1 00:00:00,000 --> 00:00:02,550 MALE SPEAKER: The following content is provided under a 2 00:00:02,550 --> 00:00:04,370 Creative Commons License. 3 00:00:04,370 --> 00:00:07,410 Your support will help MIT OpenCourseWare continue to 4 00:00:07,410 --> 00:00:11,060 offer high quality educational resources for free. 5 00:00:11,060 --> 00:00:13,960 To make a donation or view additional materials from 6 00:00:13,960 --> 00:00:18,365 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:18,365 --> 00:00:19,615 ocw.mit.edu. 8 00:00:24,741 --> 00:00:26,650 PROFESSOR: So today I want to finish up 9 00:00:26,650 --> 00:00:27,900 thinking about circuits. 10 00:00:30,090 --> 00:00:32,460 And the major topic for today is just one -- 11 00:00:32,460 --> 00:00:34,890 thinking about abstractions that we can use for thinking 12 00:00:34,890 --> 00:00:36,690 about circuits. 13 00:00:36,690 --> 00:00:40,220 The attractions are things that capitalize on linearity. 14 00:00:40,220 --> 00:00:42,560 And they include things like [UNINTELLIGIBLE] and 15 00:00:42,560 --> 00:00:43,810 superposition. 16 00:00:45,380 --> 00:00:47,430 OK so what I want to do is finish up 17 00:00:47,430 --> 00:00:48,760 thinking about circuits. 18 00:00:48,760 --> 00:00:51,900 And just to get you thinking about circuits, let's think 19 00:00:51,900 --> 00:00:52,770 about where we are. 20 00:00:52,770 --> 00:00:56,840 Last time we saw that one of the issues in thinking about 21 00:00:56,840 --> 00:01:02,380 circuit design, is the fact that every part, in principle, 22 00:01:02,380 --> 00:01:04,040 interacts with every other part. 23 00:01:04,040 --> 00:01:09,210 Which makes the design process harder than the design process 24 00:01:09,210 --> 00:01:11,580 was for things like linear systems, where we thought 25 00:01:11,580 --> 00:01:15,240 about boxes having inputs and outputs. 26 00:01:15,240 --> 00:01:17,300 In linear systems, when we thought about boxes having 27 00:01:17,300 --> 00:01:21,260 inputs and outputs, the output didn't necessarily have any 28 00:01:21,260 --> 00:01:23,230 affect on the input, unless there was an 29 00:01:23,230 --> 00:01:26,490 explicit path feedback. 30 00:01:26,490 --> 00:01:28,830 In circuits, there's feedback always. 31 00:01:28,830 --> 00:01:31,070 There's no way to avoid it. 32 00:01:31,070 --> 00:01:33,320 And in fact, that coupling can make thinking about the 33 00:01:33,320 --> 00:01:35,380 circuits very difficult. 34 00:01:35,380 --> 00:01:38,530 And we introduce that idea by just thinking about, even if 35 00:01:38,530 --> 00:01:41,110 you wanted to close a switch, you know a circuit, that's 36 00:01:41,110 --> 00:01:44,760 logically the same as adding a new part. 37 00:01:44,760 --> 00:01:47,050 And when you do that, that's going to change the currents 38 00:01:47,050 --> 00:01:49,320 and voltages everywhere. 39 00:01:49,320 --> 00:01:51,250 So that's kind of a big thing-- that's kind of the 40 00:01:51,250 --> 00:01:55,500 thing that's different about circuits from what we've been 41 00:01:55,500 --> 00:01:56,750 thinking about before. 42 00:01:59,050 --> 00:02:01,330 Because that complicates design, we'd like some way of 43 00:02:01,330 --> 00:02:02,110 dealing with it. 44 00:02:02,110 --> 00:02:04,850 And the way we introduced, and the way that you used in the 45 00:02:04,850 --> 00:02:07,280 lab, so far, has been to use a buffer. 46 00:02:07,280 --> 00:02:09,620 We can use an op-amp to make a buffer. 47 00:02:09,620 --> 00:02:13,480 And a buffer has this nice isolation property. 48 00:02:13,480 --> 00:02:16,010 In this particular circuit, it has the property that it 49 00:02:16,010 --> 00:02:18,820 copies this voltage to the output, regardless of what's 50 00:02:18,820 --> 00:02:20,070 at the output. 51 00:02:22,170 --> 00:02:24,340 That means we could put a switch here, we could change 52 00:02:24,340 --> 00:02:26,760 the light bulb, we could do anything we wanted to, and we 53 00:02:26,760 --> 00:02:30,190 would still know, because of the properties of the op-amp, 54 00:02:30,190 --> 00:02:32,677 we would still know that this voltage is going to be 8 55 00:02:32,677 --> 00:02:34,730 Volts, regardless of what we put there. 56 00:02:34,730 --> 00:02:36,890 So that's very nice, that gives us a modularity, that 57 00:02:36,890 --> 00:02:39,600 gives us a way to design things. 58 00:02:39,600 --> 00:02:41,990 It let's us design the left-- 59 00:02:41,990 --> 00:02:44,580 make a circuit that generates 8 Volts without knowing 60 00:02:44,580 --> 00:02:48,030 precisely what's going to be the ultimate 61 00:02:48,030 --> 00:02:49,820 circuit that that drives. 62 00:02:49,820 --> 00:02:53,910 So that's nice, it allows us to do modularity. 63 00:02:53,910 --> 00:02:57,650 And in many instances, that's a very good solution. 64 00:02:57,650 --> 00:02:59,310 In fact in the solution-- in the problems that you are 65 00:02:59,310 --> 00:03:00,320 working on in lab, where you're 66 00:03:00,320 --> 00:03:03,280 trying to drive a motor-- 67 00:03:03,280 --> 00:03:06,710 that's an excellent solution. 68 00:03:06,710 --> 00:03:09,200 It's not always an excellent solution. 69 00:03:09,200 --> 00:03:12,590 In some sense, it's very expensive. 70 00:03:12,590 --> 00:03:15,620 An op-amp is a complicated part. 71 00:03:15,620 --> 00:03:19,860 If you were to look inside an op-amp, there's some two dozen 72 00:03:19,860 --> 00:03:22,470 transistors in most op-amps. 73 00:03:22,470 --> 00:03:25,190 So it's not an inexpensive part, especially when you 74 00:03:25,190 --> 00:03:27,310 think about this kind of a circuit that only has three 75 00:03:27,310 --> 00:03:28,770 parts to the left. 76 00:03:28,770 --> 00:03:33,110 The op-amp is actually a more complex device than 77 00:03:33,110 --> 00:03:35,370 anything else up. 78 00:03:35,370 --> 00:03:38,700 So op-amps are wonderful, op-amps allow us to make 79 00:03:38,700 --> 00:03:41,060 buffers, buffers are wonderful, but they're not 80 00:03:41,060 --> 00:03:44,540 always the best solution for thinking about modularity. 81 00:03:44,540 --> 00:03:46,940 And, in fact, there's other ways. 82 00:03:46,940 --> 00:03:49,020 And so that's what we want to think about today is -- other 83 00:03:49,020 --> 00:03:55,800 ways for achieving modularity in circuit design. 84 00:03:55,800 --> 00:03:59,830 And the key to thinking about this, is to think about, well, 85 00:03:59,830 --> 00:04:02,540 what's the worst thing that could happen? 86 00:04:02,540 --> 00:04:08,440 If I changed this part arbitrarily, just how bad can 87 00:04:08,440 --> 00:04:09,690 things get? 88 00:04:13,290 --> 00:04:15,640 So I'll let you answer that. 89 00:04:15,640 --> 00:04:19,459 Think about that circuit, and assume that this is a 90 Volt 90 00:04:19,459 --> 00:04:22,680 power supply, 3 Ohm, 6 Ohm, but this can change-- 91 00:04:22,680 --> 00:04:23,930 R0 can change. 92 00:04:26,540 --> 00:04:30,650 I've tabulated some values of R0 and putative, corresponding 93 00:04:30,650 --> 00:04:32,270 values for V0 and I0. 94 00:04:35,130 --> 00:04:37,920 Are my putative answers right? 95 00:04:37,920 --> 00:04:40,150 So take a minute, talk to your neighbor, figure 96 00:04:40,150 --> 00:04:42,710 out how many of these-- 97 00:04:42,710 --> 00:04:47,220 let's see-- ten blue numbers are right or wrong. 98 00:05:40,613 --> 00:05:42,470 It's very quiet, you are allowed to talk to people. 99 00:07:54,460 --> 00:07:56,160 So how many of the numbers are wrong? 100 00:07:56,160 --> 00:08:00,230 So raise your hand, indicate by number fingers how many 101 00:08:00,230 --> 00:08:02,790 mistakes are in the table. 102 00:08:02,790 --> 00:08:03,820 More votes, more votes-- 103 00:08:03,820 --> 00:08:05,110 come on, come on. 104 00:08:05,110 --> 00:08:06,900 if you had talked more, you could blame it on your 105 00:08:06,900 --> 00:08:09,100 neighbor more easily, so talk quickly. 106 00:08:19,080 --> 00:08:20,330 OK. 107 00:08:24,070 --> 00:08:25,585 So does everybody agree with their neighbor? 108 00:08:29,350 --> 00:08:31,100 OK, i don't see a single right answer. 109 00:08:34,159 --> 00:08:36,429 So take 30 more seconds and think about it again. 110 00:08:36,429 --> 00:08:38,590 I don't see any right answers. 111 00:08:38,590 --> 00:08:41,261 So assume your answer's wrong. 112 00:08:41,261 --> 00:08:42,690 [CHUCKLES] 113 00:08:42,690 --> 00:08:43,380 Color blind-- 114 00:08:43,380 --> 00:08:45,180 OK, so how many of the voltages and 115 00:08:45,180 --> 00:08:46,430 currents are incorrect? 116 00:09:14,520 --> 00:09:15,950 OK, how about a re-vote? 117 00:09:15,950 --> 00:09:18,620 So how many of the blue numbers-- 118 00:09:18,620 --> 00:09:20,890 the currents and voltages, how many of V0 and I0 -- 119 00:09:20,890 --> 00:09:22,140 how many of those are incorrect? 120 00:09:25,240 --> 00:09:26,970 OK, not very many votes-- 121 00:09:26,970 --> 00:09:29,150 I'd say about 80% correct, now. 122 00:09:29,150 --> 00:09:31,020 That's definitely an improvement. 123 00:09:31,020 --> 00:09:32,300 What would have to be true? 124 00:09:32,300 --> 00:09:34,590 If I wanted to prove that some of these numbers were wrong, 125 00:09:34,590 --> 00:09:36,340 what could I do? 126 00:09:36,340 --> 00:09:39,490 Give me a condition that would have to be true if the numbers 127 00:09:39,490 --> 00:09:40,740 were correct. 128 00:09:44,616 --> 00:09:48,004 AUDIENCE: V0 has to equal I0? 129 00:09:48,004 --> 00:09:50,140 PROFESSOR: V0 has to equal I0. 130 00:09:50,140 --> 00:09:53,460 We know that this resistor better be-- 131 00:09:53,460 --> 00:09:55,080 better obey Ohm's Law. 132 00:09:55,080 --> 00:09:56,940 That's what we mean by that symbol. 133 00:09:56,940 --> 00:10:00,750 So it had better be the case that V0 is I0R0. 134 00:10:00,750 --> 00:10:02,920 So you you'd want this number to be the 135 00:10:02,920 --> 00:10:04,600 product of that and that. 136 00:10:04,600 --> 00:10:07,810 So 0 is the product of 0 and 30, that looks good. 137 00:10:07,810 --> 00:10:13,750 30 is 2 times 15, 36 is 3 times 12, 48 is 6 times 8. 