1 00:00:00 --> 00:00:03 All right. Good morning, 2 00:00:03 --> 00:00:08 all. So we take another big step 3 00:00:08 --> 00:00:16 forward today and get onto a new plane of understanding, 4 00:00:16 --> 00:00:23 if you will. In the last week and a half, 5 00:00:23 --> 00:00:31 our focus was on the storage element or storage elements 6 00:00:31 --> 00:00:40 called inductors and capacitors. And capacitors stored change 7 00:00:40 --> 00:00:46 and inductors essentially stored energy in the field, 8 00:00:46 --> 00:00:51 the magnetic flux. And the state variable for an 9 00:00:51 --> 00:00:57 inductor was the current while that for a capacitor was the 10 00:00:57 --> 00:01:02 capacitor voltage. We also looked at circuits 11 00:01:02 --> 00:01:06 containing a single storage element, we looked at RC 12 00:01:06 --> 00:01:11 circuits and we also looked at circuits containing a single 13 00:01:11 --> 00:01:14 inductor. And this was a single inductor 14 00:01:14 --> 00:01:19 with a resistor and a current source or a voltage source and 15 00:01:19 --> 00:01:22 so on. What we are going to do today 16 00:01:22 --> 00:01:27 is do what are called "second-order systems". 17 00:01:27 --> 00:01:36 18 00:01:36 --> 00:01:38 So they are on the next plane now. 19 00:01:38 --> 00:01:43 And with this second-order of systems, they are characterized 20 00:01:43 --> 00:01:47 by circuits containing two independent storage elements. 21 00:01:47 --> 00:01:52 They could be an inductor and a capacitor or two independent 22 00:01:52 --> 00:01:55 capacitors. And you will see towards the 23 00:01:55 --> 00:02:00 end what I mean by two independent capacitors. 24 00:02:00 --> 00:02:03 If I have two capacitors in parallel, they can be 25 00:02:03 --> 00:02:06 represented as a single equivalent capacitor so that 26 00:02:06 --> 00:02:08 doesn't count. It has to be two independent 27 00:02:08 --> 00:02:12 energy storage elements and resistors and voltage sources 28 00:02:12 --> 00:02:14 and so on. And what we end up getting is 29 00:02:14 --> 00:02:17 what is called "second-order dynamics". 30 00:02:17 --> 00:02:20 And much as first order circuits were represented using 31 00:02:20 --> 00:02:24 first order differential equations, this kind you end up 32 00:02:24 --> 00:02:28 getting second-order differential equations. 33 00:02:28 --> 00:02:31 Before we go into this, I would like to start 34 00:02:31 --> 00:02:36 motivating this and give you one example of why this is important 35 00:02:36 --> 00:02:38 to study. There are many, 36 00:02:38 --> 00:02:41 many examples but I will give you one. 37 00:02:41 --> 00:02:45 What I would like to do is draw your attention to our good old 38 00:02:45 --> 00:02:48 inverter driving a second inverter. 39 00:02:48 --> 00:02:52 The same circuit that we used to motivate RC studies, 40 00:02:52 --> 00:02:58 one inverter driving another. So let me draw the circuit. 41 00:02:58 --> 00:03:06 42 00:03:06 --> 00:03:10 Here is one inverter. This is, let's say, 43 00:03:10 --> 00:03:15 5 volts and this is, let's say, 2 kilo ohms. 44 00:03:15 --> 00:03:21 And I connect the output of this inverter to a second 45 00:03:21 --> 00:03:26 inverter. And what we saw in the last few 46 00:03:26 --> 00:03:32 lectures was that in this specific example there was a 47 00:03:32 --> 00:03:38 parasitic capacitor or a capacitor associated with the 48 00:03:38 --> 00:03:44 gate of this MOSFET. And that could be modeled by 49 00:03:44 --> 00:03:49 sticking a capacitor CGS between the gate of the MOSFET and 50 00:03:49 --> 00:03:53 ground. And we saw that the waveforms 51 00:03:53 --> 00:03:57 here, if I had some kind of step here. 52 00:03:57 --> 00:04:01 Let's say, for example, a step that went from high to 53 00:04:01 --> 00:04:03 low. Then out here I would have a 54 00:04:03 --> 00:04:08 transition that instead of going up rapidly like this would 55 00:04:08 --> 00:04:11 transition a little bit more slowly. 56 00:04:11 --> 00:04:15 And this transition was characterized by an RC time 57 00:04:15 --> 00:04:18 constant. And this is what gave rise to a 58 00:04:18 --> 00:04:22 delay in the eventual output. So that is what we saw 59 00:04:22 --> 00:04:27 previously, single energy storage element. 60 00:04:27 --> 00:04:31 Today what we are going to do is we are going to look at the 61 00:04:31 --> 00:04:34 same circuit, the exact same circuit, 62 00:04:34 --> 00:04:38 and have some fun with it. What we are going to say is 63 00:04:38 --> 00:04:43 look, this thing is pretty slow, so what I would like to do is 64 00:04:43 --> 00:04:47 -- why don't we go ahead and put that up. 65 00:04:47 --> 00:04:55 66 00:04:55 --> 00:05:01 What we are going to see is that the yellow waveform is the 67 00:05:01 --> 00:05:06 waveform at the input here. And the green waveform here is 68 00:05:06 --> 00:05:09 the waveform at this intermediate node. 69 00:05:09 --> 00:05:13 And notice that this waveform here is characterized by the 70 00:05:13 --> 00:05:17 slowly rising characteristics that are typical of an RC 71 00:05:17 --> 00:05:19 circuit. There are some other 72 00:05:19 --> 00:05:24 weirdnesses and so on going on here like a little bump and 73 00:05:24 --> 00:05:27 stuff like that. You can ignore all of that for 74 00:05:27 --> 00:05:31 now. It happens because of certain 75 00:05:31 --> 00:05:35 other very subtle circuit effects that you won't be 76 00:05:35 --> 00:05:38 dealing with, called Miller effects and so on 77 00:05:38 --> 00:05:41 that you won't be dealing with in 6.002. 78 00:05:41 --> 00:05:43 So focus then on this part here. 79 00:05:43 --> 00:05:46 It is pretty slow. And because of that slow 80 00:05:46 --> 00:05:51 rising, I get a very slow transition and I get some delay 81 00:05:51 --> 00:05:53 in my inverter. So you say ah-ha, 82 00:05:53 --> 00:05:59 we learned about this in 6.002, I can make it go faster. 83 00:05:59 --> 00:06:02 How can you make the circuit go faster? 84 00:06:02 --> 00:06:06 What could you do? This is rising very slowly. 85 00:06:06 --> 00:06:09 How can you make it go faster? Anybody? 86 00:06:09 --> 00:06:12 You have multiple choices, actually. 87 00:06:12 --> 00:06:15 What are your choices here? Pardon. 88 00:06:15 --> 00:06:20 Decrease the time constant. And how would you decrease the 89 00:06:20 --> 00:06:24 time constant? The capacitance is connected to 90 00:06:24 --> 00:06:30 this MOSFET gate here. I didn't want it in the first 91 00:06:30 --> 00:06:32 place but it is there, I cannot help it, 92 00:06:32 --> 00:06:35 so I can decrease the resistance. 