[MUSIC] Welcome back to Linear Circuits. This is Dr. Ferri. We're starting module two on resistive circuits. Now, in this module, we'll be concentrating on developing some analysis tools that you can use to analyze circuits like this. Specifically, we'd like to be able to solve for voltages and currents in a fairly complicated circuit. The other thing that we want to cover is, we want to introduce physical applications, including resistive circuits. This first lesson is on Kirchhoff's Voltage Law, and we will be building upon some concepts that we learned in the last module. In particular, Ohm's Law, and the idea of Loops, close paths for the current to flow. Kirchhoff's Voltage Law is presented right here. The sum of the voltages around any loop is zero. There's a shorthand notation for it that we show right here. And this is an example of a loop. Now I want to get into an analogy. That is going on a hike in a mountain range. As you go on the hike, you go up and down hills. You're gaining potential and you're losing potential as you're going around that hike. So, as you circle back or loop around back to your starting place, you have no net change of potential, no net change in elevation. It's the same thing as you go around a loop with the voltages. Right here, from this point to this point, you've gained potential, you've gained voltage. And then, from this point to that point, you've lost it. So as you go around this loop and end up back at your starting point right here, you have no net changing potential, no net changing voltage. So let's look at this example. I'm going to start right here and I'm going to go around the loop. And let me go ahead and show it is going around this way. And I'll show the arrow in the clockwise direction. Now you notice from right here to here, I'm gaining potential. And then over here, I'm gaining more potential. And then over here, I'm losing potential, and I'm losing potential again. It's kind of hard for students to remember that when they're gaining and losing potential, you have to be very cognizant about these signs right here. So I've got a little trick that I do. And the trick is when I come to a minus sign first, I subtract that term. So I would subtract this term, this term, this one I come to a plus so I add that term. So let me go ahead and write this out. So I have a minus VA minus VB plus VD plus VC is equal to zero. That's my Kirchhoff's Law The fact is, it has to sum to zero. Now I can add another loop here, and the Kirchoff's Voltage Law has to apply for this one as well. And actually I've just added two loops. I add this loop right here, and then I add a loop around the outside. So now I have three loops. And the Kirchhoff's Voltage Law has to apply for each one. Let's take a look at this loop right here. If I start right here and go around the loop this way, I get a minus V sub D and I come to the plus sign here, plus V sub e = 0. Now one thing to notice about this is that these two elements are in parallel with one another. And I know that because they share a node at this end and a they share a node at the other end. So what we've got is V sub d is equal to V sub e. Whenever I've got two elements that are in parallel with one another their voltages are the same. I'm going to do a more elaborate example. This time one with numbers and suppose in this particular example, I want to solve for V sub B. So I look at this and say okay, what equation do I need to find? And I look for a loop with V sub B in it. So here's one loop right here. So let me look at this one. There's only three terms in it. So I'm going to go ahead and do Kirchoff's voltage law in this direction. So, if I come to the plus, let me start, let me start, I'll go ahead and start right here. So, I get to the V sub b, I get to the plus first. I have a plus V sub b and then I get to the plus here. I have a plus 1 plus, but this is a minus four so plus a minus four is equal to zero. So V sub E is equal to three volts. Now, what I want to do is have you solve for V sub A. Okay, to solve for V sub A, I can do a Kirchhoff's voltage law around here. So V sub A is equal to 6 volts Now the other thing to note about this is that has got to be true for every other loop. So with these these values V sub A and V sub B being defined and everything else, then I should go back around and if I looked at the KVL around any of these other loops it should still work. They should still sum to zero. Even around this big outer loop. And how many loops do I have here? I've got one two three, and then I've got this loop right here that's four, five, six around here, and then the big one is seven. So I've got seven loops here the KVL holds for each one of those. Let's look a little bit more carefully at components that are in parallel with one another. So in this case we have two sets of parallel components. These resistors, this one right here and this one right here, are in parallel because they share the node at both sides. Similarly, over here, we have these three components here are in parallel because they share this node and this node. Now, what we've said before is that voltages across parallel elements are equal. So that means the voltage across this resistor is the same as the voltage across this resistor. And something to point out here is this voltage source right here is the voltage that is across this resistor and also the voltage that is across this current source. Another example I want to highlight is a voltage lock with a current source in there. Look at this loop right here. And I'm just going to mark it as the left loop. -10 going around this way plus this current times 100 so I'm using Ohms law. So a 100 times i1 plus 200 times i2 Is equal to zero. So that's one of my equations. Now, if I did Kirchoff's Voltage Law around here, I will go, let's see if I start right here, based on the standard convention, the passive convention on Ohm's Law, if the current is defined as going in this direction, the voltage would be positive, the voltage polarity would be like this. So in other words, I'm going into the negative side of it, -100i1. Same thing over here. This is going to be the +- side because the current's going in this way. So I've got a -300 I 3 plus the voltage drop over the source. I will call V sub I. Now as I said before a common mistake for students to make is thinking that this is zero. It is not zero. I am not sure what it is. It's a variable that I would have to solve for by solving all the other currents and then coming back and solving for that. Another example that I want to cover is a Kirchoff's voltage law with respect to open circuits. So I've got an open circuit here, and suppose I want to solve for V sub ab. V sub ab means the voltage from b to a. So from b to a. So that's v sub ab. Now once I've defined it this way, it's a little bit easier to see it, how to apply Kirchhoff's Voltage law for this example. So, if I go around here, I have a minus V sub r2, and then I come up to here and I get a plus V sub ab + V sub R 3 = 0. And if I can solve for V sub R2 through Ohm's law or something else, or if I know what these are, then I can go back and solve for this. So the key here in doing an open loop is there still is a potential to cross that open loop. You just have to define that voltage and apply KVL the standard way. To summarize, in this lesson we've gone over some key concepts. The first is the Kirchhoff's Voltage Law, which is some of the voltages around any loop is zero. Remember, we can have multiple loops in any given circuit. Special cases to remember, parallel components have the same voltage, and we looked at this example here. And note again, that a current source does not have zero voltage. And that you can see right here. This particular current source has this voltage drop across it. And then the other thing is that when we have an open circuit in our loop, we can still do the Kirchoff's voltage law around that open loop. All right, thank you very much. [MUSIC]