WEBVTT 00:00:00.000 --> 00:00:01.944 align:middle line:90% [SQUEAKING] 00:00:01.944 --> 00:00:02.916 align:middle line:90% [RUSTLING] 00:00:02.916 --> 00:00:04.860 align:middle line:90% [CLICKING] 00:00:04.860 --> 00:00:09.250 align:middle line:90% 00:00:09.250 --> 00:00:12.040 align:middle line:84% DAVID PERREAULT: OK, why don't we get started? 00:00:12.040 --> 00:00:17.320 align:middle line:84% So the last few classes, we were talking about soft switch power 00:00:17.320 --> 00:00:18.610 align:middle line:90% converters. 00:00:18.610 --> 00:00:21.910 align:middle line:84% And that's-- coverage of soft switching specifically is 00:00:21.910 --> 00:00:25.450 align:middle line:84% included in Principles of Power Electronics Chapter 24. 00:00:25.450 --> 00:00:28.060 align:middle line:84% What I'd like to do starting today 00:00:28.060 --> 00:00:31.630 align:middle line:84% is talk about a class of power converters 00:00:31.630 --> 00:00:36.250 align:middle line:84% that are often chosen because they can be soft switched, 00:00:36.250 --> 00:00:39.260 align:middle line:84% although it's not required that they be soft switched, 00:00:39.260 --> 00:00:42.760 align:middle line:84% and this is the class of power converter called 00:00:42.760 --> 00:00:44.050 align:middle line:90% resonant converters. 00:00:44.050 --> 00:00:46.720 align:middle line:84% And this is described in detail in Principles 00:00:46.720 --> 00:00:47.600 align:middle line:90% of Power Electronics. 00:00:47.600 --> 00:00:49.220 align:middle line:90% Chapter 10. 00:00:49.220 --> 00:00:51.470 align:middle line:84% And the idea of a resonant power converter 00:00:51.470 --> 00:00:57.440 align:middle line:84% is one in which we modulate our either power or current 00:00:57.440 --> 00:01:00.890 align:middle line:84% or voltage or process energy through high-frequency 00:01:00.890 --> 00:01:02.720 align:middle line:90% sinusoidal waveforms. 00:01:02.720 --> 00:01:04.280 align:middle line:84% So in a lot of the converters we do, 00:01:04.280 --> 00:01:07.280 align:middle line:84% we have some of square-ish waveform, 00:01:07.280 --> 00:01:09.110 align:middle line:90% a PWN kind of waveform. 00:01:09.110 --> 00:01:11.000 align:middle line:84% In a resonant converter, we're generally 00:01:11.000 --> 00:01:13.110 align:middle line:90% going to use sinusoids. 00:01:13.110 --> 00:01:18.110 align:middle line:84% And that's valuable in a whole bunch of applications. 00:01:18.110 --> 00:01:21.920 align:middle line:84% So, for example, if you wanted to do induction heating 00:01:21.920 --> 00:01:26.780 align:middle line:84% or wireless power transfer, or, in some cases, 00:01:26.780 --> 00:01:29.150 align:middle line:84% some kinds of ballasts for fluorescent lights, 00:01:29.150 --> 00:01:31.040 align:middle line:84% typically what one will do is generate 00:01:31.040 --> 00:01:34.970 align:middle line:84% some high-frequency, sinusoidal kinds of waveforms, 00:01:34.970 --> 00:01:37.740 align:middle line:84% and resonant converters are ideal for doing that. 00:01:37.740 --> 00:01:41.375 align:middle line:84% You can also build resonant power converters 00:01:41.375 --> 00:01:43.250 align:middle line:84% that are DC-to-DC converters, where you might 00:01:43.250 --> 00:01:47.510 align:middle line:84% go from DC into high-frequency, sinusoidal AC and then 00:01:47.510 --> 00:01:52.970 align:middle line:84% back to DC, would give you a resonant DC-to-DC converter. 00:01:52.970 --> 00:01:55.550 align:middle line:84% And there's a lot of value in this 00:01:55.550 --> 00:01:58.850 align:middle line:84% because we can build circuits that are tuned 00:01:58.850 --> 00:02:04.550 align:middle line:84% and naturally give us sinusoidal voltages and/or currents. 00:02:04.550 --> 00:02:09.830 align:middle line:84% And also, we can also typically use the frequency selectivity 00:02:09.830 --> 00:02:13.445 align:middle line:84% of resonant networks to help us modulate voltage or power. 00:02:13.445 --> 00:02:16.065 align:middle line:90% 00:02:16.065 --> 00:02:20.007 align:middle line:84% So there's a lot of different applications 00:02:20.007 --> 00:02:21.590 align:middle line:84% for resonant converters, and they have 00:02:21.590 --> 00:02:22.980 align:middle line:90% very different characteristics. 00:02:22.980 --> 00:02:25.410 align:middle line:84% In some cases, you need to run at a fixed frequency. 00:02:25.410 --> 00:02:27.410 align:middle line:84% In some cases, you need to run at variable-- 00:02:27.410 --> 00:02:31.130 align:middle line:84% or you can run at variable frequencies. 00:02:31.130 --> 00:02:33.720 align:middle line:84% It depends what the load characteristics will do. 00:02:33.720 --> 00:02:36.800 align:middle line:84% Will it be fixed in amplitude, or do you 00:02:36.800 --> 00:02:37.980 align:middle line:90% have to deliver fixed power? 00:02:37.980 --> 00:02:41.040 align:middle line:84% So there's a whole wide variety of these kind of converters. 00:02:41.040 --> 00:02:46.690 align:middle line:84% But what we'd like to do to start is lay down the basics. 00:02:46.690 --> 00:02:48.790 align:middle line:84% And on the way to doing that, I just 00:02:48.790 --> 00:02:53.230 align:middle line:84% like to remind people of some of the key characteristics 00:02:53.230 --> 00:02:55.570 align:middle line:84% of resonant networks today, particularly 00:02:55.570 --> 00:02:57.402 align:middle line:90% second order resonant networks. 00:02:57.402 --> 00:02:59.860 align:middle line:84% And so I'd like to spend a little time on this because this 00:02:59.860 --> 00:03:03.340 align:middle line:84% is going to set us up well for thinking 00:03:03.340 --> 00:03:05.650 align:middle line:84% about how we build power converters that naturally 00:03:05.650 --> 00:03:07.420 align:middle line:90% use these networks. 00:03:07.420 --> 00:03:09.520 align:middle line:84% Now, we can use different kinds of networks, 00:03:09.520 --> 00:03:13.240 align:middle line:84% but let's consider a series RLC network. 00:03:13.240 --> 00:03:20.620 align:middle line:84% So perhaps I have a voltage source vs. cosine of omega t. 00:03:20.620 --> 00:03:28.170 align:middle line:84% And we can run this into a series LRC network comprising 00:03:28.170 --> 00:03:34.080 align:middle line:84% a resistor, an inductor, and a capacitor, 00:03:34.080 --> 00:03:39.810 align:middle line:90% where I may have IL and VC. 00:03:39.810 --> 00:03:43.568 align:middle line:84% And typically, we're going to think of the resistor 00:03:43.568 --> 00:03:46.110 align:middle line:84% as the load, where we're going to want to put the load power. 00:03:46.110 --> 00:03:50.470 align:middle line:84% But let's just consider this network generally. 00:03:50.470 --> 00:03:57.940 align:middle line:84% So I might characterize this network by its input admittance. 00:03:57.940 --> 00:04:06.220 align:middle line:84% So I might look at the input admittance 00:04:06.220 --> 00:04:10.060 align:middle line:84% looking into this network as seen by the driving source. 00:04:10.060 --> 00:04:13.070 align:middle line:90% So what would be the admittance? 00:04:13.070 --> 00:04:16.930 align:middle line:84% I could write the admittance as being equal to 1 00:04:16.930 --> 00:04:25.900 align:middle line:84% over the impedance, which would be R plus k omega L plus 1 00:04:25.900 --> 00:04:31.990 align:middle line:84% over j omega C. Or I could rewrite this as 1 over R 00:04:31.990 --> 00:04:37.900 align:middle line:84% plus j times omega L minus 1 over omega C. 00:04:37.900 --> 00:04:42.250 align:middle line:84% And so we could write the magnitude of the admittance 00:04:42.250 --> 00:04:51.340 align:middle line:84% as follows, as being equal to 1 over the square root of R 00:04:51.340 --> 00:04:58.820 align:middle line:84% squared plus omega L minus 1 over omega C quantity squared. 00:04:58.820 --> 00:05:04.330 align:middle line:84% And the angle of the admittance we could see 00:05:04.330 --> 00:05:11.830 align:middle line:84% is minus the arctangent of omega L 00:05:11.830 --> 00:05:18.180 align:middle line:84% minus 1 over omega C divided by R. 00:05:18.180 --> 00:05:19.145 align:middle line:90% So all right. 00:05:19.145 --> 00:05:20.520 align:middle line:84% We have expressions for what does 00:05:20.520 --> 00:05:28.070 align:middle line:84% this admittance looks like at a value of S equals j omega. 00:05:28.070 --> 00:05:32.720 align:middle line:84% What does that look like in the frequency domain? 00:05:32.720 --> 00:05:37.200 align:middle line:84% Well, if I thought about this network, I could plot-- 00:05:37.200 --> 00:05:39.870 align:middle line:84% let's just think about the amplitude. 00:05:39.870 --> 00:05:46.610 align:middle line:84% I could think of the magnitude of YN. 00:05:46.610 --> 00:05:52.220 align:middle line:84% And I'm going to plot this on a log scale versus frequency-- 00:05:52.220 --> 00:05:53.430 align:middle line:90% angular frequency. 00:05:53.430 --> 00:05:57.620 align:middle line:84% Again, I'll plot it on a log scale. 00:05:57.620 --> 00:06:01.780 align:middle line:84% At really low frequencies, the capacitor 00:06:01.780 --> 00:06:03.110 align:middle line:90% dominates the impedance. 00:06:03.110 --> 00:06:06.070 align:middle line:90% It is the lowest admittance. 