WEBVTT 00:00:00.000 --> 00:00:08.830 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:08.830 --> 00:00:11.020 align:middle line:84% DAVID PERREAULT: Which capacitor converters, 00:00:11.020 --> 00:00:16.810 align:middle line:84% which is covered in Principles of Power Electronics chapter 6. 00:00:16.810 --> 00:00:19.480 align:middle line:90% 00:00:19.480 --> 00:00:23.680 align:middle line:84% And we saw some very interesting phenomena 00:00:23.680 --> 00:00:29.170 align:middle line:84% in terms of the effective output resistance of a switch capacitor 00:00:29.170 --> 00:00:29.870 align:middle line:90% converter. 00:00:29.870 --> 00:00:31.870 align:middle line:84% And I'd like to take a few minutes just 00:00:31.870 --> 00:00:36.340 align:middle line:84% to talk about what are the origins of that behavior. 00:00:36.340 --> 00:00:40.250 align:middle line:84% So let's just take a really simple case. 00:00:40.250 --> 00:00:46.540 align:middle line:84% Suppose I have a voltage supply that I'm going to call V sub s. 00:00:46.540 --> 00:00:47.860 align:middle line:90% And I'm going to use it. 00:00:47.860 --> 00:00:49.420 align:middle line:84% And at t equals 0 I'm going to close 00:00:49.420 --> 00:00:57.880 align:middle line:84% a switch through a resistor R to charge a capacitor, Vc. 00:00:57.880 --> 00:01:02.320 align:middle line:84% And at t equals 0 minus, we're going 00:01:02.320 --> 00:01:04.989 align:middle line:84% to say that has some initial voltage V sub 00:01:04.989 --> 00:01:06.400 align:middle line:90% i on the capacitor. 00:01:06.400 --> 00:01:08.995 align:middle line:84% So this capacitor I'm going to close the switch, 00:01:08.995 --> 00:01:10.870 align:middle line:84% and then I'm going to charge the capacitor up 00:01:10.870 --> 00:01:13.540 align:middle line:84% from V sub i to its final value, which 00:01:13.540 --> 00:01:15.580 align:middle line:90% is going to end up at V sub s. 00:01:15.580 --> 00:01:19.510 align:middle line:84% And if I plotted the behavior here, 00:01:19.510 --> 00:01:27.510 align:middle line:84% right, I could find, for example, if I plotted i sub c, 00:01:27.510 --> 00:01:29.100 align:middle line:90% what would I expect to happen? 00:01:29.100 --> 00:01:32.390 align:middle line:84% Well it's 0 before t equals 0, obviously. 00:01:32.390 --> 00:01:35.960 align:middle line:84% At t equals 0, it's going to be Vs minus Vc, which 00:01:35.960 --> 00:01:38.550 align:middle line:90% is Vi over R, right? 00:01:38.550 --> 00:01:46.490 align:middle line:84% So I'd have some initial value, Vs minus Vi over R. 00:01:46.490 --> 00:01:51.230 align:middle line:84% And then it will exponentially decay down to 0 00:01:51.230 --> 00:01:55.820 align:middle line:84% with some time constant tau is equal to RC. 00:01:55.820 --> 00:02:00.050 align:middle line:84% And so as we're charging up this capacitor, 00:02:00.050 --> 00:02:02.030 align:middle line:84% we dissipate some energy in the resistor, 00:02:02.030 --> 00:02:05.280 align:middle line:90% and we can calculate that. 00:02:05.280 --> 00:02:09.720 align:middle line:84% What would happen if I changed the resistor value here? 00:02:09.720 --> 00:02:14.040 align:middle line:84% Suppose I used a much smaller resistor. 00:02:14.040 --> 00:02:19.660 align:middle line:84% So if I did that, what would happen to the initial current? 00:02:19.660 --> 00:02:21.240 align:middle line:84% Well, if I use a smaller resistor, 00:02:21.240 --> 00:02:22.840 align:middle line:84% the initial current would be higher. 00:02:22.840 --> 00:02:23.880 align:middle line:90% So here I'll pick-- 00:02:23.880 --> 00:02:29.520 align:middle line:84% this will be Vs minus Vi over R smaller. 00:02:29.520 --> 00:02:33.130 align:middle line:84% And then it will also decay with the same RC time constant. 00:02:33.130 --> 00:02:34.350 align:middle line:90% So it'll go from 0. 00:02:34.350 --> 00:02:36.650 align:middle line:90% It'll jump up and it'll decay. 00:02:36.650 --> 00:02:39.390 align:middle line:84% Now the RC time constant will be shorter. 00:02:39.390 --> 00:02:41.240 align:middle line:90% So it'll do something like this. 00:02:41.240 --> 00:02:45.030 align:middle line:84% So tau is again equal to RC, but R is smaller 00:02:45.030 --> 00:02:47.520 align:middle line:84% than-- the yellow R is smaller than the pink R. 00:02:47.520 --> 00:02:51.700 align:middle line:84% So it starts with a higher current and decays quicker. 00:02:51.700 --> 00:02:53.180 align:middle line:90% All right? 00:02:53.180 --> 00:02:59.015 align:middle line:84% All right, well, let's calculate this stuff just analytically. 00:02:59.015 --> 00:03:03.400 align:middle line:90% So let's ask this. 00:03:03.400 --> 00:03:07.860 align:middle line:84% How much charge do I deliver in charging this capacitor? 00:03:07.860 --> 00:03:11.050 align:middle line:84% How much charge do I deliver from Vs to Vc? 00:03:11.050 --> 00:03:15.035 align:middle line:84% Or what's the equivalent, the integral of i c here, all right? 00:03:15.035 --> 00:03:19.350 align:middle line:90% 00:03:19.350 --> 00:03:22.560 align:middle line:84% So we could calculate that as delta q is 00:03:22.560 --> 00:03:24.780 align:middle line:84% the charge I'm going to deliver to the capacitor, 00:03:24.780 --> 00:03:31.710 align:middle line:84% is the integral from 0 to infinity of i cdt, right? 00:03:31.710 --> 00:03:37.410 align:middle line:84% And I can calculate that in a number of ways. 00:03:37.410 --> 00:03:42.960 align:middle line:84% I could write this i cdt as equal to dq, 00:03:42.960 --> 00:03:47.640 align:middle line:84% which is also equal to the integral of C dv, 00:03:47.640 --> 00:03:48.743 align:middle line:90% where this is-- 00:03:48.743 --> 00:03:51.760 align:middle line:90% 00:03:51.760 --> 00:03:56.590 align:middle line:84% where C dv is going from V sub i to V sub 00:03:56.590 --> 00:03:59.110 align:middle line:84% s, which is my final-- it charges up eventually 00:03:59.110 --> 00:04:00.980 align:middle line:90% to V sub s, right? 00:04:00.980 --> 00:04:09.680 align:middle line:84% So I could write this charge is just simply C times Vs minus Vi. 00:04:09.680 --> 00:04:10.180 align:middle line:90% OK? 00:04:10.180 --> 00:04:12.520 align:middle line:84% That's the amount of charge that flows out 00:04:12.520 --> 00:04:14.500 align:middle line:84% of the source on to the capacitor 00:04:14.500 --> 00:04:18.410 align:middle line:84% to get it from its initial voltage up to its final voltage 00:04:18.410 --> 00:04:20.089 align:middle line:90% Vs. OK? 00:04:20.089 --> 00:04:21.330 align:middle line:90% Any questions about that? 00:04:21.330 --> 00:04:24.200 align:middle line:90% 00:04:24.200 --> 00:04:28.148 align:middle line:84% So let's just ask, where's the energy going in this circuit? 00:04:28.148 --> 00:04:29.690 align:middle line:84% I said qualitatively, you know, we're 00:04:29.690 --> 00:04:31.250 align:middle line:84% going to dissipate some power in the resistor 00:04:31.250 --> 00:04:32.375 align:middle line:90% to charge up the capacitor. 00:04:32.375 --> 00:04:33.540 align:middle line:90% Let's calculate that. 00:04:33.540 --> 00:04:36.440 align:middle line:84% How much energy comes from the source. 00:04:36.440 --> 00:04:43.250 align:middle line:84% Delta E source-- and I'm talking about the energy out coming out 00:04:43.250 --> 00:04:46.950 align:middle line:84% of the source to charge everything-- 00:04:46.950 --> 00:04:52.710 align:middle line:84% is simply going to be Vs times delta q, which 00:04:52.710 --> 00:05:02.120 align:middle line:84% is going to be C Vs squared minus C Vs Vi. 00:05:02.120 --> 00:05:02.620 align:middle line:90% OK. 00:05:02.620 --> 00:05:04.690 align:middle line:84% That's how much energy comes out of the source 00:05:04.690 --> 00:05:07.850 align:middle line:90% to charge the capacitor. 00:05:07.850 --> 00:05:10.110 align:middle line:84% How much energy goes into the capacitor? 00:05:10.110 --> 00:05:12.440 align:middle line:90% We can again use-- 00:05:12.440 --> 00:05:15.230 align:middle line:84% oh, here we can use energy methods very clearly. 00:05:15.230 --> 00:05:19.430 align:middle line:90% Delta E cap is-- 00:05:19.430 --> 00:05:26.300 align:middle line:84% well, the energy that ends up on the cap is 1/2 C Vs squared. 00:05:26.300 --> 00:05:31.220 align:middle line:84% The energy we started on the cap before we close the switch was 00:05:31.220 --> 00:05:35.790 align:middle line:84% simply 1/2 C Vi quantity squared. 00:05:35.790 --> 00:05:39.560 align:middle line:84% And that's how much energy, and that's how much change in energy 00:05:39.560 --> 00:05:42.150 align:middle line:90% is existing on the cap. 00:05:42.150 --> 00:05:44.470 align:middle line:90% Makes sense to everybody? 00:05:44.470 --> 00:05:50.350 align:middle line:84% So that means that the lost energy, delta E loss, 00:05:50.350 --> 00:05:54.653 align:middle line:84% which, of course, some energy is coming out of the cap, 00:05:54.653 --> 00:05:56.320 align:middle line:84% some energy is coming out of the source, 00:05:56.320 --> 00:05:58.340 align:middle line:84% and some energy is ending up in the cap. 00:05:58.340 --> 00:06:02.290 align:middle line:84% The rest, obviously, delta E loss is in the resistor in this 00:06:02.290 --> 00:06:04.930 align:middle line:84% model-- there's nowhere else for it to go-- 00:06:04.930 --> 00:06:13.390 align:middle line:84% is going to be delta E source minus delta E cap, 00:06:13.390 --> 00:06:18.420 align:middle line:84% which I could write as this minus this, 00:06:18.420 --> 00:06:27.300 align:middle line:84% which is going to be C Vs squared minus C Vs Vi minus 1/2 00:06:27.300 --> 00:06:34.090 align:middle line:84% C V squared C Vs squared plus 1/2 C Vi squared. 00:06:34.090 --> 00:06:38.070 align:middle line:84% And if I collapse this down a little bit and rearrange it, 00:06:38.070 --> 00:06:46.310 align:middle line:84% what I get is 1/2 C times Vs minus Vi quantity squared. 00:06:46.310 --> 00:06:51.452 align:middle line:90% 00:06:51.452 --> 00:06:51.952 align:middle line:90% OK? 00:06:51.952 --> 00:06:55.256 align:middle line:90% 00:06:55.256 --> 00:06:59.380 align:middle line:90% And this is in the resistor? 00:06:59.380 --> 00:07:01.485 align:middle line:90% OK? 00:07:01.485 --> 00:07:02.920 align:middle line:90% Any questions about that? 00:07:02.920 --> 00:07:06.340 align:middle line:90% 00:07:06.340 --> 00:07:06.850 align:middle line:90% Yeah. 00:07:06.850 --> 00:07:10.270 align:middle line:84% AUDIENCE: So in this model, like, you 00:07:10.270 --> 00:07:13.165 align:middle line:84% didn't assume that the resistor is in the circuit? 