1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,720 continue to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,280 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,280 --> 00:00:18,450 at ocw.mit.edu. 8 00:00:22,722 --> 00:00:24,950 DENNIS FREEMAN: Hello, and welcome. 9 00:00:24,950 --> 00:00:29,480 So like last time, the important announcement 10 00:00:29,480 --> 00:00:32,090 is of course that we have an exam coming up. 11 00:00:32,090 --> 00:00:35,480 So exam 2, Wednesday, everything's 12 00:00:35,480 --> 00:00:38,210 very similar, analogous to last time, with the exception 13 00:00:38,210 --> 00:00:40,040 that we are in Walker. 14 00:00:40,040 --> 00:00:41,420 Don't go to Building 26. 15 00:00:41,420 --> 00:00:43,370 Go to Walker. 16 00:00:43,370 --> 00:00:49,210 Other than that things look very similar to before. 17 00:00:49,210 --> 00:00:52,030 The idea will be that the exam will 18 00:00:52,030 --> 00:00:54,070 focus on things since exam 1. 19 00:00:54,070 --> 00:00:57,800 However, this subject builds on itself. 20 00:00:57,800 --> 00:01:00,970 So the official coverage is 1 the 12, lectures 1 to 12, 21 00:01:00,970 --> 00:01:04,800 recitations 1 to 12, and homeworks 1 to 7. 22 00:01:04,800 --> 00:01:05,300 Questions? 23 00:01:05,300 --> 00:01:07,260 Comments? 24 00:01:07,260 --> 00:01:11,244 All the logistical issues are clear? 25 00:01:11,244 --> 00:01:12,732 AUDIENCE: I had one question. 26 00:01:12,732 --> 00:01:18,690 [INAUDIBLE] how much do we have to know about [INAUDIBLE]. 27 00:01:18,690 --> 00:01:22,646 DENNIS FREEMAN: So there was an introduction to root locus. 28 00:01:22,646 --> 00:01:24,520 Sorry, so everybody knows what root locus is? 29 00:01:27,240 --> 00:01:28,810 You can find out after class. 30 00:01:28,810 --> 00:01:30,360 You know, it will just making me feel a lot better if you 31 00:01:30,360 --> 00:01:31,500 nod your head yes. 32 00:01:31,500 --> 00:01:35,730 Root locus is the way you figure out the locus 33 00:01:35,730 --> 00:01:40,880 of possible points that a pole could move to. 34 00:01:40,880 --> 00:01:44,870 There is an enormous number of rules for thinking 35 00:01:44,870 --> 00:01:47,270 about how root locuses evolve. 36 00:01:47,270 --> 00:01:50,270 And if you're really good signals and systems person, 37 00:01:50,270 --> 00:01:52,070 you know 50 rules. 38 00:01:52,070 --> 00:01:55,880 That's not the point of this class. 39 00:01:55,880 --> 00:01:58,940 You should know that poles move around when there is feedback. 40 00:01:58,940 --> 00:02:02,880 We'll have examples of that in class today. 41 00:02:02,880 --> 00:02:05,510 You should be able to calculate how they move around 42 00:02:05,510 --> 00:02:06,770 for simple problems. 43 00:02:06,770 --> 00:02:09,680 You're not expected to know the rules. 44 00:02:09,680 --> 00:02:14,210 OK, so you shouldn't be surprised by the rules. 45 00:02:14,210 --> 00:02:17,390 If we ask you to derive one of the rules, you probably could. 46 00:02:17,390 --> 00:02:20,300 But we don't expect that you've memorized the rules. 47 00:02:20,300 --> 00:02:24,470 Does that sounds roughly correct? 48 00:02:24,470 --> 00:02:27,270 OK, other questions or comments? 49 00:02:27,270 --> 00:02:29,460 Content, administration, anything like that? 50 00:02:33,700 --> 00:02:38,220 OK, so about a week ago, Russ introduced 51 00:02:38,220 --> 00:02:41,650 the idea of using feedback for the purpose of controlling. 52 00:02:41,650 --> 00:02:45,880 And the example in lecture was controlling a robot. 53 00:02:45,880 --> 00:02:47,050 You've seen that before. 54 00:02:47,050 --> 00:02:47,550 Right? 55 00:02:47,550 --> 00:02:49,450 That's 6.01. 56 00:02:49,450 --> 00:02:51,730 And then in recitation yesterday, he 57 00:02:51,730 --> 00:02:53,890 ran over some more ways of thinking 58 00:02:53,890 --> 00:03:00,510 about more sophisticated ways to do control with feedback. 59 00:03:00,510 --> 00:03:02,490 And the point of those more sophisticated ways 60 00:03:02,490 --> 00:03:05,140 to do control is really performance. 61 00:03:05,140 --> 00:03:07,770 The idea is to enhance performance. 62 00:03:07,770 --> 00:03:11,160 And that's the theme that I want to build on today, 63 00:03:11,160 --> 00:03:13,440 today and on Tuesday. 64 00:03:13,440 --> 00:03:16,890 How do you enhance performance by using feedback? 65 00:03:16,890 --> 00:03:18,960 And we'll go through a number of examples. 66 00:03:18,960 --> 00:03:22,690 You can use feedback to increase speed and bandwidth. 67 00:03:22,690 --> 00:03:26,520 You can use feedback to change the control variable from-- 68 00:03:26,520 --> 00:03:27,570 I'll do an example-- 69 00:03:27,570 --> 00:03:29,670 speed to position. 70 00:03:29,670 --> 00:03:32,520 You can reduce distortion, reduce sensitivity 71 00:03:32,520 --> 00:03:34,290 to parameter variation. 72 00:03:34,290 --> 00:03:36,180 You can stabilize unstable systems. 73 00:03:36,180 --> 00:03:37,650 We'll do a bunch of examples. 74 00:03:37,650 --> 00:03:40,500 The point is that you can use feedback 75 00:03:40,500 --> 00:03:43,320 to accomplish a lot of different tasks 76 00:03:43,320 --> 00:03:48,550 that generally speaking improve performance. 77 00:03:48,550 --> 00:03:51,504 So with that, let me think about an op-amp. 78 00:03:51,504 --> 00:03:52,920 You all know about op-amps, right? 79 00:03:52,920 --> 00:03:56,620 That's also a 6.01 sort of thing. 80 00:03:56,620 --> 00:03:59,770 So an ideal op-amp would have a bunch of properties 81 00:03:59,770 --> 00:04:01,360 that we would like. 82 00:04:01,360 --> 00:04:02,890 We'd like it to have-- 83 00:04:02,890 --> 00:04:04,480 it we'd like it to be very high speed. 84 00:04:04,480 --> 00:04:07,630 We'd like it to have a very broad bandwidth. 85 00:04:07,630 --> 00:04:09,040 We'd like it to be fast. 86 00:04:09,040 --> 00:04:12,250 We'd like it to have a very low output impedance so it works 87 00:04:12,250 --> 00:04:15,500 like a perfect voltage source. 88 00:04:15,500 --> 00:04:17,900 We like it have a very high input impedance so 89 00:04:17,900 --> 00:04:19,399 that when we hook it up to something 90 00:04:19,399 --> 00:04:23,350 it doesn't change the something that we hooked it up to. 91 00:04:23,350 --> 00:04:25,650 There's lots of things we'd like it to have. 92 00:04:25,650 --> 00:04:28,345 Unfortunately, it's difficult to build a circuit that 93 00:04:28,345 --> 00:04:29,345 has all of those things. 94 00:04:33,000 --> 00:04:35,340 And so here is an example of that. 95 00:04:35,340 --> 00:04:39,170 Here's one of the world's most popular op-amps, the LM741, 96 00:04:39,170 --> 00:04:43,260 invented eons ago and still used today. 97 00:04:43,260 --> 00:04:47,000 The idea that's plotted here is the frequency response. 98 00:04:47,000 --> 00:04:48,620 You remember frequency response. 99 00:04:48,620 --> 00:04:50,952 What's the frequency response of an LM741? 100 00:04:53,580 --> 00:04:57,720 Well, if you plot as a function of frequency, 101 00:04:57,720 --> 00:04:59,280 how does the gain change? 102 00:04:59,280 --> 00:05:01,250 Well, the gain is enormous. 103 00:05:01,250 --> 00:05:03,630 High gain, that's what we'd like it to have. 104 00:05:03,630 --> 00:05:06,130 The gain is enormous at low frequencies, 105 00:05:06,130 --> 00:05:08,295 at 1 radian per second. 106 00:05:10,950 --> 00:05:14,550 But the gain becomes very small out at high speeds 107 00:05:14,550 --> 00:05:17,370 at the speeds we think we think about when we're thinking 108 00:05:17,370 --> 00:05:19,560 about communication systems or when we're thinking 109 00:05:19,560 --> 00:05:21,030 about computation or things. 110 00:05:21,030 --> 00:05:25,800 When we think about speeds that are electronic speeds, 111 00:05:25,800 --> 00:05:28,260 the gain is actually pretty small. 112 00:05:28,260 --> 00:05:29,670 Similarly, the phase changes. 113 00:05:29,670 --> 00:05:31,950 The phase of very low frequencies is near 0. 114 00:05:31,950 --> 00:05:33,480 That's kind of ideal. 115 00:05:33,480 --> 00:05:36,990 0 would mean in phase. 116 00:05:36,990 --> 00:05:40,440 The output reliably tells you what's the input doing. 