WEBVTT 00:00:00.000 --> 00:00:00.972 align:middle line:90% 00:00:00.972 --> 00:00:02.916 align:middle line:90% [SQUEAKING] 00:00:02.916 --> 00:00:04.374 align:middle line:90% [RUSTLING] 00:00:04.374 --> 00:00:05.832 align:middle line:90% [CLICKING] 00:00:05.832 --> 00:00:09.540 align:middle line:90% 00:00:09.540 --> 00:00:11.970 align:middle line:84% DAVID PERREAULT: OK, why don't we get started? 00:00:11.970 --> 00:00:15.750 align:middle line:84% So we've been talking about DC-to-DC converters 00:00:15.750 --> 00:00:18.900 align:middle line:84% and analyzing them and understanding 00:00:18.900 --> 00:00:22.560 align:middle line:84% what they'll do in steady state, and I'd 00:00:22.560 --> 00:00:25.950 align:middle line:84% like to today bring that maybe one step closer to things 00:00:25.950 --> 00:00:27.945 align:middle line:84% you have to think about to design one. 00:00:27.945 --> 00:00:31.380 align:middle line:84% OK, but just as a reminder, last class, 00:00:31.380 --> 00:00:34.680 align:middle line:84% we introduced what's known as the buck-boost converter, 00:00:34.680 --> 00:00:37.840 align:middle line:84% or the inverting buck-boost converter. 00:00:37.840 --> 00:00:40.650 align:middle line:90% 00:00:40.650 --> 00:00:42.980 align:middle line:84% And one version of that looks like this. 00:00:42.980 --> 00:00:45.330 align:middle line:84% We have an input voltage that I'll 00:00:45.330 --> 00:00:54.960 align:middle line:84% call V1 for example, a switch, a diode. 00:00:54.960 --> 00:00:56.020 align:middle line:90% Here's my output. 00:00:56.020 --> 00:00:58.680 align:middle line:90% That'll be V2. 00:00:58.680 --> 00:01:01.200 align:middle line:84% And then we have a filter capacitor-- in this example, 00:01:01.200 --> 00:01:05.600 align:middle line:84% I'm going to put the filter capacitor here, OK, 00:01:05.600 --> 00:01:09.150 align:middle line:84% because this is sort of the actual switching loop-- 00:01:09.150 --> 00:01:13.040 align:middle line:84% and then an inductor, where we have iL. 00:01:13.040 --> 00:01:17.060 align:middle line:84% And, right, and what we've said so far is let's 00:01:17.060 --> 00:01:24.371 align:middle line:84% think about the case where L and C are big, right, 00:01:24.371 --> 00:01:26.530 align:middle line:84% that they're approximately infinite, 00:01:26.530 --> 00:01:29.030 align:middle line:84% and that's not a bad place to start for analysis because you 00:01:29.030 --> 00:01:30.870 align:middle line:84% can learn a lot of things about that. 00:01:30.870 --> 00:01:36.350 align:middle line:84% And what that means is that iL is approximately equal to its DC 00:01:36.350 --> 00:01:42.860 align:middle line:84% value IL and vC is approximately equal to its DC value capital 00:01:42.860 --> 00:01:45.230 align:middle line:90% VC. 00:01:45.230 --> 00:01:53.720 align:middle line:84% And we said, OK, if I had to think about this converter, 00:01:53.720 --> 00:01:59.540 align:middle line:84% right, I might think about the switch voltage, v switch, 00:01:59.540 --> 00:02:03.860 align:middle line:84% and the switch current, i switch, and the diode voltage 00:02:03.860 --> 00:02:08.210 align:middle line:90% vD and the diode current iD. 00:02:08.210 --> 00:02:13.050 align:middle line:84% Right, and what we discovered about this converter last time 00:02:13.050 --> 00:02:16.110 align:middle line:84% is that if I thought of, here I have, 00:02:16.110 --> 00:02:21.130 align:middle line:84% say, an input current that I'm going to call i1 and an output 00:02:21.130 --> 00:02:27.490 align:middle line:84% current that I'm going to call i2, and if I said, 00:02:27.490 --> 00:02:30.400 align:middle line:84% OK, suppose this thing filters perfectly 00:02:30.400 --> 00:02:35.440 align:middle line:84% and i1 is approximately equal to its DC value capital I1 00:02:35.440 --> 00:02:39.310 align:middle line:84% and i2 is approximately equal to this DC value of capital I2-- 00:02:39.310 --> 00:02:41.047 align:middle line:84% and in order to really get that, you'd 00:02:41.047 --> 00:02:43.130 align:middle line:84% probably need a tiny bit of source impedance here, 00:02:43.130 --> 00:02:45.760 align:middle line:84% but let's assume that's the case-- 00:02:45.760 --> 00:02:49.210 align:middle line:84% what we saw for this converter is 00:02:49.210 --> 00:02:56.050 align:middle line:84% that IL, the DC current in the inductor, 00:02:56.050 --> 00:03:01.360 align:middle line:84% is equal to the magnitude of I1 plus the magnitude of I2, 00:03:01.360 --> 00:03:12.220 align:middle line:84% and the DC capacitor voltage VC is equal to the magnitude of V1 00:03:12.220 --> 00:03:14.360 align:middle line:90% plus the magnitude of V2. 00:03:14.360 --> 00:03:14.860 align:middle line:90% OK. 00:03:14.860 --> 00:03:19.310 align:middle line:84% And so, basically, the capacitor has 00:03:19.310 --> 00:03:22.890 align:middle line:84% to see the sum of the magnitudes of the input and output voltage, 00:03:22.890 --> 00:03:31.550 align:middle line:84% and the inductor current carries the sum of the input and output 00:03:31.550 --> 00:03:32.750 align:middle line:90% currents. 00:03:32.750 --> 00:03:34.166 align:middle line:90% OK. 00:03:34.166 --> 00:03:38.803 align:middle line:84% And then if I thought, well, OK, what is the switch voltage? 00:03:38.803 --> 00:03:42.830 align:middle line:90% 00:03:42.830 --> 00:03:46.670 align:middle line:84% OK, when the switch is off, what voltage does it block? 00:03:46.670 --> 00:03:49.880 align:middle line:84% Well, that blocks VC, and the same thing, 00:03:49.880 --> 00:03:52.880 align:middle line:84% when the diode is off, it also blocks VC. 00:03:52.880 --> 00:03:56.510 align:middle line:84% So v switch max is equal to vD max 00:03:56.510 --> 00:04:00.140 align:middle line:84% is equal to VC, which is equal to the magnitude of V1 00:04:00.140 --> 00:04:02.660 align:middle line:90% plus the magnitude of V2. 00:04:02.660 --> 00:04:09.020 align:middle line:84% And if I thought of i switch max and iD max, 00:04:09.020 --> 00:04:15.240 align:middle line:84% well, when they're on, they carry the inductor current, 00:04:15.240 --> 00:04:17.310 align:middle line:84% which is the magnitude of the input current 00:04:17.310 --> 00:04:21.420 align:middle line:84% and the output current, the DC input and output current. 00:04:21.420 --> 00:04:22.019 align:middle line:90% OK. 00:04:22.019 --> 00:04:26.160 align:middle line:84% So at least now I have a sense of what the stresses are 00:04:26.160 --> 00:04:30.460 align:middle line:84% on the diodes and transistors and the inductor and capacitor, 00:04:30.460 --> 00:04:30.960 align:middle line:90% right. 00:04:30.960 --> 00:04:32.530 align:middle line:90% Why do I care about that? 00:04:32.530 --> 00:04:35.153 align:middle line:84% Because ultimately, I have to go buy switches that can carry-- 00:04:35.153 --> 00:04:37.320 align:middle line:84% diodes that can carry the current without blowing up 00:04:37.320 --> 00:04:39.670 align:middle line:84% and can block the voltage without blowing up, 00:04:39.670 --> 00:04:41.743 align:middle line:84% and I've got to go buy capacitors that 00:04:41.743 --> 00:04:43.410 align:middle line:84% can block the voltage and inductors that 00:04:43.410 --> 00:04:44.618 align:middle line:90% can carry the current, right. 00:04:44.618 --> 00:04:46.710 align:middle line:84% So we have to start going through this analysis 00:04:46.710 --> 00:04:49.170 align:middle line:84% to say, How much voltage and current 00:04:49.170 --> 00:04:50.590 align:middle line:90% do I need in the circuit? 00:04:50.590 --> 00:04:52.090 align:middle line:90% OK. 00:04:52.090 --> 00:04:56.080 align:middle line:84% How does this circuit compare to, like, a buck converter 00:04:56.080 --> 00:04:58.030 align:middle line:90% or a boost converter? 00:04:58.030 --> 00:05:05.340 align:middle line:84% OK, well, if I took my buck converter-- 00:05:05.340 --> 00:05:07.500 align:middle line:90% OK, so here's my buck converter. 00:05:07.500 --> 00:05:11.273 align:middle line:84% In this case, I'm going to put the-- 00:05:11.273 --> 00:05:13.440 align:middle line:84% I'm going to think about the capacitor at the input, 00:05:13.440 --> 00:05:15.940 align:middle line:84% because that's where the switching current goes. 00:05:15.940 --> 00:05:21.030 align:middle line:84% OK, and here's my switch, here's my diode, 00:05:21.030 --> 00:05:22.700 align:middle line:90% and here's my inductor. 00:05:22.700 --> 00:05:23.790 align:middle line:90% I won't worry about-- 00:05:23.790 --> 00:05:26.250 align:middle line:84% I don't actually need a capacitor at the output. 00:05:26.250 --> 00:05:28.650 align:middle line:84% Although, I would typically put one. 00:05:28.650 --> 00:05:29.950 align:middle line:90% So here's v2. 00:05:29.950 --> 00:05:35.580 align:middle line:84% So I could again think of v switch and i 00:05:35.580 --> 00:05:42.150 align:middle line:90% switch and vD and iD. 00:05:42.150 --> 00:05:43.990 align:middle line:90% All right. 00:05:43.990 --> 00:05:50.660 align:middle line:84% This would be vC, and this would be iL. 00:05:50.660 --> 00:05:56.877 align:middle line:84% And again, I'm going to think of L and C big, really, really big, 00:05:56.877 --> 00:05:59.210 align:middle line:84% so that basically the inductor current and the capacitor 00:05:59.210 --> 00:06:01.620 align:middle line:84% voltage here are basically constant. 00:06:01.620 --> 00:06:02.120 align:middle line:90% OK. 00:06:02.120 --> 00:06:10.410 align:middle line:84% In that case, what I get is v switch when it's off 00:06:10.410 --> 00:06:14.520 align:middle line:84% and vD when it's off are both equal to this capacitor 00:06:14.520 --> 00:06:20.196 align:middle line:84% voltage, which is equal to v1, right. 00:06:20.196 --> 00:06:26.590 align:middle line:84% And i switch equals iD when the switch and diode are gone. 00:06:26.590 --> 00:06:32.658 align:middle line:84% They carry the inductor current, which is also equal to I2, 00:06:32.658 --> 00:06:34.200 align:middle line:84% if the inductor current is really big 00:06:34.200 --> 00:06:36.540 align:middle line:84% and I just have a DC current, right. 