WEBVTT 00:00:00.000 --> 00:00:02.425 align:middle line:90% [SQUEAKING] 00:00:02.425 --> 00:00:04.365 align:middle line:90% [RUSTLING] 00:00:04.365 --> 00:00:05.335 align:middle line:90% [CLICKING] 00:00:05.335 --> 00:00:08.913 align:middle line:90% 00:00:08.913 --> 00:00:10.830 align:middle line:84% DAVID PERREAULT: OK, why don't we get started? 00:00:10.830 --> 00:00:13.030 align:middle line:90% [SIDE CONVERSATION] 00:00:13.030 --> 00:00:15.970 align:middle line:84% We're going to switch topics today, 00:00:15.970 --> 00:00:18.730 align:middle line:84% and we're going to talk about a different class of power 00:00:18.730 --> 00:00:22.710 align:middle line:84% converter circuit called an inverter. 00:00:22.710 --> 00:00:34.390 align:middle line:84% So inverters, or DC-to-AC converters, 00:00:34.390 --> 00:00:37.313 align:middle line:84% are important for a lot of applications, right. 00:00:37.313 --> 00:00:39.730 align:middle line:84% So if you're going to build a motor drive, a lot of motors 00:00:39.730 --> 00:00:41.140 align:middle line:90% need to run off AC. 00:00:41.140 --> 00:00:43.180 align:middle line:84% If you're going to have an uninterruptible power 00:00:43.180 --> 00:00:47.200 align:middle line:84% supply to power your AC-powered devices when the grid goes out, 00:00:47.200 --> 00:00:48.030 align:middle line:90% you need it. 00:00:48.030 --> 00:00:53.020 align:middle line:84% If you're getting power from a wind turbine or, for example, 00:00:53.020 --> 00:00:56.710 align:middle line:84% a DC solar panel, you need to take that and convert it 00:00:56.710 --> 00:00:58.907 align:middle line:84% into AC in order to feed the power into the grid. 00:00:58.907 --> 00:01:00.490 align:middle line:84% So there's a lot of applications where 00:01:00.490 --> 00:01:06.250 align:middle line:84% you need to come from DC input and put out AC output. 00:01:06.250 --> 00:01:14.560 align:middle line:84% This is the topic of KPVS chapter 8, 00:01:14.560 --> 00:01:17.290 align:middle line:84% and we're going to spend the next few lectures talking 00:01:17.290 --> 00:01:19.630 align:middle line:84% about some of the issues associated 00:01:19.630 --> 00:01:22.370 align:middle line:90% with DC-to-AC conversion. 00:01:22.370 --> 00:01:25.850 align:middle line:84% Now, before we jump into that, let's 00:01:25.850 --> 00:01:30.120 align:middle line:84% just have a really brief review of expressing waveforms, 00:01:30.120 --> 00:01:32.720 align:middle line:84% periodic waveforms, in terms of Fourier series, 00:01:32.720 --> 00:01:36.380 align:middle line:84% just because it's going to be really useful in thinking about, 00:01:36.380 --> 00:01:40.310 align:middle line:84% What kinds of AC waveforms might we want to synthesize, right? 00:01:40.310 --> 00:01:44.900 align:middle line:84% So if I have a periodic waveform f of t, 00:01:44.900 --> 00:01:51.980 align:middle line:84% right, I might express that as some DC term plus n equals 1 00:01:51.980 --> 00:01:56.240 align:middle line:84% to infinity of a sub n sine of n omega 00:01:56.240 --> 00:02:04.730 align:middle line:84% 0 t plus b sub n cosine of n omega 0 t, where this 00:02:04.730 --> 00:02:07.460 align:middle line:84% is the angular frequency associated 00:02:07.460 --> 00:02:09.620 align:middle line:84% with the fundamental period capital T. 00:02:09.620 --> 00:02:15.180 align:middle line:84% So omega 0 is equal to 2 pi over the period. 00:02:15.180 --> 00:02:18.710 align:middle line:84% And then we can go find, using orthogonality, 00:02:18.710 --> 00:02:23.080 align:middle line:84% the expression to get the Fourier series coefficients. 00:02:23.080 --> 00:02:26.130 align:middle line:84% So a sub n would be equal to 2 over T 00:02:26.130 --> 00:02:34.650 align:middle line:84% the integral over t of f of t sine of n omega 0 t dt, 00:02:34.650 --> 00:02:42.450 align:middle line:84% and b sub n would simply be 2 over T integral over t f of t 00:02:42.450 --> 00:02:46.450 align:middle line:90% cosine of n omega 0 t dt. 00:02:46.450 --> 00:02:46.950 align:middle line:90% OK. 00:02:46.950 --> 00:02:49.440 align:middle line:84% So I can take any periodic waveform, 00:02:49.440 --> 00:02:55.020 align:middle line:84% break it down into some DC fundamental and harmonic 00:02:55.020 --> 00:02:58.770 align:middle line:84% description, and one way to do that is in terms of sine 00:02:58.770 --> 00:02:59.940 align:middle line:90% and cosine components. 00:02:59.940 --> 00:03:04.770 align:middle line:84% And we'll see why we picked this particular representation, 00:03:04.770 --> 00:03:06.930 align:middle line:90% shortly. 00:03:06.930 --> 00:03:07.590 align:middle line:90% OK. 00:03:07.590 --> 00:03:10.920 align:middle line:84% So it turns out that different waveforms 00:03:10.920 --> 00:03:16.600 align:middle line:84% have different special characteristics, all right. 00:03:16.600 --> 00:03:22.110 align:middle line:84% So I'd like to think about a few different kinds of waveforms, 00:03:22.110 --> 00:03:23.370 align:middle line:90% and some of this, you've seen. 00:03:23.370 --> 00:03:27.460 align:middle line:84% Some of this maybe will be a little bit less familiar. 00:03:27.460 --> 00:03:30.600 align:middle line:84% The first kind of waveform we might think of 00:03:30.600 --> 00:03:33.471 align:middle line:90% is an even waveform. 00:03:33.471 --> 00:03:38.370 align:middle line:84% Right, what does it mean for a waveform to be even? 00:03:38.370 --> 00:03:43.380 align:middle line:84% If I have T, an even waveform looks symmetric about T 00:03:43.380 --> 00:03:44.950 align:middle line:90% equals 0. 00:03:44.950 --> 00:03:45.450 align:middle line:90% OK. 00:03:45.450 --> 00:03:47.200 align:middle line:90% So it flips about T equals 0. 00:03:47.200 --> 00:03:49.600 align:middle line:84% So maybe I have a waveform that looks like this. 00:03:49.600 --> 00:03:53.940 align:middle line:84% Say this is T over 2 and this is minus T over 2. 00:03:53.940 --> 00:03:56.590 align:middle line:84% And it might look like this, for example. 00:03:56.590 --> 00:03:57.090 align:middle line:90% OK. 00:03:57.090 --> 00:04:02.820 align:middle line:84% So if I just flip it about T equals 0, nothing changes. 00:04:02.820 --> 00:04:04.460 align:middle line:90% OK. 00:04:04.460 --> 00:04:08.810 align:middle line:84% Well, why might I care about an even waveform? 00:04:08.810 --> 00:04:17.306 align:middle line:84% Because all the a-sub-n terms are 0 in that case, right. 00:04:17.306 --> 00:04:23.310 align:middle line:84% One way to think about that is, how do I find the a-sub-n terms? 00:04:23.310 --> 00:04:28.920 align:middle line:84% I take f of t, if this is f of t, 00:04:28.920 --> 00:04:31.380 align:middle line:84% and I multiply it by sine n omega t. 00:04:31.380 --> 00:04:33.420 align:middle line:84% So if I multiplied it by sine omega t, 00:04:33.420 --> 00:04:38.760 align:middle line:84% maybe I'd be multiplying it by something that looks like this. 00:04:38.760 --> 00:04:41.160 align:middle line:84% And then I take the white and the orange 00:04:41.160 --> 00:04:42.960 align:middle line:84% and multiply them and integrate them. 00:04:42.960 --> 00:04:47.780 align:middle line:84% And whatever's over here exactly is the negative 00:04:47.780 --> 00:04:51.720 align:middle line:84% of what's over here, and I get 0 in the integral 00:04:51.720 --> 00:04:52.810 align:middle line:90% when I integrate it out. 00:04:52.810 --> 00:04:58.710 align:middle line:84% So one way to think about it is that an even times an odd 00:04:58.710 --> 00:05:02.760 align:middle line:84% is odd, and then when I take the integral of an odd waveform, 00:05:02.760 --> 00:05:04.410 align:middle line:90% I get zero. 00:05:04.410 --> 00:05:05.100 align:middle line:90% OK. 00:05:05.100 --> 00:05:10.020 align:middle line:84% Another way to think about it is sines are odd waveforms, 00:05:10.020 --> 00:05:11.550 align:middle line:90% and this is an even waveform. 00:05:11.550 --> 00:05:15.060 align:middle line:84% So it makes sense that I ought to build an even waveform out 00:05:15.060 --> 00:05:18.660 align:middle line:84% of even components, right, and the cosine terms 00:05:18.660 --> 00:05:21.850 align:middle line:84% are the even components of my Fourier series. 00:05:21.850 --> 00:05:28.470 align:middle line:84% So an even waveform will only have b-sub-n terms, only 00:05:28.470 --> 00:05:31.530 align:middle line:90% cosines in DC. 00:05:31.530 --> 00:05:32.370 align:middle line:90% OK. 00:05:32.370 --> 00:05:36.870 align:middle line:84% Of course, we can also have an odd waveform, right. 00:05:36.870 --> 00:05:41.680 align:middle line:84% Even is x of t is equal to x of minus t. 00:05:41.680 --> 00:05:45.550 align:middle line:84% Right, it flips about the T-equals-0 axis. 00:05:45.550 --> 00:05:51.850 align:middle line:84% Odd is x of t is equal to minus x of minus t. 00:05:51.850 --> 00:05:52.530 align:middle line:90% OK. 00:05:52.530 --> 00:05:54.940 align:middle line:90% So what does that mean? 00:05:54.940 --> 00:06:00.220 align:middle line:84% An odd waveform would mean, if I had this, 00:06:00.220 --> 00:06:04.390 align:middle line:90% then I should have this. 00:06:04.390 --> 00:06:11.430 align:middle line:84% OK, and then I could replicate this cycle to cycle, 00:06:11.430 --> 00:06:13.570 align:middle line:84% something like this, for example, 00:06:13.570 --> 00:06:18.450 align:middle line:84% OK, so where this is minus t over 2 and this is t over 2. 00:06:18.450 --> 00:06:19.710 align:middle line:90% OK. 00:06:19.710 --> 00:06:24.610 align:middle line:84% So basically, I go across the axis and then flip down. 00:06:24.610 --> 00:06:28.740 align:middle line:84% I flip both this way and this way, and I get some symmetry. 00:06:28.740 --> 00:06:30.114 align:middle line:90% OK. 00:06:30.114 --> 00:06:34.360 align:middle line:90% And so that's a-- 00:06:34.360 --> 00:06:37.190 align:middle line:84% that characteristic, what do I have about that? 00:06:37.190 --> 00:06:42.447 align:middle line:84% Well, here, right, this is composed of only odd waveforms. 00:06:42.447 --> 00:06:44.030 align:middle line:84% I could make the same kind of argument 00:06:44.030 --> 00:06:48.690 align:middle line:84% and say that b-sub-n's ought to equal 0. 00:06:48.690 --> 00:06:49.190 align:middle line:90% OK. 00:06:49.190 --> 00:06:53.610 align:middle line:84% So I only build an odd waveform out of sine waves. 