WEBVTT 00:00:00.000 --> 00:00:08.568 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:08.568 --> 00:00:10.550 align:middle line:90% 00:00:10.550 --> 00:00:12.570 align:middle line:84% DAVID PERREAULT: OK why don't we get started? 00:00:12.570 --> 00:00:15.137 align:middle line:90% [MIXED CONVERSATIONS] 00:00:15.137 --> 00:00:17.845 align:middle line:84% DAVID PERREAULT: Spring seems to be giving everybody energy here. 00:00:17.845 --> 00:00:20.930 align:middle line:90% 00:00:20.930 --> 00:00:25.760 align:middle line:84% I wanted to continue talking about EMIs and filter design, 00:00:25.760 --> 00:00:28.490 align:middle line:84% and most of what I'm talking about 00:00:28.490 --> 00:00:33.020 align:middle line:84% is in Principles of Power Electronics chapter 26, 00:00:33.020 --> 00:00:37.220 align:middle line:84% plus some additional handouts that are going around 00:00:37.220 --> 00:00:40.250 align:middle line:90% and are also posted on Canvas. 00:00:40.250 --> 00:00:44.840 align:middle line:84% Just as a reminder, last time we talked a little bit about how 00:00:44.840 --> 00:00:47.060 align:middle line:90% one goes about measuring EMI. 00:00:47.060 --> 00:00:50.760 align:middle line:84% And we said, OK, if I have some converter-- 00:00:50.760 --> 00:00:56.660 align:middle line:84% it could be a buck converter is our canonical example here, 00:00:56.660 --> 00:01:01.070 align:middle line:84% and it's drawing some really pulsating current 00:01:01.070 --> 00:01:05.810 align:middle line:84% from its input that's going to look like sort of ix 00:01:05.810 --> 00:01:10.860 align:middle line:84% is some big pulsation at the switching frequency, right? 00:01:10.860 --> 00:01:12.450 align:middle line:84% And then I could say, well, maybe 00:01:12.450 --> 00:01:16.530 align:middle line:84% I have some input filter that maybe here I'm just going to say 00:01:16.530 --> 00:01:18.360 align:middle line:90% is some capacitor. 00:01:18.360 --> 00:01:22.470 align:middle line:84% At minimum, I always have a capacitor here. 00:01:22.470 --> 00:01:25.050 align:middle line:84% And we said, well, if I just connected 00:01:25.050 --> 00:01:28.110 align:middle line:84% this up to some source impedance, Z sub 00:01:28.110 --> 00:01:33.660 align:middle line:84% s, or some voltage source V sub s with a source impedance 00:01:33.660 --> 00:01:36.720 align:middle line:84% Z sub s, I would probably run into trouble 00:01:36.720 --> 00:01:39.300 align:middle line:84% with repeatability, because depending 00:01:39.300 --> 00:01:42.180 align:middle line:84% upon what Z sub s looked like, I would 00:01:42.180 --> 00:01:45.990 align:middle line:90% get different voltages, Vx. 00:01:45.990 --> 00:01:52.860 align:middle line:84% And maybe I'll call this current i sub y 00:01:52.860 --> 00:01:54.550 align:middle line:90% on the other side of the filter. 00:01:54.550 --> 00:02:00.240 align:middle line:84% And so it's kind of problematic from a repeatability point 00:02:00.240 --> 00:02:00.980 align:middle line:90% of view. 00:02:00.980 --> 00:02:03.000 align:middle line:84% So if I want to test to some standards 00:02:03.000 --> 00:02:06.210 align:middle line:84% I need some better way of basically 00:02:06.210 --> 00:02:10.169 align:middle line:84% controlling the impedance we're looking back into. 00:02:10.169 --> 00:02:14.510 align:middle line:84% And so to do that, when we're doing EMI testing alone, 00:02:14.510 --> 00:02:19.310 align:middle line:84% we come and then insert an additional stage in here, 00:02:19.310 --> 00:02:20.810 align:middle line:90% which I'll draw in orange. 00:02:20.810 --> 00:02:25.310 align:middle line:84% And this would look like something like this. 00:02:25.310 --> 00:02:30.170 align:middle line:84% I'll have some box that is part of my test setup, 00:02:30.170 --> 00:02:31.160 align:middle line:90% called a LISN-- 00:02:31.160 --> 00:02:33.450 align:middle line:84% Line Impedance Stabilization Network. 00:02:33.450 --> 00:02:39.280 align:middle line:84% And it will, at minimum, have some L LISN and C LISN. 00:02:39.280 --> 00:02:42.400 align:middle line:90% 00:02:42.400 --> 00:02:50.250 align:middle line:84% And then I will connect this up to some impedance analyzer 00:02:50.250 --> 00:02:53.370 align:middle line:90% that is usually 50 ohms. 00:02:53.370 --> 00:02:59.970 align:middle line:84% So if I looked at the impedance looking back this way, I would-- 00:02:59.970 --> 00:03:02.495 align:middle line:84% high EMI frequencies, when this becomes an open circuit 00:03:02.495 --> 00:03:03.870 align:middle line:84% and this becomes a short circuit, 00:03:03.870 --> 00:03:07.590 align:middle line:84% I'll look back into the spectrum analyzer 00:03:07.590 --> 00:03:09.690 align:middle line:84% impedance, which is usually 50 ohms, 00:03:09.690 --> 00:03:12.120 align:middle line:90% and I'll make my tests that way. 00:03:12.120 --> 00:03:16.020 align:middle line:84% So we get rid of the variability in the source 00:03:16.020 --> 00:03:19.770 align:middle line:84% impedance for testing purposes only by doing that. 00:03:19.770 --> 00:03:23.280 align:middle line:84% And I should say, this is actually 00:03:23.280 --> 00:03:26.640 align:middle line:90% a slight simplification. 00:03:26.640 --> 00:03:29.490 align:middle line:84% In most systems, you actually put a separate LISN 00:03:29.490 --> 00:03:32.580 align:middle line:84% on the hot lead, the power lead, and on the ground lead, 00:03:32.580 --> 00:03:36.100 align:middle line:84% and you measure the response of the LISN on both. 00:03:36.100 --> 00:03:38.580 align:middle line:84% And the usual method you do is you're 00:03:38.580 --> 00:03:43.050 align:middle line:84% looking at the frequency content of what goes into the spectrum 00:03:43.050 --> 00:03:46.570 align:middle line:84% analyzer here, typically for conducting EMI between 150 00:03:46.570 --> 00:03:48.820 align:middle line:90% kilohertz to 30 megahertz. 00:03:48.820 --> 00:03:52.540 align:middle line:84% But that again depends upon the standard you're comparing it to. 00:03:52.540 --> 00:03:55.350 align:middle line:84% Some military standards go down to much lower frequencies. 00:03:55.350 --> 00:03:56.890 align:middle line:90% And all the details. 00:03:56.890 --> 00:04:00.190 align:middle line:84% What are the LISN values, what does the test setup 00:04:00.190 --> 00:04:02.860 align:middle line:84% look like, et cetera, that's all part of a standard 00:04:02.860 --> 00:04:05.120 align:middle line:84% that you adhere to test your converter. 00:04:05.120 --> 00:04:08.392 align:middle line:90% 00:04:08.392 --> 00:04:13.940 align:middle line:84% But what we-- the basic idea is as follows. 00:04:13.940 --> 00:04:16.600 align:middle line:84% If I think about I have some ripple frequency. 00:04:16.600 --> 00:04:19.570 align:middle line:84% I'll call it ix tilde here, which 00:04:19.570 --> 00:04:21.790 align:middle line:90% might be some large signal. 00:04:21.790 --> 00:04:28.900 align:middle line:84% I put it back into my filter capacitor, or my filter, 00:04:28.900 --> 00:04:30.140 align:middle line:90% more broadly. 00:04:30.140 --> 00:04:31.930 align:middle line:90% Here is just Cf. 00:04:31.930 --> 00:04:35.710 align:middle line:84% And then that goes back into some impedance 00:04:35.710 --> 00:04:42.565 align:middle line:84% which, in the test stand only, ends up looking like our LISN. 00:04:42.565 --> 00:04:46.810 align:middle line:84% And I can measure either this current-- 00:04:46.810 --> 00:04:52.060 align:middle line:84% I'll call it iy, or the more accurately, 00:04:52.060 --> 00:04:53.860 align:middle line:84% in terms of the actual test specifications, 00:04:53.860 --> 00:04:57.370 align:middle line:84% the voltage across the LISN impedance in order 00:04:57.370 --> 00:04:59.370 align:middle line:90% to make my measurements. 00:04:59.370 --> 00:05:00.755 align:middle line:90% Any questions about that? 00:05:00.755 --> 00:05:04.400 align:middle line:90% 00:05:04.400 --> 00:05:05.620 align:middle line:90% So we said, OK. 00:05:05.620 --> 00:05:08.520 align:middle line:90% 00:05:08.520 --> 00:05:12.970 align:middle line:90% We need to design a filter here. 00:05:12.970 --> 00:05:14.890 align:middle line:84% A capacitor is not going to cut it 00:05:14.890 --> 00:05:18.370 align:middle line:84% because for typical specifications, who 00:05:18.370 --> 00:05:19.330 align:middle line:90% knows what this is? 00:05:19.330 --> 00:05:20.850 align:middle line:90% Maybe this is tens of amps. 00:05:20.850 --> 00:05:23.830 align:middle line:84% Typically, the current I'm allowed to squirt back into here 00:05:23.830 --> 00:05:26.270 align:middle line:84% is on the orders of tens of microamps. 00:05:26.270 --> 00:05:28.930 align:middle line:90% It's really small. 00:05:28.930 --> 00:05:33.640 align:middle line:84% So we usually need to design a higher-order filter than that. 00:05:33.640 --> 00:05:38.590 align:middle line:84% And so what we've thought about is building with a basic filter 00:05:38.590 --> 00:05:39.110 align:middle line:90% stage. 00:05:39.110 --> 00:05:42.460 align:middle line:84% So if here's my converter that's driving noise, 00:05:42.460 --> 00:05:44.050 align:middle line:84% and I'm going to call this ix tilde, 00:05:44.050 --> 00:05:47.470 align:middle line:84% and again, whatever this term by the converter, 00:05:47.470 --> 00:05:50.770 align:middle line:84% I will put this into some filter. 00:05:50.770 --> 00:06:01.150 align:middle line:84% And here maybe I would think of having some filter capacitor Cf 00:06:01.150 --> 00:06:05.470 align:middle line:84% and some filter inductance Lf as a basic building 00:06:05.470 --> 00:06:07.702 align:middle line:90% block for a filter. 