00:06:06.070 --> 00:06:08.830 align:middle line:84% So at really low frequencies, I might 00:06:08.830 --> 00:06:19.350 align:middle line:84% have an asymptote that goes as omega C. At really 00:06:19.350 --> 00:06:23.130 align:middle line:84% high frequencies, the inductor becomes a high impedance, 00:06:23.130 --> 00:06:28.120 align:middle line:84% so then it limits the admittance of the resonant network. 00:06:28.120 --> 00:06:37.680 align:middle line:84% So at high frequencies we might have something that goes as 1 00:06:37.680 --> 00:06:41.070 align:middle line:90% over omega L. 00:06:41.070 --> 00:06:44.640 align:middle line:84% Now, there's going to be some place where 00:06:44.640 --> 00:06:50.040 align:middle line:90% these two asymptotes intersect. 00:06:50.040 --> 00:06:51.610 align:middle line:90% Where do they intersect? 00:06:51.610 --> 00:06:57.480 align:middle line:84% Well, they intersect right where the inductive impedance 00:06:57.480 --> 00:06:59.735 align:middle line:84% and the capacitive impedance cancel, 00:06:59.735 --> 00:07:03.900 align:middle line:90% right at this magic point. 00:07:03.900 --> 00:07:06.810 align:middle line:84% If I look here, somewhere here, there's 00:07:06.810 --> 00:07:09.570 align:middle line:84% going to be some place where these two terms cancel. 00:07:09.570 --> 00:07:12.690 align:middle line:84% And where these two terms cancel the admittance 00:07:12.690 --> 00:07:14.500 align:middle line:90% goes to its maximum value. 00:07:14.500 --> 00:07:18.300 align:middle line:84% And anywhere away from that, the admittance amplitude 00:07:18.300 --> 00:07:21.690 align:middle line:84% must fall because this is a nonzero term in the denominator. 00:07:21.690 --> 00:07:24.480 align:middle line:84% And if we work out what that is-- very easy 00:07:24.480 --> 00:07:30.390 align:middle line:84% to show that that intercept point occurs at omega 00:07:30.390 --> 00:07:33.660 align:middle line:90% is equal to a value of omega 0. 00:07:33.660 --> 00:07:38.520 align:middle line:84% Omega 0 is equal to 1 over the square root of LC. 00:07:38.520 --> 00:07:40.940 align:middle line:90% All I got to do is set that-- 00:07:40.940 --> 00:07:42.610 align:middle line:84% I can set those two equal to each other 00:07:42.610 --> 00:07:45.400 align:middle line:84% and find that value of omega equals omega 0. 00:07:45.400 --> 00:07:49.256 align:middle line:84% And this is called the undamped natural frequency. 00:07:49.256 --> 00:07:57.890 align:middle line:90% 00:07:57.890 --> 00:07:59.630 align:middle line:84% And I'm going very fast because I 00:07:59.630 --> 00:08:02.887 align:middle line:84% think this should be familiar territory to everybody, 00:08:02.887 --> 00:08:05.095 align:middle line:84% but I want to make sure we get it fixed in our heads. 00:08:05.095 --> 00:08:07.770 align:middle line:90% 00:08:07.770 --> 00:08:14.860 align:middle line:84% What happens to-- at this intersection point, 00:08:14.860 --> 00:08:23.090 align:middle line:84% where in this axis is this intersection? 00:08:23.090 --> 00:08:29.270 align:middle line:84% Well, if I said, what's omega 0C, 00:08:29.270 --> 00:08:33.289 align:middle line:84% which also has to equal to 1 over omega 0L? 00:08:33.289 --> 00:08:36.716 align:middle line:90% 00:08:36.716 --> 00:08:38.840 align:middle line:84% That's going to be equal to-- omega 0C, 00:08:38.840 --> 00:08:42.320 align:middle line:84% if I plug in this value, is going to be equal to the square 00:08:42.320 --> 00:08:46.940 align:middle line:90% root of C over L. 00:08:46.940 --> 00:08:48.950 align:middle line:84% Or this is sometimes written where 00:08:48.950 --> 00:08:54.920 align:middle line:84% we define Z0 equal to the square root of L over C. 00:08:54.920 --> 00:08:57.830 align:middle line:84% And that's known as the so-called characteristic 00:08:57.830 --> 00:09:08.420 align:middle line:90% impedance of the network. 00:09:08.420 --> 00:09:10.830 align:middle line:84% So this intersection point I might 00:09:10.830 --> 00:09:17.860 align:middle line:84% write as the square root of C over L, or equal to 1 over Z0. 00:09:17.860 --> 00:09:19.270 align:middle line:90% And it's 1 over Z0. 00:09:19.270 --> 00:09:23.370 align:middle line:84% I might also call this Y0, or the characteristic admittance. 00:09:23.370 --> 00:09:28.940 align:middle line:84% So if you give me a second-order circuit like this, 00:09:28.940 --> 00:09:31.440 align:middle line:84% especially if I'm thinking about it in the frequency domain, 00:09:31.440 --> 00:09:33.420 align:middle line:84% but actually, in general, the first two 00:09:33.420 --> 00:09:35.670 align:middle line:84% things I'm going to figure out about this circuit are, 00:09:35.670 --> 00:09:37.840 align:middle line:84% what are the characteristic impedance? 00:09:37.840 --> 00:09:42.700 align:middle line:84% And what are the undamped natural frequency? 00:09:42.700 --> 00:09:44.248 align:middle line:84% So almost the first thing I'm going 00:09:44.248 --> 00:09:45.790 align:middle line:84% to do when I think about this network 00:09:45.790 --> 00:09:50.064 align:middle line:84% is calculate these two variables. 00:09:50.064 --> 00:09:52.760 align:middle line:84% The undamped natural frequency is essentially 00:09:52.760 --> 00:09:55.762 align:middle line:84% when those two elements are in resonance. 00:09:55.762 --> 00:09:57.220 align:middle line:84% Their impedance cancels each other. 00:09:57.220 --> 00:10:00.160 align:middle line:84% And we can be careful about how we define what resonance means. 00:10:00.160 --> 00:10:07.610 align:middle line:84% But we can consider it where the reactive components cancel, 00:10:07.610 --> 00:10:12.370 align:middle line:84% and I see a resistive input admittance, for example. 00:10:12.370 --> 00:10:16.870 align:middle line:84% Why do I want to think about the characteristic impedance? 00:10:16.870 --> 00:10:19.630 align:middle line:90% Well, look at it this way. 00:10:19.630 --> 00:10:21.970 align:middle line:84% One way you might think about this 00:10:21.970 --> 00:10:31.000 align:middle line:84% is that the capacitor voltage is equal to the inductor 00:10:31.000 --> 00:10:36.430 align:middle line:90% current through the-- 00:10:36.430 --> 00:10:38.680 align:middle line:84% times the impedance of the capacitor. 00:10:38.680 --> 00:10:46.790 align:middle line:84% So we might look at that and say, OK, 00:10:46.790 --> 00:10:51.710 align:middle line:84% VC, the capacitor voltage, if I'm 00:10:51.710 --> 00:10:56.980 align:middle line:84% operating at the natural frequency omega 0-- 00:10:56.980 --> 00:10:59.410 align:middle line:84% and let me just look at its magnitude-- 00:10:59.410 --> 00:11:07.900 align:middle line:84% is going to be equal to the magnitude of I sub L times 00:11:07.900 --> 00:11:16.070 align:middle line:84% the impedance, which is 1 over omega 0C at that frequency. 00:11:16.070 --> 00:11:22.220 align:middle line:84% So if I just rearrange this, I could write the magnitude of V 00:11:22.220 --> 00:11:30.560 align:middle line:84% sub C divided by the magnitude of I sub L evaluated at omega 00:11:30.560 --> 00:11:35.300 align:middle line:84% equals omega 0 only, I would see that this is going to be equal 00:11:35.300 --> 00:11:40.760 align:middle line:84% to 1 over omega 0C, which is equal to Z0. 00:11:40.760 --> 00:11:44.720 align:middle line:84% So we might think of the characteristic impedance 00:11:44.720 --> 00:11:47.270 align:middle line:84% as telling us, do I have a circuit that 00:11:47.270 --> 00:11:50.150 align:middle line:90% operates at high current-- 00:11:50.150 --> 00:11:53.120 align:middle line:84% or high voltage and low current, so its high characteristic 00:11:53.120 --> 00:11:56.240 align:middle line:84% impedance, or low voltage and high current, 00:11:56.240 --> 00:11:58.200 align:middle line:84% and that's low characteristic impedance? 00:11:58.200 --> 00:12:02.030 align:middle line:84% So at resonance, how do the capacitor 00:12:02.030 --> 00:12:04.670 align:middle line:84% voltage and inductor current relate 00:12:04.670 --> 00:12:05.910 align:middle line:90% in terms of their amplitudes? 00:12:05.910 --> 00:12:08.540 align:middle line:84% Well, they relate precisely by the characteristic impedance 00:12:08.540 --> 00:12:09.300 align:middle line:90% of the circuit. 00:12:09.300 --> 00:12:13.220 align:middle line:84% Now, all of that is true only at this one magic frequency. 00:12:13.220 --> 00:12:15.350 align:middle line:84% But we often think about operation 00:12:15.350 --> 00:12:18.970 align:middle line:84% in the vicinity of that frequency. 00:12:18.970 --> 00:12:20.430 align:middle line:90% Any questions about that? 00:12:20.430 --> 00:12:26.390 align:middle line:90% 00:12:26.390 --> 00:12:26.890 align:middle line:90% OK. 00:12:26.890 --> 00:12:31.140 align:middle line:90% 00:12:31.140 --> 00:12:35.410 align:middle line:84% How might this circuit behave under different situations? 00:12:35.410 --> 00:12:41.270 align:middle line:84% Well, I'd kind of maybe think about different regimes 00:12:41.270 --> 00:12:43.430 align:middle line:90% of operation. 00:12:43.430 --> 00:12:47.720 align:middle line:84% So one thing I might have, if I looked at this, 00:12:47.720 --> 00:12:48.916 align:middle line:90% is the following. 