00:06:53.610 --> 00:06:54.110 align:middle line:90% Why? 00:06:54.110 --> 00:06:56.690 align:middle line:84% Because sines are odd, and hence, those 00:06:56.690 --> 00:06:58.580 align:middle line:84% are the subcomponents that I get. 00:06:58.580 --> 00:06:59.080 align:middle line:90% OK. 00:06:59.080 --> 00:07:00.630 align:middle line:90% Any questions about that? 00:07:00.630 --> 00:07:04.020 align:middle line:90% 00:07:04.020 --> 00:07:07.660 align:middle line:84% So, many people will be familiar with even and odd. 00:07:07.660 --> 00:07:12.450 align:middle line:84% There's another decomposition that we can think of, 00:07:12.450 --> 00:07:16.320 align:middle line:84% which is one where we talk about something 00:07:16.320 --> 00:07:18.086 align:middle line:90% that's half-wave symmetric. 00:07:18.086 --> 00:07:26.350 align:middle line:90% 00:07:26.350 --> 00:07:28.520 align:middle line:84% And what does half-wave symmetric mean? 00:07:28.520 --> 00:07:34.090 align:middle line:84% That means x of t is equal to minus x of t minus capital T 00:07:34.090 --> 00:07:35.370 align:middle line:90% over 2. 00:07:35.370 --> 00:07:38.540 align:middle line:84% OK, so what does a half-wave symmetric waveform look like? 00:07:38.540 --> 00:07:43.850 align:middle line:84% A half-wave symmetric waveform looks like this. 00:07:43.850 --> 00:07:50.490 align:middle line:84% Suppose I have something that looks like this between 0 00:07:50.490 --> 00:07:52.080 align:middle line:90% and T over 2. 00:07:52.080 --> 00:07:53.190 align:middle line:90% OK. 00:07:53.190 --> 00:07:56.960 align:middle line:84% If I go back half a cycle, I flip it. 00:07:56.960 --> 00:07:59.600 align:middle line:84% All right, so here, I'll slide back half a cycle and I flip it, 00:07:59.600 --> 00:08:02.420 align:middle line:84% and I'll get a waveform that looks like this. 00:08:02.420 --> 00:08:03.170 align:middle line:90% OK. 00:08:03.170 --> 00:08:07.400 align:middle line:84% So here's what my half-wave symmetric waveform 00:08:07.400 --> 00:08:09.650 align:middle line:84% would look like, and so forth, OK, 00:08:09.650 --> 00:08:12.090 align:middle line:90% where this is minus T over 2. 00:08:12.090 --> 00:08:15.470 align:middle line:84% So if I go at any point in the waveform 00:08:15.470 --> 00:08:20.430 align:middle line:84% and I go back half a cycle, I flip. 00:08:20.430 --> 00:08:22.030 align:middle line:90% OK. 00:08:22.030 --> 00:08:29.560 align:middle line:84% Well, why might I be interested in half-wave symmetry? 00:08:29.560 --> 00:08:38.740 align:middle line:84% Well, that's because all of the a-sub-2-k's and b-sub-2-k's are 00:08:38.740 --> 00:08:41.450 align:middle line:90% equal to 0. 00:08:41.450 --> 00:08:46.020 align:middle line:84% OK, all the even harmonic terms are equal to 0. 00:08:46.020 --> 00:08:47.160 align:middle line:90% OK. 00:08:47.160 --> 00:08:49.030 align:middle line:90% Why would that be the case? 00:08:49.030 --> 00:08:53.760 align:middle line:84% Well, how would I find, say, the second harmonic component 00:08:53.760 --> 00:08:54.640 align:middle line:90% of this thing? 00:08:54.640 --> 00:08:56.100 align:middle line:90% I would multiply it-- 00:08:56.100 --> 00:09:01.480 align:middle line:84% for example, suppose I wanted the a-sub-2 component, 00:09:01.480 --> 00:09:04.260 align:middle line:84% I would multiply it by sine of 2 omega t, which 00:09:04.260 --> 00:09:07.590 align:middle line:84% would be multiplying it by this and multiplying it by this 00:09:07.590 --> 00:09:09.850 align:middle line:84% and then integrating over the full cycle. 00:09:09.850 --> 00:09:12.600 align:middle line:84% And you can see that the product of orange and white here, 00:09:12.600 --> 00:09:14.880 align:middle line:84% this half of the waveform, is exactly 00:09:14.880 --> 00:09:17.215 align:middle line:84% the complement of the product of the orange 00:09:17.215 --> 00:09:18.840 align:middle line:84% and white in this half of the waveform. 00:09:18.840 --> 00:09:21.210 align:middle line:84% When I integrate them, I get zero. 00:09:21.210 --> 00:09:21.720 align:middle line:90% OK. 00:09:21.720 --> 00:09:25.890 align:middle line:84% The same thing would have been true had I had a cosine wave. 00:09:25.890 --> 00:09:26.550 align:middle line:90% OK. 00:09:26.550 --> 00:09:29.310 align:middle line:84% So the point of being half-wave symmetric 00:09:29.310 --> 00:09:32.220 align:middle line:84% is that because of the orthogonality involved, 00:09:32.220 --> 00:09:39.410 align:middle line:84% all of the even terms are zero, OK, so it has no even harmonic. 00:09:39.410 --> 00:09:42.470 align:middle line:84% Now, the complement to this is something 00:09:42.470 --> 00:09:52.590 align:middle line:84% I would call a half-wave repeating, OK, 00:09:52.590 --> 00:09:56.210 align:middle line:90% and it looks like this. 00:09:56.210 --> 00:09:59.240 align:middle line:84% If I had something that was doing 00:09:59.240 --> 00:10:03.410 align:middle line:84% this in one half of the waveform for T over 2 to 0, 00:10:03.410 --> 00:10:07.580 align:middle line:84% it would just repeat half a cycle back-- that is, x of t 00:10:07.580 --> 00:10:11.990 align:middle line:84% is equal to x of t minus capital T over 2. 00:10:11.990 --> 00:10:13.181 align:middle line:90% OK. 00:10:13.181 --> 00:10:17.090 align:middle line:84% So if I have defined a period capital T, 00:10:17.090 --> 00:10:19.460 align:middle line:84% that just means the waveform repeats twice 00:10:19.460 --> 00:10:22.590 align:middle line:90% in that capital period T. OK. 00:10:22.590 --> 00:10:25.460 align:middle line:84% Or another way to think about it is 00:10:25.460 --> 00:10:28.220 align:middle line:84% the fundamental period of this waveform 00:10:28.220 --> 00:10:33.680 align:middle line:84% is actually t over 2, OK, but if we defined-- 00:10:33.680 --> 00:10:35.900 align:middle line:84% if we're looking at across some period T, 00:10:35.900 --> 00:10:46.010 align:middle line:84% this comprises only a-sub-2-k's and b-sub-2-k's. 00:10:46.010 --> 00:10:53.320 align:middle line:84% That is, a sub 2 n plus 1 and b sub 2 n plus 1 is equal to 0, 00:10:53.320 --> 00:10:53.820 align:middle line:90% right. 00:10:53.820 --> 00:10:57.740 align:middle line:90% So this has no even harmonics. 00:10:57.740 --> 00:11:00.750 align:middle line:84% This kind of waveform only has even harmonics. 00:11:00.750 --> 00:11:03.310 align:middle line:84% It has no odd harmonics or fundamental. 00:11:03.310 --> 00:11:03.810 align:middle line:90% OK. 00:11:03.810 --> 00:11:06.915 align:middle line:90% Any questions about that? 00:11:06.915 --> 00:11:09.430 align:middle line:90% 00:11:09.430 --> 00:11:09.930 align:middle line:90% Yeah? 00:11:09.930 --> 00:11:12.850 align:middle line:84% AUDIENCE: Is it only the case if we define the period to be T? 00:11:12.850 --> 00:11:13.905 align:middle line:90% DAVID PERREAULT: Yes. 00:11:13.905 --> 00:11:14.740 align:middle line:90% All right. 00:11:14.740 --> 00:11:17.190 align:middle line:90% Now, why might I do that? 00:11:17.190 --> 00:11:22.470 align:middle line:84% Because, firstly, we can sometimes 00:11:22.470 --> 00:11:24.360 align:middle line:90% decompose waveforms, right. 00:11:24.360 --> 00:11:27.690 align:middle line:84% I could say that, OK, I could come up with some waveform 00:11:27.690 --> 00:11:31.620 align:middle line:90% f of t, OK, and I could say-- 00:11:31.620 --> 00:11:38.280 align:middle line:84% so I could take some arbitrary f of t and come up with f of t 00:11:38.280 --> 00:11:45.630 align:middle line:84% is equal to some even component plus some odd component, OK, 00:11:45.630 --> 00:11:48.240 align:middle line:84% and decompose that into different parts, 00:11:48.240 --> 00:11:57.300 align:middle line:84% where f even is equal to x of t plus x of minus t over 2, 00:11:57.300 --> 00:12:03.840 align:middle line:84% and f odd is equal to x of t minus x of minus t over 2. 00:12:03.840 --> 00:12:04.620 align:middle line:90% OK. 00:12:04.620 --> 00:12:07.770 align:middle line:84% So essentially, I can decompose a waveform 00:12:07.770 --> 00:12:11.665 align:middle line:84% into one term that basically is all 00:12:11.665 --> 00:12:16.720 align:middle line:84% of the cosine components in DC and one element that 00:12:16.720 --> 00:12:19.620 align:middle line:90% is only the sine terms. 00:12:19.620 --> 00:12:20.130 align:middle line:90% OK. 00:12:20.130 --> 00:12:23.880 align:middle line:84% So I get this decomposition, and because it's 00:12:23.880 --> 00:12:28.380 align:middle line:84% sines and cosines split, the even and odd components 00:12:28.380 --> 00:12:31.530 align:middle line:84% of the waveform happen to be orthogonal. 00:12:31.530 --> 00:12:32.090 align:middle line:90% OK. 00:12:32.090 --> 00:12:39.510 align:middle line:84% Likewise, I could make a decomposition of f of t, 00:12:39.510 --> 00:12:45.780 align:middle line:84% any waveform f of t, to have a f of half-wave symmetric element 00:12:45.780 --> 00:12:51.190 align:middle line:84% plus a "f of half-wave repeating" component. 00:12:51.190 --> 00:12:55.860 align:middle line:84% And again, this is going to be the fundamental and odd 00:12:55.860 --> 00:12:56.870 align:middle line:90% harmonics. 00:12:56.870 --> 00:12:59.210 align:middle line:84% This is going to be DC and even harmonics. 00:12:59.210 --> 00:12:59.710 align:middle line:90% OK. 00:12:59.710 --> 00:13:02.820 align:middle line:84% And those are time waveform decompositions, 00:13:02.820 --> 00:13:09.510 align:middle line:84% but they're into different parts of the Fourier components. 00:13:09.510 --> 00:13:10.010 align:middle line:90% OK. 00:13:10.010 --> 00:13:11.560 align:middle line:90% Any questions about that? 00:13:11.560 --> 00:13:15.080 align:middle line:90% 00:13:15.080 --> 00:13:17.900 align:middle line:84% OK, so that's just a little kind of review, 00:13:17.900 --> 00:13:21.470 align:middle line:84% and perhaps an extension of different ways 00:13:21.470 --> 00:13:24.930 align:middle line:84% we might think of getting the components. 00:13:24.930 --> 00:13:27.200 align:middle line:84% We'll see in a while why we might 00:13:27.200 --> 00:13:29.570 align:middle line:84% think of decomposing things that way 00:13:29.570 --> 00:13:33.560 align:middle line:84% or think carefully about whether my waveform is even or odd 00:13:33.560 --> 00:13:37.070 align:middle line:90% or half-wave symmetric or not. 00:13:37.070 --> 00:13:38.312 align:middle line:90% OK. 00:13:38.312 --> 00:13:44.280 align:middle line:84% But let's start talking about an inverter. 