropriate equation to solve that.", + "OK, voltages have something to do with E and your integral of E", + "dot dl and all of that stuff, right?", + "So this is the appropriate Maxwell's equations to use.", + "And I want to find out what happens here.", + "Now remember, under LMD, I made the", + "assumption. OK, my world,", + "my playground, has del phi B by del t being", + "zero. The rate of change of flux is", + "zero. So, under these circumstances,", + "I can write this. I can break up this line", + "integral into three parts across the voltage source and across", + "the two resistors and write that down.", + "OK, and then when I can do, is now that the right-hand side", + "is zero, I can simply take this. And I know that E dot dl across", + "this element is simply VCA. This is VAB,", + "and this is VBC equals zero. OK, so when I make the", + "assumption that del phi B by del t is zero, and I go around this", + "loop, apply Maxwell's equations, what do I find?", + "I find that the sum of the voltages, VCA plus VAB plus VBC,", + "is zero. That's fantastic.", + "So now, I could say hasta la vista to this baby here.", + "And I can focus on this guy and say, Maxwell's equations,", + "this thing with squiggles and dels and all that stuff,", + "can be simplified to the sum of the voltages across a set of", + "elements in a loop in a circuit is zero.", + "OK, and this is called Kirchhoff's first first law,", + "KVL. OK, similarly,", + "in recitation section, you'll see the application of", + "Kirchhoff's current law, which comes from this be equal", + "to zero, and all the currents coming into a node being zero.", + "So, KVL and KCl directly come out of the lumped matter", + "discipline. And you can use those to solve", + "circuits like this." + ] +} \ No newline at end of file diff --git a/subs/COdQmA9g9S8.srt.sjson b/subs/COdQmA9g9S8.srt.sjson new file mode 100644 index 0000000000..a4d53ac468 --- /dev/null +++ b/subs/COdQmA9g9S8.srt.sjson @@ -0,0 +1,1961 @@ +{ + "start": [ + 0, + 5154, + 12465, + 16900, + 24332, + 31643, + 36730, + 41169, + 43953, + 48242, + 49822, + 54186, + 57723, + 61410, + 66000, + 69656, + 71895, + 76447, + 80402, + 84805, + 89358, + 93087, + 96568, + 101440, + 106234, + 109946, + 112653, + 117061, + 122205, + 125000, + 138000, + 143614, + 150000, + 157816, + 163287, + 167325, + 174490, + 179440, + 186332, + 189209, + 191619, + 195895, + 199238, + 203592, + 206469, + 210123, + 216666, + 219000, + 226000, + 229047, + 233428, + 238952, + 240000, + 251000, + 255538, + 260000, + 270000, + 273291, + 276312, + 278201, + 281600, + 285000, + 291000, + 298071, + 302123, + 305630, + 309230, + 312276, + 315600, + 321415, + 325015, + 330000, + 333771, + 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2896947, + 2901665, + 2905278, + 2911000, + 2917048, + 2922105, + 2925277, + 2931326, + 2936482, + 2941241, + 2946000 + ], + "text": [ + "Good morning. Today we move in the direction", + "that takes a big turn from the direction we have been going in", + "so far. All the devices we have had up", + "until now, resistors and voltage sources, and even your digital", + "devices like the AND gate or the inverter and so on had a very", + "specific property. We didn't dwell on that", + "property, but that property was that these were not what are", + "called memory devices. In other words,", + "the outputs at any given time are a function of the inputs", + "alone. In other words,", + "if you took your inverter or your NAND gate for that matter", + "and you build a circuit comprising 50 NAND gates", + "connected in structures that we have talked about,", + "you apply an input and boom you get an output.", + "And your output is a function of the inputs alone,", + "right? The same thing with your", + "resistors and voltage sources. At any given point in time your", + "output VO of T was some function of the input VI of T.", + "What we are going to do today is discuss a new element which", + "will introduce a whole new class of fun stuff for all of us to", + "deal with. And that is called storage.", + "In other words, the output of a circuit is now", + "going to depend not just on the inputs but it is going to depend", + "on the background or it is going to depend on where the circuit", + "has been in the past. So past is going to matter.", + "It is a very fundamental difference.", + "And what I would like to do is start by giving you folks a", + "little bit of a surprise. I am going to do a little demo", + "taking two of your inverter circuits.", + "", + "I am going to start by taking a couple of inverters.", + "Remember, I am using this structure here as an inverter.", + "And I am going to couple this to another inverter and take an", + "output C, some VS, some load resistance RL,", + "my B terminal and my A terminal.", + "So I'm going to apply some input between ground and my A", + "terminal. And for fun I want to apply a", + "square wave at the input. A square wave between zero and", + "5 volts. And this is how my time goes.", + "Let's assume that VS is 5 volts.", + "So what I am going to do is plot for you the behavior of", + "this inverter. I am going to plot for you A,", + "which would look like this. I am going to plot for you B,", + "which would be the inverted wave form.", + "And then plot C, which would be a wave form that", + "looks like this again. Let me do a plot here.", + "So this is A.", + "", + "-- and so on. Time goes this way.", + "And let's say this is between zero and 5 volts.", + "And B should be an inverted wave form that should look like", + "this.", + "", + "If all that we believe of the world so far is true then this", + "is how things should behave, so C should look like this.", + "", + "This is what the world should look like and if everything that", + "you learned about is true and correct and all of the good", + "stuff. Let me show you a little demo", + "and see if I can try to pull the rug out from under all that you", + "have learned so far and show you some surprising stuff.", + "", + "Here are the three wave forms that I showed you up here.", + "This is my A. This is my A wave form.", + "This is the B wave form. Notice that B,", + "as you expect, is an inverted form of A.", + "And this is C. We all expect this,", + "correct? But what I am going to do is", + "let me expand the time scale on this so that I can look at these", + "transitions a little bit more carefully.", + "I am just going to expand the time scale.", + "There you go. All I have done is expanded the", + "time scale and spread that out a little bit.", + "And what you see there is quite different from what you expect.", + "A is a square wave as expected, but B is stunningly different.", + "It is a zero as expected because this is a one.", + "But here I get some really strange behavior,", + "behavior that is like nothing on earth.", + "Like nothing you have seen before.", + "And then, of course, it becomes a one eventually,", + "but there's some really, really shady stuff going on", + "here. And so far you are not prepared", + "to deal with this. We have not given you the", + "facility to deal with his issue. What is the problem with this?", + "We could say who cares? What is the problem with this?", + "Let's look at the result. I am looking at this,", + "I am focusing on this piece here.", + "And notice that instead of being a sharp rise it looks like", + "this. It is going up a little bit", + "more slowly. What kind of problem would that", + "create? The problem that it creates is", + "the following. Let me play around with this", + "graph a little bit more. What I am going to do is just", + "take this output here, the C output and line it up", + "against the A output. And so I am going to line up", + "the C wave form on top of the A wave form.", + "So you can see for yourself if something really,", + "really strange and nasty is happening, I am just going to", + "move up the C wave form and line it up.", + "", + "What is happening out there? If you look carefully,", + "what you observe is that the C wave form transitions just ever", + "so slightly later than the A wave form.", + "Look here. And I claim that it is because", + "of this. Because of this,", + "the C wave form falls just a little bit later,", + "and that little thing we see out there is a delay.", + "So nothing you have learned so far prepares you for this.", + "Suddenly, instead of the output exactly following the input,", + "my output is following the input but a little bit later.", + "And it is this fact of life that things happen a little bit", + "later, is really the reason why each of you and all of us needs", + "to buy new computers every couple of years.", + "This simple basic fact. If this fact of life didn't", + "exist, you would buy one computer and be done with it for", + "life. Intel would make gobs of money", + "one year, and so would Dell and Gateway and so on,", + "and then no more. That's it.", + "This is it. But because of this a little", + "itty-bitty difference here the entire semiconductor technology", + "is charging along trying to do something about that.", + "You buy newer and newer computers each year.", + "It turns out this little itty-bitty thing here,", + "that is called the delay, the inverter delay.", + "And it happens because of a specific element that has been", + "introduced here that we have not shown you so far.", + "And a large part of the semiconductor industry and", + "follow-on courses and design and so on focuses on how could I", + "make my delay smaller, how can I get to be faster and", + "faster and faster? This relates to how fast we can", + "clock your Pentium IV. Remember it came all the way to", + "1.3 gigahertz? What's the fasted Pentium money", + "can buy today? What is the fastest P4?", + "Oh, 3.2 have come out? I don't know.", + "Ken claims 3.2. But, yeah, there you go,", + "3.2 gigahertz. It all has to do with this", + "little itty-bitty thing. You saw it for the first time", + "here. When some of you become CTOs at", + "Intel and so on, just remember that it all began", + "on October 16th with this little rinky-dink thing here.", + "What you are going to learn now is some really cool stuff that", + "has huge implications for life. So why does that happen?", + "Why did this transition happen just a little bit later?", + "The reason is that remember when this wave form reaches VT,", + "the threshold voltage of this MOSFET, this guy is going to", + "switch, right? So because of the slower rise", + "of the voltage, the VT is going to be reached a", + "small amount of time later. So I am going to hit VT", + "slightly later. And because of that this guy is", + "going to transition just a bit later because this intermediate", + "wave form B is slower. It hits VT just a little bit", + "later than if it would have made an instantaneous transition.", + "And therefore my output falls just a little bit later and this", + "gives rise to my delay in the inverter.", + "We can call that d if you would like, some delay.", + "In your course notes, this material is covered in", + "Chapters 9 and 10. That was to kind of motivate", + "why we are going to be doing all that you we will be doing.", + "Don't anybody come within a foot of this even by mistake.", + "I mean it. It is pretty deadly stuff.", + "Today we will talk about the capacitor.", + "And in the next couple of lectures I am going to tie it", + "all together and show you how this relates to that.", + "I will show you exactly how the delay happens.", + "You can compute it based on some simple principles that you", + "will learn about in the next couple of lectures.", + "What I am going to do is first of all show you,", + "I claim that that delay happens because of the presence of a", + "capacitor somewhere in there. What I will do now is take you", + "into a closer look, take a closer look at the", + "MOSFET and show you were the capacitor is.", + "This is the MOSFET that you have seen so far,", + "drain, gate and source. This is called an n-channel", + "MOSFET. And what I am going to do is", + "dissect this and show you what is actually happening,", + "what this looks like on silicon.", + "So here is my slab of silicon. It is very thin.", + "And let's say this is, I won't go into details here.", + "You will learn a lot more about this in future device classes", + "like 301 and so on, but suffice it to say I will", + "just introduce it here to give you a sense of where the", + "capacitor is. This is p-type silicon.", + "And the way you build a MOSFET is you create a couple of tubs", + "in which you dope to be n-type. The basic silicon is dope", + "p-type. And this guy here is n-type.", + "And what you do is a thin oxide layer is placed on top of that", + "and then on top of that a thin metal layer.", + "This is a metal layer. This is a thin piece of oxide,", + "silicon dioxide. And this is my P substrate.", + "Now this is a little metal layer that is really a wire on", + "top of the silicone. This metal layer could be some", + "sort of a wire that meanders around on the surface of", + "silicone. And this is a wire that", + "connects to the gate. This is the gate of my MOSFET.", + "And this guy here is the drain. And this guy here is the", + "source. And this is my gate.", + "So there is a little piece of metal here.", + "This is this piece of metal here.", + "And there is a piece of oxide and then my silicone substrate.", + "Notice that this is my oxide. When I apply a positive voltage", + "to the gate here with respect to the substrate,", + "what happens is that I draw up negative charges.", + "I draw up electrons here into this channel region and I have", + "corresponding plus type out here so that I get a view here that", + "looks like a couple of plates. And I end up with an oxide in", + "the middle. There is no connection.", + "Two plates separated by a small distance with plus q and minus q", + "on the plates. And, because of that,", + "what ends up happening here is that this piece behaves like a", + "capacitor. So a capacitor has two plates", + "with a thin insulating material in the middle with some", + "permittivity epsilon. And so I get a little piece of", + "a capacitor here. That is the capacitor that is", + "forming. I did not set out to build that", + "capacitor, but there is a capacitor nonetheless.", + "So when I apply a positive voltage at the gate,", + "negative electrons are pulled up here which forms a channel,", + "and then a current can then flow.", + "And that is how the MOSFET turns on.", + "So n-type electrons back to n-type, and I get electron flow", + "here and that gives me my channel.", + "This is just kind of devices in four minutes or less.", + "You will do an entire course on this, if you like,", + "if you take 301. What we do is to be able to", + "capture the behavior that we just saw, the funny delayed", + "behavior, we have to augment our model.", + "We have to introduce a new element.", + "So what we do is here is a MOSFET, gate,", + "drain and source. And notice here we model this", + "by putting a little capacitor, CGS between our gate and the", + "source. So this becomes a simple model", + "for our MOSFET device which is the good old gate drain source", + "device from the past with a little capacitor CGS having some", + "value for CGS in maybe ten to the minus 14 or thereabouts", + "farads. So that is a little capacitor", + "that has come about in this device that we fabricated here.", + "It is that capacitor that is at between node B and ground", + "because it is between the gate and the source of the second", + "inverter. And it is that capacitor that", + "is playing the games that we saw out there.", + "", + "So let's look at some of the behavior of an ideal linear", + "capacitor. A capacitor,", + "as I said, has a couple of plates.", + "There are a couple of plates. Between the plates is some", + "dieletric, permittivity epsilon. Let's say the area of the", + "plates is A, and let's say the plates are separated by a", + "distance D. I get some charge here,", + "let's say q. So q and minus q on the", + "capacitor. And the capacitance C is given", + "by epsilon A divided by D. Epsilon, as I said,", + "is the productivity of the dielectric.", + "So if it is free space then it would be epsilon zero which is", + "the permittivity of free space. That is the capacitance in", + "farads. And the symbol looks like this.", + "Capacitor C. Voltage v.", + "Current i. So this, much like the", + "resistor, voltage source and so on, this now becomes a primitive", + "element in your tool chest of elements like the voltage source", + "and so onn. Capacitance with the voltage v", + "across it and a current i. And I have assigned the", + "associated variables here according to the associated", + "variable discipline. A question to ask ourselves is", + "remember we said we are all now in a playground from all of", + "nature, in this playground where the lumped matter discipline", + "holds? And also remember that we said", + "that for the lumped matter discipline to hold we have to", + "make a couple of assumptions. One of those assumptions was", + "that dq/dt, for all their elements should be zero for all", + "time. So right now what about the", + "capacitor? It has got some charge q.", + "So charge must have built up somehow.", + "Does that mean that I lied all along, that we are no longer in", + "this playground, that we have been ejected from", + "the playground because of the capacitor, or are we still in", + "the circuits playground in which the lumped matter discipline", + "holds and all good things happen and so on?", + "It seems like a contradiction, doesn't it?", + "I took you from Maxwell's playgrounds to the EECS", + "playground where I said the lumped matter discipline holds.", + "And one of the foundations of the LMD was that dq/dt should be", + "zero for all time inside the elements that we are going to", + "deal with. And right now boom,", + "it's not four weeks into the course and Agarwal introduces an", + "element and it has q in it. It turns out that the capacitor", + "also adheres to the lumped matter discipline.", + "Remember the discipline says that dq/dt is zero for all time", + "within elements. So I am going to be clever.", + "What I am going to do is I want to choose element boundaries in", + "a very cleaver way. Notice that if I have q here on", + "this plate then I get minus q on the other plate.", + "So if I take the whole element, the element as a whole,", + "if I am careful in terms of how I package my boundaries,", + "if I put both my plates inside my element boundary then I still", + "do get the net charge being zero.", + "So dq/dt is indeed zero for all time provided I make sure that", + "my element has both the plates. Therefore, if you come across", + "somebody else that gives you an element that says I have an", + "idea. Let's create a new branch of", + "electrical engineering in which we model the capacitor not as", + "one element for two plates, but let's build a capacitor by", + "combining two new elements, two garbage elements called G1", + "and G2. G1 is like the top plate.", + "G2 is the bottom plate. I put them together and I get a", + "capacitor. But notice if I just pick one", + "plate then the element G1 will not adhere to the LMD.", + "It adheres to the LMD because I choose my element boundaries in", + "a way that both plates come within it.", + "So it is very fundamental and key.", + "And you can read a lot more about it in the course notes.", + "I purposely dwelt on that simple point because I think it", + "is foundational and important. And you really need to", + "understand that the capacitor does satisfy LMD.", + "We are still in the good old playground.", + "A few simple facts here. These are in the notes.", + "And you have also seen this before, I am sure.", + "I can relate the charge to the capacitance and the voltage as q", + "is equal to Cv. And q is in coulombs,", + "this is in farads and this is in volts.", + "So there is some charge q stored on the capacitor and it", + "is in coulombs and q is equal to Cv.", + "So I can differentiate this with respect to time to get the", + "current, and that becomes i=dq/dt.", + "So the current at any given time is dq/dt.", + "And so I substitute for q in terms of Cv here.", + "That is what I get. So the current i=d(Cv)/dt.", + "A 6.002 assumption, capacitance in general can be", + "time-varying. I can get time-varying", + "capacitors. In fact, there are some sensors", + "which are capacitive. And, as I talk,", + "my sound waves can change the pressure on the top plate of the", + "capacitor. And move the top plate of the", + "capacitor, thereby changing the capacitance by moving the plate.", + "Remember d here, as the plate moves closer I get", + "a higher capacitance. So we won't be dealing,", + "unless explicitly said so, with time-varying capacitances.", + "So what we can do is 6.002 allows us to write Cdv/dt.", + "So my current source capacitor is Cdv/dt.", + "I can also write down the energy, capacitors store energy.", + "E=1/2Cv^2. I am sure you have seen all", + "this before in physics and so on.", + "That is the amount of energy stored in the capacitor if it is", + "holding a charge q. Let me do a little", + "demonstration for you. They don't make glasses like", + "they used to. Our friend Lorenzo has charged", + "up this capacitor. It is a huge capacitor.", + "It is a 250 volt capacitor so it is nasty.", + "He has charged it up and has kept it there.", + "And to show you that it does contain stored charges it has", + "been sitting there holding charge.", + "Maybe the first row should go backwards, just step back for a", + "second. I think you guys would be safe", + "but I just don't want to take any chances.", + "This is holding a bunch of charge.", + "It is kind of sitting there. If I short the terminals it", + "should try to say oh, I've got a path,", + "let me get my charge out. All right.", + "Let's do it. This is always a scary moment", + "for me. And I say a little prayer", + "before I do this.", + "", + "Good? OK.", + "Gee, you guys would love to see me getting fried,", + "huh? All right.", + "Let's see.", + "", + "So it did contain charge.", + "", + "So there is a reason why Lorenzo puts one hand inside his", + "pocket when he shorts it, because there is a natural", + "tendency to hold the wire with both hands, and la,", + "la, la, la, la and put it across the capacitor.", + "By doing this you are guaranteed that you will just be", + "touching it with one hand. Hopefully you folks will", + "remember for life that a capacitor can sit around and", + "hold its charge for a while. All right.", + "That is enough of fun and games.", + "Let's get on with our business of building circuits.", + "What I am going to do is, as I promised you,", + "I am going to close the loop on that example by halfway through", + "the next lecture. I'm going take you on a bit of", + "a journey involving capacitors and resistors and involving some", + "analysis, and then we will close it all up for you at about the", + "middle of next lecture. What I would like to do next is", + "here is a new element. And let's do some fun stuff", + "with elements. Well, you know about voltage", + "sources, you know about resistors, let's put them", + "together and see how they behave.", + "Let's have a capacitor here, C, vc(t) and some current i.", + "What I am going to do, in general, whenever I have", + "something new or something strange, let's say like a", + "capacitor or some other device. It is interesting to model the", + "rest of the circuit behind it if it contains only resistors and", + "voltages and linear elements as a Thevenin equivalent.", + "So let me do that. This is R and this is vi.", + "This stuff in the back is my standard pattern,", + "voltage source in series with a resistor, and I connect that", + "across my capacitor. But remember,", + "although you saw those funny wave forms and so on,", + "the capacitor is a linear device.", + "Because you can see from here that the current relates to", + "dv/dt. That is a linear operation.", + "You don't see V squareds and Vis and things like that in", + "there. It's is a linear device.", + "Let's go back to our trusty old method, the node method.", + "If you just blindly apply the node method and simply grunge", + "through a bunch of math, you should be able to get to", + "the answer, that is for some voltage v or some form of", + "voltage vi, I should be able to figure out what vc looks like.", + "So let's do that. This is the node that is of", + "interest here with the unknown node voltage vc.", + "So let me apply the node method.", + "(vc-vi)/R is the current going this way.", + "That plus the current through the capacitor should equal zero.", + "And what is the current through the capacitor?", + "The node method tells me that, get the current in terms of the", + "element values. We know that the current is", + "given by CdvC/dt.=O. Just shuffling things around a", + "little bit, I can write RC dvc/dt+vc=vi.", + "We are writing the node equation and then getting the", + "equation that characterizes this little circuit.", + "Notice here that this has units of volts.", + "And since I have time here, this also must have units of", + "time.", + "", + "Let's go about solving this little circuit and understanding", + "how it behaves. The specific example that we", + "will look at looks like this. Let's say the capacitor voltage", + "at time T=0 is V0. This is given.", + "So at time T=0, I am telling you that the", + "capacitor contains a charge. And because of that there is a", + "voltage V0 across it. That capacitor had a voltage of", + "250 volts across it and most of the devices we deal with in", + "laptops and so on today, like the Pentium IV,", + "voltages are on the order of 1.5 volts, very small voltages.", + "So that is the value in the capacitor, the voltage.", + "That is called a state. This is called the state,", + "capacitor state. It is the state of the", + "capacitor. And I also give you that", + "vi(t)=VI. So my voltage is VI.", + "And somehow, I am not telling you how,", + "but some how it arranged to have the capacitor voltage be V0", + "at time T=0. Now I want to look to the", + "solution to this for t greater than or equal to zero.", + "And in that time my voltage vi is at some capital VI,", + "some DC voltage VI. So I am going to solve the", + "differential equation RC dvc/dt+vc=vi given these two", + "values. Input is DC voltage VI and VC0", + "is V0, the initial charge in the capacitor.", + "So from now until almost to the end of the lecture,", + "it is just going to be math by solving this very simple first", + "order differential equation. And the key here will be that", + "throughout 6.002 we will be following one method to solve", + "these. There are many methods to", + "solving differential equations, and we will follow one method.", + "That method is called the method of homogenous and", + "particular solutions. In 1802, I believe,", + "you would have learned maybe this, and certainly other", + "methods. You can use any method to solve", + "it. We will just stick to one", + "method. And this is also used in the", + "course notes. In this method what we do is", + "take the solution VC by finding two other components.", + "One is called the homogenous solution.", + "And summing that up with the particular solution.", + "And that is the total solution. So total solution is the sum of", + "the homogenous and the particular solutions.", + "And the method has three steps. As I said before,", + "we will be using this method again and again with every", + "differential equation that we encounter in this course.", + "And you won't encounter a while lot.", + "The first step we find the particular solution.", + "The second step, find the homogenous solution.", + "The total solution is the sum of the two.", + "And then find ---", + "", + "There will be some unknown constants depending on the", + "equation that you have. And in the end we simply find", + "the unknown constants by applying the initial conditions", + "that we have. Boom, boom, boom.", + "Particular. Homogenous.", + "Find constants. Three things.", + "So let's go about solving this equation and apply those three", + "conditions. Again, remember,", + "what I am doing now for the next 10 minutes or 15 minutes is", + "using math that you know about to simply solve this first order", + "of differential equations. There is nothing really new", + "that I am going to talk about here.", + "One is to find the particular solution vCP,", + "which will then be added into the vCH to get me the solution.", + "So the way you find the vCP is you find any solution that", + "satisfies this equation. This is the equation.", + "You find any solution that satisfies it.", + "And find the simplest possible solution that money can buy.", + "Find it. That's the particular solution.", + "Any solution is fine. In this case,", + "a really simple one would be vCP equals VI.", + "Let's see if a constant works. One thing you will realize in", + "differential equations is that they are actually much simpler", + "than they seem. And the reason is that almost", + "every time you have to assume you know the answer,", + "and then you are checking to see what you assumed was", + "correct. Assume the answer is this like", + "you are really smart, and then check it out and say", + "oh, yeah, that must have been the answer.", + "So here we assume that I think VI is going to work so let's try", + "it out. Substituting in here.", + "RC dvc/dt is 0. vi is a constant.", + "So I get vi equals vi, so therefore this is a", + "particular solution. Done.", + "I substitute vi here. So dvi/dt=0.", + "This vanishes and vi=VI. Bingo.", + "Therefore, VI is a solution to this equation.", + "So I am done with my vCP.", + "", + "And in general what you have to do is use trial and error.", + "By trial and error try out a bunch of solutions until you get", + "lucky. In general, again,", + "in all of 6.002 for many of the excitations a simple constant", + "usually suffices. Our second step is to find the", + "homogenous solution. And we can also do that very", + "quickly. And to do that we have to find", + "a general solution to the homogenous equation.", + "The homogenous equation is the same differential equation but", + "with the drive set to zero.", + "", + "We want to follow a set pattern to solve the differential", + "equations here, and the set pattern is find", + "vCP, vCH, find constants. And to find vCH we are also", + "going to follow a set pattern to find the homogenous solution.", + "So we set the drive to zero, so vi is set to be zero.", + "And I need to find a general solution to this.", + "As I promised earlier, diff equations are really,", + "really simple because the way we are going to solve them is we", + "are going to assume we know the answer and then go check it.", + "So let's try Ae^st. Let's try and see if this can", + "solve this particular equation for some values of A and S.", + "I am telling you that the solution is going to be of this", + "form. Assume it.", + "And then simply go ahead and find me A and S,", + "and do that by substituting it back into the equation and find", + "out the corresponding As and Ss. So let's go ahead and do that.", + "I get RC. I substitute this back up so I", + "get dAe^(st)/dt+Ae^st=0. And let me plug that in and see", + "what comes. I get RCAse^st+Ae^st=0.", + "I want to discard the trivial solution of A being 0.", + "That is a trivial solution so I will discard that.", + "And what I will do is cancel out the As from here,", + "assuming A is not zero, and cancel e^st here.", + "And what is left is RCs+1=0. What this is saying is that if", + "I can find an S such that this is true then Aest is a general", + "solution to my homogenous equation.", + "This is easy enough. And so S=-1/RC.", + "If I choose my S to be -1/RC then the simple math that I have", + "gone through shows me that this must be the solution to the", + "homogenous equation. Or in other words", + "vCH=Ae^(-t/RC). All this is saying is that", + "Ae^(-t/RC) is a solution to my homogenous equation.", + "A is an unknown constant. A is some constant.", + "I don't know what that is yet. Notice RC has popped up again.", + "And the cool thing about RC is that, this is time,", + "this also has units of time. We commonly represent RC as", + "some time constant tau, as units of time.", + "Associated with that circuit is the time constant tau,", + "which is simply RC. I commonly write this as", + "Ae^(-t/tau).", + "", + "I am very the end here. I have the particular solution", + "here. I have got the homogenous", + "solution there. I need to tell you about", + "something else. The way I found the homogenous", + "solution was in four steps. I assumed a solution of the", + "form Ae^st. I created this equation here in", + "S. This is called the", + "characteristic equation for that circuit.", + "We will see this time and time again for RC and other forms of", + "circuits. Assume a solution of this form.", + "Construct the characteristic equation.", + "Find the roots of the characteristic equation.", + "In this case it is an equation in S.", + "So this is the root. And then form the solution", + "based on that root. Four steps.", + "Ae^st, characteristic equation, root and then write down the", + "general homogenous solution. Four steps there.", + "And finally I want to write down the total solution.", + "And the total solution is simply vCP+vCH.", + "And vCP was VI and vCH was Ae^(-t/tau).", + "tau was simply RC. That is my solution.", + "Now, remember the last step. The last step was form the", + "total solution and find out the remaining constants.", + "Find out the remaining constants by using my initial", + "conditions. At t=0, I know that vC=V0.", + "I know that. And so therefore I can", + "substitute t=0 to find the constant.", + "So I know that VO=VI+A. t=0, this thing becomes 1,", + "and so I get this equation from which I get A=V0-Vi.", + "In other words, my solution vC is simply", + "VI+(VO-VI) e^(-t/tau). So the last 15 minutes have", + "just been math. No electrical engineering here,", + "but electrical engineering stopped at the point where you", + "wrote this differential equation down, went through a bunch of", + "math and came up with a solution.", + "Purely mathematically. So here I simply used math to", + "get you the solution. And, as I have been promising", + "you throughout this course, in the next lecture I will give", + "you an intuitive EE method of doing it.", + "Real electrical engineers, real EECS folks don't do it", + "this way. Real EECS folks do it", + "intuitively. And I will show you how to do", + "it in four easy seconds in the next lecture.", + "But you need to understand the foundations of how this comes", + "about, and so this is the answer.", + "You can also get the current iC is simply Cdvc/dt.", + "I won't do that for you, but you can simply", + "differentiate it and get the current.", + "So I can plot for you vC, time t, vC.", + "The intuitive way of looking at this is I have VI which is the", + "final value of the voltage. When t is infinity this part", + "goes to zero so the vC is simply VI.", + "And then there is a component V0-VI which decays according to", + "this starting out at an initial value of V0.", + "Notice when t is zero vC is V0, you can see that in the", + "equation, and so it starts out at V0 and ends up at VI.", + "I start here, I end up here.", + "And this portion V0-VI decays out over time like this.", + "And this decay is governed by the RC time constant or tau.", + "I am going to show you very quickly a couple of examples of", + "wave forms, one that goes like this and one that looks like", + "this. This is when I start with some", + "value V0 and I don't apply any input, it should decay down to", + "zero, t, t, vC, vC.", + "If I apply zero for VI then this should simply decay down to", + "nothing over time. And if I apply some VI but", + "there is no state in the capacitor then that same", + "equation is going to look like this.", + "You can go and confirm for yourselves that when I apply", + "some input but the capacitor has zero state, I start at zero,", + "I finish up at VI and my wave form looks like this.", + "There you go. That's the first one.", + "The second one where I have 5 volts on the capacitor and no", + "input. Assume that at time equals zero", + "I take away an input, short the input voltage to", + "ground for example, apply zero volts.", + "You will see the decay from 5 volts to 0 volts.", + "And in the first case I start with zero volts in my capacitor,", + "I apply input of 5 volts, and notice that at t=0 the", + "capacitor rises up to that level.", + "So notice that these circuits with capacitor and resistors are", + "typified by wave forms that are exponential rises and", + "exponential decays. We will see more of that next", + "time." + ] +} \ No newline at end of file diff --git a/subs/JB2HgohNHYQ.srt.sjson b/subs/JB2HgohNHYQ.srt.sjson new file mode 100644 index 0000000000..7722412929 --- /dev/null +++ b/subs/JB2HgohNHYQ.srt.sjson @@ -0,0 +1,1583 @@ +{ + "start": [ + 0, + 5970, + 13106, + 17766, + 24757, + 30000, + 35076, + 40246, + 44953, + 48553, + 54184, + 59630, + 63784, + 70724, + 74400, + 84900, + 94000, + 97527, + 101201, + 104507, + 108916, + 112738, + 117000, + 122135, + 127081, + 131361, + 135070, + 139635, + 143440, + 149146, + 156148, + 164925, + 174340, + 182000, + 187379, + 193497, + 199194, + 203308, + 210960, + 218484, + 224556, + 230627, + 237228, + 242486, + 248545, + 256936, + 265637, + 272940, + 279000, + 285869, + 292542, + 300000, + 304334, + 311399, + 320550, + 329059, + 339260, + 343668, + 351601, + 357184, + 361004, + 367942, + 371037, + 374995, + 377513, + 380535, + 384709, + 388163, + 392625, + 396588, + 399208, + 402861, + 405957, + 409451, + 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+ 2281235, + 2283570, + 2286797, + 2290917, + 2294762, + 2297096, + 2299774, + 2303345, + 2306984, + 2310691, + 2314948, + 2319000, + 2321611, + 2324844, + 2326958, + 2330564, + 2333611, + 2337093, + 2339891, + 2343000, + 2346802, + 2349715, + 2353761, + 2357807, + 2361933, + 2364603, + 2369620, + 2373504, + 2377954, + 2382000, + 2386315, + 2389898, + 2394050, + 2398854, + 2403000 + ], + "text": [ + "All right, let's get moving. Good morning.", + "Let me take a quick poll. So, how many of you have", + "completed Lab 4. Completed Lab 4?", + "Wow, that's great. So, how many people have begun", + "Lab 4? OK, well that's good.", + "I won't ask the last question. OK so, well I hope you're", + "having fun with this lab. Lab 4 was designed to be almost", + "like a mini-project. And, it sort of ties together a", + "lot of the content of the entire course.", + "And, it's not unlike the kind of systems that people design in", + "industry, in systems that go into a variety of devices like,", + "say, for example, digital CD players and stuff", + "like that. A lot of mixed signal stuff", + "goes in. OK, so today,", + "I'm going to continue with our discussion of energy and CMOS.", + "CMOS will be a new topic that I will introduce.", + "So, the last lecture, we spent a fair bit of time", + "talking about energy, and how to compute the energy", + "of our inverter. So, let me start from where I", + "left off, and I've given you a couple of extra pages of notes", + "today just to sort of tie it to the previous lecture.", + "Right now, I'm going to start off on page three.", + "So, what we saw last time was an inverter of this sort,", + "Vs, VIN, and we said, let's study the situation where", + "this inverter was driving a load capacitor, C.", + "Where did this load capacitor come from?", + "Well, this inverter could be driving one, or two,", + "or three, or four other larger gates, OK?", + "So, this C is lumped value of the gate capacitances of all of", + "those inverters. This may also include some", + "component due to wiring capacitance and stuff like that.", + "So, for an inverter like this, we showed in the last lecture", + "that the formula for the average power was,", + "so this was a static power independent of frequency,", + "and this was called dynamic power, and it had some bearing,", + "it's related to the frequency at which you clocked your", + "circuit. So, this was related to standby", + "power, and this to dynamic. So, what I also said is that I", + "gave you a bunch of numbers so you could compute the power", + "consumption of a chip that included 10^8 gates,", + "100 million gates, and at a frequency of 1 GHz,", + "and a bunch of other numbers. C was given to be 0.1", + "femtofarads. Femto is 10^-15.", + "So, F was 10^9. VS was 5V, and for these", + "numbers, if you plonk them down in something like this,", + "for 10^8 gates on a chip, the average power would be 10^8", + "times these two. So, this would be five squared,", + "which is 25, divided by twice.", + "RL was given to be 10 kilo-ohms, so,", + "twice, 10^4. And here we had CVS^2.", + "So, C was 10^-16, 0.1 femtofarads.", + "Vs^2 was 25, and F was 10^9.", + "So, if you commence through the numbers here,", + "what you end up getting is something that looks like this,", + "10^8 times this guy here. This is 1.25mW plus this guy", + "ends up being 2.5 microwatts. So, this should come as a bit", + "of a shocker. If I take 1.25mW,", + "and multiply that out by 10^8, this says that each gate", + "suffers a standby power loss of 1.25mW.", + "So times 10^8, I get 125kW,", + "and this guy yields 250W. OK, the 250W is manageable.", + "It's still high, and just so you don't think", + "that this is unreasonable, when the Pentium 4 first came", + "out, it was consuming 170W of power.", + "OK, you should see the heat sinks on there.", + "There's actually a huge heat sink with a fan built into the", + "top of the heat sink. OK, today it's down to more", + "reasonable numbers like 100W and so on, but when it came out it", + "was in this range. So it's high but not", + "unreasonable. But this, of course,", + "is totally wacko. OK, imagine carrying a laptop", + "around, and the sucker is blowing 125kW.", + "That'll be fun. So, clearly there's something", + "wrong here. What this is saying is that", + "this gate here consumes 125kW, there are 10^8 of these on a", + "single chip. OK, so we clearly have to do", + "something about this, otherwise the semiconductor", + "industry would fail. So, anybody have any ideas?", + "What do you think you might do here?", + "What do you think you might do to this inverter to make this", + "look better, to bring it down? What can I do?", + "Anybody? Any ideas?", + "What do you think? Well, the problem is that if I", + "look at this 125kW, well, there's a VS term here", + "and an RL term here. So, I can increase RL.", + "OK, I can make RL four times or eight times as large.", + "That'll bring the power down somewhat.", + "Can anybody think of any problem with increasing RL?", + "If I make RL really, really large,", + "will I run into other problems? Yes?", + "Exactly, the slowdown of the inverter.", + "Remember, the rise time of the inverter depends on how quickly", + "I can charge this capacitor through RL.", + "So, if I make my RL really large, I will consume less", + "standby power from hundreds of kilowatts to merely tens of", + "kilowatts. But my gates will run as slow", + "as molasses. So, clearly that's not a", + "tradeoff I would like to make. So, I can reduce my voltage to", + "maybe a volt. But that just reduces it by a", + "factor of 25, VS squared.", + "So clearly, this is not going to work.", + "I have to somehow do something else, and that will be the topic", + "of today's lecture. Also, I will dwell for a moment", + "on this term. So, if you look at the spec", + "sheet for the IBM's ASIC processor that we handed out,", + "if you recall, we talked about power", + "dissipation of 0.006 microwatts per MHz per gate.", + "OK, now you see where this is coming from.", + "Per MHz, that's because it's a multiple of f,", + "the power. Second is that it's per gate,", + "so this is the power per gate. So, as I have more gates,", + "I just have that much more power dissipation.", + "It also says power supply voltage in the range of 0.7 to", + "1.3 right next to the power expression.", + "So, you can see why they tell you all of that,", + "because both voltage, and the frequency,", + "and the number of gates come into the power of equation.", + "OK, this really simple expression here,", + "it's amazing how close this is to what people use for the", + "dynamic power in chips. OK, so as the next step,", + "what I'd like to do is, this guy, what do we do about", + "that? OK, so we've taught you to", + "build gates in a particular matter, but it's a non-starter.", + "So, how do we get rid of static power?", + "How do we get rid of static power?", + "OK, to do so, let's build up a little bit of", + "intuition. OK, so the intuition goes as", + "follows. So let's say I take my", + "inverter. Let me draw the circuit both on", + "the on state and in the off state.", + "", + "So, when VIN is high, when VIN is high,", + "I get the MOSFET turning on and has a resistance,", + "RON, and Vo is the output voltage.", + "Similarly, when VIN is low, so when VIN was high,", + "Vo was low because RON is much less than RL.", + "So, this voltage was low, while here, when VIN is low,", + "the MOSFET is off, and so I have an open circuit", + "out here. And because of that open", + "circuit, the voltage here was going to be high because VS", + "would simply appear there. So let's tailor this and see if", + "we can build up some intuition as to what to do.", + "So, when VIN is low, I don't have any static power", + "being dissipated because I don't have a connection from VS to", + "ground. OK, the current,", + "i, is zero. And, VS simply appears at the", + "output. The reason this is so is have a", + "switch here. So when this is low,", + "the switch opens up and cuts the path from power to ground.", + "This is a nice situation. Here, when VIN was high,", + "there was no switch that turns off.", + "Rather, I get a connection from VS to ground.", + "OK, so think about this situation here.", + "The insight here is, just imagine if I could do the", + "following. Imagine if I could somehow", + "magically elevate RL to be a very, very, very large number,", + "if I could make this so high as to make the power really low", + "only in the situation when the input was high,", + "OK? So, imagine if I could do", + "something like this. Imagine I could open circuit", + "this guy, RON, so when VIN was high,", + "if I could, instead of having an RL here, what if somehow I", + "could make this RL become infinity?", + "OK, so in this case, output VO would be low.", + "OK, I get many benefits by doing this.", + "One benefit is that, look, I have opened this switch", + "here so I don't have any standby current.", + "OK, the standby current is zero.", + "The second benefit is that my output gets dragged down to", + "ground, OK? Out here, my output was VS", + "multiplied by RON divided by the sum of these two.", + "Out here, I have a direct connection to ground,", + "and nothing to the power supply, VS, and so therefore I", + "have a nice, solid low. So the question is that,", + "can I get this situation? OK, that is a key insight.", + "So, imagine that somehow, when this was high,", + "I could get this to open up, much like when this was low,", + "I got this to open up. OK, so think about it.", + "So, the intuition is that what I need instead of a resistor", + "here, what if I have something like the MOSFET that I have", + "here? So, I have a MOSFET here that", + "turned off when VIN was low. OK, what if I did the", + "complementary thing? What if I put in some kind of", + "MOSFET here that would turn off when VIN was high?", + "OK, so, much like the MOSFET turned off when VIN was low down", + "here, imagine if I could find a device that could turn off when", + "VIN was high? OK, this would be on,", + "but this would be off. So the behavior of this device", + "would have to be complementary to this device.", + "So, we need some sort of a switch to introduce this new,", + "little MOSFET device with slightly different properties,", + "let me quickly review for you the properties of the MOSFET", + "that we know about, so our N channel MOSFET,", + "also called the NFET, this is what we've been seeing", + "all this while, is drawn like this.", + "I have a gate; I have a drain;", + "I have a source. And this guy is on when VGS is", + "greater than or equal to VT, OK, and off when VGS is less", + "than VT. You saw this before,", + "OK, nothing new here. So, what I need is a device", + "that behaves in a complementary manner.", + "OK, so the device is a P channel MOSFET.", + "By the way, I must point out, till about 1983-84 until the", + "early '80s, that's exactly pretty much how chips were", + "designed, OK, using an NFET for the switch", + "looking down here, and a variety of different", + "kinds of devices to be used as resistors.", + "OK, that's when technology began moving towards this new", + "kind of technology I'm going to talk about, and that", + "dramatically reducing the power consumed.", + "And, the P channel MOSFET was created, and this guy's called", + "the PFET. It's a complementary device", + "that looks as follows. OK, the difference here is", + "that, to show this is complementary,", + "I'll put a little circle here. It has a gate.", + "Just to make things a little clearer, flip the drain and", + "source terminals, and this guy is on at a", + "distinguished threshold voltage of this with the NFET device,", + "let me put an N here to say that this is the VT for the N", + "channel device. And for this guy,", + "this guy came on when VGS was greater than some voltage.", + "So, VTN could be, for example,", + "one volt. So, VGS was more than one.", + "This turned on. In this case,", + "I wanted this to turn on when VGS is some value which is lower", + "than, or much lower than, the source voltage.", + "OK, so this guy turns on when the gate voltage is higher.", + "This guy should turn on when the gate voltage is", + "significantly lower than the source voltage,", + "just the complementary behavior.", + "OK, so when VGS is less than or equal to VTP.", + "And in this case, the threshold voltage for the", + "PMOS device, say, just as an example,", + "maybe -1V. So this means that if the", + "source is at, say, 5V, OK,", + "then this device would turn on if the gate, for example,", + "using that example was less than 4V.", + "So, this is five. If the gate fell below 4V,", + "this guy would turn on. In this situation,", + "remember, if this was at zero, the gate would have to be", + "greater than 1V to turn on. In this situation,", + "the gate has to be less than 4V if the source was at five to", + "turn on. And, it's off.", + "OK, so this is a complementary device that I postulate that", + "behaves in a complementary manner.", + "So, the gate voltage rises, this guy turns on,", + "and in this situation, when the gate voltage drops", + "below the source voltage, this guy turns on.", + "OK, so when there's a rising guy that turns on in this", + "particular situation when it falls, the gate turns on and", + "shows some resistance. In this case,", + "the resistance would be RON. And to show that it's N", + "channel, let me say N. And in this case,", + "the resistance, when it turns on,", + "would be RONp to represent P channel.", + "OK, so now consider the following circuit for the", + "inverter. So, instead of my resistor,", + "I put a complementary device, OK, and that's it.", + "So all I've done here is replace my resistor with a", + "MOSFET that behaves complementary to the N channel", + "MOSFET. So this is my gate,", + "my drain. This is my source,", + "my gate, my source, and my drain.", + "OK, and this guy is called a pull up, and this guy is called", + "a pull down. OK, and the reason is that this", + "guy pulls the output to ground when it's turned on,", + "while this guy, when switched on,", + "will pull this node up to VS. So, I pull it down or pull it", + "up based on when the VIN is high or low.", + "So, let's look at the two situations.", + "So, let's say, as an example,", + "my VS is 5V, and let's say VIN in one", + "situation being 5V, and another situation being", + "equal to 0V. Let's draw the equivalent", + "circuit in both these cases. So, when VIN is high,", + "I have my usual circuit. When VIN is high,", + "this MOSFET, as before, when VIN is 5V,", + "the N channel MOSFET below is turned on, and so I have an RON", + "resistance here. But remember,", + "VIN is 5, and VS is 5V, then the voltage across the", + "source and the gate of this P channel FET is now equal,", + "five and five. OK, so this one would turn off.", + "And that's the circuit that I get.", + "The output is suitably low. In this situation,", + "if VIN is zero, what happens in this situation?", + "Here's my output. If VIN is 0V,", + "the lower device turns off. This is zero.", + "This is zero. This guy turns off,", + "and that's the situation for the N channel MOSFET.", + "How about this guy here? What happens here?", + "This is at 5. So let me just,", + "this is at 5V. OK, and VIN is at 0V.", + "OK, so therefore, the GS of this is -5V.", + "If this is zero and this is five, G, source,", + "and drain, GS is -5V, and -5V is significantly less", + "than the threshold -1V in our example.", + "So, this one will switch on. And if this one switches on,", + "what I end up getting is RONp out there.", + "So, when this one kicks in, it pulls the output high and VO", + "goes high. So, all I've done is replaced", + "my resistor with a complementary device, which switches off when", + "the input is high, and switches on when the input", + "is low. And the beauty of this is that", + "at no point, assuming all the devices are ideal here,", + "at no point do I have a short circuit between the output,", + "do I have a current path from the output to the ground from", + "the supply to ground, OK, I have this turned off or", + "this turned off. So, this type of logic", + "involving a PMOS transistor here, and the N channel", + "transistor here is called CMOS logic for, OK,", + "it's called complementary MOS logic.", + "That's what CMOS comes from. OK, so I'm sure you've read in", + "a number of places that most digital chips today use CMOS", + "technology. It comes from complementary", + "MOS, and complementary comes from the use of complementary", + "transistors: N channel, P channel, turns on when high,", + "turns off when high, turns off when low,", + "turns on when low. OK, that's exactly", + "complementary to each other. OK, so what you've seen here", + "has been the workhorse of the digital industry for the past", + "two decades, 20 years, CMOS logic.", + "OK, and even the most advanced chip from Intel has an inverter", + "that looks exactly like that. OK, if you count all the", + "inverters in the universe today, I would say a significant", + "fraction of those look exactly like that, no difference,", + "just so simple. So, the key with something like", + "that is there is no path from the power supply to the ground,", + "and so by that model, I did not consume any standby", + "power. OK, my standby power in that", + "idealized model is zero. So, let's compute P.", + "So, what is P dynamic? Let's use the method that we", + "adopted in the last lecture, and draw the equivalent", + "circuit, and compute the power. OK, so I'm going to model the", + "following situation, and assume that I drive a", + "capacitive load, C.", + "OK, and as an input, as I did the last time,", + "I'm going to assume I have some input voltage,", + "VIN, that looks like this. The cycle time,", + "T, and the frequency is 1/t, and let me assume that this is", + "T1, and this is T2. OK, and I'm assuming that T1", + "and T2 are both much larger than the respective time constants.", + "OK, the time constants when, for discharging here,", + "is C RONn, and here the relevant resistance is RONp.", + "The charging time constant is RONp times C.", + "OK, so T1 and T2 are assumed to be much greater than these two.", + "So when you look at this, there's one other benefit", + "besides the power benefit, OK, of using CMOS logic", + "compared to using NMOS. OK, it not only cuts out my", + "standby power, but there is another", + "significant advantage which is almost equal to the power", + "advantage of this kind of CMOS technology.", + "Anybody have any ideas? What's the advantage?", + "What does intuition tell you? Is CMOS going to be faster or", + "slower than NMOS? Why?", + "That's right. The key here is that the NMOS", + "design I showed you earlier was relatively slow because it took", + "me a while to charge up the load capacitor from RL.", + "In this situation, RL will become really,", + "really small; it's RONp.", + "It's roughly the same magnitude as RONm.", + "OK, if so both of these on resistances are more or less", + "equal and small, then the rise time will be of", + "the same order of magnitude as the fall time,", + "which makes this much faster than the NMOS.", + "In NMOS, my time constant was RLC, and RL was pretty large.", + "In this case it's RONp C, and RONp can be made to be very", + "small because when it's switched off, the resistance here is", + "infinity. So, in this situation,", + "if I assume T1 and T2 are much larger than the respective time", + "constants, I can go ahead and draw my equivalent circuit.", + "So, here's VS. So, for charging up,", + "let's say this one is going to a one, or to a high.", + "So, I have VS going through a resistor, RONp,", + "to a capacitor, and this thing is a switch.", + "So I have RONp, an ideal switch,", + "going to a capacitor, C, this is my V out node,", + "OK, so it's VS going through a resistance, RONp,", + "an ideal switch, to a capacitor,", + "C. That's a charging circuit.", + "For discharging, I have C, discharging through", + "an ideal switch with RONn. So, this situation,", + "I have an ideal switch, RONn.", + "OK, so that's the equivalent circuit for something like this.", + "So, in this circuit, during T1, this guy's off,", + "and this guy's on, on during T1,", + "and off otherwise. This guy is on during T2,", + "and off otherwise. OK, so just imagine,", + "this guy switches on, this guy switches off,", + "this guy switches on, this guy switches off,", + "OK? And remember,", + "this is exactly the circuit I had analyzed last time in the", + "last lecture, and the result given by v", + "double asterisk. And that result was simply", + "average power being CVS^2f. That's the exact circuit we", + "used to compute the dynamic power, CVS^2f.", + "OK, so we're done. And how did this come about?", + "This came about because the intuition here is that I'm", + "charging up the capacitor fully, and then I'm discharging the", + "capacitor through this other side, OK, and I'm consuming", + "power, dissipating power, in these two resistances during", + "charge up and during the discharge.", + "Half the power gets consumed during charge up,", + "and half during the discharge. So, I'd like to go back to", + "doing a few numbers here, and taking a look at how,", + "even with this expression, life can get pretty thorny as", + "we go ahead into the next decade.", + "OK, so for our previous example, we assumed that 10^8", + "gates, F=1 GHz, C=0.1 femtofarads,", + "VS was 5V, and I don't need RL anymore.", + "OK, why is it that I don't have any resistance component here?", + "I don't have it here because the power consumed by this", + "circuit is independent of those resistances, provided T1 and T2", + "are long enough, are much longer than the two", + "time constants, RONp C, and RONn C.", + "OK, so I don't have RL in my equation anymore.", + "I don't have any standby power. So, based on this calculation,", + "the calculation I did up there showed that I had 2.5 microwatts", + "per gate, and for 10^8 gates I had 250W for a chip with 10^8", + "gates. So, I'd like to dwell on this,", + "if you can move over to page eight in your notes,", + "here. Let me dwell on this for some", + "time, and pontificate on a few things.", + "First of all, this number,", + "as I said before, is high, but not a disaster.", + "OK, so you can't use this in laptops, but it's quite OK for a", + "desktop or a server, and so on.", + "If you just go and put your ear to a pedestal computer,", + "you'll always hear it making a sound, and that sound is because", + "of a big fan that's inside it. And, if you have a big enough", + "fan, 250W is not such a big deal.", + "But, this is certainly a real problem for mobile devices.", + "For a laptop, this is unthinkable.", + "OK, so we have to deal with this.", + "The second issue is the following, that it's 250W for", + "1GHz. Now, the fastest Pentium 4s", + "that money can by today are, what, how many GHz?", + "What's the fastest Pentium 4 you can buy today?", + "What's that? Does anybody have a 4GHz", + "Pentium 4 here? Oh, darn, you beat me.", + "Anybody have a 3? 3GHz?", + "A couple. So, I have a couple of 3GHz", + "machines, and our lab has a whole ton of them.", + "So, if Intel comes out with 4GHz machines today,", + "they've been going up by about 1GHz roughly every year for the", + "past couple of years. And, within three or four", + "years, you're going to see chips, microprocessors that are", + "in the 5-10GHz range, OK, assuming that all other", + "things stay equal, which of course they're not,", + "but just to give you some insight here,", + "if I clock these guys and build circuits that are ten times", + "faster, I very soon go up to 2.5kW, again as I said,", + "all things being equal which they're not.", + "But just to give you a sense, as I increase my frequency,", + "so does the power consumed by the chip, OK?", + "So, I really have to do something here.", + "So, if I stare at this equation, CVS^2f,", + "I want to increase f because people will buy computers if I", + "have higher frequencies. And, Intel has managed to use", + "its marketing campaigns to pretty much convince consumers", + "that high frequencies are a good thing.", + "OK, and whether they really mean anything or not,", + "that's a different issue. So, we've got this huge power", + "for assuming 5V, OK, so it turns out that", + "microprocessors, as they come out,", + "newer and newer versions run at lower and lower voltages.", + "OK, they invent technologies that use lower and lower", + "voltages, and go from VS 5V to, today, VS on the order of 1.5", + "to 1V, somewhere in that range. So the moment you do that,", + "you get a 25x reduction in power.", + "OK, so in going from 2.5kW, you would now come down to", + "something on the order of 100W, which is, again,", + "much more reasonable, again, all other things being", + "equal. It turns out that the", + "capacitance of devices also changes as you go to smaller and", + "smaller devices. And, 100W is also pretty high,", + "and still not good enough for mobile computers.", + "So, there are many, many other tricks that people", + "use to get even lower powers. One trick is to play games with", + "the clock. OK, what you do is,", + "let's say for example in some computation you are not going to", + "be using your floating point unit.", + "Or let's say I'm going to be using your integer adder unit.", + "OK, so what you can do is you can turn off the clock to those", + "devices so that those devices do not even switch when they're not", + "working. OK, if I turn off the clock to", + "a device, the device isn't even going to switch,", + "it's just going to sit there in limbo without consuming any", + "power. It's equivalent to turning off", + "both transistors. If you turn off both the PMOS", + "and NMOS somehow, OK, it's not consuming any", + "power. And by doing that,", + "you can further cut down the power.", + "So, if you can idle some of your function units,", + "it's called idling a function unit, idle a function unit for,", + "let's say, half the time. OK, you would cut down power by", + "another factor of two. We can idle,", + "then, 75% of the time, come down to 25W.", + "So, those are the classes of tricks that people play.", + "I'm going to stop here and allow the underground guide", + "folks to do the survey. But, suffice it to say that the", + "power discussion that I've gone through with you is a very high", + "level discussion as to the real thing.", + "In real life, what actually happens is that", + "there is a fair amount of standby power even for CMOS", + "logic. It turns out that although I", + "don't have a path from VS to ground for my two transistors,", + "it turns out that there are many leakage currents.", + "OK, currents leak through all kinds of places through the", + "drain of the inverter, and so on and so forth.", + "And so, there is some standby power.", + "So, let me show you a quick demo while, I guess,", + "the review handouts are going around.", + "And this shows the temperature of my CMOS inverter,", + "and as I increase the frequency, you can just watch", + "the temperature go up, and hopefully we'll blow this", + "transistor. So, I'm increasing the", + "frequency as you can see on the side here, and higher frequency", + "implies more power consumption, more temperature,", + "OK, and hopefully you will see some smoke coming out of,", + "OK, I think I blew the inverter.", + "So, the output is gone. So, it's at 110 degrees there,", + "and that blew it. Sometimes we see smoke come", + "out, but I guess today is not one of our lucky days.", + "OK, so let me stop here and have the underground guide folks", + "go through the reviews." + ] +} \ No newline at end of file diff --git a/subs/JqvKtMNz3RQ.srt.sjson b/subs/JqvKtMNz3RQ.srt.sjson new file mode 100644 index 0000000000..00137c4471 --- /dev/null +++ b/subs/JqvKtMNz3RQ.srt.sjson @@ -0,0 +1,1982 @@ +{ + "start": [ + 0, + 6936, + 17514, + 27919, + 32500, + 36293, + 40689, + 46206, + 51551, + 56206, + 60000, + 63976, + 67530, + 71507, + 76500, + 80646, + 85638, + 88599, + 93000, + 97186, + 100255, + 104302, + 108488, + 112534, + 116162, + 118813, + 123068, + 126532, + 132670, + 137520, + 143162, + 147121, + 152467, + 156921, + 160288, + 163548, + 167415, + 170371, + 173404, + 178104, + 181985, + 185422, + 188782, + 191532, + 195961, + 198558, + 202835, + 206959, + 212000, + 215637, + 219695, + 222773, + 225082, + 227740, + 229699, + 233897, + 237604, + 240823, + 246000, + 248413, + 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2840987, + 2847366, + 2854000, + 2858968, + 2862363, + 2867165, + 2869152, + 2871719, + 2875694, + 2880000, + 2883286, + 2886838, + 2890746, + 2895453, + 2898296, + 2903358, + 2912004, + 2917380, + 2923126, + 2928873, + 2939439, + 2947354, + 2953239, + 2958732, + 2962263, + 2967265, + 2972758, + 2977564, + 2983253, + 2987659, + 2991759, + 2995077, + 3001129, + 3004936, + 3009000 + ], + "text": [ + "OK, good morning all. So before we begin,", + "I just thought I'd show you a little news item that I happened", + "to read that was very relevant to what we covered recently in", + "6.002. So you recall when we did the", + "digital section a few days ago last Thursday,", + "we talked about a switch. We talked about the MOSFET", + "switch, which when turned on and off, by input signals could help", + "build gates which would then be combined in tens of millions of", + "quantities and go into chips like the Pentium 4 and AMD", + "Athlon 64, and so on it so forth.", + "So I just saw this news item that I came across,", + "and this says they are rethinking the basic", + "construction of the products. It talks about the", + "semiconductor manufacturers like AMD, Intel, and others that", + "build digital chips. They are rethinking the basic", + "construction of the products down to the architecture of the", + "transistor. That's a MOS transistor,", + "and the on/off switch inside the chip.", + "OK, now this might imply that there is a single switch inside", + "the chip, but no, there's tens of millions of", + "transistors, or tens of millions of switches inside a chip.", + "And pretty much any advancement that can be made to the basic", + "transistor can have a 10 million to 20 million times effect", + "because there are that many of them on a single chip.", + "So I thought that was very appropriate.", + "OK. Let's dive into a quick review.", + "So this week, we had begun nonlinear", + "analysis, and I just thought I'd blast through a few animations", + "that I've created, trying to give you more insight", + "into the behavior of some of the things that we have done.", + "Now first of all, as I did the last time,", + "let me try to put it in perspective most of what you've", + "learned thus far, and what we will be learning", + "today. So the past week,", + "we have been focusing on nonlinear analysis.", + "And as I pointed out, here is how this fits into the", + "big picture. So, we had our 6.002 world,", + "at what we said is that we are engineers.", + "We are going to devise our own playground in which to play with", + "our own rules. And that's our playground.", + "That's what we're going to learn about in 002,", + "and for that matter, the rest of EECS at MIT.", + "It's all within this playground here.", + "And this is the playground with lumped circuit abstraction,", + "and good old KVL, KCl, node method,", + "your basic composition rules apply within this playground", + "that directly come from Maxwell's equations because you", + "have made the lumped matter discipline assumptions.", + "OK, so then we said a large part of the playground is", + "linear, and some other much more intuitive techniques apply", + "within the linear portion of that playground,", + "techniques like the superposition,", + "Thevenin and Norton. In most exercises,", + "and quizzes, and experiments,", + "and so on that you do in real life, you can pretty much apply", + "these simple techniques. Very rarely do you have to go", + "into the node method for circuits that are more", + "complicated than single source and a couple of elements.", + "And then, there's the nonlinear part.", + "Remember, the reason I showed this is that this is the same", + "playground. OK, linear and nonlinear are", + "part of the same playground. OK, even nonlinear elements are", + "lumped circuit elements, and they follow KVL,", + "KCl, the node equation, and so on.", + "And then, last week we spent some time talking about the", + "digital abstraction. So we focused on a smaller", + "region of the playground. And the assumptions we made in", + "there were even tighter. We said that it is part of the", + "playground we shall only deal with binary values.", + "We'll digitize or lump values into highs and lows,", + "and that's where our circuits are going to be.", + "And these circuits, when looked at as a whole,", + "were nonlinear. So, this is a simple NAND gate", + "circuit. And this is the input/output", + "characteristic. So, for example,", + "if I hold B at zero, and I apply a zero to one", + "transition at A, then this is the output that I", + "will see at C. So notice, this is decidedly", + "nonlinear. Then I said that,", + "look, suppose we had to fix the input values at a given set.", + "OK, so let's say, for example,", + "I fix A at one, and B at one.", + "OK, and then look at the circuit in this situation.", + "What do I find? What I find is that the entire", + "digital set of circuits that we were looking at move over into", + "the linear space for a given set of switch settings,", + "OK? So, when I set A 1 and B 1,", + "A equal to one and B equal to one, my NAND gate becomes like", + "this. OK, it's a simple resistive", + "network with a voltage source, VS.", + "So, for a fixed set of inputs, for a given set of inputs,", + "if I don't change my inputs, then my circuit looks like a", + "linear circuit, and my good old linear analysis", + "techniques apply. So that was last week.", + "And this week, we are looking at the nonlinear", + "space. And we looked at a couple of", + "techniques in the nonlinear space, analytical techniques and", + "graphical techniques. And then, I showed you an", + "example. OK, I showed you an example", + "circuit that was something that I would like to build involving", + "the light emitting expo dweeb, my little garage door opener", + "thingamajig, and I wanted to transmit music over that light", + "beam. I also showed you that it was", + "highly distorted because it was in the nonlinear space.", + "So, today what I'm going to do is introduce a new part of the", + "playground. There's a new part of the", + "playground, and I'll show you a technique whereby by focusing on", + "this part of the playground and disciplining ourselves in the", + "kind of inputs we apply to circuits, I'm going to show you", + "that certain kinds of nonlinear circuits also move over,", + "when used in a particular way, also move into the linear", + "analysis domain. OK, so let me leave that for", + "now and go back into quickly reviewing the motivating example", + "of music that I had taken last time.", + "OK, so here was a little example.", + "So I have a music source, VI, and I apply that.", + "This device that I call the, lightheartedly,", + "the Light Emitting Expo Dweeb has a current,", + "VD, across it, or a voltage,", + "VD, across it, and a current ID through it.", + "And the light intensity, I said, was proportional to the", + "current. And because of that,", + "I was able to get the light to impinge on a receiving device,", + "which produced a current that was proportional to the", + "intensity of light falling on it.", + "And that signal would then be amplified somehow.", + "We haven't talked about all of this stuff yet.", + "This will happen next week. But let's say we somehow", + "amplify the signal and then played out through a set of", + "speakers. All right, so if I had some", + "sort of a music signal here, then I could then transmit the", + "music signal over to the side on top of this light beam.", + "But the problem, as I said the last time,", + "was that our device, the Light Emitting Expo Dweeb", + "had an exponential characteristic,", + "so that I had some trouble in getting undistorted music.", + "So, the characteristic of the VI characteristics of my device", + "looked like so. The ID versus VD curve looked", + "as follows. OK, it was decidedly nonlinear.", + "And because of that, I was getting a lot of", + "distortions in my signal, and I showed you a little trick", + "to plot, given an input waveform at a transfer function such as", + "here to plot the output function.", + "OK, let me show you another little animation that I have", + "created here for you that should give you even more intuition in", + "terms of how it happens. So, this is a characteristic I", + "showed you up here. It's on both sides,", + "but I guess it points to only one unless I shuttle back and", + "forth really fast. So on average,", + "I'll be in both places. But anyway, so here's my ID", + "versus VD characteristic. And as I said,", + "there's an exponential ID versus VD curve.", + "And I want to see what the output looks like,", + "for example, a sinusoidal input.", + "So I said, let's place the input along a little graph,", + "rotate it so, and take a sinusoid,", + "and apply a sinusoid to the input, VI, which would also", + "appear across the Light Emitting Expo Dweeb.", + "And then, what I wanted to see was how the output looked.", + "OK, so let me tell you that the output is going to look like", + "this. OK, the output is going to look", + "like so. And, a little artifice to", + "discover curves like this is to think about a point here", + "corresponding to the point on the transfer curve here,", + "because this is VD, looking at the Y intercept.", + "That's a value of ID, and that's a value of ID here.", + "And, time moves along here, and time moves along here.", + "So, I did this little animation.", + "You'd better be impressed. It took me six hours to do it.", + "So, here it goes. So, let's say I start by", + "focusing on this little point that corresponds to this point", + "on the transfer function, which then, in turn,", + "points to a time, zero, this point on my ID", + "curve. OK, I hope this works.", + "So, as my point moves down [LAUGHTER], this was fun to do,", + "I promise you. So notice that as this point", + "has the following excursion, this had the following", + "excursions here. OK, all right.", + "So let me pause that little animation there.", + "At the end of the lecture, I'll put that up again if you", + "like, and you all can come and play with it.", + "So, you can actually do this in PowerPoint.", + "It took me quite a bit of time to figure out how to do it,", + "though, but it's fun. OK, so let me show you a little", + "demo, and show you a sinusoid, and show you what the output", + "looks like if I apply a sinusoid for VI.", + "So, I'll show you ID as a function of VI when VI is a", + "sinusoid. There you go.", + "So, I applied my sinusoid VI, and this is the current that I", + "get. And notice, this is the", + "transfer function that I talked about, the ID versus VD curve of", + "my Light Emitting Expo Dweeb. And I get this highly nonlinear", + "transformation of the input as I get to the output.", + "OK, so that is a problem. And then, I also played some", + "music for you. Let's do that,", + "too. I played some music for you.", + "I applied the music as an input to the circuit,", + "and that's the output. OK, that's the output that I'm", + "observing at the amplifier. It's highly distorted.", + "OK, we can stop that. There you go.", + "OK, so that was my problem. OK, so we had covered,", + "we had gone this far last Tuesday.", + "I set the problem up for you, motivated what we had to do,", + "and showed you that I was able to transmit music over my garage", + "door opener, but I did not think I could listen to that music for", + "very long. So, I challenged all of us to", + "think about how a trick that I could use to be able to transmit", + "music and have a linear response.", + "So, did you people get time to think about it?", + "So how many people here think they know the answer?", + "It's OK, don't be modest. Go ahead.", + "Could you speak louder? Yeah, you find another", + "something, kind of element, that's got the opposite graph", + "so that when you add them together.", + "Oh, this guy wants to cheat. No.", + "He wants a new element. So, no, no new elements.", + "Pardon? Build an MP3 encoder.", + "Ah-ha, so that will happen much later.", + "Yes? Digitize the signal before you", + "send it to the LED? Digitize the signal before you", + "send it to the LED. But in some sense,", + "each of these solutions is a huge sledgehammer approach to", + "look at solving it. There's a much simpler", + "technique I can apply here. Yeah?", + "Add a voltage offset. Ah, ah-ha, that might work.", + "What else? So let's say,", + "here's my signal, right?", + "If I add a voltage offset, that will just bump the signal", + "up here. Then the curve is still", + "nonlinear. But you're getting there.", + "Well, I'll tell you what. Let's pause here.", + "Let me quit while I'm ahead. OK, so the answer here,", + "folks, is Zen. OK, what I want you to do is,", + "so, in Zen, what you have to do is you have to sit down in a", + "courtyard, and look at a rock, like a small rock on the", + "ground. And you got a focus on it till", + "the rest of Earth kind of vanishes.", + "Just focus on the rock. OK, now make like you're in a", + "courtyard, and you're looking at this little area here.", + "Just look at this. OK, and I'll give you ten", + "seconds. Sit down quietly,", + "and no sounds. Just stare at the spot here.", + "OK, make believe this is your little rock, and just stand", + "there and think about it. OK, I'll give you five seconds", + "to do that. Just stare at it.", + "And very soon, the answer should pop into your", + "heads. OK, what do you see?", + "This guy, if I focus on this really small region of the", + "graph, this small little piece looks more or less linear.", + "OK, hmm, so that should give me some insight.", + "This whole thing, the macrograph is nonlinear.", + "But I focus on a little rinky dinky piece of that graph like", + "so, that appears more or less linear.", + "If it's small enough, that appears linear.", + "So, I'm staring at this, and that appears linear.", + "The question is, how do I exploit this little", + "small, little, linear region to get a linear", + "response from my device. OK, so here's the trick that", + "I'm going to use. The little trick that I'm going", + "to use is the following. Notice that,", + "let me call this voltage at the center of this region capital", + "VD. What I can do,", + "if I take my input signal, and I just pointed out earlier,", + "I bump it up. I boost it.", + "OK, so I apply a DC offset to my input signal,", + "like so. So I apply some input signal,", + "VI, which is also equal to the VD if I look at a variable", + "across the nonlinear element. If I apply a DC offset,", + "VI, and I superimpose the music on top of that,", + "let me call my music, just to distinguish between the", + "two, capital VI, and the small vi.", + "OK, that's my music. So here's my capital VD,", + "my DC offset. And I want to superimpose my", + "music on top of that. OK, so I've gotten halfway", + "there. By superimposing my music here", + "instead of having excursions out here, I now have excursions out", + "here. OK, and so I'm using some", + "portion of the graph here. But that's still way beyond the", + "small little element there. So a second think that I do in", + "addition to boosting up the signal