WEBVTT 00:00:00.000 --> 00:00:01.988 align:middle line:90% [SQUEAKING] 00:00:01.988 --> 00:00:04.473 align:middle line:90% [RUSTLING] 00:00:04.473 --> 00:00:05.964 align:middle line:90% [CLICKING] 00:00:05.964 --> 00:00:09.250 align:middle line:90% 00:00:09.250 --> 00:00:11.290 align:middle line:84% DAVID PERREAULT: So today, I wanted 00:00:11.290 --> 00:00:16.059 align:middle line:84% to continue our discussion of resonant power converters, which 00:00:16.059 --> 00:00:21.130 align:middle line:84% are power converters, either inverters or rectifiers or DC 00:00:21.130 --> 00:00:26.110 align:middle line:84% converters, where the power is modulated 00:00:26.110 --> 00:00:28.900 align:middle line:84% through high-frequency sinusoidal wave forms. 00:00:28.900 --> 00:00:37.600 align:middle line:84% And this is again, KPVS chapter 10. 00:00:37.600 --> 00:00:43.330 align:middle line:84% And last time, we looked at the characteristics of an example 00:00:43.330 --> 00:00:46.510 align:middle line:84% resonant network, in particular, a series resonant network, 00:00:46.510 --> 00:00:50.860 align:middle line:84% where we said, OK, if we had some network that we 00:00:50.860 --> 00:00:55.945 align:middle line:84% were going to drive with a sinusoid VS cosine of omega t. 00:00:55.945 --> 00:00:58.910 align:middle line:90% 00:00:58.910 --> 00:01:01.200 align:middle line:84% And maybe this would be a series resonant network. 00:01:01.200 --> 00:01:13.070 align:middle line:84% So I'd have IL, VC and R, where we 00:01:13.070 --> 00:01:18.860 align:middle line:84% are going to deliver energy from this source into the resistor. 00:01:18.860 --> 00:01:21.770 align:middle line:84% And we said we could look at the admittance, 00:01:21.770 --> 00:01:22.863 align:middle line:90% looking into the network. 00:01:22.863 --> 00:01:28.300 align:middle line:90% 00:01:28.300 --> 00:01:31.160 align:middle line:84% And we said, that looks something like this. 00:01:31.160 --> 00:01:33.610 align:middle line:84% If I plotted the magnitude of the admittance 00:01:33.610 --> 00:01:44.800 align:middle line:84% on a log scale versus frequency on a log scale, 00:01:44.800 --> 00:01:47.980 align:middle line:84% we said, OK, at low frequencies, the impedance 00:01:47.980 --> 00:01:49.645 align:middle line:90% is dominated by the-- 00:01:49.645 --> 00:01:53.605 align:middle line:90% 00:01:53.605 --> 00:01:56.540 align:middle line:84% the admittance is dominated by the capacitor. 00:01:56.540 --> 00:01:59.530 align:middle line:90% So we get omega C. 00:01:59.530 --> 00:02:04.160 align:middle line:84% At high frequencies, it's dominated by the inductor. 00:02:04.160 --> 00:02:09.310 align:middle line:84% So I get 1 over omega L. And at some frequency, 00:02:09.310 --> 00:02:14.290 align:middle line:84% the inductance and capacitive impedances cancel each other. 00:02:14.290 --> 00:02:19.930 align:middle line:84% And that's at omega 0 is equal to 1 over the square root of LC. 00:02:19.930 --> 00:02:25.510 align:middle line:84% And that asymptote happens at 1 over Z0, 00:02:25.510 --> 00:02:29.090 align:middle line:84% which is equal to the square root of C/L. 00:02:29.090 --> 00:02:32.420 align:middle line:84% And right where these impedances cancel, 00:02:32.420 --> 00:02:34.880 align:middle line:84% the admittance goes to being 1/R. 00:02:34.880 --> 00:02:36.810 align:middle line:84% So maybe it looks like something like this. 00:02:36.810 --> 00:02:41.010 align:middle line:84% I have 1/R, where it peaks up to. 00:02:41.010 --> 00:02:43.160 align:middle line:84% So maybe my admittance characteristic 00:02:43.160 --> 00:02:45.170 align:middle line:90% looks something like this. 00:02:45.170 --> 00:02:47.330 align:middle line:90% It rises. 00:02:47.330 --> 00:02:50.335 align:middle line:84% I'll have some region about which it peaks up. 00:02:50.335 --> 00:02:53.530 align:middle line:90% 00:02:53.530 --> 00:02:55.600 align:middle line:90% And then it falls. 00:02:55.600 --> 00:02:58.360 align:middle line:90% And we said, OK, we have-- 00:02:58.360 --> 00:03:03.880 align:middle line:84% if we have a high Q network, a high-quality factor network, 00:03:03.880 --> 00:03:09.250 align:middle line:84% which we said, the quality factor Q 00:03:09.250 --> 00:03:11.860 align:middle line:84% of the network evaluated at omega 00:03:11.860 --> 00:03:20.080 align:middle line:84% equals omega 0, which we might call Q0, is equal to Z0 over R, 00:03:20.080 --> 00:03:24.940 align:middle line:84% or the square root of L/C over R. 00:03:24.940 --> 00:03:29.890 align:middle line:84% And we said that was exactly the amount by which this admittance 00:03:29.890 --> 00:03:34.060 align:middle line:84% characteristic peaks up above the intersection 00:03:34.060 --> 00:03:37.190 align:middle line:84% point of the low- and high-frequency asymptotes. 00:03:37.190 --> 00:03:38.800 align:middle line:90% So it peaks up by an amount. 00:03:38.800 --> 00:03:43.810 align:middle line:84% Q0 is equal to the square root of L/C over R. 00:03:43.810 --> 00:03:47.140 align:middle line:84% And then the other thing we said was, 00:03:47.140 --> 00:03:50.320 align:middle line:90% how broad band is that peaking? 00:03:50.320 --> 00:03:54.560 align:middle line:84% And if we took the half power point, the point-- 00:03:54.560 --> 00:03:58.160 align:middle line:84% the bandwidth over which we deliver half the power 00:03:58.160 --> 00:04:02.840 align:middle line:84% to the resistor as we would at exactly 00:04:02.840 --> 00:04:04.850 align:middle line:84% at resonance where the inductance 00:04:04.850 --> 00:04:07.290 align:middle line:84% and capacitance of impedances cancel each other, 00:04:07.290 --> 00:04:11.270 align:middle line:84% then we said, OK, that bandwidth, 00:04:11.270 --> 00:04:15.530 align:middle line:84% if I took the bandwidth divided by omega 0, 00:04:15.530 --> 00:04:18.500 align:middle line:84% the center frequency-- so the percent bandwidth is 00:04:18.500 --> 00:04:24.770 align:middle line:84% exactly equal to 1 over Q0, which is 00:04:24.770 --> 00:04:27.870 align:middle line:90% R over the square root of L/C. 00:04:27.870 --> 00:04:30.240 align:middle line:84% Now, this was all done for the series network. 00:04:30.240 --> 00:04:32.850 align:middle line:84% But we have these nice relationships 00:04:32.850 --> 00:04:35.970 align:middle line:84% which tell us how frequency selective this network is. 00:04:35.970 --> 00:04:39.690 align:middle line:84% So that's just a quick review of what we talked about last time. 00:04:39.690 --> 00:04:42.430 align:middle line:84% How are we going to use this in a power converter? 00:04:42.430 --> 00:04:45.700 align:middle line:90% 00:04:45.700 --> 00:04:48.200 align:middle line:84% Especially, if it's a high Q network, 00:04:48.200 --> 00:04:52.930 align:middle line:84% this high sensitivity to frequency 00:04:52.930 --> 00:04:56.515 align:middle line:84% means that by modulating the frequency of some drive wave 00:04:56.515 --> 00:05:00.310 align:middle line:84% form, I can very easily modulate the power going 00:05:00.310 --> 00:05:02.410 align:middle line:90% into the resistor. 00:05:02.410 --> 00:05:05.650 align:middle line:84% So using frequency modulation isn't always the way 00:05:05.650 --> 00:05:07.030 align:middle line:84% you operate a resonant converter. 00:05:07.030 --> 00:05:12.444 align:middle line:84% But it's a very common way to operate a resonant converter. 00:05:12.444 --> 00:05:15.430 align:middle line:84% So if we were to think about making an inverter, 00:05:15.430 --> 00:05:20.770 align:middle line:84% then by controlling frequency either above or below resonance, 00:05:20.770 --> 00:05:23.290 align:middle line:90% we can turn down the power. 00:05:23.290 --> 00:05:26.170 align:middle line:84% And when we're right at resonance, 00:05:26.170 --> 00:05:28.520 align:middle line:90% VS appears across the resistor. 00:05:28.520 --> 00:05:31.180 align:middle line:84% So that's the maximum voltage I can generate in the resistor 00:05:31.180 --> 00:05:33.460 align:middle line:84% because that's where the inductor and capacitive 00:05:33.460 --> 00:05:34.810 align:middle line:90% impedances are canceling. 00:05:34.810 --> 00:05:36.430 align:middle line:84% And at any other frequency, I'm going 00:05:36.430 --> 00:05:41.290 align:middle line:84% to have either a net effective inductive reactance 00:05:41.290 --> 00:05:43.400 align:middle line:84% or capacitive reactance in series. 00:05:43.400 --> 00:05:44.980 align:middle line:90% So I get a voltage divider. 00:05:44.980 --> 00:05:48.130 align:middle line:84% My admittance goes down, and I deliver less power. 00:05:48.130 --> 00:05:49.875 align:middle line:90% Any questions about that? 00:05:49.875 --> 00:05:53.170 align:middle line:90% 00:05:53.170 --> 00:05:58.930 align:middle line:84% So let's think about a basic design. 00:05:58.930 --> 00:06:01.510 align:middle line:90% Here's what we might do. 00:06:01.510 --> 00:06:04.190 align:middle line:84% How am I going to get this drive? 00:06:04.190 --> 00:06:07.030 align:middle line:84% Well, it doesn't have to be a sinusoidal drive, in part 00:06:07.030 --> 00:06:10.750 align:middle line:84% because this network is very, very frequency selective. 00:06:10.750 --> 00:06:13.300 align:middle line:84% So I could drive with some other voltage wave form 00:06:13.300 --> 00:06:20.340 align:middle line:84% and extract out the fundamental component going to the resistor. 00:06:20.340 --> 00:06:22.180 align:middle line:90% So maybe what I would do-- 00:06:22.180 --> 00:06:26.950 align:middle line:84% and this could either be called a half bridge 00:06:26.950 --> 00:06:30.310 align:middle line:84% inverter, resonant inverter, series resonant inverter. 