WEBVTT 00:00:00.000 --> 00:00:03.472 align:middle line:90% [SQUEAKING] 00:00:03.472 --> 00:00:04.464 align:middle line:90% [RUSTLING] 00:00:04.464 --> 00:00:07.936 align:middle line:90% [CLICKING] 00:00:07.936 --> 00:00:09.740 align:middle line:90% 00:00:09.740 --> 00:00:12.140 align:middle line:84% DAVID PERREAULT: OK, why don't we get started? 00:00:12.140 --> 00:00:14.840 align:middle line:84% So what I wanted to do is continue 00:00:14.840 --> 00:00:18.240 align:middle line:84% the discussion of inverters that we started last time. 00:00:18.240 --> 00:00:21.680 align:middle line:84% And what I'm going to show you is some of expansions on the way 00:00:21.680 --> 00:00:25.370 align:middle line:84% you can think about controlling the inverters to get 00:00:25.370 --> 00:00:28.130 align:middle line:84% better performance out of the waveforms. 00:00:28.130 --> 00:00:32.570 align:middle line:84% So we introduced the basic voltage source inverter 00:00:32.570 --> 00:00:36.560 align:middle line:84% structure, which is four switches, 00:00:36.560 --> 00:00:42.150 align:middle line:84% And very typically, we'll use bidirectional-conducting 00:00:42.150 --> 00:00:46.580 align:middle line:84% single-directional blocking switches like MOSFETs. 00:00:46.580 --> 00:00:47.740 align:middle line:90% Like this, for example. 00:00:47.740 --> 00:00:57.130 align:middle line:90% 00:00:57.130 --> 00:01:03.130 align:middle line:84% And if I want to synthesize some voltage Vx across the load, 00:01:03.130 --> 00:01:10.070 align:middle line:84% if I have switches S1, S2, S3, and S4, 00:01:10.070 --> 00:01:17.350 align:middle line:84% if I have the switches that are on the S1 and S2, 00:01:17.350 --> 00:01:21.580 align:middle line:84% I put positive VC across the load. 00:01:21.580 --> 00:01:26.540 align:middle line:90% If I put S2 and S3 on, I get 0. 00:01:26.540 --> 00:01:33.100 align:middle line:84% If I have S3, S4, I get negative VDC across the load. 00:01:33.100 --> 00:01:38.710 align:middle line:84% And if I have S4 and S1, I get 0 again. 00:01:38.710 --> 00:01:43.170 align:middle line:84% And we showed last time that we can 00:01:43.170 --> 00:01:49.110 align:middle line:84% use this to synthesize either plus 0 or minus repeatedly 00:01:49.110 --> 00:01:54.270 align:middle line:84% across the load, and then use that to at least, very crudely, 00:01:54.270 --> 00:01:57.310 align:middle line:84% approximate some kind of sine wave, for example. 00:01:57.310 --> 00:01:59.200 align:middle line:84% And why do I focus on a sine wave? 00:01:59.200 --> 00:02:01.140 align:middle line:84% Very often we want to synthesize something 00:02:01.140 --> 00:02:04.330 align:middle line:84% that if at least if I filter it, it will look like a sine wave 00:02:04.330 --> 00:02:07.290 align:middle line:84% because, for example, if I'm making a UPS inverter, 00:02:07.290 --> 00:02:09.030 align:middle line:84% maybe I would like to replicate what 00:02:09.030 --> 00:02:11.760 align:middle line:90% the grid does with my inverter. 00:02:11.760 --> 00:02:18.510 align:middle line:84% And when I show this as a load, what is this? 00:02:18.510 --> 00:02:20.500 align:middle line:84% I don't know, it depends on the application. 00:02:20.500 --> 00:02:22.150 align:middle line:90% It could be a motor winding. 00:02:22.150 --> 00:02:25.860 align:middle line:84% It could be some low-pass filter, 00:02:25.860 --> 00:02:28.660 align:middle line:84% for example, that's going to some load. 00:02:28.660 --> 00:02:35.810 align:middle line:84% So this could be Vx and this could be V out across the load. 00:02:35.810 --> 00:02:38.958 align:middle line:90% 00:02:38.958 --> 00:02:44.850 align:middle line:84% So we said last time that we often think, 00:02:44.850 --> 00:02:47.370 align:middle line:84% especially when we're synthesizing periodic waveforms, 00:02:47.370 --> 00:02:51.060 align:middle line:84% we often think in terms of the Fourier series 00:02:51.060 --> 00:02:56.310 align:middle line:84% than what we're doing to shape the harmonic 00:02:56.310 --> 00:02:57.990 align:middle line:90% content of the waveforms. 00:02:57.990 --> 00:03:02.820 align:middle line:84% So we said, OK, one form of the Fourier series for some signal 00:03:02.820 --> 00:03:11.340 align:middle line:84% f of t is simply equal to b0 over 2 plus the summation from 1 00:03:11.340 --> 00:03:20.790 align:middle line:84% to infinity of a sub n sine of n omega 0 t plus b 00:03:20.790 --> 00:03:26.040 align:middle line:90% sub n cosine of n omega 0 t. 00:03:26.040 --> 00:03:29.130 align:middle line:84% Where if we recall, the sine terms always 00:03:29.130 --> 00:03:33.520 align:middle line:84% odd in time, and the cosine terms are always even in time. 00:03:33.520 --> 00:03:38.160 align:middle line:84% So if I happen to want an odd waveform, 00:03:38.160 --> 00:03:43.080 align:middle line:84% then I should make the pattern f of t-- 00:03:43.080 --> 00:03:46.450 align:middle line:84% and that means I want the pattern of f of t to be odd. 00:03:46.450 --> 00:03:52.050 align:middle line:84% That means I will only have the a sub n terms. 00:03:52.050 --> 00:03:53.970 align:middle line:84% And if I want the pattern f of t to be even, 00:03:53.970 --> 00:03:56.870 align:middle line:90% I only have the b sub n terms. 00:03:56.870 --> 00:04:03.110 align:middle line:84% More importantly, we saw that we can often do interesting things 00:04:03.110 --> 00:04:05.570 align:middle line:90% with half-wave symmetry. 00:04:05.570 --> 00:04:10.430 align:middle line:84% What does half-wave symmetry mean? 00:04:10.430 --> 00:04:14.900 align:middle line:84% Half-wave symmetric waveform means that f of t 00:04:14.900 --> 00:04:22.670 align:middle line:84% is equal to minus f of t minus t over 2 where t is the period. 00:04:22.670 --> 00:04:26.450 align:middle line:84% That means if I go back half of a cycle in the waveform, 00:04:26.450 --> 00:04:27.780 align:middle line:90% the waveform is flipped. 00:04:27.780 --> 00:04:33.580 align:middle line:84% So for example, this waveform is both odd and 00:04:33.580 --> 00:04:35.150 align:middle line:90% half-wave-symmetric. 00:04:35.150 --> 00:04:41.080 align:middle line:90% 00:04:41.080 --> 00:04:41.710 align:middle line:90% Why? 00:04:41.710 --> 00:04:44.590 align:middle line:84% Because if I go back-- oh, that's a terrible drawing 00:04:44.590 --> 00:04:47.420 align:middle line:90% because this should go further. 00:04:47.420 --> 00:04:50.050 align:middle line:84% But if I go back at any point half a cycle, 00:04:50.050 --> 00:04:53.020 align:middle line:84% I flip across the axis and I get to the same point 00:04:53.020 --> 00:04:54.340 align:middle line:90% on the other side. 00:04:54.340 --> 00:04:57.580 align:middle line:84% And the reason I'm interested in half-wave symmetry 00:04:57.580 --> 00:05:01.090 align:middle line:84% is because if it's half-wave symmetry, 00:05:01.090 --> 00:05:06.885 align:middle line:84% then the a2K and the b2K terms are 0. 00:05:06.885 --> 00:05:10.427 align:middle line:90% 00:05:10.427 --> 00:05:12.010 align:middle line:84% And the reason I'm interested in doing 00:05:12.010 --> 00:05:14.385 align:middle line:84% that is suppose I want to synthesize something that looks 00:05:14.385 --> 00:05:18.860 align:middle line:84% like a sine wave, and then I want to put it through a filter, 00:05:18.860 --> 00:05:22.240 align:middle line:84% it's very hard to filter out the second harmonic 00:05:22.240 --> 00:05:24.845 align:middle line:84% because the second harmonic is only a factor of 2 00:05:24.845 --> 00:05:26.720 align:middle line:84% different than the fundamental and frequency, 00:05:26.720 --> 00:05:29.960 align:middle line:84% and I'm trying to use perhaps a low-pass filter, 00:05:29.960 --> 00:05:32.360 align:middle line:84% it just gets hard to attenuate that. 00:05:32.360 --> 00:05:36.300 align:middle line:84% If I can get rid of the second harmonic specifically, 00:05:36.300 --> 00:05:37.870 align:middle line:90% I can make my filtering easier. 00:05:37.870 --> 00:05:40.412 align:middle line:84% And if I can get rid of the-- if I make it halfway symmetric, 00:05:40.412 --> 00:05:42.700 align:middle line:84% I get rid of the second, fourth, sixth, et cetera, 00:05:42.700 --> 00:05:45.240 align:middle line:84% and I really reduce the amount of stuff 00:05:45.240 --> 00:05:46.740 align:middle line:90% I have to worry about filtering. 00:05:46.740 --> 00:05:50.470 align:middle line:84% So I can get much purer output waveforms by doing that. 00:05:50.470 --> 00:05:54.930 align:middle line:84% So if we're going to synthesize some pattern for my inverter 00:05:54.930 --> 00:05:57.720 align:middle line:90% to synthesize-- 00:05:57.720 --> 00:05:59.970 align:middle line:84% to create-- making it half-wave-symmetric 00:05:59.970 --> 00:06:01.770 align:middle line:84% is significantly advantageous if I 00:06:01.770 --> 00:06:06.000 align:middle line:84% don't want any even harmonic components in my output. 00:06:06.000 --> 00:06:16.570 align:middle line:84% And what we said was that, if I come back 00:06:16.570 --> 00:06:19.256 align:middle line:90% to this kind of filtering-- 00:06:19.256 --> 00:06:21.920 align:middle line:90% 00:06:21.920 --> 00:06:27.330 align:middle line:84% so if I think of the transfer function from Vx to Vo-- 00:06:27.330 --> 00:06:33.090 align:middle line:84% so magnitude of Vo over Vx, ideally speaking, 00:06:33.090 --> 00:06:35.310 align:middle line:84% there's going to be some kind of low-pass filter 00:06:35.310 --> 00:06:37.890 align:middle line:84% where I want to put the fundamental here 00:06:37.890 --> 00:06:45.570 align:middle line:84% so it's in the passband, but 2, 3, 4, 5, 6, et cetera. 