WEBVTT 00:00:00.000 --> 00:00:02.892 align:middle line:90% [SQUEAKING] 00:00:02.892 --> 00:00:04.338 align:middle line:90% [RUSTLING] 00:00:04.338 --> 00:00:07.230 align:middle line:90% [CLICKING] 00:00:07.230 --> 00:00:09.170 align:middle line:90% 00:00:09.170 --> 00:00:12.080 align:middle line:84% DAVID PERREAULT: OK, why don't we get started. 00:00:12.080 --> 00:00:16.970 align:middle line:84% So I'm going to start today by recapping our discussion 00:00:16.970 --> 00:00:18.470 align:middle line:90% of magnetics and transformers. 00:00:18.470 --> 00:00:19.947 align:middle line:90% And we're going to expand on it. 00:00:19.947 --> 00:00:21.530 align:middle line:84% I'll just remind you that the sections 00:00:21.530 --> 00:00:23.360 align:middle line:84% of the textbook, Principles of Power 00:00:23.360 --> 00:00:27.780 align:middle line:84% Electronics, second edition that this is in, is chapters 18. 00:00:27.780 --> 00:00:31.070 align:middle line:84% And then some of the material I'm going to talk about today is 00:00:31.070 --> 00:00:35.180 align:middle line:90% in chapters 19 through 19.4.1. 00:00:35.180 --> 00:00:36.145 align:middle line:90% So that's one thing. 00:00:36.145 --> 00:00:38.270 align:middle line:84% The other thing I'd like to remind you is now we're 00:00:38.270 --> 00:00:43.790 align:middle line:84% going to have assessments, which are effectively short little 00:00:43.790 --> 00:00:45.890 align:middle line:90% mini-quizzes. 00:00:45.890 --> 00:00:47.970 align:middle line:84% Just to remind you of the guidelines, 00:00:47.970 --> 00:00:50.940 align:middle line:84% you're not allowed to discuss them with anybody else. 00:00:50.940 --> 00:00:54.710 align:middle line:84% And the only assistance that the course staff can give you 00:00:54.710 --> 00:00:58.490 align:middle line:84% is really to answer the same kinds of questions 00:00:58.490 --> 00:01:01.390 align:middle line:90% that we would answer in an exam. 00:01:01.390 --> 00:01:03.950 align:middle line:84% And for the most part, feel free to direct 00:01:03.950 --> 00:01:07.280 align:middle line:84% any questions, clarifications just to me directly, preferably 00:01:07.280 --> 00:01:09.020 align:middle line:90% by email instead of Piazza. 00:01:09.020 --> 00:01:11.780 align:middle line:90% But I'll try to respond to both. 00:01:11.780 --> 00:01:13.590 align:middle line:90% But that's just to clarify that. 00:01:13.590 --> 00:01:16.910 align:middle line:84% So unlike homeworks, where you guys are welcome to collaborate, 00:01:16.910 --> 00:01:18.650 align:middle line:84% ask the TAs or teaching staff anything 00:01:18.650 --> 00:01:24.440 align:middle line:84% you want, please just mainly direct those questions to me. 00:01:24.440 --> 00:01:27.020 align:middle line:84% Although, the other staff can answer as well. 00:01:27.020 --> 00:01:29.365 align:middle line:84% So first of all, any questions before we get going? 00:01:29.365 --> 00:01:32.155 align:middle line:90% 00:01:32.155 --> 00:01:35.440 align:middle line:84% AUDIENCE: Is that going to be posted on Canvas after class? 00:01:35.440 --> 00:01:38.425 align:middle line:84% DAVID PERREAULT: Yes, it'll come up on Canvas or Gradescope, 00:01:38.425 --> 00:01:38.925 align:middle line:90% I guess. 00:01:38.925 --> 00:01:44.050 align:middle line:90% 00:01:44.050 --> 00:01:46.410 align:middle line:84% An by the way, these are not designed-- 00:01:46.410 --> 00:01:50.805 align:middle line:84% if you're spending 12 hours solving this problem, 00:01:50.805 --> 00:01:52.180 align:middle line:84% it's not intended to be that way. 00:01:52.180 --> 00:01:55.800 align:middle line:84% They're supposed to be short, relatively short kinds of things 00:01:55.800 --> 00:01:58.030 align:middle line:84% to check your understanding on the topic. 00:01:58.030 --> 00:01:59.640 align:middle line:84% And usually, the assessment number 00:01:59.640 --> 00:02:02.170 align:middle line:84% is of aligned with the material of the homework number. 00:02:02.170 --> 00:02:05.750 align:middle line:84% So assessment 1 covers the material in homework 1. 00:02:05.750 --> 00:02:08.320 align:middle line:90% 00:02:08.320 --> 00:02:11.770 align:middle line:84% So let's continue talking about transformers 00:02:11.770 --> 00:02:14.120 align:middle line:84% or multi-winding magnetic components. 00:02:14.120 --> 00:02:18.220 align:middle line:84% We said last time that if I took some core with a permeability 00:02:18.220 --> 00:02:20.530 align:middle line:84% and a core cross-section, a length of core, 00:02:20.530 --> 00:02:24.010 align:middle line:84% and I put two windings on it, one with N1 turns and another 00:02:24.010 --> 00:02:27.820 align:middle line:84% one with N2 turns, I would mainly get-- 00:02:27.820 --> 00:02:30.460 align:middle line:84% especially if this is a very high permeability core, 00:02:30.460 --> 00:02:34.150 align:middle line:84% I would get flux and flux density traveling around 00:02:34.150 --> 00:02:36.310 align:middle line:90% inside the core. 00:02:36.310 --> 00:02:41.830 align:middle line:84% But I might also have some flux that links winding 1, 00:02:41.830 --> 00:02:45.280 align:middle line:84% but doesn't make it over to winding 2, and links winding 2, 00:02:45.280 --> 00:02:47.500 align:middle line:84% but doesn't make it over to winding 1. 00:02:47.500 --> 00:02:51.130 align:middle line:84% Now, when the permeability of the core is really high, 00:02:51.130 --> 00:02:55.720 align:middle line:84% the amount these other components matter is less. 00:02:55.720 --> 00:02:58.280 align:middle line:84% But nonetheless, we incorporated them. 00:02:58.280 --> 00:03:00.610 align:middle line:84% And then what we did was we created a magnetic circuit 00:03:00.610 --> 00:03:05.810 align:middle line:84% model, and then worked back to what the flux linkages were 00:03:05.810 --> 00:03:08.820 align:middle line:90% for these two MMF sources. 00:03:08.820 --> 00:03:11.540 align:middle line:84% And from that, you can work back to what-- 00:03:11.540 --> 00:03:13.740 align:middle line:84% the derivative of the flux linkage is the voltage. 00:03:13.740 --> 00:03:17.150 align:middle line:84% And we can work back to the relationships 00:03:17.150 --> 00:03:22.235 align:middle line:84% between the voltages and the currents at the two terminals. 00:03:22.235 --> 00:03:27.290 align:middle line:84% And that worked out to look like an inductance matrix. 00:03:27.290 --> 00:03:29.240 align:middle line:84% So if you have an inductance, that's 00:03:29.240 --> 00:03:33.320 align:middle line:84% one parameter that models everything. 00:03:33.320 --> 00:03:35.710 align:middle line:84% If you have a two-port structure-- two inputs, 00:03:35.710 --> 00:03:36.430 align:middle line:90% two output-- 00:03:36.430 --> 00:03:42.310 align:middle line:84% or two different terminal pairs, then you get a 2 by 2 matrix. 00:03:42.310 --> 00:03:46.030 align:middle line:84% And the matrix turns out to be symmetric, as you can see. 00:03:46.030 --> 00:03:48.610 align:middle line:90% We derived this in class. 00:03:48.610 --> 00:03:51.790 align:middle line:84% And the interpretation-- and then we 00:03:51.790 --> 00:03:53.140 align:middle line:90% took this one step further. 00:03:53.140 --> 00:03:54.230 align:middle line:90% And we went quickly. 00:03:54.230 --> 00:03:56.440 align:middle line:84% We said, OK, this inductance matrix 00:03:56.440 --> 00:03:59.050 align:middle line:84% describes the terminal relations I get ideally 00:03:59.050 --> 00:04:05.086 align:middle line:84% between this voltage and current and this voltage and current. 00:04:05.086 --> 00:04:07.690 align:middle line:84% And I made the argument in class that we can 00:04:07.690 --> 00:04:10.674 align:middle line:90% make a circuit model for this. 00:04:10.674 --> 00:04:13.130 align:middle line:90% So here is the circuit model. 00:04:13.130 --> 00:04:15.640 align:middle line:84% This is what we would consider a physically based circuit 00:04:15.640 --> 00:04:17.079 align:middle line:90% model for this system. 00:04:17.079 --> 00:04:18.500 align:middle line:90% And what did we say? 00:04:18.500 --> 00:04:22.150 align:middle line:84% We said, OK, well, firstly, let's ignore 00:04:22.150 --> 00:04:27.140 align:middle line:84% the external leakage fluxes that are going outside of the core. 00:04:27.140 --> 00:04:33.720 align:middle line:84% If I have only the yellow flux component, 00:04:33.720 --> 00:04:36.770 align:middle line:84% if the core is infinitely permeable, 00:04:36.770 --> 00:04:43.730 align:middle line:84% the flux density in the core is simply 00:04:43.730 --> 00:04:46.460 align:middle line:84% related to the flux and hence the volt seconds 00:04:46.460 --> 00:04:47.200 align:middle line:90% at the windings. 00:04:47.200 --> 00:04:51.620 align:middle line:90% 00:04:51.620 --> 00:04:53.025 align:middle line:84% The voltage here and voltage here 00:04:53.025 --> 00:04:54.650 align:middle line:84% would be scaled versions of one another 00:04:54.650 --> 00:04:59.210 align:middle line:84% because they're all just linking the same flux scaled 00:04:59.210 --> 00:05:02.420 align:middle line:90% by the number of turns. 00:05:02.420 --> 00:05:07.460 align:middle line:84% But what we said was, well, H in the core 00:05:07.460 --> 00:05:11.670 align:middle line:84% is equal to B over mu in the core. 00:05:11.670 --> 00:05:15.710 align:middle line:84% So if mu in the core is infinite, then H is 0. 00:05:15.710 --> 00:05:18.590 align:middle line:84% And hence, we don't store any energy in the core. 00:05:18.590 --> 00:05:23.690 align:middle line:84% But if mu is finite, or not too big, then what happens 00:05:23.690 --> 00:05:25.580 align:middle line:84% is we get some H field in the core 00:05:25.580 --> 00:05:28.250 align:middle line:84% and B dot H is the energy density in the core. 00:05:28.250 --> 00:05:31.380 align:middle line:84% And we store energy in the physical core. 00:05:31.380 --> 00:05:35.640 align:middle line:84% And the way that pops into the circuit model 00:05:35.640 --> 00:05:38.940 align:middle line:84% is through this inductance that appears 00:05:38.940 --> 00:05:41.020 align:middle line:90% across the ideal transformer. 