138 00:10:13,750 --> 00:10:14,320 60 is-- 139 00:10:14,320 --> 00:10:17,000 well that's a little marginal. 140 00:10:17,000 --> 00:10:18,630 how about if I rearrange it a little bit, and say 141 00:10:18,630 --> 00:10:19,880 what if R is V/I? 142 00:10:23,310 --> 00:10:26,770 If I is 0, that would make the resistor infinity, so there's 143 00:10:26,770 --> 00:10:29,180 a way of thinking about that last line as correct. 144 00:10:29,180 --> 00:10:31,160 That's a little funny. 145 00:10:31,160 --> 00:10:33,280 Maybe the forward way of thinking about it is, well 146 00:10:33,280 --> 00:10:36,700 what if I made the resistor infinite? 147 00:10:36,700 --> 00:10:39,060 if I made the resistor infinite, what would be I? 148 00:10:42,380 --> 00:10:43,330 0. 149 00:10:43,330 --> 00:10:44,580 And what would be V? 150 00:10:48,170 --> 00:10:50,175 What would be the voltage difference if 151 00:10:50,175 --> 00:10:52,887 this were right, then? 152 00:10:52,887 --> 00:10:53,873 The voltage divider-- 153 00:10:53,873 --> 00:10:54,859 AUDIENCE: Oh, yeah. 154 00:10:54,859 --> 00:10:58,063 PROFESSOR: So if there's no current here, then you can use 155 00:10:58,063 --> 00:10:59,800 the voltage relationship-- the voltage-divider 156 00:10:59,800 --> 00:11:01,070 relationship here. 157 00:11:01,070 --> 00:11:02,240 6 over (3 plus 6) times 90 -- 158 00:11:02,240 --> 00:11:04,290 which is 60. 159 00:11:04,290 --> 00:11:06,820 So if you use straightforward reasoning saying, what if the 160 00:11:06,820 --> 00:11:08,150 resistor was infinitely large? 161 00:11:08,150 --> 00:11:08,990 What would happen? 162 00:11:08,990 --> 00:11:11,760 Then you would conclude that the bottom line is OK. 163 00:11:11,760 --> 00:11:12,630 If you choose-- 164 00:11:12,630 --> 00:11:18,770 if you show that R equals V over I, are you done? 165 00:11:18,770 --> 00:11:20,800 What else has to be true in order for the 166 00:11:20,800 --> 00:11:22,050 numbers to be true? 167 00:11:28,910 --> 00:11:31,370 It's not very hard. 168 00:11:31,370 --> 00:11:35,306 AUDIENCE: The voltages drop out of the [UNINTELLIGIBLE]. 169 00:11:35,306 --> 00:11:37,790 The resistor has to be the same as the-- 170 00:11:37,790 --> 00:11:39,330 PROFESSOR: So the voltages-- 171 00:11:39,330 --> 00:11:40,300 that's an instance-- 172 00:11:40,300 --> 00:11:43,310 what you just said-- is an instance of KVL. 173 00:11:43,310 --> 00:11:46,360 Basically, the voltage around all of the loops better be 0. 174 00:11:46,360 --> 00:11:48,550 The sum of the voltages around all of the loops better be 0, 175 00:11:48,550 --> 00:11:51,290 and the sum of the currents in all of those closed surfaces 176 00:11:51,290 --> 00:11:52,250 better be 0, right? 177 00:11:52,250 --> 00:11:55,540 KVL better be satisfied everywhere, and KCL better be 178 00:11:55,540 --> 00:11:57,240 satisfied everywhere. 179 00:11:57,240 --> 00:11:59,740 So, in particular, we can ask about this node. 180 00:11:59,740 --> 00:12:01,660 We could say, do the currents flowing into 181 00:12:01,660 --> 00:12:05,580 that node sum to 0? 182 00:12:05,580 --> 00:12:08,560 So if you take a line here-- let's take this line-- 183 00:12:08,560 --> 00:12:10,710 if V0 happened to be-- 184 00:12:10,710 --> 00:12:14,070 V0 happens to be the voltage here, right? 185 00:12:14,070 --> 00:12:19,730 So if V0 were 36, then that puts some voltage on this leg. 186 00:12:19,730 --> 00:12:21,353 36 would what, 54? 187 00:12:24,490 --> 00:12:26,500 So then you'd have to say-- 188 00:12:26,500 --> 00:12:28,420 you'd have to compute, then, the current through here and 189 00:12:28,420 --> 00:12:30,410 the current through there, and see if those 190 00:12:30,410 --> 00:12:32,790 currents sum to 0. 191 00:12:32,790 --> 00:12:36,890 And, in fact, if you do those calculations, you if you can 192 00:12:36,890 --> 00:12:38,830 convince yourself of those are all true. 193 00:12:38,830 --> 00:12:40,690 So the answer was 0. 194 00:12:40,690 --> 00:12:43,640 All of those answers were true. 195 00:12:43,640 --> 00:12:46,320 The point of doing the exercise was to just remind 196 00:12:46,320 --> 00:12:50,260 you about how you solve circuits, but also to let us 197 00:12:50,260 --> 00:12:53,090 look at patterns. 198 00:12:53,090 --> 00:12:55,650 The interesting thing in this problem is the pattern the 199 00:12:55,650 --> 00:13:00,140 results between the V's and the I's. 200 00:13:00,140 --> 00:13:06,290 So if I were to make a plot, in fact, those V's and I's all 201 00:13:06,290 --> 00:13:09,180 fall on a straight line. 202 00:13:09,180 --> 00:13:12,400 Well that's pretty interesting. 203 00:13:12,400 --> 00:13:17,720 So if I plot just this V versus this I-- 204 00:13:17,720 --> 00:13:19,974 so V equals 0, 30, 36, 0, 30, 36, 48, 60 -- 205 00:13:23,970 --> 00:13:26,380 and the corresponding I's, 30,15,12,6,0 -- 206 00:13:26,380 --> 00:13:29,360 30,15,12,6,0. 207 00:13:29,360 --> 00:13:33,190 Those points all fall on a straight line. 208 00:13:33,190 --> 00:13:36,820 That suggests that there's some pattern here. 209 00:13:36,820 --> 00:13:39,630 And if there's a pattern, then there might be a way to 210 00:13:39,630 --> 00:13:40,730 exploit it. 211 00:13:40,730 --> 00:13:42,720 So that's what I'm trying to develop-- 212 00:13:42,720 --> 00:13:46,840 is a way to exploit the pattern that results when 213 00:13:46,840 --> 00:13:50,470 parts of circuits interact. 214 00:13:50,470 --> 00:13:52,180 The interesting thing that we-- 215 00:13:52,180 --> 00:13:55,160 so not only is it true that there's a simple pattern, but 216 00:13:55,160 --> 00:13:59,370 it turns out that the pattern is completely independent of 217 00:13:59,370 --> 00:14:02,003 the thing that I put on the right. 218 00:14:02,003 --> 00:14:06,165 The pattern is a property of the circuit to the left. 219 00:14:08,960 --> 00:14:12,170 One way to convince yourself of that is to substitute-- 220 00:14:12,170 --> 00:14:17,120 take that resistor out, and put in a voltage source, and 221 00:14:17,120 --> 00:14:19,540 redo the problem. 222 00:14:19,540 --> 00:14:23,180 This time, instead of assuming that there's an Ohm's Law-type 223 00:14:23,180 --> 00:14:26,120 resistor here, assume there's a constant voltage, and that 224 00:14:26,120 --> 00:14:30,410 constant voltage is adjusted to 0,30,36,48,60. 225 00:14:30,410 --> 00:14:32,600 If you re-solve that problem, you get 226 00:14:32,600 --> 00:14:35,930 exactly these same currents. 227 00:14:35,930 --> 00:14:41,710 The answer continues to fall on exactly the same line. 228 00:14:41,710 --> 00:14:43,270 So that's a very interesting pattern. 229 00:14:43,270 --> 00:14:46,800 The idea is that when the circuit on the left interacts 230 00:14:46,800 --> 00:14:49,430 with the circuit on the right, regardless of what the circuit 231 00:14:49,430 --> 00:14:53,490 on the right is, you get a simple relationship between 232 00:14:53,490 --> 00:14:55,730 the voltage and current that comes out of the 233 00:14:55,730 --> 00:14:56,980 circuit on the left. 234 00:14:59,570 --> 00:15:02,510 So that motivates the idea that we can think about the 235 00:15:02,510 --> 00:15:09,260 left-hand circuit as some kind of a generic part. 236 00:15:09,260 --> 00:15:12,490 We call that generic part a one-port. 237 00:15:12,490 --> 00:15:14,750 Think about this circuit on the left-- the thing that's in 238 00:15:14,750 --> 00:15:17,100 the red box-- 239 00:15:17,100 --> 00:15:19,120 it's got at its terminals-- 240 00:15:19,120 --> 00:15:22,490 the terminals are the things that poke through the red box. 241 00:15:22,490 --> 00:15:24,530 First off it has two terminals, two terminals is 242 00:15:24,530 --> 00:15:26,760 just like all of our other parts. 243 00:15:26,760 --> 00:15:29,020 And it's like a resistor, it's like a voltage source, it's 244 00:15:29,020 --> 00:15:33,800 like a current source, it's a two terminal device. 245 00:15:33,800 --> 00:15:36,460 And just like a voltage source or a resistor or a current 246 00:15:36,460 --> 00:15:39,480 source, there's some voltage across those terminals, and 247 00:15:39,480 --> 00:15:43,470 there's some current that flows in those terminals. 248 00:15:43,470 --> 00:15:45,170 So there's some current that goes in the plus. 249 00:15:45,170 --> 00:15:47,210 And that same current comes out the minus. 250 00:15:47,210 --> 00:15:50,080 That'll be true for this circuit, just the same as it's 251 00:15:50,080 --> 00:15:53,780 true for a resistor or voltage source or anything else, the 252 00:15:53,780 --> 00:15:57,580 only difference is, the thing that's inside the box-- 253 00:15:57,580 --> 00:15:59,100 the thing that's inside the one-port-- 254 00:15:59,100 --> 00:16:03,510 is more complicated than it was for a simple resistor or 255 00:16:03,510 --> 00:16:05,080 voltage source or current source. 256 00:16:05,080 --> 00:16:08,550 So what we can think about, is this whole red box just looks 257 00:16:08,550 --> 00:16:12,320 like a super part. 258 00:16:12,320 --> 00:16:16,700 So the interesting thing that happens is, this red box 259 00:16:16,700 --> 00:16:20,300 behaves like a one-port, like a super part. 260 00:16:20,300 --> 00:16:25,510 And just like a resistor has a relationship between V and I, 261 00:16:25,510 --> 00:16:29,610 V equals IR, Or a voltage source has some kind of a 262 00:16:29,610 --> 00:16:32,520 constraint, V equals V0. 263 00:16:32,520 --> 00:16:34,250 Or a current source has some kind of a 264 00:16:34,250 --> 00:16:36,460 constraint, I equals I0. 265 00:16:36,460 --> 00:16:40,740 This funny part has a relationship between V and I 266 00:16:40,740 --> 00:16:43,340 that's given by that curve. 267 00:16:43,340 --> 00:16:47,180 In some sense, it's not very different. 268 00:16:47,180 --> 00:16:48,510 So we think that-- 269 00:16:48,510 --> 00:16:52,960 so what we want to do now is figure out the rules that 270 00:16:52,960 --> 00:16:54,780 govern the currents and voltages that 271 00:16:54,780 --> 00:16:56,030 flow through one-ports. 