WEBVTT 00:00:00.040 --> 00:00:02.460 The following content is provided under a Creative 00:00:02.460 --> 00:00:03.870 Commons license. 00:00:03.870 --> 00:00:06.910 Your support will help MIT OpenCourseWare continue to 00:00:06.910 --> 00:00:10.560 offer high-quality educational resources for free. 00:00:10.560 --> 00:00:13.460 To make a donation or view additional materials from 00:00:13.460 --> 00:00:18.090 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:18.090 --> 00:00:19.340 ocw.mit.edu. 00:00:24.190 --> 00:00:26.785 PROFESSOR: So today I want to start a new topic, circuits. 00:00:29.430 --> 00:00:30.130 No, that's good. 00:00:30.130 --> 00:00:31.390 That's good. 00:00:31.390 --> 00:00:33.662 Circuits are good. 00:00:33.662 --> 00:00:34.590 AUDIENCE: Yay. 00:00:34.590 --> 00:00:35.280 PROFESSOR: Thank you, thank you. 00:00:35.280 --> 00:00:37.880 Much better. 00:00:37.880 --> 00:00:42.290 So just to provide some perspective, I want to remind 00:00:42.290 --> 00:00:45.500 you where we are, how we got here, and where we're going. 00:00:45.500 --> 00:00:48.170 So at the beginning of the course, we promised that there 00:00:48.170 --> 00:00:49.830 were several intellectual themes that 00:00:49.830 --> 00:00:52.400 we would talk about. 00:00:52.400 --> 00:00:54.850 Probably the most important one there is 00:00:54.850 --> 00:00:57.450 designing complex systems. 00:00:57.450 --> 00:00:59.000 That's what we're really about. 00:00:59.000 --> 00:01:00.920 We would like you to be able to make 00:01:00.920 --> 00:01:02.670 very complicated systems. 00:01:02.670 --> 00:01:04.220 How do you think about parts? 00:01:04.220 --> 00:01:06.040 How do you think about connecting them? 00:01:06.040 --> 00:01:07.730 How do you think about things when you want to make 00:01:07.730 --> 00:01:11.170 something that's very complicated? 00:01:11.170 --> 00:01:13.740 Part of that is modeling. 00:01:13.740 --> 00:01:16.510 We just finished a module on modeling. 00:01:16.510 --> 00:01:19.550 So in order to make a complex system, we'd like to be able 00:01:19.550 --> 00:01:23.850 to predict how it will behave before we completely build it. 00:01:23.850 --> 00:01:26.220 Sometimes it's impossible to build the 00:01:26.220 --> 00:01:27.670 entire system before. 00:01:30.210 --> 00:01:34.030 Sometimes it's impossible to build prototypes. 00:01:34.030 --> 00:01:38.190 Sometimes you're stuck with going with the design at 00:01:38.190 --> 00:01:41.270 launch, and figuring out how it works. 00:01:41.270 --> 00:01:44.340 In those cases in specific, it's very important to be able 00:01:44.340 --> 00:01:47.850 to model it, to have some confidence that the thing's 00:01:47.850 --> 00:01:49.500 going to work. 00:01:49.500 --> 00:01:52.090 We're going to talk about augmenting physical systems 00:01:52.090 --> 00:01:52.990 with computation. 00:01:52.990 --> 00:01:55.320 That's the module we're about to begin. 00:01:55.320 --> 00:01:58.060 And we'll conclude by talking about how to build systems 00:01:58.060 --> 00:02:00.900 that are robust to change. 00:02:00.900 --> 00:02:05.240 So we started with the idea of, how do you make 00:02:05.240 --> 00:02:06.320 complicated systems? 00:02:06.320 --> 00:02:08.840 And we introduced this notion of primitives, combination, 00:02:08.840 --> 00:02:12.350 abstraction, and pattern in terms of software engineering. 00:02:12.350 --> 00:02:14.600 We did that because that's the simplest possible way of 00:02:14.600 --> 00:02:16.350 getting started. 00:02:16.350 --> 00:02:20.810 It provided a very good illustration of PCAP at the 00:02:20.810 --> 00:02:25.650 low level by thinking about Python, by thinking about the 00:02:25.650 --> 00:02:29.330 primitive structures that Python gives you, how those 00:02:29.330 --> 00:02:31.830 can be combined, how you can abstract, how you can 00:02:31.830 --> 00:02:33.450 recognize patterns. 00:02:33.450 --> 00:02:36.120 But then we also built a higher level abstraction, 00:02:36.120 --> 00:02:39.130 which was the state machine idea. 00:02:39.130 --> 00:02:42.110 There the idea was you didn't have to have state machines in 00:02:42.110 --> 00:02:45.140 order to build the brain for a robot, but it actually turns 00:02:45.140 --> 00:02:47.430 out to be easy if you do because there's 00:02:47.430 --> 00:02:49.530 a modularity there. 00:02:49.530 --> 00:02:52.550 You can figure out if each part works independent of the 00:02:52.550 --> 00:02:53.530 other parts. 00:02:53.530 --> 00:02:55.510 And then you can be pretty sure when you put it together 00:02:55.510 --> 00:02:57.750 the whole thing's going to work. 00:02:57.750 --> 00:02:59.810 So that was kind of our introduction to 00:02:59.810 --> 00:03:01.570 this notion of PCAP. 00:03:01.570 --> 00:03:03.820 Then we went on to think about signals and systems. 00:03:03.820 --> 00:03:06.730 And that was kind of our introduction to modeling. 00:03:06.730 --> 00:03:08.070 How do you make a model of something 00:03:08.070 --> 00:03:09.690 that predicts behavior? 00:03:09.690 --> 00:03:12.890 So we transitioned from thinking about, how do you 00:03:12.890 --> 00:03:16.940 structure a design to how do you think about behavior. 00:03:16.940 --> 00:03:19.350 Today we're going to start to think about circuits. 00:03:19.350 --> 00:03:24.240 Circuits are really going to a more primitive physical layer. 00:03:24.240 --> 00:03:28.210 How do you think about actually making a device? 00:03:28.210 --> 00:03:32.460 So the device that we'll think about is a thing to augment 00:03:32.460 --> 00:03:33.350 the capabilities -- 00:03:33.350 --> 00:03:36.340 the sensor capabilities of the robot. 00:03:36.340 --> 00:03:38.490 We'll think about making a light tracking system. 00:03:38.490 --> 00:03:42.420 And the idea is going to be that you'll build a head. 00:03:42.420 --> 00:03:46.480 The head has a neck, so you'll have to control the neck. 00:03:46.480 --> 00:03:48.900 The head has eyes. 00:03:48.900 --> 00:03:52.250 It will mount on top of the robot, and that'll let you 00:03:52.250 --> 00:03:55.960 drive the robot around looking for light. 00:03:55.960 --> 00:03:59.120 And you'll do that by designing a circuit. 00:03:59.120 --> 00:04:03.200 So that's kind of the game plan for the next three weeks. 00:04:03.200 --> 00:04:06.020 So today, what I want to do is introduce the notion of 00:04:06.020 --> 00:04:09.570 circuits, introduce the theory for how 00:04:09.570 --> 00:04:11.400 we think about circuits. 00:04:11.400 --> 00:04:15.100 And then in the two labs this week, the idea will be to 00:04:15.100 --> 00:04:18.410 become familiar with the practice of circuits. 00:04:18.410 --> 00:04:19.750 How do you actually build something? 00:04:19.750 --> 00:04:22.780 How do you actually make something work? 00:04:22.780 --> 00:04:23.990 So today's the theory. 00:04:23.990 --> 00:04:27.390 So the theory, the idea in circuits is to think about a 00:04:27.390 --> 00:04:34.070 physical system as the interconnection of parts and 00:04:34.070 --> 00:04:36.650 rules that connect them. 00:04:36.650 --> 00:04:39.180 In fact, the rules fall into two categories. 00:04:39.180 --> 00:04:44.490 We're going to think about the currents that go through parts 00:04:44.490 --> 00:04:47.410 and the voltage that develops across parts. 00:04:47.410 --> 00:04:50.140 And we'll see that there's a way of thinking about the 00:04:50.140 --> 00:04:53.310 behavior of the entire circuit that integrates all of those 00:04:53.310 --> 00:04:54.160 three pieces. 00:04:54.160 --> 00:04:55.670 How does the part work? 00:04:55.670 --> 00:04:59.640 How do the currents that go through the part work? 00:04:59.640 --> 00:05:02.750 And how do the voltages that are produced 00:05:02.750 --> 00:05:03.710 across the part -- 00:05:03.710 --> 00:05:06.820 how do they work? 00:05:06.820 --> 00:05:08.880 So I'll just start with two very simple examples. 00:05:08.880 --> 00:05:11.735 The first is the most trivial example you can think about. 00:05:11.735 --> 00:05:14.750 You can think about a flashlight as a circuit. 00:05:14.750 --> 00:05:17.700 You close the switch, current flows. 00:05:17.700 --> 00:05:19.450 Very simple idea. 00:05:19.450 --> 00:05:23.100 We will make a model of that that looks like this. 00:05:23.100 --> 00:05:25.600 We'll think about the battery being a source. 00:05:25.600 --> 00:05:27.640 In this case, a voltage source. 00:05:27.640 --> 00:05:31.020 We'll think about the light bulb being a resistor. 00:05:31.020 --> 00:05:32.020 We'll have two parts. 00:05:32.020 --> 00:05:34.480 We'll have to know the current-voltage relationships 00:05:34.480 --> 00:05:35.880 for both of the parts. 00:05:35.880 --> 00:05:37.670 And we'll have to know the ramifications for those 00:05:37.670 --> 00:05:40.400 currents and voltages when you put them together. 00:05:40.400 --> 00:05:42.620 Very, very simple. 00:05:42.620 --> 00:05:46.330 The other simple example that I want to illustrate is this 00:05:46.330 --> 00:05:48.790 idea of a leaky tank. 00:05:48.790 --> 00:05:52.600 Here the idea that I want to get across is that the circuit 00:05:52.600 --> 00:05:56.810 idea is quite general. 00:05:56.810 --> 00:05:58.770 When we talk about circuits, we almost always talk about 00:05:58.770 --> 00:06:01.220 electronic circuits. 00:06:01.220 --> 00:06:05.400 But the theory is by no means limited to electronics. 00:06:05.400 --> 00:06:07.870 So for example, if we think about a leaky tank, we think 00:06:07.870 --> 00:06:10.840 about a pipe spewing water into a reservoir. 00:06:10.840 --> 00:06:12.730 Maybe that's the Cambridge Reservoir. 00:06:12.730 --> 00:06:15.680 Maybe that's the water coming out of the Woburn Reservoir. 00:06:15.680 --> 00:06:18.120 Maybe that's the demand put on by the Cambridge people trying 00:06:18.120 --> 00:06:20.310 to take showers in the morning -- 00:06:20.310 --> 00:06:21.830 I don't know. 00:06:21.830 --> 00:06:24.160 So we think about flow into a tank, a 00:06:24.160 --> 00:06:25.880 reservoir, and flow out. 00:06:25.880 --> 00:06:29.790 And we can make a model for that in terms of a circuit. 00:06:29.790 --> 00:06:31.220 In the circuit there are through 00:06:31.220 --> 00:06:34.370 variables and across variables. 00:06:34.370 --> 00:06:37.510 In an electronic circuit, the through variable is current. 00:06:37.510 --> 00:06:40.000 Here the through variable is the flow rate. 00:06:40.000 --> 00:06:42.420 So it's the flow of water in and the flow of water out, 00:06:42.420 --> 00:06:43.990 represented here by these things 00:06:43.990 --> 00:06:46.840 that look like currents. 00:06:46.840 --> 00:06:51.000 And the across variable for an electronic device is voltage. 00:06:51.000 --> 00:06:53.870 The across variable for this kind of a fluidic device, the 00:06:53.870 --> 00:06:56.180 across variable is pressure. 00:06:56.180 --> 00:06:59.520 So we think as this thing gets ahead of that thing, as 00:06:59.520 --> 00:07:02.720 there's more stuff coming in than going out, the height 00:07:02.720 --> 00:07:05.870 goes up and the pressure builds. 00:07:05.870 --> 00:07:06.690 Same idea. 00:07:06.690 --> 00:07:09.070 So the point is that we'll develop the theory for 00:07:09.070 --> 00:07:11.260 circuits in terms of electronics. 00:07:11.260 --> 00:07:13.640 But you should keep in the back of your mind that it's a 00:07:13.640 --> 00:07:16.980 lot more general idea than just electronics. 00:07:16.980 --> 00:07:20.070 In fact, there are two completely distinct reasons 00:07:20.070 --> 00:07:22.660 why we even bother with circuits. 