>> We're going to continue on with our introduction to Electrical and Computer
Engineering by moving now into the Sinusoidal Steady-state,
we're analyzing circuits that involve resistors,
capacitors and inductors, that are driven by sinusoidal sources.
My name's Lee Brinton, I'm
an electrical engineering instructor in Salt Lake Community College.
We're going to start this discussion with a review of complex numbers.
We find that electrical engineers
use complex numbers just a little bit differently
than are typically taught in college algebra and trig classes.
So we're going to start out with just going back
to remembering where complex numbers come from,
we'll review rectangular and polar form
of complex numbers, we'll introduce to you,
I think for many of you will be the first time,
the complex exponential form of a complex number,
and then we'll go through some example calculations to show how
these different forms make certain types of calculations
easier than doing those same calculations in other forms.
First of all, where do complex numbers come from?
But probably, the easiest or at least an easy way to see it is to look
at the graph of the function y equals x squared minus 1,
it's simply a parabola dropped down one unit.
You can say, "Well, what are the x-intercepts there?"
To find the x-intercepts,
you of course set the equation or the function y equal to 0 and solve for x.
So we have x squared minus 1 equals 0, add one to both sides,
you get x squared equals one,
and take the square root of both sides,
and you get x equals plus or minus 1.
So sure enough, the way x-intercepts are at plus or minus one.
Well, now let's change this function to simply
changing this minus sign to a plus sign,
and we can then do the same calculations.
Now, obviously, there's no x-intercepts here.
Nonetheless, we can still set y equal to zero and try and solve for x.
We have x squared plus 1 equals 0.
Subtract one from both sides gives us x squared equals minus one.
Now take the square root of both sides and we go x equals,
we're looking for a number that when multiplied by itself equals negative one,
and in the real number system we all know there is no such number.
There's no number that when multiplied by itself
equals negative one in the real number system.
Well, that's not very satisfactory,
there's got to be a solution just because we
change the sign from minus to plus,
and all of a sudden we don't have solutions,
now there's got to be a solution.
So we define or we invented the solution to that equation.
We called it i, so we have x then is equal to plus or minus i,
where i then equals is defined as,
so let's make a third line there,
i is defined as the square root of negative one.
Now, in electrical engineering,
i is typically used to represent current,
and so electrical engineers say, "Well,
let's let j equal i or j then is equal to the square root of negative one."
Thus we have this new type of number,
it's called an imaginary number,
rather than unfortunate choice of term from my perspective,
because it's no more imaginary than the real numbers,
but you get my point,
they're numbers that exist just not in the real number system.
In fact, we're going to show that we need to introduce a new domain.
To do so, we already introduced a new domain,
is known as the imaginary number line,
and we can plot imaginary numbers on a number line
just like we can plot real numbers on the real number line.
Now, it becomes a little bit more interesting when we look at this equation.
Now what we've done is we've shifted the axis of
symmetry of the parabola off of the y-axis
over a distance of two units to the right.
Let's take this and set that equal to zero.
We have x minus minus 2 quantity squared plus 1 equals 0.
Let's read my one go, plus 1 equals 0.
Subtract one from both sides,
we get x minus 2 squared equals negative 1.
Take the square root of both sides,
we get x minus 2 equals plus or minus i,
x then equals two plus i,
and x equals 2 minus i.
Now we have x representing not a pure real number,
not a pure imaginary number,
but a combination of the two
and when you've got a real number plus an imaginary number,
we call that a complex number.
As we know, when solving equations that
involve real numbers, real coefficients,
that lead to imaginary solutions,
those solutions come as complex conjugates,
2 plus i and 2 minus i are said to be conjugates of each other because they
differ only by the sign on the imaginary part of the number.