1 00:00:00,000 --> 00:00:04,000 Good morning. OK. 2 00:00:04,000 --> 00:00:15,000 The topic for today is Energy and Power. 3 00:00:24,000 --> 00:00:28,000 Most of the time this semester, up to now at least, 4 00:00:28,000 --> 00:00:32,000 we focused a lot on speed. We have been truly speed freaks 5 00:00:32,000 --> 00:00:36,000 looking at how fast can we switch the signal, 6 00:00:36,000 --> 00:00:39,000 what does a time domain waveform look like? 7 00:00:39,000 --> 00:00:44,000 We also looked at frequency responses of circuits. 8 00:00:44,000 --> 00:00:48,000 This week we will spend on something a little bit 9 00:00:48,000 --> 00:00:53,000 different, and that relates to energy and power. 10 00:00:53,000 --> 00:00:57,000 Energy and power is gaining a lot more importance in certainly 11 00:00:57,000 --> 00:01:00,000 this decade, and will do so in the future. 12 00:01:00,000 --> 00:01:04,000 And I am going to work out a little example towards the end 13 00:01:04,000 --> 00:01:08,000 of the lecture. And there you will see that if 14 00:01:08,000 --> 00:01:11,000 you do things naively, your handheld devices, 15 00:01:11,000 --> 00:01:14,000 your cell phone, laptops and so on will just up 16 00:01:14,000 --> 00:01:18,000 and explode. You have got to be a little bit 17 00:01:18,000 --> 00:01:21,000 careful in terms of how to manage energy and power. 18 00:01:21,000 --> 00:01:25,000 Before I get into that, I just want to wrap up with a 19 00:01:25,000 --> 00:01:30,000 quick review of what we covered last week. 20 00:01:35,000 --> 00:01:39,000 We ended last week by looking at positive feedback in analog 21 00:01:39,000 --> 00:01:43,000 circuits using an op amp. And, in particular, 22 00:01:43,000 --> 00:01:46,000 we built an oscillator. 23 00:01:51,000 --> 00:01:55,000 We built an oscillator that allowed us to charge a 24 00:01:55,000 --> 00:01:58,000 capacitor. And when the voltage across the 25 00:01:58,000 --> 00:02:02,000 capacitor equaled that at the minus terminal it would flip and 26 00:02:02,000 --> 00:02:06,000 keep doing so. And at the output you would get 27 00:02:06,000 --> 00:02:08,000 a waveform that looked like this. 28 00:02:08,000 --> 00:02:11,000 You would get a square wave output. 29 00:02:11,000 --> 00:02:16,000 Now, throughout the course we have talked about getting square 30 00:02:16,000 --> 00:02:19,000 wave inputs. And this is one example of how 31 00:02:19,000 --> 00:02:23,000 you can actually produce a square wave pretty much from 32 00:02:23,000 --> 00:02:27,000 first principles using a capacitor, resistors and an op 33 00:02:27,000 --> 00:02:31,000 amp. Now, I just wanted to wrap up 34 00:02:31,000 --> 00:02:36,000 this little item here by talking about one application of an 35 00:02:36,000 --> 00:02:40,000 oscillator. And this application of the 36 00:02:40,000 --> 00:02:44,000 oscillator really nicely closed the loop on the body of 37 00:02:44,000 --> 00:02:50,000 knowledge relating to digital circuits that we have covered in 38 00:02:50,000 --> 00:02:53,000 this course. What I want to talk about 39 00:02:53,000 --> 00:02:58,000 briefly is a small digital system with a sender and a 40 00:02:58,000 --> 00:03:02,000 receiver. And the sender is sending a 41 00:03:02,000 --> 00:03:07,000 signal, the receiver receives a signal, and in this course we 42 00:03:07,000 --> 00:03:12,000 have talked about senders sending a sequence of ones and 43 00:03:12,000 --> 00:03:14,000 zeros. Say, for example, 44 00:03:14,000 --> 00:03:19,000 the sender wants to send some sort of a signal like this. 45 00:03:19,000 --> 00:03:23,000 We have seen that this is quite a legitimate signal. 46 00:03:23,000 --> 00:03:28,000 We get some kind of oscillatory behavior because of the 47 00:03:28,000 --> 00:03:34,000 inductance and capacitance associated with the wire. 48 00:03:34,000 --> 00:03:37,000 And what you have done is that you pretty much believed me when 49 00:03:37,000 --> 00:03:40,000 I said that this really corresponds to a one, 50 00:03:40,000 --> 00:03:42,000 one, zero. The sender wants to send a one, 51 00:03:42,000 --> 00:03:45,000 one, zero signal to the receiver, and the receiver gets 52 00:03:45,000 --> 00:03:46,000 it. So this is a one, 53 00:03:46,000 --> 00:03:48,000 this is a one, this is a zero. 54 00:03:48,000 --> 00:03:51,000 But if I am a receiver, I am going to look at the 55 00:03:51,000 --> 00:03:53,000 square wave. There is no such thing as 56 00:03:53,000 --> 00:03:56,000 sending a one on a wire. You cannot send a one on a 57 00:03:56,000 --> 00:03:58,000 wire. You send a voltage signal or a 58 00:03:58,000 --> 00:04:02,000 current signal on a wire. So, the receiver receives a 59 00:04:02,000 --> 00:04:05,000 voltage signal. It is going to be zero for some 60 00:04:05,000 --> 00:04:09,000 time and then maybe 5 volts or 3 volts or whatever is your high 61 00:04:09,000 --> 00:04:13,000 and then zero again. How does my receiver know it's 62 00:04:13,000 --> 00:04:15,000 a one, one, zero? Why can't it be a one, 63 00:04:15,000 --> 00:04:17,000 one, one, one, zero, zero, zero? 64 00:04:17,000 --> 00:04:20,000 It doesn't know. How does the receiver know it's 65 00:04:20,000 --> 00:04:24,000 a one, one, zero sequence and not 50 ones followed by 40 66 00:04:24,000 --> 00:04:25,000 zeros? It doesn't know. 67 00:04:25,000 --> 00:04:29,000 What we need is -- For senders to be able to 68 00:04:29,000 --> 00:04:34,000 communicate with receivers, we need some kind of agreed 69 00:04:34,000 --> 00:04:38,000 upon time when receivers sample the signal coming in and decide 70 00:04:38,000 --> 00:04:43,000 whether it's a one or a zero. They both have to agree on 71 00:04:43,000 --> 00:04:46,000 certain time bases when to look at the input. 72 00:04:46,000 --> 00:04:50,000 One way to deal with this is I can have a clock, 73 00:04:50,000 --> 00:04:54,000 a square wave signal that we call a clock in digital systems 74 00:04:54,000 --> 00:05:00,000 and ship it to the other side in the following manner. 75 00:05:06,000 --> 00:05:10,000 This clock signal can be applied to this sender and to 76 00:05:10,000 --> 00:05:14,000 this receiver. For more details on this, 77 00:05:14,000 --> 00:05:18,000 let me recommend Page 735 of the course notes that talks 78 00:05:18,000 --> 00:05:23,000 about a detailed example of the use of a clock in a digital 79 00:05:23,000 --> 00:05:27,000 system. What I can do is create a clock 80 00:05:27,000 --> 00:05:33,000 that looks like the square wave. The clock provides a notion of 81 00:05:33,000 --> 00:05:36,000 time to the circuit. And I have some kind of a clock 82 00:05:36,000 --> 00:05:39,000 signal generator. And I connect that to the 83 00:05:39,000 --> 00:05:42,000 sender and connect that to the receiver. 84 00:05:42,000 --> 00:05:46,000 And now both the receiver and the sender have a notion of 85 00:05:46,000 --> 00:05:48,000 time. And what I can do is I can tell 86 00:05:48,000 --> 00:05:53,000 my receiver, the sender and the receiver can have an agreement 87 00:05:53,000 --> 00:05:57,000 between them that says that look at the signal at your input when 88 00:05:57,000 --> 00:06:03,000 on the rising edge of the clock. Whenever the clock rises, 89 00:06:03,000 --> 00:06:07,000 when you see a rising edge look at the value in the wire and 90 00:06:07,000 --> 00:06:10,000 that's the value I sent. By doing so, 91 00:06:10,000 --> 00:06:15,000 what I can do is that the receiver can look at the signal. 