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2012-01-12 15:51:00 -05:00
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"INSTRUCTOR: Based on the previous method or the",
"previous property that you saw for linear circuits, we are",
"now ready to define the superposition method for",
"circuit analysis.",
"Note that the superposition method",
"applies to linear circuits.",
"And again, to refresh your memory, remember, we are in",
"the EECS playground that we got into by adhering to the",
"lumped matter discipline.",
"And within that, we looked at the node method and the KVL",
"KCL method that applies all over the EECS playground.",
"And then what we'll now do is look at some methods that will",
"apply to only a region of this playground where linear",
"circuits reside.",
"So the first of these methods is the superposition method.",
"And again, it applies to linear circuits.",
"So the method is very simple and inspired by the",
"superposition property that we studied for linear circuits.",
"So the method simply says this.",
"You first find the responses of the circuit to each source",
"acting alone.",
"Think of this as a divide and conquer method where you take",
"your entire circuit, and you find the responses of the",
"circuit to each source acting alone.",
"You turn off all the sources, and you find the response to",
"each source acting alone.",
"Then what you do is simply sum those individual responses.",
"Note that this technique applies to",
"independent sources only.",
"So if I have a linear circuit, then this technique applies to",
"independent sources only.",
"Now we will look at what happens when you have",
"dependent sources and so on in later lectures.",
"OK, let's move ahead.",
"The first step was find the response of the circuit to",
"each source acting alone.",
"But what does it mean for each source to act alone?",
"What it means is that you turn off all the other sources.",
"So for example, suppose we are focused on a",
"voltage source here.",
"What does it mean to turn off this voltage source?",
"In other words, if I want to set this voltage to 0, what",
"does it mean in terms of my circuit?",
"So in terms of my circuit, what this means is that I am",
"going to short the source.",
"So what you do is you short the source, and you end up",
"with a short circuit here.",
"And that is my voltage source having been set to 0.",
"And so I take all my voltage sources in the circuit and",
"simply short them out.",
"And this would be making each of the sources to act alone.",
"OK, next let's look at a current source and see what it",
"means to set a current source to 0.",
"So in this case, what you do is much as you set a voltage",
"source to 0 by shorting it out, you set the current",
"source to 0 by open-circuiting it like so.",
"So you create an open circuit in place of the current",
"source, and that's what it means to set a",
"current source to zero.",
"OK, so now we are ready to go try out an example using the",
"method of superposition.",
"And again, remember the method of superposition has",
"two parts to it.",
"In the first part, you find the responses of the circuit",
"to each source acting alone.",
"And then second, you sum up the partial responses.",
"So let's go back to our old faithful example.",
"In this example, we have two sources, a voltage source and",
"a current source.",
"And let's say our goal is to find the node voltage e taken",
"with respect to this ground.",
"And we are going to use the superposition",
"method to do the circuit.",
"So I showed the circuit on the right-hand corner here.",
"And notice that e is the node voltage that I",
"need to figure out.",
"So by the superposition method, I have two steps.",
"My first step is find the responses of the circuit to",
"each source acting alone.",
"So what I'm going to do is let me start with finding the",
"response of the circuit to the voltage source acting alone.",
"So what do I do to get a subcircuit in which the",
"voltage source is acting alone?",
"So I want you to think about this and try to figure out",
"what the resulting subcircuit is going to look like.",
"I'll give you a few seconds to do that before I do it myself.",
"All right.",
"This is the subcircuit.",
"And what I've done here is I've had to turn the current",
"source off.",
"So notice that out here I create an open circuit, and",
"that is the result of my current source",
"being turned off.",
"And this is my usual ground node.",
"And let me call this node voltage with the voltage",
"source acting alone as ev. So ev is the partial response of",
"the circuit to the voltage source acting alone.",
"Now how do we go about solving for ev?",
"Notice that we have a known pattern here.",
"We have a voltage source connected to a pair of",
"resistors, R1 and R2.",
"And ev is simply the voltage across R2.",
"Now this is a known pattern.",
"So one thing you should keep in mind is that in circuits",
"and electronics, we use patterns a lot.",
"Whenever we see a pattern, we can directly go and write down",
"the what the outward response is going to look like.",
"What I'm going to do is throughout this course, as you",
"we more and more patterns, I am going to alert",
"you to these patterns.",
"So right here is a voltage divider pattern.",
"So for the voltage divider pattern, I can write down ev",
"as being some fraction of the source voltage v. But the",
"divider action is simply R2 divide by R2 plus R2.",
"So it is simply the voltage across R2, and that is given",
"by R2 divide by R1 plus R2.",
"And this is a known pattern that you will find very",
"helpful as you analyze circuits very quickly.",
"Next let's look at I acting alone.",
"Before we do that, what do we have so far?",
"Again, recall, this is the circuit we",
"are trying to analyze.",
"We are trying to get the node voltage e.",
"And so far we have computed the partial voltage ev due to",
"the voltage source acting alone.",
"As the next step-- again, this is still step one of the",
"superposition method--",
"let's look at the partial response of the circuit due to",
"the current acting alone.",
"To get the response for the current acting alone, I need",
"to build a circuit in which the voltage",
"source is turned off.",
"I'll let you think for a couple of seconds as to what",
"the circuit will look like before I draw it out for you.",
"OK, let's go ahead.",
"So to turn off this voltage source, all I have to do is",
"replace it with a short circuit like so.",
"So notice that I have replaced the voltage source with a",
"short circuit, and I've left the current source in place.",
"I have my ground node here.",
"And notice that my goal is to find the partial voltage eI",
"that is a result of the current source acting alone.",
"So notice that I have a second pattern here.",
"And in this case, this is a current divider pattern.",
"Actually, this is a even simpler pattern.",
"I simply have a current that is flowing",
"into a pair of resistors.",
"OK, so I have a current, I, that is flowing into a pair of",
"resistors, R1 and R2.",
"And so I can compute the voltage across the pair of",
"resistors very straightforwardly.",
"So eI is simply the resistance times the current.",
"And the resistance of a paralel pair of resistors is",
"simply R1 R2 divide by R1 plus R2 times the current I.",
"OK, so now I have computed both ev and eI, the two",
"partial voltages with each of the voltages acting alone.",
"And now I am ready for step two of the",
"superposition method.",
"And the step two of the superposition method simply",
"says that you sum the two partial voltages, ev and eI.",
"So my resulting voltage e is simply the sum of the two",
"partial voltages ev and eI.",
"And I have those as R2 divide by R1 plus R2 times v. And I",
"sum that to eI.",
"And eI is R1 R2 divide by R1 plus R2 times I. And",
"there you have it.",
"Not surprisingly, this expression here is identical",
"to the expression that you obtained by using the node",
"method in the previous lecture."
