RC step response - intuition

In this video, we're going to introduce the idea of a step response. This is one of the most common occurrences in all of electronics, and it happens anytime there's some resistance and some capacitance in series. In particular, it happens billions of times a second inside every computer, so that's why we want to study this very carefully.

The step response is something that happens in a circuit when we drive the circuit with a step voltage. That's a step voltage shown right there. This response is going to be related closely to the natural response of an RC circuit. If you haven't seen that video yet, I want to encourage you to take a moment and go to watch the video on the RC natural response because I'm going to use that result right here.

Now, in the RC natural response, what we had was no energy going into a circuit. We had an RC circuit like this and there was nothing connected here, so the source was removed. We had a capacitor and a resistor, and there was a voltage on the original capacitor. There was some charge on this capacitor right here, and we worked out what's the response of the current here and what's the response of the voltage V of t across this capacitor.

There, we found that V of t equals V naught times e to the minus t over RC. So this is the natural response of an RC circuit. Now, what we're going to do in the step response is we're going to actually kick this circuit with a step. We're going to make the circuit do something, add some sort of stimulus from the outside that pushes this RC circuit in some direction, and we're going to see what that means. It's going to be related to the natural response.

VS here starts at some voltage V naught. Then at time 0 right here, it makes a step, a sharp step up to some other voltage step VS. What we want to do is find out what this circuit does. Again, we'll label this. We want to find out this V of t.

What's going to happen is, in the past before t equals 0, this circuit will be in some state, and we'll figure out what that is. Then we're going to disturb the circuit, and it's going to settle down in some new state, and that's going to be called the step response of the circuit. Our approach here is going to be to look at this. First, we're going to do this intuitively, and we'll just look at a long time ago. We'll look at a long time from now, well after the step, we'll see what the circuit's doing then, and then we'll look at what we’ll guess will happen in between.

So first we'll do this intuitively, and then we'll do it with the formal mathematics. I said we're going to break it down into three things. The first thing we're going to look at is long ago—what was this circuit doing yesterday when it was sitting here at V naught? Well, long ago, VS equals V naught. What was going to happen is some sort of current was going to flow out of here and around into this capacitor, and it's going to leave some charge here.

That charge is going to pile up on the capacitor, and we know that it's related to the voltage on the capacitor by CV. One of the things we're going to do as we analyze this circuit is track what happens to this Q. That's a good approach to thinking about what's going on here. If a long time ago VS was V naught, basically what happened? Some current flowed until, what? Until V here reached V naught.

So V equals V naught at some time in the past, V became V naught, and what happened to I? Then I went to zero. The reason we know I went to zero is because the voltage across this resistor—let me label it like this—we'll call it VR. Eventually, this side was V naught and this side became V naught. That’s zero volts across the resistor, so that means the current goes to zero.

This is the state a long time ago. I want to start sketching this. We'll do some time plots of this, and we'll make this I and this V. So we decided a long time ago V was V naught, and the current we decided was zero, so I can put that on there like that. So that's our long time ago.

Next, what we do is we go to super long time. Let's let t go to infinity. That's a long time from now, and what's the state going to be then? Well, the VS is going to be—the source voltage is going to be VS. I'll tell the part between capital letters and lower case letters here: capital letters is a fixed voltage, and VS is something that changes with time.

We can do the same sort of analysis where there's going to be some current that will flow, Q will build up until the voltage on the capacitor—there'll be some current, the voltage on the capacitor will go to VS. The same story, the voltage across the resistor then will be zero again, and that means that I will be zero again. This is for a long time from now.

So as we continue our intuition here a long time from now, VS is going to be the step voltage big V S, and what's I going to be? I we decided was going to be zero. So a long time from now out in the future, I is going to be zero again.

Okay, so now let's go back and look at what happens between. This is in—this is after the switch happens and before a long time from now. What we can guess, what we can estimate is that V naught is going to become VS somehow, and that the current's going to start at zero and it's going to do something, and it has to end up back at zero.

Okay, let's get a little more detailed. Let's make a little better guesses. The moment after the step happens, the voltage on this side goes to VS, and the voltage on this side is what? Well, the charge is a bunch of charge sitting here on the capacitor, and it hasn't had time to go anywhere. So, if that's the case, then the voltage right after the switch changes is going to still be V naught.

So that's going to be the voltage right after here is not going to jump anywhere, and that's because we physically have some charge stored on this capacitor, and it hasn’t had time to go anywhere yet. That means on this side of the resistor, just after the step happens, this is going to be V naught. Oh look, now I see. Now we have a voltage difference across here, so there's going to be a current.

All of a sudden there's going to be some current here. Let's scribble that in and see what that does. There's going to be some sort of current that happens and it's going to be I is what? It's going to be (VS - V naught) divided by R. That'll be the current; we'll label the current theorem. That’s the current we’re talking about.

Alright, so we got our current to hop up, and now there's more charge—to charge flowing onto our capacitor. So the capacitor voltage is going to start changing, more charge, more voltage, and what’s going to happen? We can estimate this; it's going to just do something like that. I can sketch that in. More and more charge will start flowing onto this capacitor, and the voltage will gradually rise until V equals V S. It's until the voltage across the resistor again is zero, and then the current will stop.

We had a sudden step of current caused by the change in the step voltage input, and then it's going to just fade out something like that until the current goes back to zero. V will go from—let's, how do I write this? Let’s go from V naught, and it will eventually become V S, and I will have a step and then go back to zero.

So that's our intuitive understanding of how this step response will look for a driven RC circuit. Next, what we'll do is we'll work this out in detail, and we'll get mathematically accurate versions of what these two curves look like.