1 00:00:00,000 --> 00:00:00,040 2 00:00:00,040 --> 00:00:02,460 The following content is provided under a Creative 3 00:00:02,460 --> 00:00:03,870 Commons license. 4 00:00:03,870 --> 00:00:06,910 Your support will help MIT OpenCourseWare continue to 5 00:00:06,910 --> 00:00:10,560 offer high-quality educational resources for free. 6 00:00:10,560 --> 00:00:13,460 To make a donation or view additional materials from 7 00:00:13,460 --> 00:00:18,090 hundreds of MIT courses, visit MIT OpenCourseWare at 8 00:00:18,090 --> 00:00:19,340 ocw.mit.edu. 9 00:00:19,340 --> 00:00:24,190 10 00:00:24,190 --> 00:00:26,785 PROFESSOR: So today I want to start a new topic, circuits. 11 00:00:26,785 --> 00:00:29,430 12 00:00:29,430 --> 00:00:30,130 No, that's good. 13 00:00:30,130 --> 00:00:31,390 That's good. 14 00:00:31,390 --> 00:00:33,662 Circuits are good. 15 00:00:33,662 --> 00:00:34,590 AUDIENCE: Yay. 16 00:00:34,590 --> 00:00:35,280 PROFESSOR: Thank you, thank you. 17 00:00:35,280 --> 00:00:37,880 Much better. 18 00:00:37,880 --> 00:00:42,290 So just to provide some perspective, I want to remind 19 00:00:42,290 --> 00:00:45,500 you where we are, how we got here, and where we're going. 20 00:00:45,500 --> 00:00:48,170 So at the beginning of the course, we promised that there 21 00:00:48,170 --> 00:00:49,830 were several intellectual themes that 22 00:00:49,830 --> 00:00:52,400 we would talk about. 23 00:00:52,400 --> 00:00:54,850 Probably the most important one there is 24 00:00:54,850 --> 00:00:57,450 designing complex systems. 25 00:00:57,450 --> 00:00:59,000 That's what we're really about. 26 00:00:59,000 --> 00:01:00,920 We would like you to be able to make 27 00:01:00,920 --> 00:01:02,670 very complicated systems. 28 00:01:02,670 --> 00:01:04,220 How do you think about parts? 29 00:01:04,220 --> 00:01:06,040 How do you think about connecting them? 30 00:01:06,040 --> 00:01:07,730 How do you think about things when you want to make 31 00:01:07,730 --> 00:01:11,170 something that's very complicated? 32 00:01:11,170 --> 00:01:13,740 Part of that is modeling. 33 00:01:13,740 --> 00:01:16,510 We just finished a module on modeling. 34 00:01:16,510 --> 00:01:19,550 So in order to make a complex system, we'd like to be able 35 00:01:19,550 --> 00:01:23,850 to predict how it will behave before we completely build it. 36 00:01:23,850 --> 00:01:26,220 Sometimes it's impossible to build the 37 00:01:26,220 --> 00:01:27,670 entire system before. 38 00:01:27,670 --> 00:01:30,210 39 00:01:30,210 --> 00:01:34,030 Sometimes it's impossible to build prototypes. 40 00:01:34,030 --> 00:01:38,190 Sometimes you're stuck with going with the design at 41 00:01:38,190 --> 00:01:41,270 launch, and figuring out how it works. 42 00:01:41,270 --> 00:01:44,340 In those cases in specific, it's very important to be able 43 00:01:44,340 --> 00:01:47,850 to model it, to have some confidence that the thing's 44 00:01:47,850 --> 00:01:49,500 going to work. 45 00:01:49,500 --> 00:01:52,090 We're going to talk about augmenting physical systems 46 00:01:52,090 --> 00:01:52,990 with computation. 47 00:01:52,990 --> 00:01:55,320 That's the module we're about to begin. 48 00:01:55,320 --> 00:01:58,060 And we'll conclude by talking about how to build systems 49 00:01:58,060 --> 00:02:00,900 that are robust to change. 50 00:02:00,900 --> 00:02:05,240 So we started with the idea of, how do you make 51 00:02:05,240 --> 00:02:06,320 complicated systems? 52 00:02:06,320 --> 00:02:08,840 And we introduced this notion of primitives, combination, 53 00:02:08,840 --> 00:02:12,350 abstraction, and pattern in terms of software engineering. 54 00:02:12,350 --> 00:02:14,600 We did that because that's the simplest possible way of 55 00:02:14,600 --> 00:02:16,350 getting started. 56 00:02:16,350 --> 00:02:20,810 It provided a very good illustration of PCAP at the 57 00:02:20,810 --> 00:02:25,650 low level by thinking about Python, by thinking about the 58 00:02:25,650 --> 00:02:29,330 primitive structures that Python gives you, how those 59 00:02:29,330 --> 00:02:31,830 can be combined, how you can abstract, how you can 60 00:02:31,830 --> 00:02:33,450 recognize patterns. 61 00:02:33,450 --> 00:02:36,120 But then we also built a higher level abstraction, 62 00:02:36,120 --> 00:02:39,130 which was the state machine idea. 63 00:02:39,130 --> 00:02:42,110 There the idea was you didn't have to have state machines in 64 00:02:42,110 --> 00:02:45,140 order to build the brain for a robot, but it actually turns 65 00:02:45,140 --> 00:02:47,430 out to be easy if you do because there's 66 00:02:47,430 --> 00:02:49,530 a modularity there. 67 00:02:49,530 --> 00:02:52,550 You can figure out if each part works independent of the 68 00:02:52,550 --> 00:02:53,530 other parts. 69 00:02:53,530 --> 00:02:55,510 And then you can be pretty sure when you put it together 70 00:02:55,510 --> 00:02:57,750 the whole thing's going to work. 71 00:02:57,750 --> 00:02:59,810 So that was kind of our introduction to 72 00:02:59,810 --> 00:03:01,570 this notion of PCAP. 73 00:03:01,570 --> 00:03:03,820 Then we went on to think about signals and systems. 74 00:03:03,820 --> 00:03:06,730 And that was kind of our introduction to modeling. 75 00:03:06,730 --> 00:03:08,070 How do you make a model of something 76 00:03:08,070 --> 00:03:09,690 that predicts behavior? 77 00:03:09,690 --> 00:03:12,890 So we transitioned from thinking about, how do you 78 00:03:12,890 --> 00:03:16,940 structure a design to how do you think about behavior. 79 00:03:16,940 --> 00:03:19,350 Today we're going to start to think about circuits. 80 00:03:19,350 --> 00:03:24,240 Circuits are really going to a more primitive physical layer. 81 00:03:24,240 --> 00:03:28,210 How do you think about actually making a device? 82 00:03:28,210 --> 00:03:32,460 So the device that we'll think about is a thing to augment 83 00:03:32,460 --> 00:03:33,350 the capabilities -- 84 00:03:33,350 --> 00:03:36,340 the sensor capabilities of the robot. 85 00:03:36,340 --> 00:03:38,490 We'll think about making a light tracking system. 86 00:03:38,490 --> 00:03:42,420 And the idea is going to be that you'll build a head. 87 00:03:42,420 --> 00:03:46,480 The head has a neck, so you'll have to control the neck. 88 00:03:46,480 --> 00:03:48,900 The head has eyes. 89 00:03:48,900 --> 00:03:52,250 It will mount on top of the robot, and that'll let you 90 00:03:52,250 --> 00:03:55,960 drive the robot around looking for light. 91 00:03:55,960 --> 00:03:59,120 And you'll do that by designing a circuit. 92 00:03:59,120 --> 00:04:03,200 So that's kind of the game plan for the next three weeks. 93 00:04:03,200 --> 00:04:06,020 So today, what I want to do is introduce the notion of 94 00:04:06,020 --> 00:04:09,570 circuits, introduce the theory for how 95 00:04:09,570 --> 00:04:11,400 we think about circuits. 96 00:04:11,400 --> 00:04:15,100 And then in the two labs this week, the idea will be to 97 00:04:15,100 --> 00:04:18,410 become familiar with the practice of circuits. 98 00:04:18,410 --> 00:04:19,750 How do you actually build something? 99 00:04:19,750 --> 00:04:22,780 How do you actually make something work? 100 00:04:22,780 --> 00:04:23,990 So today's the theory. 101 00:04:23,990 --> 00:04:27,390 So the theory, the idea in circuits is to think about a 102 00:04:27,390 --> 00:04:34,070 physical system as the interconnection of parts and 103 00:04:34,070 --> 00:04:36,650 rules that connect them. 104 00:04:36,650 --> 00:04:39,180 In fact, the rules fall into two categories. 105 00:04:39,180 --> 00:04:44,490 We're going to think about the currents that go through parts 106 00:04:44,490 --> 00:04:47,410 and the voltage that develops across parts. 107 00:04:47,410 --> 00:04:50,140 And we'll see that there's a way of thinking about the 108 00:04:50,140 --> 00:04:53,310 behavior of the entire circuit that integrates all of those 109 00:04:53,310 --> 00:04:54,160 three pieces. 110 00:04:54,160 --> 00:04:55,670 How does the part work? 111 00:04:55,670 --> 00:04:59,640 How do the currents that go through the part work? 112 00:04:59,640 --> 00:05:02,750 And how do the voltages that are produced 113 00:05:02,750 --> 00:05:03,710 across the part -- 114 00:05:03,710 --> 00:05:06,820 how do they work? 115 00:05:06,820 --> 00:05:08,880 So I'll just start with two very simple examples. 116 00:05:08,880 --> 00:05:11,735 The first is the most trivial example you can think about. 117 00:05:11,735 --> 00:05:14,750 You can think about a flashlight as a circuit. 118 00:05:14,750 --> 00:05:17,700 You close the switch, current flows. 119 00:05:17,700 --> 00:05:19,450 Very simple idea. 120 00:05:19,450 --> 00:05:23,100 We will make a model of that that looks like this. 121 00:05:23,100 --> 00:05:25,600 We'll think about the battery being a source. 122 00:05:25,600 --> 00:05:27,640 In this case, a voltage source. 123 00:05:27,640 --> 00:05:31,020 We'll think about the light bulb being a resistor. 124 00:05:31,020 --> 00:05:32,020 We'll have two parts. 125 00:05:32,020 --> 00:05:34,480 We'll have to know the current-voltage relationships 126 00:05:34,480 --> 00:05:35,880 for both of the parts. 127 00:05:35,880 --> 00:05:37,670 And we'll have to know the ramifications for those 128 00:05:37,670 --> 00:05:40,400 currents and voltages when you put them together. 129 00:05:40,400 --> 00:05:42,620 Very, very simple. 130 00:05:42,620 --> 00:05:46,330 The other simple example that I want to illustrate is this 131 00:05:46,330 --> 00:05:48,790 idea of a leaky tank. 132 00:05:48,790 --> 00:05:52,600 Here the idea that I want to get across is that the circuit 133 00:05:52,600 --> 00:05:56,810 idea is quite general. 134 00:05:56,810 --> 00:05:58,770 When we talk about circuits, we almost always talk about 135 00:05:58,770 --> 00:06:01,220 electronic circuits. 136 00:06:01,220 --> 00:06:05,400 But the theory is by no means limited to electronics. 137 00:06:05,400 --> 00:06:07,870 So for example, if we think about a leaky tank, we think 138 00:06:07,870 --> 00:06:10,840 about a pipe spewing water into a reservoir. 139 00:06:10,840 --> 00:06:12,730 Maybe that's the Cambridge Reservoir. 140 00:06:12,730 --> 00:06:15,680 Maybe that's the water coming out of the Woburn Reservoir. 141 00:06:15,680 --> 00:06:18,120 Maybe that's the demand put on by the Cambridge people trying 142 00:06:18,120 --> 00:06:20,310 to take showers in the morning -- 143 00:06:20,310 --> 00:06:21,830 I don't know. 144 00:06:21,830 --> 00:06:24,160 So we think about flow into a tank, a 145 00:06:24,160 --> 00:06:25,880 reservoir, and flow out. 146 00:06:25,880 --> 00:06:29,790 And we can make a model for that in terms of a circuit. 147 00:06:29,790 --> 00:06:31,220 In the circuit there are through 148 00:06:31,220 --> 00:06:34,370 variables and across variables. 149 00:06:34,370 --> 00:06:37,510 In an electronic circuit, the through variable is current. 150 00:06:37,510 --> 00:06:40,000 Here the through variable is the flow rate. 151 00:06:40,000 --> 00:06:42,420 So it's the flow of water in and the flow of water out, 152 00:06:42,420 --> 00:06:43,990 represented here by these things 153 00:06:43,990 --> 00:06:46,840 that look like currents. 154 00:06:46,840 --> 00:06:51,000 And the across variable for an electronic device is voltage. 155 00:06:51,000 --> 00:06:53,870 The across variable for this kind of a fluidic device, the 156 00:06:53,870 --> 00:06:56,180 across variable is pressure. 157 00:06:56,180 --> 00:06:59,520 So we think as this thing gets ahead of that thing, as 158 00:06:59,520 --> 00:07:02,720 there's more stuff coming in than going out, the height 159 00:07:02,720 --> 00:07:05,870 goes up and the pressure builds. 160 00:07:05,870 --> 00:07:06,690 Same idea. 161 00:07:06,690 --> 00:07:09,070 So the point is that we'll develop the theory for 162 00:07:09,070 --> 00:07:11,260 circuits in terms of electronics. 163 00:07:11,260 --> 00:07:13,640 But you should keep in the back of your mind that it's a 164 00:07:13,640 --> 00:07:16,980 lot more general idea than just electronics. 165 00:07:16,980 --> 00:07:20,070 In fact, there are two completely distinct reasons 166 00:07:20,070 --> 00:07:22,660 why we even bother with circuits. 167 00:07:22,660 --> 00:07:27,120 One is that they're very important to physical systems. 168 00:07:27,120 --> 00:07:29,430 If you're designing a power network, of course you have to 169 00:07:29,430 --> 00:07:32,170 think about the way the power network, the power grid works 170 00:07:32,170 --> 00:07:34,130 as a circuit. 171 00:07:34,130 --> 00:07:35,430 That's obvious. 