WEBVTT 00:00:00.000 --> 00:00:01.000 Good morning, OK. 00:00:01.000 --> 00:00:05.000 Let's get started. We have one handout today. 00:00:05.000 --> 00:00:10.000 That's your lecture notes. There's some copies still 00:00:10.000 --> 00:00:14.000 outside for those who haven't picked one up. 00:00:14.000 --> 00:00:19.000 In general, what I do is, in the lecture notes, 00:00:19.000 --> 00:00:22.000 I leave out large amounts of material. 00:00:22.000 --> 00:00:28.000 So, this will enable you to keep your hands busy while I'm 00:00:28.000 --> 00:00:34.000 lecturing and take down some notes and so on. 00:00:34.000 --> 00:00:41.000 So, don't assume that everything that I talk about is 00:00:41.000 --> 00:00:44.000 on here. Please follow along. 00:00:44.000 --> 00:00:52.000 OK, so as is my usual practice, let me start with a quick 00:00:52.000 --> 00:00:58.000 review of what we covered so far. 00:00:58.000 --> 00:01:03.000 So what we did primarily was looked at this discipline that 00:01:03.000 --> 00:01:07.000 we call the lump matter discipline, which was very 00:01:07.000 --> 00:01:13.000 similar, very reminiscent of the point mass simplification in 00:01:13.000 --> 00:01:15.000 physics. And this discipline, 00:01:15.000 --> 00:01:20.000 this set of constraints we imposed on ourselves, 00:01:20.000 --> 00:01:25.000 allowed us to move from Maxwell's equations to a very, 00:01:25.000 --> 00:01:30.000 very simple form of algebraic equations. 00:01:30.000 --> 00:01:35.000 And specifically, the discipline took two forms. 00:01:35.000 --> 00:01:42.000 One is, we said that we will deal with elements for whom the 00:01:42.000 --> 00:01:50.000 rate of change of magnetic flux is zero outside of the elements, 00:01:50.000 --> 00:01:57.000 and for whom the rate of change of charge I want to charge 00:01:57.000 --> 00:02:03.000 inside the element was zero. So, if I took any element, 00:02:03.000 --> 00:02:07.000 any element that I called a lump circuit element, 00:02:07.000 --> 00:02:11.000 like a resistor or a voltage source, and I put a black box 00:02:11.000 --> 00:02:15.000 around it, then what I'm saying is that the net charge inside 00:02:15.000 --> 00:02:18.000 that is going to be zero. And this is not true in 00:02:18.000 --> 00:02:20.000 general. We will see examples where, 00:02:20.000 --> 00:02:24.000 if you choose some piece of an element for example, 00:02:24.000 --> 00:02:27.000 there might be charge buildup, but net inside the, 00:02:27.000 --> 00:02:30.000 if I put a box around the entire element, 00:02:30.000 --> 00:02:34.000 I am going to assume that the rate of change of charge is 00:02:34.000 --> 00:02:39.000 going to be zero. So, what this did was it 00:02:39.000 --> 00:02:44.000 enabled us to create the lump circuit abstraction, 00:02:44.000 --> 00:02:49.000 where I could take elements, some element of the sort, 00:02:49.000 --> 00:02:54.000 this could be a resistor, a voltage source, 00:02:54.000 --> 00:02:57.000 or whatever, and I could now ascribe a 00:02:57.000 --> 00:03:04.000 voltage, some voltage across an element, and also some current, 00:03:04.000 --> 00:03:09.000 "i," that was going into the element. 00:03:09.000 --> 00:03:12.000 And as I go forward, when I label the voltages and 00:03:12.000 --> 00:03:16.000 currents across and through elements, I'm going to be 00:03:16.000 --> 00:03:21.000 following a convention. OK, the convention is that I'm 00:03:21.000 --> 00:03:24.000 going to label, if I label V in the following 00:03:24.000 --> 00:03:28.000 manner, then I'm going to label "i" for that element as a 00:03:28.000 --> 00:03:33.000 current flowing into the positive terminal. 00:03:33.000 --> 00:03:36.000 It's just a convention. By doing this, 00:03:36.000 --> 00:03:40.000 it turns out that the power consumed by the element is "vi" 00:03:40.000 --> 00:03:44.000 is positive. OK, so by choosing I going in 00:03:44.000 --> 00:03:48.000 this way into the positive terminal, the power consumed by 00:03:48.000 --> 00:03:51.000 the element is going to be positive. 00:03:51.000 --> 00:03:55.000 OK, so in general of even simply following this 00:03:55.000 --> 00:03:59.000 convention, when I label voltages and currents, 00:03:59.000 --> 00:04:03.000 I'll be labeling the current into an element entering in 00:04:03.000 --> 00:04:08.000 through the plus terminal. Remember, of course, 00:04:08.000 --> 00:04:12.000 if the current is going this way, let's have one amp of 00:04:12.000 --> 00:04:15.000 current flowing this way, then when I compute the 00:04:15.000 --> 00:04:18.000 current, "i" will come out to be negative. 00:04:18.000 --> 00:04:22.000 OK, so by making these assumptions, the assumptions of 00:04:22.000 --> 00:04:26.000 the lumped matter discipline, I said I was able to simplify 00:04:26.000 --> 00:04:31.000 my life tremendously. And, in particular what it did 00:04:31.000 --> 00:04:36.000 was it allowed me to take Maxwell's equations, 00:04:36.000 --> 00:04:41.000 OK, and simplify them into a very simple algebraic form, 00:04:41.000 --> 00:04:46.000 which has both a voltage law and a current law that I call 00:04:46.000 --> 00:04:51.000 Kirchhoff's voltage law, and Kirchhoff's current law. 00:04:51.000 --> 00:04:55.000 KVL simply states that if I have some circuit, 00:04:55.000 --> 00:05:01.000 and if I measured the voltages in any loop in the circuit, 00:05:01.000 --> 00:05:07.000 so if I look at the voltages in any loop, then the voltages in 00:05:07.000 --> 00:05:13.000 the loop would sum to zero. OK, so I measure voltages in 00:05:13.000 --> 00:05:16.000 the loop, and they will sum to zero. 00:05:16.000 --> 00:05:21.000 Similarly, for the current, if I take a node of a circuit, 00:05:21.000 --> 00:05:25.000 if I build the circuit, a node is a point in the 00:05:25.000 --> 00:05:29.000 circuit where multiple edges connect. 