1 00:00:02,260 --> 00:00:03,630 Welcome back to linear circuits. 2 00:00:04,640 --> 00:00:07,650 This lesson is a continuation on frequency response. 3 00:00:07,650 --> 00:00:09,150 This time were looking at Bode plots. . 4 00:00:10,310 --> 00:00:13,780 So again it, it still fits in the category of frequency response. 5 00:00:15,580 --> 00:00:19,430 In our previous lesson we introduced frequency response as a way of showing 6 00:00:19,430 --> 00:00:20,930 how a sig, how a system 7 00:00:20,930 --> 00:00:24,400 or circuit processes signals of different frequencies. 8 00:00:24,400 --> 00:00:26,870 In that particular case, we just used the linear scale. 9 00:00:28,480 --> 00:00:30,340 The only difference we're doing in this lesson 10 00:00:30,340 --> 00:00:31,910 is that we're now going to use a Bode 11 00:00:31,910 --> 00:00:34,920 plot, which is a different scale, logarithmic scale, 12 00:00:34,920 --> 00:00:36,860 as a way of showing that the frequency response. 13 00:00:38,960 --> 00:00:40,036 So, Bode plot. 14 00:00:40,036 --> 00:00:43,527 [BLANK_AUDIO] 15 00:00:43,527 --> 00:00:44,600 We use these scales. 16 00:00:44,600 --> 00:00:50,450 Again, it's logarithmic in both the, this axis as well as this axis. 17 00:00:50,450 --> 00:00:53,940 In this axis, we usually plot it on semi log paper. 18 00:00:53,940 --> 00:00:58,680 So, this scale is logarithmic here. And let me define something here. 19 00:00:58,680 --> 00:01:02,820 Let me define this distance from here to here as a decade. 20 00:01:05,850 --> 00:01:12,060 It's a distance to go from sum frequency f to 10 times the frequency. 21 00:01:14,260 --> 00:01:17,620 Call it f sub one. A particular frequency to 10 times that. 22 00:01:17,620 --> 00:01:19,500 So this is also a decade. 23 00:01:19,500 --> 00:01:22,460 Because I'm going from this frequency to this frequency. 24 00:01:22,460 --> 00:01:23,530 10 times it. 25 00:01:24,710 --> 00:01:25,620 the same thing here. 26 00:01:25,620 --> 00:01:30,480 If I went from this frequency to this frequency, same thing is also decade. 27 00:01:30,480 --> 00:01:34,600 Now, frequency is often shown in terms of radiance per second, or hertz. 28 00:01:34,600 --> 00:01:40,670 Recall that, two pi f is equal to omega. 29 00:01:40,670 --> 00:01:43,160 So that's it, remember the conversion to go between those two. 30 00:01:44,380 --> 00:01:47,020 Now on this scale, it is actually a linear scale here. 31 00:01:47,020 --> 00:01:52,150 And we often mark these in terms of 20 degree or 20 decibel differences. 32 00:01:52,150 --> 00:01:57,790 So that might be 20, 40, 60, and so on. The db stands for decibels. 33 00:01:57,790 --> 00:02:03,391 [BLANK_AUDIO] 34 00:02:03,391 --> 00:02:05,281 And we get that value, we get the 35 00:02:05,281 --> 00:02:08,501 Bode plot, by simply taking the magnitude function, 36 00:02:08,501 --> 00:02:11,721 the frequency response function and taking 20 times 37 00:02:11,721 --> 00:02:14,610 the log that's logged base ten of it. 38 00:02:14,610 --> 00:02:17,890 The angle plot is the same way we take the angle in 39 00:02:17,890 --> 00:02:22,700 degrees typically and we plot it using a logarithmic scale along this axis. 40 00:02:26,460 --> 00:02:30,470 Let's compare the linear plot and the Bode plot side-by-side. 41 00:02:30,470 --> 00:02:35,040 On the left is the linear plot, so this is a function of a magnitude of h of omega. 42 00:02:36,470 --> 00:02:39,120 And this is the angle of H of omega. 43 00:02:40,600 --> 00:02:42,670 Now every point along here, at every 44 00:02:42,670 --> 00:02:48,170 frequency, I compute the, the magnitude in decibels. 