0:01

Using complex exponential notation, some of the arithmetic operations with complex

numbers become much simpler. Now in particular, multiplication of two

complex numbers. If I write them in their complex

exponential form r1e to the j phi 1, times r2j to the j phi 2.

So, these are just two complex numbers. this is r1 times r2.

And now when I multiply exponentials, I just add the exponents.

So, this is j phi 1 plus j phi 2. So, I just add phi 1 and phi 2.

Now division becomes simpler, too. The ratio of these two numbers is just

the ratio of the prefactors r1 over r2. And then I subtract the exponent down

here in the denominator from that in the numerator.

So, this becomes j phi 1 minus phi 2. Now magnitude of the complex number re to

the j phi is very simple, too. The magnitude squared is r squared and

this term, the complex exponential part just vanishes.

Now, there's a couple of things worth saying here.

One of them is that if I take the magnitude of two complex numbers, let me

get a pen here. If I take the magnitude of two complex

numbers say z times w, that is the magnitude of z times the magnitude of w.

Now, you can prove that to yourself just by saying okay, z is a plus jb and w is c

plus jb. And just multiply it all out and

calculate it and you will see that formula is true.

Now the other thing that I want to say here is that if I take the magnitude

squared of a complex number that is, is just, it's like multiplying the complex

number by its complex conjugate. Now, the complex conjugate of this number

Ii r, and I just changed the sign of j everywhere I have it.

2:32

And so this thing becomes r squared, and then I have e to the j, and then I just

have phi minus phi. So the first one, plus phi, minus the

second one. So, this is e to the 0, and that's just

equal to 1. So this whole thing just becomes an r

squared. So, it's a simple rule then when you see

an e to the j phi that factor the modulus, the modulus and the magnitude

mean the same thing. The magnitude just becomes one.

I can take the square root of this, and instead of the magnitude squared just the

magnitude of re to the j phis just r. So, you just basically drop the complex

exponential term to get the modulus. Now, a couple of useful identifies that

you can compute by just writing down using Euler's formula.

Writing down e to the j phi equals cos phi plus j sin phi.

And then this one would cosine phi minus j sine phi.

So if you write that out, you see the cosine terms are going to add.

The sine terms will cancel, and I have a factor of 2 I have to take care of.

But this expression is just cosine of phi.

And if I subtract those two and work it out, this becomes j sine of phi.

Now, all of this has put us in the position now where we can start to talk

about something called phasors. Which we're going to use to represent

time varying signals. And what we want to talk about next is

the relationship between phasors and sinusoidal oscillation.

Now the nice thing about phasors is that it lets you keep track of both the

frequency the amplitude and the relative phase or the time delay between time

varying signals. And we're going to be looking at this

basic function a lot for the remainder of the course.

And so, we're just using Euler's identity and plugging in the exponent j omega t.

And so this factor, e to the j omega t, of course with Euler's formula, is cosine

omega t plus j sine omega t. And I'm going to show you now in a little

Matlab simulation how this e to the j omega t corresponds to circular motion in

the complex plane. As time goes on, it's just a point in the

complex plane that rotates around and the real part is the cosine and the imaginary

part is the sine. In this Matlab animation, we're going to

look at the relation between e to the j omega t, the unit phasor, and cosine and

sine, the real and imaginary parts. Now this plot here is going to plot three

different things. In this bottom left hand panel, we're

plotting e to the j omega t versus time. And re-plotting the real part along the

x-axis and the imaginary part along the y-axis.

So the very first point, for example, when we plug in t equal 0, e to the 0 is

1. So, that's going to be point right here

at the 0.1 on the real axis, 0 on the imaginary axis.

Now if I look at the real part of that, up here, it's going to be 1 at time

equals 0. So, this access is the time access

running vertically here. So, time increases as we go to the top,

and this is positive real part, negative real part.

And this is rotated, this is the projection of this point onto the

imaginary axis. So, this is time running along

horizontally and this is the imaginary part.

And so, at t equals 0, at the point 1, 0 in the imaginary plane.

So, it's 1 in the real part and 0 in the imaginary part.

As time goes on, the unit phasor is just going to rotate around in the complex

plane. So, a quarter cycle of rotation later,

the imaginary part has gone to one, and the real part has gone to zero.

So, you see the real part is not at zero, and the imaginary part is up here at one.

If I keep going, the imaginary part goes back to zero, so the y component has gone

back to zero. But now the x component is negative.

And it's maximum. Now, keep going, another quarter turn,

and I reach negative 1 along the imaginary axis, and 0 on the real axis.

And then when I complete the rotation, you come back to where you started.