00:05:29.000 --> 00:05:32.000 If I take a node, then the current coming into 00:05:32.000 --> 00:05:37.000 that node, the net current coming into a node is going to 00:05:37.000 --> 00:05:40.000 be zero. OK, so if I take any node of 00:05:40.000 --> 00:05:45.000 the circuit and sum up all the currents going into that node, 00:05:45.000 --> 00:05:51.000 they will all net sum to zero. So, notice what I've done is by 00:05:51.000 --> 00:05:55.000 this discipline, by this constraint I imposed on 00:05:55.000 --> 00:06:00.000 myself, I was able to make this incredible leap from Maxwell's 00:06:00.000 --> 00:06:04.000 equations to these really, really simple algebraic 00:06:04.000 --> 00:06:09.000 equations, KVL and KCL. And I promise you, 00:06:09.000 --> 00:06:13.000 going forward to the rest of 6.002, if this is all you know, 00:06:13.000 --> 00:06:18.000 you can pretty much solve any circuit using these two very 00:06:18.000 --> 00:06:20.000 simple relations. It's actually really, 00:06:20.000 --> 00:06:24.000 really simple. It's all very simple algebra, 00:06:24.000 --> 00:06:26.000 OK? So, just to show you an 00:06:26.000 --> 00:06:29.000 example, let me do a little demonstration. 00:06:29.000 --> 00:06:33.000 Let me build let me build a small circuit and measure some 00:06:33.000 --> 00:06:37.000 voltages for you, and show you that the voltages, 00:06:37.000 --> 00:06:42.000 indeed, add up to zero. So, here's my little circuit. 00:07:25.000 --> 00:07:29.000 So, I'm going to show you a simple circuit that looks like 00:07:29.000 --> 00:07:33.000 this, and let's go ahead and measure some voltages and 00:07:33.000 --> 00:07:35.000 currents. In terms of terminology to 00:07:35.000 --> 00:07:40.000 remember, this is called a loop. So if I start from the point C 00:07:40.000 --> 00:07:45.000 and I travel through the voltage source, come to the node A down 00:07:45.000 --> 00:07:49.000 through R1 and all the way down through R2 back to C, 00:07:49.000 --> 00:07:52.000 that's a loop. Similarly, this point A is a 00:07:52.000 --> 00:07:55.000 node where resistor R1 the voltage source V0, 00:07:55.000 --> 00:07:58.000 and R4 are connected. OK, just make sure your 00:07:58.000 --> 00:08:04.000 terminology is correct. So, what I'll do is I'll make 00:08:04.000 --> 00:08:10.000 some quick measurements for you, and show you that these KVL and 00:08:10.000 --> 00:08:14.000 KCL are indeed true. So, the circuits up there, 00:08:14.000 --> 00:08:18.000 could I have a volunteer? Any volunteer? 00:08:18.000 --> 00:08:22.000 All you have to do is write things on the board. 00:08:22.000 --> 00:08:26.000 Come on over. OK, so let me take some 00:08:26.000 --> 00:08:30.000 measurements, and why don't you write down 00:08:30.000 --> 00:08:35.000 what I measure on the board? What I'll do is, 00:08:35.000 --> 00:08:39.000 let me borrow another piece of chalk here. 00:08:39.000 --> 00:08:45.000 What I'll do is focus on this loop here, and focus on this 00:08:45.000 --> 00:08:49.000 node and make some measurements. 00:09:00.000 --> 00:09:05.000 All right, so you see the circuit up there. 00:09:05.000 --> 00:09:11.000 OK, so I get 3 volts for the voltage from C to A. 00:09:11.000 --> 00:09:16.000 so why don't you write down 3 volts? 00:09:30.000 --> 00:09:44.000 OK, so the next one is -1.6. And so that will be, 00:09:44.000 --> 00:09:50.000 I'm doing AB, V_AB. 00:09:50.000 --> 00:10:01.000 OK, and then let me do the last one. 00:10:01.000 --> 00:10:08.000 It is -1.37. The measurements, 00:10:08.000 --> 00:10:14.000 I guess, have been this way. So, what's written is V_AC. 00:10:14.000 --> 00:10:18.000 But it's OK for now. Don't worry about it. 00:10:18.000 --> 00:10:23.000 So, well, thank you. I appreciate your help here. 00:10:23.000 --> 00:10:27.000 OK, so within the bonds of experimental error, 00:10:27.000 --> 00:10:32.000 noticed that if I add up these three voltages, 00:10:32.000 --> 00:10:38.000 they nicely sum up to zero. OK, next let me focus on this 00:10:38.000 --> 00:10:41.000 node here. And at this node, 00:10:41.000 --> 00:10:44.000 let me go ahead and measure some currents. 00:10:44.000 --> 00:10:50.000 What I'll do now is change to an AC voltage so that I can go 00:10:50.000 --> 00:10:55.000 ahead and measure the current without breaking my circuit. 00:10:55.000 --> 00:10:59.000 OK, this time around, you'll get to see the 00:10:59.000 --> 00:11:04.000 measurements that I'm taking as well. 00:11:04.000 --> 00:11:09.000 So, what I have here, I guess you can see it this 00:11:09.000 --> 00:11:14.000 way. What I have here is three wires 00:11:14.000 --> 00:11:20.000 that I have pulled out from D. And this is the node D, 00:11:20.000 --> 00:11:24.000 OK? So, I have three wires coming 00:11:24.000 --> 00:11:32.000 into the node D just to make it a little bit easier for me to 00:11:32.000 --> 00:11:37.000 measure stuff. OK, so everybody keep your 00:11:37.000 --> 00:11:42.000 fingers crossed so I don't look like a fool here. 00:11:42.000 --> 00:11:46.000 I hope this works out. So, you roughly get, 00:11:46.000 --> 00:11:48.000 what's that, 10 mV. 00:11:48.000 --> 00:11:53.000 OK, so it's about 10 mV peak to peak out there, 00:11:53.000 --> 00:11:59.000 and let's say that if the waveform raises on the left-hand 00:11:59.000 --> 00:12:05.000 side, it's positive. So, it's positive 10 mV. 00:12:05.000 --> 00:12:08.000 And another positive 10 mV, so that's 20 mV. 00:12:08.000 --> 00:12:11.000 And this time, it's a negative, 00:12:11.000 --> 00:12:13.000 roughly 20, I guess, -20. 00:12:13.000 --> 00:12:17.000 So, I'm getting, in terms of currents, 00:12:17.000 --> 00:12:19.000 I have a -10, -10, I'm sorry, 00:12:19.000 --> 00:12:22.000 positive 10, positive 10, 00:12:22.000 --> 00:12:27.000 and a -20 that adds up to zero. But more interestingly, 00:12:27.000 --> 00:12:31.000 I can show you the same thing by holding this current 00:12:31.000 --> 00:12:37.000 measuring probe directly across the node. 00:12:37.000 --> 00:12:43.000 And, notice that the net current that is entering into 00:12:43.000 --> 00:12:49.000 this node here is zero. OK, so that should just show 00:12:49.000 --> 00:12:56.000 you that KCL does indeed hold in practice, and it is not just a 00:12:56.000 --> 00:13:02.000 figment of our imaginations. So, before I go on, 00:13:02.000 --> 00:13:05.000 I wanted to point one other thing out. 00:13:05.000 --> 00:13:09.000 Notice that I've written down two assumptions of the lumped 00:13:09.000 --> 00:13:11.000 matter discipline, OK? 00:13:11.000 --> 00:13:16.000 There is a total assumption of the lump matter discipline, 00:13:16.000 --> 00:13:19.000 and that assumption is, in spirit, at least, 00:13:19.000 --> 00:13:23.000 shared by the point mass simplification in physics as 00:13:23.000 --> 00:13:26.000 well. Can someone tell me what that 00:13:26.000 --> 00:13:28.000 assumption is? A total assumption, 00:13:28.000 --> 00:13:32.000 which I did not mention, which you can read in your 00:13:32.000 --> 00:13:36.000 notes in section 8.2 in the appendix, what's a total 00:13:36.000 --> 00:13:41.000 assumption that is shared in spirit with the point mass 00:13:41.000 --> 00:13:44.000 simplification? Anybody? 