272 00:16:58,270 --> 00:17:03,900 And in particular, how special was this straight line thing? 273 00:17:03,900 --> 00:17:05,990 I mean if they were always a straight line, that would be 274 00:17:05,990 --> 00:17:08,960 really easy, right? 275 00:17:08,960 --> 00:17:09,900 So the question-- 276 00:17:09,900 --> 00:17:12,650 so the next question I'd like to ask is, just how often are 277 00:17:12,650 --> 00:17:16,680 we expecting to see straight lines there? 278 00:17:16,680 --> 00:17:20,260 I already said, the primitive elements that we think about-- 279 00:17:20,260 --> 00:17:23,390 Ohm's Law, resistors, voltage sources, current sources-- 280 00:17:23,390 --> 00:17:28,109 they have straight line constraints between the 281 00:17:28,109 --> 00:17:32,170 voltages and currents that they can generate. 282 00:17:32,170 --> 00:17:33,080 So think about what I'm doing. 283 00:17:33,080 --> 00:17:35,630 I'm trying to think about a rule that's going to let me 284 00:17:35,630 --> 00:17:38,370 describe the currents and voltages into that red box 285 00:17:38,370 --> 00:17:42,340 from the previous slide, much like I would describe the 286 00:17:42,340 --> 00:17:46,680 voltages and currents in an Ohm's Law a resistor. 287 00:17:46,680 --> 00:17:48,910 I can tell you the voltage-current relationship, 288 00:17:48,910 --> 00:17:53,550 V equals IR for an Ohm's resistor, independent of what 289 00:17:53,550 --> 00:17:56,190 it's connected to. 290 00:17:56,190 --> 00:17:57,850 That's the reasoning that I'm using here. 291 00:17:57,850 --> 00:18:00,450 I'm going to try to figure out, independent of what it is 292 00:18:00,450 --> 00:18:03,150 connected to, what will be the voltage-current relationship 293 00:18:03,150 --> 00:18:05,640 for the red box? 294 00:18:05,640 --> 00:18:10,510 So the question is, when are we expecting straight lines, 295 00:18:10,510 --> 00:18:13,360 and when are we not expecting straight lines? 296 00:18:13,360 --> 00:18:15,480 So here's a simple circuit. 297 00:18:15,480 --> 00:18:18,650 Here's a super part made out of one linear resistor, one 298 00:18:18,650 --> 00:18:21,090 Ohm's Law resistor, and one voltage source. 299 00:18:21,090 --> 00:18:24,680 What's the current-voltage relationship for that part? 300 00:18:24,680 --> 00:18:29,280 What if I put a red box around the whole thing, and asked you 301 00:18:29,280 --> 00:18:32,920 to draw the I-V curve, the current voltage curve? 302 00:18:32,920 --> 00:18:35,120 Which would it look like, A,B,C,D -- 303 00:18:35,120 --> 00:18:36,670 which should have been (1), (2), (3), (4) -- 304 00:18:36,670 --> 00:18:41,380 so you can raise fingers, or none of the above? 305 00:18:41,380 --> 00:18:42,440 So take 30 seconds. 306 00:18:42,440 --> 00:18:44,985 Figure out what would be the I-V curve for this part. 307 00:21:20,050 --> 00:21:24,850 So if you map A to D into (1) to (4) -- 308 00:21:24,850 --> 00:21:28,290 which of those plots describes the current-voltage 309 00:21:28,290 --> 00:21:29,610 relationship for that circuit? 310 00:21:33,230 --> 00:21:34,890 Map A to D to (1) to (4). 311 00:21:37,480 --> 00:21:42,890 OK, you're quiet, and the success rate is smaller than 312 00:21:42,890 --> 00:21:46,240 usual, about 50% correct. 313 00:21:46,240 --> 00:21:48,800 Take 30 seconds, reconsider your answer, try to get this 314 00:21:48,800 --> 00:21:50,050 up to 85% or so. 315 00:22:13,258 --> 00:23:13,150 [CHATTER] 316 00:23:13,150 --> 00:23:15,920 So which is the best plot? 317 00:23:15,920 --> 00:23:21,140 Which plot best characterizes the circuit on the top? 318 00:23:21,140 --> 00:23:22,620 OK, that's much better. 319 00:23:22,620 --> 00:23:24,000 That's about 75%. 320 00:23:24,000 --> 00:23:27,490 Certainly not 100%, but better. 321 00:23:27,490 --> 00:23:30,970 OK so how do you think about this? 322 00:23:30,970 --> 00:23:33,960 One way to think about it is special cases. 323 00:23:33,960 --> 00:23:35,900 Can you think about any special cases that are 324 00:23:35,900 --> 00:23:37,450 particularly easy to check? 325 00:23:40,170 --> 00:23:41,680 0 -- 326 00:23:41,680 --> 00:23:46,295 OK, what equals 0? 327 00:23:46,295 --> 00:23:47,430 AUDIENCE: V or I -- 328 00:23:47,430 --> 00:23:48,490 PROFESSOR: V or I -- 329 00:23:48,490 --> 00:23:48,960 yes. 330 00:23:48,960 --> 00:23:50,340 So what if you were to make -- 331 00:23:50,340 --> 00:23:54,120 if you were to make I equal to 0? 332 00:23:54,120 --> 00:23:55,950 First off, if you made I equal to 0, what's 333 00:23:55,950 --> 00:23:58,982 special in the plots? 334 00:23:58,982 --> 00:24:03,240 If I equals 0, then you round up x-axis. 335 00:24:03,240 --> 00:24:07,470 Sorry, if you make I 0, what will V be? 336 00:24:07,470 --> 00:24:12,775 If you made I be 0, how big will be V? 337 00:24:12,775 --> 00:24:14,980 5 Volts, right? 338 00:24:14,980 --> 00:24:17,740 If you make I be 0, then there's no 339 00:24:17,740 --> 00:24:19,700 voltage across the resistor. 340 00:24:19,700 --> 00:24:22,280 So the total voltage here will be the same as the voltage 341 00:24:22,280 --> 00:24:25,690 across the voltage source, V equals 5. 342 00:24:25,690 --> 00:24:29,310 So the intersection on the x-axis should be at the point 343 00:24:29,310 --> 00:24:30,610 V equals 5. 344 00:24:30,610 --> 00:24:32,780 Makes sense? 345 00:24:32,780 --> 00:24:36,400 Now if you said V equals 0, how big would I be? 346 00:24:40,810 --> 00:24:46,660 If you set V equal to 0 -- how do you set V equal to 0? 347 00:24:46,660 --> 00:24:47,654 AUDIENCE: Negative 2.5. 348 00:24:47,654 --> 00:24:51,133 PROFESSOR: Negative 2.5, so what is negative 2.5? 349 00:24:51,133 --> 00:24:53,630 AUDIENCE: The value of I. 350 00:24:53,630 --> 00:24:55,416 PROFESSOR: So do you need-- 351 00:24:55,416 --> 00:24:59,900 how do we set V equal to 0? 352 00:24:59,900 --> 00:25:04,682 What's the circuit way of saying, set V equal to 0? 353 00:25:09,030 --> 00:25:11,530 You set V equals 0 in a circuit by-- 354 00:25:11,530 --> 00:25:12,450 AUDIENCE: Grounding. 355 00:25:12,450 --> 00:25:14,030 PROFESSOR: --grounding, by setting it-- 356 00:25:14,030 --> 00:25:15,540 by putting in a wire. 357 00:25:15,540 --> 00:25:18,516 So you run a wire from here to here. 358 00:25:18,516 --> 00:25:20,430 Voltage across a wire is always 0, right? 359 00:25:20,430 --> 00:25:22,680 So you set something to 0 by putting a wire across it. 360 00:25:22,680 --> 00:25:25,220 We call that a short circuit. 361 00:25:25,220 --> 00:25:27,990 OK, there's a short path for the voltage travel. 362 00:25:27,990 --> 00:25:34,100 So you put a short circuit here, and then I becomes V 363 00:25:34,100 --> 00:25:40,330 over R, and the only trick is that it's up, right? 364 00:25:40,330 --> 00:25:43,050 Current likes to flow down the electrochemical gradient. 365 00:25:43,050 --> 00:25:46,760 So it likes to go down the electrical gradient. 366 00:25:46,760 --> 00:25:50,580 So the current is going to go up, but the reference 367 00:25:50,580 --> 00:25:56,790 direction for this I is down, so I is minus 2.5. 368 00:25:56,790 --> 00:26:01,310 So the intersection on this axis is minus 2.5. 369 00:26:01,310 --> 00:26:03,710 So we know that it has to go through a negative on the 370 00:26:03,710 --> 00:26:09,130 bottom, and it has to go through a positive V on the 371 00:26:09,130 --> 00:26:12,670 x-axis, and the only curve that does that is (A). 372 00:26:12,670 --> 00:26:15,180 And if we wanted to be a little bit more fancy, we 373 00:26:15,180 --> 00:26:18,210 could figure out the general rule. 374 00:26:18,210 --> 00:26:20,870 We could just write an expression for Vr. 375 00:26:20,870 --> 00:26:24,040 Well by KVL, Vr is always the difference 376 00:26:24,040 --> 00:26:27,190 between V and 5 Volts. 377 00:26:27,190 --> 00:26:29,040 And we could write an expression for the current 378 00:26:29,040 --> 00:26:29,780 through the resistor. 379 00:26:29,780 --> 00:26:34,730 That's just V over R. Vr is V minus 5, R is 2. 380 00:26:34,730 --> 00:26:37,770 So I get some expression which is, lo and behold, linear in 381 00:26:37,770 --> 00:26:42,660 V. So just like that more complicated circuit that I'd 382 00:26:42,660 --> 00:26:46,860 looked at, I ended up with a straight line relationship in 383 00:26:46,860 --> 00:26:47,990 the current voltage-- 384 00:26:47,990 --> 00:26:49,980 the relationship between the current and voltage falls on a 385 00:26:49,980 --> 00:26:52,990 straight line. 386 00:26:52,990 --> 00:26:55,160 So how special is that? 387 00:26:55,160 --> 00:26:58,770 Well it actually happens pretty robustly. 388 00:26:58,770 --> 00:27:01,990 Think about what would happen if I had two parts, the 389 00:27:01,990 --> 00:27:03,130 generically-- 390 00:27:03,130 --> 00:27:06,840 so I'm thinking about these being generic boxes, inside 391 00:27:06,840 --> 00:27:07,710 could be anything. 392 00:27:07,710 --> 00:27:09,430 There could be a current source, a voltage source, a 393 00:27:09,430 --> 00:27:13,060 linear resistor, or some other one-port. 394 00:27:13,060 --> 00:27:15,590 So if I had a generic box here, and a generic box here, 395 00:27:15,590 --> 00:27:20,320 and I connected them in parallel, under the condition 396 00:27:20,320 --> 00:27:25,020 that each of the generic boxes had a straight line 397 00:27:25,020 --> 00:27:29,060 current-voltage relationship, it's easy to argue that the 398 00:27:29,060 --> 00:27:32,210 resulting relationship between current and voltage for the 399 00:27:32,210 --> 00:27:35,940 parallel combination is also a straight line. 400 00:27:35,940 --> 00:27:39,060 All you need to do is realize that if you hook two things in 401 00:27:39,060 --> 00:27:44,300 parallel, the parallel voltage is the same as V1 and V2. 