WEBVTT 00:00:00.000 --> 00:00:02.390 align:middle line:90% [SQUEAKING] 00:00:02.390 --> 00:00:04.302 align:middle line:90% [RUSTLING] 00:00:04.302 --> 00:00:05.258 align:middle line:90% [CLICKING] 00:00:05.258 --> 00:00:08.782 align:middle line:90% 00:00:08.782 --> 00:00:10.240 align:middle line:84% DAVID PERREAULT: OK, glad everybody 00:00:10.240 --> 00:00:11.282 align:middle line:90% made it through the snow. 00:00:11.282 --> 00:00:14.343 align:middle line:90% Why don't we get started? 00:00:14.343 --> 00:00:15.760 align:middle line:84% So what we're going to do is today 00:00:15.760 --> 00:00:19.720 align:middle line:84% is we're going to start talking about magnetics, because 00:00:19.720 --> 00:00:21.610 align:middle line:84% in the middle of most power converters, 00:00:21.610 --> 00:00:23.860 align:middle line:84% there's some magnetic component or more than one 00:00:23.860 --> 00:00:25.855 align:middle line:84% magnetic component, so we really ought 00:00:25.855 --> 00:00:28.930 align:middle line:84% to understand them and know how to design them. 00:00:28.930 --> 00:00:31.810 align:middle line:84% What I'm going to be talking about today in particular 00:00:31.810 --> 00:00:34.750 align:middle line:84% is in Principles of Power Electronics chapter 18, 00:00:34.750 --> 00:00:37.000 align:middle line:84% so I'll try to make sure that's in the correct reading 00:00:37.000 --> 00:00:39.790 align:middle line:90% for the homework set. 00:00:39.790 --> 00:00:43.360 align:middle line:84% And what I'd like to do is start by just reminding everybody 00:00:43.360 --> 00:00:46.930 align:middle line:84% of how this was likely treated in your introduction 00:00:46.930 --> 00:00:50.230 align:middle line:84% to physics class, straight from Maxwell's equations. 00:00:50.230 --> 00:00:53.650 align:middle line:84% And then I want to sort of step back and come up 00:00:53.650 --> 00:00:56.770 align:middle line:84% with some models that simplify things, magnetic circuit 00:00:56.770 --> 00:00:59.770 align:middle line:84% models that let you understand the behavior 00:00:59.770 --> 00:01:01.510 align:middle line:84% of magnetic components without having 00:01:01.510 --> 00:01:04.420 align:middle line:84% to write Maxwell's equations left, right, and center. 00:01:04.420 --> 00:01:05.110 align:middle line:90% OK. 00:01:05.110 --> 00:01:09.170 align:middle line:84% So let's just remind ourselves of the integral form 00:01:09.170 --> 00:01:10.640 align:middle line:90% of Maxwell's equations, right. 00:01:10.640 --> 00:01:17.830 align:middle line:84% So we have Ampere's law, right, which basically 00:01:17.830 --> 00:01:25.230 align:middle line:84% says the integral of H dot dl is equal to the integral of J dot 00:01:25.230 --> 00:01:32.500 align:middle line:84% dA plus d dt the integral of epsilon E dot dA. 00:01:32.500 --> 00:01:33.200 align:middle line:90% All right. 00:01:33.200 --> 00:01:34.820 align:middle line:90% This is displacement current. 00:01:34.820 --> 00:01:39.370 align:middle line:84% We're going to do the MQS approximation, 00:01:39.370 --> 00:01:41.260 align:middle line:84% magnetoquasistatic approximation, where 00:01:41.260 --> 00:01:45.228 align:middle line:84% we ignore the effect of changing electric fields. 00:01:45.228 --> 00:01:47.020 align:middle line:84% We're ignoring displacement current, right, 00:01:47.020 --> 00:01:50.260 align:middle line:84% so basically, this just says that if I integrate 00:01:50.260 --> 00:01:52.840 align:middle line:84% the magnetic field around a loop, 00:01:52.840 --> 00:01:55.900 align:middle line:84% it basically equals the net current that's 00:01:55.900 --> 00:01:57.790 align:middle line:90% passing through that loop. 00:01:57.790 --> 00:01:59.970 align:middle line:90% OK. 00:01:59.970 --> 00:02:01.740 align:middle line:84% The next thing we are interested in 00:02:01.740 --> 00:02:11.430 align:middle line:84% is Faraday's law, right, which says that the integral of E 00:02:11.430 --> 00:02:13.980 align:middle line:84% dot dl around a loop, the integral of the electric field 00:02:13.980 --> 00:02:19.672 align:middle line:84% around a loop, is minus the derivative of the integral of B 00:02:19.672 --> 00:02:22.170 align:middle line:84% dot dA-- that is, minus the derivative 00:02:22.170 --> 00:02:26.550 align:middle line:84% of the total magnetic flux linked by that loop. 00:02:26.550 --> 00:02:29.790 align:middle line:84% OK, that's sort of the source of induced voltage. 00:02:29.790 --> 00:02:31.185 align:middle line:90% OK. 00:02:31.185 --> 00:02:33.060 align:middle line:84% The third one we're going to be interested in 00:02:33.060 --> 00:02:40.520 align:middle line:84% is flux continuity, which is basically saying 00:02:40.520 --> 00:02:42.150 align:middle line:90% there is no magnetic charge-- 00:02:42.150 --> 00:02:47.900 align:middle line:84% that is, the integral of B dot dA, the net flux coming out 00:02:47.900 --> 00:02:51.820 align:middle line:90% of some volume, is zero. 00:02:51.820 --> 00:02:52.320 align:middle line:90% OK. 00:02:52.320 --> 00:02:54.110 align:middle line:84% So these are what we're going to be 00:02:54.110 --> 00:02:56.580 align:middle line:84% concerned with in terms of designing magnetic components. 00:02:56.580 --> 00:02:58.247 align:middle line:84% There's also Gauss's law, but we're just 00:02:58.247 --> 00:02:59.960 align:middle line:90% not going to use Gauss's law. 00:02:59.960 --> 00:03:01.712 align:middle line:90% OK. 00:03:01.712 --> 00:03:16.450 align:middle line:84% Just as a reminder here, H is magnetic field intensity, 00:03:16.450 --> 00:03:20.350 align:middle line:84% which we usually use as amperes per meter as units. 00:03:20.350 --> 00:03:32.240 align:middle line:84% B is magnetic flux density, and that's usually 00:03:32.240 --> 00:03:34.370 align:middle line:90% in units of tesla. 00:03:34.370 --> 00:03:36.194 align:middle line:90% OK. 00:03:36.194 --> 00:03:42.350 align:middle line:84% And there's some relationship between B and H, right. 00:03:42.350 --> 00:03:44.990 align:middle line:84% More generally, right, B is equal to mu 00:03:44.990 --> 00:03:49.490 align:middle line:84% 0 H plus the magnetization that relates to what's 00:03:49.490 --> 00:03:51.890 align:middle line:90% going on in the material. 00:03:51.890 --> 00:03:56.840 align:middle line:84% For our purposes, we're going to consider just soft, permeable 00:03:56.840 --> 00:03:59.780 align:middle line:84% materials, and we're going to say, OK, 00:03:59.780 --> 00:04:05.300 align:middle line:84% we can write B as being equal to mu H, where 00:04:05.300 --> 00:04:07.386 align:middle line:90% mu is the permeability. 00:04:07.386 --> 00:04:12.790 align:middle line:90% 00:04:12.790 --> 00:04:15.080 align:middle line:90% OK. 00:04:15.080 --> 00:04:18.220 align:middle line:84% And so we're assuming B and H are proportional. 00:04:18.220 --> 00:04:18.720 align:middle line:90% OK. 00:04:18.720 --> 00:04:23.820 align:middle line:84% And remember that mu 0, the permeability of free space, 00:04:23.820 --> 00:04:29.640 align:middle line:84% is 4 pi times 10 to the minus seventh henries per meter. 00:04:29.640 --> 00:04:31.125 align:middle line:90% OK. 00:04:31.125 --> 00:04:36.240 align:middle line:84% In general, I should say that this is a simplification, right, 00:04:36.240 --> 00:04:40.650 align:middle line:84% that happens when you have magnetic domains that can 00:04:40.650 --> 00:04:43.270 align:middle line:90% align due to an imposed field. 00:04:43.270 --> 00:04:48.210 align:middle line:84% More generally, one might have B is some function of H, 00:04:48.210 --> 00:04:51.120 align:middle line:84% and in fact, it might even be a function of the instantaneous H 00:04:51.120 --> 00:04:52.210 align:middle line:90% field and history. 00:04:52.210 --> 00:04:52.710 align:middle line:90% OK. 00:04:52.710 --> 00:04:56.910 align:middle line:84% So in general, the relationship between B and H 00:04:56.910 --> 00:05:01.110 align:middle line:84% can be somewhat complex, and we'll see that moving forward. 00:05:01.110 --> 00:05:02.910 align:middle line:84% But for simplicity, let's just think 00:05:02.910 --> 00:05:05.970 align:middle line:84% about having some permeability mu which 00:05:05.970 --> 00:05:09.550 align:middle line:84% is some relative permeability times mu 0. 00:05:09.550 --> 00:05:10.900 align:middle line:90% OK. 00:05:10.900 --> 00:05:15.680 align:middle line:84% So those are our basic rules that we're going to work with, 00:05:15.680 --> 00:05:19.450 align:middle line:84% and hopefully, you recall these from introductory physics 00:05:19.450 --> 00:05:22.970 align:middle line:84% or any more advanced magnetics class. 00:05:22.970 --> 00:05:28.070 align:middle line:84% Let's consider calculation of an inductance, right. 00:05:28.070 --> 00:05:29.570 align:middle line:84% So this is something that usually 00:05:29.570 --> 00:05:31.560 align:middle line:84% most introductory physics classes do. 00:05:31.560 --> 00:05:33.440 align:middle line:84% And I want to go through it in part 00:05:33.440 --> 00:05:36.450 align:middle line:84% to remind you of how we use these rules to get to it 00:05:36.450 --> 00:05:39.768 align:middle line:84% and because it'll also set up sort of the better way 00:05:39.768 --> 00:05:41.810 align:middle line:84% to do it-- or not the better way, the simpler way 00:05:41.810 --> 00:05:43.470 align:middle line:90% to handle all these things. 00:05:43.470 --> 00:05:43.970 align:middle line:90% OK. 00:05:43.970 --> 00:05:48.440 align:middle line:84% So let's suppose we just have a toroid. 00:05:48.440 --> 00:05:50.645 align:middle line:90% OK. 00:05:50.645 --> 00:05:54.990 align:middle line:84% So here we have some toroidal structure. 00:05:54.990 --> 00:05:56.600 align:middle line:90% OK. 00:05:56.600 --> 00:05:59.540 align:middle line:84% And I'm going to come up and I'm going to put a set of windings 00:05:59.540 --> 00:06:00.620 align:middle line:90% on it, right. 