WEBVTT 00:00:00.000 --> 00:00:01.952 align:middle line:90% [SQUEAKING] 00:00:01.952 --> 00:00:04.392 align:middle line:90% [RUSTLING] 00:00:04.392 --> 00:00:06.832 align:middle line:90% [CLICKING] 00:00:06.832 --> 00:00:09.280 align:middle line:90% 00:00:09.280 --> 00:00:12.160 align:middle line:90% DAVID PERREAULT: --control. 00:00:12.160 --> 00:00:18.620 align:middle line:84% And we said that we saw a few phenomena. 00:00:18.620 --> 00:00:23.620 align:middle line:84% So if I took, for example, a boost converter 00:00:23.620 --> 00:00:38.650 align:middle line:84% as my canonical example, I have V, the output voltage, 00:00:38.650 --> 00:00:42.880 align:middle line:84% as one of my state variables because the capacitor C and IL 00:00:42.880 --> 00:00:44.920 align:middle line:90% is the other state variable. 00:00:44.920 --> 00:00:50.500 align:middle line:84% And what we really control here is q of t 00:00:50.500 --> 00:00:56.230 align:middle line:84% or the local average of q of t, which we call the duty ratio. 00:00:56.230 --> 00:01:02.200 align:middle line:84% And we said we developed linearized average models 00:01:02.200 --> 00:01:07.930 align:middle line:84% for this thing with the notion that if I had some reference 00:01:07.930 --> 00:01:13.510 align:middle line:84% voltage, and I compare it to the output voltage, 00:01:13.510 --> 00:01:18.400 align:middle line:84% and I went into a compensator GC of S, 00:01:18.400 --> 00:01:21.490 align:middle line:84% and I use that to generate duty ratio 00:01:21.490 --> 00:01:28.360 align:middle line:84% and then I had my converter, which might have some IL, 00:01:28.360 --> 00:01:31.720 align:middle line:84% it might have some V out, and I then 00:01:31.720 --> 00:01:34.850 align:middle line:84% compare that output to my reference, 00:01:34.850 --> 00:01:36.940 align:middle line:90% I get an error voltage. 00:01:36.940 --> 00:01:39.670 align:middle line:84% And then I use that error voltage to generate 00:01:39.670 --> 00:01:40.628 align:middle line:90% the duty ratio I want. 00:01:40.628 --> 00:01:41.920 align:middle line:90% I stick that into my converter. 00:01:41.920 --> 00:01:47.000 align:middle line:84% And we generated some linearized average models for this thing. 00:01:47.000 --> 00:01:53.560 align:middle line:84% So I might think about the variations in all these things 00:01:53.560 --> 00:01:55.330 align:middle line:90% to linearize it. 00:01:55.330 --> 00:01:59.050 align:middle line:84% And we came up with a model for our converter, H of S, 00:01:59.050 --> 00:02:03.310 align:middle line:84% which basically converts between variations in duty ratio 00:02:03.310 --> 00:02:07.000 align:middle line:84% to variations in output voltage, and we design a converter. 00:02:07.000 --> 00:02:11.030 align:middle line:84% And when we did that, we saw a couple of things. 00:02:11.030 --> 00:02:14.870 align:middle line:84% First, we saw that the dynamics were nonlinear. 00:02:14.870 --> 00:02:17.550 align:middle line:84% That's why we had to do this linearization thing. 00:02:17.550 --> 00:02:19.800 align:middle line:84% Now, are they always nonlinear in a converter? 00:02:19.800 --> 00:02:21.690 align:middle line:90% No. 00:02:21.690 --> 00:02:23.460 align:middle line:90% But many, many converters are. 00:02:23.460 --> 00:02:26.410 align:middle line:84% And in this case, why is that nonlinearity there? 00:02:26.410 --> 00:02:28.560 align:middle line:84% The nonlinearity is there because what we're seeing 00:02:28.560 --> 00:02:33.420 align:middle line:84% is our second order responses are some kind of oscillations 00:02:33.420 --> 00:02:35.520 align:middle line:84% between the inductor and the capacitor 00:02:35.520 --> 00:02:37.180 align:middle line:90% and damped by the load resistor. 00:02:37.180 --> 00:02:40.500 align:middle line:84% And those oscillations are mediated 00:02:40.500 --> 00:02:43.620 align:middle line:84% through the switching or the modulation of the switches, 00:02:43.620 --> 00:02:45.600 align:middle line:90% which varies with time. 00:02:45.600 --> 00:02:48.690 align:middle line:84% And more to the point, the steady state value of that 00:02:48.690 --> 00:02:51.840 align:middle line:84% varies with DC input to output voltage 00:02:51.840 --> 00:02:55.060 align:middle line:90% and hence the dynamics vary. 00:02:55.060 --> 00:02:58.740 align:middle line:84% Because the modulation is mediated by the switching, 00:02:58.740 --> 00:03:00.520 align:middle line:90% we get nonlinear behavior. 00:03:00.520 --> 00:03:03.900 align:middle line:84% And what that means is that the dynamics, the small signal 00:03:03.900 --> 00:03:07.350 align:middle line:84% dynamics, physically vary with different values of, say, input 00:03:07.350 --> 00:03:08.710 align:middle line:90% voltage and output voltage. 00:03:08.710 --> 00:03:11.293 align:middle line:84% So if I have a different value of input voltage and same value 00:03:11.293 --> 00:03:14.470 align:middle line:84% of output voltage, I should expect my dynamics to vary. 00:03:14.470 --> 00:03:22.970 align:middle line:84% So one, we saw nonlinearity which arises. 00:03:22.970 --> 00:03:29.000 align:middle line:84% The other thing we saw was a pole-zero map. 00:03:29.000 --> 00:03:33.010 align:middle line:84% If I looked at H of S, what's going on in here, 00:03:33.010 --> 00:03:38.240 align:middle line:84% the one thing we saw was a right-half-plane zero 00:03:38.240 --> 00:03:40.670 align:middle line:90% in our transfer function. 00:03:40.670 --> 00:03:43.760 align:middle line:84% And we said that that right-half-plane zero 00:03:43.760 --> 00:03:47.690 align:middle line:84% comes from the underlying way in which the converter processes 00:03:47.690 --> 00:03:48.320 align:middle line:90% energy. 00:03:48.320 --> 00:03:50.040 align:middle line:90% It's not a function of the math. 00:03:50.040 --> 00:03:53.240 align:middle line:84% It's because when I change what I'm doing, 00:03:53.240 --> 00:03:55.880 align:middle line:84% the dynamics it goes through is reflective of what 00:03:55.880 --> 00:03:57.830 align:middle line:90% a right-half-plane zero does. 00:03:57.830 --> 00:04:00.980 align:middle line:84% So I'm kind of stuck with this right-half-plane zero. 00:04:00.980 --> 00:04:03.620 align:middle line:84% And then the other thing we saw was 00:04:03.620 --> 00:04:07.280 align:middle line:84% that the pole locations, which are due to the L and C 00:04:07.280 --> 00:04:09.200 align:middle line:84% and how they interact with the resistance, 00:04:09.200 --> 00:04:14.050 align:middle line:84% so it's a second-order system, tend to be very lightly damped, 00:04:14.050 --> 00:04:15.950 align:middle line:90% like this, something like that. 00:04:15.950 --> 00:04:19.970 align:middle line:84% And the amount of damping depends upon the load resistance 00:04:19.970 --> 00:04:23.350 align:middle line:84% R. So I'm really relying on this load resistance 00:04:23.350 --> 00:04:27.160 align:middle line:84% to damp the oscillations, or that makes 00:04:27.160 --> 00:04:28.600 align:middle line:90% it hard to design a controller. 00:04:28.600 --> 00:04:30.670 align:middle line:84% Lightly damp poles and a right-half-plane zero 00:04:30.670 --> 00:04:32.920 align:middle line:84% mean I'm not a very happy camper in terms 00:04:32.920 --> 00:04:34.720 align:middle line:90% of designing a good controller. 00:04:34.720 --> 00:04:38.990 align:middle line:90% So that's where we came to. 00:04:38.990 --> 00:04:41.140 align:middle line:84% We said, well, what can I do about this situation? 00:04:41.140 --> 00:04:42.890 align:middle line:84% Because this is a hard thing to deal with. 00:04:42.890 --> 00:04:44.200 align:middle line:90% You can deal with it. 00:04:44.200 --> 00:04:46.840 align:middle line:84% You can design a GC of S, a compensator that 00:04:46.840 --> 00:04:49.570 align:middle line:84% will deal with that, but it's not very pretty. 00:04:49.570 --> 00:04:56.080 align:middle line:84% One thing you can do is by choosing L and C appropriately, 00:04:56.080 --> 00:05:00.520 align:middle line:84% that helps you set dynamics the way you'd like them. 00:05:00.520 --> 00:05:02.810 align:middle line:84% The other thing you can do is change the plant. 00:05:02.810 --> 00:05:04.450 align:middle line:84% So we said, oh, it's this resistor 00:05:04.450 --> 00:05:08.090 align:middle line:90% that's damping the poles. 00:05:08.090 --> 00:05:15.440 align:middle line:84% Maybe I could go add RD and CD such 00:05:15.440 --> 00:05:18.790 align:middle line:84% that CD is a short circuit near the corner frequency 00:05:18.790 --> 00:05:20.240 align:middle line:84% where the L and C are oscillating, 00:05:20.240 --> 00:05:23.390 align:middle line:84% so RD serves to damp it, and RD doesn't vary. 00:05:23.390 --> 00:05:30.070 align:middle line:84% So by picking additions to my plant, by changing my converter, 00:05:30.070 --> 00:05:31.930 align:middle line:84% I can help make things better damped 00:05:31.930 --> 00:05:33.980 align:middle line:84% and maybe get better pole locations here. 00:05:33.980 --> 00:05:36.400 align:middle line:84% I'd have another pole location, but maybe it 00:05:36.400 --> 00:05:40.900 align:middle line:84% would be a low-order overdamped pole location. 00:05:40.900 --> 00:05:42.108 align:middle line:90% So that's one thing I can do. 00:05:42.108 --> 00:05:44.567 align:middle line:84% I don't like to do that because then I got to buy hardware. 00:05:44.567 --> 00:05:46.820 align:middle line:84% I got to buy a big capacitor and that kind of thing. 00:05:46.820 --> 00:05:53.320 align:middle line:84% The question is, what else can I do to make a better power 00:05:53.320 --> 00:05:56.800 align:middle line:90% converter controller? 00:05:56.800 --> 00:06:00.550 align:middle line:84% And that's what we're going to talk about today. 