1 00:00:00,608 --> 00:00:03,465 [BLANK_AUDIO]. 2 00:00:03,465 --> 00:00:04,197 Welcome to Linear Circuits. 3 00:00:04,197 --> 00:00:06,045 Where, today, we are going to be 4 00:00:06,045 --> 00:00:09,510 providing a supplementary section about complex numbers. 5 00:00:09,510 --> 00:00:12,890 So this is part one of a, a two part section. 6 00:00:12,890 --> 00:00:17,750 And this section will be aimed at allowing you to review complex numbers. 7 00:00:17,750 --> 00:00:21,220 to demonstrate complex numbers as, in their graphical representation, and 8 00:00:21,220 --> 00:00:24,830 then to identify complex conjugates and be able to calculate them. 9 00:00:25,850 --> 00:00:27,150 So the objectives for this lesson: first of 10 00:00:27,150 --> 00:00:28,850 all, you should be able to graph complex 11 00:00:28,850 --> 00:00:29,940 numbers on a complex plane. 12 00:00:29,940 --> 00:00:33,500 You should be able to find the complex conjugate of a complex number. 13 00:00:33,500 --> 00:00:34,560 You should be able to convert a 14 00:00:34,560 --> 00:00:37,730 complex number between rectangular and polar forms. 15 00:00:37,730 --> 00:00:40,160 And then be able to use Euler's Identity 16 00:00:40,160 --> 00:00:41,940 to find the value of a complex exponential. 17 00:00:43,110 --> 00:00:45,820 So first of all, defining complex numbers. 18 00:00:45,820 --> 00:00:49,450 Complex numbers are numbers which can be expressed in the form a plus 19 00:00:49,450 --> 00:00:54,250 i times b, for a and b being real numbers, where i represents 20 00:00:54,250 --> 00:00:55,500 the square root of the negative one. 21 00:00:56,850 --> 00:01:01,010 However, because we are already using i to represent current, we're typically going 22 00:01:01,010 --> 00:01:05,130 to use j and this is actually fairly common in the engineering fields. 23 00:01:05,130 --> 00:01:09,470 That, j is used as in place of i, but it still 24 00:01:09,470 --> 00:01:12,450 means the same thing, j is the square root of a negative 1. 25 00:01:12,450 --> 00:01:15,760 Here, a is called the real part and b the imaginary part because 26 00:01:15,760 --> 00:01:19,520 b is being multiplied by the square root of negative 1, known as 27 00:01:19,520 --> 00:01:21,420 the imaginary number. 28 00:01:21,420 --> 00:01:25,210 You can also have values that are known as purely real, which means that b is zero. 29 00:01:25,210 --> 00:01:27,050 There's no imaginary part. 30 00:01:27,050 --> 00:01:30,970 Or purely imaginary where a is zero and so there's no real part to the number. 31 00:01:32,560 --> 00:01:36,510 This is a complex plane, because complex numbers have these two 32 00:01:36,510 --> 00:01:41,310 portions, this real part and this this complex this imaginary part. 33 00:01:41,310 --> 00:01:44,890 It if we wanted to put them up to a number line, we can't just use a typical 34 00:01:44,890 --> 00:01:48,040 number line, because there's these two sections. 35 00:01:48,040 --> 00:01:50,950 So our number line now becomes a number plane. 36 00:01:50,950 --> 00:01:54,150 And it's possible to graph these values on this number plane. 37 00:01:54,150 --> 00:01:58,460 If we let one of the axes, typically the x-axis, used as the real value. 38 00:01:58,460 --> 00:02:00,910 And the other one represent the imaginary value. 39 00:02:00,910 --> 00:02:02,950 So that's what we have here. 40 00:02:04,370 --> 00:02:11,310 So as an example, if I have a number say, 2 plus j3, that would 41 00:02:11,310 --> 00:02:16,060 be graphed as going out two. And up three, right here. 42 00:02:16,060 --> 00:02:19,340 Now, there's a number of ways that we can actually graph on this plane. 