]
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"text": [
"PROFESSOR: In the previous segment, we saw that circuits",
"composed of linear elements were, themselves, linear.",
"In this segment, we will see that linear circuits or linear",
"systems can be analyzed with extremely powerful techniques.",
"We will study two of these properties of circuits,",
"homogeneity and superposition, in this segment.",
"Let's start by studying the property of homogeneity.",
"So what does homogeneity mean?",
"Supposing we have a system with a bunch of inputs one",
"output here.",
"And let's say the inputs are x1, x2, and so on.",
"And for these xes let's say the corresponding output is y.",
"Let me add a little cartoon here, just to",
"make it easy to follow.",
"As they say, a picture is worth 1,000 words.",
"So let's say, if you have apple as an input and you get",
"a different kind of apple as an output, then let's look at",
"what the property of homogeneity says.",
"If I take the same system and if I",
"multiply each of the inputs--",
"x1, x2, and the old output, y-- if I multiply each of the",
"inputs with some constant, alpha, then the property of",
"homogeneity says that the output will also be multiplied",
"by the same value, alpha.",
"So this is the homogeneity property.",
"And in terms of our picture, if you have an Apple and you",
"took a fraction of the apple out, then your output would",
"also have the same fraction of the apple out.",
"So this is the homogeneity property.",
"Next let's look at another extremely powerful technique,",
"superposition.",
"Now superposition is a property of linear circuits,",
"and let's first understand what this property means.",
"So supposing I have a system here.",
"And what I'll do is I'll apply two sets of",
"inputs to the system.",
"So in the first set, let's say I apply the a set of inputs,",
"x1a, x2a, and so on.",
"And let's say that that gives rise to the a set of outputs.",
"And then if I take the same system--",
"just to make it clear, let me mark it as s.",
"If I take the same system and I apply a b set of inputs--",
"so x1b, x2b, and so on--",
"and if I get a corresponding output, yb, then what can we",
"say about the resulting system?",
"And just to make it very clear, going back to our",
"pictures and cartoons here, let's say the a set of inputs",
"corresponds to apples, a for apples.",
"And the b set of inputs goes on to blueberries.",
"So jokingly here, if I feed a bunch of apples to the system",
"and I get applesauce at the output, and on the other hand,",
"if I feed a bunch of blueberries to the system and",
"I get blueberry sauce at the output, then let's do some fun",
"stuff here.",
"Suppose I take the same system and I feed through the system",
"the sum of both of these inputs.",
"In other words, if I feed it x1a, x2a, and so on, and I add",
"to each corresponding input the b set of inputs, so I get",
"x1b here and x2b here-- so notice that this is here, and",
"the b set of inputs is here.",
"So notice that all I've done is I've added up",
"corresponding inputs.",
"What superposition tells me is that my outputs will be a sum",
"of the corresponding inputs.",
"In other words, if I had ya with the a set of inputs",
"alone, and I had yb with the b set of inputs alone, then if I",
"sum the corresponding inputs, the a and the b inputs, then",
"my output will be a sum of the corresponding outputs as well.",
"And again, jokingly, if I feed apples and blueberries to the",
"input, then in this case, jokingly again, maybe I get",
"mixed fruit jam.",
"How does this yield something interesting for us?",
"Let me do a specific example of the property of",
"superposition and see how this will yield a very interesting",
"technique to analyze linear circuits.",
"Again, remember, both the homogeneity and superposition",
"property applies to linear circuits.",
"So let me take a specific instance of",
"superposition here.",
"And so I take some system, a linear system, and let's say I",
"have two inputs that I apply, v1 and v2.",
"And let's say my goal is to find out what the output is.",
"What I want to show you is that, by the method of",
"superposition, I can use a divide and conquer technique.",
"What I can do is I can take the same system and I can set",
"the v2 output to 0 and apply just the v1.",
"And I measure the corresponding output, y1.",
"So y1 here is the partial output, given that I have",
"applied a v1 alone.",
"Similarly, I take the same system and in this case,",
"second time around, I set v1 to 0, and I apply v2 alone.",
"And then let's say I get y2 as my output",
"with v2 acting alone.",
"Then here is something very interesting.",
"So by superposition from a previous example, recall that",
"if I took one set of inputs and a second set of inputs and",
"I summed up the corresponding values, then superposition",
"said my output would be a sum of the individual outputs with",
"just each of those applied alone.",
"In other words, if I show the v1 as 0 and then if I sum to",
"that the second set of inputs, 0 and v2 from here, so that",
"this got applied here and the first set of inputs got",
"applied here, then what superposition would say is",
"that my output would be a sum of the two partial outputs.",
"In other words, y1 would come from here and y2 would be the",
"same as the y2 with the v2 applied alone.",
"So this is an extremely powerful idea.",
"So to recap, what have I done here?",
"I have used a method of superposition to apply a set",
"of inputs in the following way.",
"I started by saying that I have a circuit and I want to",
"apply v1 and v2 to it and compute the output.",
"Now rather than trying to solve the circuit by applying",
"both v1 and v2, what I did was I broke it up into two",
"sub-problems. In the first sub-problem, what I did was I",
"applied v1 alone and set v2 to 0, and then in the second",
"sub-problem, I applied v2 alone and I set v1 to be 0.",
"And I computed the two partial outputs, y1 and y2.",
"The method of superposition then says that if I took the",
"two corresponding sets of inputs and summed them up,",
"then my output would be a sum of the two partial outputs.",
"Notice that at the input here, if you look at these two",
"inputs, the v1 plus 0 is simply the v1, v2 plus 0 is",
"simply v2, which is the same as what I have here.",
"So in other words, I was able to solve the circuit that",
"contained v1 and v2 as inputs by applying just v1 alone and",
"then v2 alone, and then by summing up the two",
"corresponding outputs.",
"This will yield an incredibly powerful method for us called",
"the method of superposition."