00:06:36.540 --> 00:06:42.360 align:middle line:84% And this is the i1, and this is i2. 00:06:42.360 --> 00:06:45.910 align:middle line:90% So that's for buck converter. 00:06:45.910 --> 00:06:47.770 align:middle line:90% What about a boost converter? 00:06:47.770 --> 00:06:50.210 align:middle line:84% Well, a boost converter looked like this. 00:06:50.210 --> 00:06:50.770 align:middle line:90% Here's v1. 00:06:50.770 --> 00:06:54.440 align:middle line:90% 00:06:54.440 --> 00:06:55.425 align:middle line:90% Here's iL. 00:06:55.425 --> 00:07:01.020 align:middle line:90% 00:07:01.020 --> 00:07:03.840 align:middle line:84% And in this case, I'm going to put the capacitor here 00:07:03.840 --> 00:07:07.590 align:middle line:84% because this pulsed current comes through the diode. 00:07:07.590 --> 00:07:09.318 align:middle line:90% OK. 00:07:09.318 --> 00:07:11.730 align:middle line:90% And here's v2. 00:07:11.730 --> 00:07:12.470 align:middle line:90% OK. 00:07:12.470 --> 00:07:24.310 align:middle line:84% So again, I have sort of v switch, i switch, vD, and iD. 00:07:24.310 --> 00:07:24.970 align:middle line:90% OK. 00:07:24.970 --> 00:07:28.930 align:middle line:84% Now, in this case, il is again constant 00:07:28.930 --> 00:07:32.260 align:middle line:84% if L and C are big, OK, so that the switch 00:07:32.260 --> 00:07:38.530 align:middle line:84% current and the diode current when they're on 00:07:38.530 --> 00:07:42.970 align:middle line:84% is equal to iL, which also equal to I1, right, because whatever 00:07:42.970 --> 00:07:45.255 align:middle line:84% the current is coming out of I1 is now 00:07:45.255 --> 00:07:46.630 align:middle line:84% the current that's being switched 00:07:46.630 --> 00:07:50.060 align:middle line:90% to the diode and the transistor. 00:07:50.060 --> 00:07:58.810 align:middle line:84% v switch is equal to vD is equal to vC, which 00:07:58.810 --> 00:08:02.250 align:middle line:90% is equal to V2 in this case. 00:08:02.250 --> 00:08:02.940 align:middle line:90% OK. 00:08:02.940 --> 00:08:06.010 align:middle line:90% So why am I going through this? 00:08:06.010 --> 00:08:09.250 align:middle line:84% Well, for either the buck or boost converter, keep in mind, 00:08:09.250 --> 00:08:12.550 align:middle line:84% in the boost converter, V2 is the big voltage, 00:08:12.550 --> 00:08:14.660 align:middle line:84% between the input and output voltage, 00:08:14.660 --> 00:08:19.000 align:middle line:84% and I1 is the bigger of the currents between the input 00:08:19.000 --> 00:08:20.920 align:middle line:90% and output current. 00:08:20.920 --> 00:08:24.960 align:middle line:84% Likewise, the switch voltage is V1. 00:08:24.960 --> 00:08:30.425 align:middle line:84% That's the bigger of the input and output voltage, right, 00:08:30.425 --> 00:08:34.340 align:middle line:84% and the current is I2, which is the bigger of the input 00:08:34.340 --> 00:08:35.309 align:middle line:90% and output current. 00:08:35.309 --> 00:08:37.549 align:middle line:84% So for either of these converters, 00:08:37.549 --> 00:08:43.549 align:middle line:84% I might write the v switch is equal to v diode, which 00:08:43.549 --> 00:08:50.930 align:middle line:84% is equal to the maximum of magnitude of V1 and V2. 00:08:50.930 --> 00:08:55.370 align:middle line:84% And this is also equal to the capacitor voltage, 00:08:55.370 --> 00:09:00.050 align:middle line:84% and the inductor current is equal to the switch current 00:09:00.050 --> 00:09:09.440 align:middle line:84% is equal to the diode current is equal to a maximum of I1 and I2. 00:09:09.440 --> 00:09:10.811 align:middle line:90% OK. 00:09:10.811 --> 00:09:13.480 align:middle line:84% So how do these two compare, right? 00:09:13.480 --> 00:09:16.990 align:middle line:84% The buck and the boost converter can only 00:09:16.990 --> 00:09:18.760 align:middle line:90% do a very specific function. 00:09:18.760 --> 00:09:20.870 align:middle line:84% One can step down, the other can step up, 00:09:20.870 --> 00:09:23.800 align:middle line:90% but they can't do both, right. 00:09:23.800 --> 00:09:30.080 align:middle line:84% But the stresses on them in either case, basically, 00:09:30.080 --> 00:09:31.830 align:middle line:84% the switch and the diode and the capacitor 00:09:31.830 --> 00:09:35.070 align:middle line:84% see the maximum of the input and output voltages, 00:09:35.070 --> 00:09:37.830 align:middle line:84% and the switch current and the diode current 00:09:37.830 --> 00:09:41.880 align:middle line:84% see the maximum of the input and output currents. 00:09:41.880 --> 00:09:46.860 align:middle line:84% If I use a buck-boost converter, I 00:09:46.860 --> 00:09:49.210 align:middle line:84% can go up or down, at least in magnitude, 00:09:49.210 --> 00:09:50.817 align:middle line:90% but now I see higher stresses. 00:09:50.817 --> 00:09:53.400 align:middle line:84% Instead of seeing the maximum of the input and output voltage, 00:09:53.400 --> 00:09:56.018 align:middle line:84% I see the sum of the input and output voltage magnitudes 00:09:56.018 --> 00:09:58.310 align:middle line:84% and the sum of the input and output current magnitudes, 00:09:58.310 --> 00:09:58.810 align:middle line:90% right. 00:09:58.810 --> 00:10:02.820 align:middle line:84% That means, for a given conversion ratio, 00:10:02.820 --> 00:10:04.950 align:middle line:84% the buck-boost converter is going 00:10:04.950 --> 00:10:07.710 align:middle line:84% to have a higher stress on the switches 00:10:07.710 --> 00:10:11.990 align:middle line:84% and the diodes and the inductors and capacitors 00:10:11.990 --> 00:10:15.600 align:middle line:84% than a buck converter or a boost converter. 00:10:15.600 --> 00:10:18.810 align:middle line:90% And why is that? 00:10:18.810 --> 00:10:20.460 align:middle line:84% In a buck-boost converter, remember, we 00:10:20.460 --> 00:10:24.630 align:middle line:84% said how it ran was basically I take energy from the input, 00:10:24.630 --> 00:10:28.427 align:middle line:84% I put it into the inductor in the first part of the cycle, 00:10:28.427 --> 00:10:30.010 align:middle line:84% and then the second part of the cycle, 00:10:30.010 --> 00:10:31.350 align:middle line:84% I take that energy out of the inductor 00:10:31.350 --> 00:10:32.580 align:middle line:90% and put it in the output. 00:10:32.580 --> 00:10:35.520 align:middle line:84% So all of the energy flowing through the circuit 00:10:35.520 --> 00:10:37.170 align:middle line:84% from the input to the output ultimately 00:10:37.170 --> 00:10:39.590 align:middle line:90% got stored in that inductor. 00:10:39.590 --> 00:10:41.210 align:middle line:90% OK. 00:10:41.210 --> 00:10:43.980 align:middle line:84% With a buck converter or a boost converter, 00:10:43.980 --> 00:10:45.260 align:middle line:90% that's not quite true. 00:10:45.260 --> 00:10:48.870 align:middle line:84% In the first part of the cycle, we turn the switch on 00:10:48.870 --> 00:10:51.420 align:middle line:84% and we store up some energy in the inductor, 00:10:51.420 --> 00:10:54.060 align:middle line:84% but we also have direct current path 00:10:54.060 --> 00:10:57.340 align:middle line:84% from the input to the output through the inductor. 00:10:57.340 --> 00:10:58.900 align:middle line:84% And in the second part of the cycle, 00:10:58.900 --> 00:11:00.483 align:middle line:84% we take the energy out of the inductor 00:11:00.483 --> 00:11:01.750 align:middle line:90% and put it in the output. 00:11:01.750 --> 00:11:04.810 align:middle line:84% So in a buck converter, only part of the energy 00:11:04.810 --> 00:11:07.810 align:middle line:84% ends up stored in the inductor, and part of it 00:11:07.810 --> 00:11:10.670 align:middle line:84% flows straight through from the input to the output. 00:11:10.670 --> 00:11:13.180 align:middle line:84% And the same thing can be said for the boost converter. 00:11:13.180 --> 00:11:17.020 align:middle line:84% The consequence of this is, is that to do a given conversion 00:11:17.020 --> 00:11:19.300 align:middle line:84% function, the buck-boost converter 00:11:19.300 --> 00:11:22.180 align:middle line:84% imposes much greater stresses on its components 00:11:22.180 --> 00:11:24.760 align:middle line:84% than the buck converter or the boost converter. 00:11:24.760 --> 00:11:27.040 align:middle line:84% All right, so we can get away with a lot 00:11:27.040 --> 00:11:32.110 align:middle line:84% of things in terms of reducing the stresses on the component 00:11:32.110 --> 00:11:35.590 align:middle line:84% by being in this sort of direct conversion approach, where I'm 00:11:35.590 --> 00:11:38.890 align:middle line:84% sort of not storing all the energy in intermediate form 00:11:38.890 --> 00:11:43.140 align:middle line:84% that I can with a buck-boost converter. 00:11:43.140 --> 00:11:46.055 align:middle line:84% Questions about that or about any of the prior material? 00:11:46.055 --> 00:11:52.998 align:middle line:90% 00:11:52.998 --> 00:11:55.290 align:middle line:84% So one of the things we're going to want to think about 00:11:55.290 --> 00:12:01.425 align:middle line:84% as we kind move from theory to thinking about design is, boy, 00:12:01.425 --> 00:12:03.300 align:middle line:84% if I'm going to do things one way or another, 00:12:03.300 --> 00:12:08.890 align:middle line:84% how can I get away with smaller voltage in current switches 00:12:08.890 --> 00:12:13.610 align:middle line:84% and less energy storage in my passive components? 00:12:13.610 --> 00:12:15.010 align:middle line:84% And some ways of doing things are 00:12:15.010 --> 00:12:16.840 align:middle line:84% going to be much better than others, OK, 00:12:16.840 --> 00:12:19.180 align:middle line:84% and that's part of the art of design, 00:12:19.180 --> 00:12:21.423 align:middle line:84% is figuring out, Well, geez, what do I need to do, 00:12:21.423 --> 00:12:22.840 align:middle line:84% and how is that going to translate 00:12:22.840 --> 00:12:26.500 align:middle line:84% into the size and loss and weight of the thing 00:12:26.500 --> 00:12:27.140 align:middle line:90% I'm designing? 