00:06:45.570 --> 00:06:47.080 align:middle line:84% I usually only want this component 00:06:47.080 --> 00:06:49.730 align:middle line:84% and I would like to filter everything else. 00:06:49.730 --> 00:06:52.868 align:middle line:84% We can get rid of these by the half-wave symmetry. 00:06:52.868 --> 00:06:54.410 align:middle line:84% And then it's just a question of what 00:06:54.410 --> 00:06:56.350 align:middle line:84% about third and fifth and so forth? 00:06:56.350 --> 00:06:58.896 align:middle line:90% 00:06:58.896 --> 00:07:02.730 align:middle line:84% What we saw last time was one more trick. 00:07:02.730 --> 00:07:08.190 align:middle line:84% We said I can synthesize 0 plus and minus. 00:07:08.190 --> 00:07:11.250 align:middle line:84% If I only switch once per cycle, I 00:07:11.250 --> 00:07:14.090 align:middle line:84% can create a waveform that looks like this. 00:07:14.090 --> 00:07:28.380 align:middle line:90% 00:07:28.380 --> 00:07:35.280 align:middle line:84% So this is pi, this is 2 pi, this is some angle omega t. 00:07:35.280 --> 00:07:41.610 align:middle line:84% If I synthesize an angle, I may have delta, pi minus delta, 00:07:41.610 --> 00:07:47.490 align:middle line:84% pi plus delta, and 2 pi minus delta. 00:07:47.490 --> 00:08:02.270 align:middle line:84% If I synthesized a waveform that looks something like this 00:08:02.270 --> 00:08:04.800 align:middle line:84% and so forth, that is half-wave-symmetric. 00:08:04.800 --> 00:08:08.240 align:middle line:90% So this is VDC. 00:08:08.240 --> 00:08:09.290 align:middle line:90% This is minus VDC. 00:08:09.290 --> 00:08:13.280 align:middle line:90% 00:08:13.280 --> 00:08:19.910 align:middle line:84% This waveform for Vx is half-wave-symmetric. 00:08:19.910 --> 00:08:22.070 align:middle line:84% And the advantage I have here is I 00:08:22.070 --> 00:08:42.860 align:middle line:84% can have S4, S1; S1, S2; S2, S3; and S3, S4; 00:08:42.860 --> 00:08:45.830 align:middle line:90% and then I'm back to S4, S1. 00:08:45.830 --> 00:08:49.550 align:middle line:84% I've switched each device on and off once per cycle, 00:08:49.550 --> 00:08:54.020 align:middle line:84% and hence, I've got the lowest switching losses 00:08:54.020 --> 00:08:57.490 align:middle line:84% that I might get for synthesizing such a pattern. 00:08:57.490 --> 00:08:59.900 align:middle line:84% So if I care a lot about the loss in switching, which 00:08:59.900 --> 00:09:03.680 align:middle line:84% I might if I'm either a very high power where I have 00:09:03.680 --> 00:09:05.840 align:middle line:84% high voltages or very high frequencies, 00:09:05.840 --> 00:09:08.420 align:middle line:84% this is a pretty nice pattern to have. 00:09:08.420 --> 00:09:10.700 align:middle line:84% And we said my one control handle 00:09:10.700 --> 00:09:15.110 align:middle line:84% in this pattern, if I use it, is this angle delta. 00:09:15.110 --> 00:09:20.280 align:middle line:84% And by picking delta, we can do a few things. 00:09:20.280 --> 00:09:23.210 align:middle line:84% One thing we said we could do is if I make delta bigger, 00:09:23.210 --> 00:09:25.610 align:middle line:84% the fundamental component gets smaller. 00:09:25.610 --> 00:09:27.602 align:middle line:90% And if I make delta smaller-- 00:09:27.602 --> 00:09:29.060 align:middle line:84% I'm sorry, if I make delta smaller, 00:09:29.060 --> 00:09:30.170 align:middle line:84% the fundamental component gets bigger. 00:09:30.170 --> 00:09:32.930 align:middle line:84% So I can control the fundamental of the output that way. 00:09:32.930 --> 00:09:35.750 align:middle line:84% The other thing we said we could do is control the harmonics. 00:09:35.750 --> 00:09:39.300 align:middle line:84% Because we said if I do it like this-- 00:09:39.300 --> 00:09:42.030 align:middle line:90% so for an odd waveform-- 00:09:42.030 --> 00:09:45.450 align:middle line:84% and I'm only picking it to be odd just because it makes 00:09:45.450 --> 00:09:49.050 align:middle line:84% the Fourier series description simpler, 00:09:49.050 --> 00:09:58.080 align:middle line:84% But for an odd waveform, what I get is the n-th harmonic 00:09:58.080 --> 00:10:00.673 align:middle line:84% component is simply going to be equal to-- 00:10:00.673 --> 00:10:03.410 align:middle line:90% 00:10:03.410 --> 00:10:06.110 align:middle line:84% which way did I-- did I do it in terms of angle or did I do it 00:10:06.110 --> 00:10:07.310 align:middle line:90% in terms of-- 00:10:07.310 --> 00:10:08.780 align:middle line:90% I guess I did it-- 00:10:08.780 --> 00:10:15.050 align:middle line:84% OK, 2 over T integral over the period of Vx 00:10:15.050 --> 00:10:19.710 align:middle line:90% of t sine of n omega t dt. 00:10:19.710 --> 00:10:23.610 align:middle line:90% 00:10:23.610 --> 00:10:29.490 align:middle line:84% By picking delta, I can choose at least one of these to mess 00:10:29.490 --> 00:10:30.420 align:middle line:90% with. 00:10:30.420 --> 00:10:35.340 align:middle line:84% And so I can calculate the amplitude of the n-th harmonic 00:10:35.340 --> 00:10:38.580 align:middle line:84% essentially as the waveform times a sine 00:10:38.580 --> 00:10:41.730 align:middle line:84% of the right frequency integrated. 00:10:41.730 --> 00:10:44.160 align:middle line:84% And the example I showed you last time 00:10:44.160 --> 00:10:46.470 align:middle line:90% is, if I thought about what-- 00:10:46.470 --> 00:10:52.140 align:middle line:84% say-- suppose I focused on the third harmonic-- so n is 3, 00:10:52.140 --> 00:10:54.485 align:middle line:84% the third harmonic would look like this. 00:10:54.485 --> 00:10:57.340 align:middle line:90% 00:10:57.340 --> 00:11:01.230 align:middle line:84% Or, I'm sorry, the sine 3 omega t would look like this. 00:11:01.230 --> 00:11:08.820 align:middle line:90% 00:11:08.820 --> 00:11:12.890 align:middle line:90% And when I picked delta-- 00:11:12.890 --> 00:11:17.410 align:middle line:84% this angle delta to be exactly 30 degrees, or pi over 6-- 00:11:17.410 --> 00:11:20.215 align:middle line:84% so this example is delta equals pi over 6. 00:11:20.215 --> 00:11:23.150 align:middle line:90% 00:11:23.150 --> 00:11:27.110 align:middle line:84% We notice that when I look at the intersection between Vx 00:11:27.110 --> 00:11:31.580 align:middle line:84% of t, and I multiply it by the pink waveform, which 00:11:31.580 --> 00:11:38.170 align:middle line:84% is sine 3 omega t, basically I get this positive area, 00:11:38.170 --> 00:11:42.850 align:middle line:84% and then I get this negative area multiplied 00:11:42.850 --> 00:11:44.390 align:middle line:90% by the white waveform. 00:11:44.390 --> 00:11:47.820 align:middle line:84% And when I multiply them and integrate them, I get 0. 00:11:47.820 --> 00:11:51.420 align:middle line:84% So I can look at this waveform and say, hey, 00:11:51.420 --> 00:11:55.230 align:middle line:84% if I pick that angle just right, the overlap 00:11:55.230 --> 00:11:59.190 align:middle line:84% with this sine 3 omega t gives me just such 00:11:59.190 --> 00:12:00.740 align:middle line:90% that I kill the third harmonic. 00:12:00.740 --> 00:12:03.830 align:middle line:90% 00:12:03.830 --> 00:12:07.970 align:middle line:84% And this kind of trick, we're going to see, 00:12:07.970 --> 00:12:12.910 align:middle line:84% we can expand in multiple ways to give us higher performance. 00:12:12.910 --> 00:12:14.980 align:middle line:84% But before I go on, is there any question 00:12:14.980 --> 00:12:17.020 align:middle line:84% about how that arises or anything 00:12:17.020 --> 00:12:18.080 align:middle line:90% up to the present point? 00:12:18.080 --> 00:12:20.040 align:middle line:84% That's the quick review of last lecture. 00:12:20.040 --> 00:12:25.170 align:middle line:90% 00:12:25.170 --> 00:12:26.340 align:middle line:90% OK. 00:12:26.340 --> 00:12:28.030 align:middle line:90% So how might I go about it? 00:12:28.030 --> 00:12:30.000 align:middle line:90% Well, the reason I did this-- 00:12:30.000 --> 00:12:32.250 align:middle line:84% a principal reason I did this was because I said, 00:12:32.250 --> 00:12:35.850 align:middle line:84% OK, I've gotten rid of the even harmonics, 00:12:35.850 --> 00:12:40.235 align:middle line:84% but I've still got to worry about 3, 5, 7, et cetera. 00:12:40.235 --> 00:12:45.990 align:middle line:84% If I picked my angle just this way, I would pin 3 to 0. 00:12:45.990 --> 00:12:48.450 align:middle line:84% So that would kill the third harmonic. 00:12:48.450 --> 00:12:53.130 align:middle line:84% And that means, since the fourth is gone by symmetry, the lowest, 00:12:53.130 --> 00:12:54.570 align:middle line:90% I have to deal with this fifth. 00:12:54.570 --> 00:12:57.330 align:middle line:84% Now I'm starting to get some nice space here. 00:12:57.330 --> 00:13:02.190 align:middle line:84% And that lets me get much more purer filtered waveforms 00:13:02.190 --> 00:13:04.950 align:middle line:84% at the output compared to this kind of pulsey shape 00:13:04.950 --> 00:13:07.570 align:middle line:90% I'm generating. 