WEBVTT

00:00:00.140 --> 00:00:04.560
>> We're going to continue on with our
introduction to Electrical and Computer

00:00:04.560 --> 00:00:08.385
Engineering by moving now into
the Sinusoidal Steady-state,

00:00:08.385 --> 00:00:10.890
we're analyzing circuits
that involve resistors,

00:00:10.890 --> 00:00:14.235
capacitors and inductors, that
are driven by sinusoidal sources.

00:00:14.235 --> 00:00:15.760
My name's Lee Brinton, I'm

00:00:15.760 --> 00:00:18.440
an electrical engineering instructor
in Salt Lake Community College.

00:00:18.440 --> 00:00:21.695
We're going to start this discussion
with a review of complex numbers.

00:00:21.695 --> 00:00:24.410
We find that electrical engineers

00:00:24.410 --> 00:00:26.540
use complex numbers just
a little bit differently

00:00:26.540 --> 00:00:31.495
than are typically taught in
college algebra and trig classes.

00:00:31.495 --> 00:00:33.740
So we're going to start
out with just going back

00:00:33.740 --> 00:00:36.450
to remembering where
complex numbers come from,

00:00:36.450 --> 00:00:38.540
we'll review rectangular and polar form

00:00:38.540 --> 00:00:40.430
of complex numbers, we'll introduce to you,

00:00:40.430 --> 00:00:42.470
I think for many of you
will be the first time,

00:00:42.470 --> 00:00:46.440
the complex exponential
form of a complex number,

00:00:46.440 --> 00:00:52.085
and then we'll go through
some example calculations to show how

00:00:52.085 --> 00:00:55.595
these different forms make
certain types of calculations

00:00:55.595 --> 00:00:59.780
easier than doing those same
calculations in other forms.

00:00:59.780 --> 00:01:02.750
First of all, where do
complex numbers come from?

00:01:02.750 --> 00:01:07.445
But probably, the easiest or
at least an easy way to see it is to look

00:01:07.445 --> 00:01:12.410
at the graph of the function
y equals x squared minus 1,

00:01:12.410 --> 00:01:15.260
it's simply a parabola
dropped down one unit.

00:01:15.260 --> 00:01:19.000
You can say, "Well, what are
the x-intercepts there?"

00:01:19.000 --> 00:01:20.265
To find the x-intercepts,

00:01:20.265 --> 00:01:25.775
you of course set the equation or
the function y equal to 0 and solve for x.

00:01:25.775 --> 00:01:30.620
So we have x squared minus 1
equals 0, add one to both sides,

00:01:30.620 --> 00:01:33.365
you get x squared equals one,

00:01:33.365 --> 00:01:35.120
and take the square root of both sides,

00:01:35.120 --> 00:01:38.390
and you get x equals plus or minus 1.

00:01:38.390 --> 00:01:42.295
So sure enough, the way x-intercepts
are at plus or minus one.

00:01:42.295 --> 00:01:45.830
Well, now let's change
this function to simply

00:01:45.830 --> 00:01:50.525
changing this minus sign to a plus sign,

00:01:50.525 --> 00:01:53.420
and we can then do the same calculations.

00:01:53.420 --> 00:01:57.315
Now, obviously, there's
no x-intercepts here.

00:01:57.315 --> 00:02:00.980
Nonetheless, we can still set y equal
to zero and try and solve for x.

00:02:00.980 --> 00:02:04.985
We have x squared plus 1 equals 0.

00:02:04.985 --> 00:02:09.215
Subtract one from both sides gives
us x squared equals minus one.

00:02:09.215 --> 00:02:13.775
Now take the square root of
both sides and we go x equals,

00:02:13.775 --> 00:02:17.500
we're looking for a number that when
multiplied by itself equals negative one,

00:02:17.500 --> 00:02:21.265
and in the real number system we
all know there is no such number.

00:02:21.265 --> 00:02:24.190
There's no number that
when multiplied by itself

00:02:24.190 --> 00:02:27.010
equals negative one in
the real number system.

