WEBVTT 00:00:00.000 --> 00:00:02.390 align:middle line:90% [SQUEAKING] 00:00:02.390 --> 00:00:04.302 align:middle line:90% [RUSTLING] 00:00:04.302 --> 00:00:06.692 align:middle line:90% [CLICKING] 00:00:06.692 --> 00:00:09.390 align:middle line:90% 00:00:09.390 --> 00:00:11.400 align:middle line:84% DAVID PERREAULT: OK, good afternoon. 00:00:11.400 --> 00:00:16.110 align:middle line:84% So last time, we started talking about power electronics 00:00:16.110 --> 00:00:17.580 align:middle line:90% with three-phase elements. 00:00:17.580 --> 00:00:20.718 align:middle line:84% We spent our time talking about just diode rectifiers. 00:00:20.718 --> 00:00:23.010 align:middle line:84% And that's actually quite important because when you're 00:00:23.010 --> 00:00:26.490 align:middle line:84% building really high-power systems, 00:00:26.490 --> 00:00:28.410 align:middle line:84% very often that's the kind of thing 00:00:28.410 --> 00:00:30.390 align:middle line:84% you have at your front end between your AC 00:00:30.390 --> 00:00:32.820 align:middle line:90% to convert to DC. 00:00:32.820 --> 00:00:35.700 align:middle line:84% At the same time, a lot of applications 00:00:35.700 --> 00:00:38.430 align:middle line:84% require high-frequency switching power conversion. 00:00:38.430 --> 00:00:40.530 align:middle line:84% So the kind of things we were talking 00:00:40.530 --> 00:00:43.050 align:middle line:84% about single-phase inverters, the question 00:00:43.050 --> 00:00:46.320 align:middle line:84% is, what do you do with a three-phase inverter? 00:00:46.320 --> 00:00:51.990 align:middle line:84% And one possibility is just to, in some cases, 00:00:51.990 --> 00:00:54.060 align:middle line:90% replicate a single-phase system. 00:00:54.060 --> 00:00:58.500 align:middle line:84% So you could imagine if I had, say, a three-phase machine 00:00:58.500 --> 00:01:03.750 align:middle line:84% with open-ended windings, maybe I 00:01:03.750 --> 00:01:10.540 align:middle line:84% would have something where I had a phase A and a phase B 00:01:10.540 --> 00:01:17.660 align:middle line:84% and a phase C. And each of them have a pair of terminals. 00:01:17.660 --> 00:01:21.010 align:middle line:84% And I say, OK, I can go build a three-phase inverter basically 00:01:21.010 --> 00:01:23.110 align:middle line:84% by building three single-phase inverters. 00:01:23.110 --> 00:01:26.183 align:middle line:84% So maybe I come in from my DC bus, VDC. 00:01:26.183 --> 00:01:30.220 align:middle line:90% 00:01:30.220 --> 00:01:36.310 align:middle line:84% And I could come in and say, OK, let me just build 00:01:36.310 --> 00:01:41.203 align:middle line:84% a single-phase inverter, which would be, say, four power 00:01:41.203 --> 00:01:41.703 align:middle line:90% MOSFETs. 00:01:41.703 --> 00:01:46.912 align:middle line:90% 00:01:46.912 --> 00:01:50.603 align:middle line:84% And I'll show the diodes on the power MOSFETs just for clarity. 00:01:50.603 --> 00:01:53.220 align:middle line:90% 00:01:53.220 --> 00:01:56.100 align:middle line:84% And I could come out, and here's one terminal, 00:01:56.100 --> 00:01:57.540 align:middle line:90% and here's the other. 00:01:57.540 --> 00:02:01.230 align:middle line:84% And I build this to drive phase A. 00:02:01.230 --> 00:02:02.763 align:middle line:90% And then I build another copy. 00:02:02.763 --> 00:02:05.460 align:middle line:90% 00:02:05.460 --> 00:02:08.460 align:middle line:84% This is phase A. I build another copy, drive phase B. 00:02:08.460 --> 00:02:13.060 align:middle line:84% And I bring it in from the same terminals. 00:02:13.060 --> 00:02:17.580 align:middle line:84% And then I build another copy, and I drive phase C. 00:02:17.580 --> 00:02:22.650 align:middle line:84% And I could again build in same DC bus. 00:02:22.650 --> 00:02:24.330 align:middle line:90% And away I go. 00:02:24.330 --> 00:02:25.980 align:middle line:90% So I have 12 devices. 00:02:25.980 --> 00:02:29.430 align:middle line:84% I just have single-phase inverters, and I run it, 00:02:29.430 --> 00:02:33.000 align:middle line:84% And in fact, sometimes people do this. 00:02:33.000 --> 00:02:35.220 align:middle line:84% And in fact, actually, at MIT right now, 00:02:35.220 --> 00:02:37.740 align:middle line:84% we're building a megawatt system that 00:02:37.740 --> 00:02:39.610 align:middle line:90% basically has this structure. 00:02:39.610 --> 00:02:42.480 align:middle line:84% It's sets of single-phase inverters 00:02:42.480 --> 00:02:46.120 align:middle line:84% that drive a three-phase system, where the machine is a custom 00:02:46.120 --> 00:02:49.660 align:middle line:84% machine and has so-called open-ended windings like this. 00:02:49.660 --> 00:02:52.540 align:middle line:90% So it works perfectly well. 00:02:52.540 --> 00:02:58.660 align:middle line:84% However, the vast majority of, say, three-phase motors 00:02:58.660 --> 00:03:02.210 align:middle line:84% are not just separate windings connected like that. 00:03:02.210 --> 00:03:09.130 align:middle line:84% Typically, if, for example, I have a three-phase motor, what 00:03:09.130 --> 00:03:13.060 align:middle line:84% I will have is either something connected in wye, where this is 00:03:13.060 --> 00:03:17.020 align:middle line:84% one motor winding, a second motor winding, and a third motor 00:03:17.020 --> 00:03:23.230 align:middle line:84% winding, where I'll have A, B, and C. 00:03:23.230 --> 00:03:26.380 align:middle line:90% I only have three terminals. 00:03:26.380 --> 00:03:29.360 align:middle line:84% Or maybe I have-- this is a wye connection. 00:03:29.360 --> 00:03:31.420 align:middle line:84% Maybe I have a delta connection where 00:03:31.420 --> 00:03:33.090 align:middle line:90% the machine is wound like this. 00:03:33.090 --> 00:03:36.220 align:middle line:90% 00:03:36.220 --> 00:03:42.370 align:middle line:84% And again, I have A, B, and C. In this case, 00:03:42.370 --> 00:03:43.610 align:middle line:90% I might have a neutral. 00:03:43.610 --> 00:03:45.520 align:middle line:90% In this case, I have no neutral. 00:03:45.520 --> 00:03:52.060 align:middle line:84% So in this case, I can't just arbitrarily apply voltages 00:03:52.060 --> 00:03:54.740 align:middle line:84% from three independent single-phase inverters. 00:03:54.740 --> 00:03:55.240 align:middle line:90% Why? 00:03:55.240 --> 00:03:58.750 align:middle line:84% Because if I accidentally connected phase A 00:03:58.750 --> 00:04:00.550 align:middle line:84% to high with one inverter and phase A 00:04:00.550 --> 00:04:03.340 align:middle line:84% to low with another inverter, I'd out my DC bus and everything 00:04:03.340 --> 00:04:05.590 align:middle line:90% would blow up. 00:04:05.590 --> 00:04:08.740 align:middle line:84% So if I'm going to drive something that's 00:04:08.740 --> 00:04:13.070 align:middle line:84% connected as a three-phase load in wye or delta, 00:04:13.070 --> 00:04:15.460 align:middle line:90% I need a different plan. 00:04:15.460 --> 00:04:18.579 align:middle line:90% OK, so what do people do? 00:04:18.579 --> 00:04:21.160 align:middle line:84% It's very much the structure that we 00:04:21.160 --> 00:04:24.050 align:middle line:84% saw with the three-phase bridge rectifier. 00:04:24.050 --> 00:04:26.350 align:middle line:84% Instead of having diodes, however, I 00:04:26.350 --> 00:04:29.920 align:middle line:90% will have active switches. 00:04:29.920 --> 00:04:32.985 align:middle line:84% So maybe I would come up and build something like this. 00:04:32.985 --> 00:04:36.550 align:middle line:90% 00:04:36.550 --> 00:04:38.157 align:middle line:90% I'll have my DC bus come in. 00:04:38.157 --> 00:04:39.490 align:middle line:90% So I'll have two terminals, VDC. 00:04:39.490 --> 00:04:44.240 align:middle line:90% 00:04:44.240 --> 00:04:47.280 align:middle line:84% I will have a first bridge leg like this. 00:04:47.280 --> 00:04:50.790 align:middle line:84% And let me make sure and show the diodes again. 00:04:50.790 --> 00:04:53.850 align:middle line:90% And this can generate VA. 00:04:53.850 --> 00:05:04.130 align:middle line:90% A second one can generate VB. 00:05:04.130 --> 00:05:11.355 align:middle line:90% And a third one can generate VC. 00:05:11.355 --> 00:05:14.740 align:middle line:90% 00:05:14.740 --> 00:05:17.170 align:middle line:84% Maybe what I'll do is I will consider 00:05:17.170 --> 00:05:20.900 align:middle line:84% the midpoint of this bus for reasons we'll talk about. 00:05:20.900 --> 00:05:29.440 align:middle line:84% And I'm going to call this node V sub R. OK, that's 00:05:29.440 --> 00:05:30.620 align:middle line:90% my reference voltage. 00:05:30.620 --> 00:05:32.530 align:middle line:84% It's just the center point of the bus. 00:05:32.530 --> 00:05:37.160 align:middle line:84% Where I ground this thing, we can discuss that. 00:05:37.160 --> 00:05:39.640 align:middle line:84% But let me just pretend that this is exactly in the middle. 00:05:39.640 --> 00:05:42.048 align:middle line:90% It's VDC over 2 from here. 00:05:42.048 --> 00:05:44.090 align:middle line:84% And we'll see why we're going to talk about that. 00:05:44.090 --> 00:05:45.530 align:middle line:90% And that's my inverter. 00:05:45.530 --> 00:05:49.490 align:middle line:84% And maybe I would connect it up to some three-phase load. 00:05:49.490 --> 00:05:51.790 align:middle line:84% So maybe it would look like this. 00:05:51.790 --> 00:05:55.000 align:middle line:90% I would have my three phases. 00:05:55.000 --> 00:05:58.040 align:middle line:90% Here's one. 00:05:58.040 --> 00:05:59.780 align:middle line:90% Here's two. 00:05:59.780 --> 00:06:01.610 align:middle line:90% And here's three. 00:06:01.610 --> 00:06:03.950 align:middle line:84% And maybe I would connect this point 00:06:03.950 --> 00:06:07.340 align:middle line:84% to what I call V sub N, or the neutral point. 00:06:07.340 --> 00:06:12.260 align:middle line:84% That's this neutral point if it was a wye-connected load. 00:06:12.260 --> 00:06:17.780 align:middle line:84% Or I could just have line to lines if there's no neutral. 00:06:17.780 --> 00:06:21.500 align:middle line:90% So what's the advantage here? 00:06:21.500 --> 00:06:25.670 align:middle line:84% Well, you'll notice that this structure only requires 00:06:25.670 --> 00:06:28.160 align:middle line:90% six stitches instead of my 12. 00:06:28.160 --> 00:06:29.900 align:middle line:90% So it's simpler. 