1 00:00:00,040 --> 00:00:02,460 The following content is provided under a Creative 2 00:00:02,460 --> 00:00:03,870 Commons license. 3 00:00:03,870 --> 00:00:06,910 Your support will help MIT OpenCourseWare continue to 4 00:00:06,910 --> 00:00:10,560 offer high-quality educational resources for free. 5 00:00:10,560 --> 00:00:13,460 To make a donation or view additional materials from 6 00:00:13,460 --> 00:00:17,390 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,390 --> 00:00:18,640 ocw.mit.edu. 8 00:00:24,660 --> 00:00:25,000 PROFESSOR: Hello. 9 00:00:25,000 --> 00:00:26,980 Welcome. 10 00:00:26,980 --> 00:00:29,690 Welcome back from spring break. 11 00:00:29,690 --> 00:00:34,120 Today what I want to do is talk about op-amps. 12 00:00:34,120 --> 00:00:38,070 And in particular what I want to do is talk about modularity 13 00:00:38,070 --> 00:00:39,030 in circuit design. 14 00:00:39,030 --> 00:00:41,510 How do you make circuits that are modular? 15 00:00:41,510 --> 00:00:45,640 We'll see that there's special problems when you think about 16 00:00:45,640 --> 00:00:48,060 modularity applied to circuits. 17 00:00:48,060 --> 00:00:51,840 And op-amps are one solution for helping us think about 18 00:00:51,840 --> 00:00:54,550 those problems. 19 00:00:54,550 --> 00:00:57,890 Before launching straight into something new though, I'd like 20 00:00:57,890 --> 00:01:02,340 to recap where we are under the assumption that you may 21 00:01:02,340 --> 00:01:07,310 not have been thinking about this exactly currently. 22 00:01:07,310 --> 00:01:13,290 So last time we took our first look at circuits. 23 00:01:13,290 --> 00:01:15,280 Circuits are very different from the kinds of things we've 24 00:01:15,280 --> 00:01:17,160 thought about before. 25 00:01:17,160 --> 00:01:20,880 Before, in programming, in linear systems theory, we 26 00:01:20,880 --> 00:01:25,380 thought about blocks that had well-defined inputs and 27 00:01:25,380 --> 00:01:27,320 well-defined outputs. 28 00:01:27,320 --> 00:01:29,310 Circuits are different. 29 00:01:29,310 --> 00:01:31,510 Circuits are all connected together. 30 00:01:31,510 --> 00:01:36,530 And circuit theory is thinking about how do you organize your 31 00:01:36,530 --> 00:01:40,560 thoughts about complicated interactions among things. 32 00:01:40,560 --> 00:01:45,630 The parts interact because they touch each other and they 33 00:01:45,630 --> 00:01:47,810 share voltages. 34 00:01:47,810 --> 00:01:49,830 Because they touch each other, they share currents. 35 00:01:49,830 --> 00:01:51,950 And you have to somehow think about that 36 00:01:51,950 --> 00:01:53,520 whole complicated system. 37 00:01:56,890 --> 00:01:59,930 We figured out that the way you think about that, you can 38 00:01:59,930 --> 00:02:03,140 think about it as three separate enterprises. 39 00:02:03,140 --> 00:02:04,290 How do you think about voltages. 40 00:02:04,290 --> 00:02:05,260 How do you think about currents. 41 00:02:05,260 --> 00:02:08,360 And how do you think about the elements. 42 00:02:08,360 --> 00:02:09,530 How do you think about voltages? 43 00:02:09,530 --> 00:02:11,610 Well, the sum of the voltages around any 44 00:02:11,610 --> 00:02:13,120 closed loop is zero. 45 00:02:13,120 --> 00:02:14,070 Done. 46 00:02:14,070 --> 00:02:14,720 KVL -- 47 00:02:14,720 --> 00:02:16,290 Kirchoff's Voltage Law. 48 00:02:16,290 --> 00:02:17,520 How do you think about currents? 49 00:02:17,520 --> 00:02:20,900 Draw any surface. 50 00:02:20,900 --> 00:02:23,670 The net current that leaves the surface is zero. 51 00:02:23,670 --> 00:02:24,350 KCL -- 52 00:02:24,350 --> 00:02:25,460 Kirchoff's Current Law. 53 00:02:25,460 --> 00:02:26,430 How do you think about the elements? 54 00:02:26,430 --> 00:02:29,130 Well it depends on the element. 55 00:02:29,130 --> 00:02:31,700 If you're thinking about a linear resistor, it's Ohm's 56 00:02:31,700 --> 00:02:34,440 Law, V equals IR. 57 00:02:34,440 --> 00:02:36,560 If you're thinking about a source, it could be a voltage 58 00:02:36,560 --> 00:02:40,640 source, then V equals v0, some fixed number. 59 00:02:40,640 --> 00:02:42,850 If you're thinking about a current source, then current 60 00:02:42,850 --> 00:02:45,840 is fixed at some fixed number. 61 00:02:45,840 --> 00:02:46,390 So it depends. 62 00:02:46,390 --> 00:02:48,840 When you're thinking about the element laws, it depends on 63 00:02:48,840 --> 00:02:51,720 what element you are thinking about. 64 00:02:51,720 --> 00:02:53,820 And then you just combine all of those, and you can solve 65 00:02:53,820 --> 00:02:56,150 the circuit. 66 00:02:56,150 --> 00:02:59,320 The only problem that arises is that those equations are 67 00:02:59,320 --> 00:03:01,750 highly redundant. 68 00:03:01,750 --> 00:03:04,200 There's a lot more KVL equations than you need. 69 00:03:04,200 --> 00:03:08,770 There's a lot more KCL equations than you need. 70 00:03:08,770 --> 00:03:12,430 And so in fact, the trick in analyzing a circuit is to 71 00:03:12,430 --> 00:03:15,650 figure out the smallest number of equations that are 72 00:03:15,650 --> 00:03:20,960 adequate, the number of equations that are necessary 73 00:03:20,960 --> 00:03:24,860 and sufficient to find a solution. 74 00:03:24,860 --> 00:03:26,440 Last time we went through three 75 00:03:26,440 --> 00:03:29,000 different ways to do that. 76 00:03:29,000 --> 00:03:32,690 We thought about what I will think of as primitive or 77 00:03:32,690 --> 00:03:36,390 element voltages and currents. 78 00:03:36,390 --> 00:03:39,450 The idea in that technique is you think about every element 79 00:03:39,450 --> 00:03:42,710 and assign to that element its voltage and its 80 00:03:42,710 --> 00:03:43,960 corresponding current. 81 00:03:47,780 --> 00:03:52,590 Then that gives you, as in this example if you've got six 82 00:03:52,590 --> 00:03:58,020 elements, that gives you six unknowns, one voltage across 83 00:03:58,020 --> 00:04:00,290 every element, one current through every element. 84 00:04:00,290 --> 00:04:05,820 So then you've got to dig up six other relationships. 85 00:04:05,820 --> 00:04:07,440 Sorry, start again. 86 00:04:07,440 --> 00:04:11,650 Six elements, six voltages, six currents, 12 unknowns, you 87 00:04:11,650 --> 00:04:13,065 need to come up with 12 equations. 88 00:04:16,579 --> 00:04:19,790 Six of them are element equations. 89 00:04:19,790 --> 00:04:20,560 So six are easy. 90 00:04:20,560 --> 00:04:23,170 Then you have to come up with six more. 91 00:04:23,170 --> 00:04:25,150 And for this particular circuit, it turns out that 92 00:04:25,150 --> 00:04:28,920 there is three independent 93 00:04:28,920 --> 00:04:31,100 Kirchoff's Voltage Law equations. 94 00:04:31,100 --> 00:04:34,560 And there are three independent KCL, Kirchoff's 95 00:04:34,560 --> 00:04:37,270 Current Law equations. 96 00:04:37,270 --> 00:04:39,080 12 equations, 12 unknowns to solve, done. 97 00:04:41,640 --> 00:04:45,480 That's actually more work than you need to do. 98 00:04:45,480 --> 00:04:48,020 So we looked at two other techniques for solving the 99 00:04:48,020 --> 00:04:49,700 same kind of circuits. 100 00:04:49,700 --> 00:04:51,265 One's called node voltages and another 101 00:04:51,265 --> 00:04:53,970 is called loop currents. 102 00:04:53,970 --> 00:04:56,760 They are duals of each other. 103 00:04:56,760 --> 00:05:00,560 In the node voltage method you figure out exactly the minimum 104 00:05:00,560 --> 00:05:02,720 number of voltages that I would have to tell you in 105 00:05:02,720 --> 00:05:07,120 order to specify all of the element voltages. 106 00:05:07,120 --> 00:05:09,790 So for example, if I told you this voltage here, and that 107 00:05:09,790 --> 00:05:11,660 voltage there, and this voltage there, and that 108 00:05:11,660 --> 00:05:15,480 voltage there, four of them, that would let you calculate 109 00:05:15,480 --> 00:05:18,510 all of the element voltages. 110 00:05:18,510 --> 00:05:21,920 But now I only have four instead of six. 111 00:05:21,920 --> 00:05:26,280 Four nodes instead of six element voltages. 112 00:05:26,280 --> 00:05:27,750 So it's smaller. 113 00:05:27,750 --> 00:05:30,720 And we talked about last time how you could find the correct 114 00:05:30,720 --> 00:05:35,390 equations that go with those node voltages. 115 00:05:35,390 --> 00:05:38,870 In the loop current method, you specify the minimum number 116 00:05:38,870 --> 00:05:44,020 of currents that would be sufficient to account for all 117 00:05:44,020 --> 00:05:44,920 of the element currents. 118 00:05:44,920 --> 00:05:47,970 And here in this circuit it required three. 119 00:05:47,970 --> 00:05:49,600 Three is smaller than six. 120 00:05:49,600 --> 00:05:52,010 There were six element currents. 121 00:05:52,010 --> 00:05:55,240 So again we've got a reduction in the number of unknowns -- 122 00:05:55,240 --> 00:05:56,965 12 4, 3. 123 00:05:59,960 --> 00:06:01,150 So that's kind of the idea. 124 00:06:01,150 --> 00:06:05,780 And just to make sure that you're all with it, think 125 00:06:05,780 --> 00:06:07,720 about this problem. 126 00:06:07,720 --> 00:06:12,700 How many of the following expressions are correct? 127 00:06:12,700 --> 00:06:14,150 Feel free to talk to your neighbor. 128 00:06:14,150 --> 00:06:16,930 At the end of about 45 seconds, I'll ask you to raise 129 00:06:16,930 --> 00:06:20,020 your hand with a number of fingers, (1) through (5) -- 130 00:06:20,020 --> 00:06:21,640 indicating which is the correct answer. 131 00:06:48,290 --> 00:06:49,820 Dead silence. 132 00:06:49,820 --> 00:06:51,070 You're allowed to talk. 133 00:08:39,390 --> 00:08:41,100 OK, so how many of the relations are true? 134 00:08:41,100 --> 00:08:43,340 Everybody raise your hand. 135 00:08:43,340 --> 00:08:45,740 Indicate a number fingers that tells me how many of the 136 00:08:45,740 --> 00:08:47,020 answers are correct. 137 00:08:49,930 --> 00:08:50,900 Come on. 138 00:08:50,900 --> 00:08:52,790 Blame it on your neighbor. 139 00:08:52,790 --> 00:08:54,330 You can say anything, right? 140 00:08:54,330 --> 00:08:57,660 It's always your neighbor's fault. 141 00:08:57,660 --> 00:08:58,200 That's very good. 142 00:08:58,200 --> 00:09:01,950 It's about 95% correct, almost all correct. 143 00:09:01,950 --> 00:09:04,260 This first equation, what is this? 144 00:09:04,260 --> 00:09:06,780 Give that a name. 145 00:09:06,780 --> 00:09:07,840 KVL. 146 00:09:07,840 --> 00:09:10,020 Is it correct? 147 00:09:10,020 --> 00:09:10,580 Sure. 148 00:09:10,580 --> 00:09:13,010 You need to figure out which path it corresponds to. 149 00:09:13,010 --> 00:09:14,180 It goes through-- 150 00:09:14,180 --> 00:09:16,640 The only v's are over here. 151 00:09:16,640 --> 00:09:18,860 So therefore it must be something that has to do with 152 00:09:18,860 --> 00:09:20,170 this circuit. 153 00:09:20,170 --> 00:09:23,850 So this says that there's 1, 2, 6, and 5. 154 00:09:23,850 --> 00:09:27,140 That's path 1, 2, 6, and 5. 155 00:09:27,140 --> 00:09:28,835 So all you really need to do is check the polarities. 156 00:09:31,700 --> 00:09:34,720 So if we took all of the variables to the same side, 157 00:09:34,720 --> 00:09:39,100 then we'd have minus v1 plus v2 plus v6 plus v5. 