WEBVTT 00:00:00.000 --> 00:00:05.324 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:05.324 --> 00:00:09.745 align:middle line:90% 00:00:09.745 --> 00:00:11.120 align:middle line:84% DAVID PERREAULT: So what I wanted 00:00:11.120 --> 00:00:17.510 align:middle line:84% to do for the next few lectures is talk a bit about control 00:00:17.510 --> 00:00:20.948 align:middle line:90% of power converters. 00:00:20.948 --> 00:00:24.650 align:middle line:84% And actually, I'm more going to talk about modeling 00:00:24.650 --> 00:00:27.290 align:middle line:84% of power converters than control of power converters 00:00:27.290 --> 00:00:32.840 align:middle line:84% because I can't really teach control theory in this class. 00:00:32.840 --> 00:00:35.777 align:middle line:84% If you haven't had a control class yet, 00:00:35.777 --> 00:00:38.360 align:middle line:84% it's really time to go out and take one because no matter what 00:00:38.360 --> 00:00:39.735 align:middle line:84% you're going to do, you're really 00:00:39.735 --> 00:00:43.970 align:middle line:84% probably going to need to understand feedback control. 00:00:43.970 --> 00:00:48.050 align:middle line:84% But what we can do in this class is show you 00:00:48.050 --> 00:00:52.520 align:middle line:84% how one can model power converters to put them-- 00:00:52.520 --> 00:00:55.730 align:middle line:84% to get a model in the format that you can use it for feedback 00:00:55.730 --> 00:00:58.850 align:middle line:90% control and do basic control. 00:00:58.850 --> 00:01:01.470 align:middle line:84% So let's think about a power converter. 00:01:01.470 --> 00:01:04.160 align:middle line:84% Let's start off with our friendly neighborhood boost 00:01:04.160 --> 00:01:05.300 align:middle line:90% converter. 00:01:05.300 --> 00:01:08.030 align:middle line:84% So maybe I have some input voltage VIN. 00:01:08.030 --> 00:01:11.880 align:middle line:90% 00:01:11.880 --> 00:01:18.200 align:middle line:84% I'll switch it with some switching waveform q of t. 00:01:18.200 --> 00:01:19.950 align:middle line:84% And then I'm going to have a diode. 00:01:19.950 --> 00:01:22.310 align:middle line:90% I will have a capacitor. 00:01:22.310 --> 00:01:23.940 align:middle line:90% And the converter ends here. 00:01:23.940 --> 00:01:29.340 align:middle line:84% And then somewhere, we usually show RL. 00:01:29.340 --> 00:01:30.530 align:middle line:90% And I have my input voltage. 00:01:30.530 --> 00:01:33.380 align:middle line:84% And I have my output voltage, VOUT. 00:01:33.380 --> 00:01:40.400 align:middle line:84% Now, I usually in my examples, just by way of an aside-- 00:01:40.400 --> 00:01:42.570 align:middle line:90% I usually show a resistive load. 00:01:42.570 --> 00:01:45.290 align:middle line:90% A lot of loads are resistive. 00:01:45.290 --> 00:01:49.220 align:middle line:84% But there's no guarantee that in an actual application, 00:01:49.220 --> 00:01:51.050 align:middle line:90% your load looks like a resistor. 00:01:51.050 --> 00:01:54.470 align:middle line:90% It could look like whatever. 00:01:54.470 --> 00:02:02.720 align:middle line:84% Maybe a lot of loads you might represent as some maybe Norton 00:02:02.720 --> 00:02:06.260 align:middle line:84% equivalent where there's some constant current component 00:02:06.260 --> 00:02:09.120 align:middle line:84% and some resistive current component. 00:02:09.120 --> 00:02:11.700 align:middle line:84% And I'm just doing the simplified 00:02:11.700 --> 00:02:15.920 align:middle line:84% case where the Norton source is 0 and the resistance is RL. 00:02:15.920 --> 00:02:18.840 align:middle line:90% 00:02:18.840 --> 00:02:20.900 align:middle line:84% This is just a way of saying in general, 00:02:20.900 --> 00:02:23.160 align:middle line:84% you would worry about exactly what 00:02:23.160 --> 00:02:27.810 align:middle line:84% the load looks like not just in terms of how much current 00:02:27.810 --> 00:02:28.470 align:middle line:90% does it draw. 00:02:28.470 --> 00:02:32.640 align:middle line:84% But exactly what the load looked like can affect the dynamics. 00:02:32.640 --> 00:02:35.490 align:middle line:84% But for simplicity, let's just talk about resistive loads, 00:02:35.490 --> 00:02:36.905 align:middle line:90% which is a very common case. 00:02:36.905 --> 00:02:42.990 align:middle line:90% 00:02:42.990 --> 00:02:45.480 align:middle line:84% What do we have to do in a typical power converter 00:02:45.480 --> 00:02:48.055 align:middle line:90% application? 00:02:48.055 --> 00:02:51.420 align:middle line:84% We want to usually regulate the output voltage 00:02:51.420 --> 00:02:53.490 align:middle line:90% to be something we want. 00:02:53.490 --> 00:03:04.020 align:middle line:84% So we want to make VOUT go to some reference voltage 00:03:04.020 --> 00:03:06.420 align:middle line:90% that I'll call VREF. 00:03:06.420 --> 00:03:08.140 align:middle line:90% VREF might be fixed-- 00:03:08.140 --> 00:03:12.190 align:middle line:84% it might be I want 12 volts output all day long-- 00:03:12.190 --> 00:03:14.760 align:middle line:84% or it may be something that you dynamically 00:03:14.760 --> 00:03:18.022 align:middle line:84% vary based on some other consideration 00:03:18.022 --> 00:03:19.480 align:middle line:84% that we're not going to talk about. 00:03:19.480 --> 00:03:21.600 align:middle line:84% So it could be a fixed or a variable reference. 00:03:21.600 --> 00:03:23.370 align:middle line:84% But for the present time, let's just 00:03:23.370 --> 00:03:25.320 align:middle line:84% assume I want to get my output on my converter 00:03:25.320 --> 00:03:27.930 align:middle line:84% to be some fixed value that I would like it to be. 00:03:27.930 --> 00:03:30.000 align:middle line:84% If I'm creating a 12-volt power supply, 00:03:30.000 --> 00:03:32.550 align:middle line:90% I want it to be 12 volts. 00:03:32.550 --> 00:03:35.820 align:middle line:84% And we have to keep the output voltage 00:03:35.820 --> 00:03:39.240 align:middle line:84% closer at the reference voltage under a wide variety 00:03:39.240 --> 00:03:39.875 align:middle line:90% of conditions. 00:03:39.875 --> 00:03:42.000 align:middle line:84% And this is where power electronics is challenging. 00:03:42.000 --> 00:03:45.225 align:middle line:84% Firstly, for variations in load-- 00:03:45.225 --> 00:03:50.280 align:middle line:90% 00:03:50.280 --> 00:03:59.700 align:middle line:84% so maybe I would have RLmin less than what RL is less than RLmax. 00:03:59.700 --> 00:04:07.980 align:middle line:84% So this would be heavy load because the lowest resistance 00:04:07.980 --> 00:04:09.450 align:middle line:90% gives me the highest power. 00:04:09.450 --> 00:04:11.360 align:middle line:90% And this might be light load. 00:04:11.360 --> 00:04:15.900 align:middle line:90% 00:04:15.900 --> 00:04:20.850 align:middle line:84% I also need to keep VOUT near Vreference in terms of-- 00:04:20.850 --> 00:04:28.620 align:middle line:90% for variations in VIN. 00:04:28.620 --> 00:04:38.040 align:middle line:84% So maybe there's a VINmin less than VIN less than VINmax 00:04:38.040 --> 00:04:39.370 align:middle line:90% that I've got to deal with. 00:04:39.370 --> 00:04:41.410 align:middle line:84% Why do I have to worry about those? 00:04:41.410 --> 00:04:44.950 align:middle line:84% Well, my load's not always going to be the same. 00:04:44.950 --> 00:04:48.000 align:middle line:84% Maybe this is controlling a voltage which 00:04:48.000 --> 00:04:50.220 align:middle line:84% goes to a bunch of microprocessors or other things 00:04:50.220 --> 00:04:52.480 align:middle line:84% or actuators and they're turning on and off. 00:04:52.480 --> 00:04:55.410 align:middle line:84% So the effective resistance, the effective power, 00:04:55.410 --> 00:04:58.240 align:middle line:84% I have to supply varies, even at fixed voltage. 00:04:58.240 --> 00:05:01.170 align:middle line:84% And here, maybe I'm powered from a battery and the battery 00:05:01.170 --> 00:05:03.730 align:middle line:84% discharges over time or I'm charging or whatever. 00:05:03.730 --> 00:05:07.260 align:middle line:84% So there's going to be some input range, input voltage 00:05:07.260 --> 00:05:09.700 align:middle line:84% range, over which this converter is designed 00:05:09.700 --> 00:05:12.550 align:middle line:84% to reject that as a disturbance and just keep the output where 00:05:12.550 --> 00:05:15.660 align:middle line:90% I want it. 00:05:15.660 --> 00:05:20.070 align:middle line:84% Now, so far, we've spent a lot of time 00:05:20.070 --> 00:05:23.550 align:middle line:84% talking about the steady-state operation. 00:05:23.550 --> 00:05:25.260 align:middle line:84% And that's usually always the first thing 00:05:25.260 --> 00:05:28.050 align:middle line:84% I look at because it tells me when things are sitting 00:05:28.050 --> 00:05:33.540 align:middle line:84% at some place, know some load resistance, some input voltage. 00:05:33.540 --> 00:05:36.780 align:middle line:84% What's it going to Do And we said for the boost 00:05:36.780 --> 00:05:45.910 align:middle line:84% converter, what I get is VOUT is equal to VIN over 1 minus D. 00:05:45.910 --> 00:05:48.460 align:middle line:84% So this is the steady-state relationship. 00:05:48.460 --> 00:05:52.090 align:middle line:84% This is in PSS, in periodic steady state. 00:05:52.090 --> 00:05:55.210 align:middle line:84% Once we've settled down, this converter ought to do this. 00:05:55.210 --> 00:06:00.046 align:middle line:84% So if I control the duty ratio of my switch, q of t, 00:06:00.046 --> 00:06:02.820 align:middle line:84% to this value, this is what I should 00:06:02.820 --> 00:06:03.905 align:middle line:90% get for an output voltage. 00:06:03.905 --> 00:06:07.085 align:middle line:90% 00:06:07.085 --> 00:06:12.220 align:middle line:84% So you might think, maybe I could just 00:06:12.220 --> 00:06:16.810 align:middle line:84% measure the input voltage, set a duty ratio, 00:06:16.810 --> 00:06:18.620 align:middle line:84% and get the output voltage I want, 00:06:18.620 --> 00:06:20.440 align:middle line:90% and that's the end of the day. 00:06:20.440 --> 00:06:22.230 align:middle line:84% But there's two problems with that. 00:06:22.230 --> 00:06:25.180 align:middle line:84% One this is only in periodic steady state. 00:06:25.180 --> 00:06:28.150 align:middle line:84% And it doesn't say how things will change transiently 00:06:28.150 --> 00:06:30.800 align:middle line:84% in between periodic steady-state conditions. 