1 00:00:00 --> 00:00:01 Good morning, all. 2 00:00:01 --> 00:00:06 Let's get going. So I guess you had your quiz 3 00:00:06 --> 00:00:11 review yesterday. I hope you guys didn't beat up 4 00:00:11 --> 00:00:15 on (name of TA) and who else was it? 5 00:00:15 --> 00:00:19 (Name of TA) too much. As you know, 6 00:00:19 --> 00:00:24 the quiz is tomorrow. And unfortunately MIT couldn't 7 00:00:24 --> 00:00:32 give us one big room so we are broken up into three rooms. 8 00:00:32 --> 00:00:38 And you will go to your room based on the first letter of 9 00:00:38 --> 00:00:44 your last name. OK, so today we shall cover a 10 00:00:44 --> 00:00:49 topic called "Large Signal Analysis". 11 00:00:49 --> 00:00:59 12 00:00:59 --> 00:01:04 So in the last couple lectures we looked at one dependent 13 00:01:04 --> 00:01:08 sources abstractly, and then we looked at an 14 00:01:08 --> 00:01:12 amplifier built using a practical dependent source 15 00:01:12 --> 00:01:16 called the MOSFET. Now, the MOSFET had to be 16 00:01:16 --> 00:01:22 operated in a given region of its operation in order to behave 17 00:01:22 --> 00:01:27 like a current source. And while it behaved like a 18 00:01:27 --> 00:01:31 current source you would get amplification or a FET 19 00:01:31 --> 00:01:36 amplifier. So that was in the past two 20 00:01:36 --> 00:01:39 lectures. What you are going to do today 21 00:01:39 --> 00:01:43 is called large signal analysis, and this is a loaded term. 22 00:01:43 --> 00:01:48 So large signal analysis means something very specific in our 23 00:01:48 --> 00:01:52 business, and I will describe to what that is. 24 00:01:52 --> 00:01:56 This analysis involves looking at a circuit containing, 25 00:01:56 --> 00:01:59 for example, a MOSFET and figuring out how 26 00:01:59 --> 00:02:04 to get that device to operate in a way that the MOSFET was always 27 00:02:04 --> 00:02:08 in saturation. So you had to figure out, 28 00:02:08 --> 00:02:12 based on parameters that you could control, 29 00:02:12 --> 00:02:16 how to establish those parameters so that the circuit 30 00:02:16 --> 00:02:20 operated in a way that the MOSFET was always in saturation. 31 00:02:20 --> 00:02:23 So large signal analysis involves that. 32 00:02:23 --> 00:02:27 And although the examples we use, use the MOSFET, 33 00:02:27 --> 00:02:33 the same kind of analysis can apply to any other device. 34 00:02:33 --> 00:02:37 Remember, your MOSFET is a primitive element that we use as 35 00:02:37 --> 00:02:40 an example in this course. There are other primitive 36 00:02:40 --> 00:02:43 elements that you can use. The course notes, 37 00:02:43 --> 00:02:46 for example, discusses a couple other 38 00:02:46 --> 00:02:49 devices. One is the "bipolar junction 39 00:02:49 --> 00:02:52 transistor", the BJT, and works through a complete 40 00:02:52 --> 00:02:56 example from start to finish involving a circuit containing a 41 00:02:56 --> 00:03:02 bipolar junction transistor. And you can do a large signal 42 00:03:02 --> 00:03:06 analysis of that device as well. It turns out that you need to 43 00:03:06 --> 00:03:09 operate that device in an interesting region of its 44 00:03:09 --> 00:03:12 operating space, and so you can conduct a large 45 00:03:12 --> 00:03:16 signal analysis of a circuit containing that device and 46 00:03:16 --> 00:03:19 figure out how best to operate that circuit. 47 00:03:19 --> 00:03:22 So that is large signal analysis, and we will do an 48 00:03:22 --> 00:03:28 example and explain how this is done using an example today. 49 00:03:28 --> 00:03:34 So to quickly review where we have been so far, 50 00:03:34 --> 00:03:40 we looked at this little structure here, 51 00:03:40 --> 00:03:44 our MOSFET amplifier. 52 00:03:44 --> 00:03:51 53 00:03:51 --> 00:03:53 Notice that when I write a voltage at a node, 54 00:03:53 --> 00:03:57 that's a short form for saying I am looking at the voltage 55 00:03:57 --> 00:04:00 between the ground node and the node at which the voltage is 56 00:04:00 --> 00:04:04 written down. So VO here and VI applied here. 57 00:04:04 --> 00:04:07 This is a very, very common circuit that we 58 00:04:07 --> 00:04:10 use. To emphasize one more point, 59 00:04:10 --> 00:04:14 in general, in the kind of circuits we look at both in this 60 00:04:14 --> 00:04:19 course and in real life, there are a few patterns that 61 00:04:19 --> 00:04:23 we use very commonly that keep repeating all the time. 62 00:04:23 --> 00:04:27 Very often you don't have to look at every possible 63 00:04:27 --> 00:04:33 permutation and combination of how things could be connected. 64 00:04:33 --> 00:04:36 This sort of connecting thing is very, very, 65 00:04:36 --> 00:04:40 very common. And you will see a lot of this 66 00:04:40 --> 00:04:43 pattern. And we do the equivalent 67 00:04:43 --> 00:04:47 circuit for this. In the equivalent circuit we 68 00:04:47 --> 00:04:51 replace the MOSFET with a dependent source provided this 69 00:04:51 --> 00:04:54 operated in the saturation region. 70 00:04:54 --> 00:04:59 So I will just say while operating under saturation the 71 00:04:59 --> 00:05:03 equivalent circuit would look like this, VO, 72 00:05:03 --> 00:05:05 VI. 73 00:05:05 --> 00:05:12 74 00:05:12 --> 00:05:22 And IDS for the dependent source was given by K/2 75 00:05:22 --> 00:05:27 (VI-VT)^2. So this was an amplifier. 76 00:05:27 --> 00:05:32 Here was the equivalent circuit while this device was in 77 00:05:32 --> 00:05:36 saturation. And to operate in saturation, 78 00:05:36 --> 00:05:41 I said that certain properties need to be true for the MOSFET. 