00:06:07.702 --> 00:06:10.390 align:middle line:84% And the notion is, I can look back. 00:06:10.390 --> 00:06:12.610 align:middle line:84% And on the other side of this filter, 00:06:12.610 --> 00:06:14.920 align:middle line:84% depending upon whether I'm-- in my test stand, 00:06:14.920 --> 00:06:16.108 align:middle line:90% I'm going to have-- 00:06:16.108 --> 00:06:17.650 align:middle line:84% I'm going to look back into something 00:06:17.650 --> 00:06:19.930 align:middle line:84% that looks like Z LISN, which at high frequencies 00:06:19.930 --> 00:06:23.060 align:middle line:84% might be 50 ohms and at lower frequencies is something else. 00:06:23.060 --> 00:06:24.790 align:middle line:84% But let me just say, this goes back 00:06:24.790 --> 00:06:27.062 align:middle line:84% into some source impedance which for the moment, 00:06:27.062 --> 00:06:29.020 align:middle line:84% for our present discussion, let me just pretend 00:06:29.020 --> 00:06:30.970 align:middle line:90% it goes back into a short. 00:06:30.970 --> 00:06:36.890 align:middle line:84% And I'm interested in this current. 00:06:36.890 --> 00:06:39.680 align:middle line:84% I'm interested in this current iy 00:06:39.680 --> 00:06:43.130 align:middle line:84% tilde, the amount that's going back into the source, 00:06:43.130 --> 00:06:45.530 align:middle line:90% in terms of ix tilde. 00:06:45.530 --> 00:06:51.020 align:middle line:84% And we said, all right, if I wanted 00:06:51.020 --> 00:06:53.030 align:middle line:84% to think about the attenuation of this filter, 00:06:53.030 --> 00:07:00.920 align:middle line:84% and maybe I could think of this as being going back into Z LISN 00:07:00.920 --> 00:07:07.910 align:middle line:84% under some other conditions if I'm in the test stand. 00:07:07.910 --> 00:07:11.240 align:middle line:84% Maybe this filter block would give me 00:07:11.240 --> 00:07:16.670 align:middle line:84% an attenuation that would go the magnitude of iy over ix tilde 00:07:16.670 --> 00:07:19.220 align:middle line:90% might look something like this. 00:07:19.220 --> 00:07:21.820 align:middle line:84% At DC, everything gets passed back. 00:07:21.820 --> 00:07:24.404 align:middle line:90% 00:07:24.404 --> 00:07:29.990 align:middle line:84% And then well above 1 over the square root of Lf Cf 00:07:29.990 --> 00:07:38.400 align:middle line:84% it will roll off as 40 dB per decade, or minus 40 00:07:38.400 --> 00:07:40.770 align:middle line:90% dB per decade. 00:07:40.770 --> 00:07:41.460 align:middle line:90% Why? 00:07:41.460 --> 00:07:45.940 align:middle line:84% Because the capacitor impedance is falling by a factor of 10, 00:07:45.940 --> 00:07:47.670 align:middle line:90% every factor of 10 in frequency. 00:07:47.670 --> 00:07:50.310 align:middle line:84% And the inductance impedance is rising 00:07:50.310 --> 00:07:53.160 align:middle line:84% by a factor of every factor of 10 impedance, factor of 10 00:07:53.160 --> 00:07:55.300 align:middle line:84% for every factor of 10 in frequency. 00:07:55.300 --> 00:07:58.530 align:middle line:84% And hence so I get a factor of 100 attenuation for every factor 00:07:58.530 --> 00:08:01.560 align:middle line:90% of 10 in frequency. 00:08:01.560 --> 00:08:04.020 align:middle line:90% And that's great. 00:08:04.020 --> 00:08:07.860 align:middle line:84% And I'm intentionally ignoring what 00:08:07.860 --> 00:08:09.300 align:middle line:84% happens in the vicinity of cutoff, 00:08:09.300 --> 00:08:12.070 align:middle line:84% because keep in mind, the cutoff some low frequency, 00:08:12.070 --> 00:08:14.970 align:middle line:84% we're going to put our PWM frequency and ripple frequency 00:08:14.970 --> 00:08:17.800 align:middle line:84% way out here somewhere so we're getting lots of attenuation. 00:08:17.800 --> 00:08:20.396 align:middle line:90% 00:08:20.396 --> 00:08:23.190 align:middle line:84% But we have to be a little bit careful 00:08:23.190 --> 00:08:27.120 align:middle line:84% because we said when we lay these filters out, first of all, 00:08:27.120 --> 00:08:32.318 align:middle line:90% there are internal parasitics. 00:08:32.318 --> 00:08:35.559 align:middle line:84% So if I go buy a filter capacitor, he's not perfect. 00:08:35.559 --> 00:08:41.230 align:middle line:84% So maybe he has some internal ESR, RESR. 00:08:41.230 --> 00:08:44.772 align:middle line:84% And what that would do is that would reduce, instead of falling 00:08:44.772 --> 00:08:46.480 align:middle line:84% off like this, it would start to fall off 00:08:46.480 --> 00:08:52.120 align:middle line:84% at minus 20 dB per decade, because above some frequency, 00:08:52.120 --> 00:08:55.000 align:middle line:84% this shunt path just looks like a resistance. 00:08:55.000 --> 00:08:59.710 align:middle line:84% And then if I keep going up, and I have some parasitic inductance 00:08:59.710 --> 00:09:04.390 align:middle line:84% here, L parasitic, then I just have a current divider 00:09:04.390 --> 00:09:06.010 align:middle line:90% between Lf and Lp. 00:09:06.010 --> 00:09:09.670 align:middle line:84% And in fact, this might even flatten out. 00:09:09.670 --> 00:09:12.700 align:middle line:84% So I might be thinking I'm going to be way down here, 00:09:12.700 --> 00:09:16.630 align:middle line:84% but really my attenuation's up here, 00:09:16.630 --> 00:09:19.580 align:middle line:84% and I don't get nearly as good a performance as I hoped. 00:09:19.580 --> 00:09:24.760 align:middle line:84% So this could be due to internal component parasitics, 00:09:24.760 --> 00:09:27.790 align:middle line:84% or it could be due to layout parasitics-- 00:09:27.790 --> 00:09:29.950 align:middle line:84% how I place everything on the circuit board. 00:09:29.950 --> 00:09:31.880 align:middle line:84% Parasitic inductance and resistance 00:09:31.880 --> 00:09:33.120 align:middle line:90% is one thing I worry about. 00:09:33.120 --> 00:09:37.730 align:middle line:84% Parasitic capacitance is another thing I worry about. 00:09:37.730 --> 00:09:45.700 align:middle line:84% So that's just a quick review of last class. 00:09:45.700 --> 00:09:50.830 align:middle line:84% But I'd like to now move forward a little bit, And let's start 00:09:50.830 --> 00:09:53.650 align:middle line:90% thinking about-- 00:09:53.650 --> 00:09:55.840 align:middle line:84% let's start thinking about what happens 00:09:55.840 --> 00:09:58.700 align:middle line:90% in the vicinity of cutoff. 00:09:58.700 --> 00:10:01.630 align:middle line:84% So if my converter is in the hundreds of kilohertz, 00:10:01.630 --> 00:10:03.250 align:middle line:90% you know, maybe this is-- 00:10:03.250 --> 00:10:05.600 align:middle line:84% this cutoff is in the kilohertz range. 00:10:05.600 --> 00:10:07.490 align:middle line:90% It depends on the system. 00:10:07.490 --> 00:10:11.080 align:middle line:84% But at any rate, my cutoff of my filter 00:10:11.080 --> 00:10:14.450 align:middle line:84% is going to be well below the switching frequency. 00:10:14.450 --> 00:10:19.140 align:middle line:84% So let's think about what's going on there. 00:10:19.140 --> 00:10:24.990 align:middle line:84% If I have some converter, and it's 00:10:24.990 --> 00:10:28.003 align:middle line:84% drawing some ripple current, and then I put in my filter-- 00:10:28.003 --> 00:10:35.785 align:middle line:90% 00:10:35.785 --> 00:10:40.590 align:middle line:84% and I'm going to call this i sub y tilde-- 00:10:40.590 --> 00:10:48.700 align:middle line:84% let me just forget about the LISN impedance for a moment. 00:10:48.700 --> 00:10:50.530 align:middle line:84% What am I going to be looking back into? 00:10:50.530 --> 00:10:53.555 align:middle line:84% That depends what I'm going to put my converter into. 00:10:53.555 --> 00:10:55.680 align:middle line:84% The LISN is only there for testing. 00:10:55.680 --> 00:10:58.680 align:middle line:84% This converter and this filter has to live in the real world 00:10:58.680 --> 00:11:03.510 align:middle line:84% where this source impedance might even be 0, for all I know 00:11:03.510 --> 00:11:07.650 align:middle line:84% So what's going to happen in the vicinity of cutoff? 00:11:07.650 --> 00:11:09.100 align:middle line:90% Well, let's think about this. 00:11:09.100 --> 00:11:18.960 align:middle line:84% If I plotted iy over ix versus frequency, 00:11:18.960 --> 00:11:20.700 align:middle line:84% and this is all on a log scale, I 00:11:20.700 --> 00:11:24.690 align:middle line:84% should say, what do we say was going to happen? 00:11:24.690 --> 00:11:29.160 align:middle line:84% At low frequency, I'm going to get a transfer function of 1, 00:11:29.160 --> 00:11:33.995 align:middle line:84% and at high frequency I'm going to go minus 40 dB per decade. 00:11:33.995 --> 00:11:37.790 align:middle line:90% 00:11:37.790 --> 00:11:41.630 align:middle line:84% The question is, what happens here at cutoff? 00:11:41.630 --> 00:11:42.860 align:middle line:90% Where is cutoff? 00:11:42.860 --> 00:11:44.810 align:middle line:90% 1 over the square root of Lf Cf. 00:11:44.810 --> 00:11:49.910 align:middle line:90% 00:11:49.910 --> 00:11:55.650 align:middle line:84% And what do I look back into at exactly the cutoff frequency? 00:11:55.650 --> 00:11:58.710 align:middle line:90% 00:11:58.710 --> 00:12:01.800 align:middle line:84% I have an inductor in parallel with the capacitor 00:12:01.800 --> 00:12:05.540 align:middle line:90% at its resonant frequency. 00:12:05.540 --> 00:12:09.770 align:middle line:84% So while I get this roll-off at high frequency, 00:12:09.770 --> 00:12:14.780 align:middle line:84% and I get all the current paths passed back at DC, basically, 00:12:14.780 --> 00:12:17.780 align:middle line:84% at the resonance frequency, Lf and Cf basically 00:12:17.780 --> 00:12:20.180 align:middle line:90% look like an open circuit. 00:12:20.180 --> 00:12:22.070 align:middle line:84% Or what's going to happen is, if I 00:12:22.070 --> 00:12:25.610 align:middle line:84% start to draw any ripple current near the resonant frequency, 00:12:25.610 --> 00:12:28.370 align:middle line:84% I'm going to get huge circulating currents 00:12:28.370 --> 00:12:29.480 align:middle line:90% in the input. 00:12:29.480 --> 00:12:33.540 align:middle line:84% And I'm going to drive a big current iy back into my input. 