WEBVTT 00:00:00.000 --> 00:00:09.160 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:09.160 --> 00:00:12.160 align:middle line:84% DAVID PERREAULT: We started talking about modeling 00:00:12.160 --> 00:00:14.140 align:middle line:90% of control systems. 00:00:14.140 --> 00:00:17.680 align:middle line:84% And really, what we want to do is 00:00:17.680 --> 00:00:21.320 align:middle line:84% have some model for the dynamics of a DC/DC converter. 00:00:21.320 --> 00:00:29.080 align:middle line:84% And the example converter that we've been using 00:00:29.080 --> 00:00:30.890 align:middle line:90% is a boost converter. 00:00:30.890 --> 00:00:33.430 align:middle line:84% So here's our example boost converter. 00:00:33.430 --> 00:00:37.900 align:middle line:84% I might have some input voltage that I'll call u. 00:00:37.900 --> 00:00:44.620 align:middle line:84% I have some switch that I'm modulating with q of t, diode, 00:00:44.620 --> 00:00:48.760 align:middle line:90% and then some output voltage. 00:00:48.760 --> 00:00:50.500 align:middle line:90% I'll call the output voltage V. 00:00:50.500 --> 00:00:56.060 align:middle line:84% And in this case, maybe I'll say, this is one state variable. 00:00:56.060 --> 00:00:59.440 align:middle line:84% V is one state variable, which is the capacitor voltage. 00:00:59.440 --> 00:01:01.600 align:middle line:84% And the other state variable in the system 00:01:01.600 --> 00:01:02.980 align:middle line:90% is the inductor current. 00:01:02.980 --> 00:01:05.170 align:middle line:84% And state variables are those variables 00:01:05.170 --> 00:01:07.930 align:middle line:84% that carry information about their history. 00:01:07.930 --> 00:01:10.040 align:middle line:84% You can't instantly change a capacitor voltage 00:01:10.040 --> 00:01:11.400 align:middle line:90% without an infinite current. 00:01:11.400 --> 00:01:13.790 align:middle line:84% You can't instantly change an inductor current 00:01:13.790 --> 00:01:15.270 align:middle line:90% without an infinite voltage. 00:01:15.270 --> 00:01:18.830 align:middle line:84% So if you know what the inductor current or capacitor 00:01:18.830 --> 00:01:21.980 align:middle line:84% voltage are at one time, you know 00:01:21.980 --> 00:01:25.520 align:middle line:84% what it's going to do because that's what carries history 00:01:25.520 --> 00:01:28.020 align:middle line:90% around with it in any system. 00:01:28.020 --> 00:01:30.230 align:middle line:84% So it's the energy storage elements. 00:01:30.230 --> 00:01:34.590 align:middle line:84% And we said we figured out, for this boost converter, 00:01:34.590 --> 00:01:39.170 align:middle line:84% if I define q of t is the modulating function, 00:01:39.170 --> 00:01:46.490 align:middle line:84% it switches between 0 and 1 for q of t. 00:01:46.490 --> 00:01:54.530 align:middle line:84% And then if I have q prime of t is equal to 1 minus q of t, 00:01:54.530 --> 00:01:59.390 align:middle line:84% if I'm modulating this at a constant duty ratio, q of t 00:01:59.390 --> 00:02:01.490 align:middle line:84% goes on and off, and I let this thing 00:02:01.490 --> 00:02:07.430 align:middle line:84% settle into periodic steady state, what we get is NPSS. 00:02:07.430 --> 00:02:14.750 align:middle line:84% We get V, the output voltage, the DC output voltage 00:02:14.750 --> 00:02:18.320 align:middle line:84% is going to be the DC input voltage divided by 1 minus D. 00:02:18.320 --> 00:02:20.900 align:middle line:84% And I'm using capital letters here to indicate these 00:02:20.900 --> 00:02:23.780 align:middle line:84% are all periodic steady state values. 00:02:23.780 --> 00:02:28.250 align:middle line:84% But we said, if I were, for example, to suddenly change 00:02:28.250 --> 00:02:34.110 align:middle line:84% the load resistance, then even if I was operating at fixed duty 00:02:34.110 --> 00:02:38.040 align:middle line:84% ratio, eventually I would have to go through some transient 00:02:38.040 --> 00:02:41.130 align:middle line:84% because, to supply a lower resistance, I need more current, 00:02:41.130 --> 00:02:42.215 align:middle line:90% more power. 00:02:42.215 --> 00:02:43.840 align:middle line:84% I can have more current from the input. 00:02:43.840 --> 00:02:44.840 align:middle line:90% Something has to change. 00:02:44.840 --> 00:02:47.310 align:middle line:84% It has to go through some transient. 00:02:47.310 --> 00:02:50.160 align:middle line:90% What we have here on the scope-- 00:02:50.160 --> 00:02:52.800 align:middle line:84% or what we have here is actually a boost converter 00:02:52.800 --> 00:02:55.090 align:middle line:90% run at fixed duty ratio. 00:02:55.090 --> 00:02:57.750 align:middle line:84% And what I'm going to do is I'm going to set 00:02:57.750 --> 00:02:59.440 align:middle line:90% this thing to single sequence. 00:02:59.440 --> 00:03:02.100 align:middle line:84% And what I have is-- on the yellow channel, 00:03:02.100 --> 00:03:04.900 align:middle line:84% I have the inductor current, just directly coupled. 00:03:04.900 --> 00:03:09.330 align:middle line:84% And on the purple channel, what I have is the output voltage, 00:03:09.330 --> 00:03:10.465 align:middle line:90% but AC coupled. 00:03:10.465 --> 00:03:12.840 align:middle line:84% So I'm just going to look at the deviations of the ripple 00:03:12.840 --> 00:03:14.220 align:middle line:90% in the AC voltage. 00:03:14.220 --> 00:03:16.710 align:middle line:84% And all I'm going to do is press this button. 00:03:16.710 --> 00:03:20.280 align:middle line:84% And that's going to add a load onto the converter. 00:03:20.280 --> 00:03:23.250 align:middle line:84% So it's going to make the load current higher 00:03:23.250 --> 00:03:25.540 align:middle line:90% or the load resistance smaller. 00:03:25.540 --> 00:03:28.374 align:middle line:90% And here we go. 00:03:28.374 --> 00:03:29.670 align:middle line:90% Oh, It didn't capture. 00:03:29.670 --> 00:03:31.370 align:middle line:90% Let's see. 00:03:31.370 --> 00:03:34.558 align:middle line:90% I don't know why that is. 00:03:34.558 --> 00:03:37.420 align:middle line:84% Oh, it would help if I actually coupled the output voltage. 00:03:37.420 --> 00:03:40.740 align:middle line:90% 00:03:40.740 --> 00:03:43.732 align:middle line:84% It seems to have-- oh, set this, set the minimum. 00:03:43.732 --> 00:03:44.440 align:middle line:90% I don't know why. 00:03:44.440 --> 00:03:46.410 align:middle line:84% All right, let me try that again. 00:03:46.410 --> 00:03:48.480 align:middle line:90% There we go. 00:03:48.480 --> 00:03:54.210 align:middle line:84% So what we can see here is that the current goes from some value 00:03:54.210 --> 00:03:56.610 align:middle line:84% and settles out to some higher value. 00:03:56.610 --> 00:03:59.910 align:middle line:84% The voltage goes through some oscillation, 00:03:59.910 --> 00:04:02.650 align:middle line:84% and then settles out to some later voltage. 00:04:02.650 --> 00:04:05.130 align:middle line:84% And the truth of the matter is we're not actually 00:04:05.130 --> 00:04:09.180 align:middle line:90% seeing the full transient here. 00:04:09.180 --> 00:04:16.275 align:middle line:84% Let me do this, try this one more time. 00:04:16.275 --> 00:04:24.620 align:middle line:90% 00:04:24.620 --> 00:04:26.340 align:middle line:84% Ah, there we see it a little bit better, 00:04:26.340 --> 00:04:28.370 align:middle line:84% except that actually what I triggered on 00:04:28.370 --> 00:04:32.096 align:middle line:90% was the step down in current. 00:04:32.096 --> 00:04:35.060 align:middle line:84% In any event, what we see is some kind 00:04:35.060 --> 00:04:36.770 align:middle line:90% of second order response. 00:04:36.770 --> 00:04:40.100 align:middle line:84% Superimposed on top of it is a ton of switching. 00:04:40.100 --> 00:04:42.650 align:middle line:84% This is going through thousands and thousands 00:04:42.650 --> 00:04:44.510 align:middle line:90% of switching cycles. 00:04:44.510 --> 00:04:47.450 align:middle line:84% And what we would like to do is some model 00:04:47.450 --> 00:04:50.233 align:middle line:84% that captures this low frequency variation, 00:04:50.233 --> 00:04:52.400 align:middle line:84% but isn't worrying about the instantaneous switching 00:04:52.400 --> 00:04:55.710 align:middle line:84% of what's going on in steady state. 00:04:55.710 --> 00:04:59.610 align:middle line:84% So I'm going to hide this, put this up. 00:04:59.610 --> 00:05:07.580 align:middle line:84% So last time, we said, well, how can we capture the low frequency 00:05:07.580 --> 00:05:10.490 align:middle line:84% content in i, l, and v without worrying 00:05:10.490 --> 00:05:12.100 align:middle line:84% about the high frequency content? 00:05:12.100 --> 00:05:24.020 align:middle line:90% 00:05:24.020 --> 00:05:26.990 align:middle line:84% We defined an operator that we called 00:05:26.990 --> 00:05:28.786 align:middle line:90% the local average operator. 00:05:28.786 --> 00:05:33.530 align:middle line:90% 00:05:33.530 --> 00:05:40.010 align:middle line:84% And this is x-bar of t is equal to 1 over t, the integral from t 00:05:40.010 --> 00:05:45.290 align:middle line:84% to minus capital T up to t of x of tau d tau. 00:05:45.290 --> 00:05:48.770 align:middle line:84% So basically, if I apply this local average 00:05:48.770 --> 00:05:51.680 align:middle line:84% to any waveform, a voltage, or a current or something like that, 00:05:51.680 --> 00:05:55.520 align:middle line:84% I'm going to look back over one capital period T, one switching 00:05:55.520 --> 00:05:58.690 align:middle line:84% cycle, and take the average over that switching cycle, 00:05:58.690 --> 00:06:00.770 align:middle line:84% so that if I have a bunch of up and down 00:06:00.770 --> 00:06:03.300 align:middle line:84% within a switching cycle, that gets filtered out. 00:06:03.300 --> 00:06:08.000 align:middle line:84% But if over time that voltage or current moves up and down, 00:06:08.000 --> 00:06:10.460 align:middle line:84% it captures the slow frequency up and down 00:06:10.460 --> 00:06:14.780 align:middle line:84% while suppressing the switching ripple and its harmonics. 