WEBVTT 00:00:00.000 --> 00:00:01.924 align:middle line:90% [SQUEAKING] 00:00:01.924 --> 00:00:04.329 align:middle line:90% [RUSTLING] 00:00:04.329 --> 00:00:05.772 align:middle line:90% [CLICKING] 00:00:05.772 --> 00:00:12.530 align:middle line:90% 00:00:12.530 --> 00:00:15.240 align:middle line:84% DAVID PERREAULT: OK, why don't we get started. 00:00:15.240 --> 00:00:18.650 align:middle line:84% So we've been talking a bit about magnetics 00:00:18.650 --> 00:00:21.650 align:middle line:84% and particularly how you model magnetics. 00:00:21.650 --> 00:00:26.270 align:middle line:84% Today, I'd like to talk a bit about magnetics loss. 00:00:26.270 --> 00:00:31.650 align:middle line:84% And we can only really scratch the surface of this topic. 00:00:31.650 --> 00:00:34.730 align:middle line:84% I mean, it actually takes up a whole chapter 00:00:34.730 --> 00:00:39.230 align:middle line:84% in Principles of Power Electronics 00:00:39.230 --> 00:00:43.170 align:middle line:84% beyond what's in the basic magnetics chapter. 00:00:43.170 --> 00:00:45.950 align:middle line:84% So it's a pretty deep topic if you really 00:00:45.950 --> 00:00:48.320 align:middle line:84% want to design very high-performance things like how 00:00:48.320 --> 00:00:50.570 align:middle line:84% you predict the losses or calculate the losses. 00:00:50.570 --> 00:00:53.270 align:middle line:84% And also, magnetics design, which 00:00:53.270 --> 00:00:58.330 align:middle line:84% is the focus of Principles of Power Electronics chapter 20, 00:00:58.330 --> 00:01:00.450 align:middle line:84% is also a pretty extensive topic. 00:01:00.450 --> 00:01:06.980 align:middle line:84% Nonetheless, you can get pretty far with some basic information. 00:01:06.980 --> 00:01:11.680 align:middle line:84% And we've put up on the web just some basic inductor design tips, 00:01:11.680 --> 00:01:15.520 align:middle line:84% and that should be enough to get you going. 00:01:15.520 --> 00:01:19.000 align:middle line:84% In today's lecture, I'm going to go into magnetics loss, 00:01:19.000 --> 00:01:24.050 align:middle line:84% and I'm going to wave my hands enough that you might start 00:01:24.050 --> 00:01:26.540 align:middle line:84% to feel a slight breeze, in the sense 00:01:26.540 --> 00:01:32.000 align:middle line:84% that I'm not going to justify or prove anything I say. 00:01:32.000 --> 00:01:34.540 align:middle line:84% My goal is just to give you enough information 00:01:34.540 --> 00:01:38.525 align:middle line:84% so that you know some things that you might look out for when 00:01:38.525 --> 00:01:39.650 align:middle line:90% you're designing magnetics. 00:01:39.650 --> 00:01:41.422 align:middle line:84% And the fact that I'm doing that is not 00:01:41.422 --> 00:01:42.880 align:middle line:84% going to prevent you from designing 00:01:42.880 --> 00:01:45.580 align:middle line:84% perfectly good inductors for your design project 00:01:45.580 --> 00:01:47.668 align:middle line:84% or design a perfectly good transformer. 00:01:47.668 --> 00:01:49.960 align:middle line:84% But when you start to really want to push the envelope, 00:01:49.960 --> 00:01:51.502 align:middle line:84% suddenly you've got to start thinking 00:01:51.502 --> 00:01:54.380 align:middle line:84% about these effects I'm going to talk about pretty carefully. 00:01:54.380 --> 00:01:57.380 align:middle line:90% 00:01:57.380 --> 00:02:01.250 align:middle line:84% So if I'm going to design a magnetic component, 00:02:01.250 --> 00:02:03.180 align:middle line:84% I really have to think about loss, 00:02:03.180 --> 00:02:06.020 align:middle line:84% and loss is important for two reasons. 00:02:06.020 --> 00:02:09.000 align:middle line:84% If you have bad magnetic components, they tend to-- 00:02:09.000 --> 00:02:10.860 align:middle line:84% the magnetic components tend to be 00:02:10.860 --> 00:02:14.860 align:middle line:84% the dominant source of loss in a converter, for the most part. 00:02:14.860 --> 00:02:17.592 align:middle line:84% So in a typical design, if you don't 00:02:17.592 --> 00:02:19.550 align:middle line:84% do a good job on the magnetics, your efficiency 00:02:19.550 --> 00:02:22.160 align:middle line:90% is going to stink. 00:02:22.160 --> 00:02:24.680 align:middle line:84% The second piece of it is is that 00:02:24.680 --> 00:02:27.170 align:middle line:84% what sets the size of the magnetic component 00:02:27.170 --> 00:02:28.160 align:middle line:90% is often the law. 00:02:28.160 --> 00:02:30.230 align:middle line:84% So what prevents you from making things smaller, 00:02:30.230 --> 00:02:32.810 align:middle line:84% often, is that if you make it too small, it'll get too hot, 00:02:32.810 --> 00:02:34.640 align:middle line:90% and you'll be out of luck. 00:02:34.640 --> 00:02:38.000 align:middle line:84% So thinking about, What are the loss mechanisms? 00:02:38.000 --> 00:02:40.730 align:middle line:84% and what to watch out for is what 00:02:40.730 --> 00:02:42.460 align:middle line:90% we want to pay attention to. 00:02:42.460 --> 00:02:45.550 align:middle line:84% Now, if I have a typical magnetic component, what do 00:02:45.550 --> 00:02:46.130 align:middle line:90% I have? 00:02:46.130 --> 00:02:50.480 align:middle line:84% I have a winding and a core, usually. 00:02:50.480 --> 00:02:53.950 align:middle line:84% So the two subsets of losses I've got to worry about 00:02:53.950 --> 00:02:55.470 align:middle line:90% are, one, the winding loss-- 00:02:55.470 --> 00:02:59.770 align:middle line:90% 00:02:59.770 --> 00:03:01.930 align:middle line:84% and sometimes this is called copper loss 00:03:01.930 --> 00:03:06.130 align:middle line:84% because most windings are made of copper. 00:03:06.130 --> 00:03:09.310 align:middle line:84% Typically copper wire is used, although aluminum wire is not 00:03:09.310 --> 00:03:12.100 align:middle line:84% infrequently used, as well, especially if you want something 00:03:12.100 --> 00:03:15.430 align:middle line:84% that's going to be very light because aluminum 00:03:15.430 --> 00:03:19.980 align:middle line:84% has better conductivity per unit mass than copper does. 00:03:19.980 --> 00:03:23.770 align:middle line:84% The second thing is, of course, core loss. 00:03:23.770 --> 00:03:26.710 align:middle line:84% And so we're going to consider each of these things 00:03:26.710 --> 00:03:32.490 align:middle line:84% today and just give you some things to watch out for. 00:03:32.490 --> 00:03:34.950 align:middle line:84% Now, at the most basic level, if we're 00:03:34.950 --> 00:03:39.300 align:middle line:84% going to think about your winding loss, 00:03:39.300 --> 00:03:40.470 align:middle line:90% you're going to wind-- 00:03:40.470 --> 00:03:41.700 align:middle line:84% so suppose we're designing an inductor. 00:03:41.700 --> 00:03:43.290 align:middle line:84% You're going to wind a bunch of turns, 00:03:43.290 --> 00:03:46.200 align:middle line:84% typically on a bobbin that's going to go on a core. 00:03:46.200 --> 00:03:48.300 align:middle line:84% So just to give you an example, if you 00:03:48.300 --> 00:03:54.780 align:middle line:84% look at this in the top right here, this is a half of a core. 00:03:54.780 --> 00:03:57.630 align:middle line:84% And then you put two of those half of a cores together, 00:03:57.630 --> 00:03:59.280 align:middle line:84% and you might grind the center post 00:03:59.280 --> 00:04:01.410 align:middle line:84% to give you a gap to store the energy in, 00:04:01.410 --> 00:04:06.140 align:middle line:84% and you would tend to wind your windings around the center post. 00:04:06.140 --> 00:04:08.370 align:middle line:84% So if you're going to have an inductor, 00:04:08.370 --> 00:04:10.540 align:middle line:84% you're going to wind it with some length of wire. 00:04:10.540 --> 00:04:17.459 align:middle line:84% So a DC-- the resistance of the wire 00:04:17.459 --> 00:04:21.720 align:middle line:84% is simply going to be rho times the length of the wire divided 00:04:21.720 --> 00:04:24.255 align:middle line:90% by the area of the wire. 00:04:24.255 --> 00:04:26.460 align:middle line:84% So you would often pick your wire gauge 00:04:26.460 --> 00:04:27.670 align:middle line:90% to give you a certain area. 00:04:27.670 --> 00:04:28.720 align:middle line:90% You'll have some length. 