158 00:09:39,100 --> 00:09:42,180 If we think about going a loop that goes through the minus 159 00:09:42,180 --> 00:09:44,140 v1, that would be this way. 160 00:09:44,140 --> 00:09:47,170 Plus v2 plus v6 plus v5. 161 00:09:47,170 --> 00:09:48,770 So that's right. 162 00:09:48,770 --> 00:09:49,720 Easy. 163 00:09:49,720 --> 00:09:51,180 What's the point of this equation? 164 00:09:51,180 --> 00:09:55,630 v6 equals e1 minus e2, what's that say? 165 00:09:55,630 --> 00:09:58,860 Why'd I ask that? 166 00:09:58,860 --> 00:10:01,300 Yeah. 167 00:10:01,300 --> 00:10:02,550 AUDIENCE: It's saying that the-- 168 00:10:05,692 --> 00:10:08,340 PROFESSOR: Exactly, it's intended to make you think 169 00:10:08,340 --> 00:10:10,510 through the relationship between the primitive 170 00:10:10,510 --> 00:10:13,920 variables, the element voltages, 171 00:10:13,920 --> 00:10:15,520 and the node voltages. 172 00:10:15,520 --> 00:10:18,660 If I told you the node voltages e1 and e2, you could 173 00:10:18,660 --> 00:10:22,570 trivially compute v6. 174 00:10:22,570 --> 00:10:25,540 You could also compute any of the other v's. 175 00:10:25,540 --> 00:10:27,250 If I told you to find v-- 176 00:10:27,250 --> 00:10:28,650 what is that? 177 00:10:28,650 --> 00:10:29,650 My eyes are not very good. 178 00:10:29,650 --> 00:10:33,800 v4, if I told you to find v4, that would be e1 minus ground. 179 00:10:33,800 --> 00:10:38,850 We assign the number 0 to ground, so that would be e1. 180 00:10:38,850 --> 00:10:42,030 So the idea, the reason I ask number (2), is to make you 181 00:10:42,030 --> 00:10:44,750 think about how you relate the node voltages 182 00:10:44,750 --> 00:10:47,280 to the element voltages. 183 00:10:47,280 --> 00:10:48,620 How about number (3), what's that? 184 00:10:53,030 --> 00:10:56,600 Well it's highly related to number (2). 185 00:10:56,600 --> 00:10:58,660 But it's different from number (2), because? 186 00:10:58,660 --> 00:10:59,600 AUDIENCE: Ohm's Law? 187 00:10:59,600 --> 00:11:02,080 PROFESSOR: Ohm's Law. 188 00:11:02,080 --> 00:11:04,820 So equation (3) is the way Ohm's Law looks when you use 189 00:11:04,820 --> 00:11:06,070 node equations. 190 00:11:07,970 --> 00:11:10,980 Ohm's Law is a little bit uglier when you use node 191 00:11:10,980 --> 00:11:14,930 equations than when you use primitive variables. 192 00:11:14,930 --> 00:11:18,090 When you use primitive variables, the same 193 00:11:18,090 --> 00:11:23,970 relationship would've simply said that v6 is R6 times i6. 194 00:11:26,550 --> 00:11:30,970 Because I'm using node voltages instead, the voltage 195 00:11:30,970 --> 00:11:36,400 across resistor six shows up as a difference. 196 00:11:36,400 --> 00:11:37,220 How about this one? 197 00:11:37,220 --> 00:11:38,470 What's that? 198 00:11:45,980 --> 00:11:47,726 True or false, this is KCL? 199 00:11:47,726 --> 00:11:48,702 AUDIENCE: False. 200 00:11:48,702 --> 00:11:49,678 AUDIENCE: False. 201 00:11:49,678 --> 00:11:52,606 AUDIENCE: --KCL. 202 00:11:52,606 --> 00:11:53,582 PROFESSOR: True. 203 00:11:53,582 --> 00:11:56,510 This is KCL, true. 204 00:11:56,510 --> 00:11:57,990 OK. 205 00:11:57,990 --> 00:12:01,005 This is KCL, false. 206 00:12:01,005 --> 00:12:02,280 Well, obviously we're numerous. 207 00:12:02,280 --> 00:12:06,900 And it's not, sort of, 100% participation, but-- 208 00:12:06,900 --> 00:12:08,830 This is not KCL. 209 00:12:08,830 --> 00:12:10,920 What is that? 210 00:12:10,920 --> 00:12:13,170 KCL says that the sum of current out 211 00:12:13,170 --> 00:12:14,710 of some closed path. 212 00:12:14,710 --> 00:12:16,410 This is mixed thing. 213 00:12:16,410 --> 00:12:18,210 i6 is one of these things. 214 00:12:18,210 --> 00:12:21,520 And IB and IC is one of these things. 215 00:12:21,520 --> 00:12:23,210 So what is the equation for? 216 00:12:26,406 --> 00:12:27,850 AUDIENCE: Incorrect. 217 00:12:27,850 --> 00:12:30,350 PROFESSOR: Incorrect, yes. 218 00:12:30,350 --> 00:12:33,428 What would be the correct way of saying equation (4)? 219 00:12:33,428 --> 00:12:34,678 AUDIENCE: i6 equals-- 220 00:12:38,308 --> 00:12:39,772 i6 is-- 221 00:12:39,772 --> 00:12:41,724 PROFESSOR: The answer to set questions according to the 222 00:12:41,724 --> 00:12:45,671 theory of lectures is go to the next slide. 223 00:12:45,671 --> 00:12:47,042 Here's the correct expression. 224 00:12:47,042 --> 00:12:48,710 Why is this correct, and why is that not correct? 225 00:12:55,310 --> 00:12:59,796 Equation (4) is intended to be how do you relate the loop 226 00:12:59,796 --> 00:13:01,885 current to the element currents? 227 00:13:06,330 --> 00:13:07,560 So i6 is an element current. 228 00:13:07,560 --> 00:13:10,900 It's the current that goes through R6. 229 00:13:10,900 --> 00:13:13,750 When we do loop currents, we have two currents going 230 00:13:13,750 --> 00:13:15,000 through R6. 231 00:13:17,580 --> 00:13:21,190 So in the loop current view, the total current that goes 232 00:13:21,190 --> 00:13:24,940 through R6 is a sum or difference of the two loop 233 00:13:24,940 --> 00:13:27,500 currents that go through R6. 234 00:13:27,500 --> 00:13:32,950 So the element current through R6, which is i6, is a sum or 235 00:13:32,950 --> 00:13:34,520 difference of loop currents. 236 00:13:34,520 --> 00:13:36,340 There are two loop currents we have to worry about. 237 00:13:36,340 --> 00:13:38,700 Which one goes through in the correct direction? 238 00:13:38,700 --> 00:13:42,530 i6 goes from left to right. 239 00:13:42,530 --> 00:13:44,960 So when we do the loop currents, we need to take 240 00:13:44,960 --> 00:13:47,650 positive as the direction from left to right. 241 00:13:47,650 --> 00:13:52,150 Well that's the direction of IC and is the opposite 242 00:13:52,150 --> 00:13:55,300 direction of IB. 243 00:13:55,300 --> 00:13:59,220 So if I want to use loop currents to specify i6, it 244 00:13:59,220 --> 00:14:01,960 would be IC minus IB. 245 00:14:01,960 --> 00:14:05,220 Everybody clear on that? 246 00:14:05,220 --> 00:14:08,970 So equation (4) is the relation between element 247 00:14:08,970 --> 00:14:13,170 currents and loop currents. 248 00:14:13,170 --> 00:14:14,420 So what's equation (5)? 249 00:14:17,270 --> 00:14:19,830 Equation (5) is Ohm's Law for loop currents. 250 00:14:24,060 --> 00:14:27,750 Again, if I were thinking about Ohm's Law for R6, I 251 00:14:27,750 --> 00:14:31,020 would have v6 equals i6 R6. 252 00:14:31,020 --> 00:14:35,340 That's the way you say it in element voltages and currents. 253 00:14:35,340 --> 00:14:41,590 Over here, the current through R6 is IC minus IB. 254 00:14:41,590 --> 00:14:43,940 So Ohm's Law looks a little bit more complicated. 255 00:14:43,940 --> 00:14:50,530 So the point is these three methods represent ways of 256 00:14:50,530 --> 00:14:56,310 figuring out a linearly independent set of unknowns 257 00:14:56,310 --> 00:14:59,110 and equations. 258 00:14:59,110 --> 00:15:00,960 They differ. 259 00:15:00,960 --> 00:15:03,870 The left hand one is probably the easiest to think about, 260 00:15:03,870 --> 00:15:05,110 especially when you're thinking about 261 00:15:05,110 --> 00:15:06,770 things like Ohm's Law. 262 00:15:06,770 --> 00:15:10,520 It's the natural way to specify the element 263 00:15:10,520 --> 00:15:12,020 relationships. 264 00:15:12,020 --> 00:15:13,920 It's the relation between the voltage and current 265 00:15:13,920 --> 00:15:17,180 through that part. 266 00:15:17,180 --> 00:15:19,350 That generally gives me a large number of 267 00:15:19,350 --> 00:15:20,600 equations and unknowns. 268 00:15:23,160 --> 00:15:27,550 You can reduce the number of unknowns by using something 269 00:15:27,550 --> 00:15:30,710 like node voltages or loop currents. 270 00:15:30,710 --> 00:15:33,980 And that gives you fewer equations to solve. 271 00:15:33,980 --> 00:15:36,010 They are completely equivalent. 272 00:15:36,010 --> 00:15:38,170 They look a little different. 273 00:15:38,170 --> 00:15:41,490 And the reason for talking about them is that when we 274 00:15:41,490 --> 00:15:44,220 think about writing a program to solve circuits 275 00:15:44,220 --> 00:15:47,350 automatically, which by the way will be the exercise in 276 00:15:47,350 --> 00:15:48,670 software lab this week. 277 00:15:51,460 --> 00:15:54,970 When we think about writing a program, writing a program is 278 00:15:54,970 --> 00:15:56,620 yet a different kind of challenge. 279 00:15:56,620 --> 00:16:01,040 What's the easiest system to automate? 280 00:16:01,040 --> 00:16:06,040 So the system that is easiest for you may or may not be the 281 00:16:06,040 --> 00:16:07,750 easiest system to automate. 282 00:16:07,750 --> 00:16:10,580 So that's the point of this week's software lab. 283 00:16:10,580 --> 00:16:14,120 We'll do a method that's closely related to the node 284 00:16:14,120 --> 00:16:15,320 voltage method. 285 00:16:15,320 --> 00:16:18,590 It's not quite node voltages. 286 00:16:18,590 --> 00:16:22,340 It's a little simpler than node voltages for a computer. 287 00:16:22,340 --> 00:16:24,300 It's a little simpler to automate. 288 00:16:24,300 --> 00:16:28,410 So we use a method that's called node voltages with 289 00:16:28,410 --> 00:16:29,660 component currents. 290 00:16:35,250 --> 00:16:36,680 OK, now I'm going to start new stuff. 291 00:16:36,680 --> 00:16:37,930 That was review. 292 00:16:40,440 --> 00:16:43,450 What I want to think about today is what is it about 293 00:16:43,450 --> 00:16:47,080 circuit design that makes it hard. 294 00:16:47,080 --> 00:16:52,040 What are the issues that make it difficult to be modular 295 00:16:52,040 --> 00:16:53,870 when we're thinking about the analysis 296 00:16:53,870 --> 00:16:55,540 and design of circuits. 297 00:16:55,540 --> 00:16:59,720 And one of the hardest things to deal with is the idea that 298 00:16:59,720 --> 00:17:04,819 in a circuit, unlike in a linear time invariant system 299 00:17:04,819 --> 00:17:08,680 of the type we talked about in the previous module, in a 300 00:17:08,680 --> 00:17:13,420 circuit the presence of every element affects the currents 301 00:17:13,420 --> 00:17:14,780 and voltages through, in 302 00:17:14,780 --> 00:17:18,450 principle, every other element. 303 00:17:18,450 --> 00:17:25,280 So if you change one thing, you change everything. 304 00:17:25,280 --> 00:17:29,700 So first off, I want to just give an example of that. 305 00:17:29,700 --> 00:17:34,190 Think about what would happen if I were trying to make a 306 00:17:34,190 --> 00:17:39,540 circuit to control the brightness of a light bulb. 307 00:17:39,540 --> 00:17:42,130 So I imagine that I've got this circuit 308 00:17:42,130 --> 00:17:44,490 and I close the switch. 309 00:17:44,490 --> 00:17:45,560 Closing the switch is 310 00:17:45,560 --> 00:17:48,280 equivalent to adding a component. 311 00:17:48,280 --> 00:17:50,910 So before I close the switch, when the switch was open, I 312 00:17:50,910 --> 00:17:53,760 have three elements, a voltage source and two resistors. 313 00:17:53,760 --> 00:17:56,520 After I close the switch, I have four elements, the 314 00:17:56,520 --> 00:17:58,380 original three plus the bulb. 315 00:17:58,380 --> 00:18:01,480 So the question is how would closing the switch 316 00:18:01,480 --> 00:18:02,730 affect V0 and I0. 