00:06:14.780 --> 00:06:18.900 align:middle line:84% And we said this operator has a bunch of properties 00:06:18.900 --> 00:06:19.930 align:middle line:90% that are interesting. 00:06:19.930 --> 00:06:22.230 align:middle line:90% One is it's linear. 00:06:22.230 --> 00:06:27.600 align:middle line:90% Two is it's time-invariant. 00:06:27.600 --> 00:06:31.530 align:middle line:84% That means this thing is an LTI operator. 00:06:31.530 --> 00:06:34.320 align:middle line:84% That means we can write a transfer function for it. 00:06:34.320 --> 00:06:35.927 align:middle line:90% It's just a linear filter. 00:06:35.927 --> 00:06:38.010 align:middle line:84% And we looked at that transfer function last time. 00:06:38.010 --> 00:06:40.770 align:middle line:84% And it, indeed, passes low frequency components 00:06:40.770 --> 00:06:44.160 align:middle line:84% and suppresses the switching frequency. 00:06:44.160 --> 00:06:46.620 align:middle line:84% A third property that we discussed 00:06:46.620 --> 00:06:53.370 align:middle line:84% is the so-called differentiation property, 00:06:53.370 --> 00:06:58.830 align:middle line:84% which says that if I took dx some signal dx of t dt 00:06:58.830 --> 00:07:02.130 align:middle line:84% and then took its local average, that ends up being 00:07:02.130 --> 00:07:07.990 align:middle line:90% equal to d/dt of x-bar of t. 00:07:07.990 --> 00:07:11.460 align:middle line:84% So taking the local average of a derivative 00:07:11.460 --> 00:07:15.190 align:middle line:84% is the same thing as the derivative of the local average. 00:07:15.190 --> 00:07:20.020 align:middle line:84% The last property is we said we can approximate 00:07:20.020 --> 00:07:25.840 align:middle line:84% x of t y of t local average of the product 00:07:25.840 --> 00:07:32.830 align:middle line:84% as being approximately equal to x-bar of t times y-bar of t 00:07:32.830 --> 00:07:43.740 align:middle line:84% if one of x and y have, first of all, small ripple, 00:07:43.740 --> 00:07:47.460 align:middle line:84% meaning it's not changing full scale within a cycle, 00:07:47.460 --> 00:07:56.940 align:middle line:84% and, two, slow variation, meaning it 00:07:56.940 --> 00:07:59.950 align:middle line:84% only changes gradually from cycle to cycle. 00:07:59.950 --> 00:08:07.470 align:middle line:84% And actually, you can relax this former case in some instances. 00:08:07.470 --> 00:08:10.930 align:middle line:84% But we saw that it doesn't always work. 00:08:10.930 --> 00:08:13.710 align:middle line:84% But if one of x and y was, say, an inductor current 00:08:13.710 --> 00:08:15.450 align:middle line:84% or a capacitor voltage, where we do 00:08:15.450 --> 00:08:18.990 align:middle line:84% expect there to be small ripple on top of a big DC, 00:08:18.990 --> 00:08:20.255 align:middle line:90% we can do that. 00:08:20.255 --> 00:08:26.760 align:middle line:90% 00:08:26.760 --> 00:08:30.810 align:middle line:84% Last time, we used this And we applied this to a circuit. 00:08:30.810 --> 00:08:33.450 align:middle line:84% So we said, we could take the boost converter circuit, 00:08:33.450 --> 00:08:36.630 align:middle line:84% and play some games, and come up with a new circuit 00:08:36.630 --> 00:08:40.270 align:middle line:84% in which all of the circuit variables were averaged. 00:08:40.270 --> 00:08:42.000 align:middle line:84% And so it gets rid of all the switching, 00:08:42.000 --> 00:08:45.870 align:middle line:84% but it carries the low frequency behavior of the circuit. 00:08:45.870 --> 00:08:47.340 align:middle line:90% And that's one way to do things. 00:08:47.340 --> 00:08:50.160 align:middle line:84% And people do it, the so-called direct circuit averaging, 00:08:50.160 --> 00:08:51.170 align:middle line:90% all the time. 00:08:51.170 --> 00:08:53.190 align:middle line:84% What I'd like to do today is to show you 00:08:53.190 --> 00:08:56.880 align:middle line:84% another way we can use this local average operator. 00:08:56.880 --> 00:09:01.480 align:middle line:84% And it's essentially the same idea. 00:09:01.480 --> 00:09:03.680 align:middle line:84% It's just implemented in a different way. 00:09:03.680 --> 00:09:06.088 align:middle line:84% Some people like one method of thinking about it. 00:09:06.088 --> 00:09:08.380 align:middle line:84% Some people like the other method of thinking about it. 00:09:08.380 --> 00:09:10.320 align:middle line:84% Frankly, when I'm doing a new converter, 00:09:10.320 --> 00:09:12.930 align:middle line:84% I usually do the method I'm going to show you today, 00:09:12.930 --> 00:09:14.860 align:middle line:90% but I like them equally. 00:09:14.860 --> 00:09:17.320 align:middle line:84% It's just the question of what's more convenient to do. 00:09:17.320 --> 00:09:18.945 align:middle line:84% And the technique I'm going to show you 00:09:18.945 --> 00:09:21.540 align:middle line:84% today is called state space averaging. 00:09:21.540 --> 00:09:23.170 align:middle line:90% And here's the idea. 00:09:23.170 --> 00:09:26.500 align:middle line:90% Suppose I wanted to know the-- 00:09:26.500 --> 00:09:29.640 align:middle line:90% 00:09:29.640 --> 00:09:32.100 align:middle line:84% suppose I wanted to know the dynamics of this circuit 00:09:32.100 --> 00:09:34.800 align:middle line:90% about how it behaved. 00:09:34.800 --> 00:09:38.550 align:middle line:84% Well, I could say, write a set of differential equations 00:09:38.550 --> 00:09:41.890 align:middle line:84% for that circuit that would describe its behavior in time. 00:09:41.890 --> 00:09:44.830 align:middle line:84% And keep in mind that this circuit really 00:09:44.830 --> 00:09:47.290 align:middle line:90% has two subcircuits. 00:09:47.290 --> 00:09:52.340 align:middle line:84% When q of t is equal to 1, the switch is on. 00:09:52.340 --> 00:09:54.985 align:middle line:84% So I could think about this circuit as being this. 00:09:54.985 --> 00:10:10.620 align:middle line:90% 00:10:10.620 --> 00:10:12.340 align:middle line:90% So the switch is on. 00:10:12.340 --> 00:10:17.920 align:middle line:84% This is a linear circuit that models that behavior. 00:10:17.920 --> 00:10:23.230 align:middle line:84% If q of t is equal to 0, I can then 00:10:23.230 --> 00:10:25.320 align:middle line:84% say the circuit really looks like this. 00:10:25.320 --> 00:10:40.761 align:middle line:90% 00:10:40.761 --> 00:10:45.330 align:middle line:84% So I could then write a differential equation, 00:10:45.330 --> 00:10:49.050 align:middle line:84% a linear differential equation, because these 00:10:49.050 --> 00:10:55.600 align:middle line:84% are all linear elements plus an input, when it's in this state. 00:10:55.600 --> 00:10:58.470 align:middle line:84% And I could write a different differential equation 00:10:58.470 --> 00:10:59.590 align:middle line:90% when it's in this state. 00:10:59.590 --> 00:11:02.640 align:middle line:84% And if I just use the right set of differential equations 00:11:02.640 --> 00:11:06.270 align:middle line:84% in each substate, I can get the behavior. 00:11:06.270 --> 00:11:07.890 align:middle line:90% So let me just write this down. 00:11:07.890 --> 00:11:10.230 align:middle line:84% And I'm going to show that we can use this switching 00:11:10.230 --> 00:11:15.705 align:middle line:84% function, q of t, where q of t varies between 1 and 0-- 00:11:15.705 --> 00:11:19.350 align:middle line:90% 00:11:19.350 --> 00:11:21.215 align:middle line:90% I'm sorry, this is 1. 00:11:21.215 --> 00:11:23.790 align:middle line:90% 00:11:23.790 --> 00:11:25.950 align:middle line:84% We can use this switching function 00:11:25.950 --> 00:11:27.930 align:middle line:84% to build up a differential equation that 00:11:27.930 --> 00:11:29.580 align:middle line:90% captures our system. 00:11:29.580 --> 00:11:31.960 align:middle line:84% So here's what I would argue we could do. 00:11:31.960 --> 00:11:37.780 align:middle line:84% We can say, what does the diL/dt look like? 00:11:37.780 --> 00:11:43.300 align:middle line:84% diL/dt would simply be the voltage across the inductor. 00:11:43.300 --> 00:11:50.950 align:middle line:84% In this state, It would just be equal to 1 over L times u. 00:11:50.950 --> 00:11:53.810 align:middle line:84% And that's true when q of t is equal to 1. 00:11:53.810 --> 00:11:56.110 align:middle line:90% So I can multiply it by q of t. 00:11:56.110 --> 00:12:00.940 align:middle line:84% When q of t is equal to 0 and 1 minus q of t is equal to 1, 00:12:00.940 --> 00:12:04.810 align:middle line:84% diL/dt is equal to u minus v. So I could say, 00:12:04.810 --> 00:12:08.110 align:middle line:90% this is plus u minus-- 00:12:08.110 --> 00:12:13.470 align:middle line:84% this is plus 1 over L u minus v times 1 minus q of t. 00:12:13.470 --> 00:12:16.321 align:middle line:90% 00:12:16.321 --> 00:12:21.340 align:middle line:84% So basically, when q of t is equal to 1, 00:12:21.340 --> 00:12:26.020 align:middle line:84% this term, diL/dt, is 1 over L u. 00:12:26.020 --> 00:12:27.970 align:middle line:90% So it matches the top circuit. 00:12:27.970 --> 00:12:32.110 align:middle line:84% And this term is gone because 1 minus q of t is 0. 00:12:32.110 --> 00:12:33.850 align:middle line:90% And this term just disappears. 00:12:33.850 --> 00:12:37.350 align:middle line:84% When q of t is equal to 0, this term disappears. 00:12:37.350 --> 00:12:39.870 align:middle line:90% And this factor becomes 1. 