1 00:00:00 --> 00:00:06 OK, good morning all. So before we begin, 2 00:00:06 --> 00:00:17 I just thought I'd show you a little news item that I happened 3 00:00:17 --> 00:00:27 to read that was very relevant to what we covered recently in 4 00:00:27 --> 6.002. 5 6.002. --> 00:00:32 So you recall when we did the 6 00:00:32 --> 00:00:36 digital section a few days ago last Thursday, 7 00:00:36 --> 00:00:40 we talked about a switch. We talked about the MOSFET 8 00:00:40 --> 00:00:46 switch, which when turned on and off, by input signals could help 9 00:00:46 --> 00:00:51 build gates which would then be combined in tens of millions of 10 00:00:51 --> 00:00:56 quantities and go into chips like the Pentium 4 and AMD 11 00:00:56 --> 00:01:00 Athlon 64, and so on it so forth. 12 00:01:00 --> 00:01:03 So I just saw this news item that I came across, 13 00:01:03 --> 00:01:07 and this says they are rethinking the basic 14 00:01:07 --> 00:01:11 construction of the products. It talks about the 15 00:01:11 --> 00:01:16 semiconductor manufacturers like AMD, Intel, and others that 16 00:01:16 --> 00:01:20 build digital chips. They are rethinking the basic 17 00:01:20 --> 00:01:25 construction of the products down to the architecture of the 18 00:01:25 --> 00:01:28 transistor. That's a MOS transistor, 19 00:01:28 --> 00:01:33 and the on/off switch inside the chip. 20 00:01:33 --> 00:01:37 OK, now this might imply that there is a single switch inside 21 00:01:37 --> 00:01:40 the chip, but no, there's tens of millions of 22 00:01:40 --> 00:01:44 transistors, or tens of millions of switches inside a chip. 23 00:01:44 --> 00:01:48 And pretty much any advancement that can be made to the basic 24 00:01:48 --> 00:01:52 transistor can have a 10 million to 20 million times effect 25 00:01:52 --> 00:01:56 because there are that many of them on a single chip. 26 00:01:56 --> 00:01:58 So I thought that was very appropriate. 27 00:01:58 --> 00:02:03 OK. Let's dive into a quick review. 28 00:02:03 --> 00:02:06 So this week, we had begun nonlinear 29 00:02:06 --> 00:02:12 analysis, and I just thought I'd blast through a few animations 30 00:02:12 --> 00:02:17 that I've created, trying to give you more insight 31 00:02:17 --> 00:02:23 into the behavior of some of the things that we have done. 32 00:02:23 --> 00:02:27 Now first of all, as I did the last time, 33 00:02:27 --> 00:02:32 let me try to put it in perspective most of what you've 34 00:02:32 --> 00:02:36 learned thus far, and what we will be learning 35 00:02:36 --> 00:02:40 today. So the past week, 36 00:02:40 --> 00:02:43 we have been focusing on nonlinear analysis. 37 00:02:43 --> 00:02:47 And as I pointed out, here is how this fits into the 38 00:02:47 --> 00:02:50 big picture. So, we had our 6.002 world, 39 00:02:50 --> 00:02:53 at what we said is that we are engineers. 40 00:02:53 --> 00:02:58 We are going to devise our own playground in which to play with 41 00:02:58 --> 00:03:01 our own rules. And that's our playground. 42 00:03:01 --> 00:03:05 That's what we're going to learn about in 002, 43 00:03:05 --> 00:03:08 and for that matter, the rest of EECS at MIT. 44 00:03:08 --> 00:03:11 It's all within this playground here. 45 00:03:11 --> 00:03:15 And this is the playground with lumped circuit abstraction, 46 00:03:15 --> 00:03:18 and good old KVL, KCl, node method, 47 00:03:18 --> 00:03:22 your basic composition rules apply within this playground 48 00:03:22 --> 00:03:26 that directly come from Maxwell's equations because you 49 00:03:26 --> 00:03:32 have made the lumped matter discipline assumptions. 50 00:03:32 --> 00:03:35 OK, so then we said a large part of the playground is 51 00:03:35 --> 00:03:39 linear, and some other much more intuitive techniques apply 52 00:03:39 --> 00:03:42 within the linear portion of that playground, 53 00:03:42 --> 00:03:45 techniques like the superposition, 54 00:03:45 --> 00:03:47 Thevenin and Norton. In most exercises, 55 00:03:47 --> 00:03:49 and quizzes, and experiments, 56 00:03:49 --> 00:03:53 and so on that you do in real life, you can pretty much apply 57 00:03:53 --> 00:03:57 these simple techniques. Very rarely do you have to go 58 00:03:57 --> 00:04:00 into the node method for circuits that are more 59 00:04:00 --> 00:04:06 complicated than single source and a couple of elements. 60 00:04:06 --> 00:04:08 And then, there's the nonlinear part. 61 00:04:08 --> 00:04:12 Remember, the reason I showed this is that this is the same 62 00:04:12 --> 00:04:14 playground. OK, linear and nonlinear are 63 00:04:14 --> 00:04:18 part of the same playground. OK, even nonlinear elements are 64 00:04:18 --> 00:04:21 lumped circuit elements, and they follow KVL, 65 00:04:21 --> 00:04:24 KCl, the node equation, and so on. 66 00:04:24 --> 00:04:27 And then, last week we spent some time talking about the 67 00:04:27 --> 00:04:32 digital abstraction. So we focused on a smaller 68 00:04:32 --> 00:04:36 region of the playground. And the assumptions we made in 69 00:04:36 --> 00:04:41 there were even tighter. We said that it is part of the 70 00:04:41 --> 00:04:45 playground we shall only deal with binary values. 71 00:04:45 --> 00:04:49 We'll digitize or lump values into highs and lows, 72 00:04:49 --> 00:04:52 and that's where our circuits are going to be. 73 00:04:52 --> 00:04:56 And these circuits, when looked at as a whole, 74 00:04:56 --> 00:05:00 were nonlinear. So, this is a simple NAND gate 75 00:05:00 --> 00:05:04 circuit. And this is the input/output 76 00:05:04 --> 00:05:06 characteristic. So, for example, 77 00:05:06 --> 00:05:10 if I hold B at zero, and I apply a zero to one 78 00:05:10 --> 00:05:14 transition at A, then this is the output that I 79 00:05:14 --> 00:05:17 will see at C. So notice, this is decidedly 80 00:05:17 --> 00:05:20 nonlinear. Then I said that, 81 00:05:20 --> 00:05:24 look, suppose we had to fix the input values at a given set. 82 00:05:24 --> 00:05:27 OK, so let's say, for example, 83 00:05:27 --> 00:05:31 I fix A at one, and B at one. 84 00:05:31 --> 00:05:34 OK, and then look at the circuit in this situation. 