93 00:06:35 --> 00:06:37 Good. Let me go ahead and do that. 94 00:06:37 --> 00:06:42 What I will do is I am going to knock this sucker out and stick 95 00:06:42 --> 00:06:46 in a new resistance that is say 50 ohms, a much smaller 96 00:06:46 --> 00:06:49 resistance. That should speed things up, 97 00:06:49 --> 00:06:51 right? That should make things go much 98 00:06:51 --> 00:06:56 faster because this is a smaller time constant because R is 99 00:06:56 --> 00:07:00 smaller, correct? OK, let's go do it. 100 00:07:00 --> 00:07:03 And let's see if we get what we expect. 101 00:07:03 --> 00:07:07 I have a little switch here. And using that switch, 102 00:07:07 --> 00:07:11 I am going to switch in this little resistance. 103 00:07:11 --> 00:07:14 Whoa, what on earth is happening out there? 104 00:07:14 --> 00:07:18 This is so much fun. What I did is I switched in a 105 00:07:18 --> 00:07:22 small resister here to decrease the time constant, 106 00:07:22 --> 00:07:27 but it looks like I got a whole bunch of crapola that I did not 107 00:07:27 --> 00:07:32 bargain for. This is certainly very fast, 108 00:07:32 --> 00:07:36 it goes up really fast, but I am not sure where it is 109 00:07:36 --> 00:07:39 going, though. Let's stare at that a little 110 00:07:39 --> 00:07:42 while longer. Let me expand the time scale 111 00:07:42 --> 00:07:44 for you. Look at this. 112 00:07:44 --> 00:07:47 Instead of a nice little smooth thing going up. 113 00:07:47 --> 00:07:50 I get something that looks like this. 114 00:07:50 --> 00:07:53 It looks something like a sinusoid. 115 00:07:53 --> 00:07:56 It looks sinusoidal, but then it is a sinusoid that 116 00:07:56 --> 00:08:01 kind of gives up and kind of gets tired and kind of goes 117 00:08:01 --> 00:08:03 away. Right? 118 00:08:03 --> 00:08:07 It kind of dies out. So nothing that you have 119 00:08:07 --> 00:08:11 learned so far has prepared you for this. 120 00:08:11 --> 00:08:15 And, trust me, when I first did some circuit 121 00:08:15 --> 00:08:20 designs myself a long, long time ago I got nailed by 122 00:08:20 --> 00:08:22 that. I looked at my circuit, 123 00:08:22 --> 00:08:28 and what ended up happening was I was noticing these sharp lines 124 00:08:28 --> 00:08:33 at all my transitions. When I looked at my scope, 125 00:08:33 --> 00:08:37 I expected to see nice little square waves but I saw these 126 00:08:37 --> 00:08:39 little nasty spikes sitting out there. 127 00:08:39 --> 00:08:42 And then when I stared at it more carefully, 128 00:08:42 --> 00:08:45 those spikes were really sinusoids that seemed to kind of 129 00:08:45 --> 00:08:48 get tired and kind of go away. So those are nasty, 130 00:08:48 --> 00:08:51 those are real and they happen all the time. 131 00:08:51 --> 00:08:54 And what we will do today is try to get into that and 132 00:08:54 --> 00:08:57 understand why that is the case. We will understand how to 133 00:08:57 --> 00:09:02 design that away. And that is a real problem, 134 00:09:02 --> 00:09:05 by the way. And the reason that is a real 135 00:09:05 --> 00:09:09 problem is the following. Look at this. 136 00:09:09 --> 00:09:12 Look down here. Because this intermediate 137 00:09:12 --> 00:09:16 voltage is meandering all over the countryside here, 138 00:09:16 --> 00:09:21 at this particular point the intermediate voltage dips quite 139 00:09:21 --> 00:09:24 low. And because it dips quite low 140 00:09:24 --> 00:09:30 look at the output. The output has a bump here. 141 00:09:30 --> 00:09:33 And it is quite possible for this output bump to now go into 142 00:09:33 --> 00:09:35 the forbidden region. Or worse. 143 00:09:35 --> 00:09:39 If this swing here was higher, this could have actually gone 144 00:09:39 --> 00:09:43 onto a one, so I would have gotten a false one pulse here. 145 00:09:43 --> 00:09:46 Instead of having a nice one to zero transition, 146 00:09:46 --> 00:09:49 I would have gotten a one to zero, oh, back to one, 147 00:09:49 --> 00:09:52 oh, back to zero and then back down to zero. 148 00:09:52 --> 00:09:57 So this is nasty stuff, really, really nasty stuff. 149 00:09:57 --> 00:10:03 What we will do is understand why that is the case today and 150 00:10:03 --> 00:10:08 see if we can explain it. What is going on here? 151 00:10:08 --> 00:10:14 What is really going on here is take a look at this circuit 152 00:10:14 --> 00:10:18 here. I will take a look at this path 153 00:10:18 --> 00:10:21 here. So this is your VS voltage 154 00:10:21 --> 00:10:25 source. Path kind of goes like this and 155 00:10:25 --> 00:10:30 around. It turns out that this circuit 156 00:10:30 --> 00:10:34 is a loop here. And when there is current flow, 157 00:10:34 --> 00:10:38 going down to basic physics you remember that I also enclose 158 00:10:38 --> 00:10:42 some amount. So there is a current flowing 159 00:10:42 --> 00:10:45 in a loop. And because of that there is an 160 00:10:45 --> 00:10:48 effective inductance here. And, in fact, 161 00:10:48 --> 00:10:52 any current flowing through a wire above a ground plane, 162 00:10:52 --> 00:10:56 for that matter, can be characterized by the 163 00:10:56 --> 00:10:59 inductance. So I can model that by sticking 164 00:10:59 --> 00:11:04 a little inductor here. So my real circuit is not 165 00:11:04 --> 00:11:07 exactly a resistor and a capacitor, but my real circuit 166 00:11:07 --> 00:11:11 is an inductor as well that comes into play because of this 167 00:11:11 --> 00:11:13 wire. Every wire, when there is a 168 00:11:13 --> 00:11:16 current flow, has an inductance associated 169 00:11:16 --> 00:11:18 with it. And because of that the real 170 00:11:18 --> 00:11:21 circuit is resistor, inductor and capacitor. 171 00:11:21 --> 00:11:23 So I end up with two storage elements now, 172 00:11:23 --> 00:11:27 and the dynamics of that are very different from that with a 173 00:11:27 --> 00:11:32 single storage element. That is just a bit of 174 00:11:32 --> 00:11:37 motivation for why our study of inductors is important. 175 00:11:37 --> 00:11:40 And I can draw a quick circuit here. 176 00:11:40 --> 00:11:45 If you look at the circuit, start from ground, 177 00:11:45 --> 00:11:49 the voltage VS and there is a resistor here. 178 00:11:49 --> 00:11:54 And then I have an inductor and then I have a capacitor. 179 00:11:54 --> 00:11:59 So it is a voltage source, resistor, inductor and 180 00:11:59 --> 00:12:04 capacitor. For this whole week we will be 181 00:12:04 --> 00:12:08 looking at circuits like this. Today what I would like to do 182 00:12:08 --> 00:12:11 is start very simple, start with the simplest 183 00:12:11 --> 00:12:15 possible form of this so that you can begin building up your 184 00:12:15 --> 00:12:18 insight and then go into more complicated cases. 