00:07:22.660 --> 00:07:27.120 One is that they're very important to physical systems. 00:07:27.120 --> 00:07:29.430 If you're designing a power network, of course you have to 00:07:29.430 --> 00:07:32.170 think about the way the power network, the power grid works 00:07:32.170 --> 00:07:34.130 as a circuit. 00:07:34.130 --> 00:07:35.430 That's obvious. 00:07:35.430 --> 00:07:37.950 In electronics, of course, if you're going to build a cell 00:07:37.950 --> 00:07:40.090 phone, you have to know how the parts interconnect 00:07:40.090 --> 00:07:40.710 electronically. 00:07:40.710 --> 00:07:42.750 That's obvious. 00:07:42.750 --> 00:07:46.540 But probably the biggest use for circuits these days is not 00:07:46.540 --> 00:07:49.800 those applications, although those are very important. 00:07:49.800 --> 00:07:53.720 Circuits are also used as models of things. 00:07:53.720 --> 00:07:58.360 So many models for complex behaviors are in 00:07:58.360 --> 00:08:00.820 fact circuit models. 00:08:00.820 --> 00:08:04.760 So in terms of electronics, the idea is that we want to 00:08:04.760 --> 00:08:06.100 get on top of electronics. 00:08:06.100 --> 00:08:08.240 We want to understand how circuits work, so we can 00:08:08.240 --> 00:08:11.170 understand things like that. 00:08:11.170 --> 00:08:16.110 If you look at how complex processors have got over my 00:08:16.110 --> 00:08:19.970 professional life, we start with my professional life down 00:08:19.970 --> 00:08:25.090 at about 1,000 transistors per processor. 00:08:25.090 --> 00:08:28.990 And today, we're up at about a billion. 00:08:28.990 --> 00:08:31.420 That's enormous. 00:08:31.420 --> 00:08:36.380 Even in the stone ages when we were designing things that had 00:08:36.380 --> 00:08:41.549 a thousand parts, we still had trouble thinking about those 00:08:41.549 --> 00:08:43.070 thousands parts all at once. 00:08:43.070 --> 00:08:44.790 We still need PCAP. 00:08:44.790 --> 00:08:48.040 We still needed ways of combining the activities of 00:08:48.040 --> 00:08:51.340 many things into a conceptual unit that was bigger. 00:08:51.340 --> 00:08:57.030 Here, it's just impossible if you don't have that. 00:08:57.030 --> 00:08:59.760 So that's one of the reasons we study circuits. 00:08:59.760 --> 00:09:01.390 And the other reason is here. 00:09:01.390 --> 00:09:04.680 So here I'm showing a model for the way 00:09:04.680 --> 00:09:05.800 a nerve cell works. 00:09:05.800 --> 00:09:09.860 This model is taken from 6.021. 00:09:09.860 --> 00:09:15.280 The idea, this comes from the study of the Hodgkin-Huxley 00:09:15.280 --> 00:09:19.810 model for neural conduction, arguably the most successful 00:09:19.810 --> 00:09:23.020 mathematical theory in biophysics. 00:09:23.020 --> 00:09:27.510 Which explains the completely non-trivial relationship 00:09:27.510 --> 00:09:33.260 between how the parts from biology works and the behavior 00:09:33.260 --> 00:09:37.460 in terms of propagated action potentials works. 00:09:37.460 --> 00:09:41.500 So the idea is that we understand how this biological 00:09:41.500 --> 00:09:43.740 system works because we think about it 00:09:43.740 --> 00:09:45.470 in terms of a circuit. 00:09:45.470 --> 00:09:51.430 That's the only successful way we have to think about that. 00:09:51.430 --> 00:09:55.430 So what I want to do then is spend today figuring out 00:09:55.430 --> 00:09:59.650 circuits At the very most primitive level. 00:09:59.650 --> 00:10:01.330 The level that I'm going to talk about in terms of 00:10:01.330 --> 00:10:04.530 circuits is roughly analogous to the level that we talked 00:10:04.530 --> 00:10:09.020 about with Python when we were thinking about how Python 00:10:09.020 --> 00:10:14.590 provides utilities for primitives, combinations, 00:10:14.590 --> 00:10:17.260 abstractions, and patterns. 00:10:17.260 --> 00:10:19.220 So I'm going to start at the very lowest level and think 00:10:19.220 --> 00:10:23.290 about, what are the basic primitives, the smallest units 00:10:23.290 --> 00:10:25.740 we'll ever think about in terms of circuits? 00:10:25.740 --> 00:10:29.450 And what are the rules by which we combine them? 00:10:29.450 --> 00:10:33.090 So I'll start with the very simplest ideas, the very 00:10:33.090 --> 00:10:35.020 simplest elements. 00:10:35.020 --> 00:10:37.720 We will oversimplify things and think about the very 00:10:37.720 --> 00:10:41.010 simplest kind of electronic elements as resistors that 00:10:41.010 --> 00:10:42.640 obey Ohm's Law, V equals iR. 00:10:45.420 --> 00:10:48.610 Voltage sources, things that maintain a constant voltage 00:10:48.610 --> 00:10:50.740 regardless of what you do. 00:10:50.740 --> 00:10:53.290 And current sources, things that maintain a constant 00:10:53.290 --> 00:10:56.810 current regardless of what you do. 00:10:56.810 --> 00:10:59.640 These things are, as I said, analogous to the primitive 00:10:59.640 --> 00:11:00.890 things that we looked at in Python. 00:11:00.890 --> 00:11:02.750 They're also analogous to the primitive things that we 00:11:02.750 --> 00:11:05.470 looked at in system functions. 00:11:05.470 --> 00:11:07.900 Can somebody think of, when we were doing difference 00:11:07.900 --> 00:11:10.880 equations, what were the primitives that we 00:11:10.880 --> 00:11:12.680 started with -- 00:11:12.680 --> 00:11:15.770 when we started to study difference equations? 00:11:15.770 --> 00:11:17.260 What's the most primitive elements 00:11:17.260 --> 00:11:18.510 that we thought about? 00:11:20.937 --> 00:11:22.320 AUDIENCE: Delay. 00:11:22.320 --> 00:11:23.340 PROFESSOR: Delay, yeah. 00:11:23.340 --> 00:11:24.740 So we thought about things like-- 00:11:27.390 --> 00:11:28.640 so we had delay. 00:11:31.810 --> 00:11:32.600 Anything else? 00:11:32.600 --> 00:11:33.330 AUDIENCE: Gain. 00:11:33.330 --> 00:11:34.300 PROFESSOR: Gain. 00:11:34.300 --> 00:11:35.550 Anything else? 00:11:38.463 --> 00:11:40.960 Add. 00:11:40.960 --> 00:11:44.130 So we had exactly three primitives. 00:11:44.130 --> 00:11:47.850 And we got pretty far with those three primitives. 00:11:47.850 --> 00:11:51.000 We learned the rules for interconnection. 00:11:51.000 --> 00:11:52.340 We didn't really make a big deal out of it. 00:11:52.340 --> 00:11:54.320 We didn't formalize it, but the rules for interconnection 00:11:54.320 --> 00:11:58.430 were something like every node has to 00:11:58.430 --> 00:12:01.810 have exactly one generator. 00:12:01.810 --> 00:12:03.830 You can't connect the output of this to 00:12:03.830 --> 00:12:06.890 this, that's illegal. 00:12:06.890 --> 00:12:09.220 Every node has to have one source. 00:12:09.220 --> 00:12:12.010 And every node can source lots of inputs. 00:12:12.010 --> 00:12:14.050 That was kind of the rules of the interconnect. 00:12:14.050 --> 00:12:15.505 The interconnects here will be a little bit more complicated. 00:12:18.340 --> 00:12:20.960 So those are the elements that we'll think about. 00:12:20.960 --> 00:12:23.370 And the first step's going to be to think about, how do they 00:12:23.370 --> 00:12:25.800 interconnect? 00:12:25.800 --> 00:12:29.160 The simplest possible interconnections are trivial. 00:12:29.160 --> 00:12:31.580 In the case of the battery, you hook up the voltage source 00:12:31.580 --> 00:12:33.280 to the resistor. 00:12:33.280 --> 00:12:35.100 The voltage source makes the voltage across 00:12:35.100 --> 00:12:37.220 this resistor 1 Volt. 00:12:37.220 --> 00:12:39.910 If we say the resistor is 1 Ohm, then there's 1 Amp 00:12:39.910 --> 00:12:40.745 current period. 00:12:40.745 --> 00:12:41.510 Done. 00:12:41.510 --> 00:12:43.750 Easy. 00:12:43.750 --> 00:12:47.160 Similarly, if we were to hook up the resistor to a current 00:12:47.160 --> 00:12:49.120 source, we would get something equally easy. 00:12:49.120 --> 00:12:51.060 Except now the current source would guarantee that the 00:12:51.060 --> 00:12:56.450 current through the resistor is an amp. 00:12:56.450 --> 00:12:59.360 Therefore, the voltage across the resistor, by Ohm's law, 00:12:59.360 --> 00:13:00.700 would be a Volt. 00:13:00.700 --> 00:13:02.830 So we would end up with the same solution for a completely 00:13:02.830 --> 00:13:04.940 different reason. 00:13:04.940 --> 00:13:06.340 Here the voltage is constrained. 00:13:06.340 --> 00:13:07.590 Here the current's constrained. 00:13:10.390 --> 00:13:16.090 Just to make sure everybody's with me, figure out, what's 00:13:16.090 --> 00:13:19.450 the current i that goes through this resistor? 00:13:19.450 --> 00:13:22.040 Slightly more complicated system. 00:13:22.040 --> 00:13:24.120 Take 20 seconds, talk to your neighbor, figure out a number 00:13:24.120 --> 00:13:25.820 between (1) and (5). 00:15:23.520 --> 00:15:24.690 OK, so what's the answer? 00:15:24.690 --> 00:15:27.630 Everybody raise your hand with a number (1) through (5). 00:15:27.630 --> 00:15:28.980 Come on, everybody vote. 00:15:28.980 --> 00:15:29.920 Come on. 00:15:29.920 --> 00:15:32.020 You can blame it on your neighbor, that's the rules. 00:15:32.020 --> 00:15:33.790 You talk to your neighbor, then you can blame dumb 00:15:33.790 --> 00:15:36.150 answers on your neighbor. 00:15:36.150 --> 00:15:41.100 OK, about 80% correct I'd say. 00:15:41.100 --> 00:15:42.410 So how do you think about this? 00:15:42.410 --> 00:15:43.450 What's going to be the current? 00:15:43.450 --> 00:15:44.420 How would you calculate the current? 00:15:44.420 --> 00:15:45.670 What do I do first? 00:15:48.462 --> 00:15:48.930 Shout. 00:15:48.930 --> 00:15:52.772 If you shout, and especially if my head's turned away I 00:15:52.772 --> 00:15:55.260 don't know who you are. 00:15:55.260 --> 00:15:56.200 AUDIENCE: Kirchhoff's law. 00:15:56.200 --> 00:15:57.240 PROFESSOR: Kirchoff's law, wonderful. 00:15:57.240 --> 00:15:57.717 Which one? 00:15:57.717 --> 00:16:00.102 There's two of them. 00:16:00.102 --> 00:16:02.964 AUDIENCE: [UNINTELLIGIBLE]. 00:16:02.964 --> 00:16:04.130 PROFESSOR: [UNINTELLIGIBLE]. 00:16:04.130 --> 00:16:05.663 What loop do you want to use? 00:16:05.663 --> 00:16:06.430 AUDIENCE: Left. 00:16:06.430 --> 00:16:07.220 PROFESSOR: Left side. 00:16:07.220 --> 00:16:10.960 So if we use the left side loop, we would conclude that 00:16:10.960 --> 00:16:12.310 there's a volt across the resistor. 00:16:12.310 --> 00:16:15.288 So the current would be? 00:16:15.288 --> 00:16:16.764 AUDIENCE: 1 Amp. 00:16:16.764 --> 00:16:17.748 PROFESSOR: An amp. 00:16:17.748 --> 00:16:21.684 Where's the current come from? 00:16:21.684 --> 00:16:23.160 AUDIENCE: The voltage. 00:16:23.160 --> 00:16:28.080 PROFESSOR: The voltage source just like before. 00:16:28.080 --> 00:16:29.880 So not quite. 00:16:29.880 --> 00:16:32.210 So the voltage source establishes this 00:16:32.210 --> 00:16:33.710 voltage would be 1. 00:16:33.710 --> 00:16:36.850 That makes this current be 1. 00:16:36.850 --> 00:16:39.220 That would be consistent with the current coming out of 00:16:39.220 --> 00:16:42.250 here, except we have to also think about that 1 Amp source. 00:16:44.810 --> 00:16:46.780 So the question is, what's does the 1 Amp source do? 00:16:46.780 --> 00:16:48.370 Nothing? 00:16:48.370 --> 00:16:52.240 It's just there sort of for decoration or for 00:16:52.240 --> 00:16:55.170 [UNINTELLIGIBLE] so that we can make an interesting 00:16:55.170 --> 00:16:56.580 question to ask in lecture? 00:16:56.580 --> 00:16:57.830 Maybe. 00:16:59.870 --> 00:17:01.040 So where's the current? 