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/courseware/static/js/video_player.js b/courseware/static/js/video_player.js
deleted file mode 100644
index 811f69eca4..0000000000
--- a/courseware/static/js/video_player.js
+++ /dev/null
@@ -1,317 +0,0 @@
-// 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/courseware/static/js/video_player.js~ b/courseware/static/js/video_player.js~
deleted file mode 100644
index 3b47fe942e..0000000000
--- a/courseware/static/js/video_player.js~
+++ /dev/null
@@ -1,260 +0,0 @@
-var load_id;
-
-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;
-}
-
-function caption_index(now) {
- // Returns the index of the current caption, given a time
- 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));
- $("#std_p6").html(caption_at(i+6));
-}
-
-function title_seek(i) {
- // Seek video forwards or backwards by i subtitles
- current=caption_index(getCurrentTime());
- new_time=caption_time_at(current+i);
-
- ytplayer.seekTo(new_time, true);
-}
-
-function updateHTML(elmId, value) {
- document.getElementById(elmId).innerHTML = value;
-}
-
-function setytplayerState(newState) {
- updateHTML("playerstate", newState);
-}
-
-function onYouTubePlayerReady(playerId) {
- ytplayer = document.getElementById("myytplayer");
- setInterval(updateytplayerInfo, 250);
- ytplayer.addEventListener("onStateChange", "onytplayerStateChange");
- ytplayer.addEventListener("onError", "onPlayerError");
- if((typeof load_id != "undefined") && (load_id != null)) {
- var id=load_id;
- load_id = null;
- loadNewVideo(id);
- }
-
-}
-
-function log_event(e) {
- //$("#eventlog").append(" ");
- //$("#eventlog").append(JSON.stringify(e));
- window['console'].log(JSON.stringify(e));
-}
-
-function seek_slide(type,oe,value) {
- //log_event([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([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(['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());
- }
- update_captions(getCurrentTime());
-
- updateHTML("videoduration", getDuration());
- updateHTML("videotime", getCurrentTime());
- updateHTML("startbytes", getStartBytes());
- updateHTML("volume", getVolume());
-}
-
-// functions for the api calls
-function loadNewVideo(id, startSeconds) {
- if (typeof ytplayer != "undefined") {
- ytplayer.loadVideoById(id, parseInt(startSeconds));
- } else {
- load_id = id;
- }
-
- $.getJSON("/static/subs/"+id+".srt.sjson", function(data) {
- captions=data;
- });
-
-}
-
-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/courseware/static/pixel.jpg b/courseware/static/pixel.jpg
deleted file mode 100644
index 8fbe381c7a..0000000000
Binary files a/courseware/static/pixel.jpg and /dev/null differ
diff --git a/courseware/static/staff/agarwal-mit-news-small.jpg b/courseware/static/staff/agarwal-mit-news-small.jpg
deleted file mode 100644
index 900038d394..0000000000
Binary files a/courseware/static/staff/agarwal-mit-news-small.jpg and /dev/null differ
diff --git a/courseware/static/staff/cjt-small.jpg b/courseware/static/staff/cjt-small.jpg
deleted file mode 100644
index 4e8537de0b..0000000000
Binary files a/courseware/static/staff/cjt-small.jpg and /dev/null differ
diff --git a/courseware/static/staff/gjs-small.jpg b/courseware/static/staff/gjs-small.jpg
deleted file mode 100644
index 4863d30b9e..0000000000
Binary files a/courseware/static/staff/gjs-small.jpg and /dev/null differ
diff --git a/courseware/static/staff/pmitros-small.jpg b/courseware/static/staff/pmitros-small.jpg
deleted file mode 100644
index ff63f6f9fd..0000000000
Binary files a/courseware/static/staff/pmitros-small.jpg and /dev/null differ
diff --git a/courseware/static/subs/2SwT6JnfCq8.srt.sjson b/courseware/static/subs/2SwT6JnfCq8.srt.sjson
deleted file mode 100644
index cbc4f06162..0000000000
--- a/courseware/static/subs/2SwT6JnfCq8.srt.sjson
+++ /dev/null
@@ -1,1934 +0,0 @@
-{
- "start": [
- 0,
- 17000,
- 20153,
- 26459,
- 34000,
- 40105,
- 43473,
- 46000,
- 57000,
- 61000,
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- 83134,
- 90000,
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- 99248,
- 103872,
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- 113280,
- 117984,
- 122767,
- 127664,
- 134613,
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- "text": [
- "",
- "All right. 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/courseware/static/subs/2vHGYdepKLw.srt.sjson b/courseware/static/subs/2vHGYdepKLw.srt.sjson
deleted file mode 100644
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- "text": [
- "Good morning, 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/courseware/static/subs/3GdMaDzIUeQ.srt.sjson b/courseware/static/subs/3GdMaDzIUeQ.srt.sjson
