WEBVTT 00:00:00.000 --> 00:00:02.440 align:middle line:90% [SQUEAKING] 00:00:02.440 --> 00:00:03.904 align:middle line:90% [RUSTLING] 00:00:03.904 --> 00:00:04.880 align:middle line:90% [CLICKING] 00:00:04.880 --> 00:00:09.290 align:middle line:90% 00:00:09.290 --> 00:00:12.710 align:middle line:84% DAVID PERREAULT: Moving on to the class material, so 00:00:12.710 --> 00:00:18.965 align:middle line:84% last class, we were talking about the AC grid and AC 00:00:18.965 --> 00:00:20.570 align:middle line:90% to DC interface. 00:00:20.570 --> 00:00:23.080 align:middle line:84% And we'll come back to that kind of topic. 00:00:23.080 --> 00:00:25.910 align:middle line:84% But what I'd like to do today is return to some topics 00:00:25.910 --> 00:00:28.520 align:middle line:84% about power converter analysis and, particularly, 00:00:28.520 --> 00:00:31.160 align:middle line:84% as related to DC to DC power conversion. 00:00:31.160 --> 00:00:34.130 align:middle line:84% Because we're going to see that that underpins a lot of kinds 00:00:34.130 --> 00:00:40.166 align:middle line:84% of power conversion, not only DC to DC but DC to AC and AC to DC. 00:00:40.166 --> 00:00:45.170 align:middle line:84% And I'd like to return to the notion of analyzing 00:00:45.170 --> 00:00:47.570 align:middle line:84% power converter circuits, particularly, 00:00:47.570 --> 00:00:50.360 align:middle line:90% in periodic steady state. 00:00:50.360 --> 00:00:54.500 align:middle line:84% And I was subtly using what I'm going to talk about right now 00:00:54.500 --> 00:00:58.200 align:middle line:84% in the first couple lectures, but I'm going to return to it. 00:00:58.200 --> 00:01:02.820 align:middle line:84% And the real question is, What tools do we 00:01:02.820 --> 00:01:05.670 align:middle line:84% use to take an arbitrary power converter 00:01:05.670 --> 00:01:11.390 align:middle line:84% and understand what kind of conversion function it can do? 00:01:11.390 --> 00:01:15.240 align:middle line:84% And let's just think about what we do for circuit analysis. 00:01:15.240 --> 00:01:17.520 align:middle line:84% Generally, one of the main things we might use is 00:01:17.520 --> 00:01:21.060 align:middle line:84% Kirchhoff's current law, which basically says, 00:01:21.060 --> 00:01:28.950 align:middle line:84% if I have a bunch of currents-- i1, i2, up to some iN-- 00:01:28.950 --> 00:01:33.060 align:middle line:84% coming into a node, the KCL just, basically, 00:01:33.060 --> 00:01:38.310 align:middle line:84% says that the summation over the j currents of i sub j 00:01:38.310 --> 00:01:40.320 align:middle line:90% has to add up to 0. 00:01:40.320 --> 00:01:43.830 align:middle line:84% And that's, basically, conservation of charge, 00:01:43.830 --> 00:01:48.954 align:middle line:84% assuming no charge builds up at any node. 00:01:48.954 --> 00:01:52.030 align:middle line:84% Well, I could take this equation and take 00:01:52.030 --> 00:01:53.690 align:middle line:90% the average of each side. 00:01:53.690 --> 00:01:57.323 align:middle line:84% So let me take the time average of this side of the equation. 00:01:57.323 --> 00:02:02.390 align:middle line:90% 00:02:02.390 --> 00:02:04.360 align:middle line:90% And doing an average-- 00:02:04.360 --> 00:02:09.130 align:middle line:84% 1 over t, the integral over t-- that's, basically, an integral. 00:02:09.130 --> 00:02:15.010 align:middle line:84% And I can interchange the order of the integral in the sum. 00:02:15.010 --> 00:02:17.680 align:middle line:84% So I could equally write the following-- 00:02:17.680 --> 00:02:22.300 align:middle line:84% the summation over j of the average values of i sub j 00:02:22.300 --> 00:02:25.030 align:middle line:90% is equal to 0. 00:02:25.030 --> 00:02:29.080 align:middle line:90% So this is average KCL. 00:02:29.080 --> 00:02:30.770 align:middle line:90% What it says is-- 00:02:30.770 --> 00:02:32.260 align:middle line:90% and it makes sense. 00:02:32.260 --> 00:02:35.710 align:middle line:84% If, instantaneously, all the currents add to 0, 00:02:35.710 --> 00:02:38.410 align:middle line:84% then the sum of the averages of the currents 00:02:38.410 --> 00:02:40.060 align:middle line:90% also have to add to 0. 00:02:40.060 --> 00:02:42.160 align:middle line:84% That's sort of-- what would I say, 00:02:42.160 --> 00:02:43.930 align:middle line:84% intuitively obvious that that's true? 00:02:43.930 --> 00:02:46.180 align:middle line:90% But it's mathematically true. 00:02:46.180 --> 00:02:48.760 align:middle line:84% And you could do the same thing for KVL, 00:02:48.760 --> 00:02:50.890 align:middle line:84% adding voltages around the loop, So I 00:02:50.890 --> 00:02:53.350 align:middle line:84% could have an average version of KVL 00:02:53.350 --> 00:02:59.080 align:middle line:84% which says that the summation of the average voltages 00:02:59.080 --> 00:03:05.340 align:middle line:84% around terminals in a loop adds to 0. 00:03:05.340 --> 00:03:08.490 align:middle line:84% So we could put these two together 00:03:08.490 --> 00:03:12.090 align:middle line:84% as two new analysis techniques that we 00:03:12.090 --> 00:03:18.900 align:middle line:84% could do when we're talking about doing circuit analysis, 00:03:18.900 --> 00:03:21.780 align:middle line:84% where we're thinking about the averages of currents 00:03:21.780 --> 00:03:27.870 align:middle line:84% going into a node or the average voltages around a loop. 00:03:27.870 --> 00:03:31.200 align:middle line:84% And just to remind you what other tools do 00:03:31.200 --> 00:03:35.190 align:middle line:84% we have at our disposal, well, we said-- 00:03:35.190 --> 00:03:40.230 align:middle line:84% previously, we said, OK, in periodic steady state, 00:03:40.230 --> 00:03:45.180 align:middle line:84% we know that if I have a capacitor 00:03:45.180 --> 00:03:49.500 align:middle line:84% and I have some current iC, the average value of iC 00:03:49.500 --> 00:03:53.790 align:middle line:84% has to be equal to 0 because iC is C dv/dt, 00:03:53.790 --> 00:03:57.510 align:middle line:84% and the average dv/dt is 0 in periodic steady state. 00:03:57.510 --> 00:04:00.570 align:middle line:84% Likewise, for an inductor, I have some voltage 00:04:00.570 --> 00:04:03.810 align:middle line:84% across the inductor, v sub L. And what I can say 00:04:03.810 --> 00:04:08.820 align:middle line:84% is the average of the inductor voltage 00:04:08.820 --> 00:04:12.150 align:middle line:84% is equal to 0 in periodic steady state. 00:04:12.150 --> 00:04:17.130 align:middle line:84% So those are four things I can use as circuit rules 00:04:17.130 --> 00:04:19.560 align:middle line:90% to help me analyze something. 00:04:19.560 --> 00:04:24.590 align:middle line:84% One more thing I might say is, if I have a power converter 00:04:24.590 --> 00:04:31.310 align:middle line:84% and I have some input power here and some output power here-- 00:04:31.310 --> 00:04:34.010 align:middle line:84% input power is coming in this way, P in, 00:04:34.010 --> 00:04:36.800 align:middle line:84% and output power is coming out this way, P out-- 00:04:36.800 --> 00:04:40.830 align:middle line:84% if I assume that the efficiency is 100%-- 00:04:40.830 --> 00:04:46.360 align:middle line:90% so if there's no loss-- 00:04:46.360 --> 00:04:49.710 align:middle line:84% then I need to have P in equals P 00:04:49.710 --> 00:04:52.380 align:middle line:84% out where we're talking about the average power flowing 00:04:52.380 --> 00:04:53.430 align:middle line:90% in and out. 00:04:53.430 --> 00:04:57.150 align:middle line:84% So that's just conservation of energy, which I often 00:04:57.150 --> 00:04:59.580 align:middle line:84% use because, even though real power converters aren't 00:04:59.580 --> 00:05:02.220 align:middle line:84% lossless, I might make an assumption 00:05:02.220 --> 00:05:07.430 align:middle line:84% for a first-order analysis that assumes they are lossless. 00:05:07.430 --> 00:05:12.140 align:middle line:84% So let's go back and think about how we might use that. 00:05:12.140 --> 00:05:17.180 align:middle line:84% And let's come back to our really simple first switching 00:05:17.180 --> 00:05:18.810 align:middle line:90% regulator example. 00:05:18.810 --> 00:05:25.010 align:middle line:84% So we said, OK, maybe I have some voltage v1. 00:05:25.010 --> 00:05:27.890 align:middle line:84% And here, I'm going to put a capacitor in parallel with v1 00:05:27.890 --> 00:05:30.430 align:middle line:84% just to illustrate that maybe there's-- 00:05:30.430 --> 00:05:32.960 align:middle line:90% 00:05:32.960 --> 00:05:37.220 align:middle line:84% some local source of charge is holding this voltage. 00:05:37.220 --> 00:05:42.620 align:middle line:84% And I will go have some single-pole double-throw switch 00:05:42.620 --> 00:05:48.290 align:middle line:84% like this, an inductor, another capacitor 00:05:48.290 --> 00:05:52.203 align:middle line:84% that I'll call c2 here, and a resistor. 00:05:52.203 --> 00:05:53.120 align:middle line:90% And I'll call this v2. 00:05:53.120 --> 00:05:57.392 align:middle line:90% 00:05:57.392 --> 00:05:59.613 align:middle line:84% In principle, I don't need capacitor c1 00:05:59.613 --> 00:06:01.280 align:middle line:84% if I have a perfect voltage source here. 00:06:01.280 --> 00:06:05.210 align:middle line:84% But I'll draw it in because we often, for filtering purposes, 00:06:05.210 --> 00:06:07.430 align:middle line:84% practically put a capacitor there, too. 00:06:07.430 --> 00:06:12.500 align:middle line:84% And we said, we can define operation of this circuit 00:06:12.500 --> 00:06:14.480 align:middle line:90% by the switch state here. 00:06:14.480 --> 00:06:17.090 align:middle line:84% So maybe when the switch is in the up position, 00:06:17.090 --> 00:06:20.387 align:middle line:84% I'll say my switching function q of t is equal to 1. 