45 00:02:48,170 --> 00:02:51,650 So over here, 92 in decibels, I get by taking 46 00:02:51,650 --> 00:02:55,899 20 times the log of it. So for example, at Omega equals 47 00:02:55,899 --> 00:03:00,511 zero, I have a value of one. Well, 20 times a log of one 48 00:03:00,511 --> 00:03:06,070 is zero, and that's where i get zero decibels right here. 49 00:03:06,070 --> 00:03:11,717 Right over here I have a value of 0.1. So, 20 times a log 50 00:03:11,717 --> 00:03:17,730 of 0.1 Is minus 20, and that's 51 00:03:17,730 --> 00:03:19,130 where I get this point here. 52 00:03:19,130 --> 00:03:21,180 So that's true of every point along the way. 53 00:03:21,180 --> 00:03:24,800 And the angle I get, really by simply moving it over. 54 00:03:24,800 --> 00:03:30,290 But I'm stretching out the scale, the frequency scale, like this. 55 00:03:30,290 --> 00:03:33,030 So, I mean, counting it, in terms of the 56 00:03:33,030 --> 00:03:37,390 logarithmic scale here, this is the value of 10, this 57 00:03:39,990 --> 00:03:42,750 would be 20, 30, 40, 50, 60, 70, 80, 90 and 100. 58 00:03:42,750 --> 00:03:47,490 Then this would be 200, 300, 400 and so on. 59 00:03:48,830 --> 00:03:52,550 So that's the difference between scales, but it's the same information. 60 00:03:52,550 --> 00:03:58,900 I'm plotting this, but this time I'm taking 20 times the log of the magnitude. 61 00:03:58,900 --> 00:04:01,970 I'm plotting that and that's the only difference 62 00:04:01,970 --> 00:04:03,610 between the Bode plot and the linear plot. 63 00:04:09,630 --> 00:04:13,414 Let's look at a first order system characteristics on a Bode plot. 64 00:04:13,414 --> 00:04:15,490 This is an RC circuit. 65 00:04:15,490 --> 00:04:25,120 This is a first order system we had looked before what the magnitude looks 66 00:04:25,120 --> 00:04:29,360 like and the magnitude function is like this, the angle function is like this. 67 00:04:29,360 --> 00:04:32,350 So if we took the magnitude and take 20 times the 68 00:04:32,350 --> 00:04:35,270 log of it and plot that and then plot the angle, 69 00:04:35,270 --> 00:04:38,030 all in the logarithmic scale, we get these two plots. 70 00:04:39,280 --> 00:04:41,342 And a couple of things that I wanted to point out here. 71 00:04:41,342 --> 00:04:45,220 One is at high frequency, if I look at 72 00:04:45,220 --> 00:04:47,979 this slope here, it looks like a straight line slope. 73 00:04:47,979 --> 00:04:53,880 And it is, it turns out it's minus 20 decibels per decade. 74 00:04:53,880 --> 00:04:56,680 because a decade is between a frequency and 75 00:04:56,680 --> 00:04:58,900 10 times that frequency, so that's one decade. 76 00:05:00,270 --> 00:05:05,150 And then we went down, if I follow this line up here, kind of 77 00:05:07,640 --> 00:05:09,030 stretch this out a bit 78 00:05:13,010 --> 00:05:15,090 It's going from here down to here. 79 00:05:15,090 --> 00:05:18,430 In fact if this kept going it would keep following that line. 80 00:05:18,430 --> 00:05:22,230 So our curve is actually asymptotic to a minus 20 decibels 81 00:05:22,230 --> 00:05:25,370 per decade line, and that's true of all first order systems. 82 00:05:26,400 --> 00:05:31,510 Now the angle, it starts out at zero And then it goes down to minus 90. 83 00:05:31,510 --> 00:05:34,560 It's asymptotic to minus 90. 84 00:05:34,560 --> 00:05:38,560 So, it goes from 0 degrees to minus 90 degrees. 85 00:05:41,040 --> 00:05:43,490 And that's, those are the characteristics of it, a first 86 00:05:43,490 --> 00:05:50,070 order system like a, RC network in this sort of configuration. 87 00:05:55,620 --> 00:05:58,160 Now lets look at a Bode Plot of an RLC Circuit. 88 00:05:58,160 --> 00:06:02,090 An RLC Circuit is what I'll call a second order system. 89 00:06:02,090 --> 00:06:04,860 Second order system because, if I were to find, look at 90 00:06:04,860 --> 00:06:08,850 the differential equation it would have a second derivative in there. 