00:07:13.165 --> 00:07:14.290 align:middle line:90% DAVID PERREAULT: No, I did. 00:07:14.290 --> 00:07:16.310 align:middle line:90% Of course I did. 00:07:16.310 --> 00:07:18.615 align:middle line:90% Right? 00:07:18.615 --> 00:07:20.770 align:middle line:84% I mean, I still have conservation of energy, right? 00:07:20.770 --> 00:07:22.690 align:middle line:90% I didn't calculate an integral. 00:07:22.690 --> 00:07:24.370 align:middle line:84% Like, I didn't write the integral 00:07:24.370 --> 00:07:25.420 align:middle line:90% in terms of the resistor. 00:07:25.420 --> 00:07:27.920 align:middle line:84% But I still, I didn't-- it's not that I didn't assume it was 00:07:27.920 --> 00:07:28.763 align:middle line:90% there. 00:07:28.763 --> 00:07:31.180 align:middle line:84% AUDIENCE: I guess I think you could have drawn the circuit 00:07:31.180 --> 00:07:33.490 align:middle line:84% and written down the same equations 00:07:33.490 --> 00:07:35.800 align:middle line:90% and arrived at the same result. 00:07:35.800 --> 00:07:38.260 align:middle line:84% DAVID PERREAULT: Absolutely right. 00:07:38.260 --> 00:07:39.970 align:middle line:90% Very good observation. 00:07:39.970 --> 00:07:42.250 align:middle line:90% So what's going on here? 00:07:42.250 --> 00:07:42.980 align:middle line:90% Did I cheat? 00:07:42.980 --> 00:07:43.910 align:middle line:90% No, I did not cheat. 00:07:43.910 --> 00:07:46.780 align:middle line:84% I just-- I'm lazy and I don't like doing hard integrals. 00:07:46.780 --> 00:07:48.860 align:middle line:84% Or even easy integrals, for that matter. 00:07:48.860 --> 00:07:50.030 align:middle line:90% What's going on here? 00:07:50.030 --> 00:07:53.000 align:middle line:90% It's exactly the picture I drew. 00:07:53.000 --> 00:07:53.500 align:middle line:90% OK. 00:07:53.500 --> 00:07:56.210 align:middle line:90% 00:07:56.210 --> 00:08:00.170 align:middle line:84% If I had some nominal value of resistor, 1 ohm, 00:08:00.170 --> 00:08:01.980 align:middle line:90% I'd get exactly this curve. 00:08:01.980 --> 00:08:03.840 align:middle line:84% And if I squared this-- this is a current. 00:08:03.840 --> 00:08:07.332 align:middle line:84% If I squared this, I could write the exponential decay, 00:08:07.332 --> 00:08:09.540 align:middle line:84% and then I could integrate it, and I'd get some loss. 00:08:09.540 --> 00:08:12.910 align:middle line:90% I would get exactly that loss. 00:08:12.910 --> 00:08:16.450 align:middle line:84% And if I change the value of resistance, I'd get this curve, 00:08:16.450 --> 00:08:19.090 align:middle line:84% and I'd get exactly the same loss. 00:08:19.090 --> 00:08:20.920 align:middle line:84% So what's being noticed here is the-- 00:08:20.920 --> 00:08:22.810 align:middle line:84% none of these answers, the amount of energy 00:08:22.810 --> 00:08:24.340 align:middle line:84% that comes out of the source, the amount of energy, 00:08:24.340 --> 00:08:26.740 align:middle line:84% the one under the cap, and the amount of energy I lose, 00:08:26.740 --> 00:08:28.990 align:middle line:84% don't depend on the resistance at all. 00:08:28.990 --> 00:08:30.890 align:middle line:90% You get the same answer. 00:08:30.890 --> 00:08:32.590 align:middle line:90% The waveforms are different. 00:08:32.590 --> 00:08:33.700 align:middle line:90% It depends. 00:08:33.700 --> 00:08:35.538 align:middle line:84% The current waveform that I get depends 00:08:35.538 --> 00:08:36.830 align:middle line:90% upon the value of the resistor. 00:08:36.830 --> 00:08:38.955 align:middle line:84% What happens if I keep making that resistor smaller 00:08:38.955 --> 00:08:40.190 align:middle line:90% and smaller and smaller? 00:08:40.190 --> 00:08:45.130 align:middle line:84% What I get is currents that have higher peaks and shorter 00:08:45.130 --> 00:08:47.800 align:middle line:84% durations, until, like, as R approaches 0, 00:08:47.800 --> 00:08:51.190 align:middle line:84% this more or less becomes a kind of impulse. 00:08:51.190 --> 00:08:54.250 align:middle line:84% But it's an impulse with an amount of charge delta 00:08:54.250 --> 00:08:57.410 align:middle line:90% q that C Vs minus Vi. 00:08:57.410 --> 00:09:01.580 align:middle line:84% And the amount of energy I lose in the resistor-- yeah, 00:09:01.580 --> 00:09:02.832 align:middle line:90% I have a smaller resistor. 00:09:02.832 --> 00:09:05.540 align:middle line:84% It's going to 0, but its current through, it's going to infinity, 00:09:05.540 --> 00:09:07.220 align:middle line:84% and it goes through it in just such a way 00:09:07.220 --> 00:09:08.762 align:middle line:84% that I lose the same amount of energy 00:09:08.762 --> 00:09:10.400 align:middle line:84% no matter what the resistance is. 00:09:10.400 --> 00:09:11.900 align:middle line:90% So it's not that I'm cheating. 00:09:11.900 --> 00:09:13.400 align:middle line:84% It's just that this is what happens 00:09:13.400 --> 00:09:15.482 align:middle line:84% when you take a limit of a resistance, go to 0, 00:09:15.482 --> 00:09:17.940 align:middle line:84% but you're getting a limit as the current goes to infinity. 00:09:17.940 --> 00:09:20.030 align:middle line:84% And if I put that into the circuit 00:09:20.030 --> 00:09:22.910 align:middle line:84% and I took that integral, and I took the limit as R goes to 0, 00:09:22.910 --> 00:09:24.830 align:middle line:90% I'd get the same answer. 00:09:24.830 --> 00:09:28.420 align:middle line:84% So I'm just lazy, and I kind of skip over the integral. 00:09:28.420 --> 00:09:29.340 align:middle line:90% OK? 00:09:29.340 --> 00:09:29.840 align:middle line:90% Yeah. 00:09:29.840 --> 00:09:30.340 align:middle line:90% Go ahead. 00:09:30.340 --> 00:09:32.392 align:middle line:84% AUDIENCE: In the case that there is a non-zero R, 00:09:32.392 --> 00:09:33.850 align:middle line:84% it makes sense that the energy that 00:09:33.850 --> 00:09:37.420 align:middle line:84% gets lost through the resistor [INAUDIBLE] R equals 0. 00:09:37.420 --> 00:09:39.520 align:middle line:90% Where is the energy lost? 00:09:39.520 --> 00:09:41.055 align:middle line:84% DAVID PERREAULT: The energy loss-- 00:09:41.055 --> 00:09:43.180 align:middle line:84% I mean, the actual distribution of the energy loss, 00:09:43.180 --> 00:09:47.230 align:middle line:84% now you're asking about idealized situations. 00:09:47.230 --> 00:09:54.620 align:middle line:84% Somewhere in this loop, I've got to dissipate energy. 00:09:54.620 --> 00:09:56.650 align:middle line:90% And so there has to be some-- 00:09:56.650 --> 00:09:58.480 align:middle line:84% it's sort of like saying if I connect 00:09:58.480 --> 00:10:02.000 align:middle line:84% a voltage source across a short circuit, what happens? 00:10:02.000 --> 00:10:05.650 align:middle line:84% Like, well, you've got to have some approximation. 00:10:05.650 --> 00:10:07.970 align:middle line:84% You've got to have some limiting resistance going to 0. 00:10:07.970 --> 00:10:11.620 align:middle line:84% So in the real world, if this switch and resistor 00:10:11.620 --> 00:10:13.057 align:middle line:84% were an ideal MOSFET or something, 00:10:13.057 --> 00:10:14.140 align:middle line:90% where would the energy go? 00:10:14.140 --> 00:10:17.710 align:middle line:84% It would depend upon where the actual resistance was 00:10:17.710 --> 00:10:20.650 align:middle line:90% distributed in the circuit. 00:10:20.650 --> 00:10:23.913 align:middle line:84% And to the extent that the resistance 00:10:23.913 --> 00:10:25.330 align:middle line:84% is in the plates of the capacitor, 00:10:25.330 --> 00:10:26.538 align:middle line:90% it will get dissipated there. 00:10:26.538 --> 00:10:31.120 align:middle line:84% But if you want to if you want to force ideality 00:10:31.120 --> 00:10:34.040 align:middle line:84% to where things are 0, then you can get squirrely results 00:10:34.040 --> 00:10:35.620 align:middle line:84% if you don't account for the fact 00:10:35.620 --> 00:10:37.960 align:middle line:90% that you have infinite currents. 00:10:37.960 --> 00:10:38.460 align:middle line:90% OK. 00:10:38.460 --> 00:10:41.880 align:middle line:90% 00:10:41.880 --> 00:10:45.420 align:middle line:84% This kind of charge, or energy-- charge 00:10:45.420 --> 00:10:49.340 align:middle line:84% in energy methods are very convenient for working things, 00:10:49.340 --> 00:10:52.980 align:middle line:84% results out where you don't want to have to integrate things. 00:10:52.980 --> 00:10:55.280 align:middle line:84% And it turns out, in fact, it doesn't 00:10:55.280 --> 00:10:57.940 align:middle line:90% matter what this element is. 00:10:57.940 --> 00:11:00.970 align:middle line:90% 00:11:00.970 --> 00:11:06.110 align:middle line:84% Basically, I could replace this with a non-linear resistor. 00:11:06.110 --> 00:11:09.093 align:middle line:84% Like, so, if the MOSFET was nonlinear, 00:11:09.093 --> 00:11:11.260 align:middle line:84% you'd get the same answer because it doesn't matter. 00:11:11.260 --> 00:11:14.230 align:middle line:84% The time waveform, the fact that I drew this 00:11:14.230 --> 00:11:16.550 align:middle line:84% as a perfect exponential, even if this resistor was 00:11:16.550 --> 00:11:20.110 align:middle line:84% a non-linear, so long as I ended up 00:11:20.110 --> 00:11:24.410 align:middle line:84% with Vc at the same voltage as Vs, 00:11:24.410 --> 00:11:26.040 align:middle line:84% I'm going to get the same dissipation. 00:11:26.040 --> 00:11:27.493 align:middle line:84% So in fact, even if this had some 00:11:27.493 --> 00:11:29.660 align:middle line:84% inductance, and it rang back and forth a little bit, 00:11:29.660 --> 00:11:30.577 align:middle line:90% and then settled down. 00:11:30.577 --> 00:11:32.590 align:middle line:90% I get the same total loss. 00:11:32.590 --> 00:11:35.170 align:middle line:90% It just doesn't matter. 00:11:35.170 --> 00:11:37.450 align:middle line:84% AUDIENCE: This is just a result of the fact that this 00:11:37.450 --> 00:11:39.580 align:middle line:90% is in some sort of [INAUDIBLE]. 00:11:39.580 --> 00:11:41.270 align:middle line:84% DAVID PERREAULT: Yeah, that's right. 00:11:41.270 --> 00:11:43.540 align:middle line:84% I mean, it satisfies conservation of energy, 00:11:43.540 --> 00:11:48.880 align:middle line:84% and it satisfies the fact that the change in the capacitor 00:11:48.880 --> 00:11:52.200 align:middle line:84% voltage determines the charge flow both here and here. 00:11:52.200 --> 00:11:55.620 align:middle line:90% 00:11:55.620 --> 00:11:57.040 align:middle line:90% Excellent observation. 