185 00:12:18 --> 00:12:22 Today what I will do is simply begin with a case where I don't 186 00:12:22 --> 00:12:26 have a resistor here and simply study a voltage source, 187 00:12:26 --> 00:12:30 an inductor and a capacitor and understand what the voltage 188 00:12:30 --> 00:12:35 looks like out here. So we look at the dynamics of a 189 00:12:35 --> 00:12:39 little system like this. Before we go on, 190 00:12:39 --> 00:12:42 I want to caution you about something. 191 00:12:42 --> 00:12:46 It is just happenstance that I have introduced for you 192 00:12:46 --> 00:12:51 capacitors based on the parasitic capacitance here and 193 00:12:51 --> 00:12:54 inductance based on parasitic inductance. 194 00:12:54 --> 00:12:59 I would hate to leave you with the impression that inductors 195 00:12:59 --> 00:13:04 and capacitors are "bad". Because when you think of a 196 00:13:04 --> 00:13:05 parasitic, you know, parasites. 197 00:13:05 --> 00:13:07 These are parasitic. You didn't expect them there, 198 00:13:07 --> 00:13:10 didn't expect this here and we got the weird behavior. 199 00:13:10 --> 00:13:12 So parasitics have a bad connotation to them. 200 00:13:12 --> 00:13:15 I do not want to leave you with a bad taste in your mouth about 201 00:13:15 --> 00:13:17 capacitors and inductors that these are just bad things. 202 00:13:17 --> 00:13:20 We just have to deal with them and deal with second-order 203 00:13:20 --> 00:13:23 differential equations and all that stuff because they're just 204 00:13:23 --> 00:13:25 bad stuff and we just have to deal with them. 205 00:13:25 --> 00:13:27 I don't want you to end up going through life hating 206 00:13:27 --> 00:13:31 capacitors and inductors. Just because of my choice of 207 00:13:31 --> 00:13:35 examples, it just happened to be introducing them as capacitors. 208 00:13:35 --> 00:13:39 I want to point out that these are fundamental lumped elements 209 00:13:39 --> 00:13:41 in their own right. They are very, 210 00:13:41 --> 00:13:45 incredibly important and useful circuits where we designed 211 00:13:45 --> 00:13:49 capacitors and inductors because we want to have them in there. 212 00:13:49 --> 00:13:53 There are many circuits that we will look at where we really 213 00:13:53 --> 00:13:56 want the inductor in there. We will design an inductor by 214 00:13:56 --> 00:14:00 wrapping wire around in a coil and get bigger inductances and 215 00:14:00 --> 00:14:04 so. Just remember that this can be 216 00:14:04 --> 00:14:08 parasitic in some cases, but in many cases it's good, 217 00:14:08 --> 00:14:12 inductors are good, so just stick with that 218 00:14:12 --> 00:14:15 thought. These are mostly good so don't 219 00:14:15 --> 00:14:17 go around hating them. All right. 220 00:14:17 --> 00:14:21 Let's go on and analyze a basic circuit like this. 221 00:14:21 --> 00:14:26 And what I would like to cover in the next hour are the 222 00:14:26 --> 00:14:30 foundations of something like that. 223 00:14:30 --> 00:14:33 I will take you through the foundations so you understand 224 00:14:33 --> 00:14:35 how it works. And, as always, 225 00:14:35 --> 00:14:39 what I am going to end up with is build up the foundations, 226 00:14:39 --> 00:14:42 help you understand why we got where we were and then help you 227 00:14:42 --> 00:14:45 build intuition. And then show you a really, 228 00:14:45 --> 00:14:49 really simple intuitive way of doing things in terms of how 229 00:14:49 --> 00:14:52 experts do it. And the real cool thing about 230 00:14:52 --> 00:14:55 EECS is that the way experts do things, things are really, 231 00:14:55 --> 00:14:59 really very simple in the end. But you need to build up some 232 00:14:59 --> 00:15:04 intuition to get there. So our circuit looks like this 233 00:15:04 --> 00:15:07 in terms of my two storage elements. 234 00:15:07 --> 00:15:11 I have a voltage vI, inductor L, capacitor C and I 235 00:15:11 --> 00:15:16 am going to look at the voltage across the capacitor and my 236 00:15:16 --> 00:15:21 current through the capacitor. So v(t) is the voltage across 237 00:15:21 --> 00:15:26 the capacitor and my current is the current through this loop 238 00:15:26 --> 00:15:31 here, which is the same as the current through the capacitor or 239 00:15:31 --> 00:15:35 the current through the inductor. 240 00:15:35 --> 00:15:38 And we are going to proceed in exactly the same manner as we 241 00:15:38 --> 00:15:41 did for first order differential equations, write the equations 242 00:15:41 --> 00:15:43 down and just boom, boom, boom, boom, 243 00:15:43 --> 00:15:47 go down the same sets of steps but just get to some place 244 00:15:47 --> 00:15:48 different. We are going to start by 245 00:15:48 --> 00:15:51 writing a node equation for this node here. 246 00:15:51 --> 00:15:54 That's the only node for which I have an unknown voltage. 247 00:15:54 --> 00:15:56 The node here is vI, so I need to find this, 248 00:15:56 --> 00:16:00 there's just one unknown node voltage. 249 00:16:00 --> 00:16:04 And I am going to need some element laws. 250 00:16:04 --> 00:16:11 For the capacitor I know the iV relation is given by the i for 251 00:16:11 --> 00:16:17 the capacitor is Cdv/dt. And just to show the capacitor 252 00:16:17 --> 00:16:23 I am just calling it dvc/dt. Similarly, for an inductor, 253 00:16:23 --> 00:16:30 L, the voltage across the inductor is given by Ldi/dt. 254 00:16:30 --> 00:16:34 So this is the vI relation for the capacitor, 255 00:16:34 --> 00:16:37 the vI relation for an inductor. 256 00:16:37 --> 00:16:42 It also suits us to write this in an integral form. 257 00:16:42 --> 00:16:48 So if I integrate both sides of this equation and I bring L down 258 00:16:48 --> 00:16:53 to this side, I end up getting something like 259 00:16:53 --> 00:17:00 this, 1/L minus infinity to t, VLdt, and that is simply iL. 260 00:17:00 --> 00:17:04 I am just simply replacing this with an integral form. 261 00:17:04 --> 00:17:09 So this is a VI relationship for the inductor and this is for 262 00:17:09 --> 00:17:13 the capacitor. So let me now go ahead and 263 00:17:13 --> 00:17:16 apply the node method for my circuit here. 264 00:17:16 --> 00:17:21 Here, for the node method, I have to equate the currents 265 00:17:21 --> 00:17:26 coming into the node or sum the currents coming into the node 266 00:17:26 --> 00:17:32 and equate that to zero. And while I do that I simply 267 00:17:32 --> 00:17:37 replace the currents by the corresponding voltages using the 268 00:17:37 --> 00:17:40 element laws. So what do I get? 269 00:17:40 --> 00:17:45 I get the current going in here to the inductor is equal to the 270 00:17:45 --> 00:17:48 current going through the capacitor. 