00:13:44.280 --> 00:13:46.370 align:middle line:84% What would be the basic structure 00:13:46.370 --> 00:13:47.820 align:middle line:90% of a three-phase inverter? 00:13:47.820 --> 00:13:52.040 align:middle line:84% Well, one basic structure might be to have some DC waveform. 00:13:52.040 --> 00:13:56.150 align:middle line:84% And of course, by the way, whether I 00:13:56.150 --> 00:13:59.930 align:middle line:84% talk about this structure as an inverter, 00:13:59.930 --> 00:14:03.290 align:middle line:84% flowing power DC to AC, or as a rectifier, flowing power 00:14:03.290 --> 00:14:06.680 align:middle line:84% AC to DC, it really just depends upon the power flow direction. 00:14:06.680 --> 00:14:11.840 align:middle line:84% And in fact, the same circuit structure can do either thing. 00:14:11.840 --> 00:14:13.490 align:middle line:90% OK. 00:14:13.490 --> 00:14:16.780 align:middle line:84% How you implement the switches, however, may be different. 00:14:16.780 --> 00:14:17.280 align:middle line:90% OK. 00:14:17.280 --> 00:14:20.810 align:middle line:84% And in fact, where did the term "inverter" come from? 00:14:20.810 --> 00:14:23.260 align:middle line:84% It came from the notion of an inverting rectifier. 00:14:23.260 --> 00:14:27.325 align:middle line:84% The first kind of power electronic solid state-- 00:14:27.325 --> 00:14:29.700 align:middle line:84% or, actually, it wasn't even solid state, it was a tube-- 00:14:29.700 --> 00:14:32.550 align:middle line:84% power electronic component that people were able to build 00:14:32.550 --> 00:14:33.828 align:middle line:90% was a rectifier. 00:14:33.828 --> 00:14:36.120 align:middle line:84% And they eventually figured out that with certain kinds 00:14:36.120 --> 00:14:39.985 align:middle line:84% of controlled rectifiers, they could go DC power to AC power. 00:14:39.985 --> 00:14:42.360 align:middle line:84% So they called that an inverting rectifier, and then they 00:14:42.360 --> 00:14:44.318 align:middle line:84% eventually just started calling it an inverter. 00:14:44.318 --> 00:14:44.818 align:middle line:90% OK. 00:14:44.818 --> 00:14:46.450 align:middle line:84% So that's where the term comes from. 00:14:46.450 --> 00:14:49.170 align:middle line:90% But here's a basic structure. 00:14:49.170 --> 00:14:55.445 align:middle line:84% OK, maybe I would have four switches, s1-- 00:14:55.445 --> 00:15:00.350 align:middle line:90% 00:15:00.350 --> 00:15:06.710 align:middle line:84% let me get my switch numberings right-- s2, s3, and s4, 00:15:06.710 --> 00:15:08.670 align:middle line:84% and we'll see why we number them this way. 00:15:08.670 --> 00:15:12.350 align:middle line:84% And here is my load, in this case, 00:15:12.350 --> 00:15:14.570 align:middle line:84% if I'm going to go from DC power to AC power. 00:15:14.570 --> 00:15:17.820 align:middle line:84% And let me call the output here vx. 00:15:17.820 --> 00:15:18.320 align:middle line:90% OK. 00:15:18.320 --> 00:15:21.320 align:middle line:84% So I'm going to come from some DC voltage, 00:15:21.320 --> 00:15:25.790 align:middle line:84% and I'm going to deliver power at AC into some voltage vx. 00:15:25.790 --> 00:15:28.420 align:middle line:90% OK. 00:15:28.420 --> 00:15:33.880 align:middle line:84% What voltages vx can I synthesize with this structure? 00:15:33.880 --> 00:15:35.570 align:middle line:90% Well, let's think about this. 00:15:35.570 --> 00:15:44.740 align:middle line:84% If I have switches on, what voltage vx do I get? 00:15:44.740 --> 00:15:52.600 align:middle line:84% Well, if s1 and s2 are on, I get plus vdc. 00:15:52.600 --> 00:15:57.920 align:middle line:84% If s2 and s3 are on, I get zero, right, 00:15:57.920 --> 00:16:00.520 align:middle line:84% because I've just shorted out the load on the bottom. 00:16:00.520 --> 00:16:06.640 align:middle line:84% If s3 and s4 are on, well, then vx becomes minus vdc. 00:16:06.640 --> 00:16:10.180 align:middle line:90% 00:16:10.180 --> 00:16:15.160 align:middle line:84% And if s4 and s1 are on, top two switches, I again 00:16:15.160 --> 00:16:16.360 align:middle line:90% get zero, right. 00:16:16.360 --> 00:16:19.120 align:middle line:84% So I sort of have two ways I can apply 00:16:19.120 --> 00:16:21.080 align:middle line:90% zero voltage across the load. 00:16:21.080 --> 00:16:23.170 align:middle line:84% I can apply a positive voltage, and I 00:16:23.170 --> 00:16:24.870 align:middle line:90% can apply a negative voltage. 00:16:24.870 --> 00:16:25.370 align:middle line:90% All right. 00:16:25.370 --> 00:16:29.140 align:middle line:84% So I can go sort of one way, the other way, and two ways, 00:16:29.140 --> 00:16:30.170 align:middle line:90% I can get zero. 00:16:30.170 --> 00:16:30.670 align:middle line:90% OK. 00:16:30.670 --> 00:16:34.660 align:middle line:84% And what we're going to see is that means given a DC voltage, 00:16:34.660 --> 00:16:36.440 align:middle line:90% I can put AC on the load. 00:16:36.440 --> 00:16:36.940 align:middle line:90% OK. 00:16:36.940 --> 00:16:43.510 align:middle line:84% And this is a very common single-phase inverter structure. 00:16:43.510 --> 00:16:46.060 align:middle line:90% If I have, say-- 00:16:46.060 --> 00:16:51.310 align:middle line:84% a typical thing is if I have an inductive and resistive load, 00:16:51.310 --> 00:16:54.300 align:middle line:84% right, maybe I would implement it this way. 00:16:54.300 --> 00:16:57.100 align:middle line:90% 00:16:57.100 --> 00:16:57.730 align:middle line:90% OK. 00:16:57.730 --> 00:17:02.740 align:middle line:84% I would have, say, a MOSFET here and a MOSFET here 00:17:02.740 --> 00:17:04.390 align:middle line:90% for my switches. 00:17:04.390 --> 00:17:07.450 align:middle line:84% And keep in mind, these guys, while I don't usually draw it, 00:17:07.450 --> 00:17:10.861 align:middle line:84% they have internal body diodes like this. 00:17:10.861 --> 00:17:11.361 align:middle line:90% OK. 00:17:11.361 --> 00:17:15.049 align:middle line:90% 00:17:15.049 --> 00:17:19.780 align:middle line:84% So this would be a typical structure, 00:17:19.780 --> 00:17:24.254 align:middle line:84% where internally I have these diodes, OK, and here's vx again. 00:17:24.254 --> 00:17:27.430 align:middle line:90% 00:17:27.430 --> 00:17:28.210 align:middle line:90% OK. 00:17:28.210 --> 00:17:31.810 align:middle line:84% Just to illustrate this kind of structure being very common, 00:17:31.810 --> 00:17:37.840 align:middle line:84% what I have here is sort of one of these things 00:17:37.840 --> 00:17:40.660 align:middle line:84% you plug into the cigarette lighter in your car 00:17:40.660 --> 00:17:44.980 align:middle line:84% to generate AC to power your laptop 00:17:44.980 --> 00:17:47.510 align:middle line:84% or something when you're on a trip or something. 00:17:47.510 --> 00:17:49.420 align:middle line:84% And you know, this is always dangerous, 00:17:49.420 --> 00:17:51.462 align:middle line:84% giving me a power converter, because usually I'll 00:17:51.462 --> 00:17:52.930 align:middle line:90% just take it apart. 00:17:52.930 --> 00:17:57.490 align:middle line:84% But what you'll see is, on one side of this thing, 00:17:57.490 --> 00:18:00.650 align:middle line:84% there's kind of four switches, and you'll see the-- in fact, 00:18:00.650 --> 00:18:02.650 align:middle line:84% we were talking about insulation pads last time. 00:18:02.650 --> 00:18:05.350 align:middle line:84% I'll pass this around and you can see the insulation pad here. 00:18:05.350 --> 00:18:07.600 align:middle line:84% And then you'll see this transformer, 00:18:07.600 --> 00:18:11.990 align:middle line:84% and you'll see a set of diodes here, which are right here. 00:18:11.990 --> 00:18:14.270 align:middle line:84% That's an isolated DC-to-DC converter. 00:18:14.270 --> 00:18:17.500 align:middle line:84% So that takes the 12 volts or 14 volts from your car battery 00:18:17.500 --> 00:18:19.610 align:middle line:90% and generates a higher voltage. 00:18:19.610 --> 00:18:22.070 align:middle line:84% And then there's four more switches, 00:18:22.070 --> 00:18:24.590 align:middle line:90% which are the ones here. 00:18:24.590 --> 00:18:29.270 align:middle line:84% Those are your four MOSFETs, and those are going to generate AC. 00:18:29.270 --> 00:18:31.880 align:middle line:84% And it goes to two plugs so you can plug in your toys 00:18:31.880 --> 00:18:35.250 align:middle line:84% and play your gaming system or whatever you want. 00:18:35.250 --> 00:18:39.410 align:middle line:84% So this is just to illustrate one very simple example 00:18:39.410 --> 00:18:43.740 align:middle line:84% of an inverter, OK, and it'll have exactly this structure, 00:18:43.740 --> 00:18:44.240 align:middle line:90% right. 00:18:44.240 --> 00:18:46.160 align:middle line:84% So we have an isolated DC-to-DC converter 00:18:46.160 --> 00:18:50.110 align:middle line:84% to get a voltage gain, and then this inverter. 00:18:50.110 --> 00:18:52.510 align:middle line:90% OK. 00:18:52.510 --> 00:18:55.300 align:middle line:84% What would this-- when I'm drawing this load vx, 00:18:55.300 --> 00:18:56.660 align:middle line:90% what am I meaning? 00:18:56.660 --> 00:18:59.080 align:middle line:84% Well, it depends what my load is, 00:18:59.080 --> 00:19:00.730 align:middle line:84% but I could imagine maybe I would 00:19:00.730 --> 00:19:11.020 align:middle line:84% have an inductor and a capacitor and some resistor that's 00:19:11.020 --> 00:19:13.150 align:middle line:90% getting my AC load, right. 00:19:13.150 --> 00:19:16.300 align:middle line:84% So this filters out higher harmonics, 00:19:16.300 --> 00:19:20.830 align:middle line:84% or this could be a machine winding where there's 00:19:20.830 --> 00:19:23.268 align:middle line:84% sort of a phase inductance from the machine 00:19:23.268 --> 00:19:24.310 align:middle line:90% winding and the resistor. 00:19:24.310 --> 00:19:27.742 align:middle line:84% It just sort of depends on what you're driving for a load, OK. 00:19:27.742 --> 00:19:29.200 align:middle line:84% Or it could be an inductive filter, 00:19:29.200 --> 00:19:32.580 align:middle line:84% and that could run into the grid, right. 