WEBVTT 00:00:00.000 --> 00:00:02.420 align:middle line:90% [SQUEAKING] 00:00:02.420 --> 00:00:04.356 align:middle line:90% [RUSTLING] 00:00:04.356 --> 00:00:06.292 align:middle line:90% [CLICKING] 00:00:06.292 --> 00:00:09.500 align:middle line:90% 00:00:09.500 --> 00:00:12.200 align:middle line:84% DAVID PERREAULT: OK, why don't we get started? 00:00:12.200 --> 00:00:15.170 align:middle line:84% So last class, we started talking 00:00:15.170 --> 00:00:17.210 align:middle line:90% about magnetic components. 00:00:17.210 --> 00:00:20.600 align:middle line:84% And this is a pretty important topic in power electronics 00:00:20.600 --> 00:00:23.420 align:middle line:84% because most power converters have magnetic components 00:00:23.420 --> 00:00:25.070 align:middle line:90% at the heart of them. 00:00:25.070 --> 00:00:30.170 align:middle line:84% And we introduced the notion of just having an inductor. 00:00:30.170 --> 00:00:33.320 align:middle line:84% So maybe we have some core where we 00:00:33.320 --> 00:00:36.320 align:middle line:84% have some permeability of the core, 00:00:36.320 --> 00:00:38.330 align:middle line:84% some cross-sectional area of the core, 00:00:38.330 --> 00:00:39.990 align:middle line:90% and some length of the core. 00:00:39.990 --> 00:00:48.020 align:middle line:84% And then we put some windings on it, maybe N turns of winding. 00:00:48.020 --> 00:00:54.560 align:middle line:84% And we have some voltage, V, and current I, 00:00:54.560 --> 00:00:59.384 align:middle line:84% defined with the associated variables convention. 00:00:59.384 --> 00:01:03.080 align:middle line:84% And we analyze this thing, and what 00:01:03.080 --> 00:01:13.610 align:middle line:84% we found is that we would, first of all, if I had a flux density 00:01:13.610 --> 00:01:17.660 align:middle line:84% B in the core, I might have B in the core 00:01:17.660 --> 00:01:23.270 align:middle line:84% is equal to mu core times H in the core, the magnetic flux 00:01:23.270 --> 00:01:25.640 align:middle line:84% density and the magnetic field strength. 00:01:25.640 --> 00:01:33.440 align:middle line:84% And we said, OK, on each turn of the winding, 00:01:33.440 --> 00:01:36.800 align:middle line:90% I'm going to have some B field. 00:01:36.800 --> 00:01:40.880 align:middle line:84% And the B field times the cross-sectional area 00:01:40.880 --> 00:01:43.345 align:middle line:84% is the magnetic flux in the core. 00:01:43.345 --> 00:01:46.180 align:middle line:90% 00:01:46.180 --> 00:01:52.330 align:middle line:84% And we said, OK, by Faraday's law, the voltage, 00:01:52.330 --> 00:01:55.060 align:middle line:90% say, on one turn-- 00:01:55.060 --> 00:01:56.505 align:middle line:90% I'll call it V1 turn-- 00:01:56.505 --> 00:02:01.690 align:middle line:90% 00:02:01.690 --> 00:02:10.320 align:middle line:84% would equal d phi/dt, with the right polarity associated 00:02:10.320 --> 00:02:11.850 align:middle line:90% with Lenz's law. 00:02:11.850 --> 00:02:15.390 align:middle line:84% And then if I have N turns, this would-- say, 00:02:15.390 --> 00:02:17.640 align:middle line:84% I have a voltage on this turn, a voltage on this turn, 00:02:17.640 --> 00:02:20.790 align:middle line:84% a voltage on this turn-- the total voltage would be 00:02:20.790 --> 00:02:24.930 align:middle line:90% equal to N times d phi core dt. 00:02:24.930 --> 00:02:30.930 align:middle line:84% Or we could write this as d/dt of N times the flux 00:02:30.930 --> 00:02:32.070 align:middle line:90% through the core. 00:02:32.070 --> 00:02:37.710 align:middle line:84% And we would often write this as d lambda/dt, where 00:02:37.710 --> 00:02:41.420 align:middle line:90% lambda is the flux linkage. 00:02:41.420 --> 00:02:46.950 align:middle line:90% 00:02:46.950 --> 00:02:48.310 align:middle line:90% That's its definition. 00:02:48.310 --> 00:02:52.350 align:middle line:84% So it's the flux linked by the net internal windings 00:02:52.350 --> 00:02:56.310 align:middle line:84% or N times the flux linked by each turn. 00:02:56.310 --> 00:03:00.870 align:middle line:84% And we also said that we derived the fact that lambda, the flux 00:03:00.870 --> 00:03:04.710 align:middle line:84% linkage, is equal to some constant L, 00:03:04.710 --> 00:03:09.900 align:middle line:84% the inductance times the current, or d lambda/dt 00:03:09.900 --> 00:03:13.155 align:middle line:84% is equal to the voltage is equal to L dI/dt. 00:03:13.155 --> 00:03:17.660 align:middle line:90% 00:03:17.660 --> 00:03:20.760 align:middle line:84% Just to show you that's true, we set up a demo. 00:03:20.760 --> 00:03:30.380 align:middle line:84% We actually-- what we have here is a toroidal inductor 00:03:30.380 --> 00:03:32.780 align:middle line:90% with a bunch of turns on it. 00:03:32.780 --> 00:03:35.300 align:middle line:84% And what we're going to do is we're 00:03:35.300 --> 00:03:37.850 align:middle line:90% going to apply a voltage to it. 00:03:37.850 --> 00:03:41.600 align:middle line:84% And what you can see right now is essentially 00:03:41.600 --> 00:03:45.260 align:middle line:84% the voltage being applied across this whole inductor. 00:03:45.260 --> 00:03:50.510 align:middle line:84% But what I did was we wound the inductor with uninsulated wires 00:03:50.510 --> 00:03:54.230 align:middle line:84% so that we could tap individual turns going around. 00:03:54.230 --> 00:04:01.760 align:middle line:84% And what you can see is if I look just on one turn, 00:04:01.760 --> 00:04:02.410 align:middle line:90% for example-- 00:04:02.410 --> 00:04:07.400 align:middle line:90% 00:04:07.400 --> 00:04:08.100 align:middle line:90% come on. 00:04:08.100 --> 00:04:12.240 align:middle line:90% 00:04:12.240 --> 00:04:14.460 align:middle line:84% It would help if I looked at the right end. 00:04:14.460 --> 00:04:17.930 align:middle line:84% If I looked on just on one turn, I'll see one voltage. 00:04:17.930 --> 00:04:20.470 align:middle line:90% 00:04:20.470 --> 00:04:21.220 align:middle line:90% OK, you know what? 00:04:21.220 --> 00:04:23.820 align:middle line:84% I think we're going to try this another time. 00:04:23.820 --> 00:04:27.250 align:middle line:84% We'll bring this demo back later. 00:04:27.250 --> 00:04:29.020 align:middle line:84% If it had worked, what you would have seen 00:04:29.020 --> 00:04:30.812 align:middle line:84% is the voltage building up around the core, 00:04:30.812 --> 00:04:32.990 align:middle line:84% which it did do in the lab 20 minutes ago. 00:04:32.990 --> 00:04:34.990 align:middle line:84% But that's the way it goes sometimes. 00:04:34.990 --> 00:04:37.540 align:middle line:84% I think we have some problem with what 00:04:37.540 --> 00:04:39.280 align:middle line:90% signal is being picked up. 00:04:39.280 --> 00:04:44.680 align:middle line:84% What we did, starting from this, is saying, well, 00:04:44.680 --> 00:04:47.800 align:middle line:84% we used Maxwell's equations to get these results. 00:04:47.800 --> 00:04:54.020 align:middle line:84% And that was fine, but it was very complicated. 00:04:54.020 --> 00:04:57.910 align:middle line:84% So then we introduced this notion of a magnetic circuit 00:04:57.910 --> 00:05:01.600 align:middle line:84% model, where we said, OK, in an electric circuit, 00:05:01.600 --> 00:05:03.590 align:middle line:90% we have KVL and KCL. 00:05:03.590 --> 00:05:06.830 align:middle line:84% We have voltage sources, sometimes called an EMF. 00:05:06.830 --> 00:05:11.320 align:middle line:84% We have current, which is our flow variable or our through 00:05:11.320 --> 00:05:12.110 align:middle line:90% variable. 00:05:12.110 --> 00:05:14.170 align:middle line:84% And we have resistances, which we 00:05:14.170 --> 00:05:18.070 align:middle line:84% talked about the resistance of a material to current flow 00:05:18.070 --> 00:05:20.060 align:middle line:84% through it when a voltage is applied. 00:05:20.060 --> 00:05:23.770 align:middle line:84% And we said, OK, we can build a magnetic circuit that 00:05:23.770 --> 00:05:26.140 align:middle line:84% describes magnetic structures, where 00:05:26.140 --> 00:05:32.020 align:middle line:84% we have an MMF, which is usually the ampere turns that's 00:05:32.020 --> 00:05:35.610 align:middle line:90% trying to drive flux. 00:05:35.610 --> 00:05:37.180 align:middle line:90% Flux is our through variable. 00:05:37.180 --> 00:05:38.890 align:middle line:90% So MMF is our across variable. 00:05:38.890 --> 00:05:40.260 align:middle line:90% Flux is our through variable. 00:05:40.260 --> 00:05:45.760 align:middle line:84% And reluctance to magnetic flux depends upon the material 00:05:45.760 --> 00:05:47.920 align:middle line:90% and the geometry. 00:05:47.920 --> 00:05:49.920 align:middle line:84% And we said just as an electrical resistance is 00:05:49.920 --> 00:05:52.320 align:middle line:84% the length of the resistor divided by the area 00:05:52.320 --> 00:05:56.010 align:middle line:84% resistor and the conductivity of the material, 00:05:56.010 --> 00:05:59.590 align:middle line:84% permeability is just conductivity to magnetic flux. 00:05:59.590 --> 00:06:02.370 align:middle line:84% And so this equivalent circuit model 00:06:02.370 --> 00:06:06.573 align:middle line:84% is very useful to very quickly making all the calculations 00:06:06.573 --> 00:06:08.490 align:middle line:84% that we would otherwise have to start applying 00:06:08.490 --> 00:06:11.240 align:middle line:90% Maxwell's equations to do. 00:06:11.240 --> 00:06:19.970 align:middle line:84% So, for example, if we sat and said let's take a core structure 00:06:19.970 --> 00:06:22.070 align:middle line:90% like this-- 00:06:22.070 --> 00:06:23.900 align:middle line:84% and this would be called a C core, 00:06:23.900 --> 00:06:25.490 align:middle line:90% where I put a gap in the core. 00:06:25.490 --> 00:06:28.190 align:middle line:84% And we saw that we would do that in part 00:06:28.190 --> 00:06:31.400 align:middle line:84% because we can make the magnetic inductance more 00:06:31.400 --> 00:06:35.040 align:middle line:84% stable to material variations, permeability variations. 