deleted file mode 100644
index 07db11ddb8..0000000000
--- a/courseware/static/subs/3GdMaDzIUeQ.srt.sjson
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- "text": [
- "",
- "OK. 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/courseware/static/subs/4TCnYYpZxEc.srt.sjson b/courseware/static/subs/4TCnYYpZxEc.srt.sjson
deleted file mode 100644
index da0f8e0a5f..0000000000
--- a/courseware/static/subs/4TCnYYpZxEc.srt.sjson
+++ /dev/null
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- "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/courseware/static/subs/9RqFFlZgf60.srt.sjson b/courseware/static/subs/9RqFFlZgf60.srt.sjson
deleted file mode 100644
index e0b4f402d0..0000000000
--- a/courseware/static/subs/9RqFFlZgf60.srt.sjson
+++ /dev/null
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- "text": [
- "Good morning, all. 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
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deleted file mode 100644
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- "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/courseware/static/subs/COdQmA9g9S8.srt.sjson b/courseware/static/subs/COdQmA9g9S8.srt.sjson
deleted file mode 100644
index a4d53ac468..0000000000
--- a/courseware/static/subs/COdQmA9g9S8.srt.sjson
+++ /dev/null
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- "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
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deleted file mode 100644
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- "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/courseware/static/subs/JqvKtMNz3RQ.srt.sjson b/courseware/static/subs/JqvKtMNz3RQ.srt.sjson
deleted file mode 100644
index 00137c4471..0000000000
--- a/courseware/static/subs/JqvKtMNz3RQ.srt.sjson
+++ /dev/null
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- "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/courseware/static/subs/Km9YIdkc2Oo.srt.sjson b/courseware/static/subs/Km9YIdkc2Oo.srt.sjson
deleted file mode 100644
index b3ba7958c1..0000000000
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- "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/courseware/static/subs/Nijya-QJ45Y.srt.sjson b/courseware/static/subs/Nijya-QJ45Y.srt.sjson
deleted file mode 100644
index bbea2f2cd9..0000000000
--- a/courseware/static/subs/Nijya-QJ45Y.srt.sjson
+++ /dev/null
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- "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/courseware/static/subs/OGtElTMJidE.srt.sjson b/courseware/static/subs/OGtElTMJidE.srt.sjson
deleted file mode 100644
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- "text": [
- "-- will try it again at the end of this lecture and you show you",
- "that stuff hopefully next time. 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
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deleted file mode 100644
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- "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/courseware/static/subs/RsJ1eg7XNVs.srt.sjson b/courseware/static/subs/RsJ1eg7XNVs.srt.sjson
deleted file mode 100644
index 621dadf3f2..0000000000
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- "text": [
- "Let's get started.",
- "",
- "Can you hear me back there? 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/courseware/static/subs/TXJIhDHtHSI.srt.sjson b/courseware/static/subs/TXJIhDHtHSI.srt.sjson
deleted file mode 100644
index 35e67f4ac2..0000000000
--- a/courseware/static/subs/TXJIhDHtHSI.srt.sjson
+++ /dev/null
@@ -1,2141 +0,0 @@
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- "text": [
- "",
- "OK. 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/courseware/static/subs/V0z_f7qxLcY.srt.sjson b/courseware/static/subs/V0z_f7qxLcY.srt.sjson
deleted file mode 100644
index 501494e55d..0000000000
--- a/courseware/static/subs/V0z_f7qxLcY.srt.sjson
+++ /dev/null
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- "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/courseware/static/subs/WT-qzgaKeGI.srt.sjson b/courseware/static/subs/WT-qzgaKeGI.srt.sjson
deleted file mode 100644
index 8094fd18c4..0000000000
--- a/courseware/static/subs/WT-qzgaKeGI.srt.sjson
+++ /dev/null
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- "text": [
- "",
- "All right. 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/courseware/static/subs/bEJ0-8pANA4.srt.sjson b/courseware/static/subs/bEJ0-8pANA4.srt.sjson
deleted file mode 100644
index 6af767801c..0000000000
--- a/courseware/static/subs/bEJ0-8pANA4.srt.sjson