00:17:01.040 --> 00:17:02.689 Where's the 1 Amp that goes through the 00:17:02.689 --> 00:17:05.184 resistor come from? 00:17:05.184 --> 00:17:07.180 AUDIENCE: [UNINTELLIGIBLE] on the right. 00:17:07.180 --> 00:17:09.176 PROFESSOR: It comes from the right. 00:17:09.176 --> 00:17:11.080 It comes from the current over here. 00:17:11.080 --> 00:17:13.130 So the idea is that if this current, ignore the voltage 00:17:13.130 --> 00:17:14.150 over here for the moment. 00:17:14.150 --> 00:17:16.560 If this current flowed through the resistor, then you'd have 00:17:16.560 --> 00:17:19.130 1 Amp going through there, and you'd have 1 Volt generated by 00:17:19.130 --> 00:17:22.069 that current which just happens to be exactly the 00:17:22.069 --> 00:17:25.900 right voltage to match the voltage 00:17:25.900 --> 00:17:28.930 from the voltage source. 00:17:28.930 --> 00:17:31.640 So if you think about this, the voltage guarantees that 00:17:31.640 --> 00:17:35.720 this is 1 Volt, but so does the current. 00:17:35.720 --> 00:17:38.980 In order to simultaneously satisfy everything, all you 00:17:38.980 --> 00:17:41.310 need to do is have all of this current go around and come 00:17:41.310 --> 00:17:44.240 down through that resistor. 00:17:44.240 --> 00:17:47.070 That will generate the volt, so there's no propensity for 00:17:47.070 --> 00:17:50.020 more current to flow out of the source because the source 00:17:50.020 --> 00:17:54.580 is 1 Volt and it's facing a circuit that's already 1 Volt. 00:17:54.580 --> 00:17:56.310 So the idea was to try to give you something that's 00:17:56.310 --> 00:18:00.260 relatively simple that you can think through on your own, but 00:18:00.260 --> 00:18:01.530 not trivial. 00:18:01.530 --> 00:18:03.060 So the answer was 1 Amp. 00:18:03.060 --> 00:18:05.870 But the 1 Amp was not for the trivial reason. 00:18:05.870 --> 00:18:10.280 The 1 Amp is because the current from the right flows 00:18:10.280 --> 00:18:14.470 through the resistor and makes the voltage be 1. 00:18:14.470 --> 00:18:15.900 So the right answer is 1. 00:18:15.900 --> 00:18:17.290 But for the reason that you might not 00:18:17.290 --> 00:18:19.490 have originally thought. 00:18:19.490 --> 00:18:22.140 But more importantly, I wanted to use that as a motivation 00:18:22.140 --> 00:18:25.013 for thinking about, how do we think about bigger circuits? 00:18:27.670 --> 00:18:30.240 So when the simple circuit, like two parts, it's no 00:18:30.240 --> 00:18:32.260 problem figuring out what the answer's going to be. 00:18:32.260 --> 00:18:35.430 But when the circuit has even three parts, it may require 00:18:35.430 --> 00:18:36.290 more thinking. 00:18:36.290 --> 00:18:38.480 And you may want to have a more structured way of 00:18:38.480 --> 00:18:39.630 thinking about the solution. 00:18:39.630 --> 00:18:40.487 Yes? 00:18:40.487 --> 00:18:43.409 AUDIENCE: What would have occurred if the current 00:18:43.409 --> 00:18:45.844 provider on the right side was 2 Amps? 00:18:45.844 --> 00:18:47.792 PROFESSOR: Great question. 00:18:47.792 --> 00:18:51.660 Had this been 2 Amps, you can't violate this voltage. 00:18:51.660 --> 00:18:54.570 So that would have been 1 Volt. 00:18:54.570 --> 00:18:57.790 So that would have been 1 Amp through the resistor. 00:18:57.790 --> 00:18:59.560 So then you're left with the problem with this guy's 00:18:59.560 --> 00:19:03.330 pushing 2 and that guy's only eating 1. 00:19:03.330 --> 00:19:06.660 But the rules for the voltage source say eat or source 00:19:06.660 --> 00:19:10.140 however much current is necessary in order to make the 00:19:10.140 --> 00:19:11.900 voltage equal to 1. 00:19:11.900 --> 00:19:16.070 So the excess amp goes through the voltage source. 00:19:16.070 --> 00:19:20.400 So the voltage source is, in fact, being supplied power 00:19:20.400 --> 00:19:23.350 rather than supplying power itself. 00:19:23.350 --> 00:19:27.400 Had this been 2 Amps, some of the power from this source 00:19:27.400 --> 00:19:30.060 would have gone into the resistor. 00:19:30.060 --> 00:19:31.730 And some of the power from this source would have 00:19:31.730 --> 00:19:34.030 actually gone into the voltage source. 00:19:34.030 --> 00:19:38.150 So if the voltage source were, for example, a model for a 00:19:38.150 --> 00:19:41.250 rechargeable battery, that rechargeable 00:19:41.250 --> 00:19:43.720 battery would be charging. 00:19:43.720 --> 00:19:44.905 Does that make sense? 00:19:44.905 --> 00:19:47.240 So if there had been a mismatch in the conditions, 00:19:47.240 --> 00:19:52.210 you still have to satisfy all the relationships from all of 00:19:52.210 --> 00:19:54.470 the sources. 00:19:54.470 --> 00:19:56.770 AUDIENCE: What if the voltage was larger? 00:19:56.770 --> 00:19:58.609 PROFESSOR: The same thing would have happened. 00:19:58.609 --> 00:20:01.265 Except now the flowing current would be in 00:20:01.265 --> 00:20:01.990 the opposite direction. 00:20:01.990 --> 00:20:04.620 Let's say that if the voltage here had been 2 Volts, then 00:20:04.620 --> 00:20:07.530 the voltage would have required that there is 2 Amps 00:20:07.530 --> 00:20:09.730 flowing here. 00:20:09.730 --> 00:20:13.590 1 Amp would come from here, but another Amp 00:20:13.590 --> 00:20:15.130 would come from here. 00:20:15.130 --> 00:20:18.020 This voltage source will supply whatever current is 00:20:18.020 --> 00:20:21.521 necessary to make its voltage law real. 00:20:26.420 --> 00:20:26.495 Ok. 00:20:26.495 --> 00:20:30.570 In fact, what we'll do now is turn toward a discussion of 00:20:30.570 --> 00:20:33.960 more complicated systems that will let you go back and in 00:20:33.960 --> 00:20:37.430 retrospect, analyze all those cases that we just did. 00:20:37.430 --> 00:20:39.240 And you'll be able to see trivially how that has to be 00:20:39.240 --> 00:20:40.460 the right answer. 00:20:40.460 --> 00:20:46.460 So what I want to do now is generate a formal structure 00:20:46.460 --> 00:20:47.960 for how you would solve circuits. 00:20:47.960 --> 00:20:49.352 Yes? 00:20:49.352 --> 00:20:53.240 AUDIENCE: So did we know of anything that could generate a 00:20:53.240 --> 00:20:56.800 current without generating a voltage, like in real life? 00:20:56.800 --> 00:20:58.310 PROFESSOR: Can anything generate a current without 00:20:58.310 --> 00:20:59.080 generating a voltage? 00:20:59.080 --> 00:21:01.130 That's a tricky question. 00:21:01.130 --> 00:21:05.280 If you think about something as generating a current, then 00:21:05.280 --> 00:21:09.700 the voltage is not necessarily determined by that part. 00:21:09.700 --> 00:21:12.030 So that's kind of illustrated here. 00:21:12.030 --> 00:21:17.000 If this guy is generating a current, this guy is not 00:21:17.000 --> 00:21:21.755 actually the element that is controlling its own voltage. 00:21:24.560 --> 00:21:27.470 In general, if you want to speak simultaneously about the 00:21:27.470 --> 00:21:32.000 current and voltage across the device, you have to know what 00:21:32.000 --> 00:21:34.600 it was connected to. 00:21:34.600 --> 00:21:36.020 Each part-- 00:21:36.020 --> 00:21:37.790 we'll get to this in a moment in case some of you are 00:21:37.790 --> 00:21:39.940 worried about launching ahead. 00:21:39.940 --> 00:21:42.300 We will cover this. 00:21:42.300 --> 00:21:44.300 This is very good motivation for figuring out what's going 00:21:44.300 --> 00:21:46.690 to happen in the next three slides. 00:21:46.690 --> 00:21:54.320 So each part gets to tell you one relationship between 00:21:54.320 --> 00:21:55.790 voltage and current. 00:21:55.790 --> 00:22:00.450 Generally speaking, that's not enough to solve for voltage 00:22:00.450 --> 00:22:01.030 and current. 00:22:01.030 --> 00:22:03.200 Voltage and current is like two unknowns. 00:22:06.300 --> 00:22:11.000 Each element relationship is one equation. 00:22:11.000 --> 00:22:14.660 So the current source gets to say current equals x, current 00:22:14.660 --> 00:22:16.000 equals 1 Amp. 00:22:16.000 --> 00:22:20.400 It doesn't get to tell you what the voltage is. 00:22:20.400 --> 00:22:22.850 So being a little more physical to try to address 00:22:22.850 --> 00:22:27.480 your question more physically, there are processes that can 00:22:27.480 --> 00:22:30.970 be extremely well modeled as current generators. 00:22:30.970 --> 00:22:35.120 In fact, many electronic semiconductor parts, like 00:22:35.120 --> 00:22:37.890 transistors, work more like a current source than like 00:22:37.890 --> 00:22:39.790 anything else. 00:22:39.790 --> 00:22:42.590 So there are devices that behave as though they were 00:22:42.590 --> 00:22:46.570 current sources, but they don't simultaneously get to 00:22:46.570 --> 00:22:49.390 tell you what is their current and what is their voltage. 00:22:49.390 --> 00:22:51.245 They only get to tell you what is their current. 00:22:55.960 --> 00:22:58.630 So let's think about now, if you had a more complicated 00:22:58.630 --> 00:23:02.930 system, how could you systematically go about 00:23:02.930 --> 00:23:05.580 finding the solution? 00:23:05.580 --> 00:23:07.430 As was mentioned earlier, there's something called 00:23:07.430 --> 00:23:08.110 Kirchhoff's law. 00:23:08.110 --> 00:23:10.060 And in fact, there's two of them. 00:23:10.060 --> 00:23:12.500 Kirchhoff's voltage law and Kirchhoff's current law. 00:23:12.500 --> 00:23:18.280 Kirchhoff's voltage law, in its most elementary form, says 00:23:18.280 --> 00:23:21.790 that if you trace the path around any closed path in a 00:23:21.790 --> 00:23:24.450 circuit, regardless of what the path is-- 00:23:24.450 --> 00:23:26.040 every closed path-- 00:23:26.040 --> 00:23:30.650 the sum of the voltages going around that closed path is 0. 00:23:30.650 --> 00:23:35.830 So for example in this circuit, the red path 00:23:35.830 --> 00:23:39.130 illustrates one closed path through the circuit. 00:23:39.130 --> 00:23:41.680 It goes up through the voltage source, down through this 00:23:41.680 --> 00:23:44.900 resistor, and then down through that resistor. 00:23:44.900 --> 00:23:49.160 Kirchhoff's voltage law says the sum of the voltages around 00:23:49.160 --> 00:23:51.990 that loop is 0. 00:23:51.990 --> 00:23:56.150 That's written mathematically here, minus v1 for here, plus 00:23:56.150 --> 00:23:59.850 v2 for here, plus v4 for here is 0. 00:23:59.850 --> 00:24:01.660 OK, where do the signs come from? 00:24:01.660 --> 00:24:05.570 The signs came from the reference directions that we 00:24:05.570 --> 00:24:09.730 assigned arbitrarily to the elements. 00:24:09.730 --> 00:24:13.280 Before I ever do a circuits question, I always assign a 00:24:13.280 --> 00:24:15.520 reference direction. 00:24:15.520 --> 00:24:17.125 Every voltage has a positive terminal 00:24:17.125 --> 00:24:19.120 and a negative terminal. 00:24:19.120 --> 00:24:22.210 And I must be consistent in order to apply these rules. 00:24:22.210 --> 00:24:27.800 These rules only work if I declare a reference direction 00:24:27.800 --> 00:24:29.320 and stick with it. 00:24:29.320 --> 00:24:31.350 If midway through a problem I flip it, I'll 00:24:31.350 --> 00:24:33.810 get the wrong answer. 00:24:33.810 --> 00:24:36.220 So the minus sign has to do with the fact that as I trace 00:24:36.220 --> 00:24:41.730 this path, I enter the minus part of this guy, but the plus 00:24:41.730 --> 00:24:43.200 part of that guy and that guy. 00:24:43.200 --> 00:24:47.250 So the sign of v1 is negated relative to the others. 00:24:50.220 --> 00:24:53.510 A different way to think about that is here, we can think 00:24:53.510 --> 00:24:56.330 that v1 is the sum of v2 and v4. 