43 00:02:19,340 --> 00:02:21,730 We could just put a dot, like this. 44 00:02:21,730 --> 00:02:28,060 Or, we could put a line going from the origin to that point. 45 00:02:28,060 --> 00:02:30,310 Or sometimes, people will actually stick an arrow 46 00:02:30,310 --> 00:02:32,750 on the end to make it a full vector. 47 00:02:32,750 --> 00:02:34,140 All of these are used. 48 00:02:34,140 --> 00:02:36,310 It just kind of depends on the application that 49 00:02:36,310 --> 00:02:38,910 you're using as to which one is most. Convenient. 50 00:02:43,110 --> 00:02:43,990 A very important thing for us to 51 00:02:43,990 --> 00:02:47,090 understand when doing when working with these 52 00:02:47,090 --> 00:02:48,790 complex numbers is this, this idea of 53 00:02:48,790 --> 00:02:53,660 a complex exponential or an imaginary exponential. 54 00:02:53,660 --> 00:02:56,450 So if I had e to the j theta. 55 00:02:56,450 --> 00:02:58,530 It's not immediately apparent what this means. 56 00:02:58,530 --> 00:03:03,060 If I had e squared, we know that that's e multiplied by itself. 57 00:03:03,060 --> 00:03:05,230 Or if I had e to the one half, we know 58 00:03:05,230 --> 00:03:08,610 that, it's the number that you multiply it by itself to 59 00:03:08,610 --> 00:03:11,420 get e. or the square root of e. 60 00:03:11,420 --> 00:03:13,630 But when you stick a complex number up there, 61 00:03:13,630 --> 00:03:16,370 an imaginary number, it's not readily apparent, but we 62 00:03:16,370 --> 00:03:18,150 can actually figure out what this means if we 63 00:03:18,150 --> 00:03:20,410 go back to the definition of the exponential function. 64 00:03:21,550 --> 00:03:26,140 The exponential function is its own derivative, is basically how it's defined. 65 00:03:26,140 --> 00:03:34,100 And it's expressed by this series. So I have this infinite sum of 66 00:03:34,100 --> 00:03:39,940 x to the n over n factorial, or in other words, this is x to the 0 over 0 67 00:03:39,940 --> 00:03:42,490 factorial, plus x to the 1 over 1 factorial, plus 68 00:03:42,490 --> 00:03:45,220 x to the 2 over 2 factorial, and so on. 69 00:03:45,220 --> 00:03:49,690 So even the j theta could be calculated by replacing this x with a j theta like this. 70 00:03:51,100 --> 00:03:54,350 And so we can expand this out in 71 00:03:54,350 --> 00:03:59,080 this format to see what this basically looks like. 72 00:03:59,080 --> 00:04:05,090 But a couple of things to note. One is if I have j squared, because j is 73 00:04:05,090 --> 00:04:10,120 the square root of negative 1, if I square it that's going to give me negative 1. 74 00:04:10,120 --> 00:04:17,150 And j cubed is going to be equal to negative 1 times j, 75 00:04:17,150 --> 00:04:24,610 or negative j, and then j to the fourth is equal to 1, and then j to the fifth 76 00:04:24,610 --> 00:04:27,460 is j, j to the sixth is negative 1 and so on. 77 00:04:27,460 --> 00:04:29,200 It just kind of keeps repeating. 78 00:04:29,200 --> 00:04:32,620 So, we can take all of these and use that 79 00:04:32,620 --> 00:04:35,270 to take this j squared and make it negative 1. 80 00:04:35,270 --> 00:04:37,190 This j q to make it a negative j. 81 00:04:37,190 --> 00:04:39,000 This j to the fourth and make it a negative 1 and so on. 82 00:04:40,710 --> 00:04:42,760 It's not really. 83 00:04:42,760 --> 00:04:46,240 Immediately noticeable as to what we gained by doing this. 84 00:04:46,240 --> 00:04:49,630 But if we go back and look at the Taylor series expansion of cosine and 85 00:04:49,630 --> 00:04:55,000 sine functions, cosine expansion looks like this and the sine looks like this. 86 00:04:56,390 --> 00:05:01,569 So if we look at the even terms, er the odd terms, we start with odd terms. 