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Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.14,0:00:04.56,Default,,0000,0000,0000,,>> We're going to continue on with our\Nintroduction to Electrical and Computer
Dialogue: 0,0:00:04.56,0:00:08.38,Default,,0000,0000,0000,,Engineering by moving now into\Nthe Sinusoidal Steady-state,
Dialogue: 0,0:00:08.38,0:00:10.89,Default,,0000,0000,0000,,we're analyzing circuits\Nthat involve resistors,
Dialogue: 0,0:00:10.89,0:00:14.24,Default,,0000,0000,0000,,capacitors and inductors, that\Nare driven by sinusoidal sources.
Dialogue: 0,0:00:14.24,0:00:15.76,Default,,0000,0000,0000,,My name's Lee Brinton, I'm
Dialogue: 0,0:00:15.76,0:00:18.44,Default,,0000,0000,0000,,an electrical engineering instructor\Nin Salt Lake Community College.
Dialogue: 0,0:00:18.44,0:00:21.70,Default,,0000,0000,0000,,We're going to start this discussion\Nwith a review of complex numbers.
Dialogue: 0,0:00:21.70,0:00:24.41,Default,,0000,0000,0000,,We find that electrical engineers
Dialogue: 0,0:00:24.41,0:00:26.54,Default,,0000,0000,0000,,use complex numbers just\Na little bit differently
Dialogue: 0,0:00:26.54,0:00:31.50,Default,,0000,0000,0000,,than are typically taught in\Ncollege algebra and trig classes.
Dialogue: 0,0:00:31.50,0:00:33.74,Default,,0000,0000,0000,,So we're going to start\Nout with just going back
Dialogue: 0,0:00:33.74,0:00:36.45,Default,,0000,0000,0000,,to remembering where\Ncomplex numbers come from,
Dialogue: 0,0:00:36.45,0:00:38.54,Default,,0000,0000,0000,,we'll review rectangular and polar form
Dialogue: 0,0:00:38.54,0:00:40.43,Default,,0000,0000,0000,,of complex numbers, we'll introduce to you,
Dialogue: 0,0:00:40.43,0:00:42.47,Default,,0000,0000,0000,,I think for many of you\Nwill be the first time,
Dialogue: 0,0:00:42.47,0:00:46.44,Default,,0000,0000,0000,,the complex exponential\Nform of a complex number,
Dialogue: 0,0:00:46.44,0:00:52.08,Default,,0000,0000,0000,,and then we'll go through\Nsome example calculations to show how
Dialogue: 0,0:00:52.08,0:00:55.60,Default,,0000,0000,0000,,these different forms make\Ncertain types of calculations
Dialogue: 0,0:00:55.60,0:00:59.78,Default,,0000,0000,0000,,easier than doing those same\Ncalculations in other forms.
Dialogue: 0,0:00:59.78,0:01:02.75,Default,,0000,0000,0000,,First of all, where do\Ncomplex numbers come from?
Dialogue: 0,0:01:02.75,0:01:07.44,Default,,0000,0000,0000,,But probably, the easiest or\Nat least an easy way to see it is to look
Dialogue: 0,0:01:07.44,0:01:12.41,Default,,0000,0000,0000,,at the graph of the function\Ny equals x squared minus 1,
Dialogue: 0,0:01:12.41,0:01:15.26,Default,,0000,0000,0000,,it's simply a parabola\Ndropped down one unit.
Dialogue: 0,0:01:15.26,0:01:19.00,Default,,0000,0000,0000,,You can say, "Well, what are\Nthe x-intercepts there?"
Dialogue: 0,0:01:19.00,0:01:20.26,Default,,0000,0000,0000,,To find the x-intercepts,
Dialogue: 0,0:01:20.26,0:01:25.78,Default,,0000,0000,0000,,you of course set the equation or\Nthe function y equal to 0 and solve for x.
Dialogue: 0,0:01:25.78,0:01:30.62,Default,,0000,0000,0000,,So we have x squared minus 1\Nequals 0, add one to both sides,
Dialogue: 0,0:01:30.62,0:01:33.36,Default,,0000,0000,0000,,you get x squared equals one,
Dialogue: 0,0:01:33.36,0:01:35.12,Default,,0000,0000,0000,,and take the square root of both sides,
Dialogue: 0,0:01:35.12,0:01:38.39,Default,,0000,0000,0000,,and you get x equals plus or minus 1.
