WEBVTT 00:00:00.000 --> 00:00:01.452 align:middle line:90% [SQUEAKING] 00:00:01.452 --> 00:00:03.388 align:middle line:90% [RUSTLING] 00:00:03.388 --> 00:00:06.776 align:middle line:90% [CLICKING] 00:00:06.776 --> 00:00:09.510 align:middle line:90% 00:00:09.510 --> 00:00:10.440 align:middle line:90% DAVID PERREAULT: OK. 00:00:10.440 --> 00:00:11.520 align:middle line:90% Thank you. 00:00:11.520 --> 00:00:16.050 align:middle line:84% Let's talk about control a little bit more. 00:00:16.050 --> 00:00:20.130 align:middle line:84% What I wanted to do today was a little bit of an example 00:00:20.130 --> 00:00:23.760 align:middle line:84% just to illustrate the trade-offs 00:00:23.760 --> 00:00:26.190 align:middle line:84% and what you need to do to think about a control 00:00:26.190 --> 00:00:27.940 align:middle line:90% design for a general converter. 00:00:27.940 --> 00:00:30.720 align:middle line:90% 00:00:30.720 --> 00:00:36.260 align:middle line:84% And the example I'm going to do is very, very garden variety. 00:00:36.260 --> 00:00:40.960 align:middle line:84% It's not very-- it's not very extreme in any dimension, 00:00:40.960 --> 00:00:44.380 align:middle line:84% So let's think about our boost converter 00:00:44.380 --> 00:00:54.380 align:middle line:84% where I have some input voltage u, an inductor, a switch, 00:00:54.380 --> 00:00:59.780 align:middle line:84% modulating with q of t, a diode, an output capacitor C, 00:00:59.780 --> 00:01:04.988 align:middle line:84% and a resistive load with the output voltage on it. 00:01:04.988 --> 00:01:08.570 align:middle line:84% And the parameters I chose for this 00:01:08.570 --> 00:01:13.160 align:middle line:84% are very, very simple, not aggressive. 00:01:13.160 --> 00:01:16.820 align:middle line:90% L is 10 microhenries. 00:01:16.820 --> 00:01:18.746 align:middle line:90% C is 50 microfarads. 00:01:18.746 --> 00:01:21.980 align:middle line:90% 00:01:21.980 --> 00:01:24.860 align:middle line:84% The switching frequency is 100 kilohertz. 00:01:24.860 --> 00:01:27.860 align:middle line:90% Pretty low. 00:01:27.860 --> 00:01:29.780 align:middle line:84% The output reference that we're going 00:01:29.780 --> 00:01:34.590 align:middle line:84% to try to make this go to via ref is equal to 24 volts. 00:01:34.590 --> 00:01:38.225 align:middle line:84% And we said some designs, that output might move around. 00:01:38.225 --> 00:01:40.850 align:middle line:84% The reference that you want to program it to might move around. 00:01:40.850 --> 00:01:42.330 align:middle line:84% Here, I'm just going to pretend it's fixed 00:01:42.330 --> 00:01:43.622 align:middle line:90% that's about the simplest case. 00:01:43.622 --> 00:01:46.195 align:middle line:90% 00:01:46.195 --> 00:01:49.210 align:middle line:84% And what are the things we said we 00:01:49.210 --> 00:01:52.750 align:middle line:84% had to deal with in terms of variations 00:01:52.750 --> 00:01:53.890 align:middle line:90% in a practical converter? 00:01:53.890 --> 00:01:56.350 align:middle line:84% Well, one of them is the input voltage. 00:01:56.350 --> 00:01:59.840 align:middle line:84% My input voltage might not always be what I wanted to. 00:01:59.840 --> 00:02:05.350 align:middle line:84% So we might say the u nominal, the nominal input voltage 00:02:05.350 --> 00:02:08.490 align:middle line:90% might be 9 volts. 00:02:08.490 --> 00:02:13.470 align:middle line:84% But we're going to design this for 8 volts less than u 00:02:13.470 --> 00:02:15.840 align:middle line:90% is less than 10 volts. 00:02:15.840 --> 00:02:18.270 align:middle line:84% Now that's a variable range, because I 00:02:18.270 --> 00:02:21.660 align:middle line:84% want to illustrate to you the impact of a variable input 00:02:21.660 --> 00:02:25.030 align:middle line:84% voltage range, but that's a pretty narrow range. 00:02:25.030 --> 00:02:28.980 align:middle line:84% I mean, many converters, 2 to 1 is very typical. 00:02:28.980 --> 00:02:30.303 align:middle line:90% 4 to 1 sometimes. 00:02:30.303 --> 00:02:32.470 align:middle line:84% Depending upon the application, even more than that. 00:02:32.470 --> 00:02:34.580 align:middle line:84% So I don't want you to get the notion that this 00:02:34.580 --> 00:02:35.800 align:middle line:90% is some extreme range. 00:02:35.800 --> 00:02:38.940 align:middle line:84% This is actually a really narrow range. 00:02:38.940 --> 00:02:41.430 align:middle line:84% The other thing we said we had to deal with 00:02:41.430 --> 00:02:43.550 align:middle line:84% was variations in the load resistor. 00:02:43.550 --> 00:02:46.722 align:middle line:90% 00:02:46.722 --> 00:02:50.230 align:middle line:84% And I'm going to use a range of 2 ohms-- 00:02:50.230 --> 00:02:51.870 align:middle line:90% that's the heaviest load-- 00:02:51.870 --> 00:02:54.330 align:middle line:84% less than R is less than 10 ohms. 00:02:54.330 --> 00:02:58.986 align:middle line:84% And I'm just going to use a pure resistive load for simplicity. 00:02:58.986 --> 00:03:02.050 align:middle line:84% So that's about-- this is the heavy load case. 00:03:02.050 --> 00:03:04.510 align:middle line:84% That's high power for a given output voltage. 00:03:04.510 --> 00:03:05.650 align:middle line:90% This is low-power case. 00:03:05.650 --> 00:03:08.230 align:middle line:84% This is about a 5-to-1 power range. 00:03:08.230 --> 00:03:10.240 align:middle line:84% All right, that's a little bit more respectable, 00:03:10.240 --> 00:03:12.300 align:middle line:84% but again, a lot of converters have 00:03:12.300 --> 00:03:14.380 align:middle line:84% to deal with very wide power ranges. 00:03:14.380 --> 00:03:16.500 align:middle line:84% And sometimes we bring in additional tricks 00:03:16.500 --> 00:03:18.870 align:middle line:84% that I'm not going to discuss to deal with really 00:03:18.870 --> 00:03:19.990 align:middle line:90% the wide load ranges. 00:03:19.990 --> 00:03:24.270 align:middle line:84% But you'll get a notion of the impact of load resistance 00:03:24.270 --> 00:03:26.340 align:middle line:90% by the load range. 00:03:26.340 --> 00:03:28.250 align:middle line:90% OK 00:03:28.250 --> 00:03:28.950 align:middle line:90% All right. 00:03:28.950 --> 00:03:31.070 align:middle line:84% So that's going to be our example. 00:03:31.070 --> 00:03:34.250 align:middle line:84% Let's step back and just remember 00:03:34.250 --> 00:03:36.740 align:middle line:84% what we did to model all this stuff. 00:03:36.740 --> 00:03:43.210 align:middle line:84% We started off with the switched equations of state. 00:03:43.210 --> 00:03:45.330 align:middle line:84% So I'm just going to copy these down just 00:03:45.330 --> 00:03:47.520 align:middle line:90% so we can see the progression. 00:03:47.520 --> 00:03:59.560 align:middle line:84% So this is the dIL dt is equal to u over L q plus u 00:03:59.560 --> 00:04:04.270 align:middle line:90% minus v over L 1 minus q. 00:04:04.270 --> 00:04:13.570 align:middle line:84% And dV dt is equal to minus 1 over Rc V plus 1 00:04:13.570 --> 00:04:25.480 align:middle line:84% over C IL minus 1 over RC V times 1 minus q. 00:04:25.480 --> 00:04:29.060 align:middle line:84% And I should have said this is q of t. 00:04:29.060 --> 00:04:35.110 align:middle line:84% So in this original model, all we've done 00:04:35.110 --> 00:04:37.120 align:middle line:84% is said broken this down into two states 00:04:37.120 --> 00:04:40.600 align:middle line:84% and use my switching function to switch back and forth 00:04:40.600 --> 00:04:41.720 align:middle line:90% between the states. 00:04:41.720 --> 00:04:44.290 align:middle line:84% And if I knew initial values for IL and dV 00:04:44.290 --> 00:04:46.930 align:middle line:84% and I knew q of t, how it was modulating, 00:04:46.930 --> 00:04:50.020 align:middle line:84% basically, at least in continuous conduction mode, 00:04:50.020 --> 00:04:53.140 align:middle line:84% I could exactly simulate the actual behavior 00:04:53.140 --> 00:04:54.760 align:middle line:90% of that converter. 00:04:54.760 --> 00:04:57.646 align:middle line:90% So if I did something like-- 00:04:57.646 --> 00:05:03.170 align:middle line:90% 00:05:03.170 --> 00:05:05.870 align:middle line:84% if I simulated that set of equations 00:05:05.870 --> 00:05:08.360 align:middle line:84% and I put a change in the load at some time 00:05:08.360 --> 00:05:14.360 align:middle line:84% and I put in a new R, I could calculate the transient there. 00:05:14.360 --> 00:05:19.950 align:middle line:84% But we said, I don't really care about all this ripple. 00:05:19.950 --> 00:05:22.170 align:middle line:84% I can't do anything about that in my controller, 00:05:22.170 --> 00:05:23.690 align:middle line:90% so we want to get rid of that. 00:05:23.690 --> 00:05:25.670 align:middle line:84% And the thing we said we were going to do 00:05:25.670 --> 00:05:31.580 align:middle line:84% was apply this notion of state space averaging. 00:05:31.580 --> 00:05:38.620 align:middle line:84% So if we apply an averaging to this-- 00:05:38.620 --> 00:05:43.360 align:middle line:84% this is x bar is equal to 1 over t the integral of t 00:05:43.360 --> 00:05:46.730 align:middle line:84% minus capital T to t of x of tau d tau. 00:05:46.730 --> 00:05:49.120 align:middle line:84% I can do that to the circuit or I 00:05:49.120 --> 00:05:50.690 align:middle line:90% can do that to these equations. 00:05:50.690 --> 00:05:54.430 align:middle line:84% Either way, I can generate an average set of equations. 00:05:54.430 --> 00:05:56.085 align:middle line:90% So I get an average model. 00:05:56.085 --> 00:06:01.250 align:middle line:90% 00:06:01.250 --> 00:06:06.230 align:middle line:84% And in this case, it was dIL bar dt 00:06:06.230 --> 00:06:16.380 align:middle line:84% is equal to 1 over L u bar minus 1 over L V bar d prime. 00:06:16.380 --> 00:06:32.610 align:middle line:84% And dV bar dt is equal to minus 1 over RC V bar plus 1 over C IL 00:06:32.610 --> 00:06:35.190 align:middle line:90% bar d prime. 00:06:35.190 --> 00:06:39.400 align:middle line:84% And so this is our state space averaged model. 00:06:39.400 --> 00:06:42.240 align:middle line:84% I could also generate a state space average circuit which 00:06:42.240 --> 00:06:45.645 align:middle line:90% would have the same dynamics. 00:06:45.645 --> 00:06:49.320 align:middle line:90% And that's helpful, Why? 00:06:49.320 --> 00:06:57.350 align:middle line:84% Because this thing here doesn't have any switching in it at all. 00:06:57.350 --> 00:07:01.060 align:middle line:84% d is how I'm varying duty ratio over time, 00:07:01.060 --> 00:07:02.660 align:middle line:90% not the switching function. 00:07:02.660 --> 00:07:04.250 align:middle line:84% It wouldn't have any switching ripple. 00:07:04.250 --> 00:07:07.600 align:middle line:84% So in fact, if I simulated this model-- 00:07:07.600 --> 00:07:10.240 align:middle line:84% first of all, I apply the averaging, which 00:07:10.240 --> 00:07:13.700 align:middle line:84% kills the switching ripple, and I simulate the average model, 00:07:13.700 --> 00:07:17.110 align:middle line:84% I will get what we see up there, the solid lines 00:07:17.110 --> 00:07:20.710 align:middle line:84% that capture the low-frequency dynamics and kill the ripple. 