117 00:05:40,440 --> 00:05:41,671 So that's good. 118 00:05:41,671 --> 00:05:43,170 But as you go to higher frequencies, 119 00:05:43,170 --> 00:05:45,090 you start to pick up a phase delay, which 120 00:05:45,090 --> 00:05:47,760 means the output is telling you what the input did 121 00:05:47,760 --> 00:05:49,800 a little while ago. 122 00:05:49,800 --> 00:05:52,020 OK, that's less good. 123 00:05:52,020 --> 00:05:55,140 So the performance in terms of providing a lot of gain 124 00:05:55,140 --> 00:05:56,790 is quite good for some frequencies, 125 00:05:56,790 --> 00:05:59,810 but not for others. 126 00:05:59,810 --> 00:06:03,250 Unfortunately, the range where it's not very good 127 00:06:03,250 --> 00:06:04,947 is most of the range. 128 00:06:04,947 --> 00:06:06,780 So if you thought about an op-amp doing even 129 00:06:06,780 --> 00:06:12,130 a very low frequency problem, like audio signal processing, 130 00:06:12,130 --> 00:06:16,260 most of the frequencies that are of interest to audio 131 00:06:16,260 --> 00:06:19,710 are in the region where the op-amp is not working too well. 132 00:06:19,710 --> 00:06:23,820 We hear sounds from about 20 Hertz to about 20 kilohertz. 133 00:06:23,820 --> 00:06:26,220 And that range is indicated by the red arrow. 134 00:06:26,220 --> 00:06:31,470 And you can see that's not where the op-amp is working ideally. 135 00:06:31,470 --> 00:06:36,040 Similarly, we would like the op-amp to be fast. 136 00:06:36,040 --> 00:06:37,660 Fast is indicated here by thinking 137 00:06:37,660 --> 00:06:40,600 about the step response. 138 00:06:40,600 --> 00:06:44,950 So if you thought about having an ideal step at the input, 139 00:06:44,950 --> 00:06:47,030 what would the output look like? 140 00:06:47,030 --> 00:06:48,790 Well, we'd like it to be an ideal step, 141 00:06:48,790 --> 00:06:50,590 but it generally isn't. 142 00:06:50,590 --> 00:06:52,300 Generally, there's some lag. 143 00:06:52,300 --> 00:06:53,980 That's associated with that phase delay 144 00:06:53,980 --> 00:06:55,511 that I talked about. 145 00:06:55,511 --> 00:06:57,010 And in fact, what I'd like you to do 146 00:06:57,010 --> 00:06:59,384 is figure out exactly how it's associated with that phase 147 00:06:59,384 --> 00:07:01,390 delay. 148 00:07:01,390 --> 00:07:04,630 Think about-- so figure out-- it will turn out 149 00:07:04,630 --> 00:07:07,150 that the step response can be characterized 150 00:07:07,150 --> 00:07:09,580 by a time constant tau. 151 00:07:09,580 --> 00:07:13,960 The time constant tau is the time required for the signal 152 00:07:13,960 --> 00:07:20,530 to get within 1 over e of this final value determined 153 00:07:20,530 --> 00:07:21,460 from this. 154 00:07:21,460 --> 00:07:23,920 I've already told you the performance parameters 155 00:07:23,920 --> 00:07:25,630 in terms of frequency. 156 00:07:25,630 --> 00:07:28,060 You should be able to use 6.003 to figure out 157 00:07:28,060 --> 00:07:31,660 what that implies about time. 158 00:07:31,660 --> 00:07:34,010 So look at your neighbor-- 159 00:07:34,010 --> 00:07:38,640 OK, look, look, look. 160 00:07:38,640 --> 00:07:40,920 And now, figure out which answer-- 161 00:07:40,920 --> 00:07:43,380 what's tau for the 741 op-amp? 162 00:07:47,853 --> 00:07:49,841 [SIDE CONVERSATION] 163 00:08:43,080 --> 00:08:44,230 OK, it's dead quiet. 164 00:08:44,230 --> 00:08:46,180 So I assume that means convergence. 165 00:08:49,996 --> 00:08:50,870 So what's the answer? 166 00:08:50,870 --> 00:08:54,490 What's the time constant tau of the 741? 167 00:08:54,490 --> 00:08:58,309 Raise your hand with a number between 1 and 5. 168 00:08:58,309 --> 00:09:00,350 Don't look at other people, just raise your hand. 169 00:09:03,110 --> 00:09:05,580 OK, I can see a couple of answers I don't like. 170 00:09:10,670 --> 00:09:13,022 How do I think about converting this representation 171 00:09:13,022 --> 00:09:13,980 to that representation? 172 00:09:13,980 --> 00:09:14,938 What should I do first? 173 00:09:19,480 --> 00:09:20,711 Shout. 174 00:09:20,711 --> 00:09:22,210 I start with the thing on the right. 175 00:09:22,210 --> 00:09:23,126 What should I do next? 176 00:09:23,126 --> 00:09:26,651 AUDIENCE: [INAUDIBLE] of like doing a 2 over tau or something 177 00:09:26,651 --> 00:09:27,150 like that? 178 00:09:27,150 --> 00:09:28,280 DENNIS FREEMAN: So we're looking for something like e 179 00:09:28,280 --> 00:09:29,449 to the minus t over tau. 180 00:09:29,449 --> 00:09:29,990 That's right. 181 00:09:32,870 --> 00:09:34,670 We're looking for something like that. 182 00:09:34,670 --> 00:09:38,036 How do I get that starting with the thing on the right? 183 00:09:38,036 --> 00:09:40,030 AUDIENCE: It's like using poles. 184 00:09:40,030 --> 00:09:42,830 DENNIS FREEMAN: Poles, what a good idea. 185 00:09:42,830 --> 00:09:46,440 How do I figure out poles from the thing on the right? 186 00:09:46,440 --> 00:09:49,640 6.003, poles, that should be like hard wire. 187 00:09:49,640 --> 00:09:50,480 Right? 188 00:09:50,480 --> 00:09:55,850 So when you hear the word 003, you should think the word pole. 189 00:09:55,850 --> 00:09:57,320 How do you associate a pole? 190 00:09:57,320 --> 00:09:59,930 Can somebody tell me what the poles of the right-hand system 191 00:09:59,930 --> 00:10:00,770 should look like? 192 00:10:03,554 --> 00:10:04,721 AUDIENCE: Where the line is. 193 00:10:04,721 --> 00:10:07,262 DENNIS FREEMAN: Where the line is-- there's a couple of lines 194 00:10:07,262 --> 00:10:07,804 there. 195 00:10:07,804 --> 00:10:08,970 AUDIENCE: The vertical line. 196 00:10:08,970 --> 00:10:10,450 DENNIS FREEMAN: The vertical line-- 197 00:10:10,450 --> 00:10:11,797 how many poles are on the right? 198 00:10:11,797 --> 00:10:13,630 How many poles are in the system represented 199 00:10:13,630 --> 00:10:14,671 by the right-hand figure? 200 00:10:14,671 --> 00:10:15,430 AUDIENCE: One. 201 00:10:15,430 --> 00:10:15,700 DENNIS FREEMAN: One. 202 00:10:15,700 --> 00:10:16,325 How many zeros? 203 00:10:18,930 --> 00:10:20,260 Zero, wonderful. 204 00:10:20,260 --> 00:10:24,010 So we have to think about a pole, just one pole. 205 00:10:24,010 --> 00:10:25,360 So we think about the s-plane. 206 00:10:25,360 --> 00:10:26,680 Right? 207 00:10:26,680 --> 00:10:27,480 Where's the pole? 208 00:10:34,677 --> 00:10:35,510 You already told me. 209 00:10:35,510 --> 00:10:37,797 AUDIENCE: 80 halves, so it was 40. 210 00:10:37,797 --> 00:10:38,630 DENNIS FREEMAN: 40-- 211 00:10:38,630 --> 00:10:39,990 OK, where's 40 over here? 212 00:10:43,510 --> 00:10:44,570 Somewhere in there. 213 00:10:44,570 --> 00:10:45,070 Got it. 214 00:10:47,620 --> 00:10:50,240 Where's the pole over here? 215 00:10:50,240 --> 00:10:51,270 It's at location 40. 216 00:10:51,270 --> 00:10:53,600 That's actually correct, well, sort of. 217 00:10:56,790 --> 00:11:00,360 Negative 40, that's exactly correct. 218 00:11:00,360 --> 00:11:05,460 So negative 40, so the pole is right here. 219 00:11:05,460 --> 00:11:10,450 OK, so the pole is at minus 40 radians per second. 220 00:11:10,450 --> 00:11:13,380 How do you know that? 221 00:11:13,380 --> 00:11:15,830 Well, that's that Bode stuff we talked about last time. 222 00:11:15,830 --> 00:11:16,760 Right? 223 00:11:16,760 --> 00:11:20,100 So if the pole were here, you would think about a vector. 224 00:11:20,100 --> 00:11:21,500 So this is the j omega axis. 225 00:11:21,500 --> 00:11:22,657 Right? 226 00:11:22,657 --> 00:11:24,740 You would think about a vector connecting the pole 227 00:11:24,740 --> 00:11:28,440 to some location on the j omega axis. 228 00:11:28,440 --> 00:11:32,395 If the location is omega small, it's going to have some-- 229 00:11:32,395 --> 00:11:34,520 that vector is going to have some length like that. 230 00:11:34,520 --> 00:11:38,070 It's going to have an angle of 0. 231 00:11:38,070 --> 00:11:40,910 And as frequency goes up, the vector 232 00:11:40,910 --> 00:11:47,060 gets longer, which means that the magnitude gets smaller 233 00:11:47,060 --> 00:11:48,920 because poles and bottom. 234 00:11:48,920 --> 00:11:51,000 Right? 235 00:11:51,000 --> 00:11:55,712 And the phase goes increasingly toward pi over 2. 236 00:11:55,712 --> 00:11:56,670 But it's in the bottom. 237 00:11:56,670 --> 00:11:59,940 So it's minus pi over 2. 238 00:11:59,940 --> 00:12:04,860 Right, so the frequency at which you get to pi over 4-- 239 00:12:04,860 --> 00:12:09,870 OK, 0pi over 2-- the frequency at which you get to pi over 4 240 00:12:09,870 --> 00:12:10,410 is-- 241 00:12:10,410 --> 00:12:12,230 so pi over 4 is here. 242 00:12:12,230 --> 00:12:14,560 Right? 