317 00:18:06,290 --> 00:18:06,970 Take a minute. 318 00:18:06,970 --> 00:18:07,870 Talk to your neighbor. 319 00:18:07,870 --> 00:18:10,655 Figure out how V0 and I0 change. 320 00:20:24,040 --> 00:20:25,600 So what's the answer? 321 00:20:25,600 --> 00:20:27,920 Everybody raise your hand, show a number of fingers equal 322 00:20:27,920 --> 00:20:30,420 to the answer. 323 00:20:30,420 --> 00:20:31,060 That's very good. 324 00:20:31,060 --> 00:20:34,540 It's 95% correct at least, maybe 100. 325 00:20:34,540 --> 00:20:36,070 OK, so the answer is (2). 326 00:20:36,070 --> 00:20:36,690 Why is the answer (2)? 327 00:20:36,690 --> 00:20:39,775 How do I figure that out? 328 00:20:39,775 --> 00:20:43,170 AUDIENCE: So the total resistance of the circuit is 329 00:20:43,170 --> 00:20:46,570 going to decrease because you're adding it in parallel. 330 00:20:46,570 --> 00:20:53,790 And then V0 is going to decrease because it's going to 331 00:20:53,790 --> 00:20:56,225 have a lower equivalent resistance. 332 00:20:56,225 --> 00:21:00,610 And because that decreases, you have I0 to-- 333 00:21:00,610 --> 00:21:03,600 PROFESSOR: Yes, I think that's exactly correct. 334 00:21:03,600 --> 00:21:06,970 So the idea was that when you add a component-- 335 00:21:06,970 --> 00:21:11,020 Let's think about the light bulb being a resistor. 336 00:21:11,020 --> 00:21:13,660 That's kind of pulled out of thin air, but I sort of 337 00:21:13,660 --> 00:21:16,120 suggested that you might do that here. 338 00:21:16,120 --> 00:21:18,810 Think about representing the light bulb as a resistor. 339 00:21:18,810 --> 00:21:21,205 Then when you close the switch, these two resistors go 340 00:21:21,205 --> 00:21:22,720 in parallel. 341 00:21:22,720 --> 00:21:26,490 When you combine two things in parallel, the result is the 342 00:21:26,490 --> 00:21:31,520 same as a resistance that has a smaller value. 343 00:21:31,520 --> 00:21:34,060 And then think about how that smaller value would interact 344 00:21:34,060 --> 00:21:36,600 with this resistor and that source. 345 00:21:36,600 --> 00:21:41,570 And you can sort of figure out that the presence of this bulb 346 00:21:41,570 --> 00:21:45,780 would reduce this voltage and increase that current. 347 00:21:45,780 --> 00:21:51,880 That's kind of a high level of reasoning given where we are. 348 00:21:51,880 --> 00:21:54,780 If you wanted to think through this a little more step by 349 00:21:54,780 --> 00:21:58,150 step, it's easy. 350 00:21:58,150 --> 00:22:00,940 You could think about figuring out what are the voltages and 351 00:22:00,940 --> 00:22:03,145 currents before and after you close the switch. 352 00:22:06,320 --> 00:22:09,830 Before you close the switch, you can just ignore this, and 353 00:22:09,830 --> 00:22:12,480 you can calculate V0 just from a voltage divider 354 00:22:12,480 --> 00:22:14,440 relationship. 355 00:22:14,440 --> 00:22:17,060 Right, that's clear? 356 00:22:17,060 --> 00:22:20,890 So you can see that when the switch is open, the voltage, 357 00:22:20,890 --> 00:22:22,155 V0, is going to be 8 Volts. 358 00:22:25,600 --> 00:22:29,550 And when the switch is open, you can figure out I0 by 359 00:22:29,550 --> 00:22:31,370 lumping these two resistors together to 360 00:22:31,370 --> 00:22:32,690 make a 3 Ohm resistor. 361 00:22:32,690 --> 00:22:34,250 And you see that I0 is 4 Amps. 362 00:22:37,360 --> 00:22:39,230 Then to figure out what happens when you close the 363 00:22:39,230 --> 00:22:40,480 switch, just repeat. 364 00:22:43,730 --> 00:22:47,140 And the algebra is a little more tedious. 365 00:22:47,140 --> 00:22:49,130 I won't try to go through it. 366 00:22:49,130 --> 00:22:51,800 By the way, the answers, the slides that I show in lecture 367 00:22:51,800 --> 00:22:54,430 are always posted on the web. 368 00:22:54,430 --> 00:22:58,300 So these slides that have the answers, they are there. 369 00:22:58,300 --> 00:23:00,820 On the web there are two handouts per lecture. 370 00:23:00,820 --> 00:23:03,290 There's an electronic version of the thing 371 00:23:03,290 --> 00:23:04,170 that we handed out. 372 00:23:04,170 --> 00:23:06,240 There's also an electronic version of my slides. 373 00:23:06,240 --> 00:23:08,880 So everything that I showed is there. 374 00:23:08,880 --> 00:23:10,580 I'm not going to go through the tedious algebra. 375 00:23:10,580 --> 00:23:12,820 It's just tedious algebra. 376 00:23:12,820 --> 00:23:15,630 But if you go through the tedious algebra, you find that 377 00:23:15,630 --> 00:23:20,240 if you represent this bulb by resistor R, V0 becomes an 378 00:23:20,240 --> 00:23:21,555 expression that looks like this. 379 00:23:24,340 --> 00:23:26,770 If you think about resistors, physical resistors, if you 380 00:23:26,770 --> 00:23:29,340 think about light bulbs being represented by a physical 381 00:23:29,340 --> 00:23:31,470 resistor, then the physical resistor has to have a 382 00:23:31,470 --> 00:23:34,340 resistance between 0 and infinity. 383 00:23:34,340 --> 00:23:36,730 And if you think about that expression, what could the 384 00:23:36,730 --> 00:23:40,020 value be if R varied between 0 and infinity? 385 00:23:40,020 --> 00:23:44,460 That expression is always less than or equal to 8 Volts, 386 00:23:44,460 --> 00:23:49,350 showing that V0 went down when you closed the switch. 387 00:23:49,350 --> 00:23:52,620 Similar tedious algebra leads to an expression like this in 388 00:23:52,620 --> 00:23:56,640 R. And if you think about how this would change as R goes 389 00:23:56,640 --> 00:23:58,930 from 0 to infinity, you see that that's always 390 00:23:58,930 --> 00:24:00,180 bigger than 4 Amps. 391 00:24:02,260 --> 00:24:04,020 So that means I0 goes up. 392 00:24:04,020 --> 00:24:06,480 So the point is you can think through this in a more 393 00:24:06,480 --> 00:24:07,580 sophisticated way. 394 00:24:07,580 --> 00:24:09,580 And we hope by the end of the course you'll all 395 00:24:09,580 --> 00:24:10,840 be able to do that. 396 00:24:10,840 --> 00:24:13,140 Or you can think through it in terms of 397 00:24:13,140 --> 00:24:14,450 just solving the circuit. 398 00:24:14,450 --> 00:24:17,516 Solve it in two cases when the bulb is there and when the 399 00:24:17,516 --> 00:24:18,690 bulb is not. 400 00:24:18,690 --> 00:24:20,710 Either way, the answer is (2). 401 00:24:20,710 --> 00:24:23,430 The V0 went down, and I0 went up. 402 00:24:23,430 --> 00:24:31,050 The point is that when I added the element, currents that 403 00:24:31,050 --> 00:24:35,150 were far away from the element still changed. 404 00:24:35,150 --> 00:24:38,460 And that's a general way circuits interact. 405 00:24:38,460 --> 00:24:40,530 They're all connected. 406 00:24:40,530 --> 00:24:43,180 So the idea, the point of doing this is that the 407 00:24:43,180 --> 00:24:45,460 addition of a new element changed the voltages and 408 00:24:45,460 --> 00:24:49,320 currents through the other element. 409 00:24:49,320 --> 00:24:52,290 That's a drag if what I was trying to do, for example, was 410 00:24:52,290 --> 00:24:58,720 design a brightness controller for the flashlight bulb. 411 00:24:58,720 --> 00:25:01,680 Imagine that what I really wanted to do was use a voltage 412 00:25:01,680 --> 00:25:04,180 divider to make 8 Volts. 413 00:25:04,180 --> 00:25:07,160 And what was in my head was I'd like that 8 Volts to be 414 00:25:07,160 --> 00:25:11,170 across the light bulb. 415 00:25:11,170 --> 00:25:12,160 That's a good idea. 416 00:25:12,160 --> 00:25:13,940 It just won't work. 417 00:25:13,940 --> 00:25:16,410 At least It won't work the way this circuit worked. 418 00:25:16,410 --> 00:25:19,600 If I just build it like so, there's an interaction between 419 00:25:19,600 --> 00:25:22,080 the bulb and the voltage divider circuits, so the 420 00:25:22,080 --> 00:25:25,760 voltage divider circuit is no longer a voltage divider. 421 00:25:25,760 --> 00:25:30,610 After, in this circuit, when I close the switch, current 422 00:25:30,610 --> 00:25:33,870 flows in this wire. 423 00:25:33,870 --> 00:25:36,650 The rule in a voltage divider is the same current has to 424 00:25:36,650 --> 00:25:39,050 flow through both resistors. 425 00:25:39,050 --> 00:25:41,350 If the same current flows through two resistors, then 426 00:25:41,350 --> 00:25:43,380 you can use the voltage divider relationship to see 427 00:25:43,380 --> 00:25:46,260 how voltage partitions between the two resistors. 428 00:25:46,260 --> 00:25:48,950 If current gets siphoned off the node between the two 429 00:25:48,950 --> 00:25:50,370 resistors, you can't use the voltage 430 00:25:50,370 --> 00:25:52,020 divider relation anymore. 431 00:25:52,020 --> 00:25:56,720 That violates the premise of the voltage divider relation. 432 00:25:56,720 --> 00:25:59,350 So when I close the switch, this is no longer a voltage 433 00:25:59,350 --> 00:26:03,840 divider, and it no longer works like a voltage divider. 434 00:26:03,840 --> 00:26:04,860 Is that clear? 435 00:26:04,860 --> 00:26:07,980 So what I'd really like is some magic circuit that I can 436 00:26:07,980 --> 00:26:12,200 put here that isolates the effect of the bulb on the 437 00:26:12,200 --> 00:26:13,790 effect of the rest of the circuit. 438 00:26:13,790 --> 00:26:16,080 And that's exactly what an op-amp does. 439 00:26:16,080 --> 00:26:18,850 That's what we're going to talk about next. 440 00:26:18,850 --> 00:26:20,370 So this magic circuit is something that 441 00:26:20,370 --> 00:26:22,750 we'll call a buffer. 442 00:26:22,750 --> 00:26:25,300 A buffer is a thing that isolates the 443 00:26:25,300 --> 00:26:27,360 left from the right. 444 00:26:27,360 --> 00:26:30,200 So the buffer is going to be something that measures the 445 00:26:30,200 --> 00:26:33,240 voltage on this side and magically generates that 446 00:26:33,240 --> 00:26:38,200 voltage over here without changing this side. 447 00:26:38,200 --> 00:26:40,490 So what we want to do now is develop some thought tools for 448 00:26:40,490 --> 00:26:43,600 how you think about op-amps. 449 00:26:43,600 --> 00:26:45,910 Op-amps are different. 450 00:26:45,910 --> 00:26:48,120 And if you just look at the picture, op-amp has to be 451 00:26:48,120 --> 00:26:49,410 different because there's too many terminals. 452 00:26:52,290 --> 00:26:54,170 It's not like a resistor that has two legs. 453 00:26:54,170 --> 00:26:55,890 It's not like a V source which has two legs. 454 00:26:55,890 --> 00:26:57,590 It's not like an I source which has two legs. 455 00:26:57,590 --> 00:26:59,890 It has three legs. 456 00:26:59,890 --> 00:27:02,230 In fact, I haven't drawn all the legs. 457 00:27:02,230 --> 00:27:03,670 There's more than three. 458 00:27:03,670 --> 00:27:06,750 There's at least five. 459 00:27:06,750 --> 00:27:08,570 So they're different. 460 00:27:08,570 --> 00:27:10,490 And the way we think about them are different. 461 00:27:10,490 --> 00:27:13,520 The key to thinking about the way an op-amp works is to 462 00:27:13,520 --> 00:27:15,390 think about a new class of elements 463 00:27:15,390 --> 00:27:18,830 called controlled elements. 