00:07:20.710 --> 00:07:23.150 align:middle line:90% And that's pretty nice. 00:07:23.150 --> 00:07:26.200 align:middle line:90% 00:07:26.200 --> 00:07:28.420 align:middle line:90% That helps me. 00:07:28.420 --> 00:07:33.520 align:middle line:84% If I thought about my original goal 00:07:33.520 --> 00:07:37.030 align:middle line:84% for building a control system, It 00:07:37.030 --> 00:07:38.710 align:middle line:90% was along the following lines. 00:07:38.710 --> 00:07:42.790 align:middle line:84% Maybe I have some reference voltage V REF. 00:07:42.790 --> 00:07:46.210 align:middle line:84% And what I could think about from a control modeling 00:07:46.210 --> 00:07:50.300 align:middle line:84% point of view is comparing that to the output voltage-- 00:07:50.300 --> 00:07:53.170 align:middle line:84% and here, I'll think about the average local average output 00:07:53.170 --> 00:07:54.310 align:middle line:90% voltage. 00:07:54.310 --> 00:08:02.820 align:middle line:84% I'll put that into some compensator, which maybe I'll 00:08:02.820 --> 00:08:05.190 align:middle line:90% call GC of s. 00:08:05.190 --> 00:08:09.750 align:middle line:90% So this is the error. 00:08:09.750 --> 00:08:12.120 align:middle line:84% Maybe I think about the local average error. 00:08:12.120 --> 00:08:14.655 align:middle line:84% And then I'm going to get a duty ratio 00:08:14.655 --> 00:08:16.530 align:middle line:84% and then I'm going to put that into the model 00:08:16.530 --> 00:08:19.740 align:middle line:90% for the converter. 00:08:19.740 --> 00:08:23.160 align:middle line:84% And the converter also has an input u, the input voltage. 00:08:23.160 --> 00:08:27.540 align:middle line:84% And out comes IL bar and V bar, which 00:08:27.540 --> 00:08:30.600 align:middle line:84% is the thing I want to control, and I feed that back. 00:08:30.600 --> 00:08:39.610 align:middle line:84% So I could take this set of equations and stick it in here, 00:08:39.610 --> 00:08:42.929 align:middle line:84% and suddenly, I would have a model for my control system. 00:08:42.929 --> 00:08:46.990 align:middle line:90% 00:08:46.990 --> 00:08:51.250 align:middle line:84% The only fly in that ointment is the fact 00:08:51.250 --> 00:08:57.010 align:middle line:84% that this average model is nonlinear. 00:08:57.010 --> 00:09:03.940 align:middle line:84% And the nonlinearity really comes in here and here. 00:09:03.940 --> 00:09:05.980 align:middle line:84% Now that doesn't mean I can't build a controller 00:09:05.980 --> 00:09:07.100 align:middle line:90% with a nonlinear model. 00:09:07.100 --> 00:09:08.420 align:middle line:90% People do that all the time. 00:09:08.420 --> 00:09:11.620 align:middle line:84% But a lot of the tools that you have for control design 00:09:11.620 --> 00:09:14.020 align:middle line:84% are based around the notion of a linear system. 00:09:14.020 --> 00:09:16.060 align:middle line:90% That's the easiest thing to do. 00:09:16.060 --> 00:09:17.720 align:middle line:90% It's the most flexible. 00:09:17.720 --> 00:09:22.390 align:middle line:84% So people generally would like to have a linear model 00:09:22.390 --> 00:09:23.650 align:middle line:90% for their system. 00:09:23.650 --> 00:09:25.720 align:middle line:84% AUDIENCE: So it's the duty cycle that 00:09:25.720 --> 00:09:27.550 align:middle line:90% introduces the nonlinearity? 00:09:27.550 --> 00:09:33.440 align:middle line:84% DAVID PERREAULT: It is the fact that the response of the system 00:09:33.440 --> 00:09:35.713 align:middle line:84% has the product of the compliment of the duty 00:09:35.713 --> 00:09:37.880 align:middle line:84% cycle and the voltage and the complement of the duty 00:09:37.880 --> 00:09:38.960 align:middle line:90% cycle and the current. 00:09:38.960 --> 00:09:43.040 align:middle line:84% So it's the fact that when I modulate duty cycle, 00:09:43.040 --> 00:09:46.400 align:middle line:84% I'm doing something that modulates how the inductor 00:09:46.400 --> 00:09:48.480 align:middle line:90% current changes. 00:09:48.480 --> 00:09:51.380 align:middle line:84% And that modulation is also proportional 00:09:51.380 --> 00:09:53.380 align:middle line:90% to the output voltage. 00:09:53.380 --> 00:09:54.910 align:middle line:90% So it's the fact that-- 00:09:54.910 --> 00:09:57.030 align:middle line:84% you can see that in here, is basically 00:09:57.030 --> 00:10:00.510 align:middle line:84% it comes time to the duty ratio and the output voltage effect 00:10:00.510 --> 00:10:03.145 align:middle line:90% how this is coming as a product. 00:10:03.145 --> 00:10:06.380 align:middle line:90% 00:10:06.380 --> 00:10:07.590 align:middle line:90% We don't like that. 00:10:07.590 --> 00:10:09.590 align:middle line:90% We want a way around it. 00:10:09.590 --> 00:10:12.320 align:middle line:90% So what do we do? 00:10:12.320 --> 00:10:13.600 align:middle line:90% We linearize. 00:10:13.600 --> 00:10:15.350 align:middle line:84% And we said, what we can do is we can say, 00:10:15.350 --> 00:10:19.160 align:middle line:84% suppose this converter is operating at some point, 00:10:19.160 --> 00:10:21.530 align:middle line:84% and all I want to know is if I move it a little bit 00:10:21.530 --> 00:10:24.510 align:middle line:84% away from that point, how will the response wiggle? 00:10:24.510 --> 00:10:26.870 align:middle line:84% I perturb the input just a tiny bit, 00:10:26.870 --> 00:10:30.440 align:middle line:84% the perturbation in the states and the output, maybe 00:10:30.440 --> 00:10:32.960 align:middle line:90% that looks close to linear. 00:10:32.960 --> 00:10:35.210 align:middle line:84% So we went through our linearization process, 00:10:35.210 --> 00:10:39.020 align:middle line:90% which we did last time. 00:10:39.020 --> 00:10:42.110 align:middle line:90% And here's the result we got. 00:10:42.110 --> 00:10:44.150 align:middle line:90% Now we're going to talk about-- 00:10:44.150 --> 00:10:47.930 align:middle line:84% instead of about the average models, 00:10:47.930 --> 00:10:51.590 align:middle line:84% we're going to talk about perturbations, things 00:10:51.590 --> 00:10:56.840 align:middle line:84% like IL tilde, which is equal to the difference 00:10:56.840 --> 00:11:00.980 align:middle line:84% between the actual local average current in the inductor 00:11:00.980 --> 00:11:03.390 align:middle line:84% and the DC current in the inductor. 00:11:03.390 --> 00:11:07.090 align:middle line:84% So it's the deviation away from the steady state. 00:11:07.090 --> 00:11:10.573 align:middle line:84% And when we did that, we said, OK, 00:11:10.573 --> 00:11:12.490 align:middle line:84% here's the kind of equations I'm going to get. 00:11:12.490 --> 00:11:18.190 align:middle line:84% I'm going to get dIL tilde dt is equal to 1 00:11:18.190 --> 00:11:31.120 align:middle line:84% over L u tilde minus d prime over L V tilde plus V 00:11:31.120 --> 00:11:34.390 align:middle line:90% over L d tilde. 00:11:34.390 --> 00:11:42.160 align:middle line:84% And dV tilde dt is equal to minus 1 00:11:42.160 --> 00:11:52.690 align:middle line:84% over RC V tilde plus D prime over C IL tilde minus IL 00:11:52.690 --> 00:11:56.380 align:middle line:90% over C d tilde. 00:11:56.380 --> 00:11:57.900 align:middle line:90% What did I do to get this? 00:11:57.900 --> 00:12:01.420 align:middle line:84% Essentially we took the nonlinear differential equations 00:12:01.420 --> 00:12:03.690 align:middle line:84% and we did a Taylor series expansion. 00:12:03.690 --> 00:12:05.890 align:middle line:84% We threw out the DC terms because we're only 00:12:05.890 --> 00:12:08.770 align:middle line:84% looking at variations and we threw out the products 00:12:08.770 --> 00:12:10.430 align:middle line:90% of small variations. 00:12:10.430 --> 00:12:12.980 align:middle line:84% So it's only valid in a limited range, 00:12:12.980 --> 00:12:16.870 align:middle line:84% but surprisingly, especially for switching power converters 00:12:16.870 --> 00:12:19.120 align:middle line:84% where the ripples are triangular, it turns out, 00:12:19.120 --> 00:12:23.410 align:middle line:84% you can get pretty far away from the operating point 00:12:23.410 --> 00:12:26.260 align:middle line:90% and it's still pretty good. 00:12:26.260 --> 00:12:28.690 align:middle line:90% So this model is pretty good. 00:12:28.690 --> 00:12:31.780 align:middle line:84% What's important about this model from our perspective 00:12:31.780 --> 00:12:34.240 align:middle line:90% is that I have-- 00:12:34.240 --> 00:12:39.050 align:middle line:84% this is a pair of linear constant coefficient 00:12:39.050 --> 00:12:40.610 align:middle line:90% differential equations. 00:12:40.610 --> 00:12:44.000 align:middle line:84% And constant coefficient in the sense 00:12:44.000 --> 00:12:49.350 align:middle line:84% that these are the coefficients to the variables. 00:12:49.350 --> 00:12:51.630 align:middle line:84% So now my variables are all small signal. 00:12:51.630 --> 00:12:55.290 align:middle line:90% These are all just constants. 00:12:55.290 --> 00:12:56.860 align:middle line:90% All right. 00:12:56.860 --> 00:13:00.080 align:middle line:84% And with this, that's a linear system. 00:13:00.080 --> 00:13:02.530 align:middle line:84% I can take a Laplace transform of it, get nice stuff, 00:13:02.530 --> 00:13:05.260 align:middle line:84% I can make transfer functions, everything else. 00:13:05.260 --> 00:13:13.060 align:middle line:84% And the only caveat is that notice, 00:13:13.060 --> 00:13:15.490 align:middle line:84% that these constants are each-- like if my-- 00:13:15.490 --> 00:13:19.360 align:middle line:84% if I decide my DC output voltage changes or, for example, 00:13:19.360 --> 00:13:21.820 align:middle line:84% I operate from a different input voltage, 00:13:21.820 --> 00:13:25.000 align:middle line:84% I'm going to get a different steady state duty ratio 00:13:25.000 --> 00:13:27.010 align:middle line:84% D, and then, hence, a different 1 00:13:27.010 --> 00:13:29.030 align:middle line:90% minus D, which is D prime here. 00:13:29.030 --> 00:13:37.790 align:middle line:84% So these constants do depend upon the load current, the input 00:13:37.790 --> 00:13:39.630 align:middle line:90% voltage, all those things. 00:13:39.630 --> 00:13:42.170 align:middle line:84% So I got to think that the solution I get 00:13:42.170 --> 00:13:44.060 align:middle line:84% is only for one point, and I've got 00:13:44.060 --> 00:13:47.180 align:middle line:84% to go find the numerical values at different points. 00:13:47.180 --> 00:13:54.020 align:middle line:84% But nonetheless, here, I have this model, which maybe 00:13:54.020 --> 00:13:55.980 align:middle line:90% is even better to put in here. 00:13:55.980 --> 00:13:59.690 align:middle line:84% The difference is, here, I might have d tilde, 00:13:59.690 --> 00:14:07.890 align:middle line:84% I might have V tilde, this is V error tilde, and so forth. 00:14:07.890 --> 00:14:10.880 align:middle line:90% And IL tilde. 00:14:10.880 --> 00:14:14.420 align:middle line:84% So that's a better model to use-- or a simpler model 00:14:14.420 --> 00:14:17.650 align:middle line:84% to use from a control design point of view. 00:14:17.650 --> 00:14:20.870 align:middle line:84% And then we said, well, how do we mostly 00:14:20.870 --> 00:14:22.320 align:middle line:90% use that kind of thing? 00:14:22.320 --> 00:14:23.840 align:middle line:84% Well, what we can do is we can say, 00:14:23.840 --> 00:14:26.660 align:middle line:84% take the Laplace transform of this thing. 