00:12:33.540 --> 00:12:36.620 align:middle line:84% So the current gain here, actually, 00:12:36.620 --> 00:12:37.770 align:middle line:90% does something like this. 00:12:37.770 --> 00:12:43.821 align:middle line:84% It goes off to infinity in this simplified case. 00:12:43.821 --> 00:12:48.590 align:middle line:84% It would do something like this, which is pretty bad. 00:12:48.590 --> 00:12:52.010 align:middle line:84% If you imagine that the converter somehow 00:12:52.010 --> 00:12:55.160 align:middle line:84% had some small amount that it was drawing 00:12:55.160 --> 00:12:57.770 align:middle line:84% at some low frequency near this cutoff, 00:12:57.770 --> 00:12:59.995 align:middle line:84% it would drive this input filter nuts. 00:12:59.995 --> 00:13:02.520 align:middle line:90% 00:13:02.520 --> 00:13:05.760 align:middle line:84% Another thing I could look at is I could say, well, 00:13:05.760 --> 00:13:11.410 align:middle line:84% what is the output impedance of the filter look like? 00:13:11.410 --> 00:13:13.840 align:middle line:84% In other words, the filter seen by the converter. 00:13:13.840 --> 00:13:17.130 align:middle line:84% If I look back into this filter, and I say, 00:13:17.130 --> 00:13:20.040 align:middle line:84% what is this impedance Zx, which is 00:13:20.040 --> 00:13:28.270 align:middle line:84% equal to Vx tilde over ix tilde, where this is Vx tilde. 00:13:28.270 --> 00:13:30.820 align:middle line:90% What does that look like? 00:13:30.820 --> 00:13:41.680 align:middle line:84% Well, I would argue in this case that at low frequencies, 00:13:41.680 --> 00:13:43.750 align:middle line:84% this is close to a short circuit, 00:13:43.750 --> 00:13:45.820 align:middle line:90% so it looks like omega Lf. 00:13:45.820 --> 00:13:48.370 align:middle line:84% And at high frequencies, this starts 00:13:48.370 --> 00:13:51.220 align:middle line:84% to look like a short circuit, so it becomes 1 over omega Cf. 00:13:51.220 --> 00:13:52.940 align:middle line:84% And you get the following behavior. 00:13:52.940 --> 00:14:06.860 align:middle line:84% I get omega Lf, I get 1 over omega Cf, 00:14:06.860 --> 00:14:09.600 align:middle line:84% and right at this cutoff frequency 00:14:09.600 --> 00:14:11.240 align:middle line:84% these two are in parallel and these are 00:14:11.240 --> 00:14:13.200 align:middle line:90% resonating against each other. 00:14:13.200 --> 00:14:16.220 align:middle line:90% One has a positive admittance. 00:14:16.220 --> 00:14:18.990 align:middle line:84% The other has a negative admittance, and they cancel. 00:14:18.990 --> 00:14:23.540 align:middle line:84% And this impedance would go off to infinity. 00:14:23.540 --> 00:14:25.740 align:middle line:84% So you're looking back into an open circuit. 00:14:25.740 --> 00:14:29.012 align:middle line:84% That makes sense why this draws kind of high current. 00:14:29.012 --> 00:14:31.220 align:middle line:84% You're trying to draw a current from an open circuit, 00:14:31.220 --> 00:14:33.512 align:middle line:84% and so you're going to get a lot of voltage ripple here 00:14:33.512 --> 00:14:37.000 align:middle line:84% and a lot of resulting current ripple back into the input. 00:14:37.000 --> 00:14:38.940 align:middle line:90% Any questions about that? 00:14:38.940 --> 00:14:43.570 align:middle line:90% 00:14:43.570 --> 00:14:45.470 align:middle line:90% So what would I do about that? 00:14:45.470 --> 00:14:48.030 align:middle line:84% What would be the kind of natural thing to do? 00:14:48.030 --> 00:14:52.210 align:middle line:90% 00:14:52.210 --> 00:14:55.816 align:middle line:84% Better put a damping leg on your filter. 00:14:55.816 --> 00:15:01.180 align:middle line:84% So while we're using this filter way out here to cut off ripple 00:15:01.180 --> 00:15:03.760 align:middle line:84% from the converter, I also have to think about what 00:15:03.760 --> 00:15:05.410 align:middle line:90% it's going to do here. 00:15:05.410 --> 00:15:07.360 align:middle line:84% Now the most natural thing to think 00:15:07.360 --> 00:15:09.310 align:middle line:84% about doing, putting in a filter, 00:15:09.310 --> 00:15:13.930 align:middle line:84% is maybe just putting a resistor-- 00:15:13.930 --> 00:15:15.500 align:middle line:90% maybe I could put a resistor-- 00:15:15.500 --> 00:15:16.960 align:middle line:84% I'll call it a damping resistor-- 00:15:16.960 --> 00:15:19.690 align:middle line:90% Rd-- right here. 00:15:19.690 --> 00:15:21.500 align:middle line:90% What would Rd do? 00:15:21.500 --> 00:15:25.730 align:middle line:84% Well if Lf and Cf are at resonance, 00:15:25.730 --> 00:15:29.395 align:middle line:84% Rd is the only thing left, because these kind 00:15:29.395 --> 00:15:30.270 align:middle line:90% of cancel each other. 00:15:30.270 --> 00:15:32.580 align:middle line:84% And my output impedance would just go to Rd. 00:15:32.580 --> 00:15:34.890 align:middle line:84% So instead of peaking off to infinity, 00:15:34.890 --> 00:15:40.260 align:middle line:84% I would peak off to some value Rd, 00:15:40.260 --> 00:15:43.680 align:middle line:84% would be the maximum of my output impedance if I did this. 00:15:43.680 --> 00:15:48.540 align:middle line:84% And likewise, if I looked at the peaking in my transfer function, 00:15:48.540 --> 00:15:51.700 align:middle line:84% it would be limited by the fact that I was damping this. 00:15:51.700 --> 00:15:54.420 align:middle line:84% And you know, maybe it does something like this. 00:15:54.420 --> 00:15:57.240 align:middle line:90% And I'm no longer unhappy. 00:15:57.240 --> 00:16:02.950 align:middle line:90% Any questions about that? 00:16:02.950 --> 00:16:04.695 align:middle line:84% Anybody see a problem with that scheme? 00:16:04.695 --> 00:16:08.490 align:middle line:90% 00:16:08.490 --> 00:16:11.280 align:middle line:84% Yeah, you're going to burn a lot of energy in this. 00:16:11.280 --> 00:16:15.270 align:middle line:84% I mean, if I come back to my original converter, 00:16:15.270 --> 00:16:20.280 align:middle line:84% I need to make Rd, typically, if I want this to be nicely damped, 00:16:20.280 --> 00:16:23.460 align:middle line:84% on the order of square root of Lf over Cf 00:16:23.460 --> 00:16:26.190 align:middle line:90% or lower for good damping. 00:16:26.190 --> 00:16:30.105 align:middle line:84% And I'm basically putting a resistor 00:16:30.105 --> 00:16:31.480 align:middle line:84% across the input of my converter. 00:16:31.480 --> 00:16:32.460 align:middle line:90% That's not very happy. 00:16:32.460 --> 00:16:35.500 align:middle line:90% 00:16:35.500 --> 00:16:39.030 align:middle line:84% So what would I do in the real world? 00:16:39.030 --> 00:16:42.410 align:middle line:84% I'd say, well, I can't accept that being a very good idea. 00:16:42.410 --> 00:16:48.200 align:middle line:84% But maybe what I could do is take this damping resistor 00:16:48.200 --> 00:16:51.290 align:middle line:84% and put it in series with a damping capacitor, Cd. 00:16:51.290 --> 00:16:53.840 align:middle line:90% 00:16:53.840 --> 00:16:54.410 align:middle line:90% OK. 00:16:54.410 --> 00:16:58.310 align:middle line:84% So I'll put Rd in series with Cd. 00:16:58.310 --> 00:16:59.490 align:middle line:90% OK, what does that do? 00:16:59.490 --> 00:17:06.589 align:middle line:84% Well, if Cd is bigger than C sub f near the cutoff, 1 00:17:06.589 --> 00:17:10.040 align:middle line:84% over the square root of Lf Cf, Cd looks like a short circuit, 00:17:10.040 --> 00:17:14.839 align:middle line:84% and it looks just like Rd is in shunt. 00:17:14.839 --> 00:17:18.113 align:middle line:84% And so I'll get this nice kind of damping. 00:17:18.113 --> 00:17:21.670 align:middle line:90% 00:17:21.670 --> 00:17:25.599 align:middle line:84% But a DC, Cd is an open circuit, so I don't get any DC current 00:17:25.599 --> 00:17:26.500 align:middle line:90% through here. 00:17:26.500 --> 00:17:28.820 align:middle line:90% And so that helps me. 00:17:28.820 --> 00:17:32.530 align:middle line:84% OK, so this Rd Cd branch is sometimes 00:17:32.530 --> 00:17:36.160 align:middle line:84% called the damping leg, And it can really 00:17:36.160 --> 00:17:39.860 align:middle line:84% do a nice job of improving your filter design. 00:17:39.860 --> 00:17:42.507 align:middle line:84% Now typically, Cd depends-- and we're 00:17:42.507 --> 00:17:44.215 align:middle line:84% going to talk about the details of this-- 00:17:44.215 --> 00:17:48.610 align:middle line:84% a typical number for Cd is usually on the order of 2 to 10 00:17:48.610 --> 00:17:53.680 align:middle line:84% times Cf, or two or greater times Cf in order 00:17:53.680 --> 00:17:55.130 align:middle line:90% to get good enough damping. 00:17:55.130 --> 00:17:56.930 align:middle line:84% But we'll get into the details of that. 00:17:56.930 --> 00:17:58.790 align:middle line:84% But before I do that, any questions? 00:17:58.790 --> 00:18:04.510 align:middle line:90% 00:18:04.510 --> 00:18:09.510 align:middle line:84% So let's think about what would happen if I put 00:18:09.510 --> 00:18:12.330 align:middle line:90% this damping branch in here. 00:18:12.330 --> 00:18:16.040 align:middle line:90% How do I pick Rd. 00:18:16.040 --> 00:18:18.650 align:middle line:84% Or another question I could ask is, 00:18:18.650 --> 00:18:21.710 align:middle line:84% what does the resonance of this filter look 00:18:21.710 --> 00:18:25.170 align:middle line:84% like as I vary Rd in this system? 00:18:25.170 --> 00:18:29.240 align:middle line:90% So let's take a look at that. 00:18:29.240 --> 00:18:30.655 align:middle line:90% I'm going to replot it here. 00:18:30.655 --> 00:18:34.055 align:middle line:90% 00:18:34.055 --> 00:18:41.802 align:middle line:84% And I'm going to look at Zx, the filter output impedance. 