00:06:00.620 --> 00:06:04.460 align:middle line:84% So here's some set of turns that I'm going to put on it, of wire, 00:06:04.460 --> 00:06:06.680 align:middle line:90% right, like this. 00:06:06.680 --> 00:06:10.850 align:middle line:84% And let's suppose I put N turns of wire on it, 00:06:10.850 --> 00:06:15.320 align:middle line:84% and I have some current flowing into it, i. 00:06:15.320 --> 00:06:19.900 align:middle line:84% Right, so there's N turns of wire, each with-- 00:06:19.900 --> 00:06:23.140 align:middle line:84% N turns of wire that has a current i through it. 00:06:23.140 --> 00:06:24.556 align:middle line:90% OK. 00:06:24.556 --> 00:06:27.310 align:middle line:84% And what I want to do is essentially 00:06:27.310 --> 00:06:32.150 align:middle line:84% find out what the inductance of this structure is, OK, 00:06:32.150 --> 00:06:35.890 align:middle line:84% and I'm going to make some approximations. 00:06:35.890 --> 00:06:39.910 align:middle line:84% I'm going to assume that this core, magnetic core, OK, has 00:06:39.910 --> 00:06:43.690 align:middle line:84% some permeability mu which is much greater than mu 0. 00:06:43.690 --> 00:06:45.940 align:middle line:84% It has some cross-sectional area that I'm 00:06:45.940 --> 00:06:48.970 align:middle line:84% going to call area of the core, some length 00:06:48.970 --> 00:06:51.250 align:middle line:84% that I'm going to call the length of the core. 00:06:51.250 --> 00:06:53.440 align:middle line:90% OK. 00:06:53.440 --> 00:06:57.360 align:middle line:84% And I'm going to assume that sort of this radius 00:06:57.360 --> 00:07:00.680 align:middle line:84% is fairly large compared to the dimension of the core, such 00:07:00.680 --> 00:07:04.620 align:middle line:84% that the fields inside the core are roughly uniform, right. 00:07:04.620 --> 00:07:06.780 align:middle line:84% So I'm kind of taking the simplest case just 00:07:06.780 --> 00:07:08.700 align:middle line:90% so we can see what's going on. 00:07:08.700 --> 00:07:09.450 align:middle line:90% OK. 00:07:09.450 --> 00:07:13.650 align:middle line:84% What do I want to do to find the inductance of this inductor? 00:07:13.650 --> 00:07:14.280 align:middle line:90% OK. 00:07:14.280 --> 00:07:17.170 align:middle line:84% I'll kind of outline it as three steps. 00:07:17.170 --> 00:07:22.230 align:middle line:84% One, I'm going to find the H field in the core, 00:07:22.230 --> 00:07:26.440 align:middle line:84% then I'm going to find the flux in the core. 00:07:26.440 --> 00:07:27.160 align:middle line:90% OK. 00:07:27.160 --> 00:07:31.510 align:middle line:84% Then I'm going to find the flux linkage in the core, lambda, 00:07:31.510 --> 00:07:34.880 align:middle line:84% and we have lambda is equal to Li. 00:07:34.880 --> 00:07:38.740 align:middle line:84% In other words, the flux linkage in the core 00:07:38.740 --> 00:07:42.010 align:middle line:84% is proportional to the current, OK, 00:07:42.010 --> 00:07:44.830 align:middle line:84% and I want to find that proportionality constant. 00:07:44.830 --> 00:07:47.530 align:middle line:84% And if you don't recall what flux linkage is, 00:07:47.530 --> 00:07:50.507 align:middle line:84% we'll come back to it, all right. 00:07:50.507 --> 00:07:52.465 align:middle line:84% So any questions about what we're trying to do? 00:07:52.465 --> 00:07:56.460 align:middle line:90% 00:07:56.460 --> 00:08:00.620 align:middle line:84% OK, so let's start about finding the H field in the core, right. 00:08:00.620 --> 00:08:03.930 align:middle line:84% So what I'm assuming is that there 00:08:03.930 --> 00:08:08.080 align:middle line:84% is going to be some H field owing to this-- 00:08:08.080 --> 00:08:11.200 align:middle line:84% a set of turns on the core with the winding, 00:08:11.200 --> 00:08:16.120 align:middle line:84% there's going to be some H field inside this core, this way. 00:08:16.120 --> 00:08:17.020 align:middle line:90% OK. 00:08:17.020 --> 00:08:19.490 align:middle line:90% So what do I know? 00:08:19.490 --> 00:08:21.908 align:middle line:84% I know that the integral of H dot 00:08:21.908 --> 00:08:29.210 align:middle line:84% dl is equal to the integral of J dot dA, right. 00:08:29.210 --> 00:08:33.640 align:middle line:84% Well, I'm going to integrate around 00:08:33.640 --> 00:08:37.990 align:middle line:84% exactly this loop like this, and how much 00:08:37.990 --> 00:08:39.955 align:middle line:90% current is piercing that loop? 00:08:39.955 --> 00:08:42.950 align:middle line:84% Right, if the loop's going through the middle of the core, 00:08:42.950 --> 00:08:50.510 align:middle line:84% well, I have i and i and i, so there's N times i going 00:08:50.510 --> 00:08:52.490 align:middle line:90% through that loop, right. 00:08:52.490 --> 00:08:57.885 align:middle line:84% So the integral of J dot dA ought to just be equal to Ni. 00:08:57.885 --> 00:08:58.385 align:middle line:90% OK. 00:08:58.385 --> 00:09:01.175 align:middle line:90% 00:09:01.175 --> 00:09:06.800 align:middle line:84% So if I then take that, I can say that, OK, H integrated 00:09:06.800 --> 00:09:09.200 align:middle line:84% around the loop, if I assume it's uniform, right, 00:09:09.200 --> 00:09:12.800 align:middle line:84% what I'm going to get is H times the length of the core 00:09:12.800 --> 00:09:21.350 align:middle line:84% is equal to Ni, or H is equal to Ni over the length of the core, 00:09:21.350 --> 00:09:26.400 align:middle line:84% OK, where this is, like, 2 pi r of the middle of the core. 00:09:26.400 --> 00:09:29.250 align:middle line:90% Any questions about that? 00:09:29.250 --> 00:09:31.890 align:middle line:84% AUDIENCE: So is the core kind of like that edge 00:09:31.890 --> 00:09:33.210 align:middle line:90% in this circular pattern? 00:09:33.210 --> 00:09:34.860 align:middle line:84% DAVID PERREAULT: Yeah, I'm assuming 00:09:34.860 --> 00:09:39.890 align:middle line:84% that the permeability of the core is really high and-- 00:09:39.890 --> 00:09:42.290 align:middle line:84% yes, so actually-- and in fact, it 00:09:42.290 --> 00:09:47.225 align:middle line:84% doesn't, in some sense, matter, right. 00:09:47.225 --> 00:09:50.030 align:middle line:84% I mean, the only assumption isn't that the core bends it. 00:09:50.030 --> 00:09:55.010 align:middle line:84% I'm just assuming it's everywhere around it. 00:09:55.010 --> 00:09:56.760 align:middle line:84% If you were worried about that assumption, 00:09:56.760 --> 00:10:00.030 align:middle line:84% you could say, well, suppose I distributed the turns exactly 00:10:00.030 --> 00:10:02.880 align:middle line:84% evenly around it the whole way, right, 00:10:02.880 --> 00:10:05.530 align:middle line:84% then everything would be sort of rotationally symmetric, 00:10:05.530 --> 00:10:07.363 align:middle line:84% so H ought to be the same everywhere, right. 00:10:07.363 --> 00:10:13.110 align:middle line:84% And then even if I didn't accept that the field I cared about 00:10:13.110 --> 00:10:16.780 align:middle line:84% was going around the core, then it would be legit. 00:10:16.780 --> 00:10:18.550 align:middle line:90% Great question. 00:10:18.550 --> 00:10:19.390 align:middle line:90% OK. 00:10:19.390 --> 00:10:21.560 align:middle line:90% So what do I got to find next? 00:10:21.560 --> 00:10:25.210 align:middle line:90% I got to find the magnetic flux. 00:10:25.210 --> 00:10:25.870 align:middle line:90% OK. 00:10:25.870 --> 00:10:30.400 align:middle line:84% Well, to find the magnetic flux, I'm 00:10:30.400 --> 00:10:34.760 align:middle line:84% going to assume that B is equal to mu core H, right. 00:10:34.760 --> 00:10:39.591 align:middle line:84% So I'm interested in finding the magnetic flux in the core. 00:10:39.591 --> 00:10:40.091 align:middle line:90% OK. 00:10:40.091 --> 00:10:42.917 align:middle line:90% 00:10:42.917 --> 00:10:46.570 align:middle line:84% And that magnetic flux is traveling around the core, OK, 00:10:46.570 --> 00:10:49.660 align:middle line:84% and it's proportional to the H field. 00:10:49.660 --> 00:10:51.220 align:middle line:90% OK. 00:10:51.220 --> 00:10:57.550 align:middle line:84% This is flux density, right, and the flux 00:10:57.550 --> 00:11:03.250 align:middle line:84% itself is equal to the B in the core 00:11:03.250 --> 00:11:06.220 align:middle line:84% times the cross-sectional area of the core, right. 00:11:06.220 --> 00:11:09.710 align:middle line:84% The net flux is the flux density times the area, OK, 00:11:09.710 --> 00:11:13.610 align:middle line:84% so the total flux traveling through this, 00:11:13.610 --> 00:11:16.340 align:middle line:84% if you want to think of the core as sort of a pipe for flux, 00:11:16.340 --> 00:11:19.870 align:middle line:84% is the B in the core times the area of the core. 00:11:19.870 --> 00:11:31.790 align:middle line:84% So I can work this out to be, OK, mu c times N over lc i 00:11:31.790 --> 00:11:34.550 align:middle line:90% times the Ac. 00:11:34.550 --> 00:11:35.060 align:middle line:90% OK. 00:11:35.060 --> 00:11:38.060 align:middle line:84% So here's the net flux in the core. 00:11:38.060 --> 00:11:38.990 align:middle line:90% OK. 00:11:38.990 --> 00:11:42.440 align:middle line:84% The flux in the core is proportional to the current 00:11:42.440 --> 00:11:45.160 align:middle line:84% with this proportionality constant. 00:11:45.160 --> 00:11:45.660 align:middle line:90% OK. 00:11:45.660 --> 00:11:48.820 align:middle line:90% Any questions about that? 00:11:48.820 --> 00:11:52.400 align:middle line:84% And so in total, I have some flux density, 00:11:52.400 --> 00:11:54.820 align:middle line:84% and so hence there's some net flux 00:11:54.820 --> 00:11:58.630 align:middle line:84% in the core that's kind of traveling around this loop. 00:11:58.630 --> 00:12:00.490 align:middle line:90% OK. 00:12:00.490 --> 00:12:01.340 align:middle line:90% All right. 00:12:01.340 --> 00:12:03.920 align:middle line:90% So what do I need next? 00:12:03.920 --> 00:12:09.590 align:middle line:84% What I want to find next is, What's the flux linkage? 00:12:09.590 --> 00:12:11.830 align:middle line:84% OK, well, what is the flux linkage? 00:12:11.830 --> 00:12:19.950 align:middle line:84% The flux linkage is the total flux linking this net winding. 00:12:19.950 --> 00:12:20.470 align:middle line:90% All right. 