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Piotr Mitros
2011-12-22 07:45:41 -05:00
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"text": [
"",
"SPEAKER 1: OK, in the previous segment, we",
"looked at a linear circuit.",
"And you saw that the equations that governed the behavior of",
"such a circuit were linear in the inputs, because the",
"components of the circuit were linear.",
"Next, let's take a look at some of the",
"properties of linearity.",
"What does linearity bias?",
"",
"In particular, linearity will bias homogeneity and",
"superposition.",
"Let's take a look at each of these in turn, and see what",
"homogeneity and superposition really means.",
"",
"So let's start with homogeneity.",
"",
"So suppose I have a system in a little box, and let's assume",
"it's a linear system.",
"I apply a set of inputs, x1, x2, and so on, to",
"produce an output y.",
"So in my facetious example here, think of",
"the inputs as apples.",
"And the processing produces a ripe apple.",
"",
"Now, what homogeneity says is this.",
"In the same system, if I feed it with fraction of apples,",
"then I will get fraction of ripe apples at the output.",
"In other words, if each of my inputs is multiplied by some",
"constant, alpha, then my outputs will also be",
"multiplied by the same fraction, alpha.",
"",
"This simply says that if I reduce my inputs by some",
"proportion, then my outputs will also reduce by the same",
"proportion So that's homogeneity.",
"",
"Next, let us look at superposition.",
"This is some really fun stuff.",
"",
"So suppose I have a linear system as before.",
"And in this case, I give it one set of inputs,",
"x1a, x2a, and so on.",
"And for fun, let the a's stand for apples.",
"So if you have a set of apples, and let's say I get",
"applesauce at the output.",
"",
"Now, if I take the same circuit, and if I feed the",
"main same circuit a different set of inputs-- x1b, x2b, and",
"so on, in this case, b stands for blueberries--",
"let's say I get blueberry jam as an output.",
"",
"So what supervision says is this.",
"If I take the same circuit, the same system, and if at",
"each input where I had previously fed",
"an apple or a blueberry.",
"The [UNINTELLIGIBLE]",
"at each of the inputs, I want to feed",
"an apple and a blueberry.",
"That's x1a + x1b, x2a + x2b, and so on.",
"Then the outputs will simply be the sum of the",
"two original outputs.",
"So they could be ya plus yb.",
"You facetiously think of it as mixed fruit jam.",
"",
"So superposition simply says that if I find the output by",
"feeding the system apples, find the output by feeding the",
"system blueberries.",
"If I fed the same system the sum of apples and blueberries,",
"then I could compute the output simply by looking at",
"the outputs of the system in which I fed just apples or",
"blueberries.",
"Let's look at a specific superposition example and get",
"a sense of what this means.",
"So here's the same system.",
"And in this case, I want to feed two voltages, v1 and v2,",
"and I want to find out the output.",
"What superposition says is this.",
"",
"I can take the same system, and set one output to zero,",
"and feed it just v1, and measure an output.",
"Similarly, I take the same system.",
"And this time around, set v1 to v0, and feed it v2, and I",
"obtain a net output.",
"",
"By the method of superposition, what it says is",
"that for the same system, if I feed it the sum of the two",
"inputs, then I can get my output simply by summing up",
"the two individual outputs.",
"In other words, I can obtain the effect of v1 and v2 acting",
"as inputs simply by taking the same circuit and applying one",
"input, setting the others to zero.",
"Measuring the output.",
"",
"Setting the second input.",
"",
"And then having all the other inputs be zero.",
"And measure the output.",
"And then take the individual outputs, y1 and y2, and simply",
"add them up.",
"And that would be the result of v1 and v2 acting together.",
"So that is a specific superposition example.",
"What it does is it allows us to solve two",
"problems, p1 and p2.",
"But each of them is a simpler problem, and simply sum the",
"two outputs of the simple problem to obtain the result.",
"",
"So method of superposition can be summarized as follows.",
"",
"You first find the response of the circuit to each source",
"acting alone.",
"And then second.",
"",
"If you want to find the response of the sources acting",
"together, then you simply sum the individual responses.",
"It is important to point out that this works only for",
"independent sources.",
"",
"You will see later how superposition works with",
"dependent sources.",
"",
"Now, in circuit position, we talked about each source",
"acting alone.",
"And for each source to act alone, I had to set all the",
"other sources to zero.",
"So what does it mean to set a source to zero?",
"So let's say, for example, I have a voltage source, as",
"shown here.",
"What does it mean to set that this voltage source to zero?",
"Well, it simply means that we will short the voltage source",
"as shown here.",
"That sets a voltage source to zero.",
"Similarly, for a current source.",
"The way you set the current source to zero is you open",
"circuit the third course.",
"",
"OK, now let's get back to our original example that we had",
"previously solved using the known method.",
"So let's go ahead and solve the same circuit using the",
"superposition method.",
"",
"OK, using the superposition method for the goal, I have to",
"find the output for each source acting alone.",
"And then I sum the outputs.",
"So let me start by finding the output with v acting alone.",
"So we'll start by setting the current source to zero.",
"And I can do that by open circuiting the current source.",
"",
"Now, as I do this, I can compute the",
"output, ev, as follows.",
"ev is simply this voltage appearing at the output",
"through a voltage divider, r1 and r2.",
"So I get r2 divided by r1 plus r2 times v. So that is a",
"component of the output as a function of v alone.",
"",
"OK, next, let me work on i acting alone.",
"So this case with i acting alone, I want to set the",
"voltage source to zero by shorting it.",
"Now, I can write ei, the component of the output that",
"relates to just a current source, as follows.",
"So I have the current flowing through here, and that current",
"flows through a parallel resistor pair.",
"And so the voltage is simply the current times the",
"effective resistance, which is simply r1 parallel r2, which",
"is given by r1, r2 divide by r1 plus r2.",
"So that gives me ei.",
"",
"So the superposition method simply says that you get the",
"effect of both sources acting simultaneously if I simply add",
"up the two components, ev and ei, like so.",
"There you go.",
"So this is due to the voltage source acting alone.",
"And this is due to the current source acting alone.",
"Notice that this equation is the same as the one you got",
"with the application of the load method.",
""
]
}