464 00:27:18,830 --> 00:27:22,340 So a controlled element is an element whose voltage current 465 00:27:22,340 --> 00:27:27,380 relationship depends somehow on a voltage and current 466 00:27:27,380 --> 00:27:30,140 measured someplace else in the circuit. 467 00:27:30,140 --> 00:27:32,470 As an example, think about a current 468 00:27:32,470 --> 00:27:35,280 controlled current source. 469 00:27:35,280 --> 00:27:38,210 That's depicted here. 470 00:27:38,210 --> 00:27:40,330 I would normally write a current 471 00:27:40,330 --> 00:27:41,510 source that was a circle. 472 00:27:41,510 --> 00:27:43,140 That means it's an independent current source. 473 00:27:43,140 --> 00:27:47,030 That means that the current is fixed. 474 00:27:47,030 --> 00:27:50,680 The little diamond thing is my way of representing the idea 475 00:27:50,680 --> 00:27:54,980 that the amount of current that comes out of this current 476 00:27:54,980 --> 00:27:59,360 source depends on something else. 477 00:27:59,360 --> 00:28:00,970 In this case it depends on IB. 478 00:28:00,970 --> 00:28:05,060 And IB happens to be a current that flows in that circuit. 479 00:28:05,060 --> 00:28:08,420 So the idea is that the current in the current source 480 00:28:08,420 --> 00:28:11,320 depends on some other current. 481 00:28:11,320 --> 00:28:13,360 It's a current controlled current source. 482 00:28:13,360 --> 00:28:17,010 It's a current source whose current is controlled by 483 00:28:17,010 --> 00:28:18,271 another current. 484 00:28:18,271 --> 00:28:19,521 Got it? 485 00:28:21,960 --> 00:28:24,070 So we'll see. 486 00:28:24,070 --> 00:28:27,240 Figure out, for this current controlled current source 487 00:28:27,240 --> 00:28:31,200 circuit, what's the ratio of Vout over Vin? 488 00:30:40,490 --> 00:30:41,680 So what's the answer? 489 00:30:41,680 --> 00:30:43,223 What's the ratio of Vout over Vin? 490 00:30:48,646 --> 00:30:50,125 100%, wonderful. 491 00:30:50,125 --> 00:30:52,590 The answer is (4). 492 00:30:52,590 --> 00:30:54,562 Easy to get. 493 00:30:54,562 --> 00:30:56,560 It's easy to get because I rigged this 494 00:30:56,560 --> 00:30:57,730 question to be easy. 495 00:30:57,730 --> 00:31:00,370 I rigged it to be easy because you can sort of figure out 496 00:31:00,370 --> 00:31:01,760 everything that's going on on the left. 497 00:31:01,760 --> 00:31:03,050 And then you can figure out everything that's 498 00:31:03,050 --> 00:31:03,860 going on on the right. 499 00:31:03,860 --> 00:31:05,860 And so there's no sort of complicated 500 00:31:05,860 --> 00:31:07,670 coupling between the two. 501 00:31:07,670 --> 00:31:09,350 So what's going on on the left, well, you 502 00:31:09,350 --> 00:31:11,160 can solve for IB. 503 00:31:11,160 --> 00:31:16,110 IB is just Vi over 1,000 Ohms. 504 00:31:16,110 --> 00:31:19,090 Then you can take that value of IB and use that to solve 505 00:31:19,090 --> 00:31:22,700 for what's going on over here. 506 00:31:22,700 --> 00:31:27,380 Over here the out is this current, 100 IB times this 507 00:31:27,380 --> 00:31:29,910 resistance, 5 Ohms. 508 00:31:29,910 --> 00:31:32,790 But we just found out that IB is VI over 1,000 Ohms. 509 00:31:32,790 --> 00:31:36,120 Substitute it in, you get half Vi. 510 00:31:36,120 --> 00:31:38,290 So the answer is number (4). 511 00:31:38,290 --> 00:31:43,450 The ratio of Vout to Vin is a half. 512 00:31:43,450 --> 00:31:47,280 So the point is these are a different kind of element, 513 00:31:47,280 --> 00:31:51,520 controlled sources, dependent sources. 514 00:31:51,520 --> 00:31:55,790 But they're not too hard to work with. 515 00:31:55,790 --> 00:32:00,350 They sort of look different structurally. 516 00:32:00,350 --> 00:32:02,370 So think about what's going on here. 517 00:32:02,370 --> 00:32:06,040 The controlled current source, the current controlled current 518 00:32:06,040 --> 00:32:10,130 source, I have to think of that as a box, because the 519 00:32:10,130 --> 00:32:12,250 current source has to know the value of IB. 520 00:32:12,250 --> 00:32:15,770 So somehow this box, this thing that's doing this 521 00:32:15,770 --> 00:32:20,180 current controlled current source, it knows about IB, and 522 00:32:20,180 --> 00:32:22,830 it knows about the current source. 523 00:32:22,830 --> 00:32:24,020 So they're somehow linked. 524 00:32:24,020 --> 00:32:27,060 So that's why I put a box around it. 525 00:32:27,060 --> 00:32:30,530 And then think about the equations that characterize 526 00:32:30,530 --> 00:32:31,420 that component. 527 00:32:31,420 --> 00:32:33,340 So now we've got a component that's got four wires 528 00:32:33,340 --> 00:32:36,690 coming out of it. 529 00:32:36,690 --> 00:32:39,000 The components we had before had two wires coming out of 530 00:32:39,000 --> 00:32:43,590 them, resistors, current sources, voltage sources. 531 00:32:43,590 --> 00:32:46,440 This kind of a component has four wires coming out of them. 532 00:32:46,440 --> 00:32:50,040 We call this kind of a component a two-port because 533 00:32:50,040 --> 00:32:54,110 we think of the left port and the right port. 534 00:32:54,110 --> 00:32:55,535 That's compared to resistor, which we 535 00:32:55,535 --> 00:32:56,785 would call a one-port. 536 00:32:59,570 --> 00:33:02,870 We think about this being a two-port. 537 00:33:02,870 --> 00:33:07,060 And there's now two equations. 538 00:33:07,060 --> 00:33:09,400 Now I've got, for that two-port, I've got two 539 00:33:09,400 --> 00:33:11,550 voltages and two currents. 540 00:33:11,550 --> 00:33:13,280 There's the voltage across the left part. 541 00:33:13,280 --> 00:33:14,840 And there's the voltage across the right part. 542 00:33:14,840 --> 00:33:16,790 And there's current through the left part. 543 00:33:16,790 --> 00:33:18,890 And there's the current through the right part. 544 00:33:18,890 --> 00:33:21,960 So with the elements we've thought about before, I had 545 00:33:21,960 --> 00:33:24,390 one voltage across it and one current through it. 546 00:33:24,390 --> 00:33:26,990 Now I've got two voltages across it and two currents. 547 00:33:26,990 --> 00:33:30,050 It's kind of twice as big. 548 00:33:30,050 --> 00:33:32,300 Not surprisingly, it takes twice as many equations. 549 00:33:34,890 --> 00:33:36,700 Ohm's Law was a single equation for 550 00:33:36,700 --> 00:33:40,130 one resistor, a one-port. 551 00:33:40,130 --> 00:33:42,440 V equals V0 was one equation for one 552 00:33:42,440 --> 00:33:44,600 component, a voltage source. 553 00:33:44,600 --> 00:33:47,320 Here, I've got one component that has 554 00:33:47,320 --> 00:33:51,590 four wires, four unknowns. 555 00:33:51,590 --> 00:33:54,170 And I get two equations. 556 00:33:54,170 --> 00:33:57,420 So for this particular dependent source, I know that 557 00:33:57,420 --> 00:34:00,340 the voltage across this pair of terminals is 0, because 558 00:34:00,340 --> 00:34:02,385 it's essentially a short circuit between the two. 559 00:34:05,240 --> 00:34:08,429 Short circuit just means connected with a wire. 560 00:34:08,429 --> 00:34:13,780 And I know that the current I2 is related to the current over 561 00:34:13,780 --> 00:34:16,380 there this way. 562 00:34:16,380 --> 00:34:19,250 The idea then is that this current controlled current 563 00:34:19,250 --> 00:34:24,130 source can be represented by a two-port, two voltages, two 564 00:34:24,130 --> 00:34:28,320 currents related by two equations. 565 00:34:28,320 --> 00:34:32,860 It's kind of structurally twice as difficult to think 566 00:34:32,860 --> 00:34:36,290 about as a one-port. 567 00:34:36,290 --> 00:34:41,230 Functionally, it's different from having two one-ports, 568 00:34:41,230 --> 00:34:44,030 because the two one-ports are coupled. 569 00:34:44,030 --> 00:34:46,350 And that's the important part is the coupling between the 570 00:34:46,350 --> 00:34:50,980 two that you can't model with a simple resistor and a simple 571 00:34:50,980 --> 00:34:54,449 constant source. 572 00:34:54,449 --> 00:34:58,570 So when we think about an op-amp, a good first model for 573 00:34:58,570 --> 00:35:01,130 an op-amp is to think about it as a voltage controlled 574 00:35:01,130 --> 00:35:02,380 voltage source. 575 00:35:04,670 --> 00:35:05,940 And so that's depicted here. 576 00:35:05,940 --> 00:35:08,550 I want to think about the op-amp, which I'll 577 00:35:08,550 --> 00:35:10,000 symbolically write this way. 578 00:35:10,000 --> 00:35:12,840 This means something that has two inputs, a plus input and 579 00:35:12,840 --> 00:35:17,220 then a minus input, and a single output, Vout. 580 00:35:17,220 --> 00:35:18,670 I can think about it as a voltage 581 00:35:18,670 --> 00:35:20,460 controlled voltage source. 582 00:35:20,460 --> 00:35:23,380 This I mean to be the element representation. 583 00:35:23,380 --> 00:35:25,380 This is how I'll draw it when I make a schematic 584 00:35:25,380 --> 00:35:27,530 diagram of a circuit. 585 00:35:27,530 --> 00:35:29,140 This is the functional form. 586 00:35:29,140 --> 00:35:30,370 This is the way I'll think about it when 587 00:35:30,370 --> 00:35:32,250 I'm analyzing it. 588 00:35:32,250 --> 00:35:35,750 I'll think about the op-amp as being a voltage controlled 589 00:35:35,750 --> 00:35:37,100 voltage source. 590 00:35:37,100 --> 00:35:42,130 This voltage source adopts a voltage that is some number K 591 00:35:42,130 --> 00:35:44,220 times the difference voltage Vplus minus Vminus. 592 00:35:46,870 --> 00:35:49,920 And the trick in op-amps is that K is typically a very big 593 00:35:49,920 --> 00:35:54,120 number, typically bigger than 10 to the 5th. 594 00:35:54,120 --> 00:35:55,970 We'll see in a minute why that's a 595 00:35:55,970 --> 00:36:02,480 frightfully useful component. 596 00:36:02,480 --> 00:36:04,950 Let's just walk through an example to see how you would 597 00:36:04,950 --> 00:36:08,180 solve a circuit that has this kind of a voltage controlled 598 00:36:08,180 --> 00:36:10,680 voltage source in it. 599 00:36:10,680 --> 00:36:13,850 So think about this circuit where I'm applying a voltage 600 00:36:13,850 --> 00:36:19,440 to the plus lead of an op-amp, and I'm wrapping the output 601 00:36:19,440 --> 00:36:24,850 back through the minus lead, through two resistors. 602 00:36:24,850 --> 00:36:26,920 And what I want to do is analyze that circuit by 603 00:36:26,920 --> 00:36:28,600 thinking about the op-amp as a voltage 604 00:36:28,600 --> 00:36:31,045 controlled voltage source. 605 00:36:34,300 --> 00:36:37,580 So I can see just by the way it's wired up that the voltage 606 00:36:37,580 --> 00:36:38,670 at the plus lead-- 607 00:36:38,670 --> 00:36:40,170 So I'm going to be thinking node voltages. 608 00:36:42,720 --> 00:36:44,630 Node voltages tend to be easy. 609 00:36:44,630 --> 00:36:46,070 I'm thinking node voltages. 610 00:36:46,070 --> 00:36:47,690 So I'm going to define all my voltages 611 00:36:47,690 --> 00:36:48,940 referenced to a ground. 612 00:36:51,550 --> 00:36:53,520 So I'll call this node ground. 613 00:36:53,520 --> 00:36:56,120 That's what the funny symbol means. 614 00:36:56,120 --> 00:37:00,790 Then this node voltage, the voltage at the plus terminal, 615 00:37:00,790 --> 00:37:02,340 is just the same as the input voltage. 