79 00:05:41 --> 00:05:47 And there are two properties that need to be true for this to 80 00:05:47 --> 00:05:52 be operating in saturation. One is that its gate to source 81 00:05:52 --> 00:05:57 voltage needs to be greater than VT, so VGS for the MOSFET should 82 00:05:57 --> 00:06:02 be greater than VT. And the second one was that the 83 00:06:02 --> 00:06:07 output voltage needed to be greater than the input voltage 84 00:06:07 --> 00:06:12 minus one threshold drop. And this was the same as VDS 85 00:06:12 --> 00:06:16 for the MOSFET, this was the same as VGS for 86 00:06:16 --> 00:06:20 the MOSFET. So what are we really saying 87 00:06:20 --> 00:06:22 here? What we are saying is that 88 00:06:22 --> 00:06:26 look, we built this circuit using a MOSFET, 89 00:06:26 --> 00:06:32 and it is up to us as engineers to choose its operating points 90 00:06:32 --> 00:06:37 in a way that these two properties hold. 91 00:06:37 --> 00:06:41 For example, to make the first condition 92 00:06:41 --> 00:06:47 true, I can discipline myself to operate such that VI is always 93 00:06:47 --> 00:06:52 greater than VT. Similarly, I can choose VS, 94 00:06:52 --> 00:06:57 RL and VI in a way that this condition is true, 95 00:06:57 --> 00:07:03 which says that the drain to source voltage across my MOSFET 96 00:07:03 --> 00:07:10 drain and source should be greater than VI minus VT. 97 00:07:10 --> 00:07:15 As an example, if VI was 2 volts and VT was, 98 00:07:15 --> 00:07:21 say, 1 volt, then what I am saying is that 99 00:07:21 --> 00:07:30 VO should be greater than or equal to 2 minus 1 or 1 volt. 100 00:07:30 --> 00:07:35 So I need to keep this high, 2, 3, 4, 5, whatever, 101 00:07:35 --> 00:07:40 a high voltage so that this guy stays in saturation. 102 00:07:40 --> 00:07:46 The relevant readings for the material that we are going to 103 00:07:46 --> 00:07:52 cover in the course notes are in 7.5.1 and 7.6. 104 00:07:52 --> 00:07:57 105 00:07:57 --> 00:08:00 So that is pretty much a review of where we were. 106 00:08:00 --> 00:08:03 We said we could build an amplifier. 107 00:08:03 --> 00:08:06 Its equivalent circuit was shown on the right. 108 00:08:06 --> 00:08:10 And, provided that, I discipline myself to operate 109 00:08:10 --> 00:08:15 in the saturation region or to have the MOSFET operating in the 110 00:08:15 --> 00:08:18 saturation region, then this would work like an 111 00:08:18 --> 00:08:22 amplifier and all would be good with the world. 112 00:08:22 --> 00:08:24 So today -- 113 00:08:24 --> 00:08:30 114 00:08:30 --> 00:08:33 -- we look at large signal analysis of a circuit. 115 00:08:33 --> 00:08:37 And an example would be this circuit up here containing a 116 00:08:37 --> 00:08:39 MOSFET. And, again, as I mentioned 117 00:08:39 --> 00:08:43 earlier, a large signal analysis is a loaded term in 6.002, 118 00:08:43 --> 00:08:47 or for that matter in circuits. And large signal analysis 119 00:08:47 --> 00:08:50 involves two steps. 120 00:08:50 --> 00:08:55 121 00:08:55 --> 00:09:00 The first step involves writing down the transfer function of 122 00:09:00 --> 00:09:02 your little circuit. In our case, 123 00:09:02 --> 00:09:05 VO is the output, VI is the input, 124 00:09:05 --> 00:09:08 so involves writing down VO versus VI. 125 00:09:08 --> 00:09:11 Simply write down the transfer function. 126 00:09:11 --> 00:09:14 In other words, the relationship between the 127 00:09:14 --> 00:09:17 output and the input for that circuit. 128 00:09:17 --> 00:09:20 And, in our case, again, we've disciplined 129 00:09:20 --> 00:09:24 ourselves to adhere to the "saturation discipline". 130 00:09:24 --> 00:09:29 And the second part of large signal analysis is to find out 131 00:09:29 --> 00:09:33 the valid input operating range. 132 00:09:33 --> 00:09:38 133 00:09:38 --> 00:09:42 Find out for the given circuit parameters, let's say I apply a 134 00:09:42 --> 00:09:46 VS and I use some value of RL and I use a given MOSFET, 135 00:09:46 --> 00:09:51 which has a given value of VT. The question then is that what 136 00:09:51 --> 00:09:56 is a valid set of input voltages that would operate the circuit 137 00:09:56 --> 00:10:00 in a way that I would be in saturation. 138 00:10:00 --> 00:10:06 And so find out the valid input range, and this would give me a 139 00:10:06 --> 00:10:09 corresponding output range -- 140 00:10:09 --> 00:10:16 141 00:10:16 --> 00:10:19 -- for saturation operation of the MOSFET. 142 00:10:19 --> 00:10:25 That is what we will dwell on in the lecture today. 143 00:10:25 --> 00:10:30 144 00:10:30 --> 00:10:34 So what we are saying here is that if I am careful with how I 145 00:10:34 --> 00:10:39 apply VI for a given value of RL and VS and for a given choice of 146 00:10:39 --> 00:10:44 my MOS transistor then I can stay within saturation provided 147 00:10:44 --> 00:10:46 I select my input voltages carefully. 148 00:10:46 --> 00:10:51 And the analysis that we will go through today will figure out 149 00:10:51 --> 00:10:54 what that range of input voltages is. 150 00:10:54 --> 00:10:58 And, again, I will use this as a motivating example, 151 00:10:58 --> 00:11:02 the MOSFET amplifier. But in general large signal 152 00:11:02 --> 00:11:06 analysis would apply to any other circuit as well. 153 00:11:06 --> 00:11:08 For example, in recitation you may learn 154 00:11:08 --> 00:11:11 about other circuits containing a MOSFET. 155 00:11:11 --> 00:11:14 And you can do a large signal analysis of other circuits 156 00:11:14 --> 00:11:19 containing a MOSFET or you might learn about some other devices 157 00:11:19 --> 00:11:22 like the bipolar junction transistor, and you could do the 158 00:11:22 --> 00:11:25 same kind of analysis for that device. 159 00:11:25 --> 00:11:30 So remember that the MOSFET amplifier here is an example. 