00:14:26.660 --> 00:14:29.990 align:middle line:84% This d dt's becomes s times IL and this becomes 00:14:29.990 --> 00:14:35.640 align:middle line:84% s times V. I can then eliminate IL from the equation, 00:14:35.640 --> 00:14:38.800 align:middle line:84% and I can actually come up with a transfer function. 00:14:38.800 --> 00:14:42.120 align:middle line:84% And if we do that, we're going to get-- 00:14:42.120 --> 00:14:46.500 align:middle line:84% if u is equal to u-- in other words, is not changing, what 00:14:46.500 --> 00:14:56.250 align:middle line:84% I get is H of s is equal to V tilde of s over d tilde of s, 00:14:56.250 --> 00:15:05.730 align:middle line:84% which is equal to minus s times V over RCD prime 00:15:05.730 --> 00:15:14.520 align:middle line:84% plus VD prime over LC divided by s 00:15:14.520 --> 00:15:23.880 align:middle line:84% squared plus 1 over RC s plus D prime squared over LC. 00:15:23.880 --> 00:15:28.650 align:middle line:84% So this is now a transfer function. 00:15:28.650 --> 00:15:33.660 align:middle line:84% And I can just stick H of s in here 00:15:33.660 --> 00:15:37.660 align:middle line:84% and get my transfer function from d to-- d tilde to V tilde, 00:15:37.660 --> 00:15:40.100 align:middle line:84% and now I'm in a position to do control design. 00:15:40.100 --> 00:15:45.050 align:middle line:90% 00:15:45.050 --> 00:15:47.150 align:middle line:84% Before I move on, are there any questions 00:15:47.150 --> 00:15:48.262 align:middle line:90% about any of these steps? 00:15:48.262 --> 00:15:50.845 align:middle line:84% These are the steps we covered in the last couple of lectures. 00:15:50.845 --> 00:15:57.850 align:middle line:90% 00:15:57.850 --> 00:15:58.350 align:middle line:90% OK. 00:15:58.350 --> 00:16:00.330 align:middle line:84% So I've taken my switched equations-- 00:16:00.330 --> 00:16:01.980 align:middle line:84% or I've derived my switch equations. 00:16:01.980 --> 00:16:03.120 align:middle line:90% And I've averaged them. 00:16:03.120 --> 00:16:04.620 align:middle line:90% Then I've linearized them. 00:16:04.620 --> 00:16:06.410 align:middle line:84% And now I have a transfer function. 00:16:06.410 --> 00:16:09.630 align:middle line:84% I can start to think about my converter control. 00:16:09.630 --> 00:16:14.105 align:middle line:84% Let's think about, what is this transfer function look like? 00:16:14.105 --> 00:16:17.520 align:middle line:90% 00:16:17.520 --> 00:16:21.240 align:middle line:84% If I plotted the poles of this circuit-- 00:16:21.240 --> 00:16:25.080 align:middle line:90% so here's the complex plane. 00:16:25.080 --> 00:16:30.225 align:middle line:84% Here's the real value of s and the imaginary value of s. 00:16:30.225 --> 00:16:35.290 align:middle line:90% 00:16:35.290 --> 00:16:40.060 align:middle line:84% What I would see is I will have one right half-plane 0 00:16:40.060 --> 00:16:42.130 align:middle line:84% because I'll have a positive value of 0, which 00:16:42.130 --> 00:16:43.850 align:middle line:90% generates a 0 in the numerator. 00:16:43.850 --> 00:16:49.540 align:middle line:84% So somewhere over here, I'm going to have a 0. 00:16:49.540 --> 00:16:53.263 align:middle line:84% And then if I plug values in here for typical parameters 00:16:53.263 --> 00:16:55.180 align:middle line:84% like I'm going to see, what we're going to see 00:16:55.180 --> 00:16:57.820 align:middle line:84% is we're going to have a pair of lightly-damped poles 00:16:57.820 --> 00:17:01.780 align:middle line:90% in the left half-plane. 00:17:01.780 --> 00:17:05.450 align:middle line:84% So this means the response is naturally oscillatory. 00:17:05.450 --> 00:17:08.500 align:middle line:84% That shouldn't surprise you because when 00:17:08.500 --> 00:17:09.970 align:middle line:84% we did a little load step there, we 00:17:09.970 --> 00:17:15.356 align:middle line:84% saw a second-order oscillatory response and settling down. 00:17:15.356 --> 00:17:15.856 align:middle line:90% OK. 00:17:15.856 --> 00:17:19.220 align:middle line:90% 00:17:19.220 --> 00:17:22.160 align:middle line:84% And again, where these poles are, 00:17:22.160 --> 00:17:24.950 align:middle line:84% we'll notice that there's a D prime squared here. 00:17:24.950 --> 00:17:27.950 align:middle line:84% So where the actual poles are depends 00:17:27.950 --> 00:17:31.010 align:middle line:90% upon the operating point. 00:17:31.010 --> 00:17:34.350 align:middle line:84% Depends upon what steady state duty ratio I'm operating at. 00:17:34.350 --> 00:17:36.450 align:middle line:84% But it looks something like this. 00:17:36.450 --> 00:17:43.140 align:middle line:84% Now you might ask, gee, you did all this math. 00:17:43.140 --> 00:17:48.060 align:middle line:84% Is this model that's popping out, is that just random? 00:17:48.060 --> 00:17:50.250 align:middle line:84% And the reason you might want to wonder that 00:17:50.250 --> 00:17:56.130 align:middle line:84% is just because this system has a right half-plane 0. 00:17:56.130 --> 00:18:03.340 align:middle line:84% Now those who designed control systems hate half-plane 0's. 00:18:03.340 --> 00:18:05.950 align:middle line:90% They are very annoying. 00:18:05.950 --> 00:18:07.540 align:middle line:90% Why are they annoying? 00:18:07.540 --> 00:18:12.150 align:middle line:84% Because usually if I take some system. 00:18:12.150 --> 00:18:14.513 align:middle line:90% And I put some gain-- 00:18:14.513 --> 00:18:15.930 align:middle line:84% forget about what this control is. 00:18:15.930 --> 00:18:17.347 align:middle line:84% Say it's just a proportional gain. 00:18:17.347 --> 00:18:19.500 align:middle line:84% And I turn up and I close the loop 00:18:19.500 --> 00:18:22.480 align:middle line:84% and I turn up the gain, what tends 00:18:22.480 --> 00:18:24.370 align:middle line:90% to happen is the poles move. 00:18:24.370 --> 00:18:26.470 align:middle line:84% The idea is I had some open loop poles, 00:18:26.470 --> 00:18:28.540 align:middle line:84% and when I put a feedback loop around them, 00:18:28.540 --> 00:18:31.050 align:middle line:84% the dynamics are going to change. 00:18:31.050 --> 00:18:34.860 align:middle line:84% And what happens is that the closed loop poles, when 00:18:34.860 --> 00:18:38.610 align:middle line:84% I close that, are somewhere else, 00:18:38.610 --> 00:18:42.030 align:middle line:84% but it tends to be that as I turn up the gain, 00:18:42.030 --> 00:18:48.080 align:middle line:84% the open loop poles tend to move towards the open loop 0. 00:18:48.080 --> 00:18:50.180 align:middle line:84% And what that means is, if I took this thing 00:18:50.180 --> 00:18:53.570 align:middle line:84% and I say I put it in a proportional feedback loop, 00:18:53.570 --> 00:18:57.070 align:middle line:84% the poles would tend to move towards the 0. 00:18:57.070 --> 00:18:59.470 align:middle line:84% Now if the poles move towards the 0 00:18:59.470 --> 00:19:01.390 align:middle line:84% and the 0's in the right half-plane, 00:19:01.390 --> 00:19:03.800 align:middle line:84% that means the thing tends to go unstable. 00:19:03.800 --> 00:19:05.770 align:middle line:84% So something with a right half-plane 0 00:19:05.770 --> 00:19:07.820 align:middle line:90% is very, very easy to make. 00:19:07.820 --> 00:19:09.970 align:middle line:90% Unstable generally. 00:19:09.970 --> 00:19:12.810 align:middle line:90% So we don't like that. 00:19:12.810 --> 00:19:16.380 align:middle line:84% But you could say, well, OK, you're mathematically 00:19:16.380 --> 00:19:17.800 align:middle line:90% finagling a bunch of stuff. 00:19:17.800 --> 00:19:20.280 align:middle line:84% Can't you just do a version of a model that's not going 00:19:20.280 --> 00:19:21.730 align:middle line:90% to have a right half-plane 0? 00:19:21.730 --> 00:19:25.210 align:middle line:84% Like, why do we have to be-- why does life have to be difficult? 00:19:25.210 --> 00:19:28.990 align:middle line:84% And the answer is the mathematical model 00:19:28.990 --> 00:19:33.340 align:middle line:84% you get actually very nicely reflects the physical dynamics 00:19:33.340 --> 00:19:34.380 align:middle line:90% of the system. 00:19:34.380 --> 00:19:37.030 align:middle line:84% It's not accidental that there's a right half-plane 0 00:19:37.030 --> 00:19:37.760 align:middle line:90% in the equations. 00:19:37.760 --> 00:19:40.450 align:middle line:84% It's intrinsic to the behavior of the system. 00:19:40.450 --> 00:19:41.170 align:middle line:90% OK 00:19:41.170 --> 00:19:46.320 align:middle line:84% What does a right half-plane 0 tend to do? 00:19:46.320 --> 00:19:53.250 align:middle line:84% If I take a system with a right half-plane 0 and I step 00:19:53.250 --> 00:19:53.980 align:middle line:90% the input-- 00:19:53.980 --> 00:20:00.660 align:middle line:84% so suppose-- here's my input, And I step my input 00:20:00.660 --> 00:20:02.600 align:middle line:90% up a little bit. 00:20:02.600 --> 00:20:05.390 align:middle line:84% What I expect to do-- eventually what I want 00:20:05.390 --> 00:20:09.410 align:middle line:84% is that the response, the output should go up, too. 00:20:09.410 --> 00:20:13.230 align:middle line:84% A system with a right half-plane 0 tends to do this. 00:20:13.230 --> 00:20:15.710 align:middle line:90% I want my output to go up from-- 00:20:15.710 --> 00:20:18.470 align:middle line:90% this is the output. 00:20:18.470 --> 00:20:21.710 align:middle line:84% I want it to go up from some level to some higher level. 00:20:21.710 --> 00:20:24.380 align:middle line:84% But what a system with a right half-plane 0 will do 00:20:24.380 --> 00:20:26.900 align:middle line:84% is as soon as I step the input, the output will actually 00:20:26.900 --> 00:20:29.540 align:middle line:84% go the wrong way, and eventually we'll 00:20:29.540 --> 00:20:34.310 align:middle line:84% settle down and go the direction I want. 00:20:34.310 --> 00:20:37.488 align:middle line:84% So right half-plane 0's are also considered nasty 00:20:37.488 --> 00:20:39.280 align:middle line:84% because you want it to go in one direction, 00:20:39.280 --> 00:20:40.822 align:middle line:84% but before it goes in that direction, 00:20:40.822 --> 00:20:44.180 align:middle line:90% it goes in some other direction. 00:20:44.180 --> 00:20:46.690 align:middle line:90% And this system's like that. 00:20:46.690 --> 00:20:49.750 align:middle line:84% But I would argue that this is like that because 00:20:49.750 --> 00:20:52.500 align:middle line:90% of the way the system behaves. 00:20:52.500 --> 00:20:56.350 align:middle line:84% Suppose I wanted to make my output voltage bigger. 00:20:56.350 --> 00:20:57.970 align:middle line:90% Forget about closing the loop. 00:20:57.970 --> 00:20:59.200 align:middle line:90% Suppose I just want to-- 00:20:59.200 --> 00:21:01.200 align:middle line:90% I'm running at some duty ratio. 00:21:01.200 --> 00:21:05.740 align:middle line:84% Suppose my input voltage is 5 volts. 00:21:05.740 --> 00:21:09.990 align:middle line:84% I'm running at 50% duty ratio, and so my output voltage 00:21:09.990 --> 00:21:11.370 align:middle line:90% goes to 10 volts-- 00:21:11.370 --> 00:21:12.520 align:middle line:90% is at 10 volts. 00:21:12.520 --> 00:21:15.820 align:middle line:84% But suppose I suddenly want to make my output voltage higher. 00:21:15.820 --> 00:21:16.415 align:middle line:90% What do I do? 00:21:16.415 --> 00:21:19.510 align:middle line:90% 00:21:19.510 --> 00:21:22.020 align:middle line:90% What do I do to my duty ratio? 