00:12:20.470 --> 00:12:21.580 align:middle line:90% What does that mean? 00:12:21.580 --> 00:12:24.240 align:middle line:84% Well, I can think about it a certain way. 00:12:24.240 --> 00:12:26.550 align:middle line:84% There's some flux coming around here, 00:12:26.550 --> 00:12:30.570 align:middle line:84% and it's piercing through the shape caused 00:12:30.570 --> 00:12:32.610 align:middle line:90% by this N-turn winding. 00:12:32.610 --> 00:12:35.730 align:middle line:84% And this helical winding, you could think of, 00:12:35.730 --> 00:12:40.560 align:middle line:84% if I dipped this whole helical winding in a bath of soapy water 00:12:40.560 --> 00:12:44.830 align:middle line:84% and I'd get some kind of surface that's spiraling up, 00:12:44.830 --> 00:12:47.670 align:middle line:84% how much net flux pierces that surface 00:12:47.670 --> 00:12:51.250 align:middle line:90% is what the flux linkage is. 00:12:51.250 --> 00:12:55.060 align:middle line:84% Another way to think about it is, OK, I 00:12:55.060 --> 00:12:58.000 align:middle line:84% have this one turn coming around, right. 00:12:58.000 --> 00:13:00.950 align:middle line:84% So it kind of came around like this into here. 00:13:00.950 --> 00:13:04.780 align:middle line:84% And so I can think of sort of this point 00:13:04.780 --> 00:13:10.210 align:middle line:84% here is kind of a loop through which the flux pierces once, 00:13:10.210 --> 00:13:12.550 align:middle line:84% and then there's another turn, the next turn, 00:13:12.550 --> 00:13:15.070 align:middle line:84% the flux pierces that once, another turn, 00:13:15.070 --> 00:13:16.670 align:middle line:90% the flux pierces that once. 00:13:16.670 --> 00:13:23.080 align:middle line:84% And so lambda, the flux linkage, the flux 00:13:23.080 --> 00:13:24.910 align:middle line:84% linking the whole N-turn winding, 00:13:24.910 --> 00:13:27.610 align:middle line:84% is just simply N, the number of turns, 00:13:27.610 --> 00:13:30.330 align:middle line:90% times the flux in the core. 00:13:30.330 --> 00:13:33.930 align:middle line:84% OK, does that make sense to everybody? 00:13:33.930 --> 00:13:36.100 align:middle line:90% So what does that mean? 00:13:36.100 --> 00:13:39.120 align:middle line:84% Lambda is, then, if I just substitute in here, I 00:13:39.120 --> 00:13:51.630 align:middle line:84% get lambda is equal to mu c Ac over Lc N squared times i. 00:13:51.630 --> 00:13:54.100 align:middle line:90% OK. 00:13:54.100 --> 00:13:58.060 align:middle line:84% And I just said the relationship between the flux 00:13:58.060 --> 00:14:05.706 align:middle line:84% linkage and the current is L, so this thing here is inductance. 00:14:05.706 --> 00:14:09.980 align:middle line:90% 00:14:09.980 --> 00:14:11.320 align:middle line:90% Any questions about that? 00:14:11.320 --> 00:14:16.090 align:middle line:90% 00:14:16.090 --> 00:14:23.610 align:middle line:84% Now, what do we mean by that, or what is the use of that? 00:14:23.610 --> 00:14:33.590 align:middle line:84% Well, voltage, the voltage on the winding is equal to L di dt, 00:14:33.590 --> 00:14:36.980 align:middle line:84% but it's also equal to d lambda dt. 00:14:36.980 --> 00:14:39.900 align:middle line:84% Right, the voltage that I see at that winding, 00:14:39.900 --> 00:14:45.710 align:middle line:84% so if I have i here, this voltage here 00:14:45.710 --> 00:14:51.290 align:middle line:84% is going to be equal to d lambda dt, where d lambda is the-- 00:14:51.290 --> 00:14:54.230 align:middle line:84% lambda is the flux linking the winding. 00:14:54.230 --> 00:14:54.740 align:middle line:90% OK. 00:14:54.740 --> 00:14:56.750 align:middle line:90% Well, why is that the case? 00:14:56.750 --> 00:14:58.940 align:middle line:84% That comes back to Faraday's law, right. 00:14:58.940 --> 00:15:04.490 align:middle line:84% We said Faraday's law said, OK, if I had, for example, 00:15:04.490 --> 00:15:09.050 align:middle line:84% one turn here coming around, the total integral of E 00:15:09.050 --> 00:15:12.200 align:middle line:84% dot dl around that gives me an MMF which 00:15:12.200 --> 00:15:14.700 align:middle line:84% is equal to the flux linking through that winding. 00:15:14.700 --> 00:15:19.680 align:middle line:84% So the voltage on one turn is d phi dt, right, 00:15:19.680 --> 00:15:22.560 align:middle line:84% because the integral of B dot dA is phi, right. 00:15:22.560 --> 00:15:28.500 align:middle line:84% The voltage on two turns total is 2 d phi dt, et cetera. 00:15:28.500 --> 00:15:36.100 align:middle line:84% So for N turns, it's N d phi dt, so it ends up being d lambda dt. 00:15:36.100 --> 00:15:38.400 align:middle line:90% That makes sense to everybody? 00:15:38.400 --> 00:15:40.710 align:middle line:84% Each turn's going to see a voltage-- 00:15:40.710 --> 00:15:45.900 align:middle line:84% this turn is going to see a v1 turn, which is d phi dt. 00:15:45.900 --> 00:15:47.700 align:middle line:84% The voltage across the whole thing 00:15:47.700 --> 00:15:53.280 align:middle line:84% is N d phi dt, or d N phi dt, or d lambda dt. 00:15:53.280 --> 00:15:54.992 align:middle line:90% OK. 00:15:54.992 --> 00:15:57.010 align:middle line:84% So when I put that together, that 00:15:57.010 --> 00:16:02.050 align:middle line:84% just gives me v is equal to L di dt, which 00:16:02.050 --> 00:16:08.290 align:middle line:84% is sort of because this is equal to d dt of Li. 00:16:08.290 --> 00:16:10.600 align:middle line:84% And we get the formula for inductance 00:16:10.600 --> 00:16:12.160 align:middle line:84% that we know and love, or formula 00:16:12.160 --> 00:16:14.980 align:middle line:84% for how an inductor behaves that we know and love. 00:16:14.980 --> 00:16:16.218 align:middle line:90% Yeah? 00:16:16.218 --> 00:16:18.260 align:middle line:84% AUDIENCE: Could you comment on the negative sign? 00:16:18.260 --> 00:16:19.050 align:middle line:90% DAVID PERREAULT: I'm sorry? 00:16:19.050 --> 00:16:20.390 align:middle line:84% AUDIENCE: Could you comment on the negative sign 00:16:20.390 --> 00:16:21.710 align:middle line:90% in Faraday's law? 00:16:21.710 --> 00:16:27.560 align:middle line:84% DAVID PERREAULT: Yeah, I mean, that's generated by Lenz's law. 00:16:27.560 --> 00:16:35.510 align:middle line:84% And what that essentially says is the sign of the voltage-- 00:16:35.510 --> 00:16:39.020 align:middle line:84% if I have a d phi dt going through, say, one loop, 00:16:39.020 --> 00:16:43.940 align:middle line:84% the sign of the voltage is such that the current induced 00:16:43.940 --> 00:16:46.070 align:middle line:84% by that voltage in an external circuit 00:16:46.070 --> 00:16:50.120 align:middle line:84% would oppose the rate of change of flux. 00:16:50.120 --> 00:16:51.800 align:middle line:90% OK. 00:16:51.800 --> 00:16:55.640 align:middle line:84% And so what that says is that for a positive current 00:16:55.640 --> 00:16:59.570 align:middle line:84% into this, I'm going to get an induced voltage this way 00:16:59.570 --> 00:17:01.840 align:middle line:90% for the d lambda dt. 00:17:01.840 --> 00:17:02.340 align:middle line:90% OK. 00:17:02.340 --> 00:17:08.040 align:middle line:84% So it basically comes back to Lenz's law. 00:17:08.040 --> 00:17:08.920 align:middle line:90% Excellent question. 00:17:08.920 --> 00:17:11.099 align:middle line:90% Any other questions? 00:17:11.099 --> 00:17:14.130 align:middle line:84% And we'll see how that plays out in the systems we're 00:17:14.130 --> 00:17:17.180 align:middle line:90% going to create. 00:17:17.180 --> 00:17:17.920 align:middle line:90% OK. 00:17:17.920 --> 00:17:20.369 align:middle line:90% So that gives me inductance. 00:17:20.369 --> 00:17:23.560 align:middle line:84% Let me point out just a couple little things about inductance. 00:17:23.560 --> 00:17:27.560 align:middle line:84% The first thing I'll note is, notice that inductance is not 00:17:27.560 --> 00:17:29.060 align:middle line:84% proportional to the number of turns, 00:17:29.060 --> 00:17:32.060 align:middle line:84% but it's proportional to the number of turns squared. 00:17:32.060 --> 00:17:33.330 align:middle line:90% Why is that? 00:17:33.330 --> 00:17:36.960 align:middle line:84% Because the H field I induce, I get N times i, 00:17:36.960 --> 00:17:38.720 align:middle line:90% so there's a factor of N there. 00:17:38.720 --> 00:17:42.170 align:middle line:84% And then the flux that gets linked, the flux linkage has 00:17:42.170 --> 00:17:45.570 align:middle line:84% another factor of N, so there's an N squared in the inductance. 00:17:45.570 --> 00:17:46.070 align:middle line:90% OK. 00:17:46.070 --> 00:17:47.528 align:middle line:84% So if I double the number of turns, 00:17:47.528 --> 00:17:50.870 align:middle line:84% I don't double the inductance, I quadruple it. 00:17:50.870 --> 00:17:52.130 align:middle line:90% All right. 00:17:52.130 --> 00:17:54.170 align:middle line:84% And in fact, a lot of manufacturers, 00:17:54.170 --> 00:17:57.140 align:middle line:84% when they give you a core, will tell you 00:17:57.140 --> 00:17:58.770 align:middle line:90% the inductance of one turn. 00:17:58.770 --> 00:18:00.145 align:middle line:84% They'll give you something that's 00:18:00.145 --> 00:18:03.665 align:middle line:84% often called the A-sub-L value, or the specific inductance, 00:18:03.665 --> 00:18:04.165 align:middle line:90% right. 00:18:04.165 --> 00:18:07.910 align:middle line:84% So it's sort of the nanohenries for one turn, right, 00:18:07.910 --> 00:18:10.520 align:middle line:84% and if you want the L for your inductor, 00:18:10.520 --> 00:18:13.680 align:middle line:84% it's equal to N squared A sub L. Right, 00:18:13.680 --> 00:18:18.080 align:middle line:84% so if A sub L is a hundred nanohenries per turn, 00:18:18.080 --> 00:18:22.055 align:middle line:84% you've got to multiply-- a hundred nanohenries at one turn, 00:18:22.055 --> 00:18:24.680 align:middle line:84% you've got to multiply it by the square of the number of turns, 00:18:24.680 --> 00:18:26.210 align:middle line:90% not the number of turns. 00:18:26.210 --> 00:18:27.930 align:middle line:90% OK. 00:18:27.930 --> 00:18:30.570 align:middle line:84% The other thing I'll note here is 00:18:30.570 --> 00:18:32.700 align:middle line:84% that notice that the inductance here 00:18:32.700 --> 00:18:35.920 align:middle line:84% is proportional to the permeability of the core, right. 