00:24:56.330 --> 00:24:58.700 That's sometimes more intuitive because if you 00:24:58.700 --> 00:25:03.280 started here, going through this path you would end up 00:25:03.280 --> 00:25:08.540 with a voltage that is v1 higher than where you started. 00:25:08.540 --> 00:25:11.580 Whereas starting here, you would end up with a voltage 00:25:11.580 --> 00:25:15.030 here that's v4 higher than where you started. 00:25:15.030 --> 00:25:17.400 And then by the time you got to here, it would be v2 plus 00:25:17.400 --> 00:25:21.970 v4 higher than where you started. 00:25:21.970 --> 00:25:25.030 You start one place and on one route, you end up v1 higher. 00:25:25.030 --> 00:25:28.030 And in the other route, you get v4 plus v2 higher. 00:25:28.030 --> 00:25:31.120 So it must be the case that v1 is the same as v2 plus v4. 00:25:33.620 --> 00:25:35.710 Those are absolutely equivalent ways of 00:25:35.710 --> 00:25:36.990 thinking about it. 00:25:36.990 --> 00:25:38.650 So those laws are equivalent. 00:25:38.650 --> 00:25:40.600 If you think about it a path, you think 00:25:40.600 --> 00:25:43.305 about some of the paths-- 00:25:43.305 --> 00:25:43.780 no. 00:25:43.780 --> 00:25:47.700 The path coinciding with the negative direction of some of 00:25:47.700 --> 00:25:49.750 the elements and the positive direction of others. 00:25:53.030 --> 00:25:58.720 OK, how many other paths are there? 00:25:58.720 --> 00:26:00.090 Take 20 seconds, talk to your neighbor. 00:26:00.090 --> 00:26:04.630 Figure out all of the possible paths for 00:26:04.630 --> 00:26:06.120 which KVL has to apply. 00:28:35.850 --> 00:28:39.450 OK, so everybody raise your hand and show a number of 00:28:39.450 --> 00:28:44.130 fingers equal to the number of KVL equations less two. 00:28:49.060 --> 00:28:51.000 Oh, very good. 00:28:51.000 --> 00:28:52.170 Virtually 100% correct. 00:28:52.170 --> 00:28:53.420 Why do you all say (5)? 00:28:55.760 --> 00:28:57.760 Which is to say 7. 00:28:57.760 --> 00:28:59.010 Why do you all say 7? 00:29:01.970 --> 00:29:03.650 So there's 3 obvious ones. 00:29:03.650 --> 00:29:05.660 I was expecting a couple of 3's. 00:29:05.660 --> 00:29:07.230 This was supposed to be-- 00:29:07.230 --> 00:29:09.930 OK, yeah, I do plot against you. 00:29:09.930 --> 00:29:12.410 I was expecting some 3's. 00:29:12.410 --> 00:29:14.770 So there's 3 obvious paths that are analogous to the 00:29:14.770 --> 00:29:15.620 first one we looked at. 00:29:15.620 --> 00:29:18.980 If I call the first path A, then there's B and C which are 00:29:18.980 --> 00:29:22.030 the excursions around here. 00:29:22.030 --> 00:29:23.850 And you can write the equations just the same. 00:29:23.850 --> 00:29:26.030 They each involve three voltages. 00:29:26.030 --> 00:29:28.620 And they each go through, some starting at the negative side 00:29:28.620 --> 00:29:32.720 and some starting at the positive side. 00:29:32.720 --> 00:29:35.230 So those are in some sense, the obvious ones. 00:29:35.230 --> 00:29:39.040 But there are others too. 00:29:39.040 --> 00:29:42.370 So one way to think about it, what I'd like you to do is 00:29:42.370 --> 00:29:44.290 enumerate all the paths through the circuit. 00:29:44.290 --> 00:29:46.640 I should have said all the paths through the circuit that 00:29:46.640 --> 00:29:53.300 go through each element one or fewer times. 00:29:53.300 --> 00:29:58.660 I don't want you to go through the same element twice. 00:29:58.660 --> 00:30:02.280 So here's another path that would go through elements at 00:30:02.280 --> 00:30:04.290 most one time. 00:30:04.290 --> 00:30:07.230 So up through here, over through here, which didn't go 00:30:07.230 --> 00:30:08.050 through any elements. 00:30:08.050 --> 00:30:10.480 Down through that element, across that, down through 00:30:10.480 --> 00:30:11.850 here, et cetera. 00:30:11.850 --> 00:30:14.090 And you get an equation for that. 00:30:14.090 --> 00:30:14.950 Here's another. 00:30:14.950 --> 00:30:16.140 Here's another. 00:30:16.140 --> 00:30:18.830 Here's another. 00:30:18.830 --> 00:30:21.220 And if you try to think about a general rule, a general rule 00:30:21.220 --> 00:30:24.410 is something like, how many of those panels can you make and 00:30:24.410 --> 00:30:29.790 piece together where the loop goes through the perimeter? 00:30:29.790 --> 00:30:31.970 You're not allowed to go through an inner place because 00:30:31.970 --> 00:30:33.430 if you went through an inner node, you'd have to 00:30:33.430 --> 00:30:34.770 go through it twice. 00:30:34.770 --> 00:30:37.550 If you wanted the path to go through an inner element, 00:30:37.550 --> 00:30:40.340 you'd have to go through that element twice. 00:30:40.340 --> 00:30:41.810 So in fact, the answer is 7. 00:30:41.810 --> 00:30:49.200 There are 7 different paths according to Kirchhoff's 00:30:49.200 --> 00:30:51.970 voltage law, all of which the sum of the voltages around 00:30:51.970 --> 00:30:53.505 those paths has to be 0. 00:30:56.080 --> 00:30:59.260 The problem is, of course, that those equations are not 00:30:59.260 --> 00:31:00.625 all linearly independent. 00:31:04.030 --> 00:31:07.280 So if you just had a general purpose equation solver-- and 00:31:07.280 --> 00:31:09.330 by the way, we'll write one of those in week 00:31:09.330 --> 00:31:11.970 8 for solving circuits. 00:31:11.970 --> 00:31:15.400 If you just passed those 7 equations into a general 00:31:15.400 --> 00:31:17.410 purpose equation solver, it would tell you there's 00:31:17.410 --> 00:31:20.010 something awry with your equations because they're not 00:31:20.010 --> 00:31:22.800 linearly independent. 00:31:22.800 --> 00:31:27.140 So you can, however, think about linearly independent in 00:31:27.140 --> 00:31:29.050 particularly simple cases. 00:31:29.050 --> 00:31:32.140 This network is a particular kind of network that we call a 00:31:32.140 --> 00:31:34.070 planar network. 00:31:34.070 --> 00:31:37.680 A planar network is one that I can draw on a sheet of paper 00:31:37.680 --> 00:31:41.570 without crossing wires. 00:31:41.570 --> 00:31:44.040 So I can draw this network without crossing wires. 00:31:44.040 --> 00:31:45.410 I'll call it planar. 00:31:45.410 --> 00:31:49.080 And it turns out that Kirchhoff's voltage laws for 00:31:49.080 --> 00:31:54.200 the innermost loops are always independent of each other. 00:31:54.200 --> 00:31:56.490 That's kind of obvious because as you go to a -- 00:31:56.490 --> 00:31:58.940 so each loop contains at least one element that some other 00:31:58.940 --> 00:32:01.460 loop didn't have. 00:32:01.460 --> 00:32:04.620 So that's kind of the reasoning for why it works. 00:32:04.620 --> 00:32:07.510 So if you think about this particular loop, which we 00:32:07.510 --> 00:32:13.370 included in the 7, you can think about that as being the 00:32:13.370 --> 00:32:17.020 sum of the loops this way, the A loop and the B loop. 00:32:17.020 --> 00:32:20.360 Because if you write KVL for the A loop and KVL for the B 00:32:20.360 --> 00:32:25.980 loop and add them, you end up deriving KVL for the more 00:32:25.980 --> 00:32:28.220 complicated path. 00:32:28.220 --> 00:32:30.190 And if you think about what's going on, it's not anything 00:32:30.190 --> 00:32:31.880 terribly magical. 00:32:31.880 --> 00:32:36.850 This path is the same as the A path added to that path, where 00:32:36.850 --> 00:32:40.900 I went through this element down when I did the A path and 00:32:40.900 --> 00:32:45.610 up when I did the B path. 00:32:45.610 --> 00:32:48.230 So those parts canceled out. 00:32:48.230 --> 00:32:50.420 That was the rule that I was talking about how I don't 00:32:50.420 --> 00:32:54.480 really want to go through the same element twice when I'm 00:32:54.480 --> 00:32:56.970 applying KVL. 00:32:56.970 --> 00:33:00.280 So the idea then is that there's a systematic way, an 00:33:00.280 --> 00:33:05.170 easy way to figure out all the KVL loops. 00:33:05.170 --> 00:33:06.765 You just think about all the possible 00:33:06.765 --> 00:33:08.360 paths through the circuit. 00:33:08.360 --> 00:33:12.210 You do have to worry about linearly independent. 00:33:12.210 --> 00:33:14.440 In the case of planar networks, that's pretty 00:33:14.440 --> 00:33:15.220 straightforward. 00:33:15.220 --> 00:33:18.510 Planar networks, you can always figure out the linearly 00:33:18.510 --> 00:33:21.770 independent KVL equations by looking at the smallest 00:33:21.770 --> 00:33:24.140 possible loops. 00:33:24.140 --> 00:33:26.960 The loops with small area. 00:33:26.960 --> 00:33:28.250 OK, so that's half of it. 00:33:28.250 --> 00:33:28.960 That's KVL. 00:33:28.960 --> 00:33:32.650 The other Kirchhoff's law is KCL, Kirchhoff's Current Law. 00:33:37.250 --> 00:33:41.760 There we are thinking of the flow of current. 00:33:41.760 --> 00:33:44.770 So the flow of current is analogous to the flow of 00:33:44.770 --> 00:33:47.560 incompressible fluid. 00:33:47.560 --> 00:33:49.960 Water, for example. 00:33:49.960 --> 00:33:52.310 If you trace the amount of water that flows through a 00:33:52.310 --> 00:33:56.560 pipe that goes into a Y, then the sum of the flows out has 00:33:56.560 --> 00:33:58.660 to equal the flow in. 00:33:58.660 --> 00:34:02.710 If that weren't true, the water would be building up. 00:34:02.710 --> 00:34:05.860 So we think about pipes as transporting the flow of water 00:34:05.860 --> 00:34:08.270 without allowing it to build up anywhere. 00:34:08.270 --> 00:34:10.956 That's precisely how we think about wires 00:34:10.956 --> 00:34:13.650 in electrical circuits. 00:34:13.650 --> 00:34:16.540 The wires allow the transport of electrons but don't allow 00:34:16.540 --> 00:34:20.770 the buildup of electrons. 00:34:20.770 --> 00:34:22.820 OK, do electrons build up? 00:34:22.820 --> 00:34:24.000 Sure. 00:34:24.000 --> 00:34:28.219 But in our idealized world, we say they don't build up in the 00:34:28.219 --> 00:34:31.130 wires, they build up in a part. 00:34:31.130 --> 00:34:33.010 And we'll have a special part that allows the 00:34:33.010 --> 00:34:34.570 electrons to build up. 00:34:34.570 --> 00:34:36.749 So we're not excluding the possibility that they build 00:34:36.749 --> 00:34:41.840 up, we're just saying that in this formalism, we don't allow 00:34:41.840 --> 00:34:44.540 the electrons to build up in the wires. 00:34:44.540 --> 00:34:47.090 So for the purpose of the wires, current in is equal to 00:34:47.090 --> 00:34:47.630 the current out. 00:34:47.630 --> 00:34:50.949 The net current in is 0. 00:34:50.949 --> 00:34:57.360 So we will think then, about the circuit having nodes. 00:34:57.360 --> 00:35:01.300 The nodes are the places where more than one element meets, 00:35:01.300 --> 00:35:03.980 two or more elements meet. 00:35:03.980 --> 00:35:07.790 And we will apply KCL at each node. 00:35:07.790 --> 00:35:10.650 So for example, in this simple circuit where I would have 00:35:10.650 --> 00:35:18.010 three parts connected in what we would call parallel, they 00:35:18.010 --> 00:35:20.330 share a node at the top and they share 00:35:20.330 --> 00:35:22.620 a node at the bottom. 00:35:22.620 --> 00:35:24.870 So even though it looks like there's multiple interconnects 00:35:24.870 --> 00:35:27.290 up here, we say that's one node. 00:35:30.430 --> 00:35:32.940 And we would say that the sum of the currents into the node 00:35:32.940 --> 00:35:35.810 is equal to the sum of the currents out. 00:35:35.810 --> 00:35:38.960 So if I lab all of the possible currents that come 00:35:38.960 --> 00:35:42.160 out of that node, I would have i1, i2, i3. 