85 00:05:34 --> 00:05:37 What do I find? What I find is that the entire 86 00:05:37 --> 00:05:42 digital set of circuits that we were looking at move over into 87 00:05:42 --> 00:05:45 the linear space for a given set of switch settings, 88 00:05:45 --> 00:05:47 OK? So, when I set A 1 and B 1, 89 00:05:47 --> 00:05:52 A equal to one and B equal to one, my NAND gate becomes like 90 00:05:52 --> 00:05:54 this. OK, it's a simple resistive 91 00:05:54 --> 00:05:56 network with a voltage source, VS. 92 00:05:56 --> 00:06:00 So, for a fixed set of inputs, for a given set of inputs, 93 00:06:00 --> 00:06:04 if I don't change my inputs, then my circuit looks like a 94 00:06:04 --> 00:06:08 linear circuit, and my good old linear analysis 95 00:06:08 --> 00:06:12 techniques apply. So that was last week. 96 00:06:12 --> 00:06:15 And this week, we are looking at the nonlinear 97 00:06:15 --> 00:06:17 space. And we looked at a couple of 98 00:06:17 --> 00:06:21 techniques in the nonlinear space, analytical techniques and 99 00:06:21 --> 00:06:25 graphical techniques. And then, I showed you an 100 00:06:25 --> 00:06:27 example. OK, I showed you an example 101 00:06:27 --> 00:06:31 circuit that was something that I would like to build involving 102 00:06:31 --> 00:06:35 the light emitting expo dweeb, my little garage door opener 103 00:06:35 --> 00:06:39 thingamajig, and I wanted to transmit music over that light 104 00:06:39 --> 00:06:43 beam. I also showed you that it was 105 00:06:43 --> 00:06:47 highly distorted because it was in the nonlinear space. 106 00:06:47 --> 00:06:52 So, today what I'm going to do is introduce a new part of the 107 00:06:52 --> 00:06:56 playground. There's a new part of the 108 00:06:56 --> 00:07:01 playground, and I'll show you a technique whereby by focusing on 109 00:07:01 --> 00:07:06 this part of the playground and disciplining ourselves in the 110 00:07:06 --> 00:07:11 kind of inputs we apply to circuits, I'm going to show you 111 00:07:11 --> 00:07:15 that certain kinds of nonlinear circuits also move over, 112 00:07:15 --> 00:07:20 when used in a particular way, also move into the linear 113 00:07:20 --> 00:07:23 analysis domain. OK, so let me leave that for 114 00:07:23 --> 00:07:28 now and go back into quickly reviewing the motivating example 115 00:07:28 --> 00:07:33 of music that I had taken last time. 116 00:07:33 --> 00:07:37 OK, so here was a little example. 117 00:07:37 --> 00:07:43 So I have a music source, VI, and I apply that. 118 00:07:43 --> 00:07:48 This device that I call the, lightheartedly, 119 00:07:48 --> 00:07:54 the Light Emitting Expo Dweeb has a current, 120 00:07:54 --> 00:07:58 VD, across it, or a voltage, 121 00:07:58 --> 00:08:05 VD, across it, and a current ID through it. 122 00:08:05 --> 00:08:10 And the light intensity, I said, was proportional to the 123 00:08:10 --> 00:08:13 current. And because of that, 124 00:08:13 --> 00:08:19 I was able to get the light to impinge on a receiving device, 125 00:08:19 --> 00:08:24 which produced a current that was proportional to the 126 00:08:24 --> 00:08:28 intensity of light falling on it. 127 00:08:28 --> 00:08:34 And that signal would then be amplified somehow. 128 00:08:34 --> 00:08:37 We haven't talked about all of this stuff yet. 129 00:08:37 --> 00:08:42 This will happen next week. But let's say we somehow 130 00:08:42 --> 00:08:46 amplify the signal and then played out through a set of 131 00:08:46 --> 00:08:49 speakers. All right, so if I had some 132 00:08:49 --> 00:08:54 sort of a music signal here, then I could then transmit the 133 00:08:54 --> 00:09:00 music signal over to the side on top of this light beam. 134 00:09:00 --> 00:09:03 But the problem, as I said the last time, 135 00:09:03 --> 00:09:08 was that our device, the Light Emitting Expo Dweeb 136 00:09:08 --> 00:09:12 had an exponential characteristic, 137 00:09:12 --> 00:09:17 so that I had some trouble in getting undistorted music. 138 00:09:17 --> 00:09:23 So, the characteristic of the VI characteristics of my device 139 00:09:23 --> 00:09:27 looked like so. The ID versus VD curve looked 140 00:09:27 --> 00:09:32 as follows. OK, it was decidedly nonlinear. 141 00:09:32 --> 00:09:36 And because of that, I was getting a lot of 142 00:09:36 --> 00:09:41 distortions in my signal, and I showed you a little trick 143 00:09:41 --> 00:09:47 to plot, given an input waveform at a transfer function such as 144 00:09:47 --> 00:09:50 here to plot the output function. 145 00:09:50 --> 00:09:55 OK, let me show you another little animation that I have 146 00:09:55 --> 00:10:00 created here for you that should give you even more intuition in 147 00:10:00 --> 00:10:06 terms of how it happens. So, this is a characteristic I 148 00:10:06 --> 00:10:08 showed you up here. It's on both sides, 149 00:10:08 --> 00:10:12 but I guess it points to only one unless I shuttle back and 150 00:10:12 --> 00:10:15 forth really fast. So on average, 151 00:10:15 --> 00:10:18 I'll be in both places. But anyway, so here's my ID 152 00:10:18 --> 00:10:21 versus VD characteristic. And as I said, 153 00:10:21 --> 00:10:24 there's an exponential ID versus VD curve. 154 00:10:24 --> 00:10:27 And I want to see what the output looks like, 155 00:10:27 --> 00:10:31 for example, a sinusoidal input. 156 00:10:31 --> 00:10:34 So I said, let's place the input along a little graph, 157 00:10:34 --> 00:10:37 rotate it so, and take a sinusoid, 158 00:10:37 --> 00:10:41 and apply a sinusoid to the input, VI, which would also 159 00:10:41 --> 00:10:44 appear across the Light Emitting Expo Dweeb. 160 00:10:44 --> 00:10:48 And then, what I wanted to see was how the output looked. 