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fischerl
2012-01-31 16:55:37 -05:00
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5
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"text": [
"SPEAKER 1: So I want 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, 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 the register R1,",
"the voltage source V0 and R4 are connected.",
"Just make sure your terminology is correct.",
"So what I'll do is let me make some quick measurements for",
"you, and show you that these KVL and KCL are indeed true.",
"So the circuit's up there.",
"So let me take some measurements.",
"And why don't you write down what I measure on the board.",
"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.",
"So you see the circuit up there.",
"[INAUDIBLE]",
"So I get three volts for the voltage from C to A. So why",
"don't you write down three volts.",
"OK, so the next one is minus 1.6.",
"And so that will be I'm doing AB, VAB.",
"And then let me do the last one.",
"And it is minus 1.37.",
"Within the bonds of experimental error, notice",
"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.",
"This time around, you'll get to see the measurements that",
"I'm taking as well.",
"What I have here is three wires that I have pulled out",
"from D. And this is the node D. So three wires coming into",
"the node D just to make it a little bit easier for me to",
"measure stuff.",
"So everybody keep your fingers crossed so I don't",
"look like a fool here.",
"Hope this works out.",
"So it's about 10 millivolts peak to peak out there.",
"Let's say that if the wave form rises on the left hand",
"side, it's positive.",
"So it's positive 10 millivolts.",
"And another positive 10 millivolts.",
"So that's 20 millivolts.",
"And this time it's a negative, roughly 20, I",
"guess, negative 20.",
"So I'm getting, in terms of currents, I have a positive",
"10, a positive 10 and a negative 20 that adds up to 0.",
"But more interestingly, I can show you the same thing by",
"holding this current measuring probe",
"directly across that node.",
"And notice that the net current that is entering into",
"this node here is zero.",
"So that should just show you that KCL does indeed hold in",
"practice, and it is not just a figment of our imaginations."
]
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"All right, this is the sub circuit.",
"And what I've done here is I've have 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 note.",
"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",
"registers, 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",
"what the output response is going to look like.",
"What I'm going to do is throughout this course, as we",
"see more and more patterns, I'm 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.",
"I can write down ev as being some fraction of the source",
"voltage v. And by the divider action, we simply R2 divide by",
"R1 plus R2.",
"So it is simply the voltage across R2, and that is given",
"by R2 divided by R1 plus R2.",
"And this is a known pattern that you'll 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're trying to analyze.",
"We are trying to get the voltage e.",
"And so far, we have computed the partial voltage ev, due to",
"the voltage source acting alone.",
"As a 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."
]
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"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": [
"SPEAKER 1: In this sequence, are going to look at a whole",
"bunch of techniques for analyzing circuits.",
"Think of these as your tool chest. So you are embarking on",
"6.002 and going on to build lots of useful electronic",
"systems.",
"But before we go and build these systems, much like a",
"carpenter who needs to go build a house, much like a",
"carpenter has to go and buy a set of tools from the local",
"store, we need to go and make sure that you have the tools",
"in your tool chest to analyze these circuits.",
"This tool chest will include things like the KVL KCL",
"method, the Node method and so on.",
"But before I begin talking about these techniques for",
"your tool chest, I'd like to do a extremely quick review--",
"as is my usual practice--",
"of what you've covered so far.",
"So remember our EECS playground.",
"We've gotten into this place by promising that we are going",
"to observe the lumped matter discipline.",
"By observing the discipline recall that the complexities",
"of Maxwell's equations and the differential equations that",
"you have to solve-- the partial differential",
"equations-- goes away.",
"And you're left with very simple linear equations.",
"And you will see a lot of that today.",
"As you make the lumped matter discipline, you end up with",
"these lumped elements, like resistors and voltage",
"sources and so on.",
"And what you do with these lumped elements is we can",
"label what are called the branch variables or the",
"terminal variables for these lumped elements, OK?",
"So for example, v is the voltage across the element.",
"And i is the current through the element.",
"The power consumed by the element is given by vi.",
"Continuing with our review, the lumped matter discipline",
"enables us to create the lumped circuit abstraction.",
"So you take these lumped elements and you connect them",
"with ideal wires.",
"So in this case, I have a resistor which is a lumped",
"element, a voltage source.",
"And I connect them with these ideal wires.",
"And what you end up with is called a lumped circuit.",
"But this is your lumped circuit abstraction.",
"Well as we made the transition from physics to EECS by",