271 00:17:48 --> 00:17:51 What is the current going the capacitor? 272 00:17:51 --> 00:17:55 In terms of its v relationship it is Cdv/dt. 273 00:17:55 --> 00:17:59 And the current going to the inductor is given by this 274 00:17:59 --> 00:18:03 relation here, which is simply 1/L minus 275 00:18:03 --> 00:18:08 infinity to t. The voltage across the 276 00:18:08 --> 00:18:14 capacitor is simply (vI-v)dt. I have just written down the 277 00:18:14 --> 00:18:17 node quotation for this node here. 278 00:18:17 --> 00:18:23 Now I will just apply a bit of math and simplify it and get the 279 00:18:23 --> 00:18:28 resulting equation. What I can do is simply 280 00:18:28 --> 00:18:33 differentiate with respect to t here. 281 00:18:33 --> 00:18:41 And get this to be Cd^2v/dt^2, the second derivative of v. 282 00:18:41 --> 00:18:47 And here what I end up getting is 1/L(vI-v). 283 00:18:47 --> 00:18:54 So I just differentiated the whole thing by d/dt here. 284 00:18:54 --> 00:19:02 And then I just move L up here. I bring d^2v/dt^2 out here. 285 00:19:02 --> 00:19:08 And then I get a minus v here, and that will be equal to, 286 00:19:08 --> 00:19:12 oh, I'm sorry. Let me leave this here. 287 00:19:12 --> 00:19:18 Bring the minus v to this side so it becomes a plus and leave 288 00:19:18 --> 00:19:22 vI on this side. So I end up getting 289 00:19:22 --> 00:19:25 LCd^2v/dt^2. I bring L up here. 290 00:19:25 --> 00:19:30 And then I take v to the other side. 291 00:19:30 --> 00:19:33 Plus v and leave vI here so I get vI. 292 00:19:33 --> 00:19:37 That is second order differential equation that 293 00:19:37 --> 00:19:40 governs the characteristics of the voltage, v. 294 00:19:40 --> 00:19:45 So much as the voltage across the capacitor was a state 295 00:19:45 --> 00:19:51 variable in our RC circuits or the current through the inductor 296 00:19:51 --> 00:19:55 was a state variable in our RL circuits, out here both the 297 00:19:55 --> 00:20:01 current through the inductor and the voltage across the capacitor 298 00:20:01 --> 00:20:06 are my two state variables. And so here I have a 299 00:20:06 --> 00:20:09 second-order equation in my voltage, v. 300 00:20:09 --> 00:20:12 Again, going through the foundations here, 301 00:20:12 --> 00:20:15 I am now going to go through a bunch of math. 302 00:20:15 --> 00:20:18 Up to here it was circuit analysis, and now I am just 303 00:20:18 --> 00:20:22 going to do math. For the next three or four 304 00:20:22 --> 00:20:25 blackboards just math. You can solve this second-order 305 00:20:25 --> 00:20:30 differential equation any which way you want. 306 00:20:30 --> 00:20:33 But just to keep things as simple as possible, 307 00:20:33 --> 00:20:36 in 6.002 I solve all the differential equations, 308 00:20:36 --> 00:20:40 it turns out we are fortunate enough we can do that, 309 00:20:40 --> 00:20:43 using the exact same method again and again and again, 310 00:20:43 --> 00:20:48 the same thing can be applied. And the method that we use to 311 00:20:48 --> 00:20:51 solve it is the method of homogenous and particular 312 00:20:51 --> 00:20:54 solutions. So the first step we are going 313 00:20:54 --> 00:20:58 to find the particular solution, vP. 314 00:20:58 --> 00:21:03 Second step we find the homogenous solution, 315 00:21:03 --> 00:21:07 vH. And the third step we are going 316 00:21:07 --> 00:21:14 to find the total solution as the sum of, v is simply the 317 00:21:14 --> 00:21:21 particular plus the homogenous solution and then solve for 318 00:21:21 --> 00:21:27 constants based on the initial conditions and the applied 319 00:21:27 --> 00:21:32 voltage. So let's write down initial 320 00:21:32 --> 00:21:34 conditions. Let's assume, 321 00:21:34 --> 00:21:37 for simplicity, that my initial conditions are 322 00:21:37 --> 00:21:42 simply the voltage across the capacitor is zero to begin and 323 00:21:42 --> 00:21:47 the current through my inductor is also zero as I begin life. 324 00:21:47 --> 00:21:51 Now, this is what is called "zero state". 325 00:21:51 --> 00:21:54 v and i are both zero, and so the response of my 326 00:21:54 --> 00:21:58 circuit for some input is going to be called ZSR. 327 00:21:58 --> 00:22:05 You've probably heard this term in one of your recitations. 328 00:22:05 --> 00:22:10 So zero state response simply says I start with my circuit at 329 00:22:10 --> 00:22:14 rest and looks at how it behaves for some given input. 330 00:22:14 --> 00:22:18 That is a little term you may end up using. 331 00:22:18 --> 00:22:22 My input next. I am going to use the following 332 00:22:22 --> 00:22:25 input. vI of t is going to be a step, 333 00:22:25 --> 00:22:31 is going to look like this. My input is at t=0 v is going 334 00:22:31 --> 00:22:35 from zero to some voltage VI and then stay at that voltage. 335 00:22:35 --> 00:22:37 It is going to be a step. Kaboom. 336 00:22:37 --> 00:22:41 And you can see why I am going with this set of variables, 337 00:22:41 --> 00:22:45 because I want make this situation as close as possible 338 00:22:45 --> 00:22:48 to the funny behavior we observed there. 339 00:22:48 --> 00:22:52 Remember we had a step, and because of the step we had 340 00:22:52 --> 00:22:56 some behavior at that node? So I will try to bring you as 341 00:22:56 --> 00:23:00 close to that. In tomorrow's lecture, 342 00:23:00 --> 00:23:04 I am going to close the loop around that and derive for you 343 00:23:04 --> 00:23:07 exactly the behavior we saw on the scope. 344 00:23:07 --> 00:23:11 And to get there I am going to be try to be as close as 345 00:23:11 --> 00:23:15 possible to the constants and other parameters in the demo. 346 00:23:15 --> 00:23:19 So VI is a step and zero state. Just in terms of notation, 347 00:23:19 --> 00:23:22 this kind of a step input occurs pretty frequently. 348 00:23:22 --> 00:23:25 And we just have a special notation for it. 349 00:23:25 --> 00:23:30 We simply call it VI is the final value here. 350 00:23:30 --> 00:23:33 And we call it u(t). So VIu(t), u(t) simply 351 00:23:33 --> 00:23:39 represents a step at time t=0, steps from zero volts to VI. 352 00:23:39 --> 00:23:44 That is just a little more notation that will come in handy 353 00:23:44 --> 00:23:47 at some point. More math now. 354 00:23:47 --> 00:23:50 Three steps, particular solution, 355 00:23:50 --> 00:23:54 homogenous solution, total solution/constants. 