616 00:37:05,890 --> 00:37:09,950 This voltage at the minus terminal looks like a voltage 617 00:37:09,950 --> 00:37:11,000 divider relationship. 618 00:37:11,000 --> 00:37:16,790 If you look at my model, it's very clear that I1 is 0, 619 00:37:16,790 --> 00:37:19,290 because there's no connection here between the 620 00:37:19,290 --> 00:37:20,980 Vplus and the Vminus. 621 00:37:20,980 --> 00:37:22,160 So I1 is 0. 622 00:37:22,160 --> 00:37:25,620 That means the total current that flows into the plus lead 623 00:37:25,620 --> 00:37:27,870 of the op-amp is 0. 624 00:37:27,870 --> 00:37:30,660 The total current that flows into the minus lead of the 625 00:37:30,660 --> 00:37:33,130 op-amp is 0. 626 00:37:33,130 --> 00:37:36,290 So because there's no current flowing in the plus and minus 627 00:37:36,290 --> 00:37:40,560 leads, I can calculate the voltage at this minus terminal 628 00:37:40,560 --> 00:37:42,440 as a voltage divider. 629 00:37:42,440 --> 00:37:46,427 And it's just R1 over the sum of R1 and R2 times V0. 630 00:37:52,500 --> 00:37:55,250 Then according to the voltage control voltage source model, 631 00:37:55,250 --> 00:37:58,240 Vout should be K times the difference 632 00:37:58,240 --> 00:38:00,960 between Vplus and Vminus. 633 00:38:03,830 --> 00:38:09,420 So I just substitute in for Vplus -- this guy, Vi, and for 634 00:38:09,420 --> 00:38:11,450 Vminus -- this guy. 635 00:38:11,450 --> 00:38:12,700 Do some algebra. 636 00:38:16,730 --> 00:38:19,510 I get this expression after some algebra. 637 00:38:19,510 --> 00:38:22,450 And then I say, yeah but I know that K's really big. 638 00:38:22,450 --> 00:38:28,800 So if K's really big, KR1 is big compared to R1. 639 00:38:28,800 --> 00:38:30,000 So I can ignore that. 640 00:38:30,000 --> 00:38:33,450 In fact, K is so big that for any reasonable choice of R1 641 00:38:33,450 --> 00:38:36,770 and R2, KR1 is even bigger than R2. 642 00:38:36,770 --> 00:38:40,030 So that means this reduces to this kind of a fraction. 643 00:38:40,030 --> 00:38:45,030 So the response of this circuit, this op-amp circuit-- 644 00:38:45,030 --> 00:38:45,640 So what did I do? 645 00:38:45,640 --> 00:38:48,080 I just took the op-amp circuit, and I modelled it as 646 00:38:48,080 --> 00:38:51,160 a voltage controlled voltage source, plugged through the 647 00:38:51,160 --> 00:38:54,210 equations, and found out that the ratio of the output 648 00:38:54,210 --> 00:38:57,190 voltage to the input voltage is R1 over R1 plus 649 00:38:57,190 --> 00:38:58,440 R2 divided by R1. 650 00:39:02,330 --> 00:39:05,150 So the idea then is that this simple circuit 651 00:39:05,150 --> 00:39:08,830 works like an amplifier. 652 00:39:08,830 --> 00:39:11,740 It's an amplifier in the sense that I can make the output 653 00:39:11,740 --> 00:39:15,480 voltage bigger than the input voltage. 654 00:39:15,480 --> 00:39:16,690 That's a very useful thing. 655 00:39:16,690 --> 00:39:20,430 In fact you'll find useful ways to use that when you do 656 00:39:20,430 --> 00:39:21,790 the design lab this week. 657 00:39:26,040 --> 00:39:29,030 So this as an amplifier. 658 00:39:29,030 --> 00:39:29,820 So here's a question. 659 00:39:29,820 --> 00:39:32,490 Make sure you follow what I just did. 660 00:39:32,490 --> 00:39:35,420 How could I choose the components R1 and R2 so that I 661 00:39:35,420 --> 00:39:36,827 make Vout equal to Vi? 662 00:40:58,832 --> 00:41:01,870 AUDIENCE: Why is this 0? 663 00:41:01,870 --> 00:41:03,120 PROFESSOR: There's no wire connecting. 664 00:41:05,565 --> 00:41:06,010 AUDIENCE: Oh. 665 00:41:06,010 --> 00:41:08,210 PROFESSOR: So there's nowhere for current to go. 666 00:41:23,910 --> 00:41:28,340 OK so how would I choose the components R1 and R2 in order 667 00:41:28,340 --> 00:41:31,200 to make the output voltage equal to the input voltage? 668 00:41:37,750 --> 00:41:38,970 Wonderful. 669 00:41:38,970 --> 00:41:42,290 So the idea is that all of these manipulations have the 670 00:41:42,290 --> 00:41:43,200 same effect. 671 00:41:43,200 --> 00:41:44,520 All you need to do is look at the 672 00:41:44,520 --> 00:41:46,520 expression that we developed. 673 00:41:46,520 --> 00:41:50,680 The expression was R1 plus R2 over R1. 674 00:41:50,680 --> 00:41:55,130 If you substitute, R1 goes to infinity, R2 equals 0, or the 675 00:41:55,130 --> 00:41:59,240 two at the same time, you get one in all of those cases. 676 00:41:59,240 --> 00:42:04,120 So that's a way that you can turn this amplifier circuit, 677 00:42:04,120 --> 00:42:07,960 that in general would make the output bigger than the input, 678 00:42:07,960 --> 00:42:10,650 into something that makes the output equal to the input. 679 00:42:15,260 --> 00:42:17,100 Now I want to turn to a simplification. 680 00:42:17,100 --> 00:42:20,840 I just dragged you through the math. 681 00:42:20,840 --> 00:42:23,400 Thinking about the op-amp as a voltage controlled voltage 682 00:42:23,400 --> 00:42:26,360 source, there's actually a shortcut. 683 00:42:26,360 --> 00:42:30,230 The shortcut is something we call the ideal op-amp. 684 00:42:30,230 --> 00:42:32,600 So what I want to do in this slide is drag you through the 685 00:42:32,600 --> 00:42:36,060 math one more time, but then we're done. 686 00:42:36,060 --> 00:42:43,150 The idea is that if you have an op-amp, a voltage 687 00:42:43,150 --> 00:42:46,510 controlled voltage source, if you represent an op-amp as a 688 00:42:46,510 --> 00:42:50,090 voltage controlled voltage source, and if K is very big, 689 00:42:50,090 --> 00:42:54,540 the effect will be to make the difference between the 690 00:42:54,540 --> 00:42:56,290 positive and negative terminals of the 691 00:42:56,290 --> 00:42:58,260 op-amp quite small. 692 00:42:58,260 --> 00:43:02,890 You can see that here by way of a simple example. 693 00:43:02,890 --> 00:43:08,180 So let me think about this case with, again, the voltage 694 00:43:08,180 --> 00:43:10,480 controlled voltage source model. 695 00:43:10,480 --> 00:43:13,430 So here I've got Vout. 696 00:43:13,430 --> 00:43:16,320 According to the voltage controlled voltage source 697 00:43:16,320 --> 00:43:18,730 model, Vout should be K times the difference 698 00:43:18,730 --> 00:43:19,980 between the two inputs. 699 00:43:22,770 --> 00:43:25,170 The positive input is clearly Vi. 700 00:43:25,170 --> 00:43:28,440 The negative input is Vo. 701 00:43:28,440 --> 00:43:31,710 One equation, two unknowns, solve for the ratio. 702 00:43:31,710 --> 00:43:34,190 The ratio of Vout to Vi is K over (1 plus K). 703 00:43:37,420 --> 00:43:38,680 That's the answer. 704 00:43:38,680 --> 00:43:42,540 Take that answer and back substitute to figure out how 705 00:43:42,540 --> 00:43:44,910 big was the difference between Vplus and Vminus. 706 00:43:47,810 --> 00:43:49,970 That's just algebra. 707 00:43:49,970 --> 00:43:54,660 And what you see is that if this is the answer, then Vplus 708 00:43:54,660 --> 00:44:00,410 minus Vminus can be written as a fraction of Vi or a similar 709 00:44:00,410 --> 00:44:02,630 fraction of Vo. 710 00:44:02,630 --> 00:44:03,302 K is big. 711 00:44:03,302 --> 00:44:05,780 K is essentially the same as K plus 1. 712 00:44:05,780 --> 00:44:08,750 K is in the denominator of both. 713 00:44:08,750 --> 00:44:11,350 What that says is the difference between the plus 714 00:44:11,350 --> 00:44:14,030 and the minus leads, the voltage between the plus and 715 00:44:14,030 --> 00:44:22,500 the minus leads, is very small if K is very large. 716 00:44:22,500 --> 00:44:24,230 OK? 717 00:44:24,230 --> 00:44:27,710 So we call that the ideal op-amp relationship. 718 00:44:27,710 --> 00:44:30,970 The utility of that is that it makes solving the op-amp 719 00:44:30,970 --> 00:44:33,480 circuits much, much easier than what 720 00:44:33,480 --> 00:44:35,350 we've just been doing. 721 00:44:35,350 --> 00:44:38,400 I've been solving the circuits by thinking about the op-amp 722 00:44:38,400 --> 00:44:41,640 as a voltage controlled voltage source. 723 00:44:41,640 --> 00:44:43,610 That's fine. 724 00:44:43,610 --> 00:44:46,700 But if I additionally know that K is very big, I can 725 00:44:46,700 --> 00:44:48,430 shortcut it. 726 00:44:48,430 --> 00:44:52,100 I can say, look, the effect of the op-amp is going to be to 727 00:44:52,100 --> 00:44:54,600 make the positive and negative inputs the same. 728 00:44:54,600 --> 00:44:57,120 If it didn't do that, think of what it would mean. 729 00:44:57,120 --> 00:45:00,460 If K is a big number, and if the output voltage is K times 730 00:45:00,460 --> 00:45:02,560 the difference, if the difference is anything other 731 00:45:02,560 --> 00:45:07,390 than epsilon, Vout has to be infinity. 732 00:45:07,390 --> 00:45:10,060 I mean if K is very big, right? 733 00:45:10,060 --> 00:45:15,160 If K is very big, the only way the output could be some 734 00:45:15,160 --> 00:45:19,530 reasonable number like a Volt would be if the difference 735 00:45:19,530 --> 00:45:22,150 between Vplus and Vminus is very small. 736 00:45:22,150 --> 00:45:24,490 OK, well let's work backwards. 737 00:45:24,490 --> 00:45:28,540 Let's start with the assumption that Vplus minus 738 00:45:28,540 --> 00:45:30,790 Vminus is very small. 739 00:45:30,790 --> 00:45:34,460 And that lets us solve the circuits very quickly. 740 00:45:34,460 --> 00:45:38,600 So for example, the same circuit that took, previously, 741 00:45:38,600 --> 00:45:43,080 a few lines to get the answer to, if I just take as a rule 742 00:45:43,080 --> 00:45:46,600 that Vplus has to be Vminus, it's a one step. 743 00:45:46,600 --> 00:45:51,130 Vplus equals Vminus, OK, Vi equals Vo, period, done. 744 00:45:51,130 --> 00:45:55,450 It's a very simple way to think about the answer to an 745 00:45:55,450 --> 00:45:57,600 op-amp circuit. 746 00:45:57,600 --> 00:46:00,760 So if the op-amp can be represented by a voltage 747 00:46:00,760 --> 00:46:04,140 controlled voltage source and if K is very large, then Vplus 748 00:46:04,140 --> 00:46:06,490 is roughly Vminus. 749 00:46:06,490 --> 00:46:10,480 Shortcut, ideal op-amp assumption. 750 00:46:10,480 --> 00:46:16,325 So, use that or ignore it depending on 751 00:46:16,325 --> 00:46:18,360 what your mood is. 752 00:46:18,360 --> 00:46:21,560 And figure out the voltage relationship for this slightly 753 00:46:21,560 --> 00:46:22,810 more complicated circuit. 754 00:48:30,540 --> 00:48:31,790 So what's the answer? 755 00:48:35,340 --> 00:48:36,360 Yes? 756 00:48:36,360 --> 00:48:38,400 No. 757 00:48:38,400 --> 00:48:40,080 How many are done? 758 00:48:40,080 --> 00:48:42,260 How many are not done? 759 00:48:42,260 --> 00:48:43,510 OK, take a minute. 760 00:48:46,290 --> 00:48:47,540 This is supposed to be easy. 761 00:48:49,610 --> 00:48:50,860 Think ideal op-amp. 762 00:50:46,680 --> 00:50:48,830 OK, what's the output? 763 00:50:48,830 --> 00:50:50,610 Everybody raise your hand, what's the output? 764 00:50:53,390 --> 00:50:56,980 More hands, more hands, more hands. 765 00:50:56,980 --> 00:51:01,860 OK, tiny number of hands, but those who showed hands are 766 00:51:01,860 --> 00:51:04,200 about 100% correct. 767 00:51:04,200 --> 00:51:06,460 I don't know how to grade that. 768 00:51:06,460 --> 00:51:08,760 Small participation, 100% among those who did 769 00:51:08,760 --> 00:51:09,760 participate. 