00:18:41.802 --> 00:18:43.260 align:middle line:84% And we're going to come back to why 00:18:43.260 --> 00:18:46.290 align:middle line:84% I'm obsessed with what the filter output impedance looks 00:18:46.290 --> 00:18:46.860 align:middle line:90% like. 00:18:46.860 --> 00:18:50.130 align:middle line:84% And that's the impedance seen looking back, 00:18:50.130 --> 00:18:51.960 align:middle line:84% what the converter looks back and sees. 00:18:51.960 --> 00:18:53.520 align:middle line:84% And in doing my analysis here, I'm 00:18:53.520 --> 00:18:55.740 align:middle line:84% just assuming that the source impedance out here 00:18:55.740 --> 00:18:57.090 align:middle line:90% looks like a short circuit. 00:18:57.090 --> 00:18:59.130 align:middle line:84% I'm making that simplifying assumption. 00:18:59.130 --> 00:19:01.920 align:middle line:90% But let's think about this. 00:19:01.920 --> 00:19:09.490 align:middle line:84% I said in the undamped case, what would I have seen? 00:19:09.490 --> 00:19:11.220 align:middle line:84% I would have seen something like this-- 00:19:11.220 --> 00:19:13.864 align:middle line:90% 1 over the square root of Lf Cf. 00:19:13.864 --> 00:19:17.860 align:middle line:90% 00:19:17.860 --> 00:19:21.290 align:middle line:90% This is omega Cf. 00:19:21.290 --> 00:19:21.790 align:middle line:90% Oops. 00:19:21.790 --> 00:19:22.290 align:middle line:90% I'm sorry. 00:19:22.290 --> 00:19:24.850 align:middle line:90% This is omega Lf. 00:19:24.850 --> 00:19:33.190 align:middle line:84% And then this is 1 over omega Cf. 00:19:33.190 --> 00:19:37.170 align:middle line:84% And this is versus frequency on a log log plot. 00:19:37.170 --> 00:19:41.374 align:middle line:90% 00:19:41.374 --> 00:19:47.470 align:middle line:84% And this is the magnitude of Zx, looking back there. 00:19:47.470 --> 00:19:53.110 align:middle line:84% If I let Rd go to infinity, it's basically 00:19:53.110 --> 00:19:54.600 align:middle line:90% like Cd is not in the circuit. 00:19:54.600 --> 00:19:57.580 align:middle line:90% 00:19:57.580 --> 00:20:01.303 align:middle line:84% So as Rd goes to infinity, what I'm going to see 00:20:01.303 --> 00:20:02.720 align:middle line:84% is something that looks like this. 00:20:02.720 --> 00:20:05.000 align:middle line:84% This is the case where Rd goes to infinity. 00:20:05.000 --> 00:20:07.365 align:middle line:84% This is just going to do what the undamped system did. 00:20:07.365 --> 00:20:15.750 align:middle line:90% 00:20:15.750 --> 00:20:19.710 align:middle line:84% So this is the case where Rd goes to infinity. 00:20:19.710 --> 00:20:23.367 align:middle line:84% So, clearly, I can't make Rd too big, or the-- 00:20:23.367 --> 00:20:25.492 align:middle line:84% even putting in the damping doesn't do me any good. 00:20:25.492 --> 00:20:28.060 align:middle line:90% 00:20:28.060 --> 00:20:31.210 align:middle line:90% Well, what happens if I make-- 00:20:31.210 --> 00:20:33.530 align:middle line:84% so, clearly, I need to make Rd small. 00:20:33.530 --> 00:20:36.170 align:middle line:84% What happens if I make Rd super small? 00:20:36.170 --> 00:20:38.510 align:middle line:84% Well, if I make Rd go to 0, what happens? 00:20:38.510 --> 00:20:43.010 align:middle line:84% If Rd was 0, i would just have Cd in parallel with Cf. 00:20:43.010 --> 00:20:46.670 align:middle line:84% So if I made Rd go to 0, what I would see 00:20:46.670 --> 00:20:49.760 align:middle line:90% is something like this. 00:20:49.760 --> 00:20:53.420 align:middle line:84% Instead of having a cut off 1 over omega Lf Cf, 00:20:53.420 --> 00:20:55.580 align:middle line:84% I'm just putting Cd in parallel with Cf 00:20:55.580 --> 00:20:57.110 align:middle line:90% so I'd have some other place-- 00:20:57.110 --> 00:20:58.080 align:middle line:90% here it is-- 00:20:58.080 --> 00:21:05.270 align:middle line:84% 1 over the square root of Lf times Cf plus Cd. 00:21:05.270 --> 00:21:07.190 align:middle line:90% And I'd have another cutoff. 00:21:07.190 --> 00:21:10.898 align:middle line:90% 00:21:10.898 --> 00:21:15.150 align:middle line:84% Actually, let me put that a little bit over to the right. 00:21:15.150 --> 00:21:17.850 align:middle line:84% Let me put it here just so you can see it more easily. 00:21:17.850 --> 00:21:19.690 align:middle line:84% So I'm going to put that cutoff right here. 00:21:19.690 --> 00:21:22.480 align:middle line:90% 00:21:22.480 --> 00:21:26.993 align:middle line:84% And what I would get is the same-- 00:21:26.993 --> 00:21:29.550 align:middle line:90% 00:21:29.550 --> 00:21:32.916 align:middle line:84% I would get a different high-frequency asymptote. 00:21:32.916 --> 00:21:38.210 align:middle line:90% 00:21:38.210 --> 00:21:41.900 align:middle line:84% Actually, let me put it right here. 00:21:41.900 --> 00:21:44.000 align:middle line:90% Be accurate. 00:21:44.000 --> 00:21:52.680 align:middle line:84% OK, so this would be 1 over omega Cf plus Cd. 00:21:52.680 --> 00:21:54.740 align:middle line:84% And I would peak up, and I'd get something that 00:21:54.740 --> 00:21:56.305 align:middle line:90% went off to infinity like this. 00:21:56.305 --> 00:22:03.690 align:middle line:90% 00:22:03.690 --> 00:22:10.830 align:middle line:84% So this is the case where Rd goes to 0 for some value of Cd 00:22:10.830 --> 00:22:12.860 align:middle line:90% that I've picked. 00:22:12.860 --> 00:22:15.510 align:middle line:84% Does that make sense to everybody? 00:22:15.510 --> 00:22:18.270 align:middle line:90% So this is a Goldilocks thing. 00:22:18.270 --> 00:22:21.510 align:middle line:84% If you make your damping resistor too big, 00:22:21.510 --> 00:22:23.710 align:middle line:90% you get bad damping. 00:22:23.710 --> 00:22:26.620 align:middle line:84% If you make your damping resistor too small, 00:22:26.620 --> 00:22:28.930 align:middle line:90% you get terrible damping. 00:22:28.930 --> 00:22:31.510 align:middle line:84% And you got to, for a given value of Cd, 00:22:31.510 --> 00:22:36.990 align:middle line:84% you've got to pick the right damping resistance. 00:22:36.990 --> 00:22:38.718 align:middle line:84% What would my output impedance look 00:22:38.718 --> 00:22:40.260 align:middle line:84% like if I picked the right one, where 00:22:40.260 --> 00:22:45.060 align:middle line:84% you can show that the best case, through a nice bit of analysis, 00:22:45.060 --> 00:22:48.170 align:middle line:84% actually goes through these two points. 00:22:48.170 --> 00:22:52.320 align:middle line:84% So if you picked Rd optimum, what you would get 00:22:52.320 --> 00:22:53.660 align:middle line:90% is something that did this. 00:22:53.660 --> 00:22:59.180 align:middle line:90% 00:22:59.180 --> 00:23:02.770 align:middle line:84% And then it would eventually fall off like this. 00:23:02.770 --> 00:23:04.220 align:middle line:90% So that's a pretty-- 00:23:04.220 --> 00:23:05.910 align:middle line:90% kind of crummy drawing. 00:23:05.910 --> 00:23:12.450 align:middle line:84% But this is sort of best Rd, the best 00:23:12.450 --> 00:23:15.780 align:middle line:84% Rd in terms of limiting how big the output impedance gets, 00:23:15.780 --> 00:23:18.840 align:middle line:84% which we could take as some kind of measure 00:23:18.840 --> 00:23:22.950 align:middle line:84% for how badly damped the filter is. 00:23:22.950 --> 00:23:26.280 align:middle line:90% Questions about that? 00:23:26.280 --> 00:23:30.690 align:middle line:84% So what do I do if I don't like that value of output resistance 00:23:30.690 --> 00:23:32.380 align:middle line:90% that I get? 00:23:32.380 --> 00:23:39.760 align:middle line:84% Well, if I make Cd bigger, I can slide this down, 00:23:39.760 --> 00:23:44.020 align:middle line:84% and then I'll pick a different value, which will then bring 00:23:44.020 --> 00:23:46.870 align:middle line:90% this intersection point down. 00:23:46.870 --> 00:23:49.630 align:middle line:90% And we'll get better damping. 00:23:49.630 --> 00:23:51.470 align:middle line:90% So I'm not going to derive this. 00:23:51.470 --> 00:23:56.290 align:middle line:84% This is actually nicely derived in one of the handouts 00:23:56.290 --> 00:23:58.270 align:middle line:90% that I've provided. 00:23:58.270 --> 00:24:04.930 align:middle line:84% But here's the final result. If I define R out as being equal 00:24:04.930 --> 00:24:09.160 align:middle line:84% to square root of Lf over Cf-- that's the characteristic 00:24:09.160 --> 00:24:13.320 align:middle line:84% impedance of my original Lf Cf branch-- 00:24:13.320 --> 00:24:20.110 align:middle line:84% and I define n as being equal to Cd over Cf-- 00:24:20.110 --> 00:24:24.330 align:middle line:84% so that's how much bigger is the damping capacitor compared 00:24:24.330 --> 00:24:25.880 align:middle line:90% to the filter capacitor-- 00:24:25.880 --> 00:24:39.510 align:middle line:84% what you can see is that Rd opt is equal to R0 times square root 00:24:39.510 --> 00:24:50.190 align:middle line:84% of 2 plus n times 4 plus 3n divided by 2n squared times 4 00:24:50.190 --> 00:24:51.310 align:middle line:90% plus n. 00:24:51.310 --> 00:24:58.350 align:middle line:90% And this gives a Zx max-- 00:24:58.350 --> 00:25:00.395 align:middle line:90% if I call this Zx max-- 00:25:00.395 --> 00:25:05.670 align:middle line:90% 00:25:05.670 --> 00:25:14.820 align:middle line:84% a value that's equal to R0 times square root of 2 plus n over n. 00:25:14.820 --> 00:25:23.640 align:middle line:84% So that means, the bigger I choose n, this goes down, 00:25:23.640 --> 00:25:29.620 align:middle line:84% and my peak output impedance of the filter goes down. 00:25:29.620 --> 00:25:30.140 align:middle line:90% All right? 00:25:30.140 --> 00:25:33.320 align:middle line:84% And I get essentially better damping. 00:25:33.320 --> 00:25:33.820 align:middle line:90% OK. 00:25:33.820 --> 00:25:34.430 align:middle line:90% Larry. 00:25:34.430 --> 00:25:38.255 align:middle line:84% AUDIENCE: Your notes say 2 times 2 plus n. 00:25:38.255 --> 00:25:40.380 align:middle line:84% DAVID PERREAULT: Whatever my notes say are correct. 