1 00:00:00 --> 00:00:09 2 00:00:09 --> 00:00:11 All right. Let's get moving. 3 00:00:11 --> 00:00:13 Good morning. 4 00:00:13 --> 00:00:17 5 00:00:17 --> 00:00:24 Today, if everything works out, we have some fun for you guys. 6 00:00:24 --> 00:00:28 I hope it works out. We'll see. 7 00:00:28 --> 00:00:34 What I am going to do today is a very major application of the 8 00:00:34 --> 00:00:39 frequency response and the frequency domain analysis of 9 00:00:39 --> 00:00:42 circuits. And this application area is 10 00:00:42 --> 00:00:47 called filters. The area of filters often times 11 00:00:47 --> 00:00:52 demands a full course or a couple of full courses all by 12 00:00:52 --> 00:00:56 itself. And filters are incredibly 13 00:00:56 --> 00:01:00 useful. They're used in virtually every 14 00:01:00 --> 00:01:03 electronic device in some form or another. 15 00:01:03 --> 00:01:08 They're used in radio tuners. We will show you a demo of that 16 00:01:08 --> 00:01:11 today. They're also used in your cell 17 00:01:11 --> 00:01:14 phones. Every single cell phone has a 18 00:01:14 --> 00:01:16 set of filters. So, for example, 19 00:01:16 --> 00:01:20 how do you pick a conversation? You pick a conversation by 20 00:01:20 --> 00:01:25 picking a certain frequency and grabbing data from there. 21 00:01:25 --> 00:01:31 They are also in wide area network wireless transmitters. 22 00:01:31 --> 00:01:33 Do we have an access point here? 23 00:01:33 --> 00:01:37 I don't see one, but you've seen wireless access 24 00:01:37 --> 00:01:41 points. Again, there they have filters 25 00:01:41 --> 00:01:44 in them. So, virtually every single 26 00:01:44 --> 00:01:49 electronic device contains a filter at some point or another. 27 00:01:49 --> 00:01:54 And so, today we will look at this major, major application of 28 00:01:54 --> 00:02:01 frequency domain analysis. Before we get into that, 29 00:02:01 --> 00:02:09 I'd like to do a bit of review. The readings for today 30 00:02:09 --> 00:02:18 correspond to Chapter 14.4.2, 14.5 and 15.2 in the course 31 00:02:18 --> 00:02:21 notes. All right. 32 00:02:21 --> 00:02:30 Let's start with the review. We looked at this circuit last 33 00:02:30 --> 00:02:33 Friday -- 34 00:02:33 --> 00:02:41 35 00:02:41 --> 00:02:46 -- where I said that for our analysis, we are going to focus 36 00:02:46 --> 00:02:49 on this small, small region of the playground. 37 00:02:49 --> 00:02:55 And what's special about this region of our playground is that 38 00:02:55 --> 00:02:59 I am going to focus on sinusoidal inputs. 39 00:02:59 --> 00:03:01 And, second, I am going to focus on the 40 00:03:01 --> 00:03:05 steady state response. How does the response look like 41 00:03:05 --> 00:03:07 if I wait a long, long time? 42 00:03:07 --> 00:03:12 And then we said that the full blown time domain analysis was 43 00:03:12 --> 00:03:13 hard. This was, remember, 44 00:03:13 --> 00:03:17 the agonizing approach? And then I taught you the 45 00:03:17 --> 00:03:21 impedance approach in the last lecture, which was blindingly 46 00:03:21 --> 00:03:23 simple. And, in that impedance 47 00:03:23 --> 00:03:28 approach, what we said we would do is -- 48 00:03:28 --> 00:03:32 I will apply the approach right now and in seconds derive the 49 00:03:32 --> 00:03:35 result for you. But the basic idea was we said 50 00:03:35 --> 00:03:39 what we are going to do is assume that we are going to 51 00:03:39 --> 00:03:42 apply inputs of the form Vi e to the j omega t. 52 00:03:42 --> 00:03:47 Wherever you see a capital and a small, there is an implicate e 53 00:03:47 --> 00:03:51 to the j omega t next to it. I'm not showing you that. 54 00:03:51 --> 00:03:55 And what I showed last time, and the class before that was 55 00:03:55 --> 00:03:59 once you find out the amplitude -- 56 00:03:59 --> 00:04:02 Once you find out the multiplier that multiplies e to 57 00:04:02 --> 00:04:05 the j omega t, it's a complex number, 58 00:04:05 --> 00:04:08 you have all the information you need. 59 00:04:08 --> 00:04:11 And once you have this, you can find out the time 60 00:04:11 --> 00:04:15 domain response by simply taking the modulus of that, 61 00:04:15 --> 00:04:19 or the amplitude and the phase of that to get the angle. 62 00:04:19 --> 00:04:22 And that gives you the time domain response. 63 00:04:22 --> 00:04:27 So, our focus has been on these quantities. 64 00:04:27 --> 00:04:31 The impedance method says what I am going to do is replace each 65 00:04:31 --> 00:04:34 of these by impedances. And then the corresponding 66 00:04:34 --> 00:04:37 impedance model looks like this. 67 00:04:37 --> 00:04:42 68 00:04:42 --> 00:04:45 Instead of R, I replace that with ZR. 69 00:04:45 --> 00:04:51 And instead of the capacitor, I am going to replace that with 70 00:04:51 --> 00:04:53 ZC. And this is my Vc. 71 00:04:53 --> 00:04:59 ZR is simply R and ZC was going to be one divided by sC where s 72 00:04:59 --> 00:05:04 was simply a shorthand notation for j omega. 73 00:05:04 --> 00:05:07 Based on this, once I converted all my 74 00:05:07 --> 00:05:12 elements into impedances, I can go ahead and apply all 75 00:05:12 --> 00:05:16 the good-old linear analysis techniques. 76 00:05:16 --> 00:05:20 I will discuss a bunch of them today. 77 00:05:20 --> 00:05:24 As an example, I could analyze this using my 78 00:05:24 --> 00:05:28 simple voltage divider relationship. 