00:06:30.800 --> 00:06:34.900 align:middle line:84% And I really care about that because if the voltage goes-- 00:06:34.900 --> 00:06:37.000 align:middle line:84% if I decide to change the load, the voltage 00:06:37.000 --> 00:06:39.370 align:middle line:90% goes wild until it settles down. 00:06:39.370 --> 00:06:42.070 align:middle line:84% I'm probably not going to be happy. 00:06:42.070 --> 00:06:43.630 align:middle line:84% The other thing about this is this 00:06:43.630 --> 00:06:46.630 align:middle line:90% is an idealized relationship. 00:06:46.630 --> 00:06:49.720 align:middle line:84% When we derived this, we did it by volt-seconds 00:06:49.720 --> 00:06:51.140 align:middle line:90% balanced across this inductor. 00:06:51.140 --> 00:06:53.570 align:middle line:84% I pretended the inductor had no resistance. 00:06:53.570 --> 00:06:55.310 align:middle line:90% I ignored the diode drop. 00:06:55.310 --> 00:06:57.190 align:middle line:84% I ignored a whole bunch of things 00:06:57.190 --> 00:07:01.670 align:middle line:84% which don't matter for getting an approximate relationship. 00:07:01.670 --> 00:07:04.190 align:middle line:84% But they do matter if I need to control this 00:07:04.190 --> 00:07:09.000 align:middle line:90% to 12 volts plus or minus 2%. 00:07:09.000 --> 00:07:11.520 align:middle line:84% The model's just not good enough. 00:07:11.520 --> 00:07:14.310 align:middle line:84% I need some better way to get my converter 00:07:14.310 --> 00:07:18.480 align:middle line:84% to get where I want it to be and maintain it there 00:07:18.480 --> 00:07:22.050 align:middle line:90% under variations like this. 00:07:22.050 --> 00:07:23.780 align:middle line:90% Any questions about that? 00:07:23.780 --> 00:07:26.710 align:middle line:90% 00:07:26.710 --> 00:07:30.460 align:middle line:84% Just to illustrate that, what I've got here 00:07:30.460 --> 00:07:37.690 align:middle line:84% is a SPICE simulation of a boost converter 00:07:37.690 --> 00:07:41.050 align:middle line:84% which is undergoing a load transient. 00:07:41.050 --> 00:07:44.500 align:middle line:84% So what's happening in this is here at some time, 00:07:44.500 --> 00:07:48.460 align:middle line:84% at actually at 1.4 milliseconds, we had a 5-ohm-- 00:07:48.460 --> 00:07:50.830 align:middle line:84% we originally had a 5-ohm load on the converter. 00:07:50.830 --> 00:07:52.900 align:middle line:84% And we put in another 5-ohm load. 00:07:52.900 --> 00:07:57.640 align:middle line:84% So it's like some new load has come in. 00:07:57.640 --> 00:08:00.100 align:middle line:84% In this example, we're just holding the duty ratio 00:08:00.100 --> 00:08:01.280 align:middle line:90% of the converter constant. 00:08:01.280 --> 00:08:05.470 align:middle line:84% So we expect that, barring a lot of second-order effects, 00:08:05.470 --> 00:08:07.180 align:middle line:84% the output voltage should go back 00:08:07.180 --> 00:08:09.850 align:middle line:84% to the same thing in periodic steady state. 00:08:09.850 --> 00:08:14.190 align:middle line:84% But clearly, if the output voltage is going to be the same 00:08:14.190 --> 00:08:16.980 align:middle line:84% and I've got smaller net resistance on it 00:08:16.980 --> 00:08:18.542 align:middle line:84% because I have the net resistance, 00:08:18.542 --> 00:08:19.500 align:middle line:90% the power has to go up. 00:08:19.500 --> 00:08:21.150 align:middle line:90% The current has to go up. 00:08:21.150 --> 00:08:22.330 align:middle line:90% So things have to change. 00:08:22.330 --> 00:08:24.750 align:middle line:84% And that's what we see happening. 00:08:24.750 --> 00:08:28.330 align:middle line:84% So this trace down here is the input current. 00:08:28.330 --> 00:08:32.309 align:middle line:84% So the input current was sitting on average around 12 amps. 00:08:32.309 --> 00:08:35.289 align:middle line:90% I go through some transient. 00:08:35.289 --> 00:08:38.380 align:middle line:90% And I end up around 24 amps. 00:08:38.380 --> 00:08:41.020 align:middle line:84% The output voltage had been sitting at 22 volts, 00:08:41.020 --> 00:08:44.890 align:middle line:84% goes through some transient, and I get back to 22 volts 00:08:44.890 --> 00:08:48.770 align:middle line:84% because it's running at fixed duty ratio. 00:08:48.770 --> 00:08:54.240 align:middle line:84% So what we see here is the stuff we've skipped over so far. 00:08:54.240 --> 00:08:59.990 align:middle line:84% We know we could calculate based on periodic steady state-- 00:08:59.990 --> 00:09:03.620 align:middle line:84% say that the output capacitor voltage and the average inductor 00:09:03.620 --> 00:09:05.880 align:middle line:90% current in both conditions. 00:09:05.880 --> 00:09:09.290 align:middle line:84% But we haven't accounted for how does it transiently 00:09:09.290 --> 00:09:11.360 align:middle line:90% go between those conditions. 00:09:11.360 --> 00:09:16.160 align:middle line:84% And that's what we have to know in order 00:09:16.160 --> 00:09:19.670 align:middle line:90% to make our system happy. 00:09:19.670 --> 00:09:22.230 align:middle line:84% And so that's what we're going to focus on. 00:09:22.230 --> 00:09:31.040 align:middle line:84% And I'd like you to notice that this response looks 00:09:31.040 --> 00:09:34.480 align:middle line:90% like a second-order system. 00:09:34.480 --> 00:09:39.470 align:middle line:84% It has a-- looks like a damped LC oscillation, sort of, which 00:09:39.470 --> 00:09:45.500 align:middle line:84% is not surprising because I've got one C and one L. 00:09:45.500 --> 00:09:48.000 align:middle line:90% And there's load on it. 00:09:48.000 --> 00:09:49.780 align:middle line:84% So energy has been taken out of it. 00:09:49.780 --> 00:09:51.160 align:middle line:90% So maybe that's not surprising. 00:09:51.160 --> 00:09:56.970 align:middle line:84% The question is, what does this thing look like? 00:09:56.970 --> 00:10:01.210 align:middle line:84% So how would I control this thing in the real world? 00:10:01.210 --> 00:10:02.600 align:middle line:90% I'd use feedback control. 00:10:02.600 --> 00:10:05.400 align:middle line:90% 00:10:05.400 --> 00:10:07.470 align:middle line:90% So what are we going to do? 00:10:07.470 --> 00:10:13.530 align:middle line:84% We're going to say I have some reference voltage, VREF, 00:10:13.530 --> 00:10:15.600 align:middle line:84% that I want my converter to go to. 00:10:15.600 --> 00:10:17.970 align:middle line:84% And I provide that voltage reference. 00:10:17.970 --> 00:10:21.570 align:middle line:84% And I'm going to compare that voltage reference 00:10:21.570 --> 00:10:24.530 align:middle line:90% to the output voltage, VOUT. 00:10:24.530 --> 00:10:33.600 align:middle line:84% And I'm going to generate an error between those two. 00:10:33.600 --> 00:10:36.996 align:middle line:84% I'll take that error and feed it into a high-gain compensator. 00:10:36.996 --> 00:10:41.890 align:middle line:90% 00:10:41.890 --> 00:10:44.200 align:middle line:84% Maybe I could represent that with a transfer 00:10:44.200 --> 00:10:46.220 align:middle line:90% function, GC of s. 00:10:46.220 --> 00:10:55.780 align:middle line:84% This might be, for example, a PID controller or something 00:10:55.780 --> 00:10:57.320 align:middle line:90% fancier-- 00:10:57.320 --> 00:10:58.850 align:middle line:90% depends on the topology. 00:10:58.850 --> 00:11:01.530 align:middle line:84% This is where having a control class-- 00:11:01.530 --> 00:11:03.140 align:middle line:84% But you might think what this does 00:11:03.140 --> 00:11:07.730 align:middle line:84% is if I have error, I amplify that error by some amount, 00:11:07.730 --> 00:11:11.270 align:middle line:84% either proportionally or look at the integral of the error 00:11:11.270 --> 00:11:14.240 align:middle line:84% or something like that, and generate an output, 00:11:14.240 --> 00:11:15.625 align:middle line:90% which I'm going to call Vd. 00:11:15.625 --> 00:11:18.200 align:middle line:90% 00:11:18.200 --> 00:11:19.860 align:middle line:90% What is this signal? 00:11:19.860 --> 00:11:22.760 align:middle line:84% This is usually a voltage, although I could represent it 00:11:22.760 --> 00:11:24.530 align:middle line:90% with a current. 00:11:24.530 --> 00:11:31.675 align:middle line:84% It's a voltage representing the duty ratio I want. 00:11:31.675 --> 00:11:35.610 align:middle line:90% 00:11:35.610 --> 00:11:38.150 align:middle line:84% Now I have a voltage representing the duty ratio. 00:11:38.150 --> 00:11:42.230 align:middle line:84% I have to put this into some PWM circuit, pulse width modulation 00:11:42.230 --> 00:11:47.270 align:middle line:84% circuit, that's going to give me q of t 00:11:47.270 --> 00:11:49.760 align:middle line:84% because if I look at up there, I'm talking 00:11:49.760 --> 00:11:51.200 align:middle line:90% about in terms of duty ratio. 00:11:51.200 --> 00:11:54.830 align:middle line:84% But the actual signal going to my switch is some q of t 00:11:54.830 --> 00:11:57.960 align:middle line:84% or some amplified version of q of t from a gate drive 00:11:57.960 --> 00:12:01.530 align:middle line:84% that's turning the switch on and off at the right duty ratio. 00:12:01.530 --> 00:12:05.700 align:middle line:84% So the input of this is some value of duty ratio I want. 00:12:05.700 --> 00:12:11.520 align:middle line:84% The output of this is some PWM waveform switching at period T 00:12:11.520 --> 00:12:17.180 align:middle line:90% with an on time that's dT. 00:12:17.180 --> 00:12:25.230 align:middle line:84% This then goes into my actual converter, 00:12:25.230 --> 00:12:29.550 align:middle line:84% which also, if you think about it, has inputs of VIN. 00:12:29.550 --> 00:12:34.215 align:middle line:84% And it's being affected by what the RL is. 00:12:34.215 --> 00:12:38.490 align:middle line:84% And then its state variables are the inductor current 00:12:38.490 --> 00:12:41.490 align:middle line:84% and the output voltage, which is also the capacitor voltage, 00:12:41.490 --> 00:12:42.280 align:middle line:90% VOUT. 00:12:42.280 --> 00:12:46.850 align:middle line:84% And that's what I'm feeding around back here. 00:12:46.850 --> 00:12:51.620 align:middle line:84% So this is a typical closed-loop control system. 