WEBVTT 00:00:00.000 --> 00:00:05.000 Good morning. Today we move in the direction 00:00:05.000 --> 00:00:12.000 that takes a big turn from the direction we have been going in 00:00:12.000 --> 00:00:16.000 so far. All the devices we have had up 00:00:16.000 --> 00:00:24.000 until now, resistors and voltage sources, and even your digital 00:00:24.000 --> 00:00:31.000 devices like the AND gate or the inverter and so on had a very 00:00:31.000 --> 00:00:36.000 specific property. We didn't dwell on that 00:00:36.000 --> 00:00:41.000 property, but that property was that these were not what are 00:00:41.000 --> 00:00:43.000 called memory devices. In other words, 00:00:43.000 --> 00:00:48.000 the outputs at any given time are a function of the inputs 00:00:48.000 --> 00:00:49.000 alone. In other words, 00:00:49.000 --> 00:00:54.000 if you took your inverter or your NAND gate for that matter 00:00:54.000 --> 00:00:57.000 and you build a circuit comprising 50 NAND gates 00:00:57.000 --> 00:01:01.000 connected in structures that we have talked about, 00:01:01.000 --> 00:01:06.000 you apply an input and boom you get an output. 00:01:06.000 --> 00:01:09.000 And your output is a function of the inputs alone, 00:01:09.000 --> 00:01:11.000 right? The same thing with your 00:01:11.000 --> 00:01:16.000 resistors and voltage sources. At any given point in time your 00:01:16.000 --> 00:01:20.000 output VO of T was some function of the input VI of T. 00:01:20.000 --> 00:01:24.000 What we are going to do today is discuss a new element which 00:01:24.000 --> 00:01:29.000 will introduce a whole new class of fun stuff for all of us to 00:01:29.000 --> 00:01:33.000 deal with. And that is called storage. 00:01:33.000 --> 00:01:36.000 In other words, the output of a circuit is now 00:01:36.000 --> 00:01:41.000 going to depend not just on the inputs but it is going to depend 00:01:41.000 --> 00:01:46.000 on the background or it is going to depend on where the circuit 00:01:46.000 --> 00:01:49.000 has been in the past. So past is going to matter. 00:01:49.000 --> 00:01:52.000 It is a very fundamental difference. 00:01:52.000 --> 00:01:57.000 And what I would like to do is start by giving you folks a 00:01:57.000 --> 00:02:02.000 little bit of a surprise. I am going to do a little demo 00:02:02.000 --> 00:02:05.000 taking two of your inverter circuits. 00:02:18.000 --> 00:02:23.000 I am going to start by taking a couple of inverters. 00:02:23.000 --> 00:02:30.000 Remember, I am using this structure here as an inverter. 00:02:30.000 --> 00:02:37.000 And I am going to couple this to another inverter and take an 00:02:37.000 --> 00:02:43.000 output C, some VS, some load resistance RL, 00:02:43.000 --> 00:02:47.000 my B terminal and my A terminal. 00:02:47.000 --> 00:02:54.000 So I'm going to apply some input between ground and my A 00:02:54.000 --> 00:02:59.000 terminal. And for fun I want to apply a 00:02:59.000 --> 00:03:06.000 square wave at the input. A square wave between zero and 00:03:06.000 --> 00:03:09.000 5 volts. And this is how my time goes. 00:03:09.000 --> 00:03:11.000 Let's assume that VS is 5 volts. 00:03:11.000 --> 00:03:15.000 So what I am going to do is plot for you the behavior of 00:03:15.000 --> 00:03:19.000 this inverter. I am going to plot for you A, 00:03:19.000 --> 00:03:23.000 which would look like this. I am going to plot for you B, 00:03:23.000 --> 00:03:26.000 which would be the inverted wave form. 00:03:26.000 --> 00:03:30.000 And then plot C, which would be a wave form that 00:03:30.000 --> 00:03:36.000 looks like this again. Let me do a plot here. 00:03:36.000 --> 00:03:39.000 So this is A. 00:03:46.000 --> 00:03:49.000 -- and so on. Time goes this way. 00:03:49.000 --> 00:03:53.000 And let's say this is between zero and 5 volts. 00:03:53.000 --> 00:03:58.000 And B should be an inverted wave form that should look like 00:03:58.000 --> 00:04:00.000 this. 00:04:11.000 --> 00:04:15.000 If all that we believe of the world so far is true then this 00:04:15.000 --> 00:04:20.000 is how things should behave, so C should look like this. 00:04:30.000 --> 00:04:33.000 This is what the world should look like and if everything that 00:04:33.000 --> 00:04:36.000 you learned about is true and correct and all of the good 00:04:36.000 --> 00:04:38.000 stuff. Let me show you a little demo 00:04:38.000 --> 00:04:41.000 and see if I can try to pull the rug out from under all that you 00:04:41.000 --> 00:04:45.000 have learned so far and show you some surprising stuff. 00:04:51.000 --> 00:04:58.000 Here are the three wave forms that I showed you up here. 00:04:58.000 --> 00:05:02.000 This is my A. This is my A wave form. 00:05:02.000 --> 00:05:05.000 This is the B wave form. Notice that B, 00:05:05.000 --> 00:05:09.000 as you expect, is an inverted form of A. 00:05:09.000 --> 00:05:12.000 And this is C. We all expect this, 00:05:12.000 --> 00:05:15.000 correct? But what I am going to do is 00:05:15.000 --> 00:05:21.000 let me expand the time scale on this so that I can look at these 00:05:21.000 --> 00:05:25.000 transitions a little bit more carefully. 00:05:25.000 --> 00:05:30.000 I am just going to expand the time scale. 00:05:30.000 --> 00:05:33.000 There you go. All I have done is expanded the 00:05:33.000 --> 00:05:37.000 time scale and spread that out a little bit. 00:05:37.000 --> 00:05:42.000 And what you see there is quite different from what you expect. 00:05:42.000 --> 00:05:48.000 A is a square wave as expected, but B is stunningly different. 00:05:48.000 --> 00:05:51.000 It is a zero as expected because this is a one. 00:05:51.000 --> 00:05:55.000 But here I get some really strange behavior, 00:05:55.000 --> 00:06:00.000 behavior that is like nothing on earth. 00:06:00.000 --> 00:06:02.000 Like nothing you have seen before. 