172 00:07:35,430 --> 00:07:37,950 In electronics, of course, if you're going to build a cell 173 00:07:37,950 --> 00:07:40,090 phone, you have to know how the parts interconnect 174 00:07:40,090 --> 00:07:40,710 electronically. 175 00:07:40,710 --> 00:07:42,750 That's obvious. 176 00:07:42,750 --> 00:07:46,540 But probably the biggest use for circuits these days is not 177 00:07:46,540 --> 00:07:49,800 those applications, although those are very important. 178 00:07:49,800 --> 00:07:53,720 Circuits are also used as models of things. 179 00:07:53,720 --> 00:07:58,360 So many models for complex behaviors are in 180 00:07:58,360 --> 00:08:00,820 fact circuit models. 181 00:08:00,820 --> 00:08:04,760 So in terms of electronics, the idea is that we want to 182 00:08:04,760 --> 00:08:06,100 get on top of electronics. 183 00:08:06,100 --> 00:08:08,240 We want to understand how circuits work, so we can 184 00:08:08,240 --> 00:08:11,170 understand things like that. 185 00:08:11,170 --> 00:08:16,110 If you look at how complex processors have got over my 186 00:08:16,110 --> 00:08:19,970 professional life, we start with my professional life down 187 00:08:19,970 --> 00:08:25,090 at about 1,000 transistors per processor. 188 00:08:25,090 --> 00:08:28,990 And today, we're up at about a billion. 189 00:08:28,990 --> 00:08:31,420 That's enormous. 190 00:08:31,420 --> 00:08:36,380 Even in the stone ages when we were designing things that had 191 00:08:36,380 --> 00:08:41,549 a thousand parts, we still had trouble thinking about those 192 00:08:41,549 --> 00:08:43,070 thousands parts all at once. 193 00:08:43,070 --> 00:08:44,790 We still need PCAP. 194 00:08:44,790 --> 00:08:48,040 We still needed ways of combining the activities of 195 00:08:48,040 --> 00:08:51,340 many things into a conceptual unit that was bigger. 196 00:08:51,340 --> 00:08:57,030 Here, it's just impossible if you don't have that. 197 00:08:57,030 --> 00:08:59,760 So that's one of the reasons we study circuits. 198 00:08:59,760 --> 00:09:01,390 And the other reason is here. 199 00:09:01,390 --> 00:09:04,680 So here I'm showing a model for the way 200 00:09:04,680 --> 00:09:05,800 a nerve cell works. 201 00:09:05,800 --> 00:09:09,860 This model is taken from 6.021. 202 00:09:09,860 --> 00:09:15,280 The idea, this comes from the study of the Hodgkin-Huxley 203 00:09:15,280 --> 00:09:19,810 model for neural conduction, arguably the most successful 204 00:09:19,810 --> 00:09:23,020 mathematical theory in biophysics. 205 00:09:23,020 --> 00:09:27,510 Which explains the completely non-trivial relationship 206 00:09:27,510 --> 00:09:33,260 between how the parts from biology works and the behavior 207 00:09:33,260 --> 00:09:37,460 in terms of propagated action potentials works. 208 00:09:37,460 --> 00:09:41,500 So the idea is that we understand how this biological 209 00:09:41,500 --> 00:09:43,740 system works because we think about it 210 00:09:43,740 --> 00:09:45,470 in terms of a circuit. 211 00:09:45,470 --> 00:09:51,430 That's the only successful way we have to think about that. 212 00:09:51,430 --> 00:09:55,430 So what I want to do then is spend today figuring out 213 00:09:55,430 --> 00:09:59,650 circuits At the very most primitive level. 214 00:09:59,650 --> 00:10:01,330 The level that I'm going to talk about in terms of 215 00:10:01,330 --> 00:10:04,530 circuits is roughly analogous to the level that we talked 216 00:10:04,530 --> 00:10:09,020 about with Python when we were thinking about how Python 217 00:10:09,020 --> 00:10:14,590 provides utilities for primitives, combinations, 218 00:10:14,590 --> 00:10:17,260 abstractions, and patterns. 219 00:10:17,260 --> 00:10:19,220 So I'm going to start at the very lowest level and think 220 00:10:19,220 --> 00:10:23,290 about, what are the basic primitives, the smallest units 221 00:10:23,290 --> 00:10:25,740 we'll ever think about in terms of circuits? 222 00:10:25,740 --> 00:10:29,450 And what are the rules by which we combine them? 223 00:10:29,450 --> 00:10:33,090 So I'll start with the very simplest ideas, the very 224 00:10:33,090 --> 00:10:35,020 simplest elements. 225 00:10:35,020 --> 00:10:37,720 We will oversimplify things and think about the very 226 00:10:37,720 --> 00:10:41,010 simplest kind of electronic elements as resistors that 227 00:10:41,010 --> 00:10:42,640 obey Ohm's Law, V equals iR. 228 00:10:42,640 --> 00:10:45,420 229 00:10:45,420 --> 00:10:48,610 Voltage sources, things that maintain a constant voltage 230 00:10:48,610 --> 00:10:50,740 regardless of what you do. 231 00:10:50,740 --> 00:10:53,290 And current sources, things that maintain a constant 232 00:10:53,290 --> 00:10:56,810 current regardless of what you do. 233 00:10:56,810 --> 00:10:59,640 These things are, as I said, analogous to the primitive 234 00:10:59,640 --> 00:11:00,890 things that we looked at in Python. 235 00:11:00,890 --> 00:11:02,750 They're also analogous to the primitive things that we 236 00:11:02,750 --> 00:11:05,470 looked at in system functions. 237 00:11:05,470 --> 00:11:07,900 Can somebody think of, when we were doing difference 238 00:11:07,900 --> 00:11:10,880 equations, what were the primitives that we 239 00:11:10,880 --> 00:11:12,680 started with -- 240 00:11:12,680 --> 00:11:15,770 when we started to study difference equations? 241 00:11:15,770 --> 00:11:17,260 What's the most primitive elements 242 00:11:17,260 --> 00:11:18,510 that we thought about? 243 00:11:18,510 --> 00:11:20,937 244 00:11:20,937 --> 00:11:22,320 AUDIENCE: Delay. 245 00:11:22,320 --> 00:11:23,340 PROFESSOR: Delay, yeah. 246 00:11:23,340 --> 00:11:24,740 So we thought about things like-- 247 00:11:24,740 --> 00:11:27,390 248 00:11:27,390 --> 00:11:28,640 so we had delay. 249 00:11:28,640 --> 00:11:31,810 250 00:11:31,810 --> 00:11:32,600 Anything else? 251 00:11:32,600 --> 00:11:33,330 AUDIENCE: Gain. 252 00:11:33,330 --> 00:11:34,300 PROFESSOR: Gain. 253 00:11:34,300 --> 00:11:35,550 Anything else? 254 00:11:35,550 --> 00:11:38,463 255 00:11:38,463 --> 00:11:40,960 Add. 256 00:11:40,960 --> 00:11:44,130 So we had exactly three primitives. 257 00:11:44,130 --> 00:11:47,850 And we got pretty far with those three primitives. 258 00:11:47,850 --> 00:11:51,000 We learned the rules for interconnection. 259 00:11:51,000 --> 00:11:52,340 We didn't really make a big deal out of it. 260 00:11:52,340 --> 00:11:54,320 We didn't formalize it, but the rules for interconnection 261 00:11:54,320 --> 00:11:58,430 were something like every node has to 262 00:11:58,430 --> 00:12:01,810 have exactly one generator. 263 00:12:01,810 --> 00:12:03,830 You can't connect the output of this to 264 00:12:03,830 --> 00:12:06,890 this, that's illegal. 265 00:12:06,890 --> 00:12:09,220 Every node has to have one source. 266 00:12:09,220 --> 00:12:12,010 And every node can source lots of inputs. 267 00:12:12,010 --> 00:12:14,050 That was kind of the rules of the interconnect. 268 00:12:14,050 --> 00:12:15,505 The interconnects here will be a little bit more complicated. 269 00:12:15,505 --> 00:12:18,340 270 00:12:18,340 --> 00:12:20,960 So those are the elements that we'll think about. 271 00:12:20,960 --> 00:12:23,370 And the first step's going to be to think about, how do they 272 00:12:23,370 --> 00:12:25,800 interconnect? 273 00:12:25,800 --> 00:12:29,160 The simplest possible interconnections are trivial. 274 00:12:29,160 --> 00:12:31,580 In the case of the battery, you hook up the voltage source 275 00:12:31,580 --> 00:12:33,280 to the resistor. 276 00:12:33,280 --> 00:12:35,100 The voltage source makes the voltage across 277 00:12:35,100 --> 00:12:37,220 this resistor 1 Volt. 278 00:12:37,220 --> 00:12:39,910 If we say the resistor is 1 Ohm, then there's 1 Amp 279 00:12:39,910 --> 00:12:40,745 current period. 280 00:12:40,745 --> 00:12:41,510 Done. 281 00:12:41,510 --> 00:12:43,750 Easy. 282 00:12:43,750 --> 00:12:47,160 Similarly, if we were to hook up the resistor to a current 283 00:12:47,160 --> 00:12:49,120 source, we would get something equally easy. 284 00:12:49,120 --> 00:12:51,060 Except now the current source would guarantee that the 285 00:12:51,060 --> 00:12:56,450 current through the resistor is an amp. 286 00:12:56,450 --> 00:12:59,360 Therefore, the voltage across the resistor, by Ohm's law, 287 00:12:59,360 --> 00:13:00,700 would be a Volt. 288 00:13:00,700 --> 00:13:02,830 So we would end up with the same solution for a completely 289 00:13:02,830 --> 00:13:04,940 different reason. 290 00:13:04,940 --> 00:13:06,340 Here the voltage is constrained. 291 00:13:06,340 --> 00:13:07,590 Here the current's constrained. 292 00:13:07,590 --> 00:13:10,390 293 00:13:10,390 --> 00:13:16,090 Just to make sure everybody's with me, figure out, what's 294 00:13:16,090 --> 00:13:19,450 the current i that goes through this resistor? 295 00:13:19,450 --> 00:13:22,040 Slightly more complicated system. 296 00:13:22,040 --> 00:13:24,120 Take 20 seconds, talk to your neighbor, figure out a number 297 00:13:24,120 --> 00:13:25,820 between (1) and (5). 298 00:13:25,820 --> 00:15:23,520 299 00:15:23,520 --> 00:15:24,690 OK, so what's the answer? 300 00:15:24,690 --> 00:15:27,630 Everybody raise your hand with a number (1) through (5). 301 00:15:27,630 --> 00:15:28,980 Come on, everybody vote. 302 00:15:28,980 --> 00:15:29,920 Come on. 303 00:15:29,920 --> 00:15:32,020 You can blame it on your neighbor, that's the rules. 304 00:15:32,020 --> 00:15:33,790 You talk to your neighbor, then you can blame dumb 305 00:15:33,790 --> 00:15:36,150 answers on your neighbor. 306 00:15:36,150 --> 00:15:41,100 OK, about 80% correct I'd say. 307 00:15:41,100 --> 00:15:42,410 So how do you think about this? 308 00:15:42,410 --> 00:15:43,450 What's going to be the current? 309 00:15:43,450 --> 00:15:44,420 How would you calculate the current? 310 00:15:44,420 --> 00:15:45,670 What do I do first? 311 00:15:45,670 --> 00:15:48,462 312 00:15:48,462 --> 00:15:48,930 Shout. 313 00:15:48,930 --> 00:15:52,772 If you shout, and especially if my head's turned away I 314 00:15:52,772 --> 00:15:55,260 don't know who you are. 315 00:15:55,260 --> 00:15:56,200 AUDIENCE: Kirchhoff's law. 316 00:15:56,200 --> 00:15:57,240 PROFESSOR: Kirchoff's law, wonderful. 317 00:15:57,240 --> 00:15:57,717 Which one? 318 00:15:57,717 --> 00:16:00,102 There's two of them. 319 00:16:00,102 --> 00:16:02,964 AUDIENCE: [UNINTELLIGIBLE]. 320 00:16:02,964 --> 00:16:04,130 PROFESSOR: [UNINTELLIGIBLE]. 321 00:16:04,130 --> 00:16:05,663 What loop do you want to use? 322 00:16:05,663 --> 00:16:06,430 AUDIENCE: Left. 323 00:16:06,430 --> 00:16:07,220 PROFESSOR: Left side. 324 00:16:07,220 --> 00:16:10,960 So if we use the left side loop, we would conclude that 325 00:16:10,960 --> 00:16:12,310 there's a volt across the resistor. 326 00:16:12,310 --> 00:16:15,288 So the current would be? 327 00:16:15,288 --> 00:16:16,764 AUDIENCE: 1 Amp. 328 00:16:16,764 --> 00:16:17,748 PROFESSOR: An amp. 329 00:16:17,748 --> 00:16:21,684 Where's the current come from? 330 00:16:21,684 --> 00:16:23,160 AUDIENCE: The voltage. 331 00:16:23,160 --> 00:16:28,080 PROFESSOR: The voltage source just like before. 332 00:16:28,080 --> 00:16:29,880 So not quite. 333 00:16:29,880 --> 00:16:32,210 So the voltage source establishes this 334 00:16:32,210 --> 00:16:33,710 voltage would be 1. 335 00:16:33,710 --> 00:16:36,850 That makes this current be 1. 336 00:16:36,850 --> 00:16:39,220 That would be consistent with the current coming out of 337 00:16:39,220 --> 00:16:42,250 here, except we have to also think about that 1 Amp source. 338 00:16:42,250 --> 00:16:44,810 339 00:16:44,810 --> 00:16:46,780 So the question is, what's does the 1 Amp source do? 340 00:16:46,780 --> 00:16:48,370 Nothing? 341 00:16:48,370 --> 00:16:52,240 It's just there sort of for decoration or for 342 00:16:52,240 --> 00:16:55,170 [UNINTELLIGIBLE] so that we can make an interesting 343 00:16:55,170 --> 00:16:56,580 question to ask in lecture? 344 00:16:56,580 --> 00:16:57,830 Maybe. 345 00:16:57,830 --> 00:16:59,870 346 00:16:59,870 --> 00:17:01,040 So where's the current? 347 00:17:01,040 --> 00:17:02,689 Where's the 1 Amp that goes through the 348 00:17:02,689 --> 00:17:05,184 resistor come from? 349 00:17:05,184 --> 00:17:07,180 AUDIENCE: [UNINTELLIGIBLE] on the right. 350 00:17:07,180 --> 00:17:09,176 PROFESSOR: It comes from the right. 351 00:17:09,176 --> 00:17:11,080 It comes from the current over here. 352 00:17:11,080 --> 00:17:13,130 So the idea is that if this current, ignore the voltage 353 00:17:13,130 --> 00:17:14,150 over here for the moment. 