00:12:39.870 --> 00:12:41.980 align:middle line:84% And I just have 1 over L u minus v. 00:12:41.980 --> 00:12:45.780 align:middle line:84% So whether I'm in this switch state or this switch state, 00:12:45.780 --> 00:12:49.710 align:middle line:84% this differential equation captures the dynamics 00:12:49.710 --> 00:12:52.040 align:middle line:90% of that inductor current. 00:12:52.040 --> 00:12:54.020 align:middle line:84% Does that make sense to everybody? 00:12:54.020 --> 00:12:56.630 align:middle line:84% I can do the same thing for the voltage. 00:12:56.630 --> 00:13:03.800 align:middle line:84% I can get dv/dt is equal to 1 over C times-- 00:13:03.800 --> 00:13:07.970 align:middle line:84% this would be minus v over r times that, minus 1 00:13:07.970 --> 00:13:19.190 align:middle line:84% over RC times q of t plus 1 over C i sub L minus v 00:13:19.190 --> 00:13:22.855 align:middle line:90% over R 1 minus q of t. 00:13:22.855 --> 00:13:26.810 align:middle line:90% 00:13:26.810 --> 00:13:29.100 align:middle line:84% So in fact, if I came up and I told 00:13:29.100 --> 00:13:32.790 align:middle line:84% you the starting state of voltage and current, inductor 00:13:32.790 --> 00:13:35.520 align:middle line:84% voltage, capacitor voltage, and inductor current, 00:13:35.520 --> 00:13:37.680 align:middle line:84% and I told you the q of t over time, 00:13:37.680 --> 00:13:40.530 align:middle line:84% I could basically numerically integrate 00:13:40.530 --> 00:13:42.870 align:middle line:84% this pair of differential equations 00:13:42.870 --> 00:13:44.670 align:middle line:84% and predict the trajectory, including 00:13:44.670 --> 00:13:47.430 align:middle line:84% all of the switching and everything else. 00:13:47.430 --> 00:13:50.490 align:middle line:84% Does that make sense to everybody? 00:13:50.490 --> 00:13:51.695 align:middle line:90% Any questions about that? 00:13:51.695 --> 00:13:54.470 align:middle line:90% 00:13:54.470 --> 00:13:56.930 align:middle line:84% So this is actually a simulation model 00:13:56.930 --> 00:14:03.950 align:middle line:84% of this converter in continuous conduction mode, 00:14:03.950 --> 00:14:05.810 align:middle line:90% including dynamics. 00:14:05.810 --> 00:14:09.680 align:middle line:84% This is a pair of differential equations, switched differential 00:14:09.680 --> 00:14:11.600 align:middle line:90% equations that capture it. 00:14:11.600 --> 00:14:14.840 align:middle line:84% Well, I could take my averaging operator 00:14:14.840 --> 00:14:17.940 align:middle line:84% and apply it to the differential equations as well. 00:14:17.940 --> 00:14:21.650 align:middle line:90% I could apply it to each side. 00:14:21.650 --> 00:14:25.460 align:middle line:84% All I would be doing is walking up and saying, 00:14:25.460 --> 00:14:28.110 align:middle line:84% OK, let me local average each side. 00:14:28.110 --> 00:14:30.800 align:middle line:84% So I'll put a local average on the left side of this. 00:14:30.800 --> 00:14:33.410 align:middle line:84% And then the local average of these sums of terms 00:14:33.410 --> 00:14:35.100 align:middle line:84% is just the sum of the local averages. 00:14:35.100 --> 00:14:36.650 align:middle line:90% So I can do this. 00:14:36.650 --> 00:14:38.460 align:middle line:90% And I can do this. 00:14:38.460 --> 00:14:41.810 align:middle line:84% And by the way, the scaling constants 00:14:41.810 --> 00:14:43.320 align:middle line:84% can come out of the local averages. 00:14:43.320 --> 00:14:44.840 align:middle line:90% So that's OK. 00:14:44.840 --> 00:14:46.310 align:middle line:90% And I can do this. 00:14:46.310 --> 00:14:48.470 align:middle line:90% And I can do this. 00:14:48.470 --> 00:14:52.480 align:middle line:84% And then I can do this, local average of all this. 00:14:52.480 --> 00:14:55.500 align:middle line:84% So all I did was I walked up and I applied that operator 00:14:55.500 --> 00:14:58.950 align:middle line:90% to each side of the equation. 00:14:58.950 --> 00:15:03.150 align:middle line:90% Well, now, I can rewrite this. 00:15:03.150 --> 00:15:10.680 align:middle line:84% And I can apply some of the properties of the local average. 00:15:10.680 --> 00:15:13.125 align:middle line:84% First of all, the local average of the derivative 00:15:13.125 --> 00:15:15.000 align:middle line:84% is just the derivative of the local averages. 00:15:15.000 --> 00:15:22.670 align:middle line:84% So I can write this as di L-bar/dt Then this would be 1 00:15:22.670 --> 00:15:28.850 align:middle line:84% over L times the local average of u. 00:15:28.850 --> 00:15:30.350 align:middle line:84% And let me rearrange this, actually. 00:15:30.350 --> 00:15:34.340 align:middle line:84% I have u times q plus u times 1 minus q. 00:15:34.340 --> 00:15:36.710 align:middle line:84% So this would just be 1 over L times u-bar. 00:15:36.710 --> 00:15:39.890 align:middle line:90% 00:15:39.890 --> 00:15:45.260 align:middle line:84% Then I have minus 1 over L or plus-- 00:15:45.260 --> 00:15:53.155 align:middle line:84% minus 1 over L the local average of v times 1 minus q. 00:15:53.155 --> 00:15:57.940 align:middle line:90% 00:15:57.940 --> 00:16:00.320 align:middle line:90% Does that make sense, everybody? 00:16:00.320 --> 00:16:02.340 align:middle line:90% So all I did was rearrange this. 00:16:02.340 --> 00:16:05.040 align:middle line:84% I put up the q and 1 minus q terms 00:16:05.040 --> 00:16:08.000 align:middle line:84% so that the u term doesn't have q in it anymore. 00:16:08.000 --> 00:16:10.760 align:middle line:84% And then it's just minus 1 over Lv times 1 minus q. 00:16:10.760 --> 00:16:12.980 align:middle line:84% And I took the local average of that. 00:16:12.980 --> 00:16:15.660 align:middle line:84% I could then do the same thing with the bottom equation. 00:16:15.660 --> 00:16:19.710 align:middle line:84% I can get dv-bar/dt, because I'm using the differentiation 00:16:19.710 --> 00:16:21.230 align:middle line:90% property, is equal to-- 00:16:21.230 --> 00:16:24.420 align:middle line:84% the v over RC term again goes up. 00:16:24.420 --> 00:16:28.700 align:middle line:84% So this is minus 1 over RC times v-bar 00:16:28.700 --> 00:16:38.120 align:middle line:84% plus 1 over c times the local average of iL times 1 minus q. 00:16:38.120 --> 00:16:40.880 align:middle line:90% 00:16:40.880 --> 00:16:44.270 align:middle line:90% Everybody down with that? 00:16:44.270 --> 00:16:51.680 align:middle line:84% Now, let me define q-prime is 1 minus q. 00:16:51.680 --> 00:17:01.500 align:middle line:84% Let me define d is equal to the local average of q of t. 00:17:01.500 --> 00:17:06.000 align:middle line:84% If I'm modulating along in a cycle, in any given cycle, 00:17:06.000 --> 00:17:10.530 align:middle line:84% the local average of q of t is the duty ratio in that cycle, 00:17:10.530 --> 00:17:12.579 align:middle line:90% neglecting small shifts in time. 00:17:12.579 --> 00:17:16.500 align:middle line:84% But this is everything slowly varying with respect to a cycle. 00:17:16.500 --> 00:17:21.810 align:middle line:84% That would mean that d-prime, which 00:17:21.810 --> 00:17:25.170 align:middle line:84% is the complement of the duty ratio, 00:17:25.170 --> 00:17:31.090 align:middle line:84% is equal to 1 minus q local average. 00:17:31.090 --> 00:17:34.540 align:middle line:84% Or it's equal to q-prime local average. 00:17:34.540 --> 00:17:39.040 align:middle line:84% So if I apply that, and then what I'm going to do 00:17:39.040 --> 00:17:45.130 align:middle line:84% is I'm going to say, OK, I can make this approximately 00:17:45.130 --> 00:17:53.410 align:middle line:90% equal to 1 over Lu-bar. 00:17:53.410 --> 00:17:55.370 align:middle line:84% Then I can make this approximately equal. 00:17:55.370 --> 00:17:59.710 align:middle line:84% If I know that xy-bar is approximately 00:17:59.710 --> 00:18:02.590 align:middle line:84% equal to x-bar y-bar if one of them 00:18:02.590 --> 00:18:04.490 align:middle line:84% has slow ripple and small variation-- well, 00:18:04.490 --> 00:18:08.560 align:middle line:84% iL does and v does because this is a capacitor voltage 00:18:08.560 --> 00:18:09.680 align:middle line:90% and inductor current. 00:18:09.680 --> 00:18:18.130 align:middle line:84% So this would be minus 1 over L v-bar d-prime. 00:18:18.130 --> 00:18:21.730 align:middle line:84% And this is approximately minus 1 00:18:21.730 --> 00:18:34.210 align:middle line:84% over RC v-bar plus 1 over c iL-bar d-prime. 00:18:34.210 --> 00:18:38.065 align:middle line:84% So this becomes a set of partial differential equations. 00:18:38.065 --> 00:18:39.440 align:middle line:84% Maybe I should just rewrite this. 00:18:39.440 --> 00:18:42.520 align:middle line:90% This is the diL-bar/dt. 00:18:42.520 --> 00:18:46.270 align:middle line:90% And this is equal to dv-bar/dt. 00:18:46.270 --> 00:18:54.130 align:middle line:84% Now, I have a set of state space averaged equations. 00:18:54.130 --> 00:18:56.870 align:middle line:84% And notice that there's no switching left in here. 00:18:56.870 --> 00:18:58.630 align:middle line:84% There's no q of t bopping up and down 00:18:58.630 --> 00:19:00.320 align:middle line:90% and turning switches on and off. 00:19:00.320 --> 00:19:02.770 align:middle line:84% This is just in terms of whatever 00:19:02.770 --> 00:19:06.630 align:middle line:84% my duty ratio is going to be tells me how this evolves. 00:19:06.630 --> 00:19:09.770 align:middle line:84% So if I knew how the commanded duty ratio varied, 00:19:09.770 --> 00:19:14.180 align:middle line:84% I would know how the local averages of i and l, iL, 00:19:14.180 --> 00:19:16.970 align:middle line:90% and v would vary. 00:19:16.970 --> 00:19:18.220 align:middle line:90% Any questions about that? 00:19:18.220 --> 00:19:22.000 align:middle line:90% 00:19:22.000 --> 00:19:24.820 align:middle line:90% So how would I use this thing? 