[BLANK_AUDIO].
Welcome to Linear Circuits.
Where, today, we are going to be
providing a supplementary section about complex numbers.
So this is part one of a, a two part section.
And this section will be aimed at allowing you to review complex numbers.
to demonstrate complex numbers as, in their graphical representation, and
then to identify complex conjugates and be able to calculate them.
So the objectives for this lesson: first of
all, you should be able to graph complex
numbers on a complex plane.
You should be able to find the complex conjugate of a complex number.
You should be able to convert a
complex number between rectangular and polar forms.
And then be able to use Euler's Identity
to find the value of a complex exponential.
So first of all, defining complex numbers.
Complex numbers are numbers which can be expressed in the form a plus
i times b, for a and b being real numbers, where i represents
the square root of the negative one.
However, because we are already using i to represent current, we're typically going
to use j and this is actually fairly common in the engineering fields.
That, j is used as in place of i, but it still
means the same thing, j is the square root of a negative 1.
Here, a is called the real part and b the imaginary part because
b is being multiplied by the square root of negative 1, known as
the imaginary number.
You can also have values that are known as purely real, which means that b is zero.
There's no imaginary part.
Or purely imaginary where a is zero and so there's no real part to the number.
This is a complex plane, because complex numbers have these two
portions, this real part and this this complex this imaginary part.
It if we wanted to put them up to a number line, we can't just use a typical
number line, because there's these two sections.
So our number line now becomes a number plane.
And it's possible to graph these values on this number plane.
If we let one of the axes, typically the x-axis, used as the real value.
And the other one represent the imaginary value.
So that's what we have here.
So as an example, if I have a number say, 2 plus j3, that would
be graphed as going out two. And up three, right here.
Now, there's a number of ways that we can actually graph on this plane.
We could just put a dot, like this.
Or, we could put a line going from the origin to that point.
Or sometimes, people will actually stick an arrow
on the end to make it a full vector.
All of these are used.
It just kind of depends on the application that
you're using as to which one is most. Convenient.
A very important thing for us to
understand when doing when working with these
complex numbers is this, this idea of
a complex exponential or an imaginary exponential.
So if I had e to the j theta.
It's not immediately apparent what this means.
If I had e squared, we know that that's e multiplied by itself.
Or if I had e to the one half, we know
that, it's the number that you multiply it by itself to
get e. or the square root of e.
But when you stick a complex number up there,
an imaginary number, it's not readily apparent, but we
can actually figure out what this means if we
go back to the definition of the exponential function.
The exponential function is its own derivative, is basically how it's defined.
And it's expressed by this series. So I have this infinite sum of
x to the n over n factorial, or in other words, this is x to the 0 over 0
factorial, plus x to the 1 over 1 factorial, plus
x to the 2 over 2 factorial, and so on.
So even the j theta could be calculated by replacing this x with a j theta like this.
And so we can expand this out in
this format to see what this basically looks like.
But a couple of things to note. One is if I have j squared, because j is
the square root of negative 1, if I square it that's going to give me negative 1.
And j cubed is going to be equal to negative 1 times j,
or negative j, and then j to the fourth is equal to 1, and then j to the fifth
is j, j to the sixth is negative 1 and so on.
It just kind of keeps repeating.
So, we can take all of these and use that
to take this j squared and make it negative 1.
This j q to make it a negative j.
This j to the fourth and make it a negative 1 and so on.
It's not really.
Immediately noticeable as to what we gained by doing this.
But if we go back and look at the Taylor series expansion of cosine and
sine functions, cosine expansion looks like this and the sine looks like this.
So if we look at the even terms, er the odd terms, we start with odd terms.
But if we go back and look at the Taylor
series expansion of cosine and sine So if we look at
the even terms, er the odd terms, of this series
and we compare those to the terms in the cosine function,
1 x squared over 2
factorial, here that x becomes a theta, x
to the fourth over factorial and so on. So we see that these odd terms correspond
to the terms of the cosine function. If we look at the even terms
and we factor out the j's.
We see that this corresponds to the sine terms.
X, x squared over 3 factorial, x to the 5
over 5 factorial, didn't quite get the j out of it.
So, factoring out the j this looks like sines, sine charts.
Typically when you have a, a sequence like
this, you're not allowed to rearrange the terms.
In the summation, if you could get different answers but there's
a certain property that some series have known
as absolute convergence, that allows you to do this.
And this is one of those times.
What this means is that we can actually group up
all of the red, and all of the green terms individually.
And do their sums on their own, and then add them together when we're done.
Which means I can take this whole thing and turn it
into the cosine of theta plus j times the sine of theta.
So what does this mean?
Well if I put this onto a complex plane, this
is the cosine of theta and this length is the sine of theta, so j sine of theta.
So every possible value.
Is going to be somewhere on this unit circle, with radius one, since it
goes up one, up to j, over to negative one, and down to negative j.
As we increase theta, it's going
counterclockwise, or anti-clockwise around this circle.
And at any time I get to 2 pi, because e to the j times 0 is
going to be this point right here at one, because the e to the 0 is 1.
If I go to e to the j times 2 pi, again the cosine of 2 pi is 1.
The sine of 2 pi is 0, so we're back to where we started.
And so it cycles over itself again and again.
So what this means is that e to
the j theta is basically telling you a direction,
it's a unit vector in some direction, and that
direction is specified by this theta, this angle theta.
Sometimes you can actually refer to this in degrees
format but typically we're just going to use radians.