770 00:51:09,760 --> 00:51:11,830 So how do I think about this? 771 00:51:11,830 --> 00:51:13,080 What's step (1)? 772 00:51:17,460 --> 00:51:19,950 All right, according to the theory of lectures, 773 00:51:19,950 --> 00:51:21,942 what is step (1)? 774 00:51:21,942 --> 00:51:23,436 AUDIENCE: Look at the previous slide. 775 00:51:23,436 --> 00:51:25,440 PROFESSOR: Look at the previous slide, yes. 776 00:51:25,440 --> 00:51:27,120 What was the previous slide? 777 00:51:27,120 --> 00:51:37,535 OK The previous slide had to do with? 778 00:51:37,535 --> 00:51:38,465 AUDIENCE: The ideal op-amps. 779 00:51:38,465 --> 00:51:39,395 PROFESSOR: Ideal op-amps. 780 00:51:39,395 --> 00:51:42,250 What's ideal op-amps say? 781 00:51:42,250 --> 00:51:44,070 Vplus equals Vminus. 782 00:51:44,070 --> 00:51:46,085 What happens here if Vplus is equal to Vminus? 783 00:51:50,940 --> 00:51:52,473 Well what's Vplus? 784 00:51:52,473 --> 00:51:53,400 AUDIENCE: 0. 785 00:51:53,400 --> 00:51:54,260 PROFESSOR: So what's Vminus? 786 00:51:54,260 --> 00:51:54,960 AUDIENCE: 0. 787 00:51:54,960 --> 00:51:56,390 PROFESSOR: 0. 788 00:51:56,390 --> 00:52:00,010 So if Vminus is 0, what do I do now? 789 00:52:09,453 --> 00:52:12,450 That bad, that hard huh? 790 00:52:12,450 --> 00:52:15,850 Well, I got a [UNINTELLIGIBLE] here in which three different 791 00:52:15,850 --> 00:52:16,710 currents can flow. 792 00:52:16,710 --> 00:52:19,570 How big is the current that can come into this node? 793 00:52:19,570 --> 00:52:21,630 What's the sum of all the currents that can 794 00:52:21,630 --> 00:52:24,060 come into that node? 795 00:52:24,060 --> 00:52:25,680 Well one could come in this way. 796 00:52:25,680 --> 00:52:27,750 One could come in that way. 797 00:52:27,750 --> 00:52:30,080 One could come in that way. 798 00:52:30,080 --> 00:52:34,024 How much current flows in the minus lead of the op-amp? 799 00:52:34,024 --> 00:52:34,452 AUDIENCE: 0. 800 00:52:34,452 --> 00:52:34,880 AUDIENCE: 0. 801 00:52:34,880 --> 00:52:36,170 PROFESSOR: None. 802 00:52:36,170 --> 00:52:38,210 No current goes into the minus. 803 00:52:38,210 --> 00:52:41,500 No current enters the op-amp through the minus lead. 804 00:52:41,500 --> 00:52:44,240 So it's the sum of three currents. 805 00:52:44,240 --> 00:52:45,490 How big is this current? 806 00:52:48,010 --> 00:52:52,595 How big is the current that flows in that lead? 807 00:52:52,595 --> 00:52:53,460 V1 -- 808 00:52:53,460 --> 00:52:55,570 V1 over 1, right? 809 00:52:55,570 --> 00:52:57,670 Ohm's Law. 810 00:52:57,670 --> 00:52:59,210 So this node is 0. 811 00:52:59,210 --> 00:53:01,590 That node is V1. 812 00:53:01,590 --> 00:53:05,110 The voltage across this resistor is V1 minus 0. 813 00:53:05,110 --> 00:53:07,030 The voltage across is V1. 814 00:53:07,030 --> 00:53:10,600 The current is V1 over R. R is 1. 815 00:53:10,600 --> 00:53:14,320 So the current that flows in this leg is V1. 816 00:53:14,320 --> 00:53:17,460 How much current flows in this leg? 817 00:53:17,460 --> 00:53:19,944 How much current flows in this leg? 818 00:53:19,944 --> 00:53:20,850 AUDIENCE: V0. 819 00:53:20,850 --> 00:53:22,910 PROFESSOR: V0. 820 00:53:22,910 --> 00:53:27,410 The idea is that the total current that flows into this 821 00:53:27,410 --> 00:53:32,350 node is V1 plus V2 plus V0. 822 00:53:32,350 --> 00:53:34,660 Solve for V0. 823 00:53:34,660 --> 00:53:37,000 V0 is minus V1 minus V2. 824 00:53:40,030 --> 00:53:40,440 Right? 825 00:53:40,440 --> 00:53:41,690 Got it? 826 00:53:43,640 --> 00:53:48,040 The idea was that this is really easy to solve if you 827 00:53:48,040 --> 00:53:51,160 use the ideal op-amp approximation. 828 00:53:51,160 --> 00:53:52,410 You can see that this is 0. 829 00:53:52,410 --> 00:53:53,480 Therefore, this is 0. 830 00:53:53,480 --> 00:53:56,520 So you have a single KCL equation. 831 00:53:56,520 --> 00:53:58,930 And the result is that this circuit looks like an 832 00:53:58,930 --> 00:54:01,480 inverting summer. 833 00:54:01,480 --> 00:54:05,060 It computes the sum of V1 and V2 and then takes the negative 834 00:54:05,060 --> 00:54:08,450 of that and presents that at the output. 835 00:54:08,450 --> 00:54:11,850 What I'm trying to motivate is that there's a whole different 836 00:54:11,850 --> 00:54:15,120 level of reasoning that you can do when you have this 837 00:54:15,120 --> 00:54:17,110 element, which is an op-amp. 838 00:54:17,110 --> 00:54:19,640 Here what we've done is we've made something that performs a 839 00:54:19,640 --> 00:54:21,980 numerical operation. 840 00:54:21,980 --> 00:54:24,780 It presents, at the output, the negative 841 00:54:24,780 --> 00:54:26,030 sum of the two inputs. 842 00:54:29,050 --> 00:54:30,680 OK another problem. 843 00:54:30,680 --> 00:54:34,720 Determine R so that V0 is twice V1 minus V2. 844 00:59:18,620 --> 00:59:19,870 So how should I choose R? 845 00:59:26,100 --> 00:59:28,030 Not very many hands. 846 00:59:28,030 --> 00:59:31,610 This is the perfect nano quiz practice, right? 847 00:59:31,610 --> 00:59:35,530 This looks like a perfect nano quiz question, right? 848 00:59:35,530 --> 00:59:36,780 Smile. 849 00:59:39,300 --> 00:59:40,550 So what's the answer? 850 00:59:42,890 --> 00:59:46,850 OK we're down to about half correct. 851 00:59:46,850 --> 00:59:48,880 This is harder. 852 00:59:48,880 --> 00:59:51,580 Basically you do exactly the same thing, it's just that 853 00:59:51,580 --> 00:59:55,090 it's a little algebraically messier. 854 00:59:55,090 --> 00:59:58,840 So not surprisingly, the first idea is to think about the 855 00:59:58,840 --> 01:00:01,080 ideal op-amp. 856 01:00:01,080 --> 01:00:03,300 We'll think about what was the voltage here, what was the 857 01:00:03,300 --> 01:00:04,970 voltage there, and then we'll equate them. 858 01:00:09,310 --> 01:00:11,220 What's the voltage at the plus lead? 859 01:00:11,220 --> 01:00:12,020 Well that's easy. 860 01:00:12,020 --> 01:00:14,840 That's just a voltage divider here. 861 01:00:14,840 --> 01:00:17,830 Since no current flows in here, I can compute the 862 01:00:17,830 --> 01:00:21,880 voltage relative to this ground as R over (1 plus R). 863 01:00:21,880 --> 01:00:23,110 That's showed here. 864 01:00:23,110 --> 01:00:24,625 Times V1. 865 01:00:27,510 --> 01:00:30,010 This one's a little bit trickier because I've got two 866 01:00:30,010 --> 01:00:32,820 sources, V2 and V0, each pumping 867 01:00:32,820 --> 01:00:36,000 current into this place. 868 01:00:36,000 --> 01:00:41,480 The way I thought about it was start with V2 and then add to 869 01:00:41,480 --> 01:00:45,280 it this component, which can be thought of as a voltage 870 01:00:45,280 --> 01:00:47,420 divider here. 871 01:00:47,420 --> 01:00:51,050 So how big is the voltage at Vminus? 872 01:00:51,050 --> 01:00:54,620 Well it's V2 plus a voltage divider here, 873 01:00:54,620 --> 01:00:57,320 which is given by this. 874 01:00:57,320 --> 01:01:00,330 That's a little tricky. 875 01:01:00,330 --> 01:01:02,930 If you're not comfortable with that step, you could also 876 01:01:02,930 --> 01:01:05,160 solve for that voltage using the node method. 877 01:01:08,080 --> 01:01:09,760 The node method will give you the same answer. 878 01:01:09,760 --> 01:01:13,390 The answer is that the voltage at the Vminus port is two 879 01:01:13,390 --> 01:01:15,930 thirds of V2 and one third of V0. 880 01:01:15,930 --> 01:01:18,060 That sort of looks right because the resistors are in a 881 01:01:18,060 --> 01:01:19,310 ratio of 2:1. 882 01:01:23,810 --> 01:01:26,750 And then using the ideal op-amp assumption, we equate 883 01:01:26,750 --> 01:01:29,880 the two, do some more algebra, and we get some relationship 884 01:01:29,880 --> 01:01:31,150 and figure out that R is 2. 885 01:01:34,560 --> 01:01:36,200 The point is that this is a 886 01:01:36,200 --> 01:01:38,210 relatively complicated circuit. 887 01:01:38,210 --> 01:01:41,360 You could have done it if I had asked you to do it with 888 01:01:41,360 --> 01:01:44,960 the voltage controlled voltage source model. 889 01:01:44,960 --> 01:01:49,140 But the algebra is already hard here with the ideal 890 01:01:49,140 --> 01:01:51,460 op-amp assumption, and with the voltage controlled voltage 891 01:01:51,460 --> 01:01:52,900 source it's even harder. 892 01:01:52,900 --> 01:01:56,900 So the idea is that the ideal op-amp assumption, the idea 893 01:01:56,900 --> 01:02:01,710 that Vplus is equal to Vminus, makes the work of calculating 894 01:02:01,710 --> 01:02:06,210 these responses significantly easier. 895 01:02:06,210 --> 01:02:07,960 Everybody's with the bottom line? 896 01:02:14,890 --> 01:02:18,630 We started with the idea that when you add an element to a 897 01:02:18,630 --> 01:02:23,990 circuit, in general, adding an element changes voltages and 898 01:02:23,990 --> 01:02:26,690 currents throughout the circuit. 899 01:02:26,690 --> 01:02:31,020 We wanted a way to make the design more modular, a way of 900 01:02:31,020 --> 01:02:36,310 adding a component without changing everything else. 901 01:02:36,310 --> 01:02:40,330 We thought about this op-amp thing. 902 01:02:40,330 --> 01:02:42,720 The model for the op-amp was voltage 903 01:02:42,720 --> 01:02:43,780 controlled voltage source. 904 01:02:43,780 --> 01:02:48,410 We inferred this ideal op-amp model. 905 01:02:48,410 --> 01:02:50,220 The ideal op-amp model was great for 906 01:02:50,220 --> 01:02:52,060 calculating the response. 907 01:02:52,060 --> 01:02:55,780 It has this one problem. 908 01:02:55,780 --> 01:02:58,350 It seems to say these circuits are identical. 909 01:03:01,870 --> 01:03:05,530 If I literally believe the ideal op-amp assumption that 910 01:03:05,530 --> 01:03:08,940 all the op-amp does is magically make the plus and 911 01:03:08,940 --> 01:03:13,600 minus terminals the same, I would conclude that this 912 01:03:13,600 --> 01:03:16,820 circuit where the input comes in the plus and the output 913 01:03:16,820 --> 01:03:22,570 wraps around to the minus generates precisely the same 914 01:03:22,570 --> 01:03:25,510 input-output relationship as this one, where the input 915 01:03:25,510 --> 01:03:30,410 comes in the minus and the output wraps around the plus. 916 01:03:30,410 --> 01:03:33,800 The ideal op-amp assumption simply says for both of those, 917 01:03:33,800 --> 01:03:37,060 Vi equals Vo. 918 01:03:37,060 --> 01:03:42,110 So I would assume from the ideal op-amp model I get the 919 01:03:42,110 --> 01:03:45,570 result that these two circuits work exactly the same way. 920 01:03:45,570 --> 01:03:46,910 Somehow that sounds wrong. 921 01:03:49,950 --> 01:03:54,540 I've got this part, and I can wire it up the 922 01:03:54,540 --> 01:03:55,760 right way and it works. 923 01:03:55,760 --> 01:03:59,440 Or I can flip the two wires, and it still works. 924 01:03:59,440 --> 01:04:01,080 Conservation of badness doesn't let 925 01:04:01,080 --> 01:04:02,160 that happen, right? 926 01:04:02,160 --> 01:04:03,875 If you do something bad, it should break. 927 01:04:06,860 --> 01:04:09,160 The ideal op-amp assumption seems to 928 01:04:09,160 --> 01:04:11,200 lead to a bogus result. 