00:06:29.900 --> 00:06:35.060 align:middle line:84% It also turns out that as compared to that solution-- 00:06:35.060 --> 00:06:38.360 align:middle line:84% let's suppose I could connect my motor either way I wanted. 00:06:38.360 --> 00:06:41.900 align:middle line:84% I could connect it in wye or delta, 00:06:41.900 --> 00:06:45.740 align:middle line:84% or I could connect it as three independent windings. 00:06:45.740 --> 00:06:49.697 align:middle line:84% Because this circuit, like this phase leg-- 00:06:49.697 --> 00:06:52.280 align:middle line:84% whatever current goes out, it returns through one of the other 00:06:52.280 --> 00:06:54.620 align:middle line:90% phase legs-- 00:06:54.620 --> 00:06:58.590 align:middle line:84% I get an advantage in terms of the utilization of the devices. 00:06:58.590 --> 00:07:02.100 align:middle line:84% So I will either require less semiconductor area 00:07:02.100 --> 00:07:05.370 align:middle line:84% to get a certain conduction loss in this structure, 00:07:05.370 --> 00:07:08.880 align:middle line:84% or I will have lower loss for the same semiconductor 00:07:08.880 --> 00:07:12.570 align:middle line:84% area-- lower conduction loss for the same semiconductor area 00:07:12.570 --> 00:07:15.060 align:middle line:84% as I will compared to the top solution. 00:07:15.060 --> 00:07:19.620 align:middle line:84% So this solution in most three-phase applications 00:07:19.620 --> 00:07:22.110 align:middle line:90% is vastly preferred. 00:07:22.110 --> 00:07:23.830 align:middle line:90% Where's the exception to that? 00:07:23.830 --> 00:07:28.710 align:middle line:84% It turns out that if I wanted to generate a certain ripple 00:07:28.710 --> 00:07:33.150 align:middle line:84% in an inductive load, like a motor inductive back 00:07:33.150 --> 00:07:36.240 align:middle line:84% emf kind of load, it turns out that you end up 00:07:36.240 --> 00:07:39.030 align:middle line:84% having to switch this bridge at a higher frequency 00:07:39.030 --> 00:07:41.950 align:middle line:84% than you do the three sets of single phase. 00:07:41.950 --> 00:07:45.030 align:middle line:84% So in a system where it's a really high-frequency output 00:07:45.030 --> 00:07:47.730 align:middle line:84% and your switching losses are heavily dominated, 00:07:47.730 --> 00:07:51.250 align:middle line:84% the top solution actually ends up starting to be advantageous. 00:07:51.250 --> 00:07:54.330 align:middle line:84% So in very specialized, high-speed machines, 00:07:54.330 --> 00:07:56.880 align:middle line:84% for example, you might go for the top solution 00:07:56.880 --> 00:07:59.200 align:middle line:84% of independent phase drive if you're allowed. 00:07:59.200 --> 00:08:03.600 align:middle line:84% But the vast majority of three-phase inverters 00:08:03.600 --> 00:08:06.060 align:middle line:84% or three-phase DC to AC converters, power 00:08:06.060 --> 00:08:07.470 align:middle line:90% could flow either direction. 00:08:07.470 --> 00:08:11.880 align:middle line:84% Where you have active control is with this three-phase bridge 00:08:11.880 --> 00:08:12.850 align:middle line:90% structure. 00:08:12.850 --> 00:08:16.170 align:middle line:90% 00:08:16.170 --> 00:08:17.975 align:middle line:90% Any questions about that so far? 00:08:17.975 --> 00:08:21.100 align:middle line:90% 00:08:21.100 --> 00:08:25.660 align:middle line:84% So one thing I wanted to note about this 00:08:25.660 --> 00:08:31.690 align:middle line:84% is that I want to pay attention that these devices are-- 00:08:31.690 --> 00:08:33.940 align:middle line:84% especially if the load is going to be inductive, 00:08:33.940 --> 00:08:36.190 align:middle line:84% so I don't actually control instantaneously, say, 00:08:36.190 --> 00:08:38.059 align:middle line:84% what the direction of that load is-- 00:08:38.059 --> 00:08:42.100 align:middle line:84% I usually have devices with reverse diodes on them, 00:08:42.100 --> 00:08:45.280 align:middle line:84% or I use MOSFETs, which have it built in because that way when 00:08:45.280 --> 00:08:47.170 align:middle line:84% I turn one device off, I can always 00:08:47.170 --> 00:08:49.070 align:middle line:84% guarantee there's a path for the current. 00:08:49.070 --> 00:08:51.790 align:middle line:84% So these are single-directional blocking, 00:08:51.790 --> 00:08:53.690 align:middle line:84% bi-directional carrying switches. 00:08:53.690 --> 00:08:57.400 align:middle line:84% So this is a kind of voltage source inverter. 00:08:57.400 --> 00:08:59.860 align:middle line:84% There are also three-phase current source inverters 00:08:59.860 --> 00:09:01.760 align:middle line:84% just as there are with single phase. 00:09:01.760 --> 00:09:06.220 align:middle line:84% But again, this is vastly the most common. 00:09:06.220 --> 00:09:12.730 align:middle line:84% Just to illustrate this, I'll show you this guy here. 00:09:12.730 --> 00:09:16.960 align:middle line:84% This is actually an inverter out of an older model Prius. 00:09:16.960 --> 00:09:19.460 align:middle line:90% You'll see the two in here. 00:09:19.460 --> 00:09:20.740 align:middle line:90% That's the DC bus coming in. 00:09:20.740 --> 00:09:22.490 align:middle line:84% That's where the capacitors and everything 00:09:22.490 --> 00:09:25.200 align:middle line:84% else would hook up and go back to the battery. 00:09:25.200 --> 00:09:30.890 align:middle line:84% And then there's phase A, phase B, phase C outputs. 00:09:30.890 --> 00:09:33.860 align:middle line:84% Each of these, what you'll see is a pile of devices in parallel 00:09:33.860 --> 00:09:35.820 align:middle line:84% in order to provide the required power. 00:09:35.820 --> 00:09:37.250 align:middle line:84% And when you look closely at this, 00:09:37.250 --> 00:09:40.010 align:middle line:84% you're going to see that there are essentially 00:09:40.010 --> 00:09:42.540 align:middle line:84% three terminal devices that go back to gate drivers. 00:09:42.540 --> 00:09:44.930 align:middle line:84% Those are the IGBTs in this older model. 00:09:44.930 --> 00:09:47.180 align:middle line:84% Today, they use silicon carbide FETs. 00:09:47.180 --> 00:09:49.910 align:middle line:84% And in antiparallel with them, because the IGBT 00:09:49.910 --> 00:09:53.180 align:middle line:84% can't carry reverse current, they have actual diodes. 00:09:53.180 --> 00:09:56.570 align:middle line:84% So in their design, the actual diode 00:09:56.570 --> 00:10:00.944 align:middle line:84% is separate from the actual switch. 00:10:00.944 --> 00:10:02.310 align:middle line:90% And so I'll pass this around. 00:10:02.310 --> 00:10:03.102 align:middle line:90% You can look at it. 00:10:03.102 --> 00:10:04.903 align:middle line:84% And by the way, if your Prius happens 00:10:04.903 --> 00:10:06.320 align:middle line:84% to be missing an inverter, I swear 00:10:06.320 --> 00:10:07.860 align:middle line:90% I had nothing to do with it. 00:10:07.860 --> 00:10:11.590 align:middle line:90% 00:10:11.590 --> 00:10:16.120 align:middle line:84% So what can I do with this inverter? 00:10:16.120 --> 00:10:18.703 align:middle line:90% Keep in mind-- go ahead, Jack. 00:10:18.703 --> 00:10:20.870 align:middle line:84% AUDIENCE: I just have a question about the inverter? 00:10:20.870 --> 00:10:22.480 align:middle line:84% Why would they choose to do IGBTs 00:10:22.480 --> 00:10:24.008 align:middle line:90% with diodes instead of MOSFET? 00:10:24.008 --> 00:10:26.050 align:middle line:84% DAVID PERREAULT: Ah, that's a very good question. 00:10:26.050 --> 00:10:29.260 align:middle line:84% Because if you think about a MOSFET, 00:10:29.260 --> 00:10:32.440 align:middle line:84% a MOSFET's on state looks like a resistance, 00:10:32.440 --> 00:10:36.610 align:middle line:84% whereas an IGBT's on state looks like a constant drop. 00:10:36.610 --> 00:10:40.000 align:middle line:84% And it turns out that at the time that was built, 00:10:40.000 --> 00:10:42.790 align:middle line:84% for a very high-voltage device, you 00:10:42.790 --> 00:10:47.530 align:middle line:84% could get lower forward drop for a given device area with an IGBT 00:10:47.530 --> 00:10:49.250 align:middle line:90% than you could with a FET. 00:10:49.250 --> 00:10:54.410 align:middle line:84% And so maybe until 5 years ago or maybe 10, 00:10:54.410 --> 00:10:58.160 align:middle line:84% it was most common to use IGBTs especially at high voltage. 00:10:58.160 --> 00:11:02.570 align:middle line:84% More recently, with wide-band gap devices, 00:11:02.570 --> 00:11:08.480 align:middle line:84% people are starting to move over to sometimes IGBTs but mainly 00:11:08.480 --> 00:11:12.653 align:middle line:84% FET kind of structures because you can get away with that. 00:11:12.653 --> 00:11:14.070 align:middle line:84% And it turns out to be lower loss. 00:11:14.070 --> 00:11:15.985 align:middle line:84% So whichever switch gives you less loss 00:11:15.985 --> 00:11:17.860 align:middle line:84% is ultimately what you want to choose to use. 00:11:17.860 --> 00:11:20.590 align:middle line:90% 00:11:20.590 --> 00:11:25.090 align:middle line:84% So let's think about what we can do with this. 00:11:25.090 --> 00:11:28.180 align:middle line:84% And keep in mind, we said one of the big advantages 00:11:28.180 --> 00:11:34.540 align:middle line:84% of having three phase is that we get natural cancellation of all 00:11:34.540 --> 00:11:35.920 align:middle line:90% the triple-N harmonics. 00:11:35.920 --> 00:11:38.890 align:middle line:84% And we're going to see that with this inverter structure. 00:11:38.890 --> 00:11:41.230 align:middle line:84% But essentially, if I think about it, 00:11:41.230 --> 00:11:42.550 align:middle line:90% I have three bridge legs. 00:11:42.550 --> 00:11:45.340 align:middle line:90% 00:11:45.340 --> 00:11:49.930 align:middle line:84% And each one of them, either the top switch can be on 00:11:49.930 --> 00:11:51.400 align:middle line:90% or the bottom switch can be on. 00:11:51.400 --> 00:11:56.320 align:middle line:84% So it's like there's 2 to the 3, or 8, possible states 00:11:56.320 --> 00:11:59.770 align:middle line:84% for these inverter switches to be on, assuming one switch is 00:11:59.770 --> 00:12:01.930 align:middle line:90% always on in each bridge leg. 00:12:01.930 --> 00:12:05.710 align:middle line:90% So eight possible states. 