00:04:28.720 --> 00:04:29.760 align:middle line:90% You wind it up. 00:04:29.760 --> 00:04:34.340 align:middle line:84% And resistivity times length over area 00:04:34.340 --> 00:04:37.460 align:middle line:90% is your wire resistance. 00:04:37.460 --> 00:04:43.460 align:middle line:84% One thing I'll just give you a heads up just to pay attention 00:04:43.460 --> 00:04:46.250 align:middle line:84% to is that, if you think about the resistivity, which 00:04:46.250 --> 00:04:48.350 align:middle line:84% is the resistivity of copper, it's 00:04:48.350 --> 00:04:50.520 align:middle line:84% actually a pretty strong function to temperature. 00:04:50.520 --> 00:04:53.780 align:middle line:84% If you raise the temperature by 100 degrees C, 00:04:53.780 --> 00:04:57.220 align:middle line:84% which is a lot, but not necessarily unreasonable, 00:04:57.220 --> 00:05:01.280 align:middle line:84% the resistivity of the wire tends to go up by about 40% 00:05:01.280 --> 00:05:03.740 align:middle line:84% because that's just the characteristic of copper. 00:05:03.740 --> 00:05:07.280 align:middle line:84% So usually, you want to use the resistivity at whatever 00:05:07.280 --> 00:05:09.140 align:middle line:84% the maximum temperature you think your wire 00:05:09.140 --> 00:05:11.515 align:middle line:84% is going to get to because that's sort of the worst case, 00:05:11.515 --> 00:05:15.800 align:middle line:84% and that's going to give you the largest resistance. 00:05:15.800 --> 00:05:18.410 align:middle line:84% So then you'd calculate your loss. 00:05:18.410 --> 00:05:21.596 align:middle line:84% If you're only considering-- this is the DC resistance. 00:05:21.596 --> 00:05:25.950 align:middle line:90% 00:05:25.950 --> 00:05:27.590 align:middle line:90% And we'll come back to that. 00:05:27.590 --> 00:05:34.380 align:middle line:84% Power is just going to be equal to IRMS quantity squared 00:05:34.380 --> 00:05:36.900 align:middle line:90% times R wire. 00:05:36.900 --> 00:05:41.350 align:middle line:84% OK if the RMS is dominated by the DC component, 00:05:41.350 --> 00:05:42.850 align:middle line:84% you have almost a current component, 00:05:42.850 --> 00:05:45.120 align:middle line:84% then RMS is close to the IDC, and you might 00:05:45.120 --> 00:05:48.720 align:middle line:90% get IDC squared times R wire. 00:05:48.720 --> 00:05:50.950 align:middle line:84% So it's pretty straightforward to calculate. 00:05:50.950 --> 00:05:55.410 align:middle line:84% And so then the trade-off you're making is, how fat do you 00:05:55.410 --> 00:05:57.550 align:middle line:90% want the wire to be? 00:05:57.550 --> 00:06:00.520 align:middle line:84% And how much length of it do you need 00:06:00.520 --> 00:06:04.065 align:middle line:90% to get your loss low enough? 00:06:04.065 --> 00:06:08.990 align:middle line:90% 00:06:08.990 --> 00:06:11.420 align:middle line:84% Something that does come up is-- and the reason 00:06:11.420 --> 00:06:14.570 align:middle line:84% why I'm talking about this wire area 00:06:14.570 --> 00:06:19.670 align:middle line:84% is because we're going to fit our turns in this window. 00:06:19.670 --> 00:06:23.600 align:middle line:84% So there's some cross-sectional area, the area of the window, 00:06:23.600 --> 00:06:27.280 align:middle line:84% that you can put your turns of wire in. 00:06:27.280 --> 00:06:29.650 align:middle line:84% And so eventually, you can only put 00:06:29.650 --> 00:06:31.300 align:middle line:84% so many turns of a certain gauge wire 00:06:31.300 --> 00:06:34.495 align:middle line:84% before you run out of window room. 00:06:34.495 --> 00:06:37.270 align:middle line:84% But then you say, well, I can use a small wire 00:06:37.270 --> 00:06:39.441 align:middle line:84% and put a lot of current through it. 00:06:39.441 --> 00:06:40.900 align:middle line:84% And yeah, maybe you can, but you've 00:06:40.900 --> 00:06:42.820 align:middle line:84% got to worry about overheating things. 00:06:42.820 --> 00:06:45.130 align:middle line:84% So something that often comes up when 00:06:45.130 --> 00:06:47.620 align:middle line:84% you're doing this is, exactly how much current can I 00:06:47.620 --> 00:06:51.091 align:middle line:90% put through that piece of wire? 00:06:51.091 --> 00:06:55.240 align:middle line:84% So if I wanted to say, what's the allowable RMS current, 00:06:55.240 --> 00:06:58.670 align:middle line:90% maybe, that my wire can carry? 00:06:58.670 --> 00:07:00.230 align:middle line:84% And that's an interesting question 00:07:00.230 --> 00:07:03.650 align:middle line:84% because there's not a single answer to that question. 00:07:03.650 --> 00:07:07.835 align:middle line:84% It's usually dictated by how hot things are going to get. 00:07:07.835 --> 00:07:11.340 align:middle line:84% And that, in turn, makes it dependent on heat transfer. 00:07:11.340 --> 00:07:14.210 align:middle line:84% Now, we haven't really talked about thermal models and heat 00:07:14.210 --> 00:07:15.020 align:middle line:90% transfer yet. 00:07:15.020 --> 00:07:17.390 align:middle line:90% And we will. 00:07:17.390 --> 00:07:22.010 align:middle line:84% But generally, people often will work by some kind 00:07:22.010 --> 00:07:23.968 align:middle line:90% of rule of thumb to start. 00:07:23.968 --> 00:07:26.010 align:middle line:84% In the end, you're going to end up with a design. 00:07:26.010 --> 00:07:28.135 align:middle line:84% You're going to find out if it gets too hot or not. 00:07:28.135 --> 00:07:31.130 align:middle line:84% But as a place to start, you might take a guess 00:07:31.130 --> 00:07:37.640 align:middle line:84% at the current density, J. That's in amps per centimeter 00:07:37.640 --> 00:07:39.652 align:middle line:90% squared or something. 00:07:39.652 --> 00:07:45.110 align:middle line:84% And the question is, how much current density, 00:07:45.110 --> 00:07:49.460 align:middle line:84% how many amps per meter squared can I put through my wire? 00:07:49.460 --> 00:07:53.540 align:middle line:90% And that number depends on size. 00:07:53.540 --> 00:07:57.230 align:middle line:84% If I have something really big, there's a long distance 00:07:57.230 --> 00:07:58.140 align:middle line:90% to get heat out. 00:07:58.140 --> 00:08:02.470 align:middle line:84% It has a very high volume-to-surface-area ratio 00:08:02.470 --> 00:08:05.850 align:middle line:84% because volume goes as a cube, and surface area only goes 00:08:05.850 --> 00:08:09.000 align:middle line:84% as the square of linear dimension. 00:08:09.000 --> 00:08:11.710 align:middle line:84% So if you have something small, it's easy to get heat out of it. 00:08:11.710 --> 00:08:14.168 align:middle line:84% If you have something big, it's hard to get heat out of it. 00:08:14.168 --> 00:08:16.860 align:middle line:84% So typically, the current densities you can use 00:08:16.860 --> 00:08:19.440 align:middle line:84% depend upon how big the thing you're building is. 00:08:19.440 --> 00:08:21.850 align:middle line:84% Smaller, you can use higher current densities. 