00:21:22.020 --> 00:21:22.980 align:middle line:90% I increase it. 00:21:22.980 --> 00:21:26.610 align:middle line:84% So suppose at some point I said, no, 10 volts isn't good enough. 00:21:26.610 --> 00:21:28.390 align:middle line:90% I want to go to 20 volts. 00:21:28.390 --> 00:21:30.200 align:middle line:90% I will step my duty ratio up. 00:21:30.200 --> 00:21:33.150 align:middle line:90% 00:21:33.150 --> 00:21:35.210 align:middle line:90% So I was running at 50%. 00:21:35.210 --> 00:21:36.838 align:middle line:84% Whatever's in my inductor current, 00:21:36.838 --> 00:21:38.380 align:middle line:84% half the time is going to the output, 00:21:38.380 --> 00:21:41.390 align:middle line:84% and that's what's supplying the resistor. 00:21:41.390 --> 00:21:44.580 align:middle line:84% Suddenly, I increased the duty ratio. 00:21:44.580 --> 00:21:47.810 align:middle line:84% Now suddenly, instead of going to the output half the time, 00:21:47.810 --> 00:21:52.070 align:middle line:84% the inductor currents going to ground half the time. 00:21:52.070 --> 00:21:56.160 align:middle line:84% So suddenly, there's less current feeding to the output. 00:21:56.160 --> 00:21:59.545 align:middle line:84% And so what's the output voltage going to do? 00:21:59.545 --> 00:22:02.380 align:middle line:84% It's going to start to fall because you're not feeding 00:22:02.380 --> 00:22:04.030 align:middle line:90% as much current to the output. 00:22:04.030 --> 00:22:07.090 align:middle line:84% Eventually, because I've turned up the duty ratio, 00:22:07.090 --> 00:22:09.490 align:middle line:84% I'm applying a positive voltage across the inductor, 00:22:09.490 --> 00:22:11.290 align:middle line:84% the inductor current is going to build up. 00:22:11.290 --> 00:22:13.902 align:middle line:84% And eventually, the inductor current will catch up 00:22:13.902 --> 00:22:15.860 align:middle line:84% and it'll feed the right current to the output. 00:22:15.860 --> 00:22:19.400 align:middle line:84% But what you're going to get is exactly a behavior like this. 00:22:19.400 --> 00:22:22.600 align:middle line:84% The output is going to droop and then it's going to come up. 00:22:22.600 --> 00:22:33.290 align:middle line:84% So the right half-plane 0 in my control model, this guy here, 00:22:33.290 --> 00:22:38.120 align:middle line:84% in fact, precisely reflects the fundamental physics 00:22:38.120 --> 00:22:41.410 align:middle line:84% of how energy is going through the converter. 00:22:41.410 --> 00:22:44.600 align:middle line:84% So you're so we're going to have to live with it. 00:22:44.600 --> 00:22:46.810 align:middle line:84% It's not a fiction of the math, it's 00:22:46.810 --> 00:22:49.730 align:middle line:90% a fundamental physical thing. 00:22:49.730 --> 00:22:52.000 align:middle line:90% Any questions? 00:22:52.000 --> 00:22:56.860 align:middle line:90% 00:22:56.860 --> 00:22:58.724 align:middle line:90% OK. 00:22:58.724 --> 00:23:01.380 align:middle line:84% So now let's think about building a controller. 00:23:01.380 --> 00:23:06.150 align:middle line:84% I want to come up here and I want to build some compensator 00:23:06.150 --> 00:23:07.430 align:middle line:90% GC of s. 00:23:07.430 --> 00:23:12.240 align:middle line:90% 00:23:12.240 --> 00:23:15.750 align:middle line:84% And the idea here is that I'm going 00:23:15.750 --> 00:23:17.653 align:middle line:84% to have some error between my reference 00:23:17.653 --> 00:23:19.570 align:middle line:84% and my output that generates an error voltage. 00:23:19.570 --> 00:23:23.760 align:middle line:84% I amplify that such that if that error voltage is positive, 00:23:23.760 --> 00:23:26.520 align:middle line:84% d gets bigger, and my error then gets smaller 00:23:26.520 --> 00:23:28.290 align:middle line:90% and I get close to what I want. 00:23:28.290 --> 00:23:31.355 align:middle line:84% So this thing, at least at DC, ought to have a high gain. 00:23:31.355 --> 00:23:34.238 align:middle line:90% 00:23:34.238 --> 00:23:36.190 align:middle line:90% So let's think about this. 00:23:36.190 --> 00:23:38.880 align:middle line:84% What do my plant poles look like here? 00:23:38.880 --> 00:23:45.870 align:middle line:84% My plant poles, I said, were one half-plane 0 and two 00:23:45.870 --> 00:23:50.635 align:middle line:84% left half-plane lightly-damped poles. 00:23:50.635 --> 00:23:53.310 align:middle line:90% 00:23:53.310 --> 00:23:54.180 align:middle line:90% All right. 00:23:54.180 --> 00:23:59.060 align:middle line:84% And there they're complex conjugates. 00:23:59.060 --> 00:24:04.160 align:middle line:84% So what would happen if I made GC of s 00:24:04.160 --> 00:24:06.560 align:middle line:90% just a proportional controller? 00:24:06.560 --> 00:24:08.690 align:middle line:84% I'm going to make this a constant gain. 00:24:08.690 --> 00:24:10.730 align:middle line:84% So I'm going to get a duty ratio that's 00:24:10.730 --> 00:24:13.350 align:middle line:90% proportional to the error. 00:24:13.350 --> 00:24:15.260 align:middle line:84% And what I would want is to turn that gain 00:24:15.260 --> 00:24:19.160 align:middle line:84% up so that a tiny error gives me a big duty ratio 00:24:19.160 --> 00:24:21.470 align:middle line:84% and I get a small steady state error. 00:24:21.470 --> 00:24:26.270 align:middle line:84% What I told you is as I change values of GC of s-- 00:24:26.270 --> 00:24:28.400 align:middle line:90% if I use one-- 00:24:28.400 --> 00:24:32.720 align:middle line:84% a first value of GC of s, if I use a first value of Kp, 00:24:32.720 --> 00:24:35.280 align:middle line:84% maybe my closed-loop poles would move here. 00:24:35.280 --> 00:24:38.653 align:middle line:84% And if I kept a higher value of Kp, they'd move here. 00:24:38.653 --> 00:24:40.820 align:middle line:84% And if you do this, you can form what's a root locus 00:24:40.820 --> 00:24:43.610 align:middle line:84% and say as a function, as I turn up Kp, where 00:24:43.610 --> 00:24:45.440 align:middle line:90% do my closed-loop poles go? 00:24:45.440 --> 00:24:48.265 align:middle line:84% And what this thing would do is it would do something like this. 00:24:48.265 --> 00:24:52.270 align:middle line:90% 00:24:52.270 --> 00:24:53.810 align:middle line:90% And then the poles would split. 00:24:53.810 --> 00:24:58.570 align:middle line:84% And so these are for different values of Kp. 00:24:58.570 --> 00:25:04.200 align:middle line:84% So that's a problem because if I turn up-- 00:25:04.200 --> 00:25:07.680 align:middle line:84% I want a high cape, because I want the error to be small, 00:25:07.680 --> 00:25:09.562 align:middle line:84% but if I make cape too big, once that pole 00:25:09.562 --> 00:25:11.520 align:middle line:84% moves to the right half-plane, this whole thing 00:25:11.520 --> 00:25:13.680 align:middle line:90% is going to oscillate. 00:25:13.680 --> 00:25:16.410 align:middle line:84% So this is the nasty thing about the right half-plane 0, 00:25:16.410 --> 00:25:17.280 align:middle line:90% is that these-- 00:25:17.280 --> 00:25:19.860 align:middle line:84% the closed-loop values will be moved 00:25:19.860 --> 00:25:22.440 align:middle line:84% from where the open loop poles are somewhere into the right 00:25:22.440 --> 00:25:25.030 align:middle line:90% half-plane. 00:25:25.030 --> 00:25:29.800 align:middle line:84% So you wouldn't want to do a proportional compensator on this 00:25:29.800 --> 00:25:31.090 align:middle line:90% just because you're going to-- 00:25:31.090 --> 00:25:32.590 align:middle line:84% either you're going to-- either it's 00:25:32.590 --> 00:25:34.990 align:middle line:84% going to have very bad stability or what's 00:25:34.990 --> 00:25:38.710 align:middle line:84% going to happen is if you your gain to be a really low value. 00:25:38.710 --> 00:25:40.510 align:middle line:84% That means that you're going to have 00:25:40.510 --> 00:25:41.980 align:middle line:84% to end up with living with a pretty 00:25:41.980 --> 00:25:44.980 align:middle line:84% big error between the output voltage you're getting 00:25:44.980 --> 00:25:46.425 align:middle line:90% and your reference voltage. 00:25:46.425 --> 00:25:47.925 align:middle line:84% It's going to be very droopy output. 00:25:47.925 --> 00:25:51.235 align:middle line:90% 00:25:51.235 --> 00:25:52.400 align:middle line:90% So what could I do? 00:25:52.400 --> 00:25:53.990 align:middle line:84% I could pick some other compensator. 00:25:53.990 --> 00:25:55.540 align:middle line:84% Now I'm not going to try to design 00:25:55.540 --> 00:25:57.190 align:middle line:84% a good compensator for you here, that's 00:25:57.190 --> 00:25:58.940 align:middle line:90% not the topic of this class. 00:25:58.940 --> 00:26:01.000 align:middle line:84% But what's the next simplest compensator 00:26:01.000 --> 00:26:04.640 align:middle line:84% maybe I could think of that would fix that problem? 00:26:04.640 --> 00:26:12.230 align:middle line:84% And I would argue you could use GC of s is equal to KI over s. 00:26:12.230 --> 00:26:13.580 align:middle line:90% This is an integral controller. 00:26:13.580 --> 00:26:15.420 align:middle line:90% 1 over s is an integrator. 00:26:15.420 --> 00:26:18.543 align:middle line:84% So what I could do is instead of making my d proportional 00:26:18.543 --> 00:26:19.960 align:middle line:84% to the error voltage, I could make 00:26:19.960 --> 00:26:22.670 align:middle line:84% it proportional to the inner integral of the error voltage. 00:26:22.670 --> 00:26:25.850 align:middle line:84% So p and I are parts of PID controllers. 00:26:25.850 --> 00:26:28.390 align:middle line:84% And the beauty of an integral controller 00:26:28.390 --> 00:26:31.930 align:middle line:84% is eventually it will integrate up the error and leave me with 0 00:26:31.930 --> 00:26:34.790 align:middle line:90% steady state error. 00:26:34.790 --> 00:26:38.650 align:middle line:84% So if I did that, I still have my plant poles-- 00:26:38.650 --> 00:26:43.990 align:middle line:84% I have two lightly-damped plant poles and a right half-plane 0. 00:26:43.990 --> 00:26:46.840 align:middle line:84% And if I did this, well, I'm also 00:26:46.840 --> 00:26:49.790 align:middle line:84% going to have another in my forward loop. 00:26:49.790 --> 00:26:54.040 align:middle line:84% I also have now the dynamics associated with the compensator. 00:26:54.040 --> 00:26:57.520 align:middle line:90% So here's his open loop pole. 00:26:57.520 --> 00:27:01.240 align:middle line:84% And when I start dialing those around, 00:27:01.240 --> 00:27:02.582 align:middle line:90% what it would look like is this. 00:27:02.582 --> 00:27:07.680 align:middle line:84% This pole would move this way, trying to cross through infinity 00:27:07.680 --> 00:27:09.300 align:middle line:90% to come back to the 0. 00:27:09.300 --> 00:27:12.900 align:middle line:84% And then this-- these poles would do something like-- 00:27:12.900 --> 00:27:15.180 align:middle line:84% they'd first swing inwards, and then 00:27:15.180 --> 00:27:19.840 align:middle line:84% they'd come outwards like this, and they do something like this. 00:27:19.840 --> 00:27:21.810 align:middle line:84% And if you take a control class, they'll 00:27:21.810 --> 00:27:24.330 align:middle line:84% tell you exactly how to calculate all these geometries 00:27:24.330 --> 00:27:26.790 align:middle line:90% and what they do. 00:27:26.790 --> 00:27:30.390 align:middle line:84% So this is-- if I add a certain value of KI, 00:27:30.390 --> 00:27:32.280 align:middle line:90% maybe my poles would sit here. 00:27:32.280 --> 00:27:36.108 align:middle line:84% If I turned up KI really big, maybe my poles would be here. 00:27:36.108 --> 00:27:38.650 align:middle line:84% And I wouldn't want that because now it would be oscillating. 