00:12:05.710 --> 00:12:12.010 align:middle line:84% If I come and show you what those states look like, 00:12:12.010 --> 00:12:13.570 align:middle line:90% they look like this. 00:12:13.570 --> 00:12:16.390 align:middle line:84% So if I think about the three inputs, 00:12:16.390 --> 00:12:19.830 align:middle line:84% I can have bottom, bottom, bottom; top, bottom, bottom; et 00:12:19.830 --> 00:12:20.330 align:middle line:90% cetera. 00:12:20.330 --> 00:12:21.740 align:middle line:90% There's eight possible states. 00:12:21.740 --> 00:12:24.320 align:middle line:90% 00:12:24.320 --> 00:12:30.880 align:middle line:84% And these are what you get for AR, BR, and CR normalized 00:12:30.880 --> 00:12:33.070 align:middle line:90% to the DC bus voltage. 00:12:33.070 --> 00:12:37.580 align:middle line:84% And then this is what you get for VAB, VBC, and VCA 00:12:37.580 --> 00:12:39.140 align:middle line:90% at the line-to-line voltages. 00:12:39.140 --> 00:12:44.440 align:middle line:84% And this is what you get for the line-to-neutral voltages 00:12:44.440 --> 00:12:46.430 align:middle line:90% on this load. 00:12:46.430 --> 00:12:52.330 align:middle line:84% And I want to emphasize that in this structure, 00:12:52.330 --> 00:12:55.750 align:middle line:84% I'm not proposing that we connect this reference 00:12:55.750 --> 00:12:57.670 align:middle line:90% voltage to the neutral. 00:12:57.670 --> 00:13:00.790 align:middle line:84% This neutral, if it's a wye-connected system, 00:13:00.790 --> 00:13:03.490 align:middle line:84% we're typically leaving this neutral floating. 00:13:03.490 --> 00:13:06.190 align:middle line:84% So it's not connected to anything. 00:13:06.190 --> 00:13:09.070 align:middle line:90% Or in a delta-connected system-- 00:13:09.070 --> 00:13:11.912 align:middle line:84% I drew this as if it were a wye-connected system. 00:13:11.912 --> 00:13:13.370 align:middle line:84% If it was a delta-connected system, 00:13:13.370 --> 00:13:16.220 align:middle line:84% there wouldn't even be a neutral. 00:13:16.220 --> 00:13:20.550 align:middle line:84% So the neutral is not connected to this reference voltage. 00:13:20.550 --> 00:13:25.220 align:middle line:84% So how do I figure out what the voltage say between VA 00:13:25.220 --> 00:13:27.390 align:middle line:90% and the neutral is? 00:13:27.390 --> 00:13:28.310 align:middle line:90% So this would be-- 00:13:28.310 --> 00:13:30.830 align:middle line:90% I'd call this VAN. 00:13:30.830 --> 00:13:35.750 align:middle line:84% Well, if I imagined that my load, which might be a motor, 00:13:35.750 --> 00:13:38.420 align:middle line:84% acts like three identical impedances, 00:13:38.420 --> 00:13:40.080 align:middle line:84% then it's just a voltage divider. 00:13:40.080 --> 00:13:44.540 align:middle line:84% So if I have one bottom switch on and two top switches on, VN 00:13:44.540 --> 00:13:46.890 align:middle line:90% goes to 2/3 of the way up. 00:13:46.890 --> 00:13:49.700 align:middle line:84% If two bottom switches are on and one top switch is 00:13:49.700 --> 00:13:52.520 align:middle line:84% on, it's 2/3 of the way to the bottom. 00:13:52.520 --> 00:13:56.120 align:middle line:84% If all the top switches are on, VN goes to the top. 00:13:56.120 --> 00:14:01.010 align:middle line:84% So the neutral just goes up and down instantaneously 00:14:01.010 --> 00:14:04.730 align:middle line:84% based on voltage division from these individual values. 00:14:04.730 --> 00:14:07.070 align:middle line:84% Now, I'm not saying you could not 00:14:07.070 --> 00:14:09.158 align:middle line:84% tie the neutral back to the reference, 00:14:09.158 --> 00:14:10.200 align:middle line:90% the center of the bridge. 00:14:10.200 --> 00:14:12.740 align:middle line:84% In fact, that would be just this circuit with half bridges 00:14:12.740 --> 00:14:14.010 align:middle line:90% instead of full bridges. 00:14:14.010 --> 00:14:19.910 align:middle line:84% It's just not common to do because you lose the three-phase 00:14:19.910 --> 00:14:20.780 align:middle line:90% harmonic-- 00:14:20.780 --> 00:14:23.180 align:middle line:84% the three-phase triple-N cancellation if you do that. 00:14:23.180 --> 00:14:25.850 align:middle line:84% You lose a lot of the benefits if you 00:14:25.850 --> 00:14:29.830 align:middle line:90% don't let the neutral float. 00:14:29.830 --> 00:14:35.930 align:middle line:84% So this just shows the voltages I can synthesize. 00:14:35.930 --> 00:14:40.820 align:middle line:84% Two of the eight states basically short out the load. 00:14:40.820 --> 00:14:44.120 align:middle line:84% Because if all the top switches are on, 00:14:44.120 --> 00:14:46.740 align:middle line:84% basically I'm tying all of these three nodes together. 00:14:46.740 --> 00:14:50.280 align:middle line:84% So they basically just short the three terminals together. 00:14:50.280 --> 00:14:53.630 align:middle line:84% If all the bottom switches are on, I do the same thing. 00:14:53.630 --> 00:14:58.340 align:middle line:84% And any other switch configuration, 00:14:58.340 --> 00:15:03.170 align:middle line:84% I get some other set of states on my inverter output. 00:15:03.170 --> 00:15:05.390 align:middle line:90% Any questions about that? 00:15:05.390 --> 00:15:08.400 align:middle line:84% AUDIENCE: So where's the voltage divider between again? 00:15:08.400 --> 00:15:13.580 align:middle line:84% DAVID PERREAULT: So for example, suppose I took-- 00:15:13.580 --> 00:15:14.910 align:middle line:90% let me draw it this way. 00:15:14.910 --> 00:15:17.550 align:middle line:90% Here's VDC. 00:15:17.550 --> 00:15:20.360 align:middle line:84% Suppose I turn the top switch on here. 00:15:20.360 --> 00:15:24.380 align:middle line:84% So that means I tie this guy to the top. 00:15:24.380 --> 00:15:34.090 align:middle line:84% If the top switch is on here, I tie this guy to the top. 00:15:34.090 --> 00:15:39.190 align:middle line:84% And if the last guy has the bottom switch on, 00:15:39.190 --> 00:15:40.790 align:middle line:90% I tie him to the bottom. 00:15:40.790 --> 00:15:44.110 align:middle line:84% So if I have two to the top and one to the bottom, 00:15:44.110 --> 00:15:47.230 align:middle line:84% this node voltage, if these green boxes are all equal, 00:15:47.230 --> 00:15:49.680 align:middle line:84% impedances will be 2/3 of the way to the top by voltage 00:15:49.680 --> 00:15:50.180 align:middle line:90% division. 00:15:50.180 --> 00:15:52.940 align:middle line:90% Does that make sense? 00:15:52.940 --> 00:15:56.790 align:middle line:84% OK, so that's the way you tend to think about it. 00:15:56.790 --> 00:15:59.700 align:middle line:84% So one way I could run this is very 00:15:59.700 --> 00:16:03.240 align:middle line:84% much like the inverse of the diode rectifier 00:16:03.240 --> 00:16:04.980 align:middle line:90% I talked about last time. 00:16:04.980 --> 00:16:10.630 align:middle line:84% If you remember last time, as the AC waveform happened 00:16:10.630 --> 00:16:14.300 align:middle line:84% and drove the diode bridge, the individual diodes turned on. 00:16:14.300 --> 00:16:19.580 align:middle line:84% If you remember, it was like 1, 2; 2, 3; 3, 4; 4, 5; 5, 6; 6, 1. 00:16:19.580 --> 00:16:24.610 align:middle line:84% So it just went around in this six-step pattern. 00:16:24.610 --> 00:16:30.520 align:middle line:84% Well, I could actively control these devices to follow through 00:16:30.520 --> 00:16:35.110 align:middle line:84% those six patterns, ignoring 000 and 111. 00:16:35.110 --> 00:16:37.040 align:middle line:84% So those are the two I wouldn't use. 00:16:37.040 --> 00:16:40.420 align:middle line:84% And this is what it would look like if you did that. 00:16:40.420 --> 00:16:46.930 align:middle line:84% You would see that, say, each half bridge in this inverter 00:16:46.930 --> 00:16:49.300 align:middle line:90% would be on half the time. 00:16:49.300 --> 00:16:53.290 align:middle line:84% So A might be on exactly like a square in phase 00:16:53.290 --> 00:16:54.620 align:middle line:90% with a sine wave. 00:16:54.620 --> 00:16:58.690 align:middle line:84% B would be on but shifted by 2 pi over 3. 00:16:58.690 --> 00:17:03.700 align:middle line:84% And C would be on with high shifted by another 2 pi over 3. 00:17:03.700 --> 00:17:08.839 align:middle line:84% So I'd have three square waves, VA, VB, and VC with respect 00:17:08.839 --> 00:17:11.599 align:middle line:90% to the reference. 00:17:11.599 --> 00:17:16.040 align:middle line:90% What would happen at the load? 00:17:16.040 --> 00:17:24.680 align:middle line:84% Well, if I looked at VAB, VBC, and VCA, that's these next-- 00:17:24.680 --> 00:17:26.130 align:middle line:90% the next three here-- 00:17:26.130 --> 00:17:28.430 align:middle line:84% and I apologize; it's not the prettiest diagram-- 00:17:28.430 --> 00:17:31.550 align:middle line:90% are the line-to-line voltages. 00:17:31.550 --> 00:17:33.800 align:middle line:84% So all I'm doing is instead of looking 00:17:33.800 --> 00:17:37.910 align:middle line:84% at what this is with respect to the reference, 00:17:37.910 --> 00:17:41.640 align:middle line:84% I'm just looking at the difference between VA and VB. 00:17:41.640 --> 00:17:45.500 align:middle line:84% And what you can see is that these three suddenly 00:17:45.500 --> 00:17:49.340 align:middle line:84% take on this six-step waveform, which looks just like what 00:17:49.340 --> 00:17:51.890 align:middle line:90% the diode rectifier was doing. 00:17:51.890 --> 00:17:55.250 align:middle line:84% It's just that now I'm imposing a DC voltage 00:17:55.250 --> 00:17:57.860 align:middle line:84% with that six-step pattern to say the machine 00:17:57.860 --> 00:18:00.830 align:middle line:90% or whatever it is I'm driving. 00:18:00.830 --> 00:18:04.420 align:middle line:90% Any questions about that? 00:18:04.420 --> 00:18:09.940 align:middle line:84% Now, if I do that, because the line-to-line voltages-- 00:18:09.940 --> 00:18:12.700 align:middle line:84% these square waves, which is the line to reference voltages, 00:18:12.700 --> 00:18:13.820 align:middle line:90% have third harmonic. 00:18:13.820 --> 00:18:17.530 align:middle line:84% But because I'm taking the difference between two of them 00:18:17.530 --> 00:18:20.260 align:middle line:84% to get the line-to-line voltage shifted by a third of a cycle, 00:18:20.260 --> 00:18:23.290 align:middle line:84% all the triple-N components disappear. 00:18:23.290 --> 00:18:28.450 align:middle line:84% So the line-to-line voltages have no triple-N components. 