1 00:00:00 --> 00:00:00,746 2 00:00:00,746 --> 00:00:06,418 So we have covered RC circuits and RL circuits, 3 00:00:06,418 --> 00:00:13,694 and today, we will spend the entire lecture on LRC circuits. 4 00:00:13,694 --> 00:00:21,093 We will only discuss them in series so that you get the basic 5 00:00:21,093 --> 00:00:25,039 idea. I have here a driving power 6 00:00:25,039 --> 00:00:30,835 supply, alternating, and here I have a 7 00:00:30,835 --> 00:00:34,301 capacitor C, self-inductor L, 8 00:00:34,301 --> 00:00:38,14 and a resistance R, this is A C, 9 00:00:38,14 --> 00:00:44,702 and that the driving voltage be V zero cosine omega T. 10 00:00:44,702 --> 00:00:51,141 We have to set up the differential equation for this, 11 00:00:51,141 --> 00:00:58,323 and I want to remind you that Kirchoff's Loop Rule does not 12 00:00:58,323 --> 00:01:02,656 hold. So the closed loop 13 00:01:02,656 --> 00:01:06,735 integral of E dot D L, in spite of what the author of 14 00:01:06,735 --> 00:01:10,499 your book wants you to believe, that is not zero. 15 00:01:10,499 --> 00:01:14,734 So how do we set it up? There are various ways that you 16 00:01:14,734 --> 00:01:17,636 can do that, I have my own discipline. 17 00:01:17,636 --> 00:01:22,028 I, in my [unintelligible], I think of this first being a, 18 00:01:22,028 --> 00:01:26,655 a battery -- by this is the plus side, and this is the minus 19 00:01:26,655 --> 00:01:31,047 side -- a current is going to flow, capacitor is going to 20 00:01:31,047 --> 00:01:35,831 charge up, electric field inside the capacitor 21 00:01:35,831 --> 00:01:39,425 is in this direction, the electric field in the 22 00:01:39,425 --> 00:01:43,955 self-inductor is always zero, because the self-inductor has 23 00:01:43,955 --> 00:01:47,002 no resistance. There's no electric field 24 00:01:47,002 --> 00:01:51,064 inside the self-inductor, no matter what some of your 25 00:01:51,064 --> 00:01:55,516 books want you to believe. Then, the electric field in the 26 00:01:55,516 --> 00:02:00,125 resistor is in this direction, and the electric field inside 27 00:02:00,125 --> 00:02:04,577 the power supply goes from plus to minus, would be in this 28 00:02:04,577 --> 00:02:09,03 direction. So if I set up the differential 29 00:02:09,03 --> 00:02:12,619 equation, I start here, I always go in the same 30 00:02:12,619 --> 00:02:16,208 direction as I, because only then is the closed 31 00:02:16,208 --> 00:02:20,734 loop integral minus L D I D T. So I go over this capacitor, 32 00:02:20,734 --> 00:02:24,167 that is V of C, then I go through the wire of 33 00:02:24,167 --> 00:02:27,678 the self-inductor. There is no electric field, 34 00:02:27,678 --> 00:02:30,721 so the integral E dot D L there is zero. 35 00:02:30,721 --> 00:02:34,154 Then I go through the resistor, so I get I R, 36 00:02:34,154 --> 00:02:38,68 and then I have here my power supply, so I get minus V zero 37 00:02:38,68 --> 00:02:42,095 cosine omega T, and that, 38 00:02:42,095 --> 00:02:47,225 now, according to Faraday's Law, equals L minus L D I D T. 39 00:02:47,225 --> 00:02:52,265 The current equals D Q D T. If the current is positive -- 40 00:02:52,265 --> 00:02:58,026 this is my positive direction -- then the charge of the capacitor 41 00:02:58,026 --> 00:03:01,806 will increase. And I also know that V of C, 42 00:03:01,806 --> 00:03:07,116 the potential difference over the capacitor is the charge on 43 00:03:07,116 --> 00:03:11,167 each one of the capacitor plates, 44 00:03:11,167 --> 00:03:14,93 divided by C. And so I substitute that in 45 00:03:14,93 --> 00:03:19,07 this equation, and I bring D L D I D T to the 46 00:03:19,07 --> 00:03:22,645 left side. That is conventionally done. 47 00:03:22,645 --> 00:03:27,162 You don't have to do that, but that's often done. 48 00:03:27,162 --> 00:03:31,772 So I get a plus L, D I D T now becomes D two Q D T 49 00:03:31,772 --> 00:03:37,418 squared -- my goal is to get everything in terms of Q -- then 50 00:03:37,418 --> 00:03:44,013 my I R become R times D Q D T, and my V of C becomes Q divided 51 00:03:44,013 --> 00:03:47,466 by C -- notice that I ranked them in order, 52 00:03:47,466 --> 00:03:50,756 D two Q D T squared, D Q D T, and then Q, 53 00:03:50,756 --> 00:03:55,443 you don't have to do that, but there is nothing wrong with 54 00:03:55,443 --> 00:04:00,459 doing that -- And then we get here, equals V zero cosine omega 55 00:04:00,459 --> 00:04:03,008 T. And this is the form in which 56 00:04:03,008 --> 00:04:07,202 most books would present this differential equation. 57 00:04:07,202 --> 00:04:13,533 And they arrive that in various ways, most books arrive at this 58 00:04:13,533 --> 00:04:17,419 equation in a completely wrong way, but they get -- anyhow, 59 00:04:17,419 --> 00:04:21,506 they end up with this equation. And so, you have to solve this 60 00:04:21,506 --> 00:04:25,19 equation, which is really beyond your present abilities, 61 00:04:25,19 --> 00:04:29,143 it's second-order differential equation, it's really part of 62 00:04:29,143 --> 00:04:31,823 eighteen oh three, so I will give you the 63 00:04:31,823 --> 00:04:34,369 solution. The basic idea being that you 64 00:04:34,369 --> 00:04:37,249 find a solution for Q as a function of time, 65 00:04:37,249 --> 00:04:39,996 and once you know Q as a function of time, 66 00:04:39,996 --> 00:04:42,14 you have, of course, the current, 67 00:04:42,14 --> 00:04:47,158 because then you take the derivative of your 68 00:04:47,158 --> 00:04:50,665 solution, and you get the current. 69 00:04:50,665 --> 00:04:55,872 I will give you the current as a function of time. 70 00:04:55,872 --> 00:05:00,867 So I, that satisfies that differential equation, 71 00:05:00,867 --> 00:05:06,712 is the V zero divided by [whistles] R squared plus omega 72 00:05:06,712 --> 00:05:12,239 L minus one over omega C squared, and the whole thing 73 00:05:12,239 --> 00:05:21,124 times cosine omega T minus phi. And the tangent of phi equals 74 00:05:21,124 --> 00:05:27,523 omega L minus one over omega C divided by R. 75 00:05:27,523 --> 00:05:34,964 We give this upstairs here a name, we call that the 76 00:05:34,964 --> 00:05:38,535 reactance. The reactance, 77 00:05:38,535 --> 00:05:47,464 and that X, or sometimes it's called xi, is omega L minus one 78 00:05:47,464 --> 00:05:50,649 over omega C. 79 00:05:50,649 --> 00:05:56,686 And the units are also ohms. We call the entire square root 80 00:05:56,686 --> 00:06:00,953 that you see here, we call that capital Z, 81 00:06:00,953 --> 00:06:07,303 which is called the impedance, so the square root of R squared 82 00:06:07,303 --> 00:06:13,132 plus that X squared equals Z, that also has units of ohm, 83 00:06:13,132 --> 00:06:16,462 and that is called the impedance. 84 00:06:16,462 --> 00:06:22,083 And so Z is an effective resistance, because this whole 85 00:06:22,083 --> 00:06:26,26 thing behaves like a resistance. 86 00:06:26,26 --> 00:06:29,906 But the resistance depends not only on R L and C, 87 00:06:29,906 --> 00:06:32,26 but also on the values of omega. 88 00:06:32,26 --> 00:06:36,285 This solution is what we call a steady-state solution, 89 00:06:36,285 --> 00:06:40,994 it is the solution that you get if you wait a certain amount of 90 00:06:40,994 --> 00:06:43,652 time. If you turn the instrument on, 91 00:06:43,652 --> 00:06:46,993 so you all of a sudden start this experiment, 92 00:06:46,993 --> 00:06:50,866 then in the beginning, you get a different solution, 93 00:06:50,866 --> 00:06:54,663 which is more complicated, you get 94 00:06:54,663 --> 00:06:58,38 transient phenomenon, but these transient phenomenon 95 00:06:58,38 --> 00:07:01,441 die out, and you end up with this solution. 96 00:07:01,441 --> 00:07:05,304 Now, there are several interesting things that you can 97 00:07:05,304 --> 00:07:08,802 see in this solution. We have to start digesting, 98 00:07:08,802 --> 00:07:10,988 this whole hour, this solution. 