00:06:20.387 --> 00:06:22.220 align:middle line:84% And when the switch is in the down position, 00:06:22.220 --> 00:06:25.160 align:middle line:84% I'll say my switching function q of t is equal to 0. 00:06:25.160 --> 00:06:27.800 align:middle line:84% And we'll see later that that's useful to mathematically model 00:06:27.800 --> 00:06:28.820 align:middle line:90% things. 00:06:28.820 --> 00:06:32.670 align:middle line:84% So this is, basically, the notion of this converter 00:06:32.670 --> 00:06:36.240 align:middle line:90% we were considering last time. 00:06:36.240 --> 00:06:40.890 align:middle line:84% In practice, how do we build this thing What I would really 00:06:40.890 --> 00:06:44.610 align:middle line:84% do is I would come back and say, OK, we'll 00:06:44.610 --> 00:06:50.070 align:middle line:90% have some voltage v1, c1. 00:06:50.070 --> 00:06:57.470 align:middle line:84% And I will have two switches like this. 00:06:57.470 --> 00:07:02.100 align:middle line:90% 00:07:02.100 --> 00:07:04.170 align:middle line:84% And I'll operate these two switches 00:07:04.170 --> 00:07:05.640 align:middle line:90% in a complementary fashion. 00:07:05.640 --> 00:07:07.470 align:middle line:84% And that serves the same function 00:07:07.470 --> 00:07:09.640 align:middle line:84% as the single-pole double-throw switch. 00:07:09.640 --> 00:07:13.650 align:middle line:84% So this switch closes when q of t is equal to 1. 00:07:13.650 --> 00:07:16.370 align:middle line:84% And this switch closes when q of t is equal to 0. 00:07:16.370 --> 00:07:19.980 align:middle line:90% 00:07:19.980 --> 00:07:25.220 align:middle line:84% Now, what would we look at in this system? 00:07:25.220 --> 00:07:31.630 align:middle line:84% Well, we can consider the voltage here that I'll call vx. 00:07:31.630 --> 00:07:33.250 align:middle line:90% Or here, it's vx. 00:07:33.250 --> 00:07:36.250 align:middle line:90% 00:07:36.250 --> 00:07:42.760 align:middle line:90% I also have vL and a current iL. 00:07:42.760 --> 00:07:49.850 align:middle line:90% 00:07:49.850 --> 00:07:52.580 align:middle line:84% And how are we going to analyze this thing? 00:07:52.580 --> 00:07:55.670 align:middle line:84% Well, let's consider periodic steady-state operation. 00:07:55.670 --> 00:07:57.170 align:middle line:84% So what I'm going to do is I'm going 00:07:57.170 --> 00:08:02.120 align:middle line:90% to switch the switches like so. 00:08:02.120 --> 00:08:03.890 align:middle line:90% Here's q of t. 00:08:03.890 --> 00:08:07.620 align:middle line:84% When the top switch is on, it's on. 00:08:07.620 --> 00:08:10.610 align:middle line:90% And I'll do that for time DT. 00:08:10.610 --> 00:08:15.140 align:middle line:84% And then I'll hold the top switch off for a time 1 minus DT 00:08:15.140 --> 00:08:17.720 align:middle line:90% and operate like this-- 00:08:17.720 --> 00:08:21.200 align:middle line:90% T plus DT and so forth. 00:08:21.200 --> 00:08:26.090 align:middle line:84% So I've got my periodic switching with some period T. So 00:08:26.090 --> 00:08:27.550 align:middle line:90% what happens in this circuit? 00:08:27.550 --> 00:08:31.870 align:middle line:90% 00:08:31.870 --> 00:08:35.590 align:middle line:84% If I were to look at this voltage vx, when I've closed 00:08:35.590 --> 00:08:39.549 align:middle line:90% this switch, vx is equal to v1. 00:08:39.549 --> 00:08:41.659 align:middle line:84% When I open this switch and close this switch, 00:08:41.659 --> 00:08:43.210 align:middle line:90% vx is equal to 0. 00:08:43.210 --> 00:08:56.880 align:middle line:84% So vx here looks just like q of t, except that it's scaled by v1 00:08:56.880 --> 00:08:58.350 align:middle line:90% and so forth. 00:08:58.350 --> 00:09:01.650 align:middle line:90% And this is value v1. 00:09:01.650 --> 00:09:02.940 align:middle line:90% This is DT. 00:09:02.940 --> 00:09:06.096 align:middle line:90% This is T. 00:09:06.096 --> 00:09:11.350 align:middle line:84% So how would I go about analyzing this circuit? 00:09:11.350 --> 00:09:15.100 align:middle line:84% Well, I can write KVL for this circuit. 00:09:15.100 --> 00:09:19.920 align:middle line:84% And maybe what I'll do is I'll think about average KVL. 00:09:19.920 --> 00:09:24.400 align:middle line:84% So I could write, for example, vx minus vL minus v2 00:09:24.400 --> 00:09:25.310 align:middle line:90% is equal to 0. 00:09:25.310 --> 00:09:30.610 align:middle line:90% 00:09:30.610 --> 00:09:33.220 align:middle line:84% And could take the average of this, which would give me 00:09:33.220 --> 00:09:38.080 align:middle line:84% the average value of vx minus the average value of vL 00:09:38.080 --> 00:09:46.430 align:middle line:84% minus the average value of v2 is equal to 0. 00:09:46.430 --> 00:09:52.100 align:middle line:84% If I think about that, though, the average value of vL 00:09:52.100 --> 00:09:54.400 align:middle line:90% is 0 in periodic steady state. 00:09:54.400 --> 00:09:58.166 align:middle line:90% 00:09:58.166 --> 00:10:02.210 align:middle line:84% So if I look at this equation, this equation 00:10:02.210 --> 00:10:04.020 align:middle line:90% is just average KVL. 00:10:04.020 --> 00:10:07.460 align:middle line:84% I'm taking the average voltage around a loop. 00:10:07.460 --> 00:10:12.590 align:middle line:84% So I'm using what's on the board behind the top board here. 00:10:12.590 --> 00:10:16.890 align:middle line:84% And then I'm saying the average value across the inductor is 0. 00:10:16.890 --> 00:10:21.530 align:middle line:84% And that gives me that the average value of v2 00:10:21.530 --> 00:10:25.340 align:middle line:84% is equal to the average value of vx. 00:10:25.340 --> 00:10:29.300 align:middle line:84% And we might write the average value of v2 of t 00:10:29.300 --> 00:10:30.530 align:middle line:90% is just capital V2. 00:10:30.530 --> 00:10:34.920 align:middle line:90% 00:10:34.920 --> 00:10:36.840 align:middle line:84% Well, what is the average value of vx? 00:10:36.840 --> 00:10:40.260 align:middle line:90% 00:10:40.260 --> 00:10:43.620 align:middle line:90% It's this average value. 00:10:43.620 --> 00:10:48.120 align:middle line:84% And it's v1 for DT of the time and 0 for 1 00:10:48.120 --> 00:10:49.770 align:middle line:90% minus DT of the time. 00:10:49.770 --> 00:10:53.820 align:middle line:84% So that's just going to be equal to D times V1. 00:10:53.820 --> 00:10:58.440 align:middle line:84% And this is the same result that we got when 00:10:58.440 --> 00:11:02.232 align:middle line:90% we analyzed the circuit before. 00:11:02.232 --> 00:11:03.690 align:middle line:84% The only thing that's any different 00:11:03.690 --> 00:11:07.740 align:middle line:84% is I've explicitly used average KVL, 00:11:07.740 --> 00:11:09.420 align:middle line:84% and I've explicitly used the fact 00:11:09.420 --> 00:11:11.933 align:middle line:84% that the average voltage across the inductor is 0. 00:11:11.933 --> 00:11:14.100 align:middle line:84% And I'm showing it to you in this very basic circuit 00:11:14.100 --> 00:11:16.860 align:middle line:84% because it gets a little trickier in much 00:11:16.860 --> 00:11:19.750 align:middle line:90% more complicated circuits. 00:11:19.750 --> 00:11:21.855 align:middle line:84% So are there any questions about this result? 00:11:21.855 --> 00:11:24.790 align:middle line:90% 00:11:24.790 --> 00:11:30.010 align:middle line:84% What that means is, then, my output voltage in this circuit, 00:11:30.010 --> 00:11:33.970 align:middle line:84% if I'm switching these functions with some period and some duty 00:11:33.970 --> 00:11:36.500 align:middle line:90% ratio, v2 will settle to DV1. 00:11:36.500 --> 00:11:39.350 align:middle line:84% D is a duty ratio that's between 0 and 1. 00:11:39.350 --> 00:11:43.280 align:middle line:84% So I can control v2 to be anything less than v1. 00:11:43.280 --> 00:11:46.133 align:middle line:84% And as we said before, ideally, everything's lossless. 00:11:46.133 --> 00:11:50.520 align:middle line:90% 00:11:50.520 --> 00:11:57.280 align:middle line:84% So let's think about a different aspect of this circuit, 00:11:57.280 --> 00:12:00.880 align:middle line:84% just to illustrate these analysis techniques. 00:12:00.880 --> 00:12:05.860 align:middle line:84% Let's take a look at, for example, this current. 00:12:05.860 --> 00:12:08.325 align:middle line:84% I'm going to call this current i sub y. 00:12:08.325 --> 00:12:13.300 align:middle line:90% 00:12:13.300 --> 00:12:15.820 align:middle line:84% So it's the current through this switch. 00:12:15.820 --> 00:12:21.010 align:middle line:84% And the thing I'm going to do is let me assume-- 00:12:21.010 --> 00:12:24.380 align:middle line:84% and we often do this when we're analyzing DC 00:12:24.380 --> 00:12:27.750 align:middle line:84% to DC converters, PWM DC [INAUDIBLE] converters. 00:12:27.750 --> 00:12:30.430 align:middle line:84% Let me assume that my inductors and my capacitors 00:12:30.430 --> 00:12:32.740 align:middle line:90% are really big. 00:12:32.740 --> 00:12:34.760 align:middle line:84% Well, what do I mean by really big? 00:12:34.760 --> 00:12:37.240 align:middle line:84% If you let it go to infinity, then it's really big. 00:12:37.240 --> 00:12:43.120 align:middle line:84% But what I'm really saying is let me assume that iL of t 00:12:43.120 --> 00:12:47.890 align:middle line:84% is approximately equal to its DC value. 