"adhering to the lumped matter discipline.",
"I had discussed in the last sequence that Maxwell's",
"equations turn into very simple algebraic equations.",
"And these are captured by Kirchoff's voltage and current",
"laws, called KVL and KCL.",
"So KVL says that for all loops in your circuit, the sum of",
"the voltages around the loops add up to zero.",
"Similarly for all the nodes in the circuit, the currents that",
"enter into a node add up to zero.",
"And in the same matter, if I just summed the currents",
"leaving a node, they also add up to zero.",
"So whether you sum the currents entering the node or",
"currents leaving a node, they would add up to zero by KCL.",
"And the beauty of this was by adhering to lump matter",
"discipline, recall the complexities Maxwell's",
"equations whether an integral form or",
"differential form went away.",
"And we were now left with extremely simple algebraic",
"expressions of the sort.",
"And you will see shortly that you will end up solving",
"extremely simple linear equations",
"as we analyze circuits.",
"So as one example here is a linear circuit.",
"It has one, two, three, four, five, six elements.",
"It's got a voltage source of a voltage v zero.",
"It's got a resistor, r one and so on and so forth.",
"So for this circuit according to our abstraction and the",
"laws of KVL and KCL, the following",
"are going to be true.",
"So for instance if I look at node a, then according to KCL,",
"the currents entering the node must be zero.",
"So in other words, the current here is ica.",
"And I can sum to that the current",
"entering from the d direction.",
"So that would be plus ida.",
"And then there is a current from the ba direction and iba.",
"Those are going to sum to zero.",
"Now I can multiply the whole thing by minus 1.",
"And that would give me the sum of the currents leaving the",
"node are also going to be zero.",
"So that is KCL.",
"Now I can also write KVL in this case for this loop, for",
"instance, here.",
"For this loop I can add up the voltages around the loop.",
"And by KVL they must add up to zero.",
"So for example if I look at the voltage vca, I can add",
"that to the next voltage here.",
"That is to vab and then add that to the",
"final voltage, vbc.",
"And by KCL--",
"so all of those must add up to be zero.",
"Now if you go back to our circuit here that you see on",
"the left hand side, I am going to show you a little demo that",
"will look at these currents, ica, ida, and iba, and also",
"look at the voltages, vcabab and bbc and actually go and",
"look at the circuit, make the measurements, and show that",
"they indeed sum up to zero.",
"Not surprising.",
"KVL and KCL really work.",
"And so the demo should also work.",
"OK.",
"So let's go and do our demo."
]
}

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"text": [
"SPEAKER 1: OK.",
"I'm now going to show you a demo of superposition in",
"action with something you see every day in real life.",
"So in this demo, I am going to have an empty",
"vat to begin with.",
"And then what I'm going to do is I'm going to pour some salt",
"water into it.",
"OK, some salt water poured into it will behave like a",
"distributed resistance.",
"And so each small volume of water can be modeled as a",
"lumped resistor.",
"And so you end up with this three-dimensional connected",
"network of resistors.",
"Now, this is a linear circuit.",
"And we will see superposition in action with this circuit.",
"So what do is I'll connect one input to it, which will be a",
"high-frequency triangular wave, like so.",
"OK, that'll be input at one part of the water.",
"And then I will input a low-frequency sinusoid at a",
"different location.",
"So those are my two inputs.",
"And as we've seen before, you can think of",
"these as V1 and V2.",
"And what I'm going to do is I'm going to observe the",
"output at a third port.",
"And my goal will be to take a look at what",
"this input looks like.",
"If superposition truly holds true, then this output should",
"really show both a component that relates to V1 and another",
"component that clearly relates to V2, if",
"superposition really worked.",
"OK?",
"So let's go on to the demo.",
"And as I said before, in the demo, you will see a vat, an",
"empty vat to begin with, into which I will add salt water.",
"And then we will take a look at what happens at the output",
"as I apply two input voltages to it.",
"This is a vat of nothing, right, and it's all empty.",
"So this is a screen of the oscilloscope now.",
"So notice that I have a sinusoid, and I have a",
"triangular wave.",
"And the output is zero.",
"And the reason is there's nothing in this vat.",
"It's empty.",
"Previously when I taught this course, I would get salt water",
"and pour salt water.",
"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.",
"OK, so our friendly demonstration expert, Lorenzo,",
"will pour some water into the vat.",
"And you should begin seeing the output begin superposition",
"now of the two.",
"You see that?",
"So you do see the sinusoidal articulation",
"and the jagged waveform.",
"And just to have some more fun, what I can do is increase",
"one of the voltages.",
"And you'll see.",
"[LAUGHTER]",
"SPEAKER 1: Now you know what would have happened if I'd",
"used Charles River water.",
"OK.",
"So my output keeps increasing as I increase the",
"corresponding waveform.",
"OK?",
"I could do this.",
"This is fun.",
"OK.",
"So let me pause there and go on to the next topic.",
"So that little demonstration showed you that even something",
"as simple as this physical entity, a vat of water,",
"behaves like a linear system.",
"And we can model that linear system as a set of resistors."