356 00:23:54 --> 00:24:00 This is almost like a mantra here, like a chorus. 357 00:24:00 --> 00:24:03 Homogenous solution we compute using a four-step method. 358 00:24:03 --> 00:24:06 And four-step method for homogenous solutions, 359 00:24:06 --> 00:24:09 it turns out that it happens to be that way for all the 360 00:24:09 --> 00:24:12 equations we will see in our course. 361 00:24:12 --> 00:24:16 The first step would be assume a solution of the form Ae^st. 362 00:24:16 --> 00:24:19 Exactly as with RCs. If you close your eyes and do 363 00:24:19 --> 00:24:23 exactly what you did for RCs you will get to where you want to 364 00:24:23 --> 00:24:25 be. You assume a solution of the 365 00:24:25 --> 00:24:27 form Ae^st. Substitute that into your 366 00:24:27 --> 00:24:31 homogenous equation. Obtain the characteristic 367 00:24:31 --> 00:24:34 equation. Solve for the roots. 368 00:24:34 --> 00:24:37 And then write down your homogenous solution. 369 00:24:37 --> 00:24:42 Same sort of steps again and again and again until you get 370 00:24:42 --> 00:24:45 bored to tears. Particular solution. 371 00:24:45 --> 00:24:49 For the particular solution, I simply need to find a 372 00:24:49 --> 00:24:53 solution, any solution, if not the most general one but 373 00:24:53 --> 00:24:57 any solution that satisfies the particular equation which 374 00:24:57 --> 00:25:03 satisfies that equation. LCd^2vP/dt^2+vP=VI. 375 00:25:03 --> 00:25:09 My input is a step and I am going to look for the solution 376 00:25:09 --> 00:25:15 for time t greater than zero. Notice that for time t less 377 00:25:15 --> 00:25:20 than or equal to zero, v is going to be zero. 378 00:25:20 --> 00:25:26 So I am looking for a solution greater than t=0. 379 00:25:26 --> 00:25:34 Here, if I substitute vP=VI, that is a particular solution. 380 00:25:34 --> 00:25:39 Because if I substitute VI here this goes to zero and then I get 381 00:25:39 --> 00:25:43 VI=VI, so this works. I promised you this was going 382 00:25:43 --> 00:25:47 to be simple. You cannot get any simpler than 383 00:25:47 --> 00:25:49 that. I have done my first step. 384 00:25:49 --> 00:25:52 I found the particular solution. 385 00:25:52 --> 00:25:57 And VI is a good enough particular solution so I will 386 00:25:57 --> 00:26:03 use it, I will take it. As my second step I am going to 387 00:26:03 --> 00:26:08 find vH or the solution to the homogenous equation. 388 00:26:08 --> 00:26:15 And the homogenous equation is simply that equation with drive 389 00:26:15 --> 00:26:18 set to zero. What I get here is 390 00:26:18 --> 00:26:23 LCd^2vH/dt^2+vH=0. That is my homogenous equation. 391 00:26:23 --> 00:26:28 I simply set the drive to be zero. 392 00:26:28 --> 00:26:32 And to find the solution here, I go through my four-step 393 00:26:32 --> 00:26:34 method. Again, in 6.002 following the 394 00:26:34 --> 00:26:38 kind of Occam's principle, we just show you the absolute 395 00:26:38 --> 00:26:41 minimum necessary to get to where you want. 396 00:26:41 --> 00:26:45 The absolute minimum necessary is it turns out that we can 397 00:26:45 --> 00:26:50 solve all our differential equations that we use here by 398 00:26:50 --> 00:26:54 using the methods of homogenous and particular solutions. 399 00:26:54 --> 00:26:58 And every homogenous solution can be solved by a four-step 400 00:26:58 --> 00:27:04 method. That is about as minimal as it 401 00:27:04 --> 00:27:09 can get. So no extraneous stuff there. 402 00:27:09 --> 00:27:13 The four-step method, four steps. 403 00:27:13 --> 00:27:21 The first step is assume a solution of the form vH=Ae^st. 404 00:27:21 --> 00:27:28 What I have noticed is that students starting out are 405 00:27:28 --> 00:27:35 usually scared of differential equations. 406 00:27:35 --> 00:27:36 I know I was when I was a student. 407 00:27:36 --> 00:27:40 And the trick with differential equations is that it is all a 408 00:27:40 --> 00:27:42 matter of psych. Just because you see some 409 00:27:42 --> 00:27:46 squigglies and squagglies and a bunch of math and so on you say 410 00:27:46 --> 00:27:49 oh, that must be hard. But differential equations are 411 00:27:49 --> 00:27:52 actually the simplest thing there is because in a large 412 00:27:52 --> 00:27:55 majority of cases the way you solve them is you assume you 413 00:27:55 --> 00:27:59 know the answer, someone tells you the answer. 414 00:27:59 --> 00:28:02 And then all you are left to do is shove the answer into the 415 00:28:02 --> 00:28:05 equation and find out the constants that makes it the 416 00:28:05 --> 00:28:07 answer. Just a matter of psych. 417 00:28:07 --> 00:28:09 Psych yourselves that this stuff is easy, 418 00:28:09 --> 00:28:11 because I am telling you what the solution is. 419 00:28:11 --> 00:28:14 All you have to do is substitute and verify. 420 00:28:14 --> 00:28:17 If you think about differential equations that way or a large 421 00:28:17 --> 00:28:20 majority of them, it really is very simple if you 422 00:28:20 --> 00:28:22 can just get past the squigglies here. 423 00:28:22 --> 00:28:26 Just get past the squigglies and then just simply stick in 424 00:28:26 --> 00:28:31 some simple stuff and it works. I mean it just cannot get any 425 00:28:31 --> 00:28:34 easier. I cannot think of any other 426 00:28:34 --> 00:28:39 field where the way you find a solution is assume you know the 427 00:28:39 --> 00:28:43 solution and stick it in. It has never made any sense to 428 00:28:43 --> 00:28:48 me but that is how it is. So we assume the solution to 429 00:28:48 --> 00:28:51 the form Ae^st, you stick it in there, 430 00:28:51 --> 00:28:55 and you have to find out the A and s that make it so. 431 00:28:55 --> 00:28:58 It cannot get any simpler than that. 432 00:28:58 --> 00:29:02 Let's stick the sucker in here and see what we can get. 433 00:29:02 --> 00:29:07 Substitute Ae^st here I get LCA, and second derivative, 434 00:29:07 --> 00:29:12 so it's s^2 e^st. And Ae^st on this one here. 435 00:29:12 --> 00:29:17 And that equals zero. And then let me just solve for 436 00:29:17 --> 00:29:22 whatever I can find. Assuming I don't take the 437 00:29:22 --> 00:29:26 trivial case A=0, I cancel these guys out. 438 00:29:26 --> 00:29:32 And what I am left with is simply LCs^2+1=0. 