92 00:06:15,000 --> 00:06:19,000 At this rising edge it sees a zero, this is vOH, 93 00:06:19,000 --> 00:06:21,000 looks up here, sees a one here, 94 00:06:21,000 --> 00:06:24,000 sees a one here and sees a zero. 95 00:06:24,000 --> 00:06:30,000 So, it correctly sampled one, one, zero at the receiving end. 96 00:06:30,000 --> 00:06:34,000 And the sender can send the same sequence here once we have 97 00:06:34,000 --> 00:06:37,000 this time base. This little brief foray 98 00:06:37,000 --> 00:06:41,000 circuits was simply to give you an application of a circuit that 99 00:06:41,000 --> 00:06:45,000 can produce a square wave. I can create a clock with a 100 00:06:45,000 --> 00:06:48,000 time base. Also, interestingly much more 101 00:06:48,000 --> 00:06:52,000 fundamental is we looked at various abstractions throughout 102 00:06:52,000 --> 00:06:55,000 the course. We talked about discretizing 103 00:06:55,000 --> 00:06:59,000 space by looking at lumped signals. 104 00:06:59,000 --> 00:07:03,000 What I also want to point out is that a clock can be viewed as 105 00:07:03,000 --> 00:07:07,000 another fundamental abstraction in the digital domain where what 106 00:07:07,000 --> 00:07:11,000 I am doing is discretizing time. What I am saying is that, 107 00:07:11,000 --> 00:07:15,000 look, in the digital domain we have already discretized value 108 00:07:15,000 --> 00:07:19,000 into zeros and ones, but we still had continuous 109 00:07:19,000 --> 00:07:22,000 time until now. And what you do in digital 110 00:07:22,000 --> 00:07:25,000 systems is to say that look, let's digitize everything, 111 00:07:25,000 --> 00:07:29,000 or rather discretize everything. 112 00:07:29,000 --> 00:07:32,000 And let's discretize time as well into these points that 113 00:07:32,000 --> 00:07:36,000 happen on the rising edge of the clock which means that the 114 00:07:36,000 --> 00:07:39,000 circuit has meaning, signals have meaning only when 115 00:07:39,000 --> 00:07:43,000 the clock is rising. That tends to discretize time 116 00:07:43,000 --> 00:07:47,000 which means that I really don't care what happens to signals in 117 00:07:47,000 --> 00:07:51,000 this time, as long as on the rising edge of the clock I get 118 00:07:51,000 --> 00:07:53,000 the right value. This concept is called 119 00:07:53,000 --> 00:07:58,000 discretizing time. And a clock lets you do that. 120 00:07:58,000 --> 00:08:01,000 Remember that in digital systems, which you will learn 121 00:08:01,000 --> 00:08:05,000 about in 004 I am really discretizing two things, 122 00:08:05,000 --> 00:08:10,000 discretizing values into zeros and ones, and at the same time 123 00:08:10,000 --> 00:08:14,000 also discretizing time into a time when I sample things and a 124 00:08:14,000 --> 00:08:17,000 time when I ignore values on the wires. 125 00:08:17,000 --> 00:08:21,000 I think you will get to clocks in 6.004 after about a month, 126 00:08:21,000 --> 00:08:26,000 so initially you would just be focusing on the statics of the 127 00:08:26,000 --> 00:08:30,000 system without worrying about any dynamic clock introduced in 128 00:08:30,000 --> 00:08:35,000 the circuit. OK, that's just a brief little 129 00:08:35,000 --> 00:08:38,000 interlude. With that let me get into 130 00:08:38,000 --> 00:08:40,000 today's topic of energy and power. 131 00:08:40,000 --> 00:08:45,000 Why is this important? The reason this is important is 132 00:08:45,000 --> 00:08:49,000 that what really determines the size of your handheld? 133 00:08:49,000 --> 00:08:52,000 You may think oh, gee, electronics in the 134 00:08:52,000 --> 00:08:55,000 handheld. Some of you may think, 135 00:08:55,000 --> 00:08:57,000 oh, the antenna in the handheld. 136 00:08:57,000 --> 00:08:59,000 No. What really, 137 00:08:59,000 --> 00:09:03,000 really determines the size and weight of your handheld devices, 138 00:09:03,000 --> 00:09:07,000 your PDAs, your cell phones, your laptops and so on is by 139 00:09:07,000 --> 00:09:10,000 and large the battery. On Page 2 I have a little 140 00:09:10,000 --> 00:09:13,000 cartoon that shows you that if we did not have you learn about 141 00:09:13,000 --> 00:09:16,000 energy and power, that's what we would all be 142 00:09:16,000 --> 00:09:18,000 doing in order to use cell phones. 143 00:09:18,000 --> 00:09:22,000 Not surprisingly the very first wireless phones ended up in 144 00:09:22,000 --> 00:09:24,000 automobiles because you had a big battery. 145 00:09:24,000 --> 00:09:29,000 And so you had these wireless phones only in cars. 146 00:09:29,000 --> 00:09:32,000 Because of a huge amount of research based on the knowledge, 147 00:09:32,000 --> 00:09:36,000 the technologies I am going to talk about in today's lecture 148 00:09:36,000 --> 00:09:40,000 and Thursday's lecture, you will see very simple and 149 00:09:40,000 --> 00:09:44,000 elegant ways of reducing the amount of battery you need to be 150 00:09:44,000 --> 00:09:47,000 able to get some kind of function out of analog or 151 00:09:47,000 --> 00:09:50,000 digital devices. I also want you to take a look 152 00:09:50,000 --> 00:09:54,000 at Page 2 of the handout that I have given you here, 153 00:09:54,000 --> 00:09:56,000 handout 63. This handout talks about the 154 00:09:56,000 --> 00:10:00,000 absolute latest in digital fabrication technology out 155 00:10:00,000 --> 00:10:04,000 there. This is not a paid commercial 156 00:10:04,000 --> 00:10:08,000 for IBM. IBM has a technology called 157 00:10:08,000 --> 00:10:11,000 CU08. It is called Blue Logic. 158 00:10:11,000 --> 00:10:14,000 It is called the Copper 08 Process. 159 00:10:14,000 --> 00:10:19,000 And in this process, if you look down on Page 1, 160 00:10:19,000 --> 00:10:23,000 for example, IBM claims that it can build up 161 00:10:23,000 --> 00:10:28,000 to 72 million gates in a single chip. 162 00:10:28,000 --> 00:10:32,000 With this technology they are able to build 70 to 80 million 163 00:10:32,000 --> 00:10:35,000 gates where a gate is, unless otherwise mentioned, 164 00:10:35,000 --> 00:10:39,000 pretty much defined as a two input NAND gate equivalent. 165 00:10:39,000 --> 00:10:42,000 So, your inverter, your NAND gate and so on count 166 00:10:42,000 --> 00:10:45,000 as a gate. And they can build close to 80 167 00:10:45,000 --> 00:10:48,000 million of these little suckers on a single chip. 168 00:10:48,000 --> 00:10:51,000 Just imagine that. And the biggest chip they can 169 00:10:51,000 --> 00:10:55,000 build is on the order of 18 to 19 millimeters on a side, 170 00:10:55,000 --> 00:10:59,000 roughly two centimeters on a side. 171 00:10:59,000 --> 00:11:02,000 On a chip that's about one square inch. 172 00:11:02,000 --> 00:11:04,000 You can put down 80 million gates. 173 00:11:04,000 --> 00:11:09,000 What is more important for today is what is on Page 2, 174 00:11:09,000 --> 00:11:12,000 actually. I have circled two things on 175 00:11:12,000 --> 00:11:15,000 Page 2. One thing that I have circled 176 00:11:15,000 --> 00:11:19,000 is power supply range in the 0.7 to 1.3 volts. 