00:06:30.310 --> 00:06:34.990 align:middle line:84% Somebody in the RF field might call it a class D power 00:06:34.990 --> 00:06:35.660 align:middle line:90% amplifier. 00:06:35.660 --> 00:06:37.360 align:middle line:84% That's another term that gets used. 00:06:37.360 --> 00:06:39.820 align:middle line:90% But it could look like this. 00:06:39.820 --> 00:06:50.660 align:middle line:84% Maybe I'll have some voltage VDC, a pair of switches. 00:06:50.660 --> 00:06:54.360 align:middle line:84% And let's assume they're MOSFETs for a second. 00:06:54.360 --> 00:06:56.510 align:middle line:84% So internally, they have these body 00:06:56.510 --> 00:07:00.110 align:middle line:84% diodes that I'll just draw for fun here. 00:07:00.110 --> 00:07:06.200 align:middle line:84% And we can say, OK, I have L. And I have some current I sub 00:07:06.200 --> 00:07:10.910 align:middle line:84% L. I have V sub C, which is going to block any DC component. 00:07:10.910 --> 00:07:15.290 align:middle line:84% And then I have my output voltage, V sub R, 00:07:15.290 --> 00:07:19.718 align:middle line:84% which I'm-- to which I'm going to deliver energy. 00:07:19.718 --> 00:07:31.481 align:middle line:84% And I might call this voltage here V sub X. 00:07:31.481 --> 00:07:34.265 align:middle line:84% So what is V sub X going to look like? 00:07:34.265 --> 00:07:37.940 align:middle line:90% 00:07:37.940 --> 00:07:42.380 align:middle line:84% If I just operate these two switches at 50% duty ratio, 00:07:42.380 --> 00:07:44.790 align:middle line:84% maybe V sub X looks something like this. 00:07:44.790 --> 00:07:48.600 align:middle line:90% It modulates between VDC and 0. 00:07:48.600 --> 00:07:52.240 align:middle line:84% At some switching frequency, that maybe varies. 00:07:52.240 --> 00:07:58.410 align:middle line:84% So this may be 2 pi over f switch. 00:07:58.410 --> 00:08:00.990 align:middle line:90% And this is in time. 00:08:00.990 --> 00:08:06.000 align:middle line:90% And this is pi over fS. 00:08:06.000 --> 00:08:11.570 align:middle line:84% So I'm going to run this, for example, at 50% duty ratio. 00:08:11.570 --> 00:08:14.470 align:middle line:84% So if I did that and I looked-- thought about-- 00:08:14.470 --> 00:08:17.770 align:middle line:90% this is my wave form VX-- 00:08:17.770 --> 00:08:20.540 align:middle line:84% what's important about VX in here? 00:08:20.540 --> 00:08:22.160 align:middle line:90% Well, I'm going to run this. 00:08:22.160 --> 00:08:24.850 align:middle line:84% I'm going to apply this to this very frequency-selective 00:08:24.850 --> 00:08:26.870 align:middle line:90% network. 00:08:26.870 --> 00:08:32.860 align:middle line:84% And let me assume that I'm going to be somewhere near resonance 00:08:32.860 --> 00:08:37.690 align:middle line:84% such that the current is going to be 00:08:37.690 --> 00:08:42.626 align:middle line:84% the voltage times the admittance here. 00:08:42.626 --> 00:08:48.940 align:middle line:84% So if I thought about this, maybe I would say, well, 00:08:48.940 --> 00:08:50.920 align:middle line:84% what does VX look like in frequency? 00:08:50.920 --> 00:08:53.260 align:middle line:84% In the frequency domain, I could think 00:08:53.260 --> 00:09:00.720 align:middle line:84% of VX of j omega, which would look like what? 00:09:00.720 --> 00:09:03.510 align:middle line:90% 00:09:03.510 --> 00:09:06.660 align:middle line:84% Well, I would have a DC term, which 00:09:06.660 --> 00:09:11.830 align:middle line:84% is basically related to the average value of the wave form. 00:09:11.830 --> 00:09:14.360 align:middle line:90% 00:09:14.360 --> 00:09:16.770 align:middle line:90% And then I would have-- 00:09:16.770 --> 00:09:25.770 align:middle line:84% at 2 pi fS, I would have some frequency content. 00:09:25.770 --> 00:09:31.760 align:middle line:84% And at minus 2 pi over fS, I'd have the same frequency content. 00:09:31.760 --> 00:09:35.010 align:middle line:84% And then there's no second harmonic, 00:09:35.010 --> 00:09:40.560 align:middle line:84% because this is half-wave symmetric, or at least the AC 00:09:40.560 --> 00:09:42.820 align:middle line:90% component's half-wave symmetric. 00:09:42.820 --> 00:09:50.565 align:middle line:84% And so then the next thing would be the third harmonic out here. 00:09:50.565 --> 00:09:53.980 align:middle line:84% And if I think about this, and if you 00:09:53.980 --> 00:09:56.620 align:middle line:84% wanted to use the exponential series, the height of this 00:09:56.620 --> 00:10:02.140 align:middle line:90% is 2 pi times VDC over pi. 00:10:02.140 --> 00:10:08.060 align:middle line:84% And here would be 2 pi VDC over 3 pi. 00:10:08.060 --> 00:10:11.350 align:middle line:84% So the third harmonic is one third of the fundamental. 00:10:11.350 --> 00:10:15.210 align:middle line:84% And it falls off as 1/n with n-odd harmonics. 00:10:15.210 --> 00:10:20.620 align:middle line:84% But what's important about this wave form is 00:10:20.620 --> 00:10:24.490 align:middle line:84% if I believe that this network is going 00:10:24.490 --> 00:10:30.270 align:middle line:84% to be doing a good job of filtering this square wave, 00:10:30.270 --> 00:10:34.180 align:middle line:84% in terms of what's important for driving current in this network, 00:10:34.180 --> 00:10:37.930 align:middle line:84% as long as this looks like a high impedance 00:10:37.930 --> 00:10:39.730 align:middle line:90% or a low admittance-- 00:10:39.730 --> 00:10:40.670 align:middle line:90% I'm looking in here. 00:10:40.670 --> 00:10:43.930 align:middle line:90% This is Y in. 00:10:43.930 --> 00:10:47.197 align:middle line:84% As long as the admittance low with 0 at DC, 00:10:47.197 --> 00:10:49.780 align:middle line:84% because of the capacitor-- so I don't really care about the DC 00:10:49.780 --> 00:10:51.040 align:middle line:90% component-- 00:10:51.040 --> 00:10:54.580 align:middle line:84% and as long as this looks like a high impedance 00:10:54.580 --> 00:10:57.640 align:middle line:84% or a low admittance at the third harmonic, 00:10:57.640 --> 00:11:01.220 align:middle line:84% the only element I really care about is the fundamental. 00:11:01.220 --> 00:11:03.970 align:middle line:90% 00:11:03.970 --> 00:11:06.850 align:middle line:84% So I could then just say, let me just 00:11:06.850 --> 00:11:11.320 align:middle line:84% think about the fundamental component driving the network. 00:11:11.320 --> 00:11:17.620 align:middle line:84% And so maybe I would mentally approximate this 00:11:17.620 --> 00:11:18.966 align:middle line:90% as the following. 00:11:18.966 --> 00:11:22.980 align:middle line:90% 00:11:22.980 --> 00:11:29.070 align:middle line:84% I call this VX1, the fundamental component, 00:11:29.070 --> 00:11:37.650 align:middle line:84% which would be VX1 of t would be equal to 2 over pi VDC. 00:11:37.650 --> 00:11:40.830 align:middle line:90% 00:11:40.830 --> 00:11:42.695 align:middle line:84% In this case, I've drawn it as a sine. 00:11:42.695 --> 00:11:44.070 align:middle line:84% So I guess it would be sine omega 00:11:44.070 --> 00:11:49.660 align:middle line:84% t, if I only thought about the fundamental component. 00:11:49.660 --> 00:11:53.680 align:middle line:84% Or I could model it exactly as VX over here 00:11:53.680 --> 00:11:55.930 align:middle line:84% and think about the square wave component. 00:11:55.930 --> 00:12:00.370 align:middle line:84% And then I run this into my network. 00:12:00.370 --> 00:12:06.100 align:middle line:90% 00:12:06.100 --> 00:12:11.560 align:middle line:84% And if I look at this then, what am I going to get? 00:12:11.560 --> 00:12:13.750 align:middle line:90% I'm going to get a current. 00:12:13.750 --> 00:12:18.070 align:middle line:84% Let me call this IL, which is also the resistor current. 00:12:18.070 --> 00:12:25.500 align:middle line:84% IL of j omega S is simply going is simply 00:12:25.500 --> 00:12:37.710 align:middle line:84% going to be equal to VX of j omega S times Y in of j omega S. 00:12:37.710 --> 00:12:42.420 align:middle line:84% In fact, if I'm going to do it, let me just be careful. 00:12:42.420 --> 00:12:45.660 align:middle line:84% If I'm just going to do this in the Laplace domain 00:12:45.660 --> 00:12:48.270 align:middle line:84% and not worry about specific frequency content, 00:12:48.270 --> 00:12:50.340 align:middle line:84% maybe I would just say this is going 00:12:50.340 --> 00:12:53.760 align:middle line:84% to be IL of S is VX of S times Y in of S. 00:12:53.760 --> 00:12:58.185 align:middle line:84% And then I can evaluate that in the frequency domain 00:12:58.185 --> 00:13:00.480 align:middle line:90% on S equals j omega. 00:13:00.480 --> 00:13:02.790 align:middle line:90% But here's the point. 00:13:02.790 --> 00:13:08.980 align:middle line:84% If everything gets filtered except for the fundamental, 00:13:08.980 --> 00:13:21.930 align:middle line:84% then I would expect IL of t will be only 00:13:21.930 --> 00:13:26.540 align:middle line:84% a sinusoid at the fundamental switching frequency. 00:13:26.540 --> 00:13:29.690 align:middle line:90% 00:13:29.690 --> 00:13:33.970 align:middle line:84% So that's what we call the fundamental harmonic 00:13:33.970 --> 00:13:35.800 align:middle line:90% approximation. 00:13:35.800 --> 00:13:39.820 align:middle line:84% What I'm assuming is this resonant tank 00:13:39.820 --> 00:13:44.680 align:middle line:84% is sufficiently frequency selective that I can ignore all 00:13:44.680 --> 00:13:46.430 align:middle line:90% the nonfundamental components. 00:13:46.430 --> 00:13:49.900 align:middle line:84% I can ignore DC perfectly, because there's no DC component. 