00:18:28.450 --> 00:18:31.292 align:middle line:84% So whatever my motor sees it's line to line, 00:18:31.292 --> 00:18:33.250 align:middle line:84% if I'm driving, say, a delta-connected machine, 00:18:33.250 --> 00:18:35.420 align:middle line:90% it will see no triple-N drive. 00:18:35.420 --> 00:18:39.220 align:middle line:84% And that's just free because I have this three-phase inverter 00:18:39.220 --> 00:18:39.740 align:middle line:90% structure. 00:18:39.740 --> 00:18:43.420 align:middle line:84% If everything's balanced, it can't generate three-phase line 00:18:43.420 --> 00:18:46.180 align:middle line:90% to line. 00:18:46.180 --> 00:18:47.340 align:middle line:90% Any questions about that? 00:18:47.340 --> 00:18:52.380 align:middle line:90% 00:18:52.380 --> 00:19:00.000 align:middle line:84% What would it do as regards the line-to-neutral voltages? 00:19:00.000 --> 00:19:03.420 align:middle line:84% Well, if I did that, we said this thing is basically 00:19:03.420 --> 00:19:05.190 align:middle line:90% a voltage divider. 00:19:05.190 --> 00:19:09.660 align:middle line:84% But I could think about it basically being-- 00:19:09.660 --> 00:19:15.740 align:middle line:84% the neutral voltage as being of the average of VA, VB, and VC. 00:19:15.740 --> 00:19:22.580 align:middle line:84% And what essentially happens is that each of VA, VB, 00:19:22.580 --> 00:19:25.130 align:middle line:84% VC has a fundamental, and then it 00:19:25.130 --> 00:19:27.590 align:middle line:84% has, for example, a third harmonic 00:19:27.590 --> 00:19:29.750 align:middle line:90% because it's a square wave. 00:19:29.750 --> 00:19:33.740 align:middle line:84% The fundamentals are 120 degrees out of phase, 00:19:33.740 --> 00:19:37.260 align:middle line:84% but that means that the third harmonics are exactly in phase. 00:19:37.260 --> 00:19:40.250 align:middle line:84% So the third harmonics of each of these square waves 00:19:40.250 --> 00:19:42.890 align:middle line:84% add at the neutral, and you get a neutral voltage 00:19:42.890 --> 00:19:46.450 align:middle line:84% that bops up and down at the third harmonic. 00:19:46.450 --> 00:19:49.750 align:middle line:84% So this has-- the neutral contains 00:19:49.750 --> 00:19:55.340 align:middle line:84% the third harmonic and actually all the triple-N harmonics. 00:19:55.340 --> 00:19:58.480 align:middle line:84% And so as a result, just as the line to line 00:19:58.480 --> 00:20:00.310 align:middle line:84% didn't have any triple-N harmonics, 00:20:00.310 --> 00:20:02.540 align:middle line:84% neither did any of the line-to-neutral voltages. 00:20:02.540 --> 00:20:04.390 align:middle line:84% So if it's wye-connected machine, 00:20:04.390 --> 00:20:06.040 align:middle line:84% the line-to-neutral voltage, which 00:20:06.040 --> 00:20:08.620 align:middle line:84% is really what's driving current in this thing, 00:20:08.620 --> 00:20:11.530 align:middle line:90% no triple-N harmonics. 00:20:11.530 --> 00:20:12.980 align:middle line:84% Does that make sense to everybody? 00:20:12.980 --> 00:20:15.690 align:middle line:90% 00:20:15.690 --> 00:20:18.120 align:middle line:84% So this kind of operation is what's 00:20:18.120 --> 00:20:21.360 align:middle line:90% known as six-step operation. 00:20:21.360 --> 00:20:27.690 align:middle line:84% At each AC cycle, I'm switching my devices as minimum as I can. 00:20:27.690 --> 00:20:35.710 align:middle line:84% And I just go through and step sequentially around in a cycle. 00:20:35.710 --> 00:20:39.570 align:middle line:84% And in fact, the Tesla egg demo that we brought, 00:20:39.570 --> 00:20:41.110 align:middle line:84% that's basically what it was doing. 00:20:41.110 --> 00:20:43.650 align:middle line:84% It had six switches, and it was just 00:20:43.650 --> 00:20:45.750 align:middle line:84% boom, boom, boom, six-stepping around 00:20:45.750 --> 00:20:48.875 align:middle line:84% to generate a rotating magnetic field which spun up the egg. 00:20:48.875 --> 00:20:52.890 align:middle line:90% 00:20:52.890 --> 00:20:55.590 align:middle line:90% That's a fine way to operate. 00:20:55.590 --> 00:20:59.850 align:middle line:84% And in fact, it generates the largest fundamental component 00:20:59.850 --> 00:21:02.290 align:middle line:84% you can generate out of this inverter. 00:21:02.290 --> 00:21:02.850 align:middle line:90% Why? 00:21:02.850 --> 00:21:04.770 align:middle line:84% Because each half bridge is basically 00:21:04.770 --> 00:21:08.340 align:middle line:90% generating a square wave-- 00:21:08.340 --> 00:21:09.720 align:middle line:84% the largest fundamental component 00:21:09.720 --> 00:21:12.030 align:middle line:84% you can generate when you're not tying 00:21:12.030 --> 00:21:15.335 align:middle line:84% the neutral back to the reference point, I should say. 00:21:15.335 --> 00:21:18.430 align:middle line:90% 00:21:18.430 --> 00:21:20.990 align:middle line:84% That's not the only way you could drive this. 00:21:20.990 --> 00:21:22.540 align:middle line:84% You could actually start thinking 00:21:22.540 --> 00:21:29.740 align:middle line:84% about doing things like harmonic elimination waveforms. 00:21:29.740 --> 00:21:33.520 align:middle line:84% Instead of generating square waves 00:21:33.520 --> 00:21:36.163 align:middle line:84% in between the top and bottom switching, 00:21:36.163 --> 00:21:38.080 align:middle line:84% you could start thinking about putting notches 00:21:38.080 --> 00:21:40.300 align:middle line:90% in those waveforms. 00:21:40.300 --> 00:21:42.910 align:middle line:84% Now, if you're going to put notches in the waveforms, 00:21:42.910 --> 00:21:45.072 align:middle line:84% you wouldn't try to eliminate the third harmonic, 00:21:45.072 --> 00:21:47.030 align:middle line:84% because the third harmonic is going to go away. 00:21:47.030 --> 00:21:49.630 align:middle line:84% You might try to eliminate the fifth and possibly 00:21:49.630 --> 00:21:51.050 align:middle line:90% the seventh harmonic. 00:21:51.050 --> 00:21:53.380 align:middle line:84% So the same kind of games we talked 00:21:53.380 --> 00:21:55.540 align:middle line:84% about playing with a single-phase inverter, 00:21:55.540 --> 00:21:57.470 align:middle line:84% you can play with three-phase inverters. 00:21:57.470 --> 00:21:59.710 align:middle line:84% It's just that you're doing each phase 00:21:59.710 --> 00:22:03.070 align:middle line:84% and then you're shifting that by a third of a cycle 00:22:03.070 --> 00:22:05.660 align:middle line:90% to get to the next phase. 00:22:05.660 --> 00:22:07.030 align:middle line:90% Any questions about that? 00:22:07.030 --> 00:22:09.780 align:middle line:90% 00:22:09.780 --> 00:22:11.580 align:middle line:84% So those are the kind of games you 00:22:11.580 --> 00:22:14.970 align:middle line:84% would play if you're at very high outputs 00:22:14.970 --> 00:22:18.210 align:middle line:84% where you don't want to switch your devices very much. 00:22:18.210 --> 00:22:20.760 align:middle line:84% But in a large part of the operating 00:22:20.760 --> 00:22:26.160 align:middle line:84% range for modern electric machines, 00:22:26.160 --> 00:22:27.850 align:middle line:90% you don't have to do that. 00:22:27.850 --> 00:22:32.490 align:middle line:84% You can do a much nicer job of synthesizing a waveform that 00:22:32.490 --> 00:22:36.000 align:middle line:90% would look very close-- 00:22:36.000 --> 00:22:38.670 align:middle line:84% that would have a fundamental content 00:22:38.670 --> 00:22:41.550 align:middle line:84% that follows very closely, for example, a sine wave, if you 00:22:41.550 --> 00:22:45.090 align:middle line:84% were driving a machine that wanted sine wave voltages, 00:22:45.090 --> 00:22:49.920 align:middle line:84% or that would generate three-phase waveforms if you 00:22:49.920 --> 00:22:52.500 align:middle line:84% wanted to make a three-phase uninterruptible power 00:22:52.500 --> 00:22:55.740 align:middle line:84% supply that would synthesize a three-phase output that 00:22:55.740 --> 00:22:58.860 align:middle line:84% would look very much like sine wave outputs. 00:22:58.860 --> 00:23:02.460 align:middle line:90% How would you do that? 00:23:02.460 --> 00:23:04.360 align:middle line:90% Well, let's think about that. 00:23:04.360 --> 00:23:06.160 align:middle line:90% What would I do? 00:23:06.160 --> 00:23:11.290 align:middle line:84% Well, what I could do is I could make each of the phase outputs, 00:23:11.290 --> 00:23:13.900 align:middle line:84% think about them independently, generate something 00:23:13.900 --> 00:23:16.930 align:middle line:84% that looked like a sine with respect to the reference, 00:23:16.930 --> 00:23:18.640 align:middle line:90% on average. 00:23:18.640 --> 00:23:21.430 align:middle line:90% Well, how would I do that? 00:23:21.430 --> 00:23:25.540 align:middle line:84% The same way we did things with DC-to-DC converters or DC-to-AC 00:23:25.540 --> 00:23:28.540 align:middle line:84% converters and single-phase DC-to-AC converters. 00:23:28.540 --> 00:23:31.630 align:middle line:84% Maybe I would take and say, OK, suppose 00:23:31.630 --> 00:23:34.780 align:middle line:84% I wanted VAR to follow some sine wave. 00:23:34.780 --> 00:23:39.490 align:middle line:84% I could create some triangle wave 00:23:39.490 --> 00:23:44.460 align:middle line:84% that's a carrier wave that does something like this. 00:23:44.460 --> 00:23:51.075 align:middle line:90% 00:23:51.075 --> 00:23:58.430 align:middle line:84% And I'm going to exaggerate the ratios between the switching 00:23:58.430 --> 00:24:03.941 align:middle line:84% frequency and the line frequency just for visualization. 00:24:03.941 --> 00:24:06.750 align:middle line:90% 00:24:06.750 --> 00:24:09.970 align:middle line:90% But here I have a triangle wave. 00:24:09.970 --> 00:24:18.020 align:middle line:84% Maybe I'll call this minus VT max and VT max. 00:24:18.020 --> 00:24:22.010 align:middle line:84% We might think of those as 1 volt, plus or minus 1 volt. 00:24:22.010 --> 00:24:24.720 align:middle line:90% And this is 0. 00:24:24.720 --> 00:24:28.570 align:middle line:84% And this we might think of as T switch. 00:24:28.570 --> 00:24:34.480 align:middle line:84% T switch is equal to 1 over the switching frequency 00:24:34.480 --> 00:24:37.795 align:middle line:90% that I'm going to switch at. 00:24:37.795 --> 00:24:43.030 align:middle line:84% And then what I could do is I could create a signal. 