00:12:47.890 --> 00:12:56.440 align:middle line:84% And v1 of t is approximately equal to the DC input. 00:12:56.440 --> 00:13:01.240 align:middle line:84% And v2 of t is approximately equal to V2. 00:13:01.240 --> 00:13:03.910 align:middle line:84% So I'm basically assuming that the capacitors are 00:13:03.910 --> 00:13:06.820 align:middle line:84% so big that there's no voltage ripple on them, 00:13:06.820 --> 00:13:13.614 align:middle line:84% and the inductor is so big that its current ripple is 0. 00:13:13.614 --> 00:13:16.100 align:middle line:90% Any questions about that? 00:13:16.100 --> 00:13:18.180 align:middle line:90% So what does that mean? 00:13:18.180 --> 00:13:22.300 align:middle line:90% What would iy look like? 00:13:22.300 --> 00:13:24.750 align:middle line:90% Well, let's plot it. 00:13:24.750 --> 00:13:30.910 align:middle line:90% 00:13:30.910 --> 00:13:33.740 align:middle line:84% When q of t is 1, this switch is closed, 00:13:33.740 --> 00:13:37.120 align:middle line:90% so iy ought to exactly equal iL. 00:13:37.120 --> 00:13:43.270 align:middle line:90% So here's iy. 00:13:43.270 --> 00:13:45.765 align:middle line:84% It's going to be equal to IL when the switch is on. 00:13:45.765 --> 00:13:50.550 align:middle line:90% 00:13:50.550 --> 00:13:57.660 align:middle line:84% And then, when the switch is off, obviously, iy goes to 0. 00:13:57.660 --> 00:14:02.230 align:middle line:84% So I have this pulsating waveform here. 00:14:02.230 --> 00:14:11.550 align:middle line:90% 00:14:11.550 --> 00:14:16.380 align:middle line:90% Now, what's the value of iy? 00:14:16.380 --> 00:14:16.960 align:middle line:90% I'm sorry. 00:14:16.960 --> 00:14:19.680 align:middle line:90% What's the value of iL here? 00:14:19.680 --> 00:14:26.490 align:middle line:84% Well, I could call this current i2. 00:14:26.490 --> 00:14:29.400 align:middle line:84% And I'm assuming, since this capacitor voltage is really, 00:14:29.400 --> 00:14:33.580 align:middle line:84% really big, I can assume the capacitor is really, really big. 00:14:33.580 --> 00:14:36.432 align:middle line:84% This capacitor ripple voltage is really small, 00:14:36.432 --> 00:14:38.890 align:middle line:84% so the ripple current through the resistor is really small. 00:14:38.890 --> 00:14:42.840 align:middle line:84% So this is approximately equal to I2. 00:14:42.840 --> 00:14:44.940 align:middle line:90% And what do I know about-- 00:14:44.940 --> 00:14:47.880 align:middle line:84% let me call this capacitor c2, and let 00:14:47.880 --> 00:14:50.760 align:middle line:90% me call this current ic2. 00:14:50.760 --> 00:14:55.750 align:middle line:84% What do I know about the average value of ic2? 00:14:55.750 --> 00:14:56.590 align:middle line:90% AUDIENCE: 0 00:14:56.590 --> 00:14:57.640 align:middle line:90% DAVID PERREAULT: 0. 00:14:57.640 --> 00:15:02.710 align:middle line:84% So that means that the average value of current I2 00:15:02.710 --> 00:15:07.000 align:middle line:84% must equal the average value of current iL. 00:15:07.000 --> 00:15:13.690 align:middle line:84% Or, put another way, iL also has to equal I2. 00:15:13.690 --> 00:15:16.703 align:middle line:84% There's no average current coming down here. 00:15:16.703 --> 00:15:19.120 align:middle line:84% So whatever the average is going through the inductor just 00:15:19.120 --> 00:15:21.550 align:middle line:90% can go to the resistor. 00:15:21.550 --> 00:15:23.270 align:middle line:84% Does that make sense to everybody? 00:15:23.270 --> 00:15:26.390 align:middle line:90% 00:15:26.390 --> 00:15:33.950 align:middle line:84% So what can I say about the average value of i sub y? 00:15:33.950 --> 00:15:38.970 align:middle line:84% Well, the average value of i sub y 00:15:38.970 --> 00:15:42.300 align:middle line:84% is whatever the average value of this waveform is. 00:15:42.300 --> 00:15:45.930 align:middle line:90% And I'll call that capital Iy. 00:15:45.930 --> 00:15:49.690 align:middle line:84% And that's going to be the average value of this PWM 00:15:49.690 --> 00:15:50.290 align:middle line:90% waveform. 00:15:50.290 --> 00:15:55.170 align:middle line:84% Well, it's I2 for a fraction D of the time and 0 for 1 00:15:55.170 --> 00:15:56.820 align:middle line:90% minus D of the time. 00:15:56.820 --> 00:16:02.430 align:middle line:84% So this must exactly be equal to D I2, which 00:16:02.430 --> 00:16:03.670 align:middle line:90% is the same thing as DIL. 00:16:03.670 --> 00:16:08.320 align:middle line:90% 00:16:08.320 --> 00:16:09.435 align:middle line:90% Any questions about that? 00:16:09.435 --> 00:16:11.940 align:middle line:90% 00:16:11.940 --> 00:16:12.840 align:middle line:90% OK. 00:16:12.840 --> 00:16:17.585 align:middle line:84% So now I know the average current coming this way. 00:16:17.585 --> 00:16:20.390 align:middle line:90% 00:16:20.390 --> 00:16:25.660 align:middle line:84% What's the average current going through this capacitor 00:16:25.660 --> 00:16:28.140 align:middle line:90% in periodic steady state? 00:16:28.140 --> 00:16:28.640 align:middle line:90% AUDIENCE: 0. 00:16:28.640 --> 00:16:29.600 align:middle line:90% DAVID PERREAULT: 0. 00:16:29.600 --> 00:16:35.945 align:middle line:84% So if I introduced this as my node, 00:16:35.945 --> 00:16:38.400 align:middle line:84% there's no average current coming here. 00:16:38.400 --> 00:16:43.100 align:middle line:84% So that must mean if this is current i1, the average of i1 00:16:43.100 --> 00:16:46.730 align:middle line:90% must equal the average of iy. 00:16:46.730 --> 00:16:51.350 align:middle line:84% Because no average current through the capacitor. 00:16:51.350 --> 00:16:58.510 align:middle line:84% So then I could say that, OK, this also, by average case cL, 00:16:58.510 --> 00:17:00.670 align:middle line:90% also has to be equal to I1. 00:17:00.670 --> 00:17:05.770 align:middle line:90% 00:17:05.770 --> 00:17:14.230 align:middle line:84% So what I can say then is that I1 is equal to D times I2. 00:17:14.230 --> 00:17:17.970 align:middle line:90% 00:17:17.970 --> 00:17:20.660 align:middle line:84% So what I've really figured out is the current conversion ratio. 00:17:20.660 --> 00:17:24.930 align:middle line:84% I could have written I2, the output current, 00:17:24.930 --> 00:17:26.839 align:middle line:84% the average output current, is equal to 1 00:17:26.839 --> 00:17:30.580 align:middle line:84% over D times the average input current. 00:17:30.580 --> 00:17:35.530 align:middle line:84% So I have this as a result, which 00:17:35.530 --> 00:17:40.000 align:middle line:84% I got by basically looking at averages of currents 00:17:40.000 --> 00:17:45.240 align:middle line:84% and knowing that the currents through the capacitors were 0. 00:17:45.240 --> 00:17:48.240 align:middle line:84% There's another way I could have gotten to that. 00:17:48.240 --> 00:17:55.680 align:middle line:90% I said that V1 is equal to what? 00:17:55.680 --> 00:17:57.190 align:middle line:90% Or V2 is equal to what? 00:17:57.190 --> 00:17:59.140 align:middle line:90% V2 is equal to DV1. 00:17:59.140 --> 00:18:03.413 align:middle line:84% So I could have said V2 is equal to DV1. 00:18:03.413 --> 00:18:07.620 align:middle line:90% 00:18:07.620 --> 00:18:13.250 align:middle line:84% So if I multiplied each side by-- 00:18:13.250 --> 00:18:15.860 align:middle line:84% so suppose I multiplied this side by I2, 00:18:15.860 --> 00:18:23.780 align:middle line:84% I could have the output power, P out, is equal to V2 times I2. 00:18:23.780 --> 00:18:27.793 align:middle line:84% Well, if I substitute this in, V2 is equal to DV1. 00:18:27.793 --> 00:18:32.228 align:middle line:90% 00:18:32.228 --> 00:18:35.210 align:middle line:84% And I2 is equal to 1 over D times I1. 00:18:35.210 --> 00:18:39.130 align:middle line:90% 00:18:39.130 --> 00:18:44.470 align:middle line:84% So that has to be, if these two cancel, has to be equal to V1I1. 00:18:44.470 --> 00:18:48.220 align:middle line:84% So another way I could have gotten at this result 00:18:48.220 --> 00:18:52.450 align:middle line:84% is, by knowing the voltage conversion ratio and the fact 00:18:52.450 --> 00:18:54.310 align:middle line:84% that power is conserved, I could have 00:18:54.310 --> 00:18:56.680 align:middle line:84% gotten to this current conversion ratio. 00:18:56.680 --> 00:19:00.070 align:middle line:84% Or the current conversion ratio is the exact inverse 00:19:00.070 --> 00:19:04.230 align:middle line:84% of the voltage conversion ratio because there's 00:19:04.230 --> 00:19:07.636 align:middle line:90% no power loss in the converter. 00:19:07.636 --> 00:19:14.360 align:middle line:90% So these tools that we have-- 00:19:14.360 --> 00:19:18.680 align:middle line:84% average KVL, average KCL, conservation of energy, 00:19:18.680 --> 00:19:20.630 align:middle line:84% periodic steady-state relations-- 00:19:20.630 --> 00:19:24.950 align:middle line:84% can let me go through a given power converter 00:19:24.950 --> 00:19:28.880 align:middle line:84% and figure out the relationships between the DC input 00:19:28.880 --> 00:19:32.280 align:middle line:84% and the DC output quantities in periodic steady state. 00:19:32.280 --> 00:19:34.520 align:middle line:84% And we'll often do that as a first step 00:19:34.520 --> 00:19:36.350 align:middle line:84% when we're looking at some power converter 00:19:36.350 --> 00:19:38.660 align:middle line:84% in order to know, What is it doing? 00:19:38.660 --> 00:19:40.560 align:middle line:90% What can it do? 00:19:40.560 --> 00:19:44.160 align:middle line:84% In this case, this converter can step down voltage. 