00:12:48.916 --> 00:12:52.050 align:middle line:90% 00:12:52.050 --> 00:12:53.645 align:middle line:90% Actually, let me do it this way. 00:12:53.645 --> 00:13:02.450 align:middle line:90% 00:13:02.450 --> 00:13:05.750 align:middle line:90% So this is omega on a log scale. 00:13:05.750 --> 00:13:10.410 align:middle line:84% This is the magnitude of YN on a log scale. 00:13:10.410 --> 00:13:12.736 align:middle line:90% And here's my asymptotes. 00:13:12.736 --> 00:13:16.520 align:middle line:90% 00:13:16.520 --> 00:13:18.980 align:middle line:90% This is omega 0. 00:13:18.980 --> 00:13:20.120 align:middle line:90% This is 1 over Z0. 00:13:20.120 --> 00:13:24.516 align:middle line:90% 00:13:24.516 --> 00:13:28.260 align:middle line:84% And what is this going to look like across frequency? 00:13:28.260 --> 00:13:30.720 align:middle line:84% Well, at really low frequencies and high frequencies 00:13:30.720 --> 00:13:32.100 align:middle line:90% it follows the asymptote. 00:13:32.100 --> 00:13:35.060 align:middle line:84% But in this middle region, we ultimately 00:13:35.060 --> 00:13:41.470 align:middle line:84% get the reactive components canceling. 00:13:41.470 --> 00:13:44.130 align:middle line:84% So right at the resonant frequency, 00:13:44.130 --> 00:13:48.190 align:middle line:84% it's going to look like 1 over R. 00:13:48.190 --> 00:13:50.220 align:middle line:84% So it kind of-- what it does kind of 00:13:50.220 --> 00:13:52.930 align:middle line:90% depends on how big 1 over R is. 00:13:52.930 --> 00:13:56.820 align:middle line:90% For a really big resistance R-- 00:13:56.820 --> 00:14:00.000 align:middle line:90% so I have R big. 00:14:00.000 --> 00:14:04.220 align:middle line:84% 1 over R is going to be down here. 00:14:04.220 --> 00:14:07.580 align:middle line:90% And I might get an asymptote. 00:14:07.580 --> 00:14:10.370 align:middle line:84% I get a third asymptote here, and I 00:14:10.370 --> 00:14:11.920 align:middle line:90% get something that does this. 00:14:11.920 --> 00:14:15.880 align:middle line:90% 00:14:15.880 --> 00:14:23.950 align:middle line:84% So this is what YN would look like for large R. 00:14:23.950 --> 00:14:28.660 align:middle line:84% Another case I might run into is, well, what if 1 over R 00:14:28.660 --> 00:14:31.930 align:middle line:90% is sort of in the vicinity of-- 00:14:31.930 --> 00:14:35.580 align:middle line:84% or R is sort of in the vicinity of the characteristic impedance? 00:14:35.580 --> 00:14:39.520 align:middle line:90% So maybe 1 over R is here. 00:14:39.520 --> 00:14:40.510 align:middle line:90% What would I see then? 00:14:40.510 --> 00:14:43.420 align:middle line:84% Well, I'd see something that just-- 00:14:43.420 --> 00:14:47.480 align:middle line:84% it's going to hit here at its maximum admittance. 00:14:47.480 --> 00:14:51.100 align:middle line:84% And so maybe its characteristic versus frequency 00:14:51.100 --> 00:14:53.090 align:middle line:84% is going to do something like this. 00:14:53.090 --> 00:14:58.790 align:middle line:84% It's going to peak up a little bit and then come down. 00:14:58.790 --> 00:15:07.240 align:middle line:84% So that's maybe some of medium value of resistance. 00:15:07.240 --> 00:15:11.650 align:middle line:90% And this kind of case-- 00:15:11.650 --> 00:15:14.200 align:middle line:84% we might think of this as what we're going to say 00:15:14.200 --> 00:15:27.820 align:middle line:84% is a medium Q. Whereas, this guy we might call very low Q. 00:15:27.820 --> 00:15:30.290 align:middle line:84% And we'll get back to what we mean by Q. 00:15:30.290 --> 00:15:32.470 align:middle line:84% And then the last case, we might run into, 00:15:32.470 --> 00:15:37.570 align:middle line:84% which is kind of an interesting one, is-- 00:15:37.570 --> 00:15:41.200 align:middle line:84% actually, maybe I should use this color. 00:15:41.200 --> 00:15:43.350 align:middle line:90% Suppose 1 over R is up here. 00:15:43.350 --> 00:15:49.150 align:middle line:90% 00:15:49.150 --> 00:15:51.640 align:middle line:90% So R is really small. 00:15:51.640 --> 00:15:55.690 align:middle line:84% Then I'm going to have a maximum case of my admittance 00:15:55.690 --> 00:15:58.960 align:middle line:84% up here well above the intersection points of these two 00:15:58.960 --> 00:15:59.775 align:middle line:90% asymptotes. 00:15:59.775 --> 00:16:01.900 align:middle line:84% And so I'm going to get some characteristics that's 00:16:01.900 --> 00:16:04.725 align:middle line:84% going to do something, at least qualitatively, like this. 00:16:04.725 --> 00:16:12.370 align:middle line:90% 00:16:12.370 --> 00:16:19.465 align:middle line:84% And this we might think of as the high Q case. 00:16:19.465 --> 00:16:20.965 align:middle line:84% Actually, let me put that over here. 00:16:20.965 --> 00:16:23.600 align:middle line:90% 00:16:23.600 --> 00:16:25.520 align:middle line:84% I'm being very qualitative about this, 00:16:25.520 --> 00:16:27.790 align:middle line:84% but this would be the high Q case. 00:16:27.790 --> 00:16:34.630 align:middle line:90% 00:16:34.630 --> 00:16:46.680 align:middle line:84% Now, really, what we're saying is, how does R compare to Z0? 00:16:46.680 --> 00:16:52.930 align:middle line:84% So in this case, this is R much bigger than Z0. 00:16:52.930 --> 00:16:54.840 align:middle line:90% This is R on the order of Z0. 00:16:54.840 --> 00:16:57.150 align:middle line:90% And this is R much less than Z0. 00:16:57.150 --> 00:17:01.530 align:middle line:84% So by knowing how your resistor compares 00:17:01.530 --> 00:17:03.450 align:middle line:84% to your characteristic impedance, 00:17:03.450 --> 00:17:06.839 align:middle line:84% you can tell what the frequency domain admittance of this thing 00:17:06.839 --> 00:17:09.040 align:middle line:84% is going to look like, at least in its amplitude. 00:17:09.040 --> 00:17:10.980 align:middle line:84% And, of course, the phase response 00:17:10.980 --> 00:17:16.040 align:middle line:84% will be very slow or very sharp correspondingly. 00:17:16.040 --> 00:17:18.024 align:middle line:90% Any questions about that? 00:17:18.024 --> 00:17:23.579 align:middle line:90% 00:17:23.579 --> 00:17:26.460 align:middle line:84% If I were to look at the frequency domain, 00:17:26.460 --> 00:17:29.250 align:middle line:84% if I were to look at, say, for example, 00:17:29.250 --> 00:17:32.820 align:middle line:84% where the poles of the admittance are-- 00:17:32.820 --> 00:17:38.370 align:middle line:84% so I plot YN of S, and then I find out where does that YN of S 00:17:38.370 --> 00:17:42.220 align:middle line:84% blow up-- at what complex frequencies it blows up-- 00:17:42.220 --> 00:17:45.257 align:middle line:84% what you would see is the following. 00:17:45.257 --> 00:17:46.590 align:middle line:90% And I'm not going to prove this. 00:17:46.590 --> 00:17:48.750 align:middle line:90% So here's the imaginary axis. 00:17:48.750 --> 00:17:50.560 align:middle line:90% Here's the real axis. 00:17:50.560 --> 00:17:58.200 align:middle line:84% And I'm plotting the poles of YN of S. 00:17:58.200 --> 00:18:01.700 align:middle line:84% What we would find is that in this green case, 00:18:01.700 --> 00:18:09.380 align:middle line:84% we would have overdamped case, particularly, actually, 00:18:09.380 --> 00:18:14.910 align:middle line:84% this is when R over Z0 is less than a half. 00:18:14.910 --> 00:18:17.650 align:middle line:90% It will be overdamped. 00:18:17.650 --> 00:18:19.990 align:middle line:84% In the intermediate case, what we're going to see 00:18:19.990 --> 00:18:22.570 align:middle line:84% is something like where the poles are underdamped, 00:18:22.570 --> 00:18:25.990 align:middle line:84% but maybe they're out here like this, something like that. 00:18:25.990 --> 00:18:29.380 align:middle line:84% And then the yellow case, where it's super peaky, 00:18:29.380 --> 00:18:31.850 align:middle line:84% we're going to see poles that are something like this. 00:18:31.850 --> 00:18:34.900 align:middle line:84% They're very close to the J omega axis. 00:18:34.900 --> 00:18:38.510 align:middle line:90% So we're sort of-- 00:18:38.510 --> 00:18:43.540 align:middle line:84% have different cases where the dynamics of this, the time 00:18:43.540 --> 00:18:48.250 align:middle line:84% domain dynamics, are related to how peaky the frequency domain 00:18:48.250 --> 00:18:49.040 align:middle line:90% response is. 00:18:49.040 --> 00:18:52.420 align:middle line:84% And those, of course, map between one another. 00:18:52.420 --> 00:18:53.895 align:middle line:90% Any questions about that? 00:18:53.895 --> 00:18:58.940 align:middle line:90% 00:18:58.940 --> 00:19:05.600 align:middle line:84% So I mentioned the so-called Q of the network. 00:19:05.600 --> 00:19:07.260 align:middle line:90% And what do I mean by that? 00:19:07.260 --> 00:19:10.975 align:middle line:84% And this is-- Q gets thrown around as a term quite a bit. 00:19:10.975 --> 00:19:13.680 align:middle line:90% 00:19:13.680 --> 00:19:17.110 align:middle line:84% And you've got to be careful in how you use it. 00:19:17.110 --> 00:19:19.710 align:middle line:90% But what do we mean generally? 00:19:19.710 --> 00:19:23.300 align:middle line:84% What is the Q of some network that we're driving? 