99 00:07:10,988 --> 00:07:13,247 It has very interesting aspects. 100 00:07:13,247 --> 00:07:16,308 For one thing, you can see that the current 101 00:07:16,308 --> 00:07:20,681 can be delayed over the driving voltage when phi is positive. 102 00:07:20,681 --> 00:07:24,324 Then the current comes later than 103 00:07:24,324 --> 00:07:26,911 the voltage. And that's the result of the 104 00:07:26,911 --> 00:07:29,304 inductor, we've discussed that before. 105 00:07:29,304 --> 00:07:33,184 But now, that's also possible that the current is leading the 106 00:07:33,184 --> 00:07:36,612 voltage, which is very hard to understand intuitively. 107 00:07:36,612 --> 00:07:40,169 That is the case when this term dominates over this one, 108 00:07:40,169 --> 00:07:43,402 then phi becomes negative, and so minus phi becomes 109 00:07:43,402 --> 00:07:45,601 positive. If minus phi is positive, 110 00:07:45,601 --> 00:07:47,8 the current is leading the voltage. 111 00:07:47,8 --> 00:07:50,387 Now you may say, "How can it possibly be? 112 00:07:50,387 --> 00:07:55,173 Does that meant that before I switch the instrument on, 113 00:07:55,173 --> 00:07:59,898 that I already have a current? Of course it doesn't mean that. 114 00:07:59,898 --> 00:08:03,23 But that's the transient solution, remember? 115 00:08:03,23 --> 00:08:07,026 When you turn something on, when you switch it on, 116 00:08:07,026 --> 00:08:11,287 this solution doesn't hold yet. This is the steady-state 117 00:08:11,287 --> 00:08:13,766 solution. So the value for I Max, 118 00:08:13,766 --> 00:08:17,949 we have always called what is front of the cosine term, 119 00:08:17,949 --> 00:08:22,288 we've always called that I Max, that value for I Max is a 120 00:08:22,288 --> 00:08:27,978 function of omega itself -- as we will analyze in detail 121 00:08:27,978 --> 00:08:31,632 today -- and of course, also, of R L and C. 122 00:08:31,632 --> 00:08:36,592 And there is one particular value for Z, and therefore for 123 00:08:36,592 --> 00:08:40,333 omega, whereby this value reaches a maximum, 124 00:08:40,333 --> 00:08:43,204 and that's what we call resonance. 125 00:08:43,204 --> 00:08:48,077 There is no value for omega for which the current is any 126 00:08:48,077 --> 00:08:50,774 higher. And so I will call here, 127 00:08:50,774 --> 00:08:53,123 the situation, as resonance. 128 00:08:53,123 --> 00:08:57,908 It is at resonance when X equals 129 00:08:57,908 --> 00:09:01,678 zero, so when omega L is one over omega C, 130 00:09:01,678 --> 00:09:06,09 so when omega is one over the square root of L C. 131 00:09:06,09 --> 00:09:11,606 And we call that the resonance frequency, and we often give a 132 00:09:11,606 --> 00:09:17,397 little subscript zero to remind you that you're dealing with the 133 00:09:17,397 --> 00:09:21,166 resonance frequency. And Z is then just R, 134 00:09:21,166 --> 00:09:25,854 because when X is zero, the omega L and the one over 135 00:09:25,854 --> 00:09:31,691 omega C eat each other up. They are not there any more, 136 00:09:31,691 --> 00:09:34,987 it's gone. And so the system behaves as if 137 00:09:34,987 --> 00:09:39,408 there were only a resistor. And so you also see that the 138 00:09:39,408 --> 00:09:42,946 maximum current that you get is then, simply, 139 00:09:42,946 --> 00:09:46,564 V zero divided by that value for R, because Z, 140 00:09:46,564 --> 00:09:48,413 the impedance, is now R. 141 00:09:48,413 --> 00:09:51,95 And in addition, if you're interested in phi, 142 00:09:51,95 --> 00:09:56,131 phi then becomes zero, so the driving voltage is then 143 00:09:56,131 --> 00:09:59,186 in phase with the current that follows. 144 00:09:59,186 --> 00:10:03,447 And so the signal that you will see 145 00:10:03,447 --> 00:10:10,652 is a cosinusoidal variation in the current, so if I have here 146 00:10:10,652 --> 00:10:17,738 the current as a function of time, and you get a signal like 147 00:10:17,738 --> 00:10:23,502 so, and this here, this period T equals your two 148 00:10:23,502 --> 00:10:29,026 pi divided by omega. So that is the -- directly 149 00:10:29,026 --> 00:10:33,23 connected to your driving frequency. 150 00:10:33,23 --> 00:10:39,595 And if the impedance Z is very low, 151 00:10:39,595 --> 00:10:43,573 then this maximum value of the current, this is what we call 152 00:10:43,573 --> 00:10:47,282 the maximum vale -- and, of course, the maximum value is 153 00:10:47,282 --> 00:10:50,721 also here, except that the cosine is minus one here, 154 00:10:50,721 --> 00:10:54,362 and the cosine is plus one here -- so if Z is very low, 155 00:10:54,362 --> 00:10:57,127 then this will be high. If Z is very high, 156 00:10:57,127 --> 00:11:00,229 this will be low. And there is only one and one 157 00:11:00,229 --> 00:11:03,398 value of Z for which the system is at resonance, 158 00:11:03,398 --> 00:11:06,298 and that is when the self-inductance and the 159 00:11:06,298 --> 00:11:11,773 capacitor eat each other up, and then you get the maximum 160 00:11:11,773 --> 00:11:17,176 possible value for the current at maximum, which is V zero over 161 00:11:17,176 --> 00:11:19,79 R. And that's the highest value 162 00:11:19,79 --> 00:11:24,671 that you could ever get them. Imagine that we have an LRC 163 00:11:24,671 --> 00:11:29,725 circuit, and we have L and R and C fixed, but we change the 164 00:11:29,725 --> 00:11:33,908 driving frequency. So we move over various values 165 00:11:33,908 --> 00:11:38,963 of Z by changing omega from a very low value to a very high 166 00:11:38,963 --> 00:11:41,752 value. If you start at a very low 167 00:11:41,752 --> 00:11:46,506 value for omega, let's say it approaches 168 00:11:46,506 --> 00:11:51,657 zero, then notice that Z goes to infinity, and so the maximum 169 00:11:51,657 --> 00:11:56,035 current becomes zero. And the person responsible for 170 00:11:56,035 --> 00:12:00,499 that is the capacitor, because if omega goes to zero, 171 00:12:00,499 --> 00:12:04,276 this goes to infinity. And that's intuitively 172 00:12:04,276 --> 00:12:09,426 pleasing, because omega zero really means you have no A C any 173 00:12:09,426 --> 00:12:13,289 more, you have D C. And with D C, 174 00:12:13,289 --> 00:12:16,836 what you're doing is, you charge up the capacitor 175 00:12:16,836 --> 00:12:20,752 when it's fully charged, no current can flow any more. 176 00:12:20,752 --> 00:12:25,185 So that's intuitively pleasing. When omega becomes very high, 177 00:12:25,185 --> 00:12:27,919 let's call it infinity, then Z, again, 178 00:12:27,919 --> 00:12:31,392 goes to infinity. So again, the maximum current, 179 00:12:31,392 --> 00:12:35,012 again goes to zero. And the person responsible for 180 00:12:35,012 --> 00:12:38,854 that is the self-inductor, because when omega goes to 181 00:12:38,854 --> 00:12:41,44 infinity, again, Z goes to infinity. 182 00:12:41,44 --> 00:12:44,693 So again, you get zero here. 183 00:12:44,693 --> 00:12:49,056 And that's also intuitively pleasing, because if you have an 184 00:12:49,056 --> 00:12:53,197 infinitely high frequency, that means the self-inductance 185 00:12:53,197 --> 00:12:57,338 puts up an enormous fight. It's ideal for a self-inductor 186 00:12:57,338 --> 00:13:01,775 to fight currents if the time over which the changes occur go 187 00:13:01,775 --> 00:13:03,328 to zero. And so, then, 188 00:13:03,328 --> 00:13:06,36 again, it says, "Sorry, you can't have any 189 00:13:06,36 --> 00:13:09,687 current." So that's also intuitively pleasing, 190 00:13:09,687 --> 00:13:13,459 that the self-inductance, then, becomes the dominant 191 00:13:13,459 --> 00:13:17,596 factor. And so what I can do now, 192 00:13:17,596 --> 00:13:21,532 I can plot the I Max as a function of omega. 