00:06:00.550 --> 00:06:03.910 align:middle line:90% And the basic idea is this. 00:06:03.910 --> 00:06:12.670 align:middle line:84% I have a converter that has two state variables. 00:06:12.670 --> 00:06:17.030 align:middle line:84% Its state variables are IL and VC. 00:06:17.030 --> 00:06:20.960 align:middle line:84% And yes, it has as its input the input voltage. 00:06:20.960 --> 00:06:25.000 align:middle line:84% And so what I've been doing is taking-- 00:06:25.000 --> 00:06:30.470 align:middle line:84% between these two, I've been feeding back the output voltage. 00:06:30.470 --> 00:06:32.030 align:middle line:84% What a control person would tell you 00:06:32.030 --> 00:06:33.738 align:middle line:84% is if you're not happy with your dynamics 00:06:33.738 --> 00:06:36.200 align:middle line:84% and you have a controllable system, 00:06:36.200 --> 00:06:37.760 align:middle line:90% feedback all of your states. 00:06:37.760 --> 00:06:39.840 align:middle line:90% Do full-state feedback. 00:06:39.840 --> 00:06:41.300 align:middle line:84% If you feedback all of your states, 00:06:41.300 --> 00:06:44.720 align:middle line:84% you can place your dynamics much better. 00:06:44.720 --> 00:06:47.720 align:middle line:84% And we're going to do a version of that 00:06:47.720 --> 00:06:50.450 align:middle line:90% called current mode control. 00:06:50.450 --> 00:06:57.077 align:middle line:84% The general concept originated in the 1960s. 00:06:57.077 --> 00:06:59.660 align:middle line:84% The version I'm going to tell you about came about in the late 00:06:59.660 --> 00:07:00.690 align:middle line:90% 1970s. 00:07:00.690 --> 00:07:02.918 align:middle line:84% And there's lots of versions of this idea. 00:07:02.918 --> 00:07:04.460 align:middle line:84% I'm telling you about what's probably 00:07:04.460 --> 00:07:07.730 align:middle line:84% the most popular one in DC-to-DC converters 00:07:07.730 --> 00:07:11.720 align:middle line:84% if you go out to buy one or to design one. 00:07:11.720 --> 00:07:14.180 align:middle line:84% A lot of the control chips use this technique. 00:07:14.180 --> 00:07:17.000 align:middle line:90% And here's the idea. 00:07:17.000 --> 00:07:20.630 align:middle line:84% For the converter, we said the input was-- well, 00:07:20.630 --> 00:07:22.550 align:middle line:90% ultimately the input is q of t. 00:07:22.550 --> 00:07:24.780 align:middle line:84% It's what's modulating the converter. 00:07:24.780 --> 00:07:34.200 align:middle line:84% Maybe I can say, OK, let me feed back the inductor current 00:07:34.200 --> 00:07:36.930 align:middle line:90% and compare that to a reference. 00:07:36.930 --> 00:07:39.010 align:middle line:84% And I'm going to call this reference IP. 00:07:39.010 --> 00:07:41.376 align:middle line:90% So this is a current reference. 00:07:41.376 --> 00:07:44.160 align:middle line:90% 00:07:44.160 --> 00:07:47.730 align:middle line:84% And based on the comparison of IL 00:07:47.730 --> 00:07:54.960 align:middle line:84% with IP in a current controller, I'm 00:07:54.960 --> 00:07:56.880 align:middle line:84% going to generate the switching function. 00:07:56.880 --> 00:07:58.380 align:middle line:84% And then what I'm going to do is I'm 00:07:58.380 --> 00:08:06.840 align:middle line:84% going to feed back the voltage and compare it to a reference 00:08:06.840 --> 00:08:11.940 align:middle line:84% and put that into a compensator GC of S. 00:08:11.940 --> 00:08:15.750 align:middle line:84% And I'm going to use GC of S to generate the current reference. 00:08:15.750 --> 00:08:18.920 align:middle line:84% So this is something called minor loop control. 00:08:18.920 --> 00:08:20.940 align:middle line:84% We're going to have a fast inner loop 00:08:20.940 --> 00:08:25.030 align:middle line:84% to control the current and then a slow outer loop 00:08:25.030 --> 00:08:26.010 align:middle line:90% to control the voltage. 00:08:26.010 --> 00:08:28.560 align:middle line:90% 00:08:28.560 --> 00:08:30.900 align:middle line:84% But we're going to feed back both current and voltage 00:08:30.900 --> 00:08:32.740 align:middle line:90% and thereby get better dynamics. 00:08:32.740 --> 00:08:34.890 align:middle line:90% That's the basic idea. 00:08:34.890 --> 00:08:37.510 align:middle line:90% Any questions about that? 00:08:37.510 --> 00:08:39.507 align:middle line:84% AUDIENCE: What does fast and slow mean? 00:08:39.507 --> 00:08:41.340 align:middle line:84% DAVID PERREAULT: Let me show you the scheme, 00:08:41.340 --> 00:08:43.382 align:middle line:84% and then you'll see what I mean by fast and slow. 00:08:43.382 --> 00:08:46.320 align:middle line:84% Remember, when we were talking about our models, 00:08:46.320 --> 00:08:50.070 align:middle line:84% for our average models up there, they're 00:08:50.070 --> 00:08:53.220 align:middle line:84% kind of good on time periods that are slow compared to half 00:08:53.220 --> 00:08:54.220 align:middle line:90% the switching frequency. 00:08:54.220 --> 00:08:58.245 align:middle line:84% So maybe you can go to 0.2 of the switching frequency. 00:08:58.245 --> 00:08:59.370 align:middle line:90% There's games you can play. 00:08:59.370 --> 00:09:01.840 align:middle line:84% But pretty much, you're looking at the average value. 00:09:01.840 --> 00:09:03.720 align:middle line:84% So you don't want your controller, 00:09:03.720 --> 00:09:06.360 align:middle line:84% GC of S in the top example, to be 00:09:06.360 --> 00:09:09.720 align:middle line:84% changing your command very quickly, certainly not 00:09:09.720 --> 00:09:11.563 align:middle line:90% within a cycle. 00:09:11.563 --> 00:09:13.230 align:middle line:84% In this scheme, what we're going to do-- 00:09:13.230 --> 00:09:16.450 align:middle line:84% that will still be true of this outer loop control. 00:09:16.450 --> 00:09:18.090 align:middle line:84% The inner loop control is actually 00:09:18.090 --> 00:09:20.970 align:middle line:84% going to run on a cycle-by-cycle basis. 00:09:20.970 --> 00:09:25.770 align:middle line:84% And that has a lot of follow-on benefits. 00:09:25.770 --> 00:09:28.560 align:middle line:84% So what we're going to do is we're 00:09:28.560 --> 00:09:30.840 align:middle line:84% going to come up with a scheme where we 00:09:30.840 --> 00:09:32.820 align:middle line:90% turn on the switch every cycle. 00:09:32.820 --> 00:09:34.680 align:middle line:84% And then I'm going to turn off the switch 00:09:34.680 --> 00:09:36.570 align:middle line:84% based on what the inductor current is 00:09:36.570 --> 00:09:38.137 align:middle line:90% doing in a particular way. 00:09:38.137 --> 00:09:39.970 align:middle line:84% And I'm going to show you the particular way 00:09:39.970 --> 00:09:41.380 align:middle line:90% we're going to examine. 00:09:41.380 --> 00:09:44.740 align:middle line:84% And so that's going to give us control over what the inductor 00:09:44.740 --> 00:09:46.220 align:middle line:90% current is doing. 00:09:46.220 --> 00:09:48.230 align:middle line:84% And then I'm going to use my outer loop 00:09:48.230 --> 00:09:51.590 align:middle line:84% to control that command to the inductor current. 00:09:51.590 --> 00:09:54.170 align:middle line:84% So the crudest way people like to think 00:09:54.170 --> 00:09:57.090 align:middle line:84% about this is I'm putting the inductor current inside a loop. 00:09:57.090 --> 00:10:00.560 align:middle line:84% So now I just get a commandable current. 00:10:00.560 --> 00:10:02.830 align:middle line:84% And then I sort of have a first-order system. 00:10:02.830 --> 00:10:04.080 align:middle line:90% Now, it's not quite that good. 00:10:04.080 --> 00:10:05.495 align:middle line:90% That's an oversimplification. 00:10:05.495 --> 00:10:07.315 align:middle line:84% I still have two state variables. 00:10:07.315 --> 00:10:08.690 align:middle line:84% It's still a second-order system, 00:10:08.690 --> 00:10:12.930 align:middle line:84% but it's not a bad way to think about it. 00:10:12.930 --> 00:10:16.760 align:middle line:84% All right, so how does this scheme work? 00:10:16.760 --> 00:10:20.550 align:middle line:84% Let me show you the basis of the scheme. 00:10:20.550 --> 00:10:24.000 align:middle line:90% It looks something like this. 00:10:24.000 --> 00:10:27.415 align:middle line:84% What I'm going to do is I'm going to create-- 00:10:27.415 --> 00:10:32.557 align:middle line:90% 00:10:32.557 --> 00:10:34.140 align:middle line:84% I'm going to have a current reference. 00:10:34.140 --> 00:10:36.950 align:middle line:90% Here's my current reference. 00:10:36.950 --> 00:10:39.800 align:middle line:90% Let me call this I peak. 00:10:39.800 --> 00:10:43.890 align:middle line:84% And this is-- within a switching cycle, 00:10:43.890 --> 00:10:48.110 align:middle line:84% this is more or less a constant defined by this compensator, 00:10:48.110 --> 00:10:49.700 align:middle line:84% but it's going to move up and down 00:10:49.700 --> 00:10:51.860 align:middle line:90% as my outer loop tells it to. 00:10:51.860 --> 00:10:55.820 align:middle line:84% Then I'm going to create a slight variation off that, 00:10:55.820 --> 00:10:59.805 align:middle line:84% and we'll explain why a little bit later. 00:10:59.805 --> 00:11:01.430 align:middle line:84% But suppose this is my switching period 00:11:01.430 --> 00:11:06.120 align:middle line:84% T. I'm going to create a slope with some slope. 00:11:06.120 --> 00:11:08.150 align:middle line:90% I'm going to call this minus MC. 00:11:08.150 --> 00:11:09.450 align:middle line:90% This is some slope-- 00:11:09.450 --> 00:11:11.000 align:middle line:90% MC is a positive number-- 00:11:11.000 --> 00:11:15.470 align:middle line:84% which is a ramp that I subtract off of my command, 00:11:15.470 --> 00:11:17.020 align:middle line:84% and I'm going to do that every cycle. 