243 00:12:14,560 --> 00:12:18,970 So the distance here, the critical frequency, 244 00:12:18,970 --> 00:12:22,810 the frequency at which the phase is pi over 4, minus pi over 4, 245 00:12:22,810 --> 00:12:26,766 is going to be the same as the frequency of the pole. 246 00:12:26,766 --> 00:12:28,390 So that's how we know it's at minus 40. 247 00:12:28,390 --> 00:12:30,770 Right? 248 00:12:30,770 --> 00:12:34,285 So we know that this then is a characterization 249 00:12:34,285 --> 00:12:34,910 of that system. 250 00:12:34,910 --> 00:12:36,060 It's not quite done. 251 00:12:36,060 --> 00:12:42,170 So what's the h of s for this system? 252 00:12:42,170 --> 00:12:45,050 If I've got a pole at minus 40, what's h of s look like? 253 00:12:50,385 --> 00:12:52,325 Shout, shout. 254 00:12:52,325 --> 00:12:56,992 These answers coming up, this is perfect practice. 255 00:12:56,992 --> 00:12:58,974 AUDIENCE: I've already talked too much. 256 00:12:58,974 --> 00:13:00,640 DENNIS FREEMAN: What's h of s look like? 257 00:13:00,640 --> 00:13:02,600 AUDIENCE: Is this something over s plus 40. 258 00:13:02,600 --> 00:13:04,810 DENNIS FREEMAN: Exactly, something over s plus 40. 259 00:13:04,810 --> 00:13:06,476 What would you like the something to be? 260 00:13:09,630 --> 00:13:11,430 We'd like it to be some k. 261 00:13:11,430 --> 00:13:14,250 We'd actually like it to be related, for example, 262 00:13:14,250 --> 00:13:15,320 to the open-loop gain. 263 00:13:20,700 --> 00:13:24,280 How big is the gain when frequencies go towards 0? 264 00:13:24,280 --> 00:13:25,500 What's the gain at 0? 265 00:13:28,120 --> 00:13:31,690 Let's say the gain of 0-- let's define the gain of 0, k0, 266 00:13:31,690 --> 00:13:36,070 let's define the gain at 0 to be-- 267 00:13:36,070 --> 00:13:37,420 so what is the gain at 0? 268 00:13:43,260 --> 00:13:46,800 10 to the 5, 2 times 10 to the 5. 269 00:13:46,800 --> 00:13:47,300 Right? 270 00:13:47,300 --> 00:13:51,540 It's a little bit above the 10 to the 5 line. 271 00:13:51,540 --> 00:13:55,640 OK, so now if I define k0 to be that gain at 0, 272 00:13:55,640 --> 00:13:57,050 what's the numerator over here? 273 00:14:00,240 --> 00:14:02,190 If the numerator over here were 1, 274 00:14:02,190 --> 00:14:07,420 what would be the low-frequency gain? 275 00:14:07,420 --> 00:14:10,180 1 over 40, right? 276 00:14:10,180 --> 00:14:12,160 To think about frequency, we think about 1 277 00:14:12,160 --> 00:14:13,720 over j omega plus 40. 278 00:14:13,720 --> 00:14:14,220 Right? 279 00:14:14,220 --> 00:14:18,280 You substitute s equals j omega to think about frequency. 280 00:14:18,280 --> 00:14:21,880 And if omega is very small, you get 1 over 40. 281 00:14:21,880 --> 00:14:22,690 Right? 282 00:14:22,690 --> 00:14:25,495 So if you wanted that to be 1, you'd have to make it 40. 283 00:14:28,932 --> 00:14:30,140 But we don't want it to be 1. 284 00:14:30,140 --> 00:14:30,980 We want it to k0. 285 00:14:30,980 --> 00:14:34,500 So you'd want it to be 40 k0. 286 00:14:34,500 --> 00:14:36,240 Or to be a little bit more general, 287 00:14:36,240 --> 00:14:41,340 we'd like this to be of the form alpha k0 divided 288 00:14:41,340 --> 00:14:42,650 by s plus alpha. 289 00:14:42,650 --> 00:14:44,850 Right? 290 00:14:44,850 --> 00:14:48,220 So now, we've got the Laplace transform. 291 00:14:48,220 --> 00:14:49,720 We can figure out the time response. 292 00:14:49,720 --> 00:14:51,178 What's the time response associated 293 00:14:51,178 --> 00:14:52,530 with that Laplace transform? 294 00:14:55,230 --> 00:14:59,240 You remember the impulse response of a system 295 00:14:59,240 --> 00:15:01,640 is related to the system function via the Laplace 296 00:15:01,640 --> 00:15:02,210 transform. 297 00:15:02,210 --> 00:15:03,091 Right? 298 00:15:03,091 --> 00:15:05,090 So all we need to do is take the inverse Laplace 299 00:15:05,090 --> 00:15:06,410 transform of that thing. 300 00:15:06,410 --> 00:15:08,450 And we get h of t. 301 00:15:08,450 --> 00:15:09,350 So what's h of t? 302 00:15:21,440 --> 00:15:26,020 OK, it's such a strain, all those weeks ago. 303 00:15:26,020 --> 00:15:29,230 e to the minus t, e to the minus something t. 304 00:15:29,230 --> 00:15:32,680 So we need something like e to the minus what? 305 00:15:32,680 --> 00:15:33,395 Alpha. 306 00:15:33,395 --> 00:15:35,140 e to the M alpha t u of t. 307 00:15:39,120 --> 00:15:40,952 And there's going to be that-- 308 00:15:40,952 --> 00:15:42,660 that thing is going to come out here too. 309 00:15:42,660 --> 00:15:43,951 And it's going to be like that. 310 00:15:47,300 --> 00:15:49,300 And then if you wanted to find the step response 311 00:15:49,300 --> 00:15:50,050 what would you do? 312 00:15:59,440 --> 00:16:00,820 h of t is the impulse response. 313 00:16:00,820 --> 00:16:02,950 How would you find the step response? 314 00:16:02,950 --> 00:16:07,634 A step is related to the impulse by integrating. 315 00:16:07,634 --> 00:16:09,800 The step response is related to the impulse response 316 00:16:09,800 --> 00:16:12,630 by, surprisingly, integrating. 317 00:16:12,630 --> 00:16:14,481 So you would integrate that thing, 318 00:16:14,481 --> 00:16:15,980 and you would end up with something. 319 00:16:15,980 --> 00:16:16,480 Right? 320 00:16:16,480 --> 00:16:19,540 So you would get something like k0 1 minus 321 00:16:19,540 --> 00:16:24,240 e to the minus alpha t u of t, something like that. 322 00:16:24,240 --> 00:16:26,450 I think that's right. 323 00:16:26,450 --> 00:16:29,675 OK, the time constant is given by alpha. 324 00:16:32,220 --> 00:16:33,840 Right? 325 00:16:33,840 --> 00:16:35,580 So what is the time constant now? 326 00:16:40,310 --> 00:16:43,190 1 over 40, right? 327 00:16:43,190 --> 00:16:45,410 So it's alpha t. 328 00:16:45,410 --> 00:16:49,170 But the time constant is t over tau. 329 00:16:49,170 --> 00:16:51,520 So the time constant is the inverse of frequency. 330 00:16:51,520 --> 00:16:52,090 Right? 331 00:16:52,090 --> 00:16:55,290 Frequency is per second. 332 00:16:55,290 --> 00:16:57,040 Time constant is seconds. 333 00:16:57,040 --> 00:16:58,900 They're reciprocals of each other. 334 00:16:58,900 --> 00:17:03,630 OK, so the idea then is that we can associate a time response 335 00:17:03,630 --> 00:17:04,410 with that. 336 00:17:04,410 --> 00:17:07,730 And that time response is pretty slow. 337 00:17:07,730 --> 00:17:11,250 A 40th of a second, 40 things per second kind of. 338 00:17:11,250 --> 00:17:12,839 That's my speed. 339 00:17:12,839 --> 00:17:14,520 Right, that's not computation speed. 340 00:17:14,520 --> 00:17:15,540 Right? 341 00:17:15,540 --> 00:17:16,910 So that's pretty slow. 342 00:17:16,910 --> 00:17:18,500 So we'd like to improve that. 343 00:17:18,500 --> 00:17:20,531 And one way that we can improve it 344 00:17:20,531 --> 00:17:21,780 is by thinking about feedback. 345 00:17:24,460 --> 00:17:27,807 So the idea would be we don't use the op-amp just out 346 00:17:27,807 --> 00:17:29,140 of the package the way it comes. 347 00:17:29,140 --> 00:17:30,348 We put it in a feedback loop. 348 00:17:33,100 --> 00:17:34,980 So for example, here you could put it 349 00:17:34,980 --> 00:17:39,870 into a non-inverting amplifier where 350 00:17:39,870 --> 00:17:43,740 you take the output divided by a voltage divider 351 00:17:43,740 --> 00:17:46,260 and feed that back in. 352 00:17:46,260 --> 00:17:49,590 In 6.003, the way we'll think about that 353 00:17:49,590 --> 00:17:51,840 is that the effectiveness plus and minus 354 00:17:51,840 --> 00:17:54,330 input is to take the difference between two signals. 355 00:17:56,697 --> 00:17:58,530 Right, we're going to take a circuit diagram 356 00:17:58,530 --> 00:18:08,621 and turn it into a integrator, gain-adder diagram, a block 357 00:18:08,621 --> 00:18:09,120 diagram. 358 00:18:09,120 --> 00:18:10,290 We're going to take a circuit and turn it 359 00:18:10,290 --> 00:18:11,700 into a block diagram. 360 00:18:11,700 --> 00:18:13,890 So the way we think about the plus and minus input 361 00:18:13,890 --> 00:18:16,040 of the op-amp is as a subtractor. 