00:06:35.040 --> 00:06:39.470 align:middle line:84% We would also do it for reasons that we 00:06:39.470 --> 00:06:44.480 align:middle line:84% tend to store energy in the gap instead of in the core material, 00:06:44.480 --> 00:06:45.660 align:middle line:90% preferably. 00:06:45.660 --> 00:06:49.430 align:middle line:90% And we go put in turns on this. 00:06:49.430 --> 00:06:57.590 align:middle line:84% So we'd have Ni, N turns of current i, and we get voltage V. 00:06:57.590 --> 00:07:02.540 align:middle line:84% And we said my magnetic circuit model would 00:07:02.540 --> 00:07:08.240 align:middle line:84% be Ni is trying to drive flux in the core. 00:07:08.240 --> 00:07:11.390 align:middle line:84% And a positive current, by the right-hand rule, 00:07:11.390 --> 00:07:14.690 align:middle line:84% is trying to throw a flux clockwise around this core. 00:07:14.690 --> 00:07:21.200 align:middle line:84% And then I would have the reluctance of the core, which 00:07:21.200 --> 00:07:24.956 align:middle line:90% is equal to lc over mu c Ac. 00:07:24.956 --> 00:07:29.798 align:middle line:90% 00:07:29.798 --> 00:07:37.660 align:middle line:84% And the reluctance of the gap is equal to lg over mu 0 Ag. 00:07:37.660 --> 00:07:43.480 align:middle line:84% And then I can calculate what the net flux traveling 00:07:43.480 --> 00:07:46.420 align:middle line:90% around this path is. 00:07:46.420 --> 00:07:49.315 align:middle line:84% And that would just be this flux, phi. 00:07:49.315 --> 00:07:53.530 align:middle line:90% 00:07:53.530 --> 00:07:55.040 align:middle line:90% So what could we say? 00:07:55.040 --> 00:07:57.370 align:middle line:90% Well, we can calculate phi. 00:07:57.370 --> 00:08:03.070 align:middle line:84% Phi is just simply equal to Ni divided 00:08:03.070 --> 00:08:06.410 align:middle line:84% by the reluctance of the core plus the reluctance of the gap. 00:08:06.410 --> 00:08:10.040 align:middle line:84% It's pretty simple resistive circuit there. 00:08:10.040 --> 00:08:13.450 align:middle line:84% And then we can say, well, lambda 00:08:13.450 --> 00:08:19.630 align:middle line:84% is the flux linkage of this MMF source. 00:08:19.630 --> 00:08:22.560 align:middle line:84% So lambda-- if I call that phi core, 00:08:22.560 --> 00:08:26.370 align:middle line:84% lambda is just equal to N times phi core, which 00:08:26.370 --> 00:08:29.490 align:middle line:84% makes it N squared over the reluctance 00:08:29.490 --> 00:08:34.980 align:middle line:84% of the core plus the reluctance of the gap times the current. 00:08:34.980 --> 00:08:43.460 align:middle line:84% And this thing here, this proportionality factor, 00:08:43.460 --> 00:08:46.550 align:middle line:84% is just the inductance L. So I can 00:08:46.550 --> 00:08:50.780 align:middle line:84% immediately from this structure get the inductance. 00:08:50.780 --> 00:08:54.630 align:middle line:84% I can also come back and say, OK, well, what's the voltage? 00:08:54.630 --> 00:09:00.110 align:middle line:84% The voltage is equal to d lambda/dt, which 00:09:00.110 --> 00:09:04.730 align:middle line:84% is N squared over the reluctance of the core plus the reluctance 00:09:04.730 --> 00:09:08.550 align:middle line:90% of the gap times di/dt. 00:09:08.550 --> 00:09:10.700 align:middle line:84% So I can come back and figure out what 00:09:10.700 --> 00:09:12.950 align:middle line:90% the voltage on this winding is. 00:09:12.950 --> 00:09:16.010 align:middle line:84% And when I do that, I have to be a little bit 00:09:16.010 --> 00:09:18.020 align:middle line:90% careful about the signs. 00:09:18.020 --> 00:09:22.100 align:middle line:84% And if you notice, the flux I'm talking about 00:09:22.100 --> 00:09:26.300 align:middle line:84% is the flux that's coming out of the positive terminal of the MMF 00:09:26.300 --> 00:09:27.950 align:middle line:84% source and the positive circuit-- 00:09:27.950 --> 00:09:32.870 align:middle line:84% relates to the voltage with the associated variables 00:09:32.870 --> 00:09:36.620 align:middle line:84% of the current going into that winding. 00:09:36.620 --> 00:09:40.310 align:middle line:84% So if I define a current and I get the MMF source, 00:09:40.310 --> 00:09:42.380 align:middle line:84% the flux coming out of the MMF source 00:09:42.380 --> 00:09:44.340 align:middle line:84% gives me the voltage with this polarity. 00:09:44.340 --> 00:09:45.965 align:middle line:90% So it's the VNi-- 00:09:45.965 --> 00:09:50.370 align:middle line:84% V times i is the power into the component. 00:09:50.370 --> 00:09:51.680 align:middle line:90% Any questions about that? 00:09:51.680 --> 00:09:54.840 align:middle line:90% 00:09:54.840 --> 00:10:00.280 align:middle line:84% So that's just a very, very quick review of last class. 00:10:00.280 --> 00:10:04.290 align:middle line:84% The other thing we may want to pay a little bit attention to 00:10:04.290 --> 00:10:08.490 align:middle line:84% and what we often use these magnetic circuits for 00:10:08.490 --> 00:10:11.610 align:middle line:84% is that we assumed in all this that B 00:10:11.610 --> 00:10:16.320 align:middle line:84% was mu H. I'm assuming that the relationship 00:10:16.320 --> 00:10:18.300 align:middle line:90% in the core material-- 00:10:18.300 --> 00:10:24.230 align:middle line:84% B is equal to mu core times H inside this magnetic core. 00:10:24.230 --> 00:10:30.740 align:middle line:84% Well, in a real core material, B is equal to mu H-- 00:10:30.740 --> 00:10:33.950 align:middle line:84% if this is H, and this is B in the core-- 00:10:33.950 --> 00:10:38.180 align:middle line:84% that's true or approximately true 00:10:38.180 --> 00:10:41.690 align:middle line:90% for rather low flux densities. 00:10:41.690 --> 00:10:43.520 align:middle line:84% Above some level that we sometimes 00:10:43.520 --> 00:10:49.430 align:middle line:84% call B SAT, or the saturation flux density, this tapers off. 00:10:49.430 --> 00:10:50.922 align:middle line:90% All the magnetic domains align. 00:10:50.922 --> 00:10:52.380 align:middle line:84% And I get something that does this. 00:10:52.380 --> 00:10:56.330 align:middle line:84% I'm simplifying a bit, but this would end up being mu 0. 00:10:56.330 --> 00:10:58.610 align:middle line:84% So in order to use this model, we 00:10:58.610 --> 00:11:01.760 align:middle line:84% know that we want to be operating down here, 00:11:01.760 --> 00:11:05.570 align:middle line:84% below the saturation flux density of the material. 00:11:05.570 --> 00:11:10.610 align:middle line:84% So I might also use this kind of model to say, at what current 00:11:10.610 --> 00:11:14.640 align:middle line:84% does this thing start to saturate given its geometry? 00:11:14.640 --> 00:11:17.120 align:middle line:90% Because B is related to flux. 00:11:17.120 --> 00:11:18.280 align:middle line:90% Any questions about that? 00:11:18.280 --> 00:11:20.920 align:middle line:90% 00:11:20.920 --> 00:11:25.130 align:middle line:84% So that was a lightning review of last class. 00:11:25.130 --> 00:11:27.890 align:middle line:84% What I'd like to do today is start 00:11:27.890 --> 00:11:30.320 align:middle line:90% talking about transformers. 00:11:30.320 --> 00:11:33.890 align:middle line:90% And what's a transformer? 00:11:33.890 --> 00:11:36.470 align:middle line:84% You might think of a transformer, or at least 00:11:36.470 --> 00:11:38.990 align:middle line:84% a two-winding transformer, as a magnetic circuit where 00:11:38.990 --> 00:11:42.110 align:middle line:84% instead of having a winding and a gap to store the energy, 00:11:42.110 --> 00:11:46.220 align:middle line:84% I have a winding and another winding to transfer energy. 00:11:46.220 --> 00:11:50.300 align:middle line:84% But a simple transformer-- maybe I'll draw it like this. 00:11:50.300 --> 00:11:52.760 align:middle line:90% Suppose I just have some core. 00:11:52.760 --> 00:11:57.170 align:middle line:84% And again, I'm going to assume it has some area of the core, 00:11:57.170 --> 00:12:00.650 align:middle line:84% permeability of the core, and some length of the core. 00:12:00.650 --> 00:12:05.540 align:middle line:84% And I'll put a first winding on it like this. 00:12:05.540 --> 00:12:13.380 align:middle line:84% And I'll say that this is N1, i1, and V1. 00:12:13.380 --> 00:12:17.260 align:middle line:84% And let me put a second winding on it 00:12:17.260 --> 00:12:28.180 align:middle line:90% that I'll call N2, i2, and V2. 00:12:28.180 --> 00:12:30.670 align:middle line:84% And notice that I'm keeping associated variables, 00:12:30.670 --> 00:12:33.370 align:middle line:90% my voltages, and currents here. 00:12:33.370 --> 00:12:35.620 align:middle line:84% Now, if I thought about this thing, 00:12:35.620 --> 00:12:38.800 align:middle line:84% what would be the magnetic circuit model. 00:12:38.800 --> 00:12:40.960 align:middle line:84% Well, I said here's an MMF source. 00:12:40.960 --> 00:12:43.630 align:middle line:84% If I have a current into this winding, I have some MMF. 00:12:43.630 --> 00:12:51.310 align:middle line:84% So my magnetic circuit would look like this, N1, i1. 