00:19:23.300 --> 00:19:24.780 align:middle line:84% This could be a mechanical system. 00:19:24.780 --> 00:19:26.590 align:middle line:84% This could be some other kind of system. 00:19:26.590 --> 00:19:29.730 align:middle line:84% It doesn't have to be a second-order electrical network. 00:19:29.730 --> 00:19:32.910 align:middle line:84% But generally, what we define is the quality factor. 00:19:32.910 --> 00:19:39.506 align:middle line:90% 00:19:39.506 --> 00:19:42.500 align:middle line:90% Q is generally defined this way. 00:19:42.500 --> 00:19:56.150 align:middle line:84% It's 2 pi times the peak energy stored under sinusoidal drive 00:19:56.150 --> 00:20:07.180 align:middle line:84% conditions divided by the energy dissipated in a cycle. 00:20:07.180 --> 00:20:10.060 align:middle line:90% 00:20:10.060 --> 00:20:13.870 align:middle line:84% And how the history of this quality factor term 00:20:13.870 --> 00:20:17.038 align:middle line:84% came to be is actually in a handout that I think is-- 00:20:17.038 --> 00:20:18.580 align:middle line:84% you should have been able to pick up, 00:20:18.580 --> 00:20:21.340 align:middle line:90% or it's also posted on Canvas. 00:20:21.340 --> 00:20:24.310 align:middle line:84% But this is the underlying modern definition 00:20:24.310 --> 00:20:26.970 align:middle line:90% of quality factor. 00:20:26.970 --> 00:20:28.470 align:middle line:84% Let's just say-- and it doesn't only 00:20:28.470 --> 00:20:30.240 align:middle line:84% apply to second-order networks like this. 00:20:30.240 --> 00:20:32.770 align:middle line:84% It applies to any network you're driving. 00:20:32.770 --> 00:20:35.470 align:middle line:84% And it also applies at any frequency 00:20:35.470 --> 00:20:37.050 align:middle line:90% you happen to be driving it. 00:20:37.050 --> 00:20:40.030 align:middle line:84% But let's think about a simple example. 00:20:40.030 --> 00:20:45.510 align:middle line:84% So suppose I was to take a sinusoidal current, 00:20:45.510 --> 00:20:51.130 align:middle line:84% and I'll call this I cosine omega T. Now 00:20:51.130 --> 00:20:53.160 align:middle line:84% I'm going to put it into some network. 00:20:53.160 --> 00:21:00.780 align:middle line:84% And let me let my network just be R and L. 00:21:00.780 --> 00:21:02.430 align:middle line:84% So I'm going to drive an LR network 00:21:02.430 --> 00:21:03.840 align:middle line:90% with a sinusoidal current. 00:21:03.840 --> 00:21:09.420 align:middle line:84% And I could ask, what is the quality factor of this network 00:21:09.420 --> 00:21:13.090 align:middle line:90% under these drive conditions? 00:21:13.090 --> 00:21:15.740 align:middle line:90% Well, it's a series RL circuit. 00:21:15.740 --> 00:21:18.640 align:middle line:84% We can just apply this definition, 00:21:18.640 --> 00:21:24.405 align:middle line:84% What's the peak energy stored in this network over the AC cycle? 00:21:24.405 --> 00:21:31.680 align:middle line:90% 00:21:31.680 --> 00:21:33.720 align:middle line:90% It's got to be-- yeah. 00:21:33.720 --> 00:21:37.150 align:middle line:84% Maximum energy is stored when the inductor current is maximum, 00:21:37.150 --> 00:21:49.220 align:middle line:84% which is I. So E stored is simply 1/2 LI squared. 00:21:49.220 --> 00:21:52.760 align:middle line:90% E stored peak is 1/2 LI squared. 00:21:52.760 --> 00:21:53.580 align:middle line:90% That's easy enough. 00:21:53.580 --> 00:21:55.840 align:middle line:90% How about the energy dissipated? 00:21:55.840 --> 00:21:59.720 align:middle line:84% Well, I would argue that this is IRMS 00:21:59.720 --> 00:22:03.870 align:middle line:84% squared R would be the power dissipated on average. 00:22:03.870 --> 00:22:10.190 align:middle line:84% So I could say E diss in one cycle could simply be the power 00:22:10.190 --> 00:22:15.920 align:middle line:84% dissipation, which is 1/2 I squared R. 00:22:15.920 --> 00:22:19.940 align:middle line:84% That's the power-- average power being dissipated in the resistor 00:22:19.940 --> 00:22:23.070 align:middle line:84% times the period would give me the energy dissipated, 00:22:23.070 --> 00:22:25.940 align:middle line:90% which would give me then-- 00:22:25.940 --> 00:22:34.660 align:middle line:84% I could write this as 1/2 I squared R times 2 pi over omega 00:22:34.660 --> 00:22:36.040 align:middle line:90% is the period. 00:22:36.040 --> 00:22:41.870 align:middle line:84% And if I rewrote that, then I would get pi 00:22:41.870 --> 00:22:46.670 align:middle line:90% I squared R divided by omega. 00:22:46.670 --> 00:22:50.630 align:middle line:84% So now let me just apply my definition of quality factor. 00:22:50.630 --> 00:22:57.380 align:middle line:84% Q is simply equal to 2 pi times the peak energy stored, 00:22:57.380 --> 00:23:04.520 align:middle line:84% which is 1/2 LI squared divided by the energy dissipated, 00:23:04.520 --> 00:23:10.640 align:middle line:84% which is pi I squared R divided by omega. 00:23:10.640 --> 00:23:18.740 align:middle line:84% And in the end, what I get is omega L over R. 00:23:18.740 --> 00:23:24.110 align:middle line:84% So what this says is if I'm driving this network, and I say, 00:23:24.110 --> 00:23:27.920 align:middle line:84% how much energy is going away into the resistance 00:23:27.920 --> 00:23:31.310 align:middle line:84% compared to the amount of energy I'm storing in this network, 00:23:31.310 --> 00:23:36.080 align:middle line:84% it's precisely omega L over R. Now, 00:23:36.080 --> 00:23:38.220 align:middle line:84% do I want that to be high or low? 00:23:38.220 --> 00:23:41.270 align:middle line:90% Well, it depends what I'm doing. 00:23:41.270 --> 00:23:46.100 align:middle line:84% If I wanted a lot of energy storage and very 00:23:46.100 --> 00:23:47.720 align:middle line:90% little dissipation-- 00:23:47.720 --> 00:23:52.910 align:middle line:84% suppose this boxed item were a model of a real inductor who 00:23:52.910 --> 00:23:56.420 align:middle line:84% has an inductance L, but he has a parasitic resistance R, 00:23:56.420 --> 00:23:59.360 align:middle line:84% this quality factor then would be a measure of how much 00:23:59.360 --> 00:24:02.900 align:middle line:84% can I store divided by how much do 00:24:02.900 --> 00:24:08.310 align:middle line:84% I lose when I store it in that inductor normalized with 2 pi. 00:24:08.310 --> 00:24:11.870 align:middle line:84% So in some sense, you could use-- for a real inductor, 00:24:11.870 --> 00:24:14.618 align:middle line:84% Q could be a measure of the goodness of the inductor. 00:24:14.618 --> 00:24:16.160 align:middle line:84% And, in fact, that's how they started 00:24:16.160 --> 00:24:20.960 align:middle line:84% using Q as a measure of goodness for inductors and capacitors. 00:24:20.960 --> 00:24:24.720 align:middle line:84% But it works out to have this definition. 00:24:24.720 --> 00:24:27.740 align:middle line:84% On the other hand, if I was using this network 00:24:27.740 --> 00:24:31.800 align:middle line:84% to transfer energy from this source into the resistor, 00:24:31.800 --> 00:24:35.550 align:middle line:84% then having dissipation in the resistor 00:24:35.550 --> 00:24:37.830 align:middle line:84% is a good thing, not a bad thing, 00:24:37.830 --> 00:24:39.740 align:middle line:84% then maybe I want the Q to be really low. 00:24:39.740 --> 00:24:41.490 align:middle line:84% And, in fact, the power factor gets better 00:24:41.490 --> 00:24:43.500 align:middle line:90% when the Q is really low. 00:24:43.500 --> 00:24:48.060 align:middle line:84% So Q came up to be a measure in a sinusoidal drive 00:24:48.060 --> 00:24:51.330 align:middle line:84% condition of energy stored in the network divided by energy 00:24:51.330 --> 00:24:54.470 align:middle line:90% dissipated over a cycle. 00:24:54.470 --> 00:24:55.585 align:middle line:90% Any questions about that? 00:24:55.585 --> 00:24:58.930 align:middle line:90% 00:24:58.930 --> 00:25:04.840 align:middle line:84% And I will have you notice that Q is, in fact, in general, 00:25:04.840 --> 00:25:06.146 align:middle line:90% a function of frequency. 00:25:06.146 --> 00:25:09.004 align:middle line:90% 00:25:09.004 --> 00:25:17.830 align:middle line:84% So it gets-- if you're going to talk about the Q of driving 00:25:17.830 --> 00:25:20.080 align:middle line:84% something, you really got to say what frequency you're 00:25:20.080 --> 00:25:22.540 align:middle line:90% driving at it, usually. 00:25:22.540 --> 00:25:25.602 align:middle line:84% And, in fact, what that tells me is 00:25:25.602 --> 00:25:28.310 align:middle line:84% that if I'm going out and buying inductors, and all things equal, 00:25:28.310 --> 00:25:30.470 align:middle line:84% if it has a given parasitic resistance, 00:25:30.470 --> 00:25:32.690 align:middle line:84% going at higher frequency gives me better energy 00:25:32.690 --> 00:25:36.110 align:middle line:84% storage per unit dissipation because my energy storage goes 00:25:36.110 --> 00:25:40.130 align:middle line:84% up with frequency, and my energy dissipation doesn't, on average. 00:25:40.130 --> 00:25:41.930 align:middle line:90% Yeah, Julia. 