00:11:17.020 --> 00:11:20.550 align:middle line:90% 00:11:20.550 --> 00:11:24.760 align:middle line:84% So that's T. Here's 2T and so forth. 00:11:24.760 --> 00:11:27.970 align:middle line:84% So I'm going to do this every cycle. 00:11:27.970 --> 00:11:31.870 align:middle line:90% Now, here's my inductor current. 00:11:31.870 --> 00:11:35.500 align:middle line:90% So I'm going to sense IL. 00:11:35.500 --> 00:11:39.982 align:middle line:84% And what I'm going to do is at the beginning of each cycle, 00:11:39.982 --> 00:11:41.440 align:middle line:84% I'm going to turn on-- this is IL-- 00:11:41.440 --> 00:11:43.500 align:middle line:84% I'm going to turn on the inductor current. 00:11:43.500 --> 00:11:45.780 align:middle line:84% I'm going to turn on the switch, I'm sorry. 00:11:45.780 --> 00:11:47.190 align:middle line:84% And the inductor current is going 00:11:47.190 --> 00:11:49.630 align:middle line:90% to rise with some slope M1. 00:11:49.630 --> 00:11:54.480 align:middle line:84% So the beginning of the cycle, I come-- 00:11:54.480 --> 00:11:59.847 align:middle line:84% q of t turns high, inductor current ramps up. 00:11:59.847 --> 00:12:02.430 align:middle line:84% And what I'm going to do is I'm going to look at that inductor 00:12:02.430 --> 00:12:04.347 align:middle line:84% current, and I'm going to compare the inductor 00:12:04.347 --> 00:12:07.120 align:middle line:90% current to Ip minus this slope. 00:12:07.120 --> 00:12:09.040 align:middle line:84% I'm going to compare it to this pink signal. 00:12:09.040 --> 00:12:16.815 align:middle line:84% And when I hit here, when the inductor current exceeds the I 00:12:16.815 --> 00:12:20.850 align:middle line:84% peak minus this compensating ramp slope, 00:12:20.850 --> 00:12:24.210 align:middle line:84% I'm going to turn off the switch and let the switch current fall 00:12:24.210 --> 00:12:26.130 align:middle line:90% like this. 00:12:26.130 --> 00:12:30.450 align:middle line:90% So this is, say, DT. 00:12:30.450 --> 00:12:32.580 align:middle line:84% The switch turns on at the beginning of the cycle. 00:12:32.580 --> 00:12:36.000 align:middle line:84% But it turns off-- instead of me setting a duty ratio explicitly, 00:12:36.000 --> 00:12:38.610 align:middle line:84% I'm going to implicitly determine the duty ratio based 00:12:38.610 --> 00:12:40.930 align:middle line:90% on this peak current. 00:12:40.930 --> 00:12:44.590 align:middle line:84% And then, beginning of the next cycle, I do the same thing, 00:12:44.590 --> 00:12:49.400 align:middle line:84% I turn the switch on, it rises, it falls. 00:12:49.400 --> 00:12:52.340 align:middle line:84% And I get the same kind of general behavior. 00:12:52.340 --> 00:12:55.480 align:middle line:84% It's just that instead of saying turn 00:12:55.480 --> 00:12:58.450 align:middle line:84% the do switch off after some time, DT, 00:12:58.450 --> 00:13:00.640 align:middle line:84% I'm going to say turn it off implicitly when it hits 00:13:00.640 --> 00:13:03.750 align:middle line:90% some current that I command. 00:13:03.750 --> 00:13:05.045 align:middle line:90% Any questions about that? 00:13:05.045 --> 00:13:13.330 align:middle line:90% 00:13:13.330 --> 00:13:15.010 align:middle line:90% Why do I like that? 00:13:15.010 --> 00:13:17.110 align:middle line:84% Well, one thing we're going to see 00:13:17.110 --> 00:13:19.760 align:middle line:84% is that we're going to get better control dynamics. 00:13:19.760 --> 00:13:23.020 align:middle line:84% And I should say, by the way, this one, this slope, 00:13:23.020 --> 00:13:24.700 align:middle line:90% is minus M2. 00:13:24.700 --> 00:13:28.060 align:middle line:84% And why am I talking about M1, M2, minus MC? 00:13:28.060 --> 00:13:29.950 align:middle line:84% That's because this same picture applies 00:13:29.950 --> 00:13:31.010 align:middle line:90% to any kind of converter. 00:13:31.010 --> 00:13:37.930 align:middle line:84% For the boost, M1 is just equal to the input voltage over L, 00:13:37.930 --> 00:13:42.790 align:middle line:84% and M2 is equal to the output voltage minus the input 00:13:42.790 --> 00:13:45.152 align:middle line:84% voltage over L. But I could have drawn the same picture 00:13:45.152 --> 00:13:47.110 align:middle line:84% if it were a buck converter or the same picture 00:13:47.110 --> 00:13:50.970 align:middle line:90% if it were some other converter. 00:13:50.970 --> 00:13:52.477 align:middle line:90% So why am I doing this? 00:13:52.477 --> 00:13:53.560 align:middle line:90% Well, a couple of reasons. 00:13:53.560 --> 00:13:55.810 align:middle line:84% One, we're going to see it gives us much nicer dynamics. 00:13:55.810 --> 00:13:56.310 align:middle line:90% Why? 00:13:56.310 --> 00:13:59.310 align:middle line:84% Because I'm basically explicitly controlling 00:13:59.310 --> 00:14:00.625 align:middle line:90% the inductor current. 00:14:00.625 --> 00:14:02.250 align:middle line:84% And then the other thing that's kind of 00:14:02.250 --> 00:14:08.900 align:middle line:84% nice is if I did duty ratio control, what 00:14:08.900 --> 00:14:10.150 align:middle line:90% is the inductor current doing? 00:14:10.150 --> 00:14:10.692 align:middle line:90% I don't know. 00:14:10.692 --> 00:14:12.520 align:middle line:84% If I put a small resistance on it, 00:14:12.520 --> 00:14:14.040 align:middle line:84% it's just going to keep cranking up the current 00:14:14.040 --> 00:14:15.207 align:middle line:90% until it does what it wants. 00:14:15.207 --> 00:14:18.530 align:middle line:84% Here, I'm telling the switch to turn off 00:14:18.530 --> 00:14:20.040 align:middle line:90% if the current gets too big. 00:14:20.040 --> 00:14:25.530 align:middle line:84% So this gives me automatic current control. 00:14:25.530 --> 00:14:27.630 align:middle line:84% My currents can't get out of control. 00:14:27.630 --> 00:14:29.820 align:middle line:84% Now, in a boost converter, that's not quite true 00:14:29.820 --> 00:14:31.770 align:middle line:84% because if I put a short circuit on this, 00:14:31.770 --> 00:14:35.160 align:middle line:84% there's nothing I'm going to do because the switch won't control 00:14:35.160 --> 00:14:36.040 align:middle line:90% the inductor current. 00:14:36.040 --> 00:14:38.457 align:middle line:84% But as long as the output voltage is bigger than the input 00:14:38.457 --> 00:14:43.650 align:middle line:84% voltage, by having this peak current reference, 00:14:43.650 --> 00:14:47.040 align:middle line:84% I can always limit the amount of current the converter 00:14:47.040 --> 00:14:48.810 align:middle line:84% is processing and hence prevent it 00:14:48.810 --> 00:14:50.120 align:middle line:90% from overheating, for example. 00:14:50.120 --> 00:14:53.720 align:middle line:90% 00:14:53.720 --> 00:14:59.570 align:middle line:84% All right, so essentially, this technique gives me 00:14:59.570 --> 00:15:03.700 align:middle line:84% cycle-by-cycle current control over the switch-- 00:15:03.700 --> 00:15:04.790 align:middle line:90% over the inductor current. 00:15:04.790 --> 00:15:07.960 align:middle line:84% Now, I have to sense the inductor current because I'm 00:15:07.960 --> 00:15:10.510 align:middle line:90% controlling IL here. 00:15:10.510 --> 00:15:13.730 align:middle line:84% Sometimes I do literally sense the inductor current. 00:15:13.730 --> 00:15:17.890 align:middle line:84% Sometimes, keep in mind, you're only looking at the current 00:15:17.890 --> 00:15:19.270 align:middle line:90% when the switch is on. 00:15:19.270 --> 00:15:22.570 align:middle line:84% So sometimes people will come and put a little resistor 00:15:22.570 --> 00:15:24.580 align:middle line:84% right here or use the switch resistance 00:15:24.580 --> 00:15:27.490 align:middle line:84% itself and sense the current in the switch. 00:15:27.490 --> 00:15:30.340 align:middle line:84% It gives you the same thing as far as this comparison 00:15:30.340 --> 00:15:32.210 align:middle line:90% is concerned. 00:15:32.210 --> 00:15:34.345 align:middle line:90% Any questions about that? 00:15:34.345 --> 00:15:36.950 align:middle line:90% 00:15:36.950 --> 00:15:38.040 align:middle line:90% Yeah. 00:15:38.040 --> 00:15:40.590 align:middle line:84% AUDIENCE: How would you adjust the peak current level 00:15:40.590 --> 00:15:41.947 align:middle line:90% if your load changes? 00:15:41.947 --> 00:15:43.780 align:middle line:84% DAVID PERREAULT: Ah, that's a good question. 00:15:43.780 --> 00:15:45.030 align:middle line:90% So what's going to happen? 00:15:45.030 --> 00:15:47.550 align:middle line:84% I'm going to do the same kind of deal. 00:15:47.550 --> 00:15:49.980 align:middle line:90% If the load voltage-- 00:15:49.980 --> 00:15:51.970 align:middle line:84% well, let me go back to the right controller. 00:15:51.970 --> 00:15:54.850 align:middle line:90% 00:15:54.850 --> 00:15:57.370 align:middle line:84% If the load voltage is too low, too 00:15:57.370 --> 00:16:00.130 align:middle line:84% low means below the reference, the error voltage 00:16:00.130 --> 00:16:05.320 align:middle line:84% gets bigger, this compensator, this outer loop compensator 00:16:05.320 --> 00:16:06.040 align:middle line:90% amplifies it. 00:16:06.040 --> 00:16:07.360 align:middle line:90% Maybe it integrates the error. 00:16:07.360 --> 00:16:10.590 align:middle line:84% It will make the peak command get bigger. 00:16:10.590 --> 00:16:15.265 align:middle line:84% If this peak command moves up, so will the switching time. 00:16:15.265 --> 00:16:17.140 align:middle line:84% And then the inductor will have more current, 00:16:17.140 --> 00:16:19.182 align:middle line:84% and eventually the inductor will feed the output, 00:16:19.182 --> 00:16:21.730 align:middle line:84% and it will bring the output voltage up. 00:16:21.730 --> 00:16:25.720 align:middle line:84% So the ultimate output voltage is really controlled 00:16:25.720 --> 00:16:28.360 align:middle line:84% by this, looking at the error between the output voltage what 00:16:28.360 --> 00:16:30.130 align:middle line:84% I wanted, just like I did before. 