00:12:51.310 --> 00:12:55.570 align:middle line:84% Then I have the reluctance of the core 00:12:55.570 --> 00:13:00.700 align:middle line:84% that I have to worry about, so let me put that in. 00:13:00.700 --> 00:13:02.500 align:middle line:84% Here's the reluctance of the core. 00:13:02.500 --> 00:13:07.340 align:middle line:84% And now I have another MMF source, N2 i2. 00:13:07.340 --> 00:13:11.380 align:middle line:84% But now I've got to be careful because what's 00:13:11.380 --> 00:13:15.100 align:middle line:84% the relative polarity of what this MMF source is going 00:13:15.100 --> 00:13:18.040 align:middle line:84% to do to drive flux and what this MMF source is 00:13:18.040 --> 00:13:19.220 align:middle line:90% going to do to drive flux? 00:13:19.220 --> 00:13:24.070 align:middle line:84% And in this case, if I have a positive current 00:13:24.070 --> 00:13:25.900 align:middle line:84% into the winding, by the right-hand rule, 00:13:25.900 --> 00:13:28.890 align:middle line:84% it's trying to push flux clockwise around the core, 00:13:28.890 --> 00:13:29.390 align:middle line:90% right? 00:13:29.390 --> 00:13:33.620 align:middle line:84% So I could imagine some core flux 00:13:33.620 --> 00:13:39.320 align:middle line:84% being driven this way, phi core, from this positive current, i1, 00:13:39.320 --> 00:13:40.110 align:middle line:90% here. 00:13:40.110 --> 00:13:45.140 align:middle line:84% Notice that the way I drew this, if I put a positive current i2 00:13:45.140 --> 00:13:47.120 align:middle line:84% here, by the right-hand rule, it's 00:13:47.120 --> 00:13:51.530 align:middle line:84% also trying to drive flux in the same direction. 00:13:51.530 --> 00:13:58.160 align:middle line:84% That means in my magnetic circuit model here, 00:13:58.160 --> 00:14:01.080 align:middle line:90% I would have N2 i2. 00:14:01.080 --> 00:14:04.970 align:middle line:84% And notice that N2 and i2 and N1 and i1 00:14:04.970 --> 00:14:08.540 align:middle line:84% are trying to push flux in the same direction around the core. 00:14:08.540 --> 00:14:11.550 align:middle line:84% And you could say that's correct. 00:14:11.550 --> 00:14:13.430 align:middle line:90% It is correct. 00:14:13.430 --> 00:14:15.210 align:middle line:90% Why did it work out this way? 00:14:15.210 --> 00:14:17.810 align:middle line:84% Because you could look at it either one way or another. 00:14:17.810 --> 00:14:20.570 align:middle line:84% Because of the relative way I drew the polarity 00:14:20.570 --> 00:14:22.370 align:middle line:84% of this winding, which way it was wrapped 00:14:22.370 --> 00:14:25.010 align:middle line:84% on the core, or equivalently, which way I 00:14:25.010 --> 00:14:26.990 align:middle line:90% defined voltage and current. 00:14:26.990 --> 00:14:30.270 align:middle line:84% I could define it some other way. 00:14:30.270 --> 00:14:33.270 align:middle line:84% I could define voltage V2 the other way, in which case-- 00:14:33.270 --> 00:14:34.950 align:middle line:84% and i2 the other way, in which case 00:14:34.950 --> 00:14:37.030 align:middle line:84% I'd have to flip that MMF source. 00:14:37.030 --> 00:14:38.820 align:middle line:84% So the polarities matter a lot in this, 00:14:38.820 --> 00:14:40.270 align:middle line:90% so you ought to think about it. 00:14:40.270 --> 00:14:41.940 align:middle line:84% The other thing I will say is when 00:14:41.940 --> 00:14:45.480 align:middle line:84% we talk about a transformer, at least a two-winding transformer, 00:14:45.480 --> 00:14:48.630 align:middle line:84% you'll often see something called the dot convention. 00:14:48.630 --> 00:14:51.035 align:middle line:84% And so somewhere in the structure, 00:14:51.035 --> 00:14:52.410 align:middle line:84% maybe you'll see this, and you'll 00:14:52.410 --> 00:14:53.702 align:middle line:90% see it in the schematic symbol. 00:14:53.702 --> 00:14:56.130 align:middle line:84% There's going to be a dot here and a dot here. 00:14:56.130 --> 00:15:02.060 align:middle line:84% What that means is in practice current 00:15:02.060 --> 00:15:06.350 align:middle line:90% into the dot throw flux-- 00:15:06.350 --> 00:15:10.320 align:middle line:84% try to throw flux the same direction around the core. 00:15:10.320 --> 00:15:14.610 align:middle line:84% So that just is saying something about the relative definitions 00:15:14.610 --> 00:15:21.590 align:middle line:84% of the winding polarities just is denoted by the dots. 00:15:21.590 --> 00:15:24.193 align:middle line:84% So just think, currents into the dot 00:15:24.193 --> 00:15:25.860 align:middle line:84% throw flux the same way around the core. 00:15:25.860 --> 00:15:29.870 align:middle line:90% 00:15:29.870 --> 00:15:37.035 align:middle line:84% OK, so this is my simple circuit model, 00:15:37.035 --> 00:15:38.410 align:middle line:84% and I have reluctance to the core 00:15:38.410 --> 00:15:42.320 align:middle line:90% as l core over mu core A core. 00:15:42.320 --> 00:15:44.890 align:middle line:84% What could I say about this magnetic structure 00:15:44.890 --> 00:15:46.990 align:middle line:90% as to what it would do? 00:15:46.990 --> 00:15:50.710 align:middle line:84% Well, let me start by considering the idealized case 00:15:50.710 --> 00:15:54.350 align:middle line:90% where mu core is really big. 00:15:54.350 --> 00:15:55.810 align:middle line:90% It's a really big number. 00:15:55.810 --> 00:15:58.750 align:middle line:84% If mu core is really big or if the area of core is really big, 00:15:58.750 --> 00:16:01.840 align:middle line:84% the reluctance of the core is really small. 00:16:01.840 --> 00:16:04.760 align:middle line:84% And for many transformer designs, 00:16:04.760 --> 00:16:07.585 align:middle line:90% that's what you would do. 00:16:07.585 --> 00:16:09.578 align:middle line:84% You'd try to approximate it that way. 00:16:09.578 --> 00:16:10.870 align:middle line:90% So let's see what we would get. 00:16:10.870 --> 00:16:13.400 align:middle line:90% What would I get for voltage V1? 00:16:13.400 --> 00:16:27.280 align:middle line:84% Well, V1 is equal to d/dt of N1 times the flux in the core 00:16:27.280 --> 00:16:30.100 align:middle line:90% because V is equal to d lambda-- 00:16:30.100 --> 00:16:32.628 align:middle line:90% V1 is d lambda 1 dt. 00:16:32.628 --> 00:16:40.820 align:middle line:84% V2 is simply equal to d/dt of N2 times the flux in the core. 00:16:40.820 --> 00:16:43.090 align:middle line:84% And that's because notice in both cases, 00:16:43.090 --> 00:16:44.590 align:middle line:90% the flux in the core-- 00:16:44.590 --> 00:16:49.960 align:middle line:84% I've defined the flux clockwise here. 00:16:49.960 --> 00:16:52.540 align:middle line:84% So the flux of the core is the flux out 00:16:52.540 --> 00:16:54.500 align:middle line:90% of the top of this MMF source. 00:16:54.500 --> 00:16:57.780 align:middle line:84% It's also out of the positive terminal of this MMF source. 00:16:57.780 --> 00:17:00.990 align:middle line:84% So that's why the signs are the same here. 00:17:00.990 --> 00:17:02.940 align:middle line:90% That make sense to everybody? 00:17:02.940 --> 00:17:04.780 align:middle line:90% So what does this say overall? 00:17:04.780 --> 00:17:08.619 align:middle line:84% Well, I could pull the N1 and N2 out and divide the two. 00:17:08.619 --> 00:17:11.339 align:middle line:84% And what this relationship tells me 00:17:11.339 --> 00:17:14.310 align:middle line:84% is that V1 and V2 are proportional 00:17:14.310 --> 00:17:18.930 align:middle line:84% because V1 is proportional to the flux in the core 00:17:18.930 --> 00:17:20.772 align:middle line:84% by the number of turns, and V2 is 00:17:20.772 --> 00:17:22.980 align:middle line:84% proportional flux in the core by the number of turns. 00:17:22.980 --> 00:17:30.170 align:middle line:84% What I get is V2 over V1 is equal to N2 over N1. 00:17:30.170 --> 00:17:37.130 align:middle line:84% So this is one characteristic that I get in this system. 00:17:37.130 --> 00:17:40.380 align:middle line:90% Make sense to everybody? 00:17:40.380 --> 00:17:43.475 align:middle line:84% Well, what else can I say about this system? 00:17:43.475 --> 00:17:46.830 align:middle line:90% 00:17:46.830 --> 00:17:54.300 align:middle line:84% N1 i1 plus N2 i2 is the total MMF here, 00:17:54.300 --> 00:17:58.560 align:middle line:84% and that's got to be equal to phi core times the reluctance 00:17:58.560 --> 00:18:00.520 align:middle line:90% of the core. 00:18:00.520 --> 00:18:03.885 align:middle line:90% That's just magnetic KVL here. 00:18:03.885 --> 00:18:06.980 align:middle line:90% 00:18:06.980 --> 00:18:09.930 align:middle line:90% Well, all right, what can I say? 00:18:09.930 --> 00:18:10.940 align:middle line:90% Well, if I said-- 00:18:10.940 --> 00:18:16.730 align:middle line:84% if reluctance of the core was really, really small, goes to 0, 00:18:16.730 --> 00:18:20.100 align:middle line:84% then this right hand ought to go to 0. 00:18:20.100 --> 00:18:24.590 align:middle line:84% So what I have is in the limit, as mu of the core 00:18:24.590 --> 00:18:33.860 align:middle line:84% goes to infinity, what I get is N1 i1 plus N2 i2 is equal to 0. 00:18:33.860 --> 00:18:41.440 align:middle line:84% Or equivalently, i2 over i1 is equal to minus N1 over N2. 00:18:41.440 --> 00:18:47.106 align:middle line:90% 00:18:47.106 --> 00:18:50.210 align:middle line:84% So what I have is two sets of relations here, 00:18:50.210 --> 00:18:53.410 align:middle line:84% one between the voltages and the sets of terminals 00:18:53.410 --> 00:18:57.570 align:middle line:84% and one between the currents of the sets of terminals. 