]
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"PROFESSOR: In the previous segment, we looked at the",
"Thevenin method, which applied to linear circuits.",
"And in that method, I could take an arbitrary network, n,",
"containing a whole bunch of sources and resistors where",
"the current sources are voltage sources.",
"And I could replace that with its Thevenin equivalent, where",
"the Thevenin equivalent pattern was given by a voltage",
"source VTH.",
"in series with the resistance RTH.",
"Now one of the things that I want you to think about in",
"circuits is that whenever you have circuits containing",
"voltages and properties involving voltages, you can",
"immediately think of complimentary properties",
"involving current sources.",
"So much as I was able to replace an arbitrary network n",
"with its Thevenin equivalent, you can think about replacing",
"the arbitrary network n with some pattern involving a",
"current source.",
"And indeed, that method where you replace an arbitrary",
"network with a pattern involving a current source is",
"called the Norton method.",
"So in the Norton method, what we do is we are able to",
"replace an arbitrary network n with its Norton equivalent,",
"where the Norton equivalent contains the current source IN",
"in parallel with the resistance RTH.",
"The current source in parallel with RTH is a Norton pattern,",
"and you should contrast it with the corresponding",
"Thevenin pattern.",
"So a voltage source in series with a resistor is a Thevenin",
"pattern, and a current source in parallel with the",
"resistance is the Norton pattern.",
"So in the case of the Norton pattern, which contains the",
"current source IN and the Norton resistance RN, in the",
"same manner that you defined the Thevenin equivalent, we",
"can derive the Norton equivalent.",
"So to obtain the current IN, what you do is you obtain the",
"short circuit current seen at the port.",
"So recall for the Thevenin method, I open-circuited the",
"port here, and measured the voltage, and that was my VTH.",
"So the Norton method, what I do is I short circuit this",
"port and measure the current I. That current is IN, or the",
"Norton current.",
"And then I can get a Norton resistance RN--",
"oh, I should say RN here-- the Norton resistance RN, which,",
"incidentally, is identical to the Thevenin resistance RTH.",
"I can get that in the same manner as I got RTH, which is",
"I shut off all the independent voltage sources and current",
"sources and simply take the resistance measure at the",
"port, of course, without the short.",
"I go ahead and measure the resistance seen at the port",
"with all the voltages and currents turned off.",
"OK?",
"That would be my Norton resistance, RN.",
"So again, we stated this pattern, a current source IN",
"parallel to a Norton resistance RN, where this",
"corresponds to the Thevenin equivalent, where I have VTH",
"and RTH in series.",
"So the Norton is the dual of the Thevenin.",
"OK?",
"Where the Thevenin is the voltage in series with the",
"resistance, and its dual is the Norton equivalent, which",
"is the current source in parallel with resistance.",
"An immediate question that comes to mind is how are the",
"Thevenin and Norton equivalents",
"related to each other?",
"After all, they are the duals of one another.",
"And so notice the Thevenin equivalent on the left-hand",
"side and the Norton equivalent on the right-hand side.",
"The resistances RTH and the Norton",
"resistance RN are equal.",
"And then we can also relate the Norton current IN and the",
"Thevenin voltage as follows.",
"IN is VTH divided by RTH.",
"And so if you take the Thevenin voltage and divide",
"that by the Thevenin resistance, you get the Norton",
"current, IN.",
"So this very quickly shows you how you can go from the Norton",
"to the Thevenin.",
"One question I am often asked is when do I use the Thevenin",
"method and when do I use the Norton method?",
"Well, you know, that comes from experience.",
"Like you look at circuits in one of our earlier examples",
"where the external network contained a voltage source in",
"series with a resistance, it made sense to take the rest of",
"the network and turn that into Thevenin equivalent.",
"However, if I have a basic circuit containing a current",
"source and a resistance in parallel, then it might be",
"easier to take my external network and convert that into",
"a Norton equivalent.",
"So, really, when you want to get things in",
"parallel, think Norton.",
"When you want to get things in series to simplify analysis,",
"think Thevenin."
