1 00:00:02,670 --> 00:00:05,360 Welcome back to our course on linear circuits. 2 00:00:05,360 --> 00:00:07,240 And today, we're going to be talking about impedance. 3 00:00:07,240 --> 00:00:11,450 And trying to apply the things that we learned last time about phasers into a 4 00:00:11,450 --> 00:00:14,460 practical application. Something that we're doing for circuit 5 00:00:14,460 --> 00:00:17,070 analysis, and seeing how useful it can be. 6 00:00:17,070 --> 00:00:20,830 The aim for this lesson is to identify impedances, which is this mathematical 7 00:00:20,830 --> 00:00:26,404 tool for analyzing linear circuits. That have have sinusoidal. 8 00:00:26,404 --> 00:00:29,250 So again, in our previous lesson, we talked about sinusoids. 9 00:00:29,250 --> 00:00:33,070 I reviewed some of the properties of those, as well as defined phasors. 10 00:00:33,070 --> 00:00:37,578 And found how to calculate them. So, now we get to move onto impedance. 11 00:00:37,578 --> 00:00:43,310 Impedance is as I mentioned before, somewhat analogous to resistance. 12 00:00:43,310 --> 00:00:49,150 And it will become a major point that going forward and doing search analysis 13 00:00:49,150 --> 00:00:54,580 and techniques in AC systems. For this lesson, our objective is to be 14 00:00:54,580 --> 00:00:58,920 able to describe impedance. Calculate the impedance of resistors, 15 00:00:58,920 --> 00:01:03,400 capacitors, and conductors and identify the relationship between the voltage and 16 00:01:03,400 --> 00:01:06,120 the current based upon an impedance value. 17 00:01:08,990 --> 00:01:16,230 So, first of all, the defining impedance. Impedance is, and you can see how similar 18 00:01:16,230 --> 00:01:21,500 it is to resistance here. It's a way of relating the voltage and 19 00:01:21,500 --> 00:01:25,300 the current. When our voltage or current sources are 20 00:01:25,300 --> 00:01:30,130 sinusoidal. And so, here I've showed, and whenever I 21 00:01:30,130 --> 00:01:33,689 use these block letters, this is corresponding to phasor. 22 00:01:33,689 --> 00:01:39,700 So, now I have a phasor for my voltage, and a phasor for my current. 23 00:01:39,700 --> 00:01:45,150 Impedance is not a phasor, but it relates the phasers to one another. 24 00:01:45,150 --> 00:01:47,120 So remember, the phasers are complex numbers. 25 00:01:47,120 --> 00:01:51,060 And so, if v is a complex number, and i is a complex number, and I know what they 26 00:01:51,060 --> 00:01:53,720 are. Then, it's possible for me to find some 27 00:01:53,720 --> 00:01:56,810 complex number z, that makes this equation work. 28 00:01:58,570 --> 00:02:01,610 So then, we think about the ratio. And so, you'll notice that this is very 29 00:02:01,610 --> 00:02:06,290 similar to Ohm's law, which Ohm's law stated that E is equal to I times R. 30 00:02:08,760 --> 00:02:13,318 And so, that's really kind of what it is, is R impedance is going to be a relation 31 00:02:13,318 --> 00:02:18,009 of our phasor voltage, to our phasor current. 32 00:02:19,600 --> 00:02:25,829 So, let's replace this arbitrary impedance, this arbitrary z device with a 33 00:02:25,829 --> 00:02:31,638 resistor, and see what we get. Well, if v is equal to some magnitude Vm 34 00:02:31,638 --> 00:02:39,720 times the cosine of omega t plus theta, and I know what my R is. 35 00:02:41,540 --> 00:02:51,240 Then, we can calculate my current i using i is equal to v divided by R. 36 00:02:52,320 --> 00:02:59,972 So, i's then going to be just Vm divided by R times the cosine of omega t plus 37 00:02:59,972 --> 00:03:03,140 theta. because it really doesn't care what's 38 00:03:03,140 --> 00:03:05,950 going on in time or any of this type of changing stuff. 39 00:03:05,950 --> 00:03:11,980 All it really needs to know is that this function is going to be scaled by R, and 40 00:03:11,980 --> 00:03:14,035 then I get my current out. So, very simple. 41 00:03:14,035 --> 00:03:23,380 Well, if I put these into phasor term, then V is going to be Vm arc theta, and I 42 00:03:23,380 --> 00:03:32,820 is going to be im arc theta which is Vm over R arc theta. 43 00:03:32,820 --> 00:03:39,547 So taking the ratio, I discovered that V over I, which is z, is simply R, no 44 00:03:39,547 --> 00:03:40,669 resistance. So, let's clear all of this up. 45 00:03:40,669 --> 00:03:55,620 And here we see the phasor for the voltage, the phasor for the current, and 46 00:03:55,620 --> 00:03:58,670 that z sub R is equal to R. Just the resistance. 47 00:03:58,670 --> 00:04:02,270 Very simple. In this case, our impedance. 