Dialogue: 0,0:01:38.39,0:01:42.30,Default,,0000,0000,0000,,So sure enough, the way x-intercepts\Nare at plus or minus one.
Dialogue: 0,0:01:42.30,0:01:45.83,Default,,0000,0000,0000,,Well, now let's change\Nthis function to simply
Dialogue: 0,0:01:45.83,0:01:50.52,Default,,0000,0000,0000,,changing this minus sign to a plus sign,
Dialogue: 0,0:01:50.52,0:01:53.42,Default,,0000,0000,0000,,and we can then do the same calculations.
Dialogue: 0,0:01:53.42,0:01:57.32,Default,,0000,0000,0000,,Now, obviously, there's\Nno x-intercepts here.
Dialogue: 0,0:01:57.32,0:02:00.98,Default,,0000,0000,0000,,Nonetheless, we can still set y equal\Nto zero and try and solve for x.
Dialogue: 0,0:02:00.98,0:02:04.98,Default,,0000,0000,0000,,We have x squared plus 1 equals 0.
Dialogue: 0,0:02:04.98,0:02:09.22,Default,,0000,0000,0000,,Subtract one from both sides gives\Nus x squared equals minus one.
Dialogue: 0,0:02:09.22,0:02:13.78,Default,,0000,0000,0000,,Now take the square root of\Nboth sides and we go x equals,
Dialogue: 0,0:02:13.78,0:02:17.50,Default,,0000,0000,0000,,we're looking for a number that when\Nmultiplied by itself equals negative one,
Dialogue: 0,0:02:17.50,0:02:21.26,Default,,0000,0000,0000,,and in the real number system we\Nall know there is no such number.
Dialogue: 0,0:02:21.26,0:02:24.19,Default,,0000,0000,0000,,There's no number that\Nwhen multiplied by itself
Dialogue: 0,0:02:24.19,0:02:27.01,Default,,0000,0000,0000,,equals negative one in\Nthe real number system.
Dialogue: 0,0:02:27.01,0:02:28.90,Default,,0000,0000,0000,,Well, that's not very satisfactory,
Dialogue: 0,0:02:28.90,0:02:30.61,Default,,0000,0000,0000,,there's got to be\Na solution just because we
Dialogue: 0,0:02:30.61,0:02:34.17,Default,,0000,0000,0000,,change the sign from minus to plus,
Dialogue: 0,0:02:34.17,0:02:35.73,Default,,0000,0000,0000,,and all of a sudden we\Ndon't have solutions,
Dialogue: 0,0:02:35.73,0:02:36.84,Default,,0000,0000,0000,,now there's got to be a solution.
Dialogue: 0,0:02:36.84,0:02:40.75,Default,,0000,0000,0000,,So we define or we invented\Nthe solution to that equation.
Dialogue: 0,0:02:40.75,0:02:46.91,Default,,0000,0000,0000,,We called it i, so we have x then\Nis equal to plus or minus i,
Dialogue: 0,0:02:47.09,0:02:51.75,Default,,0000,0000,0000,,where i then equals is defined as,
Dialogue: 0,0:02:51.75,0:02:53.54,Default,,0000,0000,0000,,so let's make a third line there,
Dialogue: 0,0:02:53.54,0:02:58.89,Default,,0000,0000,0000,,i is defined as the square root\Nof negative one.
Dialogue: 0,0:02:58.89,0:03:01.08,Default,,0000,0000,0000,,Now, in electrical engineering,
Dialogue: 0,0:03:01.08,0:03:04.46,Default,,0000,0000,0000,,i is typically used to represent current,
Dialogue: 0,0:03:04.46,0:03:06.41,Default,,0000,0000,0000,,and so electrical engineers say, "Well,
Dialogue: 0,0:03:06.41,0:03:12.20,Default,,0000,0000,0000,,let's let j equal i or j then is equal\Nto the square root of negative one."
Dialogue: 0,0:03:12.20,0:03:15.20,Default,,0000,0000,0000,,Thus we have this new type of number,
Dialogue: 0,0:03:15.20,0:03:16.90,Default,,0000,0000,0000,,it's called an imaginary number,
Dialogue: 0,0:03:16.90,0:03:20.18,Default,,0000,0000,0000,,rather than unfortunate choice\Nof term from my perspective,
Dialogue: 0,0:03:20.18,0:03:23.18,Default,,0000,0000,0000,,because it's no more imaginary\Nthan the real numbers,
Dialogue: 0,0:03:23.18,0:03:25.22,Default,,0000,0000,0000,,but you get my point,
Dialogue: 0,0:03:25.22,0:03:29.32,Default,,0000,0000,0000,,they're numbers that exist just\Nnot in the real number system.