354 00:17:14,150 --> 00:17:16,560 If this current flowed through the resistor, then you'd have 355 00:17:16,560 --> 00:17:19,130 1 Amp going through there, and you'd have 1 Volt generated by 356 00:17:19,130 --> 00:17:22,069 that current which just happens to be exactly the 357 00:17:22,069 --> 00:17:25,900 right voltage to match the voltage 358 00:17:25,900 --> 00:17:28,930 from the voltage source. 359 00:17:28,930 --> 00:17:31,640 So if you think about this, the voltage guarantees that 360 00:17:31,640 --> 00:17:35,720 this is 1 Volt, but so does the current. 361 00:17:35,720 --> 00:17:38,980 In order to simultaneously satisfy everything, all you 362 00:17:38,980 --> 00:17:41,310 need to do is have all of this current go around and come 363 00:17:41,310 --> 00:17:44,240 down through that resistor. 364 00:17:44,240 --> 00:17:47,070 That will generate the volt, so there's no propensity for 365 00:17:47,070 --> 00:17:50,020 more current to flow out of the source because the source 366 00:17:50,020 --> 00:17:54,580 is 1 Volt and it's facing a circuit that's already 1 Volt. 367 00:17:54,580 --> 00:17:56,310 So the idea was to try to give you something that's 368 00:17:56,310 --> 00:18:00,260 relatively simple that you can think through on your own, but 369 00:18:00,260 --> 00:18:01,530 not trivial. 370 00:18:01,530 --> 00:18:03,060 So the answer was 1 Amp. 371 00:18:03,060 --> 00:18:05,870 But the 1 Amp was not for the trivial reason. 372 00:18:05,870 --> 00:18:10,280 The 1 Amp is because the current from the right flows 373 00:18:10,280 --> 00:18:14,470 through the resistor and makes the voltage be 1. 374 00:18:14,470 --> 00:18:15,900 So the right answer is 1. 375 00:18:15,900 --> 00:18:17,290 But for the reason that you might not 376 00:18:17,290 --> 00:18:19,490 have originally thought. 377 00:18:19,490 --> 00:18:22,140 But more importantly, I wanted to use that as a motivation 378 00:18:22,140 --> 00:18:25,013 for thinking about, how do we think about bigger circuits? 379 00:18:25,013 --> 00:18:27,670 380 00:18:27,670 --> 00:18:30,240 So when the simple circuit, like two parts, it's no 381 00:18:30,240 --> 00:18:32,260 problem figuring out what the answer's going to be. 382 00:18:32,260 --> 00:18:35,430 But when the circuit has even three parts, it may require 383 00:18:35,430 --> 00:18:36,290 more thinking. 384 00:18:36,290 --> 00:18:38,480 And you may want to have a more structured way of 385 00:18:38,480 --> 00:18:39,630 thinking about the solution. 386 00:18:39,630 --> 00:18:40,487 Yes? 387 00:18:40,487 --> 00:18:43,409 AUDIENCE: What would have occurred if the current 388 00:18:43,409 --> 00:18:45,844 provider on the right side was 2 Amps? 389 00:18:45,844 --> 00:18:47,792 PROFESSOR: Great question. 390 00:18:47,792 --> 00:18:51,660 Had this been 2 Amps, you can't violate this voltage. 391 00:18:51,660 --> 00:18:54,570 So that would have been 1 Volt. 392 00:18:54,570 --> 00:18:57,790 So that would have been 1 Amp through the resistor. 393 00:18:57,790 --> 00:18:59,560 So then you're left with the problem with this guy's 394 00:18:59,560 --> 00:19:03,330 pushing 2 and that guy's only eating 1. 395 00:19:03,330 --> 00:19:06,660 But the rules for the voltage source say eat or source 396 00:19:06,660 --> 00:19:10,140 however much current is necessary in order to make the 397 00:19:10,140 --> 00:19:11,900 voltage equal to 1. 398 00:19:11,900 --> 00:19:16,070 So the excess amp goes through the voltage source. 399 00:19:16,070 --> 00:19:20,400 So the voltage source is, in fact, being supplied power 400 00:19:20,400 --> 00:19:23,350 rather than supplying power itself. 401 00:19:23,350 --> 00:19:27,400 Had this been 2 Amps, some of the power from this source 402 00:19:27,400 --> 00:19:30,060 would have gone into the resistor. 403 00:19:30,060 --> 00:19:31,730 And some of the power from this source would have 404 00:19:31,730 --> 00:19:34,030 actually gone into the voltage source. 405 00:19:34,030 --> 00:19:38,150 So if the voltage source were, for example, a model for a 406 00:19:38,150 --> 00:19:41,250 rechargeable battery, that rechargeable 407 00:19:41,250 --> 00:19:43,720 battery would be charging. 408 00:19:43,720 --> 00:19:44,905 Does that make sense? 409 00:19:44,905 --> 00:19:47,240 So if there had been a mismatch in the conditions, 410 00:19:47,240 --> 00:19:52,210 you still have to satisfy all the relationships from all of 411 00:19:52,210 --> 00:19:54,470 the sources. 412 00:19:54,470 --> 00:19:56,770 AUDIENCE: What if the voltage was larger? 413 00:19:56,770 --> 00:19:58,609 PROFESSOR: The same thing would have happened. 414 00:19:58,609 --> 00:20:01,265 Except now the flowing current would be in 415 00:20:01,265 --> 00:20:01,990 the opposite direction. 416 00:20:01,990 --> 00:20:04,620 Let's say that if the voltage here had been 2 Volts, then 417 00:20:04,620 --> 00:20:07,530 the voltage would have required that there is 2 Amps 418 00:20:07,530 --> 00:20:09,730 flowing here. 419 00:20:09,730 --> 00:20:13,590 1 Amp would come from here, but another Amp 420 00:20:13,590 --> 00:20:15,130 would come from here. 421 00:20:15,130 --> 00:20:18,020 This voltage source will supply whatever current is 422 00:20:18,020 --> 00:20:21,521 necessary to make its voltage law real. 423 00:20:21,521 --> 00:20:26,420 424 00:20:26,420 --> 00:20:26,495 Ok. 425 00:20:26,495 --> 00:20:30,570 In fact, what we'll do now is turn toward a discussion of 426 00:20:30,570 --> 00:20:33,960 more complicated systems that will let you go back and in 427 00:20:33,960 --> 00:20:37,430 retrospect, analyze all those cases that we just did. 428 00:20:37,430 --> 00:20:39,240 And you'll be able to see trivially how that has to be 429 00:20:39,240 --> 00:20:40,460 the right answer. 430 00:20:40,460 --> 00:20:46,460 So what I want to do now is generate a formal structure 431 00:20:46,460 --> 00:20:47,960 for how you would solve circuits. 432 00:20:47,960 --> 00:20:49,352 Yes? 433 00:20:49,352 --> 00:20:53,240 AUDIENCE: So did we know of anything that could generate a 434 00:20:53,240 --> 00:20:56,800 current without generating a voltage, like in real life? 435 00:20:56,800 --> 00:20:58,310 PROFESSOR: Can anything generate a current without 436 00:20:58,310 --> 00:20:59,080 generating a voltage? 437 00:20:59,080 --> 00:21:01,130 That's a tricky question. 438 00:21:01,130 --> 00:21:05,280 If you think about something as generating a current, then 439 00:21:05,280 --> 00:21:09,700 the voltage is not necessarily determined by that part. 440 00:21:09,700 --> 00:21:12,030 So that's kind of illustrated here. 441 00:21:12,030 --> 00:21:17,000 If this guy is generating a current, this guy is not 442 00:21:17,000 --> 00:21:21,755 actually the element that is controlling its own voltage. 443 00:21:21,755 --> 00:21:24,560 444 00:21:24,560 --> 00:21:27,470 In general, if you want to speak simultaneously about the 445 00:21:27,470 --> 00:21:32,000 current and voltage across the device, you have to know what 446 00:21:32,000 --> 00:21:34,600 it was connected to. 447 00:21:34,600 --> 00:21:36,020 Each part-- 448 00:21:36,020 --> 00:21:37,790 we'll get to this in a moment in case some of you are 449 00:21:37,790 --> 00:21:39,940 worried about launching ahead. 450 00:21:39,940 --> 00:21:42,300 We will cover this. 451 00:21:42,300 --> 00:21:44,300 This is very good motivation for figuring out what's going 452 00:21:44,300 --> 00:21:46,690 to happen in the next three slides. 453 00:21:46,690 --> 00:21:54,320 So each part gets to tell you one relationship between 454 00:21:54,320 --> 00:21:55,790 voltage and current. 455 00:21:55,790 --> 00:22:00,450 Generally speaking, that's not enough to solve for voltage 456 00:22:00,450 --> 00:22:01,030 and current. 457 00:22:01,030 --> 00:22:03,200 Voltage and current is like two unknowns. 458 00:22:03,200 --> 00:22:06,300 459 00:22:06,300 --> 00:22:11,000 Each element relationship is one equation. 460 00:22:11,000 --> 00:22:14,660 So the current source gets to say current equals x, current 461 00:22:14,660 --> 00:22:16,000 equals 1 Amp. 462 00:22:16,000 --> 00:22:20,400 It doesn't get to tell you what the voltage is. 463 00:22:20,400 --> 00:22:22,850 So being a little more physical to try to address 464 00:22:22,850 --> 00:22:27,480 your question more physically, there are processes that can 465 00:22:27,480 --> 00:22:30,970 be extremely well modeled as current generators. 466 00:22:30,970 --> 00:22:35,120 In fact, many electronic semiconductor parts, like 467 00:22:35,120 --> 00:22:37,890 transistors, work more like a current source than like 468 00:22:37,890 --> 00:22:39,790 anything else. 469 00:22:39,790 --> 00:22:42,590 So there are devices that behave as though they were 470 00:22:42,590 --> 00:22:46,570 current sources, but they don't simultaneously get to 471 00:22:46,570 --> 00:22:49,390 tell you what is their current and what is their voltage. 472 00:22:49,390 --> 00:22:51,245 They only get to tell you what is their current. 473 00:22:51,245 --> 00:22:55,960 474 00:22:55,960 --> 00:22:58,630 So let's think about now, if you had a more complicated 475 00:22:58,630 --> 00:23:02,930 system, how could you systematically go about 476 00:23:02,930 --> 00:23:05,580 finding the solution? 477 00:23:05,580 --> 00:23:07,430 As was mentioned earlier, there's something called 478 00:23:07,430 --> 00:23:08,110 Kirchhoff's law. 479 00:23:08,110 --> 00:23:10,060 And in fact, there's two of them. 480 00:23:10,060 --> 00:23:12,500 Kirchhoff's voltage law and Kirchhoff's current law. 481 00:23:12,500 --> 00:23:18,280 Kirchhoff's voltage law, in its most elementary form, says 482 00:23:18,280 --> 00:23:21,790 that if you trace the path around any closed path in a 483 00:23:21,790 --> 00:23:24,450 circuit, regardless of what the path is-- 484 00:23:24,450 --> 00:23:26,040 every closed path-- 485 00:23:26,040 --> 00:23:30,650 the sum of the voltages going around that closed path is 0. 486 00:23:30,650 --> 00:23:35,830 So for example in this circuit, the red path 487 00:23:35,830 --> 00:23:39,130 illustrates one closed path through the circuit. 488 00:23:39,130 --> 00:23:41,680 It goes up through the voltage source, down through this 489 00:23:41,680 --> 00:23:44,900 resistor, and then down through that resistor. 490 00:23:44,900 --> 00:23:49,160 Kirchhoff's voltage law says the sum of the voltages around 491 00:23:49,160 --> 00:23:51,990 that loop is 0. 492 00:23:51,990 --> 00:23:56,150 That's written mathematically here, minus v1 for here, plus 493 00:23:56,150 --> 00:23:59,850 v2 for here, plus v4 for here is 0. 494 00:23:59,850 --> 00:24:01,660 OK, where do the signs come from? 495 00:24:01,660 --> 00:24:05,570 The signs came from the reference directions that we 496 00:24:05,570 --> 00:24:09,730 assigned arbitrarily to the elements. 497 00:24:09,730 --> 00:24:13,280 Before I ever do a circuits question, I always assign a 498 00:24:13,280 --> 00:24:15,520 reference direction. 499 00:24:15,520 --> 00:24:17,125 Every voltage has a positive terminal 500 00:24:17,125 --> 00:24:19,120 and a negative terminal. 501 00:24:19,120 --> 00:24:22,210 And I must be consistent in order to apply these rules. 502 00:24:22,210 --> 00:24:27,800 These rules only work if I declare a reference direction 503 00:24:27,800 --> 00:24:29,320 and stick with it. 504 00:24:29,320 --> 00:24:31,350 If midway through a problem I flip it, I'll 505 00:24:31,350 --> 00:24:33,810 get the wrong answer. 506 00:24:33,810 --> 00:24:36,220 So the minus sign has to do with the fact that as I trace 507 00:24:36,220 --> 00:24:41,730 this path, I enter the minus part of this guy, but the plus 508 00:24:41,730 --> 00:24:43,200 part of that guy and that guy. 509 00:24:43,200 --> 00:24:47,250 So the sign of v1 is negated relative to the others. 510 00:24:47,250 --> 00:24:50,220 511 00:24:50,220 --> 00:24:53,510 A different way to think about that is here, we can think 512 00:24:53,510 --> 00:24:56,330 that v1 is the sum of v2 and v4. 513 00:24:56,330 --> 00:24:58,700 That's sometimes more intuitive because if you 514 00:24:58,700 --> 00:25:03,280 started here, going through this path you would end up 515 00:25:03,280 --> 00:25:08,540 with a voltage that is v1 higher than where you started. 516 00:25:08,540 --> 00:25:11,580 Whereas starting here, you would end up with a voltage 517 00:25:11,580 --> 00:25:15,030 here that's v4 higher than where you started. 518 00:25:15,030 --> 00:25:17,400 And then by the time you got to here, it would be v2 plus 519 00:25:17,400 --> 00:25:21,970 v4 higher than where you started. 520 00:25:21,970 --> 00:25:25,030 You start one place and on one route, you end up v1 higher. 521 00:25:25,030 --> 00:25:28,030 And in the other route, you get v4 plus v2 higher. 522 00:25:28,030 --> 00:25:31,120 So it must be the case that v1 is the same as v2 plus v4. 