00:27:38.650 --> 00:27:42.660 align:middle line:84% So I can still make this thing unstable by using 00:27:42.660 --> 00:27:44.250 align:middle line:90% too much integral gain. 00:27:44.250 --> 00:27:49.920 align:middle line:84% But even with a small integral gain, I can still get to-- 00:27:49.920 --> 00:27:53.820 align:middle line:84% 0 error here gives me a duty ratio, which gets my output, 00:27:53.820 --> 00:27:57.030 align:middle line:84% so my output voltage will track my reference even 00:27:57.030 --> 00:28:00.482 align:middle line:90% with a small integral gain. 00:28:00.482 --> 00:28:03.780 align:middle line:90% Any questions about that? 00:28:03.780 --> 00:28:04.280 align:middle line:90% Yeah, Lowry? 00:28:04.280 --> 00:28:06.280 align:middle line:84% AUDIENCE: How do you know which line you're on-- 00:28:06.280 --> 00:28:07.470 align:middle line:90% [INAUDIBLE] line you're on? 00:28:07.470 --> 00:28:12.830 align:middle line:84% DAVID PERREAULT: How do I know which value on the line? 00:28:12.830 --> 00:28:15.023 align:middle line:90% AUDIENCE: I guess-- oh, I see. 00:28:15.023 --> 00:28:16.940 align:middle line:84% Each of them represents a very different pole. 00:28:16.940 --> 00:28:17.340 align:middle line:90% DAVID PERREAULT: Yeah. 00:28:17.340 --> 00:28:18.550 align:middle line:90% No, each of these-- 00:28:18.550 --> 00:28:22.830 align:middle line:84% each of these points-- these two points represent one gain of KI, 00:28:22.830 --> 00:28:25.740 align:middle line:84% these two points represent some bigger value of KI, 00:28:25.740 --> 00:28:26.740 align:middle line:90% and so forth. 00:28:26.740 --> 00:28:30.125 align:middle line:84% So I can pick a value of KI that would keep my poles over here. 00:28:30.125 --> 00:28:36.510 align:middle line:90% 00:28:36.510 --> 00:28:37.240 align:middle line:90% All right. 00:28:37.240 --> 00:28:40.500 align:middle line:84% So-- I wouldn't do this in the real world, 00:28:40.500 --> 00:28:43.030 align:middle line:84% but maybe I could say, well, if I'm really careful, 00:28:43.030 --> 00:28:45.030 align:middle line:84% maybe I can get away with an integral controller 00:28:45.030 --> 00:28:48.020 align:middle line:84% on this thing and it'll do the right thing. 00:28:48.020 --> 00:28:53.940 align:middle line:84% So now let's start thinking about doing this design. 00:28:53.940 --> 00:28:55.603 align:middle line:84% Maybe I'll use an integral controller 00:28:55.603 --> 00:28:57.520 align:middle line:84% because it's the easiest thing I can think of. 00:28:57.520 --> 00:29:01.775 align:middle line:84% Let's start talking about my open-loop poles and 0's. 00:29:01.775 --> 00:29:05.010 align:middle line:90% 00:29:05.010 --> 00:29:10.440 align:middle line:84% Firstly, we got to remember that this transfer function varies 00:29:10.440 --> 00:29:12.330 align:middle line:90% with operating condition. 00:29:12.330 --> 00:29:20.640 align:middle line:84% So if I'm at a different value of load resistance, 00:29:20.640 --> 00:29:24.600 align:middle line:84% I'm going to have a different 0 location and I'm 00:29:24.600 --> 00:29:27.180 align:middle line:90% going to have a different-- 00:29:27.180 --> 00:29:32.340 align:middle line:84% this doesn't show very cleanly that the pole locations will 00:29:32.340 --> 00:29:35.250 align:middle line:84% differ, except that it's in this term, which actually relates 00:29:35.250 --> 00:29:37.230 align:middle line:90% to the damping. 00:29:37.230 --> 00:29:41.760 align:middle line:84% So if I calculate this transfer function's poles 00:29:41.760 --> 00:29:44.910 align:middle line:84% and 0's for different values of R 00:29:44.910 --> 00:29:47.520 align:middle line:84% at the nominal input resistance, this 00:29:47.520 --> 00:29:50.930 align:middle line:90% is what you're going to see. 00:29:50.930 --> 00:29:53.130 align:middle line:90% The poles in the right half 0-- 00:29:53.130 --> 00:29:54.868 align:middle line:84% right half-plane, these are the-- 00:29:54.868 --> 00:29:56.660 align:middle line:84% I'm sorry, the 0's in the right half-plane, 00:29:56.660 --> 00:29:59.090 align:middle line:84% the location of the 0 moves with different values 00:29:59.090 --> 00:30:01.190 align:middle line:90% of the load resistance. 00:30:01.190 --> 00:30:05.490 align:middle line:84% If I look at the locations of the poles, 00:30:05.490 --> 00:30:07.240 align:middle line:84% they move with the load resistance. 00:30:07.240 --> 00:30:09.780 align:middle line:84% And what will happen is as R gets bigger, 00:30:09.780 --> 00:30:11.960 align:middle line:84% this thing gets less and less damped. 00:30:11.960 --> 00:30:14.670 align:middle line:84% The open-loop poles are moving towards the J omega 00:30:14.670 --> 00:30:17.660 align:middle line:90% axis, which is right here. 00:30:17.660 --> 00:30:18.837 align:middle line:90% Why does it do that? 00:30:18.837 --> 00:30:21.170 align:middle line:84% Well, if you think about back to the original converter, 00:30:21.170 --> 00:30:23.740 align:middle line:90% what's happening? 00:30:23.740 --> 00:30:27.320 align:middle line:84% Essentially, L and C are oscillating. 00:30:27.320 --> 00:30:30.820 align:middle line:84% They slosh energy back and forth dynamically. 00:30:30.820 --> 00:30:33.700 align:middle line:84% Not just directly because they're not directly connected, 00:30:33.700 --> 00:30:36.067 align:middle line:84% but modulated through the duty ratio. 00:30:36.067 --> 00:30:37.900 align:middle line:84% They're still trading energy back and forth, 00:30:37.900 --> 00:30:39.442 align:middle line:84% and that's the source of oscillation. 00:30:39.442 --> 00:30:43.530 align:middle line:84% That's why it's second-order and can be underdamped. 00:30:43.530 --> 00:30:45.950 align:middle line:90% The resistor damps that. 00:30:45.950 --> 00:30:49.700 align:middle line:84% So it's some funny, modulated version of the LC oscillation 00:30:49.700 --> 00:30:51.470 align:middle line:90% with a parallel resistor. 00:30:51.470 --> 00:30:54.660 align:middle line:84% So basically, it's like a parallel resonant tank, 00:30:54.660 --> 00:30:58.000 align:middle line:84% and the resistor is what's damping that thing. 00:30:58.000 --> 00:31:00.770 align:middle line:84% If you make the R big, the damping is small, 00:31:00.770 --> 00:31:02.160 align:middle line:90% and it oscillates more. 00:31:02.160 --> 00:31:05.630 align:middle line:84% So it shouldn't be surprising that the location 00:31:05.630 --> 00:31:08.810 align:middle line:84% of the open loop poles gets closer to the J omega axis 00:31:08.810 --> 00:31:13.980 align:middle line:84% and less damped as R gets bigger. 00:31:13.980 --> 00:31:16.590 align:middle line:84% So I might really have a trouble if my R went to 100, 00:31:16.590 --> 00:31:19.900 align:middle line:90% but I'm only making it go to 10. 00:31:19.900 --> 00:31:22.780 align:middle line:84% Noticing that also, things vary with input voltage 00:31:22.780 --> 00:31:25.240 align:middle line:84% because if I think about input voltage, 00:31:25.240 --> 00:31:30.370 align:middle line:84% D prime here is changing, So that's also 00:31:30.370 --> 00:31:33.970 align:middle line:90% changing the pole locations. 00:31:33.970 --> 00:31:35.012 align:middle line:90% In fact, very directly. 00:31:35.012 --> 00:31:36.470 align:middle line:84% And in fact, you can see that here. 00:31:36.470 --> 00:31:38.080 align:middle line:84% So this is a plot of what happens 00:31:38.080 --> 00:31:45.260 align:middle line:84% when I vary both R and input voltage, and hence, duty ratio. 00:31:45.260 --> 00:31:48.580 align:middle line:84% And all I did was I clunked the pole locations into MATLAB 00:31:48.580 --> 00:31:51.250 align:middle line:90% and said, plot it for me. 00:31:51.250 --> 00:31:53.285 align:middle line:90% So any questions about that? 00:31:53.285 --> 00:31:55.860 align:middle line:90% 00:31:55.860 --> 00:31:57.800 align:middle line:84% So any compensator I'm going to have to build, 00:31:57.800 --> 00:31:59.550 align:middle line:84% the typical thing you do-- you could build 00:31:59.550 --> 00:32:01.650 align:middle line:84% a variable compensator that did different things 00:32:01.650 --> 00:32:05.010 align:middle line:84% for different operating points, but the most typical thing to do 00:32:05.010 --> 00:32:07.500 align:middle line:84% in a DC converter is to say, I know 00:32:07.500 --> 00:32:10.040 align:middle line:90% my plant characteristics vary. 00:32:10.040 --> 00:32:12.300 align:middle line:84% Let me design one compensator that'll 00:32:12.300 --> 00:32:14.560 align:middle line:90% work for any location on that. 00:32:14.560 --> 00:32:16.830 align:middle line:84% That's the simplest thing you can do. 00:32:16.830 --> 00:32:21.510 align:middle line:84% And so let's see what would happen if we tried to do that. 00:32:21.510 --> 00:32:25.470 align:middle line:84% What I'm going to do is let me put 00:32:25.470 --> 00:32:28.877 align:middle line:90% this system in a closed loop. 00:32:28.877 --> 00:32:30.210 align:middle line:90% I'm going to use a compensator-- 00:32:30.210 --> 00:32:31.850 align:middle line:84% I'm going to use this compensator. 00:32:31.850 --> 00:32:34.610 align:middle line:84% I'm going to use an integral controller. 00:32:34.610 --> 00:32:37.320 align:middle line:84% And I'm going to use my plant dynamics there. 00:32:37.320 --> 00:32:40.130 align:middle line:84% And then for different values of KI, 00:32:40.130 --> 00:32:42.228 align:middle line:84% I'm going to take the closed loop, 00:32:42.228 --> 00:32:44.270 align:middle line:84% I'm going to find the closed-loop pole locations, 00:32:44.270 --> 00:32:46.380 align:middle line:84% then I'm just going to plot them. 00:32:46.380 --> 00:32:50.680 align:middle line:84% And I'm just going to do that physically in MATLAB. 00:32:50.680 --> 00:32:51.690 align:middle line:90% Here we go. 00:32:51.690 --> 00:32:55.210 align:middle line:84% So the open-loop pole locations are the ends here. 00:32:55.210 --> 00:32:57.790 align:middle line:84% And this is for a load resistance of 2 ohms. 00:32:57.790 --> 00:32:59.170 align:middle line:84% This is one end of my load range. 00:32:59.170 --> 00:33:01.090 align:middle line:84% This is the heavy load end of my load range. 00:33:01.090 --> 00:33:10.090 align:middle line:84% We can see that for KI, because I have this integral gain of 5, 00:33:10.090 --> 00:33:12.730 align:middle line:84% I'm going to have a pole here that's due to the compensator, 00:33:12.730 --> 00:33:16.340 align:middle line:84% and then the poles that's due to the plant are going to be here. 00:33:16.340 --> 00:33:18.940 align:middle line:84% And the right half-plane 0 is out here. 00:33:18.940 --> 00:33:22.240 align:middle line:84% So that system ought to at least be stable. 00:33:22.240 --> 00:33:25.210 align:middle line:84% And I plotted this for a bunch of different values, increasing 00:33:25.210 --> 00:33:26.150 align:middle line:90% values of KI. 00:33:26.150 --> 00:33:28.720 align:middle line:84% So this root locus that I was showing you, 00:33:28.720 --> 00:33:31.887 align:middle line:84% I've calculated for a bunch of different values of KI. 00:33:31.887 --> 00:33:33.220 align:middle line:90% So you can see what will happen. 