- [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.