177 00:11:19,000 --> 00:11:23,000 Notice that that voltage, the power supply voltage for 178 00:11:23,000 --> 00:11:28,000 these chips is significantly lower than the 5 volts that we 179 00:11:28,000 --> 00:11:34,000 have been normally talking about in this course. 180 00:11:34,000 --> 00:11:37,000 When in doubt our problems have used 5 volts. 181 00:11:37,000 --> 00:11:42,000 But notice that in this technology they're talking about 182 00:11:42,000 --> 00:11:47,000 using voltages for the power supply VS in the range of 0.7 to 183 00:00:01,300 --> 00:11:49,000 Why is it so much lower? 184 00:11:49,000 --> 00:11:53,000 Well, you will find out. The second thing I've circled 185 00:11:53,000 --> 00:11:57,000 is something called power dissipation. 186 00:11:57,000 --> 00:12:01,000 And you say power dissipation is said to be 0.006 microwatts 187 00:12:01,000 --> 00:12:07,000 per megahertz per gate. It says power dissipation is 6 188 00:12:07,000 --> 00:12:09,000 nanowatts per megahertz per gate. 189 00:12:09,000 --> 00:12:13,000 What that says is that each gate off your circuit will 190 00:12:13,000 --> 00:12:17,000 dissipate this much power at a 1 megahertz frequency. 191 00:12:17,000 --> 00:12:21,000 And the implication of that is that you should be able to 192 00:12:21,000 --> 00:12:25,000 convert that single number to the power dissipation in any 193 00:12:25,000 --> 00:12:29,000 chip that you might build depending on the number of gates 194 00:12:29,000 --> 00:12:32,000 that you have, the frequency you run the 195 00:12:32,000 --> 00:12:38,000 circuit at, the voltage that you use and so on and so forth. 196 00:12:38,000 --> 00:12:41,000 By the end of today's lecture you will be able to take this 197 00:12:41,000 --> 00:12:44,000 number and correlate that into the power dissipation of any 198 00:12:44,000 --> 00:12:47,000 chip that you might want to build with this. 199 00:12:47,000 --> 00:12:49,000 That just serves as the motivation that by the end of 200 00:12:49,000 --> 00:12:53,000 this lecture you will understand how to very quickly in five 201 00:12:53,000 --> 00:12:55,000 seconds or less, boom, given a chip, 202 00:12:55,000 --> 00:12:59,000 oh, yeah, that should consume about 30 watts of power. 203 00:12:59,000 --> 00:13:02,000 And what you will also do, based on some examples here, 204 00:13:02,000 --> 00:13:07,000 estimate the power of not the Pentium IV but a chip following 205 00:13:07,000 --> 00:13:10,000 the Pentium VI, let's call it the Pentium V 206 00:13:10,000 --> 00:13:13,000 would consume if it ran at 1 gigahertz. 207 00:13:13,000 --> 00:13:17,000 We will come up with some absolutely shocking numbers 208 00:13:17,000 --> 00:13:21,000 based on what you have learned. With that kind of motivation 209 00:13:21,000 --> 00:13:25,000 let me get into talking about some theory and get into the 210 00:13:25,000 --> 00:13:29,000 foundations of energy and power. 211 00:13:34,000 --> 00:13:36,000 Let's go to Page 3. 212 00:13:46,000 --> 00:13:50,000 To drive the theoretical discussion, I would like to 213 00:13:50,000 --> 00:13:54,000 focus on the energy dissipated in a MOSFET gate. 214 00:13:54,000 --> 00:13:59,000 And fundamentally we will talk about looking at energy and 215 00:13:59,000 --> 00:14:03,000 power in circuits containing switches, resistors and 216 00:14:03,000 --> 00:14:07,000 capacitors. The MOSFET gate is simply an 217 00:14:07,000 --> 00:14:10,000 illustrative example to drive the theory. 218 00:14:10,000 --> 00:14:13,000 But fundamentally what I am going to show you, 219 00:14:13,000 --> 00:14:17,000 or lead you through today, I will tell you how to compute 220 00:14:17,000 --> 00:14:21,000 the power and energy when you have capacitors, 221 00:14:21,000 --> 00:14:25,000 resistors, voltage sources and switches in your circuit. 222 00:14:25,000 --> 00:14:28,000 We will look at a circuit that looks like this. 223 00:14:28,000 --> 00:14:32,000 Your vanilla inverter circuit. 224 00:14:48,000 --> 00:14:50,000 My inverter. I apply some vIN signal here. 225 00:14:50,000 --> 00:14:54,000 It could be a square wave. It could be some sequence of 226 00:14:54,000 --> 00:14:57,000 ones and zeros. And this is an inverter that we 227 00:14:57,000 --> 00:14:59,000 all know and love. And this guy here is, 228 00:14:59,000 --> 00:15:04,000 stuck in a capacitor here. And this capacitor is meant to 229 00:15:04,000 --> 00:15:10,000 model the input gate capacitance of whatever this inverter drives 230 00:15:10,000 --> 00:15:16,000 plus any capacitance of the wire leading up to that gate and so 231 00:15:16,000 --> 00:15:19,000 on. It is just a lumped capacitor 232 00:15:19,000 --> 00:15:24,000 that I have stuck on there. I am interested in determining 233 00:15:24,000 --> 00:15:28,000 a few things. One is what we call the standby 234 00:15:28,000 --> 00:15:31,000 power. You will see all these terms 235 00:15:31,000 --> 00:15:33,000 being used in cell phones and so on. 236 00:15:33,000 --> 00:15:36,000 In your cell phone, your cell phone manufacturer 237 00:15:36,000 --> 00:15:39,000 gives you two numbers. Of course both are over 238 00:15:39,000 --> 00:15:41,000 exaggerations, but they give you two numbers 239 00:15:41,000 --> 00:15:44,000 nonetheless. One number is the number of 240 00:15:44,000 --> 00:15:47,000 days that the cell phone battery will last when in standby mode, 241 00:15:47,000 --> 00:15:49,000 right? That's exactly where standby 242 00:15:49,000 --> 00:15:51,000 comes from. In standby mode, 243 00:15:51,000 --> 00:15:54,000 how much power does your cell phone or how long will the 244 00:15:54,000 --> 00:15:58,000 battery last, that's the standby power. 245 00:15:58,000 --> 00:16:02,000 And the second thing is what we call active use power. 246 00:16:02,000 --> 00:16:07,000 Active use is when you are making a phone call and so on, 247 00:16:07,000 --> 00:16:11,000 what is the power consumed? And there again your 248 00:16:11,000 --> 00:16:17,000 manufacturer of your cell phone will give you a much smaller 249 00:16:17,000 --> 00:16:21,000 number for the active use power of your cell phone. 250 00:16:21,000 --> 00:16:26,000 What I am going to do is assume for discussion that the inverter 251 00:16:26,000 --> 00:16:33,000 is driven by a square wave signal of the following sort. 252 00:16:33,000 --> 00:16:37,000 This is vIN. And I am going to drive this 253 00:16:37,000 --> 00:16:44,000 with a signal of this sort. The period applied at the 254 00:16:44,000 --> 00:16:50,000 input, so I am switching the inverter on and off, 255 00:16:50,000 --> 00:16:56,000 on and off, on and off. And T1 seconds for the high, 256 00:16:56,000 --> 00:17:01,000 T2 seconds for the low. This is the inverter, 257 00:17:01,000 --> 00:17:06,000 this is the input signal, and we'll keep coming back to 258 00:17:06,000 --> 00:17:10,000 that again and again. Rather than directly taking 259 00:17:10,000 --> 00:17:16,000 this circuit and analyzing its power, I would like to do things 260 00:17:16,000 --> 00:17:21,000 in a slightly roundabout manner. What I would like to do is show 261 00:17:21,000 --> 00:17:26,000 you some very simple circuits and analyze their standby and 262 00:17:26,000 --> 00:17:31,000 active powers. And then show you that this 263 00:17:31,000 --> 00:17:37,000 circuit simply is a combination of some of the simple things 264 00:17:37,000 --> 00:17:39,000 that you have seen. Example 1. 