00:06:02.000 --> 00:06:05.000 And then, of course, it becomes a one eventually, 00:06:05.000 --> 00:06:09.000 but there's some really, really shady stuff going on 00:06:09.000 --> 00:06:11.000 here. And so far you are not prepared 00:06:11.000 --> 00:06:14.000 to deal with this. We have not given you the 00:06:14.000 --> 00:06:18.000 facility to deal with his issue. What is the problem with this? 00:06:18.000 --> 00:06:22.000 We could say who cares? What is the problem with this? 00:06:22.000 --> 00:06:25.000 Let's look at the result. I am looking at this, 00:06:25.000 --> 00:06:29.000 I am focusing on this piece here. 00:06:29.000 --> 00:06:34.000 And notice that instead of being a sharp rise it looks like 00:06:34.000 --> 00:06:36.000 this. It is going up a little bit 00:06:36.000 --> 00:06:40.000 more slowly. What kind of problem would that 00:06:40.000 --> 00:06:44.000 create? The problem that it creates is 00:06:44.000 --> 00:06:47.000 the following. Let me play around with this 00:06:47.000 --> 00:06:52.000 graph a little bit more. What I am going to do is just 00:06:52.000 --> 00:06:56.000 take this output here, the C output and line it up 00:06:56.000 --> 00:07:02.000 against the A output. And so I am going to line up 00:07:02.000 --> 00:07:05.000 the C wave form on top of the A wave form. 00:07:05.000 --> 00:07:09.000 So you can see for yourself if something really, 00:07:09.000 --> 00:07:13.000 really strange and nasty is happening, I am just going to 00:07:13.000 --> 00:07:17.000 move up the C wave form and line it up. 00:07:22.000 --> 00:07:28.000 What is happening out there? If you look carefully, 00:07:28.000 --> 00:07:33.000 what you observe is that the C wave form transitions just ever 00:07:33.000 --> 00:07:37.000 so slightly later than the A wave form. 00:07:37.000 --> 00:07:41.000 Look here. And I claim that it is because 00:07:41.000 --> 00:07:43.000 of this. Because of this, 00:07:43.000 --> 00:07:47.000 the C wave form falls just a little bit later, 00:07:47.000 --> 00:07:52.000 and that little thing we see out there is a delay. 00:07:52.000 --> 00:07:59.000 So nothing you have learned so far prepares you for this. 00:07:59.000 --> 00:08:03.000 Suddenly, instead of the output exactly following the input, 00:08:03.000 --> 00:08:07.000 my output is following the input but a little bit later. 00:08:07.000 --> 00:08:12.000 And it is this fact of life that things happen a little bit 00:08:12.000 --> 00:08:17.000 later, is really the reason why each of you and all of us needs 00:08:17.000 --> 00:08:20.000 to buy new computers every couple of years. 00:08:20.000 --> 00:08:24.000 This simple basic fact. If this fact of life didn't 00:08:24.000 --> 00:08:28.000 exist, you would buy one computer and be done with it for 00:08:28.000 --> 00:08:31.000 life. Intel would make gobs of money 00:08:31.000 --> 00:08:35.000 one year, and so would Dell and Gateway and so on, 00:08:35.000 --> 00:08:36.000 and then no more. That's it. 00:08:36.000 --> 00:08:39.000 This is it. But because of this a little 00:08:39.000 --> 00:08:43.000 itty-bitty difference here the entire semiconductor technology 00:08:43.000 --> 00:08:46.000 is charging along trying to do something about that. 00:08:46.000 --> 00:08:49.000 You buy newer and newer computers each year. 00:08:49.000 --> 00:08:52.000 It turns out this little itty-bitty thing here, 00:08:52.000 --> 00:08:54.000 that is called the delay, the inverter delay. 00:08:54.000 --> 00:08:58.000 And it happens because of a specific element that has been 00:08:58.000 --> 00:09:03.000 introduced here that we have not shown you so far. 00:09:03.000 --> 00:09:06.000 And a large part of the semiconductor industry and 00:09:06.000 --> 00:09:11.000 follow-on courses and design and so on focuses on how could I 00:09:11.000 --> 00:09:15.000 make my delay smaller, how can I get to be faster and 00:09:15.000 --> 00:09:18.000 faster and faster? This relates to how fast we can 00:09:18.000 --> 00:09:22.000 clock your Pentium IV. Remember it came all the way to 00:09:22.000 --> 00:09:26.000 1.3 gigahertz? What's the fasted Pentium money 00:09:26.000 --> 00:09:30.000 can buy today? What is the fastest P4? 00:09:30.000 --> 00:09:32.000 Oh, 3.2 have come out? I don't know. 00:09:32.000 --> 00:09:35.000 Ken claims 3.2. But, yeah, there you go, 00:09:35.000 --> 00:09:38.000 3.2 gigahertz. It all has to do with this 00:09:38.000 --> 00:09:42.000 little itty-bitty thing. You saw it for the first time 00:09:42.000 --> 00:09:45.000 here. When some of you become CTOs at 00:09:45.000 --> 00:09:49.000 Intel and so on, just remember that it all began 00:09:49.000 --> 00:09:53.000 on October 16th with this little rinky-dink thing here. 00:09:53.000 --> 00:09:58.000 What you are going to learn now is some really cool stuff that 00:09:58.000 --> 00:10:04.000 has huge implications for life. So why does that happen? 00:10:04.000 --> 00:10:09.000 Why did this transition happen just a little bit later? 00:10:09.000 --> 00:10:15.000 The reason is that remember when this wave form reaches VT, 00:10:15.000 --> 00:10:21.000 the threshold voltage of this MOSFET, this guy is going to 00:10:21.000 --> 00:10:25.000 switch, right? So because of the slower rise 00:10:25.000 --> 00:10:30.000 of the voltage, the VT is going to be reached a 00:10:30.000 --> 00:10:35.000 small amount of time later. So I am going to hit VT 00:10:35.000 --> 00:10:39.000 slightly later. And because of that this guy is 00:10:39.000 --> 00:10:43.000 going to transition just a bit later because this intermediate 00:10:43.000 --> 00:10:47.000 wave form B is slower. It hits VT just a little bit 00:10:47.000 --> 00:10:51.000 later than if it would have made an instantaneous transition. 00:10:51.000 --> 00:10:56.000 And therefore my output falls just a little bit later and this 00:10:56.000 --> 00:11:00.000 gives rise to my delay in the inverter. 00:11:00.000 --> 00:11:06.000 We can call that d if you would like, some delay. 00:11:06.000 --> 00:11:13.000 In your course notes, this material is covered in 00:11:13.000 --> 00:11:20.000 Chapters 9 and 10. That was to kind of motivate 00:11:20.000 --> 00:11:30.000 why we are going to be doing all that you we will be doing. 00:11:30.000 --> 00:11:35.000 Don't anybody come within a foot of this even by mistake. 00:11:35.000 --> 00:11:39.000 I mean it. It is pretty deadly stuff. 00:11:39.000 --> 00:11:43.000 Today we will talk about the capacitor. 00:11:43.000 --> 00:11:49.000 And in the next couple of lectures I am going to tie it 00:11:49.000 --> 00:11:55.000 all together and show you how this relates to that. 00:11:55.000 --> 00:12:01.000 I will show you exactly how the delay happens. 