0:00:00.140,0:00:04.560
>> We're going to continue on with our[br]introduction to Electrical and Computer

0:00:04.560,0:00:08.385
Engineering by moving now into[br]the Sinusoidal Steady-state,

0:00:08.385,0:00:10.890
we're analyzing circuits[br]that involve resistors,

0:00:10.890,0:00:14.235
capacitors and inductors, that[br]are driven by sinusoidal sources.

0:00:14.235,0:00:15.760
My name's Lee Brinton, I'm

0:00:15.760,0:00:18.440
an electrical engineering instructor[br]in Salt Lake Community College.

0:00:18.440,0:00:21.695
We're going to start this discussion[br]with a review of complex numbers.

0:00:21.695,0:00:24.410
We find that electrical engineers

0:00:24.410,0:00:26.540
use complex numbers just[br]a little bit differently

0:00:26.540,0:00:31.495
than are typically taught in[br]college algebra and trig classes.

0:00:31.495,0:00:33.740
So we're going to start[br]out with just going back

0:00:33.740,0:00:36.450
to remembering where[br]complex numbers come from,

0:00:36.450,0:00:38.540
we'll review rectangular and polar form

0:00:38.540,0:00:40.430
of complex numbers, we'll introduce to you,

0:00:40.430,0:00:42.470
I think for many of you[br]will be the first time,

0:00:42.470,0:00:46.440
the complex exponential[br]form of a complex number,

0:00:46.440,0:00:52.085
and then we'll go through[br]some example calculations to show how

0:00:52.085,0:00:55.595
these different forms make[br]certain types of calculations

0:00:55.595,0:00:59.780
easier than doing those same[br]calculations in other forms.

0:00:59.780,0:01:02.750
First of all, where do[br]complex numbers come from?

0:01:02.750,0:01:07.445
But probably, the easiest or[br]at least an easy way to see it is to look

0:01:07.445,0:01:12.410
at the graph of the function[br]y equals x squared minus 1,

0:01:12.410,0:01:15.260
it's simply a parabola[br]dropped down one unit.

0:01:15.260,0:01:19.000
You can say, "Well, what are[br]the x-intercepts there?"

0:01:19.000,0:01:20.265
To find the x-intercepts,

0:01:20.265,0:01:25.775
you of course set the equation or[br]the function y equal to 0 and solve for x.

0:01:25.775,0:01:30.620
So we have x squared minus 1[br]equals 0, add one to both sides,

0:01:30.620,0:01:33.365
you get x squared equals one,

0:01:33.365,0:01:35.120
and take the square root of both sides,

0:01:35.120,0:01:38.390
and you get x equals plus or minus 1.

0:01:38.390,0:01:42.295
So sure enough, the way x-intercepts[br]are at plus or minus one.

0:01:42.295,0:01:45.830
Well, now let's change[br]this function to simply

0:01:45.830,0:01:50.525
changing this minus sign to a plus sign,

0:01:50.525,0:01:53.420
and we can then do the same calculations.

0:01:53.420,0:01:57.315
Now, obviously, there's[br]no x-intercepts here.

0:01:57.315,0:02:00.980
Nonetheless, we can still set y equal[br]to zero and try and solve for x.

0:02:00.980,0:02:04.985
We have x squared plus 1 equals 0.

0:02:04.985,0:02:09.215
Subtract one from both sides gives[br]us x squared equals minus one.

0:02:09.215,0:02:13.775
Now take the square root of[br]both sides and we go x equals,

0:02:13.775,0:02:17.500
we're looking for a number that when[br]multiplied by itself equals negative one,

0:02:17.500,0:02:21.265
and in the real number system we[br]all know there is no such number.

0:02:21.265,0:02:24.190
There's no number that[br]when multiplied by itself

0:02:24.190,0:02:27.010
equals negative one in[br]the real number system.

0:02:27.010,0:02:28.900
Well, that's not very satisfactory,

0:02:28.900,0:02:30.610
there's got to be[br]a solution just because we

0:02:30.610,0:02:34.170
change the sign from minus to plus,

0:02:34.170,0:02:35.730
and all of a sudden we[br]don't have solutions,

0:02:35.730,0:02:36.840
now there's got to be a solution.

0:02:36.840,0:02:40.750
So we define or we invented[br]the solution to that equation.

0:02:40.750,0:02:46.910
We called it i, so we have x then[br]is equal to plus or minus i,

0:02:47.090,0:02:51.750
where i then equals is defined as,

0:02:51.750,0:02:53.535
so let's make a third line there,

0:02:53.535,0:02:58.890
i is defined as the square root[br]of negative one.

0:02:58.890,0:03:01.079
Now, in electrical engineering,

0:03:01.079,0:03:04.460
i is typically used to represent current,

0:03:04.460,0:03:06.410
and so electrical engineers say, "Well,

0:03:06.410,0:03:12.205
let's let j equal i or j then is equal[br]to the square root of negative one."

0:03:12.205,0:03:15.200
Thus we have this new type of number,

0:03:15.200,0:03:16.895
it's called an imaginary number,

0:03:16.895,0:03:20.180
rather than unfortunate choice[br]of term from my perspective,

0:03:20.180,0:03:23.180
because it's no more imaginary[br]than the real numbers,

0:03:23.180,0:03:25.220
but you get my point,

0:03:25.220,0:03:29.315
they're numbers that exist just[br]not in the real number system.

0:03:29.315,0:03:33.050
In fact, we're going to show that[br]we need to introduce a new domain.

0:03:33.050,0:03:38.290
To do so, we already[br]introduced a new domain,

0:03:38.290,0:03:40.830
is known as the imaginary number line,

0:03:40.830,0:03:43.730
and we can plot imaginary[br]numbers on a number line

0:03:43.730,0:03:47.000
just like we can plot real numbers[br]on the real number line.

0:03:47.000,0:03:52.040
Now, it becomes a little bit more[br]interesting when we look at this equation.

0:03:52.040,0:03:56.210
Now what we've done is[br]we've shifted the axis of

0:03:56.210,0:03:58.130
symmetry of the parabola off of the y-axis

0:03:58.130,0:04:00.980
over a distance of two units to the right.

0:04:00.980,0:04:04.655
Let's take this and set that equal to zero.

0:04:04.655,0:04:15.045
We have x minus minus 2 quantity[br]squared plus 1 equals 0.

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

0:04:18.690,0:04:20.279
Subtract one from both sides,

0:04:20.279,0:04:25.980
we get x minus 2 squared equals negative 1.

0:04:25.980,0:04:27.540
Take the square root of both sides,

0:04:27.540,0:04:34.200
we get x minus 2 equals plus or minus i,

0:04:34.200,0:04:40.425
x then equals two plus i,

0:04:40.425,0:04:45.570
and x equals 2 minus i.

0:04:45.570,0:04:49.020
Now we have x representing[br]not a pure real number,

0:04:49.020,0:04:50.640
not a pure imaginary number,

0:04:50.640,0:04:52.155
but a combination of the two

0:04:52.155,0:04:55.260
and when you've got a real number[br]plus an imaginary number,

0:04:55.260,0:04:58.620
we call that a complex number.

0:04:58.620,0:05:02.750
As we know, when solving equations that

0:05:02.750,0:05:07.210
involve real numbers, real coefficients,

0:05:07.210,0:05:11.215
that lead to imaginary solutions,

0:05:11.215,0:05:14.510
those solutions come as complex conjugates,

0:05:15.340,0:05:21.500
2 plus i and 2 minus i are said to be[br]conjugates of each other because they

0:05:21.500,0:05:28.620
differ only by the sign on[br]the imaginary part of the number.