00:25:40.380 --> 00:25:41.650 align:middle line:90% 2 times 2 plus n. 00:25:41.650 --> 00:25:43.500 align:middle line:90% Yes. 00:25:43.500 --> 00:25:47.220 align:middle line:84% Now by the way, these are nice equations. 00:25:47.220 --> 00:25:49.505 align:middle line:84% There's a very clever analysis that shows all this. 00:25:49.505 --> 00:25:54.545 align:middle line:90% 00:25:54.545 --> 00:25:56.787 align:middle line:84% And certainly, this is well known, 00:25:56.787 --> 00:25:58.120 align:middle line:90% and it's in one of the handouts. 00:25:58.120 --> 00:26:01.170 align:middle line:90% 00:26:01.170 --> 00:26:08.010 align:middle line:84% The nice thing about this is that you can say, well, 00:26:08.010 --> 00:26:12.040 align:middle line:84% how small do I or how well the damp do I want this thing, 00:26:12.040 --> 00:26:14.760 align:middle line:84% I will pick n big enough to give me that, 00:26:14.760 --> 00:26:17.580 align:middle line:84% and then that gives me my value of damping capacitor, 00:26:17.580 --> 00:26:21.480 align:middle line:84% and away I go, and I get enough, make my output impedance 00:26:21.480 --> 00:26:22.900 align:middle line:90% small or my system well damped. 00:26:22.900 --> 00:26:25.980 align:middle line:90% 00:26:25.980 --> 00:26:29.010 align:middle line:84% Now the thing we don't like about doing 00:26:29.010 --> 00:26:32.130 align:middle line:84% this is you say, well, why don't I just make n really big, 00:26:32.130 --> 00:26:34.060 align:middle line:84% and then this will be really well damped. 00:26:34.060 --> 00:26:38.250 align:middle line:84% Well, the reason is, I've got to pay for that capacitor. 00:26:38.250 --> 00:26:41.640 align:middle line:84% If I'm choosing Cd as 10 times Cf, 00:26:41.640 --> 00:26:46.050 align:middle line:84% well, that's expensive, in terms of cost 00:26:46.050 --> 00:26:47.880 align:middle line:84% but also in terms of volume, because it 00:26:47.880 --> 00:26:51.480 align:middle line:84% has to store 10 times the energy of Cf, right? 00:26:51.480 --> 00:26:54.660 align:middle line:84% On the other hand, Cd and Cf, if I come back 00:26:54.660 --> 00:26:57.990 align:middle line:84% to this original design, Cf is important 00:26:57.990 --> 00:27:00.810 align:middle line:84% at the ripple frequency We were talking about-- 00:27:00.810 --> 00:27:03.720 align:middle line:84% up here, we were talking about, well, I really 00:27:03.720 --> 00:27:06.450 align:middle line:84% care about the parasitics of this filter capacitor 00:27:06.450 --> 00:27:09.340 align:middle line:84% in order to get good, high-frequency attenuation. 00:27:09.340 --> 00:27:12.690 align:middle line:84% So Cf's job, the filter capacitor's job, 00:27:12.690 --> 00:27:15.380 align:middle line:84% is really to help me attenuate the high ripple frequency. 00:27:15.380 --> 00:27:19.530 align:middle line:84% He wants to be a really good capacitor for high frequency. 00:27:19.530 --> 00:27:25.470 align:middle line:84% On the other hand, Cd is really operative near the cutoff, which 00:27:25.470 --> 00:27:27.520 align:middle line:90% is orders of magnitude lower. 00:27:27.520 --> 00:27:29.740 align:middle line:84% So Cd doesn't have to be a great capacitor. 00:27:29.740 --> 00:27:33.430 align:middle line:84% In fact, if Cd had parasitic inductance, 00:27:33.430 --> 00:27:34.960 align:middle line:90% I probably don't care that much. 00:27:34.960 --> 00:27:36.932 align:middle line:84% In fact, if he has parasitic resistance, 00:27:36.932 --> 00:27:38.640 align:middle line:84% I still may not care that much, because I 00:27:38.640 --> 00:27:41.730 align:middle line:84% can use that parasitic resistance as part of my filter 00:27:41.730 --> 00:27:42.730 align:middle line:90% damping leg. 00:27:42.730 --> 00:27:48.210 align:middle line:84% So this might be, for example, an electrolytic capacitor, 00:27:48.210 --> 00:27:51.550 align:middle line:84% whereas this might be film or ceramic capacitor. 00:27:51.550 --> 00:27:54.180 align:middle line:84% So the jobs of these two capacitors are different, 00:27:54.180 --> 00:27:55.570 align:middle line:90% and the construction of them. 00:27:55.570 --> 00:27:57.240 align:middle line:84% And you can get a lot of energy storage 00:27:57.240 --> 00:28:01.020 align:middle line:84% in electrolytic, for example, as compared to a ceramic or a film 00:28:01.020 --> 00:28:03.860 align:middle line:90% capacitor typically. 00:28:03.860 --> 00:28:05.240 align:middle line:90% Any questions about that? 00:28:05.240 --> 00:28:11.130 align:middle line:90% 00:28:11.130 --> 00:28:11.630 align:middle line:90% All right. 00:28:11.630 --> 00:28:17.080 align:middle line:84% So what we're going to do is we're going to say, jeez, 00:28:17.080 --> 00:28:18.850 align:middle line:84% how well damped do I want this to be? 00:28:18.850 --> 00:28:21.820 align:middle line:84% First of all, design my filter Lf 00:28:21.820 --> 00:28:25.310 align:middle line:84% and Cf to attenuate the ripple frequency. 00:28:25.310 --> 00:28:28.940 align:middle line:84% Then I'll say, how well damped do I need 00:28:28.940 --> 00:28:30.920 align:middle line:90% this to be down at the cutoff? 00:28:30.920 --> 00:28:35.740 align:middle line:84% And I will say, OK, I will pick n to be big enough, 00:28:35.740 --> 00:28:38.420 align:middle line:84% the ratio of Cd over Cf to be big enough, 00:28:38.420 --> 00:28:40.550 align:middle line:84% and then out of that will-- out of this will 00:28:40.550 --> 00:28:43.593 align:middle line:84% drop what filter damping resistor I need. 00:28:43.593 --> 00:28:45.010 align:middle line:84% And then we kind of have a filter. 00:28:45.010 --> 00:28:48.660 align:middle line:90% 00:28:48.660 --> 00:28:53.430 align:middle line:84% By the way, if you don't like remembering these equations, 00:28:53.430 --> 00:28:57.065 align:middle line:84% it's not too hard to simulate or just calculate for a few values, 00:28:57.065 --> 00:28:58.390 align:middle line:90% and then you'd zero in on it. 00:28:58.390 --> 00:29:00.333 align:middle line:90% It's just the Goldilocks thing. 00:29:00.333 --> 00:29:03.000 align:middle line:84% It's too hot, it's too cold, you find the one that's just right, 00:29:03.000 --> 00:29:03.917 align:middle line:90% and then you're there. 00:29:03.917 --> 00:29:08.770 align:middle line:90% 00:29:08.770 --> 00:29:14.770 align:middle line:84% I should say, this isn't the only way to damp a filter. 00:29:14.770 --> 00:29:18.490 align:middle line:90% If I thought of a-- 00:29:18.490 --> 00:29:24.840 align:middle line:84% if I thought of a different damping mechanism, right, 00:29:24.840 --> 00:29:28.010 align:middle line:84% I could come back to the same basic filter structure. 00:29:28.010 --> 00:29:31.310 align:middle line:90% Here's Cf. 00:29:31.310 --> 00:29:31.810 align:middle line:90% OK. 00:29:31.810 --> 00:29:34.920 align:middle line:90% Here's Lf. 00:29:34.920 --> 00:29:37.680 align:middle line:84% And he's going to be coming back into the source impedance, which 00:29:37.680 --> 00:29:39.510 align:middle line:84% are just presented as a short circuit. 00:29:39.510 --> 00:29:44.380 align:middle line:84% And he's going to be driven by some converter ripple ix tilde. 00:29:44.380 --> 00:29:47.250 align:middle line:84% And I want to worry about what iy tilde here 00:29:47.250 --> 00:29:50.490 align:middle line:84% is, this current gain of the filter. 00:29:50.490 --> 00:29:52.710 align:middle line:84% I could come back to this and say, you know what? 00:29:52.710 --> 00:29:55.650 align:middle line:84% Instead of using a shunt damping leg like that, 00:29:55.650 --> 00:29:58.650 align:middle line:84% in parallel with Cf, maybe I could 00:29:58.650 --> 00:30:01.810 align:middle line:90% go get a damping leg like this. 00:30:01.810 --> 00:30:08.460 align:middle line:90% I could have Ld and Rd here. 00:30:08.460 --> 00:30:10.590 align:middle line:84% To the extent that this is a short circuit, 00:30:10.590 --> 00:30:14.950 align:middle line:84% this now also gives me parallel damping. 00:30:14.950 --> 00:30:20.900 align:middle line:84% In this case, I would make, if this is Lf, in this case, 00:30:20.900 --> 00:30:27.310 align:middle line:84% I would make Ld less than Lf, so that nearer the cutoff 00:30:27.310 --> 00:30:30.070 align:middle line:84% frequency, 1 over the square root of Lf Cf, 00:30:30.070 --> 00:30:32.260 align:middle line:84% Ld now looks like a short circuit, 00:30:32.260 --> 00:30:38.770 align:middle line:84% and it's Rd that's damping Lf and Cf. 00:30:38.770 --> 00:30:41.050 align:middle line:84% When I get to high enough frequencies, 00:30:41.050 --> 00:30:42.403 align:middle line:90% Ld starts to look inductive. 00:30:42.403 --> 00:30:45.070 align:middle line:84% It looks like a higher impedance than Rd, and I get my roll-off. 00:30:45.070 --> 00:30:48.520 align:middle line:90% 00:30:48.520 --> 00:30:52.600 align:middle line:84% Now this is also a very popular damping scheme. 00:30:52.600 --> 00:30:56.615 align:middle line:84% And you can do the same thing for this analysis. 00:30:56.615 --> 00:30:58.740 align:middle line:84% There's always going to be, for a given value of Ld 00:30:58.740 --> 00:31:01.470 align:middle line:84% relative to Lf, there's an optimum value of Rd 00:31:01.470 --> 00:31:03.350 align:middle line:90% to give you a good damping. 00:31:03.350 --> 00:31:08.120 align:middle line:84% The downside to this filter-damping strategy 00:31:08.120 --> 00:31:17.360 align:middle line:84% is that, whereas this additional shunt path only kind of improved 00:31:17.360 --> 00:31:21.080 align:middle line:84% your filtering well above cutoff, here, 00:31:21.080 --> 00:31:27.110 align:middle line:84% because Ld is small, you're kind of hurting your filter impedance 00:31:27.110 --> 00:31:30.020 align:middle line:84% This path's impedance gone down compared 00:31:30.020 --> 00:31:31.140 align:middle line:90% to this path's impedance. 00:31:31.140 --> 00:31:37.760 align:middle line:84% So this doesn't do as good a job slightly above cutoff 00:31:37.760 --> 00:31:39.480 align:middle line:90% because Ld is small. 