00:05:41.020 --> 00:05:43.536 align:middle line:90% 00:05:43.536 --> 00:05:47.380 align:middle line:84% And so basically, the difference in having infinite permeability, 00:05:47.380 --> 00:05:49.380 align:middle line:84% where you have no energy storage or equivalently 00:05:49.380 --> 00:05:51.300 align:middle line:84% an infinite magnetizing inductance that 00:05:51.300 --> 00:05:54.030 align:middle line:84% carries no current, is I get a finite magnetizing 00:05:54.030 --> 00:05:57.930 align:middle line:84% inductance that has a current that ends up 00:05:57.930 --> 00:06:00.450 align:middle line:84% being a mismatch between the ideal 00:06:00.450 --> 00:06:04.500 align:middle line:84% turns ratio of the transformer or the ideal transformation 00:06:04.500 --> 00:06:06.330 align:middle line:90% ratio of an ideal transformer. 00:06:06.330 --> 00:06:09.960 align:middle line:84% And this magnetizing inductance is really 00:06:09.960 --> 00:06:13.500 align:middle line:84% the reason why we can't apply DC, 00:06:13.500 --> 00:06:15.600 align:middle line:84% because I can only have some finite flux 00:06:15.600 --> 00:06:19.130 align:middle line:84% density before the transformer saturates. 00:06:19.130 --> 00:06:24.690 align:middle line:84% Then we said OK, because I have flux that links winding 1 00:06:24.690 --> 00:06:27.070 align:middle line:84% but may not make it over to winding 2, 00:06:27.070 --> 00:06:31.790 align:middle line:84% I get some leakage inductance on this side. 00:06:31.790 --> 00:06:33.610 align:middle line:84% And because I have flux linking winding 00:06:33.610 --> 00:06:37.450 align:middle line:84% 2 that doesn't link winding 1, I have some leakage inductance 00:06:37.450 --> 00:06:38.860 align:middle line:90% on this side. 00:06:38.860 --> 00:06:46.510 align:middle line:84% And basically, 1/2 L leakage 1 I1 squared is precisely 00:06:46.510 --> 00:06:49.810 align:middle line:84% the energy that's stored in space by the magnetic field 00:06:49.810 --> 00:06:53.250 align:middle line:90% associated with leakage flux 1. 00:06:53.250 --> 00:07:00.240 align:middle line:84% And likewise, 1/2 L leakage 2 I2 squared is precisely the energy 00:07:00.240 --> 00:07:02.790 align:middle line:84% that's stored out in space of fields that link winding 2 00:07:02.790 --> 00:07:04.130 align:middle line:90% but do not link winding 1. 00:07:04.130 --> 00:07:07.020 align:middle line:90% 00:07:07.020 --> 00:07:11.580 align:middle line:84% Basically, if I have some ideal transformation ratio-- 00:07:11.580 --> 00:07:13.890 align:middle line:90% I'll call this V1-prime. 00:07:13.890 --> 00:07:15.930 align:middle line:90% And this is V2-prime-- 00:07:15.930 --> 00:07:20.330 align:middle line:84% which would be exactly proportional to one another 00:07:20.330 --> 00:07:21.960 align:middle line:84% by the turns ratio of the transformer, 00:07:21.960 --> 00:07:26.666 align:middle line:84% the leakage inductance has caused some mismatch. 00:07:26.666 --> 00:07:28.720 align:middle line:84% So first of all, any questions about that? 00:07:28.720 --> 00:07:31.480 align:middle line:90% 00:07:31.480 --> 00:07:37.270 align:middle line:84% So this model is a standard physical model 00:07:37.270 --> 00:07:38.930 align:middle line:90% for a two-winding transformer. 00:07:38.930 --> 00:07:41.080 align:middle line:84% And these parasitics each represent something 00:07:41.080 --> 00:07:43.740 align:middle line:84% very specific physically in the world. 00:07:43.740 --> 00:07:49.130 align:middle line:90% 00:07:49.130 --> 00:07:51.950 align:middle line:84% The other thing I will note about this 00:07:51.950 --> 00:07:59.330 align:middle line:84% is that we can get these parameters of the circuit model. 00:07:59.330 --> 00:08:01.550 align:middle line:84% And how do I know the circuit model is correct? 00:08:01.550 --> 00:08:06.280 align:middle line:84% Well, if you go back and find the relationships between V1 00:08:06.280 --> 00:08:11.690 align:middle line:84% and V2 using the circuit model, it'll turn into exactly this, 00:08:11.690 --> 00:08:15.580 align:middle line:84% so long as you use these parameters. 00:08:15.580 --> 00:08:21.980 align:middle line:84% So what this says is that this circuit model can represent 00:08:21.980 --> 00:08:23.430 align:middle line:90% this inductance matrix. 00:08:23.430 --> 00:08:26.510 align:middle line:84% And the inductance matrix represents the terminal 00:08:26.510 --> 00:08:29.570 align:middle line:84% relationships of the device, neglecting 00:08:29.570 --> 00:08:32.850 align:middle line:90% loss and everything else. 00:08:32.850 --> 00:08:36.720 align:middle line:84% I didn't say this last time, but you can also go naturally 00:08:36.720 --> 00:08:40.494 align:middle line:84% relate the circuit model parameters back to L11, 00:08:40.494 --> 00:08:43.799 align:middle line:84% L22, and the mutual inductance, LM. 00:08:43.799 --> 00:08:47.580 align:middle line:84% And again, you'll just see that this representation 00:08:47.580 --> 00:08:49.320 align:middle line:84% and this representation are precisely 00:08:49.320 --> 00:08:51.670 align:middle line:84% the same, as long as you have the right parameters. 00:08:51.670 --> 00:08:54.231 align:middle line:90% 00:08:54.231 --> 00:08:56.070 align:middle line:84% And then the last thing I'll note 00:08:56.070 --> 00:08:58.410 align:middle line:84% is you ought to be careful whenever you're 00:08:58.410 --> 00:09:03.270 align:middle line:84% doing your own analysis like this to recognize that you have 00:09:03.270 --> 00:09:05.670 align:middle line:84% to be careful about relating-- remember, 00:09:05.670 --> 00:09:08.580 align:middle line:84% the flux that's of interest for the flux linkage 00:09:08.580 --> 00:09:11.490 align:middle line:84% to relate to voltage on the electrical side 00:09:11.490 --> 00:09:12.900 align:middle line:84% is related to the flux coming out 00:09:12.900 --> 00:09:16.110 align:middle line:84% of the positive terminal of the MMF source. 00:09:16.110 --> 00:09:18.000 align:middle line:84% So you've got to pay attention to your signs, 00:09:18.000 --> 00:09:19.792 align:middle line:84% especially when you have multiple windings. 00:09:19.792 --> 00:09:24.090 align:middle line:84% And that comes back to representing Lenz's law. 00:09:24.090 --> 00:09:27.070 align:middle line:84% So let me stop there and just ask if there's any questions. 00:09:27.070 --> 00:09:31.500 align:middle line:90% 00:09:31.500 --> 00:09:36.810 align:middle line:84% So that was a lightning review of last class. 00:09:36.810 --> 00:09:40.460 align:middle line:84% What I'd like to do today is expand out 00:09:40.460 --> 00:09:42.210 align:middle line:84% our discussion of transformers in a couple 00:09:42.210 --> 00:09:44.590 align:middle line:90% of different directions. 00:09:44.590 --> 00:09:48.220 align:middle line:84% One relates to how we might model a transformer. 00:09:48.220 --> 00:09:55.135 align:middle line:84% This circuit model, as I said, is a physically based model, 00:09:55.135 --> 00:10:00.010 align:middle line:84% in that every parasitic component represents 00:10:00.010 --> 00:10:01.840 align:middle line:84% some physical energy stored somewhere 00:10:01.840 --> 00:10:04.370 align:middle line:90% in space in the device. 00:10:04.370 --> 00:10:08.550 align:middle line:84% And that's mostly what we want to do. 00:10:08.550 --> 00:10:11.360 align:middle line:84% I mean, that would be a typical thing we'd want to understand. 00:10:11.360 --> 00:10:15.770 align:middle line:84% But recognize that in terms of-- if all I cared about 00:10:15.770 --> 00:10:18.350 align:middle line:84% was matching the terminal relations, 00:10:18.350 --> 00:10:22.730 align:middle line:84% I just wanted a black box inside my white box 00:10:22.730 --> 00:10:28.130 align:middle line:84% there that mapped voltage and current at terminal port 1 00:10:28.130 --> 00:10:32.900 align:middle line:84% and voltage and current at port 2, what that's captured by 00:10:32.900 --> 00:10:38.060 align:middle line:84% is that's fully captured by this inductance matrix description. 00:10:38.060 --> 00:10:40.250 align:middle line:84% Now, how many independent parameters 00:10:40.250 --> 00:10:42.175 align:middle line:84% does my inductance matrix description have? 00:10:42.175 --> 00:10:48.780 align:middle line:90% 00:10:48.780 --> 00:10:50.670 align:middle line:90% It really has three parameters. 00:10:50.670 --> 00:10:57.900 align:middle line:84% It has L11, L22, and then LM because an inductance matrix, 00:10:57.900 --> 00:11:00.330 align:middle line:84% because a magnetic component like this 00:11:00.330 --> 00:11:05.340 align:middle line:84% is a reciprocal element, the matrix describing it ends up 00:11:05.340 --> 00:11:06.520 align:middle line:90% having to be symmetric. 00:11:06.520 --> 00:11:09.630 align:middle line:84% And you can see that these two elements-- it's not 00:11:09.630 --> 00:11:12.360 align:middle line:84% just random that they ended up being the same thing. 00:11:12.360 --> 00:11:16.440 align:middle line:84% It has to turn out that way because of reciprocity. 00:11:16.440 --> 00:11:22.320 align:middle line:84% Well, how many independent parameters 00:11:22.320 --> 00:11:25.570 align:middle line:90% does my circuit model have? 00:11:25.570 --> 00:11:32.600 align:middle line:84% Well, if I picked physical turns ratios-- 00:11:32.600 --> 00:11:34.912 align:middle line:84% I know I had 10 turns in the primary and 20 turns 00:11:34.912 --> 00:11:35.620 align:middle line:90% on the secondary. 00:11:35.620 --> 00:11:40.090 align:middle line:84% I pick 1 to 2 as the turns ratio of the ideal transformer-- 00:11:40.090 --> 00:11:41.890 align:middle line:84% well then this has just the right amount. 00:11:41.890 --> 00:11:42.663 align:middle line:90% It has L11-- 00:11:42.663 --> 00:11:45.210 align:middle line:90% 00:11:45.210 --> 00:11:45.710 align:middle line:90% I'm sorry. 00:11:45.710 --> 00:11:50.330 align:middle line:84% It has L leakage 1, L leakage 2, and the magnetizing inductance. 