00:24:43.030 --> 00:24:44.133 align:middle line:90% Let me call this-- 00:24:44.133 --> 00:24:49.270 align:middle line:90% 00:24:49.270 --> 00:24:59.960 align:middle line:90% I'll call this VAR ref. 00:24:59.960 --> 00:25:02.000 align:middle line:84% This is the reference voltage to which 00:25:02.000 --> 00:25:05.540 align:middle line:84% I want within a switching cycle VA 00:25:05.540 --> 00:25:11.260 align:middle line:84% to follow with respect to the reference. 00:25:11.260 --> 00:25:12.560 align:middle line:90% So what would I do? 00:25:12.560 --> 00:25:14.320 align:middle line:84% I'd then put this into a comparator 00:25:14.320 --> 00:25:17.185 align:middle line:90% if I called this V triangle. 00:25:17.185 --> 00:25:19.690 align:middle line:90% 00:25:19.690 --> 00:25:22.750 align:middle line:84% So I could then just create a comparator 00:25:22.750 --> 00:25:29.290 align:middle line:84% and say, OK, into this terminal, I'm going to put VAR ref. 00:25:29.290 --> 00:25:33.970 align:middle line:84% Into this terminal, I'm going to put V triangle. 00:25:33.970 --> 00:25:39.130 align:middle line:84% And then out of this comes q1 of t, which is equal to-- 00:25:39.130 --> 00:25:42.970 align:middle line:84% which is equal to 1 minus q4 of t. 00:25:42.970 --> 00:25:51.460 align:middle line:84% And this is q1, so q1 of t goes here, and q4 goes here. 00:25:51.460 --> 00:25:54.432 align:middle line:84% So whenever this switch is high-- 00:25:54.432 --> 00:25:57.015 align:middle line:84% whenever this switch is on, this switch is off and vice versa. 00:25:57.015 --> 00:26:00.340 align:middle line:90% 00:26:00.340 --> 00:26:01.700 align:middle line:90% Why would I do that? 00:26:01.700 --> 00:26:07.530 align:middle line:84% If you think about it, if Vref is all the way at the top, 00:26:07.530 --> 00:26:10.930 align:middle line:84% that means the top switch is always on. 00:26:10.930 --> 00:26:13.680 align:middle line:84% If VA is near the bottom, that means 00:26:13.680 --> 00:26:16.200 align:middle line:90% the bottom switch is always on. 00:26:16.200 --> 00:26:18.420 align:middle line:84% If the reference is right here in the middle, 00:26:18.420 --> 00:26:21.880 align:middle line:84% this triangle wave goes from a plus and minus, 00:26:21.880 --> 00:26:24.630 align:middle line:84% I would have it switching with 50% duty 00:26:24.630 --> 00:26:27.220 align:middle line:90% cycle between top and bottom. 00:26:27.220 --> 00:26:30.570 align:middle line:84% And on average, with respect to VR, this would be 0. 00:26:30.570 --> 00:26:34.680 align:middle line:84% So I can linearly change the average voltage synthesized 00:26:34.680 --> 00:26:39.750 align:middle line:84% by VA with respect to VR over one switching period based 00:26:39.750 --> 00:26:43.630 align:middle line:90% on this triangle intercept PWM. 00:26:43.630 --> 00:26:46.490 align:middle line:84% Does that make sense to everybody? 00:26:46.490 --> 00:26:50.190 align:middle line:84% Now, I should say I'm showing this with balanced triangles. 00:26:50.190 --> 00:26:52.940 align:middle line:84% You could do this with any other kind of waveform, 00:26:52.940 --> 00:26:55.160 align:middle line:84% like a sawtooth waveform or something like that, 00:26:55.160 --> 00:26:58.010 align:middle line:84% where at least there's a proportionality in time 00:26:58.010 --> 00:27:02.690 align:middle line:84% between the height and how long the comparator trips for. 00:27:02.690 --> 00:27:05.300 align:middle line:84% But balanced triangles tend to give you 00:27:05.300 --> 00:27:07.730 align:middle line:84% better switching harmonic content 00:27:07.730 --> 00:27:10.940 align:middle line:90% than nonbalanced triangle waves. 00:27:10.940 --> 00:27:14.660 align:middle line:84% Does that make sense to everybody? 00:27:14.660 --> 00:27:20.300 align:middle line:84% So suppose I had a machine and I wanted 00:27:20.300 --> 00:27:31.340 align:middle line:84% to create some waveform that looked like something like VAR 00:27:31.340 --> 00:27:32.790 align:middle line:90% on average. 00:27:32.790 --> 00:27:35.480 align:middle line:84% So I'm going to draw this bar, meaning average 00:27:35.480 --> 00:27:37.940 align:middle line:90% over one switching cycle. 00:27:37.940 --> 00:27:45.860 align:middle line:84% I would like this to be V sub m sine of omega line t, 00:27:45.860 --> 00:27:51.920 align:middle line:84% for example, where I'm assuming that omega line is much, 00:27:51.920 --> 00:27:55.040 align:middle line:90% much less than 2 pi f switch. 00:27:55.040 --> 00:27:58.950 align:middle line:84% That is, this is a very slowly varying sine wave. 00:27:58.950 --> 00:28:10.227 align:middle line:84% So what I'm saying is I have some triangle wave like this. 00:28:10.227 --> 00:28:11.310 align:middle line:90% Maybe I'll draw it bigger. 00:28:11.310 --> 00:28:14.480 align:middle line:90% 00:28:14.480 --> 00:28:17.150 align:middle line:84% This is my PWM waveform, and I'm trying 00:28:17.150 --> 00:28:21.640 align:middle line:84% to generate a sine wave that maybe there's 00:28:21.640 --> 00:28:25.340 align:middle line:84% a hundred switching cycles or something like that 00:28:25.340 --> 00:28:27.270 align:middle line:90% per line cycle. 00:28:27.270 --> 00:28:34.040 align:middle line:84% So this is 2 pi over omega switch. 00:28:34.040 --> 00:28:40.110 align:middle line:84% And this is 2 pi over omega line versus time. 00:28:40.110 --> 00:28:42.740 align:middle line:84% Does that make sense to everybody? 00:28:42.740 --> 00:28:45.590 align:middle line:84% And as long as-- and I should say, 00:28:45.590 --> 00:28:48.335 align:middle line:84% what I'm trying to do is I need my-- 00:28:48.335 --> 00:28:51.800 align:middle line:90% 00:28:51.800 --> 00:28:59.760 align:middle line:84% if this is V sub m sine of omega lt, 00:28:59.760 --> 00:29:05.870 align:middle line:84% I need V sub m scaled such that at least the-- 00:29:05.870 --> 00:29:07.380 align:middle line:90% I'm sorry, this is VA-- 00:29:07.380 --> 00:29:10.340 align:middle line:90% this is the reference voltage-- 00:29:10.340 --> 00:29:12.780 align:middle line:90% is less than VDC over 2. 00:29:12.780 --> 00:29:17.870 align:middle line:84% So let me clarify how we're going to do this. 00:29:17.870 --> 00:29:21.320 align:middle line:84% And I apologize, that's probably not very clear. 00:29:21.320 --> 00:29:23.510 align:middle line:84% What I'm going to do is I'm going 00:29:23.510 --> 00:29:38.840 align:middle line:84% to make VAR ref equal to V sub m divided by VDC over 2 times 00:29:38.840 --> 00:29:39.995 align:middle line:90% V sub triangle. 00:29:39.995 --> 00:29:43.280 align:middle line:90% 00:29:43.280 --> 00:29:49.280 align:middle line:84% And then I'll make that sine of omega lt. 00:29:49.280 --> 00:29:52.520 align:middle line:84% So this thing-- this thing is the thing-- 00:29:52.520 --> 00:29:56.930 align:middle line:84% this is VA ref that I'm going to compare to VT max. 00:29:56.930 --> 00:29:58.550 align:middle line:90% So this is-- 00:29:58.550 --> 00:30:02.480 align:middle line:90% I'm sorry, this is VT max. 00:30:02.480 --> 00:30:06.470 align:middle line:84% So if VT max was 1 volt, just for simplicity, 00:30:06.470 --> 00:30:09.530 align:middle line:84% I'm going to make V sub m over VDC 00:30:09.530 --> 00:30:11.920 align:middle line:84% over 2 the height of my reference waveform. 00:30:11.920 --> 00:30:14.735 align:middle line:90% 00:30:14.735 --> 00:30:16.360 align:middle line:90% So I can create-- 00:30:16.360 --> 00:30:18.750 align:middle line:84% if I do this, then what I'm going to get 00:30:18.750 --> 00:30:28.020 align:middle line:84% is VAR as a result of running this thing into this comparator 00:30:28.020 --> 00:30:32.130 align:middle line:84% network because this is my triangle away 00:30:32.130 --> 00:30:33.310 align:middle line:90% from my controller. 00:30:33.310 --> 00:30:36.990 align:middle line:84% The peak of my voltage is switching between plus VDC 00:30:36.990 --> 00:30:39.400 align:middle line:90% over 2 and minus VDC over 2. 00:30:39.400 --> 00:30:45.030 align:middle line:84% So what I'm going to get is VAR is going 00:30:45.030 --> 00:30:51.930 align:middle line:90% to be equal to Vm over VDC-- 00:30:51.930 --> 00:31:03.840 align:middle line:84% it's going to be equal to Vm sine of omega lt, 00:31:03.840 --> 00:31:07.580 align:middle line:84% as long as V sub m is less than VDC over 2. 00:31:07.580 --> 00:31:11.940 align:middle line:90% 00:31:11.940 --> 00:31:12.840 align:middle line:90% Is that clear? 00:31:12.840 --> 00:31:14.885 align:middle line:84% Or should I try to explain that better? 00:31:14.885 --> 00:31:18.690 align:middle line:90% 00:31:18.690 --> 00:31:22.290 align:middle line:84% I can modulate this thing up and down 00:31:22.290 --> 00:31:25.020 align:middle line:84% such that what would be the biggest sine 00:31:25.020 --> 00:31:27.510 align:middle line:84% wave I could create without distortion with this scheme, 00:31:27.510 --> 00:31:29.760 align:middle line:84% or at least with what I've told you so far, 00:31:29.760 --> 00:31:34.050 align:middle line:84% would be if V sub m was actually equal to VDC 00:31:34.050 --> 00:31:42.070 align:middle line:84% over 2, which would mean the top of this waveform 00:31:42.070 --> 00:31:44.870 align:middle line:84% would exactly touch the top of the triangle wave. 00:31:44.870 --> 00:31:46.960 align:middle line:84% So I wouldn't have any period in which 00:31:46.960 --> 00:31:50.470 align:middle line:90% I'm exceeding duty ratio of 1. 00:31:50.470 --> 00:31:58.730 align:middle line:84% The problem here is if my reference waveform exceeds 00:31:58.730 --> 00:32:00.740 align:middle line:84% the height of the triangle wave, then 00:32:00.740 --> 00:32:03.230 align:middle line:84% I'm going to get some time period where I stop switching, 00:32:03.230 --> 00:32:08.210 align:middle line:84% and I just have the top switch on constantly 00:32:08.210 --> 00:32:09.830 align:middle line:84% or the bottom switch on constantly. 00:32:09.830 --> 00:32:12.680 align:middle line:90% And then I get some distortion. 00:32:12.680 --> 00:32:16.520 align:middle line:84% By keeping my reference always below the height of the triangle 00:32:16.520 --> 00:32:24.710 align:middle line:84% wave, I kind of keep the no distortion relative at least 00:32:24.710 --> 00:32:29.000 align:middle line:84% over a switching cycle, averaged over a switching cycle. 