00:33:33.220 --> 00:33:37.430 align:middle line:84% If I make KI too big, I will get instability. 00:33:37.430 --> 00:33:40.910 align:middle line:90% 00:33:40.910 --> 00:33:43.110 align:middle line:84% Now that's for one end of my load range, 00:33:43.110 --> 00:33:47.300 align:middle line:84% but keep in mind, the starting place of my poles-- 00:33:47.300 --> 00:33:50.230 align:middle line:84% my open-loop pole's dependent upon my load resistance. 00:33:50.230 --> 00:33:52.580 align:middle line:84% If I started with the other end of my load range-- 00:33:52.580 --> 00:33:58.090 align:middle line:84% 10 ohms, light load, this is what it would look like. 00:33:58.090 --> 00:34:03.560 align:middle line:84% For KI equals 5, I have a dominant pole here. 00:34:03.560 --> 00:34:06.990 align:middle line:84% This is the closed-loop pole location. 00:34:06.990 --> 00:34:08.540 align:middle line:90% And this is-- 00:34:08.540 --> 00:34:11.349 align:middle line:84% I have a pair of super lightly-damped poles here. 00:34:11.349 --> 00:34:14.255 align:middle line:90% 00:34:14.255 --> 00:34:19.929 align:middle line:84% And I could then say, all right, let me calculate 00:34:19.929 --> 00:34:21.590 align:middle line:90% the closed-loop pole locations. 00:34:21.590 --> 00:34:23.800 align:middle line:84% That's how-- once I have the feedback loop, 00:34:23.800 --> 00:34:27.489 align:middle line:84% if I kick the output, that's how it'll oscillate. 00:34:27.489 --> 00:34:30.190 align:middle line:90% With my control system in place. 00:34:30.190 --> 00:34:33.880 align:middle line:84% And I could calculate that for every possible value 00:34:33.880 --> 00:34:36.699 align:middle line:84% of R and input voltage that I might deal with. 00:34:36.699 --> 00:34:40.700 align:middle line:84% And if you do that, this is what you get. 00:34:40.700 --> 00:34:47.900 align:middle line:84% So you could argue that no matter what here, it's 00:34:47.900 --> 00:34:50.000 align:middle line:90% at least stable. 00:34:50.000 --> 00:34:52.139 align:middle line:84% All the poles are in the left half-plane. 00:34:52.139 --> 00:34:54.800 align:middle line:84% This is the J omega axis on the right here. 00:34:54.800 --> 00:34:59.030 align:middle line:84% And I'm going to have a dominant first-order pole, 00:34:59.030 --> 00:35:01.730 align:middle line:84% and then I'm going to have some lightly-damped higher-order 00:35:01.730 --> 00:35:04.220 align:middle line:90% poles. 00:35:04.220 --> 00:35:07.160 align:middle line:90% Any questions about that? 00:35:07.160 --> 00:35:10.010 align:middle line:84% So I could at least say I built a stable system that 00:35:10.010 --> 00:35:11.740 align:middle line:90% won't blow up-- 00:35:11.740 --> 00:35:14.323 align:middle line:84% hopefully-- for any of the loads or input voltages 00:35:14.323 --> 00:35:15.615 align:middle line:90% I'm going to put on this thing. 00:35:15.615 --> 00:35:18.660 align:middle line:90% 00:35:18.660 --> 00:35:20.940 align:middle line:90% OK. 00:35:20.940 --> 00:35:22.300 align:middle line:90% How stable is it? 00:35:22.300 --> 00:35:29.790 align:middle line:84% And this is where me and your average control textbook 00:35:29.790 --> 00:35:32.060 align:middle line:84% break paths in terms of what we're happy with. 00:35:32.060 --> 00:35:34.980 align:middle line:90% 00:35:34.980 --> 00:35:37.290 align:middle line:84% A typical measure for those who do control 00:35:37.290 --> 00:35:39.720 align:middle line:84% are things either like the phase margin you have 00:35:39.720 --> 00:35:41.970 align:middle line:84% or the gain margin you have on your loop. 00:35:41.970 --> 00:35:43.950 align:middle line:84% So what you do is you look at the loop gain 00:35:43.950 --> 00:35:48.030 align:middle line:84% and you plot that out and you can look at the open-loop loop 00:35:48.030 --> 00:35:50.560 align:middle line:90% gain with a controller. 00:35:50.560 --> 00:35:52.770 align:middle line:90% This is what it looks like. 00:35:52.770 --> 00:35:55.290 align:middle line:84% So this would tell you at 10 ohms, which 00:35:55.290 --> 00:35:57.870 align:middle line:84% is the most lightly damped point in 00:35:57.870 --> 00:36:03.870 align:middle line:84% that, this says we have a gain margin of 17 dB 00:36:03.870 --> 00:36:07.770 align:middle line:84% and a phase margin of almost 90 degrees. 00:36:07.770 --> 00:36:11.080 align:middle line:84% So any control guy will tell you that's a super stable system. 00:36:11.080 --> 00:36:13.840 align:middle line:84% You're not you're not going to be unstable, no worries. 00:36:13.840 --> 00:36:16.020 align:middle line:90% And they're right. 00:36:16.020 --> 00:36:18.170 align:middle line:84% The output is not going to blow up. 00:36:18.170 --> 00:36:21.760 align:middle line:84% Whether it's acceptable or not is another question, 00:36:21.760 --> 00:36:23.510 align:middle line:84% and that's what you've got to think about. 00:36:23.510 --> 00:36:24.780 align:middle line:90% And so let's look at this. 00:36:24.780 --> 00:36:28.180 align:middle line:84% Let's look at some load step responses. 00:36:28.180 --> 00:36:31.570 align:middle line:84% So what we do is we have the closed-loop control, 00:36:31.570 --> 00:36:33.760 align:middle line:84% and then we pick the load, and we 00:36:33.760 --> 00:36:36.350 align:middle line:84% see how does the voltage and current respond? 00:36:36.350 --> 00:36:39.970 align:middle line:84% In this case, we're stepping up the-- 00:36:39.970 --> 00:36:41.530 align:middle line:84% we're stepping down the resistance 00:36:41.530 --> 00:36:46.330 align:middle line:84% to see a load step from light load to heavy load, 00:36:46.330 --> 00:36:48.670 align:middle line:84% and then from heavy load to light load. 00:36:48.670 --> 00:36:52.120 align:middle line:84% So here's one from one input end to the other. 00:36:52.120 --> 00:36:56.440 align:middle line:84% And we can see that here's the voltage response, 00:36:56.440 --> 00:36:58.488 align:middle line:84% and here's the current response here. 00:36:58.488 --> 00:36:59.780 align:middle line:90% And it pretty much settles out. 00:36:59.780 --> 00:37:03.160 align:middle line:84% I think this is a simulation thing out here. 00:37:03.160 --> 00:37:07.450 align:middle line:84% So you could say, that's not too badly damped. 00:37:07.450 --> 00:37:09.370 align:middle line:90% I don't mind that so much. 00:37:09.370 --> 00:37:10.860 align:middle line:90% It's all right. 00:37:10.860 --> 00:37:14.485 align:middle line:90% It's not great, but it's stable. 00:37:14.485 --> 00:37:16.830 align:middle line:84% But that's going from light load to heavy load. 00:37:16.830 --> 00:37:21.590 align:middle line:90% 00:37:21.590 --> 00:37:23.420 align:middle line:84% What happens-- oh, and I should say, 00:37:23.420 --> 00:37:25.830 align:middle line:84% that's the actual switch circuit model. 00:37:25.830 --> 00:37:28.350 align:middle line:84% What is the average model say will happen? 00:37:28.350 --> 00:37:29.113 align:middle line:90% There it is. 00:37:29.113 --> 00:37:31.280 align:middle line:84% So the average model-- the thing I want to point out 00:37:31.280 --> 00:37:33.740 align:middle line:84% is even though that's a pretty big load step, 00:37:33.740 --> 00:37:36.590 align:middle line:84% the average model does a really good job of capturing 00:37:36.590 --> 00:37:38.030 align:middle line:90% what goes on. 00:37:38.030 --> 00:37:40.460 align:middle line:84% So you can really get behind averaging 00:37:40.460 --> 00:37:43.400 align:middle line:84% as a way to get a simple look at what your system is doing. 00:37:43.400 --> 00:37:47.240 align:middle line:84% And in fact, if you calculated what the oscillation dynamics 00:37:47.240 --> 00:37:48.740 align:middle line:84% looks like, the small signal model 00:37:48.740 --> 00:37:49.990 align:middle line:90% still looks pretty darn good. 00:37:49.990 --> 00:37:53.690 align:middle line:90% 00:37:53.690 --> 00:37:54.540 align:middle line:90% All right. 00:37:54.540 --> 00:37:57.368 align:middle line:84% Well, what about if I go in the other direction? 00:37:57.368 --> 00:37:58.160 align:middle line:90% I'm going to take-- 00:37:58.160 --> 00:38:00.450 align:middle line:84% I have heavy load, a lot of load on the converter. 00:38:00.450 --> 00:38:03.440 align:middle line:84% Now I'm going to step the load light. 00:38:03.440 --> 00:38:05.690 align:middle line:90% Well, here we go, boom. 00:38:05.690 --> 00:38:08.120 align:middle line:84% The first thing you'll notice, by the way, 00:38:08.120 --> 00:38:14.690 align:middle line:84% is when the load went away, what happens to the output voltage? 00:38:14.690 --> 00:38:16.820 align:middle line:90% It rockets up. 00:38:16.820 --> 00:38:17.803 align:middle line:90% Why does it rocket up? 00:38:17.803 --> 00:38:19.220 align:middle line:84% Because there was all this current 00:38:19.220 --> 00:38:22.890 align:middle line:84% in the inductor and the load resistance-- the load went away. 00:38:22.890 --> 00:38:25.280 align:middle line:84% And so all that energy that was in the inductor 00:38:25.280 --> 00:38:27.260 align:middle line:90% ends up in the output capacitor. 00:38:27.260 --> 00:38:30.850 align:middle line:84% And that output voltage swings way up. 00:38:30.850 --> 00:38:33.150 align:middle line:84% And there ain't a lot you can do about that. 00:38:33.150 --> 00:38:35.800 align:middle line:84% The energy in the inductor has to go somewhere, 00:38:35.800 --> 00:38:38.590 align:middle line:84% and it goes into the output voltage. 00:38:38.590 --> 00:38:43.480 align:middle line:84% So in the real world, if I didn't want to see a ginormous 00:38:43.480 --> 00:38:45.697 align:middle line:84% spike on my output voltage-- like a 10-volt spike 00:38:45.697 --> 00:38:48.280 align:middle line:84% in my output voltage, I better throw a bigger output capacitor 00:38:48.280 --> 00:38:52.320 align:middle line:84% on there, or use a smaller inductor. 00:38:52.320 --> 00:38:55.650 align:middle line:84% So when you're thinking in your design projects about how 00:38:55.650 --> 00:38:59.610 align:middle line:84% to size LNC and you want some cut-off, 00:38:59.610 --> 00:39:03.780 align:middle line:84% using a 1-Henry inductor and a picofarad load filter capacitor 00:39:03.780 --> 00:39:07.650 align:middle line:84% is probably a bad idea because you've got to eat the energy 00:39:07.650 --> 00:39:10.080 align:middle line:84% in that inductor, in a load transient. 00:39:10.080 --> 00:39:13.715 align:middle line:84% So word to the wise, you got to you got to size those very 00:39:13.715 --> 00:39:15.840 align:middle line:84% wisely and make sure you have enough energy storage 00:39:15.840 --> 00:39:17.730 align:middle line:84% in your capacitor that he's not going 00:39:17.730 --> 00:39:21.310 align:middle line:84% to be too unhappy to absorb the energy from the inductor 00:39:21.310 --> 00:39:24.291 align:middle line:90% in a transient. 00:39:24.291 --> 00:39:27.330 align:middle line:84% But, more to the point, what else goes on here? 00:39:27.330 --> 00:39:31.750 align:middle line:84% Here's what happens to the energy in the inductor. 00:39:31.750 --> 00:39:34.320 align:middle line:84% The inductor current-- this is the inductor current here, 00:39:34.320 --> 00:39:36.180 align:middle line:90% it goes blam! 00:39:36.180 --> 00:39:37.450 align:middle line:90% It goes down to 0. 