00:19:44.160 --> 00:19:46.940 align:middle line:84% This converter can create a v2 that's smaller than v1. 00:19:46.940 --> 00:19:50.060 align:middle line:84% But it couldn't, for example, create a voltage 00:19:50.060 --> 00:19:53.765 align:middle line:84% that was bigger than v1 because the average analysis tells me 00:19:53.765 --> 00:19:54.265 align:middle line:90% so. 00:19:54.265 --> 00:19:57.070 align:middle line:90% 00:19:57.070 --> 00:19:58.320 align:middle line:90% Any questions about that? 00:19:58.320 --> 00:20:06.840 align:middle line:90% 00:20:06.840 --> 00:20:10.020 align:middle line:84% One thing I should say about doing this kind of analysis-- 00:20:10.020 --> 00:20:13.260 align:middle line:84% because I'm doing the world's simplest case here. 00:20:13.260 --> 00:20:15.750 align:middle line:84% And you'll need to use these techniques 00:20:15.750 --> 00:20:17.300 align:middle line:90% on all kinds of converters. 00:20:17.300 --> 00:20:21.180 align:middle line:84% The thing that often messes people up, 00:20:21.180 --> 00:20:23.760 align:middle line:84% the thing that will make you make a mistake and get the wrong 00:20:23.760 --> 00:20:25.677 align:middle line:84% answer in your analysis, the number-one thing, 00:20:25.677 --> 00:20:27.820 align:middle line:90% is the following-- 00:20:27.820 --> 00:20:32.870 align:middle line:84% these statements are all about average currents and voltages. 00:20:32.870 --> 00:20:36.560 align:middle line:84% Do not mistake the average for the instantaneous. 00:20:36.560 --> 00:20:44.680 align:middle line:84% So, in other words, I can say that the average value of iy, 00:20:44.680 --> 00:20:48.130 align:middle line:84% which is equal to the average value of I1, which we calculated 00:20:48.130 --> 00:20:52.630 align:middle line:90% over here, is D times I2. 00:20:52.630 --> 00:20:54.320 align:middle line:84% But that's true of the average value. 00:20:54.320 --> 00:20:55.840 align:middle line:90% But here's iy. 00:20:55.840 --> 00:20:58.240 align:middle line:84% It's almost never equal to its average value, 00:20:58.240 --> 00:21:00.020 align:middle line:84% except at these instantaneous points. 00:21:00.020 --> 00:21:03.040 align:middle line:84% So if you take and say, oh, I know the average value, 00:21:03.040 --> 00:21:05.530 align:middle line:84% but then you substitute it for the instantaneous value, 00:21:05.530 --> 00:21:08.150 align:middle line:84% you're going to get strange results. 00:21:08.150 --> 00:21:10.330 align:middle line:84% So just keep in mind, separate out 00:21:10.330 --> 00:21:12.430 align:middle line:84% in your mind, when you're talking about averages 00:21:12.430 --> 00:21:15.130 align:middle line:84% and when you're talking about instantaneous waveforms. 00:21:15.130 --> 00:21:18.460 align:middle line:84% And the waveforms in the circuit that, obviously, the average is 00:21:18.460 --> 00:21:21.693 align:middle line:84% equal to the instantaneous is if you have an inductor current 00:21:21.693 --> 00:21:23.610 align:middle line:84% or capacitor voltage, and the ripple is small. 00:21:23.610 --> 00:21:31.780 align:middle line:90% 00:21:31.780 --> 00:21:33.220 align:middle line:90% What would I do-- 00:21:33.220 --> 00:21:35.240 align:middle line:84% how would I implement this converter? 00:21:35.240 --> 00:21:38.120 align:middle line:84% And we're going to talk a lot about switch implementation. 00:21:38.120 --> 00:21:41.350 align:middle line:84% What I might typically do in this kind of design 00:21:41.350 --> 00:21:42.646 align:middle line:90% is the following. 00:21:42.646 --> 00:21:49.270 align:middle line:90% 00:21:49.270 --> 00:21:53.530 align:middle line:84% I might get a MOSFET for a switch, a power 00:21:53.530 --> 00:21:58.283 align:middle line:84% MOSFET for a switch, and make my other switch a diode. 00:21:58.283 --> 00:22:03.532 align:middle line:90% 00:22:03.532 --> 00:22:06.840 align:middle line:84% And then I can turn on this switch 00:22:06.840 --> 00:22:08.700 align:middle line:90% by controlling its gate voltage. 00:22:08.700 --> 00:22:11.730 align:middle line:84% So I can modulate this switch on and off. 00:22:11.730 --> 00:22:13.950 align:middle line:84% And when I turn this switch off, the diode 00:22:13.950 --> 00:22:16.260 align:middle line:84% will carry the positive inductor current, 00:22:16.260 --> 00:22:18.080 align:middle line:90% and I can convert energy. 00:22:18.080 --> 00:22:20.130 align:middle line:84% And this is a converter that will 00:22:20.130 --> 00:22:22.460 align:middle line:84% let me clearly take energy from this side 00:22:22.460 --> 00:22:23.460 align:middle line:90% and put it to this side. 00:22:23.460 --> 00:22:24.730 align:middle line:90% I've got a resistor over here. 00:22:24.730 --> 00:22:30.280 align:middle line:90% 00:22:30.280 --> 00:22:34.720 align:middle line:84% And this, by the way, is known as a buck converter, 00:22:34.720 --> 00:22:37.690 align:middle line:84% if I haven't mentioned that, because it bucks the input 00:22:37.690 --> 00:22:42.250 align:middle line:84% voltage down, or it steps the input voltage down. 00:22:42.250 --> 00:22:44.830 align:middle line:84% Let's talk about what else I could 00:22:44.830 --> 00:22:49.030 align:middle line:84% do as another example of using these tools. 00:22:49.030 --> 00:22:52.125 align:middle line:84% Suppose I was to take this kind of converter. 00:22:52.125 --> 00:22:53.500 align:middle line:84% And let's forget about the switch 00:22:53.500 --> 00:22:54.820 align:middle line:90% implementation for a moment. 00:22:54.820 --> 00:22:56.560 align:middle line:84% But suppose I was to come over here 00:22:56.560 --> 00:23:02.560 align:middle line:84% and say, OK, I have a source over here and a load over here. 00:23:02.560 --> 00:23:05.410 align:middle line:84% What happens if I just switch those two things? 00:23:05.410 --> 00:23:06.620 align:middle line:90% I'll switch the two ports. 00:23:06.620 --> 00:23:08.880 align:middle line:84% I'll switch where the source is and where the load is. 00:23:08.880 --> 00:23:12.220 align:middle line:90% 00:23:12.220 --> 00:23:15.280 align:middle line:90% So let's draw that case. 00:23:15.280 --> 00:23:17.640 align:middle line:90% And I'll keep making the-- 00:23:17.640 --> 00:23:20.850 align:middle line:84% I'll make the left side v2 and the right side v1. 00:23:20.850 --> 00:23:23.280 align:middle line:90% So here we go. 00:23:23.280 --> 00:23:24.090 align:middle line:90% Here's c2. 00:23:24.090 --> 00:23:26.740 align:middle line:90% 00:23:26.740 --> 00:23:29.540 align:middle line:84% I put my resistor over here now-- 00:23:29.540 --> 00:23:30.040 align:middle line:90% v2. 00:23:30.040 --> 00:23:34.772 align:middle line:90% 00:23:34.772 --> 00:23:39.290 align:middle line:84% And I'll go back to my single-pole double-throw switch 00:23:39.290 --> 00:23:42.906 align:middle line:84% notation, like this, just for simplicity. 00:23:42.906 --> 00:23:47.510 align:middle line:90% 00:23:47.510 --> 00:23:50.270 align:middle line:90% Here's c1 one. 00:23:50.270 --> 00:23:52.400 align:middle line:90% And here's v1. 00:23:52.400 --> 00:23:55.090 align:middle line:84% And I'm going to make the same kind of assumption. 00:23:55.090 --> 00:24:00.450 align:middle line:84% Let me assume that the inductor is really big, 00:24:00.450 --> 00:24:04.910 align:middle line:84% the capacitors are really big, and the ripple current 00:24:04.910 --> 00:24:08.030 align:middle line:84% in the inductor and the ripple capacitor voltage 00:24:08.030 --> 00:24:10.080 align:middle line:90% are both small. 00:24:10.080 --> 00:24:12.050 align:middle line:84% The capacitor voltage ripple is small. 00:24:12.050 --> 00:24:15.240 align:middle line:90% 00:24:15.240 --> 00:24:18.725 align:middle line:84% I will, again, redefine this voltage vx here. 00:24:18.725 --> 00:24:22.590 align:middle line:90% 00:24:22.590 --> 00:24:26.950 align:middle line:84% But maybe just for fun, I'll redefine my switching function. 00:24:26.950 --> 00:24:29.340 align:middle line:84% You could have done it either way. 00:24:29.340 --> 00:24:32.308 align:middle line:84% In this case, I'm going to say q of t is equal to 1 00:24:32.308 --> 00:24:33.975 align:middle line:84% when the switch is in the down position. 00:24:33.975 --> 00:24:36.600 align:middle line:90% 00:24:36.600 --> 00:24:41.940 align:middle line:84% And q of t is equal to 0 when the switch 00:24:41.940 --> 00:24:43.770 align:middle line:90% is in the up position. 00:24:43.770 --> 00:24:45.670 align:middle line:84% Now, why am I choosing to define that way? 00:24:45.670 --> 00:24:47.350 align:middle line:84% I could do it either way I wanted. 00:24:47.350 --> 00:24:48.750 align:middle line:90% It doesn't matter. 00:24:48.750 --> 00:24:51.690 align:middle line:84% I happened to define it this way because in the kind of converter 00:24:51.690 --> 00:24:56.010 align:middle line:84% we're doing, that's the way you usually define it. 00:24:56.010 --> 00:24:58.590 align:middle line:84% So let's think about what happens here. 00:24:58.590 --> 00:25:01.950 align:middle line:84% I'm going to do the same thing I did before. 00:25:01.950 --> 00:25:04.380 align:middle line:90% So here's q of t. 00:25:04.380 --> 00:25:07.860 align:middle line:84% It's going to be 1 for some fraction of the time DT 00:25:07.860 --> 00:25:12.300 align:middle line:84% and 0 for the remainder of the cycle T and so forth. 00:25:12.300 --> 00:25:16.670 align:middle line:90% 00:25:16.670 --> 00:25:24.590 align:middle line:84% What's going to happen to vx in this case? 00:25:24.590 --> 00:25:26.520 align:middle line:90% vx is going to do the following. 