160 00:11:30 --> 00:11:34 I will be using that as a driving example to explain large 161 00:11:34 --> 00:11:36 signal analysis. So the first step, 162 00:11:36 --> 00:11:39 as I mentioned earlier, is to get the VO versus VI. 163 00:11:39 --> 00:11:44 And in general for some circuit that you build the output will 164 00:11:44 --> 00:11:47 not even be a voltage. There are certain circuits 165 00:11:47 --> 00:11:51 where the output might be some kind of a current. 166 00:11:51 --> 00:11:55 Let's say I am building some kind of a circuit where I would 167 00:11:55 --> 00:11:59 like the output current or the current through some edge of the 168 00:11:59 --> 00:12:04 circuit to depend on some input. In that case the transfer 169 00:12:04 --> 00:12:08 function would be the output current versus VI. 170 00:12:08 --> 00:12:13 And if I had an input current here it would be output current 171 00:12:13 --> 00:12:16 versus input current, you know, whatever the given 172 00:12:16 --> 00:12:20 problem tells you. So this is under the saturation 173 00:12:20 --> 00:12:23 discipline. And I will not rederive this 174 00:12:23 --> 00:12:26 for you. You can apply a good old 175 00:12:26 --> 00:12:30 technique like the analytical method. 176 00:12:30 --> 00:12:34 Or you can use the graphical method to get the appropriate 177 00:12:34 --> 00:12:37 answer here. I wanted to point out in a 178 00:12:37 --> 00:12:41 quick aside that why do we care about graphical analysis? 179 00:12:41 --> 00:12:45 Once you have the analytical method, why do you care about 180 00:12:45 --> 00:12:48 the graphical method? And a student asked me a 181 00:12:48 --> 00:12:52 question after lecture last Thursday, and it occurred to me 182 00:12:52 --> 00:12:58 that it's not obvious why you need the graphical method. 183 00:12:58 --> 00:13:01 So it turns out that often times you do not have an 184 00:13:01 --> 00:13:05 equation describing the device. So let's say, 185 00:13:05 --> 00:13:07 for example, I am a manufacturer. 186 00:13:07 --> 00:13:10 Let's say I am AMD. As AMD I sit down and my 187 00:13:10 --> 00:13:13 semiconductor division builds a MOSFET. 188 00:13:13 --> 00:13:17 And when you build a MOSFET your experiments and your 189 00:13:17 --> 00:13:22 fabrication division often times doesn't give you an equation 190 00:13:22 --> 00:13:25 with the MOSFET. They build something and then 191 00:13:25 --> 00:13:30 you look at it and you experiment with it. 192 00:13:30 --> 00:13:33 You apply various input voltages and you measure 193 00:13:33 --> 00:13:35 currents and output voltages and so on. 194 00:13:35 --> 00:13:39 And so what you end up getting is a graph that describes the 195 00:13:39 --> 00:13:43 behavior of the MOSFET. And you have seen this in your 196 00:13:43 --> 00:13:46 lab as well, your 2N7000 or was it 2000? 197 00:13:46 --> 7000. 198 7000. --> 00:13:47 So your 2N7000, 199 00:13:47 --> 00:13:51 the MOSFET you use in the lab also gives you a data sheet. 200 00:13:51 --> 00:13:54 And in that data sheet you see a bunch of curves. 201 00:13:54 --> 00:13:59 So very often devices come with data sheets. 202 00:13:59 --> 00:14:03 And when you have a data sheet but no equation then you can 203 00:14:03 --> 00:14:06 apply the graphical method and solve your circuits. 204 00:14:06 --> 00:14:09 In this example, assuming I can apply the 205 00:14:09 --> 00:14:13 analytical method, here was the expression that I 206 00:14:13 --> 00:14:17 had derived for you in the last lecture. 207 00:14:17 --> 00:14:24 208 00:14:24 --> 00:14:30 So VO was related to VI using the square law relationship. 209 00:14:30 --> 00:14:38 And we can plot and do other fun stuff with this equation. 210 00:14:38 --> 00:14:42 So here is the input voltage VI. 211 00:14:42 --> 00:14:47 That is my VT. So notice that VO is VS. 212 00:14:47 --> 00:14:55 This is true when VI greater than or equal to VT and VO 213 00:14:55 --> 00:15:02 greater than or equal to VI minus VT. 214 00:15:02 --> 00:15:07 So these are the constraints of the saturation discipline. 215 00:15:07 --> 00:15:13 And in our particular situation when VI was less than VT output 216 00:15:13 --> 00:15:17 would simply be VS. If VI is less than VT the 217 00:15:17 --> 00:15:23 MOSFET would turn off, switch off, and I would have no 218 00:15:23 --> 00:15:28 current flowing through RL, and VS would appear at the 219 00:15:28 --> 00:15:31 output. So until VT, 220 00:15:31 --> 00:15:36 I have VS, and then following that I get the square law 221 00:15:36 --> 00:15:39 behavior articulated by this equation. 222 00:15:39 --> 00:15:44 And this was simply VS-K/2 (VI-VT)(RL^2). 223 00:15:44 --> 00:15:57 224 00:15:57 --> 00:16:03 So that's the first part. You have seen this before. 225 00:16:03 --> 00:16:09 The transfer function shows that I have a square law 226 00:16:09 --> 00:16:16 dependence between VI and VO. So now I can embark on the 227 00:16:16 --> 00:16:24 second step of my large signal analysis, and my goal is to find 228 00:16:24 --> 00:16:32 the valid input operating range. So what does that mean? 229 00:16:32 --> 00:16:38 What I am looking to do here is, for this little circuit, 230 00:16:38 --> 00:16:42 is drain, source, gate, VI, VO, 231 00:16:42 --> 00:16:46 RL and VS. What I am looking to do is that 232 00:16:46 --> 00:16:52 given the value of the supply VS, RL and a MOSFET, 233 00:16:52 --> 00:16:58 in our case given a MOSFET implies that it is a given value 234 00:16:58 --> 00:17:05 of K and a given value of VT for that MOSFET. 235 00:17:05 --> 00:17:09 So what I am going to do is find out, let's assume VI is my 236 00:17:09 --> 00:17:13 free variable here. So my goal will be to find out 237 00:17:13 --> 00:17:18 the range of VI for which this device stays in saturation. 