00:08:21.850 --> 00:08:26.620 align:middle line:84% A typical number might be on the order of 500 amps per centimeter 00:08:26.620 --> 00:08:28.510 align:middle line:90% squared. 00:08:28.510 --> 00:08:31.130 align:middle line:84% And we're talking about RMS values. 00:08:31.130 --> 00:08:33.650 align:middle line:90% But again, it depends on scale. 00:08:33.650 --> 00:08:37.780 align:middle line:84% This chart, which I've pulled out of KPVS-- 00:08:37.780 --> 00:08:42.460 align:middle line:84% what it shows is, if I took this kind of core, which 00:08:42.460 --> 00:08:47.960 align:middle line:84% is known as an RM core, or Rectangular Modular core, RM 4 00:08:47.960 --> 00:08:49.870 align:middle line:90% core is really small. 00:08:49.870 --> 00:08:51.380 align:middle line:90% RM 14 core is the biggest. 00:08:51.380 --> 00:08:55.210 align:middle line:84% It's maybe getting close to the size of my fist. 00:08:55.210 --> 00:09:00.270 align:middle line:84% And this shows, if I have only winding loss, 00:09:00.270 --> 00:09:03.440 align:middle line:84% how hot will the thing get based on measured data? 00:09:03.440 --> 00:09:07.690 align:middle line:84% And what we can see is maybe you have a few-hundred watts per 00:09:07.690 --> 00:09:09.640 align:middle line:84% centimeter squared for a big element 00:09:09.640 --> 00:09:11.540 align:middle line:84% and maybe much higher for a small element. 00:09:11.540 --> 00:09:14.200 align:middle line:84% And this is for a 25-degree C rise. 00:09:14.200 --> 00:09:17.270 align:middle line:84% On the one hand, maybe you can take more than a 25-degree C 00:09:17.270 --> 00:09:17.770 align:middle line:90% rise. 00:09:17.770 --> 00:09:19.285 align:middle line:90% It's not unusual to have 50. 00:09:19.285 --> 00:09:21.160 align:middle line:84% On the other hand, you also have other losses 00:09:21.160 --> 00:09:22.340 align:middle line:90% besides the winding loss. 00:09:22.340 --> 00:09:24.820 align:middle line:84% So maybe this is kind of representative 00:09:24.820 --> 00:09:27.080 align:middle line:84% of what might be allowable for current densities. 00:09:27.080 --> 00:09:31.285 align:middle line:84% And so you can see my number of 500 amps per centimeter squared 00:09:31.285 --> 00:09:32.060 align:middle line:90% is right here. 00:09:32.060 --> 00:09:35.260 align:middle line:84% So typically, you might start your design off and say, 00:09:35.260 --> 00:09:38.080 align:middle line:84% if it's big, maybe I'll use 100 or 200 amps per centimeter 00:09:38.080 --> 00:09:39.550 align:middle line:90% squared as my guess. 00:09:39.550 --> 00:09:42.040 align:middle line:84% If it's smaller, maybe I'll go to 500 amps 00:09:42.040 --> 00:09:44.830 align:middle line:84% per centimeter squared or even a little bit higher. 00:09:44.830 --> 00:09:46.900 align:middle line:84% And then what's really going to limit me 00:09:46.900 --> 00:09:50.210 align:middle line:90% is, how hot does the thing get? 00:09:50.210 --> 00:09:51.370 align:middle line:90% Any questions about that? 00:09:51.370 --> 00:09:54.975 align:middle line:90% 00:09:54.975 --> 00:09:55.475 align:middle line:90% OK. 00:09:55.475 --> 00:09:58.080 align:middle line:90% 00:09:58.080 --> 00:09:59.170 align:middle line:90% That's all well and good. 00:09:59.170 --> 00:10:01.410 align:middle line:90% And that's a basic notion. 00:10:01.410 --> 00:10:05.160 align:middle line:84% Here's how I calculate my winding loss. 00:10:05.160 --> 00:10:08.077 align:middle line:84% I want to fit a lot of turns in a small cross-sectional area, 00:10:08.077 --> 00:10:09.660 align:middle line:84% so I've got to pick an A wire, and I'm 00:10:09.660 --> 00:10:12.840 align:middle line:84% going to be limited by how much current I can carry in that wire 00:10:12.840 --> 00:10:15.750 align:middle line:84% by some number that I make an estimate at 00:10:15.750 --> 00:10:18.386 align:middle line:84% and then go figure out how hot it gets afterwards. 00:10:18.386 --> 00:10:21.030 align:middle line:90% 00:10:21.030 --> 00:10:24.522 align:middle line:84% There are, however, other effects. 00:10:24.522 --> 00:10:30.730 align:middle line:84% And chief among these is what happens at high frequency. 00:10:30.730 --> 00:10:35.020 align:middle line:84% So all of this calculation of the resistance 00:10:35.020 --> 00:10:36.380 align:middle line:90% is the DC resistance. 00:10:36.380 --> 00:10:38.560 align:middle line:84% This is what happens if I put a constant current 00:10:38.560 --> 00:10:39.960 align:middle line:90% through the wire. 00:10:39.960 --> 00:10:44.910 align:middle line:84% If I put a current that varies very quickly in time 00:10:44.910 --> 00:10:47.130 align:middle line:84% through a big wire, what happens is 00:10:47.130 --> 00:10:51.430 align:middle line:84% I don't get to use the full wire cross-section. 00:10:51.430 --> 00:10:54.660 align:middle line:90% There are two effects. 00:10:54.660 --> 00:11:03.060 align:middle line:84% One is known as skin effect that I'm going to talk about now. 00:11:03.060 --> 00:11:04.570 align:middle line:90% And here's the idea. 00:11:04.570 --> 00:11:08.310 align:middle line:90% Suppose I had some wire. 00:11:08.310 --> 00:11:11.060 align:middle line:90% Here's some length of wire. 00:11:11.060 --> 00:11:12.500 align:middle line:84% And I put some current through it. 00:11:12.500 --> 00:11:16.580 align:middle line:84% Now, a DC-- what I expect is that current, 00:11:16.580 --> 00:11:19.910 align:middle line:84% if it's going down the wire, to distribute itself 00:11:19.910 --> 00:11:22.970 align:middle line:84% evenly across the volume of the wire, evenly 00:11:22.970 --> 00:11:26.290 align:middle line:84% across the cross-section of the wire. 00:11:26.290 --> 00:11:27.782 align:middle line:90% And that's what you get. 00:11:27.782 --> 00:11:30.490 align:middle line:84% The current will just spread out and be carried evenly everywhere 00:11:30.490 --> 00:11:32.490 align:middle line:90% in the wire. 00:11:32.490 --> 00:11:38.430 align:middle line:84% But now I have this AC current going into the wire. 00:11:38.430 --> 00:11:41.780 align:middle line:84% And that AC current develops a magnetic field. 00:11:41.780 --> 00:11:45.130 align:middle line:84% So the current that's in the middle of the wire 00:11:45.130 --> 00:11:51.780 align:middle line:84% is going to generate an H field in the wire 00:11:51.780 --> 00:11:53.860 align:middle line:84% and, hence, a B field in the wire. 00:11:53.860 --> 00:11:57.090 align:middle line:84% And that represents magnetic flux being carried around 00:11:57.090 --> 00:12:01.110 align:middle line:90% inside the wire. 00:12:01.110 --> 00:12:03.420 align:middle line:84% But what do we know about something 00:12:03.420 --> 00:12:08.570 align:middle line:84% that generates a dB/dt, a change in magnetic flux density? 00:12:08.570 --> 00:12:13.250 align:middle line:84% We know by Faraday's law that that generates a voltage. 