00:13:49.900 --> 00:13:52.630 align:middle line:84% But whatever third harmonic shows up, 00:13:52.630 --> 00:13:55.510 align:middle line:84% I just assume the admittance down to the third harmonic 00:13:55.510 --> 00:13:56.360 align:middle line:90% is really low. 00:13:56.360 --> 00:14:00.200 align:middle line:90% 00:14:00.200 --> 00:14:03.290 align:middle line:84% For example, if my fundamental were here, 00:14:03.290 --> 00:14:06.260 align:middle line:84% maybe my second harmonic is here. 00:14:06.260 --> 00:14:08.210 align:middle line:84% My third harmonic is way down here. 00:14:08.210 --> 00:14:09.920 align:middle line:84% You started off at a third of the height. 00:14:09.920 --> 00:14:13.170 align:middle line:84% And then it's got this much bigger attenuation. 00:14:13.170 --> 00:14:16.070 align:middle line:84% Maybe I just ignore the higher harmonic content of the current 00:14:16.070 --> 00:14:20.030 align:middle line:84% and just pretend this perfect sinusoid. 00:14:20.030 --> 00:14:22.582 align:middle line:90% Any questions about that? 00:14:22.582 --> 00:14:24.040 align:middle line:84% AUDIENCE: You're saying we can make 00:14:24.040 --> 00:14:25.810 align:middle line:84% this approximation because we're close 00:14:25.810 --> 00:14:27.000 align:middle line:90% to that resonant frequency? 00:14:27.000 --> 00:14:28.250 align:middle line:90% DAVID PERREAULT: That's right. 00:14:28.250 --> 00:14:31.630 align:middle line:84% And there are times when that approximation breaks down. 00:14:31.630 --> 00:14:33.190 align:middle line:90% And we'll talk about that. 00:14:33.190 --> 00:14:38.230 align:middle line:84% But to first order, for basic analysis of resonant converters, 00:14:38.230 --> 00:14:40.780 align:middle line:84% we often make the assumption that 00:14:40.780 --> 00:14:43.840 align:middle line:84% at least as far as the current I'm going to get 00:14:43.840 --> 00:14:47.680 align:middle line:84% is concerned, I can just focus on the fundamental 00:14:47.680 --> 00:14:50.560 align:middle line:84% and just think of that current as being 00:14:50.560 --> 00:14:54.640 align:middle line:84% a sine wave with a magnitude that's determined 00:14:54.640 --> 00:15:00.210 align:middle line:84% by the magnitude of the voltage I'm driving with times whatever 00:15:00.210 --> 00:15:05.790 align:middle line:84% the amplitude is here, in terms of the admittance. 00:15:05.790 --> 00:15:08.270 align:middle line:90% Any questions about that? 00:15:08.270 --> 00:15:11.240 align:middle line:84% And then, of course, the power is then I squared R. 00:15:11.240 --> 00:15:13.910 align:middle line:90% So we get a lot of attenuation. 00:15:13.910 --> 00:15:15.390 align:middle line:84% We get a very steep characteristic 00:15:15.390 --> 00:15:16.800 align:middle line:90% of that very frequency. 00:15:16.800 --> 00:15:19.660 align:middle line:90% 00:15:19.660 --> 00:15:20.260 align:middle line:90% All right. 00:15:20.260 --> 00:15:22.023 align:middle line:84% So we get an approximate sinusoid. 00:15:22.023 --> 00:15:24.730 align:middle line:90% 00:15:24.730 --> 00:15:26.860 align:middle line:90% So I could say-- 00:15:26.860 --> 00:15:33.670 align:middle line:84% I could write then here that the component at the fundamental-- 00:15:33.670 --> 00:15:37.660 align:middle line:90% I have IL of t here. 00:15:37.660 --> 00:15:42.430 align:middle line:84% But I could relate that back and forth with IL 00:15:42.430 --> 00:15:53.250 align:middle line:84% of j omega S, which is equal to VX1, 00:15:53.250 --> 00:15:56.280 align:middle line:84% the amplitude of the fundamental, 00:15:56.280 --> 00:16:04.560 align:middle line:84% times Y in of j omega S. And it's the fact that this-- 00:16:04.560 --> 00:16:06.690 align:middle line:84% the admittance is so frequency selective 00:16:06.690 --> 00:16:08.045 align:middle line:90% that lets me modulate power. 00:16:08.045 --> 00:16:10.610 align:middle line:90% 00:16:10.610 --> 00:16:12.200 align:middle line:90% All right. 00:16:12.200 --> 00:16:14.580 align:middle line:84% Let's think about how we could do that. 00:16:14.580 --> 00:16:16.730 align:middle line:84% We could modulate power, actually, 00:16:16.730 --> 00:16:20.670 align:middle line:84% by going in either direction off of resonance. 00:16:20.670 --> 00:16:22.490 align:middle line:90% I would get maximum-- 00:16:22.490 --> 00:16:25.430 align:middle line:84% according to this, if I put my switching frequency right here, 00:16:25.430 --> 00:16:28.250 align:middle line:84% I'd get maximum power for a given voltage. 00:16:28.250 --> 00:16:31.520 align:middle line:84% If I go in either direction, the admittance goes down, 00:16:31.520 --> 00:16:35.960 align:middle line:90% and I get less power. 00:16:35.960 --> 00:16:38.300 align:middle line:84% And the maximum voltage I can apply 00:16:38.300 --> 00:16:42.080 align:middle line:84% with this particular network is the voltage across the resistor 00:16:42.080 --> 00:16:44.385 align:middle line:84% is equal to the input voltage, or the-- 00:16:44.385 --> 00:16:46.510 align:middle line:84% I should say, the fundamental of the input voltage. 00:16:46.510 --> 00:16:49.090 align:middle line:90% 00:16:49.090 --> 00:16:49.730 align:middle line:90% All right. 00:16:49.730 --> 00:16:51.610 align:middle line:90% Well, what would I choose to do? 00:16:51.610 --> 00:16:54.460 align:middle line:84% Very often, what we're going to do 00:16:54.460 --> 00:16:58.420 align:middle line:84% is we're going to run this above resonance, at least 00:16:58.420 --> 00:16:59.570 align:middle line:90% for modern devices. 00:16:59.570 --> 00:17:01.750 align:middle line:84% And when I talk about resonant converters, 00:17:01.750 --> 00:17:05.980 align:middle line:84% they don't have to be soft switched. 00:17:05.980 --> 00:17:08.530 align:middle line:84% But frequently, you choose this kind of approach 00:17:08.530 --> 00:17:10.960 align:middle line:84% because you can make it soft switched. 00:17:10.960 --> 00:17:16.480 align:middle line:84% And if we operate above resonance, 00:17:16.480 --> 00:17:20.726 align:middle line:84% is the input admittance look inductive or capacitive? 00:17:20.726 --> 00:17:23.890 align:middle line:90% 00:17:23.890 --> 00:17:27.081 align:middle line:84% Or the input impedance, does it look inductive or capacitive? 00:17:27.081 --> 00:17:29.973 align:middle line:90% 00:17:29.973 --> 00:17:31.600 align:middle line:90% It looks inductive. 00:17:31.600 --> 00:17:33.700 align:middle line:84% As I go to higher and higher frequency, 00:17:33.700 --> 00:17:36.785 align:middle line:84% the magnitude of the impedance of the inductor 00:17:36.785 --> 00:17:37.660 align:middle line:90% keeps getting bigger. 00:17:37.660 --> 00:17:39.370 align:middle line:84% And the capacitor keeps getting smaller. 00:17:39.370 --> 00:17:44.140 align:middle line:84% So it starts to look inductive once I get above resonance. 00:17:44.140 --> 00:17:49.000 align:middle line:84% What that means is that relative to the phase of the voltage, 00:17:49.000 --> 00:17:50.770 align:middle line:84% the phase of the current is going 00:17:50.770 --> 00:17:54.280 align:middle line:84% to be phase shifted with a lagging current. 00:17:54.280 --> 00:17:58.810 align:middle line:84% So let's just think about what this would look like. 00:17:58.810 --> 00:18:02.625 align:middle line:90% Here's my half bridge again. 00:18:02.625 --> 00:18:06.560 align:middle line:90% 00:18:06.560 --> 00:18:10.370 align:middle line:90% And again, I'm going to-- 00:18:10.370 --> 00:18:14.390 align:middle line:84% let me just draw in the body diode of these FETs just 00:18:14.390 --> 00:18:16.990 align:middle line:90% for clarity. 00:18:16.990 --> 00:18:18.350 align:middle line:90% And I'm going to draw-- 00:18:18.350 --> 00:18:25.460 align:middle line:84% and here's L, C, and R. And when I'm talking about I sub L, 00:18:25.460 --> 00:18:33.840 align:middle line:84% I can think of the same thing as being the same as I sub R. 00:18:33.840 --> 00:18:36.810 align:middle line:84% And here is the voltage that I'm synthesizing. 00:18:36.810 --> 00:18:41.590 align:middle line:84% I'll call this V sub X. So V sub X looks like this. 00:18:41.590 --> 00:18:49.170 align:middle line:90% 00:18:49.170 --> 00:18:52.990 align:middle line:90% Just let me do this. 00:18:52.990 --> 00:19:02.110 align:middle line:84% V sub X looks like this, where I'm controlling this switching 00:19:02.110 --> 00:19:02.610 align:middle line:90% period. 00:19:02.610 --> 00:19:06.720 align:middle line:90% 00:19:06.720 --> 00:19:09.480 align:middle line:84% That should be-- supposed to be 50% duty ratio. 00:19:09.480 --> 00:19:12.360 align:middle line:90% So here's 2 pi over fS. 00:19:12.360 --> 00:19:14.730 align:middle line:90% And this is t. 00:19:14.730 --> 00:19:21.960 align:middle line:84% So if I'm above resonance, so if 2 pi fS 00:19:21.960 --> 00:19:25.890 align:middle line:84% is greater than omega 0, what I expect 00:19:25.890 --> 00:19:32.670 align:middle line:84% is this net admittance to look inductive, 00:19:32.670 --> 00:19:36.270 align:middle line:84% which means that I sub L should lag 00:19:36.270 --> 00:19:40.872 align:middle line:90% the fundamental of this voltage. 00:19:40.872 --> 00:19:43.770 align:middle line:84% Well, what does the fundamental of this voltage look like? 00:19:43.770 --> 00:19:45.330 align:middle line:90% This is VDC. 00:19:45.330 --> 00:19:51.240 align:middle line:90% So this is V sub X. And-- 00:19:51.240 --> 00:19:52.020 align:middle line:90% of t. 00:19:52.020 --> 00:19:57.060 align:middle line:84% What his fundamental looks like is something like this. 00:19:57.060 --> 00:20:03.095 align:middle line:84% V sub X1 is in phase with V sub X. And it looks like this. 00:20:03.095 --> 00:20:07.950 align:middle line:90% 00:20:07.950 --> 00:20:13.530 align:middle line:84% So the fundamental of V sub X has an amplitude. 