is shrink it.", + "Think of boost and shrink, BS.", + "So what I want to do is boost up the signal using a DC offset,", + "and shrink the sucker. OK, so I'm going to go with a", + "small signal and bump it up. OK, so now what happens is that", + "small signal in its excursions, only uses that little portion", + "of the graph. OK, again, remember:", + "bump and shrink, bump and shrink,", + "two things, boost and shrink. So what do you think of that", + "trick? So, by doing that,", + "what happens is that signal that has excursions here will", + "produce a corresponding response in this region,", + "OK? And I argue that since this is", + "more or less like a straight line, I invoke Zen here,", + "and argue that this little signal now gets transformed,", + "and I get a linear response. OK: boost and shrink.", + "So in terms of my circuit, let me draw it out for you.", + "My Light Emitting Expo Dweeb, and this whole signal was what", + "I used to call V capital I, and that's made up of two", + "components now, a bump offset,", + "and a shrunk voltage VI. It shrunk, so therefore I've", + "used the small v and small i, like, really,", + "really small. In the same manner,", + "I get a VD ID across the LED, and the corresponding values", + "here will also have a DC offset and a small response.", + "Let me call that ID plus I small d.", + "I'll do all this mathematically in a second as well,", + "but first let me do it completely intuitively so you", + "get some insight into what's going on.", + "And, VD is simply capital VD plus small vd.", + "OK, and this is the same as VI, I, and VI.", + "OK, so what have I done? I've done two things.", + "I have said, as an engineer,", + "OK, I care about getting music across my garage door opener.", + "And I'll do what it takes to do that.", + "OK, so as an engineer, I'll do two things.", + "I'm going to bump my signal up and shrink it.", + "And the bumping and shrinking, and I do it like this.", + "I shrink my signal, the music signal here,", + "and add a DC offset. OK, and I claim that the music", + "I listened on the other side now, provided I have enough", + "amplification there, is going to be undistorted.", + "OK, so far I've showing this to you completely intuitively using", + "little sketches, no math.", + "I promise you, I'll give you a bunch of math", + "in a few seconds, but just get the basic idea,", + "and get the intuition behind it.", + "So let's go back to our demo and take a look.", + "So remember, BS, right, bump and shrink.", + "So what I'm going to do is first of all,", + "let me bump up the signal. So, what I'll do is I want to", + "add an offset to my input, and let me bump it up.", + "Let me shrink it first. It'll make the point a little", + "clearer. So, the big input,", + "green, is a big input. Let me shrink it.", + "", + "OK, so I've made my input small, and in the middle of that", + "picture out there, you see the region of the", + "transfer curve that's being articulated.", + "OK, this region of the curve is being articulated by the small", + "signal. It's a much smaller signal.", + "And the output is still distorted because I have to do", + "two things: bump and shrink. I've only shrunk.", + "OK, let me bump it up now. What's the yellow curve?", + "It's going to get linear. It's going to get proportional", + "to the input. Then I'm bumping it up now.", + "I can make it smaller, make it even smaller,", + "there you go. Isn't that fantastic?", + "So, I'm making nature do my bidding here,", + "OK? So, this is one of those,", + "when I learned electronics and so on many, many years ago,", + "this was one of those really big ah-ha moments for me,", + "saying, wow, that stuff is cool.", + "It's something that I couldn't think about myself,", + "and it's not obvious, and by being disciplined and", + "creative in how I use circuits, I can do really,", + "really cool things. OK, remember this as a big", + "ah-ha moment for you. So, here's my little signal", + "that I've shrunk and bumped up, and my output is a sinusoid,", + "and not this funny, distorted waveform.", + "And notice that this is the region of the curve that is", + "being articulated. So, I can make the signal even", + "smaller if I like. OK, and what I'd like to do", + "next is play music for you, and if you don't believe your", + "eyes, you can at least believe your ears.", + "Let me go to the distorted signal again,", + "switch to music, and raise it up.", + "OK, now what we'll do is shrink the music signal and then bump", + "it up. Can I turn the volume down a", + "little bit? That's good.", + "OK, so if I shrunk the volume a little bit, and let me bump it", + "up, now. [MUSIC PLAYS] Just remember", + "this as a big ah-ha moment. OK, the signal is really,", + "really small. I like that.", + "I like the enthusiasm. OK, so the signal's very small,", + "and I get a more or less linear response.", + "OK. All right, so that's intuition,", + "and the approach that I've taken is called,", + "it's variously called small signal analysis,", + "incremental analysis, small signal method,", + "small signal discipline, whatever you want.", + "", + "OK, this simply says that by boosting and shrinking my", + "signal, I get a response that's more or less linear even when I", + "have a nonlinear device. And this technique is called", + "the small signal approach. So, just to focus on that a", + "little bit longer, switch to page five of your", + "notes and let me draw something out for you.", + "", + "OK, so what I have here, this is my offset VD,", + "and from the VD offset I have my little signal V small d,", + "and the total signal is called V capital D.", + "Offset, small signal, and that's my total signal.", + "OK, notice the offset is all capital.", + "The total signal is small v capital D, and the music or the", + "small signal is small v small d. Similarly, the output is going", + "to look like this, and here I get an offset in the", + "output ID. I get a corresponding signal,", + "I small d, and I get a total signal, I capital D,", + "OK? The cool thing to notice is", + "that the signal here, the output signal here", + "corresponding to the input signal, the music signal,", + "VD, is small I small D, and that is more or less", + "linear. OK, and I can even plot the", + "signal like so. This is my input,", + "v capital D. That's T.", + "This is VD, V small d. That is my total input.", + "And similarly, I have an output.", + "And this is my output ID. And, that looks like this,", + "I capital D, small i small d,", + "total signal I capital D. OK, so that's the small signal", + "method. So, let me summarize that for", + "you.", + "", + "There are three steps to the method.", + "So, first of all, operate at some DC offset.", + "This is also called DC bias, and in that example it's VDID.", + "OK, so I choose an operating point that bumps up the", + "operation in some region of interest.", + "The second step is to superimpose small signal on top", + "of VD, capital V capital D, to superimpose a small signal,", + "and the third step is observe the response --", + "-- and the response, small i small d,", + "that's the music part of the response, ID,", + "is approximately linear. OK, three steps to the method", + "here, and just remember this notation.", + "And, my notation in the small signal model is as follows.", + "My total signal ID is the sum of two signals,", + "I capital D plus small i small d.", + "This is called the total signal.", + "That's called the DC offset. And this is the superimposed", + "small signal. OK, total signal,", + "DC offset, plus the small signal.", + "And sometimes, especially when doing math,", + "and so on, we may oftentimes represent ID as a delta,", + "I capital D, OK, to show that ID is", + "incremental change in the value of I capital D.", + "And because of that, this method is also often", + "called the incremental method, incremental analysis.", + "OK, so far what I've done is given you some intuition.", + "I've developed a small, simple method,", + "given you some insight into why we use this method,", + "and also shown you some demonstrations that show that", + "when I bump and shrink, and observe the response,", + "I do get a more or less linear response.", + "So let me now do this mathematically and show you that", + "mathematically, you can also derive your", + "response to be a linear response.", + "This is page seven. So, I know that ID is some", + "function of the diode voltage. F was my nonlinear function.", + "OK, so my function F was a nonlinear function.", + "So therefore, ID was nonlinearly related to", + "VD. So, let's do the math.", + "So as a first step, what we did was replace VD by a", + "DC offset, the small signal method, a DC offset,", + "plus a small incremental change.", + "OK, by doing the math, let me simply use the delta VD", + "notation to show you that I'm dealing with small increments,", + "and also because in the mathematics community,", + "when you learn about some of these techniques,", + "they will use the incremental change notation,", + "which is the delta VD notation. In electrical engineering,", + "we use a small v, small d notation.", + "So, this is a large DC offset, and this is a small change", + "about that offset. So, you folks have taken math", + "courses before, and been looking at finding out", + "the value of a function, which is a small change for an", + "input value, which is a small change about a big input value", + "or a big DC point is Taylor's expansion.", + "OK, so let's use Taylor's series expansion,", + "OK, and substitute VD plus delta VD into this,", + "and see what ID looks like. Again, let me tell you where", + "I'm going with this. ID equals F of VD.", + "This is a nonlinear function, OK?", + "I claim that by replacing VD, the input, with the DC offset", + "plus a small value, the resulting response to the", + "small value will be linear, OK?", + "So what I'm going to do next is replace VD with this sum here,", + "and then do the math, and show you that the response", + "corresponding, or the change in ID", + "corresponding to the change in VD is going to be linear.", + "All right, so let's expand this function using Taylor's series", + "near the DC offset point, capital V capital D.", + "OK, so ID is simply, by Taylor's series,", + "I want to find out a value of the function close to V capital", + "D. OK, so I take the value of the", + "function at that point, and then I add a few terms in", + "my Taylor's series expansion. The first term is simply the", + "good old Taylor's series stuff. OK, the first term is the first", + "derivative of the function times the change.", + "And then, the second one is second derivative.", + "", + "OK, and then I get higher order terms.", + "So this is nothing new here. This is good old Taylor series", + "expansion, and again, let me tell you where I'm", + "going. I want to look at the response", + "for an input that looks like this, and I want to show you at", + "the end of the day that the response in ID,", + "the effect on ID of using an input like this is as if that", + "effect, the incremental change is linearly related to the small", + "input, delta VD. So here's my Taylor's series", + "expansion for delta V. Now remember,", + "I told you that delta VD is much, much smaller than V", + "capital D. OK, it's a very,", + "very small quantity. But that quantity is really", + "very small. Then what I'm going to get is", + "that my output is, I can begin to ignore my second", + "order terms. OK, delta VD is very,", + "very, very small. Then, what I'm going to do is", + "that ignore higher order terms. So I'll go and ignore higher", + "order terms. They'll all go to zero.", + "Remember, I can do this because by design I've chosen delta VD", + "to be very, very, very small.", + "OK, remember, we are engineers.", + "I've chosen it in a way that this is very small.", + "OK, so I'm telling you that's the case, and under those", + "conditions, I can ignore second higher order terms,", + "in which case I am left with this expression here.", + "So let me rewrite this. Let me rewrite this down here.", + "", + "OK, I've just copied this turnout, I've ignored all these", + "terms here, and so I have a more or less equal to sign that", + "remains. So what I'm going to do is when", + "I apply a small input of this form to a large DC offset,", + "my output is also going to look like some output offset with a", + "change in the output offset. And let me call the output", + "offset I capital D, and some small change in the", + "output delta ID. OK, we'll make sure we can", + "convince ourselves that this is indeed the case.", + "Notice that this guy here, F of capital V capital D is a", + "constant. That's a constant with respect", + "to the incremental change, delta VD.", + "Similarly, this part here is a constant.", + "Notice that this term here is the first derivative of the", + "function evaluated at the DC bias point, capital V capital D.", + "OK, so this term is also a constant with respect to delta", + "VD. So notice, then,", + "I have a constant term plus a constant term multiplying a", + "small change, delta VD.", + "So what I can do next is, in this case,", + "given that I have a constant term on both sides,", + "and on this side it's a time varying term,", + "what I can do is equate the two constant terms.", + "I can go ahead and equate these two terms.", + "Remember, I have a constant plus a time varying term,", + "OK, if I'm assuming here that delta VD, my little music signal", + "is a time varying term. So, this constant will equal", + "this, so ID must equal F of VD. And I know that's the case", + "because the function evaluated at the DC offset gives me the DC", + "current ID. And similarly,", + "ID is equal to that component. Delta ID is equal to D,", + "F of --", + "", + "OK, so my incremental change in the output is the first", + "derivative multiplied by the small change in the current.", + "OK, so I'm pretty much done. So, therefore,", + "notice that delta ID is proportional to delta VD.", + "OK, and that's what I had set out to show.", + "Remember, I had set out to show that provided my input is a", + "small excursion around a large DC offset,", + "then my output could also be a large DC offset with a small", + "excursion on top of it where the two excursions,", + "the input excursion and the output excursion would be", + "linearly related like so. OK, and the method is very", + "simple. I simply expanded the function", + "about that point, that DC point,", + "neglected higher order terms, and notice that my incremental", + "term was simply the derivative plus the incremental change,", + "a derivative times the incremental change in the input.", + "Move onto page nine, and I'd like to give you a", + "quick graphical interpretation of this.", + "So I gave an intuitive explanation earlier.", + "This is a mathematical explanation that shows you that", + "the input could be linearly related to the output,", + "provided, the outputs would be linearly related to the input,", + "provided the input has a DC offset, and small excursions", + "about that DC offset. So, let me give you some", + "intuition in what you've really done here, using a little graph", + "here. So, I'm going to plot ID versus", + "VD, and notice that I have some point here, V capital D,", + "I capital D. That's my DC bias.", + "So, I have some DC bias point here.", + "OK, what is this? That is simply the slope of the", + "curve at that point. OK, it's the slope of this", + "curve evaluated at this point. So this guy here is simply the", + "slope of this curve evaluated at ID VD.", + "OK, now, what I care about is this point here,", + "and this point here. So let's say that this is delta", + "VD, all right, and that corresponds to this", + "point here. So what I've done is taken the", + "slope and multiplied that by delta VD.", + "So I've taken the slope, and multiplied it by delta VD,", + "OK, and that gives me this component here.", + "OK, and so, this is the point that I'm going to get.", + "So in other words, what I've done is approximated", + "point A using the Taylor trick by the point B.", + "OK, so this is a point, A, which is what I really want,", + "and I've approximated that by taking the slope of the function", + "at V capital D, and multiplying that by the", + "change in the input to get the corresponding Y offset,", + "and that's the point that I get.", + "And notice that if I make this delta VD small enough,", + "then the error between these two points becomes smaller and", + "smaller. So back to our example,", + "so ID was a e to the BVD. This was the relation for our", + "Expo Dweeb, and let me just plug in the values.", + "So, ID plus small id. Notice, I'm just shuttling back", + "and forth between the notation delta VD, and small v small d.", + "", + "OK, and so that is given by a e to the BVD, oops,", + "plus, I'm just writing that equation up there.", + "Let me call this equation X. And so, I get the second term", + "is the derivative, ab times e to the BVD times", + "delta VD, small VD, and equating this term that the", + "DC offset. Notice that this is the DC", + "offset in the output, and the small signal,", + "ID is, further notice that in this particular example,", + "what's that? a e to the BVD.", + "That's simply ID again. It just happens to be that way", + "in this example. So, I get ID times BVD.", + "So, for my input, small id, my incremental change", + "in the output is some ID times B times VD.", + "And notice that this is a constant.", + "And because that is a constant, my small signal behavior ID is", + "going to be linearly related to the signal, VD,", + "the input signal VD. OK, in the last three minutes,", + "I'd like to give you one additional insight.", + "So what we've shown so far is if I have an offset and a small", + "change above it, then my output ID will be", + "linearly related to my input. Now let's stare at this thing", + "again. Let me rewrite it.", + "It's some constant IDB times VD.", + "So, where have we seen such an expression before?", + "OK, where ID was some constant times VD.", + "OK, remember, I equals V divided by R:", + "Ohm's law. What I want to show you now is", + "how we constantly keep simplifying our lives.", + "The moment we hit some complication and things get too", + "painful to analyze, as engineers,", + "we come up with some clever tricks to make an analysis and", + "use of circuits simple again. And so, notice that this is", + "similar to some, one by RD VD,", + "where RD is simply one over IDB.", + "I'm just defining this to be RD.", + "And what that means is that I can take a nonlinear circuit", + "that looks like this. OK, and what I can do is", + "replace this by its incremental equivalent, and build what is", + "called a small signal circuit. And I'll just introduce it", + "here. And we will revisit the circuit", + "in much more gory detail a couple of weeks from now.", + "So, what I can do is build a small signal circuit where I", + "have all the small signal variables, and replace a", + "nonlinear device by a simple little resistor whose value is", + "given by IDB. OK, so therefore,", + "what I can do is take my nonlinear circuit,", + "and for small, incremental changes,", + "replace that circuit with this equivalent small signal circuit,", + "and go back to doing simple stuff again.", + "Thank you." + ] +} \ No newline at end of file diff --git a/subs/Km9YIdkc2Oo.srt.sjson b/subs/Km9YIdkc2Oo.srt.sjson new file mode 100644 index 0000000000..b3ba7958c1 --- /dev/null +++ b/subs/Km9YIdkc2Oo.srt.sjson @@ -0,0 +1,1838 @@ +{ + "start": [ + 0, + 10000, + 12000, + 19000, + 22870, + 28675, + 32207, + 37193, + 38910, + 42833, + 47738, + 52724, + 57956, + 62466, + 67228, + 70205, + 75392, + 80070, + 83387, + 89000, + 93295, + 96517, + 98879, + 102244, + 105896, + 110263, + 114057, + 117852, + 122636, + 124896, + 129886, + 134594, + 139301, + 143538, + 149000, + 153008, + 155814, + 160023, + 163431, + 167306, + 171381, + 174521, + 178129, + 182222, + 194000, + 202000, + 205806, + 207870, + 212415, + 216038, + 221519, + 227000, + 242000, + 247265, + 253391, + 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2789943, + 2798971, + 2808000, + 2813641, + 2817333, + 2823829, + 2829702, + 2833787, + 2837361, + 2840170, + 2843872, + 2850000, + 2858708, + 2866194, + 2873986, + 2883000, + 2884764, + 2887449, + 2889597, + 2891976, + 2894968, + 2897960, + 2902563, + 2906859, + 2912000, + 2916628, + 2919509, + 2923439, + 2926845, + 2931473, + 2936974, + 2943000, + 2946781, + 2949156, + 2953729, + 2955487, + 2960412, + 2963930, + 2966304, + 2971000 + ], + "text": [ + "", + "Good morning.", + "", + "All right. Today we are going to take a", + "fresh look at some of the stuff we covered in the last two", + "lectures. And the graph I want you to", + "keep in mind as we go through this lecture in terms of what to", + "expect. This was time.", + "And last Tuesday's lecture we covered some stuff.", + "I talked about a method for the sinusoidal response which was", + "agony, I warned you it will be agony, and then towards the end", + "I showed you another method that was quite a bit easier but still", + "pretty hard. And I promised you that today", + "there will be a new method which is going to be so easy ,", + "actually almost trite. Just imagine.", + "I am going to make a statement right now that I think you will", + "all find hard to believe. What I am going to say is just", + "imagine your RLC circuit, your resistor,", + "inductor and capacitor, a parallel form or series form.", + "Imagine that you could write down the characteristic equation", + "for that by observation in 30 seconds or less.", + "Just imagine that. By observation,", + "boom, write down the characteristic equation for", + "virtually any RLC circuit or RC circuit or whatever.", + "And we all know that once you have the characteristic equation", + "you could very easily go from there to the time domain", + "response intuitively or to the sinusoidal steady-state", + "response, too. So just keep that thought in", + "mind. Imagine 30 seconds.", + "And that is what you should expect in today's lecture.", + "Students often ask me, if this stuff is actually so", + "easy why do you take us through this tortuous path?", + "Are we just mean? Do we just want you show you", + "how hard things are and then show the easy way?", + "I have argued with myself every year as to whether to just go", + "ahead and give the easy path and that's it.", + "But I think the reason we cover the basic foundations is that it", + "gives you a level of insight that you would not have", + "otherwise gotten if I directly jumped into the easy method.", + "So you need to understand the foundations and you need to have", + "seen that at least once. And second, once you do", + "something the hard way, you appreciate all the more the", + "easy method. All right.", + "Today we cover what is called \"The Impedance Model\".", + "", + "First let me do a review just because of the large amount of", + "content in the last two lectures.", + "I did them using view graphs. I usually don't like to do", + "that, but even then it was quite rushed.", + "So let me quickly summarize for you kind of the main points.", + "We have been looking at, on Tuesday, the sinusoidal --", + "", + "--looking at the sinusoidal steady state response.", + "Also fondly denoted as SSS. And the readings for this were", + "Chapters 14.1 and 14.2. what we said was if you took", + "this example circuit and we fed as input cosine of omega t,", + "we have an R and a C, and let's say we cared about", + "the output response and we cared about the capacitor voltage.", + "What we talked about was focused on the sinusoidal", + "steady-state response. And what that meant was first", + "of all focus on steady-state. In other words,", + "just to capture the steady-state behavior when t", + "goes to infinity after a long period of time.", + "And for most of the circuits that we consider,", + "because of the R or presence of any resistance,", + "the homogenous response usually would die out because the", + "homogenous response is usually of the form minus t by tau.", + "And as t goes to infinity this term tends to go to zero.", + "We are just looking at the steady-state.", + "And therefore, because of the circuits we", + "looked at, we can ignore the homogenous response.", + "All we are left to do is to find the particular response to", + "sinusoids of this form. And second was focus on", + "sinusoids. We said the reason for this was", + "that, let's say we did not care particularly", + "What happened when I just turned on my amplifier.", + "I just turned on my amplifier, often times you see some", + "distorted sound coming out for a few seconds and then hear a much", + "clearer sound. And that initial part is due to", + "the transient response. And let's say we don't care", + "about that. We care about the steady state.", + "Second we focus on sinusoids because based on the Fourier", + "series experience that you had previously, we can represent", + "repeated signals as a sum of sines.", + "And therefore it is important to understand the behavior of", + "these circuits when the input is a sinusoid.", + "And what was important was this introduced a new way of looking", + "at circuits, and that was the frequency viewpoint.", + "When we looked at transient responses, we plotted response", + "as a function of time. And when we look at sinusoidal", + "steady-state, it becomes interesting to plot", + "the response as a function of the frequency,", + "a function of omega.", + "", + "What I will do is draw a little chart for you to sort of", + "visualize the various processes we have been going through.", + "We can liken obtaining the sinusoidal steady-state response", + "to following these steps. Here is my input.", + "What I did as a first step was fed my input to a usual circuit", + "model. My elements were lumped", + "elements, built the circuit and wrote down the VI relationship", + "for the element. As a second step I set up the", + "differential equation.", + "", + "This was the first of four steps, set up a differential", + "equation. And then the path that I took", + "first was fraught with real nightmarish trig.", + "", + "By the end of the day it would still yield an answer.", + "It could be a nightmare. But I would get something", + "cosine omega t plus something, some phase.", + "I could grunge through the trig.", + "And I gave up halfway in class here, but you could grunge", + "through it if you would like. And you would get the answer to", + "be some sinusoid with some amplitude and some phase.", + "So Vi cosine omega t would produce the response that was", + "something cosine omega t plus some phase.", + "We said this was too painful so let's punt this.", + "Instead, what we said we would do is take a detour,", + "take an easier path. And the easier path looked like", + "this. I said let's sneak in --", + "", + "-- Vie^(j omega t) drive. That is just imagine,", + "do the math as if you had fed in not a Vi cosine omega t but a", + "Vie^(j omega t). And from Euler's relation you", + "know that the real part is Vi cosine omega t.", + "So we said that I am going to sneak in this thing,", + "find the response and just take the real part of that because", + "the real part of the input gives me this.", + "So this is my \"sneaky path\". And what I did there,", + "as soon as we fed in the e^(j omega t), because of the", + "property of exponentials, the e^(j omega t) cancelled out", + "in my equation. And what was left was some", + "fairly simple complex algebra. And at the end of the day,", + "after I grunged through some fairly simple complex algebra,", + "I ended up with some response that looked like this.", + "Vpe^(j omega t). What I would find is that for", + "the input Vie^(j omega t), I would get a response Vpe^(j", + "omega t). And then what I said we would", + "do is take the real part. Why take the real part?", + "Because this is a fake, a sneaky input.", + "The input I really care about is the real part of the sneaky", + "input. So this is my sneaky output.", + "And what I care about is the real part of the sneaky output.", + "That is sort of the inverse superposition argument that I", + "made on Tuesday that if what I care about is the real part of", + "this input, then I just take the real part and get the output", + "that I care about. So I take the real part.", + "Notice that Vp here, in the examples we did,", + "we did an RC example. The Vp here was a complex", + "number. So I could represent that", + "complex number as, in many ways.", + "This is e^(j omega t). I could represent Vp in an", + "amplitude, as a phasor, actually polar coordinates.", + "I can say that the equivalent to Vpe to the j angle Vp.", + "Vp is a complex number. If you look at the complex", + "appendix in your course notes, I can represent a complex", + "number as an amplitude multiplied by e raised to j", + "times some phase. It's simple complex algebra.", + "And then what I could do here is take the real part of that.", + "And when I took the real part of that what came about was that", + "this was simply Vp. Notice that the angle Vp goes", + "in here so it becomes j times omega t plus angle Vp.", + "It is Vp amplitude times e raised to j omega t plus j angle", + "Vp. And the real part of that is", + "simply Vp cosine of that stuff. What I end up getting here is", + "Vp cosine omega t plus Vp. The cool thing to notice was", + "that once I found out this response here,", + "I could immediately write down the output based on Vp.", + "In other words, once I had Vp,", + "I could stop right there in my math.", + "I got Vp very quickly here. This step produced Vp very", + "quickly, after two algebraic steps.", + "And then from here I could directly write down the answer", + "as homogenous of Vp cosine omega t plus angle Vp.", + "Boom, right there. So this was a much shorter", + "path. And here I just described to", + "you how this yields an expression for Vp and angle Vp.", + "And for our example Vp was 1/(1+j omega RC).", + "And we often times write a shorthand notation 1+sRC,", + "where S is simply j omega. We commonly jump back and forth", + "between the shorthand notation S and j omega.", + "S has some other fundamental, has another fundamental", + "significance you will learn about in future courses,", + "but for now S is simply a short form for j omega.", + "This was the path that we took. There is a hard path and an", + "easier path. Today I am going to claim that", + "even this was too hard. There is an even easier path.", + "And today what I am going to show you is that from here we", + "are going to take one step and get here.", + "I am going to show you today that we won't do this,", + "we won't do this, not this, not this,", + "none of this. One step and then we are going", + "to get the answer. So let's do that.", + "", + "Before we jump into the impedance method and get into", + "doing that, I just would like to plot for you this function here", + "just so we can understand a little bit better exactly what", + "is going on. As I mentioned to you,", + "the output vO for our circuit there was simply Vp cosine of", + "omega T plus angle Vp. Oh, that's Vp so this one", + "should be Vi here. I am showing you Vp so there is", + "a Vi in there. Vp/Vi=1/(1+j omega RC).", + "This is a complex number, and it is simply a number that", + "when multiplied with Vi gives me the output.", + "This is also called a transfer function and represented as H(j", + "omega). This guy is a transfer", + "function, much like the gain of my amplifier.", + "Which when multiplied by the input to get me the output.", + "This guy is a complex multiplier which when multiplied", + "by Vi gives me Vp. And as such we call it a", + "transfer function H(j omega). And we can plot this function.", + "Notice that this a function of omega.", + "Remember we are taking the frequency domain view,", + "so where has time vanished? Remember that we are taking the", + "steady state view. So we are saying in the steady", + "state, if I wait long enough this is how my circuit is going", + "to behave, this is how a circuit is going to behave.", + "And the transient responses have died away and I have time", + "in my output here so my output is a cosine.", + "But that in itself is not very interesting.", + "It is a cosine of some amplitude and has some phase.", + "What we will plot is we are going to plot this property", + "here, Vp as a function of the frequency.", + "Vp is frequency dependent. As an example,", + "I could plot the absolute value of Vp/Vi, the modulus of that", + "versus omega. And notice that when omega is", + "zero again intuitive ways of plotting this is to look at the", + "value at zero and look at the value at large omega.", + "For small omega, omega goes to zero this is one,", + "so it starts off here. And when omega is very large", + "then it is much bigger than one here, so this goes down.", + "Far away this one looks like 1/omega RC.", + "And this function, assuming I have linear scales", + "on my X and Y axes looks like this.", + "We also commonly plot this using log-log scales.", + "And when you do log-log scales you get a straight line here,", + "and then you actually get a straight line of slope minus one", + "because the log of this gives you a line with a constant", + "slope, it's a slope of negative one so it becomes a straight", + "line going down. The other interesting thing to", + "realize is that this magnitude is simply one by one plus omega", + "squared R squared C squared, the square root of this.", + "That's the magnitude here. And notice when omega equals", + "1/RC, this thing, the denominator becomes one by", + "square root of 2. Somewhere here when omega", + "equals 1/RC The output is one by square", + "root 2 times the input. It's an interesting point.", + "And this is called the \"break frequency\".", + "You can view it as a frequency where I am getting this", + "transition from one to a lower value, and it is where the", + "output is one by square root two times the value of the input.", + "Now you can think back on the demo we showed you earlier.", + "And in the demo remember that as I increased the frequency of", + "my input sinusoid my output kept becoming smaller and smaller and", + "smaller. And you notice that you can see", + "this dying out or decaying of the amplitude as I increase my", + "omega. Let me go back.", + "What you have done is that, we're going to apply a bunch of", + "sinusoids to the same circuit and plot the frequency response,", + "the ratio of the output versus input as a function of", + "frequency. And kept applying a variety of", + "frequencies. So you can listen to the", + "frequencies as they go by, and we will plot the amplitude", + "up on the screen for you. Just for fun we are going to", + "play frequencies between, say, 10 hertz and 20 kilohertz.", + "It will be fun for you to figure out at what point you", + "stop hearing the frequencies. We are going to play from 10", + "hertz to 20 kilohertz. And figure out where your ears", + "cut out. That will tell you what the", + "break frequency of your ear is.", + "", + "You can see the amplitude being articulated.", + "The bottom figure is the phase. This is the frequency axis.", + "This is the amplitude, log-log scales.", + "", + "I am not sure about you but I cannot hear anymore.", + "If you bring your canine friends to class it is quite", + "possible that they would go berserk somewhere here.", + "As I promised you, when I plot this on a log-log", + "scale I get a straight line here and a straight line out there as", + "well and the bottom line gives you the phase.", + "Now, what you can also do is you can also go to Websim.", + "Websim is now linked on your course homepage.", + "You can go to Websim and you can play with various L and C", + "and R values. And if you plot frequency", + "response, if you click on the frequency response button,", + "boom, it will give you frequency responses for your", + "circuit that look exactly like that.", + "You can go and play around with that.", + "Thank you. All right.", + "As the next step I promised to show you an easier path.", + "And let's build some insight.", + "", + "Is there a simpler way to get where we would like to get?", + "In particular, is there a simpler way to get", + "Vp? Let's focus on Vp.", + "Why Vp? Because remember Vp was the", + "complex amplitude of e to the j omega t.", + "And once I know Vp then I know this expression here.", + "Also notice that this here, the denominator is simply the", + "characteristic equation for, I wonder how many of you", + "noticed it, is simply the characteristic equation for the", + "RC circuit. If I can write down Vp,", + "I can write down the characteristic equation,", + "it will be in the denominator. I can also write down the", + "frequency response very easily by taking the magnitude and", + "phase of Vp. So Vp has all the information", + "humankind needs for those circuits.", + "Is there a simpler way to get Vp?", + "To bring some insight, let's go ahead and write down", + "--", + "", + "Let's stare at this for a while longer and see if light bulbs go", + "off in our minds. Of course, I could write this", + "as Vi/(1+sRC). I just replaced the shorthand", + "notation for a j omega. And I simply divide by SC", + "throughout. So I get Vi times,", + "I simply divide by SC throughout.", + "Here is Vi. I have one by SC,", + "one by SC plus R. Light bulbs beginning to go", + "off?", + "", + "The form we have here is 1/SC, some function of my capacitance", + "divided by something connected to my capacitance plus R.", + "This is Vi multiplied by something connected to", + "capacitance divided by something connected to capacitance plus R.", + "And remember your circuit.", + "", + "What is that reminiscent of? What does that remind you of?", + "Voltage divider? Hmm.", + "There is some voltage divider thing going on here.", + "I just cannot quite pin it. It is something about the", + "capacitor, capacitor plus booster, some voltage divider", + "thingamajig happening here. We will try to figure that out.", + "What I will do is replace those terms with something called Zc.", + "Zc plus Zr. If I can find out the Zr and Zc", + "somehow, I can write down the Vp by inspection by the voltage", + "divider action, by some generalization of the", + "good old Ohm's law that I know about.", + "Let's proceed further and see if we can make some kind of a", + "connection between this and this.", + "If I can make the connection then boom, I'm done.", + "I will just use voltage dividers and I am home.", + "", + "OK, so let's play around and see.", + "There is something in there. By now you should know that we", + "are very close. There is something going on in", + "there. I just need to get that spark.", + "I just need to make that spark so I can bridge the gap between", + "something that is really easy versus where I am.", + "Let's take a look at the resistor.", + "", + "I have my resistor with the voltage vR across it and a", + "current iR. Remember to get to any sort of", + "steady state you are going to be dealing with the drives of the", + "form vI e to the j omega t, exponential drives.", + "And by taking the real part, I know I get the input,", + "and the real part of the output gives me the actual output.", + "Let's say my iR is simply Ire^st and my vR is Vre^st.", + "The S is, again, a shorthand notation for j", + "omega. If my current Ire^st of the", + "exponential form shown there and here is Vr, I need to find out", + "what relates Vr and Ir for the element relationship for the", + "resistor to hold. In general, Ir and Vr are", + "complex numbers. For the resistor,", + "I know that Vr=RIr. And I substitute using my", + "complex drives here. So it is Vre^st=RIre^st.", + "I am just substituting for these drives,", + "Ohm's law should apply, and I cancel off e^st.", + "And so I get Vr=RIr. Interesting.", + "For the resistor I find that, based on the fundamental", + "principles of resistor action, the complex amplitude of the", + "voltage simply relates to the complex amplitude of the input", + "by the proportionality factor R. In other words,", + "for the resistor -- Just as the time domain V and I", + "were related by the proportionality constant R,", + "the complex amplitudes Vr and Ir are also related in the same", + "way. That's interesting.", + "Now let's look at the capacitor.", + "", + "Some current ic flowing through it and a voltage vc.", + "Let's say the current is Ice^st and the voltage is Vce^st.", + "Let's plug these into the element law for the capacitor", + "and see if we can find out a way of relating vc and ic.", + "I know that ic is simply Cdvc/dt.", + "So I replace this with Ice^st=Cd/dt(vce^st),", + "which is simply Ice^st=CsVce^st.", + "So I can cancel this out again. Interesting.", + "Ic=CsVc. Very interesting.", + "What is interesting here? Notice that in the time domain", + "Ic=Cdvc/dt, the element law for the capacitor.", + "So I said let's use exponential drives, Ice^st,", + "Vce^st, that's an exponential drive, and try to find out what", + "the relationship between the complex amplitudes are.", + "I plug them and what do I find? I find that if my input is", + "Vce^st, and Vc is the amplitude of the input,", + "then the current is simply given by something multiplied", + "Vc. It's very similar in form to", + "what I saw here. The resistor,", + "Vr=RIr. For the capacitor,", + "Vc=Ic/sc. 1/sc kind of plays the role of", + "R. In other words,", + "the complex amplitudes around the capacitor are related by Vc", + "equals some constant times Ic. Almost like a funny Ohm's law", + "kind of relationship where Vc and IC are complex amplitudes.", + "For the inductor it is the same way, iL, vL and L.", + "Let's say iL=Ile^st and vL=Vle^st.", + "Substitute the values for the inductor into its element", + "relationship as well. I know that vL=LdiL/dt.", + "Therefore, substituting the complex amplitudes is L.", + "And diL/dt will simply be Ilse^st.", + "So I cancel out the exponentials.", + "The reason we're able to do all of this is simply the remarkable", + "beauty of exponentials. Exponentials are absolutely", + "stunningly beautiful. The reason is that when I", + "differentiate them what I get back is the exponential times", + "some constant, and the constant was in its", + "numerator multiplying t. And that's the beauty of", + "exponentials. If this was a sine then I would", + "get cosine and a sine. With exponentials these cancel", + "out and what I am left with is something that is LsIl.", + "Again, for the inductor, the voltage across the inductor", + "relates to some constant Ls here times Il.", + "This is absolutely stunning and almost looks like a form of", + "Ohm's law here. What I am going to do is let's", + "give this the name Zr. Let's give this 1/sC the name", + "Zc. And let's give this the name", + "ZL. It kind of behaves like a", + "resistor, so the resistor simply becomes Zr.", + "And 1/sC behaved like a resistor so I called it Zc.", + "And this is a ZL. These are called \"impedances\".", + "", + "In other words, for a capacitor,", + "as far as complex inputs and outputs are concerned,", + "if Vc and Ic is fed to it, the capacitor can be replaced", + "by an impedance Zc where I can write the relationship between", + "Vc and Ic as Vc=ZcIc. Where Zc is simply one by sc.", + "Similarly, for an inductor --", + "", + "-- I can write its impedance ZL as sL and I get Vl=ZLIl.", + "And finally for a resistor it is pretty simple.", + "", + "What I am saying is that if I am in the region of the", + "playground, if I constrain myself in the region of the", + "playground where my inputs are something Vi e to the j omega t", + "or exponentials, in that little region of the", + "playground now, I am focusing more and more on", + "small parts of the playground so I am kind of boxed in right now.", + "In that region of the playground this applies.", + "In that region of the playground, I can replace", + "resistors by impedances, capacitors with impedances of", + "value 1/sC. And within that playground the", + "beauty of analysis there is that in that region of the playground", + "where the inputs are of the form Vie^st, it turns out that the", + "element laws are simply generalizations of Ohm's law.", + "That is absolutely stunning. It is one of the biggest", + "hallelujah moments in learning circuits.", + "This is really big. And I think this is almost as", + "big as the realization that you can take a nonlinear circuit,", + "operate it at a given operating point, and you can sit around", + "doing Zen things, looking at small perturbations", + "in there, those are going to be linearly related.", + "This is one of the big hallelujah moments in 6.002.", + "And this is of the same magnitude as the small signal", + "response being linear. It is something that is", + "completely non-intuitive. It is something that you just", + "would not have known until you had seen it happen.", + "The same way here. This is very important so I", + "will repeat it again. I have boxed myself into this", + "small region of the playground where all I care about are", + "sinusoidal inputs and steady-state responses.", + "So there I focus on complex inputs, Vi e to the j omega t.", + "And I have just shown you that I can replace inductors,", + "capacitors, resistors with their impedances.", + "And the amplitudes of the corresponding signals around", + "them are related by just a simple Ohm's law like", + "relationship using impedances. I am sort of boxed into this", + "playground, right? In my playground it is all", + "about e to the ij omega t. e to the ij omega t is implicit", + "everywhere. I just don't show it.", + "If I want to talk to somebody else outside but within MIT in", + "this small region, it's all e to the ij omega t in", + "there. If I want to talk to somebody", + "outside, get out of MIT, get out of this playground,", + "what else do I have to do? I have to take the real part.", + "Don't forget that. Remember that,", + "take for example Vc here, so Vc is this,", + "so implicit in all of this is that if I measure Vc at some", + "place it is really going to be Vce to the j omega t.", + "And if we the cosine, the real part,", + "then I have to take a real part of this.", + "And the real part of that would Vc cosine of omega t angle Vc.", + "This piece here kind of goes unsaid.", + "We will agree that we have to do it, but we just skip that", + "step because it is obvious. We just deal with Vcs and Vls", + "now. So a new notation certainly", + "sneaked by you, and that notation looks like a", + "big letter and a small letter. Remember you have seen vL,", + "this is the total behavior, you have seen vl,", + "that's a small signal behavior, and now you see this,", + "Vl, capital V small l. And we also have DC,", + "we have labeled operating point values as VL,", + "capital V, capital L. We have one thing left so", + "nobody go out there inventing something new because we would", + "be in trouble. This is capital V,", + "small l, and this is simply \"complex amplitude\" in the small", + "boxed region of my playground where good things happen and", + "exponentials fly. Whenever someone gives you a", + "variable, capital V, small l, remember it's a", + "complex amplitude, a complex number,", + "and you know how to get to the time domain from there.", + "You take that number, take the real part,", + "multiple the number by e to the j omega t and take the real", + "part, which is tantamount to magnitude cosine omega t plus", + "angle of that number. Actually, you know what?", + "Let's send this up.", + "", + "Back to an example.", + "", + "Oh, I'm sorry. I'm sorry.", + "This is not good. This is my time domain circuit.", + "Remember this was my time domain circuit.", + "A vI input. A vC output.", + "I wanted to analyze this. What I am telling you now is", + "let's box ourselves in this impedance playground.", + "And in the impedance playground the input becomes the complex", + "amplitude of the input, my resistance gets replaced by", + "a box Zr, my capacitor gets replaced by a box Zc.", + "And the voltage I care about here is Vc.", + "Zr = R and Zc=1/sC. Now, there we go.", + "I can write down Vc using a voltage divider action as Vc is", + "simply Zc/(Zc+Zr), done, times Vi of course.", + "And that gives me 1/sC divided by 1/sC+R and multiplying", + "throughout by sC I get 1/1+sCR where S is j omega.", + "Just cannot get any simpler. How long did I take to do this?", + "30 seconds. Where I spent a whole lecture", + "on Tuesday grinding through first trig, giving up halfway", + "and collapsing, and then showing you the sneaky", + "path which was still pretty painful, but 30 seconds,", + "boom. This stuff is spectacularly", + "beautiful. The really cool thing here is", + "that in this impedance domain for linear circuits all your", + "good old tricks apply. Your Thevenin,", + "your Norton, your superposition,", + "name it and it applies for this linear circuit.", + "If you close your eyes and make believe that Zr is like an R and", + "simply apply all the techniques you have learned so far in this", + "linear playground. Just a little hack at the end", + "where this is the complex amplitude.", + "And if you want to go to the time domain part then you do the", + "usual thing. Modulus Vc cosine omega t plus", + "angle Vc. Just remember that.", + "That's the jump to get back to the time domain.", + "Just to show you that this not just works for one little", + "rinky-dink circuit here, let me take a more complicated", + "circuit. If I believe in my own BS,", + "I should be able to apply this theory to my series RLC,", + "the big painful circuit that we did differential equations for", + "about a week ago. Let's do it.", + "", + "I have an inductor, a capacitor and a resistor.", + "What I am going to do is replace this with the impedance", + "model. Input Vi.", + "Let's say this was vI. Let's say I cared about vR.", + "L, C and R. The impedance model would", + "simply be Vi. What's the impedance of an", + "inductor? SL.", + "And for the capacitor it is 1/sC.", + "And for a resistor it is simply R.", + "And just remember, if I can find out VR then for", + "an input cosine of the form Vi cosine omega t the output will", + "given by |Vr| cosine of omega t plus angle Vr.", + "Just remember this last step. But Vr itself is trivially", + "determined. It is the voltage divider", + "action again times Vi. And the voltage divider action", + "is in the denominator I sum these thingamajigs,", + "so ZL+ZC+ZR, ZR in the numerator.", + "And Zr is simply R. ZL is sL.", + "Zc is 1/sC. And R is R.", + "Vi. And I multiply through by,", + "in this particular situation, by s/L.", + "I want to get it into the same form as you've seen before.", + "Multiply throughout, the numerator and denominator", + "by s/L, what do I get? I get RS/L and out here I end", + "up getting S squared plus 1/LC, and I get plus R/L S.", + "I am done. Look at that.", + "Well, a little more than 30 seconds.", + "Maybe a minute. What is this?", + "Where have you seen this before?", + "The denominator of this expression here?", + "Ah, characteristic equation for the RLC.", + "Remember I promised you in the beginning that when we come to", + "the end of the day using a simple one-minute expression I", + "am going to write down the characteristic equation?", + "Boom, here is what I get. Did somebody hear an echo in", + "there? Notice that just by doing a", + "simple voltage divider thingamajig, I got this", + "expression. And now I can write down the", + "frequency response by replacing s is equal to j omega.", + "Even more beautiful and what is even more stunningly pretty here", + "is that remember the intuitive method I taught you about?", + "The characteristic equation gives you alpha,", + "omega nought, omega d and Q.", + "And based on those we can sketch even the time domain", + "response. Guess what?", + "RLC circuits are pass\u00e9 now. You can just write this thing", + "down and you're done, 30 seconds or less.", + "No DEs, no trig, no nothing.", + "OK." + ] +} \ No newline at end of file diff --git a/subs/Nijya-QJ45Y.srt.sjson b/subs/Nijya-QJ45Y.srt.sjson new file mode 100644 index 0000000000..bbea2f2cd9 --- /dev/null +++ b/subs/Nijya-QJ45Y.srt.sjson @@ -0,0 +1,1934 @@ +{ + "start": [ + 0, + 11000, + 14680, + 20333, + 27431, + 34267, + 41425, + 45859, + 49288, + 53722, + 59076, + 64063, + 68760, + 74274, + 77542, + 83566, + 90000, + 99000, + 106000, + 109400, + 114900, + 119000, + 125000, + 129693, + 136617, + 141780, + 148000, + 153230, + 156845, + 160078, + 163692, + 166451, + 171872, + 176532, + 183000, + 187372, + 190729, + 195102, + 197600, + 202207, + 206891, + 211000, + 218000, + 222837, + 226911, + 230900, + 236077, + 241000, + 244540, + 249045, + 252344, + 257091, + 261597, + 264816, + 267551, + 270850, + 276000, + 279918, + 283562, + 286037, + 289131, + 292981, + 296487, + 300423, + 305114, + 307850, + 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3018587, + 3023234, + 3028000 + ], + "text": [ + "", + "All right. Let's get started.", + "I guess this watch is a couple minutes fast.", + "First a quick announcement. In case you have forgotten,", + "your lab notebooks are due tomorrow with the post-lab", + "exercises for the first lab. OK, so I am going to continue", + "with amplifiers today. And to just give you a sense of", + "where we headed, we have this five lecture", + "sequence covering different aspects of amplifiers with", + "dependent sources and showed how we could build an amplifier with", + "it on Tuesday. Today I am going to show you a", + "real device that implements a dependent source.", + "And then next Tuesday we will talk about analysis of an", + "amplifier. Wednesday is our quiz.", + "Thursday and the Tuesday after that we then talk about small", + "signal analysis and small signal use of the amplifier.", + "Today we will talk about the MOSFET amplifier.", + "", + "So let's start with a quick review.", + "And in the last lecture, I showed you that I could build", + "a amplifier using a dependent source.", + "", + "And a dependent source worked as follows.", + "Let's say I had a circuit and I connected a dependent source", + "into the circuit. Let's say in this example I", + "have a current source. So this is some circuit.", + "And the current i is a function of some parameter in the", + "circuit. That's why this is a dependent", + "source. This is a dependent current", + "source. So it could be that I have some", + "element inside. And I measure,", + "I sample the voltage across the element or between any two", + "points in the circuit. And, in this little example", + "here, this current could be dependent on that voltage.", + "So notice that although I showed you the two terminals of", + "the dependent source that carried a current,", + "there is another implicit port, another implicit terminal", + "there. And that terminal there is", + "called the \"control port\" of the dependent source at which I", + "apply a voltage or current that will control the value of the", + "current source. As a quick aside.", + "", + "There is a small glitch with the tools in your tool chest.", + "We talked about the superposition technique where", + "you were taught to turn on one source at a time,", + "for a linear circuit one source at a time, and then sum up the", + "responses to all the sources acting one at a time.", + "Well, what do you do about dependent sources?", + "A dependent source is a source. And we have to modify the", + "superposition statement just a little bit.", + "And for details you can look at Section 3.5.1 of your course", + "notes on the details and some examples on how to do this.", + "So the approach is very simple, actually.", + "The approach is, for the purpose of", + "superposition, to not treat your dependent", + "source as sources that you turn on and turn off.", + "So what you do is when you do superposition with dependent", + "sources simply leave all your dependent sources in the", + "circuit. Just leave them in there and", + "turn on and off only your independent sources.", + "So look at the response of the circuit by turning on your", + "independent sources one at a time and summing up the", + "responses. And your dependent sources stay", + "within the circuit and simply analyze them as you do anything", + "else. So essentially what it says is", + "that just be a little cautious when you have dependent sources,", + "but the basic method applies almost without any change.", + "The readings for today's lecture are Section 7.3 to 7.6.", + "", + "So since we are going to build up on the dependent source", + "amplifier, let me start with a quick review of that amplifier.", + "We built our amplifier as follows.", + "", + "We connected our dependent source in the following manner.", + "And the current through the dependent source in the example", + "we took was related to an input voltage vI.", + "", + "So some voltage vI. And so these two were the", + "control port of the dependent source and a vI was applied", + "there. And I showed you a simple", + "amplifier built with a dependent source that behaved in this", + "manner. And again I will keep reminding", + "you, just remember that the dependent source is actually", + "this box here, the control port and the output", + "port. And commonly we don't", + "explicitly show the control port for those dependent sources for", + "which the control port does not have any other affect on the", + "circuit, like it doesn't draw any current or things like that.", + "So in this particular example we said that this behaved in the", + "following manner for vI greater than or equal to 1 volt and iD", + "was zero otherwise.", + "", + "So we can analyze the circuit to figure out what vO is going", + "to look like. And a simple application of KVL", + "at this loop here, again, you know,", + "when I say this loop here, I am pointing at something", + "here. That is the VS source that is", + "implicitly across these two nodes.", + "Again, this is a shorthand notation where this little up", + "arrow here implies that I have a voltage source connected between", + "these two terminals here. And so there is a loop here", + "that involves VS. So Vo is simply VS minus the", + "drop across this resistor. So it's VS minus the drop", + "across this resistor gives me vO.", + "And the drop across the resistor is simply iD RL.", + "iD is the current here and that's the drop across the", + "resistor. And I could get the explicit", + "relationship of vO versus vI by substituting for iD as vI minus", + "one all squared. So vO relates to vI in the", + "following manner. Nothing new so far.", + "I have pretty much reviewed what we did the last time.", + "Here is where we take our next step forward with some new", + "material. Up to now I have talked as a", + "theoretician would where I said just imagine that you had", + "spherical cow or something like that.", + "Here I just asked you to imagine this ideal dependent", + "source, control port and an output port, and it behaved in", + "this manner. So as a next step what I would", + "like to do is show you a practical dependent source which", + "turns out to be a little bit more complicated than this", + "idealized dependent source that I showed you in many dimensions.", + "Real life tends to impose a bunch of practical constraints", + "on you, and we will look at those in a second.", + "If I could find a dependent source that looked like this --", + "", + "We had a control port A prime and output port B prime.", + "And I looked at some examples where the current through the", + "dependent current source was some function of the input", + "voltage. This is a \"voltage controlled", + "current source\". What I am going to do is talk", + "about a device that can give me this behavior or some close", + "approximation to it. It turns out that under certain", + "conditions the MOSFET that you have already looked at behaves", + "in this manner. The MOSFET that you've seen", + "sort of behaves like this. And let me show you under what", + "conditions the MOSFET behaves in that manner.", + "Let me create some room for myself.", + "Notice that I need a control port, needed an output port.", + "And I am going to view my MOSFET in a slightly different", + "manner than you have seen before.", + "I draw these two terminals here.", + "And this was a three terminal MOSFET.", + "This was my drain, my gate and my source terminal.", + "It was a three terminal device, but what I do is I view the", + "MOSFET slightly differently. I will just use this terminal", + "to be common across both the gate and the drain.", + "And so this voltage here is vGS.", + "I am just using the source port, the source terminal along", + "with the gate as a terminal pair.", + "I am using the same source along with the drain as another", + "terminal pair. So I have a vDS out there and I", + "have some current iDS that flows out here.", + "Notice that when I view the MOSFET in this manner I have", + "accomplished my first step, which is I seem to have a box", + "which has a port here and a port here.", + "And I also explained to you that a MOSFET behaves in a", + "particular manner. For one, the output port", + "behaved as an open circuit under certain conditions when --", + "This was vGS, G, drain and source.", + "When vGS was less than a threshold voltage VT this MOSFET", + "had an equivalent circuit that looked like this.", + "So when vGS was less than some threshold voltage VT then there", + "was an open circuit between the drain and the source.", + "And you saw this before. So far nothing new here.", + "However, when vGS is greater than or equal to VT --", + "", + "vGS was greater than VT. The MOSFET behavior we looked", + "at earlier showed that this behaved either like a short", + "circuit in the simplest form or in a slightly more detailed form", + "it behaved like a resistor. We call that the SR model of", + "the MOSFET. So when vGS was greater than VT", + "we said that a simple way to approximate MOSFET behavior was", + "to view this as a resistor connected between the drain and", + "the source. That was our SR model use of", + "the MOSFET. It turns out that we kind of", + "lied. We were sort of looking at the", + "MOSFET in a really funny way. And I shone the light on the", + "MOSFET in a really, really clever way.", + "Well, I shouldn't say clever. A really, really tricky way.", + "And tricked you into believing that it was just a resistor.", + "And we constrained how you use the MOSFET.", + "So that behavior was indeed a resistive behavior.", + "But it turns out that in real life the behavior of the MOSFET", + "between the drain and the source terminals is much more", + "complicated than the limited form in which you saw it.", + "So today what I am going to do is take the wraps off the", + "complete MOSFET and show you its full behavior in all its gory", + "glory. And I will spend a bit of time", + "on that to clearly emphasize under what conditions the MOSFET", + "behaves like a resistor, as you saw when you did digital", + "circuits, or behaves differently in other domains of use.", + "Let me pause for a second and leave this space blank here.", + "And let's do some investigations.", + "", + "Let me leave this here. I won't draw in anything yet.", + "You will figure out what it looks like yourselves under", + "certain conditions. What I will do next is apply", + "some voltages on a MOSFET and observe the current versus vDS", + "behavior and plot that on a scope and take a look at it.", + "What I am going to do --", + "", + "-- is figure out what iDS looks like for --", + "", + "Remember iG into the gate for 6.002 is always going to be", + "zero. In much more detailed analyses", + "of the MOSFET, in future courses you may see", + "slightly more complex behavior. But as far as we are concerned", + "it is an open circuit looking into the gate.", + "So I am going to apply a vGS across the MOSFET,", + "apply a vDS across the MOSFET and plot iDS versus vDS.", + "First let me show you what you already know.", + "What you already know --", + "", + "This is vDS. I will just keep doing as much", + "as I can of what you already know.", + "And then when I do some new stuff I will tell you", + "explicitly. You've seen this before.", + "The MOSFET behaves like an open circuit when vGS less than VT.", + "That is when vG is less than a threshold voltage VT,", + "I have zero current flowing through the MOSFET.", + "And when vGS was greater than VT then the S model of the", + "MOSFET the switch model simply said that look,", + "we can model the D2S as a short circuit.", + "You saw this in your labs and you saw that it was a very,", + "very small resistance between the drain and the source and it", + "kind of looked like a short circuit.", + "But then we said well, that's not quite it.", + "There is some resistance. And so we said a slightly more", + "accurate model would have this line droop a little bit to imply", + "that there was some resistance R_on between the drain and the", + "source, so vDS iDS. So this was when vGS less than", + "VT and vGS greater than or equal to VT.", + "I have some resistance. And that showed me a straight", + "line kind of like behavior. And I showed you that behavior.", + "So far absolutely nothing new. Now what I have plotted there", + "for you is that behavior. Up here notice that this is the", + "vDS axis, this is the iDS axis. I am plotting iDS versus vDS.", + "And when vGS -- The gate voltage is more than a", + "threshold, notice that I see what looks like something more", + "or less like a straight line. And this is a straight line", + "with some slope, more or less a straight line", + "implying resistive behavior. And we also had some fun and", + "games here. We said hey,", + "what if I turn vGS off? Boom.", + "That would be my iDS of zero implying that the MOSFET behaved", + "like an open circuit between the drain and the source.", + "I applied a positive vGS more than VT and it began to look", + "like a resistor. Open circuit,", + "resistor, open circuit, resistor, OK?", + "Up until now nothing new. So you shouldn't have learned", + "anything at all that is new until now in today's lecture.", + "Now watch. What I am going to do is,", + "as I said, I kind of lied all this time and I just showed you", + "this behavior. And what I have been doing all", + "along is very carefully using a very small value of vDS.", + "Notice it's a small values of vDS.", + "I haven't told you what it looks like as vDS increases.", + "Well, let's go try it out. We have a scope here.", + "We have the MOSFET here. Now, I am not sure what is", + "going to happen now. You may see smoke or have an", + "explosion, who knows what? But look up there for a second.", + "I am just going to increase vDS and you can figure out what", + "happens for yourselves. I increase vDS.", + "Whoa, what a liar. Agarwal is a liar.", + "I have been kind of tricking you.", + "I have been putting -- Covering up all this part here", + "and showing you just this region of the curve for small values of", + "vDS. But as I increase vDS this is", + "nothing that looks even close to that of resistive behavior.", + "So what's happening here? What's happening is that as I", + "increase my vDS the iDS curve tails off and saturates at some", + "value of current. Notice it saturates at some", + "value of current. And so I am going to look at", + "this region of behavior. Notice that what we have looked", + "at so far was the behavior for small vDS.", + "It kind of looks resistive. But when I pump up the vDS,", + "really whack this node really hard with a much larger vDS the", + "guy says, oh, I give up.", + "And the current saturates out and flattens out and holds the", + "value steady at some value. So what's that behavior look", + "like? What is my horizontal line", + "above the X axis in terms of V I elements?", + "What is that behavior like? Current source,", + "exactly. So this is current source like", + "behavior. And so let me start by drawing", + "you a little model and explaining it in more detail.", + "What happens is that under certain conditions,", + "and the conditions are the following, when vDS,", + "that is my drain to source voltage is greater than or equal", + "to vGS minus VT. When my drain voltage goes", + "above vGS minus VT, so if vGS is 3 volts and if VT", + "is 1 volt, then if vDS goes above 2 volts,", + "if I am hammering the drain of the MOSFET with a higher voltage", + "then this guy says I give up, can't show you nice restive", + "behavior, and the current saturates out and it doesn't", + "allow you draw any more current than a maximum value.", + "And that's the current source behavior.", + "This one behaves like a current source.", + "And the current iDS is given by the following expression.", + "", + "The current is given by iDS is equal to a constant K divide by", + "two times (vGS-VT) all squared. Kind of reminiscent of the", + "carefully chosen dependent source example,", + "just that this one here is VT. This model, which applies when", + "vGS is greater than VT, the MOSFET has to be on and the", + "drain to source voltage in the MOSFET must be larger than some", + "value, and that value is vGS minus VT then this guy begins to", + "behave like a current source. This model of the MOSFET is", + "called the \"switch current source model\".", + "", + "So in the region of the MOSFET characteristics where vGS is", + "greater than VT and the drain to source voltage is larger than", + "vGS minus VT, the MOSFET behaved like a", + "current source between its drain and source terminals.", + "And in that part we model the MOSFET as a current source.", + "And so not surprisingly that part of the model is called the", + "SCS model in contrast with the SR model where we had a", + "resistor. Again, remember,", + "this is not meant to be conflicting.", + "It is not like gee, how can the MOSFET look like a", + "resistor, and then suddenly what happens it becomes a current", + "source. Well, the two regions are", + "different. It is not that it is behaving", + "as a current source for the same parameters, no.", + "When vDS is less than this right-hand side it does behave", + "resistive. The SR model applies.", + "But increase vDS beyond a point, the current saturates and", + "the SCS applies like so. So let's draw.", + "The SCS behavior can be drawn here vDS and iDS.", + "As I mentioned to you, for small values of vDS,", + "let's say I pick some value of vGS, let's say vGS3,", + "some value vGS, it is going to look like a", + "resistor until vDS becomes equal to vGS3 minus VT.", + "And after that it saturates out and begins to look like a", + "current source. And this point is where vDS", + "becomes equal to vGS minus VT. And this way is when this equal", + "sign becomes a greater than sign, vDS becomes larger then I", + "move into this part of the curve.", + "Similarly, for various other values of vGS it will look like", + "this --", + "", + "-- and so on. And it behaved like an open", + "circuit as before when vGS less than VT.", + "When vGS less than VT it is still behaving like an open", + "circuit. And so as I increase my vGS,", + "provided I keep my vDS greater than vGS minus VT,", + "I get current source like behavior.", + "And notice that this is increasing vGS.", + "I have purposely drawn these curves at greater distances from", + "each other to imply that it is a nonlinear relationship in that", + "if I increase vGS by some amount, the increase in vDS is", + "related to the square of vGS. It is vGS minus VT all squared.", + "So I get a family of curves of that look like this.", + "And this is in the region of operation where vDS equals vGS", + "minus VT. And this applies in this regime", + "where vDS less than vGS minus VT.", + "This region of operation is called, as you might expect,", + "the \"saturation region\".", + "", + "We say the MOSFET has been hammered, the MOSFET has been", + "walloped, the MOSFET is in saturation.", + "So the MOSFET is in saturation. This region,", + "corresponding to this, is called the triode region.", + "", + "This is really very simple. All we are doing is saying that", + "when vDS is increased beyond a certain limit,", + "given my vGS minus VT, the MOSFET begins to behave", + "like a current source. It cannot draw any more", + "current. It limits the current to a", + "given value like a current source.", + "But on the left-hand side of this it behaves in a resistive", + "manner. So what I would like to do is", + "--", + "", + "What I will do is, we've plotted for you,", + "for the MOSFET, all its characteristics in its", + "full glory for a whole bunch of values of vGS and a whole bunch", + "of values of vDS. And let me stare at those", + "curves with you for a few seconds and walk you through", + "them. So what do I have here?", + "One of these curves corresponds to a given value of vGS.", + "This may be vGS equals 2 volts. This is vDS,", + "the drain to source voltage, and this is the current.", + "So focus on this curve for now. In the beginning I hid the", + "right-hand side behavior from you and showed you just the", + "resistive behavior out here. When I increase vDS to be much", + "larger the curve saturated and I got the saturation region", + "operation of the MOSFET. And notice as I increase my", + "value of vGS the saturation current also increases according", + "to a square law behavior. So these are the entire curves", + "of the MOSFET. Finally the truth comes out.", + "And notice that when vDS is less than vGS minus VT,", + "I have more or less resistive behavior.", + "But when vDS is greater than vGS minus VT I get current", + "source like behavior. So one question you may ask is", + "when do I use one model or the other?", + "When do I use the SR model and when do I use the SCS model?", + "If you want to do a real detailed analysis then you can", + "use the SR model when vDS is less than vGS minus VT.", + "And you would use this model when vDS is greater than or", + "equal to vGS minus VT. That is simple enough.", + "In 6.002, to eliminate confusion we constrain how we", + "look at things a little bit more stringently.", + "And what we do is that for our entire digital analysis,", + "for the entire digital world we focus on the SR model.", + "And I will tell you why in a second.", + "So for all digital circuits, invertors, look at power of", + "invertors, look at delay, a bunch of other things,", + "we will be using the SR model in 6.002.", + "And I will tell you why in a second.", + "And for analog -- That is for amplifier designs", + "and situations like that, we will be operating the MOSFET", + "in a saturation region. And I will talk about that in a", + "second. What I am saying here is that", + "in 6.002, when we do analog designs, we are going to", + "discipline ourselves to using the MOSFET only in this region.", + "We are going to constrain ourselves to play in only this", + "region of the playground where vDS is quite large.", + "Why? Because I am asking you to.", + "I am saying let's play in that part of the playground and keep", + "your vDS high. And so the MOSFET is going to", + "be operating somewhere in here. So we can apply just the SCS", + "model, just the current source behavior in that region.", + "There is another important reason, which I will get to in a", + "second. And for digital designs we will", + "simply use the SR model. And it turns out that this is", + "realistic because in the digital designs that you have you seen", + "and will be seeing in this course, the pull down MOSFET is", + "on, or when these pull down MOSFETs are on,", + "the output voltage is pulled down close to ground.", + "So vDS is very, very small.", + "So it does make sense that this model apply.", + "And when we talk about amplifiers, I am asking you to", + "follow this discipline. I will tell you why in a", + "second. I am saying analog designs", + "follow this discipline that I call the saturation discipline.", + "It says simply operate the MOSFET operating in saturation", + "as a current source. We will look at an amplifier in", + "a second, and I will tell you why.", + "", + "Now let's do a MOSFET amplifier.", + "Remember my amplifier had an input port and an output port.", + "And in general in our use we are going to have a common", + "ground. And we have a VS and a ground", + "here as well. That is the power port of the", + "amplifier. The input port and the output", + "port.", + "", + "And let me redraw the circuit putting a MOSFET in place of the", + "current source, RL, VS, vO, drain,", + "gate, source, vI.", + "So my input is vI. Again, the MOSFET output is vO.", + "And I have a resistor RL. Hey, we've seen that before.", + "It turns out this is not surprising.", + "You've seen this before. This was our primitive inverter", + "circuit. So what's different here?", + "We showed you the circuit as an inverter.", + "What's different here is that when we look at MOSFET behavior", + "as a current source, this behaves like an amplifier.", + "In other words, when vDS is greater than some", + "value then this behaves like a current source.", + "When vDS is small, in other words,", + "in the digital design when vDS was small here,", + "because when the MOSFET was on it pulled the voltage down to", + "ground, we could view this behavior as a resistor.", + "And exactly the same thing, it is an amplifier.", + "And with digital designs, I was driving it with 5 volts", + "and 0 volts and that was it, rail to rail.", + "As an amplifier, what I am doing now is looking", + "at a small region of its behavior when vDS is greater", + "than vGS minus VT. What I am saying is that for", + "amplification let's follow the saturation discipline.", + "And the reason is that when this behaves like a current", + "source, what I have shown you is that if this behaves like a", + "current source I have shown you that this expression up here", + "gives you amplification. In last lecture we plotted a", + "bunch of values for vO versus vI, and we saw that we were", + "getting amplification. For a small change in vI,", + "I was getting a larger change in vO, and that was when I had", + "the equation for a current source in there.", + "And so we know for a fact that if I can operate this as a", + "current source, with a reasonable choice of", + "values here, I am going to be able to get amplification.", + "What I haven't told you is if this is operated in the linear", + "region, in fact, you do not get amplification.", + "I won't cover that, but you can check that out in", + "your course notes as a discussion or you can try it out", + "for yourself. Replace this with the SR model", + "for small vDS and you can show yourselves that you don't get", + "any amplification. In order to get the", + "amplification we are telling ourselves let's focus on this", + "part of the playground where vDS is greater than or equal to vGS", + "minus VT. And for vGS greater than or", + "equal to VT. So when vGS is greater than VT", + "the MOSFET is on. Further, when vDS is large,", + "larger than vGS minus VT this behaves like a current source.", + "So we have now created a small playground for ourselves where", + "we can build lots of fun little amplifiers and other circuits.", + "And provided our circuits follow the saturation discipline", + "where for the MOSFET or MOSFETs in the circuit these expressions", + "are true then the MOSFETs are going to be in saturation,", + "the current source model applies, and I will be indeed", + "getting saturation. In future courses you may", + "actually see the MOSFET used in other regimes of operation for a", + "variety of reasons. But in 6.002 when we talk about", + "amplifiers and so on we will be adopting the saturation", + "discipline. And your homework problems and", + "so on will state that. Assume that the MOSFETs are in", + "saturation. What that means is that you can", + "begin to model them as a current source and simply analyze their", + "behavior accordingly. One minor nit.", + "Note that vDS for the MOSFET is the same as vO.", + "And vGS for the MOSFET is the same as vI.", + "So if you see me jumping back and forth using vOs and vIs or", + "vDSs and vGSs they are the same thing in this circuit.", + "If you are dealing with circuits with many MOSFETs then", + "you will have vDS1s and vGS1s and so on and so forth.", + "But for this simple circuit, vO and vDS are the same,", + "vI and vGS are the same. So we could go ahead and", + "analyze that circuit. What I do to analyze the", + "circuit, I am telling you this. I am telling you that the", + "MOSFET is behaving in saturation.", + "I am telling you this. We have disciplined ourselves", + "to say that in that circuit the MOSFET is in saturation.", + "As soon as we tell you that we can then go ahead and analyze", + "that circuit. And to analyze that circuit", + "what you will do is simply replace the MOSFET with its", + "equivalent model, and that looks like this.", + "Since you have been told that it is in saturation,", + "we can replace the MOSFET with its current source model.", + "", + "And the current iDS for the MOSFET is given by K/2(vI-VT)^2.", + "And it is always good to write the constraints under which you", + "are implicitly working close by. So the constraints are one,", + "vGS is greater than or equal to VT, vDS is greater than or equal", + "to vGS minus VT. These constraints immediately", + "follow from a statement of the type we are operating under the", + "saturation discipline or the MOSFET