00:12:13.250 --> 00:12:15.160 align:middle line:84% And that voltage that you generate 00:12:15.160 --> 00:12:17.590 align:middle line:84% is going to cause, essentially, eddy currents 00:12:17.590 --> 00:12:19.570 align:middle line:90% to circulate within the wire. 00:12:19.570 --> 00:12:21.790 align:middle line:84% And they're going to try to reject 00:12:21.790 --> 00:12:26.060 align:middle line:84% the change in magnetic flux density in the wire. 00:12:26.060 --> 00:12:29.050 align:middle line:84% So what you're going to get is induced eddy currents 00:12:29.050 --> 00:12:30.860 align:middle line:90% that do something like this. 00:12:30.860 --> 00:12:36.250 align:middle line:84% They're going to come up the wire in the middle 00:12:36.250 --> 00:12:38.303 align:middle line:84% and go down the wire on the outside. 00:12:38.303 --> 00:12:41.365 align:middle line:90% 00:12:41.365 --> 00:12:45.292 align:middle line:84% And so what's going to happen is, in net, 00:12:45.292 --> 00:12:47.750 align:middle line:84% the current in the middle of the wire will start to cancel. 00:12:47.750 --> 00:12:51.214 align:middle line:90% 00:12:51.214 --> 00:12:55.170 align:middle line:84% And so what you will get if you go to a high enough frequency is 00:12:55.170 --> 00:12:59.850 align:middle line:84% that the net currents will only be carried, essentially-- 00:12:59.850 --> 00:13:02.220 align:middle line:84% if I just said, where is the net current-- 00:13:02.220 --> 00:13:07.920 align:middle line:84% will be carried only in a band around the surface of the wire. 00:13:07.920 --> 00:13:10.390 align:middle line:84% So instead of getting the full wire cross-section 00:13:10.390 --> 00:13:13.180 align:middle line:84% at high frequency, you will only get 00:13:13.180 --> 00:13:15.910 align:middle line:90% to use the surface of the wire. 00:13:15.910 --> 00:13:18.040 align:middle line:84% That's what's known as skin effect. 00:13:18.040 --> 00:13:19.340 align:middle line:90% Why does this happen? 00:13:19.340 --> 00:13:21.610 align:middle line:84% It happens exactly for these eddy current reasons. 00:13:21.610 --> 00:13:24.593 align:middle line:84% To actually solve things, you generate 00:13:24.593 --> 00:13:26.510 align:middle line:84% what's known as a magnetic diffusion equation. 00:13:26.510 --> 00:13:28.300 align:middle line:84% You solve it, and you can solve for, 00:13:28.300 --> 00:13:30.070 align:middle line:84% what is the current density profile 00:13:30.070 --> 00:13:31.815 align:middle line:84% going in from the surface of the wire? 00:13:31.815 --> 00:13:35.748 align:middle line:90% 00:13:35.748 --> 00:13:42.390 align:middle line:84% This is kind of annoying because I got a certain amount of wire, 00:13:42.390 --> 00:13:44.800 align:middle line:84% and I wanted to use it to carry current. 00:13:44.800 --> 00:13:47.700 align:middle line:84% And now what I'm telling you is, if the frequency is high enough, 00:13:47.700 --> 00:13:49.140 align:middle line:90% I don't get to use all the wire. 00:13:49.140 --> 00:13:51.598 align:middle line:84% And I've just got this dead space in the middle of my wire, 00:13:51.598 --> 00:13:53.310 align:middle line:90% and I'm not getting to use it. 00:13:53.310 --> 00:13:56.370 align:middle line:84% Well, I said that's at high frequency and low frequency. 00:13:56.370 --> 00:13:57.730 align:middle line:90% What do I mean by that? 00:13:57.730 --> 00:14:00.600 align:middle line:84% There's something called the skin depth, which falls out 00:14:00.600 --> 00:14:02.260 align:middle line:90% of this analysis I mentioned. 00:14:02.260 --> 00:14:08.460 align:middle line:84% And the formula for skin depth, which is written as delta, 00:14:08.460 --> 00:14:10.980 align:middle line:84% is equal to the square root of the conductivity, 00:14:10.980 --> 00:14:13.920 align:middle line:84% in this case of the copper, divided 00:14:13.920 --> 00:14:20.850 align:middle line:84% by pi times the permeability, in this case of the copper, 00:14:20.850 --> 00:14:23.490 align:middle line:90% and the frequency. 00:14:23.490 --> 00:14:26.700 align:middle line:84% This skin depth is a characteristic length 00:14:26.700 --> 00:14:30.630 align:middle line:84% that says, how quickly does the fields and currents fall off 00:14:30.630 --> 00:14:32.450 align:middle line:90% inside the wire? 00:14:32.450 --> 00:14:35.780 align:middle line:84% So if you take an electromagnetic analysis-- 00:14:35.780 --> 00:14:38.300 align:middle line:84% you've probably done this analysis before. 00:14:38.300 --> 00:14:42.607 align:middle line:84% But what it says is, notice that this is a function of frequency. 00:14:42.607 --> 00:14:44.690 align:middle line:84% The skin depth decreases as 1 over the square root 00:14:44.690 --> 00:14:45.730 align:middle line:90% of frequency. 00:14:45.730 --> 00:14:48.545 align:middle line:84% So essentially, if the skin depth 00:14:48.545 --> 00:14:51.950 align:middle line:84% is on the order of the diameter of my wire, 00:14:51.950 --> 00:14:53.990 align:middle line:84% I pretty much carry it approximately 00:14:53.990 --> 00:14:56.630 align:middle line:84% constant across the surface of the wire. 00:14:56.630 --> 00:15:01.390 align:middle line:84% If my skin depth gets small compared to the wire, 00:15:01.390 --> 00:15:03.640 align:middle line:84% that's what I'm only going to carry it in the surface. 00:15:03.640 --> 00:15:06.743 align:middle line:84% And it doesn't just get carried-- 00:15:06.743 --> 00:15:10.560 align:middle line:90% 00:15:10.560 --> 00:15:13.120 align:middle line:84% here, I'm giving you a circular cross-section. 00:15:13.120 --> 00:15:16.590 align:middle line:84% Let's just think of a rectangular cross-section. 00:15:16.590 --> 00:15:21.180 align:middle line:84% And I would say, OK, at DC, if I looked 00:15:21.180 --> 00:15:24.900 align:middle line:84% at the current density versus the wire dimension between, say, 00:15:24.900 --> 00:15:29.413 align:middle line:84% 0 and the width of the wire, I had single-sided conduction, 00:15:29.413 --> 00:15:30.830 align:middle line:84% which I'll talk about in a minute, 00:15:30.830 --> 00:15:34.830 align:middle line:84% at DC, what you'd get is a uniform current density. 00:15:34.830 --> 00:15:41.150 align:middle line:84% This is J versus this depth into the wire. 00:15:41.150 --> 00:15:45.280 align:middle line:90% This is at DC. 00:15:45.280 --> 00:15:47.200 align:middle line:84% At high frequency AC, what happens 00:15:47.200 --> 00:15:50.480 align:middle line:84% is you get a lot of current on the surface. 00:15:50.480 --> 00:15:52.576 align:middle line:84% And it falls off in an exponential. 00:15:52.576 --> 00:15:55.865 align:middle line:90% 00:15:55.865 --> 00:16:01.045 align:middle line:84% And this goes as sort of e to the minus x over delta. 00:16:01.045 --> 00:16:03.733 align:middle line:84% It falls off exponentially with a length constant 00:16:03.733 --> 00:16:04.650 align:middle line:90% that's the skin depth. 00:16:04.650 --> 00:16:07.160 align:middle line:90% 00:16:07.160 --> 00:16:09.090 align:middle line:84% So when do you have to worry about this? 00:16:09.090 --> 00:16:12.110 align:middle line:84% When your wire is on the dimension-- 00:16:12.110 --> 00:16:15.820 align:middle line:84% or larger, significantly larger, than dimension of the skin 00:16:15.820 --> 00:16:16.320 align:middle line:90% depth. 00:16:16.320 --> 00:16:17.555 align:middle line:84% So you just calculate the skin depth, 00:16:17.555 --> 00:16:18.800 align:middle line:84% and you'll find out if you're going 00:16:18.800 --> 00:16:20.008 align:middle line:90% to have a lot of skin effect. 00:16:20.008 --> 00:16:24.930 align:middle line:90% 00:16:24.930 --> 00:16:29.820 align:middle line:84% Interestingly, if you have a case where 00:16:29.820 --> 00:16:31.350 align:middle line:90% the width of the wire-- 00:16:31.350 --> 00:16:34.710 align:middle line:84% I'm having a foil where I'm carrying current into the board 00:16:34.710 --> 00:16:35.430 align:middle line:90% now. 00:16:35.430 --> 00:16:38.460 align:middle line:84% So this is the notion there's a foil going into the board. 