00:20:13.530 --> 00:20:15.570 align:middle line:90% If the amplitude of-- 00:20:15.570 --> 00:20:22.710 align:middle line:84% if the peak value of VX is VDC, this value 00:20:22.710 --> 00:20:27.130 align:middle line:84% is going to be 2 over pi times VDC. 00:20:27.130 --> 00:20:29.730 align:middle line:90% It's just a square wave. 00:20:29.730 --> 00:20:34.030 align:middle line:84% That's what you get for a square wave. 00:20:34.030 --> 00:20:38.760 align:middle line:84% And what I've said is that the inductor current should lag 00:20:38.760 --> 00:20:41.050 align:middle line:90% the fundamental of the voltage. 00:20:41.050 --> 00:20:43.073 align:middle line:90% So this is VX1. 00:20:43.073 --> 00:20:48.380 align:middle line:90% 00:20:48.380 --> 00:20:50.210 align:middle line:90% The fundamentals should lag. 00:20:50.210 --> 00:20:52.360 align:middle line:84% So what I should see then is a current 00:20:52.360 --> 00:20:53.865 align:middle line:90% that does something like this. 00:20:53.865 --> 00:21:03.750 align:middle line:90% 00:21:03.750 --> 00:21:11.760 align:middle line:84% So here is IL of t for under this condition. 00:21:11.760 --> 00:21:15.110 align:middle line:90% Any questions about that? 00:21:15.110 --> 00:21:17.990 align:middle line:84% Now, why am I interested in this case? 00:21:17.990 --> 00:21:20.060 align:middle line:84% Because let's think about the switching 00:21:20.060 --> 00:21:23.473 align:middle line:90% transitions on this inverter. 00:21:23.473 --> 00:21:26.860 align:middle line:90% 00:21:26.860 --> 00:21:30.040 align:middle line:84% Let's think about what's happening here. 00:21:30.040 --> 00:21:38.830 align:middle line:84% If I call this S1 and this S2, in this time period, S1 is on. 00:21:38.830 --> 00:21:41.280 align:middle line:84% And in this time period, S2 is on. 00:21:41.280 --> 00:21:44.680 align:middle line:90% 00:21:44.680 --> 00:21:46.150 align:middle line:90% Make sense, everybody? 00:21:46.150 --> 00:21:50.530 align:middle line:84% Let's focus in right on this switching transition 00:21:50.530 --> 00:21:53.680 align:middle line:84% right here, right when I'm going to go from S1 to S2. 00:21:53.680 --> 00:21:56.440 align:middle line:90% And let me zoom in on that. 00:21:56.440 --> 00:21:57.555 align:middle line:90% So here I am. 00:21:57.555 --> 00:22:02.000 align:middle line:90% 00:22:02.000 --> 00:22:03.870 align:middle line:90% I have S1 on. 00:22:03.870 --> 00:22:10.620 align:middle line:90% 00:22:10.620 --> 00:22:13.320 align:middle line:84% At some point, I'm going to turn S1 off. 00:22:13.320 --> 00:22:16.680 align:middle line:90% 00:22:16.680 --> 00:22:18.960 align:middle line:90% And I'm going to initiate-- 00:22:18.960 --> 00:22:20.130 align:middle line:90% I'm going to turn S1 off. 00:22:20.130 --> 00:22:22.290 align:middle line:84% And then I'm going to turn S2 on. 00:22:22.290 --> 00:22:24.525 align:middle line:84% But I need some dead time between them, 00:22:24.525 --> 00:22:26.400 align:middle line:84% because otherwise, I could accidentally short 00:22:26.400 --> 00:22:28.850 align:middle line:90% out my DC bus. 00:22:28.850 --> 00:22:33.115 align:middle line:84% And in that time period, what is the sign of the current IL? 00:22:33.115 --> 00:22:37.040 align:middle line:90% 00:22:37.040 --> 00:22:39.620 align:middle line:90% It's greater than 0. 00:22:39.620 --> 00:22:44.060 align:middle line:84% So if I were to think about this switching transition-- 00:22:44.060 --> 00:22:48.440 align:middle line:84% and let me just think about this particular time. 00:22:48.440 --> 00:22:52.370 align:middle line:84% What I'm going to do at that particular time-- 00:22:52.370 --> 00:22:55.230 align:middle line:90% here's VDC. 00:22:55.230 --> 00:23:00.650 align:middle line:84% If I were to think about this, what I'm doing is at that time, 00:23:00.650 --> 00:23:02.660 align:middle line:90% I'm opening this switch. 00:23:02.660 --> 00:23:04.770 align:middle line:84% And then this switch is already open. 00:23:04.770 --> 00:23:05.900 align:middle line:90% But there's a diode here. 00:23:05.900 --> 00:23:08.420 align:middle line:90% 00:23:08.420 --> 00:23:11.090 align:middle line:84% And then there's an output current that's 00:23:11.090 --> 00:23:13.765 align:middle line:90% IL that's greater than 0. 00:23:13.765 --> 00:23:16.520 align:middle line:90% 00:23:16.520 --> 00:23:22.190 align:middle line:84% Now, the thing I haven't drawn in this picture 00:23:22.190 --> 00:23:26.390 align:middle line:84% is the fact that these devices actually have some capacitance. 00:23:26.390 --> 00:23:29.700 align:middle line:84% And I could go add additional capacitance. 00:23:29.700 --> 00:23:34.010 align:middle line:84% So that's either the device capacitance itself 00:23:34.010 --> 00:23:38.830 align:middle line:84% or maybe I'm adding external capacitance. 00:23:38.830 --> 00:23:42.082 align:middle line:84% Now, if I put capacitors across the device, 00:23:42.082 --> 00:23:44.290 align:middle line:84% what kind of switching do you think I'm going to get? 00:23:44.290 --> 00:23:47.940 align:middle line:84% Zero-voltage switching or zero current switching? 00:23:47.940 --> 00:23:49.320 align:middle line:90% Zero-voltage switching, right? 00:23:49.320 --> 00:23:52.080 align:middle line:84% So because when I turn this switch off, 00:23:52.080 --> 00:23:56.300 align:middle line:84% when I pop the switch off, when I go to turn the switch off, 00:23:56.300 --> 00:23:57.860 align:middle line:90% there's a capacitor here. 00:23:57.860 --> 00:24:01.100 align:middle line:84% And there's a capacitor here, which 00:24:01.100 --> 00:24:03.890 align:middle line:84% are going to keep the voltage across the turning 00:24:03.890 --> 00:24:08.190 align:middle line:84% off switch low while he's turning off. 00:24:08.190 --> 00:24:12.620 align:middle line:84% So I can turn off the top switch at very low loss. 00:24:12.620 --> 00:24:15.470 align:middle line:84% What's going to happen during this time period? 00:24:15.470 --> 00:24:18.630 align:middle line:84% So imagine I turn this switch off, and this switch is off. 00:24:18.630 --> 00:24:23.300 align:middle line:84% So now I have neither switch is on. 00:24:23.300 --> 00:24:26.550 align:middle line:84% Well, I have a positive current here. 00:24:26.550 --> 00:24:30.650 align:middle line:84% The total voltage VDC across these two capacitors 00:24:30.650 --> 00:24:32.710 align:middle line:90% is going to be constant. 00:24:32.710 --> 00:24:34.730 align:middle line:84% And if these two capacitors are equal, 00:24:34.730 --> 00:24:36.350 align:middle line:84% that means actually the current has 00:24:36.350 --> 00:24:44.866 align:middle line:84% to split between this capacitor and this capacitor. 00:24:44.866 --> 00:24:47.920 align:middle line:84% And so that means the voltage on the top capacitor 00:24:47.920 --> 00:24:49.150 align:middle line:90% is going to charge up. 00:24:49.150 --> 00:24:51.310 align:middle line:84% And the voltage across the bottom capacitor 00:24:51.310 --> 00:24:53.990 align:middle line:90% is going to charge down. 00:24:53.990 --> 00:24:56.043 align:middle line:84% So then if I were to look at this voltage, 00:24:56.043 --> 00:24:58.210 align:middle line:84% I'm going to zoom in right at this time period, when 00:24:58.210 --> 00:25:00.560 align:middle line:90% I have both S1 and S2 off. 00:25:00.560 --> 00:25:02.710 align:middle line:84% What I'm going to see is a transition 00:25:02.710 --> 00:25:04.470 align:middle line:84% that's going to do something like this. 00:25:04.470 --> 00:25:10.651 align:middle line:90% 00:25:10.651 --> 00:25:14.020 align:middle line:84% And then when this voltage hits 0-- 00:25:14.020 --> 00:25:16.870 align:middle line:90% and this is voltage VX-- 00:25:16.870 --> 00:25:20.440 align:middle line:84% when this voltage hits 0, the diode has got to turn on. 00:25:20.440 --> 00:25:25.180 align:middle line:84% So if this was S1 on, even if I don't turn-- even if I don't 00:25:25.180 --> 00:25:29.750 align:middle line:84% turn this bottom switch on yet, when this hits 0, 00:25:29.750 --> 00:25:34.970 align:middle line:90% I'm going to get S2/D2 is on. 00:25:34.970 --> 00:25:36.620 align:middle line:84% The diode will turn on naturally. 00:25:36.620 --> 00:25:38.740 align:middle line:84% And then I can turn on the switch with 0 voltage. 00:25:38.740 --> 00:25:41.260 align:middle line:90% 00:25:41.260 --> 00:25:45.010 align:middle line:84% So if I were to zoom in on this switching transition, 00:25:45.010 --> 00:25:46.900 align:middle line:84% if I put some capacitance here and I 00:25:46.900 --> 00:25:49.300 align:middle line:84% put some dead time between S1 and S2, 00:25:49.300 --> 00:25:52.660 align:middle line:84% I can turn off this switch at 0 voltage. 00:25:52.660 --> 00:25:55.090 align:middle line:84% And I can turn on this switch at 0 voltage 00:25:55.090 --> 00:25:57.420 align:middle line:90% because the diode will catch it. 00:25:57.420 --> 00:25:59.860 align:middle line:84% Does that make sense to everybody? 00:25:59.860 --> 00:26:03.280 align:middle line:84% Now, here's the cool thing about a resonant converter. 00:26:03.280 --> 00:26:05.538 align:middle line:84% What's the other transition-- or in a series 00:26:05.538 --> 00:26:07.580 align:middle line:84% resonant converter-- what's the other transition? 00:26:07.580 --> 00:26:10.990 align:middle line:90% I want to go from bottom to top. 00:26:10.990 --> 00:26:13.390 align:middle line:84% When I want to go from bottom to top, 00:26:13.390 --> 00:26:16.200 align:middle line:84% the inductor current is now negative. 00:26:16.200 --> 00:26:18.580 align:middle line:84% That means the inductor current's coming this way. 00:26:18.580 --> 00:26:21.210 align:middle line:84% When I turn off the bottom switch, 00:26:21.210 --> 00:26:22.740 align:middle line:90% this capacitor will charge. 