238 00:17:18 --> 00:17:21 And I will use a couple of methods to do that, 239 00:17:21 --> 00:17:25 and I will use both a combination of a graphical 240 00:17:25 --> 00:17:30 method to give you intuition and then apply analytical analysis 241 00:17:30 --> 00:17:36 to get down to specific answers. So let's start with the 242 00:17:36 --> 00:17:39 intuitive part. So here is VI, 243 00:17:39 --> 00:17:43 VO. I will use the transfer curve 244 00:17:43 --> 00:17:49 VO versus VI to help build intuition here. 245 00:17:49 --> 00:17:59 246 00:17:59 --> 00:18:03 So that is what it looks like. So the first step, 247 00:18:03 --> 00:18:07 looking at this graph, we know that this point here, 248 00:18:07 --> 00:18:12 that VI needs to be greater than VT to satisfy the first 249 00:18:12 --> 00:18:16 equation. Let me just write down both 250 00:18:16 --> 00:18:20 equations here. So VI greater than or equal to 251 00:18:20 --> 00:18:24 VT is one of them, and VO is greater than VI minus 252 00:18:24 --> 00:18:31 VT is a second equation. And just remember that this is 253 00:18:31 --> 00:18:36 the same as VDS and this is the same as VGS. 254 00:18:36 --> 00:18:42 So VI must be greater than VT for the MOSFET to turn on. 255 00:18:42 --> 00:18:48 And so therefore the valid operating range starts at this 256 00:18:48 --> 00:18:55 point and is somewhere up here. So the first part is pretty 257 00:18:55 --> 00:18:59 easy. Somewhere here -- 258 00:18:59 --> 00:19:02 Somewhere at that point, my output voltage VO. 259 00:19:02 --> 00:19:05 I'm not quite sure what that point is. 260 00:19:05 --> 00:19:09 My output voltage VO, as this keeps falling down, 261 00:19:09 --> 00:19:14 my output voltage VO goes lower than one threshold below VI. 262 00:19:14 --> 00:19:18 And at that point it goes into its triode region, 263 00:19:18 --> 00:19:21 and I need to find out what that point is. 264 00:19:21 --> 00:19:26 So somewhere here I go into my triode region and begin to show 265 00:19:26 --> 00:19:30 a different behavior than the amplifying square law 266 00:19:30 --> 00:19:36 relationship there and go into my triode behavior. 267 00:19:36 --> 00:19:39 So I need to find out what this point is. 268 00:19:39 --> 00:19:45 Once I find out what that point is then this will be my valid 269 00:19:45 --> 00:19:49 operating range. So let's figure out what that 270 00:19:49 --> 00:19:53 point is. At that point the following is 271 00:19:53 --> 00:19:56 true. Certainly VI is greater than 272 00:19:56 --> 00:19:59 VT. And at that point the output 273 00:19:59 --> 00:20:06 goes below one threshold, the input minus one threshold. 274 00:20:06 --> 00:20:13 So at this point the following is true, VO is equal to VI minus 275 00:20:13 --> 00:20:17 VT. At that point the output 276 00:20:17 --> 00:20:21 voltage is equal to the input minus VT. 277 00:20:21 --> 00:20:28 And if the output goes lower then it will violate this 278 00:20:28 --> 00:20:33 equation here. It is no longer greater than 279 00:20:33 --> 00:20:38 that number. So how do we find out what this 280 00:20:38 --> 00:20:41 point is? The principle intuition. 281 00:20:41 --> 00:20:47 Let's draw some lines here. Let's assume that VI and VT use 282 00:20:47 --> 00:20:49 the same scale, say, volts. 283 00:20:49 --> 00:20:56 So if I draw a straight line at 45 degrees then that is a curve 284 00:20:56 --> 00:21:01 representing VI equals VO. We all know that. 285 00:21:01 --> 00:21:06 No big shakes. So the line at 45 degrees here 286 00:21:06 --> 00:21:10 is the line at which VI equal VO. 287 00:21:10 --> 00:21:15 And if I take that line now, the VI equals VO line, 288 00:21:15 --> 00:21:20 and I begin translating it to the right. 289 00:21:20 --> 00:21:25 So let's take a line here. Let's take a line there. 290 00:21:25 --> 00:21:33 That line will be simply equal to VO equals VI minus VT. 291 00:21:33 --> 00:21:36 I have translated that to the right. 292 00:21:36 --> 00:21:40 And so this line is simply VO equals VI minus VT. 293 00:21:40 --> 00:21:46 So this line is a locus of points at which VO is equal to 294 00:21:46 --> 00:21:49 this value. This minus VT shows up as a 295 00:21:49 --> 00:21:54 translation to the right. So I take my VO equals VI line, 296 00:21:54 --> 00:22:00 translate that to the right and it becomes VO equals VI minus 297 00:22:00 --> 00:22:05 VT. Elementary geometry 101 or 298 00:22:05 --> 00:22:10 whatever. So what do we have here? 299 00:22:10 --> 00:22:17 Above this we have the condition VO greater than or 300 00:22:17 --> 00:22:24 equal to VI minus VT, and below that we have VO less 301 00:22:24 --> 00:22:31 than VI minus VT. If we look at this graph here, 302 00:22:31 --> 00:22:35 this is the valid input operating range. 303 00:22:35 --> 00:22:41 Starting at this point greater than VT, and at this point my 304 00:22:41 --> 00:22:45 output equals VO equals VI minus VT. 305 00:22:45 --> 00:22:51 This must be the valid operating range for the input 306 00:22:51 --> 00:22:56 here to here. And correspondingly the outputs 307 00:22:56 --> 00:23:02 are from here to this point to here like so. 308 00:23:02 --> 00:23:06 So this is my valid input operating range and this is my 309 00:23:06 --> 00:23:11 valid output operating range or the corresponding valid output 310 00:23:11 --> 00:23:14 operating range. So what does this say? 311 00:23:14 --> 00:23:19 What this is saying is that if I, as the designer of the 312 00:23:19 --> 00:23:24 circuit, am disciplined enough to apply inputs that are in this 313 00:23:24 --> 00:23:30 range, VT to some value here, graphically shown here. 314 00:23:30 --> 00:23:34 Then my MOSFET will remain in saturation. 315 00:23:34 --> 00:23:40 And correspondingly my outputs will go between VS and some 316 00:23:40 --> 00:23:45 value here. So hopefully that gives you 317 00:23:45 --> 00:23:49 some of the intuition behind how we get it. 318 00:23:49 --> 00:23:56 And let's continue. Let me label this point X. 319 00:23:56 --> 00:24:04 320 00:24:04 --> 00:24:09 So continuing with two to get the valid operating range. 