193 00:13:21,532 --> 00:13:24,736 So here is omega, and here is I Max, 194 00:13:24,736 --> 00:13:28,856 and we already agreed that when omega is zero, 195 00:13:28,856 --> 00:13:33,158 then I Max is zero. But when omega is very high, 196 00:13:33,158 --> 00:13:37,369 it's also zero. But when omega is at resonance, 197 00:13:37,369 --> 00:13:42,953 omega zero, which is one over the square root of L C -- notice 198 00:13:42,953 --> 00:13:48,259 that R has nothing to do with the resonant frequency, 199 00:13:48,259 --> 00:13:52,056 it's really determined by L and C, because it's the side, 200 00:13:52,056 --> 00:13:55,649 it's the X that you want to make zero, and X is only a 201 00:13:55,649 --> 00:13:59,785 function of L and C -- at this frequency, we have a value here 202 00:13:59,785 --> 00:14:03,649 which is V zero divided by R. And so the curve that you're 203 00:14:03,649 --> 00:14:06,361 going to see, which we call the resonance 204 00:14:06,361 --> 00:14:10,497 curve, is something like this. You start out with an extremely 205 00:14:10,497 --> 00:14:13,141 small current, you go through resonance, 206 00:14:13,141 --> 00:14:16,666 we have a high current, and then at high frequencies, 207 00:14:16,666 --> 00:14:20,317 again, you go down to zero. 208 00:14:20,317 --> 00:14:25,376 And so the left part, when you are below resonance, 209 00:14:25,376 --> 00:14:31,345 it's really the capacitance which is the dominant guy in the 210 00:14:31,345 --> 00:14:35,594 whole game -- and phi, by the way, is here, 211 00:14:35,594 --> 00:14:41,665 less than zero -- here it is the inductor that plays the key 212 00:14:41,665 --> 00:14:47,533 role, and here phi equals larger than zero, and right here, 213 00:14:47,533 --> 00:14:50,77 phi, and only there, phi is zero, 214 00:14:50,77 --> 00:14:56,645 only when you're exactly at resonance. 215 00:14:56,645 --> 00:15:01,022 I'd like to show you some numerical results, 216 00:15:01,022 --> 00:15:06,418 and for that I have a transparency -- it's also on the 217 00:15:06,418 --> 00:15:11,101 web, so you don't have to copy the numbers, uh, 218 00:15:11,101 --> 00:15:17,31 you can download them -- these are just some numerical numbers 219 00:15:17,31 --> 00:15:23,52 which I want to digest with you, so that you get a feeling for 220 00:15:23,52 --> 00:15:27,202 the effect, that you see it in 221 00:15:27,202 --> 00:15:30,6 front of your own eyes, what is happening, 222 00:15:30,6 --> 00:15:34,412 how this curve evolves. We have here a given R, 223 00:15:34,412 --> 00:15:38,141 L, and C, ten, five times ten to the minus two 224 00:15:38,141 --> 00:15:42,45 Henry, and three times ten to the minus seven farads. 225 00:15:42,45 --> 00:15:46,843 The resonant frequency is a little over eight thousand 226 00:15:46,843 --> 00:15:50,82 radians per second, you see it here in kiloHertz, 227 00:15:50,82 --> 00:15:55,13 and you see here the impedance -- and what I do here, 228 00:15:55,13 --> 00:15:59,804 I have a driving frequency which is ten percent 229 00:15:59,804 --> 00:16:03,443 below the resonance frequency. And I calculate for you, 230 00:16:03,443 --> 00:16:06,207 here omega L, which is three hundred sixty 231 00:16:06,207 --> 00:16:10,116 seven ohms, and one over omega C, which is four fifty three 232 00:16:10,116 --> 00:16:12,206 ohms. You are a little bit below 233 00:16:12,206 --> 00:16:15,306 resonance, and so C dominates. And you can see, 234 00:16:15,306 --> 00:16:18,744 indeed, that this ohm value is larger than this one. 235 00:16:18,744 --> 00:16:22,923 And so out of that pops a value for X, out of that pops a value 236 00:16:22,923 --> 00:16:25,215 for Z. Notice that X is eighty six, 237 00:16:25,215 --> 00:16:29,057 and Z is only a hair larger than eighty 238 00:16:29,057 --> 00:16:31,984 six, because this R almost doesn't add to Z, 239 00:16:31,984 --> 00:16:35,728 because you get here the square root of ten squared plus 240 00:16:35,728 --> 00:16:38,792 eighty-six squared, that is almost eighty six. 241 00:16:38,792 --> 00:16:42,128 It becomes eighty seven. And then you see that the 242 00:16:42,128 --> 00:16:46,145 current, the maximum current, which is this value for V zero 243 00:16:46,145 --> 00:16:48,732 divided by, uh, the Z, by eighty seven, 244 00:16:48,732 --> 00:16:50,91 becomes oh point one one amperes. 245 00:16:50,91 --> 00:16:53,77 And now, the system is driven at resonance, 246 00:16:53,77 --> 00:16:57,582 and notice that it's exactly characteristic for resonance 247 00:16:57,582 --> 00:17:01,788 that omega L and one over omega C have the 248 00:17:01,788 --> 00:17:04,606 same value. They are not there any more, 249 00:17:04,606 --> 00:17:07,136 they're gone. And so X becomes zero, 250 00:17:07,136 --> 00:17:10,966 so the impedance becomes ohm -- ten ohms, which is the 251 00:17:10,966 --> 00:17:15,446 resistance, and so the maximum current is now V zero divided by 252 00:17:15,446 --> 00:17:19,131 R, which is one amperes. And when you're ten percent 253 00:17:19,131 --> 00:17:22,383 over resonance, then the self-inductor becomes 254 00:17:22,383 --> 00:17:25,924 to be more powerful than the capacitor, and again, 255 00:17:25,924 --> 00:17:29,609 your current is substantially down, 256 00:17:29,609 --> 00:17:34,46 in this, case, eight times lower than at 257 00:17:34,46 --> 00:17:39,434 resonance. We define, at a height of oh 258 00:17:39,434 --> 00:17:47,02 point seven times the value at resonance, we define a width of 259 00:17:47,02 --> 00:17:51,621 this curve. And this width is given in 260 00:17:51,621 --> 00:17:59,332 terms of delta omega. And that width -- 261 00:17:59,332 --> 00:18:04,729 and I will give you the answer without mathematical proof, 262 00:18:04,729 --> 00:18:09,273 it's not so difficult, but it's a little bit of a 263 00:18:09,273 --> 00:18:13,061 headache -- that value is R divided by L. 264 00:18:13,061 --> 00:18:17,038 So the larger R is, the broader it becomes. 265 00:18:17,038 --> 00:18:22,435 So if we look at delta omega, for the numbers that we have 266 00:18:22,435 --> 00:18:27,642 there, the numbers of the transparency -- so this is for 267 00:18:27,642 --> 00:18:32,944 for the numbers that we have there, 268 00:18:32,944 --> 00:18:38,386 we have delta omega, would be R, which is ten ohms, 269 00:18:38,386 --> 00:18:42,957 divided by five times ten to the minus two, 270 00:18:42,957 --> 00:18:48,181 and that is about two hundred radians per second. 271 00:18:48,181 --> 00:18:54,385 We define Q not as charge -- don't never confuse that with 272 00:18:54,385 --> 00:18:59,935 charge -- we call that the quality of the resonance, 273 00:18:59,935 --> 00:19:04,942 and the quality is defined as omega 274 00:19:04,942 --> 00:19:09,736 zero divided by delta omega. Now, omega zero itself is one 275 00:19:09,736 --> 00:19:14,699 over the square root of L C, and delta omega as R divided by 276 00:19:14,699 --> 00:19:17,306 L. And so that makes the quality 277 00:19:17,306 --> 00:19:21,007 one over R times the square root of L over C. 278 00:19:21,007 --> 00:19:24,792 And the quality is the measure for omega zero, 279 00:19:24,792 --> 00:19:28,073 which is this, what I'm pointing at now, 280 00:19:28,073 --> 00:19:31,185 divided by delta omega, which is this. 281 00:19:31,185 --> 00:19:35,979 So if the quality is high, this peak is relatively narrow, 282 00:19:35,979 --> 00:19:39,422 and if the quality is low, 283 00:19:39,422 --> 00:19:43,022 it's relatively wide. You may ask yourself the 284 00:19:43,022 --> 00:19:48,063 question, why do we define delta omega at seventy percent of the 285 00:19:48,063 --> 00:19:51,663 maximum current at resonance? Why not at half? 286 00:19:51,663 --> 00:19:55,823 There's a good reason for that, because, in practice, 287 00:19:55,823 --> 00:20:00,624 we are more interested in power than that we are in currents. 288 00:20:00,624 --> 00:20:03,824 And power is proportional with I squared. 