00:13:07.570 --> 00:13:10.030 align:middle line:90% But what else could I do? 00:13:10.030 --> 00:13:13.870 align:middle line:84% Maybe I can play that game and get rid of more harmonics 00:13:13.870 --> 00:13:16.070 align:middle line:90% by switching more. 00:13:16.070 --> 00:13:19.180 align:middle line:84% So I've only switched each device once per cycle, 00:13:19.180 --> 00:13:20.930 align:middle line:84% but suppose I'm willing to give up on that 00:13:20.930 --> 00:13:23.550 align:middle line:84% and switch more times per cycle on and off. 00:13:23.550 --> 00:13:26.570 align:middle line:84% That means I can create more times 00:13:26.570 --> 00:13:32.300 align:middle line:84% when I have positive or 0 or negative in any given time. 00:13:32.300 --> 00:13:35.750 align:middle line:84% I would like to do that perhaps in a way that helps me get 00:13:35.750 --> 00:13:37.430 align:middle line:90% rid of some higher harmonics. 00:13:37.430 --> 00:13:39.380 align:middle line:84% So now that I've gotten rid of the third, 00:13:39.380 --> 00:13:41.300 align:middle line:84% my lowest are the fifth and the seventh. 00:13:41.300 --> 00:13:45.320 align:middle line:84% Maybe I could figure out some way to get rid of the fifth. 00:13:45.320 --> 00:13:47.540 align:middle line:90% Well, how could I do that? 00:13:47.540 --> 00:13:51.800 align:middle line:84% Let's think of a slightly more sophisticated pulsing waveform 00:13:51.800 --> 00:13:53.450 align:middle line:90% than the white one. 00:13:53.450 --> 00:13:55.580 align:middle line:90% Here's-- let me do this. 00:13:55.580 --> 00:13:58.085 align:middle line:84% Suppose I came and I said, you know what? 00:13:58.085 --> 00:14:01.220 align:middle line:90% 00:14:01.220 --> 00:14:04.020 align:middle line:90% If I put a 0 right here-- 00:14:04.020 --> 00:14:05.640 align:middle line:90% so let me do the same thing. 00:14:05.640 --> 00:14:10.610 align:middle line:90% I will put a positive step here. 00:14:10.610 --> 00:14:14.390 align:middle line:84% But what I'm going to do is I'm going to look at this point. 00:14:14.390 --> 00:14:18.860 align:middle line:84% And if I cut out a little bit on each side of this, 00:14:18.860 --> 00:14:21.290 align:middle line:84% I don't change the fact that the third harmonic 00:14:21.290 --> 00:14:24.470 align:middle line:84% is gone because as long as I make 00:14:24.470 --> 00:14:27.410 align:middle line:84% sure the amount of positive in this multiple 00:14:27.410 --> 00:14:30.260 align:middle line:84% of the third harmonic balance is the amount of negative area 00:14:30.260 --> 00:14:33.290 align:middle line:84% with the third harmonic sine wave, 00:14:33.290 --> 00:14:35.580 align:middle line:84% I will maintain 0 third harmonic. 00:14:35.580 --> 00:14:38.330 align:middle line:84% So maybe I could do something like this. 00:14:38.330 --> 00:14:40.645 align:middle line:84% I will create a pulse waveform that does this. 00:14:40.645 --> 00:14:43.920 align:middle line:90% 00:14:43.920 --> 00:14:49.490 align:middle line:84% And maybe I'll do that again on this side. 00:14:49.490 --> 00:14:53.240 align:middle line:84% And I could-- and I would now want 00:14:53.240 --> 00:14:55.830 align:middle line:84% to keep my waveform half-wave-symmetric, 00:14:55.830 --> 00:14:59.530 align:middle line:84% so I'll do the same thing on the other side and do this. 00:14:59.530 --> 00:15:04.580 align:middle line:90% 00:15:04.580 --> 00:15:05.280 align:middle line:90% All right. 00:15:05.280 --> 00:15:07.220 align:middle line:84% Now if I've done this in a right way 00:15:07.220 --> 00:15:12.590 align:middle line:84% and I just make this symmetric about 60 degrees, 00:15:12.590 --> 00:15:14.540 align:middle line:84% and I put in some other angle-- maybe 00:15:14.540 --> 00:15:19.070 align:middle line:84% I'll call that capital delta, capital delta, 00:15:19.070 --> 00:15:24.320 align:middle line:84% I haven't introduced any third harmonic again 00:15:24.320 --> 00:15:28.250 align:middle line:84% because the multiple of this pink-colored waveform 00:15:28.250 --> 00:15:33.737 align:middle line:84% or orangey-colored waveform with this sine 3 omega 00:15:33.737 --> 00:15:35.570 align:middle line:84% t, the interval of it's still going to be 0, 00:15:35.570 --> 00:15:37.310 align:middle line:90% it's still orthogonal. 00:15:37.310 --> 00:15:39.910 align:middle line:90% Any questions about that? 00:15:39.910 --> 00:15:41.540 align:middle line:90% What do I do with delta? 00:15:41.540 --> 00:15:45.820 align:middle line:84% I can pick that-- what do I do with capital delta here? 00:15:45.820 --> 00:15:51.110 align:middle line:84% Maybe I can pick that so the fifth harmonic goes away. 00:15:51.110 --> 00:15:53.630 align:middle line:84% And in fact, I'll show you the one result-- 00:15:53.630 --> 00:15:56.880 align:middle line:84% and this is just a very simple example. 00:15:56.880 --> 00:15:59.440 align:middle line:90% This is what you would get. 00:15:59.440 --> 00:16:01.860 align:middle line:84% And in this case, I'm going to pick this waveform capital 00:16:01.860 --> 00:16:05.670 align:middle line:90% delta to be exactly 12 degrees. 00:16:05.670 --> 00:16:08.670 align:middle line:84% And because my artistry is not so good, 00:16:08.670 --> 00:16:11.290 align:middle line:84% I've decided to resort to the board-- 00:16:11.290 --> 00:16:15.005 align:middle line:84% I mean to the figure from the book. 00:16:15.005 --> 00:16:15.880 align:middle line:90% But this is the same. 00:16:15.880 --> 00:16:17.160 align:middle line:84% You can see this is the third-- this 00:16:17.160 --> 00:16:18.660 align:middle line:90% is how I cancel third harmonic. 00:16:18.660 --> 00:16:21.730 align:middle line:90% Here's sine 5 omega t. 00:16:21.730 --> 00:16:25.680 align:middle line:84% And if I multiply this original one pulse per half cycle 00:16:25.680 --> 00:16:27.390 align:middle line:84% waveform by sine 5 omega t, you can 00:16:27.390 --> 00:16:29.370 align:middle line:84% see that the negative area doesn't 00:16:29.370 --> 00:16:31.140 align:middle line:84% cancel the positive area, and hence, I 00:16:31.140 --> 00:16:34.184 align:middle line:90% have some fifth harmonic. 00:16:34.184 --> 00:16:39.780 align:middle line:84% But, if I then put in my nulls right here, like I said, 00:16:39.780 --> 00:16:41.950 align:middle line:84% centered about 60 degrees, it turns out, 00:16:41.950 --> 00:16:45.030 align:middle line:84% if I use that capital delta being 12 degrees, 00:16:45.030 --> 00:16:48.630 align:middle line:84% I can make the negative area exactly cancel the positive area 00:16:48.630 --> 00:16:50.520 align:middle line:90% and kill the fifth harmonic. 00:16:50.520 --> 00:16:53.083 align:middle line:84% So if I use this waveform-- and it's, again, 00:16:53.083 --> 00:16:54.750 align:middle line:84% half-wave-symmetric, it has three pulses 00:16:54.750 --> 00:16:57.940 align:middle line:84% in each half-cycle instead of one pulse in each half-cycle, 00:16:57.940 --> 00:17:00.960 align:middle line:84% I could not only keep the third harmonic killed, 00:17:00.960 --> 00:17:03.018 align:middle line:84% but also kill off the fifth, which 00:17:03.018 --> 00:17:05.060 align:middle line:84% would then leave the lowest harmonic the seventh. 00:17:05.060 --> 00:17:09.250 align:middle line:90% 00:17:09.250 --> 00:17:11.470 align:middle line:84% That makes it easier for this filter 00:17:11.470 --> 00:17:13.670 align:middle line:84% to kill it, or equivalently, for the same filter, 00:17:13.670 --> 00:17:17.920 align:middle line:84% I get more purer sine wave at my actual output over here. 00:17:17.920 --> 00:17:19.605 align:middle line:90% Any questions about that? 00:17:19.605 --> 00:17:22.300 align:middle line:90% 00:17:22.300 --> 00:17:24.849 align:middle line:84% This kind of trick where I'm going 00:17:24.849 --> 00:17:28.510 align:middle line:84% to introduce new pulses in each half-cycle 00:17:28.510 --> 00:17:32.080 align:middle line:90% is called harmonic elimination. 00:17:32.080 --> 00:17:39.670 align:middle line:90% 00:17:39.670 --> 00:17:43.030 align:middle line:84% And we can often view it as exactly through this method, 00:17:43.030 --> 00:17:48.460 align:middle line:84% is how do I synthesize pulses that satisfy this orthogonality 00:17:48.460 --> 00:17:49.305 align:middle line:90% with each harmonic? 00:17:49.305 --> 00:17:51.930 align:middle line:90% 00:17:51.930 --> 00:17:54.465 align:middle line:90% Now, is this the best I can do? 00:17:54.465 --> 00:17:56.940 align:middle line:84% Well, first of all, if I synthesize this waveform, 00:17:56.940 --> 00:17:59.855 align:middle line:84% how many times do you think I switch each device per cycle? 00:17:59.855 --> 00:18:03.430 align:middle line:90% 00:18:03.430 --> 00:18:06.552 align:middle line:84% Without working it out, you can essentially-- 00:18:06.552 --> 00:18:08.260 align:middle line:84% each time you get a pulse per half-cycle, 00:18:08.260 --> 00:18:10.390 align:middle line:84% you have to add on some switching transitions, 00:18:10.390 --> 00:18:12.610 align:middle line:84% so you get higher switching losses. 00:18:12.610 --> 00:18:19.990 align:middle line:84% So what I'm really doing is I'm trading switching loss for lower 00:18:19.990 --> 00:18:23.110 align:middle line:90% filtered output components. 00:18:23.110 --> 00:18:28.940 align:middle line:84% Now, is this as good as I can do to get rid of the fifth? 00:18:28.940 --> 00:18:29.440 align:middle line:90% No. 00:18:29.440 --> 00:18:32.800 align:middle line:90% It turns out that-- 00:18:32.800 --> 00:18:36.560 align:middle line:84% and you can write equations and start solving them to do this. 00:18:36.560 --> 00:18:41.690 align:middle line:84% And I've given-- in KPVS chapter 8, there's two references. 