362 00:18:19,020 --> 00:18:23,670 And then this k of s that characterized-- so this thing, 363 00:18:23,670 --> 00:18:29,280 right, this thing here, we plug-in there just as a box. 364 00:18:29,280 --> 00:18:32,730 And the effect of the resistors is to divide the output signal 365 00:18:32,730 --> 00:18:35,680 by some constant beta. 366 00:18:35,680 --> 00:18:40,930 So we can reduce the circuit to a 6.003 equivalent. 367 00:18:40,930 --> 00:18:42,430 And then we can use Black's equation 368 00:18:42,430 --> 00:18:44,900 and all those other things that we do in 6.003. 369 00:18:44,900 --> 00:18:47,760 So we can write an expression for the closed-loop system 370 00:18:47,760 --> 00:18:50,800 the looks like k of s over 1 plus beta k of s. 371 00:18:50,800 --> 00:18:53,480 Make sense? 372 00:18:53,480 --> 00:18:58,160 So then, as we've already seen, the op-amp by itself 373 00:18:58,160 --> 00:19:01,680 looks like a pole at minus 40. 374 00:19:01,680 --> 00:19:04,520 So we can substitute this expression 375 00:19:04,520 --> 00:19:07,760 into that expression for the closed loop. 376 00:19:07,760 --> 00:19:09,530 And we get a new closed loop. 377 00:19:09,530 --> 00:19:17,030 So h of s, which had been alpha k0 over s plus alpha becomes-- 378 00:19:17,030 --> 00:19:29,740 the new value is alpha k0 over s plus alpha plus alpha beta k0. 379 00:19:29,740 --> 00:19:33,130 OK, so the point is that we can think about the way 380 00:19:33,130 --> 00:19:37,000 the feedback changes the system function 381 00:19:37,000 --> 00:19:40,060 by just formulating what's the circuit look like in terms 382 00:19:40,060 --> 00:19:45,610 of a 6.003 block diagram. 383 00:19:45,610 --> 00:19:50,250 So now, the effect of that, it looks as though what happens 384 00:19:50,250 --> 00:19:55,020 is the pole went from here to here. 385 00:19:55,020 --> 00:19:58,740 How big of a change is that? 386 00:19:58,740 --> 00:20:02,850 What's the biggest change in the position of the pole that you 387 00:20:02,850 --> 00:20:04,590 can achieve using feedback? 388 00:20:24,600 --> 00:20:26,568 AUDIENCE: [INAUDIBLE] distance. 389 00:20:26,568 --> 00:20:27,721 DENNIS FREEMAN: I'm sorry. 390 00:20:27,721 --> 00:20:30,012 AUDIENCE: [? Is ?] [? beta ?] the [INAUDIBLE] distance. 391 00:20:30,012 --> 00:20:31,992 Or where are they located? 392 00:20:31,992 --> 00:20:34,200 DENNIS FREEMAN: I want a factor by which you multiply 393 00:20:34,200 --> 00:20:36,480 alpha to find the biggest place that you 394 00:20:36,480 --> 00:20:38,840 could have put the new pole. 395 00:20:45,681 --> 00:20:46,680 I didn't say that right. 396 00:20:46,680 --> 00:20:47,679 I didn't say that right. 397 00:20:47,679 --> 00:20:49,630 I want to know where is the new pole. 398 00:20:49,630 --> 00:20:51,780 Sorry, I confused myself. 399 00:20:51,780 --> 00:20:53,220 Where is the new pole? 400 00:20:53,220 --> 00:20:55,200 How left can you put the new-- 401 00:20:55,200 --> 00:20:58,310 how left can you get the new pole to be? 402 00:20:58,310 --> 00:21:01,210 What's the furtherest-- left is good, right? 403 00:21:01,210 --> 00:21:02,180 Left is good. 404 00:21:02,180 --> 00:21:04,880 Everybody knows that, right? 405 00:21:04,880 --> 00:21:07,860 So right is bad. 406 00:21:07,860 --> 00:21:11,580 So small distances in the s-plane 407 00:21:11,580 --> 00:21:15,750 correspond to slow responses. 408 00:21:15,750 --> 00:21:18,180 Big distances correspond to fast responses. 409 00:21:18,180 --> 00:21:20,630 That's what we saw in part one. 410 00:21:20,630 --> 00:21:23,154 So we're trying to push the pole that way as far as we can. 411 00:21:23,154 --> 00:21:24,320 How far can we get it to go? 412 00:21:26,960 --> 00:21:28,460 OK, everybody raise your hands. 413 00:21:28,460 --> 00:21:32,650 The answer is-- and again, I see lots of answers 414 00:21:32,650 --> 00:21:35,430 that I don't quite like. 415 00:21:35,430 --> 00:21:38,210 With a minor perturbation, all those people who I don't like 416 00:21:38,210 --> 00:21:41,910 can become right by saying-- 417 00:21:41,910 --> 00:21:44,580 so a lot of people have said this, which I don't quite like. 418 00:21:44,580 --> 00:21:46,640 What's the biggest distance that I can move it? 419 00:21:50,990 --> 00:21:53,130 OK, new vote-- 420 00:21:53,130 --> 00:21:54,750 I didn't like tw0. 421 00:21:54,750 --> 00:21:57,060 And so therefore, I will change my answer to four. 422 00:21:57,060 --> 00:21:58,810 Wrong, no. 423 00:22:02,250 --> 00:22:03,440 How big can beta be? 424 00:22:09,360 --> 00:22:10,290 How big can beta be? 425 00:22:16,660 --> 00:22:19,230 What's the biggest possible value of beta? 426 00:22:19,230 --> 00:22:20,549 One. 427 00:22:20,549 --> 00:22:22,090 So how far can I push it to the left? 428 00:22:24,940 --> 00:22:28,670 So I can push it this far. 429 00:22:28,670 --> 00:22:29,470 Right? 430 00:22:29,470 --> 00:22:32,170 So if I make beta 1, that's as far as it goes. 431 00:22:32,170 --> 00:22:34,550 Beta 0 is kind of the rightmost position. 432 00:22:34,550 --> 00:22:36,260 Beta 1 is the leftmost position. 433 00:22:36,260 --> 00:22:37,730 I can get it that high. 434 00:22:37,730 --> 00:22:38,810 But that's very good. 435 00:22:38,810 --> 00:22:39,740 Right? 436 00:22:39,740 --> 00:22:44,250 1 plus k0-- k0 is a number like 10 to the 5th. 437 00:22:44,250 --> 00:22:46,730 So I can get at like 10 to the 5th bigger than it was. 438 00:22:46,730 --> 00:22:48,050 That's enormous. 439 00:22:48,050 --> 00:22:52,100 That means that I can stretch the frequency response. 440 00:22:52,100 --> 00:22:54,530 If the pole is the thing that determines the frequency 441 00:22:54,530 --> 00:22:56,600 response by that construction up there, 442 00:22:56,600 --> 00:22:59,190 I can stretch the frequency response by 10 to the 5th. 443 00:22:59,190 --> 00:23:01,550 That's huge. 444 00:23:01,550 --> 00:23:04,550 Pure win, right? 445 00:23:04,550 --> 00:23:06,737 Well, that doesn't sound right. 446 00:23:06,737 --> 00:23:09,070 So I've stretched the region where the gain is constant. 447 00:23:09,070 --> 00:23:11,720 I've stretched the region where the phase is close to 0. 448 00:23:11,720 --> 00:23:15,650 It's a 100% win except that I've cheated 449 00:23:15,650 --> 00:23:20,330 when I drew the plot because I normalized it by the DC gain. 450 00:23:20,330 --> 00:23:22,280 DC, I probably didn't define that yet. 451 00:23:22,280 --> 00:23:23,855 DC is direct current. 452 00:23:23,855 --> 00:23:26,570 It has nothing to do with current or direct. 453 00:23:26,570 --> 00:23:28,760 DC means zero frequency. 454 00:23:28,760 --> 00:23:32,770 I have no idea why we insist on using that word, but we do. 455 00:23:32,770 --> 00:23:35,740 DC means zero frequency. 456 00:23:35,740 --> 00:23:38,930 OK, so you just do that automatic mapping in your head. 457 00:23:38,930 --> 00:23:42,464 Every time I say DC, I mean zero frequency. 458 00:23:42,464 --> 00:23:43,880 Everybody else talks that way too. 459 00:23:43,880 --> 00:23:49,250 So the DC gain, I've divided the magnitude by the DC gain, 460 00:23:49,250 --> 00:23:51,960 h of j0. 461 00:23:51,960 --> 00:23:54,800 So this is the relative frequency magnitude, 462 00:23:54,800 --> 00:24:02,160 h of j omega, divided by the magnitude at omega equals 0. 463 00:24:02,160 --> 00:24:07,810 Had I not done that, the picture would have looked different. 464 00:24:07,810 --> 00:24:09,460 If you think about what happens-- 465 00:24:09,460 --> 00:24:13,500 so the top curve here shows what happens when the-- 466 00:24:13,500 --> 00:24:18,970 shows the response curve for a 741 op-amp with no feedback. 467 00:24:18,970 --> 00:24:21,591 It has a pole at 40. 468 00:24:21,591 --> 00:24:25,320 It has a pole at minus 40 to be a little more correct. 469 00:24:25,320 --> 00:24:27,600 So there's a single pole at minus 40. 470 00:24:27,600 --> 00:24:28,980 That corresponds to beta 0. 471 00:24:28,980 --> 00:24:32,000 That's open loop, no feedback. 472 00:24:32,000 --> 00:24:35,920 OK, and what you can see is the DC gain 473 00:24:35,920 --> 00:24:38,700 is 2 times 10 to the 5th. 474 00:24:38,700 --> 00:24:43,160 And there's an AC gain defined by the cutoff frequency. 475 00:24:43,160 --> 00:24:45,920 If you compare this expression and that expression, 476 00:24:45,920 --> 00:24:48,830 the interesting thing that happens 477 00:24:48,830 --> 00:24:54,580 is that the DC response depends critically on beta. 478 00:24:57,554 --> 00:25:03,020 But the high-frequency asymptote doesn't depend on beta at all. 479 00:25:03,020 --> 00:25:08,190 The high-frequency asymptote of this is alpha k0 over s. 480 00:25:08,190 --> 00:25:10,850 The high-frequency asymptote of this is alpha k0 over s. 481 00:25:10,850 --> 00:25:14,040 They're the same high-frequency asymptote. 