00:39:37.450 --> 00:39:40.590 align:middle line:84% Basically, the output voltage gets big. 00:39:40.590 --> 00:39:42.705 align:middle line:84% The converter says, oh my god, turn-- 00:39:42.705 --> 00:39:45.530 align:middle line:90% 00:39:45.530 --> 00:39:46.940 align:middle line:90% what can I do? 00:39:46.940 --> 00:39:50.120 align:middle line:84% The most I can do is turn my duty ratio to 0. 00:39:50.120 --> 00:39:53.960 align:middle line:84% And I turn my duty ratio to 0, and all that energy 00:39:53.960 --> 00:39:56.330 align:middle line:84% in the inductor ends up in the capacitor. 00:39:56.330 --> 00:39:59.640 align:middle line:84% And then it starts to switch again. 00:39:59.640 --> 00:40:02.440 align:middle line:84% But you notice here, things are clipped at 0. 00:40:02.440 --> 00:40:05.640 align:middle line:84% In this time period, this converter 00:40:05.640 --> 00:40:08.165 align:middle line:84% is actually in discontinuous conduction mode. 00:40:08.165 --> 00:40:13.434 align:middle line:90% 00:40:13.434 --> 00:40:17.930 align:middle line:84% So actually, that average model isn't quite right anymore. 00:40:17.930 --> 00:40:21.990 align:middle line:84% Now if we simulate this, we actually do pretty well, 00:40:21.990 --> 00:40:25.460 align:middle line:84% but the simulation-- actually, we stick it in a diode to keep 00:40:25.460 --> 00:40:26.000 align:middle line:90% the-- 00:40:26.000 --> 00:40:27.800 align:middle line:90% we clamped it so-- 00:40:27.800 --> 00:40:32.960 align:middle line:84% in the simulation so that it wouldn't have the inductor 00:40:32.960 --> 00:40:34.430 align:middle line:90% current go negative. 00:40:34.430 --> 00:40:35.540 align:middle line:90% Why? 00:40:35.540 --> 00:40:38.160 align:middle line:84% Because this has a switch in a diode in it. 00:40:38.160 --> 00:40:44.310 align:middle line:84% Now if you had two switches, this average model's perfect, 00:40:44.310 --> 00:40:45.600 align:middle line:90% it works all the time. 00:40:45.600 --> 00:40:47.540 align:middle line:90% It is still clipped. 00:40:47.540 --> 00:40:50.120 align:middle line:84% In our system, we can never have duty ratios greater than 1 00:40:50.120 --> 00:40:53.820 align:middle line:84% or less than 0, so there is some clipping that goes on. 00:40:53.820 --> 00:40:56.990 align:middle line:84% But, the other thing I wanted to point out-- 00:40:56.990 --> 00:41:00.633 align:middle line:84% and this is related to what we said earlier. 00:41:00.633 --> 00:41:02.300 align:middle line:84% I showed you we have the gain margin of, 00:41:02.300 --> 00:41:08.120 align:middle line:84% what, 17 db and the phase margin was 90 degrees, that's 00:41:08.120 --> 00:41:09.560 align:middle line:90% perfectly stable. 00:41:09.560 --> 00:41:11.190 align:middle line:84% Anybody would say that's very stable. 00:41:11.190 --> 00:41:14.760 align:middle line:84% But look at the oscillation in the output voltage here. 00:41:14.760 --> 00:41:17.807 align:middle line:90% Like, to my mind-- 00:41:17.807 --> 00:41:19.890 align:middle line:84% I'm sorry, this is the oscillation in the current, 00:41:19.890 --> 00:41:21.848 align:middle line:84% but there's also an oscillation in the voltage. 00:41:21.848 --> 00:41:23.820 align:middle line:90% That's very ringy. 00:41:23.820 --> 00:41:27.390 align:middle line:84% I wouldn't like that at all in a real converter, 00:41:27.390 --> 00:41:30.070 align:middle line:84% especially if I'm feeding some sensitive load, 00:41:30.070 --> 00:41:34.580 align:middle line:84% and when I set the load, the output voltage is going erargh. 00:41:34.580 --> 00:41:37.720 align:middle line:84% And what you can-- and where does this oscillation come from? 00:41:37.720 --> 00:41:44.120 align:middle line:84% This oscillation comes from these lightly-damped poles. 00:41:44.120 --> 00:41:47.420 align:middle line:84% The dominant pole here makes it really stable. 00:41:47.420 --> 00:41:49.370 align:middle line:84% It's not going to-- the output's not 00:41:49.370 --> 00:41:51.660 align:middle line:84% going to walk off and go to infinity or something, 00:41:51.660 --> 00:41:54.570 align:middle line:84% so that's why the model says it's stable. 00:41:54.570 --> 00:41:57.500 align:middle line:84% Nonetheless, these lightly-damped poles 00:41:57.500 --> 00:41:59.820 align:middle line:84% still cause oscillation in the output. 00:41:59.820 --> 00:42:03.860 align:middle line:84% Now I chose on purpose an extreme example 00:42:03.860 --> 00:42:06.020 align:middle line:90% that makes it look pretty bad. 00:42:06.020 --> 00:42:08.420 align:middle line:84% What you can see sometimes is that you'll 00:42:08.420 --> 00:42:10.730 align:middle line:84% have something that's very, very stable, 00:42:10.730 --> 00:42:13.640 align:middle line:84% but there'll be some really lightly-damped pole, 00:42:13.640 --> 00:42:17.930 align:middle line:84% maybe not even a little bit down, but way down. 00:42:17.930 --> 00:42:20.000 align:middle line:84% And what you often see when you do that, 00:42:20.000 --> 00:42:22.370 align:middle line:84% if you leave this lightly-damped pole hanging there, 00:42:22.370 --> 00:42:29.510 align:middle line:84% is that your output voltage will look fine on a macro scale, 00:42:29.510 --> 00:42:33.650 align:middle line:84% but then you zoom in on it, and you find it's wobbling by 10 00:42:33.650 --> 00:42:37.370 align:middle line:84% or 20 or 30 millivolts on top of your 24-volt output, 00:42:37.370 --> 00:42:40.340 align:middle line:84% and then you have to ask the question, is that tolerable? 00:42:40.340 --> 00:42:44.120 align:middle line:84% And maybe I don't care about 30 millivolts of ripple 00:42:44.120 --> 00:42:47.250 align:middle line:84% at 20 kilohertz because there's some lightly-damped pole pair. 00:42:47.250 --> 00:42:48.740 align:middle line:90% I could live with that. 00:42:48.740 --> 00:42:50.240 align:middle line:84% But there's other applications where 00:42:50.240 --> 00:42:51.782 align:middle line:84% they tell you, oh, by the way, you've 00:42:51.782 --> 00:42:54.080 align:middle line:84% got to have your output voltage rock solid, 00:42:54.080 --> 00:42:57.200 align:middle line:84% and some lightly-damped pole pair 00:42:57.200 --> 00:42:59.720 align:middle line:84% that's going to get excited by noise and just womble 00:42:59.720 --> 00:43:02.660 align:middle line:84% around is probably not acceptable. 00:43:02.660 --> 00:43:06.790 align:middle line:84% So in addition to making your control loops stable, 00:43:06.790 --> 00:43:11.320 align:middle line:84% I would argue you've got to think about how precise does it 00:43:11.320 --> 00:43:13.180 align:middle line:84% have to track the output voltage? 00:43:13.180 --> 00:43:16.490 align:middle line:84% How much will it wiggle when something happens to it? 00:43:16.490 --> 00:43:17.690 align:middle line:90% And is that acceptable? 00:43:17.690 --> 00:43:20.110 align:middle line:84% And in power supplies, very often your job 00:43:20.110 --> 00:43:24.260 align:middle line:84% is to make the world clean, solid voltage no matter what. 00:43:24.260 --> 00:43:27.370 align:middle line:84% And in those cases, you've got to think very carefully 00:43:27.370 --> 00:43:30.530 align:middle line:84% about how you want to deal with the closed-loop poles. 00:43:30.530 --> 00:43:34.120 align:middle line:84% Maybe you want to make sure that no poles are super 00:43:34.120 --> 00:43:37.030 align:middle line:84% lightly damped even if they're pretty far down 00:43:37.030 --> 00:43:39.380 align:middle line:84% and not contributing that much to stability, 00:43:39.380 --> 00:43:43.620 align:middle line:84% maybe they'll make your output response unpleasant. 00:43:43.620 --> 00:43:45.610 align:middle line:90% So, that was me talking a lot. 00:43:45.610 --> 00:43:47.600 align:middle line:84% Are there any questions about those examples? 00:43:47.600 --> 00:43:50.610 align:middle line:90% 00:43:50.610 --> 00:43:54.180 align:middle line:84% AUDIENCE: What devices would require a more stable output 00:43:54.180 --> 00:43:54.803 align:middle line:90% voltage? 00:43:54.803 --> 00:43:56.970 align:middle line:84% DAVID PERREAULT: If you were building a power supply 00:43:56.970 --> 00:43:58.890 align:middle line:84% to feed a sensitive instrumentation 00:43:58.890 --> 00:44:05.280 align:middle line:84% amplifier, for example, because if you have some instrumentation 00:44:05.280 --> 00:44:07.470 align:middle line:84% amplifier that's picking up some tiny signal, 00:44:07.470 --> 00:44:12.360 align:middle line:84% I don't know, for an EKG or whatever it is, that amplifier 00:44:12.360 --> 00:44:14.910 align:middle line:84% has some power supply rejection ratio, 00:44:14.910 --> 00:44:18.240 align:middle line:84% meaning the output of the amplifier 00:44:18.240 --> 00:44:21.520 align:middle line:84% will give you the response to the input, 00:44:21.520 --> 00:44:23.940 align:middle line:84% but there's a tiny little response to the power supply 00:44:23.940 --> 00:44:25.310 align:middle line:90% variation. 00:44:25.310 --> 00:44:28.350 align:middle line:84% And that little response may be bad. 00:44:28.350 --> 00:44:31.700 align:middle line:84% So if your power supply is wiggling around, 00:44:31.700 --> 00:44:33.810 align:middle line:84% a little bit of that gets into the output, 00:44:33.810 --> 00:44:35.070 align:middle line:90% and so you don't like that. 00:44:35.070 --> 00:44:39.630 align:middle line:84% If what I'm feeding is a resistor, maybe you don't care. 00:44:39.630 --> 00:44:41.760 align:middle line:84% But so it depends on the application, 00:44:41.760 --> 00:44:46.220 align:middle line:84% but a lot of the time, if it's a power supply for instrumentation 00:44:46.220 --> 00:44:48.590 align:middle line:84% amplifier or for an RF transmitter 00:44:48.590 --> 00:44:51.050 align:middle line:84% or for a receiver or something like that, you've 00:44:51.050 --> 00:44:54.460 align:middle line:90% got to keep it very clean. 00:44:54.460 --> 00:44:55.250 align:middle line:90% Yeah, Jack? 00:44:55.250 --> 00:44:58.210 align:middle line:84% AUDIENCE: Is there a phase margin for the transfer function 00:44:58.210 --> 00:45:00.893 align:middle line:90% from D to V. 00:45:00.893 --> 00:45:01.810 align:middle line:90% DAVID PERREAULT: Yeah. 00:45:01.810 --> 00:45:03.250 align:middle line:90% That's true, yes. 00:45:03.250 --> 00:45:06.645 align:middle line:84% AUDIENCE: That's why you showed us a load step that that phase 00:45:06.645 --> 00:45:07.520 align:middle line:90% margin was different. 00:45:07.520 --> 00:45:08.600 align:middle line:90% DAVID PERREAULT: That's right. 00:45:08.600 --> 00:45:11.020 align:middle line:84% But on the other hand, from a small signal point of view, 00:45:11.020 --> 00:45:14.050 align:middle line:90% all the pole locations are-- 00:45:14.050 --> 00:45:17.170 align:middle line:84% once I have a linear system, the pole locations 00:45:17.170 --> 00:45:20.490 align:middle line:84% are the same for any input to any-- 00:45:20.490 --> 00:45:21.180 align:middle line:90% for any input. 00:45:21.180 --> 00:45:24.895 align:middle line:90% 00:45:24.895 --> 00:45:28.340 align:middle line:84% That's a good question, but the phase margin-- 00:45:28.340 --> 00:45:29.590 align:middle line:90% AUDIENCE: I guess [INAUDIBLE]. 