00:11:57.040 --> 00:12:00.000 align:middle line:90% 00:12:00.000 --> 00:12:02.280 align:middle line:90% What does this imply? 00:12:02.280 --> 00:12:05.940 align:middle line:84% Well, what is the efficiency of charging this capacitor? 00:12:05.940 --> 00:12:07.850 align:middle line:84% If I thought, OK, I brought this capacitor 00:12:07.850 --> 00:12:11.480 align:middle line:84% from some initial voltage up to Vs. 00:12:11.480 --> 00:12:15.950 align:middle line:84% What fraction of the energy I sourced got to the load, right? 00:12:15.950 --> 00:12:18.470 align:middle line:84% So if I thought of the capacitor as the load. 00:12:18.470 --> 00:12:21.890 align:middle line:84% So I could think of the charging efficiency 00:12:21.890 --> 00:12:29.720 align:middle line:84% as being equal to the energy delta E cap-- 00:12:29.720 --> 00:12:31.820 align:middle line:84% that's the energy that went to the capacitor-- 00:12:31.820 --> 00:12:33.040 align:middle line:90% over delta E source. 00:12:33.040 --> 00:12:36.120 align:middle line:90% 00:12:36.120 --> 00:12:38.410 align:middle line:90% And what is that? 00:12:38.410 --> 00:12:45.570 align:middle line:84% Well, I could write that as equal to 1/2 C Vs squared minus 00:12:45.570 --> 00:12:54.490 align:middle line:84% 1/2 C Vi quantity squared divided by delta E source is C 00:12:54.490 --> 00:12:56.750 align:middle line:90% Vs squared minus C Vs Vi. 00:12:56.750 --> 00:13:04.424 align:middle line:90% 00:13:04.424 --> 00:13:11.455 align:middle line:84% And I can divide out the capacitance here in this. 00:13:11.455 --> 00:13:11.955 align:middle line:90% OK. 00:13:11.955 --> 00:13:16.050 align:middle line:90% 00:13:16.050 --> 00:13:22.990 align:middle line:84% So I can write this as equal to-- if I factor out 1/2 Vs 00:13:22.990 --> 00:13:26.320 align:middle line:84% squared, or if I divide top and bottom by Vs squared, 00:13:26.320 --> 00:13:40.530 align:middle line:84% I can write this as 1/2 times 1 minus Vi over Vs quantity 00:13:40.530 --> 00:13:49.065 align:middle line:84% squared divided by 1 minus Vi over Vs, 00:13:49.065 --> 00:14:00.230 align:middle line:84% which I could write as 1/2 times 1 plus Vi over Vs. 00:14:00.230 --> 00:14:02.540 align:middle line:90% That makes sense to everybody? 00:14:02.540 --> 00:14:04.250 align:middle line:84% And in fact, actually, you can write it 00:14:04.250 --> 00:14:05.880 align:middle line:90% in a slightly different way. 00:14:05.880 --> 00:14:14.810 align:middle line:84% If I thought of delta V as being equal to Vs minus Vi-- 00:14:14.810 --> 00:14:17.040 align:middle line:84% that's how much I'm charging up the cap-- 00:14:17.040 --> 00:14:19.650 align:middle line:84% this can also be written-- and I won't go through the math-- 00:14:19.650 --> 00:14:27.380 align:middle line:90% this can also be written as 1-- 00:14:27.380 --> 00:14:27.880 align:middle line:90% sorry. 00:14:27.880 --> 00:14:35.522 align:middle line:90% 1 minus 1/2 of delta V over Vs. 00:14:35.522 --> 00:14:38.230 align:middle line:90% 00:14:38.230 --> 00:14:41.030 align:middle line:84% So why am I dragging you through this? 00:14:41.030 --> 00:14:43.870 align:middle line:84% What I want to understand is how does this charging 00:14:43.870 --> 00:14:45.280 align:middle line:90% efficiency go? 00:14:45.280 --> 00:14:50.446 align:middle line:84% Well, if I charge the capacitor from 0, 00:14:50.446 --> 00:14:52.390 align:middle line:84% what would be the charging efficiency? 00:14:52.390 --> 00:14:55.920 align:middle line:84% Well, if Vi was 0, this would go to 0. 00:14:55.920 --> 00:14:57.485 align:middle line:90% This would go to this. 00:14:57.485 --> 00:14:59.610 align:middle line:84% The numerator would go to 1, the button denominator 00:14:59.610 --> 00:15:00.193 align:middle line:90% would go to 1. 00:15:00.193 --> 00:15:02.670 align:middle line:84% My charging efficiency would be 50%. 00:15:02.670 --> 00:15:06.450 align:middle line:84% So if I charge the capacitor up from 0 to final value, 00:15:06.450 --> 00:15:08.910 align:middle line:84% I put 1/2 C V squared on the capacitor, 00:15:08.910 --> 00:15:12.240 align:middle line:84% and I dissipate half C V squared in the resistor. 00:15:12.240 --> 00:15:15.240 align:middle line:84% So it's not a very happy energy transfer. 00:15:15.240 --> 00:15:19.220 align:middle line:84% I lose half my energy in charging up that capacitor. 00:15:19.220 --> 00:15:23.870 align:middle line:84% On the other hand, suppose Vi was-- 00:15:23.870 --> 00:15:28.220 align:middle line:84% suppose the initial voltage on the capacitor was half of Vs. 00:15:28.220 --> 00:15:32.930 align:middle line:84% If I do this calculation, this is a half of 1 plus 0.5, 00:15:32.930 --> 00:15:34.950 align:middle line:90% or it becomes 0.75. 00:15:34.950 --> 00:15:40.310 align:middle line:84% So if I charge half way up from Vs to Vs, 00:15:40.310 --> 00:15:44.900 align:middle line:84% I have 75% energy efficiency in charging that. 00:15:44.900 --> 00:15:48.590 align:middle line:84% And in the limit as Vi approaches Vs, 00:15:48.590 --> 00:15:51.460 align:middle line:90% the efficiency goes to 1. 00:15:51.460 --> 00:15:54.940 align:middle line:84% So if I want to have high efficiency, 00:15:54.940 --> 00:15:57.970 align:middle line:84% I only want to charge the capacitor a little. 00:15:57.970 --> 00:16:00.280 align:middle line:84% And the way you could think about that 00:16:00.280 --> 00:16:02.330 align:middle line:90% is, what's this happening? 00:16:02.330 --> 00:16:05.590 align:middle line:84% When I have the capacitor voltage is very small 00:16:05.590 --> 00:16:08.618 align:middle line:84% and the Vs is very big, I've got a big voltage 00:16:08.618 --> 00:16:10.660 align:middle line:84% across this resistor, I'm dumping a lot of energy 00:16:10.660 --> 00:16:12.610 align:middle line:84% in the resistor compared to the capacitor. 00:16:12.610 --> 00:16:15.370 align:middle line:84% Whereas, if Vc is very close to Vs, 00:16:15.370 --> 00:16:17.840 align:middle line:84% I only have a tiny voltage across the resistor. 00:16:17.840 --> 00:16:20.500 align:middle line:84% Most of the energy goes into the capacitor. 00:16:20.500 --> 00:16:23.620 align:middle line:84% So if I charge a capacitor only a little bit, 00:16:23.620 --> 00:16:27.640 align:middle line:84% I only lose a small fraction of the energy. 00:16:27.640 --> 00:16:29.060 align:middle line:84% Does that make sense to everybody? 00:16:29.060 --> 00:16:33.210 align:middle line:90% 00:16:33.210 --> 00:16:41.080 align:middle line:84% I should have noted, by the way, that there 00:16:41.080 --> 00:16:47.830 align:middle line:84% is a way to make this circuit have greater efficiency. 00:16:47.830 --> 00:16:53.060 align:middle line:84% And what that would be is if I put an inductor in series here, 00:16:53.060 --> 00:16:57.440 align:middle line:84% and I slosh charge from Vs onto Vc, 00:16:57.440 --> 00:17:00.710 align:middle line:84% but let Vc finish and then reopen 00:17:00.710 --> 00:17:05.960 align:middle line:84% the switch when Vc is at some voltage greater than Vs-- 00:17:05.960 --> 00:17:09.530 align:middle line:84% so this limit on the efficiency that it 00:17:09.530 --> 00:17:13.130 align:middle line:84% doesn't matter how I charge it is only true if Vc ends up 00:17:13.130 --> 00:17:18.619 align:middle line:84% at Vs. If I make Vc somehow end up greater than Vs-- 00:17:18.619 --> 00:17:21.680 align:middle line:84% for example, through a resident sloshing of energy-- 00:17:21.680 --> 00:17:24.709 align:middle line:84% then I can get higher efficiency. 00:17:24.709 --> 00:17:26.839 align:middle line:84% But in a switch capacitor converter, 00:17:26.839 --> 00:17:28.280 align:middle line:84% or in a converter where I'm going 00:17:28.280 --> 00:17:32.750 align:middle line:84% to let the final value of the capacitor go to Vs, 00:17:32.750 --> 00:17:34.910 align:middle line:84% the efficiency is fixed by the charge 00:17:34.910 --> 00:17:38.550 align:middle line:90% transfer in the initial value. 00:17:38.550 --> 00:17:39.755 align:middle line:90% Any questions about that? 00:17:39.755 --> 00:17:46.136 align:middle line:90% 00:17:46.136 --> 00:17:49.460 align:middle line:84% So how does this relate to what we were talking about last time? 00:17:49.460 --> 00:17:53.620 align:middle line:84% Let's remind ourselves of the circuit that we saw last time. 00:17:53.620 --> 00:17:58.025 align:middle line:84% And it was a two to one switch capacitor step-down converter. 00:17:58.025 --> 00:17:59.650 align:middle line:84% When we say two to one, we're referring 00:17:59.650 --> 00:18:01.270 align:middle line:84% to voltage conversion, because that's 00:18:01.270 --> 00:18:03.790 align:middle line:90% just the way people do it. 00:18:03.790 --> 00:18:06.075 align:middle line:84% So here's the circuit we're talking about. 00:18:06.075 --> 00:18:08.770 align:middle line:90% 00:18:08.770 --> 00:18:12.445 align:middle line:84% Here's Vs. I'm going to have a first switch 1. 00:18:12.445 --> 00:18:14.320 align:middle line:84% I'm going to have a capacitor where I'm going 00:18:14.320 --> 00:18:17.200 align:middle line:90% to have some voltage Vc on it. 00:18:17.200 --> 00:18:19.420 align:middle line:90% Another switch 1. 00:18:19.420 --> 00:18:22.090 align:middle line:84% And then this is going to go to some big capacitor. 00:18:22.090 --> 00:18:24.310 align:middle line:90% I'll call it C big. 00:18:24.310 --> 00:18:27.080 align:middle line:84% That's going to hold the voltage. 00:18:27.080 --> 00:18:31.900 align:middle line:84% And he's loaded with some current, I out. 00:18:31.900 --> 00:18:35.380 align:middle line:84% And then this is one charging state, 00:18:35.380 --> 00:18:41.230 align:middle line:84% and then the other charging state looks like this. 00:18:41.230 --> 00:18:43.120 align:middle line:90% This is switch 2. 00:18:43.120 --> 00:18:44.310 align:middle line:90% And switch 2. 00:18:44.310 --> 00:18:47.370 align:middle line:90% 00:18:47.370 --> 00:18:55.030 align:middle line:84% So what we do is, in, say, the switch 2 state-- 00:18:55.030 --> 00:18:55.550 align:middle line:90% I'm sorry. 00:18:55.550 --> 00:18:59.290 align:middle line:84% In the switch 1 state, we charge from Vs, 00:18:59.290 --> 00:19:04.030 align:middle line:84% charging this capacitor C up by the difference between Vs 00:19:04.030 --> 00:19:04.860 align:middle line:90% and V out. 00:19:04.860 --> 00:19:09.136 align:middle line:90% 00:19:09.136 --> 00:19:13.330 align:middle line:84% And then in the 2 state, we discharge C into the output 00:19:13.330 --> 00:19:15.460 align:middle line:84% so that the output current is going 00:19:15.460 --> 00:19:20.300 align:middle line:84% to be always twice the average input current. 