321 00:24:09 --> 00:24:14 I have shown you intuitively where that point is, 322 00:24:14 --> 00:24:20 but what I will do next is actually compute that for you. 323 00:24:20 --> 00:24:24 It is a pretty simple computation. 324 00:24:24 --> 00:24:29 Note that point X is the intersection of two curves VO 325 00:24:29 --> 00:24:36 equals VI minus VT. And the second curve is VO 326 00:24:36 --> 00:24:44 equals VS minus K divide by 2, VI minus V2 all squared RL. 327 00:24:44 --> 00:24:52 So the point X iss at the intersection of these curves, 328 00:24:52 --> 00:24:59 and I can very easily get that as follows. 329 00:24:59 --> 00:25:06 What I will do is I will simply substitute for VI minus VT from 330 00:25:06 --> 00:25:10 this equation here and then solve for it, 331 00:25:10 --> 00:25:17 so I get VO equals VS minus K divide by 2 VO squared RL. 332 00:25:17 --> 00:25:21 And so this gives me a quadratic in VO. 333 00:25:21 --> 00:25:25 And I can solve for it pretty easily. 334 00:25:25 --> 00:25:34 And I get for a quadratic AX squared plus BX plus C equals 0. 335 00:25:34 --> 00:25:38 The solution is given by VO is minus B plus/minus square root 336 00:25:38 --> 00:25:41 of B squared minus 4AC divided by 2A. 337 00:25:41 --> 00:25:45 And so I am just going to get those numbers here. 338 00:25:45 --> 00:25:49 So the coefficient of VO, that is B, is minus 1. 339 00:25:49 --> 00:25:53 Take the positive root because we are up in the positive 340 00:25:53 --> 00:25:57 voltages here. And square root of B squared, 341 00:25:57 --> 00:26:06 that is 1, minus 4AC. So I get a plus 4 times K 342 00:26:06 --> 00:26:17 divide by 2 RL. And 2A is simply 2 times K 343 00:26:17 --> 00:26:30 divided by 2 times RL. So that is what I get. 344 00:26:30 --> 00:26:35 That gives me VO. So it tells me that VO, 345 00:26:35 --> 00:26:42 at the point where the output just equals one threshold drop 346 00:26:42 --> 00:26:50 below VI is given by the other circuit perimeter such as VS, 347 00:26:50 --> 00:26:55 RL and so on. Oh, I am missing a VS here. 348 00:26:55 --> 00:27:02 I just forgot the VS up here. That is my VO. 349 00:27:02 --> 00:27:12 So what is VI equal to? Remember that at this point VO 350 00:27:12 --> 00:27:22 equals VI minus VT, so VI is simply VT plus -- 351 00:27:22 --> 00:27:40 352 00:27:40 --> 00:27:43 I have not taught you anything earth shattering here. 353 00:27:43 --> 00:27:47 I have just done some grubby math here to solve these two 354 00:27:47 --> 00:27:49 equations. So this is a straight line at 355 00:27:49 --> 00:27:53 45 degrees from VT and this is the transfer function. 356 00:27:53 --> 00:27:55 And I need to find the intersection. 357 00:27:55 --> 00:27:59 And the intersection is given by this point. 358 00:27:59 --> 00:28:04 359 00:28:04 --> 00:28:10 So that point, VI being VT plus something, 360 00:28:10 --> 00:28:15 is simply the second dot on the X axis. 361 00:28:15 --> 00:28:20 So therefore I am pretty much done. 362 00:28:20 --> 00:28:28 My valid input range for VI goes from VT. 363 00:28:28 --> 00:28:32 So it starts at VT. That is where the transistor 364 00:28:32 --> 00:28:36 just turns on. And then goes all the way to 365 00:28:36 --> 00:28:43 this point, VT plus minus 1 plus square root of 1 plus 2 K RL VS 366 00:28:43 --> 00:28:46 K RL. So this is my valid operating 367 00:28:46 --> 00:28:49 range. And again remember I won't 368 00:28:49 --> 00:28:54 dwell on this equation because, in some sense, 369 00:28:54 --> 00:28:59 you will get a different set of limits for other devices, 370 00:28:59 --> 00:29:05 for other circuits containing a MOSFET. 371 00:29:05 --> 00:29:08 Or, for that matter, for other outputs that one may 372 00:29:08 --> 00:29:11 be focusing on. So what is more important here 373 00:29:11 --> 00:29:16 is not so much the results that you see but the process that I 374 00:29:16 --> 00:29:19 have gone through. So what is more important here 375 00:29:19 --> 00:29:22 is how did I get here? And the way I got here was 376 00:29:22 --> 00:29:26 looked at the graph and said look, the MOSFET is in 377 00:29:26 --> 00:29:31 saturation in that regime. And I am finding the bounding 378 00:29:31 --> 00:29:35 points of the regime of saturation operation. 379 00:29:35 --> 00:29:39 So now, as an engineer, I can say that hey, 380 00:29:39 --> 00:29:43 look, if you build a MOSFET circuit like so, 381 00:29:43 --> 00:29:48 with a given value of RL, a given MOSFET and a given VS, 382 00:29:48 --> 00:29:52 then if you limit yourself you are operating with input 383 00:29:52 --> 00:29:56 voltages in this range thou shalt be happy. 384 00:29:56 --> 00:30:01 If you go beyond that range then you will be violating the 385 00:30:01 --> 00:30:07 saturation discipline. So the corresponding output 386 00:30:07 --> 00:30:08 range -- 387 00:30:08 --> 00:30:18 388 00:30:18 --> 00:30:22 I can write the corresponding output range, 389 00:30:22 --> 00:30:28 and that goes from VS, when the input is at VT the 390 00:30:28 --> 00:30:36 output is at VS and goes from VS down to the input minus VT. 391 00:30:36 --> 00:30:39 Which is simply minus 1 plus -- 392 00:30:39 --> 00:30:51 393 00:30:51 --> 00:30:55 Let me go back and quickly show you a little MOSFET circuit, 394 00:30:55 --> 00:30:58 amplified circuit so you can stare at a real transfer curve 395 00:30:58 --> 00:31:03 yourselves. And indeed convince yourselves 396 00:31:03 --> 00:31:10 that roughly at the point where proportionately shown in the 397 00:31:10 --> 00:31:17 curve up there the MOSFET indeed goes into its triode region and 398 00:31:17 --> 00:31:22 begins heading out of its saturation region. 