is in saturation.", + "Let me just mark this equation as A, and we will refer to it", + "again.", + "", + "So with this new little circuit with the MOSFET working as a", + "current source, let's go ahead and analyze our", + "amplifier. Notice that to analyze the", + "circuit I have a current source. It's a dependent current source", + "where the current depends on the square of the input.", + "So I want to go and analyze it. This is a nonlinear circuit.", + "So I can apply any one of the methods that we talked about", + "last week for nonlinear circuits.", + "To analyze it I will go ahead and use the analytical method.", + "And my goal will be to obtain vO versus vI.", + "Again, remember where are we here?", + "The MOSFET circuit operating in saturation so I can replace this", + "with a current source. It is nonlinear.", + "And so I can apply one of the two methods, the analytical", + "method or the graphical method. Let's do both and start with", + "the analytical method. The analytical method simply", + "says go forth, apply the node method and", + "solve. Simple stuff.", + "Let's go ahead and do that. Node method.", + "I have a single node here that is of interest.", + "I know the voltage vI at this node.", + "I know the voltage VS at this node.", + "So the only unknown is here at vO.", + "So I will go ahead and do that. Let me go ahead and equate the", + "currents into the node to be zero.", + "So the currents out of the node here are iDS.", + "And that was equal the current into that same node.", + "So iDS must equal VS minus vO divided by RL.", + "iDS=VS-vO/RL. For later reference,", + "let me call that B. Simplifying,", + "what I can do is, we know that iDS is given by", + "K/2(vI-VT)^2. So I replace iDS with this", + "expression and I multiply that by RL.", + "So I get K/2(vI-VT)RL. So iDS gets multiplied by RL", + "and I get vO on this side and VS remains out here.", + "All I have done is multiplied both sides by RL.", + "So it is RL iDS, taken RL iDS to this side,", + "that is here, I get the minus sign,", + "and VS stays here, vO comes here.", + "So that is my final expression. Remember this is true under", + "certain conditions. I will keep hammering that home", + "because some of the most common errors made by people is in", + "forgetting the constraints under which this was obtained.", + "And the constraint under which this was obtained is the", + "saturation discipline. And that was true when vGS for", + "a MOSFET was greater than or equal to VT and vDS for a MOSFET", + "was greater than or equal to vGS minus VT.", + "I also know that for vGS less than VT, vO=VS.", + "So when vGS is less than VT then this one turns off.", + "That's why it is the SCS model, switch current source model.", + "When vGS is less than zero it turns off and VS directly", + "appears at vO. I would like to stare at this", + "constraint with you for a second, vDS greater than or", + "equal to vGS minus VT here. And vDS is simply vO.", + "I want to rewrite this constraint in terms of iDS.", + "It will come in handy. So iDS is K/2(vI-VT)^2.", + "This is vI-VT. So vI-VT is simply square root", + "of 2iDS/K. In other words,", + "I can write iDS less than or equal to K/2vO^2.", + "So this constraint expressed in terms of iDS is simply iDS less", + "than or equal to K/2vO^2.", + "", + "So all I've done here is analyzed this nonlinear circuit.", + "I can also analyze it using the graphical method.", + "And in order to do that, for my nonlinear circuit,", + "in order to do that, all I have to do is plot.", + "Let's have iDS here and vDS here.", + "And as we did with a nonlinear expo dweeb, what I do is I plot", + "the device characteristics iDS versus vDS.", + "The device characteristics under saturation look like this,", + "so vGS increasing. iDS versus vDS has a bunch of", + "curves that look like current sources of increasing values.", + "That simply reflects equation A.", + "And then I superimpose on top of that the expression that", + "comes up due to equation B which is iDS equals,", + "let me write that down here, iDS equals VS/RL - vO/RL.", + "That's B. And let me plot that.", + "That is a straight line relationship between iDS and vO.", + "And so when vO is zero iDS is VS/RL.", + "And when iDS is zero vO equals VS.", + "Remember, vO and vDS are the same.", + "So this is what I get. This is the straight line", + "corresponding to equation B here.", + "", + "And, as before, we just find the point where", + "the two intersect. Let's say I am given some value", + "of vGS. And let's say I am given some", + "known value of vDS. So for that I can go ahead and", + "find out the corresponding value of iDS from this graph.", + "Just as I told you when we did the expo dweeb stuff,", + "this line here is called a load line.", + "You will be seeing that again and again and again where we", + "have the equation corresponding to the one shown here,", + "the equation written for the output loop superimposed on the", + "device characteristics. That's called a load line.", + "So I can get this point corresponding to the operating", + "point of the MOSFET for this iDS, vDS and vGS by using the", + "graphical method. In the next lecture we are", + "going to look at, given a device of this sort,", + "how do we figure out the boundaries of valid operation so", + "that the MOSFET stays in saturation?" + ] +} \ No newline at end of file diff --git a/subs/OGtElTMJidE.srt.sjson b/subs/OGtElTMJidE.srt.sjson new file mode 100644 index 0000000000..60216e74e2 --- /dev/null +++ b/subs/OGtElTMJidE.srt.sjson @@ -0,0 +1,1841 @@ +{ + "start": [ + 0, + 4800, + 9523, + 13561, + 16000, + 29000, + 34380, + 39852, + 43044, + 47330, + 52254, + 58000, + 63257, + 68247, + 71990, + 76891, + 81168, + 84465, + 89811, + 94000, + 98776, + 101576, + 106764, + 110470, + 113847, + 118705, + 124184, + 128959, + 132034, + 135352, + 139560, + 143687, + 148219, + 151462, + 156798, + 160412, + 163424, + 168673, + 172287, + 176504, + 181662, + 185657, + 190792, + 196403, + 201919, + 205818, + 212000, + 216500, + 220375, + 226625, + 231750, + 238125, + 244000, + 246615, + 250230, + 255000, + 259384, + 264153, + 269000, + 273889, + 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For today we are going to start", + "with nonlinear analysis. Before we do that I wanted to", + "do a little bit of review.", + "", + "I wanted to give you the past three weeks in perspective and", + "show you how all of these things fit into the grand scheme of", + "things. We began by building a great", + "little playground, and within that playground we", + "said that by enforcing upon ourselves the lumped matter", + "discipline we created the lumped circuit abstraction.", + "So within that playfield we assumed that we had dq by dt and", + "d phi by dt to be 0 so that gave us as the lumped circuit", + "abstraction. And within that lumped circuit", + "abstraction, within this playground we looked at several", + "methods of analyzing circuits, including the KVL,", + "KCL method. We also learned the method", + "involving composing resistors, the voltage dividers and so on", + "and solving circuits intuitively.", + "And we also looked at the node method, which is kind of the", + "workhorse of the circuits industry.", + "So when in doubt apply the node method and it will get you where", + "you want to go. Now, we also said that this is", + "good, here is our playground. We said hey,", + "if we focus on those circuits that are linear we come to the", + "left part of our playground. And we said that for linear", + "circuits in this part of the playground we can further use a", + "couple of techniques, a few techniques,", + "superposition, Thevenin, Norton and so on.", + "So these techniques allow you to very quickly analyze", + "complicated circuits, especially when you're looking", + "to find a single current, or voltage or some parameter of", + "interest. Whenever you see,", + "if you see a circuit containing multiple voltage sources or two", + "or more voltage sources or current sources,", + "as a first step think superposition.", + "And so these are very powerful techniques that let you analyze", + "very complicated circuits very effectively.", + "After we did this we said, oh, let me draw another", + "playground here. This is another piece of our", + "playground. And if these are linear circuit", + "then this half of the playground is nonlinear circuits.", + "And we said that if you further focus on discretized values,", + "if you discretized values and focused only on circuits that", + "dealt with binary signals, highs and lows,", + "then we came into this small regime of the playground.", + "And notice that digital circuits are,", + "by their very nature, nonlinear.", + "Remember the circuit, A, B, this was one of our NOR", + "gate circuits? And if you look at transfer", + "functions, that is if I plot, let's say for example,", + "for some combination of input values.", + "Let's say I plot v in verses v out.", + "Let's say, for example, I turned this guy off by", + "setting B to 0 and then I simply apply a low to high transition", + "at v in, then what I would see at the output is a transfer", + "function of the following sort where as v in changes the output", + "switches at some point and then stays at a low value.", + "So when v in is low v out is high and v in and high v out is", + "low. So that's kind of the v out", + "versus v in when B is set at 0. So notice that this is a", + "nonlinear curve. This is not a straight line.", + "It's a nonlinear curve. And so therefore in the digital", + "domain we see highly nonlinear functions that look like this", + "and so on. However, take a look at this", + "circuit. Suppose I focus on the circuit", + "for a given set of switch settings.", + "Let's say, for example, I focus on the circuit when A", + "and B are both 1s. For a given set of switch", + "settings, notice that I'm going to be either in this region or", + "in this region. Notice that this region is a", + "straight line. So if I focus on let's say both", + "A and B at once then I get something like this.", + "", + "And in this situation, for a given set of switch", + "settings, notice that my digital circuit now can be analyzed", + "using linear techniques. So therefore my digital gets", + "moved into the linear domain for a given set of switch settings.", + "", + "So if I fix my switch settings and look at the circuit then", + "each circuit, for a given set of switch", + "settings, is comprised of voltage sources and some", + "resistors and it's a linear circuit.", + "Again, I can go back and apply all my linear techniques to", + "virtually all the digital circuits that you will be", + "dealing with in 6.002. Again, remember if I fix my", + "switch settings, if I fix the inputs then the", + "output can be determined using linear techniques.", + "Because the digital circuits we're showing you in 6.002", + "simply comprise linear elements like voltage sources and", + "resistors and so on. You'll see some more later.", + "But you can apply your linear techniques and analyze them.", + "The cool thing here is that with just two weeks of stuff", + "that you've learned in 6.002, you are well on our way to", + "being able to analyze certain classes of digital circuits for", + "a given set of switch settings and many, many,", + "many linear circuits. What we will do today is focus", + "on nonlinear circuits. So we look at this space.", + "Notice again that up until now we've dealt with these three", + "methods, which apply to all circuits within this playground,", + "the lumped circuit playground. And the subset of that is the", + "linear domain. And we can analyze linear", + "circuits in this way. And digital circuits,", + "for a given set of switch settings, also fall within this", + "category. So notice that you can go ahead", + "and analyze the digital circuits using superposition or other", + "techniques like that. The next big step for us is to", + "begin our analysis of nonlinear circuits today.", + "The important thing to remember is that nonlinear circuits are", + "also within this big playground in which we are going under the", + "lumped matter discipline. So nonlinear circuits are also", + "lumped circuits. And therefore because we are in", + "that playground we can use any one of our techniques,", + "KVL, KCL or the node method to analyze nonlinear circuits.", + "So if you see a nonlinear circuit, don't get daunted.", + "Just remember this is meant to be simple stuff.", + "So let me simply write down the node equation and analyze it.", + "There is really nothing new in today's lecture.", + "I'm just going to show you a nonlinear circuit and analyzing", + "using techniques that you already know.", + "Today nonlinear circuits. And we look at several methods", + "of analyzing nonlinear circuits. We look at the \"Analytic", + "Method\". We look at a \"Graphical", + "Method\". You will look at a \"Piecewise", + "Linear Method\" in the book.", + "", + "I won't be covering this in lecture.", + "You can read Section 4.4 for the piecewise linear method.", + "In this method you take your curves and you approximate them", + "with a bunch of straight line segments, kind of like the v", + "out, v in curve I've shown you there, and analyze the circuit", + "using linear techniques within any given straight line segment.", + "We will also do incremental analysis.", + "This is also called small signal analysis.", + "So I will cover these two today, I will introduce this one", + "today, and wrap that up during the next lecture.", + "", + "Let's start with a simple example.", + "", + "So I have some voltage, V, some voltage source V.", + "And I have some resistor, R.", + "And I have a fictitious device here that I labeled D.", + "Let's call this fictitious device the \"Expo Dweeb\".", + "I purposely chose a funky name because this is a fictitious", + "device. Let's call it the Expo Dweeb.", + "And let me write down the associated variables for this", + "device as follows. iD is the current flowing into", + "this terminal and vD is the voltage across this device.", + "So this is a nonlinear device.", + "", + "And this device is characterized by the following", + "equation. Much like resistors were", + "characterized by an iV relation, V is equal to iR,", + "or i is equal to V/R. This device is also", + "characterized by the following element relationship.", + "It's a e raised to bvD. So there is an exponentiation", + "here. Again, this is a fictitious", + "device. And I'll show some funky things", + "that it does in a second. It's a very simple relation.", + "It's an exponential relation where the current relates to the", + "exponentiated value of the voltage vD across the element.", + "So I can plot iD versus vD for this element as follows.", + "Notice that when vD is 0 iD is a, so I have a here,", + "and it looks like this. It's a funny device,", + "a fictitious device. So when vD is 0,", + "I have some current flowing the device, and as vD increases I", + "get an exponential increase in the current through that device.", + "This device is funny in the sense that it is not a passive", + "device in that notice that when vD and iD are positive the", + "product is positive, which is fine,", + "which says that it is consuming power.", + "On the other hand, on the left-hand side notice", + "that the vI relation is negative, which means that when", + "I put a negative voltage on it, it can still sustain a positive", + "current. This must imply that the device", + "is producing power. But for the purpose of a", + "nonlinear analysis we don't have to worry about that.", + "Let's just do it mathematically and find out what it looks like.", + "So back to this again. I have a voltage source,", + "a resistor and my Expo Dweeb connected in that manner.", + "Now, again, reflect on this pattern.", + "A voltage source or a current source, a resistor and some", + "device. This is a very standard pattern", + "you will see again and again and again.", + "In particular, if you look at this device,", + "it's a nonlinear device here. And facing the nonlinear device", + "is a voltage source in series with a resistor.", + "And the reason I say that this is an incredibly important", + "pairing is the following. Notice that if on the left-hand", + "side I had any linear circuit and I had a single nonlinear", + "element in that circuit. Notice that by a Thevenin", + "reduction that you've learned you can take this entire mess.", + "If all you care about is the behavior of the nonlinear", + "device, for the purpose of analyzing this nonlinear device,", + "you can take this entire linear circuit, no matter how", + "complicated it is, voltage sources,", + "current sources, resistors and a bunch of other", + "funky stuff, you can boil all of that down to a Thevenin", + "equivalent, a voltage and a resistor in series.", + "So we can trick you. We can give you a complicated", + "circuit and say ah-ha, tell me what the current is", + "through this device if I apply some voltage,", + "3 volts there. What you can do is you can say", + "ah-ha, I don't care what happens here so I'm just going to", + "replace the whole thing with a Thevenin equivalent.", + "And you've done your homework now and you can calculate", + "Thevenin equivalents for circuits.", + "And simply replace this and then go ahead and solve the", + "circuit. Again, remember we are", + "engineers. We are looking for answers.", + "We are looking to build interesting systems.", + "And, in general, we like to take the simplest", + "path possible to the solution. So simplify your lives and", + "create a simple Thevenin coupled to a nonlinear device and then", + "you will be rolling. When we talk about a variety of", + "other circuits, nonlinear circuits,", + "time-varying circuits and so on in the rest of this course,", + "we will look at this pattern again and again and again and", + "again until we are blue in the face.", + "And, just remember, the reason we keep looking at", + "this pattern is that whenever you have some big linear mess", + "connected to some interesting device what you can do is if all", + "you care about is analyzing the behavior of that device,", + "you can take this linear mess and simply figure out the", + "Thevenin equivalent, or the Norton equivalent if you", + "like and replace this whole thing with its equivalent and", + "then go ahead and analyze it. So boil an arbitrarily circuit", + "down to a very simple pattern of this sort.", + "What this means is because of this brilliant Thevenin", + "simplification, going forward through the rest", + "of this course we will mostly deal with very simple circuits", + "like this, voltage source, resistor and the device.", + "That's it. Very, very, very rarely will", + "you see multiple sources and lots of resistors in a circuit.", + "It's usually going to be simple stuff.", + "And remember how we got here, by making a Thevenin", + "simplification of a linear mess. All right.", + "If in homeworks or quizzes or in real life,", + "or in many examples of real life, if you find that you have", + "to deal with a lot of grunge and a lot of mess,", + "step back and think a little bit.", + "Try to use intuition and see if you can simplify things using", + "some clever trick or method. Method 1 of analysis.", + "Let's go ahead and analyze this pattern here,", + "this template circuit, if you will,", + "a voltage source a resistor and a nonlinear device.", + "This is the analytical method. And remember the node method", + "applies, so let me go ahead and apply the node method.", + "To apply the node method, what do I do?", + "I first have to select a ground node.", + "Let me insulate this as my ground node.", + "Let me label all the nodes with their voltages.", + "So this node has voltage V and this node has label the capital", + "D. So let me go ahead and analyze", + "this using the node method. So the node method says for", + "each of the nodes in the circuit whose voltage is not known go", + "ahead and write down KCL implicitly applying the element", + "relationships to replace the current values with the voltage", + "values. Let's start with the current", + "going in that direction. Current going from the vD node", + "through resistor R, which looks as follows,", + "vD - V divided by R. That's a current going that", + "way. And the current going down is", + "iD. In general, when I apply the", + "node method, I don't write iD here but I go ahead and write", + "the element relation ae to the bvD here.", + "Then I get an equation in vD and I just solve the mode", + "voltage. However, just to make a couple", + "of extra points later, let me go ahead and do that in", + "two steps, write down this and then go ahead and write down iD", + "separately as ae to the bvD. Again, remember,", + "don't get confused here. In a node method,", + "I don't write down a second step.", + "I directly write down ae to bvD in place of iD.", + "I get one equation in vD, I go solve it.", + "Just for fun today, I'm taking two steps here,", + "writing iD and explicitly putting down iD as ae to the", + "bvD. Now, that's it.", + "I mean this is all there is to it.", + "You guys can now go ahead and analyze nonlinear circuits.", + "You get a bunch of equations, a bunch of unknowns,", + "go solve. I have two equations here.", + "vD and iD are my unknowns and I can just go ahead and solve for", + "them. Now, in general with nonlinear", + "circuits, often times it's hard to get a closed form solution so", + "you may have to use a bunch of methods.", + "You can try a closed form solution or you can try", + "numerical solutions or you can do trial and error.", + "In this case, I'll just go ahead and tell", + "you. Suppose I choose V as 1 volt,", + "R is 1 ohm and b is 1 over volt and a is \u017a amps for those", + "values, approximately vD is roughly 0.5 volts and iD is", + "roughly 0.4 volts. You can do this by using trial", + "and error or other methods. In 6.002 we don't dwell on", + "working too hard to solve equations of this sort.", + "If you cannot substitute this in here and solve it directly,", + "we don't ask you to go and learn numerical method and the", + "techniques and so on to solve it.", + "But just remember that you can use trial and error or you can", + "use back substitution and other techniques that you will learn", + "in future numerical methods classes and apply it here.", + "But suffice it to say that, for here we can stick with", + "trial and error if you like. And for these values,", + "vD and iD are 0.5 and approximately 0.4.", + "You're done. It's really that simple.", + "Yes. Oh, I'm sorry.", + "Good catch. I know there is one person", + "that's not sleeping here. Good.", + "So, as I said, there's not a whole lot to it.", + "Whether it's a nonlinear circuit or a linear circuit and", + "as long as I am inside this playground here where the lumped", + "circuit abstraction holds, I can apply my node equations", + "and then go ahead and solve it.", + "", + "Let me show you a few more methods so we can articulate", + "your repertoire of tools for nonlinear circuits.", + "And I'd like to show you a graphical technique.", + "I personally rarely use a graphical technique to solve", + "circuits. And why am I sharing this with", + "you? It turns out that often times", + "by looking at things graphically you can get some better insights", + "into circuit behavior. You can also show cool demos", + "when you show graphs of responses kind of playing with", + "each other and so on. So this is fun for getting", + "intuition and things like that. Graphically all I'm really", + "going to do is solve those two equations graphically.", + "So I'm going to plot equation one.", + "Let me rewrite equation one as follows.", + "iD is --", + "", + "I'm just rewriting equation one as follows.", + "V/R - vD/R. And I can also draw the second", + "guy --", + "", + "OK, I can do this as well. I can do an iD versus vD plot.", + "And in this particular situation, you've seen this", + "already, that's my iD versus vD curve right there.", + "And I can do the same for this one here.", + "So this equation establishes the following straight line", + "relationship. It says that when vD is 0,", + "iD is V/R. So that's here.", + "And similarly when iD is 0 then vD is equal to V so I get", + "something here. So that's my straight line", + "relationship corresponding to this equation here.", + "So what I can do is I can simply solve these by", + "superimposing the two curves on the same vD, iD template here", + "and finding the intersection of the curves.", + "So I can take this curve corresponding to two and I can", + "take this curve corresponding to one, and this is V/R and this is", + "V, 0, and I can find the intersection point.", + "This curve here, for reasons that will be", + "obvious about three weeks from now, is called the load line.", + "It's called the load line. You will understand why that is", + "so in a later lecture. So I've given you a template on", + "Page 6 to boil these two down into one equation.", + "So there, again, you can substitute the values", + "for V is 1 volt and R is 1 and so on and so forth and get the", + "same kind of result as you did previously.", + "", + "So there is really nothing new here.", + "All I've done in the second method is combined the two", + "equations graphically and found the solution by looking at where", + "the two curves intersect.", + "", + "At the start of the lecture I also told you that you may want", + "to go and check out the piecewise linear technique --", + "", + "-- in Section 4.4 of the course notes.", + "", + "All right. For today let me do a third", + "method called \"Incremental Analysis\".", + "", + "This technique is also called the small signal method.", + "", + "I'm going to show you, before I go into the method,", + "in today's lecture what I'll do is I'll give you a motivating", + "example for why we need the small signal approach.", + "I'll give you a motivating example and show you a little", + "demo. And then I will close with", + "showing you a problem with applying a standard approach,", + "and I'll ask you to see if you can figure out a way to handle", + "it in time for next lecture. So let me give you the", + "motivation here.", + "", + "So here is what I want to do. Many of you have seen one of", + "those electric eye garage door openers, right?", + "You have a receiver at one end and you have some kind of a", + "light beam at the other, and when you walk through it", + "stops, or rather it cuts the circuit and stops the door from", + "closing. And when no one is going", + "through it maintains a connection and lets the door", + "close. So what we did is we went to", + "Home Depot, or one of those stores, and bought a very", + "standard device that essentially produces some response when", + "light impinges on it. And my goal will be to see if I", + "can send music over the light beam using a simple garage door", + "opener device. So here is the little circuit", + "that I will do. We actually went there and", + "built this. I will also show you a demo.", + "", + "Here is my time-varying voltage, vI(t),", + "and this is some music signal.", + "", + "And get some music signal. And I want to connect this to", + "this device, which is a device found in garage door openers.", + "I am going to call it a LED. If you like,", + "you can view it as, this is very similar to our", + "Expo Dweeb. This is called a \"Light", + "Emitting Expo Dweeb\". That's why it is LED.", + "So what the LED does is, as I apply this voltage across", + "it, that same voltage appears across the Light Emitting Expo", + "Dweeb. And there is some current that", + "flows through the device. And for our analysis we will", + "assume that this device virtually has an identical iD", + "characteristic to the Expo Dweeb just that it emits light.", + "So when I pass a current through it, it emits light.", + "And the light intensity is proportional to the current that", + "flows through. So it emits light and light", + "intensity, LD, is proportional to iD.", + "", + "Here is my little light emitting device,", + "which when current flows through it, itproduces light", + "because its intensity is proportional to the current.", + "And what I will do is I will stick in the receiver here.", + "Think of it as a photo resistor or some other device where I am", + "going to connect that in a circuit.", + "I am not going to spend too much time on this side.", + "I'm going to focus on the left-hand side here.", + "And let's say I have some kind of amplifier and speakers and so", + "on and so forth. Suffice it to say that when the", + "light falls on this device PR that iR that goes through here", + "is proportional to the received light intensity.", + "So if the current is proportional to the received", + "light intensity then I amplify that signal in my amplifier and", + "I get the music playing out here.", + "And notice that the following chain of dependences apply.", + "So I have an input music signal VI.", + "That gets converted to some iD. These are all time-varying", + "signals, so VI is a time-varying signal and so is iD.", + "And iD gets converted to light of some intensity LD.", + "This in turn gets attenuated somewhat and is received at the", + "photo resistor. And I get some intensity LR", + "impinging on that device there. And that in turn produces a", + "current iR and then iR is amplified and goes through a", + "speaker and so on and produces sound.", + "Notice that using this chain I've taken a music signal here", + "and I am playing it here. And just imagine that this is", + "your garage door opener device here where the light emitted is", + "being articulated by the voltage signal VI.", + "And received here. So notice that if I cut this,", + "if I stick something in here and block it then I get no", + "response here, but if I take my hand away then", + "I do get some response. But this is fine.", + "This should work. You could try this at home if", + "you'd like. If you have a garage door", + "opener, just stick a little circuit like this and it should", + "simply work. We have a problem,", + "though. The problem is that,", + "as I said, I'm using the Expo Dweeb here, the light emitting", + "Expo Dweeb, and its characteristics are as follows.", + "iD is exponentially related to the voltage vD,", + "so this is nonlinear.", + "", + "And that's a real problem. Because this is nonlinear,", + "I am going to get a distorted output.", + "Let me show you a little wave form, a little graph to show you", + "how the distortion happens and then show you a little demo", + "showing you the distortion. Let me graphically show you the", + "kind of distortion that is happening here,", + "and I will do it by drawing the following graph.", + "", + "So this is the vD, iD curve for our device.", + "And what I'm going to plot for you is if I have a time-varying", + "vD voltage, I just want to see what the time-varying iD current", + "looks like. And a trick to plot that is to", + "take your input voltage like so. And let's say I apply a", + "sinusoid. So I am just taking a", + "time-varying sinusoidal voltage and rotating the plot 90 degrees", + "like so, so I can see where these points correspond to on", + "that curve. So what this says is that at", + "some point here, for example,", + "where vI, at this point and time, vI is here.", + "Notice vI and vD are the same thing because vI is applied", + "across vD. vI directly applies across the", + "device, and so vI equals vD at all time.", + "So this voltage here corresponds to this voltage,", + "it corresponds to this current and then I can find out what the", + "current is for that voltage. By using the same artifice I", + "can plot the output current iD like so.", + "So for this value I get some current here.", + "And so at time T0 I start here. And notice that as this signal", + "moves up here, I can find out the", + "corresponding values of iD by looking at where a straight line", + "intersects here and plotting the values here.", + "I have a nice little graphical animation to show you this.", + "Hopefully, the laptop will work tomorrow and we can check that.", + "I am doing nothing new here. Just showing you a trick to be", + "able to plot vI versus v out relationships,", + "or vI or versus other relationships based on some kind", + "of a transfer function. So what you end up getting is", + "something that looks like this. Why is that?", + "Notice that this curve here corresponds to the signal.", + "As this signal moves from here to here, this point moves from", + "here to here and that corresponds to this iD.", + "When this moves from here to here that corresponds to a point", + "moving from this part of the curve to here,", + "and that looks like so. And then for the whole negative", + "incursion, notice that the whole negative incursion moves here,", + "so for that entire negative incursion I get an output that", + "looks like this. Notice that this device has", + "completely cut off and hammered negative going signals.", + "What it's done is that rather than giving me a nice little", + "negative spike incursion here, or excursion here,", + "what this is doing is that it is taking this excursion and", + "simply slamming it down to this value here.", + "And then again, when I go back up,", + "I get this peak here. So notice that what was a nice", + "little sinusoid out there gets hammered and squished into this", + "funny curve here. What this device is doing is", + "for positive values it tends to produce exponentially greater", + "current so I get boom, high-rising peaks corresponding", + "to these two, and for negative going voltages", + "it simply compresses them to a low positive value here.", + "And that's what I see here corresponding to negative", + "excursion. So notice that what this will", + "do, if I view sound, if I input sound here,", + "and sound has negative going excursions it will simply", + "scrunch them. But more or less let the", + "positive things through. And that is going to give rise", + "to a bunch of distortion in my signal.", + "So I would like to show you a little demo.", + "Actually, we've gone ahead and built a little device like this.", + "We have an honest to goodness little device costing,", + "I don't know, 50 cents or $1 or something,", + "which is a little voltage, it's a device that emits light", + "proportional to the current flowing through it.", + "I have a receiver. And I am going to play some", + "music, and you will listen to the output here.", + "And hopefully you should see a bunch of distortion because of", + "that effect that I showed you.", + "", + "And what I will do is, before we do that,", + "you will see two curves up there.", + "The yellow, I believe is the vI, is the input,", + "and the green, I believe, is a signal", + "proportionate to -- The other way around.", + "Oh, I see. So green is the input.", + "So green, the lower one is the input and the upper one is the", + "distorted output. So we are going to play some", + "sound through it, music through it and you can", + "listen, through a little CD player.", + "", + "So a couple of things. The good news is that it works.", + "However, I doubt that music artists will come to my studio", + "to record if this is the quality of what I produce.", + "Do notice that there are hardly any negative going excursions in", + "that curve up there, right?", + "All the negative ones have been like scrunched up down into a", + "flat line there, and that's the reason I get", + "this distortion. And just to prove to you that I", + "am indeed using a garage door opener device and not faking it", + "here, I am going to just shut the signal off by stopping the", + "light using a piece of paper here.", + "So notice that this device here is the little device that has a", + "light beam going through the center, and I am going to take", + "this piece of paper, can you turn it up?", + "", + "So let's have some fun with this.", + "If I were to put this piece of paper halfway down,", + "I should get half the intensity, right.", + "So my sound should diminish in volume a little bit.", + "Maybe that will work. Let's see if it works.", + "Nothing to do with 002 but it's just fun.", + "Louder. You can make it loud.", + "", + "Too much coffee. My hand is shaking.", + "I guess you did see the lowering of volume,", + "right?", + "", + "OK. Just way too much coffee,", + "and so my hand was shaking too fast imposing its own sine wave", + "on top of the signal. What did I show you?", + "This was garbage, right?", + "We had a nice little signal input, and the output was", + "completely distorted because I was playing sound over this and", + "this is what happened. Switch to Page 9.", + "Now, this is what I would have liked to have happened.", + "On Page 9 what I would have liked to see happen is this.", + "", + "Suppose I had a light emitting device that looked linear,", + "a straight line where the current was linearly related to", + "vD. Then what I would see,", + "if I had a sinusoid here then I would get a sinusoid here.", + "No distortion there, right?", + "If only things were like I wanted them, if I had a linear", + "device, but I don't have a linear device.", + "I have an Expo Dweeb. Now you know why I call it a", + "dweeb. Well, I'd like a linear device", + "and it's exponential. But this is what I would like.", + "", + "And if I had this I wouldn't show it to you today.", + "If I had this my music would go through without any distortion", + "and I wouldn't have to run cables through my attic.", + "I could just use my garage door opener to play signals from my", + "bedroom and living room and so on, right?", + "So the key thing here is how do I get this?", + "And what I would like you to do is think about it yourselves.", + "What I am given is something like this.", + "This about it yourselves, you know, what would you do?", + "See if you can come to me before lecture tomorrow or", + "Thursday and tell me the answer, OK?" + ] +} \ No newline at end of file diff --git a/subs/R4KxlqsuZ0A.srt.sjson b/subs/R4KxlqsuZ0A.srt.sjson new file mode 100644 index 0000000000..a315dafa99 --- /dev/null +++ b/subs/R4KxlqsuZ0A.srt.sjson @@ -0,0 +1,1901 @@ +{ + "start": [ + 0, + 7000, + 10623, + 15455, + 19823, + 24283, + 28000, + 34641, + 38027, + 45581, + 51702, + 56000, + 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+ 2858608, + 2863727, + 2868006, + 2871699, + 2876062, + 2881480, + 2886026, + 2891710, + 2897393, + 2900803, + 2907210, + 2913613, + 2921080, + 2925416, + 2929510, + 2934569, + 2943000, + 2951000, + 2957000, + 2958111, + 2961370, + 2965814, + 2968185, + 2972481, + 2976925, + 2980627, + 2984625, + 2988623, + 2992103, + 2997000 + ], + "text": [ + "", + "We have put some of the quiz stats here.", + "The mean was about 75%. And I must tell you that that", + "is very impressive. I guess MIT undergrads never", + "cease to amaze me. And this was not an easy quiz.", + "This was a relatively hard quiz.", + "And that average implies that you guys did well on a", + "relatively hard quiz. Good.", + "Let's get back to our final lecture on amplifiers and small", + "signal circuits. And as always let me start with", + "a review. Very quickly --", + "", + "-- we came up with a notation to represent small signals.", + "And our notation looked like this.", + "Our total variable was small and capital, and this was a DC", + "bias and this was a small signal.", + "", + "This is also called the operating point.", + "And the small signal is also called the incremental signal.", + "In general, if you have some function, some variable of", + "interest in the circuit, say a total variable V out,", + "let's say it relates to some input variable as F of VI.", + "So mathematically we can find out V out by simply finding the", + "slope of this function at the operating point and then", + "multiplying it by the incremental change in the input.", + "Gold standard math. So we do the slope of this", + "function and evaluate it at the operating point.", + "So this would give us the slope of the function.", + "And multiply that by small VI, which is incremental change.", + "This is standard math. What this will tell you is", + "given a small change in VI this function gives you,", + "this expression gives you the small change in V out.", + "And in lecture we have pretty much used this method so far,", + "used the math to get to where we wanted it to be.", + "And then the way we provided biasing and so on was for our", + "amplifier in particular we had a bias voltage,", + "some small signal value, VS.", + "And this was output which was also given to be some output", + "operating point plus a small change, which was a change in", + "the output voltage. So what we have done here is", + "mathematically computed small V out.", + "And what I am showing you here is to get the same effect in a", + "circuit is you build your circuit and replace what used to", + "be a total variable with a DC bias plus a small change.", + "And then you will get your output here.", + "And this output will relate to this input using this", + "expression.", + "", + "So this is more review. To continue on with the math", + "review, for our amplifier VO was given to be VS-K/2(vI-VT)^2 RL.", + "So this was the output versus input relationship for the", + "amplifier. And mathematically I could get", + "the small change in the output VO by simply differentiating", + "this function with respect to VI, evaluating that function,", + "at capital VI and multiplying by the small change in the", + "input. And the resulting expression", + "that we got for small VO --", + "", + "-- was simply minus K, this was our DC value,", + "and RL times small VI. So we derived all of this the", + "last time. So nothing new so far.", + "So my small signal output was some function given by", + "K(VI-VT)RL times small vi. And notice that this is how VI", + "relates to VO. And this is a constant with", + "respect to VI. V capital I is a DC bias,", + "so this is a constant. So therefore this is the linear", + "relationship that we had set out to get.", + "This term here, for reasons we will see today,", + "this term here K(VI-VT) is called gm.", + "Transconductance. We will look at it in more", + "detail a little later.", + "", + "Even more review.", + "", + "So I can draw the transfer function and plot VO versus VI.", + "Another way to graphically view what is going on is by plotting", + "the load line curve for this circuit, so this is VI.", + "And I said we draw that by first plotting the --", + "", + "These were our MOSFET curves. And we know that at some point", + "the MOSFET gets into saturation, so this curve was iDS=K/2 VO^2.", + "And to the right side of the curve the MOSFET is in", + "saturation. And we said we will adhere to", + "the saturation discipline and operate in this regime.", + "When the MOSFET gets into this region it is in its triode", + "region. And then we could draw the load", + "line here. The load line codified the", + "following relationship, iDS=VS/RL-VO/RL.", + "This was a load line. So I have superimposed a load", + "line on the device characteristics,", + "and I am going to show you a little demonstration based on", + "that at this point. So these curves were drawn for", + "increasing values of VI. And if I choose some operating", + "point here then this point would correspond to some bias,", + "this bias point would correspond to some input voltage", + "VI, a corresponding output bias VO and a corresponding current", + "iDS. So iDS capitals,", + "VO capitals, VI capitals represent the", + "operating point values for our little circuit.", + "So far there is nothing new. One thing we stopped the last", + "time by pointing out that the gain of our amplifier,", + "this is the gain, -K(VI-VT)RL.", + "That is the gain A of the amplifier.", + "That gain related to VI. A gain was proportional to", + "VI-VT. So therefore if I increased VI,", + "I would get more gain. So the question is how do we", + "choose a bias point? And in our particular example,", + "let's say we are free to play around with VI.", + "So we play around with VI and I can choose various bias points.", + "So where do you set the bias point?", + "What are the various characteristics of the circuit", + "that relate to my bias point? Well, first,", + "of course, is gain. The gain depends on how I", + "choose VI. I will show you that in a", + "moment. The second important thing,", + "in other words, if I choose a bias point that", + "is a small VI then my gain is going to be smaller.", + "If I choose a bias point that's at a much higher value of VI,", + "I get a bigger gain. The second important", + "consideration is operating range.", + "", + "Notice that if I choose a bias point here then as the input", + "changes -- Notice VI in this graph goes up", + "or down, and I would be traversing and following", + "different lines here in my MOSFET characteristic.", + "And as VI increases the operating point would come up", + "here and so on. So if about this operating", + "point I varied my input voltage VI then, so let's say about this", + "operating point, if my input VI,", + "my small signal VI varied about a small range then", + "correspondingly the output value would vary about this part of my", + "load line. So notice now that the", + "operating range, how far can VI vary before the", + "MOSFET goes out of its saturation discipline?", + "Well, on the low side my VI can come down to here.", + "And we looked at the operating ranges for an amplifier.", + "And I can come all the way down to VT.", + "At that point the output will come here.", + "Similarly at the high end VI could get up to a high value.", + "And we computed that value in the last lecture.", + "And the corresponding value of the input would be here.", + "So in some sense I can traverse all the way from here to here", + "and have the MOSFET remain in saturation.", + "Remember we are not talking about linearity right now,", + "just about the valid operating range based on my definition", + "which is that the MOSFET should stay in saturation.", + "So if I chose my operating point here then I get this range", + "here. And, on the other hand,", + "if I chose my operating point to be here, for negative", + "excursions of the input signal I have a very small amount before", + "I hit cutoff. So if I chose my operating", + "point here then for negative traversals of VI about the", + "operating point I very quickly hit cutoff.", + "So if I want symmetric swings then this is the best that I can", + "do in terms of the valid input operating range if I want", + "symmetric swings given that this is my bias point.", + "On the other hand, if I chose my bias point", + "somewhere here, or very carefully chose my bias", + "point then my input can vary on a much wider region and still", + "get symmetric swings. And so therefore the choice of", + "bias point also influences the maximum swing range of my input", + "signal. I shouldn't call this operating", + "range. I should call it input swing", + "range. We defined the valid input", + "operating range as the range for which the amplifier satisfied", + "the saturation discipline. So the two key issues,", + "gain and the input swing. Let me show you a quick demo", + "and try to point out on a graph some of the characteristics that", + "relate to the matter we have been talking about so far.", + "So what I show here are these curves for the MOSFET.", + "This is VO and this iDS. This is the zero point.", + "Ignore this line down here. This line up here corresponds", + "to the output voltage VO. What I am going to do now is,", + "through some careful circuit hacking, I'm going to show show", + "you a load line and show you the bias point, and show you how the", + "bias point can be moved up and down by changing the input", + "voltage which changes the corresponding output voltage.", + "", + "It is hardly visible out there.", + "", + "Is it there? OK.", + "It is not really clear, but notice that as I increase", + "my input, I am increasing my input.", + "My output keeps coming down. And I hope your eyesight is", + "better than mine because I don't see a dot up there.", + "I am amazed. This is the first time this has", + "happened to me. That's OK.", + "All right. As you can see,", + "as I change the input value the output operating point changes,", + "and the dot out there traverses, articulates a load", + "line. I guess I have to believe that", + "there is a dot out there. Next what I will do is show you", + "some more fun stuff. What I will do is instead of", + "having just a dot by having a DC voltage, let me apply an input", + "sinusoid. So if I apply an input sinusoid", + "at some bias then I should see an articulation of the", + "corresponding region of the load line corresponding to the input.", + "So, as you can see here, now the bottom line,", + "here is my input and this is my output.", + "And notice that this the region of the load line articulated", + "when the input is of this magnitude.", + "Now let's have some fun. As I increase my input,", + "you can see that a larger portion of the load line is", + "articulated, right? There you go.", + "And as I decrease my input a smaller region of the load line", + "is articulated. Let's leave it here for a", + "moment. And what I will do next,", + "this is the region here that we are looking at,", + "let me increase the bias. If I increase the bias,", + "if I increase VI, what do you think should happen", + "to this line here? Well, if I increase the bias,", + "the line should go up, right?", + "Because remember the dot? The dot is in the middle of", + "this thing here. If I increase the bias this", + "should move up here. So that line moves up.", + "Do you expect anything else to happen to that line?", + "Pardon? It increases,", + "exactly. If I increase the bias point to", + "here then this must also increase because my gain has", + "increased. Let me do that.", + "So let me increase the input bias.", + "Indeed notice that the region of the load line articulated is", + "larger now. Let me decrease the bias.", + "And notice that because the gain is smaller the little", + "segment shown is also smaller. I have shown you two things so", + "far. One is that I as I increase my", + "bias the line indeed rises up corresponding to a higher value", + "for the input operating point. And the second is that I get a", + "larger swing in the output as I increase the bias.", + "Just to show that for those like me who were visually", + "challenged in terms of viewing that little dot up there,", + "let me get some audio so you can actually hear the sinusoidal", + "tone. It is a big annoying.", + "", + "As I reduce the bias the gain is decreased.", + "As I increase the bias you can see that the gain is increased", + "and the tone is louder. Let's have some more fun and", + "let's play some music now. And what I am going to show you", + "with the music -- The reason I play the music is", + "not just for fun. Well, it's 85% fun and 15%", + "learning. Can we turn it on for a second?", + "What I would like to do is, as we play the music,", + "the reason I am playing the music for that 15% is so you can", + "listen to distortion. I want you to listen to the", + "distortion. That is when the articulation", + "is here you are not going to get much distortion.", + "But as I get into cutoff you should be getting a bunch of", + "distortion. Similarly, as you get into the", + "triode region you should also be getting distortion because the", + "amplification from being somewhat nonlinear here becomes", + "highly nonlinear at those two points.", + "So let's just play the signal. So volume increases,", + "or rather the amplitude increases by increasing the", + "bias. Now you should hear the volume", + "go down and distortion.", + "", + "So notice now that the bias point is way down here.", + "So the gain is very low, and plus there is a distortion", + "because of cutoff. Now what I will do is blast it", + "up here, and you will see that the volume has gone up but then", + "you see distortion again. Let's see if you can stand the", + "volume here.", + "", + "Even the CD doesn't like that.", + "", + "Notice that as I went up here the volume kept increasing", + "because the gain kept increasing, but as I got into", + "the triode region I began to lose my gain because,", + "remember, the amplifier doesn't have gain in the triode region,", + "MOSFET in its triode region, and we also get a bunch of", + "distortion out there. Finally, it turns out that as", + "people are building amplifiers --", + "I think this was in the mid to late `50s and `60s and so on.", + "They said man, electrical engineers are not", + "going to get their thing right. So they invented a new kind of", + "music which was much more tolerant to distortion.", + "And I will play that music for you.", + "It is called hard rock. I challenge you to tell me it", + "is distorting.", + "", + "Sounds good to me.", + "", + "OK. All right.", + "That'll do it. Thank you.", + "I hope there are no hard rock musicians in here who will come", + "and beat me up after lecture or something.", + "All right. Believe it or not most of that", + "was review. There is nothing new today", + "besides some fun and games and so on.", + "I will give you a breather for five seconds before jumping into", + "something even more fun.", + "", + "I want you to look at the middle board here.", + "And, as I told you in the beginning of 6.002,", + "engineering is about building useful systems.", + "Engineering is not about showing off at math or saying", + "man, I am really cool in math and stuff.", + "Engineering is about building useful systems,", + "and you want to find the simplest, easiest,", + "cheapest way to get there. Unlike deep areas of math and", + "theory and so on, the beauty is in the", + "simplicity. So the aesthetics are in how", + "simply can we make things and still get to where we want to", + "be? All through the course what you", + "will be seeing happening again and again and again is when", + "things begin to get too grovelly in terms of math,", + "we will step back and say oops, we are engineers,", + "remember? Let's find a much simpler way", + "to do it and use intuition. So time and time and time", + "again, I am going to take you on a simpler path where you can", + "solve things by inspection by pure intuition.", + "Most circuit designers do that. So take a look at this.", + "I don't like this nasty differentiation here.", + "That's getting into late high school calculus and so on.", + "Let's avoid the math and let's see if you find some way of", + "doing it that is even much more simpler.", + "And that is what I will do next and show you what is called the", + "small signal circuit view. A purely circuit way of", + "developing the small signal model.", + "So let me just start by drawing the large signal equivalent", + "circuit for you. I will draw it here for reasons", + "that will be obvious at the end of the class.", + "", + "All right. This is the large signal", + "equivalent circuit model for our MOSFET amplifier.", + "VS and here is my current source.", + "iDS relates to the square of VI minus VT.", + "So stare at that for a second. And that is a nonlinear", + "circuit. iDS relates to the square of VI", + "minus VT. Let me start by making the", + "following claim. Let me shoot from the hip here", + "and make the following grand claim, and then I will show you", + "how I can prove that claim. The grand claim I am about to", + "make says the following. A bunch of little devices here.", + "It is a nonlinear circuit. Just suppose for a moment we do", + "a Gedanken experiment. Suppose I replace each of my", + "circuit elements here with its linearized element equivalent.", + "In other words, here is a VS source,", + "here is a dependent current source, let me replace them with", + "their linear equivalent circuit models.", + "In other words, with their corresponding small", + "signal element models. And I will show you what those", + "are in a second. The resistor has a", + "corresponding small signal element.", + "The dependent current source has a corresponding small signal", + "behavioral element model. And what I am going to do is", + "keep the same circuit connections and simply pull out", + "the large signal model for the element and replace it with a", + "small signal element model. And by the nature of the small", + "signal model they are all going to be linear.", + "So what I am going to be left with is a linear circuit with", + "simple linear circuit elements in there.", + "And then once I have a linear circuit, I should be able to", + "analyze that linear circuit using methods 1,", + "2 and 3, superposition, Thevenin, node method and so", + "on. And certainly the intuitive", + "methods like superposition and Thevenin, which make life a lot", + "easier for me with linear models, and thereby get the", + "function that I am looking for very quickly.", + "Again, my claim is that I can replace each of these large", + "signal models by just small signal equivalents and then just", + "analyze the resultant circuit. And I claim that I should be", + "able to get the same answer. That's a claim.", + "All right? So what I will do is give you", + "an informal proof for why I can do that.", + "And I also ask you to refer to Section 8.2.1 of the course", + "notes to go through the foundations of the small circuit", + "model in more detail. The intuition is that,", + "remember KVL and KCL? I can write down KVL and KCL", + "for every loop in that circuit and every node in that circuit.", + "If I do KVL and KCL, I will end up with something", + "like this. For the input loop I get VI", + "something or the other applying KVL.", + "For the output loop I get V out something or the other.", + "And then applying KCL I get some other equation in iDS.", + "So here are my KVL and KCL equations for that circuit.", + "Now, KVL and KCL are simply a different representation of the", + "circuit because within those KVL and KCL is encoded the topology", + "of the circuit. Remember each KVL equation", + "represents a loop and each KCL equation represents how nodes", + "are connected together. So KVL and KCL equations encode", + "within them the topology of my circuit.", + "What I do next is, say, I replace each of these", + "with the bias plus the small signal, so I get the bias plus", + "the small signal and keep the equations the same.", + "", + "All I have done in my big set of KVL, KCL equations,", + "I have simply replaced the total variable with the large", + "signal variable and the small signal quantity.", + "Then comes a key trick. The key trick is that because", + "the bias point variables, they are a valid solution to", + "the circuit. The circuit is in this", + "quiescent state, and those are valid solutions", + "to circuit. So therefore I can cancel them", + "out. So the VI, the large signal", + "values can be cancelled out leaving just small signal", + "variables in there. So from the KVL,", + "KCL equations I can cancel out the large signal values,", + "the DC bias points because they satisfy the KVL and KCL", + "themselves. In other words,", + "I could have written VI plus V out and so on.", + "Since they are satisfied I just strike out the large signal", + "variable from both sides of each of these equations,", + "so what is left is the same KVL, KCL equations but with", + "small variables in place of the big variables.", + "What that should tell you, this informal proof should tell", + "you is that the small signal variables should then satisfy", + "the same form of the KVL, KCL equations that the total", + "variables satisfy. And because the KVL,", + "KCL equations are a reflection of the topology of the circuit,", + "what that says is that the small signal variables must also", + "satisfy KVL and KCL. And since these arrive from the", + "small signal elements that says that I can replace the big", + "elements with the small elements and KVL and KCL will hold for", + "the resulting circuit. This is a very quick breeze", + "through, an informal proof to show that I can replace the big", + "elements with the corresponding little element models and then", + "simply apply linear techniques. Refer to Section 8.2.1 for more", + "foundations and more discussion about the foundations for why we", + "can do this. That brings up the small signal", + "circuit method. The circuit method for small", + "signal analysis has three steps. The first step is find", + "operating point by using LS. First you analyze your large", + "signal circuit and find the operating point.", + "You have to do this, because remember,", + "the small signal models depend on the operating point values.", + "Remember the gain of our amplifier depended on the bias", + "point. Second step is develop small", + "signal models of elements. Second step is take each of the", + "elements in your circuit and find their equivalent small", + "circuit model for each of the elements.", + "Third step is replace original elements with their small signal", + "model elements. Third step is simply take the", + "large elements and replace them with their small signal", + "equivalent models. Then analyze resulting circuit,", + "and that circuit will be a linear circuit.", + "So let's do an example. I will just use the amplifier", + "as an example of this method. And convince you that you are", + "going to get the same expression at the end, but just so,", + "so simply without even the smallest amount of grubby math.", + "Three steps. The first step is to find the", + "operating point using the large signal model.", + "And let me just do that here. I get my V out =", + "VS-K/2(VI-VT)^2 RL. Let me just write down that out", + "here. Don't worry about copying that", + "down. It is on the last page of your", + "notes. The first step of the method", + "simply applies the large signal model and finds out the behavior", + "of that circuit to find out what the bias point values are.", + "The second step is to develop the small signal model of my", + "elements. How do I go about developing", + "the small signal models of elements?", + "Let's start with the MOSFET. The large signal model for the", + "MOSFET looks like this.", + "", + "Here is my Vgs. This is my gate.", + "This is my drain. This is my source.", + "And I know my iDS to be K/2(Vgs-VT)^2.", + "So this is the large signal model for the MOSFET,", + "again in saturation. I am talking about all of these", + "models are under the saturation discipline.", + "So Vgs relates to iDS in the following way for the MOSFET.", + "That is iDS, is K/2 and that is my square", + "law relationship. So what is a corresponding", + "small signal model? I go ahead and start with this.", + "The corresponding small signal model simply says that iDS", + "relates to Vgs in the following way.", + "All I have to do is find a small signal equivalent where I", + "need to find out, given a small change in the", + "input Vgs, what is the small change in the iDS?", + "So I can apply my standard trick to a much simpler", + "expression here, which is iDS simply,", + "I differentiate this function with respect to Vgs.", + "", + "So I don't completely eliminate the math here,", + "but it is a much simpler problem here.", + "At Vgs equals the bias point times small vgs.", + "I can find the small change in iDS corresponding to a small", + "change in the input using this expression.", + "That gives me iDS as simply K(Vgs-VT) vgs.", + "I call this gm, and I will tell you why in a", + "second. So what does this expression", + "say? This expression says that if I", + "have a small change in Vgs then this will be my small change in", + "iDS. Notice that the resulting small", + "signal model is also a dependent current source.", + "It is a voltage controlled dependent current source.", + "So the output is the current, and it is a dependent current", + "source and it depends on the input voltage.", + "The good news is that notice that this one,", + "this expression here gm is a constant related to the bias", + "point values. Therefore, notice that the", + "small signal model for the MOSFET in saturation,", + "not surprisingly, is a linear voltage controlled", + "current source according to the following expression.", + "So iDS=gm Vgs. Gm is a representation for", + "K(Vgs-VT) and is called a transconductance.", + "It is called a transconductance because it, in some sense,", + "deflects the conductance properties of this based on the", + "input. So it is a transconductance.", + "So this value is called Vgs. Therefore, I can build the", + "small signal model as follows. Vgs is a voltage controlled", + "current source and iDS is simply gm Vgs.", + "So this is my gate, drain, source.", + "", + "So that is the small signal model for my MOSFET.", + "As a next step what are the other elements in my circuit?", + "Let's see. I have a voltage source and I", + "have a resistor, so let me find out the", + "corresponding small signal model for a DC supply VS.", + "This is Page 7. I will do it mathematically for", + "you, but often times it is always good to do a sanity check", + "using intuition. Let me ask you,", + "the large signal for a DC supply looks like this.", + "", + "The element law for a voltage source is VS equals some capital", + "VS. It is a constant voltage.", + "So what do you expect to be the small signal model for a voltage", + "source? In other words,", + "for a small change, suppose I have a small change", + "in the current, by how much should the output", + "VS change? It shouldn't change.", + "It is a voltage source. So what does intuition tell you", + "is a small signal model for the voltage source?", + "A short. So the key here is that a", + "voltage source behaves like a short circuit for small", + "perturbations. In other words,", + "if I change the current flowing through it by a small amount", + "somehow, the output is still going to held at VS.", + "In other words, small signals are simply going", + "to scoot through this voltage source without having any impact", + "whatsoever on the voltage. Or mathematically I could also", + "do small vs is del by del IS of VS evaluated at IS equals some", + "capital IS times small IS. And therefore VS equals zero.", + "What that means is that the small signal model for my", + "voltage source is simply a short circuit.", + "", + "So in a small circuit voltage sources appear like a short", + "circuit. Finally, I have a resistor,", + "my resistor R. Let me find out its", + "corresponding small signal model.", + "The large signal model looks like this R, VR,", + "IR. And I know that VR is simply", + "RIR. And to find the small signal", + "equivalent I do del of IRR divided by del IR for IR", + "calculated at some constant value times small IR.", + "What I am looking to do is to find out what is the change in", + "the voltage across R for a small perturbation in the current?", + "Again, let me exhort you to rely on intuition to at least", + "sanity check your answers. So what do you think this", + "should look like? It's a resistor and I have a", + "small change in the current, by what do you expect the", + "voltage to change? Think about,", + "for the next five seconds, what the small signal model for", + "this should look like and then I will go ahead and write down the", + "answer.", + "", + "So differentiating I simply get RIR.", + "In other words, for a resistor the small signal", + "model is the resistor itself.", + "", + "So what I have done so far, let me just take you through", + "where we are right now, give you the big picture there.", + "I began by suggesting that looking to find an even simpler", + "way to do small signal analysis. I gave you an informal proof to", + "show that if I had small signal element models for all of my", + "elements, I could simply replace them in the circuit and then do", + "a corresponding linear circuit analysis phase to get the result", + "I am looking for. There are three steps to the", + "method. As a first step we began by", + "finding small signal models for each of our elements.", + "For the nonlinear MOSFET the small signal model was a linear", + "dependent current source. For a voltage source the", + "corresponding small signal model was a short circuit.", + "Again, that makes sense intuitively if I change the", + "current through a voltage source by a small amount.", + "By how much does the voltage change?", + "It is a voltage source, silly.", + "The voltage doesn't change. So the small signal V,", + "the small change in the voltage is zero, and that is the same", + "thing as a short circuit. For a resistor by how much does", + "the voltage change if I change the current by a small amount?", + "Well, it will change by R times the current change,", + "and that is the property of a resistor, R.", + "As a final step what I would like to do, on Page 8,", + "I'd like to very quickly draw for you the small signal circuit", + "and then analyze it. This is the large signal", + "circuit. That is a large signal circuit.", + "And let me draw the small signal circuit.", + "And the method says simply pluck out, gouge out each of", + "these elements. And simply replace each of", + "these nasty nonlinear elements with the corresponding small", + "signal linear equivalents. So let's do that.", + "Remember, for the input you replace input with its small", + "signal voltage because I am telling you that it's sourcing a", + "small change in VI. So that is VI.", + "And then I replace a short for VS.", + "I replace an R for RL because it is an RL itself for the small", + "signal model. And then for the dependent", + "source, we discovered that the dependent source was a linear", + "dependent source given where ids=gmvi.", + "Remember, this was my small VO. Here you go.", + "I have a small signal circuit here where I have simply created", + "that by replacing each of the big elements by little", + "rinky-dink elements. Now these are all linear", + "elements so I can do a really simple linear analysis.", + "What method shall we use? Well, this is so simple.", + "I will just go ahead and use the node method.", + "So applying the node method at the node with voltage VO,", + "what I will do is the current going up, VO divided by RL", + "equals the current going down iDS.", + "And so the current going up is VO divided by RL and the current", + "going down is -- Oops, I should have done this.", + "The total current going out is zero, so the sum of these two is", + "zero. That is my good old node method", + "here. And I know that iDS is simply", + "gmvi equals zero. So right there I have the", + "relationship between VO and VI. So VO is simply minus gmviRL.", + "And remember gm was simply K VI minus VT.", + "", + "We are done, OK?", + "What have we here? I created a linear circuit", + "which simply comprised small signal models for each of my big", + "elements. And then I simply did a", + "straightforward linear analysis using any one of the linear", + "techniques I knew about. This is simple enough so I", + "apply the node method. And I've got the equation at", + "this node, simplified it and I directly got the answer.", + "In one or two steps I directly gave you the output as a", + "function of the input. It can't get any simpler.", + "Thank you." + ] +} \ No newline at end of file diff --git a/subs/RsJ1eg7XNVs.srt.sjson b/subs/RsJ1eg7XNVs.srt.sjson new file mode 100644 index 0000000000..621dadf3f2 --- /dev/null +++ b/subs/RsJ1eg7XNVs.srt.sjson @@ -0,0 +1,1991 @@ +{ + "start": [ + 0, + 4000, + 11000, + 15900, + 18350, + 23716, + 29783, + 34486, + 41473, + 45144, + 49644, + 56750, + 61746, + 66103, + 70080, + 75384, + 81256, + 84760, + 90064, + 93250, + 96342, + 100770, + 105031, + 107704, + 111297, + 114806, + 118566, + 122577, + 128011, + 133631, + 137446, + 143468, + 146579, + 152000, + 156450, + 160277, + 164727, + 168866, + 172458, + 176596, + 180813, + 185877, + 190885, + 197172, + 201327, + 206762, + 210729, + 214785, + 218775, + 222500, + 226157, + 230214, + 233206, + 238727, + 244363, + 249363, + 253818, + 257090, + 262727, + 264545, + 267363, + 272363, + 279222, + 283666, + 288333, + 295222, + 301154, + 307464, + 312309, + 317605, + 323464, + 330000, + 333038, + 337523, + 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Loud and clear.", + "OK. Let's get started.", + "Before I begin, just a couple of announcements.", + "Brad Buren is one of our students here and he needs a", + "note-taker. It's a paid position.", + "So if you are interested you can stop by after class and see", + "him. He's sitting right here out", + "there, OK? Second, just a reminder that", + "6.002 does have prerequisites. And the prerequisites are 8.02", + "and 18.03. So with that let me start off", + "with the usual. Do a quick review of what we've", + "done so far. So we started out life looking", + "at the laws of physics and Maxwell's equations and so on.", + "And those were way too hard so we said let's make life easy for", + "ourselves. So we chose to play in this", + "playground in which we said we shall adhere to the lumped", + "matter discipline. OK?", + "The LMD. So we are in that playground.", + "So this entire course, and for that matter large parts", + "of EECS are within that playground, within which the", + "lumped matter discipline applies.", + "So as soon as we jumped into the playground,", + "the LMD playground, we could take Maxwell's", + "equations and abstract them out into two very,", + "very simple rules. And the very simple rules were", + "KVL and KCL. KVL simply said that I can sum", + "the voltages in any loop in a circuit and the result then", + "would be zero. Similarly, I can sum the", + "currents that enter or exit any node and the sum will also be", + "zero. So what you can now do is,", + "if you feel like, you can go around and brag.", + "Oh, yeah, we use Maxwell's equations in everyday life and,", + "yeah, it's good stuff. And the key is that this is", + "really an encapsulation of Maxwell's equations within this", + "playground that we are in. So I talked about the first", + "method of circuit analysis in the last lecture.", + "And that method simply took the, wrote KVL for all the", + "loops, wrote KCL for all the nodes and wrote element vi", + "relationships. And together gave you a big", + "bunch of equations. And you sat down and grunged", + "through the equations and you solved for branch voltages and", + "currents. So we reviewed a second method", + "of circuit analysis. And I'll simply call it circuit", + "composition. The basic idea behind this", + "method was to learn some simple rules of how resistors add and", + "conductances add and so on and so forth and look at a circuit", + "and simplify the circuit by making series simplifications", + "when the resistors are in series and so on and so forth,", + "and compose it and play around with it till we end up with the", + "current, the voltages that we are looking for.", + "This is the intuitive method. And so a section in Chapter 2,", + "I believe, of the course notes discusses several examples using", + "this method and attempts to make a little bit formal the", + "intuitive approach that is applied in this method.", + "So we then looked at the node method.", + "And the node method was simply a particular way of applying KVL", + "and KCL. Node method,", + "remember? We took a ground node.", + "Then we labeled the nodes of the remaining voltages with", + "respect to that ground. Then we wrote KCL for each of", + "the nodes. And when we wrote KCL for each", + "of the nodes, remember, KVL was implicit in", + "this expression that we used for each of the currents that were", + "exiting each node. So if Ej was a node voltage,", + "then Ej minus Ei multiplied by the conductance Gi was the", + "current that was going through one of those,", + "I should call it Gij. This is a conductance that", + "connects nodes i and j. That gave us the KVL that fed", + "into the same system. So these are three methods.", + "The node method, by the way, is sort of the", + "workhorse of the 6.002 industry. And for that matter for all of", + "the circuits industry. When in doubt,", + "apply the mode method, you'll be OK.", + "That applies to linear circuits, nonlinear circuits,", + "what have you. What I'm going to do today is", + "go through two more methods. So notice that the first few", + "lectures of this course, the first three lectures simply", + "comprise transitioning you from the world of physics to the", + "world of EECS. And then two lectures on giving", + "you a bag of tricks. So we start you off with the", + "sort of tools, your mallets and chisels and so", + "on and so forth. And these five methods are your", + "tools. We'll look at two methods", + "today. One method is called the method", + "of superposition and the second method is called the Thevenin", + "method. And these methods apply only to", + "linear circuits. So we look at the subset of", + "circuits that are linear, and these two methods apply to", + "only those circuits. These are methods that combined", + "with intuition really enables you to solve very interesting", + "circuits very, very quickly.", + "So let me do an example using a usual node method.", + "And then jump into introducing the superposition methods and", + "Thevenin methods using that same example.", + "So let me draw you an example circuit here.", + "", + "So, again, I'm using this example, I will use this example", + "to introduce the method of superposition and the Thevenin", + "method. So what I'm going to do is", + "start off the usual way and analyze the circuit using a", + "method that you know now, the node method.", + "And what I'll do is write down the node equations for this by", + "applying the node method. So if you recall the node", + "method. I choose a ground node.", + "I'm going to choose this node. It's got both the voltage", + "source connected to it, and it's also got many other", + "edges impinging on it. So I'm going to choose that as", + "my ground node and I'm going to label the other nodes with their", + "voltages. So this is an unknown.", + "I'll label it as e. I guess we just have one", + "unknown e. And I know the voltage of this", + "node, and that is simply V. Since it's V,", + "there's a voltage source between the ground node and that", + "node. So what I can do next is that I", + "can write down the node equation for this node and then go from", + "there. So let me go ahead and do that.", + "So let me sum up the currents going outside,", + "going outwards. So I have e minus v divide by", + "R1, I have e minus zero divide by R2, and I have minus i equals", + "zero. This is a node equation.", + "The first thing I want you to observe is that this equation is", + "linear in V and i. What I mean by linear is that", + "you don't see terms like Vi or V-squared and things like that.", + "It's some constant times V plus some constant times i equals", + "some other constant. So that's quite nice.", + "So I'm going to rearrange the terms in the following manner.", + "I'll move the known sources to the right-hand side and collect", + "the coefficients of e on this side, so I get one by R1 plus", + "one by R2 over here.", + "", + "So stare at this for a moment and notice again here I have e,", + "my unknown node voltage, there is some constant", + "multiplier, and that equals some function of V summed up with", + "some function of i. And, again, notice that this is", + "a linear combination of V and i. No multiplication terms and so", + "on and so forth. This is a pretty standard form", + "in which we will represent equations quite often.", + "And just to label it, this is often labeled G as the", + "conductance matrix. Of course this is e,", + "our unknown node voltages, and this is a linear sum of", + "sources. So this is a very standard way", + "that we will represent equations.", + "We did that last week as well, or rather on Tuesday where I", + "took a conductance matrix, multiplied that by a column", + "vector of unknown node voltages and equated that to some linear", + "combination of my source voltages.", + "The reason the circuit is linear is that I have only", + "linear elements in the circuit. I don't have any nonlinear", + "elements. And because of that I can", + "rewrite this in the following manner.", + "I'm just going to express e as a function of V and i and bring", + "it over to this side. So it's some function of i.", + "So I get R1 R2 divide by R1 plus R2.", + "And I bring R1 R2 to this side. That's what I get.", + "So stare at this for a few seconds, very common form.", + "My unknown node voltage is equal to this stuff on the", + "right-hand side. The stuff on the right-hand", + "side has a term multiplying the source voltage V and some other", + "term multiplying the current I. And if I were to put this in", + "sort of symbol-like form my unknown node voltage is some", + "constant times V1 plus some constant times,", + "is of the form constant times the source current,", + "constant times the source voltage and so on.", + "The units of As and Vs are different because in this case A", + "has no units because V is a voltage.", + "And so is e. In this case V has units of", + "resistance. So that V times i gives me a", + "voltage. So stare at this equation for a", + "few seconds and this should help us build up some insight that", + "will allow us to write down the answer almost by inspection.", + "I'm going to show you a method now, in a few minutes,", + "which will allow you to write down the answer e just by", + "starring at the circuit without having to go through node", + "equations