00:19:20.300 --> 00:19:21.870 align:middle line:90% And we came up with a model. 00:19:21.870 --> 00:19:25.560 align:middle line:84% We derived a model for everything inside of this box. 00:19:25.560 --> 00:19:28.910 align:middle line:84% So I could take everything in here 00:19:28.910 --> 00:19:31.490 align:middle line:84% and come up with an equivalent average circuit 00:19:31.490 --> 00:19:33.220 align:middle line:90% model, a DC average model. 00:19:33.220 --> 00:19:34.760 align:middle line:90% It would look like this. 00:19:34.760 --> 00:19:36.920 align:middle line:90% I'd have V in. 00:19:36.920 --> 00:19:40.370 align:middle line:90% 00:19:40.370 --> 00:19:44.016 align:middle line:84% And the rest of it I can model with a two to one transformer. 00:19:44.016 --> 00:19:52.650 align:middle line:90% 00:19:52.650 --> 00:19:57.090 align:middle line:90% And then a resistor, R eq. 00:19:57.090 --> 00:20:02.300 align:middle line:84% And in the slow-switching limit, this 00:20:02.300 --> 00:20:08.930 align:middle line:84% is 1 over 4 fc, where this is the switching frequency. 00:20:08.930 --> 00:20:12.400 align:middle line:84% And this is everything inside that box. 00:20:12.400 --> 00:20:15.690 align:middle line:90% 00:20:15.690 --> 00:20:18.670 align:middle line:84% And then I can model the output resistance. 00:20:18.670 --> 00:20:22.440 align:middle line:90% Output current, I out. 00:20:22.440 --> 00:20:25.380 align:middle line:90% And this is V out. 00:20:25.380 --> 00:20:26.700 align:middle line:90% That makes sense to everybody? 00:20:26.700 --> 00:20:29.764 align:middle line:90% Everybody recall this? 00:20:29.764 --> 00:20:34.710 align:middle line:84% So notice that this is the slow switching limit, 00:20:34.710 --> 00:20:41.640 align:middle line:84% and the slow switching limit, R eq on the log scale, 00:20:41.640 --> 00:20:54.730 align:middle line:84% versus F switch on a log scale, goes as 1 over 4 F C. 00:20:54.730 --> 00:21:00.590 align:middle line:90% And why does it do that? 00:21:00.590 --> 00:21:07.890 align:middle line:84% Because for a given load current, 00:21:07.890 --> 00:21:11.080 align:middle line:84% the charge transfer on and off of this capacitance 00:21:11.080 --> 00:21:13.610 align:middle line:84% is exactly determined by the load current. 00:21:13.610 --> 00:21:15.700 align:middle line:84% I transfer a certain amount of charge per cycle 00:21:15.700 --> 00:21:19.090 align:middle line:84% through the cap with delta q coming out of Vs 00:21:19.090 --> 00:21:22.420 align:middle line:84% and ultimately 2 delta q going to the output. 00:21:22.420 --> 00:21:27.660 align:middle line:84% And as I make that current bigger and bigger, that 00:21:27.660 --> 00:21:31.200 align:middle line:84% capacitor, if I make the current bigger and bigger, 00:21:31.200 --> 00:21:33.870 align:middle line:84% that capacitor has to charge up and down more, 00:21:33.870 --> 00:21:35.520 align:middle line:90% and my efficiency should droop. 00:21:35.520 --> 00:21:38.340 align:middle line:84% Or equivalently, as I make C bigger and bigger, 00:21:38.340 --> 00:21:41.290 align:middle line:84% for a given charge transfer, his voltage swing shrinks. 00:21:41.290 --> 00:21:43.930 align:middle line:90% 00:21:43.930 --> 00:21:45.495 align:middle line:84% Or if I turn up switching frequency, 00:21:45.495 --> 00:21:47.010 align:middle line:90% then the amount of time-- 00:21:47.010 --> 00:21:50.220 align:middle line:84% I have to transfer less charge in each switching cycle. 00:21:50.220 --> 00:21:53.620 align:middle line:84% And the capacitor voltage doesn't move as much. 00:21:53.620 --> 00:21:56.500 align:middle line:84% And the resistance goes down, the charging loss goes down. 00:21:56.500 --> 00:21:57.210 align:middle line:90% Why? 00:21:57.210 --> 00:21:59.460 align:middle line:84% Because essentially, in each cycle, 00:21:59.460 --> 00:22:03.180 align:middle line:84% I charge and discharge that capacitor less. 00:22:03.180 --> 00:22:07.320 align:middle line:84% So this reflection that the output resistance or the loss 00:22:07.320 --> 00:22:09.660 align:middle line:84% due to charging and discharging this resistance, 00:22:09.660 --> 00:22:13.710 align:middle line:84% this capacitor through the switch resistance, is inversely 00:22:13.710 --> 00:22:19.410 align:middle line:84% varied with frequency, or increases with load current, 00:22:19.410 --> 00:22:22.680 align:middle line:84% that the loss is exactly related to how much up and down 00:22:22.680 --> 00:22:24.430 align:middle line:90% I'm charging the capacitor. 00:22:24.430 --> 00:22:27.370 align:middle line:90% 00:22:27.370 --> 00:22:28.605 align:middle line:90% Any questions about that? 00:22:28.605 --> 00:22:33.170 align:middle line:90% 00:22:33.170 --> 00:22:37.480 align:middle line:84% And then the only thing we said last time was, well, 00:22:37.480 --> 00:22:39.610 align:middle line:84% this isn't the slow switching limit where I have 00:22:39.610 --> 00:22:41.250 align:middle line:90% this kind of charging, right? 00:22:41.250 --> 00:22:45.130 align:middle line:84% The capacitor charges-- the capacitor current 00:22:45.130 --> 00:22:48.970 align:middle line:84% does this, and gets to its final charge state, 00:22:48.970 --> 00:22:53.950 align:middle line:84% right, assuming that C big is so big that it's staying constant. 00:22:53.950 --> 00:22:57.130 align:middle line:84% If I turn up the switching frequency enough, 00:22:57.130 --> 00:22:59.440 align:middle line:84% I get on to a time scale that's short 00:22:59.440 --> 00:23:01.130 align:middle line:90% compared to this time constant. 00:23:01.130 --> 00:23:04.030 align:middle line:90% Maybe it's only doing this. 00:23:04.030 --> 00:23:07.480 align:middle line:84% The current never charging or discharging the capacitor 00:23:07.480 --> 00:23:08.660 align:middle line:90% voltage very much. 00:23:08.660 --> 00:23:12.325 align:middle line:84% And the capacitor voltage isn't moving around very much. 00:23:12.325 --> 00:23:15.380 align:middle line:84% And in that case, I would get to a new limit. 00:23:15.380 --> 00:23:18.830 align:middle line:84% And we said the resistance in the fast switching limit-- 00:23:18.830 --> 00:23:20.480 align:middle line:90% so this is the SSL-- 00:23:20.480 --> 00:23:22.990 align:middle line:90% 00:23:22.990 --> 00:23:25.750 align:middle line:90% would look like this. 00:23:25.750 --> 00:23:35.970 align:middle line:84% And in the FSL, R eq ends up being 4-- 00:23:35.970 --> 00:23:39.810 align:middle line:90% I'm sorry-- 2R switch. 00:23:39.810 --> 00:23:44.250 align:middle line:84% And that makes sense, because in the fast-switching limit, 00:23:44.250 --> 00:23:48.360 align:middle line:84% I'm basically always charging the output through two switches. 00:23:48.360 --> 00:23:51.330 align:middle line:84% And so I ought to see two switch resistances 00:23:51.330 --> 00:23:52.870 align:middle line:90% in the fast-switching limit. 00:23:52.870 --> 00:23:54.900 align:middle line:84% And if I said, well, what does it 00:23:54.900 --> 00:24:00.990 align:middle line:84% look like across the switching frequency for a given 00:24:00.990 --> 00:24:03.300 align:middle line:84% load across the switching frequency, 00:24:03.300 --> 00:24:06.112 align:middle line:90% it does something like this. 00:24:06.112 --> 00:24:08.320 align:middle line:84% It's one thing in the fast-switching-- slow-switching 00:24:08.320 --> 00:24:08.820 align:middle line:90% limit. 00:24:08.820 --> 00:24:11.682 align:middle line:84% It's another thing in the fast-switching limit. 00:24:11.682 --> 00:24:15.460 align:middle line:90% Any questions about that? 00:24:15.460 --> 00:24:17.320 align:middle line:84% So if I'm going to draw more current, 00:24:17.320 --> 00:24:19.575 align:middle line:90% I'm going to get more droop. 00:24:19.575 --> 00:24:21.990 align:middle line:84% If I draw more current, there's more droop 00:24:21.990 --> 00:24:24.640 align:middle line:84% across this equivalent resistance. 00:24:24.640 --> 00:24:28.980 align:middle line:84% Or if I turn up frequency until I hit the fast-switching limit, 00:24:28.980 --> 00:24:30.630 align:middle line:84% I can make that resistance smaller 00:24:30.630 --> 00:24:33.700 align:middle line:90% and make the droop smaller. 00:24:33.700 --> 00:24:35.630 align:middle line:84% Let's see a demonstration of this. 00:24:35.630 --> 00:24:37.750 align:middle line:84% So I'm going to pass this to Dave Otten, who's 00:24:37.750 --> 00:24:40.860 align:middle line:84% built us a really beautiful demo to illustrate this property. 00:24:40.860 --> 00:24:50.180 align:middle line:90% 00:24:50.180 --> 00:24:51.932 align:middle line:90% DAVID OTTEN: OK. 00:24:51.932 --> 00:24:55.280 align:middle line:90% So right now, I'm going to-- 00:24:55.280 --> 00:24:57.920 align:middle line:84% the purple waveform which I'm about to turn off 00:24:57.920 --> 00:25:01.160 align:middle line:84% is my input supply, and the green waveform 00:25:01.160 --> 00:25:03.330 align:middle line:90% is the output of this converter. 00:25:03.330 --> 00:25:05.540 align:middle line:84% And right now I have no load on it, OK? 00:25:05.540 --> 00:25:09.780 align:middle line:84% So let me turn off the purple so you can see it. 00:25:09.780 --> 00:25:13.610 align:middle line:84% Now the yellow waveform is the top of our flying capacitor. 00:25:13.610 --> 00:25:16.550 align:middle line:84% So you can see that it sometimes is up at the supply. 00:25:16.550 --> 00:25:18.570 align:middle line:90% Sometimes it's at the output. 00:25:18.570 --> 00:25:22.010 align:middle line:84% And then the blue waveform is the bottom 00:25:22.010 --> 00:25:23.520 align:middle line:90% of the flying capacitor. 00:25:23.520 --> 00:25:26.220 align:middle line:84% Sometimes it's ground and sometimes it's at the middle, 00:25:26.220 --> 00:25:26.720 align:middle line:90% OK? 00:25:26.720 --> 00:25:30.970 align:middle line:84% So now I'm going to make sure we can see the green. 00:25:30.970 --> 00:25:35.155 align:middle line:84% And I'm going to turn on a small load. 00:25:35.155 --> 00:25:40.340 align:middle line:84% And you can see that the output voltage came down as a result. 00:25:40.340 --> 00:25:42.460 align:middle line:84% And if you look at the yellow waveform, you know, 00:25:42.460 --> 00:25:46.040 align:middle line:84% you can sort of see an RC kind of a time constant on it. 00:25:46.040 --> 00:25:50.620 align:middle line:84% And it looks like we could switch sooner. 