523 00:25:31,120 --> 00:25:33,620 524 00:25:33,620 --> 00:25:35,710 Those are absolutely equivalent ways of 525 00:25:35,710 --> 00:25:36,990 thinking about it. 526 00:25:36,990 --> 00:25:38,650 So those laws are equivalent. 527 00:25:38,650 --> 00:25:40,600 If you think about it a path, you think 528 00:25:40,600 --> 00:25:43,305 about some of the paths-- 529 00:25:43,305 --> 00:25:43,780 no. 530 00:25:43,780 --> 00:25:47,700 The path coinciding with the negative direction of some of 531 00:25:47,700 --> 00:25:49,750 the elements and the positive direction of others. 532 00:25:49,750 --> 00:25:53,030 533 00:25:53,030 --> 00:25:58,720 OK, how many other paths are there? 534 00:25:58,720 --> 00:26:00,090 Take 20 seconds, talk to your neighbor. 535 00:26:00,090 --> 00:26:04,630 Figure out all of the possible paths for 536 00:26:04,630 --> 00:26:06,120 which KVL has to apply. 537 00:26:06,120 --> 00:28:35,850 538 00:28:35,850 --> 00:28:39,450 OK, so everybody raise your hand and show a number of 539 00:28:39,450 --> 00:28:44,130 fingers equal to the number of KVL equations less two. 540 00:28:44,130 --> 00:28:49,060 541 00:28:49,060 --> 00:28:51,000 Oh, very good. 542 00:28:51,000 --> 00:28:52,170 Virtually 100% correct. 543 00:28:52,170 --> 00:28:53,420 Why do you all say (5)? 544 00:28:53,420 --> 00:28:55,760 545 00:28:55,760 --> 00:28:57,760 Which is to say 7. 546 00:28:57,760 --> 00:28:59,010 Why do you all say 7? 547 00:28:59,010 --> 00:29:01,970 548 00:29:01,970 --> 00:29:03,650 So there's 3 obvious ones. 549 00:29:03,650 --> 00:29:05,660 I was expecting a couple of 3's. 550 00:29:05,660 --> 00:29:07,230 This was supposed to be-- 551 00:29:07,230 --> 00:29:09,930 OK, yeah, I do plot against you. 552 00:29:09,930 --> 00:29:12,410 I was expecting some 3's. 553 00:29:12,410 --> 00:29:14,770 So there's 3 obvious paths that are analogous to the 554 00:29:14,770 --> 00:29:15,620 first one we looked at. 555 00:29:15,620 --> 00:29:18,980 If I call the first path A, then there's B and C which are 556 00:29:18,980 --> 00:29:22,030 the excursions around here. 557 00:29:22,030 --> 00:29:23,850 And you can write the equations just the same. 558 00:29:23,850 --> 00:29:26,030 They each involve three voltages. 559 00:29:26,030 --> 00:29:28,620 And they each go through, some starting at the negative side 560 00:29:28,620 --> 00:29:32,720 and some starting at the positive side. 561 00:29:32,720 --> 00:29:35,230 So those are in some sense, the obvious ones. 562 00:29:35,230 --> 00:29:39,040 But there are others too. 563 00:29:39,040 --> 00:29:42,370 So one way to think about it, what I'd like you to do is 564 00:29:42,370 --> 00:29:44,290 enumerate all the paths through the circuit. 565 00:29:44,290 --> 00:29:46,640 I should have said all the paths through the circuit that 566 00:29:46,640 --> 00:29:53,300 go through each element one or fewer times. 567 00:29:53,300 --> 00:29:58,660 I don't want you to go through the same element twice. 568 00:29:58,660 --> 00:30:02,280 So here's another path that would go through elements at 569 00:30:02,280 --> 00:30:04,290 most one time. 570 00:30:04,290 --> 00:30:07,230 So up through here, over through here, which didn't go 571 00:30:07,230 --> 00:30:08,050 through any elements. 572 00:30:08,050 --> 00:30:10,480 Down through that element, across that, down through 573 00:30:10,480 --> 00:30:11,850 here, et cetera. 574 00:30:11,850 --> 00:30:14,090 And you get an equation for that. 575 00:30:14,090 --> 00:30:14,950 Here's another. 576 00:30:14,950 --> 00:30:16,140 Here's another. 577 00:30:16,140 --> 00:30:18,830 Here's another. 578 00:30:18,830 --> 00:30:21,220 And if you try to think about a general rule, a general rule 579 00:30:21,220 --> 00:30:24,410 is something like, how many of those panels can you make and 580 00:30:24,410 --> 00:30:29,790 piece together where the loop goes through the perimeter? 581 00:30:29,790 --> 00:30:31,970 You're not allowed to go through an inner place because 582 00:30:31,970 --> 00:30:33,430 if you went through an inner node, you'd have to 583 00:30:33,430 --> 00:30:34,770 go through it twice. 584 00:30:34,770 --> 00:30:37,550 If you wanted the path to go through an inner element, 585 00:30:37,550 --> 00:30:40,340 you'd have to go through that element twice. 586 00:30:40,340 --> 00:30:41,810 So in fact, the answer is 7. 587 00:30:41,810 --> 00:30:49,200 There are 7 different paths according to Kirchhoff's 588 00:30:49,200 --> 00:30:51,970 voltage law, all of which the sum of the voltages around 589 00:30:51,970 --> 00:30:53,505 those paths has to be 0. 590 00:30:53,505 --> 00:30:56,080 591 00:30:56,080 --> 00:30:59,260 The problem is, of course, that those equations are not 592 00:30:59,260 --> 00:31:00,625 all linearly independent. 593 00:31:00,625 --> 00:31:04,030 594 00:31:04,030 --> 00:31:07,280 So if you just had a general purpose equation solver-- and 595 00:31:07,280 --> 00:31:09,330 by the way, we'll write one of those in week 596 00:31:09,330 --> 00:31:11,970 8 for solving circuits. 597 00:31:11,970 --> 00:31:15,400 If you just passed those 7 equations into a general 598 00:31:15,400 --> 00:31:17,410 purpose equation solver, it would tell you there's 599 00:31:17,410 --> 00:31:20,010 something awry with your equations because they're not 600 00:31:20,010 --> 00:31:22,800 linearly independent. 601 00:31:22,800 --> 00:31:27,140 So you can, however, think about linearly independent in 602 00:31:27,140 --> 00:31:29,050 particularly simple cases. 603 00:31:29,050 --> 00:31:32,140 This network is a particular kind of network that we call a 604 00:31:32,140 --> 00:31:34,070 planar network. 605 00:31:34,070 --> 00:31:37,680 A planar network is one that I can draw on a sheet of paper 606 00:31:37,680 --> 00:31:41,570 without crossing wires. 607 00:31:41,570 --> 00:31:44,040 So I can draw this network without crossing wires. 608 00:31:44,040 --> 00:31:45,410 I'll call it planar. 609 00:31:45,410 --> 00:31:49,080 And it turns out that Kirchhoff's voltage laws for 610 00:31:49,080 --> 00:31:54,200 the innermost loops are always independent of each other. 611 00:31:54,200 --> 00:31:56,490 That's kind of obvious because as you go to a -- 612 00:31:56,490 --> 00:31:58,940 so each loop contains at least one element that some other 613 00:31:58,940 --> 00:32:01,460 loop didn't have. 614 00:32:01,460 --> 00:32:04,620 So that's kind of the reasoning for why it works. 615 00:32:04,620 --> 00:32:07,510 So if you think about this particular loop, which we 616 00:32:07,510 --> 00:32:13,370 included in the 7, you can think about that as being the 617 00:32:13,370 --> 00:32:17,020 sum of the loops this way, the A loop and the B loop. 618 00:32:17,020 --> 00:32:20,360 Because if you write KVL for the A loop and KVL for the B 619 00:32:20,360 --> 00:32:25,980 loop and add them, you end up deriving KVL for the more 620 00:32:25,980 --> 00:32:28,220 complicated path. 621 00:32:28,220 --> 00:32:30,190 And if you think about what's going on, it's not anything 622 00:32:30,190 --> 00:32:31,880 terribly magical. 623 00:32:31,880 --> 00:32:36,850 This path is the same as the A path added to that path, where 624 00:32:36,850 --> 00:32:40,900 I went through this element down when I did the A path and 625 00:32:40,900 --> 00:32:45,610 up when I did the B path. 626 00:32:45,610 --> 00:32:48,230 So those parts canceled out. 627 00:32:48,230 --> 00:32:50,420 That was the rule that I was talking about how I don't 628 00:32:50,420 --> 00:32:54,480 really want to go through the same element twice when I'm 629 00:32:54,480 --> 00:32:56,970 applying KVL. 630 00:32:56,970 --> 00:33:00,280 So the idea then is that there's a systematic way, an 631 00:33:00,280 --> 00:33:05,170 easy way to figure out all the KVL loops. 632 00:33:05,170 --> 00:33:06,765 You just think about all the possible 633 00:33:06,765 --> 00:33:08,360 paths through the circuit. 634 00:33:08,360 --> 00:33:12,210 You do have to worry about linearly independent. 635 00:33:12,210 --> 00:33:14,440 In the case of planar networks, that's pretty 636 00:33:14,440 --> 00:33:15,220 straightforward. 637 00:33:15,220 --> 00:33:18,510 Planar networks, you can always figure out the linearly 638 00:33:18,510 --> 00:33:21,770 independent KVL equations by looking at the smallest 639 00:33:21,770 --> 00:33:24,140 possible loops. 640 00:33:24,140 --> 00:33:26,960 The loops with small area. 641 00:33:26,960 --> 00:33:28,250 OK, so that's half of it. 642 00:33:28,250 --> 00:33:28,960 That's KVL. 643 00:33:28,960 --> 00:33:32,650 The other Kirchhoff's law is KCL, Kirchhoff's Current Law. 644 00:33:32,650 --> 00:33:37,250 645 00:33:37,250 --> 00:33:41,760 There we are thinking of the flow of current. 646 00:33:41,760 --> 00:33:44,770 So the flow of current is analogous to the flow of 647 00:33:44,770 --> 00:33:47,560 incompressible fluid. 648 00:33:47,560 --> 00:33:49,960 Water, for example. 649 00:33:49,960 --> 00:33:52,310 If you trace the amount of water that flows through a 650 00:33:52,310 --> 00:33:56,560 pipe that goes into a Y, then the sum of the flows out has 651 00:33:56,560 --> 00:33:58,660 to equal the flow in. 652 00:33:58,660 --> 00:34:02,710 If that weren't true, the water would be building up. 653 00:34:02,710 --> 00:34:05,860 So we think about pipes as transporting the flow of water 654 00:34:05,860 --> 00:34:08,270 without allowing it to build up anywhere. 655 00:34:08,270 --> 00:34:10,956 That's precisely how we think about wires 656 00:34:10,956 --> 00:34:13,650 in electrical circuits. 657 00:34:13,650 --> 00:34:16,540 The wires allow the transport of electrons but don't allow 658 00:34:16,540 --> 00:34:20,770 the buildup of electrons. 659 00:34:20,770 --> 00:34:22,820 OK, do electrons build up? 660 00:34:22,820 --> 00:34:24,000 Sure. 661 00:34:24,000 --> 00:34:28,219 But in our idealized world, we say they don't build up in the 662 00:34:28,219 --> 00:34:31,130 wires, they build up in a part. 663 00:34:31,130 --> 00:34:33,010 And we'll have a special part that allows the 664 00:34:33,010 --> 00:34:34,570 electrons to build up. 665 00:34:34,570 --> 00:34:36,749 So we're not excluding the possibility that they build 666 00:34:36,749 --> 00:34:41,840 up, we're just saying that in this formalism, we don't allow 667 00:34:41,840 --> 00:34:44,540 the electrons to build up in the wires. 668 00:34:44,540 --> 00:34:47,090 So for the purpose of the wires, current in is equal to 669 00:34:47,090 --> 00:34:47,630 the current out. 670 00:34:47,630 --> 00:34:50,949 The net current in is 0. 671 00:34:50,949 --> 00:34:57,360 So we will think then, about the circuit having nodes. 672 00:34:57,360 --> 00:35:01,300 The nodes are the places where more than one element meets, 673 00:35:01,300 --> 00:35:03,980 two or more elements meet. 674 00:35:03,980 --> 00:35:07,790 And we will apply KCL at each node. 675 00:35:07,790 --> 00:35:10,650 So for example, in this simple circuit where I would have 676 00:35:10,650 --> 00:35:18,010 three parts connected in what we would call parallel, they 677 00:35:18,010 --> 00:35:20,330 share a node at the top and they share 678 00:35:20,330 --> 00:35:22,620 a node at the bottom. 679 00:35:22,620 --> 00:35:24,870 So even though it looks like there's multiple interconnects 680 00:35:24,870 --> 00:35:27,290 up here, we say that's one node. 681 00:35:27,290 --> 00:35:30,430 682 00:35:30,430 --> 00:35:32,940 And we would say that the sum of the currents into the node 683 00:35:32,940 --> 00:35:35,810 is equal to the sum of the currents out. 684 00:35:35,810 --> 00:35:38,960 So if I lab all of the possible currents that come 685 00:35:38,960 --> 00:35:42,160 out of that node, I would have i1, i2, i3. 686 00:35:42,160 --> 00:35:43,685 i1 goes through the first one, the second one 687 00:35:43,685 --> 00:35:45,100 and the third one. 688 00:35:45,100 --> 00:35:48,300 And so I would conclude from Kirchhoff's current law that 689 00:35:48,300 --> 00:35:51,340 the sum of i1, i2, and i3 is 0. 690 00:35:51,340 --> 00:35:51,720 OK. 691 00:35:51,720 --> 00:35:53,670 Easy, right? 692 00:35:53,670 --> 00:35:59,380 As I said, we're going to make an abstraction where the 693 00:35:59,380 --> 00:36:03,620 electrons don't build up in the wires. 694 00:36:03,620 --> 00:36:06,190 They don't even build up in the parts. 695 00:36:06,190 --> 00:36:07,330 They do get stored in the parts. 