and so on. The more and more methods I", + "teach you, the more you will be able to do a lot of this", + "completely by yourselves. In this particular example it's", + "a relatively simple circuit but these methods would be", + "particularly useful when you have more complicated", + "situations. But before I go on let me spend", + "a few minutes pontificating on linearity.", + "So that's a linear circuit. And this equation gives me the", + "unknown node voltage e as a linear sum of source voltages", + "and source currents. Linearity implies two", + "properties, the property of homogeneity and also gives vice", + "to the property of superposition.", + "Let's do homogeneity first. What this says is if I have a", + "circuit, some circuit and I feed it some sort of inputs,", + "A, then let's say my output is S.", + "If you're feeling hungry think of these as apples and the", + "circuit converts them into applesauce.", + "So what homogeneity says is that what I can do is if I take", + "each of my apples and instead of feeding it an entire apple what", + "if I give it three-quarters of an apple?", + "Say I multiple all my inputs by some constant alpha,", + "three-quarters. What that says is that at the", + "output instead of getting one full bottle of applesauce I'm", + "going to get three-quarters of a bottle of apple sauce.", + "So if I proportionately reduce all the inputs and if this is a", + "linear circuit then so shall my output be reduced in the same", + "proportion. So that's homogeneity.", + "Next, let's look at superposition.", + "", + "The property of superposition says the following.", + "The same kind of circuit. If I feed it apples then I get", + "applesauce. I take the same circuit,", + "and this time around if I feed the circuit a different set of", + "inputs, say blueberries. And let's say my output,", + "oops, let me do it this way. So as my output I get blueberry", + "sauce, if such exists. So apples applesauce,", + "blueberries give me blueberry sauce.", + "Then what I'm going to get if I mix up the two,", + "so let's say I take my circuit, the same circuit with a set of", + "inputs and in this example one output.", + "Let's say I mix up my inputs and some of my inputs in the", + "following way, here I feed an A1 plus B1 and", + "here A2 plus B2 and so on then at the output I am going to get", + "a mush of apple sauce and blueberry sauce.", + "All this says is that if I apply just apples I get", + "applesauce. If I apply just blueberries I", + "get blueberry sauce. Then if I were to figure out", + "how this blender would have worked had I fed in the", + "combinations of apples and blueberries, then for the", + "purposes of understanding that blender all I could have done", + "was taken by two outputs and just mixed them up together", + "myself and that's exactly what I'd get.", + "So if I sum up the inputs my outputs would also be the sum of", + "the outputs with the inputs applied by themselves.", + "So let me take this here and munge around with hit for a few", + "seconds and get something interesting out of it.", + "So notice two inputs, two inputs, outputs.", + "In your notes I've given you another template for the next", + "set of scribbles I'm going to make here.", + "So use the next set of templates on page three.", + "What I'm going to do here is something very simple,", + "set one output to zero and feed a voltage V1.", + "So that's feed a voltage V1 and set the other output to zero.", + "And let's say I get Y1 as an output.", + "And in this case I set the first voltage to zero and feed a", + "different voltage V2 on the second input.", + "And let's say my output is Y2. This is just a particular", + "application of the superposition principle I just outlined.", + "Apply V1 set one output to zero.", + "Apply V2 set the original output to zero.", + "Then what I'm going to find is that the answer will simply look", + "like this, just replace for As and Bs what I just did and we", + "get V1 and zero here and we get zero and V2 here.", + "And as my output I'm going to get exactly the sum Y1 plus Y2.", + "This is simply a particular application of superposition", + "where what I'm saying is the following.", + "If you look at this circuit here effectively what have I", + "done? Effectively what I've done is", + "apply the voltage V1 on one input and a voltage V2 on the", + "other input. V1 here.", + "V2 here. And the output is Y1 plus Y2.", + "What I'm saying is look backwards now.", + "What I'm saying is that the whole components of the output", + "Y1 plus Y2 could individually be derived in the following manner.", + "I could get the component Y1 by simply applying one of the", + "voltages and setting the other to zero.", + "I can get the other component Y2 by setting yet another input", + "to zero and applying the voltage V2 to get Y2.", + "And sum then up and that's my answer.", + "This will become a lot clearer with an example.", + "Again, remember if I have a bunch of inputs applied to a", + "circuit, V1, V2 and so on, and I get some output then what", + "this is saying is that I can alternatively find out the", + "answer by applying just one voltage, setting all the others", + "to zero, measuring the output, apply a second voltage,", + "set all inputs to zero, measure the output and sum of", + "applesauce and blueberry sauce and there you get the answer.", + "Let's do an example. And before we go into that I", + "talked about setting voltage sources and current sources to", + "zero. First of all,", + "what does it mean to set a voltage source to zero?", + "This is the same as this. Setting a voltage source to", + "zero is simply replacing the voltage source with a short,", + "and setting a current source to zero simply implies an open", + "circuit. So when I say zero that source,", + "if it's a voltage source short it, if it's a current source", + "open it.", + "", + "I can take any two nodes in the world and measure the potential", + "difference across them. So there may be some potential", + "difference across these set by the circuit that I haven't shown", + "you on this side. There might be some other", + "circuit that is controlling the voltage of these two nodes.", + "The same with the short. What's V going to be?", + "But there is a V. It's zero.", + "So that's method four, method of superposition.", + "And this method says that the output of a circuit --", + "", + "Again, remember I'm focusing on linear circuits.", + "Remember, I have this playground where LMD applies.", + "And within that playground I'm playing in the south goal area.", + "In the south goal area, in that subset of the", + "playground circuits are linear. So in that part of the", + "playground superposition applies because there circuits are", + "linear. So the output of a circuit is", + "determined by summing up the responses to each source acting", + "alone.", + "", + "Now, in this statement here this source stands for", + "independent source. I haven't talked about", + "independent versus dependent sources.", + "We'll talk about dependent sources a few weeks from today.", + "And just so you don't get confused, for dependent sources", + "you will be looking at Section 3.3.3 of your course notes to", + "see how superposition works with dependent sources.", + "But remember we haven't covered dependent sources yet.", + "We will be covering them about two weeks from now.", + "So let's go back to our example and apply the method of", + "superposition to an example. So the method says sum up the", + "outputs of each of the sub-circuits where I'm applying", + "one source acting alone. So let me just do this here.", + "Let me start with the circuit. And let me start with shutting", + "I off. So I have voltage V --", + "", + "I have R2. And I'm shutting I off.", + "So I have replaced this with an open circuit.", + "So I is zero. Let me call the node voltage eV", + "to reflect that component of the node voltage that arises due to", + "V acting alone. And you should look at this", + "pattern here and very quickly be able to write the answer for", + "patterns like this voltage, the two resistors.", + "That's called a resistive divider.", + "It will appear again and again and again.", + "And eV is simply V times R2 divided by R1 plus R2.", + "That's still my ground node. So the voltage here is simply", + "this voltage divided by the two resistors to give you the", + "current multiplied by R2 to give you the voltage across this R.", + "Remember this pattern. You apply voltage divider", + "patterns probably more times than any other pattern that you", + "might imagine. So that's with the V acting", + "alone. Now, let me do I acting alone.", + "So for I acting alone --", + "", + "And what I do this time around is replace this with a short,", + "replace the voltage source to the short.", + "And let me call this voltage eI for the component of the voltage", + "due to the current I. And eI, in this case,", + "is simply given by yet another pattern here,", + "the current across a pair or resistors is simply the", + "effective resistance multiplied by the current so it's i and the", + "effective resistance is R1, R2 or R1 plus R2.", + "That's eI. That's a component that node", + "due to the current I. Now, so the method says that.", + "Then take these components, sum them up and there you have", + "the answer. So E is simply ev plus ei.", + "The components of V and I acting alone,", + "just simply V times R2 divided by R1 plus R2 plus R1,", + "R2. There we go.", + "Fortunately, the fates have been kind to us", + "and the answer is the same as the answer we obtained with the", + "node method. No surprise here.", + "So this is actually an incredibly simple method.", + "So you can take a very complex circuit.", + "What have you really done here? You can take a very complex", + "circuit and you can solve a very complex circuit by breaking it", + "down into many simple individual sub problems.", + "You will do this in EECS time and time and time again.", + "Whether it's in software systems or hardware systems or", + "what have you, you're often times building", + "complicated systems. Remember doom on this side?", + "And the way and when you put these things together,", + "let's say a large software system, is you don't write the", + "whole piece of software starting main and grunge down.", + "You build a lot of little components and tie the", + "components together. In the same manner here you", + "take a big circuit and you find its behavior for each source", + "acting alone. Lots of little inky dinky", + "simple little circuits. And you will see examples in", + "your homework where you're given a big circuit or because it set", + "all the Is to zero and the other Vs to zero the whole circuit", + "almost vanishes and all that you're left with is a little", + "resistor or two. So this is the very,", + "very powerful method. I'd like to do a little", + "demonstration for you. And what I'm going to show you", + "is the demo is a vat of water. Actually, I'll tell you what it", + "is in a second. But assume it is salt water for", + "now. I'll apply two voltages.", + "In this case I'm going to apply a sinusoid.", + "That's not very good. A sinusoid and a triangular", + "wave. And what I'm going to do is", + "measure the response at this site.", + "Now, this is a vat of salt water.", + "And I'm going to tell you it behaves like a linear system.", + "If you view each little particle, or each little", + "cubic-centimeter or whatever of water, it'll behave like little", + "resistor. So this vat of salt water", + "behaves like big distributed resistor in the following", + "manner.", + "", + "And so on. This of this big mesh of little", + "resistors, but it's all resistors.", + "It's a linear circuit. So I'm going to apply two", + "voltages, a triangular and a sinusoid, and we're going to", + "observe the output. And what do you expect to see", + "there? You will see the superposition", + "of the two, which is you'll see a sinusoid.", + "And then you'll see the jagged triangular thing articulating", + "the sinusoid pattern. What I'm going to do right now,", + "don't put any water yet. This is the vat of nothing", + "right now. It's all empty.", + "Can we show the screen on this side?", + "The oscilloscope screen?", + "", + "OK. Oh, there you go.", + "So this is the screen of the oscilloscope now.", + "Notice that I have a sinusoid and I have a triangular wave and", + "the output is zero. And the reason is there is", + "nothing in this vat. It's empty.", + "So previously when I taught this course I would get", + "saltwater and pour saltwater. Then we discovered a much", + "better source of water that conducted electricity like one", + "real mean fluid. Cambridge water.", + "It just works very pleasantly. It just conducts electricity", + "like nothing at all. And I've been thinking of using", + "Charles River water next time and see what happens,", + "although there we'd probably get some biological organisms", + "doing strange things at you. But go ahead.", + "Our friendly demonstration expert, Lorenzo,", + "will pour some water into the vat.", + "And you should begin seeing the output being a superposition of", + "the two. So as he pours,", + "there you go, do you see that?", + "So you do see the sinusoidal articulation and the jagged wave", + "form. And just to have some more fun,", + "what I can do is increase one of the voltages.", + "And you'll see --", + "", + "Now you know what would have happened if I had used Charles", + "River water. So my output keeps increasing", + "as I increase the corresponding wave form.", + "", + "I could do this, this is fun.", + "So let me pause there and go onto the next topic.", + "So that little demonstration showed you that even something", + "as simple as this physical entity vat of water behaves like", + "a linear system, and we can model that linear", + "system as a set of resistors. Unbeknownst to you,", + "right now, in the past ten seconds I introduced a new", + "concept. It's called subliminal", + "advertising. So one of the things we do in", + "EE a lot is model real systems. So often times if I wanted to", + "look at the behavior of salt, behavior of a vat of water,", + "I can model it as a set of resistors for certain kinds of", + "activities. Just hold that thought for some", + "time later in your careers. All right.", + "That's method four, the superposition method.", + "Remember, it is methods like this that will make your life", + "really, really, really easy.", + "If you find that you are having to do a lot of grunging homework", + "or something, just step back and think", + "superposition, think Thevenin or think", + "composition rule. There must be a simpler way", + "usually. Let's do the next method.", + "This is called the Thevenin method.", + "To derive this method let me start by applying superposition", + "to some circuit. So let's say I have some", + "arbitrary network N. Assume it's a linear network", + "and the network has a whole bunch of goodies in it.", + "It has a bunch of resistors, it has a bunch of voltage", + "sources, and it has a bunch of current sources.", + "Many current sources. Many voltage sources.", + "Many resistors. Some jumbled voltage sources,", + "current sources and resistors. And I look at two nodes in this", + "network. Here are two nodes in the", + "network, two points in the network were elements connect.", + "I'm looking at those two nodes and all I want to do is the", + "following. I want to figure out if I take", + "a rinky-dinky little current source and apply it there,", + "all I want to figure out is what is V and what is I.", + "There is this mongo box out here, a black box of resistors,", + "voltage source and current sources, too many to count.", + "I pick two nodes, apply a current source,", + "and all I care about is what is the voltage that I will measure", + "by applying it here. Notice the current here will be", + "I because the current here is I. And I apply it here.", + "I want to measure what the voltage is.", + "Now, with the insight you've obtained from superposition,", + "you should be able to jump up and state the form of the", + "answer. So by superposition we know the", + "following. We know that the effect of the", + "circuit will be the same as the sum of components being added", + "up. Sum of component,", + "sum of component, a bunch of components added up.", + "Each component will be the response of one source acting", + "alone. So if I can figure out the", + "effect of one source acting alone and put that down here,", + "and do the same thing for all the sources, that's what I will", + "get. So for the source Vm it's a", + "linear circuit. So I know that my answer is", + "going to be, in the final answer is going to be a Vm term and", + "it's going to be multiplied by some alpha M term.", + "I know that. It's a linear circuit so I know", + "that the answer shall have a term Vm multiplied by some", + "constant. Simple, I know that.", + "Similarly, the same is true for, oh, this is the term Vm.", + "And what I can do is I can measure just this effect by", + "setting all the other sources to zero.", + "So I can set all the other current sources to zero and all", + "voltage sources, except for this one,", + "and I can get that answer. So, similarly,", + "for every voltage source I am going to get a term.", + "So for every single voltage source, M1, M2,", + "M3 and so on I'm going to get such a term and they're all", + "going to sum up. Similarly, I'm going to get a", + "term for In. And I know there will be an In", + "term, and I know it's going to be some constant beta", + "multiplying In. In this example of ours here,", + "in this example, remember alpha was this and", + "beta was this constant here. There's some constant beta,", + "some constant alpha. And because I have a whole", + "bunch of current sources there's going to be such a term for each", + "one of them. And each one of these terms,", + "Vm, In will be the voltage I would see here if I set all the", + "other Vms to zero and I set all the other current sources,", + "except for that one to zero. What am I missing?", + "Is that it? The response here,", + "V here. Am I missing anything here?", + "Is that it? Now, don't all yell at once.", + "What am I missing? Current source i,", + "exactly. So if I have a current source i", + "then there's an effect of this current as well.", + "And so I write down i there, too.", + "It's going to be some constant multiplying I.", + "And that constant is going to look like a resistor,", + "right, because this circuit contains current sources,", + "voltage sources and resistors. If I've shorted all my voltage", + "sources and opened all my current sources,", + "what's left in here? Just a whole caboodle full of", + "Rs. It's just going to look like", + "some resistance R. And that's what I get here.", + "So this is what V is going to look like and that's a form.", + "So let's take a look at these components.", + "", + "Let's focus on the easy part first.", + "What does this look like? This component looks like an I,", + "it looks like a current and has some resistance.", + "What is that resistance given by?", + "Supposing I gave you this network and this currency source", + "and I asked you tell me R. How would you measure R?", + "What you would do is open all the current sources,", + "short all the voltage sources, put a ohmmeter in there and", + "measure the resistance R. That's R.", + "OK, so we understand this term. What about this term here?", + "Can someone tell me the units of this term here,", + "this big thing here? Voltage.", + "This is a voltage. This is a voltage.", + "iR is a voltage. So this does behave like a", + "voltage. And it behaves like some", + "voltage V. So notice that as far as this", + "current I is concerned the rest of the universe looks like a", + "resistor and a voltage source behaving in some manner.", + "And let me just call it Vth for now, and you'll know why in a", + "second.", + "", + "The voltage has a form, some voltage plus Ri.", + "So, in other words, as far as this I is concerned", + "this whole network here N full of all the nice stuff is", + "indistinguishable to this I here.", + "So my I is sitting out there injecting a current into two", + "nodes. If I am i, I'm looking at this,", + "this network looks no different than a voltage source in series", + "with the resistor R. Notice that the equation for", + "this simple circuit is this, so I is given by V minus Vth", + "divided by R. Just remember.", + "", + "It's a circuit. In other words,", + "Agarwal sitting here cannot tell the difference if I'm", + "measuring the voltage here between a circuit that looks", + "like a Vth in series to the resistor or this huge mess of", + "voltage sources and current sources and so on.", + "Now, we will talk about Vth and R.", + "R is called the resistance of the network as seen from the", + "port with all the sources shut off.", + "And similarly Vth, what is Vth?", + "Vth is the open circuit voltage.", + "In other words, if I apply the voltage here", + "this is the response of all the current sources and all the", + "voltage sources acting together. So it's as if I took this out", + "and simply measured my V here as if I didn't exist,", + "correct? Because this is the component", + "of i. So if I opened i and measured", + "V, I would get that big term on the left-hand side.", + "That's my Vth. So that inspires the next", + "method called the Thevenin method.", + "", + "In this method what I'm going to do is take some circuit,", + "I'm on Page 9, with a mess of stuff.", + "It's a big mess of stuff. And if I care to look at its", + "impact on something else that I add from the outside then as far", + "as the outside world is concerned this is", + "indistinguishable from a circuit that looks like this.", + "", + "So what I can do is if I want to figure out what's happening", + "here then, for the purpose of my analysis, this simple network", + "here with R and Vth becomes a surrogate for this entire mess.", + "So for the purpose of finding out the behavior at this point,", + "I can take this huge mess and replace it with its Thevenin", + "surrogate or Thevenin equivalent.", + "This is called the Thevenin equivalent of this big network.", + "Let me do an example that will make the method completely", + "clear. Again, remember in EECS,", + "most of our lives are about how can we make things so simple as", + "being able to be analyzed by inspection?", + "And so this is a method that takes you further down that", + "path. So let me use the same circuit", + "that I've been using before, my voltage V,", + "R1, R2. This is an R.", + "I'm 55 minutes fast so we have another three or four minutes.", + "So this is my circuit. And let's say all I care about", + "is finding out i1. That's all I care about.", + "And what I'm going to do is I'm going to box this up and see if", + "I can replace that with its Thevenin equivalent.", + "So I'm going to box that up.", + "", + "What I'm saying is that I'm going to box it up and replace", + "it with this Thevenin equivalent.", + "I don't know what Vth and R are at this point.", + "I'm just calling it Rth for fun.", + "I don't know what these two values are, but if I knew what", + "these two values were I can determine I really trivially as", + "follows. I can get i1 as simply V minus", + "Vth divided by R1 plus Rth. So if I knew Vth and Rth,", + "I can write down i1 by inspection in that manner.", + "So next, finally, how do I get Vth and Rth?", + "You get Rth by looking at this network and shutting off all the", + "voltage sources and measuring the resistance there.", + "So I short my voltage source, that's R1.", + "Oops, wrong way. I need to look this way.", + "So looking this way, that's what I get.", + "So what's Rth? Rth is simply R2.", + "So I have opened my current source.", + "Similarly, for Vth, remember all I want to do is", + "look at the two nodes, step back, put a voltmeter", + "there, measure the voltage, that's my open circuit voltage.", + "So the way I do it is I take the circuit and simply measure", + "the voltage there. That's R2.", + "That's my current capital I. And I simply want to measure", + "the open circuit voltage here, which is what?", + "Just simply if I stand back and I kind of gingerly measure the", + "voltage here without disturbing anything, I simply get IR2.", + "So Vth is IR2 and Rth is R2 and here is the formula for the", + "current in this branch when I apply a voltage source and a", + "resistor R1 to this little circuit here.", + "OK, let's pause and let me summarize this in about ten", + "seconds. I had this circuit here.", + "I wanted to find out i1. So what I said I'd do is take", + "this complicated mess, well, it's not a complicated", + "mess but assume it is, and replace with it a", + "resistance Rth got by turning off all the sources.", + "And the voltage in series, Vth, which I get simply by", + "pulling this thing out, taking my input,", + "this part out and simply measuring the open circuit", + "voltage out there, Vth.", + "And then I replaced the whole network with this new network", + "that they call the Thevenin network, and voila,", + "I get the answer in a second." + ] +} \ No newline at end of file diff --git a/subs/TXJIhDHtHSI.srt.sjson b/subs/TXJIhDHtHSI.srt.sjson new file mode 100644 index 0000000000..35e67f4ac2 --- /dev/null +++ b/subs/TXJIhDHtHSI.srt.sjson @@ -0,0 +1,2141 @@ +{ + "start": [ + 0, + 6000, + 8000, + 17000, + 23875, + 29000, + 34818, + 40436, + 45352, + 50770, + 54783, + 60000, + 64846, + 67855, + 70529, + 75208, + 79470, + 84401, + 86908, + 93702, + 97276, + 104936, + 112212, + 120000, + 129043, + 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Good morning.", + "", + "In the last lecture I did a little demonstration for you", + "where I showed you a pair of inverters.", + "And showed you that the output of the first inverter looked", + "weird, certainly not like anything we have seen thus far.", + "It looked like a slow rising transition like this.", + "And using that motivation we have begun our study of RC", + "circuits. And in particular for today the", + "lecture is titled \"Digital Circuit Speed\".", + "We are going to look at the fundamentals of digital circuit", + "speed. And it all boils down to an RC", + "delay. By the end of the lecture,", + "I am going to show you two numbers that you can look at a", + "circuit and obtain by observation, multiply them out", + "and you will get a good idea of the speed at which a circuit", + "will run. It is pretty amazing.", + "So as a quick review -- The relevant section for this", + "is Chapter 10.4. As a review,", + "we said to understand things like this we need to develop the", + "foundations for RC circuits. And the example I covered was", + "that of a very simple circuit that looked like this --", + "An RC circuit of this form. And I also showed you that for", + "an input of the form, input that steps from zero to", + "VI at time T equal to zero. And assuming that the capacitor", + "state at time T equals zero was zero.", + "What this means is that the capacitor starts from rest,", + "so at time T=0, oops, this is VI,", + "I'm sorry. So we assume that the capacitor", + "starts from rest. At time T=0 I apply a VI step,", + "capital VI. And then I want to look at how", + "the voltage across the capacitor behaves.", + "And we did a bunch of analysis. And at the end of the day,", + "in the final demo in the lecture last time I showed you", + "that the capacitor would behave like this.", + "It would start off at, oops.", + "I am sorry. This should be,", + "let's assume that started off at VO.", + "We get a different equation for zero.", + "So let's say the capacitor started off at VO,", + "in which case VC at time T=0 is VO as expected.", + "And we showed that the output would look something like this.", + "After a long period of time this would come up to VI and", + "this rise had a time constant of tau=RC.", + "So we wrote the equation for this waveform.", + "And this is the case when VI is greater than VO.", + "I would like you to stare at the circuit and this result here", + "to get more intuition on what is going on.", + "At time T=0, VC starts off at VO as expected", + "because I am telling you that is the case, that is initial", + "condition. It starts off at VO.", + "Then this one steps to VI. There is no infinite transition", + "anywhere here, and so the capacitor holds its", + "voltage at VO, at time T=0.", + "And then the VI here, which is greater than VO,", + "begins to charge the capacitor up, charge it through this", + "resistor. And so therefore the capacitor", + "charges up. After a long period of time,", + "from the basic foundations of capacitors, we know that the", + "capacitor appears like a long-term open circuit to DC.", + "This is a DC voltage VI. So it appears like an open", + "circuit. So after a long period of time", + "VI appears at the end. And from here to here I have an", + "exponential rise that is typified by an equation of the", + "form -t/RC. This kind of waveform rising", + "from a smaller value to a higher value is typified by this", + "expression. We saw the expression when we", + "developed the equations last time.", + "On the other hand, if the input was such that VI", + "was smaller than VO, so let's say VI was smaller", + "than VO then what will happen is that the capacitor voltage would", + "start off at VO, because I am telling you that", + "is the initial condition, and would then decay in this", + "manner to the final value of VI which is the input.", + "Instead of going up this way it decays down to the final value", + "applied to the circuit. Again, the time constant is RC.", + "But this is typified by a form, this is exponential rise and", + "this guy e^-t/RC is an exponential decay.", + "The key thing to remember is that when you have RC circuits", + "of this form, the waveforms that you get are", + "either each of the e^-t/RC or 1-e^-t/RC.", + "So you can now begin to see how waveforms such as that come", + "about. We will do an example and sit", + "down and compute the inverter delay.", + "And notice that this waveform here is very typical or", + "corresponds to this waveform that we see here.", + "Here I am starting at VO. And assuming this axis starts", + "off at zero, this one starts very close to zero and then", + "rises up to some final value. So far I have reviewed some", + "material for you that I covered the last time.", + "As a second step, I would like to give you a much", + "more intuitive approach -- -- that doesn't involve solving", + "any differential equations. And the reason I do this is", + "that most experienced circuit designers do not sit down and", + "write differential equations each time they see an RC", + "circuit. When you are starting out and", + "you see an RC circuit, you say node method and you", + "write the differential equation, but experienced people don't do", + "that. They look at it and they can", + "sketch the waveform out by inspection.", + "And I will show you how to do that.", + "It is indeed incredibly simple once I give you some intuition.", + "Throughout the rest of this course, I will be showing you", + "many such examples where initially I develop the", + "foundations of stuff and then show you an intuitive approach", + "that very quickly lets you either get the final answer or", + "at least sanity check the answer that you have gotten.", + "And this is how experienced circuit designers deal with", + "stuff. How many people here have seen", + "this movie Bend it Like Beckham? So you know this Beckham", + "character doesn't think about how he is going to curve the", + "ball. He just does it and it happens.", + "He doesn't sit down writing differential equations to find", + "out the projectile trajectory and all of that stuff.", + "You just kind of do it. These series of intuitions I am", + "going to give you is going to be in line with the Bend it Like", + "Beckham kind of intuition. And this one in particular I", + "would like to do in honor of one of your recitation instructions", + "Professor David Perreault. And so this piece of intuition", + "is going to be termed \"Practice it Like Perreault\".", + "Watch what I do with the other names.", + "Professor David Perreault is really a world expert in", + "designing really incredible power supplies for very,", + "very small chips and so on. He doesn't start writing", + "differential equations to do this stuff.", + "He looks at it and sketches it out.", + "Let me show you how he would do this.", + "Suppose I have my circuit like before, VI, R and C,", + "and I am telling you that VC(0)=VO.", + "And my input VI is a step that looks like this.", + "VI is a step. How would Professor Perreault", + "do this? Let's do it completely by", + "intuition. No math here.", + "All right. We know that I have told you", + "that this guy starts off at VO. I am telling you that.", + "You know it is going to start at VO.", + "And there is no impulse or huge infinite transition,", + "and so the capacitor starts off at VO.", + "We also know from basic capacitor properties that after", + "a long period of time, in the steady state,", + "this is but a DC voltage. If you apply a DC and here is", + "my capacitor. After a long period of time", + "this guy is going to look like an open circuit.", + "It is going to charge up to some value and then is going to", + "look like an open circuit. Because if it didn't,", + "you would keep charging it and its voltage would keep", + "increasing. That doesn't happen,", + "it looks like an open circuit. So it looks like an open", + "circuit in the long run. The voltage across it must be", + "capital VI. If I don't have current flowing", + "in the circuit then the only way that can happen is --", + "This open circuit. Capital VI appears across the", + "capacitor. Well, after a long period of", + "time I know that the output must look like this.", + "In this case, I have assumed VI is greater", + "than VO. So you have two points of your", + "curve, VO and VI after a long period of time.", + "And, as I told you earlier, with capacitors you get two", + "kinds of curves. Two things.", + "What you do is go zoop. There you go.", + "You're done. And this has an exponential", + "rise. This is with the form", + "1-e^-t/RC. So we can write an equation for", + "that as follows. VC we know has something to do", + "with minus t/RC. This is of that form,", + "so there has to be that term in there somewhere.", + "And I start off with VO. At time T=0 this is one and", + "this is one, so this term becomes a zero.", + "At time T=0 that becomes a zero so I get VO here.", + "I am going to make sure this stuff stays zero at time T=0,", + "so I start off with VO. Now, as time wears on what", + "happens here? This voltage here,", + "VI-VO, if you look at this difference.", + "That is exponentially decaying over time.", + "And so therefore all I have to do here is write VI-VO.", + "There is the answer. I know the form of the curve.", + "I am just fitting an expression that meets this form.", + "This starts off at VO. When time T=0 this second", + "expression is zero and so it is VO.", + "And this difference here decays down to zero.", + "And this difference here, VI-VO is multiplied by this", + "term here and that is what I get.", + "And you can confirm this. At time T=0 this is zero.", + "At time T infinity this goes to zero, this goes to zero leaving", + "a one, and VO and minus VO cancel and I get a VI.", + "Virtually any such simple voltage source,", + "current source, resistor, capacitor,", + "circuit for most inputs like steps and so on can be analyzed", + "in this manner. Initial value,", + "final value, it's simple.", + "And just to show you that this is simple, I am going to label", + "this expression this way. It is of the form 1-e^-t/RC.", + "Just remember that. Now, by the same token,", + "what if VI had been smaller than VO?", + "Then that is simple, too.", + "I would have had my VI being here.", + "VI would have been here. And that is of the form.", + "In this particular situation, here is my VI,", + "my starting value and I do this.", + "And just to label that, let me label that this way.", + "I just told you that for RC circuits you go this way or you", + "go this way. So it is down here.", + "I get some kind of an exponential decay.", + "And, like before, think of this one.", + "This one has VI as a base value here.", + "And the difference between the two is VO minus VI.", + "And that difference decays. So I have a VI out here,", + "and this difference decays so I get VO-VI and that decays in", + "this form. So I get an exponential decay", + "of this difference here. Just stare at it for a while", + "longer. You should be able to just go", + "and knock it off like this, just like Professor Perreault", + "would. No differential equations.", + "Just write it down by looking at the curve.", + "Let's keep these two in mind, OK, these forms?", + "One is the 1-e^-t/RC form and the e^-t/RC.", + "Both have a time constant RC. Let me just make this a dashed", + "line just to be on the safe side here.", + "", + "That is our first piece of intuition.", + "And, as I pointed out before, in problems you face in life or", + "in ones that we give you, feel free to use the intuitive", + "method. Or what you can do is apply the", + "mathematical method and then check your answer by using your", + "intuition. What I would like to do next is", + "apply what you have learned so far to figure out what we set", + "out to figure out, which is the delay of my", + "inverter. I had promised you that by the", + "end of this lecture I was going to close the loop on that little", + "demo. I was going to close the loop", + "for you on this little circuit that we had looked at,", + "one inverter driving another inverter.", + "This was A, this was inverter X, and this was my node B.", + "The green curve you see out there, the middle one has a", + "transition shown up there. And what I am going to do next", + "is use the results we have gotten so far to compute a", + "number. We are going to compute a delay", + "number both for a rising transition.", + "We will call that delay DR for rising transition.", + "And we will compute a delay for the falling transition DF.", + "Remember, that this is the input that falls down sharply.", + "The intermediate node B rises much more slowly.", + "And because this rises much more slowly this guy here falls", + "a little after this transition here, and so there is a delay.", + "And I am going to apply what we have learned so far and do an", + "example for you and figure out what that delay is.", + "This is an absolute foundational calculation done in", + "building digital circuits all the time.", + "It is remarkable that something so simple is used in designing", + "even the most complex of circuits to obtain very quick", + "ideas of what my delay will look like when I have some subcircuit", + "driving some other piece of subcircuit.", + "Let me just draw a few equivalent circuits for you.", + "The internal circuit looks like this.", + "", + "This is my inverter X, A, my node B.", + "And notice that I have this capacitor CGS.", + "Since I am interested in this node, let me show you that,", + "this capacitor explicitly, it is because of this capacitor", + "here that arises because of this MOSFET here between the gate and", + "the source. And that capacitor gives rise", + "to this RC thing that we are seeing.", + "This is RL, this is RL, VS, VS.", + "And let's say, just as up there,", + "at time T=0 I get a transition like so, a falling transition", + "from say 5 volts to 0 volts at the node A.", + "This is VA here. That is shown up there.", + "And VB --", + "", + "We had expected that VB would look like this.", + "We expected VB to be instantaneous and looking like", + "that, but instead because of the capacitor VB looks like this.", + "And remember, again, this is of the form", + "1-e^-t/RC. And we will write down the", + "answers by inspection. From this let me draw the", + "connection to circuit delay by showing you another little graph", + "here t, VB, zero. And what I am going to show", + "you, this is 5 volts. And so the output goes like", + "this from close to zero to 5 volts.", + "It is close to zero. Because, at least with the", + "inverters we have been seeing in lab and so on,", + "the RON for the inverter is very, very small compared RL.", + "So it is virtually zero down here.", + "And so what is the delay? I mentioned there are two", + "delays of interest. One is the rising delay.", + "That is the logical value at the end, if I wait a long enough", + "period of time, is a logical one.", + "Delay is simply defined as starting from here how long does", + "this output take to get to a valid one?", + "At what voltage here can I say that this transition corresponds", + "to a logical one? At what voltage here can I say", + "that that represents a valid one?", + "Any ideas? Yes.", + "It depends on the discipline, bingo.", + "So it depends on the discipline.", + "Now let's get more specific. Since it depends on the", + "discipline, at what value based on something in the discipline", + "can I say this thing is a logical one?", + "This is an output remember. VOH, bingo.", + "There is some VOH somewhere. And it takes some amount of", + "time to get to a valid logical one output, ergo there is your", + "delay. This is tR.", + "And I call this the rising delay of the inverter X.", + "It is interesting that the rising delay of inverter X,", + "based on our model, depends on the parameters of", + "this inverter and the parameters of whatever it is driving.", + "So remember that the delay is not necessarily just the", + "property of the inverter itself, but it depends on the context.", + "If I stick my inverter before another inverter like this,", + "it is the capacitance on that inverter by our model that tells", + "me what the delay is going to look like, of course in addition", + "to RL. And we will do the math in a", + "few seconds. By the same token,", + "if I had this wire connecting not to one inverter but going to", + "ten other inverters, I expect to have a capacitance", + "equal to ten times CGS. And so therefore this thing", + "should rise even more slowly, correct?", + "The more capacitance on here the slower it rises up.", + "Simple. If I put more and more load on", + "this line by putting more and more MOSFETs on that line,", + "more and more inverters this will rise slower.", + "In our example I just have one, so let's go ahead and compute", + "the delay. This is called the rising delay", + "of X. That says that for this node", + "here to go from its output value to a valid one,", + "which is VOH how long does it take?", + "Notice that if this capacitor was zero then you would have", + "seen an instantaneous transition.", + "If you have an instantaneous transition then notice that the", + "rising delay was zero. That was the model we had", + "looked at up until learning about capacitors.", + "So let's go ahead and compute the number.", + "I can draw an equivalent circuit for computing a rising", + "delay. The equivalent circuit for the", + "rising delay looks like the following.", + "The VS voltage source, with a resistor RL and a", + "capacitor CGS, because when I turn this guy", + "off, this guy has gone off, and so as far as the rise time", + "of this node is concerned I can look at this circuit,", + "ground through CGS through RL through VS back to ground.", + "And just for simplicity, let me draw this in a form that", + "we understand.", + "", + "CGS. Let me use this as my ground", + "node. And this is the voltage VB.", + "And this is RL. And V is simply VS once that", + "transition happens. My other equations here,", + "VI=VS. And what is VB(0)?", + "VB(0) is at what value does this node start out?", + "Notice that for simplicity here if this RON is much,", + "much smaller than RL, then this node would be very", + "close to ground. So I will just go ahead and say", + "that VB at T=0 is approximately zero.", + "And then what I want to find out is what does the value look", + "like for time starting from zero and then going forward?", + "Well, we have become experts at this now.", + "", + "Let's do the intuition here. Start off with zero.", + "That's good. Because my initial value is", + "zero, I start off here. What is the final value?", + "After a long time, since this is a DC voltage,", + "what would be the value at VB after a long time?", + "Pardon? VS.", + "If I wait long enough then it is going to be at VS.", + "This is greater than the initial value,", + "so we're done. That is my 1-e^-t/RC form.", + "It took me three seconds there. It's pretty cool.", + "We could add the expression for this.", + "And the expression was I take my starting value,", + "which is zero, and I add to that this", + "difference VS and I multiply that by this form.", + "There we go. And remember I get this from", + "that rising form up here. V0=0, this is zero,", + "so it is simply VI times that, and VI=VS.", + "I really would like you to get this intuition.", + "If I had two choices, one is that you understand the", + "intuition and are able to sketch that versus in your sleep be", + "able to solve the differential equation and get to the answer.", + "I would much rather you get the intuition, if it is one or the", + "other. It is very simple.", + "Start off at zero, I go chuck, and boom,", + "I get to VS and this is my 1-e^-t/RC form.", + "I need to compute tR. And tR is the time that this", + "takes to get to VOH.", + "", + "For what value of time, for what T, does VB reach VOH?", + "I want to find tR. What's tR?", + "From that equation, that simply tells me the", + "trajectory of VB as a function of time.", + "And so I need to find out what is T for which VB is VOH?", + "I write VOH=VS (1-e^-t/RC). So after a rise time my output", + "is going to be VOH. And so let me go ahead and find", + "tR. Let's see.", + "I bring this to this left-hand side and divide VOH by VS,", + "and then I move things around and what I end up getting is", + "-tR/RC and on the other side I get ln(1-VOH/VS).", + "Divide VOH by VS, that is this,", + "move this to the other side, and move e^-t/RC to this side.", + "And take logarithms on both sides.", + "This is what I get. tR is therefore -RLCGS", + "ln(1-VOH/VS). That is my rise time.", + "You can just do this by inspection.", + "It is just so awfully simple. Just to give to some intuition", + "with numbers and so on. Let's say that RL=1K,", + "VS=5 volts, VOH=4 volts, CGS=0.1 pF.", + "This happens so often that we often time call it \"puff\".", + "0.1 puff. It is pF, it's called puff.", + "If it is nF, I don't know why they didn't", + "call it \"nuff\". They just call it nanofarads.", + "TR for these numbers gets to be one times ten to the three times", + "point one times ten to the minus twelve for pico-farads", + "ln(1-4/5). And if you do the math you get", + "this down to 0.16 nanoseconds. This means that if I had an", + "inverter like that droving another inverter then my output", + "transition would be delayed by 0.16 nanoseconds.", + "Trust me, when Intel builds microprocessors or when Broadcom", + "builds its cable modem chips, they have to do this one way or", + "the other using a computer tool or by hand for virtually every", + "little subcircuit in their chip. That is how you get the delays", + "or some approximation thereof. What I want you also to do is,", + "for no particular reason, I will just compute for you the", + "following quantity RLCGS. The time constant of that", + "circuit for no reason at all. I am just going to compute it", + "and stick it here. And RLCGS 1 K times 1 pF is", + "simply 0.1 nanoseconds. I am just writing it and", + "sticking it there for no particular reason.", + "The next step let's do the falling delay,", + "DF. That is the rising delay.", + "And, although I didn't show this to you in the demo,", + "there is a corresponding delay of the fall time.", + "It doesn't fall instantly, but rather it falls rather", + "slowly. Let's draw the equivalent", + "circuit for when the node X falls.", + "Notice that in my inverters here, this node starts off being", + "at VS. This is high.", + "And this is going to fall because when I turn this", + "transistor on it is going to pull this node to ground or it", + "is going to fall down. And what is the equivalent", + "circuit? The equivalent circuit is that", + "ground through capacitor to this node.", + "At this node I have RON connecting to ground and I have", + "RL connecting to ground through VS.", + "Let me draw that little circuit for you.", + "Remember life begins and ends on storage elements,", + "so I will draw them first. My storage element CGS.", + "That is VB. And, as I said,", + "this is node X, it goes from RON to ground,", + "and it also goes through RL through VS to ground.", + "And in this particular situation VB of zero for the", + "following delay, VB starts off at VS so VB of", + "zero is VS. And the final output I am not", + "sure yet. What is the final value of the", + "voltage at this node? I don't know that yet.", + "I need to compute that. So what I will do is whenever", + "you see something like this, a capacitor connecting to", + "linear stuff, or a nonlinear element", + "connecting to linear stuff. For no apparent reason you", + "should at least think about what?", + "Think Thevenin, exactly.", + "And then see if you can use the Thevenin method to simplify your", + "life. Capacitor, a bunch of stuff", + "here, I need to find out the initial value.", + "Oh, I know that. That is VS.", + "Done. I need to find the final value", + "using my intuitive method. For the final value,", + "I could do it just by looking at this, but I wanted to throw", + "in Thevenin. Hey, let me try to the Thevenin", + "equivalent and see if that makes my life any easier.", + "VTH. The Thevenin method says that", + "you can replace this circuit here with a Thevenin equivalent", + "of the sort for the purpose of determining what happens at this", + "node given that that is linear.", + "", + "So I need to find out that for the purpose of determining what", + "happens at the node X. I have to replace this with its", + "Thevenin equivalent. And I now need to find out RTH", + "and VTH. So I get RTH by looking in", + "here, shorting this guy and looking at the resistance.", + "So I look in like this, then I short this guy here and", + "I get RL in parallel with RON because this one shorts to", + "ground. So RTH is simply RL in parallel", + "with RON. This is a convenient notation", + "for RL being in parallel with RON.", + "And you all know the value of that.", + "It is another one of our very simple patterns like voltage", + "divider and so on. Resistances in parallel can be", + "computed as RL RON divided by RL plus RON.", + "What is VTH? VTH is the open circuit voltage", + "here. If I take out this capacitor,", + "I want to find out what the voltage here is.", + "Ah-ha, voltage divider. VS, the voltage divider here,", + "RL and RON. I could write this down as VS", + "times RON/(RL+RON). Remember you will see again and", + "again and again and again in 6.002 or any circuit stuff that", + "you do, you will see them all over Thevenin.", + "Voltage dividers, current dividers,", + "resistances in series, resistances in parallel,", + "RC thing-a-ma-jigs like this. So if you just remember those", + "10 to 15 intuitive patterns then you are pretty much set for", + "life. It just comes on again and", + "again and again. Parallel resistors.", + "Voltage dividers. You should be able to write", + "down a voltage divider in your sleep.", + "So this is what I have. Let me now write down", + "intuitively what I expect the node X to do just by inspection.", + "Let's see. What is the initial value of", + "the voltage across the capacitor, intuitive method?", + "This is how Professor Perreault would do it, remember?", + "He would start off by saying ah-ha, initial value is VS", + "because I am told it is VS. I start off with VS.", + "And so I start off here. What is the value after a long,", + "long time based on this circuit here?", + "V Thevenin. After a long time this is a DC", + "voltage because that is a DC voltage.", + "The capacitor looks like an open circuit after a long time.", + "And VTH appears there so it is simply V Thevenin.", + "", + "And then when you see those two, boy, I love doing this,", + "you go like this. That is the coolest part.", + "And then I am done. It is so simple.", + "Three seconds or less, I am able to tell you what the", + "delay of an inverter is purely by intuition,", + "completely intuitively. I mean I haven't done any", + "solving. It is just by observation.", + "Took this circuit, made my life easy,", + "Thevenin, looked at RTH, VTH and then sketched it by", + "inspection. Again, if you find that things", + "are really, really, really simple don't be", + "surprised. Once you get some conceptual", + "understanding things are indeed very simple.", + "You can eliminate a lot of math just by staring at things", + "attempting to build up the intuition.", + "As a next step what I can do is write down the expression for", + "VB. And I write down the expression", + "from a falling transition. How do I do it?", + "What was it? What is the method?", + "I take the lowest value of interest here.", + "That is VTH. And then I add to that this", + "difference decaying exponentially.", + "And that difference is simply VS-VTH.", + "And that decays exponentially. This form is the e^-t/RC form.", + "And, boom, I am done. Many of you are wondering,", + "Professor Agarwal, if life was so simple,", + "why on earth did you have us mess around with those", + "differential equations to get here?", + "You show us differential equations and then you don't use", + "them anymore. Well, that is a good question.", + "The answer to that is that you need to understand the", + "foundations. Once you understand the", + "foundations you can find simplifying techniques to get to", + "where you need to be, but you need to understand the", + "foundations. You need to at least see why", + "things are the way they are at least once.", + "Understand the foundations and then find intuitive ways of", + "getting your answers. So now my falling delay here", + "is, I start off with VOS and I need to get all the way down to", + "what value to compute. At some point here,", + "this is a valid one, at some point VB becomes a", + "valid zero for the output. And that is when I stop my tF", + "block. What is the value here for this", + "to be a valid zero? Don't all yell at once.", + "VOL. I simply had to figure out what", + "is the value of time, this is Page 7,", + "for which this expression decays down to VOL.", + "So it is VTH+(VS-VTH) e^-tF/RC. Then I simplify this.", + "How do I do that? VOL-VTH.", + "Then I divide that by VS-VTH. So VOL-VTH.", + "Divide that by VS-VTH. Take logarithms on both sides", + "and then multiply by RC. So I get tF is -RC log of that.", + "This is R Thevenin and this is CGS.", + "How did I get this? VOL-VTH divided by VS-VTH.", + "Take logs on both sides. And then multiply throughout by", + "-1/-RC and I get my tF. Done.", + "Let's do it for the same set numbers, just that we add an RON", + "of 10 ohms. I will do this for RON of 10", + "ohms and compute the value for you.", + "tF=-RTH. RTH is RON parallel RL.", + "This is 10 ohms. That is 1K.", + "So 10 ohms in parallel with 1K is approximately 10 ohms.", + "So let me just use approximately 10 ohms.", + "1 pF, that is RC times ln of VOL.", + "Oh, I need to give you a VOL. Let's say my discipline has VOL", + "being 1 volt. And so therefore I end up", + "getting a VOL-VTH divided by VS-VTH.", + "Since RON is much, much, much smaller than RL,", + "since RON is 10 ohms and this is 1K, most of VS will drop", + "across RL. This is a hundred times", + "smaller. Compared to VOL,", + "which is 1 volt, VTH is very,", + "very small. VTH will be on the order of", + "0.05, and so therefore I simply write down VOL here and say VTH", + "is approximately zero, and I get VS-VTH.", + "This is approximately 5. So let me just say this is", + "approximately. And if you do it you will get", + "1.6 pico-seconds. Again, just for fun,", + "let me write the corresponding RC time constant for the", + "circuit, which is RTHCGS. So RTH is approximately 10 ohms", + "and CGS is 1 pF, so this is 1 picosecond.", + "", + "Now you will understand why I have been writing this time", + "constant down. It turns out that the time", + "constant is a very, very important number.", + "So you see an RC circuit, and you compute its time", + "constant for an RLC connection like this, it is the series", + "resistance times the capacitor. The time constant is a very", + "important number. And usually the circuit delays", + "are in the neighborhood of the time constant value.", + "In this case this is 1 pS. That is 1.6 pS.", + "And in this case we had 0.1 nS and 0.16 nS.", + "So the time constant itself is a good indicator of what your", + "delays are going to be like. If you have no time,", + "you are sloshing your cereal down in the morning and you need", + "to know how long the delay of the inverter very quickly,", + "you have three seconds. Just do the RC and that is a", + "good first approximation. What I would like to do next in", + "the last three or four minutes is set up a little demo for you", + "for your recitation, and then your recitation will", + "cover it.", + "", + "This is a true story. This really,", + "really happened. In this West Coast school,", + "which shall remain nameless, they had a chip,", + "they built a chip. And the chip had a bunch of", + "pins, as you might imagine. And the pin,", + "as you have a trace on a board, a wire on a board there are", + "some capacitance attached to wires, between the wire and", + "ground. And that is a capacitor.", + "And they just called it a load capacitance.", + "It could have been 0.1 pF or 0.01 pF or something like that.", + "What they found when they built this chip --", + "What they found was that the voltage here they expected to", + "look like this, this computer science", + "abstraction and so on, zero to one transition,", + "boom, it should look like this. But for the reasons we saw", + "today the observed transition was much slower and looked like", + "this. So the students said ah-ha,", + "let's speed up this chip. We can speed up the chip by", + "looking at the RL and RON of my driving inverters.", + "And if I make RL small -- Notice if I make RL small my", + "delay is small. If I make RON small my falling", + "delay is small. So let's make really small RLs", + "and RONs and let's all have fun. Unfortunately,", + "what they observed was that by making RL and RON both small,", + "the RC time constant small they expected to see a much sharper", + "rise time. And this was the original.", + "But what really happened was -- They expected this to get", + "faster and kind of look like this, but what happened was", + "disaster struck. What they observed was", + "something like that. This is a real-life story.", + "And so instead of getting something like this they go", + "something like this. And why is that a problem?", + "That is a problem because notice when I expect to be at a", + "zero, I got some spikes that went higher than VIL into the", + "forbidden region and did bad things to me.", + "So let me show you a little demo and show you that that's", + "exactly how the circuit is behaving.", + "", + "Notice that this is what I expect but this is what I see.", + "Look at the purple curve here. Notice these spikes that are", + "showing up there. This is true.", + "They saw it happen. And why is this happening?", + "It turns out that what was happening was that the two pins", + "were next to each other. And I will show you a little", + "demonstration here. Let's see if you can figure out", + "why this was happening. Think of these as two pins and", + "the pins are close together. I am just modeling the two pins", + "with a role of wire. And what I am going to do is --", + "", + "I am going to separate the wires and keep them far apart.", + "It is like keeping my pins far apart.", + "Hey, guess what happened? Those nasty spikes went away.", + "But then I cannot keep my pins 1 meter apart on a chip.", + "Your laptops are going to look 20 yards long.", + "You want the pins to be very close to each other so that you", + "can have many pins on chips and therefore have very small", + "systems. But then look,", + "I get the spikes. Any idea why that is happening?", + "Why is that when the pins are close together I get those", + "spikes? Any ideas?", + "Somewhat? We just learned about", + "capacitors, so this must have to do with capacitors.", + "There is this parasitic capacitor between the pins,", + "exactly. Here is what is happening.", + "Here is what I expect. I expect a nice square wave at", + "the output. But instead I have a pin next", + "to me. And I have a faster wave form", + "driving it. And so therefore there is a", + "parasitic capacitor here. And because of that I get", + "something called \"crosstalk\". And the model for crosstalk is", + "some resultant resistance with the parasitic capacitor and I", + "get those spikes. And the 6.002 experts saw the", + "solution. They said how do we fix this", + "problem? 6.002 experts said the way we", + "fix this problem if it is slow it may be better.", + "Instead of having sharp transitions let me drive it with", + "slower transitions. Let's switch to the demo again.", + "You will see this in recitation, but I will show you", + "the demo very quickly. I have a sharp transition of", + "the input, which is that yellow thing out there.", + "I am going to make the transition slower.", + "Switch to a triangular wave. And you will notice the spikes", + "go away. Oh, no.", + "That is the wrong one. The other one.", + "There you go. The moment I switch to a slower", + "transition boom, the spikes go away.", + "You want to switch back to square?", + "There you go. The 6.002 experts saw the", + "solution. Slower transitions.", + "And you will do this example in detail in Section tomorrow.", + "Thank you." + ] +} \ No newline at end of file diff --git a/subs/V0z_f7qxLcY.srt.sjson b/subs/V0z_f7qxLcY.srt.sjson new file mode 100644 index 0000000000..501494e55d --- /dev/null +++ b/subs/V0z_f7qxLcY.srt.sjson @@ -0,0 +1,2222 @@ +{ + "start": [ + 0, + 4000, + 8500, + 17166, + 23000, + 41000, + 46304, + 51608, + 54652, + 59000, + 62665, + 68163, + 72504, + 77810, + 79353, + 84562, + 89000, + 92043, + 96126, + 99838, + 104366, + 106445, + 111048, + 113943, + 118174, + 123000, + 126188, + 131867, + 136151, + 139738, + 144221, + 150000, + 154737, + 157788, + 161000, + 166000, + 174821, + 179000, + 183000, + 189201, + 194826, + 202471, + 210980, + 215263, + 218953, + 222559, + 227170, + 230021, + 233878, + 238155, + 242201, + 244891, + 248070, + 252961, + 256141, + 260461, + 264375, + 270000, + 273588, + 277529, + 281118, + 284355, + 289000, + 294000, + 300000, + 306000, + 312790, + 318873, + 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+ 3121222, + 3123608, + 3125703, + 3128264, + 3130301, + 3132687, + 3135132, + 3138333, + 3143000 + ], + "text": [ + "", + "All right. Good morning all.", + "Today we embark on another new chapter in what we do.", + "And the topic is going to be --", + "", + "We will talk about this thing called an Operational Amplifier.", + "Before I get into the lecture, I want to point out a couple of", + "things. One is that you are going to", + "hear about two big words in today's lecture.", + "Two big and incredibly important words.", + "And I want to mention those words to you right now so that", + "when I come to them in lecture you can say OK,", + "I better pay really close attention, these are important", + "words. All right.", + "One of them is abstraction. The second one is feedback.", + "Two incredibly important concepts.", + "Abstraction, you have been seeing a couple", + "times during this course, once in the beginning where we", + "abstracted out Maxwell's equations by focusing on a", + "smaller playground and simply using KVL, KCL in place of those", + "equations. A big abstraction.", + "It turns out that almost all of EECS is based upon abstractions", + "at various levels. In the first lecture,", + "I also showed you the layer upon layer of abstraction that", + "we built to be able to build interesting systems.", + "The second big thing is feedback.", + "And I am going to relate this to anti-lock breaks in cars.", + "And so, you can wait and see how we do that.", + "It's an incredibly important concept.", + "Before we dive into the amplifier abstraction,", + "let me first talk about something that you know.", + "Start with something that you know and then lead up into the", + "operational amplifier and its circuits.", + "You know about the MOSFET amplifier.", + "", + "The MOSFET amplifier that you know about looked like this.", + "It was based on a MOSFET.", + "", + "There was a VS supply. There was a vI input,", + "a vO output and, as I said, a VS supply.", + "So, this was a MOSFET circuit that you've seen before.", + "One way of viewing this is that this circuit has three major", + "ports. This here is the input port", + "with voltage vI. This here, between the drain", + "terminal and the ground, is the output port.", + "I take the output between the drain terminal and ground.", + "And, finally, we have a third port,", + "which is this one. It is called the power port.", + "I apply VS between this terminal here and the ground", + "terminal. And that gives us the power", + "port. This device here was a three", + "port device. Input port or control port,", + "output port and a power port. And so we looked at the circuit", + "and did a whole bunch of analyses of it.", + "Then what I can do at this point, now that you've seen", + "this, it's often times interesting to think about", + "abstracting this out into some kind of a building block.", + "Much like in software, you write a procedure and you", + "abstract out the internal details of the procedure in the", + "procedure declaration and in the call that you make.", + "In the same way, we can take this little device", + "here and abstract that out into the following abstraction.", + "", + "We could abstract that out as a device that looks like this.", + "", + "I have my input port, I have my output port and I", + "have my power port. So, I can apply VS here.", + "Notice that I've taken these six terminals here,", + "one, two, three, four, five and six,", + "and put a box around it. And just exposed the terminals", + "to you. And I need to tell you a little", + "bit more about the internal properties, but suffice it to", + "say that you can begin working with this little block.", + "An even simpler version of this for many applications might just", + "look like this, vI and vO where there is a", + "ground that is shared among them that is implicit in this", + "picture. And vI and vO can simply be the", + "node voltages at these nodes. This is a progressively more", + "abstract representation of this amplifier.", + "What we can do is, provided we know,", + "we can abstract out the relevant properties of this", + "block and expose them outside. And the relevant properties", + "might well be that, let's say here the properties", + "may be that I in is always zero. I can also express to you the", + "gain of this amplifier. I may also be able to tell you", + "the Thevenin equivalent for the output.", + "There are some properties that I can give you that will let you", + "use this building block abstractly.", + "Today, what we will do is introduce a powerful abstraction", + "of a type of amplifier. This is called the operational", + "amplifier or \"op amp\" for short. What I am going to do is give", + "you a slightly more involved building block than the one I", + "have shown you there. But suffice it to say that the", + "idea is going to be the same. This building block looks like", + "this. This building block has an", + "input port. This building block also has a", + "port in which to connect power or the power port.", + "And the way I am going to connect power,", + "I am going to connect a plus VS supply here.", + "That is going to be my ground node.", + "And I am going to connect a minus VS supply to this node", + "here. So, these voltages are both VS.", + "I want to apply a plus VS here and a negative VS out here.", + "And I am going to take the output between the ground node", + "and the output node of the operational amplifier and call", + "that a vO. This is the output port.", + "So, input port and output port and a power port.", + "Think of this as a pattern where I have an input port", + "across which I connect the input.", + "I have a power port across which I connect a plus VS,", + "minus VS supply, and then I take the output", + "terminal and take a ground terminal, which is defined by", + "external components of my circuitry, and use this as my", + "reference node. Remember ground is just a", + "reference node. I am going to use this as a", + "reference node. These two are equal in", + "magnitude. And take this as my output.", + "And when I do something like this, I can build an even", + "simpler, so this is an abstract differential input amplifier.", + "In other words, this amplifier is going to", + "amplify whatever I apply at the input.", + "A slightly more abstract representation of this looks", + "like this. vOUT and plus/minus vIN.", + "This is a slightly more abstract representation where,", + "remember, we are going to draw this again and again,", + "maybe at least 38 or 39 times in this course.", + "And, remember, each time you draw it,", + "remember that there is an implicit power port,", + "a plus/minus supply that is connected which we don't show.", + "And I remember when I first learned about it a long time ago", + "there was a confusion in me initially.", + "How does this work? Where is the power coming from?", + "Just remember that power comes from a plus/minus supply,", + "and we just don't show that in this abstraction.", + "Now, the details, a lot of details are in Chapter", + "16 of your course notes. That's the reading for that.", + "The other thing is that there are some other key properties of", + "this amplifier. And let me discuss those very", + "quickly. First of all,", + "I can draw a circuit model for the amplifier.", + "Make some room for myself here. And this is a circuit model for", + "what we call the ideal operational amplifier.", + "And the circuit model is going to look like this.", + "This is an abstract device. And, in terms of analyzing how", + "this behaves in a circuit, I am going to show you this", + "abstract circuit that looks as follows.", + "Some input v is applied at these two terminals here.", + "And this terminal is called my v plus terminal and this is", + "called my v minus terminal, so this corresponds to these", + "two terminals. I am telling you that the", + "current going in is going to be zero, so i plus is going to be", + "zero and i minus is going to be zero.", + "i plus is the current in here and i minus is the current into", + "the v minus terminal, and both these currents are", + "going to be zero in this device here.", + "The output is going to look like this.", + "Let me just call it vOUT to be consistent with this here.", + "And taken with ground as my reference.", + "The output is simply Av. In other words,", + "what I am doing is I am going to model this as a device that", + "has a dependent source at its output.", + "And the dependent source here is a voltage controlled voltage", + "source. It is a dependent source,", + "it is a voltage controlled voltage source such that the", + "output voltage is A times the voltage v across its input.", + "This is actually very simple. Think of these three terminals", + "I have shown you here. I applied input across these.", + "And the output is going to be A times whatever I applied.", + "And A is going to tend towards infinity.", + "A is going to be huge. And specific values for A might", + "be a hundred thousand or a million or things of that sort.", + "Huge A in this abstract amplifier.", + "In addition to that, the other properties are that", + "it is going to have infinite input resistance.", + "That means looking in this looks like an open circuit.", + "The fact that this is open here implies the infinite input", + "resistance across this port. What about the output here?", + "Remember, this is a voltage source.", + "And we have a zero output resistance, which means that no", + "matter how the load affects this, as I apply a load this is", + "going to behave like an ideal voltage source and keep holding", + "the voltage constant based on whatever the function I", + "establish here. And A is virtually infinite.", + "Let me pause there for a few seconds and just dwell on this", + "so you just understand what the basic device is.", + "Following this basic definition, I am just going to", + "build a whole bunch of fun little circuits.", + "The analysis will be pretty straightforward,", + "but this is a big conceptual leap here where there is some", + "circuitry inside. Containing resistors,", + "MOSFETs, a whole bunch of stuff in there.", + "I am not telling you what is inside it.", + "Much like I could build an abstract amplifier,", + "I could put an abstract box around the amplifier you saw", + "earlier, I want to put a box around some circuitry.", + "I am not telling you what the circuitry is.", + "And, if you are curious, you should look at page 581 of", + "your course notes. There is an example solved.", + "The example is for a differential amplifier.", + "This is the small signal analysis chapter.", + "That differential amplifier that's solved in that example is", + "usually the first stage in an operational amplifier circuit.", + "That differential amplifier is the first stage at the input.", + "And that differential amplifier, as the name implies,", + "amplifies not a single voltage but amplifies a differential", + "voltage. Note that this guy amplifies", + "the voltage difference between these two terminals.", + "That's v here. And v is simply the same as v", + "plus minus v minus. It's the node voltage here", + "minus the node voltage here. That is what's amplified.", + "It amplifies a difference. Therefore, it is called a", + "difference amplifier or a differential amplifier.", + "And so that input stage is what is inside the op amp.", + "It's got a bunch of other circuitry like level shifters", + "and so on. And at the output it has got a", + "buffer. At at the output it has", + "something that is reminiscent of the source follower circuit that", + "you learned about in recitations, solved an example", + "in the course notes and in your homework as well.", + "And you solved a variant of the source follower on your quiz as", + "well in problem two. So, a circuit that looks like", + "that appears at the output. Remember, for the source", + "follower, the resistance looking in from the output was very,", + "very small. You have seen some of the", + "pieces that go inside the amplifier, but we will deal with", + "this as a building block and simply represent it using this", + "abstract little circuit. To dwell on this a little", + "longer, this little device here is the workhorse of the analog", + "industry. Much like your primitive gate", + "abstraction, your inverter and NAND gate and so on,", + "much as your primitive inverter or NAND gate was from the", + "foundations of the digital industry.", + "Remember we learned how to build this little abstract", + "device called a NAND gate or an inverter?", + "We noticed that those form the foundations of very complicated", + "microprocessors. Those were the building blocks", + "of the digital industry. In the same way,", + "this little beast here is the building block of the analog", + "industry. Just to give you an analogy", + "from software, think of this abstract little", + "device as a library routine from a library of functions when you", + "program in C++ or whatever. Can someone give me an example", + "of an incredibly popular routine that we use all the time that", + "may be called the workhorse of the software industry?", + "Pardon? An abstraction,", + "an abstract procedure. One example might be something", + "like a printf. Printf is an abstract name for", + "a procedure that goes and does something for you.", + "It is amazing how we take the lowly printf for granted.", + "I stick my printf into my program, it includes the", + "standard IO library and it goes and prints a value.", + "You won't believe how complicated the printf is.", + "As you go into learning more advanced software subjects,", + "implementing the printf is a nightmare.", + "It is horrendously complicated. Just imagine.", + "You give it a string and it has to go and print that on your", + "terminal or on your Windows system or whatever.", + "Think of the complicated steps it has to go through.", + "But, as far as you're concerned, it's simple.", + "Just print out something and you're done.", + "The same way. Think of this as the printf of", + "the analog business. It is really simple,", + "and the analysis is going to be incredibly simple,", + "it will be mind-bogglingly simple, but inside it,", + "heavens forbid if you look inside it.", + "Tell you what, go into to S-T-D-I-O dot in one", + "of the library routines and just pore through printf.", + "The world's worst horrendous macros are in there.", + "I mean it is just nasty. The same way inside the op amp,", + "it is nasty. You don't want to go there.", + "Much like in your C programming in your classes,", + "you were able to use printf without fully knowing how it was", + "implemented. Probably some MIT god or some", + "key graduate implemented it, but once it was implemented you", + "just used it based on simple abstract rules as to how it", + "behaved. You didn't have to know what", + "was inside it to use it. The same way with the", + "operational amplifier. So, just think of printf when", + "you see this and just imagine how simple it is going to be to", + "use it. You may think that I spend way", + "too much time, ten minutes dwelling on this", + "abstract concept, but I like to dwell on things", + "that I think are incredibly important.", + "The concept of abstraction is very important.", + "And it's not just in software. The concept of abstraction", + "pervades all of EECS. And if I were to give you a", + "project to say go and ask every professor what is the one word", + "that you think best describes all of EECS?", + "Just pick one word. Go ask every single professor", + "you know. What is a single word?", + "If you were to characterize all of EECS with just one word,", + "what might that word be? In my mind, it is the A word,", + "abstraction. It is all over.", + "If you do a grep on all the words used by all your", + "professors in your four years here, I promise you the first", + "one will be know. And the second one will be", + "abstraction. Check it out.", + "See if what I am saying is true or not.", + "It is all over the place. In 6.001, how many times do you", + "think the word abstraction was used in 6.001?", + "It's all over the map. It's the A word all over.", + "Imagine your shock when you see it being used in 002 because the", + "same concept applies. We build more complicated", + "systems by abstracting out the details of lesser objects,", + "and then using those to build the more complicated systems.", + "Abstraction is a very powerful mechanism of dealing with", + "complexity. Next step is how do I go about", + "using the op amp? Let me show you how it looks on", + "a scope. What I am going to do is apply", + "input to the op amp, I am going to look at the", + "output, place the resistor RL to ground and look at the output.", + "And here I am going to apply a plus VS and out here a minus VS.", + "Again, remember that a plus VS simply looks like this and a", + "minus VS simply looks like this. It's just an inverted VS", + "applied here so I get a minus VS at this input.", + "First of all, what I would like to do is as I", + "change vIN, I am going to plot for you how vOUT looks.", + "vIN and this is vO. I am going to plot vIN in terms", + "of microvolts and vO in volts. vIN is going to have a very", + "very small, the scale is going to be in microvolts because", + "remember the gain of this is huge.", + "It's on the order of ten to the sixth.", + "It's huge. Small changes in vIN are going", + "to cause massive changes in vO. I have a very fine scale on the", + "X axis. What is going to happen if I", + "somehow magically make vIN exactly zero?", + "If I short these two terminals, if this was a completely ideal", + "op amp, which it never is, if it's a completely ideal op", + "amp, then my output should be zero.", + "As I increase my vIN the output should be A times vIN.", + "For some small value of vIN, small v, let's say one", + "microvolt, the output should be one volt.", + "A is a constant so this would look like a straight line.", + "And let's say my supply voltages are 12 volts minus 12", + "volts, if this were an ideal amplifier and I didn't have to", + "worry about the supply, this would just go on extending", + "forever. But I have a plus 12 volt", + "supply and a minus 12 volt supply.", + "My output cannot go past those limits.", + "And so, therefore, my output kind of flattens out", + "at these two points. And it is called hitting the", + "rails. Output goes up and you hear a", + "thunk