00:12:51.620 --> 00:12:54.830 align:middle line:84% If VOUT gets smaller than the reference voltage, 00:12:54.830 --> 00:12:55.730 align:middle line:90% Verror goes up. 00:12:55.730 --> 00:12:56.810 align:middle line:90% I amplify that. 00:12:56.810 --> 00:12:58.790 align:middle line:90% My duty ratio goes up. 00:12:58.790 --> 00:13:02.000 align:middle line:84% And when my duty ratio goes up, my output voltage goes up 00:13:02.000 --> 00:13:03.470 align:middle line:90% and I close the loop. 00:13:03.470 --> 00:13:08.544 align:middle line:84% And eventually, hopefully this settles to VOUT close to VREF. 00:13:08.544 --> 00:13:10.685 align:middle line:84% Any questions about that picture? 00:13:10.685 --> 00:13:13.210 align:middle line:90% 00:13:13.210 --> 00:13:14.860 align:middle line:84% Just out of curiosity, how many people 00:13:14.860 --> 00:13:19.050 align:middle line:84% here have had either a classical or state space control class? 00:13:19.050 --> 00:13:21.580 align:middle line:90% 00:13:21.580 --> 00:13:22.730 align:middle line:90% Good fraction. 00:13:22.730 --> 00:13:23.470 align:middle line:90% Good. 00:13:23.470 --> 00:13:26.120 align:middle line:84% Let's make that fraction 100% next year 00:13:26.120 --> 00:13:28.620 align:middle line:84% because you really want to know this, no matter what you do. 00:13:28.620 --> 00:13:36.900 align:middle line:90% 00:13:36.900 --> 00:13:41.850 align:middle line:84% This is a standard feedback system. 00:13:41.850 --> 00:13:44.715 align:middle line:90% What we need to do this-- 00:13:44.715 --> 00:13:47.790 align:middle line:84% if I put it in the wrong compensator and I put feedback, 00:13:47.790 --> 00:13:51.290 align:middle line:84% I could make this thing oscillate and go unstable. 00:13:51.290 --> 00:13:56.968 align:middle line:84% We've all seen the effects of bad feedback, 00:13:56.968 --> 00:13:58.760 align:middle line:84% like when you put your microphone too close 00:13:58.760 --> 00:13:59.630 align:middle line:90% to your speaker. 00:13:59.630 --> 00:14:01.170 align:middle line:84% And suddenly, the thing goes crazy. 00:14:01.170 --> 00:14:04.100 align:middle line:84% And you can certainly make a power converter go crazy 00:14:04.100 --> 00:14:07.008 align:middle line:90% if you're not careful. 00:14:07.008 --> 00:14:10.010 align:middle line:84% And what we need to know in order 00:14:10.010 --> 00:14:16.490 align:middle line:84% to make this work is some model that basically says, as I'm 00:14:16.490 --> 00:14:19.100 align:middle line:84% changing my duty ratio or something else, 00:14:19.100 --> 00:14:24.955 align:middle line:84% like the load resistor, how does this dynamically vary? 00:14:24.955 --> 00:14:28.150 align:middle line:84% If I know that, then I can design a control loop 00:14:28.150 --> 00:14:30.530 align:middle line:90% for the thing. 00:14:30.530 --> 00:14:32.740 align:middle line:84% And so what I'd like to talk about 00:14:32.740 --> 00:14:36.610 align:middle line:84% is how do we build this model, this dynamic model, 00:14:36.610 --> 00:14:38.620 align:middle line:84% instead of a periodic steady-state model. 00:14:38.620 --> 00:14:40.420 align:middle line:84% What does the dynamic model of this thing 00:14:40.420 --> 00:14:43.330 align:middle line:84% look like from variations in duty ratio, 00:14:43.330 --> 00:14:47.240 align:middle line:84% for example, to the variations in the output voltage? 00:14:47.240 --> 00:14:50.780 align:middle line:84% And coming back to this picture, we 00:14:50.780 --> 00:14:54.780 align:middle line:84% can see that what we have here is a bunch of switching. 00:14:54.780 --> 00:14:57.260 align:middle line:84% Here's the switching ripple, the triangular ripple current 00:14:57.260 --> 00:14:58.370 align:middle line:90% in the inductor. 00:14:58.370 --> 00:15:00.227 align:middle line:84% And it goes through some oscillations. 00:15:00.227 --> 00:15:02.060 align:middle line:84% And then it settles down, and the same thing 00:15:02.060 --> 00:15:03.270 align:middle line:90% with the capacitor voltage. 00:15:03.270 --> 00:15:07.160 align:middle line:84% Well, I'm not particularly interested in this switching 00:15:07.160 --> 00:15:09.270 align:middle line:90% ripple. 00:15:09.270 --> 00:15:11.520 align:middle line:84% Why am I not interested in it is because notice 00:15:11.520 --> 00:15:15.780 align:middle line:84% that the dynamical behavior is at a very low frequency compared 00:15:15.780 --> 00:15:18.440 align:middle line:90% to the switching ripple. 00:15:18.440 --> 00:15:21.440 align:middle line:84% And I can't really do anything about the switching ripple, 00:15:21.440 --> 00:15:22.160 align:middle line:90% anyways. 00:15:22.160 --> 00:15:24.875 align:middle line:84% In other words, that's intrinsic to the converter. 00:15:24.875 --> 00:15:26.500 align:middle line:84% I'm going to put some energy in and I'm 00:15:26.500 --> 00:15:27.690 align:middle line:90% going to process it through. 00:15:27.690 --> 00:15:28.732 align:middle line:90% I'm going to take it out. 00:15:28.732 --> 00:15:32.810 align:middle line:84% If I'm trying to, say, vary my duty ratio on the time 00:15:32.810 --> 00:15:36.900 align:middle line:84% scale of this switching, I'm very, very fast, 00:15:36.900 --> 00:15:40.220 align:middle line:84% I'm probably going to make the converter do unhappy things. 00:15:40.220 --> 00:15:43.580 align:middle line:84% My whole notion was that at least within a cycle, 00:15:43.580 --> 00:15:48.920 align:middle line:84% duty ratio is not something that I'm swinging around wildly. 00:15:48.920 --> 00:15:50.900 align:middle line:84% So I'm interested in the behavior 00:15:50.900 --> 00:15:54.290 align:middle line:84% of this circuit for frequencies relatively low 00:15:54.290 --> 00:15:57.035 align:middle line:84% compared to the switching frequency. 00:15:57.035 --> 00:15:59.690 align:middle line:84% So what I want to do is generate models 00:15:59.690 --> 00:16:03.320 align:middle line:84% that capture this average behavior, 00:16:03.320 --> 00:16:06.110 align:middle line:84% like these low-frequency oscillations, 00:16:06.110 --> 00:16:09.040 align:middle line:90% and ignore the switching ripple. 00:16:09.040 --> 00:16:10.405 align:middle line:90% Everybody follow that? 00:16:10.405 --> 00:16:15.934 align:middle line:90% 00:16:15.934 --> 00:16:19.310 align:middle line:84% So let me think about this in a couple of pieces. 00:16:19.310 --> 00:16:22.840 align:middle line:84% Well, first of all, let's just talk about this PWM block. 00:16:22.840 --> 00:16:24.230 align:middle line:90% We've talked about this before. 00:16:24.230 --> 00:16:26.030 align:middle line:84% This isn't part of the average model. 00:16:26.030 --> 00:16:29.700 align:middle line:84% But essentially, what would the PWM block look like? 00:16:29.700 --> 00:16:32.680 align:middle line:90% 00:16:32.680 --> 00:16:38.530 align:middle line:84% Basically, what I would have is maybe I 00:16:38.530 --> 00:16:41.720 align:middle line:84% would generate some triangle wave. 00:16:41.720 --> 00:16:49.820 align:middle line:84% This could be a sawtooth, for example, with a period T. 00:16:49.820 --> 00:16:51.130 align:middle line:90% And I'll just generate this. 00:16:51.130 --> 00:16:54.070 align:middle line:84% And maybe for simplicity, I'll just say this is a 1-volt 00:16:54.070 --> 00:16:56.860 align:middle line:90% signal. 00:16:56.860 --> 00:17:00.600 align:middle line:84% And I will compare that to this duty ratio signal. 00:17:00.600 --> 00:17:01.630 align:middle line:90% So here's Vd. 00:17:01.630 --> 00:17:05.800 align:middle line:90% 00:17:05.800 --> 00:17:08.530 align:middle line:90% And Vd is-- maybe it's changing. 00:17:08.530 --> 00:17:10.369 align:middle line:90% But it's changing very slowly. 00:17:10.369 --> 00:17:12.940 align:middle line:90% So here's Vd. 00:17:12.940 --> 00:17:15.560 align:middle line:84% And I'm just going to run this into a comparator. 00:17:15.560 --> 00:17:19.310 align:middle line:90% I'm going to call this VTRI. 00:17:19.310 --> 00:17:28.820 align:middle line:84% And I'll compare VTRI to Vd and generate q of t. 00:17:28.820 --> 00:17:31.340 align:middle line:84% So all I'm going to do is-- at the beginning of each cycle, 00:17:31.340 --> 00:17:35.750 align:middle line:84% what you can see is if this is time, 00:17:35.750 --> 00:17:37.640 align:middle line:84% at the beginning of each cycle, q of t 00:17:37.640 --> 00:17:46.470 align:middle line:84% is going to go high until the triangle exceeds the V sub d, 00:17:46.470 --> 00:17:48.510 align:middle line:90% in which case I'll go low. 00:17:48.510 --> 00:17:52.350 align:middle line:84% And then I'll go high at the beginning of the next cycle. 00:17:52.350 --> 00:17:57.540 align:middle line:84% And I'll stay high until I hit the end, until I hit 00:17:57.540 --> 00:18:00.720 align:middle line:90% the comparison, and so forth-- 00:18:00.720 --> 00:18:07.300 align:middle line:84% same again in the next cycle, and so forth. 00:18:07.300 --> 00:18:12.090 align:middle line:84% So by comparing my V sub d to a linearly rising element, 00:18:12.090 --> 00:18:14.490 align:middle line:84% I can generate something where this is-- 00:18:14.490 --> 00:18:17.310 align:middle line:84% whatever V sub d was here, that's-- represents some duty 00:18:17.310 --> 00:18:18.030 align:middle line:90% ratio. 00:18:18.030 --> 00:18:20.235 align:middle line:84% That's going to be dT at this point. 00:18:20.235 --> 00:18:22.110 align:middle line:84% And I'm going to have dT at this point, which 00:18:22.110 --> 00:18:23.970 align:middle line:90% might be different, et cetera. 00:18:23.970 --> 00:18:27.570 align:middle line:84% And each cycle, I get a duty ratio 00:18:27.570 --> 00:18:31.560 align:middle line:84% that's related to V sub d during that cycle assuming V sub d 00:18:31.560 --> 00:18:34.980 align:middle line:90% doesn't vary very quickly. 00:18:34.980 --> 00:18:38.280 align:middle line:84% So this is how I build the PWM generator. 00:18:38.280 --> 00:18:41.220 align:middle line:90% That's this block here. 00:18:41.220 --> 00:18:42.490 align:middle line:90% That's not how I model it. 00:18:42.490 --> 00:18:43.900 align:middle line:90% That's just how I implement it. 00:18:43.900 --> 00:18:48.360 align:middle line:84% So from a mental perspective, we can 00:18:48.360 --> 00:18:51.090 align:middle line:84% worry about going from some desired duty 00:18:51.090 --> 00:18:54.210 align:middle line:84% ratio to some response or the local average duty ratio 00:18:54.210 --> 00:18:56.670 align:middle line:84% to some response where I'm assuming 00:18:56.670 --> 00:18:59.880 align:middle line:84% that Vd is only varying very slowly within any given 00:18:59.880 --> 00:19:01.950 align:middle line:90% switching cycle. 