00:45:29.590 --> 00:45:31.090 align:middle line:84% DAVID PERREAULT: I'd have to think-- 00:45:31.090 --> 00:45:32.960 align:middle line:84% I'd have to think about whether that's true. 00:45:32.960 --> 00:45:35.305 align:middle line:84% But it's also true that you would still see that-- 00:45:35.305 --> 00:45:37.390 align:middle line:84% if I did a tiny little load step, 00:45:37.390 --> 00:45:39.560 align:middle line:84% you'd still see that kind of thing. 00:45:39.560 --> 00:45:42.620 align:middle line:84% And if I did a tiny reference step 00:45:42.620 --> 00:45:45.000 align:middle line:84% and came back to what I'm treating as the input, 00:45:45.000 --> 00:45:45.985 align:middle line:90% you'd still see that. 00:45:45.985 --> 00:45:46.610 align:middle line:90% AUDIENCE: Yeah. 00:45:46.610 --> 00:45:49.027 align:middle line:84% I guess that's [INAUDIBLE] phase margin that's [INAUDIBLE] 00:45:49.027 --> 00:45:50.232 align:middle line:90% is pretty [INAUDIBLE]. 00:45:50.232 --> 00:45:51.440 align:middle line:90% DAVID PERREAULT: What's that? 00:45:51.440 --> 00:45:53.330 align:middle line:84% AUDIENCE: [INAUDIBLE] 90 degrees is pretty [INAUDIBLE]. 00:45:53.330 --> 00:45:56.180 align:middle line:84% DAVID PERREAULT: Right, but the only thing that's telling you 00:45:56.180 --> 00:45:57.960 align:middle line:84% is what the dominant pole is doing. 00:45:57.960 --> 00:45:59.270 align:middle line:90% It is very damped. 00:45:59.270 --> 00:46:03.200 align:middle line:84% It will-- the majority of the transition 00:46:03.200 --> 00:46:07.580 align:middle line:90% will reflect that dominant pole. 00:46:07.580 --> 00:46:09.350 align:middle line:84% So the reason this looks damped is 00:46:09.350 --> 00:46:12.170 align:middle line:84% because it's dominated by this guy down here. 00:46:12.170 --> 00:46:14.960 align:middle line:90% Let me put on-- 00:46:14.960 --> 00:46:16.271 align:middle line:90% right? 00:46:16.271 --> 00:46:19.750 align:middle line:84% But that doesn't mean there's not some small oscillations. 00:46:19.750 --> 00:46:21.360 align:middle line:84% So the small oscillations aren't going 00:46:21.360 --> 00:46:22.960 align:middle line:90% to make your circuit blow up. 00:46:22.960 --> 00:46:25.382 align:middle line:84% And that's why they say, oh, it's great, 00:46:25.382 --> 00:46:26.340 align:middle line:90% it's very, very stable. 00:46:26.340 --> 00:46:27.460 align:middle line:90% And it's true. 00:46:27.460 --> 00:46:30.480 align:middle line:84% The problem is, if these wiggling around, 00:46:30.480 --> 00:46:32.340 align:middle line:84% even if it's very, very small, might 00:46:32.340 --> 00:46:35.540 align:middle line:84% be unacceptable as a response at the output, 00:46:35.540 --> 00:46:39.030 align:middle line:84% that's the point I wanted to make to make. 00:46:39.030 --> 00:46:40.770 align:middle line:90% Does that answer your question? 00:46:40.770 --> 00:46:43.668 align:middle line:90% OK. 00:46:43.668 --> 00:46:46.860 align:middle line:84% So this is just to give you an idea of the way people-- 00:46:46.860 --> 00:46:49.470 align:middle line:84% first of all, the process people use to get a model, 00:46:49.470 --> 00:46:51.900 align:middle line:84% and then the way they think about what you might use that 00:46:51.900 --> 00:46:55.125 align:middle line:84% model for, understanding if some of you have had control system 00:46:55.125 --> 00:46:56.375 align:middle line:90% design and some of you hadn't. 00:46:56.375 --> 00:46:58.900 align:middle line:90% 00:46:58.900 --> 00:47:03.760 align:middle line:84% You notice the question I dodged is, well, 00:47:03.760 --> 00:47:06.970 align:middle line:84% if this is a bad compensator and this is a bad compensator, 00:47:06.970 --> 00:47:10.006 align:middle line:90% what's a good compensator? 00:47:10.006 --> 00:47:15.540 align:middle line:84% And the answer to that is, there's 00:47:15.540 --> 00:47:18.510 align:middle line:90% a lot of things you can do. 00:47:18.510 --> 00:47:21.930 align:middle line:84% One of them-- if I just wanted to use something 00:47:21.930 --> 00:47:26.700 align:middle line:84% like an integral controller or a more fancy version that's not 00:47:26.700 --> 00:47:29.850 align:middle line:84% just an integral controller, but something that was a good duty 00:47:29.850 --> 00:47:33.320 align:middle line:84% ratio control scheme, one thing I could do 00:47:33.320 --> 00:47:38.757 align:middle line:84% is change my plant so that it had friendlier dynamics. 00:47:38.757 --> 00:47:41.090 align:middle line:84% I already said, if you don't want the voltage overshoot, 00:47:41.090 --> 00:47:42.450 align:middle line:90% I'd use a bigger capacitor. 00:47:42.450 --> 00:47:46.970 align:middle line:84% I can also do things to the actual converter 00:47:46.970 --> 00:47:49.730 align:middle line:84% to make it friendlier for my dynamics. 00:47:49.730 --> 00:47:56.900 align:middle line:84% And what we saw as one example was that-- 00:47:56.900 --> 00:47:59.150 align:middle line:84% well, what's really happening is this resistor 00:47:59.150 --> 00:48:04.824 align:middle line:84% is damping the LC oscillation modulated through the switching. 00:48:04.824 --> 00:48:07.680 align:middle line:84% But the problem is, when R gets big, it's not well-damped. 00:48:07.680 --> 00:48:10.180 align:middle line:84% Well, what happens if I walk up to this thing and I say, 00:48:10.180 --> 00:48:11.230 align:middle line:90% you know what? 00:48:11.230 --> 00:48:13.610 align:middle line:84% I will throw a damping leg on the output of this. 00:48:13.610 --> 00:48:16.245 align:middle line:84% I'll put another resistor in here, RD. 00:48:16.245 --> 00:48:20.680 align:middle line:90% 00:48:20.680 --> 00:48:23.260 align:middle line:90% And here, I have see C big. 00:48:23.260 --> 00:48:26.380 align:middle line:90% 00:48:26.380 --> 00:48:27.830 align:middle line:84% When we talk about filter design, 00:48:27.830 --> 00:48:30.038 align:middle line:84% you'll also see these see these kind of damping legs. 00:48:30.038 --> 00:48:32.580 align:middle line:84% And you've seen it in snubbers, too, right? 00:48:32.580 --> 00:48:38.190 align:middle line:84% The idea is, at frequencies near the oscillation point with L 00:48:38.190 --> 00:48:41.130 align:middle line:84% and C, if C looks like a short circuit, 00:48:41.130 --> 00:48:45.380 align:middle line:84% then RD is in parallel with R and damps this. 00:48:45.380 --> 00:48:48.020 align:middle line:84% And then if R gets big or small, it always 00:48:48.020 --> 00:48:49.470 align:middle line:90% has RD in parallel with it. 00:48:49.470 --> 00:48:53.750 align:middle line:84% So if this was 2 ohms and I was happy with the response I saw 00:48:53.750 --> 00:48:57.680 align:middle line:84% for 2-ohm damping, then maybe this damping leg added 00:48:57.680 --> 00:49:03.740 align:middle line:84% on to my converter would satisfy me. 00:49:03.740 --> 00:49:06.530 align:middle line:84% And I also have some more big capacitance here 00:49:06.530 --> 00:49:10.200 align:middle line:84% to help me absorb larger voltage transients. 00:49:10.200 --> 00:49:12.810 align:middle line:84% So one thing I can do if I don't like my dynamics, 00:49:12.810 --> 00:49:15.630 align:middle line:84% and you might not, either because the overshoots are too 00:49:15.630 --> 00:49:18.540 align:middle line:84% big or because it's not as well-damped as you want, 00:49:18.540 --> 00:49:22.170 align:middle line:84% as you can do things like add damping legs or other things 00:49:22.170 --> 00:49:23.010 align:middle line:90% to the plant-- 00:49:23.010 --> 00:49:24.510 align:middle line:84% to the actual converter or change 00:49:24.510 --> 00:49:28.230 align:middle line:84% the component values so that it is better damped, 00:49:28.230 --> 00:49:31.860 align:middle line:84% or that it has friendlier response. 00:49:31.860 --> 00:49:33.460 align:middle line:90% Nothing wrong with that. 00:49:33.460 --> 00:49:35.720 align:middle line:90% Do it all the time. 00:49:35.720 --> 00:49:37.610 align:middle line:90% The upside is it works. 00:49:37.610 --> 00:49:40.100 align:middle line:84% The downside is, now I've got to go back out 00:49:40.100 --> 00:49:43.790 align:middle line:84% and buy an extra capacitor and a damping resistor-- 00:49:43.790 --> 00:49:45.520 align:middle line:90% it's more expensive. 00:49:45.520 --> 00:49:47.330 align:middle line:84% There's some things you can't change. 00:49:47.330 --> 00:49:49.530 align:middle line:84% I mean, some things you must change in the plant. 00:49:49.530 --> 00:49:51.560 align:middle line:84% If I want to reduce that amount, that voltage 00:49:51.560 --> 00:49:53.660 align:middle line:84% is going to swing up when the load steps down, 00:49:53.660 --> 00:49:55.500 align:middle line:84% I better go get more capacitance. 00:49:55.500 --> 00:49:57.930 align:middle line:84% There's no way around it because there's nowhere 00:49:57.930 --> 00:50:02.450 align:middle line:84% else for the energy to go unless I'm going to somehow dump it. 00:50:02.450 --> 00:50:04.460 align:middle line:84% So sometimes you have to change things. 00:50:04.460 --> 00:50:07.520 align:middle line:84% But you don't tend to add hardware 00:50:07.520 --> 00:50:10.775 align:middle line:90% if you can do it in control. 00:50:10.775 --> 00:50:14.470 align:middle line:84% And the way I might think about doing something 00:50:14.470 --> 00:50:16.470 align:middle line:84% like that, I'd certainly use a bigger capacitor. 00:50:16.470 --> 00:50:19.220 align:middle line:84% The other thing I would probably do in this kind of converter 00:50:19.220 --> 00:50:21.870 align:middle line:90% is a different control scheme. 00:50:21.870 --> 00:50:23.810 align:middle line:84% You notice that in this control scheme, 00:50:23.810 --> 00:50:28.250 align:middle line:84% we reduced everything down to a transfer function 00:50:28.250 --> 00:50:31.070 align:middle line:84% from duty ratio to voltage and I just ignored what the inductor 00:50:31.070 --> 00:50:32.980 align:middle line:90% current was doing. 00:50:32.980 --> 00:50:34.600 align:middle line:84% Well, it turns out that control guy 00:50:34.600 --> 00:50:37.200 align:middle line:84% will tell you, well, you should be doing full-state feedback. 00:50:37.200 --> 00:50:40.060 align:middle line:84% In other words, don't just feedback V, but feedback I, 00:50:40.060 --> 00:50:43.210 align:middle line:84% and use that knowledge to improve the control so 00:50:43.210 --> 00:50:47.470 align:middle line:84% that you basically don't make your poles worse, 00:50:47.470 --> 00:50:48.650 align:middle line:90% but you make them better. 00:50:48.650 --> 00:50:52.690 align:middle line:84% And it turns out that just by doing better control, by looking 00:50:52.690 --> 00:50:56.200 align:middle line:84% at the inductor current and the voltage, 00:50:56.200 --> 00:50:58.210 align:middle line:84% I can make a better controller that 00:50:58.210 --> 00:51:02.360 align:middle line:84% will lead to a much nicer-- more nicely damped response. 00:51:02.360 --> 00:51:05.830 align:middle line:84% And we will talk about that-- that technique is called 00:51:05.830 --> 00:51:08.440 align:middle line:84% current mode control or current program control, 00:51:08.440 --> 00:51:11.110 align:middle line:84% and that will be the topic of next class. 00:51:11.110 --> 00:51:13.110 align:middle line:90% Have a great day. 00:51:13.110 --> 00:51:16.000 align:middle line:90%