[MUSIC] Welcome back to Linear Circuits. This is Dr. Ferri. This lesson is on series and parallel resistors. The ultimate objective is to take a fairly complicated circuit and simplify it, so it's easier to analyze. Well, the first step in doing that is to simplify very simple combinations, series resistors and parallel resistors. Now in series, again, would mean that we have the same current going through both resistors, that's what it means to be in series. And in parallel we have the resistors connected at both ends. The derivations here will depend on a few things. One is Ohm's law which we've seen before, the Kirchhoff's current law and the Kirchhoff's voltage law. Now let's start out with what we mean by equivalent resistance. Equivalent resistance means that I want to replace this combination a resistors with one resistor and the equivalent in terms of the current and voltage. So, if I send the current through this combination of resistors it give's me a voltage shot. Well, if I take that same current through this one it gives me the same voltage drop. So that's what means to be equivalent. In terms of the parallel combination, again, I have this current that I'm sending through that set of resistors and it gives me this voltage. So, this single resistor will give me that same current and voltage relationship. Let's start out with resistors in series. And we want to be able to derive this. This is the formula here. Actually we just add them up. The equivalent is r1 plus r2. So going back to this current voltage relationship that we need them to be equal. Let's defined the voltage of V1 and V2 across this two resistors. That means that the total voltage V ab = V1 + V2 and then if I use Ohm's law and make a substitution in terms of this resistors times the current into there and then I can just simplify the expression. And what I now have here is this relationship, a current voltage relationship where if I say that Req is equal to R1 + R2 I get Ohm's law. So what we've just done is derive the two resistors in series. It generalizes to this case right here where I have how many resistors I have in series I just add them up. So if I have ten resistors I just add up ten resistors. As an example, I got three resistors here, I add them up, I get 40 Ohms as my equivalent resistance. Now let's look at resistors in parallel. This one is a little bit more complicated. The formula is more complicated. For two resistors like this, I end up with one over the sum of their inverses. And with two resistors it actually simplifies to an easier formula to remember. That is the product over the sum. Let's start with the current voltage relationships in both cases. Remember they have to be equal, so in the first case I'm going to break this current into the two. I want an i2 breaking up this way. And then I can do the KCL rate there at that node and I get this equation and I'm going to solve for i1. Now also I can do a KVL over here which is just, simplifies to the case here where I've got two resistors in parallel. Their voltages are equal. And I've used Ohm's law here for the voltage of each side. Now, I can take this expression right here, and substitute it into this expression right here. And I get the following. What's drawn right here goes down to here. And then I try to solve for i2, so once I solve for i2 I can go back to this expression where I use Vab again. So this expression is Vab is equal to this part I just brought that down. And then I made the substitution in here for i 2 and I get this expression. This right here is the same as this expression right there and if I simplified it a little bit more I can put it into this form right here. Now let's look at a couple of special cases. One is shorted resistors. So it's equivalent of having a resistance of 0 in parallel with a resistor. And if I plug that into my formula because I have the product over the sum. R equals 0 on this side means that the equivalent resistance across here is 0. And that makes sense because all the current is going to want to flow right through this way instead of through the other branch. Because it's lower resistance, it's in fact zero resistance. Now let's generalize it to K resistors in parallel right here. It's just one over the sum of their inverses, however many I have, that's what I have in there. And for two resistors it simplifies to the product over the sum. With K resistors, you kind of have to look through this equation right here. So these examples I've got, several sets of resistors in parallel, and I want to replace each one with a equivalent. So it's the product over the sum, the product will be 200. Over, the sum would be 30, and that's how many Ohms I've got. In this case, it's actually simpler, because I've got two resistors that are the same. If they're the same, then if I put them in that formula, I'm going to get half the resistance here. Now, it makes sense. If you look at this, this is a smaller value for the equivalent than either of these. And the reason it makes sense, is because when the current goes along here, it has two paths for it to follow. That means there's less of a smaller total resistance for the current to follow than if this route. Then if there were just one resistor here. And if I've got three in parallel then I put that into the formula. I'm going to have 1 over 10 plus 1 over 20 plus 1 over 40 all in the denominator, and if I find that value it's going to be 5.71 Ohms which is smaller than any of the individual ones. Now in summary, the key concepts we've gone through is resistors in series, this is my formula here. Resistors in parallel, that's my formula. And remember for two resistors it reduces down to this form. And the other thing that we've talked about is when I have a short circuit in parallel with the resistor then the equivalent resistance is zero. Thank you. [SOUND]