1 00:00:00 --> 00:00:04 2 00:00:04 --> 00:00:08 All right. Good morning all. 3 00:00:08 --> 00:00:17 Today we embark on another new chapter in what we do. 4 00:00:17 --> 00:00:23 And the topic is going to be -- 5 00:00:23 --> 00:00:41 6 00:00:41 --> 00:00:46 We will talk about this thing called an Operational Amplifier. 7 00:00:46 --> 00:00:51 Before I get into the lecture, I want to point out a couple of 8 00:00:51 --> 00:00:54 things. One is that you are going to 9 00:00:54 --> 00:00:59 hear about two big words in today's lecture. 10 00:00:59 --> 00:01:02 Two big and incredibly important words. 11 00:01:02 --> 00:01:08 And I want to mention those words to you right now so that 12 00:01:08 --> 00:01:12 when I come to them in lecture you can say OK, 13 00:01:12 --> 00:01:17 I better pay really close attention, these are important 14 00:01:17 --> 00:01:19 words. All right. 15 00:01:19 --> 00:01:24 One of them is abstraction. The second one is feedback. 16 00:01:24 --> 00:01:29 Two incredibly important concepts. 17 00:01:29 --> 00:01:32 Abstraction, you have been seeing a couple 18 00:01:32 --> 00:01:36 times during this course, once in the beginning where we 19 00:01:36 --> 00:01:39 abstracted out Maxwell's equations by focusing on a 20 00:01:39 --> 00:01:44 smaller playground and simply using KVL, KCL in place of those 21 00:01:44 --> 00:01:46 equations. A big abstraction. 22 00:01:46 --> 00:01:51 It turns out that almost all of EECS is based upon abstractions 23 00:01:51 --> 00:01:53 at various levels. In the first lecture, 24 00:01:53 --> 00:01:58 I also showed you the layer upon layer of abstraction that 25 00:01:58 --> 00:02:03 we built to be able to build interesting systems. 26 00:02:03 --> 00:02:06 The second big thing is feedback. 27 00:02:06 --> 00:02:11 And I am going to relate this to anti-lock breaks in cars. 28 00:02:11 --> 00:02:16 And so, you can wait and see how we do that. 29 00:02:16 --> 00:02:19 It's an incredibly important concept. 30 00:02:19 --> 00:02:24 Before we dive into the amplifier abstraction, 31 00:02:24 --> 00:02:30 let me first talk about something that you know. 32 00:02:30 --> 00:02:34 Start with something that you know and then lead up into the 33 00:02:34 --> 00:02:37 operational amplifier and its circuits. 34 00:02:37 --> 00:02:41 You know about the MOSFET amplifier. 35 00:02:41 --> 00:02:46 36 00:02:46 --> 00:02:54 The MOSFET amplifier that you know about looked like this. 37 00:02:54 --> 00:02:59 It was based on a MOSFET. 38 00:02:59 --> 00:03:03 39 00:03:03 --> 00:03:09 There was a VS supply. There was a vI input, 40 00:03:09 --> 00:03:14 a vO output and, as I said, a VS supply. 41 00:03:14 --> 00:03:22 So, this was a MOSFET circuit that you've seen before. 42 00:03:22 --> 00:03:30 One way of viewing this is that this circuit has three major 43 00:03:30 --> 00:03:35 ports. This here is the input port 44 00:03:35 --> 00:03:38 with voltage vI. This here, between the drain 45 00:03:38 --> 00:03:42 terminal and the ground, is the output port. 46 00:03:42 --> 00:03:47 I take the output between the drain terminal and ground. 47 00:03:47 --> 00:03:50 And, finally, we have a third port, 48 00:03:50 --> 00:03:53 which is this one. It is called the power port. 49 00:03:53 --> 00:03:58 I apply VS between this terminal here and the ground 50 00:03:58 --> 00:04:02 terminal. And that gives us the power 51 00:04:02 --> 00:04:04 port. This device here was a three 52 00:04:04 --> 00:04:08 port device. Input port or control port, 53 00:04:08 --> 00:04:12 output port and a power port. And so we looked at the circuit 54 00:04:12 --> 00:04:16 and did a whole bunch of analyses of it. 55 00:04:16 --> 00:04:20 Then what I can do at this point, now that you've seen 56 00:04:20 --> 00:04:24 this, it's often times interesting to think about 57 00:04:24 --> 00:04:30 abstracting this out into some kind of a building block. 58 00:04:30 --> 00:04:33 Much like in software, you write a procedure and you 59 00:04:33 --> 00:04:37 abstract out the internal details of the procedure in the 60 00:04:37 --> 00:04:41 procedure declaration and in the call that you make. 61 00:04:41 --> 00:04:44 In the same way, we can take this little device 62 00:04:44 --> 00:04:49 here and abstract that out into the following abstraction. 63 00:04:49 --> 00:04:54 64 00:04:54 --> 00:05:00 We could abstract that out as a device that looks like this. 65 00:05:00 --> 00:05:06 66 00:05:06 --> 00:05:12 I have my input port, I have my output port and I 67 00:05:12 --> 00:05:18 have my power port. So, I can apply VS here. 68 00:05:18 --> 00:05:25 Notice that I've taken these six terminals here, 69 00:05:25 --> 00:05:30 one, two, three, four, five and six, 70 00:05:30 --> 00:05:37 and put a box around it. And just exposed the terminals 71 00:05:37 --> 00:05:39 to you. And I need to tell you a little 72 00:05:39 --> 00:05:43 bit more about the internal properties, but suffice it to 73 00:05:43 --> 00:05:46 say that you can begin working with this little block. 74 00:05:46 --> 00:05:51 An even simpler version of this for many applications might just 75 00:05:51 --> 00:05:53 look like this, vI and vO where there is a 76 00:05:53 --> 00:05:57 ground that is shared among them that is implicit in this 77 00:05:57 --> 00:06:01 picture. And vI and vO can simply be the 78 00:06:01 --> 00:06:05 node voltages at these nodes. This is a progressively more 79 00:06:05 --> 00:06:08 abstract representation of this amplifier. 80 00:06:08 --> 00:06:11 What we can do is, provided we know, 81 00:06:11 --> 00:06:14 we can abstract out the relevant properties of this 82 00:06:14 --> 00:06:19 block and expose them outside. And the relevant properties 83 00:06:19 --> 00:06:22 might well be that, let's say here the properties 84 00:06:22 --> 00:06:27 may be that I in is always zero. I can also express to you the 85 00:06:27 --> 00:06:33 gain of this amplifier. I may also be able to tell you 86 00:06:33 --> 00:06:37 the Thevenin equivalent for the output. 87 00:06:37 --> 00:06:44 There are some properties that I can give you that will let you 88 00:06:44 --> 00:06:48 use this building block abstractly. 