Main content

## Electrical engineering

### Course: Electrical engineering > Unit 2

Lesson 4: Natural and forced response- Capacitor i-v equations
- A capacitor integrates current
- Capacitor i-v equation in action
- Inductor equations
- Inductor kickback (1 of 2)
- Inductor kickback (2 of 2)
- Inductor i-v equation in action
- RC natural response - intuition
- RC natural response - derivation
- RC natural response - example
- RC natural response
- RC step response - intuition
- RC step response setup (1 of 3)
- RC step response solve (2 of 3)
- RC step response example (3 of 3)
- RC step response
- RL natural response
- Sketching exponentials
- Sketching exponentials - examples
- LC natural response intuition 1
- LC natural response intuition 2
- LC natural response derivation 1
- LC natural response derivation 2
- LC natural response derivation 3
- LC natural response derivation 4
- LC natural response example
- LC natural response
- LC natural response - derivation
- RLC natural response - intuition
- RLC natural response - derivation
- RLC natural response - variations

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# RC step response - intuition

An RC circuit responds to a step of voltage with a smooth transition between the starting and ending voltage. The RC step response is a fundamental behavior of all digital circuits. Created by Willy McAllister.

## Want to join the conversation?

- In this circuit, what is the direction of the current? First, it goes from left to right starting at source voltage and it stops at Capacitor. Then, it goes in backwards such as right to left? Is the direction changing? In RC Natural Response, I think it would go backward when the switch turns off.(4 votes)
- Hello Yws,

There many things happening in this video.

First, Willy assumes you have seen the previous video covering the natural response of an RC circuit. Here we have a "source less" circuit which is to say there is no voltage supply. We assume an initial condition where the capacitor has magically been charged to some voltage. Starting at "time zero" the current flows out of the capacitor until all of the energy has been turned to waste heat in the resistor.

That was the smaller circuit on the right...

On the left we have a new circuit. This time it includes an external source. Also, we assume the capacitor starts out with zero voltage (more advanced analysis will start the capacitor with a non-zero voltage). In this circuit the current flows from the source to the capacitor. As it does this the resistor get hot. Later after the capacitor is fully charged the current stops flowing and the resistor cools off.

Please leave a comment below if you have any questions.

Regards,

APD(5 votes)

- For some reason I am having trouble understanding where the expression "i = (Vs - Vo) / R" came from. How was that derived?(1 vote)
- At7:30I'm talking about what happens to the current the instant after the voltage step happens. A good way find a current is to use Ohm's Law on a nearby resistor. You usually are given the resistance value. In this case it is a symbol name, R. And you hope (pray) you can figure out the voltage on either end of the resistor.

In this case we can find the two voltages on R. On the left is the step voltage, Vs. On the right is the original voltage on the capacitor, Vo. So the voltage difference is Vs-Vo, and the current at that moment is i = (Vs-Vo)/R.(2 votes)

## Video transcript

- [Voiceover] In this
video, we're gonna introduce the idea of a step response. This is one of the most common occurrences in all of electronics,
and it happens any time there's some resistance and
some capacitance in series, and in particular, it
happens billions of times a second inside every computer. So, that's why we want to
study this very carefully. So, 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, and this response is
gonna be related closely to the natural response of an RC circuit, and 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, 'cuz I'm gonna 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, and we had a C and an R, 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. And 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, and now, what we're gonna
do in the step response is we're gonna actually kick
this circuit with a step. We're gonna make the circuit do something, add some sort of stimulus from the outside that pushes this RC
circuit in some direction, and we're gonna see what that means, and it's gonna be related
to the natural response. So, VS here starts at
some voltage V naught, then it times zero right here, it makes a step, a sharp step, up to some other voltage step, VS, and what we wanna do is we wanna see what this circuit does, and
again, we'll label this. We wanna find out this V of T. So, what's gonna happen is, in the past, before T equals zero, this
circuit will be in some state, and we'll figure out what that is, and then we're gonna disturb the circuit and it's gonna settle
down in some new state, and that's gonna be called the
step response of the circuit. So, our approach here is
gonna be to look at this, first, we're gonna 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 guess at
what happens in between. So, first, we'll do this intuitively, and then we'll do it with
the formal mathematics. So, I said we're gonna break
it down into three things. The first thing we're
gonna look at is long ago. What was the circuit doing yesterday, when it was sitting here at V naught? Well, long ago, VS equals V naught, and what was gonna happen
is some sort of current was gonna flow out of here, and around into this capacitor, and it's gonna leave some charge here. That charge is gonna
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 gonna
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. So, if a long time ago, VS was V zero, basically, what happened? Some current flowed until
V here reached V zero, so V equals V zero, at some time in the past, V became V naught, and what happened to I? Then, I went to zero, and 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, and that's zero volts across the resistor, so that means the current goes to zero, so this is the state a long time ago. I wanna start sketching this. We'll do some time plots of this, and we'll make this I and this V, and 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 gonna be then? Well, the VS is gonna be-- the source voltage is gonna be VS. Tell apart between capital letters and lowercase letters here. Capital letters is a fixed voltage, and VS is something
that changes with time, and we can do the same sort of analysis. There's gonna be some
current that will flow, Q will build up until the
voltage on the capacitor is... There'll be some current. The voltage on the
capacitor will go to VS, and 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
gonna be the step voltage, big V, S, and what's I gonna be? I, we decided, was gonna be zero. So, a long time from
now, out of the future, I is gonna be zero again. Okay. So now, let's go back and
look at what happens between. Okay, this is after the switch happens, and before a long time from now. And what we can guess,
what we can estimate, is that V naught is
gonna become VS somehow, and that the current's
gonna start at zero, and it's gonna do something, and it has to end up back at zero. Okay, let's get a little more detailed. Let's make some 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, there's 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 gonna still be V naught. So, that's gonna be the
voltage right after here is not gonna 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 gonna be V naught. Oh, look, now, see? Now, we have a voltage
difference across here, so there's gonna be a current. All of a sudden, there's
gonna be some current here. Let's scribble that in,
let's see what that does. There's gonna be some sort
of current that happens, and it's gonna be I is what? It's gonna be VS minus
V naught divided by R. That'll be the current. We'll label the current there. That's the current we're talking about. All right, so we got
our current to hop up, and now there's more charge,
there's charge flowing onto our capacitor, so
the capacitor voltage is gonna start changing. More charge, more voltage. And what's gonna happen? We can estimate this. It's gonna 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 VS, 'til 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 gonna just fade
out, something like that, until the current goes back to zero. V will go from... How do I write this? It'll go from V naught, and
it'll eventually become VS, 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, and next what we'll do is
we'll work this out in detail, and we'll get mathematically accurate versions of what these
two currents look like.