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Complex exponential magnitude

A closer look at the complex exponential term in Euler's Formula. We see that it represents a complex number, a distance of 1 from the origin of the complex plane. Created by Willy McAllister.

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Video transcript

- [Voiceover] In this video we're gonna talk a bunch about this fantastic number e to the j omega t. And one of the coolest things that's gonna happen here, we're gonna bring together what we know about complex numbers and this exponential form of complex numbers and sines and cosines as a function of time. And what we're gonna end up with is the idea of a number that spins. I think this is really one of the coolest things in electronics and it's really at the essence of all signal processing theory. So this number here, e to the j omega t, this is based on Euler's formula. Just as a reminder, Euler's formula is e to the j, we'll use theta as our variable, equals cosine theta plus j times sine of theta. That's one form of Euler's formula. And the other form is with a negative up in the exponent. We say e to the minus j theta equals cosine theta minus j sine theta. Now if I go and plot this, what it looks like is this. So if I plot this on a complex plane, and that's a plane that has a real axis and an imaginary axis. Now we remember that j is the variable that we use for the imaginary unit. A j squared is equal to minus one. And we use that in electrical engineering instead of i. One way to express this number is by plotting it on this complex plane. And so if I pick out a location for some complex number and I say, oh, okay, this is the x-coordinate is cosine of theta. And what's this coordinate here? On the j-axis, that's sine theta. And if I draw a line right through our number, this is the angle theta. So this is one of the representations of complex numbers is this Euler's formula or the exponential form. And we can represent it this way here. This notation is challenging. I can't help but every time I look at this I start to do e to the complex to something and everything I know about taking exponents and things like that, it kind of confuses me in my head. But what I've done over time is basically say, e to the j anything, that whole thing is a complex number and this is what that complex number looks like right there. So let's take a look at some of the properties of this complex number. One of the things we can ask is what is the magnitude of e to the j theta? If I put magnitude absolute value or magnitude bars around that. And what that says is what is this value here for r? And we can figure that out using the Pythagorean Theorem. What we know is this squared equals x value squared which is cosine plus the y value squared which is sine theta. So it equals cosine squared of theta plus sine squared of theta. Okay, that's just, we just applied the Pythagorean Theorem to this right triangle right here, this right triangle. Now from trigonometry, from trigonometry we know what? What's the value of this? Cosine squared plus sine squared for any angle equals one. So that tells us that e to the j theta magnitude squared is equal to one or that e to the j theta magnitude is also one. All right, so we can write down e to the j theta, the magnitude of that is equal to one. Let's go back to our friend up here. That says that the length of this vector, this is a complex number, that is distance one away from the origin. So we'll tuck that away. We know that the magnitude of e to the j theta is one and I can actually go over here and draw now a circle on here like this and if I put the circle right through there, there's the unit circle and it has a radius of one. So I know that for any value of theta that my complex number e to the j theta, it's gonna be somewhere on this yellow circle. So e to the j theta is somewhere on this circle and the angle is what? The angle is right here. It's whatever is multiplied by j up in that exponent. Anything that multiplies by j, that's an angle. So what if I wanted something that wasn't on the unit circle? What if I wanted something that was farther away from the origin than that? Well what I would do is I would say, I would take A, some amplitude, e to the j theta, and this amplitude would expand the length of that vector. So, I would say, if I wanted to know how far away it is, well, the magnitude of e to the j theta is one, and the magnitude of A is A. So this equals A. And if I was to sketch in a circle for that, let's say A was a little bit bigger than one, it would be a little bit bigger out, and this would have a value here. That would be a value of A, the radius would be A. Pretty flexible notation. So with this notation we can represent any number in the complex plane with this kind of format here. We'll take a little break here and in the next video we'll put in for theta, we'll put in an argument that has to do with time and we'll see what happens to this complex exponential.