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Video transcript
- [Voiceover] Now we're going to cover a really important circuit in electronics, it's the resistor capacitor circuit, or RC circuit. And in particular, in this video, we're going to talk about the natural response of an RC circuit. The natural response is what happens when you put some initial energy into the circuit. And in the case of an RC circuit, it represents charge that's stored on the capacitor at the beginning of the analysis. We're going to go through an intuitive description of what happens in this circuit when this switch is changed from this position, where we're connecting a battery to the RC combination, and then we're going to basically move the switch over to here, and watch what happens as that capacitor discharges back through the resistor. And that discharge pattern is called the natural response. So let's look at the circuit as it's sitting here right now. Let's say that the switch has been in this position for a really long time. Now sometime in the past, there was some sort of a current that flowed out of here. That current flowed through the resistor, and on to this capacitor, right here, and that left us with some charge. There's some charge here. What that looks like, let's do a close up picture of the capacitor. As charge flowed in here, it piled up on this side of the capacitor, and there's a corresponding negative charge that collected on the other side. This amount of charge matches this amount of charge. So that's what we mean when we say that q, or charge, is collecting on a capacitor. And for a capacitor structure, we know that the amount of charge here, is equal to the capacitance value times the voltage. So as more and more charge accumulates here, the voltage goes up. C is fixed, it's a property of the capacitor. If charge goes up, then v goes up. So let's go back over here and see what happens as this current flows onto the capacitor, charge accumulates, and eventually, this voltage here will rise up to be plus and minus v naught. And that that point, when this point is at v naught, and this point is at v naught, the current through the resistor stops, so the current here will be zero, and the voltage at this point will be v naught. So I want to plot that right now. We'll begin our sketch of what this response looks like. So this will be our time axis. This will be v of t. Which is this voltage here. And this will be i of t. And we're going to label i of t to be this current here. And you'll see why I picked that direction in a minute. So, at the beginning, before I throw the switch. We're going to throw the switch in this direction right here in a minute. But before we do that, we have v naught on the capacitor, and we have i of t equals to zero, so let me fill those in. Before time equals zero. This is equal to v naught. And this current down here is zero. So now we're ready to throw the switch. Let me do that. We'll erase this. And then put the switch in this position here. And now let me clean this up. I'm going to erase the battery here. The battery's done it's job for us, which was to initially put some charge on our capacitor, so we don't need this anymore. So here's our simplified circuit. And what we have here, there's a bunch of charge stored up on C, and what's going to happen? It's going to start rushing out of the capacitor, and going back through the resistor, to basically come over here and neutralize the charge on the bottom plate. What's going to happen is over here, the charge is going to leave, it's going to go over through the resistor, it's going to come back in here, and these pluses and minuses will then be neutralized. And that means if there's no net charge, the voltage is going to go to zero. So we can draw that. What we're going to say is after all this charge rushes out of here, goes through the resistor, and ends up on the back side of the capacitor, neutralizing the charge in the circuit, that means the voltage in the circuit is going to be zero. So we can draw that. We draw the long time from now out here. It's going to sketch into something like that. It's going to be low. Likewise, the current is going to end up at zero, when we're finished here. And that's what happens with every natural response. It basically, the energy in the circuit is allowed to die to zero. So let's see if we can fill in what's going to happen in between. While this voltage here started at v naught, and it's going to end up at zero. The amount of current flowing out of here is going to be proportional to the voltage across this resistor, right? The amount of current flowing out is one over r times v. So if v is high, then i is going to be high. And as v goes down, as we use up our charge, the current is going to go down. So I can sort of guess what's going to happen here, it's going to basically come down, and end up somehow at zero, after a while. Now we don't know how long it's going to take yet, but it's going to have some shape like this. So let's look at what the current is going to do. The current right now, when the switch was thrown, is at zero. As soon as that switch is thrown, as soon as we connect it right here, all of a sudden, we have v naught on this side of the resistor, and zero on this side of the resistor. And so there's going to be a sharp current increase going through this resistor here. So we expect this to go up like that. To jump up to some value. What we think the current is going to look like, the current is actually over here. It's directly proportional to the voltage, so as the voltage goes down, the current is going to go down. And we can take a guess, it'll have the same shape as the voltage, so it's going to go down something like that. Until it dies out to zero. So this is our forecast for what the natural response of an RC circuit looks like, and in the next video, we will actually derive really precise expressions for what those two curves look like.