00:25:41.930 --> 00:25:44.090 align:middle line:84% AUDIENCE: Does this definition for Q 00:25:44.090 --> 00:25:46.700 align:middle line:84% still work if the drive signal is AC 00:25:46.700 --> 00:25:49.990 align:middle line:84% but not sinusoidal with a square [INAUDIBLE] driver or something? 00:25:49.990 --> 00:25:50.907 align:middle line:90% DAVID PERREAULT: Yeah. 00:25:50.907 --> 00:25:53.450 align:middle line:84% Usually, they don't-- usually, you define it under sinusoidal 00:25:53.450 --> 00:25:56.800 align:middle line:90% drive conditions. 00:25:56.800 --> 00:25:59.020 align:middle line:84% I'm not sure if anybody ever came up 00:25:59.020 --> 00:26:04.130 align:middle line:84% with some other definition, but usually that's what's intended. 00:26:04.130 --> 00:26:08.100 align:middle line:84% Sinusoids are sort of for LTI systems or their eigenfunction. 00:26:08.100 --> 00:26:09.790 align:middle line:84% So that's what we usually worry about. 00:26:09.790 --> 00:26:14.340 align:middle line:90% 00:26:14.340 --> 00:26:16.830 align:middle line:84% So that's the definition of Q. How 00:26:16.830 --> 00:26:21.960 align:middle line:84% does that relate to when we talk about the Q of-- 00:26:21.960 --> 00:26:24.450 align:middle line:84% if I'm talking about the Q of this network here, 00:26:24.450 --> 00:26:27.090 align:middle line:84% and I'm saying it's high Q or low Q network? 00:26:27.090 --> 00:26:30.250 align:middle line:84% And this is where we got to be careful about what we mean by Q. 00:26:30.250 --> 00:26:31.800 align:middle line:84% This is the underlying definition. 00:26:31.800 --> 00:26:34.650 align:middle line:84% But often, when we talk about the quality factor of a network, 00:26:34.650 --> 00:26:37.710 align:middle line:84% especially, say, a second-order resonant network, 00:26:37.710 --> 00:26:42.240 align:middle line:84% what's meant by that is the quality factor, if you happen 00:26:42.240 --> 00:26:45.390 align:middle line:84% to be driving it at its resonant frequency, or in this case, 00:26:45.390 --> 00:26:48.377 align:middle line:90% omega equals omega 0. 00:26:48.377 --> 00:26:50.460 align:middle line:84% So when you're talking about the Q of the network, 00:26:50.460 --> 00:26:53.670 align:middle line:84% you say, let me drive at its resonant frequency 00:26:53.670 --> 00:26:55.730 align:middle line:84% and talk about what its Q is there. 00:26:55.730 --> 00:27:01.820 align:middle line:90% 00:27:01.820 --> 00:27:06.500 align:middle line:84% So why don't we go calculate the Q of this network? 00:27:06.500 --> 00:27:09.432 align:middle line:84% And I'm going to calculate it at omega equals omega 0. 00:27:09.432 --> 00:27:11.390 align:middle line:84% So when I'm talking about the Q of the network, 00:27:11.390 --> 00:27:14.390 align:middle line:84% when I say "of the network," what you should be thinking 00:27:14.390 --> 00:27:17.930 align:middle line:90% of is Q of omega equals omega 0. 00:27:17.930 --> 00:27:21.990 align:middle line:84% You could find the Q at some other drive frequency, 00:27:21.990 --> 00:27:24.940 align:middle line:84% and that would be the quality factor of driving that network. 00:27:24.940 --> 00:27:27.630 align:middle line:84% But it wouldn't be what people refer to as the Q of the network 00:27:27.630 --> 00:27:30.498 align:middle line:90% itself. 00:27:30.498 --> 00:27:32.910 align:middle line:84% And that's sort of-- and the reason I say that is 00:27:32.910 --> 00:27:36.256 align:middle line:90% because it's sort of implicit-- 00:27:36.256 --> 00:27:39.510 align:middle line:90% 00:27:39.510 --> 00:27:45.795 align:middle line:84% it's sort of an implicit condition when you're talking 00:27:45.795 --> 00:27:46.920 align:middle line:90% about the Q of the network. 00:27:46.920 --> 00:27:49.220 align:middle line:90% So let's talk about this. 00:27:49.220 --> 00:27:50.400 align:middle line:90% So here I am. 00:27:50.400 --> 00:27:57.080 align:middle line:84% I'm going to drive a VS cosine omega T. 00:27:57.080 --> 00:28:03.270 align:middle line:84% And I'm going to drive this network at omega equals omega 0. 00:28:03.270 --> 00:28:05.450 align:middle line:90% So let me actually-- 00:28:05.450 --> 00:28:06.600 align:middle line:90% let's be careful. 00:28:06.600 --> 00:28:13.940 align:middle line:84% I'm going to talk about this Q of omega 0 T. 00:28:13.940 --> 00:28:14.960 align:middle line:90% I have RLC-- 00:28:14.960 --> 00:28:20.690 align:middle line:90% 00:28:20.690 --> 00:28:28.690 align:middle line:84% I sub L, V sub C. At omega equals omega 0, what is I sub L? 00:28:28.690 --> 00:28:35.830 align:middle line:90% 00:28:35.830 --> 00:28:40.480 align:middle line:84% Well, I sub L, I would argue, at the natural resonant frequency, 00:28:40.480 --> 00:28:44.130 align:middle line:90% the impedance of C and L cancel. 00:28:44.130 --> 00:28:50.190 align:middle line:84% And my admittance is 1 over R, or my impedance is just R. 00:28:50.190 --> 00:28:57.840 align:middle line:84% So then this should just be VS over R cosine of omega T-- 00:28:57.840 --> 00:28:58.530 align:middle line:90% omega 0T. 00:28:58.530 --> 00:29:04.130 align:middle line:90% 00:29:04.130 --> 00:29:13.020 align:middle line:84% And at that magic frequency, because the capacitor current 00:29:13.020 --> 00:29:18.950 align:middle line:90% is exactly driven-- 00:29:18.950 --> 00:29:22.170 align:middle line:84% the capacitor voltage is exactly driven by the inductor current, 00:29:22.170 --> 00:29:27.285 align:middle line:84% these two at that frequency are exactly 90 degrees out of phase. 00:29:27.285 --> 00:29:31.920 align:middle line:84% So what that means is that when IL is peaking, 00:29:31.920 --> 00:29:37.650 align:middle line:90% VC is at its 0 crossing. 00:29:37.650 --> 00:29:45.010 align:middle line:84% So we could very easily show that the maximum energy 00:29:45.010 --> 00:29:50.790 align:middle line:84% storage would occur when IL was at its peak. 00:29:50.790 --> 00:29:55.140 align:middle line:84% So, for example, at omega 0 T equals 0. 00:29:55.140 --> 00:29:57.690 align:middle line:84% The circuit also stores maximum energy 00:29:57.690 --> 00:29:59.310 align:middle line:84% when the inductor current is at 0 00:29:59.310 --> 00:30:01.600 align:middle line:84% and the capacitor voltage is at his peak. 00:30:01.600 --> 00:30:04.050 align:middle line:84% And it would be the same stored energy 00:30:04.050 --> 00:30:05.970 align:middle line:84% because these two these two elements 00:30:05.970 --> 00:30:09.750 align:middle line:84% are essentially just flopping energy between one another, 00:30:09.750 --> 00:30:12.430 align:middle line:90% and all the power-- 00:30:12.430 --> 00:30:14.980 align:middle line:84% the net power-- is going into the resistor. 00:30:14.980 --> 00:30:29.890 align:middle line:84% So we could say the peak energy stored is just going to be 00:30:29.890 --> 00:30:39.250 align:middle line:84% simply 1/2 L IP squared, which would be 1/2 L VS squared over R 00:30:39.250 --> 00:30:39.750 align:middle line:90% squared. 00:30:39.750 --> 00:30:45.140 align:middle line:90% 00:30:45.140 --> 00:30:46.585 align:middle line:90% at omega equal omega 0. 00:30:46.585 --> 00:30:49.850 align:middle line:90% 00:30:49.850 --> 00:30:51.436 align:middle line:90% The energy dissipated. 00:30:51.436 --> 00:30:55.430 align:middle line:90% 00:30:55.430 --> 00:30:57.260 align:middle line:90% Well, we can calculate it. 00:30:57.260 --> 00:30:59.690 align:middle line:84% As long as I know the current, I have the same energy 00:30:59.690 --> 00:31:01.110 align:middle line:90% as up there. 00:31:01.110 --> 00:31:05.810 align:middle line:84% That's going to be equal to pi times I squared, 00:31:05.810 --> 00:31:11.000 align:middle line:84% which is VS squared over R squared 00:31:11.000 --> 00:31:22.910 align:middle line:84% times R divided by omega, which is pi VS squared over R omega. 00:31:22.910 --> 00:31:29.120 align:middle line:84% So then if I were going to calculate Q evaluated at omega 00:31:29.120 --> 00:31:32.540 align:middle line:84% equals omega 0, which I'm going to call-- to designate that, I'm 00:31:32.540 --> 00:31:34.850 align:middle line:90% going to call that Q0. 00:31:34.850 --> 00:31:40.190 align:middle line:84% The Q of the circuit I'll call Q0 to distinguish it 00:31:40.190 --> 00:31:41.900 align:middle line:84% from the Q of the circuit operating 00:31:41.900 --> 00:31:44.930 align:middle line:90% at some other frequency. 00:31:44.930 --> 00:31:50.190 align:middle line:84% We could write this as 2 pi times the peak energy stored. 00:31:50.190 --> 00:31:58.440 align:middle line:84% It's 1/2 L VS squared over R squared divided by the energy 00:31:58.440 --> 00:32:05.870 align:middle line:84% dissipated, which is pi VS squared over R omega. 00:32:05.870 --> 00:32:10.580 align:middle line:84% And that's R omega 0 because we're at omega 0. 00:32:10.580 --> 00:32:14.100 align:middle line:84% And if I write that out, what I'm going to get is this. 00:32:14.100 --> 00:32:16.760 align:middle line:90% I get the two pi's canceling. 00:32:16.760 --> 00:32:20.550 align:middle line:90% I get the VS squareds canceling. 00:32:20.550 --> 00:32:23.240 align:middle line:84% I get this R cancels with this one. 00:32:23.240 --> 00:32:30.860 align:middle line:84% So I get omega 0 L divided by R, which we just said was-- 00:32:30.860 --> 00:32:36.810 align:middle line:84% omega 0 L, if you remember, is Z0, my characteristic impedance. 