79 00:05:28 --> 00:05:34 Vc is simply ZC divided by ZC plus ZR times Vi. 80 00:05:34 --> 00:05:40 And that, in turn, is, well, let's say I divide 81 00:05:40 --> 00:05:47 this by Vi so I can get the response relation, 82 00:05:47 --> 00:05:55 is ZC divided by ZC plus ZR. And ZC I know to be one by j 83 00:05:55 --> 00:06:01 omega C, plus R. And multiplying throughout by j 84 00:06:01 --> 00:06:10 omega C, I get one divided by one plus j omega CR. 85 00:06:10 --> 00:06:14 It's incredibly simple. This is simply called the 86 00:06:14 --> 00:06:19 frequency response. And it's a transfer function 87 00:06:19 --> 00:06:24 representing the relationship between the output complex 88 00:06:24 --> 00:06:30 amplitude with the input. We can also plot this. 89 00:06:30 --> 00:06:33 Notice that in our entire analysis we have not only 90 00:06:33 --> 00:06:37 assumed sinusoidal input, but we're also saying that let 91 00:06:37 --> 00:06:40 us look at this only in the steady state. 92 00:06:40 --> 00:06:44 So, we will wait for time to be really, really large, 93 00:06:44 --> 00:06:48 and then look at the response. And so, therefore, 94 00:06:48 --> 00:06:52 we will plot the response not as a function of time, 95 00:06:52 --> 00:06:56 but rather we are going to plot the response as a function of 96 00:06:56 --> 00:06:59 omega. What we are going to say is I 97 00:06:59 --> 00:07:03 am going to input a sinusoid and my output is going to be some 98 00:07:03 --> 00:07:06 other sinusoid. And since I'm waiting for a 99 00:07:06 --> 00:07:09 long time to look at the output, time doesn't make sense 100 00:07:09 --> 00:07:11 anymore. Rather, my free variable is 101 00:07:11 --> 00:07:14 going to be my frequency, so I am going to change the 102 00:07:14 --> 00:07:17 frequency of the input that I apply. 103 00:07:17 --> 00:07:20 And so, I am going to plot this as a function of omega. 104 00:07:20 --> 00:07:24 This represents a completely complimentary view of circuits, 105 00:07:24 --> 00:07:27 the time domain view and then there is a frequency domain 106 00:07:27 --> 00:07:31 view. The frequency domain view says 107 00:07:31 --> 00:07:37 how did this circuit behave as I apply sinusoids of differing 108 00:07:37 --> 00:07:42 frequencies? I can plot that relationship in 109 00:07:42 --> 00:07:47 a graph like this, and this relationship is simply 110 00:07:47 --> 00:07:51 given by a parameter edge the transfer function, 111 00:07:51 --> 00:07:56 it's a function of omega. And I can also plot the 112 00:07:56 --> 00:08:02 absolute value of that. And let's take a look at what 113 00:08:02 --> 00:08:05 it looks like. So, I can look at functions 114 00:08:05 --> 00:08:08 like this and very quickly plot the response. 115 00:08:08 --> 00:08:13 I am going to do a whole bunch of plots just by staring at 116 00:08:13 --> 00:08:17 circuits and staring at expressions like this. 117 00:08:17 --> 00:08:20 And you will see a number of them today. 118 00:08:20 --> 00:08:23 First of all, the way you plot these is look 119 00:08:23 --> 00:08:27 for the values where omega is very small and when omega is 120 00:08:27 --> 00:08:31 very large. When omega is very, 121 00:08:31 --> 00:08:36 very small this term goes away. And so, for very small values 122 00:08:36 --> 00:08:39 of omega the output is simply one. 123 00:08:39 --> 00:08:42 Vc by Vi is simply one. This part goes away. 124 00:08:42 --> 00:08:46 What happens when omega is very, very large? 125 00:08:46 --> 00:08:50 When omega is really large, this part dominates, 126 00:08:50 --> 00:08:54 is much greater than one. If I ignore one in relation to 127 00:08:54 --> 00:08:59 this guy and take the absolute value of that then I simply get 128 00:08:59 --> 00:09:05 one divided by omega CR when omega is very large. 129 00:09:05 --> 00:09:10 So, when omega is very large, I get a decay of the form one 130 00:09:10 --> 00:09:14 over omega CR. I know the value for small 131 00:09:14 --> 00:09:18 omega, and it looks like this for very large omega. 132 00:09:18 --> 00:09:23 And, if you plot it out, this is how it's going to look 133 00:09:23 --> 00:09:27 like. Let's stare at this form for a 134 00:09:27 --> 00:09:32 little while longer. And let's plot some properties 135 00:09:32 --> 00:09:34 off it. First of all, 136 00:09:34 --> 00:09:40 you notice something else. When omega CR equals one then, 137 00:09:40 --> 00:09:44 in other words, when omega equals one by RC, 138 00:09:44 --> 00:09:49 notice that the output is given by one plus j. 139 00:09:49 --> 00:09:55 And the absolute value of that is simply one divided the square 140 00:09:55 --> 00:09:58 root of two. So, in other words, 141 00:09:58 --> 00:10:04 when omega is one by RC -- When omega is one by CR then 142 00:10:04 --> 00:10:10 the output is one by square root two times its value when omega 143 00:10:10 --> 00:10:14 is very, very small. So, that is one little piece of 144 00:10:14 --> 00:10:18 information. If you look at the form of 145 00:10:18 --> 00:10:23 this, I would like you to stare at it for a few minutes and try 146 00:10:23 --> 00:10:28 to understand what this represents. 147 00:10:28 --> 00:10:32 This says that for very low frequencies the response is 148 00:10:32 --> 00:10:36 virtually the same as the input in amplitude. 149 00:10:36 --> 00:10:39 In other words, if I apply some very low 150 00:10:39 --> 00:10:44 frequency sinusoid of some amplitude then the output 151 00:10:44 --> 00:10:48 amplitude is going to be same as that amplitude. 