91 00:06:08,850 --> 00:06:09,850 So I call it second order. 92 00:06:14,060 --> 00:06:17,630 So, this particular RLC circuit has this transfer function. 93 00:06:17,630 --> 00:06:20,300 If I find the magnitude, and then take 20 times 94 00:06:20,300 --> 00:06:22,560 the log of it and plot it, I get this. 95 00:06:22,560 --> 00:06:24,980 And, if I take the angle, I get this. 96 00:06:24,980 --> 00:06:26,670 What are the characteristics here? 97 00:06:27,730 --> 00:06:31,800 Well, at high frequency, if I look at this slope. 98 00:06:31,800 --> 00:06:34,300 This slope is minus 40 decibels per decade, 99 00:06:36,110 --> 00:06:37,830 and that's because this is a second order, 100 00:06:40,810 --> 00:06:45,360 a second order circuit. I'll get minus 40 db per decade. 101 00:06:45,360 --> 00:06:49,030 And also with this RLC, or second order circuit, I look at the angle. 102 00:06:49,030 --> 00:06:50,310 And the angle also is different. 103 00:06:50,310 --> 00:06:54,290 It goes from zero degrees to minus 180 degrees. 104 00:06:58,440 --> 00:07:05,036 Now the curve here, this comes to about, right around one over the square root of 105 00:07:05,036 --> 00:07:11,950 LC, is what I'll call the cross-over frequency, one over the square root of LC. 106 00:07:11,950 --> 00:07:15,018 That's often called the resonate frequency. 107 00:07:15,018 --> 00:07:23,234 [SOUND] 108 00:07:23,234 --> 00:07:30,990 Now a second order systems in RLC circuits don't always look like this. 109 00:07:30,990 --> 00:07:34,326 You notice the resonate frequency didn't involve R. 110 00:07:34,326 --> 00:07:39,600 And the R is clearly is in this function, so it does affect what it will look like. 111 00:07:39,600 --> 00:07:43,670 And the effect of R is really shown better by looking at 112 00:07:43,670 --> 00:07:48,440 a case where this is, this is the value of very, large value 113 00:07:48,440 --> 00:07:52,410 of R. So I'll say plot for large R. 114 00:07:52,410 --> 00:07:52,910 I'm 115 00:07:56,830 --> 00:08:04,830 going to redraw this, frequency response but for a very small value of R. 116 00:08:04,830 --> 00:08:05,780 We'll show the difference. 117 00:08:08,500 --> 00:08:11,775 So this is for a, small R. 118 00:08:11,775 --> 00:08:16,730 [SOUND]. 119 00:08:16,730 --> 00:08:20,395 And what we see is that it peaks up here. 120 00:08:20,395 --> 00:08:23,820 [SOUND]. 121 00:08:23,820 --> 00:08:24,320 And 122 00:08:26,130 --> 00:08:27,510 that's really a resonant peak. 123 00:08:30,880 --> 00:08:35,830 And what happens is that as R gets smaller, this peak gets higher. 124 00:08:38,670 --> 00:08:41,180 That peak goes higher as R gets smaller. 125 00:08:43,500 --> 00:08:51,195 Now the, the value of the high frequency is still minus 40 db per decade 126 00:08:51,195 --> 00:08:53,620 [SOUND]. 127 00:08:53,620 --> 00:08:57,880 The angle still goes from zero to minus 180 degrees. 128 00:08:57,880 --> 00:08:59,605 But it does so a little bit sharper. 129 00:08:59,605 --> 00:09:05,390 [SOUND] 130 00:09:05,390 --> 00:09:09,410 And what we see is this, this has to do with damping. 131 00:09:10,780 --> 00:09:11,280 What 132 00:09:13,840 --> 00:09:20,490 we call the damping ratio goes down as R goes down. 133 00:09:20,490 --> 00:09:24,160 And so the peak goes up, causing a resonance. 134 00:09:24,160 --> 00:09:26,220 What does it mean, this peak? 135 00:09:27,250 --> 00:09:33,270 What it means is that at low frequency the output amplitude is 136 00:09:33,270 --> 00:09:35,640 going to be about be about the same as the input amplitude. 137 00:09:35,640 --> 00:09:39,070 But there becomes a frequency range in here where the output 138 00:09:39,070 --> 00:09:41,680 amplitude is actually larger than the input. 