00:26:22.740 --> 00:26:26.020 align:middle line:84% This capacitor will discharge until the top diode turns on. 00:26:26.020 --> 00:26:27.270 align:middle line:90% And I'll reverse the process. 00:26:27.270 --> 00:26:30.780 align:middle line:84% And I'll also switch the bottom device with 0 voltage turn off. 00:26:30.780 --> 00:26:34.060 align:middle line:84% And the top device with 0 voltage turn on. 00:26:34.060 --> 00:26:37.530 align:middle line:90% Does that make sense, everybody? 00:26:37.530 --> 00:26:40.020 align:middle line:84% So it's a long way of saying, because I 00:26:40.020 --> 00:26:43.320 align:middle line:84% have this sinusoidal current, approximately sinusoidal 00:26:43.320 --> 00:26:45.100 align:middle line:90% current, the current-- 00:26:45.100 --> 00:26:46.680 align:middle line:90% and I've made it inductive. 00:26:46.680 --> 00:26:49.860 align:middle line:84% I phase delayed it with respect to where the voltage is. 00:26:49.860 --> 00:26:53.640 align:middle line:84% I get naturally the opportunity to do zero-voltage switching. 00:26:53.640 --> 00:26:57.300 align:middle line:84% So in most modern series resonant converters, 00:26:57.300 --> 00:26:59.460 align:middle line:84% because we're interested in zero-voltage switching, 00:26:59.460 --> 00:27:02.220 align:middle line:84% we will operate them up in this range, 00:27:02.220 --> 00:27:05.760 align:middle line:84% above resonance by some degree, and modulate power 00:27:05.760 --> 00:27:10.200 align:middle line:84% over this range, by going down the curve this way. 00:27:10.200 --> 00:27:14.490 align:middle line:84% Now, it turns out, if I wanted to go the other way, 00:27:14.490 --> 00:27:16.495 align:middle line:84% I could actually do zero-current switching. 00:27:16.495 --> 00:27:18.120 align:middle line:84% If I wanted to operate below resonance, 00:27:18.120 --> 00:27:19.860 align:middle line:84% I could do zero-current switching. 00:27:19.860 --> 00:27:23.550 align:middle line:84% But we're often interested in getting zero-voltage switching 00:27:23.550 --> 00:27:27.180 align:middle line:84% because I can get rid of losses associated with the device 00:27:27.180 --> 00:27:28.450 align:middle line:90% capacitor switching. 00:27:28.450 --> 00:27:31.085 align:middle line:90% 00:27:31.085 --> 00:27:31.960 align:middle line:90% Questions about that? 00:27:31.960 --> 00:27:36.050 align:middle line:90% 00:27:36.050 --> 00:27:36.600 align:middle line:90% All right. 00:27:36.600 --> 00:27:41.074 align:middle line:84% So we can get zero-voltage switching. 00:27:41.074 --> 00:27:45.810 align:middle line:84% And by doing that, we can go at pretty high frequencies. 00:27:45.810 --> 00:27:48.600 align:middle line:84% And then we can use frequency modulation, 00:27:48.600 --> 00:27:51.460 align:middle line:84% frequency control to control the power into our load. 00:27:51.460 --> 00:27:52.720 align:middle line:90% Now, what is our load? 00:27:52.720 --> 00:27:55.320 align:middle line:84% Well, I haven't said what this load is. 00:27:55.320 --> 00:27:57.160 align:middle line:84% This load could be any number of things. 00:27:57.160 --> 00:28:00.060 align:middle line:84% It could be, for example, the equivalent 00:28:00.060 --> 00:28:02.220 align:middle line:84% resistance of an induction-heating load, 00:28:02.220 --> 00:28:03.005 align:middle line:90% for example. 00:28:03.005 --> 00:28:05.965 align:middle line:90% 00:28:05.965 --> 00:28:12.280 align:middle line:84% And thereby, by modulating frequency, I modulate power. 00:28:12.280 --> 00:28:16.640 align:middle line:84% And that's how an induction heater can work. 00:28:16.640 --> 00:28:19.300 align:middle line:84% And I can run at high frequencies because of that. 00:28:19.300 --> 00:28:24.780 align:middle line:84% Now, what's nice about this series resonant inverter 00:28:24.780 --> 00:28:27.000 align:middle line:90% circuit? 00:28:27.000 --> 00:28:30.630 align:middle line:84% Well, the first thing I said is we get zero-voltage switching 00:28:30.630 --> 00:28:33.760 align:middle line:84% for our devices, which is really helpful for efficiency. 00:28:33.760 --> 00:28:36.480 align:middle line:90% 00:28:36.480 --> 00:28:43.730 align:middle line:84% Notice also that if I modulate power, how do I-- let's assume R 00:28:43.730 --> 00:28:44.730 align:middle line:90% is fixed for the moment. 00:28:44.730 --> 00:28:46.070 align:middle line:90% It's a fixed resistor. 00:28:46.070 --> 00:28:48.980 align:middle line:84% Let's assume I modulate the power 00:28:48.980 --> 00:28:52.580 align:middle line:90% by modulating the frequency. 00:28:52.580 --> 00:28:57.910 align:middle line:84% I get lower power by making this resistor current lower. 00:28:57.910 --> 00:29:00.790 align:middle line:84% But that also means that the device currents and the inductor 00:29:00.790 --> 00:29:03.790 align:middle line:84% current and the capacitor current also got lower. 00:29:03.790 --> 00:29:10.420 align:middle line:84% So this kind of characteristic has the nice habit that-- 00:29:10.420 --> 00:29:15.850 align:middle line:84% has the nice habit that as I'm reducing 00:29:15.850 --> 00:29:18.850 align:middle line:84% power, all my currents in my devices and components 00:29:18.850 --> 00:29:20.320 align:middle line:90% all go down with-- 00:29:20.320 --> 00:29:21.410 align:middle line:90% as I reduce power. 00:29:21.410 --> 00:29:24.280 align:middle line:84% So I can keep efficiency high as my power is 00:29:24.280 --> 00:29:26.140 align:middle line:90% getting lower and lower. 00:29:26.140 --> 00:29:30.730 align:middle line:84% And that's a nice characteristic of a series resonant converter. 00:29:30.730 --> 00:29:33.970 align:middle line:84% Another nice thing is these device 00:29:33.970 --> 00:29:38.025 align:middle line:84% voltages are both limited to the DC input voltage. 00:29:38.025 --> 00:29:44.633 align:middle line:90% 00:29:44.633 --> 00:29:48.500 align:middle line:84% However, a slight warning, if I were 00:29:48.500 --> 00:29:51.450 align:middle line:84% to look at, say, the capacitor voltage, 00:29:51.450 --> 00:29:56.750 align:middle line:84% if I were to ask about VC, VC can actually 00:29:56.750 --> 00:29:59.780 align:middle line:84% be much bigger than the peak value of VX 00:29:59.780 --> 00:30:02.990 align:middle line:84% because these two components are in resonance. 00:30:02.990 --> 00:30:04.640 align:middle line:84% And in fact, if you remember, we said 00:30:04.640 --> 00:30:07.670 align:middle line:84% that the capacitor voltage in a series resonant network 00:30:07.670 --> 00:30:10.430 align:middle line:90% can be Q times-- 00:30:10.430 --> 00:30:16.290 align:middle line:84% Q of omega 0 times the fundamental input voltage. 00:30:16.290 --> 00:30:21.390 align:middle line:84% So if I have a Q of 10, then whatever the fundamental of this 00:30:21.390 --> 00:30:27.190 align:middle line:84% voltage is, if I ran at resonance, VC would be 10X that. 00:30:27.190 --> 00:30:29.560 align:middle line:84% Or if Q is 100, it would be 100X that. 00:30:29.560 --> 00:30:31.100 align:middle line:90% So you've got to be careful. 00:30:31.100 --> 00:30:32.800 align:middle line:84% The device voltages are nicely limited. 00:30:32.800 --> 00:30:35.920 align:middle line:84% But you've got to be very careful about these two 00:30:35.920 --> 00:30:37.990 align:middle line:84% components, can have pretty big voltages on them, 00:30:37.990 --> 00:30:42.230 align:middle line:84% depending upon the quality factor of the network. 00:30:42.230 --> 00:30:43.850 align:middle line:84% Now, we would like quality factor 00:30:43.850 --> 00:30:48.830 align:middle line:84% to be high so that we get sinusoidal currents. 00:30:48.830 --> 00:30:52.010 align:middle line:84% If you imagine that the quality factor of this was really low-- 00:30:52.010 --> 00:30:55.010 align:middle line:84% that is, resistance was really big, for example-- 00:30:55.010 --> 00:30:58.520 align:middle line:84% then instead of looking like this pretty curve, 00:30:58.520 --> 00:31:01.470 align:middle line:90% it would look like this. 00:31:01.470 --> 00:31:05.330 align:middle line:84% And then maybe I don't do a good job of attenuating my harmonics. 00:31:05.330 --> 00:31:07.160 align:middle line:84% And my current wave forms get distorted. 00:31:07.160 --> 00:31:09.950 align:middle line:84% And then perhaps I don't get the right phase shifts 00:31:09.950 --> 00:31:12.300 align:middle line:84% for zero-voltage switching and all that. 00:31:12.300 --> 00:31:15.230 align:middle line:84% So you want Q to be high enough that you 00:31:15.230 --> 00:31:18.680 align:middle line:84% get good attenuation of harmonics and low enough 00:31:18.680 --> 00:31:21.830 align:middle line:84% that you don't have big stresses on these components 00:31:21.830 --> 00:31:27.090 align:middle line:84% and big parasitic losses in those components. 00:31:27.090 --> 00:31:28.055 align:middle line:90% Questions about that? 00:31:28.055 --> 00:31:34.230 align:middle line:90% 00:31:34.230 --> 00:31:39.000 align:middle line:84% Now, what is a challenge with this? 00:31:39.000 --> 00:31:43.360 align:middle line:90% Well, this works beautifully. 00:31:43.360 --> 00:31:45.700 align:middle line:84% And I should say one other thing. 00:31:45.700 --> 00:31:50.040 align:middle line:84% There's a limit that relates how far above resonance I 00:31:50.040 --> 00:31:52.740 align:middle line:84% have to be to the size of these capacitors 00:31:52.740 --> 00:31:54.290 align:middle line:84% and the amplitude of the current. 