439 00:29:32 --> 00:29:36 In other words, what I end up getting is B, 440 00:29:36 --> 00:29:38 s^2=-1/LC. My first step was, 441 00:29:38 --> 00:29:43 I am giving you solutions, stick them in there, 442 00:29:43 --> 00:29:48 assume a solution of this form. Second step is get the 443 00:29:48 --> 00:29:53 characteristic equation. And the way you get the 444 00:29:53 --> 00:29:59 characteristic equation is that you simply stick this guy in 445 00:29:59 --> 00:30:04 there. And what you end up getting is 446 00:30:04 --> 00:30:09 some equation in s^2. Do you remember what you got 447 00:30:09 --> 00:30:13 for first order circuits? What s was? 448 00:30:13 --> 00:30:17 What is s? For first order circuits, 449 00:30:17 --> 00:30:21 what did you get as a characteristic equation? 450 00:30:21 --> 00:30:24 s+1/RC=0. The same thing. 451 00:30:24 --> 00:30:30 Just remember to blindly apply the steps. 452 00:30:30 --> 00:30:33 It will lead you to the answer. This is called the 453 00:30:33 --> 00:30:37 "characteristic equation". This is incredibly important. 454 00:30:37 --> 00:30:41 You will see in about a couple weeks from now that once you 455 00:30:41 --> 00:30:44 write the characteristic equation down for a circuit, 456 00:30:44 --> 00:30:48 it tells you all there is to know about the circuit. 457 00:30:48 --> 00:30:51 And often times you can stop solving right here. 458 00:30:51 --> 00:30:54 To experienced circuit designers this tells me 459 00:30:54 --> 00:30:59 everything there is to know. This is really key. 460 00:30:59 --> 00:31:02 That's why it's called a characteristic equation. 461 00:31:02 --> 00:31:05 I believe in problem number three of the homework that will 462 00:31:05 --> 00:31:09 be coming out this week, that is exactly what you are 463 00:31:09 --> 00:31:11 going to do. I am going to give you a 464 00:31:11 --> 00:31:15 circuit, ask you to get to the characteristic equation quickly 465 00:31:15 --> 00:31:18 and then from there intuit the solution. 466 00:31:18 --> 00:31:21 Write the characteristic equation and then just intuit 467 00:31:21 --> 00:31:25 solution, it's that simple. So, step A, assume a solution 468 00:31:25 --> 00:31:27 of the form, step B, write the characteristic 469 00:31:27 --> 00:31:33 equation down. And let me just simplify that a 470 00:31:33 --> 00:31:37 little bit. I go ahead and find my roots. 471 00:31:37 --> 00:31:42 And my roots here, remember that j is the square 472 00:31:42 --> 00:31:47 root of minus one. And so what I end up getting 473 00:31:47 --> 00:31:53 is, my two roots here are, plus j square root of 1/LC and 474 00:31:53 --> 00:31:59 minus j square root of 1/LC. Two roots. 475 00:31:59 --> 00:32:03 And just as a shorthand notation, much like I had a 476 00:32:03 --> 00:32:08 shorthand notation for RC, what was my shorthand notation 477 00:32:08 --> 00:32:09 for RC? Tau. 478 00:32:09 --> 00:32:15 Just as tau was big in first order, we have a corresponding 479 00:32:15 --> 00:32:20 thing that is big in second order and that is omega nought. 480 00:32:20 --> 00:32:24 Omega nought is simply square root 1/LC. 481 00:32:24 --> 00:32:28 Just as tau was RC, omega nought is a shorthand 482 00:32:28 --> 00:32:33 here. And so s is simply plus or 483 00:32:33 --> 00:32:40 minus j omega nought. Notice that in this equation 484 00:32:40 --> 00:32:48 here, if you take the square root of LC there that has units 485 00:32:48 --> 00:32:56 of time, so one divided by that has units of frequency. 486 00:32:56 --> 00:33:04 Notice that this guy is a frequency in radians. 487 00:33:04 --> 00:33:09 I end up getting my roots of the homogenous equation, 488 00:33:09 --> 00:33:13 and that is my third step. And as my fourth step, 489 00:33:13 --> 00:33:18 I simply write down the homogenous solution as 490 00:33:18 --> 00:33:23 substituting s with its roots and writing the most general 491 00:33:23 --> 00:33:29 possible form of the solution, and that would be A1e^(j omega 492 00:33:29 --> 00:33:34 nought t)+A2e^(-j omega nought t). 493 00:33:34 --> 00:33:36 Done. Some constant times this 494 00:33:36 --> 00:33:38 solution plus some other constant times, 495 00:33:38 --> 00:33:41 the other solution. Plus zero omega nought. 496 00:33:41 --> 00:33:44 Remember it comes from here, Ae^st. 497 00:33:44 --> 00:33:48 I assume the solution of this form, so my solution in this 498 00:33:48 --> 00:33:52 most general case would be s being j omega nought in one 499 00:33:52 --> 00:33:55 case, minus j omega nought in the other case, 500 00:33:55 --> 00:34:00 and I sum the two to get the most general solution. 501 00:34:00 --> 00:34:05 502 00:34:05 --> 00:34:10 So blasting ahead. I now have my homogenous 503 00:34:10 --> 00:34:14 solution. And as my third step of 504 00:34:14 --> 00:34:21 solution to differential equations I write down the total 505 00:34:21 --> 00:34:27 solution, v=vP+vH, particular plus the homogenous 506 00:34:27 --> 00:34:32 solutions. And v=VI, was my particular 507 00:34:32 --> 00:34:38 solution, +A1e^(j omega nought t)+A2e^(-j omega nought t) is my 508 00:34:38 --> 00:34:41 complete solution. The final step, 509 00:34:41 --> 00:34:46 write down the total solution and find the constants from the 510 00:34:46 --> 00:34:51 initial conditions. To find the constants from the 511 00:34:51 --> 00:34:54 initial conditions, let's start with, 512 00:34:54 --> 00:34:59 the voltage is zero to begin with. 513 00:34:59 --> 00:35:03 This equation governs the characteristics of v, 514 00:35:03 --> 00:35:08 so I need to find the initial conditions. 515 00:35:08 --> 00:35:12 First of all, I know that know that v(0)=0. 516 00:35:12 --> 00:35:17 From there I substitute t=0. And so this goes to one, 517 00:35:17 --> 00:35:21 this goes to one, and I end up getting 518 00:35:21 --> 00:35:25 0=VI+A1+A2. That is my first expression. 519 00:35:25 --> 00:35:31 And then I am also given that i(0)=0. 520 00:35:31 --> 00:35:37 And so I can get that as well. How do I get i? 521 00:35:37 --> 00:35:41 This is v. I know that i=Cdv/dt, 522 00:35:41 --> 00:35:47 so I can get i by simply multiplying by C and 523 00:35:47 --> 00:35:52 differentiating this with respect to t. 524 00:35:52 --> 00:36:00 I get C, this guy vanishes so I get d/dt of this. 525 00:36:00 --> 00:36:07 So it is CA1(j omega nought) e^(j omega nought t)+CA2(-j 526 00:36:07 --> 00:36:12 omega nought)e^(-j omega nought t). 