482 00:25:14,040 --> 00:25:17,430 What happened was that as I increased the feedback, 483 00:25:17,430 --> 00:25:20,800 I lopped off the low-frequency gain, 484 00:25:20,800 --> 00:25:23,500 which had the effect of making the frequency look 485 00:25:23,500 --> 00:25:24,850 like it got very big. 486 00:25:27,390 --> 00:25:28,360 So pure win? 487 00:25:28,360 --> 00:25:29,030 No. 488 00:25:29,030 --> 00:25:30,050 Almost pure win? 489 00:25:30,050 --> 00:25:31,030 Yes. 490 00:25:31,030 --> 00:25:37,130 OK, I've traded-- the trick is I've traded gain for bandwidth. 491 00:25:37,130 --> 00:25:38,990 That's important because the folks 492 00:25:38,990 --> 00:25:40,550 who know how to microfabricate things 493 00:25:40,550 --> 00:25:45,060 know how to make things with very high gain. 494 00:25:45,060 --> 00:25:48,910 Making things that are fast is harder. 495 00:25:48,910 --> 00:25:50,890 This is a way that we can take advantage 496 00:25:50,890 --> 00:25:54,820 of the relative simplicity of making things with high gain 497 00:25:54,820 --> 00:25:58,870 to make them behave as though they're fast. 498 00:25:58,870 --> 00:26:01,590 OK, you can see the same thing if you think 499 00:26:01,590 --> 00:26:04,860 about the response in time. 500 00:26:04,860 --> 00:26:06,540 If you think about the response in time, 501 00:26:06,540 --> 00:26:10,349 we're going to go from e to the minus alpha t to-- 502 00:26:10,349 --> 00:26:11,640 I guess I didn't write it down. 503 00:26:11,640 --> 00:26:13,264 It's going to be e to the minus alpha 1 504 00:26:13,264 --> 00:26:18,620 plus alpha beta k0 t, same factor. 505 00:26:18,620 --> 00:26:22,250 It's going to get 10 to the 5th times faster. 506 00:26:22,250 --> 00:26:25,070 Again, it looks like pure win. 507 00:26:25,070 --> 00:26:27,320 But that's because I've cheated because I'm 508 00:26:27,320 --> 00:26:32,360 plotting via this normalized final value that 509 00:26:32,360 --> 00:26:35,440 depends on beta. 510 00:26:35,440 --> 00:26:39,520 If beta were 0, the final value is 2 times 10 the 5th. 511 00:26:39,520 --> 00:26:43,240 If beta is anything else, the final value is smaller. 512 00:26:43,240 --> 00:26:45,970 If I plot it not normalized, you get 513 00:26:45,970 --> 00:26:47,930 a kind of different picture. 514 00:26:47,930 --> 00:26:53,200 So if you were to use the op-amp open loop, 515 00:26:53,200 --> 00:26:55,600 the response has a time constant of 1 over 40, 516 00:26:55,600 --> 00:26:58,260 as we said before. 517 00:26:58,260 --> 00:27:04,930 If you include feedback, it decreases the final value, 518 00:27:04,930 --> 00:27:10,660 but has the property that it does not change this slope. 519 00:27:10,660 --> 00:27:14,760 So if you think about this, this is an analytic expression 520 00:27:14,760 --> 00:27:18,320 for the closed-loop step response. 521 00:27:18,320 --> 00:27:22,776 I get that by integrating one of these expressions. 522 00:27:22,776 --> 00:27:24,150 I didn't write the beta one down. 523 00:27:24,150 --> 00:27:27,810 I need the beta equivalent one of these. 524 00:27:27,810 --> 00:27:29,310 If you right the beta equivalent one 525 00:27:29,310 --> 00:27:34,060 of these, which is on a previous slide, and integrate it, you 526 00:27:34,060 --> 00:27:37,720 you get this expression. 527 00:27:37,720 --> 00:27:41,070 And the if you differentiate this expression with regard 528 00:27:41,070 --> 00:27:43,920 to time to find the slope, the slope at 0 529 00:27:43,920 --> 00:27:50,260 is unchanged by beta because the exponent here, 1 plus beta k0, 530 00:27:50,260 --> 00:27:52,550 has the same form as the denominator here. 531 00:27:52,550 --> 00:27:54,540 So when you differentiate it, this comes down. 532 00:27:54,540 --> 00:27:55,307 The two cancel. 533 00:27:55,307 --> 00:27:57,640 And you end up with a slope that is independent of beta. 534 00:27:57,640 --> 00:28:00,550 The effect of that is that if you change the final value, 535 00:28:00,550 --> 00:28:03,110 it appears to go much faster. 536 00:28:03,110 --> 00:28:03,750 So pure win? 537 00:28:03,750 --> 00:28:05,250 No. 538 00:28:05,250 --> 00:28:05,950 Apparent win? 539 00:28:05,950 --> 00:28:07,300 Yes. 540 00:28:07,300 --> 00:28:11,630 We've traded gain for speed. 541 00:28:11,630 --> 00:28:15,550 OK, so that's ways that we can use feedback 542 00:28:15,550 --> 00:28:18,930 to improve the performance of an op-amp. 543 00:28:18,930 --> 00:28:22,010 OK, the big performance parameters 544 00:28:22,010 --> 00:28:25,220 that I talked about today were bandwidth and speed. 545 00:28:25,220 --> 00:28:27,170 You could do the same kind of analysis 546 00:28:27,170 --> 00:28:29,000 with many different kinds of metrics 547 00:28:29,000 --> 00:28:31,430 for op-amps like output impedance, input impedance, 548 00:28:31,430 --> 00:28:32,760 distortion reduction. 549 00:28:32,760 --> 00:28:35,150 We'll do distortion next time. 550 00:28:35,150 --> 00:28:40,490 | there's lots of other things that feedback improves. 551 00:28:40,490 --> 00:28:43,250 I've illustrated it with two. 552 00:28:43,250 --> 00:28:47,000 In both of those, as will be the case in all of the others, 553 00:28:47,000 --> 00:28:51,339 the effective feedback is to trade gain 554 00:28:51,339 --> 00:28:52,630 for something else of interest. 555 00:28:55,440 --> 00:28:58,440 OK, the next example I want to think about 556 00:28:58,440 --> 00:29:02,240 is thinking about a motor controller 557 00:29:02,240 --> 00:29:05,790 where what I want to think about is 558 00:29:05,790 --> 00:29:10,230 changing the way we think about the input-output relationship. 559 00:29:10,230 --> 00:29:11,550 So I take a motor. 560 00:29:11,550 --> 00:29:14,290 That's my motor. 561 00:29:14,290 --> 00:29:17,880 And what I want to do is have it control position. 562 00:29:17,880 --> 00:29:20,310 So it's not too easy to see probably because it's 563 00:29:20,310 --> 00:29:21,720 such a tiny little motor. 564 00:29:21,720 --> 00:29:23,345 Actually, it's not a tiny little motor. 565 00:29:23,345 --> 00:29:25,980 It's an enormous motor. 566 00:29:25,980 --> 00:29:27,600 So the red bar, that's the shaft. 567 00:29:27,600 --> 00:29:29,600 Right? 568 00:29:29,600 --> 00:29:34,890 And so the first question I want to think about is what's 569 00:29:34,890 --> 00:29:37,395 the relationship for the motor? 570 00:29:40,480 --> 00:29:42,400 What's the relationship for the motor 571 00:29:42,400 --> 00:29:47,568 for the transformation from voltage to angle of the shaft? 572 00:29:47,568 --> 00:29:49,526 So what's the relation between v of t and theta 573 00:29:49,526 --> 00:29:52,340 of t for DC motor? 574 00:29:52,340 --> 00:29:54,440 This is very much the same problem 575 00:29:54,440 --> 00:29:58,790 you worked in in the head-turning problem in 6.01. 576 00:29:58,790 --> 00:30:01,460 So you should be able to remember this answer from 6.01. 577 00:30:01,460 --> 00:30:03,680 What's the relationship between the voltage into a DC 578 00:30:03,680 --> 00:30:05,540 motor and the angle out? 579 00:30:05,540 --> 00:30:10,764 Is it one, two, three, four, or none of the above? 580 00:30:10,764 --> 00:30:11,680 Talk to your neighbor. 581 00:30:11,680 --> 00:30:12,846 Figure out the right answer. 582 00:30:15,923 --> 00:30:17,919 [SIDE CONVERSATION] 583 00:31:18,232 --> 00:31:20,440 So what's the answer-- one, two, three, four, or none 584 00:31:20,440 --> 00:31:21,040 of the above? 585 00:31:25,210 --> 00:31:27,080 Everybody votes. 586 00:31:27,080 --> 00:31:29,120 You're on such a streak. 587 00:31:29,120 --> 00:31:32,690 More than half the answers, I really don't like. 588 00:31:32,690 --> 00:31:34,730 OK, so we'll do a demo, and we'll figure out. 589 00:31:34,730 --> 00:31:37,370 So the question is, what is the relationship 590 00:31:37,370 --> 00:31:39,680 between voltage and speed? 591 00:31:39,680 --> 00:31:41,240 So I've got a motor, and I've got 592 00:31:41,240 --> 00:31:44,237 a knob that controls the voltage that goes to the motor. 593 00:31:44,237 --> 00:31:46,070 OK, so now, I turn the motor on, and nothing 594 00:31:46,070 --> 00:31:50,390 happens because I preset the voltage to be 0. 