00:18:41.690 --> 00:18:44.200 align:middle line:84% There's a classic Patel and Hoft reference 00:18:44.200 --> 00:18:46.460 align:middle line:84% that introduced this general idea, 00:18:46.460 --> 00:18:49.750 align:middle line:84% and then there's a more recent Chasen reference 00:18:49.750 --> 00:18:53.350 align:middle line:84% that really goes into detail about the best ways to do this. 00:18:53.350 --> 00:18:56.710 align:middle line:84% Is each time you put in one pulse and a half-cycle, 00:18:56.710 --> 00:19:01.160 align:middle line:84% you can ultimately pick one harmonic to eliminate. 00:19:01.160 --> 00:19:02.740 align:middle line:84% And when we talk about one harmonic, 00:19:02.740 --> 00:19:04.890 align:middle line:84% we're keeping everything half-wave-symmetric, 00:19:04.890 --> 00:19:08.810 align:middle line:84% so we care about one odd harmonic. 00:19:08.810 --> 00:19:11.250 align:middle line:84% So we're assuming our evens are all gone. 00:19:11.250 --> 00:19:12.480 align:middle line:90% I only have odds. 00:19:12.480 --> 00:19:15.050 align:middle line:84% So I could actually pick this waveform 00:19:15.050 --> 00:19:18.050 align:middle line:84% because I have 2 degrees of freedom-- 00:19:18.050 --> 00:19:19.850 align:middle line:84% the little delta in this example, 00:19:19.850 --> 00:19:23.660 align:middle line:84% gamma, I could kill the fifth and the second, 00:19:23.660 --> 00:19:26.180 align:middle line:84% fifth and the seventh, or the third and the fifth, 00:19:26.180 --> 00:19:28.400 align:middle line:84% but only have two pulses and a half-cycle, 00:19:28.400 --> 00:19:30.890 align:middle line:84% and so have fewer switching transitions than I 00:19:30.890 --> 00:19:32.870 align:middle line:90% do in this example waveform. 00:19:32.870 --> 00:19:36.950 align:middle line:84% And generally, the rule is, for each pulse per half-cycle, 00:19:36.950 --> 00:19:41.510 align:middle line:84% I get to pick one odd harmonic to eliminate. 00:19:41.510 --> 00:19:45.210 align:middle line:90% Any questions about that? 00:19:45.210 --> 00:19:48.430 align:middle line:84% AUDIENCE: Is there like a practical limit to how high 00:19:48.430 --> 00:19:50.010 align:middle line:90% of a harmonic [INAUDIBLE]? 00:19:50.010 --> 00:19:53.310 align:middle line:84% DAVID PERREAULT: Yeah, that is an excellent question. 00:19:53.310 --> 00:20:01.680 align:middle line:84% There is because if I want to eliminate the 97th harmonic, 00:20:01.680 --> 00:20:05.700 align:middle line:84% what that means is I've got to control the timing resolution 00:20:05.700 --> 00:20:10.680 align:middle line:84% just so that I make that multiple with the 97th harmonic 00:20:10.680 --> 00:20:11.770 align:middle line:90% integrate at 0. 00:20:11.770 --> 00:20:16.080 align:middle line:84% So the higher the number of harmonics I want to eliminate, 00:20:16.080 --> 00:20:19.840 align:middle line:84% the more precise the timing requirements become. 00:20:19.840 --> 00:20:22.580 align:middle line:84% So you can imagine, how would you do this? 00:20:22.580 --> 00:20:24.570 align:middle line:84% You'd have a lookup table that steps along 00:20:24.570 --> 00:20:28.200 align:middle line:84% and it says at every time slice, which switches should be on? 00:20:28.200 --> 00:20:31.440 align:middle line:84% And the number of time slices I need 00:20:31.440 --> 00:20:33.390 align:middle line:84% to break it up into accurately enough 00:20:33.390 --> 00:20:35.580 align:middle line:84% to murder those low-frequency harmonics gets 00:20:35.580 --> 00:20:36.940 align:middle line:90% worse and worse and worse. 00:20:36.940 --> 00:20:40.980 align:middle line:84% The other limit is, eventually I switch so many times that it's-- 00:20:40.980 --> 00:20:45.600 align:middle line:84% that my switching loss goes crazy and I don't like it. 00:20:45.600 --> 00:20:48.480 align:middle line:84% The other thing I'll point out is 00:20:48.480 --> 00:20:50.850 align:middle line:84% if I took this to its logical conclusion-- 00:20:50.850 --> 00:20:54.120 align:middle line:84% so suppose I came off and I said, OK, 00:20:54.120 --> 00:20:56.310 align:middle line:84% I want to get rid of a bunch of harmonics. 00:20:56.310 --> 00:20:59.250 align:middle line:84% So imagine my pulse waveform looked like this, 00:20:59.250 --> 00:21:00.468 align:middle line:90% and I'm just going to-- 00:21:00.468 --> 00:21:02.010 align:middle line:84% I'm not going to be quantitative now, 00:21:02.010 --> 00:21:03.690 align:middle line:84% but maybe I would have something that 00:21:03.690 --> 00:21:09.030 align:middle line:84% looked like this, a pulse and another pulse, another pulse. 00:21:09.030 --> 00:21:11.740 align:middle line:84% And these are all very controlled widths. 00:21:11.740 --> 00:21:13.740 align:middle line:90% And then I do this. 00:21:13.740 --> 00:21:18.630 align:middle line:90% 00:21:18.630 --> 00:21:19.702 align:middle line:90% Something like that. 00:21:19.702 --> 00:21:21.910 align:middle line:84% I don't think I have the right number of pulses here, 00:21:21.910 --> 00:21:25.530 align:middle line:84% but I have something that's shaped to generate 00:21:25.530 --> 00:21:29.460 align:middle line:84% a set of pulses that make this, at least at low frequencies when 00:21:29.460 --> 00:21:32.940 align:middle line:84% I filter it, look closer to a sine wave. 00:21:32.940 --> 00:21:34.440 align:middle line:84% Maybe these middle pulses would tend 00:21:34.440 --> 00:21:38.106 align:middle line:84% to be bigger than these edge pulses. 00:21:38.106 --> 00:21:41.280 align:middle line:84% And I pick the pulsing times and positions 00:21:41.280 --> 00:21:44.230 align:middle line:84% exactly to give me this magic cancellation. 00:21:44.230 --> 00:21:47.940 align:middle line:84% What I am doing, if I think about this in the frequency 00:21:47.940 --> 00:21:54.300 align:middle line:84% domain, is I've got my fundamental that I want. 00:21:54.300 --> 00:21:58.900 align:middle line:84% I've got some filter that's going to then cut off. 00:21:58.900 --> 00:22:02.490 align:middle line:84% And what I'm doing by doing this is I'm 00:22:02.490 --> 00:22:06.070 align:middle line:84% eliminating a bunch of the low-order harmonics. 00:22:06.070 --> 00:22:09.413 align:middle line:84% But you can see that this has a bunch of high-frequency content. 00:22:09.413 --> 00:22:11.830 align:middle line:84% I'm switching up and down and up and down and up and down. 00:22:11.830 --> 00:22:14.730 align:middle line:84% That suggests to you there's a bunch of high frequency stuff 00:22:14.730 --> 00:22:15.870 align:middle line:90% there. 00:22:15.870 --> 00:22:21.900 align:middle line:84% So if I look at the original waveform Vx, this waveform-- 00:22:21.900 --> 00:22:25.440 align:middle line:84% In fact, if I looked at its total harmonic distortion, 00:22:25.440 --> 00:22:27.090 align:middle line:90% it actually is higher. 00:22:27.090 --> 00:22:29.970 align:middle line:84% What I'm doing is I'm killing off the low-frequency stuff 00:22:29.970 --> 00:22:33.660 align:middle line:84% and then I'm really growing the high-frequency stuff. 00:22:33.660 --> 00:22:36.780 align:middle line:84% So it's not like my waveform actually 00:22:36.780 --> 00:22:38.590 align:middle line:90% has less harmonic content. 00:22:38.590 --> 00:22:40.750 align:middle line:84% It actually has considerably more. 00:22:40.750 --> 00:22:44.160 align:middle line:84% And depending upon how you choose to-- 00:22:44.160 --> 00:22:48.330 align:middle line:84% what angles you choose to kill off your low-order harmonics, 00:22:48.330 --> 00:22:51.340 align:middle line:84% you will end up with more or less stuff at high frequencies. 00:22:51.340 --> 00:22:54.450 align:middle line:84% But undoubtedly, you're going to get more high-frequency stuff 00:22:54.450 --> 00:22:57.934 align:middle line:84% because you've got a lot of switching going on. 00:22:57.934 --> 00:23:02.820 align:middle line:84% The thing about that is, is if I think about a low-pass filter, 00:23:02.820 --> 00:23:07.390 align:middle line:84% it's easy for me to murder the high frequencies in a cut-off. 00:23:07.390 --> 00:23:11.590 align:middle line:84% It's hard for me to eliminate stuff that's near cut-off. 00:23:11.590 --> 00:23:15.210 align:middle line:84% So I'm making the challenge of basically pushing the frequency 00:23:15.210 --> 00:23:22.220 align:middle line:84% content up and keeping the stuff that's hard to filter low. 00:23:22.220 --> 00:23:26.740 align:middle line:84% So I'm not getting rid of high-frequency content, 00:23:26.740 --> 00:23:28.240 align:middle line:84% I'm actually increasing it, but what 00:23:28.240 --> 00:23:31.850 align:middle line:84% I am is getting rid of the stuff that's hardest to filter. 00:23:31.850 --> 00:23:33.520 align:middle line:90% Any questions about that? 00:23:33.520 --> 00:23:37.270 align:middle line:90% 00:23:37.270 --> 00:23:37.840 align:middle line:90% OK. 00:23:37.840 --> 00:23:42.120 align:middle line:84% So that's one strategy I can use. 00:23:42.120 --> 00:23:44.400 align:middle line:90% What else could I do? 00:23:44.400 --> 00:23:49.420 align:middle line:84% And actually, in my original waveform here that we showed, 00:23:49.420 --> 00:23:52.720 align:middle line:84% we saw a little bit of a hint of this. 00:23:52.720 --> 00:23:55.210 align:middle line:90% And here's the idea. 00:23:55.210 --> 00:24:02.550 align:middle line:84% Suppose I have some waveform x of t 00:24:02.550 --> 00:24:11.020 align:middle line:84% that I might express as the summation of n 00:24:11.020 --> 00:24:20.440 align:middle line:84% equals 1 to infinity x of n sine of n omega 0 t plus phi sub n. 00:24:20.440 --> 00:24:23.290 align:middle line:90% It has some Fourier series. 