00:12:27.140 --> 00:12:27.640 align:middle line:90% OK. 00:12:27.640 --> 00:12:31.000 align:middle line:84% And I want to take some initial steps today 00:12:31.000 --> 00:12:37.020 align:middle line:84% towards thinking about how we might do that, OK, and one 00:12:37.020 --> 00:12:39.020 align:middle line:84% of the things we're going to have to do in order 00:12:39.020 --> 00:12:41.480 align:middle line:84% to do that is we're going to have to throw out 00:12:41.480 --> 00:12:44.750 align:middle line:90% this assumption, right. 00:12:44.750 --> 00:12:47.450 align:middle line:84% For purposes of figuring out my conversion ratio 00:12:47.450 --> 00:12:51.260 align:middle line:84% and those kind of things, I can imagine that my inductors 00:12:51.260 --> 00:12:52.710 align:middle line:90% and capacitors are really big. 00:12:52.710 --> 00:12:54.620 align:middle line:84% That's a perfectly valid thing to do, 00:12:54.620 --> 00:12:56.390 align:middle line:90% and we do it all the time. 00:12:56.390 --> 00:13:00.470 align:middle line:84% But if I'm going to go build this thing, first of all, 00:13:00.470 --> 00:13:03.780 align:middle line:84% I can't go out and buy a one-henry inductor very easily. 00:13:03.780 --> 00:13:06.633 align:middle line:84% And the bigger the inductance I buy, 00:13:06.633 --> 00:13:09.050 align:middle line:84% what we're going to see is the more energy it has to store 00:13:09.050 --> 00:13:10.842 align:middle line:84% and the bigger it is and the lossier it is, 00:13:10.842 --> 00:13:11.690 align:middle line:90% and everything else. 00:13:11.690 --> 00:13:15.470 align:middle line:84% So I need to start figuring out, Well, 00:13:15.470 --> 00:13:21.170 align:middle line:84% how big does L and C have to be in order to achieve my goals? 00:13:21.170 --> 00:13:24.422 align:middle line:84% And that's what I'd like to talk about today. 00:13:24.422 --> 00:13:25.880 align:middle line:84% And the first thing we've got to do 00:13:25.880 --> 00:13:30.710 align:middle line:84% is figure out, Well, what is the impact of finite component 00:13:30.710 --> 00:13:33.200 align:middle line:90% values on the converter? 00:13:33.200 --> 00:13:35.112 align:middle line:90% OK. 00:13:35.112 --> 00:13:38.470 align:middle line:84% And what we're going to see is there's 00:13:38.470 --> 00:13:40.465 align:middle line:90% a few things that might-- 00:13:40.465 --> 00:13:42.340 align:middle line:84% one of the things I have to do is figure out, 00:13:42.340 --> 00:13:44.860 align:middle line:84% in any of these converters, what value of L 00:13:44.860 --> 00:13:47.450 align:middle line:84% do I pick, what value of C do I pick? 00:13:47.450 --> 00:13:50.390 align:middle line:84% How I figure those out depends upon a few things. 00:13:50.390 --> 00:13:50.890 align:middle line:90% OK. 00:13:50.890 --> 00:13:54.760 align:middle line:84% Things that might determine that, first of all, 00:13:54.760 --> 00:13:58.360 align:middle line:84% is ripple, how much voltage and current ripple 00:13:58.360 --> 00:14:01.150 align:middle line:84% are permissible in the circuit, and we're 00:14:01.150 --> 00:14:03.370 align:middle line:90% going to focus on ripple today. 00:14:03.370 --> 00:14:09.577 align:middle line:84% There's also the question of transient dynamics. 00:14:09.577 --> 00:14:11.660 align:middle line:84% We'll look at that a little bit later in the term. 00:14:11.660 --> 00:14:13.243 align:middle line:84% And that can also impact the decisions 00:14:13.243 --> 00:14:16.630 align:middle line:84% you make, but it's sort of ripple first, transient dynamics 00:14:16.630 --> 00:14:17.460 align:middle line:90% second. 00:14:17.460 --> 00:14:20.500 align:middle line:84% And the other thing that's is going to come into play 00:14:20.500 --> 00:14:25.250 align:middle line:84% is, How big and lossy and expensive are the components? 00:14:25.250 --> 00:14:27.770 align:middle line:84% And we'll touch on that today a little bit. 00:14:27.770 --> 00:14:32.310 align:middle line:84% We'll at least get sort of a rough measure of how that goes. 00:14:32.310 --> 00:14:33.510 align:middle line:90% OK. 00:14:33.510 --> 00:14:38.800 align:middle line:84% So let's start to think about how we 00:14:38.800 --> 00:14:43.930 align:middle line:90% might start sizing components. 00:14:43.930 --> 00:14:45.680 align:middle line:84% In order to do that, we need to figure out 00:14:45.680 --> 00:14:48.340 align:middle line:84% a way to calculate ripple in the components. 00:14:48.340 --> 00:14:51.030 align:middle line:84% OK, so here's the game we're going to play. 00:14:51.030 --> 00:14:54.320 align:middle line:84% Let's think about the boost converter, for example. 00:14:54.320 --> 00:14:55.952 align:middle line:90% OK. 00:14:55.952 --> 00:14:57.900 align:middle line:90% So here's my boost converter. 00:14:57.900 --> 00:15:16.910 align:middle line:90% 00:15:16.910 --> 00:15:21.980 align:middle line:84% And I'm going to worry about what the ripple, for example, 00:15:21.980 --> 00:15:24.770 align:middle line:84% on the capacitor voltage is, and I'm also 00:15:24.770 --> 00:15:28.250 align:middle line:84% going to worry about what the ripple in the inductor current 00:15:28.250 --> 00:15:29.270 align:middle line:90% is. 00:15:29.270 --> 00:15:30.440 align:middle line:90% OK. 00:15:30.440 --> 00:15:34.550 align:middle line:84% If I make an L and C really big, then the ripple goes to zero, 00:15:34.550 --> 00:15:37.970 align:middle line:84% but I can't afford to make them too big because the components 00:15:37.970 --> 00:15:39.390 align:middle line:90% will be too big and too lossy. 00:15:39.390 --> 00:15:39.890 align:middle line:90% OK. 00:15:39.890 --> 00:15:42.655 align:middle line:90% So what does this look like? 00:15:42.655 --> 00:15:45.250 align:middle line:90% 00:15:45.250 --> 00:15:52.750 align:middle line:84% Let me plot for a second what iD would look like. 00:15:52.750 --> 00:15:56.960 align:middle line:90% And what do I mean by iD? 00:15:56.960 --> 00:16:00.390 align:middle line:90% Let's look at this current. 00:16:00.390 --> 00:16:03.470 align:middle line:90% 00:16:03.470 --> 00:16:05.460 align:middle line:90% OK. 00:16:05.460 --> 00:16:06.520 align:middle line:90% Why? 00:16:06.520 --> 00:16:11.170 align:middle line:84% Because that's going to be a key to figuring out 00:16:11.170 --> 00:16:13.790 align:middle line:84% how much ripple I end up with on the capacitor voltage. 00:16:13.790 --> 00:16:15.340 align:middle line:84% So what happens in this converter? 00:16:15.340 --> 00:16:18.610 align:middle line:84% In the first part of the cycle, q of T is high. 00:16:18.610 --> 00:16:22.330 align:middle line:84% And the switch is on, and iD is 0. 00:16:22.330 --> 00:16:26.000 align:middle line:84% And then the second part of the cycle-- 00:16:26.000 --> 00:16:27.410 align:middle line:84% so that goes on for some time DT. 00:16:27.410 --> 00:16:30.310 align:middle line:90% 00:16:30.310 --> 00:16:32.340 align:middle line:90% OK. 00:16:32.340 --> 00:16:36.425 align:middle line:84% And in the second part of the cycle, the output goes to v2. 00:16:36.425 --> 00:16:39.420 align:middle line:90% Right, so the-- 00:16:39.420 --> 00:16:41.210 align:middle line:90% I'm sorry. 00:16:41.210 --> 00:16:44.090 align:middle line:84% In the first part of the cycle, iD is 0 00:16:44.090 --> 00:16:50.000 align:middle line:84% because iL is going through the switch, 00:16:50.000 --> 00:16:53.750 align:middle line:84% and the second part of the cycle, iD is equal to iL, right. 00:16:53.750 --> 00:16:58.100 align:middle line:84% So this, I'm just going to call IL, capital IL. 00:16:58.100 --> 00:16:58.670 align:middle line:90% OK. 00:16:58.670 --> 00:17:03.530 align:middle line:84% Now, I'm kind of assuming that L is really, really big here. 00:17:03.530 --> 00:17:07.650 align:middle line:84% I'm going to ignore the ripple current in the inductor 00:17:07.650 --> 00:17:11.970 align:middle line:84% in order to calculate the ripple voltage on the capacitor. 00:17:11.970 --> 00:17:14.790 align:middle line:84% All right, and it turns out that you can do that freely 00:17:14.790 --> 00:17:16.599 align:middle line:84% and you'll still get the right answer, OK, 00:17:16.599 --> 00:17:18.359 align:middle line:84% and I'll show you that in a moment. 00:17:18.359 --> 00:17:20.071 align:middle line:90% OK. 00:17:20.071 --> 00:17:23.829 align:middle line:90% But this is what ID looks like. 00:17:23.829 --> 00:17:24.579 align:middle line:90% OK. 00:17:24.579 --> 00:17:28.770 align:middle line:90% So here's T, and this is ID. 00:17:28.770 --> 00:17:30.030 align:middle line:90% OK. 00:17:30.030 --> 00:17:35.200 align:middle line:90% Well, what's ID's average value? 00:17:35.200 --> 00:17:37.900 align:middle line:84% Right, if I were to calculate the average value 00:17:37.900 --> 00:17:42.410 align:middle line:84% of the diode current ID, that would simply 00:17:42.410 --> 00:17:47.060 align:middle line:84% be the average of this pulsed waveform, which is basically 00:17:47.060 --> 00:17:49.105 align:middle line:90% 1 minus D times IL. 00:17:49.105 --> 00:17:52.590 align:middle line:90% 00:17:52.590 --> 00:17:54.502 align:middle line:90% OK. 00:17:54.502 --> 00:17:58.010 align:middle line:84% So maybe what I would do is say, OK, 00:17:58.010 --> 00:18:00.560 align:middle line:84% let me come up with a model for this converter, 00:18:00.560 --> 00:18:02.240 align:middle line:84% that I don't really care about what's 00:18:02.240 --> 00:18:04.180 align:middle line:90% going on over here, right. 00:18:04.180 --> 00:18:07.420 align:middle line:84% In terms of figuring out the capacitor voltage ripple, 00:18:07.420 --> 00:18:09.160 align:middle line:84% all I care is about what current's 00:18:09.160 --> 00:18:13.900 align:middle line:84% coming through the diode and driving this output circuit. 00:18:13.900 --> 00:18:14.440 align:middle line:90% OK. 00:18:14.440 --> 00:18:19.570 align:middle line:84% So I might model this this way, model this 00:18:19.570 --> 00:18:24.310 align:middle line:84% as some ripple current iD, OK, that is then 00:18:24.310 --> 00:18:32.430 align:middle line:84% going into my capacitor vC and my load resistor. 