sound and you hit the rails.", + "When you play with op amps in your next lab,", + "if you listen really, really carefully you may hear", + "it. So, this saturates out.", + "Not surprisingly, this region where the output", + "saturates at the supply is called the saturation region.", + "Remember, don't confuse it with-- It's not the same as your", + "saturation in the MOSFET. It is a totally different", + "thing. It is just happenstance that we", + "call this saturation. And if you would like to think", + "about it, you can think of it as the thunk region.", + "That's probably more appropriate to distinguish it", + "from the saturation region in the MOSFET.", + "And, not surprisingly, this one is called the active", + "region. And it is in this region that", + "we use the op amp. Here it has hit the rails and", + "is kind of dangling out there. It's not much use to us.", + "It's in this active region that we use it because this is where", + "the gain is seen. Now, it turns out that this is", + "a very high gain device. It is very skittish.", + "This gain is kind of a really funny thing.", + "It's dependent on a bunch of factors.", + "This could be temperature dependent.", + "This gain here and this curve is just completely skittish.", + "It could depend on temperature. It could depend on time of day.", + "It could depend on what medication this amplifier is on.", + "It could depend on its mood swings.", + "Who knows what? This is kind of unstable.", + "And A in particular is highly unstable.", + "It is going to be big, that's for sure,", + "but it could be ten to the six, on a rainy day it might be two", + "times ten to the six. If it feeling sleepy it may be", + "point five times ten to the sixth.", + "It is big but I cannot rely on it.", + "Let me show you an example. I want to show you this curve", + "for this MOSFET, apply an input and plotting the", + "output. What I will do is take a look", + "at this curve. Then what I am going to do is", + "use a heat gun to heat the op amp and you are going to see", + "this vary all over the map. If you still remember last", + "week, some of you may remember that from some place in a", + "similar situation where the gm for the MOSFETs you were given", + "was also dependent on temperature and stuff like that.", + "It is a very common occurrence. And that is certainly the case", + "for the MOSFET.", + "", + "Let's apply input. Let's do this.", + "This is vIN versus vOUT for the amplifier.", + "Notice that this is plus 12 volts, this is minus 12 volts.", + "It is about two volts per division.", + "This axis here is in microvolts, I believe.", + "For a very small change, for a few tens of microvolts,", + "I have an incredibly high gain. Notice that this has an", + "incredibly high gain here. The gain is the slope of this", + "line, almost a vertical line. What I am going to do next,", + "is to have some fun, is I am going to heat the op", + "amp. To show you that A is kind of", + "really skittish and also the fact that it doesn't quite hit", + "zero, it does all kinds of weird things, I am going to heat the", + "op amp. And then let's take a look at", + "how that curve fluctuates.", + "", + "What you saw there was that the op amp began to behave really", + "weirdly as I heated it. Instead of doing this it", + "sometimes did this really weirdly, like getting an offset", + "from the center and so on. And it does a bunch of other", + "weird things, but we won't go into those", + "details. It's not relevant for this", + "course. But the point is that the gain", + "and the offset at the input are dependent on temperature.", + "And we look for ways to make it less dependent on temperature.", + "As the next step, what I would like to do is", + "build a circuit. This is model equivalent of", + "your Hello World program. We are going to use the printf", + "and build a small program on the printf.", + "You don't have to worry about how printf is implemented,", + "just that we can build very highly interesting circuits with", + "this horrendously complicated function based on a simple", + "abstraction of the device. The circuit that we will build", + "is called a noninverting amplifier.", + "", + "From now on, I am not going to show you the", + "plus/minus VS. I am not going to show the", + "power port, but it is in there. It's hidden under the", + "abstraction layer. This is my op amp.", + "And I am going to build the following circuit.", + "This is my v plus and this is my v minus.", + "What I am going to do is for the v plus I shall apply a vIN.", + "Let me talk a little bit about ground as well.", + "Ground is commonly taken as the point at which I connect my VS", + "and minus VS supply. It is kind of at the midpoint.", + "And if VS and minus VS are very carefully tuned then the output", + "is also going to be at that same ground reference when the input", + "is zero. So, the ground is defined as", + "the point at which I connect my plus/minus VS supplies.", + "I apply my vIN out here. Then what I am going to do,", + "here is my output vO. I am going to have a resistive", + "divider to ground here and label these R1 and R2.", + "And what I am going to do here is feed this back to the input,", + "to the v minus input. I am going to sample the", + "voltage here and feed that into here.", + "So, this is my abstract model and this is my Hello World", + "program. What we are going to do is", + "simply analyze how this little program behaves.", + "So, my equivalent circuit model.", + "The way to analyze these is after one or two of these", + "examples, you will be able to directly analyze this just by", + "looking at it, by inspection.", + "But, much as we did for the other pieces,", + "let me grunge through drawing the equivalent circuit and", + "grinding through the analysis, and then show you the much", + "simpler way of doing it. And even here,", + "even with this grinding analysis, it is going to be", + "pretty simple in any case. So, I will replace the op amp", + "with its equivalent circuit model.", + "Its equivalent circuit was v plus, v minus.", + "", + "So, that was the equivalent circuit model of the operational", + "amplifier, just this piece. I draw that for you.", + "Then what I am going to do is I connect my v in here.", + "And, remember, I have an R1,", + "R2 resistive divider here. And this one gets connected to", + "this terminal there. I also know that i plus is", + "zero. I also know that i minus is", + "zero. All I've done is simply", + "replaced the amplifier with its equivalent circuit.", + "Let's go ahead and analyze that circuit now.", + "Let's go ahead and analyze that circuit.", + "And it's going to be pretty simple, actually.", + "What I am going to show you is the hard way of doing it.", + "I will show you a much easier way, but the hard way itself is", + "pathetically easy. What I want to do is find vO in", + "terms of vIN. And there will be a bunch of", + "other factors thrown in, including things like R1 and", + "R2, A and stuff like that. Let's go and analyze it.", + "vO, let's look at that circuit. By the way, let me take 30", + "seconds and make a little speech at this point.", + "When you see circuits like this, and I saw this happen in", + "quiz two as well, for some reason,", + "when you see a new kind of circuit, don't completely go", + "berserk or freeze or whatever. There is just no reason to.", + "You know the node method. The node method is the", + "workhorse of our business. When in doubt apply the node", + "method. It will simply work.", + "Don't freeze. Don't think oh,", + "man, I need to apply a pattern that I know already.", + "I must have seen this somewhere.", + "When in doubt boom, apply the node method.", + "This circuit here, all I have here is one unknown", + "node voltage. I know the voltage of v plus,", + "I need to compute the voltage vO.", + "There are two unknowns, vO is an unknown and the", + "voltage here at v minus is another unknown.", + "This is a very simple circuit involving a dependent voltage", + "controlled voltage source, and you need to find out vO and", + "v minus using the node method. Just apply it.", + "It's simple. Don't freeze.", + "Just look at it and say I can do it and apply the node method.", + "It will simply work. So, let's do that.", + "What I can do here is vO is A times v plus minus v minus.", + "This is actually really simple. And then, if I take v plus", + "here, I know v plus is simply vIN so I will just make that", + "substitution right away. So, v plus is simply vIN.", + "What is v minus? v minus here is vO --", + "", + "What is v plus? I'm sorry, v minus.", + "v minus is simply the voltage that is between R1 and R2.", + "Notice that no current flows in to the v minus node.", + "There is no current flowing in. Voltage at v minus is simply", + "the voltage given by the resistive divider,", + "which is vO times R2 divided by R1 plus R2.", + "Stare at that for another second.", + "The voltage at this node here is simply given by the resistive", + "divider. Because no current is flowing", + "in this direction. And no current flows in because", + "I am telling you there is no current there based on my", + "abstraction. I am telling you i minus is", + "zero. That voltage is simply the", + "voltage at this resistive divider.", + "And so I can simplify it further and write this as vO.", + "So I get, there is a one here. And I move this thing over to", + "this side so I get one plus A times R2 divided by R1 plus R2.", + "And that is equal to AvIN. And simplifying it some more,", + "I get vO is AvIN divided by one plus AR2 divided by R1 plus R2.", + "Notice how simple this is, and this is the hard method.", + "All I have done is analyze the circuit using the basic circuit", + "analysis principle that you learned the first week of the", + "course, and I have the output for you.", + "I just noted very carefully what the relationships were", + "between the various elements in the abstraction.", + "Notice here that I am told that A is extremely large.", + "A is on the order of ten to the six and so on.", + "And suppose it is the case that, let me write that down", + "again. vO is AvIN, one plus AR2,", + "R2. Suppose R1 and R2 are more or", + "less comparable and A is ten to the six, it's a huge number,", + "so this whole number is much, much greater than one.", + "If it is much huger than one, what I can do is I can then", + "write this as follows. I can say that this is more or", + "less equal to AvIN divided by AR2 divided by R1 plus R2.", + "I am ignoring the one here. As soon as I do that,", + "notice I can cancel out A and I get vO to be approximately equal", + "to vIN times R1 plus R2 divided by R2.", + "Notice now that when the gain is very large the output is a", + "function of the input multiplied by some number.", + "The beauty of this thing here is that when A is very large,", + "or this expression is very large, A cancels out and there", + "is no A in this relationship. This means that even though the", + "basic amplifier was very skittish, the output here", + "relates to the input based on components that I have control", + "over. These are soldiers in my army.", + "I control them. So, to give you a sense of some", + "numbers here, suppose A was ten to the six.", + "And I choose R1 to be 9R. And R to be some R.", + "Then vO is ten to the sixth vIN divided by one plus ten to the", + "six R divided by 9R plus R. So, that is ten to the six vIN", + "divided by one plus ten to the six divided by ten.", + "All right. If I ignore the one here,", + "the ten to the six and ten to the six cancel out,", + "this ends up giving me 10vIN. So, I get a really nice", + "amplifier whose output is simply ten times the input and", + "determined solely by some resistor values.", + "Let me show you another quick demo this time and show you the", + "amplifier again, but with resistors connected", + "like that. And then I show you that I want", + "to heat the amplifier to the wazoo, the op amp to the wazoo,", + "but vO is going to be absolutely rock solid.", + "Let's try that out.", + "", + "This time around, this is the transfer function,", + "the vO versus vIN. And notice that this time", + "around I have similar scales on the X and Y axes,", + "and this has a slope of 10. This is the point where the", + "amplifier saturates at plus 12 volts, and this is minus 12", + "volts, and this point here is a zero.", + "So, this is vIN, vOUT, plus 12,", + "minus 12 and this slope is 10. What I am going to do now is", + "heat the op amp to the wazoo and this ain't going to change", + "because it's my external resistors that control it", + "independent of the value of A, provided A continues to be very", + "large. I am just articulating the", + "vOUT, vIN curve. And let me start heating the op", + "amp.", + "", + "Notice that it's pretty stable. It doesn't change because it is", + "independent of the amplifier values.", + "What I have done now is by connecting these resistors in", + "this way, I have a nice amplifier with a gain of ten.", + "The question you may ask yourselves is why?", + "There is this little sucker in there that wants to shoot things", + "up by ten to the sixth. Wants to knock things off the", + "one rail or the negative rail. Why is it that it's behaving", + "like a docile lamb here and giving us a nice little factor", + "of ten gain no matter what I do to it?", + "Why is it doing that? What is the intuition behind", + "it? I will draw something on the", + "board, but for the next ten seconds I want you think about", + "it. See if you can come up with", + "some insight as to why is it doing that.", + "Why is it exactly ten? Why isn't the ten to the sixth", + "kind of killing me somehow? Why am I getting exactly ten no", + "matter what happens? See if you can come up with", + "some intuition and then I will show you how it works.", + "I will redraw the circuit in the meantime.", + "", + "Let me see if I can give you some intuition.", + "This is my circuit, and let's say this is R and", + "this is R. As an example,", + "let's assume that the input is 5 volts, vIN is 5 volts.", + "If R and R are equal, what should the output be?", + "It's R and R, so it's R1 plus R2 divided by", + "R2, right? It's 2R divided by R,", + "so it has a gain of two. My amplifier has a gain of two", + "because R1 plus R2 divided by R2, which is my gain,", + "is R plus R divided by R equals two.", + "So, this will be 10 volts. If that is 10 volts this is", + "going to be 5 volts, correct?", + "This R and R, voltage divider,", + "this is five, so I get 5 volts here.", + "This is v plus. This is v minus.", + "I get R and R, 5 volts here,", + "that's how the circuit looks. Now let's understand what is", + "going on. And listen very carefully.", + "This is going to be a key insight that I hope you will", + "carry with you for the rest of your lives.", + "This is really, really key.", + "What you are going to see is, I think, the third big ah-ha", + "moment in 6.002. Like small signal analysis,", + "like the frequency domain stuff we saw, I think this is the", + "third big one in the next 30 or 40 seconds, things that are", + "completely either not necessarily intuitive but are", + "just spectacular in terms of what they can do for you.", + "Let's see. Let's suppose that because I am", + "heating it, let's suppose that A suddenly tends to increase.", + "It wants to increase because I have heated it.", + "A is saying I want to get out this mold here and starts to", + "break through its shackles here. Let's say, as a Gedanken", + "experiment, that it tries to shoot up this to 12 volts.", + "It tries to push it up higher. This is just a Gedanken", + "experiment. The up arrow says that the", + "increase in A is trying to push up vO momentarily.", + "Let's see what happens. It is trying to push up vO", + "momentarily, so let's say this goes to 12 hypothetically.", + "If that goes to 12, what should this volt node go", + "to? Six, exactly.", + "This goes to 6 volts. If that goes to six,", + "what does v minus go to? 6 volts again.", + "So, v minus goes to 6 volts. Now at the input I have 5 volts", + "at v plus and 6 volts at v minus, so where should the", + "output go? The output should go down", + "because the voltage of the negative terminal is higher.", + "And so the output is A times v plus minus v minus.", + "And because this has gone down, this has gone up here it is", + "going to try to pull the output down.", + "That is going to pull the output down let's say to 9 volts", + "or something. Cachunk, there is a big battle", + "going on here. A has gone up,", + "it has boosted it up to 12, but the moment that goes to 12,", + "this goes to 6, this goes to 6,", + "and the op amp output has to go down to 9 volts now because this", + "input is higher here. If this goes to 9,", + "this goes to 4.5. If that goes to 4.5,", + "this goes to 4.5. What happens now?", + "If this goes to 4.5, what happens?", + "It wants to go back up. Can't it make up its mind?", + "This guy wants to go back up now because v plus is higher", + "than v minus. What am I seeing here?", + "This whole circuit here behaves like my little son,", + "my 9-year-old. If say do this,", + "he wants to do the exact opposite.", + "So, there is a trick in how you make them do things for you.", + "Look at this. Because of this arrangement of", + "the circuit when A tries to push the output up,", + "the rest of the circuit tries to pull it back down to where it", + "used to be. If the circuit tries not to", + "follow the true path, the rest of the circuit tries", + "to whack it into shape so it follows a true path.", + "And what's happening is because, in this arrangement,", + "I have fed back a portion of the output to the negative", + "input. I have fed back some of the", + "output to the negative input. And by providing this feedback", + "of a portion of the output to the negative input,", + "I have arranged it in a way that I have something called", + "negative feedback. What negative feedback does is", + "that if this wanted to go wild and crazy, the circuit provides", + "it with some negative feedback like you just saw.", + "Feedback, a big word. If you take a poll of all the", + "EECS faculty, I suspect that feedback would", + "rank at least as the ninth or tenth most important word in the", + "EECS. If abstract is number one,", + "I think this would rank like a nine or a ten or something.", + "So, that's the reason why it worked.", + "In the last couple of minutes, let me give you some insight,", + "based on something that you know, on how feedback works.", + "This is a road here. Let's look at anti lock breaks.", + "This is my tire. And let's say I have a set of", + "disk brakes here. As the car is moving forward,", + "if I apply the brakes the tire stops rolling,", + "but if I apply the breaks too hard it can lock up the tire and", + "the whole car can skid. The way anti lock breaks work", + "is as follows. There is a controller that sits", + "here. And there is a little person", + "looking at the wheel and seeing is it turning.", + "So, this is a feedback. And it is saying is it turning?", + "Yes. Or, is it not turning?", + "No. All this person watching the", + "tire is doing is saying is it turning or is it not turning.", + "That is it. That is a negative feedback.", + "And so, if it is no and if it is yes.", + "If it is yes then what this does is it applies the brakes", + "even more strongly. It is turning so I can apply", + "more brakes. But if it says oops,", + "it stopped turning, what it does is it simply", + "releases, the controller releases the brakes.", + "And when the controller releases the brakes this one", + "tends to loosen up a little bit and the tire starts turning", + "again. So, this way you are constantly", + "keeping the tire in its region of critical friction so that it", + "is constantly moving. And static friction applies to", + "how hard you can brake and it doesn't start skidding.", + "In fact, if you take your car out, and I don't say you do", + "this. Let's say go onto the Charles", + "River in the dead of winter and you drive on the lake and you", + "slam your anti lock brakes on, on an icy patch,", + "you will notice that there is a constant sound that looks like", + "something is vibrating in there. That is exactly what is", + "happening. Oops, the tire is locked.", + "Release the brakes. The wheel is turning.", + "Jam the brakes on. That is exactly what is", + "happening. The same way as out there,", + "you notice that oops, the output is going up,", + "pull it down, oops, it's going down,", + "pull it up. So, there is constant negative", + "feedback that is keeping the output stable.", + "A very important concept. And I will ask your recitation", + "instructors to cover the very simple method that is on page 9." + ] +} \ No newline at end of file diff --git a/subs/WT-qzgaKeGI.srt.sjson b/subs/WT-qzgaKeGI.srt.sjson new file mode 100644 index 0000000000..8094fd18c4 --- /dev/null +++ b/subs/WT-qzgaKeGI.srt.sjson @@ -0,0 +1,1892 @@ +{ + "start": [ + 0, + 9000, + 11571, + 13000, + 17000, + 24138, + 28000, + 34000, + 39311, + 42950, + 47475, + 52885, + 56131, + 60435, + 63657, + 68371, + 71200, + 74028, + 76464, + 80942, + 85342, + 91000, + 93717, + 97837, + 101081, + 104061, + 109321, + 114668, + 121733, + 129977, + 138688, + 141177, + 150199, + 153000, + 161000, + 166009, + 169830, + 175009, + 179000, + 181796, + 185697, + 187685, + 192101, + 193868, + 197401, + 201743, + 203878, + 208000, + 212386, + 215676, + 219551, + 222915, + 227448, + 231323, + 235490, + 239000, + 242831, + 245484, + 248210, + 251747, + 255578, + 259631, + 262800, + 267000, + 271217, + 274551, + 277000, + 282000, + 285428, + 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Let's get moving.", + "Good morning.", + "", + "Today, if everything works out, we have some fun for you guys.", + "I hope it works out. We'll see.", + "What I am going to do today is a very major application of the", + "frequency response and the frequency domain analysis of", + "circuits. And this application area is", + "called filters. The area of filters often times", + "demands a full course or a couple of full courses all by", + "itself. And filters are incredibly", + "useful. They're used in virtually every", + "electronic device in some form or another.", + "They're used in radio tuners. We will show you a demo of that", + "today. They're also used in your cell", + "phones. Every single cell phone has a", + "set of filters. So, for example,", + "how do you pick a conversation? You pick a conversation by", + "picking a certain frequency and grabbing data from there.", + "They are also in wide area network wireless transmitters.", + "Do we have an access point here?", + "I don't see one, but you've seen wireless access", + "points. Again, there they have filters", + "in them. So, virtually every single", + "electronic device contains a filter at some point or another.", + "And so, today we will look at this major, major application of", + "frequency domain analysis. Before we get into that,", + "I'd like to do a bit of review. The readings for today", + "correspond to Chapter 14.4.2, 14.5 and 15.2 in the course", + "notes. All right.", + "Let's start with the review. We looked at this circuit last", + "Friday --", + "", + "-- where I said that for our analysis, we are going to focus", + "on this small, small region of the playground.", + "And what's special about this region of our playground is that", + "I am going to focus on sinusoidal inputs.", + "And, second, I am going to focus on the", + "steady state response. How does the response look like", + "if I wait a long, long time?", + "And then we said that the full blown time domain analysis was", + "hard. This was, remember,", + "the agonizing approach? And then I taught you the", + "impedance approach in the last lecture, which was blindingly", + "simple. And, in that impedance", + "approach, what we said we would do is --", + "I will apply the approach right now and in seconds derive the", + "result for you. But the basic idea was we said", + "what we are going to do is assume that we are going to", + "apply inputs of the form Vi e to the j omega t.", + "Wherever you see a capital and a small, there is an implicate e", + "to the j omega t next to it. I'm not showing you that.", + "And what I showed last time, and the class before that was", + "once you find out the amplitude --", + "Once you find out the multiplier that multiplies e to", + "the j omega t, it's a complex number,", + "you have all the information you need.", + "And once you have this, you can find out the time", + "domain response by simply taking the modulus of that,", + "or the amplitude and the phase of that to get the angle.", + "And that gives you the time domain response.", + "So, our focus has been on these quantities.", + "The impedance method says what I am going to do is replace each", + "of these by impedances. And then the corresponding", + "impedance model looks like this.", + "", + "Instead of R, I replace that with ZR.", + "And instead of the capacitor, I am going to replace that with", + "ZC. And this is my Vc.", + "ZR is simply R and ZC was going to be one divided by sC where s", + "was simply a shorthand notation for j omega.", + "Based on this, once I converted all my", + "elements into impedances, I can go ahead and apply all", + "the good-old linear analysis techniques.", + "I will discuss a bunch of them today.", + "As an example, I could analyze this using my", + "simple voltage divider relationship.", + "Vc is simply ZC divided by ZC plus ZR times Vi.", + "And that, in turn, is, well, let's say I divide", + "this by Vi so I can get the response relation,", + "is ZC divided by ZC plus ZR. And ZC I know to be one by j", + "omega C, plus R. And multiplying throughout by j", + "omega C, I get one divided by one plus j omega CR.", + "It's incredibly simple. This is simply called the", + "frequency response. And it's a transfer function", + "representing the relationship between the output complex", + "amplitude with the input. We can also plot this.", + "Notice that in our entire analysis we have not only", + "assumed sinusoidal input, but we're also saying that let", + "us look at this only in the steady state.", + "So, we will wait for time to be really, really large,", + "and then look at the response. And so, therefore,", + "we will plot the response not as a function of time,", + "but rather we are going to plot the response as a function of", + "omega. What we are going to say is I", + "am going to input a sinusoid and my output is going to be some", + "other sinusoid. And since I'm waiting for a", + "long time to look at the output, time doesn't make sense", + "anymore. Rather, my free variable is", + "going to be my frequency, so I am going to change the", + "frequency of the input that I apply.", + "And so, I am going to plot this as a function of omega.", + "This represents a completely complimentary view of circuits,", + "the time domain view and then there is a frequency domain", + "view. The frequency domain view says", + "how did this circuit behave as I apply sinusoids of differing", + "frequencies? I can plot that relationship in", + "a graph like this, and this relationship is simply", + "given by a parameter edge the transfer function,", + "it's a function of omega. And I can also plot the", + "absolute value of that. And let's take a look at what", + "it looks like. So, I can look at functions", + "like this and very quickly plot the response.", + "I am going to do a whole bunch of plots just by staring at", + "circuits and staring at expressions like this.", + "And you will see a number of them today.", + "First of all, the way you plot these is look", + "for the values where omega is very small and when omega is", + "very large. When omega is very,", + "very small this term goes away. And so, for very small values", + "of omega the output is simply one.", + "Vc by Vi is simply one. This part goes away.", + "What happens when omega is very, very large?", + "When omega is really large, this part dominates,", + "is much greater than one. If I ignore one in relation to", + "this guy and take the absolute value of that then I simply get", + "one divided by omega CR when omega is very large.", + "So, when omega is very large, I get a decay of the form one", + "over omega CR. I know the value for small", + "omega, and it looks like this for very large omega.", + "And, if you plot it out, this is how it's going to look", + "like. Let's stare at this form for a", + "little while longer. And let's plot some properties", + "off it. First of all,", + "you notice something else. When omega CR equals one then,", + "in other words, when omega equals one by RC,", + "notice that the output is given by one plus j.", + "And the absolute value of that is simply one divided the square", + "root of two. So, in other words,", + "when omega is one by RC -- When omega is one by CR then", + "the output is one by square root two times its value when omega", + "is very, very small. So, that is one little piece of", + "information. If you look at the form of", + "this, I would like you to stare at it for a few minutes and try", + "to understand what this represents.", + "This says that for very low frequencies the response is", + "virtually the same as the input in amplitude.", + "In other words, if I apply some very low", + "frequency sinusoid of some amplitude then the output", + "amplitude is going to be same as that amplitude.", + "And that's a one. Now, it also says when I apply", + "a very high frequency, at very high frequencies it", + "decays. So, this graph which says I am", + "going to pass low frequencies without any attenuation,", + "without hammering it, but I am going to clobber high", + "frequencies and give you a very low amplitude signal at the", + "output but pass through, almost without attenuation,", + "the input at low frequencies. And so this is an example of", + "what is called a low pass filter or LPF.", + "What this is saying is that this little circuit here acts", + "like a low pass filter. It's a low pass filter because", + "it passes low frequencies without attenuation but kills", + "high frequencies. If I take some music,", + "and you will do experiments with this in lab.", + "When is lab three? People are doing lab three", + "right now, right? Lab three is going on right now", + "and early next week as well. And, in lab three,", + "you will play with looking at the response to music of", + "different types of filters. If apply some music here,", + "you will see that the output will pass low frequencies but", + "really attenuate high frequencies.", + "You will hear a lot of the low sounding base and so on but", + "attenuate a lot of the high frequencies.", + "All right. The other thing that I", + "encourage you to do is Websim has built in pages for a large", + "number of such circuits. You can go in there and play", + "with the values of RC, or L for that matter,", + "for a variety of circuits. And, if you click on frequency", + "response, you actually get both the amplitude response and the", + "phase as well. You can play with various", + "values of RLC and see how the frequency response looks like", + "for each of the circuits. As a next step,", + "what I would like to do is just give you a sense of how", + "impedances combine. This won't be very surprising", + "given that they behave just like resistors, but it's good to go", + "through it nonetheless. Suppose, just to build some", + "insight, suppose I had two resistors in series.", + "All right. R1 and R2.", + "And this was my A and B terminals respectively.", + "And let's say the complex amplitude of the voltage was Vab", + "across this. Then I could relate,", + "let's say Iab was the current, I can relate these resistances.", + "Or, I could relate Vab and Iab as follows.", + "Simply Vab divided by Iab equals R1 plus R2.", + "I know that. And the same thing applies to R", + "viewed as an impedance. It's still impedance R,", + "and so this one still goes ahead and applies.", + "The second thing I can try is the circuit of this form.", + "A, B, and I have an R1 and an L in this case.", + "And what I can do is, in the impedance model,", + "I can view this as an impedance of value j omega L.", + "And I can also combine them to get the impedance between A and", + "B. Much as I got a resistance", + "between A and B, I can get an impedance between", + "A and B as Vab divided by Iab. And that will be given by ZR1", + "plus ZL, and that is simply R1 plus j omega L.", + "Similarly, I can do an even more complicated circuit.", + "So, resistance. And here I have a capacitor in", + "series with the resistance, and then I apply inductor to", + "it. This is A, B,", + "Iab and plus, minus Vab.", + "And let me call this R1 and let me call this R2 and this is C", + "and L. I can go about combining these", + "in much the same manner that I combine my resistances in the", + "series parallel simplifications. I can define an impedance Zab", + "between the A and B terminals as ZR1 plus Z of this combination,", + "impedance of this combination, which is simply impedance of C", + "and that of R2 in parallel with each other.", + "I get Zc in parallel with ZR2. Notice that this notation", + "simply says that look at the impedance of the capacitor in", + "parallel with a resistor. And then, finally,", + "I add to that the series impedance of the inductor ZL.", + "Exactly as you would have done for resistances,", + "if all of these resistances you would have said R of this piece", + "plus the R of the parallel combination plus the R of", + "whatever was here. This time around we have", + "impedances. And replacing this with the", + "values, this is R1. I know for ZL it's j omega L.", + "And so, for ZL, parallel ZR2 it is given by", + "ZCZR2 divided by ZC plus ZR2, which is simply R1 here and j", + "omega L. And let me just substitute the", + "values here. I know that ZR2 is simply R2,", + "ZC is one by j omega C, and then one by j omega C plus", + "R2. And I can go ahead and simplify", + "that further and get my impedance Zab.", + "Notice how simple analysis has become.", + "Using this technique, using the impedance method", + "we've managed to convert our analysis from solving", + "differential equations to going back to algebra.", + "A large part of what we do in circuits is see how we can get", + "back to really simple algebra and try to be clever about how", + "we do things. So, this is as far as analysis", + "is concerned. In the next five minutes,", + "I want to give you some insight into how you can build different", + "kinds of impedances.", + "", + "And I won't go into too much detail but give some insight", + "into how you can get a sense for the kind of filters you want to", + "design. Or, at the very least,", + "given a filter, how can you very quickly get", + "some insight into what kind of filter it is,", + "how it performs, what its frequency response is", + "and so on. And, this time around,", + "this piece of intuition will be in honor of Umans.", + "And back to our Bend it Like Beckham series,", + "I call this \"Unleash it like Umans\".", + "What experts in the field do is they don't go about sitting", + "around writing differential equations, but rather use a lot", + "of insight into how to solve these things.", + "And so in honor of Umans, I will label this unleash it", + "like Umans. Let's get some insight into how", + "the response of various elements look like.", + "Let's take, for example, I have some impedance Z.", + "Let's say this could be a resistor, it could be an", + "inductor or it could be a capacitor.", + "Let's take a look at what the frequency response of just these", + "elements look like. In other words,", + "what are the frequency dependents of Z itself?", + "Let me just plot the impedance of each of these elements as a", + "function of frequency. Let me just take the absolute", + "value of their impedance. Notice that it's a complex", + "number. For the inductor it's j omega", + "L. And let me take the absolute", + "value omega L in that case and plot it for you.", + "And use that to develop some insight.", + "Let's do a simple case first. If Z is a resistance of value R", + "then no matter what the frequency my value is going to", + "be R. If I have an inductor of value", + "L then the impedance is going to look like j omega L,", + "and so I am going to omega L for that.", + "And the dependence of that simply says that for low omega", + "the impedance is very small. For omega zero the impedance is", + "zero and it increases linearly with omega.", + "So, it's omega L for the inductor.", + "Impedance increases linerally as I increase the frequency.", + "What about for the capacitor? For the capacitor,", + "the impedance is one divided by j omega C.", + "And so, therefore, I get the dependence being", + "related to omega C. Which says that for very high", + "frequencies impedance is very low, but for very low", + "frequencies the impedance is very high and I get a behavior", + "pattern that looks something like this.", + "It goes as one by omega C. As omega is very large,", + "my impedance is very small. If omega is very small,", + "my impedance goes towards that of an open circuit.", + "This is not surprising. You've known this before,", + "right? That a capacitor behaves like", + "an open circuit for DC. An inductor behaves like a", + "short circuit for DC. Notice that zero frequency here", + "corresponds to DC. The capacitor looks like an", + "open circuit for DC, very high impedance.", + "The inductor looks like a short circuit for DC,", + "very low impedance. And the opposite is true at", + "very high frequencies. While R is a constant", + "throughout. Let's use this to build some", + "insight into how our circuits might look.", + "Let me do this example.", + "", + "Let's say I have a Vi and I measure the response across the", + "resistor.", + "", + "So, I measure Vr divided by Vi and take the absolute value and", + "take a look at how it's going to look like.", + "I want you to stare at this for me and help me with what the", + "response is going to look like. Let's take incredibly high", + "frequencies. At very high frequencies,", + "this has a very high frequency, what do the capacitor look like", + "to very high frequencies? Is it an open or is it a short?", + "A short circuit. At very high frequencies the", + "capacitor looks like a short circuit.", + "Then Vi simply appears across the resistor,", + "which means that at very high frequencies the output is very", + "close to the input. At very low frequencies what", + "happens? At very low frequencies the", + "capacitor looks like an open circuit.", + "If this looks like an open circuit then very little voltage", + "will drop across this resistor here because most of it is going", + "to drop across the capacitor. What is going to happen is,", + "for very low values, I am going to be looking at", + "something out here. And, because of that,", + "my response looks like this. And this is of a different form", + "than the one you saw earlier. In this case,", + "I pass high frequencies but attenuate low frequencies.", + "Not surprisingly, this is called a high pass", + "filter.", + "", + "You need to begin to be able to think about capacitors and", + "inductors in terms of their high and low frequency properties.", + "And, if you develop that intuition, once you develop the", + "intuition about capacitors and inductors and their frequency", + "relationship, that will be a big step forward", + "in 002. If you get that insight,", + "you will go a long way in terms of knowing how to tackle", + "problems and being able to quickly sketch responses.", + "Yes.", + "", + "In the case of, if we get something like j", + "omega L, what you can do is take the limit as omega goes to zero.", + "If it is omega L then notice that it is going to start", + "linear. And, on the other hand,", + "if when you get very high frequencies, for example,", + "if you get one by something omega C then this is a", + "hyperbolic relationship, so it is going to go ahead", + "looking like this. So, you can take a look at a", + "lot of these functions at their very low values and see how they", + "look like at that point. All right.", + "The next one I would like to draw for you is something that", + "looks like this.", + "", + "Let's say, for example, I have an inductor L and a", + "resistor R and I want to see what that looks like.", + "In this particular example, I have H, take the absolute", + "value. So, what is this going to look", + "like? I am going to look at the value", + "across the resistor here. Here what I am going to find is", + "that at very low frequencies this guy is a short circuit.", + "Since this guy is a short circuit, all the voltage drops", + "across the resistor so it's going to look like this.", + "And, at very high frequencies, what I am going to find is that", + "the inductor is going to appear like an open circuit.", + "And so, therefore, all the voltage is going to", + "pretty much drop across the inductor.", + "It will be R divided by something plus omega L.", + "So, at high frequencies this guy is going to taper off to", + "zero and is going to look like this.", + "And this is back to my low pass filter.", + "Just to go back to a question asked earlier,", + "how do you know what this looks like?", + "I can very quickly write down the expression for H of j omega.", + "This is simply going to be R divided by R plus if this is VR.", + "VR is simply R divided by one by j omega C.", + "I multiply it out by j omega C in the numerator and the", + "denominator. I'm going to find j omega C", + "here and I am going to get one by j omega C here.", + "And what is going to happen with something like this is that", + "as omega becomes very small then I am going to ignore this.", + "When omega becomes very small, I can ignore this with respect", + "to one, and I get R j omega C. Given that, is what I've drawn", + "here correct or wrong? This goes away with respect to", + "one. I am left with R j omega C,", + "right? For very low frequencies.", + "Given what I have drawn here, is that correct or is that", + "wrong? Well, it's hard to say.", + "For very, very low frequencies it starts out being linear", + "because it's an omega relationship,", + "and then it goes up like this and then goes out there.", + "Let me go onto another example. Let me do another example here", + "which is something like -- I need to make sure I don't", + "make a mistake here. If I get R j omega C by R j", + "omega C, you know what, this ends up being a first", + "order system, and so is going to look like", + "this. I blew it there.", + "Back to this system here. If I have an L and an R and I", + "look at this equation to look at what happens across L,", + "you can plot that again. And for very low frequencies it", + "is going to be zero amplitude here and for very high", + "frequencies this is going to be an open circuit,", + "and so the response is going to look something like this.", + "That's going to end up being your high pass filter.", + "As another example, I would like to do a series RLC", + "circuit --", + "", + "-- and try to get you some sense of what that output looks", + "like. Let's use our intuition and", + "first write down what this looks like and then go and do some", + "math and see if the math corresponds to what our", + "intuition tells us. I want to plot Vr with respect", + "to Vi. I want to plot it there.", + "For something like this, what happens at very low", + "frequencies? We are just looking to get", + "very, very crudely what this graph is going to look like.", + "Very, very crudely what this graph is going to look like.", + "Given that I am taking the voltage across VR,", + "what happens at very low frequencies?", + "At incredibly low frequencies, the inductor looks like a short", + "circuit, but the capacitor looks like open circuit.", + "An open circuit in series with a short circuit that ends up", + "looking like an open circuit. And so, therefore,", + "all my voltage falls across VR. Now, what happens at very high", + "frequencies? At very high frequencies the", + "capacitor looks like a short. But the inductor looks like an", + "open circuit now for very high frequencies, correct?", + "Just remember, capacitor is short for high", + "frequencies inductor open for high frequencies.", + "So, this ends up having a very high impedance.", + "At very high frequencies this guy has a very high impedance.", + "And, because of that, for a high value of frequency,", + "I end up going in that manner. This behavior has the effect of", + "the capacitor here. And for very high frequencies I", + "get the effect of the inductor. And so this means that I have", + "very low values for low frequencies, very low values for", + "high frequencies. And, as the frequency", + "increases, I do something like this.", + "I keep building up, then the inductor begins to", + "play a role, and then I taper off again.", + "This kind of a filter where I kill low and high frequencies", + "and pass intermediate frequencies is called a band", + "pass filter, BPF. This means that it passes", + "frequencies in some band. Let's get some more insight on", + "this by writing down the equations.", + "So, Vr divided by Vi is simply R.", + "Using the impedance relation it is R divided by j omega L plus", + "one divided by j omega C plus R. I am going to use this equation", + "later, so let me stash it away on my stack and put a little", + "notation there. I am going to multiply", + "throughout by j omega C. And what I end up getting is j", + "omega RC divided by one plus R j omega RC, and then here,", + "I get j times j is minus one, so I get minus omega squared.", + "Let me rewrite it this way. I get minus omega squared.", + "So, j j is minus one, omega times omega is omega", + "squared, and then I get an LC. That's what I end up getting.", + "And if I take the absolute value here, I end up getting,", + "back to your complex algebra, the square root of this real", + "value squared plus imaginary value squared.", + "So, one minus omega squared LC plus omega RC squared.", + "This is from, you can look it up in your", + "complex algebra appendix in the course notes.", + "It's simply omega RC here, then square of the real value", + "plus the square of the imaginary value, and take the square root", + "of that. By staring at this,", + "you can notice that you realize a really important property.", + "When omega equals LC. I'm sorry.", + "When omega equals one divided by LC, what happens?", + "Sorry, square root of LC. When omega is one divided by", + "square root of LC then omega squared times LC becomes one.", + "When this is true then this becomes one, and one and one", + "cancel out. And, not only that,", + "when these cancel out, these two cancel out at that", + "point, so I end up getting a one, which means that when omega", + "equals omega nought equals one by square root of LC and I end", + "up getting a value that is one. It's pretty amazing.", + "Which means that if I drive this at omega nought,", + "if my sinusoid has a frequency omega nought where omega nought", + "is one by square root of LC, if I'm sitting here and this is", + "a black box on the right-hand side, and I drive this at a", + "frequency omega nought equals one divided by square root of", + "LC, what does this entire circuit look like to me?", + "I'm sitting there, the black box here.", + "I'm driving it at omega nought equals one by square root of LC", + "at that frequency. What does that circuit look", + "like? Yes.", + "It looks like a resistor. It's pretty amazing.", + "It means that even though I have an L and a C here,", + "if I happen to drive this at omega nought then the circuit", + "looks purely resistive and it seems to give me the same input", + "appearing at the output. In other words,", + "the effect of these two cancels out.", + "And that aspect is called driving the circuit at its", + "resonance point. Resonance is when you're", + "driving the circuit at omega nought equals one by a square", + "root of LC.", + "", + "I will very quickly sketch for you a couple of other ways of", + "looking at circuits. Supposing I looked at this", + "value here, Vlc, I looked at the value across", + "the inductor and the capacitor, what will the frequency", + "response look like? I am looking at the voltage", + "across the inductor and the capacitor in series.", + "Let's see. Let's go back to our usual", + "mantra. Think about Steve Umans when", + "you do this. What would he do?", + "He would say ah-ha, at very low frequencies the", + "capacitor is going to look like an open circuit.", + "In my voltage divider, I am measuring the voltage", + "across an open circuit, so the entire Vi must drop", + "across the inductor and capacitor.", + "Similarly, at very high frequencies the inductor looks", + "like an open circuit now, so it looks like this.", + "At very high frequencies inductor is an open circuit.", + "And, again, I'm looking at the voltage divider across the near", + "infinite resistance, impedance, so I get a high", + "value here as well. Well, in the middle the value", + "dips and I get something like this.", + "So, this thing is called a band stop filter.", + "Here I can nail any specific frequency, as long as the", + "frequency falls in roughly that regime.", + "Yet another example.", + "", + "The reason I'm working on so many examples is that to", + "experts, a large part of what they do is look at a circuit and", + "boom, give a rough form of how it looks like.", + "That can get you half the way there in most of what you're", + "going to do. How did this look like?", + "If I take the voltage Vo versus Vi, let's take a look.", + "At very low frequencies, the inductor looks like a short", + "circuit, correct? I am talking the voltage across", + "a short circuit, so it looks like this.", + "At very high frequencies, I am taking a voltage across a", + "parallel combination, but the capacitor is now a", + "short circuit. So, that looks like a", + "capacitor. This looks like an inductor out", + "here and this is a capacitor holding sway here.", + "And so, somewhere in the middle it goes up and comes down like", + "that. So, it's a band pass filter.", + "What is amazing is that you can take fairly complicated", + "circuits, and just by doing a quick analysis of what happens", + "at very low frequencies, what happens at very high", + "frequencies, you can roughly sketch the response.", + "And then what you should do, in addition to that,", + "is if it's a second order circuit, just assume that it's", + "going to do something interesting at its resonance", + "frequency, at omega nought equals one by square root of LC.", + "Something interesting is going to happen.", + "Check it out. And for circuits that are first", + "order, RC or RL, the important number is the", + "time constant RC. Usually, when you're driving it", + "at one by RC, omega equals one by RC then", + "what happens is that you often times end up getting a value", + "that is one by square root two times the input value in the", + "circuits we looked at here. Next, what I am going to do is", + "talk about a major, major application of filters.", + "And that is an AM receiver. Let me do Radios 101 for 30", + "seconds. These guys have an antenna.", + "You take a ground here. You pick up a signal at your", + "antenna. There is an implied ground as", + "well. And what you do,", + "as a first step, is you begin processing the", + "signal now. What we place right there is a", + "little filter that looks like this.", + "It is a inductor and a capacitor in parallel.", + "And this capacitor is really your tuner that you can tune to", + "radio frequencies. And then what you have here is", + "a bunch of other processing and end up with your speaker.", + "And the processing that happens here is you have a demodulator,", + "you have an amplifier and a bunch of other things that let's", + "not worry about them for now. What we do here is the antenna", + "picks up a signal. So, in some sense,", + "this part of the circuit here is your source.", + "I could replace it with its Thevenin equivalent as follows.", + "", + "So, the front end of your radio looks like a Vi,", + "R, L and a C. Where have you seen this", + "before? Right there.", + "That's the front end of radios. Let me tell you why I need a", + "band pass filter in a radio out here.", + "The way life works is as follows.", + "I have my frequency. Let me do this not in radians", + "but in kilohertz for now, and let me plot your radio", + "signal strength. In the Boston area,", + "the signals go between 540 kilohertz and they go all the", + "way to 1600 kilohertz. In some areas we have begun to", + "use the 1700 extra band as well for some new stations.", + "This is the frequency range of interest.", + "If you look at your radio tuner, you will see 540", + "kilohertz all the way up to 1600 and you can tune your AM radio.", + "The way it works is that each station is given 10 kilohertz of", + "spectrum here. And so, this is at 1000", + "kilohertz, 1010 kilohertz and so on.", + "And each station transmits its signal in plus or minus 5", + "kilohertz around that point. And this station transmits it", + "here and this station transmits it here and so on.", + "This is 1030. This guy is WBZ News Radio", + "1030, for those of you who listen to it.", + "What happens is that at 10 kilohertz, each station gets 10", + "kilohertz, and so WBZ transmits in the 10 kilohertz around 1030.", + "Notice that each of these signals transmitted by radio", + "stations happen within small bands.", + "Now, you will learn a lot more about modulation and how do you", + "get a signal to go in a small band and all that stuff.", + "You will learn about that in 6.003.", + "For now, don't worry about how I did all of this.", + "How do you listen to that station?", + "The way you listen to that station is you put a low pass", + "filter here. You put a low pass filter that", + "does the following. Let's say I want to hear WBZ", + "1030.", + "", + "If I pass this entire signal through that filter.", + "And if I arrange to have the omega nought of my filter at", + "1030. If I can arrange to have the", + "omega nought at 1030 then this is the response of my filter.", + "And I am going to pick out this guy and cut out everything else.", + "I am just going to get this.", + "", + "Let's listen to the station for some time.", + "", + "So, you can see I can tune to the station WBUL." + ] +} \ No newline at end of file diff --git a/subs/bEJ0-8pANA4.srt.sjson b/subs/bEJ0-8pANA4.srt.sjson new file mode 100644 index 0000000000..6af767801c --- /dev/null +++ b/subs/bEJ0-8pANA4.srt.sjson @@ -0,0 +1,1352 @@ +{ + "start": [ + 6549, + 25220, + 36170, + 38000, + 66990, + 74980, + 81730, + 87810, + 95000, + 102880, + 106970, + 114460, + 124340, + 127750, + 142670, + 149770, + 155040, + 157120, + 165410, + 170790, + 176260, + 177680, + 183490, + 191130, + 197870, + 203380, + 212819, + 221180, + 228370, + 231500, + 252890, + 261099, + 265129, + 272349, + 279279, + 282370, + 297039, + 305580, + 308789, + 323599, + 332669, + 335279, + 342559, + 347889, + 352439, + 359800, + 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+ 3085680, + 3095490, + 3097040, + 3102430, + 3109270, + 3113760, + 3116600, + 3122660, + 3128980, + 3133360 + ], + "text": [ + "All right, good morning. So today, we are going to talk about what is both a basic device", + "in itself, the amplifier, and it also serves as a real key example of both nonlinear analysis", + "and small signal analysis.", + "So, today, dependent sources and amplifiers. So, let me first spend a few seconds just", + "pointing out to you some of the key points from our previous lectures. I also want to", + "point out that each chapter in the course notes has a summary at the end of it.", + "And if you take a quick scan of the summary at the end of each chapter, it highlights", + "the major takeaway points from each chapter. It stresses what's important, and if you have", + "to remember a few things, what are those things to remember? So, to quickly review, we talked", + "about a few primitive elements: resistors, voltage sources, and so on.", + "And by now, you should have the facility to play around with these device elements. And", + "then we talked about the Node method, and this is kind of the workhorse of 6.002. When", + "in doubt, use the Node method.", + "OK, and this will work both for linear circuits and nonlinear circuits. OK, so if you see", + "a problem, or if you see a situation in real life that requires analysis, then as a first", + "step, you should try to think of whether you could apply some of the key intuitive shortcut", + "methods, superposition.", + "One of my favorites, the Thevenin method, the Norton method, or the method that involves", + "composition, that is very quickly analyzing circuits that have resistors in series and", + "parallel. OK, so if you can apply one of these quick, intuitive, shortcut methods, go do", + "so.", + "If you can't, then usually you can resort to the Node method irrespective of whether", + "the circuit is linear or nonlinear. So the last week was focused on the nonlinear method", + "or nonlinear circuits, and we spent the first lecture talking about a straightforward application", + "of the Node method, which gave us a bunch of nonlinear equations that we had to solve.", + "In the last lecture, we talked about the small signal trick. What we said is if you look", + "at the whole space of nonlinear circuits, then within that space, if we focus on small", + "variations, small perturbations about an operating point, then even the behavior of nonlinear", + "circuits in that small regime would be linear.", + "So small signal method. And as an example, I showed you how I could take a highly nonlinear", + "device like the garage door opener LED, and using that, build a pretty nice transmitter", + "that would transmit music.", + "And as long as we kept the signal small, and operated the device in a region where its", + "transfer curve was relatively smooth, and I biased, or set the operating point appropriately,", + "I would get a linear, small signal response.", + "OK. So today, we're going to do a couple things. We're going to look at dependent sources.", + "And the reading for this is section 2.6 of your course notes. And, the dependent source", + "will be a new element in your tool chest.", + "We will also do amplifiers, and amplifiers are in section 7.1 and section 7.2 of your", + "course notes. So, before I begin with dependent sources, I'm just a huge believer in motivating", + "things with real world examples.", + "OK, so let me start by motivating: why we need an amplifier? Why do we need to do things", + "like this? Or why do we even bother? And, spend a few minutes really getting you to", + "appreciate that amplification is fundamental.", + "OK, it's as foundational to life as high fat potato chips and stuff like that. So, let's", + "do some basic examples here. So first, let me talk about, why do we need to amplify signals.", + "Why amplify? Why do we care about building an amplifier? So, an amplifier, think of a", + "little box, and apply some sort of small input.", + "And I get a larger output. In this example, this may be a voltage with a swing of 10 mV,", + "and in this case, the output might be another voltage with a swing of, say, 100 mV. And", + "commonly, the amplifier, in addition to an input and an output, input port and output", + "port, may also contain the power port, OK, so that I can apply a power supply to the", + "amplifier because commonly as an amplifier signal, I'm looking for a power gain as well,", + "an increase in the power provided by the output.", + "So, that's an abstract definition of an amplifier, and let's take a look at an example of why", + "we may need this. So let's say I have a small, useful signal, and let's say the signal has", + "1 mV peak to peak.", + "And, I'm looking to transmit the signal over a wire to some other point. But let's say", + "that in this environment, I get a bunch of noise that is in a noisy environment. And", + "in this environment, let's assume that some noise may get superimposed.", + "And if I have a 1 mV signal, and 10 mV of noise, then what I end up with at the output", + "is something that looks like this. And it's really hard to distinguish my 1 mV signal", + "from that large amount of noise.", + "On the other hand, if I do the following, if I took the signal and passed the signal", + "to an amplifier, and I amplified the signal to be a much larger version of the same signal,", + "let's say in this particular situation 100 mV peak to peak signal.", + "OK, so I magnified the signal by a factor of 100. OK, let's say it's a linear amplifier,", + "I linearly amplified signal to be 100 mV, then in that case, if I had a noise on top", + "of this, it's going to be less discernible.", + "The signal will look like this.", + "OK, my 10 mV noise would add on to it. But, this is still pretty decent. I can still recognize", + "the input. And so, this is one application of amplification. If I need to send something", + "from point A to point B as an analog signal, then an amplified signal is less prone to", + "noise attacks than a small signal.", + "Not surprisingly, a large number of devices that are used in everyday life have amplifiers", + "built into them. So, get a little cell phone, and virtually every single cell phone contains", + "an amplifier. By the way, this is an all digital cell phone.", + "It's a Kyocera, I forget the number now. It's completely digital. OK, although they say", + "it's completely digital, it turns out that a significant fraction of the circuitry is", + "analog, in particular, so digital is sort of a marketing term to say that there's something", + "special about this.", + "But remember, there's a bunch of analog stuff. So, here's my little antenna from the cell", + "phone. OK, and typically the first thing that happens to a signal as it comes out of the", + "antenna in your cell phone is, look at cell phone circuits, or cell phone systems would", + "be something that looks like this, OK, this, and may have a label LNA.", + "If someone were to take a guess at what LNA might stand for? What's that? Linear amplifier.", + "That's pretty good. So that's LNA. Close enough. A is correct. It's amplifier. What does L", + "and N stand for? Low noise.", + "OK, so this stands for low noise amplifier. So, I get a really rinky dinky small signal", + "here, and then the low noise amplifier amplifies a signal. And in real cell phones, and for", + "that matter, in your 802.11b, or 802.11a, or 802.11g wireless cards, same thing.", + "Antenna, low noise amplifier, and then you may have a bunch of processing. And commonly,", + "you have a bunch of analog processing. And then, you convert the analog to a digital", + "signal. OK, I recall last week I asked somebody in class here, how would we transmit the signal", + "from point A to point B without it being impacted way too much by noise, and he said, oh, go", + "digital.", + "Good point. OK, so if I go digital, I can transfer the signal without noise being a", + "real factor. But the analog to digital converters need the signal strengths to be a given value", + "before it can chop it up into digital levels.", + "OK, so an amplifier is very fundamental. OK, and so in this case, what may be a signal", + "of a few tens of microvolts to be amplified to some large enough value that it can be", + "further processed. So, that's application of amplification in the analog domain.", + "Let me talk about amplification in the digital domain. So, that's in the analog domain. This", + "amplification is in the domain that I have both analog and digital. OK, and now let me", + "talk about amplification in the digital domain, OK? I'm going to argue that amplification", + "is absolutely foundational to the digital domain.", + "OK, the digital abstraction would not occur if I did not have basic amplification. OK,", + "and the next minute and 37 seconds I will prove that to you, OK? So, let's do so. So,", + "let's suppose I have a very simple digital system, and the system simply contains a pair", + "of inverters.", + "So, if I send a one here, it's a zero here and a one here, which is a very simple, trivial,", + "digital system. And here's the input. Here's the output. And we said that for digital systems", + "of this sort to work, they have to follow a static discipline.", + "OK, our signals and our circuits must follow a discipline for them all to work together.", + "And, the discipline we described comprised of signals adhering to certain voltage thresholds", + "so that all the components in the system could agree on what comprised a zero, and what comprised", + "a one, OK? So the way we did that was we said that you would have a threshold called VIH,", + "V input high, and another threshold called VIL, V input low.", + "OK, and we said that this circuit must recognize signals that are higher than VIH, 3 V for", + "example as a one, and simultaneously, any signal that has a voltage level less than", + "VIL, say, two volts, should be recognized as a zero.", + "That was the input constraint. On the output, it had a similar set of constraints, where", + "we had tougher constraints on devices, where we said that the output had to satisfy a output", + "low constraint, output high constraint.", + "What this said is that for this circuit to be called a good digital circuit that satisfies", + "the static discipline, signals that were ones here should be recognized as such. And if", + "I am producing a one as an output, then the signal level should be higher than VOH.", + "Similarly, if the signal's a zero, then it should be less than VOL. So as an example,", + "this may be 2 V, this may be 3 V, and this may be 4 V, and this may be 1 V. OK, so input,", + "I should recognize 2 V and less as a zero, but at the output I have to produce a very,", + "very low value, 1 V.", + "So, I have some noise margin. So as an example, say if I made a plot of the input/output,", + "so I get my VIL here and VIH here. This is time. This would comprise a valid digital", + "signal: zero, one, zero, one, and so on.", + "OK, now, I had a tougher set of constraints at the output. I would have VOL, VOH. So,", + "at the output, OK, I'm required to stretch the ones and zeros to be further apart from", + "each other so that I get noise margin, and the corresponding signal for our little circuit", + "there would look like so.", + "Right, if this is a valid input, then this would be the corresponding, valid output.", + "OK, and need I say more? OK, you can see that, intuitively, look, there's amplification happening", + "here, and the reason is that VOL is chosen to be less than VIL, and VOH is higher than", + "VIH.", + "So therefore, the signal has to be stretched. The signal has to be amplified. OK, and what's", + "the minimum amplification needed for the system to work? The minimum amplification is if I", + "had a signal that looked like this.", + "OK, that barely skimmed the VIL, VIH level. OK, so if signal were this high peak to peak,", + "VIH minus VIL, and what's the absolute minimum signal at the output? It would look something", + "like this. OK, barely skimming VOL and VOH, OK, so the corresponding output level would", + "be VOH minus VOL.", + "OK, so this is the absolute minimum amplification that my digital circuit has to provide. OK,", + "and notice, VOH is larger than VIH. VOL is smaller than VIL. Therefore, this quantity", + "needs to be greater than one.", + "OK, so I've shown you both a simple, graphical, intuitive explanation, and this is a slightly", + "more formal proof that even the digital circuit really requires to have amplification built", + "into it, if it is to satisfy valid static disciplines.", + "Yes? Yes. The question is, is that the same as gain? Good question. Yes, the term amplification", + "has many, many variants. You could say gain. You could say amplification. You could say", + "increase in signal strength, and so on and so forth.", + "And in fact, when talking about low noise amplifiers, people sometimes talk about having", + "the low noise, high gain amplifier at the input stage. OK, so let me pause there in", + "terms of motivation. So, I believe I've motivated every which way: pure analog, analog/digital,", + "and digital.", + "OK, so I've covered every single base here. And so, we need amplification. OK, so let's", + "look at how to build a fundamental, primitive device called the amplifier. Before we do", + "that, however, let me take a quick detour.", + "It will be convenient for me, as I show you how to build an amplifier, to introduce a", + "new device, a new element, called the dependent source. OK, let me introduce a new device", + "for your arsenal of devices, along with resistors, You learned about a MOSFET, a switch, voltage", + "source, current source, and now a dependent source.", + "So, a dependent source looks like this, OK, has an output port, and has a control port.", + "So, a dependent source in its simplest form has two ports: an input port and an output", + "port. Remember, a port is a convenient pairing of terminals, and I apply signals to such", + "terminal pairs.", + "But this is a abstract diagram for a dependent source, and to get a little bit more specific,", + "let me show you an example of a dependent source. So, let's say, here's my input, and", + "I label the terminal variables for the input.", + "VC is the voltage applied to the input, and IC is the current into this terminal here.", + "And, here is the symbol for the dependent source. Much like a current source or a voltage", + "source has a circle around it, the corresponding symbol for a dependent source is like so.", + "So this example, for instance, is a dependent, current source. I can apply the corresponding", + "output variables, I0, OK, and I can say that the current, I, is some function. In this", + "example, I've designed the example that the current through the current source, I, is", + "some function of the input voltage or the control voltage, VC.", + "OK, so notice that the current through a current source, the current through this current source,", + "I, is some function of another variable. OK, in this example, it's the voltage across its", + "control port.", + "Not surprisingly, this device is called a voltage controlled current source -- -- or", + "a VCCS. So, in like manner I can also devise other forms of sources. You can think of this", + "is a device where a voltage controls an output current.", + "You can think of all other combinations, current controlling current, voltage controlling voltage,", + "current controlling voltage, and so on. So, another example, I give you another dependent", + "source, and in this situation, my output current is controlled by an input current, VC.", + "IC rather. And I claim that I for this one is some function of a current, IC. OK, it's", + "another dependent source where the output current for its output port is related to", + "the current, IC. And, this is a current controlled current source.", + "OK, it's a current controlled current source. And, if I had lots of time on my hands, and", + "I was wanting to kill time, I'd sit around drawing for you, other types of dependent", + "sources. I would draw for you a current controlled voltage sourced, and I could also draw for", + "you a voltage controlled voltage source.", + "OK, so that's an abstract diagram for such a source. And so, let's do a few examples", + "involving elements like this. To begin, just so you can build up your intuition, let me", + "start by doing a very simple circuit, involving an independent current source, OK, just so we can relate back to what we've", + "been doing so far.", + "So, let's say I have some resistor, and I have a standard current source with current", + "I nought. This is an independent current source. Remember the circle? And, some resistor, R,", + "and let's say I care about the voltage across the resistor.", + "OK, so I have a current I nought flowing through it. So, I can very quickly write down VR as,", + "simply, I0 R. OK, it's the drop across the resistor when a current I nought flows through", + "it. OK, so this is what you've been used to doing.", + "Correspondingly, I can do an example with a dependent current source. And, as an example,", + "I'll use a voltage controlled current source. OK, a voltage controlled current source is", + "a dependent current source whose output current depends on the voltage applied at the control", + "port of the current source.", + "So let me build a little circuit. OK, so here's my current. And let's say it's VC IC for the", + "control port, and similarly, let's say my current I here is some function of the control", + "port voltage. And let's say, to be specific, there is some K over VC, some function.", + "OK, there are a variety of dependent sources that can be built, and here's a hypothetical", + "device where the output current is mathematically related to the input in the following manner.", + "So, let me build a circuit of the following form.", + "So, let's add the resistor, R, and here's my circuit, OK? And, as before, let me look", + "to figuring out what VR is. So, notice that I have to supply some voltage at the input", + "so that the output can depend on the input because right now I don't know what the input", + "here.", + "So what I'll do is let me apply VR over here. OK, so let me make this connection. OK, let", + "me make the connection from here to here. What I've done is I've applied VR at the control", + "port of the dependent current source.", + "OK, and I often draw a circuit like this. This looks pretty messy. I will often draw", + "the circuit like so: R, VR. OK, short form circuit drawing would look like this. This", + "is a complete drawing that I show you the explicit connections of the control port,", + "but oftentimes, when the control port does not have any other impact in the circuit,", + "you can eliminate, don't explicitly show the control port.", + "Rather, you can simply show the dependence of the output current on whatever circuit", + "variable you have in mind. So, you can draw the diamond like this, and see its current", + "is some function of VR. VR in this is case is K divided by VR, OK? OK, so let's go ahead and analyze this little", + "circuit here, and look at what this might give us.", + "Our goal, as before, is to find out the value, VR. So, in this case, let's apply the Node", + "method to this node, and sum the currents into that node to be zero. OK, so sum the", + "currents going into that node to be zero.", + "The current going down is simply VR divided by R. OK, and that is equal to the current", + "that is going out of the node. And so that is equal to F of VR. And I know that F of", + "VR is given by K divided by VR.", + "OK, a simple application of the Node method. So then, I collect VR's on the left hand side,", + "and I get VR squared is K times R, OK, and VR is simply the square root of KR. There", + "you go: I'm done. OK, I've gone ahead an applied the Node method to this, and when have to", + "figure out the current here, I simply reflect the fact that it depends on VR like so, and", + "I just go ahead and solve the circuit.", + "Remember, the workhorse of the circuit industry, the Node method, when in doubt, apply it.", + "It simply works. And notice, this is a nonlinear circuit. OK, the dependence is nonlinear,", + "and I get the response like so.", + "So, to plug in some numbers, supposing K was 10 to the minus 3 amperes per volt, and R", + "was one kilo ohm, then I can plug the numbers in and the kilo here cancels with the 10 to", + "the minus 3, and I get VR equals 1 V.", + "OK, this simply says, if I build a circuit like this, then this voltage here will be", + "1 V. So, again, as long as you remember that the dependent source is simply another little", + "circuit element, OK, and you usually draw just the output port for dependent sources,", + "and reflect the way that the control affects the current, that'll suffice, and you get,", + "through the application of the Node method, the variable you're interested in.", + "Let's do another example,", + "OK, of another fun current source, a voltage controlled current source, and look at it", + "this way. So, let's say I have a resistor, and I have a current source, a resistor, RL,", + "and this goes to some, I apply a VS here.", + "Remember this short form notation; that's simply applying a supply VS between that node", + "and the ground. OK, and let us say the current IV through the device is some function of", + "the current at its control port.", + "OK, so I'm not going to show you that. But remember that the device already looks like", + "this, that there is a control port here. I'm not showing that to you. And let us say that", + "I apply some voltage, VI, to the input port.", + "The reason we often don't show the input port is for many practical dependent sources, the", + "input has no other effect on the circuit. So, for example, in this case, the input has", + "infinite resistance looking in.", + "So therefore, if I apply a VI here, it doesn't draw any current from VI. I simply apply the", + "voltage, VI. It doesn't affect the circuit in any other way except in terms of how it", + "controls the current ID.", + "So let's say the current ID is some function of VI because VI is applied at the control", + "port. OK, and as I pointed out before, I oftentimes, just for clarity, just to show this dependent", + "source explicitly.", + "OK, so let's work the example. So as I said, I'm going to choose ID to be F of VI, and", + "let's pick some specific parameters here. Let's say it's K by two VI minus one, both", + "squared. OK, and let's say this is true for VI less than equal to one volt.", + "And let us also say that ID equals zero for VI less than one volt. OK, it's a dependent", + "source, and it can have various forms of dependences on the input. And, I just picked an example", + "of some hypothetical, or as yet, hypothetical dependent source, the current through which", + "is related to the input using a square law relation, VI minus one all squared as long", + "as VI is greater than one.", + "And if VI is less than one, then the current is simply zero, it shuts off. So, I can go", + "ahead and apply. So, let's say I want to find out V0 versus VI. So, I care about finding", + "out V0. V0 is the voltage of this node with respect to ground.", + "OK, so it's a slightly more complicated circuit than you saw up here, than you saw up there.", + "So, let's go ahead and do this example. Start by applying the workhorse of the circuits", + "business, the Node method, and let's start with doing this for VI.", + "Let's first do it for VI greater than one, notice the behavior of this is different for", + "different ranges of VI. So let's first do it for VI greater than or equal to one and", + "apply the Node method. Node method says sum the currents going into this node; we know", + "the voltage at this node.", + "It's VI. We know the voltage at this node. It's VS. OK, the only unknown is V nought.", + "And so, let's go ahead and write the node equations for that node. So, the current going", + "up, let me simply equate the current going up to the current that has been supplied by", + "this particular node here.", + "And, that should equate that the two of them should sum to zero, the current going up plus", + "the current going down should sum to zero. So, I get V0 minus VS divided by R. That's", + "the current going up.", + "Plus, the current going down must sum to zero, plus ID must sum to zero. And ID is going", + "to be K divided by two VI minus one all squared. That must equal zero. Straightforward application", + "of Node method, current going up plus the current going down at this node should equal", + "zero because the total current leaving the node must be zero, OK? So I can go ahead and", + "simplify this, multiply it throughout by, I call this RL here.", + "So, multiply it throughout by RL, and move all of this to the other side, so I get VS", + "divided by RL, multiply it throughout by RL. I get VS at this side. I take this term to", + "the other side. This becomes a minus.", + "RL multiplies here, so I get KRL. That's the expression I get. V nought is VS minus KRL", + "divided by two times VI minus one all squared. Let me put a box around this because I will", + "be referring to this more times in 6.002 for a variety of reasons than probably any other", + "equation on Earth.", + "OK, this is the first time you saw it. You saw it here. OK, mark it down. You'll smile", + "every other time you look at it in quizzes, and you will find out why this comes up very", + "often in 6.002. So, I'll just give you a few seconds to savor this big moment in your 6.002", + "life.", + "All right, OK, so it's pretty simple actually. I mean, there's really not much. A lot of", + "this stuff is just a plain old, simple application of the Node method, and things just fall out.", + "It's just so simple.", + "So, the V nought, I apply the Node method, I get V nought for this nonlinear circuit.", + "I can also it for VI less than one. For VI less than one, when VI is less than one, what", + "happens? ID is zero. OK, since ID is zero, think of this as an open circuit.", + "OK, so there's no voltage drop across RL. And, this voltage V nought is equal to VS.", + "So, I like to see things in pictures. I'm not an equations kind of person. I'm much", + "more of a graphical person.", + "So, let me draw a little graph to show how V nought, to see the form of V nought, and", + "then let's study that little system a little bit more carefully. So, this is page seven,", + "and we plot V nought versus VI for you.", + "And let's take a look at how this really simple circuit looks. This has got nothing. It's", + "got an RL resistor connected to a supply, and a dependent current source, and I apply", + "some voltage VI at the input.", + "It's a very, very simple circuit. So, let's see. So as long as VI is less than one, the output stays at VS. OK, that makes intuitive", + "sense, right? As long as the current here is zero, this is like an open circuit here.", + "If this is an open circuit, then effectively, V nought is simply the voltage VS. V nought", + "simply appears here. If you want to grunge through KVL and KCL, go ahead. VS minus RL", + "times the current is V nought, and the current is zero so it's, yes.", + "So, this is simply VS. When VI goes above one volt, fun stuff begins to happen. OK,", + "when V nought goes above one volt, then this equation applies because VI is greater than", + "one. This equation applies.", + "And, when VI is a one, one minus one is zero. This term cancels out, so this is VS. OK,", + "phew! So, I start off here. As VI increases, what happens now? As VI increases, this term", + "here becomes increasingly negative, OK, subtracting from VS.", + "OK, so I get some behavior like this. V nought begins to drop. And it makes intuitive sense,", + "right? As ID begins to increase, the voltage here will begin to drop because I'm drawing", + "more and more current through RL.", + "I'm dropping more and more across RL. So more and more drops across RL, so V nought begins", + "to drop too. So, it looks something like this. I'll show you a little demo, but my claim", + "is that you have just seen an amplifier.", + "Whoa. You just saw an amplifier. So, I snuck an amplifier by you, OK? So, I just snuck an amplifier past you. I'll show", + "you why in a second. So, let's take a look at this waveform here. Let's not worry about", + "what happens way down here.", + "We'll talk about that a little later. But, look at this curve here. I claim there is", + "amplification in the following sense. Focus on some change in the input voltage, delta", + "VI, OK, and for that change in input voltage, I get some change in the output voltage.", + "OK, for some change in the input voltage, delta VI, I get some change in the output", + "voltage. And guess what? In this, at least the way I have drawn it, delta V nought divided", + "by delta VI, if I can find regions of the curve where this is greater than one, then", + "I have amplification.", + "OK, so what's that saying? What that's saying is that if I apply some voltage here, OK,", + "and I change that voltage by a small amount from, let's say, 2 V to 2.1. OK, I am going", + "to find the output voltage.", + "Let's say I go from 2 V to 2.1 here. OK, abstractly out there, I might have an output that goes", + "from three to, let's say, two V perhaps. OK, so for a 0.1 change here, I'm going to get", + "a bigger drop here, so from 3 V to 2 V, giving me an amplification in this little circuit.", + "OK, so we'll see this again and again, and you'll really understand it. So, I have a", + "small change in the input, and I have a corresponding larger change in the output. So, I've shown", + "you an amplifier.", + "I haven't shown you a linear amplifier. There's an extra charge for that. OK, that'll happen", + "later. OK, all I've shown you so far is an amplifier, and this happens to be a crummy", + "amplifier. It's a nonlinear amplifier because, notice, this is not linear.", + "It's a nice little curve, and so it's not linear. But, I promised you an amplifier,", + "and I'm cheap, and that's all you get for now. OK, we'll see linear stuff later, but", + "for now, I have a little amplifier.", + "So, let's do some real numbers, and plot some numbers down, and also look at a demo. So,", + "let's do an example. Let's say VS is 10 V, that the K is two milliamps per V squared,", + "and let's say RL is five kilo-ohms, OK? So, let me substitute these values into that equation,", + "and I get V nought is, VS is ten.", + "So, it's ten minus, KRL divided by two. So, K is two milliamps. Two milliamps times five", + "kilo-ohms is ten divided by two gives me five, and VI minus one squared. That's what I have.", + "I just plug in a bunch of numbers, and that's what I get.", + "So, what I'll do is let me just do a little table for you, and plot using real numbers,", + "simply plot those values for you.", + "So when VI is zero, my current is zero, and I get - oh, that equation doesn't apply, by", + "the way; that applies when VI is greater than one. Ok. So as long as VI is less than one,", + "my output is simply VS, the output is simply ten volts.", + "Ok, so all the way up to one, my output V is - all the units are all volts - that's", + "what I get.", + "I come down to 2, I plug in two for VI, 2 minus 1 is a 1. So it's 1 squared. 