139 00:09:41,680 --> 00:09:45,240 Remember, zero db corresponds to an amplitude of one. 140 00:09:45,240 --> 00:09:49,690 So that means the output amplitude is equal to the input amplitude. 141 00:09:49,690 --> 00:09:52,540 When we're greater than one, when we're greater than zero, the 142 00:09:52,540 --> 00:09:59,210 output amplitude is greater than the input amplitude in the resonance range. 143 00:10:07,180 --> 00:10:09,430 And then it goes down even more so. 144 00:10:10,480 --> 00:10:13,170 So the output just looks like it's, it's a bigger amplitude. 145 00:10:13,170 --> 00:10:16,850 Bigger bigger sinusoid than the input amplitude. 146 00:10:20,020 --> 00:10:21,860 Now, let's look at an example. 147 00:10:21,860 --> 00:10:26,150 Trying to figure out what the output is to a particular input using the Bode plot. 148 00:10:28,170 --> 00:10:31,500 So this is my input to my circuit. 149 00:10:31,500 --> 00:10:37,050 Ant this is my Bode plot which is the, shows the transfer function in there. 150 00:10:38,530 --> 00:10:40,570 I've got three different components to my input. 151 00:10:40,570 --> 00:10:42,600 I want to analyse them each individually. 152 00:10:43,740 --> 00:10:47,236 So we use the same thing while, if I look at the 153 00:10:47,236 --> 00:10:53,400 output amplitude is equal to the input amplitude times the, transfer function 154 00:10:53,400 --> 00:10:58,460 at that frequency the magnitude of it and the angle of the output is 155 00:10:58,460 --> 00:11:03,499 equal to the angle of the transfer function at that frequency. 156 00:11:04,980 --> 00:11:08,650 That's how we figure out what the steady state response is to a sine wave. 157 00:11:08,650 --> 00:11:10,550 Well I've got three different components here. 158 00:11:10,550 --> 00:11:13,520 The first being dc which corresponds to omega equals zero. 159 00:11:17,760 --> 00:11:20,000 So I have to figure out what the amplitude 160 00:11:20,000 --> 00:11:24,070 is, at or the magnitude is at omega equals zero. 161 00:11:24,070 --> 00:11:25,920 Well I don't actually plot omega equals 162 00:11:25,920 --> 00:11:28,830 zero here because it's a logarithmic scale. 163 00:11:28,830 --> 00:11:33,100 So this is 10, and if I went out equal distance here, this would be one. 164 00:11:33,100 --> 00:11:38,000 And then this would be 0.1, 0.001, and you can 165 00:11:38,000 --> 00:11:41,100 see, I don't actually plot omega equals zero on this plot. 166 00:11:41,100 --> 00:11:42,960 But what I see is that this, this is 167 00:11:42,960 --> 00:11:45,840 approximately zero dB, and it looks like it's going to stay zero dB. 168 00:11:46,880 --> 00:11:52,920 So I will say 20 times the log of h of zero is zero db. 169 00:11:52,920 --> 00:11:56,570 And I have to 170 00:11:56,570 --> 00:12:01,230 back out what h of zero is in order to use this formula here. 171 00:12:01,230 --> 00:12:08,437 So h of zero, magnitude is equal to 10 raise to the zero over 20. 172 00:12:12,190 --> 00:12:17,080 So I take 0 divided by twenty and then raise both sides by a 173 00:12:17,080 --> 00:12:22,142 power of 10, and that equals what? So that particular thing 174 00:12:22,142 --> 00:12:27,220 says that if I put a DC component of one I get out a DC component of one. 175 00:12:28,470 --> 00:12:30,880 Now, let's look at the next frequency component 176 00:12:30,880 --> 00:12:33,400 that's 100 and this is radiance per second. 177 00:12:35,880 --> 00:12:40,280 At 100, again this also showing radiance per second, I should mark that. 178 00:12:42,670 --> 00:12:48,070 At 100 I'm right here. So I look right there and right there. 