289 00:20:03,824 --> 00:20:08,785 And so when you square this, you get point five. 290 00:20:08,785 --> 00:20:12,971 And point five means, then, that this is really the 291 00:20:12,971 --> 00:20:17,241 width at half-power. And so that's the reason why we 292 00:20:17,241 --> 00:20:21,595 chose the oh point seven times the maximum current at 293 00:20:21,595 --> 00:20:24,609 resonance. It's really the half-power 294 00:20:24,609 --> 00:20:27,539 width. Resonance can be destructive. 295 00:20:27,539 --> 00:20:30,637 Uh, imagine, if you have a very high-Q 296 00:20:30,637 --> 00:20:33,986 system, if you're slightly off-resonance, 297 00:20:33,986 --> 00:20:38,424 there's almost no current, no power dissipated in your 298 00:20:38,424 --> 00:20:42,192 resistor, and now, you come, 299 00:20:42,192 --> 00:20:44,316 all of a sudden, on the resonance, 300 00:20:44,316 --> 00:20:47,728 you can an enormous current, and that means there's an 301 00:20:47,728 --> 00:20:50,497 enormous power dissipation in your resistor, 302 00:20:50,497 --> 00:20:52,685 and you can burn out your resistor. 303 00:20:52,685 --> 00:20:56,033 You can destroy your circuits, if you're not careful. 304 00:20:56,033 --> 00:20:59,832 And next lecture -- and Monday, I will also discuss with you 305 00:20:59,832 --> 00:21:03,823 some med- mechanical resonances. Mechanical systems can also go 306 00:21:03,823 --> 00:21:06,72 into [unintelligible] can also be destructive. 307 00:21:06,72 --> 00:21:10,068 At certain frequencies, the systems behave -- call it 308 00:21:10,068 --> 00:21:12,901 k- violently, they respond extremely strongly 309 00:21:12,901 --> 00:21:16,061 to their input frequency, 310 00:21:16,061 --> 00:21:19,649 and things can break. Humans also have resonance 311 00:21:19,649 --> 00:21:22,932 frequencies, you can call them, if you want, 312 00:21:22,932 --> 00:21:26,52 emotional resonances. All have sensitive nerves. 313 00:21:26,52 --> 00:21:30,642 Someone makes a particular remark, go through the roof. 314 00:21:30,642 --> 00:21:34,154 Also, falling in love, when you think about it, 315 00:21:34,154 --> 00:21:37,207 is a resonance phenomenon, and that, too, 316 00:21:37,207 --> 00:21:40,643 can be rather destructive. As many of us know. 317 00:21:40,643 --> 00:21:44,841 But now I would like to demonstrate to you the resonance 318 00:21:44,841 --> 00:21:49,669 curves -- I'm going to choose particular 319 00:21:49,669 --> 00:21:52,066 values of, um, R, L, and C, 320 00:21:52,066 --> 00:21:56,401 which I can change, and then I will show you the 321 00:21:56,401 --> 00:21:59,536 current as a function of frequency. 322 00:21:59,536 --> 00:22:03,501 And these are the values that I have chosen. 323 00:22:03,501 --> 00:22:07,743 Again, this is on the Web, you can download it, 324 00:22:07,743 --> 00:22:10,694 so you don't have to copy it now. 325 00:22:10,694 --> 00:22:17,979 And I will change the -- the light setting so that we can 326 00:22:17,979 --> 00:22:21,798 also enjoy the demonstration. The idea being that, 327 00:22:21,798 --> 00:22:26,475 for these values that I have there, in the first line you see 328 00:22:26,475 --> 00:22:29,515 R sixty ohms, and the self-inductance is 329 00:22:29,515 --> 00:22:33,022 fifty milliHenry, and the capacitance is point 330 00:22:33,022 --> 00:22:36,296 three microfarads. So that's a given there. 331 00:22:36,296 --> 00:22:39,648 And I give you here the resonance frequency, 332 00:22:39,648 --> 00:22:43,077 eight thousand, in terms of omega radians per 333 00:22:43,077 --> 00:22:47,676 second, this is the resonance frequency in Hertz -- and just 334 00:22:47,676 --> 00:22:52,415 in case you're interested, I gave you the Q 335 00:22:52,415 --> 00:22:56,396 value there as well. And what I'm going to do now 336 00:22:56,396 --> 00:23:01,539 for you, is I'm going to sweep the input frequency from zero to 337 00:23:01,539 --> 00:23:04,442 sixteen thousand radians per second. 338 00:23:04,442 --> 00:23:08,423 So my omega can go from zero to sixteen thousand. 339 00:23:08,423 --> 00:23:11,741 And I leave the values as they are, here. 340 00:23:11,741 --> 00:23:16,136 So I'm going to sweep, sweep over this eight thousand. 341 00:23:16,136 --> 00:23:20,45 And so you're going to see that curve. 342 00:23:20,45 --> 00:23:23,839 Except that I'm show -- I'm going to show you I as a 343 00:23:23,839 --> 00:23:25,966 function of frequency, not I Max. 344 00:23:25,966 --> 00:23:29,356 And I is oscillating, because there's a cosine term. 345 00:23:29,356 --> 00:23:31,749 And so, for instance, if I were here, 346 00:23:31,749 --> 00:23:35,537 with this value for omega, you would see then that it goes 347 00:23:35,537 --> 00:23:38,329 up, it goes down, it goes up, it goes down, 348 00:23:38,329 --> 00:23:41,453 it goes up, and it goes down. And when I'm here, 349 00:23:41,453 --> 00:23:44,643 you will see this. And keep that in mind when you 350 00:23:44,643 --> 00:23:48,432 look at the curve that you're going to see there -- and so 351 00:23:48,432 --> 00:23:53,429 it's only the envelope, then, that is the I Max. 352 00:23:53,429 --> 00:23:58,482 But you actually see the entire current as a function of 353 00:23:58,482 --> 00:24:01,79 frequency. And I am going to do that, 354 00:24:01,79 --> 00:24:06,384 then, for all these four values that you see there. 355 00:24:06,384 --> 00:24:11,622 So, let's first change the light so that we get an optimum 356 00:24:11,622 --> 00:24:15,573 situation for you. And now, I will show you. 357 00:24:15,573 --> 00:24:20,81 Already, the results of the first line -- so these are the 358 00:24:20,81 --> 00:24:25,074 values that you see there. 359 00:24:25,074 --> 00:24:27,885 And I go -- I sl- I go very slowly. 360 00:24:27,885 --> 00:24:30,944 Now omega is zero here, omega goes up, 361 00:24:30,944 --> 00:24:34,334 I go through resonance, and omega is here, 362 00:24:34,334 --> 00:24:39,13 the value would be sixteen thousand radians per second that 363 00:24:39,13 --> 00:24:42,52 we have here. And it sweeps back and forth 364 00:24:42,52 --> 00:24:45,248 between zero and sixteen thousand. 365 00:24:45,248 --> 00:24:48,803 So you see a dramatic increase at resonance. 366 00:24:48,803 --> 00:24:53,02 So now what I'm going to do, I'm going to double the 367 00:24:53,02 --> 00:24:57,422 self-inductance. And when you double the 368 00:24:57,422 --> 00:24:59,705 self-inductance, this one goes up, 369 00:24:59,705 --> 00:25:02,887 omega zero goes down. So all I want you to see, 370 00:25:02,887 --> 00:25:06,76 that the resonance frequency, which is here -- that's the 371 00:25:06,76 --> 00:25:10,219 maximum -- that the resonance frequency will shift. 372 00:25:10,219 --> 00:25:12,916 Because if L goes up by a factor of two, 373 00:25:12,916 --> 00:25:16,928 the resonant frequency will come down by the square root of 374 00:25:16,928 --> 00:25:19,349 two. And so I am going to increase L 375 00:25:19,349 --> 00:25:23,43 by a factor of two -- let me make sure that I have the right 376 00:25:23,43 --> 00:25:27,856 know here -- and this is my L. Notice that the 377 00:25:27,856 --> 00:25:31,576 f- the resonant frequency is now at a lower value, 378 00:25:31,576 --> 00:25:34,765 it is here. Also notice that the peak value 379 00:25:34,765 --> 00:25:38,864 at resonance has not changed. Because the peak value at 380 00:25:38,864 --> 00:25:41,977 resonance, you see on the blackboard here, 381 00:25:41,977 --> 00:25:45,621 is V zero over R. So as long as I don't change R, 382 00:25:45,621 --> 00:25:49,264 that doesn't change. It's only the frequency that 383 00:25:49,264 --> 00:25:51,846 changes. Omega zero is one over the 384 00:25:51,846 --> 00:25:54,275 square root of L C. That changes. 