00:12:01.000 --> 00:12:04.000 You can compute it based on some simple principles that you 00:12:04.000 --> 00:12:07.000 will learn about in the next couple of lectures. 00:12:07.000 --> 00:12:11.000 What I am going to do is first of all show you, 00:12:11.000 --> 00:12:14.000 I claim that that delay happens because of the presence of a 00:12:14.000 --> 00:12:18.000 capacitor somewhere in there. What I will do now is take you 00:12:18.000 --> 00:12:21.000 into a closer look, take a closer look at the 00:12:21.000 --> 00:12:24.000 MOSFET and show you were the capacitor is. 00:12:24.000 --> 00:12:27.000 This is the MOSFET that you have seen so far, 00:12:27.000 --> 00:12:34.000 drain, gate and source. This is called an n-channel 00:12:34.000 --> 00:12:39.000 MOSFET. And what I am going to do is 00:12:39.000 --> 00:12:47.000 dissect this and show you what is actually happening, 00:12:47.000 --> 00:12:51.000 what this looks like on silicon. 00:12:51.000 --> 00:13:00.000 So here is my slab of silicon. It is very thin. 00:13:00.000 --> 00:13:03.000 And let's say this is, I won't go into details here. 00:13:03.000 --> 00:13:08.000 You will learn a lot more about this in future device classes 00:13:08.000 --> 00:13:12.000 like 301 and so on, but suffice it to say I will 00:13:12.000 --> 00:13:16.000 just introduce it here to give you a sense of where the 00:13:16.000 --> 00:13:19.000 capacitor is. This is p-type silicon. 00:13:19.000 --> 00:13:24.000 And the way you build a MOSFET is you create a couple of tubs 00:13:24.000 --> 00:13:28.000 in which you dope to be n-type. The basic silicon is dope 00:13:28.000 --> 00:13:33.000 p-type. And this guy here is n-type. 00:13:33.000 --> 00:13:39.000 And what you do is a thin oxide layer is placed on top of that 00:13:39.000 --> 00:13:44.000 and then on top of that a thin metal layer. 00:13:44.000 --> 00:13:50.000 This is a metal layer. This is a thin piece of oxide, 00:13:50.000 --> 00:13:56.000 silicon dioxide. And this is my P substrate. 00:13:56.000 --> 00:14:00.000 Now this is a little metal layer that is really a wire on 00:14:00.000 --> 00:14:03.000 top of the silicone. This metal layer could be some 00:14:03.000 --> 00:14:07.000 sort of a wire that meanders around on the surface of 00:14:07.000 --> 00:14:10.000 silicone. And this is a wire that 00:14:10.000 --> 00:14:13.000 connects to the gate. This is the gate of my MOSFET. 00:14:13.000 --> 00:14:17.000 And this guy here is the drain. And this guy here is the 00:14:17.000 --> 00:14:19.000 source. And this is my gate. 00:14:19.000 --> 00:14:22.000 So there is a little piece of metal here. 00:14:22.000 --> 00:14:25.000 This is this piece of metal here. 00:14:25.000 --> 00:14:31.000 And there is a piece of oxide and then my silicone substrate. 00:14:31.000 --> 00:14:37.000 Notice that this is my oxide. When I apply a positive voltage 00:14:37.000 --> 00:14:41.000 to the gate here with respect to the substrate, 00:14:41.000 --> 00:14:46.000 what happens is that I draw up negative charges. 00:14:46.000 --> 00:14:52.000 I draw up electrons here into this channel region and I have 00:14:52.000 --> 00:14:58.000 corresponding plus type out here so that I get a view here that 00:14:58.000 --> 00:15:05.000 looks like a couple of plates. And I end up with an oxide in 00:15:05.000 --> 00:15:08.000 the middle. There is no connection. 00:15:08.000 --> 00:15:14.000 Two plates separated by a small distance with plus q and minus q 00:15:14.000 --> 00:15:17.000 on the plates. And, because of that, 00:15:17.000 --> 00:15:22.000 what ends up happening here is that this piece behaves like a 00:15:22.000 --> 00:15:26.000 capacitor. So a capacitor has two plates 00:15:26.000 --> 00:15:30.000 with a thin insulating material in the middle with some 00:15:30.000 --> 00:15:35.000 permittivity epsilon. And so I get a little piece of 00:15:35.000 --> 00:15:38.000 a capacitor here. That is the capacitor that is 00:15:38.000 --> 00:15:40.000 forming. I did not set out to build that 00:15:40.000 --> 00:15:43.000 capacitor, but there is a capacitor nonetheless. 00:15:43.000 --> 00:15:46.000 So when I apply a positive voltage at the gate, 00:15:46.000 --> 00:15:49.000 negative electrons are pulled up here which forms a channel, 00:15:49.000 --> 00:15:51.000 and then a current can then flow. 00:15:51.000 --> 00:15:53.000 And that is how the MOSFET turns on. 00:15:53.000 --> 00:15:57.000 So n-type electrons back to n-type, and I get electron flow 00:15:57.000 --> 00:16:00.000 here and that gives me my channel. 00:16:00.000 --> 00:16:04.000 This is just kind of devices in four minutes or less. 00:16:04.000 --> 00:16:08.000 You will do an entire course on this, if you like, 00:16:08.000 --> 00:16:11.000 if you take 301. What we do is to be able to 00:16:11.000 --> 00:16:15.000 capture the behavior that we just saw, the funny delayed 00:16:15.000 --> 00:16:18.000 behavior, we have to augment our model. 00:16:18.000 --> 00:16:21.000 We have to introduce a new element. 00:16:21.000 --> 00:16:24.000 So what we do is here is a MOSFET, gate, 00:16:24.000 --> 00:16:28.000 drain and source. And notice here we model this 00:16:28.000 --> 00:16:33.000 by putting a little capacitor, CGS between our gate and the 00:16:33.000 --> 00:16:37.000 source. So this becomes a simple model 00:16:37.000 --> 00:16:43.000 for our MOSFET device which is the good old gate drain source 00:16:43.000 --> 00:16:48.000 device from the past with a little capacitor CGS having some 00:16:48.000 --> 00:16:53.000 value for CGS in maybe ten to the minus 14 or thereabouts 00:16:53.000 --> 00:16:56.000 farads. So that is a little capacitor 00:16:56.000 --> 00:17:03.000 that has come about in this device that we fabricated here. 00:17:03.000 --> 00:17:07.000 It is that capacitor that is at between node B and ground 00:17:07.000 --> 00:17:12.000 because it is between the gate and the source of the second 00:17:12.000 --> 00:17:15.000 inverter. And it is that capacitor that 00:17:15.000 --> 00:17:20.000 is playing the games that we saw out there. 