1 00:00:02,220 --> 00:00:04,260 Welcome back to our course on Linear Circuits. 2 00:00:04,260 --> 00:00:07,080 At this point, you've finished the first three modules. 3 00:00:07,080 --> 00:00:11,890 So now, we've covered most of the basic circuit elements that we'll be looking at 4 00:00:11,890 --> 00:00:14,170 and learned a little bit about some analysis techniques. 5 00:00:14,170 --> 00:00:16,520 Now, we're going to start moving on to some more advanced topics. 6 00:00:16,520 --> 00:00:19,470 This module's going to be covering frequency analysis. 7 00:00:19,470 --> 00:00:23,890 This was mentioned before. We will start doing our math a little bit 8 00:00:23,890 --> 00:00:26,680 differently, but a lot of the material that was covered in the first three 9 00:00:26,680 --> 00:00:30,720 modules will turn out to be very useful going forward and we'll see how it can 10 00:00:30,720 --> 00:00:33,450 continue to be applied in the new material. 11 00:00:33,450 --> 00:00:37,720 This particular lesson talks about sinusoids and phasors. 12 00:00:37,720 --> 00:00:41,340 And so, we will be reviewing the properties of sinusoids We'll be 13 00:00:41,340 --> 00:00:44,625 describing phasors and what they are, as well as how to use them. 14 00:00:44,625 --> 00:00:51,000 This particular lesson uses a fair bit of information from complex numbers. 15 00:00:51,000 --> 00:00:54,330 So if you don't remember complex numbers, it might be good to go and review. 16 00:00:54,330 --> 00:00:57,120 or if, as you're going through the lesson, if there's something you don't 17 00:00:57,120 --> 00:01:00,120 quite understand, go back and review that and you should be able to, to follow 18 00:01:00,120 --> 00:01:03,692 along pretty easily. In this module, we'll start with 19 00:01:03,692 --> 00:01:07,720 sinusoids and phasors and understanding those topics will allow us to start 20 00:01:07,720 --> 00:01:14,260 talking about impedance which is somewhat analogous to resistance but for circuits 21 00:01:14,260 --> 00:01:17,650 that has sinusoidal inputs. And then, we'll look at doing circuit 22 00:01:17,650 --> 00:01:23,100 analysis in AC systems, so systems with this Is a frequency, the sinusoidal 23 00:01:23,100 --> 00:01:27,320 frequency behaviors. The objectives for this lesson are to 24 00:01:27,320 --> 00:01:32,920 identify the properties of sinusoids like amplitude, frequency period, and so on. 25 00:01:32,920 --> 00:01:37,485 Describe the properties of sinusoids for capacitors and inductors. 26 00:01:37,485 --> 00:01:43,100 Find phasors of sinusoidal functions and then seeing how we can use phasors To add 27 00:01:43,100 --> 00:01:48,420 two sinusoid functions today. First well looking at the sinusoidal 28 00:01:48,420 --> 00:01:51,750 curve. Lets go through some definitions and some 29 00:01:51,750 --> 00:01:56,390 important things that we should note. First of all,we have a function v of t 30 00:01:56,390 --> 00:01:59,668 which is equal to Vm cosine of omega t plus theta. 31 00:01:59,668 --> 00:02:07,600 So this Vm is our peak value and it identifies the highest passive point as 32 00:02:07,600 --> 00:02:11,330 well as the lowest negative point of this centered sinusoid. 33 00:02:13,770 --> 00:02:17,370 We also have this theta which is a phase angle and we'll kind of see how that 34 00:02:17,370 --> 00:02:20,830 relates to the time or this plot here soon. 35 00:02:22,010 --> 00:02:26,770 This T here is our period. And basically the period is just the 36 00:02:26,770 --> 00:02:31,720 amount of time between two corresponding points on our sinusoid. 37 00:02:31,720 --> 00:02:33,900 So, in this case, I picked the two maximal points. 