RC natural response intuition (1 of 3)

Now we're going to cover a really important circuit in electronics: it's the resistor-capacitor circuit, or RC circuit. 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. 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 onto this capacitor right here, and that left us with some charge. There's some charge here, and what that looks like... Let's do a close-up picture of the capacitor. As charge flows 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. 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_KN. At that point, when this point's at V_KN and this point is at V_0, the current through the resistor stops. So, the current here will be zero, and the voltage at this point will be V.

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(t), which is this voltage here, and this will be I(t). We're going to label I(t) to be this current here, and you'll see why I picked that direction in a minute.

All right, so at the beginning, before I throw the switch, before we're going to throw the switch in this direction right here in a minute, we have V_KN on the capacitor, and we have I(t) equal to zero. So, let me fill those in. Before time equals zero, this is equal to V_KN, 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 has done its 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 is a bunch of charge stored on C. What's going to happen is 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, come back in here, and these pluses and minuses will then be neutralized. 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 could draw that. We draw the long time from now out here; it's going to sketch in as 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. 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.

Well, this voltage here started at V_K0, 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 1/R times V. So, if V is high, then I is going to be high. As V goes down, as we use up our charge, the current's 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's going to do. The current right now, 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_K0 on this side of the resistor and zero on this side of the resistor. 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's 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's going to have pretty much 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.