00:35:42.160 --> 00:35:43.685 i1 goes through the first one, the second one 00:35:43.685 --> 00:35:45.100 and the third one. 00:35:45.100 --> 00:35:48.300 And so I would conclude from Kirchhoff's current law that 00:35:48.300 --> 00:35:51.340 the sum of i1, i2, and i3 is 0. 00:35:51.340 --> 00:35:51.720 OK. 00:35:51.720 --> 00:35:53.670 Easy, right? 00:35:53.670 --> 00:35:59.380 As I said, we're going to make an abstraction where the 00:35:59.380 --> 00:36:03.620 electrons don't build up in the wires. 00:36:03.620 --> 00:36:06.190 They don't even build up in the parts. 00:36:06.190 --> 00:36:07.330 They do get stored in the parts. 00:36:07.330 --> 00:36:10.230 That's a little confusing, we'll come back to that. 00:36:10.230 --> 00:36:12.540 If they don't build up in the parts, then the current that 00:36:12.540 --> 00:36:14.720 goes in this leg has to come out that leg. 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 00:36:23.930 --> 00:36:27.030 up with another equation down here, which turns out to be 00:36:27.030 --> 00:36:28.900 precisely the same as the one at the top. 00:36:33.380 --> 00:36:35.790 Everybody's happy with that? 00:36:35.790 --> 00:36:37.550 So we're thinking about this just the way we would think 00:36:37.550 --> 00:36:39.650 about water flow. 00:36:39.650 --> 00:36:42.840 If there's water flow into a part, it better be coming out. 00:36:42.840 --> 00:36:46.750 If there's water flow in a pipe, the water that goes into 00:36:46.750 --> 00:36:49.830 the pipe better come out of the pipe someplace. 00:36:49.830 --> 00:36:51.700 So here is an arbitrary network 00:36:51.700 --> 00:36:53.340 made out of four parts. 00:36:53.340 --> 00:36:57.070 How many linearly independent KCL equations are there? 00:38:20.810 --> 00:38:24.740 So how many linearly independent KCL equations are 00:38:24.740 --> 00:38:26.280 in that network? 00:38:26.280 --> 00:38:30.280 Everyone raise your hand, some number of KCL equations. 00:38:34.125 --> 00:38:36.250 OK, I'm seeing a bigger variety. 00:38:36.250 --> 00:38:38.170 I see (1)'s, (2)'s, and (3)'s. 00:38:38.170 --> 00:38:39.200 I don't see any (4)'s. 00:38:39.200 --> 00:38:40.515 That's probably good. 00:38:43.400 --> 00:38:46.600 So how do you think about the number of linearly independent 00:38:46.600 --> 00:38:47.850 KCL equations? 00:38:52.945 --> 00:38:55.860 So the first thing to do is to label things. 00:38:59.430 --> 00:39:01.660 So you have to have reference directions before you can sort 00:39:01.660 --> 00:39:03.250 of think about things. 00:39:03.250 --> 00:39:04.540 So we have four elements. 00:39:04.540 --> 00:39:08.900 We would be expecting to see four element currents. 00:39:08.900 --> 00:39:10.800 The same current that goes into an element has 00:39:10.800 --> 00:39:12.460 to come out of it. 00:39:12.460 --> 00:39:14.080 So there's element current 1, 2, 3, and 4. 00:39:17.490 --> 00:39:21.230 There are three nodes, so we might be 00:39:21.230 --> 00:39:24.680 expecting three KCL equations. 00:39:24.680 --> 00:39:28.190 Here's one node from which you would conclude that the sum of 00:39:28.190 --> 00:39:31.520 i1 and i2 better be 0. 00:39:31.520 --> 00:39:33.520 Here's a node from which you would conclude that the 00:39:33.520 --> 00:39:38.890 current in i2 better be i3 plus i4. 00:39:38.890 --> 00:39:41.440 And here is a node from which you would conclude that i1 00:39:41.440 --> 00:39:45.230 plus i3 plus i4 is 0. 00:39:45.230 --> 00:39:48.380 So I can write one KCL equation for every node, 00:39:48.380 --> 00:39:50.840 that's not surprising. 00:39:50.840 --> 00:39:52.820 But if you look at those equations, you'll see that 00:39:52.820 --> 00:39:54.390 they're not linearly independent. 00:39:54.390 --> 00:39:57.710 In fact, if you solve this one for i2-- 00:39:57.710 --> 00:39:59.150 it's already solved for i2. 00:39:59.150 --> 00:40:03.910 Stick that answer up here, you get i3 plus i4 added to i1 is 00:40:03.910 --> 00:40:06.590 0, which is just the same as that equation. 00:40:06.590 --> 00:40:11.540 So of those three equations, only two of them are linearly 00:40:11.540 --> 00:40:13.090 independent. 00:40:13.090 --> 00:40:14.525 The answer to that problem was (2). 00:40:18.820 --> 00:40:20.920 And there's a pattern. 00:40:20.920 --> 00:40:23.690 So think about the pattern in terms of figuring out the 00:40:23.690 --> 00:40:26.320 number of linearly independent KCL equations that are in a 00:40:26.320 --> 00:40:27.735 slightly more complicated network. 00:42:00.512 --> 00:42:05.880 So what's the answer here, how many KCL 00:42:05.880 --> 00:42:07.270 equations are in this network? 00:42:10.040 --> 00:42:11.290 Wow. 00:42:13.160 --> 00:42:18.020 Well, I'm not getting any of the answers I would have said. 00:42:18.020 --> 00:42:20.776 What does that mean? 00:42:20.776 --> 00:42:23.550 Ah, I'm forgetting to add 2. 00:42:23.550 --> 00:42:24.270 That's my problem. 00:42:24.270 --> 00:42:27.370 OK, now I'm getting some of the answers that I would 00:42:27.370 --> 00:42:27.980 expect to get. 00:42:27.980 --> 00:42:30.080 OK, got it. 00:42:30.080 --> 00:42:31.730 I confused myself. 00:42:31.730 --> 00:42:33.910 OK, the vast majority say (1). 00:42:33.910 --> 00:42:35.640 How do you get that? 00:42:35.640 --> 00:42:36.890 Which is 3. 00:42:42.574 --> 00:42:45.740 So again, you think of how in this circuit there are four 00:42:45.740 --> 00:42:51.880 nodes, A, B, C, D. So we can think about writing a KCL 00:42:51.880 --> 00:42:53.060 equation for each one. 00:42:53.060 --> 00:42:56.450 If we go to A, A has three currents coming out of it -- 00:42:56.450 --> 00:42:57.360 1, 2, 3. 00:42:57.360 --> 00:43:02.910 So the sum of those has to be 0, et cetera. 00:43:02.910 --> 00:43:05.010 And if you think about those equations, they're not 00:43:05.010 --> 00:43:06.450 linearly independent either. 00:43:09.010 --> 00:43:11.250 If you work through the math, you see that there's exactly 00:43:11.250 --> 00:43:15.330 one of those equations that you can eliminate. 00:43:15.330 --> 00:43:17.350 So you're left with three linearly 00:43:17.350 --> 00:43:20.250 independent KCL equations. 00:43:20.250 --> 00:43:22.210 And so there's a pattern emerging here. 00:43:22.210 --> 00:43:23.460 Somebody see the pattern? 00:43:25.540 --> 00:43:26.520 1 minus. 00:43:26.520 --> 00:43:27.770 Can somebody prove the pattern? 00:43:30.560 --> 00:43:31.520 So there's a pattern here. 00:43:31.520 --> 00:43:35.810 The pattern is take the number of nodes and the number of 00:43:35.810 --> 00:43:37.910 independent KCL equations as one less. 00:43:41.480 --> 00:43:43.398 So the challenge is, can you prove it? 00:43:46.390 --> 00:43:47.640 And by the theory of lectures-- 00:43:50.858 --> 00:43:51.356 AUDIENCE: Yes. 00:43:51.356 --> 00:43:53.350 PROFESSOR: Yes. 00:43:53.350 --> 00:43:55.510 And by a corollary of the theory of lectures, the way 00:43:55.510 --> 00:43:57.010 you would prove it is? 00:43:57.010 --> 00:43:57.970 AUDIENCE: On the next slide. 00:43:57.970 --> 00:43:58.980 PROFESSOR: On the next slide. 00:43:58.980 --> 00:44:00.230 Exactly. 00:44:02.960 --> 00:44:04.210 So how do I prove it? 00:44:06.958 --> 00:44:08.930 Yeah? 00:44:08.930 --> 00:44:11.395 AUDIENCE: Whenever you take minus 1, you just add all the 00:44:11.395 --> 00:44:12.650 [UNINTELLIGIBLE] together [UNINTELLIGIBLE]. 00:44:12.650 --> 00:44:12.910 PROFESSOR: Yeah. 00:44:12.910 --> 00:44:15.312 So there's something special about the last one. 00:44:15.312 --> 00:44:17.304 Why should there be something special about the last one. 00:44:17.304 --> 00:44:19.296 AUDIENCE: Because the circuit's closed. 00:44:19.296 --> 00:44:22.504 PROFESSOR: Because the circuit's closed. 00:44:22.504 --> 00:44:23.760 That's right. 00:44:23.760 --> 00:44:26.230 So the idea is to sort of generalize the way 00:44:26.230 --> 00:44:27.480 we think about KCL. 00:44:29.690 --> 00:44:30.710 So we start with a circuit. 00:44:30.710 --> 00:44:33.840 We think about having four nodes here. 00:44:33.840 --> 00:44:37.270 It's certainly the case that KCL holds for each node. 00:44:37.270 --> 00:44:41.376 So here's KCL for that node. 00:44:41.376 --> 00:44:46.830 But now if you think about KCL for this node, and then add 00:44:46.830 --> 00:44:51.180 them, that looks like a KCL equation. 00:44:53.850 --> 00:44:57.200 But it applies to a super node. 00:44:57.200 --> 00:45:02.520 Imagine the node defined by the black box, and think about 00:45:02.520 --> 00:45:05.293 the net currents into or out of the black node. 00:45:08.470 --> 00:45:13.940 This current i2, which leaves the red node, enters the green 00:45:13.940 --> 00:45:17.890 node, but doesn't go through the surface of the 00:45:17.890 --> 00:45:20.520 black node at all. 00:45:20.520 --> 00:45:23.600 That's exactly the current that's subtracted out when we 00:45:23.600 --> 00:45:26.986 added the red equation to the green equation. 00:45:26.986 --> 00:45:28.940 Does that make sense? 00:45:28.940 --> 00:45:34.090 So KCL says, oh, if all the currents at a node have to sum 00:45:34.090 --> 00:45:40.380 to 0, and if elements have the same current coming out and 00:45:40.380 --> 00:45:48.560 going in, then if you draw a box around an element, what 00:45:48.560 --> 00:45:51.000 goes into the element is the same as what 00:45:51.000 --> 00:45:52.240 comes out of the element. 00:45:52.240 --> 00:45:55.340 It doesn't change the net current through the surface. 00:45:55.340 --> 00:45:58.890 So the generalization of the KCL equation, KCL says the sum 00:45:58.890 --> 00:46:00.450 of the currents into a node is 0. 00:46:00.450 --> 00:46:06.690 The generalization says take any closed path in a circuit, 00:46:06.690 --> 00:46:11.630 the sum of the currents going across that closed path is 0. 00:46:11.630 --> 00:46:16.600 So if we apply that rule again, think about node 3. 00:46:16.600 --> 00:46:20.470 If we add the result of node 3 to the black node, which was 00:46:20.470 --> 00:46:25.170 the sum of 1 and 2, we get the new green curve. 00:46:25.170 --> 00:46:27.340 We get the new green equation. 00:46:27.340 --> 00:46:29.720 And what that says is the sum of the currents going across 00:46:29.720 --> 00:46:33.120 the green super node-- 00:46:33.120 --> 00:46:36.530 OK, so what's going on? i1 is coming out of it, i4 is coming 00:46:36.530 --> 00:46:38.330 out of it, i5 is coming out of it. 00:46:38.330 --> 00:46:41.540 So the sum of i1, i4, and i5 has to be 0. 00:46:46.570 --> 00:46:49.030 Well, KCL says the sum of the currents coming out of a 00:46:49.030 --> 00:46:50.890 node must be 0. 00:46:50.890 --> 00:46:54.990 The super KCL says the sum of the currents coming out of any 00:46:54.990 --> 00:47:00.180 closed region is also 0. 00:47:00.180 --> 00:47:03.840 But the interesting thing about this closed region is 00:47:03.840 --> 00:47:06.425 that it encloses all but one of the nodes. 00:47:09.230 --> 00:47:11.310 That's always true. 00:47:11.310 --> 00:47:14.150 Regardless of the system, regardless of the circuit, you 00:47:14.150 --> 00:47:19.090 can always draw a line that will isolate one node from all 00:47:19.090 --> 00:47:22.030 the others. 00:47:22.030 --> 00:47:25.825 So what that proves is that you can always write KCL for 00:47:25.825 --> 00:47:28.920 this node in terms of KCL for those nodes. 00:47:31.870 --> 00:47:31.965 Ok. 00:47:31.965 --> 00:47:34.770 So there's a generalization then that says that you can 00:47:34.770 --> 00:47:40.120 always write KCL for every node. 00:47:40.120 --> 00:47:42.130 They will always be linearly dependent. 