00:19:01.950 --> 00:19:05.272 align:middle line:84% And if your compensator lets it very, 00:19:05.272 --> 00:19:07.355 align:middle line:84% very quickly, you're probably going to be unhappy. 00:19:07.355 --> 00:19:12.570 align:middle line:90% 00:19:12.570 --> 00:19:15.160 align:middle line:84% So that gives me a way to implement this. 00:19:15.160 --> 00:19:18.300 align:middle line:84% And I should say, by the way, that if I go buy a PWM 00:19:18.300 --> 00:19:22.020 align:middle line:84% chip for a typical DC-to-DC converter or controller chip, 00:19:22.020 --> 00:19:27.570 align:middle line:84% I'll usually get a reference, some kind of error amplifier, 00:19:27.570 --> 00:19:31.830 align:middle line:84% op-amp, and other things to generate the compensator. 00:19:31.830 --> 00:19:33.790 align:middle line:90% I'll get the PWM chip. 00:19:33.790 --> 00:19:36.480 align:middle line:84% And in fact, this triangle also serves as my clock. 00:19:36.480 --> 00:19:38.100 align:middle line:84% So I can set the clock frequency. 00:19:38.100 --> 00:19:41.830 align:middle line:84% I can set T. And then maybe it gives me a gate driver. 00:19:41.830 --> 00:19:45.180 align:middle line:84% So I'll get all this stuff in one chip 00:19:45.180 --> 00:19:48.430 align:middle line:90% very often for a simple design. 00:19:48.430 --> 00:19:50.960 align:middle line:84% Of course, you can also do a lot of this stuff digitally 00:19:50.960 --> 00:19:56.160 align:middle line:84% these days, depending upon what kind of processor you use. 00:19:56.160 --> 00:19:58.920 align:middle line:90% But this doesn't tell me-- 00:19:58.920 --> 00:20:02.280 align:middle line:84% this tells me how did I get from duty ratio to the q of t. 00:20:02.280 --> 00:20:07.116 align:middle line:84% But it doesn't tell me how I model this whole block. 00:20:07.116 --> 00:20:10.830 align:middle line:84% And that's what I'd like to talk about now. 00:20:10.830 --> 00:20:13.860 align:middle line:84% And we said we are not interested 00:20:13.860 --> 00:20:15.750 align:middle line:84% in this high-frequency ripple, but only 00:20:15.750 --> 00:20:19.240 align:middle line:90% the low-frequency behavior. 00:20:19.240 --> 00:20:22.200 align:middle line:84% And towards that end, I'd like to introduce what's 00:20:22.200 --> 00:20:24.350 align:middle line:90% known as a local average. 00:20:24.350 --> 00:20:33.065 align:middle line:84% So the local average operator is this. 00:20:33.065 --> 00:20:37.710 align:middle line:84% If I had some signal for any signal x of t, 00:20:37.710 --> 00:20:40.080 align:middle line:84% some time-varying waveform x of t-- could be a voltage, 00:20:40.080 --> 00:20:42.580 align:middle line:84% could be a current, whatever I wanted-- 00:20:42.580 --> 00:20:46.990 align:middle line:84% I can define a local average x bar of t 00:20:46.990 --> 00:20:53.590 align:middle line:84% is defined as being equal to 1 over T integral between t 00:20:53.590 --> 00:20:59.060 align:middle line:84% minus capital T and t of x of tau d tau. 00:20:59.060 --> 00:21:02.470 align:middle line:84% This is the local average of a signal. 00:21:02.470 --> 00:21:04.310 align:middle line:90% And what is this local average? 00:21:04.310 --> 00:21:06.790 align:middle line:90% It's just a moving average. 00:21:06.790 --> 00:21:10.510 align:middle line:84% So if I'm at my signal time right now at time T, 00:21:10.510 --> 00:21:11.890 align:middle line:84% what I'm going to do is I'm going 00:21:11.890 --> 00:21:17.350 align:middle line:84% to look back over 1 capital T and take 00:21:17.350 --> 00:21:20.170 align:middle line:84% the average of the waveform just over that time. 00:21:20.170 --> 00:21:22.420 align:middle line:84% And then if I'm at another time, I'll look back 00:21:22.420 --> 00:21:25.030 align:middle line:84% and I'll take the average over at that time. 00:21:25.030 --> 00:21:33.520 align:middle line:84% So if I had some time-varying waveform x of t and I said, 00:21:33.520 --> 00:21:42.100 align:middle line:90% here's t1, maybe I'd go back. 00:21:42.100 --> 00:21:45.400 align:middle line:84% And here's t1 minus capital T. Maybe 00:21:45.400 --> 00:21:48.650 align:middle line:84% I would take the average of this waveform over that time. 00:21:48.650 --> 00:21:56.590 align:middle line:84% And then if I'm at some other time t2, I'd go back one cycle. 00:21:56.590 --> 00:21:59.836 align:middle line:90% And this is t minus T2-- 00:21:59.836 --> 00:22:03.670 align:middle line:84% t2 minus capital T. And I'd look at the average over this time 00:22:03.670 --> 00:22:04.790 align:middle line:90% period, et cetera. 00:22:04.790 --> 00:22:06.490 align:middle line:84% So at any given point, I'm going to look 00:22:06.490 --> 00:22:11.740 align:middle line:84% at what I've been doing over the last capital T 00:22:11.740 --> 00:22:13.630 align:middle line:90% and take that average. 00:22:13.630 --> 00:22:15.330 align:middle line:90% Why am I doing that? 00:22:15.330 --> 00:22:19.210 align:middle line:84% Well, because if I set capital T to be my switching period, 00:22:19.210 --> 00:22:22.600 align:middle line:84% I can look at what the average of this variable 00:22:22.600 --> 00:22:28.930 align:middle line:84% is over one switching cycle at any point in time. 00:22:28.930 --> 00:22:32.180 align:middle line:90% 00:22:32.180 --> 00:22:35.840 align:middle line:84% And we can think of this as basically tracking 00:22:35.840 --> 00:22:41.000 align:middle line:84% the low-frequency variations and rejecting the switching ripple. 00:22:41.000 --> 00:22:43.040 align:middle line:84% And we can think about it qualitatively. 00:22:43.040 --> 00:22:45.150 align:middle line:84% And we can think about it quantitatively. 00:22:45.150 --> 00:22:47.750 align:middle line:84% Any questions about the idea of what we 00:22:47.750 --> 00:22:49.320 align:middle line:90% want to do with this operator? 00:22:49.320 --> 00:22:54.650 align:middle line:90% 00:22:54.650 --> 00:22:58.250 align:middle line:84% So what are the properties of this operator? 00:22:58.250 --> 00:23:01.030 align:middle line:90% 00:23:01.030 --> 00:23:07.880 align:middle line:84% The first is that it's LTI, linear and time-invariant 00:23:07.880 --> 00:23:12.940 align:middle line:84% So when I say it's linear, I mean the following-- 00:23:12.940 --> 00:23:20.850 align:middle line:84% that if I took a times x of t plus b times y of t 00:23:20.850 --> 00:23:26.860 align:middle line:84% and then I took the local average of this whole signal, 00:23:26.860 --> 00:23:32.470 align:middle line:84% what I would get is a times x bar of t 00:23:32.470 --> 00:23:34.960 align:middle line:90% plus b times y bar of t. 00:23:34.960 --> 00:23:38.440 align:middle line:90% 00:23:38.440 --> 00:23:39.310 align:middle line:90% That's nice. 00:23:39.310 --> 00:23:45.550 align:middle line:84% It's also time-invariant, by which 00:23:45.550 --> 00:23:52.930 align:middle line:84% I mean if I took a signal x and shifted it by some time t0 00:23:52.930 --> 00:23:58.020 align:middle line:84% and then I took the local average of that shifted signal, 00:23:58.020 --> 00:24:03.660 align:middle line:90% it's equal to x bar shifted. 00:24:03.660 --> 00:24:06.690 align:middle line:84% It's equal to x bar of t shifted by t0. 00:24:06.690 --> 00:24:08.340 align:middle line:84% So the local average of the shift 00:24:08.340 --> 00:24:11.130 align:middle line:84% is the shift of the local average. 00:24:11.130 --> 00:24:13.750 align:middle line:90% That make sense to everybody? 00:24:13.750 --> 00:24:15.670 align:middle line:84% So now why am I worrying that it's 00:24:15.670 --> 00:24:18.313 align:middle line:84% a linear time-invariant operator? 00:24:18.313 --> 00:24:20.230 align:middle line:84% That means I can actually represent this thing 00:24:20.230 --> 00:24:23.726 align:middle line:90% with a transfer function. 00:24:23.726 --> 00:24:25.820 align:middle line:84% As an operator, it's basically a filter 00:24:25.820 --> 00:24:27.710 align:middle line:90% that I can put on my circuit. 00:24:27.710 --> 00:24:33.480 align:middle line:84% So this is just a way of describing a filter. 00:24:33.480 --> 00:24:36.070 align:middle line:84% Here's what the filter transfer function looks like. 00:24:36.070 --> 00:24:38.590 align:middle line:84% Here's the operator definition and time domain. 00:24:38.590 --> 00:24:41.380 align:middle line:84% Here's what it looks like in the frequency domain. 00:24:41.380 --> 00:24:44.610 align:middle line:84% What we can see is that at low frequencies, near DC, 00:24:44.610 --> 00:24:45.970 align:middle line:90% it has a gain of 1. 00:24:45.970 --> 00:24:49.050 align:middle line:84% It passes through the low-frequency content. 00:24:49.050 --> 00:24:51.540 align:middle line:84% But this particular filter just happens 00:24:51.540 --> 00:24:55.290 align:middle line:84% to have zero amplitudes at the switching frequency, twice 00:24:55.290 --> 00:24:57.580 align:middle line:84% the switching frequency, three times, et cetera. 00:24:57.580 --> 00:25:00.420 align:middle line:84% So it very precisely keeps the low-frequency stuff 00:25:00.420 --> 00:25:02.510 align:middle line:84% and murders all the switching ripple. 00:25:02.510 --> 00:25:05.160 align:middle line:90% 00:25:05.160 --> 00:25:08.440 align:middle line:84% Now, could I pick some other filter that captures this idea? 00:25:08.440 --> 00:25:09.880 align:middle line:90% Sure. 00:25:09.880 --> 00:25:12.120 align:middle line:84% But we just happened to pick this one 00:25:12.120 --> 00:25:14.580 align:middle line:84% because it's a good one for our purpose, which 00:25:14.580 --> 00:25:16.200 align:middle line:84% is to keep the low-frequency content 00:25:16.200 --> 00:25:18.670 align:middle line:84% and get rid of the switching ripple and its harmonics. 00:25:18.670 --> 00:25:21.360 align:middle line:90% 00:25:21.360 --> 00:25:24.990 align:middle line:84% You'll also notice that the phase of this thing 00:25:24.990 --> 00:25:26.792 align:middle line:90% is basically representing-- 00:25:26.792 --> 00:25:28.500 align:middle line:84% it's a linear phase response-- basically, 00:25:28.500 --> 00:25:29.800 align:middle line:90% a delay of half a cycle. 00:25:29.800 --> 00:25:30.420 align:middle line:90% Why? 00:25:30.420 --> 00:25:35.640 align:middle line:84% Because if I'm looking now, I'm really asking, 00:25:35.640 --> 00:25:38.100 align:middle line:84% what's my signal over the last capital T? 00:25:38.100 --> 00:25:43.020 align:middle line:84% So I'm, on average, looking back half a t in duration. 