00:18:57.570 --> 00:19:00.510 align:middle line:84% These two sets of equations that I've starred 00:19:00.510 --> 00:19:03.780 align:middle line:84% are known as the ideal transformer relations. 00:19:03.780 --> 00:19:06.710 align:middle line:84% This is what an ideal transformer would do. 00:19:06.710 --> 00:19:09.086 align:middle line:84% So we would often draw that like this. 00:19:09.086 --> 00:19:13.580 align:middle line:90% 00:19:13.580 --> 00:19:16.230 align:middle line:84% Usually, they often put bars here. 00:19:16.230 --> 00:19:19.700 align:middle line:84% That's to symbolize the fact that there's a core there. 00:19:19.700 --> 00:19:22.490 align:middle line:84% We will have our dots, which are telling us 00:19:22.490 --> 00:19:24.880 align:middle line:90% about voltage polarities. 00:19:24.880 --> 00:19:30.130 align:middle line:84% And I would indicate the number of turns N1 to N2. 00:19:30.130 --> 00:19:36.158 align:middle line:84% And then I would have V1, i1, and V2 i2. 00:19:36.158 --> 00:19:39.100 align:middle line:90% 00:19:39.100 --> 00:19:41.680 align:middle line:84% The other way you can look at the dots 00:19:41.680 --> 00:19:49.240 align:middle line:84% is the voltages have the same polarity 00:19:49.240 --> 00:19:52.500 align:middle line:84% if the positive terminals at the dot. 00:19:52.500 --> 00:19:55.510 align:middle line:84% So this is saying which way is the voltage? 00:19:55.510 --> 00:20:00.140 align:middle line:84% Is it-- positive terminal here and positive 00:20:00.140 --> 00:20:04.390 align:middle line:84% terminal here means the sign of V1 and V2 is the same. 00:20:04.390 --> 00:20:11.490 align:middle line:84% So if you see this symbol, very often 00:20:11.490 --> 00:20:14.520 align:middle line:84% that's used to designate an ideal transformer 00:20:14.520 --> 00:20:16.410 align:middle line:90% that has these relations. 00:20:16.410 --> 00:20:19.080 align:middle line:84% And if you look at this, this is saying that the voltage scale 00:20:19.080 --> 00:20:21.000 align:middle line:90% is one way. 00:20:21.000 --> 00:20:24.670 align:middle line:84% So the voltage with its positive terminal at the dot this way 00:20:24.670 --> 00:20:27.520 align:middle line:84% is proportional to the voltage here. 00:20:27.520 --> 00:20:32.630 align:middle line:84% The current, whatever current comes into this dot, 00:20:32.630 --> 00:20:35.510 align:middle line:84% a scaled version of that current must come out of this dot 00:20:35.510 --> 00:20:39.360 align:middle line:84% because i2 over i2 has a minus sign here. 00:20:39.360 --> 00:20:41.510 align:middle line:84% So current goes into one dot, comes out 00:20:41.510 --> 00:20:44.900 align:middle line:84% the other dot is the way people think about it-- 00:20:44.900 --> 00:20:47.300 align:middle line:90% with the right scaling. 00:20:47.300 --> 00:20:49.430 align:middle line:84% If you look at this carefully, this also 00:20:49.430 --> 00:20:54.150 align:middle line:84% means V1 i1 equals minus V2 i2 at all times. 00:20:54.150 --> 00:20:57.270 align:middle line:84% So an ideal transformer stores no energy. 00:20:57.270 --> 00:21:00.560 align:middle line:84% Whatever power flows in one side, flows out the other, 00:21:00.560 --> 00:21:04.080 align:middle line:84% but the voltages and currents are scaled. 00:21:04.080 --> 00:21:05.480 align:middle line:84% So that's often the way you use-- 00:21:05.480 --> 00:21:07.320 align:middle line:84% if you need to scale up and down a voltage-- 00:21:07.320 --> 00:21:10.310 align:middle line:84% if you scale up the voltage, it scales down the current, 00:21:10.310 --> 00:21:14.250 align:middle line:84% gives you this transformation that you want. 00:21:14.250 --> 00:21:15.470 align:middle line:90% Any questions about that? 00:21:15.470 --> 00:21:21.766 align:middle line:90% 00:21:21.766 --> 00:21:24.400 align:middle line:84% So the first thing you're going to want 00:21:24.400 --> 00:21:26.200 align:middle line:84% to think about when you see this symbol 00:21:26.200 --> 00:21:29.133 align:middle line:84% is that you've got an ideal transformer. 00:21:29.133 --> 00:21:30.550 align:middle line:84% And these equations, which I think 00:21:30.550 --> 00:21:33.193 align:middle line:84% are in chapter 1 of Principles of Power Electronics, 00:21:33.193 --> 00:21:35.110 align:middle line:84% are just worth memorizing if you don't already 00:21:35.110 --> 00:21:36.068 align:middle line:90% have them in your head. 00:21:36.068 --> 00:21:38.840 align:middle line:90% 00:21:38.840 --> 00:21:45.140 align:middle line:84% Let's now start talking about what real transformers do 00:21:45.140 --> 00:21:48.560 align:middle line:84% because it's not quite as simple and pretty as that. 00:21:48.560 --> 00:21:50.450 align:middle line:90% In the real world-- 00:21:50.450 --> 00:21:52.040 align:middle line:90% the reason this is-- 00:21:52.040 --> 00:21:54.590 align:middle line:84% and if I draw this symbol, usually what I'll mean 00:21:54.590 --> 00:21:57.746 align:middle line:90% is an ideal transformer. 00:21:57.746 --> 00:22:00.790 align:middle line:90% 00:22:00.790 --> 00:22:02.720 align:middle line:84% Well, real transformers aren't ideal. 00:22:02.720 --> 00:22:04.870 align:middle line:90% Well, how are they not ideal? 00:22:04.870 --> 00:22:07.160 align:middle line:84% They're not ideal in a lot of ways. 00:22:07.160 --> 00:22:09.160 align:middle line:84% But let's think about, first of all, 00:22:09.160 --> 00:22:13.780 align:middle line:84% some of the principal parasitics that show up in these things. 00:22:13.780 --> 00:22:16.510 align:middle line:84% Well, where would I get the first one? 00:22:16.510 --> 00:22:23.080 align:middle line:84% The first principal parasitic kind of comes from this equation 00:22:23.080 --> 00:22:24.700 align:middle line:90% here. 00:22:24.700 --> 00:22:27.550 align:middle line:84% What did I do to make this assumption? 00:22:27.550 --> 00:22:30.330 align:middle line:84% What did I do to get to this current relation? 00:22:30.330 --> 00:22:33.970 align:middle line:84% I assumed that phi c times the reluctance of the core 00:22:33.970 --> 00:22:39.170 align:middle line:84% was 0 or the core reluctance was 0. 00:22:39.170 --> 00:22:43.870 align:middle line:84% Well, the core reluctance isn't really 0. 00:22:43.870 --> 00:22:49.720 align:middle line:84% This is actually some non-zero value. 00:22:49.720 --> 00:22:52.660 align:middle line:84% That means that the current relations 00:22:52.660 --> 00:22:54.040 align:middle line:90% are kind of mismatched. 00:22:54.040 --> 00:22:58.930 align:middle line:84% I don't really get i2 over i1 being some perfect scaling, 00:22:58.930 --> 00:23:02.205 align:middle line:84% but I get a mismatch that's exactly by this amount. 00:23:02.205 --> 00:23:05.190 align:middle line:90% 00:23:05.190 --> 00:23:10.040 align:middle line:84% So what does that look like in circuit terms? 00:23:10.040 --> 00:23:13.730 align:middle line:84% Well, suppose I took my transformer 00:23:13.730 --> 00:23:17.940 align:middle line:84% and I just put a winding on one side-- 00:23:17.940 --> 00:23:20.760 align:middle line:84% I have winding on both sides, but I open circuit the second 00:23:20.760 --> 00:23:21.430 align:middle line:90% winding. 00:23:21.430 --> 00:23:23.680 align:middle line:84% I just don't connect it to anything. 00:23:23.680 --> 00:23:27.860 align:middle line:84% Then what do I know about i2 if it's disconnected? 00:23:27.860 --> 00:23:30.530 align:middle line:90% i2 is 0 if it's disconnected. 00:23:30.530 --> 00:23:33.710 align:middle line:84% The ideal transformer relations here 00:23:33.710 --> 00:23:40.940 align:middle line:84% would tell you that because i2 is 0, hence i1 must be 0. 00:23:40.940 --> 00:23:44.250 align:middle line:84% But if we just thought about this thing, 00:23:44.250 --> 00:23:47.483 align:middle line:84% well, if I open circuit the second one, I can ignore it. 00:23:47.483 --> 00:23:49.400 align:middle line:84% For practical purposes, it's not really there. 00:23:49.400 --> 00:23:51.530 align:middle line:84% But I do have a winding in the core. 00:23:51.530 --> 00:23:55.020 align:middle line:84% That looks kind of like an inductor, doesn't it? 00:23:55.020 --> 00:23:56.653 align:middle line:90% Well, so what would I expect-- 00:23:56.653 --> 00:23:59.940 align:middle line:90% 00:23:59.940 --> 00:24:03.060 align:middle line:84% What would I expect in a real system? 00:24:03.060 --> 00:24:05.140 align:middle line:90% Well, it's exactly-- 00:24:05.140 --> 00:24:06.250 align:middle line:90% I already have the model. 00:24:06.250 --> 00:24:07.958 align:middle line:84% I have already have the model right here. 00:24:07.958 --> 00:24:09.750 align:middle line:84% All I've done is made i2 0, which 00:24:09.750 --> 00:24:12.100 align:middle line:90% means this MMF source is 0. 00:24:12.100 --> 00:24:14.730 align:middle line:84% And I have this flux source, N1 i1, 00:24:14.730 --> 00:24:16.330 align:middle line:90% and a reluctance of the core. 00:24:16.330 --> 00:24:21.185 align:middle line:84% So what I would really get if i2 is 0-- 00:24:21.185 --> 00:24:27.530 align:middle line:90% 00:24:27.530 --> 00:24:28.770 align:middle line:90% What would I get? 00:24:28.770 --> 00:24:33.170 align:middle line:84% I would get phi in the core would really 00:24:33.170 --> 00:24:42.080 align:middle line:84% be equal to N1 i1 divided by the reluctance of the core, which 00:24:42.080 --> 00:24:47.000 align:middle line:84% would give me lambda 1 would be N1 squared 00:24:47.000 --> 00:24:51.160 align:middle line:84% over the reluctance of the core times i1. 00:24:51.160 --> 00:24:57.830 align:middle line:84% This is just an inductance that we're going to call L mu 1. 