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+  "text": [
+    "", 
+    "SPEAKER 1: OK, in the previous segment, we", 
+    "looked at a linear circuit.", 
+    "And you saw that the equations that governed the behavior of", 
+    "such a circuit were linear in the inputs, because the", 
+    "components of the circuit were linear.", 
+    "Next, let's take a look at some of the", 
+    "properties of linearity.", 
+    "What does linearity bias?", 
+    "", 
+    "In particular, linearity will bias homogeneity and", 
+    "superposition.", 
+    "Let's take a look at each of these in turn, and see what", 
+    "homogeneity and superposition really means.", 
+    "", 
+    "So let's start with homogeneity.", 
+    "", 
+    "So suppose I have a system in a little box, and let's assume", 
+    "it's a linear system.", 
+    "I apply a set of inputs, x1, x2, and so on, to", 
+    "produce an output y.", 
+    "So in my facetious example here, think of", 
+    "the inputs as apples.", 
+    "And the processing produces a ripe apple.", 
+    "", 
+    "Now, what homogeneity says is this.", 
+    "In the same system, if I feed it with fraction of apples,", 
+    "then I will get fraction of ripe apples at the output.", 
+    "In other words, if each of my inputs is multiplied by some", 
+    "constant, alpha, then my outputs will also be", 
+    "multiplied by the same fraction, alpha.", 
+    "", 
+    "This simply says that if I reduce my inputs by some", 
+    "proportion, then my outputs will also reduce by the same", 
+    "proportion So that's homogeneity.", 
+    "", 
+    "Next, let us look at superposition.", 
+    "This is some really fun stuff.", 
+    "", 
+    "So suppose I have a linear system as before.", 
+    "And in this case, I give it one set of inputs,", 
+    "x1a, x2a, and so on.", 
+    "And for fun, let the a's stand for apples.", 
+    "So if you have a set of apples, and let's say I get", 
+    "applesauce at the output.", 
+    "", 
+    "Now, if I take the same circuit, and if I feed the", 
+    "main same circuit a different set of inputs-- x1b, x2b, and", 
+    "so on, in this case, b stands for blueberries--", 
+    "let's say I get blueberry jam as an output.", 
+    "", 
+    "So what supervision says is this.", 
+    "If I take the same circuit, the same system, and if at", 
+    "each input where I had previously fed", 
+    "an apple or a blueberry.", 
+    "The [UNINTELLIGIBLE]", 
+    "at each of the inputs, I want to feed", 
+    "an apple and a blueberry.", 
+    "That's x1a + x1b, x2a + x2b, and so on.", 
+    "Then the outputs will simply be the sum of the", 
+    "two original outputs.", 
+    "So they could be ya plus yb.", 
+    "You facetiously think of it as mixed fruit jam.", 
+    "", 
+    "So superposition simply says that if I find the output by", 
+    "feeding the system apples, find the output by feeding the", 
+    "system blueberries.", 
+    "If I fed the same system the sum of apples and blueberries,", 
+    "then I could compute the output simply by looking at", 
+    "the outputs of the system in which I fed just apples or", 
+    "blueberries.", 
+    "Let's look at a specific superposition example and get", 
+    "a sense of what this means.", 
+    "So here's the same system.", 
+    "And in this case, I want to feed two voltages, v1 and v2,", 
+    "and I want to find out the output.", 
+    "What superposition says is this.", 
+    "", 
+    "I can take the same system, and set one output to zero,", 
+    "and feed it just v1, and measure an output.", 
+    "Similarly, I take the same system.", 
+    "And this time around, set v1 to v0, and feed it v2, and I", 
+    "obtain a net output.", 
+    "", 
+    "By the method of superposition, what it says is", 
+    "that for the same system, if I feed it the sum of the two", 
+    "inputs, then I can get my output simply by summing up", 
+    "the two individual outputs.", 
+    "In other words, I can obtain the effect of v1 and v2 acting", 
+    "as inputs simply by taking the same circuit and applying one", 
+    "input, setting the others to zero.", 
+    "Measuring the output.", 
+    "", 
+    "Setting the second input.", 
+    "", 
+    "And then having all the other inputs be zero.", 
+    "And measure the output.", 
+    "And then take the individual outputs, y1 and y2, and simply", 
+    "add them up.", 
+    "And that would be the result of v1 and v2 acting together.", 
+    "So that is a specific superposition example.", 
+    "What it does is it allows us to solve two", 
+    "problems, p1 and p2.", 
+    "But each of them is a simpler problem, and simply sum the", 
+    "two outputs of the simple problem to obtain the result.", 
+    "", 
+    "So method of superposition can be summarized as follows.", 
+    "", 
+    "You first find the response of the circuit to each source", 
+    "acting alone.", 
+    "And then second.", 
+    "", 
+    "If you want to find the response of the sources acting", 
+    "together, then you simply sum the individual responses.", 
+    "It is important to point out that this works only for", 
+    "independent sources.", 
+    "", 
+    "You will see later how superposition works with", 
+    "dependent sources.", 
+    "", 
+    "Now, in circuit position, we talked about each source", 
+    "acting alone.", 
+    "And for each source to act alone, I had to set all the", 
+    "other sources to zero.", 
+    "So what does it mean to set a source to zero?", 
+    "So let's say, for example, I have a voltage source, as", 
+    "shown here.", 
+    "What does it mean to set that this voltage source to zero?", 
+    "Well, it simply means that we will short the voltage source", 
+    "as shown here.", 
+    "That sets a voltage source to zero.", 
+    "Similarly, for a current source.", 
+    "The way you set the current source to zero is you open", 
+    "circuit the third course.", 
+    "", 
+    "OK, now let's get back to our original example that we had", 
+    "previously solved using the known method.", 
+    "So let's go ahead and solve the same circuit using the", 
+    "superposition method.", 
+    "", 
+    "OK, using the superposition method for the goal, I have to", 
+    "find the output for each source acting alone.", 
+    "And then I sum the outputs.", 
+    "So let me start by finding the output with v acting alone.", 
+    "So we'll start by setting the current source to zero.", 
+    "And I can do that by open circuiting the current source.", 
+    "", 
+    "Now, as I do this, I can compute the", 
+    "output, ev, as follows.", 
+    "ev is simply this voltage appearing at the output", 
+    "through a voltage divider, r1 and r2.", 
+    "So I get r2 divided by r1 plus r2 times v. So that is a", 
+    "component of the output as a function of v alone.", 
+    "", 
+    "OK, next, let me work on i acting alone.", 
+    "So this case with i acting alone, I want to set the", 
+    "voltage source to zero by shorting it.", 
+    "Now, I can write ei, the component of the output that", 
+    "relates to just a current source, as follows.", 
+    "So I have the current flowing through here, and that current", 
+    "flows through a parallel resistor pair.", 
+    "And so the voltage is simply the current times the", 
+    "effective resistance, which is simply r1 parallel r2, which", 
+    "is given by r1, r2 divide by r1 plus r2.", 
+    "So that gives me ei.", 
+    "", 
+    "So the superposition method simply says that you get the", 
+    "effect of both sources acting simultaneously if I simply add", 
+    "up the two components, ev and ei, like so.", 
+    "There you go.", 
+    "So this is due to the voltage source acting alone.", 
+    "And this is due to the current source acting alone.", 
+    "Notice that this equation is the same as the one you got", 
+    "with the application of the load method.", 
+    ""
+  ]
+}
\ No newline at end of file