265 00:17:39,000 --> 00:17:45,000 I would like to take a simple circuit that looks like this. 266 00:17:45,000 --> 00:17:49,000 A voltage source V applied across a resistor R, 267 00:17:49,000 --> 00:17:53,000 some current I. And if I apply a voltage across 268 00:17:53,000 --> 00:17:57,000 this resistor, that voltage would simply 269 00:17:57,000 --> 00:18:03,000 appear across the resistor. And the power is simply given 270 00:18:03,000 --> 00:18:07,000 by VI which is simply V squared divided by R. 271 00:18:07,000 --> 00:18:11,000 This is 6.002 101 in the very first chapter. 272 00:18:11,000 --> 00:18:16,000 That is the power that is dissipated by this resistor, 273 00:18:16,000 --> 00:18:21,000 simply V squared divided by R. That's the power dissipated by 274 00:18:21,000 --> 00:18:25,000 the resistor. Where does that power come 275 00:18:25,000 --> 00:18:28,000 from? The voltage source supplies the 276 00:18:28,000 --> 00:18:33,000 power. So, this guy here supplies this 277 00:18:33,000 --> 00:18:37,000 power and this guy here dissipates it. 278 00:18:37,000 --> 00:18:41,000 What is the energy that I dissipate in T time? 279 00:18:41,000 --> 00:18:46,000 Remember, power is the rate of energy dissipation. 280 00:18:46,000 --> 00:18:52,000 And so energy is simply power multiplied by time. 281 00:18:52,000 --> 00:18:57,000 For a circuit like this, energy dissipated in time T is 282 00:18:57,000 --> 00:19:02,000 simply VIT. For our gate remember we have 283 00:19:02,000 --> 00:19:07,000 two situations. We have VS, we have RL we have 284 00:19:07,000 --> 00:19:10,000 RON, vO and vIN. So, vIN is high. 285 00:19:10,000 --> 00:19:15,000 If vIN is high with respect to ground then RON, 286 00:19:15,000 --> 00:19:19,000 the switch is on, and this is the circuit that I 287 00:19:19,000 --> 00:19:22,000 see. In this situation the power 288 00:19:22,000 --> 00:19:30,000 consumed is simply V squared divided by the resistance here. 289 00:19:30,000 --> 00:19:38,000 It is simply VS squared divided by RL plus RON. 290 00:19:38,000 --> 00:19:45,000 Let me mark that with an asterisk. 291 00:19:45,000 --> 00:19:56,000 I will refer to this later. Similarly, when vIN is low the 292 00:19:56,000 --> 00:20:02,000 MOSFET is off. And the power is simply zero. 293 00:20:02,000 --> 00:20:06,000 I have no current flowing down and the power is zero. 294 00:20:06,000 --> 00:20:09,000 Absolutely basic stuff. Absolutely basic. 295 00:20:09,000 --> 00:20:12,000 So, the power, when I have the MOSFET on, 296 00:20:12,000 --> 00:20:15,000 for the kind of inverters you have seen so far, 297 00:20:15,000 --> 00:20:19,000 this is the power consumed by the inverter. 298 00:20:19,000 --> 00:20:23,000 And this asterisk here is simply to say hold that thought, 299 00:20:23,000 --> 00:20:26,000 we will get back to it a little later. 300 00:20:26,000 --> 00:20:30,000 Let me work out a second example. 301 00:20:30,000 --> 00:20:37,000 In this second example, I would like to consider the 302 00:20:37,000 --> 00:20:44,000 following circuit, a voltage source VS with a 303 00:20:44,000 --> 00:20:52,000 strange arrangement of switches, S1 with a resistance R1, 304 00:20:52,000 --> 00:21:02,000 a capacitor C in this manner, a switch S2 and a resistor R2. 305 00:21:02,000 --> 00:21:05,000 For now don't worry about how the circuit comes about. 306 00:21:05,000 --> 00:21:08,000 Just assume that I have drawn the circuit for you. 307 00:21:08,000 --> 00:21:12,000 And what I want to do is compute the power under certain 308 00:21:12,000 --> 00:21:14,000 conditions. Notice that if this is off and 309 00:21:14,000 --> 00:21:18,000 this is off, there is no current flowing either in this loop or 310 00:21:18,000 --> 00:21:22,000 this loop, and the power dissipated by the circuit is 311 00:21:22,000 --> 00:21:24,000 zero. But there are some arrangement 312 00:21:24,000 --> 00:21:28,000 of switches for which I do consume power. 313 00:21:28,000 --> 00:21:33,000 And so let me show you that arrangement of switches. 314 00:21:33,000 --> 00:21:39,000 And what I am going to do is assume that the switches open 315 00:21:39,000 --> 00:21:43,000 and close with the following periodic cycles. 316 00:21:43,000 --> 00:21:49,000 Let's assume that when this is high S1 is closed and S2 is 317 00:21:49,000 --> 00:21:55,000 open, and when this is low assume that S1 is open, 318 00:21:55,000 --> 00:21:59,000 S2 is closed. And let's assume this is T, 319 00:21:59,000 --> 00:22:04,000 this is T1, this is T2. That sequence should be 320 00:22:04,000 --> 00:22:09,000 reminiscent of this input that I am feeding to this inverter. 321 00:22:09,000 --> 00:22:13,000 All I am telling you here is that I am giving you the 322 00:22:13,000 --> 00:22:15,000 circuit. I want to compute the power 323 00:22:15,000 --> 00:22:20,000 consumption of the circuit. And what I am telling you is 324 00:22:20,000 --> 00:22:24,000 that with the frequency, with a time period of capital 325 00:22:24,000 --> 00:22:28,000 T, for the first T1 seconds this switch is closed and that is 326 00:22:28,000 --> 00:22:32,000 open. So, this circuit applies. 327 00:22:32,000 --> 00:22:39,000 In the second half of the clock this switch is open so this 328 00:22:39,000 --> 00:22:44,000 circuit applies. And what I am interested in 329 00:22:44,000 --> 00:22:51,000 finding out is what is the energy dissipated in each cycle 330 00:22:51,000 --> 00:22:57,000 of time capital T? And I also want to find out the 331 00:22:57,000 --> 00:23:02,000 average power. Just spend about five seconds 332 00:23:02,000 --> 00:23:07,000 just staring at this and kind of intuit what is going on here. 333 00:23:07,000 --> 00:23:11,000 I start by putting a voltage source here and I close the 334 00:23:11,000 --> 00:23:13,000 switch. That is open. 335 00:23:13,000 --> 00:23:16,000 Start by closing this, what happens? 336 00:23:16,000 --> 00:23:20,000 When I close the switch VS is going to charge up this 337 00:23:20,000 --> 00:23:23,000 capacitor. I get current flowing through 338 00:23:23,000 --> 00:23:30,000 my resistor, so I am going to be charging up this capacitor here. 339 00:23:30,000 --> 00:23:33,000 Then let's say I allow T1 to be as large as possible, 340 00:23:33,000 --> 00:23:36,000 and so this capacitor is going to be charged up to all of VS. 341 00:23:36,000 --> 00:23:40,000 After a long time this guy gets to be VS in the capacitor. 342 00:23:40,000 --> 00:23:43,000 And as it is charging up I have current flow through the 343 00:23:43,000 --> 00:23:46,000 resistor, so it is sitting there dissipating power. 344 00:23:46,000 --> 00:23:49,000 Notice that this sucker does not dissipate energy. 345 00:23:49,000 --> 00:23:52,000 It simply stores energy. So, the energy supplied by the 346 00:23:52,000 --> 00:23:55,000 voltage source comes in, some of it gets stored in the 347 00:23:55,000 --> 00:23:58,000 capacitor and some of it is being dissipated by the 348 00:23:58,000 --> 00:24:02,000 resistor. That gets me to the end of T1. 349 00:24:02,000 --> 00:24:06,000 At the end of T2 I open the switch and close this switch. 350 00:24:06,000 --> 00:24:10,000 When I close the switch I have some energy on the capacitor, 351 00:24:10,000 --> 00:24:13,000 and the voltage across the capacitor begins to drive a 352 00:24:13,000 --> 00:24:16,000 current through this resistor R2. 