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- "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/courseware/static/subs/bX8i2yECWaU.srt.sjson b/courseware/static/subs/bX8i2yECWaU.srt.sjson
deleted file mode 100644
index 654b895b62..0000000000
--- a/courseware/static/subs/bX8i2yECWaU.srt.sjson
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- "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/courseware/static/subs/dyxcCoUgETU.srt.sjson b/courseware/static/subs/dyxcCoUgETU.srt.sjson
deleted file mode 100644
index 5be1bda72f..0000000000
--- a/courseware/static/subs/dyxcCoUgETU.srt.sjson
+++ /dev/null
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- "text": [
- "All right. 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
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deleted file mode 100644
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- "text": [
- "",
- "SPEAKER 1: OK, in the previous segment, we",
- "looked at a linear circuit.",
- "And you saw that the equations that governed the behavior of",
- "such a circuit were linear in the inputs, because the",
- "components of the circuit were linear.",
- "Next, let's take 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
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- "text": [
- "All right. 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/courseware/static/subs/v6vqWasIHaw.srt.sjson b/courseware/static/subs/v6vqWasIHaw.srt.sjson
deleted file mode 100644
index 9909b360ba..0000000000
--- a/courseware/static/subs/v6vqWasIHaw.srt.sjson
+++ /dev/null
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- "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/courseware/static/subs/wNuBD4PYWvs.srt.sjson b/courseware/static/subs/wNuBD4PYWvs.srt.sjson
deleted file mode 100644
index 479fc4b80b..0000000000
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- "text": [
- "Good morning. 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/courseware/static/subs/ypX20WnHNQw.srt.sjson b/courseware/static/subs/ypX20WnHNQw.srt.sjson
deleted file mode 100644
index 6aa9eef172..0000000000
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- "text": [
- "All right. 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/courseware/static/subs/zpzcLzD2dV4.srt.sjson b/courseware/static/subs/zpzcLzD2dV4.srt.sjson
deleted file mode 100644
index 3ab511a3e5..0000000000
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- "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
diff --git a/settings.py b/settings.py
index f080316198..2de81fe2be 100644
--- a/settings.py
+++ b/settings.py
@@ -1,8 +1,3 @@
-# Django settings for mitx project.
-
-execfile("../settings.py")
-
-
DEFAULT_FROM_EMAIL = 'pmitros@csail.mit.edu'
HTTPS = 'on'
@@ -134,3 +129,6 @@ LOGGING = {
},
}
}
+
+# Django settings for mitx project.
+execfile("../settings.py")
diff --git a/staticbook/__init__.py b/staticbook/__init__.py
new file mode 100644
index 0000000000..e69de29bb2
diff --git a/staticbook/models.py b/staticbook/models.py
new file mode 100644
index 0000000000..71a8362390
--- /dev/null
+++ b/staticbook/models.py
@@ -0,0 +1,3 @@
+from django.db import models
+
+# Create your models here.
diff --git a/staticbook/tests.py b/staticbook/tests.py
new file mode 100644
index 0000000000..501deb776c
--- /dev/null
+++ b/staticbook/tests.py
@@ -0,0 +1,16 @@
+"""
+This file demonstrates writing tests using the unittest module. These will pass
+when you run "manage.py test".
+
+Replace this with more appropriate tests for your application.
+"""
+
+from django.test import TestCase
+
+
+class SimpleTest(TestCase):
+ def test_basic_addition(self):
+ """
+ Tests that 1 + 1 always equals 2.
+ """
+ self.assertEqual(1 + 1, 2)
diff --git a/staticbook/views.py b/staticbook/views.py
new file mode 100644
index 0000000000..b4665577e1
--- /dev/null
+++ b/staticbook/views.py
@@ -0,0 +1,9 @@
+# Create your views here.
+from djangomako.shortcuts import render_to_response, render_to_string
+from django.shortcuts import redirect
+import os
+from django.conf import settings
+from django.http import Http404
+
+def index(request, page):
+ return render_to_response('staticbook.html',{'page':int(page)})
diff --git a/temp.py b/temp.py
deleted file mode 100644
index c184420193..0000000000
--- a/temp.py
+++ /dev/null
@@ -1,2 +0,0 @@
-q={0: {'answer': 13.0, 'tolerance': 0.050000000000000003, 'lid': 0, 'type': 'numericalresponse', 'id': 'resistor_0'}, 1: {'answer': 0.30769230769200001, 'tolerance': 0.050000000000000003, 'lid': 1, 'type': 'numericalresponse', 'id': 'resistor_1'}}
-a={u'resistor_0': u'', u'resistor_1': u''}