696 00:36:07,330 --> 00:36:10,230 That's a little confusing, we'll come back to that. 697 00:36:10,230 --> 00:36:12,540 If they don't build up in the parts, then the current that 698 00:36:12,540 --> 00:36:14,720 goes in this leg has to come out that leg. 699 00:36:14,720 --> 00:36:17,920 700 00:36:17,920 --> 00:36:23,930 If that's true, then i1 is i4, i2 is i5, i3 is i6, and we end 701 00:36:23,930 --> 00:36:27,030 up with another equation down here, which turns out to be 702 00:36:27,030 --> 00:36:28,900 precisely the same as the one at the top. 703 00:36:28,900 --> 00:36:33,380 704 00:36:33,380 --> 00:36:35,790 Everybody's happy with that? 705 00:36:35,790 --> 00:36:37,550 So we're thinking about this just the way we would think 706 00:36:37,550 --> 00:36:39,650 about water flow. 707 00:36:39,650 --> 00:36:42,840 If there's water flow into a part, it better be coming out. 708 00:36:42,840 --> 00:36:46,750 If there's water flow in a pipe, the water that goes into 709 00:36:46,750 --> 00:36:49,830 the pipe better come out of the pipe someplace. 710 00:36:49,830 --> 00:36:51,700 So here is an arbitrary network 711 00:36:51,700 --> 00:36:53,340 made out of four parts. 712 00:36:53,340 --> 00:36:57,070 How many linearly independent KCL equations are there? 713 00:36:57,070 --> 00:38:20,810 714 00:38:20,810 --> 00:38:24,740 So how many linearly independent KCL equations are 715 00:38:24,740 --> 00:38:26,280 in that network? 716 00:38:26,280 --> 00:38:30,280 Everyone raise your hand, some number of KCL equations. 717 00:38:30,280 --> 00:38:34,125 718 00:38:34,125 --> 00:38:36,250 OK, I'm seeing a bigger variety. 719 00:38:36,250 --> 00:38:38,170 I see (1)'s, (2)'s, and (3)'s. 720 00:38:38,170 --> 00:38:39,200 I don't see any (4)'s. 721 00:38:39,200 --> 00:38:40,515 That's probably good. 722 00:38:40,515 --> 00:38:43,400 723 00:38:43,400 --> 00:38:46,600 So how do you think about the number of linearly independent 724 00:38:46,600 --> 00:38:47,850 KCL equations? 725 00:38:47,850 --> 00:38:52,945 726 00:38:52,945 --> 00:38:55,860 So the first thing to do is to label things. 727 00:38:55,860 --> 00:38:59,430 728 00:38:59,430 --> 00:39:01,660 So you have to have reference directions before you can sort 729 00:39:01,660 --> 00:39:03,250 of think about things. 730 00:39:03,250 --> 00:39:04,540 So we have four elements. 731 00:39:04,540 --> 00:39:08,900 We would be expecting to see four element currents. 732 00:39:08,900 --> 00:39:10,800 The same current that goes into an element has 733 00:39:10,800 --> 00:39:12,460 to come out of it. 734 00:39:12,460 --> 00:39:14,080 So there's element current 1, 2, 3, and 4. 735 00:39:14,080 --> 00:39:17,490 736 00:39:17,490 --> 00:39:21,230 There are three nodes, so we might be 737 00:39:21,230 --> 00:39:24,680 expecting three KCL equations. 738 00:39:24,680 --> 00:39:28,190 Here's one node from which you would conclude that the sum of 739 00:39:28,190 --> 00:39:31,520 i1 and i2 better be 0. 740 00:39:31,520 --> 00:39:33,520 Here's a node from which you would conclude that the 741 00:39:33,520 --> 00:39:38,890 current in i2 better be i3 plus i4. 742 00:39:38,890 --> 00:39:41,440 And here is a node from which you would conclude that i1 743 00:39:41,440 --> 00:39:45,230 plus i3 plus i4 is 0. 744 00:39:45,230 --> 00:39:48,380 So I can write one KCL equation for every node, 745 00:39:48,380 --> 00:39:50,840 that's not surprising. 746 00:39:50,840 --> 00:39:52,820 But if you look at those equations, you'll see that 747 00:39:52,820 --> 00:39:54,390 they're not linearly independent. 748 00:39:54,390 --> 00:39:57,710 In fact, if you solve this one for i2-- 749 00:39:57,710 --> 00:39:59,150 it's already solved for i2. 750 00:39:59,150 --> 00:40:03,910 Stick that answer up here, you get i3 plus i4 added to i1 is 751 00:40:03,910 --> 00:40:06,590 0, which is just the same as that equation. 752 00:40:06,590 --> 00:40:11,540 So of those three equations, only two of them are linearly 753 00:40:11,540 --> 00:40:13,090 independent. 754 00:40:13,090 --> 00:40:14,525 The answer to that problem was (2). 755 00:40:14,525 --> 00:40:18,820 756 00:40:18,820 --> 00:40:20,920 And there's a pattern. 757 00:40:20,920 --> 00:40:23,690 So think about the pattern in terms of figuring out the 758 00:40:23,690 --> 00:40:26,320 number of linearly independent KCL equations that are in a 759 00:40:26,320 --> 00:40:27,735 slightly more complicated network. 760 00:40:27,735 --> 00:42:00,512 761 00:42:00,512 --> 00:42:05,880 So what's the answer here, how many KCL 762 00:42:05,880 --> 00:42:07,270 equations are in this network? 763 00:42:07,270 --> 00:42:10,040 764 00:42:10,040 --> 00:42:11,290 Wow. 765 00:42:11,290 --> 00:42:13,160 766 00:42:13,160 --> 00:42:18,020 Well, I'm not getting any of the answers I would have said. 767 00:42:18,020 --> 00:42:20,776 What does that mean? 768 00:42:20,776 --> 00:42:23,550 Ah, I'm forgetting to add 2. 769 00:42:23,550 --> 00:42:24,270 That's my problem. 770 00:42:24,270 --> 00:42:27,370 OK, now I'm getting some of the answers that I would 771 00:42:27,370 --> 00:42:27,980 expect to get. 772 00:42:27,980 --> 00:42:30,080 OK, got it. 773 00:42:30,080 --> 00:42:31,730 I confused myself. 774 00:42:31,730 --> 00:42:33,910 OK, the vast majority say (1). 775 00:42:33,910 --> 00:42:35,640 How do you get that? 776 00:42:35,640 --> 00:42:36,890 Which is 3. 777 00:42:36,890 --> 00:42:42,574 778 00:42:42,574 --> 00:42:45,740 So again, you think of how in this circuit there are four 779 00:42:45,740 --> 00:42:51,880 nodes, A, B, C, D. So we can think about writing a KCL 780 00:42:51,880 --> 00:42:53,060 equation for each one. 781 00:42:53,060 --> 00:42:56,450 If we go to A, A has three currents coming out of it -- 782 00:42:56,450 --> 00:42:57,360 1, 2, 3. 783 00:42:57,360 --> 00:43:02,910 So the sum of those has to be 0, et cetera. 784 00:43:02,910 --> 00:43:05,010 And if you think about those equations, they're not 785 00:43:05,010 --> 00:43:06,450 linearly independent either. 786 00:43:06,450 --> 00:43:09,010 787 00:43:09,010 --> 00:43:11,250 If you work through the math, you see that there's exactly 788 00:43:11,250 --> 00:43:15,330 one of those equations that you can eliminate. 789 00:43:15,330 --> 00:43:17,350 So you're left with three linearly 790 00:43:17,350 --> 00:43:20,250 independent KCL equations. 791 00:43:20,250 --> 00:43:22,210 And so there's a pattern emerging here. 792 00:43:22,210 --> 00:43:23,460 Somebody see the pattern? 793 00:43:23,460 --> 00:43:25,540 794 00:43:25,540 --> 00:43:26,520 1 minus. 795 00:43:26,520 --> 00:43:27,770 Can somebody prove the pattern? 796 00:43:27,770 --> 00:43:30,560 797 00:43:30,560 --> 00:43:31,520 So there's a pattern here. 798 00:43:31,520 --> 00:43:35,810 The pattern is take the number of nodes and the number of 799 00:43:35,810 --> 00:43:37,910 independent KCL equations as one less. 800 00:43:37,910 --> 00:43:41,480 801 00:43:41,480 --> 00:43:43,398 So the challenge is, can you prove it? 802 00:43:43,398 --> 00:43:46,390 803 00:43:46,390 --> 00:43:47,640 And by the theory of lectures-- 804 00:43:47,640 --> 00:43:50,858 805 00:43:50,858 --> 00:43:51,356 AUDIENCE: Yes. 806 00:43:51,356 --> 00:43:53,350 PROFESSOR: Yes. 807 00:43:53,350 --> 00:43:55,510 And by a corollary of the theory of lectures, the way 808 00:43:55,510 --> 00:43:57,010 you would prove it is? 809 00:43:57,010 --> 00:43:57,970 AUDIENCE: On the next slide. 810 00:43:57,970 --> 00:43:58,980 PROFESSOR: On the next slide. 811 00:43:58,980 --> 00:44:00,230 Exactly. 812 00:44:00,230 --> 00:44:02,960 813 00:44:02,960 --> 00:44:04,210 So how do I prove it? 814 00:44:04,210 --> 00:44:06,958 815 00:44:06,958 --> 00:44:08,930 Yeah? 816 00:44:08,930 --> 00:44:11,395 AUDIENCE: Whenever you take minus 1, you just add all the 817 00:44:11,395 --> 00:44:12,650 [UNINTELLIGIBLE] together [UNINTELLIGIBLE]. 818 00:44:12,650 --> 00:44:12,910 PROFESSOR: Yeah. 819 00:44:12,910 --> 00:44:15,312 So there's something special about the last one. 820 00:44:15,312 --> 00:44:17,304 Why should there be something special about the last one. 821 00:44:17,304 --> 00:44:19,296 AUDIENCE: Because the circuit's closed. 822 00:44:19,296 --> 00:44:22,504 PROFESSOR: Because the circuit's closed. 823 00:44:22,504 --> 00:44:23,760 That's right. 824 00:44:23,760 --> 00:44:26,230 So the idea is to sort of generalize the way 825 00:44:26,230 --> 00:44:27,480 we think about KCL. 826 00:44:27,480 --> 00:44:29,690 827 00:44:29,690 --> 00:44:30,710 So we start with a circuit. 828 00:44:30,710 --> 00:44:33,840 We think about having four nodes here. 829 00:44:33,840 --> 00:44:37,270 It's certainly the case that KCL holds for each node. 830 00:44:37,270 --> 00:44:41,376 So here's KCL for that node. 831 00:44:41,376 --> 00:44:46,830 But now if you think about KCL for this node, and then add 832 00:44:46,830 --> 00:44:51,180 them, that looks like a KCL equation. 833 00:44:51,180 --> 00:44:53,850 834 00:44:53,850 --> 00:44:57,200 But it applies to a super node. 835 00:44:57,200 --> 00:45:02,520 Imagine the node defined by the black box, and think about 836 00:45:02,520 --> 00:45:05,293 the net currents into or out of the black node. 837 00:45:05,293 --> 00:45:08,470 838 00:45:08,470 --> 00:45:13,940 This current i2, which leaves the red node, enters the green 839 00:45:13,940 --> 00:45:17,890 node, but doesn't go through the surface of the 840 00:45:17,890 --> 00:45:20,520 black node at all. 841 00:45:20,520 --> 00:45:23,600 That's exactly the current that's subtracted out when we 842 00:45:23,600 --> 00:45:26,986 added the red equation to the green equation. 843 00:45:26,986 --> 00:45:28,940 Does that make sense? 844 00:45:28,940 --> 00:45:34,090 So KCL says, oh, if all the currents at a node have to sum 845 00:45:34,090 --> 00:45:40,380 to 0, and if elements have the same current coming out and 846 00:45:40,380 --> 00:45:48,560 going in, then if you draw a box around an element, what 847 00:45:48,560 --> 00:45:51,000 goes into the element is the same as what 848 00:45:51,000 --> 00:45:52,240 comes out of the element. 849 00:45:52,240 --> 00:45:55,340 It doesn't change the net current through the surface. 850 00:45:55,340 --> 00:45:58,890 So the generalization of the KCL equation, KCL says the sum 851 00:45:58,890 --> 00:46:00,450 of the currents into a node is 0. 852 00:46:00,450 --> 00:46:06,690 The generalization says take any closed path in a circuit, 853 00:46:06,690 --> 00:46:11,630 the sum of the currents going across that closed path is 0. 854 00:46:11,630 --> 00:46:16,600 So if we apply that rule again, think about node 3. 855 00:46:16,600 --> 00:46:20,470 If we add the result of node 3 to the black node, which was 856 00:46:20,470 --> 00:46:25,170 the sum of 1 and 2, we get the new green curve. 857 00:46:25,170 --> 00:46:27,340 We get the new green equation. 858 00:46:27,340 --> 00:46:29,720 And what that says is the sum of the currents going across 859 00:46:29,720 --> 00:46:33,120 the green super node-- 860 00:46:33,120 --> 00:46:36,530 OK, so what's going on? i1 is coming out of it, i4 is coming 861 00:46:36,530 --> 00:46:38,330 out of it, i5 is coming out of it. 862 00:46:38,330 --> 00:46:41,540 So the sum of i1, i4, and i5 has to be 0. 863 00:46:41,540 --> 00:46:46,570 864 00:46:46,570 --> 00:46:49,030 Well, KCL says the sum of the currents coming out of a 865 00:46:49,030 --> 00:46:50,890 node must be 0. 866 00:46:50,890 --> 00:46:54,990 The super KCL says the sum of the currents coming out of any 867 00:46:54,990 --> 00:47:00,180 closed region is also 0. 868 00:47:00,180 --> 00:47:03,840 But the interesting thing about this closed region is 869 00:47:03,840 --> 00:47:06,425 that it encloses all but one of the nodes. 870 00:47:06,425 --> 00:47:09,230 871 00:47:09,230 --> 00:47:11,310 That's always true. 872 00:47:11,310 --> 00:47:14,150 Regardless of the system, regardless of the circuit, you 873 00:47:14,150 --> 00:47:19,090 can always draw a line that will isolate one node from all 874 00:47:19,090 --> 00:47:22,030 the others. 875 00:47:22,030 --> 00:47:25,825 So what that proves is that you can always write KCL for 876 00:47:25,825 --> 00:47:28,920 this node in terms of KCL for those nodes. 877 00:47:28,920 --> 00:47:31,870 878 00:47:31,870 --> 00:47:31,965 Ok. 879 00:47:31,965 --> 00:47:34,770 So there's a generalization then that says that you can 880 00:47:34,770 --> 00:47:40,120 always write KCL for every node. 881 00:47:40,120 --> 00:47:42,130 They will always be linearly dependent. 882 00:47:42,130 --> 00:47:43,580 So you can always throw away one. 883 00:47:43,580 --> 00:47:47,780 884 00:47:47,780 --> 00:47:49,690 So in some sense now, we're done. 885 00:47:49,690 --> 00:47:51,840 We've just finished circuit theory. 886 00:47:51,840 --> 00:47:55,060 We talked about how every element has to have a law. 887 00:47:55,060 --> 00:47:56,470 A resistor is Ohm's Law. 888 00:47:56,470 --> 00:47:58,820 A voltage source says that the voltage across the terminals 889 00:47:58,820 --> 00:48:00,295 is always a constant. 