00:17:27.000 --> 00:17:31.000 So let's look at some of the behavior of an ideal linear 00:17:31.000 --> 00:17:34.000 capacitor. A capacitor, 00:17:34.000 --> 00:17:38.000 as I said, has a couple of plates. 00:17:38.000 --> 00:17:44.000 There are a couple of plates. Between the plates is some 00:17:44.000 --> 00:17:51.000 dieletric, permittivity epsilon. Let's say the area of the 00:17:51.000 --> 00:17:57.000 plates is A, and let's say the plates are separated by a 00:17:57.000 --> 00:18:02.000 distance D. I get some charge here, 00:18:02.000 --> 00:18:06.000 let's say q. So q and minus q on the 00:18:06.000 --> 00:18:10.000 capacitor. And the capacitance C is given 00:18:10.000 --> 00:18:15.000 by epsilon A divided by D. Epsilon, as I said, 00:18:15.000 --> 00:18:19.000 is the productivity of the dielectric. 00:18:19.000 --> 00:18:25.000 So if it is free space then it would be epsilon zero which is 00:18:25.000 --> 00:18:32.000 the permittivity of free space. That is the capacitance in 00:18:32.000 --> 00:18:35.000 farads. And the symbol looks like this. 00:18:35.000 --> 00:18:38.000 Capacitor C. Voltage v. 00:18:38.000 --> 00:18:40.000 Current i. So this, much like the 00:18:40.000 --> 00:18:46.000 resistor, voltage source and so on, this now becomes a primitive 00:18:46.000 --> 00:18:52.000 element in your tool chest of elements like the voltage source 00:18:52.000 --> 00:18:56.000 and so onn. Capacitance with the voltage v 00:18:56.000 --> 00:19:01.000 across it and a current i. And I have assigned the 00:19:01.000 --> 00:19:05.000 associated variables here according to the associated 00:19:05.000 --> 00:19:08.000 variable discipline. A question to ask ourselves is 00:19:08.000 --> 00:19:13.000 remember we said we are all now in a playground from all of 00:19:13.000 --> 00:19:17.000 nature, in this playground where the lumped matter discipline 00:19:17.000 --> 00:19:20.000 holds? And also remember that we said 00:19:20.000 --> 00:19:23.000 that for the lumped matter discipline to hold we have to 00:19:23.000 --> 00:19:29.000 make a couple of assumptions. One of those assumptions was 00:19:29.000 --> 00:19:33.000 that dq/dt, for all their elements should be zero for all 00:19:33.000 --> 00:19:36.000 time. So right now what about the 00:19:36.000 --> 00:19:39.000 capacitor? It has got some charge q. 00:19:39.000 --> 00:19:42.000 So charge must have built up somehow. 00:19:42.000 --> 00:19:47.000 Does that mean that I lied all along, that we are no longer in 00:19:47.000 --> 00:19:51.000 this playground, that we have been ejected from 00:19:51.000 --> 00:19:56.000 the playground because of the capacitor, or are we still in 00:19:56.000 --> 00:20:01.000 the circuits playground in which the lumped matter discipline 00:20:01.000 --> 00:20:06.000 holds and all good things happen and so on? 00:20:06.000 --> 00:20:09.000 It seems like a contradiction, doesn't it? 00:20:09.000 --> 00:20:12.000 I took you from Maxwell's playgrounds to the EECS 00:20:12.000 --> 00:20:17.000 playground where I said the lumped matter discipline holds. 00:20:17.000 --> 00:20:21.000 And one of the foundations of the LMD was that dq/dt should be 00:20:21.000 --> 00:20:26.000 zero for all time inside the elements that we are going to 00:20:26.000 --> 00:20:28.000 deal with. And right now boom, 00:20:28.000 --> 00:20:32.000 it's not four weeks into the course and Agarwal introduces an 00:20:32.000 --> 00:20:38.000 element and it has q in it. It turns out that the capacitor 00:20:38.000 --> 00:20:41.000 also adheres to the lumped matter discipline. 00:20:41.000 --> 00:20:45.000 Remember the discipline says that dq/dt is zero for all time 00:20:45.000 --> 00:20:48.000 within elements. So I am going to be clever. 00:20:48.000 --> 00:20:52.000 What I am going to do is I want to choose element boundaries in 00:20:52.000 --> 00:20:55.000 a very cleaver way. Notice that if I have q here on 00:20:55.000 --> 00:21:00.000 this plate then I get minus q on the other plate. 00:21:00.000 --> 00:21:04.000 So if I take the whole element, the element as a whole, 00:21:04.000 --> 00:21:09.000 if I am careful in terms of how I package my boundaries, 00:21:09.000 --> 00:21:14.000 if I put both my plates inside my element boundary then I still 00:21:14.000 --> 00:21:17.000 do get the net charge being zero. 00:21:17.000 --> 00:21:22.000 So dq/dt is indeed zero for all time provided I make sure that 00:21:22.000 --> 00:21:27.000 my element has both the plates. Therefore, if you come across 00:21:27.000 --> 00:21:32.000 somebody else that gives you an element that says I have an 00:21:32.000 --> 00:21:36.000 idea. Let's create a new branch of 00:21:36.000 --> 00:21:40.000 electrical engineering in which we model the capacitor not as 00:21:40.000 --> 00:21:44.000 one element for two plates, but let's build a capacitor by 00:21:44.000 --> 00:21:48.000 combining two new elements, two garbage elements called G1 00:21:48.000 --> 00:21:51.000 and G2. G1 is like the top plate. 00:21:51.000 --> 00:21:55.000 G2 is the bottom plate. I put them together and I get a 00:21:55.000 --> 00:21:58.000 capacitor. But notice if I just pick one 00:21:58.000 --> 00:22:03.000 plate then the element G1 will not adhere to the LMD. 00:22:03.000 --> 00:22:08.000 It adheres to the LMD because I choose my element boundaries in 00:22:08.000 --> 00:22:11.000 a way that both plates come within it. 00:22:11.000 --> 00:22:14.000 So it is very fundamental and key. 00:22:14.000 --> 00:22:18.000 And you can read a lot more about it in the course notes. 00:22:18.000 --> 00:22:23.000 I purposely dwelt on that simple point because I think it 00:22:23.000 --> 00:22:29.000 is foundational and important. And you really need to 00:22:29.000 --> 00:22:33.000 understand that the capacitor does satisfy LMD. 00:22:33.000 --> 00:22:37.000 We are still in the good old playground. 00:22:37.000 --> 00:22:41.000 A few simple facts here. These are in the notes. 00:22:41.000 --> 00:22:45.000 And you have also seen this before, I am sure. 00:22:45.000 --> 00:22:51.000 I can relate the charge to the capacitance and the voltage as q 00:22:51.000 --> 00:22:55.000 is equal to Cv. And q is in coulombs, 00:22:55.000 --> 00:23:00.000 this is in farads and this is in volts. 