00:25:50.620 --> 00:25:53.620 align:middle line:84% If I increase the load a little bit-- 00:25:53.620 --> 00:25:56.162 align:middle line:90% so this is twice as much. 00:25:56.162 --> 00:25:59.870 align:middle line:84% You can see that that RC time constant might be faster. 00:25:59.870 --> 00:26:02.620 align:middle line:84% And you can also see that the output changed by quite a bit 00:26:02.620 --> 00:26:04.820 align:middle line:90% in terms of its output droop. 00:26:04.820 --> 00:26:09.070 align:middle line:84% So now I'm going to start turning up the frequency. 00:26:09.070 --> 00:26:13.930 align:middle line:84% And we can watch the output increase. 00:26:13.930 --> 00:26:17.980 align:middle line:84% So there's the frequency coming up. 00:26:17.980 --> 00:26:20.560 align:middle line:84% Let me change the timescale a little bit 00:26:20.560 --> 00:26:22.260 align:middle line:90% so we can see it better now. 00:26:22.260 --> 00:26:27.160 align:middle line:84% It's less clear, exactly, what the RC time constant is here. 00:26:27.160 --> 00:26:30.290 align:middle line:84% Maybe it's settling in the time that we give it, 00:26:30.290 --> 00:26:32.450 align:middle line:84% but maybe it hasn't quite made it yet. 00:26:32.450 --> 00:26:36.720 align:middle line:84% And you can see that the output voltage is rising. 00:26:36.720 --> 00:26:39.970 align:middle line:90% Oops, went the wrong way. 00:26:39.970 --> 00:26:43.060 align:middle line:84% So I can turn it up by quite a bit. 00:26:43.060 --> 00:26:46.030 align:middle line:84% And this is sort of-- so I went from 5 kilohertz 00:26:46.030 --> 00:26:48.290 align:middle line:90% to 250 kilohertz. 00:26:48.290 --> 00:26:51.310 align:middle line:90% So-- whoops. 00:26:51.310 --> 00:26:54.160 align:middle line:90% Sorry, I changed the wrong knob. 00:26:54.160 --> 00:26:55.210 align:middle line:90% Here we go. 00:26:55.210 --> 00:26:59.500 align:middle line:84% So at this high frequency, you can 00:26:59.500 --> 00:27:02.030 align:middle line:84% see that there really isn't much ripple on the output. 00:27:02.030 --> 00:27:04.270 align:middle line:84% We're doing a pretty good job of switching it 00:27:04.270 --> 00:27:07.030 align:middle line:84% fast enough so that the capacitor voltage doesn't 00:27:07.030 --> 00:27:07.900 align:middle line:90% change. 00:27:07.900 --> 00:27:11.860 align:middle line:84% So this is the most efficient point for this converter, 00:27:11.860 --> 00:27:14.500 align:middle line:84% particularly if you don't count the power that 00:27:14.500 --> 00:27:16.600 align:middle line:84% goes into the gate drives and the power that goes 00:27:16.600 --> 00:27:18.100 align:middle line:90% into switching the switches. 00:27:18.100 --> 00:27:22.100 align:middle line:84% That decreases the efficiency a little bit. 00:27:22.100 --> 00:27:24.870 align:middle line:84% But in terms of the power part of it, this is its best shot. 00:27:24.870 --> 00:27:30.055 align:middle line:90% 00:27:30.055 --> 00:27:31.430 align:middle line:84% DAVID PERREAULT: So any questions 00:27:31.430 --> 00:27:33.920 align:middle line:84% about the fact that we can really 00:27:33.920 --> 00:27:36.930 align:middle line:84% get some benefit by turning up the switching frequency? 00:27:36.930 --> 00:27:39.830 align:middle line:84% We drop the resistance, which increases our converter 00:27:39.830 --> 00:27:42.710 align:middle line:84% efficiency, neglecting gate charge 00:27:42.710 --> 00:27:44.600 align:middle line:90% losses and that sort of thing. 00:27:44.600 --> 00:27:45.943 align:middle line:90% And we would keep doing that. 00:27:45.943 --> 00:27:48.110 align:middle line:84% We'd usually like to keep doing that until we get up 00:27:48.110 --> 00:27:49.180 align:middle line:90% here somewhere. 00:27:49.180 --> 00:27:51.755 align:middle line:90% So we're at maximum efficiency. 00:27:51.755 --> 00:27:54.500 align:middle line:84% We wouldn't want to go out here, because there are other losses 00:27:54.500 --> 00:27:55.470 align:middle line:90% that I'm not counting. 00:27:55.470 --> 00:27:57.990 align:middle line:84% So there's always an optimum switching frequency. 00:27:57.990 --> 00:28:00.328 align:middle line:84% And the nice thing is is the load decreases. 00:28:00.328 --> 00:28:02.120 align:middle line:84% We can often change the switching frequency 00:28:02.120 --> 00:28:04.730 align:middle line:84% to keep efficiency really high in these converters 00:28:04.730 --> 00:28:07.910 align:middle line:90% across a very wide power range. 00:28:07.910 --> 00:28:09.655 align:middle line:90% Any questions about that? 00:28:09.655 --> 00:28:13.160 align:middle line:90% 00:28:13.160 --> 00:28:18.230 align:middle line:84% So let's consider a few other things about switch capacitor 00:28:18.230 --> 00:28:19.310 align:middle line:90% converters. 00:28:19.310 --> 00:28:23.610 align:middle line:84% One thing I'd like you to notice is, in fact, 00:28:23.610 --> 00:28:26.660 align:middle line:84% that if you have the right switch implementation, 00:28:26.660 --> 00:28:30.020 align:middle line:84% switch capacitor converters are bi-directional in their power 00:28:30.020 --> 00:28:32.480 align:middle line:84% flow capability, just like other converters. 00:28:32.480 --> 00:28:35.420 align:middle line:84% Remember, we could run a buck converter essentially backwards 00:28:35.420 --> 00:28:39.620 align:middle line:84% and turn it into a boost Converter That same trick works 00:28:39.620 --> 00:28:41.960 align:middle line:84% with switching power converters-- works with switch 00:28:41.960 --> 00:28:43.270 align:middle line:90% capacitor converters. 00:28:43.270 --> 00:28:46.140 align:middle line:84% So let me just choose another converter. 00:28:46.140 --> 00:28:50.060 align:middle line:84% And what I'm going to do is I'm going to go take the output, 00:28:50.060 --> 00:28:54.280 align:middle line:84% put it over here, take the source, put it over here. 00:28:54.280 --> 00:28:55.730 align:middle line:90% I'll draw it backwards. 00:28:55.730 --> 00:28:58.580 align:middle line:84% And the only other thing is I'll change the voltage polarity here 00:28:58.580 --> 00:28:59.960 align:middle line:90% just so you can-- 00:28:59.960 --> 00:29:02.310 align:middle line:90% just for clarity in the circuit. 00:29:02.310 --> 00:29:04.410 align:middle line:84% And what would we get if I did that? 00:29:04.410 --> 00:29:08.520 align:middle line:84% It would look something like this. 00:29:08.520 --> 00:29:10.980 align:middle line:90% So OK. 00:29:10.980 --> 00:29:12.517 align:middle line:90% Let me draw. 00:29:12.517 --> 00:29:13.350 align:middle line:90% I'll keep my source. 00:29:13.350 --> 00:29:15.450 align:middle line:84% And I'll keep my source on my load 00:29:15.450 --> 00:29:18.630 align:middle line:84% and I'll flip the circuit around. 00:29:18.630 --> 00:29:20.140 align:middle line:90% So here we go. 00:29:20.140 --> 00:29:23.550 align:middle line:90% Here's Vs. OK. 00:29:23.550 --> 00:29:25.260 align:middle line:90% I'm going to have-- 00:29:25.260 --> 00:29:29.730 align:middle line:84% I'm first going to go through with a switch and a 1, 00:29:29.730 --> 00:29:34.230 align:middle line:90% and then up, switch 2. 00:29:34.230 --> 00:29:38.970 align:middle line:84% And then I'm going to have a capacitor. 00:29:38.970 --> 00:29:42.030 align:middle line:84% And I'm going to draw it as Vc, like this. 00:29:42.030 --> 00:29:45.710 align:middle line:84% And then I'm going to have a switch that's a 1. 00:29:45.710 --> 00:29:53.150 align:middle line:84% And I've got to come to the output. 00:29:53.150 --> 00:29:54.835 align:middle line:90% So I'm going to have a C big. 00:29:54.835 --> 00:29:58.597 align:middle line:90% 00:29:58.597 --> 00:30:00.840 align:middle line:84% And I'm going to have a load current on this side. 00:30:00.840 --> 00:30:05.448 align:middle line:90% 00:30:05.448 --> 00:30:10.980 align:middle line:84% And now-- so what am I going to get here? 00:30:10.980 --> 00:30:13.300 align:middle line:84% I've got to draw my other switch. 00:30:13.300 --> 00:30:16.800 align:middle line:90% So he's ending up on this side. 00:30:16.800 --> 00:30:19.680 align:middle line:84% So let me get just get this right. 00:30:19.680 --> 00:30:21.800 align:middle line:84% Switch 1, and switch 2 goes here. 00:30:21.800 --> 00:30:27.410 align:middle line:90% 00:30:27.410 --> 00:30:29.330 align:middle line:84% Switch 1 comes from the source and comes here. 00:30:29.330 --> 00:30:29.960 align:middle line:90% Yes. 00:30:29.960 --> 00:30:32.360 align:middle line:90% So 2 and 2, like this. 00:30:32.360 --> 00:30:33.200 align:middle line:90% OK? 00:30:33.200 --> 00:30:37.310 align:middle line:84% So what am I doing in this circuit? 00:30:37.310 --> 00:30:38.310 align:middle line:90% 1, 1, 2, 2. 00:30:38.310 --> 00:30:39.240 align:middle line:90% It's the same thing. 00:30:39.240 --> 00:30:41.900 align:middle line:84% All I did was basically flip the internal circuit around 00:30:41.900 --> 00:30:45.620 align:middle line:84% and kept the source and load in the same place. 00:30:45.620 --> 00:30:48.810 align:middle line:84% So what are the two switch states here? 00:30:48.810 --> 00:30:54.110 align:middle line:84% If I have 1 and 1, what I'm doing is this. 00:30:54.110 --> 00:30:57.040 align:middle line:84% If switch 1s are on, what I got is this. 00:30:57.040 --> 00:31:02.280 align:middle line:90% 00:31:02.280 --> 00:31:05.510 align:middle line:90% And here's my C big. 00:31:05.510 --> 00:31:06.010 align:middle line:90% All right. 00:31:06.010 --> 00:31:07.010 align:middle line:90% And here's my I out. 00:31:07.010 --> 00:31:11.710 align:middle line:90% 00:31:11.710 --> 00:31:14.340 align:middle line:84% And if I have my switch, so this is state 1. 00:31:14.340 --> 00:31:17.200 align:middle line:90% 00:31:17.200 --> 00:31:36.860 align:middle line:84% If I have state 2, essentially, I am doing this. 00:31:36.860 --> 00:31:49.420 align:middle line:90% 00:31:49.420 --> 00:31:50.020 align:middle line:90% OK. 00:31:50.020 --> 00:31:51.380 align:middle line:90% So what happens? 00:31:51.380 --> 00:31:53.830 align:middle line:84% Like how might I think about the charge transfers 00:31:53.830 --> 00:31:55.095 align:middle line:90% here in states 1 and 2? 00:31:55.095 --> 00:31:58.000 align:middle line:90% 00:31:58.000 --> 00:32:00.520 align:middle line:84% In this state, state 2, what I'm going 00:32:00.520 --> 00:32:05.830 align:middle line:84% to do, if I have a active load over here, I out, 00:32:05.830 --> 00:32:09.190 align:middle line:84% is I'm basically in state 2, I'm basically just charging 00:32:09.190 --> 00:32:11.440 align:middle line:90% this capacitor from the input. 