152 00:10:48 --> 00:10:52 And that's a one. Now, it also says when I apply 153 00:10:52 --> 00:10:56 a very high frequency, at very high frequencies it 154 00:10:56 --> 00:11:01 decays. So, this graph which says I am 155 00:11:01 --> 00:11:06 going to pass low frequencies without any attenuation, 156 00:11:06 --> 00:11:12 without hammering it, but I am going to clobber high 157 00:11:12 --> 00:11:18 frequencies and give you a very low amplitude signal at the 158 00:11:18 --> 00:11:24 output but pass through, almost without attenuation, 159 00:11:24 --> 00:11:30 the input at low frequencies. And so this is an example of 160 00:11:30 --> 00:11:36 what is called a low pass filter or LPF. 161 00:11:36 --> 00:11:41 What this is saying is that this little circuit here acts 162 00:11:41 --> 00:11:47 like a low pass filter. It's a low pass filter because 163 00:11:47 --> 00:11:52 it passes low frequencies without attenuation but kills 164 00:11:52 --> 00:11:56 high frequencies. If I take some music, 165 00:11:56 --> 00:12:02 and you will do experiments with this in lab. 166 00:12:02 --> 00:12:04 When is lab three? People are doing lab three 167 00:12:04 --> 00:12:07 right now, right? Lab three is going on right now 168 00:12:07 --> 00:12:10 and early next week as well. And, in lab three, 169 00:12:10 --> 00:12:13 you will play with looking at the response to music of 170 00:12:13 --> 00:12:17 different types of filters. If apply some music here, 171 00:12:17 --> 00:12:20 you will see that the output will pass low frequencies but 172 00:12:20 --> 00:12:22 really attenuate high frequencies. 173 00:12:22 --> 00:12:26 You will hear a lot of the low sounding base and so on but 174 00:12:26 --> 00:12:30 attenuate a lot of the high frequencies. 175 00:12:30 --> 00:12:33 All right. The other thing that I 176 00:12:33 --> 00:12:38 encourage you to do is Websim has built in pages for a large 177 00:12:38 --> 00:12:43 number of such circuits. You can go in there and play 178 00:12:43 --> 00:12:48 with the values of RC, or L for that matter, 179 00:12:48 --> 00:12:53 for a variety of circuits. And, if you click on frequency 180 00:12:53 --> 00:12:59 response, you actually get both the amplitude response and the 181 00:12:59 --> 00:13:04 phase as well. You can play with various 182 00:13:04 --> 00:13:09 values of RLC and see how the frequency response looks like 183 00:13:09 --> 00:13:12 for each of the circuits. As a next step, 184 00:13:12 --> 00:13:17 what I would like to do is just give you a sense of how 185 00:13:17 --> 00:13:21 impedances combine. This won't be very surprising 186 00:13:21 --> 00:13:27 given that they behave just like resistors, but it's good to go 187 00:13:27 --> 00:13:32 through it nonetheless. Suppose, just to build some 188 00:13:32 --> 00:13:37 insight, suppose I had two resistors in series. 189 00:13:37 --> 00:13:39 All right. R1 and R2. 190 00:13:39 --> 00:13:44 And this was my A and B terminals respectively. 191 00:13:44 --> 00:13:50 And let's say the complex amplitude of the voltage was Vab 192 00:13:50 --> 00:13:53 across this. Then I could relate, 193 00:13:53 --> 00:14:01 let's say Iab was the current, I can relate these resistances. 194 00:14:01 --> 00:14:05 Or, I could relate Vab and Iab as follows. 195 00:14:05 --> 00:14:10 Simply Vab divided by Iab equals R1 plus R2. 196 00:14:10 --> 00:14:15 I know that. And the same thing applies to R 197 00:14:15 --> 00:14:20 viewed as an impedance. It's still impedance R, 198 00:14:20 --> 00:14:27 and so this one still goes ahead and applies. 199 00:14:27 --> 00:14:34 The second thing I can try is the circuit of this form. 200 00:14:34 --> 00:14:40 A, B, and I have an R1 and an L in this case. 201 00:14:40 --> 00:14:46 And what I can do is, in the impedance model, 202 00:14:46 --> 00:14:53 I can view this as an impedance of value j omega L. 203 00:14:53 --> 00:15:02 And I can also combine them to get the impedance between A and 204 00:15:02 --> 00:15:06 B. Much as I got a resistance 205 00:15:06 --> 00:15:11 between A and B, I can get an impedance between 206 00:15:11 --> 00:15:18 A and B as Vab divided by Iab. And that will be given by ZR1 207 00:15:18 --> 00:15:23 plus ZL, and that is simply R1 plus j omega L. 208 00:15:23 --> 00:15:30 Similarly, I can do an even more complicated circuit. 209 00:15:30 --> 00:15:36 So, resistance. And here I have a capacitor in 210 00:15:36 --> 00:15:43 series with the resistance, and then I apply inductor to 211 00:15:43 --> 00:15:46 it. This is A, B, 212 00:15:46 --> 00:15:49 Iab and plus, minus Vab. 213 00:15:49 --> 00:15:57 And let me call this R1 and let me call this R2 and this is C 214 00:15:57 --> 00:16:02 and L. I can go about combining these 215 00:16:02 --> 00:16:08 in much the same manner that I combine my resistances in the 216 00:16:08 --> 00:16:13 series parallel simplifications. I can define an impedance Zab 217 00:16:13 --> 00:16:19 between the A and B terminals as ZR1 plus Z of this combination, 218 00:16:19 --> 00:16:25 impedance of this combination, which is simply impedance of C 219 00:16:25 --> 00:16:30 and that of R2 in parallel with each other. 220 00:16:30 --> 00:16:34 I get Zc in parallel with ZR2. Notice that this notation 221 00:16:34 --> 00:16:38 simply says that look at the impedance of the capacitor in 222 00:16:38 --> 00:16:41 parallel with a resistor. And then, finally, 223 00:16:41 --> 00:16:45 I add to that the series impedance of the inductor ZL. 224 00:16:45 --> 00:16:49 Exactly as you would have done for resistances, 225 00:16:49 --> 00:16:53 if all of these resistances you would have said R of this piece 226 00:16:53 --> 00:16:57 plus the R of the parallel combination plus the R of 227 00:16:57 --> 00:17:03 whatever was here. This time around we have 228 00:17:03 --> 00:17:08 impedances. And replacing this with the 229 00:17:08 --> 00:17:15 values, this is R1. I know for ZL it's j omega L. 230 00:17:15 --> 00:17:21 And so, for ZL, parallel ZR2 it is given by 231 00:17:21 --> 00:17:29 ZCZR2 divided by ZC plus ZR2, which is simply R1 here and j 232 00:17:29 --> 00:17:35 omega L. And let me just substitute the 233 00:17:35 --> 00:17:40 values here. I know that ZR2 is simply R2, 234 00:17:40 --> 00:17:46 ZC is one by j omega C, and then one by j omega C plus 235 00:17:46 --> 00:17:50 R2. And I can go ahead and simplify 236 00:17:50 --> 00:17:54 that further and get my impedance Zab. 237 00:17:54 --> 00:18:00 Notice how simple analysis has become. 