385 00:25:54,275 --> 00:25:57,387 It is now here. So I have increased L by a 386 00:25:57,387 --> 00:26:01,679 factor of two, I can bring it back to my 387 00:26:01,679 --> 00:26:04,966 original resonance frequency by now changing C. 388 00:26:04,966 --> 00:26:08,967 If you increase by a factor of two, all you have to do is 389 00:26:08,967 --> 00:26:13,04 decrease C by a factor of two, and you're back at the same 390 00:26:13,04 --> 00:26:15,898 resonance. So I'm going to make C down -- 391 00:26:15,898 --> 00:26:19,256 C lower by a factor of two, which I'm doing now, 392 00:26:19,256 --> 00:26:22,543 and if you look, now, here, and you have a good 393 00:26:22,543 --> 00:26:26,901 memory, you will see that the resonant frequency is back where 394 00:26:26,901 --> 00:26:29,331 it was. Again, V zero over R is not 395 00:26:29,331 --> 00:26:33,104 changed, and the resonant frequency is 396 00:26:33,104 --> 00:26:36,208 back here, even though L is now twice is higher, 397 00:26:36,208 --> 00:26:39,444 and C is twice lower. To show you the effect of R, 398 00:26:39,444 --> 00:26:42,548 I will double R now, and I will leave everything 399 00:26:42,548 --> 00:26:46,049 else alone, so the resonance frequency will stay here, 400 00:26:46,049 --> 00:26:49,747 but of course the maximum current -- this high value will 401 00:26:49,747 --> 00:26:51,795 now come down, because, you see, 402 00:26:51,795 --> 00:26:54,701 at resonance, this value is v zero divided by 403 00:26:54,701 --> 00:26:57,409 R, and since R goes up by a factor of two, 404 00:26:57,409 --> 00:27:01,569 you will see that the maximum current will go 405 00:27:01,569 --> 00:27:05,541 down by a factor of two. And so, I go now from sixty -- 406 00:27:05,541 --> 00:27:08,556 from fifty -- oh, no, I go from sixty to a 407 00:27:08,556 --> 00:27:12,822 hundred, I don't double it -- I can't go any further than a 408 00:27:12,822 --> 00:27:15,47 hundred. But you'll see a substantial 409 00:27:15,47 --> 00:27:18,411 reduction. So if you're ready for this -- 410 00:27:18,411 --> 00:27:22,236 remember this height -- and now you see hundred ohms, 411 00:27:22,236 --> 00:27:25,545 and it is much lower. It was this high before, 412 00:27:25,545 --> 00:27:29,002 and now it's only here. But notice the resonance 413 00:27:29,002 --> 00:27:33,455 frequency has not changed. So this is an extremely 414 00:27:33,455 --> 00:27:35,474 interesting behavior, and every time, 415 00:27:35,474 --> 00:27:37,887 you have to think through what is happening, 416 00:27:37,887 --> 00:27:40,804 you have not much intuition for it, you're not alone, 417 00:27:40,804 --> 00:27:43,216 I don't have much intuition for that either. 418 00:27:43,216 --> 00:27:46,526 But with these rather simple equations -- they're really not 419 00:27:46,526 --> 00:27:49,555 that difficult -- here, this is really the heart of the 420 00:27:49,555 --> 00:27:52,192 equation, and then, of course, your tangent phi, 421 00:27:52,192 --> 00:27:55,39 in case you're interested in the phase lag -- that they're 422 00:27:55,39 --> 00:27:58,587 really not that difficult. I will not hold you responsible 423 00:27:58,587 --> 00:28:02,009 for being able to derive the r- solution. 424 00:28:02,009 --> 00:28:05,42 I give you the solution. But with that solution, 425 00:28:05,42 --> 00:28:08,541 you can do a lot, and you can understand the 426 00:28:08,541 --> 00:28:11,226 behavior of these circuits quite well. 427 00:28:11,226 --> 00:28:13,765 Now I'm going to do a demonstration, 428 00:28:13,765 --> 00:28:16,596 and I warn you, you have to very closely 429 00:28:16,596 --> 00:28:19,643 follow, step-by-step, what I'm going to do. 430 00:28:19,643 --> 00:28:23,925 Because if you miss one small step, you're lost for the next 431 00:28:23,925 --> 00:28:26,174 twelve minutes. Completely lost. 432 00:28:26,174 --> 00:28:30,528 You will just see nice things, but you don't know what you're 433 00:28:30,528 --> 00:28:34,782 looking at. So follow me closely. 434 00:28:34,782 --> 00:28:39,725 I have an LRC circuit. And the LRC circuit is right 435 00:28:39,725 --> 00:28:41,505 here. This is my L, 436 00:28:41,505 --> 00:28:44,965 oh point one Henry, these are my Cs, 437 00:28:44,965 --> 00:28:49,514 and this is my R, it's a two-hundred watt light 438 00:28:49,514 --> 00:28:52,183 bulb. I have an LRC circuit. 439 00:28:52,183 --> 00:28:56,435 And I will give you the values fro L, for R, 440 00:28:56,435 --> 00:29:00,389 and for C. I'm going to drive it at sixty 441 00:29:00,389 --> 00:29:04,812 Hertz. So omega equals three hundred 442 00:29:04,812 --> 00:29:07,601 and seventy seven radians per second. 443 00:29:07,601 --> 00:29:11,63 And whatever you're going to see, for the next twelve 444 00:29:11,63 --> 00:29:14,419 minutes, that is not going to change. 445 00:29:14,419 --> 00:29:18,836 That frequency is a given. I simply plug it into the wall, 446 00:29:18,836 --> 00:29:23,485 and so my driving voltage is hundred and ten times the square 447 00:29:23,485 --> 00:29:28,289 root of two, times the cosine of omega T, and it is this omega. 448 00:29:28,289 --> 00:29:33,016 So it's the hundred ten volts that comes out of 449 00:29:33,016 --> 00:29:38,932 the wall, so to speak. My light bulb R is sixty ohms 450 00:29:38,932 --> 00:29:43,92 when it is hot, and it is a two hundred watt 451 00:29:43,92 --> 00:29:46,82 bulb. So if it's -- bright 452 00:29:46,82 --> 00:29:52,505 two-hundred watt bulb. The self-inductance L is oh 453 00:29:52,505 --> 00:29:57,609 point one Henry, and the capacitance is eight 454 00:29:57,609 --> 00:30:01,437 microfarads. So this is God-given. 455 00:30:01,437 --> 00:30:08,209 With these three values, I can calculate what Z is, 456 00:30:08,209 --> 00:30:13,36 and Z is three hundred ohms. In case you're interested, 457 00:30:13,36 --> 00:30:18,51 I can also give you omega L, that is thirty-eight ohms, 458 00:30:18,51 --> 00:30:23,946 and I can give you one over omega C, that is three hundred 459 00:30:23,946 --> 00:30:28,047 and thirty two ohms. So notice that this is, 460 00:30:28,047 --> 00:30:33,77 by far, the dominant player in the game, compared to omega L. 461 00:30:33,77 --> 00:30:39,97 In case you're interested in the resonance frequency, 462 00:30:39,97 --> 00:30:44,22 omega zero is one over the square root of L C, 463 00:30:44,22 --> 00:30:49,321 and that is about eleven hundred and twenty radians per 464 00:30:49,321 --> 00:30:51,399 second. In other words, 465 00:30:51,399 --> 00:30:54,705 my driver is nowhere near resonance. 466 00:30:54,705 --> 00:30:59,051 I am way below resonance. When you're way below 467 00:30:59,051 --> 00:31:02,546 resonance, it is the C that dominates. 468 00:31:02,546 --> 00:31:06,797 Look at the three hundred and thirty two ohms, 469 00:31:06,797 --> 00:31:11,897 and compare that with the thirty-eight. 470 00:31:11,897 --> 00:31:16,322 I can calculate, now, what the maximum current 471 00:31:16,322 --> 00:31:18,485 is, that we get, I Max. 472 00:31:18,485 --> 00:31:21,238 That is V zero, divided by Z, 473 00:31:21,238 --> 00:31:25,269 that impedance, so that is hundred and ten 474 00:31:25,269 --> 00:31:29,792 square root two, divided by three hundred ohms, 475 00:31:29,792 --> 00:31:33,036 and that is oh point five amperes. 