00:23:00.000 --> 00:23:06.000 So there is some charge q stored on the capacitor and it 00:23:06.000 --> 00:23:10.000 is in coulombs and q is equal to Cv. 00:23:10.000 --> 00:23:17.000 So I can differentiate this with respect to time to get the 00:23:17.000 --> 00:23:21.000 current, and that becomes i=dq/dt. 00:23:21.000 --> 00:23:28.000 So the current at any given time is dq/dt. 00:23:28.000 --> 00:23:32.000 And so I substitute for q in terms of Cv here. 00:23:32.000 --> 00:23:36.000 That is what I get. So the current i=d(Cv)/dt. 00:23:36.000 --> 00:23:41.000 A 6.002 assumption, capacitance in general can be 00:23:41.000 --> 00:23:44.000 time-varying. I can get time-varying 00:23:44.000 --> 00:23:48.000 capacitors. In fact, there are some sensors 00:23:48.000 --> 00:23:51.000 which are capacitive. And, as I talk, 00:23:51.000 --> 00:23:57.000 my sound waves can change the pressure on the top plate of the 00:23:57.000 --> 00:24:02.000 capacitor. And move the top plate of the 00:24:02.000 --> 00:24:08.000 capacitor, thereby changing the capacitance by moving the plate. 00:24:08.000 --> 00:24:13.000 Remember d here, as the plate moves closer I get 00:24:13.000 --> 00:24:17.000 a higher capacitance. So we won't be dealing, 00:24:17.000 --> 00:24:23.000 unless explicitly said so, with time-varying capacitances. 00:24:23.000 --> 00:24:29.000 So what we can do is 6.002 allows us to write Cdv/dt. 00:24:29.000 --> 00:24:33.000 So my current source capacitor is Cdv/dt. 00:24:33.000 --> 00:24:39.000 I can also write down the energy, capacitors store energy. 00:24:39.000 --> 00:24:42.000 E=1/2Cv^2. I am sure you have seen all 00:24:42.000 --> 00:24:46.000 this before in physics and so on. 00:24:46.000 --> 00:24:52.000 That is the amount of energy stored in the capacitor if it is 00:24:52.000 --> 00:24:56.000 holding a charge q. Let me do a little 00:24:56.000 --> 00:25:02.000 demonstration for you. They don't make glasses like 00:25:02.000 --> 00:25:07.000 they used to. Our friend Lorenzo has charged 00:25:07.000 --> 00:25:11.000 up this capacitor. It is a huge capacitor. 00:25:11.000 --> 00:25:15.000 It is a 250 volt capacitor so it is nasty. 00:25:15.000 --> 00:25:19.000 He has charged it up and has kept it there. 00:25:19.000 --> 00:25:25.000 And to show you that it does contain stored charges it has 00:25:25.000 --> 00:25:30.000 been sitting there holding charge. 00:25:30.000 --> 00:25:36.000 Maybe the first row should go backwards, just step back for a 00:25:36.000 --> 00:25:40.000 second. I think you guys would be safe 00:25:40.000 --> 00:25:44.000 but I just don't want to take any chances. 00:25:44.000 --> 00:25:48.000 This is holding a bunch of charge. 00:25:48.000 --> 00:25:54.000 It is kind of sitting there. If I short the terminals it 00:25:54.000 --> 00:25:58.000 should try to say oh, I've got a path, 00:25:58.000 --> 00:26:02.000 let me get my charge out. All right. 00:26:02.000 --> 00:26:05.000 Let's do it. This is always a scary moment 00:26:05.000 --> 00:26:08.000 for me. And I say a little prayer 00:26:08.000 --> 00:26:10.000 before I do this. 00:26:19.000 --> 00:26:19.000 Good? OK. Gee, you guys would love to see me getting fried, 00:26:21.000 --> 00:26:22.000 huh? All right. 00:26:22.000 --> 00:26:23.000 Let's see. 00:26:45.000 --> 00:26:47.000 So it did contain charge. 00:27:00.000 --> 00:27:04.000 So there is a reason why Lorenzo puts one hand inside his 00:27:04.000 --> 00:27:08.000 pocket when he shorts it, because there is a natural 00:27:08.000 --> 00:27:12.000 tendency to hold the wire with both hands, and la, 00:27:12.000 --> 00:27:16.000 la, la, la, la and put it across the capacitor. 00:27:16.000 --> 00:27:21.000 By doing this you are guaranteed that you will just be 00:27:21.000 --> 00:27:25.000 touching it with one hand. Hopefully you folks will 00:27:25.000 --> 00:27:29.000 remember for life that a capacitor can sit around and 00:27:29.000 --> 00:27:33.000 hold its charge for a while. All right. 00:27:33.000 --> 00:27:35.000 That is enough of fun and games. 00:27:35.000 --> 00:27:39.000 Let's get on with our business of building circuits. 00:27:39.000 --> 00:27:41.000 What I am going to do is, as I promised you, 00:27:41.000 --> 00:27:46.000 I am going to close the loop on that example by halfway through 00:27:46.000 --> 00:27:49.000 the next lecture. I'm going take you on a bit of 00:27:49.000 --> 00:27:53.000 a journey involving capacitors and resistors and involving some 00:27:53.000 --> 00:27:57.000 analysis, and then we will close it all up for you at about the 00:27:57.000 --> 00:28:02.000 middle of next lecture. What I would like to do next is 00:28:02.000 --> 00:28:06.000 here is a new element. And let's do some fun stuff 00:28:06.000 --> 00:28:10.000 with elements. Well, you know about voltage 00:28:10.000 --> 00:28:14.000 sources, you know about resistors, let's put them 00:28:14.000 --> 00:28:17.000 together and see how they behave. 00:28:17.000 --> 00:28:21.000 Let's have a capacitor here, C, vc(t) and some current i. 00:28:21.000 --> 00:28:25.000 What I am going to do, in general, whenever I have 00:28:25.000 --> 00:28:30.000 something new or something strange, let's say like a 00:28:30.000 --> 00:28:36.000 capacitor or some other device. It is interesting to model the 00:28:36.000 --> 00:28:41.000 rest of the circuit behind it if it contains only resistors and 00:28:41.000 --> 00:28:46.000 voltages and linear elements as a Thevenin equivalent. 00:28:46.000 --> 00:28:50.000 So let me do that. This is R and this is vi. 00:28:50.000 --> 00:28:54.000 This stuff in the back is my standard pattern, 00:28:54.000 --> 00:28:59.000 voltage source in series with a resistor, and I connect that 00:28:59.000 --> 00:29:03.000 across my capacitor. But remember, 00:29:03.000 --> 00:29:07.000 although you saw those funny wave forms and so on, 00:29:07.000 --> 00:29:10.000 the capacitor is a linear device. 00:29:10.000 --> 00:29:15.000 Because you can see from here that the current relates to 00:29:15.000 --> 00:29:18.000 dv/dt. That is a linear operation. 00:29:18.000 --> 00:29:23.000 You don't see V squareds and Vis and things like that in 00:29:23.000 --> 00:29:25.000 there. It's is a linear device. 00:29:25.000 --> 00:29:32.000 Let's go back to our trusty old method, the node method. 