402 00:27:44,300 --> 00:27:46,210 So all the voltages are the same. 403 00:27:46,210 --> 00:27:47,220 And the currents add. 404 00:27:47,220 --> 00:27:50,270 So Ip, the current end of the parallel combination is the 405 00:27:50,270 --> 00:27:51,520 sum of the two I's. 406 00:27:53,760 --> 00:27:56,860 So if you think about the relationship between V1 and 407 00:27:56,860 --> 00:28:00,730 I1, and the relationship between V2 and I2, you could 408 00:28:00,730 --> 00:28:02,760 derive the relationship between Ip 409 00:28:02,760 --> 00:28:07,020 and Vp by just adding. 410 00:28:07,020 --> 00:28:11,710 V1 equals v2 equals Vp, that's what we saw here. 411 00:28:11,710 --> 00:28:13,650 And the current at the bottom is the sum of 412 00:28:13,650 --> 00:28:14,900 the other two currents. 413 00:28:17,480 --> 00:28:20,170 So, in particular, if this was a straight line, and that was 414 00:28:20,170 --> 00:28:22,490 a straight line, the sum of two straight lines is a 415 00:28:22,490 --> 00:28:24,830 straight line, right? 416 00:28:24,830 --> 00:28:28,090 If you add two straight lines, you get a new straight line. 417 00:28:28,090 --> 00:28:31,200 So what this shows is that if you started with parts that 418 00:28:31,200 --> 00:28:34,630 were themselves straight lines, parallel combinations 419 00:28:34,630 --> 00:28:39,960 would generate a new part with another straight line. 420 00:28:39,960 --> 00:28:42,390 Same sort of thing happens if the two parts we're in series. 421 00:28:45,200 --> 00:28:48,100 In series we would have Iseries -- 422 00:28:48,100 --> 00:28:50,820 it's the same as I1 equals I2-- 423 00:28:50,820 --> 00:28:54,300 so that's the equivalent of the y-axes, where previously 424 00:28:54,300 --> 00:28:55,660 we had equivalents of the x-axes. 425 00:28:58,370 --> 00:29:03,250 And now if both of these boxes had linear I-V curves, I would 426 00:29:03,250 --> 00:29:07,290 now add horizontally rather than adding vertically. 427 00:29:07,290 --> 00:29:08,840 But I have the same result. 428 00:29:08,840 --> 00:29:13,470 If the two, individual one-ports had linear I-V 429 00:29:13,470 --> 00:29:19,280 curves, and if I add horizontally, I'll get a new 430 00:29:19,280 --> 00:29:20,750 linear I-V curve. 431 00:29:23,642 --> 00:29:26,910 And in fact, if you start-- 432 00:29:26,910 --> 00:29:31,830 if you put any combination of parts that have linear I-V 433 00:29:31,830 --> 00:29:37,350 curves together to form a new circuit-- 434 00:29:37,350 --> 00:29:41,470 a new one-port, the I-V curve for that new one-port will be 435 00:29:41,470 --> 00:29:43,280 a linear function. 436 00:29:43,280 --> 00:29:46,530 And the way to see that, is to think about linear equations. 437 00:29:49,160 --> 00:29:52,570 Remember when we solve a circuit, we have to have one 438 00:29:52,570 --> 00:29:57,385 law for every element, Like Ohm's Law-- it's V equals IR, 439 00:29:57,385 --> 00:30:00,480 or voltage source is V equals V0 or something like that-- 440 00:30:00,480 --> 00:30:03,970 we need one law for every element. 441 00:30:03,970 --> 00:30:07,570 And then we have KVL and KCL. 442 00:30:07,570 --> 00:30:11,310 Well all of the-- if we start with the assumption that each 443 00:30:11,310 --> 00:30:14,210 component has a straight line relationship between voltage 444 00:30:14,210 --> 00:30:16,760 and current, which is true for linear resistors, voltage 445 00:30:16,760 --> 00:30:19,000 sources, and current sources. 446 00:30:19,000 --> 00:30:23,020 Those equations that describe those parts are what we call 447 00:30:23,020 --> 00:30:25,270 linear equations. 448 00:30:25,270 --> 00:30:33,570 Linear equation is an equation where the function of the 449 00:30:33,570 --> 00:30:35,760 unknowns is a quote -- 450 00:30:35,760 --> 00:30:36,950 linear function. 451 00:30:36,950 --> 00:30:38,920 It has the form-- 452 00:30:38,920 --> 00:30:42,750 some constant times an unknown, plus some other 453 00:30:42,750 --> 00:30:46,140 constant times some other unknown, plus, in principle, 454 00:30:46,140 --> 00:30:48,310 any number of those. 455 00:30:48,310 --> 00:30:49,923 But it has to be a linear function, so you're not 456 00:30:49,923 --> 00:30:52,950 allowed to have things like V-squared in there. 457 00:30:52,950 --> 00:30:55,480 But if you think about things like voltage sources and 458 00:30:55,480 --> 00:30:59,130 current sources and Ohm's Law, they don't have squares in 459 00:30:59,130 --> 00:31:02,180 there, they're linear equations. 460 00:31:02,180 --> 00:31:05,000 And the idea then, when we're solving for the I-V 461 00:31:05,000 --> 00:31:08,810 relationship for this new one-port thing, all we're 462 00:31:08,810 --> 00:31:12,760 doing is we're solving a system of linear equation. 463 00:31:12,760 --> 00:31:16,820 Sort of in the abstract, the idea is that we write down all 464 00:31:16,820 --> 00:31:19,140 of the component equations inside the box, we write down 465 00:31:19,140 --> 00:31:22,570 all of the relevant KVL and KCL, and then we solve. 466 00:31:22,570 --> 00:31:24,570 So they're all going to have-- 467 00:31:24,570 --> 00:31:26,850 each one of those equations is going to be linear, so it's 468 00:31:26,850 --> 00:31:32,200 going to be something like, say, a0x0 if x is my unknown. 469 00:31:32,200 --> 00:31:35,270 'x' could be a current or a voltage. 470 00:31:35,270 --> 00:31:40,080 And then I might have sum of x1, then I might have sum of 471 00:31:40,080 --> 00:31:43,020 x2, and I might have a whole bunch of things, and they all 472 00:31:43,020 --> 00:31:48,790 add together to be some constant, like that. 473 00:31:48,790 --> 00:31:51,550 So that might represent part one-- 474 00:31:51,550 --> 00:31:55,270 that might represent one of the components in the box. 475 00:31:55,270 --> 00:31:57,310 Then I would have another component, which would be some 476 00:31:57,310 --> 00:31:59,075 other linear equation. 477 00:32:06,490 --> 00:32:10,930 Then I'd have a KCL equation, well KCL is easy, because that 478 00:32:10,930 --> 00:32:13,410 only has currents in it, and the multipliers are 479 00:32:13,410 --> 00:32:15,260 all 1 or minus 1. 480 00:32:15,260 --> 00:32:16,670 Right, so that's clearly linear. 481 00:32:16,670 --> 00:32:18,300 And I have KVL equations. 482 00:32:18,300 --> 00:32:19,840 Those are linear. 483 00:32:19,840 --> 00:32:24,260 And the point is, that when you solve linear equations, 484 00:32:24,260 --> 00:32:26,240 you get a new linear equation. 485 00:32:26,240 --> 00:32:30,200 So think about solving this by the method of substitution. 486 00:32:30,200 --> 00:32:34,060 I could figure out what is x0 here. 487 00:32:34,060 --> 00:32:36,860 If I use this equation to figure out x0. 488 00:32:36,860 --> 00:32:41,000 x0 would be a linear function of all the other x's. 489 00:32:41,000 --> 00:32:45,290 So then if I plug that linear function into here, I replace 490 00:32:45,290 --> 00:32:48,420 this linear equation with a new linear equation with one 491 00:32:48,420 --> 00:32:50,530 fewer unknown. 492 00:32:50,530 --> 00:32:53,980 if I keep doing that, I just keep replacing linear 493 00:32:53,980 --> 00:32:55,780 equations with other linear equations. 494 00:32:55,780 --> 00:32:59,350 When I'm all done, I'm left with a new linear equation. 495 00:32:59,350 --> 00:33:05,030 That's why, if you start with parts that linear I-V curves, 496 00:33:05,030 --> 00:33:08,230 you'll end up with a straight line I-V curve. 497 00:33:08,230 --> 00:33:12,520 So that idea that the I-V function is a straight line is 498 00:33:12,520 --> 00:33:13,380 quite robust. 499 00:33:13,380 --> 00:33:18,420 It will happen anytime you go to circuit out of ideal 500 00:33:18,420 --> 00:33:21,360 parts-- whereby ideal parts, I mean Ohm's Law resistors, 501 00:33:21,360 --> 00:33:25,190 voltage sources, and current sources. 502 00:33:25,190 --> 00:33:30,310 So that has a very interesting circuit interpretation. 503 00:33:30,310 --> 00:33:34,160 If I know that an arbitrary circuit can be represented by 504 00:33:34,160 --> 00:33:35,090 a straight line-- 505 00:33:35,090 --> 00:33:36,680 if the I-V curve can be represented 506 00:33:36,680 --> 00:33:38,350 by a straight line-- 507 00:33:38,350 --> 00:33:43,180 well that generates an equivalent set. 508 00:33:43,180 --> 00:33:46,160 There's obviously more than one circuit that could 509 00:33:46,160 --> 00:33:48,960 generate the same straight line. 510 00:33:48,960 --> 00:33:53,480 Here's a circuit that can generate that straight line. 511 00:33:53,480 --> 00:33:56,610 In fact, it will always be true that I can generate a 512 00:33:56,610 --> 00:33:59,870 circuit with one voltage source and one Ohm's Law 513 00:33:59,870 --> 00:34:05,130 resistor that will mimic the behavior of any arbitrary 514 00:34:05,130 --> 00:34:07,630 combination of resistors, voltage 515 00:34:07,630 --> 00:34:09,940 sources, and current sources. 516 00:34:09,940 --> 00:34:11,850 All I need to do is think about, you take the 517 00:34:11,850 --> 00:34:17,170 complicated thing, figure out its straight line plot, now 518 00:34:17,170 --> 00:34:20,909 you read off some critical numbers from this plot. 519 00:34:20,909 --> 00:34:25,500 You say, OK, well what if the current were 0? 520 00:34:25,500 --> 00:34:28,630 Well if the current were 0, I'd be on this axis. 521 00:34:28,630 --> 00:34:34,320 If the current were 0 in the circuit, V would be V0. 522 00:34:34,320 --> 00:34:37,110 So that means you look over here, you figure out the 523 00:34:37,110 --> 00:34:41,020 x-intercept, and the x-intercept is the value of 524 00:34:41,020 --> 00:34:42,270 the voltage source. 525 00:34:46,130 --> 00:34:52,190 Similarly, if you figure out the rate of growth of I -- 526 00:34:52,190 --> 00:34:57,240 so more generally, if you solve for I, I will be the 527 00:34:57,240 --> 00:35:01,110 difference between V and V0 divided by R, that's just 528 00:35:01,110 --> 00:35:04,540 Ohm's law for the resistor. 529 00:35:04,540 --> 00:35:09,580 And that then lets you figure out the law for the slope. 530 00:35:09,580 --> 00:35:12,710 The slope over here is going to turn out to be one over 531 00:35:12,710 --> 00:35:13,260 this resistor-- 532 00:35:13,260 --> 00:35:15,130 I should have had-- this should be R0, 533 00:35:15,130 --> 00:35:16,530 these two should match. 