00:11:50.330 --> 00:11:51.830 align:middle line:84% And I could have put the magnetizing 00:11:51.830 --> 00:11:53.455 align:middle line:84% either on the primary and the secondary 00:11:53.455 --> 00:11:55.010 align:middle line:90% and scaled it in the right way. 00:11:55.010 --> 00:11:59.930 align:middle line:84% But it, as long as I'm using the physical turns ratio, 00:11:59.930 --> 00:12:01.430 align:middle line:84% has the right amount of parameters. 00:12:01.430 --> 00:12:05.790 align:middle line:84% There's a 1 to 1 mapping between the two models. 00:12:05.790 --> 00:12:12.400 align:middle line:84% However, you might not always want to use the physical turns 00:12:12.400 --> 00:12:13.650 align:middle line:90% ratio. 00:12:13.650 --> 00:12:16.590 align:middle line:84% I might walk up to you and hand you the transformer. 00:12:16.590 --> 00:12:17.650 align:middle line:90% Here's this transformer. 00:12:17.650 --> 00:12:19.650 align:middle line:90% I bought it on DigiKey. 00:12:19.650 --> 00:12:21.370 align:middle line:90% Come back with the model for it. 00:12:21.370 --> 00:12:23.953 align:middle line:84% But you're not allowed to cut it apart because it's expensive. 00:12:23.953 --> 00:12:25.570 align:middle line:90% And I don't want to break it. 00:12:25.570 --> 00:12:28.230 align:middle line:84% Well, how do you know what the actual turns 00:12:28.230 --> 00:12:30.660 align:middle line:84% ratio, like physically the number of wires on each side 00:12:30.660 --> 00:12:31.160 align:middle line:90% are? 00:12:31.160 --> 00:12:33.060 align:middle line:90% Well, you don't. 00:12:33.060 --> 00:12:42.330 align:middle line:84% So in terms of this model, the turns ratio is also a parameter. 00:12:42.330 --> 00:12:44.610 align:middle line:84% So in this circuit model, I really 00:12:44.610 --> 00:12:47.260 align:middle line:84% have four parameters I can fiddle with. 00:12:47.260 --> 00:12:49.440 align:middle line:84% And if I knew this one to be the physical one, 00:12:49.440 --> 00:12:55.206 align:middle line:84% then all the other things turn into physical quantities. 00:12:55.206 --> 00:12:58.290 align:middle line:84% But in terms of modeling things, I 00:12:58.290 --> 00:13:03.840 align:middle line:84% could always go back and choose some non-physical value 00:13:03.840 --> 00:13:08.430 align:middle line:84% and still create a circuit model that correctly matched 00:13:08.430 --> 00:13:10.860 align:middle line:84% the inductance matrix model because here I 00:13:10.860 --> 00:13:11.770 align:middle line:90% have four parameters. 00:13:11.770 --> 00:13:13.140 align:middle line:90% There, I only have three. 00:13:13.140 --> 00:13:15.850 align:middle line:84% And in fact, sometimes, we do that. 00:13:15.850 --> 00:13:20.400 align:middle line:84% We may do that because we don't know 00:13:20.400 --> 00:13:22.380 align:middle line:90% some parameter like this one. 00:13:22.380 --> 00:13:25.000 align:middle line:84% Or we may do it for convenience purposes. 00:13:25.000 --> 00:13:26.730 align:middle line:84% And what I'm doing is I'm throwing out 00:13:26.730 --> 00:13:29.028 align:middle line:84% the knowledge of where energy is being stored exactly. 00:13:29.028 --> 00:13:30.570 align:middle line:84% And I'm just coming up with something 00:13:30.570 --> 00:13:33.840 align:middle line:84% that mathematically gives me the right terminal relations. 00:13:33.840 --> 00:13:35.140 align:middle line:90% So why might I do that? 00:13:35.140 --> 00:13:39.810 align:middle line:84% So I might argue that I could create a circuit model that 00:13:39.810 --> 00:13:42.660 align:middle line:90% was really just like this-- 00:13:42.660 --> 00:13:50.046 align:middle line:90% LA, LB, an ideal transformer-- 00:13:50.046 --> 00:13:53.030 align:middle line:90% 00:13:53.030 --> 00:13:54.120 align:middle line:90% I don't know what I used. 00:13:54.120 --> 00:13:56.910 align:middle line:90% I think I just did to Nx to Ny-- 00:13:56.910 --> 00:14:01.470 align:middle line:90% 00:14:01.470 --> 00:14:05.520 align:middle line:90% and then LC. 00:14:05.520 --> 00:14:13.620 align:middle line:84% And I would call this V1 and I1 and this V2 and I2. 00:14:13.620 --> 00:14:17.050 align:middle line:90% 00:14:17.050 --> 00:14:20.665 align:middle line:84% And now, if I put this in a black box, 00:14:20.665 --> 00:14:22.040 align:middle line:84% I can just say, OK, I'm not going 00:14:22.040 --> 00:14:24.630 align:middle line:84% to worry about whether this physically represents things. 00:14:24.630 --> 00:14:29.400 align:middle line:84% But now, I have four parameters to play with. 00:14:29.400 --> 00:14:32.860 align:middle line:84% And I can use those four parameters any way I want. 00:14:32.860 --> 00:14:34.360 align:middle line:90% Well, what might I do? 00:14:34.360 --> 00:14:37.500 align:middle line:84% Maybe I might just want to say, it's easy for me 00:14:37.500 --> 00:14:40.440 align:middle line:84% to think about the voltage scaling for a 1 to 1 00:14:40.440 --> 00:14:41.980 align:middle line:90% transformer. 00:14:41.980 --> 00:14:43.800 align:middle line:84% The ideal transformer in the middle 00:14:43.800 --> 00:14:46.710 align:middle line:90% here has some turns ratio on it. 00:14:46.710 --> 00:14:49.920 align:middle line:90% Maybe I just make that 1 to 1. 00:14:49.920 --> 00:14:54.340 align:middle line:84% If I do that, I can then just create a model that has a 1 to 1 00:14:54.340 --> 00:14:58.660 align:middle line:84% transform, even if my real device does not. 00:14:58.660 --> 00:15:02.720 align:middle line:84% I could create a circuit model where I force this parameter. 00:15:02.720 --> 00:15:05.770 align:middle line:84% I arbitrarily pick this to be 1 to 1. 00:15:05.770 --> 00:15:10.420 align:middle line:84% And then I say, I'm going to have three other parameters 00:15:10.420 --> 00:15:13.930 align:middle line:84% that now I'm going to pick those three parameters to match 00:15:13.930 --> 00:15:15.360 align:middle line:90% the terminal relations. 00:15:15.360 --> 00:15:16.820 align:middle line:90% What would I get in this case? 00:15:16.820 --> 00:15:22.230 align:middle line:84% What I would get is L11 minus LM would be LA. 00:15:22.230 --> 00:15:27.190 align:middle line:90% LM would be LB. 00:15:27.190 --> 00:15:33.340 align:middle line:90% And L22 minus LM would be LC. 00:15:33.340 --> 00:15:35.920 align:middle line:84% So the nice thing about this circuit model 00:15:35.920 --> 00:15:40.510 align:middle line:84% is if I know the inductance matrix 00:15:40.510 --> 00:15:42.940 align:middle line:84% and all I know is the inductance matrix, 00:15:42.940 --> 00:15:46.840 align:middle line:84% this is a perfectly good model for a transformer. 00:15:46.840 --> 00:15:50.830 align:middle line:84% The parameters may not represent anything physical. 00:15:50.830 --> 00:15:52.870 align:middle line:84% But it still gives me the right relationships 00:15:52.870 --> 00:16:00.940 align:middle line:90% between V1, I1, and V2, I2. 00:16:00.940 --> 00:16:02.340 align:middle line:90% Any questions about that? 00:16:02.340 --> 00:16:05.800 align:middle line:90% 00:16:05.800 --> 00:16:09.970 align:middle line:84% So this is a trick that's often played to make circuits easier 00:16:09.970 --> 00:16:10.840 align:middle line:90% to think about. 00:16:10.840 --> 00:16:12.290 align:middle line:90% And then I can go remap it. 00:16:12.290 --> 00:16:13.957 align:middle line:84% If I'm actually designing a transformer, 00:16:13.957 --> 00:16:17.210 align:middle line:84% I can go remap it and get the actual physical model. 00:16:17.210 --> 00:16:20.020 align:middle line:84% But I don't need to use a physical model all the time. 00:16:20.020 --> 00:16:22.833 align:middle line:84% This is sometimes called the T model. 00:16:22.833 --> 00:16:24.250 align:middle line:84% Or they're all called the T model. 00:16:24.250 --> 00:16:25.750 align:middle line:90% But this is the T model. 00:16:25.750 --> 00:16:27.200 align:middle line:90% This is 1 to 1. 00:16:27.200 --> 00:16:31.050 align:middle line:90% 00:16:31.050 --> 00:16:32.940 align:middle line:84% There are other choices we might also make. 00:16:32.940 --> 00:16:35.670 align:middle line:84% And we'll see an example of that shortly. 00:16:35.670 --> 00:16:38.135 align:middle line:90% So any questions about that? 00:16:38.135 --> 00:16:43.570 align:middle line:90% 00:16:43.570 --> 00:16:49.960 align:middle line:84% So let's start talking about constraints on the parameters. 00:16:49.960 --> 00:16:55.720 align:middle line:84% I said I have three independent parameters. 00:16:55.720 --> 00:16:58.220 align:middle line:84% But can the parameters be anything I want? 00:16:58.220 --> 00:17:01.480 align:middle line:84% Well, no, they can't, not for a real device anyways. 00:17:01.480 --> 00:17:06.250 align:middle line:84% Let's just start thinking about if I just handed you 00:17:06.250 --> 00:17:12.310 align:middle line:84% an inductor, L, off-shelf, real inductor, 00:17:12.310 --> 00:17:15.690 align:middle line:84% what can I say about the inductance of this thing? 00:17:15.690 --> 00:17:19.680 align:middle line:84% Well, I would argue that, in a real inductor, at least 00:17:19.680 --> 00:17:27.060 align:middle line:84% large signal wise, the inductance has to be positive. 00:17:27.060 --> 00:17:28.530 align:middle line:90% Why is that? 00:17:28.530 --> 00:17:31.170 align:middle line:84% Because if I apply a positive voltage So 00:17:31.170 --> 00:17:33.190 align:middle line:84% suppose I applied a step in voltage. 00:17:33.190 --> 00:17:38.330 align:middle line:90% Here's V. Or here's V1. 00:17:38.330 --> 00:17:39.690 align:middle line:90% What should I1 do? 00:17:39.690 --> 00:17:44.900 align:middle line:84% Well, when I step the voltage from 0 to 1, I1 should go from 0 00:17:44.900 --> 00:17:47.392 align:middle line:90% and ramp up. 00:17:47.392 --> 00:17:49.820 align:middle line:84% And since V is positive and I is positive, 00:17:49.820 --> 00:17:52.020 align:middle line:84% I've got energy going into this device. 00:17:52.020 --> 00:18:01.660 align:middle line:84% And this slope, dI/dT, is 1 over L. 00:18:01.660 --> 00:18:05.560 align:middle line:84% What would happen if the inductance was negative? 