00:32:11.440 --> 00:32:18.980 align:middle line:84% So this capacitor Vc is going to charge up to Vn or to Vs. 00:32:18.980 --> 00:32:21.950 align:middle line:90% 00:32:21.950 --> 00:32:24.280 align:middle line:84% Then, in state 1, what I'm going to do 00:32:24.280 --> 00:32:28.500 align:middle line:84% is I'm going to discharge him like this. 00:32:28.500 --> 00:32:31.660 align:middle line:90% 00:32:31.660 --> 00:32:33.730 align:middle line:84% So I'm putting something that's charged 00:32:33.730 --> 00:32:35.770 align:middle line:84% to Vs, in series with Vs, and I'm 00:32:35.770 --> 00:32:37.720 align:middle line:90% discharging it into the output. 00:32:37.720 --> 00:32:40.375 align:middle line:84% So the output ideally ought to charge up. 00:32:40.375 --> 00:32:42.250 align:middle line:84% If I had no load current, what will happen is 00:32:42.250 --> 00:32:45.490 align:middle line:84% the output will charge up to 2 Vs or 2 Vn-- 00:32:45.490 --> 00:32:47.506 align:middle line:84% let me call this Vn just for consistency. 00:32:47.506 --> 00:32:53.476 align:middle line:90% 00:32:53.476 --> 00:32:57.640 align:middle line:84% So this becomes a 1 to 2 converter. 00:32:57.640 --> 00:32:58.190 align:middle line:90% All right? 00:32:58.190 --> 00:33:01.150 align:middle line:84% I can do the same kind of analysis on this converter 00:33:01.150 --> 00:33:04.090 align:middle line:84% that I did on the converter from last class. 00:33:04.090 --> 00:33:07.870 align:middle line:84% And if I did that, we would get this equivalent circuit model. 00:33:07.870 --> 00:33:11.740 align:middle line:90% I would get Vn. 00:33:11.740 --> 00:33:14.020 align:middle line:84% The equivalent circuit model for this whole thing 00:33:14.020 --> 00:33:15.045 align:middle line:90% would be like this. 00:33:15.045 --> 00:33:17.890 align:middle line:90% 00:33:17.890 --> 00:33:18.915 align:middle line:90% 1 to 2. 00:33:18.915 --> 00:33:24.100 align:middle line:90% 00:33:24.100 --> 00:33:33.190 align:middle line:84% And then I would have an R eq that would be equal to 1 00:33:33.190 --> 00:33:38.860 align:middle line:90% over F switch C in SSL. 00:33:38.860 --> 00:33:40.600 align:middle line:84% And then here's my load current, I out. 00:33:40.600 --> 00:33:43.350 align:middle line:90% 00:33:43.350 --> 00:33:45.360 align:middle line:90% It's the same kind of analysis. 00:33:45.360 --> 00:33:46.948 align:middle line:84% We do the same charge transfer thing. 00:33:46.948 --> 00:33:47.740 align:middle line:90% We get an equation. 00:33:47.740 --> 00:33:52.950 align:middle line:84% We see that this circuit model right here matches what goes on 00:33:52.950 --> 00:33:56.220 align:middle line:90% in the slow-switching limit. 00:33:56.220 --> 00:33:58.055 align:middle line:84% Does that make sense to everybody? 00:33:58.055 --> 00:34:01.860 align:middle line:90% 00:34:01.860 --> 00:34:05.010 align:middle line:84% The neat thing about this is that actually the model 00:34:05.010 --> 00:34:05.855 align:middle line:90% hasn't changed. 00:34:05.855 --> 00:34:10.212 align:middle line:90% 00:34:10.212 --> 00:34:11.420 align:middle line:90% I'm using idealized switches. 00:34:11.420 --> 00:34:14.449 align:middle line:84% Let's just say I assume they're perfect switches that can block 00:34:14.449 --> 00:34:15.690 align:middle line:90% and carry bidirectionally. 00:34:15.690 --> 00:34:17.719 align:middle line:84% So I have a full switch implementation, 00:34:17.719 --> 00:34:19.980 align:middle line:84% full four-quadrant switching implementation. 00:34:19.980 --> 00:34:22.630 align:middle line:90% 00:34:22.630 --> 00:34:27.100 align:middle line:84% I could take this circuit model and reflect this R equivalent 00:34:27.100 --> 00:34:30.050 align:middle line:90% through the transformer. 00:34:30.050 --> 00:34:32.780 align:middle line:84% And what would I do when I go through the transformer? 00:34:32.780 --> 00:34:37.520 align:middle line:84% I have to scale impedances by the square of the terms ratio. 00:34:37.520 --> 00:34:41.810 align:middle line:84% So another model I could develop for this, an equivalent 00:34:41.810 --> 00:34:44.580 align:middle line:90% to this model, is the following. 00:34:44.580 --> 00:34:49.460 align:middle line:84% I could put a resistor here that would be scaled by 1/4. 00:34:49.460 --> 00:35:03.470 align:middle line:84% So I'd have 1 over 4 F switch C. 1 to 2. 00:35:03.470 --> 00:35:05.910 align:middle line:84% And this would be my alternative circuit model. 00:35:05.910 --> 00:35:08.470 align:middle line:90% And now here I have I out. 00:35:08.470 --> 00:35:13.012 align:middle line:90% 00:35:13.012 --> 00:35:14.260 align:middle line:90% So here's my point. 00:35:14.260 --> 00:35:17.360 align:middle line:84% This is just another way to draw the circuit model, same circuit 00:35:17.360 --> 00:35:17.860 align:middle line:90% model. 00:35:17.860 --> 00:35:19.440 align:middle line:84% I've just reflected the resistance 00:35:19.440 --> 00:35:20.440 align:middle line:90% through the transformer. 00:35:20.440 --> 00:35:21.982 align:middle line:84% And if you look at this, you're going 00:35:21.982 --> 00:35:25.560 align:middle line:84% to see it's exactly equal to the one up there. 00:35:25.560 --> 00:35:28.090 align:middle line:90% It's 1 over 4 F switch C. 00:35:28.090 --> 00:35:31.870 align:middle line:84% So the energy loss doesn't really 00:35:31.870 --> 00:35:34.143 align:middle line:84% depend upon which way we're transferring power 00:35:34.143 --> 00:35:35.060 align:middle line:90% through the converter. 00:35:35.060 --> 00:35:37.000 align:middle line:84% In other words, am I sticking current in here 00:35:37.000 --> 00:35:40.030 align:middle line:84% and delivering energy to here or vice versa? 00:35:40.030 --> 00:35:42.710 align:middle line:84% I'm going to get the same efficiency in either direction-- 00:35:42.710 --> 00:35:45.130 align:middle line:84% so long as my switch implementations and so forth can 00:35:45.130 --> 00:35:47.780 align:middle line:90% support that. 00:35:47.780 --> 00:35:51.150 align:middle line:90% Any questions about that? 00:35:51.150 --> 00:35:53.927 align:middle line:84% So once you've analyzed the 2 to 1, 00:35:53.927 --> 00:35:55.260 align:middle line:90% you've also analyzed the 1 to 2. 00:35:55.260 --> 00:35:56.385 align:middle line:90% You just don't know it yet. 00:35:56.385 --> 00:35:59.220 align:middle line:84% We could have just, like, swapped the source and the load 00:35:59.220 --> 00:36:02.580 align:middle line:84% and been done, and then I would have probably slid the resistor 00:36:02.580 --> 00:36:05.408 align:middle line:84% through the transformer just to put it into standard form, 00:36:05.408 --> 00:36:06.575 align:middle line:90% and we would have been done. 00:36:06.575 --> 00:36:13.210 align:middle line:90% 00:36:13.210 --> 00:36:20.970 align:middle line:84% Let's talk about-- let's talk about challenges 00:36:20.970 --> 00:36:21.760 align:middle line:90% of this converter. 00:36:21.760 --> 00:36:24.610 align:middle line:84% If I thought about building this converter, 00:36:24.610 --> 00:36:32.980 align:middle line:84% OK, let's think about where the physical components are big. 00:36:32.980 --> 00:36:41.120 align:middle line:84% This capacitor here is my energy transfer capacitor. 00:36:41.120 --> 00:36:43.900 align:middle line:84% And we've already said, if I can make him a bigger 00:36:43.900 --> 00:36:48.760 align:middle line:84% energy transfer capacitor, the bigger C gets, 00:36:48.760 --> 00:36:50.300 align:middle line:90% the smaller the resistance gets. 00:36:50.300 --> 00:36:51.970 align:middle line:84% So it's good to have this capacitor be 00:36:51.970 --> 00:36:54.543 align:middle line:84% pretty big relative to the amount of power 00:36:54.543 --> 00:36:55.460 align:middle line:90% I'm pulling out of it. 00:36:55.460 --> 00:36:58.640 align:middle line:84% So I can be close to the fast-switching limit. 00:36:58.640 --> 00:37:01.550 align:middle line:84% On the other hand, I've treated this capacitor here 00:37:01.550 --> 00:37:02.970 align:middle line:90% as if it was infinite. 00:37:02.970 --> 00:37:06.920 align:middle line:84% So if I do nothing else, usually this capacitor C 00:37:06.920 --> 00:37:09.080 align:middle line:90% big is big compared to C. 00:37:09.080 --> 00:37:12.560 align:middle line:84% And his job here, if you notice, is that he's 00:37:12.560 --> 00:37:13.700 align:middle line:90% holding up the output. 00:37:13.700 --> 00:37:16.740 align:middle line:84% And one of the two states, in this state, 00:37:16.740 --> 00:37:19.620 align:middle line:84% there really is nothing else making V out 00:37:19.620 --> 00:37:21.810 align:middle line:90% constant except for C big. 00:37:21.810 --> 00:37:25.080 align:middle line:84% He has to be big enough so that V out's 00:37:25.080 --> 00:37:29.180 align:middle line:84% not going to droop while this guy is in his other state. 00:37:29.180 --> 00:37:30.860 align:middle line:84% Does that make sense to everybody? 00:37:30.860 --> 00:37:33.320 align:middle line:84% So when you size the capacitors in this thing, 00:37:33.320 --> 00:37:35.390 align:middle line:84% if I built this circuit, usually C big 00:37:35.390 --> 00:37:38.840 align:middle line:84% would be substantially bigger than C, which is in some sense 00:37:38.840 --> 00:37:42.080 align:middle line:84% a shame, because I'm using C big only as a filter 00:37:42.080 --> 00:37:44.810 align:middle line:84% but not to transfer energy through the converter. 00:37:44.810 --> 00:37:47.180 align:middle line:84% So the question is, is there something clever 00:37:47.180 --> 00:37:51.380 align:middle line:84% I could do about this to make better use of my capacitor 00:37:51.380 --> 00:37:52.880 align:middle line:90% energy storage? 00:37:52.880 --> 00:37:54.790 align:middle line:90% And the answer is, there is. 00:37:54.790 --> 00:37:56.420 align:middle line:90% And here's the trick. 00:37:56.420 --> 00:37:58.465 align:middle line:84% Suppose I was to build this circuit. 00:37:58.465 --> 00:38:01.425 align:middle line:90% 00:38:01.425 --> 00:38:04.620 align:middle line:90% And let me just-- 00:38:04.620 --> 00:38:06.150 align:middle line:90% let me just draw-- 00:38:06.150 --> 00:38:07.540 align:middle line:90% I'll redraw this circuit. 