00:25:43.020 --> 00:25:46.350 align:middle line:84% If I didn't like that, I could make my local average 00:25:46.350 --> 00:25:51.120 align:middle line:84% between t minus T over 2 and t plus T over 2, capital T over 2. 00:25:51.120 --> 00:25:52.830 align:middle line:84% And then I wouldn't have any delay in it. 00:25:52.830 --> 00:25:55.723 align:middle line:90% But I also wouldn't be causal. 00:25:55.723 --> 00:25:57.390 align:middle line:84% Now, I'm not saying we're actually going 00:25:57.390 --> 00:25:59.970 align:middle line:90% to physically build a filter. 00:25:59.970 --> 00:26:03.570 align:middle line:84% But this is an operator that lets 00:26:03.570 --> 00:26:07.440 align:middle line:84% me think about how I'd process my signals so that I could 00:26:07.440 --> 00:26:10.590 align:middle line:84% keep the low-frequency content and get rid of the switching 00:26:10.590 --> 00:26:12.560 align:middle line:90% ripple. 00:26:12.560 --> 00:26:13.765 align:middle line:90% Any questions about that? 00:26:13.765 --> 00:26:21.300 align:middle line:90% 00:26:21.300 --> 00:26:27.030 align:middle line:84% So we're going to see how to use this kind of filter 00:26:27.030 --> 00:26:28.325 align:middle line:90% to do interesting things. 00:26:28.325 --> 00:26:31.620 align:middle line:90% 00:26:31.620 --> 00:26:36.385 align:middle line:84% There's a few ways we can use it. 00:26:36.385 --> 00:26:38.850 align:middle line:90% And they give similar results. 00:26:38.850 --> 00:26:42.010 align:middle line:90% But they're useful. 00:26:42.010 --> 00:26:45.310 align:middle line:84% But let me just, lastly, talk about a couple 00:26:45.310 --> 00:26:50.050 align:middle line:84% of other properties of this operator. 00:26:50.050 --> 00:26:53.140 align:middle line:84% And the first that's a very useful property 00:26:53.140 --> 00:26:54.985 align:middle line:90% is the differentiation property. 00:26:54.985 --> 00:27:00.970 align:middle line:90% 00:27:00.970 --> 00:27:02.980 align:middle line:84% And this is one of the reasons why we 00:27:02.980 --> 00:27:04.450 align:middle line:90% picked this particular filter. 00:27:04.450 --> 00:27:09.700 align:middle line:90% If I took some signal dx dt-- 00:27:09.700 --> 00:27:11.020 align:middle line:90% so some time-varying signal-- 00:27:11.020 --> 00:27:12.820 align:middle line:84% I looked at its derivative, and I 00:27:12.820 --> 00:27:15.880 align:middle line:84% said, give me the local average of the derivative 00:27:15.880 --> 00:27:19.720 align:middle line:84% of this signal, It's pretty easy to show 00:27:19.720 --> 00:27:25.900 align:middle line:84% that this is the same as the derivative of x bar of t. 00:27:25.900 --> 00:27:27.970 align:middle line:84% That is, the derivative of the local average 00:27:27.970 --> 00:27:31.627 align:middle line:84% is the local average of the derivative, 00:27:31.627 --> 00:27:33.210 align:middle line:84% or the local average of the derivative 00:27:33.210 --> 00:27:34.570 align:middle line:84% is the derivative of the local average. 00:27:34.570 --> 00:27:35.440 align:middle line:90% Why is that true? 00:27:35.440 --> 00:27:37.990 align:middle line:84% Because basically, I'm taking the derivative of this thing. 00:27:37.990 --> 00:27:40.960 align:middle line:84% I can interchange the order of differentiation and integration. 00:27:40.960 --> 00:27:43.490 align:middle line:90% I get back the same thing. 00:27:43.490 --> 00:27:45.690 align:middle line:84% And this property turns out to be 00:27:45.690 --> 00:27:48.335 align:middle line:84% a very useful one for modeling, as we'll see in a minute. 00:27:48.335 --> 00:27:52.720 align:middle line:90% 00:27:52.720 --> 00:28:04.640 align:middle line:84% So how might I use this average model or average operator? 00:28:04.640 --> 00:28:06.860 align:middle line:84% And the way I'd like to show you to use it 00:28:06.860 --> 00:28:09.670 align:middle line:90% today is on a circuit itself. 00:28:09.670 --> 00:28:12.620 align:middle line:84% So suppose I came up and I gave you a boost converter 00:28:12.620 --> 00:28:16.910 align:middle line:84% and I said, well, gee, could I apply that average model 00:28:16.910 --> 00:28:21.360 align:middle line:84% to the boost converter itself, to the circuit? 00:28:21.360 --> 00:28:24.780 align:middle line:84% And let's think about what that means. 00:28:24.780 --> 00:28:29.580 align:middle line:84% If I thought about any circuit, what defines its topology? 00:28:29.580 --> 00:28:31.110 align:middle line:90% What defines its topology? 00:28:31.110 --> 00:28:32.250 align:middle line:90% What's connected to what? 00:28:32.250 --> 00:28:37.560 align:middle line:84% The structure of it is really the set of KVL and KCL equations 00:28:37.560 --> 00:28:40.620 align:middle line:90% that exist for that circuit. 00:28:40.620 --> 00:28:49.020 align:middle line:84% So any circuit has some set of KVL equations, which essentially 00:28:49.020 --> 00:28:54.613 align:middle line:84% are the sum of Vij is equal to 0 around a bunch of loops. 00:28:54.613 --> 00:28:57.030 align:middle line:84% So there's a whole bunch of equations that have this form. 00:28:57.030 --> 00:29:00.680 align:middle line:90% 00:29:00.680 --> 00:29:06.320 align:middle line:84% And likewise, there's a set of KCL equations, the sum of i 00:29:06.320 --> 00:29:10.330 align:middle line:90% sub k's, that are equal to 0. 00:29:10.330 --> 00:29:15.690 align:middle line:84% So I could take any circuit and write down a list of KVL and KCL 00:29:15.690 --> 00:29:20.130 align:middle line:84% equations that pertain to that circuit, one 00:29:20.130 --> 00:29:22.862 align:middle line:84% for every voltage loop, one for every node. 00:29:22.862 --> 00:29:25.320 align:middle line:84% And we can go back and say-- figure out which ones of those 00:29:25.320 --> 00:29:28.410 align:middle line:90% are independent, and so forth. 00:29:28.410 --> 00:29:31.470 align:middle line:84% Well, what happens if I came up and said, 00:29:31.470 --> 00:29:35.970 align:middle line:84% let me take a KVL or KCL equation or, more generally, 00:29:35.970 --> 00:29:38.220 align:middle line:84% the ones that represent my circuit 00:29:38.220 --> 00:29:42.300 align:middle line:90% and average those things? 00:29:42.300 --> 00:29:48.090 align:middle line:84% Well, all I would do is if I took this and I averaged it-- 00:29:48.090 --> 00:29:53.730 align:middle line:84% when I say average, I mean apply this operator-- 00:29:53.730 --> 00:29:54.660 align:middle line:90% what would I do? 00:29:54.660 --> 00:29:56.400 align:middle line:84% I'd take the local average of each side. 00:29:56.400 --> 00:29:58.470 align:middle line:90% The local average of 0 is 0. 00:29:58.470 --> 00:30:03.110 align:middle line:84% This is just the local average of the sum, which I already 00:30:03.110 --> 00:30:06.545 align:middle line:84% said, because of linearity, is the sum of the local averages. 00:30:06.545 --> 00:30:13.640 align:middle line:84% So this gives me, if I average it, the sum of Vij local average 00:30:13.640 --> 00:30:15.050 align:middle line:90% is equal to 0. 00:30:15.050 --> 00:30:19.330 align:middle line:84% And this gives me the sum of i sub k local average, which is 0. 00:30:19.330 --> 00:30:22.008 align:middle line:90% 00:30:22.008 --> 00:30:24.400 align:middle line:84% So what I'm really saying here is 00:30:24.400 --> 00:30:28.210 align:middle line:84% if I have some circuit like my boost converter up there, 00:30:28.210 --> 00:30:34.930 align:middle line:84% I could apply averaging to that circuit or all the KVL and KCL 00:30:34.930 --> 00:30:35.680 align:middle line:90% equations. 00:30:35.680 --> 00:30:38.110 align:middle line:84% And I could get another circuit that's 00:30:38.110 --> 00:30:42.560 align:middle line:84% average that has the same connections in it. 00:30:42.560 --> 00:30:43.300 align:middle line:90% Why? 00:30:43.300 --> 00:30:45.880 align:middle line:84% Because for every KVL equation in this circuit, 00:30:45.880 --> 00:30:48.050 align:middle line:84% there's an identical KVL equation in this circuit. 00:30:48.050 --> 00:30:50.020 align:middle line:84% And it just has averaged voltages. 00:30:50.020 --> 00:30:52.960 align:middle line:84% And likewise, every KCL in this circuit 00:30:52.960 --> 00:30:56.500 align:middle line:84% generates a new KCL circuit just with averaged currents. 00:30:56.500 --> 00:30:59.770 align:middle line:84% So I can get a new circuit with the same topology 00:30:59.770 --> 00:31:01.305 align:middle line:90% for which averages pertain. 00:31:01.305 --> 00:31:03.920 align:middle line:90% 00:31:03.920 --> 00:31:05.090 align:middle line:90% Why does this work? 00:31:05.090 --> 00:31:07.710 align:middle line:84% This is the-- similar to what we were saying. 00:31:07.710 --> 00:31:10.140 align:middle line:84% Remember, we talked about, well, there's 00:31:10.140 --> 00:31:14.200 align:middle line:84% average KVL and average KCL over all time 00:31:14.200 --> 00:31:15.900 align:middle line:90% from periodic steady state. 00:31:15.900 --> 00:31:18.250 align:middle line:84% Well, I can also talk about it over a cycle. 00:31:18.250 --> 00:31:21.040 align:middle line:84% If KVL holds instantaneously, then it 00:31:21.040 --> 00:31:22.970 align:middle line:84% ought to hold for the local averages. 00:31:22.970 --> 00:31:24.970 align:middle line:84% And likewise, if KCL into a circuit node 00:31:24.970 --> 00:31:27.670 align:middle line:84% holds instantaneously, it ought to hold 00:31:27.670 --> 00:31:29.060 align:middle line:90% averaged over the last cycle. 00:31:29.060 --> 00:31:32.570 align:middle line:84% So it makes sense that you should be able to do this. 00:31:32.570 --> 00:31:35.740 align:middle line:84% And the key point here is because these equations 00:31:35.740 --> 00:31:38.140 align:middle line:84% have the same form as these equations, 00:31:38.140 --> 00:31:41.410 align:middle line:84% you get the same exact circuit back, the same exact circuit 00:31:41.410 --> 00:31:43.100 align:middle line:90% topology back. 00:31:43.100 --> 00:31:45.370 align:middle line:84% What the elements in that circuit topology do 00:31:45.370 --> 00:31:47.140 align:middle line:90% is another question. 00:31:47.140 --> 00:31:49.990 align:middle line:84% We haven't said how are voltages and currents 00:31:49.990 --> 00:31:52.120 align:middle line:90% of the elements related. 