00:32:29.000 --> 00:32:30.085 align:middle line:90% Any questions about that? 00:32:30.085 --> 00:32:34.070 align:middle line:90% 00:32:34.070 --> 00:32:37.550 align:middle line:84% So when we do this, and I say, OK, I'm 00:32:37.550 --> 00:32:42.710 align:middle line:84% going to do this for phase A, I do the same thing for phase B, 00:32:42.710 --> 00:32:45.170 align:middle line:84% except I shift it by a third of a line cycle, 00:32:45.170 --> 00:32:48.270 align:middle line:84% and phase C is shifted by a third of a line cycle, 00:32:48.270 --> 00:32:51.590 align:middle line:84% then at least as far as the local averages, 00:32:51.590 --> 00:32:55.130 align:middle line:84% I'm creating a voltage A with respect 00:32:55.130 --> 00:32:56.888 align:middle line:84% to reference that follows a sine wave, 00:32:56.888 --> 00:32:58.430 align:middle line:84% a voltage B that follows a sine wave, 00:32:58.430 --> 00:33:00.150 align:middle line:84% a voltage C that follows a sine wave. 00:33:00.150 --> 00:33:03.110 align:middle line:84% And yes, each of these nodes is going up and down, 00:33:03.110 --> 00:33:06.680 align:middle line:84% but if this load is inductive, it will filter that, 00:33:06.680 --> 00:33:10.350 align:middle line:84% and then the currents will look very sinusoidal. 00:33:10.350 --> 00:33:12.240 align:middle line:84% Does that make sense to everybody? 00:33:12.240 --> 00:33:15.150 align:middle line:84% So I can really generate something 00:33:15.150 --> 00:33:17.820 align:middle line:84% that looks like a sinusoidal, at least on average, 00:33:17.820 --> 00:33:21.900 align:middle line:84% output, just the way I would do it with a single-phase inverter. 00:33:21.900 --> 00:33:25.403 align:middle line:84% So this is the same trick of triangle intercept of PWM 00:33:25.403 --> 00:33:26.570 align:middle line:90% that we talked about before. 00:33:26.570 --> 00:33:31.460 align:middle line:90% 00:33:31.460 --> 00:33:32.670 align:middle line:90% Any questions? 00:33:32.670 --> 00:33:36.230 align:middle line:90% 00:33:36.230 --> 00:33:38.650 align:middle line:90% OK, so that's nice. 00:33:38.650 --> 00:33:43.610 align:middle line:84% This technique is very good when we 00:33:43.610 --> 00:33:47.790 align:middle line:84% are trying to synthesize a waveform with a lot of purity. 00:33:47.790 --> 00:33:50.090 align:middle line:84% The six-step waveform I showed you before 00:33:50.090 --> 00:33:51.680 align:middle line:84% had a ton of harmonic content, a ton 00:33:51.680 --> 00:33:54.150 align:middle line:84% of low-frequency harmonic content in it. 00:33:54.150 --> 00:33:57.950 align:middle line:84% This PWM waveform really only has 00:33:57.950 --> 00:34:01.250 align:middle line:84% content as long as you stay where your reference never 00:34:01.250 --> 00:34:04.280 align:middle line:84% exceeds the height of the triangle. 00:34:04.280 --> 00:34:08.929 align:middle line:84% It really only has of harmonic content near the switching 00:34:08.929 --> 00:34:11.040 align:middle line:90% frequency and its harmonics. 00:34:11.040 --> 00:34:14.810 align:middle line:84% There's some difference with the line frequency and the carrier 00:34:14.810 --> 00:34:17.025 align:middle line:84% frequency stuff, but it's all pushed up 00:34:17.025 --> 00:34:18.150 align:middle line:90% to the switching frequency. 00:34:18.150 --> 00:34:19.817 align:middle line:84% So if I have a high switching frequency, 00:34:19.817 --> 00:34:23.120 align:middle line:84% I get a very nice purity waveform, at least averaged 00:34:23.120 --> 00:34:24.115 align:middle line:90% over a switching cycle. 00:34:24.115 --> 00:34:26.968 align:middle line:90% 00:34:26.968 --> 00:34:30.199 align:middle line:84% But I said, what's the largest waveform 00:34:30.199 --> 00:34:33.110 align:middle line:84% I can synthesize that way, at least with what I've told you 00:34:33.110 --> 00:34:38.699 align:middle line:84% so far, is I can make this voltage V sub m up 00:34:38.699 --> 00:34:40.800 align:middle line:90% to VDC over 2. 00:34:40.800 --> 00:34:45.840 align:middle line:84% So this inverter can switch and it can synthesize-- 00:34:45.840 --> 00:34:46.590 align:middle line:90% I could think of-- 00:34:46.590 --> 00:34:52.199 align:middle line:84% I could think of this as synthesizing VAR, VBR, and VCR, 00:34:52.199 --> 00:34:55.620 align:middle line:84% which is going to also give me the same thing for VAN, VBN, 00:34:55.620 --> 00:34:59.280 align:middle line:84% and VCN, because there's no triple-N content here, 00:34:59.280 --> 00:35:08.670 align:middle line:84% of being V sub m sine omega line t so long as Vm is less than 00:35:08.670 --> 00:35:10.700 align:middle line:90% or equal to VDC over 2. 00:35:10.700 --> 00:35:14.530 align:middle line:90% 00:35:14.530 --> 00:35:17.350 align:middle line:84% If I'm going to synthesize a sine wave like that, 00:35:17.350 --> 00:35:22.533 align:middle line:84% we often define something called m or the modulation index. 00:35:22.533 --> 00:35:27.670 align:middle line:90% 00:35:27.670 --> 00:35:30.610 align:middle line:84% Sometimes this is called the depth of modulation 00:35:30.610 --> 00:35:38.530 align:middle line:84% also, which is essentially V sub m divided by VDC over 2. 00:35:38.530 --> 00:35:43.910 align:middle line:84% This is how big a sine wave am I generating? 00:35:43.910 --> 00:35:47.090 align:middle line:84% How big is this sine wave relative to VDC 00:35:47.090 --> 00:35:48.770 align:middle line:84% over 2 or relative to the maximum 00:35:48.770 --> 00:35:50.370 align:middle line:90% I can do without distorting? 00:35:50.370 --> 00:35:53.240 align:middle line:90% 00:35:53.240 --> 00:35:55.940 align:middle line:84% If at least with what I've told you so far, 00:35:55.940 --> 00:36:02.953 align:middle line:84% I make m greater than 1 or V sub m bigger than VDC over 2, 00:36:02.953 --> 00:36:04.370 align:middle line:84% then what will happen is I'll stop 00:36:04.370 --> 00:36:05.870 align:middle line:84% switching for part of the waveform, 00:36:05.870 --> 00:36:08.240 align:middle line:90% and I'll start distorting. 00:36:08.240 --> 00:36:10.840 align:middle line:90% Any questions about that? 00:36:10.840 --> 00:36:16.530 align:middle line:90% 00:36:16.530 --> 00:36:19.030 align:middle line:90% Well, that's fine. 00:36:19.030 --> 00:36:21.990 align:middle line:84% And through this whole period, at least 00:36:21.990 --> 00:36:24.840 align:middle line:84% averaged over a switching cycle, if I'm 00:36:24.840 --> 00:36:28.260 align:middle line:84% doing this sinusoidal or sine triangle PWM, 00:36:28.260 --> 00:36:31.230 align:middle line:84% this neutral point averaged over switching cycle 00:36:31.230 --> 00:36:33.510 align:middle line:84% always has 0 average voltage, in other words, 00:36:33.510 --> 00:36:37.870 align:middle line:84% equal to the reference voltage, halfway between the bus voltage. 00:36:37.870 --> 00:36:40.530 align:middle line:84% So I'm not getting any third harmonic 00:36:40.530 --> 00:36:41.870 align:middle line:90% swinging around on the neutral. 00:36:41.870 --> 00:36:47.900 align:middle line:90% 00:36:47.900 --> 00:36:50.480 align:middle line:90% Why do I think about this? 00:36:50.480 --> 00:36:53.210 align:middle line:84% Because if you think about an electric machine, 00:36:53.210 --> 00:36:55.130 align:middle line:84% and this is a typical application 00:36:55.130 --> 00:36:57.770 align:middle line:84% for this kind of thing, the back emf 00:36:57.770 --> 00:36:59.990 align:middle line:84% tends to be proportional to speed. 00:36:59.990 --> 00:37:04.730 align:middle line:84% And how much current I want to drive in the motor-- 00:37:04.730 --> 00:37:06.650 align:middle line:84% if I want more current, I try to drive 00:37:06.650 --> 00:37:09.740 align:middle line:84% more voltage, which will drive more current into the windings. 00:37:09.740 --> 00:37:13.610 align:middle line:84% So very often, the amplitude of the voltage, 00:37:13.610 --> 00:37:17.000 align:middle line:84% if I thought about some machine, which is just sitting 00:37:17.000 --> 00:37:20.810 align:middle line:90% here connected like this-- 00:37:20.810 --> 00:37:23.060 align:middle line:84% and maybe I'm just trying to generate-- 00:37:23.060 --> 00:37:28.790 align:middle line:84% I'm trying to use my inverter to generate essentially three 00:37:28.790 --> 00:37:31.310 align:middle line:84% voltages, which are then going to drive current 00:37:31.310 --> 00:37:34.250 align:middle line:90% into these phases. 00:37:34.250 --> 00:37:39.710 align:middle line:84% At low speeds, the back emf voltage of the motor is low, 00:37:39.710 --> 00:37:42.200 align:middle line:84% and I want low-- or I want low current. 00:37:42.200 --> 00:37:44.870 align:middle line:84% I want these voltage waveforms-- these are now 00:37:44.870 --> 00:37:46.170 align:middle line:90% at the line frequency. 00:37:46.170 --> 00:37:52.300 align:middle line:84% These are at V sub m sine of omega line t. 00:37:52.300 --> 00:37:55.110 align:middle line:84% I want these voltages to be small. 00:37:55.110 --> 00:37:59.250 align:middle line:84% But as speed goes up and the back emf internal to the machine 00:37:59.250 --> 00:38:02.920 align:middle line:84% goes up, I want a bigger V sub m, for example. 00:38:02.920 --> 00:38:05.040 align:middle line:90% So very often I can-- 00:38:05.040 --> 00:38:07.200 align:middle line:84% basically, by changing the V sub m, 00:38:07.200 --> 00:38:09.120 align:middle line:84% by changing the amplitude of my reference, 00:38:09.120 --> 00:38:11.460 align:middle line:84% I change these voltages' amplitudes, 00:38:11.460 --> 00:38:12.900 align:middle line:84% and then I change how much current 00:38:12.900 --> 00:38:14.290 align:middle line:90% I'm driving into my load. 00:38:14.290 --> 00:38:16.540 align:middle line:84% So I often want to modulate that. 00:38:16.540 --> 00:38:18.630 align:middle line:84% And typically, as I'm going up in speed 00:38:18.630 --> 00:38:20.550 align:middle line:84% or I'm going up in power or both, 00:38:20.550 --> 00:38:23.913 align:middle line:84% I'm trying to make V sub m bigger. 00:38:23.913 --> 00:38:25.330 align:middle line:84% Does that make sense to everybody? 00:38:25.330 --> 00:38:28.180 align:middle line:90% 00:38:28.180 --> 00:38:33.670 align:middle line:84% Now, the challenge is eventually I run out of headroom. 