00:24:57.830 --> 00:25:01.730 align:middle line:84% It's the magnetizing inductance of the core reference 00:25:01.730 --> 00:25:04.010 align:middle line:90% to the first winding. 00:25:04.010 --> 00:25:06.808 align:middle line:84% So essentially, if I have nothing 00:25:06.808 --> 00:25:08.600 align:middle line:84% connected to the second winding, this thing 00:25:08.600 --> 00:25:11.180 align:middle line:90% just looks like an inductor. 00:25:11.180 --> 00:25:15.530 align:middle line:84% And what I'm going to get is, if I just differentiate this, 00:25:15.530 --> 00:25:18.560 align:middle line:84% I'm going to get something that's V1 is equal to L 00:25:18.560 --> 00:25:22.760 align:middle line:90% mu 1 times di1/dt. 00:25:22.760 --> 00:25:27.890 align:middle line:84% I'm going to get a current into the first winding that 00:25:27.890 --> 00:25:30.470 align:middle line:84% essentially is the current into an inductor. 00:25:30.470 --> 00:25:32.850 align:middle line:84% And why is there a current into the inductor? 00:25:32.850 --> 00:25:36.720 align:middle line:84% Because I have flux that's being generated in the core. 00:25:36.720 --> 00:25:39.800 align:middle line:84% That flux is storing energy in magnetic form, 00:25:39.800 --> 00:25:42.410 align:middle line:84% and that has to come out being represented somehow 00:25:42.410 --> 00:25:44.888 align:middle line:90% electrically. 00:25:44.888 --> 00:25:48.670 align:middle line:84% So what would I do to represent that-- 00:25:48.670 --> 00:25:50.360 align:middle line:90% I have finite flux in the core. 00:25:50.360 --> 00:25:53.230 align:middle line:84% The flux would have been 0 or negligible 00:25:53.230 --> 00:25:57.430 align:middle line:84% if the reluctance of the core was 0, but it's not. 00:25:57.430 --> 00:26:02.110 align:middle line:84% So what I could say then is that maybe I could 00:26:02.110 --> 00:26:04.300 align:middle line:90% model this thing this way. 00:26:04.300 --> 00:26:09.070 align:middle line:84% I could model the transformer as an ideal transformer, N1 to N2. 00:26:09.070 --> 00:26:17.050 align:middle line:90% 00:26:17.050 --> 00:26:21.370 align:middle line:90% Here's V1 and i1. 00:26:21.370 --> 00:26:22.570 align:middle line:90% And here's V2 and i2. 00:26:22.570 --> 00:26:27.550 align:middle line:90% 00:26:27.550 --> 00:26:29.740 align:middle line:84% But maybe I could just say, all right, 00:26:29.740 --> 00:26:34.390 align:middle line:84% let me model this as an extra inductor. 00:26:34.390 --> 00:26:38.290 align:middle line:84% This is my ideal transformer here. 00:26:38.290 --> 00:26:44.450 align:middle line:84% Let me model this behavior as another inductor. 00:26:44.450 --> 00:26:47.720 align:middle line:84% And I'm going to call this inductor L mu 1. 00:26:47.720 --> 00:26:53.150 align:middle line:84% And I'm going to call this current i mu 1. 00:26:53.150 --> 00:26:56.690 align:middle line:84% So if the secondary side is disconnected, 00:26:56.690 --> 00:26:59.600 align:middle line:84% the ideal transformer has no current going into it, 00:26:59.600 --> 00:27:03.120 align:middle line:84% but I still get a magnetizing current. 00:27:03.120 --> 00:27:06.180 align:middle line:84% And the way I think about that magnetizing current is 00:27:06.180 --> 00:27:09.570 align:middle line:84% the energy stored in this so-called magnetizing 00:27:09.570 --> 00:27:12.330 align:middle line:84% inductance, this fictitious inductance that I'm using 00:27:12.330 --> 00:27:14.830 align:middle line:84% to model the behavior of an actual device, 00:27:14.830 --> 00:27:20.010 align:middle line:84% 1/2 half L mu 1 i mu 1 squared models the energy stored 00:27:20.010 --> 00:27:22.710 align:middle line:90% in the core. 00:27:22.710 --> 00:27:27.260 align:middle line:84% So the fact that I have non-zero flux 00:27:27.260 --> 00:27:34.220 align:middle line:84% to generate my induced voltage on the other winding 00:27:34.220 --> 00:27:37.005 align:middle line:84% means that I'm going to end up with some energy stored 00:27:37.005 --> 00:27:37.505 align:middle line:90% in the core. 00:27:37.505 --> 00:27:41.270 align:middle line:84% L mu 1 is not infinite, and I get some finite current here 00:27:41.270 --> 00:27:43.280 align:middle line:90% that's storing energy. 00:27:43.280 --> 00:27:45.100 align:middle line:90% Any questions about that? 00:27:45.100 --> 00:27:48.090 align:middle line:90% 00:27:48.090 --> 00:27:50.970 align:middle line:84% So if I have a real transformer, I often 00:27:50.970 --> 00:27:53.460 align:middle line:84% think of it as an ideal transformer 00:27:53.460 --> 00:27:57.480 align:middle line:84% plus some magnetizing inductance that I can often 00:27:57.480 --> 00:27:59.820 align:middle line:84% think of as a completely separate inductor 00:27:59.820 --> 00:28:03.830 align:middle line:84% to model the behavior of the actual device. 00:28:03.830 --> 00:28:07.310 align:middle line:90% 00:28:07.310 --> 00:28:15.800 align:middle line:84% By the way, I put this on the primary side, in the V1 i1 side, 00:28:15.800 --> 00:28:19.470 align:middle line:84% as opposed to the so-called secondary side, the V2 i2 side. 00:28:19.470 --> 00:28:21.980 align:middle line:84% I could have done it the other way. 00:28:21.980 --> 00:28:26.795 align:middle line:84% I could have modeled the transformer equally like this. 00:28:26.795 --> 00:28:29.882 align:middle line:90% 00:28:29.882 --> 00:28:41.910 align:middle line:90% N1 to N2 and had V1, i1, V2, i2. 00:28:41.910 --> 00:28:45.830 align:middle line:84% And I could have placed my magnetizing element 00:28:45.830 --> 00:28:48.130 align:middle line:90% on this side. 00:28:48.130 --> 00:28:51.070 align:middle line:90% This would have been L mu 2. 00:28:51.070 --> 00:28:52.900 align:middle line:84% The magnetizing inductance referred 00:28:52.900 --> 00:28:54.670 align:middle line:84% to the secondary side, which would 00:28:54.670 --> 00:28:57.790 align:middle line:90% have been equal to N2 squared-- 00:28:57.790 --> 00:29:02.650 align:middle line:84% or N2 over N1 quantity squared L mu 1, 00:29:02.650 --> 00:29:09.330 align:middle line:84% or equal to N2 squared over the reluctance of the core. 00:29:09.330 --> 00:29:12.750 align:middle line:84% So I could place this fictitious component 00:29:12.750 --> 00:29:15.750 align:middle line:84% on either side of the core because whichever 00:29:15.750 --> 00:29:19.270 align:middle line:84% place I place it, it's modeling the same phenomena, 00:29:19.270 --> 00:29:21.315 align:middle line:90% the energy stored in the core-- 00:29:21.315 --> 00:29:23.520 align:middle line:84% the energy associated with the flux 00:29:23.520 --> 00:29:25.970 align:middle line:90% being thrown around the core. 00:29:25.970 --> 00:29:26.895 align:middle line:90% Yeah. 00:29:26.895 --> 00:29:32.770 align:middle line:84% AUDIENCE: So but when you have both sides conducting current, 00:29:32.770 --> 00:29:36.580 align:middle line:84% wouldn't you have a different value in total? 00:29:36.580 --> 00:29:37.790 align:middle line:90% Wouldn't you have to do both? 00:29:37.790 --> 00:29:39.330 align:middle line:90% DAVID PERREAULT: No. 00:29:39.330 --> 00:29:41.580 align:middle line:90% No, because think about it. 00:29:41.580 --> 00:29:45.450 align:middle line:84% Suppose I still have the secondary disconnected. 00:29:45.450 --> 00:29:48.630 align:middle line:84% If I apply a voltage to the primary side, 00:29:48.630 --> 00:29:52.350 align:middle line:84% I get a scaled voltage applied to the secondary. 00:29:52.350 --> 00:29:56.970 align:middle line:84% I get some current, I mu 2, which 00:29:56.970 --> 00:30:02.700 align:middle line:84% is going to be N2 over N1 times V1. 00:30:02.700 --> 00:30:09.933 align:middle line:90% And then the d/dt of that-- 00:30:09.933 --> 00:30:12.600 align:middle line:84% or the integral of that is going to give me this current, sorry. 00:30:12.600 --> 00:30:15.390 align:middle line:90% 00:30:15.390 --> 00:30:17.310 align:middle line:84% But there's nowhere for this current 00:30:17.310 --> 00:30:19.015 align:middle line:84% to flow except out of this dot, which 00:30:19.015 --> 00:30:21.140 align:middle line:84% means then there's a current flowing into this dot. 00:30:21.140 --> 00:30:23.730 align:middle line:90% 00:30:23.730 --> 00:30:26.480 align:middle line:84% And if I look at what this current coming in is, 00:30:26.480 --> 00:30:29.840 align:middle line:84% it looks exactly the same as if I had placed it on this side 00:30:29.840 --> 00:30:31.970 align:middle line:84% and had nothing on the secondary side. 00:30:31.970 --> 00:30:37.220 align:middle line:84% So ultimately, this magnetizing inductance really 00:30:37.220 --> 00:30:40.170 align:middle line:84% just reflects the current mismatch 00:30:40.170 --> 00:30:43.550 align:middle line:84% that's phi core times reluctance of the core. 00:30:43.550 --> 00:30:46.850 align:middle line:84% And I can place it either place, but I 00:30:46.850 --> 00:30:48.890 align:middle line:90% don't need to place it both. 00:30:48.890 --> 00:30:50.810 align:middle line:90% It's one or the other. 00:30:50.810 --> 00:30:53.300 align:middle line:84% If I apply a voltage here, I'll get a certain current 00:30:53.300 --> 00:30:55.500 align:middle line:90% into L mu 1. 00:30:55.500 --> 00:30:57.180 align:middle line:90% And if I scale-- 00:30:57.180 --> 00:31:00.740 align:middle line:84% instead I put it on the other side, and I scale it right, 00:31:00.740 --> 00:31:03.230 align:middle line:84% it's just coming out of the ideal transformer on this dot 00:31:03.230 --> 00:31:05.660 align:middle line:84% and into the ideal transformer on that dot. 00:31:05.660 --> 00:31:09.295 align:middle line:84% AUDIENCE: So does that mean that the inductance 00:31:09.295 --> 00:31:10.670 align:middle line:84% of the transformer doesn't depend 00:31:10.670 --> 00:31:13.031 align:middle line:84% on the magnitude of the currents? 