]
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"text": [
"PROFESSOR: So far in 6.002, you've seen how by making some",
"simplifications of Maxwell's equations, we've ended up with",
"very simple algebraic laws that can be",
"used to analyze circuits.",
"And in this little playground of ours,",
"really good things happen.",
"This is our EECS playground, where by making some",
"simplifications, we've been able to really come up with",
"very simple techniques for analyzing circuits.",
"So today, as we build up our arsenal of techniques for",
"analyzing circuits, we'll look at a further simplification.",
"We're going to look at a part of this playground in which",
"linear circuits reside.",
"These kinds of circuits allow us to use some even more",
"powerful techniques to analyze them.",
"These new techniques that you will learn today include",
"superposition, Thevenin and Norton.",
"It's pretty amazing to think of these techniques as the",
"bazookas of EECS.",
"OK, let me start with a quick review as is my usual practice",
"of what we've seen so far.",
"So the circuit analysis method you've seen so far include,",
"first of all, the KVL, KCL method.",
"In this method, the KVL, KCL, and the element rules are used",
"to analyze circuits.",
"The KVL method, as you've seen before, is one where you take",
"the various voltages in a loop in a circuit and they should",
"all sum to zero.",
"Similarly, the currents at any given node,",
"must also sum to zero.",
"That's KCL.",
"And so you apply these KVL, KCL and element rules to",
"circuits, and then go solve them.",
"But we did see that they ended up with a very large number of",
"equations that were quite messy to resolve.",
"The second technique that we looked at last week included",
"circuit decomposition rules, where you applied some of",
"these rules to compose resistors and other elements,",
"and solved circuits that way.",
"And then, we looked at the node method, where you chose a",
"ground in a circuit.",
"And then you applied KCL at each of the nodes, the",
"reference to the ground node.",
"And what was interesting is that the KVL was implicit in",
"the pattern that you used in the node method, where the",
"pattern was the ei minus ej, divided by the resistance of a",
"given [UNINTELLIGIBLE].",
"And this was a pattern that you really borrowed from KVL,",
"and this was implicitly used in the node method.",
"The node method, of course, is the workhorse of 6.002 and can",
"be applied to all kinds of circuits.",
"It really applies in the entire playground of ours.",
"So when in doubt, simply go ahead and",
"apply the node method.",
"OK, as I promised, we'll now look at the smaller region of",
"the playground, and see if some interesting things happen",
"in that playground.",
"So to give you a quick review, let me start by looking at",
"some basic properties of linear circuits.",
"Following that, I will look at the superposition tool kit for",
"your arsenal of techniques to analyze circuits.",
"And then I will talk about the Thevenin method.",
"And finally, I'll talk about the Norton method.",
"OK, so let's get started with linear circuits and see what",
"linearity means.",
"So I want to do an example using the following little",
"circuit, in which I have a voltage source",
"and a current source.",
"And my goal here is to find the voltage at this node e.",
"And I'm going to pick a ground node here and find the voltage",
"e referenced to this ground node that I've just",
"marked out for you.",
"So to analyze a circuit, you're supposed to write down",
"the node equations.",
"And write down the mode equations for the node.",
"That's the only node for which we have an unknown.",
"Recall that I have another node here, but because that is",
"connected by a voltage source to ground, its voltage is",
"simply V.",
"OK, let's start by writing the node equations for the node",
"with unknown voltage e.",
"So let's start by looking at the current that goes from the",
"node with voltage e to the left.",
"And so I take e minus V divided by R1.",
"That is the current headed in the direction from e to v.",
"And then I add the current that is going down this edge.",
"That is just simply e divided by R2.",
"And then I take the current that is",
"headed in this direction.",
"And that is simply minus I. According to the KCL rule,",
"they all sum up to 0.",
"So the first thing you will notice as you write the node",
"equation right away is that this equation here shows you",
"that it is linear in the various voltages and currents",
"e, V, and I. In other words, you don't have terms, no terms",
"that are eV, or V squared, or VI, and so on.",
"So this circuit here is",
"characterized by the node equation.",
"And notice that it is linear in the e, V, and I terms.",
"OK, let's go ahead and look at this in some more detail.",
"And I'm going to simplify the equation a little bit more to",
"really get a much clearer sense of the linearity here.",
"So I've written down the node equation.",
"And I'm going to simplify by rearranging some of the terms",
"in the equation.",
"So let me collect the various coefficients.",
"So let me write down e on this side, and look at and collect",
"all the terms that multiply e.",
"So I see a 1 by R1 term here.",
"Let me write that down.",
"I see a 1 by R2 term.",
"And so e is my unknown.",
"Let we collect the knowns on the right-hand side.",
"So one of the knowns is I. So let me move it to the",
"right-hand side and put that there.",
"The other known is this one here, V divided by R1.",
"I'm going to take it to the right-hand side.",
"It becomes plus V divided by R1.",
"So let me introduce a little bit of nomenclature here.",
"So notice I've rearranged the terms. And I get some",
"coefficients multiplying e, and that is a sum of the",
"sources, V and I, where V has been scaled by",
"1 divided by R1.",
"So when you look at these terms, let's understand a",
"couple of things.",
"Let's understand a couple of things right away.",
"So if you look at the coefficient of the unknown e,",
"in general when you have a much larger circuit, it is not",
"a single value, but rather it is a matrix of values.",
"And it's just good to know these terms. It's called a",
"conductance matrix.",
"Similarly, it's usually given the symbol G.",
"Similarly, the node voltages becomes a column vector of",
"node voltages, and it's given the label e.",
"And then, on the right-hand side, you have the linear sum",
"of sources, and that is given the term S. And so you get the",
"form G times e equals S, where G is a matrix usually, e is a",
"column vector.",
"And similarly, the sum of sources ends up being a column",
"vector as well.",
"So with that bit of cultural background in terms of what",
"these things are called in general, let me",
"simplify them further.",
"And what I get now is--",
"let's try to have e on the left-hand side all by itself,",
"and divide both sides by the coefficient of e.",
"And this guy here, if I simplify it, becomes R1 R2",
"divided by R2 plus R1.",
"So when I multiply out both sides by that, what I end up",
"getting is I have a V here, and I have the I here.",
"So what is the coefficient of V?",
"So I multiply both sides by R1 R2.",
"And so R1 and R1 cancel out.",
"So I end up with R2 at the top.",
"And I get R1 plus R2 down here.",
"Similarly, so what do you think we get for I here?",
"For I, I end up with R1 R2 divided by R1",
"plus R2 down here.",
"So notice that the linearity becomes much more obvious at",
"this point where my unknown e is some constant coefficient",
"times the voltage v and some constant coefficient times the",
"current I.",
"In other words, e is of the following form. e is some",
"constant coefficient times my first source, some constant",
"coefficient times V2, and so on.",
"Plus some other coefficient times I1, some other",
"coefficient times I2, and so on.",
"So if you look at this form here, notice that this is",
"linear in the sources VIs and IIs."