00:19:24.820 --> 00:19:30.640 align:middle line:84% I could come up and say, all right, I had my converter here. 00:19:30.640 --> 00:19:33.340 align:middle line:84% What I really wanted to do is say, OK, I 00:19:33.340 --> 00:19:35.273 align:middle line:84% have some reference voltage, v-ref. 00:19:35.273 --> 00:19:38.850 align:middle line:90% 00:19:38.850 --> 00:19:43.320 align:middle line:90% And I compared that to-- 00:19:43.320 --> 00:19:45.960 align:middle line:84% in this case, I'm going to say it's a local average version 00:19:45.960 --> 00:19:48.450 align:middle line:90% of the output voltage, v-bar. 00:19:48.450 --> 00:19:53.040 align:middle line:84% And I'm going to generate some v-error bar. 00:19:53.040 --> 00:19:56.586 align:middle line:84% OK I'm going to run that into some compensator. 00:19:56.586 --> 00:20:00.360 align:middle line:90% 00:20:00.360 --> 00:20:02.200 align:middle line:90% That could be gc of s. 00:20:02.200 --> 00:20:07.290 align:middle line:84% Think of this as, for example, a lead lag controller, or a PID 00:20:07.290 --> 00:20:09.540 align:middle line:84% controller, or something like that, something 00:20:09.540 --> 00:20:15.640 align:middle line:84% that has high gain so that it's going to amplify the error. 00:20:15.640 --> 00:20:19.290 align:middle line:84% And out of this comes some signal that I'm going to call 00:20:19.290 --> 00:20:21.310 align:middle line:90% d, the duty ratio I want. 00:20:21.310 --> 00:20:24.157 align:middle line:84% Now, that might be some voltage representing the duty ratio 00:20:24.157 --> 00:20:25.740 align:middle line:84% or current representing the duty ratio 00:20:25.740 --> 00:20:27.160 align:middle line:90% on an actual implementation. 00:20:27.160 --> 00:20:28.650 align:middle line:84% But from a theoretical perspective, 00:20:28.650 --> 00:20:32.070 align:middle line:84% the output of this compensator is just the duty ratio I want. 00:20:32.070 --> 00:20:34.355 align:middle line:84% I need to stick this into the plant. 00:20:34.355 --> 00:20:37.958 align:middle line:90% 00:20:37.958 --> 00:20:39.750 align:middle line:84% That's what a control person would call it. 00:20:39.750 --> 00:20:41.190 align:middle line:90% We'd call it the converter. 00:20:41.190 --> 00:20:43.874 align:middle line:90% 00:20:43.874 --> 00:20:47.780 align:middle line:84% And of course, it has another input, which is, 00:20:47.780 --> 00:20:50.240 align:middle line:90% in this case, u, or u-bar. 00:20:50.240 --> 00:20:52.580 align:middle line:84% And out of this thing, out of this model 00:20:52.580 --> 00:20:54.860 align:middle line:90% would come iL-bar and v-bar. 00:20:54.860 --> 00:20:58.210 align:middle line:90% 00:20:58.210 --> 00:21:01.950 align:middle line:84% And so what I needed was a dynamic model for this thing. 00:21:01.950 --> 00:21:03.240 align:middle line:90% Well, guess what? 00:21:03.240 --> 00:21:08.150 align:middle line:84% Here is a differential equation model 00:21:08.150 --> 00:21:10.970 align:middle line:84% for the behavior of the converter. 00:21:10.970 --> 00:21:14.250 align:middle line:84% And I could just stick this inside of here. 00:21:14.250 --> 00:21:18.260 align:middle line:84% And this is saying, if I put in d, what happens to iL and v? 00:21:18.260 --> 00:21:19.693 align:middle line:90% And this model is my dynamics. 00:21:19.693 --> 00:21:23.705 align:middle line:90% 00:21:23.705 --> 00:21:26.180 align:middle line:90% Any questions about that? 00:21:26.180 --> 00:21:27.355 align:middle line:90% No switching involved. 00:21:27.355 --> 00:21:30.230 align:middle line:90% 00:21:30.230 --> 00:21:38.240 align:middle line:84% Now, I should have noted, last time, we drew a circuit model. 00:21:38.240 --> 00:21:41.777 align:middle line:84% We didn't put a pair state space equation in here. 00:21:41.777 --> 00:21:43.110 align:middle line:90% We actually had a circuit model. 00:21:43.110 --> 00:21:44.902 align:middle line:84% What was that circuit model from last time? 00:21:44.902 --> 00:21:47.580 align:middle line:90% It really looked like this. 00:21:47.580 --> 00:21:52.700 align:middle line:90% It had u-bar iL-bar. 00:21:52.700 --> 00:21:56.510 align:middle line:90% 00:21:56.510 --> 00:22:00.050 align:middle line:84% It had a voltage source here, which 00:22:00.050 --> 00:22:05.990 align:middle line:84% was d-prime v-bar and a dependent current source, 00:22:05.990 --> 00:22:09.620 align:middle line:90% which was d-prime iL-bar. 00:22:09.620 --> 00:22:17.420 align:middle line:90% And then it had v-bar here. 00:22:17.420 --> 00:22:20.240 align:middle line:84% If you write the differential equation 00:22:20.240 --> 00:22:22.280 align:middle line:84% for this circuit, this nonlinear circuit 00:22:22.280 --> 00:22:24.590 align:middle line:84% that we got by circuit averaging, 00:22:24.590 --> 00:22:29.750 align:middle line:84% it will give you exactly this set of differential equations. 00:22:29.750 --> 00:22:31.900 align:middle line:90% diL/dt is u. 00:22:31.900 --> 00:22:35.720 align:middle line:84% L diL/dt is u-bar minus d-prime v-bar. 00:22:35.720 --> 00:22:37.370 align:middle line:90% That's exactly this equation. 00:22:37.370 --> 00:22:40.220 align:middle line:90% 00:22:40.220 --> 00:22:44.265 align:middle line:84% So I can describe it with circuit averaging. 00:22:44.265 --> 00:22:46.140 align:middle line:84% I can describe it with state space averaging. 00:22:46.140 --> 00:22:47.723 align:middle line:84% This just gives me the equation for it 00:22:47.723 --> 00:22:51.860 align:middle line:84% so I can go back and forth between the two as I want. 00:22:51.860 --> 00:22:52.680 align:middle line:90% They both work. 00:22:52.680 --> 00:22:54.138 align:middle line:84% There are other circuit models that 00:22:54.138 --> 00:22:56.580 align:middle line:84% represent the same differential equations, of course. 00:22:56.580 --> 00:22:58.340 align:middle line:84% You can do things different ways. 00:22:58.340 --> 00:23:00.770 align:middle line:84% But the underlying dynamics will always be the same, 00:23:00.770 --> 00:23:03.344 align:middle line:90% if you apply averaging. 00:23:03.344 --> 00:23:04.975 align:middle line:90% Any questions about that? 00:23:04.975 --> 00:23:09.630 align:middle line:90% 00:23:09.630 --> 00:23:12.970 align:middle line:84% Now, what is the headache of this approach? 00:23:12.970 --> 00:23:14.098 align:middle line:90% Or it's not a headache. 00:23:14.098 --> 00:23:15.640 align:middle line:84% It's nothing to do with the approach. 00:23:15.640 --> 00:23:19.950 align:middle line:84% I shouldn't say-- this is a fundamental-- 00:23:19.950 --> 00:23:24.840 align:middle line:84% this is a fundamental property of the system, 00:23:24.840 --> 00:23:29.130 align:middle line:84% is that this is a nonlinear circuit and this is a nonlinear 00:23:29.130 --> 00:23:30.260 align:middle line:90% differential equations. 00:23:30.260 --> 00:23:31.140 align:middle line:90% Why? 00:23:31.140 --> 00:23:34.980 align:middle line:84% Because I have a product between my state variable, 00:23:34.980 --> 00:23:38.320 align:middle line:84% my output voltage, and my control input, the duty ratio. 00:23:38.320 --> 00:23:41.320 align:middle line:84% And likewise, I have a-- or 1 minus the duty ratio. 00:23:41.320 --> 00:23:43.980 align:middle line:84% And here, I have a product between my other state variable 00:23:43.980 --> 00:23:46.820 align:middle line:90% and 1 minus the duty ratio. 00:23:46.820 --> 00:23:48.946 align:middle line:84% And that makes this system nonlinear. 00:23:48.946 --> 00:23:53.130 align:middle line:90% 00:23:53.130 --> 00:23:53.880 align:middle line:90% Why do I care? 00:23:53.880 --> 00:23:55.380 align:middle line:90% Why is that a pain in the butt? 00:23:55.380 --> 00:23:56.690 align:middle line:90% Well, what would I like to do? 00:23:56.690 --> 00:23:59.390 align:middle line:84% If I go back to standard control theory, what I'd really 00:23:59.390 --> 00:24:02.720 align:middle line:84% like is perhaps some kind of transfer function that says, 00:24:02.720 --> 00:24:05.790 align:middle line:84% if I change the duty ratio, what does the output voltage do? 00:24:05.790 --> 00:24:07.457 align:middle line:84% Because if I'm going to write everything 00:24:07.457 --> 00:24:12.320 align:middle line:84% in terms of a transfer function, I can apply linear control 00:24:12.320 --> 00:24:15.320 align:middle line:90% theory and life is good. 00:24:15.320 --> 00:24:17.520 align:middle line:84% How do I usually get the transfer function? 00:24:17.520 --> 00:24:21.170 align:middle line:84% I take the Laplace transform of this stuff, which 00:24:21.170 --> 00:24:23.090 align:middle line:84% converts this differential equation 00:24:23.090 --> 00:24:24.430 align:middle line:90% into an algebraic equation. 00:24:24.430 --> 00:24:29.540 align:middle line:84% diL/dt becomes just S iL, where S is just the Laplace variable. 00:24:29.540 --> 00:24:31.895 align:middle line:84% But this is kind of a pain in the ass in this equation. 00:24:31.895 --> 00:24:32.395 align:middle line:90% Why? 00:24:32.395 --> 00:24:36.070 align:middle line:90% 00:24:36.070 --> 00:24:37.733 align:middle line:90% Taking the Laplace of a product. 00:24:37.733 --> 00:24:41.810 align:middle line:90% 00:24:41.810 --> 00:24:44.060 align:middle line:84% Anybody recall what you have to do when you take 00:24:44.060 --> 00:24:47.280 align:middle line:90% the product with a Laplace? 00:24:47.280 --> 00:24:49.110 align:middle line:90% You end up with a convolution. 00:24:49.110 --> 00:24:50.960 align:middle line:90% And that's not very pretty. 00:24:50.960 --> 00:24:55.650 align:middle line:84% So I can't just write a transfer function for this. 00:24:55.650 --> 00:24:58.080 align:middle line:84% I can't boil this thing down into a transfer function 00:24:58.080 --> 00:25:03.660 align:middle line:84% because this Laplace of the product is a problem. 