00:19:32.580 --> 00:19:34.260 align:middle line:84% So that's the basic notion of what 00:19:34.260 --> 00:19:36.900 align:middle line:90% the structure of an inverter is. 00:19:36.900 --> 00:19:39.930 align:middle line:84% Why don't we think about, How could we 00:19:39.930 --> 00:19:45.550 align:middle line:84% well approximate a sinusoid by switching these transistors, 00:19:45.550 --> 00:19:47.130 align:middle line:90% S1 to S4? 00:19:47.130 --> 00:19:48.870 align:middle line:84% And the case I'd like to start with 00:19:48.870 --> 00:19:50.470 align:middle line:90% is perhaps the simplest one. 00:19:50.470 --> 00:19:53.520 align:middle line:84% I'd like to switch each switch on and off 00:19:53.520 --> 00:19:56.250 align:middle line:90% only once per AC output cycle. 00:19:56.250 --> 00:19:58.530 align:middle line:84% And let's imagine for the moment that what I want 00:19:58.530 --> 00:20:00.870 align:middle line:84% is something that approximates a sine wave at the output 00:20:00.870 --> 00:20:03.960 align:middle line:84% or crudely approximates a sine wave at the output. 00:20:03.960 --> 00:20:04.590 align:middle line:90% OK. 00:20:04.590 --> 00:20:08.400 align:middle line:84% Why am I focusing on switching only once per cycle? 00:20:08.400 --> 00:20:11.520 align:middle line:84% Well, the number of times I switch per cycle 00:20:11.520 --> 00:20:13.520 align:middle line:84% is going to have to do with my switching losses. 00:20:13.520 --> 00:20:16.170 align:middle line:84% Right, we talked about switching losses. 00:20:16.170 --> 00:20:20.340 align:middle line:84% And so especially at very high power levels, 00:20:20.340 --> 00:20:22.320 align:middle line:84% I really want to minimize the number of times 00:20:22.320 --> 00:20:28.170 align:middle line:84% I switch per cycle in order to reduce those switching losses. 00:20:28.170 --> 00:20:29.385 align:middle line:90% OK. 00:20:29.385 --> 00:20:32.400 align:middle line:84% So the obsession with not switching very often 00:20:32.400 --> 00:20:35.230 align:middle line:84% comes from mitigating those losses, 00:20:35.230 --> 00:20:37.500 align:middle line:84% and so I want to just treat the simplest case. 00:20:37.500 --> 00:20:39.150 align:middle line:84% And this is actually what one might 00:20:39.150 --> 00:20:42.240 align:middle line:84% do at very high power levels, where you really don't 00:20:42.240 --> 00:20:46.040 align:middle line:90% want to switch very often, OK-- 00:20:46.040 --> 00:20:48.980 align:middle line:84% or a very high frequency levels, I should say. 00:20:48.980 --> 00:20:49.900 align:middle line:90% OK. 00:20:49.900 --> 00:20:52.450 align:middle line:84% So let's think about what we might do. 00:20:52.450 --> 00:20:55.315 align:middle line:84% Let me plot things in terms of electrical angle. 00:20:55.315 --> 00:20:58.897 align:middle line:90% 00:20:58.897 --> 00:21:01.155 align:middle line:90% OK, so here is-- 00:21:01.155 --> 00:21:09.740 align:middle line:90% 00:21:09.740 --> 00:21:13.410 align:middle line:84% here is-- so there's pi and 2 pi. 00:21:13.410 --> 00:21:14.640 align:middle line:90% This is omega t. 00:21:14.640 --> 00:21:16.110 align:middle line:90% This is electrical angle. 00:21:16.110 --> 00:21:18.350 align:middle line:84% And what I was hoping to synthesize 00:21:18.350 --> 00:21:20.275 align:middle line:90% is some sine wave, right. 00:21:20.275 --> 00:21:22.000 align:middle line:84% So maybe it would look like this. 00:21:22.000 --> 00:21:26.710 align:middle line:90% 00:21:26.710 --> 00:21:28.840 align:middle line:84% An ideal sine wave, if I could synthesize it, 00:21:28.840 --> 00:21:30.200 align:middle line:90% would look like this. 00:21:30.200 --> 00:21:31.550 align:middle line:90% OK. 00:21:31.550 --> 00:21:35.810 align:middle line:84% Now, I clearly can't synthesize that with my inverter, 00:21:35.810 --> 00:21:37.470 align:middle line:90% but what could I do? 00:21:37.470 --> 00:21:42.650 align:middle line:84% Well, I can generate a positive voltage, right, so maybe what 00:21:42.650 --> 00:21:47.420 align:middle line:84% I would do is-- let's just suppose, 00:21:47.420 --> 00:21:52.220 align:middle line:84% starting some time here, I will apply plus vdc. 00:21:52.220 --> 00:21:55.900 align:middle line:84% Maybe I'll do that at some electrical angle delta. 00:21:55.900 --> 00:21:56.560 align:middle line:90% OK. 00:21:56.560 --> 00:22:01.510 align:middle line:84% So in this time period, I will synthesize plus vdc, 00:22:01.510 --> 00:22:04.020 align:middle line:84% and what I'm going to do is I'm going to do this. 00:22:04.020 --> 00:22:08.320 align:middle line:84% I'm going to do this between delta and pi minus delta. 00:22:08.320 --> 00:22:09.706 align:middle line:90% OK. 00:22:09.706 --> 00:22:12.250 align:middle line:84% And how would I do that in this time period? 00:22:12.250 --> 00:22:17.080 align:middle line:84% What switch pattern would let me synthesize plus vdc? 00:22:17.080 --> 00:22:18.010 align:middle line:90% AUDIENCE: s1s2. 00:22:18.010 --> 00:22:22.610 align:middle line:84% DAVID PERREAULT: s1s2 will give me plus vdc. 00:22:22.610 --> 00:22:27.322 align:middle line:84% Now I want to synthesize zero in this time period, 00:22:27.322 --> 00:22:28.780 align:middle line:84% and I'm going to do that between pi 00:22:28.780 --> 00:22:30.660 align:middle line:90% minus delta and pi plus delta. 00:22:30.660 --> 00:22:33.420 align:middle line:90% 00:22:33.420 --> 00:22:33.920 align:middle line:90% OK. 00:22:33.920 --> 00:22:38.020 align:middle line:84% I can get zero just by leaving switch 2 on 00:22:38.020 --> 00:22:41.800 align:middle line:84% and going to switch s3 so I can have s2s3. 00:22:41.800 --> 00:22:42.670 align:middle line:90% OK. 00:22:42.670 --> 00:22:46.960 align:middle line:84% Now I'm in the negative half of the cycle, so maybe I want to-- 00:22:46.960 --> 00:22:52.470 align:middle line:84% maybe I want to make this minus vdc, right. 00:22:52.470 --> 00:22:59.650 align:middle line:84% So I will then switch here, and from pi plus delta 00:22:59.650 --> 00:23:06.430 align:middle line:84% to 2 pi minus delta, I will have minus vdc. 00:23:06.430 --> 00:23:10.860 align:middle line:84% And I get that with what switch pattern? 00:23:10.860 --> 00:23:12.330 align:middle line:90% s3s4. 00:23:12.330 --> 00:23:15.900 align:middle line:84% So I've turned off S2, and now I've turned on s4. 00:23:15.900 --> 00:23:19.350 align:middle line:84% And then I can get back to synthesizing zero with s4s1. 00:23:19.350 --> 00:23:22.650 align:middle line:90% 00:23:22.650 --> 00:23:32.190 align:middle line:84% Right, so here's my pattern, OK, and it looks like this. 00:23:32.190 --> 00:23:32.690 align:middle line:90% OK. 00:23:32.690 --> 00:23:35.270 align:middle line:84% Does that make sense to everybody? 00:23:35.270 --> 00:23:37.230 align:middle line:90% So what have I done? 00:23:37.230 --> 00:23:41.990 align:middle line:84% Each switch turns on-- if I look over at 2 pi, one AC cycle, 00:23:41.990 --> 00:23:44.210 align:middle line:84% each switch turns on once per cycle, 00:23:44.210 --> 00:23:46.950 align:middle line:84% each switch turns off once per cycle. 00:23:46.950 --> 00:23:51.503 align:middle line:84% OK, so that's the minimum I can sort of do and synthesize 00:23:51.503 --> 00:23:52.420 align:middle line:90% this kind of waveform. 00:23:52.420 --> 00:23:54.070 align:middle line:90% Now, what can you tell me? 00:23:54.070 --> 00:24:02.000 align:middle line:84% I drew a sine wave here, right, and I 00:24:02.000 --> 00:24:06.540 align:middle line:84% want in some measure for my synthesized AC output voltage 00:24:06.540 --> 00:24:08.670 align:middle line:90% of my inverter, this vx-- 00:24:08.670 --> 00:24:10.610 align:middle line:90% right, this is vx-- 00:24:10.610 --> 00:24:15.925 align:middle line:84% to try to approximate that sine wave in some fashion. 00:24:15.925 --> 00:24:19.850 align:middle line:90% 00:24:19.850 --> 00:24:23.340 align:middle line:90% This is v sine omega t. 00:24:23.340 --> 00:24:26.630 align:middle line:84% Well, sine omega t is odd, right. 00:24:26.630 --> 00:24:29.750 align:middle line:84% So if I want to do a good job with as least kind 00:24:29.750 --> 00:24:32.330 align:middle line:84% of unwanted harmonic content as I can, 00:24:32.330 --> 00:24:35.600 align:middle line:84% it makes sense that because sine is odd, 00:24:35.600 --> 00:24:39.050 align:middle line:84% I also ought to use only odd components. 00:24:39.050 --> 00:24:42.110 align:middle line:84% Right, so I ought to synthesize it with an odd waveform. 00:24:42.110 --> 00:24:44.160 align:middle line:84% And what do I know about this waveform? 00:24:44.160 --> 00:24:45.140 align:middle line:90% This waveform is odd. 00:24:45.140 --> 00:24:48.440 align:middle line:84% And I should-- maybe I'll just draw it out here, right. 00:24:48.440 --> 00:24:52.760 align:middle line:84% It is indeed-- this is minus delta. 00:24:52.760 --> 00:24:58.040 align:middle line:84% This white waveform here, vx, is indeed odd, right. 00:24:58.040 --> 00:25:01.880 align:middle line:84% It reflects-- if I flip it across T equals 0 and I flip it, 00:25:01.880 --> 00:25:03.830 align:middle line:90% I get the same thing, 00:25:03.830 --> 00:25:06.360 align:middle line:84% So what I know is this white waveform, it's odd. 00:25:06.360 --> 00:25:08.990 align:middle line:84% It's comprising only sine components. 00:25:08.990 --> 00:25:11.720 align:middle line:84% So it comprises only sine omega t, 00:25:11.720 --> 00:25:14.390 align:middle line:84% some amount of sine of 3 omega t, some amount of sine-- 00:25:14.390 --> 00:25:15.850 align:middle line:90% I'm sorry. 00:25:15.850 --> 00:25:19.940 align:middle line:84% It has only sine terms, as far as I have told you so far. 00:25:19.940 --> 00:25:22.342 align:middle line:84% OK, there's no cosine terms in this thing. 00:25:22.342 --> 00:25:23.800 align:middle line:84% So that means it's good because I'm 00:25:23.800 --> 00:25:25.690 align:middle line:84% kind of building it out of its-- the things 00:25:25.690 --> 00:25:27.220 align:middle line:90% I would want to build it out of. 00:25:27.220 --> 00:25:28.470 align:middle line:90% Any questions about that? 00:25:28.470 --> 00:25:32.560 align:middle line:90% 00:25:32.560 --> 00:25:35.480 align:middle line:84% What else did I do in this waveform? 