Sketching exponentials

Now I want to show you a really useful manual skill that you can use when you have voltages that look like exponentials. We're going to talk about this exponential curve here that's generated as part of the natural response of this RC circuit.

We worked out that the voltage across here, this voltage V of t on the capacitor, its natural response is equal to V0 V KN * e to the minus t over RC. This value V KN is the starting voltage that our input source provides, and then it immediately steps down to zero. This circuit then does its natural response. We get a current coming out of this capacitor flowing around in a circle like this, and that's the natural response of this RC circuit.

I want to look at the properties of this function right here. It's got some interesting properties. I'll write it right here: V KN * e to the minus t over RC. The first thing we can look at is this V value. That's this value here. This is V KN. In this particular chart, V was equal to 1 volt. That's what it originally charged up to.

Now we want to look at this point right here, right when it goes through time equals zero and the curve starts to drop. We know how high it is; it's V volts high. What I want to look at now is what's the slope right at this point. The slope of a curve is the derivative of a curve evaluated at this time, at time equals zero.

Let's take the derivative of our function right here, so it's D/Dt (V KN * e to the minus t over RC), and that equals V KN * (-1/RC * e to the minus t over RC). That's the derivative of this exponential for all time. Now we evaluate it at t = 0 and we get V KN / RC * e to the 0.

Since e to the 0 is 1, that equals the slope at time equals zero. That's the value right here. That's what that slope looks like; that's that tangent line to the curve.

Okay, now the next thing I want to do is actually take this line and extend it all the way down until it crosses this axis, until it crosses the time axis. The next question we're going to ask is: What is this point right here?

Let me move up a little bit. So now we have a line; we've defined a line. That means we have an equation of a line. The equation of a line is y = slope * x + b.

B is the intercept on the voltage axis. M is the slope. I can plug in my values here for our chart. Y is the voltage axis. We know what the slope is; it's right, it's sitting right here, it's -V KN / RC. We multiply by time and we add the Y intercept, which we know is V KN.

Alright, and now what I want to do is find out what time the voltage equals zero for this orange line. So we're going to plug in zero for volts and work out what time. So 0 = V KN * (-1/RC * t) + V KN.

If I divide both sides by V KN, I get 0 on this side and this term on this side, so I can say 0 = -1/RC * t + 1. We want to isolate t, so we'll take the one on the other side and multiply by RC.

So one more step: -1 = -1/RC * t, and in the end, we come up with t = RC. T equals RC is that time right there. That's how many seconds after the step that this line hits the time axis.

Notice something here: There's no mention of V KN in here. There's no V KN; it's not here. It divided out in this step back here; it's one of our steps that disappeared going from this step to this step here.

So independent of how high this thing starts — it could be high or it could be low — the slope of this line always goes right through the time equals RC. One more thing we want to work out is: at time equals RC, what is the value of the exponential? What is this voltage? What is this voltage here when time equals RC?

We can use our equation again. We can plug into our equation and find out how to do that. If I go back to V = V KN * e to the minus t over RC, and this time we're going to plug in RC.

So V equals V KN * e to the minus (RC / RC) or equals V KN * e to the minus one. Now the value of e is roughly equal to 2.7, and the value of 1/e is roughly equal to 0.37, or another way I could say that is 37%.

So in the end, this voltage right here is about 37% of V KN. That's that value right there. So these are two little things we're going to tuck away in our head: The time for that line to hit the time axis is RC, and if I want to know the value of where the exponential actually is at time equals RC, it's roughly 37% of where it started from.

Okay, that's the basic idea, and in the next video, I'll show you how to use these ideas to sketch exponentials really quickly.