890 00:48:00,295 --> 00:48:02,730 A current source says that the current through the current 891 00:48:02,730 --> 00:48:04,530 source is always a constant. 892 00:48:04,530 --> 00:48:07,380 So every element tells you one law. 893 00:48:07,380 --> 00:48:10,170 894 00:48:10,170 --> 00:48:12,760 We know how to think about KVL. 895 00:48:12,760 --> 00:48:18,930 So we know the rule for how the across variables behave. 896 00:48:18,930 --> 00:48:21,230 What's the aggregate behavior of all the across variables? 897 00:48:21,230 --> 00:48:24,140 Well, KVL has to be satisfied for every possible loop. 898 00:48:24,140 --> 00:48:26,430 The loops don't have to be independent. 899 00:48:26,430 --> 00:48:28,620 You have to worry about whether they're independent. 900 00:48:28,620 --> 00:48:31,020 The only simple rule we came up with-- we'll come up with 901 00:48:31,020 --> 00:48:32,415 another one in a moment. 902 00:48:32,415 --> 00:48:35,340 The only simple rule that we came up with was for planar 903 00:48:35,340 --> 00:48:38,410 circuits, where the innermost loops were linearly 904 00:48:38,410 --> 00:48:39,720 independent of each other. 905 00:48:39,720 --> 00:48:44,280 And you have to write KCL for all the nodes, except one. 906 00:48:44,280 --> 00:48:45,530 One of them never matters. 907 00:48:45,530 --> 00:48:48,110 908 00:48:48,110 --> 00:48:50,210 So in some sense, we're done. 909 00:48:50,210 --> 00:48:52,450 What we would do to solve the circuit, 910 00:48:52,450 --> 00:48:54,310 think about every element. 911 00:48:54,310 --> 00:48:59,240 For every element assign a voltage, a reference voltage. 912 00:48:59,240 --> 00:49:02,570 For every element, assign a current. 913 00:49:02,570 --> 00:49:04,460 Make sure they go in the right direction. 914 00:49:04,460 --> 00:49:07,460 We always define currents to go down 915 00:49:07,460 --> 00:49:09,070 the potential gradient. 916 00:49:09,070 --> 00:49:11,920 They always go in the directions through the element 917 00:49:11,920 --> 00:49:14,840 from the positive to the negative. 918 00:49:14,840 --> 00:49:18,610 So for every element, assign a current and a voltage. 919 00:49:18,610 --> 00:49:22,420 We have 6 elements, that's 12 unknowns. 920 00:49:22,420 --> 00:49:25,370 Now we dig and we find 12 equations. 921 00:49:25,370 --> 00:49:28,980 In this particular circuit, we found those 12 equations. 922 00:49:28,980 --> 00:49:31,505 There were three KCL equations, one for each of the 923 00:49:31,505 --> 00:49:32,550 inner loops. 924 00:49:32,550 --> 00:49:35,420 There were three KCL equations, one for each node 925 00:49:35,420 --> 00:49:38,080 except one. 926 00:49:38,080 --> 00:49:41,020 There were 5 Ohm's law equations, one for each one of 927 00:49:41,020 --> 00:49:41,750 the resistors. 928 00:49:41,750 --> 00:49:45,390 There was one source equation for the voltage source. 929 00:49:45,390 --> 00:49:49,120 12 equations, 12 unknowns, we're done. 930 00:49:49,120 --> 00:49:52,900 The only problem is a lot of equations. 931 00:49:52,900 --> 00:49:54,500 It's not a very complicated circuit. 932 00:49:54,500 --> 00:49:56,430 We've only got 6 elements. 933 00:49:56,430 --> 00:49:59,580 I tried to motivate this in terms of studying networks 934 00:49:59,580 --> 00:50:04,390 that had 10 to the 9 elements. 935 00:50:04,390 --> 00:50:06,750 This technique is not particularly great 936 00:50:06,750 --> 00:50:08,920 at 10 to the 9. 937 00:50:08,920 --> 00:50:11,240 It would probably work. 938 00:50:11,240 --> 00:50:13,130 But we would probably be interested in 939 00:50:13,130 --> 00:50:15,970 finding simpler ways. 940 00:50:15,970 --> 00:50:20,430 So there are simpler ways you might imagine, and we'll 941 00:50:20,430 --> 00:50:24,680 discuss two of them just very briefly. 942 00:50:24,680 --> 00:50:27,160 The dumb way that I just talked about is what we call 943 00:50:27,160 --> 00:50:30,190 primitive variables, element variables. 944 00:50:30,190 --> 00:50:33,600 If you write all the element variables, v1, v2, v3, v4, v5 945 00:50:33,600 --> 00:50:37,480 v6, all of the element currents, i1, i2, i3, i4, i5, 946 00:50:37,480 --> 00:50:40,630 i6, write all the equations, you can solve it. 947 00:50:40,630 --> 00:50:43,620 However, if you're judicious, you can figure out a smaller 948 00:50:43,620 --> 00:50:46,600 number of unknowns and a correspondingly smaller number 949 00:50:46,600 --> 00:50:47,340 of equations. 950 00:50:47,340 --> 00:50:49,640 One method is called the node method. 951 00:50:49,640 --> 00:50:52,180 952 00:50:52,180 --> 00:50:54,400 When we're thinking about the individual elements, the thing 953 00:50:54,400 --> 00:50:56,465 that matters is the voltage across the element. 954 00:50:56,465 --> 00:50:59,230 955 00:50:59,230 --> 00:51:02,130 However, that's not the easy way to 956 00:51:02,130 --> 00:51:06,140 write the circuit equations. 957 00:51:06,140 --> 00:51:10,110 A much easier way is not to tell me the voltage across an 958 00:51:10,110 --> 00:51:13,360 element, but instead tell me the voltage associated with 959 00:51:13,360 --> 00:51:14,610 each of the nodes. 960 00:51:14,610 --> 00:51:18,710 961 00:51:18,710 --> 00:51:23,400 If I tell you the voltage associated with every node, 962 00:51:23,400 --> 00:51:25,930 the important thing about that way of defining the variables 963 00:51:25,930 --> 00:51:29,040 is that you're guaranteed that from those variables, you can 964 00:51:29,040 --> 00:51:32,360 tell me the voltage across every part. 965 00:51:32,360 --> 00:51:35,630 So for example, in this circuit, this voltage source-- 966 00:51:35,630 --> 00:51:42,140 so if I call this one ground, we'll always have a magic node 967 00:51:42,140 --> 00:51:43,410 called ground. 968 00:51:43,410 --> 00:51:47,490 It is not special in the least. 969 00:51:47,490 --> 00:51:50,780 It's just the reference voltage. 970 00:51:50,780 --> 00:51:51,610 I'll come back to that. 971 00:51:51,610 --> 00:51:54,400 I'll say words in a minute about what the reference is. 972 00:51:54,400 --> 00:51:58,100 We always get to declare one node to be ground. 973 00:51:58,100 --> 00:52:01,270 We get one free node. 974 00:52:01,270 --> 00:52:04,830 It's a node whose voltage we don't care about because it's 975 00:52:04,830 --> 00:52:06,380 the reference for all voltages. 976 00:52:06,380 --> 00:52:09,260 It's a node whose current we don't care about because we 977 00:52:09,260 --> 00:52:12,370 get to throw away one node when we do current equations. 978 00:52:12,370 --> 00:52:15,400 So we have one special mode called ground, about which we 979 00:52:15,400 --> 00:52:16,660 don't care too much. 980 00:52:16,660 --> 00:52:18,900 Except that it's the most important node in the circuit. 981 00:52:18,900 --> 00:52:20,650 Except for that, we don't care about it. 982 00:52:20,650 --> 00:52:23,430 So this guy's ground. 983 00:52:23,430 --> 00:52:26,080 We think about its voltage being 0. 984 00:52:26,080 --> 00:52:30,360 Then this voltage supply makes that node be v0. 985 00:52:30,360 --> 00:52:32,250 I don't know what that is, so I'll call it e1. 986 00:52:32,250 --> 00:52:36,340 And I don't know what that is, so I'll call it e2. 987 00:52:36,340 --> 00:52:39,600 So if I tell you the voltage on all of those nodes, ground 988 00:52:39,600 --> 00:52:44,550 voltage is 0, the top voltage is v0, the left voltage is e1, 989 00:52:44,550 --> 00:52:45,670 the right voltage is e2. 990 00:52:45,670 --> 00:52:51,700 From those four numbers, 0 and 3 nontrivial numbers, you can 991 00:52:51,700 --> 00:52:54,940 find all of the component voltages. 992 00:52:54,940 --> 00:52:59,460 So for example, the voltage v6, the voltage across R6 is 993 00:52:59,460 --> 00:53:00,710 e2 minus e1. 994 00:53:00,710 --> 00:53:03,420 995 00:53:03,420 --> 00:53:06,740 The voltage v4, the voltage across the R4 996 00:53:06,740 --> 00:53:10,470 resistor is e1 minus 0. 997 00:53:10,470 --> 00:53:15,690 So if I tell you all the node voltages, you can tell me all 998 00:53:15,690 --> 00:53:17,250 of the element voltages. 999 00:53:17,250 --> 00:53:20,530 And in general, there's fewer nodes than there are 1000 00:53:20,530 --> 00:53:21,560 components. 1001 00:53:21,560 --> 00:53:23,820 OK, that's great. 1002 00:53:23,820 --> 00:53:26,990 So instead of naming the volts across the elements, we'll 1003 00:53:26,990 --> 00:53:30,115 name the voltages at the nodes because there's fewer of them. 1004 00:53:30,115 --> 00:53:32,750 1005 00:53:32,750 --> 00:53:36,620 Then, all we need to do in the node method is write the 1006 00:53:36,620 --> 00:53:42,780 minimum number of KCL equations. 1007 00:53:42,780 --> 00:53:46,060 We know we only have two unknowns, e1 and e2. 1008 00:53:46,060 --> 00:53:47,970 And it turns out-- and you can prove this, but I 1009 00:53:47,970 --> 00:53:50,990 won't prove it today. 1010 00:53:50,990 --> 00:53:54,000 It turns out that you need two KCL equations. 1011 00:53:54,000 --> 00:53:57,670 Two unknowns, e1, e2, two KCL equations. 1012 00:53:57,670 --> 00:54:01,400 And it turns out those two KCL equations are exactly the KCL 1013 00:54:01,400 --> 00:54:03,400 equations associated with the two nodes. 1014 00:54:03,400 --> 00:54:05,950 1015 00:54:05,950 --> 00:54:10,990 So the current leaving e1, so KCL at e1 -- 1016 00:54:10,990 --> 00:54:14,780 well, there's a current that goes that way. 1017 00:54:14,780 --> 00:54:19,080 Well, that's the voltage drop in going from e1 to v0, e1 1018 00:54:19,080 --> 00:54:21,380 minus v0, divided by R2. 1019 00:54:21,380 --> 00:54:23,610 That's Ohm's law. 1020 00:54:23,610 --> 00:54:28,600 So this term represents the current going up that leg plus 1021 00:54:28,600 --> 00:54:30,930 the current that goes through this leg, which is e1 1022 00:54:30,930 --> 00:54:33,780 minus e2 over R6. 1023 00:54:33,780 --> 00:54:36,450 Plus the current going in that leg, which is e1 1024 00:54:36,450 --> 00:54:39,250 minus 0 over R4. 1025 00:54:39,250 --> 00:54:40,980 The sum of those three currents better be 0. 1026 00:54:40,980 --> 00:54:43,640 1027 00:54:43,640 --> 00:54:46,460 Analogously, the sum of the currents at this node must be 1028 00:54:46,460 --> 00:54:50,070 0, and the equation looks virtually the same. 1029 00:54:50,070 --> 00:54:56,230 Because v0 is known, so it didn't add an unknown. 1030 00:54:56,230 --> 00:55:00,250 v0 was set by the voltage, by the voltage source. 1031 00:55:00,250 --> 00:55:04,000 So I have two equations, two unknowns, solved. 1032 00:55:04,000 --> 00:55:04,630 Done. 1033 00:55:04,630 --> 00:55:07,040 So rather than solving 12 equations and 12 unknowns, I 1034 00:55:07,040 --> 00:55:09,300 can do it with two equations and two unknowns. 1035 00:55:09,300 --> 00:55:11,640 That's called the node method. 1036 00:55:11,640 --> 00:55:15,610 One of the most interesting theories about circuits is 1037 00:55:15,610 --> 00:55:19,110 that every simplification that you can think about for 1038 00:55:19,110 --> 00:55:22,250 voltage has an analogous simplification that you can 1039 00:55:22,250 --> 00:55:23,050 think about in current. 1040 00:55:23,050 --> 00:55:24,030 That's called duality. 1041 00:55:24,030 --> 00:55:26,430 We won't do that because it's kind of complicated. 1042 00:55:26,430 --> 00:55:27,790 But it's kind of a cute result. 1043 00:55:27,790 --> 00:55:29,500 If you can think of a simplification that works in 1044 00:55:29,500 --> 00:55:31,920 voltage, then there is an analogous one, and 1045 00:55:31,920 --> 00:55:33,200 you can prove it. 1046 00:55:33,200 --> 00:55:36,860 In fact, you can formally derive what it must have been. 1047 00:55:36,860 --> 00:55:40,090 1048 00:55:40,090 --> 00:55:42,970 This is a rule for how you can simplify things by thinking 1049 00:55:42,970 --> 00:55:45,115 about voltages in aggregate. 1050 00:55:45,115 --> 00:55:47,640 1051 00:55:47,640 --> 00:55:50,480 Rather than thinking about the element voltages, think about 1052 00:55:50,480 --> 00:55:51,890 the node voltages. 1053 00:55:51,890 --> 00:55:56,510 The analogous current law is rather than thinking about the 1054 00:55:56,510 --> 00:55:59,770 currents through the elements, the element currents, think 1055 00:55:59,770 --> 00:56:01,950 about loop currents. 1056 00:56:01,950 --> 00:56:04,110 OK, that's a little bizarre. 1057 00:56:04,110 --> 00:56:07,430 So we name the loop, the current that flows in this 1058 00:56:07,430 --> 00:56:12,590 loop, IA, the current that flows in this loop, IB, and 1059 00:56:12,590 --> 00:56:14,780 the current hat flows in this loop, IC. 1060 00:56:14,780 --> 00:56:16,280 What on earth is he doing? 