476 00:31:33,036 --> 00:31:39,23 And now I can tell you how much power is dissipated in the light 477 00:31:39,23 --> 00:31:44,899 bulb, which is I square R. So in the light bulb, 478 00:31:44,899 --> 00:31:49,344 which is the only component that has a resistance -- well, 479 00:31:49,344 --> 00:31:52,23 maybe the L has, also, a little bit of 480 00:31:52,23 --> 00:31:56,987 resistance, but we will ignore that for now -- so in the light 481 00:31:56,987 --> 00:32:01,744 bulb, the average power over one full cycle of my oscillations 482 00:32:01,744 --> 00:32:06,111 equals the mean value between I squared R, time averaged, 483 00:32:06,111 --> 00:32:10,556 the time average value of cosine omega T squared is always 484 00:32:10,556 --> 00:32:15,391 one-half, so this one-half that comes here is the result of the 485 00:32:15,391 --> 00:32:20,293 fact that the I there has a cosine omega T, 486 00:32:20,293 --> 00:32:23,698 and you square that, you get one half, 487 00:32:23,698 --> 00:32:26,919 if you average it over oscillations, 488 00:32:26,919 --> 00:32:30,047 so now I get I Max squared times R. 489 00:32:30,047 --> 00:32:34,648 So this is one-half times my oh point five squared, 490 00:32:34,648 --> 00:32:40,262 and the R of the light bulb is sixty, and this turns out to be 491 00:32:40,262 --> 00:32:43,114 seven point five watts. Nothing. 492 00:32:43,114 --> 00:32:46,887 Why is it nothing? Well, that's the way we 493 00:32:46,887 --> 00:32:50,157 designed this system. 494 00:32:50,157 --> 00:32:54,847 We're way below resonance. So if I plug the system in, 495 00:32:54,847 --> 00:32:59,36 which I will do for you, this two-hundred watt light 496 00:32:59,36 --> 00:33:03,518 bulb will only dissipate seven and a half watts, 497 00:33:03,518 --> 00:33:07,589 and you won't see anything. You ready for this? 498 00:33:07,589 --> 00:33:09,447 Three, two, one, zero. 499 00:33:09,447 --> 00:33:12,101 Physics works. You see nothing. 500 00:33:12,101 --> 00:33:14,756 Light bulb, not working. Great. 501 00:33:14,756 --> 00:33:19,977 So now comes the question, what could we do 502 00:33:19,977 --> 00:33:24,097 to get the system back on resonance? 503 00:33:24,097 --> 00:33:31,162 Well, we know that our driving frequency is three hundred and 504 00:33:31,162 --> 00:33:34,93 seventy seven radians per second. 505 00:33:34,93 --> 00:33:41,052 And so what we could do, we could change the C in the 506 00:33:41,052 --> 00:33:46,351 circuit, and we could change L in the circuit. 507 00:33:46,351 --> 00:33:51,446 If we increase C, or we increase L, 508 00:33:51,446 --> 00:33:57,477 then the resonance frequency will shift from eleven twenty 509 00:33:57,477 --> 00:34:01,392 down, make L larger, or make C larger, 510 00:34:01,392 --> 00:34:05,624 then, obviously, you shift your resonance 511 00:34:05,624 --> 00:34:10,491 frequency down. And my goal is to shift it down 512 00:34:10,491 --> 00:34:16,099 to three hundred and seventy seven radians per second. 513 00:34:16,099 --> 00:34:21,856 I first want to make a curve for you here, 514 00:34:21,856 --> 00:34:26,312 very roughly, of I Max as a function of 515 00:34:26,312 --> 00:34:30,417 frequency, omega, and that's the de- 516 00:34:30,417 --> 00:34:36,632 demonstration I just did. When this was eleven hundred 517 00:34:36,632 --> 00:34:43,551 and twenty, we had about oh point five amperes at R value of 518 00:34:43,551 --> 00:34:50,939 three hundred and seventy seven. So this is one point over I Max 519 00:34:50,939 --> 00:34:56,966 versus omega curve. I can now ask the question, 520 00:34:56,966 --> 00:35:01,04 if we drive the system at resonance, what then, 521 00:35:01,04 --> 00:35:04,14 would have been the maximum current? 522 00:35:04,14 --> 00:35:07,062 We're not driving it at resonance. 523 00:35:07,062 --> 00:35:10,693 But suppose we had driven it at resonance. 524 00:35:10,693 --> 00:35:15,387 Well, at resonance -- so at resonance, I Max is V zero 525 00:35:15,387 --> 00:35:19,107 divided by R. There is no resi- there is no 526 00:35:19,107 --> 00:35:24,333 self-inductor, and there is no capacitance at 527 00:35:24,333 --> 00:35:27,055 resonance. And so that would then be 528 00:35:27,055 --> 00:35:29,778 hundred and ten, square root of two, 529 00:35:29,778 --> 00:35:33,045 divided by sixty. But that is two point six 530 00:35:33,045 --> 00:35:35,846 amperes. That is substantially larger 531 00:35:35,846 --> 00:35:39,347 than one-half. So this value here is two point 532 00:35:39,347 --> 00:35:42,536 six amperes. It is too bad that I can't go 533 00:35:42,536 --> 00:35:47,437 there, because I am stuck to me three hundred and seventy seven, 534 00:35:47,437 --> 00:35:49,927 and I'm not going to change that. 535 00:35:49,927 --> 00:35:55,606 Throughout the demonstration, I stick to my driver here. 536 00:35:55,606 --> 00:35:59,218 So keep in mind that the driver is always here, 537 00:35:59,218 --> 00:36:02,988 I don't change that. So the resonance curve -- we 538 00:36:02,988 --> 00:36:06,601 call this the resonance curve -- is very broad, 539 00:36:06,601 --> 00:36:10,449 something like this. And I believe the Q is around 540 00:36:10,449 --> 00:36:13,119 2, you can check that for yourself. 541 00:36:13,119 --> 00:36:17,124 I can calculate what the power is, that the bulb is 542 00:36:17,124 --> 00:36:21,13 dissipating at resonance. Well, that is my one-half, 543 00:36:21,13 --> 00:36:26,235 which is the time-average cosine squared -- 544 00:36:26,235 --> 00:36:29,674 my I Max is now point -- two point six amperes, 545 00:36:29,674 --> 00:36:33,936 so now I get two point six squared -- and my resistance is 546 00:36:33,936 --> 00:36:37,6 still sixty ohms -- and this is two hundred watts. 547 00:36:37,6 --> 00:36:41,264 What a coincidence, that's the way we designed the 548 00:36:41,264 --> 00:36:44,405 experiment. So if we could be at resonance, 549 00:36:44,405 --> 00:36:48,443 the light bulb would be happy like a clam at high tide, 550 00:36:48,443 --> 00:36:53,079 it's exactly two hundred watts. It wants two point six amperes, 551 00:36:53,079 --> 00:36:56,742 that's what it wants. Then it is a 552 00:36:56,742 --> 00:37:01,075 two hundred watt light bulb. And so we want to make the 553 00:37:01,075 --> 00:37:04,284 light bulb happy. And how can we do that? 554 00:37:04,284 --> 00:37:08,457 Well, we can either increase L, or we can increase C. 555 00:37:08,457 --> 00:37:12,869 And I want to catch two birds with one stone during this 556 00:37:12,869 --> 00:37:16,239 demonstration, I'm going to teach you a way 557 00:37:16,239 --> 00:37:20,411 that you can increase L, something that we have never 558 00:37:20,411 --> 00:37:22,738 discussed before. I have here, 559 00:37:22,738 --> 00:37:26,75 ferromagnetic material, which has a value for kappa 560 00:37:26,75 --> 00:37:32,274 somewhere around ten, twelve -- let's say it is ten 561 00:37:32,274 --> 00:37:37,568 for now -- so that means that if I bring that iron core inside, 562 00:37:37,568 --> 00:37:41,068 that I could make this go up to one Henry. 563 00:37:41,068 --> 00:37:45,081 Because remember, the meaning of self-inductance 564 00:37:45,081 --> 00:37:49,349 is the flux divided by the current in the solenoid. 565 00:37:49,349 --> 00:37:53,704 And so if you bring in ferromagnetic material with a 566 00:37:53,704 --> 00:37:57,204 kappa M of ten, then the flux goes up by a 567 00:37:57,204 --> 00:38:01,591 factor of ten. The current remains the same, 568 00:38:01,591 --> 00:38:04,759 and therefore your self-inductance goes up by a 569 00:38:04,759 --> 00:38:07,788 factor of ten. So you see that you can make a 570 00:38:07,788 --> 00:38:11,369 variable self-inductance by bringing in ferromagnetic 571 00:38:11,369 --> 00:38:13,71 material, and by removing it again. 572 00:38:13,71 --> 00:38:17,084 And so the question, now, is, how high should L be 573 00:38:17,084 --> 00:38:20,045 to get on resonance? Well, that's very easy, 574 00:38:20,045 --> 00:38:23,625 because we know that one over the square root of L C, 575 00:38:23,625 --> 00:38:27,343 we want to make that three hundred and seventy [break], 576 00:38:27,343 --> 00:38:30,992 that's our driver. But we know that 577 00:38:30,992 --> 00:38:33,806 C is eight microfarads, that's a God-given, 578 00:38:33,806 --> 00:38:37,891 we're not going to change that. And so you can easily show now 579 00:38:37,891 --> 00:38:41,575 that if we could only make L oh point eight eight Henry, 580 00:38:41,575 --> 00:38:45,661 we should be back on resonance. Then we have three hundred and 581 00:38:45,661 --> 00:38:48,943 seventy seven radians per second as our resonance. 