238 00:18:00 --> 00:18:03 Using this technique, using the impedance method 239 00:18:03 --> 00:18:07 we've managed to convert our analysis from solving 240 00:18:07 --> 00:18:11 differential equations to going back to algebra. 241 00:18:11 --> 00:18:15 A large part of what we do in circuits is see how we can get 242 00:18:15 --> 00:18:20 back to really simple algebra and try to be clever about how 243 00:18:20 --> 00:18:23 we do things. So, this is as far as analysis 244 00:18:23 --> 00:18:26 is concerned. In the next five minutes, 245 00:18:26 --> 00:18:31 I want to give you some insight into how you can build different 246 00:18:31 --> 00:18:34 kinds of impedances. 247 00:18:34 --> 00:18:43 248 00:18:43 --> 00:18:46 And I won't go into too much detail but give some insight 249 00:18:46 --> 00:18:50 into how you can get a sense for the kind of filters you want to 250 00:18:50 --> 00:18:52 design. Or, at the very least, 251 00:18:52 --> 00:18:55 given a filter, how can you very quickly get 252 00:18:55 --> 00:18:58 some insight into what kind of filter it is, 253 00:18:58 --> 00:19:01 how it performs, what its frequency response is 254 00:19:01 --> 00:19:05 and so on. And, this time around, 255 00:19:05 --> 00:19:09 this piece of intuition will be in honor of Umans. 256 00:19:09 --> 00:19:14 And back to our Bend it Like Beckham series, 257 00:19:14 --> 00:19:17 I call this "Unleash it like Umans". 258 00:19:17 --> 00:19:23 What experts in the field do is they don't go about sitting 259 00:19:23 --> 00:19:28 around writing differential equations, but rather use a lot 260 00:19:28 --> 00:19:34 of insight into how to solve these things. 261 00:19:34 --> 00:19:39 And so in honor of Umans, I will label this unleash it 262 00:19:39 --> 00:19:43 like Umans. Let's get some insight into how 263 00:19:43 --> 00:19:47 the response of various elements look like. 264 00:19:47 --> 00:19:51 Let's take, for example, I have some impedance Z. 265 00:19:51 --> 00:19:56 Let's say this could be a resistor, it could be an 266 00:19:56 --> 00:20:01 inductor or it could be a capacitor. 267 00:20:01 --> 00:20:05 Let's take a look at what the frequency response of just these 268 00:20:05 --> 00:20:07 elements look like. In other words, 269 00:20:07 --> 00:20:10 what are the frequency dependents of Z itself? 270 00:20:10 --> 00:20:14 Let me just plot the impedance of each of these elements as a 271 00:20:14 --> 00:20:17 function of frequency. Let me just take the absolute 272 00:20:17 --> 00:20:21 value of their impedance. Notice that it's a complex 273 00:20:21 --> 00:20:23 number. For the inductor it's j omega 274 00:20:23 --> 00:20:25 L. And let me take the absolute 275 00:20:25 --> 00:20:30 value omega L in that case and plot it for you. 276 00:20:30 --> 00:20:33 And use that to develop some insight. 277 00:20:33 --> 00:20:38 Let's do a simple case first. If Z is a resistance of value R 278 00:20:38 --> 00:20:43 then no matter what the frequency my value is going to 279 00:20:43 --> 00:20:46 be R. If I have an inductor of value 280 00:20:46 --> 00:20:51 L then the impedance is going to look like j omega L, 281 00:20:51 --> 00:20:55 and so I am going to omega L for that. 282 00:20:55 --> 00:21:00 And the dependence of that simply says that for low omega 283 00:21:00 --> 00:21:06 the impedance is very small. For omega zero the impedance is 284 00:21:06 --> 00:21:09 zero and it increases linearly with omega. 285 00:21:09 --> 00:21:12 So, it's omega L for the inductor. 286 00:21:12 --> 00:21:16 Impedance increases linerally as I increase the frequency. 287 00:21:16 --> 00:21:20 What about for the capacitor? For the capacitor, 288 00:21:20 --> 00:21:23 the impedance is one divided by j omega C. 289 00:21:23 --> 00:21:27 And so, therefore, I get the dependence being 290 00:21:27 --> 00:21:33 related to omega C. Which says that for very high 291 00:21:33 --> 00:21:38 frequencies impedance is very low, but for very low 292 00:21:38 --> 00:21:45 frequencies the impedance is very high and I get a behavior 293 00:21:45 --> 00:21:49 pattern that looks something like this. 294 00:21:49 --> 00:21:54 It goes as one by omega C. As omega is very large, 295 00:21:54 --> 00:22:00 my impedance is very small. If omega is very small, 296 00:22:00 --> 00:22:03 my impedance goes towards that of an open circuit. 297 00:22:03 --> 00:22:06 This is not surprising. You've known this before, 298 00:22:06 --> 00:22:09 right? That a capacitor behaves like 299 00:22:09 --> 00:22:12 an open circuit for DC. An inductor behaves like a 300 00:22:12 --> 00:22:15 short circuit for DC. Notice that zero frequency here 301 00:22:15 --> 00:22:18 corresponds to DC. The capacitor looks like an 302 00:22:18 --> 00:22:21 open circuit for DC, very high impedance. 303 00:22:21 --> 00:22:24 The inductor looks like a short circuit for DC, 304 00:22:24 --> 00:22:27 very low impedance. And the opposite is true at 305 00:22:27 --> 00:22:31 very high frequencies. While R is a constant 306 00:22:31 --> 00:22:34 throughout. Let's use this to build some 307 00:22:34 --> 00:22:37 insight into how our circuits might look. 308 00:22:37 --> 00:22:40 Let me do this example. 309 00:22:40 --> 00:22:45 310 00:22:45 --> 00:22:51 Let's say I have a Vi and I measure the response across the 311 00:22:51 --> 00:22:53 resistor. 312 00:22:53 --> 00:23:00 313 00:23:00 --> 00:23:05 So, I measure Vr divided by Vi and take the absolute value and 314 00:23:05 --> 00:23:08 take a look at how it's going to look like. 315 00:23:08 --> 00:23:13 I want you to stare at this for me and help me with what the 316 00:23:13 --> 00:23:18 response is going to look like. Let's take incredibly high 317 00:23:18 --> 00:23:21 frequencies. At very high frequencies, 318 00:23:21 --> 00:23:26 this has a very high frequency, what do the capacitor look like 319 00:23:26 --> 00:23:32 to very high frequencies? Is it an open or is it a short? 