1061 00:56:16,280 --> 00:56:20,250 Well, the element voltages are some linear combination of 1062 00:56:20,250 --> 00:56:24,070 those loop currents. 1063 00:56:24,070 --> 00:56:26,460 And in fact, the coefficients in the linear combination are 1064 00:56:26,460 --> 00:56:27,660 one and minus one. 1065 00:56:27,660 --> 00:56:32,320 So the element current I4, the current that flows through the 1066 00:56:32,320 --> 00:56:37,780 R4 resistor is the sum of IA coming down minus IC, 1067 00:56:37,780 --> 00:56:39,670 which is going up. 1068 00:56:39,670 --> 00:56:45,500 So there's a way of thinking about each element current as 1069 00:56:45,500 --> 00:56:49,790 a sum or difference of the loop currents. 1070 00:56:49,790 --> 00:56:52,180 Everybody get that? 1071 00:56:52,180 --> 00:56:54,600 So instead of thinking about the individual element 1072 00:56:54,600 --> 00:56:56,190 currents, I think about the loop currents. 1073 00:56:56,190 --> 00:57:00,900 And now, I need to write three KVL equations. 1074 00:57:00,900 --> 00:57:06,730 So in the node method, I named the nodes and had to write two 1075 00:57:06,730 --> 00:57:08,480 KCL equations. 1076 00:57:08,480 --> 00:57:11,205 Here, I named the loop currents and I have to write 1077 00:57:11,205 --> 00:57:15,590 three KVL equations, one for each loop. 1078 00:57:15,590 --> 00:57:16,760 It's completely analogous. 1079 00:57:16,760 --> 00:57:19,420 If you write out a sentence, what did you do? 1080 00:57:19,420 --> 00:57:21,760 I assigned a voltage to every node, and I wrote 1081 00:57:21,760 --> 00:57:22,910 KCL of all the nodes. 1082 00:57:22,910 --> 00:57:26,040 Then if you turn the word "current" into the word 1083 00:57:26,040 --> 00:57:30,540 "voltage," the word "node" into the word "loop," you 1084 00:57:30,540 --> 00:57:33,920 derive this new method. 1085 00:57:33,920 --> 00:57:37,340 So this says that if I write KVL at the A loop, think about 1086 00:57:37,340 --> 00:57:40,580 spinning around this loop, as I go up through the voltage 1087 00:57:40,580 --> 00:57:43,910 source, so I go in the negative terminal here. 1088 00:57:43,910 --> 00:57:46,090 So that's minus v0. 1089 00:57:46,090 --> 00:57:48,940 As I go down through this resistor, I have to use Ohm's 1090 00:57:48,940 --> 00:57:51,770 law, so that's R2 times the down current. 1091 00:57:51,770 --> 00:57:57,610 Well, the down current is IA down minus IB up. 1092 00:57:57,610 --> 00:57:59,380 So I went up through here, down through here. 1093 00:57:59,380 --> 00:58:01,160 Now I go down through this one. 1094 00:58:01,160 --> 00:58:04,580 When I go down through that one, according to Ohm's law, 1095 00:58:04,580 --> 00:58:07,530 that's R4 times the current through that element. 1096 00:58:07,530 --> 00:58:13,490 That current-- well, it's IA down and it's IC up. 1097 00:58:13,490 --> 00:58:18,650 So this is the KVL equation for that loop. 1098 00:58:18,650 --> 00:58:23,030 I write two more of them, and I end up with three equations 1099 00:58:23,030 --> 00:58:25,590 and three unknowns. 1100 00:58:25,590 --> 00:58:29,410 Both the node method and the loop method resulted in a lot 1101 00:58:29,410 --> 00:58:33,330 fewer equations than the primitives did. 1102 00:58:33,330 --> 00:58:38,400 I had 12 primitive unknowns, 6 voltages and 6 currents. 1103 00:58:38,400 --> 00:58:42,670 In the node method, I get the number of independent nodes as 1104 00:58:42,670 --> 00:58:44,910 the number of equations and unknowns, which is less than 1105 00:58:44,910 --> 00:58:46,520 the number of primitive variables. 1106 00:58:46,520 --> 00:58:49,280 In the loop method, I have the number of independent loops. 1107 00:58:49,280 --> 00:58:51,840 1108 00:58:51,840 --> 00:58:55,100 Which is again, smaller. 1109 00:58:55,100 --> 00:58:57,220 So the idea then is that we have a couple of ways to think 1110 00:58:57,220 --> 00:59:00,360 about solving circuits. 1111 00:59:00,360 --> 00:59:04,450 Fundamentally, all we have are the element relationships and 1112 00:59:04,450 --> 00:59:05,846 the rules for combination. 1113 00:59:05,846 --> 00:59:09,180 Oh, this is starting to sound like PCAP, primitives and 1114 00:59:09,180 --> 00:59:10,380 combinations. 1115 00:59:10,380 --> 00:59:14,860 So the primitives are, how does the element constrain the 1116 00:59:14,860 --> 00:59:17,260 voltages and currents? 1117 00:59:17,260 --> 00:59:19,470 We know three of those, Ohm's law, voltage 1118 00:59:19,470 --> 00:59:21,480 source, current source. 1119 00:59:21,480 --> 00:59:24,380 And what are the rules for combination? 1120 00:59:24,380 --> 00:59:27,010 Well, the currents add to the node, and the voltages add 1121 00:59:27,010 --> 00:59:30,070 around loops. 1122 00:59:30,070 --> 00:59:33,720 OK, just to make sure you've absorb all that, figure out 1123 00:59:33,720 --> 00:59:35,535 the current I for this circuit. 1124 00:59:35,535 --> 01:02:31,840 1125 01:02:31,840 --> 01:02:33,810 OK, what's a good way to start? 1126 01:02:33,810 --> 01:02:38,830 What should I do to start thinking about calculating I? 1127 01:02:38,830 --> 01:02:40,190 OK, bad way. 1128 01:02:40,190 --> 01:02:42,520 Assign voltages and currents to everything. 1129 01:02:42,520 --> 01:02:46,150 4 elements, that's 4 voltages, 4 currents. 1130 01:02:46,150 --> 01:02:47,620 That's 8 unknowns. 1131 01:02:47,620 --> 01:02:48,990 Find 8 equations, solve. 1132 01:02:48,990 --> 01:02:50,200 That'll work. 1133 01:02:50,200 --> 01:02:50,980 Bad way. 1134 01:02:50,980 --> 01:02:52,230 What's a better way? 1135 01:02:52,230 --> 01:02:56,146 1136 01:02:56,146 --> 01:02:57,396 OK, [UNINTELLIGIBLE PHRASE]. 1137 01:02:57,396 --> 01:03:00,986 1138 01:03:00,986 --> 01:03:01,954 AUDIENCE: [UNINTELLIGIBLE PHRASE]. 1139 01:03:01,954 --> 01:03:03,890 PROFESSOR: It was on the previous sheet. 1140 01:03:03,890 --> 01:03:06,820 [UNINTELLIGIBLE PHRASE]. 1141 01:03:06,820 --> 01:03:08,020 AUDIENCE: KVL [UNINTELLIGIBLE]. 1142 01:03:08,020 --> 01:03:09,660 PROFESSOR: KVL for where? 1143 01:03:09,660 --> 01:03:11,616 AUDIENCE: Loops. 1144 01:03:11,616 --> 01:03:12,594 PROFESSOR: Which loop? 1145 01:03:12,594 --> 01:03:13,572 AUDIENCE: Left loop. 1146 01:03:13,572 --> 01:03:16,506 PROFESSOR: So do KVL on the left loop? 1147 01:03:16,506 --> 01:03:17,973 AUDIENCE: Yes. 1148 01:03:17,973 --> 01:03:19,440 PROFESSOR: OK, that's good. 1149 01:03:19,440 --> 01:03:22,090 But you have to tell me how to assign variables. 1150 01:03:22,090 --> 01:03:24,273 Do you want 8 primitive variables? 1151 01:03:24,273 --> 01:03:28,540 1152 01:03:28,540 --> 01:03:30,920 8 primitive variables are v1, i1, v2, i2, v3, i3, v4, i4. 1153 01:03:30,920 --> 01:03:36,130 1154 01:03:36,130 --> 01:03:38,540 So that's what I mean by primitive variables. 1155 01:03:38,540 --> 01:03:42,700 Or element variables is another word for it. 1156 01:03:42,700 --> 01:03:44,555 What's a better way than using element variables? 1157 01:03:44,555 --> 01:03:47,552 1158 01:03:47,552 --> 01:03:48,000 Yeah. 1159 01:03:48,000 --> 01:03:49,908 AUDIENCE: Create 2 loop equations. 1160 01:03:49,908 --> 01:03:52,902 PROFESSOR: Create 2 loop equations, that's fantastic. 1161 01:03:52,902 --> 01:03:54,950 AUDIENCE: I1 for the first loop, I2 for the second loop. 1162 01:03:54,950 --> 01:03:58,750 PROFESSOR: So if you do I1 going around here, then I1 is 1163 01:03:58,750 --> 01:04:05,915 actually I. And if you do I2 going around here, what's I2? 1164 01:04:05,915 --> 01:04:10,673 1165 01:04:10,673 --> 01:04:11,670 AUDIENCE: [INAUDIBLE]. 1166 01:04:11,670 --> 01:04:16,490 PROFESSOR: So if I think about I2 spinning around this loop, 1167 01:04:16,490 --> 01:04:20,920 so the sum of I1 and I2 goes through that box. 1168 01:04:20,920 --> 01:04:23,200 But the only current that goes through this box is? 1169 01:04:23,200 --> 01:04:26,556 1170 01:04:26,556 --> 01:04:27,806 AUDIENCE: [INAUDIBLE]. 1171 01:04:27,806 --> 01:04:29,993 1172 01:04:29,993 --> 01:04:31,957 PROFESSOR: So the suggestion is that I think about-- 1173 01:04:31,957 --> 01:04:43,250 1174 01:04:43,250 --> 01:04:49,790 so if I have I1 here, but I know that's I. Then I can see 1175 01:04:49,790 --> 01:04:51,310 immediately that since the only current that 1176 01:04:51,310 --> 01:04:52,380 goes through here-- 1177 01:04:52,380 --> 01:04:54,100 so if I have I1 and I2. 1178 01:04:54,100 --> 01:04:56,010 That was a very clever idea. 1179 01:04:56,010 --> 01:04:58,090 If you have I1 and I2, the only current that goes through 1180 01:04:58,090 --> 01:05:02,470 here is I. So I1 must've been I. 1181 01:05:02,470 --> 01:05:04,640 The only current that goes over here 1182 01:05:04,640 --> 01:05:07,390 must've been this guy. 1183 01:05:07,390 --> 01:05:11,340 So this must be minus 10. 1184 01:05:11,340 --> 01:05:14,130 So I could redo that this way. 1185 01:05:14,130 --> 01:05:19,130 I could say I've got 10 going that way. 1186 01:05:19,130 --> 01:05:21,410 That make sense? 1187 01:05:21,410 --> 01:05:26,340 So now I only have one unknown which is I. So that's a very 1188 01:05:26,340 --> 01:05:28,310 clever way of doing it. 1189 01:05:28,310 --> 01:05:31,720 So what I could do is showed here. 1190 01:05:31,720 --> 01:05:35,180 I have I going around one loop and I have 10 1191 01:05:35,180 --> 01:05:37,950 going around that loop. 1192 01:05:37,950 --> 01:05:39,950 That completely specifies all the currents. 1193 01:05:39,950 --> 01:05:43,080 So now all I need to do is write KVL for 1194 01:05:43,080 --> 01:05:45,980 these different cases. 1195 01:05:45,980 --> 01:05:50,810 So if I write KVL for the left loop, then I get going up 1196 01:05:50,810 --> 01:05:55,160 through here, that's minus 15, and going down through here, 1197 01:05:55,160 --> 01:05:58,360 going to the right through this guy is 3I. 1198 01:05:58,360 --> 01:06:03,090 Going down through this guy is 2 times I plus 10. 1199 01:06:03,090 --> 01:06:05,450 Both of these are going down, so you have to add them. 1200 01:06:05,450 --> 01:06:10,010 1201 01:06:10,010 --> 01:06:11,960 So I get one equation and one unknown. 1202 01:06:11,960 --> 01:06:15,160 And when I solve it, I get minus one. 1203 01:06:15,160 --> 01:06:16,410 That make sense? 1204 01:06:16,410 --> 01:06:18,470 1205 01:06:18,470 --> 01:06:19,900 There's an analogous way you could have 1206 01:06:19,900 --> 01:06:21,150 done it with one node. 1207 01:06:21,150 --> 01:06:24,080 1208 01:06:24,080 --> 01:06:28,130 You could have said that the circuit has a single node and 1209 01:06:28,130 --> 01:06:31,860 figured out KCL for that one node. 1210 01:06:31,860 --> 01:06:34,170 KCL would be the sum of the currents here. 1211 01:06:34,170 --> 01:06:35,770 There's a current that goes that way, that 1212 01:06:35,770 --> 01:06:36,880 way, and that way. 1213 01:06:36,880 --> 01:06:39,240 And again, you end up with 1 equation and 1 unknown. 1214 01:06:39,240 --> 01:06:40,069 Yes? 1215 01:06:40,069 --> 01:06:41,319 AUDIENCE: [UNINTELLIGIBLE PHRASE]. 1216 01:06:41,319 --> 01:06:45,558 1217 01:06:45,558 --> 01:06:46,808 PROFESSOR: Correct. 1218 01:06:46,808 --> 01:06:49,051 1219 01:06:49,051 --> 01:06:51,430 If I thought about this current going this way, it 1220 01:06:51,430 --> 01:06:53,890 would be minus 10. 1221 01:06:53,890 --> 01:06:59,050 If I flipped the direction, then it's plus 10. 1222 01:06:59,050 --> 01:07:03,820 So the loop current has the property that it's the only 1223 01:07:03,820 --> 01:07:05,290 current through this element. 1224 01:07:05,290 --> 01:07:07,220 So that has to match. 1225 01:07:07,220 --> 01:07:11,234 It's one of two currents that go through this element. 1226 01:07:11,234 --> 01:07:13,186 AUDIENCE: You said that everything that 1227 01:07:13,186 --> 01:07:15,626 [UNINTELLIGIBLE PHRASE]. 1228 01:07:15,626 --> 01:07:17,578 PROFESSOR: This loop is [UNINTELLIGIBLE]. 1229 01:07:17,578 --> 01:07:18,066 Yes. 1230 01:07:18,066 --> 01:07:20,018 AUDIENCE: So why is the [UNINTELLIGIBLE PHRASE]. 1231 01:07:20,018 --> 01:07:23,434 1232 01:07:23,434 --> 01:07:24,410 PROFESSOR: Correct. 1233 01:07:24,410 --> 01:07:26,880 I want to have this picture now. 1234 01:07:26,880 --> 01:07:29,570 So if I'm doing it with loops, I have two loops. 1235 01:07:29,570 --> 01:07:32,390 1236 01:07:32,390 --> 01:07:35,100 The current through this element is just I. The current 1237 01:07:35,100 --> 01:07:37,720 through this element is just I. The current through this 1238 01:07:37,720 --> 01:07:40,020 element is just 10. 1239 01:07:40,020 --> 01:07:42,040 The current through this element-- well, the sum of 1240 01:07:42,040 --> 01:07:43,560 these two currents go through that element. 