00:24:23.290 --> 00:24:27.160 align:middle line:84% Then if I had some delayed version of this waveform-- 00:24:27.160 --> 00:24:32.380 align:middle line:84% let me call this x of t minus t1, 00:24:32.380 --> 00:24:34.072 align:middle line:84% what would his content look like? 00:24:34.072 --> 00:24:35.530 align:middle line:84% Well, I just go to this expression, 00:24:35.530 --> 00:24:37.840 align:middle line:90% I substitute in t minus t1. 00:24:37.840 --> 00:24:41.410 align:middle line:84% So I would get, OK, this is the summation of n 00:24:41.410 --> 00:24:51.950 align:middle line:84% equals 1 to infinity of x sub n sine of n omega 0 t minus t1-- 00:24:51.950 --> 00:25:00.760 align:middle line:84% so I'll say this is n omega 0 t minus n omega 0 t1 plus phi n. 00:25:00.760 --> 00:25:11.220 align:middle line:84% So if I define some delta phi 1, the phase 00:25:11.220 --> 00:25:13.650 align:middle line:84% shift at the fundamental associated 00:25:13.650 --> 00:25:21.780 align:middle line:84% with this time shift t1, that would simply be omega 0 t1. 00:25:21.780 --> 00:25:25.410 align:middle line:84% That is, if I'm shifting by a time t1, 00:25:25.410 --> 00:25:34.450 align:middle line:84% I'm shifting the fundamental by a phase omega 0 t1. 00:25:34.450 --> 00:25:37.220 align:middle line:84% What am I shifting the n-th harmonic by? 00:25:37.220 --> 00:25:38.650 align:middle line:90% This is the fundamental shift. 00:25:38.650 --> 00:25:45.940 align:middle line:90% 00:25:45.940 --> 00:25:52.870 align:middle line:84% The n-th harmonic, delta phi n, is equal to n delta phi 1. 00:25:52.870 --> 00:25:58.650 align:middle line:84% So this is the n-th harmonic phase shift. 00:25:58.650 --> 00:26:02.620 align:middle line:90% 00:26:02.620 --> 00:26:07.420 align:middle line:84% And so here, you can see, because the n-th harmonic-- 00:26:07.420 --> 00:26:14.480 align:middle line:84% this thing-- and maybe I should be careful, define delta phi 1 00:26:14.480 --> 00:26:17.240 align:middle line:84% as being minus omega t1, and maybe this 00:26:17.240 --> 00:26:19.380 align:middle line:90% is minus n omega delta phi 1. 00:26:19.380 --> 00:26:22.550 align:middle line:90% So this is delta phi 1-- 00:26:22.550 --> 00:26:25.350 align:middle line:90% or n delta phi 1 is the amount. 00:26:25.350 --> 00:26:28.770 align:middle line:90% I'm shifting the n-th harmonic. 00:26:28.770 --> 00:26:30.940 align:middle line:84% Does that makes sense to everybody? 00:26:30.940 --> 00:26:33.350 align:middle line:90% So why am I telling you this? 00:26:33.350 --> 00:26:36.020 align:middle line:84% Suppose I did some game like this-- 00:26:36.020 --> 00:26:40.850 align:middle line:84% and I would have said that this is maybe not the most common way 00:26:40.850 --> 00:26:44.030 align:middle line:84% to play this game, but you can actually 00:26:44.030 --> 00:26:46.010 align:middle line:84% find people who have built converters this way, 00:26:46.010 --> 00:26:47.500 align:middle line:90% especially for high power. 00:26:47.500 --> 00:26:50.120 align:middle line:84% Suppose I had some DC input voltage. 00:26:50.120 --> 00:26:53.300 align:middle line:90% 00:26:53.300 --> 00:26:55.450 align:middle line:90% And I will build an inverter. 00:26:55.450 --> 00:27:00.740 align:middle line:90% 00:27:00.740 --> 00:27:02.690 align:middle line:90% Here's the first inverter. 00:27:02.690 --> 00:27:04.850 align:middle line:84% And I'll give him an output voltage Vx1. 00:27:04.850 --> 00:27:10.220 align:middle line:90% 00:27:10.220 --> 00:27:16.460 align:middle line:84% And now let me build a second inverter for switches. 00:27:16.460 --> 00:27:19.070 align:middle line:90% 00:27:19.070 --> 00:27:24.080 align:middle line:84% And I'm going to give him an output that's Vx2. 00:27:24.080 --> 00:27:26.420 align:middle line:84% And what I'll do is I'll transform or couple 00:27:26.420 --> 00:27:27.955 align:middle line:90% these two outputs like this. 00:27:27.955 --> 00:27:32.570 align:middle line:90% 00:27:32.570 --> 00:27:37.400 align:middle line:84% And let me just imagine, this is a one-to-one transformer. 00:27:37.400 --> 00:27:40.260 align:middle line:84% And here's another one-to-one transformer. 00:27:40.260 --> 00:27:45.810 align:middle line:90% 00:27:45.810 --> 00:27:46.410 align:middle line:90% OK. 00:27:46.410 --> 00:27:47.140 align:middle line:90% So this is Vx2. 00:27:47.140 --> 00:27:50.027 align:middle line:90% 00:27:50.027 --> 00:27:51.360 align:middle line:90% That's a one-to-one transformer. 00:27:51.360 --> 00:27:59.520 align:middle line:84% That means that this total Vx is simply equal to Vx1 plus Vx2. 00:27:59.520 --> 00:28:02.700 align:middle line:84% So I've got a first inverter generating 00:28:02.700 --> 00:28:05.760 align:middle line:84% a first pattern and a second inverter 00:28:05.760 --> 00:28:07.990 align:middle line:90% generating a second pattern. 00:28:07.990 --> 00:28:11.660 align:middle line:84% So what is the harmonic-- what is the content? 00:28:11.660 --> 00:28:13.090 align:middle line:90% If I said that-- 00:28:13.090 --> 00:28:16.210 align:middle line:90% 00:28:16.210 --> 00:28:23.692 align:middle line:84% suppose I say Vx1 is equal to the summation of n equals 1 00:28:23.692 --> 00:28:25.150 align:middle line:84% to infinity-- let's assume I'm just 00:28:25.150 --> 00:28:34.840 align:middle line:84% doing odd waveforms of V sub n sine n omega 0 t plus phi sub n. 00:28:34.840 --> 00:28:37.690 align:middle line:90% 00:28:37.690 --> 00:28:48.220 align:middle line:84% That means that Vx2, if I shift by t1, Vx2 is just 00:28:48.220 --> 00:28:52.150 align:middle line:84% going to be summation of n equals 1 to infinity of V sub n 00:28:52.150 --> 00:29:00.500 align:middle line:84% sine n omega 0 t plus n delta phi 1 00:29:00.500 --> 00:29:08.095 align:middle line:84% plus phi n where delta phi 1 is just equal to omega 0 t1. 00:29:08.095 --> 00:29:10.700 align:middle line:90% 00:29:10.700 --> 00:29:11.460 align:middle line:90% All right. 00:29:11.460 --> 00:29:14.946 align:middle line:84% So now I have these two waveforms and I add them. 00:29:14.946 --> 00:29:18.470 align:middle line:84% So now I'm going to get Vx is simply 00:29:18.470 --> 00:29:31.940 align:middle line:84% equal to the summation of Vn times sine of n omega 0 t1-- 00:29:31.940 --> 00:29:39.560 align:middle line:84% I'm sorry, omega 0 t plus 5 phi n plus sine n omega 0 t plus phi 00:29:39.560 --> 00:29:48.040 align:middle line:90% n plus delta plus n delta phi 1. 00:29:48.040 --> 00:29:49.050 align:middle line:90% All right. 00:29:49.050 --> 00:29:54.840 align:middle line:90% Well, what happens if I make-- 00:29:54.840 --> 00:29:57.150 align:middle line:84% the difference between-- the phase difference 00:29:57.150 --> 00:30:00.990 align:middle line:84% between this waveform and this waveform 00:30:00.990 --> 00:30:05.880 align:middle line:90% is precisely n delta phi 1. 00:30:05.880 --> 00:30:10.810 align:middle line:84% At each harmonic, I've got a phase shift that's-- 00:30:10.810 --> 00:30:12.380 align:middle line:84% the n-th harmonic has a phase shift 00:30:12.380 --> 00:30:14.690 align:middle line:90% that's n times delta phi 1. 00:30:14.690 --> 00:30:20.337 align:middle line:84% Well, what happens when n delta phi 1 is pi, or 180 degrees? 00:30:20.337 --> 00:30:21.295 align:middle line:90% What's going to happen? 00:30:21.295 --> 00:30:25.280 align:middle line:90% 00:30:25.280 --> 00:30:28.770 align:middle line:84% This is going to be 180 degrees out of this 00:30:28.770 --> 00:30:31.610 align:middle line:90% and they're going to cancel. 00:30:31.610 --> 00:30:36.500 align:middle line:84% So if I'm clever, if I go build two waveforms that 00:30:36.500 --> 00:30:42.100 align:middle line:84% are shifted in time by just the right magic amount such 00:30:42.100 --> 00:30:45.010 align:middle line:84% that when I add them, there are 180 degrees of-- the harmonic 00:30:45.010 --> 00:30:47.540 align:middle line:84% that I care about is 180 degrees out of phase, 00:30:47.540 --> 00:30:49.780 align:middle line:90% it's going to cancel. 00:30:49.780 --> 00:30:53.806 align:middle line:84% This trick is called harmonic cancellation. 00:30:53.806 --> 00:31:04.090 align:middle line:90% 00:31:04.090 --> 00:31:07.510 align:middle line:90% Let's go see an example of this. 00:31:07.510 --> 00:31:09.250 align:middle line:84% So what I've done is let's pretend 00:31:09.250 --> 00:31:13.090 align:middle line:84% I've synthesized those two waveforms. 00:31:13.090 --> 00:31:15.940 align:middle line:84% And suppose the two waveforms I've 00:31:15.940 --> 00:31:18.160 align:middle line:84% synthesized with each of those bridges, 00:31:18.160 --> 00:31:20.330 align:middle line:84% each switch is switching once per cycle, 00:31:20.330 --> 00:31:23.500 align:middle line:84% so it's got a delta of 30 degrees, and I do the first guy. 00:31:23.500 --> 00:31:25.870 align:middle line:84% And then I do the second guy, but I'm 00:31:25.870 --> 00:31:29.620 align:middle line:90% shifting him by 36 degrees. 00:31:29.620 --> 00:31:32.540 align:middle line:84% I'm offsetting him by 36 degrees. 00:31:32.540 --> 00:31:34.263 align:middle line:90% What's 5 times 36? 00:31:34.263 --> 00:31:38.190 align:middle line:90% 00:31:38.190 --> 00:31:39.900 align:middle line:90% 180. 