5 times", + "1 squared is five. And 10 minus 5 is 5. I get 5 out here.", + "Ok? And then, I can go and do the math. If it's 2.1, I get 4 volts here. And 2.2, I get", + "2.8. And so on. Ok, notice that the .1 volt change here resulted in a - go up a .1 change", + "here results in a minus 1 volt change there. So input went up by .1 volts, my output...", + "kerplunked down by one whole volt.", + "Ok, so a small change here resulted in a bigger change there. And that's the amplification", + "that I am claiming here.", + "Ok? So. Let me show you a small demo of a small device that I built involving such a", + "dependent source.", + "Let's do it here.", + "Ok. So on my s axis, here is VI. And VO is on my y axis. And focus on this little point", + "here.", + "Ok? Right now my VI is zero. And so therefore that's my output. I will gradually increase", + "VI, we're going to watch the output and see how it behaves. Pretty much like the little", + "graph I drew for you.", + "So I'm increasing VI. Ok, notice that invitially the current source is off, the dependent source", + "is off, so I move straight down. Ok, nothing happens. Until I hit a value at which the", + "the current source begins to come on. And then I begin to see a drop in the output as", + "the current source begins to conduct current.", + "So you see that as I increase the voltage VI, boom! You see the huge drop.", + "Ok, notice that for a small change, I'm now getting a big drop in the output. Ok?", + "So let's pause here for a second, and in the last couple of minutes, I want to cover one", + "last point.", + "Notice that the curve I've shown you up there looks like the curve up here: it goes kaboom!", + "and drops according to come kind of formulation that I've shown you here.", + "In the last couple of minutes, let me discuss a small point that's a practical issue.", + "In the curve that I showed you, in the mathematics that I gave you, if you just go by the math.", + "So what I'm about to show you will differentiate a mathematician from an electrical engineer.", + "Mathematicians would have taken the curve, and reported it like this. This is zero. Mathematically, that equation says that starting", + "here, this current simply goes down. But if I told you this device that I have here, this", + "dependent source is a practical dependent source, a device that I have physically built,", + "and I also say that it's a passive device. In other words, it cannot produce power. It's", + "like a little resistor. It doesn't produce power. It's a passive device.", + "So if it's a passive device, if I tell you that, then you'll say, Something doesn't make", + "sense here. Mathematically, it says it should look like this. But what's special about a", + "point down here? The point down here says the output has gone negative. This is zero", + "here. The output has gone negative. And my current source is still supplying a current.", + "Ok, so up here, the voltage across the device is positive, and it's supplying a current,", + "so it's consuming power like a resistor. Like all bad little resistors do, they burn power.", + "So here, on the other hand, my output is going to be negative, but it's still sitting there", + "sinking current. Ok, because VO is negative, but my current is still in the same direction,", + "what has now happened is the device has begun to supply power.", + "So mathematically, this curve says the device has begun to supply power.", + "It turns out it's not a practical device, it's a passive device. So it cannot go here.", + "So what happens is that somewhere along here, our model breaks down. The equation I've shown", + "you for the current source, where is it?", + "This model breaks down.", + "When V0 becomes very small, the model breaks down and it no longer behaves like a current", + "source.", + "It begins to behave more and more like a resistor. And what happens realistically, is that the", + "output goes down and then kind of becomes a zero-hugging line. I'll show you that in", + "a second.", + "And it doesn't - for this particular device it doesn't really go down here.", + "Let me just show you that part. So notice for that device, mathematically, just by that", + "model, it should have just gone through the floor and it points through to the corridor", + "below, but this is a prcatical device.", + "So notice that the model breaks down. And that's what begins to happen.", + "Ok, the device stops behaving like a dependent current source, rather it behaves like some", + "corny old resistor or something like that and saturates out. Ok." + ] +} \ No newline at end of file diff --git a/subs/bX8i2yECWaU.srt.sjson b/subs/bX8i2yECWaU.srt.sjson new file mode 100644 index 0000000000..654b895b62 --- /dev/null +++ b/subs/bX8i2yECWaU.srt.sjson @@ -0,0 +1,1538 @@ +{ + "start": [ + 0, + 5000, + 11368, + 14894, + 20466, + 27403, + 34000, + 41431, + 47137, + 54170, + 62000, + 70235, + 76411, + 84509, + 90000, + 97721, + 104430, + 110759, + 116455, + 123828, + 130033, + 136897, + 143498, + 148382, + 154455, + 160000, + 165145, + 171025, + 177430, + 184417, + 191960, + 198207, + 202685, + 209167, + 217731, + 225323, + 232529, + 239477, + 246683, + 252214, + 259357, + 266500, + 270785, + 275785, + 284000, + 288117, + 292529, + 296573, + 304109, + 314668, + 319947, + 330506, + 338000, + 343213, + 348545, + 353639, + 360037, + 367133, + 374199, + 381666, + 387266, + 395133, + 402539, + 408792, + 415988, + 424806, + 432928, + 443038, + 450000, + 456553, + 463553, + 466531, + 471595, + 480234, + 487373, + 491937, + 495954, + 500975, 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2806333, + 2811571, + 2813761, + 2817857, + 2823000, + 2828523, + 2832849, + 2838004, + 2843619, + 2866130, + 2916000, + 2921000 + ], + "text": [ + "", + "I will be replacing Professor Agarwal today because he is", + "away. I am one of the recitation", + "instructors for those of you who have not seen me.", + "We will talk today about a neat application of RC networks and", + "expand those to application in MOS memory systems.", + "To connect with everything, we will get back to the basic", + "circuit that we have been discussing so far.", + "And you recall the circuit that we have been studying,", + "the canonical RC with an input voltage function of t.", + "And we had specified that we solved this problem for the case", + "of a step input or a condition in which a t=0.", + "At t greater or equal to zero vI is equal to some capital VI", + "value that for now on is constant.", + "And the other condition that we discussed was the value of the", + "voltage on the capacitor that would exist at time t=0.", + "Let's call that vc(0). And in general there is some", + "finite value here. It can be zero or it can be", + "different from zero. Given that, we learned how to", + "write down directly, without messing around with", + "differential equations, the answer for the voltage on", + "the capacitor vc(t), let me define also my vc right", + "here, is equal to VI, the final value,", + "plus vc(0), the initial value on the capacitor,", + "minus the final value, e^-t/RC.", + "This is our standard equation to which we plug in,", + "and it's either a rising exponential if VI is larger than", + "VC or a decaying exponential if VI is a smaller value than VC.", + "This should all be familiar. And, again, as pointed out in", + "the notes, the reading for today is 10.3 and for the new material", + "you should look at Chapter 11 where we discuss memory.", + "This is where we stood as of last time.", + "Now, I would like to discuss a little bit more about the", + "storage of charge in capacitors. And how we can take advantage", + "of that for storing logic state. One of the things that I am", + "sure you must be aware of is that one of the perhaps most", + "massively produced chips is actually the so-called DRAM", + "which you find in every PC and every computer that exists", + "anywhere. This DRAM is dynamic random", + "access memory in which we can store a state and come back and", + "look at it at any time later, provided we don't power off our", + "machine. The logic state in the basic", + "memory elements, of which instead there are", + "close to 1 giga elements per chip, are stored on capacitors.", + "And so we will play a little bit with that concept today.", + "And, although we're not going to discuss the specific example", + "of the DRAM, the basic elements of the DRAM you will see", + "actually in a demo shortly. So that's the general response", + "of this network that I have here to an input VI that happens at", + "t=0. Now, the one thing that you", + "recognize immediately is that it really doesn't matter what the", + "value of VI was for t less than zero.", + "What really counts is the value of VI at t=0.", + "And that's the value that we're interested in.", + "Now, there is an implicit statement in that.", + "And that statement is that somehow that network appears", + "like this at t=0. So, there has to be some switch", + "there, and you will see that, that basically starts my", + "condition to that at t=0. And so the history of VI really", + "doesn't matter. The response following that", + "equation that we have there will depend on the initial value", + "which is vc(0) here. Now that is the voltage on the", + "capacitor at that time. And then assuming that VI is a", + "value that is larger than vc(0) will have a rising exponential", + "that will come to this value. And this is the time constant", + "RC and this is time. So, the capacitor starts with", + "some voltage here and goes to a new voltage that is imposed by", + "the input for time greater than zero.", + "We can define at any one time, say this time,", + "this time, this time, this time the state of the", + "capacitor. The state.", + "What is the state of the capacitor?", + "The state is the summary of all inputs that are relevant to", + "predicting the future. If I know the state of the", + "capacitor this time, I can predict what it is going", + "to go given a response VI here in the future.", + "So, predicts the future. Now, what is the state variable", + "on the capacitor? What is actually stored on the", + "capacitor? You can say,", + "well, what is stored is voltage.", + "The real physical quantity that is stored is the charge q which", + "is for linear capacitors related to the voltage,", + "let me actually write it correctly, vc like this.", + "So, the real state variable is this.", + "", + "But for a linear capacitor, since there is one-to-one", + "relationship between the two, v is also a state variable.", + "OK, so let's then go back to our original circuit.", + "What we have is --", + "", + "-- vc(t), so that's the future value of the voltage on the", + "capacitor, is a function of vc(0), the initial value and the", + "variable input now in the future time.", + "And for the case of vI(t) being constant VI for t greater or", + "equal than zero we have the equation that we just described.", + "Nothing new.", + "", + "All the past inputs to the capacitor for time t less or", + "equal to zero is summarized in this value.", + "And vi being constant the future is predicted from that.", + "So, that's the concept of the state.", + "There is an initial state on the capacitor.", + "And then there is a final state that will be reached when", + "equilibrium actually is achieved.", + "There is a fair amount of discussion in the text,", + "and we don't go in great detail here, but it is both convenient", + "for analysis and also it's interesting in many cases to", + "look at the response of a linear network for two different", + "conditions. So, we're interested in two", + "cases. One is the so-called zero state", + "response. Now, what is the zero state", + "response? It's the response to a", + "condition in which we impose an input and impose also the", + "condition that the initial value, initial state of the", + "capacitor is zero. So then we ask how does it", + "respond to vi(t)? So, starting with a capacitor", + "at zero state what is the response?", + "It allows us to decouple the initial conditions from the", + "response to the input. Now, you will see that this is", + "actually very useful. The second condition to which", + "we're also very interested is the so-called zero input", + "response. What is that?", + "That is vi(t)=0. Now, it's the condition under", + "which there is no input. vi(t)=0.", + "The question here is how does it relax?", + "We're starting with an initial state.", + "So, how this state relaxes out in the circuit.", + "Now, the zero state response, this one here is Z so called SR", + "for our case, which I will write like this,", + "vC, ZSR is simply a rising exponential.", + "We start from zero and we go to VI.", + "So, it's VI-VI e^-t/RC. So, that's the ZSR.", + "The ZIR, the zero input response is like this.", + "It's the decay of the initial voltage on the capacitor to zero", + "or to equilibrium. Starting from vC(0) we're", + "decaying like this. Now, do you see something", + "that's rather obvious from what's on the board in terms of", + "ZIR and ZSR and the final complete answer which is there?", + "", + "They are specific cases, but how do they relate to the", + "full answer? It's the sum.", + "It's the superposition of the two.", + "What basically we see here -- And that's actually a general", + "statement, is that vC = vC,ZSR + vC,ZIR.", + "Now, you may say this is trivial because we started from", + "that, ended back in that from some very simple observations.", + "However, we are not always solving networks for responses", + "that are steps. The input voltage may be a", + "ramp. We did that in recitation.", + "Or, it could be an impulse. Or, it can be a more", + "complicated function. Having this observation in", + "place actually allows us to solve the problem rather neatly.", + "If I have time at the end, I might come back to this.", + "So, this is the same equation as I started with,", + "arrived at from a principle of superposition of two different", + "solutions. One application of state which", + "can be, since we have energy storage element here,", + "the capacitor, which can be stored on the", + "capacitor is in memory. And you may ask,", + "so why do we need a memory node to perform logic?", + "Well, there are cases in which a result depends on previous", + "results. So, a computation proceeds in", + "time. In order to do that,", + "we need to store intermediate results and proceed forward.", + "One good example is if you're doing a continuous summation,", + "say, on your calculator, you keep putting things in the", + "memory. The M+ button,", + "right? And you keep adding a series of", + "numbers. Every time we store the sum of", + "the previous operation we add another number and so on.", + "Clearly we need some way of storing state.", + "For a complete computing system, we need combinational", + "logic and we need memory. In fact, these are the two", + "basic elements that are essential for any kind of", + "computing system. We need to remember", + "intermediate results. We need to remember transient", + "inputs. And that's the role that all", + "these enormous amount of memory that comes to play in computers", + "is doing. The basic memory abstraction is", + "as follows.", + "", + "Imagine a block which needs to be populated by transistor,", + "resistor, capacitor, whatever elements.", + "And it has a control input, which we will call the store.", + "It has a state input that we will call dIN and has an output", + "dOUT. When we're telling this", + "element, OK, now it's time to store, it looks at the input dIN", + "and stores it for, in principle,", + "an infinite amount of time. If we were to make a drawing of", + "this, of what this looks like, let's suppose,", + "let me do all this in one axis. So, time moves this way.", + "Let's suppose that we have an input dIN that looks like this,", + "and the store command comes in the form of a logic.", + "Let's actually suggest here this is logic one,", + "this is logic zero. And, although this is not", + "absolutely necessary, let's also define that the", + "store command comes in the form of a logic one at this store", + "input. Store, let's say,", + "looks like this.", + "", + "What does the output look like then in this particular case?", + "Assuming that the output was dOUT, the stored element was", + "zero prior to the store, then the output would look like", + "this. This is dOUT.", + "As you can see, it would remember the one that", + "it saw at this point. In fact, it would do that", + "irrespective of what was stored in this memory cell.", + "For example, suppose it was storing one and", + "the output didn't change, it's still one.", + "If it was storing a zero, it would flip to a one.", + "If we had another store, let's say here,", + "then what happens? Then it would go back down to", + "zero because now we sampled an input that is zero and we", + "flipped the state. That's what a memory --", + "", + "-- element or cell would do for us.", + "It would remember the output state.", + "And, not only that, but in principle it should be", + "undisturbable. In other words,", + "I may do something to this dOUT but it should not flip the", + "state. And that comes about quite a", + "bit. Because in actual integrated", + "circuit memory there is lots and lots and lots of nearest", + "neighbors to this cell which, when they're flipped,", + "have a cross-coupling to the cell.", + "The cell must be designed robust enough that it doesn't", + "flip, that no coupling actually occurs.", + "All right. Now we're going to try to apply", + "what we've learned so far to invent a basic memory element.", + "And, believe it or not, this is the key to the DRAM.", + "", + "Let's implement this in a circuit.", + "", + "Suppose I have a switch here like this.", + "", + "And I will put a capacitor. I take my dOUT here.", + "This is dIN. And the switch is operated by a", + "command here that we will call store.", + "When store is one it goes up. When store is zero it is down", + "here. That's capacitor C.", + "This is the storage node.", + "", + "What are we actually storing in this case?", + "Let's suppose that this voltage here is 5 volts.", + "I flip the switch up to one and I flip it back down to zero.", + "What's the voltage in this capacitor here?", + "5 volts. Now the capacitor is at 5", + "volts, I put dIN to ground, flip the switch back up and", + "then back down to its known storing condition.", + "What's the voltage in the capacitor?", + "It's zero, exactly. So, it does store the value of", + "the voltage that it saw, five or zero,", + "high and low. It stores it because it stores", + "charge. That's actually the physical", + "quantity that's stored. It's manifested as a voltage,", + "which we see. All right.", + "Now, is this, oh, before I move from here.", + "What is the basic cell in a DRAM, one that you go out and", + "buy by the billions of cells? It's actually this.", + "The only difference is that this switch here is replaced", + "with a MOSFET.", + "", + "And that's all it is. So, a MOSFET plays the role of", + "the switch. When the gate is high this is a", + "resistor and connects the input to the capacitor.", + "And when the gate voltage is below the threshold voltage this", + "is an open, as we've seen, and it isolates the transistor", + "from the output. So, that's the basic memory", + "element.", + "", + "And, as I said, it's the key to a DRAM.", + "OK. Now let's consider a little bit", + "the conditions of operation of this thing.", + "Let me draw the circuit in two conditions.", + "One in which it is storing, one in which it is sampling and", + "one in which it is storing. Not to redraw this thing.", + "Assuming that I have a MOSFET there, I would have the on", + "resistance in place here when store=1.", + "Now, in principle, the output is connected to --", + "", + "-- some load resistance. We'll talk a little bit more", + "about this load resistance in a minute.", + "This is the situation when we are at store=1 situation.", + "For example, let's suppose that dIN is 5", + "volts. Now, what is the situation for", + "store=0? It's very simple.", + "We have the capacitor C and dOUT and here we have a", + "resistance. The switch is open.", + "This is store=0 condition. What we have in this case is we", + "have a problem similar to what I was discussing earlier.", + "It is a ZIR, if you like,", + "situation. And this you can think of as a", + "ZSR if we're starting with zero charge on the capacitor,", + "but I'm interested in this part.", + "In this case, I am starting with a vC(0)=5", + "volts. And I'm asking myself how long", + "will this cell hold the value? And, in fact,", + "that is actually what happens in a dynamic RAM.", + "The value on the capacitor is not stored forever.", + "In fact, that's why we call it dynamic because we have to come", + "back and restore it every once in a while.", + "For how long are we going to store the charge?", + "What's the response of vc for t greater than zero after the", + "switch flicked? It's very simple.", + "It's vc is equal to 5 volts e to the minus t over RC,", + "right? That's the response.", + "We have a decay. And applying to the things we", + "know. We start from 5 volts,", + "let's say here, I have a decay going down", + "towards zero, at some point we are going to", + "cross the threshold for high. The only period in which I have", + "a valid output, if the capacitor was storing a", + "one, is this period here. This is the only period in", + "which I have valid stored one because, once I go beyond", + "capital T here, I have crossed the legal limit,", + "threshold for discriminating a high output.", + "And from then on the output is no longer valid.", + "So, this memory is good provided time is less than", + "capital T. It's not a case in which the", + "capacitor can hold charge forever.", + "In fact, we can calculate, that is we can solve for T in", + "this particular case. It's in your notes.", + "Nothing really profound. T is equal to minus RC log VOH", + "over 5 volts. So, this is basically what the", + "response is going to be. Now, there is an implicit", + "assumption here, which is that the store pulse", + "width is much, much larger than RON C.", + "In other words, when we want to store a one", + "here starting from zero, we better charge it all the way", + "up to 5 volts in the time that our switch is connected here.", + "And what is the relevant time constant?", + "It's going to be the RON C. In fact, it's actually the RON", + "parallel RL with C. But typically RON is much,", + "much less than RL so we don't have to worry about that.", + "Dominant time constant is RON C.", + "So, provided these things are happening, we have a memory.", + "Now, we can try to improve things a little bit.", + "We see here that we will have a decay to an invalid state in", + "time T. How can we improve things?", + "One way to improve things are the buffer.", + "Here is our memory element again.", + "Here is the capacitor. This is the storing node.", + "Now I am going to put the buffering effect.", + "I am going to put two buffers here.", + "Two invertors, I should say,", + "because if I am storing a one here I want to be able to see a", + "one here as well. And, in this case,", + "what I am looking at is the RIN of the buffer.", + "And, in principle, I have out here the RL.", + "Now, this is better because if RIN is much larger than RL then", + "the time T, in this case, is much larger than the case", + "without buffer. So, we buffer the effect of VL.", + "This could be one of these neat circuits we saw in recitation", + "like a source faller, for example,", + "or it can be just an inverter in which case you just see the", + "input of a transistor. So, now this condition can be", + "satisfied. Let me give you some cases", + "which are some numbers that are typical for a dynamic RAM.", + "Typical times we're talking about is RIN on order of 1", + "gigaohm and storage node capacitor on order of 1", + "femtofarad to one picofarad. Now, if you can do the math in", + "your head, which is just multiplication,", + "you will see that the time constant, the RC is between 1", + "millisecond to 1 microsecond. And for DRAMs,", + "actually, we try to be in the order of milliseconds.", + "These are the times we're talking about.", + "If I have this kind of circuit, somehow there has got to be", + "additional circuitry that comes back, samples the voltage here", + "and restores it. And that is actually what is", + "happening in a DRAM. And my laptop is working there", + "and its DRAM keeps getting refreshed every,", + "say, millisecond or whatever the condition is.", + "But, in our case, we are going to do a slightly", + "different case in which we will create a static memory.", + "Let's actually look at, first of all,", + "the case of the discharge. Pay attention to,", + "let me actually break the loop here.", + "This is my capacitor. This is a resistor that is in", + "series with a capacitor like you see here.", + "Actually, I am going to keep that resistor in series with the", + "capacitor, even in this case, because I have it for my second", + "part of my example. I charge the capacitor to 5", + "volts. And you can see here this", + "lights up, I hope everybody can see it, proportional to the", + "voltage that I have here. From here on it's all logic", + "levels. So, the intensity of light here", + "will always be the same. It's either lit or it's not", + "lit. Right now I am charging the", + "capacitor. In fact, let's see.", + "Maybe I can discharge the capacitor first.", + "", + "Here the capacitor is discharged.", + "As you can see, the input is zero,", + "the output is a one, and then the output of this", + "inverter here is a one. I have two inverters in series.", + "And I am going to charge the capacitor.", + "I charged it to 5 volts and this lit up, this is off of", + "course, that's an inverter, this is a valid zero,", + "produce a valid one. And now I am going to take the", + "input out. As you can see it's stored.", + "In fact, we have to wait for a very long time.", + "We don't have enough time to wait for this to discharge,", + "so instead what I am going to do now is I am going to add also", + "the resistor. Now I am going to flip the", + "resistor in parallel with the capacitor to imitate what", + "happens when we have an input resistance.", + "You saw that there was a discharge of the capacitor.", + "This input level went down. Voltage here flipped over to a", + "one. Let me do it again now with a", + "resistor in place. Storing charge on the", + "capacitor. That's the store command.", + "Now, don't store. I have less,", + "about a second. The element here is 20,000", + "microfarads and 100 ohms which gives me a time constant of two", + "seconds. Assuming a VOC of the order of,", + "let's say, I don't know what it is for this case,", + "2.5, the log would be about 0.5, so it cuts basically the", + "time to about one. So, it lasts about one second,", + "if my math is all correct. It's actually a little longer", + "than a second, excuse me, but the point is", + "that the charge is gone. Now, notice,", + "however, that there is something I can do here,", + "which is that suppose I take the switch or a switch and bring", + "it back and provide a path from the output to the input here.", + "And this switch is open when this is closed and closed when", + "this is open. So, this basically is the", + "compliment of store. What I am doing now is I put a", + "charge here, it produces a valid one at this point,", + "and then I am feeding this valid one back to the input.", + "As you can see, this will now allow me,", + "even though I have a high resistance, to store the value", + "for a long time. In this case,", + "what I am going to do is I am going to connect the output,", + "as you can see here. And I have my resistor in.", + "And I am storing zero here, storing 5 volts.", + "Now I am going to flip the switch.", + "Basically, I mean the don't store, don't look case.", + "You notice this dims a little bit.", + "Sorry. No, I want the resistor in.", + "", + "There. Yes.", + "OK, so the output remain value. This dimmed a little bit but", + "the output has remained OK. All right.", + "So, we've provided a feedback. Now we've created a static", + "memory. This will hold charge for as", + "long as the circuit is powered up.", + "Now, there is still one little problem that I have with this", + "kind of configuration. And that is if I disturb this", + "output the charge may, the state may change.", + "So, for example, let's say that I have --", + "I disturbed it by coming close to it, so let's charge it again.", + "", + "OK. I flipped the switch.", + "I flipped the state from the output.", + "That is an invalid condition. I shouldn't be able to do that.", + "How do I avoid that? How can I avoid this problem", + "that you just saw?", + "", + "Well, I need yet another buffer.", + "The answer is in your notes. If I don't take the output here", + "but rather take the output here, or if I don't want an inverted", + "output, if I don't want an inverted output,", + "I could put yet another element there.", + "Then the situation would be fine.", + "In this case, let me do it again.", + "Charge.", + "", + "Why isn't this lit?", + "", + "A bad one?", + "", + "Now, of course we disturbed the input.", + "Now, of course I can do anything I want here.", + "Nothing happens, but you may say this is a", + "trivial case because this is already zero.", + "So, I am going to change the state.", + "Here's is the changed state. See.", + "I can show this. Nothing happens up there.", + "So, this is an interesting situation in which I am", + "buffering the output so that the output does not feed back to the", + "input. And, by and large,", + "in designing circuits this is something that we do.", + "Now, in the remaining three minutes there is an example that", + "we have. Can we put the laptop here?", + "", + "OK, so here is an example of how memory can be put together", + "now to create something a little bit more complicated.", + "And you can see the memory cells that we were discussing", + "here. There's four of them,", + "so this is a four bit memory. There is a decoder at the", + "beginning here which decodes the address of each cell,", + "so the input here will tell me which cell I need to address.", + "Let's look at the truth table. This is the truth table for the", + "decoder. As you can see,", + "depending on the address that I have here, this is zero,", + "one, two and three in a binary system, only A,", + "B, C or D is up, is high.", + "Which means that this end operation here only allows the", + "input that is presented to all of the cells,", + "what is going through the AND gate here to appear at the", + "output. If, for example,", + "we have a one, zero, the only end input that", + "is going to be high is going to be this one.", + "And that means the only cell that will look at the input when", + "the store comes up is going to be this one here.", + "At that point it will store whatever is on the input cell", + "because that's an AND operation. That is a simple example of a", + "memory. And following that simple", + "arrangement you can build incredibly large memory systems.", + "So, that's all I had for today. And I will see you on Tuesday." + ] +} \ No newline at end of file diff --git a/subs/dyxcCoUgETU.srt.sjson b/subs/dyxcCoUgETU.srt.sjson new file mode 100644 index 0000000000..5be1bda72f --- /dev/null +++ b/subs/dyxcCoUgETU.srt.sjson @@ -0,0 +1,2063 @@ +{ + "start": [ + 0, + 2925, + 8395, + 13738, + 19844, + 23915, + 31420, + 38435, + 42250, + 46228, + 49476, + 54022, + 58000, + 63000, + 77000, + 79282, + 82602, + 86682, + 89933, + 93599, + 96919, + 101000, + 104492, + 107920, + 109634, + 113317, + 116174, + 119539, + 123021, + 126995, + 130229, + 133328, + 137101, + 140536, + 143433, + 147273, + 151450, + 156355, + 165165, + 170898, + 178728, + 186000, + 191400, + 197942, + 202823, + 208119, + 213000, + 219875, + 225125, + 229375, + 236500, + 242000, + 245788, + 251173, + 255560, + 260146, + 264035, + 269419, + 276000, + 284494, + 289500, + 297387, + 303000, + 307273, + 309631, + 312873, + 316042, + 319873, + 323042, + 326873, + 330115, + 332989, + 335863, + 341267, + 344409, + 349060, 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Good morning.", + "Good morning. So, we have some fun stuff for", + "today's lecture, and as far as the final is", + "concerned and so on, I'd like you to forget about", + "anything we do today, absolutely.", + "So, get your mind to become a blank, and forget anything you", + "hear in today's lecture. So, what I'm going to show you", + "today will hopefully completely blow your minds.", + "And I'm not talking about controlled substances or", + "anything. So what I'm going to do is show", + "you a few things that behave completely and spectacularly", + "differently than how you expect them to.", + "And, today's lecture is appropriately called --", + "", + "OK. So, we're going to violate the", + "abstraction barrier here, and do some fun things.", + "And, the important thing to realize is that in all of 6.002,", + "we have, after all, based on some assumptions we", + "made at the beginning of the course like lumped matter", + "discipline and so on, we have landed ourselves in", + "this playground called the playground of 6.002.", + "And, within that playground, certain ground rules apply.", + "OK, and our entire course depended on those assumptions", + "being true. So, for example,", + "the first assumption we made that brought us from Maxwell's", + "equations to the lumped matter discipline was,", + "or rather the circuit abstraction, was a lumped matter", + "discipline. And there were three tenets of", + "the lumped matter discipline. One is that the rate of change", + "of flux was going to be zero within our circuits,", + "not inside elements, but in the circuit itself,", + "and second, the dq by dt was going to be zero outside the", + "elements, and third, something we did not dwell upon", + "in the course, but it's certainly present in", + "the course notes is that the speeds of signals that we are", + "going to consider are going to be much slower than the speed of", + "light. OK, so we're going to be", + "working in a realm where we are going to be well slower than the", + "speed of light. OK, so starting with that,", + "let me walk you through some examples and some fun stuff.", + "So, the first case is called the Double Take.", + "So, let me sketch out a small little circuit for you,", + "and take a look at the expected behavior, and then show you what", + "really happens in real life. So, the first case,", + "I have a voltage source, and what I'm going to do is", + "make a transition from a zero to a one.", + "Think of it as a step input, and through a Thevenin like", + "resistance, I want to feed it to a circuit.", + "The circuit will go to an inverter.", + "This node goes to an inverter, and goes through some other", + "circuits within our own design here.", + "So, again, remember, a step input here,", + "and this input goes through a Thevenin like resistance,", + "or is applied to some other circuit elements.", + "So, if I apply a step here, what do you expect?", + "You expect that, so let me call that VI,", + "and let me call that Vo. So, if I plot VI as a function", + "of time, and let's say this step input happens at t=0.", + "So let's say this is t=0 here, and let's say this is a 5V", + "step. So, I expect that this input", + "here is going to go to, VI here, is going to go to 5V", + "at t=0. What do I expect at Vo?", + "At Vo, based on our circuit abstraction, I get a step input", + "here. I should get a step of some", + "magnitude here, depending on what's connected", + "in this direction. And let's simply say that", + "what's connected here is an inverter, and maybe other", + "inverters at the other side. So essentially,", + "as far as this node is concerned, it's got some wires", + "connected to it. And at the end of the wires,", + "it has an open circuit, an open circuit,", + "for example, like the gate input of this", + "inverter. So what do you expect at V", + "nought? A step input here,", + "and at V nought I see an open circuit.", + "OK, so I expect the same step at V nought: 5V.", + "So, that's what we've prepared you for, OK?", + "But, the fun thing that we're going to see,", + "so this is what you expect, and I'll show you a little demo", + "that is going to show you something very different.", + "What you're going to see is not this.", + "OK, you're not going to be seeing that.", + "Rather, I'm going to show you something that looks like this.", + "", + "So, at t=0, I do see Vo looking like a step, and approximately", + "halfway through, decides, ah,", + "well never mind, and flattens out,", + "OK, then says, oh, OK, and zoom,", + "it goes back up to 5V. So, it sort of does a bit of a", + "double take up there saying, hey, what's going on here?", + "And zoom, jumps up to 5V, and then it's five as you", + "expect. OK, so this is some finite", + "amount of time that looks like that.", + "OK, so try to understand what's going on.", + "So let me show you a quick little demo.", + "So that's the input VI. OK, so that's the input VI that", + "you expect, and I won't do anything to my circuit at this", + "point. And, go ahead.", + "So, let's see what happens now. There you go.", + "So now, I'm showing you the output here at Vo.", + "So at VI, there's a nice little step, and at Vo,", + "notice that I get something that behaves like this.", + "OK, and I promise you, nothing we've taught you in", + "6.002 prepares you for this. OK, and as I mentioned at the", + "beginning of this lecture, it would behoove you to forget", + "about everything you learn in today's lecture for the next two", + "weeks at least. So what's going on here?", + "Any ideas? Anybody?", + "Any thoughts? So what's up with my circuit", + "here? It says, oh,", + "OK, a step. It starts off and says,", + "oh, never mind, and then meanders along at 2.5V", + "and then says oh, step, yes, I remember,", + "and then boom, it jumps up to 5V.", + "So, any theories? Any guesses?", + "Any wild guesses? OK, so let me draw you a little", + "bit more of a detailed circuit, and see if you can explain", + "what's going on here. So, the circuit that I've drawn", + "there is not quite the circuit I have at least in terms of my", + "wires. So, what I have is something", + "that looks like this, VI, and this is going to step", + "to 5V. I do have a resistance,", + "R. This is Vo, this does go to an", + "inverter. But what is also happening is", + "that I have a long wire. OK, you see this guy here?", + "We had one of our union folks stretch out along the floor", + "here. We have a really long wire that", + "connects to the Vo node, and there's also a long", + "corresponding ground. So, this wire is a coaxial", + "cable that is used for Ethernet and such like.", + "It's got a core that carries a signal, and around the core is", + "shielding that is the ground. OK, so that goes a long way,", + "and at the end, it is open.", + "OK, it's an open circuit at the end.", + "I haven't connected anything out there: open circuit.", + "So, you know, something's happening here", + "that's making the circuit behave like this.", + "So, this is VI. At Vo --", + "", + "So at Vo I'm getting this funny behavior.", + "OK, so does anybody want to take the next piece of clues", + "here, does anybody want to take a stab at guessing what might be", + "going on here? Yes?", + "Ah, we have a shill in the audience here.", + "So, the theory is that the step here, think of it as an", + "electromagnetic pulse that goes from zero to five,", + "and things in real life don't travel instantaneously.", + "So, there's something with a wave that flies down,", + "and the wave goes to the end, flips, and then comes back,", + "and then establishes the full voltage here.", + "So that is indeed at the root of what's going on.", + "And let me put it in layman's terms and then describe the", + "details of what's going on here. OK, so the way to view what's", + "going on is that I have this long wire.", + "OK, in the very first lecture, I started off by saying wires", + "are ideal. OK, ideal wires are such that I", + "can transmit signals on them. Wires are small so that the", + "propagation time of signals is inconsequential compared to the", + "rise times and fall times of the signals of interest.", + "By having this really long cable here, I have clearly", + "violated that assumption, which is the wires are really,", + "really long here. OK, and so I somehow need to", + "model what the wire is doing to my circuit when I don't have a", + "small wire. So what actually happens,", + "the way to view it is the following.", + "So, although this is a wire, to understand the mechanics of", + "what's going on, I really have to model it much", + "more accurately, OK?", + "And, the way to model a wire like this is that notice that", + "every small element of a wire has associated with it some", + "inductance. OK, so let's take a small", + "segment of the coax cable here. The coax cable is a small core", + "surrounded by a metallic shield. OK, that's a ground.", + "And so, when I have a wire surrounded by a metallic shield,", + "that also has the capacitance, OK, inductance and capacitance.", + "So this small segment can be modeled as a really small", + "inductance, and a really tiny capacitance.", + "Similarly, the next segment can be modeled as a tiny inductance", + "and a capacitance. There is also a resistance", + "here, but let's assume that the resistance is zero for our", + "model, and also the parallel resistance is also infinity.", + "OK, so it's an inductor, capacitor, and really the", + "situation that I have is not a pair of ideal wires,", + "but really a really, really small inductance,", + "and a small capacitance in parallel.", + "So, it's more of a set of distributed elements that I have", + "here. Notice that in my lump circuit", + "abstraction, when we talked about the RLC model for the wire", + "between two inverters, we lumped it.", + "We lumped this thing into a model that looked like this.", + "OK, we lumped the resistance into a source resistance.", + "We lumped all the inductors into a lumped inductor.", + "We lumped all the capacitances into a lumped capacitance.", + "OK, but in this situation, I can do this when the signal", + "speeds of interest are much, much, much slower than the", + "speed of light than the propagation speeds of", + "electromagnetic signals. In this case,", + "that is not quite true. And so, therefore,", + "we have to model it much more exactly.", + "We need to see what's going on. So, what's happening here is", + "that at t=0, I get this step. So, think of that as a pulse of", + "energy, and the instant it comes here, and instantaneously this", + "guy looks like a voltage divider, OK?", + "I've chosen my resistance, R, here to match the", + "instantaneous impedance looking in, which is also R.", + "I've arranged it to be that way.", + "So, instantaneously, the point at which the pulse", + "appears at this point, looking down here looks like", + "another resistor to this pulse. OK, therefore,", + "when I start out, I start out going up and", + "pausing at 2.5 because instantaneously,", + "this looks like a resistance, R.", + "So instantaneously, it's a voltage divider,", + "R, and so it's 2.5 here, instantaneously.", + "OK, then what happens? Then those little pulse", + "propagates down. What does it mean for a pulse", + "of energy to propagate down? Well, it begins sending a", + "current through the inductor, begins charging up the", + "capacitor, current here, so that's what I mean by saying", + "that the pulse of energy goes down.", + "OK, it's a step that sends current to the inductor and", + "charges of the capacitors, and that wave front moves out", + "here and comes all the way here. What happens there?", + "Well, think about it. Supposing you stand here,", + "and you hold a long string in your hand somehow,", + "and just do this Gedanken experiment.", + "It's not easy to do. And so, let's say you somehow", + "have the long string that you're holding onto,", + "and the string on the other side is not connected to", + "anything. OK, just imagine this", + "experiment. OK, and what you do is you", + "suddenly raise the string up at your end by about a foot.", + "What are you going to see happen?", + "So instantaneously, the string is up here,", + "but the rest of the string is down a foot below.", + "And then you see this wave propagate down the string,", + "right? So here's a string.", + "I lift this thing, and you see this wave propagate", + "all the way down, the one foot wave propagate all", + "the way down until you come here.", + "What happens here? So, out here,", + "the string is down here, the wave propagates out here", + "and pulls it up to one. And then what?", + "There's nothing connected there, so the string is zipped", + "up, but it's got the energy. OK, where does energy go?", + "Well, it continues going up, and sends a wave back.", + "OK, so just think of a string that you pull up like this and", + "propagates down, boom, hits the other end,", + "reverses, and comes back at me. OK, you can look at a", + "complementary situation, not the same as this,", + "but complementary by taking a string, tying it to a door,", + "and lifting it up. It's not the same situation.", + "It's a complementary situation where it's tied down.", + "Tying down a string is tantamount to shorting the ends", + "here. OK, in that case what you'll", + "see happen: as the wave goes down, at the end the string", + "can't move, so the wave goes and flips around and comes back.", + "Try it out at home. Take a long piece of string,", + "tie it up there, do this, OK?", + "And you'll see the wave go out, flip, and then come back at", + "you. So, if your friends see you", + "tying a long piece of string doing this, hopefully they won't", + "think you're nuts or something. OK, so the same way here:", + "this thing flies down, OK, there's no way to dissipate", + "the energy here, so this thing continues up.", + "And then, what I'm going to see happen is the wave move back.", + "OK, the wave begins to move back, and that's another 2.5V,", + "resulting in a net 5V at this terminal.", + "That wave begins to blast back, OK, and then when it comes back", + "here, after some amount of time, it raises this to 5V,", + "and that's what you see happen here.", + "So, this is a wave going down, and then after a time,", + "2t, it goes back up to 5V. That's a return wave.", + "It's 2t because to get down here is t seconds,", + "and then t seconds to come back, which is why we have 2t.", + "OK, that is why you see that pulse at 2.5.", + "OK, so I'd like to show you a few more things here.", + "Clearly we don't want that in our circuits.", + "Could someone tell me what problem would happen if my", + "signals looked like this in my digital circuits?", + "Instead of being nice little steps, if there was a little", + "thing in the middle and then a step, what's the problem with", + "signals like this? In digital circuits,", + "what did it violate? Yeah?", + "Exactly. This little sucker here is", + "meandering out in the forbidden region for all of 2T.", + "Can't do that. OK, can't have that.", + "Well, so we need to fix the problem because this is real", + "life. OK, but what if you and your", + "buddy were signaling each other but using digital signals from", + "one dorm room to another maybe a few hundred feet down?", + "Your circuit isn't going to work because the signal's going", + "to meander around in the forbidden region for some time.", + "So, any ideas what might you do?", + "Yeah? Put a resistor on the end.", + "OK, trick the circuit. So, what you can do,", + "and I'm going to show you a little demo here,", + "what you can do is the reason I got this wave propagating back,", + "was that there was nothing to absorb the energy.", + "So instead, what if I put another resistor here,", + "R? So, as far as a burst of energy", + "is concerned, it says, oh,", + "yeah, it just looks the same. It's R, and goes and dissipates", + "in this resistor, R, and guess what?", + "I don't have any wave going back, and I'm done.", + "So, what I'm going to find, then, is that out here,", + "this goes up to 5V, but out here,", + "I will have a signal that starts out and goes up to 2.5,", + "and that's it. OK, I lift it up,", + "it goes down, it goes to 2.5 because in the", + "lumped model that you've been dealing with,", + "it's a resistor R, a resistor R to ground,", + "and you're taking the connection here or here.", + "So, it's your standard lumped model, your voltage resistive", + "divider, and it just simply works.", + "Yeah, that's it. So, this is the end of the", + "cable. OK, if somehow you could watch", + "this and that at the same time, so what I'm going to do,", + "and this is a resistor, R, I'm just going to plug it", + "in. OK, if the fates are smiling at", + "me, what should you see there? What should happen is that the", + "second jump from 2.5 to 5 should simply go away.", + "It should just go to 2.5. Let's try that.", + "There you go. I take it out,", + "it jumps back up. OK, so all I've done here is", + "put in a resistor at the end, and I'm still measuring the", + "voltage here. So, that's one solution.", + "One solution is to put a resistor here.", + "So, I absorb the energy, and the resistance has to be", + "equal to the instantaneous impedance looking in.", + "And the instantaneous impedance, for many of these", + "cables is 50 ohms. It's called a characteristic", + "impedance. OK, you'll learn a lot more", + "about it if you take 6.014. That course starts out with", + "assuming that things are distributed in that matter.", + "OK, so if you want to design multi-gigahertz chips,", + "it turns out that if you have signals that are traveling", + "around at edge speeds in the 0.1-1 nanosecond range,", + "remember, light travels roughly one nanosecond a foot.", + "And if the signals are roughly of interest are 0.1 nanoseconds,", + "then if the chips are one inch in size, right there,", + "the propagation speed of a signal across a chip is 0.1", + "nanoseconds. OK, so today,", + "we have to deal with these issues and try to figure out", + "what to do about them. OK, so that's one solution that", + "somebody pointed out. There is a second solution.", + "Anybody else have a second solution for me?", + "And then there's a third solution, too.", + "So it's OK. You can give me either the", + "second or the third solution. It doesn't matter.", + "Anybody? You have two to choose from,", + "come on. Yeah?", + "You can do that, yeah.", + "So we could define the problem away by saying this transition", + "is such that my high is below 2.5.", + "So, once it goes above 2.5, who cares what it does?", + "That's a good point. That's solution number four,", + "and that works. OK, so I still need two and", + "three. Put a diode in there?", + "Yeah, I guess you could. If the diode had the same kind", + "of impedance looking in, it kind of may work.", + "That's solution 4.2. I'm still waiting for solution", + "two and three. Pardon?", + "Cut off the cable? Exactly.", + "So, the solution says, work on a different problem.", + "And that is solution number two.", + "OK, so the idea is, the root of all evil,", + "this long wire, which is why I had this thing", + "here. So instead, if I had short", + "wires, then what will happen is if it's a very small wire,", + "it'll look like this. And the wire's small enough.", + "I will see an itty-bitty thingamajig out there,", + "but not a whole lot. By the way, the fun thing is", + "that you can actually calculate the speed of light,", + "the experiment I just showed you.", + "Can we put that up again? No, the big one.", + "So, in the experiment that I showed you, this distance was", + "about 500 nanoseconds, OK?", + "This distance was 500 nanoseconds this time interval.", + "The length of this cable is about 500 feet,", + "somewhere around 500 feet. So you can figure out the speed", + "of light. What's the speed of light?", + "So, this is about 500 nanoseconds, and this cable is", + "roughly 500 feet. What's the speed of light?", + "Roughly a foot per nanosecond. So, would you believe that in", + "6.002 we've figured out the speed of light from a simple", + "experiment? All right, so let's do the next", + "experiment now. Let's take out the long cable,", + "and connect a short cable instead.", + "So, what I'm going to do is disconnect the long cable,", + "and instead, connect a small cable.", + "It's still relatively long, but much shorter than the 500", + "foot cable. So what you should see happen", + "now is that the little step should not be this big,", + "but much, much smaller. So, take a look up there.", + "There you go. OK, so with this thingamajig,", + "the little blip there is very small.", + "And of course, if I make it even smaller,", + "then that can virtually vanish. OK, so that is solution number", + "two. So, we've done one,", + "two, four, 4.2. So, what's solution number", + "three? One more solution.", + "Pardon? So, another solution we", + "mentioned is we change this resistance.", + "and that will work, if I make this very,", + "very low, then I'll get much closer to 5V here.", + "Yeah, that's a possibility. That's solution six I guess.", + "So what was solution number three?", + "And you all should be able to solve this.", + "You guys know the answer. OK, you folks should be able to", + "solve this. Yes?", + "Ah, clock. So, what I can do is just as", + "was pointed out, that I leveraged my abstraction", + "by changing my VOH and VIH thresholds.", + "So that'll work. The alternative thing is to use", + "a clock. A clock is a distinguished", + "signal that I send around in my digital circuit,", + "OK? So all I do is if I arrange it", + "such that my clock doesn't happen in this vicinity,", + "but rather, my clock happens late enough, then I'm going to", + "sample and look at my signals only on the rising and falling", + "edges of the clock, in which case I won't be", + "looking at the signal, but the signal is doing weird", + "things. OK, so a decent clock would", + "also solve the problem. OK, any last minute questions", + "before we go onto the next one? OK, the next problem that we're", + "going to look at is titled the Double Dip.", + "OK, so what I'm going to do here is our Vs power supply,", + "and what I'm going to do is feed the power supply to an", + "inverter. OK, so we've been doing this", + "all along; Vs, I feed the supply to an", + "inverter. And what I'm also going to do", + "is, so this is ground, and I'm going to feed it to,", + "so feed the power supply connection to a couple of", + "inverters. OK, and what I'm going to do is", + "apply some sort of a signal to this inverter,", + "and I'm going to observe, and I'm going to look at this", + "signal here. So, the abstraction should tell", + "you that here's a power supply. This is 5V, or whatever the", + "supply voltage is to these two inverters.", + "That should be fine, and feed some sort of input to", + "this inverter, OK, and the output here should", + "be simply determined by this input.", + "This signal can have absolutely no bearing on this output.", + "OK, and let's look at that and actually confirm it.", + "So, I build a circuit like this, and we look at this", + "output, and initially there should not be any,", + "it should simply work fine. OK, so it should work now,", + "right? OK.", + "So what you have here, this input here is the input", + "that I'm feeding to this inverter.", + "That is a straight line. Is that the power supply?", + "It doesn't matter? OK, so I believe this is the,", + "we'll check in a few minutes, but I suspect this is the power", + "supply, and this guy here is the output looking here.", + "So, the green one is the look here part.", + "So, there must have been a one-to-zero transition here,", + "and that's all fine. So, so far, so good.", + "OK, no problem so far. Now what I'm going to do is I'm", + "going to do something to the circuit that as far as", + "abstraction is concerned, it doesn't show up on the", + "circuit. OK, it's below the abstraction", + "layer. OK, I'm going to do something,", + "and suddenly, some things are going to", + "happen. Look up there.", + "The circuit hasn't changed. It's the same circuit.", + "I've done nothing to the circuit.", + "OK, look at the green output. I've done nothing to the", + "circuit that is visible here. OK, it's below the radar screen", + "here. It's below the abstraction", + "barrier. But, look at the disaster here.", + "OK, in particular, the spikes going up are not so", + "much of a problem. Because of the static", + "discipline, if I am at five or six or seven,", + "it doesn't matter as long as I am higher than VOH.", + "So as long as I'm higher than VOH I don't have a problem.", + "But the problems are these repeated dips.", + "OK, the dips are a problem here, which is why I labeled", + "this experiment the Double Dip. OK, the dips are bad because if", + "they are large enough, they can then group the output", + "down into the forbidden region, or worse yet,", + "make it look like a zero. OK, so you're not prepared for", + "this. So what I'm going to do is tell", + "you what I did to the circuit, and then ask you to help me", + "figure it out. So all I did was applied a load", + "resistance to this, I think of 50 ohms or some RL.", + "I just applied a load resistor. And this inverter here,", + "I believe, is a CMOS inverter that looks, OK?", + "So I have this input applied to this inverter,", + "and all I did is I applied an RL load here.", + "And notice that the load here should not really change what's", + "happening if this is an ideal inverter, OK,", + "the load here should simply draw some current but really", + "should not change any other property.", + "OK, so just remember, what's the signal doing?", + "The signal is high. This guy turns on,", + "and current flows like this. So, let's say I had some sort", + "of a capacitor here. This charges like this,", + "and when it's slow, the PFET is on,", + "and current flows through here down here.", + "And then when this goes high, this guy goes off,", + "and this guy turns on. OK, so the current flows out", + "this way and this charges through this guy.", + "When I turn it off, the P fret turns on and draws", + "current from the top. OK, so do we have any theories", + "as to why I'm getting that messy stuff, the dips and the spikes,", + "on the output of this inverter? So why does this inverter care", + "what the load of this inverter is?", + "I mean, who cares? So, put your thinking caps on.", + "Any theories? You guys did pretty well with", + "the previous one. And this is much easier,", + "actually. Need a better power supply;", + "OK, so what I'm going to do is I'm going to replace the power", + "supply, and instead, use a much bigger power supply", + "at 5V. A big, mongo power supply that", + "can supply 100 amps, and guess what,", + "I've made the changes, but guess what,", + "I still see the spikes. Good try, but it didn't work", + "out. Good try, good try.", + "What next? Any other solutions?", + "Yes? So dips are because of the", + "resistance, and the spikes are because of the inductance?", + "You're half correct. So, which one is it?", + "So, dips are because of resistances, and spikes are", + "because of inductances. You're half correct.", + "It turns out that both the dips and the spikes are because of", + "inductances. OK, but be that as it may,", + "let me give you the next clue here, and then see if you can", + "come closer to the answer. So, what I've done here is I've", + "made this wire really, really long.", + "OK, it's a really long wire, OK, but it's a thick wire,", + "so it's a long, long, thick wire.", + "So it's not the resistance. It's really,", + "really thick and mongo, and it's a long wire,", + "so a signal wire above a ground plane behaves like an inductor.", + "And so here, it has the capacitance to,", + "but in this case it's inductance.", + "It's inductance here. So, I'll give you another ten", + "seconds to think about it and then tell you the answer.", + "But despite the inductance here, it turns out if I take out", + "this resistor, the problem goes away.", + "Look, I take out the resistor, the problem goes away.", + "Yes, there is an inductor here. OK, I take out this resistor,", + "problem goes away. I put the resistor back in,", + "boom. Yes?", + "OK, pretty good. That's 86 points.", + "So here's what's going on. There's an inductor here,", + "and when I put a 50 ohm resistor here,", + "I put this resistor. When the PFET turns on,", + "it draws a current. OK, it's going to draw a", + "current. It draws a current;", + "remember that across an inductor, I have a drop.", + "And the drop relates to the di/dt.", + "Remember, for a capacitor, the current is Cdv/dt.", + "For the inductor, the voltage across the inductor", + "is Ldi/dt. So, if di/dt,", + "from switching a large current through the inductor every", + "cycle, OK, big di/dt, di/dt is large.", + "I've made it large by having a very small RL,", + "so, you know, pulling a big current through", + "every few, whatever, every cycle,", + "and then stopping it. And so therefore,", + "I'm getting these big drops across this inductor that relate", + "to Ldi/dt. In other words,", + "the power supply here is fine. While you guys were watching,", + "I switched to the huge, mongo power supply,", + "and so this voltage is fine. But then this voltage after the", + "wire is the problem. So, this voltage here doesn't", + "look like this anymore. Rather, it has spikes that go", + "down, for example, and when I switch the other", + "way, they go up. OK, so therefore,", + "what I end up having here is big spikes on this power supply.", + "And when this guy's power supply goes wacko,", + "then I see the spikes on its output as well.", + "OK, so what are the solutions for that?", + "Any solutions here? What can I do to fix the", + "problem? Pardon?", + "Stop using the, exactly.", + "When in doubt, do something else.", + "Build a different design. So what I could do is this is", + "pretty dumb, using a long wire. And so, no, but trust me,", + "oftentimes you go to the store room and they give you a big", + "roll of wire, and you're too lazy to cut a", + "piece out. Use the whole roll,", + "and use the two ends, and connect it in,", + "OK? So, if I had a much shorter", + "piece of wire, then that can solve my problem.", + "But again, remember, what's small to you may not be", + "small to the circuit. OK, so let's say,", + "for example, I'm Intel, and I'm building a", + "10 GHz Pentium 6 processor. OK, it's 0.1 nanosecond is my", + "cycle time. There, even a small,", + "itty bitty wire can be a real problem.", + "OK, and so therefore, distributing power throughout a", + "one inch chip that's clocking at 10 GHz is a really,", + "really hard problem. And our own David Perreault,", + "who is doing one of our sections, is one of the world's", + "experts in this field. Distributing power,", + "something as simple as, how do I get 1V in a stable", + "manner to every single device on my chip?", + "It's a hard problem. OK, so now, you have to begin", + "feeding your power supply connections much like RC", + "circuits, OK, and you have to solve some hard", + "problems to be able to simply distribute power decently", + "throughout your circuit. So, what else can I do?", + "Yeah? Say it again?", + "Ah, I can do that. I could use different wires to", + "connect each of the inverters. That's a good point.", + "So here, the coupling happens because I connect the two", + "inverters way out here. So instead, I use a different", + "cable. I hadn't thought of that.", + "That's a creative solution. OK, so in fact,", + "if you build a chip, so we built this chip called", + "RAW in our group, and it has on the order of 10", + "million gates. And this chip we built with", + "IBM's technology, and it turns out that you don't", + "send power supply in through a pin and then connect that 1.5V", + "supply to all your gates. What you do is from that pin,", + "you then build special power supply buffering trees.", + "And each tree, each leaf of the tree drives a", + "subcircuit. In other words,", + "if this is a chip, you have lots and lots of gates", + "throughout your chip. What you do not do is bring in", + "a power supply like this, and then connect.", + "You don't do that. That's the worst possible thing", + "you can do. It's an absolute disaster for", + "the reason just brought up. OK, so instead what you do is", + "divide up the chip into, say, four quadrants.", + "OK, in our case, we have 16 quadrants.", + "And then what you do is from this point, you take one wire", + "that goes to this quadrant, one wire that comes here,", + "one here, and one here, so that you're getting the", + "power supply very close to the source, and you have different", + "connections going to each quadrant so that switching in", + "this quadrant will not affect this guy because of the", + "inductance of this lead here. OK, and if you hadn't taken", + "6.002, you'd have been arguing with IBM, I don't want 16 wires.", + "I want just one wire. OK, so there are other", + "solutions, of course. There's a couple more solution.", + "One is that what you can do is part of the problem here is that", + "all my transitions are really, really sharp.", + "OK, so di/dt is very, very large.", + "So, there's a whole new technique in design of digital", + "and analog circuits, which talks about,", + "maybe I should call it waveform engineering, OK,", + "or edge engineering. OK, it's also called edge", + "smoothing. The idea is that rather than", + "have very sharp edges in your circuit, you try to have", + "smoother edges. And when you have smoother", + "edges, OK, then your di/dt is now going to be less.", + "It's not going to be very, very high.", + "Rather, your delta I is spread out over a longer period of", + "time. Of course, that means the", + "circuits may have to run a little slower,", + "but that can also solve the problem.", + "And in fact, that same smoothing of the", + "waveforms was also the solution you saw in the capacitive", + "coupling we saw a month and a half ago.", + "And let me show you the demo, and then close up.", + "Not working? OK, that's OK.", + "It doesn't matter. So if you remember the demo", + "from the lecture about a month and a half ago in capacitors,", + "I talked about a chip with two pins, and there was this", + "capacitive coupling between the pins.", + "And because of this, if this waveform is switching,", + "then because of this coupling, you will end up getting,", + "if this is the signal here, you will end up getting spikes", + "on this pin because of the signaling of the other pin.", + "And that's good old capacitive coupling.", + "OK, and to eliminate this, what you can do is much like", + "with the inductance system, if you, rather than having", + "sharp transitions on this pin, if you have smooth transitions", + "that look like this, then what you can do is you'll", + "now spread delta V from here to here over a longer delta T.", + "OK, delta T has become longer, and because of that,", + "you end up getting much better behavior, and you don't end up", + "getting these spikes. So therefore,", + "if you want to build really, really fast circuits,", + "you have to be really careful. You can build fast circuits,", + "but watch out for them fast edges.", + "OK, fast edges are nasty. They kill you.", + "That's something to remember as you build the next generation of", + "circuits. Well, thank you all.", + "I had a blast, and I hope you guys had fun", + "too. Thank you." + ] +} \ No newline at end of file diff --git a/subs/fDtrSZ69xII.srt.sjson b/subs/fDtrSZ69xII.srt.sjson new file mode 100644 index 0000000000..54fa279eb1 --- /dev/null +++ b/subs/fDtrSZ69xII.srt.sjson @@ -0,0 +1,533 @@ +{ + "start": [ + 0, + 760, + 5550, + 7730, + 15040, + 22780, + 26410, + 31490, + 35630, + 37015, + 39890, + 47340, + 49390, + 53330, + 56170, + 59924, + 61630, + 65640, + 72340, + 75120, + 80660, + 83550, + 88110, + 92500, + 96538, + 99170, + 103910, + 113990, + 118090, + 121540, + 125630, + 128211, + 131510, + 135680, + 139700, + 142190, + 145380, + 149430, + 151048, + 154330, + 159440, + 164260, + 166990, + 170886, + 174520, + 176040, + 179664, + 190760, + 196600, + 201060, + 203958, + 207444, + 211940, + 217210, + 220450, + 223430, + 224640, + 226290, + 228162, + 233830, + 238620, + 239640, + 242880, + 246200, + 249490, + 257300, + 260600, + 262290, + 265810, + 269740, + 275500, + 278270, + 281910, + 284090, + 286260, + 289730, + 293659, + 296010, + 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7730, + 15040, + 22780, + 26410, + 31490, + 35630, + 37015, + 39890, + 47340, + 49390, + 53330, + 56170, + 59924, + 61630, + 65640, + 72340, + 75120, + 80660, + 83550, + 88110, + 92500, + 96538, + 99170, + 103910, + 113990, + 118090, + 121540, + 125630, + 128211, + 131510, + 135680, + 139700, + 142190, + 145380, + 149430, + 151048, + 154330, + 159440, + 164260, + 166990, + 170886, + 174520, + 176040, + 179664, + 190760, + 196600, + 201060, + 203958, + 207444, + 211940, + 217210, + 220450, + 223430, + 224640, + 226290, + 228162, + 233830, + 238620, + 239640, + 242880, + 246200, + 249490, + 257300, + 260600, + 262290, + 265810, + 269740, + 275500, + 278270, + 281910, + 284090, + 286260, + 289730, + 293659, + 296010, + 298570, + 309794, + 315430, + 318030, + 322270, + 324090, + 326892, + 331472, + 340490, + 347140, + 349891, + 355780, + 363420, + 366200, + 367450, + 370890, + 372140, + 375180, + 378400, + 379900, + 383420, + 384510, + 390320, + 393190, + 397940, + 401926, + 407770, + 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a look at some of the", + "properties of linearity.", + "What does linearity bias?", + "", + "In particular, linearity will bias homogeneity and", + "superposition.", + "Let's take a look at each of these in turn, and see what", + "homogeneity and superposition really means.", + "", + "So let's start with homogeneity.", + "", + "So suppose I have a system in a little box, and let's assume", + "it's a linear system.", + "I apply a set of inputs, x1, x2, and so on, to", + "produce an output y.", + "So in my facetious example here, think of", + "the inputs as apples.", + "And the processing produces a ripe apple.", + "", + "Now, what homogeneity says is this.", + "In the same system, if I feed it with fraction of apples,", + "then I will get fraction of ripe apples at the output.", + "In other words, if each of my inputs is multiplied by some", + "constant, alpha, then my outputs will also be", + "multiplied by the same fraction, alpha.", + "", + "This simply says that if I reduce my inputs by some", + "proportion, then my outputs will also reduce by the same", + "proportion So that's homogeneity.", + "", + "Next, let us look at superposition.", + "This is some really fun stuff.", + "", + "So suppose I have a linear system as before.", + "And in this case, I give it one set of inputs,", + "x1a, x2a, and so on.", + "And for fun, let the a's stand for apples.", + "So if you have a set of apples, and let's say I get", + "applesauce at the output.", + "", + "Now, if I take the same circuit, and if I feed the", + "main same circuit a different set of inputs-- x1b, x2b, and", + "so on, in this case, b stands for blueberries--", + "let's say I get blueberry jam as an output.", + "", + "So what supervision says is this.", + "If I take the same circuit, the same system, and if at", + "each input where I had previously fed", + "an apple or a blueberry.", + "The [UNINTELLIGIBLE]", + "at each of the inputs, I want to feed", + "an apple and a blueberry.", + "That's x1a + x1b, x2a + x2b, and so on.", + "Then the outputs will simply be the sum of the", + "two original outputs.", + "So they could be ya plus yb.", + "You facetiously think of it as mixed fruit jam.", + "", + "So superposition simply says that if I find the output by", + "feeding the system apples, find the output by feeding the", + "system blueberries.", + "If I fed the same system the sum of apples and blueberries,", + "then I could compute the output simply by looking at", + "the outputs of the system in which I fed just apples or", + "blueberries.", + "Let's look at a specific superposition example and get", + "a sense of what this means.", + "So here's the same system.", + "And in this case, I want to feed two voltages, v1 and v2,", + "and I want to find out the output.", + "What superposition says is this.", + "", + "I can take the same system, and set one output to zero,", + "and feed it just v1, and measure an output.", + "Similarly, I take the same system.", + "And this time around, set v1 to v0, and feed it v2, and I", + "obtain a net output.", + "", + "By the method of superposition, what it says is", + "that for the same system, if I feed it the sum of the two", + "inputs, then I can get my output simply by summing up", + "the two individual outputs.", + "In other words, I can obtain the effect of v1 and v2 acting", + "as inputs simply by taking the same circuit and applying one", + "input, setting the others to zero.", + "Measuring the output.", + "", + "Setting the second input.", + "", + "And then having all the other inputs be zero.", + "And measure the output.", + "And then take the individual outputs, y1 and y2, and simply", + "add them up.", + "And that would be the result of v1 and v2 acting together.", + "So that is a specific superposition example.", + "What it does is it allows us to solve two", + "problems, p1 and p2.", + "But each of them is a simpler problem, and simply sum the", + "two outputs of the simple problem to obtain the result.", + "", + "So method of superposition can be summarized as follows.", + "", + "You first find the response of the circuit to each source", + "acting alone.", + "And then second.", + "", + "If you want to find the response of the sources acting", + "together, then you simply sum the individual responses.", + "It is important to point out that this works only for", + "independent sources.", + "", + "You will see later how superposition works with", + "dependent sources.", + "", + "Now, in circuit position, we talked about each source", + "acting alone.", + "And for each source to act alone, I had to set all the", + "other sources to zero.", + "So what does it mean to set a source to zero?", + "So let's say, for example, I have a voltage source, as", + "shown here.", + "What does it mean to set that this voltage source to zero?", + "Well, it simply means that we will short the voltage source", + "as shown here.", + "That sets a voltage source to zero.", + "Similarly, for a current source.", + "The way you set the current source to zero is you open", + "circuit the third course.", + "", + "OK, now let's get back to our original example that we had", + "previously solved using the known method.", + "So let's go ahead and solve the same circuit using the", + "superposition method.", + "", + "OK, using the superposition method for the goal, I have to", + "find the output for each source acting alone.", + "And then I sum the outputs.", + "So let me start by finding the output with v acting alone.", + "So we'll start by setting the current source to zero.", + "And I can do that by open circuiting the current source.", + "", + "Now, as I do this, I can compute the", + "output, ev, as follows.", + "ev is simply this voltage appearing at the output", + "through a voltage divider, r1 and r2.", + "So I get r2 divided by r1 plus r2 times v. So that is a", + "component of the output as a function of v alone.", + "", + "OK, next, let me work on i acting alone.", + "So this case with i acting alone, I want to set the", + "voltage source to zero by shorting it.", + "Now, I can write ei, the component of the output that", + "relates to just a current source, as follows.", + "So I have the current flowing through here, and that current", + "flows through a parallel resistor pair.", + "And so the voltage is simply the current times the", + "effective resistance, which is simply r1 parallel r2, which", + "is given by r1, r2 divide by r1 plus r2.", + "So that gives me ei.", + "", + "So the superposition method simply says that you get the", + "effect of both sources acting simultaneously