1241 01:07:43,560 --> 01:07:46,730 1242 01:07:46,730 --> 01:07:49,512 Does that make sense? 1243 01:07:49,512 --> 01:07:50,762 AUDIENCE: [INAUDIBLE] 1244 01:07:50,762 --> 01:07:55,428 1245 01:07:55,428 --> 01:07:57,893 PROFESSOR: This loop current is just a fraction of the 1246 01:07:57,893 --> 01:08:01,344 current in the whole system. 1247 01:08:01,344 --> 01:08:05,550 So this loop current goes through this element and 1248 01:08:05,550 --> 01:08:07,150 contributes to this element. 1249 01:08:07,150 --> 01:08:08,400 But so does that one. 1250 01:08:08,400 --> 01:08:12,074 1251 01:08:12,074 --> 01:08:15,150 OK, if you're still confused, you should try to get it 1252 01:08:15,150 --> 01:08:17,170 straightened out in one of the software labs or the hardware 1253 01:08:17,170 --> 01:08:18,670 lab, or talk to me after lecture. 1254 01:08:18,670 --> 01:08:20,760 But the idea is to decompose in the 1255 01:08:20,760 --> 01:08:22,720 case of the node voltages. 1256 01:08:22,720 --> 01:08:25,380 Think about the element voltages in terms of 1257 01:08:25,380 --> 01:08:27,000 differences in the node voltages. 1258 01:08:27,000 --> 01:08:30,580 1259 01:08:30,580 --> 01:08:32,439 In the case of the loop currents, think about the 1260 01:08:32,439 --> 01:08:36,100 element currents in terms of a sum of loop currents. 1261 01:08:36,100 --> 01:08:40,029 1262 01:08:40,029 --> 01:08:43,600 OK, so the answer is minus 1 regardless of how you do it. 1263 01:08:43,600 --> 01:08:46,345 1264 01:08:46,345 --> 01:08:46,472 Ok. 1265 01:08:46,472 --> 01:08:49,439 The remaining thing I want to do today is think about 1266 01:08:49,439 --> 01:08:52,140 abstraction. 1267 01:08:52,140 --> 01:08:55,180 We've talked about the primitives, which are things 1268 01:08:55,180 --> 01:08:59,750 like resistors, voltage sources, and current sources. 1269 01:08:59,750 --> 01:09:03,100 Means of combinations, that's KVL and KCL. 1270 01:09:03,100 --> 01:09:05,090 Now we want to think about abstraction. 1271 01:09:05,090 --> 01:09:07,700 And the first abstraction that we'll talk about is, how do 1272 01:09:07,700 --> 01:09:11,510 you think about one element that represents 1273 01:09:11,510 --> 01:09:13,680 more than one element? 1274 01:09:13,680 --> 01:09:15,740 This is the same thing that we did when we thought about 1275 01:09:15,740 --> 01:09:18,770 linear systems, when we did signals and systems. 1276 01:09:18,770 --> 01:09:23,270 We started with R's and K's and pluses, and we made single 1277 01:09:23,270 --> 01:09:29,020 boxes that had lots of R's and pluses and gains in them. 1278 01:09:29,020 --> 01:09:32,590 What was the name of the thing that was inside the box? 1279 01:09:32,590 --> 01:09:36,270 If we combined lots of R's, gains, and pluses into a 1280 01:09:36,270 --> 01:09:39,860 single box, what would we call the thing that's in the box? 1281 01:09:39,860 --> 01:09:41,019 AUDIENCE: [INAUDIBLE]. 1282 01:09:41,019 --> 01:09:41,789 PROFESSOR: Shout again. 1283 01:09:41,789 --> 01:09:42,260 AUDIENCE: System function. 1284 01:09:42,260 --> 01:09:44,930 PROFESSOR: System function. 1285 01:09:44,930 --> 01:09:45,019 Right? 1286 01:09:45,019 --> 01:09:47,609 So we started with boxes that only had 1287 01:09:47,609 --> 01:09:49,580 things like R's in them. 1288 01:09:49,580 --> 01:09:53,920 But eventually, we got boxes that looked like much more 1289 01:09:53,920 --> 01:10:00,000 complicated things like that. 1290 01:10:00,000 --> 01:10:02,170 We thought about a system function which was a 1291 01:10:02,170 --> 01:10:06,580 generalized box, that could have lots of R's, or lots of 1292 01:10:06,580 --> 01:10:08,890 gains, or lots of pluses in it. 1293 01:10:08,890 --> 01:10:14,810 And that was a way of abstracting complicated 1294 01:10:14,810 --> 01:10:17,140 systems so they looked like simple systems. 1295 01:10:17,140 --> 01:10:18,900 What we want to do here is the same thing for circuits. 1296 01:10:18,900 --> 01:10:21,530 We want to have a single element, a single circuit 1297 01:10:21,530 --> 01:10:25,620 element, that represents many circuit elements. 1298 01:10:25,620 --> 01:10:29,240 And the simplest case of that is for series in parallel 1299 01:10:29,240 --> 01:10:31,050 combinations of resistors. 1300 01:10:31,050 --> 01:10:34,890 It's very simple to think about how if you had two Ohm's 1301 01:10:34,890 --> 01:10:39,220 law devices connected in series, you could replace 1302 01:10:39,220 --> 01:10:42,490 those two with a single resistor. 1303 01:10:42,490 --> 01:10:45,520 And the voltage-current relationships measured at the 1304 01:10:45,520 --> 01:10:48,460 outside of the box would be the same. 1305 01:10:48,460 --> 01:10:51,140 1306 01:10:51,140 --> 01:10:53,930 That's how we think about an abstraction in circuits. 1307 01:10:53,930 --> 01:10:59,730 When is it that you can draw a box around a piece of a 1308 01:10:59,730 --> 01:11:03,590 circuit and think about that as one element? 1309 01:11:03,590 --> 01:11:06,350 The very simplest cases, the series combination of two 1310 01:11:06,350 --> 01:11:09,940 resistors, same sort of thing happens for the parallel 1311 01:11:09,940 --> 01:11:11,830 combination. 1312 01:11:11,830 --> 01:11:15,870 And that simple abstraction makes some things very easy. 1313 01:11:15,870 --> 01:11:19,370 What would be the equivalent resistance for a complicated 1314 01:11:19,370 --> 01:11:21,350 system like that? 1315 01:11:21,350 --> 01:11:24,070 Well, that's easy. 1316 01:11:24,070 --> 01:11:29,440 All you need to do is think about successively reducing 1317 01:11:29,440 --> 01:11:32,170 the pieces. 1318 01:11:32,170 --> 01:11:35,050 Here I'm thinking about that having four resistors. 1319 01:11:35,050 --> 01:11:39,020 I can just successively apply series and parallel in order 1320 01:11:39,020 --> 01:11:41,760 to reduce that, make it less complicated. 1321 01:11:41,760 --> 01:11:47,660 So I can think about combining these two in series to get, 1322 01:11:47,660 --> 01:11:50,025 instead of two 1 Ohm resistors, one 2 Ohm resistor. 1323 01:11:50,025 --> 01:11:52,580 1324 01:11:52,580 --> 01:11:56,640 Then I can think about these two 2 Ohm resistors being 1325 01:11:56,640 --> 01:11:59,170 equivalently one parallel 1 Ohm resistor. 1326 01:11:59,170 --> 01:12:02,810 1327 01:12:02,810 --> 01:12:06,170 And so this whole thing looks as though it's just 2 Ohms 1328 01:12:06,170 --> 01:12:09,560 from the outside world. 1329 01:12:09,560 --> 01:12:11,220 That's what we mean by an abstraction. 1330 01:12:11,220 --> 01:12:15,430 What we're trying to do and what we will do over the next 1331 01:12:15,430 --> 01:12:18,380 two weeks, is we'll think about ways of combining 1332 01:12:18,380 --> 01:12:22,630 circuits so that we can reduce the complexity this way. 1333 01:12:22,630 --> 01:12:25,250 1334 01:12:25,250 --> 01:12:28,070 Another convenient way of thinking about reducing the 1335 01:12:28,070 --> 01:12:31,120 work that you need to do is to think about common patterns 1336 01:12:31,120 --> 01:12:33,130 that result. 1337 01:12:33,130 --> 01:12:37,740 PCAP, Primitives, Combinations, Abstractions. 1338 01:12:37,740 --> 01:12:40,120 So the series of parallel idea was an abstraction. 1339 01:12:40,120 --> 01:12:42,330 A pattern, here's a common pattern. 1340 01:12:42,330 --> 01:12:48,930 If you've got two resistors in series, if the same current 1341 01:12:48,930 --> 01:12:52,250 flows through two resistors, then there's a way of very 1342 01:12:52,250 --> 01:12:56,170 simply calculating the voltage that falls across each. 1343 01:12:56,170 --> 01:13:01,090 So you can think about the sum resistor, R1 plus R2 since 1344 01:13:01,090 --> 01:13:03,290 they're in series. 1345 01:13:03,290 --> 01:13:05,190 So that allows you then to compute the 1346 01:13:05,190 --> 01:13:08,390 current from the voltage. 1347 01:13:08,390 --> 01:13:11,240 Then the voltage that falls across this guy is by Ohm's 1348 01:13:11,240 --> 01:13:16,000 law, just the current times its resistor, 1349 01:13:16,000 --> 01:13:18,150 which is like that. 1350 01:13:18,150 --> 01:13:19,970 And similarly with this one. 1351 01:13:19,970 --> 01:13:24,560 So you can see that some fraction of this voltage v 1352 01:13:24,560 --> 01:13:26,930 occurs across the v1 terminal. 1353 01:13:26,930 --> 01:13:30,100 And some different fraction appears across the v2 1354 01:13:30,100 --> 01:13:34,770 terminal, such that the sum of the fractions is, of course, 1355 01:13:34,770 --> 01:13:38,280 v. That's what has to happen for the two. 1356 01:13:38,280 --> 01:13:40,260 And there's a proportional drop. 1357 01:13:40,260 --> 01:13:43,350 The bigger R1, the bigger is the proportion of the voltage 1358 01:13:43,350 --> 01:13:45,860 that falls across R1. 1359 01:13:45,860 --> 01:13:49,620 So it's a simple way of thinking about how voltage 1360 01:13:49,620 --> 01:13:51,120 drops across two resistors. 1361 01:13:51,120 --> 01:13:53,610 There's a completely analogous way of thinking about how 1362 01:13:53,610 --> 01:13:58,690 current splits between two resistors. 1363 01:13:58,690 --> 01:14:02,440 Here the result looks virtually the same, except it 1364 01:14:02,440 --> 01:14:06,710 has kind of the unintuitive property that most of the 1365 01:14:06,710 --> 01:14:11,710 current goes through the resistor that is the smallest. 1366 01:14:11,710 --> 01:14:19,910 So you get a bigger current in i1 in proportion to the R2. 1367 01:14:19,910 --> 01:14:23,040 So it works very much like the voltage case, except that it 1368 01:14:23,040 --> 01:14:26,060 has this inversion in it, that the current likes to go 1369 01:14:26,060 --> 01:14:29,870 through the smaller resistor. 1370 01:14:29,870 --> 01:14:33,670 OK, so last problem. 1371 01:14:33,670 --> 01:14:38,880 Using those kinds of ideas, think about how you could 1372 01:14:38,880 --> 01:14:42,855 compute the voltage v0 and determine what's the answer. 1373 01:14:42,855 --> 01:16:30,480 1374 01:16:30,480 --> 01:16:33,910 So what's the easy way to think about this answer? 1375 01:16:33,910 --> 01:16:35,160 What do I do first? 1376 01:16:35,160 --> 01:16:38,912 1377 01:16:38,912 --> 01:16:40,409 AUDIENCE: Superposition. 1378 01:16:40,409 --> 01:16:41,407 PROFESSOR: So superposition? 1379 01:16:41,407 --> 01:16:43,902 That's one thing. 1380 01:16:43,902 --> 01:16:46,397 AUDIENCE: Simplify [UNINTELLIGIBLE]. 1381 01:16:46,397 --> 01:16:47,395 PROFESSOR: Simplify. 1382 01:16:47,395 --> 01:16:51,886 So what's a good simplification? 1383 01:16:51,886 --> 01:16:52,884 Collapse? 1384 01:16:52,884 --> 01:16:56,127 AUDIENCE: You can put the two [UNINTELLIGIBLE] in series and 1385 01:16:56,127 --> 01:16:57,375 treat them as one. 1386 01:16:57,375 --> 01:17:01,780 PROFESSOR: So you can treat this as a series combination, 1387 01:17:01,780 --> 01:17:06,025 and you can replace the series of 1 and 3 with a? 1388 01:17:06,025 --> 01:17:06,995 AUDIENCE: [UNINTELLIGIBLE PHRASE]. 1389 01:17:06,995 --> 01:17:07,480 AUDIENCE: 4.. 1390 01:17:07,480 --> 01:17:08,935 PROFESSOR: 4. 1391 01:17:08,935 --> 01:17:12,330 So this can be replaced by four [UNINTELLIGIBLE] 1392 01:17:12,330 --> 01:17:12,815 resistor. 1393 01:17:12,815 --> 01:17:13,300 Now what? 1394 01:17:13,300 --> 01:17:15,725 AUDIENCE: You can do the same on the parallel one. 1395 01:17:15,725 --> 01:17:18,343 PROFESSOR: So you can replace the parallel of the 6 1396 01:17:18,343 --> 01:17:19,506 and a 12 with a? 1397 01:17:19,506 --> 01:17:20,342 AUDIENCE: 4. 1398 01:17:20,342 --> 01:17:22,653 PROFESSOR: Amazing -- with a 4. 1399 01:17:22,653 --> 01:17:25,599 So there's a 4 there and there's a 4 there. 1400 01:17:25,599 --> 01:17:26,090 And the answer is? 1401 01:17:26,090 --> 01:17:27,072 AUDIENCE: It's half. 1402 01:17:27,072 --> 01:17:28,545 PROFESSOR: Half of whatever it was by voltage 1403 01:17:28,545 --> 01:17:30,020 [UNINTELLIGIBLE] relationship. 1404 01:17:30,020 --> 01:17:32,760 So you think about this becoming that. 1405 01:17:32,760 --> 01:17:35,120 You think about the parallel becoming that. 1406 01:17:35,120 --> 01:17:38,200 You get a simple divide by 2 voltage divider. 1407 01:17:38,200 --> 01:17:46,660 So the answer is 7 and 1/2, which was the middle answer. 1408 01:17:46,660 --> 01:17:51,780 And so what we did today was basically a whirlwind tour of 1409 01:17:51,780 --> 01:17:54,790 the theory of circuits. 1410 01:17:54,790 --> 01:17:58,060 And the goal for the rest of the week is to go to the lab 1411 01:17:58,060 --> 01:18:00,580 and do the same sort of thing with practical where you build 1412 01:18:00,580 --> 01:18:03,440 a circuit, and try to use some of these ideas to understand 1413 01:18:03,440 --> 01:18:04,690 what it does. 1414 01:18:04,690 --> 01:18:13,295