00:18:35.920 --> 00:18:38.170 align:middle line:84% And we kind of made the approximation at the beginning 00:18:38.170 --> 00:18:41.520 align:middle line:84% that the permeability of the core is large compared to mu 0, 00:18:41.520 --> 00:18:44.915 align:middle line:84% in order to make this simplified calculation work. 00:18:44.915 --> 00:18:47.610 align:middle line:90% 00:18:47.610 --> 00:18:49.830 align:middle line:84% The problem with having a high permeability-- 00:18:49.830 --> 00:18:53.190 align:middle line:84% and that could be a thousand or more-- 00:18:53.190 --> 00:18:57.330 align:middle line:84% is that it's not a very stable number, right. 00:18:57.330 --> 00:19:01.350 align:middle line:84% It could be plus or minus 25%, even in the ideal case, 00:19:01.350 --> 00:19:03.252 align:middle line:84% for a lot of core materials, and that's 00:19:03.252 --> 00:19:05.460 align:middle line:84% before you talk about temperature or flux or anything 00:19:05.460 --> 00:19:06.752 align:middle line:90% else going and changing, right. 00:19:06.752 --> 00:19:09.030 align:middle line:84% So this is not a great way to make 00:19:09.030 --> 00:19:15.050 align:middle line:84% a very stable valued inductance, so we might want to think about, 00:19:15.050 --> 00:19:19.130 align:middle line:84% How do we deal with the variability that often turns up 00:19:19.130 --> 00:19:20.750 align:middle line:84% in the permeability of core materials 00:19:20.750 --> 00:19:22.320 align:middle line:90% when we design inductors? 00:19:22.320 --> 00:19:24.640 align:middle line:90% OK, and we'll see that. 00:19:24.640 --> 00:19:25.850 align:middle line:90% Any questions? 00:19:25.850 --> 00:19:29.240 align:middle line:90% 00:19:29.240 --> 00:19:29.740 align:middle line:90% OK. 00:19:29.740 --> 00:19:31.810 align:middle line:84% Now, before I jump into, How do we 00:19:31.810 --> 00:19:35.290 align:middle line:84% do this quicker than writing Maxwell's equations? 00:19:35.290 --> 00:19:39.100 align:middle line:84% let's consider one more example, and what 00:19:39.100 --> 00:19:44.425 align:middle line:84% I'd like to consider now is an example with a gap in the core. 00:19:44.425 --> 00:19:44.925 align:middle line:90% OK. 00:19:44.925 --> 00:19:45.880 align:middle line:90% So I'm going to have-- 00:19:45.880 --> 00:19:48.340 align:middle line:84% I'll draw it as a C core this time. 00:19:48.340 --> 00:19:51.850 align:middle line:84% It doesn't have much to do with the actual shape of the core. 00:19:51.850 --> 00:19:54.025 align:middle line:90% OK. 00:19:54.025 --> 00:19:55.275 align:middle line:90% So here's my core. 00:19:55.275 --> 00:20:03.850 align:middle line:90% 00:20:03.850 --> 00:20:04.390 align:middle line:90% OK. 00:20:04.390 --> 00:20:08.055 align:middle line:84% So I'm going to put N turns on this core. 00:20:08.055 --> 00:20:12.750 align:middle line:90% 00:20:12.750 --> 00:20:16.610 align:middle line:84% I'm going to put a current i in here again. 00:20:16.610 --> 00:20:21.710 align:middle line:84% I'm going to assume that this core, the actual core section, 00:20:21.710 --> 00:20:26.150 align:middle line:84% has a length of core and a cross-sectional area core 00:20:26.150 --> 00:20:29.390 align:middle line:84% and a permeability of the core which again I'm going to assume 00:20:29.390 --> 00:20:31.110 align:middle line:90% is very high. 00:20:31.110 --> 00:20:33.280 align:middle line:84% Then I'm going to have some gap here, 00:20:33.280 --> 00:20:35.220 align:middle line:84% and maybe that gap has nothing in it. 00:20:35.220 --> 00:20:36.400 align:middle line:90% Maybe it's just air. 00:20:36.400 --> 00:20:38.560 align:middle line:84% It's the permeability of free space, right. 00:20:38.560 --> 00:20:40.560 align:middle line:84% So maybe I'll assume that this is 00:20:40.560 --> 00:20:44.190 align:middle line:84% the area of the gap, this spacing, which 00:20:44.190 --> 00:20:46.930 align:middle line:84% might be the treated the same as the area of the core. 00:20:46.930 --> 00:20:48.540 align:middle line:84% It has a permeability which might 00:20:48.540 --> 00:20:50.520 align:middle line:84% be the permeability of free space, 00:20:50.520 --> 00:20:52.900 align:middle line:84% and something that's the length of the gap. 00:20:52.900 --> 00:20:56.460 align:middle line:84% All right, just this is the length of the gap here, lg. 00:20:56.460 --> 00:20:57.540 align:middle line:90% OK. 00:20:57.540 --> 00:21:00.580 align:middle line:90% Everybody see that setup? 00:21:00.580 --> 00:21:03.550 align:middle line:84% So how would I analyze this structure? 00:21:03.550 --> 00:21:04.360 align:middle line:90% OK. 00:21:04.360 --> 00:21:09.810 align:middle line:84% Well, I would follow the same exact pattern, OK, 00:21:09.810 --> 00:21:14.400 align:middle line:84% except that because part of the structure has high permeability 00:21:14.400 --> 00:21:18.060 align:middle line:84% and part of it has low permeability, 00:21:18.060 --> 00:21:21.920 align:middle line:84% finding the H field is a little bit trickier, 00:21:21.920 --> 00:21:24.910 align:middle line:84% or finding the flux is a little bit trickier. 00:21:24.910 --> 00:21:29.213 align:middle line:84% So if I were to follow my first step, with Ampere's law, 00:21:29.213 --> 00:21:30.630 align:middle line:84% right, what I'm going to do is I'm 00:21:30.630 --> 00:21:33.110 align:middle line:84% going to get the integral of H dot 00:21:33.110 --> 00:21:40.030 align:middle line:84% dl is going to be the H in the core times the length 00:21:40.030 --> 00:21:46.430 align:middle line:84% of the core plus the H in the gap times the length of the gap. 00:21:46.430 --> 00:21:49.340 align:middle line:84% OK, and that's going to be equal to Ni. 00:21:49.340 --> 00:21:52.430 align:middle line:90% 00:21:52.430 --> 00:21:53.480 align:middle line:90% OK. 00:21:53.480 --> 00:21:55.100 align:middle line:90% All right. 00:21:55.100 --> 00:21:58.370 align:middle line:84% The question is, How do I deal with the H in the core 00:21:58.370 --> 00:22:01.430 align:middle line:84% and the H in the gap, or how are they related? 00:22:01.430 --> 00:22:11.470 align:middle line:84% OK, where we get that is we come back to flux continuity, right. 00:22:11.470 --> 00:22:15.670 align:middle line:84% Flux is neither created nor destroyed, right. 00:22:15.670 --> 00:22:18.450 align:middle line:90% There's no magnetic charge. 00:22:18.450 --> 00:22:22.340 align:middle line:84% So that means whatever flux here comes out 00:22:22.340 --> 00:22:26.830 align:middle line:84% of the core, whatever flux enters a little pillbox 00:22:26.830 --> 00:22:29.290 align:middle line:84% at the edge of this core, must also pop out the other side 00:22:29.290 --> 00:22:31.825 align:middle line:90% into the gap, right. 00:22:31.825 --> 00:22:36.130 align:middle line:84% So what that roughly means is that the B 00:22:36.130 --> 00:22:38.440 align:middle line:84% in the core times the area of the core 00:22:38.440 --> 00:22:43.570 align:middle line:84% must equal the B in the gap times the area of the gap. 00:22:43.570 --> 00:22:45.282 align:middle line:90% OK. 00:22:45.282 --> 00:22:48.200 align:middle line:90% And if I then assume-- 00:22:48.200 --> 00:22:51.367 align:middle line:84% let me just assume, for sake of argument, that the way this flux 00:22:51.367 --> 00:22:53.450 align:middle line:84% goes, I'm just going to treat the area of the core 00:22:53.450 --> 00:22:56.950 align:middle line:84% and the area of the gap the same. 00:22:56.950 --> 00:22:57.490 align:middle line:90% OK. 00:22:57.490 --> 00:23:00.550 align:middle line:84% So then I can just say, OK, if Ac 00:23:00.550 --> 00:23:06.470 align:middle line:84% equals Ag, then that means the B in the core is equal to the-- 00:23:06.470 --> 00:23:09.240 align:middle line:90% 00:23:09.240 --> 00:23:12.830 align:middle line:84% B in the core is equal to B in the gap. 00:23:12.830 --> 00:23:15.230 align:middle line:90% OK. 00:23:15.230 --> 00:23:19.080 align:middle line:84% Actually, I don't even have to make that approximation. 00:23:19.080 --> 00:23:22.930 align:middle line:90% Let's just leave it like this. 00:23:22.930 --> 00:23:25.480 align:middle line:84% Actually, it's even better if I leave it like this. 00:23:25.480 --> 00:23:26.940 align:middle line:90% OK. 00:23:26.940 --> 00:23:28.890 align:middle line:84% The B in the core times the area of the core 00:23:28.890 --> 00:23:32.460 align:middle line:84% is equal to B in the gap times the area of the gap, which is 00:23:32.460 --> 00:23:38.890 align:middle line:90% also equal to the total flux. 00:23:38.890 --> 00:23:41.080 align:middle line:84% If there's some flux running around this loop, 00:23:41.080 --> 00:23:42.440 align:middle line:90% it looks like this. 00:23:42.440 --> 00:23:46.030 align:middle line:84% Here's the total flux that's traveling around this loop. 00:23:46.030 --> 00:23:48.460 align:middle line:84% And so all I'm saying is whatever flux is coming down 00:23:48.460 --> 00:23:53.582 align:middle line:84% this pipe in the core is jumping out into the gap, right, 00:23:53.582 --> 00:23:58.250 align:middle line:84% so I can say that each of these is equal to the flux, OK, 00:23:58.250 --> 00:24:01.170 align:middle line:90% in the core. 00:24:01.170 --> 00:24:01.840 align:middle line:90% All right. 00:24:01.840 --> 00:24:08.940 align:middle line:84% So then maybe I can write this as mu core H core 00:24:08.940 --> 00:24:14.610 align:middle line:84% times A core is equal to mu gap or mu 0 00:24:14.610 --> 00:24:18.180 align:middle line:84% times H in the gap times the area of the gap 00:24:18.180 --> 00:24:20.470 align:middle line:90% is equal to phi in the core. 00:24:20.470 --> 00:24:21.290 align:middle line:90% OK. 00:24:21.290 --> 00:24:28.660 align:middle line:84% So I could then say, OK, Hc is equal to phi in the core divided 00:24:28.660 --> 00:24:34.600 align:middle line:84% by mu c Ac, and H in the gap is equal to the flux 00:24:34.600 --> 00:24:41.320 align:middle line:84% in the core divided by mu c dot mu 0 area of the gap. 00:24:41.320 --> 00:24:41.820 align:middle line:90% OK. 00:24:41.820 --> 00:24:45.900 align:middle line:84% So now let me just go substitute that in here, 00:24:45.900 --> 00:24:47.670 align:middle line:90% and what am I going to get? 00:24:47.670 --> 00:24:49.950 align:middle line:84% I'm going to get Hc, which is this, 00:24:49.950 --> 00:24:52.530 align:middle line:90% so I'm going to get phi c-- 00:24:52.530 --> 00:24:54.520 align:middle line:84% I'll just call it phi-- eh, I'll call it phi 00:24:54.520 --> 00:25:14.870 align:middle line:84% c-- phi c times Lc over mu c Ac plus phi c times Lg over mu 0 00:25:14.870 --> 00:25:18.120 align:middle line:90% Ag is equal to Ni. 00:25:18.120 --> 00:25:24.070 align:middle line:90% 00:25:24.070 --> 00:25:25.090 align:middle line:90% OK. 00:25:25.090 --> 00:25:28.410 align:middle line:90% So let's use that. 00:25:28.410 --> 00:25:33.810 align:middle line:84% So now what I'm going to get is the following. 