00:25:03.660 --> 00:25:08.370 align:middle line:84% So the question is, what can I do? 00:25:08.370 --> 00:25:11.190 align:middle line:84% And the answer is, it's nonlinear 00:25:11.190 --> 00:25:15.600 align:middle line:84% and it has nonlinear dynamics, meaning it truly is not linear. 00:25:15.600 --> 00:25:17.280 align:middle line:90% That's the way it is. 00:25:17.280 --> 00:25:22.000 align:middle line:84% But we could find out what it does for small variations. 00:25:22.000 --> 00:25:25.320 align:middle line:84% In fact, in my original example, I just showed you 00:25:25.320 --> 00:25:26.280 align:middle line:90% the output voltage. 00:25:26.280 --> 00:25:29.080 align:middle line:84% And I did this little bump, but it was around 0. 00:25:29.080 --> 00:25:31.090 align:middle line:84% That's because I was AC coupling it. 00:25:31.090 --> 00:25:33.570 align:middle line:84% I was showing you the deviation in the output voltage 00:25:33.570 --> 00:25:37.000 align:middle line:90% from its steady state value. 00:25:37.000 --> 00:25:40.540 align:middle line:84% And what I can go do is I can say, OK, 00:25:40.540 --> 00:25:42.850 align:middle line:84% if I believe my converter is going to be operating 00:25:42.850 --> 00:25:45.280 align:middle line:84% in periodic steady state somewhere, 00:25:45.280 --> 00:25:47.920 align:middle line:84% and then I put in a small perturbation away from that 00:25:47.920 --> 00:25:51.340 align:middle line:84% steady state-- in other words, maybe I step my reference 00:25:51.340 --> 00:25:54.260 align:middle line:90% voltage just a tiny bit-- 00:25:54.260 --> 00:25:56.100 align:middle line:90% it should respond. 00:25:56.100 --> 00:25:57.810 align:middle line:90% And the dynamics are nonlinear. 00:25:57.810 --> 00:26:04.080 align:middle line:84% But for small variations, they're pretty close to linear. 00:26:04.080 --> 00:26:06.560 align:middle line:84% So what I might do is I might go and say, 00:26:06.560 --> 00:26:15.020 align:middle line:84% all right, suppose that all of my variables in my system, 00:26:15.020 --> 00:26:24.820 align:middle line:84% x of t, was equal to some DC value, 00:26:24.820 --> 00:26:29.020 align:middle line:84% sometimes called the bias or the steady state, 00:26:29.020 --> 00:26:33.786 align:middle line:84% plus some AC variation, small variation. 00:26:33.786 --> 00:26:37.420 align:middle line:90% 00:26:37.420 --> 00:26:43.030 align:middle line:84% And then I could express, well, the small variations are simply 00:26:43.030 --> 00:26:44.830 align:middle line:90% the total variable minus-- 00:26:44.830 --> 00:26:49.270 align:middle line:84% it's the deviation of the total variable from the steady state 00:26:49.270 --> 00:26:52.280 align:middle line:90% operating point. 00:26:52.280 --> 00:26:55.240 align:middle line:84% And what I can go do is I can express 00:26:55.240 --> 00:26:56.800 align:middle line:90% all my variables that way. 00:26:56.800 --> 00:27:02.680 align:middle line:84% I can say, OK, u is equal to some DC value plus any variation 00:27:02.680 --> 00:27:04.960 align:middle line:90% I put on the input voltage. 00:27:04.960 --> 00:27:10.270 align:middle line:84% v is equal to some DC voltage plus some small variation, where 00:27:10.270 --> 00:27:12.670 align:middle line:84% this is the deviation away from-- 00:27:12.670 --> 00:27:17.650 align:middle line:84% this is the AC-coupled version, essentially. 00:27:17.650 --> 00:27:22.540 align:middle line:84% iL is equal to some DC iL plus some iL 00:27:22.540 --> 00:27:25.030 align:middle line:84% tilde, some small variation, et cetera. 00:27:25.030 --> 00:27:29.920 align:middle line:84% And what can we say mathematically? 00:27:29.920 --> 00:27:37.390 align:middle line:84% Suppose I had some nonlinear differential equation, dx/dt, 00:27:37.390 --> 00:27:41.680 align:middle line:84% where x is some variable or some vector of variables-- 00:27:41.680 --> 00:27:43.480 align:middle line:84% and this is a vector of equations-- 00:27:43.480 --> 00:27:52.000 align:middle line:84% is equal to some function of the state variables, the inputs, 00:27:52.000 --> 00:27:55.510 align:middle line:84% and could be a function of time, too. 00:27:55.510 --> 00:27:59.948 align:middle line:84% So this is some arbitrarily nonlinear differential equation. 00:27:59.948 --> 00:28:01.740 align:middle line:84% It could be a vector differential equation, 00:28:01.740 --> 00:28:03.970 align:middle line:84% like this has two state variables. 00:28:03.970 --> 00:28:06.540 align:middle line:84% So x could be a vector of state variables, in general. 00:28:06.540 --> 00:28:09.540 align:middle line:84% And I could have a vector of inputs, r. 00:28:09.540 --> 00:28:13.310 align:middle line:84% And suppose I had this and I had some steady state operating 00:28:13.310 --> 00:28:13.810 align:middle line:90% point. 00:28:13.810 --> 00:28:22.560 align:middle line:84% I have f of x and capital R and t is equal to 0. 00:28:22.560 --> 00:28:26.070 align:middle line:90% This is an operating point. 00:28:26.070 --> 00:28:33.960 align:middle line:84% This is a nonlinear diffy q, or system 00:28:33.960 --> 00:28:35.500 align:middle line:90% of differential equations. 00:28:35.500 --> 00:28:38.610 align:middle line:90% 00:28:38.610 --> 00:28:40.560 align:middle line:90% Why is this an operating point? 00:28:40.560 --> 00:28:45.090 align:middle line:84% Because what I'm saying is, if I put f of capital 00:28:45.090 --> 00:28:47.400 align:middle line:84% x and capital r in here, I get 0, 00:28:47.400 --> 00:28:50.380 align:middle line:84% which means dx/dt is 0, which means it's not going anywhere. 00:28:50.380 --> 00:28:51.400 align:middle line:90% It sits there. 00:28:51.400 --> 00:28:56.820 align:middle line:84% So this equation is precisely defining some operating point 00:28:56.820 --> 00:28:59.565 align:middle line:84% where it's just going to sit, some periodic steady state 00:28:59.565 --> 00:29:00.815 align:middle line:90% value where it's going to sit. 00:29:00.815 --> 00:29:04.375 align:middle line:90% 00:29:04.375 --> 00:29:08.770 align:middle line:84% And what we can do is we can linearize this guy. 00:29:08.770 --> 00:29:10.750 align:middle line:90% What do I mean by that? 00:29:10.750 --> 00:29:15.580 align:middle line:84% Let me just do the simplest 1D version of this idea. 00:29:15.580 --> 00:29:23.740 align:middle line:84% Suppose I have one dimension, x, and I have f of x. 00:29:23.740 --> 00:29:29.910 align:middle line:84% And this has some function, some nonlinear-- 00:29:29.910 --> 00:29:31.520 align:middle line:90% this is not a straight line-- 00:29:31.520 --> 00:29:33.150 align:middle line:90% function. 00:29:33.150 --> 00:29:40.080 align:middle line:90% Well, this is f of x. 00:29:40.080 --> 00:29:43.350 align:middle line:84% Notice that there's some place where f of x goes through 0. 00:29:43.350 --> 00:29:45.515 align:middle line:90% And I could call that x. 00:29:45.515 --> 00:29:48.426 align:middle line:90% 00:29:48.426 --> 00:29:54.630 align:middle line:84% And so the variation, dx/dt, is exactly 00:29:54.630 --> 00:29:56.550 align:middle line:84% defined by this function near here. 00:29:56.550 --> 00:29:59.850 align:middle line:84% But maybe near this operating point, I could come and say, 00:29:59.850 --> 00:30:00.690 align:middle line:90% you know what? 00:30:00.690 --> 00:30:03.500 align:middle line:84% Let me just approximate him with a tangent line. 00:30:03.500 --> 00:30:07.330 align:middle line:90% 00:30:07.330 --> 00:30:11.170 align:middle line:84% And I will approximate f of x right near this operating 00:30:11.170 --> 00:30:14.480 align:middle line:90% point with a straight line. 00:30:14.480 --> 00:30:22.100 align:middle line:84% And then I can look at, if I have some variable x, 00:30:22.100 --> 00:30:28.820 align:middle line:84% the distance here to some other point. 00:30:28.820 --> 00:30:31.640 align:middle line:84% This distance I might call x-tilde. 00:30:31.640 --> 00:30:36.050 align:middle line:84% The deviation between x and its operating point 00:30:36.050 --> 00:30:37.260 align:middle line:90% is just the variation. 00:30:37.260 --> 00:30:40.100 align:middle line:84% So I can worry about this differential equation, 00:30:40.100 --> 00:30:43.400 align:middle line:90% variations about this point. 00:30:43.400 --> 00:30:46.430 align:middle line:84% And how am I going to approximate that? 00:30:46.430 --> 00:30:51.740 align:middle line:84% I could approximate that as dx/dt, 00:30:51.740 --> 00:30:58.670 align:middle line:84% which I could write as capital d capital X/dt plus dx-tilde/dt 00:30:58.670 --> 00:31:00.870 align:middle line:90% dt. 00:31:00.870 --> 00:31:03.080 align:middle line:90% So x is capital-- 00:31:03.080 --> 00:31:04.350 align:middle line:90% this is x. 00:31:04.350 --> 00:31:08.330 align:middle line:90% This is capital X plus x-tilde. 00:31:08.330 --> 00:31:18.770 align:middle line:84% This is just 0 because this is the derivative of a constant. 00:31:18.770 --> 00:31:20.750 align:middle line:90% It's going to be equal to-- 00:31:20.750 --> 00:31:23.480 align:middle line:90% well, dx/dt was f of x R and t. 00:31:23.480 --> 00:31:27.180 align:middle line:84% I'm just going to substitute x plus x-tilde et cetera in here. 00:31:27.180 --> 00:31:38.840 align:middle line:84% What I'm going to get is f of x R and t plus df/dx evaluated 00:31:38.840 --> 00:31:41.563 align:middle line:90% at x and R for-- 00:31:41.563 --> 00:31:47.990 align:middle line:90% 00:31:47.990 --> 00:31:54.470 align:middle line:84% evaluated times x minus capital X, which is actually 00:31:54.470 --> 00:32:03.290 align:middle line:84% just times x-tilde, plus df/dR evaluated 00:32:03.290 --> 00:32:10.100 align:middle line:90% at x and R times R-tilde. 00:32:10.100 --> 00:32:13.310 align:middle line:84% So all I'm doing is instead of using f of x, 00:32:13.310 --> 00:32:14.930 align:middle line:90% I'm doing a Fourier-- 00:32:14.930 --> 00:32:18.260 align:middle line:84% I'm doing a Taylor expansion of this thing 00:32:18.260 --> 00:32:22.105 align:middle line:84% and looking at just the derivatives at that point. 