320 00:23:32 --> 00:23:35 A short circuit. At very high frequencies the 321 00:23:35 --> 00:23:38 capacitor looks like a short circuit. 322 00:23:38 --> 00:23:41 Then Vi simply appears across the resistor, 323 00:23:41 --> 00:23:45 which means that at very high frequencies the output is very 324 00:23:45 --> 00:23:49 close to the input. At very low frequencies what 325 00:23:49 --> 00:23:51 happens? At very low frequencies the 326 00:23:51 --> 00:23:54 capacitor looks like an open circuit. 327 00:23:54 --> 00:23:58 If this looks like an open circuit then very little voltage 328 00:23:58 --> 00:24:03 will drop across this resistor here because most of it is going 329 00:24:03 --> 00:24:09 to drop across the capacitor. What is going to happen is, 330 00:24:09 --> 00:24:13 for very low values, I am going to be looking at 331 00:24:13 --> 00:24:17 something out here. And, because of that, 332 00:24:17 --> 00:24:23 my response looks like this. And this is of a different form 333 00:24:23 --> 00:24:27 than the one you saw earlier. In this case, 334 00:24:27 --> 00:24:32 I pass high frequencies but attenuate low frequencies. 335 00:24:32 --> 00:24:36 Not surprisingly, this is called a high pass 336 00:24:36 --> 00:24:38 filter. 337 00:24:38 --> 00:24:44 338 00:24:44 --> 00:24:47 You need to begin to be able to think about capacitors and 339 00:24:47 --> 00:24:51 inductors in terms of their high and low frequency properties. 340 00:24:51 --> 00:24:55 And, if you develop that intuition, once you develop the 341 00:24:55 --> 00:24:59 intuition about capacitors and inductors and their frequency 342 00:24:59 --> 00:25:01 relationship, that will be a big step forward 343 00:25:01 --> 00:25:04 in 002. If you get that insight, 344 00:25:04 --> 00:25:07 you will go a long way in terms of knowing how to tackle 345 00:25:07 --> 00:25:10 problems and being able to quickly sketch responses. 346 00:25:10 --> 00:25:11 Yes. 347 00:25:11 --> 00:25:22 348 00:25:22 --> 00:25:25 In the case of, if we get something like j 349 00:25:25 --> 00:25:30 omega L, what you can do is take the limit as omega goes to zero. 350 00:25:30 --> 00:25:33 If it is omega L then notice that it is going to start 351 00:25:33 --> 00:25:36 linear. And, on the other hand, 352 00:25:36 --> 00:25:39 if when you get very high frequencies, for example, 353 00:25:39 --> 00:25:43 if you get one by something omega C then this is a 354 00:25:43 --> 00:25:47 hyperbolic relationship, so it is going to go ahead 355 00:25:47 --> 00:25:50 looking like this. So, you can take a look at a 356 00:25:50 --> 00:25:55 lot of these functions at their very low values and see how they 357 00:25:55 --> 00:25:59 look like at that point. All right. 358 00:25:59 --> 00:26:02 The next one I would like to draw for you is something that 359 00:26:02 --> 00:26:03 looks like this. 360 00:26:03 --> 00:26:08 361 00:26:08 --> 00:26:12 Let's say, for example, I have an inductor L and a 362 00:26:12 --> 00:26:16 resistor R and I want to see what that looks like. 363 00:26:16 --> 00:26:20 In this particular example, I have H, take the absolute 364 00:26:20 --> 00:26:23 value. So, what is this going to look 365 00:26:23 --> 00:26:27 like? I am going to look at the value 366 00:26:27 --> 00:26:32 across the resistor here. Here what I am going to find is 367 00:26:32 --> 00:26:36 that at very low frequencies this guy is a short circuit. 368 00:26:36 --> 00:26:40 Since this guy is a short circuit, all the voltage drops 369 00:26:40 --> 00:26:44 across the resistor so it's going to look like this. 370 00:26:44 --> 00:26:49 And, at very high frequencies, what I am going to find is that 371 00:26:49 --> 00:26:52 the inductor is going to appear like an open circuit. 372 00:26:52 --> 00:26:56 And so, therefore, all the voltage is going to 373 00:26:56 --> 00:27:00 pretty much drop across the inductor. 374 00:27:00 --> 00:27:03 It will be R divided by something plus omega L. 375 00:27:03 --> 00:27:08 So, at high frequencies this guy is going to taper off to 376 00:27:08 --> 00:27:11 zero and is going to look like this. 377 00:27:11 --> 00:27:14 And this is back to my low pass filter. 378 00:27:14 --> 00:27:17 Just to go back to a question asked earlier, 379 00:27:17 --> 00:27:20 how do you know what this looks like? 380 00:27:20 --> 00:27:25 I can very quickly write down the expression for H of j omega. 381 00:27:25 --> 00:27:32 This is simply going to be R divided by R plus if this is VR. 382 00:27:32 --> 00:27:36 VR is simply R divided by one by j omega C. 383 00:27:36 --> 00:27:41 I multiply it out by j omega C in the numerator and the 384 00:27:41 --> 00:27:45 denominator. I'm going to find j omega C 385 00:27:45 --> 00:27:49 here and I am going to get one by j omega C here. 386 00:27:49 --> 00:27:55 And what is going to happen with something like this is that 387 00:27:55 --> 00:28:02 as omega becomes very small then I am going to ignore this. 388 00:28:02 --> 00:28:07 When omega becomes very small, I can ignore this with respect 389 00:28:07 --> 00:28:12 to one, and I get R j omega C. Given that, is what I've drawn 390 00:28:12 --> 00:28:16 here correct or wrong? This goes away with respect to 391 00:28:16 --> 00:28:19 one. I am left with R j omega C, 392 00:28:19 --> 00:28:21 right? For very low frequencies. 