00:31:39.480 --> 00:31:44.510 align:middle line:84% So what will happen is, instead of getting-- 00:31:44.510 --> 00:31:53.630 align:middle line:84% if I looked at iy over ix, iy over ix, instead of cutting off 00:31:53.630 --> 00:31:56.930 align:middle line:84% if minus 40 dB per decade, what's going to happen is I'm 00:31:56.930 --> 00:32:00.470 align:middle line:84% going to cut off at minus 20 dB per decade 00:32:00.470 --> 00:32:03.110 align:middle line:84% because all I have is this damping resistor being 00:32:03.110 --> 00:32:03.930 align:middle line:90% important. 00:32:03.930 --> 00:32:07.310 align:middle line:84% And then, once Ld comes in, then I'll start falling off, 00:32:07.310 --> 00:32:10.890 align:middle line:84% and I will eventually roll off at minus 40 dB per decade. 00:32:10.890 --> 00:32:13.100 align:middle line:84% But my roll-off is not going to be as good. 00:32:13.100 --> 00:32:14.310 align:middle line:90% So what do I have to do? 00:32:14.310 --> 00:32:17.420 align:middle line:84% I have to pick a correspondingly lower cutoff frequency 00:32:17.420 --> 00:32:21.410 align:middle line:84% to account for what Ld is doing to my filtering. 00:32:21.410 --> 00:32:25.440 align:middle line:84% So that's the downside of this inductive damping scheme. 00:32:25.440 --> 00:32:27.890 align:middle line:84% The upside to the inductive damping scheme 00:32:27.890 --> 00:32:33.640 align:middle line:84% is that whereas Cd costs me a lot of energy storage 00:32:33.640 --> 00:32:37.150 align:middle line:84% because he's got to block the full input voltage, 00:32:37.150 --> 00:32:39.550 align:middle line:84% Ld is small and carries no DC current, 00:32:39.550 --> 00:32:42.040 align:middle line:84% so it can be a pretty small inductor. 00:32:42.040 --> 00:32:44.600 align:middle line:84% So they both have their upsides and downsides. 00:32:44.600 --> 00:32:46.310 align:middle line:90% They're both widely used. 00:32:46.310 --> 00:32:47.920 align:middle line:90% Just depends on the situation. 00:32:47.920 --> 00:32:49.030 align:middle line:90% Try them both out. 00:32:49.030 --> 00:32:50.710 align:middle line:84% In one of the handouts that I provided, 00:32:50.710 --> 00:32:53.810 align:middle line:84% I have actually shown all kinds of different filters, 00:32:53.810 --> 00:32:57.670 align:middle line:84% some higher-order versions than this. 00:32:57.670 --> 00:33:01.600 align:middle line:84% But typically, an Lc filter as a building block 00:33:01.600 --> 00:33:03.050 align:middle line:90% is a pretty good structure. 00:33:03.050 --> 00:33:06.045 align:middle line:90% 00:33:06.045 --> 00:33:09.740 align:middle line:90% Any questions about that? 00:33:09.740 --> 00:33:10.790 align:middle line:90% Larry. 00:33:10.790 --> 00:33:14.160 align:middle line:84% AUDIENCE: Do we have to consider any of this at the power stage? 00:33:14.160 --> 00:33:15.868 align:middle line:84% Because you also have notes saying-- 00:33:15.868 --> 00:33:16.910 align:middle line:90% DAVID PERREAULT: Oh, yes. 00:33:16.910 --> 00:33:18.237 align:middle line:90% Well, yes. 00:33:18.237 --> 00:33:20.070 align:middle line:84% Actually, let me answer that exact question. 00:33:20.070 --> 00:33:23.240 align:middle line:90% The answer is yes, you do. 00:33:23.240 --> 00:33:26.720 align:middle line:90% Why was I talking-- 00:33:26.720 --> 00:33:29.010 align:middle line:84% I kind of focused in this picture. 00:33:29.010 --> 00:33:32.000 align:middle line:84% I focused on what the output impedance of the filter 00:33:32.000 --> 00:33:33.330 align:middle line:90% looks like. 00:33:33.330 --> 00:33:38.460 align:middle line:84% Well, let's think about this for a second. 00:33:38.460 --> 00:33:40.775 align:middle line:90% Suppose I have some power stage. 00:33:40.775 --> 00:33:43.850 align:middle line:90% 00:33:43.850 --> 00:33:45.600 align:middle line:90% Here's my power stage. 00:33:45.600 --> 00:33:47.130 align:middle line:90% This is my power converter. 00:33:47.130 --> 00:33:52.040 align:middle line:84% And he's got some you know R out here that he's 00:33:52.040 --> 00:33:55.100 align:middle line:90% controlling some V out on. 00:33:55.100 --> 00:33:57.827 align:middle line:84% And he's being powered from an input voltage 00:33:57.827 --> 00:33:58.910 align:middle line:90% that I'm going to call Vn. 00:33:58.910 --> 00:34:02.250 align:middle line:90% 00:34:02.250 --> 00:34:06.150 align:middle line:84% And then I'm going to have some filter. 00:34:06.150 --> 00:34:09.210 align:middle line:84% And let me consider a simple filter. 00:34:09.210 --> 00:34:10.513 align:middle line:90% Here's Cf. 00:34:10.513 --> 00:34:14.550 align:middle line:90% 00:34:14.550 --> 00:34:15.755 align:middle line:90% Here's Lf. 00:34:15.755 --> 00:34:18.780 align:middle line:90% 00:34:18.780 --> 00:34:19.949 align:middle line:90% Let me put-- 00:34:19.949 --> 00:34:24.780 align:middle line:84% I'm just going to use a simple resistive damping leg, Rd, 00:34:24.780 --> 00:34:28.410 align:middle line:90% just for simplicity's sake. 00:34:28.410 --> 00:34:31.650 align:middle line:84% And now this is going to get connected to my power source. 00:34:31.650 --> 00:34:35.969 align:middle line:90% So here is Vn. 00:34:35.969 --> 00:34:39.000 align:middle line:84% Or I'm going to call this Vs and Zs. 00:34:39.000 --> 00:34:41.699 align:middle line:84% But again, let me just consider the case where Zs is 0, 00:34:41.699 --> 00:34:42.595 align:middle line:90% for simplicity. 00:34:42.595 --> 00:34:45.460 align:middle line:90% 00:34:45.460 --> 00:34:46.449 align:middle line:90% All right. 00:34:46.449 --> 00:34:48.580 align:middle line:84% What does this power converter do? 00:34:48.580 --> 00:34:51.280 align:middle line:84% Let's assume he's a perfect power converter. 00:34:51.280 --> 00:34:55.810 align:middle line:84% He's going to go and keep the output voltage at the reference 00:34:55.810 --> 00:34:58.820 align:middle line:84% no matter what the input voltage does. 00:34:58.820 --> 00:35:00.450 align:middle line:90% That's his job. 00:35:00.450 --> 00:35:01.658 align:middle line:90% Input voltage varies. 00:35:01.658 --> 00:35:02.200 align:middle line:90% I don't care. 00:35:02.200 --> 00:35:07.138 align:middle line:84% I'm going to keep 12 volts at the output no matter what. 00:35:07.138 --> 00:35:11.010 align:middle line:84% So what would I say that he does? 00:35:11.010 --> 00:35:14.400 align:middle line:84% Well, he's a constant power load in that case. 00:35:14.400 --> 00:35:16.550 align:middle line:84% No matter what the input voltage is doing, 00:35:16.550 --> 00:35:20.840 align:middle line:84% I want to draw V reference squared over R0 if this is 00:35:20.840 --> 00:35:25.455 align:middle line:90% some fixed load at the output. 00:35:25.455 --> 00:35:28.090 align:middle line:90% 00:35:28.090 --> 00:35:35.710 align:middle line:84% So if I thought of this current as being some i n, What 00:35:35.710 --> 00:35:43.090 align:middle line:84% I get is i n times Vn is equal to whatever my power output 00:35:43.090 --> 00:35:47.100 align:middle line:84% power is going to be, which is V out squared over R out. 00:35:47.100 --> 00:35:50.190 align:middle line:90% 00:35:50.190 --> 00:35:56.790 align:middle line:84% Or I could write this as in terms of the-- 00:35:56.790 --> 00:35:58.710 align:middle line:84% if I reference, write it in terms of voltage, 00:35:58.710 --> 00:36:00.570 align:middle line:90% I could write this as V out-- 00:36:00.570 --> 00:36:05.580 align:middle line:84% I'm sorry, Vn is equal to P out over i n. 00:36:05.580 --> 00:36:08.640 align:middle line:84% And I might evaluate I might consider that 00:36:08.640 --> 00:36:13.320 align:middle line:84% as some operating point at which I'm sitting Vn and i n. 00:36:13.320 --> 00:36:15.230 align:middle line:90% So if I were to plot that-- 00:36:15.230 --> 00:36:18.780 align:middle line:90% 00:36:18.780 --> 00:36:20.280 align:middle line:90% let's plot that. 00:36:20.280 --> 00:36:28.180 align:middle line:84% What we're going to get is Here's Vn, here's i n. 00:36:28.180 --> 00:36:31.120 align:middle line:84% This converter, his job at his input port 00:36:31.120 --> 00:36:37.700 align:middle line:84% is basically to maintain the system on this curve, 00:36:37.700 --> 00:36:43.670 align:middle line:84% where Vn is equal to P out over i n. 00:36:43.670 --> 00:36:47.510 align:middle line:90% 00:36:47.510 --> 00:36:49.880 align:middle line:84% And if he's going to sit at some operating point-- 00:36:49.880 --> 00:36:51.230 align:middle line:90% maybe that's here. 00:36:51.230 --> 00:36:55.790 align:middle line:84% Maybe I have some DC operating point Vn and capital 00:36:55.790 --> 00:37:00.820 align:middle line:90% I n that we're going to sit at. 00:37:00.820 --> 00:37:04.800 align:middle line:84% So how does this behave for variations? 00:37:04.800 --> 00:37:09.760 align:middle line:84% Suppose my input voltage wiggles a little bit. 00:37:09.760 --> 00:37:11.820 align:middle line:90% What does my input current do? 00:37:11.820 --> 00:37:17.050 align:middle line:84% Well, it's exactly the kind of linearization we caught before. 00:37:17.050 --> 00:37:20.080 align:middle line:84% I can look at what this tangent line looks like. 00:37:20.080 --> 00:37:29.640 align:middle line:84% And I can write d Vn, D i n evaluated at Vn i n 00:37:29.640 --> 00:37:35.850 align:middle line:84% is equal to minus p 0 over i n squared. 00:37:35.850 --> 00:37:39.120 align:middle line:84% All I did was differentiate this, and evaluate it 00:37:39.120 --> 00:37:40.390 align:middle line:90% at the operating point. 00:37:40.390 --> 00:37:46.620 align:middle line:84% And this is equal to what I'm going to call r sub l. 00:37:46.620 --> 00:37:48.480 align:middle line:90% This is r sub l. 00:37:48.480 --> 00:37:55.030 align:middle line:84% This slope is r sub L. Does that make sense to everybody? 