This is known as Euler's identity, that e to the j
theta is equal to the cosine of theta plus j sine theta.
It's an extraordinarily useful identity for working with complex numbers, and I
had a professor that often referred to it as a deathbed identity.
It's something that you should remember on your deathbed.
It's that important, it's that useful in engineering applications.
Now that we have this idea of
this imaginary exponential, what happens if we have
a complex exponential something that has a real part as well as an imaginary part?
If I have e to the alpha plus j theta,
well we know that if I add two exponentials that's the
same thing as multiplying the two individual terms together, so this
is e to the alpha times e to the j theta.
Instead of using e to the alpha, we'll replace that
with an a, and let it be its own term.
And if I
use a little bit of notation, I'm going to convert
this and say a to the e, or a times e
to the j theta will be a angle theta, or a
arg theta, sometimes how that's referred to, this little angle thing.
What does this give us?
Well a lets us know the length of
the vector and the theta tells us the direction.
Because we're just taking the, the univector that's specified by this,
e to the j theta, which basically tells us, it
gives us a line and some direction specified by theta.
And then we're making that line longer.
By multiplying it by a, or shorter, depending on
whether a is greater than or less than zero.
So the length of this is a, and that means that if I look at my real part and my
imaginary part, my real part being a and my imaginary
part being b, a is going to be equal to a
times the cosine of theta, and this comes from basic trigonometry.
B is going to be a times the sine
of theta, or capital A times the sine of theta.
That means that we can convert back and forth between what we
know is rectangular form which is the a plus j b format.
To pull our form which is this amplitude a, with an angle of theta.
We can calculate a by
taking the square root of a squared plus b squared,
because this is basically just making use of, the pythagorean theorem.
That a squared plus b square has to equal this capital A squared.
So if I do this a squared plus b squared I get a.
For the angle theta, it's a little bit trickier.
Because what we have
is that the tangent of theta
is equal to the opposite over the adjacent.
And this is a right angle, so the opposite being b, over the adjacent a.
So if I want to find theta I need
to do the inverse tangent operation, Which is arctangent,
on this side, which gives me theta, is equal
to the arc tangent of b divided by a.
However consider,
let's let B equal 2 and A equal 1. That means the B
divided by A is equal to 2, but it B were equal to 2 And A
negative 1. B divided by A still gives us 2.
Well, what's the difference?
Well, in one circumstance, if I have B2 A1
we're somewhere over here, and I have B negative
2, A negative 1, we're somewhere over here.
There are two different complex numbers so how do
we tell which one we need to use well you
need to look at the sign of b and a
because the arc tangent will give you two equivalent solutions.
One's over here and one's over here.
Actually the arc tension is going to give you
something that is on the, the right hand side.
As its results.
You're, if you find that you're actually not
in one of those two sections, if you're somewhere
over here, you need to basically add pi to it to rotate it around to the other side.
You'd see the same thing if you had a negative
and positive that it'll be one, in of these two quadrants.
So it's very important to pay close attention.
To, to the, that when you're doing the arctangent operation.
When your doing polar to rectangular, it's a little bit more straightforward.
Because it, for finding a, you're just going to do a times the cosine of theta.
And for finding b, you just do a times the sine of theta.
And this just comes from what we saw before.
Finally, we're going to talk about the complex conjugate.
Now the complex conjugate is where if I have a complex number, typically for
complex numbers we're going to use z as the variable we use to represent them.
So z is equal to a plus jb.
It's complex conjugate is a minus jb.
And if we look at what that looks like graphically, it's basically
taking the, this point or this vector, and we're reflecting it across
the real axis. So what was Z is now down here.
The complex conjugate will be referred to typically as Z star.
Sometimes you'll see it written as Z bar.
with a little line over the z as opposed to the star.
it really doesn't matter too much just so long as you know it's a complex conjugate.
All we're doing is taking this plus j and making it a minus j.
If we look at that in polar format, if I went up theta and to this point,
if I wanted to find this point, we're basically just
doing the same Thing is the same distance from the origin
but we need to go the opposite direction so we're going
to have it be a with an angle of minus theta.
The real trick to doing this is you just replace j with a minus j.
Anytime you have a complex with a j and you
replace it with a minus j you'll be all right.
And that's not actually a trivial.
Solution that's not actually a trivial result.
For example, if we look at, this with the polar form,
this is equivalent to A times E to the J theta.
So if I want to find Z star, that's going to be A times E and we
have a J here so multiply it, make it a negative J, to the minus J theta.
Which is equivalent to A with an angle of minus theta.
And you can actually do a very complicated thing.
You can have a plus jb
divided by c plus jd times e minus jf.
Maybe we could even add in an, a g. Minus jh.
And you could actually take this whole thing, and anywhere
you see a j replace it with a minus j.
So it'd be a minus jb, divided by c, minus jd, e plus
jf, plus g, plus jh And that would be the complex conjugate of the previous one.
In order to get a nice, simple, clean solution that was in some a plus jb
form, you still have to do a lot
of calculation, but it's still the complex conjugate.
So it's quite a handy thing.
It's very simple to calculate complex conjugates using that.
To summarize, we graphed values on the
complex plane.
We showed graph, complex exponentials in polar form.
And calculated the complex conjugate.
And so, this leads us on to the
next section, where we will actually do arithmetic operations.
Addition, subraction, multiplication and division using complex numbers.
So, until then.