393 00:28:21 --> 00:28:26 Given what I have drawn here, is that correct or is that 394 00:28:26 --> 00:28:30 wrong? Well, it's hard to say. 395 00:28:30 --> 00:28:36 For very, very low frequencies it starts out being linear 396 00:28:36 --> 00:28:40 because it's an omega relationship, 397 00:28:40 --> 00:28:47 and then it goes up like this and then goes out there. 398 00:28:47 --> 00:28:54 Let me go onto another example. Let me do another example here 399 00:28:54 --> 00:29:01 which is something like -- I need to make sure I don't 400 00:29:01 --> 00:29:05 make a mistake here. If I get R j omega C by R j 401 00:29:05 --> 00:29:09 omega C, you know what, this ends up being a first 402 00:29:09 --> 00:29:12 order system, and so is going to look like 403 00:29:12 --> 00:29:14 this. I blew it there. 404 00:29:14 --> 00:29:19 Back to this system here. If I have an L and an R and I 405 00:29:19 --> 00:29:23 look at this equation to look at what happens across L, 406 00:29:23 --> 00:29:29 you can plot that again. And for very low frequencies it 407 00:29:29 --> 00:29:34 is going to be zero amplitude here and for very high 408 00:29:34 --> 00:29:38 frequencies this is going to be an open circuit, 409 00:29:38 --> 00:29:43 and so the response is going to look something like this. 410 00:29:43 --> 00:29:48 That's going to end up being your high pass filter. 411 00:29:48 --> 00:29:53 As another example, I would like to do a series RLC 412 00:29:53 --> 00:29:55 circuit -- 413 00:29:55 --> 00:30:10 414 00:30:10 --> 00:30:14 -- and try to get you some sense of what that output looks 415 00:30:14 --> 00:30:17 like. Let's use our intuition and 416 00:30:17 --> 00:30:22 first write down what this looks like and then go and do some 417 00:30:22 --> 00:30:26 math and see if the math corresponds to what our 418 00:30:26 --> 00:30:31 intuition tells us. I want to plot Vr with respect 419 00:30:31 --> 00:30:34 to Vi. I want to plot it there. 420 00:30:34 --> 00:30:38 For something like this, what happens at very low 421 00:30:38 --> 00:30:41 frequencies? We are just looking to get 422 00:30:41 --> 00:30:46 very, very crudely what this graph is going to look like. 423 00:30:46 --> 00:30:51 Very, very crudely what this graph is going to look like. 424 00:30:51 --> 00:30:55 Given that I am taking the voltage across VR, 425 00:30:55 --> 00:31:00 what happens at very low frequencies? 426 00:31:00 --> 00:31:05 At incredibly low frequencies, the inductor looks like a short 427 00:31:05 --> 00:31:09 circuit, but the capacitor looks like open circuit. 428 00:31:09 --> 00:31:14 An open circuit in series with a short circuit that ends up 429 00:31:14 --> 00:31:18 looking like an open circuit. And so, therefore, 430 00:31:18 --> 00:31:23 all my voltage falls across VR. Now, what happens at very high 431 00:31:23 --> 00:31:26 frequencies? At very high frequencies the 432 00:31:26 --> 00:31:32 capacitor looks like a short. But the inductor looks like an 433 00:31:32 --> 00:31:36 open circuit now for very high frequencies, correct? 434 00:31:36 --> 00:31:39 Just remember, capacitor is short for high 435 00:31:39 --> 00:31:42 frequencies inductor open for high frequencies. 436 00:31:42 --> 00:31:46 So, this ends up having a very high impedance. 437 00:31:46 --> 00:31:50 At very high frequencies this guy has a very high impedance. 438 00:31:50 --> 00:31:54 And, because of that, for a high value of frequency, 439 00:31:54 --> 00:31:59 I end up going in that manner. This behavior has the effect of 440 00:31:59 --> 00:32:04 the capacitor here. And for very high frequencies I 441 00:32:04 --> 00:32:10 get the effect of the inductor. And so this means that I have 442 00:32:10 --> 00:32:14 very low values for low frequencies, very low values for 443 00:32:14 --> 00:32:18 high frequencies. And, as the frequency 444 00:32:18 --> 00:32:21 increases, I do something like this. 445 00:32:21 --> 00:32:25 I keep building up, then the inductor begins to 446 00:32:25 --> 00:32:30 play a role, and then I taper off again. 447 00:32:30 --> 00:32:36 This kind of a filter where I kill low and high frequencies 448 00:32:36 --> 00:32:42 and pass intermediate frequencies is called a band 449 00:32:42 --> 00:32:46 pass filter, BPF. This means that it passes 450 00:32:46 --> 00:32:53 frequencies in some band. Let's get some more insight on 451 00:32:53 --> 00:32:57 this by writing down the equations. 452 00:32:57 --> 00:33:02 So, Vr divided by Vi is simply R. 453 00:33:02 --> 00:33:10 Using the impedance relation it is R divided by j omega L plus 454 00:33:10 --> 00:33:18 one divided by j omega C plus R. I am going to use this equation 455 00:33:18 --> 00:33:26 later, so let me stash it away on my stack and put a little 456 00:33:26 --> 00:33:31 notation there. I am going to multiply 457 00:33:31 --> 00:33:38 throughout by j omega C. And what I end up getting is j 458 00:33:38 --> 00:33:44 omega RC divided by one plus R j omega RC, and then here, 459 00:33:44 --> 00:33:51 I get j times j is minus one, so I get minus omega squared. 460 00:33:51 --> 00:33:59 Let me rewrite it this way. I get minus omega squared. 461 00:33:59 --> 00:34:04 So, j j is minus one, omega times omega is omega 462 00:34:04 --> 00:34:10 squared, and then I get an LC. That's what I end up getting. 463 00:34:10 --> 00:34:16 And if I take the absolute value here, I end up getting, 464 00:34:16 --> 00:34:23 back to your complex algebra, the square root of this real 465 00:34:23 --> 00:34:29 value squared plus imaginary value squared. 466 00:34:29 --> 00:34:34 So, one minus omega squared LC plus omega RC squared. 467 00:34:34 --> 00:34:38 This is from, you can look it up in your 468 00:34:38 --> 00:34:42 complex algebra appendix in the course notes. 469 00:34:42 --> 00:34:47 It's simply omega RC here, then square of the real value 470 00:34:47 --> 00:34:54 plus the square of the imaginary value, and take the square root 471 00:34:54 --> 00:34:56 of that. By staring at this, 472 00:34:56 --> 00:35:04 you can notice that you realize a really important property. 