00:29:32.000 --> 00:29:36.000 If you just blindly apply the node method and simply grunge 00:29:36.000 --> 00:29:40.000 through a bunch of math, you should be able to get to 00:29:40.000 --> 00:29:44.000 the answer, that is for some voltage v or some form of 00:29:44.000 --> 00:29:49.000 voltage vi, I should be able to figure out what vc looks like. 00:29:49.000 --> 00:29:52.000 So let's do that. This is the node that is of 00:29:52.000 --> 00:29:56.000 interest here with the unknown node voltage vc. 00:29:56.000 --> 00:29:58.000 So let me apply the node method. 00:29:58.000 --> 00:30:03.000 (vc-vi)/R is the current going this way. 00:30:03.000 --> 00:30:09.000 That plus the current through the capacitor should equal zero. 00:30:09.000 --> 00:30:14.000 And what is the current through the capacitor? 00:30:14.000 --> 00:30:20.000 The node method tells me that, get the current in terms of the 00:30:20.000 --> 00:30:25.000 element values. We know that the current is 00:30:25.000 --> 00:30:31.000 given by CdvC/dt.=O. Just shuffling things around a 00:30:31.000 --> 00:30:35.000 little bit, I can write RC dvc/dt+vc=vi. 00:30:35.000 --> 00:30:41.000 We are writing the node equation and then getting the 00:30:41.000 --> 00:30:47.000 equation that characterizes this little circuit. 00:30:47.000 --> 00:30:51.000 Notice here that this has units of volts. 00:30:51.000 --> 00:30:58.000 And since I have time here, this also must have units of 00:30:58.000 --> 00:31:00.000 time. 00:31:06.000 --> 00:31:12.000 Let's go about solving this little circuit and understanding 00:31:12.000 --> 00:31:16.000 how it behaves. The specific example that we 00:31:16.000 --> 00:31:22.000 will look at looks like this. Let's say the capacitor voltage 00:31:22.000 --> 00:31:25.000 at time T=0 is V0. This is given. 00:31:25.000 --> 00:31:30.000 So at time T=0, I am telling you that the 00:31:30.000 --> 00:31:36.000 capacitor contains a charge. And because of that there is a 00:31:36.000 --> 00:31:40.000 voltage V0 across it. That capacitor had a voltage of 00:31:40.000 --> 00:31:45.000 250 volts across it and most of the devices we deal with in 00:31:45.000 --> 00:31:48.000 laptops and so on today, like the Pentium IV, 00:31:48.000 --> 00:31:53.000 voltages are on the order of 1.5 volts, very small voltages. 00:31:53.000 --> 00:31:57.000 So that is the value in the capacitor, the voltage. 00:31:57.000 --> 00:32:02.000 That is called a state. This is called the state, 00:32:02.000 --> 00:32:05.000 capacitor state. It is the state of the 00:32:05.000 --> 00:32:08.000 capacitor. And I also give you that 00:32:08.000 --> 00:32:10.000 vi(t)=VI. So my voltage is VI. 00:32:10.000 --> 00:32:14.000 And somehow, I am not telling you how, 00:32:14.000 --> 00:32:19.000 but some how it arranged to have the capacitor voltage be V0 00:32:19.000 --> 00:32:22.000 at time T=0. Now I want to look to the 00:32:22.000 --> 00:32:28.000 solution to this for t greater than or equal to zero. 00:32:28.000 --> 00:32:35.000 And in that time my voltage vi is at some capital VI, 00:32:35.000 --> 00:32:41.000 some DC voltage VI. So I am going to solve the 00:32:41.000 --> 00:32:48.000 differential equation RC dvc/dt+vc=vi given these two 00:32:48.000 --> 00:32:53.000 values. Input is DC voltage VI and VC0 00:32:53.000 --> 00:33:00.000 is V0, the initial charge in the capacitor. 00:33:00.000 --> 00:33:03.000 So from now until almost to the end of the lecture, 00:33:03.000 --> 00:33:08.000 it is just going to be math by solving this very simple first 00:33:08.000 --> 00:33:12.000 order differential equation. And the key here will be that 00:33:12.000 --> 00:33:16.000 throughout 6.002 we will be following one method to solve 00:33:16.000 --> 00:33:19.000 these. There are many methods to 00:33:19.000 --> 00:33:23.000 solving differential equations, and we will follow one method. 00:33:23.000 --> 00:33:27.000 That method is called the method of homogenous and 00:33:27.000 --> 00:33:31.000 particular solutions. In 1802, I believe, 00:33:31.000 --> 00:33:36.000 you would have learned maybe this, and certainly other 00:33:36.000 --> 00:33:40.000 methods. You can use any method to solve 00:33:40.000 --> 00:33:43.000 it. We will just stick to one 00:33:43.000 --> 00:33:46.000 method. And this is also used in the 00:33:46.000 --> 00:33:50.000 course notes. In this method what we do is 00:33:50.000 --> 00:33:55.000 take the solution VC by finding two other components. 00:33:55.000 --> 00:34:00.000 One is called the homogenous solution. 00:34:00.000 --> 00:34:03.000 And summing that up with the particular solution. 00:34:03.000 --> 00:34:08.000 And that is the total solution. So total solution is the sum of 00:34:08.000 --> 00:34:11.000 the homogenous and the particular solutions. 00:34:11.000 --> 00:34:15.000 And the method has three steps. As I said before, 00:34:15.000 --> 00:34:19.000 we will be using this method again and again with every 00:34:19.000 --> 00:34:23.000 differential equation that we encounter in this course. 00:34:23.000 --> 00:34:26.000 And you won't encounter a while lot. 00:34:26.000 --> 00:34:31.000 The first step we find the particular solution. 00:34:31.000 --> 00:34:39.000 The second step, find the homogenous solution. 00:34:39.000 --> 00:34:47.000 The total solution is the sum of the two. 00:34:47.000 --> 00:34:51.000 And then find --- 00:34:56.000 --> 00:34:58.000 There will be some unknown constants depending on the 00:34:58.000 --> 00:35:02.000 equation that you have. And in the end we simply find 00:35:02.000 --> 00:35:06.000 the unknown constants by applying the initial conditions 00:35:06.000 --> 00:35:09.000 that we have. Boom, boom, boom. 00:35:09.000 --> 00:35:10.000 Particular. Homogenous. 00:35:10.000 --> 00:35:13.000 Find constants. Three things. 00:35:13.000 --> 00:35:17.000 So let's go about solving this equation and apply those three 00:35:17.000 --> 00:35:19.000 conditions. Again, remember, 00:35:19.000 --> 00:35:24.000 what I am doing now for the next 10 minutes or 15 minutes is 00:35:24.000 --> 00:35:29.000 using math that you know about to simply solve this first order 00:35:29.000 --> 00:35:35.000 of differential equations. There is nothing really new 00:35:35.000 --> 00:35:39.000 that I am going to talk about here. 00:35:39.000 --> 00:35:43.000 One is to find the particular solution vCP, 00:35:43.000 --> 00:35:49.000 which will then be added into the vCH to get me the solution. 