929 01:04:11,200 --> 01:04:13,570 It seems to say these two are the same. 930 01:04:13,570 --> 01:04:16,810 So what I'd like to do is think about that for a moment. 931 01:04:16,810 --> 01:04:19,160 We want to be comfortable with the assumption that we make. 932 01:04:19,160 --> 01:04:22,480 The ideal op-amp assumption makes analysis really easy, 933 01:04:22,480 --> 01:04:26,090 but we'd like to understand exactly what we're assuming. 934 01:04:26,090 --> 01:04:27,550 This just doesn't sound right. 935 01:04:27,550 --> 01:04:28,880 OK, so let's back up. 936 01:04:31,590 --> 01:04:33,360 Let's not do the ideal op-amp assumption. 937 01:04:33,360 --> 01:04:36,320 Let's Instead say that we use the voltage controlled voltage 938 01:04:36,320 --> 01:04:38,240 source model. 939 01:04:38,240 --> 01:04:39,980 So what I've done here is substitute 940 01:04:39,980 --> 01:04:41,760 into the left circuit. 941 01:04:41,760 --> 01:04:47,890 So this circuit is shown here and the other over here. 942 01:04:47,890 --> 01:04:50,520 All I've done is connected the input either to the plus or to 943 01:04:50,520 --> 01:04:55,640 the minus port and wrapped the other one around. 944 01:04:55,640 --> 01:04:58,180 But now I'm analyzing it using the voltage controlled voltage 945 01:04:58,180 --> 01:05:00,100 source model. 946 01:05:00,100 --> 01:05:01,540 And I've done the tedious algebra. 947 01:05:01,540 --> 01:05:02,840 And I don't think there's any mistakes 948 01:05:02,840 --> 01:05:05,190 in the tedious algebra. 949 01:05:05,190 --> 01:05:07,690 In one case I get K over (1 plus K), which for 950 01:05:07,690 --> 01:05:10,541 large K is about 1. 951 01:05:10,541 --> 01:05:15,150 In the other case I get minus K over (1 minus K), which for 952 01:05:15,150 --> 01:05:17,460 K large is about 1. 953 01:05:17,460 --> 01:05:21,090 The ideal op-amp assumption says that it doesn't matter. 954 01:05:21,090 --> 01:05:22,690 The voltage controlled voltage source model 955 01:05:22,690 --> 01:05:24,640 says it doesn't matter. 956 01:05:24,640 --> 01:05:27,130 And Freeman thinks this doesn't make any sense. 957 01:05:30,210 --> 01:05:32,830 So what's going on here? 958 01:05:32,830 --> 01:05:39,770 What's going on is that we've actually made an enormous leap 959 01:05:39,770 --> 01:05:42,420 in thinking about the op-amp even as a voltage controlled 960 01:05:42,420 --> 01:05:43,670 voltage source. 961 01:05:45,530 --> 01:05:48,370 The two models that we talked about, the voltage controlled 962 01:05:48,370 --> 01:05:51,270 voltage source and the ideal op-amp model, are great for 963 01:05:51,270 --> 01:05:53,900 calculating answers. 964 01:05:53,900 --> 01:05:58,410 They are not so good at providing a mechanism for how 965 01:05:58,410 --> 01:06:00,300 the op-amp is actually working. 966 01:06:00,300 --> 01:06:04,300 Think about just the ideal op-amp assumption. 967 01:06:04,300 --> 01:06:07,020 It's great to think, OK, the op-amp is going to do whatever 968 01:06:07,020 --> 01:06:09,770 magic is necessary to make these two leads the same. 969 01:06:12,580 --> 01:06:15,350 Well how does it do that? 970 01:06:15,350 --> 01:06:17,580 The thing about the ideal op-amp assumption is that it 971 01:06:17,580 --> 01:06:21,730 doesn't tell you the mechanism by which that happens. 972 01:06:21,730 --> 01:06:27,530 What does the op-amp actually do in order to get 973 01:06:27,530 --> 01:06:28,730 Vplus equal to Vminus? 974 01:06:28,730 --> 01:06:31,660 What's actually going on inside the op-amp? 975 01:06:31,660 --> 01:06:36,650 The ideal op-amp assumption is very good for analysis and not 976 01:06:36,650 --> 01:06:39,000 very good with mechanism. 977 01:06:39,000 --> 01:06:40,120 What's the mechanism? 978 01:06:40,120 --> 01:06:42,170 What's the op-amp actually doing? 979 01:06:45,080 --> 01:06:48,120 What it's really doing is moving charge around. 980 01:06:50,800 --> 01:06:51,220 8.02 -- 981 01:06:51,220 --> 01:06:53,480 charge. 982 01:06:53,480 --> 01:06:56,010 So if you're going to have a change in voltage, there has 983 01:06:56,010 --> 01:06:57,900 to be motion of charge. 984 01:06:57,900 --> 01:07:00,870 And that's what's missing from the voltage controlled voltage 985 01:07:00,870 --> 01:07:03,520 model and from the ideal op-amp model. 986 01:07:03,520 --> 01:07:05,380 How do you think about moving charge around? 987 01:07:05,380 --> 01:07:07,180 Well it's the same as moving water around. 988 01:07:07,180 --> 01:07:10,050 Somehow, I think it's all those years of playing with 989 01:07:10,050 --> 01:07:13,870 water when I was a little kid, I find my intuition about 990 01:07:13,870 --> 01:07:16,260 water is better than my intuition about charge. 991 01:07:16,260 --> 01:07:19,045 So let me start by thinking about the intuition for water. 992 01:07:22,310 --> 01:07:26,030 The way a water tank works is if the flow in is different 993 01:07:26,030 --> 01:07:27,320 from the flow out, the height changes. 994 01:07:30,010 --> 01:07:31,870 That's continuity. 995 01:07:31,870 --> 01:07:37,800 So if the water is conserved, if there's not significant 996 01:07:37,800 --> 01:07:40,570 evaporation over the duration of this experiment, or if 997 01:07:40,570 --> 01:07:43,610 molecules of water do not spontaneously disappear or 998 01:07:43,610 --> 01:07:48,220 appear, under either of those two assumptions, then the 999 01:07:48,220 --> 01:07:50,560 change in the height is proportional to the difference 1000 01:07:50,560 --> 01:07:52,400 between the rate at which it's coming in and the rate at 1001 01:07:52,400 --> 01:07:53,490 which it's coming out. 1002 01:07:53,490 --> 01:07:55,340 If the rate at which it's going out is equal to the rate 1003 01:07:55,340 --> 01:07:57,590 at which it's going in, there's no change in height. 1004 01:07:57,590 --> 01:07:59,360 If it's coming in faster, it's going up. 1005 01:07:59,360 --> 01:08:02,070 If it's coming in slower, it's going down. 1006 01:08:02,070 --> 01:08:05,040 All easy, right? 1007 01:08:05,040 --> 01:08:06,410 Charge works the same way. 1008 01:08:09,720 --> 01:08:12,790 The thing that accumulates charge in an electronic 1009 01:08:12,790 --> 01:08:15,230 circuit is a capacitor. 1010 01:08:15,230 --> 01:08:18,990 And there's a direct analogy between thinking about the way 1011 01:08:18,990 --> 01:08:23,569 the flux of water generates height and the way flux of 1012 01:08:23,569 --> 01:08:27,200 charge generates volts. 1013 01:08:27,200 --> 01:08:33,680 So we can think about the flux of charge, that's current. 1014 01:08:33,680 --> 01:08:38,990 If there's a net flux, if there's a bigger current into 1015 01:08:38,990 --> 01:08:42,330 a capacitor then there is out, the voltage on that 1016 01:08:42,330 --> 01:08:44,640 capacitor goes up. 1017 01:08:44,640 --> 01:08:47,420 If there's a smaller current in than goes out, the voltage 1018 01:08:47,420 --> 01:08:50,609 on the capacitor goes down just like the height 1019 01:08:50,609 --> 01:08:51,940 in a tank of water. 1020 01:08:54,470 --> 01:08:57,060 We can make a much more realistic model for the way an 1021 01:08:57,060 --> 01:09:01,460 op-amp works by explicitly making a representation for 1022 01:09:01,460 --> 01:09:05,899 how the charge flows. 1023 01:09:05,899 --> 01:09:07,890 And that's shown here. 1024 01:09:07,890 --> 01:09:09,930 I'm going to take the voltage controlled voltage source 1025 01:09:09,930 --> 01:09:15,290 model but explicitly make a representation for the output 1026 01:09:15,290 --> 01:09:17,550 charging up. 1027 01:09:17,550 --> 01:09:18,590 What's the op-amp do? 1028 01:09:18,590 --> 01:09:22,080 The op-amp senses the voltage at the input and the output 1029 01:09:22,080 --> 01:09:27,270 and does something to change the voltage at the output. 1030 01:09:27,270 --> 01:09:32,229 The thing it does is if the positive voltage is greater 1031 01:09:32,229 --> 01:09:35,680 than the negative voltage, it pumps charge 1032 01:09:35,680 --> 01:09:39,359 into the output node. 1033 01:09:39,359 --> 01:09:43,689 If the opposite occurs, it sucks charge out. 1034 01:09:43,689 --> 01:09:48,950 Now it's important, this is not an accurate depiction of 1035 01:09:48,950 --> 01:09:50,260 what's inside an op-amp. 1036 01:09:50,260 --> 01:09:55,190 This is the model for an op-amp. 1037 01:09:55,190 --> 01:09:58,210 I don't want to lead any of you to think that this is 1038 01:09:58,210 --> 01:09:59,620 literally what's in an op-amp. 1039 01:09:59,620 --> 01:10:02,110 This is not literally what's in an op-amp. 1040 01:10:02,110 --> 01:10:06,060 Here is much more literally what's in an op-amp. 1041 01:10:06,060 --> 01:10:09,120 This is the schematic diagram of a 709 1042 01:10:09,120 --> 01:10:11,410 which is a Widlar circuit. 1043 01:10:11,410 --> 01:10:14,060 It's a complicated transistor circuit. 1044 01:10:14,060 --> 01:10:15,310 It's ingenious. 1045 01:10:17,330 --> 01:10:19,660 This is how you build a circuit that has the 1046 01:10:19,660 --> 01:10:22,530 remarkable property of the ideal op-amp circuit. 1047 01:10:22,530 --> 01:10:25,065 It's not perfectly obvious from here. 1048 01:10:25,065 --> 01:10:29,270 And even here this doesn't give Widlar enough credit. 1049 01:10:29,270 --> 01:10:31,000 Here's what he really did. 1050 01:10:31,000 --> 01:10:36,120 He designed masks to shield semiconductor materials so 1051 01:10:36,120 --> 01:10:38,710 that you could turn them into transistors so that it would 1052 01:10:38,710 --> 01:10:40,390 turn into an op-amp. 1053 01:10:40,390 --> 01:10:42,100 We're not going to worry about this. 1054 01:10:42,100 --> 01:10:45,560 This is two levels of abstraction more complex than 1055 01:10:45,560 --> 01:10:47,370 we're going to worry about. 1056 01:10:47,370 --> 01:10:50,410 We're just going to say, I don't know what's in there, 1057 01:10:50,410 --> 01:10:51,960 but this is the way it behaves. 1058 01:10:51,960 --> 01:10:56,870 This is intended to be a behavioral model for how an 1059 01:10:56,870 --> 01:10:57,380 op-amp works. 1060 01:10:57,380 --> 01:10:58,670 And in fact, that's an important 1061 01:10:58,670 --> 01:11:03,290 idea that we use circuits-- 1062 01:11:03,290 --> 01:11:06,440 I use circuits on a daily basis not because I design 1063 01:11:06,440 --> 01:11:08,280 semiconductor devices but because I work 1064 01:11:08,280 --> 01:11:09,970 on biological issues. 1065 01:11:09,970 --> 01:11:12,830 I actually study hearing. 1066 01:11:12,830 --> 01:11:17,500 And we make circuit models for the way biological parts work. 1067 01:11:17,500 --> 01:11:20,490 And that helps us to understand the way the 1068 01:11:20,490 --> 01:11:21,810 biological part works. 1069 01:11:21,810 --> 01:11:26,400 For example, here is a model taken from 6.021, where we try 1070 01:11:26,400 --> 01:11:31,060 to understand how a nerve propagates an action potential 1071 01:11:31,060 --> 01:11:32,490 by making a circuit model. 1072 01:11:35,160 --> 01:11:36,540 That's the same thing we're doing here. 1073 01:11:36,540 --> 01:11:41,100 We're making a circuit model for how the op-amp works. 1074 01:11:41,100 --> 01:11:42,550 it's not intended to be literally 1075 01:11:42,550 --> 01:11:43,590 what's inside the op-amp. 1076 01:11:43,590 --> 01:11:46,110 It's intended to be a model that let's us think about the 1077 01:11:46,110 --> 01:11:48,940 behavior of the op-amp. 1078 01:11:48,940 --> 01:11:51,510 This lets us understand now why it's different when you 1079 01:11:51,510 --> 01:11:52,760 flip the wires. 