48 00:04:02,270 --> 00:04:05,070 Impedance has a complex number, but in this case it just happens to be real, 49 00:04:05,070 --> 00:04:09,070 with a resistance, resistant element. It's just simply R. 50 00:04:09,070 --> 00:04:11,890 So, pretty simple to calculate. Let's look at one that's a little bit 51 00:04:11,890 --> 00:04:14,640 more tricky. What's the impedance of an inductor? 52 00:04:15,730 --> 00:04:21,040 So again, we're going to be using some of these ideas of the behavior that we see 53 00:04:21,040 --> 00:04:29,026 in the circuits, so we know that with the inductor, v is L di/dt, right? 54 00:04:29,026 --> 00:04:36,550 And so again, we're going to assume that we have phasers for inputs. 55 00:04:36,550 --> 00:04:40,740 And then, we're going to see if our voltage, for example, is a sinusoid, and 56 00:04:40,740 --> 00:04:43,380 our current's a sinusoid, we have some known inductance. 57 00:04:43,380 --> 00:04:46,080 We want to see the behavior between them and put them into phaser form. 58 00:04:46,080 --> 00:04:54,310 So, this is what we see. If i is this function here, it has a 59 00:04:54,310 --> 00:04:57,890 phaser like this. If V is a function, but we can calculate 60 00:04:57,890 --> 00:05:03,100 these based on L. And so, if you plug in the i here for the 61 00:05:03,100 --> 00:05:05,658 function i, we discover that V is equal to this. 62 00:05:05,658 --> 00:05:11,572 For our phaser, it's going to be LI sub m omega arc theta plus pi halves. 63 00:05:11,572 --> 00:05:16,570 Now, I'm not going to go through the derivation of how exactly this came out. 64 00:05:16,570 --> 00:05:20,180 In order to do it, you need to apply a few trigonometric entities. 65 00:05:20,180 --> 00:05:21,960 But I encourage you to try it as an exercise. 66 00:05:21,960 --> 00:05:24,840 And if you're having trouble understanding where this came from go 67 00:05:24,840 --> 00:05:29,395 ahead and post to the forums. If I take the ratio of those two 68 00:05:29,395 --> 00:05:36,270 impedances, I discover that we get this L omega arc pi halves, which turns out is 69 00:05:36,270 --> 00:05:43,920 equal to j omega L. So that means if I have an impedance, if 70 00:05:43,920 --> 00:05:50,000 I want to calculate an impedence for an inductor, it's going to be this j omega 71 00:05:50,000 --> 00:05:53,550 L. So a few important things to note, first 72 00:05:53,550 --> 00:05:57,870 of all, this frequency is important. With resistors there was no frequency, it 73 00:05:57,870 --> 00:06:00,710 was completely invarient of the frequency. 74 00:06:00,710 --> 00:06:03,240 We also notice that it's purely imaginary. 75 00:06:03,240 --> 00:06:07,380 There's no real term to this, that's all an imaginary thing. 76 00:06:07,380 --> 00:06:12,800 So, what does this kind of mean? Well, if you think about it this way, as 77 00:06:12,800 --> 00:06:19,300 I put a voltage onto this inductor here, it's going to cause current to start to 78 00:06:19,300 --> 00:06:23,970 want to flow more. As I increase my voltage, my current's 79 00:06:23,970 --> 00:06:27,920 going to try and match it. As I start to decrease my voltage, then 80 00:06:27,920 --> 00:06:31,780 the current's going to kind of resist that change, but is eventually kind of 81 00:06:31,780 --> 00:06:34,720 going to go down. And so again, if we go back and remember 82 00:06:34,720 --> 00:06:38,530 from the phasers how we saw kind of the space difference between the current and 83 00:06:38,530 --> 00:06:41,830 the voltage and a capacitor. The same thing happens with an inductor. 84 00:06:42,950 --> 00:06:45,860 And so, you can go back and take a look and see the behavior. 85 00:06:45,860 --> 00:06:51,880 And identify that when we have these complex terms that leads us to these 86 00:06:51,880 --> 00:06:56,860 phase shifts. And we also have attenuations based upon 87 00:06:56,860 --> 00:07:00,612 these frequencies for capacitors and inductors. 88 00:07:00,612 --> 00:07:06,290 So to recap, inductors are purely imaginative impedances. 89 00:07:06,290 --> 00:07:09,864 They scale based on frequency, and they're positive-imaginary, and 90 00:07:09,864 --> 00:07:15,190 positive-imaginary values means that the current lags the voltage. 91 00:07:15,190 --> 00:07:19,629 If it were negative-imaginary, current would lead the voltage, like a capacitor. 