00:13:44.000 --> 00:13:49.000 A total assumption to be made here is that in all the signals 00:13:49.000 --> 00:13:53.000 that we will study in this course, we've made the 00:13:53.000 --> 00:13:57.000 assumption that the signal speeds of interest, 00:13:57.000 --> 00:14:01.000 transition speeds, and so on, are much slower than 00:14:01.000 --> 00:14:05.000 the speed of light. OK, that my signal transition 00:14:05.000 --> 00:14:11.000 speeds of interest are much slower than the speed of light. 00:14:11.000 --> 00:14:14.000 Remember, the laws of motion, the discrete laws of motion 00:14:14.000 --> 00:14:18.000 break down if your objects begin moving at the speed of light. 00:14:18.000 --> 00:14:22.000 OK, the same token here, our lump circuit abstraction 00:14:22.000 --> 00:14:24.000 breaks down if we approach the speed of light. 00:14:24.000 --> 00:14:28.000 And there are follow on courses that talk about waveguides and 00:14:28.000 --> 00:14:32.000 other distributed analysis techniques that deal with 00:14:32.000 --> 00:14:36.000 signals that travel close to speeds of light. 00:14:36.000 --> 00:14:41.000 OK, so with that, let me go on to talking about 00:14:41.000 --> 00:14:48.000 method one of circuit analysis. This is called the basic KVL 00:14:48.000 --> 00:14:53.000 KCL method. So just based on those two 00:14:53.000 --> 00:15:00.000 simple algebraic relations, I can analyze very interesting 00:15:00.000 --> 00:15:05.000 and complicated circuits. The method goes as follows. 00:15:05.000 --> 00:15:09.000 So, let's say our goal is, given a circuit like this, 00:15:09.000 --> 00:15:12.000 our goal is to solve it. OK, in this course, 00:15:12.000 --> 00:15:15.000 we will do two kinds of things: analysis and synthesis. 00:15:15.000 --> 00:15:17.000 Analysis says, given a circuit, 00:15:17.000 --> 00:15:20.000 OK, what can you tell me about the circuit? 00:15:20.000 --> 00:15:24.000 OK, so we'll solve existing circuits for all the voltages 00:15:24.000 --> 00:15:26.000 and currents, voltages across elements, 00:15:26.000 --> 00:15:30.000 and currents through those elements. 00:15:30.000 --> 00:15:32.000 Synthesis says, given a function, 00:15:32.000 --> 00:15:35.000 I may ask you to go and build circuits. 00:15:35.000 --> 00:15:40.000 OK, so for analysis here, we can apply this method that I 00:15:40.000 --> 00:15:44.000 want to show you. And the idea here is that, 00:15:44.000 --> 00:15:48.000 given a circuit like this, let us figure out all the 00:15:48.000 --> 00:15:53.000 voltages and currents that are a function of the way these 00:15:53.000 --> 00:15:57.000 elements are connected. So, the basic KVL and KCL 00:15:57.000 --> 00:16:02.000 method has the following steps. The first step is to write down 00:16:02.000 --> 00:16:08.000 the element VI relationships. OK, right down the element VI 00:16:08.000 --> 00:16:10.000 relationships for all the elements. 00:16:10.000 --> 00:16:14.000 The second step is write KCL for all the nodes, 00:16:14.000 --> 00:16:19.000 and the third step is to write KVL for all the loops in the 00:16:19.000 --> 00:16:20.000 circuit. That's it. 00:16:20.000 --> 00:16:23.000 Just go ahead and write down element rules, 00:16:23.000 --> 00:16:27.000 KVL, and KCL, and then go ahead and solve the 00:16:27.000 --> 00:16:29.000 circuit. So, what we'll do, 00:16:29.000 --> 00:16:33.000 we'll do an example, of course. 00:16:33.000 --> 00:16:39.000 But, just as a refresher, we've looked at a bunch of 00:16:39.000 --> 00:16:43.000 elements so far, and for the resistor, 00:16:43.000 --> 00:16:50.000 the element relation says that V is pi R, where R is the 00:16:50.000 --> 00:16:56.000 resistance of the element here. For a voltage source, 00:16:56.000 --> 00:17:01.000 V is equal to V nought. That's the element 00:17:01.000 --> 00:17:06.000 relationship. And for a current source, 00:17:06.000 --> 00:17:12.000 the element is the relation is, "i" is simply the current 00:17:12.000 --> 00:17:19.000 flowing through the element. OK, so these are some of the 00:17:19.000 --> 00:17:25.000 simple element rules for the devices that the current source, 00:17:25.000 --> 00:17:30.000 voltage source, and the resistor. 00:17:30.000 --> 00:17:34.000 So let's go ahead and solve this simple circuit. 00:17:34.000 --> 00:17:40.000 And what I'll do is go ahead and solve the circuit for you. 00:17:40.000 --> 00:17:46.000 OK, if you turn to page five of your notes, I'm going to go 00:17:46.000 --> 00:17:53.000 ahead and edit the circuit here. You can scribble the values on 00:17:53.000 --> 00:17:58.000 your notes on page five. OK, so as a first step of my 00:17:58.000 --> 00:18:03.000 KVL KCL method, I need to write down all my 00:18:03.000 --> 00:18:08.000 element VI relationships. So, before I do that, 00:18:08.000 --> 00:18:13.000 let me go ahead and label all the voltages and currents that 00:18:13.000 --> 00:18:17.000 are unknowns in the circuit. So, let me label the voltages 00:18:17.000 --> 00:18:22.000 and currents associated with the voltage source as here. 00:18:22.000 --> 00:18:26.000 Notice, I continue to follow this convention where whenever I 00:18:26.000 --> 00:18:31.000 label voltages and currents for an element, I will show the 00:18:31.000 --> 00:18:35.000 current going into the positive terminal of the element 00:18:35.000 --> 00:18:40.000 variable, OK, after element variable voltage. 00:18:40.000 --> 00:18:42.000 So here, I have V nought and I nought. 00:18:42.000 --> 00:18:47.000 Let me pause here for five seconds and show you a point of 00:18:47.000 --> 00:18:49.000 confusion that happens sometimes. 