00:16:30.130 --> 00:16:31.960 align:middle line:84% I just have an intermediate value 00:16:31.960 --> 00:16:34.600 align:middle line:84% where it controls the current at which the inductor is 00:16:34.600 --> 00:16:36.960 align:middle line:90% going to run to get there. 00:16:36.960 --> 00:16:40.260 align:middle line:84% And if you think about it, the inductor current times, 00:16:40.260 --> 00:16:43.140 align:middle line:84% the input voltage is the power I'm drawing. 00:16:43.140 --> 00:16:46.812 align:middle line:84% So it's in some indirect way controlling the amount of power 00:16:46.812 --> 00:16:48.395 align:middle line:84% that I'm going to draw from the input. 00:16:48.395 --> 00:16:51.840 align:middle line:90% 00:16:51.840 --> 00:16:56.340 align:middle line:84% All right, so one thing it gives us is a cycle-by-cycle current 00:16:56.340 --> 00:16:56.920 align:middle line:90% limiting. 00:16:56.920 --> 00:16:59.820 align:middle line:84% The other thing it's going to give us is better dynamics. 00:16:59.820 --> 00:17:02.010 align:middle line:84% The question is, how do we analyze 00:17:02.010 --> 00:17:03.310 align:middle line:90% the dynamics of this thing? 00:17:03.310 --> 00:17:04.510 align:middle line:90% What did I do before? 00:17:04.510 --> 00:17:06.260 align:middle line:84% The thing I did before was we started-- 00:17:06.260 --> 00:17:09.900 align:middle line:84% we averaged the average circuit, either the circuit or the state 00:17:09.900 --> 00:17:11.960 align:middle line:84% space model, and came up with a duty-- 00:17:11.960 --> 00:17:14.069 align:middle line:84% and then we linearized it, and we came up 00:17:14.069 --> 00:17:17.040 align:middle line:84% with a conversion between the duty ratio 00:17:17.040 --> 00:17:20.130 align:middle line:84% and the local average inductor current and local average output 00:17:20.130 --> 00:17:21.440 align:middle line:90% voltage. 00:17:21.440 --> 00:17:25.170 align:middle line:84% And we could try to do that for this scheme. 00:17:25.170 --> 00:17:27.630 align:middle line:84% It's a little bit complicated, and it turns out 00:17:27.630 --> 00:17:29.620 align:middle line:84% it's mathematically very difficult. 00:17:29.620 --> 00:17:31.950 align:middle line:84% You kind of walk into a wall of math 00:17:31.950 --> 00:17:34.500 align:middle line:84% if you try to play this game directly. 00:17:34.500 --> 00:17:38.360 align:middle line:84% But there are clever ways to get around it. 00:17:38.360 --> 00:17:44.360 align:middle line:84% I've already got a model for how duty ratio relates 00:17:44.360 --> 00:17:47.010 align:middle line:84% to the local averages of the output voltages 00:17:47.010 --> 00:17:49.890 align:middle line:84% and the local average of the inductor current. 00:17:49.890 --> 00:17:53.630 align:middle line:84% If I can relate the peak current to the duty ratio, 00:17:53.630 --> 00:17:55.280 align:middle line:90% then I'm kind of in business. 00:17:55.280 --> 00:17:58.610 align:middle line:84% I can just kind of swap it out and create a new model. 00:17:58.610 --> 00:17:59.860 align:middle line:90% And that's what people do. 00:17:59.860 --> 00:18:03.410 align:middle line:84% It's much more simple, or it's more or less the only way 00:18:03.410 --> 00:18:06.900 align:middle line:84% to go about it that I know of that works out nicely. 00:18:06.900 --> 00:18:09.080 align:middle line:90% And so here's the idea. 00:18:09.080 --> 00:18:13.700 align:middle line:84% This picture I drew is actually a periodic steady state picture. 00:18:13.700 --> 00:18:15.710 align:middle line:90% I used capital D here. 00:18:15.710 --> 00:18:21.050 align:middle line:84% I'm assuming it does something every cycle, DT. 00:18:21.050 --> 00:18:24.650 align:middle line:84% Let's just draw a slightly better picture 00:18:24.650 --> 00:18:27.380 align:middle line:90% for purposes of analysis. 00:18:27.380 --> 00:18:29.040 align:middle line:90% OK, so here we go. 00:18:29.040 --> 00:18:31.700 align:middle line:90% 00:18:31.700 --> 00:18:32.870 align:middle line:90% Here's my waveform. 00:18:32.870 --> 00:18:38.510 align:middle line:90% Here's my peak current, I peak. 00:18:38.510 --> 00:18:44.600 align:middle line:84% And then I have my so-called compensating ramp, which 00:18:44.600 --> 00:18:46.760 align:middle line:90% falls with some slope minus Mc. 00:18:46.760 --> 00:18:52.080 align:middle line:90% 00:18:52.080 --> 00:18:54.760 align:middle line:84% And now I have my inductor current. 00:18:54.760 --> 00:18:57.070 align:middle line:84% Let's suppose the inductor current starts here. 00:18:57.070 --> 00:19:06.120 align:middle line:84% It rises up with slope M1, hits the intersection, that's DT. 00:19:06.120 --> 00:19:10.950 align:middle line:84% And then it falls with some slope M2 or minus M2. 00:19:10.950 --> 00:19:15.010 align:middle line:84% But I'm actually treating this thing dynamically. 00:19:15.010 --> 00:19:16.980 align:middle line:84% I'm not necessarily saying this end point is 00:19:16.980 --> 00:19:19.420 align:middle line:90% the same as the starting point. 00:19:19.420 --> 00:19:25.005 align:middle line:84% So this is a dynamic picture of what happens over one cycle. 00:19:25.005 --> 00:19:27.670 align:middle line:90% 00:19:27.670 --> 00:19:31.630 align:middle line:84% And what I want to do is I want to relate somehow 00:19:31.630 --> 00:19:37.150 align:middle line:84% this duty ratio in this cycle to what the local average 00:19:37.150 --> 00:19:38.625 align:middle line:90% of the inductor current is. 00:19:38.625 --> 00:19:42.510 align:middle line:90% 00:19:42.510 --> 00:19:44.522 align:middle line:84% And I'm not going to do that by local averaging. 00:19:44.522 --> 00:19:46.230 align:middle line:84% What I'm going to do is I'm going to make 00:19:46.230 --> 00:19:48.930 align:middle line:90% a geometric approximation. 00:19:48.930 --> 00:19:53.340 align:middle line:84% I'm going to say, OK, let me look over this window 00:19:53.340 --> 00:19:57.270 align:middle line:84% and say the average of this waveform, which is basically 00:19:57.270 --> 00:20:03.600 align:middle line:84% the average here, the average of this current 00:20:03.600 --> 00:20:07.710 align:middle line:90% is the local average. 00:20:07.710 --> 00:20:14.640 align:middle line:84% So I'm saying IL of t is approximately 00:20:14.640 --> 00:20:21.450 align:middle line:84% equal to the average of IL over this window. 00:20:21.450 --> 00:20:27.060 align:middle line:84% Strictly speaking, this is equal to IL of capital T, 00:20:27.060 --> 00:20:31.320 align:middle line:84% if this is time T. And it's not quite the same thing 00:20:31.320 --> 00:20:33.750 align:middle line:84% anywhere else, but it turns out it's 00:20:33.750 --> 00:20:37.600 align:middle line:84% pretty-- it's not a bad geometric approximation. 00:20:37.600 --> 00:20:44.820 align:middle line:84% Now, if I wanted to be really theoretically pure about this, 00:20:44.820 --> 00:20:48.030 align:middle line:84% I might actually-- instead of looking over this window, 00:20:48.030 --> 00:20:53.100 align:middle line:84% which is centered at 0.5T, I might look at a window centered 00:20:53.100 --> 00:20:56.850 align:middle line:84% at dT and look around here and it extend into the different 00:20:56.850 --> 00:20:57.880 align:middle line:90% switching cycles. 00:20:57.880 --> 00:21:00.030 align:middle line:84% But it turns out for practical purposes, 00:21:00.030 --> 00:21:02.080 align:middle line:84% you really more or less get the same-- 00:21:02.080 --> 00:21:04.360 align:middle line:84% for most cases, you more or less get the same thing. 00:21:04.360 --> 00:21:06.930 align:middle line:84% So the approximation I'm going to make 00:21:06.930 --> 00:21:10.650 align:middle line:84% is good enough given all the other approximations 00:21:10.650 --> 00:21:11.540 align:middle line:90% in the system. 00:21:11.540 --> 00:21:14.460 align:middle line:84% So I'm just going to average it over this window. 00:21:14.460 --> 00:21:17.610 align:middle line:84% But do people see what the basic idea I'm going to use? 00:21:17.610 --> 00:21:23.610 align:middle line:84% I'm going to approximate IL bar as the average of this waveform. 00:21:23.610 --> 00:21:26.883 align:middle line:84% And then that's a function of the duty ratio in that cycle. 00:21:26.883 --> 00:21:29.550 align:middle line:84% And then I'm going to swap that into my local average duty ratio 00:21:29.550 --> 00:21:31.640 align:middle line:90% equation. 00:21:31.640 --> 00:21:33.390 align:middle line:84% Actually, I'm going to linearize it first, 00:21:33.390 --> 00:21:34.807 align:middle line:84% but that's neither here nor there. 00:21:34.807 --> 00:21:35.800 align:middle line:90% So let's do that. 00:21:35.800 --> 00:21:41.850 align:middle line:84% What is my approximation to IL bar? 00:21:41.850 --> 00:21:44.370 align:middle line:90% Well, let's see. 00:21:44.370 --> 00:21:54.180 align:middle line:84% The peak of this waveform here is I peak minus Mc dT. 00:21:54.180 --> 00:21:58.650 align:middle line:90% That would be, if I wanted-- 00:21:58.650 --> 00:22:03.870 align:middle line:84% if I just use this value, that would be basically 00:22:03.870 --> 00:22:06.930 align:middle line:90% if the local-- 00:22:06.930 --> 00:22:08.808 align:middle line:84% if the waveform-- if the orange waveform 00:22:08.808 --> 00:22:10.350 align:middle line:84% looked like this green waveform, that 00:22:10.350 --> 00:22:12.610 align:middle line:84% would be exactly the local average in the cycle. 