89 00:06:48 --> 00:06:55 Today, what we will do is introduce a powerful abstraction 90 00:06:55 --> 00:07:01 of a type of amplifier. This is called the operational 91 00:07:01 --> 00:07:09 amplifier or "op amp" for short. What I am going to do is give 92 00:07:09 --> 00:07:15 you a slightly more involved building block than the one I 93 00:07:15 --> 00:07:21 have shown you there. But suffice it to say that the 94 00:07:21 --> 00:07:28 idea is going to be the same. This building block looks like 95 00:07:28 --> 00:07:31 this. This building block has an 96 00:07:31 --> 00:07:37 input port. This building block also has a 97 00:07:37 --> 00:07:43 port in which to connect power or the power port. 98 00:07:43 --> 00:07:47 And the way I am going to connect power, 99 00:07:47 --> 00:07:52 I am going to connect a plus VS supply here. 100 00:07:52 --> 00:07:56 That is going to be my ground node. 101 00:07:56 --> 00:08:03 And I am going to connect a minus VS supply to this node 102 00:08:03 --> 00:08:08 here. So, these voltages are both VS. 103 00:08:08 --> 00:08:14 I want to apply a plus VS here and a negative VS out here. 104 00:08:14 --> 00:08:21 And I am going to take the output between the ground node 105 00:08:21 --> 00:08:27 and the output node of the operational amplifier and call 106 00:08:27 --> 00:08:32 that a vO. This is the output port. 107 00:08:32 --> 00:08:35 So, input port and output port and a power port. 108 00:08:35 --> 00:08:39 Think of this as a pattern where I have an input port 109 00:08:39 --> 00:08:42 across which I connect the input. 110 00:08:42 --> 00:08:46 I have a power port across which I connect a plus VS, 111 00:08:46 --> 00:08:49 minus VS supply, and then I take the output 112 00:08:49 --> 00:08:54 terminal and take a ground terminal, which is defined by 113 00:08:54 --> 00:08:58 external components of my circuitry, and use this as my 114 00:08:58 --> 00:09:02 reference node. Remember ground is just a 115 00:09:02 --> 00:09:06 reference node. I am going to use this as a 116 00:09:06 --> 00:09:09 reference node. These two are equal in 117 00:09:09 --> 00:09:11 magnitude. And take this as my output. 118 00:09:11 --> 00:09:16 And when I do something like this, I can build an even 119 00:09:16 --> 00:09:20 simpler, so this is an abstract differential input amplifier. 120 00:09:20 --> 00:09:23 In other words, this amplifier is going to 121 00:09:23 --> 00:09:26 amplify whatever I apply at the input. 122 00:09:26 --> 00:09:30 A slightly more abstract representation of this looks 123 00:09:30 --> 00:09:34 like this. vOUT and plus/minus vIN. 124 00:09:34 --> 00:09:38 This is a slightly more abstract representation where, 125 00:09:38 --> 00:09:42 remember, we are going to draw this again and again, 126 00:09:42 --> 00:09:46 maybe at least 38 or 39 times in this course. 127 00:09:46 --> 00:09:49 And, remember, each time you draw it, 128 00:09:49 --> 00:09:52 remember that there is an implicit power port, 129 00:09:52 --> 00:09:58 a plus/minus supply that is connected which we don't show. 130 00:09:58 --> 00:10:02 And I remember when I first learned about it a long time ago 131 00:10:02 --> 00:10:05 there was a confusion in me initially. 132 00:10:05 --> 00:10:08 How does this work? Where is the power coming from? 133 00:10:08 --> 00:10:13 Just remember that power comes from a plus/minus supply, 134 00:10:13 --> 00:10:16 and we just don't show that in this abstraction. 135 00:10:16 --> 00:10:20 Now, the details, a lot of details are in Chapter 136 00:10:20 --> 00:10:24 16 of your course notes. That's the reading for that. 137 00:10:24 --> 00:10:28 The other thing is that there are some other key properties of 138 00:10:28 --> 00:10:35 this amplifier. And let me discuss those very 139 00:10:35 --> 00:10:37 quickly. First of all, 140 00:10:37 --> 00:10:44 I can draw a circuit model for the amplifier. 141 00:10:44 --> 00:10:52 Make some room for myself here. And this is a circuit model for 142 00:10:52 --> 00:11:00 what we call the ideal operational amplifier. 143 00:11:00 --> 00:11:03 And the circuit model is going to look like this. 144 00:11:03 --> 00:11:08 This is an abstract device. And, in terms of analyzing how 145 00:11:08 --> 00:11:12 this behaves in a circuit, I am going to show you this 146 00:11:12 --> 00:11:15 abstract circuit that looks as follows. 147 00:11:15 --> 00:11:19 Some input v is applied at these two terminals here. 148 00:11:19 --> 00:11:23 And this terminal is called my v plus terminal and this is 149 00:11:23 --> 00:11:27 called my v minus terminal, so this corresponds to these 150 00:11:27 --> 00:11:32 two terminals. I am telling you that the 151 00:11:32 --> 00:11:36 current going in is going to be zero, so i plus is going to be 152 00:11:36 --> 00:11:39 zero and i minus is going to be zero. 153 00:11:39 --> 00:11:44 i plus is the current in here and i minus is the current into 154 00:11:44 --> 00:11:48 the v minus terminal, and both these currents are 155 00:11:48 --> 00:11:51 going to be zero in this device here. 156 00:11:51 --> 00:11:54 The output is going to look like this. 157 00:11:54 --> 00:12:00 Let me just call it vOUT to be consistent with this here. 158 00:12:00 --> 00:12:03 And taken with ground as my reference. 159 00:12:03 --> 00:12:07 The output is simply Av. In other words, 160 00:12:07 --> 00:12:12 what I am doing is I am going to model this as a device that 161 00:12:12 --> 00:12:16 has a dependent source at its output. 162 00:12:16 --> 00:12:22 And the dependent source here is a voltage controlled voltage 163 00:12:22 --> 00:12:25 source. It is a dependent source, 164 00:12:25 --> 00:12:30 it is a voltage controlled voltage source such that the 165 00:12:30 --> 00:12:37 output voltage is A times the voltage v across its input. 166 00:12:37 --> 00:12:42 This is actually very simple. Think of these three terminals 167 00:12:42 --> 00:12:46 I have shown you here. I applied input across these. 168 00:12:46 --> 00:12:51 And the output is going to be A times whatever I applied. 169 00:12:51 --> 00:12:54 And A is going to tend towards infinity. 170 00:12:54 --> 00:12:59 A is going to be huge. And specific values for A might 171 00:12:59 --> 00:13:05 be a hundred thousand or a million or things of that sort. 172 00:13:05 --> 00:13:09 Huge A in this abstract amplifier. 173 00:13:09 --> 00:13:15 In addition to that, the other properties are that 174 00:13:15 --> 00:13:21 it is going to have infinite input resistance. 175 00:13:21 --> 00:13:30 That means looking in this looks like an open circuit. 176 00:13:30 --> 00:13:35 The fact that this is open here implies the infinite input 177 00:13:35 --> 00:13:40 resistance across this port. What about the output here? 178 00:13:40 --> 00:13:44 Remember, this is a voltage source. 179 00:13:44 --> 00:13:49 And we have a zero output resistance, which means that no 180 00:13:49 --> 00:13:55 matter how the load affects this, as I apply a load this is 181 00:13:55 --> 00:14:01 going to behave like an ideal voltage source and keep holding 182 00:14:01 --> 00:14:06 the voltage constant based on whatever the function I 183 00:14:06 --> 00:14:10 establish here. And A is virtually infinite. 184 00:14:10 --> 00:14:14 Let me pause there for a few seconds and just dwell on this 185 00:14:14 --> 00:14:17 so you just understand what the basic device is. 186 00:14:17 --> 00:14:20 Following this basic definition, I am just going to 187 00:14:20 --> 00:14:23 build a whole bunch of fun little circuits. 