92 00:07:22,490 --> 00:07:25,500 This puts all of them together. So, if you've gone through and you've 93 00:07:25,500 --> 00:07:29,970 tried to derive them, this is kind of your little go-to page to make sure that 94 00:07:29,970 --> 00:07:34,170 you got the right answers. We already did the, the derivation for 95 00:07:34,170 --> 00:07:38,039 the resistors, and discovered that the impedance of a resistor is simply R. 96 00:07:39,290 --> 00:07:43,195 And because of this behavior, we see that the voltages and the currents are 97 00:07:43,195 --> 00:07:46,340 in-phase. And in all these pictures, I know it's a 98 00:07:46,340 --> 00:07:50,830 bit small but red corresponds to currents and blue corresponds to voltages. 99 00:07:50,830 --> 00:07:57,060 So for resistors, voltages and currents are zero in the same places and there is 100 00:07:57,060 --> 00:07:59,229 maximum and minimum values in the same places. 101 00:08:01,850 --> 00:08:04,800 And so is the frequency invariant. If I make this frequency much, much 102 00:08:04,800 --> 00:08:08,780 higher, it doesn't matter. The red curve is going to have the same 103 00:08:08,780 --> 00:08:13,740 amplitude as it does here. that's not, not an issue. 104 00:08:13,740 --> 00:08:19,260 Looking, looking at a capacitor, we see that, as we saw in the previous lesson, 105 00:08:19,260 --> 00:08:23,860 our current leads voltage. There's been a bit of a phase shift, 106 00:08:23,860 --> 00:08:28,520 turns out that the impedance for capacitors with one over j omega C. 107 00:08:28,520 --> 00:08:32,370 And this is a negative imaginary number. If you don't understand why it's 108 00:08:32,370 --> 00:08:34,360 negative, go back and look at complex numbers. 109 00:08:34,360 --> 00:08:36,532 But, it's important that you kind of understand. 110 00:08:36,532 --> 00:08:43,710 Also as the frequency increases, the voltage attenuate which means the voltage 111 00:08:43,710 --> 00:08:46,040 gets smaller. If I have a very very high frequency 112 00:08:46,040 --> 00:08:49,420 system my voltages are going to be smaller in the capacitors. 113 00:08:49,420 --> 00:08:51,885 And why is this? We know that the voltage can't 114 00:08:51,885 --> 00:08:56,520 instanteously change on a capacitor, and the voltage basically is desribing the 115 00:08:56,520 --> 00:09:00,790 amount of charge tha'ts on the plates. So, if we want to keep a high voltage 116 00:09:00,790 --> 00:09:04,640 with a high frequency, we need a lot of current to put charge on to this place 117 00:09:04,640 --> 00:09:08,370 very very quickly. And so, it's sometimes good to go through 118 00:09:08,370 --> 00:09:11,640 and try and get an intuitive understanding of why this behavior 119 00:09:11,640 --> 00:09:14,550 occurs. Now, I'm just going to go and quickly 120 00:09:14,550 --> 00:09:20,270 recap for the inductors. The inductance gives us this j omega L 121 00:09:20,270 --> 00:09:24,530 for our impedance. The current now lags the voltage in an 122 00:09:24,530 --> 00:09:28,690 inductor, and current attenuates for high frequency. 123 00:09:28,690 --> 00:09:32,100 And you can see why it's kind of analogous to the voltage attenuating for 124 00:09:32,100 --> 00:09:35,476 high frequencies with capacitive circuits. 125 00:09:35,476 --> 00:09:38,035 so this is kind of a good table, good point of reference. 126 00:09:38,035 --> 00:09:42,780 I encourage you to go through and do some of these analyses yourself following the 127 00:09:42,780 --> 00:09:45,980 structure that's been presented here. And see if you can understand where all 128 00:09:45,980 --> 00:09:47,790 of these things are coming from. It's a good exercise. 129 00:09:49,850 --> 00:09:54,130 To summarize, we defined impedance and we calculated impedance of linear devices 130 00:09:54,130 --> 00:09:56,152 and even define, derive where it comes from. 131 00:09:56,152 --> 00:10:01,450 Well, I must say it's kind of a tryst summary and inductor. 132 00:10:01,450 --> 00:10:04,960 And then, we describe the relationships between the current and the voltage given 133 00:10:04,960 --> 00:10:08,350 from the known impedance. In the next lesson, we'll start using all 134 00:10:08,350 --> 00:10:12,090 of the things we've learned here. And look at our analysis techniques that 135 00:10:12,090 --> 00:10:16,070 we learned in previous modules, and put them together and see what works. 136 00:10:16,070 --> 00:10:20,090 What doesn't and what we can, what we can do going forward. 137 00:10:20,090 --> 00:10:22,830 So, look forward to seeing you next time and seeing you on the forums. 138 00:10:22,830 --> 00:10:25,980 Because you'll probably have some questions on the material that we covered 139 00:10:25,980 --> 00:10:28,180 in these previous two lessons. Until then.