00:22:12.610 --> 00:22:15.480 align:middle line:84% But what I really need to do is kind 00:22:15.480 --> 00:22:18.810 align:middle line:84% of subtract off the average of this area 00:22:18.810 --> 00:22:20.290 align:middle line:90% and the average of this area. 00:22:20.290 --> 00:22:24.060 align:middle line:84% And then I'll have the average underneath 00:22:24.060 --> 00:22:26.430 align:middle line:90% the inductor current curve. 00:22:26.430 --> 00:22:28.740 align:middle line:84% So if I do that, then I'm going to get, 00:22:28.740 --> 00:22:35.310 align:middle line:84% OK, I've got to subtract off the area of this triangle here. 00:22:35.310 --> 00:22:39.510 align:middle line:84% So that would be minus 1/2 the base, 00:22:39.510 --> 00:22:45.833 align:middle line:84% which is dT times the height, which is dT times M1. 00:22:45.833 --> 00:22:51.220 align:middle line:90% 00:22:51.220 --> 00:22:54.420 align:middle line:84% And likewise-- and by the way, I better divide that by T 00:22:54.420 --> 00:22:55.710 align:middle line:90% because I'm taking an average. 00:22:55.710 --> 00:22:56.760 align:middle line:90% So this is 1 over 2T. 00:22:56.760 --> 00:23:00.030 align:middle line:90% 00:23:00.030 --> 00:23:04.740 align:middle line:84% And I better subtract off the area of this triangle divided 00:23:04.740 --> 00:23:10.131 align:middle line:84% by capital T. So this would be minus 1 over 2T, 00:23:10.131 --> 00:23:16.450 align:middle line:84% 1 minus d quantity squared T squared M2. 00:23:16.450 --> 00:23:23.050 align:middle line:84% So if I rewrite that, IL bar is approximately equal to I peak 00:23:23.050 --> 00:23:37.120 align:middle line:84% minus Mc dT minus 1/2 d squared TM1 minus 1/2 1 minus d squared 00:23:37.120 --> 00:23:39.670 align:middle line:90% TM2. 00:23:39.670 --> 00:23:42.655 align:middle line:84% And this is the substitution I'm going to make. 00:23:42.655 --> 00:23:46.190 align:middle line:90% 00:23:46.190 --> 00:23:49.390 align:middle line:84% Or it's the basis of the substitution I'm going to make. 00:23:49.390 --> 00:23:52.760 align:middle line:90% Any questions about that? 00:23:52.760 --> 00:23:56.120 align:middle line:84% So I've got a geometric approximation relating 00:23:56.120 --> 00:24:00.260 align:middle line:84% the duty ratio in the cycle, the peak current in the cycle, 00:24:00.260 --> 00:24:03.250 align:middle line:84% and the local average in the cycle. 00:24:03.250 --> 00:24:05.410 align:middle line:90% So let me linearize this. 00:24:05.410 --> 00:24:10.102 align:middle line:90% 00:24:10.102 --> 00:24:11.810 align:middle line:84% I'm going to linearize this because we're 00:24:11.810 --> 00:24:14.300 align:middle line:84% going to substitute it into our linearized equations, 00:24:14.300 --> 00:24:20.210 align:middle line:84% and I'm going to solve for d tilde. 00:24:20.210 --> 00:24:28.851 align:middle line:84% And if I do that, the result you get is-- 00:24:28.851 --> 00:24:31.640 align:middle line:84% oh, and then the last thing I guess I should do 00:24:31.640 --> 00:24:37.790 align:middle line:84% is let me substitute in, in this case, u-- 00:24:37.790 --> 00:24:41.750 align:middle line:84% I'm sorry, M1 is equal to u over L. 00:24:41.750 --> 00:24:46.640 align:middle line:84% And M2 is equal to V minus u over L. 00:24:46.640 --> 00:24:48.690 align:middle line:84% And if I make that substitution, I do it, 00:24:48.690 --> 00:24:52.620 align:middle line:84% what I get for my boost converter is d tilde is equal 00:24:52.620 --> 00:25:04.710 align:middle line:84% to 1 over McT I peak tilde minus IL tilde minus D squared minus D 00:25:04.710 --> 00:25:18.220 align:middle line:84% prime squared over 2LMC, u tilde minus d prime squared over 2LMC, 00:25:18.220 --> 00:25:20.410 align:middle line:90% V tilde. 00:25:20.410 --> 00:25:26.530 align:middle line:84% So this is a linearized version of this guy for the boost 00:25:26.530 --> 00:25:27.550 align:middle line:90% converter. 00:25:27.550 --> 00:25:30.070 align:middle line:84% If you wanted to do a buck boost converter, 00:25:30.070 --> 00:25:32.510 align:middle line:84% you'd need different expressions for M1 and M2, 00:25:32.510 --> 00:25:34.760 align:middle line:84% and you'd do the same process, and you'd linearize it. 00:25:34.760 --> 00:25:36.690 align:middle line:84% You'd get a different expression. 00:25:36.690 --> 00:25:38.570 align:middle line:90% Any questions about that? 00:25:38.570 --> 00:25:41.870 align:middle line:90% 00:25:41.870 --> 00:25:44.990 align:middle line:84% OK, so what's the point of doing this? 00:25:44.990 --> 00:25:48.230 align:middle line:84% The point of doing this is I can now 00:25:48.230 --> 00:25:52.910 align:middle line:84% substitute in for d tilde in my existing equations. 00:25:52.910 --> 00:25:55.220 align:middle line:84% And remember, d tilde was my control variable 00:25:55.220 --> 00:25:58.590 align:middle line:84% in all my existing state space averaged linearized equations. 00:25:58.590 --> 00:26:03.940 align:middle line:84% If I substituted for d tilde, I get rid of d tilde. 00:26:03.940 --> 00:26:07.090 align:middle line:84% And all this other stuff is already 00:26:07.090 --> 00:26:09.855 align:middle line:84% in the equations except for I peak tilde. 00:26:09.855 --> 00:26:14.400 align:middle line:90% 00:26:14.400 --> 00:26:17.940 align:middle line:84% Does that make sense to everybody? 00:26:17.940 --> 00:26:23.010 align:middle line:84% So I'm going to now get a new set of linearized state space 00:26:23.010 --> 00:26:26.100 align:middle line:84% equations, except instead of detailed in the equations, 00:26:26.100 --> 00:26:28.510 align:middle line:84% I'm going to have a control variable Ip tilde. 00:26:28.510 --> 00:26:32.490 align:middle line:90% That's what I needed here. 00:26:32.490 --> 00:26:37.700 align:middle line:84% So basically, I can take this out and swap it 00:26:37.700 --> 00:26:50.460 align:middle line:84% for an H2 of S that's basically equal to V tilde over I 00:26:50.460 --> 00:26:50.990 align:middle line:90% peak tilde. 00:26:50.990 --> 00:26:55.950 align:middle line:90% 00:26:55.950 --> 00:27:00.040 align:middle line:84% So I basically got my whole inner loop going. 00:27:00.040 --> 00:27:03.460 align:middle line:84% And now if I have Ip tilde and V tilde, 00:27:03.460 --> 00:27:05.625 align:middle line:84% I can now giant design my outer loop. 00:27:05.625 --> 00:27:08.770 align:middle line:90% 00:27:08.770 --> 00:27:10.960 align:middle line:84% What was the point of doing this substitution 00:27:10.960 --> 00:27:12.770 align:middle line:90% and getting my new H2 of S? 00:27:12.770 --> 00:27:17.540 align:middle line:84% If I look at H2 of S, I look at that transfer function, 00:27:17.540 --> 00:27:19.810 align:middle line:90% what am I going to find? 00:27:19.810 --> 00:27:23.230 align:middle line:84% I'm still going to find the right-half-plane zero. 00:27:23.230 --> 00:27:24.470 align:middle line:90% I can't get rid of that. 00:27:24.470 --> 00:27:28.420 align:middle line:84% That's related to how the converter behaves. 00:27:28.420 --> 00:27:33.730 align:middle line:84% If I step I peak tilde, I'm still going to see-- 00:27:33.730 --> 00:27:36.970 align:middle line:84% I'm still going to see funny behavior in the output voltage 00:27:36.970 --> 00:27:38.460 align:middle line:84% before the output voltage response. 00:27:38.460 --> 00:27:40.640 align:middle line:84% And my right-half-plane zero doesn't go away. 00:27:40.640 --> 00:27:43.720 align:middle line:84% The beautiful thing is, however, what 00:27:43.720 --> 00:27:48.280 align:middle line:84% I will tend to see is instead of two really lightly damped poles, 00:27:48.280 --> 00:27:51.865 align:middle line:84% I'll tend to see a dominant pole and then a much higher frequency 00:27:51.865 --> 00:27:52.365 align:middle line:90% pole. 00:27:52.365 --> 00:27:56.020 align:middle line:90% 00:27:56.020 --> 00:27:57.080 align:middle line:90% Why? 00:27:57.080 --> 00:27:59.050 align:middle line:84% Because essentially the way people 00:27:59.050 --> 00:28:03.040 align:middle line:84% like to think of it is this inner loop is making the current 00:28:03.040 --> 00:28:04.830 align:middle line:90% do what you want. 00:28:04.830 --> 00:28:08.040 align:middle line:84% And hence, the dominant pole is more or less 00:28:08.040 --> 00:28:10.920 align:middle line:84% governed by how the current from that inductor 00:28:10.920 --> 00:28:13.800 align:middle line:84% feeds into the output capacitor and charges up and down 00:28:13.800 --> 00:28:17.080 align:middle line:90% the output capacitor. 00:28:17.080 --> 00:28:20.210 align:middle line:84% So it's not that I have a first-order system. 00:28:20.210 --> 00:28:21.440 align:middle line:90% There is a second pole here. 00:28:21.440 --> 00:28:26.030 align:middle line:84% It's just dominated by this low frequency pole. 00:28:26.030 --> 00:28:29.510 align:middle line:84% And that's a much easier thing to control. 00:28:29.510 --> 00:28:31.815 align:middle line:90% Any questions about that? 00:28:31.815 --> 00:28:34.330 align:middle line:90% 00:28:34.330 --> 00:28:37.708 align:middle line:84% So right now we can go design GC of S. 00:28:37.708 --> 00:28:39.250 align:middle line:84% I still have a right-half-plane zero, 00:28:39.250 --> 00:28:41.080 align:middle line:84% but it's a much easier plan to compensate, 00:28:41.080 --> 00:28:45.040 align:middle line:84% and I can put my dynamics much more nicely than I could before. 00:28:45.040 --> 00:28:48.580 align:middle line:84% And I get free current limiting in the process. 00:28:48.580 --> 00:28:54.760 align:middle line:84% And in fact, if you're designing most kinds of PWM-DC converters, 00:28:54.760 --> 00:28:56.560 align:middle line:84% most of the control chips on the market 00:28:56.560 --> 00:28:58.940 align:middle line:84% are going to use current mode control. 