161 00:10:48 --> 00:10:52 OK, so let me tell you that the output is going to look like 162 00:10:52 --> 00:10:55 this. OK, the output is going to look 163 00:10:55 --> 00:10:57 like so. And, a little artifice to 164 00:10:57 --> 00:11:01 discover curves like this is to think about a point here 165 00:11:01 --> 00:11:05 corresponding to the point on the transfer curve here, 166 00:11:05 --> 00:11:08 because this is VD, looking at the Y intercept. 167 00:11:08 --> 00:11:14 That's a value of ID, and that's a value of ID here. 168 00:11:14 --> 00:11:17 And, time moves along here, and time moves along here. 169 00:11:17 --> 00:11:20 So, I did this little animation. 170 00:11:20 --> 00:11:24 You'd better be impressed. It took me six hours to do it. 171 00:11:24 --> 00:11:27 So, here it goes. So, let's say I start by 172 00:11:27 --> 00:11:31 focusing on this little point that corresponds to this point 173 00:11:31 --> 00:11:35 on the transfer function, which then, in turn, 174 00:11:35 --> 00:11:38 points to a time, zero, this point on my ID 175 00:11:38 --> 00:11:44 curve. OK, I hope this works. 176 00:11:44 --> 00:11:56 So, as my point moves down [LAUGHTER], this was fun to do, 177 00:11:56 --> 00:12:02 I promise you. So notice that as this point 178 00:12:02 --> 00:12:05 has the following excursion, this had the following 179 00:12:05 --> 00:12:08 excursions here. OK, all right. 180 00:12:08 --> 00:12:11 So let me pause that little animation there. 181 00:12:11 --> 00:12:15 At the end of the lecture, I'll put that up again if you 182 00:12:15 --> 00:12:18 like, and you all can come and play with it. 183 00:12:18 --> 00:12:21 So, you can actually do this in PowerPoint. 184 00:12:21 --> 00:12:26 It took me quite a bit of time to figure out how to do it, 185 00:12:26 --> 00:12:30 though, but it's fun. OK, so let me show you a little 186 00:12:30 --> 00:12:34 demo, and show you a sinusoid, and show you what the output 187 00:12:34 --> 00:12:39 looks like if I apply a sinusoid for VI. 188 00:12:39 --> 00:12:43 So, I'll show you ID as a function of VI when VI is a 189 00:12:43 --> 00:12:44 sinusoid. There you go. 190 00:12:44 --> 00:12:49 So, I applied my sinusoid VI, and this is the current that I 191 00:12:49 --> 00:12:51 get. And notice, this is the 192 00:12:51 --> 00:12:56 transfer function that I talked about, the ID versus VD curve of 193 00:12:56 --> 00:13:05 my Light Emitting Expo Dweeb. And I get this highly nonlinear 194 00:13:05 --> 00:13:15 transformation of the input as I get to the output. 195 00:13:15 --> 00:13:25 OK, so that is a problem. And then, I also played some 196 00:13:25 --> 00:13:30 music for you. Let's do that, 197 00:13:30 --> 00:13:35 too. I played some music for you. 198 00:13:35 --> 00:13:39 I applied the music as an input to the circuit, 199 00:13:39 --> 00:13:44 and that's the output. OK, that's the output that I'm 200 00:13:44 --> 00:13:49 observing at the amplifier. It's highly distorted. 201 00:13:49 --> 00:13:52 OK, we can stop that. There you go. 202 00:13:52 --> 00:13:56 OK, so that was my problem. OK, so we had covered, 203 00:13:56 --> 00:14:01 we had gone this far last Tuesday. 204 00:14:01 --> 00:14:05 I set the problem up for you, motivated what we had to do, 205 00:14:05 --> 00:14:10 and showed you that I was able to transmit music over my garage 206 00:14:10 --> 00:14:15 door opener, but I did not think I could listen to that music for 207 00:14:15 --> 00:14:18 very long. So, I challenged all of us to 208 00:14:18 --> 00:14:23 think about how a trick that I could use to be able to transmit 209 00:14:23 --> 00:14:25 music and have a linear response. 210 00:14:25 --> 00:14:30 So, did you people get time to think about it? 211 00:14:30 --> 00:14:34 So how many people here think they know the answer? 212 00:14:34 --> 00:14:38 It's OK, don't be modest. Go ahead. 213 00:14:38 --> 00:14:42 Could you speak louder? Yeah, you find another 214 00:14:42 --> 00:14:47 something, kind of element, that's got the opposite graph 215 00:14:47 --> 00:14:51 so that when you add them together. 216 00:14:51 --> 00:14:54 Oh, this guy wants to cheat. No. 217 00:14:54 --> 00:15:00 He wants a new element. So, no, no new elements. 218 00:15:00 --> 00:15:02 Pardon? Build an MP3 encoder. 219 00:15:02 --> 00:15:05 Ah-ha, so that will happen much later. 220 00:15:05 --> 00:15:08 Yes? Digitize the signal before you 221 00:15:08 --> 00:15:12 send it to the LED? Digitize the signal before you 222 00:15:12 --> 00:15:15 send it to the LED. But in some sense, 223 00:15:15 --> 00:15:20 each of these solutions is a huge sledgehammer approach to 224 00:15:20 --> 00:15:24 look at solving it. There's a much simpler 225 00:15:24 --> 00:15:26 technique I can apply here. Yeah? 226 00:15:26 --> 00:15:32 Add a voltage offset. Ah, ah-ha, that might work. 227 00:15:32 --> 00:15:34 What else? So let's say, 228 00:15:34 --> 00:15:36 here's my signal, right? 229 00:15:36 --> 00:15:41 If I add a voltage offset, that will just bump the signal 230 00:15:41 --> 00:15:43 up here. Then the curve is still 231 00:15:43 --> 00:15:46 nonlinear. But you're getting there. 232 00:15:46 --> 00:15:50 Well, I'll tell you what. Let's pause here. 233 00:15:50 --> 00:15:55 Let me quit while I'm ahead. OK, so the answer here, 234 00:15:55 --> 00:15:58 folks, is Zen. OK, what I want you to do is, 235 00:15:58 --> 00:16:03 so, in Zen, what you have to do is you have to sit down in a 236 00:16:03 --> 00:16:08 courtyard, and look at a rock, like a small rock on the 237 00:16:08 --> 00:16:12 ground. And you got a focus on it till 238 00:16:12 --> 00:16:15 the rest of Earth kind of vanishes. 239 00:16:15 --> 00:16:18 Just focus on the rock. OK, now make like you're in a 240 00:16:18 --> 00:16:22 courtyard, and you're looking at this little area here. 241 00:16:22 --> 00:16:25 Just look at this. OK, and I'll give you ten 242 00:16:25 --> 00:16:26 seconds. Sit down quietly, 243 00:16:26 --> 00:16:29 and no sounds. Just stare at the spot here. 244 00:16:29 --> 00:16:33 OK, make believe this is your little rock, and just stand 245 00:16:33 --> 00:16:38 there and think about it. OK, I'll give you five seconds 246 00:16:38 --> 00:16:41 to do that. Just stare at it. 247 00:16:41 --> 00:16:45 And very soon, the answer should pop into your 248 00:16:45 --> 00:16:47 heads. OK, what do you see? 249 00:16:47 --> 00:16:52 This guy, if I focus on this really small region of the 250 00:16:52 --> 00:16:57 graph, this small little piece looks more or less linear. 