00:16:38.460 --> 00:16:42.370 align:middle line:84% Or you can do something similar here. 00:16:42.370 --> 00:16:48.490 align:middle line:84% Interestingly, the loss is exactly the same, in this case, 00:16:48.490 --> 00:16:52.540 align:middle line:84% as if I carried all the current in one skin depth. 00:16:52.540 --> 00:16:55.570 align:middle line:84% And it's just an interesting thing you can derive. 00:16:55.570 --> 00:16:59.080 align:middle line:84% So you can think about it from a cross-sectional area carrying 00:16:59.080 --> 00:17:00.080 align:middle line:90% capability. 00:17:00.080 --> 00:17:04.240 align:middle line:84% You can imagine, even though I had w of width of wire, 00:17:04.240 --> 00:17:07.030 align:middle line:84% I can only use perhaps delta of it. 00:17:07.030 --> 00:17:09.819 align:middle line:84% And I can just calculate the AC resistance 00:17:09.819 --> 00:17:13.339 align:middle line:90% as if I had the skin depth. 00:17:13.339 --> 00:17:16.780 align:middle line:84% So I would go back, and instead of carrying-- 00:17:16.780 --> 00:17:18.250 align:middle line:84% having the area of the wire, which 00:17:18.250 --> 00:17:20.770 align:middle line:84% might be the height of the wire times the width of the wire, 00:17:20.770 --> 00:17:24.560 align:middle line:84% I'd have the height of the wire times delta. 00:17:24.560 --> 00:17:26.780 align:middle line:84% Or another way to express that is, 00:17:26.780 --> 00:17:33.650 align:middle line:84% if w is much greater than delta, if the width of my wire 00:17:33.650 --> 00:17:36.650 align:middle line:84% is much greater than the skin depth, 00:17:36.650 --> 00:17:40.100 align:middle line:84% then I can write the AC resistance as being 00:17:40.100 --> 00:17:43.010 align:middle line:84% equal to the DC resistance, which is what I calculated 00:17:43.010 --> 00:17:50.810 align:middle line:90% over there, times w over delta. 00:17:50.810 --> 00:17:54.790 align:middle line:84% So if my width of my wire is 10 skin depths, 00:17:54.790 --> 00:17:59.850 align:middle line:84% my AC resistance is 10x the DC resistance. 00:17:59.850 --> 00:18:02.142 align:middle line:84% And from this, my AC resistance goes up 00:18:02.142 --> 00:18:03.600 align:middle line:84% as the square root of the frequency 00:18:03.600 --> 00:18:08.410 align:middle line:84% of what I'm carrying for when I get into this regime. 00:18:08.410 --> 00:18:11.900 align:middle line:90% Any questions about that? 00:18:11.900 --> 00:18:18.070 align:middle line:84% This chart here just tends to show, 00:18:18.070 --> 00:18:20.570 align:middle line:84% this is the field versus the width. 00:18:20.570 --> 00:18:22.660 align:middle line:84% So basically, we've got a current source 00:18:22.660 --> 00:18:26.108 align:middle line:84% coming here and carrying current. 00:18:26.108 --> 00:18:28.150 align:middle line:84% And then the question is, where in this conductor 00:18:28.150 --> 00:18:29.650 align:middle line:90% does the current return? 00:18:29.650 --> 00:18:31.798 align:middle line:84% This shows the field inside the conductor, 00:18:31.798 --> 00:18:33.590 align:middle line:84% and then this shows the associated current. 00:18:33.590 --> 00:18:38.137 align:middle line:84% So if I have a width of this conductor that's 00:18:38.137 --> 00:18:40.720 align:middle line:84% thin compared to a skin depth or on the order of a skin depth, 00:18:40.720 --> 00:18:43.700 align:middle line:84% yeah, the current is almost evenly distributed across. 00:18:43.700 --> 00:18:45.400 align:middle line:84% But once I turn up the frequency, 00:18:45.400 --> 00:18:47.442 align:middle line:84% and the width is much bigger than the skin depth, 00:18:47.442 --> 00:18:49.720 align:middle line:84% the current falls off exponentially 00:18:49.720 --> 00:18:53.510 align:middle line:84% from the edge of the conductor where there's high field. 00:18:53.510 --> 00:18:59.240 align:middle line:84% So when I say, oh, Why is the current sticking to the surface 00:18:59.240 --> 00:19:02.360 align:middle line:84% of the wire? the question is, does it stick to this side, 00:19:02.360 --> 00:19:04.200 align:middle line:90% or does it stick to this side? 00:19:04.200 --> 00:19:06.890 align:middle line:84% It sticks to whatever side of the conductor 00:19:06.890 --> 00:19:09.695 align:middle line:84% has high magnetic field imposed on it. 00:19:09.695 --> 00:19:11.320 align:middle line:84% The way you would analyze this is you'd 00:19:11.320 --> 00:19:12.710 align:middle line:90% say, OK, let me analyze this. 00:19:12.710 --> 00:19:14.980 align:middle line:84% Where are the magnetic fields around the conductor? 00:19:14.980 --> 00:19:16.850 align:middle line:84% Where there's high magnetic fields, 00:19:16.850 --> 00:19:21.130 align:middle line:84% that's the surface of the conductor to which the current 00:19:21.130 --> 00:19:23.900 align:middle line:90% is going to get compressed. 00:19:23.900 --> 00:19:25.140 align:middle line:90% Any questions about that? 00:19:25.140 --> 00:19:27.170 align:middle line:90% 00:19:27.170 --> 00:19:29.670 align:middle line:84% Now, I don't know about you, but that kind of bends my brain 00:19:29.670 --> 00:19:30.940 align:middle line:90% a little bit. 00:19:30.940 --> 00:19:35.860 align:middle line:84% So I thought the nicest thing is to actually see this for real. 00:19:35.860 --> 00:19:39.130 align:middle line:84% And you can see that here, the H field-- 00:19:39.130 --> 00:19:41.750 align:middle line:84% this is the H field, and this is the current density. 00:19:41.750 --> 00:19:43.125 align:middle line:84% We can't see the current density, 00:19:43.125 --> 00:19:45.070 align:middle line:84% but maybe we can look at how the H fields drop 00:19:45.070 --> 00:19:47.260 align:middle line:84% inside the conductor, and that will 00:19:47.260 --> 00:19:55.010 align:middle line:84% relate to how the currents distribute inside the 00:19:55.010 --> 00:19:56.540 align:middle line:84% PROFESSOR 1: So to demonstrate this, 00:19:56.540 --> 00:20:02.180 align:middle line:84% we have this giant block of aluminum here, 00:20:02.180 --> 00:20:05.870 align:middle line:84% and the aluminum forms a one-turn secondary 00:20:05.870 --> 00:20:07.410 align:middle line:90% on this transformer. 00:20:07.410 --> 00:20:10.440 align:middle line:84% So here, you might be able to see there's a core here. 00:20:10.440 --> 00:20:12.690 align:middle line:84% We have a winding over here that we're going to drive. 00:20:12.690 --> 00:20:17.490 align:middle line:84% But this is one turn around the inside of this transformer. 00:20:17.490 --> 00:20:19.860 align:middle line:84% And so we're going to have current flowing in there. 00:20:19.860 --> 00:20:22.190 align:middle line:84% And it's hard for us to measure the current, 00:20:22.190 --> 00:20:24.630 align:middle line:84% but the current generates a magnetic field. 00:20:24.630 --> 00:20:27.620 align:middle line:84% And so we have this magnetic field probe over here, 00:20:27.620 --> 00:20:30.865 align:middle line:84% and we're going to move the magnetic field probe. 00:20:30.865 --> 00:20:33.970 align:middle line:84% Right now, the end of the probe is right on the inside edge 00:20:33.970 --> 00:20:34.470 align:middle line:90% here. 00:20:34.470 --> 00:20:37.370 align:middle line:84% And we're going to move it about halfway across the block, 00:20:37.370 --> 00:20:39.440 align:middle line:84% and then we're going to move it back again. 