87 00:05:01,569 --> 00:05:04,018 But if we go back and look at the Taylor 88 00:05:04,018 --> 00:05:07,336 series expansion of cosine and sine So if we look at 89 00:05:07,336 --> 00:05:10,259 the even terms, er the odd terms, of this series 90 00:05:10,259 --> 00:05:12,825 and we compare those to the terms in the cosine function, 91 00:05:12,825 --> 00:05:16,490 1 x squared over 2 92 00:05:16,490 --> 00:05:21,260 factorial, here that x becomes a theta, x 93 00:05:21,260 --> 00:05:26,010 to the fourth over factorial and so on. So we see that these odd terms correspond 94 00:05:26,010 --> 00:05:32,196 to the terms of the cosine function. If we look at the even terms 95 00:05:32,196 --> 00:05:39,228 and we factor out the j's. 96 00:05:39,228 --> 00:05:44,878 We see that this corresponds to the sine terms. 97 00:05:44,878 --> 00:05:47,920 X, x squared over 3 factorial, x to the 5 98 00:05:47,920 --> 00:05:51,930 over 5 factorial, didn't quite get the j out of it. 99 00:05:51,930 --> 00:05:56,150 So, factoring out the j this looks like sines, sine charts. 100 00:05:56,150 --> 00:05:58,530 Typically when you have a, a sequence like 101 00:05:58,530 --> 00:06:01,480 this, you're not allowed to rearrange the terms. 102 00:06:01,480 --> 00:06:05,000 In the summation, if you could get different answers but there's 103 00:06:05,000 --> 00:06:07,300 a certain property that some series have known 104 00:06:07,300 --> 00:06:09,810 as absolute convergence, that allows you to do this. 105 00:06:09,810 --> 00:06:11,660 And this is one of those times. 106 00:06:11,660 --> 00:06:13,250 What this means is that we can actually group up 107 00:06:13,250 --> 00:06:16,370 all of the red, and all of the green terms individually. 108 00:06:17,660 --> 00:06:22,300 And do their sums on their own, and then add them together when we're done. 109 00:06:23,500 --> 00:06:24,980 Which means I can take this whole thing and turn it 110 00:06:24,980 --> 00:06:27,480 into the cosine of theta plus j times the sine of theta. 111 00:06:29,840 --> 00:06:30,790 So what does this mean? 112 00:06:30,790 --> 00:06:33,050 Well if I put this onto a complex plane, this 113 00:06:35,410 --> 00:06:41,480 is the cosine of theta and this length is the sine of theta, so j sine of theta. 114 00:06:41,480 --> 00:06:43,840 So every possible value. 115 00:06:43,840 --> 00:06:48,820 Is going to be somewhere on this unit circle, with radius one, since it 116 00:06:48,820 --> 00:06:52,710 goes up one, up to j, over to negative one, and down to negative j. 117 00:06:54,750 --> 00:06:57,100 As we increase theta, it's going 118 00:06:57,100 --> 00:07:00,340 counterclockwise, or anti-clockwise around this circle. 119 00:07:01,390 --> 00:07:06,040 And at any time I get to 2 pi, because e to the j times 0 is 120 00:07:06,040 --> 00:07:10,290 going to be this point right here at one, because the e to the 0 is 1. 121 00:07:10,290 --> 00:07:15,793 If I go to e to the j times 2 pi, again the cosine of 2 pi is 1. 122 00:07:15,793 --> 00:07:20,010 The sine of 2 pi is 0, so we're back to where we started. 123 00:07:20,010 --> 00:07:22,090 And so it cycles over itself again and again. 124 00:07:23,390 --> 00:07:24,360 So what this means is that e to 125 00:07:24,360 --> 00:07:28,050 the j theta is basically telling you a direction, 126 00:07:28,050 --> 00:07:30,130 it's a unit vector in some direction, and that 127 00:07:30,130 --> 00:07:32,810 direction is specified by this theta, this angle theta. 128 00:07:34,270 --> 00:07:36,960 Sometimes you can actually refer to this in degrees 129 00:07:36,960 --> 00:07:38,550 format but typically we're just going to use radians. 