00:25:33.810 --> 00:25:37.770 align:middle line:84% I can write phi in the core is simply 00:25:37.770 --> 00:25:47.880 align:middle line:84% equal to Ni divided by this bracketed expression, Lc over mu 00:25:47.880 --> 00:25:54.150 align:middle line:90% c Ac plus lg over mu 0 Ag. 00:25:54.150 --> 00:25:54.650 align:middle line:90% OK. 00:25:54.650 --> 00:25:59.520 align:middle line:84% And then I can just say, OK, that's fine. 00:25:59.520 --> 00:26:02.660 align:middle line:84% That means the flux linkage, lambda, is simply 00:26:02.660 --> 00:26:07.190 align:middle line:84% N times the flux in the core, right, and that's going to be N 00:26:07.190 --> 00:26:25.280 align:middle line:84% squared divided by lc over mu c Ac plus lg over mu 0 Ag times i. 00:26:25.280 --> 00:26:32.900 align:middle line:84% And this thing, again, OK, is what we call inductance, 00:26:32.900 --> 00:26:36.235 align:middle line:90% L. Any questions about that? 00:26:36.235 --> 00:26:39.050 align:middle line:90% 00:26:39.050 --> 00:26:41.330 align:middle line:84% So we've again used Maxwell's equations 00:26:41.330 --> 00:26:44.090 align:middle line:84% to come up with the inductance of our structure. 00:26:44.090 --> 00:26:44.590 align:middle line:90% OK. 00:26:44.590 --> 00:26:49.220 align:middle line:84% I'll point out a couple of things about this. 00:26:49.220 --> 00:26:56.900 align:middle line:90% Suppose if-- as long as I have-- 00:26:56.900 --> 00:27:03.380 align:middle line:84% if I have Ac is on the same scale as Ag-- 00:27:03.380 --> 00:27:06.080 align:middle line:84% in other words, the gap area and the core area 00:27:06.080 --> 00:27:07.700 align:middle line:90% are about the same thing-- 00:27:07.700 --> 00:27:16.460 align:middle line:84% as if lg over mu g-- over mu 0 is much greater 00:27:16.460 --> 00:27:20.480 align:middle line:90% than l core over mu core. 00:27:20.480 --> 00:27:23.930 align:middle line:84% And remember that mu core, I'm assuming a very high 00:27:23.930 --> 00:27:25.980 align:middle line:90% permeability relative to mu 0. 00:27:25.980 --> 00:27:29.000 align:middle line:84% And if it's much higher than the length of the gap, 00:27:29.000 --> 00:27:32.570 align:middle line:84% then what I get is L is approximately equal to N 00:27:32.570 --> 00:27:38.330 align:middle line:90% squared over lg over mu 0 Ag. 00:27:38.330 --> 00:27:38.990 align:middle line:90% OK. 00:27:38.990 --> 00:27:43.390 align:middle line:84% In other words, this term no longer matters. 00:27:43.390 --> 00:27:46.570 align:middle line:84% And then I get an inductance that doesn't really 00:27:46.570 --> 00:27:50.020 align:middle line:84% depend upon the exact value of the core permeability 00:27:50.020 --> 00:27:52.940 align:middle line:84% because it's dominated by this term. 00:27:52.940 --> 00:27:57.550 align:middle line:84% OK, so I'm setting my inductance based on geometry, say, based 00:27:57.550 --> 00:28:01.060 align:middle line:84% on the area and length of the gap rather than any material 00:28:01.060 --> 00:28:03.430 align:middle line:84% parameter, and that could be a very desirable thing 00:28:03.430 --> 00:28:08.100 align:middle line:84% because now some variation doesn't matter anymore. 00:28:08.100 --> 00:28:09.230 align:middle line:90% Any questions about that? 00:28:09.230 --> 00:28:13.440 align:middle line:90% 00:28:13.440 --> 00:28:17.460 align:middle line:84% What does that mean as a practical matter if I do this, 00:28:17.460 --> 00:28:19.710 align:middle line:90% if I get into this situation? 00:28:19.710 --> 00:28:24.860 align:middle line:84% What I'm really saying is that the magnetic energy stored-- 00:28:24.860 --> 00:28:28.330 align:middle line:84% right, I get an energy storage of 1/2 Li squared-- 00:28:28.330 --> 00:28:32.070 align:middle line:84% right, we all know that the energy stored in an inductor, 00:28:32.070 --> 00:28:34.030 align:middle line:84% a linear inductor, is 1/2 Li squared, right? 00:28:34.030 --> 00:28:36.340 align:middle line:90% Where is that energy stored? 00:28:36.340 --> 00:28:41.255 align:middle line:84% It's stored in the magnetic fields, right, 00:28:41.255 --> 00:28:43.960 align:middle line:84% but where in this structure is the energy stored? 00:28:43.960 --> 00:28:48.040 align:middle line:84% Well, the magnetic energy storage density is equal 00:28:48.040 --> 00:28:48.540 align:middle line:90% to 1/2-- 00:28:48.540 --> 00:28:51.060 align:middle line:90% 00:28:51.060 --> 00:28:54.420 align:middle line:84% or, actually, the total magnetic energy stored is going to be 1/2 00:28:54.420 --> 00:28:58.920 align:middle line:84% the integral of the volume B dot H dv. 00:28:58.920 --> 00:28:59.420 align:middle line:90% OK. 00:28:59.420 --> 00:29:04.580 align:middle line:84% In other words, it's the product of the B and H fields 00:29:04.580 --> 00:29:08.270 align:middle line:84% integrated gives me the total energy stored. 00:29:08.270 --> 00:29:09.420 align:middle line:90% OK. 00:29:09.420 --> 00:29:13.540 align:middle line:84% Now, if the area of the core and the area of the gap 00:29:13.540 --> 00:29:17.890 align:middle line:84% are the same, right, we know that the B 00:29:17.890 --> 00:29:22.110 align:middle line:84% in the core and the B and the gap are the same, right. 00:29:22.110 --> 00:29:28.550 align:middle line:84% But the H is B over mu, and that means 00:29:28.550 --> 00:29:30.680 align:middle line:84% that the H field in the core is going 00:29:30.680 --> 00:29:35.800 align:middle line:84% to be really small compared to the H field in the gap. 00:29:35.800 --> 00:29:36.300 align:middle line:90% OK. 00:29:36.300 --> 00:29:38.160 align:middle line:84% So the flux is sort of continuous. 00:29:38.160 --> 00:29:40.680 align:middle line:84% It doesn't change as you're going around, 00:29:40.680 --> 00:29:43.860 align:middle line:84% OK, the flux density, but the magnetic field strength 00:29:43.860 --> 00:29:44.920 align:middle line:90% does change. 00:29:44.920 --> 00:29:49.680 align:middle line:84% And what that means is you'll have very high magnetic field 00:29:49.680 --> 00:29:52.290 align:middle line:90% strength in the gap. 00:29:52.290 --> 00:29:55.380 align:middle line:84% So if I'm drawing my magnetic field H, 00:29:55.380 --> 00:30:01.850 align:middle line:84% I'll have a lot of field in the gap and very little in the core. 00:30:01.850 --> 00:30:02.530 align:middle line:90% OK. 00:30:02.530 --> 00:30:05.420 align:middle line:84% And as a consequence, all the energy is actually-- 00:30:05.420 --> 00:30:08.980 align:middle line:84% or the dominant portion of the energy is stored in the gap. 00:30:08.980 --> 00:30:10.488 align:middle line:84% And that's why I sort of don't care 00:30:10.488 --> 00:30:13.030 align:middle line:84% about what the core is doing, because all the energy's stored 00:30:13.030 --> 00:30:17.050 align:middle line:84% here, if I've met that approximation. 00:30:17.050 --> 00:30:20.110 align:middle line:84% So we often do that, put a gap in it, 00:30:20.110 --> 00:30:23.440 align:middle line:84% firstly because it makes us insensitive 00:30:23.440 --> 00:30:25.210 align:middle line:84% to the actual material parameters, which 00:30:25.210 --> 00:30:26.180 align:middle line:90% can vary a lot. 00:30:26.180 --> 00:30:29.110 align:middle line:84% And, B, as it turns out, because of the limitations 00:30:29.110 --> 00:30:32.080 align:middle line:84% on how much flux you can have, you 00:30:32.080 --> 00:30:33.520 align:middle line:84% can get a lot more energy storage 00:30:33.520 --> 00:30:35.960 align:middle line:90% if you put a gap in the core. 00:30:35.960 --> 00:30:36.460 align:middle line:90% OK. 00:30:36.460 --> 00:30:38.230 align:middle line:84% And what we're really doing is we're 00:30:38.230 --> 00:30:41.080 align:middle line:84% kind of focusing the magnetic field down into this space 00:30:41.080 --> 00:30:42.600 align:middle line:90% and storing the energy here. 00:30:42.600 --> 00:30:45.460 align:middle line:90% 00:30:45.460 --> 00:30:46.635 align:middle line:90% Any questions about that? 00:30:46.635 --> 00:30:50.540 align:middle line:90% 00:30:50.540 --> 00:30:55.550 align:middle line:84% Now, as you might imagine, we can do these calculations 00:30:55.550 --> 00:30:58.130 align:middle line:84% and use Maxwell's equations and get to all these results. 00:30:58.130 --> 00:31:01.750 align:middle line:84% And that's great, but it's kind of tedious, right. 00:31:01.750 --> 00:31:06.220 align:middle line:84% It would be nice if there was a much faster way 00:31:06.220 --> 00:31:08.860 align:middle line:84% to do these calculations, both to figure out 00:31:08.860 --> 00:31:11.200 align:middle line:84% things like inductances, but also to figure out, 00:31:11.200 --> 00:31:13.810 align:middle line:84% What are the magnetic fluxes here and there? 00:31:13.810 --> 00:31:16.390 align:middle line:84% What are what are the flux linkages 00:31:16.390 --> 00:31:17.720 align:middle line:90% I get on different windings? 00:31:17.720 --> 00:31:20.262 align:middle line:84% I might have a transformer with more than one winding, right. 00:31:20.262 --> 00:31:22.690 align:middle line:90% How can I do that easily? 00:31:22.690 --> 00:31:25.480 align:middle line:84% And the way we do that is using something 00:31:25.480 --> 00:31:29.210 align:middle line:84% called a magnetic circuit model, OK. 00:31:29.210 --> 00:31:41.045 align:middle line:90% 00:31:41.045 --> 00:31:44.360 align:middle line:90% And here's the idea. 00:31:44.360 --> 00:31:50.017 align:middle line:84% Let me come back to this equation here, right. 