00:20:39.440 --> 00:20:43.070 align:middle line:84% Now, when we move it from the edge to the middle, 00:20:43.070 --> 00:20:47.180 align:middle line:84% we're going to excite our transformer at 60 hertz, so 00:20:47.180 --> 00:20:49.140 align:middle line:90% relatively low frequency. 00:20:49.140 --> 00:20:51.330 align:middle line:84% And so the skin depth will be quite long, 00:20:51.330 --> 00:20:56.810 align:middle line:84% and so we should get good current in the block. 00:20:56.810 --> 00:21:00.240 align:middle line:84% And we'll see that it goes quite deep into the block. 00:21:00.240 --> 00:21:02.570 align:middle line:84% Then when we return, we're going to turn up 00:21:02.570 --> 00:21:05.100 align:middle line:90% the frequency to 240 hertz. 00:21:05.100 --> 00:21:08.600 align:middle line:84% And then we're going to see how that compares to the 60 hertz 00:21:08.600 --> 00:21:10.460 align:middle line:84% result. So the first time, I'm just 00:21:10.460 --> 00:21:13.835 align:middle line:84% going to run it so that you can just see how the apparatus is 00:21:13.835 --> 00:21:14.335 align:middle line:90% working. 00:21:14.335 --> 00:21:18.076 align:middle line:90% 00:21:18.076 --> 00:21:22.610 align:middle line:84% So I'm using a stepper motor here to move the probe. 00:21:22.610 --> 00:21:25.685 align:middle line:84% And so you can see that it's slowly pulling out. 00:21:25.685 --> 00:21:28.010 align:middle line:84% And then when we get about halfway, 00:21:28.010 --> 00:21:31.460 align:middle line:90% then it's going to go back. 00:21:31.460 --> 00:21:33.530 align:middle line:84% It's changed the frequency at this point, 00:21:33.530 --> 00:21:35.420 align:middle line:84% and then it's going to go back in again. 00:21:35.420 --> 00:21:38.285 align:middle line:84% So now I'll change the display so you could look at the scope. 00:21:38.285 --> 00:21:52.570 align:middle line:90% 00:21:52.570 --> 00:21:56.110 align:middle line:84% It takes a second for VJ to lock. 00:21:56.110 --> 00:21:58.610 align:middle line:90% So now this is the-- 00:21:58.610 --> 00:22:00.460 align:middle line:84% it's the strength of the magnetic field, 00:22:00.460 --> 00:22:04.746 align:middle line:84% which is proportional to the strength of the current. 00:22:04.746 --> 00:22:06.390 align:middle line:90% So as it's going in-- 00:22:06.390 --> 00:22:08.590 align:middle line:84% or as it's going from the edge to the middle, 00:22:08.590 --> 00:22:11.080 align:middle line:84% you can see that the magnitude is going down. 00:22:11.080 --> 00:22:13.175 align:middle line:84% It has kind of an exponential look to it. 00:22:13.175 --> 00:22:16.930 align:middle line:90% 00:22:16.930 --> 00:22:19.270 align:middle line:84% And now it's changed frequency, so you 00:22:19.270 --> 00:22:21.855 align:middle line:84% can see that the amplitude is smaller right away. 00:22:21.855 --> 00:22:29.890 align:middle line:90% 00:22:29.890 --> 00:22:32.380 align:middle line:84% So I've tried to adjust it so that the two amplitudes are 00:22:32.380 --> 00:22:36.010 align:middle line:84% the same at the outside edge and so you 00:22:36.010 --> 00:22:41.050 align:middle line:84% can see what the relative distances are for the two 00:22:41.050 --> 00:22:42.080 align:middle line:90% different frequencies. 00:22:42.080 --> 00:22:45.346 align:middle line:84% The green one is 60 hertz, and the yellow one is 240 hertz. 00:22:45.346 --> 00:22:48.340 align:middle line:90% 00:22:48.340 --> 00:22:49.935 align:middle line:84% OK, I'll give you back your screen. 00:22:49.935 --> 00:22:55.288 align:middle line:90% 00:22:55.288 --> 00:22:56.580 align:middle line:90% DAVID PERREAULT: Thanks, Steve. 00:22:56.580 --> 00:23:00.780 align:middle line:84% So you could see that the skin depth-- 00:23:00.780 --> 00:23:05.880 align:middle line:84% that's the exponential length constant with which things 00:23:05.880 --> 00:23:07.260 align:middle line:84% decayed-- was about a factor of 2 00:23:07.260 --> 00:23:09.760 align:middle line:84% different in those two examples because the frequencies were 00:23:09.760 --> 00:23:12.240 align:middle line:90% about a factor of 4 different. 00:23:12.240 --> 00:23:15.190 align:middle line:84% So what do I hope that you're going to take away from that? 00:23:15.190 --> 00:23:17.815 align:middle line:84% What I hope you're going to take away is you're going to think, 00:23:17.815 --> 00:23:19.650 align:middle line:84% when you design something, oh, if I'm 00:23:19.650 --> 00:23:28.140 align:middle line:84% trying to carry high-frequency currents, 00:23:28.140 --> 00:23:30.720 align:middle line:84% I better think about how big my conductor is 00:23:30.720 --> 00:23:32.040 align:middle line:90% relative to the skin depth. 00:23:32.040 --> 00:23:34.720 align:middle line:84% And maybe I can use the DC resistance, 00:23:34.720 --> 00:23:39.810 align:middle line:84% or maybe I ought to figure out what the skin depth is 00:23:39.810 --> 00:23:43.260 align:middle line:84% and calculate the AC resistance of my winding. 00:23:43.260 --> 00:23:45.600 align:middle line:84% And by the way, I put in the notes-- 00:23:45.600 --> 00:23:50.650 align:middle line:84% this approximation is for if w is much bigger than the skin 00:23:50.650 --> 00:23:51.150 align:middle line:90% depth. 00:23:51.150 --> 00:23:53.760 align:middle line:84% In the lecture notes, I put what the approximation 00:23:53.760 --> 00:23:57.040 align:middle line:84% looks like if it's in some intermediate value. 00:23:57.040 --> 00:24:02.160 align:middle line:84% So it's not like there's a frequency-dependent AC 00:24:02.160 --> 00:24:04.542 align:middle line:84% resistance, but it's important because sometimes what you 00:24:04.542 --> 00:24:06.000 align:middle line:84% can find is, you're trying to carry 00:24:06.000 --> 00:24:07.480 align:middle line:84% large currents at high frequencies, 00:24:07.480 --> 00:24:09.688 align:middle line:84% you find out that your AC resistance is significantly 00:24:09.688 --> 00:24:12.390 align:middle line:84% bigger than you would have gotten if you just 00:24:12.390 --> 00:24:15.890 align:middle line:90% calculated this value. 00:24:15.890 --> 00:24:17.215 align:middle line:90% Any questions about that? 00:24:17.215 --> 00:24:20.250 align:middle line:90% 00:24:20.250 --> 00:24:21.990 align:middle line:90% Yeah. 00:24:21.990 --> 00:24:24.780 align:middle line:84% AUDIENCE: Is this [INAUDIBLE] of having one big conductor 00:24:24.780 --> 00:24:27.280 align:middle line:90% with many parallel conductors? 00:24:27.280 --> 00:24:30.070 align:middle line:90% DAVID PERREAULT: Yes and no. 00:24:30.070 --> 00:24:31.880 align:middle line:90% Yes, you can. 00:24:31.880 --> 00:24:35.380 align:middle line:84% But you just can't have parallel conductors because, indeed, what 00:24:35.380 --> 00:24:38.150 align:middle line:84% is one big conductor except a bunch of parallel conductors? 00:24:38.150 --> 00:24:42.220 align:middle line:84% But what you can use is something called litz wire. 00:24:42.220 --> 00:24:44.210 align:middle line:84% And here's a picture of litz wire. 00:24:44.210 --> 00:24:48.220 align:middle line:84% So what you do is-- if you want to carry a lot of current, 00:24:48.220 --> 00:24:49.990 align:middle line:84% so you need a lot of cross-section, 00:24:49.990 --> 00:24:52.250 align:middle line:84% but your skin depth is small, what do you do? 00:24:52.250 --> 00:24:54.950 align:middle line:84% You get a bunch of insulated strands. 