534 00:35:16,530 --> 00:35:22,170 So the slope of this line is one over that resistor. 535 00:35:22,170 --> 00:35:25,980 So in general, regardless of how complicated the box is, 536 00:35:25,980 --> 00:35:29,110 figure out the straight line and I-V curves, read off the 537 00:35:29,110 --> 00:35:33,070 x-axis, read off the slope, and that lets you construct a 538 00:35:33,070 --> 00:35:36,590 simple circuit that has the same Vi curve. 539 00:35:36,590 --> 00:35:40,390 We call that circuit the Thevenin Equivalent. 540 00:35:40,390 --> 00:35:44,470 What that means is, you can think about a complicated 541 00:35:44,470 --> 00:35:47,980 circuit, regardless of how many parts, by a circuit it 542 00:35:47,980 --> 00:35:50,270 just has two. 543 00:35:50,270 --> 00:35:54,620 That's an abstraction that lets us think more simply 544 00:35:54,620 --> 00:36:00,230 about complicated circuits even if there's no buffers. 545 00:36:00,230 --> 00:36:03,990 This is true always. 546 00:36:03,990 --> 00:36:05,650 OK, it's not true-- 547 00:36:05,650 --> 00:36:08,980 it's not it's not the case that if I change the thing 548 00:36:08,980 --> 00:36:10,460 that I put here-- 549 00:36:10,460 --> 00:36:13,550 if I change the thing I put here, this relationship is 550 00:36:13,550 --> 00:36:14,350 still true. 551 00:36:14,350 --> 00:36:20,900 It's the equivalent of Ohm's Law for complicated circuits. 552 00:36:20,900 --> 00:36:23,010 Ohm's Law is what I get if what's in the 553 00:36:23,010 --> 00:36:26,050 box is a single resistor. 554 00:36:26,050 --> 00:36:28,760 If what's in the box is complicated, more generally 555 00:36:28,760 --> 00:36:33,170 this is what I get, the Thevenin Equivalent. 556 00:36:33,170 --> 00:36:34,970 Of course there's lots of circuits that 557 00:36:34,970 --> 00:36:38,380 have this I-V curve. 558 00:36:38,380 --> 00:36:41,320 So a different one is a current 559 00:36:41,320 --> 00:36:43,490 source with a resistor. 560 00:36:43,490 --> 00:36:46,570 That's called a Norton Equivalent. 561 00:36:46,570 --> 00:36:47,930 You do the same sort of thing. 562 00:36:47,930 --> 00:36:52,190 What would happen if here, if I set V to be 0? 563 00:36:52,190 --> 00:36:54,030 Well how do I make V be 0? 564 00:36:54,030 --> 00:36:55,980 I make V be 0 -- 565 00:36:55,980 --> 00:36:57,720 Did I write that on the slide? 566 00:36:57,720 --> 00:36:58,520 no I didn't write it on the slide. 567 00:36:58,520 --> 00:37:01,480 I make V be 0 by putting a short circuit here. 568 00:37:01,480 --> 00:37:03,220 I connect a wire. 569 00:37:03,220 --> 00:37:07,450 If I put a wire here, then how big is the current I? 570 00:37:10,710 --> 00:37:14,870 Well if I put a wire here, it's easier for this current 571 00:37:14,870 --> 00:37:17,300 to go through the wire than it is to go 572 00:37:17,300 --> 00:37:19,600 through that resistor. 573 00:37:19,600 --> 00:37:23,580 So all of this current goes through the wire. 574 00:37:23,580 --> 00:37:26,800 All of this current is I0. 575 00:37:26,800 --> 00:37:30,570 So if I put a short circuit across V -- 576 00:37:30,570 --> 00:37:34,660 if I make V be 0, how big is i? 577 00:37:34,660 --> 00:37:36,430 Minus I0 right? 578 00:37:36,430 --> 00:37:38,260 All the current I0 goes through, it 579 00:37:38,260 --> 00:37:41,140 just goes to backwards. 580 00:37:41,140 --> 00:37:44,400 So that's how I get this point. 581 00:37:44,400 --> 00:37:49,130 So if I wanted to replace some complicated circuit with a 582 00:37:49,130 --> 00:37:54,830 Norton Equivalent, I would take the complicated system, 583 00:37:54,830 --> 00:38:01,270 to figure out the I-V curve, read off the intercept on this 584 00:38:01,270 --> 00:38:07,030 axis, change the sign-- 585 00:38:07,030 --> 00:38:09,370 that's the most confusing part, by the way. 586 00:38:09,370 --> 00:38:12,840 This is the error that you all make. 587 00:38:12,840 --> 00:38:15,460 There's a minus sign in the current relationship. 588 00:38:15,460 --> 00:38:19,370 We like to draw-- it just makes us feel good to have the 589 00:38:19,370 --> 00:38:22,260 arrow go up. 590 00:38:22,260 --> 00:38:24,420 And if you make the arrow go up, it's in the wrong 591 00:38:24,420 --> 00:38:30,710 direction to I. So when we did the Thevenin equivalent, there 592 00:38:30,710 --> 00:38:34,150 was no sign from it, this was V0. 593 00:38:34,150 --> 00:38:36,700 So the voltage on the voltage source is equal to the 594 00:38:36,700 --> 00:38:38,140 x-intercept. 595 00:38:38,140 --> 00:38:41,015 But when we do the Norton, the current in the current source 596 00:38:41,015 --> 00:38:46,280 is minus the y-intercept. 597 00:38:46,280 --> 00:38:47,650 Same idea, though. 598 00:38:47,650 --> 00:38:48,250 OK? 599 00:38:48,250 --> 00:38:51,590 So the idea is that Thevenin and Norton equivalent circuits 600 00:38:51,590 --> 00:38:54,150 are equivalent, in the sense that they 601 00:38:54,150 --> 00:38:57,200 generate the same voltage. 602 00:38:57,200 --> 00:38:59,160 that the more complicated circuit did. 603 00:38:59,160 --> 00:39:02,060 So that means for thinking about the circuit we can 604 00:39:02,060 --> 00:39:06,580 ignore the complicated stuff, and just know two numbers, V0 605 00:39:06,580 --> 00:39:08,486 and R0, or I0 and R0. 606 00:39:14,550 --> 00:39:21,060 So one more step, what this all means is, that if you can 607 00:39:21,060 --> 00:39:25,180 represent the current voltage relationship for an arbitrary 608 00:39:25,180 --> 00:39:30,120 circuit in terms of a straight line, that means that when 609 00:39:30,120 --> 00:39:32,820 we're trying to characterize an arbitrary circuit, it 610 00:39:32,820 --> 00:39:34,580 doesn't matter how complicated it is. 611 00:39:34,580 --> 00:39:38,480 It could have 100 parts in it. 612 00:39:38,480 --> 00:39:41,830 Regardless of how complicated it is, I only need to measure 613 00:39:41,830 --> 00:39:46,060 two things in order to fully characterize it. 614 00:39:46,060 --> 00:39:49,870 If a circuit is made out of linear resistors, voltage 615 00:39:49,870 --> 00:39:54,600 sources, and current sources, three special linear parts-- 616 00:39:54,600 --> 00:39:58,110 if a circuit is composed entirely out of linear parts, 617 00:39:58,110 --> 00:40:00,490 doesn't matter how many parts are in it, there could 100, 618 00:40:00,490 --> 00:40:02,130 there could be 1,000-- 619 00:40:02,130 --> 00:40:05,180 I only need to measure two things to get a complete 620 00:40:05,180 --> 00:40:06,610 description. 621 00:40:06,610 --> 00:40:09,800 And that's because two points determine a straight line. 622 00:40:09,800 --> 00:40:14,820 I know, by having proved it by using linear algebra, I know 623 00:40:14,820 --> 00:40:17,330 by using linear algebra that the solution 624 00:40:17,330 --> 00:40:18,530 is a straight line. 625 00:40:18,530 --> 00:40:20,840 I know from geometry that two points 626 00:40:20,840 --> 00:40:22,160 determine a straight line. 627 00:40:22,160 --> 00:40:24,830 I only need to find out two points. 628 00:40:24,830 --> 00:40:28,250 So by convention, the easiest two points is usually the 629 00:40:28,250 --> 00:40:29,740 simplest cases. 630 00:40:29,740 --> 00:40:33,530 Set the voltage to 0 and set the current to be 0. 631 00:40:33,530 --> 00:40:36,810 So that motivates the idea that if I have an arbitrary 632 00:40:36,810 --> 00:40:41,240 circuit, if I want to figure out this reduced complexity 633 00:40:41,240 --> 00:40:42,900 abstraction-- 634 00:40:42,900 --> 00:40:45,100 say I want to make a Thevenin Equivalent-- 635 00:40:45,100 --> 00:40:48,460 what I would do is first to ask the question, how big is 636 00:40:48,460 --> 00:40:52,330 the open circuit voltage? 637 00:40:52,330 --> 00:40:54,210 Open circuit means there's no 638 00:40:54,210 --> 00:40:56,420 connection between the terminals. 639 00:40:56,420 --> 00:40:59,380 That means the current is 0. 640 00:40:59,380 --> 00:41:01,476 If the current is 0, I'm on the x-axis. 641 00:41:04,060 --> 00:41:06,610 So over here I'm on the x-axis, so I'm thinking about 642 00:41:06,610 --> 00:41:09,000 the red point. 643 00:41:09,000 --> 00:41:12,830 Open circuit over here means there's no connection here, 644 00:41:12,830 --> 00:41:16,590 which means the current is 0, and all I need to ask is-- 645 00:41:16,590 --> 00:41:18,960 regardless of how complicated it is the circuit-- 646 00:41:18,960 --> 00:41:23,100 how big is the voltage that I would measure here? 647 00:41:23,100 --> 00:41:25,690 So in this circuit, if I have a-- 648 00:41:29,630 --> 00:41:32,430 yeah, I confused myself for a moment. 649 00:41:32,430 --> 00:41:37,440 if this current is 0, then this voltage drop is 0. 650 00:41:37,440 --> 00:41:41,040 This voltage source prescribes the voltage between these 651 00:41:41,040 --> 00:41:45,930 nodes to be 1 Volt, so V0 is 1 Volt. 652 00:41:45,930 --> 00:41:47,710 So I just fell one point. 653 00:41:47,710 --> 00:41:53,480 I set I to be 0, I found the open circuit voltage. 654 00:41:53,480 --> 00:41:55,440 Then I need to find one other point, because I 655 00:41:55,440 --> 00:41:56,740 only to find two. 656 00:41:56,740 --> 00:42:00,590 So I'll find the short circuit current. 657 00:42:00,590 --> 00:42:03,540 Imagine that I put a wire between the input and the 658 00:42:03,540 --> 00:42:07,230 output, and I'll compute how much current that circuit 659 00:42:07,230 --> 00:42:09,870 generates in that wire. 660 00:42:09,870 --> 00:42:12,640 Again the only confusing part is that the reference 661 00:42:12,640 --> 00:42:14,540 directions are backwards to the way you might have 662 00:42:14,540 --> 00:42:16,220 expected them to be. 663 00:42:16,220 --> 00:42:20,020 If I put a short circuit here, then the voltage between these 664 00:42:20,020 --> 00:42:21,400 nodes is still 1 Volt. 665 00:42:21,400 --> 00:42:24,540 That's what the voltage source always says. 666 00:42:24,540 --> 00:42:29,100 So the current that flows through this wire, the short 667 00:42:29,100 --> 00:42:34,160 circuit current, is just this V over that R, except it's in 668 00:42:34,160 --> 00:42:35,410 the negative direction. 