00:35:49.000 --> 00:35:54.000 So the way you find the vCP is you find any solution that 00:35:54.000 --> 00:36:00.000 satisfies this equation. This is the equation. 00:36:00.000 --> 00:36:03.000 You find any solution that satisfies it. 00:36:03.000 --> 00:36:07.000 And find the simplest possible solution that money can buy. 00:36:07.000 --> 00:36:10.000 Find it. That's the particular solution. 00:36:10.000 --> 00:36:13.000 Any solution is fine. In this case, 00:36:13.000 --> 00:36:16.000 a really simple one would be vCP equals VI. 00:36:16.000 --> 00:36:21.000 Let's see if a constant works. One thing you will realize in 00:36:21.000 --> 00:36:25.000 differential equations is that they are actually much simpler 00:36:25.000 --> 00:36:30.000 than they seem. And the reason is that almost 00:36:30.000 --> 00:36:33.000 every time you have to assume you know the answer, 00:36:33.000 --> 00:36:37.000 and then you are checking to see what you assumed was 00:36:37.000 --> 00:36:40.000 correct. Assume the answer is this like 00:36:40.000 --> 00:36:44.000 you are really smart, and then check it out and say 00:36:44.000 --> 00:36:46.000 oh, yeah, that must have been the answer. 00:36:46.000 --> 00:36:51.000 So here we assume that I think VI is going to work so let's try 00:36:51.000 --> 00:36:53.000 it out. Substituting in here. 00:36:53.000 --> 00:36:55.000 RC dvc/dt is 0. vi is a constant. 00:36:55.000 --> 00:36:59.000 So I get vi equals vi, so therefore this is a 00:36:59.000 --> 00:37:02.000 particular solution. Done. 00:37:02.000 --> 00:37:05.000 I substitute vi here. So dvi/dt=0. 00:37:05.000 --> 00:37:08.000 This vanishes and vi=VI. Bingo. 00:37:08.000 --> 00:37:12.000 Therefore, VI is a solution to this equation. 00:37:12.000 --> 00:37:15.000 So I am done with my vCP. 00:37:22.000 --> 00:37:23.000 And in general what you have to do is use trial and error. 00:37:23.000 --> 00:37:24.000 By trial and error try out a bunch of solutions until you get 00:37:24.000 --> 00:37:24.000 lucky. In general, again, in all of 6.002 for many of the excitations a simple constant 00:37:25.000 --> 00:37:26.000 usually suffices. Our second step is to find the 00:37:26.000 --> 00:37:27.000 homogenous solution. And we can also do that very 00:37:27.000 --> 00:37:27.000 quickly. And to do that we have to find a general solution to the homogenous equation. 00:37:29.000 --> 00:37:52.000 The homogenous equation is the same differential equation but 00:37:52.000 --> 00:38:04.000 with the drive set to zero. 00:38:11.000 --> 00:38:15.000 We want to follow a set pattern to solve the differential 00:38:15.000 --> 00:38:19.000 equations here, and the set pattern is find 00:38:19.000 --> 00:38:23.000 vCP, vCH, find constants. And to find vCH we are also 00:38:23.000 --> 00:38:29.000 going to follow a set pattern to find the homogenous solution. 00:38:29.000 --> 00:38:34.000 So we set the drive to zero, so vi is set to be zero. 00:38:34.000 --> 00:38:38.000 And I need to find a general solution to this. 00:38:38.000 --> 00:38:43.000 As I promised earlier, diff equations are really, 00:38:43.000 --> 00:38:49.000 really simple because the way we are going to solve them is we 00:38:49.000 --> 00:38:55.000 are going to assume we know the answer and then go check it. 00:38:55.000 --> 00:39:00.000 So let's try Ae^st. Let's try and see if this can 00:39:00.000 --> 00:39:05.000 solve this particular equation for some values of A and S. 00:39:05.000 --> 00:39:10.000 I am telling you that the solution is going to be of this 00:39:10.000 --> 00:39:12.000 form. Assume it. 00:39:12.000 --> 00:39:16.000 And then simply go ahead and find me A and S, 00:39:16.000 --> 00:39:21.000 and do that by substituting it back into the equation and find 00:39:21.000 --> 00:39:27.000 out the corresponding As and Ss. So let's go ahead and do that. 00:39:27.000 --> 00:39:36.000 I get RC. I substitute this back up so I 00:39:36.000 --> 00:39:50.000 get dAe^(st)/dt+Ae^st=0. And let me plug that in and see 00:39:50.000 --> 00:40:00.000 what comes. I get RCAse^st+Ae^st=0. 00:40:00.000 --> 00:40:05.000 I want to discard the trivial solution of A being 0. 00:40:05.000 --> 00:40:11.000 That is a trivial solution so I will discard that. 00:40:11.000 --> 00:40:16.000 And what I will do is cancel out the As from here, 00:40:16.000 --> 00:40:21.000 assuming A is not zero, and cancel e^st here. 00:40:21.000 --> 00:40:28.000 And what is left is RCs+1=0. What this is saying is that if 00:40:28.000 --> 00:40:34.000 I can find an S such that this is true then Aest is a general 00:40:34.000 --> 00:40:40.000 solution to my homogenous equation. 00:40:40.000 --> 00:40:44.000 This is easy enough. And so S=-1/RC. 00:40:44.000 --> 00:40:51.000 If I choose my S to be -1/RC then the simple math that I have 00:40:51.000 --> 00:40:57.000 gone through shows me that this must be the solution to the 00:40:57.000 --> 00:41:02.000 homogenous equation. Or in other words 00:41:02.000 --> 00:41:07.000 vCH=Ae^(-t/RC). All this is saying is that 00:41:07.000 --> 00:41:12.000 Ae^(-t/RC) is a solution to my homogenous equation. 00:41:12.000 --> 00:41:16.000 A is an unknown constant. A is some constant. 00:41:16.000 --> 00:41:21.000 I don't know what that is yet. Notice RC has popped up again. 00:41:21.000 --> 00:41:26.000 And the cool thing about RC is that, this is time, 00:41:26.000 --> 00:41:33.000 this also has units of time. We commonly represent RC as 00:41:33.000 --> 00:41:38.000 some time constant tau, as units of time. 00:41:38.000 --> 00:41:45.000 Associated with that circuit is the time constant tau, 00:41:45.000 --> 00:41:50.000 which is simply RC. I commonly write this as 00:41:50.000 --> 00:41:53.000 Ae^(-t/tau). 