00:25:26.520 --> 00:25:32.870 align:middle line:90% 00:25:32.870 --> 00:25:36.760 align:middle line:84% When the switch is in the one position, vx is 0. 00:25:36.760 --> 00:25:42.320 align:middle line:90% 00:25:42.320 --> 00:25:45.620 align:middle line:84% When q of t is 0 and the switch is in the up position, vx is V2. 00:25:45.620 --> 00:25:48.520 align:middle line:90% 00:25:48.520 --> 00:25:51.570 align:middle line:84% So it's going to switch oppositely this time. 00:25:51.570 --> 00:25:54.520 align:middle line:90% 00:25:54.520 --> 00:25:57.220 align:middle line:90% And this is V2. 00:25:57.220 --> 00:26:00.050 align:middle line:90% 00:26:00.050 --> 00:26:03.015 align:middle line:84% And again, I'm assuming c2 is really big so that V2 00:26:03.015 --> 00:26:04.390 align:middle line:84% has no ripple, and I can write it 00:26:04.390 --> 00:26:06.855 align:middle line:84% as a capital letter-- that is, as a DC quantity. 00:26:06.855 --> 00:26:09.760 align:middle line:90% 00:26:09.760 --> 00:26:12.610 align:middle line:90% So what can I say here? 00:26:12.610 --> 00:26:16.610 align:middle line:84% What do I know about the average voltage vx? 00:26:16.610 --> 00:26:23.484 align:middle line:90% 00:26:23.484 --> 00:26:24.398 align:middle line:90% AUDIENCE: It's v1. 00:26:24.398 --> 00:26:25.773 align:middle line:84% DAVID PERREAULT: It has to be v1. 00:26:25.773 --> 00:26:26.497 align:middle line:90% Why? 00:26:26.497 --> 00:26:28.080 align:middle line:84% Let me define-- this time, I'll define 00:26:28.080 --> 00:26:29.710 align:middle line:90% the inductor voltage this way. 00:26:29.710 --> 00:26:34.080 align:middle line:84% Here's v sub L. So if I have v1 minus v sub L minus v sub 00:26:34.080 --> 00:26:37.290 align:middle line:84% x is 0, the average value, capital V1, 00:26:37.290 --> 00:26:40.320 align:middle line:84% minus the average value of vL minus the average value of vx 00:26:40.320 --> 00:26:41.220 align:middle line:90% is 0. 00:26:41.220 --> 00:26:43.140 align:middle line:90% The average value of vL is 0. 00:26:43.140 --> 00:26:46.560 align:middle line:84% So hence, v1 must equal vx, on average. 00:26:46.560 --> 00:26:51.330 align:middle line:84% So that means that if I was to plot the-- 00:26:51.330 --> 00:26:54.390 align:middle line:90% I'm sorry, this is vx-- 00:26:54.390 --> 00:26:58.830 align:middle line:84% if I was to find the average value of vx, 00:26:58.830 --> 00:27:02.880 align:middle line:84% the average value of vx is equal to-- 00:27:02.880 --> 00:27:14.910 align:middle line:84% if this is DT and this is T, it must be equal to 1 minus D V2. 00:27:14.910 --> 00:27:17.440 align:middle line:90% And it's exactly what's said. 00:27:17.440 --> 00:27:22.170 align:middle line:84% This also ends up having to equal-- 00:27:22.170 --> 00:27:29.470 align:middle line:84% the average value of vx, which we know is 1 minus D V2 00:27:29.470 --> 00:27:31.900 align:middle line:90% also has to be equal to v1. 00:27:31.900 --> 00:27:42.340 align:middle line:84% Or I could rewrite this as V2 is equal to V1 over 1 minus D. 00:27:42.340 --> 00:27:44.740 align:middle line:84% And remember, D is some duty ratio. 00:27:44.740 --> 00:27:45.790 align:middle line:90% It's a fraction of time. 00:27:45.790 --> 00:27:50.255 align:middle line:84% So I have 0 is less than D is less than 1. 00:27:50.255 --> 00:27:52.890 align:middle line:90% 00:27:52.890 --> 00:27:56.820 align:middle line:84% Does that make sense to everybody? 00:27:56.820 --> 00:28:00.430 align:middle line:90% So what does this converter do? 00:28:00.430 --> 00:28:05.780 align:middle line:84% Well, that means that if 0 is less than D is less than 1, 00:28:05.780 --> 00:28:11.060 align:middle line:84% that means that if D was equal to 0, V2 would be in V1. 00:28:11.060 --> 00:28:13.160 align:middle line:84% And as I make D bigger and bigger and bigger, 00:28:13.160 --> 00:28:15.550 align:middle line:84% this denominator gets smaller and smaller and smaller. 00:28:15.550 --> 00:28:17.870 align:middle line:84% And V2 gets bigger, bigger and bigger and bigger. 00:28:17.870 --> 00:28:26.650 align:middle line:84% So what I get is v1 is less than v2 is less than infinity. 00:28:26.650 --> 00:28:30.790 align:middle line:84% What this says is this version of the circuit, 00:28:30.790 --> 00:28:34.990 align:middle line:84% I can make v2 anything bigger than v1 00:28:34.990 --> 00:28:40.630 align:middle line:84% that I want in periodic steady state. 00:28:40.630 --> 00:28:41.840 align:middle line:90% That's kind of interesting. 00:28:41.840 --> 00:28:43.660 align:middle line:90% This is no longer-- 00:28:43.660 --> 00:28:46.720 align:middle line:84% before, I picture this circuit over here 00:28:46.720 --> 00:28:49.360 align:middle line:84% as something where I synthesize an average value, 00:28:49.360 --> 00:28:50.720 align:middle line:90% and I just filter it. 00:28:50.720 --> 00:28:53.240 align:middle line:84% That's not quite what's happening over here. 00:28:53.240 --> 00:28:58.120 align:middle line:84% Or maybe it's not as simple to think about it that way. 00:28:58.120 --> 00:29:01.390 align:middle line:84% But I can get this conversion result. 00:29:01.390 --> 00:29:05.200 align:middle line:84% I could also figure out what it's doing about the currents. 00:29:05.200 --> 00:29:08.920 align:middle line:84% So if I said, OK, I have i1 over here. 00:29:08.920 --> 00:29:13.360 align:middle line:84% And, on average, this would be i1 approximately 00:29:13.360 --> 00:29:15.160 align:middle line:90% equal to capital I1. 00:29:15.160 --> 00:29:21.817 align:middle line:84% And I have i2 over here capital I2, what should I have? 00:29:21.817 --> 00:29:23.650 align:middle line:84% Well, I think what I'm going to end up with, 00:29:23.650 --> 00:29:32.360 align:middle line:84% it has to be that I2 ought to equal I1 times 1 minus D. 00:29:32.360 --> 00:29:41.250 align:middle line:84% Because that means that V2 times I2 ought to equal V1 times I1. 00:29:41.250 --> 00:29:45.430 align:middle line:84% So I can infer this current conversion ratio. 00:29:45.430 --> 00:29:48.690 align:middle line:84% So the output voltage, V2, has to be bigger 00:29:48.690 --> 00:29:50.250 align:middle line:90% than the input voltage. 00:29:50.250 --> 00:29:54.270 align:middle line:84% And the output current, I2, has to be smaller than the input 00:29:54.270 --> 00:29:58.290 align:middle line:90% current to get 100% efficiency. 00:29:58.290 --> 00:30:01.200 align:middle line:90% Any questions about that? 00:30:01.200 --> 00:30:06.560 align:middle line:90% 00:30:06.560 --> 00:30:08.990 align:middle line:90% What makes this work? 00:30:08.990 --> 00:30:13.400 align:middle line:84% Well, I'm going to make an assumption here. 00:30:13.400 --> 00:30:17.150 align:middle line:84% And I've kind of been assuming this, that v1 is greater than 0, 00:30:17.150 --> 00:30:19.140 align:middle line:84% and v2 is greater than 0, and so forth. 00:30:19.140 --> 00:30:23.930 align:middle line:84% If we looked at building this converter just the way I looked 00:30:23.930 --> 00:30:25.760 align:middle line:84% at implementing this other converter, 00:30:25.760 --> 00:30:27.860 align:middle line:84% the way you would typically implement this thing 00:30:27.860 --> 00:30:29.360 align:middle line:90% is like this. 00:30:29.360 --> 00:30:42.890 align:middle line:84% You would perhaps come and say, OK, here is v2. 00:30:42.890 --> 00:30:44.960 align:middle line:84% And what I would usually do is I'd 00:30:44.960 --> 00:30:48.820 align:middle line:90% build my switch pair like this. 00:30:48.820 --> 00:30:51.220 align:middle line:84% I'd make the bottom switch a MOSFET 00:30:51.220 --> 00:30:52.860 align:middle line:90% and the top switch a diode. 00:30:52.860 --> 00:30:58.520 align:middle line:90% 00:30:58.520 --> 00:31:01.990 align:middle line:90% And I put v1 over here. 00:31:01.990 --> 00:31:07.630 align:middle line:84% And basically, the reason I made the switching function q of t 00:31:07.630 --> 00:31:10.750 align:middle line:84% that way is because this switch is now the, quote unquote, 00:31:10.750 --> 00:31:14.860 align:middle line:84% "active" switch, or in the simplest implementation, 00:31:14.860 --> 00:31:17.030 align:middle line:90% this is the active switch. 00:31:17.030 --> 00:31:21.250 align:middle line:84% And I switch these two switches, so when the switch turns on, 00:31:21.250 --> 00:31:23.320 align:middle line:84% the diode turns off, and vice versa. 00:31:23.320 --> 00:31:26.110 align:middle line:84% And v2 will grow to be bigger than v1. 00:31:26.110 --> 00:31:29.442 align:middle line:84% So if I need a big voltage, that's the way-- 00:31:29.442 --> 00:31:31.900 align:middle line:84% or I have a small battery, and I need a much higher voltage 00:31:31.900 --> 00:31:34.810 align:middle line:84% to run my stuff, this is a great converter to do it. 00:31:34.810 --> 00:31:42.640 align:middle line:84% For that reason, it's called a boost converter 00:31:42.640 --> 00:31:44.530 align:middle line:84% because it can take a small input voltage 00:31:44.530 --> 00:31:48.000 align:middle line:84% and boost it to a bigger output voltage. 