527 00:36:12 --> 00:36:21 From here I am given that that is zero, and so therefore this 528 00:36:21 --> 00:36:26 guy becomes a one, this guy becomes a one, 529 00:36:26 --> 00:36:34 j omega nought, j omega nought cancel out. 530 00:36:34 --> 00:36:43 What I end up getting is A1=A2. From the second initial 531 00:36:43 --> 00:36:50 condition I get A1=A2. From these two, 532 00:36:50 --> 00:36:58 if I substitute here for A2, I get VI + 2A1 = 0, 533 00:36:58 --> 00:37:05 or A1=-VI/2. That is also equal to A2. 534 00:37:05 --> 00:37:13 Therefore, my total solution now can be written in terms of 535 00:37:13 --> 00:37:19 the actual values of the constants I have obtained. 536 00:37:19 --> 00:37:24 I get VI-VI/2. So A1 and A2 are equal. 537 00:37:24 --> 00:37:33 I just pull them outside. I pull VI-2 outside and I stick 538 00:37:33 --> 00:37:38 these two guys in parenthesis in. 539 00:37:38 --> 00:37:46 Again, I promised you no more circuits from here on until the 540 00:37:46 --> 00:37:52 very last board or something like that. 541 00:37:52 --> 00:37:57 It is all math, so not much else happening 542 00:37:57 --> 00:38:01 there. More math. 543 00:38:01 --> 00:38:06 If you would like, I could skip all the way to the 544 00:38:06 --> 00:38:13 end and show you the answer. But I just love to write 545 00:38:13 --> 00:38:19 equations on the board so let me just go through that. 546 00:38:19 --> 00:38:25 I am going to simplify this a little further here. 547 00:38:25 --> 00:38:31 And we should remember this form by the Euler relation, 548 00:38:31 --> 00:38:38 ejx=cos x+j sin x. And by the same token, 549 00:38:38 --> 00:38:47 (e^jx + e^-jx)/2=cos x. You all should know this from 550 00:38:47 --> 00:38:54 the Euler relation. So were are using this guy 551 00:38:54 --> 00:39:04 here, ej^x + e^-jx=2cos x. And so this one is 2 cosine of 552 00:39:04 --> 00:39:08 omega nought t, 2 and 2 cancel out, 553 00:39:08 --> 00:39:16 and what I am left with is v(t)=VI-VI cos( omega nought t). 554 00:39:16 --> 00:39:24 And the current is Cdv/dt, which is simply CVI sin( omega 555 00:39:24 --> 00:39:30 nought t). Just remember that omega nought 556 00:39:30 --> 00:39:36 is the square root of 1/LC. We are done. 557 00:39:36 --> 00:39:44 In fact, I did not give that answer the importance that was 558 00:39:44 --> 00:39:48 due so let me just draw. 559 00:39:48 --> 00:39:57 560 00:39:57 --> 00:40:00 There. That is better. 561 00:40:00 --> 00:40:02 Enough math. In a nutshell, 562 00:40:02 --> 00:40:06 what did we do. We wrote the node method, 563 00:40:06 --> 00:40:10 it's a very simple circuit, to write down the equation 564 00:40:10 --> 00:40:15 governing that circuit. And then we grunged through a 565 00:40:15 --> 00:40:18 bunch of math. Not a whole lot here. 566 00:40:18 --> 00:40:23 It is pretty simple. And ended up with a relation 567 00:40:23 --> 00:40:28 that says the voltage across the capacitor for a step input, 568 00:40:28 --> 00:40:33 assuming zero state, is a constant VI-VI cos omega 569 00:40:33 --> 00:40:36 t. Notice that even though I have 570 00:40:36 --> 00:40:39 a step input, the circuit dynamics are such 571 00:40:39 --> 00:40:42 that I get a cosine in there. You can begin to see where 572 00:40:42 --> 00:40:44 these cosines are coming from now. 573 00:40:44 --> 00:40:47 They come in here. And if you recall the example I 574 00:40:47 --> 00:40:50 showed you earlier of the inverter circuit, 575 00:40:50 --> 00:40:52 remember there was a cosine that decayed, 576 00:40:52 --> 00:40:56 that was sort of losing energy and kind of dying out? 577 00:40:56 --> 00:41:00 So you can see where the cosines are coming from. 578 00:41:00 --> 00:41:11 And just to draw you a little sketch here. 579 00:41:11 --> 00:41:26 Let me draw v and i for you and let me plot omega t, 580 00:41:26 --> 00:41:35 pi/2, pi and so on. Let me plot VI. 581 00:41:35 --> 00:41:41 When time t=0, VI=0, cosine omega t is one, 582 00:41:41 --> 00:41:48 and so VI-VI=0. That is simply a cosine that 583 00:41:48 --> 00:41:56 starts out at zero here, and at pi I get cosine omega t 584 00:41:56 --> 00:42:02 is minus one, so I get plus VI on the other 585 00:42:02 --> 00:42:08 side. So I end up at +2VI. 586 00:42:08 --> 00:42:13 At this point the voltage is here. 587 00:42:13 --> 00:42:19 And notice that this guy looks like this. 588 00:42:19 --> 00:42:27 It is a cosine that is translated up so that its mean 589 00:42:27 --> 00:42:37 value is not zero but VI. It is just a translation up of 590 00:42:37 --> 00:42:42 a cosine. Similarly, in this case for the 591 00:42:42 --> 00:42:48 current it is a sinusoidal characteristic. 592 00:42:48 --> 00:42:56 And it looks something like this where the peak is given by 593 00:42:56 --> 00:43:00 CVI, oh, I messed up. 594 00:43:00 --> 00:43:08 595 00:43:08 --> 00:43:14 When I differentiated this is missed the omega nought out 596 00:43:14 --> 00:43:15 there. 597 00:43:15 --> 00:43:25 598 00:43:25 --> 00:43:30 What I would like to do now -- This is the form of the output 599 00:43:30 --> 00:43:33 for a step input. What I would like to do next is 600 00:43:33 --> 00:43:36 show you a demo. But before I show you a demo, 601 00:43:36 --> 00:43:40 I always found it strange that I have a step input and then I 602 00:43:40 --> 00:43:44 have two little elements, how can I get a sine coming out 603 00:43:44 --> 00:43:47 of the output? I would like to get some 604 00:43:47 --> 00:43:50 intuition as to why things behave the way they are. 605 00:43:50 --> 00:43:54 I could go and pray to find out, but let me just give you 606 00:43:54 --> 00:43:58 some very basic insight as to why this behaves the way it 607 00:43:58 --> 00:44:02 does. Let me draw the circuit for you 608 00:44:02 --> 00:44:05 here. And this is my inductor L and 609 00:44:05 --> 00:44:08 capacitance C. Remember this is v. 610 00:44:08 --> 00:44:13 Let me just walk you through what is happening there and get 611 00:44:13 --> 00:44:17 you to understand this. Now, you have seen sines occur 612 00:44:17 --> 00:44:20 before. If you go and write down the 613 00:44:20 --> 00:44:24 equation of motion of a pendulum, you know, 614 00:44:24 --> 00:44:28 you have a pendulum, you move it to one side, 615 00:44:28 --> 00:44:31 let go. It is also governed by 616 00:44:31 --> 00:44:35 sinusoidal characteristics. And you will find that the 617 00:44:35 --> 00:44:39 equation governing its motion is very much of the same form, 618 00:44:39 --> 00:44:43 and you get the sinusoid where you have energy that is sloshing 619 00:44:43 --> 00:44:47 back and forth between maximum potential energy to maximum 620 00:44:47 --> 00:44:51 kinetic energy and zero potential energy back to maximum 621 00:44:51 --> 00:44:53 potential energy, zero kinetic. 