353 00:24:16,000 --> 00:24:19,000 And now the capacitor supplies its stored energy, 354 00:24:19,000 --> 00:24:23,000 and its stored energy then begins to dissipate through 355 00:24:23,000 --> 00:24:26,000 resistor R2. And if T2 is very long then all 356 00:24:26,000 --> 00:24:30,000 the charge in the capacitor drains out. 357 00:24:30,000 --> 00:24:35,000 And the voltage in the capacitor at the end will be 358 00:24:35,000 --> 00:24:38,000 zero. So, that is just sort of a high 359 00:24:38,000 --> 00:24:42,000 level description of what goes on. 360 00:24:42,000 --> 00:24:47,000 Now let's go ahead and compute from first principles the 361 00:24:47,000 --> 00:24:51,000 energetics of this little circuit. 362 00:24:51,000 --> 00:24:57,000 Let's look at the entire period capital T, and as a first step 363 00:24:57,000 --> 00:25:03,000 look at T1. When T1 is in place S1 is 364 00:25:03,000 --> 00:25:08,000 closed and S2 is open. Accordingly, 365 00:25:08,000 --> 00:25:13,000 the circuit that applies looks like this. 366 00:25:13,000 --> 00:25:19,000 I have VS, S1 is closed, so that is closed, 367 00:25:19,000 --> 00:25:26,000 and I have this resistance R1, I have this capacitance C, 368 00:25:26,000 --> 00:25:33,000 some voltage VC across the capacitor. 369 00:25:33,000 --> 00:25:37,000 You can go ahead and assume that VC of zero is zero. 370 00:25:37,000 --> 00:25:43,000 That I start off my life with no voltage across the capacitor. 371 00:25:43,000 --> 00:25:47,000 First of all, let me plot the waveforms and 372 00:25:47,000 --> 00:25:53,000 write the expressions down and then compute the energy supplied 373 00:25:53,000 --> 00:26:00,000 by the voltage source and then look at where the energy goes. 374 00:26:00,000 --> 00:26:03,000 You all know, or should know by now, 375 00:26:03,000 --> 00:26:07,000 if I plot VC as a function of time, remember, 376 00:26:07,000 --> 00:26:11,000 this is really easy to do. VC as a function of time goes 377 00:26:11,000 --> 00:26:15,000 like this. At time T equal to zero I am 378 00:26:15,000 --> 00:26:19,000 telling you that the capacitor voltage is zero. 379 00:26:19,000 --> 00:26:23,000 I am telling you that. So, it is at zero. 380 00:26:23,000 --> 00:26:29,000 And then the capacitor charges up until it reaches VS. 381 00:26:29,000 --> 00:26:33,000 I also know that after a long time this will be VS, 382 00:26:33,000 --> 00:26:40,000 after a long time that will be VS, and between those two I have 383 00:26:40,000 --> 00:26:43,000 a rising function that looks like this. 384 00:26:43,000 --> 00:26:47,000 I can similarly plot the current for you. 385 00:26:47,000 --> 00:26:52,000 At time T equal to zero instantaneously the capacitor 386 00:26:52,000 --> 00:26:57,000 looks like a short, and so the current that I start 387 00:26:57,000 --> 00:27:03,000 off with is going to be VS divided by R1. 388 00:27:03,000 --> 00:27:06,000 The voltage across the capacitor is zero. 389 00:27:06,000 --> 00:27:09,000 All the voltage falls across the resistor R1. 390 00:27:09,000 --> 00:27:14,000 So, VS divided by R1 is the initial instantaneous current. 391 00:27:14,000 --> 00:27:17,000 And after a long time, because VC reaches VS, 392 00:27:17,000 --> 00:27:20,000 the current is going to be zero. 393 00:27:20,000 --> 00:27:24,000 And between those two points I get an exponential decay. 394 00:27:24,000 --> 00:27:31,000 I could very quickly write down the expression for the current. 395 00:27:31,000 --> 00:27:37,000 And that is simply the initial value VS divided by R1 times the 396 00:27:37,000 --> 00:27:43,000 exponential decay minus T divided by the time constant for 397 00:27:43,000 --> 00:27:47,000 the circuit R1C. You have seen this stuff 398 00:27:47,000 --> 00:27:50,000 before. Here comes the part that we 399 00:27:50,000 --> 00:27:55,000 care about for now. Let's find out what is the 400 00:27:55,000 --> 00:28:00,000 total energy provided by the source. 401 00:28:00,000 --> 00:28:05,000 When dealing with energy computations you have to be 402 00:28:05,000 --> 00:28:11,000 incredibly careful of these words here, supply, 403 00:28:11,000 --> 00:28:17,000 provided versus dissipated. Dissipated implies that the 404 00:28:17,000 --> 00:28:24,000 resistor is burning energy. Provided means that the source 405 00:28:24,000 --> 00:28:30,000 is supplying that energy. So, energy provided by source 406 00:28:30,000 --> 00:28:35,000 during T1. Let's go ahead and compute that 407 00:28:35,000 --> 00:28:38,000 very quickly. The energy supplied by the 408 00:28:38,000 --> 00:28:42,000 source is simply the voltage across the source multiplied by 409 00:28:42,000 --> 00:28:45,000 the current being supplied by the source. 410 00:28:45,000 --> 00:28:48,000 This is i. Remember, by associated 411 00:28:48,000 --> 00:28:52,000 variables convention, if I have a voltage across some 412 00:28:52,000 --> 00:28:57,000 element and the current into the element is positive then that 413 00:28:57,000 --> 00:29:02,000 element dissipates power. If the voltage here is, 414 00:29:02,000 --> 00:29:07,000 say, 1 volt and it is supplying current, if the i is out in the 415 00:29:07,000 --> 00:29:11,000 other direction then it is supplying power. 416 00:29:11,000 --> 00:29:15,000 In this case, the current i is going to be on 417 00:29:15,000 --> 00:29:21,000 the outside, heading outside. The total energy is going to be 418 00:29:21,000 --> 00:29:25,000 the instantaneous power integrated over time, 419 00:29:25,000 --> 00:29:30,000 and that is simply VS. Remember, the instantaneous 420 00:29:30,000 --> 00:29:35,000 power is VS times the current i, so the instantaneous power is 421 00:29:35,000 --> 00:29:38,000 simply VS times i, that is the instantaneous 422 00:29:38,000 --> 00:29:41,000 power. To get the energy provided by 423 00:29:41,000 --> 00:29:44,000 source and some time, I have to integrate that 424 00:29:44,000 --> 00:29:48,000 instantaneous power over the period of interest T1. 425 00:29:48,000 --> 00:29:53,000 That gives me the energy supplied by the source during 426 00:29:53,000 --> 00:29:55,000 T1. And let me go ahead and 427 00:29:55,000 --> 00:30:00,000 substitute for i with this expression here. 428 00:30:00,000 --> 00:30:08,000 It is VS times i, and i is VS divided by R1 times 429 00:30:08,000 --> 00:30:14,000 this expression here. That gives me 430 00:30:14,000 --> 00:30:21,000 (VS^2/R1)e^(-t/R1C) dt. Let me carry out the 431 00:30:21,000 --> 00:30:26,000 integration there. I get -1/RC, 432 00:30:26,000 --> 00:30:35,000 so I get this outside. And I also get to write down, 433 00:30:35,000 --> 00:30:40,000 oops, let me do that a little bit more carefully. 434 00:30:40,000 --> 00:30:46,000 VS^2/R1 simply comes out and I get a -R1C in the numerator. 435 00:30:46,000 --> 00:30:53,000 If I differentiate it then I get R1C in the denominator. 436 00:30:53,000 --> 00:30:58,000 I have an integral that comes up here. 437 00:30:58,000 --> 00:31:03,000 And then I write down e^-t/R1C, zero and T1. 