[MUSIC] Welcome back to Linear Circuits. This is Dr. Ferri. This lesson is on linearity and superposition. The objective is to first define linearity in circuits. And one of the goals here is to introduce superposition to solve circuit problems. So we have a circuit that looks something like this where we've got a lot of different sources in there. We want to find a sort of simple way of analyzing a circuit like that. This lesson builds upon linear functions that you've seen from math classes in the past, linear functions. And the other thing that we'll be building on is basic circuit solution methods using the KVL, the KCL, mesh node analysis, voltage divider, current divider, anything that we've already covered is fair game in this lesson. Let's start out with linear circuit elements. Right here I show two things. One is just a linear function. f(x)=mx where the independent variable is x and then our slope is m. And similarly we've got the equation iR, V equation for a resistor where my independent variable is i and V is my dependent variable and then R is my slope. So you could see that the resistor it has the same sort of behavior as this. This is just the equation of a line that goes through the origin. Let's look at some properties that we're going to need to prove linearity. First of all is this sort of scaling factor. The fact that if I have, this is for a circuit with an input i into this resistor. And if I input i, I get a V is equal to Ri. Now, the thing is, what happens if I multiply i by a scale, or factor a? So instead of i, I put in i times a, maybe 2 times i or something like that. Then, the output here is a times i. In other words, the same scaling factor here occurs for the output V. Same scaling factor. So if I double the i, I double the corresponding V. Now the other property we have is an additive property where, if I have two different inputs, i1 and i2, and they correspond to two different voltages, Ri1 and Ri2, then what happens if I put in a different input current, which is the sum of those two? Well then the corresponding output is the sum of the voltages for the first two cases. So in other words, I summed these up and that's my new input. Then the same thing is my output is the sum of the corresponding outputs there. These two together are known as a superposition property I want to apply this to circuits. We just looked at particular element, a resistor. We said a resistor is linear. Now we want to show that the same thing holds in circuits, where this time the input is the input to our circuit. So in this particular case, I show a circuit with an input source right here with 5 volts, and 2 as my corresponding output to that particular input. What happens if I scale the input? So in other words in this case, I multiply the input to the circuit by 2. Well the corresponding output is also multiplied by 2. And this would be true if I multiplied it by any other factor. Even if I negated it I would have the same multiplier or scalar on the output. Now let's look at the other property, the additive property and apply that to circuits. In this case, I've got that same circuit and I'm looking at an input 5 and a corresponding output 2. If instead I had a different input into this same circuit. In this case I'm putting a sinusoid into that circuit and I'm getting out a sinusoid output. Now sinusoids are very common in circuits. We see them, for example, in house circuits because the voltage that you measure, that you get out of a house plug is a sinusoid. So in this particular case I've got this input that gives me this output. This input gives me this output. So what happens if I want to sum those two individual input? 5 + 10 cos(t) and that's my voltage source. Well I just have to sum the corresponding outputs. This output here and this output come to here. And this is known as the superposition property. Where the output, the out, is equal to the sum of the corresponding outputs for each individual value into V sub s. This leads to superposition in circuits. Now in that last case we looked at superposition in terms of different inputs into the same source. Now, we're going to be looking at how to handle different sources. So the superposition method in circuits is to zero out all sources but one. So here we've got three sources, one, two, and three. And I handle them individually, I zero out all the sources but one, finding out to that particular source and then repeat this procedure for each source, and then at the end, some of the corresponding outputs. So in this particular case, this is the output that I would want to find corresponding to each one of the individual sources, and then I sum the corresponding outputs for each of those sources. Now let's look at this in more detail, it'll make more sense. Now let's do the analysis for this particular circuit. The first step is to zero out the sources. So once let me say I'm going to start with this source right there. I'm going to leave that in there, so this is number 1 is my first source. So in this particular case, I need to zero out this voltage source, and I'd replace it with a short circuit to zero out a voltage source. Because a short circuit means there is zero voltage drop across it. So I have a zero source. And I'm going to call this V sub 01. The output due to my first source. And then I've open circuited this current source because an open circuit means that there's zero current. So that zero's out the source. So there's a lot of different ways of solving this problem it goes back to all the different ways that we've analyzed circuits so far. I can use mesh analysis, I can use node analysis, I can use a combination of divider law a couple of times. Any way I want to solve it, I could, and in this particular case, if you solve it, I get the value of 0.2564 volts, now the next step is to go to another one of my sources. This one, I'll go ahead and look at this source right here, another voltage source. So in other words I'll have to draw the circuit again. This time I'm zeroing out this source right there. So I'm replace it with a short circuit. All voltage sources get replaced by short circuits. And then I've got the 2 volts here. This time I'm going to call the output V sub 02. And again, I've removed the current source because they ran out of current source means that you open circuit it. If I solve this problem, again, I can use any method I want probably what I do is combine these resistors in parallel with one another and then I just got a series combination of resistors and I can solve for this voltage, and in this particular case I'm going to solve for the voltage, 0.6122 volts. Okay, then the third case is I finally go back and look for that last source. And I redraw the circuit, zeroing out everything but that last source. So in this case, I've replaced these two voltage sources with short circuits. And I'm going to call my output here V sub 03, and if I solve for that I get 0.6122 volts. So what I now have is V sub 01. I solved for the output here with respect to this source and only this source. V sub 02 is the output with respect to this source, and only that source. V sub 03 then is the output with respect to this source. And the last step is to sum the corresponding outputs. So this zero which is the output of all these sources, when they're all in the circuit, is equal to the sum of these terms. And that's equal to 0.2564. And it just turns out in this case, two of these canceled one another, it doesn't have to be that way. The procedure is that you just find this corresponding output and then sum them together. To summarize we've gone over some key concepts. First of all, resistors are linear elements. We used the idea of a linear function to show that a resistor's IV characteristics follow that of what a linear function is. The second thing is, we looked at linearity in circuits and this time we're looking at the input to the circuit we need some sort of source, and then the output in this case, for example, a voltage. It's whatever output we're interested in that circuit. And what we found is that the sum of the corresponding outputs for each different value V sub s is how we can find V sub 0. So if I have different values of V sub s right there and I find the corresponding outputs for each on of them then I can just sum them together to find the corresponding true output when the input in the sum of all those terms. So I just summed the corresponding outputs for each different value for V sub s. Then the other case is we used what was called the superposition method in circuits. And that's a case where I have multiple sources and I do the same thing right here. I sum the corresponding outputs for each V sub s. So for each V sub s I find the value of that output and then at the end I go back and sum them all together. And that's a superposition method. All right, I want to thank you and I'd like to encourage you to go to the forums to ask and answer questions. [MUSIC]