1080 01:11:55,070 --> 01:11:58,580 Let's start with the original configuration where I put the 1081 01:11:58,580 --> 01:12:01,880 voltage in the plus lead, and I wrap the output back around 1082 01:12:01,880 --> 01:12:02,750 to the minus lead. 1083 01:12:02,750 --> 01:12:04,800 What's going to happen? 1084 01:12:04,800 --> 01:12:08,550 What's the op-amp actually do? 1085 01:12:08,550 --> 01:12:10,900 Imagine that things had been stable. 1086 01:12:10,900 --> 01:12:12,020 This was 0. 1087 01:12:12,020 --> 01:12:13,430 The output was 0. 1088 01:12:13,430 --> 01:12:15,980 Everybody was happy. 1089 01:12:15,980 --> 01:12:20,670 And now all of a sudden the input voltages steps up. 1090 01:12:20,670 --> 01:12:22,300 What happens? 1091 01:12:22,300 --> 01:12:24,440 The input voltage steps up. 1092 01:12:24,440 --> 01:12:26,440 The voltage source suddenly generates a 1093 01:12:26,440 --> 01:12:28,720 huge positive voltage. 1094 01:12:28,720 --> 01:12:33,330 And that starts to put current into the capacitor that 1095 01:12:33,330 --> 01:12:37,340 represents the voltage at the output of the op-amp. 1096 01:12:37,340 --> 01:12:41,200 So as a result of this voltage being higher than where it 1097 01:12:41,200 --> 01:12:46,260 was, current is being dumped by the op-amp into this 1098 01:12:46,260 --> 01:12:49,310 capacitor, and the output voltage starts to increase. 1099 01:12:49,310 --> 01:12:50,560 That's shown it red. 1100 01:12:53,000 --> 01:12:56,310 As the capacitor voltage gets bigger, as the output voltage 1101 01:12:56,310 --> 01:12:59,510 increases, the difference between Vplus and Vminus gets 1102 01:12:59,510 --> 01:13:05,450 smaller, and the rate at which charge is flowing into the 1103 01:13:05,450 --> 01:13:08,270 capacitor slows. 1104 01:13:08,270 --> 01:13:11,560 And in fact, as the voltage at the output gets very close to 1105 01:13:11,560 --> 01:13:15,060 the voltage at the input, the rate of current into the 1106 01:13:15,060 --> 01:13:17,530 capacitor goes to 0. 1107 01:13:17,530 --> 01:13:19,520 And the output voltage stabilizes 1108 01:13:19,520 --> 01:13:22,450 at the input voltage. 1109 01:13:22,450 --> 01:13:25,120 The same thing happens in reverse if I were to change 1110 01:13:25,120 --> 01:13:27,455 input voltage and make it go negative. 1111 01:13:30,000 --> 01:13:35,500 If the input voltage went negative, then K times Vplus 1112 01:13:35,500 --> 01:13:38,290 minus Vminus would be a big negative number and it would 1113 01:13:38,290 --> 01:13:41,810 suck current out. 1114 01:13:41,810 --> 01:13:45,490 The voltage controlled voltage source would suck current out 1115 01:13:45,490 --> 01:13:49,270 of the capacitor and make the voltage fall. 1116 01:13:49,270 --> 01:13:50,360 The voltage would fall. 1117 01:13:50,360 --> 01:13:52,680 The absolute difference between the plus and minus 1118 01:13:52,680 --> 01:13:54,660 port would get smaller. 1119 01:13:54,660 --> 01:13:58,080 The current flowing would become less. 1120 01:13:58,080 --> 01:14:01,600 And it would again stabilize when the output is 1121 01:14:01,600 --> 01:14:02,850 equal to the input. 1122 01:14:05,670 --> 01:14:10,310 Contrast that to what would happen if I put the input into 1123 01:14:10,310 --> 01:14:13,450 the minus port. 1124 01:14:13,450 --> 01:14:17,480 If I put the input into the minus port, and the input goes 1125 01:14:17,480 --> 01:14:24,450 through a step, then the input going positive makes the 1126 01:14:24,450 --> 01:14:26,360 voltage controlled voltage source generate a 1127 01:14:26,360 --> 01:14:27,870 big negative voltage. 1128 01:14:27,870 --> 01:14:30,390 It sucks current out. 1129 01:14:30,390 --> 01:14:31,735 So the input went positive. 1130 01:14:35,160 --> 01:14:36,590 And the output goes negative. 1131 01:14:36,590 --> 01:14:39,540 It goes the wrong way. 1132 01:14:39,540 --> 01:14:42,190 That's bad. 1133 01:14:42,190 --> 01:14:46,480 So here, if I put the input into the minus lead, a 1134 01:14:46,480 --> 01:14:49,830 positive transition of the input leads to the output 1135 01:14:49,830 --> 01:14:51,910 going the wrong way. 1136 01:14:51,910 --> 01:14:56,050 And as it goes the wrong way, the drive to make it go the 1137 01:14:56,050 --> 01:14:58,470 wrong way gets bigger. 1138 01:14:58,470 --> 01:14:59,530 It's a runaway system. 1139 01:14:59,530 --> 01:15:02,670 It's positive feedback. 1140 01:15:02,670 --> 01:15:06,520 So the idea is that by supporting the input into the 1141 01:15:06,520 --> 01:15:09,840 negative lead instead of into the positive lead, leads to a 1142 01:15:09,840 --> 01:15:13,780 positive feedback situation in which a small change at the 1143 01:15:13,780 --> 01:15:19,190 input makes the output go the wrong way, convinces the 1144 01:15:19,190 --> 01:15:21,750 op-amp that things are getting worse, so it makes even more 1145 01:15:21,750 --> 01:15:25,450 current, which makes it go even more the wrong way. 1146 01:15:25,450 --> 01:15:31,050 And the same thing happens with the flip situation. 1147 01:15:31,050 --> 01:15:33,660 The idea then is that by thinking 1148 01:15:33,660 --> 01:15:35,010 about the flow of current-- 1149 01:15:35,010 --> 01:15:36,160 What's the op-amp actually doing? 1150 01:15:36,160 --> 01:15:38,910 The op-amp is actually sensing the difference between the 1151 01:15:38,910 --> 01:15:42,330 voltage at the positive port and the minus port, and it's 1152 01:15:42,330 --> 01:15:45,360 sourcing current that changes the output 1153 01:15:45,360 --> 01:15:48,540 voltage to go up or down. 1154 01:15:48,540 --> 01:15:56,520 You need to wire the op-amp so that ultimately the output 1155 01:15:56,520 --> 01:15:58,450 equals the input. 1156 01:15:58,450 --> 01:16:01,840 Otherwise the ideal op-amp condition that Vplus equals 1157 01:16:01,840 --> 01:16:04,670 Vminus will ever be attained. 1158 01:16:04,670 --> 01:16:08,870 So the left one is what we would refer to as a stable 1159 01:16:08,870 --> 01:16:11,350 feedback situation. 1160 01:16:11,350 --> 01:16:14,580 You can think about that as analogous to thinking about a 1161 01:16:14,580 --> 01:16:16,950 ball in a valley. 1162 01:16:16,950 --> 01:16:19,850 So we have a valley, and we have a ball. 1163 01:16:19,850 --> 01:16:22,430 The ball's going to roll down here eventually. 1164 01:16:22,430 --> 01:16:23,970 It's stable. 1165 01:16:23,970 --> 01:16:26,270 Bop it a little bit to the right, it'll roll back into 1166 01:16:26,270 --> 01:16:26,700 the valley. 1167 01:16:26,700 --> 01:16:28,130 Bop it a little to the left, it'll roll 1168 01:16:28,130 --> 01:16:29,260 back into the valley. 1169 01:16:29,260 --> 01:16:32,670 That's as opposed to, when we've wired up the wrong way, 1170 01:16:32,670 --> 01:16:36,010 it's like we have an unstable equilibrium. 1171 01:16:36,010 --> 01:16:39,470 It's as though we're trying to put the ball there. 1172 01:16:39,470 --> 01:16:43,200 Bop it a little to right, it runs away to the right. 1173 01:16:43,200 --> 01:16:45,680 Bop it a little to the left, it runs away to the left. 1174 01:16:45,680 --> 01:16:48,730 That's unstable. 1175 01:16:48,730 --> 01:16:50,760 There is a metastability plane. 1176 01:16:50,760 --> 01:16:54,420 If you actually balanced it exactly, exactly, exactly at 1177 01:16:54,420 --> 01:16:58,530 the right place, it would stay there. 1178 01:16:58,530 --> 01:17:00,290 That's what the solution is telling you. 1179 01:17:00,290 --> 01:17:04,370 That was the minus K over (1 minus K). 1180 01:17:04,370 --> 01:17:07,140 The fact that the equations had a solution is correct, 1181 01:17:07,140 --> 01:17:10,150 it's just an unstable solution. 1182 01:17:10,150 --> 01:17:15,530 The tiniest little disturbance will make it go awry. 1183 01:17:15,530 --> 01:17:18,770 The idea then is that the ideal op-amp 1184 01:17:18,770 --> 01:17:19,990 assumption is fine. 1185 01:17:19,990 --> 01:17:21,080 It's a good thing to use. 1186 01:17:21,080 --> 01:17:23,380 It doesn't tell you about the mechanism. 1187 01:17:23,380 --> 01:17:25,460 If you think a little bit more about the way the physics 1188 01:17:25,460 --> 01:17:30,170 works, you have to wire the circuit up so that it has 1189 01:17:30,170 --> 01:17:35,740 negative feedback in order to get a stable equilibrium. 1190 01:17:35,740 --> 01:17:38,230 So if you understand that, and you just make sure that you've 1191 01:17:38,230 --> 01:17:40,680 hooked it up so that it has negative feedback, then you 1192 01:17:40,680 --> 01:17:42,600 can use the ideal op-amp assumption. 1193 01:17:42,600 --> 01:17:44,290 Everything's just fine. 1194 01:17:44,290 --> 01:17:46,500 And in the interest of time, I'm going to just skip over 1195 01:17:46,500 --> 01:17:48,840 this because we've already talked about all of this. 1196 01:17:48,840 --> 01:17:50,650 It's just another example. 1197 01:17:50,650 --> 01:17:53,750 The last thing I want to mention is just that in order 1198 01:17:53,750 --> 01:17:58,140 to work this magic, the op-amp has to 1199 01:17:58,140 --> 01:17:59,455 get power from somewhere. 1200 01:18:03,670 --> 01:18:06,640 And so that means that it's not a three-terminal device. 1201 01:18:06,640 --> 01:18:09,590 It's not the kind of device I've been drawing so far that 1202 01:18:09,590 --> 01:18:10,990 has got power pins too. 1203 01:18:13,640 --> 01:18:16,930 The way the op-amp works is it takes power from those power 1204 01:18:16,930 --> 01:18:21,630 pins to be able to force the output to a voltage that'll 1205 01:18:21,630 --> 01:18:24,780 make the input the Vplus equal to Vminus. 1206 01:18:24,780 --> 01:18:26,750 That's the mechanism by which it works. 1207 01:18:26,750 --> 01:18:30,150 And that'll have important consequences when we build the 1208 01:18:30,150 --> 01:18:34,820 robot head, because what it means is that we're not going 1209 01:18:34,820 --> 01:18:37,480 to be able to generate arbitrary voltages at the 1210 01:18:37,480 --> 01:18:39,020 output of an op-amp. 1211 01:18:39,020 --> 01:18:41,600 We're going to be limited to generating voltages that are 1212 01:18:41,600 --> 01:18:44,510 between the power rails. 1213 01:18:44,510 --> 01:18:47,630 So if we were to supply plus and minus 10 Volts to the 1214 01:18:47,630 --> 01:18:50,780 power pins, we would not be expecting to be able to 1215 01:18:50,780 --> 01:18:53,800 generate 20 Volts at the output. 1216 01:18:53,800 --> 01:18:55,650 So that's an important implication when we do the 1217 01:18:55,650 --> 01:18:57,560 design lab. 1218 01:18:57,560 --> 01:19:01,260 In summary, what we've done is we've showed how circuits can 1219 01:19:01,260 --> 01:19:05,600 be a pain to make modular, because in principle, adding 1220 01:19:05,600 --> 01:19:08,950 one component changes voltages and currents everywhere. 1221 01:19:08,950 --> 01:19:11,580 But there's a way using op-amps to have this buffering 1222 01:19:11,580 --> 01:19:15,420 idea that lets us logically separate parts of circuits so 1223 01:19:15,420 --> 01:19:17,562 the one can control the other. 1224 01:19:17,562 --> 01:19:18,812 Have a good day.