00:31:13.031 --> 00:31:14.830 align:middle line:84% DAVID PERREAULT: The magnetizing inductance 00:31:14.830 --> 00:31:16.330 align:middle line:84% has nothing to do with the magnitude 00:31:16.330 --> 00:31:18.610 align:middle line:90% of the currents, that's right. 00:31:18.610 --> 00:31:20.320 align:middle line:90% And the same way-- 00:31:20.320 --> 00:31:23.050 align:middle line:84% I mean, ideally speaking-- the same way 00:31:23.050 --> 00:31:25.810 align:middle line:84% in the inductor, the inductance value I get 00:31:25.810 --> 00:31:27.170 align:middle line:90% doesn't relate to the currents. 00:31:27.170 --> 00:31:30.810 align:middle line:84% It's just the scaling between flux linkage and current. 00:31:30.810 --> 00:31:33.130 align:middle line:84% Yeah, so it doesn't depend on the currents. 00:31:33.130 --> 00:31:35.730 align:middle line:84% And if I had some other load out here, 00:31:35.730 --> 00:31:38.740 align:middle line:84% I could just keep modeling everything I'm doing. 00:31:38.740 --> 00:31:44.910 align:middle line:84% I just need one extra element to model that parasitic behavior. 00:31:44.910 --> 00:31:45.750 align:middle line:90% Great question. 00:31:45.750 --> 00:31:46.810 align:middle line:90% Any other questions? 00:31:46.810 --> 00:31:50.810 align:middle line:90% 00:31:50.810 --> 00:31:56.720 align:middle line:84% OK, so that's one practical nonideality. 00:31:56.720 --> 00:31:59.270 align:middle line:84% There's a second practical nonideality 00:31:59.270 --> 00:32:02.650 align:middle line:84% that we also want to think about very often. 00:32:02.650 --> 00:32:03.870 align:middle line:90% And what is that? 00:32:03.870 --> 00:32:14.030 align:middle line:84% Well, if I had, again, an infinitely permeable core 00:32:14.030 --> 00:32:18.710 align:middle line:84% or a permeability that was so much higher than the surrounding 00:32:18.710 --> 00:32:21.410 align:middle line:84% universe that I could ignore any flux that 00:32:21.410 --> 00:32:25.160 align:middle line:84% goes anywhere except in the core, that would be great. 00:32:25.160 --> 00:32:30.180 align:middle line:84% But the problem is, as we said, permeabilities are 1,000 00:32:30.180 --> 00:32:32.750 align:middle line:84% or 10,000 mu 0 or something like that. 00:32:32.750 --> 00:32:34.720 align:middle line:84% Magnetic circuits are kind of leaky. 00:32:34.720 --> 00:32:37.740 align:middle line:84% The flux pipes are very leaky pipes. 00:32:37.740 --> 00:32:44.550 align:middle line:84% What that means is if I applied some Ni over here, 00:32:44.550 --> 00:32:47.480 align:middle line:84% I will indeed get some flux going around the core, 00:32:47.480 --> 00:32:50.690 align:middle line:84% but I will get an additional component of flux 00:32:50.690 --> 00:32:53.720 align:middle line:90% that might link winding 1. 00:32:53.720 --> 00:32:55.730 align:middle line:90% But you know what? 00:32:55.730 --> 00:33:00.570 align:middle line:84% It can come out of the core and link back this way. 00:33:00.570 --> 00:33:03.990 align:middle line:84% So I'm going to get some leakage flux I'm 00:33:03.990 --> 00:33:06.510 align:middle line:84% going to call phi leakage 1, which 00:33:06.510 --> 00:33:12.120 align:middle line:84% links the primary turn, primary winding but eventually 00:33:12.120 --> 00:33:15.540 align:middle line:84% either here or here or somewhere else jumps out of the core 00:33:15.540 --> 00:33:20.250 align:middle line:84% and doesn't make it over to the second winding. 00:33:20.250 --> 00:33:23.550 align:middle line:84% Likewise, you could imagine that I 00:33:23.550 --> 00:33:27.360 align:middle line:84% could have some flux that would link the second winding 00:33:27.360 --> 00:33:30.660 align:middle line:90% but then jump out of the core-- 00:33:30.660 --> 00:33:33.300 align:middle line:84% and I'll call that phi leakage 2-- 00:33:33.300 --> 00:33:35.360 align:middle line:84% and never links the primary winding. 00:33:35.360 --> 00:33:38.935 align:middle line:90% 00:33:38.935 --> 00:33:45.490 align:middle line:84% And this parasitic leakage flux from one winding to the other 00:33:45.490 --> 00:33:50.480 align:middle line:84% imposes additional nonidealities on the behavior of the system. 00:33:50.480 --> 00:33:56.980 align:middle line:84% And it becomes particularly worse in a percentage basis 00:33:56.980 --> 00:33:58.970 align:middle line:84% if the core permeability is high. 00:33:58.970 --> 00:34:02.830 align:middle line:84% But appreciate, even if I made the core permeability infinite, 00:34:02.830 --> 00:34:05.510 align:middle line:84% there is still some flux that's going to jump out. 00:34:05.510 --> 00:34:08.820 align:middle line:84% It just may not be a big fraction of it. 00:34:08.820 --> 00:34:10.614 align:middle line:84% Does that make sense to everybody? 00:34:10.614 --> 00:34:13.230 align:middle line:90% 00:34:13.230 --> 00:34:17.540 align:middle line:84% So the question is, how would I model that kind of behavior? 00:34:17.540 --> 00:34:21.449 align:middle line:90% 00:34:21.449 --> 00:34:23.860 align:middle line:84% Ah, something I forgot to tell you. 00:34:23.860 --> 00:34:27.600 align:middle line:84% So we're going to model this behavior in one second. 00:34:27.600 --> 00:34:31.870 align:middle line:84% There's something I forgot to tell you about the magnetizing-- 00:34:31.870 --> 00:34:34.600 align:middle line:84% inductance of the magnetizing flux, which I better 00:34:34.600 --> 00:34:35.699 align:middle line:90% not forget to tell you. 00:34:35.699 --> 00:34:43.870 align:middle line:90% 00:34:43.870 --> 00:34:45.340 align:middle line:90% One of the things-- 00:34:45.340 --> 00:34:50.889 align:middle line:84% one of the aspects of this magnetizing behavior 00:34:50.889 --> 00:34:54.794 align:middle line:84% is that it's a reason you can't use transformers at DC. 00:34:54.794 --> 00:34:58.040 align:middle line:90% 00:34:58.040 --> 00:35:02.960 align:middle line:84% So we said we want to convert voltage. 00:35:02.960 --> 00:35:06.900 align:middle line:84% Why did Westinghouse and Tesla win over Edison? 00:35:06.900 --> 00:35:10.430 align:middle line:84% It's because Edison didn't have a great way to convert voltage 00:35:10.430 --> 00:35:11.630 align:middle line:90% from one level to another. 00:35:11.630 --> 00:35:13.750 align:middle line:84% He couldn't step down voltage very easily. 00:35:13.750 --> 00:35:16.370 align:middle line:90% 00:35:16.370 --> 00:35:18.660 align:middle line:84% You can do it with AC, but it's hard to do with DC. 00:35:18.660 --> 00:35:19.170 align:middle line:90% Why is that? 00:35:19.170 --> 00:35:22.340 align:middle line:84% What would happen if I applied a constant voltage 00:35:22.340 --> 00:35:24.660 align:middle line:84% across this primary of this transformer? 00:35:24.660 --> 00:35:26.016 align:middle line:90% AUDIENCE: It would short? 00:35:26.016 --> 00:35:28.060 align:middle line:84% DAVID PERREAULT: Well, what would happen 00:35:28.060 --> 00:35:32.110 align:middle line:84% is I'd have a continuously increasing magnetizing current 00:35:32.110 --> 00:35:33.850 align:middle line:90% forever. 00:35:33.850 --> 00:35:35.360 align:middle line:90% V is equal to L di/dt. 00:35:35.360 --> 00:35:37.840 align:middle line:84% That would mean if I have a constant voltage here, 00:35:37.840 --> 00:35:41.340 align:middle line:84% I'd have a constant di/dt in this inductor. 00:35:41.340 --> 00:35:49.090 align:middle line:84% Or looked at another way, V is equal to L mu 1 di/dt. 00:35:49.090 --> 00:35:57.300 align:middle line:84% That means i1 is equal to 1 over L mu, the integral of Vdt. 00:35:57.300 --> 00:36:03.330 align:middle line:90% 00:36:03.330 --> 00:36:07.860 align:middle line:84% Keep in mind that the flux linkage-- 00:36:07.860 --> 00:36:09.360 align:middle line:84% another way to look at it is, what's 00:36:09.360 --> 00:36:11.260 align:middle line:90% the flux linkage in the core? 00:36:11.260 --> 00:36:13.650 align:middle line:90% Lambda is equal to Li. 00:36:13.650 --> 00:36:21.840 align:middle line:84% But lambda is equal to N phi, or say N1 phi-- lambda 00:36:21.840 --> 00:36:26.700 align:middle line:84% 1 is equal to N1 phi, which is equal to N1, 00:36:26.700 --> 00:36:31.110 align:middle line:84% the area of the core times the flux density in the core. 00:36:31.110 --> 00:36:37.080 align:middle line:84% And this is equal to the integral of V1dt. 00:36:37.080 --> 00:36:39.930 align:middle line:90% That's what lambda 1 is. 00:36:39.930 --> 00:36:46.650 align:middle line:84% This means that if I have a limit, a saturation flux density 00:36:46.650 --> 00:36:49.590 align:middle line:84% limit on my core, B core has to be less than B sat 00:36:49.590 --> 00:36:54.990 align:middle line:84% or it saturates, and as somebody said, shorts out, what 00:36:54.990 --> 00:37:02.100 align:middle line:84% I need is the integral of V1dt has to be less than 00:37:02.100 --> 00:37:09.300 align:middle line:84% or equal to N1 B core times Ac, area of the core. 00:37:09.300 --> 00:37:14.340 align:middle line:84% And what this means is if I have a sinusoidal voltage, 00:37:14.340 --> 00:37:16.203 align:middle line:84% that means this integral is going to go up, 00:37:16.203 --> 00:37:18.370 align:middle line:84% and then it's going to go down and then go negative. 