399 00:31:22 --> 00:31:27 Notice that here that is the same curve, the transfer 400 00:31:27 --> 00:31:32 function. And the amplified output is at 401 00:31:32 --> 00:31:36 VS until input reaches a threshold voltage VT. 402 00:31:36 --> 00:31:41 And once input goes beyond VT the output begins to drop 403 00:31:41 --> 00:31:45 precipitously. And at some point here this 404 00:31:45 --> 00:31:49 begins to go into its triode region. 405 00:31:49 --> 00:31:53 And what I am going to do is simply increase the input 406 00:31:53 --> 00:31:59 voltage VI, and you will see that the output them begins to 407 00:31:59 --> 00:32:05 go into its triode region. It keeps dropping. 408 00:32:05 --> 00:32:10 And, as you can see, the output begins to go into a 409 00:32:10 --> 00:32:16 space where the gain is no longer more than 1. 410 00:32:16 --> 00:32:21 And this is a triode region of MOSFET operation. 411 00:32:21 --> 00:32:29 So the MOSFET is in saturation, things are going great. 412 00:32:29 --> 00:32:35 As I increase my VI notice at some point I begin to go out of 413 00:32:35 --> 00:32:38 my saturation region of the MOSFET. 414 00:32:38 --> 00:32:43 And somewhere here I go from the saturation region and 415 00:32:43 --> 00:32:47 transition into the triode region. 416 00:32:47 --> 00:32:52 And this value shown here gives you the corresponding input 417 00:32:52 --> 00:32:59 voltage and the output voltage. Other practical devices like 418 00:32:59 --> 00:33:03 bipolar junction transistors or MOSFETs and other circuits and 419 00:33:03 --> 00:33:06 so on can be subjected to a similar analysis. 420 00:33:06 --> 00:33:10 And you can find out the valid operating regions for that 421 00:33:10 --> 00:33:13 device as well, or for that circuit. 422 00:33:13 --> 00:33:17 So as a next step what I would like to do -- 423 00:33:17 --> 00:33:22 424 00:33:22 --> 00:33:25 Out here I began by looking at the transfer function, 425 00:33:25 --> 00:33:29 the VO versus VI curve, and used that to drive the 426 00:33:29 --> 00:33:34 intuition behind how we calculated the bounding regions. 427 00:33:34 --> 00:33:38 You can do the same kind of analysis intuitively looking at 428 00:33:38 --> 00:33:42 yet another curve, another set of graphs that you 429 00:33:42 --> 00:33:45 are familiar with, and that is a load line 430 00:33:45 --> 00:33:48 characteristic. And it is interesting to get 431 00:33:48 --> 00:33:52 two interpretations. And you can use whichever one 432 00:33:52 --> 00:33:57 you feel comfortable with. So I will do two alternatively 433 00:33:57 --> 00:34:01 and show you another set of curves that you can use to get 434 00:34:01 --> 00:34:03 that. 435 00:34:03 --> 00:34:10 436 00:34:10 --> 00:34:16 Here I am going to plot IDS versus VDS, which is the same as 437 00:34:16 --> 00:34:19 VO. This was the load line graph 438 00:34:19 --> 00:34:25 that we had seen earlier. And, just for our reference, 439 00:34:25 --> 00:34:31 remember that VI must be greater than VT for saturation 440 00:34:31 --> 00:34:35 operation. Similarly VO should be greater 441 00:34:35 --> 00:34:40 than or equal to VI minus VT for saturation operation. 442 00:34:40 --> 00:34:45 Those are my limits. The way we got the load line 443 00:34:45 --> 00:34:50 graph was we superimposed the load line equation over the 444 00:34:50 --> 00:34:55 device characteristics. And so let me plot the device 445 00:34:55 --> 00:35:00 characteristics in the saturation region. 446 00:35:00 --> 00:35:06 Remember that this constraint could be related to the current 447 00:35:06 --> 00:35:12 as I derived for you in the last lecture as follows. 448 00:35:12 --> 00:35:18 IDS being less than or equal to K divided by 2 VO squared. 449 00:35:18 --> 00:35:22 So in terms of my IDS versus VDS relation, 450 00:35:22 --> 00:35:28 this lateral constraint is equivalent to IDS being less 451 00:35:28 --> 00:35:35 than K by 2 VO squared. So this is that equation. 452 00:35:35 --> 00:35:40 So this line is IDS equals K by 2 VO squared. 453 00:35:40 --> 00:35:48 And in this region I have the valid operating region where IDS 454 00:35:48 --> 00:35:56 is less than that quality. So here are all my other curves 455 00:35:56 --> 00:36:02 for various values VGS. So here are my devices curves, 456 00:36:02 --> 00:36:06 IDS versus VDS. Remember that these curves come 457 00:36:06 --> 00:36:08 down like this, for the MOSFET, 458 00:36:08 --> 00:36:11 right? Just that we focus on the 459 00:36:11 --> 00:36:16 right-hand side because that is where the MOSFET is in 460 00:36:16 --> 00:36:19 saturation. And on this side the MOSFET is 461 00:36:19 --> 00:36:23 in its triode region, and we discipline ourselves not 462 00:36:23 --> 00:36:30 to operate the MOSFET such that it is in its triode region. 463 00:36:30 --> 00:36:34 So those were the device characteristics. 464 00:36:34 --> 00:36:39 And then I could plot my load line equation. 465 00:36:39 --> 00:36:42 My load line equation, if you recall, 466 00:36:42 --> 00:36:48 was IDS = VS/RL - VO/RL. This was simply obtained by 467 00:36:48 --> 00:36:54 writing KVL at the loop containing the output node and 468 00:36:54 --> 00:36:58 the supply VS. Notice there that VO is equal 469 00:36:58 --> 00:37:06 to VS minus IDS times RL. And that is simply by dividing 470 00:37:06 --> 00:37:12 by RL on both sides and moving IDS to the left-hand side we get 471 00:37:12 --> 00:37:16 this equation. And this equation gives rise to 472 00:37:16 --> 00:37:22 a curve that looks like this. And what is this point here? 473 00:37:22 --> 00:37:27 This point is where VO is 0. So when VO is 0 my IDS is 474 00:37:27 --> 00:37:34 simply VS divided by RL. And this point is obtained when 475 00:37:34 --> 00:37:39 IDS is 0. And under those conditions VS 476 00:37:39 --> 00:37:45 and VO are equal so this is VS. This is my saturation region 477 00:37:45 --> 00:37:52 and this is the triode region. This was another interesting 478 00:37:52 --> 00:37:56 graph. We often times fondly call it 479 00:37:56 --> 00:38:01 the load line graph. So here is a load line 480 00:38:01 --> 00:38:06 superimposed on the MOSFET device IDS versus VDS curves for 481 00:38:06 --> 00:38:10 a variety of values of VGS. So by looking at this curve, 482 00:38:10 --> 00:38:14 we can also intuitively determine the valid operating 483 00:38:14 --> 00:38:17 range. So what are the two points 484 00:38:17 --> 00:38:20 here? I will let you stare at it for 485 00:38:20 --> 00:38:24 a couple of seconds yourselves to figure out what two points 486 00:38:24 --> 00:38:28 here bound the valid operating range of the MOSFET, 487 00:38:28 --> 00:38:33 the valid operating range of the circuit. 