00:38:07.540 --> 00:38:08.315 align:middle line:90% So here's Vn. 00:38:08.315 --> 00:38:14.510 align:middle line:90% 00:38:14.510 --> 00:38:16.650 align:middle line:90% And here's my circuit. 00:38:16.650 --> 00:38:18.660 align:middle line:84% I'm going to redraw this just up here. 00:38:18.660 --> 00:38:25.940 align:middle line:90% And I get-- here's 2 Vc. 00:38:25.940 --> 00:38:28.580 align:middle line:90% 00:38:28.580 --> 00:38:30.020 align:middle line:90% Here's 1. 00:38:30.020 --> 00:38:31.520 align:middle line:90% I'm sorry. 00:38:31.520 --> 00:38:33.080 align:middle line:90% This is switch 1. 00:38:33.080 --> 00:38:34.650 align:middle line:90% This is switch 1. 00:38:34.650 --> 00:38:35.945 align:middle line:90% I also have a switch 2. 00:38:35.945 --> 00:38:38.950 align:middle line:90% 00:38:38.950 --> 00:38:40.125 align:middle line:90% And I have a switch 2. 00:38:40.125 --> 00:38:43.410 align:middle line:90% 00:38:43.410 --> 00:38:47.360 align:middle line:84% And he comes down to this ground input, right? 00:38:47.360 --> 00:38:50.575 align:middle line:84% And then, I have C big over here. 00:38:50.575 --> 00:38:54.940 align:middle line:90% 00:38:54.940 --> 00:38:55.860 align:middle line:90% And here's my output. 00:38:55.860 --> 00:38:58.456 align:middle line:90% 00:38:58.456 --> 00:39:01.370 align:middle line:84% And I'm going to deliver energy here. 00:39:01.370 --> 00:39:03.370 align:middle line:90% What is this current? 00:39:03.370 --> 00:39:04.880 align:middle line:84% What does this current look like? 00:39:04.880 --> 00:39:06.120 align:middle line:90% Let me call this-- 00:39:06.120 --> 00:39:07.122 align:middle line:90% I'll call this ix1. 00:39:07.122 --> 00:39:10.600 align:middle line:90% 00:39:10.600 --> 00:39:13.090 align:middle line:90% What does ix1 look like? 00:39:13.090 --> 00:39:18.760 align:middle line:84% Well, let me think about, in the fast-switching limit, what 00:39:18.760 --> 00:39:19.790 align:middle line:90% ix1 looks like. 00:39:19.790 --> 00:39:22.840 align:middle line:84% Well, when I'm in the fast-switching limit, 00:39:22.840 --> 00:39:27.810 align:middle line:84% I'm dumping some charge off of Vc into the output. 00:39:27.810 --> 00:39:30.100 align:middle line:84% And in the fast-switching limit, this capacitor 00:39:30.100 --> 00:39:32.120 align:middle line:84% is not charging or discharging much. 00:39:32.120 --> 00:39:34.940 align:middle line:84% So the current flow is limited by these switch resistances. 00:39:34.940 --> 00:39:39.330 align:middle line:84% So in the fast-switching limit, ix1 looks like this. 00:39:39.330 --> 00:39:45.610 align:middle line:90% 00:39:45.610 --> 00:39:47.010 align:middle line:90% So this is state 1. 00:39:47.010 --> 00:39:52.850 align:middle line:90% 00:39:52.850 --> 00:39:58.100 align:middle line:84% And then in state 2, he's just 0 because in stage 2, 00:39:58.100 --> 00:40:00.440 align:middle line:90% the switch is open. 00:40:00.440 --> 00:40:04.790 align:middle line:84% And so I dump a bunch of charge into C big in state 1, 00:40:04.790 --> 00:40:06.090 align:middle line:90% and then nothing in state 2. 00:40:06.090 --> 00:40:11.450 align:middle line:84% So C big's left to take this total charge 00:40:11.450 --> 00:40:15.700 align:middle line:84% and then deliver the average current to the output. 00:40:15.700 --> 00:40:17.860 align:middle line:84% Well, what would happen if I came along and said, 00:40:17.860 --> 00:40:18.460 align:middle line:90% you know what? 00:40:18.460 --> 00:40:22.175 align:middle line:90% Let me build a second converter. 00:40:22.175 --> 00:40:23.550 align:middle line:84% I'm going to build the complement 00:40:23.550 --> 00:40:25.560 align:middle line:90% to the first converter. 00:40:25.560 --> 00:40:29.280 align:middle line:84% So I'm going to build something that looks like this. 00:40:29.280 --> 00:40:30.610 align:middle line:90% He'll be my orange converter. 00:40:30.610 --> 00:40:34.560 align:middle line:84% So maybe what I'll do is I'll build a second converter exactly 00:40:34.560 --> 00:40:36.040 align:middle line:90% like this. 00:40:36.040 --> 00:40:40.290 align:middle line:84% And I'm going to make this turn on in state 2. 00:40:40.290 --> 00:40:44.630 align:middle line:84% I'm going to have another capacitor, Vc prime. 00:40:44.630 --> 00:40:46.670 align:middle line:90% OK. 00:40:46.670 --> 00:40:47.920 align:middle line:90% Here's state 2. 00:40:47.920 --> 00:40:54.310 align:middle line:90% 00:40:54.310 --> 00:40:57.220 align:middle line:84% And now I'll deliver his charge to the output. 00:40:57.220 --> 00:41:00.430 align:middle line:90% And I'll call this ix2. 00:41:00.430 --> 00:41:04.940 align:middle line:84% Now in state 1 I'm going to do the complement. 00:41:04.940 --> 00:41:05.990 align:middle line:90% So I'm going to do this. 00:41:05.990 --> 00:41:09.170 align:middle line:84% And here's-- I'm going to have this guy in state 1 and this guy 00:41:09.170 --> 00:41:09.955 align:middle line:90% in state 1. 00:41:09.955 --> 00:41:13.060 align:middle line:90% 00:41:13.060 --> 00:41:13.840 align:middle line:90% All right. 00:41:13.840 --> 00:41:17.260 align:middle line:84% So all I've done is I've built a copy of this circuit, 00:41:17.260 --> 00:41:20.260 align:middle line:84% but I've switched the switches exactly oppositely. 00:41:20.260 --> 00:41:22.720 align:middle line:84% Does that make sense to everybody? 00:41:22.720 --> 00:41:24.970 align:middle line:90% So what is ix? 00:41:24.970 --> 00:41:27.860 align:middle line:84% Maybe I should call them ix prime here. 00:41:27.860 --> 00:41:29.450 align:middle line:90% What is ix prime? 00:41:29.450 --> 00:41:30.620 align:middle line:90% I guess I call them ix1. 00:41:30.620 --> 00:41:32.500 align:middle line:90% I guess I will call this ix2. 00:41:32.500 --> 00:41:38.710 align:middle line:84% I have ix1. ix2 is going to be the exact complement of ix1 00:41:38.710 --> 00:41:42.450 align:middle line:84% because he happens in the opposite portion of the cycle. 00:41:42.450 --> 00:41:48.030 align:middle line:84% And then if I said, well, what's the total current coming in? 00:41:48.030 --> 00:41:52.350 align:middle line:90% What is this current ix? 00:41:52.350 --> 00:41:54.720 align:middle line:90% Well, ix is just now doing this. 00:41:54.720 --> 00:41:56.880 align:middle line:90% He's following ix1. 00:41:56.880 --> 00:41:58.660 align:middle line:90% He's following ix2. 00:41:58.660 --> 00:42:05.160 align:middle line:84% So he's almost a constant current into the output. 00:42:05.160 --> 00:42:07.950 align:middle line:84% And so this capacitor is only filtering 00:42:07.950 --> 00:42:10.120 align:middle line:84% the difference between this and the load current. 00:42:10.120 --> 00:42:13.170 align:middle line:84% The load current I l, or I out, is just 00:42:13.170 --> 00:42:16.120 align:middle line:90% the average of this current. 00:42:16.120 --> 00:42:19.215 align:middle line:84% So I out looks like this, all right? 00:42:19.215 --> 00:42:22.750 align:middle line:90% 00:42:22.750 --> 00:42:27.140 align:middle line:84% And so this circuit really doesn't need C big anymore. 00:42:27.140 --> 00:42:30.330 align:middle line:84% He only needs a tiny capacitor here. 00:42:30.330 --> 00:42:32.280 align:middle line:84% So I could take this giant capacitor 00:42:32.280 --> 00:42:36.120 align:middle line:84% and divvy them up among this guy, if this is C prime 00:42:36.120 --> 00:42:39.540 align:middle line:84% and this guy is C. And basically, 00:42:39.540 --> 00:42:43.020 align:middle line:84% instead of having an energy transfer or capacitor 00:42:43.020 --> 00:42:47.300 align:middle line:84% and a giant filter capacitor, I can have two pretty big energy 00:42:47.300 --> 00:42:50.010 align:middle line:84% transfer capacitors and a tiny filter capacitor. 00:42:50.010 --> 00:42:53.780 align:middle line:84% And I can get the same behavior, the same performance, 00:42:53.780 --> 00:42:56.420 align:middle line:84% for much less capacitance or lower output resistance 00:42:56.420 --> 00:42:59.600 align:middle line:90% for the same total capacitance. 00:42:59.600 --> 00:43:03.100 align:middle line:90% Any questions about that? 00:43:03.100 --> 00:43:05.150 align:middle line:84% This trick is very commonly used. 00:43:05.150 --> 00:43:06.890 align:middle line:90% This is called interleaving. 00:43:06.890 --> 00:43:09.980 align:middle line:90% 00:43:09.980 --> 00:43:15.590 align:middle line:84% So this is an interleaved 1 to 2 step-up switch capacitor 00:43:15.590 --> 00:43:16.980 align:middle line:90% converter. 00:43:16.980 --> 00:43:20.490 align:middle line:84% And it just has the benefit that you can get the same performance 00:43:20.490 --> 00:43:22.200 align:middle line:84% for much less total capacitance because I 00:43:22.200 --> 00:43:25.340 align:middle line:84% don't need to have some filtering-- any filtering going 00:43:25.340 --> 00:43:25.840 align:middle line:90% on. 00:43:25.840 --> 00:43:29.535 align:middle line:84% And I should say, by the way, this is a 2-phase circuit, 00:43:29.535 --> 00:43:30.660 align:middle line:90% with states 1 and states 2. 00:43:30.660 --> 00:43:31.902 align:middle line:90% You can have more phases. 00:43:31.902 --> 00:43:33.360 align:middle line:84% There are other games you can play. 00:43:33.360 --> 00:43:34.840 align:middle line:90% But this is the basic one. 00:43:34.840 --> 00:43:36.420 align:middle line:90% What's the penalty you pay? 00:43:36.420 --> 00:43:39.330 align:middle line:84% To do this, I more or less have twice as many energy transfer 00:43:39.330 --> 00:43:41.350 align:middle line:84% capacitors and twice as many switches. 00:43:41.350 --> 00:43:44.190 align:middle line:84% And in a world where the count of the components matters, 00:43:44.190 --> 00:43:45.900 align:middle line:90% that's a pain in the butt. 00:43:45.900 --> 00:43:49.590 align:middle line:84% On the other hand, if I cared about the size of my energy 00:43:49.590 --> 00:43:52.860 align:middle line:84% transfer capacitors, this thing is wonderful. 00:43:52.860 --> 00:43:55.320 align:middle line:84% And this is a very common technique, 00:43:55.320 --> 00:43:57.720 align:middle line:84% is interleaving your switch capacitor converters 00:43:57.720 --> 00:44:01.860 align:middle line:84% to alleviate the need for big either input 00:44:01.860 --> 00:44:06.290 align:middle line:84% filter capacitors or output filter capacitors. 00:44:06.290 --> 00:44:07.040 align:middle line:90% Questions? 