188 00:14:23 --> 00:14:26 The analysis will be pretty straightforward, 189 00:14:26 --> 00:14:30 but this is a big conceptual leap here where there is some 190 00:14:30 --> 00:14:33 circuitry inside. Containing resistors, 191 00:14:33 --> 00:14:36 MOSFETs, a whole bunch of stuff in there. 192 00:14:36 --> 00:14:39 I am not telling you what is inside it. 193 00:14:39 --> 00:14:42 Much like I could build an abstract amplifier, 194 00:14:42 --> 00:14:46 I could put an abstract box around the amplifier you saw 195 00:14:46 --> 00:14:50 earlier, I want to put a box around some circuitry. 196 00:14:50 --> 00:14:53 I am not telling you what the circuitry is. 197 00:14:53 --> 00:14:57 And, if you are curious, you should look at page 581 of 198 00:14:57 --> 00:15:02 your course notes. There is an example solved. 199 00:15:02 --> 00:15:07 The example is for a differential amplifier. 200 00:15:07 --> 00:15:11 This is the small signal analysis chapter. 201 00:15:11 --> 00:15:17 That differential amplifier that's solved in that example is 202 00:15:17 --> 00:15:23 usually the first stage in an operational amplifier circuit. 203 00:15:23 --> 00:15:30 That differential amplifier is the first stage at the input. 204 00:15:30 --> 00:15:34 And that differential amplifier, as the name implies, 205 00:15:34 --> 00:15:39 amplifies not a single voltage but amplifies a differential 206 00:15:39 --> 00:15:42 voltage. Note that this guy amplifies 207 00:15:42 --> 00:15:46 the voltage difference between these two terminals. 208 00:15:46 --> 00:15:50 That's v here. And v is simply the same as v 209 00:15:50 --> 00:15:54 plus minus v minus. It's the node voltage here 210 00:15:54 --> 00:16:00 minus the node voltage here. That is what's amplified. 211 00:16:00 --> 00:16:03 It amplifies a difference. Therefore, it is called a 212 00:16:03 --> 00:16:06 difference amplifier or a differential amplifier. 213 00:16:06 --> 00:16:10 And so that input stage is what is inside the op amp. 214 00:16:10 --> 00:16:13 It's got a bunch of other circuitry like level shifters 215 00:16:13 --> 00:16:16 and so on. And at the output it has got a 216 00:16:16 --> 00:16:18 buffer. At at the output it has 217 00:16:18 --> 00:16:23 something that is reminiscent of the source follower circuit that 218 00:16:23 --> 00:16:26 you learned about in recitations, solved an example 219 00:16:26 --> 00:16:31 in the course notes and in your homework as well. 220 00:16:31 --> 00:16:35 And you solved a variant of the source follower on your quiz as 221 00:16:35 --> 00:16:38 well in problem two. So, a circuit that looks like 222 00:16:38 --> 00:16:42 that appears at the output. Remember, for the source 223 00:16:42 --> 00:16:46 follower, the resistance looking in from the output was very, 224 00:16:46 --> 00:16:49 very small. You have seen some of the 225 00:16:49 --> 00:16:53 pieces that go inside the amplifier, but we will deal with 226 00:16:53 --> 00:16:57 this as a building block and simply represent it using this 227 00:16:57 --> 00:17:02 abstract little circuit. To dwell on this a little 228 00:17:02 --> 00:17:06 longer, this little device here is the workhorse of the analog 229 00:17:06 --> 00:17:10 industry. Much like your primitive gate 230 00:17:10 --> 00:17:14 abstraction, your inverter and NAND gate and so on, 231 00:17:14 --> 00:17:18 much as your primitive inverter or NAND gate was from the 232 00:17:18 --> 00:17:21 foundations of the digital industry. 233 00:17:21 --> 00:17:25 Remember we learned how to build this little abstract 234 00:17:25 --> 00:17:30 device called a NAND gate or an inverter? 235 00:17:30 --> 00:17:34 We noticed that those form the foundations of very complicated 236 00:17:34 --> 00:17:38 microprocessors. Those were the building blocks 237 00:17:38 --> 00:17:41 of the digital industry. In the same way, 238 00:17:41 --> 00:17:45 this little beast here is the building block of the analog 239 00:17:45 --> 00:17:47 industry. Just to give you an analogy 240 00:17:47 --> 00:17:51 from software, think of this abstract little 241 00:17:51 --> 00:17:55 device as a library routine from a library of functions when you 242 00:17:55 --> 00:18:01 program in C++ or whatever. Can someone give me an example 243 00:18:01 --> 00:18:06 of an incredibly popular routine that we use all the time that 244 00:18:06 --> 00:18:11 may be called the workhorse of the software industry? 245 00:18:11 --> 00:18:13 Pardon? An abstraction, 246 00:18:13 --> 00:18:17 an abstract procedure. One example might be something 247 00:18:17 --> 00:18:21 like a printf. Printf is an abstract name for 248 00:18:21 --> 00:18:25 a procedure that goes and does something for you. 249 00:18:25 --> 00:18:31 It is amazing how we take the lowly printf for granted. 250 00:18:31 --> 00:18:34 I stick my printf into my program, it includes the 251 00:18:34 --> 00:18:37 standard IO library and it goes and prints a value. 252 00:18:37 --> 00:18:40 You won't believe how complicated the printf is. 253 00:18:40 --> 00:18:43 As you go into learning more advanced software subjects, 254 00:18:43 --> 00:18:46 implementing the printf is a nightmare. 255 00:18:46 --> 00:18:48 It is horrendously complicated. Just imagine. 256 00:18:48 --> 00:18:52 You give it a string and it has to go and print that on your 257 00:18:52 --> 00:18:55 terminal or on your Windows system or whatever. 258 00:18:55 --> 00:19:00 Think of the complicated steps it has to go through. 259 00:19:00 --> 00:19:02 But, as far as you're concerned, it's simple. 260 00:19:02 --> 00:19:05 Just print out something and you're done. 261 00:19:05 --> 00:19:08 The same way. Think of this as the printf of 262 00:19:08 --> 00:19:10 the analog business. It is really simple, 263 00:19:10 --> 00:19:13 and the analysis is going to be incredibly simple, 264 00:19:13 --> 00:19:17 it will be mind-bogglingly simple, but inside it, 265 00:19:17 --> 00:19:19 heavens forbid if you look inside it. 266 00:19:19 --> 00:19:22 Tell you what, go into to S-T-D-I-O dot in one 267 00:19:22 --> 00:19:25 of the library routines and just pore through printf. 268 00:19:25 --> 00:19:30 The world's worst horrendous macros are in there. 269 00:19:30 --> 00:19:33 I mean it is just nasty. The same way inside the op amp, 270 00:19:33 --> 00:19:35 it is nasty. You don't want to go there. 