00:31:50.017 --> 00:31:52.999 align:middle line:90% 00:31:52.999 --> 00:31:57.360 align:middle line:84% This sort of looks like the following, roughly speaking. 00:31:57.360 --> 00:32:03.480 align:middle line:84% This equation kind of looks like v is equal to-- 00:32:03.480 --> 00:32:18.045 align:middle line:84% I'm sorry, I is equal to v divided by R1 plus R2, right. 00:32:18.045 --> 00:32:19.960 align:middle line:84% If I have a resistance, remember, 00:32:19.960 --> 00:32:23.530 align:middle line:84% if I have a resistance of a resistor, 00:32:23.530 --> 00:32:25.220 align:middle line:90% how do I calculate that? 00:32:25.220 --> 00:32:27.888 align:middle line:84% It's the length of the resistive structure 00:32:27.888 --> 00:32:29.680 align:middle line:84% divided by the cross-sectional area divided 00:32:29.680 --> 00:32:30.770 align:middle line:90% by the conductivity. 00:32:30.770 --> 00:32:34.730 align:middle line:84% It's length over conductivity times cross-sectional area. 00:32:34.730 --> 00:32:36.430 align:middle line:84% That's just the sort of thing here, 00:32:36.430 --> 00:32:39.490 align:middle line:84% except that instead of having conductivity, 00:32:39.490 --> 00:32:42.470 align:middle line:90% I have permeability. 00:32:42.470 --> 00:32:43.290 align:middle line:90% All right. 00:32:43.290 --> 00:32:47.630 align:middle line:84% So this sort of looks like I is equal to v over R1 00:32:47.630 --> 00:32:50.180 align:middle line:90% plus the quantity R1 plus R2. 00:32:50.180 --> 00:32:55.490 align:middle line:84% And notice that that flux is passing through the core 00:32:55.490 --> 00:32:58.330 align:middle line:90% and the gap in series. 00:32:58.330 --> 00:33:03.840 align:middle line:84% So if I thought of flux as kind of like being like a current, 00:33:03.840 --> 00:33:08.490 align:middle line:84% maybe I could make an equivalent circuit model that talks about 00:33:08.490 --> 00:33:13.320 align:middle line:84% fluxes and currents-- or MMFs, as we'll see in a second-- 00:33:13.320 --> 00:33:19.572 align:middle line:84% and use that as a means of quickly calculating things. 00:33:19.572 --> 00:33:20.958 align:middle line:90% OK. 00:33:20.958 --> 00:33:22.730 align:middle line:90% So let's take a look at that. 00:33:22.730 --> 00:33:25.360 align:middle line:90% 00:33:25.360 --> 00:33:31.690 align:middle line:84% In our model, we're going to define Ni, quantities of Ni-- 00:33:31.690 --> 00:33:34.780 align:middle line:90% 00:33:34.780 --> 00:33:37.750 align:middle line:84% or maybe I should write this in yellow-- 00:33:37.750 --> 00:33:43.570 align:middle line:84% as being what's known as a-- and sometimes we use the symbol F, 00:33:43.570 --> 00:33:44.530 align:middle line:90% script F-- 00:33:44.530 --> 00:33:55.520 align:middle line:84% as being what we call an magnetomotive force, or MMF. 00:33:55.520 --> 00:33:57.700 align:middle line:90% OK. 00:33:57.700 --> 00:34:00.900 align:middle line:84% In some sense, the N turns with i 00:34:00.900 --> 00:34:03.120 align:middle line:84% in there are sort of the thing that's 00:34:03.120 --> 00:34:06.455 align:middle line:90% driving flux in the system. 00:34:06.455 --> 00:34:08.620 align:middle line:90% Right, no i, no flux. 00:34:08.620 --> 00:34:10.130 align:middle line:90% No turns, no flux. 00:34:10.130 --> 00:34:14.090 align:middle line:84% So the thing that's trying to push flux is this Ni. 00:34:14.090 --> 00:34:15.770 align:middle line:90% It's the magnetomotive force. 00:34:15.770 --> 00:34:19.010 align:middle line:84% This is sort of like our voltage. 00:34:19.010 --> 00:34:20.305 align:middle line:90% Then I have flux. 00:34:20.305 --> 00:34:23.389 align:middle line:90% 00:34:23.389 --> 00:34:27.960 align:middle line:84% This is just simply flux, right, which 00:34:27.960 --> 00:34:29.580 align:middle line:84% I'm going to think of as something 00:34:29.580 --> 00:34:32.350 align:middle line:84% as I've sort of expressed it, is flowing. 00:34:32.350 --> 00:34:35.570 align:middle line:84% That's what's going through the magnetic circuit. 00:34:35.570 --> 00:34:36.980 align:middle line:84% And then the last thing I'm going 00:34:36.980 --> 00:34:39.770 align:middle line:84% to have is something I'm going to call-- 00:34:39.770 --> 00:34:41.420 align:middle line:90% I'm going to use script x-- 00:34:41.420 --> 00:34:47.280 align:middle line:84% a reluctance, which is lx, the length over mu x times Ax. 00:34:47.280 --> 00:34:49.206 align:middle line:84% This, I'm going to call it a reluctance. 00:34:49.206 --> 00:34:52.040 align:middle line:90% 00:34:52.040 --> 00:34:54.445 align:middle line:84% This is sort of like a resistance, 00:34:54.445 --> 00:35:00.950 align:middle line:84% right, where in some sense, mu is 00:35:00.950 --> 00:35:04.880 align:middle line:84% sort of conductivity for magnetic flux, 00:35:04.880 --> 00:35:07.113 align:middle line:84% all right, because reluctance gets 00:35:07.113 --> 00:35:08.780 align:middle line:84% big-- or resistance gets big when I have 00:35:08.780 --> 00:35:11.960 align:middle line:90% higher length of the structure. 00:35:11.960 --> 00:35:14.577 align:middle line:84% Resistance gets small when I get bigger area of the structure, 00:35:14.577 --> 00:35:16.160 align:middle line:84% and then there's the conductivity term 00:35:16.160 --> 00:35:17.540 align:middle line:90% in an electrical resistor. 00:35:17.540 --> 00:35:21.200 align:middle line:84% Here, its resistance to flux, and we 00:35:21.200 --> 00:35:23.450 align:middle line:90% would call this a reluctance. 00:35:23.450 --> 00:35:24.680 align:middle line:90% OK. 00:35:24.680 --> 00:35:27.820 align:middle line:84% Does that make sense to everybody? 00:35:27.820 --> 00:35:29.205 align:middle line:90% So here's the idea. 00:35:29.205 --> 00:35:33.530 align:middle line:90% 00:35:33.530 --> 00:35:48.370 align:middle line:84% If I have this structure, where I'm going to have N and i, 00:35:48.370 --> 00:35:54.700 align:middle line:84% and that's going to drive a flux around this loop, 00:35:54.700 --> 00:36:02.050 align:middle line:84% the way I would model this is with an MMF, Ni. 00:36:02.050 --> 00:36:05.140 align:middle line:84% That's what's trying to push flux through the circuit. 00:36:05.140 --> 00:36:09.310 align:middle line:84% It has to travel through the core, 00:36:09.310 --> 00:36:12.340 align:middle line:84% and I have a reluctance of the core which 00:36:12.340 --> 00:36:17.340 align:middle line:84% is l, the length of the core, over the permeability 00:36:17.340 --> 00:36:19.380 align:middle line:84% of the core and the area of the core. 00:36:19.380 --> 00:36:20.160 align:middle line:90% OK. 00:36:20.160 --> 00:36:23.670 align:middle line:84% Then it has to go through the gap, right. 00:36:23.670 --> 00:36:25.560 align:middle line:84% The flux is going around the gap, 00:36:25.560 --> 00:36:27.810 align:middle line:84% and I get the reluctance of the gap 00:36:27.810 --> 00:36:31.590 align:middle line:84% is equal to the length of the gap over mu 0 divided 00:36:31.590 --> 00:36:32.650 align:middle line:90% by the area of the gap. 00:36:32.650 --> 00:36:33.150 align:middle line:90% OK. 00:36:33.150 --> 00:36:40.720 align:middle line:84% And what I'm going to get is, flowing around this loop is phi, 00:36:40.720 --> 00:36:42.930 align:middle line:90% is flux. 00:36:42.930 --> 00:36:45.970 align:middle line:84% Does that make sense to everybody? 00:36:45.970 --> 00:36:50.200 align:middle line:84% So this is what's known as a magnetic circuit model. 00:36:50.200 --> 00:36:54.790 align:middle line:84% Right, so if you tell me how much Ni and total reluctance 00:36:54.790 --> 00:36:57.550 align:middle line:84% is flowing through, I can calculate flux, 00:36:57.550 --> 00:37:01.540 align:middle line:84% and it turns out exactly into that equation, right. 00:37:01.540 --> 00:37:04.830 align:middle line:84% Flux is Ni divided by the sum of the reluctance of the core 00:37:04.830 --> 00:37:06.080 align:middle line:90% and the reluctance of the gap. 00:37:06.080 --> 00:37:07.580 align:middle line:90% It's exactly that equation. 00:37:07.580 --> 00:37:11.620 align:middle line:84% Everything else fell out from it. 00:37:11.620 --> 00:37:14.650 align:middle line:84% Does that make sense to everybody? 00:37:14.650 --> 00:37:18.880 align:middle line:84% So to the extent that we can break our structures into pieces 00:37:18.880 --> 00:37:22.090 align:middle line:84% of sort of flux pipe, if you will, or high permeability-- 00:37:22.090 --> 00:37:24.042 align:middle line:84% they're sort of like our wires, if you will, 00:37:24.042 --> 00:37:26.500 align:middle line:84% they may have some resistance, but they're like our wires-- 00:37:26.500 --> 00:37:31.000 align:middle line:84% and then some high-reluctance, low-permeability regions 00:37:31.000 --> 00:37:33.260 align:middle line:84% which flux is being forced to cross, 00:37:33.260 --> 00:37:36.680 align:middle line:84% this lumped model is pretty good, 00:37:36.680 --> 00:37:38.690 align:middle line:84% and it's a lot easier than starting 00:37:38.690 --> 00:37:41.180 align:middle line:90% to write Maxwell's equations. 00:37:41.180 --> 00:37:41.680 align:middle line:90% OK. 00:37:41.680 --> 00:37:44.490 align:middle line:84% Now, what does it take for a circuit model to work? 00:37:44.490 --> 00:37:49.340 align:middle line:84% Right, I mean, I've sort of told you what the across variable is. 00:37:49.340 --> 00:37:50.930 align:middle line:90% That's MMF. 00:37:50.930 --> 00:37:53.210 align:middle line:84% The through variable is flux, and I 00:37:53.210 --> 00:37:55.010 align:middle line:90% have my reluctance elements. 00:37:55.010 --> 00:37:57.690 align:middle line:84% But what do I need to make a circuit model work? 00:37:57.690 --> 00:38:00.270 align:middle line:84% I mean, why should it work for the general case? 00:38:00.270 --> 00:38:02.280 align:middle line:84% Well, what do I have in an electrical circuit? 00:38:02.280 --> 00:38:07.070 align:middle line:90% I have KVL and KCL, right. 