00:44:07.040 --> 00:44:11.530 align:middle line:90% 00:44:11.530 --> 00:44:14.620 align:middle line:84% From a DC perspective, all we do is basically 00:44:14.620 --> 00:44:16.480 align:middle line:84% put two of those circuits in parallel, 00:44:16.480 --> 00:44:19.190 align:middle line:84% and I get half the output resistance essentially. 00:44:19.190 --> 00:44:21.850 align:middle line:84% But what I've gotten rid of is the filter capacitor 00:44:21.850 --> 00:44:25.100 align:middle line:90% that's not in that DC model. 00:44:25.100 --> 00:44:28.460 align:middle line:84% Now I should say, I've kind of spent all my time 00:44:28.460 --> 00:44:34.670 align:middle line:84% talking about 2 to 1 or 1 to 2 switch capacitor converters. 00:44:34.670 --> 00:44:37.070 align:middle line:84% There are a lot of different kind of switch capacitor 00:44:37.070 --> 00:44:37.590 align:middle line:90% converters. 00:44:37.590 --> 00:44:43.980 align:middle line:84% You can expand out to almost any rational conversion ratio. 00:44:43.980 --> 00:44:46.910 align:middle line:84% And keep in mind, as we said, the rational conversion 00:44:46.910 --> 00:44:49.250 align:middle line:84% ratio gives you perfect conversion in current-- 00:44:49.250 --> 00:44:53.310 align:middle line:84% you know, the this current is exactly twice this current, 00:44:53.310 --> 00:44:53.810 align:middle line:90% right? 00:44:53.810 --> 00:44:55.435 align:middle line:84% So you get a perfect current conversion 00:44:55.435 --> 00:44:59.150 align:middle line:84% ratio and an approximate voltage conversion ratio. 00:44:59.150 --> 00:45:01.550 align:middle line:84% But you can almost get any ratio, 00:45:01.550 --> 00:45:05.160 align:middle line:84% by picking the switching states in the capacitors, 00:45:05.160 --> 00:45:07.460 align:middle line:84% you can get almost any ratio here 00:45:07.460 --> 00:45:11.320 align:middle line:84% that's rational-- like 2 to 1, 3 to 1, 4 to 1, 3 to 2, 00:45:11.320 --> 00:45:14.570 align:middle line:90% you name it, OK? 00:45:14.570 --> 00:45:17.780 align:middle line:84% There are bounds, and there's different ways 00:45:17.780 --> 00:45:19.430 align:middle line:90% to expand on this basic idea. 00:45:19.430 --> 00:45:22.010 align:middle line:84% Let me just show you a couple for-- 00:45:22.010 --> 00:45:23.630 align:middle line:90% out of many. 00:45:23.630 --> 00:45:25.340 align:middle line:90% Here's one idea. 00:45:25.340 --> 00:45:26.880 align:middle line:90% Suppose I did this. 00:45:26.880 --> 00:45:33.850 align:middle line:84% Suppose I had my input source Vn, and I did this. 00:45:33.850 --> 00:45:38.890 align:middle line:84% I will transfer energy to a capacitor 00:45:38.890 --> 00:45:45.275 align:middle line:84% here, and then to here, and then to here. 00:45:45.275 --> 00:45:52.700 align:middle line:90% 00:45:52.700 --> 00:45:56.180 align:middle line:84% And then I'll have my C big here. 00:45:56.180 --> 00:45:57.040 align:middle line:90% All right. 00:45:57.040 --> 00:46:01.640 align:middle line:84% So when I close this and switch 1, 1, 1, 1, 00:46:01.640 --> 00:46:06.020 align:middle line:84% I'm going to charge these capacitors 00:46:06.020 --> 00:46:08.130 align:middle line:84% in series between the input and the output. 00:46:08.130 --> 00:46:10.255 align:middle line:84% So I'll get a charge flow that does this. 00:46:10.255 --> 00:46:13.570 align:middle line:90% 00:46:13.570 --> 00:46:18.290 align:middle line:84% And I'll charge up all my capacitors to the output. 00:46:18.290 --> 00:46:21.580 align:middle line:84% And then I can add another set of switches that are like this. 00:46:21.580 --> 00:46:29.990 align:middle line:90% 00:46:29.990 --> 00:46:39.510 align:middle line:84% And then maybe these guys will discharge in parallel 00:46:39.510 --> 00:46:40.240 align:middle line:90% to the output. 00:46:40.240 --> 00:46:43.620 align:middle line:84% So I'll get charge flows that flow out of this capacitor 00:46:43.620 --> 00:46:48.065 align:middle line:84% like this out of this guy, like this, out of this guy like this. 00:46:48.065 --> 00:46:52.050 align:middle line:84% So I'll charge them in series and discharge them in parallel 00:46:52.050 --> 00:46:53.140 align:middle line:90% to the output. 00:46:53.140 --> 00:47:02.280 align:middle line:84% So in the final state, these capacitors will charge to V out. 00:47:02.280 --> 00:47:04.470 align:middle line:84% And then so I would get, in the state 1, 00:47:04.470 --> 00:47:09.340 align:middle line:84% I would get V out plus V out plus V out plus V out. 00:47:09.340 --> 00:47:12.500 align:middle line:90% So four V out at the input. 00:47:12.500 --> 00:47:16.960 align:middle line:84% So this is a 4 to 1 step-down series parallel switch capacitor 00:47:16.960 --> 00:47:19.280 align:middle line:90% converter. 00:47:19.280 --> 00:47:22.100 align:middle line:84% So we charge the capacitors in series. 00:47:22.100 --> 00:47:23.580 align:middle line:90% We discharge them in parallel. 00:47:23.580 --> 00:47:25.220 align:middle line:84% Or, if I turn this around and swap 00:47:25.220 --> 00:47:30.700 align:middle line:84% where the source and load were, it would be a 1 to 4 converter. 00:47:30.700 --> 00:47:31.860 align:middle line:90% That's one kind. 00:47:31.860 --> 00:47:34.580 align:middle line:90% 00:47:34.580 --> 00:47:35.960 align:middle line:90% There are many others. 00:47:35.960 --> 00:47:37.002 align:middle line:90% And I'm kind out of time. 00:47:37.002 --> 00:47:38.585 align:middle line:84% So I'm not going to draw any of those. 00:47:38.585 --> 00:47:40.100 align:middle line:84% There are others listed in the book. 00:47:40.100 --> 00:47:41.370 align:middle line:90% And I show another one. 00:47:41.370 --> 00:47:43.350 align:middle line:84% I show a ladder converter in my notes. 00:47:43.350 --> 00:47:49.157 align:middle line:84% What distinguishes these converters is how many switches 00:47:49.157 --> 00:47:51.740 align:middle line:84% and how many capacitors do you need to do a certain conversion 00:47:51.740 --> 00:47:52.290 align:middle line:90% function. 00:47:52.290 --> 00:47:54.020 align:middle line:90% Some are better than others. 00:47:54.020 --> 00:47:57.780 align:middle line:84% What are the voltage ratings of the switches and capacitors? 00:47:57.780 --> 00:48:01.250 align:middle line:84% How much energy storage do you need to do a given conversion 00:48:01.250 --> 00:48:01.970 align:middle line:90% function? 00:48:01.970 --> 00:48:04.410 align:middle line:84% And whether they're the same or not. 00:48:04.410 --> 00:48:07.100 align:middle line:84% So in this circuit, for example, this switch 00:48:07.100 --> 00:48:10.470 align:middle line:84% has different ratings than the other switches for example. 00:48:10.470 --> 00:48:13.520 align:middle line:84% So you might choose this one if you really 00:48:13.520 --> 00:48:16.640 align:middle line:84% cared about all the capacitors having the same voltage, 00:48:16.640 --> 00:48:19.820 align:middle line:84% but you weren't so worried about the switches being different. 00:48:19.820 --> 00:48:22.250 align:middle line:84% You might choose a different one if you 00:48:22.250 --> 00:48:23.550 align:middle line:90% had different constraints. 00:48:23.550 --> 00:48:26.570 align:middle line:84% So there's a bunch of different converters that are popular-- 00:48:26.570 --> 00:48:29.900 align:middle line:84% series, parallel like this, one ladder, Dixon. 00:48:29.900 --> 00:48:32.120 align:middle line:84% There's a whole variety of them, that you 00:48:32.120 --> 00:48:36.310 align:middle line:84% pick the one that matches the design regime you're in. 00:48:36.310 --> 00:48:38.130 align:middle line:84% If you're in an integrated circuit, 00:48:38.130 --> 00:48:39.643 align:middle line:90% maybe you care about one thing. 00:48:39.643 --> 00:48:41.310 align:middle line:84% If you're really high voltage, maybe you 00:48:41.310 --> 00:48:43.700 align:middle line:90% care about another thing. 00:48:43.700 --> 00:48:47.400 align:middle line:90% Any questions about that? 00:48:47.400 --> 00:48:53.130 align:middle line:84% I should also say that while I'm showing you DC to DC converters, 00:48:53.130 --> 00:48:56.070 align:middle line:84% you can also make switch-capacitor inverters 00:48:56.070 --> 00:48:58.770 align:middle line:90% or switch-capacitor rectifiers. 00:48:58.770 --> 00:49:02.880 align:middle line:84% And in fact I haven't drawn the right circuit. 00:49:02.880 --> 00:49:03.940 align:middle line:90% I'm kind out of time. 00:49:03.940 --> 00:49:05.700 align:middle line:90% It's in the notes. 00:49:05.700 --> 00:49:09.870 align:middle line:84% But if you can take a circuit that has an AC-driven input with 00:49:09.870 --> 00:49:15.240 align:middle line:84% diodes, and charges the capacitors in one part 00:49:15.240 --> 00:49:17.640 align:middle line:84% of the cycle, and then discharges them in the other 00:49:17.640 --> 00:49:21.360 align:middle line:84% in just such a manner that you shuffle charge up and get 00:49:21.360 --> 00:49:24.870 align:middle line:84% a rectifier that multiplies the voltage-- 00:49:24.870 --> 00:49:27.360 align:middle line:84% The so-called famous Cockcroft-Walton voltage 00:49:27.360 --> 00:49:31.200 align:middle line:84% multiplier is one example of a switch capacitor rectifier. 00:49:31.200 --> 00:49:34.110 align:middle line:84% And those are very often used in high-voltage power supplies 00:49:34.110 --> 00:49:36.430 align:middle line:84% where you want to generate very high voltages. 00:49:36.430 --> 00:49:38.250 align:middle line:84% And that's why, in fact, Cockcroft-Walton 00:49:38.250 --> 00:49:39.870 align:middle line:84% got the Nobel Prize for splitting 00:49:39.870 --> 00:49:43.410 align:middle line:84% the atom with really huge voltages generated by just such 00:49:43.410 --> 00:49:44.480 align:middle line:90% a rectifier. 00:49:44.480 --> 00:49:47.050 align:middle line:84% So while I've spent my time talking about switch capacitor 00:49:47.050 --> 00:49:51.420 align:middle line:84% DC to conversion, you can do inversion and rectification. 00:49:51.420 --> 00:49:55.690 align:middle line:84% And they all follow the same fundamental principles. 00:49:55.690 --> 00:49:57.880 align:middle line:84% So I'm kind out of time to talk about this, 00:49:57.880 --> 00:50:00.470 align:middle line:84% but I'd be happy to take any final questions. 00:50:00.470 --> 00:50:05.025 align:middle line:90% 00:50:05.025 --> 00:50:05.525 align:middle line:90% OK. 00:50:05.525 --> 00:50:09.310 align:middle line:84% In the next class we will take up a new topic. 00:50:09.310 --> 00:50:11.200 align:middle line:90% Have a great day. 00:50:11.200 --> 00:50:15.000 align:middle line:90%