271 00:19:35 --> 00:19:38 Much like in your C programming in your classes, 272 00:19:38 --> 00:19:42 you were able to use printf without fully knowing how it was 273 00:19:42 --> 00:19:45 implemented. Probably some MIT god or some 274 00:19:45 --> 00:19:48 key graduate implemented it, but once it was implemented you 275 00:19:48 --> 00:19:52 just used it based on simple abstract rules as to how it 276 00:19:52 --> 00:19:54 behaved. You didn't have to know what 277 00:19:54 --> 00:19:57 was inside it to use it. The same way with the 278 00:19:57 --> 00:20:02 operational amplifier. So, just think of printf when 279 00:20:02 --> 00:20:06 you see this and just imagine how simple it is going to be to 280 00:20:06 --> 00:20:09 use it. You may think that I spend way 281 00:20:09 --> 00:20:12 too much time, ten minutes dwelling on this 282 00:20:12 --> 00:20:15 abstract concept, but I like to dwell on things 283 00:20:15 --> 00:20:18 that I think are incredibly important. 284 00:20:18 --> 00:20:21 The concept of abstraction is very important. 285 00:20:21 --> 00:20:25 And it's not just in software. The concept of abstraction 286 00:20:25 --> 00:20:30 pervades all of EECS. And if I were to give you a 287 00:20:30 --> 00:20:34 project to say go and ask every professor what is the one word 288 00:20:34 --> 00:20:38 that you think best describes all of EECS? 289 00:20:38 --> 00:20:41 Just pick one word. Go ask every single professor 290 00:20:41 --> 00:20:44 you know. What is a single word? 291 00:20:44 --> 00:20:48 If you were to characterize all of EECS with just one word, 292 00:20:48 --> 00:20:52 what might that word be? In my mind, it is the A word, 293 00:20:52 --> 00:20:54 abstraction. It is all over. 294 00:20:54 --> 00:20:58 If you do a grep on all the words used by all your 295 00:20:58 --> 00:21:03 professors in your four years here, I promise you the first 296 00:21:03 --> 00:21:08 one will be know. And the second one will be 297 00:21:08 --> 00:21:10 abstraction. Check it out. 298 00:21:10 --> 00:21:13 See if what I am saying is true or not. 299 00:21:13 --> 00:21:17 It is all over the place. In 6.001, how many times do you 300 00:21:17 --> 00:21:21 think the word abstraction was used in 6.001? 301 00:21:21 --> 00:21:25 It's all over the map. It's the A word all over. 302 00:21:25 --> 00:21:30 Imagine your shock when you see it being used in 002 because the 303 00:21:30 --> 00:21:35 same concept applies. We build more complicated 304 00:21:35 --> 00:21:40 systems by abstracting out the details of lesser objects, 305 00:21:40 --> 00:21:45 and then using those to build the more complicated systems. 306 00:21:45 --> 00:21:50 Abstraction is a very powerful mechanism of dealing with 307 00:21:50 --> 00:21:54 complexity. Next step is how do I go about 308 00:21:54 --> 00:21:58 using the op amp? Let me show you how it looks on 309 00:21:58 --> 00:22:03 a scope. What I am going to do is apply 310 00:22:03 --> 00:22:09 input to the op amp, I am going to look at the 311 00:22:09 --> 00:22:17 output, place the resistor RL to ground and look at the output. 312 00:22:17 --> 00:22:25 And here I am going to apply a plus VS and out here a minus VS. 313 00:22:25 --> 00:22:32 Again, remember that a plus VS simply looks like this and a 314 00:22:32 --> 00:22:40 minus VS simply looks like this. It's just an inverted VS 315 00:22:40 --> 00:22:45 applied here so I get a minus VS at this input. 316 00:22:45 --> 00:22:50 First of all, what I would like to do is as I 317 00:22:50 --> 00:22:57 change vIN, I am going to plot for you how vOUT looks. 318 00:22:57 --> 00:23:03 vIN and this is vO. I am going to plot vIN in terms 319 00:23:03 --> 00:23:08 of microvolts and vO in volts. vIN is going to have a very 320 00:23:08 --> 00:23:14 very small, the scale is going to be in microvolts because 321 00:23:14 --> 00:23:17 remember the gain of this is huge. 322 00:23:17 --> 00:23:21 It's on the order of ten to the sixth. 323 00:23:21 --> 00:23:25 It's huge. Small changes in vIN are going 324 00:23:25 --> 00:23:32 to cause massive changes in vO. I have a very fine scale on the 325 00:23:32 --> 00:23:36 X axis. What is going to happen if I 326 00:23:36 --> 00:23:39 somehow magically make vIN exactly zero? 327 00:23:39 --> 00:23:45 If I short these two terminals, if this was a completely ideal 328 00:23:45 --> 00:23:50 op amp, which it never is, if it's a completely ideal op 329 00:23:50 --> 00:23:53 amp, then my output should be zero. 330 00:23:53 --> 00:24:00 As I increase my vIN the output should be A times vIN. 331 00:24:00 --> 00:24:03 For some small value of vIN, small v, let's say one 332 00:24:03 --> 00:24:06 microvolt, the output should be one volt. 333 00:24:06 --> 00:24:10 A is a constant so this would look like a straight line. 334 00:24:10 --> 00:24:14 And let's say my supply voltages are 12 volts minus 12 335 00:24:14 --> 00:24:19 volts, if this were an ideal amplifier and I didn't have to 336 00:24:19 --> 00:24:23 worry about the supply, this would just go on extending 337 00:24:23 --> 00:24:25 forever. But I have a plus 12 volt 338 00:24:25 --> 00:24:27 supply and a minus 12 volt supply. 339 00:24:27 --> 00:24:32 My output cannot go past those limits. 340 00:24:32 --> 00:24:35 And so, therefore, my output kind of flattens out 341 00:24:35 --> 00:24:39 at these two points. And it is called hitting the 342 00:24:39 --> 00:24:42 rails. Output goes up and you hear a 343 00:24:42 --> 00:24:45 thunk sound and you hit the rails. 344 00:24:45 --> 00:24:49 When you play with op amps in your next lab, 345 00:24:49 --> 00:24:53 if you listen really, really carefully you may hear 346 00:24:53 --> 00:24:55 it. So, this saturates out. 347 00:24:55 --> 00:24:59 Not surprisingly, this region where the output 348 00:24:59 --> 00:25:05 saturates at the supply is called the saturation region. 349 00:25:05 --> 00:25:09 Remember, don't confuse it with-- It's not the same as your 350 00:25:09 --> 00:25:13 saturation in the MOSFET. It is a totally different 351 00:25:13 --> 00:25:16 thing. It is just happenstance that we 352 00:25:16 --> 00:25:20 call this saturation. And if you would like to think 353 00:25:20 --> 00:25:24 about it, you can think of it as the thunk region. 354 00:25:24 --> 00:25:27 That's probably more appropriate to distinguish it 355 00:25:27 --> 00:25:32 from the saturation region in the MOSFET. 356 00:25:32 --> 00:25:36 And, not surprisingly, this one is called the active 357 00:25:36 --> 00:25:39 region. And it is in this region that 358 00:25:39 --> 00:25:43 we use the op amp. Here it has hit the rails and 359 00:25:43 --> 00:25:48 is kind of dangling out there. It's not much use to us. 360 00:25:48 --> 00:25:53 It's in this active region that we use it because this is where 361 00:25:53 --> 00:25:58 the gain is seen. Now, it turns out that this is 362 00:25:58 --> 00:26:01 a very high gain device. It is very skittish. 363 00:26:01 --> 00:26:05 This gain is kind of a really funny thing. 364 00:26:05 --> 00:26:08 It's dependent on a bunch of factors. 365 00:26:08 --> 00:26:10 This could be temperature dependent. 366 00:26:10 --> 00:26:15 This gain here and this curve is just completely skittish. 