00:47:42.130 --> 00:47:43.580 So you can always throw away one. 00:47:47.780 --> 00:47:49.690 So in some sense now, we're done. 00:47:49.690 --> 00:47:51.840 We've just finished circuit theory. 00:47:51.840 --> 00:47:55.060 We talked about how every element has to have a law. 00:47:55.060 --> 00:47:56.470 A resistor is Ohm's Law. 00:47:56.470 --> 00:47:58.820 A voltage source says that the voltage across the terminals 00:47:58.820 --> 00:48:00.295 is always a constant. 00:48:00.295 --> 00:48:02.730 A current source says that the current through the current 00:48:02.730 --> 00:48:04.530 source is always a constant. 00:48:04.530 --> 00:48:07.380 So every element tells you one law. 00:48:10.170 --> 00:48:12.760 We know how to think about KVL. 00:48:12.760 --> 00:48:18.930 So we know the rule for how the across variables behave. 00:48:18.930 --> 00:48:21.230 What's the aggregate behavior of all the across variables? 00:48:21.230 --> 00:48:24.140 Well, KVL has to be satisfied for every possible loop. 00:48:24.140 --> 00:48:26.430 The loops don't have to be independent. 00:48:26.430 --> 00:48:28.620 You have to worry about whether they're independent. 00:48:28.620 --> 00:48:31.020 The only simple rule we came up with-- we'll come up with 00:48:31.020 --> 00:48:32.415 another one in a moment. 00:48:32.415 --> 00:48:35.340 The only simple rule that we came up with was for planar 00:48:35.340 --> 00:48:38.410 circuits, where the innermost loops were linearly 00:48:38.410 --> 00:48:39.720 independent of each other. 00:48:39.720 --> 00:48:44.280 And you have to write KCL for all the nodes, except one. 00:48:44.280 --> 00:48:45.530 One of them never matters. 00:48:48.110 --> 00:48:50.210 So in some sense, we're done. 00:48:50.210 --> 00:48:52.450 What we would do to solve the circuit, 00:48:52.450 --> 00:48:54.310 think about every element. 00:48:54.310 --> 00:48:59.240 For every element assign a voltage, a reference voltage. 00:48:59.240 --> 00:49:02.570 For every element, assign a current. 00:49:02.570 --> 00:49:04.460 Make sure they go in the right direction. 00:49:04.460 --> 00:49:07.460 We always define currents to go down 00:49:07.460 --> 00:49:09.070 the potential gradient. 00:49:09.070 --> 00:49:11.920 They always go in the directions through the element 00:49:11.920 --> 00:49:14.840 from the positive to the negative. 00:49:14.840 --> 00:49:18.610 So for every element, assign a current and a voltage. 00:49:18.610 --> 00:49:22.420 We have 6 elements, that's 12 unknowns. 00:49:22.420 --> 00:49:25.370 Now we dig and we find 12 equations. 00:49:25.370 --> 00:49:28.980 In this particular circuit, we found those 12 equations. 00:49:28.980 --> 00:49:31.505 There were three KCL equations, one for each of the 00:49:31.505 --> 00:49:32.550 inner loops. 00:49:32.550 --> 00:49:35.420 There were three KCL equations, one for each node 00:49:35.420 --> 00:49:38.080 except one. 00:49:38.080 --> 00:49:41.020 There were 5 Ohm's law equations, one for each one of 00:49:41.020 --> 00:49:41.750 the resistors. 00:49:41.750 --> 00:49:45.390 There was one source equation for the voltage source. 00:49:45.390 --> 00:49:49.120 12 equations, 12 unknowns, we're done. 00:49:49.120 --> 00:49:52.900 The only problem is a lot of equations. 00:49:52.900 --> 00:49:54.500 It's not a very complicated circuit. 00:49:54.500 --> 00:49:56.430 We've only got 6 elements. 00:49:56.430 --> 00:49:59.580 I tried to motivate this in terms of studying networks 00:49:59.580 --> 00:50:04.390 that had 10 to the 9 elements. 00:50:04.390 --> 00:50:06.750 This technique is not particularly great 00:50:06.750 --> 00:50:08.920 at 10 to the 9. 00:50:08.920 --> 00:50:11.240 It would probably work. 00:50:11.240 --> 00:50:13.130 But we would probably be interested in 00:50:13.130 --> 00:50:15.970 finding simpler ways. 00:50:15.970 --> 00:50:20.430 So there are simpler ways you might imagine, and we'll 00:50:20.430 --> 00:50:24.680 discuss two of them just very briefly. 00:50:24.680 --> 00:50:27.160 The dumb way that I just talked about is what we call 00:50:27.160 --> 00:50:30.190 primitive variables, element variables. 00:50:30.190 --> 00:50:33.600 If you write all the element variables, v1, v2, v3, v4, v5 00:50:33.600 --> 00:50:37.480 v6, all of the element currents, i1, i2, i3, i4, i5, 00:50:37.480 --> 00:50:40.630 i6, write all the equations, you can solve it. 00:50:40.630 --> 00:50:43.620 However, if you're judicious, you can figure out a smaller 00:50:43.620 --> 00:50:46.600 number of unknowns and a correspondingly smaller number 00:50:46.600 --> 00:50:47.340 of equations. 00:50:47.340 --> 00:50:49.640 One method is called the node method. 00:50:52.180 --> 00:50:54.400 When we're thinking about the individual elements, the thing 00:50:54.400 --> 00:50:56.465 that matters is the voltage across the element. 00:50:59.230 --> 00:51:02.130 However, that's not the easy way to 00:51:02.130 --> 00:51:06.140 write the circuit equations. 00:51:06.140 --> 00:51:10.110 A much easier way is not to tell me the voltage across an 00:51:10.110 --> 00:51:13.360 element, but instead tell me the voltage associated with 00:51:13.360 --> 00:51:14.610 each of the nodes. 00:51:18.710 --> 00:51:23.400 If I tell you the voltage associated with every node, 00:51:23.400 --> 00:51:25.930 the important thing about that way of defining the variables 00:51:25.930 --> 00:51:29.040 is that you're guaranteed that from those variables, you can 00:51:29.040 --> 00:51:32.360 tell me the voltage across every part. 00:51:32.360 --> 00:51:35.630 So for example, in this circuit, this voltage source-- 00:51:35.630 --> 00:51:42.140 so if I call this one ground, we'll always have a magic node 00:51:42.140 --> 00:51:43.410 called ground. 00:51:43.410 --> 00:51:47.490 It is not special in the least. 00:51:47.490 --> 00:51:50.780 It's just the reference voltage. 00:51:50.780 --> 00:51:51.610 I'll come back to that. 00:51:51.610 --> 00:51:54.400 I'll say words in a minute about what the reference is. 00:51:54.400 --> 00:51:58.100 We always get to declare one node to be ground. 00:51:58.100 --> 00:52:01.270 We get one free node. 00:52:01.270 --> 00:52:04.830 It's a node whose voltage we don't care about because it's 00:52:04.830 --> 00:52:06.380 the reference for all voltages. 00:52:06.380 --> 00:52:09.260 It's a node whose current we don't care about because we 00:52:09.260 --> 00:52:12.370 get to throw away one node when we do current equations. 00:52:12.370 --> 00:52:15.400 So we have one special mode called ground, about which we 00:52:15.400 --> 00:52:16.660 don't care too much. 00:52:16.660 --> 00:52:18.900 Except that it's the most important node in the circuit. 00:52:18.900 --> 00:52:20.650 Except for that, we don't care about it. 00:52:20.650 --> 00:52:23.430 So this guy's ground. 00:52:23.430 --> 00:52:26.080 We think about its voltage being 0. 00:52:26.080 --> 00:52:30.360 Then this voltage supply makes that node be v0. 00:52:30.360 --> 00:52:32.250 I don't know what that is, so I'll call it e1. 00:52:32.250 --> 00:52:36.340 And I don't know what that is, so I'll call it e2. 00:52:36.340 --> 00:52:39.600 So if I tell you the voltage on all of those nodes, ground 00:52:39.600 --> 00:52:44.550 voltage is 0, the top voltage is v0, the left voltage is e1, 00:52:44.550 --> 00:52:45.670 the right voltage is e2. 00:52:45.670 --> 00:52:51.700 From those four numbers, 0 and 3 nontrivial numbers, you can 00:52:51.700 --> 00:52:54.940 find all of the component voltages. 00:52:54.940 --> 00:52:59.460 So for example, the voltage v6, the voltage across R6 is 00:52:59.460 --> 00:53:00.710 e2 minus e1. 00:53:03.420 --> 00:53:06.740 The voltage v4, the voltage across the R4 00:53:06.740 --> 00:53:10.470 resistor is e1 minus 0. 00:53:10.470 --> 00:53:15.690 So if I tell you all the node voltages, you can tell me all 00:53:15.690 --> 00:53:17.250 of the element voltages. 00:53:17.250 --> 00:53:20.530 And in general, there's fewer nodes than there are 00:53:20.530 --> 00:53:21.560 components. 00:53:21.560 --> 00:53:23.820 OK, that's great. 00:53:23.820 --> 00:53:26.990 So instead of naming the volts across the elements, we'll 00:53:26.990 --> 00:53:30.115 name the voltages at the nodes because there's fewer of them. 00:53:32.750 --> 00:53:36.620 Then, all we need to do in the node method is write the 00:53:36.620 --> 00:53:42.780 minimum number of KCL equations. 00:53:42.780 --> 00:53:46.060 We know we only have two unknowns, e1 and e2. 00:53:46.060 --> 00:53:47.970 And it turns out-- and you can prove this, but I 00:53:47.970 --> 00:53:50.990 won't prove it today. 00:53:50.990 --> 00:53:54.000 It turns out that you need two KCL equations. 00:53:54.000 --> 00:53:57.670 Two unknowns, e1, e2, two KCL equations. 00:53:57.670 --> 00:54:01.400 And it turns out those two KCL equations are exactly the KCL 00:54:01.400 --> 00:54:03.400 equations associated with the two nodes. 00:54:05.950 --> 00:54:10.990 So the current leaving e1, so KCL at e1 -- 00:54:10.990 --> 00:54:14.780 well, there's a current that goes that way. 00:54:14.780 --> 00:54:19.080 Well, that's the voltage drop in going from e1 to v0, e1 00:54:19.080 --> 00:54:21.380 minus v0, divided by R2. 00:54:21.380 --> 00:54:23.610 That's Ohm's law. 00:54:23.610 --> 00:54:28.600 So this term represents the current going up that leg plus 00:54:28.600 --> 00:54:30.930 the current that goes through this leg, which is e1 00:54:30.930 --> 00:54:33.780 minus e2 over R6. 00:54:33.780 --> 00:54:36.450 Plus the current going in that leg, which is e1 00:54:36.450 --> 00:54:39.250 minus 0 over R4. 00:54:39.250 --> 00:54:40.980 The sum of those three currents better be 0. 00:54:43.640 --> 00:54:46.460 Analogously, the sum of the currents at this node must be 00:54:46.460 --> 00:54:50.070 0, and the equation looks virtually the same. 00:54:50.070 --> 00:54:56.230 Because v0 is known, so it didn't add an unknown. 00:54:56.230 --> 00:55:00.250 v0 was set by the voltage, by the voltage source. 00:55:00.250 --> 00:55:04.000 So I have two equations, two unknowns, solved. 00:55:04.000 --> 00:55:04.630 Done. 00:55:04.630 --> 00:55:07.040 So rather than solving 12 equations and 12 unknowns, I 00:55:07.040 --> 00:55:09.300 can do it with two equations and two unknowns. 00:55:09.300 --> 00:55:11.640 That's called the node method. 00:55:11.640 --> 00:55:15.610 One of the most interesting theories about circuits is 00:55:15.610 --> 00:55:19.110 that every simplification that you can think about for 00:55:19.110 --> 00:55:22.250 voltage has an analogous simplification that you can 00:55:22.250 --> 00:55:23.050 think about in current. 00:55:23.050 --> 00:55:24.030 That's called duality. 00:55:24.030 --> 00:55:26.430 We won't do that because it's kind of complicated. 00:55:26.430 --> 00:55:27.790 But it's kind of a cute result. 00:55:27.790 --> 00:55:29.500 If you can think of a simplification that works in 00:55:29.500 --> 00:55:31.920 voltage, then there is an analogous one, and 00:55:31.920 --> 00:55:33.200 you can prove it. 00:55:33.200 --> 00:55:36.860 In fact, you can formally derive what it must have been. 00:55:40.090 --> 00:55:42.970 This is a rule for how you can simplify things by thinking 00:55:42.970 --> 00:55:45.115 about voltages in aggregate. 00:55:47.640 --> 00:55:50.480 Rather than thinking about the element voltages, think about 00:55:50.480 --> 00:55:51.890 the node voltages. 00:55:51.890 --> 00:55:56.510 The analogous current law is rather than thinking about the 00:55:56.510 --> 00:55:59.770 currents through the elements, the element currents, think 00:55:59.770 --> 00:56:01.950 about loop currents. 