00:25:35.480 --> 00:25:40.660 align:middle line:84% Well, you notice that this half of the waveform, 00:25:40.660 --> 00:25:47.270 align:middle line:84% for the negative sign, Is exactly the flip. 00:25:47.270 --> 00:25:49.130 align:middle line:84% I come back half a cycle and I flip it 00:25:49.130 --> 00:25:51.460 align:middle line:84% for the first half of the cycle, right. 00:25:51.460 --> 00:25:53.910 align:middle line:90% What characteristic has that? 00:25:53.910 --> 00:25:58.130 align:middle line:90% 00:25:58.130 --> 00:26:00.000 align:middle line:90% It's half-wave symmetric. 00:26:00.000 --> 00:26:03.530 align:middle line:84% So what that means is that this white waveform that I've 00:26:03.530 --> 00:26:06.920 align:middle line:84% synthesized has no even harmonics, right. 00:26:06.920 --> 00:26:09.890 align:middle line:84% So if I did a Fourier decomposition 00:26:09.890 --> 00:26:13.470 align:middle line:84% on this white waveform, what I know is it has-- 00:26:13.470 --> 00:26:16.760 align:middle line:84% well, it has no DC, it has no cosine terms, 00:26:16.760 --> 00:26:18.416 align:middle line:90% and it has no even harmonics. 00:26:18.416 --> 00:26:20.780 align:middle line:84% Right, so the lowest harmonic component 00:26:20.780 --> 00:26:24.890 align:middle line:84% could only be the third, and then the fifth and the seventh 00:26:24.890 --> 00:26:26.330 align:middle line:90% and so forth. 00:26:26.330 --> 00:26:26.990 align:middle line:90% OK. 00:26:26.990 --> 00:26:32.710 align:middle line:84% So the reason I chose this pattern just this way, 00:26:32.710 --> 00:26:34.960 align:middle line:84% for both being odd, because I was trying to-- happened 00:26:34.960 --> 00:26:36.760 align:middle line:84% to be trying to match a sine wave, 00:26:36.760 --> 00:26:39.740 align:middle line:84% but more importantly, that I made it a half-wave symmetric, 00:26:39.740 --> 00:26:41.920 align:middle line:84% I've gotten rid of even harmonic components. 00:26:41.920 --> 00:26:44.440 align:middle line:84% And if I imagined that I was going to come up here 00:26:44.440 --> 00:26:52.510 align:middle line:84% and say, oh, you know, I'm going to take this vx here and try 00:26:52.510 --> 00:26:56.800 align:middle line:84% to synthesize some output voltage vac by filtering it, 00:26:56.800 --> 00:26:59.410 align:middle line:84% if I can get rid of my even harmonics, 00:26:59.410 --> 00:27:02.110 align:middle line:84% I can more easily filter that waveform, right. 00:27:02.110 --> 00:27:06.280 align:middle line:84% When we thought about sort of DC-to-DC converters, 00:27:06.280 --> 00:27:10.130 align:middle line:84% I'm trying to separate out DC from any AC stuff. 00:27:10.130 --> 00:27:12.520 align:middle line:84% That's kind of easy, right, because they're 00:27:12.520 --> 00:27:16.000 align:middle line:84% kind of infinitely separated in frequency, or at least on a log 00:27:16.000 --> 00:27:17.740 align:middle line:90% scale. 00:27:17.740 --> 00:27:20.140 align:middle line:84% If I'm trying to separate out some fundamental 00:27:20.140 --> 00:27:23.290 align:middle line:84% that I want to create, and I want to get my fundamental here 00:27:23.290 --> 00:27:25.270 align:middle line:84% but I've got second harmonic here, 00:27:25.270 --> 00:27:28.990 align:middle line:84% I need a very good filter to keep 1 omega t 00:27:28.990 --> 00:27:31.200 align:middle line:90% and kill 2 omega t, right. 00:27:31.200 --> 00:27:33.470 align:middle line:84% Well, by making it half-wave symmetric, 00:27:33.470 --> 00:27:34.920 align:middle line:90% I don't need to kill 2 omega t. 00:27:34.920 --> 00:27:37.130 align:middle line:90% I just need to kill 3 omega t. 00:27:37.130 --> 00:27:37.680 align:middle line:90% All right. 00:27:37.680 --> 00:27:41.270 align:middle line:84% So there's a large motivation to control the harmonic 00:27:41.270 --> 00:27:43.610 align:middle line:84% content of your waveforms by picking 00:27:43.610 --> 00:27:46.340 align:middle line:84% waveform symmetries, and hence the interest 00:27:46.340 --> 00:27:49.410 align:middle line:84% in half-wave symmetric waveforms. 00:27:49.410 --> 00:27:50.220 align:middle line:90% Questions? 00:27:50.220 --> 00:27:54.080 align:middle line:90% 00:27:54.080 --> 00:27:56.530 align:middle line:90% OK. 00:27:56.530 --> 00:27:58.320 align:middle line:90% Why did I bother? 00:27:58.320 --> 00:28:01.150 align:middle line:84% And by the way, I should've said, 00:28:01.150 --> 00:28:07.090 align:middle line:84% if I had made my angle delta equals 0, OK, at delta equals 0, 00:28:07.090 --> 00:28:09.482 align:middle line:84% this would just be a sine wave, right. 00:28:09.482 --> 00:28:13.210 align:middle line:84% So at delta equals 0, what I would 00:28:13.210 --> 00:28:21.490 align:middle line:84% get is I would get my voltage v x would simply equal vdc 00:28:21.490 --> 00:28:29.480 align:middle line:84% times summation n equals 1 to infinity of 4 00:28:29.480 --> 00:28:36.540 align:middle line:84% over pi n sine of n omega 0 t, right. 00:28:36.540 --> 00:28:37.680 align:middle line:90% Where did I get that from? 00:28:37.680 --> 00:28:41.100 align:middle line:84% That is just the Fourier series for a square wave. 00:28:41.100 --> 00:28:41.600 align:middle line:90% OK. 00:28:41.600 --> 00:28:43.230 align:middle line:90% You can look it up in any book. 00:28:43.230 --> 00:28:46.400 align:middle line:84% And so what that says is that, first of all, 00:28:46.400 --> 00:28:49.970 align:middle line:84% it's a sine wave series because it's odd, right, 00:28:49.970 --> 00:28:53.360 align:middle line:84% so I knew all the even terms went away. 00:28:53.360 --> 00:28:55.680 align:middle line:84% Because it's half wave symmetric, 00:28:55.680 --> 00:28:58.530 align:middle line:84% a square wave is half-wave symmetric, then-- 00:28:58.530 --> 00:29:01.770 align:middle line:84% or 50% duty cycle square wave is half-wave symmetric-- 00:29:01.770 --> 00:29:10.040 align:middle line:84% then it doesn't have any-- this is "n odd only" summation. 00:29:10.040 --> 00:29:13.730 align:middle line:84% And that means it's 4 over pi sine of omega t 00:29:13.730 --> 00:29:17.040 align:middle line:84% plus 4 over 3 pi sine 3 omega t, and so forth. 00:29:17.040 --> 00:29:19.640 align:middle line:84% It has harmonics that sort of fall off as 1 over n, 00:29:19.640 --> 00:29:22.070 align:middle line:90% but only odd components. 00:29:22.070 --> 00:29:25.195 align:middle line:84% Does that make sense to everybody? 00:29:25.195 --> 00:29:25.695 align:middle line:90% OK. 00:29:25.695 --> 00:29:28.550 align:middle line:90% 00:29:28.550 --> 00:29:32.720 align:middle line:84% I came in and I introduced this angle delta, right, 00:29:32.720 --> 00:29:37.640 align:middle line:84% and I said what you could do for delta equals zero and so forth. 00:29:37.640 --> 00:29:38.920 align:middle line:90% Why do I have it there? 00:29:38.920 --> 00:29:41.030 align:middle line:84% What could I do with my angle delta? 00:29:41.030 --> 00:29:44.000 align:middle line:84% Right, I have my one control variable that I can use, 00:29:44.000 --> 00:29:49.110 align:middle line:84% and I'm still switching each switch only once per cycle. 00:29:49.110 --> 00:29:49.610 align:middle line:90% OK. 00:29:49.610 --> 00:29:51.440 align:middle line:90% Well, what can I do? 00:29:51.440 --> 00:29:54.020 align:middle line:84% I can basically vary delta between zero-- 00:29:54.020 --> 00:29:57.520 align:middle line:84% that's my perfect square wave-- and something less than pi 00:29:57.520 --> 00:29:58.040 align:middle line:90% over 2. 00:29:58.040 --> 00:29:58.540 align:middle line:90% OK. 00:29:58.540 --> 00:30:06.720 align:middle line:84% And I can really use delta to do kind of two things. 00:30:06.720 --> 00:30:09.940 align:middle line:90% One, I can vary the fundamental. 00:30:09.940 --> 00:30:14.480 align:middle line:90% 00:30:14.480 --> 00:30:16.926 align:middle line:84% And two, I can control harmonics. 00:30:16.926 --> 00:30:20.480 align:middle line:90% 00:30:20.480 --> 00:30:22.700 align:middle line:84% I can't do both at the same time, 00:30:22.700 --> 00:30:25.010 align:middle line:84% but I can use that as a control variable 00:30:25.010 --> 00:30:28.740 align:middle line:84% without switching more times per cycle. 00:30:28.740 --> 00:30:30.010 align:middle line:90% OK. 00:30:30.010 --> 00:30:33.150 align:middle line:84% What would I do in terms of varying the fundamental? 00:30:33.150 --> 00:30:37.230 align:middle line:84% Very often, if I'm driving a motor or something, 00:30:37.230 --> 00:30:40.613 align:middle line:84% right, how hard I'm driving the motor 00:30:40.613 --> 00:30:42.780 align:middle line:84% kind of has to do with the amplitude of the waveform 00:30:42.780 --> 00:30:44.700 align:middle line:84% I'm driving it with, right, so if I 00:30:44.700 --> 00:30:48.070 align:middle line:84% can have some means of amplitude control, that's a good thing. 00:30:48.070 --> 00:30:48.570 align:middle line:90% OK. 00:30:48.570 --> 00:30:52.260 align:middle line:84% Let's just think about, What does the fundamental of here 00:30:52.260 --> 00:30:54.690 align:middle line:90% look like, right? 00:30:54.690 --> 00:31:05.940 align:middle line:84% So vx, I said we can express vx as the sum of odd harmonics 00:31:05.940 --> 00:31:10.150 align:middle line:84% only, right, odd harmonic sine terms only. 00:31:10.150 --> 00:31:12.000 align:middle line:90% What is v1? 00:31:12.000 --> 00:31:17.280 align:middle line:84% Right, so vx of t, I could express 00:31:17.280 --> 00:31:25.770 align:middle line:84% as being v1 sine of omega t plus v3 sine 00:31:25.770 --> 00:31:37.748 align:middle line:84% of 3 omega t plus v5 sine 5 omega t, and so forth, right. 00:31:37.748 --> 00:31:39.720 align:middle line:84% But if what I'm mainly interested in 00:31:39.720 --> 00:31:42.960 align:middle line:84% is controlling the fundamental as the thing I'm driving, 00:31:42.960 --> 00:31:44.500 align:middle line:90% what is that fundamental? 00:31:44.500 --> 00:31:46.080 align:middle line:90% Well, we come back-- 00:31:46.080 --> 00:31:49.290 align:middle line:84% basically, we come back to this expression 00:31:49.290 --> 00:31:55.460 align:middle line:84% here, OK, to figure out what v1 is. 00:31:55.460 --> 00:31:57.240 align:middle line:90% So why don't we write v1? 