669 00:42:38,640 --> 00:42:41,320 And the way you can think about that is that the slope 670 00:42:41,320 --> 00:42:43,460 of this curve has to be positive. 671 00:42:43,460 --> 00:42:45,710 I need-- because of the way Ohm's Law works-- 672 00:42:45,710 --> 00:42:48,780 increasing the voltage better tends to increase the current, 673 00:42:48,780 --> 00:42:51,090 because that's what Ohm's Law resistors do. 674 00:42:51,090 --> 00:42:54,230 So I need to have this slope be positive if it's going to 675 00:42:54,230 --> 00:42:56,720 be Ohm's Law. 676 00:42:56,720 --> 00:43:00,350 So I characterize just those two points, and then the 677 00:43:00,350 --> 00:43:04,570 resistance is simply the ratio of the two. 678 00:43:04,570 --> 00:43:07,560 So the resistance is related to the slope. 679 00:43:07,560 --> 00:43:09,240 It's 1 over the slope, you don't need 680 00:43:09,240 --> 00:43:10,380 to worry about that. 681 00:43:10,380 --> 00:43:13,250 The resistance is always V over I, so you just take the 682 00:43:13,250 --> 00:43:15,900 open circuit voltage and divide by the short circuit 683 00:43:15,900 --> 00:43:20,150 current, minus sign, and you get that the 684 00:43:20,150 --> 00:43:21,530 resistor must be 2 Ohms. 685 00:43:21,530 --> 00:43:23,200 Is that all clear? 686 00:43:23,200 --> 00:43:26,760 The idea is that we're trying to build an abstraction that 687 00:43:26,760 --> 00:43:31,000 lets us simplify the way we think about circuits, without 688 00:43:31,000 --> 00:43:32,250 introducing buffers. 689 00:43:34,450 --> 00:43:39,900 So that means then that these two circuits are equivalent to 690 00:43:39,900 --> 00:43:43,110 that circuit, in the sense that they all share 691 00:43:43,110 --> 00:43:45,110 the same I-V curve. 692 00:43:45,110 --> 00:43:47,640 If you substituted one server for the other, you couldn't 693 00:43:47,640 --> 00:43:52,230 tell from outside the red box which was on the 694 00:43:52,230 --> 00:43:56,970 inside of the red box. 695 00:43:56,970 --> 00:43:58,830 OK so I'll do an example now. 696 00:43:58,830 --> 00:44:01,050 Think about, what if I want to find the equivalent-- 697 00:44:01,050 --> 00:44:03,620 the Thevenin equivalent for this circuit. 698 00:44:03,620 --> 00:44:06,200 I just do the things I just told you to do. 699 00:44:06,200 --> 00:44:09,910 First thing I think about is, what's the 700 00:44:09,910 --> 00:44:12,535 open circuit voltage? 701 00:44:12,535 --> 00:44:14,740 So If I think about open circuit, there's 702 00:44:14,740 --> 00:44:17,290 no connection here. 703 00:44:17,290 --> 00:44:20,430 That means this current is 0. 704 00:44:20,430 --> 00:44:22,450 That means that the voltage that develops 705 00:44:22,450 --> 00:44:25,140 is the voltage divider. 706 00:44:25,140 --> 00:44:26,670 So I get 7 and a 1/2 volts-- 707 00:44:26,670 --> 00:44:31,750 3 over (1 plus 3) times 10, right. 708 00:44:31,750 --> 00:44:33,660 So that gives me one point-- 709 00:44:33,660 --> 00:44:36,620 that tells me the voltage source for the Thevenin 710 00:44:36,620 --> 00:44:37,940 equivalent. 711 00:44:37,940 --> 00:44:40,580 Then for the second point, I want to think about the short 712 00:44:40,580 --> 00:44:41,790 circuit current. 713 00:44:41,790 --> 00:44:46,550 So I consider putting a wire here, and then I compute the 714 00:44:46,550 --> 00:44:49,990 amount of current that flows in that wire. 715 00:44:49,990 --> 00:44:54,610 And in this circuit, this wire shorts out that resistor, so 716 00:44:54,610 --> 00:44:57,180 all the current goes through this wire, and none of the 717 00:44:57,180 --> 00:44:59,720 current goes through that resistor. 718 00:44:59,720 --> 00:45:02,450 So that means that the total current that flows is 10 Volts 719 00:45:02,450 --> 00:45:06,870 divided by 1 Ohm, which is 10 Amps. 720 00:45:06,870 --> 00:45:09,190 And then I know that the equivalent resistance is the 721 00:45:09,190 --> 00:45:11,980 ratio of the open circuit voltage to the short circuit 722 00:45:11,980 --> 00:45:15,630 current, except I have to worry about the minus sign. 723 00:45:15,630 --> 00:45:19,190 And so I end up with the equivalent resistance being 724 00:45:19,190 --> 00:45:22,100 the 7.5 Volts, which is the open circuit voltage, divided 725 00:45:22,100 --> 00:45:25,760 by 10 Amps, so I get 7.5 Ohms. 726 00:45:25,760 --> 00:45:29,280 And so the answer then, is that here is the circuit I 727 00:45:29,280 --> 00:45:31,970 started with, here's the Thevenin equivalent circuit, 728 00:45:31,970 --> 00:45:33,970 they're identical in the sense that they have 729 00:45:33,970 --> 00:45:36,390 the same I-V curve. 730 00:45:36,390 --> 00:45:39,300 And you can just sort of see why that has to be true. 731 00:45:39,300 --> 00:45:43,020 If you think about, here's the two circuits, and if you think 732 00:45:43,020 --> 00:45:47,490 about the simple cases, if you set I to be 0, the voltages 733 00:45:47,490 --> 00:45:49,590 better be the same. 734 00:45:49,590 --> 00:45:52,800 Well over here it's 7.5 by the voltage divider, over here 735 00:45:52,800 --> 00:45:55,155 it's 7.5 by the fact that there's no current going 736 00:45:55,155 --> 00:45:57,460 through that resistor. 737 00:45:57,460 --> 00:46:00,050 And the short circuit current better be the same. 738 00:46:00,050 --> 00:46:02,080 So over here, if you short this out, you're going to get 739 00:46:02,080 --> 00:46:04,090 10 Amps, over here if you short it out, you're going to 740 00:46:04,090 --> 00:46:05,160 get 10 Amps. 741 00:46:05,160 --> 00:46:06,550 So you can always go back and forth. 742 00:46:06,550 --> 00:46:09,680 The point is, that you can substitute the simpler circuit 743 00:46:09,680 --> 00:46:12,350 for the more complex circuit, because they have 744 00:46:12,350 --> 00:46:15,160 the same I-V curve. 745 00:46:15,160 --> 00:46:21,120 OK, just to make sure that you're following me, here's a 746 00:46:21,120 --> 00:46:26,780 question that has to do with taking the same circuit, but 747 00:46:26,780 --> 00:46:29,950 considering the Thevenin equivalent-- 748 00:46:29,950 --> 00:46:31,980 Thevenin or Norton equivalent-- 749 00:46:31,980 --> 00:46:33,350 at three different ports. 750 00:46:33,350 --> 00:46:35,300 Here I'm thinking about what would happen if I looked in 751 00:46:35,300 --> 00:46:41,380 terminal A, what if I looked in terminal B, or looked in 752 00:46:41,380 --> 00:46:42,640 terminal C? 753 00:46:42,640 --> 00:46:46,540 Figure out the Thevenin and Norton parameters, and see if 754 00:46:46,540 --> 00:46:47,790 there's an error in the table. 755 00:52:09,860 --> 00:52:11,213 So how many errors are in the table? 756 00:52:17,670 --> 00:52:20,890 About 50% correct again. 757 00:52:20,890 --> 00:52:22,640 So which entry don't you like. 758 00:52:25,604 --> 00:52:27,086 AUDIENCE: 2D? 759 00:52:27,086 --> 00:52:30,060 PROFESSOR: That's exactly right, so 2D is wrong. 760 00:52:30,060 --> 00:52:32,002 How do I figure out 1A? 761 00:52:32,002 --> 00:52:35,180 How do I figure out V0 for the a circuit? 762 00:52:39,784 --> 00:52:43,600 So the definition of V0 is the open circuit voltage. 763 00:52:43,600 --> 00:52:47,220 So I need to figure out, for the a circuit, how big would 764 00:52:47,220 --> 00:52:51,770 be the voltage across A if there was no current flowing 765 00:52:51,770 --> 00:52:55,430 in the leg of A. Everybody clear on that? 766 00:52:55,430 --> 00:52:57,510 So the thing that I would do is, I would think about, 767 00:52:57,510 --> 00:52:59,470 there's no connection here. 768 00:52:59,470 --> 00:53:00,440 What's the voltage here? 769 00:53:00,440 --> 00:53:02,810 So how would I calculate that? 770 00:53:02,810 --> 00:53:03,613 Yeah. 771 00:53:03,613 --> 00:53:06,028 AUDIENCE: You would use the current-divider relationship 772 00:53:06,028 --> 00:53:10,616 to figure out that current flowing through that is going 773 00:53:10,616 --> 00:53:11,824 to be 4 Amps. 774 00:53:11,824 --> 00:53:14,722 PROFESSOR: Precisely, so first I need to take-- here are the 775 00:53:14,722 --> 00:53:18,065 current sources, so I have two resistor lengths, so that's a 776 00:53:18,065 --> 00:53:21,240 perfect set up for the current divider. 777 00:53:21,240 --> 00:53:25,380 So the amount of current that goes in this leg compared to 778 00:53:25,380 --> 00:53:30,360 that leg is the ratio of this resistance to the sum of 779 00:53:30,360 --> 00:53:32,460 resistances, right? 780 00:53:32,460 --> 00:53:35,020 And if you work that out, you're going to get 4 Amps 781 00:53:35,020 --> 00:53:37,550 coming through here. 782 00:53:37,550 --> 00:53:40,870 So then after you know the current through this leg, it's 783 00:53:40,870 --> 00:53:43,700 an easy matter to take the current and turn it into a 784 00:53:43,700 --> 00:53:48,590 voltage, so the voltage at the a port is 4 Amps times 5 Ohms 785 00:53:48,590 --> 00:53:49,945 is 20 Volts. 786 00:53:49,945 --> 00:53:51,195 Is that clear? 787 00:53:53,810 --> 00:53:58,650 And similarly, but with a different answer, the voltage 788 00:53:58,650 --> 00:54:03,860 at the B terminal is the same 4 Amps, but now times 10 Ohms. 789 00:54:03,860 --> 00:54:06,040 So that's how we got 40. 790 00:54:06,040 --> 00:54:08,940 And at C, it's the same 4 Amps, but now it's times the 791 00:54:08,940 --> 00:54:12,150 sum, 4 times 15 is 60. 792 00:54:12,150 --> 00:54:13,630 Everybody's happy about that? 793 00:54:13,630 --> 00:54:18,330 So the point is that when you generate an equivalent 794 00:54:18,330 --> 00:54:22,610 circuit, it depends upon which set of terminals you're using. 795 00:54:22,610 --> 00:54:25,050 You can't just take a circuit and say, give me the Thevenin 796 00:54:25,050 --> 00:54:25,540 Equivalent. 