00:31:39.900 --> 00:31:44.285 align:middle line:84% So then when I shift them, then I add them, 00:31:44.285 --> 00:31:47.020 align:middle line:84% in the bottom waveform you see the result here. 00:31:47.020 --> 00:31:51.210 align:middle line:84% And this waveform-- actually, that 00:31:51.210 --> 00:31:54.210 align:middle line:84% looks a lot more like a sine wave, doesn't it? 00:31:54.210 --> 00:31:58.380 align:middle line:84% That's because I've cancelled the fifth harmonic. 00:31:58.380 --> 00:32:01.800 align:middle line:84% In fact, not only have I canceled the fifth harmonic, 00:32:01.800 --> 00:32:04.130 align:middle line:84% I've canceled the 10th, although we didn't have 10th. 00:32:04.130 --> 00:32:07.480 align:middle line:84% I've also canceled the 15th, the 20th, the 25th, and everything. 00:32:07.480 --> 00:32:10.470 align:middle line:84% So basically, unlike harmonic elimination 00:32:10.470 --> 00:32:12.840 align:middle line:84% where I'm playing this game by getting something that 00:32:12.840 --> 00:32:14.970 align:middle line:84% has no low-frequency content but has 00:32:14.970 --> 00:32:19.600 align:middle line:84% a bunch of high-frequency stuff, harmonic cancelation 00:32:19.600 --> 00:32:21.670 align:middle line:84% cancels out some waveform you care 00:32:21.670 --> 00:32:25.390 align:middle line:84% about plus multiples of that, so it looks 00:32:25.390 --> 00:32:29.260 align:middle line:90% a lot more like a sine wave. 00:32:29.260 --> 00:32:30.865 align:middle line:84% I've now taken each of my devices-- 00:32:30.865 --> 00:32:32.740 align:middle line:84% I've still only switched them once per cycle, 00:32:32.740 --> 00:32:34.200 align:middle line:84% I've just got twice as many devices 00:32:34.200 --> 00:32:35.707 align:middle line:84% that are half the size to process 00:32:35.707 --> 00:32:36.790 align:middle line:90% the given amount of power. 00:32:36.790 --> 00:32:40.890 align:middle line:84% So I've of paid for this trick with a little bit of complexity. 00:32:40.890 --> 00:32:49.540 align:middle line:84% But I get a waveform out that is very much closer to pure 00:32:49.540 --> 00:32:54.026 align:middle line:84% and much easier to filter to get what you want. 00:32:54.026 --> 00:32:58.070 align:middle line:84% Any questions about that as a trick? 00:32:58.070 --> 00:32:58.834 align:middle line:90% Yeah? 00:32:58.834 --> 00:33:00.417 align:middle line:84% AUDIENCE: [INAUDIBLE] seems like you'd 00:33:00.417 --> 00:33:03.170 align:middle line:84% be really careful about whether we actually end up 00:33:03.170 --> 00:33:05.060 align:middle line:90% out of phase with one another. 00:33:05.060 --> 00:33:06.620 align:middle line:84% If you're in a high-power situation 00:33:06.620 --> 00:33:10.100 align:middle line:84% where your wires are heating up a bit, does that change this? 00:33:10.100 --> 00:33:11.970 align:middle line:84% Do have to worry about that sort of thing? 00:33:11.970 --> 00:33:13.928 align:middle line:84% DAVID PERREAULT: Yeah, I wouldn't worry so much 00:33:13.928 --> 00:33:17.900 align:middle line:84% about the wires heating up, but your broader point 00:33:17.900 --> 00:33:20.810 align:middle line:90% is a good one. 00:33:20.810 --> 00:33:26.420 align:middle line:84% Just as I'm kind of limited by timing angles with this guy, 00:33:26.420 --> 00:33:30.230 align:middle line:84% how much I phase-shifted, if I don't control that very well, 00:33:30.230 --> 00:33:33.440 align:middle line:84% if there's delays between the circuit, 00:33:33.440 --> 00:33:38.700 align:middle line:84% depending upon how much absolute time that is, I'm limited-- 00:33:38.700 --> 00:33:41.780 align:middle line:84% essentially, this harmonic is limited by how accurately I 00:33:41.780 --> 00:33:43.310 align:middle line:90% control that t1. 00:33:43.310 --> 00:33:50.250 align:middle line:84% So I want to be very good about shifting this the right amount. 00:33:50.250 --> 00:33:56.880 align:middle line:84% Now, I will say that this kind of trick, 00:33:56.880 --> 00:34:01.180 align:middle line:84% I do need some better hardware to do it. 00:34:01.180 --> 00:34:03.160 align:middle line:90% I can't synthesize this level. 00:34:03.160 --> 00:34:06.387 align:middle line:84% This level is sometimes called a three-level waveform. 00:34:06.387 --> 00:34:08.429 align:middle line:84% And you say, well, there's not three levels here, 00:34:08.429 --> 00:34:11.460 align:middle line:84% but the way they count it is one-half of this waveform 00:34:11.460 --> 00:34:15.480 align:middle line:84% is 0 plus VDC over 2 and plus VDC. 00:34:15.480 --> 00:34:18.370 align:middle line:90% One polar side has three levels. 00:34:18.370 --> 00:34:20.370 align:middle line:84% So they call that a three-level waveform. 00:34:20.370 --> 00:34:22.840 align:middle line:84% With a three-level waveform, I can cancel-- 00:34:22.840 --> 00:34:25.260 align:middle line:84% I can choose to cancel a harmonic. 00:34:25.260 --> 00:34:28.469 align:middle line:84% Now, what we'll see later is sometimes 00:34:28.469 --> 00:34:30.060 align:middle line:84% I might choose the third, sometimes I 00:34:30.060 --> 00:34:32.800 align:middle line:84% might choose the fifth, it depends on what I'm doing. 00:34:32.800 --> 00:34:36.750 align:middle line:84% And in fact, even if you come back to this original waveform 00:34:36.750 --> 00:34:47.690 align:middle line:84% that I drew, this waveform, the original white waveform that had 00:34:47.690 --> 00:34:53.530 align:middle line:84% those 30-degree shifts, what I could really view that is 00:34:53.530 --> 00:34:58.210 align:middle line:84% one-half of this bridge generating a square wave 00:34:58.210 --> 00:35:02.170 align:middle line:84% and the other half of the bridge generating a square wave 00:35:02.170 --> 00:35:07.180 align:middle line:84% that's 60 degrees out of phase at the fundamental and sends 00:35:07.180 --> 00:35:08.350 align:middle line:90% three times-- 00:35:08.350 --> 00:35:11.770 align:middle line:84% at three times the frequency at the third harmonic-- or six, 00:35:11.770 --> 00:35:15.850 align:middle line:84% ninth, 12th, et cetera, it'll be 180 degrees out of phase. 00:35:15.850 --> 00:35:19.420 align:middle line:84% So in fact, the pulse waveform, the once-- 00:35:19.420 --> 00:35:21.490 align:middle line:84% one pulse-per-cycle waveform that I showed you, 00:35:21.490 --> 00:35:23.740 align:middle line:84% you could really view it as a cancellation 00:35:23.740 --> 00:35:26.200 align:middle line:84% because I have half-bridges that are each generating 00:35:26.200 --> 00:35:29.185 align:middle line:84% square waves, and I get a harmonic cancellation 00:35:29.185 --> 00:35:31.643 align:middle line:84% between those square waves, killing the triple n harmonics. 00:35:31.643 --> 00:35:34.830 align:middle line:90% 00:35:34.830 --> 00:35:36.990 align:middle line:84% But yes, as I go up in frequency, 00:35:36.990 --> 00:35:38.280 align:middle line:90% it becomes harder to do. 00:35:38.280 --> 00:35:42.870 align:middle line:84% And I need fancier circuitry to cancel 00:35:42.870 --> 00:35:45.370 align:middle line:84% lots and lots of harmonics, but sometimes it's worth it. 00:35:45.370 --> 00:35:47.370 align:middle line:84% And people-- as we'll see, people 00:35:47.370 --> 00:35:52.830 align:middle line:84% actually do quite fancy circuits to help them do that. 00:35:52.830 --> 00:35:55.680 align:middle line:90% Other questions? 00:35:55.680 --> 00:35:58.620 align:middle line:84% AUDIENCE: In this example, the third harmonic is still 00:35:58.620 --> 00:36:01.658 align:middle line:84% cancelled because of the shape of the first two? 00:36:01.658 --> 00:36:02.950 align:middle line:90% DAVID PERREAULT: Yeah, exactly. 00:36:02.950 --> 00:36:05.710 align:middle line:84% That's the beautiful thing about this. 00:36:05.710 --> 00:36:09.160 align:middle line:84% Is once I've cancelled the waveform, 00:36:09.160 --> 00:36:12.010 align:middle line:84% and then I take some cancellation and then I add it 00:36:12.010 --> 00:36:15.280 align:middle line:84% to some other waveform, I can't introduce new harmonics. 00:36:15.280 --> 00:36:19.400 align:middle line:84% I'm just taking some existing set of Fourier coefficients. 00:36:19.400 --> 00:36:24.880 align:middle line:84% The most that can happen is they add up and stay constant. 00:36:24.880 --> 00:36:26.850 align:middle line:84% In other words, if I take half-sized waveforms 00:36:26.850 --> 00:36:28.600 align:middle line:84% and add them up, the most I'm going to get 00:36:28.600 --> 00:36:29.810 align:middle line:90% is back to the original. 00:36:29.810 --> 00:36:30.760 align:middle line:84% Other than that, I'm just going to get 00:36:30.760 --> 00:36:32.320 align:middle line:90% some degree of cancellation. 00:36:32.320 --> 00:36:35.510 align:middle line:84% So I don't introduce new frequencies by doing this trick, 00:36:35.510 --> 00:36:37.250 align:middle line:90% I just kill frequencies. 00:36:37.250 --> 00:36:38.080 align:middle line:90% So yes. 00:36:38.080 --> 00:36:40.900 align:middle line:84% So my original white waveform, because it's 00:36:40.900 --> 00:36:45.250 align:middle line:84% harmonic cancelation of the half-bridges, 00:36:45.250 --> 00:36:49.510 align:middle line:84% I have no triple n harmonics, no 3, 6, 9, 12, 15, 18, 00:36:49.510 --> 00:36:50.210 align:middle line:90% and so forth. 00:36:50.210 --> 00:36:55.170 align:middle line:84% I don't have any evens because of the half-wave symmetry. 