00:18:05.560 --> 00:18:09.130 align:middle line:84% If the inductance was negative, instead of doing this, 00:18:09.130 --> 00:18:10.375 align:middle line:90% it would do this. 00:18:10.375 --> 00:18:12.860 align:middle line:90% 00:18:12.860 --> 00:18:14.860 align:middle line:84% This would be 1 over L. It would be sloping down 00:18:14.860 --> 00:18:16.190 align:middle line:90% because L is negative. 00:18:16.190 --> 00:18:19.330 align:middle line:90% So this is L less than 0. 00:18:19.330 --> 00:18:23.110 align:middle line:84% And energy, voltage times current, would be negative. 00:18:23.110 --> 00:18:24.452 align:middle line:90% And energy would be coming out. 00:18:24.452 --> 00:18:26.410 align:middle line:84% So I'd apply a positive voltage and my inductor 00:18:26.410 --> 00:18:28.420 align:middle line:84% would start squirting energy back at me. 00:18:28.420 --> 00:18:30.500 align:middle line:84% I'd love to have one of those things. 00:18:30.500 --> 00:18:32.710 align:middle line:84% I could make quite a bit of money off that. 00:18:32.710 --> 00:18:35.440 align:middle line:84% But unfortunately, conservation of energy 00:18:35.440 --> 00:18:37.635 align:middle line:84% tells me I can't get that, at least large signal. 00:18:37.635 --> 00:18:40.510 align:middle line:90% 00:18:40.510 --> 00:18:42.515 align:middle line:84% So what do we know about an inductor? 00:18:42.515 --> 00:18:44.390 align:middle line:84% We know the inductance is a practical matter. 00:18:44.390 --> 00:18:46.527 align:middle line:84% The inductance has to be positive for anything 00:18:46.527 --> 00:18:48.610 align:middle line:84% you're going to wind up with wires and cores, that 00:18:48.610 --> 00:18:52.660 align:middle line:84% doesn't have a source of energy inside it. 00:18:52.660 --> 00:18:57.740 align:middle line:84% Well, what could I say about my transformer parameters? 00:18:57.740 --> 00:19:01.720 align:middle line:84% Well, you can make a similar argument 00:19:01.720 --> 00:19:03.700 align:middle line:84% because what I really did here was 00:19:03.700 --> 00:19:10.520 align:middle line:84% I said dI/dT is equal to 1 over L V. 00:19:10.520 --> 00:19:13.030 align:middle line:84% That's the argument I just made about my signal terminal 00:19:13.030 --> 00:19:15.350 align:middle line:90% inductor. 00:19:15.350 --> 00:19:18.260 align:middle line:84% What could I say about my inductance matrix? 00:19:18.260 --> 00:19:21.590 align:middle line:84% If I have a two-terminal device, what I get 00:19:21.590 --> 00:19:24.620 align:middle line:90% is V is equal to L dI/dT. 00:19:24.620 --> 00:19:32.420 align:middle line:84% This equation here is simply V, a vector equation, 00:19:32.420 --> 00:19:39.100 align:middle line:84% is equal to L, where this is an inductance matrix, times I, 00:19:39.100 --> 00:19:41.030 align:middle line:84% where that's a vector of currents. 00:19:41.030 --> 00:19:47.115 align:middle line:84% So it's I dot, I should say, dI/dT. 00:19:47.115 --> 00:19:49.060 align:middle line:90% I can do the same thing. 00:19:49.060 --> 00:19:58.590 align:middle line:84% And I could write, OK, that means that d/dT of I1 and I2 00:19:58.590 --> 00:20:04.350 align:middle line:84% is equal to L inverse times the voltage of vectors V1 and V2. 00:20:04.350 --> 00:20:08.910 align:middle line:90% 00:20:08.910 --> 00:20:10.610 align:middle line:90% And what does that look like? 00:20:10.610 --> 00:20:13.190 align:middle line:84% This is going to look like the inverse of that matrix. 00:20:13.190 --> 00:20:15.860 align:middle line:84% So that would be 1 over the determinant, 00:20:15.860 --> 00:20:21.220 align:middle line:84% L11 L22 minus LM quantity squared 00:20:21.220 --> 00:20:29.800 align:middle line:84% times L22 L11 minus LM minus LM times the vector 00:20:29.800 --> 00:20:31.210 align:middle line:90% of voltages V1 and V2. 00:20:31.210 --> 00:20:34.464 align:middle line:90% 00:20:34.464 --> 00:20:37.040 align:middle line:90% So this is the [INAUDIBLE]. 00:20:37.040 --> 00:20:41.450 align:middle line:84% Suppose I made V2 negative, which would be basically V20, 00:20:41.450 --> 00:20:45.230 align:middle line:84% which would be me dropping a short on the secondary side, 00:20:45.230 --> 00:20:47.280 align:middle line:84% and then I apply a voltage over here. 00:20:47.280 --> 00:20:48.500 align:middle line:90% What should happen? 00:20:48.500 --> 00:20:49.510 align:middle line:90% Well, the same thing. 00:20:49.510 --> 00:20:52.010 align:middle line:84% If I have a short over here and I apply a voltage over here, 00:20:52.010 --> 00:20:54.220 align:middle line:84% I just ought to see some inductance there. 00:20:54.220 --> 00:20:56.780 align:middle line:84% And I should see energy ramping up into it. 00:20:56.780 --> 00:20:59.150 align:middle line:90% Well, that only happens-- 00:20:59.150 --> 00:21:01.880 align:middle line:84% L22 is a positive number because that's 00:21:01.880 --> 00:21:04.640 align:middle line:84% the inductance measured on the secondary side 00:21:04.640 --> 00:21:08.970 align:middle line:90% with the primary open. 00:21:08.970 --> 00:21:11.970 align:middle line:84% This thing has to be a positive number. 00:21:11.970 --> 00:21:14.880 align:middle line:84% Or another way to put it is this inductance matrix 00:21:14.880 --> 00:21:18.660 align:middle line:84% has to be positive semi-definite. 00:21:18.660 --> 00:21:23.820 align:middle line:84% So what I get is a requirement that LM, the magnitude of LM, 00:21:23.820 --> 00:21:27.810 align:middle line:84% has to be less than the square root of L11 L22 00:21:27.810 --> 00:21:32.135 align:middle line:90% for any physical transformer. 00:21:32.135 --> 00:21:34.430 align:middle line:84% In the more winding case that we'll see, 00:21:34.430 --> 00:21:37.730 align:middle line:84% it just means the matrix has to be positive semidefinite. 00:21:37.730 --> 00:21:40.840 align:middle line:90% 00:21:40.840 --> 00:21:46.350 align:middle line:84% Sometimes, we define a coupling coefficient, k. 00:21:46.350 --> 00:21:51.770 align:middle line:84% And k is defined as equal to LM over the square root of L11 L22. 00:21:51.770 --> 00:21:54.360 align:middle line:90% 00:21:54.360 --> 00:21:58.470 align:middle line:84% LM being equal to the square root of L11 L22 just 00:21:58.470 --> 00:22:00.030 align:middle line:90% is basically perfect coupling. 00:22:00.030 --> 00:22:01.060 align:middle line:90% What does that mean? 00:22:01.060 --> 00:22:03.870 align:middle line:84% That means there is essentially no leakage flux. 00:22:03.870 --> 00:22:06.700 align:middle line:84% All the flux in the primary is linking the secondary. 00:22:06.700 --> 00:22:09.710 align:middle line:90% 00:22:09.710 --> 00:22:15.030 align:middle line:84% So k is sometimes used to say, in some sense, how much leakage 00:22:15.030 --> 00:22:17.260 align:middle line:90% is in the transformer. 00:22:17.260 --> 00:22:19.810 align:middle line:84% But more broadly, what we can say is, 00:22:19.810 --> 00:22:21.970 align:middle line:84% yeah, I've got three parameters-- 00:22:21.970 --> 00:22:23.800 align:middle line:90% L11, L22, and LM. 00:22:23.800 --> 00:22:25.420 align:middle line:84% But there are some limits on what 00:22:25.420 --> 00:22:28.410 align:middle line:84% you can get in terms of the relationships in the real world. 00:22:28.410 --> 00:22:32.250 align:middle line:90% 00:22:32.250 --> 00:22:33.395 align:middle line:90% Any questions about that? 00:22:33.395 --> 00:22:37.660 align:middle line:90% 00:22:37.660 --> 00:22:38.160 align:middle line:90% OK. 00:22:38.160 --> 00:22:41.660 align:middle line:90% 00:22:41.660 --> 00:22:45.463 align:middle line:90% Let's start to expand upon this. 00:22:45.463 --> 00:22:46.880 align:middle line:84% We started with an inductor, which 00:22:46.880 --> 00:22:50.200 align:middle line:84% is basically a one-terminal device, one-port device. 00:22:50.200 --> 00:22:52.280 align:middle line:84% Then we did a transformer, a basic transformer, 00:22:52.280 --> 00:22:53.900 align:middle line:90% which is a two-port device. 00:22:53.900 --> 00:22:56.658 align:middle line:84% But we're not really limited to only two ports. 00:22:56.658 --> 00:22:58.200 align:middle line:84% We can have as many ports as we want. 00:22:58.200 --> 00:23:01.460 align:middle line:90% And as we'll see, we often do. 00:23:01.460 --> 00:23:10.190 align:middle line:84% So let's think about what might happen if I have more ports. 00:23:10.190 --> 00:23:17.660 align:middle line:84% And it turns out now, naturally, because you have more terminal 00:23:17.660 --> 00:23:20.760 align:middle line:84% pairs, there's many more ways you can configure things. 00:23:20.760 --> 00:23:22.880 align:middle line:84% But I'm just going to show you two 00:23:22.880 --> 00:23:26.940 align:middle line:90% that are very basic versions. 00:23:26.940 --> 00:23:29.030 align:middle line:84% And what I'm going to show you is the equivalent 00:23:29.030 --> 00:23:31.670 align:middle line:84% of the ideal transformer model for these two basic versions 00:23:31.670 --> 00:23:32.545 align:middle line:90% that are very common. 00:23:32.545 --> 00:23:37.018 align:middle line:90% 00:23:37.018 --> 00:23:39.230 align:middle line:90% Suppose I did this. 00:23:39.230 --> 00:23:43.260 align:middle line:84% All I did to create this transformer 00:23:43.260 --> 00:23:45.600 align:middle line:84% is I took something that had one winding on 00:23:45.600 --> 00:23:47.720 align:middle line:84% and added a second winding on the core path. 00:23:47.720 --> 00:23:51.340 align:middle line:84% Well, what happens if I add a third winding on the core path? 00:23:51.340 --> 00:23:58.560 align:middle line:84% So I could say, OK, here's my transformer, here's my core. 00:23:58.560 --> 00:24:03.615 align:middle line:90% And I'll have N1 turns here. 00:24:03.615 --> 00:24:10.330 align:middle line:90% 00:24:10.330 --> 00:24:13.125 align:middle line:90% I'll have N2 turns here. 00:24:13.125 --> 00:24:19.540 align:middle line:90% 00:24:19.540 --> 00:24:23.990 align:middle line:84% And notice that positive voltage here throws flux 00:24:23.990 --> 00:24:25.230 align:middle line:90% around the core this way. 00:24:25.230 --> 00:24:26.700 align:middle line:90% So I have a dot here. 00:24:26.700 --> 00:24:29.790 align:middle line:84% Likewise, by the right-hand rule, I have a dot over here. 00:24:29.790 --> 00:24:31.285 align:middle line:90% Let me add one more. 00:24:31.285 --> 00:24:34.890 align:middle line:90% 00:24:34.890 --> 00:24:38.760 align:middle line:90% I'll have N3 turns on this one. 00:24:38.760 --> 00:24:42.160 align:middle line:90% I'll have I3 and V3. 