00:37:55.030 --> 00:37:58.340 align:middle line:90% 00:37:58.340 --> 00:38:02.870 align:middle line:84% Anybody notice anything about this resistance here, r sub l, 00:38:02.870 --> 00:38:03.490 align:middle line:90% this slope? 00:38:03.490 --> 00:38:04.490 align:middle line:90% AUDIENCE: It's negative. 00:38:04.490 --> 00:38:06.350 align:middle line:90% DAVID PERREAULT: It's negative. 00:38:06.350 --> 00:38:09.380 align:middle line:84% So this negative sign is the source 00:38:09.380 --> 00:38:11.546 align:middle line:84% of a lot of pain and embarrassment, 00:38:11.546 --> 00:38:15.140 align:middle line:84% because essentially, from a small signal perspective, 00:38:15.140 --> 00:38:17.870 align:middle line:84% the power converter looks like a negative resistance 00:38:17.870 --> 00:38:19.790 align:middle line:90% to the source. 00:38:19.790 --> 00:38:23.310 align:middle line:84% And negative resistances tend to make you unhappy because why? 00:38:23.310 --> 00:38:25.350 align:middle line:84% Because they can make things oscillate. 00:38:25.350 --> 00:38:26.000 align:middle line:90% All right? 00:38:26.000 --> 00:38:33.340 align:middle line:84% So if I then thought of what this converter looks like, 00:38:33.340 --> 00:38:39.670 align:middle line:84% OK, I might have, looking this way, what I see 00:38:39.670 --> 00:38:44.290 align:middle line:90% is rl is less than 0. 00:38:44.290 --> 00:38:48.140 align:middle line:84% Looking this way, I see a negative resistor. 00:38:48.140 --> 00:38:50.220 align:middle line:90% What do I see looking this way? 00:38:50.220 --> 00:38:53.190 align:middle line:90% I see Lf, Cf, and Rd. 00:38:53.190 --> 00:38:56.030 align:middle line:84% So if I were to redraw this, what I have 00:38:56.030 --> 00:39:08.330 align:middle line:84% is essentially Cf, Lf, and Rd in parallel. 00:39:08.330 --> 00:39:11.290 align:middle line:84% And so this, we've been calling-- 00:39:11.290 --> 00:39:15.540 align:middle line:84% this output impedance we've been calling Zx, right? 00:39:15.540 --> 00:39:19.780 align:middle line:90% 00:39:19.780 --> 00:39:21.800 align:middle line:84% And he's looking in the other direction. 00:39:21.800 --> 00:39:26.170 align:middle line:84% He's looking into rl less than 0. 00:39:26.170 --> 00:39:31.600 align:middle line:84% And this is my converter and his load, 00:39:31.600 --> 00:39:35.450 align:middle line:84% in the case where the converter is perfect. 00:39:35.450 --> 00:39:36.240 align:middle line:90% All right. 00:39:36.240 --> 00:39:38.110 align:middle line:90% So what's going to be true? 00:39:38.110 --> 00:39:42.830 align:middle line:84% Well, basically Rd and rl in this example are in parallel. 00:39:42.830 --> 00:39:47.300 align:middle line:84% So the net resistance, R net, the effect 00:39:47.300 --> 00:39:49.820 align:middle line:84% of resistance damping the Lc tank, 00:39:49.820 --> 00:39:57.530 align:middle line:84% is equal to Rd rl over Rd plus rl. 00:39:57.530 --> 00:39:58.983 align:middle line:90% Our net is greater than 0. 00:39:58.983 --> 00:40:01.730 align:middle line:90% 00:40:01.730 --> 00:40:07.850 align:middle line:84% Our net is greater than 0 for the magnitude of Zx-- 00:40:07.850 --> 00:40:17.426 align:middle line:84% or I should say, for Rd is less than the magnitude of rl. 00:40:17.426 --> 00:40:20.555 align:middle line:84% So if I'm looking in here, and that slope-- 00:40:20.555 --> 00:40:23.840 align:middle line:90% 00:40:23.840 --> 00:40:27.860 align:middle line:84% this slope is minus 1 ohms, I'd better 00:40:27.860 --> 00:40:30.930 align:middle line:90% make sure Rd is less than 1 ohm. 00:40:30.930 --> 00:40:33.800 align:middle line:84% Otherwise, basically this LCR stage 00:40:33.800 --> 00:40:36.550 align:middle line:90% is going to start to oscillate. 00:40:36.550 --> 00:40:38.750 align:middle line:84% And that's not going to be happy. 00:40:38.750 --> 00:40:40.980 align:middle line:84% And this is true in this instance, 00:40:40.980 --> 00:40:42.730 align:middle line:84% but it's true in a lot of other instances, 00:40:42.730 --> 00:40:44.313 align:middle line:84% where you have to deal with these kind 00:40:44.313 --> 00:40:46.690 align:middle line:84% of self-regulating loads, is they just tend 00:40:46.690 --> 00:40:48.590 align:middle line:90% to destabilize your system. 00:40:48.590 --> 00:40:50.620 align:middle line:84% And in this case, the system we're destabilizing 00:40:50.620 --> 00:40:51.560 align:middle line:90% is the input filter. 00:40:51.560 --> 00:40:54.560 align:middle line:84% But in other systems, you can destabilize other things. 00:40:54.560 --> 00:40:57.460 align:middle line:84% So in a DC power system, you might 00:40:57.460 --> 00:41:01.220 align:middle line:84% worry about what it's doing to long inductive 00:41:01.220 --> 00:41:06.270 align:middle line:84% runs of cable or capacitive filters on things. 00:41:06.270 --> 00:41:07.970 align:middle line:84% You can start to make oscillations 00:41:07.970 --> 00:41:11.540 align:middle line:90% in constant power loads. 00:41:11.540 --> 00:41:15.250 align:middle line:84% So generally, the reason we obsess over 00:41:15.250 --> 00:41:17.800 align:middle line:84% what's our output impedance of our filter 00:41:17.800 --> 00:41:20.710 align:middle line:84% is because I need to keep it small enough 00:41:20.710 --> 00:41:22.690 align:middle line:84% that when it looks into a negative input 00:41:22.690 --> 00:41:26.470 align:middle line:84% impedance-- effectively a power converter-- the whole system is 00:41:26.470 --> 00:41:28.550 align:middle line:90% going to be stable. 00:41:28.550 --> 00:41:33.410 align:middle line:84% And here I've shown you in the simplest case for Rd. 00:41:33.410 --> 00:41:38.090 align:middle line:84% But generally what we need, the more general statement of this, 00:41:38.090 --> 00:41:39.713 align:middle line:90% is that we need the maximum-- 00:41:39.713 --> 00:41:43.730 align:middle line:90% 00:41:43.730 --> 00:41:47.210 align:middle line:84% output impedance of my filter across all frequencies 00:41:47.210 --> 00:41:53.450 align:middle line:84% ought to be well below the magnitude of P 00:41:53.450 --> 00:41:56.155 align:middle line:84% out minus P out over I n squared. 00:41:56.155 --> 00:41:59.445 align:middle line:90% 00:41:59.445 --> 00:42:02.270 align:middle line:90% So I've got to pick my damping. 00:42:02.270 --> 00:42:06.110 align:middle line:84% And that's going to drive me to go pick some minimum-- 00:42:06.110 --> 00:42:08.810 align:middle line:84% if I were to use this capacitive damping scheme, 00:42:08.810 --> 00:42:12.110 align:middle line:84% that's going to drive me to pick some minimum value 00:42:12.110 --> 00:42:15.530 align:middle line:84% of the damping capacitance so that I can minimize 00:42:15.530 --> 00:42:20.540 align:middle line:84% the maximum value of my output, my filter output impedance, 00:42:20.540 --> 00:42:23.190 align:middle line:90% across frequency. 00:42:23.190 --> 00:42:24.931 align:middle line:84% Does that make sense to everybody? 00:42:24.931 --> 00:42:27.760 align:middle line:90% 00:42:27.760 --> 00:42:34.630 align:middle line:84% Now just to drive home in your head that this isn't some-- 00:42:34.630 --> 00:42:38.930 align:middle line:84% just some kind of mathematical funniness, 00:42:38.930 --> 00:42:42.230 align:middle line:84% Muncie has been very kind to set up an actual power converter 00:42:42.230 --> 00:42:43.300 align:middle line:90% with an actual filter. 00:42:43.300 --> 00:42:48.910 align:middle line:90% 00:42:48.910 --> 00:42:53.810 align:middle line:84% And this is actually a buck converter 00:42:53.810 --> 00:42:56.120 align:middle line:90% with an Lc input filter. 00:42:56.120 --> 00:42:59.120 align:middle line:84% And then what she has done is made sure 00:42:59.120 --> 00:43:02.940 align:middle line:84% that we can basically put a switch on this damping leg. 00:43:02.940 --> 00:43:07.790 align:middle line:84% We can put the damping leg in or we can take the damping leg out. 00:43:07.790 --> 00:43:20.090 align:middle line:84% And if we power this up, that's the input voltage 00:43:20.090 --> 00:43:21.350 align:middle line:90% to the converter. 00:43:21.350 --> 00:43:24.140 align:middle line:90% So it's actually on the-- 00:43:24.140 --> 00:43:30.710 align:middle line:84% it's exactly the voltage Vn here that we're seeing. 00:43:30.710 --> 00:43:32.940 align:middle line:84% This is the input voltage that we're looking at-- 00:43:32.940 --> 00:43:35.070 align:middle line:90% Vn. 00:43:35.070 --> 00:43:41.810 align:middle line:84% If we have the filter in place, Vn basically sits at Vs. 00:43:41.810 --> 00:43:50.620 align:middle line:84% But if I take out that damping leg, that's what happens. 00:43:50.620 --> 00:43:53.050 align:middle line:84% The input voltage is still fixed, 00:43:53.050 --> 00:43:55.920 align:middle line:90% but this voltage is going crazy. 00:43:55.920 --> 00:43:58.278 align:middle line:84% And you say, well, I could live with that. 00:43:58.278 --> 00:43:58.820 align:middle line:90% I don't care. 00:43:58.820 --> 00:43:59.590 align:middle line:90% Well, it depends. 00:43:59.590 --> 00:44:02.560 align:middle line:84% Sometimes you can hear a lot of noise. 00:44:02.560 --> 00:44:07.180 align:middle line:84% This is just with the converter at some value, 00:44:07.180 --> 00:44:08.360 align:middle line:90% some value of load. 00:44:08.360 --> 00:44:10.480 align:middle line:84% But imagine this thing we're trying to drive, 00:44:10.480 --> 00:44:12.830 align:middle line:84% draw current varying near that frequency, 00:44:12.830 --> 00:44:15.130 align:middle line:90% it would go really crazy. 00:44:15.130 --> 00:44:17.480 align:middle line:84% So you really can't tolerate this kind of thing. 00:44:17.480 --> 00:44:19.570 align:middle line:84% And you would see this in the out-- 00:44:19.570 --> 00:44:21.920 align:middle line:84% converter response at the output and so forth. 