00:37:18.370 --> 00:37:21.660 align:middle line:84% There's a limit of the volt seconds 00:37:21.660 --> 00:37:26.940 align:middle line:84% I can apply to the core that's limited by this saturation flux 00:37:26.940 --> 00:37:27.440 align:middle line:90% density. 00:37:27.440 --> 00:37:30.390 align:middle line:90% 00:37:30.390 --> 00:37:34.380 align:middle line:84% So one important consequence of this parasitic-- 00:37:34.380 --> 00:37:36.450 align:middle line:84% and I apologize for not mentioning it earlier-- 00:37:36.450 --> 00:37:39.840 align:middle line:84% is the fact that there's a relationship 00:37:39.840 --> 00:37:43.770 align:middle line:84% between the number of turns and the saturation flux density 00:37:43.770 --> 00:37:46.650 align:middle line:84% and the cross-sectional area and the maximum volt seconds 00:37:46.650 --> 00:37:52.320 align:middle line:84% you can apply on the winding before you get it out 00:37:52.320 --> 00:37:53.700 align:middle line:90% of the saturation range. 00:37:53.700 --> 00:37:55.770 align:middle line:84% Once you get it out of the saturation range, 00:37:55.770 --> 00:37:58.460 align:middle line:90% the permeability goes away. 00:37:58.460 --> 00:38:00.730 align:middle line:84% And then more or less the transformer shorts out, 00:38:00.730 --> 00:38:03.585 align:middle line:84% and you get huge magnetizing currents, and bad stuff happens. 00:38:03.585 --> 00:38:06.115 align:middle line:90% 00:38:06.115 --> 00:38:10.306 align:middle line:84% So that's the other consequence of this particular parasitic. 00:38:10.306 --> 00:38:13.960 align:middle line:90% 00:38:13.960 --> 00:38:19.960 align:middle line:84% The second parasitic-- first of all, any questions about that? 00:38:19.960 --> 00:38:20.460 align:middle line:90% Yeah. 00:38:20.460 --> 00:38:22.793 align:middle line:84% AUDIENCE: Are there any applications where you could use 00:38:22.793 --> 00:38:25.650 align:middle line:84% this like a switch supply, where you're applying a DC voltage 00:38:25.650 --> 00:38:28.470 align:middle line:84% for a little bit and then you switch it off and so that way 00:38:28.470 --> 00:38:29.580 align:middle line:90% you're-- 00:38:29.580 --> 00:38:32.820 align:middle line:84% like, the volt seconds that you have the transformer on 00:38:32.820 --> 00:38:35.310 align:middle line:84% is few enough that you don't reach that saturation field 00:38:35.310 --> 00:38:37.840 align:middle line:84% but you're still using it for your DC voltage? 00:38:37.840 --> 00:38:41.118 align:middle line:90% Is there any use for that? 00:38:41.118 --> 00:38:42.720 align:middle line:90% DAVID PERREAULT: Sort of. 00:38:42.720 --> 00:38:46.590 align:middle line:84% The notion of applying a voltage for a little while 00:38:46.590 --> 00:38:50.140 align:middle line:84% and storing some energy in this magnetizing inductance, 00:38:50.140 --> 00:38:52.080 align:middle line:84% which is sort of what you just said, 00:38:52.080 --> 00:38:55.470 align:middle line:84% is exactly what you do in certain kinds of converters. 00:38:55.470 --> 00:38:57.630 align:middle line:84% So in some kinds of converters, we will actually 00:38:57.630 --> 00:39:00.090 align:middle line:84% use this parasitic component to our advantage, 00:39:00.090 --> 00:39:02.750 align:middle line:84% and we'll talk about how you design for that. 00:39:02.750 --> 00:39:06.560 align:middle line:90% But that's a limited time. 00:39:06.560 --> 00:39:11.820 align:middle line:84% The integral of V1dt is limited by the volt seconds. 00:39:11.820 --> 00:39:14.150 align:middle line:84% You still can't apply it forever. 00:39:14.150 --> 00:39:15.690 align:middle line:84% And yes, we will use it that way. 00:39:15.690 --> 00:39:19.250 align:middle line:84% So sometimes parasitic is a bad thing. 00:39:19.250 --> 00:39:20.490 align:middle line:90% Sometimes it's a good thing. 00:39:20.490 --> 00:39:22.550 align:middle line:84% And we'll see using it both ways. 00:39:22.550 --> 00:39:24.240 align:middle line:90% Great question. 00:39:24.240 --> 00:39:25.390 align:middle line:90% Any other questions? 00:39:25.390 --> 00:39:28.220 align:middle line:90% 00:39:28.220 --> 00:39:30.410 align:middle line:84% OK, so I apologize for forgetting 00:39:30.410 --> 00:39:33.080 align:middle line:84% to mention the other impact of this parasitic. 00:39:33.080 --> 00:39:42.360 align:middle line:84% Let's come back to the fact that in addition to this parasitic, 00:39:42.360 --> 00:39:46.430 align:middle line:84% I have some flux that links winding 00:39:46.430 --> 00:39:48.650 align:middle line:84% 1 that doesn't link winding 2 and some flux that 00:39:48.650 --> 00:39:51.920 align:middle line:84% may link winding 2 that doesn't link winding 1. 00:39:51.920 --> 00:39:53.590 align:middle line:90% Well, how would I model that? 00:39:53.590 --> 00:39:56.600 align:middle line:90% 00:39:56.600 --> 00:39:59.480 align:middle line:84% Well, what does the magnetic circuit for that look 00:39:59.480 --> 00:40:01.330 align:middle line:90% like once I add things in? 00:40:01.330 --> 00:40:05.634 align:middle line:90% Well, I still have N1 i1. 00:40:05.634 --> 00:40:08.780 align:middle line:90% 00:40:08.780 --> 00:40:13.340 align:middle line:84% And I still have my reluctance of my core. 00:40:13.340 --> 00:40:14.896 align:middle line:90% And I have N2 i2. 00:40:14.896 --> 00:40:17.990 align:middle line:90% 00:40:17.990 --> 00:40:22.160 align:middle line:84% But then what I'm doing when I'm saying, 00:40:22.160 --> 00:40:24.890 align:middle line:84% oh, there's some flux that's going to come up here and jump 00:40:24.890 --> 00:40:27.110 align:middle line:84% out of the core, I'm sort of saying 00:40:27.110 --> 00:40:30.830 align:middle line:84% there's some path for flux to flow 00:40:30.830 --> 00:40:33.652 align:middle line:84% that doesn't go through the other winding. 00:40:33.652 --> 00:40:35.535 align:middle line:84% In what I've drawn so far, anything 00:40:35.535 --> 00:40:38.670 align:middle line:84% that goes through my circuit must go around the loop here. 00:40:38.670 --> 00:40:46.040 align:middle line:84% And I have phi core I'm imagining doing this. 00:40:46.040 --> 00:40:49.280 align:middle line:84% What I'm saying now is, oh, well, I 00:40:49.280 --> 00:40:56.060 align:middle line:84% can have some flux that comes up here through this MMF source 00:40:56.060 --> 00:40:58.580 align:middle line:84% but returns through some other path 00:40:58.580 --> 00:41:01.700 align:middle line:84% that I'm going to call phi leakage 1. 00:41:01.700 --> 00:41:05.210 align:middle line:84% And I'm going to have a second flux 00:41:05.210 --> 00:41:12.860 align:middle line:84% path that I'll call reluctance leakage 2 00:41:12.860 --> 00:41:15.080 align:middle line:90% that's going to go through-- 00:41:15.080 --> 00:41:17.910 align:middle line:84% flux may come down here and return up here. 00:41:17.910 --> 00:41:21.020 align:middle line:84% And I'm going to call that phi leakage 2. 00:41:21.020 --> 00:41:23.900 align:middle line:84% So this is my extended magnetic circuit model, 00:41:23.900 --> 00:41:25.400 align:middle line:84% accounting for the fact that there's 00:41:25.400 --> 00:41:28.490 align:middle line:84% other paths for flux to flow and not link 00:41:28.490 --> 00:41:31.590 align:middle line:90% the two windings together. 00:41:31.590 --> 00:41:32.735 align:middle line:90% Any questions about that? 00:41:32.735 --> 00:41:36.800 align:middle line:90% 00:41:36.800 --> 00:41:43.260 align:middle line:84% So if I add this on, what would I get? 00:41:43.260 --> 00:41:46.280 align:middle line:84% Well, that's the magnetic circuit model. 00:41:46.280 --> 00:41:49.490 align:middle line:90% What is phi leakage 1? 00:41:49.490 --> 00:41:55.400 align:middle line:84% Phi leakage 1 is simply equal to N1 i1 00:41:55.400 --> 00:41:57.510 align:middle line:84% over whatever this reluctance is. 00:41:57.510 --> 00:41:59.375 align:middle line:84% So I'll call this reluctance leakage 1. 00:41:59.375 --> 00:42:02.270 align:middle line:90% 00:42:02.270 --> 00:42:12.080 align:middle line:84% Phi leakage 2 is equal to N2 i2 over reluctance leakage 2. 00:42:12.080 --> 00:42:15.740 align:middle line:84% And by the way, sometimes since these have big air path 00:42:15.740 --> 00:42:18.380 align:middle line:84% components, sometimes it's tricky to figure out 00:42:18.380 --> 00:42:21.770 align:middle line:84% what the values of leakage reluctance 1 00:42:21.770 --> 00:42:23.072 align:middle line:90% and leakage reluctance 2 are. 00:42:23.072 --> 00:42:24.530 align:middle line:84% But nonetheless, there's some value 00:42:24.530 --> 00:42:26.900 align:middle line:84% that you can figure out one way or another. 00:42:26.900 --> 00:42:32.300 align:middle line:84% And then we have phi core, which is unchanged. 00:42:32.300 --> 00:42:41.790 align:middle line:84% It's N1 i2 plus N2 i2 divided by the reluctance of the core. 00:42:41.790 --> 00:42:44.620 align:middle line:84% So that's my magnetic circuit model for the whole thing. 00:42:44.620 --> 00:42:45.815 align:middle line:90% Any questions about that? 00:42:45.815 --> 00:42:49.320 align:middle line:90% 00:42:49.320 --> 00:42:50.920 align:middle line:90% So what do I get? 00:42:50.920 --> 00:42:55.607 align:middle line:84% Well, I can basically figure out what the-- 00:42:55.607 --> 00:42:59.250 align:middle line:90% 00:42:59.250 --> 00:43:07.260 align:middle line:84% I can basically figure out what the net flux 00:43:07.260 --> 00:43:08.740 align:middle line:90% linkage on winding 1 is. 00:43:08.740 --> 00:43:11.190 align:middle line:90% What's lambda 1? 