00:24:54.950 --> 00:24:58.060 align:middle line:84% So each of these strands is insulated from one another. 00:24:58.060 --> 00:25:02.688 align:middle line:84% But then you can't just parallel them because of the way 00:25:02.688 --> 00:25:03.980 align:middle line:90% the magnetic field will behave. 00:25:03.980 --> 00:25:05.110 align:middle line:84% Basically, the current will still 00:25:05.110 --> 00:25:07.430 align:middle line:84% be carried in the outer ones and not in the inner ones. 00:25:07.430 --> 00:25:09.160 align:middle line:84% But what they do is they basically 00:25:09.160 --> 00:25:12.490 align:middle line:90% weave these windings together. 00:25:12.490 --> 00:25:14.900 align:middle line:84% In the simplest thing, you can twist it. 00:25:14.900 --> 00:25:17.590 align:middle line:84% But the key thing is you want-- sometimes the conductor is 00:25:17.590 --> 00:25:19.880 align:middle line:84% on the inside, and sometimes it's on the outside. 00:25:19.880 --> 00:25:22.300 align:middle line:84% So if you imagine that you weave your windings 00:25:22.300 --> 00:25:25.600 align:middle line:84% in a pattern called litz transposition such 00:25:25.600 --> 00:25:28.570 align:middle line:84% that each piece of conductor spends an equal amount of time 00:25:28.570 --> 00:25:31.450 align:middle line:84% on the surface, and in the middle, and the center, 00:25:31.450 --> 00:25:33.250 align:middle line:84% then if you want to think about it, 00:25:33.250 --> 00:25:34.570 align:middle line:84% the current doesn't know which way to go. 00:25:34.570 --> 00:25:36.610 align:middle line:84% They all look equal, so it goes through equal through everybody, 00:25:36.610 --> 00:25:37.970 align:middle line:90% and everybody's happy. 00:25:37.970 --> 00:25:41.153 align:middle line:84% So you'll see a lot of high-frequency designs 00:25:41.153 --> 00:25:42.820 align:middle line:84% where you're carrying high currents that 00:25:42.820 --> 00:25:46.390 align:middle line:84% will use litz wire as the conductor for precisely 00:25:46.390 --> 00:25:47.280 align:middle line:90% that purpose. 00:25:47.280 --> 00:25:49.030 align:middle line:84% And when you want to make really big wire, 00:25:49.030 --> 00:25:50.470 align:middle line:90% you make smaller litz bundles. 00:25:50.470 --> 00:25:52.480 align:middle line:84% And then, as you see down here, you 00:25:52.480 --> 00:25:54.550 align:middle line:84% weave those together or twist those together 00:25:54.550 --> 00:25:56.150 align:middle line:90% in order to get this effect. 00:25:56.150 --> 00:25:58.660 align:middle line:84% And there are papers that show you, well, 00:25:58.660 --> 00:26:01.860 align:middle line:84% how small your litz bundles have to be, 00:26:01.860 --> 00:26:03.610 align:middle line:84% but the first thing you might think of is, 00:26:03.610 --> 00:26:05.470 align:middle line:84% all you're dealing with is skin effect, 00:26:05.470 --> 00:26:08.890 align:middle line:84% is you would like your litz wire to have a diameter that's 00:26:08.890 --> 00:26:12.620 align:middle line:84% on the order of a skin depth is a good place to start, 00:26:12.620 --> 00:26:14.990 align:middle line:84% if all you're thinking about is skin effect. 00:26:14.990 --> 00:26:16.190 align:middle line:90% Excellent question. 00:26:16.190 --> 00:26:17.490 align:middle line:90% Any other questions? 00:26:17.490 --> 00:26:20.850 align:middle line:90% 00:26:20.850 --> 00:26:24.130 align:middle line:90% So skin effect is a pain. 00:26:24.130 --> 00:26:26.020 align:middle line:84% But it turns out that there's something 00:26:26.020 --> 00:26:29.590 align:middle line:84% even worse than skin effect that can occur, 00:26:29.590 --> 00:26:32.310 align:middle line:84% and that's known as proximity effect. 00:26:32.310 --> 00:26:45.110 align:middle line:90% 00:26:45.110 --> 00:26:47.220 align:middle line:90% What is proximity effect? 00:26:47.220 --> 00:26:51.190 align:middle line:84% Well, proximity effect is when we get eddy currents 00:26:51.190 --> 00:26:55.000 align:middle line:84% in a wire due to fields from some other piece of conductor 00:26:55.000 --> 00:26:56.500 align:middle line:90% nearby. 00:26:56.500 --> 00:26:58.820 align:middle line:90% And I'm still generating loss. 00:26:58.820 --> 00:27:04.780 align:middle line:84% So imagine that-- and here's an example. 00:27:04.780 --> 00:27:06.430 align:middle line:84% Imagine I have a piece of conductor, 00:27:06.430 --> 00:27:08.590 align:middle line:84% and maybe this is a foil conductor carrying 00:27:08.590 --> 00:27:12.490 align:middle line:90% some current in my-- 00:27:12.490 --> 00:27:15.392 align:middle line:84% I'm carrying current in this direction. 00:27:15.392 --> 00:27:17.830 align:middle line:84% And because of where this conductor is sitting, 00:27:17.830 --> 00:27:20.350 align:middle line:84% it's sitting inside some H field, 00:27:20.350 --> 00:27:24.340 align:middle line:84% like here, some external H field on each surface. 00:27:24.340 --> 00:27:27.160 align:middle line:84% And this conductor could be carrying no net current. 00:27:27.160 --> 00:27:28.450 align:middle line:90% It's just sitting there. 00:27:28.450 --> 00:27:30.520 align:middle line:84% But because of some other current nearby, 00:27:30.520 --> 00:27:31.790 align:middle line:90% it has a field on it. 00:27:31.790 --> 00:27:33.190 align:middle line:90% Well, what should I effect. 00:27:33.190 --> 00:27:37.520 align:middle line:84% I'm trying to punch each field through the conductor. 00:27:37.520 --> 00:27:40.210 align:middle line:84% And again, that's going to generate 00:27:40.210 --> 00:27:45.680 align:middle line:84% some eddy current trying to reject the flux from the center. 00:27:45.680 --> 00:27:48.760 align:middle line:84% So what that's going to do is that's essentially 00:27:48.760 --> 00:27:51.910 align:middle line:90% going to generate a current. 00:27:51.910 --> 00:27:54.850 align:middle line:84% It's going to generate an eddy current that 00:27:54.850 --> 00:28:02.530 align:middle line:84% goes in on this surface and comes out on this surface. 00:28:02.530 --> 00:28:03.430 align:middle line:90% Why? 00:28:03.430 --> 00:28:05.560 align:middle line:84% Because that's going to generate-- 00:28:05.560 --> 00:28:07.630 align:middle line:84% if this is an AC current, the dB/dt 00:28:07.630 --> 00:28:09.430 align:middle line:84% generates these circulating currents 00:28:09.430 --> 00:28:11.907 align:middle line:84% inside the conductor that try to reject 00:28:11.907 --> 00:28:12.990 align:middle line:90% the field from the center. 00:28:12.990 --> 00:28:15.930 align:middle line:90% 00:28:15.930 --> 00:28:20.160 align:middle line:84% Here in this plot you can see, these are solutions. 00:28:20.160 --> 00:28:22.590 align:middle line:84% Here's where I have a piece of conductor, 00:28:22.590 --> 00:28:26.050 align:middle line:84% and there's vertical fields on each surface. 00:28:26.050 --> 00:28:28.120 align:middle line:84% So the conductor is not carrying any net current. 00:28:28.120 --> 00:28:31.590 align:middle line:84% And what we can see is we get eddy currents that decay away 00:28:31.590 --> 00:28:37.940 align:middle line:84% into the middle of the conductor with the same skin depth 00:28:37.940 --> 00:28:39.030 align:middle line:90% relationship. 