251 00:16:57 --> 00:17:02 OK, hmm, so that should give me some insight. 252 00:17:02 --> 00:17:05 This whole thing, the macrograph is nonlinear. 253 00:17:05 --> 00:17:09 But I focus on a little rinky dinky piece of that graph like 254 00:17:09 --> 00:17:11 so, that appears more or less linear. 255 00:17:11 --> 00:17:14 If it's small enough, that appears linear. 256 00:17:14 --> 00:17:18 So, I'm staring at this, and that appears linear. 257 00:17:18 --> 00:17:21 The question is, how do I exploit this little 258 00:17:21 --> 00:17:24 small, little, linear region to get a linear 259 00:17:24 --> 00:17:27 response from my device. OK, so here's the trick that 260 00:17:27 --> 00:17:31 I'm going to use. The little trick that I'm going 261 00:17:31 --> 00:17:35 to use is the following. Notice that, 262 00:17:35 --> 00:17:42 let me call this voltage at the center of this region capital 263 00:17:42 --> 00:17:43 VD. What I can do, 264 00:17:43 --> 00:17:50 if I take my input signal, and I just pointed out earlier, 265 00:17:50 --> 00:17:52 I bump it up. I boost it. 266 00:17:52 --> 00:17:57 OK, so I apply a DC offset to my input signal, 267 00:17:57 --> 00:18:02 like so. So I apply some input signal, 268 00:18:02 --> 00:18:07 VI, which is also equal to the VD if I look at a variable 269 00:18:07 --> 00:18:12 across the nonlinear element. If I apply a DC offset, 270 00:18:12 --> 00:18:16 VI, and I superimpose the music on top of that, 271 00:18:16 --> 00:18:20 let me call my music, just to distinguish between the 272 00:18:20 --> 00:18:23 two, capital VI, and the small vi. 273 00:18:23 --> 00:18:27 OK, that's my music. So here's my capital VD, 274 00:18:27 --> 00:18:32 my DC offset. And I want to superimpose my 275 00:18:32 --> 00:18:35 music on top of that. OK, so I've gotten halfway 276 00:18:35 --> 00:18:38 there. By superimposing my music here 277 00:18:38 --> 00:18:43 instead of having excursions out here, I now have excursions out 278 00:18:43 --> 00:18:46 here. OK, and so I'm using some 279 00:18:46 --> 00:18:50 portion of the graph here. But that's still way beyond the 280 00:18:50 --> 00:18:55 small little element there. So a second think that I do in 281 00:18:55 --> 00:19:00 addition to boosting up the signal is shrink it. 282 00:19:00 --> 00:19:02 Think of boost and shrink, BS. 283 00:19:02 --> 00:19:08 So what I want to do is boost up the signal using a DC offset, 284 00:19:08 --> 00:19:13 and shrink the sucker. OK, so I'm going to go with a 285 00:19:13 --> 00:19:19 small signal and bump it up. OK, so now what happens is that 286 00:19:19 --> 00:19:24 small signal in its excursions, only uses that little portion 287 00:19:24 --> 00:19:27 of the graph. OK, again, remember: 288 00:19:27 --> 00:19:30 bump and shrink, bump and shrink, 289 00:19:30 --> 00:19:37 two things, boost and shrink. So what do you think of that 290 00:19:37 --> 00:19:38 trick? So, by doing that, 291 00:19:38 --> 00:19:43 what happens is that signal that has excursions here will 292 00:19:43 --> 00:19:46 produce a corresponding response in this region, 293 00:19:46 --> 00:19:49 OK? And I argue that since this is 294 00:19:49 --> 00:19:53 more or less like a straight line, I invoke Zen here, 295 00:19:53 --> 00:19:57 and argue that this little signal now gets transformed, 296 00:19:57 --> 00:20:02 and I get a linear response. OK: boost and shrink. 297 00:20:02 --> 00:20:06 So in terms of my circuit, let me draw it out for you. 298 00:20:06 --> 00:20:12 My Light Emitting Expo Dweeb, and this whole signal was what 299 00:20:12 --> 00:20:16 I used to call V capital I, and that's made up of two 300 00:20:16 --> 00:20:19 components now, a bump offset, 301 00:20:19 --> 00:20:23 and a shrunk voltage VI. It shrunk, so therefore I've 302 00:20:23 --> 00:20:27 used the small v and small i, like, really, 303 00:20:27 --> 00:20:31 really small. In the same manner, 304 00:20:31 --> 00:20:37 I get a VD ID across the LED, and the corresponding values 305 00:20:37 --> 00:20:41 here will also have a DC offset and a small response. 306 00:20:41 --> 00:20:44 Let me call that ID plus I small d. 307 00:20:44 --> 00:20:49 I'll do all this mathematically in a second as well, 308 00:20:49 --> 00:20:54 but first let me do it completely intuitively so you 309 00:20:54 --> 00:20:57 get some insight into what's going on. 310 00:20:57 --> 00:21:03 And, VD is simply capital VD plus small vd. 311 00:21:03 --> 00:21:06 OK, and this is the same as VI, I, and VI. 312 00:21:06 --> 00:21:09 OK, so what have I done? I've done two things. 313 00:21:09 --> 00:21:11 I have said, as an engineer, 314 00:21:11 --> 00:21:16 OK, I care about getting music across my garage door opener. 315 00:21:16 --> 00:21:19 And I'll do what it takes to do that. 316 00:21:19 --> 00:21:22 OK, so as an engineer, I'll do two things. 317 00:21:22 --> 00:21:25 I'm going to bump my signal up and shrink it. 318 00:21:25 --> 00:21:29 And the bumping and shrinking, and I do it like this. 319 00:21:29 --> 00:21:33 I shrink my signal, the music signal here, 320 00:21:33 --> 00:21:38 and add a DC offset. OK, and I claim that the music 321 00:21:38 --> 00:21:41 I listened on the other side now, provided I have enough 322 00:21:41 --> 00:21:45 amplification there, is going to be undistorted. 323 00:21:45 --> 00:21:49 OK, so far I've showing this to you completely intuitively using 324 00:21:49 --> 00:21:50 little sketches, no math. 325 00:21:50 --> 00:21:53 I promise you, I'll give you a bunch of math 326 00:21:53 --> 00:21:56 in a few seconds, but just get the basic idea, 327 00:21:56 --> 00:22:00 and get the intuition behind it. 328 00:22:00 --> 00:22:04 So let's go back to our demo and take a look. 329 00:22:04 --> 00:22:08 So remember, BS, right, bump and shrink. 