if I simply add", + "up the two components, ev and ei, like so.", + "There you go.", + "So this is due to the voltage source acting alone.", + "And this is due to the current source acting alone.", + "Notice that this equation is the same as the one you got", + "with the application of the load method.", + "" + ] +} \ No newline at end of file diff --git a/subs/ke3SL_R92ys.srt.sjson b/subs/ke3SL_R92ys.srt.sjson new file mode 100644 index 0000000000..e45e6945b9 --- /dev/null +++ b/subs/ke3SL_R92ys.srt.sjson @@ -0,0 +1,2063 @@ +{ + "start": [ + 0, + 1586, + 2000, + 9000, + 14852, + 19711, + 27000, + 31810, + 36300, + 39507, + 44398, + 49369, + 52095, + 56505, + 59873, + 65311, + 68614, + 72412, + 76458, + 78110, + 81990, + 84633, + 90000, + 94142, + 98598, + 102741, + 107274, + 110557, + 115247, + 119000, + 123008, + 126943, + 131248, + 134812, + 136000, + 148000, + 151976, + 157248, + 161410, + 164000, + 176000, + 178000, + 183000, + 187153, + 191397, + 195822, + 200608, + 205484, + 210000, + 214132, + 217144, + 221137, + 224640, + 228632, + 232415, + 234586, + 239000, + 243636, + 248636, + 252000, + 264000, + 267692, + 272538, + 275812, + 278875, + 282812, + 286375, + 289437, + 292937, + 295250, + 298750, + 301441, + 303913, + 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Good morning,", + "all.", + "", + "You have two handouts, lecture notes and an article on", + "mixed signal chips. A mixed signal stands for", + "circuits that have both analog and digital components to them.", + "The reason I am giving you the handout is that Lab 4 and also", + "your last homework involve designing and building a mixed", + "signal circuit. It's a real fun exercise.", + "And I just wanted to tell you that from past experience people", + "who have taken 6.002 often view the last lab as the single most", + "fun thing they did in all of 6.002.", + "So, as you go into Lab 4, you should be telling yourself", + "I should be having fun, I should be having,", + "I should be having fun. You have to positively psych", + "yourself. Otherwise, it's going to go by.", + "And then you're going to say boy, that was fun,", + "I wish I had savored the moment as I was doing it.", + "All right. Let's see.", + "What do we do today? Today's lecture is actually", + "going to be a fair amount of fun.", + "We are going to blast through a bunch of fun things.", + "And some things that you will be quite unprepared for.", + "Until now, in the last two lectures with op amps we talked", + "about negative feedback. That is applying some portion", + "of the output voltage to the negative input so that I could", + "control this high strung device, my op amp.", + "Today, what we are going to do is try to get a handle on what", + "happens if we use positive feedback.", + "It's the usual curious child. You tell them to do this,", + "and of course they're going to try to do this as well.", + "And we are going to try to do that and see what happens and", + "look to see if we can build some useful circuits.", + "Today --", + "", + "As motivation, let me do a quick review of a", + "circuit that should now become affixed in your brains in a", + "standard pattern. This is a circuit that gives", + "you negative feedback.", + "", + "R1 and R2.", + "", + "And I apply a vIN. By now you should be able to", + "look at this pattern. And this is your inverting", + "amplifier pattern. So, you should be able to write", + "down by inspection this is simply vIN or the minus vIN", + "times R2 divided by R1. This is an amplifier whose gain", + "is controlled by the ratio of R2 and R1.", + "This is a negative feedback circuit because it is always fun", + "to do the intuition thing and say that look,", + "if this voltage tends to go more positive than I care then", + "this negative input goes more positive than I care.", + "If that goes more positive then the negative input v minus", + "becomes more positive in the plus input which yanks the", + "output down. So, there is a nice", + "counteracting force that keeps the output stable.", + "Let's look at this circuit. Being curious engineers,", + "let's look at the opposite here where I give myself some", + "positive feedback in this op amp.", + "", + "And it is going to be interesting to analyze this", + "because what we find out on the face of it is not quite actually", + "how it behaves. We are going to spend most of", + "the lecture today on understanding the dynamics of", + "circuits that look like this and to see if we can build some fun", + "and interesting circuits and systems based on this kind of", + "positive feedback. It is positive feedback because", + "I am feeding back a portion of the output to the positive", + "input. And you should be able to stare", + "at this and already begin to intuit what should happen to", + "this. Let's think about it.", + "This is zero. Remember, with positive", + "feedback, the famous v plus is equal to v minus method doesn't", + "apply anymore. Let's apply very simple", + "analyses. If this is zero,", + "let's say for example that this output tends to go a little bit", + "more positive. This output,", + "due to some noise or perturbation,", + "tends to go up a little bit. If that goes up a little bit", + "then because of feedback this node tends to go up a little", + "bit. If this node tends to go up a", + "little bit this exacerbates the positive input here and this one", + "goes cachunk, whacks into the positive rail.", + "Let's take the other point of view and look at it intuitively.", + "What if this one tries to droop a little bit?", + "If it droops a little bit then the input at the plus terminal", + "droops a little bit. If that tends to go down a", + "little bit, that makes the output droop further and it goes", + "and hits into the negative rail. I can see that this circuit", + "wants to hammer into the positive rail or hammer into the", + "negative rail because of the positive feedback.", + "It is like if you give incredibly positive feedback all", + "the time, and by positive feedback I mean feedback", + "encouraging the child to do whatever the child is doing.", + "It could be if he does bad stuff you give a lot of positive", + "feedback or good stuff you give a lot of positive feedback then", + "you are guaranteed to have a very good child or a very bad", + "child. You are not going to have", + "anybody in the middle. Same way here.", + "By giving positive feedback you're going to drive this into", + "the positive rail or drive this into the negative rail.", + "Now, I am going to analyze this in two steps.", + "First I am going to analyze this using a method you've seen", + "before which is replace the op amp with its equivalent circuit", + "and analyze it statically. And by analyzing it statically", + "we are going to show that the simple static analysis will", + "yield the following expression. I put this in quotes,", + "well, for a reason you will see shortly.", + "When I apply a plain and simple static analysis here is what I", + "find. Let's go ahead with the", + "analysis and see what is basically different about these", + "two. And, first of all,", + "I will confirm for you that our naive analysis we have seen so", + "far will give rise to that expression.", + "So, let's go ahead and analyze that circuit.", + "And to analyze that circuit what I will do is replace the op", + "amp with its equivalent circuit. If you remember the op amp is", + "characterized by the following circuit, A times v+ minus v-,", + "vOUT. This is the equivalent circuit", + "of my op amp. And let me just impose that", + "external circuit on this op amp. I have grounded my v- terminal.", + "My v+ terminal goes through a resistor and a supply,", + "the v into ground, it's the resistance R1.", + "This terminal goes to the output through a resistor R2.", + "So, this is the equivalent circuit.", + "And I can apply the same good-old techniques I have", + "learned about all through this course to this circuit and see", + "what vOUT looks like. Very simply,", + "vOUT is this expression here A times v+ minus v-.", + "And because of my ground connection v- is zero.", + "Then let me go ahead and replace v+ with the voltage that", + "relates vOUT and vIN. What is v+?", + "v+ is simply the current through this part of the", + "circuit, the current flowing here times the resistance R1.", + "That gives me the drop across R1.", + "And to that I add vIN and that will give me V+.", + "And then of course I multiply this by the gain here.", + "So, let me write down that expression.", + "The current through this is simply vOUT minus vIN.", + "That is the voltage drop between these two points.", + "I divide that by the resistance R1 plus R2.", + "That gives me the current flowing through here.", + "That times R1 is the drop across resistor R1.", + "And to that I add vIN and that gives me the voltage v+.", + "So, this is v+. That is simply vIN plus the", + "drop across the resistance R1. Let me shuffle things around", + "and put all the vOUT terms on this side here.", + "I get a 1+ for that vOUT and let me move AR1 divided by R1", + "plus R2 to the left-hand side. And I pick up a minus sign.", + "So, I get AR1 divide by R1 plus R2.", + "I pick up that. And on the left-hand sign I end", + "up with vIN, and my vIN here is a function of the vIN that I", + "have here. I have an A multiplying both", + "the vINs. And then I get a one for this", + "vIN here and there is a minus sign, so I get a minus R1", + "divided by R1+R2. That is the expression that I", + "have. Let me go ahead and simplify", + "that a little further and move this whole thing down here.", + "That gives me my expression as a function of vIN.", + "What I will do is, let me continue here.", + "vOUT=vIN A(1-R1/(R1+R2)). By the way, you may be", + "wondering why I am going through so laboriously what is seemingly", + "a very simple exercise. The reason I want to do is it I", + "want to very carefully show you that the result produced by this", + "exercise is exactly that. No magic here.", + "No cheating. We are going to get exactly", + "that. And then stare at it and say", + "huh, how did that happen? And then we are going to try to", + "figure out how it actually behaves following that.", + "I divide this by 1-AR1/(R1+R2). And by now you should be", + "familiar with the technique of ignoring small numbers when I", + "have a big number next to it. So, AR1/(R1+R2) can be very", + "much larger than one because A is very large.", + "So, I can ignore my one there. And then what I am going to do", + "is multiply the numerator and denominator by R1+R2.", + "Oh, this A and this A is going to cancel out.", + "This A and this A will then cancel out.", + "And then I multiply the numerator and denominator by", + "R1+R2, so this R1+R2 vanishes. I get R1+R2 here.", + "R1+R2 minus R1 is simply R2. And then down here I get a R1", + "and then I have a minus sign out there.", + "Notice that vOUT we have found to be equal to vIN R2 divided by", + "R1. That is not wrong.", + "That is correct. Technically that is correct.", + "But you will see in a few seconds that in practice that", + "that's rarely what you are going to see happen.", + "And we will try to understand why that is so.", + "What we have done so far, if you stare at these two", + "panels here, first of all, we know that the inverting", + "amplifier has the expression for vOUT up there.", + "And through this laborious exercise we have also shown that", + "even with positive feedback, if I take a static view of the", + "circuit -- If I take a snapshot of the", + "circuit and simply analyze it as a static circuit,", + "I get the same expression vOUT. But what we are going to do is", + "when I explain to you that look, a small perturbation in vOUT is", + "going to drive the op amp to the positive and negative rail,", + "that is where the insight begins to show.", + "That if everything were magical and I could somehow exactly keep", + "things just so that will be true.", + "I will be able to build that positive feedback circuit where", + "the output is equal to R2/R1 vIN.", + "But remember even the slightly amount of perturbation is going", + "to send the op amp scurrying off to the positive rail or the", + "negative rail. How do we analyze that?", + "How do we analyze the behavior of a circuit that based on a", + "small perturbation begins to move one place or another?", + "We want to analyze the dynamics of the op amp.", + "And to analyze the dynamics what I need to do is give you a", + "slightly more detailed view of the operational amplifier.", + "If the operational amplifier is not moving instantaneously", + "between the plus and minus rail, I need to give you a more", + "detailed model that encapsulates the behavior of the op amp.", + "And so let me do that. If you want to study the", + "dynamics of an op amp -- By dynamics I mean how an op", + "amp moves as I perturb the input or the output and so on.", + "To capture the dynamics of the op amp we build a slightly more", + "involved circuit, so v+ and v-.", + "", + "This is what we've seen before, two terminals and dependent", + "source that amplifies the difference input here by a large", + "amount. Instead what we are going to do", + "here is something slightly different and interpose the", + "following circuit in the middle here.", + "This is a model of the dynamics of an op amp.", + "We are going to impose a small RC circuit in here.", + "This is R. This is C.", + "And I am going to call the voltage across the capacitor v*.", + "Notice what I have done is rather than say this is Av+", + "minus v- I am breaking it apart in two dependent sources,", + "the first dependent source, which is simply v+ minus v-,", + "and there is a RC time constant surrounding it and then here I", + "simply add on my gain Av*. Notice that if it turned out", + "that the resistance here, for example,", + "was zero then v+ minus v- would appear across v* and this would", + "be A(v+ - v-), what you have seen before.", + "It is always good to take a look at circuits and look at", + "what happens when some component goes to an extreme value.", + "This would give you your basic op amp circuit.", + "What I would like to do next is analyze the following circuit to", + "understand how positive and negative feedback work together.", + "And by understanding that then be able to explain how a", + "positive feedback circuit works or a negative feedback circuit", + "works. Here is what I will do.", + "This part simply corresponds to my positive feedback circuit,", + "R2, R1. So, that is my positive", + "feedback circuit. And I will do the same thing on", + "this side.", + "", + "All I am doing is applying both a positive feedback through R2", + "and R1 and negative feedback through R4 and R3 and", + "representing the dynamics of the op amp and then standing back", + "and ee, all right, let's see what happens to you.", + "So, I am sticking positive feedback, negative feedback,", + "the dynamics of the op amp here and let's see what happens.", + "What I would like to do is impose this circuit on top of", + "this op amp model. To save myself some effort,", + "let me just go ahead and modify this circuit directly.", + "I get an R2 here, an R1 here, and then up here I", + "get an R4, R3 here. The math is going to be just a", + "little bit grubby but the result is actually pretty spectacular.", + "So, all I have done is replace the op amp with its internal", + "circuit out here. And now we are going to take a", + "look at what happens to op amp dynamics when there is a small", + "perturbation. Let's develop an equation of", + "this circuit containing a capacitor using techniques that", + "we already know. Just to give you some insight", + "into what you're going to see, notice that if I make a small", + "perturbation in the voltage across the capacitor,", + "let's say I make a small perturbation to the capacitor", + "voltage let's say by applying some initial condition kind of", + "thing onto the capacitor. Then let's say that the output", + "changes to some value K. So, the change on the capacitor", + "must have been K divided by A. And what you are going to see", + "is what happens to the op amp when the initial condition on", + "the capacitor is such that this output gets perturbed to the", + "value K. Let's write an equation for", + "this little circuit and see what happens.", + "Recall our goal was to understand what happens when I", + "perturbed the output a little bit.", + "Here I perturbed the output such that its value goes to K.", + "And I can perturb the output by changing what happens at the", + "capacitor. Let me write the equation for", + "this circuit now and then to understand what happens to this", + "capacitor circuit if I let go after giving it a small", + "perturbation. What I am going to do is let me", + "start by writing the good old equation for this little circuit", + "here. And that equation is simply the", + "voltage here v+ minus v- equals the voltage across the RC.", + "So, v+ minus v- will be equal to the voltage drop across the", + "resistor plus that across the capacitor.", + "The voltage across the capacitor is v*.", + "The voltage across the resistor is the current through the", + "capacitor C dv*/dt times R. So, v* plus RC dv/dt is equal", + "to v+ minus v-. RC dv*/dt plus v* is v+ minus", + "v-. You have done this millions of", + "times before, but yet again.", + "This voltage here is equal to the drop across these two,", + "and the drop across these two is v*, the drop across C,", + "plus the current through the capacitor C dv/dt times the", + "resistance R. Or you can apply the node", + "method as well and get the same expression.", + "Now, we also know here that vO divided by A is v*.", + "I can go ahead and replace this guy here, v* by vO divided by A.", + "RC/A dvO/dt. Recall, I want the dynamics of", + "vO so let me just get an expression in vO.", + "So, I get vO divided by A plus v+ minus v- equals.", + "Now, I want an expression in vO, an equation in vO,", + "so I need to express v+ and v- in terms of vO.", + "What are these expressions? The expression for v- is vO and", + "this voltage divider, so it's vOR3/(R3+R4).", + "And just for simplicity, let me call this some constant", + "gamma minus. This is some fraction", + "R3/(R3+R4). And let me call that fraction", + "gamma minus. Similarly, v+ is vO R1/(R1+R2).", + "And let me call that gamma plus.", + "All I am doing is replacing v+ and v- in terms of vO.", + "So, effectively, what I have here is v+ is some", + "fraction of vO. That's the best intuitive way", + "of thinking about it, some fraction of vO.", + "And v- is some fraction of vO as well.", + "And I just stick these. I now have an expression in vO.", + "Don't get psyched by gamma plus and gamma minus.", + "Simply read this as if it is an F1 and F2 if you would like.", + "So, vO times some fraction minus vO times some other", + "fraction. I am feeding back some fraction", + "of the output to the positive and to the negative terminals.", + "Then, just moving things around a little bit,", + "dividing throughout by A divided by RC.", + "So, I divided by A divided by RC.", + "Plus vO divided by RC. And what I am going to do here", + "in a second, vO gamma plus minus gamma minus.", + "And I have multiplied by A divided by RC throughout.", + "Finally, collecting all the vO terms I get vO times one divided", + "by RC plus A divided by RC. I got a plus sign here so I", + "will just reverse these two guys in there, gamma minus minus", + "gamma plus equals zero. All I have done here is simply", + "grunged through some math to express this equation in terms", + "of vO. And just to make it even", + "simpler, I will just replace this thing by one divided by T,", + "much as we did for first order equations.", + "What I end up with is dvO/dt+vO/T=0.", + "", + "Despite all the grubbiness, I end up with something that is", + "very, very familiar to all of us.", + "I went through a bunch of gyrations to substitute for v+,", + "v- and v*, but at the end of the day I got the simple", + "expression which was dvO/dt+vO/T=0.", + "Where capital T is the time constant of the circuit,", + "and the time constant of the circuit relates to the", + "expression in there 1/RC+A/RC(gamma minus - gamma", + "plus). The gamma minus and gamma plus", + "are the respective portions of the output fed back to the", + "negative input and the positive input.", + "Now, as we all know, based on very simple intuition", + "that we can completely predict the behavior of a first order of", + "an RC circuit once we know what the initial condition of the", + "capacitor is and once you know the time constant.", + "That's it. We know, we are masters at the", + "fact that the capacitor is going to behave like this.", + "It is going to be exponential. And I do know that the time", + "constant capital T. What's here?", + "It is simply the initial condition.", + "There is no drive input. I am not driving this with any", + "input here. There is no input drive", + "anywhere here. This is simply the natural", + "dynamics of the system. And, recall,", + "I start off with bumping the capacitor voltage such that the", + "output starts off being K. That is it.", + "You should be able to write down this expression and the", + "form of the response simply based on this.", + "So, this is what I bumped up the output to be by perturbing", + "the capacitor voltage. My output response based on", + "this equation is going to look like that.", + "Let's try to understand what that means.", + "It is actually quite a lot of fun.", + "How do we plot that response? You all learned that the way to", + "plot the response is plot the initial value,", + "plot the final value, and go cachoock,", + "right? It's pretty simple.", + "I am going to start at K. I know that.", + "I am going to start at K and I am going to go and find out what", + "the steady state value is. Here is where the interesting", + "stuff comes in. The final value on the", + "capacitor depends a lot on whether T is positive or", + "negative. In my RC circuits that I looked", + "at what was T? In the very simple RC circuit", + "we looked at what was capital T? What was the time constant?", + "RC. This was RC.", + "This was a positive quantity. When capital T is positive my", + "output is going to look like this.", + "When T is positive. And T is positive when this", + "expression is positive. And if A is so large that I can", + "ignore the 1/RC term, if A is very,", + "very large and I can ignore the left-hand term here then T is", + "positive when gamma minus is greater than gamma plus.", + "So, when gamma minus is greater than gamma plus,", + "I have a stable circuit, this is the good-old stuff we", + "have seen before. Now things begin to make sense.", + "Intuitively, what am I saying here?", + "All the gammas and other pieces of crapola aside,", + "what am I really saying here in English?", + "What I am saying here is that if the portion of the output fed", + "to the negative input is greater than that fed to the positive", + "input then I have net negative feedback.", + "I have net negative feedback. I am feeding the output back to", + "both the positive and negative inputs.", + "And if my negative input has a stronger effect then I am going", + "to see the op amp output decay down to a value that I expect", + "which is going to be zero. Notice that since I am not", + "applying any input here, I expect the stable point for", + "this to be output going to zero. I don't have any input there.", + "Let's take a look at another situation.", + "What happens when the opposite is true?", + "What happens when gamma minus is less than gamma plus?", + "When I feedback more, what happens when I do this,", + "when gamma plus is greater than gamma minus?", + "The opposite is true. This means that I am feeding", + "back more to the positive input. A bigger proportion goes to the", + "positive than the negative. What happens then?", + "Then what happens is capital T becomes negative.", + "We cannot see this happening on the RC circuit because capital T", + "is equal to RC, but here we have a more", + "complicated circuit and capital T can go negative.", + "If capital T goes negative then this whole thing in the exponent", + "there goes positive. If that goes positive what", + "should the output look like? It should take off into", + "never-never land. There we go.", + "I start off at zero and a make a small perturbation,", + "and the output should go as t divided by capital T.", + "The dynamics of this it goes berserk, so it is net positive", + "feedback. This is called a stable", + "situation. This is unstable.", + "What happens when capital T goes to infinity?", + "When capital T goes to infinity, spend five seconds", + "thinking about what it means physically.", + "What does it mean for the time constant of an RC circuit to go", + "to infinity? That means that your R and C", + "are very, very, very large.", + "That means that circuit is going to be very,", + "very sluggish. Think elephant.", + "A big time constant. I want to move a leg.", + "It takes a while to do that. Think big.", + "Big time constant. So, everything is going to", + "happen really slowly. It's like moving in molasses.", + "Big time constant. Everything is going to happen", + "really, really slowly. If gamma minus is greater than", + "gamma plus with a huge time constant it is going to look", + "like this. And the output is going to look", + "like this. I make T even larger.", + "All right. It is going to like this.", + "", + "I make these so large that T tends to zero,", + "T tends to infinity in which case I get this situation.", + "The output goes dah. OK?", + "Very slow. Very lethargic.", + "Big time constant. T tends to infinity.", + "And so if this is stable, this is unstable,", + "this is called corresponding neutral.", + "And there is a mechanical analog to all of this.", + "You can show that this situation is akin to let's say I", + "had a physical well of the sort and I had a ball in there.", + "I let the ball go. Then the ball will come down", + "here and settle down in a stable state.", + "Any small perturbation of the ball will get it to come down", + "and settle down here. The unstable situation is this", + "situation where I have a ball sitting up here where any small", + "perturbation will get it to zip down to a positive rail or to a", + "negative rail. So, this is an unstable", + "equilibrium situation. And exactly the reason we got", + "this analysis in the static situation is that this can", + "happen. If I do this circuit here and", + "don't perturb it then I could get the output sitting at zero,", + "but the slightest perturbation, boom, it is going to fall down", + "or go up. What about the neutral", + "equilibrium state? That can be modeled like a", + "table top and the ball is here. It doesn't matter where you go.", + "There you are. How many people saw the", + "Buckaroo Bonzi thing? Possibly well before your time.", + "OK. I have this table here.", + "No matter what I do to it, it just goes and settles down", + "where it is, and that is neutral equilibrium.", + "But what this gives you is a fun view of the dynamics of the", + "operational amplifier as I make small perturbations to it.", + "And the even more interesting thing here is you have the tools", + "based on your first order RC analysis to analyze the dynamics", + "of a simple op amp circuit. OK, so much for theory.", + "Now let's get to some action here.", + "All right. Fine.", + "That is really pretty, good and so on,", + "but what can you do for me? What good does this property do", + "for me? What can I build?", + "What we will do is look at the op amp circuit and focus on the", + "situation where I have net positive feedback.", + "In particular just look at this circuit with R1 and R2 and send", + "both to infinity. So, I have no negative feedback", + "and I ground this terminal here and take a look at what happens", + "to a circuit with positive feedback and see if I can build", + "some interesting circuits. What you are going to do is", + "build on a circuit called the basic comparator.", + "What is that? If I have an op amp that looks", + "like this, and remember a VS rail and minus VS supply there,", + "this is v+, this is v-, I can build a very basic", + "comparator by doing the following.", + "All the circuits I am going to show you are going to build on", + "this basic little circuit. What I am going to do is", + "consider applying an input to the v- terminal,", + "applying some sort of an input and taking a look at how the", + "output behaves. So, I apply some input vIN.", + "And if I just do that, if this is v+ minus v- here", + "then I am going to get something that goes like this.", + "That is when this is positive here then this guy is going to", + "go to the VS rail and this guy is going to go to the minus VS", + "rail. In terms of the,", + "if I plot the same thing, in terms of vIN,", + "and this is vOUT, if I plot the thing in terms of", + "vIN then notice that as vIN increases this guy should go to", + "a negative rail. So, in terms of vIN it looks", + "like this. What this says is that as the", + "input becomes more and more positive applied to v- then the", + "output goes to minus VS, and if the input becomes more", + "and more negative then the output goes to VS.", + "This is what is called a very basic comparator circuit.", + "It compares the two inputs and goes up if the input is in one", + "direction and goes to the other rail if the input is in the", + "opposite direction. So supposing I feed this- I can", + "plot this is a function of time. Let's say I plot vIN.", + "Let's say I feed some vIN here. Let me just call this.", + "I feed some vIN to this circuit here, then what do you expect", + "the output to look like, the output wave form?", + "For all positive vINs the output is negative.", + "So, my output vO is going to be negative as long as vIN is", + "positive. And when vIN becomes negative", + "this one shoots up and behaves like this.", + "This is minus VS. That is plus VS.", + "This is my input vIN. Then this guy is going to be my", + "output. As vIN is positive output slams", + "to the negative rail. When vIN becomes negative the", + "output slams to the positive rail.", + "So, that is quite nice. And so such a circuit is pretty", + "useful to me. Let's say, for example,", + "I want to build a little digital circuit that is fed ones", + "and zeros. I can use a comparator to turn", + "my vIN voltage into a sequence of ones and zeros.", + "When vIN is positive I produce a zero and when vIN is negative", + "I produce a one. I can get this one,", + "zero, one, zero sequence coming out corresponding to the values", + "of vIN being greater or less than zero.", + "Now, one problem with something like this is that this circuit", + "can be quite messy in the following situation.", + "Suppose I superimpose a small amount of noise in vIN.", + "In particular, let's say that I have some", + "amount of noise on vIN. I get a bunch of noise sitting", + "around here. What happens is that at this", + "point where the value goes negative, I do bump up.", + "But when for a second I have my input going above zero again --", + "-- this output comes down again and out here it goes up again.", + "I get this nasty behavior at the point where the input is", + "around zero. When the input is around zero,", + "the input is meandering around zero because of noise,", + "I get a huge amount of up and down glitches on the output.", + "That's not very nice. And we will do a little circuit", + "that attempts to fix that little problem.", + "What we are going to do is use positive feedback.", + "And I am going to build you a circuit that shows that we can", + "eliminate this for small noise on the input.", + "So, let's build the following circuit.", + "So I still feed vi to the negative input,", + "but this time around I give it some positive feedback.", + "So, I give it some positive feedback.", + "And what I am going to do is feedback a portion of vO to the", + "positive input. This is positive feedback.", + "And, in particular, let's assume that VS equals 12", + "volts. And to the negative one I", + "connect -VS. This guy is going to go between", + "12 and -12. And correspondingly because", + "these two are equal this one is going to go between 6 and -6.", + "This is going to be a 12 or -12.", + "Remember, the top rail and the bottom rail.", + "And this one is going to be a +6 or -6.", + "And let's understand how this circuit works when I apply an", + "input vIN. Let's start by saying that", + "assume my input is zero for a moment.", + "And let's say my output starts off being 12 volts.", + "The output is 12 volts then the input here is going to be 6", + "volts. In this case v+ is going to be", + "6 volts. The output is 12,", + "v+ is going to be 6 volts. And my circuit is sitting out", + "there doing nothing. Now, this started off being", + "zero. Let's say vIN increases.", + "As vIN begins to increase what happens?", + "Well, nothing until vIN reaches 6 volts.", + "Since this is 6, vIN has to go up to 6 volts,", + "has to equal this voltage before I can flip the circuit.", + "What happens when vIN is greater than 6 volts,", + "if vIN goes above 6 then I have more voltage on a negative", + "terminal than the positive so the op amp flips its state.", + "And vO gets to -12 volts. When vi goes above 6,", + "vO gets to 12 volts. And what does v+ go to?", + "In this state v+ goes to half of -12 which is -6 volts.", + "Now, this guy is sitting at -6 and this guy is sitting at -12.", + "If this one keeps rising nothing happens,", + "so output can stay at -12. So I am pretty safe.", + "Then let's say v begins to come down.", + "As v begins to come down, does anything happen when v", + "gets to 6 again? If v is equal to 6 what", + "happens? Nothing because this is at -6", + "now. So, there is still a huge net", + "negative voltage here from v+ to v-.", + "And so therefore I sit at -12. Oh, well, I keep coming down", + "until I reach -6. When I reach -6 here these two", + "become equal. And what happens when this", + "becomes less than -6? v- becomes less than -6.", + "If this one goes below this voltage, this is -6 and this is", + "-7. There is a net positive voltage", + "between v+ and v-, so this output swings to the", + "positive rail like so. We will spend a lot more time", + "on this in the next few minutes to really hammer the point home.", + "What is interesting about this is that even though the moment", + "vi became more than 6, I swung to the positive rail,", + "and then I had to go all the way back down to -6 before I", + "could change state. I had to go way down before it", + "could flip again. How can we make use of that?", + "Well, let me draw you a little vi versus vO diagram and then", + "talk about how that can be useful to us.", + "This is vi, this is vO, this is zero.", + "Let's say this is 12, -12, -6, +6.", + "Let's plot that on the screen and see what it looks like.", + "As I told you, the output was at 12 volts to", + "begin with and my input was at zero.", + "So, my input kept increasing. When the input hit +6 what", + "happened to my output? My output swung down to -12.", + "As the input kept increasing nothing happened.", + "This was step one, this was step two,", + "step three. My input kept increasing and", + "output stayed at -12 volts. Then what I said was well,", + "let's bring the input down. So, my input began to go down,", + "step four, became more and more negative.", + "Nothing happened until I reached -6.", + "When I reached -6 I swung positive, step five.", + "Again, one, two, three, four,", + "five. I am going up here.", + "It came up here. And nothing happens until I", + "reach -6, but at -6 boom, I switch to the positive rail.", + "And as I get more and more negative I stay there.", + "Then again, as I start increasing again,", + "nothing happens until I reach +6.", + "Think of that as your seventh step.", + "What is spectacular about this is that I seem to have a circuit", + "that now has some knowledge of where it came.", + "If it is coming from here it switches at +6,", + "but if it is coming from here it switches at -6.", + "So, there seems to be sort of a lag in the behavior of the", + "circuit or some memory property in the circuit.", + "This kind of behavior is called hysteresis.", + "The word comes from magnetic circuits where,", + "or rather elements that you're trying to magnetize.", + "Where if you take a magnet and move it over a piece of metal it", + "may leave some residual magnetism in it.", + "And, in the same way, that is called hysteresis.", + "Same way here. As the voltage increases it", + "seems to leave some residual in the circuit so that it effects", + "when it shifts. The good news with this is that", + "now, if I take the same kind of noisy wave form that I had", + "before and do this -- If this is vi then what is", + "going to happen is for vO I am going to be negative at this", + "point. Nothing happens here because I", + "have to get to -6 or +6 before something happens.", + "Out here I get to -6 and I switch state and go up to +12.", + "And then this one comes up above -6 very slightly out", + "there. Nothing happens because the", + "next change will happen only when the input goes to +6.", + "So, if eventually the input gets to +6 and then I am going", + "to change state again. It is actually a really cool", + "property and something that is completely non-obvious.", + "In the last 30 seconds let me show you a quick demo.", + "And, based on this property of hysteresis, I have actually", + "built a little circuit. Let me do that first.", + "Notice here that I am showing you the input on the X axis vi", + "and vO on the Y axis. Notice how the output switches", + "at +6 volts and switches at a -6 volts to +12 or -12.", + "That's the hysteresis property. And we can actually use this", + "property to build a clock circuit, which is on page 9,", + "build an oscillator that sits there and oscillates by itself.", + "And you will see details of that in recitation tomorrow." + ] +} \ No newline at end of file diff --git a/subs/v6vqWasIHaw.srt.sjson b/subs/v6vqWasIHaw.srt.sjson new file mode 100644 index 0000000000..9909b360ba --- /dev/null +++ b/subs/v6vqWasIHaw.srt.sjson @@ -0,0 +1,2057 @@ +{ + "start": [ + 0, + 25000, + 30000, + 35108, + 40869, + 46739, + 50000, + 66000, + 71349, + 74559, + 79588, + 85901, + 90395, + 95077, + 99110, + 103885, + 109085, + 113967, + 119692, + 123078, + 126394, + 130697, + 134929, + 138315, + 142406, + 146286, + 149248, + 152000, + 158000, + 164495, + 170152, + 175390, + 180000, + 185963, + 190734, + 196590, + 202228, + 208409, + 216000, + 220943, + 226161, + 231105, + 235591, + 242000, + 246222, + 250524, + 255464, + 260085, + 263112, + 267494, + 273589, + 277565, + 281541, + 286996, + 291434, + 298000, + 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3039952, + 3044679, + 3048494, + 3053221, + 3058000 + ], + "text": [ + "", + "All right. Good morning.", + "Let's get started. So the last lecture we showed", + "you how to go digital. The fact that going digital had", + "some key benefits for us. And what we'll do today is go", + "inside the digital gate.", + "", + "Let's do a quick review. We began life by observing", + "nature. We said those Maxwell's", + "equations are tough. Let's simplify our lives by", + "discretizing or lumping matter. So we got the lumped circuit", + "abstraction. Then we had this noise problem", + "here. In order to be able to handle", + "that let's do some more discretization,", + "some more lumping. So we said let's discretize", + "values and deal with two levels, a high and a low.", + "That's where the binary voltage levels come up,", + "a high level and a low level. And then we said that in", + "discretizing it we have to make some assumptions.", + "We have to impose some constraints on ourselves.", + "Just as with the lumped matter discipline, we imposed a couple", + "of constraints in going from the continuous matter world to a", + "lumped matter world. Similarly, we have to impose", + "some discipline on ourselves, some constraints on ourselves", + "in going from the continuous value regime to the digital", + "value regime. And that discipline is called", + "the static discipline.", + "", + "And what the static discipline says is that if you have senders", + "and receivers in a digital system then they all need to", + "adhere to some standard. If I was a sender I had to", + "adhere to some tough output standards.", + "I had to be sure to shift values that exceeded some high", + "voltage threshold. And if I was sending a low", + "value I had to make sure my values were lower than some", + "output low voltage threshold. Similarly, if I was the", + "receiver then I had to guarantee to recognize as a one all", + "voltages that where above some input high voltage threshold.", + "And similarly I had to guarantee to recognize as a zero", + "voltages that were below some input low voltage threshold.", + "So provided senders and receivers in a system adhere to", + "these voltage levels, to this discipline then they", + "would all very comfortably work correctly in a digital system.", + "Then we also said that once you deal with such values,", + "one you deal with digital values we can now postulate a", + "bunch of digital elements that process these values in a manner", + "very reminiscent of our analog circuits where we get analog", + "signals. And you've already learned how", + "to process analog signals. You've learned about resistor", + "dividers and so on and so forth. You feed in an analog signal", + "and you get an output analog signal as well.", + "Now, here the resistor in the analog domain,", + "elements like resistors and voltage sources were the symbols", + "that you dealt with. Here, in the digital domain,", + "the primitive elements that we will be using are called gates.", + "As one example, this is called the NAND gate.", + "So we looked at the AND gate in the previous lecture.", + "This is an example of another gate called the NAND gate.", + "The NAND gate has the following truth table.", + "Our two inputs A and B and this output C.", + "And the NAND gate works as follows.", + "The output -- In English I can describe its", + "properties as the output is a high at all times when at least", + "one of these inputs is a low value.", + "So it's high whenever at least one input is a low.", + "So it's high here. It's high here.", + "Oops, it's high here, high here.", + "And when, oops. And when both inputs are a high", + "the output is a low. This is a NAND gate.", + "Notice that these are exactly complimentary to the AND gate.", + "The AND gate outputs were 0-0-0-1.", + "And the AND gate symbol looked like this.", + "In general, notice that this little bubble here,", + "it's called a bubble. That bubble implies a negation,", + "an inversion. So we take the AND gate,", + "invert the output and negate the output and you get the NAND", + "gate. So these elements are", + "combinational gates. And in combinational gates they", + "adhere to two properties. One is that they must satisfy", + "the static discipline.", + "", + "All the systems, all the elements in our", + "repertoire in the digital domain need to satisfy the static", + "discipline. And the properties of a", + "combinational gate are that its outputs are a function of inputs", + "alone.", + "", + "In other words, it doesn't store any state or", + "doesn't store any history inside it.", + "You can figure out its output just by looking at the inputs at", + "that instant. Think of it as a completely", + "transparent entity where its output reflects some function of", + "the inputs at every instant of time.", + "", + "So I'll show you an example of a digital circuit.", + "", + "So much as I could interconnect resistors and voltage sources", + "and current sources to build analog circuits,", + "I can now build digital circuits using primitive", + "elements such as these. So, for example,", + "I could build a simple circuit that looked like this,", + "two inputs A and B here, I get an output.", + "And I feed that to another NAND gate with another input C.", + "This device is called an inverter.", + "The inverter simply flips the sense of the input.", + "So if C is a 1 the output is a 0, if C is a 0 the output", + "becomes a 1. It's an inverter.", + "It simply inverts its input. Yet another primitive device.", + "And this is my output D. So there are three gates in", + "this design. And I can quickly write down", + "what the output looks like using some very simple Boolean algebra", + "or dealing with Boolean values here.", + "So for AND gate the output is A and B.", + "Remember dot is a short form for and.", + "But there's a negation, inversion, so represent", + "inversions with a bar. So my output is A dot B bar.", + "There is a C here. So this is my output C bar.", + "And this is a NAND gate. So it takes one input A dot B.", + "It takes the second input C bar and ANDs those and inverts them.", + "So that's the output. So there are three gates in", + "this example. So you can think of building", + "very complicated circuits containing large numbers of", + "gates. In fact, the microprocessors", + "that you use in your laptop contain a large number of gates.", + "Can someone guess how many gates are in the Pentium IV,", + "roughly? Approximate,", + "how many? How many gates in a Pentium IV?", + "40 million. 100 million.", + "In the Pentium IV you have on the order of 20 million gates.", + "20 million gates in the Pentium IV.", + "And life begins in 002. Here you learn about onsies and", + "twosies, and in the real world you will be dealing with tens of", + "millions of gates. But this is for the Pentium IV.", + "My research group at Laboratory for Computer Science built a", + "chip called the Raw chip. And this chip has 3 million", + "gates. And so there are several", + "undergraduate students involved in this project in their third", + "year, and they're beginning to deal with millions of gates.", + "So the key thing to remember is that 002 provides the", + "foundations where you make the switch from the analog signal to", + "the digital signal or from continuous matter to lumped", + "matter. And learn about the foundations", + "of these primitive elements. And by the end of this course", + "you will begin dealing with small systems,", + "analog systems that contain on the order of 10 to 20 primitive", + "elements. You will also begin dealing", + "with small digital systems that contain tens of gates.", + "In your final project you will build a mixed signal circuit", + "involving an audio playback system.", + "You will have digital data stored in a memory chip and you", + "will build a circuit to extract that data, filter it and then", + "convert it to the analog domain and then play it on a set of", + "speakers. And that has on the order of", + "about 50 to 100 primitive elements.", + "So by the end of 002 you will have learned to deal with", + "hundreds of elements. And then you will take other", + "courses like 004 and so on where you will then make the leap to", + "learn further abstractions that will take you from subsystems to", + "systems with millions of gates. So the key is to manage the", + "complexity of dealing with millions of gates it's all about", + "abstractions. You have to build abstractions", + "and double abstractions so you can deal with complexity.", + "So the rest of EECS will take you from three gates to 20", + "million gates and software systems that operate on 20", + "million gates or whatever. So there is still a ways to go.", + "Lorenzo, our friend has gone to bring a demonstration that we", + "forgot to bring today. That will show you that little", + "digital circuit in a mock up form.", + "So what's today's lecture about?", + "Today's lecture is going to be about what's inside a gate?", + "", + "How to build a gate. Once you build a gate you can", + "then put millions of them into computer systems or analog", + "systems or other sorts of systems.", + "And what we'll do here is understand what's inside this", + "abstraction. This is an abstract element", + "that looks like a little circle and a line with some stuff", + "inside it, with some properties. But someone's got to build", + "that. It doesn't come from nature.", + "You don't go and harvest gates from trees, you got to go build", + "that, and someone has got to do that.", + "So what to learn here is how do we go about building a gate?", + "And here you will see practically how do you deal with", + "voltage thresholds that satisfy a given static discipline?", + "So before I jump into building a gate, let me try to build up", + "some intuition. As is my usual practice,", + "I'd love to get you to build some intuition as to how to", + "build a gate. And then we'll go through the", + "mechanics of doing it. So to build intuition,", + "let me show you an analogous situation in fluids.", + "So let's say I have a cauldron of water.", + "This is like a power supply. And I need to feed this fluid", + "down at some output source. And what I do in the middle is", + "put in a couple of taps, faucets, all right?", + "And so what do these guys do? Under what condition do you", + "have fluid flow out of the tube at the other end?", + "You will have fluid flow if -- So let me call this A and B.", + "If A is on and B is on then C has water.", + "Otherwise, if both A and B are not on then C has no water.", + "So this is already beginning to sound like a AND gate,", + "correct, where you get water only if A and B are both turned", + "on. So we're going to use this", + "insight, a stream of some flow and I put things to obstruct the", + "flow. And when both the obstructions", + "are lifted I get the output. I want to use that intuition to", + "build an AND gate. Similarly, I could build a", + "system that allows me to build the following structure --", + "", + "So in this scenario let me call this --", + "", + "-- the signal of A and B here. And in this situation under", + "what conditions, provided the power supply has", + "water, under what conditions do I get water out?", + "In this situation, it is I get water if A or B are", + "turned on. So I don't need to turn both A", + "and B on. If either one of them is on,", + "I'm going to get fluid flow here.", + "So this will help us build the inside to build the OR gate.", + "So that's an analogy involving items we see in everyday life.", + "Let me now move into the electrical domain.", + "In the electrical domain my analogy would be something like", + "this. Let's say I have a power supply", + "and I have two switches A and B. And I build a little circuit", + "that connects this voltage source across the bulb using a", + "couple of switches. In this case,", + "the bulb is on if both switches A and B are on.", + "My bulb turns on. If I switch either one of them", + "off my bulb turns off. So notice that I can begin", + "implementing things like this if I had this element.", + "I had sources already. I know how to deal with bulbs.", + "I model them as resistors. So I need to do something about", + "this new element called a \"switch\".", + "So let me build an abstract device.", + "I'll tell you how to do that in real life in a second.", + "So if I had the switch I could build things like this.", + "I could put switches in series in a circuit and get myself", + "something that looks like a AND function.", + "So let me go ahead and build an equivalent circuit for a switch.", + "So the switch has a couple of terminals here and I have a", + "control. Switches have a control and", + "they have a pair of terminals. And the equivalent circuit for", + "this looks like this. This is for my switch.", + "So when control is a 0. Then my switch is open to give", + "me an open circuit in the circuit that I've shown you", + "here. And, by the same token,", + "if my control is a 1 then --", + "", + "-- I have a connection between in and out.", + "And this is a short circuit. So, in other words,", + "if my switch has 0 at its control, I'll talk about how to", + "get that, I have an open circuit, and if it's a 1 then I", + "have a short circuit. This is a switch going on and", + "off. Now, in traditional switches", + "mechanical pressure is my control signal.", + "If I apply mechanical pressure my switch could turn on.", + "And if I take away the mechanical pressure then I could", + "get an off situation. So let's for now imagine that", + "we have a switch. I still haven't told you how I", + "am going to get a switch in real life.", + "Let's imagine you have a switch.", + "It's a three terminal device. There's a control thingamajig", + "coming in. Input and an output.", + "So let's build the following little circuit containing a", + "switch. So what I'm going to do,", + "I will take a resistance RL and plug it in here.", + "", + "And connect my power supply like so.", + "So the little circuit that I build has a resistor.", + "And I connect the switch in this pattern and I get a VS.", + "Lorenzo, you can set that up there if you'd like.", + "No problem. So I get a VS here.", + "Now, a couple of lectures ago I told you that 6.002,", + "and for that matter, 004 and many of our other", + "courses deal with combinations of elements.", + "And we often deal with the same kinds of combinations again and", + "again and again. We see the same sorts of", + "patterns happening, and we need to begin to learn", + "to identify these patterns. This is an incredibly common", + "pattern. You'll see this pattern more", + "times in 6.002 than any other pattern, I promise you.", + "A power supply connected to a resistor and connected to a", + "couple of terminals of some interesting device.", + "I promise there will be at least one such pattern on the", + "quiz, for example. These patterns are incredibly", + "common. So let's take a look at the", + "interesting properties of this pattern.", + "Since this pattern occurs so commonly, I am going to create a", + "short form. I have already created a short", + "form which is this ground node here.", + "By putting ground 0 all I'm really saying is that there is a", + "wire connecting these two and that's my ground.", + "So I already have a short form here.", + "My second short form is when I connect a power supply to a", + "node. Then what I'm going to do is", + "come up with yet another short form that looks like this,", + "an up arrow with the voltage written there.", + "This symbol simply says that this node is connected to a", + "power supply with voltage, or a voltage source voltage VS.", + "So I just have come up with a slightly simpler representation", + "for the little pattern that I have.", + "Now let's take a look at the properties of this little", + "system. Let's first look at what", + "happens when C is 0. When C is 0,", + "let me draw the equivalent circuit for this using the open", + "circuit out there.", + "", + "That's what I get, OK?", + "So when C is 0, if VS is a high voltage,", + "let's say 5 volts, what do you expect at the", + "output if C is a 0? This voltage VS appears at V", + "out because this is an open circuit here.", + "Remember, RL and this little device form a voltage divider.", + "But since it's an open circuit its resistance is infinity.", + "And so therefore in this resistor divider all the voltage", + "falls across this open circuit. So, in this case,", + "v out is a 1 or a high voltage. But let's take a look at what", + "happens when C is a 1. In this situation,", + "I have my RL, that's what I have.", + "It's a short circuit at the switch and C is a 1.", + "So what's the voltage v out in this case?", + "Not surprisingly, since I've shorted this node to", + "ground the voltage at this point is 0.", + "So if I have low voltage that's corresponding to logical 0s that", + "corresponds to a 0. So I can build a simple truth", + "table for C and use logical symbols here.", + "So when C is a 0 I get a high at the output and when C is 1 I", + "get a low at the output. Have you seen a device that", + "behaves like this so far? That's a little inverter.", + "That's the exact behavior of an inverter.", + "So this thing I've written here is a truth table for an", + "inverter. So notice with just a simple", + "little switch and a resistor, I have managed to build an", + "inverter. Before I go on,", + "I guess we have some things to show you.", + "And let me pause for a couple of seconds and do that.", + "First of all, what I want to show you is the", + "following idea. So as I was preparing for this", + "lecture last night I said, now here I am telling the 6.002", + "gang that you need to learn about analog circuits and", + "resistors and all of that stuff, and you also need to learn", + "about digital systems and all of that stuff.", + "And I said, because these two are very commonplace and often", + "times they occur together. So I said well,", + "if I really believe in my own BS then there should be", + "something around me where I can find both of them", + "instantaneously. So I said let me do the", + "following experiment. Let me close my eyes and reach", + "out and see what I touch. So I closed my eyes,", + "reached out, and guess what?", + "I touched the lonely mouse. The mouse.", + "So I said let me see what is in side the mouse.", + "And if I believe in my BS we should find analog,", + "little components and digital components in there,", + "right? So let's see what is inside the", + "mouse. All right.", + "There we go. Don't try this at home,", + "as with many other things we do in lecture.", + "", + "Come on. Show me what I want to see.", + "OK, here we go. Not bad.", + "Let me show you what we have here in this poor shattered", + "mouse. That's my finger,", + "silly. You should recognize this", + "little resistor here. That thing with the little", + "bands, oh, here we go. We'll use this.", + "That's a resistor. And you'll see capacitors in", + "about four weeks. That's a capacitor.", + "And there is a digital IC here. That's a digital IC.", + "That contains a bunch of gates inside it.", + "So this mouse has not made a liar out of me.", + "So what I just showed you was a little device that we use in", + "everyday life that has both analog components and digital", + "components. A large number of devices that", + "we use in daily life are this way.", + "You can do the same thing to your laptop.", + "You could go try it out. And you will find a bunch of", + "analog components and a bunch of digital components.", + "And you really, really need to understand the", + "whole caboodle here. Let me show you a fun little", + "demo involving gates. Now, I want you to be very", + "careful here. Lots of caveats here.", + "If your grandmother asks you how big is a gate don't say this", + "big. This is how big gates used to", + "be, I would say, when they were first invented.", + "When they built gates out of discrete vacuum tubes and so on,", + "this is how big a gate used to be.", + "This is roughly that big. Today in a chip,", + "in a small VLSI, very large scaled integrated", + "circuit in a chip, which is about 1 cm on the", + "side, how many gates do you think I can fit in a thumbnail", + "sized chip? Any guesses?", + "With today's technology, how many gates can I fit on a", + "chip? It has to be more than a", + "million because I just told you that Pentium IV was 20 million", + "and that was a year ago. How many?", + "40 million is a good guess. So on the order of 40 to 80", + "million gates in a 1 square centimeter.", + "Intel just announced that they will be shipping a chip", + "containing 1 billion switches. Remember, this whole thing is a", + "gate, right? Inverter, a resistor and a", + "switch. This thing is a switch.", + "So Intel is going to be shipping something containing a", + "billion of those little elements.", + "Just keep those large numbers in mind.", + "So here is a little circuit that I showed you here,", + "A, B, the NAND gate, the NAND gate at the output and", + "the inverter. So this output A is going to be", + "1 whenever either A or B is off. So the output is a 1 in this", + "case when both A and B are off. I turn A to 1,", + "output is still a 1. So the moment I turn both of", + "these inputs into a 1, these are 1s,", + "the output goes to 0. That's behavior for NAND gate.", + "If I switch any one of the inputs to a 0 the output should", + "go to a 1. Similarly, for the inverter", + "here, when the input is a 0 the output is a 1.", + "And when I switch it so should the output.", + "Now imagine a circuit, a little chip containing", + "billions of these devices. And just imagine all of these", + "1s and 0s flying around. So one simple switch in the", + "input, like a click of a keystroke could actually cause a", + "billion signals in your circuit to be flipping around.", + "And that causes some fun stuff to happen, which we will learn", + "about a few months from now. But for now that's a quick show", + "of a little circuit that looks like that.", + "Let me go back to talking about building other types of gates.", + "", + "So that was an inverter. So now you know.", + "You're almost halfway to being able to build a Pentium IV.", + "You've come all the way from nature to gates.", + "And Pentium IV contains 20 million of them so you now know", + "how gates are built. So that's an inverter.", + "Let's look at how we can build other forms of gates.", + "To build another gate let me do this.", + "", + "How about this pattern? If I build a pattern like this", + "with A and B coming in here and I put two switches with their", + "inputs in and out, so two switches in series.", + "Let's write down the truth table for what this looks like.", + "Let's see. When A and B are both 0,", + "what should the output be? These are both off so the", + "output is directly VS which is a high.", + "When either of these switches is off 0-1 or 1-0.", + "If either switch is off then this node is cut off from", + "ground. There is no current flowing", + "here. So this entire voltage drops", + "across this infinite resistance here, and so I get 1s at the", + "output as well. If both switches are on what", + "happens? If both A and B are on then I", + "get a short circuit to ground and my output is a 0.", + "So can someone tell me what gate this is?", + "Awesome. We just build a NAND gate.", + "This is unbelievable. Five lectures and you've", + "already come all the way from nature to the primitive building", + "blocks of microprocessors. It's pretty amazing.", + "So what about this one here?", + "", + "What's this? I haven't told you this before", + "but if an AND gate becomes a NAND gate, this is kind of an OR", + "arrangement, what should an OR become?", + "NOR. It's all completely logical.", + "So you can go home and practice a truth table for this.", + "A, B and C. I'll just fill in one of the", + "rows. So in this particular", + "situation, if both A and B are 0, if A is 0 and B is 0,", + "both the switches are off, so it's as if this little", + "sucker here is cut off from ground and VS falls across from", + "C to ground here and the output is a 1, so on and so forth.", + "So I can build other interesting forms of gates.", + "So let's say I build something that looks like this.", + "", + "I build something like this.", + "", + "You can write the truth table for this or you can look at this", + "and write down the function that this one supports.", + "Notice that this output here is going to be a high only when", + "both of these are not connected to ground.", + "And if you stare at it some more the function this one", + "presents, this is my AND function.", + "Suppose this one didn't exist, that would be my AND function.", + "But because this one exists that's in an OR configuration", + "and so I get a C. And so because of that I get", + "something that looks like this. So this is my A dot B,", + "this is my plus because of a parallel here,", + "and ultimately this caused an inversion in this gate.", + "So the primitive pattern has a generic inversion built into the", + "output. That is why they commonly end", + "up building NAND gates and NOR gates and so on as the simplest", + "gates. We don't build AND gates and OR", + "gates. How can I convert this one to", + "an AND gate? Anybody?", + "Put an inverter on the output. So what I can do is take this", + "little sucker here, put an inverter here and I get", + "an AND gate. So the real primitives in", + "circuits tend to be NANDs and NORs.", + "", + "OK. So the real practical among you", + "should be saying at this point all right, all right,", + "I buy this, if there existed a switch.", + "I know exactly how to go from nature to building Pentium IVs", + "if there exists a switch. So that the obvious next step", + "for me is to show you a switch, a physical switch device.", + "And to introduce a switch device, let me show you a three", + "terminal element. Remember, the switch has three", + "terminals, an input, output and something called the", + "control, C. So I'm going to introduce a new", + "primitive element called \"The MOSFET Device\".", + "MOSFET stands for metal-oxide semiconductor field-effect", + "transistor. This is shortened to FET or", + "transistor. Now I'm going to show you that", + "this works like a switch. And before I do that,", + "in fact, let me do that first. Then I'll show you something", + "else. So this device has the", + "following symbol. It has a terminal called a", + "gate, the drain and the source. Gate, drain and source.", + "Three terminals. This is the primitive element", + "that forms virtually every electronic component built", + "today. This is the foundation of the", + "universe. So this little MOSFET device,", + "we can look at how it behaves. I'll show you this thing on the", + "screen in a second, but this guy behaves very much", + "like this device I was postulating earlier.", + "Let's take a look at this device on the scope.", + "To do so let me label some voltages and currents.", + "So let me label this voltage as vDS.", + "Let me label this voltage as vGS between the gate and the", + "source. And let me label the current", + "coming into this node iG. In this device,", + "the physical device that I'm going to show you,", + "the current going into the gate is always 0.", + "So iG is always going to be 0 for 6.002.", + "In real life there is some leakage and so on.", + "But in 6.002 for now we deal with a very simple abstract", + "model, iG is 0. And let me label the current", + "here as iDS. To be correct with the", + "nomenclation, the current into node D should", + "be labeled iD, but because iG is 0 iD flows", + "out through the source as well, so I would simply call it iDS", + "just so that I can show that vDS and iDS are the two voltages and", + "currents that I am going to deal with.", + "So that's my little device here.", + "And notice that the source terminal is common.", + "I use the source both for the control GS and I use the source", + "for the drain as well. So you can view this as input,", + "view this as out, and you can view this,", + "if you like, as the control abstractly.", + "So let me show you a plot of how this behaves.", + "To understand how it behaves, I can draw an equivalent", + "circuit for it. So in this particular", + "situation, if its behavior is characterized by the voltage", + "applied to vGS. Much like the control on the", + "switch, vGS is my control. So if vGS is 0,", + "oh, I'm sorry. If vGS is greater than or equal", + "to some threshold voltage VT -- So vGS, the voltage applied", + "here is greater than some voltage, VT, a threshold", + "voltage, or the pressure of the switch is greater than some", + "threshold pressure then this guy behaves like a short circuit.", + "This is iDS, this is my drain and this is my", + "source. So if the voltage applied", + "between the gate and the source is higher than some threshold", + "then this behaves like a short circuit.", + "Similarly, if the voltage vGS is less than some threshold VT", + "then in that situation --", + "", + "-- I get an open circuit. And when I have an open circuit", + "between D and S then the current iDS is going to be 0.", + "So this is the idealized model. And this idealized model is", + "called \"the switch model of the MOSFET\".", + "The switch model or the S model of the MOSFET.", + "Well, if you want to see the internals of the MOSFET,", + "I won't cover that in lecture or recitation.", + "You can look at the section, I believe Section 6.7 of the", + "course notes. That has the internal structure", + "of the MOSFET and how you physically construct such a", + "device. So what I can do here is step", + "back and stare at the device for a second or two.", + "And what it says is that if I apply a lot of pressure,", + "if vGS is greater than a threshold VT then I get a short", + "circuit here just like my switch.", + "When in doubt think faucet. If you put pressure on the", + "faucet, think of this as closing, and when I open it,", + "when vGS goes less than VD, less than a threshold,", + "I take off the pressure and then it becomes an open circuit.", + "So I can plot the following.", + "", + "Much like I plotted the iV characteristics of two terminal", + "elements, I can plot the iV characteristics of this three", + "terminal element in the following way.", + "I can focus on two terminals and look at vDS and iDS for that", + "terminal pair and draw the curves for how it will behave as", + "I change vGS that I applied. So what I'm going to show you", + "is that if vGS is less than a threshold then this behaves like", + "a open circuit. So no matter what the voltage", + "is the current is 0. Similarly, if vGS greater than", + "equal to some threshold voltage then I get the behavior iV curve", + "of a short circuit where the current can be anything and", + "controlled by external forces like in any short circuit.", + "So let me show you on the screen.", + "Lorenzo has kindly put the graph up already.", + "So I'm showing the iV curve of a switch.", + "Notice that when vGS is greater than VT, greater than a", + "threshold I get the vertical line corresponding to a short", + "circuit. Is it this one?", + "This one. There we go.", + "So what I'm going to do here is I'm going to reduce vGS to below", + "VT. What should you see happening?", + "The curve, from being a short circuit, should hammer down to", + "becoming an open circuit. That's the curve for an open", + "circuit as I drew out there for you.", + "VGS pressure ain't enough. Lots of pressure,", + "boom, it's a short circuit. I really like to think of this", + "pressure analogy if I get confused whenever I look at a", + "MOS transistor and I need to look at vGS and so on I always", + "think vGS is greater than VT. Lots of pressure on the switch", + "it turns on. Just remember that,", + "and then you won't forget this vGS thing here.", + "So that's the behavior of a switch.", + "And so viola, there's our switch.", + "", + "So I've given you a three terminal element that is a", + "switch that is controlled like a mechanical switch.", + "So I can build a, if I replace --", + "", + "This was my switch earlier. And what I can do is replace", + "this with my MOSFET and that's what I get.", + "And I won't bother showing you this is your inverter.", + "All of that has replaced the abstract switch with a physical", + "switch which behaves as shown in the graph up there.", + "And so I apply an input here and I take the output here.", + "So as 6.002 you could look at this and say ah-ha,", + "that is an inverter. When you go to 004 what you", + "will do is build this triangle and a circle around it and you", + "will ignore what's inside and just look at that.", + "So in 002 we showed you that the internals look like a", + "pattern with a MOSFET and a resistor, but it's really the", + "abstract inverter looking in from the outside.", + "I'm just going to close the loop inside the digital gate,", + "and this was inside your little inverter with a resistor and a", + "switch. Let me continue with this for a", + "little longer here --", + "", + "-- and do something that we like to do a lot,", + "which is plot what are called input / output curves.", + "So let's say the voltage applied here is v in and let's", + "call this v out. For fun let's plot a v in", + "versus v out for this inverter. So when input is a 0,", + "let's say VT is 1 volt for the inverter.", + "The threshold voltage is 1 volt.", + "The threshold pressure is 1 volt.", + "So when input is a 0, and let's say VS is 5 volts.", + "So when the input is a 0, this guy is turned off.", + "So what's the output? What's the output voltage?", + "If this is turned off, what's the output voltage?", + "It's the supply. The supply directly shows up", + "here. And so as long as the input is", + "0 the output is at 5 volts. And this is true until the", + "input reaches 1 volt. As long as the input is less", + "than 1 volt my output stays high.", + "And then when my input exceeds or hits 1 volt then at that", + "point the switch turns on and the MOSFET turns on and shorts", + "the output to ground in which case boom, this is what I get.", + "And then, no matter how much I increase the input,", + "my switch stays on and the output follows a zero volts at", + "the output. So this is my v in versus v out", + "curve for the inverter. One of the interesting things", + "that we do a lot is see whether this satisfies some voltage", + "threshold. So let's say I have a VOL of", + "0.5 volts, VOH of 4.5, VIL of 0.9 and VIH of 4.1", + "volts. So VOL says in its low value is", + "the output less than 0.5? Yup, output less than 0.5.", + "In its high is it more than 4.5?", + "Yup, it's more than 4.5. Does it recognize all values", + "below VIL as a low input? Yup.", + "So anything below 0.9 or 1 for that matter is viewed as a low.", + "That's good. So these pass.", + "And high, anything above 4.1, is that treated as a high?", + "Yes. So anything above 4.1 is", + "treated as a high and the output goes low.", + "So therefore this inverter that I've designed for you here", + "satisfies the static discipline and this inverter can be used in", + "circuits or other devices that conform to this value here.", + "In your recitation, you will look at a slightly", + "more detailed model of the switch where the switch behaves", + "like a resistor." + ] +} \ No newline at end of file diff --git a/subs/wNuBD4PYWvs.srt.sjson b/subs/wNuBD4PYWvs.srt.sjson new file mode 100644 index 0000000000..479fc4b80b --- /dev/null +++ b/subs/wNuBD4PYWvs.srt.sjson @@ -0,0 +1,2168 @@ +{ + "start": [ + 0, + 4137, + 15000, + 24000, + 28142, + 32865, + 36511, + 39991, + 44051, + 48028, + 53000, + 57514, + 60548, + 64840, + 68170, + 71426, + 74829, + 78012, + 81712, + 85560, + 90000, + 95000, + 99992, + 103715, + 106000, + 111000, + 115016, + 118351, + 122979, + 126393, + 128820, + 131400, + 136027, + 139213, + 143310, + 147406, + 151553, + 156750, + 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OK.", + "The topic for today is Energy and Power.", + "", + "Most of the time this semester, up to now at least,", + "we focused a lot on speed. We have been truly speed freaks", + "looking at how fast can we switch the signal,", + "what does a time domain waveform look like?", + "We also looked at frequency responses of circuits.", + "This week we will spend on something a little bit", + "different, and that relates to energy and power.", + "Energy and power is gaining a lot more importance in certainly", + "this decade, and will do so in the future.", + "And I am going to work out a little example towards the end", + "of the lecture. And there you will see that if", + "you do things naively, your handheld devices,", + "your cell phone, laptops and so on will just up", + "and explode. You have got to be a little bit", + "careful in terms of how to manage energy and power.", + "Before I get into that, I just want to wrap up with a", + "quick review of what we covered last week.", + "", + "We ended last week by looking at positive feedback in analog", + "circuits using an op amp. And, in particular,", + "we built an oscillator.", + "", + "We built an oscillator that allowed us to charge a", + "capacitor. And when the voltage across the", + "capacitor equaled that at the minus terminal it would flip and", + "keep doing so. And at the output you would get", + "a waveform that looked like this.", + "You would get a square wave output.", + "Now, throughout the course we have talked about getting square", + "wave inputs. And this is one example of how", + "you can actually produce a square wave pretty much from", + "first principles using a capacitor, resistors and an op", + "amp. Now, I just wanted to wrap up", + "this little item here by talking about one application of an", + "oscillator. And this application of the", + "oscillator really nicely closed the loop on the body of", + "knowledge relating to digital circuits that we have covered in", + "this course. What I want to talk about", + "briefly is a small digital system with a sender and a", + "receiver. And the sender is sending a", + "signal, the receiver receives a signal, and in this course we", + "have talked about senders sending a sequence of ones and", + "zeros. Say, for example,", + "the sender wants to send some sort of a signal like this.", + "We have seen that this is quite a legitimate signal.", + "We get some kind of oscillatory behavior because of the", + "inductance and capacitance associated with the wire.", + "And what you have done is that you pretty much believed me when", + "I said that this really corresponds to a one,", + "one, zero. The sender wants to send a one,", + "one, zero signal to the receiver, and the receiver gets", + "it. So this is a one,", + "this is a one, this is a zero.", + "But if I am a receiver, I am going to look at the", + "square wave. There is no such thing as", + "sending a one on a wire. You cannot send a one on a", + "wire. You send a voltage signal or a", + "current signal on a wire. So, the receiver receives a", + "voltage signal. It is going to be zero for some", + "time and then maybe 5 volts or 3 volts or whatever is your high", + "and then zero again. How does my receiver know it's", + "a one, one, zero? Why can't it be a one,", + "one, one, one, zero, zero, zero?", + "It doesn't know. How does the receiver know it's", + "a one, one, zero sequence and not 50 ones followed by 40", + "zeros? It doesn't know.", + "What we need is -- For senders to be able to", + "communicate with receivers, we need some kind of agreed", + "upon time when receivers sample the signal coming in and decide", + "whether it's a one or a zero. They both have to agree on", + "certain time bases when to look at the input.", + "One way to deal with this is I can have a clock,", + "a square wave signal that we call a clock in digital systems", + "and ship it to the other side in the following manner.", + "", + "This clock signal can be applied to this sender and to", + "this receiver. For more details on this,", + "let me recommend Page 735 of the course notes that talks", + "about a detailed example of the use of a clock in a digital", + "system. What I can do is create a clock", + "that looks like the square wave. The clock provides a notion of", + "time to the circuit. And I have some kind of a clock", + "signal generator. And I connect that to the", + "sender and connect that to the receiver.", + "And now both the receiver and the sender have a notion of", + "time. And what I can do is I can tell", + "my receiver, the sender and the receiver can have an agreement", + "between them that says that look at the signal at your input when", + "on the rising edge of the clock. Whenever the clock rises,", + "when you see a rising edge look at the value in the wire and", + "that's the value I sent. By doing so,", + "what I can do is that the receiver can look at the signal.", + "At this rising edge it sees a zero, this is vOH,", + "looks up here, sees a one here,", + "sees a one here and sees a zero.", + "So, it correctly sampled one, one, zero at the receiving end.", + "And the sender can send the same sequence here once we have", + "this time base. This little brief foray", + "circuits was simply to give you an application of a circuit that", + "can produce a square wave. I can create a clock with a", + "time base. Also, interestingly much more", + "fundamental is we looked at various abstractions throughout", + "the course. We talked about discretizing", + "space by looking at lumped signals.", + "What I also want to point out is that a clock can be viewed as", + "another fundamental abstraction in the digital domain where what", + "I am doing is discretizing time. What I am saying is that,", + "look, in the digital domain we have already discretized value", + "into zeros and ones, but we still had continuous", + "time until now. And what you do in digital", + "systems is to say that look, let's digitize everything,", + "or rather discretize everything.", + "And let's discretize time as well into these points that", + "happen on the rising edge of the clock which means that the", + "circuit has meaning, signals have meaning only when", + "the clock is rising. That tends to discretize time", + "which means that I really don't care what happens to signals in", + "this time, as long as on the rising edge of the clock I get", + "the right value. This concept is called", + "discretizing time. And a clock lets you do that.", + "Remember that in digital systems, which you will learn", + "about in 004 I am really discretizing two things,", + "discretizing values into zeros and