367 00:26:15 --> 00:26:20 It could depend on temperature. It could depend on time of day. 368 00:26:20 --> 00:26:24 It could depend on what medication this amplifier is on. 369 00:26:24 --> 00:26:27 It could depend on its mood swings. 370 00:26:27 --> 00:26:32 Who knows what? This is kind of unstable. 371 00:26:32 --> 00:26:34 And A in particular is highly unstable. 372 00:26:34 --> 00:26:36 It is going to be big, that's for sure, 373 00:26:36 --> 00:26:40 but it could be ten to the six, on a rainy day it might be two 374 00:26:40 --> 00:26:43 times ten to the six. If it feeling sleepy it may be 375 00:26:43 --> 00:26:46 point five times ten to the sixth. 376 00:26:46 --> 00:26:48 It is big but I cannot rely on it. 377 00:26:48 --> 00:26:51 Let me show you an example. I want to show you this curve 378 00:26:51 --> 00:26:54 for this MOSFET, apply an input and plotting the 379 00:26:54 --> 00:26:56 output. What I will do is take a look 380 00:26:56 --> 00:27:00 at this curve. Then what I am going to do is 381 00:27:00 --> 00:27:04 use a heat gun to heat the op amp and you are going to see 382 00:27:04 --> 00:27:07 this vary all over the map. If you still remember last 383 00:27:07 --> 00:27:11 week, some of you may remember that from some place in a 384 00:27:11 --> 00:27:15 similar situation where the gm for the MOSFETs you were given 385 00:27:15 --> 00:27:18 was also dependent on temperature and stuff like that. 386 00:27:18 --> 00:27:22 It is a very common occurrence. And that is certainly the case 387 00:27:22 --> 00:27:24 for the MOSFET. 388 00:27:24 --> 00:27:31 389 00:27:31 --> 00:27:34 Let's apply input. Let's do this. 390 00:27:34 --> 00:27:38 This is vIN versus vOUT for the amplifier. 391 00:27:38 --> 00:27:44 Notice that this is plus 12 volts, this is minus 12 volts. 392 00:27:44 --> 00:27:47 It is about two volts per division. 393 00:27:47 --> 00:27:52 This axis here is in microvolts, I believe. 394 00:27:52 --> 00:27:57 For a very small change, for a few tens of microvolts, 395 00:27:57 --> 00:28:03 I have an incredibly high gain. Notice that this has an 396 00:28:03 --> 00:28:07 incredibly high gain here. The gain is the slope of this 397 00:28:07 --> 00:28:11 line, almost a vertical line. What I am going to do next, 398 00:28:11 --> 00:28:14 is to have some fun, is I am going to heat the op 399 00:28:14 --> 00:28:16 amp. To show you that A is kind of 400 00:28:16 --> 00:28:20 really skittish and also the fact that it doesn't quite hit 401 00:28:20 --> 00:28:24 zero, it does all kinds of weird things, I am going to heat the 402 00:28:24 --> 00:28:27 op amp. And then let's take a look at 403 00:28:27 --> 00:28:30 how that curve fluctuates. 404 00:28:30 --> 00:29:00 405 00:29:00 --> 00:29:04 What you saw there was that the op amp began to behave really 406 00:29:04 --> 00:29:07 weirdly as I heated it. Instead of doing this it 407 00:29:07 --> 00:29:11 sometimes did this really weirdly, like getting an offset 408 00:29:11 --> 00:29:15 from the center and so on. And it does a bunch of other 409 00:29:15 --> 00:29:18 weird things, but we won't go into those 410 00:29:18 --> 00:29:21 details. It's not relevant for this 411 00:29:21 --> 00:29:23 course. But the point is that the gain 412 00:29:23 --> 00:29:29 and the offset at the input are dependent on temperature. 413 00:29:29 --> 00:29:33 And we look for ways to make it less dependent on temperature. 414 00:29:33 --> 00:29:37 As the next step, what I would like to do is 415 00:29:37 --> 00:29:40 build a circuit. This is model equivalent of 416 00:29:40 --> 00:29:44 your Hello World program. We are going to use the printf 417 00:29:44 --> 00:29:47 and build a small program on the printf. 418 00:29:47 --> 00:29:51 You don't have to worry about how printf is implemented, 419 00:29:51 --> 00:29:56 just that we can build very highly interesting circuits with 420 00:29:56 --> 00:30:00 this horrendously complicated function based on a simple 421 00:30:00 --> 00:30:08 abstraction of the device. The circuit that we will build 422 00:30:08 --> 00:30:14 is called a noninverting amplifier. 423 00:30:14 --> 00:30:20 424 00:30:20 --> 00:30:22 From now on, I am not going to show you the 425 00:30:22 --> 00:30:25 plus/minus VS. I am not going to show the 426 00:30:25 --> 00:30:28 power port, but it is in there. It's hidden under the 427 00:30:28 --> 00:30:31 abstraction layer. This is my op amp. 428 00:30:31 --> 00:30:35 And I am going to build the following circuit. 429 00:30:35 --> 00:30:38 This is my v plus and this is my v minus. 430 00:30:38 --> 00:30:43 What I am going to do is for the v plus I shall apply a vIN. 431 00:30:43 --> 00:30:47 Let me talk a little bit about ground as well. 432 00:30:47 --> 00:30:52 Ground is commonly taken as the point at which I connect my VS 433 00:30:52 --> 00:30:58 and minus VS supply. It is kind of at the midpoint. 434 00:30:58 --> 00:31:03 And if VS and minus VS are very carefully tuned then the output 435 00:31:03 --> 00:31:08 is also going to be at that same ground reference when the input 436 00:31:08 --> 00:31:11 is zero. So, the ground is defined as 437 00:31:11 --> 00:31:16 the point at which I connect my plus/minus VS supplies. 438 00:31:16 --> 00:31:20 I apply my vIN out here. Then what I am going to do, 439 00:31:20 --> 00:31:24 here is my output vO. I am going to have a resistive 440 00:31:24 --> 00:31:30 divider to ground here and label these R1 and R2. 441 00:31:30 --> 00:31:36 And what I am going to do here is feed this back to the input, 442 00:31:36 --> 00:31:40 to the v minus input. I am going to sample the 443 00:31:40 --> 00:31:44 voltage here and feed that into here. 444 00:31:44 --> 00:31:49 So, this is my abstract model and this is my Hello World 445 00:31:49 --> 00:31:53 program. What we are going to do is 446 00:31:53 --> 00:31:59 simply analyze how this little program behaves. 447 00:31:59 --> 00:32:00 So, my equivalent circuit model. 448 00:32:00 --> 00:32:03 The way to analyze these is after one or two of these 449 00:32:03 --> 00:32:07 examples, you will be able to directly analyze this just by 450 00:32:07 --> 00:32:09 looking at it, by inspection. 451 00:32:09 --> 00:32:11 But, much as we did for the other pieces, 452 00:32:11 --> 00:32:14 let me grunge through drawing the equivalent circuit and 453 00:32:14 --> 00:32:18 grinding through the analysis, and then show you the much 454 00:32:18 --> 00:32:20 simpler way of doing it. And even here, 455 00:32:20 --> 00:32:23 even with this grinding analysis, it is going to be 456 00:32:23 --> 00:32:26 pretty simple in any case. So, I will replace the op amp 457 00:32:26 --> 00:32:30 with its equivalent circuit model. 458 00:32:30 --> 00:32:33 Its equivalent circuit was v plus, v minus. 