330 00:22:08 --> 00:22:12 So what I'm going to do is first of all, 331 00:22:12 --> 00:22:18 let me bump up the signal. So, what I'll do is I want to 332 00:22:18 --> 00:22:23 add an offset to my input, and let me bump it up. 333 00:22:23 --> 00:22:28 Let me shrink it first. It'll make the point a little 334 00:22:28 --> 00:22:32 clearer. So, the big input, 335 00:22:32 --> 00:22:35 green, is a big input. Let me shrink it. 336 00:22:35 --> 00:22:46 337 00:22:46 --> 00:22:49 OK, so I've made my input small, and in the middle of that 338 00:22:49 --> 00:22:52 picture out there, you see the region of the 339 00:22:52 --> 00:22:54 transfer curve that's being articulated. 340 00:22:54 --> 00:22:58 OK, this region of the curve is being articulated by the small 341 00:22:58 --> 00:23:00 signal. It's a much smaller signal. 342 00:23:00 --> 00:23:03 And the output is still distorted because I have to do 343 00:23:03 --> 00:23:08 two things: bump and shrink. I've only shrunk. 344 00:23:08 --> 00:23:13 OK, let me bump it up now. What's the yellow curve? 345 00:23:13 --> 00:23:20 It's going to get linear. It's going to get proportional 346 00:23:20 --> 00:23:25 to the input. Then I'm bumping it up now. 347 00:23:25 --> 00:23:30 I can make it smaller, make it even smaller, 348 00:23:30 --> 00:23:34 there you go. Isn't that fantastic? 349 00:23:34 --> 00:23:37 So, I'm making nature do my bidding here, 350 00:23:37 --> 00:23:40 OK? So, this is one of those, 351 00:23:40 --> 00:23:44 when I learned electronics and so on many, many years ago, 352 00:23:44 --> 00:23:48 this was one of those really big ah-ha moments for me, 353 00:23:48 --> 00:23:51 saying, wow, that stuff is cool. 354 00:23:51 --> 00:23:55 It's something that I couldn't think about myself, 355 00:23:55 --> 00:23:59 and it's not obvious, and by being disciplined and 356 00:23:59 --> 00:24:03 creative in how I use circuits, I can do really, 357 00:24:03 --> 00:24:07 really cool things. OK, remember this as a big 358 00:24:07 --> 00:24:11 ah-ha moment for you. So, here's my little signal 359 00:24:11 --> 00:24:15 that I've shrunk and bumped up, and my output is a sinusoid, 360 00:24:15 --> 00:24:18 and not this funny, distorted waveform. 361 00:24:18 --> 00:24:22 And notice that this is the region of the curve that is 362 00:24:22 --> 00:24:26 being articulated. So, I can make the signal even 363 00:24:26 --> 00:24:29 smaller if I like. OK, and what I'd like to do 364 00:24:29 --> 00:24:33 next is play music for you, and if you don't believe your 365 00:24:33 --> 00:24:38 eyes, you can at least believe your ears. 366 00:24:38 --> 00:24:43 Let me go to the distorted signal again, 367 00:24:43 --> 00:24:48 switch to music, and raise it up. 368 00:24:48 --> 00:24:56 OK, now what we'll do is shrink the music signal and then bump 369 00:24:56 --> 00:25:01 it up. Can I turn the volume down a 370 00:25:01 --> 00:25:06 little bit? That's good. 371 00:25:06 --> 00:25:13 OK, so if I shrunk the volume a little bit, and let me bump it 372 00:25:13 --> 00:25:17 up, now. [MUSIC PLAYS] Just remember 373 00:25:17 --> 00:25:23 this as a big ah-ha moment. OK, the signal is really, 374 00:25:23 --> 00:25:26 really small. I like that. 375 00:25:26 --> 00:25:33 I like the enthusiasm. OK, so the signal's very small, 376 00:25:33 --> 00:25:38 and I get a more or less linear response. 377 00:25:38 --> 00:25:42 OK. All right, so that's intuition, 378 00:25:42 --> 00:25:46 and the approach that I've taken is called, 379 00:25:46 --> 00:25:50 it's variously called small signal analysis, 380 00:25:50 --> 00:25:54 incremental analysis, small signal method, 381 00:25:54 --> 00:25:59 small signal discipline, whatever you want. 382 00:25:59 --> 00:26:08 383 00:26:08 --> 00:26:13 OK, this simply says that by boosting and shrinking my 384 00:26:13 --> 00:26:20 signal, I get a response that's more or less linear even when I 385 00:26:20 --> 00:26:25 have a nonlinear device. And this technique is called 386 00:26:25 --> 00:26:31 the small signal approach. So, just to focus on that a 387 00:26:31 --> 00:26:36 little bit longer, switch to page five of your 388 00:26:36 --> 00:26:42 notes and let me draw something out for you. 389 00:26:42 --> 00:26:47 390 00:26:47 --> 00:26:52 OK, so what I have here, this is my offset VD, 391 00:26:52 --> 00:26:59 and from the VD offset I have my little signal V small d, 392 00:26:59 --> 00:27:06 and the total signal is called V capital D. 393 00:27:06 --> 00:27:10 Offset, small signal, and that's my total signal. 394 00:27:10 --> 00:27:14 OK, notice the offset is all capital. 395 00:27:14 --> 00:27:19 The total signal is small v capital D, and the music or the 396 00:27:19 --> 00:27:25 small signal is small v small d. Similarly, the output is going 397 00:27:25 --> 00:27:30 to look like this, and here I get an offset in the 398 00:27:30 --> 00:27:35 output ID. I get a corresponding signal, 399 00:27:35 --> 00:27:40 I small d, and I get a total signal, I capital D, 400 00:27:40 --> 00:27:43 OK? The cool thing to notice is 401 00:27:43 --> 00:27:48 that the signal here, the output signal here 402 00:27:48 --> 00:27:54 corresponding to the input signal, the music signal, 403 00:27:54 --> 00:27:59 VD, is small I small D, and that is more or less 404 00:27:59 --> 00:28:02 linear. OK, and I can even plot the 405 00:28:02 --> 00:28:10 signal like so. This is my input, 406 00:28:10 --> 00:28:15 v capital D. That's T. 407 00:28:15 --> 00:28:26 This is VD, V small d. That is my total input. 408 00:28:26 --> 00:28:36 And similarly, I have an output. 409 00:28:36 --> 00:28:44 And this is my output ID. And, that looks like this, 410 00:28:44 --> 00:28:49 I capital D, small i small d, 411 00:28:49 --> 00:28:58 total signal I capital D. OK, so that's the small signal 412 00:28:58 --> 00:29:02 method. So, let me summarize that for 413 00:29:02 --> 00:29:03 you. 414 00:29:03 --> 00:29:12 415 00:29:12 --> 00:29:16 There are three steps to the method. 