00:38:07.070 --> 00:38:10.410 align:middle line:84% Well, let's come back over to Maxwell's equations. 00:38:10.410 --> 00:38:15.005 align:middle line:84% Basically, flux continuity says the integral of B 00:38:15.005 --> 00:38:19.866 align:middle line:84% dot dA going into some region as zero, right. 00:38:19.866 --> 00:38:23.440 align:middle line:84% That's essentially saying the net flux going into any piece 00:38:23.440 --> 00:38:27.190 align:middle line:90% of space must add up to zero. 00:38:27.190 --> 00:38:31.675 align:middle line:84% That is exactly the magnetic circuit equivalent of KCL, 00:38:31.675 --> 00:38:32.175 align:middle line:90% right. 00:38:32.175 --> 00:38:35.190 align:middle line:84% And in fact, if I think about KVL and KCL 00:38:35.190 --> 00:38:38.740 align:middle line:84% for electric circuits, they also derive from Maxwell's equations. 00:38:38.740 --> 00:38:40.830 align:middle line:84% They're actually approximations, right. 00:38:40.830 --> 00:38:44.190 align:middle line:84% KCL in the electric circuit, I'm thinking that there's just 00:38:44.190 --> 00:38:45.220 align:middle line:90% no charge buildup. 00:38:45.220 --> 00:38:47.550 align:middle line:84% It's a conservation-of-charge thing here. 00:38:47.550 --> 00:38:53.660 align:middle line:84% Here, I'm using flux continuity to justify KCL. 00:38:53.660 --> 00:38:55.440 align:middle line:90% What am I doing here? 00:38:55.440 --> 00:38:58.430 align:middle line:84% I'm saying that the sum of the Ni's, if you 00:38:58.430 --> 00:39:03.320 align:middle line:84% will, around the loop, or the sum of Ni times phi times Rc 00:39:03.320 --> 00:39:09.200 align:middle line:84% plus phi times Rg, is giving me Ni, that is essentially 00:39:09.200 --> 00:39:11.110 align:middle line:90% Ampere's law. 00:39:11.110 --> 00:39:16.500 align:middle line:84% Right, Ampere's law is saying, How does the MMF that I generate 00:39:16.500 --> 00:39:18.460 align:middle line:90% relate to the flux I get? 00:39:18.460 --> 00:39:18.960 align:middle line:90% OK. 00:39:18.960 --> 00:39:25.770 align:middle line:84% So this sum around this loop is exactly what 00:39:25.770 --> 00:39:28.790 align:middle line:90% Ampere's law is doing. 00:39:28.790 --> 00:39:29.810 align:middle line:90% OK. 00:39:29.810 --> 00:39:36.260 align:middle line:84% So this is a pretty good tool for handling the general case. 00:39:36.260 --> 00:39:36.760 align:middle line:90% OK. 00:39:36.760 --> 00:39:39.860 align:middle line:84% Let me stop there and just see what questions we have. 00:39:39.860 --> 00:39:42.110 align:middle line:84% Well, before I do that, so let me just clarify, right. 00:39:42.110 --> 00:39:52.160 align:middle line:84% In an electric circuit, I have an EMF, 00:39:52.160 --> 00:39:54.150 align:middle line:84% and I'm saying, in my magnetic circuit, 00:39:54.150 --> 00:39:56.840 align:middle line:90% what I'm going to have is-- 00:39:56.840 --> 00:40:00.770 align:middle line:84% or maybe I'll draw it this way, in yellow-- 00:40:00.770 --> 00:40:03.080 align:middle line:90% I have my MMF Ni. 00:40:03.080 --> 00:40:08.450 align:middle line:90% 00:40:08.450 --> 00:40:12.740 align:middle line:84% In an electric circuit, I have current. 00:40:12.740 --> 00:40:15.800 align:middle line:84% Whereas, in my magnetic circuit, I 00:40:15.800 --> 00:40:22.100 align:middle line:84% have flux, and in my electric circuit, I have resistance. 00:40:22.100 --> 00:40:28.580 align:middle line:84% R is equal to lx over sigma Ax for my resistor, 00:40:28.580 --> 00:40:31.640 align:middle line:84% and in my magnetic circuit, I have a reluctance 00:40:31.640 --> 00:40:36.630 align:middle line:90% which is lx over mu x Ax. 00:40:36.630 --> 00:40:37.130 align:middle line:90% OK. 00:40:37.130 --> 00:40:40.970 align:middle line:84% So these are just the mappings between my electric circuit 00:40:40.970 --> 00:40:49.600 align:middle line:84% and my magnetic circuit, and I have KVL and KCL in each case. 00:40:49.600 --> 00:40:50.100 align:middle line:90% OK. 00:40:50.100 --> 00:40:51.017 align:middle line:90% So let me pause there. 00:40:51.017 --> 00:40:55.200 align:middle line:84% Are there any questions about this model? 00:40:55.200 --> 00:40:59.013 align:middle line:84% AUDIENCE: What exactly is the area of air gap that's in this? 00:40:59.013 --> 00:41:00.680 align:middle line:84% DAVID PERREAULT: It doesn't have to be-- 00:41:00.680 --> 00:41:03.360 align:middle line:90% 00:41:03.360 --> 00:41:09.740 align:middle line:84% if I imagined that this gap was really small, OK, 00:41:09.740 --> 00:41:15.200 align:middle line:84% so I have a tiny gap and a big, say, cross-sectional area, 00:41:15.200 --> 00:41:21.430 align:middle line:84% most of the flux just kind of goes straight across, right, 00:41:21.430 --> 00:41:27.210 align:middle line:84% like this, and maybe I get a tiny bit 00:41:27.210 --> 00:41:31.150 align:middle line:90% of fringing out at the corners. 00:41:31.150 --> 00:41:31.650 align:middle line:90% OK. 00:41:31.650 --> 00:41:34.428 align:middle line:84% So for the most part, if I ask, Where is flux going? 00:41:34.428 --> 00:41:35.220 align:middle line:90% there's not a lot-- 00:41:35.220 --> 00:41:38.670 align:middle line:84% I mean, flux always leaves the surface perpendicularly, right. 00:41:38.670 --> 00:41:41.250 align:middle line:84% So there's not a lot of flux that's 00:41:41.250 --> 00:41:44.310 align:middle line:84% bending out so that the effective area through which 00:41:44.310 --> 00:41:47.970 align:middle line:84% this is going in the gap is about the same as the core. 00:41:47.970 --> 00:41:50.280 align:middle line:84% Now, if I had a really big gap, what 00:41:50.280 --> 00:41:55.210 align:middle line:84% I get is a bunch of flux doing this kind of thing, 00:41:55.210 --> 00:41:57.800 align:middle line:84% right, so then the effective area 00:41:57.800 --> 00:42:02.690 align:middle line:84% through which it's traveling out in the free space 00:42:02.690 --> 00:42:05.470 align:middle line:84% is effectively bigger than it is in the core. 00:42:05.470 --> 00:42:08.570 align:middle line:84% But if I consider this case where I have a very short gap-- 00:42:08.570 --> 00:42:10.580 align:middle line:84% you know, I can ignore fringing, and so 00:42:10.580 --> 00:42:13.550 align:middle line:84% I just treat area core as the area gap. 00:42:13.550 --> 00:42:16.130 align:middle line:84% If you said I suddenly have a tiny bit of core material 00:42:16.130 --> 00:42:19.220 align:middle line:84% and a lot of stuff going on, that's very hard to calculate. 00:42:19.220 --> 00:42:21.800 align:middle line:84% Magnetic circuits tend to work well 00:42:21.800 --> 00:42:27.460 align:middle line:84% when I have sort of a lot of high-permeability material 00:42:27.460 --> 00:42:32.110 align:middle line:84% to carry my flux and little spaces 00:42:32.110 --> 00:42:34.630 align:middle line:84% like high-reluctance regions to which it's going 00:42:34.630 --> 00:42:36.200 align:middle line:90% to jump across short distances. 00:42:36.200 --> 00:42:39.760 align:middle line:84% That's the best case for these magnetic circuits. 00:42:39.760 --> 00:42:41.390 align:middle line:90% Great question. 00:42:41.390 --> 00:42:42.450 align:middle line:90% Other questions? 00:42:42.450 --> 00:42:46.070 align:middle line:90% 00:42:46.070 --> 00:42:49.930 align:middle line:84% So what does this kind of thing let us do? 00:42:49.930 --> 00:42:54.370 align:middle line:84% Well, it means that you can say, for example, suppose I suddenly 00:42:54.370 --> 00:42:56.870 align:middle line:84% said, you know what, I'm going to build this structure. 00:42:56.870 --> 00:43:01.440 align:middle line:90% 00:43:01.440 --> 00:43:05.350 align:middle line:84% I'll say the whole thing has a cross-sectional area A. 00:43:05.350 --> 00:43:09.760 align:middle line:84% But here I go, and I'm going to put my winding on the center leg 00:43:09.760 --> 00:43:10.313 align:middle line:90% here. 00:43:10.313 --> 00:43:11.980 align:middle line:84% And this is actually a kind of structure 00:43:11.980 --> 00:43:15.240 align:middle line:84% you might build, in fact, for something, right. 00:43:15.240 --> 00:43:17.650 align:middle line:90% So I have N, and I have i. 00:43:17.650 --> 00:43:24.580 align:middle line:84% And I might say, OK, I have some dimension A here, 00:43:24.580 --> 00:43:27.000 align:middle line:84% and I know the cross-sectional area of the whole thing, 00:43:27.000 --> 00:43:28.110 align:middle line:90% and so forth. 00:43:28.110 --> 00:43:32.770 align:middle line:84% How could I find the flux and inductance of this structure? 00:43:32.770 --> 00:43:36.570 align:middle line:84% I might recognize that flux is going to come up here. 00:43:36.570 --> 00:43:40.050 align:middle line:84% Some of it's going to go this way, and I'll have phi 1. 00:43:40.050 --> 00:43:42.810 align:middle line:84% And I'll have some flux that goes this way, 00:43:42.810 --> 00:43:44.590 align:middle line:90% and I'll have phi 2. 00:43:44.590 --> 00:43:51.440 align:middle line:84% And this flux in the middle will be phi 1 plus phi 2, right. 00:43:51.440 --> 00:43:53.430 align:middle line:90% Can I calculate that structure? 00:43:53.430 --> 00:43:54.200 align:middle line:90% Yeah. 00:43:54.200 --> 00:43:55.290 align:middle line:90% Why? 00:43:55.290 --> 00:43:56.730 align:middle line:90% Because I could say. 00:43:56.730 --> 00:44:05.040 align:middle line:84% OK, I have Ni, OK, and then I say, 00:44:05.040 --> 00:44:07.970 align:middle line:84% there's this block of material and I'm only drawing this 2D. 00:44:07.970 --> 00:44:10.410 align:middle line:84% There's this block of material going up the middle, 00:44:10.410 --> 00:44:13.470 align:middle line:84% and maybe I'll call that sort of-- 00:44:13.470 --> 00:44:16.170 align:middle line:84% I don't know what I called it my notes. 00:44:16.170 --> 00:44:17.623 align:middle line:90% Maybe I call that R3. 00:44:17.623 --> 00:44:20.220 align:middle line:90% 00:44:20.220 --> 00:44:22.220 align:middle line:84% And then there's some path through the core that 00:44:22.220 --> 00:44:24.150 align:middle line:90% goes this way and this way. 00:44:24.150 --> 00:44:27.450 align:middle line:90% Maybe I'll call that Rc1. 