473 00:35:04 --> 00:35:08 When omega equals LC. I'm sorry. 474 00:35:08 --> 00:35:14 When omega equals one divided by LC, what happens? 475 00:35:14 --> 00:35:21 Sorry, square root of LC. When omega is one divided by 476 00:35:21 --> 00:35:30 square root of LC then omega squared times LC becomes one. 477 00:35:30 --> 00:35:34 When this is true then this becomes one, and one and one 478 00:35:34 --> 00:35:37 cancel out. And, not only that, 479 00:35:37 --> 00:35:41 when these cancel out, these two cancel out at that 480 00:35:41 --> 00:35:47 point, so I end up getting a one, which means that when omega 481 00:35:47 --> 00:35:52 equals omega nought equals one by square root of LC and I end 482 00:35:52 --> 00:35:58 up getting a value that is one. It's pretty amazing. 483 00:35:58 --> 00:36:01 Which means that if I drive this at omega nought, 484 00:36:01 --> 00:36:06 if my sinusoid has a frequency omega nought where omega nought 485 00:36:06 --> 00:36:11 is one by square root of LC, if I'm sitting here and this is 486 00:36:11 --> 00:36:16 a black box on the right-hand side, and I drive this at a 487 00:36:16 --> 00:36:20 frequency omega nought equals one divided by square root of 488 00:36:20 --> 00:36:24 LC, what does this entire circuit look like to me? 489 00:36:24 --> 00:36:29 I'm sitting there, the black box here. 490 00:36:29 --> 00:36:33 I'm driving it at omega nought equals one by square root of LC 491 00:36:33 --> 00:36:36 at that frequency. What does that circuit look 492 00:36:36 --> 00:36:36 like? Yes. 493 00:36:36 --> 00:36:40 It looks like a resistor. It's pretty amazing. 494 00:36:40 --> 00:36:43 It means that even though I have an L and a C here, 495 00:36:43 --> 00:36:47 if I happen to drive this at omega nought then the circuit 496 00:36:47 --> 00:36:51 looks purely resistive and it seems to give me the same input 497 00:36:51 --> 00:36:54 appearing at the output. In other words, 498 00:36:54 --> 00:36:58 the effect of these two cancels out. 499 00:36:58 --> 00:37:02 And that aspect is called driving the circuit at its 500 00:37:02 --> 00:37:05 resonance point. Resonance is when you're 501 00:37:05 --> 00:37:10 driving the circuit at omega nought equals one by a square 502 00:37:10 --> 00:37:12 root of LC. 503 00:37:12 --> 00:37:21 504 00:37:21 --> 00:37:27 I will very quickly sketch for you a couple of other ways of 505 00:37:27 --> 00:37:32 looking at circuits. Supposing I looked at this 506 00:37:32 --> 00:37:37 value here, Vlc, I looked at the value across 507 00:37:37 --> 00:37:43 the inductor and the capacitor, what will the frequency 508 00:37:43 --> 00:37:48 response look like? I am looking at the voltage 509 00:37:48 --> 00:37:53 across the inductor and the capacitor in series. 510 00:37:53 --> 00:37:57 Let's see. Let's go back to our usual 511 00:37:57 --> 00:38:01 mantra. Think about Steve Umans when 512 00:38:01 --> 00:38:03 you do this. What would he do? 513 00:38:03 --> 00:38:07 He would say ah-ha, at very low frequencies the 514 00:38:07 --> 00:38:10 capacitor is going to look like an open circuit. 515 00:38:10 --> 00:38:14 In my voltage divider, I am measuring the voltage 516 00:38:14 --> 00:38:18 across an open circuit, so the entire Vi must drop 517 00:38:18 --> 00:38:20 across the inductor and capacitor. 518 00:38:20 --> 00:38:25 Similarly, at very high frequencies the inductor looks 519 00:38:25 --> 00:38:30 like an open circuit now, so it looks like this. 520 00:38:30 --> 00:38:35 At very high frequencies inductor is an open circuit. 521 00:38:35 --> 00:38:42 And, again, I'm looking at the voltage divider across the near 522 00:38:42 --> 00:38:47 infinite resistance, impedance, so I get a high 523 00:38:47 --> 00:38:52 value here as well. Well, in the middle the value 524 00:38:52 --> 00:38:56 dips and I get something like this. 525 00:38:56 --> 00:39:02 So, this thing is called a band stop filter. 526 00:39:02 --> 00:39:07 Here I can nail any specific frequency, as long as the 527 00:39:07 --> 00:39:12 frequency falls in roughly that regime. 528 00:39:12 --> 00:39:15 Yet another example. 529 00:39:15 --> 00:39:20 530 00:39:20 --> 00:39:23 The reason I'm working on so many examples is that to 531 00:39:23 --> 00:39:27 experts, a large part of what they do is look at a circuit and 532 00:39:27 --> 00:39:30 boom, give a rough form of how it looks like. 533 00:39:30 --> 00:39:33 That can get you half the way there in most of what you're 534 00:39:33 --> 00:39:37 going to do. How did this look like? 535 00:39:37 --> 00:39:43 If I take the voltage Vo versus Vi, let's take a look. 536 00:39:43 --> 00:39:49 At very low frequencies, the inductor looks like a short 537 00:39:49 --> 00:39:54 circuit, correct? I am talking the voltage across 538 00:39:54 --> 00:40:00 a short circuit, so it looks like this. 539 00:40:00 --> 00:40:04 At very high frequencies, I am taking a voltage across a 540 00:40:04 --> 00:40:09 parallel combination, but the capacitor is now a 541 00:40:09 --> 00:40:12 short circuit. So, that looks like a 542 00:40:12 --> 00:40:16 capacitor. This looks like an inductor out 543 00:40:16 --> 00:40:20 here and this is a capacitor holding sway here. 544 00:40:20 --> 00:40:25 And so, somewhere in the middle it goes up and comes down like 545 00:40:25 --> 00:40:30 that. So, it's a band pass filter. 546 00:40:30 --> 00:40:33 What is amazing is that you can take fairly complicated 547 00:40:33 --> 00:40:36 circuits, and just by doing a quick analysis of what happens 548 00:40:36 --> 00:40:39 at very low frequencies, what happens at very high 549 00:40:39 --> 00:40:42 frequencies, you can roughly sketch the response. 