459 00:32:33 --> 00:32:44 460 00:32:44 --> 00:32:49 So, that was the equivalent circuit model of the operational 461 00:32:49 --> 00:32:53 amplifier, just this piece. I draw that for you. 462 00:32:53 --> 00:32:59 Then what I am going to do is I connect my v in here. 463 00:32:59 --> 00:33:02 And, remember, I have an R1, 464 00:33:02 --> 00:33:09 R2 resistive divider here. And this one gets connected to 465 00:33:09 --> 00:33:14 this terminal there. I also know that i plus is 466 00:33:14 --> 00:33:18 zero. I also know that i minus is 467 00:33:18 --> 00:33:22 zero. All I've done is simply 468 00:33:22 --> 00:33:30 replaced the amplifier with its equivalent circuit. 469 00:33:30 --> 00:33:34 Let's go ahead and analyze that circuit now. 470 00:33:34 --> 00:33:38 Let's go ahead and analyze that circuit. 471 00:33:38 --> 00:33:42 And it's going to be pretty simple, actually. 472 00:33:42 --> 00:33:48 What I am going to show you is the hard way of doing it. 473 00:33:48 --> 00:33:54 I will show you a much easier way, but the hard way itself is 474 00:33:54 --> 00:33:59 pathetically easy. What I want to do is find vO in 475 00:33:59 --> 00:34:04 terms of vIN. And there will be a bunch of 476 00:34:04 --> 00:34:08 other factors thrown in, including things like R1 and 477 00:34:08 --> 00:34:12 R2, A and stuff like that. Let's go and analyze it. 478 00:34:12 --> 00:34:16 vO, let's look at that circuit. By the way, let me take 30 479 00:34:16 --> 00:34:20 seconds and make a little speech at this point. 480 00:34:20 --> 00:34:24 When you see circuits like this, and I saw this happen in 481 00:34:24 --> 00:34:27 quiz two as well, for some reason, 482 00:34:27 --> 00:34:31 when you see a new kind of circuit, don't completely go 483 00:34:31 --> 00:34:36 berserk or freeze or whatever. There is just no reason to. 484 00:34:36 --> 00:34:39 You know the node method. The node method is the 485 00:34:39 --> 00:34:43 workhorse of our business. When in doubt apply the node 486 00:34:43 --> 00:34:45 method. It will simply work. 487 00:34:45 --> 00:34:47 Don't freeze. Don't think oh, 488 00:34:47 --> 00:34:50 man, I need to apply a pattern that I know already. 489 00:34:50 --> 00:34:52 I must have seen this somewhere. 490 00:34:52 --> 00:34:56 When in doubt boom, apply the node method. 491 00:34:56 --> 00:35:00 This circuit here, all I have here is one unknown 492 00:35:00 --> 00:35:03 node voltage. I know the voltage of v plus, 493 00:35:03 --> 00:35:06 I need to compute the voltage vO. 494 00:35:06 --> 00:35:10 There are two unknowns, vO is an unknown and the 495 00:35:10 --> 00:35:14 voltage here at v minus is another unknown. 496 00:35:14 --> 00:35:18 This is a very simple circuit involving a dependent voltage 497 00:35:18 --> 00:35:23 controlled voltage source, and you need to find out vO and 498 00:35:23 --> 00:35:27 v minus using the node method. Just apply it. 499 00:35:27 --> 00:35:31 It's simple. Don't freeze. 500 00:35:31 --> 00:35:36 Just look at it and say I can do it and apply the node method. 501 00:35:36 --> 00:35:40 It will simply work. So, let's do that. 502 00:35:40 --> 00:35:45 What I can do here is vO is A times v plus minus v minus. 503 00:35:45 --> 00:35:50 This is actually really simple. And then, if I take v plus 504 00:35:50 --> 00:35:55 here, I know v plus is simply vIN so I will just make that 505 00:35:55 --> 00:36:01 substitution right away. So, v plus is simply vIN. 506 00:36:01 --> 00:36:10 What is v minus? v minus here is vO -- 507 00:36:10 --> 00:36:25 508 00:36:25 --> 00:36:27 What is v plus? I'm sorry, v minus. 509 00:36:27 --> 00:36:33 v minus is simply the voltage that is between R1 and R2. 510 00:36:33 --> 00:36:37 Notice that no current flows in to the v minus node. 511 00:36:37 --> 00:36:42 There is no current flowing in. Voltage at v minus is simply 512 00:36:42 --> 00:36:46 the voltage given by the resistive divider, 513 00:36:46 --> 00:36:50 which is vO times R2 divided by R1 plus R2. 514 00:36:50 --> 00:36:52 Stare at that for another second. 515 00:36:52 --> 00:36:58 The voltage at this node here is simply given by the resistive 516 00:36:58 --> 00:37:02 divider. Because no current is flowing 517 00:37:02 --> 00:37:06 in this direction. And no current flows in because 518 00:37:06 --> 00:37:10 I am telling you there is no current there based on my 519 00:37:10 --> 00:37:14 abstraction. I am telling you i minus is 520 00:37:14 --> 00:37:16 zero. That voltage is simply the 521 00:37:16 --> 00:37:19 voltage at this resistive divider. 522 00:37:19 --> 00:37:23 And so I can simplify it further and write this as vO. 523 00:37:23 --> 00:37:28 So I get, there is a one here. And I move this thing over to 524 00:37:28 --> 00:37:35 this side so I get one plus A times R2 divided by R1 plus R2. 525 00:37:35 --> 00:37:42 And that is equal to AvIN. And simplifying it some more, 526 00:37:42 --> 00:37:51 I get vO is AvIN divided by one plus AR2 divided by R1 plus R2. 527 00:37:51 --> 00:38:00 Notice how simple this is, and this is the hard method. 528 00:38:00 --> 00:38:05 All I have done is analyze the circuit using the basic circuit 529 00:38:05 --> 00:38:09 analysis principle that you learned the first week of the 530 00:38:09 --> 00:38:12 course, and I have the output for you. 531 00:38:12 --> 00:38:17 I just noted very carefully what the relationships were 532 00:38:17 --> 00:38:20 between the various elements in the abstraction. 533 00:38:20 --> 00:38:25 Notice here that I am told that A is extremely large. 534 00:38:25 --> 00:38:30 A is on the order of ten to the six and so on. 535 00:38:30 --> 00:38:36 And suppose it is the case that, let me write that down 536 00:38:36 --> 00:38:40 again. vO is AvIN, one plus AR2, 537 00:38:40 --> 00:38:44 R2. Suppose R1 and R2 are more or 538 00:38:44 --> 00:38:52 less comparable and A is ten to the six, it's a huge number, 539 00:38:52 --> 00:39:00 so this whole number is much, much greater than one. 540 00:39:00 --> 00:39:06 If it is much huger than one, what I can do is I can then 541 00:39:06 --> 00:39:13 write this as follows. I can say that this is more or 542 00:39:13 --> 00:39:20 less equal to AvIN divided by AR2 divided by R1 plus R2. 543 00:39:20 --> 00:39:26 I am ignoring the one here. As soon as I do that, 544 00:39:26 --> 00:39:34 notice I can cancel out A and I get vO to be approximately equal 545 00:39:34 --> 00:39:40 to vIN times R1 plus R2 divided by R2. 546 00:39:40 --> 00:39:45 Notice now that when the gain is very large the output is a 547 00:39:45 --> 00:39:49 function of the input multiplied by some number. 