00:31:57.240 --> 00:32:03.560 align:middle line:84% v1 in this expression would simply be equal to 2 over T-- 00:32:03.560 --> 00:32:06.700 align:middle line:84% in this case, I'm doing it in electrical angle of 2 pi-- 00:32:06.700 --> 00:32:22.516 align:middle line:84% integral over 0 to 2 pi vx sine of omega 0 t d omega t, right. 00:32:22.516 --> 00:32:25.210 align:middle line:84% Because I have this-- so I'm going to multiply that 00:32:25.210 --> 00:32:26.360 align:middle line:90% by the sine wave. 00:32:26.360 --> 00:32:28.275 align:middle line:84% Actually, conveniently, I've drawn the sine wave up there, 00:32:28.275 --> 00:32:28.775 align:middle line:90% right. 00:32:28.775 --> 00:32:33.790 align:middle line:84% I can just say that's going to be equal to-- 00:32:33.790 --> 00:32:36.880 align:middle line:84% this is 1 over pi, but I can do it over only half the cycle, 00:32:36.880 --> 00:32:41.020 align:middle line:84% and I can get two over pi the integral 00:32:41.020 --> 00:32:50.520 align:middle line:84% from delta to pi minus delta of vdc sine of omega 0 t 00:32:50.520 --> 00:32:54.450 align:middle line:90% d omega 0 t, right. 00:32:54.450 --> 00:32:56.416 align:middle line:84% All I'm doing is I'm multiplying-- 00:32:56.416 --> 00:33:00.910 align:middle line:90% 00:33:00.910 --> 00:33:05.230 align:middle line:84% essentially, I'm multiplying this green waveform, or a unit 00:33:05.230 --> 00:33:12.890 align:middle line:84% height version of this green waveform, by the white waveform 00:33:12.890 --> 00:33:15.000 align:middle line:90% and integrating it to find v1. 00:33:15.000 --> 00:33:17.600 align:middle line:84% And because the two halves of the integral are the same, 00:33:17.600 --> 00:33:20.360 align:middle line:84% I can just do it over half a cycle and double it. 00:33:20.360 --> 00:33:21.968 align:middle line:90% OK. 00:33:21.968 --> 00:33:24.470 align:middle line:84% And if I do that, that becomes very convenient 00:33:24.470 --> 00:33:26.430 align:middle line:84% because this just becomes minus cosine. 00:33:26.430 --> 00:33:42.170 align:middle line:84% So what I get is I get 2 over pi vdc cosine of delta minus 00:33:42.170 --> 00:33:48.900 align:middle line:84% cosine of pi minus delta, right, which just gives me-- 00:33:48.900 --> 00:33:51.840 align:middle line:84% I could rewrite this as being equal to 4 00:33:51.840 --> 00:33:58.120 align:middle line:90% vdc over pi cosine of delta. 00:33:58.120 --> 00:33:59.420 align:middle line:90% That makes sense to everybody? 00:33:59.420 --> 00:34:02.950 align:middle line:90% 00:34:02.950 --> 00:34:05.380 align:middle line:90% So what am I saying? 00:34:05.380 --> 00:34:10.179 align:middle line:84% If I made delta equal 0, that's my square wave case, right. 00:34:10.179 --> 00:34:13.489 align:middle line:84% I just get a fundamental that's 4 over pi vdc, right. 00:34:13.489 --> 00:34:17.380 align:middle line:84% That's exactly what I said before for the square wave case. 00:34:17.380 --> 00:34:21.100 align:middle line:84% As I keep increasing delta, I make 00:34:21.100 --> 00:34:23.739 align:middle line:84% it nonzero further into the cycle this way 00:34:23.739 --> 00:34:28.690 align:middle line:84% and further into the cycle that way, I reduce my fundamental. 00:34:28.690 --> 00:34:29.500 align:middle line:90% Why? 00:34:29.500 --> 00:34:33.370 align:middle line:84% Because basically, as delta becomes bigger, 00:34:33.370 --> 00:34:35.530 align:middle line:84% I'm reducing the amount of overlap 00:34:35.530 --> 00:34:37.653 align:middle line:84% between the white waveform and the green waveform. 00:34:37.653 --> 00:34:39.070 align:middle line:84% And when I multiply and integrate, 00:34:39.070 --> 00:34:42.770 align:middle line:84% I get a smaller number, and that goes as the cosine of delta. 00:34:42.770 --> 00:34:43.490 align:middle line:90% OK. 00:34:43.490 --> 00:34:47.940 align:middle line:84% So what I can do is, if what I cared about mainly 00:34:47.940 --> 00:34:53.010 align:middle line:84% was the fundamental amplitude of my output, 00:34:53.010 --> 00:35:00.350 align:middle line:84% I can modulate that for a fixed DC voltage by modulating delta. 00:35:00.350 --> 00:35:00.850 align:middle line:90% OK. 00:35:00.850 --> 00:35:06.080 align:middle line:84% I have a means of controlling the fundamental amplitude. 00:35:06.080 --> 00:35:09.020 align:middle line:84% Does that make sense to everybody? 00:35:09.020 --> 00:35:12.520 align:middle line:84% I want to drive my motor easier, I use a bigger delta, 00:35:12.520 --> 00:35:15.150 align:middle line:84% I drive it with a little less fundamental amplitude. 00:35:15.150 --> 00:35:17.180 align:middle line:84% If I want more fundamental amplitude, 00:35:17.180 --> 00:35:19.158 align:middle line:84% I use a smaller delta, and the most I can do 00:35:19.158 --> 00:35:21.200 align:middle line:84% is a square wave where I get a fundamental that's 00:35:21.200 --> 00:35:22.400 align:middle line:90% 4 over pi times vdc. 00:35:22.400 --> 00:35:25.450 align:middle line:90% 00:35:25.450 --> 00:35:26.200 align:middle line:90% Questions? 00:35:26.200 --> 00:35:29.900 align:middle line:90% 00:35:29.900 --> 00:35:33.350 align:middle line:84% What else could I do with this thing? 00:35:33.350 --> 00:35:36.380 align:middle line:84% Well, I could pick delta to control the fundamental. 00:35:36.380 --> 00:35:38.810 align:middle line:84% Another way to control the fundamental 00:35:38.810 --> 00:35:43.170 align:middle line:84% would be to directly control vdc, right. 00:35:43.170 --> 00:35:46.710 align:middle line:84% So in that inverter I'm passing around, right, they 00:35:46.710 --> 00:35:48.742 align:middle line:84% have an isolated DC-to-DC converter. 00:35:48.742 --> 00:35:49.450 align:middle line:90% Well, guess what? 00:35:49.450 --> 00:35:52.050 align:middle line:84% They can use that if they want to control 00:35:52.050 --> 00:35:53.770 align:middle line:90% the vdc that they get, right. 00:35:53.770 --> 00:35:57.565 align:middle line:84% So if you have a DC-to-DC converter before your inverter, 00:35:57.565 --> 00:35:59.940 align:middle line:84% you get to control this because you have a converter that 00:35:59.940 --> 00:36:01.670 align:middle line:90% can control it. 00:36:01.670 --> 00:36:02.190 align:middle line:90% All right. 00:36:02.190 --> 00:36:04.190 align:middle line:84% So maybe I don't need to use delta 00:36:04.190 --> 00:36:05.400 align:middle line:90% to control the fundamental. 00:36:05.400 --> 00:36:08.060 align:middle line:84% Maybe I can do something else with it. 00:36:08.060 --> 00:36:11.330 align:middle line:84% Well, the other thing I can do is harmonic control. 00:36:11.330 --> 00:36:21.030 align:middle line:84% Let's ask the question, What does v3 look like? 00:36:21.030 --> 00:36:30.420 align:middle line:84% Well, v3 is equal to 2 over 2 pi integral from 0 00:36:30.420 --> 00:36:45.890 align:middle line:84% to 2 pi of vx of t times sine of 3 omega 0 t d omega 0 t. 00:36:45.890 --> 00:36:51.620 align:middle line:84% And likewise, so what I'm doing in this case 00:36:51.620 --> 00:36:56.412 align:middle line:84% is I'm going to multiply this waveform by sine 3 omega t, 00:36:56.412 --> 00:36:58.370 align:middle line:84% right, so I'm going to multiply it by something 00:36:58.370 --> 00:36:59.253 align:middle line:90% that looks like this. 00:36:59.253 --> 00:37:08.690 align:middle line:90% 00:37:08.690 --> 00:37:12.140 align:middle line:84% Right, and then I'm-- that's horribly asymmetric. 00:37:12.140 --> 00:37:14.660 align:middle line:84% But I'm going to multiply it by sine 3 omega 0 t 00:37:14.660 --> 00:37:16.640 align:middle line:90% and then integrate it, right. 00:37:16.640 --> 00:37:21.450 align:middle line:84% Well, OK, because that's again half-wave symmetric, 00:37:21.450 --> 00:37:24.470 align:middle line:84% I can write that as simply being equal to-- 00:37:24.470 --> 00:37:27.980 align:middle line:84% this is 1 over pi, but then I can double it and only 00:37:27.980 --> 00:37:29.540 align:middle line:90% do it over half the cycle. 00:37:29.540 --> 00:37:36.500 align:middle line:84% And I get 2 over pi the integral from delta 00:37:36.500 --> 00:37:48.320 align:middle line:84% to pi minus delta vdc sine of 3 omega 0 t d omega 0 t. 00:37:48.320 --> 00:37:49.976 align:middle line:90% OK. 00:37:49.976 --> 00:37:52.580 align:middle line:84% It's the same game all over again, but what I get 00:37:52.580 --> 00:38:09.125 align:middle line:84% is 4 vdc over 3 pi times the cosine of 3 delta. 00:38:09.125 --> 00:38:12.090 align:middle line:90% 00:38:12.090 --> 00:38:14.970 align:middle line:84% OK, that's just the result of that integral. 00:38:14.970 --> 00:38:15.790 align:middle line:90% OK. 00:38:15.790 --> 00:38:18.270 align:middle line:90% So I can again-- 00:38:18.270 --> 00:38:20.250 align:middle line:90% you know, I can-- 00:38:20.250 --> 00:38:22.950 align:middle line:84% once I've determined delta, I determined the fundamental 00:38:22.950 --> 00:38:25.260 align:middle line:84% and I determined the third harmonic, right. 00:38:25.260 --> 00:38:29.580 align:middle line:90% But what can I do with this? 00:38:29.580 --> 00:38:33.810 align:middle line:84% Well, if delta was 30 degrees, or pi over 6, 00:38:33.810 --> 00:38:35.450 align:middle line:84% what would be the cosine of 3 delta? 00:38:35.450 --> 00:38:38.020 align:middle line:90% 00:38:38.020 --> 00:38:41.540 align:middle line:90% What's the cosine of 90 degrees? 00:38:41.540 --> 00:38:43.070 align:middle line:90% Zip, right. 00:38:43.070 --> 00:38:49.570 align:middle line:84% So if I pick delta is equal to 30 degrees, v3 goes to 0. 00:38:49.570 --> 00:38:52.930 align:middle line:90% 00:38:52.930 --> 00:38:54.910 align:middle line:90% That's kind of nice, right. 00:38:54.910 --> 00:38:56.720 align:middle line:90% Why is that nice? 00:38:56.720 --> 00:38:59.110 align:middle line:84% I'm trying to make something that looks like a sine wave 00:38:59.110 --> 00:38:59.890 align:middle line:90% and has-- 00:38:59.890 --> 00:39:03.200 align:middle line:84% kind of limit the harmonic content so that I can filter it, 00:39:03.200 --> 00:39:04.630 align:middle line:90% right. 00:39:04.630 --> 00:39:07.610 align:middle line:84% By picking delta's 30 degrees, I can make v3 to go to 0. 00:39:07.610 --> 00:39:09.060 align:middle line:90% What am I doing there? 