130 00:07:40,330 --> 00:07:42,450 This is known as Euler's identity, that e to the j 131 00:07:42,450 --> 00:07:45,630 theta is equal to the cosine of theta plus j sine theta. 132 00:07:45,630 --> 00:07:49,820 It's an extraordinarily useful identity for working with complex numbers, and I 133 00:07:49,820 --> 00:07:53,760 had a professor that often referred to it as a deathbed identity. 134 00:07:53,760 --> 00:07:55,920 It's something that you should remember on your deathbed. 135 00:07:55,920 --> 00:07:59,250 It's that important, it's that useful in engineering applications. 136 00:08:01,640 --> 00:08:02,630 Now that we have this idea of 137 00:08:02,630 --> 00:08:05,080 this imaginary exponential, what happens if we have 138 00:08:05,080 --> 00:08:09,590 a complex exponential something that has a real part as well as an imaginary part? 139 00:08:09,590 --> 00:08:11,480 If I have e to the alpha plus j theta, 140 00:08:12,640 --> 00:08:15,050 well we know that if I add two exponentials that's the 141 00:08:15,050 --> 00:08:17,880 same thing as multiplying the two individual terms together, so this 142 00:08:17,880 --> 00:08:20,930 is e to the alpha times e to the j theta. 143 00:08:20,930 --> 00:08:23,260 Instead of using e to the alpha, we'll replace that 144 00:08:23,260 --> 00:08:26,280 with an a, and let it be its own term. 145 00:08:26,280 --> 00:08:26,880 And if I 146 00:08:26,880 --> 00:08:29,140 use a little bit of notation, I'm going to convert 147 00:08:29,140 --> 00:08:31,590 this and say a to the e, or a times e 148 00:08:31,590 --> 00:08:35,080 to the j theta will be a angle theta, or a 149 00:08:35,080 --> 00:08:39,430 arg theta, sometimes how that's referred to, this little angle thing. 150 00:08:40,600 --> 00:08:41,690 What does this give us? 151 00:08:41,690 --> 00:08:44,200 Well a lets us know the length of 152 00:08:44,200 --> 00:08:46,250 the vector and the theta tells us the direction. 153 00:08:48,200 --> 00:08:52,042 Because we're just taking the, the univector that's specified by this, 154 00:08:52,042 --> 00:08:55,380 e to the j theta, which basically tells us, it 155 00:08:55,380 --> 00:08:58,210 gives us a line and some direction specified by theta. 156 00:08:58,210 --> 00:09:01,010 And then we're making that line longer. 157 00:09:01,010 --> 00:09:03,190 By multiplying it by a, or shorter, depending on 158 00:09:03,190 --> 00:09:04,510 whether a is greater than or less than zero. 159 00:09:04,510 --> 00:09:12,060 So the length of this is a, and that means that if I look at my real part and my 160 00:09:12,060 --> 00:09:14,790 imaginary part, my real part being a and my imaginary 161 00:09:14,790 --> 00:09:17,150 part being b, a is going to be equal to a 162 00:09:17,150 --> 00:09:22,050 times the cosine of theta, and this comes from basic trigonometry. 163 00:09:22,050 --> 00:09:25,270 B is going to be a times the sine 164 00:09:25,270 --> 00:09:27,370 of theta, or capital A times the sine of theta. 165 00:09:28,600 --> 00:09:31,660 That means that we can convert back and forth between what we 166 00:09:31,660 --> 00:09:35,120 know is rectangular form which is the a plus j b format. 167 00:09:35,120 --> 00:09:40,490 To pull our form which is this amplitude a, with an angle of theta. 168 00:09:40,490 --> 00:09:42,400 We can calculate a by 169 00:09:42,400 --> 00:09:44,700 taking the square root of a squared plus b squared, 170 00:09:44,700 --> 00:09:49,070 because this is basically just making use of, the pythagorean theorem. 171 00:09:49,070 --> 00:09:53,840 That a squared plus b square has to equal this capital A squared. 