00:28:58.940 --> 00:29:01.000 align:middle line:84% This version that I'm showing you 00:29:01.000 --> 00:29:02.980 align:middle line:84% is called peak current mode control 00:29:02.980 --> 00:29:06.310 align:middle line:84% because you're controlling the peak of the inductor current. 00:29:06.310 --> 00:29:08.120 align:middle line:90% And that's the most common kind. 00:29:08.120 --> 00:29:09.640 align:middle line:90% There's also valley control. 00:29:09.640 --> 00:29:13.060 align:middle line:84% And there's a few different versions of this general idea, 00:29:13.060 --> 00:29:14.745 align:middle line:84% but this is the most popular one. 00:29:14.745 --> 00:29:21.082 align:middle line:90% 00:29:21.082 --> 00:29:22.710 align:middle line:90% So that's one trick. 00:29:22.710 --> 00:29:23.980 align:middle line:90% Any questions up to now? 00:29:23.980 --> 00:29:26.650 align:middle line:90% 00:29:26.650 --> 00:29:32.350 align:middle line:84% AUDIENCE: Is there an easy way to understand why [INAUDIBLE]? 00:29:32.350 --> 00:29:34.940 align:middle line:84% DAVID PERREAULT: Excellent question. 00:29:34.940 --> 00:29:36.740 align:middle line:90% I'll cover that one next. 00:29:36.740 --> 00:29:38.280 align:middle line:90% Any other questions? 00:29:38.280 --> 00:29:41.786 align:middle line:90% 00:29:41.786 --> 00:29:49.240 align:middle line:84% OK, yeah, why did I even bother with this ramping MC signal? 00:29:49.240 --> 00:29:51.250 align:middle line:90% Why not just use I peak? 00:29:51.250 --> 00:29:55.030 align:middle line:90% And in some cases, you can. 00:29:55.030 --> 00:29:58.005 align:middle line:84% But here's the problem with doing that. 00:29:58.005 --> 00:30:02.252 align:middle line:90% 00:30:02.252 --> 00:30:06.250 align:middle line:84% In some cases, if I do that, this is the situation 00:30:06.250 --> 00:30:08.080 align:middle line:90% I can find myself in. 00:30:08.080 --> 00:30:11.890 align:middle line:84% I'm going along, and suppose I just have I peak, 00:30:11.890 --> 00:30:16.120 align:middle line:84% and I get rid of the compensating ramp. 00:30:16.120 --> 00:30:18.100 align:middle line:84% So I'm at the beginning of my cycle. 00:30:18.100 --> 00:30:19.640 align:middle line:90% And here's my inductor current. 00:30:19.640 --> 00:30:23.650 align:middle line:84% I turn on my switch, I go up, and I hit the top. 00:30:23.650 --> 00:30:27.620 align:middle line:84% And I trip, so this is my duty ratio in this cycle. 00:30:27.620 --> 00:30:28.720 align:middle line:90% And I turn off. 00:30:28.720 --> 00:30:32.290 align:middle line:84% And then I turn off till the end of the cycle. 00:30:32.290 --> 00:30:33.770 align:middle line:90% And here's the end of the cycle. 00:30:33.770 --> 00:30:36.340 align:middle line:84% Here's capital T. And now I turn on my inductor current on, 00:30:36.340 --> 00:30:39.190 align:middle line:90% and he ramps up. 00:30:39.190 --> 00:30:40.180 align:middle line:90% He hits here. 00:30:40.180 --> 00:30:41.960 align:middle line:84% Here's my duty ratio in this cycle. 00:30:41.960 --> 00:30:46.310 align:middle line:84% And then he turns off, and I hit the end of the cycle, 2T. 00:30:46.310 --> 00:30:49.990 align:middle line:84% And you can see that I don't settle down 00:30:49.990 --> 00:30:52.510 align:middle line:84% into periodic steady state operation at the switching 00:30:52.510 --> 00:30:54.070 align:middle line:90% period. 00:30:54.070 --> 00:30:57.930 align:middle line:84% In this example, it repeats every other cycle. 00:30:57.930 --> 00:31:00.661 align:middle line:90% 00:31:00.661 --> 00:31:03.470 align:middle line:84% In some examples, it won't settle down at all. 00:31:03.470 --> 00:31:05.530 align:middle line:84% It'll just bungle around chaotically. 00:31:05.530 --> 00:31:08.440 align:middle line:90% I mean, literally chaotically. 00:31:08.440 --> 00:31:09.980 align:middle line:90% Now, is that a problem? 00:31:09.980 --> 00:31:13.035 align:middle line:84% Well, I mean, at the highest order, it's not. 00:31:13.035 --> 00:31:14.660 align:middle line:84% I'm still controlling the peak current. 00:31:14.660 --> 00:31:17.440 align:middle line:84% My output loop will still regulate and everything, 00:31:17.440 --> 00:31:19.940 align:middle line:84% but it is problematic from other perspectives. 00:31:19.940 --> 00:31:25.060 align:middle line:84% One perspective is my ripple in this case is a heck of a lot 00:31:25.060 --> 00:31:30.040 align:middle line:84% bigger than if I were at some fixed intermediate duty ratio. 00:31:30.040 --> 00:31:33.030 align:middle line:90% So I get much bigger ripple. 00:31:33.030 --> 00:31:35.790 align:middle line:84% The control is jittery because it's bouncing around depending 00:31:35.790 --> 00:31:38.940 align:middle line:90% upon the operating condition. 00:31:38.940 --> 00:31:42.450 align:middle line:84% I have lower frequency content in my output. 00:31:42.450 --> 00:31:44.580 align:middle line:84% It's bigger ripple, and it's lower frequencies 00:31:44.580 --> 00:31:46.210 align:middle line:90% that can make a mess of things. 00:31:46.210 --> 00:31:49.390 align:middle line:84% So people don't like this behavior. 00:31:49.390 --> 00:31:52.850 align:middle line:84% This is sometimes called subharmonic oscillation. 00:31:52.850 --> 00:31:59.580 align:middle line:84% Well, this behavior where it's rippling at a subharmonic 00:31:59.580 --> 00:32:01.210 align:middle line:84% is called a subharmonic oscillation. 00:32:01.210 --> 00:32:02.370 align:middle line:84% But as I mentioned, in some cases, 00:32:02.370 --> 00:32:03.495 align:middle line:90% it can actually be chaotic. 00:32:03.495 --> 00:32:06.120 align:middle line:90% 00:32:06.120 --> 00:32:09.080 align:middle line:84% Generally, this issue is called the ripple instability. 00:32:09.080 --> 00:32:11.610 align:middle line:84% This ripple doesn't settle down into periodic steady state 00:32:11.610 --> 00:32:12.930 align:middle line:90% behavior. 00:32:12.930 --> 00:32:17.790 align:middle line:84% And what people figured out-- and the first paper I knew about 00:32:17.790 --> 00:32:21.360 align:middle line:84% this was out of the Caltech group in the late 00:32:21.360 --> 00:32:23.550 align:middle line:84% '70s, but other people thought of it too-- 00:32:23.550 --> 00:32:27.330 align:middle line:84% was to stick this compensating ramp in, which if you put enough 00:32:27.330 --> 00:32:29.520 align:middle line:84% compensating ramp in makes that behavior go away, 00:32:29.520 --> 00:32:31.470 align:middle line:84% and it'll settle down just like this case, 00:32:31.470 --> 00:32:34.920 align:middle line:90% to a kind of a fixed ripple. 00:32:34.920 --> 00:32:37.046 align:middle line:90% OK. 00:32:37.046 --> 00:32:40.950 align:middle line:84% AUDIENCE: Did we cover how the MC is calculated? 00:32:40.950 --> 00:32:44.430 align:middle line:84% DAVID PERREAULT: No, but I will in about 30 seconds. 00:32:44.430 --> 00:32:47.640 align:middle line:90% Excellent question. 00:32:47.640 --> 00:32:51.030 align:middle line:84% By the way, you might say, why did they think to do this? 00:32:51.030 --> 00:32:52.410 align:middle line:90% Well, A, they were clever. 00:32:52.410 --> 00:32:55.620 align:middle line:84% But B, think about it-- in most power converters, 00:32:55.620 --> 00:33:01.012 align:middle line:84% you have a ramp generator to set your clock frequency. 00:33:01.012 --> 00:33:02.470 align:middle line:84% That's how you get your oscillator. 00:33:02.470 --> 00:33:03.880 align:middle line:84% I'm going to turn on every cycle. 00:33:03.880 --> 00:33:05.440 align:middle line:90% How do I know what that is? 00:33:05.440 --> 00:33:07.380 align:middle line:84% I generate a ramp that it hits the threshold, 00:33:07.380 --> 00:33:09.540 align:middle line:90% and that sets my clock period. 00:33:09.540 --> 00:33:11.440 align:middle line:84% So they had this ramp signal sticking around. 00:33:11.440 --> 00:33:13.815 align:middle line:84% So they just basically took a little section of that ramp 00:33:13.815 --> 00:33:16.350 align:middle line:84% signal and added or subtracted it from I peak, 00:33:16.350 --> 00:33:19.290 align:middle line:84% and they found that fixed their problem. 00:33:19.290 --> 00:33:22.680 align:middle line:84% So that's how they got to the idea, I think. 00:33:22.680 --> 00:33:23.845 align:middle line:90% Or that's why it's-- 00:33:23.845 --> 00:33:25.720 align:middle line:84% I shouldn't say that's how they got the idea. 00:33:25.720 --> 00:33:27.780 align:middle line:84% That's why it's a natural trick to play 00:33:27.780 --> 00:33:32.040 align:middle line:84% because you have the signal around at your disposal. 00:33:32.040 --> 00:33:35.220 align:middle line:84% But the question is, how big do I 00:33:35.220 --> 00:33:37.890 align:middle line:84% have to make this MC to get this behavior instead 00:33:37.890 --> 00:33:39.360 align:middle line:90% of this behavior? 00:33:39.360 --> 00:33:43.836 align:middle line:90% And let's analyze that. 00:33:43.836 --> 00:33:45.195 align:middle line:90% So let me take-- 00:33:45.195 --> 00:33:48.420 align:middle line:90% 00:33:48.420 --> 00:33:52.080 align:middle line:84% let me analyze-- let me first start and say, OK, 00:33:52.080 --> 00:33:55.190 align:middle line:84% what would my periodic steady state behavior be? 00:33:55.190 --> 00:33:56.360 align:middle line:90% Here is I peak. 00:33:56.360 --> 00:34:00.210 align:middle line:90% 00:34:00.210 --> 00:34:04.590 align:middle line:90% Here is my compensating ramp. 00:34:04.590 --> 00:34:07.390 align:middle line:90% Here's period T. 00:34:07.390 --> 00:34:09.724 align:middle line:84% So if my compensating ramp looks like this-- 00:34:09.724 --> 00:34:12.290 align:middle line:90% 00:34:12.290 --> 00:34:15.590 align:middle line:90% this is a slope minus MC-- 00:34:15.590 --> 00:34:18.710 align:middle line:84% maybe I could say my periodic steady state 00:34:18.710 --> 00:34:20.670 align:middle line:90% behavior looks like this. 00:34:20.670 --> 00:34:24.110 align:middle line:90% I start off somewhere. 