00:24:42.160 --> 00:24:46.490 align:middle line:90% 00:24:46.490 --> 00:24:50.390 align:middle line:84% Well, what would be the magnetic circuit model for this thing 00:24:50.390 --> 00:24:54.350 align:middle line:84% if this had some reluctance of the core? 00:24:54.350 --> 00:24:56.250 align:middle line:90% It would look like this. 00:24:56.250 --> 00:25:06.594 align:middle line:84% I would have N1 I1 N2 I2 and N3 I3. 00:25:06.594 --> 00:25:10.010 align:middle line:90% 00:25:10.010 --> 00:25:13.670 align:middle line:84% And then I would have some reluctance 00:25:13.670 --> 00:25:16.220 align:middle line:84% of the core, which is equal to the length of the core over mu 00:25:16.220 --> 00:25:18.060 align:middle line:90% of the core A of the core. 00:25:18.060 --> 00:25:23.270 align:middle line:84% And then I would have some flux in the core, which 00:25:23.270 --> 00:25:24.085 align:middle line:90% is doing this. 00:25:24.085 --> 00:25:28.040 align:middle line:90% 00:25:28.040 --> 00:25:32.990 align:middle line:84% So all I've done is added one more MMF source. 00:25:32.990 --> 00:25:39.250 align:middle line:84% What would be the terminal relations of this thing? 00:25:39.250 --> 00:25:41.860 align:middle line:84% Well, if I do this approximation, 00:25:41.860 --> 00:25:46.260 align:middle line:84% I'm pretending there's no leakage flux at all. 00:25:46.260 --> 00:25:55.990 align:middle line:84% That means that lambda 1 is equal to N1 phi core. 00:25:55.990 --> 00:25:59.860 align:middle line:84% Lambda 2 is equal to N2 phi core. 00:25:59.860 --> 00:26:04.232 align:middle line:90% Lambda 3 is equal to 3 phi core. 00:26:04.232 --> 00:26:06.360 align:middle line:84% So what I get is then phi core is 00:26:06.360 --> 00:26:10.980 align:middle line:84% equal to lambda 1 over N1, which is 00:26:10.980 --> 00:26:16.960 align:middle line:84% equal to lambda 2 over N2, which is equal to lambda 3 over N3. 00:26:16.960 --> 00:26:22.530 align:middle line:84% Or if I differentiated this, what I would get is V1 over N1 00:26:22.530 --> 00:26:28.890 align:middle line:84% has to equal V2 over N2, which is equal to V3 over 3. 00:26:28.890 --> 00:26:38.520 align:middle line:84% So what I've got here is, by adding my third winning, 00:26:38.520 --> 00:26:41.520 align:middle line:90% I get another-- 00:26:41.520 --> 00:26:44.160 align:middle line:90% just another scale voltage. 00:26:44.160 --> 00:26:45.900 align:middle line:84% And we often do that because we often 00:26:45.900 --> 00:26:48.720 align:middle line:84% want some ratiometric sets of voltages 00:26:48.720 --> 00:26:52.210 align:middle line:84% that our transformer might create. 00:26:52.210 --> 00:26:55.240 align:middle line:90% What about the currents? 00:26:55.240 --> 00:26:59.040 align:middle line:84% Well, let me do the simplest case here. 00:26:59.040 --> 00:27:04.350 align:middle line:90% If mu core goes to infinity-- 00:27:04.350 --> 00:27:07.440 align:middle line:84% or actually, let me just do magnetic circuit 00:27:07.440 --> 00:27:09.300 align:middle line:90% KVL around this loop. 00:27:09.300 --> 00:27:17.170 align:middle line:84% What I get is N1 I1 plus N2 I2 plus N3 I3-- 00:27:17.170 --> 00:27:20.930 align:middle line:84% that's sum of these voltages-- must equal phi core R core. 00:27:20.930 --> 00:27:23.730 align:middle line:90% 00:27:23.730 --> 00:27:24.665 align:middle line:90% Everybody buy that? 00:27:24.665 --> 00:27:27.220 align:middle line:90% 00:27:27.220 --> 00:27:27.850 align:middle line:90% OK. 00:27:27.850 --> 00:27:31.270 align:middle line:84% Well, if reluctance of the core equals 0-- 00:27:31.270 --> 00:27:33.400 align:middle line:84% or equivalently, that's because the permeability 00:27:33.400 --> 00:27:35.200 align:middle line:90% went to infinity. 00:27:35.200 --> 00:27:37.630 align:middle line:84% So the reluctance of the core is 0-- 00:27:37.630 --> 00:27:40.480 align:middle line:84% then the right-hand side of this becomes 0. 00:27:40.480 --> 00:27:45.550 align:middle line:84% And at reluctance of the core goes to 0, what I get 00:27:45.550 --> 00:27:54.170 align:middle line:84% is N1 I1 plus N2 I2 plus N3 I3 is equal to 0. 00:27:54.170 --> 00:28:00.580 align:middle line:84% So the voltage relationship just scales from 2 to 3, 00:28:00.580 --> 00:28:03.363 align:middle line:84% but the current relationship doesn't. 00:28:03.363 --> 00:28:04.780 align:middle line:84% The current relationship just says 00:28:04.780 --> 00:28:07.960 align:middle line:84% the sum of the NI's into the windings have to be 0. 00:28:07.960 --> 00:28:10.990 align:middle line:84% And what that turns out to mean is all the energy flowing 00:28:10.990 --> 00:28:13.228 align:middle line:84% in two of the ports must come out 00:28:13.228 --> 00:28:14.770 align:middle line:84% the third of the port instantaneously 00:28:14.770 --> 00:28:16.440 align:middle line:90% for this idealized version. 00:28:16.440 --> 00:28:21.150 align:middle line:90% 00:28:21.150 --> 00:28:22.190 align:middle line:90% Questions about that? 00:28:22.190 --> 00:28:24.980 align:middle line:90% 00:28:24.980 --> 00:28:28.880 align:middle line:84% What would I get if I started including parasitics? 00:28:28.880 --> 00:28:33.890 align:middle line:84% So what would I get if I started having a non-zero reluctance 00:28:33.890 --> 00:28:38.390 align:middle line:84% or I started having leakage fluxes from the windings, 00:28:38.390 --> 00:28:39.480 align:middle line:90% that kind of thing? 00:28:39.480 --> 00:28:43.160 align:middle line:84% Well, what I would get is instead of a 2 00:28:43.160 --> 00:28:49.130 align:middle line:84% by 2 inductance matrix, I would get a 3 by 3 inductance matrix. 00:28:49.130 --> 00:28:53.130 align:middle line:84% And that would capture the parasitic behavior. 00:28:53.130 --> 00:28:56.970 align:middle line:84% This version, this idealized version, 00:28:56.970 --> 00:28:58.940 align:middle line:90% we might often draw this way. 00:28:58.940 --> 00:29:02.500 align:middle line:90% 00:29:02.500 --> 00:29:06.980 align:middle line:84% I'll have multiple windings, all illustrated on the same core. 00:29:06.980 --> 00:29:08.650 align:middle line:90% So if you see this symbol-- 00:29:08.650 --> 00:29:19.750 align:middle line:84% and this is N1, N2, N3 with V1, V2, V3-- 00:29:19.750 --> 00:29:27.190 align:middle line:84% that's what-- this is the ideal relations for this structure, 00:29:27.190 --> 00:29:30.250 align:middle line:84% where all the windings are illustrated on this single core 00:29:30.250 --> 00:29:31.630 align:middle line:90% path. 00:29:31.630 --> 00:29:36.460 align:middle line:84% And because we can talk about which way each 00:29:36.460 --> 00:29:38.740 align:middle line:84% of the currents in each of these windings 00:29:38.740 --> 00:29:41.230 align:middle line:84% throw flux around the core, we can draw the dots 00:29:41.230 --> 00:29:42.530 align:middle line:90% of the individual windings. 00:29:42.530 --> 00:29:45.580 align:middle line:90% And you see this like this. 00:29:45.580 --> 00:29:47.280 align:middle line:90% Any questions about that? 00:29:47.280 --> 00:29:50.400 align:middle line:90% 00:29:50.400 --> 00:29:52.830 align:middle line:84% What would happen if I included non-idealities? 00:29:52.830 --> 00:29:54.820 align:middle line:90% I'd surround this thing-- 00:29:54.820 --> 00:29:56.790 align:middle line:84% perhaps, one way to do it is I could 00:29:56.790 --> 00:30:01.230 align:middle line:84% surround this thing with additional inductances 00:30:01.230 --> 00:30:02.880 align:middle line:90% to match with the parasitics. 00:30:02.880 --> 00:30:06.090 align:middle line:90% And we'll talk more about that. 00:30:06.090 --> 00:30:06.840 align:middle line:90% Questions? 00:30:06.840 --> 00:30:10.980 align:middle line:90% 00:30:10.980 --> 00:30:15.510 align:middle line:84% So let's think about other possibilities, however. 00:30:15.510 --> 00:30:19.020 align:middle line:84% This is perhaps the most common way 00:30:19.020 --> 00:30:21.600 align:middle line:84% to build a three-winding, or five-winding, 00:30:21.600 --> 00:30:24.073 align:middle line:90% or whatever transformer. 00:30:24.073 --> 00:30:25.740 align:middle line:84% Well, what we're going to get is a bunch 00:30:25.740 --> 00:30:29.100 align:middle line:84% of voltages at the terminals that are ideally ratiometrically 00:30:29.100 --> 00:30:32.160 align:middle line:84% related because that's the most common thing of what we want. 00:30:32.160 --> 00:30:37.770 align:middle line:84% We'll see next week cases where we 00:30:37.770 --> 00:30:41.760 align:middle line:84% might want to do that for converter design purposes. 00:30:41.760 --> 00:30:44.310 align:middle line:84% But it's not the only thing you could do. 00:30:44.310 --> 00:30:47.738 align:middle line:84% Imagine this alternative structure. 00:30:47.738 --> 00:30:49.030 align:middle line:90% I might call this series-wound. 00:30:49.030 --> 00:30:52.100 align:middle line:84% They're all wound on the same series core path. 00:30:52.100 --> 00:30:54.310 align:middle line:84% What if I did something-- instead of that, 00:30:54.310 --> 00:30:56.350 align:middle line:84% suppose I did something like this. 00:30:56.350 --> 00:30:59.975 align:middle line:84% Suppose I built a core structure that looked like this. 00:30:59.975 --> 00:31:07.390 align:middle line:90% 00:31:07.390 --> 00:31:12.470 align:middle line:90% And I put here's winding 1. 00:31:12.470 --> 00:31:16.670 align:middle line:90% And here's V1, I1. 00:31:16.670 --> 00:31:23.940 align:middle line:90% Here's winding 2, V2, I2. 00:31:23.940 --> 00:31:31.010 align:middle line:90% And here's winding 3, V3, I3. 00:31:31.010 --> 00:31:33.800 align:middle line:84% So my drawing is not very good here. 00:31:33.800 --> 00:31:37.640 align:middle line:84% But what you can see is now I have three legs. 00:31:37.640 --> 00:31:43.020 align:middle line:84% And each leg has his own winding. 