438 00:31:03,000 --> 00:31:07,000 So, this R1 and this R1 cancel out. 439 00:31:07,000 --> 00:31:13,000 And I end up getting something that looks like this. 440 00:31:13,000 --> 00:31:18,000 I get CVS^2. And so there is a minus sign 441 00:31:18,000 --> 00:31:23,000 out here, so at zero this thing goes to a one, 442 00:31:23,000 --> 00:31:29,000 so I get a one. And because of minus sign I get 443 00:31:29,000 --> 00:31:35,000 e to the -T1/R1C. All I have done here is simply 444 00:31:35,000 --> 00:31:40,000 go through the math to do this integration here. 445 00:31:40,000 --> 00:31:45,000 What I am also going to do is assume that if T1, 446 00:31:45,000 --> 00:31:49,000 if the time that the switch is closed is much, 447 00:31:49,000 --> 00:31:54,000 much bigger than the time constant of the circuit, 448 00:31:54,000 --> 00:31:59,000 T1 is much, much greater than R1C, if this is much, 449 00:31:59,000 --> 00:32:05,000 much greater than R1C then this term goes to zero. 450 00:32:05,000 --> 00:32:09,000 And this becomes more or less equal to CVS^2. 451 00:32:09,000 --> 00:32:15,000 What do we have here? What we have here is that if I 452 00:32:15,000 --> 00:32:22,000 let the switch stay closed for a long time and S to be open then 453 00:32:22,000 --> 00:32:30,000 the voltage source is going to supply some amount of energy. 454 00:32:30,000 --> 00:32:34,000 That energy will equal CVS^2. The voltage across the 455 00:32:34,000 --> 00:32:40,000 capacitor will be VS and all that energy would have been 456 00:32:40,000 --> 00:32:44,000 supplied by this guy. Let me pose the following 457 00:32:44,000 --> 00:32:48,000 conundrum here. If the voltage across the 458 00:32:48,000 --> 00:32:52,000 capacitor is VS, because we know the energy 459 00:32:52,000 --> 00:32:56,000 stored in the capacitor is half CV^2. 460 00:32:56,000 --> 00:33:02,000 So, the energy in the capacitor is half CVS^2. 461 00:33:02,000 --> 00:33:06,000 At the end of the day, since the voltage across the 462 00:33:06,000 --> 00:33:10,000 capacitor is VS, ΩCV@2 is the energy stored 463 00:33:10,000 --> 00:33:11,000 here. But we know, 464 00:33:11,000 --> 00:33:16,000 from this calculation, the source has supplied CVS^2. 465 00:33:16,000 --> 00:33:19,000 Source has supplied twice that energy. 466 00:33:19,000 --> 00:33:24,000 This guy has supplied twice that energy and only half of 467 00:33:24,000 --> 00:33:30,000 that is stored here. Who ate up the other half? 468 00:33:30,000 --> 00:33:32,000 The resistor, exactly. 469 00:33:32,000 --> 00:33:36,000 The resistor has walloped half the energy. 470 00:33:36,000 --> 00:33:41,000 Let me just show it to you. It dissipated ΩCVS^2. 471 00:33:41,000 --> 00:33:47,000 It's pretty interesting. It's a pretty simple result. 472 00:33:47,000 --> 00:33:53,000 If T1 is very large compared to time constant then half the 473 00:33:53,000 --> 00:33:59,000 energy is in the capacitor and half of it has been burned by 474 00:33:59,000 --> 00:34:03,000 R1. This energy has not been 475 00:34:03,000 --> 00:34:05,000 burned. It is simply stored. 476 00:34:05,000 --> 00:34:08,000 It is stored by the capacitor. 477 00:34:17,000 --> 00:34:21,000 And if you do simple energy conservation arithmetic here, 478 00:34:21,000 --> 00:34:25,000 the energy dissipated in the resistor plus that stored in the 479 00:34:25,000 --> 00:34:30,000 capacitor equals the energy supplied by the source. 480 00:34:30,000 --> 00:34:33,000 All right. Let's go to T2 now. 481 00:34:33,000 --> 00:34:37,000 At T2, S2 is closed and S1 is open. 482 00:34:37,000 --> 00:34:44,000 Let's look at the second part of the cycle when S1 is open and 483 00:34:44,000 --> 00:34:49,000 S2 is closed. And what is going to happen now 484 00:34:49,000 --> 00:34:56,000 is the left-hand part of the circuit can be ignored and I can 485 00:34:56,000 --> 00:35:01,000 focus on this part. So, S2 is closed. 486 00:35:01,000 --> 00:35:05,000 This is RC, my capacitor, this is vC. 487 00:35:05,000 --> 00:35:09,000 This is the circuit of interest. 488 00:35:09,000 --> 00:35:12,000 What is the initial condition on this? 489 00:35:12,000 --> 00:35:16,000 What is the value of vC initially? 490 00:35:16,000 --> 00:35:22,000 Start off, because remember, I allowed this capacity to 491 00:35:22,000 --> 00:35:27,000 charge up fully, and so initially I have VS on 492 00:35:27,000 --> 00:35:32,000 the capacitor. And so the energy on the 493 00:35:32,000 --> 00:35:36,000 capacitor initially is ΩCVS^2. That is the energy on the 494 00:35:36,000 --> 00:35:39,000 capacitor. This time around I won't go 495 00:35:39,000 --> 00:35:44,000 through an integration process like that, but you can if you 496 00:35:44,000 --> 00:35:49,000 like, and do it in a much similar manner to say that now 497 00:35:49,000 --> 00:35:54,000 let's suppose that T2 is much greater than this time constant. 498 00:35:54,000 --> 00:36:00,000 If T2 is much greater than R2C, this time constant. 499 00:36:00,000 --> 00:36:04,000 If that time is much greater than this entire, 500 00:36:04,000 --> 00:36:11,000 the initial voltage VS drives a current through the resistor, 501 00:36:11,000 --> 00:36:17,000 and after some amount of time the voltage across the capacitor 502 00:36:17,000 --> 00:36:24,000 goes to zero and all the energy in the capacitor gets dissipated 503 00:36:24,000 --> 00:36:28,000 in R. So, if T2 is much greater than 504 00:36:28,000 --> 00:36:35,000 R2C then energy dissipated in R2 is simply ΩCVS^2. 505 00:36:35,000 --> 00:36:38,000 Notice that the energy dissipated in R1, 506 00:36:38,000 --> 00:36:43,000 in the first half cycle is ΩCVS^2 and the second half cycle 507 00:36:43,000 --> 00:36:48,000 during T2, if T2 is large enough, all this energy gets 508 00:36:48,000 --> 00:36:53,000 dissipated in this resistor R2. And I have that expression 509 00:36:53,000 --> 00:36:55,000 here. 510 00:37:00,000 --> 00:37:07,000 So let me just say that this is E1 and let me say that this is 511 00:37:07,000 --> 00:37:10,000 E2. So, E1 is dissipated in the 512 00:37:10,000 --> 00:37:17,000 resistor and E2 is dissipated in R2 in the second half cycle. 513 00:37:17,000 --> 00:37:23,000 A couple of interesting things to note at this point. 514 00:37:23,000 --> 00:37:30,000 One is that E1 and E2 are independent of R. 515 00:37:30,000 --> 00:37:33,000 If the time constant is small enough compared to the time that 516 00:37:33,000 --> 00:37:37,000 I charge the capacitor then half the energy gets lots in the 517 00:37:37,000 --> 00:37:40,000 resistor, and that is simply ΩCVS^2. 518 00:37:40,000 --> 00:37:43,000 And if I let this discharge completely it doesn't matter 519 00:37:43,000 --> 00:37:46,000 what resistor I am discharging it through. 520 00:37:46,000 --> 00:37:49,000 That's the intuition. If I have certain energy here 521 00:37:49,000 --> 00:37:52,000 and I let it discharge completely it doesn't matter 522 00:37:52,000 --> 00:37:55,000 what this resistor is. Small or large, 523 00:37:55,000 --> 00:37:58,000 it doesn't matter. All this energy gets dissipated 524 00:37:58,000 --> 00:38:02,000 there. The rate at which the energy 525 00:38:02,000 --> 00:38:05,000 gets dissipated will change depending on R2. 526 00:38:05,000 --> 00:38:09,000 If R2 is very small then I get a burst of power initially and 527 00:38:09,000 --> 00:38:14,000 then a rapid decay after that, but if R2 is very large then I 528 00:38:14,000 --> 00:38:16,000 have a much slower release of energy. 