797 00:54:25,540 --> 00:54:27,040 You have to say, give me the Thevenin 798 00:54:27,040 --> 00:54:29,190 equivalent looking somewhere. 799 00:54:29,190 --> 00:54:32,390 So I could look in the A port, the B port, or the C port, and 800 00:54:32,390 --> 00:54:34,850 I get different V0's. 801 00:54:34,850 --> 00:54:36,100 How would I compute the I0? 802 00:54:42,188 --> 00:54:45,182 Short circuit current-- 803 00:54:45,182 --> 00:54:49,370 so what I would do is I would short this out. 804 00:54:49,370 --> 00:54:52,543 When I do that, all the current flows in this leg and 805 00:54:52,543 --> 00:54:55,550 one of the current flows that leg. 806 00:54:55,550 --> 00:54:57,890 So that means I have, equivalently, 10 Ohms in 807 00:54:57,890 --> 00:55:00,440 parallel with 10 Ohms. 808 00:55:00,440 --> 00:55:03,310 Then by the current divider, how much current 809 00:55:03,310 --> 00:55:06,230 goes down one leg? 810 00:55:06,230 --> 00:55:07,480 Half of it. 811 00:55:09,910 --> 00:55:14,380 I get a different answer over here, because now I short out 812 00:55:14,380 --> 00:55:17,660 that node, which shorts out that resistor, so now I get a 813 00:55:17,660 --> 00:55:20,530 different ratio of resistors. 814 00:55:20,530 --> 00:55:22,880 It's not the same as the first, so I know that this 815 00:55:22,880 --> 00:55:25,640 answer can't be 5. 816 00:55:25,640 --> 00:55:28,630 And, in fact, if you work it out, the answer is 20 over 3. 817 00:55:31,860 --> 00:55:38,140 And finally, if you short here, then you know that that 818 00:55:38,140 --> 00:55:42,600 short circuit shorts out both this series 819 00:55:42,600 --> 00:55:44,940 combination and that one. 820 00:55:44,940 --> 00:55:47,270 So all of the 10 Amps goes through that, so 821 00:55:47,270 --> 00:55:48,790 you get that 10 Amps. 822 00:55:48,790 --> 00:55:52,150 Then how do you get R0, which is the ratio of V0 over I0. 823 00:55:55,370 --> 00:55:58,560 So if you take V0 over I0, you get 4. 824 00:55:58,560 --> 00:56:01,460 Here if you do right answer, you get 6, and here if you do 825 00:56:01,460 --> 00:56:03,840 that you get 6. 826 00:56:03,840 --> 00:56:06,380 The point is, that the Thevenin 827 00:56:06,380 --> 00:56:08,680 equivalent you get is different. 828 00:56:08,680 --> 00:56:11,320 The Norton equivalent that you get is different, depending on 829 00:56:11,320 --> 00:56:12,570 which ports you're looking at. 830 00:56:15,460 --> 00:56:20,980 OK there's two reasons for thinking about this. 831 00:56:20,980 --> 00:56:21,820 One-- 832 00:56:21,820 --> 00:56:24,530 so why am I thinking about all of these equivalent circuits? 833 00:56:24,530 --> 00:56:28,120 So I wanted to have an abstraction that was useful 834 00:56:28,120 --> 00:56:29,950 for thinking about how parts interact. 835 00:56:32,520 --> 00:56:35,800 I wanted a way of thinking about what would happen if I 836 00:56:35,800 --> 00:56:40,300 changed the load on the circuit without having to 837 00:56:40,300 --> 00:56:42,920 recalculate all the voltages and currents 838 00:56:42,920 --> 00:56:44,100 throughout the circuit. 839 00:56:44,100 --> 00:56:47,260 And so this Thevenin and Norton idea is a 840 00:56:47,260 --> 00:56:48,820 way of doing that. 841 00:56:48,820 --> 00:56:51,290 That's important from a practical sense, because when 842 00:56:51,290 --> 00:56:52,660 you buy a part-- 843 00:56:52,660 --> 00:56:55,910 when you buy an electronic part, they tell you how it 844 00:56:55,910 --> 00:56:58,940 works by telling you the Thevenin equivalent or the 845 00:56:58,940 --> 00:57:01,150 Norton equivalent, or whatever is the easy way 846 00:57:01,150 --> 00:57:03,280 to think about it. 847 00:57:03,280 --> 00:57:05,030 So there's a practical reason-- 848 00:57:05,030 --> 00:57:07,970 when you buy an op-amp, they tell you how good the op-amp 849 00:57:07,970 --> 00:57:10,960 is by telling you how big is the equivalent resistance at 850 00:57:10,960 --> 00:57:12,370 the output. 851 00:57:12,370 --> 00:57:15,470 So it has a practical value, because it lets you-- it's the 852 00:57:15,470 --> 00:57:19,020 way you specify an electronic part. 853 00:57:19,020 --> 00:57:21,910 You can't -- when the manufacturer makes a part, 854 00:57:21,910 --> 00:57:24,670 they can't know what you're going to do with it. 855 00:57:24,670 --> 00:57:28,180 So they tell you how it works by telling you something about 856 00:57:28,180 --> 00:57:30,030 the equivalent circuit. 857 00:57:30,030 --> 00:57:32,480 There's also a different reason for thinking about 858 00:57:32,480 --> 00:57:36,110 this, and that is because it's conceptually simplifying to 859 00:57:36,110 --> 00:57:39,390 think about Thevenin and Norton equivalents. 860 00:57:39,390 --> 00:57:41,530 So here's an example that's very much like the first 861 00:57:41,530 --> 00:57:43,340 problem that I worked out. 862 00:57:43,340 --> 00:57:45,670 What would be the effect of closing this switch 863 00:57:45,670 --> 00:57:48,770 on the current I? 864 00:57:48,770 --> 00:57:51,990 We solved the problem very much like this last time, and 865 00:57:51,990 --> 00:57:55,170 one way you can solve it is you figure out I in two cases, 866 00:57:55,170 --> 00:57:57,190 when the switch is open and when the switch is closed. 867 00:57:57,190 --> 00:57:59,010 And you figure out whether went up or down, and you know 868 00:57:59,010 --> 00:57:59,990 the answer. 869 00:57:59,990 --> 00:58:01,910 The point is, that if you think about this in terms of 870 00:58:01,910 --> 00:58:04,380 equivalent circuits, it's completely trivial. 871 00:58:07,110 --> 00:58:08,730 If I think about what would happen-- 872 00:58:08,730 --> 00:58:12,740 I'm interested in what happens at I, what I do is split the 873 00:58:12,740 --> 00:58:15,203 circuit into two pieces, the stuff to the left of I, and 874 00:58:15,203 --> 00:58:17,070 the stuff to the right to of I. I make a Thevenin 875 00:58:17,070 --> 00:58:19,500 equivalent for both of them, and then I jam together the 876 00:58:19,500 --> 00:58:22,780 two Thevenin equivalents. 877 00:58:22,780 --> 00:58:27,230 What I get, in detail, is showed here. 878 00:58:27,230 --> 00:58:31,240 So if I look left, I see a 20 Volt source with 879 00:58:31,240 --> 00:58:34,070 a 4, 4, and a 2. 880 00:58:34,070 --> 00:58:37,970 That has a Thevenin equivalent that is showed here, which is 881 00:58:37,970 --> 00:58:40,670 independent of the state of the switch. 882 00:58:40,670 --> 00:58:43,930 So the same Thevenin occurs on the two sides, switch open, or 883 00:58:43,930 --> 00:58:46,070 switch closed. 884 00:58:46,070 --> 00:58:49,040 If i make a Thevenin over here, there's no sources. 885 00:58:49,040 --> 00:58:51,670 So it's just going to be a resistance. 886 00:58:51,670 --> 00:58:54,133 So when the switch is open, I have to 2 Ohms, when the 887 00:58:54,133 --> 00:58:55,383 switch is closed, I have 1 Ohm. 888 00:58:58,910 --> 00:59:03,620 I don't even need to figure out what are these part values 889 00:59:03,620 --> 00:59:07,490 to see that, if I make this resistance smaller, which 890 00:59:07,490 --> 00:59:12,096 happens if I close the switch, the current goes up. 891 00:59:12,096 --> 00:59:15,370 I mean it's true that I showed here that the Thevenin voltage 892 00:59:15,370 --> 00:59:17,370 is 10 and the Thevenin resistor is 4. 893 00:59:17,370 --> 00:59:19,720 I don't even care. 894 00:59:19,720 --> 00:59:21,760 Regardless of what the Thevenin voltage was, 895 00:59:21,760 --> 00:59:23,800 regardless of what the Thevenin current is, it's 896 00:59:23,800 --> 00:59:26,270 going to be the same when the switch is open and closed. 897 00:59:26,270 --> 00:59:29,710 And that's a powerful statement. 898 00:59:29,710 --> 00:59:33,050 I know it's the same, so when I close this which, all I've 899 00:59:33,050 --> 00:59:35,990 really done is I've made that resistor smaller. 900 00:59:35,990 --> 00:59:40,620 The net series resistance is down, the current is up. 901 00:59:40,620 --> 00:59:43,610 So I have two reasons for thinking about Thevenin and 902 00:59:43,610 --> 00:59:44,290 Norton equivalence. 903 00:59:44,290 --> 00:59:48,720 One is practical value, the other is conceptual 904 00:59:48,720 --> 00:59:49,760 simplicity. 905 00:59:49,760 --> 00:59:52,440 It lets you simplify way you think about a circuit, and 906 00:59:52,440 --> 00:59:54,740 it's a way of gaining intuition without ever even 907 00:59:54,740 --> 00:59:56,310 solving the equations. 908 00:59:56,310 --> 01:00:00,120 Most circuit designers don't solve certain equations. 909 01:00:00,120 --> 01:00:03,020 They know what it's going to do just by looking at it, and 910 01:00:03,020 --> 01:00:05,150 this is the kind of reasoning that they use. 911 01:00:08,130 --> 01:00:10,310 OK, there's one more topic-- 912 01:00:10,310 --> 01:00:13,520 the idea of Thevenins and Nortons really derived from 913 01:00:13,520 --> 01:00:16,320 linear algebra. 914 01:00:16,320 --> 01:00:19,730 The basic parts of most interest are linear. 915 01:00:19,730 --> 01:00:21,710 And when you put together a system out of linear parts, 916 01:00:21,710 --> 01:00:24,410 you get a linear system. 917 01:00:24,410 --> 01:00:28,090 There's one more consequence of linearity that is terribly 918 01:00:28,090 --> 01:00:32,020 useful, and that's the idea of super position. 919 01:00:32,020 --> 01:00:35,000 If you have a system of equations that is linear, and 920 01:00:35,000 --> 01:00:39,710 if you have multiple sources, things are very simple. 921 01:00:39,710 --> 01:00:42,050 You can see that over here. 922 01:00:42,050 --> 01:00:45,310 If I had multiple sources, the system of equations that I get 923 01:00:45,310 --> 01:00:47,540 wouldn't look quite like this. 924 01:00:47,540 --> 01:00:49,720 These terms, the constant terms, they're the ones that 925 01:00:49,720 --> 01:00:51,810 come from the drives, the voltage 926 01:00:51,810 --> 01:00:53,120 sources, the current sources. 