]
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"text": [
"PROFESSOR: OK, so i1 is very quickly calculated by taking",
"the difference of V and VTH and dividing that by the sum",
"of the two resistances.",
"So the current i1, flowing in this direction, is given by V",
"minus VTH divided by R1 plus RTH.",
"And there I have it.",
"Notice that by converting the arbitrary network N into the",
"Thevenin equivalent, I turn this into a really trivial",
"problem where I could write down i1 simply by inspection.",
"Finally, before we move off the topic of the Thevenin",
"network and the Thevenin method, it is worthwhile",
"staring at this pattern here comprising the voltage source",
"in series with the resistance and the equation v equals vTH",
"plus RTHi that governs the vi of the Thevenin pattern.",
"So it's worthwhile looking at this graphically as well.",
"So I'm going to show you a little graph here and plot the",
"iv relation for this Thevenin pattern.",
"And usually the way you draw these iv relations is you try",
"to find out an open-circuit voltage.",
"In other words, try to find out of the voltage when the",
"current is 0, so open-circuit voltage when i is 0.",
"So in this case, when i is set to 0 and there's no current,",
"then the open-circuit voltage here is simply",
"vTH so v equals vTH.",
"So in other words, when the current is 0,",
"my voltage is vTH.",
"So I found one point on the iv relation for the Thevenin",
"equivalent pattern.",
"Next, let me find out the short-circuit current.",
"And for the short circuit, v is set to 0.",
"So in this case, I apply a short circuit here, and I'm",
"going to measure the current.",
"So the current in this case, i is given by minus",
"vTH divided by RTH.",
"Notice that because vTH is positive here, the current",
"flows down in this direction and is given by vTH divided by",
"RTH, and that is in a direction opposite to i.",
"And so therefore, I get the minus sign.",
"So my current is given by minus vTH divided by RTH.",
"And so for 0 volts, I get some current here, and that is",
"given by vTH divided by RTH.",
"And so this voltage here is my open-circuit voltage, and this",
"current here is my short-circuit current, OK?",
"So this is my short-circuit current, and that's my",
"open-circuit voltage, and my iv relation is drawn by simply",
"connecting the two points.",
"And the slope of this curve is given by 1 divided by RTH.",
"So here again, to very quickly summarize, I've drawn the iv",
"relation for this definite pattern, which has a vTH in",
"series with an RTH and found two points on the iv relation.",
"One is on the voltage axis, and one is on the current axis",
"by measuring the open-circuit voltage and the short-circuit",
"current, and then drawing a line through them."
]
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"SPEAKER 1: At this point, I'd like to do a quick summary of",
"all that we have covered in 6.002 so far.",
"We began life in 6.002 by discretizing matter into",
"lumped elements.",
"And we agreed to observe the lumped matter discipline.",
"What this did was it turned real life elements, as our",
"little pickle on which we performed the fun experiment a",
"little while ago.",
"And we were able to turn that into lumped elements, such as",
"a resistor, which we could use in circuits to obtain",
"properties such as voltages and currents.",
"And when we took these lumped elements and put them together",
"into circuits, we got the lumped circuit abstraction.",
"All right.",
"So once we built these circuits containing these",
"lumped elements, we then went on to develop an arsenal of",
"tools that we could use to analyze circuits.",
"So notice that the fun part of 6.002 doubled",
"has really not begun.",
"We just made some assumptions that allowed us to make the",
"transition from physics and the complications of Maxwell's",
"equations into some very simple territories where",
"algebraic equations rule.",
"And then, we sat down and developed some analysis",
"techniques for the circuits that we built.",
"Notice we really haven't built anything useful or",
"done any fun stuff.",
"We are still developing our tools for the",
"circuits that we created.",
"So we developed a bunch of circuit analysis methods.",
"We started off with the KVL, KCL method.",
"And in this method, you took a circuit.",
"And you wrote the KVL, KCL equations and also the I-V",
"relations for each of the elements.",
"And then you got a whole series of linear equations.",
"And we went and solved them.",
"The second technique we learned was applying",
"simplification rules, where you combine resistances in",
"series and so on and so forth, a resistance in parallel or a",
"voltage in series, and then used those to",
"simplify the circuit.",
"The third method that we developed was the node method.",
"The node method is really the workhorse of our industry.",
"When in doubt, use the node method.",
"The node method applies to all circuits and, in fact, is used",
"for linear or nonlinear circuits.",
"And by and large, when in doubt, we just",
"apply the node method.",
"So if you look at these techniques here, they apply to",
"any network.",
"And then, we looked at another class of techniques, the",
"superposition, Thevenin, and Norton.",
"In the superposition method, we simplified a circuit into",
"subcircuits by shutting off all sources save for one.",
"And then we found the subcircuit result.",
"And then we added up all the results.",
"That was the superposition method.",
"And then we learned about the Thevenin and Norton methods.",
"The Thevenin method, it determined a Thevenin pattern",
"circuit to replace an arbitrary network, while the",
"Norton method replaced an arbitrary circuit with the",
"Norton pattern.",
"Now, don't forget that the superposition, Thevenin, and",
"Norton methods apply only to linear circuits.",
"So these techniques, superposition, Thevenin and",
"Norton, are extremely powerful.",
"And they will work with linear networks.",
"And we will see a lot of situations where we have",
"linear networks.",
"And so these are very important techniques.",
"And then, of course, for nonlinear networks, or for",
"networks that contain nonlinear elements, we will",
"largely use the node method, where the node method is",
"really the workhorse of our industry.",
"Now people often ask me, when do I use which technique?",
"You've taught us the KVL, KCL method, the node method, the",
"Norton method.",
"Where do I apply these techniques?",
"The way to think about it is just imagine if you are a",
"carpenter, and you have a tool chest full of tools.",
"You know, you have a chisel.",
"You have a screwdriver.",
"And you have my favorite tool, a chainsaw.",
"So you have all of these tools.",
"And if you asked a carpenter, well, where do you use which",
"tool, what do you expect the carpenter would respond?",
"The carpenter would say, well, that's why you pay a carpenter",
"the big bucks.",
"They know what tool to use where, and they're practiced",
"in using it.",
"Similarly, as engineers, as you get more practice, you",
"will begin using these tools and beginning to figure out",
"where to use what tool.",
"But fundamentally, the superposition, Thevenin, and",
"Norton methods apply to linear networks.",
"And the node method applies anywhere.",
"So when in doubt, just use the node method.",
"And much like a carpenter would use a chainsaw to chop",
"down a tree, you can use your superposition method to",
"analyze circuits that contain a number of sources, where you",
"can break it down into very simple subcircuits.",
"But of course, the circuit has to be linear to begin with.",
"So really, you will develop experience in terms of where",
"to apply which method.",
"But again, don't forget that the node method will apply in",
"any situation."