00:31:48.000 --> 00:31:51.910 align:middle line:84% In this case, my source is on the right, 00:31:51.910 --> 00:31:54.110 align:middle line:84% and my load resistor is on the left. 00:31:54.110 --> 00:31:56.050 align:middle line:84% So you can infer that power is going 00:31:56.050 --> 00:31:58.720 align:middle line:90% to flow from right to left. 00:31:58.720 --> 00:32:03.460 align:middle line:84% Very often, people like to draw these things with power flowing 00:32:03.460 --> 00:32:05.860 align:middle line:84% left to right, maybe because that's the way people read 00:32:05.860 --> 00:32:07.058 align:middle line:90% or I don't know. 00:32:07.058 --> 00:32:08.600 align:middle line:84% There's nothing fundamental about it. 00:32:08.600 --> 00:32:09.940 align:middle line:90% You can do it any way you want. 00:32:09.940 --> 00:32:13.450 align:middle line:84% But it might look a little bit more familiar to some people 00:32:13.450 --> 00:32:18.510 align:middle line:84% if you drew it the other way, like this. 00:32:18.510 --> 00:32:30.320 align:middle line:90% 00:32:30.320 --> 00:32:32.190 align:middle line:90% And we have a boost converter. 00:32:32.190 --> 00:32:38.490 align:middle line:90% 00:32:38.490 --> 00:32:44.080 align:middle line:84% Let's think about what this converter is really doing. 00:32:44.080 --> 00:32:47.400 align:middle line:84% How is this thing working, really? 00:32:47.400 --> 00:32:50.906 align:middle line:84% What's going on with its operation? 00:32:50.906 --> 00:32:53.840 align:middle line:90% 00:32:53.840 --> 00:32:57.660 align:middle line:84% We can look at it from a couple of perspectives. 00:32:57.660 --> 00:32:59.940 align:middle line:90% One way to look at it is this. 00:32:59.940 --> 00:33:06.092 align:middle line:84% I'm showing you the voltage vx here. 00:33:06.092 --> 00:33:12.420 align:middle line:84% Why don't I look at the current iL. 00:33:12.420 --> 00:33:18.860 align:middle line:84% And I'm going to draw the current iL this way 00:33:18.860 --> 00:33:21.290 align:middle line:90% in this example. 00:33:21.290 --> 00:33:25.920 align:middle line:84% So when the switch is in the down position, what happens? 00:33:25.920 --> 00:33:30.816 align:middle line:84% v1 is, basically, applied across the inductor. 00:33:30.816 --> 00:33:33.620 align:middle line:84% So if v1's is applied across the inductor, 00:33:33.620 --> 00:33:36.630 align:middle line:84% what happens to the inductor current? 00:33:36.630 --> 00:33:38.940 align:middle line:84% We've said the ripple is small, but let's 00:33:38.940 --> 00:33:41.193 align:middle line:84% just now not think this inductor's infinitely big. 00:33:41.193 --> 00:33:43.110 align:middle line:84% What's going to happen to the inductor current 00:33:43.110 --> 00:33:44.130 align:middle line:90% during that time? 00:33:44.130 --> 00:33:47.460 align:middle line:84% Well, if I'm applying a positive voltage across the inductor, 00:33:47.460 --> 00:33:51.150 align:middle line:90% I gotta have a constant di/dt. 00:33:51.150 --> 00:33:54.690 align:middle line:84% So in that first part of the cycle, 00:33:54.690 --> 00:33:56.780 align:middle line:90% if I were to plot i sub L-- 00:33:56.780 --> 00:33:59.870 align:middle line:84% and this is, I should say i sub L of t. 00:33:59.870 --> 00:34:01.610 align:middle line:90% I'm looking at its ripple now. 00:34:01.610 --> 00:34:02.930 align:middle line:90% I'm including its ripple. 00:34:02.930 --> 00:34:04.970 align:middle line:84% In the first part of the cycle, when the switch 00:34:04.970 --> 00:34:07.700 align:middle line:84% is in the down position, I'm applying a voltage 00:34:07.700 --> 00:34:09.199 align:middle line:90% across the inductor. 00:34:09.199 --> 00:34:11.639 align:middle line:84% The inductor current is ramping up. 00:34:11.639 --> 00:34:13.310 align:middle line:90% What does that mean? 00:34:13.310 --> 00:34:17.060 align:middle line:84% I'm storing energy in the inductor. 00:34:17.060 --> 00:34:19.460 align:middle line:84% I'm taking energy out of the input. 00:34:19.460 --> 00:34:24.139 align:middle line:84% I have a positive current coming out of the input. 00:34:24.139 --> 00:34:27.500 align:middle line:84% And it's basically going into this inductor. 00:34:27.500 --> 00:34:30.920 align:middle line:90% So power is being drawn from v1. 00:34:30.920 --> 00:34:32.219 align:middle line:90% I'm charging up the inductor. 00:34:32.219 --> 00:34:36.750 align:middle line:90% I had 1/2 Li small squared here. 00:34:36.750 --> 00:34:41.360 align:middle line:84% Now I have 1/2 Li slightly bigger squared here. 00:34:41.360 --> 00:34:43.329 align:middle line:84% So I put energy into this inductor. 00:34:43.329 --> 00:34:45.860 align:middle line:90% 00:34:45.860 --> 00:34:48.139 align:middle line:84% What happens in the second half of the cycle? 00:34:48.139 --> 00:34:51.290 align:middle line:84% In the second half of the cycle, I 00:34:51.290 --> 00:34:57.400 align:middle line:84% reconnect the other end of the inductor to the v2. 00:34:57.400 --> 00:35:00.000 align:middle line:84% Now, is v2 bigger or smaller than v1? 00:35:00.000 --> 00:35:04.050 align:middle line:90% 00:35:04.050 --> 00:35:04.710 align:middle line:90% Bigger, right? 00:35:04.710 --> 00:35:06.730 align:middle line:84% Because we said this is a boost converter. 00:35:06.730 --> 00:35:08.160 align:middle line:90% It makes a big v2. 00:35:08.160 --> 00:35:10.650 align:middle line:84% That means, in the second part of the cycle, 00:35:10.650 --> 00:35:13.680 align:middle line:84% when I have the switch in the up position, 00:35:13.680 --> 00:35:18.360 align:middle line:84% the voltage across the inductor ought to be negative. 00:35:18.360 --> 00:35:21.450 align:middle line:84% And so that means I ought to have a negative constant voltage 00:35:21.450 --> 00:35:25.950 align:middle line:84% across the inductor, and the current ought to ramp down. 00:35:25.950 --> 00:35:28.020 align:middle line:84% So in the second part of the cycle, 00:35:28.020 --> 00:35:31.560 align:middle line:84% the current's going to ramp down like this. 00:35:31.560 --> 00:35:35.540 align:middle line:90% And so what does that mean? 00:35:35.540 --> 00:35:37.900 align:middle line:84% If the current in the inductor is decreasing, 00:35:37.900 --> 00:35:40.970 align:middle line:84% is energy going into the inductor or out of the inductor? 00:35:40.970 --> 00:35:42.360 align:middle line:90% AUDIENCE: Out. 00:35:42.360 --> 00:35:44.610 align:middle line:84% DAVID PERREAULT: Energy is coming out of the inductor. 00:35:44.610 --> 00:35:45.770 align:middle line:90% So that's true. 00:35:45.770 --> 00:35:48.090 align:middle line:84% So we're taking it-- in the first part of the cycle, 00:35:48.090 --> 00:35:50.432 align:middle line:90% we put energy into the inductor. 00:35:50.432 --> 00:35:51.890 align:middle line:84% The second part of the cycle, we're 00:35:51.890 --> 00:35:53.465 align:middle line:84% taking energy out of the inductor. 00:35:53.465 --> 00:35:55.770 align:middle line:84% Well, where is that energy going? 00:35:55.770 --> 00:35:58.550 align:middle line:84% Well, it's got to be going to the output. 00:35:58.550 --> 00:36:01.660 align:middle line:84% I have current flowing this way into the output. 00:36:01.660 --> 00:36:04.520 align:middle line:90% 00:36:04.520 --> 00:36:06.530 align:middle line:84% The other thing that's true is, even 00:36:06.530 --> 00:36:08.960 align:middle line:84% in the second part of the cycle, there's still current 00:36:08.960 --> 00:36:13.520 align:middle line:84% coming from the input through the inductor to the output. 00:36:13.520 --> 00:36:17.693 align:middle line:84% So I also have power flowing from the input to the output. 00:36:17.693 --> 00:36:19.110 align:middle line:84% So in the first half of the cycle, 00:36:19.110 --> 00:36:20.527 align:middle line:84% I'm taking energy out of the input 00:36:20.527 --> 00:36:22.560 align:middle line:90% and putting it in the inductor. 00:36:22.560 --> 00:36:24.120 align:middle line:84% In the second part of the cycle, I'm 00:36:24.120 --> 00:36:26.280 align:middle line:84% taking energy out of the inductor, 00:36:26.280 --> 00:36:27.960 align:middle line:84% plus more energy out of the input, 00:36:27.960 --> 00:36:30.070 align:middle line:90% and putting it to the output. 00:36:30.070 --> 00:36:33.760 align:middle line:84% And it just works out that when I do this periodic steady-state 00:36:33.760 --> 00:36:37.690 align:middle line:84% analysis thing, what this thing has to do when it settles down 00:36:37.690 --> 00:36:40.120 align:middle line:84% such that the end of the cycle is just like the beginning 00:36:40.120 --> 00:36:43.330 align:middle line:84% of the cycle and the next cycle looks exactly the same-- 00:36:43.330 --> 00:36:44.590 align:middle line:90% does the same thing-- 00:36:44.590 --> 00:36:46.690 align:middle line:84% that the output voltage has to be bigger 00:36:46.690 --> 00:36:49.710 align:middle line:90% than the input voltage. 00:36:49.710 --> 00:36:54.110 align:middle line:84% Now, I don't know about you, but when I first saw that, 00:36:54.110 --> 00:36:55.880 align:middle line:90% I thought it was kind of cool. 00:36:55.880 --> 00:36:57.790 align:middle line:84% It's very easy to make a voltage divider. 00:36:57.790 --> 00:37:00.600 align:middle line:84% This is kind of like a voltage undivider. 00:37:00.600 --> 00:37:02.005 align:middle line:90% It makes bigger voltages. 00:37:02.005 --> 00:37:05.140 align:middle line:84% Well, let's take a look at this in practice. 00:37:05.140 --> 00:37:12.660 align:middle line:84% Montsy set us up a nice demo of an actual boost converter. 00:37:12.660 --> 00:37:14.216 align:middle line:90% And we'll get to see-- 00:37:14.216 --> 00:37:16.476 align:middle line:90% [HUMS] 00:37:16.476 --> 00:37:24.620 align:middle line:90% 00:37:24.620 --> 00:37:27.520 align:middle line:90% --a converter running. 