00:44:21.920 --> 00:44:25.720 align:middle line:84% So you really do have to seriously account 00:44:25.720 --> 00:44:28.510 align:middle line:84% not only for the fact that you need to damp the LNC, 00:44:28.510 --> 00:44:30.550 align:middle line:84% but you need to account for the negative input 00:44:30.550 --> 00:44:33.610 align:middle line:84% impedance of the converter in order 00:44:33.610 --> 00:44:36.530 align:middle line:90% to get acceptable response. 00:44:36.530 --> 00:44:37.750 align:middle line:90% Any question about that? 00:44:37.750 --> 00:44:48.140 align:middle line:90% 00:44:48.140 --> 00:44:59.500 align:middle line:84% Now what I've told you so far is necessary. 00:44:59.500 --> 00:45:02.860 align:middle line:84% You'd better design your filter for that. 00:45:02.860 --> 00:45:05.380 align:middle line:84% But it comes out to be a little bit trickier too, 00:45:05.380 --> 00:45:07.390 align:middle line:84% because we've talked about modeling 00:45:07.390 --> 00:45:08.980 align:middle line:90% and control of your converter. 00:45:08.980 --> 00:45:12.590 align:middle line:84% We've said, OK, I want some certain transfer functions. 00:45:12.590 --> 00:45:16.750 align:middle line:84% We use averaging to derive kind of local average models, 00:45:16.750 --> 00:45:21.160 align:middle line:84% and you get your transfer functions and all that. 00:45:21.160 --> 00:45:24.790 align:middle line:84% When we did all that, we assumed that you were 00:45:24.790 --> 00:45:28.630 align:middle line:90% working from a perfect source. 00:45:28.630 --> 00:45:30.800 align:middle line:84% The picture we drew for our boost converter 00:45:30.800 --> 00:45:32.590 align:middle line:84% was just you as a voltage source. 00:45:32.590 --> 00:45:34.640 align:middle line:90% It's just zero-impedance source. 00:45:34.640 --> 00:45:37.270 align:middle line:84% We pretended there was no filter there. 00:45:37.270 --> 00:45:39.380 align:middle line:84% In the real world, there is a filter there. 00:45:39.380 --> 00:45:43.760 align:middle line:84% And that filter can indeed impact your converter. 00:45:43.760 --> 00:45:51.770 align:middle line:84% If I come back to the baseline here, 00:45:51.770 --> 00:45:59.350 align:middle line:90% maybe I don't have my LISN here. 00:45:59.350 --> 00:46:01.180 align:middle line:90% That's fine. 00:46:01.180 --> 00:46:03.520 align:middle line:84% There's no LISN when I'm running in the real world. 00:46:03.520 --> 00:46:06.190 align:middle line:84% And maybe my source impedance is low, 00:46:06.190 --> 00:46:12.632 align:middle line:84% but I still have some converter and filter. 00:46:12.632 --> 00:46:14.090 align:middle line:84% And who knows what this looks like? 00:46:14.090 --> 00:46:18.195 align:middle line:84% This looks like whatever thing you drew here. 00:46:18.195 --> 00:46:25.720 align:middle line:84% But I could replace this whole thing with the source voltage 00:46:25.720 --> 00:46:33.070 align:middle line:84% Vs and an output impedance Zx, the output impedance seen 00:46:33.070 --> 00:46:34.070 align:middle line:90% from my filter. 00:46:34.070 --> 00:46:39.280 align:middle line:84% So this is really the model of my source and filter. 00:46:39.280 --> 00:46:42.760 align:middle line:84% It's the source voltage in series with the output 00:46:42.760 --> 00:46:43.820 align:middle line:90% impedance of my filter. 00:46:43.820 --> 00:46:47.518 align:middle line:84% That's a Thevenin model for my system. 00:46:47.518 --> 00:46:51.620 align:middle line:84% And my assumption in designing the converter and controller 00:46:51.620 --> 00:46:55.860 align:middle line:90% was that Zs was 0, or Zx was 0. 00:46:55.860 --> 00:46:59.682 align:middle line:84% But Zx isn't 0 once I've designed my filter. 00:46:59.682 --> 00:47:03.380 align:middle line:84% How close do I need to make it towards 0? 00:47:03.380 --> 00:47:09.130 align:middle line:84% Well, what I would like to do is have 00:47:09.130 --> 00:47:12.610 align:middle line:84% the case where all of my models that I used 00:47:12.610 --> 00:47:15.040 align:middle line:84% to derive my converter, or control of transfer function, 00:47:15.040 --> 00:47:16.270 align:middle line:90% all of that stuff is valid. 00:47:16.270 --> 00:47:19.155 align:middle line:84% It's not affected by the filter I designed later. 00:47:19.155 --> 00:47:22.170 align:middle line:90% 00:47:22.170 --> 00:47:24.240 align:middle line:84% And a sufficient-- not necessary, 00:47:24.240 --> 00:47:28.150 align:middle line:84% but a sufficient condition to doing that is two things. 00:47:28.150 --> 00:47:30.785 align:middle line:84% And I'm not going to prove this to you. 00:47:30.785 --> 00:47:35.650 align:middle line:90% 00:47:35.650 --> 00:47:37.750 align:middle line:84% But there's some very clever analysis. 00:47:37.750 --> 00:47:40.510 align:middle line:84% People have shown this, and in particular, 00:47:40.510 --> 00:47:45.100 align:middle line:84% Erickson and Maksimovic at University of Colorado Boulder, 00:47:45.100 --> 00:47:47.510 align:middle line:84% did some very nice analysis of this. 00:47:47.510 --> 00:47:51.190 align:middle line:84% And a sufficient conversion-- a sufficient condition 00:47:51.190 --> 00:47:52.220 align:middle line:90% is the following. 00:47:52.220 --> 00:47:58.870 align:middle line:84% First, if I assume my converter operates perfectly like this, 00:47:58.870 --> 00:48:01.150 align:middle line:84% we would like the output impedance 00:48:01.150 --> 00:48:03.730 align:middle line:84% of my filter, Zx, the maximum value, 00:48:03.730 --> 00:48:06.970 align:middle line:84% to be much less than the magnitude of the negative input 00:48:06.970 --> 00:48:10.170 align:middle line:84% resistance if I have perfect control. 00:48:10.170 --> 00:48:13.860 align:middle line:84% And that would be perfectly the case 00:48:13.860 --> 00:48:15.720 align:middle line:84% if I had a perfect converter that 00:48:15.720 --> 00:48:18.220 align:middle line:84% had infinite bandwidth and controlling the output anyways. 00:48:18.220 --> 00:48:20.680 align:middle line:84% So I kind of need to meet that requirement. 00:48:20.680 --> 00:48:23.204 align:middle line:90% 00:48:23.204 --> 00:48:25.040 align:middle line:84% But what they showed is, in additionally, 00:48:25.040 --> 00:48:26.720 align:middle line:84% a sufficient condition, although not 00:48:26.720 --> 00:48:30.920 align:middle line:84% necessary to getting a stable system, but sufficient to be 00:48:30.920 --> 00:48:32.990 align:middle line:84% that you can ignore the design of the filter 00:48:32.990 --> 00:48:34.820 align:middle line:84% or decouple the design of the filter 00:48:34.820 --> 00:48:37.790 align:middle line:84% from the design of your controller for your converter, 00:48:37.790 --> 00:48:40.880 align:middle line:84% is to also say, if I imagined, instead of having 00:48:40.880 --> 00:48:43.040 align:middle line:84% a perfect controller over the output, 00:48:43.040 --> 00:48:47.220 align:middle line:84% I just regulated my converter at fixed-duty ratio. 00:48:47.220 --> 00:48:49.170 align:middle line:84% And I looked at the input impedance 00:48:49.170 --> 00:48:52.320 align:middle line:84% of the converter at fixed-duty ratio across frequency-- 00:48:52.320 --> 00:48:54.900 align:middle line:84% and that's going to be variable across frequency-- 00:48:54.900 --> 00:48:59.790 align:middle line:84% at every frequency, that input impedance under fixed-duty ratio 00:48:59.790 --> 00:49:02.700 align:middle line:84% ought to be smaller than the output impedance 00:49:02.700 --> 00:49:06.490 align:middle line:84% at the same frequency looking back into my output filter. 00:49:06.490 --> 00:49:08.200 align:middle line:84% If you put those two conditions together, 00:49:08.200 --> 00:49:12.687 align:middle line:84% you get a complete sufficient condition 00:49:12.687 --> 00:49:14.770 align:middle line:84% that you can ignore the effect of the input filter 00:49:14.770 --> 00:49:16.228 align:middle line:84% on the rest of your control design. 00:49:16.228 --> 00:49:18.844 align:middle line:90% 00:49:18.844 --> 00:49:23.160 align:middle line:84% But a good starting place for all 00:49:23.160 --> 00:49:26.770 align:middle line:84% that is just to make sure that under perfect control, 00:49:26.770 --> 00:49:29.410 align:middle line:84% that your filter is well enough damped 00:49:29.410 --> 00:49:31.000 align:middle line:84% to deal with the negative resistance 00:49:31.000 --> 00:49:32.210 align:middle line:90% of a perfect controller. 00:49:32.210 --> 00:49:33.730 align:middle line:84% So that's the place I would start 00:49:33.730 --> 00:49:36.790 align:middle line:90% in thinking about any design. 00:49:36.790 --> 00:49:40.540 align:middle line:84% So that's a little bit about filter design and filter 00:49:40.540 --> 00:49:41.327 align:middle line:90% damping. 00:49:41.327 --> 00:49:42.910 align:middle line:84% The last thing I should have mentioned 00:49:42.910 --> 00:49:46.270 align:middle line:84% was, well, what if this Lc filter isn't enough? 00:49:46.270 --> 00:49:50.200 align:middle line:84% You can cascade Lc filter stages and get 00:49:50.200 --> 00:49:51.920 align:middle line:90% more and more attenuation. 00:49:51.920 --> 00:49:57.220 align:middle line:84% And again, there's rules for how you want to cascade those filter 00:49:57.220 --> 00:49:57.760 align:middle line:90% impedances. 00:49:57.760 --> 00:50:00.010 align:middle line:84% And again, those are in the handouts. 00:50:00.010 --> 00:50:02.250 align:middle line:84% Any last questions before we wrap up for the day? 00:50:02.250 --> 00:50:06.710 align:middle line:90% 00:50:06.710 --> 00:50:10.340 align:middle line:84% OK, next class we'll talk about one other additional item that's 00:50:10.340 --> 00:50:12.230 align:middle line:90% very important in filter design. 00:50:12.230 --> 00:50:14.200 align:middle line:90% Have a great day. 00:50:14.200 --> 00:50:18.000 align:middle line:90%