00:31:54.290 --> 00:31:58.950 align:middle line:84% Basically, this integral is basically 00:31:58.950 --> 00:32:01.980 align:middle line:84% underneath the orange curve of the inductor current. 00:32:01.980 --> 00:32:05.130 align:middle line:84% Between here and where it crosses 0 00:32:05.130 --> 00:32:08.520 align:middle line:84% is exactly the charge that I can deliver onto these capacitors 00:32:08.520 --> 00:32:09.870 align:middle line:90% for soft switching. 00:32:09.870 --> 00:32:11.940 align:middle line:84% So I need a certain amount of phase shift 00:32:11.940 --> 00:32:14.250 align:middle line:84% and/or a certain amount of current amplitude 00:32:14.250 --> 00:32:16.590 align:middle line:84% in order to get zero-voltage soft switching if I'm 00:32:16.590 --> 00:32:20.130 align:middle line:84% going to use this load network to do my soft switching. 00:32:20.130 --> 00:32:23.200 align:middle line:84% There are other ways to do it as well. 00:32:23.200 --> 00:32:26.080 align:middle line:84% But that's something you got to pay attention to. 00:32:26.080 --> 00:32:29.480 align:middle line:90% But here's a challenge. 00:32:29.480 --> 00:32:35.900 align:middle line:84% Suppose now I don't control this resistor very well. 00:32:35.900 --> 00:32:38.240 align:middle line:84% If this was an induction heater thing, 00:32:38.240 --> 00:32:40.190 align:middle line:84% the equivalent resistance I might see 00:32:40.190 --> 00:32:45.260 align:middle line:84% might depend upon the thing I've put on my network 00:32:45.260 --> 00:32:49.602 align:middle line:84% to heat, how much energy I'm going to dump into it. 00:32:49.602 --> 00:32:54.260 align:middle line:84% If this resistor gets too small, the quality factor 00:32:54.260 --> 00:32:56.420 align:middle line:90% of the network gets really high. 00:32:56.420 --> 00:32:58.830 align:middle line:84% And then I'd better stay away from resonance. 00:32:58.830 --> 00:33:01.910 align:middle line:84% Otherwise, I'll get huge voltages on my components-- 00:33:01.910 --> 00:33:04.760 align:middle line:90% on my inductor and capacitor. 00:33:04.760 --> 00:33:09.770 align:middle line:84% On the other hand, if I make R too big, 00:33:09.770 --> 00:33:13.760 align:middle line:84% suddenly the network quality factor gets too low. 00:33:13.760 --> 00:33:16.580 align:middle line:84% And I get this frequency attenuation. 00:33:16.580 --> 00:33:18.240 align:middle line:90% I don't get good attenuation. 00:33:18.240 --> 00:33:21.110 align:middle line:84% I've got to go to really high frequencies to attenuate things. 00:33:21.110 --> 00:33:23.720 align:middle line:84% So a challenge of these resonant networks 00:33:23.720 --> 00:33:27.380 align:middle line:84% is keeping the load resistance in an acceptable range 00:33:27.380 --> 00:33:28.770 align:middle line:90% for what you're trying to do. 00:33:28.770 --> 00:33:33.510 align:middle line:90% And that's a trick of design. 00:33:33.510 --> 00:33:34.700 align:middle line:90% Any questions about that? 00:33:34.700 --> 00:33:41.840 align:middle line:90% 00:33:41.840 --> 00:33:48.930 align:middle line:84% I should say that I've shown you a series resonant network. 00:33:48.930 --> 00:33:51.710 align:middle line:90% You can use other topologies. 00:33:51.710 --> 00:33:54.230 align:middle line:84% And they give you different characteristics 00:33:54.230 --> 00:33:57.120 align:middle line:84% in terms of the behavior you'll get of the resonant network. 00:33:57.120 --> 00:33:58.953 align:middle line:84% So let me just show you a different example. 00:33:58.953 --> 00:34:02.780 align:middle line:90% 00:34:02.780 --> 00:34:04.820 align:middle line:84% I could do the full dual of this thing. 00:34:04.820 --> 00:34:06.900 align:middle line:84% But let me show you a different kind. 00:34:06.900 --> 00:34:08.580 align:middle line:84% This is a series resonant converter. 00:34:08.580 --> 00:34:10.420 align:middle line:84% Here's a parallel resonant inverter. 00:34:10.420 --> 00:34:13.520 align:middle line:90% 00:34:13.520 --> 00:34:14.904 align:middle line:90% I'll have VDC again. 00:34:14.904 --> 00:34:19.480 align:middle line:90% 00:34:19.480 --> 00:34:25.060 align:middle line:84% And for simplicity, let me put in C infinity. 00:34:25.060 --> 00:34:28.480 align:middle line:84% This is just a blocking capacitor. 00:34:28.480 --> 00:34:31.060 align:middle line:84% So I'm going to get a voltage on the other side of this 00:34:31.060 --> 00:34:32.699 align:middle line:90% that I'm going to call VX prime. 00:34:32.699 --> 00:34:37.719 align:middle line:90% 00:34:37.719 --> 00:34:41.739 align:middle line:84% VX prime is, again, just going to be a square wave. 00:34:41.739 --> 00:34:45.730 align:middle line:84% And it's going to go from-- it's going to go between VDC over 2 00:34:45.730 --> 00:34:47.534 align:middle line:90% and minus VDC over 2. 00:34:47.534 --> 00:34:51.460 align:middle line:90% 00:34:51.460 --> 00:34:53.199 align:middle line:84% I'm putting a blocking capacitor in there 00:34:53.199 --> 00:34:56.739 align:middle line:84% just to extract out the AC component, which 00:34:56.739 --> 00:34:59.790 align:middle line:90% I'm using to drive the network. 00:34:59.790 --> 00:35:03.720 align:middle line:84% And then here's my parallel resonant network. 00:35:03.720 --> 00:35:12.315 align:middle line:90% 00:35:12.315 --> 00:35:16.260 align:middle line:84% This would be called a parallel resonant inverter. 00:35:16.260 --> 00:35:18.152 align:middle line:84% Instead of having a blocking capacitor here, 00:35:18.152 --> 00:35:19.860 align:middle line:84% I could have done this in different ways, 00:35:19.860 --> 00:35:21.760 align:middle line:84% including using a full bridge, et cetera. 00:35:21.760 --> 00:35:23.980 align:middle line:84% But I just want to illustrate this. 00:35:23.980 --> 00:35:26.380 align:middle line:84% Now, why is this a parallel resonant network? 00:35:26.380 --> 00:35:39.700 align:middle line:84% If I thought of VX prime as a voltage source, 00:35:39.700 --> 00:35:45.185 align:middle line:84% if I make VX prime 0, L, C, and R here are in parallel. 00:35:45.185 --> 00:35:48.460 align:middle line:84% So this is a parallel resonant network 00:35:48.460 --> 00:35:50.410 align:middle line:84% driven by a voltage source inserted 00:35:50.410 --> 00:35:55.033 align:middle line:84% in series with the resonant inductor here. 00:35:55.033 --> 00:35:57.450 align:middle line:84% So that's how somebody would think about this and in terms 00:35:57.450 --> 00:35:59.100 align:middle line:90% of a resonant converter. 00:35:59.100 --> 00:36:06.120 align:middle line:84% And how might I think about the behavior of this? 00:36:06.120 --> 00:36:08.700 align:middle line:90% Well, maybe I could plot-- 00:36:08.700 --> 00:36:11.480 align:middle line:90% VC and VR are the same thing. 00:36:11.480 --> 00:36:18.150 align:middle line:84% So maybe I could plot the magnitude of V sub R 00:36:18.150 --> 00:36:26.530 align:middle line:84% over VX prime versus frequency for this network. 00:36:26.530 --> 00:36:30.750 align:middle line:84% So I'm looking at the transfer function between here and here 00:36:30.750 --> 00:36:33.060 align:middle line:90% for this network. 00:36:33.060 --> 00:36:36.630 align:middle line:84% At DC, clearly, VR equals VC, or at low frequency, 00:36:36.630 --> 00:36:38.310 align:middle line:84% because the inductor becomes a short. 00:36:38.310 --> 00:36:43.300 align:middle line:84% At really high frequency, this rolls off at 40 dB per decade. 00:36:43.300 --> 00:36:45.030 align:middle line:84% So what you're going to get is something 00:36:45.030 --> 00:36:47.140 align:middle line:90% that does something like this. 00:36:47.140 --> 00:36:48.870 align:middle line:90% It's going to go for one. 00:36:48.870 --> 00:36:51.520 align:middle line:90% It's going to peak up. 00:36:51.520 --> 00:36:53.920 align:middle line:90% And then it's going to fall. 00:36:53.920 --> 00:36:58.150 align:middle line:84% And this peak, if the network quality factor here-- 00:36:58.150 --> 00:37:01.810 align:middle line:84% and quality factor here is now R over the square root of L/C. 00:37:01.810 --> 00:37:04.540 align:middle line:90% It's the other way around. 00:37:04.540 --> 00:37:07.180 align:middle line:84% This peak is going to be near omega 0, which 00:37:07.180 --> 00:37:11.836 align:middle line:84% is 1 over the square root of LC again. 00:37:11.836 --> 00:37:16.740 align:middle line:84% So again, we get this high-frequency selectivity 00:37:16.740 --> 00:37:22.440 align:middle line:84% of the voltage gain from my drive, 00:37:22.440 --> 00:37:25.410 align:middle line:84% which has a fixed amplitude set by the DC voltage, 00:37:25.410 --> 00:37:28.030 align:middle line:84% to the voltage I put across my resistor. 00:37:28.030 --> 00:37:34.360 align:middle line:84% And again, if I'm operating up here in this regime, 00:37:34.360 --> 00:37:37.880 align:middle line:84% I'm going to have an inductive input impedance coming in. 00:37:37.880 --> 00:37:40.930 align:middle line:84% So again, I can soft switch these devices, 00:37:40.930 --> 00:37:44.296 align:middle line:84% zero-voltage switch these devices. 00:37:44.296 --> 00:37:47.170 align:middle line:84% And now I can, again, by controlling frequency 00:37:47.170 --> 00:37:49.270 align:middle line:84% of the drive, control the amplitude 00:37:49.270 --> 00:37:54.040 align:middle line:84% across my resistor and hence control power. 00:37:54.040 --> 00:37:55.170 align:middle line:90% Any questions about that? 00:37:55.170 --> 00:37:58.230 align:middle line:90% 00:37:58.230 --> 00:38:01.200 align:middle line:84% Now, how does this network differ from the series 00:38:01.200 --> 00:38:02.200 align:middle line:90% resonant network? 00:38:02.200 --> 00:38:04.530 align:middle line:90% Well, notice one thing. 