00:02:27.010 --> 00:02:28.900
Well, that's not very satisfactory,

00:02:28.900 --> 00:02:30.610
there's got to be
a solution just because we

00:02:30.610 --> 00:02:34.170
change the sign from minus to plus,

00:02:34.170 --> 00:02:35.730
and all of a sudden we
don't have solutions,

00:02:35.730 --> 00:02:36.840
now there's got to be a solution.

00:02:36.840 --> 00:02:40.750
So we define or we invented
the solution to that equation.

00:02:40.750 --> 00:02:46.910
We called it i, so we have x then
is equal to plus or minus i,

00:02:47.090 --> 00:02:51.750
where i then equals is defined as,

00:02:51.750 --> 00:02:53.535
so let's make a third line there,

00:02:53.535 --> 00:02:58.890
i is defined as the square root
of negative one.

00:02:58.890 --> 00:03:01.079
Now, in electrical engineering,

00:03:01.079 --> 00:03:04.460
i is typically used to represent current,

00:03:04.460 --> 00:03:06.410
and so electrical engineers say, "Well,

00:03:06.410 --> 00:03:12.205
let's let j equal i or j then is equal
to the square root of negative one."

00:03:12.205 --> 00:03:15.200
Thus we have this new type of number,

00:03:15.200 --> 00:03:16.895
it's called an imaginary number,

00:03:16.895 --> 00:03:20.180
rather than unfortunate choice
of term from my perspective,

00:03:20.180 --> 00:03:23.180
because it's no more imaginary
than the real numbers,

00:03:23.180 --> 00:03:25.220
but you get my point,

00:03:25.220 --> 00:03:29.315
they're numbers that exist just
not in the real number system.

00:03:29.315 --> 00:03:33.050
In fact, we're going to show that
we need to introduce a new domain.

00:03:33.050 --> 00:03:38.290
To do so, we already
introduced a new domain,

00:03:38.290 --> 00:03:40.830
is known as the imaginary number line,

00:03:40.830 --> 00:03:43.730
and we can plot imaginary
numbers on a number line

00:03:43.730 --> 00:03:47.000
just like we can plot real numbers
on the real number line.

00:03:47.000 --> 00:03:52.040
Now, it becomes a little bit more
interesting when we look at this equation.

00:03:52.040 --> 00:03:56.210
Now what we've done is
we've shifted the axis of

00:03:56.210 --> 00:03:58.130
symmetry of the parabola off of the y-axis

00:03:58.130 --> 00:04:00.980
over a distance of two units to the right.

00:04:00.980 --> 00:04:04.655
Let's take this and set that equal to zero.

00:04:04.655 --> 00:04:15.045
We have x minus minus 2 quantity
squared plus 1 equals 0.

00:04:15.045 --> 00:04:18.690
Let's read my one go, plus 1 equals 0.

00:04:18.690 --> 00:04:20.279
Subtract one from both sides,

00:04:20.279 --> 00:04:25.980
we get x minus 2 squared equals negative 1.

00:04:25.980 --> 00:04:27.540
Take the square root of both sides,

00:04:27.540 --> 00:04:34.200
we get x minus 2 equals plus or minus i,

00:04:34.200 --> 00:04:40.425
x then equals two plus i,

00:04:40.425 --> 00:04:45.570
and x equals 2 minus i.

00:04:45.570 --> 00:04:49.020
Now we have x representing
not a pure real number,

00:04:49.020 --> 00:04:50.640
not a pure imaginary number,

00:04:50.640 --> 00:04:52.155
but a combination of the two

00:04:52.155 --> 00:04:55.260
and when you've got a real number
plus an imaginary number,

00:04:55.260 --> 00:04:58.620
we call that a complex number.

00:04:58.620 --> 00:05:02.750
As we know, when solving equations that

00:05:02.750 --> 00:05:07.210
involve real numbers, real coefficients,

00:05:07.210 --> 00:05:11.215
that lead to imaginary solutions,

00:05:11.215 --> 00:05:14.510
those solutions come as complex conjugates,

00:05:15.340 --> 00:05:21.500
2 plus i and 2 minus i are said to be
conjugates of each other because they

00:05:21.500 --> 00:05:28.620
differ only by the sign on
the imaginary part of the number.