00:56:01.950 --> 00:56:04.110 OK, that's a little bizarre. 00:56:04.110 --> 00:56:07.430 So we name the loop, the current that flows in this 00:56:07.430 --> 00:56:12.590 loop, IA, the current that flows in this loop, IB, and 00:56:12.590 --> 00:56:14.780 the current hat flows in this loop, IC. 00:56:14.780 --> 00:56:16.280 What on earth is he doing? 00:56:16.280 --> 00:56:20.250 Well, the element voltages are some linear combination of 00:56:20.250 --> 00:56:24.070 those loop currents. 00:56:24.070 --> 00:56:26.460 And in fact, the coefficients in the linear combination are 00:56:26.460 --> 00:56:27.660 one and minus one. 00:56:27.660 --> 00:56:32.320 So the element current I4, the current that flows through the 00:56:32.320 --> 00:56:37.780 R4 resistor is the sum of IA coming down minus IC, 00:56:37.780 --> 00:56:39.670 which is going up. 00:56:39.670 --> 00:56:45.500 So there's a way of thinking about each element current as 00:56:45.500 --> 00:56:49.790 a sum or difference of the loop currents. 00:56:49.790 --> 00:56:52.180 Everybody get that? 00:56:52.180 --> 00:56:54.600 So instead of thinking about the individual element 00:56:54.600 --> 00:56:56.190 currents, I think about the loop currents. 00:56:56.190 --> 00:57:00.900 And now, I need to write three KVL equations. 00:57:00.900 --> 00:57:06.730 So in the node method, I named the nodes and had to write two 00:57:06.730 --> 00:57:08.480 KCL equations. 00:57:08.480 --> 00:57:11.205 Here, I named the loop currents and I have to write 00:57:11.205 --> 00:57:15.590 three KVL equations, one for each loop. 00:57:15.590 --> 00:57:16.760 It's completely analogous. 00:57:16.760 --> 00:57:19.420 If you write out a sentence, what did you do? 00:57:19.420 --> 00:57:21.760 I assigned a voltage to every node, and I wrote 00:57:21.760 --> 00:57:22.910 KCL of all the nodes. 00:57:22.910 --> 00:57:26.040 Then if you turn the word "current" into the word 00:57:26.040 --> 00:57:30.540 "voltage," the word "node" into the word "loop," you 00:57:30.540 --> 00:57:33.920 derive this new method. 00:57:33.920 --> 00:57:37.340 So this says that if I write KVL at the A loop, think about 00:57:37.340 --> 00:57:40.580 spinning around this loop, as I go up through the voltage 00:57:40.580 --> 00:57:43.910 source, so I go in the negative terminal here. 00:57:43.910 --> 00:57:46.090 So that's minus v0. 00:57:46.090 --> 00:57:48.940 As I go down through this resistor, I have to use Ohm's 00:57:48.940 --> 00:57:51.770 law, so that's R2 times the down current. 00:57:51.770 --> 00:57:57.610 Well, the down current is IA down minus IB up. 00:57:57.610 --> 00:57:59.380 So I went up through here, down through here. 00:57:59.380 --> 00:58:01.160 Now I go down through this one. 00:58:01.160 --> 00:58:04.580 When I go down through that one, according to Ohm's law, 00:58:04.580 --> 00:58:07.530 that's R4 times the current through that element. 00:58:07.530 --> 00:58:13.490 That current-- well, it's IA down and it's IC up. 00:58:13.490 --> 00:58:18.650 So this is the KVL equation for that loop. 00:58:18.650 --> 00:58:23.030 I write two more of them, and I end up with three equations 00:58:23.030 --> 00:58:25.590 and three unknowns. 00:58:25.590 --> 00:58:29.410 Both the node method and the loop method resulted in a lot 00:58:29.410 --> 00:58:33.330 fewer equations than the primitives did. 00:58:33.330 --> 00:58:38.400 I had 12 primitive unknowns, 6 voltages and 6 currents. 00:58:38.400 --> 00:58:42.670 In the node method, I get the number of independent nodes as 00:58:42.670 --> 00:58:44.910 the number of equations and unknowns, which is less than 00:58:44.910 --> 00:58:46.520 the number of primitive variables. 00:58:46.520 --> 00:58:49.280 In the loop method, I have the number of independent loops. 00:58:51.840 --> 00:58:55.100 Which is again, smaller. 00:58:55.100 --> 00:58:57.220 So the idea then is that we have a couple of ways to think 00:58:57.220 --> 00:59:00.360 about solving circuits. 00:59:00.360 --> 00:59:04.450 Fundamentally, all we have are the element relationships and 00:59:04.450 --> 00:59:05.846 the rules for combination. 00:59:05.846 --> 00:59:09.180 Oh, this is starting to sound like PCAP, primitives and 00:59:09.180 --> 00:59:10.380 combinations. 00:59:10.380 --> 00:59:14.860 So the primitives are, how does the element constrain the 00:59:14.860 --> 00:59:17.260 voltages and currents? 00:59:17.260 --> 00:59:19.470 We know three of those, Ohm's law, voltage 00:59:19.470 --> 00:59:21.480 source, current source. 00:59:21.480 --> 00:59:24.380 And what are the rules for combination? 00:59:24.380 --> 00:59:27.010 Well, the currents add to the node, and the voltages add 00:59:27.010 --> 00:59:30.070 around loops. 00:59:30.070 --> 00:59:33.720 OK, just to make sure you've absorb all that, figure out 00:59:33.720 --> 00:59:35.535 the current I for this circuit. 01:02:31.840 --> 01:02:33.810 OK, what's a good way to start? 01:02:33.810 --> 01:02:38.830 What should I do to start thinking about calculating I? 01:02:38.830 --> 01:02:40.190 OK, bad way. 01:02:40.190 --> 01:02:42.520 Assign voltages and currents to everything. 01:02:42.520 --> 01:02:46.150 4 elements, that's 4 voltages, 4 currents. 01:02:46.150 --> 01:02:47.620 That's 8 unknowns. 01:02:47.620 --> 01:02:48.990 Find 8 equations, solve. 01:02:48.990 --> 01:02:50.200 That'll work. 01:02:50.200 --> 01:02:50.980 Bad way. 01:02:50.980 --> 01:02:52.230 What's a better way? 01:02:56.146 --> 01:02:57.396 OK, [UNINTELLIGIBLE PHRASE]. 01:03:00.986 --> 01:03:01.954 AUDIENCE: [UNINTELLIGIBLE PHRASE]. 01:03:01.954 --> 01:03:03.890 PROFESSOR: It was on the previous sheet. 01:03:03.890 --> 01:03:06.820 [UNINTELLIGIBLE PHRASE]. 01:03:06.820 --> 01:03:08.020 AUDIENCE: KVL [UNINTELLIGIBLE]. 01:03:08.020 --> 01:03:09.660 PROFESSOR: KVL for where? 01:03:09.660 --> 01:03:11.616 AUDIENCE: Loops. 01:03:11.616 --> 01:03:12.594 PROFESSOR: Which loop? 01:03:12.594 --> 01:03:13.572 AUDIENCE: Left loop. 01:03:13.572 --> 01:03:16.506 PROFESSOR: So do KVL on the left loop? 01:03:16.506 --> 01:03:17.973 AUDIENCE: Yes. 01:03:17.973 --> 01:03:19.440 PROFESSOR: OK, that's good. 01:03:19.440 --> 01:03:22.090 But you have to tell me how to assign variables. 01:03:22.090 --> 01:03:24.273 Do you want 8 primitive variables? 01:03:28.540 --> 01:03:30.920 8 primitive variables are v1, i1, v2, i2, v3, i3, v4, i4. 01:03:36.130 --> 01:03:38.540 So that's what I mean by primitive variables. 01:03:38.540 --> 01:03:42.700 Or element variables is another word for it. 01:03:42.700 --> 01:03:44.555 What's a better way than using element variables? 01:03:47.552 --> 01:03:48.000 Yeah. 01:03:48.000 --> 01:03:49.908 AUDIENCE: Create 2 loop equations. 01:03:49.908 --> 01:03:52.902 PROFESSOR: Create 2 loop equations, that's fantastic. 01:03:52.902 --> 01:03:54.950 AUDIENCE: I1 for the first loop, I2 for the second loop. 01:03:54.950 --> 01:03:58.750 PROFESSOR: So if you do I1 going around here, then I1 is 01:03:58.750 --> 01:04:05.915 actually I. And if you do I2 going around here, what's I2? 01:04:10.673 --> 01:04:11.670 AUDIENCE: [INAUDIBLE]. 01:04:11.670 --> 01:04:16.490 PROFESSOR: So if I think about I2 spinning around this loop, 01:04:16.490 --> 01:04:20.920 so the sum of I1 and I2 goes through that box. 01:04:20.920 --> 01:04:23.200 But the only current that goes through this box is? 01:04:26.556 --> 01:04:27.806 AUDIENCE: [INAUDIBLE]. 01:04:29.993 --> 01:04:31.957 PROFESSOR: So the suggestion is that I think about-- 01:04:43.250 --> 01:04:49.790 so if I have I1 here, but I know that's I. Then I can see 01:04:49.790 --> 01:04:51.310 immediately that since the only current that 01:04:51.310 --> 01:04:52.380 goes through here-- 01:04:52.380 --> 01:04:54.100 so if I have I1 and I2. 01:04:54.100 --> 01:04:56.010 That was a very clever idea. 01:04:56.010 --> 01:04:58.090 If you have I1 and I2, the only current that goes through 01:04:58.090 --> 01:05:02.470 here is I. So I1 must've been I. 01:05:02.470 --> 01:05:04.640 The only current that goes over here 01:05:04.640 --> 01:05:07.390 must've been this guy. 01:05:07.390 --> 01:05:11.340 So this must be minus 10. 01:05:11.340 --> 01:05:14.130 So I could redo that this way. 01:05:14.130 --> 01:05:19.130 I could say I've got 10 going that way. 01:05:19.130 --> 01:05:21.410 That make sense? 01:05:21.410 --> 01:05:26.340 So now I only have one unknown which is I. So that's a very 01:05:26.340 --> 01:05:28.310 clever way of doing it. 01:05:28.310 --> 01:05:31.720 So what I could do is showed here. 01:05:31.720 --> 01:05:35.180 I have I going around one loop and I have 10 01:05:35.180 --> 01:05:37.950 going around that loop. 01:05:37.950 --> 01:05:39.950 That completely specifies all the currents. 01:05:39.950 --> 01:05:43.080 So now all I need to do is write KVL for 01:05:43.080 --> 01:05:45.980 these different cases. 01:05:45.980 --> 01:05:50.810 So if I write KVL for the left loop, then I get going up 01:05:50.810 --> 01:05:55.160 through here, that's minus 15, and going down through here, 01:05:55.160 --> 01:05:58.360 going to the right through this guy is 3I. 01:05:58.360 --> 01:06:03.090 Going down through this guy is 2 times I plus 10. 01:06:03.090 --> 01:06:05.450 Both of these are going down, so you have to add them. 01:06:10.010 --> 01:06:11.960 So I get one equation and one unknown. 01:06:11.960 --> 01:06:15.160 And when I solve it, I get minus one. 01:06:15.160 --> 01:06:16.410 That make sense? 01:06:18.470 --> 01:06:19.900 There's an analogous way you could have 01:06:19.900 --> 01:06:21.150 done it with one node. 01:06:24.080 --> 01:06:28.130 You could have said that the circuit has a single node and 01:06:28.130 --> 01:06:31.860 figured out KCL for that one node. 01:06:31.860 --> 01:06:34.170 KCL would be the sum of the currents here. 01:06:34.170 --> 01:06:35.770 There's a current that goes that way, that 01:06:35.770 --> 01:06:36.880 way, and that way. 01:06:36.880 --> 01:06:39.240 And again, you end up with 1 equation and 1 unknown. 01:06:39.240 --> 01:06:40.069 Yes? 01:06:40.069 --> 01:06:41.319 AUDIENCE: [UNINTELLIGIBLE PHRASE]. 01:06:45.558 --> 01:06:46.808 PROFESSOR: Correct. 01:06:49.051 --> 01:06:51.430 If I thought about this current going this way, it 01:06:51.430 --> 01:06:53.890 would be minus 10. 01:06:53.890 --> 01:06:59.050 If I flipped the direction, then it's plus 10. 01:06:59.050 --> 01:07:03.820 So the loop current has the property that it's the only 01:07:03.820 --> 01:07:05.290 current through this element. 01:07:05.290 --> 01:07:07.220 So that has to match. 01:07:07.220 --> 01:07:11.234 It's one of two currents that go through this element. 01:07:11.234 --> 01:07:13.186 AUDIENCE: You said that everything that 01:07:13.186 --> 01:07:15.626 [UNINTELLIGIBLE PHRASE]. 01:07:15.626 --> 01:07:17.578 PROFESSOR: This loop is [UNINTELLIGIBLE]. 01:07:17.578 --> 01:07:18.066 Yes. 01:07:18.066 --> 01:07:20.018 AUDIENCE: So why is the [UNINTELLIGIBLE PHRASE]. 01:07:23.434 --> 01:07:24.410 PROFESSOR: Correct. 01:07:24.410 --> 01:07:26.880 I want to have this picture now. 01:07:26.880 --> 01:07:29.570 So if I'm doing it with loops, I have two loops. 01:07:32.390 --> 01:07:35.100 The current through this element is just I. The current 01:07:35.100 --> 01:07:37.720 through this element is just I. The current through this 01:07:37.720 --> 01:07:40.020 element is just 10. 