Dialogue: 0,0:03:29.32,0:03:33.05,Default,,0000,0000,0000,,In fact, we're going to show that\Nwe need to introduce a new domain.
Dialogue: 0,0:03:33.05,0:03:38.29,Default,,0000,0000,0000,,To do so, we already\Nintroduced a new domain,
Dialogue: 0,0:03:38.29,0:03:40.83,Default,,0000,0000,0000,,is known as the imaginary number line,
Dialogue: 0,0:03:40.83,0:03:43.73,Default,,0000,0000,0000,,and we can plot imaginary\Nnumbers on a number line
Dialogue: 0,0:03:43.73,0:03:47.00,Default,,0000,0000,0000,,just like we can plot real numbers\Non the real number line.
Dialogue: 0,0:03:47.00,0:03:52.04,Default,,0000,0000,0000,,Now, it becomes a little bit more\Ninteresting when we look at this equation.
Dialogue: 0,0:03:52.04,0:03:56.21,Default,,0000,0000,0000,,Now what we've done is\Nwe've shifted the axis of
Dialogue: 0,0:03:56.21,0:03:58.13,Default,,0000,0000,0000,,symmetry of the parabola off of the y-axis
Dialogue: 0,0:03:58.13,0:04:00.98,Default,,0000,0000,0000,,over a distance of two units to the right.
Dialogue: 0,0:04:00.98,0:04:04.66,Default,,0000,0000,0000,,Let's take this and set that equal to zero.
Dialogue: 0,0:04:04.66,0:04:15.04,Default,,0000,0000,0000,,We have x minus minus 2 quantity\Nsquared plus 1 equals 0.
Dialogue: 0,0:04:15.04,0:04:18.69,Default,,0000,0000,0000,,Let's read my one go, plus 1 equals 0.
Dialogue: 0,0:04:18.69,0:04:20.28,Default,,0000,0000,0000,,Subtract one from both sides,
Dialogue: 0,0:04:20.28,0:04:25.98,Default,,0000,0000,0000,,we get x minus 2 squared equals negative 1.
Dialogue: 0,0:04:25.98,0:04:27.54,Default,,0000,0000,0000,,Take the square root of both sides,
Dialogue: 0,0:04:27.54,0:04:34.20,Default,,0000,0000,0000,,we get x minus 2 equals plus or minus i,
Dialogue: 0,0:04:34.20,0:04:40.42,Default,,0000,0000,0000,,x then equals two plus i,
Dialogue: 0,0:04:40.42,0:04:45.57,Default,,0000,0000,0000,,and x equals 2 minus i.
Dialogue: 0,0:04:45.57,0:04:49.02,Default,,0000,0000,0000,,Now we have x representing\Nnot a pure real number,
Dialogue: 0,0:04:49.02,0:04:50.64,Default,,0000,0000,0000,,not a pure imaginary number,
Dialogue: 0,0:04:50.64,0:04:52.16,Default,,0000,0000,0000,,but a combination of the two
Dialogue: 0,0:04:52.16,0:04:55.26,Default,,0000,0000,0000,,and when you've got a real number\Nplus an imaginary number,
Dialogue: 0,0:04:55.26,0:04:58.62,Default,,0000,0000,0000,,we call that a complex number.
Dialogue: 0,0:04:58.62,0:05:02.75,Default,,0000,0000,0000,,As we know, when solving equations that
Dialogue: 0,0:05:02.75,0:05:07.21,Default,,0000,0000,0000,,involve real numbers, real coefficients,
Dialogue: 0,0:05:07.21,0:05:11.22,Default,,0000,0000,0000,,that lead to imaginary solutions,
Dialogue: 0,0:05:11.22,0:05:14.51,Default,,0000,0000,0000,,those solutions come as complex conjugates,
Dialogue: 0,0:05:15.34,0:05:21.50,Default,,0000,0000,0000,,2 plus i and 2 minus i are said to be\Nconjugates of each other because they
Dialogue: 0,0:05:21.50,0:05:28.62,Default,,0000,0000,0000,,differ only by the sign on\Nthe imaginary part of the number.