00:43:11.190 --> 00:43:16.860 align:middle line:84% Lambda 1 is equal to N1 times the total flux coming out 00:43:16.860 --> 00:43:23.010 align:middle line:84% of the positive terminal of the MMF source, N1 i1. 00:43:23.010 --> 00:43:28.590 align:middle line:84% So that should be N1 times phi leakage 1 plus phi 00:43:28.590 --> 00:43:33.990 align:middle line:84% core, which would be equal to N1 squared 00:43:33.990 --> 00:43:38.640 align:middle line:84% over reluctance leakage 1 times i1 00:43:38.640 --> 00:43:44.490 align:middle line:84% plus N1 times phi core, which is N1 squared 00:43:44.490 --> 00:43:50.490 align:middle line:84% over the reluctance of the core i1 plus N1 N2 00:43:50.490 --> 00:43:55.410 align:middle line:84% over the reluctance of the core, i2. 00:43:55.410 --> 00:44:03.210 align:middle line:84% Lambda 2, which is the flux linking the second MMF source, 00:44:03.210 --> 00:44:10.030 align:middle line:84% is equal to N2 times phi leakage 2 plus phi core. 00:44:10.030 --> 00:44:12.860 align:middle line:84% It's the same story all over again. 00:44:12.860 --> 00:44:15.760 align:middle line:84% I'll just substitute in my values here. 00:44:15.760 --> 00:44:23.140 align:middle line:84% And what I get is this is going to be equal to N2 squared 00:44:23.140 --> 00:44:28.870 align:middle line:84% over reluctance leakage 2 plus N2 times 00:44:28.870 --> 00:44:34.360 align:middle line:84% phi core, which is N2 squared over the reluctance of the core. 00:44:34.360 --> 00:44:35.530 align:middle line:90% This is times i2. 00:44:35.530 --> 00:44:37.480 align:middle line:90% This is times i2. 00:44:37.480 --> 00:44:44.390 align:middle line:84% Plus N1 N2 over the reluctance of the core, i2. 00:44:44.390 --> 00:44:49.570 align:middle line:84% So that's two equations expressing the two flux 00:44:49.570 --> 00:44:52.630 align:middle line:90% linkages in terms of i1 and i2. 00:44:52.630 --> 00:44:55.210 align:middle line:90% I can put that in a matrix form. 00:44:55.210 --> 00:44:58.810 align:middle line:84% And that would just be lambda 1, lambda 2, 00:44:58.810 --> 00:45:03.160 align:middle line:84% which is equal to essentially the integral of V1 00:45:03.160 --> 00:45:06.800 align:middle line:90% and the integral of V2 dt. 00:45:06.800 --> 00:45:09.430 align:middle line:90% 00:45:09.430 --> 00:45:16.810 align:middle line:84% And that would be equal to N1 squared over reluctance leakage 00:45:16.810 --> 00:45:25.390 align:middle line:84% 1 plus N1 squared over the reluctance of the core plus N1 00:45:25.390 --> 00:45:31.900 align:middle line:84% N2 over the reluctance of the core times i1 i2, 00:45:31.900 --> 00:45:33.490 align:middle line:90% so matrix equation. 00:45:33.490 --> 00:45:34.580 align:middle line:90% I get the same thing. 00:45:34.580 --> 00:45:39.790 align:middle line:84% I get N1 N2 over the reluctance of the core times N2 squared 00:45:39.790 --> 00:45:43.990 align:middle line:84% over the reluctance of the core plus N2 squared over reluctance 00:45:43.990 --> 00:45:45.250 align:middle line:90% leakage 2. 00:45:45.250 --> 00:45:48.390 align:middle line:84% That's the whole matrix equation. 00:45:48.390 --> 00:45:52.560 align:middle line:84% So instead of the integral of-- or instead of V1 00:45:52.560 --> 00:45:54.720 align:middle line:84% is equal to something times di1/dt, 00:45:54.720 --> 00:45:59.130 align:middle line:84% I have a matrix equation between the voltage 00:45:59.130 --> 00:46:01.230 align:middle line:84% on the-- the current into the two windings 00:46:01.230 --> 00:46:03.360 align:middle line:84% and the voltages on the two windings. 00:46:03.360 --> 00:46:07.980 align:middle line:84% If I differentiated this equation to get V1 and V2, 00:46:07.980 --> 00:46:19.050 align:middle line:84% I could write this as V1 V2 is equal to L11, LM, LM, 00:46:19.050 --> 00:46:28.430 align:middle line:90% and L22 times i1, i2, dot. 00:46:28.430 --> 00:46:30.180 align:middle line:84% Or maybe I should just get rid of the dot, 00:46:30.180 --> 00:46:35.070 align:middle line:84% and I'll just write this as lambda 1 and lambda 2. 00:46:35.070 --> 00:46:40.340 align:middle line:84% So there's now a matrix equation relating flux linkages 00:46:40.340 --> 00:46:42.260 align:middle line:90% and currents. 00:46:42.260 --> 00:46:43.450 align:middle line:90% Any questions about that? 00:46:43.450 --> 00:46:50.430 align:middle line:90% 00:46:50.430 --> 00:46:52.300 align:middle line:90% What do I want to come back to? 00:46:52.300 --> 00:46:54.630 align:middle line:84% Let me come back to a equivalent circuit 00:46:54.630 --> 00:46:57.540 align:middle line:90% model associated with this. 00:46:57.540 --> 00:46:59.760 align:middle line:84% I gave you an equivalent circuit model. 00:46:59.760 --> 00:47:02.010 align:middle line:90% What happens to my-- 00:47:02.010 --> 00:47:04.890 align:middle line:90% 00:47:04.890 --> 00:47:08.050 align:middle line:84% what happens to my voltage and current relationships? 00:47:08.050 --> 00:47:10.080 align:middle line:90% And here's what you can do. 00:47:10.080 --> 00:47:13.260 align:middle line:84% You can take this model or this model 00:47:13.260 --> 00:47:16.230 align:middle line:84% here and come up with an equivalent circuit that 00:47:16.230 --> 00:47:19.060 align:middle line:84% has the same relationship between voltages and currents. 00:47:19.060 --> 00:47:20.610 align:middle line:90% And it looks like this. 00:47:20.610 --> 00:47:29.470 align:middle line:84% I have an ideal transformer N1 to N2. 00:47:29.470 --> 00:47:32.530 align:middle line:84% And then I have a magnetizing inductance. 00:47:32.530 --> 00:47:39.645 align:middle line:84% L mu 1 is equal to N1 squared over reluctance of the core. 00:47:39.645 --> 00:47:42.330 align:middle line:90% 00:47:42.330 --> 00:47:49.750 align:middle line:84% Then I have a leakage inductance that we're 00:47:49.750 --> 00:47:54.370 align:middle line:84% going to call L leakage 1, which is equal to N1 squared 00:47:54.370 --> 00:47:57.310 align:middle line:90% over reluctance leakage 1. 00:47:57.310 --> 00:48:02.610 align:middle line:84% And then I'm going to have a second parasitic inductance, L 00:48:02.610 --> 00:48:07.200 align:middle line:84% leakage 2, which is equal to N2 squared over reluctance leakage 00:48:07.200 --> 00:48:09.300 align:middle line:90% 2. 00:48:09.300 --> 00:48:17.800 align:middle line:84% And I put these together, and then I have V2 i2 and V1 i1. 00:48:17.800 --> 00:48:22.490 align:middle line:90% 00:48:22.490 --> 00:48:26.720 align:middle line:84% So all that happens, if I include the fact that 00:48:26.720 --> 00:48:30.650 align:middle line:84% is in this drawing there's some additional leakage flux 00:48:30.650 --> 00:48:35.390 align:middle line:84% paths that go out of the area that I ideally would have, 00:48:35.390 --> 00:48:41.980 align:middle line:84% is that I get two independent inductors here that get 00:48:41.980 --> 00:48:44.860 align:middle line:84% added on to my full transformer model. 00:48:44.860 --> 00:48:48.280 align:middle line:84% The energy stored in this kind of purple inductor 00:48:48.280 --> 00:48:51.850 align:middle line:84% here relates to the magnetic energy 00:48:51.850 --> 00:48:56.380 align:middle line:84% that's stored in the field associated with this flux path. 00:48:56.380 --> 00:48:59.950 align:middle line:84% Likewise, the energy stored in this inductance 00:48:59.950 --> 00:49:02.950 align:middle line:84% is exactly the energy stored in the flux that links 00:49:02.950 --> 00:49:05.590 align:middle line:90% winding 2 but not winding 1. 00:49:05.590 --> 00:49:09.610 align:middle line:90% And so what this does is-- 00:49:09.610 --> 00:49:14.080 align:middle line:84% whereas the magnetizing inductance created a mismatch 00:49:14.080 --> 00:49:16.450 align:middle line:84% in the current relationships between the primary and 00:49:16.450 --> 00:49:18.070 align:middle line:84% secondary currents deviating from 00:49:18.070 --> 00:49:22.060 align:middle line:84% my ideal current relationships, the leakage inductances, 00:49:22.060 --> 00:49:26.560 align:middle line:84% so-called, that I would add on mess up the ideal voltage 00:49:26.560 --> 00:49:27.700 align:middle line:90% transformation. 00:49:27.700 --> 00:49:33.790 align:middle line:84% Because the voltage, if I call this V1 prime and V2 prime, 00:49:33.790 --> 00:49:36.460 align:middle line:84% these follow the ideal voltage relations. 00:49:36.460 --> 00:49:38.980 align:middle line:84% But the voltage, the actual voltages of the transformer 00:49:38.980 --> 00:49:42.130 align:middle line:90% terminals do not. 00:49:42.130 --> 00:49:47.120 align:middle line:84% So this is what we would consider a transformer 00:49:47.120 --> 00:49:51.170 align:middle line:84% model that has an ideal transformer at its heart 00:49:51.170 --> 00:49:58.760 align:middle line:84% but then incorporates three additional inductances to model 00:49:58.760 --> 00:50:01.700 align:middle line:84% the energy stored in the core and the energy storage 00:50:01.700 --> 00:50:05.740 align:middle line:84% and the leakage fields around the transformer. 00:50:05.740 --> 00:50:06.950 align:middle line:90% So I apologize. 00:50:06.950 --> 00:50:08.710 align:middle line:90% I've run over there. 00:50:08.710 --> 00:50:12.070 align:middle line:84% I'll take any last quick question before we wrap up, 00:50:12.070 --> 00:50:16.000 align:middle line:84% or I'll happily answer questions after class. 00:50:16.000 --> 00:50:16.970 align:middle line:90% OK, have a great day. 00:50:16.970 --> 00:50:19.800 align:middle line:90% We'll pick this up on Wednesday. 00:50:19.800 --> 00:50:25.000 align:middle line:90%