00:32:22.105 --> 00:32:25.440 align:middle line:90% 00:32:25.440 --> 00:32:27.490 align:middle line:90% What's f of x R and t? 00:32:27.490 --> 00:32:32.010 align:middle line:84% This is just 0 because I just told you this thing was 0. 00:32:32.010 --> 00:32:40.770 align:middle line:84% And so I just get df/dx and df/dR times x-tilde and R-tilde. 00:32:40.770 --> 00:32:42.810 align:middle line:84% Does that make sense to everybody? 00:32:42.810 --> 00:32:45.630 align:middle line:84% And all I'm really doing is I'm linearizing-- 00:32:45.630 --> 00:32:48.160 align:middle line:84% I'm approximating this nonlinear curve. 00:32:48.160 --> 00:32:49.980 align:middle line:84% If I have my deviations are small enough, 00:32:49.980 --> 00:32:51.397 align:middle line:84% I can't really tell the difference 00:32:51.397 --> 00:32:54.240 align:middle line:84% between the curvy f of x and the straight line orange 00:32:54.240 --> 00:32:56.790 align:middle line:90% approximation. 00:32:56.790 --> 00:32:58.830 align:middle line:90% That make sense to everybody? 00:32:58.830 --> 00:33:03.510 align:middle line:84% To do this, I do require that f is differentiable near here. 00:33:03.510 --> 00:33:07.320 align:middle line:84% But our systems are, so life is good. 00:33:07.320 --> 00:33:10.500 align:middle line:84% Any questions about that idea generally? 00:33:10.500 --> 00:33:13.480 align:middle line:90% 00:33:13.480 --> 00:33:16.280 align:middle line:90% So that's messy to think about. 00:33:16.280 --> 00:33:20.290 align:middle line:84% But actually, it's quite simple in the real world. 00:33:20.290 --> 00:33:22.420 align:middle line:84% In the real world, what am I going to do? 00:33:22.420 --> 00:33:24.340 align:middle line:84% In the real world, all I'm going to do 00:33:24.340 --> 00:33:26.380 align:middle line:84% is I'm going to come up to this equation 00:33:26.380 --> 00:33:32.350 align:middle line:84% and substitute in these expansions and factor it. 00:33:32.350 --> 00:33:34.360 align:middle line:90% Life is easy. 00:33:34.360 --> 00:33:35.960 align:middle line:90% So let me do that. 00:33:35.960 --> 00:33:39.220 align:middle line:84% So what I have is I have diL-bar/dt. 00:33:39.220 --> 00:33:41.200 align:middle line:84% And I said, iL-- oh, I should have 00:33:41.200 --> 00:33:43.180 align:middle line:84% said these are the local averages because I'm 00:33:43.180 --> 00:33:45.490 align:middle line:90% working with a local averages. 00:33:45.490 --> 00:33:48.790 align:middle line:90% diL-bar/dt is equal to-- 00:33:48.790 --> 00:33:53.080 align:middle line:90% 00:33:53.080 --> 00:33:59.410 align:middle line:84% is equal to d/dt of iL plus diL-tilde/dt. 00:33:59.410 --> 00:34:02.260 align:middle line:90% 00:34:02.260 --> 00:34:07.090 align:middle line:84% And we said this was 0 because iL is a constant. 00:34:07.090 --> 00:34:09.127 align:middle line:84% And that's going to be equal to what? 00:34:09.127 --> 00:34:11.710 align:middle line:84% That's going to be equal to-- all I've got to do is substitute 00:34:11.710 --> 00:34:12.543 align:middle line:90% the stuff in there-- 00:34:12.543 --> 00:34:28.540 align:middle line:84% 1 over L u plus u-tilde minus 1 over L capital V plus v-tilde 00:34:28.540 --> 00:34:30.489 align:middle line:90% times-- 00:34:30.489 --> 00:34:32.650 align:middle line:90% it's 1 minus d-- 00:34:32.650 --> 00:34:34.900 align:middle line:84% I'm sorry, it's d-prime, which is 1 minus d. 00:34:34.900 --> 00:34:40.090 align:middle line:90% That's 1 minus d minus d-tilde. 00:34:40.090 --> 00:34:43.150 align:middle line:90% And that's the top equation. 00:34:43.150 --> 00:34:52.489 align:middle line:84% And I get dv-bar/dt, which is equal to d/dt of capital V, 00:34:52.489 --> 00:34:57.200 align:middle line:84% which we said was 0, plus dV-tilde/dt, 00:34:57.200 --> 00:35:12.380 align:middle line:84% which is equal to minus 1 over RC v plus v-tilde plus 1 over c 00:35:12.380 --> 00:35:22.430 align:middle line:84% times iL plus iL-tilde times 1 minus d minus d-tilde. 00:35:22.430 --> 00:35:26.750 align:middle line:90% 00:35:26.750 --> 00:35:28.730 align:middle line:90% I apologize for all the algebra. 00:35:28.730 --> 00:35:30.030 align:middle line:90% But guess what? 00:35:30.030 --> 00:35:31.590 align:middle line:90% You can't do this without it. 00:35:31.590 --> 00:35:32.930 align:middle line:90% So that's life. 00:35:32.930 --> 00:35:34.370 align:middle line:84% All right, so what can I do here? 00:35:34.370 --> 00:35:36.740 align:middle line:90% Let me just refactor this. 00:35:36.740 --> 00:35:38.670 align:middle line:84% And here's how I'm going to factor it. 00:35:38.670 --> 00:35:43.010 align:middle line:84% I'm going to put all of the terms that are just operating 00:35:43.010 --> 00:35:44.923 align:middle line:90% point capital letters together. 00:35:44.923 --> 00:35:46.340 align:middle line:84% Then I'm going to have mixed ones. 00:35:46.340 --> 00:35:48.048 align:middle line:84% And then I'm going to have all terms that 00:35:48.048 --> 00:35:50.400 align:middle line:90% are products of tildes. 00:35:50.400 --> 00:35:52.150 align:middle line:84% So let me just rewrite this for you. 00:35:52.150 --> 00:36:03.370 align:middle line:84% So what I get is diL-tilde/dt is equal to 1 over L u minus 1 over 00:36:03.370 --> 00:36:14.560 align:middle line:84% L times d-prime v, so 1 over L times u minus d-prime v. 00:36:14.560 --> 00:36:16.220 align:middle line:84% Then I'm going to get mixed terms. 00:36:16.220 --> 00:36:21.610 align:middle line:84% I'm going to get plus 1 over L u-tilde 00:36:21.610 --> 00:36:27.460 align:middle line:90% minus 1 over L v d-tilde. 00:36:27.460 --> 00:36:33.490 align:middle line:84% Oops, it's going to be plus 1 over L v d-tilde 00:36:33.490 --> 00:36:41.260 align:middle line:84% minus 1 over L d-prime, which is 1 minus capital D, v-tilde. 00:36:41.260 --> 00:36:48.250 align:middle line:84% And then I'm going to get plus 1 over L v-tilde d-tilde. 00:36:48.250 --> 00:36:51.170 align:middle line:84% Last equation, and then life will get easier. 00:36:51.170 --> 00:36:52.850 align:middle line:84% I won't have to write any equations. 00:36:52.850 --> 00:37:03.880 align:middle line:84% It's dv-tilde/dt is going to be equal to minus 1 over RC minus-- 00:37:03.880 --> 00:37:06.430 align:middle line:90% I'm sorry, it's going to be-- 00:37:06.430 --> 00:37:11.470 align:middle line:84% it's going to be 1 over C times minus v 00:37:11.470 --> 00:37:19.550 align:middle line:90% over R plus iL times 1 minus d-- 00:37:19.550 --> 00:37:24.090 align:middle line:90% 00:37:24.090 --> 00:37:33.840 align:middle line:84% that's the all-caps-- term minus 1 over RC v-tilde plus 1 00:37:33.840 --> 00:37:48.540 align:middle line:84% over C d-prime over C iL-tilde plus or minus iL over C d-tilde, 00:37:48.540 --> 00:37:56.100 align:middle line:84% and then minus 1 over C iL-tilde d-tilde. 00:37:56.100 --> 00:37:58.260 align:middle line:90% I apologize for all that math. 00:37:58.260 --> 00:38:02.180 align:middle line:84% But let me get to why I'm telling you this. 00:38:02.180 --> 00:38:05.600 align:middle line:90% Let's look at this factor. 00:38:05.600 --> 00:38:07.530 align:middle line:90% And I wrote u here. 00:38:07.530 --> 00:38:09.230 align:middle line:84% I should have been very careful to write 00:38:09.230 --> 00:38:15.590 align:middle line:84% capital U. These are all DC periodic steady state values. 00:38:15.590 --> 00:38:17.950 align:middle line:90% This is the DC input voltage. 00:38:17.950 --> 00:38:20.780 align:middle line:84% This is the DC output voltage at an operating point. 00:38:20.780 --> 00:38:24.800 align:middle line:84% Or should I say, in periodic steady state local averages? 00:38:24.800 --> 00:38:29.100 align:middle line:90% What is 1 minus d times v? 00:38:29.100 --> 00:38:34.740 align:middle line:84% Well, if I came back over to my boost converter here, 00:38:34.740 --> 00:38:38.670 align:middle line:90% 1 minus d times v is just u. 00:38:38.670 --> 00:38:43.860 align:middle line:90% So this whole thing is 0. 00:38:43.860 --> 00:38:47.070 align:middle line:84% Likewise, this is the output current, capital V 00:38:47.070 --> 00:38:51.630 align:middle line:90% over R, capital IL times-- 00:38:51.630 --> 00:38:55.680 align:middle line:84% this should have been d-prime is simply 00:38:55.680 --> 00:39:00.700 align:middle line:84% the current going into the output from the inductor. 00:39:00.700 --> 00:39:03.360 align:middle line:84% This is also 0, if you worked it out. 00:39:03.360 --> 00:39:06.193 align:middle line:84% So basically, all capital terms in this thing, 00:39:06.193 --> 00:39:08.610 align:middle line:84% if you're dealing with the periodic steady state operating 00:39:08.610 --> 00:39:12.240 align:middle line:84% points of a converter, are going to drop out and go away. 00:39:12.240 --> 00:39:14.000 align:middle line:84% If they don't, you did your math wrong. 00:39:14.000 --> 00:39:16.940 align:middle line:90% 00:39:16.940 --> 00:39:19.760 align:middle line:90% What about these guys over here? 00:39:19.760 --> 00:39:22.070 align:middle line:84% This is the small variation in the output 00:39:22.070 --> 00:39:24.390 align:middle line:84% voltage times the small variation in the duty ratio. 00:39:24.390 --> 00:39:27.620 align:middle line:84% I'm going to argue that this is a small thing 00:39:27.620 --> 00:39:28.710 align:middle line:90% and this is a small thing. 00:39:28.710 --> 00:39:30.410 align:middle line:90% This is small squared. 00:39:30.410 --> 00:39:32.820 align:middle line:84% I'm going to pretend he's approximately 0. 00:39:32.820 --> 00:39:35.870 align:middle line:84% Likewise, this is small times small. 