00:32:36.810 --> 00:32:44.790 align:middle line:84% So this is Z0 over R is the quality factor of the circuit. 00:32:44.790 --> 00:32:52.120 align:middle line:84% So it immediately comes back to this case. 00:32:52.120 --> 00:32:58.600 align:middle line:84% If X0 over R is big, where R is much smaller than Z0-- 00:32:58.600 --> 00:32:59.950 align:middle line:90% I'm up here. 00:32:59.950 --> 00:33:04.010 align:middle line:84% And so exactly comes back to this. 00:33:04.010 --> 00:33:09.230 align:middle line:84% So if I were to plot this, then, maybe I would come here 00:33:09.230 --> 00:33:13.295 align:middle line:84% and I'd say, OK, let me do the high Q case. 00:33:13.295 --> 00:33:15.890 align:middle line:90% 00:33:15.890 --> 00:33:18.810 align:middle line:84% And I would have this is 1 over Z0. 00:33:18.810 --> 00:33:23.960 align:middle line:84% Here's 1 over R. So maybe what I have is 00:33:23.960 --> 00:33:34.445 align:middle line:84% something that does this, peaks up like this. 00:33:34.445 --> 00:33:37.990 align:middle line:90% 00:33:37.990 --> 00:33:43.490 align:middle line:84% And exactly what we're saying is that the ratio-- 00:33:43.490 --> 00:33:45.812 align:middle line:84% I didn't draw that in a very pretty fashion. 00:33:45.812 --> 00:33:47.770 align:middle line:84% Maybe it comes down a little quicker like this. 00:33:47.770 --> 00:33:52.890 align:middle line:90% 00:33:52.890 --> 00:34:00.420 align:middle line:84% What we're saying is that the exact amount 00:34:00.420 --> 00:34:03.400 align:middle line:84% I'm going to peak up, the ratio-- 00:34:03.400 --> 00:34:05.850 align:middle line:90% this is 1 over Z0. 00:34:05.850 --> 00:34:09.150 align:middle line:84% This is 1 over R. The amount that I peak up 00:34:09.150 --> 00:34:12.810 align:middle line:84% above the intersection, above the two asymptotes intersecting, 00:34:12.810 --> 00:34:20.909 align:middle line:84% is exactly Q of omega 0, which is Q0 here. 00:34:20.909 --> 00:34:25.170 align:middle line:84% So what we can say is that if I have-- 00:34:25.170 --> 00:34:28.139 align:middle line:84% and this is why I immediately-- another reason I immediately 00:34:28.139 --> 00:34:29.370 align:middle line:90% like to calculate Z0. 00:34:29.370 --> 00:34:31.469 align:middle line:84% If I'm in a series resonant circuit, 00:34:31.469 --> 00:34:35.940 align:middle line:84% I know that the quality factor of the circuit is Z0 over R, 00:34:35.940 --> 00:34:38.670 align:middle line:84% and the amount I'm going to peak up 00:34:38.670 --> 00:34:41.190 align:middle line:84% above the intersection of the asymptotes in terms 00:34:41.190 --> 00:34:45.210 align:middle line:84% of its admittance is exactly the quality factor of the circuit, 00:34:45.210 --> 00:34:47.040 align:middle line:90% or Z0 over R. 00:34:47.040 --> 00:34:49.067 align:middle line:84% So if Z0 is much bigger than R, I 00:34:49.067 --> 00:34:50.400 align:middle line:90% know I'm going to peak up a lot. 00:34:50.400 --> 00:34:57.250 align:middle line:84% If Z0 is smaller, it's going to be a flatter frequency response. 00:34:57.250 --> 00:34:58.470 align:middle line:90% Any questions about that? 00:34:58.470 --> 00:35:02.720 align:middle line:90% 00:35:02.720 --> 00:35:06.540 align:middle line:84% I'd like to show you one last thing that comes up. 00:35:06.540 --> 00:35:09.230 align:middle line:84% And I should say we could have considered 00:35:09.230 --> 00:35:10.438 align:middle line:90% a parallel resonant network. 00:35:10.438 --> 00:35:12.480 align:middle line:84% We could have considered other resonant networks. 00:35:12.480 --> 00:35:15.800 align:middle line:84% I don't want you to think that they're all precisely the same, 00:35:15.800 --> 00:35:18.610 align:middle line:84% but they have a lot of characteristics in common. 00:35:18.610 --> 00:35:21.955 align:middle line:90% 00:35:21.955 --> 00:35:24.372 align:middle line:84% And, in fact, if I had done the parallel resonant network, 00:35:24.372 --> 00:35:25.788 align:middle line:84% actually what you would have found 00:35:25.788 --> 00:35:27.310 align:middle line:84% is the quality factor is exactly-- 00:35:27.310 --> 00:35:29.970 align:middle line:84% instead of being Z0 over R, it turns out to be R over Z0. 00:35:29.970 --> 00:35:32.490 align:middle line:90% It's exactly the inverse. 00:35:32.490 --> 00:35:35.220 align:middle line:84% But there's a lot of value in figuring out the quality 00:35:35.220 --> 00:35:37.680 align:middle line:84% factor of the network because it tells you 00:35:37.680 --> 00:35:39.965 align:middle line:84% exactly how much is it going to peak up. 00:35:39.965 --> 00:35:42.960 align:middle line:90% 00:35:42.960 --> 00:35:47.260 align:middle line:84% But we could ask ourselves one more question. 00:35:47.260 --> 00:35:50.620 align:middle line:84% I mentioned that-- and I haven't gotten 00:35:50.620 --> 00:35:54.000 align:middle line:84% to why we're going to get soft switching out of this. 00:35:54.000 --> 00:35:55.580 align:middle line:90% We'll get to that. 00:35:55.580 --> 00:36:02.180 align:middle line:84% But I might be interested in how frequency-selective this network 00:36:02.180 --> 00:36:04.390 align:middle line:90% is. 00:36:04.390 --> 00:36:04.890 align:middle line:90% Why? 00:36:04.890 --> 00:36:08.030 align:middle line:84% Because if I was going to drive this at variable frequency-- 00:36:08.030 --> 00:36:11.750 align:middle line:84% suppose I could control omega over a range-- 00:36:11.750 --> 00:36:13.670 align:middle line:84% the impedance of this guy, especially 00:36:13.670 --> 00:36:16.340 align:middle line:84% if it's high Q, the admittance, and hence, 00:36:16.340 --> 00:36:19.050 align:middle line:84% the impedance changes very quickly with frequency. 00:36:19.050 --> 00:36:22.130 align:middle line:84% If I move off the resonant frequency, 00:36:22.130 --> 00:36:26.130 align:middle line:84% the admittance drops pretty quickly if it's high Q. 00:36:26.130 --> 00:36:29.063 align:middle line:84% So I might be able to modulate how much current I'm 00:36:29.063 --> 00:36:31.230 align:middle line:84% going to get and how much power I'm going to deliver 00:36:31.230 --> 00:36:33.017 align:middle line:90% by modulating frequency. 00:36:33.017 --> 00:36:35.350 align:middle line:84% In fact, that's what we often do in resonant converters. 00:36:35.350 --> 00:36:36.900 align:middle line:90% Not always, but often. 00:36:36.900 --> 00:36:45.120 align:middle line:84% So one thing I might be interested in is how broad-- 00:36:45.120 --> 00:36:47.640 align:middle line:84% if I said I'm going to peak up, how broad 00:36:47.640 --> 00:36:51.900 align:middle line:90% is this peaking in frequency? 00:36:51.900 --> 00:36:55.650 align:middle line:84% So why don't we go calculate how broad that peaking and frequency 00:36:55.650 --> 00:36:56.550 align:middle line:90% is. 00:36:56.550 --> 00:37:03.540 align:middle line:84% And the measure I might use to capture that is the following. 00:37:03.540 --> 00:37:06.000 align:middle line:84% I might look at this and say, OK, 00:37:06.000 --> 00:37:10.740 align:middle line:84% suppose I'm going to drive with a voltage, over what frequency 00:37:10.740 --> 00:37:13.980 align:middle line:84% range do I get at least half the power I would get exactly 00:37:13.980 --> 00:37:15.040 align:middle line:90% at resonance? 00:37:15.040 --> 00:37:17.882 align:middle line:90% 00:37:17.882 --> 00:37:19.470 align:middle line:90% So what would that be? 00:37:19.470 --> 00:37:21.620 align:middle line:84% If I'm going to get half the power, 00:37:21.620 --> 00:37:23.840 align:middle line:84% that means the current has to go down 00:37:23.840 --> 00:37:26.060 align:middle line:84% by the 1 over the square root of 2 00:37:26.060 --> 00:37:29.480 align:middle line:90% because then I get pi squared-- 00:37:29.480 --> 00:37:30.690 align:middle line:90% give me a half. 00:37:30.690 --> 00:37:38.540 align:middle line:84% So maybe I'm looking for the frequency range over which I 00:37:38.540 --> 00:37:43.730 align:middle line:84% have the magnitude of the admittance drops down to 1 over 00:37:43.730 --> 00:37:46.460 align:middle line:90% the square root of 2R. 00:37:46.460 --> 00:37:48.640 align:middle line:84% Because the admittance 1 over the square root of 2R, 00:37:48.640 --> 00:37:50.970 align:middle line:84% then the magnitude of the current is going to be 1 over 00:37:50.970 --> 00:37:53.553 align:middle line:84% the square root of 2-- go down as 1 over the square root of 2, 00:37:53.553 --> 00:37:55.750 align:middle line:84% and my power is going to go down by half. 00:37:55.750 --> 00:38:11.620 align:middle line:84% And so then I might end up with some frequency range here, 00:38:11.620 --> 00:38:14.800 align:middle line:84% say between-- say this is some frequency range between omega 00:38:14.800 --> 00:38:23.380 align:middle line:84% 2 and omega 1, where over this half power bandwidth, 00:38:23.380 --> 00:38:25.840 align:middle line:84% I will get at least half the power as I would 00:38:25.840 --> 00:38:28.903 align:middle line:90% at the resonant frequency. 00:38:28.903 --> 00:38:30.320 align:middle line:84% Does that make sense to everybody? 