00:31:43.020 --> 00:31:46.620 align:middle line:84% That means that the flux going through this here, 00:31:46.620 --> 00:31:50.040 align:middle line:84% or the flux linkage here, isn't directly related 00:31:50.040 --> 00:31:51.450 align:middle line:90% to these two flux linkages. 00:31:51.450 --> 00:31:53.320 align:middle line:90% Or it's not 1 to 1. 00:31:53.320 --> 00:31:56.910 align:middle line:84% It's not just scaled because here I only had one flux 00:31:56.910 --> 00:31:58.180 align:middle line:90% path going around the core. 00:31:58.180 --> 00:32:01.020 align:middle line:90% Ideally, here was my flux path. 00:32:01.020 --> 00:32:03.630 align:middle line:90% There was only one flux path. 00:32:03.630 --> 00:32:08.298 align:middle line:84% Here, there's a flux path here, and a different one here, 00:32:08.298 --> 00:32:09.340 align:middle line:90% and a different one here. 00:32:09.340 --> 00:32:14.480 align:middle line:84% And I get some constraints, but they're not the same fluxes. 00:32:14.480 --> 00:32:18.870 align:middle line:84% What does that mean in terms of the terminal relationships? 00:32:18.870 --> 00:32:21.230 align:middle line:84% Well, here's a magnetic circuit that I 00:32:21.230 --> 00:32:22.320 align:middle line:90% could draw for this thing. 00:32:22.320 --> 00:32:25.940 align:middle line:84% I could say, OK, maybe I will have 00:32:25.940 --> 00:32:31.550 align:middle line:90% N1 I1 and some reluctance 1. 00:32:31.550 --> 00:32:36.470 align:middle line:84% Here's some reluctance that's the middle path, reluctance 2 00:32:36.470 --> 00:32:39.110 align:middle line:90% and N2 I2. 00:32:39.110 --> 00:32:42.020 align:middle line:84% And here's the right half, N3 I3. 00:32:42.020 --> 00:32:46.920 align:middle line:90% 00:32:46.920 --> 00:32:52.625 align:middle line:84% And I could have phi 1, phi 2, and phi 3. 00:32:52.625 --> 00:32:58.970 align:middle line:90% 00:32:58.970 --> 00:33:03.510 align:middle line:84% What would be the terminal relations of this guy? 00:33:03.510 --> 00:33:09.300 align:middle line:84% Well, in this case, if I ignore leakage flux, 00:33:09.300 --> 00:33:12.580 align:middle line:84% what do I know about these three fluxes? 00:33:12.580 --> 00:33:15.330 align:middle line:84% Well, all these three fluxes come up into this top node. 00:33:15.330 --> 00:33:17.940 align:middle line:90% And the fluxes must add to 0. 00:33:17.940 --> 00:33:23.370 align:middle line:84% So what I get is phi 1 plus phi 2 plus phi 3 00:33:23.370 --> 00:33:27.360 align:middle line:90% has to equal 0 by magnetic KCL. 00:33:27.360 --> 00:33:29.850 align:middle line:90% There's no magnetic charge. 00:33:29.850 --> 00:33:34.500 align:middle line:84% Well, that means that lambda 1 over N1 00:33:34.500 --> 00:33:42.490 align:middle line:84% plus lambda 2 over N2 plus lambda 3 over N3 has to equal 0. 00:33:42.490 --> 00:33:46.530 align:middle line:84% Or if I differentiate this equation, what I'm going to get 00:33:46.530 --> 00:33:57.090 align:middle line:84% is V1 over N1 plus V2 over N2 plus V3 over N3 equals 0. 00:33:57.090 --> 00:34:03.310 align:middle line:84% So I no longer get the voltages on the winding scaling. 00:34:03.310 --> 00:34:06.720 align:middle line:84% What I get is the sum of the scaled voltages have to be 0. 00:34:06.720 --> 00:34:10.449 align:middle line:90% 00:34:10.449 --> 00:34:13.850 align:middle line:84% What can I say about the currents? 00:34:13.850 --> 00:34:21.440 align:middle line:84% Well, if I let reluctance 1, reluctance 2, and reluctance 3 00:34:21.440 --> 00:34:26.780 align:middle line:84% go to 0, if I assume I have an infinite permeability core, 00:34:26.780 --> 00:34:30.409 align:middle line:84% then all I get is these three MMF sources. 00:34:30.409 --> 00:34:33.530 align:middle line:84% And if I'm going to avoid infinite fluxes in the core, 00:34:33.530 --> 00:34:34.920 align:middle line:90% what would that require? 00:34:34.920 --> 00:34:39.290 align:middle line:84% That would require that N3 I3, N2 I1, and N1 I1 00:34:39.290 --> 00:34:41.120 align:middle line:90% equal each other. 00:34:41.120 --> 00:34:51.270 align:middle line:84% So I would get N1 I1 equals N2 I2 is equal to N3 I3. 00:34:51.270 --> 00:34:55.790 align:middle line:84% So in this parallel-wound transformer structure-- 00:34:55.790 --> 00:34:58.100 align:middle line:84% or parallel magnetic paths instead of series 00:34:58.100 --> 00:35:05.350 align:middle line:84% magnetic paths, the currents in the winding scale equally, 00:35:05.350 --> 00:35:10.120 align:middle line:84% ideally, and the sum of the scaled voltages adds to 0. 00:35:10.120 --> 00:35:13.480 align:middle line:84% It's exactly the dual of this structure. 00:35:13.480 --> 00:35:15.320 align:middle line:84% In fact, these are structural duals, 00:35:15.320 --> 00:35:19.360 align:middle line:84% so between series and parallel, so you get dual relationships 00:35:19.360 --> 00:35:22.510 align:middle line:90% between voltages and currents. 00:35:22.510 --> 00:35:25.410 align:middle line:90% Any questions about that? 00:35:25.410 --> 00:35:27.345 align:middle line:84% AUDIENCE: So it's not intuitive why 00:35:27.345 --> 00:35:28.845 align:middle line:84% you had a reluctance for each source 00:35:28.845 --> 00:35:31.940 align:middle line:84% and have them between the two sources. 00:35:31.940 --> 00:35:32.970 align:middle line:90% [INAUDIBLE] 00:35:32.970 --> 00:35:33.720 align:middle line:90% good for? 00:35:33.720 --> 00:35:35.250 align:middle line:90% DAVID PERREAULT: Well, I just-- 00:35:35.250 --> 00:35:36.640 align:middle line:90% yeah, that's a good question. 00:35:36.640 --> 00:35:40.000 align:middle line:84% I just said, OK, let me take this first section, 00:35:40.000 --> 00:35:42.390 align:middle line:84% like maybe this section of the core, 00:35:42.390 --> 00:35:45.970 align:middle line:84% and ascribe one reluctance that this winding's on. 00:35:45.970 --> 00:35:48.990 align:middle line:84% And let me ascribe a second section of the core, 00:35:48.990 --> 00:35:52.203 align:middle line:84% like here, that this has wound on. 00:35:52.203 --> 00:35:54.370 align:middle line:84% And then the third section of the core is over here. 00:35:54.370 --> 00:35:56.200 align:middle line:90% And that's the third reluctance. 00:35:56.200 --> 00:35:57.940 align:middle line:84% You could break it up a different way, 00:35:57.940 --> 00:36:00.440 align:middle line:84% but that's a one way you could think about it. 00:36:00.440 --> 00:36:03.820 align:middle line:84% So I've got three windings and three flux 00:36:03.820 --> 00:36:07.355 align:middle line:84% pipes that are all going between the top and the bottom. 00:36:07.355 --> 00:36:08.230 align:middle line:90% Does that make sense? 00:36:08.230 --> 00:36:12.770 align:middle line:90% 00:36:12.770 --> 00:36:16.100 align:middle line:84% This structure is less used than the other structure, 00:36:16.100 --> 00:36:17.760 align:middle line:84% but it's actually quite widely used. 00:36:17.760 --> 00:36:18.260 align:middle line:90% Why? 00:36:18.260 --> 00:36:20.240 align:middle line:84% Because there are certain cases where maybe I 00:36:20.240 --> 00:36:22.190 align:middle line:84% have different parts of my circuit 00:36:22.190 --> 00:36:24.600 align:middle line:84% and I want to force them to all have the same current. 00:36:24.600 --> 00:36:26.900 align:middle line:84% So if I have N1 equal N2 equal N3, 00:36:26.900 --> 00:36:30.530 align:middle line:84% I can force currents to split or be matched. 00:36:30.530 --> 00:36:32.840 align:middle line:84% So if I want to match voltages, that's 00:36:32.840 --> 00:36:34.500 align:middle line:90% a great transformer structure. 00:36:34.500 --> 00:36:36.260 align:middle line:84% If I want to force currents to be matched, 00:36:36.260 --> 00:36:38.570 align:middle line:84% this is a great transformer structure. 00:36:38.570 --> 00:36:40.620 align:middle line:84% It just depends what you want to do. 00:36:40.620 --> 00:36:42.480 align:middle line:84% And of course, with many windings, 00:36:42.480 --> 00:36:44.830 align:middle line:84% there's all kinds of stuff you can do. 00:36:44.830 --> 00:36:46.680 align:middle line:84% There's all kinds of magnetic structures 00:36:46.680 --> 00:36:51.150 align:middle line:84% you can and might build for different applications. 00:36:51.150 --> 00:36:54.090 align:middle line:84% I should say, by the way, however, notice that, 00:36:54.090 --> 00:36:56.150 align:middle line:84% because there's no single flux path, 00:36:56.150 --> 00:36:59.235 align:middle line:84% there's not really a good way to indicate the dot convention 00:36:59.235 --> 00:36:59.735 align:middle line:90% here. 00:36:59.735 --> 00:37:02.450 align:middle line:90% 00:37:02.450 --> 00:37:05.120 align:middle line:84% There's something where the currents into the winding scale, 00:37:05.120 --> 00:37:09.810 align:middle line:84% but we don't usually represent those with dots. 00:37:09.810 --> 00:37:11.347 align:middle line:84% People usually draw the structure 00:37:11.347 --> 00:37:12.430 align:middle line:90% or they do something else. 00:37:12.430 --> 00:37:14.190 align:middle line:84% There's not a really perfect schematic, 00:37:14.190 --> 00:37:17.315 align:middle line:84% widely accepted schematic relationship between those. 00:37:17.315 --> 00:37:22.000 align:middle line:90% 00:37:22.000 --> 00:37:25.302 align:middle line:90% Let's talk about the non-ideal. 00:37:25.302 --> 00:37:27.760 align:middle line:84% These are the equivalent of ideal transformer relationships 00:37:27.760 --> 00:37:29.093 align:middle line:90% for a three-winding transformer. 00:37:29.093 --> 00:37:30.910 align:middle line:84% You could extend that for N windings. 00:37:30.910 --> 00:37:35.140 align:middle line:84% If I had N windings on one core or N parallel core sections, 00:37:35.140 --> 00:37:40.390 align:middle line:84% you just expand those relationships naturally. 00:37:40.390 --> 00:37:44.630 align:middle line:84% What would I get if I included parasitics? 00:37:44.630 --> 00:37:49.060 align:middle line:84% I would get a 3 by 3 inductance matrix. 