243 lines
16 KiB
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243 lines
16 KiB
HTML
<ul class="tabs">
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<li class="active">English (main)</li>
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<li>French</li>
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<li>English v2</li>
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<li>+</li>
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</ul>
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<textarea class="captions">
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{
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"text": [
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"SUBJECT 1: The various methods\nI'm going to show you--",
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"in particular, the first method\none and method two for",
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"solving non-linear equations--",
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"are really just particular\nways of solving a pair of",
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"equations where at least\none of each have some",
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"non-linearity to them.",
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"So let's start with the\ngraphical method.",
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"And my circuit is on the\nright-hand side.",
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"And I'm showing you the same\nvoltage source resistor--",
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"the Thevenin pattern--",
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"connected to Device D. And\nbelow that, I have the",
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"equation iD equals ae\nraised to bvD, which",
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"is the device equation.",
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"So as before, let me\ngo ahead and do all",
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"the first few steps.",
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"I go ahead and write the\nnode equation at vD.",
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"And I got that.",
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"I go in and substitute, as I\nalways do, in the node method",
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"for the current, using\nthe device relation.",
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"And just for fun here, let me\nkeep that separate for now.",
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"In the analytical method, notice\nthat you ended up with",
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"these two equations, and\nyou had to solve",
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"for these two unknowns.",
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"And we did that using\nanalytical methods.",
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"In this video, I'm going to\nsolve these using the",
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"graphical method.",
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"In order to do so here's what\nI'm going to do I want to",
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"start by rearranging the terms\nin my equation 1 to make it a",
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"little bit more convenient\nto draw the graph.",
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"So notice in equation 2, I\nhave iD equals something.",
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"And then I have an expression\nin vD and iD.",
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"So what I'd like to do is let\nme start by taking this",
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"equation here and expressing it\nin more of a standard form",
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"so I can get something symmetric\nto equation 2.",
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"So let me express this by\npulling iD to the left-hand",
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"side all by itself, and so I\nget something like this.",
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"So I get iD on the left-hand\nside, and then I'm going to",
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"move vD minus V divided by\nr do the right hand side.",
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"So let's start with minus V\ndivided by R, and when I move",
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"that to the right-hand side, it\nbecomes V divided by R. And",
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"then when I move vD over R to\nthe right-hand side, I get",
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"minus vD over R. And I've simply\nnot done much here.",
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"I just have done some\nrearranging of the terms in",
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"equation 1.",
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"And since I haven't done\nanything unique and different,",
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"I'm just going to label\nthis as 1 prime.",
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"So I just got this from equation\n1, and I just labeled",
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"that as 1 prime.",
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"Continuing with the method--",
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"and now I'm summarizing for you\nequations 1 prime and 2--",
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"we want both of them for iD on\nthe left-hand side expressed",
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"as a function of vD.",
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"And in equation 2,\nit is non-linear.",
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"So the graphical method can\nbe summarized as follows.",
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"To start, I want you to\nnotice something.",
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"Essentially all we're trying to\ndo is find a solution, find",
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"a value for vD and iD that\nsatisfies both equations, 1",
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"prime and 2.",
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"that's all we're trying to do.",
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"It's just math here.",
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"There's no circuits here.",
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"We're just doing some relatively\nsimple math.",
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"We just have to figure out what\niD and vD are, and we'll",
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"use the graphical method.",
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"So to do the graphical method,\nwhat I'm going to do is I'm",
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"going to plot the two equations\nin a pair of graphs.",
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"And let me first start by\nplotting equation 2.",
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"And in equation 2, I'm going\nto plot iD equals a",
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"raised to ae bvD.",
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"I'm going to plot iD equals\na times e raised to bvD.",
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"That's my equation 2.",
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"And this is the plot\nthat I get for it.",
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"It's a plot that I had before.",
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"So that's my equation 2.",
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"Notice that this plot here is\nsimply the constraint on iD",
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"and vD imposed by the device.",
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"This little circuit here\ncontaining the Thevenin",
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"equivalent and connected to\nthe device, equation 2 is",
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"simply the constraint imposed\nby the device.",
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"The device properties are such\nthat it is going to constrain",
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"iD and vD into some\nrelationship.",
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"This constraint that I've\nplotted here as equation 2 is",
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"simply the constraint imposed\nby this device.",
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"OK, next, let me\ngo to 1 prime.",
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"And in 1 prime, let\nme go ahead and",
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"plot the 1 prime equation.",
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"And that equation, as we\nrewrite that here, is V",
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"divided by R minus vD divided\nby R. So I'm just going to",
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"plot that equation for you\nin a second graph.",
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"So how do I plot this?",
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"So notice here that when vD is\n0, then iD is V divided by R.",
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"So that is one point on\nthe straight line.",
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"Notice that this is an equation\nfor a straight line.",
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"It's a linear relationship\nbetween iD and vD.",
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"Next, when iD is 0--\nso when iD is 0--",
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"then notice that R and R can be\ncanceled out, and V will be",
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"equal to vD.",
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"So when iD is 0, V\nand vD are equal.",
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"So therefore, vD equals V. So\nthis is the line when iD is 0.",
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"And for that, vD equals V. So\nthen I get this [? long, ?]",
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"straight line for the\nrelationship between vD and iD",
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"according to 1 prime.",
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"So what's the slope\nof this line here?",
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"Can you tell me what the\nslope of this line is.",
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"Let me give you a few seconds\nto think about it.",
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null,
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"OK, from the equation 1 prime--\nfrom this equation--",
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"the slope is simply given by the\ncoefficient of vD, since",
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"the slope is negative, which\nis why the line is inclined",
