Sketching exponentials - examples

Now we're going to take the ideas from the last video and learn how to sketch in these exponentials really rapidly. Now I want to move this up, and we'll do some a couple of examples.

Here's an example circuit I've already set up. It's an RC circuit. This is 1000 ohms, and this is a 2 microfarad capacitor, and this voltage source provides a step voltage to us. It's going to start out at 1 volt and then step rapidly down to zero volts at time equals zero, and then again at ten milliseconds, it'll step up again to one volt and continue on that way. This could be a clock in a circuit in a digital circuit, or it could be any kind of a digital signal.

And what we want to know is we want to find this voltage right here. What does that look like? We're going to use what we know about RC waveforms to tell us what's going to happen here really quickly. We don't need to use a computer to simulate this.

All right, so first thing I need to know is what's the RC time constant. So RC equals 1000 k times 2 microfarads, and that equals 2 k times k is plus three, micro is minus six. So this is 2 times 10 to the minus 3, or 2 milliseconds.

So I can actually go over right here to my chart and I can put a spot right here at 2 milliseconds, right there. There's five milliseconds, there's five, there's about two. And real rapidly what I know is that that line is going to look roughly like this. It's going to start, of course, it's going to start right here, and that line's going to go right through here.

So let me see if I can just sketch that in. That's about what that is going to look like. Now, on the other side when it goes up, when it goes up, it's going to start here and it's going to go up to where it's going to go. Two more milliseconds, one, two is 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 is going to happen over here. As soon as that goes down, I go two milliseconds over, it's going to 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 of 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 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, 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 up; the exponential is going to go through that. So it's going to come down and then curve off like this, and that's going to 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 going to 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 is reaching its high level and is 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 what the change here, the change here is 4 microfarads. This is still 1k and this is 4 microfarads.

So RC, let's do it again real fast. RC equals 1k times 4 microfarads, and that equals four milliseconds. So now I know that the time constant is four milliseconds. That means I know that my straight line sketch, here's five milliseconds, it goes through right about here.

I start up here; I can draw, I can sketch in a straight line like that. It goes roughly like that. Same thing over here, I can go out 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 sketched the line in.

And now we'll go back and do our 37 trick, which is 37 is right about there, and we'll come up; it goes through right there. And if I sketch it in, now what's going to happen is it's going to go through that and something over there is going to happen.

I don't know if it's going to finish; it might, it might not. This one's going to go up same way. Let's do 37 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, and we'll see how that looks. Let's move it up, and we did pretty well.

So notice how down here, down in this area here, we sort of got a… that's pretty accurate. I missed a little bit up 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 then the other one is this point right here, that's about 37 percent of where it started from, 37 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.