00:36:55.170 --> 00:37:01.080 align:middle line:84% And then if I do this trick, I kill off all the 5n harmonics. 00:37:01.080 --> 00:37:02.540 align:middle line:90% And you can keep that game up. 00:37:02.540 --> 00:37:05.020 align:middle line:84% For a more complicated circuit, I'd kill even more, 00:37:05.020 --> 00:37:07.880 align:middle line:90% and the waveform just becomes-- 00:37:07.880 --> 00:37:12.300 align:middle line:84% it starts to become very small steppy, close to a sine wave, 00:37:12.300 --> 00:37:14.590 align:middle line:84% and you get very beautiful waveforms. 00:37:14.590 --> 00:37:17.290 align:middle line:90% 00:37:17.290 --> 00:37:19.090 align:middle line:90% Yeah? 00:37:19.090 --> 00:37:22.617 align:middle line:84% AUDIENCE: Does the 6-degree displacement matter? 00:37:22.617 --> 00:37:23.950 align:middle line:90% DAVID PERREAULT: Does the which? 00:37:23.950 --> 00:37:25.408 align:middle line:84% AUDIENCE: Because it's a sine wave, 00:37:25.408 --> 00:37:26.960 align:middle line:90% but it's displaced by 6 degrees. 00:37:26.960 --> 00:37:27.460 align:middle line:90% So-- 00:37:27.460 --> 00:37:29.954 align:middle line:90% DAVID PERREAULT: It's a wave-- 00:37:29.954 --> 00:37:30.780 align:middle line:90% oh, yeah. 00:37:30.780 --> 00:37:33.080 align:middle line:84% So actually, if you look in the lecture notes, 00:37:33.080 --> 00:37:35.360 align:middle line:84% instead of displacing it by 36 degrees, 00:37:35.360 --> 00:37:39.050 align:middle line:84% I displace one by plus 18 and one by minus 18. 00:37:39.050 --> 00:37:41.910 align:middle line:84% And you get the same cancelation and then it stays odd. 00:37:41.910 --> 00:37:42.410 align:middle line:90% Yeah. 00:37:42.410 --> 00:37:45.690 align:middle line:84% So, yeah, I don't have to have 0 plus a delayed. 00:37:45.690 --> 00:37:48.110 align:middle line:84% I can have minus half the delay and plus half the delay 00:37:48.110 --> 00:37:50.980 align:middle line:84% and it still stays as an odd waveform. 00:37:50.980 --> 00:37:51.800 align:middle line:90% Excellent question. 00:37:51.800 --> 00:37:54.170 align:middle line:84% AUDIENCE: So do you also use this trick to just increase 00:37:54.170 --> 00:37:55.955 align:middle line:90% the AC output voltage? 00:37:55.955 --> 00:37:57.330 align:middle line:84% DAVID PERREAULT: You could, yeah. 00:37:57.330 --> 00:37:59.330 align:middle line:84% So, yeah, what would be what would 00:37:59.330 --> 00:38:02.480 align:middle line:90% be another benefit of this? 00:38:02.480 --> 00:38:07.880 align:middle line:84% The Vx has twice the amplitude of VDC I could synthesize. 00:38:07.880 --> 00:38:09.260 align:middle line:84% Another way-- and that's just for 00:38:09.260 --> 00:38:14.000 align:middle line:84% this particular implementation, which is only one way to do it. 00:38:14.000 --> 00:38:16.520 align:middle line:84% We will see next class other circuits 00:38:16.520 --> 00:38:19.070 align:middle line:90% that their benefit is they-- 00:38:19.070 --> 00:38:22.610 align:middle line:84% for a given input voltage, they use lower-voltage switches 00:38:22.610 --> 00:38:24.060 align:middle line:90% to get the same output. 00:38:24.060 --> 00:38:27.920 align:middle line:84% So there's a lot of ways you can take advantage 00:38:27.920 --> 00:38:29.940 align:middle line:90% of the complexity you inherit. 00:38:29.940 --> 00:38:31.850 align:middle line:84% The downside to this one is principally 00:38:31.850 --> 00:38:35.630 align:middle line:84% that I have transformers that are rated for the output-- 00:38:35.630 --> 00:38:39.290 align:middle line:84% that are rated for the frequency I'm outputting. 00:38:39.290 --> 00:38:40.790 align:middle line:84% In some cases I might not like that, 00:38:40.790 --> 00:38:44.660 align:middle line:84% but there's other circuits that get around that. 00:38:44.660 --> 00:38:47.150 align:middle line:84% Let me point out that this notion-- 00:38:47.150 --> 00:38:52.750 align:middle line:84% this cancellation notion is a very widespread one. 00:38:52.750 --> 00:38:55.600 align:middle line:84% It gets used in a lot of kinds of circuits. 00:38:55.600 --> 00:38:58.890 align:middle line:84% Let me show you another one that maybe 00:38:58.890 --> 00:39:05.060 align:middle line:84% is completely different application space, 00:39:05.060 --> 00:39:11.440 align:middle line:90% but suppose I had a-- 00:39:11.440 --> 00:39:17.005 align:middle line:84% suppose I had a buck converter that looked like this. 00:39:17.005 --> 00:39:21.332 align:middle line:90% 00:39:21.332 --> 00:39:25.080 align:middle line:90% And here's my output capacitor. 00:39:25.080 --> 00:39:26.650 align:middle line:90% And here's my load. 00:39:26.650 --> 00:39:30.780 align:middle line:84% And maybe this output capacitor is a microprocessor. 00:39:30.780 --> 00:39:33.090 align:middle line:84% So maybe this is the 12-volt input, 00:39:33.090 --> 00:39:36.490 align:middle line:84% and I'm delivering voltage to my microprocessor. 00:39:36.490 --> 00:39:42.520 align:middle line:84% This current here is a triangle wave. 00:39:42.520 --> 00:39:45.690 align:middle line:84% So maybe this current looks something like this. 00:39:45.690 --> 00:39:59.100 align:middle line:84% It's got some DC value and it's got some ripple 00:39:59.100 --> 00:40:00.940 align:middle line:90% at the switching frequency. 00:40:00.940 --> 00:40:03.900 align:middle line:90% 00:40:03.900 --> 00:40:08.410 align:middle line:84% Maybe, if I came in with a second unit-- 00:40:08.410 --> 00:40:12.990 align:middle line:84% and I will build another parallel guy here, 00:40:12.990 --> 00:40:19.830 align:middle line:84% and I will switch him with some phase shift 00:40:19.830 --> 00:40:21.840 align:middle line:84% and he'll have an output current. 00:40:21.840 --> 00:40:24.090 align:middle line:84% And maybe I make this output current-- 00:40:24.090 --> 00:40:25.235 align:middle line:90% here's T over 2. 00:40:25.235 --> 00:40:28.338 align:middle line:90% 00:40:28.338 --> 00:40:33.600 align:middle line:84% And here's-- gotta be careful here, here's 3T over 2. 00:40:33.600 --> 00:40:35.860 align:middle line:90% Here's 2T. 00:40:35.860 --> 00:40:38.020 align:middle line:90% Maybe I will shift this one. 00:40:38.020 --> 00:40:42.145 align:middle line:84% So that he starts half a cycle later like this. 00:40:42.145 --> 00:40:52.320 align:middle line:90% 00:40:52.320 --> 00:40:57.080 align:middle line:84% What total current goes into the output and capacitor? 00:40:57.080 --> 00:41:00.210 align:middle line:84% Well, it's the sum of this one and this one. 00:41:00.210 --> 00:41:05.610 align:middle line:84% But if I basically make orange and pink shifted 00:41:05.610 --> 00:41:08.130 align:middle line:84% by 180 degrees of the fundamental, 00:41:08.130 --> 00:41:11.340 align:middle line:84% that means the fundamental ripple will cancel. 00:41:11.340 --> 00:41:15.900 align:middle line:84% If there's second harmonic component in the pink waveform, 00:41:15.900 --> 00:41:18.120 align:middle line:84% the second harmonic of the orange waveform 00:41:18.120 --> 00:41:21.580 align:middle line:84% will be shifted by 360 degrees at the second harmonic. 00:41:21.580 --> 00:41:24.360 align:middle line:84% So the fundamentals will cancel, the second harmonics 00:41:24.360 --> 00:41:25.380 align:middle line:90% will reinforce. 00:41:25.380 --> 00:41:27.480 align:middle line:90% And the net ripple-- 00:41:27.480 --> 00:41:29.550 align:middle line:84% I'm not even going to try to draw it here, 00:41:29.550 --> 00:41:31.950 align:middle line:90% but it would basically be-- 00:41:31.950 --> 00:41:36.680 align:middle line:84% have a fundamental that was at twice the switching frequency. 00:41:36.680 --> 00:41:40.190 align:middle line:84% So I get something that's both smaller amplitude-- 00:41:40.190 --> 00:41:42.860 align:middle line:84% and you can show that the amplitude will be, at most, 00:41:42.860 --> 00:41:44.180 align:middle line:90% half-- 00:41:44.180 --> 00:41:47.150 align:middle line:84% and twice the frequency, which makes 00:41:47.150 --> 00:41:51.640 align:middle line:84% it four times as easy for that capacitor to filter it. 00:41:51.640 --> 00:41:55.220 align:middle line:84% So this same kind of trick of harmonic cancellation-- here, 00:41:55.220 --> 00:41:57.720 align:middle line:84% I'm cancelling the fundamentals because what I would like is 00:41:57.720 --> 00:41:59.110 align:middle line:90% a DC output current. 00:41:59.110 --> 00:42:01.660 align:middle line:84% This technique is called interleaving. 00:42:01.660 --> 00:42:05.520 align:middle line:90% 00:42:05.520 --> 00:42:09.180 align:middle line:84% It's very widely used, and in fact, most motherboards, 00:42:09.180 --> 00:42:11.920 align:middle line:84% if you look at them, they don't have a buck converter 00:42:11.920 --> 00:42:13.170 align:middle line:90% or something driving the load. 00:42:13.170 --> 00:42:15.900 align:middle line:84% They'll have a whole bunch of buck converters lined up, 00:42:15.900 --> 00:42:18.540 align:middle line:84% and they phase-shift them all just such 00:42:18.540 --> 00:42:21.750 align:middle line:84% that the first n harmonics will go away, 00:42:21.750 --> 00:42:23.820 align:middle line:84% and the ripple frequency will be much higher 00:42:23.820 --> 00:42:26.280 align:middle line:84% than the switching frequency of the individual converters. 00:42:26.280 --> 00:42:28.230 align:middle line:84% That also means that you can play 00:42:28.230 --> 00:42:30.780 align:middle line:84% with response speeds of that because one 00:42:30.780 --> 00:42:35.260 align:middle line:84% will react much quicker than if they were all in phase. 