00:42:02.000 --> 00:42:08.000 I am very the end here. I have the particular solution 00:42:08.000 --> 00:42:11.000 here. I have got the homogenous 00:42:11.000 --> 00:42:16.000 solution there. I need to tell you about 00:42:16.000 --> 00:42:21.000 something else. The way I found the homogenous 00:42:21.000 --> 00:42:27.000 solution was in four steps. I assumed a solution of the 00:42:27.000 --> 00:42:32.000 form Ae^st. I created this equation here in 00:42:32.000 --> 00:42:34.000 S. This is called the 00:42:34.000 --> 00:42:38.000 characteristic equation for that circuit. 00:42:38.000 --> 00:42:44.000 We will see this time and time again for RC and other forms of 00:42:44.000 --> 00:42:47.000 circuits. Assume a solution of this form. 00:42:47.000 --> 00:42:51.000 Construct the characteristic equation. 00:42:51.000 --> 00:42:55.000 Find the roots of the characteristic equation. 00:42:55.000 --> 00:43:00.000 In this case it is an equation in S. 00:43:00.000 --> 00:43:05.000 So this is the root. And then form the solution 00:43:05.000 --> 00:43:08.000 based on that root. Four steps. 00:43:08.000 --> 00:43:15.000 Ae^st, characteristic equation, root and then write down the 00:43:15.000 --> 00:43:21.000 general homogenous solution. Four steps there. 00:43:21.000 --> 00:43:28.000 And finally I want to write down the total solution. 00:43:28.000 --> 00:43:33.000 And the total solution is simply vCP+vCH. 00:43:33.000 --> 00:43:38.000 And vCP was VI and vCH was Ae^(-t/tau). 00:43:38.000 --> 00:43:43.000 tau was simply RC. That is my solution. 00:43:43.000 --> 00:43:50.000 Now, remember the last step. The last step was form the 00:43:50.000 --> 00:43:59.000 total solution and find out the remaining constants. 00:43:59.000 --> 00:44:05.000 Find out the remaining constants by using my initial 00:44:05.000 --> 00:44:10.000 conditions. At t=0, I know that vC=V0. 00:44:10.000 --> 00:44:15.000 I know that. And so therefore I can 00:44:15.000 --> 00:44:20.000 substitute t=0 to find the constant. 00:44:20.000 --> 00:44:26.000 So I know that VO=VI+A. t=0, this thing becomes 1, 00:44:26.000 --> 00:44:35.000 and so I get this equation from which I get A=V0-Vi. 00:44:35.000 --> 00:44:38.000 In other words, my solution vC is simply 00:44:38.000 --> 00:44:43.000 VI+(VO-VI) e^(-t/tau). So the last 15 minutes have 00:44:43.000 --> 00:44:47.000 just been math. No electrical engineering here, 00:44:47.000 --> 00:44:53.000 but electrical engineering stopped at the point where you 00:44:53.000 --> 00:44:58.000 wrote this differential equation down, went through a bunch of 00:44:58.000 --> 00:45:03.000 math and came up with a solution. 00:45:03.000 --> 00:45:07.000 Purely mathematically. So here I simply used math to 00:45:07.000 --> 00:45:11.000 get you the solution. And, as I have been promising 00:45:11.000 --> 00:45:16.000 you throughout this course, in the next lecture I will give 00:45:16.000 --> 00:45:19.000 you an intuitive EE method of doing it. 00:45:19.000 --> 00:45:23.000 Real electrical engineers, real EECS folks don't do it 00:45:23.000 --> 00:45:26.000 this way. Real EECS folks do it 00:45:26.000 --> 00:45:30.000 intuitively. And I will show you how to do 00:45:30.000 --> 00:45:35.000 it in four easy seconds in the next lecture. 00:45:35.000 --> 00:45:41.000 But you need to understand the foundations of how this comes 00:45:41.000 --> 00:45:44.000 about, and so this is the answer. 00:45:44.000 --> 00:45:49.000 You can also get the current iC is simply Cdvc/dt. 00:45:49.000 --> 00:45:53.000 I won't do that for you, but you can simply 00:45:53.000 --> 00:45:57.000 differentiate it and get the current. 00:45:57.000 --> 00:46:02.000 So I can plot for you vC, time t, vC. 00:46:02.000 --> 00:46:08.000 The intuitive way of looking at this is I have VI which is the 00:46:08.000 --> 00:46:15.000 final value of the voltage. When t is infinity this part 00:46:15.000 --> 00:46:19.000 goes to zero so the vC is simply VI. 00:46:19.000 --> 00:46:26.000 And then there is a component V0-VI which decays according to 00:46:26.000 --> 00:46:32.000 this starting out at an initial value of V0. 00:46:32.000 --> 00:46:38.000 Notice when t is zero vC is V0, you can see that in the 00:46:38.000 --> 00:46:45.000 equation, and so it starts out at V0 and ends up at VI. 00:46:45.000 --> 00:46:49.000 I start here, I end up here. 00:46:49.000 --> 00:46:55.000 And this portion V0-VI decays out over time like this. 00:46:55.000 --> 00:47:04.000 And this decay is governed by the RC time constant or tau. 00:47:04.000 --> 00:47:10.000 I am going to show you very quickly a couple of examples of 00:47:10.000 --> 00:47:17.000 wave forms, one that goes like this and one that looks like 00:47:17.000 --> 00:47:21.000 this. This is when I start with some 00:47:21.000 --> 00:47:28.000 value V0 and I don't apply any input, it should decay down to 00:47:28.000 --> 00:47:32.000 zero, t, t, vC, vC. 00:47:32.000 --> 00:47:37.000 If I apply zero for VI then this should simply decay down to 00:47:37.000 --> 00:47:41.000 nothing over time. And if I apply some VI but 00:47:41.000 --> 00:47:45.000 there is no state in the capacitor then that same 00:47:45.000 --> 00:47:48.000 equation is going to look like this. 00:47:48.000 --> 00:47:53.000 You can go and confirm for yourselves that when I apply 00:47:53.000 --> 00:47:58.000 some input but the capacitor has zero state, I start at zero, 00:47:58.000 --> 00:48:04.000 I finish up at VI and my wave form looks like this. 00:48:04.000 --> 00:48:07.000 There you go. That's the first one. 00:48:07.000 --> 00:48:13.000 The second one where I have 5 volts on the capacitor and no 00:48:13.000 --> 00:48:16.000 input. Assume that at time equals zero 00:48:16.000 --> 00:48:21.000 I take away an input, short the input voltage to 00:48:21.000 --> 00:48:25.000 ground for example, apply zero volts. 00:48:25.000 --> 00:48:31.000 You will see the decay from 5 volts to 0 volts. 00:48:31.000 --> 00:48:37.000 And in the first case I start with zero volts in my capacitor, 00:48:37.000 --> 00:48:42.000 I apply input of 5 volts, and notice that at t=0 the 00:48:42.000 --> 00:48:45.000 capacitor rises up to that level. 00:48:45.000 --> 00:48:51.000 So notice that these circuits with capacitor and resistors are 00:48:51.000 --> 00:48:56.000 typified by wave forms that are exponential rises and 00:48:56.000 --> 00:49:01.000 exponential decays. We will see more of that next 00:49:01.000 --> 00:49:04.000 time.