00:38:33.670 --> 00:38:36.790 align:middle line:84% Suppose the internal back emf of these motors 00:38:36.790 --> 00:38:38.560 align:middle line:90% is getting really big. 00:38:38.560 --> 00:38:41.410 align:middle line:84% So I need a pretty big voltage V sub m 00:38:41.410 --> 00:38:44.470 align:middle line:90% to drive current into my load. 00:38:44.470 --> 00:38:47.770 align:middle line:84% Once V sub m equals VDC over 2, suddenly I'm 00:38:47.770 --> 00:38:49.450 align:middle line:84% going to start distorting, right? 00:38:49.450 --> 00:38:51.590 align:middle line:84% I'm not going to be generating just sine waves. 00:38:51.590 --> 00:38:55.980 align:middle line:84% I'm going to have low frequency harmonic content in my output. 00:38:55.980 --> 00:38:58.050 align:middle line:84% So the question is, is there a way 00:38:58.050 --> 00:39:03.070 align:middle line:84% you can extend the range over which 00:39:03.070 --> 00:39:06.400 align:middle line:84% you can generate, if you will, pure sine 00:39:06.400 --> 00:39:09.520 align:middle line:84% waves at the output of your motor 00:39:09.520 --> 00:39:12.490 align:middle line:84% and generate like no low frequency, line 00:39:12.490 --> 00:39:14.440 align:middle line:90% frequency harmonic content? 00:39:14.440 --> 00:39:17.330 align:middle line:84% And the answer is there's a trick for doing this. 00:39:17.330 --> 00:39:19.750 align:middle line:84% It can be done in a variety of different ways, 00:39:19.750 --> 00:39:24.070 align:middle line:84% but I just thought I'd mention it because in the motors world, 00:39:24.070 --> 00:39:26.930 align:middle line:90% this is a very common trick. 00:39:26.930 --> 00:39:29.420 align:middle line:90% And the idea is this. 00:39:29.420 --> 00:39:32.155 align:middle line:90% Suppose if I thought-- 00:39:32.155 --> 00:39:36.850 align:middle line:90% 00:39:36.850 --> 00:39:39.160 align:middle line:84% if I thought of what the duty ratio-- 00:39:39.160 --> 00:39:42.010 align:middle line:90% if I thought of this switch q1-- 00:39:42.010 --> 00:39:46.270 align:middle line:84% if I thought of this as q1, the average value of q1 over a cycle 00:39:46.270 --> 00:39:51.340 align:middle line:84% as being the duty ratio of this switch, so modulation index 00:39:51.340 --> 00:39:52.990 align:middle line:84% of 1, the duty ratio of the top switch 00:39:52.990 --> 00:39:57.000 align:middle line:84% goes to 1, with what I've told you so far. 00:39:57.000 --> 00:40:01.570 align:middle line:84% So I could talk about the duty ratio of this half bridge, 00:40:01.570 --> 00:40:03.850 align:middle line:84% just like it was a buck converter, 00:40:03.850 --> 00:40:07.560 align:middle line:84% where I'm talking about on state fraction of the top switch. 00:40:07.560 --> 00:40:10.450 align:middle line:90% 00:40:10.450 --> 00:40:15.460 align:middle line:84% The way this would look if I did my sine triangle PWM would be d1 00:40:15.460 --> 00:40:27.710 align:middle line:84% of t would be equal to 1/2 plus m over 2 sine of omega line t. 00:40:27.710 --> 00:40:31.540 align:middle line:84% So if I did my sine triangle PWM, if m was one, 00:40:31.540 --> 00:40:34.960 align:middle line:84% duty ratio goes to 1 when the sine wave is peak, 00:40:34.960 --> 00:40:39.170 align:middle line:84% and it goes to 0 when sine wave is at minimum. 00:40:39.170 --> 00:40:41.300 align:middle line:84% Does that make sense to everybody? 00:40:41.300 --> 00:40:45.860 align:middle line:84% And then if I was looking at d3, it would be this same three-- 00:40:45.860 --> 00:40:47.010 align:middle line:90% actually, it's not d3. 00:40:47.010 --> 00:40:48.950 align:middle line:84% I forget what the middle switch-- 00:40:48.950 --> 00:40:51.281 align:middle line:90% top switch-- it is d3. 00:40:51.281 --> 00:41:03.530 align:middle line:84% d3 would be equal to 1/2 plus m over 2 sine omega lt minus 2 pi 00:41:03.530 --> 00:41:06.140 align:middle line:90% over 3, for example. 00:41:06.140 --> 00:41:07.660 align:middle line:84% Does that make sense to everybody? 00:41:07.660 --> 00:41:11.240 align:middle line:84% And d5, which would be the third bridge leg, 00:41:11.240 --> 00:41:21.080 align:middle line:84% would have 1/2 plus m over 2 sine omega lt plus 2 pi over 3. 00:41:21.080 --> 00:41:25.810 align:middle line:84% So this is what would result with this triangle intercept 00:41:25.810 --> 00:41:28.400 align:middle line:90% PWM done on the three phases. 00:41:28.400 --> 00:41:33.950 align:middle line:84% And what we said was basically the average voltage 00:41:33.950 --> 00:41:36.230 align:middle line:90% is proportional-- 00:41:36.230 --> 00:41:38.870 align:middle line:84% the average voltage at that node with respect to ground 00:41:38.870 --> 00:41:41.120 align:middle line:84% is proportional to the duty ratio. 00:41:41.120 --> 00:41:43.160 align:middle line:84% With respect to the midpoint, there's 00:41:43.160 --> 00:41:46.880 align:middle line:84% this one half of the bus voltage in there. 00:41:46.880 --> 00:41:53.510 align:middle line:84% The question would be I have a trouble that I'm limited 00:41:53.510 --> 00:41:58.180 align:middle line:90% to making m less than 1. 00:41:58.180 --> 00:42:01.870 align:middle line:84% Because when m exactly equals one, the peak of this 00:42:01.870 --> 00:42:06.650 align:middle line:84% drives the duty ratio of one just touches the top. 00:42:06.650 --> 00:42:09.790 align:middle line:84% If I try to make m 1.5, what would happen 00:42:09.790 --> 00:42:12.670 align:middle line:84% is for some part of the cycle, duty ratio basically 00:42:12.670 --> 00:42:14.390 align:middle line:84% saturates at 1, the top switches on, 00:42:14.390 --> 00:42:16.150 align:middle line:90% and I get a bunch of distortion. 00:42:16.150 --> 00:42:19.460 align:middle line:90% Is that clear to everybody? 00:42:19.460 --> 00:42:21.540 align:middle line:90% So here's the trick people use. 00:42:21.540 --> 00:42:23.780 align:middle line:90% It's kind of a cool trick. 00:42:23.780 --> 00:42:27.530 align:middle line:84% There's different ways to implement this trick. 00:42:27.530 --> 00:42:29.250 align:middle line:84% So this isn't the way you have to do it. 00:42:29.250 --> 00:42:32.570 align:middle line:90% But suppose I did this. 00:42:32.570 --> 00:42:47.846 align:middle line:84% I could take and add m over 12 sine of 3 omega line t 00:42:47.846 --> 00:42:50.236 align:middle line:90% to each of these waveforms. 00:42:50.236 --> 00:43:00.020 align:middle line:90% 00:43:00.020 --> 00:43:02.510 align:middle line:90% Why would I do that? 00:43:02.510 --> 00:43:09.680 align:middle line:84% Well, because when I think about sine, if I looked at this duty 00:43:09.680 --> 00:43:12.420 align:middle line:90% ratio, it would look like this. 00:43:12.420 --> 00:43:14.120 align:middle line:90% If I just took-- 00:43:14.120 --> 00:43:19.050 align:middle line:84% here, I would have a half, and I'm doing something like this. 00:43:19.050 --> 00:43:23.360 align:middle line:90% 00:43:23.360 --> 00:43:27.480 align:middle line:84% What I can do is I can add in a third harmonic component. 00:43:27.480 --> 00:43:29.270 align:middle line:84% What does a third harmonic component do? 00:43:29.270 --> 00:43:30.655 align:middle line:90% It does something like this. 00:43:30.655 --> 00:43:36.000 align:middle line:90% 00:43:36.000 --> 00:43:46.020 align:middle line:84% So the sum of this piece, which is this, and this piece, which 00:43:46.020 --> 00:43:48.630 align:middle line:84% is this, has some distorted shape 00:43:48.630 --> 00:43:50.240 align:middle line:90% which does something like this. 00:43:50.240 --> 00:43:55.970 align:middle line:90% 00:43:55.970 --> 00:44:04.020 align:middle line:84% And what that means is I can turn up m just bigger than 1, 00:44:04.020 --> 00:44:06.980 align:middle line:84% and it turns out to be about 1.15, 00:44:06.980 --> 00:44:12.920 align:middle line:84% before basically the peak will hit VDC over 2 00:44:12.920 --> 00:44:17.720 align:middle line:84% or before my duty ratio will saturate at 1. 00:44:17.720 --> 00:44:23.140 align:middle line:84% And so that gives me an extra 15% of voltage I can synthesize. 00:44:23.140 --> 00:44:24.830 align:middle line:84% And the way I'm synthesizing it is I'm 00:44:24.830 --> 00:44:28.480 align:middle line:84% adding third harmonic distortion in. 00:44:28.480 --> 00:44:31.960 align:middle line:84% So I'm adding some third harmonic to VAR, to VBR, 00:44:31.960 --> 00:44:34.150 align:middle line:90% and VCR. 00:44:34.150 --> 00:44:36.370 align:middle line:90% But why doesn't that matter? 00:44:36.370 --> 00:44:39.910 align:middle line:84% Because the third harmonic, that third harmonic component 00:44:39.910 --> 00:44:42.250 align:middle line:84% that I drew in yellow, just appears at the neutral. 00:44:42.250 --> 00:44:44.780 align:middle line:84% And the line-to-line voltages don't have it. 00:44:44.780 --> 00:44:47.550 align:middle line:90% 00:44:47.550 --> 00:44:50.310 align:middle line:84% So that's a technique that's sometimes called 00:44:50.310 --> 00:44:52.320 align:middle line:90% third-harmonic injection PWM. 00:44:52.320 --> 00:44:57.870 align:middle line:84% You can do it by adding third harmonic onto your-- 00:44:57.870 --> 00:45:01.500 align:middle line:84% instead of doing this, you add some third harmonic 00:45:01.500 --> 00:45:04.290 align:middle line:84% onto that thing, and then you get an ability 00:45:04.290 --> 00:45:09.390 align:middle line:84% to expand your duty ratio range or expand 00:45:09.390 --> 00:45:14.120 align:middle line:84% the fundamental voltage you can synthesize without distortion. 00:45:14.120 --> 00:45:16.460 align:middle line:90% Any questions about that? 00:45:16.460 --> 00:45:17.120 align:middle line:90% Yeah, Jack. 00:45:17.120 --> 00:45:19.940 align:middle line:84% AUDIENCE: Can you explain again how it's-- 00:45:19.940 --> 00:45:21.210 align:middle line:90% like VDC is fixed. 00:45:21.210 --> 00:45:25.010 align:middle line:90% So how is this increasing-- 00:45:25.010 --> 00:45:26.602 align:middle line:90% overcoming back emf? 00:45:26.602 --> 00:45:27.560 align:middle line:90% DAVID PERREAULT: Right. 00:45:27.560 --> 00:45:32.990 align:middle line:84% Well, let's assume the back emf is at the fundamental, right? 