38 00:02:33,900 --> 00:02:38,880 You could use the two minimal points or even if you look at these two points at 39 00:02:38,880 --> 00:02:41,900 the zero axis that correspond to the function going down. 40 00:02:41,900 --> 00:02:43,348 Any two points that correspond to one another. 41 00:02:43,348 --> 00:02:48,200 You mention, you measured the distance and time between them and that gives you 42 00:02:48,200 --> 00:02:50,480 your period. And from that you can calculate your 43 00:02:50,480 --> 00:02:54,260 frequency where the frequency is equal to 1 over that value. 44 00:02:56,190 --> 00:03:00,870 Frequency could also be thought of as the number of cycles, number of times you go 45 00:03:00,870 --> 00:03:07,480 through one sin-, one complete sinusoid. Every second, measured in Hertz, where 1 46 00:03:07,480 --> 00:03:14,130 hertz is equal to 1 over 1 second. Now, all of these things are kind of just 47 00:03:14,130 --> 00:03:17,070 looking at the function itself. We actually have an angular frequency 48 00:03:17,070 --> 00:03:25,375 where we kind of think of this, distance, as corresponding to 2 pi radians. 49 00:03:25,375 --> 00:03:31,300 Or 360 degrees. And so the angular frequency tells us how 50 00:03:31,300 --> 00:03:34,230 many degrees, or how many radians we go through every second. 51 00:03:35,600 --> 00:03:39,826 And so we can calculate that by just multiplying our frequency f by 2 pi to 52 00:03:39,826 --> 00:03:43,660 get radians per second. You could also do this in, in degrees per 53 00:03:43,660 --> 00:03:46,134 second, I suppose. But typically, it'll be radians per 54 00:03:46,134 --> 00:03:49,000 second, another phase angle. Phase angle is a little hard to wrap your 55 00:03:49,000 --> 00:03:53,380 head around, but if you look at this function here, you can see that if were a 56 00:03:53,380 --> 00:03:58,360 real cosign function, basic cosign function it would have maximal value for 57 00:03:58,360 --> 00:04:01,750 time t equals 0. But it's been shifted to the right by 58 00:04:01,750 --> 00:04:05,820 some small amount. The phase angle measures how much of a 59 00:04:05,820 --> 00:04:09,360 phase, how many degrees it has been shifted over. 60 00:04:09,360 --> 00:04:16,034 So we can calculate that as negative 2 pi times tp over t, where that t is the 61 00:04:16,034 --> 00:04:24,040 period, and tp is the point where that first p occurs, and so, that's how we 62 00:04:24,040 --> 00:04:26,950 find a phase angle. Our phase angles are going to be negative 63 00:04:26,950 --> 00:04:31,070 if we're shifting to the right and it will be positive if we're shifting to the 64 00:04:31,070 --> 00:04:37,170 left. So, let's see how sinusoids behave if we 65 00:04:37,170 --> 00:04:41,450 put them through a real element. So, we'll use a capacitor. 66 00:04:41,450 --> 00:04:46,320 If our input is for example voltage going through this capacitor and we know that 67 00:04:46,320 --> 00:04:52,026 the relationship between voltage and current in the capacitors i equals Cc dv 68 00:04:52,026 --> 00:04:55,910 dt. So if I have this sinusoidal input V of 69 00:04:55,910 --> 00:05:01,760 C, I can calculate my i of c by taking the derivative and scaling it by C, which 70 00:05:01,760 --> 00:05:06,630 gives us this. So let's think about what's kind of going 71 00:05:06,630 --> 00:05:12,440 on. As my current goes up, that corresponds 72 00:05:12,440 --> 00:05:17,670 to putting more charge onto these plates. But anywhere my current is positive, I'm 73 00:05:17,670 --> 00:05:24,400 putting charge onto my capacitive plates, which means that my voltage is 74 00:05:24,400 --> 00:05:27,490 increasing. But at a certain point, when my current 75 00:05:27,490 --> 00:05:34,220 starts to go negative, now, I'm removing the charge from my plates. 