01:07:40.020 --> 01:07:42.040 The current through this element-- well, the sum of 01:07:42.040 --> 01:07:43.560 these two currents go through that element. 01:07:46.730 --> 01:07:49.512 Does that make sense? 01:07:49.512 --> 01:07:50.762 AUDIENCE: [INAUDIBLE] 01:07:55.428 --> 01:07:57.893 PROFESSOR: This loop current is just a fraction of the 01:07:57.893 --> 01:08:01.344 current in the whole system. 01:08:01.344 --> 01:08:05.550 So this loop current goes through this element and 01:08:05.550 --> 01:08:07.150 contributes to this element. 01:08:07.150 --> 01:08:08.400 But so does that one. 01:08:12.074 --> 01:08:15.150 OK, if you're still confused, you should try to get it 01:08:15.150 --> 01:08:17.170 straightened out in one of the software labs or the hardware 01:08:17.170 --> 01:08:18.670 lab, or talk to me after lecture. 01:08:18.670 --> 01:08:20.760 But the idea is to decompose in the 01:08:20.760 --> 01:08:22.720 case of the node voltages. 01:08:22.720 --> 01:08:25.380 Think about the element voltages in terms of 01:08:25.380 --> 01:08:27.000 differences in the node voltages. 01:08:30.580 --> 01:08:32.439 In the case of the loop currents, think about the 01:08:32.439 --> 01:08:36.100 element currents in terms of a sum of loop currents. 01:08:40.029 --> 01:08:43.600 OK, so the answer is minus 1 regardless of how you do it. 01:08:46.345 --> 01:08:46.472 Ok. 01:08:46.472 --> 01:08:49.439 The remaining thing I want to do today is think about 01:08:49.439 --> 01:08:52.140 abstraction. 01:08:52.140 --> 01:08:55.180 We've talked about the primitives, which are things 01:08:55.180 --> 01:08:59.750 like resistors, voltage sources, and current sources. 01:08:59.750 --> 01:09:03.100 Means of combinations, that's KVL and KCL. 01:09:03.100 --> 01:09:05.090 Now we want to think about abstraction. 01:09:05.090 --> 01:09:07.700 And the first abstraction that we'll talk about is, how do 01:09:07.700 --> 01:09:11.510 you think about one element that represents 01:09:11.510 --> 01:09:13.680 more than one element? 01:09:13.680 --> 01:09:15.740 This is the same thing that we did when we thought about 01:09:15.740 --> 01:09:18.770 linear systems, when we did signals and systems. 01:09:18.770 --> 01:09:23.270 We started with R's and K's and pluses, and we made single 01:09:23.270 --> 01:09:29.020 boxes that had lots of R's and pluses and gains in them. 01:09:29.020 --> 01:09:32.590 What was the name of the thing that was inside the box? 01:09:32.590 --> 01:09:36.270 If we combined lots of R's, gains, and pluses into a 01:09:36.270 --> 01:09:39.860 single box, what would we call the thing that's in the box? 01:09:39.860 --> 01:09:41.019 AUDIENCE: [INAUDIBLE]. 01:09:41.019 --> 01:09:41.789 PROFESSOR: Shout again. 01:09:41.789 --> 01:09:42.260 AUDIENCE: System function. 01:09:42.260 --> 01:09:44.930 PROFESSOR: System function. 01:09:44.930 --> 01:09:45.019 Right? 01:09:45.019 --> 01:09:47.609 So we started with boxes that only had 01:09:47.609 --> 01:09:49.580 things like R's in them. 01:09:49.580 --> 01:09:53.920 But eventually, we got boxes that looked like much more 01:09:53.920 --> 01:10:00.000 complicated things like that. 01:10:00.000 --> 01:10:02.170 We thought about a system function which was a 01:10:02.170 --> 01:10:06.580 generalized box, that could have lots of R's, or lots of 01:10:06.580 --> 01:10:08.890 gains, or lots of pluses in it. 01:10:08.890 --> 01:10:14.810 And that was a way of abstracting complicated 01:10:14.810 --> 01:10:17.140 systems so they looked like simple systems. 01:10:17.140 --> 01:10:18.900 What we want to do here is the same thing for circuits. 01:10:18.900 --> 01:10:21.530 We want to have a single element, a single circuit 01:10:21.530 --> 01:10:25.620 element, that represents many circuit elements. 01:10:25.620 --> 01:10:29.240 And the simplest case of that is for series in parallel 01:10:29.240 --> 01:10:31.050 combinations of resistors. 01:10:31.050 --> 01:10:34.890 It's very simple to think about how if you had two Ohm's 01:10:34.890 --> 01:10:39.220 law devices connected in series, you could replace 01:10:39.220 --> 01:10:42.490 those two with a single resistor. 01:10:42.490 --> 01:10:45.520 And the voltage-current relationships measured at the 01:10:45.520 --> 01:10:48.460 outside of the box would be the same. 01:10:51.140 --> 01:10:53.930 That's how we think about an abstraction in circuits. 01:10:53.930 --> 01:10:59.730 When is it that you can draw a box around a piece of a 01:10:59.730 --> 01:11:03.590 circuit and think about that as one element? 01:11:03.590 --> 01:11:06.350 The very simplest cases, the series combination of two 01:11:06.350 --> 01:11:09.940 resistors, same sort of thing happens for the parallel 01:11:09.940 --> 01:11:11.830 combination. 01:11:11.830 --> 01:11:15.870 And that simple abstraction makes some things very easy. 01:11:15.870 --> 01:11:19.370 What would be the equivalent resistance for a complicated 01:11:19.370 --> 01:11:21.350 system like that? 01:11:21.350 --> 01:11:24.070 Well, that's easy. 01:11:24.070 --> 01:11:29.440 All you need to do is think about successively reducing 01:11:29.440 --> 01:11:32.170 the pieces. 01:11:32.170 --> 01:11:35.050 Here I'm thinking about that having four resistors. 01:11:35.050 --> 01:11:39.020 I can just successively apply series and parallel in order 01:11:39.020 --> 01:11:41.760 to reduce that, make it less complicated. 01:11:41.760 --> 01:11:47.660 So I can think about combining these two in series to get, 01:11:47.660 --> 01:11:50.025 instead of two 1 Ohm resistors, one 2 Ohm resistor. 01:11:52.580 --> 01:11:56.640 Then I can think about these two 2 Ohm resistors being 01:11:56.640 --> 01:11:59.170 equivalently one parallel 1 Ohm resistor. 01:12:02.810 --> 01:12:06.170 And so this whole thing looks as though it's just 2 Ohms 01:12:06.170 --> 01:12:09.560 from the outside world. 01:12:09.560 --> 01:12:11.220 That's what we mean by an abstraction. 01:12:11.220 --> 01:12:15.430 What we're trying to do and what we will do over the next 01:12:15.430 --> 01:12:18.380 two weeks, is we'll think about ways of combining 01:12:18.380 --> 01:12:22.630 circuits so that we can reduce the complexity this way. 01:12:25.250 --> 01:12:28.070 Another convenient way of thinking about reducing the 01:12:28.070 --> 01:12:31.120 work that you need to do is to think about common patterns 01:12:31.120 --> 01:12:33.130 that result. 01:12:33.130 --> 01:12:37.740 PCAP, Primitives, Combinations, Abstractions. 01:12:37.740 --> 01:12:40.120 So the series of parallel idea was an abstraction. 01:12:40.120 --> 01:12:42.330 A pattern, here's a common pattern. 01:12:42.330 --> 01:12:48.930 If you've got two resistors in series, if the same current 01:12:48.930 --> 01:12:52.250 flows through two resistors, then there's a way of very 01:12:52.250 --> 01:12:56.170 simply calculating the voltage that falls across each. 01:12:56.170 --> 01:13:01.090 So you can think about the sum resistor, R1 plus R2 since 01:13:01.090 --> 01:13:03.290 they're in series. 01:13:03.290 --> 01:13:05.190 So that allows you then to compute the 01:13:05.190 --> 01:13:08.390 current from the voltage. 01:13:08.390 --> 01:13:11.240 Then the voltage that falls across this guy is by Ohm's 01:13:11.240 --> 01:13:16.000 law, just the current times its resistor, 01:13:16.000 --> 01:13:18.150 which is like that. 01:13:18.150 --> 01:13:19.970 And similarly with this one. 01:13:19.970 --> 01:13:24.560 So you can see that some fraction of this voltage v 01:13:24.560 --> 01:13:26.930 occurs across the v1 terminal. 01:13:26.930 --> 01:13:30.100 And some different fraction appears across the v2 01:13:30.100 --> 01:13:34.770 terminal, such that the sum of the fractions is, of course, 01:13:34.770 --> 01:13:38.280 v. That's what has to happen for the two. 01:13:38.280 --> 01:13:40.260 And there's a proportional drop. 01:13:40.260 --> 01:13:43.350 The bigger R1, the bigger is the proportion of the voltage 01:13:43.350 --> 01:13:45.860 that falls across R1. 01:13:45.860 --> 01:13:49.620 So it's a simple way of thinking about how voltage 01:13:49.620 --> 01:13:51.120 drops across two resistors. 01:13:51.120 --> 01:13:53.610 There's a completely analogous way of thinking about how 01:13:53.610 --> 01:13:58.690 current splits between two resistors. 01:13:58.690 --> 01:14:02.440 Here the result looks virtually the same, except it 01:14:02.440 --> 01:14:06.710 has kind of the unintuitive property that most of the 01:14:06.710 --> 01:14:11.710 current goes through the resistor that is the smallest. 01:14:11.710 --> 01:14:19.910 So you get a bigger current in i1 in proportion to the R2. 01:14:19.910 --> 01:14:23.040 So it works very much like the voltage case, except that it 01:14:23.040 --> 01:14:26.060 has this inversion in it, that the current likes to go 01:14:26.060 --> 01:14:29.870 through the smaller resistor. 01:14:29.870 --> 01:14:33.670 OK, so last problem. 01:14:33.670 --> 01:14:38.880 Using those kinds of ideas, think about how you could 01:14:38.880 --> 01:14:42.855 compute the voltage v0 and determine what's the answer. 01:16:30.480 --> 01:16:33.910 So what's the easy way to think about this answer? 01:16:33.910 --> 01:16:35.160 What do I do first? 01:16:38.912 --> 01:16:40.409 AUDIENCE: Superposition. 01:16:40.409 --> 01:16:41.407 PROFESSOR: So superposition? 01:16:41.407 --> 01:16:43.902 That's one thing. 01:16:43.902 --> 01:16:46.397 AUDIENCE: Simplify [UNINTELLIGIBLE]. 01:16:46.397 --> 01:16:47.395 PROFESSOR: Simplify. 01:16:47.395 --> 01:16:51.886 So what's a good simplification? 01:16:51.886 --> 01:16:52.884 Collapse? 01:16:52.884 --> 01:16:56.127 AUDIENCE: You can put the two [UNINTELLIGIBLE] in series and 01:16:56.127 --> 01:16:57.375 treat them as one. 01:16:57.375 --> 01:17:01.780 PROFESSOR: So you can treat this as a series combination, 01:17:01.780 --> 01:17:06.025 and you can replace the series of 1 and 3 with a? 01:17:06.025 --> 01:17:06.995 AUDIENCE: [UNINTELLIGIBLE PHRASE]. 01:17:06.995 --> 01:17:07.480 AUDIENCE: 4.. 01:17:07.480 --> 01:17:08.935 PROFESSOR: 4. 01:17:08.935 --> 01:17:12.330 So this can be replaced by four [UNINTELLIGIBLE] 01:17:12.330 --> 01:17:12.815 resistor. 01:17:12.815 --> 01:17:13.300 Now what? 01:17:13.300 --> 01:17:15.725 AUDIENCE: You can do the same on the parallel one. 01:17:15.725 --> 01:17:18.343 PROFESSOR: So you can replace the parallel of the 6 01:17:18.343 --> 01:17:19.506 and a 12 with a? 01:17:19.506 --> 01:17:20.342 AUDIENCE: 4. 01:17:20.342 --> 01:17:22.653 PROFESSOR: Amazing -- with a 4. 01:17:22.653 --> 01:17:25.599 So there's a 4 there and there's a 4 there. 01:17:25.599 --> 01:17:26.090 And the answer is? 01:17:26.090 --> 01:17:27.072 AUDIENCE: It's half. 01:17:27.072 --> 01:17:28.545 PROFESSOR: Half of whatever it was by voltage 01:17:28.545 --> 01:17:30.020 [UNINTELLIGIBLE] relationship. 01:17:30.020 --> 01:17:32.760 So you think about this becoming that. 01:17:32.760 --> 01:17:35.120 You think about the parallel becoming that. 01:17:35.120 --> 01:17:38.200 You get a simple divide by 2 voltage divider. 01:17:38.200 --> 01:17:46.660 So the answer is 7 and 1/2, which was the middle answer. 01:17:46.660 --> 01:17:51.780 And so what we did today was basically a whirlwind tour of 01:17:51.780 --> 01:17:54.790 the theory of circuits. 01:17:54.790 --> 01:17:58.060 And the goal for the rest of the week is to go to the lab 01:17:58.060 --> 01:18:00.580 and do the same sort of thing with practical where you build 01:18:00.580 --> 01:18:03.440 a circuit, and try to use some of these ideas to understand 01:18:03.440 --> 01:18:04.690 what it does.