00:45:32.990 --> 00:45:40.040 align:middle line:84% So what I have to worry about is how much fundamental can I 00:45:40.040 --> 00:45:41.870 align:middle line:90% create here? 00:45:41.870 --> 00:45:43.910 align:middle line:84% The vacuum is at the fundamental. 00:45:43.910 --> 00:45:46.880 align:middle line:84% How much fundamental can I create here? 00:45:46.880 --> 00:45:51.140 align:middle line:84% And the problem with not doing this third-harmonic distortion 00:45:51.140 --> 00:45:53.390 align:middle line:84% trick is that if I try to turn up the duty ratio more, 00:45:53.390 --> 00:45:55.280 align:middle line:84% I get a lot of low frequency harmonics. 00:45:55.280 --> 00:45:57.950 align:middle line:84% I get third and fifth, other content 00:45:57.950 --> 00:46:00.170 align:middle line:84% in here, which is then going to drive low frequency 00:46:00.170 --> 00:46:03.040 align:middle line:90% content in the machine. 00:46:03.040 --> 00:46:07.780 align:middle line:84% This trick, what it lets me do is I basically 00:46:07.780 --> 00:46:12.346 align:middle line:84% am going to create the fundamental, 00:46:12.346 --> 00:46:15.260 align:middle line:84% plus I'm going to create some third harmonic here, 00:46:15.260 --> 00:46:17.840 align:middle line:84% third harmonic here, and third harmonic here. 00:46:17.840 --> 00:46:19.995 align:middle line:84% But the third harmonic elements all cancel. 00:46:19.995 --> 00:46:21.370 align:middle line:84% They just show up at the neutral, 00:46:21.370 --> 00:46:23.980 align:middle line:84% and the neutral goes up and down by the third harmonic. 00:46:23.980 --> 00:46:29.470 align:middle line:84% So I can keep this component pure 00:46:29.470 --> 00:46:33.250 align:middle line:84% because I'm not saturating my comparison anymore. 00:46:33.250 --> 00:46:36.460 align:middle line:84% I'm not making this signal go out of the range 00:46:36.460 --> 00:46:40.210 align:middle line:84% just for that little bit, for that extra roughly 15% 00:46:40.210 --> 00:46:42.200 align:middle line:90% of modulation index over 1. 00:46:42.200 --> 00:46:44.895 align:middle line:90% So I can go up to m of 1.15. 00:46:44.895 --> 00:46:45.770 align:middle line:90% Does that make sense? 00:46:45.770 --> 00:46:48.760 align:middle line:90% 00:46:48.760 --> 00:46:55.300 align:middle line:84% Now, that gives me a little bit more head room 00:46:55.300 --> 00:46:57.440 align:middle line:84% to synthesize a little bit more voltage. 00:46:57.440 --> 00:46:59.770 align:middle line:84% And we're often constrained by how much 00:46:59.770 --> 00:47:02.260 align:middle line:84% voltage we can synthesize with our inverter. 00:47:02.260 --> 00:47:05.440 align:middle line:84% What do I do if I'm not happy about that? 00:47:05.440 --> 00:47:10.080 align:middle line:84% Well, I could just keep turning up m beyond 1. 00:47:10.080 --> 00:47:13.010 align:middle line:84% What happens if I turn up m beyond 1? 00:47:13.010 --> 00:47:19.690 align:middle line:84% Well, if I turn up m beyond 1, and my reference waveform 00:47:19.690 --> 00:47:24.070 align:middle line:84% was going to swing up here like that, what I would do 00:47:24.070 --> 00:47:29.200 align:middle line:84% is I would get a bunch of PWM switching 00:47:29.200 --> 00:47:33.730 align:middle line:84% until the middle of the cycle or something where I just flat top. 00:47:33.730 --> 00:47:38.140 align:middle line:84% And so my PWM waveform for each phase might look like this. 00:47:38.140 --> 00:47:39.550 align:middle line:90% It's PWMing. 00:47:39.550 --> 00:47:41.350 align:middle line:84% And then it'll just sit at the top. 00:47:41.350 --> 00:47:43.840 align:middle line:90% And it'll start PWMing again. 00:47:43.840 --> 00:47:46.090 align:middle line:84% And if I keep pushing it up and I 00:47:46.090 --> 00:47:49.130 align:middle line:84% keep making my reference waveform bigger 00:47:49.130 --> 00:47:53.640 align:middle line:84% and bigger and bigger, the limit of that is m goes to infinity. 00:47:53.640 --> 00:47:56.580 align:middle line:84% It's just that I'm just going to generate a square wave. 00:47:56.580 --> 00:47:58.680 align:middle line:84% He's going to be on for half the cycle. 00:47:58.680 --> 00:48:00.330 align:middle line:84% He's going to be on for half the cycle. 00:48:00.330 --> 00:48:02.240 align:middle line:84% And you're back to the six-step waveform 00:48:02.240 --> 00:48:05.626 align:middle line:90% that I talked about before. 00:48:05.626 --> 00:48:08.320 align:middle line:84% And in that case, then I really do generate 00:48:08.320 --> 00:48:10.100 align:middle line:90% the maximum fundamental. 00:48:10.100 --> 00:48:14.440 align:middle line:84% The maximum fundamental is basically VAR is a square wave. 00:48:14.440 --> 00:48:18.670 align:middle line:84% It's VDC over 2 times 4 over pi is 00:48:18.670 --> 00:48:22.630 align:middle line:84% the fundamental of a square wave, of amplitude 00:48:22.630 --> 00:48:25.480 align:middle line:90% plus and minus VDC over 2. 00:48:25.480 --> 00:48:28.400 align:middle line:84% That's the biggest fundamental I can synthesize. 00:48:28.400 --> 00:48:34.150 align:middle line:84% And that sets an upper bound of how the fundamental voltage 00:48:34.150 --> 00:48:35.300 align:middle line:90% I can drive here. 00:48:35.300 --> 00:48:38.500 align:middle line:84% The only problem I get with doing that is that waveform has 00:48:38.500 --> 00:48:40.270 align:middle line:84% third harmonic, which will cancel, 00:48:40.270 --> 00:48:44.420 align:middle line:84% but it also has 5th and 7th and 11th and 13th. 00:48:44.420 --> 00:48:48.367 align:middle line:84% And I will get harmonic currents flowing in these phase windings 00:48:48.367 --> 00:48:50.200 align:middle line:84% when those harmonic currents probably aren't 00:48:50.200 --> 00:48:53.750 align:middle line:90% helping me drive the machine. 00:48:53.750 --> 00:48:55.970 align:middle line:84% So I can synthesize something that gives me 00:48:55.970 --> 00:48:59.240 align:middle line:84% nice, effectively-- at least on a local average basis, 00:48:59.240 --> 00:49:02.450 align:middle line:84% average over switching cycle-- sine waves up 00:49:02.450 --> 00:49:06.320 align:middle line:84% until the amplitude equals VDC over 2. 00:49:06.320 --> 00:49:10.760 align:middle line:84% If I use my third harmonic injection trick, 00:49:10.760 --> 00:49:15.660 align:middle line:84% I can get it up to VDC over 2 times 1.15. 00:49:15.660 --> 00:49:18.440 align:middle line:84% And then if I don't care about distortion, 00:49:18.440 --> 00:49:22.170 align:middle line:84% I'm going to let a bunch of low frequency harmonics exist. 00:49:22.170 --> 00:49:26.000 align:middle line:84% I can just drive up the modulation index even further 00:49:26.000 --> 00:49:29.270 align:middle line:84% and get up to 4 over pi VDC over 2. 00:49:29.270 --> 00:49:31.670 align:middle line:90% And people do that. 00:49:31.670 --> 00:49:34.940 align:middle line:84% In fact, in the Prius, the Toyota guys eventually 00:49:34.940 --> 00:49:37.580 align:middle line:84% published a paper saying, yeah, we ran out of headroom 00:49:37.580 --> 00:49:39.620 align:middle line:84% and we wanted to run our motor faster 00:49:39.620 --> 00:49:43.640 align:middle line:84% for higher speed, so we just let it saturate and generate 00:49:43.640 --> 00:49:44.520 align:middle line:90% the harmonics. 00:49:44.520 --> 00:49:47.000 align:middle line:84% And the motor still worked, and we were OK. 00:49:47.000 --> 00:49:49.910 align:middle line:90% But you pay a price for that. 00:49:49.910 --> 00:49:51.900 align:middle line:90% So people do do this. 00:49:51.900 --> 00:49:54.380 align:middle line:84% But if you want your waveforms to be pure and not 00:49:54.380 --> 00:49:57.398 align:middle line:84% have low frequency harmonics and generate that noise 00:49:57.398 --> 00:49:59.940 align:middle line:84% in your load, whatever it is, with that low frequency content 00:49:59.940 --> 00:50:02.695 align:middle line:84% in your load, you've got to keep your modulation index down. 00:50:02.695 --> 00:50:05.270 align:middle line:90% 00:50:05.270 --> 00:50:07.940 align:middle line:90% Any questions? 00:50:07.940 --> 00:50:08.810 align:middle line:90% Yeah. 00:50:08.810 --> 00:50:12.200 align:middle line:84% AUDIENCE: Is it possible to step up VDC with a [INAUDIBLE] 00:50:12.200 --> 00:50:12.780 align:middle line:90% before? 00:50:12.780 --> 00:50:14.000 align:middle line:90% DAVID PERREAULT: Absolutely. 00:50:14.000 --> 00:50:16.130 align:middle line:84% And in fact, that's what people do. 00:50:16.130 --> 00:50:18.300 align:middle line:90% So why is this a concern? 00:50:18.300 --> 00:50:21.740 align:middle line:84% Because if you're in a Prius, you've got this bus voltage. 00:50:21.740 --> 00:50:24.120 align:middle line:84% And that bus voltage is your battery voltage. 00:50:24.120 --> 00:50:27.208 align:middle line:84% And that might vary over some range. 00:50:27.208 --> 00:50:29.750 align:middle line:84% And then you're trying to deal with this motor whose back emf 00:50:29.750 --> 00:50:32.190 align:middle line:84% varies over some range and you need to drive it. 00:50:32.190 --> 00:50:34.730 align:middle line:84% If you don't like that, then you put another stage 00:50:34.730 --> 00:50:36.770 align:middle line:84% in between your battery and your converter, 00:50:36.770 --> 00:50:40.050 align:middle line:84% and you can control the DC bus voltage to go up higher. 00:50:40.050 --> 00:50:42.890 align:middle line:84% And then you're only limited by how high 00:50:42.890 --> 00:50:45.140 align:middle line:90% voltage these switches are. 00:50:45.140 --> 00:50:46.680 align:middle line:84% So you're always limited by that. 00:50:46.680 --> 00:50:50.690 align:middle line:84% But yes, people do actually control this bus voltage 00:50:50.690 --> 00:50:53.390 align:middle line:84% from the battery voltage, for example, in an electric vehicle 00:50:53.390 --> 00:50:55.020 align:middle line:90% to be something higher. 00:50:55.020 --> 00:50:58.490 align:middle line:84% And they might control it dynamically if they had to, 00:50:58.490 --> 00:51:02.720 align:middle line:84% depending upon the operating condition they want. 00:51:02.720 --> 00:51:03.240 align:middle line:90% All right. 00:51:03.240 --> 00:51:05.180 align:middle line:90% Well, have a great day. 00:51:05.180 --> 00:51:08.950 align:middle line:84% We will pick up a new topic next class. 00:51:08.950 --> 00:51:12.000 align:middle line:90%