416 00:29:16 --> 00:29:21 So, first of all, operate at some DC offset. 417 00:29:21 --> 00:29:27 This is also called DC bias, and in that example it's VDID. 418 00:29:27 --> 00:29:33 OK, so I choose an operating point that bumps up the 419 00:29:33 --> 00:29:39 operation in some region of interest. 420 00:29:39 --> 00:29:45 The second step is to superimpose small signal on top 421 00:29:45 --> 00:29:53 of VD, capital V capital D, to superimpose a small signal, 422 00:29:53 --> 00:30:00 and the third step is observe the response -- 423 00:30:00 --> 00:30:06 -- and the response, small i small d, 424 00:30:06 --> 00:30:14 that's the music part of the response, ID, 425 00:30:14 --> 00:30:23 is approximately linear. OK, three steps to the method 426 00:30:23 --> 00:30:32 here, and just remember this notation. 427 00:30:32 --> 00:30:39 And, my notation in the small signal model is as follows. 428 00:30:39 --> 00:30:46 My total signal ID is the sum of two signals, 429 00:30:46 --> 00:30:50 I capital D plus small i small d. 430 00:30:50 --> 00:30:54 This is called the total signal. 431 00:30:54 --> 00:31:03 That's called the DC offset. And this is the superimposed 432 00:31:03 --> 00:31:07 small signal. OK, total signal, 433 00:31:07 --> 00:31:12 DC offset, plus the small signal. 434 00:31:12 --> 00:31:17 And sometimes, especially when doing math, 435 00:31:17 --> 00:31:24 and so on, we may oftentimes represent ID as a delta, 436 00:31:24 --> 00:31:29 I capital D, OK, to show that ID is 437 00:31:29 --> 00:31:37 incremental change in the value of I capital D. 438 00:31:37 --> 00:31:41 And because of that, this method is also often 439 00:31:41 --> 00:31:46 called the incremental method, incremental analysis. 440 00:31:46 --> 00:31:51 OK, so far what I've done is given you some intuition. 441 00:31:51 --> 00:31:55 I've developed a small, simple method, 442 00:31:55 --> 00:31:59 given you some insight into why we use this method, 443 00:31:59 --> 00:32:05 and also shown you some demonstrations that show that 444 00:32:05 --> 00:32:09 when I bump and shrink, and observe the response, 445 00:32:09 --> 00:32:15 I do get a more or less linear response. 446 00:32:15 --> 00:32:21 So let me now do this mathematically and show you that 447 00:32:21 --> 00:32:27 mathematically, you can also derive your 448 00:32:27 --> 00:32:32 response to be a linear response. 449 00:32:32 --> 00:32:38 This is page seven. So, I know that ID is some 450 00:32:38 --> 00:32:47 function of the diode voltage. F was my nonlinear function. 451 00:32:47 --> 00:32:54 OK, so my function F was a nonlinear function. 452 00:32:54 --> 00:33:00 So therefore, ID was nonlinearly related to 453 00:33:00 --> 00:33:03 VD. So, let's do the math. 454 00:33:03 --> 00:33:07 So as a first step, what we did was replace VD by a 455 00:33:07 --> 00:33:11 DC offset, the small signal method, a DC offset, 456 00:33:11 --> 00:33:13 plus a small incremental change. 457 00:33:13 --> 00:33:17 OK, by doing the math, let me simply use the delta VD 458 00:33:17 --> 00:33:21 notation to show you that I'm dealing with small increments, 459 00:33:21 --> 00:33:25 and also because in the mathematics community, 460 00:33:25 --> 00:33:28 when you learn about some of these techniques, 461 00:33:28 --> 00:33:32 they will use the incremental change notation, 462 00:33:32 --> 00:33:38 which is the delta VD notation. In electrical engineering, 463 00:33:38 --> 00:33:41 we use a small v, small d notation. 464 00:33:41 --> 00:33:46 So, this is a large DC offset, and this is a small change 465 00:33:46 --> 00:33:50 about that offset. So, you folks have taken math 466 00:33:50 --> 00:33:54 courses before, and been looking at finding out 467 00:33:54 --> 00:33:59 the value of a function, which is a small change for an 468 00:33:59 --> 00:34:04 input value, which is a small change about a big input value 469 00:34:04 --> 00:34:09 or a big DC point is Taylor's expansion. 470 00:34:09 --> 00:34:12 OK, so let's use Taylor's series expansion, 471 00:34:12 --> 00:34:16 OK, and substitute VD plus delta VD into this, 472 00:34:16 --> 00:34:21 and see what ID looks like. Again, let me tell you where 473 00:34:21 --> 00:34:24 I'm going with this. ID equals F of VD. 474 00:34:24 --> 00:34:28 This is a nonlinear function, OK? 475 00:34:28 --> 00:34:33 I claim that by replacing VD, the input, with the DC offset 476 00:34:33 --> 00:34:37 plus a small value, the resulting response to the 477 00:34:37 --> 00:34:40 small value will be linear, OK? 478 00:34:40 --> 00:34:46 So what I'm going to do next is replace VD with this sum here, 479 00:34:46 --> 00:34:50 and then do the math, and show you that the response 480 00:34:50 --> 00:34:53 corresponding, or the change in ID 481 00:34:53 --> 00:34:58 corresponding to the change in VD is going to be linear. 482 00:34:58 --> 00:35:04 All right, so let's expand this function using Taylor's series 483 00:35:04 --> 00:35:10 near the DC offset point, capital V capital D. 484 00:35:10 --> 00:35:13 OK, so ID is simply, by Taylor's series, 485 00:35:13 --> 00:35:19 I want to find out a value of the function close to V capital 486 00:35:19 --> 00:35:21 D. OK, so I take the value of the 487 00:35:21 --> 00:35:26 function at that point, and then I add a few terms in 488 00:35:26 --> 00:35:34 my Taylor's series expansion. The first term is simply the 489 00:35:34 --> 00:35:43 good old Taylor's series stuff. OK, the first term is the first 490 00:35:43 --> 00:35:49 derivative of the function times the change. 491 00:35:49 --> 00:35:57 And then, the second one is second derivative. 492 00:35:57 --> 00:36:09 493 00:36:09 --> 00:36:11 OK, and then I get higher order terms. 