595 00:31:50,390 --> 00:31:52,610 Now, I turn the voltage to the right 596 00:31:52,610 --> 00:31:55,800 to make the voltage a bit positive, 597 00:31:55,800 --> 00:32:08,930 more positive, back towards 0, at 0, negative, more negative. 598 00:32:08,930 --> 00:32:15,444 OK, so what's the relationship between voltage and angle? 599 00:32:15,444 --> 00:32:18,250 AUDIENCE: Is that velocity or voltage? 600 00:32:18,250 --> 00:32:20,350 DENNIS FREEMAN: v is voltage, sorry. 601 00:32:20,350 --> 00:32:21,360 v is voltage. 602 00:32:21,360 --> 00:32:23,660 Theta is angle. 603 00:32:23,660 --> 00:32:25,220 So here's 0. 604 00:32:25,220 --> 00:32:27,590 Here's 90. 605 00:32:27,590 --> 00:32:30,110 Here's 0, pi over 2, pi. 606 00:32:30,110 --> 00:32:30,904 Right? 607 00:32:30,904 --> 00:32:32,570 The question is, what's the relationship 608 00:32:32,570 --> 00:32:34,287 between the voltage input of the motor 609 00:32:34,287 --> 00:32:35,870 and the angle of the shaft the output? 610 00:32:38,720 --> 00:32:39,845 And now, the answer is-- 611 00:32:42,910 --> 00:32:45,415 OK, the number correct has not changed. 612 00:32:49,360 --> 00:32:51,970 So tell me in words-- forget the one, two, three, four-- 613 00:32:51,970 --> 00:32:55,640 what's the relationship between voltage and angle? 614 00:33:02,410 --> 00:33:04,060 What is [INAUDIBLE] of the voltage? 615 00:33:04,060 --> 00:33:05,176 AUDIENCE: [INAUDIBLE] 616 00:33:05,176 --> 00:33:06,550 DENNIS FREEMAN: Angular velocity. 617 00:33:06,550 --> 00:33:10,540 So how do I get angular velocity? 618 00:33:10,540 --> 00:33:12,082 AUDIENCE: [? Measure ?] of the angle. 619 00:33:12,082 --> 00:33:14,415 DENNIS FREEMAN: So I'm giving you the angle, theta of t. 620 00:33:14,415 --> 00:33:15,740 How do I get angular velocity? 621 00:33:15,740 --> 00:33:18,932 AUDIENCE: Derivative of the angle-- the derivative. 622 00:33:18,932 --> 00:33:20,640 DENNIS FREEMAN: Take the derivative, yes. 623 00:33:20,640 --> 00:33:28,287 So what I want is that theta dot proportional to v. OK? 624 00:33:28,287 --> 00:33:29,870 Which of those relationships say that? 625 00:33:29,870 --> 00:33:31,790 None of them. 626 00:33:31,790 --> 00:33:36,020 OK, it was a trick question of course. 627 00:33:36,020 --> 00:33:40,970 So what do I want to have as my model for the DC motor? 628 00:33:40,970 --> 00:33:47,210 I want my model to take a voltage in and give me theta 629 00:33:47,210 --> 00:33:48,290 out. 630 00:33:48,290 --> 00:33:49,680 What should be in the model? 631 00:33:57,090 --> 00:33:59,380 k, it's always good to throw a k in. 632 00:33:59,380 --> 00:34:02,410 That always works. 633 00:34:02,410 --> 00:34:05,070 Excellent answer. 634 00:34:05,070 --> 00:34:06,750 k always works, sure. 635 00:34:06,750 --> 00:34:09,460 In any physical system, you never get one, right? 636 00:34:09,460 --> 00:34:10,135 Yeah. 637 00:34:10,135 --> 00:34:11,622 AUDIENCE: [INAUDIBLE] 638 00:34:11,622 --> 00:34:14,080 DENNIS FREEMAN: You're going to need an integrator, exactly 639 00:34:14,080 --> 00:34:14,989 right. 640 00:34:14,989 --> 00:34:16,572 So it's going to need something like I 641 00:34:16,572 --> 00:34:20,800 think I call it a gamma, a being the accumulator guy. 642 00:34:20,800 --> 00:34:21,549 Right? 643 00:34:21,549 --> 00:34:24,239 A different way of saying that would be-- 644 00:34:24,239 --> 00:34:25,719 or I write it in terms of Laplace 645 00:34:25,719 --> 00:34:29,130 transforms, I would get gamma over s. 646 00:34:29,130 --> 00:34:37,710 So if I put v in, and I want to get theta out, 647 00:34:37,710 --> 00:34:39,937 right, I need to put it through an integrator. 648 00:34:42,540 --> 00:34:46,949 So the idea is that if I put a constant voltage in-- say 649 00:34:46,949 --> 00:34:48,370 the voltage is 0-- 650 00:34:48,370 --> 00:34:50,000 theta was flat. 651 00:34:50,000 --> 00:34:52,770 But if I put a voltage at the blue line, 652 00:34:52,770 --> 00:34:54,820 theta increased at some rate. 653 00:34:54,820 --> 00:34:56,880 And if I changed it to red, it increased faster 654 00:34:56,880 --> 00:34:59,010 just like an integrator would do. 655 00:34:59,010 --> 00:35:01,380 So what I want to do is think about the motor 656 00:35:01,380 --> 00:35:03,310 being an integrator. 657 00:35:03,310 --> 00:35:06,570 OK, that lets me then cast the motor, which 658 00:35:06,570 --> 00:35:10,517 is a physical thing, into 6.003 terms again. 659 00:35:10,517 --> 00:35:12,100 So now, I can think about the problem. 660 00:35:12,100 --> 00:35:15,250 I'd really like it not to be a speed controller. 661 00:35:15,250 --> 00:35:18,740 I'd really like it to be a position controller. 662 00:35:18,740 --> 00:35:21,760 So feedback to the rescue-- 663 00:35:21,760 --> 00:35:25,235 what I'll do is I'll put it in a loop. 664 00:35:25,235 --> 00:35:26,610 So the idea is going to be that I 665 00:35:26,610 --> 00:35:31,710 take my motor, which looks like an integrator, gamma a, 666 00:35:31,710 --> 00:35:35,550 feedback a signal that's proportional to the angle, 667 00:35:35,550 --> 00:35:40,290 which I derive by having a potentiometer strapped 668 00:35:40,290 --> 00:35:42,810 to the back of the shaft-- 669 00:35:42,810 --> 00:35:43,590 so there's a pot. 670 00:35:43,590 --> 00:35:46,886 I've taken off the little stop so that it spins around 671 00:35:46,886 --> 00:35:47,760 doesn't hit the stop. 672 00:35:47,760 --> 00:35:48,090 Right? 673 00:35:48,090 --> 00:35:50,298 Otherwise, it would be kind of bad for the first part 674 00:35:50,298 --> 00:35:51,450 of the demo. 675 00:35:51,450 --> 00:35:53,940 So it's a surgically altered potentiometer 676 00:35:53,940 --> 00:35:55,860 that spins around forever. 677 00:35:55,860 --> 00:35:59,710 But over most of the range, it's reporting angle. 678 00:35:59,710 --> 00:36:02,830 So that's the way I get this feedback loop in. 679 00:36:02,830 --> 00:36:05,200 And then I put it into this thing, 680 00:36:05,200 --> 00:36:08,110 which is really just a 741 wired up 681 00:36:08,110 --> 00:36:09,980 like I showed in the first example, 682 00:36:09,980 --> 00:36:11,980 so that it would take the difference between two 683 00:36:11,980 --> 00:36:16,240 things, the desired input, which is a potentiometer that I'm 684 00:36:16,240 --> 00:36:21,509 turning here, and the actual position, which 685 00:36:21,509 --> 00:36:24,050 is the potentiometer up here, the same kind of potentiometer, 686 00:36:24,050 --> 00:36:26,930 just one that I turn, the other that the motor turns. 687 00:36:26,930 --> 00:36:29,420 And the op-amp figures out the difference, 688 00:36:29,420 --> 00:36:32,870 multiplies it by a gain, alpha, and presents it to the motor. 689 00:36:32,870 --> 00:36:35,300 That's the idea. 690 00:36:35,300 --> 00:36:39,380 And if you just think about 003, it's pretty easy to see that 691 00:36:39,380 --> 00:36:43,940 you can represent this relationship by a single pole. 692 00:36:43,940 --> 00:36:45,566 Right, there's some annoying constants. 693 00:36:45,566 --> 00:36:47,940 Right, you have to worry about the gain of the amplifier, 694 00:36:47,940 --> 00:36:50,160 the gain of the motor, and how much feedback. 695 00:36:50,160 --> 00:36:51,950 That's alpha, beta, gamma. 696 00:36:51,950 --> 00:36:55,960 But pretty much, it's just a pole. 697 00:36:55,960 --> 00:37:00,890 So what happens then as you change the gain? 698 00:37:00,890 --> 00:37:03,930 Well, if the gain is 0, the pole-- 699 00:37:03,930 --> 00:37:11,060 so if the gain, beta, is 0, the pole is at 0. 700 00:37:11,060 --> 00:37:15,380 OK, it works just like an integrator. 701 00:37:15,380 --> 00:37:17,190 That's what we said before. 702 00:37:17,190 --> 00:37:19,160 That's a sanity check where you check something 703 00:37:19,160 --> 00:37:20,451 you already know the answer to. 704 00:37:20,451 --> 00:37:21,320 Right? 705 00:37:21,320 --> 00:37:24,740 So if you set beta to 0, you get that the transformation 706 00:37:24,740 --> 00:37:26,630 is an integrator, good. 707 00:37:26,630 --> 00:37:29,620 If you set beta and anything else, what happens? 708 00:37:29,620 --> 00:37:33,140 Well, the pole goes screaming out that way. 709 00:37:33,140 --> 00:37:33,950 That's good, right? 710 00:37:33,950 --> 00:37:37,250 Remember good is that way. 711 00:37:37,250 --> 00:37:38,120 Right? 