488 00:38:33 --> 00:38:36 I will start. One is this point, 489 00:38:36 --> 00:38:43 because at this point the output is VS and VGS has just 490 00:38:43 --> 00:38:49 begun to equal VT. So think about where the second 491 00:38:49 --> 00:38:55 point is for valid operation. It is here, and, 492 00:38:55 --> 00:39:02 somewhere along that load line. Remember the load line is a 493 00:39:02 --> 00:39:06 constraint that must be met by the output VO. 494 00:39:06 --> 00:39:11 It is the constraint imposed by KVL on the output. 495 00:39:11 --> 00:39:17 So the output is constrained to operate in this regime for 496 00:39:17 --> 00:39:22 various values of VGS. So as the output keeps going 497 00:39:22 --> 00:39:27 from here all the way here, at some point I exit my 498 00:39:27 --> 00:39:32 saturation region. And that other point is given 499 00:39:32 --> 00:39:36 by this one. So notice that this is the 500 00:39:36 --> 00:39:40 curve that bounds. On the left-hand side of this 501 00:39:40 --> 00:39:43 the MOSFET is no longer in saturation. 502 00:39:43 --> 00:39:49 It is on the right-hand side, and so therefore this is the 503 00:39:49 --> 00:39:52 valid operating region. 504 00:39:52 --> 00:40:00 505 00:40:00 --> 00:40:04 Here to here. This is good. 506 00:40:04 --> 00:40:09 This is VS. That is good to know. 507 00:40:09 --> 00:40:17 And for this point I know that VI, which is VGS, 508 00:40:17 --> 00:40:23 equals VT. I know VO is equal to VS. 509 00:40:23 --> 00:40:30 And IDS, at this point, is 0. 510 00:40:30 --> 00:40:37 So VO and IDS being VS and 0 correspondingly are the output 511 00:40:37 --> 00:40:42 operating perimeters when VI equals VD. 512 00:40:42 --> 00:40:48 So that is one point. And let's find out what this 513 00:40:48 --> 00:40:53 point is. At that point I get my output 514 00:40:53 --> 00:41:03 just entering the range of the MOSFET triode region operation. 515 00:41:03 --> 00:41:09 Notice that this point is the intersection of two curves, 516 00:41:09 --> 00:41:15 this line and this curve. So this curve here is given by 517 00:41:15 --> 00:41:19 IDS equals K divided by 2 VO squared. 518 00:41:19 --> 00:41:23 And this is my load line equation. 519 00:41:23 --> 00:41:30 So that is VS divided by RL minus VO divided by RL. 520 00:41:30 --> 00:41:32 That's it. So I won't go ahead and solve 521 00:41:32 --> 00:41:35 that for you. You can go and check it out and 522 00:41:35 --> 00:41:39 convince yourselves that if you solve these two equations and 523 00:41:39 --> 00:41:43 find out the VO for this, it should be the same VO that 524 00:41:43 --> 00:41:46 you obtained using the other graph. 525 00:41:46 --> 00:41:57 526 00:41:57 --> 00:42:01 What I have done here, obtaining the valid regions of 527 00:42:01 --> 00:42:04 operation is no different from what I did here. 528 00:42:04 --> 00:42:08 The two are alternative approaches to getting to the 529 00:42:08 --> 00:42:11 same place. Just that over the years what I 530 00:42:11 --> 00:42:16 have discovered is that there are one class of people that are 531 00:42:16 --> 00:42:21 output transfer function people, this graph, and another set of 532 00:42:21 --> 00:42:27 people that are load line people that like to think that way. 533 00:42:27 --> 00:42:31 I have always been a transfer function person myself, 534 00:42:31 --> 00:42:35 but some of you may be load line people and so you can use 535 00:42:35 --> 00:42:39 that to drive your intuition. It is pretty amazing. 536 00:42:39 --> 00:42:43 As we get into this business and keep going down the path, 537 00:42:43 --> 00:42:48 it is amazing how some people really kind of get the load line 538 00:42:48 --> 00:42:53 thing and others feel much more comfortable with the transfer 539 00:42:53 --> 00:42:55 function. So pick whatever you want. 540 00:42:55 --> 00:43:00 So what we have so far is we have conducted a large signal 541 00:43:00 --> 00:43:06 analysis of a MOSFET amplifier. It is an analysis of a circuit, 542 00:43:06 --> 00:43:09 and we found two things. One is the transfer function 543 00:43:09 --> 00:43:13 under saturation operation, and we found the valid input 544 00:43:13 --> 00:43:17 operating ranges and the corresponding valid output 545 00:43:17 --> 00:43:19 operating ranges for the circuit. 546 00:43:19 --> 00:43:23 In the last five or six minutes let me talk about a couple of 547 00:43:23 --> 00:43:26 other issues. And the first issue is what we 548 00:43:26 --> 00:43:31 have done so far is intuitively and mathematically shown you 549 00:43:31 --> 00:43:36 what the valid regions are. Now you are thinking that's 550 00:43:36 --> 00:43:41 fine, but how do I get there? This region is good, 551 00:43:41 --> 00:43:44 VT through that other point, that's good, 552 00:43:44 --> 00:43:49 but how do I get there? How do I make my amplifier 553 00:43:49 --> 00:43:54 operate in that region? The answer is pretty simple, 554 00:43:54 --> 00:44:00 and let me drive the intuition again using a graph. 