ones, and at the same time", + "also discretizing time into a time when I sample things and a", + "time when I ignore values on the wires.", + "I think you will get to clocks in 6.004 after about a month,", + "so initially you would just be focusing on the statics of the", + "system without worrying about any dynamic clock introduced in", + "the circuit. OK, that's just a brief little", + "interlude. With that let me get into", + "today's topic of energy and power.", + "Why is this important? The reason this is important is", + "that what really determines the size of your handheld?", + "You may think oh, gee, electronics in the", + "handheld. Some of you may think,", + "oh, the antenna in the handheld.", + "No. What really,", + "really determines the size and weight of your handheld devices,", + "your PDAs, your cell phones, your laptops and so on is by", + "and large the battery. On Page 2 I have a little", + "cartoon that shows you that if we did not have you learn about", + "energy and power, that's what we would all be", + "doing in order to use cell phones.", + "Not surprisingly the very first wireless phones ended up in", + "automobiles because you had a big battery.", + "And so you had these wireless phones only in cars.", + "Because of a huge amount of research based on the knowledge,", + "the technologies I am going to talk about in today's lecture", + "and Thursday's lecture, you will see very simple and", + "elegant ways of reducing the amount of battery you need to be", + "able to get some kind of function out of analog or", + "digital devices. I also want you to take a look", + "at Page 2 of the handout that I have given you here,", + "handout 63. This handout talks about the", + "absolute latest in digital fabrication technology out", + "there. This is not a paid commercial", + "for IBM. IBM has a technology called", + "CU08. It is called Blue Logic.", + "It is called the Copper 08 Process.", + "And in this process, if you look down on Page 1,", + "for example, IBM claims that it can build up", + "to 72 million gates in a single chip.", + "With this technology they are able to build 70 to 80 million", + "gates where a gate is, unless otherwise mentioned,", + "pretty much defined as a two input NAND gate equivalent.", + "So, your inverter, your NAND gate and so on count", + "as a gate. And they can build close to 80", + "million of these little suckers on a single chip.", + "Just imagine that. And the biggest chip they can", + "build is on the order of 18 to 19 millimeters on a side,", + "roughly two centimeters on a side.", + "On a chip that's about one square inch.", + "You can put down 80 million gates.", + "What is more important for today is what is on Page 2,", + "actually. I have circled two things on", + "Page 2. One thing that I have circled", + "is power supply range in the 0.7 to 1.3 volts.", + "Notice that that voltage, the power supply voltage for", + "these chips is significantly lower than the 5 volts that we", + "have been normally talking about in this course.", + "When in doubt our problems have used 5 volts.", + "But notice that in this technology they're talking about", + "using voltages for the power supply VS in the range of 0.7 to", + "1.3. Why is it so much lower?", + "Well, you will find out. The second thing I've circled", + "is something called power dissipation.", + "And you say power dissipation is said to be 0.006 microwatts", + "per megahertz per gate. It says power dissipation is 6", + "nanowatts per megahertz per gate.", + "What that says is that each gate off your circuit will", + "dissipate this much power at a 1 megahertz frequency.", + "And the implication of that is that you should be able to", + "convert that single number to the power dissipation in any", + "chip that you might build depending on the number of gates", + "that you have, the frequency you run the", + "circuit at, the voltage that you use and so on and so forth.", + "By the end of today's lecture you will be able to take this", + "number and correlate that into the power dissipation of any", + "chip that you might want to build with this.", + "That just serves as the motivation that by the end of", + "this lecture you will understand how to very quickly in five", + "seconds or less, boom, given a chip,", + "oh, yeah, that should consume about 30 watts of power.", + "And what you will also do, based on some examples here,", + "estimate the power of not the Pentium IV but a chip following", + "the Pentium VI, let's call it the Pentium V", + "would consume if it ran at 1 gigahertz.", + "We will come up with some absolutely shocking numbers", + "based on what you have learned. With that kind of motivation", + "let me get into talking about some theory and get into the", + "foundations of energy and power.", + "", + "Let's go to Page 3.", + "", + "To drive the theoretical discussion, I would like to", + "focus on the energy dissipated in a MOSFET gate.", + "And fundamentally we will talk about looking at energy and", + "power in circuits containing switches, resistors and", + "capacitors. The MOSFET gate is simply an", + "illustrative example to drive the theory.", + "But fundamentally what I am going to show you,", + "or lead you through today, I will tell you how to compute", + "the power and energy when you have capacitors,", + "resistors, voltage sources and switches in your circuit.", + "We will look at a circuit that looks like this.", + "Your vanilla inverter circuit.", + "", + "My inverter. I apply some vIN signal here.", + "It could be a square wave. It could be some sequence of", + "ones and zeros. And this is an inverter that we", + "all know and love. And this guy here is,", + "stuck in a capacitor here. And this capacitor is meant to", + "model the input gate capacitance of whatever this inverter drives", + "plus any capacitance of the wire leading up to that gate and so", + "on. It is just a lumped capacitor", + "that I have stuck on there. I am interested in determining", + "a few things. One is what we call the standby", + "power. You will see all these terms", + "being used in cell phones and so on.", + "In your cell phone, your cell phone manufacturer", + "gives you two numbers. Of course both are over", + "exaggerations, but they give you two numbers", + "nonetheless. One number is the number of", + "days that the cell phone battery will last when in standby mode,", + "right? That's exactly where standby", + "comes from. In standby mode,", + "how much power does your cell phone or how long will the", + "battery last, that's the standby power.", + "And the second thing is what we call active use power.", + "Active use is when you are making a phone call and so on,", + "what is the power consumed? And there again your", + "manufacturer of your cell phone will give you a much smaller", + "number for the active use power of your cell phone.", + "What I am going to do is assume for discussion that the inverter", + "is driven by a square wave signal of the following sort.", + "This is vIN. And I am going to drive this", + "with a signal of this sort. The period applied at the", + "input, so I am switching the inverter on and off,", + "on and off, on and off. And T1 seconds for the high,", + "T2 seconds for the low. This is the inverter,", + "this is the input signal, and we'll keep coming back to", + "that again and again. Rather than directly taking", + "this circuit and analyzing its power, I would like to do things", + "in a slightly roundabout manner. What I would like to do is show", + "you some very simple circuits and analyze their standby and", + "active powers. And then show you that this", + "circuit simply is a combination of some of the simple things", + "that you have seen. Example 1.", + "I would like to take a simple circuit that looks like this.", + "A voltage source V applied across a resistor R,", + "some current I. And if I apply a voltage across", + "this resistor, that voltage would simply", + "appear across the resistor. And the power is simply given", + "by VI which is simply V squared divided by R.", + "This is 6.002 101 in the very first chapter.", + "That is the power that is dissipated by this resistor,", + "simply V squared divided by R. That's the power dissipated by", + "the resistor. Where does that power come", + "from? The voltage source supplies the", + "power. So, this guy here supplies this", + "power and this guy here dissipates it.", + "What is the energy that I dissipate in T time?", + "Remember, power is the rate of energy dissipation.", + "And so energy is simply power multiplied by time.", + "For a circuit like this, energy dissipated in time T is", + "simply VIT. For our gate remember we have", + "two situations. We have VS, we have RL we have", + "RON, vO and vIN. So, vIN is high.", + "If vIN is high with respect to ground then RON,", + "the switch is on, and this is the circuit that I", + "see. In this situation the power", + "consumed is simply V squared divided by the resistance here.", + "It is simply VS squared divided by RL plus RON.", + "Let me mark that with an asterisk.", + "I will refer to this later. Similarly, when vIN is low the", + "MOSFET is off. And the power is simply zero.", + "I have no current flowing down and the power is zero.", + "Absolutely basic stuff. Absolutely basic.", + "So, the power, when I have the MOSFET on,", + "for the kind of inverters you have seen so far,", + "this is the power consumed by the inverter.", + "And this asterisk here is simply to say hold that thought,", + "we will get back to it a little later.", + "Let me work out a second example.", + "In this second example, I would like to consider the", + "following circuit, a voltage source VS with a", + "strange arrangement of switches, S1 with a resistance R1,", + "a capacitor C in this manner, a switch S2 and a resistor R2.", + "For now don't worry about how the circuit comes about.", + "Just assume that I have drawn the circuit for you.", + "And what I want to do is compute the power under certain", + "conditions. Notice that if this is off and", + "this is off, there is no current flowing either in this loop or", + "this loop, and the power dissipated by the circuit is", + "zero. But there are some arrangement", + "of switches for which I do consume power.", + "And so let me show you that arrangement of switches.", + "And what I am going to do is assume that the switches open", + "and close with the following periodic cycles.", + "Let's assume that when this is high S1 is closed and S2 is", + "open, and when this is low assume that S1 is open,", + "S2 is closed. And let's assume this is T,", + "this is T1, this is T2. That sequence should be", + "reminiscent of this input that I am feeding to this inverter.", + "All I am telling you here is that I am giving you the", + "circuit. I want to compute the power", + "consumption of the circuit. And what I am telling you is", + "that with the frequency, with a time period of capital", + "T, for the first T1 seconds this switch is closed and that is", + "open. So, this circuit applies.", + "In the second half of the clock this switch is open so this", + "circuit applies. And what I am interested in", + "finding out is what is the energy dissipated in each cycle", + "of time capital T? And I also want to find out the", + "average power. Just spend about five seconds", + "just staring at this and kind of intuit what is going on here.", + "I start by putting a voltage source here and I close the", + "switch. That is open.", + "Start by closing this, what happens?", + "When I close the switch VS is going to charge up this", + "capacitor. I get current flowing through", + "my resistor, so I am going to be charging up this capacitor here.", + "Then let's say I allow T1 to be as large as possible,", + "and so this capacitor is going to be charged up to all of VS.", + "After a long time this guy gets to be VS in the capacitor.", + "And as it is charging up I have current flow through the", + "resistor, so it is sitting there dissipating power.", + "Notice that this sucker does not dissipate energy.", + "It simply stores energy. So, the energy supplied by the", + "voltage source comes in, some of it gets stored in the", + "capacitor and some of it is being dissipated by the", + "resistor. That gets me to the end of T1.", + "At the end of T2 I open the switch and close this switch.", + "When I close the switch I have some energy on the capacitor,", + "and the voltage across the capacitor begins to drive a", + "current through this resistor R2.", + "And now the capacitor supplies its stored energy,", + "and its stored energy then begins to dissipate through", + "resistor R2. And if T2 is very long then all", + "the charge in the capacitor drains out.", + "And the voltage in the capacitor at the end will be", + "zero. So, that is just sort of a high", + "level description of what goes on.", + "Now let's go ahead and compute from first principles the", + "energetics of this little circuit.", + "Let's look at the entire period capital T, and as a first step", + "look at T1. When T1 is in place S1 is", + "closed and S2 is open. Accordingly,", + "the circuit that applies looks like this.", + "I have VS, S1 is closed, so that is closed,", + "and I have this resistance R1, I have this capacitance C,", + "some voltage VC across the capacitor.", + "You can go ahead and assume that VC of zero is zero.", + "That I start off my life with no voltage across the capacitor.", + "First of all, let me plot the waveforms and", + "write the expressions down and then compute the energy supplied", + "by the voltage source and then look at where the energy goes.", + "You all know, or should know by now,", + "if I plot VC as a function of time, remember,", + "this is really easy to do. VC as a function of time goes", + "like this. At time T equal to zero I am", + "telling you that the capacitor voltage is zero.", + "I am telling you that. So, it is at zero.", + "And then the capacitor charges up until it reaches VS.", + "I also know that after a long time this will be VS,", + "after a long time that will be VS, and between those two I have", + "a rising function that looks like this.", + "I can similarly plot the current for you.", + "At time T equal to zero instantaneously the capacitor", + "looks like a short, and so the current that I start", + "off with is going to be VS divided by R1.", + "The voltage across the capacitor is zero.", + "All the voltage falls across the resistor R1.", + "So, VS divided by R1 is the initial instantaneous current.", + "And after a long time, because VC reaches VS,", + "the current is going to be zero.", + "And between those two points I get an exponential decay.", + "I could very quickly write down the expression for the current.", + "And that is simply the initial value VS divided by R1 times the", + "exponential decay minus T divided by the time constant for", + "the circuit R1C. You have seen this stuff", + "before. Here comes the part that we", + "care about for now. Let's find out what is the", + "total energy provided by the source.", + "When dealing with energy computations you have to be", + "incredibly careful of these words here, supply,", + "provided versus dissipated. Dissipated implies that the", + "resistor is burning energy. Provided means that the source", + "is supplying that energy. So, energy provided by source", + "during T1. Let's go ahead and compute that", + "very quickly. The energy supplied by the", + "source is simply the voltage across the source multiplied by", + "the current being supplied by the source.", + "This is i. Remember, by associated", + "variables convention, if I have a voltage across some", + "element and the current into the element is positive then that", + "element dissipates power. If the voltage here is,", + "say, 1 volt and it is supplying current, if the i is out in the", + "other direction then it is supplying power.", + "In this case, the current i is going to be on", + "the outside, heading outside. The total energy is going to be", + "the instantaneous power integrated over time,", + "and that is simply VS. Remember, the instantaneous", + "power is VS times the current i, so the instantaneous power is", + "simply VS times i, that is the instantaneous", + "power. To get the energy provided by", + "source and some time, I have to integrate that", + "instantaneous power over the period of interest T1.", + "That gives me the energy supplied by the source during", + "T1. And let me go ahead and", + "substitute for i with this expression here.", + "It is VS times i, and i is VS divided by R1 times", + "this expression here. That gives me", + "(VS^2/R1)e^(-t/R1C) dt. Let me carry out the", + "integration there. I get -1/RC,", + "so I get this outside. And I also get to write down,", + "oops, let me do that a little bit more carefully.", + "VS^2/R1 simply comes out and I get a -R1C in the numerator.", + "If I differentiate it then I get R1C in the denominator.", + "I have an integral that comes up here.", + "And then I write down e^-t/R1C, zero and T1.", + "So, this R1 and this R1 cancel out.", + "And I end up getting something that looks like this.", + "I get CVS^2. And so there is a minus sign", + "out here, so at zero this thing goes to a one,", + "so I get a one. And because of minus sign I get", + "e to the -T1/R1C. All I have done here is simply", + "go through the math to do this integration here.", + "What I am also going to do is assume that if T1,", + "if the time that the switch is closed is much,", + "much bigger than the time constant of the circuit,", + "T1 is much, much greater than R1C, if this is much,", + "much greater than R1C then this term goes to zero.", + "And this becomes more or less equal to CVS^2.", + "What do we have here? What we have here is that if I", + "let the switch stay closed for a long time and S to be open then", + "the voltage source is going to supply some amount of energy.", + "That energy will equal CVS^2. The voltage across the", + "capacitor will be VS and all that energy would have been", + "supplied by this guy. Let me pose the following", + "conundrum here. If the voltage across the", + "capacitor is VS, because we know the energy", + "stored in the capacitor is half CV^2.", + "So, the energy in the capacitor is half CVS^2.", + "At the end of the day, since the voltage across the", + "capacitor is VS, \u00bdCV@2 is the energy stored", + "here. But we know,", + "from this calculation, the source has supplied CVS^2.", + "Source has supplied twice that energy.", + "This guy has supplied twice that energy and only half of", + "that is stored here. Who ate up the other half?", + "The resistor, exactly.", + "The resistor has walloped half the energy.", + "Let me just show it to you. It dissipated \u00bdCVS^2.", + "It's pretty interesting. It's a pretty simple result.", + "If T1 is very large compared to time constant then half the", + "energy is in the capacitor and half of it has been burned by", + "R1. This energy has not been", + "burned. It is simply stored.", + "It is stored by the capacitor.", + "", + "And if you do simple energy conservation arithmetic here,", + "the energy dissipated in the resistor plus that stored in the", + "capacitor equals the energy supplied by the source.", + "All right. Let's go to T2 now.", + "At T2, S2 is closed and S1 is open.", + "Let's look at the second part of the cycle when S1 is open and", + "S2 is closed. And what is going to happen now", + "is the left-hand part of the circuit can be ignored and I can", + "focus on this part. So, S2 is closed.", + "This is RC, my capacitor, this is vC.", + "This is the circuit of interest.", + "What is the initial condition on this?", + "What is the value of vC initially?", + "Start off, because remember, I allowed this capacity to", + "charge up fully, and so initially I have VS on", + "the capacitor. And so the energy on the", + "capacitor initially is \u00bdCVS^2. That is the energy on the", + "capacitor. This time around I won't go", + "through an integration process like that, but you can if you", + "like, and do it in a much similar manner to say that now", + "let's suppose that T2 is much greater than this time constant.", + "If T2 is much greater than R2C, this time constant.", + "If that time is much greater than this entire,", + "the initial voltage VS drives a current through the resistor,", + "and after some amount of time the voltage across the capacitor", + "goes to zero and all the energy in the capacitor gets dissipated", + "in R. So, if T2 is much greater than", + "R2C then energy dissipated in R2 is simply \u00bdCVS^2.", + "Notice that the energy dissipated in R1,", + "in the first half cycle is \u00bdCVS^2 and the second half cycle", + "during T2, if T2 is large enough, all this energy gets", + "dissipated in this resistor R2. And I have that expression", + "here.", + "", + "So let me just say that this is E1 and let me say that this is", + "E2. So, E1 is dissipated in the", + "resistor and E2 is dissipated in R2 in the second half cycle.", + "A couple of interesting things to note at this point.", + "One is that E1 and E2 are independent of R.", + "If the time constant is small enough compared to the time that", + "I charge the capacitor then half the energy gets lots in the", + "resistor, and that is simply \u00bdCVS^2.", + "And if I let this discharge completely it doesn't matter", + "what resistor I am discharging it through.", + "That's the intuition. If I have certain energy here", + "and I let it discharge completely it doesn't matter", + "what this resistor is. Small or large,", + "it doesn't matter. All this energy gets dissipated", + "there. The rate at which the energy", + "gets dissipated will change depending on R2.", + "If R2 is very small then I get a burst of power initially and", + "then a rapid decay after that, but if R2 is very large then I", + "have a much slower release of energy.", + "But suffice it to say that the energy dissipated,", + "the total energy in T2 is simply \u00bdCVS^2.", + "All right.", + "", + "Let's put T1 and T2 together and look at the total energy", + "dissipated --", + "", + "Total energy dissipated. E is simply E1 plus E2.", + "Dissipated in each cycle. Assuming T1 and T2 are much", + "larger than the respective time constants.", + "And I know that this is \u00bdCVS^2, \u00bdCVS^2, so this is simply", + "CVS^2. If I have an arrangement of", + "switches and capacitors like that, I charge the capacitor,", + "discharge the capacitor, charge the capacitor,", + "discharge the capacitor. What it is saying is that in a", + "charge/discharge cycle I am using up CVS^2 of energy.", + "\u00bdCVS^2 when I charge it up and \u00bdCVS^2 when I discharge it.", + "That is what I get. Let's compute the average power", + "dissipated, P average in a cycle is simply E/T where T is the", + "period of the square wave sequence that I have shown you", + "out there. This is simply CVS^2 divided by", + "T. If the period of the square", + "wave is capital T, I can express that as a", + "frequency. Let's say for example the", + "period of the square wave is T, so let's say the frequency of", + "the square wave is simply 1/T. I can also express this as", + "C(VS^2)f.", + "", + "What does this say? Let me mark that as a thing to", + "remember, the second thing to remember.", + "One was the power that was the static power.", + "And second is this power relating to this frequency f and", + "the charging and discharging of the capacitor in that little", + "circuit shown up there. So, this average power is", + "CVS^2f. What this is saying is that if", + "f is high, if I have high frequency of charging and", + "discharging the capacitor then I am charging and discharging much", + "more frequently so I am going to consume more power.", + "Notice that at any given time there is no direct connection", + "between the power supply and the ground.", + "What I am doing is my capacitor is an intermediary.", + "I am dumping some charge in the capacitor and the capacitor is", + "dumping the charge into ground. It behaves like a switch to", + "capacitor. And what it is doing is it is", + "being charged and discharged at frequency f.", + "So, it makes sense that the amount of average current that I", + "am pumping through relates to the frequency at which I am", + "charging and discharging the capacitor.", + "And similarly the average power also relates to the value of the", + "capacitor. If C is larger I dissipate more", + "energy. And the same way with the", + "voltage. If the voltage is higher then", + "the power in that period, or the average power relates to", + "CVS^2. Spend a few seconds staring at", + "the two expressions. This power here relating to", + "just this connection between the power supply and ground and that", + "power out there relating to charging and discharging", + "capacitors. Let's get back to our inverter", + "right now.", + "", + "This is our inverter circuit. Let us say that I drive the", + "input with the waveform shown here.", + "Well, I go back to the same situation as here.", + "I drive the input with a square wave, with T1 and T2 as the high", + "time and the low time. The equivalent circuit for this", + "is not exactly what we saw there.", + "The equivalent circuit for this would look like this.", + "I have a VS. And the VS supply is connected", + "through RL, VS connected through RL to a capacitor C.", + "This is my voltage vO. So, VS is always connected to", + "ground through this resistor and capacitor in this manner.", + "And then I have a resistor here RON corresponding to that", + "MOSFET. And there I am switching it on", + "and off in a way that it is on during T1 and off during T2.", + "So, the situation here is a bit different from that simple", + "situation I computed there. Much like I computed the power", + "dissipation in that circuit, I can go ahead and compute the", + "total power dissipated in this circuit.", + "I won't do it here. The algebra tends to be a big", + "more grubbier than what I have been through.", + "And suffice it to say that you can show that the average power", + "is given by (VS^2)/(2(RL+RON))+(CVS^2)f", + "(RL^2)/(RL+RON)^2.", + "", + "OK? And for details I suggest that", + "you look at section 12.3 of the course notes.", + "Section 12.3 goes through the algebra to compute the total", + "power dissipated by this specific circuit,", + "and here is the expression we get.", + "And let's take the specific situation where RL is much", + "greater than RON. If RL is much greater than RON", + "then I can ignore this RON here.", + "", + "And I get this. And out here,", + "if I ignore RON, then RL and RL will cancel out", + "and I get CVS^2f. If I ignore RON compared to RL", + "this is the expression I get. Now you can see why I went", + "through those two examples. This is exactly the power", + "consumed by the connection between power supply and ground.", + "And this CVS^2f is the power consumed in charging and", + "discharging the capacitor. If you look at the circuit here", + "it is consuming two kinds of power.", + "One kind of power is due to the current flowing directly from VS", + "through RL and RON to ground. Oh, this also assumes,", + "by the way, that T1 is equal to T2.", + "", + "So, in this circuit there are two kinds of power.", + "One is the power when the switch is on and I have a", + "current flowing from VS to RL to ground.", + "Notice I get an extra factor of two in the denominator here.", + "And that two comes about because the connection to ground", + "only happens half the time. It's half that power out there", + "because I am connected to ground only when the switch is on.", + "And that happens only half the time, and so therefore I get the", + "VS^2/2RL. And then CVS^2f is simply the", + "power that I consumed because I am charging and discharging the", + "capacitor C. Notice that in this inverter", + "circuit there are two kinds of power.", + "One is called the standby power which is static power being", + "consumed by the circuit, and the second power is the", + "dynamic power because the circuit is switching up and", + "down. This relates to star and this", + "relates to the double star. And to demonstrate that,", + "I have a little demonstration here that has an inverter.", + "And I am going to up the frequency of the square way of", + "driving the inverter. I am going to show you a few", + "numbers so hang on for two minutes after this demo.", + "I will give you some numbers, but I want you to go ahead and", + "compute the numbers based on what we have seen here.", + "And you will get suitably impressed, I promise you.", + "This is the input fed to the inverter.", + "This is the output of the inverter.", + "Notice that the output of the inverter reflects some sort of", + "an RC time constant because of the output driving the", + "capacitor, and the same way here.", + "I start off by showing you that on the left-hand side I am", + "simply measuring the power being consumed by the circuit.", + "Notice that the power being consumed is expressed by the", + "needle being at this point here. This is a very low frequency so", + "this is almost all standby power consumed by the inverter.", + "The inverter is on half the time, and when it is on it is", + "consuming power. What I am going to now is", + "increase the frequency. As I increase the frequency", + "driving the inverter what should happen to this needle?", + "As I increase the frequency there that waveform should", + "become closer and closer together.", + "And what should happen to the needle?", + "That should begin to go up. If I increase the frequency it", + "should consume more and more power and the needle should", + "start going up. So, let me do that for you.", + "In terms of numbers there it is on top of the four on the scale", + "in the middle. I am going to increase the", + "frequency very slowly. Unfortunately,", + "the sampling scope messes up the waveform.", + "Ignore the waveform for now. Just look at the meter as I", + "increase the frequency.", + "", + "Notice that I have increased the frequency by about a factor", + "of 2 or 3. And notice here that this meter", + "has moved. The needle has moved to the", + "right. And I can keep doing that and", + "the needle keeps moving to the right as I am consuming more and", + "more power because I'm driving the inverter faster and faster", + "and faster. That should convince you that", + "there is a standby power and there is some power component", + "related to frequency. This relates to your standby", + "power in your cell phone. This relates to active use.", + "Let me show you some numbers, and you can plug those numbers", + "in yourself and see how much power this converter is going to", + "consume and see if it makes sense.", + "Assume that I have a chip with 10^8 gates.", + "F is 1 gigahertz. That is 10^9.", + "Assume C is 0.1 femtofarads which is 10^-16 farads.", + "Assume VS is 5 volts. Assume RL is 10 kilo ohms.", + "Use these numbers. Plug these numbers in here and", + "get a sense if our modern-day circuitry used that inverter,", + "what would be the power consumed by a chip that contains", + "10^8 of these gates? You will find out that you may", + "have to use a nuclear power reactor to actually drive that", + "chip, but go check it out for yourselves.", + "In the next lecture we will see then how do our cell phones", + "work, how does life go on despite this horrendous", + "calculation here." + ] +} \ No newline at end of file diff --git a/subs/ypX20WnHNQw.srt.sjson b/subs/ypX20WnHNQw.srt.sjson new file mode 100644 index 0000000000..6aa9eef172 --- /dev/null +++ b/subs/ypX20WnHNQw.srt.sjson @@ -0,0 +1,2141 @@ +{ + "start": [ + 0, + 3552, + 8341, + 16837, + 23016, + 31512, + 40933, + 46380, + 51304, + 57380, + 62166, + 66416, + 71250, + 74500, + 79416, + 82333, + 87000, + 96000, + 98622, + 103390, + 107761, + 112450, + 115549, + 120000, + 123033, + 126257, + 128911, + 132451, + 134916, + 137318, + 140731, + 144207, + 148000, + 151410, + 156372, + 158232, + 161100, + 165829, + 168465, + 172496, + 178000, + 186000, + 190600, + 195546, + 201527, + 206127, + 212223, + 218319, + 224545, + 229818, + 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Good morning,", + "all. So we take another big step", + "forward today and get onto a new plane of understanding,", + "if you will. In the last week and a half,", + "our focus was on the storage element or storage elements", + "called inductors and capacitors. And capacitors stored change", + "and inductors essentially stored energy in the field,", + "the magnetic flux. And the state variable for an", + "inductor was the current while that for a capacitor was the", + "capacitor voltage. We also looked at circuits", + "containing a single storage element, we looked at RC", + "circuits and we also looked at circuits containing a single", + "inductor. And this was a single inductor", + "with a resistor and a current source or a voltage source and", + "so on. What we are going to do today", + "is do what are called \"second-order systems\".", + "", + "So they are on the next plane now.", + "And with this second-order of systems, they are characterized", + "by circuits containing two independent storage elements.", + "They could be an inductor and a capacitor or two independent", + "capacitors. And you will see towards the", + "end what I mean by two independent capacitors.", + "If I have two capacitors in parallel, they can be", + "represented as a single equivalent capacitor so that", + "doesn't count. It has to be two independent", + "energy storage elements and resistors and voltage sources", + "and so on. And what we end up getting is", + "what is called \"second-order dynamics\".", + "And much as first order circuits were represented using", + "first order differential equations, this kind you end up", + "getting second-order differential equations.", + "Before we go into this, I would like to start", + "motivating this and give you one example of why this is important", + "to study. There are many,", + "many examples but I will give you one.", + "What I would like to do is draw your attention to our good old", + "inverter driving a second inverter.", + "The same circuit that we used to motivate RC studies,", + "one inverter driving another. So let me draw the circuit.", + "", + "Here is one inverter. This is, let's say,", + "5 volts and this is, let's say, 2 kilo ohms.", + "And I connect the output of this inverter to a second", + "inverter. And what we saw in the last few", + "lectures was that in this specific example there was a", + "parasitic capacitor or a capacitor associated with the", + "gate of this MOSFET. And that could be modeled by", + "sticking a capacitor CGS between the gate of the MOSFET and", + "ground. And we saw that the waveforms", + "here, if I had some kind of step here.", + "Let's say, for example, a step that went from high to", + "low. Then out here I would have a", + "transition that instead of going up rapidly like this would", + "transition a little bit more slowly.", + "And this transition was characterized by an RC time", + "constant. And this is what gave rise to a", + "delay in the eventual output. So that is what we saw", + "previously, single energy storage element.", + "Today what we are going to do is we are going to look at the", + "same circuit, the exact same circuit,", + "and have some fun with it. What we are going to say is", + "look, this thing is pretty slow, so what I would like to do is", + "-- why don't we go ahead and put that up.", + "", + "What we are going to see is that the yellow waveform is the", + "waveform at the input here. And the green waveform here is", + "the waveform at this intermediate node.", + "And notice that this waveform here is characterized by the", + "slowly rising characteristics that are typical of an RC", + "circuit. There are some other", + "weirdnesses and so on going on here like a little bump and", + "stuff like that. You can ignore all of that for", + "now. It happens because of certain", + "other very subtle circuit effects that you won't be", + "dealing with, called Miller effects and so on", + "that you won't be dealing with in 6.002.", + "So focus then on this part here.", + "It is pretty slow. And because of that slow", + "rising, I get a very slow transition and I get some delay", + "in my inverter. So you say ah-ha,", + "we learned about this in 6.002, I can make it go faster.", + "How can you make the circuit go faster?", + "What could you do? This is rising very slowly.", + "How can you make it go faster? Anybody?", + "You have multiple choices, actually.", + "What are your choices here? Pardon.", + "Decrease the time constant. And how would you decrease the", + "time constant? The capacitance is connected to", + "this MOSFET gate here. I didn't want it in the first", + "place but it is there, I cannot help it,", + "so I can decrease the resistance.", + "Good. Let me go ahead and do that.", + "What I will do is I am going to knock this sucker out and stick", + "in a new resistance that is say 50 ohms, a much smaller", + "resistance. That should speed things up,", + "right? That should make things go much", + "faster because this is a smaller time constant because R is", + "smaller, correct? OK, let's go do it.", + "And let's see if we get what we expect.", + "I have a little switch here. And using that switch,", + "I am going to switch in this little resistance.", + "Whoa, what on earth is happening out there?", + "This is so much fun. What I did is I switched in a", + "small resister here to decrease the time constant,", + "but it looks like I got a whole bunch of crapola that I did not", + "bargain for. This is certainly very fast,", + "it goes up really fast, but I am not sure where it is", + "going, though. Let's stare at that a little", + "while longer. Let me expand the time scale", + "for you. Look at this.", + "Instead of a nice little smooth thing going up.", + "I get something that looks like this.", + "It looks something like a sinusoid.", + "It looks sinusoidal, but then it is a sinusoid that", + "kind of gives up and kind of gets tired and kind of goes", + "away. Right?", + "It kind of dies out. So nothing that you have", + "learned so far has prepared you for this.", + "And, trust me, when I first did some circuit", + "designs myself a long, long time ago I got nailed by", + "that. I looked at my circuit,", + "and what ended up happening was I was noticing these sharp lines", + "at all my transitions. When I looked at my scope,", + "I expected to see nice little square waves but I saw these", + "little nasty spikes sitting out there.", + "And then when I stared at it more carefully,", + "those spikes were really sinusoids that seemed to kind of", + "get tired and kind of go away. So those are nasty,", + "those are real and they happen all the time.", + "And what we will do today is try to get into that and", + "understand why that is the case. We will understand how to", + "design that away. And that is a real problem,", + "by the way. And the reason that is a real", + "problem is the following. Look at this.", + "Look down here. Because this intermediate", + "voltage is meandering all over the countryside here,", + "at this particular point the intermediate voltage dips quite", + "low. And because it dips quite low", + "look at the output. The output has a bump here.", + "And it is quite possible for this output bump to now go into", + "the forbidden region. Or worse.", + "If this swing here was higher, this could have actually gone", + "onto a one, so I would have gotten a false one pulse here.", + "Instead of having a nice one to zero transition,", + "I would have gotten a one to zero, oh, back to one,", + "oh, back to zero and then back down to zero.", + "So this is nasty stuff, really, really nasty stuff.", + "What we will do is understand why that is the case today and", + "see if we can explain it. What is going on here?", + "What is really going on here is take a look at this circuit", + "here. I will take a look at this path", + "here. So this is your VS voltage", + "source. Path kind of goes like this and", + "around. It turns out that this circuit", + "is a loop here. And when there is current flow,", + "going down to basic physics you remember that I also enclose", + "some amount. So there is a current flowing", + "in a loop. And because of that there is an", + "effective inductance here. And, in fact,", + "any current flowing through a wire above a ground plane,", + "for that matter, can be characterized by the", + "inductance. So I can model that by sticking", + "a little inductor here. So my real circuit is not", + "exactly a resistor and a capacitor, but my real circuit", + "is an inductor as well that comes into play because of this", + "wire. Every wire, when there is a", + "current flow, has an inductance associated", + "with it. And because of that the real", + "circuit is resistor, inductor and capacitor.", + "So I end up with two storage elements now,", + "and the dynamics of that are very different from that with a", + "single storage element. That is just a bit of", + "motivation for why our study of inductors is important.", + "And I can draw a quick circuit here.", + "If you look at the circuit, start from ground,", + "the voltage VS and there is a resistor here.", + "And then I have an inductor and then I have a capacitor.", + "So it is a voltage source, resistor, inductor and", + "capacitor. For this whole week we will be", + "looking at circuits like this. Today what I would like to do", + "is start very simple, start with the simplest", + "possible form of this so that you can begin building up your", + "insight and then go into more complicated cases.", + "Today what I will do is simply begin with a case where I don't", + "have a resistor here and simply study a voltage source,", + "an inductor and a capacitor and understand what the voltage", + "looks like out here. So we look at the dynamics of a", + "little system like this. Before we go on,", + "I want to caution you about something.", + "It is just happenstance that I have introduced for you", + "capacitors based on the parasitic capacitance here and", + "inductance based on parasitic inductance.", + "I would hate to leave you with the impression that inductors", + "and capacitors are \"bad\". Because when you think of a", + "parasitic, you know, parasites.", + "These are parasitic. You didn't expect them there,", + "didn't expect this here and we got the weird behavior.", + "So parasitics have a bad connotation to them.", + "I do not want to leave you with a bad taste in your mouth about", + "capacitors and inductors that these are just bad things.", + "We just have to deal with them and deal with second-order", + "differential equations and all that stuff because they're just", + "bad stuff and we just have to deal with them.", + "I don't want you to end up going through life hating", + "capacitors and inductors. Just because of my choice of", + "examples, it just happened to be introducing them as capacitors.", + "I want to point out that these are fundamental lumped elements", + "in their own right. They are very,", + "incredibly important and useful circuits where we designed", + "capacitors and inductors because we want to have them in there.", + "There are many circuits that we will look at where we really", + "want the inductor in there. We will design an inductor by", + "wrapping wire around in a coil and get bigger inductances and", + "so. Just remember that this can be", + "parasitic in some cases, but in many cases it's good,", + "inductors are good, so just stick with that", + "thought. These are mostly good so don't", + "go around hating them. All right.", + "Let's go on and analyze a basic circuit like this.", + "And what I would like to cover in the next hour are the", + "foundations of something like that.", + "I will take you through the foundations so you understand", + "how it works. And, as always,", + "what I am going to end up with is build up the foundations,", + "help you understand why we got where we were and then help you", + "build intuition. And then show you a really,", + "really simple intuitive way of doing things in terms of how", + "experts do it. And the real cool thing about", + "EECS is that the way experts do things, things are really,", + "really very simple in the end. But you need to build up some", + "intuition to get there. So our circuit looks like this", + "in terms of my two storage elements.", + "I have a voltage vI, inductor L, capacitor C and I", + "am going to look at the voltage across the capacitor and my", + "current through the capacitor. So v(t) is the voltage across", + "the capacitor and my current is the current through this loop", + "here, which is the same as the current through the capacitor or", + "the current through the inductor.", + "And we are going to proceed in exactly the same manner as we", + "did for first order differential equations, write the equations", + "down and just boom, boom, boom, boom,", + "go down the same sets of steps but just get to some place", + "different. We are going to start by", + "writing a node equation for this node here.", + "That's the only node for which I have an unknown voltage.", + "The node here is vI, so I need to find this,", + "there's just one unknown node voltage.", + "And I am going to need some element laws.", + "For the capacitor I know the iV relation is given by the i for", + "the capacitor is Cdv/dt. And just to show the capacitor", + "I am just calling it dvc/dt. Similarly, for an inductor,", + "L, the voltage across the inductor is given by Ldi/dt.", + "So this is the vI relation for the capacitor,", + "the vI relation for an inductor.", + "It also suits us to write this in an integral form.", + "So if I integrate both sides of this equation and I bring L down", + "to this side, I end up getting something like", + "this, 1/L minus infinity to t, VLdt, and that is simply iL.", + "I am just simply replacing this with an integral form.", + "So this is a VI relationship for the inductor and this is for", + "the capacitor. So let me now go ahead and", + "apply the node method for my circuit here.", + "Here, for the node method, I have to equate the currents", + "coming into the node or sum the currents coming into the node", + "and equate that to zero. And while I do that I simply", + "replace the currents by the corresponding voltages using the", + "element laws. So what do I get?", + "I get the current going in here to the inductor is equal to the", + "current going through the capacitor.", + "What is the current going the capacitor?", + "In terms of its v relationship it is Cdv/dt.", + "And the current going to the inductor is given by this", + "relation here, which is simply 1/L minus", + "infinity to t. The voltage across the", + "capacitor is simply (vI-v)dt. I have just written down the", + "node quotation for this node here.", + "Now I will just apply a bit of math and simplify it and get the", + "resulting equation. What I can do is simply", + "differentiate with respect to t here.", + "And get this to be Cd^2v/dt^2, the second derivative of v.", + "And here what I end up getting is 1/L(vI-v).", + "So I just differentiated the whole thing by d/dt here.", + "And then I just move L up here. I bring d^2v/dt^2 out here.", + "And then I get a minus v here, and that will be equal to,", + "oh, I'm sorry. Let me leave this here.", + "Bring the minus v to this side so it becomes a plus and leave", + "vI on this side. So I end up getting", + "LCd^2v/dt^2. I bring L up here.", + "And then I take v to the other side.", + "Plus v and leave vI here so I get vI.", + "That is second order differential equation that", + "governs the characteristics of the voltage, v.", + "So much as the voltage across the capacitor was a state", + "variable in our RC circuits or the current through the inductor", + "was a state variable in our RL circuits, out here both the", + "current through the inductor and the voltage across the capacitor", + "are my two state variables. And so here I have a", + "second-order equation in my voltage, v.", + "Again, going through the foundations here,", + "I am now going to go through a bunch of math.", + "Up to here it was circuit analysis, and now I am just", + "going to do math. For the next three or four", + "blackboards just math. You can solve this second-order", + "differential equation any which way you want.", + "But just to keep things as simple as possible,", + "in 6.002 I solve all the differential equations,", + "it turns out we are fortunate enough we can do that,", + "using the exact same method again and again and again,", + "the same thing can be applied. And the method that we use to", + "solve it is the method of homogenous and particular", + "solutions. So the first step we are going", + "to find the particular solution, vP.", + "Second step we find the homogenous solution,", + "vH. And the third step we are going", + "to find the total solution as the sum of, v is simply the", + "particular plus the homogenous solution and then solve for", + "constants based on the initial conditions and the applied", + "voltage. So let's write down initial", + "conditions. Let's assume,", + "for simplicity, that my initial conditions are", + "simply the voltage across the capacitor is zero to begin and", + "the current through my inductor is also zero as I begin life.", + "Now, this is what is called \"zero state\".", + "v and i are both zero, and so the response of my", + "circuit for some input is going to be called ZSR.", + "You've probably heard this term in one of your recitations.", + "So zero state response simply says I start with my circuit at", + "rest and looks at how it behaves for some given input.", + "That is a little term you may end up using.", + "My input next. I am going to use the following", + "input. vI of t is going to be a step,", + "is going to look like this. My input is at t=0 v is going", + "from zero to some voltage VI and then stay at that voltage.", + "It is going to be a step. Kaboom.", + "And you can see why I am going with this set of variables,", + "because I want make this situation as close as possible", + "to the funny behavior we observed there.", + "Remember we had a step, and because of the step we had", + "some behavior at that node? So I will try to bring you as", + "close to that. In tomorrow's lecture,", + "I am going to close the loop around that and derive for you", + "exactly the behavior we saw on the scope.", + "And to get there I am going to be try to be as close as", + "possible to the constants and other parameters in the demo.", + "So VI is a step and zero state. Just in terms of notation,", + "this kind of a step input occurs pretty frequently.", + "And we just have a special notation for it.", + "We simply call it VI is the final value here.", + "And we call it u(t). So VIu(t), u(t) simply", + "represents a step at time t=0, steps from zero volts to VI.", + "That is just a little more notation that will come in handy", + "at some point. More math now.", + "Three steps, particular solution,", + "homogenous solution, total solution/constants.", + "This is almost like a mantra here, like a chorus.", + "Homogenous solution we compute using a four-step method.", + "And four-step method for homogenous solutions,", + "it turns out that it happens to be that way for all the", + "equations we will see in our course.", + "The first step would be assume a solution of the form Ae^st.", + "Exactly as with RCs. If you close your eyes and do", + "exactly what you did for RCs you will get to where you want to", + "be. You assume a solution of the", + "form Ae^st. Substitute that into your", + "homogenous equation. Obtain the characteristic", + "equation. Solve for the roots.", + "And then write down your homogenous solution.", + "Same sort of steps again and again and again until you get", + "bored to tears. Particular solution.", + "For the particular solution, I simply need to find a", + "solution, any solution, if not the most general one but", + "any solution that satisfies the particular equation which", + "satisfies that equation. LCd^2vP/dt^2+vP=VI.", + "My input is a step and I am going to look for the solution", + "for time t greater than zero. Notice that for time t less", + "than or equal to zero, v is going to be zero.", + "So I am looking for a solution greater than t=0.", + "Here, if I substitute vP=VI, that is a particular solution.", + "Because if I substitute VI here this goes to zero and then I get", + "VI=VI, so this works. I promised you this was going", + "to be simple. You cannot get any simpler than", + "that. I have done my first step.", + "I found the particular solution.", + "And VI is a good enough particular solution so I will", + "use it, I will take it. As my second step I am going to", + "find vH or the solution to the homogenous equation.", + "And the homogenous equation is simply that equation with drive", + "set to zero. What I get here is", + "LCd^2vH/dt^2+vH=0. That is my homogenous equation.", + "I simply set the drive to be zero.", + "And to find the solution here, I go through my four-step", + "method. Again, in 6.002 following the", + "kind of Occam's principle, we just show you the absolute", + "minimum necessary to get to where you want.", + "The absolute minimum necessary is it turns out that we can", + "solve all our differential equations that we use here by", + "using the methods of homogenous and particular solutions.", + "And every homogenous solution can be solved by a four-step", + "method. That is about as minimal as it", + "can get. So no extraneous stuff there.", + "The four-step method, four steps.", + "The first step is assume a solution of the form vH=Ae^st.", + "What I have noticed is that students starting out are", + "usually scared of differential equations.", + "I know I was when I was a student.", + "And the trick with differential equations is that it is all a", + "matter of psych. Just because you see some", + "squigglies and squagglies and a bunch of math and so on you say", + "oh, that must be hard. But differential equations are", + "actually the simplest thing there is because in a large", + "majority of cases the way you solve them is you assume you", + "know the answer, someone tells you the answer.", + "And then all you are left to do is shove the answer into the", + "equation and find out the constants that makes it the", + "answer. Just a matter of psych.", + "Psych yourselves that this stuff is easy,", + "because I am telling you what the solution is.", + "All you have to do is substitute and verify.", + "If you think about differential equations that way or a large", + "majority of them, it really is very simple if you", + "can just get past the squigglies here.", + "Just get past the squigglies and then just simply stick in", + "some simple stuff and it works. I mean it just cannot get any", + "easier. I cannot think of any other", + "field where the way you find a solution is assume you know the", + "solution and stick it in. It has never made any sense to", + "me but that is how it is. So we assume the solution to", + "the form Ae^st, you stick it in there,", + "and you have to find out the A and s that make it so.", + "It cannot get any simpler than that.", + "Let's stick the sucker in here and see what we can get.", + "Substitute Ae^st here I get LCA, and second derivative,", + "so it's s^2 e^st. And Ae^st on this one here.", + "And that equals zero. And then let me just solve for", + "whatever I can find. Assuming I don't take the", + "trivial case A=0, I cancel these guys out.", + "And what I am left with is simply LCs^2+1=0.", + "In other words, what I end up getting is B,", + "s^2=-1/LC. My first step was,", + "I am giving you solutions, stick them in there,", + "assume a solution of this form. Second step is get the", + "characteristic equation. And the way you get the", + "characteristic equation is that you simply stick this guy in", + "there. And what you end up getting is", + "some equation in s^2. Do you remember what you got", + "for first order circuits? What s was?", + "What is s? For first order circuits,", + "what did you get as a characteristic equation?", + "s+1/RC=0. The same thing.", + "Just remember to blindly apply the steps.", + "It will lead you to the answer. This is called the", + "\"characteristic equation\". This is incredibly important.", + "You will see in about a couple weeks from now that once you", + "write the characteristic equation down for a circuit,", + "it tells you all there is to know about the circuit.", + "And often times you can stop solving right here.", + "To experienced circuit designers this tells me", + "everything there is to know. This is really key.", + "That's why it's called a characteristic equation.", + "I believe in problem number three of the homework that will", + "be coming out this week, that is exactly what you are", + "going to do. I am going to give you a", + "circuit, ask you to get to the characteristic equation quickly", + "and then from there intuit the solution.", + "Write the characteristic equation and then just intuit", + "solution, it's that simple. So, step A, assume a solution", + "of the form, step B, write the characteristic", + "equation down. And let me just simplify that a", + "little bit. I go ahead and find my roots.", + "And my roots here, remember that j is the square", + "root of minus one. And so what I end up getting", + "is, my two roots here are, plus j square root of 1/LC and", + "minus j square root of 1/LC. Two roots.", + "And just as a shorthand notation, much like I had a", + "shorthand notation for RC, what was my shorthand notation", + "for RC? Tau.", + "Just as tau was big in first order, we have a corresponding", + "thing that is big in second order and that is omega nought.", + "Omega nought is simply square root 1/LC.", + "Just as tau was RC, omega nought is a shorthand", + "here. And so s is simply plus or", + "minus j omega nought. Notice that in this equation", + "here, if you take the square root of LC there that has units", + "of time, so one divided by that has units of frequency.", + "Notice that this guy is a frequency in radians.", + "I end up getting my roots of the homogenous equation,", + "and that is my third step. And as my fourth step,", + "I simply write down the homogenous solution as", + "substituting s with its roots and writing the most general", + "possible form of the solution, and that would be A1e^(j omega", + "nought t)+A2e^(-j omega nought t).", + "Done. Some constant times this", + "solution plus some other constant times,", + "the other solution. Plus zero omega nought.", + "Remember it comes from here, Ae^st.", + "I assume the solution of this form, so my solution in this", + "most general case would be s being j omega nought in one", + "case, minus j omega nought in the other case,", + "and I sum the two to get the most general solution.", + "", + "So blasting ahead. I now have my homogenous", + "solution. And as my third step of", + "solution to differential equations I write down the total", + "solution, v=vP+vH, particular plus the homogenous", + "solutions. And v=VI, was my particular", + "solution, +A1e^(j omega nought t)+A2e^(-j omega nought t) is my", + "complete solution. The final step,", + "write down the total solution and find the constants from the", + "initial conditions. To find the constants from the", + "initial conditions, let's start with,", + "the voltage is zero to begin with.", + "This equation governs the characteristics of v,", + "so I need to find the initial conditions.", + "First of all, I know that know that v(0)=0.", + "From there I substitute t=0. And so this goes to one,", + "this goes to one, and I end up getting", + "0=VI+A1+A2. That is my first expression.", + "And then I am also given that i(0)=0.", + "And so I can get that as well. How do I get i?", + "This is v. I know that i=Cdv/dt,", + "so I can get i by simply multiplying by C and", + "differentiating this with respect to t.", + "I get C, this guy vanishes so I get d/dt of this.", + "So it is CA1(j omega nought) e^(j omega nought t)+CA2(-j", + "omega nought)e^(-j omega nought t).", + "From here I am given that that is zero, and so therefore this", + "guy becomes a one, this guy becomes a one,", + "j omega nought, j omega nought cancel out.", + "What I end up getting is A1=A2. From the second initial", + "condition I get A1=A2. From these two,", + "if I substitute here for A2, I get VI + 2A1 = 0,", + "or A1=-VI/2. That is also equal to A2.", + "Therefore, my total solution now can be written in terms of", + "the actual values of the constants I have obtained.", + "I get VI-VI/2. So A1 and A2 are equal.", + "I just pull them outside. I pull VI-2 outside and I stick", + "these two guys in parenthesis in.", + "Again, I promised you no more circuits from here on until the", + "very last board or something like that.", + "It is all math, so not much else happening", + "there. More math.", + "If you would like, I could skip all the way to the", + "end and show you the answer. But I just love to write", + "equations on the board so let me just go through that.", + "I am going to simplify this a little further here.", + "And we should remember this form by the Euler relation,", + "ejx=cos x+j sin x. And by the same token,", + "(e^jx + e^-jx)/2=cos x. You all should know this from", + "the Euler relation. So were are using this guy", + "here, ej^x + e^-jx=2cos x. And so this one is 2 cosine of", + "omega nought t, 2 and 2 cancel out,", + "and what I am left with is v(t)=VI-VI cos( omega nought t).", + "And the current is Cdv/dt, which is simply CVI sin( omega", + "nought t). Just remember that omega nought", + "is the square root of 1/LC. We are done.", + "In fact, I did not give that answer the importance that was", + "due so let me just draw.", + "", + "There. That is better.", + "Enough math. In a nutshell,", + "what did we do. We wrote the node method,", + "it's a very simple circuit, to write down the equation", + "governing that circuit. And then we grunged through a", + "bunch of math. Not a whole lot here.", + "It is pretty simple. And ended up with a relation", + "that says the voltage across the capacitor for a step input,", + "assuming zero state, is a constant VI-VI cos omega", + "t. Notice that even though I have", + "a step input, the circuit dynamics are such", + "that I get a cosine in there. You can begin to see where", + "these cosines are coming from now.", + "They come in here. And if you recall the example I", + "showed you earlier of the inverter circuit,", + "remember there was a cosine that decayed,", + "that was sort of losing energy and kind of dying out?", + "So you can see where the cosines are coming from.", + "And just to draw you a little sketch here.", + "Let me draw v and i for you and let me plot omega t,", + "pi/2, pi and so on. Let me plot VI.", + "When time t=0, VI=0, cosine omega t is one,", + "and so VI-VI=0. That is simply a cosine that", + "starts out at zero here, and at pi I get cosine omega t", + "is minus one, so I get plus VI on the other", + "side. So I end up at +2VI.", + "At this point the voltage is here.", + "And notice that this guy looks like this.", + "It is a cosine that is translated up so that its mean", + "value is not zero but VI. It is just a translation up of", + "a cosine. Similarly, in this case for the", + "current it is a sinusoidal characteristic.", + "And it looks something like this where the peak is given by", + "CVI, oh, I messed up.", + "", + "When I differentiated this is missed the omega nought out", + "there.", + "", + "What I would like to do now -- This is the form of the output", + "for a step input. What I would like to do next is", + "show you a demo. But before I show you a demo,", + "I always found it strange that I have a step input and then I", + "have two little elements, how can I get a sine coming out", + "of the output? I would like to get some", + "intuition as to why things behave the way they are.", + "I could go and pray to find out, but let me just give you", + "some very basic insight as to why this behaves the way it", + "does. Let me draw the circuit for you", + "here. And this is my inductor L and", + "capacitance C. Remember this is v.", + "Let me just walk you through what is happening there and get", + "you to understand this. Now, you have seen sines occur", + "before. If you go and write down the", + "equation of motion of a pendulum, you know,", + "you have a pendulum, you move it to one side,", + "let go. It is also governed by", + "sinusoidal characteristics. And you will find that the", + "equation governing its motion is very much of the same form,", + "and you get the sinusoid where you have energy that is sloshing", + "back and forth between maximum potential energy to maximum", + "kinetic energy and zero potential energy back to maximum", + "potential energy, zero kinetic.", + "So it is energy sloshing back and forth.", + "The same way here. Capacitors and inductors store", + "energy. Let's walk through and see what", + "happens. I start off with both of them", + "having the stage zero, zero current,", + "zero voltage. I apply a step here.", + "Boom, the step comes instanteously to VI.", + "I notice that the capacitor voltage cannot change instantly", + "unless there is an infinite pulse of a sort,", + "so this guy cannot change instantly.", + "And so its voltage starts off being zero.", + "So the entire voltage here, KVL must be true no matter", + "what. They are absolutely fundamental", + "principles from Maxwell's equations.", + "KVL must hold, which means that the entire", + "voltage VI must appear across the inductor.", + "I put a big voltage across the inductor and its current begins", + "to build up. There you go.", + "A voltage across the inductor, its current begins to build up.", + "As its current begins to build up that current must flow", + "through the capacitor, too.", + "And as current flows through a capacitor it is depositing", + "charge into the capacitor. As the capacitor begins to get", + "charge deposited on it, its voltage begins to rise.", + "Let's see what happens here. Its voltage keeps rising.", + "At some point, the voltage across the", + "capacitor is equal to VI. But then VI equals this VI", + "here. So when the two become VI,", + "the inductor has zero volts across it.", + "So there is no longer a potential difference that is", + "increasing the current in that direction.", + "At that point, at pi divided by 2,", + "I have some current going into the inductor so there is no", + "longer a pressure that is forcing more current through the", + "inductor because this voltage reaches VI.", + "But remember capacitors like to sit around holding voltages.", + "Just remember that demo. That rinky-dink capacitor sat", + "there stubbornly holding its voltage.", + "And it had a huge spark towards the end.", + "It just sat there holding its voltage.", + "In the same manner, inductors love to sit around", + "holding a current. They will do whatever they can", + "to keep the current going through them.", + "It has got the current going through.", + "And few forces on earth can change that.", + "And so therefore, even though the capacitor", + "voltage is VI and the voltage drop across the inductor is", + "zero, it still keeps supplying a current.", + "It has got the current. It's got inertia.", + "It keeps going. It is like a runaway train.", + "You may not be pushing the train from the back,", + "but once it is running it has got kinetic energy and is going", + "to run no matter what for a least some more time,", + "even if you take away the force on the train.", + "So I have taken away the force on the punching more current", + "through, but it has kinetic energy.", + "It has current flowing through it so it continues to supply a", + "current. Because it continues to supply", + "the current the capacitor voltage keeps increasing.", + "This is a subtle insight which is absolutely spectacular that", + "with zero volts across it, it still keeps pumping that", + "current. Capacitor voltage has gone up.", + "And guess what? The voltage on this side is", + "higher now but this guy is still pumping a current.", + "Man, I have been born to do this, you know,", + "I shall pump a current. However, because the voltage", + "has now gone up here gradually the current begins to diminish.", + "So the capacitor is concerned. You pump a current into me,", + "my voltage goes up. At some point,", + "like a runaway train, it comes to a halt.", + "The current through the capacitor drains and now goes to", + "zero and the capacitor voltage reaches 2VI.", + "So this is at 2VI now and this is at VI.", + "Now the situation is not in equilibrium.", + "At this point there is zero current through it,", + "but guess what? I have a VI pumping in this", + "direction now. I have the same VI punching in", + "this direction. So guess what?", + "Its current must now build up in this direction and its", + "current begins to build up in that direction.", + "That begins to discharge the capacitor and the capacitor then", + "goes on to a negative, or the current goes down to a", + "maximum negative current, and this process continues.", + "What you are seeing here is energy.", + "It is sloshing back and forth between the two,", + "and that is kind of a key. I will just quickly put up a", + "demo that you can watch as you are walking out.", + "With a step input, notice the green is the voltage", + "across the capacitor and the orange is the current through", + "the capacitor." + ] +} \ No newline at end of file diff --git a/subs/zpzcLzD2dV4.srt.sjson b/subs/zpzcLzD2dV4.srt.sjson new file mode 100644 index 0000000000..3ab511a3e5 --- /dev/null +++ b/subs/zpzcLzD2dV4.srt.sjson @@ -0,0 +1,572 @@ +{ + "start": [ + 0, + 1176, + 6660, + 11910, + 18130, + 21870, + 25890, + 29310, + 31300, + 35050, + 38130, + 40690, + 44320, + 47910, + 50490, + 53430, + 59720, + 64300, + 68200, + 71920, + 75196, + 83400, + 88850, + 91850, + 95680, + 96630, + 101150, + 104760, + 109530, + 115689, + 122270, + 126810, + 131150, + 133790, + 140560, + 147800, + 153010, + 156860, + 162357, + 164010, + 166990, + 170490, + 173520, + 176870, + 183290, + 187140, + 191490, + 196180, + 201730, + 206460, + 209050, + 214100, + 215590, + 218280, + 220180, + 224170, + 227720, + 228970, + 233510, + 235715, + 238260, + 241690, + 244694, + 249930, + 254050, + 258720, + 263340, + 268055, + 271880, + 276940, + 278190, + 281560, + 286860, + 288910, + 294310, + 295770, + 299090, + 301330, + 303870, + 307960, + 310460, + 313896, + 317010, + 318260, + 324480, + 326600, + 330280, + 336500, + 342630, + 345310, + 346330, + 346920, + 350060, + 350810, + 353190, + 359180, + 365600, + 369410, + 372760, + 374640, + 379380, + 383320, + 387100, + 394000, + 395080, + 398780, + 402600, + 405330, + 407410, + 417210, + 423730, + 427250, + 428420, + 436400, + 441790, + 444180, + 445430, + 448630, + 452510, + 455710, + 463570, + 465430, + 466680, + 470480, + 474490, + 480380, + 482740, + 487990, + 490800, + 493030, + 498370, + 504070, + 505660, + 507760, + 510610, + 511650, + 514809, + 518190, + 521630, + 525350, + 529710, + 533790, + 535040, + 538120, + 543420, + 545920, + 548725, + 549975, + 553010, + 556820, + 564220, + 570020, + 575360, + 576120, + 582960, + 589840, + 593700, + 598650, + 604560, + 606850, + 608650, + 612260, + 616070, + 619336, + 621680, + 622480, + 624430, + 627700, + 630220, + 634778, + 638070, + 641765, + 645120, + 648206, + 651920, + 653550, + 654990, + 655830, + 659170, + 664330, + 668605, + 676080, + 677945, + 679720, + 685200, + 690660, + 694700, + 697520 + ], + "end": [ + 1176, + 6660, + 11910, + 18130, + 21870, + 25890, + 29310, + 31300, + 35050, + 38130, + 40690, + 44320, + 47910, + 50490, + 53430, + 59720, + 64300, + 68200, + 71920, + 75196, + 83400, + 88850, + 91850, + 95680, + 96630, + 101150, + 104760, + 109530, + 115689, + 122270, + 126810, + 131150, + 133790, + 140560, + 147800, + 153010, + 156860, + 162357, + 164010, + 166990, + 170490, + 173520, + 176870, + 183290, + 187140, + 191490, + 196180, + 201730, + 206460, + 209050, + 214100, + 215590, + 218280, + 220180, + 224170, + 227720, + 228970, + 233510, + 235715, + 238260, + 241690, + 244694, + 249930, + 254050, + 258720, + 263340, + 268055, + 271880, + 276940, + 278190, + 281560, + 286860, + 288910, + 294310, + 295770, + 299090, + 301330, + 303870, + 307960, + 310460, + 313896, + 317010, + 318260, + 324480, + 326600, + 330280, + 336500, + 342630, + 345310, + 346330, + 346920, + 350060, + 350810, + 353190, + 359180, + 365600, + 369410, + 372760, + 374640, + 379380, + 383320, + 387100, + 394000, + 395080, + 398780, + 402600, + 405330, + 407410, + 417210, + 423730, + 427250, + 428420, + 436400, + 441790, + 444180, + 445430, + 448630, + 452510, + 455710, + 463570, + 465430, + 466680, + 470480, + 474490, + 480380, + 482740, + 487990, + 490800, + 493030, + 498370, + 504070, + 505660, + 507760, + 510610, + 511650, + 514809, + 518190, + 521630, + 525350, + 529710, + 533790, + 535040, + 538120, + 543420, + 545920, + 548725, + 549975, + 553010, + 556820, + 564220, + 570020, + 575360, + 576120, + 582960, + 589840, + 593700, + 598650, + 604560, + 606850, + 608650, + 612260, + 616070, + 619336, + 621680, + 622480, + 624430, + 627700, + 630220, + 634778, + 638070, + 641765, + 645120, + 648206, + 651920, + 653550, + 654990, + 655830, + 659170, + 664330, + 668605, + 676080, + 677945, + 679720, + 685200, + 690660, + 694700, + 697520, + 701800 + ], + "text": [ + "", + "SPEAKER 1: In this unit, we're going to study several new", + "tools in our tool chest. These include the superposition", + "method, the Thevenin method, and the Norton method.", + "The recommended reading is chapter three of the textbook.", + "", + "Before we begin, let's do a quick review of what we've", + "seen so far.", + "So recall we've been building up our tool chest, a tool", + "chest of techniques that can analyze circuits.", + "We've seen several.", + "First, we've seen the KVL/KCL method, where, given the", + "KVL/KCL rules, you would write a bunch of equations to", + "circuits and go and solve them.", + "Then, you looked at the circuit composition rules", + "method, where you simplified circuits into more composed", + "circuits and analyzed them much more simply.", + "The third method was the Node method.", + "The Node method is the workhorse of 6.002 and pretty", + "much all of circuit analysis.", + "The key idea behind the Node method is that KCL at the", + "nodes is applied using all voltages referenced to ground,", + "where the ground is taken as zero.", + "That enables substantial simplicity in", + "how the Node method--", + "in how the rules are applied, giving you way better and", + "efficient techniques.", + "So these are three methods in your tool chest.", + "So today, we are going to look at how to build up your tool", + "chest. And I'm going to introduce a new technique, or", + "a set of two techniques, the superposition method, the", + "Thevenin method, and the Norton method.", + "I'll start with the superposition method.", + "Now, these methods apply only to linear circuits.", + "Now recall, when we did the analysis of various circuits--", + "recall that we had looked at this playground of ours.", + "And in this playground, all our techniques applied.", + "Now it turns out, there's a region of the playground in", + "which circuits are linear.", + "It's a subset of the playground.", + "Within this playground, some extremely", + "powerful tools apply.", + "Imagine that, within this playground, you are able to", + "use a bazooka as a weapon, while you can't use it outside", + "this little piece of playground.", + "This subset of the playground that we've been used to so far", + "is the playground containing only linear circuits.", + "So in this unit, I would like to introduce linear circuits.", + "Then I'll talk about the properties of linearity.", + "What does it buy us?", + "And then I will talk about the superposition method.", + "The superposition method applies", + "only for linear circuits.", + "And we'll look at how that can substantially", + "simplify circuit analysis.", + "So in this video segment, let's start with introducing", + "linear circuits.", + "", + "I would like to introduce a technique with an example.", + "", + "This is a very simple circuit containing two sources, the", + "voltage source and the current source.", + "I'm going to apply the Node method to analyze this.", + "So according to the Node method, I am going to start", + "with introducing a ground node.", + "So I want to pick this node as my ground node.", + "And the reason I'm doing that is, as is our usual practice,", + "this ground node connects to one terminal of both of our", + "sources, and many edges of the circuit come", + "together at that point.", + "", + "The voltage at this node here is V, because there's a", + "voltage source connected there.", + "And the voltage at this node is labeled e. e is my unknown.", + "And my goal is to find e.", + "", + "So how do I apply the Node method?", + "", + "I'm going to start by adding a node equation for the node e.", + "", + "So here is my node equation.", + "How do I get that?", + "Let's look at the first term.", + "", + "The first term, e minus V divided by", + "R1, arises as follows.", + "So recall, I have to apply KCL at the node with voltage e,", + "and sum the currents coming into the node,", + "and set them to 0.", + "So--", + "I'm sorry.", + "I'm summing the currents leaving the node and setting", + "them to be 0.", + "I could do either.", + "So there's a current leaving this node in this direction.", + "In that direction, the current leaving that node is e minus", + "the voltage V divided by R1.", + "So it's e minus V divided by R1 is the current", + "leaving this node.", + "The current leaving this node through R2 is simply e divided", + "by R2, and that's this piece.", + "Finally, the current leaving this node in this direction is", + "simply minus I. And these three currents sum to 0.", + "OK.", + "That gives us the node equation, or the", + "node at voltage e.", + "You've already noticed some very interesting things in", + "this equation.", + "notice that the equation is linear in e, V, and I. Notice", + "that you don't see any terms involving the product eV, or V", + "squared, VI, and so on.", + "So [UNINTELLIGIBLE PHRASE]", + "this equation is linear in e, V, and I.", + "Let's go on and stir this equation a little more.", + "And I'm going to rearrange the terms.", + "So I take this equation--", + "", + "I'm going to take this equation and rearrange the", + "terms a little bit.", + "And if you look at the equation here, 1 by R1, is", + "multiplying the inverse.", + "We divide by R1.", + "", + "And that's shown here.", + "And then, similarly, 1 divided by R2 is multiplying e here.", + "And I show that here.", + "I move I to the right hand side, so that's here.", + "Similarly, we divide by R1.", + "Shows up here.", + "So by rearranging terms, I show the unknown voltage e to", + "be some function of the voltages V and I.", + "This turned out to be a canonic", + "representation of equations.", + "And in this representation, in this case, I", + "have a single voltage.", + "But in general, this could be multiple voltages, which I", + "could collect into a matrix form", + "This matrix here is called the conductance matrix.", + "I could stick my vector of node voltages here.", + "And on the right hand side, I have a linear sum of sources,", + "where V and I are my voltage and current sources,", + "respectively.", + "", + "In matrix form, this matrix is commonly called G. The vector", + "of unknown voltages is called e.", + "And on the right hand side, I have the", + "linear sum of sources.", + "", + "So let's take it one step further.", + "And in this equation, here is what I am doing.", + "I have taken this component and I have moved it over to", + "the right hand side, exposing e all by itself.", + "OK.", + "As I simplify the equation, here is what I get.", + "Notice that when I move this to the side, my V here is now", + "multiplied by R2 divided by R1 plus R2.", + "And I is multiplied by R1 and R2 divided by R1 plus R2.", + "So all I've done here is taken this part and simply moved it", + "to the right hand side and divided the right", + "hand side with that.", + "And this is what I get.", + "This should make it very clear that my unknown node voltage e", + "is a straight linear--", + "is a linear sum of the sources.", + "V is my source.", + "I is a source.", + "And I have some constant multiplier", + "multiplying those voltages.", + "So this should make it clear that this is a constant.", + "This is a constant.", + "And I have a V and a I, and so I get a linear sum of sources.", + "", + "I can do it a little bit more clearly.", + "Looks like I've scribbled around, unfortunately, so", + "let's see if this eraser thingy works.", + "Ah.", + "There you go.", + "That works pretty nicely.", + "And let me get some color back into my pen here.", + "And so notice that in general, for any circuit, any node", + "voltage e can be represented as a linear combination of all", + "the sources, whether they're voltage", + "sources or current sources.", + "But a1, a2, b1, b2 and so on are constants that are", + "independent of the source voltages and source currents.", + "And so, in general, I end up with my unknown being a linear", + "combination of the various source voltages.", + "" + ] +} \ No newline at end of file