712 00:37:38,120 --> 00:37:39,550 That's not always true. 713 00:37:39,550 --> 00:37:42,080 But when you're trying to make something work fast, 714 00:37:42,080 --> 00:37:42,990 that's the rule. 715 00:37:42,990 --> 00:37:43,620 Right? 716 00:37:43,620 --> 00:37:46,380 Fast is over there. 717 00:37:46,380 --> 00:37:52,769 So the idea then would be that if I hook up this circuit-- 718 00:37:52,769 --> 00:37:54,810 right, I take the feedback, run it into an op-amp 719 00:37:54,810 --> 00:37:57,390 and multiply it, use that to control the motor instead 720 00:37:57,390 --> 00:38:00,210 of my turning the knob, then this thing 721 00:38:00,210 --> 00:38:05,730 should have the behavior that the pole goes from the origin 722 00:38:05,730 --> 00:38:07,680 to the right. 723 00:38:07,680 --> 00:38:12,230 So the question is, if that happened, what would happen? 724 00:38:12,230 --> 00:38:18,040 OK, just doing the same kind of analysis we did for the op-amp, 725 00:38:18,040 --> 00:38:19,910 the step response goes from being 726 00:38:19,910 --> 00:38:24,510 the response of a pole, which is an integrator, to being 727 00:38:24,510 --> 00:38:29,890 this kind of a response, which has a final value. 728 00:38:29,890 --> 00:38:34,180 That converts it from being voltage to velocity 729 00:38:34,180 --> 00:38:36,790 into voltage to angle. 730 00:38:36,790 --> 00:38:38,940 Does that make sense? 731 00:38:38,940 --> 00:38:45,480 It used to when the feedback was 0, the pole was at 0. 732 00:38:45,480 --> 00:38:49,070 So the output continuously rose. 733 00:38:49,070 --> 00:38:52,070 There was no steady-state position. 734 00:38:52,070 --> 00:38:55,460 But now with feedback, I get a step response 735 00:38:55,460 --> 00:38:58,130 that has a final value. 736 00:38:58,130 --> 00:39:00,140 The effect of moving the pole is the same 737 00:39:00,140 --> 00:39:02,590 as the effect of moving the pole. 738 00:39:02,590 --> 00:39:05,750 In the op-amp example, I get a step response 739 00:39:05,750 --> 00:39:10,450 that becomes increasingly like a step. 740 00:39:10,450 --> 00:39:13,840 It goes from being a sluggish step to a fast step. 741 00:39:13,840 --> 00:39:18,520 OK, so that's the way I use 003 to use feedback to fix 742 00:39:18,520 --> 00:39:19,360 the motor. 743 00:39:19,360 --> 00:39:21,280 And I've done that. 744 00:39:21,280 --> 00:39:26,650 So I have to move the input to the amplifier from my pot 745 00:39:26,650 --> 00:39:36,410 to the op-amp, which is there hopefully. 746 00:39:36,410 --> 00:39:42,120 So now when I turn it on, it becomes a position. 747 00:39:42,120 --> 00:39:45,390 OK, and I can test that by turning the pot. 748 00:39:45,390 --> 00:39:49,200 And in fact, when I turn the pot, the motor moves. 749 00:39:49,200 --> 00:39:50,380 Wonderful, voila, I won. 750 00:39:50,380 --> 00:39:50,880 Right? 751 00:39:50,880 --> 00:39:51,410 Great. 752 00:39:51,410 --> 00:39:53,300 Yes, applause please. 753 00:39:53,300 --> 00:39:53,900 Come on. 754 00:39:53,900 --> 00:39:54,510 [APPLAUSE] 755 00:39:54,510 --> 00:39:56,850 Exactly, exactly. 756 00:39:56,850 --> 00:40:00,000 OK, well, it's pretty wimpy actually 757 00:40:00,000 --> 00:40:02,520 because I've got this motor that weighs about, I don't know, 758 00:40:02,520 --> 00:40:03,600 three pounds. 759 00:40:03,600 --> 00:40:08,460 And it's exerting about a quarter of an ounce of torque. 760 00:40:08,460 --> 00:40:10,399 It's not really very impressive. 761 00:40:10,399 --> 00:40:11,190 So how do I fix it? 762 00:40:15,340 --> 00:40:16,090 What do I do? 763 00:40:16,090 --> 00:40:17,990 How do I fix it? 764 00:40:17,990 --> 00:40:19,500 I have a wimpy controller. 765 00:40:19,500 --> 00:40:20,030 Yes. 766 00:40:20,030 --> 00:40:20,863 AUDIENCE: More gain. 767 00:40:20,863 --> 00:40:22,870 DENNIS FREEMAN: More gain, of course. 768 00:40:22,870 --> 00:40:26,660 So remember that response. 769 00:40:26,660 --> 00:40:29,020 So now, I turn off the motor and increase the gain 770 00:40:29,020 --> 00:40:31,100 by a factor of three. 771 00:40:31,100 --> 00:40:38,440 And the answer is much better, well, maybe. 772 00:40:38,440 --> 00:40:41,170 So what do I do? 773 00:40:41,170 --> 00:40:42,860 More gain, of course. 774 00:40:42,860 --> 00:40:46,060 So another factor of three-- 775 00:40:46,060 --> 00:40:51,340 much better, well, maybe. 776 00:40:51,340 --> 00:40:52,790 So another factor of three. 777 00:40:56,030 --> 00:40:57,970 OK, another factor of three. 778 00:41:02,150 --> 00:41:04,190 Hmm, another factor of three. 779 00:41:08,150 --> 00:41:10,920 Some of you close may hear that there's a little buzzing sound 780 00:41:10,920 --> 00:41:11,420 now. 781 00:41:11,420 --> 00:41:12,290 I can hear it. 782 00:41:12,290 --> 00:41:13,840 That's the op-amp killing itself. 783 00:41:19,460 --> 00:41:20,580 OK, it's not working. 784 00:41:20,580 --> 00:41:21,420 Why is it not? 785 00:41:21,420 --> 00:41:23,157 Is it working? 786 00:41:23,157 --> 00:41:23,990 I just gave it away. 787 00:41:23,990 --> 00:41:24,740 It's not working. 788 00:41:24,740 --> 00:41:26,480 How do I know it's not working? 789 00:41:26,480 --> 00:41:28,740 Here's my theory. 790 00:41:28,740 --> 00:41:30,020 Here's my answer. 791 00:41:30,020 --> 00:41:32,840 How do they match or mismatch? 792 00:41:32,840 --> 00:41:34,250 Is it an angle controller? 793 00:41:34,250 --> 00:41:34,910 Yes or no? 794 00:41:34,910 --> 00:41:36,555 Yes. 795 00:41:36,555 --> 00:41:37,930 AUDIENCE: Didn't we just do this? 796 00:41:37,930 --> 00:41:40,130 DENNIS FREEMAN: Yeah, it's an angle controller. 797 00:41:40,130 --> 00:41:43,030 What's different about this behavior and that behavior? 798 00:41:47,258 --> 00:41:48,540 AUDIENCE: There's oscillation. 799 00:41:48,540 --> 00:41:50,040 DENNIS FREEMAN: There's oscillation. 800 00:41:50,040 --> 00:41:52,360 Right, this thing, turn it back on. 801 00:41:52,360 --> 00:41:54,580 Let the thing kill itself. 802 00:41:54,580 --> 00:41:56,280 There's an oscillation. 803 00:41:56,280 --> 00:41:59,646 Turn the gain down so it doesn't kill itself quite as quickly. 804 00:41:59,646 --> 00:42:02,570 It's oscillating. 805 00:42:02,570 --> 00:42:05,710 Watch the oscillation. 806 00:42:05,710 --> 00:42:09,190 Think about characterizing the oscillation. 807 00:42:09,190 --> 00:42:11,950 Make the gain smaller. 808 00:42:11,950 --> 00:42:12,450 That's bad. 809 00:42:16,540 --> 00:42:17,080 Ignore that. 810 00:42:21,210 --> 00:42:23,765 Think about the oscillation. 811 00:42:23,765 --> 00:42:25,770 It's still oscillating. 812 00:42:25,770 --> 00:42:31,322 Factor of three smaller, it's still oscillating. 813 00:42:31,322 --> 00:42:32,280 That doesn't oscillate. 814 00:42:32,280 --> 00:42:32,821 What's wrong? 815 00:42:35,980 --> 00:42:39,490 OK, so the $64,000 question, what did I do wrong? 816 00:42:39,490 --> 00:42:41,530 Right, the goal is to make a model of the motor, 817 00:42:41,530 --> 00:42:43,810 analyze the model, figure out how to put feedback around it, 818 00:42:43,810 --> 00:42:44,476 make it perfect. 819 00:42:44,476 --> 00:42:45,282 Yes. 820 00:42:45,282 --> 00:42:46,782 AUDIENCE: The model on the board was 821 00:42:46,782 --> 00:42:48,275 created without any inertia. 822 00:42:48,275 --> 00:42:49,650 DENNIS FREEMAN: I made too simple 823 00:42:49,650 --> 00:42:51,510 of a model for the motor. 824 00:42:51,510 --> 00:42:53,280 I ignored inertia. 825 00:42:53,280 --> 00:42:56,880 I ignored all manner of things. 826 00:42:56,880 --> 00:42:59,470 I ignored friction. 827 00:42:59,470 --> 00:43:03,300 So how can you tell that I ignored those things? 828 00:43:03,300 --> 00:43:04,980 So one way you can tell-- 829 00:43:04,980 --> 00:43:06,240 let's see. 830 00:43:06,240 --> 00:43:07,350 Is it off or on? 831 00:43:07,350 --> 00:43:07,850 It's on. 832 00:43:07,850 --> 00:43:09,390 Or turn it off. 833 00:43:09,390 --> 00:43:11,310 Go back to the original configuration 834 00:43:11,310 --> 00:43:14,320 where it's a speed controller. 835 00:43:14,320 --> 00:43:18,030 Turn the speed on a little bit. 836 00:43:18,030 --> 00:43:19,470 Turn it off. 837 00:43:19,470 --> 00:43:22,020 Grab it with my fingers. 838 00:43:22,020 --> 00:43:23,190 Turn it on. 839 00:43:23,190 --> 00:43:23,950 Now, release it. 840 00:43:27,020 --> 00:43:29,338 What did you see? 841 00:43:29,338 --> 00:43:30,620 Grab it with my fingers. 