494 00:36:11 --> 00:36:16 So this is nothing new here. This is good old Taylor series 495 00:36:16 --> 00:36:19 expansion, and again, let me tell you where I'm 496 00:36:19 --> 00:36:22 going. I want to look at the response 497 00:36:22 --> 00:36:27 for an input that looks like this, and I want to show you at 498 00:36:27 --> 00:36:30 the end of the day that the response in ID, 499 00:36:30 --> 00:36:34 the effect on ID of using an input like this is as if that 500 00:36:34 --> 00:36:39 effect, the incremental change is linearly related to the small 501 00:36:39 --> 00:36:44 input, delta VD. So here's my Taylor's series 502 00:36:44 --> 00:36:47 expansion for delta V. Now remember, 503 00:36:47 --> 00:36:52 I told you that delta VD is much, much smaller than V 504 00:36:52 --> 00:36:54 capital D. OK, it's a very, 505 00:36:54 --> 00:36:59 very small quantity. But that quantity is really 506 00:36:59 --> 00:37:03 very small. Then what I'm going to get is 507 00:37:03 --> 00:37:07 that my output is, I can begin to ignore my second 508 00:37:07 --> 00:37:09 order terms. OK, delta VD is very, 509 00:37:09 --> 00:37:13 very, very small. Then, what I'm going to do is 510 00:37:13 --> 00:37:18 that ignore higher order terms. So I'll go and ignore higher 511 00:37:18 --> 00:37:20 order terms. They'll all go to zero. 512 00:37:20 --> 00:37:25 Remember, I can do this because by design I've chosen delta VD 513 00:37:25 --> 00:37:29 to be very, very, very small. 514 00:37:29 --> 00:37:32 OK, remember, we are engineers. 515 00:37:32 --> 00:37:37 I've chosen it in a way that this is very small. 516 00:37:37 --> 00:37:42 OK, so I'm telling you that's the case, and under those 517 00:37:42 --> 00:37:48 conditions, I can ignore second higher order terms, 518 00:37:48 --> 00:37:53 in which case I am left with this expression here. 519 00:37:53 --> 00:38:00 So let me rewrite this. Let me rewrite this down here. 520 00:38:00 --> 00:38:15 521 00:38:15 --> 00:38:19 OK, I've just copied this turnout, I've ignored all these 522 00:38:19 --> 00:38:24 terms here, and so I have a more or less equal to sign that 523 00:38:24 --> 00:38:27 remains. So what I'm going to do is when 524 00:38:27 --> 00:38:32 I apply a small input of this form to a large DC offset, 525 00:38:32 --> 00:38:37 my output is also going to look like some output offset with a 526 00:38:37 --> 00:38:43 change in the output offset. And let me call the output 527 00:38:43 --> 00:38:49 offset I capital D, and some small change in the 528 00:38:49 --> 00:38:54 output delta ID. OK, we'll make sure we can 529 00:38:54 --> 00:38:59 convince ourselves that this is indeed the case. 530 00:38:59 --> 00:39:05 Notice that this guy here, F of capital V capital D is a 531 00:39:05 --> 00:39:12 constant. That's a constant with respect 532 00:39:12 --> 00:39:17 to the incremental change, delta VD. 533 00:39:17 --> 00:39:23 Similarly, this part here is a constant. 534 00:39:23 --> 00:39:31 Notice that this term here is the first derivative of the 535 00:39:31 --> 00:39:41 function evaluated at the DC bias point, capital V capital D. 536 00:39:41 --> 00:39:45 OK, so this term is also a constant with respect to delta 537 00:39:45 --> 00:39:47 VD. So notice, then, 538 00:39:47 --> 00:39:52 I have a constant term plus a constant term multiplying a 539 00:39:52 --> 00:39:54 small change, delta VD. 540 00:39:54 --> 00:39:57 So what I can do next is, in this case, 541 00:39:57 --> 00:40:02 given that I have a constant term on both sides, 542 00:40:02 --> 00:40:05 and on this side it's a time varying term, 543 00:40:05 --> 00:40:11 what I can do is equate the two constant terms. 544 00:40:11 --> 00:40:14 I can go ahead and equate these two terms. 545 00:40:14 --> 00:40:18 Remember, I have a constant plus a time varying term, 546 00:40:18 --> 00:40:22 OK, if I'm assuming here that delta VD, my little music signal 547 00:40:22 --> 00:40:26 is a time varying term. So, this constant will equal 548 00:40:26 --> 00:40:31 this, so ID must equal F of VD. And I know that's the case 549 00:40:31 --> 00:40:36 because the function evaluated at the DC offset gives me the DC 550 00:40:36 --> 00:40:39 current ID. And similarly, 551 00:40:39 --> 00:40:46 ID is equal to that component. Delta ID is equal to D, 552 00:40:46 --> 00:40:48 F of -- 553 00:40:48 --> 00:41:00 554 00:41:00 --> 00:41:04 OK, so my incremental change in the output is the first 555 00:41:04 --> 00:41:09 derivative multiplied by the small change in the current. 556 00:41:09 --> 00:41:13 OK, so I'm pretty much done. So, therefore, 557 00:41:13 --> 00:41:17 notice that delta ID is proportional to delta VD. 558 00:41:17 --> 00:41:21 OK, and that's what I had set out to show. 559 00:41:21 --> 00:41:26 Remember, I had set out to show that provided my input is a 560 00:41:26 --> 00:41:31 small excursion around a large DC offset, 561 00:41:31 --> 00:41:35 then my output could also be a large DC offset with a small 562 00:41:35 --> 00:41:39 excursion on top of it where the two excursions, 563 00:41:39 --> 00:41:43 the input excursion and the output excursion would be 564 00:41:43 --> 00:41:48 linearly related like so. OK, and the method is very 565 00:41:48 --> 00:41:51 simple. I simply expanded the function 566 00:41:51 --> 00:41:53 about that point, that DC point, 567 00:41:53 --> 00:41:58 neglected higher order terms, and notice that my incremental 568 00:41:58 --> 00:42:03 term was simply the derivative plus the incremental change, 569 00:42:03 --> 00:42:09 a derivative times the incremental change in the input. 570 00:42:09 --> 00:42:12 Move onto page nine, and I'd like to give you a 571 00:42:12 --> 00:42:16 quick graphical interpretation of this. 572 00:42:16 --> 00:42:19 So I gave an intuitive explanation earlier. 