00:18:32.430 --> 00:18:35.780 align:middle line:90% 00:18:35.780 --> 00:18:36.280 align:middle line:90% OK. 00:18:36.280 --> 00:18:37.770 align:middle line:90% Any questions about that? 00:18:37.770 --> 00:18:41.400 align:middle line:90% 00:18:41.400 --> 00:18:45.140 align:middle line:84% Now, if I design this circuit, I'm 00:18:45.140 --> 00:18:47.900 align:middle line:84% interested in the ripple on the capacitor, 00:18:47.900 --> 00:18:52.460 align:middle line:84% and what I'm likely to do is put the switching frequency, 00:18:52.460 --> 00:18:54.590 align:middle line:90% T is equal to 1 over-- 00:18:54.590 --> 00:18:57.630 align:middle line:84% this is equal to 1 over the switching frequency, right. 00:18:57.630 --> 00:19:01.580 align:middle line:84% What I'm likely to do is to make this rC such 00:19:01.580 --> 00:19:04.020 align:middle line:84% that it filters that ripple current. 00:19:04.020 --> 00:19:07.100 align:middle line:84% I've got this pulsating current in, and what I really want 00:19:07.100 --> 00:19:11.820 align:middle line:84% is vC to be almost constant, and hence the v2 is almost constant, 00:19:11.820 --> 00:19:12.320 align:middle line:90% right. 00:19:12.320 --> 00:19:18.200 align:middle line:84% So that means that sort of the AC component of this pulsating 00:19:18.200 --> 00:19:21.860 align:middle line:84% waveform, we really want it to go through the capacitor, 00:19:21.860 --> 00:19:27.140 align:middle line:84% almost exclusively, and the DC component of this current, 00:19:27.140 --> 00:19:30.380 align:middle line:84% 1 minus D IL, is what goes to the output, right. 00:19:30.380 --> 00:19:32.760 align:middle line:84% Well, what do I need to make that happen? 00:19:32.760 --> 00:19:39.630 align:middle line:84% What I need is 2 pi f switch ought to be much greater than 1 00:19:39.630 --> 00:19:45.420 align:middle line:84% over RC, right, so that this rC network filters the pulsating 00:19:45.420 --> 00:19:48.270 align:middle line:84% component of this current, and so that 00:19:48.270 --> 00:19:49.890 align:middle line:84% gets the DC current going this way 00:19:49.890 --> 00:19:52.020 align:middle line:84% and the ripple current going this way. 00:19:52.020 --> 00:19:52.780 align:middle line:90% OK. 00:19:52.780 --> 00:19:54.095 align:middle line:90% Any questions about that? 00:19:54.095 --> 00:19:57.460 align:middle line:90% 00:19:57.460 --> 00:20:01.090 align:middle line:84% AUDIENCE: How did you get that, that 1 over RC? 00:20:01.090 --> 00:20:05.860 align:middle line:84% DAVID PERREAULT: OK, so the time constant of this is RC. 00:20:05.860 --> 00:20:12.100 align:middle line:84% That means its pole is at s is equal minus 1 over RC. 00:20:12.100 --> 00:20:18.610 align:middle line:84% And so I'm putting the switching frequency, the ripple frequency, 00:20:18.610 --> 00:20:20.950 align:middle line:84% far beyond that so that this becomes essentially 00:20:20.950 --> 00:20:23.920 align:middle line:84% a low-pass filter, so I'm putting it well 00:20:23.920 --> 00:20:24.705 align:middle line:90% beyond the cutoff. 00:20:24.705 --> 00:20:28.230 align:middle line:90% 00:20:28.230 --> 00:20:32.190 align:middle line:90% OK, so suppose I do that. 00:20:32.190 --> 00:20:34.110 align:middle line:90% OK. 00:20:34.110 --> 00:20:36.480 align:middle line:84% Maybe what I should be thinking about is, 00:20:36.480 --> 00:20:40.962 align:middle line:84% if I can make the approximation that all the ripple current is 00:20:40.962 --> 00:20:43.170 align:middle line:84% going to the capacitor and all the DC current's going 00:20:43.170 --> 00:20:46.200 align:middle line:84% to the resistor, and I only want to focus on the ripple, 00:20:46.200 --> 00:20:48.340 align:middle line:84% why don't I think about, with the ripple current, 00:20:48.340 --> 00:20:50.640 align:middle line:84% let me make an AC model for this thing 00:20:50.640 --> 00:20:53.520 align:middle line:84% and just subtract off the DC component? 00:20:53.520 --> 00:20:55.023 align:middle line:90% OK, I know none of the ripple-- 00:20:55.023 --> 00:20:57.690 align:middle line:84% I know none of the DC current's going into the capacitor, right. 00:20:57.690 --> 00:21:00.750 align:middle line:84% We sort of know that from the fact 00:21:00.750 --> 00:21:02.580 align:middle line:90% of the periodic steady state. 00:21:02.580 --> 00:21:03.120 align:middle line:90% OK. 00:21:03.120 --> 00:21:07.710 align:middle line:84% So maybe what I could do is make a new model, 00:21:07.710 --> 00:21:17.350 align:middle line:84% that I'm going to call iD tilde, which is exactly just the AC 00:21:17.350 --> 00:21:20.480 align:middle line:90% component of this waveform ID. 00:21:20.480 --> 00:21:20.980 align:middle line:90% OK. 00:21:20.980 --> 00:21:23.890 align:middle line:84% So I'm going to subtract off the DC value 00:21:23.890 --> 00:21:26.480 align:middle line:84% and just keep the AC value, and what's that going to be? 00:21:26.480 --> 00:21:28.022 align:middle line:84% Well, in the first part of the cycle, 00:21:28.022 --> 00:21:34.420 align:middle line:84% it's going to be 0 minus 1 minus D IL, so this will be minus 1 00:21:34.420 --> 00:21:40.040 align:middle line:84% minus D IL, OK, in the first part of the cycle. 00:21:40.040 --> 00:21:42.920 align:middle line:84% In the second part of the cycle, it 00:21:42.920 --> 00:21:45.890 align:middle line:84% would be ID minus 1 minus D IL, which would 00:21:45.890 --> 00:21:50.710 align:middle line:90% make it D IL, OK, like this. 00:21:50.710 --> 00:21:55.750 align:middle line:90% 00:21:55.750 --> 00:21:58.240 align:middle line:90% Right, and this is DT. 00:21:58.240 --> 00:22:00.680 align:middle line:90% This is T. OK. 00:22:00.680 --> 00:22:03.700 align:middle line:84% So this is the AC component of the ripple current 00:22:03.700 --> 00:22:05.980 align:middle line:90% coming through the diode. 00:22:05.980 --> 00:22:11.310 align:middle line:90% So here we have iD tilde. 00:22:11.310 --> 00:22:12.110 align:middle line:90% OK. 00:22:12.110 --> 00:22:14.840 align:middle line:84% And where that's really going, it's 00:22:14.840 --> 00:22:16.910 align:middle line:90% all going into the capacitor. 00:22:16.910 --> 00:22:18.600 align:middle line:90% Right, that's my approximation. 00:22:18.600 --> 00:22:21.110 align:middle line:84% All the ripple current goes into the capacitor. 00:22:21.110 --> 00:22:23.270 align:middle line:90% OK. 00:22:23.270 --> 00:22:26.300 align:middle line:84% If I make this approximation, then, I 00:22:26.300 --> 00:22:30.470 align:middle line:84% ought to be able to calculate the ripple voltage 00:22:30.470 --> 00:22:33.310 align:middle line:90% on the capacitor, vC tilde. 00:22:33.310 --> 00:22:37.620 align:middle line:90% 00:22:37.620 --> 00:22:38.495 align:middle line:90% Does that make sense? 00:22:38.495 --> 00:22:41.080 align:middle line:90% 00:22:41.080 --> 00:22:44.500 align:middle line:90% So what should happen? 00:22:44.500 --> 00:22:46.343 align:middle line:90% Well, if I have-- 00:22:46.343 --> 00:22:50.860 align:middle line:90% 00:22:50.860 --> 00:22:55.590 align:middle line:84% in the first part of the cycle, right, here's vC tilde. 00:22:55.590 --> 00:22:59.375 align:middle line:90% 00:22:59.375 --> 00:22:59.875 align:middle line:90% OK. 00:22:59.875 --> 00:23:03.540 align:middle line:84% In the first part of the cycle, I've 00:23:03.540 --> 00:23:09.150 align:middle line:84% got a negative constant current through the capacitor, right, 00:23:09.150 --> 00:23:12.527 align:middle line:84% so basically, the capacitor voltage 00:23:12.527 --> 00:23:14.610 align:middle line:84% ought to be falling in the first part of the cycle 00:23:14.610 --> 00:23:22.310 align:middle line:84% linearly, OK, for DT, and then should 00:23:22.310 --> 00:23:27.067 align:middle line:84% be rising in the second part of the cycle for DT. 00:23:27.067 --> 00:23:27.567 align:middle line:90% OK. 00:23:27.567 --> 00:23:31.383 align:middle line:90% 00:23:31.383 --> 00:23:33.180 align:middle line:84% This is what the ripple current-- this 00:23:33.180 --> 00:23:35.580 align:middle line:84% is what the ripple voltage vC tilde is 00:23:35.580 --> 00:23:43.290 align:middle line:84% going to look like, all right, under my approximation. 00:23:43.290 --> 00:23:44.010 align:middle line:90% All right. 00:23:44.010 --> 00:23:46.140 align:middle line:90% Well, what does that mean? 00:23:46.140 --> 00:23:48.780 align:middle line:84% Well, it turns out that if I have a triangle wave, 00:23:48.780 --> 00:23:53.350 align:middle line:84% its midpoint is halfway between the two extremes. 00:23:53.350 --> 00:23:59.640 align:middle line:84% So this, I might call delta vC peak-to-peak, 00:23:59.640 --> 00:24:01.650 align:middle line:84% and it turns out, what that means is 00:24:01.650 --> 00:24:06.570 align:middle line:84% this will be delta vC peak-to-peak over 2, 00:24:06.570 --> 00:24:11.520 align:middle line:84% and this will be delta vC peak-to-peak over 2. 00:24:11.520 --> 00:24:12.270 align:middle line:90% OK. 00:24:12.270 --> 00:24:14.200 align:middle line:90% Well, what is this value? 00:24:14.200 --> 00:24:19.175 align:middle line:84% And I should have said this is DT and this is T, all right. 00:24:19.175 --> 00:24:21.690 align:middle line:90% 00:24:21.690 --> 00:24:25.650 align:middle line:90% iC is equal to C dvC dt. 00:24:25.650 --> 00:24:32.010 align:middle line:84% Right, delta vC peak-to-peak ought to be equal to 1 over C. 00:24:32.010 --> 00:24:38.040 align:middle line:84% The integral, I could do it from here to here, 00:24:38.040 --> 00:24:51.150 align:middle line:84% for example, between DT and T of iC dt, 00:24:51.150 --> 00:24:52.500 align:middle line:90% which works out to be what? 00:24:52.500 --> 00:24:56.203 align:middle line:84% That's 1 over C. The current is constant at D IL. 00:24:56.203 --> 00:25:00.240 align:middle line:90% 00:25:00.240 --> 00:25:09.645 align:middle line:84% And that happens for a time 1 minus D T. All right, 00:25:09.645 --> 00:25:12.260 align:middle line:84% so this is delta vC peak-to-peak. 00:25:12.260 --> 00:25:16.025 align:middle line:90% 00:25:16.025 --> 00:25:17.025 align:middle line:90% Does everybody buy that? 00:25:17.025 --> 00:25:20.470 align:middle line:90% 00:25:20.470 --> 00:25:24.574 align:middle line:90% So what can I say, then? 00:25:24.574 --> 00:25:28.490 align:middle line:84% The capacitor ripple voltage under these approximations 00:25:28.490 --> 00:25:32.250 align:middle line:84% is inversely proportional to the capacitor value, right. 