00:18:49.000 --> 00:18:53.000 Often times, people confuse between what is 00:18:53.000 --> 00:18:57.000 called the variable that is associated with the element 00:18:57.000 --> 00:19:01.000 versus the element value. OK, notice that here, 00:19:01.000 --> 00:19:07.000 capital V nought is the voltage that this voltage source 00:19:07.000 --> 00:19:12.000 provides, while this name here, v nought, is simply a variable 00:19:12.000 --> 00:19:17.000 that we've used to label the voltage across that element. 00:19:17.000 --> 00:19:21.000 So, similarly, I can label v1 as the voltage 00:19:21.000 --> 00:19:26.000 across the resistor, and i1 is the current flowing 00:19:26.000 --> 00:19:30.000 through the resistor. So this method of labeling, 00:19:30.000 --> 00:19:36.000 where I follow the convention, that the current flows into the 00:19:36.000 --> 00:19:44.000 positive terminal is called the associated variables discipline. 00:19:44.000 --> 00:19:48.000 I was trying to use the word discipline in situations where 00:19:48.000 --> 00:19:50.000 you have a choice, OK, and of a variety of 00:19:50.000 --> 00:19:54.000 possible choices, you pick one as the convention. 00:19:54.000 --> 00:19:56.000 OK, so here, as a convention, 00:19:56.000 --> 00:20:00.000 we use the associated variables discipline, and use that method 00:20:00.000 --> 00:20:04.000 to consistently label the unknown voltages and currents in 00:20:04.000 --> 00:20:09.000 our circuits. OK, so let me continue the 00:20:09.000 --> 00:20:13.000 labeling here, v4, i4, i3, v3 here, 00:20:13.000 --> 00:20:16.000 and v2 and i2, v5, and i5. 00:20:16.000 --> 00:20:22.000 I think that's it. So, I've gone ahead and labeled 00:20:22.000 --> 00:20:27.000 all my unknowns. So each of these voltages and 00:20:27.000 --> 00:20:35.000 currents are the voltages and currents associated with each of 00:20:35.000 --> 00:20:40.000 the elements. And my goal is to solve for 00:20:40.000 --> 00:20:44.000 these. OK, so in terms of our solution 00:20:44.000 --> 00:20:49.000 here, let's follow the method that I outlined for you. 00:20:49.000 --> 00:20:54.000 So, as the first step I am simply going to go ahead and 00:20:54.000 --> 00:21:00.000 write down all the element VI relationships. 00:21:00.000 --> 00:21:05.000 OK, so as a first step, I'm going to go ahead and write 00:21:05.000 --> 00:21:12.000 down all the VI relationships. So, can someone yell out for me 00:21:12.000 --> 00:21:16.000 the VI relationship for the voltage source? 00:21:16.000 --> 00:21:20.000 OK, good. So, v0 is capital V nought, 00:21:20.000 --> 00:21:25.000 that is that the variable V nought is simply equal to the 00:21:25.000 --> 00:21:29.000 voltage, v0. Similarly, I can write the 00:21:29.000 --> 00:21:32.000 others. v1 is i1, R1. 00:21:32.000 --> 00:21:36.000 v2 is i2, R2, and so on. 00:21:36.000 --> 00:21:39.000 OK, and I have one, two, three, four, 00:21:39.000 --> 00:21:44.000 five, six elements. So, I will get six such 00:21:44.000 --> 00:21:48.000 equations. Step two, I'm going to go ahead 00:21:48.000 --> 00:21:52.000 and write KCL for the nodes in my system. 00:21:52.000 --> 00:21:57.000 So, let me start with node A. So, for node A, 00:21:57.000 --> 00:22:05.000 let me take as positive the currents going out of the node. 00:22:05.000 --> 00:22:10.000 So, I get i nought flowing out, plus i1 flowing out, 00:22:10.000 --> 00:22:16.000 plus i4 flowing out, and they must sum to zero for 00:22:16.000 --> 00:22:20.000 node A. Then, I can go ahead and do the 00:22:20.000 --> 00:22:24.000 other nodes, let's say, for example, 00:22:24.000 --> 00:22:28.000 I do node B. For node B, I have i2 going 00:22:28.000 --> 00:22:31.000 out. That's positive, 00:22:31.000 --> 00:22:36.000 i3, and i1 is coming in, so I get -i1 equals zero. 00:22:36.000 --> 00:22:39.000 OK, so I have one, two, three, four, 00:22:39.000 --> 00:22:44.000 I have four nodes. OK, so I would get four 00:22:44.000 --> 00:22:47.000 equations. It turns out that the fourth 00:22:47.000 --> 00:22:53.000 equation is not independent. You can derive it from the 00:22:53.000 --> 00:22:56.000 others. So, I get three independent 00:22:56.000 --> 00:23:01.000 equations out of this. I can then write KVL. 00:23:01.000 --> 00:23:05.000 And for KVL, I just go down my loops here. 00:23:05.000 --> 00:23:10.000 And let me go through this first loop here in this manner. 00:23:10.000 --> 00:23:15.000 OK, and a simple trick that I use, you have to be incredibly 00:23:15.000 --> 00:23:20.000 careful when you go through this in keeping your minuses and 00:23:20.000 --> 00:23:23.000 pluses correct. Otherwise you can get 00:23:23.000 --> 00:23:26.000 hopelessly muddled. Once you label it, 00:23:26.000 --> 00:23:32.000 you need to be sure that you get all your minuses and pluses 00:23:32.000 --> 00:23:34.000 correct. So, for KVL, 00:23:34.000 --> 00:23:38.000 what I'd like to do is, let's say I start at C, 00:23:38.000 --> 00:23:40.000 and from C I'm going to go to A. 00:23:40.000 --> 00:23:44.000 For A I go to B, and from B I'm going to come 00:23:44.000 --> 00:23:46.000 back to C. OK, that's how I traverse my 00:23:46.000 --> 00:23:49.000 loop. And, the trick that I'm going 00:23:49.000 --> 00:23:52.000 to follow is, as my finger walks through that 00:23:52.000 --> 00:23:57.000 loop, I'm going to label the voltage as the first sign that I 00:23:57.000 --> 00:24:02.000 see for that voltage. OK, so I'm going to start with 00:24:02.000 --> 00:24:05.000 C, and I go up. I start by punching into the 00:24:05.000 --> 00:24:08.000 voltage source element, and then punch into it, 00:24:08.000 --> 00:24:10.000 I hit the minus sign for the V nought. 00:24:10.000 --> 00:24:14.000 OK, so I'm just going to write down minus V nought, 00:24:14.000 --> 00:24:18.000 plus then I go through and as I come up to A and go down to B, 00:24:18.000 --> 00:24:21.000 I punch to the plus sign of the V1. 00:24:21.000 --> 00:24:24.000 So, that's plus V1. And then I punch into the plus 00:24:24.000 --> 00:24:26.000 sign of the V2, and so I get plus V2, 00:24:26.000 --> 00:24:30.000 and that is zero. OK, good. 00:24:30.000 --> 00:24:33.000 So, that matches what you have in your notes as well. 00:24:33.000 --> 00:24:36.000 So, this is the first equation. Similarly, I can go through my 00:24:36.000 --> 00:24:40.000 other loops and write down equations for each of the loops. 