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### 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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# Sketching exponentials - examples

Sketch exponentials with different time constants. Created by Willy McAllister.

## Want to join the conversation?

- @6:27"that this waveform is at risk of not making it

Current transcript segment all the way down to the final value. Before it's asked to turn up and go around the other way" why? due to the time constant?(2 votes)- Yes. The time constant is slow relative to the time interval between switching events. The voltage waveform tends to look like a "sawtooth" with curved edges. With a fast time constant the voltage looks like a square wave with rounded sides and flat top/bottom.(2 votes)

- I didn't understand at6:29minutes of the video. What do you mean by saying, "Before it's asked to turn up and go around the other way"(1 vote)
- What we see here is the aqua output signal, v_c, rising toward its potential final value, 1V, but not making it. The orange input signal, v_s, transitions from 1V to 0V before the output voltage completes its exponential rise. That causes the rising waveform to reverse direction and become a falling waveform.

So what I meant by "it's asked to turn around" is that the input v_s switched states from high to low prior to the output v_c reaching its final state.(2 votes)

- In the second example, what would happen if we draw out more periods?

What I mean is: at`t=0.02 s`

, the voltage did not make it "all the way up". Thus, since now it starts a little lower than at`t=0s`

, does it mean that in the next period it would make it "all the way down", until zero at`t=0.03s`

?

I'm not sure whether it would make it all the way down to zero, but my intuition says, it would for sure go a bit lower than it went at`t=0.01s`

.

In that case, at the next peak (`t=0.04s`

) it would be even a bit lower than it was at`t=0.02s`

. And so on, the peaks would be gradually lower.

I guess there would be some limit value (maybe zero, maybe something else) to which the peaks at`k*0.02s`

"converge"?

Or would, at some point the voltage "bounce" somehow and get back to`1.0 V`

and the cycle start all over again?(1 vote) - Sorry, again is the same video as the previous, under a different title?(0 votes)
- No.

In the first video, Will shows us the concept, the idea, and utility of sketching exponentials. Then, in the second video, he shows us examples to elucidate how this simple thing can be useful and quite precise.(4 votes)

## Video transcript

- [Voiceover] Now we're
gonna take the ideas from the last video and
learn how to sketch in these exponentials really rapidly. Now I gonna move this up and
we'll do a couple of examples. Here's an example circuit
I have already set up. It's an RC circuit, this is 1000 ohms, and this is two microfarad capacitor. And this voltage source
provides a step voltage to us. It's gonna start out at one
volt, and then step rapidly down to zero volts at times equals zero. And then again at ten milliseconds, it'll step up again to one volt and continue that way. This can be a clock in a
circuit, in a digital circuit, or it can be any kind of a digital signal. What we want to know, is we want to find this voltage right here. What does that look like? We're gonna use what we
know about RC waveforms to tell us what's gonna
happen here, really quickly. We don't need to use a
computer to simulate this. So, first thing I need to know, is what's the RC time constants? RC equals one thousand, one K, times two microfarads, and that equals two K times, K is plus three, micro is minus six, so this is two times
ten to the minus three or two milliseconds. So I can actually go over here to my chart and I can put a spot right
here at two milliseconds. Right there, there's five milliseconds, there's five, there's about
two and rapidly what I know, is that that line is gonna look roughly, like this. It's gonna start, of course,
it's gonna start right here and that line's gonna
go right through here. So, let's see if I can sketch that in. That's about what that's gonna look like. Now, on the other side, when it goes up, it's gonna start here and it's gonna go up to where? It's gonna go two more milliseconds. One, two. It's about two more milliseconds. That's the final value of that voltage. And I can draw another
line here like this, I can sketch that in. And the same thing's
gonna happen over here. As soon as that goes down,
I go two milliseconds over. It's gonna go right through about there. Start here. And down I go. Now, let me get a little more exact. What I want to do now is do a
little better job sketching in this exponential curve. And we know that it's about
37 percent, right about there. Here's about 37 percent of the starting voltage, right there. So that now I just use my sketching skills and I just smooth in some kind of curve that looks like that. And on the other side,
I come down 37 percent, which is roughly right about there. And I can draw, again, I
can just sketch this in. And that'll come up to some
value there, finishing value. And the same thing here,
I go 37 percent up, the exponential's gonna go through that. So it's gonna come down and
then curve off like this. And that's gonna be my estimate. Now, I didn't use a computer to do that. How do we know if we got it right? Well, I did use a computer once, so here's what it's gonna look like. Let me turn that on. And we'll move it up a little bit. And we can compare what we got. I simulated this in Excel,
just using the equations to get a really precise answer. And what you notice, is our
sketch looks pretty good, it looks pretty close to that. So what we're comparing,
is compare this, to that. And that's pretty similar, that's not bad. And that tells me that our output signal, our voltage on the capacitor, is following the digital
signal, fairly well. And it's reaching its high level. And it's rapidly coming
down as fast as it can. So, let me do one more,
let me do one more problem and we'll change the capacitor value. Now, the change here is four microferads. This is still one K. This is four microferads. So, RC, let's do it again real fast. RC equals one K times four microfarads and that equals four milliseconds. So now I know that the time
constant is four milliseconds. That means that I know
that straight line sketch, here's five milliseconds, it
goes through right about here. I start up here. I can sketch in a
straight line, like that. It goes roughly like that. Same thing over here. I can go over another four milliseconds. Up like that. And sketch in my slope. And we'll do the third one over here. Here's four milliseconds after the change, is right about there. And I sketch the line in. And now we'll go back and
do our 37 percent trick. Which is 37 percent, is right about there. And we'll come up, it
goes through right there. And if I sketch it in now,
what's gonna happen is, it's gonna go through that and something over there's gonna happen. I don't know if it's gonna finish. It might, it might not. This one's gonna go up, same way. Let's do 37 percent down. It's about there. Accurate enough. And we'll sketch in this line. And then we'll do the
same thing over here, up 37 percent. And down we go. Now, let me show you the
answer that I calculated with the computer. We'll see how that looks. Let's move it up. And we did pretty well. Notice how down here, notice
how down in this area here, we sort of got it, that's pretty accurate. I missed a little bit out here. That's okay, that's okay. But with that time constant,
we can notice right away that this waveform is
at risk of not making it all the way down to the final value. Before it's asked to turn up
and go around the other way. So that kind of intuition,
that kind of understanding of an exponential curve,
really makes you very quick. You can do this manually. And in your head, you can sketch out what these exponentials look like. The time point here, this line right here, that's RC seconds after it drops. That's the basic thing. And the other one is
this point right here. That's about 37 percent of where it started from. Thirty seven percent to go. So that's just a nice little skill to have when you're looking at exponentials, which you do all the time if
you're designing computers. Or any other kind of circuit that has sudden changes in
voltage like we had here.