00:25:32.250 --> 00:25:34.520 align:middle line:84% So if I were to come up to you and say, 00:25:34.520 --> 00:25:37.310 align:middle line:84% you know what, I'm only going to allow a hundred 00:25:37.310 --> 00:25:41.660 align:middle line:84% millivolts of ripple in my output, 00:25:41.660 --> 00:25:45.240 align:middle line:84% or some value, other value of delta vC 00:25:45.240 --> 00:25:49.230 align:middle line:84% peak-to-peak, this equation tells me what that should be, 00:25:49.230 --> 00:25:49.902 align:middle line:90% right. 00:25:49.902 --> 00:25:53.240 align:middle line:84% This just says, basically, C has to be 00:25:53.240 --> 00:25:58.220 align:middle line:84% greater than or equal to D times 1 00:25:58.220 --> 00:26:06.590 align:middle line:84% minus D IL T over delta vC peak-to-peak. 00:26:06.590 --> 00:26:08.438 align:middle line:90% All right. 00:26:08.438 --> 00:26:12.560 align:middle line:84% So if you tell me, OK, what's my load current and my switching 00:26:12.560 --> 00:26:14.280 align:middle line:90% period and my duty ratio? 00:26:14.280 --> 00:26:16.970 align:middle line:84% this tells me, How big does the capacitor have 00:26:16.970 --> 00:26:19.340 align:middle line:84% to be in order to reduce the output ripple down 00:26:19.340 --> 00:26:22.620 align:middle line:84% to an acceptable value that I set? 00:26:22.620 --> 00:26:24.965 align:middle line:90% Any questions on that? 00:26:24.965 --> 00:26:28.080 align:middle line:90% 00:26:28.080 --> 00:26:36.390 align:middle line:84% Now, it turns out that I could do the same kind of calculation 00:26:36.390 --> 00:26:40.840 align:middle line:90% for the inductor current ripple. 00:26:40.840 --> 00:26:43.300 align:middle line:84% All right, and I should say, by the way, 00:26:43.300 --> 00:26:46.300 align:middle line:84% that in a typical converter, I usually 00:26:46.300 --> 00:26:48.790 align:middle line:84% am not allowing very much capacitor voltage ripple. 00:26:48.790 --> 00:26:52.210 align:middle line:84% Right, if I want a DC-to-DC converter, this has to be small, 00:26:52.210 --> 00:26:55.300 align:middle line:84% and we'll come into what we mean by, How small is small? 00:26:55.300 --> 00:26:55.960 align:middle line:90% OK. 00:26:55.960 --> 00:26:59.840 align:middle line:84% Inductor current ripple, maybe we can allow a bit more, 00:26:59.840 --> 00:27:02.130 align:middle line:84% but we can do the same trick to calculate it, right. 00:27:02.130 --> 00:27:05.690 align:middle line:84% So if I were to come up here and say, OK, 00:27:05.690 --> 00:27:10.260 align:middle line:84% let me go through the same really quick procedure, 00:27:10.260 --> 00:27:12.430 align:middle line:90% OK, so here we go. 00:27:12.430 --> 00:27:15.960 align:middle line:84% Here's my converter, and what I'm really 00:27:15.960 --> 00:27:19.960 align:middle line:84% going to think about now is, What is this voltage? 00:27:19.960 --> 00:27:28.870 align:middle line:84% Suppose I define this voltage vX here, 00:27:28.870 --> 00:27:33.190 align:middle line:84% OK, I could come up with a model that looks like this. 00:27:33.190 --> 00:27:42.380 align:middle line:84% I have v1 and then an inductor and then a voltage 00:27:42.380 --> 00:27:43.997 align:middle line:84% that I'll call-- a pulsating voltage 00:27:43.997 --> 00:27:45.080 align:middle line:90% that I'm going to call vX. 00:27:45.080 --> 00:27:50.510 align:middle line:90% 00:27:50.510 --> 00:27:52.650 align:middle line:90% And what does vX look like? 00:27:52.650 --> 00:27:56.030 align:middle line:90% vX looks like this. 00:27:56.030 --> 00:28:02.200 align:middle line:84% In the first part of the cycle, in the first part of the cycle, 00:28:02.200 --> 00:28:09.160 align:middle line:84% when the switch is on, vX is 0 until some time DT. 00:28:09.160 --> 00:28:12.970 align:middle line:84% In the second part of the cycle, the switch is off 00:28:12.970 --> 00:28:15.370 align:middle line:84% and the diode is on, and it's equal to vC 00:28:15.370 --> 00:28:17.030 align:middle line:90% which is the same as v2, right. 00:28:17.030 --> 00:28:19.130 align:middle line:90% So here I go. 00:28:19.130 --> 00:28:22.720 align:middle line:90% Here's capital VC. 00:28:22.720 --> 00:28:24.790 align:middle line:90% OK, so the switch voltage-- 00:28:24.790 --> 00:28:27.460 align:middle line:84% I'm sorry, the switch voltage comes up here. 00:28:27.460 --> 00:28:27.960 align:middle line:90% OK. 00:28:27.960 --> 00:28:31.960 align:middle line:90% 00:28:31.960 --> 00:28:37.080 align:middle line:84% And I could then say, OK, this is vX of T in my ripple model, 00:28:37.080 --> 00:28:39.980 align:middle line:84% and the average voltage here, which 00:28:39.980 --> 00:28:43.010 align:middle line:84% is the same thing as the average value of v1 00:28:43.010 --> 00:28:50.150 align:middle line:84% is 1 minus D times VC, right, is the average voltage 00:28:50.150 --> 00:28:52.420 align:middle line:90% of this pulsating waveform. 00:28:52.420 --> 00:28:53.550 align:middle line:90% OK. 00:28:53.550 --> 00:28:56.190 align:middle line:84% I could then say, all right, this 00:28:56.190 --> 00:28:59.290 align:middle line:84% is what the circuit model looks like. 00:28:59.290 --> 00:29:01.930 align:middle line:84% Let me just consider its AC values. 00:29:01.930 --> 00:29:04.440 align:middle line:84% Right, let me consider the ripple current only 00:29:04.440 --> 00:29:05.580 align:middle line:90% in this thing. 00:29:05.580 --> 00:29:07.510 align:middle line:90% So, OK, what happens? 00:29:07.510 --> 00:29:09.190 align:middle line:90% What's the ripple in capital V1? 00:29:09.190 --> 00:29:11.160 align:middle line:90% It's nothing, right. 00:29:11.160 --> 00:29:15.870 align:middle line:84% Then I have my inductor, and this is L with a current iL. 00:29:15.870 --> 00:29:23.210 align:middle line:90% 00:29:23.210 --> 00:29:24.110 align:middle line:90% All right. 00:29:24.110 --> 00:29:25.590 align:middle line:90% Here's iL. 00:29:25.590 --> 00:29:29.540 align:middle line:84% This will be iL tilde, the AC component only. 00:29:29.540 --> 00:29:36.090 align:middle line:84% This thing, I may have a ripple voltage that looks like this. 00:29:36.090 --> 00:29:44.295 align:middle line:84% In the first part of the cycle, this is minus 1 minus D VC. 00:29:44.295 --> 00:29:50.580 align:middle line:90% 00:29:50.580 --> 00:29:52.560 align:middle line:84% And the second part of the cycle, it's 00:29:52.560 --> 00:29:56.920 align:middle line:84% VC minus 1 minus D VC, which is D VC. 00:29:56.920 --> 00:30:04.350 align:middle line:84% OK, so this is what vX tilde looks like. 00:30:04.350 --> 00:30:07.205 align:middle line:84% OK, and here's vX tilde, my circuit. 00:30:07.205 --> 00:30:10.530 align:middle line:90% 00:30:10.530 --> 00:30:17.670 align:middle line:84% All right, and of course, v sub L is just minus vX tilde, right. 00:30:17.670 --> 00:30:23.720 align:middle line:84% So then I can figure out that, OK, basically, 00:30:23.720 --> 00:30:28.950 align:middle line:84% the negative of this voltage is the voltage on the inductor. 00:30:28.950 --> 00:30:36.100 align:middle line:84% Right, so v sub L is equal to L di dt. 00:30:36.100 --> 00:30:40.420 align:middle line:84% So delta iL peak-to-peak is simply 00:30:40.420 --> 00:30:45.890 align:middle line:84% going to be 1 over L the integral 00:30:45.890 --> 00:30:51.360 align:middle line:90% from 0 to dt of v sub L dt. 00:30:51.360 --> 00:30:52.200 align:middle line:90% OK. 00:30:52.200 --> 00:30:54.760 align:middle line:90% And so what am I expecting? 00:30:54.760 --> 00:30:58.540 align:middle line:84% I'm putting vX is negative constant voltage, 00:30:58.540 --> 00:31:01.650 align:middle line:84% so that means vL is a positive constant voltage 00:31:01.650 --> 00:31:03.130 align:middle line:90% in the first part of the cycle. 00:31:03.130 --> 00:31:05.490 align:middle line:84% And I should get an inductor ripple current here 00:31:05.490 --> 00:31:10.360 align:middle line:84% that does this, and then in the second part of the cycle, 00:31:10.360 --> 00:31:11.480 align:middle line:90% it does this. 00:31:11.480 --> 00:31:14.290 align:middle line:90% So again, the voltage ripple-- 00:31:14.290 --> 00:31:18.490 align:middle line:84% I'm sorry, the inductor current ripple iL tilde 00:31:18.490 --> 00:31:20.930 align:middle line:90% is going to be triangular. 00:31:20.930 --> 00:31:30.250 align:middle line:84% And this amount here Is going to be delta iL peak-to-peak, 00:31:30.250 --> 00:31:33.550 align:middle line:84% which I can calculate that as 1 over L-- 00:31:33.550 --> 00:31:39.460 align:middle line:84% this is going to be 1 minus D VC, which 00:31:39.460 --> 00:31:41.260 align:middle line:90% goes on for a time period DT. 00:31:41.260 --> 00:31:46.335 align:middle line:90% 00:31:46.335 --> 00:31:46.835 align:middle line:90% OK. 00:31:46.835 --> 00:31:52.760 align:middle line:84% And what I get in the end is that L 00:31:52.760 --> 00:31:59.760 align:middle line:84% ought to be greater than or equal to 1 minus D times D 00:31:59.760 --> 00:32:06.856 align:middle line:84% times VC T over delta iL peak-to-peak. 00:32:06.856 --> 00:32:11.760 align:middle line:84% Right, so if I have a limitation on the allowable ripple current 00:32:11.760 --> 00:32:14.420 align:middle line:84% in the inductor, I can calculate, 00:32:14.420 --> 00:32:16.820 align:middle line:84% for a given operating condition, how big 00:32:16.820 --> 00:32:19.640 align:middle line:90% that inductor needs to be. 00:32:19.640 --> 00:32:23.810 align:middle line:84% And the only assumption I made was that I ignored the ripple 00:32:23.810 --> 00:32:27.800 align:middle line:90% in v2 when I calculated this. 00:32:27.800 --> 00:32:28.790 align:middle line:90% OK. 00:32:28.790 --> 00:32:30.355 align:middle line:84% Are there any questions about that? 00:32:30.355 --> 00:32:36.130 align:middle line:90% 00:32:36.130 --> 00:32:39.310 align:middle line:84% Now, interesting, I made some approximations, right. 