Eulers formula magnitude

In this video, we're going to talk a bunch about this fantastic number e to the J Omega T. One of the coolest things that's going to happen here, we're going to bring together what we know about complex numbers and this exponential form of complex numbers and signs and cos signs as a function of time. What we're going to 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 the essence of all signal processing theory.

So this number here, e to the J Omega T, 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 sine 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 sine Theta minus J sine Theta.

Now, if I go and plot this, what it looks like is this. 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. 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. If I pick out a location for some complex number and I say, "Oh, okay, this is the x coordinate; it's cosine of Theta," and what's this coordinate here on the J axis? On the J axis, that's sine Theta. 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.

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 the complex to something, and everything I know about exponential, taking exponents and things like that, 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 this complex number looks like right there.

So let's take a look at some of the properties of this complex number. All right, 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, what that says is, "What is this value here for R?" We can figure that out using the Pythagorean theorem. What we know is this squared equals x value squared, which is cosine squared, plus the y value squared, which is sine squared.

So it equals cosine squared Theta plus sine squared Theta. Okay, that's just we just applied the Pythagorean theorem to this right triangle right here, this right triangle. Now, from trigonometry, we know what 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.

Right, so we 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.

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, my complex number e to the J Theta is going to 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 was not on the unit circle? What if I wanted something that was farther away from the origin than that?

What I would do is I would say I would take 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 1, and the magnitude of A is A. So this equals A.

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 an argument that has to do with time, and we'll see what happens to this complex exponential.