00:24:40.000 --> 00:24:43.000 OK, and the convention that I like to follow is as I go 00:24:43.000 --> 00:24:45.000 through the loop, I write down as a sign for the 00:24:45.000 --> 00:24:49.000 voltage the first sign that I counter for that element. 00:24:49.000 --> 00:24:51.000 OK, you can do the exact opposite, if you want, 00:24:51.000 --> 00:24:54.000 just to be different. But, as long as you stay 00:24:54.000 --> 00:24:57.000 consistent, you'll be OK. All right, so in the same 00:24:57.000 --> 00:25:00.000 manner here, there are four loops that I can have, 00:25:00.000 --> 00:25:03.000 so four equations. Again, one of them turns out to 00:25:03.000 --> 00:25:08.000 be dependent on the others. So I end up getting three 00:25:08.000 --> 00:25:12.000 independent equations. So, I get a total of 12 00:25:12.000 --> 00:25:15.000 equations. I get 12 equations. 00:25:15.000 --> 00:25:20.000 There are six elements, OK, voltage source, 00:25:20.000 --> 00:25:24.000 and five resistors. So, there are six unknown 00:25:24.000 --> 00:25:27.000 voltages, and six unknown currents. 00:25:27.000 --> 00:25:33.000 So, I have 12 equations, and 12 unknowns. 00:25:33.000 --> 00:25:38.000 OK, I can take all of the equations and put them through a 00:25:38.000 --> 00:25:42.000 big crank, and sit there and grind. 00:25:42.000 --> 00:25:47.000 And if I was really cruel, I'd give this as a homework 00:25:47.000 --> 00:25:53.000 problem, and have you grind, and grind, and grind until you 00:25:53.000 --> 00:25:57.000 get your six voltages and six currents. 00:25:57.000 --> 00:26:01.000 OK, it works. OK, so you get 12 equations, 00:26:01.000 --> 00:26:07.000 and this method just works. However, notice that this is 00:26:07.000 --> 00:26:10.000 quite a grubby method. It's quite grungy. 00:26:10.000 --> 00:26:14.000 I get 12 equations, and it's quite a pain even for 00:26:14.000 --> 00:26:18.000 a simple circuit like this. However, suffice it to say that 00:26:18.000 --> 00:26:22.000 this fundamental method is one step away from Maxwell's 00:26:22.000 --> 00:26:25.000 equations, simply works. OK? 00:26:25.000 --> 00:26:28.000 So what you'll do is the rest of this lecture, 00:26:28.000 --> 00:26:33.000 I'll introduce you to a couple more methods. 00:26:33.000 --> 00:26:40.000 One is an intuitive method, and another one called the node 00:26:40.000 --> 00:26:46.000 method is a little bit more formal, but is much more, 00:26:46.000 --> 00:26:51.000 I guess, terse Than the KVL KCL method. 00:26:51.000 --> 00:26:56.000 Method 2. So the relevant section to read 00:26:56.000 --> 00:00:02.400 in the course notes is section 00:27:02.000 --> 00:27:06.000 One of the things that I will be stressing this semester is 00:27:06.000 --> 00:27:08.000 intuition. What you'll find is that as you 00:27:08.000 --> 00:27:11.000 become EECS majors, and so on, and go on, 00:27:11.000 --> 00:27:15.000 or if you talk to your TAs or your professors and so on, 00:27:15.000 --> 00:27:19.000 you will find that very rarely do they actually go ahead and 00:27:19.000 --> 00:27:21.000 apply the formal methods of analysis. 00:27:21.000 --> 00:27:25.000 OK, by and large, engineers are able to look at a 00:27:25.000 --> 00:27:28.000 circuit and simply by observation write down an 00:27:28.000 --> 00:27:31.000 answer. And usually in the past, 00:27:31.000 --> 00:27:35.000 what we have tried to do is kind of ignore that process and 00:27:35.000 --> 00:27:37.000 told our students, look, we teach you all the 00:27:37.000 --> 00:27:40.000 formal methods, and you will develop your own 00:27:40.000 --> 00:27:44.000 intuition and be able to do it. What we'll try to do this term 00:27:44.000 --> 00:27:47.000 is try to stress the intuitive methods, and try to show you how 00:27:47.000 --> 00:27:51.000 the intuitive process goes, so you can very quickly solve 00:27:51.000 --> 00:27:54.000 many of these circuits simply by inspection. 00:27:54.000 --> 00:27:57.000 OK, so this method that I'm going to show you here is one 00:27:57.000 --> 00:28:02.000 such an intuitive method. And I'll call it element 00:28:02.000 --> 00:28:08.000 combination tools. OK, for many simple circuits, 00:28:08.000 --> 00:28:14.000 you can solve them very quickly by applying this method. 00:28:14.000 --> 00:28:18.000 The components of this method are these. 00:28:18.000 --> 00:28:24.000 I learned about how to compose a bunch of elements. 00:28:24.000 --> 00:28:27.000 So, let's say, for example, 00:28:27.000 --> 00:28:32.000 I have a set of resistors, R1 through RN, 00:28:32.000 --> 00:28:38.000 in series. OK, you can use KVL and KCL to 00:28:38.000 --> 00:28:44.000 show that this is equivalent to a single resistor whose value is 00:28:44.000 --> 00:28:48.000 given by the sum of the resistances. 00:28:48.000 --> 00:28:54.000 OK, so if I have resistors in series, then effectively it's 00:28:54.000 --> 00:28:59.000 the same as if there was a single resistor whose value is 00:28:59.000 --> 00:29:07.000 the sum of all the resistances. OK, you can look at the course 00:29:07.000 --> 00:29:11.000 notes for a proof for derivation of this fact. 00:29:11.000 --> 00:29:16.000 Similarly, if I have resistances in parallel, 00:29:16.000 --> 00:29:20.000 so let me call them conductances. 00:29:20.000 --> 00:29:24.000 A conductance is the reciprocal of a resistance. 00:29:24.000 --> 00:29:31.000 If resistance is measured in ohms, conductance is measured in 00:29:31.000 --> 00:29:36.000 mhos, M-H-O-S. OK, so that's the conductance 00:29:36.000 --> 00:29:39.000 is G1, G2, and G3. And effectively, 00:29:39.000 --> 00:29:44.000 this is the same as having a single conductance whose 00:29:44.000 --> 00:29:49.000 effective value is given by the sum of the conductances. 00:29:49.000 --> 00:29:53.000 OK, the conductances in parallel add, 00:29:53.000 --> 00:30:00.000 and resistances in series add. Similarly, for voltage sources, 00:30:00.000 --> 00:30:06.000 if I have voltage sources in series, then they are tantamount 00:30:06.000 --> 00:30:10.000 to the sum of the voltages. And similarly, 00:30:10.000 --> 00:30:14.000 for currents, if I have currents in parallel, 00:30:14.000 --> 00:30:19.000 then they can be viewed as a single current source, 00:30:19.000 --> 00:30:24.000 whose currents are the sum of the individual parallel 00:30:24.000 --> 00:30:30.000 currents. So, let's do a quick example. 