00:31:52.120 --> 00:31:56.770 align:middle line:84% All we've said is that the circuit topology stays the same. 00:31:56.770 --> 00:31:58.200 align:middle line:90% Any questions about that? 00:31:58.200 --> 00:32:01.800 align:middle line:90% 00:32:01.800 --> 00:32:04.450 align:middle line:84% Let's start talking about circuit elements. 00:32:04.450 --> 00:32:08.163 align:middle line:90% What do I have in this circuit? 00:32:08.163 --> 00:32:10.080 align:middle line:84% If I come back and look at my boost converter, 00:32:10.080 --> 00:32:12.960 align:middle line:84% I have inductors, capacitors, resistors, and then switches 00:32:12.960 --> 00:32:16.610 align:middle line:90% and sources and switches. 00:32:16.610 --> 00:32:19.930 align:middle line:84% Well, the local average of a source 00:32:19.930 --> 00:32:21.680 align:middle line:84% is just the local average of a source. 00:32:21.680 --> 00:32:25.160 align:middle line:90% So VIN would become VIN bar. 00:32:25.160 --> 00:32:26.900 align:middle line:90% But what about a resistor? 00:32:26.900 --> 00:32:28.670 align:middle line:90% What about my load resistor? 00:32:28.670 --> 00:32:31.010 align:middle line:84% Well, if I took my load resistor, 00:32:31.010 --> 00:32:35.270 align:middle line:90% what I really have is V-- 00:32:35.270 --> 00:32:43.130 align:middle line:84% I have i sub R and V sub R. And what I get is V sub R 00:32:43.130 --> 00:32:46.550 align:middle line:90% is equal to i sub R times R. 00:32:46.550 --> 00:32:52.350 align:middle line:84% Well, what happens if I take the local average of that guy? 00:32:52.350 --> 00:32:55.562 align:middle line:84% I'm just going to take the local average of this equation. 00:32:55.562 --> 00:32:56.520 align:middle line:90% So I'll come over here. 00:32:56.520 --> 00:32:58.050 align:middle line:90% And I'll average the left. 00:32:58.050 --> 00:32:59.280 align:middle line:90% And I'll average the right. 00:32:59.280 --> 00:33:02.130 align:middle line:84% And I can pull R out of the equation. 00:33:02.130 --> 00:33:06.630 align:middle line:84% So then my average element ought to just be this. 00:33:06.630 --> 00:33:12.870 align:middle line:84% I have V sub R bar, i sub R bar, and R 00:33:12.870 --> 00:33:14.940 align:middle line:84% because my relationship is simply 00:33:14.940 --> 00:33:20.790 align:middle line:84% V sub R bar is equal to i sub R bar R, which is exactly 00:33:20.790 --> 00:33:23.810 align:middle line:90% the relation of a resistor. 00:33:23.810 --> 00:33:27.260 align:middle line:84% So that just says if I had a resistor in my original circuit 00:33:27.260 --> 00:33:30.110 align:middle line:84% and I look at the local average of the constitutive relation 00:33:30.110 --> 00:33:34.220 align:middle line:84% of the two things connecting the nodes, if it's a resistor, 00:33:34.220 --> 00:33:36.850 align:middle line:90% it just stays being a resistor. 00:33:36.850 --> 00:33:39.540 align:middle line:90% That make sense? 00:33:39.540 --> 00:33:41.535 align:middle line:84% Well, what happens if I have an inductor? 00:33:41.535 --> 00:33:44.250 align:middle line:90% 00:33:44.250 --> 00:33:49.320 align:middle line:84% I have V sub L, i sub L. And the relationship 00:33:49.320 --> 00:33:54.240 align:middle line:84% is V sub L is equal to L di sub L dt. 00:33:54.240 --> 00:33:58.740 align:middle line:84% Well, if I average this guy, I'm going 00:33:58.740 --> 00:34:02.160 align:middle line:84% to take the local average here and the local average here. 00:34:02.160 --> 00:34:04.370 align:middle line:84% This shouldn't have been a local average. 00:34:04.370 --> 00:34:05.800 align:middle line:90% And this would give me what? 00:34:05.800 --> 00:34:10.320 align:middle line:84% Well, the relationship is V sub L bar. 00:34:10.320 --> 00:34:14.010 align:middle line:84% Now, I can pull the L out because of linearity. 00:34:14.010 --> 00:34:15.900 align:middle line:84% And the local average of the derivative 00:34:15.900 --> 00:34:17.733 align:middle line:84% becomes the derivative of the local average. 00:34:17.733 --> 00:34:19.570 align:middle line:84% I said that was my property over here. 00:34:19.570 --> 00:34:26.310 align:middle line:84% And what I get is then-- is equal to L d dt of iL bar, which 00:34:26.310 --> 00:34:36.020 align:middle line:84% is just an inductor of value L. Does that make sense? 00:34:36.020 --> 00:34:39.840 align:middle line:84% I could do the same thing with a capacitor. 00:34:39.840 --> 00:34:42.780 align:middle line:84% So the beautiful thing here is if I 00:34:42.780 --> 00:34:46.469 align:middle line:84% go to average my circuit, the resistors, inductors, 00:34:46.469 --> 00:34:51.340 align:middle line:84% capacitors, sources just become the local averages 00:34:51.340 --> 00:34:52.370 align:middle line:90% of themselves. 00:34:52.370 --> 00:34:55.805 align:middle line:90% 00:34:55.805 --> 00:34:59.420 align:middle line:84% Well, that's nice because what I'm suggesting 00:34:59.420 --> 00:35:02.840 align:middle line:84% is I can come to this circuit and take its local average 00:35:02.840 --> 00:35:05.180 align:middle line:84% and come up with a new circuit that does the same thing, 00:35:05.180 --> 00:35:08.230 align:middle line:90% but without switching ripple. 00:35:08.230 --> 00:35:11.680 align:middle line:90% That make sense to everybody? 00:35:11.680 --> 00:35:15.705 align:middle line:84% Let me see how I would do this here. 00:35:15.705 --> 00:35:19.980 align:middle line:90% 00:35:19.980 --> 00:35:22.410 align:middle line:84% But in order to do it, what I haven't said 00:35:22.410 --> 00:35:25.720 align:middle line:90% is what happens to the switches. 00:35:25.720 --> 00:35:27.925 align:middle line:84% And that's a little trickier to think about. 00:35:27.925 --> 00:35:34.130 align:middle line:90% 00:35:34.130 --> 00:35:36.800 align:middle line:84% To do that, let me make something that's a little bit 00:35:36.800 --> 00:35:39.910 align:middle line:90% circuit model-friendly. 00:35:39.910 --> 00:35:42.320 align:middle line:84% So let me come up with a different model for my boost 00:35:42.320 --> 00:35:43.150 align:middle line:90% converter. 00:35:43.150 --> 00:35:50.510 align:middle line:84% My boost converter had an input voltage, an inductor, 00:35:50.510 --> 00:35:53.760 align:middle line:90% L of that with a current iL. 00:35:53.760 --> 00:36:01.050 align:middle line:84% And then it had a capacitor, C, and a resistor, R, 00:36:01.050 --> 00:36:05.605 align:middle line:84% with an output voltage VOUT on it. 00:36:05.605 --> 00:36:08.825 align:middle line:84% And then it had the switch in the diode. 00:36:08.825 --> 00:36:11.496 align:middle line:90% 00:36:11.496 --> 00:36:14.520 align:middle line:84% And I could represent everything with the switch in the diode. 00:36:14.520 --> 00:36:17.490 align:middle line:84% But maybe it would be easier from a modeling perspective 00:36:17.490 --> 00:36:20.580 align:middle line:84% if I just came back and modeled the switch in diode 00:36:20.580 --> 00:36:23.220 align:middle line:84% with equivalent dependent sources. 00:36:23.220 --> 00:36:25.535 align:middle line:84% And here's how I'm going to argue-- 00:36:25.535 --> 00:36:27.785 align:middle line:84% and let me-- there's a couple of ways I could do this. 00:36:27.785 --> 00:36:30.830 align:middle line:90% 00:36:30.830 --> 00:36:32.990 align:middle line:90% Maybe the switch-- 00:36:32.990 --> 00:36:37.280 align:middle line:84% I have to worry about what's the voltage, Vx, here 00:36:37.280 --> 00:36:40.260 align:middle line:84% and the current, ix, at this node. 00:36:40.260 --> 00:36:42.790 align:middle line:90% Maybe this is Vx and this is ix. 00:36:42.790 --> 00:36:45.530 align:middle line:90% 00:36:45.530 --> 00:36:52.400 align:middle line:84% Maybe I would model this as a dependent voltage source. 00:36:52.400 --> 00:36:56.540 align:middle line:84% So what would this voltage source be? 00:36:56.540 --> 00:36:59.840 align:middle line:84% I could write this voltage source, dependent source, 00:36:59.840 --> 00:37:08.950 align:middle line:84% as being 1 minus q times VOUT because notice 00:37:08.950 --> 00:37:11.610 align:middle line:90% this voltage, Vx, pulsates. 00:37:11.610 --> 00:37:14.320 align:middle line:84% When the switch is on, when q is high, it's 0. 00:37:14.320 --> 00:37:16.600 align:middle line:84% And when switch is off, it's equal to VOUT. 00:37:16.600 --> 00:37:18.780 align:middle line:84% So in continuous conduction mode, 00:37:18.780 --> 00:37:21.900 align:middle line:84% this is a perfectly legitimate representation 00:37:21.900 --> 00:37:23.730 align:middle line:84% of what the voltage is doing there. 00:37:23.730 --> 00:37:26.700 align:middle line:84% So I'm replacing the switch with something 00:37:26.700 --> 00:37:31.740 align:middle line:84% that generates the same voltage, Vx, as the switch does. 00:37:31.740 --> 00:37:35.250 align:middle line:84% I'm going to replace the current diode 00:37:35.250 --> 00:37:39.960 align:middle line:84% with a dependent current source that equals id. 00:37:39.960 --> 00:37:45.330 align:middle line:84% And that would be 1 minus q times iL. 00:37:45.330 --> 00:37:47.850 align:middle line:84% When the switch is on, this current 00:37:47.850 --> 00:37:49.650 align:middle line:90% is 0 because the diode is off. 00:37:49.650 --> 00:37:52.500 align:middle line:84% When the switch is off, this dependent current source 00:37:52.500 --> 00:37:55.120 align:middle line:90% is equal to iL. 00:37:55.120 --> 00:37:57.480 align:middle line:84% So I've got one dependent voltage source 00:37:57.480 --> 00:37:59.340 align:middle line:84% and one dependent current source, which 00:37:59.340 --> 00:38:01.650 align:middle line:84% impose the same constraints as what's 00:38:01.650 --> 00:38:07.080 align:middle line:84% happening at this node as the original switch in diode. 00:38:07.080 --> 00:38:08.405 align:middle line:90% Any questions about that? 00:38:08.405 --> 00:38:11.290 align:middle line:90% 00:38:11.290 --> 00:38:13.096 align:middle line:90% Yeah, Laurie? 00:38:13.096 --> 00:38:19.010 align:middle line:84% STUDENT: So [INAUDIBLE] is your choice [INAUDIBLE]? 00:38:19.010 --> 00:38:21.100 align:middle line:90% DAVID PERREAULT: No. 00:38:21.100 --> 00:38:22.840 align:middle line:84% In fact, I could have modeled this 00:38:22.840 --> 00:38:24.100 align:middle line:84% with a dependent current source and that 00:38:24.100 --> 00:38:25.150 align:middle line:90% with a dependent voltage source. 