548 00:39:49 --> 00:39:54 The beauty of this thing here is that when A is very large, 549 00:39:54 --> 00:39:59 or this expression is very large, A cancels out and there 550 00:39:59 --> 00:40:06 is no A in this relationship. This means that even though the 551 00:40:06 --> 00:40:11 basic amplifier was very skittish, the output here 552 00:40:11 --> 00:40:16 relates to the input based on components that I have control 553 00:40:16 --> 00:40:20 over. These are soldiers in my army. 554 00:40:20 --> 00:40:25 I control them. So, to give you a sense of some 555 00:40:25 --> 00:40:29 numbers here, suppose A was ten to the six. 556 00:40:29 --> 00:40:35 And I choose R1 to be 9R. And R to be some R. 557 00:40:35 --> 00:40:44 Then vO is ten to the sixth vIN divided by one plus ten to the 558 00:40:44 --> 00:40:52 six R divided by 9R plus R. So, that is ten to the six vIN 559 00:40:52 --> 00:41:01 divided by one plus ten to the six divided by ten. 560 00:41:01 --> 00:41:05 All right. If I ignore the one here, 561 00:41:05 --> 00:41:11 the ten to the six and ten to the six cancel out, 562 00:41:11 --> 00:41:17 this ends up giving me 10vIN. So, I get a really nice 563 00:41:17 --> 00:41:23 amplifier whose output is simply ten times the input and 564 00:41:23 --> 00:41:30 determined solely by some resistor values. 565 00:41:30 --> 00:41:34 Let me show you another quick demo this time and show you the 566 00:41:34 --> 00:41:37 amplifier again, but with resistors connected 567 00:41:37 --> 00:41:41 like that. And then I show you that I want 568 00:41:41 --> 00:41:45 to heat the amplifier to the wazoo, the op amp to the wazoo, 569 00:41:45 --> 00:41:48 but vO is going to be absolutely rock solid. 570 00:41:48 --> 00:41:51 Let's try that out. 571 00:41:51 --> 00:42:00 572 00:42:00 --> 00:42:03 This time around, this is the transfer function, 573 00:42:03 --> 00:42:07 the vO versus vIN. And notice that this time 574 00:42:07 --> 00:42:11 around I have similar scales on the X and Y axes, 575 00:42:11 --> 00:42:15 and this has a slope of 10. This is the point where the 576 00:42:15 --> 00:42:20 amplifier saturates at plus 12 volts, and this is minus 12 577 00:42:20 --> 00:42:23 volts, and this point here is a zero. 578 00:42:23 --> 00:42:25 So, this is vIN, vOUT, plus 12, 579 00:42:25 --> 00:42:31 minus 12 and this slope is 10. What I am going to do now is 580 00:42:31 --> 00:42:36 heat the op amp to the wazoo and this ain't going to change 581 00:42:36 --> 00:42:40 because it's my external resistors that control it 582 00:42:40 --> 00:42:45 independent of the value of A, provided A continues to be very 583 00:42:45 --> 00:42:47 large. I am just articulating the 584 00:42:47 --> 00:42:51 vOUT, vIN curve. And let me start heating the op 585 00:42:51 --> 00:42:53 amp. 586 00:42:53 --> 00:43:01 587 00:43:01 --> 00:43:06 Notice that it's pretty stable. It doesn't change because it is 588 00:43:06 --> 00:43:09 independent of the amplifier values. 589 00:43:09 --> 00:43:14 What I have done now is by connecting these resistors in 590 00:43:14 --> 00:43:18 this way, I have a nice amplifier with a gain of ten. 591 00:43:18 --> 00:43:22 The question you may ask yourselves is why? 592 00:43:22 --> 00:43:27 There is this little sucker in there that wants to shoot things 593 00:43:27 --> 00:43:32 up by ten to the sixth. Wants to knock things off the 594 00:43:32 --> 00:43:36 one rail or the negative rail. Why is it that it's behaving 595 00:43:36 --> 00:43:39 like a docile lamb here and giving us a nice little factor 596 00:43:39 --> 00:43:41 of ten gain no matter what I do to it? 597 00:43:41 --> 00:43:44 Why is it doing that? What is the intuition behind 598 00:43:44 --> 00:43:46 it? I will draw something on the 599 00:43:46 --> 00:43:50 board, but for the next ten seconds I want you think about 600 00:43:50 --> 00:43:51 it. See if you can come up with 601 00:43:51 --> 00:43:54 some insight as to why is it doing that. 602 00:43:54 --> 00:43:57 Why is it exactly ten? Why isn't the ten to the sixth 603 00:43:57 --> 00:44:01 kind of killing me somehow? Why am I getting exactly ten no 604 00:44:01 --> 00:44:04 matter what happens? See if you can come up with 605 00:44:04 --> 00:44:07 some intuition and then I will show you how it works. 606 00:44:07 --> 00:44:10 I will redraw the circuit in the meantime. 607 00:44:10 --> 00:44:28 608 00:44:28 --> 00:44:30 Let me see if I can give you some intuition. 609 00:44:30 --> 00:44:34 This is my circuit, and let's say this is R and 610 00:44:34 --> 00:44:36 this is R. As an example, 611 00:44:36 --> 00:44:42 let's assume that the input is 5 volts, vIN is 5 volts. 612 00:44:42 --> 00:44:46 If R and R are equal, what should the output be? 613 00:44:46 --> 00:44:50 It's R and R, so it's R1 plus R2 divided by 614 00:44:50 --> 00:44:53 R2, right? It's 2R divided by R, 615 00:44:53 --> 00:44:59 so it has a gain of two. My amplifier has a gain of two 616 00:44:59 --> 00:45:03 because R1 plus R2 divided by R2, which is my gain, 617 00:45:03 --> 00:45:09 is R plus R divided by R equals two. 618 00:45:09 --> 00:45:13 So, this will be 10 volts. If that is 10 volts this is 619 00:45:13 --> 00:45:16 going to be 5 volts, correct? 620 00:45:16 --> 00:45:18 This R and R, voltage divider, 621 00:45:18 --> 00:45:22 this is five, so I get 5 volts here. 622 00:45:22 --> 00:45:24 This is v plus. This is v minus. 623 00:45:24 --> 00:45:27 I get R and R, 5 volts here, 624 00:45:27 --> 00:45:32 that's how the circuit looks. Now let's understand what is 625 00:45:32 --> 00:45:34 going on. And listen very carefully. 626 00:45:34 --> 00:45:37 This is going to be a key insight that I hope you will 627 00:45:37 --> 00:45:40 carry with you for the rest of your lives. 628 00:45:40 --> 00:45:41 This is really, really key. 629 00:45:41 --> 00:45:45 What you are going to see is, I think, the third big ah-ha 630 00:45:45 --> 00:45:47 moment in 6.002. Like small signal analysis, 631 00:45:47 --> 00:45:51 like the frequency domain stuff we saw, I think this is the 632 00:45:51 --> 00:45:54 third big one in the next 30 or 40 seconds, things that are 633 00:45:54 --> 00:45:57 completely either not necessarily intuitive but are 634 00:45:57 --> 00:46:02 just spectacular in terms of what they can do for you. 635 00:46:02 --> 00:46:04 Let's see. Let's suppose that because I am 636 00:46:04 --> 00:46:08 heating it, let's suppose that A suddenly tends to increase. 637 00:46:08 --> 00:46:11 It wants to increase because I have heated it. 638 00:46:11 --> 00:46:15 A is saying I want to get out this mold here and starts to 639 00:46:15 --> 00:46:19 break through its shackles here. Let's say, as a Gedanken 640 00:46:19 --> 00:46:22 experiment, that it tries to shoot up this to 12 volts. 641 00:46:22 --> 00:46:26 It tries to push it up higher. This is just a Gedanken 642 00:46:26 --> 00:46:30 experiment. The up arrow says that the 643 00:46:30 --> 00:46:34 increase in A is trying to push up vO momentarily. 644 00:46:34 --> 00:46:38 Let's see what happens. It is trying to push up vO 645 00:46:38 --> 00:46:43 momentarily, so let's say this goes to 12 hypothetically. 646 00:46:43 --> 00:46:47 If that goes to 12, what should this volt node go 647 00:46:47 --> 00:46:49 to? Six, exactly. 