00:38:04.530 --> 00:38:10.260 align:middle line:84% In this series resonant network, the largest voltage 00:38:10.260 --> 00:38:12.990 align:middle line:84% I could ever put on VR is the fundamental 00:38:12.990 --> 00:38:16.050 align:middle line:84% that I drive with, because it's just 00:38:16.050 --> 00:38:18.090 align:middle line:84% a voltage divider between the resistor 00:38:18.090 --> 00:38:21.030 align:middle line:90% and the reactive components. 00:38:21.030 --> 00:38:25.650 align:middle line:84% Here, I actually have-- because of the resonance 00:38:25.650 --> 00:38:28.870 align:middle line:84% between L and C, I can actually get voltage gain. 00:38:28.870 --> 00:38:33.750 align:middle line:84% I can get a voltage that's much bigger than my input voltage. 00:38:33.750 --> 00:38:36.030 align:middle line:84% So if I want voltage gain out of the thing, 00:38:36.030 --> 00:38:40.520 align:middle line:84% maybe this parallel network looks good. 00:38:40.520 --> 00:38:44.100 align:middle line:84% And how much this peaks up is a function of resistance. 00:38:44.100 --> 00:38:45.695 align:middle line:90% So this is for R big. 00:38:45.695 --> 00:38:48.650 align:middle line:90% 00:38:48.650 --> 00:38:51.860 align:middle line:90% Maybe I'll call that R big. 00:38:51.860 --> 00:38:55.280 align:middle line:90% Maybe this is R medium. 00:38:55.280 --> 00:38:58.030 align:middle line:84% If I had a big resistor, it might look like this. 00:38:58.030 --> 00:39:04.590 align:middle line:90% 00:39:04.590 --> 00:39:08.240 align:middle line:84% And if I had a small resistor, maybe it'd look like this. 00:39:08.240 --> 00:39:12.570 align:middle line:90% 00:39:12.570 --> 00:39:14.200 align:middle line:90% But here's the point. 00:39:14.200 --> 00:39:17.480 align:middle line:84% I can get voltage gain, at least if I have the right resistance 00:39:17.480 --> 00:39:17.980 align:middle line:90% value. 00:39:17.980 --> 00:39:19.630 align:middle line:90% And sometimes we use this. 00:39:19.630 --> 00:39:23.670 align:middle line:84% So this kind of network actually is often 00:39:23.670 --> 00:39:26.940 align:middle line:84% what's used if you're going to have a fluorescent bulb. 00:39:26.940 --> 00:39:30.150 align:middle line:84% So if you think about it, like one of these fluorescent bulbs 00:39:30.150 --> 00:39:32.140 align:middle line:90% here, what is it? 00:39:32.140 --> 00:39:36.212 align:middle line:84% It's a gas of-- it's a tube with low-pressure gas in it. 00:39:36.212 --> 00:39:41.630 align:middle line:84% If the lamp is off, that has a really high impedance, 00:39:41.630 --> 00:39:45.230 align:middle line:84% because it's just two electrodes with some gas in between it. 00:39:45.230 --> 00:39:48.350 align:middle line:84% And you've got to get a really, really big voltage on it 00:39:48.350 --> 00:39:51.570 align:middle line:84% to break down the gas and so-called strike the lamp. 00:39:51.570 --> 00:39:53.870 align:middle line:90% Well, how can I do it? 00:39:53.870 --> 00:39:56.810 align:middle line:84% If I take this thing and I put this thing as an open circuit 00:39:56.810 --> 00:39:58.880 align:middle line:84% and I drive this half bridge right 00:39:58.880 --> 00:40:00.980 align:middle line:84% near the resonant frequency of L and C, 00:40:00.980 --> 00:40:04.790 align:middle line:84% I'm going to build up a really big capacitor voltage here. 00:40:04.790 --> 00:40:07.910 align:middle line:84% That capacitor voltage could be tens of times 00:40:07.910 --> 00:40:11.390 align:middle line:90% as big as my DC voltage. 00:40:11.390 --> 00:40:15.140 align:middle line:84% Eventually, the fluorescent lamp breaks over. 00:40:15.140 --> 00:40:17.450 align:middle line:90% And then you've ionized the gas. 00:40:17.450 --> 00:40:19.640 align:middle line:84% And then it looks like a resistor but not 00:40:19.640 --> 00:40:21.630 align:middle line:84% a huge resistor, a small resistor. 00:40:21.630 --> 00:40:25.250 align:middle line:84% And I go from the green curve maybe down to the-- 00:40:25.250 --> 00:40:26.930 align:middle line:84% oh, sorry-- I go from the green curve 00:40:26.930 --> 00:40:29.120 align:middle line:84% maybe down to the white or orange curve. 00:40:29.120 --> 00:40:32.360 align:middle line:84% And then I can modulate the amount of power and light 00:40:32.360 --> 00:40:35.165 align:middle line:90% by moving frequency. 00:40:35.165 --> 00:40:37.490 align:middle line:84% So this kind of network would be at the core 00:40:37.490 --> 00:40:41.200 align:middle line:84% of many fluorescent lamp ballast drivers. 00:40:41.200 --> 00:40:45.010 align:middle line:84% And we're taking advantage of the resonant voltage gain I 00:40:45.010 --> 00:40:46.820 align:middle line:90% can get from this high Q tank. 00:40:46.820 --> 00:40:50.580 align:middle line:84% And it's particularly high Q when the lamp isn't struck. 00:40:50.580 --> 00:40:52.290 align:middle line:84% So sometimes I want the big voltage, 00:40:52.290 --> 00:40:56.540 align:middle line:84% even though I got to rate this capacitor and inductor for it. 00:40:56.540 --> 00:40:57.655 align:middle line:90% Any questions about that? 00:40:57.655 --> 00:41:00.860 align:middle line:90% 00:41:00.860 --> 00:41:05.690 align:middle line:84% So I can get frequency-dependent voltage gain and power control 00:41:05.690 --> 00:41:07.910 align:middle line:90% in either of these networks. 00:41:07.910 --> 00:41:14.150 align:middle line:84% This network has the opposite characteristics of this network, 00:41:14.150 --> 00:41:19.310 align:middle line:84% in the sense that if I make R too big rather than too small, 00:41:19.310 --> 00:41:22.430 align:middle line:90% I get very peaky resonances. 00:41:22.430 --> 00:41:25.940 align:middle line:84% And if I make R too small, this thing 00:41:25.940 --> 00:41:30.470 align:middle line:84% starts to give me poor frequency selectivity and hence 00:41:30.470 --> 00:41:33.590 align:middle line:90% poor attenuation. 00:41:33.590 --> 00:41:35.990 align:middle line:84% People choose one or the other, depending 00:41:35.990 --> 00:41:37.520 align:middle line:90% upon what they're trying to do. 00:41:37.520 --> 00:41:39.480 align:middle line:84% Or I could choose something in between. 00:41:39.480 --> 00:41:42.530 align:middle line:84% So, for example, one thing people might do 00:41:42.530 --> 00:41:45.920 align:middle line:84% is say, OK, let me have a network. 00:41:45.920 --> 00:41:49.310 align:middle line:84% And maybe I'll make it look like this. 00:41:49.310 --> 00:41:55.730 align:middle line:84% I'll have two Cs and an R. So maybe I'll call this C1 and C2, 00:41:55.730 --> 00:41:57.170 align:middle line:90% and this is R. 00:41:57.170 --> 00:42:03.170 align:middle line:84% So if R is small, then it shorts out C2. 00:42:03.170 --> 00:42:04.110 align:middle line:90% And I can ignore that. 00:42:04.110 --> 00:42:06.240 align:middle line:84% And it looks like a parallel resonant network. 00:42:06.240 --> 00:42:11.570 align:middle line:84% If R is really big and I run to a high enough frequency, 00:42:11.570 --> 00:42:13.700 align:middle line:84% this branch looks sort of inductive 00:42:13.700 --> 00:42:15.387 align:middle line:90% when I'm above this resonance. 00:42:15.387 --> 00:42:17.970 align:middle line:84% And then I have the equivalent of a parallel resonant network. 00:42:17.970 --> 00:42:21.710 align:middle line:84% This would be called an LCC or a series 00:42:21.710 --> 00:42:24.230 align:middle line:90% parallel resonant network. 00:42:24.230 --> 00:42:27.220 align:middle line:84% And that kind of network's used too, 00:42:27.220 --> 00:42:30.960 align:middle line:84% to balance what happens at large and small resistances, 00:42:30.960 --> 00:42:31.790 align:middle line:90% for example. 00:42:31.790 --> 00:42:33.950 align:middle line:84% It has some sort of hybrid combination 00:42:33.950 --> 00:42:35.960 align:middle line:90% of series and parallel. 00:42:35.960 --> 00:42:39.230 align:middle line:84% You can also have LLC, where there's two inductors and one 00:42:39.230 --> 00:42:39.830 align:middle line:90% capacitor. 00:42:39.830 --> 00:42:41.660 align:middle line:84% That's another, actually, extremely common 00:42:41.660 --> 00:42:42.920 align:middle line:90% resonant network. 00:42:42.920 --> 00:42:45.380 align:middle line:84% So the easiest thing to get your brain around 00:42:45.380 --> 00:42:48.220 align:middle line:90% is the series and parallel kind. 00:42:48.220 --> 00:42:49.870 align:middle line:84% There are other kinds that people 00:42:49.870 --> 00:42:52.240 align:middle line:84% use because they give particular characteristics 00:42:52.240 --> 00:42:54.730 align:middle line:84% for different ranges of load resistance 00:42:54.730 --> 00:42:59.970 align:middle line:84% or for different ranges of operating power, for example. 00:42:59.970 --> 00:43:01.130 align:middle line:90% Any questions about that? 00:43:01.130 --> 00:43:03.990 align:middle line:90% 00:43:03.990 --> 00:43:06.750 align:middle line:90% Now, a lot of converters-- 00:43:06.750 --> 00:43:10.080 align:middle line:84% I mentioned things like, where do they use these things? 00:43:10.080 --> 00:43:11.820 align:middle line:90% Induction heating is common. 00:43:11.820 --> 00:43:15.372 align:middle line:84% Driving fluorescent lamps, where you want sinusoidal power 00:43:15.372 --> 00:43:16.330 align:middle line:90% delivery into the lamp. 00:43:16.330 --> 00:43:19.680 align:middle line:84% That's usually pretty low frequency. 00:43:19.680 --> 00:43:23.910 align:middle line:84% Wireless power transfer, if I'm going to go across a link 00:43:23.910 --> 00:43:28.320 align:middle line:90% where I want sinusoidal fields. 