00:34:24.110 --> 00:34:29.739 align:middle line:90% I ramp up with slope M1. 00:34:29.739 --> 00:34:34.000 align:middle line:84% Then I ramp down with slope minus M2. 00:34:34.000 --> 00:34:37.489 align:middle line:90% 00:34:37.489 --> 00:34:39.610 align:middle line:84% And I end up exactly where I started 00:34:39.610 --> 00:34:40.860 align:middle line:90% at the beginning of the cycle. 00:34:40.860 --> 00:34:43.280 align:middle line:84% This is periodic steady state behavior. 00:34:43.280 --> 00:34:46.639 align:middle line:84% So this is what I'm going to call DT. 00:34:46.639 --> 00:34:48.889 align:middle line:84% This is the behavior-- this yellow waveform is 00:34:48.889 --> 00:34:52.510 align:middle line:90% the behavior I'm looking for. 00:34:52.510 --> 00:34:57.970 align:middle line:84% Let's suppose my inductor at the beginning of this cycle started 00:34:57.970 --> 00:34:59.300 align:middle line:90% off a little bit high. 00:34:59.300 --> 00:35:02.200 align:middle line:90% So suppose he started off here. 00:35:02.200 --> 00:35:04.540 align:middle line:84% He has some error, some deviation 00:35:04.540 --> 00:35:07.905 align:middle line:84% from periodic steady state that I'm going to call delta I sub n. 00:35:07.905 --> 00:35:09.280 align:middle line:84% So at the beginning of the cycle, 00:35:09.280 --> 00:35:12.490 align:middle line:90% he starts off a little bit high. 00:35:12.490 --> 00:35:14.560 align:middle line:90% What would happen? 00:35:14.560 --> 00:35:16.690 align:middle line:84% Well, he would ramp up at the same slope 00:35:16.690 --> 00:35:20.140 align:middle line:84% because the output voltage is the same. 00:35:20.140 --> 00:35:21.930 align:middle line:90% So this would still be M1. 00:35:21.930 --> 00:35:27.150 align:middle line:84% He would hit the trip point here at something 00:35:27.150 --> 00:35:31.100 align:middle line:84% I'll call d star T. It's not the steady state value. 00:35:31.100 --> 00:35:41.800 align:middle line:84% And then he would fall at minus M2, like this. 00:35:41.800 --> 00:35:44.993 align:middle line:84% And then he would end up with some error 00:35:44.993 --> 00:35:47.410 align:middle line:84% at the end of the cycle or the beginning of the next cycle 00:35:47.410 --> 00:35:52.130 align:middle line:90% that I'll call delta I n plus 1. 00:35:52.130 --> 00:35:54.720 align:middle line:84% So if he starts off here, he's going to end off here. 00:35:54.720 --> 00:35:57.380 align:middle line:84% And I have a different error from periodic steady state 00:35:57.380 --> 00:35:59.140 align:middle line:84% at the beginning of the next cycle. 00:35:59.140 --> 00:36:01.620 align:middle line:90% Does that make sense? 00:36:01.620 --> 00:36:06.420 align:middle line:84% So let's see if we can quantify what that looks like. 00:36:06.420 --> 00:36:11.040 align:middle line:84% And again, this was done by this group, this Middlebrook group. 00:36:11.040 --> 00:36:13.080 align:middle line:84% And by the way, it's in a paper that we've 00:36:13.080 --> 00:36:15.330 align:middle line:90% posted on Canvas for you. 00:36:15.330 --> 00:36:19.980 align:middle line:84% And this behavior is also described in a book chapter 00:36:19.980 --> 00:36:24.020 align:middle line:84% that we've also posted on Canvas for you. 00:36:24.020 --> 00:36:25.520 align:middle line:90% Here's the idea. 00:36:25.520 --> 00:36:30.170 align:middle line:84% Let's quantify what delta I sub n is. 00:36:30.170 --> 00:36:36.830 align:middle line:84% I could write delta I sub n in terms of this time difference, D 00:36:36.830 --> 00:36:44.930 align:middle line:84% minus d star T times, let's see-- 00:36:44.930 --> 00:36:50.660 align:middle line:84% I go D minus d star T. I go up slope M1 00:36:50.660 --> 00:36:54.170 align:middle line:84% and then backwards, so up slope MC. 00:36:54.170 --> 00:36:59.540 align:middle line:84% So this is going to be D1 times d1 times is M1 plus MC. 00:36:59.540 --> 00:37:06.850 align:middle line:84% So I could write this delta In because these waveforms are 00:37:06.850 --> 00:37:13.900 align:middle line:84% parallel in terms of M1 plus MC times d minus d star T. Does 00:37:13.900 --> 00:37:17.160 align:middle line:90% that make sense to everybody? 00:37:17.160 --> 00:37:23.970 align:middle line:84% I could write this difference, delta In plus 1 also 00:37:23.970 --> 00:37:28.030 align:middle line:90% in terms of D minus d star t. 00:37:28.030 --> 00:37:28.530 align:middle line:90% Why? 00:37:28.530 --> 00:37:31.870 align:middle line:84% Because these two lines are parallel. 00:37:31.870 --> 00:37:41.190 align:middle line:84% And what I've really got is this is M1-- 00:37:41.190 --> 00:37:43.770 align:middle line:90% oops, I'm sorry-- M1 plus MC. 00:37:43.770 --> 00:37:45.113 align:middle line:90% I could write this-- 00:37:45.113 --> 00:37:49.880 align:middle line:90% 00:37:49.880 --> 00:37:59.730 align:middle line:84% I could write this in terms of D minus d star T times MC 00:37:59.730 --> 00:38:00.600 align:middle line:90% minus M2. 00:38:00.600 --> 00:38:08.060 align:middle line:90% 00:38:08.060 --> 00:38:10.590 align:middle line:84% They're both related to this time difference. 00:38:10.590 --> 00:38:15.340 align:middle line:84% It's just I'm kind of running up and down different slopes. 00:38:15.340 --> 00:38:18.410 align:middle line:84% Does that make sense to everybody? 00:38:18.410 --> 00:38:21.230 align:middle line:84% So if I take the ratio of these two things, 00:38:21.230 --> 00:38:26.490 align:middle line:84% I could write this as delta In plus 1 over delta In. 00:38:26.490 --> 00:38:29.180 align:middle line:84% That's what I get in one cycle is just 00:38:29.180 --> 00:38:30.770 align:middle line:84% going to be this ratio, which I can 00:38:30.770 --> 00:38:33.980 align:middle line:84% write as the bottom over the top here, 00:38:33.980 --> 00:38:41.120 align:middle line:90% which is minus M2 minus MC. 00:38:41.120 --> 00:38:42.930 align:middle line:84% Actually, I'll write it this way. 00:38:42.930 --> 00:38:51.295 align:middle line:84% It's MC minus M2 divided by MC plus M1. 00:38:51.295 --> 00:38:54.680 align:middle line:90% 00:38:54.680 --> 00:38:57.740 align:middle line:84% OK, that make sense to everybody? 00:38:57.740 --> 00:39:06.860 align:middle line:84% That means that if I started off with some error delta I0, 00:39:06.860 --> 00:39:15.110 align:middle line:84% I could write delta In as that ratio, which 00:39:15.110 --> 00:39:17.592 align:middle line:90% I could write as minus-- 00:39:17.592 --> 00:39:19.050 align:middle line:84% and I'm going to write this the way 00:39:19.050 --> 00:39:26.250 align:middle line:84% they often write it in papers-- minus M2 minus MC over M1 00:39:26.250 --> 00:39:29.930 align:middle line:90% plus MC to the n-th power. 00:39:29.930 --> 00:39:33.300 align:middle line:90% 00:39:33.300 --> 00:39:36.330 align:middle line:84% Let me make sure I've gotten that right. 00:39:36.330 --> 00:39:39.150 align:middle line:90% Yeah, M1 plus MC. 00:39:39.150 --> 00:39:43.590 align:middle line:84% So can anybody tell me what the condition on this factor 00:39:43.590 --> 00:39:48.425 align:middle line:84% is to make an initial error delta I0 go to 0? 00:39:48.425 --> 00:39:52.360 align:middle line:90% 00:39:52.360 --> 00:39:55.190 align:middle line:84% Yeah, this ratio-- what I require-- 00:39:55.190 --> 00:40:00.640 align:middle line:84% this suggests that I need the magnitude of minus M2 00:40:00.640 --> 00:40:06.700 align:middle line:84% minus MC over M1 plus MC to be less than 1. 00:40:06.700 --> 00:40:09.550 align:middle line:90% This is the-- 00:40:09.550 --> 00:40:15.580 align:middle line:84% I've got a discrete time model for what's going to have-- 00:40:15.580 --> 00:40:22.070 align:middle line:84% and it's going to decay if this term is less than 1. 00:40:22.070 --> 00:40:27.680 align:middle line:84% So what that usually says is that if I make MC bigger, 00:40:27.680 --> 00:40:29.780 align:middle line:90% the numerator gets smaller. 00:40:29.780 --> 00:40:31.590 align:middle line:84% And actually, the denominator gets bigger. 00:40:31.590 --> 00:40:34.820 align:middle line:84% So a bigger MC will make this decay more quickly. 00:40:34.820 --> 00:40:37.730 align:middle line:84% And in fact, if I made MC exactly equal to M2, 00:40:37.730 --> 00:40:41.150 align:middle line:90% it would be gone in one cycle. 00:40:41.150 --> 00:40:43.190 align:middle line:84% Does that make sense to everybody? 00:40:43.190 --> 00:40:46.730 align:middle line:84% So what we'll do then, and we can look this up. 00:40:46.730 --> 00:40:49.760 align:middle line:84% It depends on the converter, what 00:40:49.760 --> 00:40:51.350 align:middle line:90% M1 and M2 are in the converter. 00:40:51.350 --> 00:40:54.110 align:middle line:84% But we will pick this compensating ramp slope 00:40:54.110 --> 00:40:55.440 align:middle line:90% to be big enough-- 00:40:55.440 --> 00:40:59.670 align:middle line:84% MC is a positive number to be big enough 00:40:59.670 --> 00:41:02.160 align:middle line:84% that my ripple will decay away in a reasonable number 00:41:02.160 --> 00:41:02.670 align:middle line:90% of cycles. 00:41:02.670 --> 00:41:06.802 align:middle line:84% Maybe I make this magnitude less than 0.5 or 0.7 or something 00:41:06.802 --> 00:41:08.135 align:middle line:90% like that, so it'll settle down. 00:41:08.135 --> 00:41:11.990 align:middle line:90% 00:41:11.990 --> 00:41:13.875 align:middle line:90% Any questions about that? 00:41:13.875 --> 00:41:18.290 align:middle line:90% 00:41:18.290 --> 00:41:23.570 align:middle line:84% Now, I should be a little bit careful in what I'm saying. 