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"the following way.",
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"So the slope here is simply\nminus 1 divided by R.",
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"So that is my constraint\nthat relates vD to iD.",
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"And where did that constraint\ncome from?",
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"That constraint is the\nconstraint on iD and vD that",
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"has been imposed by the\nrest of the circuit.",
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"So if the first constraint was\nimposed by the device, then",
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"the second constraint is\nimposed by the Thevenin",
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"equivalent that is connected\nto the device.",
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"The Thevenin equivalent was\nthat V and R in series and",
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"those two impose a Thevenin\nconstraint on the terminal",
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"pair that relates vD and iD.",
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"So now my next step is,\ngiven these two graphs",
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"for iD versus vD--",
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"and clearly those are\nmy two constraints.",
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"Graph 2 says that iD and vD\nmust be somewhere on this",
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"trajectory.",
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"Graph 1 prime says, well, I'm\nnot going to let vD and iD be",
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"anywhere else but\non this curve.",
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"That's it.",
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"Both of them are fighting with\neach other and telling each",
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"other, nope, I'm not going to\nallow you to do anything.",
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"You have to be on my curve.",
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"So in this case, I have two\ncurves, and I need a point",
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"that satisfies both curves.",
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"And that is easy enough to do.",
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"And I simply have to go\nand satisfy both these",
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"constraints, and that will\ngive me the answer",
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"for vD versus iD.",
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"Before I do that, so I can go\nand solve it for you, let me",
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"go and pick some values for\nthe various parameters.",
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"So as before, I'm going to pick\nV equals 1 volt, R equals",
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"1 ohm, a, a quarter of an amp,\nand b to be one volt inverse.",
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"So I'll pick the same parameters\nas I had done when",
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"I did the analytical method.",
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"Next, what I'll do is I'll\nsubstitute these parameters",
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"into the two equations 1 prime\nand 2 and rewrite them with",
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"the parameters substituted.",
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"So for 1 prime, I get iD.",
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"Since V is 1 and R is 1, I get\nV divided by R equals 1.",
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"And then since R is\n1, I get minus vD.",
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"So this is 1 prime, once I've\nsubstituted the values.",
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"And then for equation\n2, what do I get?",
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"I get iD equals ae\nraised to bvD.",
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"a is 1/4, so I write\nthat down.",
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"b is 1, and so I\nget vD up here.",
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"So I have my two equations in\nterms of the parameters I have",
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"chosen, and now I can go ahead\nand plot the two equations and",
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"see where they intersect.",
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"I've given you the form of the\ngraph here, and so let me go",
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"ahead and plot this.",
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"Let me go ahead and plot 2\nfirst. 2 looks like this,",
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"where this point is 1/4.",
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"That was the a point.",
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"As I said before, this curve\nis simply my Equation 2.",
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"Then let me go ahead and\nplot equation 1 prime.",
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"And as you recall, that looked\nsomething like this where the",
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"vD intercept was V and the\ny-intercept was given by V",
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"divided by R.",
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"And in this case, it was\n1, and V was also 1.",
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"So those are my two curves, and\nhere's the point where the",
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"two coincide.",
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"And then I just have to go and\nfind the values of iD and vD",
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"where they two intersect.",
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"So at this point, it will be\n0.56 volts, and this point",
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"will be 0.44 amps.",
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"It's the same as what\nI calculated in",
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"the analytical method.",
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"So let me go ahead and write\nthat down. iD equals, in this",
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"case, 0.44 amps, and vD\nequals 0.56 volts.",
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"So basically, I've just\ntaken the two graphs,",
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"superimposed them.",
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"This was 1 prime, and the\nnon-linear one was related to",
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"Equation 2.",
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"So before I jump off to one\nother thing, I can define",
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"something for you.",
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"Notice this curve here.",
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"This is a straight line that\nreflects the Thevenin",
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"constraint that I apply on\nmy non-linear device.",
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"And I mentioned earlier we're\ngoing to do this again and",
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"again and again.",
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"I think you will see this at\nleast 10 more times in this",
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"course where I take a Thevenin\nequivalent of the following",
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"form, some voltage V, some R,\nand apply that in series",
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"across something interesting.",
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"So this line that I see here is\nthe constraint imposed by",
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"the Thevenin equivalent.",
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"And you can see that from\nthe equation 1 prime.",
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"Now, there's a name\nfor this line.",
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"So this line is called the "load\nline." You will see more",
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"reasons for this later.",
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"But this line is called\nthe "load line."",
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"Let me show you one other little\ntrick, just in case you",
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"didn't completely get how I\nsolved the graphical method.",
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"But one little trick here.",
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"If you like PowerPoint,\nyou will enjoy this.",
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"And if you don't like\nPowerPoint, you",
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"will hate me for it.",
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"So if you look at these\ncurves, I have my two",
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"equations in iD-- the 1 prime\nand 2-- and I've plotted them",
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"for you here.",
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"On the left-hand side,\nI plotted equation 2.",
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"Right-hand side, I plotted\nequation 1.",
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"And fundamentally, all that the\ngraphical method is doing",
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"is simply superimposing\nthe two graphs.",
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"And by superimposing the two\ngraphs, it is finding the",
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"point where the two\ncurves intersect.",
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"And this is what you get.",
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"So let me do that again.",
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"So I take one of the curves and\nsuperimpose it on top of",
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"the other curve given the\nsame axes and the same",
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"scales for the axes.",
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"And then I go ahead and find\nthe solution for the point",
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"where the two intersect.",
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null
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]
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}
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</textarea>
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