]
}

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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."
]
}

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"text": [
"SPEAKER 1: So before I wrap up the node method, I wanted to",
"take some time and show you something interesting, largely",
"for your cultural interest. This is not required for the",
"course, and so this is optional.",
"I want to show you how these equations are really solved in",
"real life and how computers can be used to do this for us.",
"So if you go back to step four of the node method, you recall",
"we had these two equations and two unknowns, where the",
"unknowns were e1 and e2.",
"I can turn these into matrix form, like so, where I have",
"matrix times a column vector equals another column vector,",
"and where I get a two-by-two matrix called the",
"conductivity matrix.",
"I get a column vector of the unknown node voltages.",
"And I get a column vector of the source voltages.",
"So I've taken my system of equations and converted them",
"into the matrix notation, like so.",
"And the way this works is each of these equations can be",
"obtained by doing a row column multiply.",
"So if I multiply the first row with this column here, I take",
"e1, multiply that by G1 plus G2 plus G3, I get",
"the first term here.",
"I take e2, multiply that by minus G3, I",
"get the second term.",
"And I equate that to G1V0.",
"Similarly, to get the second equation, I would take the",
"second row and multiply that by the same column vector.",
"Now, let's take a look at how things like",
"this might be solved.",
"So here is my matrix equation again, with a two-by-two",
"matrix multiplying a column vector equal",
"another column vector.",
"So we can use many sophisticated techniques to",
"analyze systems like this.",
"And if you've done courses in linear algebra, you would've",
"noticed that linear algebra spends a lot of time trying to",
"analyze systems of this sort, where A is some matrix, x is a",
"column vector of unknowns, and b is a",
"column vector of knowns.",
"And so you often have to go and solve for the column",
"vector of unknowns.",
"Now, with circuit analysis, if I have a circuit containing",
"millions of elements, I would have a column vector here,",
"which had a million elements.",
"And so clearly we would love to use computers to do that.",
"And not surprisingly, that is what to",
"do in circuit analysis.",
"In fact, the lab that you work on in 6002-X has the schematic",
"capture system where you can build circuits.",
"But for the analysis, the computer programs that are",
"used to analyze the voltages and so on, and show them to",
"you, use methods very similar to this, where you use these",
"sophisticated linear algebraic solvers that solve systems of",
"equations of the form Ax equals b.",
"So back to our system.",
"I can go ahead and solve the system.",
"I won't bore you with the details of",
"the agony of algebra.",
"I've gone ahead and solved these equations for you.",
"And so I get e1 and e2 are expressed as fairly big",
"equations of the sort.",
"So one interesting fact that I mentioned to you was you end",
"up with systems of equations that you",
"can solve with computer.",
"There's one other interesting fact.",
"Notice that the circuit that we used, our demo",
"circuit, was linear.",
"It was composed of linear elements and so on.",
"And so notice here that e1 is also related to a set of",
"linear terms. So notice that this equation here is linear",
"in V0, i1, V0, and so on.",
"And it is all in linear terms. Also note that there are no",
"negatives in the denominator.",
"So the fact that this is linear in the constant sources",
"will be used in a later sequence that we will cover in",
"a few days.",
"So watch out for that.",
"So just to wrap up and just go through the motions here, let",
"me just go ahead and solve for e2 just for kicks.",
"And I'll do it for values G1, G5, and then so on and so",
"forth, given here, and for i1 chosen to be 0, just for fun.",
"And so if I plug these numbers into the equation for e2 that",
"I just showed you, I'm going to get e2 as a function of V0,",
"as follows.",
"Then e2 is going to be 0.6 times V0.",
"And so if I pick V0 to be three volts, then e2 is going",
"to be 1.8 volts.",
"So there you have it."
]
}