00:39:09.060 --> 00:39:11.800 align:middle line:84% If I come back here to this picture, 00:39:11.800 --> 00:39:15.550 align:middle line:84% I said I'm multiplying the white waveform by the blue waveform, 00:39:15.550 --> 00:39:19.640 align:middle line:84% but notice I drew, actually, delta is exactly 30 degrees. 00:39:19.640 --> 00:39:21.160 align:middle line:84% What happens when it's 30 degrees 00:39:21.160 --> 00:39:26.590 align:middle line:84% is this positive area in the multiplication is sort 00:39:26.590 --> 00:39:29.500 align:middle line:84% of a half-sine bump and a half-sine bump, 00:39:29.500 --> 00:39:34.380 align:middle line:84% exactly cancels this in each in each half cycle. 00:39:34.380 --> 00:39:40.240 align:middle line:84% And when I do that, boom, the third harmonic goes away. 00:39:40.240 --> 00:39:41.760 align:middle line:90% All right. 00:39:41.760 --> 00:39:44.075 align:middle line:84% So what am I left with if I do that? 00:39:44.075 --> 00:39:46.930 align:middle line:90% 00:39:46.930 --> 00:39:50.840 align:middle line:84% Right, if I thought about my system, 00:39:50.840 --> 00:39:55.880 align:middle line:90% suppose I put up some filter. 00:39:55.880 --> 00:40:02.360 align:middle line:84% Like, here's some filter, and here are some vac 00:40:02.360 --> 00:40:03.750 align:middle line:90% that I want, for example. 00:40:03.750 --> 00:40:10.580 align:middle line:84% So here's vx, and here is some vac that's filtered, 00:40:10.580 --> 00:40:11.690 align:middle line:90% that I might care about. 00:40:11.690 --> 00:40:17.810 align:middle line:84% Or in some cases, I might care about ix, which is also related 00:40:17.810 --> 00:40:19.370 align:middle line:90% to vx by some filtering. 00:40:19.370 --> 00:40:21.500 align:middle line:84% OK, it depends what I'm interested in. 00:40:21.500 --> 00:40:26.510 align:middle line:84% But what I can think about that is taking some values of vx 00:40:26.510 --> 00:40:29.240 align:middle line:84% and then running it through a filter transfer function. 00:40:29.240 --> 00:40:31.460 align:middle line:84% That might be vac over vx, and maybe I 00:40:31.460 --> 00:40:33.480 align:middle line:84% can make it look like some cutoff, right. 00:40:33.480 --> 00:40:36.540 align:middle line:90% Well, what do I generally get? 00:40:36.540 --> 00:40:42.110 align:middle line:84% I get a fundamental, then I get a second harmonic 00:40:42.110 --> 00:40:44.960 align:middle line:84% and a third harmonic and a fourth harmonic and a fifth 00:40:44.960 --> 00:40:50.580 align:middle line:84% harmonic and a sixth harmonic, right, so 1, 2, 3, 4, 5, 6. 00:40:50.580 --> 00:40:57.590 align:middle line:84% I want to put the fundamental within the cutoff of my filter 00:40:57.590 --> 00:41:00.060 align:middle line:84% because I'm trying to get fundamental to the output, 00:41:00.060 --> 00:41:01.638 align:middle line:84% but I don't want I want to filter off 00:41:01.638 --> 00:41:02.680 align:middle line:90% all the harmonics, right. 00:41:02.680 --> 00:41:07.780 align:middle line:84% Well, naturally, by half-wave symmetry, 00:41:07.780 --> 00:41:09.920 align:middle line:90% I've gotten rid of 2, 4, and 6. 00:41:09.920 --> 00:41:12.603 align:middle line:84% So I've killed these just by how I've 00:41:12.603 --> 00:41:14.020 align:middle line:84% picked the pattern of the waveform 00:41:14.020 --> 00:41:16.150 align:middle line:90% to be half wave symmetric. 00:41:16.150 --> 00:41:16.870 align:middle line:90% OK. 00:41:16.870 --> 00:41:24.510 align:middle line:84% Now, if I magically go pick a delta of 30, 00:41:24.510 --> 00:41:28.560 align:middle line:90% then I kill off this guy. 00:41:28.560 --> 00:41:30.510 align:middle line:90% OK. 00:41:30.510 --> 00:41:33.490 align:middle line:84% By picking delta exactly 30, I kill off the third, 00:41:33.490 --> 00:41:36.360 align:middle line:84% and so the lowest contents I have to deal with 00:41:36.360 --> 00:41:39.720 align:middle line:90% are the fifth and the seventh. 00:41:39.720 --> 00:41:42.160 align:middle line:84% It's a heck of a lot easier to filter the fifth 00:41:42.160 --> 00:41:44.240 align:middle line:90% than it is the second or third. 00:41:44.240 --> 00:41:44.740 align:middle line:90% OK. 00:41:44.740 --> 00:41:48.550 align:middle line:84% So I can get much cleaner output voltage waveforms, 00:41:48.550 --> 00:41:50.410 align:middle line:84% even though I'm not switching very often, 00:41:50.410 --> 00:41:53.910 align:middle line:84% by being very clever in how I pick my switching angles. 00:41:53.910 --> 00:41:57.640 align:middle line:90% 00:41:57.640 --> 00:41:58.815 align:middle line:90% Any questions about that? 00:41:58.815 --> 00:42:04.520 align:middle line:90% 00:42:04.520 --> 00:42:06.140 align:middle line:90% And we are going to see-- 00:42:06.140 --> 00:42:14.820 align:middle line:84% and this can all be related back to some games about how 00:42:14.820 --> 00:42:19.260 align:middle line:84% I'm picking the precise waveform I synthesize 00:42:19.260 --> 00:42:22.020 align:middle line:84% with the states I have, which are basically 00:42:22.020 --> 00:42:25.190 align:middle line:90% plus vdc minus vdc and zero. 00:42:25.190 --> 00:42:25.690 align:middle line:90% OK. 00:42:25.690 --> 00:42:28.460 align:middle line:90% 00:42:28.460 --> 00:42:32.630 align:middle line:84% Let me just give you a little bit of extra kind 00:42:32.630 --> 00:42:34.950 align:middle line:90% of color about inverters. 00:42:34.950 --> 00:42:36.920 align:middle line:84% And we're going to talk about expanding out 00:42:36.920 --> 00:42:39.707 align:middle line:84% on this in a lot of different dimensions as we move forward, 00:42:39.707 --> 00:42:41.540 align:middle line:84% but I wanted to give you sort of an idea of, 00:42:41.540 --> 00:42:45.240 align:middle line:90% like, what's the fundamental-- 00:42:45.240 --> 00:42:50.310 align:middle line:84% no pun intended-- smallest thing I can do to get nice waveforms? 00:42:50.310 --> 00:42:52.030 align:middle line:90% OK. 00:42:52.030 --> 00:42:56.950 align:middle line:84% One thing relates to how I control 00:42:56.950 --> 00:42:58.383 align:middle line:84% these switches in the real world, 00:42:58.383 --> 00:42:59.800 align:middle line:84% and I mention this because there's 00:42:59.800 --> 00:43:01.390 align:middle line:84% the theoretically controlling the switches 00:43:01.390 --> 00:43:03.440 align:middle line:84% and then there's the practical considerations. 00:43:03.440 --> 00:43:04.210 align:middle line:90% OK. 00:43:04.210 --> 00:43:07.840 align:middle line:84% If I come to this thing, suppose I put this kind of filter 00:43:07.840 --> 00:43:12.190 align:middle line:84% in here, right, so suppose this is equal to this box 00:43:12.190 --> 00:43:13.720 align:middle line:90% that I'm drawing. 00:43:13.720 --> 00:43:16.780 align:middle line:84% So basically, I've got an inductive load, right. 00:43:16.780 --> 00:43:19.890 align:middle line:84% So if this load is somewhat inductive-- 00:43:19.890 --> 00:43:22.390 align:middle line:84% or maybe it's resistive-- this is what a motor winding might 00:43:22.390 --> 00:43:23.020 align:middle line:90% look like. 00:43:23.020 --> 00:43:25.860 align:middle line:90% 00:43:25.860 --> 00:43:29.384 align:middle line:84% I've got to be careful never to open a circuit that winding, 00:43:29.384 --> 00:43:30.360 align:middle line:90% right. 00:43:30.360 --> 00:43:36.240 align:middle line:84% So suppose I have s1 and s2 on, right, 00:43:36.240 --> 00:43:39.087 align:middle line:84% and I wanted to do that switching pattern. 00:43:39.087 --> 00:43:40.670 align:middle line:84% What's the next thing I'm going to do? 00:43:40.670 --> 00:43:44.760 align:middle line:90% 00:43:44.760 --> 00:43:49.550 align:middle line:84% What would be my next state after s1 and s2 is on? 00:43:49.550 --> 00:43:54.830 align:middle line:84% s2s3, right, so I'm going to turn off s1 and turn on s3. 00:43:54.830 --> 00:43:55.640 align:middle line:90% OK. 00:43:55.640 --> 00:43:58.070 align:middle line:84% Now, when you think about doing that in the real world, 00:43:58.070 --> 00:44:00.110 align:middle line:84% you've got to be a little bit careful, right. 00:44:00.110 --> 00:44:04.190 align:middle line:84% If I ever turned on s1 and s3 together, 00:44:04.190 --> 00:44:05.900 align:middle line:84% unfortunate things would happen, right. 00:44:05.900 --> 00:44:09.703 align:middle line:84% That's called a shoot-through, and if you do it too long, 00:44:09.703 --> 00:44:11.120 align:middle line:84% you will kill the switches, right, 00:44:11.120 --> 00:44:12.843 align:middle line:84% because you'll short-circuit the DC bus 00:44:12.843 --> 00:44:14.760 align:middle line:84% and you can source a lot of current into that. 00:44:14.760 --> 00:44:18.110 align:middle line:84% So you've got to make sure s1 and s3 are never on together 00:44:18.110 --> 00:44:20.570 align:middle line:84% and s2 and s4 are never on together. 00:44:20.570 --> 00:44:22.400 align:middle line:84% At the same time, you really, really 00:44:22.400 --> 00:44:24.270 align:middle line:84% don't want to open-circuit that load. 00:44:24.270 --> 00:44:27.380 align:middle line:84% But the nice thing about this structure-- and by the way, 00:44:27.380 --> 00:44:30.440 align:middle line:84% this is sometimes called a, quote, unquote, "VSI," 00:44:30.440 --> 00:44:34.070 align:middle line:84% or Voltage Source Inverter, because you're coming from a DC 00:44:34.070 --> 00:44:38.090 align:middle line:90% voltage, OK, and generating AC-- 00:44:38.090 --> 00:44:41.870 align:middle line:90% is that I can have s1 and s2 on. 00:44:41.870 --> 00:44:45.410 align:middle line:90% First, I turn off s1. 00:44:45.410 --> 00:44:51.020 align:middle line:84% If I turn off s1, say this current's positive, 00:44:51.020 --> 00:44:53.030 align:middle line:90% he still has to flow somewhere. 00:44:53.030 --> 00:44:57.920 align:middle line:84% But he can just commutate from s1 into this diode, right, 00:44:57.920 --> 00:45:01.760 align:middle line:90% and so I don't have to worry. 00:45:01.760 --> 00:45:05.510 align:middle line:84% And then once the diode's on, I can turn on s3, right. 00:45:05.510 --> 00:45:10.640 align:middle line:84% So basically, I will have my q1 of t. 00:45:10.640 --> 00:45:14.810 align:middle line:84% I'll turn the switch 1 off, and then I'll have q3 of t. 00:45:14.810 --> 00:45:20.600 align:middle line:84% And I'll turn him on after some so-called dead-time delay 00:45:20.600 --> 00:45:22.760 align:middle line:90% between the two switches. 