00:30:30.000 --> 00:30:36.000 So let's do this example. So, let's say I have a circuit 00:30:36.000 --> 00:30:41.000 that looks like this, and three resistances. 00:30:41.000 --> 00:30:47.000 And let's say all I care about is the current, 00:30:47.000 --> 00:30:50.000 I, that flows through this wire. 00:30:50.000 --> 00:30:54.000 All I care about is that current. 00:30:54.000 --> 00:31:02.000 Of course, you can go ahead and write KVL and KCL. 00:31:02.000 --> 00:31:06.000 You will get four equations, and there are four unknowns. 00:31:06.000 --> 00:31:09.000 And you can solve it. But, I can apply my element 00:31:09.000 --> 00:31:12.000 combination rules, and very quickly figure out 00:31:12.000 --> 00:31:16.000 what the current I is, using the following technique. 00:31:16.000 --> 00:31:19.000 So, what I can do is, I can, first of all, 00:31:19.000 --> 00:31:23.000 take this circuit. And, I can compose these two 00:31:23.000 --> 00:31:27.000 resistances and show that the circuit is equivalent as far as 00:31:27.000 --> 00:31:30.000 this current, I, is concerned to the 00:31:30.000 --> 00:31:33.000 following circuit, R1. 00:31:33.000 --> 00:31:39.000 And I take the sum of the two conductances, 00:31:39.000 --> 00:31:46.000 OK, and that comes out to be R1, R2, R3, R2 plus R3. 00:31:46.000 --> 00:31:54.000 And then, I can further simplify it, and I get a single 00:31:54.000 --> 00:32:01.000 resistance, whose value is given by R1 plus R2, 00:32:01.000 --> 00:32:06.000 R3, R3. OK, I'm just simplifying the 00:32:06.000 --> 00:32:08.000 circuit. Now, from this circuit, 00:32:08.000 --> 00:32:11.000 I can get the answer that I need. 00:32:11.000 --> 00:32:15.000 I is simply the voltage, V, divided by R1 plus. 00:32:15.000 --> 00:32:20.000 OK, so in situations like this where I'm looking for a single 00:32:20.000 --> 00:32:25.000 current, I can very quickly get to the answer by applying some 00:32:25.000 --> 00:32:28.000 of these element combination rules. 00:32:28.000 --> 00:32:34.000 And, I can get rid of having to go through formal steps. 00:32:34.000 --> 00:32:36.000 So, in general, whenever you encounter a 00:32:36.000 --> 00:32:41.000 circuit, by and large attempt to use intuitive methods to solve 00:32:41.000 --> 00:32:43.000 it. And go to a formal method only 00:32:43.000 --> 00:32:47.000 if some intuitive method fails. Even in your homework, 00:32:47.000 --> 00:32:51.000 by and large, the homeworks are not meant to 00:32:51.000 --> 00:32:54.000 be grungy. OK, if you find a lot of grunge 00:32:54.000 --> 00:32:57.000 in your homework, just remember you're probably 00:32:57.000 --> 00:33:04.000 not using some intuitive method. OK, so just be cautious. 00:33:04.000 --> 00:33:11.000 All right, so let me go on to the third method of circuit 00:33:11.000 --> 00:33:19.000 analysis, and the third method is called the node method. 00:33:19.000 --> 00:33:28.000 So, the node method is simply a specific application of the KVL 00:33:28.000 --> 00:33:36.000 KCL method and results in a much, much more compact form of 00:33:36.000 --> 00:33:42.000 the final equations. If there's one method that you 00:33:42.000 --> 00:33:46.000 have to remember for life, then I would say just remember 00:33:46.000 --> 00:33:48.000 this method. OK, the node method is a 00:33:48.000 --> 00:33:50.000 workhorse of the easiest industry. 00:33:50.000 --> 00:33:55.000 OK, if there's one method that you want to consistently apply, 00:33:55.000 --> 00:33:58.000 then this is the one to remember. 00:33:58.000 --> 00:34:01.000 So, let me quickly outline for you to method, 00:34:01.000 --> 00:34:04.000 and then work out an example for you. 00:34:04.000 --> 00:34:09.000 The first step of the node method will be to select a 00:34:09.000 --> 00:34:14.000 reference or a ground node. This is the symbol for a ground 00:34:14.000 --> 00:34:16.000 node. The ground node simply says 00:34:16.000 --> 00:34:21.000 that I'm going to denote voltages at that point to be 00:34:21.000 --> 00:34:26.000 zero, and measure all my other voltages with reference to that 00:34:26.000 --> 00:34:30.000 point. So, I'm going to select a 00:34:30.000 --> 00:34:36.000 ground node in my circuit. Second, I want to label the 00:34:36.000 --> 00:34:41.000 remaining voltages with respect to the ground node. 00:34:41.000 --> 00:34:48.000 So, label voltages for all the other nodes with respect to the 00:34:48.000 --> 00:34:52.000 ground node. Next, write KCL for each of the 00:34:52.000 --> 00:34:57.000 nodes write KCL. OK, but don't write KCL for the 00:34:57.000 --> 00:35:02.000 ground node. Remember, if you have N nodes, 00:35:02.000 --> 00:35:07.000 the node equations will give you N-1 independent equations. 00:35:07.000 --> 00:35:12.000 So, write KCL for the nodes, but don't do so for the ground 00:35:12.000 --> 00:35:15.000 node. Then, solve for the node 00:35:15.000 --> 00:35:18.000 voltages. So, let's say when we label 00:35:18.000 --> 00:35:21.000 voltages. I want to be labeling them as E 00:35:21.000 --> 00:35:26.000 something or the other. So, solve for the unknown node 00:35:26.000 --> 00:35:31.000 voltages. And then, once I know all the 00:35:31.000 --> 00:35:37.000 voltages associated with the nodes, I can then back solve for 00:35:37.000 --> 00:35:41.000 all the branch voltages and currents. 00:35:41.000 --> 00:35:47.000 OK, once I know all the node voltages, I can then go ahead 00:35:47.000 --> 00:35:52.000 and figure out all the branch voltages and the branch 00:35:52.000 --> 00:35:56.000 currents. So, let's go ahead and apply 00:35:56.000 --> 00:36:02.000 this method, and work out an example. 00:36:02.000 --> 00:36:05.000 Again, remember, if there's one method that you 00:36:05.000 --> 00:36:08.000 should remember, it's the node method. 