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## AC circuit analysis

# Complex exponential magnitude

## 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.