842 00:43:30,620 --> 00:43:32,130 Turn it on. 843 00:43:32,130 --> 00:43:32,630 Let go. 844 00:43:37,080 --> 00:43:40,410 We're trying to poke a hole in this. 845 00:43:40,410 --> 00:43:41,700 There is my model. 846 00:43:41,700 --> 00:43:44,130 My model is wrong. 847 00:43:44,130 --> 00:43:47,250 I claim I just told you the key experiment for figuring out 848 00:43:47,250 --> 00:43:48,360 why my model is wrong. 849 00:43:48,360 --> 00:43:51,030 Now you have to figure out how to interpret that experiment. 850 00:43:51,030 --> 00:43:59,580 So what does my theory say would happen if I turn on the power, 851 00:43:59,580 --> 00:44:00,970 hold the bar, and release it? 852 00:44:04,730 --> 00:44:07,642 AUDIENCE: [INAUDIBLE]. 853 00:44:07,642 --> 00:44:09,100 DENNIS FREEMAN: It should instantly 854 00:44:09,100 --> 00:44:11,500 reach the terminal velocity. 855 00:44:11,500 --> 00:44:12,607 What's it doing? 856 00:44:12,607 --> 00:44:14,690 AUDIENCE: I has just like a rise time [INAUDIBLE]. 857 00:44:14,690 --> 00:44:16,730 DENNIS FREEMAN: It's ramping up. 858 00:44:16,730 --> 00:44:19,510 That's what's wrong with my model. 859 00:44:19,510 --> 00:44:21,950 OK, so the model is too simple. 860 00:44:21,950 --> 00:44:23,650 It's not just an integrator. 861 00:44:23,650 --> 00:44:27,150 It's a slow integrator. 862 00:44:27,150 --> 00:44:30,210 OK, I can model that too. 863 00:44:30,210 --> 00:44:34,530 If I model the slow integration, I 864 00:44:34,530 --> 00:44:36,660 can think about the motor doesn't really 865 00:44:36,660 --> 00:44:37,980 behave like an integrator. 866 00:44:37,980 --> 00:44:43,330 It works like a sluggish integrator. 867 00:44:43,330 --> 00:44:46,520 That might be a lot easier to see if I write that in s. 868 00:44:46,520 --> 00:44:55,140 So that's the same as gamma p over s plus gamma. 869 00:44:55,140 --> 00:44:58,790 No, s times-- no, close, wrong. 870 00:45:05,200 --> 00:45:07,290 So it used to be gamma over s. 871 00:45:07,290 --> 00:45:09,420 Now, it's gamma over s s plus p. 872 00:45:12,150 --> 00:45:18,960 OK, so I'm replacing what had been an integrator here 873 00:45:18,960 --> 00:45:21,540 with a sluggish integrator. 874 00:45:21,540 --> 00:45:25,640 Sluggish is another pole. 875 00:45:25,640 --> 00:45:30,230 So instead of representing it by a single pole at the origin, 876 00:45:30,230 --> 00:45:31,910 I represent it by two poles. 877 00:45:31,910 --> 00:45:35,510 If you calculate the step response 878 00:45:35,510 --> 00:45:38,057 for the single pole at the origin, 879 00:45:38,057 --> 00:45:39,140 you get the straight line. 880 00:45:39,140 --> 00:45:41,800 That was model one. 881 00:45:41,800 --> 00:45:44,140 If you calculate the step response for the modified 882 00:45:44,140 --> 00:45:48,420 model, that's this one. 883 00:45:48,420 --> 00:45:51,280 Ultimately, they become parallel. 884 00:45:51,280 --> 00:45:54,760 So the speed becomes the same. 885 00:45:54,760 --> 00:45:56,830 But there's sluggishness because of inertia 886 00:45:56,830 --> 00:46:00,580 and so forth so that it takes a while for the real motor 887 00:46:00,580 --> 00:46:01,610 to achieve that. 888 00:46:01,610 --> 00:46:06,940 So you can see that in the model two response, because the angle 889 00:46:06,940 --> 00:46:12,030 does not immediately change when I let go of it, 890 00:46:12,030 --> 00:46:13,950 it takes a while for it to start to change. 891 00:46:13,950 --> 00:46:15,910 That's inertia. 892 00:46:15,910 --> 00:46:25,000 So if I put that model, model two, into the feedback loop, 893 00:46:25,000 --> 00:46:27,760 then I make a different prediction about the way things 894 00:46:27,760 --> 00:46:30,400 should work. 895 00:46:30,400 --> 00:46:31,790 That prediction is shown here. 896 00:46:31,790 --> 00:46:32,290 Right? 897 00:46:32,290 --> 00:46:35,020 That's just thinking about system functions, 898 00:46:35,020 --> 00:46:38,189 saying that I've got two poles, trying to figure out 899 00:46:38,189 --> 00:46:40,480 where the poles are, figuring out where the time domain 900 00:46:40,480 --> 00:46:42,979 response is, integrating it to turn it into a step response. 901 00:46:42,979 --> 00:46:47,860 Do all those things, and you get a response. 902 00:46:47,860 --> 00:46:51,820 The point is that the new model looks like two poles. 903 00:46:51,820 --> 00:46:55,810 And when you analyze the root locus, 904 00:46:55,810 --> 00:47:00,870 those poles move when you change the feedback. 905 00:47:00,870 --> 00:47:04,490 So as I change the feedback number, beta, as beta 906 00:47:04,490 --> 00:47:08,150 goes from 0 to big, the poles go toward each other 907 00:47:08,150 --> 00:47:10,490 and then split off. 908 00:47:10,490 --> 00:47:12,620 That's just math. 909 00:47:12,620 --> 00:47:14,870 That's the math that was showed over here. 910 00:47:14,870 --> 00:47:19,640 All I did was write the second model up there, 911 00:47:19,640 --> 00:47:24,690 figure out the system function, find the poles of the system 912 00:47:24,690 --> 00:47:25,440 function. 913 00:47:25,440 --> 00:47:27,125 They depend on beta. 914 00:47:27,125 --> 00:47:29,530 And that dependence is shown here. 915 00:47:29,530 --> 00:47:32,070 And so the response then, that's where 916 00:47:32,070 --> 00:47:33,570 the oscillations come from. 917 00:47:33,570 --> 00:47:36,930 If the gain is big enough, I get a pair 918 00:47:36,930 --> 00:47:40,810 of poles that are off the real axis. 919 00:47:40,810 --> 00:47:43,900 The fact that they have an imaginary component, the poles, 920 00:47:43,900 --> 00:47:45,880 means that there's an oscillation. 921 00:47:45,880 --> 00:47:49,870 And what happens is I change the feedback is not 922 00:47:49,870 --> 00:47:51,890 that the oscillation goes away. 923 00:47:51,890 --> 00:47:57,160 In fact, the real part of the envelope doesn't change. 924 00:47:57,160 --> 00:47:59,200 That's roughly what I was seeing here. 925 00:47:59,200 --> 00:48:01,750 If you watched what was going on in the closed-loop 926 00:48:01,750 --> 00:48:06,230 configuration, it always oscillated for a couple 927 00:48:06,230 --> 00:48:06,890 of seconds-- 928 00:48:06,890 --> 00:48:09,680 like a second or two seconds or three seconds. 929 00:48:09,680 --> 00:48:12,290 That second or two seconds didn't change. 930 00:48:12,290 --> 00:48:14,540 What did change was the bumps-- 931 00:48:14,540 --> 00:48:16,910 how many bumps there were in the two seconds. 932 00:48:16,910 --> 00:48:20,090 And that's what the theory says. 933 00:48:20,090 --> 00:48:25,910 Because the real part of the imaginary pair of poles 934 00:48:25,910 --> 00:48:29,190 doesn't change, the envelope doesn't change. 935 00:48:29,190 --> 00:48:31,760 But because the imaginary part does change, 936 00:48:31,760 --> 00:48:34,740 the oscillation frequency does change. 937 00:48:34,740 --> 00:48:38,280 So the result then is something whose envelope doesn't really 938 00:48:38,280 --> 00:48:38,780 improve. 939 00:48:41,570 --> 00:48:45,410 So that model then to a large extent does 940 00:48:45,410 --> 00:48:48,720 predict the behavior that I have measured. 941 00:48:48,720 --> 00:48:50,810 And then what that motivates is if I 942 00:48:50,810 --> 00:48:53,120 want to make the motor work better, 943 00:48:53,120 --> 00:48:57,440 I need some even better controller 944 00:48:57,440 --> 00:48:59,960 because my simple proportional controller, that's 945 00:48:59,960 --> 00:49:02,500 as good as it's going to do. 946 00:49:02,500 --> 00:49:07,420 OK, so what I tried to illustrate today 947 00:49:07,420 --> 00:49:09,190 was ways of thinking about feedback 948 00:49:09,190 --> 00:49:11,020 to enhance performance. 949 00:49:11,020 --> 00:49:12,550 We looked at an op-amp and saw how 950 00:49:12,550 --> 00:49:15,040 you could think about feedback increasing the bandwidth 951 00:49:15,040 --> 00:49:16,720 or making it faster. 952 00:49:16,720 --> 00:49:18,220 We looked at a motor and found out 953 00:49:18,220 --> 00:49:20,511 how you could think about changing the motor from being 954 00:49:20,511 --> 00:49:24,616 a voltage-controlled velocity to a voltage-controlled position. 955 00:49:24,616 --> 00:49:25,990 And we saw how the feedback could 956 00:49:25,990 --> 00:49:27,400 be used for both of those. 957 00:49:27,400 --> 00:49:31,300 Next time, we'll look at a couple other examples. 958 00:49:31,300 --> 00:49:32,850 Thanks.