529 00:38:16,000 --> 00:38:20,000 But suffice it to say that the energy dissipated, 530 00:38:20,000 --> 00:38:23,000 the total energy in T2 is simply ΩCVS^2. 531 00:38:23,000 --> 00:38:25,000 All right. 532 00:38:32,000 --> 00:38:37,000 Let's put T1 and T2 together and look at the total energy 533 00:38:37,000 --> 00:38:39,000 dissipated -- 534 00:38:48,000 --> 00:38:55,000 Total energy dissipated. E is simply E1 plus E2. 535 00:38:55,000 --> 00:39:02,000 Dissipated in each cycle. Assuming T1 and T2 are much 536 00:39:02,000 --> 00:39:06,000 larger than the respective time constants. 537 00:39:06,000 --> 00:39:11,000 And I know that this is ΩCVS^2, ΩCVS^2, so this is simply 538 00:39:11,000 --> 00:39:14,000 CVS^2. If I have an arrangement of 539 00:39:14,000 --> 00:39:19,000 switches and capacitors like that, I charge the capacitor, 540 00:39:19,000 --> 00:39:23,000 discharge the capacitor, charge the capacitor, 541 00:39:23,000 --> 00:39:28,000 discharge the capacitor. What it is saying is that in a 542 00:39:28,000 --> 00:39:34,000 charge/discharge cycle I am using up CVS^2 of energy. 543 00:39:34,000 --> 00:39:40,000 ΩCVS^2 when I charge it up and ΩCVS^2 when I discharge it. 544 00:39:40,000 --> 00:39:46,000 That is what I get. Let's compute the average power 545 00:39:46,000 --> 00:39:54,000 dissipated, P average in a cycle is simply E/T where T is the 546 00:39:54,000 --> 00:40:00,000 period of the square wave sequence that I have shown you 547 00:40:00,000 --> 00:40:06,000 out there. This is simply CVS^2 divided by 548 00:40:06,000 --> 00:40:09,000 T. If the period of the square 549 00:40:09,000 --> 00:40:14,000 wave is capital T, I can express that as a 550 00:40:14,000 --> 00:40:18,000 frequency. Let's say for example the 551 00:40:18,000 --> 00:40:25,000 period of the square wave is T, so let's say the frequency of 552 00:40:25,000 --> 00:40:33,000 the square wave is simply 1/T. I can also express this as 553 00:40:33,000 --> 00:40:35,000 C(VS^2)f. 554 00:40:43,000 --> 00:40:49,000 What does this say? Let me mark that as a thing to 555 00:40:49,000 --> 00:40:53,000 remember, the second thing to remember. 556 00:40:53,000 --> 00:41:00,000 One was the power that was the static power. 557 00:41:00,000 --> 00:41:04,000 And second is this power relating to this frequency f and 558 00:41:04,000 --> 00:41:08,000 the charging and discharging of the capacitor in that little 559 00:41:08,000 --> 00:41:11,000 circuit shown up there. So, this average power is 560 00:41:11,000 --> 00:41:14,000 CVS^2f. What this is saying is that if 561 00:41:14,000 --> 00:41:17,000 f is high, if I have high frequency of charging and 562 00:41:17,000 --> 00:41:22,000 discharging the capacitor then I am charging and discharging much 563 00:41:22,000 --> 00:41:27,000 more frequently so I am going to consume more power. 564 00:41:27,000 --> 00:41:30,000 Notice that at any given time there is no direct connection 565 00:41:30,000 --> 00:41:33,000 between the power supply and the ground. 566 00:41:33,000 --> 00:41:36,000 What I am doing is my capacitor is an intermediary. 567 00:41:36,000 --> 00:41:40,000 I am dumping some charge in the capacitor and the capacitor is 568 00:41:40,000 --> 00:41:43,000 dumping the charge into ground. It behaves like a switch to 569 00:41:43,000 --> 00:41:46,000 capacitor. And what it is doing is it is 570 00:41:46,000 --> 00:41:49,000 being charged and discharged at frequency f. 571 00:41:49,000 --> 00:41:52,000 So, it makes sense that the amount of average current that I 572 00:41:52,000 --> 00:41:56,000 am pumping through relates to the frequency at which I am 573 00:41:56,000 --> 00:42:00,000 charging and discharging the capacitor. 574 00:42:00,000 --> 00:42:05,000 And similarly the average power also relates to the value of the 575 00:42:05,000 --> 00:42:09,000 capacitor. If C is larger I dissipate more 576 00:42:09,000 --> 00:42:12,000 energy. And the same way with the 577 00:42:12,000 --> 00:42:15,000 voltage. If the voltage is higher then 578 00:42:15,000 --> 00:42:20,000 the power in that period, or the average power relates to 579 00:42:20,000 --> 00:42:24,000 CVS^2. Spend a few seconds staring at 580 00:42:24,000 --> 00:42:29,000 the two expressions. This power here relating to 581 00:42:29,000 --> 00:42:34,000 just this connection between the power supply and ground and that 582 00:42:34,000 --> 00:42:39,000 power out there relating to charging and discharging 583 00:42:39,000 --> 00:42:42,000 capacitors. Let's get back to our inverter 584 00:42:42,000 --> 00:42:44,000 right now. 585 00:42:55,000 --> 00:43:00,000 This is our inverter circuit. Let us say that I drive the 586 00:43:00,000 --> 00:43:03,000 input with the waveform shown here. 587 00:43:03,000 --> 00:43:08,000 Well, I go back to the same situation as here. 588 00:43:08,000 --> 00:43:14,000 I drive the input with a square wave, with T1 and T2 as the high 589 00:43:14,000 --> 00:43:19,000 time and the low time. The equivalent circuit for this 590 00:43:19,000 --> 00:43:22,000 is not exactly what we saw there. 591 00:43:22,000 --> 00:43:27,000 The equivalent circuit for this would look like this. 592 00:43:27,000 --> 00:43:34,000 I have a VS. And the VS supply is connected 593 00:43:34,000 --> 00:43:40,000 through RL, VS connected through RL to a capacitor C. 594 00:43:40,000 --> 00:43:47,000 This is my voltage vO. So, VS is always connected to 595 00:43:47,000 --> 00:43:55,000 ground through this resistor and capacitor in this manner. 596 00:43:55,000 --> 00:44:02,000 And then I have a resistor here RON corresponding to that 597 00:44:02,000 --> 00:44:09,000 MOSFET. And there I am switching it on 598 00:44:09,000 --> 00:44:17,000 and off in a way that it is on during T1 and off during T2. 599 00:44:17,000 --> 00:44:25,000 So, the situation here is a bit different from that simple 600 00:44:25,000 --> 00:44:33,000 situation I computed there. Much like I computed the power 601 00:44:33,000 --> 00:44:39,000 dissipation in that circuit, I can go ahead and compute the 602 00:44:39,000 --> 00:44:43,000 total power dissipated in this circuit. 603 00:44:43,000 --> 00:44:48,000 I won't do it here. The algebra tends to be a big 604 00:44:48,000 --> 00:44:53,000 more grubbier than what I have been through. 605 00:44:53,000 --> 00:44:59,000 And suffice it to say that you can show that the average power 606 00:44:59,000 --> 00:45:03,000 is given by (VS^2)/(2(RL+RON))+(CVS^2)f 607 00:45:03,000 --> 00:45:07,000 (RL^2)/(RL+RON)^2. 608 00:45:12,000 --> 00:45:14,000 OK? And for details I suggest that 609 00:45:14,000 --> 00:45:18,000 you look at section 12.3 of the course notes. 610 00:45:18,000 --> 00:45:23,000 Section 12.3 goes through the algebra to compute the total 611 00:45:23,000 --> 00:45:26,000 power dissipated by this specific circuit, 612 00:45:26,000 --> 00:45:30,000 and here is the expression we get. 613 00:45:30,000 --> 00:45:37,000 And let's take the specific situation where RL is much 614 00:45:37,000 --> 00:45:43,000 greater than RON. If RL is much greater than RON 615 00:45:43,000 --> 00:45:48,000 then I can ignore this RON here. 616 00:45:55,000 --> 00:45:57,000 And I get this. And out here, 617 00:45:57,000 --> 00:46:02,000 if I ignore RON, then RL and RL will cancel out 618 00:46:02,000 --> 00:46:07,000 and I get CVS^2f. If I ignore RON compared to RL 619 00:46:07,000 --> 00:46:12,000 this is the expression I get. Now you can see why I went 620 00:46:12,000 --> 00:46:17,000 through those two examples. This is exactly the power 621 00:46:17,000 --> 00:46:22,000 consumed by the connection between power supply and ground. 