00:37:27.520 --> 00:37:29.740 align:middle line:84% By the way, this particular converter 00:37:29.740 --> 00:37:32.710 align:middle line:84% was designed by an undergraduate at MIT who is now a famous power 00:37:32.710 --> 00:37:39.700 align:middle line:84% electronics designer, as well as a high school student, so pretty 00:37:39.700 --> 00:37:42.586 align:middle line:84% early design for the high school student. 00:37:42.586 --> 00:37:47.600 align:middle line:84% So let's see if we can get waveforms up here. 00:37:47.600 --> 00:37:51.180 align:middle line:84% So what do we see on the screen up here? 00:37:51.180 --> 00:37:54.660 align:middle line:84% My orange, which is the inductor current, 00:37:54.660 --> 00:37:58.120 align:middle line:84% is actually the deep blue waveform on the right. 00:37:58.120 --> 00:38:01.120 align:middle line:84% The yellow is exactly what you're seeing. 00:38:01.120 --> 00:38:05.400 align:middle line:84% It's the voltage across the bottom switch. 00:38:05.400 --> 00:38:09.510 align:middle line:84% And the input voltage is the light blue. 00:38:09.510 --> 00:38:12.750 align:middle line:84% And the output voltage is the purple. 00:38:12.750 --> 00:38:17.310 align:middle line:84% So what you can see is that we are-- 00:38:17.310 --> 00:38:20.310 align:middle line:84% and we don't have all the waveforms quite 00:38:20.310 --> 00:38:23.550 align:middle line:84% lined up here in terms of the magnitudes 00:38:23.550 --> 00:38:25.520 align:middle line:90% and where the voltages are. 00:38:25.520 --> 00:38:26.815 align:middle line:90% But we can do that. 00:38:26.815 --> 00:38:33.530 align:middle line:90% 00:38:33.530 --> 00:38:34.260 align:middle line:90% All right. 00:38:34.260 --> 00:38:37.130 align:middle line:84% But the point is, in the first part of the cycle, 00:38:37.130 --> 00:38:40.670 align:middle line:84% I grab energy from the input, and I put it in the inductor. 00:38:40.670 --> 00:38:42.190 align:middle line:90% Inductor current ramps up. 00:38:42.190 --> 00:38:44.025 align:middle line:84% Second part of the cycle, that energy, 00:38:44.025 --> 00:38:45.650 align:middle line:84% and more energy from the input's, being 00:38:45.650 --> 00:38:47.090 align:middle line:90% thrown to the output. 00:38:47.090 --> 00:38:48.680 align:middle line:90% And I rinse and repeat that. 00:38:48.680 --> 00:38:52.730 align:middle line:84% And the thing that lets me get the voltage higher at the output 00:38:52.730 --> 00:38:55.820 align:middle line:84% than the input, outside of the fact that the math says so, 00:38:55.820 --> 00:39:01.100 align:middle line:84% is that when I turn the bottom switch off, 00:39:01.100 --> 00:39:04.280 align:middle line:84% when I turn the switch from here to here, 00:39:04.280 --> 00:39:05.930 align:middle line:84% or if I turn the bottom switch off, 00:39:05.930 --> 00:39:08.780 align:middle line:84% this inductor current doesn't want to go to 0. 00:39:08.780 --> 00:39:10.340 align:middle line:90% And v is equal to L di/dt. 00:39:10.340 --> 00:39:11.870 align:middle line:84% The inductor will generate whatever 00:39:11.870 --> 00:39:14.850 align:middle line:84% voltage it has to to keep the current flowing. 00:39:14.850 --> 00:39:17.420 align:middle line:84% And so what it does is it generates 00:39:17.420 --> 00:39:22.800 align:middle line:84% enough voltage to deliver energy to the top of the device. 00:39:22.800 --> 00:39:26.270 align:middle line:90% I could-- let's see, do I have-- 00:39:26.270 --> 00:39:31.610 align:middle line:84% if I looked at this waveform, if I turn this switch off 00:39:31.610 --> 00:39:35.400 align:middle line:84% and I have a positive current going this way, 00:39:35.400 --> 00:39:38.040 align:middle line:84% even if the diode's off, this inductor 00:39:38.040 --> 00:39:40.500 align:middle line:84% is going to generate a big negative voltage 00:39:40.500 --> 00:39:42.660 align:middle line:90% to try to keep current flowing. 00:39:42.660 --> 00:39:44.460 align:middle line:84% And it's going to generate voltage 00:39:44.460 --> 00:39:47.800 align:middle line:90% until it turns this diode on. 00:39:47.800 --> 00:39:50.430 align:middle line:84% So current can then keep flowing from this lower input 00:39:50.430 --> 00:39:52.350 align:middle line:84% voltage to the higher output voltage 00:39:52.350 --> 00:39:58.170 align:middle line:84% via this negative voltage across the inductor. 00:39:58.170 --> 00:40:02.690 align:middle line:84% So this ability of the inductor to force current flow 00:40:02.690 --> 00:40:05.600 align:middle line:84% is what lets me generate the output voltage that's 00:40:05.600 --> 00:40:07.920 align:middle line:90% higher than the input voltage. 00:40:07.920 --> 00:40:09.510 align:middle line:90% Yeah? 00:40:09.510 --> 00:40:11.510 align:middle line:84% AUDIENCE: When you first turn on the [INAUDIBLE] 00:40:11.510 --> 00:40:13.997 align:middle line:84% why does the current ramp linearly when you touch it? 00:40:13.997 --> 00:40:14.830 align:middle line:90% DAVID PERREAULT: Ah. 00:40:14.830 --> 00:40:17.993 align:middle line:84% Because if I have this bottom switch on-- that's 00:40:17.993 --> 00:40:18.910 align:middle line:90% an excellent question. 00:40:18.910 --> 00:40:23.220 align:middle line:84% Why does the current ramp up linearly in the inductor? 00:40:23.220 --> 00:40:27.430 align:middle line:84% If this switch is on, I've imposed 0 voltage here. 00:40:27.430 --> 00:40:31.020 align:middle line:90% I've got v1 across the inductor. 00:40:31.020 --> 00:40:34.143 align:middle line:90% So I got v is equal to L di/dt. 00:40:34.143 --> 00:40:37.610 align:middle line:90% 00:40:37.610 --> 00:40:44.390 align:middle line:84% That means that di/dt is equal to 1 00:40:44.390 --> 00:40:48.200 align:middle line:84% over L times v, which, in this case, is v1. 00:40:48.200 --> 00:40:51.620 align:middle line:84% So the slope of that current is simply the input voltage 00:40:51.620 --> 00:40:53.350 align:middle line:90% divided by the inductance. 00:40:53.350 --> 00:40:56.530 align:middle line:84% If the inductance is big, that slope is small, 00:40:56.530 --> 00:41:00.030 align:middle line:84% but I'm nonetheless putting energy in the inductor. 00:41:00.030 --> 00:41:01.320 align:middle line:90% Yeah? 00:41:01.320 --> 00:41:02.960 align:middle line:84% AUDIENCE: I have a follow-up question. 00:41:02.960 --> 00:41:05.960 align:middle line:90% 00:41:05.960 --> 00:41:10.900 align:middle line:84% How do the slopes know to change with the duty cycle? 00:41:10.900 --> 00:41:12.310 align:middle line:84% It seems like magic that it stays 00:41:12.310 --> 00:41:13.518 align:middle line:90% in the periodic steady state. 00:41:13.518 --> 00:41:15.948 align:middle line:90% 00:41:15.948 --> 00:41:17.990 align:middle line:84% DAVID PERREAULT: First of all, not all converters 00:41:17.990 --> 00:41:22.140 align:middle line:84% that you could conceive of will reach steady state. 00:41:22.140 --> 00:41:25.573 align:middle line:84% And this kind of average analysis we're doing 00:41:25.573 --> 00:41:27.240 align:middle line:84% is one of the things you use to do that. 00:41:27.240 --> 00:41:29.930 align:middle line:84% So I've seen converters where people had some conversion idea, 00:41:29.930 --> 00:41:33.200 align:middle line:84% but it turns out that, no matter what you do in the converter, 00:41:33.200 --> 00:41:34.637 align:middle line:84% it has a positive average voltage 00:41:34.637 --> 00:41:37.220 align:middle line:84% across the inductor, which means the inductor current ramps up 00:41:37.220 --> 00:41:39.795 align:middle line:90% until something blows up. 00:41:39.795 --> 00:41:45.200 align:middle line:84% What happens, in this case, is suppose the output voltage 00:41:45.200 --> 00:41:46.675 align:middle line:90% was too small. 00:41:46.675 --> 00:41:49.210 align:middle line:90% 00:41:49.210 --> 00:41:52.900 align:middle line:84% Well, in that case, then, the inductor current 00:41:52.900 --> 00:41:57.570 align:middle line:84% would ramp up with v1 over L slope. 00:41:57.570 --> 00:42:00.360 align:middle line:84% It would ramp down but not with as great a slope, 00:42:00.360 --> 00:42:04.130 align:middle line:84% and the inductor current would keep increasing. 00:42:04.130 --> 00:42:07.050 align:middle line:84% And it would start to pump more and more charge to the output 00:42:07.050 --> 00:42:10.500 align:middle line:84% until the output voltage comes up. 00:42:10.500 --> 00:42:14.630 align:middle line:84% So what happens is there's a feedback 00:42:14.630 --> 00:42:18.740 align:middle line:84% loop that's driving the inductor current to keep increasing. 00:42:18.740 --> 00:42:22.130 align:middle line:84% And that drives the output voltage up further, 00:42:22.130 --> 00:42:25.000 align:middle line:84% for a given resistance, until you hit some equilibrium. 00:42:25.000 --> 00:42:26.750 align:middle line:84% And that equilibrium is the output voltage 00:42:26.750 --> 00:42:30.410 align:middle line:84% is now big enough that the negative volt 00:42:30.410 --> 00:42:31.970 align:middle line:84% seconds on the inductor is the same 00:42:31.970 --> 00:42:35.340 align:middle line:90% as the positive volt seconds. 00:42:35.340 --> 00:42:38.760 align:middle line:84% So will all converters that you could think of, 00:42:38.760 --> 00:42:41.220 align:middle line:84% all combinations of switches, inductors, and capacitors 00:42:41.220 --> 00:42:41.800 align:middle line:90% do this? 00:42:41.800 --> 00:42:42.600 align:middle line:90% No. 00:42:42.600 --> 00:42:44.670 align:middle line:90% Some will just blow up on you. 00:42:44.670 --> 00:42:47.910 align:middle line:84% Others will come to a nice, happy 00:42:47.910 --> 00:42:49.840 align:middle line:90% periodic steady-state operation. 