76 00:05:35,310 --> 00:05:39,240 So now, my voltage is going to drop. So what do we observe here? 77 00:05:39,240 --> 00:05:46,200 Well, we observe that my current is going to go up before my voltage does, which is 78 00:05:46,200 --> 00:05:49,468 to say that we're, well, we're going to say that the current leads the voltage. 79 00:05:49,468 --> 00:05:53,360 So there's a little bit of phase shift between these two curves. 80 00:05:55,390 --> 00:06:00,170 We also notice that the voltage reach, reaches its peak value where the current 81 00:06:00,170 --> 00:06:04,410 reaches its 0 value. Well, that makes sense because of this 82 00:06:04,410 --> 00:06:09,732 derivative property, the slope of this l curve right here is going to be 0. 83 00:06:09,732 --> 00:06:14,720 So the currents going to be 0. So it's kind of the behavior that we're 84 00:06:14,720 --> 00:06:18,950 going to observe when we're working with sinusoids in capacitors and it's also 85 00:06:18,950 --> 00:06:27,010 important notice this omega here. As my frequency increases, my current 86 00:06:27,010 --> 00:06:30,630 increases, so what does that means? Well, that means if I make this frequency 87 00:06:30,630 --> 00:06:34,010 really really fast. If I want to get that to the same voltage 88 00:06:34,010 --> 00:06:39,110 values, I need bigger currents. Because I have less time to put charge 89 00:06:39,110 --> 00:06:44,800 onto the plates. And so, as we increase our frequency, the 90 00:06:44,800 --> 00:06:48,060 relative size of this current is going to be smaller. 91 00:06:48,060 --> 00:06:50,910 Or sorry, the relative size of the current's going to be larger, with 92 00:06:50,910 --> 00:06:53,650 respect to the voltage as the dot omega increases. 93 00:06:56,180 --> 00:06:58,542 Now, we're going to talk about this thing called phasors. 94 00:06:58,542 --> 00:07:05,810 A phasor is where we take a sinusoid and we re-encode it as a complex number, and 95 00:07:05,810 --> 00:07:08,350 that allows us to do some math in a very simple way. 96 00:07:08,350 --> 00:07:12,000 As opposed to having to use all of the sinusoidal functions and a bunch of 97 00:07:12,000 --> 00:07:14,980 trigonometric identities. We can simply just do complex number 98 00:07:14,980 --> 00:07:18,350 arithmetic, and we can do some really neat things. 99 00:07:18,350 --> 00:07:24,773 So, in this example, we have this cosine function here, described by the function 100 00:07:24,773 --> 00:07:29,450 Vm cosine 2000 pi minus pi 3rds. So here we see the frequency is 1000 101 00:07:29,450 --> 00:07:33,660 hertz. So our angular frequency this would 102 00:07:33,660 --> 00:07:38,115 actually be 2000 pi t. [UNKNOWN] the angular frequency here is 103 00:07:38,115 --> 00:07:43,460 2000 pi. We also notice that there's been a phase 104 00:07:43,460 --> 00:07:47,490 shift that's going to be negative because we've shifted it to the right. 105 00:07:47,490 --> 00:07:51,097 And how much have we shifted it? Well, it's pi thirds radians. 106 00:07:51,097 --> 00:07:53,490 What I'm going to do is, I'm going to take this. 107 00:07:53,490 --> 00:07:58,720 And I'm going to take its amplitude. Put it right here. 108 00:07:58,720 --> 00:08:01,290 Then take its phase. Which is minus pi thirds. 109 00:08:02,430 --> 00:08:05,984 And put it here. And this becomes a complex number. 110 00:08:05,984 --> 00:08:11,240 Where Vm is the amplitude, and minus pi 3rds is the phase, so if we plot that on 111 00:08:11,240 --> 00:08:15,250 a complex plane, it looks like this. We could also describe this using 112 00:08:15,250 --> 00:08:21,960 rectangular coordinates. so here, because this is 60 degrees, this 113 00:08:21,960 --> 00:08:25,480 distance is going to be half of Vm, and this distance is going to be minus the 114 00:08:25,480 --> 00:08:28,910 square root of 3 over 2vm. We should also notice that here there's a 115 00:08:28,910 --> 00:08:34,030 j in the rectangular, so 1 half Vm minus j squared 3 over 2. 