00:38:30.320 --> 00:38:32.940 align:middle line:90% 00:38:32.940 --> 00:38:36.130 align:middle line:84% So why don't we go find this half power bandwidth. 00:38:36.130 --> 00:38:40.560 align:middle line:90% 00:38:40.560 --> 00:38:42.290 align:middle line:90% What will I see then? 00:38:42.290 --> 00:38:45.930 align:middle line:84% Well, what I'm looking for is if I'm looking for an admittance 00:38:45.930 --> 00:38:48.000 align:middle line:84% that's 1 over the square root of 2R-- 00:38:48.000 --> 00:38:51.150 align:middle line:84% I want these two magic points here-- 00:38:51.150 --> 00:38:56.160 align:middle line:84% and I look at my admittance equation, 00:38:56.160 --> 00:38:59.580 align:middle line:84% the magnitude of YN I want to be 1 over the square root of 2R. 00:38:59.580 --> 00:39:02.380 align:middle line:90% 00:39:02.380 --> 00:39:05.260 align:middle line:90% Here's the magnitude of YN. 00:39:05.260 --> 00:39:13.260 align:middle line:84% What I really want is this guy to be R, or minus R. 00:39:13.260 --> 00:39:18.030 align:middle line:84% Because then this will be R squared plus R squared gives me 00:39:18.030 --> 00:39:18.690 align:middle line:90% 2R squared. 00:39:18.690 --> 00:39:23.130 align:middle line:84% Square root, it gives me square root of 2R in the denominator. 00:39:23.130 --> 00:39:25.570 align:middle line:84% Does that make sense to everybody? 00:39:25.570 --> 00:39:30.610 align:middle line:84% So what I could say is then what I want is-- 00:39:30.610 --> 00:39:35.320 align:middle line:84% I want the two frequencies to be omega L minus 1 00:39:35.320 --> 00:39:40.250 align:middle line:84% over omega C, the magnitude of that 00:39:40.250 --> 00:39:48.890 align:middle line:84% to equal R. This is going to define omega 1 and omega 2. 00:39:48.890 --> 00:39:50.245 align:middle line:90% Any questions about that? 00:39:50.245 --> 00:39:53.720 align:middle line:90% 00:39:53.720 --> 00:39:56.810 align:middle line:90% So let's just do that. 00:39:56.810 --> 00:39:59.630 align:middle line:90% I could find-- let me just-- 00:39:59.630 --> 00:40:06.320 align:middle line:84% what I can say is let's find the case where I have omega 1L minus 00:40:06.320 --> 00:40:11.700 align:middle line:84% 1 over omega 1C is equal to minus R. 00:40:11.700 --> 00:40:15.980 align:middle line:84% I'm picking the lower of the two frequencies because this is 00:40:15.980 --> 00:40:17.520 align:middle line:84% going getting bigger with frequency. 00:40:17.520 --> 00:40:23.180 align:middle line:84% So I want the one that's minus R for omega 1. 00:40:23.180 --> 00:40:25.250 align:middle line:84% So if I multiply through by this, 00:40:25.250 --> 00:40:33.920 align:middle line:84% then I get omega 1 squared LC minus-- 00:40:33.920 --> 00:40:35.720 align:middle line:90% or plus-- I'm sorry-- 00:40:35.720 --> 00:40:42.660 align:middle line:84% omega 1 RC minus 1 is equal to 0. 00:40:42.660 --> 00:40:45.040 align:middle line:84% So that's the equation that's going to define it. 00:40:45.040 --> 00:40:54.000 align:middle line:84% That gives me omega 1 is equal to minus RC plus or minus 00:40:54.000 --> 00:41:03.760 align:middle line:84% the square root of RC quantity squared minus 4 LC all over 2 00:41:03.760 --> 00:41:08.550 align:middle line:84% LC, which gives me minus R over 2L. 00:41:08.550 --> 00:41:15.296 align:middle line:90% 00:41:15.296 --> 00:41:21.580 align:middle line:84% This is going to have to be plus to yield a positive frequency. 00:41:21.580 --> 00:41:25.130 align:middle line:90% The square root of 1 over LC-- 00:41:25.130 --> 00:41:29.810 align:middle line:90% 00:41:29.810 --> 00:41:30.310 align:middle line:90% oops. 00:41:30.310 --> 00:41:30.810 align:middle line:90% I'm sorry. 00:41:30.810 --> 00:41:31.938 align:middle line:90% This should be plus-- 00:41:31.938 --> 00:41:34.530 align:middle line:90% 00:41:34.530 --> 00:41:45.710 align:middle line:84% 1 over 4 L of C divided by R over 2L quantity squared. 00:41:45.710 --> 00:41:46.590 align:middle line:90% And that's omega. 00:41:46.590 --> 00:41:48.800 align:middle line:84% This is the expression for omega 1, 00:41:48.800 --> 00:41:54.590 align:middle line:84% which is exactly the frequency where I'm sitting right here-- 00:41:54.590 --> 00:41:56.840 align:middle line:84% a factor of 1 over square root of 2 00:41:56.840 --> 00:41:59.850 align:middle line:90% down from the peak admittance. 00:41:59.850 --> 00:42:01.055 align:middle line:90% Any questions about that? 00:42:01.055 --> 00:42:03.790 align:middle line:90% 00:42:03.790 --> 00:42:05.980 align:middle line:90% What would I do if I got the-- 00:42:05.980 --> 00:42:07.780 align:middle line:84% would I do to get the second one? 00:42:07.780 --> 00:42:21.410 align:middle line:84% I would have omega 2L minus 1 over omega 2C equal positive R. 00:42:21.410 --> 00:42:24.120 align:middle line:84% Let me just skip through the algebra because we can do 00:42:24.120 --> 00:42:24.620 align:middle line:90% the algebra. 00:42:24.620 --> 00:42:26.990 align:middle line:84% I just wanted you to see how you get it. 00:42:26.990 --> 00:42:34.300 align:middle line:84% What we get is exactly equal to R over 2L. 00:42:34.300 --> 00:42:48.400 align:middle line:84% So omega 2 is equal to R over 2L plus the square root of 1 over 00:42:48.400 --> 00:42:56.470 align:middle line:84% LC plus 1 over R over 2L quantity squared. 00:42:56.470 --> 00:43:02.580 align:middle line:84% So notice, just as an aside, omega 1 and omega 2 00:43:02.580 --> 00:43:05.620 align:middle line:84% aren't quite centered at 1 over the square root of LC. 00:43:05.620 --> 00:43:07.060 align:middle line:90% It's slightly off of that. 00:43:07.060 --> 00:43:09.750 align:middle line:84% But if R becomes small, it's high Q, 00:43:09.750 --> 00:43:13.140 align:middle line:84% then it's very close to centered at omega 0. 00:43:13.140 --> 00:43:15.700 align:middle line:84% But here's the point I wanted to make. 00:43:15.700 --> 00:43:19.800 align:middle line:90% What is omega 2 minus omega 1? 00:43:19.800 --> 00:43:22.470 align:middle line:84% Omega 2 minus omega 1-- what I'm trying to do 00:43:22.470 --> 00:43:26.700 align:middle line:84% is calculate what is this width in frequency 00:43:26.700 --> 00:43:33.485 align:middle line:84% over which I have at least half the power as I do at the peak? 00:43:33.485 --> 00:43:36.310 align:middle line:90% 00:43:36.310 --> 00:43:38.380 align:middle line:84% That's sometimes called the half-power bandwidth. 00:43:38.380 --> 00:43:43.480 align:middle line:84% And if I get that, that's just going to equal to R over 2L. 00:43:43.480 --> 00:43:44.740 align:middle line:90% OK? 00:43:44.740 --> 00:43:45.260 align:middle line:90% I'm sorry. 00:43:45.260 --> 00:43:52.820 align:middle line:84% It's just equal to R over L. Because R over 2L minus R over 00:43:52.820 --> 00:43:56.856 align:middle line:84% 2L, and these other two terms cancel. 00:43:56.856 --> 00:43:58.010 align:middle line:90% Well, what happens-- 00:43:58.010 --> 00:44:01.120 align:middle line:84% I mean, the absolute bandwidth maybe isn't as interesting. 00:44:01.120 --> 00:44:03.400 align:middle line:84% But how about I go normalize that 00:44:03.400 --> 00:44:05.880 align:middle line:84% to the natural resonant frequency? 00:44:05.880 --> 00:44:12.100 align:middle line:84% So why don't I just take this and normalize it to omega 0? 00:44:12.100 --> 00:44:14.810 align:middle line:90% That's the percentage bandwidth. 00:44:14.810 --> 00:44:18.650 align:middle line:84% If I think of this as sort of my center frequency of interest, 00:44:18.650 --> 00:44:23.910 align:middle line:84% this is the percentage of this over which I get half the power. 00:44:23.910 --> 00:44:29.730 align:middle line:84% And so all I got to do is put omega 0 here in the denominator. 00:44:29.730 --> 00:44:36.630 align:middle line:84% And this is exactly equal to R over omega 0L is just Z0, 00:44:36.630 --> 00:44:42.607 align:middle line:84% which is equal to 1 over Q. 1 over Q0, I should say, 00:44:42.607 --> 00:44:43.190 align:middle line:90% to be careful. 00:44:43.190 --> 00:44:47.150 align:middle line:90% 00:44:47.150 --> 00:44:50.470 align:middle line:84% So what's the conclusion of that? 00:44:50.470 --> 00:44:55.356 align:middle line:84% What I'm saying here is if I say what is the bandwidth-- 00:44:55.356 --> 00:45:00.370 align:middle line:90% 00:45:00.370 --> 00:45:09.350 align:middle line:84% If I wanted to say what is omega 2 minus omega 1 over omega 0, 00:45:09.350 --> 00:45:13.090 align:middle line:84% the percentage bandwidth is exactly equal to 1 over Q0. 00:45:13.090 --> 00:45:16.250 align:middle line:90% 00:45:16.250 --> 00:45:20.750 align:middle line:84% So if I thought about this network, 00:45:20.750 --> 00:45:24.600 align:middle line:84% suppose I could dial the resistance 00:45:24.600 --> 00:45:28.800 align:middle line:84% and just look what happened to my response? 00:45:28.800 --> 00:45:34.470 align:middle line:84% As I make my resistor smaller, Q naught goes up. 00:45:34.470 --> 00:45:38.470 align:middle line:84% As Q naught goes up, two things happen. 