00:42:35.260 --> 00:42:37.890 align:middle line:84% If I put these two converters in phase, all I would get 00:42:37.890 --> 00:42:40.200 align:middle line:84% is twice the ripple current at the fundamental. 00:42:40.200 --> 00:42:43.290 align:middle line:84% Instead, I get, at most, half at twice the frequency. 00:42:43.290 --> 00:42:47.040 align:middle line:84% And I can play that game out to n different converters 00:42:47.040 --> 00:42:47.820 align:middle line:90% interleaved. 00:42:47.820 --> 00:42:54.980 align:middle line:84% So, this trick of harmonic cancellation is really neat. 00:42:54.980 --> 00:42:56.000 align:middle line:90% It's widely used. 00:42:56.000 --> 00:42:57.680 align:middle line:90% It's used in DC-DC converters. 00:42:57.680 --> 00:43:01.420 align:middle line:84% It's also used to kill off unwanted harmonics in inverters. 00:43:01.420 --> 00:43:04.253 align:middle line:84% Harmonic elimination is also used. 00:43:04.253 --> 00:43:06.670 align:middle line:84% There's nothing that says you couldn't do the two of those 00:43:06.670 --> 00:43:07.190 align:middle line:90% together. 00:43:07.190 --> 00:43:10.390 align:middle line:84% You saw that, Instead of using that just one 00:43:10.390 --> 00:43:14.290 align:middle line:84% pulse per half-cycle, I could use some pulses per half-cycle 00:43:14.290 --> 00:43:17.560 align:middle line:84% to get-- eliminate some harmonics, use some phase shift, 00:43:17.560 --> 00:43:20.350 align:middle line:84% eliminate some more, and I can intermix those two 00:43:20.350 --> 00:43:25.110 align:middle line:84% and they get very nicely, easily-filtered waveforms. 00:43:25.110 --> 00:43:28.580 align:middle line:90% Any questions about that? 00:43:28.580 --> 00:43:29.080 align:middle line:90% OK. 00:43:29.080 --> 00:43:32.410 align:middle line:84% Just to demonstrate all this, Rafa 00:43:32.410 --> 00:43:34.510 align:middle line:84% has been kind enough to set up a demo. 00:43:34.510 --> 00:43:38.230 align:middle line:84% So I'll let her show you some actual experimental waveforms 00:43:38.230 --> 00:43:41.080 align:middle line:84% from an inverter that was built by a student who 00:43:41.080 --> 00:43:43.270 align:middle line:90% took this class. 00:43:43.270 --> 00:43:46.880 align:middle line:84% RAFA ISLAM: OK, so, what are we looking at-- am I too loud? 00:43:46.880 --> 00:43:47.410 align:middle line:90% Sorry. 00:43:47.410 --> 00:43:51.100 align:middle line:84% What are we looking at, at the top is-- 00:43:51.100 --> 00:43:51.890 align:middle line:90% guess what? 00:43:51.890 --> 00:43:53.500 align:middle line:90% What inverter is it? 00:43:53.500 --> 00:43:56.830 align:middle line:84% I s an inverter board that has four switches. 00:43:56.830 --> 00:44:00.010 align:middle line:84% And we're probing at the output voltage. 00:44:00.010 --> 00:44:04.390 align:middle line:84% So, we had a DC input voltage of 15 volt, 00:44:04.390 --> 00:44:06.190 align:middle line:90% and then we're switching it. 00:44:06.190 --> 00:44:09.040 align:middle line:84% And as you can see, as we saw in class, 00:44:09.040 --> 00:44:12.970 align:middle line:84% it uses the three levels, 0, positive, negative, 00:44:12.970 --> 00:44:17.860 align:middle line:84% to create something that somehow resembles a sinusoid. 00:44:17.860 --> 00:44:21.080 align:middle line:84% And as you can see, it has half-wave symmetry, 00:44:21.080 --> 00:44:25.420 align:middle line:84% so we do not have any a2K, b2KR, the even harmonics. 00:44:25.420 --> 00:44:28.130 align:middle line:84% So we only have to worry about the odd harmonics. 00:44:28.130 --> 00:44:31.520 align:middle line:84% Now let's-- at the bottom down here, 00:44:31.520 --> 00:44:34.370 align:middle line:84% we are looking at the FFT of the voltage waveform. 00:44:34.370 --> 00:44:39.710 align:middle line:84% So the voltage magnitudes at different harmonics. 00:44:39.710 --> 00:44:48.360 align:middle line:84% Now, if I move my cursor, at the first spike where it happens, 00:44:48.360 --> 00:44:52.590 align:middle line:84% it's really hard to see, but you can see on the top-right 00:44:52.590 --> 00:44:55.170 align:middle line:84% that it's happening at 600 hertz, which 00:44:55.170 --> 00:44:57.390 align:middle line:84% means that's our switching frequency, 00:44:57.390 --> 00:45:00.090 align:middle line:90% And it's the fundamental. 00:45:00.090 --> 00:45:05.563 align:middle line:84% And then, let's see where the next spike happens. 00:45:05.563 --> 00:45:09.670 align:middle line:90% 00:45:09.670 --> 00:45:13.240 align:middle line:84% The second-highest spike happens at 2.9 kilohertz, 00:45:13.240 --> 00:45:16.100 align:middle line:84% which is very close to the fifth harmonic. 00:45:16.100 --> 00:45:17.090 align:middle line:90% What does that mean? 00:45:17.090 --> 00:45:19.820 align:middle line:84% That means we do not have any third harmonic. 00:45:19.820 --> 00:45:23.420 align:middle line:84% So we eliminated the third harmonic from the circuit. 00:45:23.420 --> 00:45:27.360 align:middle line:90% Now, let's do another thing. 00:45:27.360 --> 00:45:30.190 align:middle line:90% It's going crazy, but let's see. 00:45:30.190 --> 00:45:30.700 align:middle line:90% OK. 00:45:30.700 --> 00:45:34.840 align:middle line:84% Now, like we saw in the class, if we introduce pulse-- 00:45:34.840 --> 00:45:38.200 align:middle line:84% or notches on both sides of the waveform, 00:45:38.200 --> 00:45:40.070 align:middle line:90% it still have a symmetric. 00:45:40.070 --> 00:45:42.740 align:middle line:84% So we're still doing the same thing. 00:45:42.740 --> 00:45:48.130 align:middle line:84% But interestingly, if we move to the second high spike, 00:45:48.130 --> 00:45:54.230 align:middle line:84% it's almost about 4.2 kilohertz, which is the seventh harmonic. 00:45:54.230 --> 00:45:58.150 align:middle line:84% So we erased-- we eliminated third and fifth harmonic. 00:45:58.150 --> 00:46:01.900 align:middle line:84% Now we can take this a step further. 00:46:01.900 --> 00:46:05.850 align:middle line:90% Let me run it, and stop it. 00:46:05.850 --> 00:46:07.188 align:middle line:90% OK. 00:46:07.188 --> 00:46:08.080 align:middle line:90% Yes. 00:46:08.080 --> 00:46:12.830 align:middle line:84% So now if move all the way back, that 00:46:12.830 --> 00:46:17.000 align:middle line:84% will be the second spike occurs at 5.3 kilohertz, which 00:46:17.000 --> 00:46:18.660 align:middle line:90% is like the ninth harmonics. 00:46:18.660 --> 00:46:23.430 align:middle line:84% So we eliminated all the lower third, fifth, seventh harmonics. 00:46:23.430 --> 00:46:27.110 align:middle line:84% You will notice that they're getting a bit-- so the higher 00:46:27.110 --> 00:46:29.340 align:middle line:84% harmonics are getting higher in magnitude, 00:46:29.340 --> 00:46:32.900 align:middle line:84% but that's OK because we care about the lower harmonics 00:46:32.900 --> 00:46:37.970 align:middle line:84% more because it's easier to get rid of-- the higher harmonics 00:46:37.970 --> 00:46:40.100 align:middle line:84% will need smaller filter components 00:46:40.100 --> 00:46:42.540 align:middle line:84% or smaller devices for applications. 00:46:42.540 --> 00:46:43.210 align:middle line:90% That's all. 00:46:43.210 --> 00:46:45.340 align:middle line:90% 00:46:45.340 --> 00:46:46.340 align:middle line:90% DAVID PERREAULT: Thanks. 00:46:46.340 --> 00:46:49.880 align:middle line:84% So I guess what we've seen today is 00:46:49.880 --> 00:46:54.620 align:middle line:84% harmonic elimination where we introduce extra pulses carefully 00:46:54.620 --> 00:46:55.730 align:middle line:90% timed. 00:46:55.730 --> 00:46:57.980 align:middle line:84% We've also seen harmonic cancellation 00:46:57.980 --> 00:47:00.770 align:middle line:84% where we take multiple waveforms and shift them and add them up 00:47:00.770 --> 00:47:04.490 align:middle line:84% to a murder the harmonics we don't like. 00:47:04.490 --> 00:47:07.130 align:middle line:90% Those can be both used together. 00:47:07.130 --> 00:47:09.830 align:middle line:84% What we'll start seeing next time is firstly, well, 00:47:09.830 --> 00:47:12.660 align:middle line:84% other ways we can do this kind of harmonic cancellation 00:47:12.660 --> 00:47:16.085 align:middle line:84% or harmonic elimination game, more sophisticated circuits. 00:47:16.085 --> 00:47:18.085 align:middle line:84% And after that, we'll start talking about, well, 00:47:18.085 --> 00:47:19.380 align:middle line:90% what happens if-- 00:47:19.380 --> 00:47:21.870 align:middle line:84% we've posited this on the whole thing 00:47:21.870 --> 00:47:24.180 align:middle line:84% I want to make as a sine wave, which is often true. 00:47:24.180 --> 00:47:26.740 align:middle line:84% If I'm going to run a lot of motors, I want a sine wave. 00:47:26.740 --> 00:47:29.903 align:middle line:84% If I'm trying to generate something 00:47:29.903 --> 00:47:32.070 align:middle line:84% for the AC grid, which is supposed to be sinusoidal, 00:47:32.070 --> 00:47:32.862 align:middle line:90% I want a sine wave. 00:47:32.862 --> 00:47:34.945 align:middle line:84% But there's times where we don't want a sine wave, 00:47:34.945 --> 00:47:36.310 align:middle line:90% so what do we do in those cases? 00:47:36.310 --> 00:47:40.700 align:middle line:84% And we'll pick those topics up next class, so have a great day. 00:47:40.700 --> 00:47:43.000 align:middle line:90%