00:32:39.310 --> 00:32:42.117 align:middle line:84% When I calculated the capacitor voltage ripple, 00:32:42.117 --> 00:32:43.700 align:middle line:84% I ignored the inductor current ripple, 00:32:43.700 --> 00:32:45.783 align:middle line:84% and when I calculated the inductor current ripple, 00:32:45.783 --> 00:32:47.430 align:middle line:84% I ignored the capacitor voltage ripple. 00:32:47.430 --> 00:32:49.480 align:middle line:90% Does it matter that I did that? 00:32:49.480 --> 00:32:51.020 align:middle line:90% The answer is no. 00:32:51.020 --> 00:32:51.760 align:middle line:90% Why? 00:32:51.760 --> 00:33:00.820 align:middle line:84% Because I pretended that IL was constant at a constant current 00:33:00.820 --> 00:33:03.836 align:middle line:84% when I calculated this waveform, right. 00:33:03.836 --> 00:33:05.820 align:middle line:90% Well, what does IL look like? 00:33:05.820 --> 00:33:11.630 align:middle line:84% IL has some ripple in it and so this triangular ripple falling. 00:33:11.630 --> 00:33:13.730 align:middle line:84% The only thing that would have happened 00:33:13.730 --> 00:33:16.010 align:middle line:84% had I included that is that I would 00:33:16.010 --> 00:33:19.330 align:middle line:90% have had a little ripple here. 00:33:19.330 --> 00:33:20.731 align:middle line:90% OK. 00:33:20.731 --> 00:33:22.520 align:middle line:84% And if I included that and then I 00:33:22.520 --> 00:33:24.980 align:middle line:84% calculated the voltage characteristic from that, 00:33:24.980 --> 00:33:31.547 align:middle line:84% all that would have happened is instead of getting a slope 00:33:31.547 --> 00:33:33.380 align:middle line:84% or instead of getting a ripple waveform that 00:33:33.380 --> 00:33:35.120 align:middle line:84% looked like a perfect triangle wave 00:33:35.120 --> 00:33:38.450 align:middle line:84% like I drew here, all that would have happened 00:33:38.450 --> 00:33:41.360 align:middle line:84% is I would have gotten a different version 00:33:41.360 --> 00:33:43.015 align:middle line:84% with a slightly different curvature, 00:33:43.015 --> 00:33:45.140 align:middle line:84% and the peak to peak ripple would have been exactly 00:33:45.140 --> 00:33:46.110 align:middle line:90% the same thing. 00:33:46.110 --> 00:33:46.830 align:middle line:90% OK. 00:33:46.830 --> 00:33:49.133 align:middle line:84% So I can make these approximations. 00:33:49.133 --> 00:33:50.550 align:middle line:84% Ignore the inductor current ripple 00:33:50.550 --> 00:33:53.640 align:middle line:84% for calculating the capacitor voltage ripple, and vice versa, 00:33:53.640 --> 00:33:56.590 align:middle line:84% and I still get the right answer. 00:33:56.590 --> 00:34:01.030 align:middle line:84% And what those have given me is a means 00:34:01.030 --> 00:34:03.340 align:middle line:84% to get to, How big a capacitor do 00:34:03.340 --> 00:34:05.860 align:middle line:84% I need to get to a certain capacitor voltage ripple, 00:34:05.860 --> 00:34:07.512 align:middle line:84% and how big an inductor do I need 00:34:07.512 --> 00:34:09.345 align:middle line:84% to get to a certain inductor current ripple? 00:34:09.345 --> 00:34:12.420 align:middle line:90% 00:34:12.420 --> 00:34:13.230 align:middle line:90% Questions? 00:34:13.230 --> 00:34:17.159 align:middle line:90% 00:34:17.159 --> 00:34:20.310 align:middle line:84% So I would say that if I was going to sit down and design 00:34:20.310 --> 00:34:28.310 align:middle line:84% a converter, I might start to think about, Hey, 00:34:28.310 --> 00:34:30.904 align:middle line:90% how much ripple is tolerable? 00:34:30.904 --> 00:34:34.100 align:middle line:84% Right, that's sort of an application thing, right? 00:34:34.100 --> 00:34:37.219 align:middle line:84% I'm building this boost converter to power some thing 00:34:37.219 --> 00:34:42.530 align:middle line:84% or other, and perhaps the specifications 00:34:42.530 --> 00:34:45.139 align:middle line:84% I'm giving you for the system is you're 00:34:45.139 --> 00:34:47.630 align:middle line:84% allowed a hundred millivolts of ripple on top 00:34:47.630 --> 00:34:52.602 align:middle line:84% of whatever your 50 volts of output is, or something. 00:34:52.602 --> 00:34:54.060 align:middle line:84% Somebody's going to tell you you're 00:34:54.060 --> 00:34:56.489 align:middle line:84% going to have this much voltage ripple 00:34:56.489 --> 00:35:00.270 align:middle line:84% or this much current ripple allowed into the input. 00:35:00.270 --> 00:35:00.780 align:middle line:90% OK. 00:35:00.780 --> 00:35:02.697 align:middle line:84% Now, one way to get those is just 00:35:02.697 --> 00:35:04.030 align:middle line:90% to make these components bigger. 00:35:04.030 --> 00:35:05.820 align:middle line:84% We'll see other ways to reduce those 00:35:05.820 --> 00:35:08.200 align:middle line:84% by adding higher-order filters and that kind of thing. 00:35:08.200 --> 00:35:11.735 align:middle line:84% But this is a pretty good place to start to get L and C values. 00:35:11.735 --> 00:35:15.150 align:middle line:84% OK, and you notice, in either case, 00:35:15.150 --> 00:35:21.420 align:middle line:84% what I ended up with is triangular-shaped ripples, OK, 00:35:21.420 --> 00:35:23.460 align:middle line:84% because essentially we're assuming that 00:35:23.460 --> 00:35:25.140 align:middle line:90% the current's just-- 00:35:25.140 --> 00:35:29.160 align:middle line:84% the ripple current's just going into the corresponding passive 00:35:29.160 --> 00:35:30.540 align:middle line:84% component, the constant current's 00:35:30.540 --> 00:35:32.998 align:middle line:84% going to a passive component, and I get a triangular ripple 00:35:32.998 --> 00:35:33.540 align:middle line:90% waveform. 00:35:33.540 --> 00:35:34.740 align:middle line:90% OK. 00:35:34.740 --> 00:35:38.950 align:middle line:84% Well, we can actually use that to our advantage. 00:35:38.950 --> 00:35:42.686 align:middle line:84% We can kind of say, OK, any such component-- 00:35:42.686 --> 00:35:45.750 align:middle line:90% 00:35:45.750 --> 00:35:50.020 align:middle line:84% suppose I have some waveform I'm going to call x of t, OK, 00:35:50.020 --> 00:35:55.150 align:middle line:84% and maybe it has some DC value X that I'll 00:35:55.150 --> 00:35:59.110 align:middle line:84% call capital X. This could be a capacitor voltage or an inductor 00:35:59.110 --> 00:35:59.950 align:middle line:90% current. 00:35:59.950 --> 00:36:03.160 align:middle line:84% And then maybe it has some triangular ripple on it 00:36:03.160 --> 00:36:05.050 align:middle line:90% like this. 00:36:05.050 --> 00:36:05.550 align:middle line:90% OK. 00:36:05.550 --> 00:36:10.936 align:middle line:90% 00:36:10.936 --> 00:36:15.535 align:middle line:84% And so then I'll have some peak value. 00:36:15.535 --> 00:36:18.960 align:middle line:90% 00:36:18.960 --> 00:36:26.700 align:middle line:84% I'll call this x peak, and I'll call this x min. 00:36:26.700 --> 00:36:27.584 align:middle line:90% OK. 00:36:27.584 --> 00:36:35.690 align:middle line:84% And If this is delta x peak-to-peak, 00:36:35.690 --> 00:36:38.900 align:middle line:84% it turns out that for triangular ripple-- 00:36:38.900 --> 00:36:42.810 align:middle line:84% any triangle wave, like this, the average 00:36:42.810 --> 00:36:45.920 align:middle line:84% is exactly halfway between the peaks. 00:36:45.920 --> 00:36:53.030 align:middle line:84% That means that this is delta x peak-to-peak over 2 00:36:53.030 --> 00:36:57.840 align:middle line:84% and this is delta x peak-to-peak over 2. 00:36:57.840 --> 00:37:00.770 align:middle line:84% All right, that's just a property of triangle waves. 00:37:00.770 --> 00:37:02.300 align:middle line:90% OK. 00:37:02.300 --> 00:37:04.210 align:middle line:90% Why am I getting to that? 00:37:04.210 --> 00:37:07.940 align:middle line:84% Because we might define the ripple ratio. 00:37:07.940 --> 00:37:08.440 align:middle line:90% OK. 00:37:08.440 --> 00:37:15.910 align:middle line:90% 00:37:15.910 --> 00:37:19.680 align:middle line:84% And I'm going to call this script R sub x, 00:37:19.680 --> 00:37:24.600 align:middle line:84% and I'm going to define this to be equal to delta x peak-to-peak 00:37:24.600 --> 00:37:26.650 align:middle line:90% divided by 2. 00:37:26.650 --> 00:37:32.040 align:middle line:84% That's sort of the middle, the average to the peak, normalized 00:37:32.040 --> 00:37:37.180 align:middle line:90% to X, right. 00:37:37.180 --> 00:37:38.890 align:middle line:84% So this is sort of-- you might think 00:37:38.890 --> 00:37:41.920 align:middle line:84% of this as being like a percentage ripple or a fraction 00:37:41.920 --> 00:37:43.160 align:middle line:90% ripple. 00:37:43.160 --> 00:37:43.660 align:middle line:90% OK. 00:37:43.660 --> 00:37:45.410 align:middle line:90% Why am I interested in that? 00:37:45.410 --> 00:37:48.910 align:middle line:84% I'm interested in that because a hundred millivolts of ripple 00:37:48.910 --> 00:37:50.835 align:middle line:90% on top of half a volt is a lot. 00:37:50.835 --> 00:37:53.210 align:middle line:84% A hundred millivolts of ripple on top of a hundred volts, 00:37:53.210 --> 00:37:54.580 align:middle line:90% maybe it's not so big, right? 00:37:54.580 --> 00:37:59.080 align:middle line:84% So we might want to normalize the ripple we consider 00:37:59.080 --> 00:38:01.390 align:middle line:84% to the average value of any waveform, inductor 00:38:01.390 --> 00:38:03.520 align:middle line:90% current or capacitor voltage. 00:38:03.520 --> 00:38:04.840 align:middle line:90% OK. 00:38:04.840 --> 00:38:10.900 align:middle line:84% So people often talk about the ripple ratio of the waveforms, 00:38:10.900 --> 00:38:13.030 align:middle line:84% or it's sort of a version of the percentage ripple 00:38:13.030 --> 00:38:14.830 align:middle line:90% if you multiply by a hundred. 00:38:14.830 --> 00:38:16.180 align:middle line:90% OK. 00:38:16.180 --> 00:38:18.940 align:middle line:84% I will say that if you look in the literature, 00:38:18.940 --> 00:38:21.440 align:middle line:84% there's a skew in how people define things. 