00:38:25.150 --> 00:38:26.483 align:middle line:90% It would have been totally fine. 00:38:26.483 --> 00:38:28.830 align:middle line:84% It would have done the same thing. 00:38:28.830 --> 00:38:31.390 align:middle line:84% There's a lot of ways I could do this. 00:38:31.390 --> 00:38:33.910 align:middle line:84% And in fact, I don't even have to do it. 00:38:33.910 --> 00:38:38.000 align:middle line:84% I could operate everything I was going to say with the switches. 00:38:38.000 --> 00:38:39.580 align:middle line:84% But really, what I'm interested in 00:38:39.580 --> 00:38:41.530 align:middle line:84% is, how does the switch in the diode 00:38:41.530 --> 00:38:44.050 align:middle line:84% control the waveforms in the circuit 00:38:44.050 --> 00:38:48.100 align:middle line:84% and cause currents to be fed to the capacitor and voltages 00:38:48.100 --> 00:38:49.400 align:middle line:90% to be fed to the inductor? 00:38:49.400 --> 00:38:52.850 align:middle line:84% And this is just a simpler way to get it. 00:38:52.850 --> 00:38:56.095 align:middle line:84% Does that make sense to everybody? 00:38:56.095 --> 00:39:00.250 align:middle line:84% So this is still a switching circuit. 00:39:00.250 --> 00:39:03.250 align:middle line:84% In fact, if I simulated this circuit and I knew q of t 00:39:03.250 --> 00:39:03.970 align:middle line:90% from some-- 00:39:03.970 --> 00:39:07.120 align:middle line:84% I ran a controller and I had some q of t, 00:39:07.120 --> 00:39:08.530 align:middle line:90% I could simulate this. 00:39:08.530 --> 00:39:13.340 align:middle line:84% And it would generate exactly this switching waveform. 00:39:13.340 --> 00:39:16.180 align:middle line:84% If I told you q of t and then q of t 00:39:16.180 --> 00:39:18.940 align:middle line:84% is the constant duty ratio here and I simulated 00:39:18.940 --> 00:39:23.560 align:middle line:84% this circuit with that q of t, I would-- and I changed the load 00:39:23.560 --> 00:39:26.970 align:middle line:84% resistance, I would get exactly that response. 00:39:26.970 --> 00:39:30.070 align:middle line:84% So as long as I'm in continuous conduction mode, 00:39:30.070 --> 00:39:33.800 align:middle line:84% this model represents that circuit. 00:39:33.800 --> 00:39:37.510 align:middle line:84% But I want to get rid of the switching. 00:39:37.510 --> 00:39:40.830 align:middle line:84% So let me just go average this circuit. 00:39:40.830 --> 00:39:43.470 align:middle line:84% I'm going to apply that operator to that circuit. 00:39:43.470 --> 00:39:44.700 align:middle line:90% And what do I know? 00:39:44.700 --> 00:39:47.760 align:middle line:84% Well, I know that if I apply the operator to the circuit, 00:39:47.760 --> 00:39:51.790 align:middle line:84% the topology of the new circuit doesn't change. 00:39:51.790 --> 00:39:54.240 align:middle line:90% Why? 00:39:54.240 --> 00:39:55.920 align:middle line:84% Because when I apply averaging, I 00:39:55.920 --> 00:39:58.200 align:middle line:84% get the same set of KVL and KCL equations. 00:39:58.200 --> 00:40:01.390 align:middle line:84% So the topology ought to be the same. 00:40:01.390 --> 00:40:04.530 align:middle line:84% And what that means is I can take the same circuit. 00:40:04.530 --> 00:40:11.070 align:middle line:84% And I can just replace VIN by VIN bar, iL by iL bar, 00:40:11.070 --> 00:40:12.990 align:middle line:90% VOUT by VOUT bar. 00:40:12.990 --> 00:40:16.050 align:middle line:84% And L, C, and R all stay the same 00:40:16.050 --> 00:40:17.580 align:middle line:84% because I've told you what happens 00:40:17.580 --> 00:40:23.720 align:middle line:84% to the I-V characteristics of my linear time-invariant 00:40:23.720 --> 00:40:25.332 align:middle line:90% components. 00:40:25.332 --> 00:40:27.740 align:middle line:84% The only thing I haven't said what happens 00:40:27.740 --> 00:40:31.920 align:middle line:84% is to these two components I've drawn in pink. 00:40:31.920 --> 00:40:35.210 align:middle line:84% And so what I really have to do is take the local average 00:40:35.210 --> 00:40:41.240 align:middle line:84% of the product of VOUT and the switching waveform and iL 00:40:41.240 --> 00:40:44.270 align:middle line:90% and the switching waveform. 00:40:44.270 --> 00:40:46.580 align:middle line:84% And that's ugly because now I have 00:40:46.580 --> 00:40:49.130 align:middle line:90% the local average of a product. 00:40:49.130 --> 00:40:53.000 align:middle line:84% And I've got to go think about that a little bit. 00:40:53.000 --> 00:40:56.090 align:middle line:84% Now, let me tell you something about the local average 00:40:56.090 --> 00:40:58.640 align:middle line:90% of products. 00:40:58.640 --> 00:41:03.230 align:middle line:84% One property that this operator doesn't have, in general, 00:41:03.230 --> 00:41:11.600 align:middle line:84% is that typically, if I have x of t y of t 00:41:11.600 --> 00:41:20.840 align:middle line:84% and I take the local average of that, in general, it's not 00:41:20.840 --> 00:41:25.850 align:middle line:90% equal to x bar of t y bar of t. 00:41:25.850 --> 00:41:30.490 align:middle line:84% You can't just arbitrarily do that. 00:41:30.490 --> 00:41:32.240 align:middle line:90% How can I prove it to you? 00:41:32.240 --> 00:41:38.400 align:middle line:84% Suppose I had two signals that were like this. 00:41:38.400 --> 00:41:39.825 align:middle line:90% Here's x of t. 00:41:39.825 --> 00:41:45.840 align:middle line:90% 00:41:45.840 --> 00:41:49.500 align:middle line:84% Suppose x of t is just a 50% duty ratio square wave 00:41:49.500 --> 00:41:50.130 align:middle line:90% like this. 00:41:50.130 --> 00:41:51.690 align:middle line:90% Here's 0.5t. 00:41:51.690 --> 00:41:52.980 align:middle line:90% Here's t. 00:41:52.980 --> 00:41:56.650 align:middle line:90% This is x of t. 00:41:56.650 --> 00:42:00.810 align:middle line:84% What is x bar of t for that signal? 00:42:00.810 --> 00:42:05.907 align:middle line:84% And suppose this goes on forever and this is of height 1. 00:42:05.907 --> 00:42:06.990 align:middle line:90% It's just going to be 1/2. 00:42:06.990 --> 00:42:09.870 align:middle line:84% At any given time, if I slide a window over this 00:42:09.870 --> 00:42:11.580 align:middle line:84% and I take the average, I've always 00:42:11.580 --> 00:42:13.930 align:middle line:90% got it high half the time. 00:42:13.930 --> 00:42:20.470 align:middle line:84% So x bar of t is just equal to 0.5. 00:42:20.470 --> 00:42:23.970 align:middle line:84% Well, what happens if I made its complement, y bar of t? 00:42:23.970 --> 00:42:30.120 align:middle line:90% 00:42:30.120 --> 00:42:32.800 align:middle line:90% I'm sorry, here's y of t. 00:42:32.800 --> 00:42:35.920 align:middle line:84% y of t is exactly the complement of x of t. 00:42:35.920 --> 00:42:44.375 align:middle line:84% So what I would argue is that y bar of t is also equal to 0.5. 00:42:44.375 --> 00:42:46.000 align:middle line:84% It's just-- happens to be out of phase. 00:42:46.000 --> 00:42:48.730 align:middle line:90% 00:42:48.730 --> 00:42:50.730 align:middle line:90% Well, what's x of t y of t? 00:42:50.730 --> 00:42:56.440 align:middle line:90% 00:42:56.440 --> 00:42:59.050 align:middle line:84% I think I heard somebody say that it's 0. 00:42:59.050 --> 00:43:01.250 align:middle line:90% One of them is 0 all the time. 00:43:01.250 --> 00:43:03.160 align:middle line:90% And that's "zip." 00:43:03.160 --> 00:43:07.120 align:middle line:84% So if I took the local average of x bar-- 00:43:07.120 --> 00:43:10.120 align:middle line:84% if I took the local average of x times y, 00:43:10.120 --> 00:43:12.430 align:middle line:84% that would have to be equal to 0. 00:43:12.430 --> 00:43:20.420 align:middle line:84% But if I took x bar y bar, that would actually be equal to 0.25. 00:43:20.420 --> 00:43:23.560 align:middle line:84% So that's proof that in general, I can't just 00:43:23.560 --> 00:43:26.110 align:middle line:84% take the local average of the product 00:43:26.110 --> 00:43:28.360 align:middle line:84% and expect it to be the product of the local averages. 00:43:28.360 --> 00:43:32.280 align:middle line:84% That just doesn't work, generally. 00:43:32.280 --> 00:43:35.520 align:middle line:84% And I care because I have this local average 00:43:35.520 --> 00:43:36.760 align:middle line:90% of a product over here. 00:43:36.760 --> 00:43:42.240 align:middle line:84% But imagine the reason this failed 00:43:42.240 --> 00:43:45.810 align:middle line:84% was because when y was high, x was low, and vice versa. 00:43:45.810 --> 00:43:49.750 align:middle line:84% They were both doing crazy stuff. 00:43:49.750 --> 00:43:54.310 align:middle line:84% What happens if, perhaps, one of x of t and y of t 00:43:54.310 --> 00:43:57.760 align:middle line:84% look like a constant, roughly speaking? 00:43:57.760 --> 00:43:59.720 align:middle line:90% And what do I mean by constant? 00:43:59.720 --> 00:44:01.720 align:middle line:84% I mean that it has two properties. 00:44:01.720 --> 00:44:09.690 align:middle line:84% If x or y has, one, a small ripple-- 00:44:09.690 --> 00:44:13.060 align:middle line:90% 00:44:13.060 --> 00:44:18.310 align:middle line:84% in other words, it's not varying full-scale within the cycle-- 00:44:18.310 --> 00:44:26.140 align:middle line:84% and, two, slow variation, meaning it's not one thing 00:44:26.140 --> 00:44:29.078 align:middle line:84% in one cycle and jumping to some massive thing in another cycle, 00:44:29.078 --> 00:44:31.120 align:middle line:84% massively different in another cycle, it's just-- 00:44:31.120 --> 00:44:33.850 align:middle line:90% the two waveforms just-- 00:44:33.850 --> 00:44:36.220 align:middle line:84% one of the two just-- it may move, 00:44:36.220 --> 00:44:37.590 align:middle line:90% but it only moves a little bit-- 00:44:37.590 --> 00:44:40.312 align:middle line:90% 00:44:40.312 --> 00:44:46.870 align:middle line:84% then I can think within any given interval, one of those two 00:44:46.870 --> 00:44:50.210 align:middle line:84% is constant and then like linearity. 00:44:50.210 --> 00:44:53.860 align:middle line:84% Then I can just take the product of the local averages. 00:44:53.860 --> 00:44:58.090 align:middle line:84% So in this case, I can have xy bar 00:44:58.090 --> 00:45:00.590 align:middle line:84% is approximately equal to x bar y bar. 00:45:00.590 --> 00:45:02.800 align:middle line:84% So the only reason this failed is 00:45:02.800 --> 00:45:05.900 align:middle line:84% because they're both flapping up and down in counterpoint. 