00:45:22.760 --> 00:45:23.260 align:middle line:90% OK. 00:45:23.260 --> 00:45:26.050 align:middle line:84% Or even if the current was coming this way, 00:45:26.050 --> 00:45:30.245 align:middle line:84% if I turn off s1, the diode's on, and then I can turn on s3 00:45:30.245 --> 00:45:32.120 align:middle line:84% and it'll just turn off this diode naturally. 00:45:32.120 --> 00:45:34.600 align:middle line:84% Right, so one thing I want to emphasize 00:45:34.600 --> 00:45:36.220 align:middle line:84% is that in the real world, you're 00:45:36.220 --> 00:45:40.780 align:middle line:84% always having some controls to make sure you're not 00:45:40.780 --> 00:45:42.730 align:middle line:90% shooting through your switches. 00:45:42.730 --> 00:45:45.100 align:middle line:90% OK. 00:45:45.100 --> 00:45:46.870 align:middle line:84% That's just a practical detail that you 00:45:46.870 --> 00:45:50.030 align:middle line:84% take care of when you lay out the controls for your inverter. 00:45:50.030 --> 00:45:51.210 align:middle line:90% Any questions about that? 00:45:51.210 --> 00:45:55.350 align:middle line:90% 00:45:55.350 --> 00:45:57.475 align:middle line:84% AUDIENCE: What about other switches with no diodes? 00:45:57.475 --> 00:45:58.933 align:middle line:84% DAVID PERREAULT: Ah, well, then you 00:45:58.933 --> 00:46:00.750 align:middle line:84% could get unhappy really quick, right. 00:46:00.750 --> 00:46:03.070 align:middle line:90% So if I had IGBTs-- 00:46:03.070 --> 00:46:07.000 align:middle line:84% and by the way, IGBTs are very common for driving inverters. 00:46:07.000 --> 00:46:09.390 align:middle line:84% What they will do is they will go put external diodes 00:46:09.390 --> 00:46:13.290 align:middle line:84% across them, because otherwise you will blow them up. 00:46:13.290 --> 00:46:15.580 align:middle line:84% Right, so in fact, later, I'll bring 00:46:15.580 --> 00:46:17.830 align:middle line:84% in an inverter module for a Prius, 00:46:17.830 --> 00:46:21.040 align:middle line:84% and you can actually see the IGBTs, 00:46:21.040 --> 00:46:22.780 align:middle line:84% and right next to them are the diodes 00:46:22.780 --> 00:46:24.410 align:middle line:90% that they put in to do it. 00:46:24.410 --> 00:46:25.670 align:middle line:90% That's an old Prius inverter. 00:46:25.670 --> 00:46:30.850 align:middle line:84% The new Prius inverters uses silicon carbide FETs, I think. 00:46:30.850 --> 00:46:33.117 align:middle line:90% So that's one way to do this. 00:46:33.117 --> 00:46:35.450 align:middle line:84% And as I said, this is called a voltage source inverter. 00:46:35.450 --> 00:46:39.160 align:middle line:84% It's not the only way you can build an inverter, OK. 00:46:39.160 --> 00:46:40.990 align:middle line:90% What would be another way? 00:46:40.990 --> 00:46:44.530 align:middle line:90% Well, here's another trick. 00:46:44.530 --> 00:46:50.050 align:middle line:84% Suppose I took a DC voltage source 00:46:50.050 --> 00:46:54.310 align:middle line:84% and I put it in series with L big, 00:46:54.310 --> 00:46:56.650 align:middle line:84% and I'll make this inductor so big 00:46:56.650 --> 00:47:01.690 align:middle line:84% that this I becomes approximately equal to IDC, 00:47:01.690 --> 00:47:02.330 align:middle line:90% right. 00:47:02.330 --> 00:47:04.630 align:middle line:84% If I have a huge inductor here, I can sort of 00:47:04.630 --> 00:47:07.420 align:middle line:84% make a voltage source in series with an inductor look, at least 00:47:07.420 --> 00:47:09.920 align:middle line:84% on a short time scale, like a current source, right. 00:47:09.920 --> 00:47:14.500 align:middle line:84% So I might think of this thing as being a current source 00:47:14.500 --> 00:47:15.000 align:middle line:90% now, right. 00:47:15.000 --> 00:47:21.980 align:middle line:84% So here I have some current IDC, and now I 00:47:21.980 --> 00:47:27.320 align:middle line:84% might want to create an AC current from that. 00:47:27.320 --> 00:47:30.620 align:middle line:84% I could do that again with a set of switches. 00:47:30.620 --> 00:47:31.340 align:middle line:90% OK. 00:47:31.340 --> 00:47:34.900 align:middle line:84% Maybe, however, what I would want is switches that do this. 00:47:34.900 --> 00:47:49.700 align:middle line:90% 00:47:49.700 --> 00:47:51.895 align:middle line:84% I'm picking a different switch type just for fun. 00:47:51.895 --> 00:47:56.050 align:middle line:90% 00:47:56.050 --> 00:47:56.680 align:middle line:90% OK. 00:47:56.680 --> 00:48:02.030 align:middle line:84% These switches can carry unidirectional current. 00:48:02.030 --> 00:48:04.420 align:middle line:84% But they can block voltage in both directions, 00:48:04.420 --> 00:48:09.050 align:middle line:84% right, because they will not ever carry current that way, 00:48:09.050 --> 00:48:11.210 align:middle line:84% and if I try to put a reverse voltage on them, 00:48:11.210 --> 00:48:14.420 align:middle line:84% this diode will block and everything will be happy. 00:48:14.420 --> 00:48:15.140 align:middle line:90% OK. 00:48:15.140 --> 00:48:18.350 align:middle line:84% So then I could have a current source essentially 00:48:18.350 --> 00:48:20.840 align:middle line:84% going in, and then instead of having a filter that 00:48:20.840 --> 00:48:23.150 align:middle line:84% looks inductive, maybe I will have 00:48:23.150 --> 00:48:30.460 align:middle line:84% a filter that looks capacitive, something like this. 00:48:30.460 --> 00:48:34.110 align:middle line:90% 00:48:34.110 --> 00:48:37.950 align:middle line:90% And here I can have iac or-- 00:48:37.950 --> 00:48:41.250 align:middle line:90% or vac or an iac. 00:48:41.250 --> 00:48:41.790 align:middle line:90% OK. 00:48:41.790 --> 00:48:44.610 align:middle line:84% So what I'm going to do is I can switch this DC 00:48:44.610 --> 00:48:49.270 align:middle line:84% current into the load this way by having these two switches on. 00:48:49.270 --> 00:48:53.650 align:middle line:84% I can switch this current into the load the other way 00:48:53.650 --> 00:48:57.270 align:middle line:90% by having these two switches on. 00:48:57.270 --> 00:49:02.300 align:middle line:84% Or if I turn these two switches on, the load gets no current, 00:49:02.300 --> 00:49:03.920 align:middle line:84% or if I switch these two switches on, 00:49:03.920 --> 00:49:06.480 align:middle line:90% the load gets no current. 00:49:06.480 --> 00:49:07.520 align:middle line:90% That make sense? 00:49:07.520 --> 00:49:11.760 align:middle line:84% So instead of synthesizing some pulsed DC voltage that's 00:49:11.760 --> 00:49:15.700 align:middle line:84% positive and negative, I can synthesize a pulsed AC current 00:49:15.700 --> 00:49:17.020 align:middle line:90% that's positive and negative. 00:49:17.020 --> 00:49:21.570 align:middle line:84% This would be called a current source inverter. 00:49:21.570 --> 00:49:25.320 align:middle line:84% Now, in practical applications, voltage source inverters, 00:49:25.320 --> 00:49:26.820 align:middle line:84% especially at low powers, tend to be 00:49:26.820 --> 00:49:30.010 align:middle line:84% much more common because they're simpler to realize. 00:49:30.010 --> 00:49:32.160 align:middle line:84% You don't need a big inductor and everything else. 00:49:32.160 --> 00:49:35.220 align:middle line:84% People do, however, at high power, sometimes 00:49:35.220 --> 00:49:40.200 align:middle line:84% like current source inverters because if these switches fail, 00:49:40.200 --> 00:49:43.050 align:middle line:84% you don't immediately short everything out, like 00:49:43.050 --> 00:49:45.618 align:middle line:84% from a DC voltage source, and get a huge pulse of current. 00:49:45.618 --> 00:49:47.910 align:middle line:84% Things take time to ramp up through this current, which 00:49:47.910 --> 00:49:51.000 align:middle line:84% lets you blow fuses or shut things down or whatever. 00:49:51.000 --> 00:49:52.950 align:middle line:84% So at high power, sometimes people 00:49:52.950 --> 00:49:55.230 align:middle line:84% like versions of current source inverters, 00:49:55.230 --> 00:49:58.590 align:middle line:84% but more frequently, people use voltage source inverters. 00:49:58.590 --> 00:50:00.000 align:middle line:90% And interestingly, by the way-- 00:50:00.000 --> 00:50:04.140 align:middle line:84% I talked about dead time, where these two switches have 00:50:04.140 --> 00:50:08.760 align:middle line:84% to be off at the same time for a little while 00:50:08.760 --> 00:50:11.940 align:middle line:84% and we let current go through the diodes. 00:50:11.940 --> 00:50:16.890 align:middle line:84% Here, I'd better never have a time when all four switches are 00:50:16.890 --> 00:50:20.160 align:middle line:84% off, because then I'd be open-circuiting this guy, right. 00:50:20.160 --> 00:50:24.750 align:middle line:84% So what I might do is, if this switch and this switch were on, 00:50:24.750 --> 00:50:29.200 align:middle line:84% I will then briefly turn this one on also, 00:50:29.200 --> 00:50:31.220 align:middle line:84% and then I can turn this one off, 00:50:31.220 --> 00:50:34.570 align:middle line:84% right, so I have overlap in my switch-on times instead 00:50:34.570 --> 00:50:37.900 align:middle line:84% of dead time in my switch-on times. 00:50:37.900 --> 00:50:38.410 align:middle line:90% OK. 00:50:38.410 --> 00:50:41.830 align:middle line:84% But you can use, apply all the same concepts I talked about, 00:50:41.830 --> 00:50:45.070 align:middle line:84% about synthesizing pulsed-voltage waveforms, 00:50:45.070 --> 00:50:47.470 align:middle line:84% to synthesize pulsed-current waveforms. 00:50:47.470 --> 00:50:50.200 align:middle line:84% We'll spend most of our time talking about voltage source 00:50:50.200 --> 00:50:53.200 align:middle line:84% inverters, but I just wanted you to know there are other ways 00:50:53.200 --> 00:50:55.260 align:middle line:90% to play these games. 00:50:55.260 --> 00:50:57.515 align:middle line:84% Any final questions before we wrap up for the day? 00:50:57.515 --> 00:51:00.600 align:middle line:90% 00:51:00.600 --> 00:51:02.060 align:middle line:90% OK, we'll pick this up tomorrow. 00:51:02.060 --> 00:51:03.950 align:middle line:90% Have a great day. 00:51:03.950 --> 00:51:07.000 align:middle line:90%