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"text": [
"PROFESSOR: In this segment, we will develop a second method",
"that will work for linear circuits.",
"This method is called the Thevenin method, and its",
"complementary method, the Norton method.",
"So in order to develop this method for linear circuits,",
"suppose you have some arbitrary network, N. And",
"let's say in this arbitrary network you have a whole bunch",
"of resistors, you have a whole bunch of voltage sources, Vm,",
"and a whole bunch of current sources as well.",
"OK, so you have a bunch of voltage sources.",
"And these are all independent voltage sources for now.",
"V1, V2, V3, all the way through VM.",
"You have a whole bunch of current sources, I1, I2, I3,",
"all the way through to In.",
"Suppose I am interested in applying some kind of",
"excitation due to an external current source with current i.",
"And let us say that all I care to do is to determine the",
"voltage, V. So the setup here is I have an arbitrary",
"network, N, with a ton of resistors and voltage sources",
"and current sources, all independent.",
"And I apply an external excitation, and",
"it's a current i.",
"And just assume that all I want to do is to determine the",
"voltage, V. Assume that I don't care about any internal",
"voltages, Vi, or any internal currents, Ij, and so on.",
"Let's say I don't care about any of",
"those internal currents.",
"All I care about is the voltage, V.",
"So in a situation like this where you care about some",
"voltage or current at some point, you can use the",
"property of linearity and the resulting superposition",
"property to develop an extremely powerful method to",
"get you to your answer very quickly.",
"So in order to develop this method, let me use the",
"superposition technique and try to figure out the voltage,",
"V. So the voltage V will be some combination of the linear",
"sources in this linear circuit.",
"So let's say, for example, I want to figure out the",
"contribution to the voltage V due to the voltage source, Vm.",
"What I would then do is I would set i to 0, I would set",
"all the currents, I1 through In, to 0, and I would set all",
"the voltages, V1 through Vm minus 1, to 0 as well, and",
"just look at the impact on V due to Vm acting alone.",
"So the impact on V due to Vm acting alone, and notice that",
"once all the currents have been turned off and all the",
"other voltages have been turned off, I'm left with a",
"big resistor network.",
"And I'm going to get some multiplier here.",
"Let's label that alpha m.",
"So that will be the result of the source Vm acting alone on",
"the output voltage, V.",
"Similarly, I can do the same thing for all the voltages.",
"I can do it for V1 acting alone.",
"I can do it for V2 acting alone, and so on and so forth.",
"For each of those voltages, V1, V2, and so on, I get a",
"term that looks like this.",
"I get alpha 1 V1, alpha 2 V2, and so on and so forth, all",
"the way through alpha m Vm.",
"And so what I get is a sum over m of all of these terms.",
"By applying V1 acting alone and V2 acting alone and so on,",
"I get a term, alpha 1 V1, alpha 2 V2, and so on.",
"Next, I'll move on to the currents, and in this case,",
"supposing I set all the currents, say, for In--",
"I'm going to set the small i to 0, and I'm going to set all",
"the Vm's to 0, and I'm going to set all the currents, I1",
"through In minus 1, to 0.",
"So in that situation, I'm going to get some factor, Bn,",
"that applies to In.",
"So I've got some factor Bn, beta n, that applies to In.",
"And that is the result of In acting alone.",
"Similarly, I can have I1 acting alone, I2 acting alone,",
"and so on, and for each of them, I get a term, beta 1 I1",
"plus beta 2 I2 and so on and so forth.",
"And I sum them all up as well over all n.",
"So with i acting alone, I'm going to set all of the other",
"In's and Vm's to 0, and just have i acting alone.",
"And so therefore I will get an i, and I'm going to have some",
"multiplier.",
"And in this case, notice that when all the Vm's and all the",
"In's are set to 0, I'm just left with a big resistor",
"network here.",
"And so it will look like a big effective resistor looking",
"into this port.",
"And so let me write that down as some R multiplied",
"by the current i.",
"So notice now that the voltage V is a sum of a bunch of",
"components that arise from the voltages acting alone, the",
"currents acting alone, and also the external excitation",
"acting alone.",
"And just recall, how did I get this?",
"How did I get each of these terms?",
"So this term which sums up all the voltages I got by setting",
"all the In's to 0 and by setting the external",
"excitation to 0.",
"Similarly, I got the second term here by setting all the",
"voltage sources to 0, and also by setting the external",
"excitation to 0.",
"And finally, this last term, I got that by setting all the",
"currents to 0 and all the voltages to 0.",
"So what we're going to do now is by using this insight,",
"we're going to build up a new technique.",
"And to give you a preview to that technique, I'm going to",
"show you a trick that will enable you to replace this",
"network, N, by something very, very simple.",
"In fact, I'm going to show you that we're going to be able to",
"replace that entire network by some voltage source and some",
"resistor acting in series.",
"And this will be able to replace this entire network,",
"N, for the purpose of computing the voltage, V. So",
"just think of that as a preview of upcoming",
"attractions.",
"So moving on, let's take a look at the equation that we",
"just developed and understand that a little bit more.",
"So if you look at the equation here that we developed, V is",
"some component related to Vm's, some component related",
"to In's, and some component related to the external",
"excitation, i.",
"So if you look at this term here, if you look at the alpha",
"m term here, that has no units.",
"It has no units because I have a voltage on the left-hand",
"side and it's multiplying a voltage.",
"Similarly, beta and In must result in a voltage, and so",
"therefore beta n must have resistance units.",
"And then similarly, R that is multiplying i must have",
"resistance units as well.",
"So notice that if you look at the Vm-related term and the",
"In-related term, then what you see is that together they are",