179 00:12:49,400 --> 00:12:51,010 My magnitude is, 180 00:12:56,590 --> 00:13:00,300 and this is the magnitude I'll say in db which is 181 00:13:00,300 --> 00:13:03,280 20 times the log of that is equal to zero db. 182 00:13:04,620 --> 00:13:08,570 And what we already found above, that means that h. 183 00:13:08,570 --> 00:13:14,770 That's going to equal one, and the angle was zero. 184 00:13:20,610 --> 00:13:25,395 Okay now let me go to my last component, 185 00:13:25,395 --> 00:13:29,745 3000, 20 times the log of h at 186 00:13:29,745 --> 00:13:34,939 3000 is equal to. Let me go to 3000. 187 00:13:34,939 --> 00:13:40,240 So this is 1000, 2000, this is 3000. So 188 00:13:40,240 --> 00:13:45,810 following up to here, I get 3000. And that looks that is 10 decibels. 189 00:13:48,680 --> 00:13:54,340 And over here, at 3000, that is right 190 00:13:54,340 --> 00:13:58,350 here, that looks like that's about minus 70 degrees. 191 00:14:00,990 --> 00:14:02,140 So we get 10 decibels. 192 00:14:04,300 --> 00:14:06,310 If I want to find what h is, 193 00:14:11,030 --> 00:14:15,690 that's equal to 10 to the, and then it's this number divide that number. 194 00:14:16,820 --> 00:14:21,082 So it's 10 divided by 20, and that is 195 00:14:21,082 --> 00:14:25,970 3.16. So putting 196 00:14:25,970 --> 00:14:30,930 it all together, my output, I'll call it the out of T is the output 197 00:14:30,930 --> 00:14:36,110 corresponding to the one by itself, which is one, plus the output 198 00:14:36,110 --> 00:14:41,300 corresponding to this cosign by itself, and the magnitude. 199 00:14:42,360 --> 00:14:47,320 the output amplitude is, the input amplitude times this 200 00:14:47,320 --> 00:14:49,286 which is one so the amplitude stays the same. 201 00:14:49,286 --> 00:14:54,654 Cosine 100 T, the original, the original phase 202 00:14:54,654 --> 00:14:59,890 lag, or phase is minus 45. And I haven't changed it here. 203 00:15:02,740 --> 00:15:08,830 So in other words this, these two components are passed through 204 00:15:08,830 --> 00:15:13,460 my circuit without really being changed. And then the last component of my input, 205 00:15:16,820 --> 00:15:19,710 the amplitude of the input is one. 206 00:15:19,710 --> 00:15:22,620 I multiply it by the magnitude of the transfer function. 207 00:15:22,620 --> 00:15:25,494 Magnitude of the frequency response at that point which is 3.16. 208 00:15:27,730 --> 00:15:35,270 Cosine same frequency 3000, and this time my phase is minus 70. 209 00:15:35,270 --> 00:15:40,680 So, what's interesting is at resonance if I put a sine wave at 210 00:15:40,680 --> 00:15:45,070 resonance, my output amplitude is actually is larger than my, my ampli, amplitude. 211 00:15:49,120 --> 00:15:51,960 So in summary, a frequency response is 212 00:15:51,960 --> 00:15:55,390 a plot of the transfer function versus frequency. 213 00:15:55,390 --> 00:15:57,580 In our last lesson we were looking at plotting 214 00:15:57,580 --> 00:16:00,930 it first in the linear, as a linear plot. 215 00:16:00,930 --> 00:16:02,130 This case it's a Bode plot. 216 00:16:03,160 --> 00:16:07,050 Bode plot is just fr-, frequency response on a log scale. 217 00:16:07,050 --> 00:16:09,390 The units are in decibels or dB. 218 00:16:09,390 --> 00:16:13,810 An RC circuit, we found that the magnitude goes down by 20 dB per decade. 219 00:16:13,810 --> 00:16:14,250 And the phase 220 00:16:14,250 --> 00:16:15,488 goes from zero to minus 90. 221 00:16:15,488 --> 00:16:21,490 An RLC circuit, we found the magnitude goes down by 40 decibels per decade. 222 00:16:21,490 --> 00:16:24,210 And the phase goes from zero to minus 180. 223 00:16:24,210 --> 00:16:28,390 And what we found is that an RLC circuit with low damping has a resonant peak and 224 00:16:28,390 --> 00:16:33,122 that affects the output. The output would have a higher amplitude 225 00:16:33,122 --> 00:16:39,410 at resonance than the input. In our next lesson we'll do a lab demo of 226 00:16:39,410 --> 00:16:43,860 an RLC circuit frequency response. I'll see you online, Thank you.