927 01:00:53,120 --> 01:00:56,040 So if I had two drives, I would get something else here. 928 01:00:56,040 --> 01:00:59,790 I would get plus a at N+1 for example-- 929 01:00:59,790 --> 01:01:02,650 source one, source two. 930 01:01:02,650 --> 01:01:06,471 And I would get source one, source two. 931 01:01:06,471 --> 01:01:10,850 So the idea is that, if I have a system that has multiple 932 01:01:10,850 --> 01:01:16,200 sources, I know from the structure of the linear 933 01:01:16,200 --> 01:01:20,846 equations, there is just more constant terms. 934 01:01:20,846 --> 01:01:22,480 OK, well what's that mean? 935 01:01:22,480 --> 01:01:25,720 That means you can just use linear algebra to see that the 936 01:01:25,720 --> 01:01:29,790 answer to this problem is the sum of the answers to that 937 01:01:29,790 --> 01:01:33,970 problem plus the answers to that problem. 938 01:01:33,970 --> 01:01:37,050 That's just linear algebra. 939 01:01:37,050 --> 01:01:40,130 What that means in terms of circuits is, I can figure out 940 01:01:40,130 --> 01:01:42,640 the response to a circuit by turning on the 941 01:01:42,640 --> 01:01:44,590 sources one at a time. 942 01:01:44,590 --> 01:01:47,360 That's called superposition. 943 01:01:47,360 --> 01:01:50,440 And generally speaking, it's a lot easier than solving the 944 01:01:50,440 --> 01:01:52,630 circuit out the long way. 945 01:01:52,630 --> 01:01:57,180 So here I've got two sources So say I wanted to compute I, 946 01:01:57,180 --> 01:02:01,040 in response to V0 and I0. 947 01:02:01,040 --> 01:02:05,930 What I would do, is I would turn off the I0 source, and 948 01:02:05,930 --> 01:02:08,420 calculate the voltage that results just 949 01:02:08,420 --> 01:02:10,400 from the voltage source. 950 01:02:10,400 --> 01:02:13,610 That's the same as setting this to 0 and finding the 951 01:02:13,610 --> 01:02:14,860 response to this. 952 01:02:18,240 --> 01:02:22,820 So if I want to set I to 0, the way you set I to be 0 is 953 01:02:22,820 --> 01:02:25,790 open circuit. 954 01:02:25,790 --> 01:02:29,160 If you open circuit something, there's no current going to 955 01:02:29,160 --> 01:02:30,410 flow through it. 956 01:02:30,410 --> 01:02:34,960 So I replace the current source with an open circuit, 957 01:02:34,960 --> 01:02:38,860 and compute the i that would be result when the current 958 01:02:38,860 --> 01:02:40,970 source isn't there. 959 01:02:40,970 --> 01:02:42,540 OK, well that's easy. 960 01:02:42,540 --> 01:02:44,480 If their current source we're not there, this 961 01:02:44,480 --> 01:02:45,500 would be open circuit. 962 01:02:45,500 --> 01:02:48,630 The current, i, would just be the total voltage going 963 01:02:48,630 --> 01:02:50,770 through R1 and R2. 964 01:02:50,770 --> 01:02:56,890 So the answer, I1, the first component of the current I, 965 01:02:56,890 --> 01:03:00,840 would be V0, the result of the voltage source divided by the 966 01:03:00,840 --> 01:03:02,850 sum of R1 and R2. 967 01:03:02,850 --> 01:03:03,840 Now that's not the whole answer. 968 01:03:03,840 --> 01:03:07,580 That would be the whole answer if I0 weren't there. 969 01:03:07,580 --> 01:03:11,280 So now I have to worry about the other case. 970 01:03:11,280 --> 01:03:13,810 What if only I0 were there? 971 01:03:13,810 --> 01:03:16,860 Well now I have to set V to be 0. 972 01:03:16,860 --> 01:03:20,060 Well setting V to 0 is not open circuiting it. 973 01:03:20,060 --> 01:03:22,806 You can't just reach in, grab the voltage source, and throw 974 01:03:22,806 --> 01:03:27,220 it away, because that'll make the current 0. 975 01:03:27,220 --> 01:03:31,900 If I want the voltage to be 0, I have to short circuit it. 976 01:03:31,900 --> 01:03:35,610 So what I do then, is I leave the current source alone, and 977 01:03:35,610 --> 01:03:38,030 I replace the voltage source with a short circuit, 978 01:03:38,030 --> 01:03:41,780 guaranteeing that V is 0. 979 01:03:41,780 --> 01:03:46,160 Then I ask, how big is I2, the component of I that results 980 01:03:46,160 --> 01:03:48,730 from the current source? 981 01:03:48,730 --> 01:03:52,680 Well if I've short circuited this, then I just get a 982 01:03:52,680 --> 01:03:55,200 current divider. 983 01:03:55,200 --> 01:03:58,720 This current has to do with how readily the current 984 01:03:58,720 --> 01:04:01,030 divides between R1 and R2. 985 01:04:01,030 --> 01:04:04,170 The amount that goes through the R1 side is in proportion 986 01:04:04,170 --> 01:04:07,670 to R2 make R2 bigger, more of it goes through R1. 987 01:04:07,670 --> 01:04:10,120 Standard current divider, except for the 988 01:04:10,120 --> 01:04:11,370 slippery minus sign. 989 01:04:14,460 --> 01:04:17,960 So if I only had the current source, the current I2 would 990 01:04:17,960 --> 01:04:22,150 have been current divider operating I0 with the slippery 991 01:04:22,150 --> 01:04:23,280 minus sign. 992 01:04:23,280 --> 01:04:26,770 So that means, by superposition, that the result 993 01:04:26,770 --> 01:04:29,475 of having both sources on is just the sum of those answers. 994 01:04:32,390 --> 01:04:35,570 That's a very big simplification. 995 01:04:35,570 --> 01:04:39,050 If you've got multiple sources in the circuit, you can think 996 01:04:39,050 --> 01:04:41,540 about them all at once. 997 01:04:41,540 --> 01:04:44,410 And if you really good at writing linear equations and 998 01:04:44,410 --> 01:04:48,760 solving them with pencils, you'll get the right answer. 999 01:04:48,760 --> 01:04:51,820 But there's an enormous simplification if you just 1000 01:04:51,820 --> 01:04:55,950 turn off all but one, and do them one at a time. 1001 01:04:55,950 --> 01:04:57,680 So here's a problem. 1002 01:04:57,680 --> 01:05:00,770 Here's a very simple circuit, compute V 1003 01:05:00,770 --> 01:05:02,940 by using super position. 1004 01:05:02,940 --> 01:05:04,350 V is the voltage across the resistor. 1005 01:05:04,350 --> 01:05:07,420 How big would V be if you used the idea of super position? 1006 01:06:52,790 --> 01:06:54,040 So what's the answer? 1007 01:07:03,190 --> 01:07:08,010 50% correct, roughly speaking. 1008 01:07:08,010 --> 01:07:09,200 OK I want to use superposition. 1009 01:07:09,200 --> 01:07:12,580 So how big would the voltage v be, if all I had was the 1010 01:07:12,580 --> 01:07:14,910 voltage source? 1011 01:07:14,910 --> 01:07:15,430 One. 1012 01:07:15,430 --> 01:07:17,000 How big would the voltage be if i only 1013 01:07:17,000 --> 01:07:19,106 had the current source? 1014 01:07:19,106 --> 01:07:21,510 Ah, got half of you. 1015 01:07:21,510 --> 01:07:24,250 That explains the 50% correct. 1016 01:07:24,250 --> 01:07:25,693 How big would the voltage be, if I only 1017 01:07:25,693 --> 01:07:27,970 had the current source? 1018 01:07:27,970 --> 01:07:29,560 It's tempting to say (1). 1019 01:07:29,560 --> 01:07:34,390 I say tempting, because that's the wrong answer. 1020 01:07:34,390 --> 01:07:36,810 [LAUGHTER] 1021 01:07:36,810 --> 01:07:39,530 What is wrong about the answer (1)? 1022 01:07:39,530 --> 01:07:44,464 What does the voltage do to the current source? 1023 01:07:44,464 --> 01:07:46,776 The voltage due to the current source is the voltage that the 1024 01:07:46,776 --> 01:07:48,020 current source would have generated if the voltage 1025 01:07:48,020 --> 01:07:50,730 source weren't there. 1026 01:07:50,730 --> 01:07:52,335 If the voltage source weren't there, then-- 1027 01:07:55,500 --> 01:07:57,050 half of you got it right, so half of you 1028 01:07:57,050 --> 01:07:58,300 can shout the answer. 1029 01:08:01,340 --> 01:08:04,600 So if the voltage source weren't there, the voltage 1030 01:08:04,600 --> 01:08:05,920 source would be 0. 1031 01:08:05,920 --> 01:08:09,920 If the voltage source were 0, then V would be 0. 1032 01:08:09,920 --> 01:08:13,290 So if the voltage due to the voltage source is 1, the 1033 01:08:13,290 --> 01:08:15,910 voltage due to the current source is 0. 1034 01:08:15,910 --> 01:08:17,310 The sum of the two is 1. 1035 01:08:17,310 --> 01:08:18,180 The answer is 1. 1036 01:08:18,180 --> 01:08:19,899 OK? 1037 01:08:19,899 --> 01:08:21,149 Make sense? 1038 01:08:24,420 --> 01:08:26,689 So, very closely related problem. 1039 01:08:26,689 --> 01:08:28,510 What's the current I? 1040 01:08:28,510 --> 01:08:29,760 Solve that by superposition. 1041 01:10:08,110 --> 01:10:12,800 So how big is the current I according to superposition? 1042 01:10:12,800 --> 01:10:18,244 Wonderful, so how big is the current i generated by the 1043 01:10:18,244 --> 01:10:21,070 voltage source? 1044 01:10:21,070 --> 01:10:22,892 1. 1045 01:10:22,892 --> 01:10:26,286 How big is the I generated by the current source? 1046 01:10:26,286 --> 01:10:28,980 Negative 1, the sum is 0. 1047 01:10:28,980 --> 01:10:30,990 All right, so those two problems were trivial by 1048 01:10:30,990 --> 01:10:31,395 superposition. 1049 01:10:31,395 --> 01:10:33,840 They're not too hard by non-superposition. 1050 01:10:33,840 --> 01:10:36,280 But the point is, that they're trivial by super position. 1051 01:10:36,280 --> 01:10:38,220 So what we saw today-- 1052 01:10:38,220 --> 01:10:41,700 the goal for today was to generate some abstractions 1053 01:10:41,700 --> 01:10:44,730 that let you think about the way parts interact with each 1054 01:10:44,730 --> 01:10:48,480 other, because that's a central issue when you're 1055 01:10:48,480 --> 01:10:51,700 thinking about circuit design And we built the idea of 1056 01:10:51,700 --> 01:10:54,100 Thevenins and Nortons and super positions. 1057 01:10:54,100 --> 01:10:55,740 That was the basic idea. 1058 01:10:55,740 --> 01:10:57,450 And the sub-theme-- 1059 01:10:57,450 --> 01:11:01,450 or maybe I should say the major theme is really that. 1060 01:11:01,450 --> 01:11:03,110 It's the importance of linear algebra. 1061 01:11:03,110 --> 01:11:07,610 So the take home message is probably, take 18.06. 1062 01:11:07,610 --> 01:11:11,250 So this was all about the application of 18.06 to the 1063 01:11:11,250 --> 01:11:13,230 solving of circuits. 1064 01:11:13,230 --> 01:11:14,480 See you later.