00:39:35.870 --> 00:39:37.920 align:middle line:84% I'm going to say he's approximately 0. 00:39:37.920 --> 00:39:41.570 align:middle line:84% Basically, these terms are precisely 00:39:41.570 --> 00:39:45.350 align:middle line:84% the curvature away in this nonlinear function 00:39:45.350 --> 00:39:47.150 align:middle line:90% from the straight line. 00:39:47.150 --> 00:39:48.830 align:middle line:84% And I'm saying, if I get close enough, 00:39:48.830 --> 00:39:50.520 align:middle line:90% I can ignore that curvature. 00:39:50.520 --> 00:39:54.110 align:middle line:84% If I make my variations away from that point small enough, 00:39:54.110 --> 00:39:59.940 align:middle line:84% I can't tell the difference, which leaves me with an equation 00:39:59.940 --> 00:40:05.940 align:middle line:84% in terms of diL-tilde/dt, which is in terms of two-- 00:40:05.940 --> 00:40:07.680 align:middle line:84% basically, it's in terms of this. 00:40:07.680 --> 00:40:10.330 align:middle line:90% I'm going to have constants. 00:40:10.330 --> 00:40:14.670 align:middle line:84% This is 1 over L. This is 1 over L times capital V. 00:40:14.670 --> 00:40:17.140 align:middle line:84% So capital V is my periodic steady state value. 00:40:17.140 --> 00:40:20.430 align:middle line:84% That's just a constant, as far as the variations are concerned. 00:40:20.430 --> 00:40:24.990 align:middle line:84% This is minus 1 over L. This is another constant. 00:40:24.990 --> 00:40:28.590 align:middle line:84% d-prime is my steady state duty ratio. 00:40:28.590 --> 00:40:31.980 align:middle line:84% And d-tilde is my variation away from that. 00:40:31.980 --> 00:40:36.420 align:middle line:84% Or d-prime is 1 minus my steady state duty ratio. 00:40:36.420 --> 00:40:38.770 align:middle line:84% And I can do the same thing with these other terms. 00:40:38.770 --> 00:40:43.970 align:middle line:84% These, as far as this equation is concerned, are all constants. 00:40:43.970 --> 00:40:47.090 align:middle line:84% Does that make sense to everybody? 00:40:47.090 --> 00:40:53.595 align:middle line:84% So I would argue then that this beast, this beastie-- 00:40:53.595 --> 00:40:59.020 align:middle line:90% 00:40:59.020 --> 00:41:01.330 align:middle line:90% if I take away all the stuff-- 00:41:01.330 --> 00:41:04.090 align:middle line:84% I throw away all the stuff, all I 00:41:04.090 --> 00:41:08.920 align:middle line:84% have is small signal variables times constants added up. 00:41:08.920 --> 00:41:13.420 align:middle line:84% So this is also a system of differential equations. 00:41:13.420 --> 00:41:16.180 align:middle line:84% But as far as a given periodic steady state 00:41:16.180 --> 00:41:19.810 align:middle line:84% operating condition, it's a linear constant coefficient 00:41:19.810 --> 00:41:22.090 align:middle line:90% differential equations. 00:41:22.090 --> 00:41:25.600 align:middle line:84% This is the friendly kind of differential equations, 00:41:25.600 --> 00:41:26.840 align:middle line:90% the ones we love. 00:41:26.840 --> 00:41:29.020 align:middle line:90% Well, why do I like that? 00:41:29.020 --> 00:41:31.540 align:middle line:84% Because I could take that set of differential 00:41:31.540 --> 00:41:34.360 align:middle line:90% equations and come back here. 00:41:34.360 --> 00:41:37.270 align:middle line:84% And instead of worrying about v-error, 00:41:37.270 --> 00:41:41.560 align:middle line:84% I could worry about v-error variation. 00:41:41.560 --> 00:41:44.140 align:middle line:84% I could worry about the variation 00:41:44.140 --> 00:41:49.392 align:middle line:84% in duty cycle, the variation in my inductor current away 00:41:49.392 --> 00:41:51.850 align:middle line:84% from-- this variations away from the steady state operating 00:41:51.850 --> 00:41:54.280 align:middle line:84% point, the variation in voltage away 00:41:54.280 --> 00:41:55.930 align:middle line:84% from my steady state operating point. 00:41:55.930 --> 00:41:59.140 align:middle line:84% And this would be feeding back v-tilde. 00:41:59.140 --> 00:42:03.190 align:middle line:84% And suddenly, I stick these differential equations in here. 00:42:03.190 --> 00:42:06.380 align:middle line:90% And these are linear. 00:42:06.380 --> 00:42:08.120 align:middle line:84% And suddenly, my life became easy 00:42:08.120 --> 00:42:10.950 align:middle line:84% because now I can apply all my linear control theory to it. 00:42:10.950 --> 00:42:14.490 align:middle line:90% 00:42:14.490 --> 00:42:15.440 align:middle line:90% Questions about that? 00:42:15.440 --> 00:42:23.010 align:middle line:90% 00:42:23.010 --> 00:42:27.150 align:middle line:84% So how might I do this in practice? 00:42:27.150 --> 00:42:28.870 align:middle line:90% A lot of things I can do. 00:42:28.870 --> 00:42:30.720 align:middle line:84% And I should say, by the way, there's 00:42:30.720 --> 00:42:33.930 align:middle line:84% nothing that prevents me from doing the original nonlinear 00:42:33.930 --> 00:42:36.150 align:middle line:90% stuff and doing a nonlinear-- 00:42:36.150 --> 00:42:39.660 align:middle line:84% having a nonlinear plan and considering Lyapunov controllers 00:42:39.660 --> 00:42:42.240 align:middle line:90% or anything I want to consider. 00:42:42.240 --> 00:42:46.660 align:middle line:84% It's just complicated and perhaps unnecessarily so. 00:42:46.660 --> 00:42:51.390 align:middle line:84% By doing this linearization trick around an operating point, 00:42:51.390 --> 00:42:55.170 align:middle line:84% I get a set of linear constant coefficient differential 00:42:55.170 --> 00:42:59.880 align:middle line:84% equations that then give me nice behavior. 00:42:59.880 --> 00:43:02.760 align:middle line:84% The only constraint, the only fly in this ointment 00:43:02.760 --> 00:43:06.420 align:middle line:84% is the differential equations, the coefficients in it 00:43:06.420 --> 00:43:08.320 align:middle line:90% depend on the operating point. 00:43:08.320 --> 00:43:12.730 align:middle line:84% So I've got to say, geez, I'm operating at d-prime is 0.5. 00:43:12.730 --> 00:43:14.890 align:middle line:90% My steady state is at 50%. 00:43:14.890 --> 00:43:17.630 align:middle line:84% It's for this boost conversion ratio. 00:43:17.630 --> 00:43:20.870 align:middle line:84% And then I'll get the dynamics for that boost conversion ratio, 00:43:20.870 --> 00:43:22.740 align:middle line:84% for small deviations away from that. 00:43:22.740 --> 00:43:26.510 align:middle line:84% But if I suddenly change duty ratio to 75% and d-prime 00:43:26.510 --> 00:43:30.440 align:middle line:84% became 0.25, the equations are different 00:43:30.440 --> 00:43:32.510 align:middle line:90% and the dynamics are different, 00:43:32.510 --> 00:43:35.042 align:middle line:84% So the fact that this is nonlinear, 00:43:35.042 --> 00:43:37.500 align:middle line:84% it means, at different operating points, the dynamics vary. 00:43:37.500 --> 00:43:39.090 align:middle line:90% Well, you're stuck with that. 00:43:39.090 --> 00:43:40.170 align:middle line:90% It's a nonlinear system. 00:43:40.170 --> 00:43:42.890 align:middle line:90% That's what it does. 00:43:42.890 --> 00:43:45.350 align:middle line:90% But if I'm interested in-- 00:43:45.350 --> 00:43:47.180 align:middle line:84% jeez, I kicked it a little bit, I 00:43:47.180 --> 00:43:50.180 align:middle line:84% changed my duty ratio a little, or I stepped my load, 00:43:50.180 --> 00:43:53.660 align:middle line:84% when I get close to that second operating point, 00:43:53.660 --> 00:43:56.020 align:middle line:84% I can predict what the dynamics are very exactly. 00:43:56.020 --> 00:43:58.655 align:middle line:90% 00:43:58.655 --> 00:44:01.850 align:middle line:84% And by the way, even though this is nonlinear, 00:44:01.850 --> 00:44:06.050 align:middle line:84% it turns out that it's pretty good. 00:44:06.050 --> 00:44:09.810 align:middle line:84% The dynamics are quite nicely captured. 00:44:09.810 --> 00:44:12.150 align:middle line:90% So how might I deal with this? 00:44:12.150 --> 00:44:13.140 align:middle line:90% Well, I could do this. 00:44:13.140 --> 00:44:14.430 align:middle line:90% What would I do? 00:44:14.430 --> 00:44:20.750 align:middle line:84% I could turn this whole thing into to a transfer function, 00:44:20.750 --> 00:44:21.350 align:middle line:90% for example. 00:44:21.350 --> 00:44:23.780 align:middle line:84% Like over here, what I would really like 00:44:23.780 --> 00:44:32.450 align:middle line:84% is some transfer function, perhaps H of s, where H of s 00:44:32.450 --> 00:44:38.900 align:middle line:84% goes from d-tilde to v-tilde, for example. 00:44:38.900 --> 00:44:41.480 align:middle line:84% It's a transfer function from variations in duty ratio 00:44:41.480 --> 00:44:46.490 align:middle line:84% to variations in output voltage if I kept my input 00:44:46.490 --> 00:44:49.670 align:middle line:90% voltage constant, for example. 00:44:49.670 --> 00:44:50.970 align:middle line:90% Well, how would I do that? 00:44:50.970 --> 00:44:54.050 align:middle line:84% I'd just take the Laplace transform of this stuff, 00:44:54.050 --> 00:44:57.530 align:middle line:84% all my i sub L-tildes would become S i sub L-tildes. 00:44:57.530 --> 00:44:59.470 align:middle line:90% This would become S v-tilde. 00:44:59.470 --> 00:45:01.970 align:middle line:84% And then I could look at this thing, and I could go in here, 00:45:01.970 --> 00:45:05.900 align:middle line:84% and I could multiply this by S. I could then 00:45:05.900 --> 00:45:09.740 align:middle line:84% substitute my S iL-tilde in here and iL-tiles would go away. 00:45:09.740 --> 00:45:14.630 align:middle line:84% And I'd get everything that relates d-tilde and v-tilde. 