00:38:21.440 --> 00:38:23.050 align:middle line:84% Some people define the ripple ratio 00:38:23.050 --> 00:38:25.540 align:middle line:84% as being delta x peak-to-peak over X, 00:38:25.540 --> 00:38:28.060 align:middle line:84% and some people define it as delta x peak-to-peak over 2 00:38:28.060 --> 00:38:30.760 align:middle line:84% over X. You've just got to be careful of what's 00:38:30.760 --> 00:38:33.850 align:middle line:84% being referred to when you read the literature. 00:38:33.850 --> 00:38:36.250 align:middle line:84% I kind of like this definition, A, 00:38:36.250 --> 00:38:38.930 align:middle line:84% because it's what used in the text. 00:38:38.930 --> 00:38:40.540 align:middle line:84% But one of the nice things about it 00:38:40.540 --> 00:38:45.720 align:middle line:84% is you can then say x sub peak is just simply 00:38:45.720 --> 00:38:52.380 align:middle line:84% equal to the DC value times 1 plus the ripple ratio of x, 00:38:52.380 --> 00:38:55.980 align:middle line:84% all right, and so it's convenient to use 00:38:55.980 --> 00:38:59.680 align:middle line:84% this form of the definition of ripple ratio. 00:38:59.680 --> 00:39:00.870 align:middle line:90% Any questions about that? 00:39:00.870 --> 00:39:04.380 align:middle line:90% 00:39:04.380 --> 00:39:11.350 align:middle line:84% OK, so how might I think about using this kind of definition? 00:39:11.350 --> 00:39:14.970 align:middle line:84% Well, I might talk about sort of-- 00:39:14.970 --> 00:39:18.900 align:middle line:84% I might talk about the ripple ratio and the capacitor voltage 00:39:18.900 --> 00:39:26.280 align:middle line:84% as being delta vC peak-to-peak over 2 divided by VC, 00:39:26.280 --> 00:39:28.740 align:middle line:84% and the ripple ratio and the inductor current 00:39:28.740 --> 00:39:32.490 align:middle line:84% as being delta iL peak-to-peak over 2 00:39:32.490 --> 00:39:35.490 align:middle line:84% divided by the DC inductor current. 00:39:35.490 --> 00:39:36.852 align:middle line:90% OK. 00:39:36.852 --> 00:39:40.310 align:middle line:84% And we can then use the ripple ratio 00:39:40.310 --> 00:39:45.200 align:middle line:84% that I'm talking about to relate sort of the average values 00:39:45.200 --> 00:39:49.400 align:middle line:84% to actual peak values because we know the ripple is triangular. 00:39:49.400 --> 00:39:50.670 align:middle line:90% OK. 00:39:50.670 --> 00:39:54.570 align:middle line:84% There's another way we can use this, too. 00:39:54.570 --> 00:39:56.830 align:middle line:84% There's a way to start thinking about, 00:39:56.830 --> 00:39:59.040 align:middle line:84% as I change the ripple ratio of my converter, 00:39:59.040 --> 00:40:01.140 align:middle line:84% How much ripple is allowed in my converter? 00:40:01.140 --> 00:40:04.040 align:middle line:90% How big does the converter get? 00:40:04.040 --> 00:40:05.520 align:middle line:90% OK. 00:40:05.520 --> 00:40:07.140 align:middle line:90% I've come up with expressions. 00:40:07.140 --> 00:40:10.400 align:middle line:84% I need this big an inductor or this big a capacitor 00:40:10.400 --> 00:40:13.280 align:middle line:84% if I knew those factors, but it might 00:40:13.280 --> 00:40:16.400 align:middle line:84% be nice to be able to express that a little bit better. 00:40:16.400 --> 00:40:19.230 align:middle line:84% How does that relate to the size of the component? 00:40:19.230 --> 00:40:21.800 align:middle line:90% 00:40:21.800 --> 00:40:25.010 align:middle line:84% That's a very complicated question in general, right, 00:40:25.010 --> 00:40:32.170 align:middle line:84% saying, How physically big is my inductor capacitor going to be? 00:40:32.170 --> 00:40:34.680 align:middle line:84% That's a hard question to answer because-- and we're 00:40:34.680 --> 00:40:37.650 align:middle line:84% going to see that when we start to design the components 00:40:37.650 --> 00:40:43.900 align:middle line:84% themselves, but it's reasonable to expect that to first order, 00:40:43.900 --> 00:40:47.860 align:middle line:84% the size of your component, the physical size of your component, 00:40:47.860 --> 00:40:51.170 align:middle line:84% ought to be related to the amount of energy it stores. 00:40:51.170 --> 00:40:54.470 align:middle line:84% Right, if I have to store twice as much energy 00:40:54.470 --> 00:40:57.920 align:middle line:84% and I get a certain energy density, if you imagine, 00:40:57.920 --> 00:41:01.910 align:middle line:84% of storage, then twice as much energy storage 00:41:01.910 --> 00:41:06.280 align:middle line:84% ought to be twice as much size, the first order, right? 00:41:06.280 --> 00:41:11.440 align:middle line:84% So engineers often use energy storage of the components 00:41:11.440 --> 00:41:14.500 align:middle line:84% as a very crude stand-in for how physically big 00:41:14.500 --> 00:41:16.480 align:middle line:84% the components are going to be, and I 00:41:16.480 --> 00:41:19.000 align:middle line:84% want to emphasize that it's a very crude stand-in. 00:41:19.000 --> 00:41:23.210 align:middle line:84% It's not true, necessarily, that it's a-- 00:41:23.210 --> 00:41:25.720 align:middle line:84% it's exactly proportional relationship or something 00:41:25.720 --> 00:41:26.220 align:middle line:90% like that. 00:41:26.220 --> 00:41:29.000 align:middle line:84% It's not, but it's at least a rough measure that you 00:41:29.000 --> 00:41:31.000 align:middle line:90% can get your hands around. 00:41:31.000 --> 00:41:31.794 align:middle line:90% OK. 00:41:31.794 --> 00:41:36.160 align:middle line:84% So let's think about what that might mean. 00:41:36.160 --> 00:41:37.570 align:middle line:90% OK. 00:41:37.570 --> 00:41:44.120 align:middle line:84% Well, let's think about the energy storage of the capacitor, 00:41:44.120 --> 00:41:44.620 align:middle line:90% right. 00:41:44.620 --> 00:41:47.620 align:middle line:84% I've got an expression here for the energy 00:41:47.620 --> 00:41:49.600 align:middle line:90% storage of the capacitor. 00:41:49.600 --> 00:41:51.400 align:middle line:90% OK. 00:41:51.400 --> 00:41:57.670 align:middle line:84% So we said that C has to be of this size, OK, 00:41:57.670 --> 00:42:05.580 align:middle line:84% and in my boost converter, IL is the same thing as I1, right. 00:42:05.580 --> 00:42:08.610 align:middle line:84% So what I get is, for a boost converter, then, 00:42:08.610 --> 00:42:15.060 align:middle line:84% I get C is greater than or equal to D times 1 minus D-- 00:42:15.060 --> 00:42:18.840 align:middle line:84% IL is the same thing as capital I1. 00:42:18.840 --> 00:42:24.930 align:middle line:84% This is the switching period divided by delta vC peak-to-peak 00:42:24.930 --> 00:42:27.150 align:middle line:90% over 2-- 00:42:27.150 --> 00:42:29.460 align:middle line:84% I'm sorry, delta vC peak-to-peak. 00:42:29.460 --> 00:42:30.120 align:middle line:90% OK. 00:42:30.120 --> 00:42:33.860 align:middle line:90% Let me substitute this in. 00:42:33.860 --> 00:42:37.730 align:middle line:84% So that means C has got to be greater than or equal to D 00:42:37.730 --> 00:42:51.760 align:middle line:84% times 1 minus D I1 T divided by 2 VC, which is also 00:42:51.760 --> 00:43:00.040 align:middle line:84% equal to V2, RC, ripple ratio of the capacitor. 00:43:00.040 --> 00:43:00.540 align:middle line:90% OK. 00:43:00.540 --> 00:43:04.040 align:middle line:84% So this is the same thing, just expressed 00:43:04.040 --> 00:43:06.950 align:middle line:84% in terms of input and output quantities 00:43:06.950 --> 00:43:10.410 align:middle line:84% and the ripple ratio on the capacitor. 00:43:10.410 --> 00:43:13.370 align:middle line:84% How much percentage ripple am I going to allow in the capacitor? 00:43:13.370 --> 00:43:14.335 align:middle line:90% Everybody buy that? 00:43:14.335 --> 00:43:17.230 align:middle line:90% 00:43:17.230 --> 00:43:19.300 align:middle line:84% If I then came back and said, OK, 00:43:19.300 --> 00:43:22.220 align:middle line:84% how much energy is stored in the capacitor? 00:43:22.220 --> 00:43:28.780 align:middle line:84% energy in the capacitor is simply equal to 1/2 C vC peak 00:43:28.780 --> 00:43:30.400 align:middle line:90% squared, right. 00:43:30.400 --> 00:43:32.215 align:middle line:84% If I've got to go buy a capacitor 00:43:32.215 --> 00:43:34.090 align:middle line:84% and I want to know how much energy is stored, 00:43:34.090 --> 00:43:35.530 align:middle line:84% I care about the peak voltage I'm 00:43:35.530 --> 00:43:36.850 align:middle line:84% going to put on the capacitor because that's what 00:43:36.850 --> 00:43:38.290 align:middle line:90% the capacitor is rated for. 00:43:38.290 --> 00:43:40.490 align:middle line:84% And I care about the capacitance value I'm using, 00:43:40.490 --> 00:43:42.670 align:middle line:84% and that energy is 1/2 C vC squared. 00:43:42.670 --> 00:43:44.940 align:middle line:90% Everybody buy that? 00:43:44.940 --> 00:43:47.825 align:middle line:90% So what is that going to be? 00:43:47.825 --> 00:43:50.750 align:middle line:90% 00:43:50.750 --> 00:43:56.750 align:middle line:84% Well, all that's going to be is vC is equal to 1/2-- 00:43:56.750 --> 00:43:59.660 align:middle line:84% C, I'm just going to substitute in this here, 00:43:59.660 --> 00:44:11.840 align:middle line:84% so it's going to be D times 1 minus D I1 T over 2 V2 RC. 00:44:11.840 --> 00:44:14.720 align:middle line:84% And v peak squared on the capacitor 00:44:14.720 --> 00:44:20.480 align:middle line:84% is simply equal to v2 times 1 plus the ripple ratio 00:44:20.480 --> 00:44:23.970 align:middle line:84% on the capacitor, quantity squared. 00:44:23.970 --> 00:44:25.470 align:middle line:90% All right, this is v peak. 00:44:25.470 --> 00:44:28.850 align:middle line:84% It's v2 times 1 plus the ripple ratio. 00:44:28.850 --> 00:44:29.570 align:middle line:90% OK. 00:44:29.570 --> 00:44:35.420 align:middle line:84% If I were then to rewrite this, what I would get is D times-- 00:44:35.420 --> 00:44:46.890 align:middle line:84% if I look at this, by the way, D times 1 minus D I1 is I2-- 00:44:46.890 --> 00:44:50.502 align:middle line:84% right, this is the same thing as I2-- 00:44:50.502 --> 00:44:55.140 align:middle line:90% T over 2-- 00:44:55.140 --> 00:44:56.610 align:middle line:90% I'm sorry, over 4. 