555 00:44:00 --> 00:44:13 556 00:44:13 --> 00:44:15 So this is a graph. And I showed you that -- 557 00:44:15 --> 00:44:22 558 00:44:22 --> 00:44:25 That was my valid region here. Take a 45 degree line, 559 00:44:25 --> 00:44:28 find out where it intersects the transfer function, 560 00:44:28 --> 00:44:32 then this is the valid region here, VT through that 561 00:44:32 --> 00:44:36 coordinating function that we developed out there. 562 00:44:36 --> 00:44:45 If I have an input that looks like so, some input whose 563 00:44:45 --> 00:44:54 gyrations fall within this range, will constantly keep the 564 00:44:54 --> 00:45:01 MOSFET in saturation. And the corresponding output 565 00:45:01 --> 00:45:06 will look like this. If my input is in this range, 566 00:45:06 --> 00:45:09 my output will be within this range. 567 00:45:09 --> 00:45:12 And how do I get my input to be here? 568 00:45:12 --> 00:45:17 Let's say I have a sinusoid that is 1 volt peak to peak or 569 00:45:17 --> 00:45:21 whatever. How do I get my sinusoid up 570 00:45:21 --> 00:45:24 there? Well, you have learned the 571 00:45:24 --> 00:45:28 trick on how to boost things. Remember boost? 572 00:45:28 --> 00:45:33 All you have to do is boost up your signal by some value 573 00:45:33 --> 00:45:39 capital VI. And the way you do that is as 574 00:45:39 --> 00:45:42 follows. VS, RL, VO. 575 00:45:42 --> 00:45:49 What you do is you apply a DC offset to your input. 576 00:45:49 --> 00:45:56 You take your sinusoid and boost it up so that all the 577 00:45:56 --> 00:46:04 gyrations of the input are in the valid range. 578 00:46:04 --> 00:46:06 This is my input, some VA. 579 00:46:06 --> 00:46:13 Then I apply some DC offset capital VI given by this value 580 00:46:13 --> 00:46:16 here. And boost up the interesting 581 00:46:16 --> 00:46:20 input? My interesting input is the VA. 582 00:46:20 --> 00:46:27 And I boost it up by capital VI so that this guy is always in 583 00:46:27 --> 00:46:32 saturation. I would like to show you a 584 00:46:32 --> 00:46:36 little demo now. I am going to show you an input 585 00:46:36 --> 00:46:40 that is a triangular wave. And what we will do is I'll 586 00:46:40 --> 00:46:44 play with a wide variety of offset voltages. 587 00:46:44 --> 00:46:49 This guy is a triangular wave. And what I am going to do is 588 00:46:49 --> 00:46:54 apply a triangular wave and we'll look at the output and 589 00:46:54 --> 00:46:59 convince ourselves that I get amplification when VI is big 590 00:46:59 --> 00:47:05 enough that the input goes into a valid operating range. 591 00:47:05 --> 00:47:09 And we will look at a variety of ranges here. 592 00:47:09 --> 00:47:13 You can put it a little larger. 593 00:47:13 --> 00:47:23 594 00:47:23 --> 00:47:28 OK. So the triangular wave is my 595 00:47:28 --> 00:47:31 input. And this is my output. 596 00:47:31 --> 00:47:34 This looks nothing like a triangular wave. 597 00:47:34 --> 00:47:38 And the reason is that I do not have the right offset. 598 00:47:38 --> 00:47:42 So what I will do is gradually increase the offset on the 599 00:47:42 --> 00:47:45 MOSFET. So at this point the offset is 600 00:47:45 --> 00:47:47 very low, a very small near zero offset. 601 00:47:47 --> 00:47:50 And so therefore my output is a disaster. 602 00:47:50 --> 00:47:53 My MOSFET is not in saturation all the time. 603 00:47:53 --> 00:47:57 So what I will do here is apply some sort of offset. 604 00:47:57 --> 00:48:01 Is this the one? We want to switch. 605 00:48:01 --> 00:48:06 This is the input. You can see I am applying an 606 00:48:06 --> 00:48:12 offset by bumping and boosting up the input. 607 00:48:12 --> 00:48:20 608 00:48:20 --> 00:48:23 I don't have clipping happening at both ends, 609 00:48:23 --> 00:48:26 but I get something. And I get amplification. 610 00:48:26 --> 00:48:30 Now let me apply way too much of an offset. 611 00:48:30 --> 00:48:33 With this offset I am kind of operating here. 612 00:48:33 --> 00:48:37 What I will do now is apply an even higher offset so that this 613 00:48:37 --> 00:48:40 triangular wave begins to move here. 614 00:48:40 --> 00:48:44 If I apply a very high offset what I am doing is overdriving 615 00:48:44 --> 00:48:47 the amplifier, boosting it so high that the 616 00:48:47 --> 00:48:51 MOSFET is going to go into its triode region and you are going 617 00:48:51 --> 00:48:54 to see that I won't have any gain. 618 00:48:54 --> 00:48:58 My output is going to shrink noticeably if I overdrive the 619 00:48:58 --> 00:49:03 input. You will notice the input going 620 00:49:03 --> 00:49:09 higher and higher. Pull the trigger point down. 621 00:49:09 --> 00:49:13 There you go. Notice that as I boost up my 622 00:49:13 --> 00:49:20 input even higher notice that the output is a really small 623 00:49:20 --> 00:49:26 image of what the right input should be. 624 00:49:26 --> 00:49:30 The right answer here, of course, is that I apply some 625 00:49:30 --> 00:49:34 right amount of offset to boost up the input into the right 626 00:49:34 --> 00:49:38 regime so that the output is seen to be some amplified 627 00:49:38 --> 00:49:42 version of this input. So I showed you three things. 628 00:49:42 --> 00:49:46 One is very little offset. That was like so, 629 00:49:46 --> 00:49:49 as the thing comes down. A very high offset, 630 00:49:49 --> 00:49:53 it gets killed again. And the right amount of offset. 631 00:49:53 --> 00:49:58 But notice that we still have a problem, even with the right 632 00:49:58 --> 00:50:02 offset. The output is not linearly 633 00:50:02 --> 00:50:04 related to the input. It is nonlinear. 634 00:50:04 --> 00:50:09 And the answer to get a linear response is good old small 635 00:50:09 --> 00:50:12 signal stuff. And we will be looking at the 636 00:50:12 --> 50:15 small signal part in the next lecture.