00:45:14.630 --> 00:45:16.640 align:middle line:84% I'll just write down that solution for you 00:45:16.640 --> 00:45:18.000 align:middle line:90% because I'm running out of time. 00:45:18.000 --> 00:45:21.605 align:middle line:84% But if you do that, here's what you get. 00:45:21.605 --> 00:45:23.480 align:middle line:84% First of all, any questions about the process 00:45:23.480 --> 00:45:26.870 align:middle line:90% I just mentioned? 00:45:26.870 --> 00:45:29.365 align:middle line:84% So what I would get is something like this. 00:45:29.365 --> 00:45:33.290 align:middle line:90% 00:45:33.290 --> 00:45:42.190 align:middle line:84% I would get H of s is equal to v-tilde over d-tilde, which 00:45:42.190 --> 00:45:59.070 align:middle line:84% is equal to minus s times iL over C plus v d-prime over LC 00:45:59.070 --> 00:46:06.630 align:middle line:84% divided by s squared plus 1 over RC s plus d-prime squared 00:46:06.630 --> 00:46:09.500 align:middle line:90% over LC. 00:46:09.500 --> 00:46:14.510 align:middle line:84% So this is a second order differential-- 00:46:14.510 --> 00:46:16.570 align:middle line:84% this is a second order transfer function. 00:46:16.570 --> 00:46:20.240 align:middle line:84% It has two poles, S squared in the denominator. 00:46:20.240 --> 00:46:24.850 align:middle line:84% So this characteristic equation is second order. 00:46:24.850 --> 00:46:26.363 align:middle line:90% Well, we expect that. 00:46:26.363 --> 00:46:27.530 align:middle line:90% It's got to be second order. 00:46:27.530 --> 00:46:28.060 align:middle line:90% Why? 00:46:28.060 --> 00:46:30.010 align:middle line:84% Because what are my energy storage elements 00:46:30.010 --> 00:46:31.390 align:middle line:90% in my converter? 00:46:31.390 --> 00:46:33.310 align:middle line:84% My energy storage elements in my converter 00:46:33.310 --> 00:46:35.390 align:middle line:84% are one inductor and one capacitor. 00:46:35.390 --> 00:46:36.760 align:middle line:90% So it is second order. 00:46:36.760 --> 00:46:39.610 align:middle line:90% Great, I expect that. 00:46:39.610 --> 00:46:41.825 align:middle line:84% These have two left half plane zeros. 00:46:41.825 --> 00:46:43.450 align:middle line:84% I can tell because all the coefficients 00:46:43.450 --> 00:46:46.630 align:middle line:84% are positive on the characteristic equation. 00:46:46.630 --> 00:46:49.040 align:middle line:84% It has one right half plane zero. 00:46:49.040 --> 00:46:52.070 align:middle line:84% We'll come back to talking about that. 00:46:52.070 --> 00:46:53.720 align:middle line:90% So that's great. 00:46:53.720 --> 00:46:58.750 align:middle line:90% I get a behavior I'm expecting. 00:46:58.750 --> 00:47:02.800 align:middle line:84% Notice that the pole locations move around 00:47:02.800 --> 00:47:04.100 align:middle line:90% depending upon duty ratio. 00:47:04.100 --> 00:47:10.226 align:middle line:84% So if I'm sitting here and I'm coming from 5 to 10 volts DC, 00:47:10.226 --> 00:47:12.100 align:middle line:90% d-prime would be 0.5. 00:47:12.100 --> 00:47:14.410 align:middle line:84% And I would get one set of pole locations. 00:47:14.410 --> 00:47:18.150 align:middle line:84% If I was coming from 5 to 20 volts, 00:47:18.150 --> 00:47:19.830 align:middle line:84% I'd have a different value of d-prime. 00:47:19.830 --> 00:47:21.970 align:middle line:84% And I'd get a different set of pole locations. 00:47:21.970 --> 00:47:27.130 align:middle line:84% So the poles do move around at different operating points. 00:47:27.130 --> 00:47:29.100 align:middle line:90% But that's life. 00:47:29.100 --> 00:47:30.630 align:middle line:90% I can deal with that. 00:47:30.630 --> 00:47:31.990 align:middle line:90% How accurate is this? 00:47:31.990 --> 00:47:33.070 align:middle line:90% Well, I plugged in-- 00:47:33.070 --> 00:47:34.960 align:middle line:84% I won't bother writing it on the board. 00:47:34.960 --> 00:47:38.940 align:middle line:84% I plugged in values for an example. 00:47:38.940 --> 00:47:43.860 align:middle line:84% And it's precisely the example I showed you. 00:47:43.860 --> 00:47:46.085 align:middle line:84% It's precisely the example I showed you-- 00:47:46.085 --> 00:47:51.840 align:middle line:90% 00:47:51.840 --> 00:47:55.590 align:middle line:84% precisely the example I showed you last class, in fact, 00:47:55.590 --> 00:47:56.826 align:middle line:90% this transient. 00:47:56.826 --> 00:48:01.980 align:middle line:90% 00:48:01.980 --> 00:48:07.420 align:middle line:84% This is we get a step, we step-- in this case, we step the load. 00:48:07.420 --> 00:48:10.080 align:middle line:84% So I calculate everything for the new, 00:48:10.080 --> 00:48:14.310 align:middle line:84% the final value of what the inductor current would be. 00:48:14.310 --> 00:48:16.257 align:middle line:84% And I just plug into that equation. 00:48:16.257 --> 00:48:17.840 align:middle line:84% And what do you get for that equation? 00:48:17.840 --> 00:48:23.450 align:middle line:84% You get a 0 at 35 kiloradians per second and poles at minus 00:48:23.450 --> 00:48:27.720 align:middle line:84% 4,000 plus or minus J 16, 279 radians per second. 00:48:27.720 --> 00:48:30.140 align:middle line:84% That gives an oscillation period that ought 00:48:30.140 --> 00:48:32.840 align:middle line:90% to be about 0.4 milliseconds. 00:48:32.840 --> 00:48:37.100 align:middle line:84% Well, if you look at this, that's pretty good. 00:48:37.100 --> 00:48:40.840 align:middle line:84% That's about what we got, 1.6 to 2. 00:48:40.840 --> 00:48:43.340 align:middle line:84% So even though this is a pretty large step-- 00:48:43.340 --> 00:48:46.570 align:middle line:90% it's a factor of 2 in load-- 00:48:46.570 --> 00:48:49.150 align:middle line:84% I can predict the transient response, 00:48:49.150 --> 00:48:51.880 align:middle line:84% the average transient response, with this nice little linearized 00:48:51.880 --> 00:48:52.380 align:middle line:90% model. 00:48:52.380 --> 00:48:55.990 align:middle line:84% That means, if I put this thing in a feedback controller, 00:48:55.990 --> 00:48:59.170 align:middle line:84% I ought to be able to build a compensator that 00:48:59.170 --> 00:49:01.000 align:middle line:84% will control the closed loop dynamics 00:49:01.000 --> 00:49:04.220 align:middle line:90% and give me the response I want. 00:49:04.220 --> 00:49:09.470 align:middle line:84% So that is state space averaging in its application, 00:49:09.470 --> 00:49:10.910 align:middle line:90% and then linearization. 00:49:10.910 --> 00:49:12.830 align:middle line:84% So I'm going to average my circuit. 00:49:12.830 --> 00:49:15.330 align:middle line:84% And I get a nonlinear average model. 00:49:15.330 --> 00:49:17.690 align:middle line:84% Then I'm going to linearize the differential equations. 00:49:17.690 --> 00:49:20.770 align:middle line:84% And that gives me a nice linear constant coefficient model 00:49:20.770 --> 00:49:22.520 align:middle line:84% that I can model with a transfer function, 00:49:22.520 --> 00:49:24.930 align:middle line:84% at least for a given operating point. 00:49:24.930 --> 00:49:26.500 align:middle line:90% Any questions before we wrap up? 00:49:26.500 --> 00:49:27.000 align:middle line:90% Yeah? 00:49:27.000 --> 00:49:28.920 align:middle line:84% STUDENT: When you said [INAUDIBLE], 00:49:28.920 --> 00:49:31.470 align:middle line:84% were you showing us some simulation or-- 00:49:31.470 --> 00:49:34.200 align:middle line:84% DAVID PERREAULT: Oh, what's this? 00:49:34.200 --> 00:49:36.250 align:middle line:90% Where did I get that number? 00:49:36.250 --> 00:49:37.420 align:middle line:90% STUDENT: Yeah. 00:49:37.420 --> 00:49:40.060 align:middle line:84% DAVID PERREAULT: What I did was-- 00:49:40.060 --> 00:49:41.540 align:middle line:90% well, it's not on the graph. 00:49:41.540 --> 00:49:44.770 align:middle line:84% What I did was I plugged into this. 00:49:44.770 --> 00:49:52.050 align:middle line:84% And I got the values of S. And the imaginary component 00:49:52.050 --> 00:49:57.520 align:middle line:84% of the pole locations is the damped natural frequency. 00:49:57.520 --> 00:50:00.240 align:middle line:84% And that should correspond to the oscillation 00:50:00.240 --> 00:50:04.410 align:middle line:90% period of a transient response. 00:50:04.410 --> 00:50:07.920 align:middle line:84% So if I took an RLC circuit and I put a step on it, 00:50:07.920 --> 00:50:11.927 align:middle line:84% I would see a transient with the pole locations indicated. 00:50:11.927 --> 00:50:14.010 align:middle line:84% And all I did was said, what's the pole locations? 00:50:14.010 --> 00:50:16.470 align:middle line:84% And how does that match that transient? 00:50:16.470 --> 00:50:17.192 align:middle line:90% Yeah? 00:50:17.192 --> 00:50:18.900 align:middle line:84% STUDENT: When did you choose to linearize 00:50:18.900 --> 00:50:21.570 align:middle line:84% throughout the starting or ending operating points? 00:50:21.570 --> 00:50:23.627 align:middle line:84% DAVID PERREAULT: Well, OK, so I've 00:50:23.627 --> 00:50:25.710 align:middle line:84% got to linearize around the ending operating point 00:50:25.710 --> 00:50:28.530 align:middle line:84% because I'm asking how does it vary about where I'm going. 00:50:28.530 --> 00:50:32.100 align:middle line:84% And it's going to settle down to that place. 00:50:32.100 --> 00:50:34.922 align:middle line:84% Asking what the dynamics are about where I'm leaving from 00:50:34.922 --> 00:50:36.630 align:middle line:84% doesn't tell me very much because I'm not 00:50:36.630 --> 00:50:37.180 align:middle line:90% going to be there. 00:50:37.180 --> 00:50:38.263 align:middle line:90% I'm going away from there. 00:50:38.263 --> 00:50:42.220 align:middle line:90% 00:50:42.220 --> 00:50:44.420 align:middle line:90% Any last questions? 00:50:44.420 --> 00:50:47.150 align:middle line:84% So we'll take up control of power converters 00:50:47.150 --> 00:50:49.600 align:middle line:90% again next class. 00:50:49.600 --> 00:50:52.000 align:middle line:90%