00:45:38.470 --> 00:45:41.430 align:middle line:84% First of all, this thing gets peakier 00:45:41.430 --> 00:45:46.740 align:middle line:84% in terms of the amount my resonant peak goes up. 00:45:46.740 --> 00:45:49.860 align:middle line:84% But at the same time, that resonant peak 00:45:49.860 --> 00:45:51.425 align:middle line:84% gets sharper and sharper and sharper. 00:45:51.425 --> 00:45:55.078 align:middle line:90% 00:45:55.078 --> 00:45:58.750 align:middle line:84% And so I really want to figure out the Q of the network 00:45:58.750 --> 00:46:05.080 align:middle line:84% because I want to know, is it doing one of these things on me 00:46:05.080 --> 00:46:08.560 align:middle line:84% where I don't get much peaking and I don't 00:46:08.560 --> 00:46:10.510 align:middle line:90% get much frequency selectivity? 00:46:10.510 --> 00:46:14.350 align:middle line:84% I have to work over a really wide bandwidth in order 00:46:14.350 --> 00:46:17.890 align:middle line:84% to modulate, say, power going into the resistor? 00:46:17.890 --> 00:46:21.300 align:middle line:84% Or do I get something that's really, really sharp? 00:46:21.300 --> 00:46:27.165 align:middle line:90% 00:46:27.165 --> 00:46:31.098 align:middle line:84% But I get two effects out of that. 00:46:31.098 --> 00:46:33.140 align:middle line:84% So I just got have to recognize, those two things 00:46:33.140 --> 00:46:34.260 align:middle line:90% go hand in hand. 00:46:34.260 --> 00:46:40.010 align:middle line:84% Bigger peaking, also corresponds to more sharpness. 00:46:40.010 --> 00:46:42.940 align:middle line:90% I may want the peaking. 00:46:42.940 --> 00:46:45.820 align:middle line:90% Why might I want the peaking? 00:46:45.820 --> 00:46:51.610 align:middle line:84% Because what we said was, if I work it out, 00:46:51.610 --> 00:46:54.760 align:middle line:84% the more peaky it is, the higher the voltage 00:46:54.760 --> 00:46:56.770 align:middle line:84% stress on that capacitor, the more voltage 00:46:56.770 --> 00:46:59.830 align:middle line:84% I'm putting across the capacitor in the inductor. 00:46:59.830 --> 00:47:04.330 align:middle line:84% And, in some cases, we can use that to get voltage gain. 00:47:04.330 --> 00:47:06.730 align:middle line:84% In fact, if I were thinking about a lamp ballast, 00:47:06.730 --> 00:47:08.560 align:middle line:84% it's not a series network in that case, 00:47:08.560 --> 00:47:12.110 align:middle line:84% it's a parallel network, but I might actually use that. 00:47:12.110 --> 00:47:15.820 align:middle line:84% So if you want to like, break over a fluorescent lamp 00:47:15.820 --> 00:47:16.363 align:middle line:90% ballast-- 00:47:16.363 --> 00:47:17.780 align:middle line:84% it's more or less an open circuit. 00:47:17.780 --> 00:47:20.197 align:middle line:84% You want to break down the gases, the voltage of the gases 00:47:20.197 --> 00:47:23.080 align:middle line:84% to make it strike, you can use the resonant gain 00:47:23.080 --> 00:47:25.270 align:middle line:84% of your network to get voltages that are much 00:47:25.270 --> 00:47:27.020 align:middle line:90% bigger than your drive voltage. 00:47:27.020 --> 00:47:31.948 align:middle line:84% And, in fact, at resonance, the voltage on that capacitor 00:47:31.948 --> 00:47:33.490 align:middle line:84% in this network-- and I don't think-- 00:47:33.490 --> 00:47:36.700 align:middle line:84% I'm not sure if I exactly showed you this, but it's easy to show. 00:47:36.700 --> 00:47:40.960 align:middle line:84% If I go right to resonance, the voltage on this capacitor 00:47:40.960 --> 00:47:45.490 align:middle line:84% is precisely Q0 times the source voltage 00:47:45.490 --> 00:47:48.250 align:middle line:90% at the resonant frequency. 00:47:48.250 --> 00:47:50.380 align:middle line:84% So maybe I would want some situation where 00:47:50.380 --> 00:47:53.518 align:middle line:90% I can generate large voltages. 00:47:53.518 --> 00:47:55.060 align:middle line:84% Well, the resonance helps me do that. 00:47:55.060 --> 00:47:57.180 align:middle line:84% But to get a large voltage relative to the input, 00:47:57.180 --> 00:47:59.075 align:middle line:90% I need a higher Q0. 00:47:59.075 --> 00:48:00.450 align:middle line:84% On the other hand, I'm only going 00:48:00.450 --> 00:48:02.825 align:middle line:84% to get that over a narrower and narrower frequency range. 00:48:02.825 --> 00:48:05.750 align:middle line:90% 00:48:05.750 --> 00:48:06.870 align:middle line:90% On the other hand-- 00:48:06.870 --> 00:48:10.160 align:middle line:84% so on the one hand, maybe I want high Q 00:48:10.160 --> 00:48:12.350 align:middle line:84% because I get large voltage gain. 00:48:12.350 --> 00:48:17.360 align:middle line:84% Or because I get large frequency selectivity, 00:48:17.360 --> 00:48:20.810 align:middle line:84% like higher Q means I don't have to move very much in frequency 00:48:20.810 --> 00:48:22.640 align:middle line:90% to modulate power. 00:48:22.640 --> 00:48:26.370 align:middle line:84% But at the same time, what did we say that meant? 00:48:26.370 --> 00:48:28.500 align:middle line:84% That means when we're getting towards resonance, 00:48:28.500 --> 00:48:30.900 align:middle line:90% that means I'm storing-- 00:48:30.900 --> 00:48:33.210 align:middle line:84% the proportion of energy I'm storing 00:48:33.210 --> 00:48:35.460 align:middle line:84% compared to that amount I'm putting into the resistor, 00:48:35.460 --> 00:48:37.200 align:middle line:90% it gets bigger. 00:48:37.200 --> 00:48:40.440 align:middle line:84% So if my if my network Q is too high, 00:48:40.440 --> 00:48:42.690 align:middle line:84% that means I get big voltage and current 00:48:42.690 --> 00:48:46.320 align:middle line:84% stresses on my capacitor and my inductor respectively. 00:48:46.320 --> 00:48:48.540 align:middle line:84% And it means I have to store a lot of energy 00:48:48.540 --> 00:48:50.397 align:middle line:84% in the circuit for the amount I'm 00:48:50.397 --> 00:48:51.605 align:middle line:90% delivering into the resistor. 00:48:51.605 --> 00:48:54.255 align:middle line:90% 00:48:54.255 --> 00:48:56.130 align:middle line:84% So we have-- what we're going to see 00:48:56.130 --> 00:48:58.320 align:middle line:84% is as we start talking about power converters that 00:48:58.320 --> 00:49:04.050 align:middle line:84% use this characteristic, low Q means, eh, 00:49:04.050 --> 00:49:06.210 align:middle line:84% maybe I don't get the frequency selectivity I want, 00:49:06.210 --> 00:49:09.870 align:middle line:84% and I can't use frequency to modulate things very well. 00:49:09.870 --> 00:49:13.320 align:middle line:84% If Q is too high, I get large stresses 00:49:13.320 --> 00:49:14.790 align:middle line:84% on my inductor and capacitor, or I 00:49:14.790 --> 00:49:16.410 align:middle line:84% have to store a large amount of energy 00:49:16.410 --> 00:49:18.910 align:middle line:84% for the amount of energy I put into the resistor. 00:49:18.910 --> 00:49:20.890 align:middle line:90% That's not so good. 00:49:20.890 --> 00:49:23.410 align:middle line:84% And in the medium, maybe I get something 00:49:23.410 --> 00:49:27.670 align:middle line:84% where I get nice ability to modulate the amplitude 00:49:27.670 --> 00:49:30.250 align:middle line:84% with frequency over a reasonable frequency range 00:49:30.250 --> 00:49:35.800 align:middle line:84% without putting too much energy storage in my network. 00:49:35.800 --> 00:49:36.765 align:middle line:90% Questions about that? 00:49:36.765 --> 00:49:40.780 align:middle line:90% 00:49:40.780 --> 00:49:43.180 align:middle line:84% So what I want you to take away from today, 00:49:43.180 --> 00:49:45.520 align:middle line:84% and this is just done in the example of a series 00:49:45.520 --> 00:49:48.197 align:middle line:84% resonant circuit, first of all, when you're 00:49:48.197 --> 00:49:49.780 align:middle line:84% handed a circuit like this, and you're 00:49:49.780 --> 00:49:51.280 align:middle line:84% thinking about sinusoidal drive, you 00:49:51.280 --> 00:49:54.070 align:middle line:84% want to calculate characteristic impedance 00:49:54.070 --> 00:49:57.020 align:middle line:90% and natural resonant frequency. 00:49:57.020 --> 00:49:58.960 align:middle line:84% And the next thing you want to get after that 00:49:58.960 --> 00:50:00.400 align:middle line:90% is quality factor-- 00:50:00.400 --> 00:50:03.040 align:middle line:84% Q0, quality factor of the circuit. 00:50:03.040 --> 00:50:05.740 align:middle line:84% Because those are going to tell you 00:50:05.740 --> 00:50:11.770 align:middle line:84% essentially how peaky or sharp this whole thing is. 00:50:11.770 --> 00:50:14.020 align:middle line:84% And then the second thing we can take away 00:50:14.020 --> 00:50:16.420 align:middle line:84% from that is what kind of bandwidths 00:50:16.420 --> 00:50:18.760 align:middle line:84% do I need to operate to modulate power? 00:50:18.760 --> 00:50:20.710 align:middle line:84% And what kind of stress ranges am I 00:50:20.710 --> 00:50:23.820 align:middle line:84% going to get compared to my input drive in the network? 00:50:23.820 --> 00:50:26.490 align:middle line:90% 00:50:26.490 --> 00:50:29.920 align:middle line:90% We will take this up next class. 00:50:29.920 --> 00:50:32.010 align:middle line:90% Go out and have a beautiful day. 00:50:32.010 --> 00:50:33.950 align:middle line:90% Nice day out. 00:50:33.950 --> 00:50:37.000 align:middle line:90%