573 00:42:19 --> 00:42:24 This is a mathematical explanation that shows you that 574 00:42:24 --> 00:42:28 the input could be linearly related to the output, 575 00:42:28 --> 00:42:33 provided, the outputs would be linearly related to the input, 576 00:42:33 --> 00:42:38 provided the input has a DC offset, and small excursions 577 00:42:38 --> 00:42:43 about that DC offset. So, let me give you some 578 00:42:43 --> 00:42:50 intuition in what you've really done here, using a little graph 579 00:42:50 --> 00:42:54 here. So, I'm going to plot ID versus 580 00:42:54 --> 00:43:00 VD, and notice that I have some point here, V capital D, 581 00:43:00 --> 00:43:03 I capital D. That's my DC bias. 582 00:43:03 --> 00:43:08 So, I have some DC bias point here. 583 00:43:08 --> 00:43:14 OK, what is this? That is simply the slope of the 584 00:43:14 --> 00:43:20 curve at that point. OK, it's the slope of this 585 00:43:20 --> 00:43:28 curve evaluated at this point. So this guy here is simply the 586 00:43:28 --> 00:43:34 slope of this curve evaluated at ID VD. 587 00:43:34 --> 00:43:41 OK, now, what I care about is this point here, 588 00:43:41 --> 00:43:50 and this point here. So let's say that this is delta 589 00:43:50 --> 00:43:57 VD, all right, and that corresponds to this 590 00:43:57 --> 00:44:03 point here. So what I've done is taken the 591 00:44:03 --> 00:44:07 slope and multiplied that by delta VD. 592 00:44:07 --> 00:44:12 So I've taken the slope, and multiplied it by delta VD, 593 00:44:12 --> 00:44:17 OK, and that gives me this component here. 594 00:44:17 --> 00:44:22 OK, and so, this is the point that I'm going to get. 595 00:44:22 --> 00:44:27 So in other words, what I've done is approximated 596 00:44:27 --> 00:44:34 point A using the Taylor trick by the point B. 597 00:44:34 --> 00:44:38 OK, so this is a point, A, which is what I really want, 598 00:44:38 --> 00:44:44 and I've approximated that by taking the slope of the function 599 00:44:44 --> 00:44:48 at V capital D, and multiplying that by the 600 00:44:48 --> 00:44:53 change in the input to get the corresponding Y offset, 601 00:44:53 --> 00:44:55 and that's the point that I get. 602 00:44:55 --> 00:45:00 And notice that if I make this delta VD small enough, 603 00:45:00 --> 00:45:05 then the error between these two points becomes smaller and 604 00:45:05 --> 00:45:10 smaller. So back to our example, 605 00:45:10 --> 00:45:16 so ID was a e to the BVD. This was the relation for our 606 00:45:16 --> 00:45:21 Expo Dweeb, and let me just plug in the values. 607 00:45:21 --> 00:45:26 So, ID plus small id. Notice, I'm just shuttling back 608 00:45:26 --> 00:45:34 and forth between the notation delta VD, and small v small d. 609 00:45:34 --> 00:45:43 610 00:45:43 --> 00:45:51 OK, and so that is given by a e to the BVD, oops, 611 00:45:51 --> 00:45:59 plus, I'm just writing that equation up there. 612 00:45:59 --> 00:46:08 Let me call this equation X. And so, I get the second term 613 00:46:08 --> 00:46:13 is the derivative, ab times e to the BVD times 614 00:46:13 --> 00:46:18 delta VD, small VD, and equating this term that the 615 00:46:18 --> 00:46:22 DC offset. Notice that this is the DC 616 00:46:22 --> 00:46:27 offset in the output, and the small signal, 617 00:46:27 --> 00:46:33 ID is, further notice that in this particular example, 618 00:46:33 --> 00:46:37 what's that? a e to the BVD. 619 00:46:37 --> 00:46:45 That's simply ID again. It just happens to be that way 620 00:46:45 --> 00:46:50 in this example. So, I get ID times BVD. 621 00:46:50 --> 00:46:56 So, for my input, small id, my incremental change 622 00:46:56 --> 00:47:03 in the output is some ID times B times VD. 623 00:47:03 --> 00:47:07 And notice that this is a constant. 624 00:47:07 --> 00:47:15 And because that is a constant, my small signal behavior ID is 625 00:47:15 --> 00:47:20 going to be linearly related to the signal, VD, 626 00:47:20 --> 00:47:27 the input signal VD. OK, in the last three minutes, 627 00:47:27 --> 00:47:34 I'd like to give you one additional insight. 628 00:47:34 --> 00:47:38 So what we've shown so far is if I have an offset and a small 629 00:47:38 --> 00:47:42 change above it, then my output ID will be 630 00:47:42 --> 00:47:47 linearly related to my input. Now let's stare at this thing 631 00:47:47 --> 00:47:49 again. Let me rewrite it. 632 00:47:49 --> 00:47:51 It's some constant IDB times VD. 633 00:47:51 --> 00:47:55 So, where have we seen such an expression before? 634 00:47:55 --> 00:48:00 OK, where ID was some constant times VD. 635 00:48:00 --> 00:48:03 OK, remember, I equals V divided by R: 636 00:48:03 --> 00:48:06 Ohm's law. What I want to show you now is 637 00:48:06 --> 00:48:10 how we constantly keep simplifying our lives. 638 00:48:10 --> 00:48:15 The moment we hit some complication and things get too 639 00:48:15 --> 00:48:18 painful to analyze, as engineers, 640 00:48:18 --> 00:48:23 we come up with some clever tricks to make an analysis and 641 00:48:23 --> 00:48:32 use of circuits simple again. And so, notice that this is 642 00:48:32 --> 00:48:37 similar to some, one by RD VD, 643 00:48:37 --> 00:48:43 where RD is simply one over IDB. 644 00:48:43 --> 00:48:48 I'm just defining this to be RD. 645 00:48:48 --> 00:48:59 And what that means is that I can take a nonlinear circuit 646 00:48:59 --> 00:49:07 that looks like this. OK, and what I can do is 647 00:49:07 --> 00:49:13 replace this by its incremental equivalent, and build what is 648 00:49:13 --> 00:49:18 called a small signal circuit. And I'll just introduce it 649 00:49:18 --> 00:49:22 here. And we will revisit the circuit 650 00:49:22 --> 00:49:27 in much more gory detail a couple of weeks from now. 651 00:49:27 --> 00:49:32 So, what I can do is build a small signal circuit where I 652 00:49:32 --> 00:49:37 have all the small signal variables, and replace a 653 00:49:37 --> 00:49:43 nonlinear device by a simple little resistor whose value is 654 00:49:43 --> 00:49:47 given by IDB. OK, so therefore, 655 00:49:47 --> 00:49:51 what I can do is take my nonlinear circuit, 656 00:49:51 --> 00:49:55 and for small, incremental changes, 657 00:49:55 --> 00:50:01 replace that circuit with this equivalent small signal circuit, 658 00:50:01 --> 00:50:04 and go back to doing simple stuff again. 659 00:50:04 --> 50:07 Thank you.