00:37:49.060 --> 00:37:51.353 align:middle line:90% So what do I get, in this case? 00:37:51.353 --> 00:37:54.910 align:middle line:90% 00:37:54.910 --> 00:37:57.430 align:middle line:84% Well, what I would get in the general case 00:37:57.430 --> 00:38:12.760 align:middle line:84% is V1 V2 V3 is equal to some inductance matrix, L11 L22 L33. 00:38:12.760 --> 00:38:19.740 align:middle line:84% Because it's reciprocal, I would have a symmetric matrix. 00:38:19.740 --> 00:38:24.845 align:middle line:84% So what I would have is L21 would equal L12. 00:38:24.845 --> 00:38:28.340 align:middle line:90% L31 would equal L13. 00:38:28.340 --> 00:38:40.730 align:middle line:84% And L32 would equal L23 times d/dT I1 I2 I3. 00:38:40.730 --> 00:38:46.370 align:middle line:84% Maybe I would write it as I1 I2 I3 dot. 00:38:46.370 --> 00:38:52.920 align:middle line:84% So this is V is equal to L matrix I dot for dI/dT. 00:38:52.920 --> 00:38:57.460 align:middle line:90% 00:38:57.460 --> 00:39:02.414 align:middle line:84% How many independent parameters do I have in this structure? 00:39:02.414 --> 00:39:05.620 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:39:05.620 --> 00:39:08.050 align:middle line:90% DAVID PERREAULT: Six, yes. 00:39:08.050 --> 00:39:10.940 align:middle line:84% I have my three self-inductances. 00:39:10.940 --> 00:39:12.640 align:middle line:84% Those self-inductances are-- this 00:39:12.640 --> 00:39:15.790 align:middle line:84% is the inductance at port 1 with nothing else connected to it. 00:39:15.790 --> 00:39:17.290 align:middle line:84% So if I just measured it like it was 00:39:17.290 --> 00:39:19.873 align:middle line:84% an inductor from one port with everything else open circuited. 00:39:19.873 --> 00:39:21.130 align:middle line:90% I get L11. 00:39:21.130 --> 00:39:25.030 align:middle line:84% Same thing for measuring L2, same thing for measuring port 3. 00:39:25.030 --> 00:39:29.620 align:middle line:84% Then I have L21, which is equal to L12, L31, which 00:39:29.620 --> 00:39:35.170 align:middle line:84% is equal to L13, and L32, which is equal to L23. 00:39:35.170 --> 00:39:42.630 align:middle line:84% So if I have a single inductor, I have one free parameter. 00:39:42.630 --> 00:39:46.350 align:middle line:84% If I have a 2 by 2, I have three parameters. 00:39:46.350 --> 00:39:51.080 align:middle line:84% If I have a 3 by 3, I have six parameters. 00:39:51.080 --> 00:39:53.330 align:middle line:84% What would happen if I had a four-winding transformer 00:39:53.330 --> 00:39:54.163 align:middle line:90% in the general case? 00:39:54.163 --> 00:39:57.290 align:middle line:90% Well, I'd add a V4 and an I4. 00:39:57.290 --> 00:40:02.660 align:middle line:84% And then I'd basically add one more bottom row 00:40:02.660 --> 00:40:04.850 align:middle line:84% of independent parameters, four parameters. 00:40:04.850 --> 00:40:07.500 align:middle line:90% And I'd have 10. 00:40:07.500 --> 00:40:14.460 align:middle line:84% So it turns out that an N-winding transformer ends up 00:40:14.460 --> 00:40:19.356 align:middle line:84% with N times N plus 1 over 2 independent parameters. 00:40:19.356 --> 00:40:22.520 align:middle line:90% 00:40:22.520 --> 00:40:28.375 align:middle line:84% So it's 1, 3, 6, 10, and so forth. 00:40:28.375 --> 00:40:31.190 align:middle line:90% 00:40:31.190 --> 00:40:34.160 align:middle line:84% That gets kind of ugly, kind of quick. 00:40:34.160 --> 00:40:36.390 align:middle line:90% But that's the way it is. 00:40:36.390 --> 00:40:38.280 align:middle line:90% That's the general case. 00:40:38.280 --> 00:40:40.250 align:middle line:84% And this inductance matrix, still, 00:40:40.250 --> 00:40:41.690 align:middle line:84% for the same reason I said, it has 00:40:41.690 --> 00:40:43.606 align:middle line:90% to be positive semi-definite. 00:40:43.606 --> 00:40:47.120 align:middle line:90% 00:40:47.120 --> 00:40:48.400 align:middle line:90% Any questions about that? 00:40:48.400 --> 00:40:53.300 align:middle line:90% 00:40:53.300 --> 00:40:54.850 align:middle line:90% So now, what might I want? 00:40:54.850 --> 00:40:59.990 align:middle line:90% 00:40:59.990 --> 00:41:02.240 align:middle line:90% I might want the equivalent. 00:41:02.240 --> 00:41:05.420 align:middle line:84% I've already said, OK, I have an inductance matrix. 00:41:05.420 --> 00:41:13.220 align:middle line:84% And if I come up to my device and I measure V1, and I1, 00:41:13.220 --> 00:41:15.890 align:middle line:84% and have all the other I0 open circuits, I can get L11. 00:41:15.890 --> 00:41:20.960 align:middle line:84% I can do a bunch of measurements on my device 00:41:20.960 --> 00:41:25.220 align:middle line:84% and figure out what these independent parameters are. 00:41:25.220 --> 00:41:27.560 align:middle line:90% That can be tricky. 00:41:27.560 --> 00:41:29.060 align:middle line:84% If you don't do it carefully, you'll 00:41:29.060 --> 00:41:30.800 align:middle line:84% get stuff that's not quite right. 00:41:30.800 --> 00:41:35.590 align:middle line:84% But you can get the terminal relationships. 00:41:35.590 --> 00:41:37.920 align:middle line:84% And so I can always get the inductance matrix 00:41:37.920 --> 00:41:40.100 align:middle line:84% by doing measurements on a given device. 00:41:40.100 --> 00:41:43.290 align:middle line:90% 00:41:43.290 --> 00:41:45.620 align:middle line:90% How would I get a circuit model? 00:41:45.620 --> 00:41:48.980 align:middle line:84% What would my circuit model look like? 00:41:48.980 --> 00:41:51.760 align:middle line:84% Well, that's a trickier question. 00:41:51.760 --> 00:41:56.380 align:middle line:84% It's possible to do a physical model 00:41:56.380 --> 00:42:02.170 align:middle line:84% like this model with actual physical parameters. 00:42:02.170 --> 00:42:05.630 align:middle line:84% But that turns out to very often be quite tricky. 00:42:05.630 --> 00:42:10.540 align:middle line:84% And as you can see, this already has a bunch of parameters to it. 00:42:10.540 --> 00:42:12.230 align:middle line:90% But what might we do? 00:42:12.230 --> 00:42:14.770 align:middle line:84% Suppose I wanted to just say, hey, 00:42:14.770 --> 00:42:17.570 align:middle line:84% I'm not going to worry about getting a physical model. 00:42:17.570 --> 00:42:21.430 align:middle line:84% Just give me a model that has just enough independent 00:42:21.430 --> 00:42:26.680 align:middle line:84% parameters to represent the three-port, or four-port, 00:42:26.680 --> 00:42:30.430 align:middle line:90% or five-port network I have. 00:42:30.430 --> 00:42:34.480 align:middle line:84% Now, there's, in general, a lot of ways you could do that. 00:42:34.480 --> 00:42:39.190 align:middle line:84% I'm going to show you one that I particularly like. 00:42:39.190 --> 00:42:43.170 align:middle line:84% It's called the extended cantilever model. 00:42:43.170 --> 00:42:45.330 align:middle line:84% It's by no means the only way to do it, 00:42:45.330 --> 00:42:48.270 align:middle line:84% but it's a very clean means of doing it. 00:42:48.270 --> 00:42:51.960 align:middle line:84% So when we do circuit modeling for all kinds of devices, 00:42:51.960 --> 00:42:54.540 align:middle line:84% we often use the extended cantilever model. 00:42:54.540 --> 00:42:56.473 align:middle line:84% And it goes a little bit like this. 00:42:56.473 --> 00:42:59.280 align:middle line:90% 00:42:59.280 --> 00:43:02.975 align:middle line:84% Suppose I had a one-terminal device. 00:43:02.975 --> 00:43:05.600 align:middle line:90% 00:43:05.600 --> 00:43:08.870 align:middle line:84% Maybe I would come up here and I'd say, OK, here's 00:43:08.870 --> 00:43:10.820 align:middle line:90% my one-terminal device. 00:43:10.820 --> 00:43:14.390 align:middle line:84% The inductance-- and I'm going to-- 00:43:14.390 --> 00:43:17.940 align:middle line:84% maybe I'll just draw this with a ground reference 00:43:17.940 --> 00:43:19.400 align:middle line:90% here for simplicity. 00:43:19.400 --> 00:43:23.820 align:middle line:84% This is my inductance L. But maybe I'll call it L11. 00:43:23.820 --> 00:43:26.640 align:middle line:84% And I'm going to use little l11 to distinguish it 00:43:26.640 --> 00:43:28.680 align:middle line:90% from my inductance matrix value. 00:43:28.680 --> 00:43:35.400 align:middle line:84% But if I just had a single device with V1 and I1, 00:43:35.400 --> 00:43:37.140 align:middle line:84% I'd need one parameter that I'm going 00:43:37.140 --> 00:43:38.670 align:middle line:84% to call little l11, which is just 00:43:38.670 --> 00:43:40.550 align:middle line:84% the inductance of that one-turn winding. 00:43:40.550 --> 00:43:44.345 align:middle line:90% 00:43:44.345 --> 00:43:49.730 align:middle line:84% So I have one parameter here that captures what I want. 00:43:49.730 --> 00:43:52.360 align:middle line:90% 00:43:52.360 --> 00:43:54.880 align:middle line:84% How many more parameters would I need in general 00:43:54.880 --> 00:44:00.140 align:middle line:84% to model a two-port device, two-winding transformer? 00:44:00.140 --> 00:44:02.533 align:middle line:90% 00:44:02.533 --> 00:44:03.200 align:middle line:90% AUDIENCE: Three. 00:44:03.200 --> 00:44:04.740 align:middle line:90% DAVID PERREAULT: Three, right? 00:44:04.740 --> 00:44:11.840 align:middle line:84% So I somehow need to extend this model so that I would have 00:44:11.840 --> 00:44:15.080 align:middle line:90% two additional free parameters. 00:44:15.080 --> 00:44:17.300 align:middle line:84% And if I want to keep it as simple as possible, 00:44:17.300 --> 00:44:20.073 align:middle line:84% I want only two additional free parameters. 00:44:20.073 --> 00:44:21.740 align:middle line:84% And again, I'm stepping back and saying, 00:44:21.740 --> 00:44:23.960 align:middle line:84% I don't care if their physical parameters. 00:44:23.960 --> 00:44:27.920 align:middle line:84% I just want them to capture the terminal 00:44:27.920 --> 00:44:29.600 align:middle line:90% relations of the device. 00:44:29.600 --> 00:44:35.000 align:middle line:84% So maybe I would come up and say, OK, I'm 00:44:35.000 --> 00:44:41.270 align:middle line:84% going to go add one more inductance. 00:44:41.270 --> 00:44:42.980 align:middle line:90% I'll call this L12. 