582 00:38:48,943 --> 00:38:52,694 And you shouldn't be surprised that omega L is then three 583 00:38:52,694 --> 00:38:56,512 hundred and thirty two ohms. Of course it is three hundred 584 00:38:56,512 --> 00:39:00,329 and thirty two ohms, because it's going to 585 00:39:00,329 --> 00:39:04,224 eat up that one over omega C, they're going to cancel each 586 00:39:04,224 --> 00:39:06,548 other. That's another way you could 587 00:39:06,548 --> 00:39:10,784 have calculated the value of L, by simply saying omega L has to 588 00:39:10,784 --> 00:39:14,884 be three hundred and thirty two. So there are various ways to 589 00:39:14,884 --> 00:39:18,505 get your resonance frequency. And so what I have done, 590 00:39:18,505 --> 00:39:21,922 if I make L oh point eight eight Henry, I shift the 591 00:39:21,922 --> 00:39:25,748 resonance frequency to this value, and my resonance curve 592 00:39:25,748 --> 00:39:29,438 now will look like this. But this height is going to be 593 00:39:29,438 --> 00:39:33,059 the same, because the height is only 594 00:39:33,059 --> 00:39:36,989 determined by V zero and R. And so what I'm going to do now 595 00:39:36,989 --> 00:39:40,918 is I'm going to turn on the instrument where it was before, 596 00:39:40,918 --> 00:39:43,967 [unintelligible] this was the resonance curve, 597 00:39:43,967 --> 00:39:47,558 you won't see any light. And I'm going to move the bar 598 00:39:47,558 --> 00:39:51,081 in, I probably don't have to move it in all they way, 599 00:39:51,081 --> 00:39:54,401 and then the light bulb will go through resonance, 600 00:39:54,401 --> 00:39:58,33 the system will go through resonance, you will get your two 601 00:39:58,33 --> 00:40:01,311 point six amperes, and the light bulb will be 602 00:40:01,311 --> 00:40:03,411 happy. And so let's do that now. 603 00:40:03,411 --> 00:40:07,137 All right? So here, system is now -- 604 00:40:07,137 --> 00:40:09,583 current is going. Sixty Hertz, 605 00:40:09,583 --> 00:40:13,882 hundred and ten volts. [unintelligible] square as we 606 00:40:13,882 --> 00:40:18,435 call that, and it's way off resonance, seven and a half 607 00:40:18,435 --> 00:40:22,145 watts, the light bulb doesn't even feel warm. 608 00:40:22,145 --> 00:40:27,119 And now I bring the -- this enormous piece of iron in there, 609 00:40:27,119 --> 00:40:31,672 and when I shove it in, the self-inductance will slowly 610 00:40:31,672 --> 00:40:36,478 go up, the resonance frequency will shirt, and the current 611 00:40:36,478 --> 00:40:40,525 through the system will increase. 612 00:40:40,525 --> 0. And there we go. 613 0. --> 00:40:41,759 614 00:40:41,759 --> 00:40:44,156 There it is. I'm on resonance now. 615 00:40:44,156 --> 00:40:46,48 So the light bulb is quite happy. 616 00:40:46,48 --> 00:40:50,183 I want to do one more thing. I'm going to double the 617 00:40:50,183 --> 00:40:53,233 capacitance. And what happens when I double 618 00:40:53,233 --> 00:40:55,412 the capacitance? The frequency, 619 00:40:55,412 --> 00:40:59,189 the resonance frequency goes further down that it was 620 00:40:59,189 --> 00:41:02,892 already, because remember, when you increase L or C, 621 00:41:02,892 --> 00:41:05,289 the resonance frequency goes down. 622 00:41:05,289 --> 00:41:08,049 So I'm going to double the capacitance. 623 00:41:08,049 --> 00:41:12,478 So that curve that you see on the 624 00:41:12,478 --> 00:41:16,582 left there, has shifted, now, even further to the left. 625 00:41:16,582 --> 00:41:19,318 And the light bulb is not very happy. 626 00:41:19,318 --> 00:41:23,421 The light bulb is -- the system is off resonance again. 627 00:41:23,421 --> 00:41:27,525 But if I increase C -- I doubled it -- I can decrease L 628 00:41:27,525 --> 00:41:31,097 by a factor of two, all I have to do is pull the 629 00:41:31,097 --> 00:41:33,984 iron slowly out. And then the resonance 630 00:41:33,984 --> 00:41:38,088 frequency comes back at three hundred and seventy seven 631 00:41:38,088 --> 00:41:40,216 radians per second. Watch it. 632 00:41:40,216 --> 00:41:44,32 Back on resonance. Amazing, isn't it? 633 00:41:44,32 --> 00:41:46,358 Physics works. So I showed you this 634 00:41:46,358 --> 00:41:48,997 demonstration, and I spent so much time on it 635 00:41:48,997 --> 00:41:52,655 because I wanted you to be able to appreciate -- I want you to 636 00:41:52,655 --> 00:41:54,933 be able to see through these equations. 637 00:41:54,933 --> 00:41:57,692 The answer, in physics, in general, lies in the 638 00:41:57,692 --> 00:42:01,469 equations, but it only works if you build up a certain amount of 639 00:42:01,469 --> 00:42:03,988 understanding. And that is not always easy, 640 00:42:03,988 --> 00:42:05,907 and the demonstrations help that. 641 00:42:05,907 --> 00:42:08,725 Seeing is believing. And you may have to go over 642 00:42:08,725 --> 00:42:11,543 this at home in order to digest it a little bit. 643 00:42:11,543 --> 00:42:16,101 I don't expect you to get all this just in one lecture, 644 00:42:16,101 --> 00:42:19,256 of course not. We have a new impedance here, 645 00:42:19,256 --> 00:42:23,367 we have the reactance here, we have the idea that omega L 646 00:42:23,367 --> 00:42:26,523 eats up omega C, then we get into resonance, 647 00:42:26,523 --> 00:42:29,458 all these phenomenon take time to digest. 648 00:42:29,458 --> 00:42:33,642 There are many key practical applications in LRC circuits, 649 00:42:33,642 --> 00:42:37,312 more than you may think. Your radio and your TV are 650 00:42:37,312 --> 00:42:40,101 systems that you, without realizing it, 651 00:42:40,101 --> 00:42:43,477 you tune them to resonance. There's an antenna, 652 00:42:43,477 --> 00:42:48,285 and that receives many, many stations at all different 653 00:42:48,285 --> 00:42:51,217 frequencies, and you are changing -- in general, 654 00:42:51,217 --> 00:42:54,213 you change the capacitor. And when you change the 655 00:42:54,213 --> 00:42:57,894 capacitor in the LRC circuit, you are changing the resonance 656 00:42:57,894 --> 00:43:01,701 frequency, and at that resonance frequency, the system is very 657 00:43:01,701 --> 00:43:05,008 sensitive, drives a very high current, and it selects, 658 00:43:05,008 --> 00:43:07,754 then, a particular station at that frequency. 659 00:43:07,754 --> 00:43:11,498 That's what you're doing with your TV, and that's what you're 660 00:43:11,498 --> 00:43:14,493 doing with your radio. You're changing a variable 661 00:43:14,493 --> 00:43:16,864 capacitor. Another application which is 662 00:43:16,864 --> 00:43:20,919 quite common are metal detectors. 663 00:43:20,919 --> 00:43:25,817 Metal detectors are resonant circuits, they are set at 664 00:43:25,817 --> 00:43:30,252 resonance, and then, when you bring metal nearby, 665 00:43:30,252 --> 00:43:35,149 you bring them off resonance, and then the alarm goes. 666 00:43:35,149 --> 00:43:39,862 In general, they have two coils -- this is one coil, 667 00:43:39,862 --> 00:43:43,743 and this is another coil. Inductance L one, 668 00:43:43,743 --> 00:43:46,792 resistor R one, capacitance C one, 669 00:43:46,792 --> 00:43:49,472 L two, R two, C 670 00:43:49,472 --> 00:43:51,704 two. And so there's a current, 671 00:43:51,704 --> 00:43:55,939 I one, running through this one, a current I two running 672 00:43:55,939 --> 00:43:59,25 through that one. To set up the differential 673 00:43:59,25 --> 00:44:02,33 equations for this system is not so easy. 674 00:44:02,33 --> 00:44:05,795 You get two differential equations, of course, 675 00:44:05,795 --> 00:44:09,183 one for this system, and one for this system. 