116 00:08:34,030 --> 00:08:37,810 Here, j is just equal to the square root of negative 1. 117 00:08:37,810 --> 00:08:41,910 We're going to use the square root of negative 1 as j instead of i, which you 118 00:08:41,910 --> 00:08:45,215 might be more familiar with because we're already using i is current. 119 00:08:45,215 --> 00:08:48,535 And it turns out that in a lot of engineering, people use J instead of I. 120 00:08:48,535 --> 00:08:52,790 So, just don't let that startle you. Anytime you see a j, just think it's an 121 00:08:52,790 --> 00:08:57,510 i. Now, it's pretty simple to go from here, 122 00:08:57,510 --> 00:08:59,910 and if we wanted to go back, the same thing applies. 123 00:08:59,910 --> 00:09:04,310 We find the amplitude of this as well as the phase here, and then you just go 124 00:09:04,310 --> 00:09:08,630 back. Vm cosine of whatever your frequency is, 125 00:09:08,630 --> 00:09:12,730 and then your phase. Notice that the phasor has an implied 126 00:09:12,730 --> 00:09:14,565 frequency. There is a frequency there. 127 00:09:14,565 --> 00:09:18,615 But nowhere here in this complex number do we say what the frequency is. 128 00:09:18,615 --> 00:09:23,160 But it's very important because if the frequencies don't match, you can't do the 129 00:09:23,160 --> 00:09:27,330 basic arithmetic. Now let's see how these can be useful. 130 00:09:27,330 --> 00:09:33,142 Suppose that I wanted to add 7 cosine of 120 pi t plus 30 degrees plus 3 cosine 131 00:09:33,142 --> 00:09:36,940 120 pi t minus 60 degrees. You wanted to add these two functions 132 00:09:36,940 --> 00:09:41,150 together, you'd probably have to use a lot of trigonometric identities trying to 133 00:09:41,150 --> 00:09:45,610 combine these can be a real pain. But it turns out that you can do it very 134 00:09:45,610 --> 00:09:52,040 simply by using phasors. So as we already saw going from this time 135 00:09:52,040 --> 00:09:58,120 function tho this phasor is simply take in this amplitude and this phase angle, 136 00:09:58,120 --> 00:10:02,460 and going like this. The same thing for V2, which gives us 137 00:10:02,460 --> 00:10:06,398 these two vectors. Again, in our complex plane. 138 00:10:06,398 --> 00:10:12,205 If I want to add these two vectors together, it's kind of the same thing as 139 00:10:12,205 --> 00:10:17,190 concatenating this vector on the end of this one or this one on the end of this 140 00:10:17,190 --> 00:10:19,430 one. So we're going to get some vector that 141 00:10:19,430 --> 00:10:23,595 kind of goes out like this. It turns out that when you want to add 142 00:10:23,595 --> 00:10:26,990 phasors together, add complex numbers together, it's easier to put them into 143 00:10:26,990 --> 00:10:31,380 rectangular format, and then convert back to the phasor format. 144 00:10:31,380 --> 00:10:33,515 So just quickly so see how you would do that. 145 00:10:33,515 --> 00:10:39,735 for 7 arc 30, because this is a 30 60 90 triangle. 146 00:10:39,735 --> 00:10:48,118 V1 is just going to be so the square root of 3 divided by 2 times 7 plus j times 1 147 00:10:48,118 --> 00:10:53,574 half, times 7. And if you're kind of confused as where 148 00:10:53,574 --> 00:10:59,030 this come, came from, just go back to your trigonometry. 149 00:10:59,030 --> 00:11:04,050 And look at 30, 60, 90 triangles and it'll be pretty simple to see where it 150 00:11:04,050 --> 00:11:06,845 comes from. So it's just a 30, 60, 90 triangle. 151 00:11:06,845 --> 00:11:09,407 So you just convert them into rectangular. 152 00:11:09,407 --> 00:11:12,660 You add them together. And then you convert them back into this 153 00:11:12,660 --> 00:11:16,465 polar form, kind of this amplitude and face type of a thing. 