00:41:23.570 --> 00:41:27.710 align:middle line:84% And by the way, what this means is if I had MC equal to 0, 00:41:27.710 --> 00:41:30.900 align:middle line:84% this would be minus M2 or M1 for the boost converter. 00:41:30.900 --> 00:41:32.840 align:middle line:84% What that means is this magnitude 00:41:32.840 --> 00:41:36.230 align:middle line:84% would be greater than 1 if it was d greater than 0.5. 00:41:36.230 --> 00:41:40.370 align:middle line:84% So if you took a boost converter and you didn't put any slope 00:41:40.370 --> 00:41:43.838 align:middle line:84% compensation-- if you're only running at 25% duty ratio, fine, 00:41:43.838 --> 00:41:45.380 align:middle line:84% you won't have any ripple instability 00:41:45.380 --> 00:41:46.560 align:middle line:90% because it'll settle down. 00:41:46.560 --> 00:41:48.920 align:middle line:84% If you get near 50% duty ratio or above, 00:41:48.920 --> 00:41:51.500 align:middle line:84% it'll start doing this subharmonic oscillation 00:41:51.500 --> 00:41:54.040 align:middle line:90% or ripple instability stuff. 00:41:54.040 --> 00:41:57.580 align:middle line:84% This is sometimes called the compensating ramp 00:41:57.580 --> 00:41:59.580 align:middle line:84% or the stabilizing ramp for that reason. 00:41:59.580 --> 00:42:02.600 align:middle line:90% 00:42:02.600 --> 00:42:07.430 align:middle line:84% Now, why don't I want to make MC really big? 00:42:07.430 --> 00:42:10.640 align:middle line:84% Well, keep in mind, if we came back 00:42:10.640 --> 00:42:22.340 align:middle line:84% to this original model, notice that MC 00:42:22.340 --> 00:42:26.360 align:middle line:90% appears in this substitution. 00:42:26.360 --> 00:42:31.940 align:middle line:84% My local average linearized dynamics of my converter 00:42:31.940 --> 00:42:35.180 align:middle line:90% are not independent of MC. 00:42:35.180 --> 00:42:41.090 align:middle line:84% And when I told you, oh, by the way, I get nicely-- 00:42:41.090 --> 00:42:43.880 align:middle line:84% I get a really low frequency pole and then a high frequency 00:42:43.880 --> 00:42:48.530 align:middle line:84% pole both on the real axis, that's when MC is small. 00:42:48.530 --> 00:42:55.850 align:middle line:84% As I make MC bigger and bigger, what happens 00:42:55.850 --> 00:42:59.490 align:middle line:84% is these closed-- these open loop poles move. 00:42:59.490 --> 00:43:04.670 align:middle line:84% And if I make MC too big, I get undamped poles again. 00:43:04.670 --> 00:43:08.600 align:middle line:84% So basically, the higher this value of MC, 00:43:08.600 --> 00:43:12.080 align:middle line:84% the closer the converter transfer functions 00:43:12.080 --> 00:43:14.360 align:middle line:84% and current mode control look like they 00:43:14.360 --> 00:43:16.855 align:middle line:84% do for duty ratio control or voltage mode control. 00:43:16.855 --> 00:43:20.274 align:middle line:90% 00:43:20.274 --> 00:43:22.220 align:middle line:90% So what do we do? 00:43:22.220 --> 00:43:27.560 align:middle line:84% We would usually do this kind of ripple instability analysis 00:43:27.560 --> 00:43:29.720 align:middle line:84% and pick an MC that was big enough 00:43:29.720 --> 00:43:32.900 align:middle line:84% to make this settle down and make me happy but not 00:43:32.900 --> 00:43:34.340 align:middle line:90% much bigger. 00:43:34.340 --> 00:43:36.077 align:middle line:90% And then I'd use that value-- 00:43:36.077 --> 00:43:37.910 align:middle line:84% I'd plunk it into here in this average model 00:43:37.910 --> 00:43:40.280 align:middle line:84% and find out where my pole locations are 00:43:40.280 --> 00:43:43.350 align:middle line:90% and go on from there. 00:43:43.350 --> 00:43:45.150 align:middle line:90% I could also make-- 00:43:45.150 --> 00:43:49.860 align:middle line:84% obviously, I could make my compensating ramp slope depend 00:43:49.860 --> 00:43:52.720 align:middle line:90% upon M1 or M2 in some way. 00:43:52.720 --> 00:43:54.780 align:middle line:84% People don't usually go to that range. 00:43:54.780 --> 00:43:56.280 align:middle line:84% If you've got to have your converter 00:43:56.280 --> 00:43:57.750 align:middle line:84% operating in a really wide range, 00:43:57.750 --> 00:43:59.250 align:middle line:84% maybe you start to play those games. 00:43:59.250 --> 00:44:02.400 align:middle line:84% But generally, you're just picking some slope 00:44:02.400 --> 00:44:07.200 align:middle line:84% and setting it with a resistor or something. 00:44:07.200 --> 00:44:08.070 align:middle line:90% Questions? 00:44:08.070 --> 00:44:14.670 align:middle line:90% 00:44:14.670 --> 00:44:16.076 align:middle line:90% Let me summarize. 00:44:16.076 --> 00:44:18.610 align:middle line:90% 00:44:18.610 --> 00:44:21.640 align:middle line:84% We didn't like the fact that we had these very lightly damped 00:44:21.640 --> 00:44:23.610 align:middle line:90% polls. 00:44:23.610 --> 00:44:29.830 align:middle line:84% So we did full state feedback, and we feed back 00:44:29.830 --> 00:44:31.400 align:middle line:90% current on this inner loop. 00:44:31.400 --> 00:44:33.880 align:middle line:84% And then we use the outer loop to set the current reference 00:44:33.880 --> 00:44:35.120 align:middle line:90% and do the outer loop. 00:44:35.120 --> 00:44:37.965 align:middle line:84% And that's the most common way people do things. 00:44:37.965 --> 00:44:44.550 align:middle line:90% 00:44:44.550 --> 00:44:48.150 align:middle line:84% I'll show you a demonstration of doing this that Mansi was 00:44:48.150 --> 00:44:49.990 align:middle line:90% kind enough to set up for us. 00:44:49.990 --> 00:44:55.050 align:middle line:84% And all I'm doing is this is again our demo boost converter 00:44:55.050 --> 00:44:56.040 align:middle line:90% here. 00:44:56.040 --> 00:44:59.760 align:middle line:84% And it's actually using current mode control. 00:44:59.760 --> 00:45:03.000 align:middle line:84% And the notion is what we have it set up 00:45:03.000 --> 00:45:09.550 align:middle line:84% is that we can turn on and off this compensating ramp. 00:45:09.550 --> 00:45:13.030 align:middle line:90% So here we go. 00:45:13.030 --> 00:45:14.840 align:middle line:90% Let me power up the converter. 00:45:14.840 --> 00:45:19.070 align:middle line:90% 00:45:19.070 --> 00:45:21.560 align:middle line:90% And that's pretty nice. 00:45:21.560 --> 00:45:24.090 align:middle line:84% I got more or less periodic steady state operation. 00:45:24.090 --> 00:45:26.720 align:middle line:84% There's actually a tiny bit of jitter in the duty ratio here, 00:45:26.720 --> 00:45:29.750 align:middle line:84% but it's basically switching once per cycle 00:45:29.750 --> 00:45:31.730 align:middle line:90% and nicely settled down. 00:45:31.730 --> 00:45:36.550 align:middle line:84% What happens if I turn off the compensating ramp? 00:45:36.550 --> 00:45:38.830 align:middle line:84% Every cycle is doing something different, 00:45:38.830 --> 00:45:43.600 align:middle line:84% and I can single sequence this so you can see one shot. 00:45:43.600 --> 00:45:46.270 align:middle line:84% What you can see is sometimes it has a long duty ratio. 00:45:46.270 --> 00:45:48.140 align:middle line:84% Sometimes it has a short duty ratio. 00:45:48.140 --> 00:45:49.880 align:middle line:84% It kind of pips around like this. 00:45:49.880 --> 00:45:52.630 align:middle line:84% It's not unstable in the sense that my converter 00:45:52.630 --> 00:45:53.843 align:middle line:90% is going to blow up. 00:45:53.843 --> 00:45:55.510 align:middle line:84% But what it does mean is that the ripple 00:45:55.510 --> 00:45:57.050 align:middle line:90% is all over the place. 00:45:57.050 --> 00:46:00.340 align:middle line:84% And I should say that this is a function-- 00:46:00.340 --> 00:46:04.240 align:middle line:84% if I change the voltage, this tends to-- 00:46:04.240 --> 00:46:07.450 align:middle line:84% if I have low-- if I have a low boosting ratio, 00:46:07.450 --> 00:46:09.680 align:middle line:84% it goes away, even without a compensating ramp. 00:46:09.680 --> 00:46:12.400 align:middle line:84% But the higher I go, because of that ripple instability, 00:46:12.400 --> 00:46:15.250 align:middle line:84% I get some really kind of funky behavior. 00:46:15.250 --> 00:46:17.385 align:middle line:84% If I throw in my compensating ramp-- 00:46:17.385 --> 00:46:24.990 align:middle line:90% 00:46:24.990 --> 00:46:27.040 align:middle line:84% I've just done something bad to my converter. 00:46:27.040 --> 00:46:31.600 align:middle line:90% 00:46:31.600 --> 00:46:39.144 align:middle line:84% There, I get nice steady state operation. 00:46:39.144 --> 00:46:42.040 align:middle line:84% So that is current mode control for you. 00:46:42.040 --> 00:46:44.970 align:middle line:84% I wanted to tell you about it because in the context 00:46:44.970 --> 00:46:47.460 align:middle line:84% of DC-to-DC converters, it's probably the most widely 00:46:47.460 --> 00:46:48.910 align:middle line:90% used control technique. 00:46:48.910 --> 00:46:51.120 align:middle line:84% And to analyze it, you do everything we talked about 00:46:51.120 --> 00:46:53.910 align:middle line:84% so far, and you take one more step further 00:46:53.910 --> 00:46:56.100 align:middle line:90% to get a current mode model. 00:46:56.100 --> 00:46:58.360 align:middle line:84% Any final questions before we wrap up? 00:46:58.360 --> 00:47:01.150 align:middle line:90% 00:47:01.150 --> 00:47:05.320 align:middle line:84% So that was our very brief sequence on modeling and control 00:47:05.320 --> 00:47:06.500 align:middle line:90% of power converters. 00:47:06.500 --> 00:47:08.890 align:middle line:84% We will take up a new topic next class. 00:47:08.890 --> 00:47:10.800 align:middle line:90% Have a great day. 00:47:10.800 --> 00:47:15.000 align:middle line:90%