00:44:42.980 --> 00:44:45.920 align:middle line:90% 00:44:45.920 --> 00:44:54.130 align:middle line:84% And then I'm going to add an ideal transformer 00:44:54.130 --> 00:45:01.290 align:middle line:84% and give it a transformation ratio of 1 to N2. 00:45:01.290 --> 00:45:05.760 align:middle line:84% And then I'll call this V2 and I2. 00:45:05.760 --> 00:45:11.690 align:middle line:90% 00:45:11.690 --> 00:45:18.350 align:middle line:84% So now, this structure has a second parameter 00:45:18.350 --> 00:45:21.820 align:middle line:90% and a third parameter. 00:45:21.820 --> 00:45:23.350 align:middle line:84% Does that make sense to everybody? 00:45:23.350 --> 00:45:25.880 align:middle line:84% So clearly, I have three parameters here. 00:45:25.880 --> 00:45:28.000 align:middle line:84% If I pick the parameters right, I can match a 2 00:45:28.000 --> 00:45:30.280 align:middle line:90% by 2 inductance matrix. 00:45:30.280 --> 00:45:33.830 align:middle line:84% Does that make sense to everybody? 00:45:33.830 --> 00:45:36.080 align:middle line:90% What is this model? 00:45:36.080 --> 00:45:38.900 align:middle line:84% Well, this model, how does this model just-- 00:45:38.900 --> 00:45:40.647 align:middle line:84% recognize this is connected here. 00:45:40.647 --> 00:45:42.480 align:middle line:84% I'm just drawing it for simplicity that way. 00:45:42.480 --> 00:45:48.320 align:middle line:84% This model is just like this model, except that all I've done 00:45:48.320 --> 00:45:51.590 align:middle line:90% is I've made L leakage 1 0. 00:45:51.590 --> 00:45:53.420 align:middle line:90% So then I have this. 00:45:53.420 --> 00:45:56.400 align:middle line:90% This is one inductance. 00:45:56.400 --> 00:45:58.250 align:middle line:84% And then I took L leakage 2 and I put it 00:45:58.250 --> 00:46:00.810 align:middle line:84% on the other side of the transformer. 00:46:00.810 --> 00:46:06.120 align:middle line:84% So my three parameters are this, the turns ratio, and this guy. 00:46:06.120 --> 00:46:08.610 align:middle line:90% And I've set this one to 0. 00:46:08.610 --> 00:46:10.950 align:middle line:90% So it's a non-physical thing. 00:46:10.950 --> 00:46:16.290 align:middle line:84% But it gives me the right set of parameters to model any 2 by 2. 00:46:16.290 --> 00:46:18.750 align:middle line:84% So that's sometimes called the cantilever model, 00:46:18.750 --> 00:46:23.466 align:middle line:84% the so-named because you have this L of inductances. 00:46:23.466 --> 00:46:29.480 align:middle line:90% 00:46:29.480 --> 00:46:30.210 align:middle line:90% That's fine. 00:46:30.210 --> 00:46:32.208 align:middle line:90% I've got three total parameters. 00:46:32.208 --> 00:46:34.250 align:middle line:84% I can model any two-winding transformer that way. 00:46:34.250 --> 00:46:36.083 align:middle line:84% They're not going to be physical parameters. 00:46:36.083 --> 00:46:39.260 align:middle line:84% And by the way, I should have warned you before, 00:46:39.260 --> 00:46:46.580 align:middle line:84% if I have this model for a 2 by 2, what I know is that all of L 00:46:46.580 --> 00:46:50.540 align:middle line:84% leakage 1 L mu 1 and L leakage 2 all have 00:46:50.540 --> 00:46:51.830 align:middle line:90% to be positive inductances. 00:46:51.830 --> 00:46:52.530 align:middle line:90% Why? 00:46:52.530 --> 00:46:55.560 align:middle line:84% Because they each represent some energy stored in space. 00:46:55.560 --> 00:46:59.480 align:middle line:84% So those are physical values of inductance representing 00:46:59.480 --> 00:47:01.050 align:middle line:90% physical energy storage. 00:47:01.050 --> 00:47:03.720 align:middle line:84% And this turns ratio is the physical turns ratio. 00:47:03.720 --> 00:47:06.980 align:middle line:84% If I do anything else, if I arbitrarily 00:47:06.980 --> 00:47:09.740 align:middle line:84% pick one of these models, there's no guarantee 00:47:09.740 --> 00:47:11.630 align:middle line:84% that the inductances in my parameter 00:47:11.630 --> 00:47:13.887 align:middle line:84% end up being positive or anything else. 00:47:13.887 --> 00:47:15.470 align:middle line:84% They're going to be whatever they need 00:47:15.470 --> 00:47:20.400 align:middle line:90% to match the terminal behavior. 00:47:20.400 --> 00:47:26.400 align:middle line:84% So you can easily get some negative inductance value 00:47:26.400 --> 00:47:31.350 align:middle line:84% in your model if you're not using a physical model. 00:47:31.350 --> 00:47:33.518 align:middle line:84% And so don't get concerned if you see that. 00:47:33.518 --> 00:47:35.060 align:middle line:84% And that's just because you're trying 00:47:35.060 --> 00:47:39.110 align:middle line:84% to pick parameters to really match what the inductance matrix 00:47:39.110 --> 00:47:41.280 align:middle line:90% description does. 00:47:41.280 --> 00:47:43.560 align:middle line:84% And whatever those circuit parameters turn out to be, 00:47:43.560 --> 00:47:49.700 align:middle line:84% you'll get the right inductance matrix description. 00:47:49.700 --> 00:47:54.750 align:middle line:84% So this is great for a two-winding transformer. 00:47:54.750 --> 00:47:57.060 align:middle line:84% I have three parameters for my two-winding transformer. 00:47:57.060 --> 00:48:01.320 align:middle line:84% How many do I need for my three-winding transformer? 00:48:01.320 --> 00:48:02.530 align:middle line:90% I need six total. 00:48:02.530 --> 00:48:04.560 align:middle line:84% That means I need three new ones. 00:48:04.560 --> 00:48:14.370 align:middle line:84% So if I'm going to add on a V3 and I3, 00:48:14.370 --> 00:48:15.970 align:middle line:90% I need three more parameters. 00:48:15.970 --> 00:48:17.310 align:middle line:90% So what do I add on? 00:48:17.310 --> 00:48:22.290 align:middle line:84% I add on one transformer ratio as a parameter, 1 to N3. 00:48:22.290 --> 00:48:25.790 align:middle line:90% So here's one new parameter. 00:48:25.790 --> 00:48:27.400 align:middle line:90% And now, I need two more. 00:48:27.400 --> 00:48:32.750 align:middle line:84% Well, I can model the other two just with an inductance 00:48:32.750 --> 00:48:39.220 align:middle line:84% from here to here, which I'll call L23, 00:48:39.220 --> 00:48:44.490 align:middle line:84% and inductance from here to here, which I'll call L13. 00:48:44.490 --> 00:48:48.180 align:middle line:84% And that gives me my six parameters. 00:48:48.180 --> 00:48:51.690 align:middle line:84% And I can tie this one to the same reference point. 00:48:51.690 --> 00:48:54.780 align:middle line:84% So you can see where this is going. 00:48:54.780 --> 00:49:00.270 align:middle line:84% Every time I want to add a new term, a new port, 00:49:00.270 --> 00:49:06.960 align:middle line:84% I add in one transformer with some transformation ratio, 00:49:06.960 --> 00:49:09.330 align:middle line:90% and then a leakage-- 00:49:09.330 --> 00:49:12.120 align:middle line:84% quote unquote, "leakage--" to each other internal node 00:49:12.120 --> 00:49:13.200 align:middle line:90% in the circuit. 00:49:13.200 --> 00:49:14.520 align:middle line:90% And I can expand out. 00:49:14.520 --> 00:49:17.790 align:middle line:84% And I'll always have exactly the right number of parameters 00:49:17.790 --> 00:49:23.466 align:middle line:84% to match the n by n inductance matrix. 00:49:23.466 --> 00:49:27.610 align:middle line:84% Does that make sense to everybody? 00:49:27.610 --> 00:49:29.300 align:middle line:90% So I'm running out of time. 00:49:29.300 --> 00:49:31.720 align:middle line:84% But what I'll tell you about this model-- and the details 00:49:31.720 --> 00:49:35.080 align:middle line:90% are all in the text. 00:49:35.080 --> 00:49:38.170 align:middle line:84% The nice thing about this model is, firstly, it 00:49:38.170 --> 00:49:40.900 align:middle line:84% turns out there's always a set of measurements 00:49:40.900 --> 00:49:44.530 align:middle line:84% you can do on your circuit that will reveal directly 00:49:44.530 --> 00:49:48.020 align:middle line:90% what these ratios are. 00:49:48.020 --> 00:49:49.973 align:middle line:84% So for example, I can find L11 just 00:49:49.973 --> 00:49:52.390 align:middle line:84% by open circuiting everything and measuring the inductance 00:49:52.390 --> 00:49:53.540 align:middle line:90% at this port. 00:49:53.540 --> 00:49:56.620 align:middle line:84% If I put a voltage here and I measure all the voltages 00:49:56.620 --> 00:49:59.780 align:middle line:84% at the other terminals, that gives N2, N3, and so forth. 00:49:59.780 --> 00:50:02.200 align:middle line:84% There's a set of measurements I can do that will always 00:50:02.200 --> 00:50:04.120 align:middle line:90% reveal these parameters. 00:50:04.120 --> 00:50:06.370 align:middle line:84% So if I want to create a model for a physical device, 00:50:06.370 --> 00:50:07.900 align:middle line:90% I can do that. 00:50:07.900 --> 00:50:10.720 align:middle line:84% It also turns out that all these parameters 00:50:10.720 --> 00:50:15.350 align:middle line:84% can be directly related back to the inductance matrix values. 00:50:15.350 --> 00:50:17.690 align:middle line:84% And I put the equations in there. 00:50:17.690 --> 00:50:21.510 align:middle line:90% And it's exactly how you do it. 00:50:21.510 --> 00:50:23.840 align:middle line:84% But the nice thing is, if I have an inductance matrix 00:50:23.840 --> 00:50:25.340 align:middle line:84% description of the thing I wanted, 00:50:25.340 --> 00:50:27.390 align:middle line:84% I can create a circuit model that has that. 00:50:27.390 --> 00:50:29.240 align:middle line:84% This is a non-physical circuit model, 00:50:29.240 --> 00:50:33.100 align:middle line:84% but it matches the terminal relationships. 00:50:33.100 --> 00:50:34.400 align:middle line:90% So I'm out of time. 00:50:34.400 --> 00:50:37.780 align:middle line:84% We've expanded from inductance to 00:50:37.780 --> 00:50:39.800 align:middle line:90% n-winding magnetic structures. 00:50:39.800 --> 00:50:41.640 align:middle line:84% Are there any questions before we wrap up? 00:50:41.640 --> 00:50:46.610 align:middle line:90% 00:50:46.610 --> 00:50:47.600 align:middle line:90% OK, have a great day. 00:50:47.600 --> 00:50:49.780 align:middle line:90% And we'll see you tomorrow. 00:50:49.780 --> 00:50:54.000 align:middle line:90%