154 00:11:16,465 --> 00:11:21,340 So let's see what we get. Adding them together gives us a phases 155 00:11:21,340 --> 00:11:29,650 format of 7.62 arc 6.8, right here. And since I already have this phasor 156 00:11:29,650 --> 00:11:33,845 format, all I have to do is take the amplitude pop it down there. 157 00:11:33,845 --> 00:11:38,790 The angular frequency here is going to be the same as my 2 inputs. 158 00:11:38,790 --> 00:11:41,180 And this is one of the reasons that phasors have this implied frequency and 159 00:11:41,180 --> 00:11:44,550 is so important. So that just goes back where it was, and 160 00:11:44,550 --> 00:11:49,800 then, our phase comes back to 6.8. So we see that we've been able to add 161 00:11:49,800 --> 00:11:53,930 these two cosines functions together which would be really difficult to do 162 00:11:55,030 --> 00:11:58,190 using our normal functions. We put them into phasors and math is a 163 00:11:58,190 --> 00:12:02,510 lot easier to do. Couple of comments about phasors. 164 00:12:02,510 --> 00:12:06,460 First of all, you cannot compare phasors with different frequencies. 165 00:12:06,460 --> 00:12:11,480 They just don't mesh together. and there's a lot of ways you can go and 166 00:12:11,480 --> 00:12:15,089 practice that, try taking two functions together, you can try and add them as 167 00:12:15,089 --> 00:12:19,400 phases and see what you get. Also, you cannot multiply phasors 168 00:12:19,400 --> 00:12:22,140 together and expect it to give you the same thing as multiplying the two 169 00:12:22,140 --> 00:12:24,830 functions together. For example, if I have the cosine of x 170 00:12:24,830 --> 00:12:27,496 and I wanted to multiply by x, it's going to be cosine squared of x. 171 00:12:27,496 --> 00:12:38,528 And going to trigonometric identity turns out that that's 1 half plus 1 half of the 172 00:12:38,528 --> 00:12:44,040 cosine of 2x. But if you did this in phasor format, 173 00:12:44,040 --> 00:12:53,343 this would be 1 arc 0 degrees times 1 arc 0 degrees and those that are just 1 times 174 00:12:53,343 --> 00:12:57,440 1 complex numbers are just 1. So the result would be 1 over 0 degrees, 175 00:12:57,440 --> 00:13:00,625 which would say that multiplying these two things together just gives you the 176 00:13:00,625 --> 00:13:03,900 cosine of x. So clearly, they're not the same. 177 00:13:03,900 --> 00:13:06,970 But doing this multiplication of phasors has its own meaning. 178 00:13:06,970 --> 00:13:10,465 You just can't kind of interpret it that way. 179 00:13:10,465 --> 00:13:13,590 And especially, taking the ratios of phasors will be very useful. 180 00:13:13,590 --> 00:13:19,420 And so, we'll see in the next section where that's used for analysis, and ways 181 00:13:19,420 --> 00:13:24,250 that that's actually applied. So to summarize, we reviewed sinusoidal 182 00:13:24,250 --> 00:13:28,360 properties, and identified the sinusoid behavior in linear devices and found 183 00:13:28,360 --> 00:13:32,250 phasors and used them to add sinusoids together, which is much easier than just 184 00:13:32,250 --> 00:13:37,650 adding the functions together themselves. There's probably going to be some 185 00:13:37,650 --> 00:13:40,870 questions about this material, it's a lot of things to cover very quickly. 186 00:13:40,870 --> 00:13:43,850 So if there are any questions that you have go to forums, post in there, we'll 187 00:13:43,850 --> 00:13:46,659 be sure to try and make sure everything is clear to you. 188 00:13:46,659 --> 00:13:50,740 In our next lesson we will be defining a, this thing called impedance, which is a 189 00:13:50,740 --> 00:13:55,700 property that describes how currents and voltages relate to each other when we 190 00:13:55,700 --> 00:13:59,001 have sinusoidal systems. So, until then.