00:44:27.450 --> 00:44:32.240 align:middle line:84% And there's some gap here, and I'll call that Rg1. 00:44:32.240 --> 00:44:35.400 align:middle line:84% Right, and the same on the other side, 00:44:35.400 --> 00:44:42.790 align:middle line:84% there's some Rc2 and some R gap 2, right. 00:44:42.790 --> 00:44:46.450 align:middle line:84% And then I could say, OK, well, you know, 00:44:46.450 --> 00:44:49.150 align:middle line:84% KVL, KCL, and this magnetic circuit all 00:44:49.150 --> 00:44:51.800 align:middle line:84% apply, and I can find the total flux phi. 00:44:51.800 --> 00:44:54.610 align:middle line:84% So I could just say, look, the net reluctance 00:44:54.610 --> 00:44:59.260 align:middle line:84% looking up into here is simply equal to r net, 00:44:59.260 --> 00:45:07.870 align:middle line:84% is simply equal to R3 plus Rc1 plus R gap 00:45:07.870 --> 00:45:14.530 align:middle line:84% 1 in parallel with Rc2 plus R gap 2. 00:45:14.530 --> 00:45:18.440 align:middle line:84% Right, I now have the net reluctance. 00:45:18.440 --> 00:45:22.480 align:middle line:84% I then get phi as being-- the phi in the center leg as being 00:45:22.480 --> 00:45:29.700 align:middle line:90% equal to Ni divided by R net. 00:45:29.700 --> 00:45:34.080 align:middle line:84% And then I can find lambda is equal to N 00:45:34.080 --> 00:45:42.950 align:middle line:84% squared over R net times i, and this is the inductance, right. 00:45:42.950 --> 00:45:44.940 align:middle line:84% So, boom, just by knowing the geometry 00:45:44.940 --> 00:45:46.940 align:middle line:84% of each of these things, I calculate inductance, 00:45:46.940 --> 00:45:49.090 align:middle line:90% and I'm done. 00:45:49.090 --> 00:45:49.610 align:middle line:90% All right. 00:45:49.610 --> 00:45:52.300 align:middle line:84% And I didn't have to start writing Maxwell's equations 00:45:52.300 --> 00:45:54.770 align:middle line:90% or anything else. 00:45:54.770 --> 00:45:56.581 align:middle line:84% Does that make sense to everybody? 00:45:56.581 --> 00:45:59.950 align:middle line:90% 00:45:59.950 --> 00:46:03.400 align:middle line:84% Now, this is fully as legitimate as electric circuits, 00:46:03.400 --> 00:46:07.300 align:middle line:84% driving it down from Maxwell's equations. 00:46:07.300 --> 00:46:14.020 align:middle line:84% I will say that there are some kind of limitations 00:46:14.020 --> 00:46:17.630 align:middle line:84% that make it harder to use than an electric circuit. 00:46:17.630 --> 00:46:18.740 align:middle line:90% Why is that? 00:46:18.740 --> 00:46:22.330 align:middle line:84% If I go build some electric circuit like this, 00:46:22.330 --> 00:46:26.666 align:middle line:84% I go get a battery and some resistors and stuff, 00:46:26.666 --> 00:46:29.360 align:middle line:84% it's a pretty good bet that my current's flowing 00:46:29.360 --> 00:46:30.410 align:middle line:90% around this loop, right. 00:46:30.410 --> 00:46:34.940 align:middle line:84% The difference in electrical conductivity 00:46:34.940 --> 00:46:37.730 align:middle line:84% between my conductors and my insulators 00:46:37.730 --> 00:46:40.320 align:middle line:84% might be, like, 12 orders of magnitude, right. 00:46:40.320 --> 00:46:45.350 align:middle line:84% So if I have "10 to the 12th" difference in conductivity 00:46:45.350 --> 00:46:48.170 align:middle line:84% between my wire and the stuff that's around it, 00:46:48.170 --> 00:46:50.830 align:middle line:84% all the current goes through the wire, right. 00:46:50.830 --> 00:46:54.355 align:middle line:84% So KCL is pretty darn good, right. 00:46:54.355 --> 00:46:57.610 align:middle line:90% 00:46:57.610 --> 00:46:59.980 align:middle line:84% On the other hand, in a magnetic circuit, 00:46:59.980 --> 00:47:02.020 align:middle line:90% what's guiding the flux? 00:47:02.020 --> 00:47:04.540 align:middle line:84% What makes this reluctance element work? 00:47:04.540 --> 00:47:06.600 align:middle line:84% Well, instead of conductivity, which is guiding 00:47:06.600 --> 00:47:10.410 align:middle line:84% where electrical currents go, I have permeability 00:47:10.410 --> 00:47:13.600 align:middle line:84% determining where magnetic flux goes. 00:47:13.600 --> 00:47:17.140 align:middle line:84% And typical magnetic materials that you use in power 00:47:17.140 --> 00:47:22.220 align:middle line:84% applications might be 10 to the 3 or 10 to the 4th times mu 0, 00:47:22.220 --> 00:47:25.730 align:middle line:90% OK, and not 10 to the 12th. 00:47:25.730 --> 00:47:29.000 align:middle line:84% And so what that means is that the things that are my 00:47:29.000 --> 00:47:31.820 align:middle line:90% conductors, like my cores-- 00:47:31.820 --> 00:47:34.430 align:middle line:84% right, my conductors of the magnetic flux are my cores-- 00:47:34.430 --> 00:47:38.570 align:middle line:84% only have 10 to the 3 or 10 to the 4th times the magnetic 00:47:38.570 --> 00:47:42.200 align:middle line:84% conductivity as the space around it. 00:47:42.200 --> 00:47:45.502 align:middle line:84% So magnetic circuits are a kind of leaky, if you will, right. 00:47:45.502 --> 00:47:47.210 align:middle line:84% It'd be like building an electric circuit 00:47:47.210 --> 00:47:49.652 align:middle line:84% where your insulators are crummy and charge 00:47:49.652 --> 00:47:51.860 align:middle line:84% is kind of like-- currents are kind of flowing around 00:47:51.860 --> 00:47:53.310 align:middle line:90% in other places, too. 00:47:53.310 --> 00:47:55.850 align:middle line:84% So magnetic circuits tend to be more 00:47:55.850 --> 00:48:01.340 align:middle line:84% approximate than electrical circuits in doing calculations. 00:48:01.340 --> 00:48:02.720 align:middle line:90% OK. 00:48:02.720 --> 00:48:06.060 align:middle line:84% And it becomes particularly hard when you start to say, 00:48:06.060 --> 00:48:10.440 align:middle line:84% OK, I've got really big gaps and flux is going out here a lot, 00:48:10.440 --> 00:48:11.190 align:middle line:90% right. 00:48:11.190 --> 00:48:13.720 align:middle line:84% Then it's not that the model is wrong, 00:48:13.720 --> 00:48:16.260 align:middle line:84% it's just hard to figure out what the correct area is 00:48:16.260 --> 00:48:24.365 align:middle line:84% and what the correct reluctances and so forth are, right. 00:48:24.365 --> 00:48:25.760 align:middle line:84% So magnetic circuits, you've just 00:48:25.760 --> 00:48:28.350 align:middle line:84% got to think of them as being a little bit leaky. 00:48:28.350 --> 00:48:28.850 align:middle line:90% OK. 00:48:28.850 --> 00:48:31.640 align:middle line:84% And so typically, what we're relying on 00:48:31.640 --> 00:48:36.980 align:middle line:84% is sections of core that are very low reluctance and then 00:48:36.980 --> 00:48:41.750 align:middle line:84% small pieces that are fairly high reluctance in order to make 00:48:41.750 --> 00:48:43.165 align:middle line:90% the calculations very accurate. 00:48:43.165 --> 00:48:44.790 align:middle line:84% But you can do them one way or another, 00:48:44.790 --> 00:48:46.957 align:middle line:84% and they're also a great thinking tool even when you 00:48:46.957 --> 00:48:47.810 align:middle line:90% can't. 00:48:47.810 --> 00:48:52.130 align:middle line:84% The other thing I will say is that in an electrical circuit, 00:48:52.130 --> 00:48:55.460 align:middle line:84% our unit of conductivity is sigma, right. 00:48:55.460 --> 00:48:58.940 align:middle line:84% We have J equals sigma E, like Ohm's law, right. 00:48:58.940 --> 00:49:01.010 align:middle line:84% And materials that are conductive, 00:49:01.010 --> 00:49:04.700 align:middle line:84% they tend to follow that law over a pretty wide range 00:49:04.700 --> 00:49:07.920 align:middle line:84% of electric fields and current densities-- not always, right. 00:49:07.920 --> 00:49:09.740 align:middle line:84% Like, you could run into limits of that. 00:49:09.740 --> 00:49:12.710 align:middle line:84% But in magnetic structures, what we'll 00:49:12.710 --> 00:49:16.460 align:middle line:84% see is, if I have the relationship between B and H, 00:49:16.460 --> 00:49:21.630 align:middle line:84% right, it tends to do something like this, that we'll see. 00:49:21.630 --> 00:49:24.850 align:middle line:84% If this is the mu of the core material, 00:49:24.850 --> 00:49:30.450 align:middle line:84% it's above some saturation flux density, right, 00:49:30.450 --> 00:49:32.550 align:middle line:84% this drops off and starts to become mu 0. 00:49:32.550 --> 00:49:34.470 align:middle line:84% Basically, all the magnetic domains align, 00:49:34.470 --> 00:49:36.970 align:middle line:84% and it doesn't behave very well anymore. 00:49:36.970 --> 00:49:40.140 align:middle line:84% And so my approximation of thinking 00:49:40.140 --> 00:49:42.780 align:middle line:84% of just something as some permeability mu core 00:49:42.780 --> 00:49:46.480 align:middle line:84% only applies over some ranges of flux densities. 00:49:46.480 --> 00:49:49.900 align:middle line:84% So I might have to go back and put that into my thinking, too. 00:49:49.900 --> 00:49:53.010 align:middle line:84% I can't use these out to infinite flux densities the way 00:49:53.010 --> 00:49:54.090 align:middle line:90% I might think-- 00:49:54.090 --> 00:49:57.000 align:middle line:84% I might be less hesitant to do in a conventional electrical 00:49:57.000 --> 00:49:57.850 align:middle line:90% circuit. 00:49:57.850 --> 00:50:00.120 align:middle line:84% So there are some limitations on this model, 00:50:00.120 --> 00:50:03.090 align:middle line:84% but, boy, does it make your life easy, OK. 00:50:03.090 --> 00:50:07.780 align:middle line:84% And we're going to take great advantage of it moving forward. 00:50:07.780 --> 00:50:09.080 align:middle line:90% Any questions? 00:50:09.080 --> 00:50:12.710 align:middle line:90% 00:50:12.710 --> 00:50:13.530 align:middle line:90% OK, great. 00:50:13.530 --> 00:50:16.610 align:middle line:84% We will take up more on magnetics and magnetic circuits 00:50:16.610 --> 00:50:17.600 align:middle line:90% tomorrow-- 00:50:17.600 --> 00:50:19.930 align:middle line:90% sorry, Monday. 00:50:19.930 --> 00:50:25.000 align:middle line:90%