"independent of the external excitation, and the behave",
"like a voltage.",
"Similarly, if you look at the right-hand term here, Ri,",
"that's a resistance multiplied by a current, and of course,",
"that also has resulting voltage units.",
"So let's look at the first part here, the part comprising",
"the Vm's and the In's.",
"That behaves like a voltage, and it is independent of",
"external excitation.",
"And so what I'm going to do is I am going to call it VTH.",
"And don't worry about why I'm calling it VTH right now, but",
"let's assume that I call it VTH.",
"So that's a first component that is independent of",
"external excitation and behaves like a voltage.",
"Now if you look at the second component, as I said earlier,",
"the second component clearly depends on the current, i.",
"But if you look at the multiplier R, then that is",
"also independent of external excitation and",
"behaves like a resistor.",
"So let's call that RTH.",
"So again, if you look at the term that gives you V, it will",
"now comprise a portion that we've called VTH.",
"And it has a second component that relates to the external",
"excitation, i, by is multiplied by something we",
"call RTH, where the RTH is independent of the external",
"excitation, i.",
"So what I've just done is I've written down the excitation as",
"a sum of a voltage here that I decided to call VTH, and I've",
"also said that the second part is RTH times i.",
"And that is equal to the voltage, V. So notice V equals",
"VTH plus RTH i.",
"So in other words, if you look at this expression here, as",
"far as the external world is concerned-- and again, for the",
"purpose of computing the voltage, our network N is",
"completely indistinguishable from another network.",
"What does that network look like?",
"So this whole network, N, can be replaced by the following",
"network that I'm going to show you in a quick second.",
"So I have a voltage, VTH, here, and I have",
"a resistance, RTH.",
"And this is my voltage, V. And what I've done here is,",
"according to this equation, V equals VTH plus RTH i--",
"I have simply expressed that in circuit form as a little",
"pattern showing a VTH and then with an RTH in series.",
"And this is easy to see, that when i is 0, the voltage at",
"the output here is VTH.",
"And then if I have a short circuit where V is 0, then the",
"current through that short circuit will simply be given",
"by VTH divided by RTH.",
"And so this little pattern here can now replace my",
"network N for the purpose of computing V.",
"This network is called the Thevenin equivalent network.",
"And so I can take this network, replace the arbitrary",
"network N with this network, the Thevenin equivalent",
"network, and then I can connect my external",
"excitation, i, and then compute the voltage V that",
"appears across the terminals.",
"In other words, what I've done is, using the Thevenin method,",
"I can take an arbitrary network N. I can figure out",
"the VTH and RTH that relate solely to that network's",
"properties, and then replace that network with",
"VTH and RTH in series.",
"And the series connection is called the",
"Thevenin equivalent network.",
"So you will see this is another pattern.",
"This is called the Thevenin pattern.",
"So in circuits and electronics courses, you will see the",
"Thevenin pattern very often.",
"And in this pattern, you have a voltage source in series",
"with a resistor.",
"And again, it's easy to see that when you have I being 0,",
"then the Thevenin voltage, VTH, appears at the output",
"terminal as a VTH.",
"And similarly, if you short the terminal pair here where V",
"is 0, then your current is going to be given by the",
"expression, VTH divided by RTH.",
"And since the current is going to be in the direction",
"opposite to this current, i, there will be a negative.",
"But essentially you can see why this expression, V equals",
"VTH plus RTH i, directly relates to the Thevenin",
"pattern with the voltage source in series with the",
"resistance."
]
}

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"text": [
"PROFESSOR: Let's move on from here, now that we've seen the",
"Thevenin pattern.",
"What I'd like to do is continue analyzing this a",
"little bit more.",
"As I said earlier, the Thevenin pattern replaces the",
"arbitrary network, N. And VTH here in the Thevenin network",
"is simply the open circuit voltage seen at",
"the terminal pair.",
"In other words, if I looked at network N and I simply",
"measured the voltage at the terminal pair, it has to be an",
"open circuit voltage.",
"That is, without anything connected there, then whatever",
"voltage I measure there would be the Thevenin voltage.",
"Notice that you can see the same thing here.",
"If I measure the voltage at that point, without any",
"external excitation, with an open circuit here, then VTH",
"would directly appear at the terminal, and I would be able",
"to measure the Thevenin voltage.",
"Similarly, what is RTH ?",
"RTH is simply the resistance of the network seen at the",
"terminal there, with all the voltages, Vm's, and all the",
"currents, In's, set to 0.",
"In other words, the way you can find the Thevenin",
"resistance of a network is simply set all the voltages",
"engines and all the currents to 0, and then go ahead and",
"simply measure the resistance looking into",
"that terminal pair.",
"And that will give you RTH.",
"Again, as a quick summary, given any network n, and this",
"case a network with independent voltage and",
"current sources, I can find a Thevenin equivalent which",
"comprises a Thevenin voltage, VTH, and a",
"resistance, RTH, as follows.",
"I open-circuit the connection.",
"In other words, I don't connect anything to the",
"terminal pair when I'm measuring the voltage V. I",
"measure the open circuit voltage there,",
"and that is my VTH.",
"Then I turn off all my current sources and voltage sources",
"inside the network N, and simply measure its resistance.",
"And that is RTH.",
"So that's how I can get RTH and VTH by measurement.",
"And for the purpose of computing some property such",
"as the voltage when an external excitation is",
"applied, I can use those values of VTH and RTH to",
"replace the entire arbitrary network N with this Thevenin",
"equivalent pattern.",
"This is a very, very powerful technique, and we will see",
"some examples of this shortly.",