648 00:46:49 --> 00:46:52 This goes to 6 volts. If that goes to six, 649 00:46:52 --> 00:46:56 what does v minus go to? 6 volts again. 650 00:46:56 --> 00:47:02 So, v minus goes to 6 volts. Now at the input I have 5 volts 651 00:47:02 --> 00:47:06 at v plus and 6 volts at v minus, so where should the 652 00:47:06 --> 00:47:09 output go? The output should go down 653 00:47:09 --> 00:47:13 because the voltage of the negative terminal is higher. 654 00:47:13 --> 00:47:17 And so the output is A times v plus minus v minus. 655 00:47:17 --> 00:47:22 And because this has gone down, this has gone up here it is 656 00:47:22 --> 00:47:25 going to try to pull the output down. 657 00:47:25 --> 00:47:29 That is going to pull the output down let's say to 9 volts 658 00:47:29 --> 00:47:34 or something. Cachunk, there is a big battle 659 00:47:34 --> 00:47:35 going on here. A has gone up, 660 00:47:35 --> 00:47:39 it has boosted it up to 12, but the moment that goes to 12, 661 00:47:39 --> 00:47:41 this goes to 6, this goes to 6, 662 00:47:41 --> 00:47:45 and the op amp output has to go down to 9 volts now because this 663 00:47:45 --> 00:47:48 input is higher here. If this goes to 9, 664 00:47:48 --> 00:47:51 this goes to 4.5. If that goes to 4.5, 665 00:47:51 --> 00:47:53 this goes to 4.5. What happens now? 666 00:47:53 --> 00:47:55 If this goes to 4.5, what happens? 667 00:47:55 --> 00:48:00 It wants to go back up. Can't it make up its mind? 668 00:48:00 --> 00:48:04 This guy wants to go back up now because v plus is higher 669 00:48:04 --> 00:48:07 than v minus. What am I seeing here? 670 00:48:07 --> 00:48:11 This whole circuit here behaves like my little son, 671 00:48:11 --> 00:48:14 my 9-year-old. If say do this, 672 00:48:14 --> 00:48:17 he wants to do the exact opposite. 673 00:48:17 --> 00:48:21 So, there is a trick in how you make them do things for you. 674 00:48:21 --> 00:48:25 Look at this. Because of this arrangement of 675 00:48:25 --> 00:48:29 the circuit when A tries to push the output up, 676 00:48:29 --> 00:48:34 the rest of the circuit tries to pull it back down to where it 677 00:48:34 --> 00:48:39 used to be. If the circuit tries not to 678 00:48:39 --> 00:48:43 follow the true path, the rest of the circuit tries 679 00:48:43 --> 00:48:48 to whack it into shape so it follows a true path. 680 00:48:48 --> 00:48:52 And what's happening is because, in this arrangement, 681 00:48:52 --> 00:48:57 I have fed back a portion of the output to the negative 682 00:48:57 --> 00:49:01 input. I have fed back some of the 683 00:49:01 --> 00:49:06 output to the negative input. And by providing this feedback 684 00:49:06 --> 00:49:09 of a portion of the output to the negative input, 685 00:49:09 --> 00:49:14 I have arranged it in a way that I have something called 686 00:49:14 --> 00:49:18 negative feedback. What negative feedback does is 687 00:49:18 --> 00:49:23 that if this wanted to go wild and crazy, the circuit provides 688 00:49:23 --> 00:49:27 it with some negative feedback like you just saw. 689 00:49:27 --> 00:49:32 Feedback, a big word. If you take a poll of all the 690 00:49:32 --> 00:49:35 EECS faculty, I suspect that feedback would 691 00:49:35 --> 00:49:39 rank at least as the ninth or tenth most important word in the 692 00:49:39 --> 00:49:41 EECS. If abstract is number one, 693 00:49:41 --> 00:49:46 I think this would rank like a nine or a ten or something. 694 00:49:46 --> 00:49:48 So, that's the reason why it worked. 695 00:49:48 --> 00:49:53 In the last couple of minutes, let me give you some insight, 696 00:49:53 --> 00:49:57 based on something that you know, on how feedback works. 697 00:49:57 --> 00:50:02 This is a road here. Let's look at anti lock breaks. 698 00:50:02 --> 00:50:05 This is my tire. And let's say I have a set of 699 00:50:05 --> 00:50:09 disk brakes here. As the car is moving forward, 700 00:50:09 --> 00:50:13 if I apply the brakes the tire stops rolling, 701 00:50:13 --> 00:50:18 but if I apply the breaks too hard it can lock up the tire and 702 00:50:18 --> 00:50:22 the whole car can skid. The way anti lock breaks work 703 00:50:22 --> 00:50:26 is as follows. There is a controller that sits 704 00:50:26 --> 00:50:30 here. And there is a little person 705 00:50:30 --> 00:50:34 looking at the wheel and seeing is it turning. 706 00:50:34 --> 00:50:39 So, this is a feedback. And it is saying is it turning? 707 00:50:39 --> 00:50:42 Yes. Or, is it not turning? 708 00:50:42 --> 00:50:45 No. All this person watching the 709 00:50:45 --> 00:50:50 tire is doing is saying is it turning or is it not turning. 710 00:50:50 --> 00:50:54 That is it. That is a negative feedback. 711 00:50:54 --> 00:50:59 And so, if it is no and if it is yes. 712 00:50:59 --> 00:51:03 If it is yes then what this does is it applies the brakes 713 00:51:03 --> 00:51:06 even more strongly. It is turning so I can apply 714 00:51:06 --> 00:51:08 more brakes. But if it says oops, 715 00:51:08 --> 00:51:11 it stopped turning, what it does is it simply 716 00:51:11 --> 00:51:15 releases, the controller releases the brakes. 717 00:51:15 --> 00:51:18 And when the controller releases the brakes this one 718 00:51:18 --> 00:51:23 tends to loosen up a little bit and the tire starts turning 719 00:51:23 --> 00:51:25 again. So, this way you are constantly 720 00:51:25 --> 00:51:30 keeping the tire in its region of critical friction so that it 721 00:51:30 --> 00:51:34 is constantly moving. And static friction applies to 722 00:51:34 --> 00:51:37 how hard you can brake and it doesn't start skidding. 723 00:51:37 --> 00:51:41 In fact, if you take your car out, and I don't say you do 724 00:51:41 --> 00:51:42 this. Let's say go onto the Charles 725 00:51:42 --> 00:51:46 River in the dead of winter and you drive on the lake and you 726 00:51:46 --> 00:51:49 slam your anti lock brakes on, on an icy patch, 727 00:51:49 --> 00:51:52 you will notice that there is a constant sound that looks like 728 00:51:52 --> 00:51:55 something is vibrating in there. That is exactly what is 729 00:51:55 --> 00:51:57 happening. Oops, the tire is locked. 730 00:51:57 --> 00:52:01 Release the brakes. The wheel is turning. 731 00:52:01 --> 00:52:03 Jam the brakes on. That is exactly what is 732 00:52:03 --> 00:52:05 happening. The same way as out there, 733 00:52:05 --> 00:52:08 you notice that oops, the output is going up, 734 00:52:08 --> 00:52:10 pull it down, oops, it's going down, 735 00:52:10 --> 00:52:12 pull it up. So, there is constant negative 736 00:52:12 --> 00:52:15 feedback that is keeping the output stable. 737 00:52:15 --> 00:52:18 A very important concept. And I will ask your recitation 738 00:52:18 --> 52:21 instructors to cover the very simple method that is on page 9.