622 00:46:22,000 --> 00:46:26,000 And this CVS^2f is the power consumed in charging and 623 00:46:26,000 --> 00:46:32,000 discharging the capacitor. If you look at the circuit here 624 00:46:32,000 --> 00:46:35,000 it is consuming two kinds of power. 625 00:46:35,000 --> 00:46:40,000 One kind of power is due to the current flowing directly from VS 626 00:46:40,000 --> 00:46:44,000 through RL and RON to ground. Oh, this also assumes, 627 00:46:44,000 --> 00:46:48,000 by the way, that T1 is equal to T2. 628 00:46:55,000 --> 00:46:57,000 So, in this circuit there are two kinds of power. 629 00:46:57,000 --> 00:46:59,000 One is the power when the switch is on and I have a 630 00:46:59,000 --> 00:47:02,000 current flowing from VS to RL to ground. 631 00:47:02,000 --> 00:47:05,000 Notice I get an extra factor of two in the denominator here. 632 00:47:05,000 --> 00:47:09,000 And that two comes about because the connection to ground 633 00:47:09,000 --> 00:47:13,000 only happens half the time. It's half that power out there 634 00:47:13,000 --> 00:47:17,000 because I am connected to ground only when the switch is on. 635 00:47:17,000 --> 00:47:21,000 And that happens only half the time, and so therefore I get the 636 00:47:21,000 --> 00:47:24,000 VS^2/2RL. And then CVS^2f is simply the 637 00:47:24,000 --> 00:47:28,000 power that I consumed because I am charging and discharging the 638 00:47:28,000 --> 00:47:31,000 capacitor C. Notice that in this inverter 639 00:47:31,000 --> 00:47:34,000 circuit there are two kinds of power. 640 00:47:34,000 --> 00:47:38,000 One is called the standby power which is static power being 641 00:47:38,000 --> 00:47:41,000 consumed by the circuit, and the second power is the 642 00:47:41,000 --> 00:47:44,000 dynamic power because the circuit is switching up and 643 00:47:44,000 --> 00:47:47,000 down. This relates to star and this 644 00:47:47,000 --> 00:47:50,000 relates to the double star. And to demonstrate that, 645 00:47:50,000 --> 00:47:53,000 I have a little demonstration here that has an inverter. 646 00:47:53,000 --> 00:47:57,000 And I am going to up the frequency of the square way of 647 00:47:57,000 --> 00:48:01,000 driving the inverter. I am going to show you a few 648 00:48:01,000 --> 00:48:04,000 numbers so hang on for two minutes after this demo. 649 00:48:04,000 --> 00:48:08,000 I will give you some numbers, but I want you to go ahead and 650 00:48:08,000 --> 00:48:11,000 compute the numbers based on what we have seen here. 651 00:48:11,000 --> 00:48:14,000 And you will get suitably impressed, I promise you. 652 00:48:14,000 --> 00:48:16,000 This is the input fed to the inverter. 653 00:48:16,000 --> 00:48:19,000 This is the output of the inverter. 654 00:48:19,000 --> 00:48:22,000 Notice that the output of the inverter reflects some sort of 655 00:48:22,000 --> 00:48:25,000 an RC time constant because of the output driving the 656 00:48:25,000 --> 00:48:29,000 capacitor, and the same way here. 657 00:48:29,000 --> 00:48:33,000 I start off by showing you that on the left-hand side I am 658 00:48:33,000 --> 00:48:38,000 simply measuring the power being consumed by the circuit. 659 00:48:38,000 --> 00:48:43,000 Notice that the power being consumed is expressed by the 660 00:48:43,000 --> 00:48:48,000 needle being at this point here. This is a very low frequency so 661 00:48:48,000 --> 00:48:53,000 this is almost all standby power consumed by the inverter. 662 00:48:53,000 --> 00:48:57,000 The inverter is on half the time, and when it is on it is 663 00:48:57,000 --> 00:49:01,000 consuming power. What I am going to now is 664 00:49:01,000 --> 00:49:04,000 increase the frequency. As I increase the frequency 665 00:49:04,000 --> 00:49:07,000 driving the inverter what should happen to this needle? 666 00:49:07,000 --> 00:49:10,000 As I increase the frequency there that waveform should 667 00:49:10,000 --> 00:49:12,000 become closer and closer together. 668 00:49:12,000 --> 00:49:15,000 And what should happen to the needle? 669 00:49:15,000 --> 00:49:18,000 That should begin to go up. If I increase the frequency it 670 00:49:18,000 --> 00:49:21,000 should consume more and more power and the needle should 671 00:49:21,000 --> 00:49:24,000 start going up. So, let me do that for you. 672 00:49:24,000 --> 00:49:28,000 In terms of numbers there it is on top of the four on the scale 673 00:49:28,000 --> 00:49:31,000 in the middle. I am going to increase the 674 00:49:31,000 --> 00:49:34,000 frequency very slowly. Unfortunately, 675 00:49:34,000 --> 00:49:37,000 the sampling scope messes up the waveform. 676 00:49:37,000 --> 00:49:41,000 Ignore the waveform for now. Just look at the meter as I 677 00:49:41,000 --> 00:49:44,000 increase the frequency. 678 00:49:49,000 --> 00:49:52,000 Notice that I have increased the frequency by about a factor 679 00:49:52,000 --> 00:49:54,000 of 2 or 3. And notice here that this meter 680 00:49:54,000 --> 00:49:56,000 has moved. The needle has moved to the 681 00:49:56,000 --> 00:49:58,000 right. And I can keep doing that and 682 00:49:58,000 --> 00:50:01,000 the needle keeps moving to the right as I am consuming more and 683 00:50:01,000 --> 00:50:04,000 more power because I'm driving the inverter faster and faster 684 00:50:04,000 --> 00:50:07,000 and faster. That should convince you that 685 00:50:07,000 --> 00:50:11,000 there is a standby power and there is some power component 686 00:50:11,000 --> 00:50:15,000 related to frequency. This relates to your standby 687 00:50:15,000 --> 00:50:18,000 power in your cell phone. This relates to active use. 688 00:50:18,000 --> 00:50:22,000 Let me show you some numbers, and you can plug those numbers 689 00:50:22,000 --> 00:50:26,000 in yourself and see how much power this converter is going to 690 00:50:26,000 --> 00:50:29,000 consume and see if it makes sense. 691 00:50:29,000 --> 00:50:33,000 Assume that I have a chip with 10^8 gates. 692 00:50:33,000 --> 00:50:38,000 F is 1 gigahertz. That is 10^9. 693 00:50:38,000 --> 00:50:47,000 Assume C is 0.1 femtofarads which is 10^-16 farads. 694 00:50:47,000 --> 00:50:56,000 Assume VS is 5 volts. Assume RL is 10 kilo ohms. 695 00:50:56,000 --> 00:51:06,000 Use these numbers. Plug these numbers in here and 696 00:51:06,000 --> 00:51:16,000 get a sense if our modern-day circuitry used that inverter, 697 00:51:16,000 --> 00:51:26,000 what would be the power consumed by a chip that contains 698 00:51:26,000 --> 00:51:35,000 10^8 of these gates? You will find out that you may 699 00:51:35,000 --> 00:51:45,000 have to use a nuclear power reactor to actually drive that 700 00:51:45,000 --> 00:51:52,000 chip, but go check it out for yourselves. 701 00:51:52,000 --> 00:52:03,000 In the next lecture we will see then how do our cell phones 702 00:52:03,000 --> 00:52:11,000 work, how does life go on despite this horrendous 703 00:52:11,000 --> 00:52:14,000 calculation here.