00:45:05.900 --> 00:45:07.700 align:middle line:84% If only one of them is flapping up and down 00:45:07.700 --> 00:45:10.900 align:middle line:84% and the other is constant, at least constant-- 00:45:10.900 --> 00:45:15.430 align:middle line:84% not changing that much in a given capital T interval, 00:45:15.430 --> 00:45:18.910 align:middle line:90% I could make that approximation. 00:45:18.910 --> 00:45:21.855 align:middle line:84% Does that make sense to everybody? 00:45:21.855 --> 00:45:28.470 align:middle line:84% So this might be true of if x or y is an inductor 00:45:28.470 --> 00:45:31.650 align:middle line:90% current or a capacitor voltage. 00:45:31.650 --> 00:45:35.437 align:middle line:84% We know that even in that example I was showing you, yeah, 00:45:35.437 --> 00:45:36.270 align:middle line:90% there's some ripple. 00:45:36.270 --> 00:45:41.040 align:middle line:84% But it's on top of a big DC offset. 00:45:41.040 --> 00:45:44.710 align:middle line:84% It's not really violating that approximation very much. 00:45:44.710 --> 00:45:52.530 align:middle line:84% So here, maybe I could say that 1 minus q-- 00:45:52.530 --> 00:45:57.060 align:middle line:84% the product of 1 minus q times VOUT 00:45:57.060 --> 00:46:02.790 align:middle line:84% is approximately equal to 1 minus q times VOUT bar 00:46:02.790 --> 00:46:08.910 align:middle line:84% and 1 minus q times iL is approximately equal to 1 00:46:08.910 --> 00:46:11.255 align:middle line:90% minus q bar times iL bar. 00:46:11.255 --> 00:46:15.330 align:middle line:90% 00:46:15.330 --> 00:46:21.765 align:middle line:84% Let me define d as being equal to q bar of t. 00:46:21.765 --> 00:46:26.020 align:middle line:90% 00:46:26.020 --> 00:46:27.900 align:middle line:84% In any given cycle, if I'm assuming 00:46:27.900 --> 00:46:30.030 align:middle line:84% my duty ratio is changing slowly, 00:46:30.030 --> 00:46:32.640 align:middle line:84% it's about equal to the local average of q of t 00:46:32.640 --> 00:46:35.580 align:middle line:84% because that's what my PWM modulator over there was doing. 00:46:35.580 --> 00:46:38.120 align:middle line:90% It was giving me a duty ratio. 00:46:38.120 --> 00:46:42.480 align:middle line:84% Now, this is precisely only true when I'm looking at capital T. 00:46:42.480 --> 00:46:43.590 align:middle line:90% And I look back. 00:46:43.590 --> 00:46:46.500 align:middle line:84% And I get d for that cycle is exactly here 00:46:46.500 --> 00:46:48.100 align:middle line:84% if I did my local average right here. 00:46:48.100 --> 00:46:50.610 align:middle line:84% But if d of t is changing only slowly, 00:46:50.610 --> 00:46:55.740 align:middle line:84% that's a pretty good model to within half a cycle. 00:46:55.740 --> 00:46:59.200 align:middle line:90% So then I could replace this. 00:46:59.200 --> 00:47:07.710 align:middle line:84% I could say 1 minus q is equal to 1 minus d times VOUT bar. 00:47:07.710 --> 00:47:11.920 align:middle line:84% And this is equal to 1 minus d times iL bar. 00:47:11.920 --> 00:47:14.260 align:middle line:90% And I can average this circuit. 00:47:14.260 --> 00:47:17.310 align:middle line:84% And what I'm going to get is this. 00:47:17.310 --> 00:47:23.760 align:middle line:90% VIN bar iL bar-- 00:47:23.760 --> 00:47:31.500 align:middle line:84% this becomes a source 1 minus d times VOUT bar. 00:47:31.500 --> 00:47:44.310 align:middle line:84% And this becomes 1 minus d times iL bar CR VOUT bar. 00:47:44.310 --> 00:47:49.230 align:middle line:90% This circuit has no switching. 00:47:49.230 --> 00:47:51.660 align:middle line:84% d is a continuously varying variable 00:47:51.660 --> 00:47:54.090 align:middle line:84% that my controller might generate. 00:47:54.090 --> 00:47:56.970 align:middle line:84% There's no switching left in this circuit. 00:47:56.970 --> 00:47:58.622 align:middle line:90% It does vary in time. 00:47:58.622 --> 00:47:59.580 align:middle line:90% It's a dynamic circuit. 00:47:59.580 --> 00:48:02.293 align:middle line:84% It has an L and a C. And if d changes over time, 00:48:02.293 --> 00:48:04.210 align:middle line:84% the voltages and currents in this will change. 00:48:04.210 --> 00:48:06.090 align:middle line:84% But I've killed the switching ripple, 00:48:06.090 --> 00:48:11.090 align:middle line:84% essentially applying my local average operator to the circuit. 00:48:11.090 --> 00:48:16.110 align:middle line:90% 00:48:16.110 --> 00:48:18.320 align:middle line:90% That make sense to everybody? 00:48:18.320 --> 00:48:21.840 align:middle line:84% Let's take a look at what happens if I do that. 00:48:21.840 --> 00:48:29.480 align:middle line:84% This is a simulation of this exact circuit 00:48:29.480 --> 00:48:33.030 align:middle line:84% for the same transient, except that I'm simulating this thing. 00:48:33.030 --> 00:48:38.390 align:middle line:84% It's in that little box that says "average model." 00:48:38.390 --> 00:48:39.960 align:middle line:90% There is a diode there. 00:48:39.960 --> 00:48:42.210 align:middle line:84% That diode is just providing a constant drop 00:48:42.210 --> 00:48:45.080 align:middle line:84% because keep in mind my approximate models didn't really 00:48:45.080 --> 00:48:48.030 align:middle line:84% include constant diode drops or anything like that. 00:48:48.030 --> 00:48:50.310 align:middle line:84% So that diode there is not doing any switching. 00:48:50.310 --> 00:48:52.230 align:middle line:84% It's providing a constant DC offset. 00:48:52.230 --> 00:48:54.530 align:middle line:84% But this is the transient you get when you change 00:48:54.530 --> 00:48:57.500 align:middle line:90% the resistance in that circuit. 00:48:57.500 --> 00:48:59.390 align:middle line:90% Here's the voltage. 00:48:59.390 --> 00:49:00.750 align:middle line:90% And here's the current. 00:49:00.750 --> 00:49:03.380 align:middle line:90% It's right up the middle. 00:49:03.380 --> 00:49:07.070 align:middle line:84% So basically, I've got a circuit that has second-order response. 00:49:07.070 --> 00:49:12.500 align:middle line:84% And it tracks the average values of my waveforms-- 00:49:12.500 --> 00:49:14.630 align:middle line:90% gotten rid of the switching. 00:49:14.630 --> 00:49:21.420 align:middle line:90% So this circuit could be-- 00:49:21.420 --> 00:49:23.730 align:middle line:90% it's not in a controller form. 00:49:23.730 --> 00:49:26.910 align:middle line:84% But it's a circuit that tracks this whole thing 00:49:26.910 --> 00:49:28.150 align:middle line:90% without any switching. 00:49:28.150 --> 00:49:30.740 align:middle line:90% 00:49:30.740 --> 00:49:31.690 align:middle line:90% Questions about that? 00:49:31.690 --> 00:49:36.940 align:middle line:90% 00:49:36.940 --> 00:49:41.670 align:middle line:84% Now, we're going to take this up next class. 00:49:41.670 --> 00:49:49.670 align:middle line:84% But I wanted to show you the basis for doing this. 00:49:49.670 --> 00:49:52.840 align:middle line:84% I want to say this is really useful for building 00:49:52.840 --> 00:49:54.725 align:middle line:84% control models for converters because now I 00:49:54.725 --> 00:49:55.600 align:middle line:90% have something that-- 00:49:55.600 --> 00:49:58.780 align:middle line:84% I've gotten rid of all the complicated switching stuff. 00:49:58.780 --> 00:50:01.750 align:middle line:84% But it still gives me the low-frequency dynamics 00:50:01.750 --> 00:50:04.660 align:middle line:84% that I want to deal with my controller. 00:50:04.660 --> 00:50:09.430 align:middle line:84% I will warn you that this system is nonlinear. 00:50:09.430 --> 00:50:10.370 align:middle line:90% Why? 00:50:10.370 --> 00:50:13.930 align:middle line:84% Because you notice d is my control variable here. 00:50:13.930 --> 00:50:16.900 align:middle line:84% And it multiplies the output voltage. 00:50:16.900 --> 00:50:18.770 align:middle line:84% And it multiplies the inductor current. 00:50:18.770 --> 00:50:21.160 align:middle line:84% So I have a product of my control input and my state 00:50:21.160 --> 00:50:21.830 align:middle line:90% variables. 00:50:21.830 --> 00:50:24.100 align:middle line:84% This is not a linear system from-- in terms 00:50:24.100 --> 00:50:27.020 align:middle line:84% of going from duty ratio to output. 00:50:27.020 --> 00:50:29.660 align:middle line:84% So I still have nonlinearity I have to deal with. 00:50:29.660 --> 00:50:31.880 align:middle line:90% And that's just life. 00:50:31.880 --> 00:50:34.820 align:middle line:84% We'll talk about how to deal with that. 00:50:34.820 --> 00:50:37.940 align:middle line:84% It's very good for doing low-frequency controller 00:50:37.940 --> 00:50:40.110 align:middle line:90% modeling. 00:50:40.110 --> 00:50:41.940 align:middle line:84% It's not useful for other things. 00:50:41.940 --> 00:50:46.790 align:middle line:84% So for example, if I asked you to calculate power dissipation 00:50:46.790 --> 00:50:49.650 align:middle line:84% of components, you cannot use this model. 00:50:49.650 --> 00:50:50.550 align:middle line:90% Why? 00:50:50.550 --> 00:50:53.260 align:middle line:84% Because what would the power dissipation in the switch be? 00:50:53.260 --> 00:50:54.718 align:middle line:84% The power dissipation of the switch 00:50:54.718 --> 00:50:56.730 align:middle line:84% would be the switch voltage times the switch 00:50:56.730 --> 00:50:59.950 align:middle line:84% current, both of which are pulsing up and down together. 00:50:59.950 --> 00:51:03.640 align:middle line:84% It violates this assumption I was talking about. 00:51:03.640 --> 00:51:05.790 align:middle line:84% In fact, the voltage and current on the switch 00:51:05.790 --> 00:51:07.970 align:middle line:90% actually look a lot like that. 00:51:07.970 --> 00:51:10.250 align:middle line:84% So I can use this average model to do things 00:51:10.250 --> 00:51:11.640 align:middle line:90% like control dynamics. 00:51:11.640 --> 00:51:13.040 align:middle line:90% That's perfect for it. 00:51:13.040 --> 00:51:15.140 align:middle line:84% I can't use it to answer questions 00:51:15.140 --> 00:51:18.930 align:middle line:84% like dissipation or other kinds of things. 00:51:18.930 --> 00:51:22.140 align:middle line:90% So I am well out of time today. 00:51:22.140 --> 00:51:25.440 align:middle line:84% But we will take up the notion of control modeling next class. 00:51:25.440 --> 00:51:27.350 align:middle line:90% Have a great day. 00:51:27.350 --> 00:51:32.000 align:middle line:90%