172 00:09:53,840 --> 00:09:56,660 So if I do this a squared plus b squared I get a. 173 00:09:57,830 --> 00:10:01,010 For the angle theta, it's a little bit trickier. 174 00:10:01,010 --> 00:10:02,000 Because what we have 175 00:10:04,030 --> 00:10:05,340 is that the tangent of theta 176 00:10:07,380 --> 00:10:09,990 is equal to the opposite over the adjacent. 177 00:10:11,450 --> 00:10:17,180 And this is a right angle, so the opposite being b, over the adjacent a. 178 00:10:17,180 --> 00:10:18,550 So if I want to find theta I need 179 00:10:18,550 --> 00:10:22,800 to do the inverse tangent operation, Which is arctangent, 180 00:10:22,800 --> 00:10:25,510 on this side, which gives me theta, is equal 181 00:10:25,510 --> 00:10:29,550 to the arc tangent of b divided by a. 182 00:10:30,940 --> 00:10:32,270 However consider, 183 00:10:32,270 --> 00:10:37,700 let's let B equal 2 and A equal 1. That means the B 184 00:10:37,700 --> 00:10:43,470 divided by A is equal to 2, but it B were equal to 2 And A 185 00:10:43,470 --> 00:10:49,290 negative 1. B divided by A still gives us 2. 186 00:10:49,290 --> 00:10:50,530 Well, what's the difference? 187 00:10:50,530 --> 00:10:53,440 Well, in one circumstance, if I have B2 A1 188 00:10:53,440 --> 00:10:57,945 we're somewhere over here, and I have B negative 189 00:10:57,945 --> 00:11:02,236 2, A negative 1, we're somewhere over here. 190 00:11:03,730 --> 00:11:07,490 There are two different complex numbers so how do 191 00:11:07,490 --> 00:11:08,920 we tell which one we need to use well you 192 00:11:08,920 --> 00:11:12,060 need to look at the sign of b and a 193 00:11:13,280 --> 00:11:17,230 because the arc tangent will give you two equivalent solutions. 194 00:11:17,230 --> 00:11:18,670 One's over here and one's over here. 195 00:11:18,670 --> 00:11:20,120 Actually the arc tension is going to give you 196 00:11:20,120 --> 00:11:22,949 something that is on the, the right hand side. 197 00:11:24,940 --> 00:11:26,180 As its results. 198 00:11:26,180 --> 00:11:29,770 You're, if you find that you're actually not 199 00:11:29,770 --> 00:11:31,390 in one of those two sections, if you're somewhere 200 00:11:31,390 --> 00:11:38,080 over here, you need to basically add pi to it to rotate it around to the other side. 201 00:11:39,150 --> 00:11:42,040 You'd see the same thing if you had a negative 202 00:11:42,040 --> 00:11:45,330 and positive that it'll be one, in of these two quadrants. 203 00:11:45,330 --> 00:11:47,590 So it's very important to pay close attention. 204 00:11:47,590 --> 00:11:50,324 To, to the, that when you're doing the arctangent operation. 205 00:11:52,060 --> 00:11:54,510 When your doing polar to rectangular, it's a little bit more straightforward. 206 00:11:54,510 --> 00:12:00,380 Because it, for finding a, you're just going to do a times the cosine of theta. 207 00:12:00,380 --> 00:12:03,680 And for finding b, you just do a times the sine of theta. 208 00:12:04,760 --> 00:12:06,169 And this just comes from what we saw before. 209 00:12:09,360 --> 00:12:11,560 Finally, we're going to talk about the complex conjugate. 