00:36:08.000 --> 00:36:11.000 OK, and when in doubt, consistently apply the node 00:36:11.000 --> 00:36:15.000 method and it will work whether your circuit is linear or 00:36:15.000 --> 00:36:20.000 nonlinear, if the resistors are built in the US or the USSR it 00:36:20.000 --> 00:36:23.000 doesn't matter. OK, the node method will simply 00:36:23.000 --> 00:36:26.000 work, linear or nonlinear, OK? 00:36:26.000 --> 00:36:30.000 So, what I'm going to do is I'm going to build a circuit that's 00:36:30.000 --> 00:36:36.000 my old faithful. It's our old faithful, 00:36:36.000 --> 00:36:45.000 plus I'll make it a little bit more complicated by adding in 00:36:45.000 --> 00:36:52.000 the current source. So, let's go have some fun. 00:36:52.000 --> 00:36:59.000 Let's do this. So here's my voltage source, 00:36:59.000 --> 00:37:04.000 as before. OK, what I'll do is for fun, 00:37:04.000 --> 00:37:13.000 add a current source out there. And, you can convince 00:37:13.000 --> 00:37:20.000 yourselves that if you go ahead and apply the KVL KCL method, 00:37:20.000 --> 00:37:25.000 it'll really be a mess of equations. 00:37:25.000 --> 00:37:28.000 OK, so R1, R3, R4, R2, R5. 00:37:28.000 --> 00:37:36.000 OK, so let's follow our method and just plug and chug here. 00:37:36.000 --> 00:37:39.000 So let's apply the first step. I select a ground node. 00:37:39.000 --> 00:37:42.000 It's a reference node from which I'll measure all my other 00:37:42.000 --> 00:37:44.000 voltages. OK, now without knowing 00:37:44.000 --> 00:37:48.000 anything about the node method, try to use intuition as to 00:37:48.000 --> 00:37:51.000 which node you should choose as a ground node. 00:37:51.000 --> 00:37:55.000 Remember, you want to label the ground node with the voltage 00:37:55.000 --> 00:37:58.000 zero, and measure all the other voltages with respect to that 00:37:58.000 --> 00:38:02.000 node. OK, a usual trick is to pick a 00:38:02.000 --> 00:38:07.000 node which has the largest number of elements connected to 00:38:07.000 --> 00:38:11.000 it as the ground node. OK, and in particular, 00:38:11.000 --> 00:38:16.000 you will find out later it's useful to pick a node in which 00:38:16.000 --> 00:38:20.000 all your voltage sources, the maximum number of your 00:38:20.000 --> 00:38:23.000 voltage sources are also connected. 00:38:23.000 --> 00:38:27.000 OK, so in this instance, I'm going to choose this as my 00:38:27.000 --> 00:38:32.000 ground node. OK, that's my first step. 00:38:32.000 --> 00:38:38.000 I chose that as my ground node. And I'm going to label that as 00:38:38.000 --> 00:38:41.000 having a voltage zero. Second step, 00:38:41.000 --> 00:38:47.000 I'll label voltages of the other branches with respect to 00:38:47.000 --> 00:38:52.000 the ground node. OK, so what I'll do is add this 00:38:52.000 --> 00:38:55.000 node here. So I'm going to label that 00:38:55.000 --> 00:39:00.000 voltage E1. These are my unknowns. 00:39:00.000 --> 00:39:05.000 Remember, node method, because my node voltages are my 00:39:05.000 --> 00:39:09.000 unknowns. So, I label this as E1. 00:39:09.000 --> 00:39:14.000 I label this one as my unknown voltage, E2. 00:39:14.000 --> 00:39:19.000 What about this one here? Is that voltage unknown? 00:39:19.000 --> 00:39:23.000 No. I know what the voltage is 00:39:23.000 --> 00:39:28.000 because I know that this node is at a voltage, 00:39:28.000 --> 00:39:33.000 V0, higher than the ground node. 00:39:33.000 --> 00:39:38.000 OK, notice that to go from here to here, I directly go through a 00:39:38.000 --> 00:39:42.000 voltage source. And so, this node has voltage 00:39:42.000 --> 00:39:45.000 V0. And I'll simply write down V0. 00:39:45.000 --> 00:39:50.000 OK, try to simplify the number of steps that you have to go 00:39:50.000 --> 00:39:55.000 through, so directly go ahead and write down the voltage, 00:39:55.000 --> 00:39:58.000 V0, for that node. What I will also do, 00:39:58.000 --> 00:40:02.000 is for convenience, I'm going to write down, 00:40:02.000 --> 00:40:08.000 I'm going to use conductances. So I'm going to use GI in the 00:40:08.000 --> 00:40:12.000 place of one by RI, and write down a bunch of node 00:40:12.000 --> 00:40:14.000 equations. OK, so step one, 00:40:14.000 --> 00:40:19.000 I've chosen my ground node. Step two, I've labeled my node 00:40:19.000 --> 00:40:23.000 voltages, E, OK? I've done that with two of my 00:40:23.000 --> 00:40:27.000 steps. Now, let me go ahead and -- 00:40:41.000 --> 00:40:44.000 OK, so let me go ahead and apply step three. 00:40:44.000 --> 00:40:50.000 And, step three says go ahead and apply KCL for each of the 00:40:50.000 --> 00:40:54.000 nodes at which you have an unknown node voltage. 00:40:54.000 --> 00:40:59.000 And then that will give you your equations. 00:40:59.000 --> 00:41:02.000 So let me start by applying KCL at E1. 00:41:02.000 --> 00:41:06.000 So, let me write KCL at E1. I do one more thing. 00:41:06.000 --> 00:41:09.000 Notice, I don't have any currents there. 00:41:09.000 --> 00:41:14.000 OK, so how do I write KCL? KCL simply says the sum of 00:41:14.000 --> 00:41:18.000 currents into a node is zero again, remember, 00:41:18.000 --> 00:41:23.000 by the lump matter discipline. So, if I don't have currents in 00:41:23.000 --> 00:41:28.000 there, so the trick that I adopt is that to write KCL, 00:41:28.000 --> 00:41:34.000 I use the node voltages, and implicitly substitute for 00:41:34.000 --> 00:41:37.000 the node voltages, divide by the elemental the 00:41:37.000 --> 00:41:41.000 resistance, for instance, so I take the node voltages, 00:41:41.000 --> 00:41:44.000 and divide by the resistance, get the current. 00:41:44.000 --> 00:41:48.000 OK, so I implicitly apply element relationships to get the 00:41:48.000 --> 00:41:51.000 node currents. So, the example that make it 00:41:51.000 --> 00:41:55.000 clear, so I take node E1 and, again, for currents going out 00:41:55.000 --> 00:42:00.000 I'm going to assume to have, to be positive. 00:42:00.000 --> 00:42:05.000 So, the current going up is E1 minus V nought, 00:42:05.000 --> 00:42:09.000 divide by R1, so I multiplied by the G1. 00:42:09.000 --> 00:42:16.000 That's the current going up. Plus, the current going down is 00:42:16.000 --> 00:42:22.000 E1 minus zero where the ground node potential is zero, 00:42:22.000 --> 00:42:28.000 G2, OK, plus the current that is going to resistor R3, 00:42:28.000 --> 00:42:35.000 which is simply E1 minus E2, divide by R3. 00:42:35.000 --> 00:42:37.000 So, E1 minus E2, divide by R3, 00:42:37.000 --> 00:42:40.000 or multiplied by G3 is equal to zero. 00:42:40.000 --> 00:42:44.000 OK, see how I got this? This is simply KCL, 00:42:44.000 --> 00:42:49.000 but to get my currents, I simply take the differences 00:42:49.000 --> 00:42:54.000 of voltages across elements, and divide by the element of 00:42:54.000 --> 00:42:58.000 resistance, and I get the currents. 