00:42:49.840 --> 00:42:53.540 align:middle line:84% And that's usually the kind of converters we aim for. 00:42:53.540 --> 00:42:55.450 align:middle line:90% Excellent question. 00:42:55.450 --> 00:42:56.540 align:middle line:90% Other questions? 00:42:56.540 --> 00:42:58.778 align:middle line:90% 00:42:58.778 --> 00:43:00.570 align:middle line:84% AUDIENCE: Theoretically, does the frequency 00:43:00.570 --> 00:43:02.112 align:middle line:84% at which you switch make a difference 00:43:02.112 --> 00:43:04.010 align:middle line:90% if your duty cycle's the same. 00:43:04.010 --> 00:43:06.380 align:middle line:84% DAVID PERREAULT: That's an excellent question. 00:43:06.380 --> 00:43:08.360 align:middle line:84% The question is, does the frequency 00:43:08.360 --> 00:43:10.190 align:middle line:84% make a difference as long as you're 00:43:10.190 --> 00:43:11.340 align:middle line:90% doing the right duty cycle? 00:43:11.340 --> 00:43:13.850 align:middle line:84% And the answer is, on the one hand, no. 00:43:13.850 --> 00:43:19.010 align:middle line:84% Nothing about this said what the period capital 00:43:19.010 --> 00:43:22.960 align:middle line:90% T is in this mode of operation. 00:43:22.960 --> 00:43:31.390 align:middle line:84% What does change is, however, this ripple. 00:43:31.390 --> 00:43:35.492 align:middle line:84% So if I made T bigger, I'd get more current ripple. 00:43:35.492 --> 00:43:37.075 align:middle line:84% And I started off the analysis saying, 00:43:37.075 --> 00:43:39.380 align:middle line:84% oh, I'm going to assume all my ripples are small. 00:43:39.380 --> 00:43:44.680 align:middle line:84% Well, how big L has to be in order for my ripple to be, 00:43:44.680 --> 00:43:47.690 align:middle line:84% quote unquote, "small" depends upon the period. 00:43:47.690 --> 00:43:50.290 align:middle line:84% So we'll talk a lot about, how do I pick my period, 00:43:50.290 --> 00:43:52.700 align:middle line:84% how do I pick my inductors, all those things. 00:43:52.700 --> 00:43:56.230 align:middle line:84% So it matters in the practical sense 00:43:56.230 --> 00:43:58.100 align:middle line:84% but, if I had big enough components, 00:43:58.100 --> 00:44:01.730 align:middle line:90% not in the theoretical sense. 00:44:01.730 --> 00:44:05.520 align:middle line:90% Any other questions? 00:44:05.520 --> 00:44:09.065 align:middle line:84% Let me just note a couple of other things. 00:44:09.065 --> 00:44:13.165 align:middle line:90% 00:44:13.165 --> 00:44:14.540 align:middle line:84% And what I'm about to tell you is 00:44:14.540 --> 00:44:17.370 align:middle line:84% completely unimportant from a practical perspective. 00:44:17.370 --> 00:44:20.820 align:middle line:84% But it's an interesting thing to observe. 00:44:20.820 --> 00:44:24.190 align:middle line:84% Let me come back to this buck converter here. 00:44:24.190 --> 00:44:26.100 align:middle line:84% This is a stepping down converter. 00:44:26.100 --> 00:44:30.240 align:middle line:84% If I asked, What is the actual thing 00:44:30.240 --> 00:44:33.720 align:middle line:90% that this switch is doing-- 00:44:33.720 --> 00:44:38.910 align:middle line:84% this has some v switch and i switch-- 00:44:38.910 --> 00:44:41.873 align:middle line:84% it turns out that if you go analyze this-- 00:44:41.873 --> 00:44:43.290 align:middle line:84% and I put it in the lecture notes, 00:44:43.290 --> 00:44:45.540 align:middle line:84% even though I'm not going to go through it in detail-- 00:44:45.540 --> 00:44:48.450 align:middle line:84% if you turn around and ask, What is this switch physically doing, 00:44:48.450 --> 00:44:53.400 align:middle line:84% it's absorbing power in DC and voltage and current components 00:44:53.400 --> 00:44:54.785 align:middle line:90% on the switch. 00:44:54.785 --> 00:44:56.160 align:middle line:84% So the current through the switch 00:44:56.160 --> 00:44:58.190 align:middle line:84% has a DC term and a bunch of AC terms. 00:44:58.190 --> 00:45:00.695 align:middle line:90% 00:45:00.695 --> 00:45:02.500 align:middle line:90% And so does the voltage. 00:45:02.500 --> 00:45:06.460 align:middle line:84% It turns out if I look to the DC voltage times the DC current, 00:45:06.460 --> 00:45:10.210 align:middle line:84% the switch is absorbing power from the DC waveforms. 00:45:10.210 --> 00:45:15.070 align:middle line:84% And it's generating AC waveforms and the AC components 00:45:15.070 --> 00:45:16.350 align:middle line:90% of those waveforms. 00:45:16.350 --> 00:45:19.450 align:middle line:84% So in some sense, this switch, what it's really doing 00:45:19.450 --> 00:45:22.540 align:middle line:84% is it's taking power in DC waveforms 00:45:22.540 --> 00:45:25.120 align:middle line:84% and turning it into power to AC waveforms. 00:45:25.120 --> 00:45:28.120 align:middle line:84% If I ask what this switch was doing down here, 00:45:28.120 --> 00:45:32.060 align:middle line:84% this guy down here, he's actually doing the opposite. 00:45:32.060 --> 00:45:34.910 align:middle line:84% So I can think of this as, quote unquote, an "inverting switch." 00:45:34.910 --> 00:45:36.970 align:middle line:90% It takes DC and turns it into-- 00:45:36.970 --> 00:45:40.150 align:middle line:84% power in DC waveforms and turns it into power in AC waveforms. 00:45:40.150 --> 00:45:42.070 align:middle line:84% And this switch does the exact opposite 00:45:42.070 --> 00:45:44.090 align:middle line:84% and turns it back into DC waveforms. 00:45:44.090 --> 00:45:45.727 align:middle line:84% And between the two of them, they 00:45:45.727 --> 00:45:47.060 align:middle line:90% give you the voltage conversion. 00:45:47.060 --> 00:45:51.400 align:middle line:84% So the switch in our power converter could also be thought 00:45:51.400 --> 00:45:55.660 align:middle line:84% of as a funny kind of "change where the energy is 00:45:55.660 --> 00:45:59.000 align:middle line:84% in the waveforms it's processing" kind of device. 00:45:59.000 --> 00:46:01.810 align:middle line:84% Now, that's not necessarily important or helpful 00:46:01.810 --> 00:46:04.600 align:middle line:84% for analyzing power converters, but it's just 00:46:04.600 --> 00:46:07.180 align:middle line:84% an interesting fact to understand the function of what 00:46:07.180 --> 00:46:08.060 align:middle line:90% the switch is doing. 00:46:08.060 --> 00:46:11.320 align:middle line:84% It's moving energy around in frequency 00:46:11.320 --> 00:46:12.890 align:middle line:90% without losing any of it. 00:46:12.890 --> 00:46:15.410 align:middle line:90% 00:46:15.410 --> 00:46:21.470 align:middle line:84% The last thing I'll note is just that we can view-- and, again, 00:46:21.470 --> 00:46:23.690 align:middle line:84% you don't have to view things this way-- 00:46:23.690 --> 00:46:27.740 align:middle line:84% but we might view both the buck converter and the boost 00:46:27.740 --> 00:46:34.740 align:middle line:84% converter as two connections of one structure. 00:46:34.740 --> 00:46:37.250 align:middle line:84% All I did was, really, flip it around. 00:46:37.250 --> 00:46:40.190 align:middle line:84% But I could block this off and just 00:46:40.190 --> 00:46:49.330 align:middle line:84% think of this structure from here, here, and here-- 00:46:49.330 --> 00:46:51.550 align:middle line:84% I'm sorry, here, here, and here-- 00:46:51.550 --> 00:46:57.730 align:middle line:84% as being this cell that has one capacitor, one inductor, 00:46:57.730 --> 00:47:00.190 align:middle line:84% and one single-pole double-throw switch. 00:47:00.190 --> 00:47:04.130 align:middle line:84% And that's also buried into this circuit-- 00:47:04.130 --> 00:47:08.970 align:middle line:84% one inductor, one capacitor, and one single-pole double-throw 00:47:08.970 --> 00:47:10.080 align:middle line:90% switch. 00:47:10.080 --> 00:47:12.030 align:middle line:84% And the only reason I mention that is 00:47:12.030 --> 00:47:14.970 align:middle line:84% because you can build other kinds of converters that 00:47:14.970 --> 00:47:16.440 align:middle line:84% have different connections to this 00:47:16.440 --> 00:47:19.310 align:middle line:84% that will give you different power converter functions. 00:47:19.310 --> 00:47:23.010 align:middle line:84% And we will introduce one of those next class. 00:47:23.010 --> 00:47:26.790 align:middle line:84% So what I wanted you to take away from this class 00:47:26.790 --> 00:47:29.580 align:middle line:84% was, first of all, what are the kind of techniques 00:47:29.580 --> 00:47:31.950 align:middle line:84% that we can use to analyze power converters. 00:47:31.950 --> 00:47:35.340 align:middle line:84% There's average voltage, average current, average KVL, 00:47:35.340 --> 00:47:39.440 align:middle line:84% average KCL, inductor and capacitor constraints. 00:47:39.440 --> 00:47:41.190 align:middle line:84% And we can use that to say, How does power 00:47:41.190 --> 00:47:42.535 align:middle line:90% flow through the converters? 00:47:42.535 --> 00:47:44.910 align:middle line:84% And I wanted to start talking about some of the functions 00:47:44.910 --> 00:47:45.490 align:middle line:90% we can do. 00:47:45.490 --> 00:47:47.865 align:middle line:84% So we've seen now a buck converter and a boost converter. 00:47:47.865 --> 00:47:52.330 align:middle line:84% And next time, we'll move on and look at some other approaches. 00:47:52.330 --> 00:47:54.380 align:middle line:90% Have a great day. 00:47:54.380 --> 00:47:58.000 align:middle line:90%