550 00:40:42 --> 00:40:45 And then what you should do, in addition to that, 551 00:40:45 --> 00:40:48 is if it's a second order circuit, just assume that it's 552 00:40:48 --> 00:40:51 going to do something interesting at its resonance 553 00:40:51 --> 00:40:54 frequency, at omega nought equals one by square root of LC. 554 00:40:54 --> 00:40:57 Something interesting is going to happen. 555 00:40:57 --> 00:41:01 Check it out. And for circuits that are first 556 00:41:01 --> 00:41:05 order, RC or RL, the important number is the 557 00:41:05 --> 00:41:09 time constant RC. Usually, when you're driving it 558 00:41:09 --> 00:41:12 at one by RC, omega equals one by RC then 559 00:41:12 --> 00:41:17 what happens is that you often times end up getting a value 560 00:41:17 --> 00:41:22 that is one by square root two times the input value in the 561 00:41:22 --> 00:41:26 circuits we looked at here. Next, what I am going to do is 562 00:41:26 --> 00:41:32 talk about a major, major application of filters. 563 00:41:32 --> 00:41:41 And that is an AM receiver. Let me do Radios 101 for 30 564 00:41:41 --> 00:41:48 seconds. These guys have an antenna. 565 00:41:48 --> 00:41:57 You take a ground here. You pick up a signal at your 566 00:41:57 --> 00:42:02 antenna. There is an implied ground as 567 00:42:02 --> 00:42:03 well. And what you do, 568 00:42:03 --> 00:42:07 as a first step, is you begin processing the 569 00:42:07 --> 00:42:10 signal now. What we place right there is a 570 00:42:10 --> 00:42:12 little filter that looks like this. 571 00:42:12 --> 00:42:16 It is a inductor and a capacitor in parallel. 572 00:42:16 --> 00:42:20 And this capacitor is really your tuner that you can tune to 573 00:42:20 --> 00:42:24 radio frequencies. And then what you have here is 574 00:42:24 --> 00:42:30 a bunch of other processing and end up with your speaker. 575 00:42:30 --> 00:42:36 And the processing that happens here is you have a demodulator, 576 00:42:36 --> 00:42:42 you have an amplifier and a bunch of other things that let's 577 00:42:42 --> 00:42:48 not worry about them for now. What we do here is the antenna 578 00:42:48 --> 00:42:51 picks up a signal. So, in some sense, 579 00:42:51 --> 00:42:56 this part of the circuit here is your source. 580 00:42:56 --> 00:43:03 I could replace it with its Thevenin equivalent as follows. 581 00:43:03 --> 00:43:09 582 00:43:09 --> 00:43:13 So, the front end of your radio looks like a Vi, 583 00:43:13 --> 00:43:17 R, L and a C. Where have you seen this 584 00:43:17 --> 00:43:19 before? Right there. 585 00:43:19 --> 00:43:26 That's the front end of radios. Let me tell you why I need a 586 00:43:26 --> 00:43:31 band pass filter in a radio out here. 587 00:43:31 --> 00:43:35 The way life works is as follows. 588 00:43:35 --> 00:43:41 I have my frequency. Let me do this not in radians 589 00:43:41 --> 00:43:48 but in kilohertz for now, and let me plot your radio 590 00:43:48 --> 00:43:52 signal strength. In the Boston area, 591 00:43:52 --> 00:43:59 the signals go between 540 kilohertz and they go all the 592 00:43:59 --> 00:44:06 way to 1600 kilohertz. In some areas we have begun to 593 00:44:06 --> 00:44:10 use the 1700 extra band as well for some new stations. 594 00:44:10 --> 00:44:13 This is the frequency range of interest. 595 00:44:13 --> 00:44:17 If you look at your radio tuner, you will see 540 596 00:44:17 --> 00:44:22 kilohertz all the way up to 1600 and you can tune your AM radio. 597 00:44:22 --> 00:44:27 The way it works is that each station is given 10 kilohertz of 598 00:44:27 --> 00:44:31 spectrum here. And so, this is at 1000 599 00:44:31 --> 00:44:36 kilohertz, 1010 kilohertz and so on. 600 00:44:36 --> 00:44:43 And each station transmits its signal in plus or minus 5 601 00:44:43 --> 00:44:50 kilohertz around that point. And this station transmits it 602 00:44:50 --> 00:44:57 here and this station transmits it here and so on. 603 00:44:57 --> 00:45:02 This is 1030. This guy is WBZ News Radio 604 00:45:02 --> 00:45:06 1030, for those of you who listen to it. 605 00:45:06 --> 00:45:12 What happens is that at 10 kilohertz, each station gets 10 606 00:45:12 --> 00:45:18 kilohertz, and so WBZ transmits in the 10 kilohertz around 1030. 607 00:45:18 --> 00:45:23 Notice that each of these signals transmitted by radio 608 00:45:23 --> 00:45:28 stations happen within small bands. 609 00:45:28 --> 00:45:31 Now, you will learn a lot more about modulation and how do you 610 00:45:31 --> 00:45:34 get a signal to go in a small band and all that stuff. 611 00:45:34 --> 6.003. You will learn about that in 612 6.003. --> 00:45:36 613 00:45:36 --> 00:45:39 For now, don't worry about how I did all of this. 614 00:45:39 --> 00:45:41 How do you listen to that station? 615 00:45:41 --> 00:45:44 The way you listen to that station is you put a low pass 616 00:45:44 --> 00:45:47 filter here. You put a low pass filter that 617 00:45:47 --> 00:45:49 does the following. Let's say I want to hear WBZ 618 00:45:49 --> 1030. 619 1030. --> 00:45:51 620 00:45:51 --> 00:45:56 621 00:45:56 --> 00:46:00 If I pass this entire signal through that filter. 622 00:46:00 --> 00:46:04 And if I arrange to have the omega nought of my filter at 623 00:46:04 --> 1030. 624 1030. --> 00:46:07 If I can arrange to have the 625 00:46:07 --> 00:46:11 omega nought at 1030 then this is the response of my filter. 626 00:46:11 --> 00:46:17 And I am going to pick out this guy and cut out everything else. 627 00:46:17 --> 00:46:20 I am just going to get this. 628 00:46:20 --> 00:46:40 629 00:46:40 --> 00:46:42 Let's listen to the station for some time. 630 00:46:42 --> 00:47:52 631 00:47:52 --> 47:55 So, you can see I can tune to the station WBUL.