00:44:56.610 --> 00:44:58.420 align:middle line:90% 2 times 2 is over 4. 00:44:58.420 --> 00:45:01.920 align:middle line:84% And then this is going to be v2 squared divided by V2 00:45:01.920 --> 00:45:08.400 align:middle line:84% is just v2, and then times 1 plus the ripple ratio 00:45:08.400 --> 00:45:10.360 align:middle line:90% squared over the ripple ratio. 00:45:10.360 --> 00:45:11.300 align:middle line:90% OK. 00:45:11.300 --> 00:45:14.630 align:middle line:84% And I'll rewrite one more thing, V2 times I2 00:45:14.630 --> 00:45:17.660 align:middle line:90% is just the output power, right. 00:45:17.660 --> 00:45:29.210 align:middle line:84% So this is going to be D times P out times T divided by 4 times 1 00:45:29.210 --> 00:45:32.920 align:middle line:84% plus the ripple ratio squared over the ripple ratio. 00:45:32.920 --> 00:45:33.420 align:middle line:90% All right. 00:45:33.420 --> 00:45:36.230 align:middle line:90% I apologize for all the math. 00:45:36.230 --> 00:45:39.860 align:middle line:84% This is the energy stored in the capacitor. 00:45:39.860 --> 00:45:40.820 align:middle line:90% OK. 00:45:40.820 --> 00:45:43.760 align:middle line:90% So why am I telling you this? 00:45:43.760 --> 00:45:49.990 align:middle line:84% This tells me, as a very broad function of ripple and operating 00:45:49.990 --> 00:45:52.120 align:middle line:84% power and everything, How much energy 00:45:52.120 --> 00:45:54.940 align:middle line:84% do I have to go put in that capacitor which I can use 00:45:54.940 --> 00:45:56.890 align:middle line:84% as a crude measure of how big and expensive 00:45:56.890 --> 00:45:59.000 align:middle line:90% that capacitor is going to be? 00:45:59.000 --> 00:45:59.500 align:middle line:90% OK. 00:45:59.500 --> 00:46:03.460 align:middle line:84% So what makes my capacitor requirement bigger? 00:46:03.460 --> 00:46:06.460 align:middle line:84% Its proportion-- the energy storage, which I'm claiming 00:46:06.460 --> 00:46:08.797 align:middle line:84% ought to be proportional to size, 00:46:08.797 --> 00:46:10.380 align:middle line:84% it's proportional to the output power. 00:46:10.380 --> 00:46:12.300 align:middle line:84% That's sort of physically reasonable, right. 00:46:12.300 --> 00:46:14.970 align:middle line:84% If I double the power, maybe I need twice as much energy 00:46:14.970 --> 00:46:17.500 align:middle line:84% storage to do the same conversion function. 00:46:17.500 --> 00:46:18.000 align:middle line:90% OK. 00:46:18.000 --> 00:46:22.450 align:middle line:84% It's proportional to D. Well, keep in mind, 00:46:22.450 --> 00:46:26.680 align:middle line:84% the bigger D is, the bigger the voltage conversion ratio is, 00:46:26.680 --> 00:46:30.400 align:middle line:84% right, because v2 is v1 over 1 minus D. 00:46:30.400 --> 00:46:34.120 align:middle line:84% So the more I'm stepping up voltage, the bigger D I need, 00:46:34.120 --> 00:46:37.780 align:middle line:84% so if I want to do a bigger step-up conversion, that 00:46:37.780 --> 00:46:42.010 align:middle line:84% means I need more energy storage in my capacitor. 00:46:42.010 --> 00:46:48.210 align:middle line:84% Also notice that this function over here 00:46:48.210 --> 00:46:50.500 align:middle line:84% is some function of the ripple ratio. 00:46:50.500 --> 00:46:53.490 align:middle line:84% It turns out, for most ranges of ripple ratio which I care about, 00:46:53.490 --> 00:46:56.700 align:middle line:84% which are much less than 1, making ripple ratio smaller 00:46:56.700 --> 00:47:00.180 align:middle line:90% makes this fraction bigger. 00:47:00.180 --> 00:47:03.090 align:middle line:84% So if I tell you I need less ripple, 00:47:03.090 --> 00:47:07.020 align:middle line:84% smaller ripple ratio, suddenly my energy storage got bigger. 00:47:07.020 --> 00:47:08.010 align:middle line:90% All right. 00:47:08.010 --> 00:47:11.220 align:middle line:84% So that says, if I want really nice input and output waveforms, 00:47:11.220 --> 00:47:14.710 align:middle line:84% I'm going to need more energy storage. 00:47:14.710 --> 00:47:15.210 align:middle line:90% OK. 00:47:15.210 --> 00:47:18.960 align:middle line:84% Lastly, notice that this is the switching period here, or 1 00:47:18.960 --> 00:47:20.680 align:middle line:90% over the switching frequency. 00:47:20.680 --> 00:47:23.190 align:middle line:84% The energy storage I need is proportional to the switching 00:47:23.190 --> 00:47:25.440 align:middle line:84% period, or 1 over the switching frequency. 00:47:25.440 --> 00:47:29.680 align:middle line:84% That means if I switch slowly, I need very big capacitors. 00:47:29.680 --> 00:47:32.440 align:middle line:84% And I showed you this for the capacitor in the lecture notes. 00:47:32.440 --> 00:47:35.410 align:middle line:84% I put the equivalent expression for the inductor. 00:47:35.410 --> 00:47:37.110 align:middle line:84% Basically, if you just substitute 00:47:37.110 --> 00:47:39.180 align:middle line:84% ripple ratio of the inductor for a capacitor, 00:47:39.180 --> 00:47:42.100 align:middle line:90% you get the same result. 00:47:42.100 --> 00:47:42.600 align:middle line:90% OK. 00:47:42.600 --> 00:47:45.410 align:middle line:84% So what that says is either the inductor or capacitor are going 00:47:45.410 --> 00:47:47.250 align:middle line:84% to get bigger and bigger and bigger 00:47:47.250 --> 00:47:51.800 align:middle line:84% if I either reduce my switching frequency 00:47:51.800 --> 00:47:55.915 align:middle line:84% or reduce the amount of ripple, OK, 00:47:55.915 --> 00:47:59.800 align:middle line:84% and that's what gives me a means to start saying, geez, 00:47:59.800 --> 00:48:01.690 align:middle line:90% I'd better limit-- 00:48:01.690 --> 00:48:05.830 align:middle line:84% if I want to limit this physical size or the cost of this thing, 00:48:05.830 --> 00:48:08.710 align:middle line:84% I'm going to have to start making some decisions about how 00:48:08.710 --> 00:48:10.750 align:middle line:90% I pick my components. 00:48:10.750 --> 00:48:13.100 align:middle line:90% OK. 00:48:13.100 --> 00:48:16.760 align:middle line:84% And that just gives you a first-order place 00:48:16.760 --> 00:48:19.760 align:middle line:84% to be in terms of picking component values. 00:48:19.760 --> 00:48:21.450 align:middle line:90% And this is only one thing. 00:48:21.450 --> 00:48:22.490 align:middle line:90% This is based on ripple. 00:48:22.490 --> 00:48:23.990 align:middle line:84% We're also going to talk about based 00:48:23.990 --> 00:48:26.750 align:middle line:84% on things like transient performance 00:48:26.750 --> 00:48:28.730 align:middle line:84% and that kind of thing, but this gives you 00:48:28.730 --> 00:48:30.860 align:middle line:84% at least a place to start for picking 00:48:30.860 --> 00:48:33.530 align:middle line:84% component values to get down to certain allowed levels 00:48:33.530 --> 00:48:36.400 align:middle line:90% of ripple. 00:48:36.400 --> 00:48:37.910 align:middle line:90% Any questions about that? 00:48:37.910 --> 00:48:42.117 align:middle line:90% 00:48:42.117 --> 00:48:44.450 align:middle line:84% AUDIENCE: These results are all for the boost converter? 00:48:44.450 --> 00:48:46.960 align:middle line:84% DAVID PERREAULT: These results are for the boost converter. 00:48:46.960 --> 00:48:50.650 align:middle line:84% You can derive exactly equivalent results for the buck 00:48:50.650 --> 00:48:54.490 align:middle line:84% converter or any other converter you want, and people often do. 00:48:54.490 --> 00:48:58.150 align:middle line:84% I will say that if I said, How much EC or EL is 00:48:58.150 --> 00:49:00.880 align:middle line:84% required for a boost converter to certain conversion ratio? 00:49:00.880 --> 00:49:03.460 align:middle line:84% and I went back and did it for the buck-boost converter 00:49:03.460 --> 00:49:08.110 align:middle line:84% that we started with, it would be significantly worse, 00:49:08.110 --> 00:49:09.880 align:middle line:90% much more energy storage. 00:49:09.880 --> 00:49:12.355 align:middle line:84% And what that would mean is I'm going 00:49:12.355 --> 00:49:14.480 align:middle line:84% to end up with a bigger, more expensive converter,. 00:49:14.480 --> 00:49:17.172 align:middle line:84% And what this comes down to is you're competing, right. 00:49:17.172 --> 00:49:18.880 align:middle line:84% Like, the other guy's designing something 00:49:18.880 --> 00:49:20.890 align:middle line:84% and you're designing something, and yours 00:49:20.890 --> 00:49:23.230 align:middle line:84% is smaller and more efficient because you've 00:49:23.230 --> 00:49:26.410 align:middle line:84% used less energy storage to do the same conversion function. 00:49:26.410 --> 00:49:29.480 align:middle line:84% You know, you could win or lose based on those decisions. 00:49:29.480 --> 00:49:31.850 align:middle line:84% But yes, you can derive similar results, 00:49:31.850 --> 00:49:33.620 align:middle line:90% and the typical thing is true. 00:49:33.620 --> 00:49:37.060 align:middle line:84% The more conversion transformation you're doing, 00:49:37.060 --> 00:49:38.800 align:middle line:84% you tend to require more energy storage. 00:49:38.800 --> 00:49:41.530 align:middle line:84% The lower the ripple you need, you require more energy storage. 00:49:41.530 --> 00:49:44.470 align:middle line:84% The slower you switch or the higher the power, 00:49:44.470 --> 00:49:48.100 align:middle line:90% you require more energy storage. 00:49:48.100 --> 00:49:49.280 align:middle line:90% Any other questions? 00:49:49.280 --> 00:49:55.060 align:middle line:90% 00:49:55.060 --> 00:49:57.850 align:middle line:84% OK, next time, we'll start thinking a little bit more 00:49:57.850 --> 00:50:01.560 align:middle line:84% about conversion ripple and what can happen in extreme cases. 00:50:01.560 --> 00:50:03.450 align:middle line:90% Have a great day. 00:50:03.450 --> 00:50:08.000 align:middle line:90%