210 00:12:11,560 --> 00:12:15,560 Now the complex conjugate is where if I have a complex number, typically for 211 00:12:15,560 --> 00:12:19,540 complex numbers we're going to use z as the variable we use to represent them. 212 00:12:19,540 --> 00:12:21,740 So z is equal to a plus jb. 213 00:12:21,740 --> 00:12:24,570 It's complex conjugate is a minus jb. 214 00:12:26,310 --> 00:12:30,510 And if we look at what that looks like graphically, it's basically 215 00:12:30,510 --> 00:12:34,580 taking the, this point or this vector, and we're reflecting it across 216 00:12:34,580 --> 00:12:38,060 the real axis. So what was Z is now down here. 217 00:12:38,060 --> 00:12:41,190 The complex conjugate will be referred to typically as Z star. 218 00:12:41,190 --> 00:12:43,690 Sometimes you'll see it written as Z bar. 219 00:12:43,690 --> 00:12:46,280 with a little line over the z as opposed to the star. 220 00:12:46,280 --> 00:12:49,520 it really doesn't matter too much just so long as you know it's a complex conjugate. 221 00:12:50,790 --> 00:12:55,020 All we're doing is taking this plus j and making it a minus j. 222 00:12:55,020 --> 00:12:59,790 If we look at that in polar format, if I went up theta and to this point, 223 00:12:59,790 --> 00:13:01,970 if I wanted to find this point, we're basically just 224 00:13:01,970 --> 00:13:05,810 doing the same Thing is the same distance from the origin 225 00:13:05,810 --> 00:13:07,760 but we need to go the opposite direction so we're going 226 00:13:07,760 --> 00:13:10,190 to have it be a with an angle of minus theta. 227 00:13:12,060 --> 00:13:15,530 The real trick to doing this is you just replace j with a minus j. 228 00:13:15,530 --> 00:13:17,650 Anytime you have a complex with a j and you 229 00:13:17,650 --> 00:13:20,330 replace it with a minus j you'll be all right. 230 00:13:20,330 --> 00:13:22,400 And that's not actually a trivial. 231 00:13:22,400 --> 00:13:24,620 Solution that's not actually a trivial result. 232 00:13:24,620 --> 00:13:28,260 For example, if we look at, this with the polar form, 233 00:13:28,260 --> 00:13:31,470 this is equivalent to A times E to the J theta. 234 00:13:32,990 --> 00:13:37,480 So if I want to find Z star, that's going to be A times E and we 235 00:13:37,480 --> 00:13:41,540 have a J here so multiply it, make it a negative J, to the minus J theta. 236 00:13:42,620 --> 00:13:45,680 Which is equivalent to A with an angle of minus theta. 237 00:13:45,680 --> 00:13:47,540 And you can actually do a very complicated thing. 238 00:13:47,540 --> 00:13:50,960 You can have a plus jb 239 00:13:50,960 --> 00:13:58,150 divided by c plus jd times e minus jf. 240 00:13:58,150 --> 00:14:03,750 Maybe we could even add in an, a g. Minus jh. 241 00:14:03,750 --> 00:14:07,640 And you could actually take this whole thing, and anywhere 242 00:14:07,640 --> 00:14:09,730 you see a j replace it with a minus j. 243 00:14:09,730 --> 00:14:16,106 So it'd be a minus jb, divided by c, minus jd, e plus 244 00:14:16,106 --> 00:14:24,390 jf, plus g, plus jh And that would be the complex conjugate of the previous one. 245 00:14:25,690 --> 00:14:29,560 In order to get a nice, simple, clean solution that was in some a plus jb 246 00:14:29,560 --> 00:14:30,620 form, you still have to do a lot 247 00:14:30,620 --> 00:14:33,700 of calculation, but it's still the complex conjugate. 248 00:14:33,700 --> 00:14:35,190 So it's quite a handy thing. 249 00:14:35,190 --> 00:14:37,789 It's very simple to calculate complex conjugates using that. 250 00:14:39,170 --> 00:14:41,140 To summarize, we graphed values on the 251 00:14:41,140 --> 00:14:41,860 complex plane. 252 00:14:41,860 --> 00:14:44,660 We showed graph, complex exponentials in polar form. 253 00:14:44,660 --> 00:14:46,840 And calculated the complex conjugate. 254 00:14:46,840 --> 00:14:48,230 And so, this leads us on to the 255 00:14:48,230 --> 00:14:51,220 next section, where we will actually do arithmetic operations. 256 00:14:51,220 --> 00:14:54,950 Addition, subraction, multiplication and division using complex numbers. 257 00:14:54,950 --> 00:14:55,340 So, until then.