00:42:58.000 --> 00:43:01.000 OK, so I can similarly write KCL at E2. 00:43:01.000 --> 00:43:06.000 So, at KCL at E2, again, let me go outwards. 00:43:06.000 --> 00:43:14.000 So, the current going up is E2 minus V nought multiplied by G4. 00:43:14.000 --> 00:43:22.000 The current going left is E2 minus E1 divided by R3 or 00:43:22.000 --> 00:43:29.000 multiplied by G3. The current going down is E2 00:43:29.000 --> 00:43:37.000 minus zero multiplied by G5. And, the current going down is 00:43:37.000 --> 00:43:40.000 -I1. OK, you've got to be careful 00:43:40.000 --> 00:43:46.000 with your polarities here. So all the currents going out 00:43:46.000 --> 00:43:50.000 sum to zero. And here are the currents that 00:43:50.000 --> 00:43:56.000 are going out at this point. So what I do next is I can move 00:43:56.000 --> 00:44:01.000 the constant terms to the left-hand side and collect my 00:44:01.000 --> 00:44:07.000 unknowns. So, let me write them out here. 00:44:07.000 --> 00:44:14.000 So, let's say I get E1 here, OK, and from this equation, 00:44:14.000 --> 00:44:20.000 I have a V nought, G1, which comes out here. 00:44:20.000 --> 00:44:26.000 So, minus V nought G1 comes over to the other side. 00:44:26.000 --> 00:44:34.000 And, let me collect all the values that multiply E1. 00:44:34.000 --> 00:44:39.000 So I get, G1 is one example. I have G2, and I have G3. 00:44:39.000 --> 00:44:43.000 And then, for E2, I have minus G3. 00:44:43.000 --> 00:44:49.000 OK, so I'll simply express this as the element voltages 00:44:49.000 --> 00:44:55.000 multiplied by some terms in parentheses, and I put my 00:44:55.000 --> 00:44:59.000 external sources on the right hand side. 00:44:59.000 --> 00:45:06.000 Similarly, I go ahead and do the same thing here. 00:45:06.000 --> 00:45:10.000 In this instance, let me move my sources to the 00:45:10.000 --> 00:45:13.000 right. So, I get I1 coming out there, 00:45:13.000 --> 00:45:16.000 and I get V nought G4 coming out there. 00:45:16.000 --> 00:45:21.000 By the way, I just want to mention to you that if you're 00:45:21.000 --> 00:45:25.000 looking to fall asleep, this is a good time to do so 00:45:25.000 --> 00:45:30.000 because as soon as I write down these two equations, 00:45:30.000 --> 00:45:36.000 OK, from now on it's nap time. There's nothing new that you're 00:45:36.000 --> 00:45:40.000 going to learn from here on. It's just Anant Agarwal having 00:45:40.000 --> 00:45:43.000 fun at the blackboard, pushing symbols around. 00:45:43.000 --> 00:45:46.000 So, once you write down these two node equations, 00:45:46.000 --> 00:45:48.000 the rest of it is just grubby math. 00:45:48.000 --> 00:45:52.000 So, let me just have some fun. So let me just go ahead and do 00:45:52.000 --> 00:45:54.000 that. So, I moved my voltages and 00:45:54.000 --> 00:45:58.000 currents to the other side. And let me collect all the 00:45:58.000 --> 00:46:01.000 coefficients for E1 here. So, E1 minus G3, 00:46:01.000 --> 00:46:04.000 and that's it, I guess. 00:46:04.000 --> 00:46:08.000 OK, and then I'll do the same for E2. 00:46:08.000 --> 00:46:12.000 So, I get G4, and I get G3, 00:46:12.000 --> 00:46:18.000 and I get G5. OK, so notice here that I have 00:46:18.000 --> 00:46:22.000 two equations, and two unknowns. 00:46:22.000 --> 00:46:29.000 OK, the two equations are on the right hand side, 00:46:29.000 --> 00:46:35.000 I have some voltages and currents which are my dry 00:46:35.000 --> 00:46:43.000 voltages and dry currents. OK, so actually this is getting 00:46:43.000 --> 00:46:46.000 quite boring. I'm going to pause here, 00:46:46.000 --> 00:46:52.000 and talk about something else. So, you can take this and you 00:46:52.000 --> 00:46:56.000 can put it in matrix form, so I've done that for you on 00:46:56.000 --> 00:47:00.000 page ten. It's all matrix form. 00:47:00.000 --> 00:47:03.000 Yeah, I know that. You can use any technique to 00:47:03.000 --> 00:47:07.000 solve it, use algebraic techniques, use linear algebraic 00:47:07.000 --> 00:47:09.000 methods to solve it, use a computer, 00:47:09.000 --> 00:47:11.000 whatever you want. And, computers, 00:47:11.000 --> 00:47:15.000 when computers analyze circuits, they write down these 00:47:15.000 --> 00:47:18.000 equations, and deal with solving matrices. 00:47:18.000 --> 00:47:21.000 So, when you take the linear algebra across, 00:47:21.000 --> 00:47:25.000 how many people here have taken a linear algebra class? 00:47:25.000 --> 00:47:30.000 How many people here have heard of Gaussian elimination? 00:47:30.000 --> 00:47:34.000 How can more people have heard of Gaussian elimination than 00:47:34.000 --> 00:47:37.000 took a linear algebra class? Well anyway, 00:47:37.000 --> 00:47:42.000 so now you know why you took those linear algebra classes. 00:47:42.000 --> 00:47:47.000 And so, if I just collected these into matrix form -- 00:48:06.000 --> 00:48:09.000 OK, so I just simply expressed those two equations in linear 00:48:09.000 --> 00:48:12.000 algebraic form, and here's my column vector of 00:48:12.000 --> 00:48:15.000 unknowns, and you can apply any of the techniques you've learned 00:48:15.000 --> 00:48:17.000 in linear algebra to solve for this. 00:48:17.000 --> 00:48:20.000 Gaussian elimination works. OK, and in computer, 00:48:20.000 --> 00:48:23.000 people doing research in computer techniques, 00:48:23.000 --> 00:48:26.000 or solving such equations simply deals with huge equations 00:48:26.000 --> 00:48:28.000 like this, building computer programs that, 00:48:28.000 --> 00:48:33.000 given equations like this, can go ahead and solve them. 00:48:33.000 --> 00:48:36.000 OK, so let me stop here and reemphasize that what you've 00:48:36.000 --> 00:48:39.000 done is made a huge leap from Maxwell's equations to using the 00:48:39.000 --> 00:48:43.000 lump matter discipline to KVL and KCL, which ended up giving a 00:48:43.000 --> 00:48:46.000 simple algebraic equation to solve, and not having to worry 00:48:46.000 --> 00:48:49.000 about partial differential equations that were the form of 00:48:49.000 --> 00:48:52.000 Maxwell's equations.