00:43:28.320 --> 00:43:30.180 align:middle line:84% Another example, however, is just 00:43:30.180 --> 00:43:32.680 align:middle line:90% straight DC to DC conversion. 00:43:32.680 --> 00:43:36.060 align:middle line:84% I didn't say what this resistor was. 00:43:36.060 --> 00:43:40.170 align:middle line:84% But imagine I use a network like this one. 00:43:40.170 --> 00:43:41.370 align:middle line:90% I take DC. 00:43:41.370 --> 00:43:44.740 align:middle line:84% I chop it up, generate a square wave. 00:43:44.740 --> 00:43:52.510 align:middle line:84% If I have a high Q tank, then this current 00:43:52.510 --> 00:43:54.362 align:middle line:84% becomes sinusoidal, or close to it. 00:43:54.362 --> 00:43:56.320 align:middle line:84% And that's where I make my fundamental harmonic 00:43:56.320 --> 00:43:57.500 align:middle line:90% approximation. 00:43:57.500 --> 00:43:59.680 align:middle line:84% And by the way, for power transfer, 00:43:59.680 --> 00:44:03.400 align:middle line:84% if this current is sinusoidal, we 00:44:03.400 --> 00:44:07.360 align:middle line:84% said we can only transfer average power through voltages 00:44:07.360 --> 00:44:09.490 align:middle line:84% and currents at the same frequency. 00:44:09.490 --> 00:44:13.480 align:middle line:84% So even though VX has a bunch of harmonic content, 00:44:13.480 --> 00:44:18.400 align:middle line:84% as long as this network filters to make the current sinusoidal, 00:44:18.400 --> 00:44:20.830 align:middle line:84% then the only component of this thing 00:44:20.830 --> 00:44:29.200 align:middle line:84% that matters is the fundamental component of the drive 00:44:29.200 --> 00:44:34.030 align:middle line:84% voltage because no other harmonic 00:44:34.030 --> 00:44:37.330 align:middle line:84% component delivers average power through the network. 00:44:37.330 --> 00:44:39.520 align:middle line:84% So the fact that I'm mediating power 00:44:39.520 --> 00:44:45.280 align:middle line:84% through a sinusoidal, in this case, current is very valuable. 00:44:45.280 --> 00:44:47.750 align:middle line:84% So how would I build a DC-DC converter? 00:44:47.750 --> 00:44:53.170 align:middle line:84% Well, maybe I'll use an inverter, create a square wave, 00:44:53.170 --> 00:44:57.710 align:middle line:84% use a high Q network to extract a sinusoid. 00:44:57.710 --> 00:45:01.420 align:middle line:84% And then I can use that to control voltage gain, because I 00:45:01.420 --> 00:45:04.700 align:middle line:84% can control the amplitude of the current by varying frequency. 00:45:04.700 --> 00:45:08.140 align:middle line:84% And then I just run it into a rectifier. 00:45:08.140 --> 00:45:10.900 align:middle line:90% So maybe I take this thing. 00:45:10.900 --> 00:45:11.925 align:middle line:90% And I say, OK. 00:45:11.925 --> 00:45:15.690 align:middle line:90% 00:45:15.690 --> 00:45:18.610 align:middle line:90% I say, let's build a converter. 00:45:18.610 --> 00:45:23.110 align:middle line:90% 00:45:23.110 --> 00:45:24.970 align:middle line:84% And this is going to be soft switched, which 00:45:24.970 --> 00:45:27.400 align:middle line:84% is one of the reasons why I'm doing it. 00:45:27.400 --> 00:45:30.910 align:middle line:84% I have L, C. And then what I'll do 00:45:30.910 --> 00:45:33.520 align:middle line:84% is maybe I'll just put a transformer here. 00:45:33.520 --> 00:45:44.110 align:middle line:84% And then on the other side of this transformer, 00:45:44.110 --> 00:45:47.200 align:middle line:90% I'll put a rectifier-- 00:45:47.200 --> 00:45:49.150 align:middle line:84% in this case, a full bridge rectifier-- 00:45:49.150 --> 00:45:52.800 align:middle line:90% and make that V out. 00:45:52.800 --> 00:45:56.560 align:middle line:90% And this is transformer. 00:45:56.560 --> 00:45:57.560 align:middle line:90% And this is transformer. 00:45:57.560 --> 00:46:02.770 align:middle line:84% So if this is, for example, I sub L, 00:46:02.770 --> 00:46:06.850 align:middle line:84% maybe this is then IL prime, which 00:46:06.850 --> 00:46:10.990 align:middle line:84% is exactly just I sub L scaled by the turns ratio 00:46:10.990 --> 00:46:12.480 align:middle line:84% or the inverse of the turns ratio. 00:46:12.480 --> 00:46:15.850 align:middle line:90% 00:46:15.850 --> 00:46:18.182 align:middle line:90% What does this network-- 00:46:18.182 --> 00:46:20.140 align:middle line:84% if I were to say, What does this impedance look 00:46:20.140 --> 00:46:21.130 align:middle line:90% like, looking in here? 00:46:21.130 --> 00:46:24.440 align:middle line:90% 00:46:24.440 --> 00:46:29.930 align:middle line:84% well, if IL is a sinusoid, basically, I'm 00:46:29.930 --> 00:46:31.850 align:middle line:84% going to take a sinusoidal current. 00:46:31.850 --> 00:46:33.770 align:middle line:84% And then I'm going to rectify it. 00:46:33.770 --> 00:46:37.250 align:middle line:84% So then IX here would look like what? 00:46:37.250 --> 00:46:43.885 align:middle line:84% It would look like exactly like a rectified sinusoid 00:46:43.885 --> 00:46:45.890 align:middle line:90% of amplitude. 00:46:45.890 --> 00:46:48.650 align:middle line:90% Let me call this I sub L. 00:46:48.650 --> 00:46:52.090 align:middle line:84% So if I sub L has amplitude I sub L, 00:46:52.090 --> 00:46:56.880 align:middle line:84% then this just looks like a rectified version. 00:46:56.880 --> 00:46:59.440 align:middle line:90% The DC current-- 00:46:59.440 --> 00:47:00.790 align:middle line:90% I'll call this IX. 00:47:00.790 --> 00:47:03.830 align:middle line:84% The average value of IX-- and I'm running out of time. 00:47:03.830 --> 00:47:05.600 align:middle line:84% So I don't have chance to work this out. 00:47:05.600 --> 00:47:09.670 align:middle line:84% The average value of IX is simply going to be equal to 2 00:47:09.670 --> 00:47:13.910 align:middle line:90% over pi times IL. 00:47:13.910 --> 00:47:14.570 align:middle line:90% I'm sorry. 00:47:14.570 --> 00:47:15.430 align:middle line:90% Yeah, times IL. 00:47:15.430 --> 00:47:18.785 align:middle line:90% 00:47:18.785 --> 00:47:25.450 align:middle line:84% So in some sense, what am I going to see here? 00:47:25.450 --> 00:47:28.260 align:middle line:84% I'm going to deliver some average current into my output. 00:47:28.260 --> 00:47:33.120 align:middle line:90% 00:47:33.120 --> 00:47:36.390 align:middle line:84% What is the voltage that appears? 00:47:36.390 --> 00:47:41.990 align:middle line:84% If this is IL prime, let me plot IL prime for you. 00:47:41.990 --> 00:47:47.250 align:middle line:90% 00:47:47.250 --> 00:47:49.325 align:middle line:90% IL prime maybe looks like this. 00:47:49.325 --> 00:47:59.230 align:middle line:90% 00:47:59.230 --> 00:48:04.350 align:middle line:84% What does this voltage look like? 00:48:04.350 --> 00:48:09.080 align:middle line:90% Let me call this V sub y. 00:48:09.080 --> 00:48:12.390 align:middle line:84% Well, when IL prime is positive, I have V out. 00:48:12.390 --> 00:48:14.750 align:middle line:84% When L prime is negative, I have minus V out. 00:48:14.750 --> 00:48:18.265 align:middle line:84% So the voltage he sees looks like this. 00:48:18.265 --> 00:48:24.425 align:middle line:90% 00:48:24.425 --> 00:48:25.300 align:middle line:90% Does that make sense? 00:48:25.300 --> 00:48:27.920 align:middle line:90% 00:48:27.920 --> 00:48:31.400 align:middle line:84% So the current into the rectifier 00:48:31.400 --> 00:48:33.390 align:middle line:90% generates a square wave voltage. 00:48:33.390 --> 00:48:35.432 align:middle line:84% But I'm only transferring power in this network. 00:48:35.432 --> 00:48:36.890 align:middle line:84% The fact that there's a square wave 00:48:36.890 --> 00:48:41.430 align:middle line:84% voltage here and here doesn't matter. 00:48:41.430 --> 00:48:43.610 align:middle line:84% It's the fundamental component of that voltage that 00:48:43.610 --> 00:48:45.330 align:middle line:90% matters for power transfer. 00:48:45.330 --> 00:48:49.310 align:middle line:84% The fundamental of this orange square wave is-- 00:48:49.310 --> 00:48:51.500 align:middle line:84% basically, it looks something like this. 00:48:51.500 --> 00:48:54.890 align:middle line:90% 00:48:54.890 --> 00:48:58.860 align:middle line:84% It's exactly in phase with the current. 00:48:58.860 --> 00:49:02.670 align:middle line:84% So essentially, this whole rectifier network 00:49:02.670 --> 00:49:04.470 align:middle line:84% reflected through this transformer 00:49:04.470 --> 00:49:09.030 align:middle line:84% looks like an equivalent resistance. 00:49:09.030 --> 00:49:13.150 align:middle line:84% And I'm back exactly to this original model I had. 00:49:13.150 --> 00:49:15.150 align:middle line:84% And the only thing I've done there is I've 00:49:15.150 --> 00:49:18.600 align:middle line:84% said, I don't care about the harmonics of this, 00:49:18.600 --> 00:49:22.800 align:middle line:84% because as long as the current's close to sinusoidal, 00:49:22.800 --> 00:49:27.280 align:middle line:84% almost all the power is in the fundamental. 00:49:27.280 --> 00:49:29.770 align:middle line:90% So I've run out of time there. 00:49:29.770 --> 00:49:31.360 align:middle line:90% But I'll come back next class. 00:49:31.360 --> 00:49:32.980 align:middle line:84% And we can talk a little bit more 00:49:32.980 --> 00:49:37.510 align:middle line:84% about how we might model DC-DC converters in this way. 00:49:37.510 --> 00:49:39.190 align:middle line:84% But let me ask, are there any questions 00:49:39.190 --> 00:49:40.550 align:middle line:90% before we wrap up for the day? 00:49:40.550 --> 00:49:43.130 align:middle line:90% 00:49:43.130 --> 00:49:43.630 align:middle line:90% All right. 00:49:43.630 --> 00:49:45.740 align:middle line:90% I will see you tomorrow. 00:49:45.740 --> 00:49:49.000 align:middle line:90%