"OK.",
"So we are ready to define the Thevenin method.",
"So in the Thevenin method, you are given some network N,",
"maybe a horrendous network with a whole bunch of voltage",
"sources and current sources and resistors and",
"so on and so forth.",
"And you have some other external",
"network, E. E for external.",
"Then what I can do is, by the Thevenin method, I can break",
"this down into a simpler problem as follows.",
"What I would do as a first step is I would replace the",
"network N with this Thevenin equivalent.",
"And then as my second step, I would solve with the external",
"network, E.",
"So in other words, what I would do is I would replace",
"the network N with its Thevenin equivalent as my",
"first step.",
"And then I would go ahead and solve with the external",
"network E in my second step.",
"And notice that I can compute RTH and VTH for the Thevenin",
"equivalent using the discussion from a few minutes",
"ago, where I can measure it by measuring the open circuit",
"voltage and the effective resistance of the network with",
"all sources turned off.",
"Or I can also compute it by using superposition and",
"turning off each of the voltages one by one.",
"I can do it in either way.",
"So this is the Thevenin method.",
"Again, as a first step, I replace the network N with a",
"Thevenin equivalent, and then I solve for the external",
"network, E.",
"Now you might wonder, when is this useful?",
"So it turns out that in many cases where all you're",
"interested in is some value, V for example, and you don't",
"really care about what's going on inside this horrendous",
"network N, then you can simplify your life",
"tremendously by replacing that entire network with its",
"Thevenin equivalent, and then solving the circuit for the",
"voltage V based on that Thevenin equivalent.",
"Again, this is a process where we divide up the problem into",
"two simpler components, finding a Thevenin and then",
"solving the resulting circuit.",
"So let us now move on to an example and go to our old",
"faithful little circuit here.",
"And in this example, let us say that my goal here is to",
"find out the current, I1.",
"All I care about is to find out the current, I1.",
"Now of course I can go ahead and apply the Norton",
"method and do it.",
"But let us say that in this case I want to apply the",
"Thevenin method and use this as an example, and I'm going",
"to show you how simple it can get.",
"So to apply the Thevenin method, remember the first",
"part was to identify some sub-network and replace that",
"by its Thevenin equivalent.",
"So what I'm going to do is I'm going to identify this network",
"here as the arbitrary network N whose Thevenin equivalent",
"I'm going to find.",
"And then I'm going to view the network on the left-hand side,",
"the voltage and resistance, as the network E. And the reason",
"I'm going to do that is, notice that if I replace",
"network N with a Thevenin equivalent, which would be",
"simply a resistance and a voltage, notice that the",
"resulting circuit would be very simple to analyze,",
"because it would have pair of voltages and a pair of",
"resistances in series.",
"Let me we go ahead and find the Thevenin equivalent for",
"network one in step one of the Thevenin method.",
"So I'm going to replace network N by its Thevenin",
"equivalent.",
"So my network E is going to stay the same.",
"So E hasn't changed.",
"I'm trying to figure out the value of I1.",
"And I'm going to replace the network N with this Thevenin",
"equivalent.",
"So the Thevenin equivalent will fit in here into this",
"box, and let me draw the Thevenin pattern, just to be",
"clear, where this is R Thevenin",
"and this is V Thevenin.",
"And we'll replace network N with its Thevenin equivalent,",
"and in order to do that, I need to find all the values of",
"RTH and VTH.",
"And let me go ahead and do that next.",
"Continuing with step one of the Thevenin method, I have my",
"network N. And I need to find out VTH and RTH for that",
"network N. And again, as a reminder, this is what I have",
"for VTH and RTH.",
"So let me first find out the VTH.",
"And as I told you earlier, VTH is simply the open circuit",
"voltage of the network N. So how do I find out what the",
"open circuit voltage of that network N is?",
"Well, it's pretty straightforward.",
"Let me draw the network here.",
"Let's have an R2 here, and I have a current here.",
"And the open circuit voltage is my Thevenin voltage, VTH.",
"So here's your simple pattern.",
"I have a current I that is flowing into a resistance R2.",
"And so therefore VTH is simply I times R2.",
"So that gives me VTH.",
"So the second step, I need to find out RTH.",
"And the way I'm going to find out RTH is I'm going to turn",
"off all the independent sources inside network N, and",
"then measure the resistance looking in.",
"So this was my network from before.",
"This corresponds to this network.",
"I'll do the same now.",
"So I'm going to put my network in there.",
"And the difference is that I'm going to turn off all the",
"voltages and currents that are",
"independent inside that network.",
"So as I draw it out, R2 stays the same.",
"But here, what I'm going to do is I'm going to turn off my",
"current source and open-circuit that.",
"So I'm going to turn off my current source, and so now,",
"RTH will simply be the resistance looking in.",
"That is easy enough to do.",
"So RTH is simply R2.",
"So there you have it.",
"I've computed VTH and RTH, and that lets me develop my",
"Thevenin network, and I'm ready for step two.",
"So as I said earlier, VTH and RTH are given",
"by these two values.",
"And what I'm going to do is I'm going to replace the",
"network N with its Thevenin equivalent.",
"Network E stays the same.",
"I show that here.",
"That hasn't changed.",
"But now what I'm going to do is I'm going to replace the",
"network N with its Thevenin equivalent like so.",
"But I get the RTH in series with the VTH, and the VTH is",
"given by IR2 and RTH is given by R2.",
"So in step two, I now need to solve with the external",
"network E applied to the Thevenin equivalent network",
"that as replaced the original network N. So this is, again,",
"a very simple circuit, and recall all that I",
"cared about was I1.",
"And so, let me let you think for a couple of seconds about",
"what the value of I1 is before I go and do it myself."
]
}