622 00:44:53 --> 00:44:56 So it is energy sloshing back and forth. 623 00:44:56 --> 00:45:01 The same way here. Capacitors and inductors store 624 00:45:01 --> 00:45:04 energy. Let's walk through and see what 625 00:45:04 --> 00:45:06 happens. I start off with both of them 626 00:45:06 --> 00:45:09 having the stage zero, zero current, 627 00:45:09 --> 00:45:11 zero voltage. I apply a step here. 628 00:45:11 --> 00:45:14 Boom, the step comes instanteously to VI. 629 00:45:14 --> 00:45:18 I notice that the capacitor voltage cannot change instantly 630 00:45:18 --> 00:45:22 unless there is an infinite pulse of a sort, 631 00:45:22 --> 00:45:24 so this guy cannot change instantly. 632 00:45:24 --> 00:45:29 And so its voltage starts off being zero. 633 00:45:29 --> 00:45:32 So the entire voltage here, KVL must be true no matter 634 00:45:32 --> 00:45:34 what. They are absolutely fundamental 635 00:45:34 --> 00:45:36 principles from Maxwell's equations. 636 00:45:36 --> 00:45:38 KVL must hold, which means that the entire 637 00:45:38 --> 00:45:40 voltage VI must appear across the inductor. 638 00:45:40 --> 00:45:44 I put a big voltage across the inductor and its current begins 639 00:45:44 --> 00:45:45 to build up. There you go. 640 00:45:45 --> 00:45:49 A voltage across the inductor, its current begins to build up. 641 00:45:49 --> 00:45:52 As its current begins to build up that current must flow 642 00:45:52 --> 00:45:54 through the capacitor, too. 643 00:45:54 --> 00:45:57 And as current flows through a capacitor it is depositing 644 00:45:57 --> 00:46:02 charge into the capacitor. As the capacitor begins to get 645 00:46:02 --> 00:46:07 charge deposited on it, its voltage begins to rise. 646 00:46:07 --> 00:46:12 Let's see what happens here. Its voltage keeps rising. 647 00:46:12 --> 00:46:16 At some point, the voltage across the 648 00:46:16 --> 00:46:21 capacitor is equal to VI. But then VI equals this VI 649 00:46:21 --> 00:46:24 here. So when the two become VI, 650 00:46:24 --> 00:46:29 the inductor has zero volts across it. 651 00:46:29 --> 00:46:32 So there is no longer a potential difference that is 652 00:46:32 --> 00:46:35 increasing the current in that direction. 653 00:46:35 --> 00:46:37 At that point, at pi divided by 2, 654 00:46:37 --> 00:46:42 I have some current going into the inductor so there is no 655 00:46:42 --> 00:46:46 longer a pressure that is forcing more current through the 656 00:46:46 --> 00:46:49 inductor because this voltage reaches VI. 657 00:46:49 --> 00:46:53 But remember capacitors like to sit around holding voltages. 658 00:46:53 --> 00:46:57 Just remember that demo. That rinky-dink capacitor sat 659 00:46:57 --> 00:47:01 there stubbornly holding its voltage. 660 00:47:01 --> 00:47:03 And it had a huge spark towards the end. 661 00:47:03 --> 00:47:05 It just sat there holding its voltage. 662 00:47:05 --> 00:47:09 In the same manner, inductors love to sit around 663 00:47:09 --> 00:47:12 holding a current. They will do whatever they can 664 00:47:12 --> 00:47:14 to keep the current going through them. 665 00:47:14 --> 00:47:17 It has got the current going through. 666 00:47:17 --> 00:47:19 And few forces on earth can change that. 667 00:47:19 --> 00:47:22 And so therefore, even though the capacitor 668 00:47:22 --> 00:47:26 voltage is VI and the voltage drop across the inductor is 669 00:47:26 --> 00:47:30 zero, it still keeps supplying a current. 670 00:47:30 --> 00:47:32 It has got the current. It's got inertia. 671 00:47:32 --> 00:47:34 It keeps going. It is like a runaway train. 672 00:47:34 --> 00:47:37 You may not be pushing the train from the back, 673 00:47:37 --> 00:47:41 but once it is running it has got kinetic energy and is going 674 00:47:41 --> 00:47:43 to run no matter what for a least some more time, 675 00:47:43 --> 00:47:46 even if you take away the force on the train. 676 00:47:46 --> 00:47:49 So I have taken away the force on the punching more current 677 00:47:49 --> 00:47:52 through, but it has kinetic energy. 678 00:47:52 --> 00:47:55 It has current flowing through it so it continues to supply a 679 00:47:55 --> 00:47:57 current. Because it continues to supply 680 00:47:57 --> 00:48:02 the current the capacitor voltage keeps increasing. 681 00:48:02 --> 00:48:05 This is a subtle insight which is absolutely spectacular that 682 00:48:05 --> 00:48:09 with zero volts across it, it still keeps pumping that 683 00:48:09 --> 00:48:11 current. Capacitor voltage has gone up. 684 00:48:11 --> 00:48:14 And guess what? The voltage on this side is 685 00:48:14 --> 00:48:17 higher now but this guy is still pumping a current. 686 00:48:17 --> 00:48:20 Man, I have been born to do this, you know, 687 00:48:20 --> 00:48:24 I shall pump a current. However, because the voltage 688 00:48:24 --> 00:48:29 has now gone up here gradually the current begins to diminish. 689 00:48:29 --> 00:48:33 So the capacitor is concerned. You pump a current into me, 690 00:48:33 --> 00:48:36 my voltage goes up. At some point, 691 00:48:36 --> 00:48:39 like a runaway train, it comes to a halt. 692 00:48:39 --> 00:48:44 The current through the capacitor drains and now goes to 693 00:48:44 --> 00:48:47 zero and the capacitor voltage reaches 2VI. 694 00:48:47 --> 00:48:50 So this is at 2VI now and this is at VI. 695 00:48:50 --> 00:48:53 Now the situation is not in equilibrium. 696 00:48:53 --> 00:48:57 At this point there is zero current through it, 697 00:48:57 --> 00:49:01 but guess what? I have a VI pumping in this 698 00:49:01 --> 00:49:05 direction now. I have the same VI punching in 699 00:49:05 --> 00:49:07 this direction. So guess what? 700 00:49:07 --> 00:49:11 Its current must now build up in this direction and its 701 00:49:11 --> 00:49:14 current begins to build up in that direction. 702 00:49:14 --> 00:49:18 That begins to discharge the capacitor and the capacitor then 703 00:49:18 --> 00:49:22 goes on to a negative, or the current goes down to a 704 00:49:22 --> 00:49:26 maximum negative current, and this process continues. 705 00:49:26 --> 00:49:30 What you are seeing here is energy. 706 00:49:30 --> 00:49:33 It is sloshing back and forth between the two, 707 00:49:33 --> 00:49:37 and that is kind of a key. I will just quickly put up a 708 00:49:37 --> 00:49:40 demo that you can watch as you are walking out. 709 00:49:40 --> 00:49:44 With a step input, notice the green is the voltage 710 00:49:44 --> 00:49:48 across the capacitor and the orange is the current through 711 00:49:48 --> 49:51 the capacitor.