676 00:44:09,183 --> 00:44:13,88 The problem -- but at the same time the trick -- is that there 677 00:44:13,88 --> 00:44:18,5 is a mutual inductance between the two, because when this one 678 00:44:18,5 --> 00:44:23,717 produces a current I two, there is a magnetic flux going 679 00:44:23,717 --> 00:44:26,691 through this one. That's why we call it mutual 680 00:44:26,691 --> 00:44:30,789 inductance, which we have never really dealt with in any detail 681 00:44:30,789 --> 00:44:32,706 in eight oh two, and we won't. 682 00:44:32,706 --> 00:44:36,871 But I want to mention that this differential equation here is an 683 00:44:36,871 --> 00:44:38,92 equation in I one, L one, R one, 684 00:44:38,92 --> 00:44:42,291 C one, but also contains a term N times D I two D T. 685 00:44:42,291 --> 00:44:45,86 And this differential equation is an equation in L two, 686 00:44:45,86 --> 00:44:48,438 R two, C two, I two, but also contains a 687 00:44:48,438 --> 00:44:53,925 term, N times D I one D T. And so now you get two coupled 688 00:44:53,925 --> 00:44:56,945 differential equations. It's not just one in I one, 689 00:44:56,945 --> 00:44:59,725 and one in I two. It is one in I one and I two, 690 00:44:59,725 --> 00:45:02,141 and it is another one in I one and I two. 691 00:45:02,141 --> 00:45:05,646 And that is a pain in the neck, and it's not easy to solve. 692 00:45:05,646 --> 00:45:08,546 But there are mathematicians who can do all that, 693 00:45:08,546 --> 00:45:10,842 and they solved that for us physicists. 694 00:45:10,842 --> 00:45:13,44 In any case, what is important that clearly, 695 00:45:13,44 --> 00:45:16,219 since you have two coils, you get two resonance 696 00:45:16,219 --> 00:45:19,36 frequencies, not just one, but you have two resonance 697 00:45:19,36 --> 00:45:21,596 frequencies. And at least one of those 698 00:45:21,596 --> 00:45:25,024 resonance frequencies depends very 699 00:45:25,024 --> 00:45:27,245 strongly on the mutual inductance. 700 00:45:27,245 --> 00:45:30,745 And so the system is tuned to set at one of these two 701 00:45:30,745 --> 00:45:33,909 resonance frequencies. And now someone brings in 702 00:45:33,909 --> 00:45:36,197 metal. You bring in a chu- chunk of 703 00:45:36,197 --> 00:45:38,486 metal. And in this metal the metal 704 00:45:38,486 --> 00:45:41,716 will experience the magnetic field, flux changes, 705 00:45:41,716 --> 00:45:44,072 A C of course, always, in the metal, 706 00:45:44,072 --> 00:45:46,697 it will start building up eddy currents. 707 00:45:46,697 --> 00:45:50,534 And so the eddy currents will change the magnetic coupling 708 00:45:50,534 --> 00:45:55,214 between these two loops. And so M changes. 709 00:45:55,214 --> 00:45:58,149 And when M changes, you go off resonance, 710 00:45:58,149 --> 00:46:01,671 and bingo, your alarm goes off. Clearly, and very 711 00:46:01,671 --> 00:46:04,752 unfortunately, this never works for plastic 712 00:46:04,752 --> 00:46:06,954 bombs. You must have conducting 713 00:46:06,954 --> 00:46:10,549 material that you bring in, so it must be a metal. 714 00:46:10,549 --> 00:46:14,731 These systems have very high Q, very high Q means they are 715 00:46:14,731 --> 00:46:17,96 extremely sensitive. If you have very high Q, 716 00:46:17,96 --> 00:46:22,435 that curve is extremely narrow, and so the slightest change in 717 00:46:22,435 --> 00:46:26,398 M, you go off resonance, and you get 718 00:46:26,398 --> 00:46:29,116 your alarm. And so they come in various 719 00:46:29,116 --> 00:46:31,548 forms, and in various applications. 720 00:46:31,548 --> 00:46:33,909 At the airport, it is very simple. 721 00:46:33,909 --> 00:46:36,198 You walk through these two coils. 722 00:46:36,198 --> 00:46:40,418 You may not have realized that, but you simply walk through. 723 00:46:40,418 --> 00:46:43,637 You walk through, like, there's one coil here, 724 00:46:43,637 --> 00:46:46,57 one coil there, and that's how they detect 725 00:46:46,57 --> 00:46:49,288 whether you have any, uh, metal on you. 726 00:46:49,288 --> 00:46:53,437 The one -- the metal detectors that are used to search your 727 00:46:53,437 --> 00:46:57,73 body, including the ones that are used to 728 00:46:57,73 --> 00:47:01,434 search the ground for coins -- uh, when I was a kid, 729 00:47:01,434 --> 00:47:04,629 I went often to the beach in the Netherlands, 730 00:47:04,629 --> 00:47:08,478 and there were often people who were having this metal 731 00:47:08,478 --> 00:47:11,238 detectors, they were looking for coins. 732 00:47:11,238 --> 00:47:14,87 And those metal detectors, basically the same idea, 733 00:47:14,87 --> 00:47:18,792 always two coils -- this is this coil, and this is this 734 00:47:18,792 --> 00:47:22,859 coil, and when metal comes nearby, you have a system that 735 00:47:22,859 --> 00:47:25,909 goes off resonance, you get a difference -- 736 00:47:25,909 --> 00:47:29,541 magnetic coupling between the two 737 00:47:29,541 --> 00:47:32,186 coils, and the system goes into alarm. 738 00:47:32,186 --> 00:47:36,405 And I'd like to demonstrate that, for wh- for which I need a 739 00:47:36,405 --> 00:47:38,693 student. Any student who wants to 740 00:47:38,693 --> 00:47:41,839 volunteer, I have here a metal detector, and, 741 00:47:41,839 --> 00:47:44,842 thank goodness, we don't have to go through 742 00:47:44,842 --> 00:47:47,774 metal detectors yet, at MIT -- you want to 743 00:47:47,774 --> 00:47:49,776 volunteer? Boy, you're brave, 744 00:47:49,776 --> 00:47:53,923 you're all the way at the end of the -- can't wait what you 745 00:47:53,923 --> 00:47:57,498 have in store for us. Oh, you look like you're made 746 00:47:57,498 --> 00:48:00,287 of steel. OK, you -- I hope you left all 747 00:48:00,287 --> 00:48:04,722 your secret weapons up there, 748 00:48:04,722 --> 00:48:07,48 did you? [unintelligible]. 749 00:48:07,48 --> 00:48:14,43 So this is a -- this is a metal detector, and if you opened this 750 00:48:14,43 --> 00:48:20,056 up here, you would see two coils, one in the center, 751 00:48:20,056 --> 00:48:23,255 and one near the edge. [beep]. 752 00:48:23,255 --> 00:48:28,991 And we have to -- I have to calibrate it first -- OK. 753 00:48:28,991 --> 00:48:33,183 Oh, you left your cigarettes out there? 754 00:48:33,183 --> 00:48:41,296 They -- they won't record it. Student: Definitely not 755 00:48:41,296 --> 00:48:44,647 working. No, that's probably not 756 00:48:44,647 --> 00:48:47,674 working. Well, just a second, 757 00:48:47,674 --> 0. may have to recalibrate this. 758 0. --> 00:48:50,918 759 00:48:50,918 --> 00:48:55,566 Pretty quiet. Doesn't look like you have any 760 00:48:55,566 --> 00:49:00,215 metal there. I'm pr- fairly sure I turned it 761 00:49:00,215 --> 00:49:03,026 on. They don't always work, 762 00:49:03,026 --> 00:49:06,053 do they? Marcos, you have any 763 00:49:06,053 --> 00:49:11,701 [unintelligible]? [beep] Uh-uh. 764 00:49:11,701 --> 00:49:16,182 It's working. No, it's not working very well. 765 00:49:16,182 --> 00:49:19,645 It's working well here. [laughter]. 766 00:49:19,645 --> 00:49:25,144 Wonder what you got there. Are you metal -- is this not 767 00:49:25,144 --> 00:49:29,115 made of metal, all these things in here? 768 00:49:29,115 --> 00:49:35,225 Student: I have lots of metal on me, and [unintelligible] I'm 769 00:49:35,225 --> 00:49:40,113 surprised it's only going off at the belt buckle, 770 00:49:40,113 --> 00:49:44,289 it's weird. Oh, you may be a weird person, 771 00:49:44,289 --> 00:49:49,38 that may be [laughter]. [beeping]. 772 00:49:49,38 --> 00:49:53,67 Can you turn around? Oh, that's your watch. 773 00:49:53,67 --> 00:49:55,815 [beeping]. [laughter]. 774 00:49:55,815 --> 00:50:01,331 You'd better turn around again. Ah, there are the keys. 775 00:50:01,331 --> 00:50:06,234 You're not made of steel, but you're a great guy, 776 00:50:06,234 --> 00:50:09,4 thank you very much. [applause]. 777 00:50:09,4 --> 00:50:11,954 Today is Wednesday, right? 778 00:50:11,954 --> 50:17 See you Friday.