00:28:39.030 --> 00:28:42.370 align:middle line:90% 00:28:42.370 --> 00:28:44.990 align:middle line:84% Why is proximity effect particularly bad? 00:28:44.990 --> 00:28:47.442 align:middle line:84% Because I'm inducing loss in this conductor, 00:28:47.442 --> 00:28:49.900 align:middle line:84% even though I'm not getting to use that conductor to really 00:28:49.900 --> 00:28:52.400 align:middle line:90% carry any current. 00:28:52.400 --> 00:28:56.290 align:middle line:84% Well, why should I care about this proximity effect? 00:28:56.290 --> 00:29:00.580 align:middle line:84% Because imagine I have some core, 00:29:00.580 --> 00:29:03.460 align:middle line:84% like a pot core, where I have some center post, 00:29:03.460 --> 00:29:10.690 align:middle line:84% and then I have this window where I'm winding currents. 00:29:10.690 --> 00:29:14.670 align:middle line:90% So here is my core. 00:29:14.670 --> 00:29:18.540 align:middle line:84% So I'm looking at a cross-section of the core, 00:29:18.540 --> 00:29:19.890 align:middle line:90% like this. 00:29:19.890 --> 00:29:23.330 align:middle line:84% And suppose this is a transformer. 00:29:23.330 --> 00:29:27.960 align:middle line:84% Then maybe I wrap my conductor in and back out. 00:29:27.960 --> 00:29:32.750 align:middle line:84% So maybe I have some foil conductor or a set of wire 00:29:32.750 --> 00:29:39.040 align:middle line:84% conductors that's going in here and carrying current in this way 00:29:39.040 --> 00:29:43.360 align:middle line:84% and then is wrapped around, and it's coming out here. 00:29:43.360 --> 00:29:45.790 align:middle line:90% And I have a layer of that. 00:29:45.790 --> 00:29:47.480 align:middle line:90% [INAUDIBLE] 00:29:47.480 --> 00:29:51.160 align:middle line:84% And maybe I have a second layer because I'm winding out, 00:29:51.160 --> 00:29:52.560 align:middle line:90% and I have another second layer. 00:29:52.560 --> 00:29:56.190 align:middle line:90% 00:29:56.190 --> 00:30:01.310 align:middle line:84% Going in, going in, going in, going in, and coming out. 00:30:01.310 --> 00:30:05.390 align:middle line:90% 00:30:05.390 --> 00:30:06.300 align:middle line:90% It's doing that. 00:30:06.300 --> 00:30:08.480 align:middle line:84% And maybe over here I have some secondaries, 00:30:08.480 --> 00:30:11.070 align:middle line:84% so maybe the secondary carries an equal and opposite current. 00:30:11.070 --> 00:30:12.695 align:middle line:84% Maybe it carries twice as much current. 00:30:12.695 --> 00:30:14.100 align:middle line:90% It's a 2-to-1 transformer. 00:30:14.100 --> 00:30:18.980 align:middle line:84% So here, I have two dots, two dots coming out, 00:30:18.980 --> 00:30:21.510 align:middle line:84% two dots coming out, two dots coming out. 00:30:21.510 --> 00:30:24.120 align:middle line:90% And same thing on this side-- 00:30:24.120 --> 00:30:28.010 align:middle line:84% it's two dots going in and so forth. 00:30:28.010 --> 00:30:30.590 align:middle line:84% So I'm plotting the current density, 00:30:30.590 --> 00:30:34.490 align:middle line:84% and these are the net currents that are being carried. 00:30:34.490 --> 00:30:36.710 align:middle line:90% So what happens? 00:30:36.710 --> 00:30:39.980 align:middle line:84% Inside my core-- if this is my core, 00:30:39.980 --> 00:30:43.250 align:middle line:84% if this is high-permeability material, there is no H field. 00:30:43.250 --> 00:30:47.480 align:middle line:90% 00:30:47.480 --> 00:30:52.430 align:middle line:84% So if I execute Ampere's law, integral of H.dl 00:30:52.430 --> 00:30:55.550 align:middle line:84% around, say, a loop here, the only field that appears 00:30:55.550 --> 00:30:57.330 align:middle line:90% is in the window. 00:30:57.330 --> 00:30:59.340 align:middle line:84% And what that means is, in this picture, 00:30:59.340 --> 00:31:00.750 align:middle line:84% if I plotted the H field, I ought 00:31:00.750 --> 00:31:06.490 align:middle line:84% to have some H field here that's associated with this current 00:31:06.490 --> 00:31:07.670 align:middle line:90% from the middle. 00:31:07.670 --> 00:31:11.710 align:middle line:84% And if I looped around two layers of winding, 00:31:11.710 --> 00:31:17.230 align:middle line:90% I get two H fields and so forth. 00:31:17.230 --> 00:31:20.590 align:middle line:84% So what's happening is I go across the window, 00:31:20.590 --> 00:31:21.673 align:middle line:90% my field builds up. 00:31:21.673 --> 00:31:24.340 align:middle line:84% In the first layer, I've got one unit of H. In the second layer, 00:31:24.340 --> 00:31:26.298 align:middle line:84% I've got two units of H. If I had three layers, 00:31:26.298 --> 00:31:27.940 align:middle line:90% I'd have three units of H. 00:31:27.940 --> 00:31:31.420 align:middle line:84% This layer is only carrying one unit of current, 00:31:31.420 --> 00:31:40.470 align:middle line:84% but you can see that it has an average of 1.5, or it has-- 00:31:40.470 --> 00:31:41.040 align:middle line:90% I'm sorry. 00:31:41.040 --> 00:31:44.260 align:middle line:84% It has a field that's imposed on it 00:31:44.260 --> 00:31:47.780 align:middle line:84% that's bigger than the net current it's carrying. 00:31:47.780 --> 00:31:51.650 align:middle line:84% So this is an example where fields owing to this layer 00:31:51.650 --> 00:31:53.420 align:middle line:90% are hitting this layer. 00:31:53.420 --> 00:31:56.740 align:middle line:84% Fields owing from these two layers would hit the next layer. 00:31:56.740 --> 00:31:58.760 align:middle line:84% And what happens is then I've got this effect 00:31:58.760 --> 00:32:01.820 align:middle line:84% where I'm imposing fields on my conductor, 00:32:01.820 --> 00:32:06.740 align:middle line:84% and I'm going to get eddy currents in the second layer 00:32:06.740 --> 00:32:07.960 align:middle line:90% that's going to cause loss. 00:32:07.960 --> 00:32:10.580 align:middle line:90% 00:32:10.580 --> 00:32:13.040 align:middle line:84% And the problem with that is, especially 00:32:13.040 --> 00:32:15.560 align:middle line:84% if I have very many layers of winding, 00:32:15.560 --> 00:32:20.690 align:middle line:84% and I have very high AC current components, that induced loss 00:32:20.690 --> 00:32:22.220 align:middle line:90% gets reflected-- 00:32:22.220 --> 00:32:24.740 align:middle line:84% the way it looks electrically is there's a much higher AC 00:32:24.740 --> 00:32:26.030 align:middle line:90% resistance to the winding. 00:32:26.030 --> 00:32:29.960 align:middle line:84% So you could calculate your wire resistance, 00:32:29.960 --> 00:32:31.970 align:middle line:84% your theoretical wire or DC wire resistance 00:32:31.970 --> 00:32:34.020 align:middle line:90% and say it's 10 milliohms. 00:32:34.020 --> 00:32:37.140 align:middle line:84% And then you could think, oh, but if I include skin effect, 00:32:37.140 --> 00:32:41.660 align:middle line:84% it's 20 or 30 milliohms because I have a bunch of skin effect. 00:32:41.660 --> 00:32:44.218 align:middle line:84% And then you figure out what the proximity effect is. 00:32:44.218 --> 00:32:46.010 align:middle line:84% If you have a bunch of layers, and the loss 00:32:46.010 --> 00:32:47.870 align:middle line:84% can go as the square of number of layers, 00:32:47.870 --> 00:32:50.840 align:middle line:90% you could have 300 milliohms. 00:32:50.840 --> 00:32:53.563 align:middle line:84% So whereas your DC resistance might only 00:32:53.563 --> 00:32:55.980 align:middle line:84% have been 10 milliohms, suddenly you've got 300 milliohms. 00:32:55.980 --> 00:33:01.170 align:middle line:84% That's a factor of 30 times the loss you were expecting. 00:33:01.170 --> 00:33:02.730 align:middle line:90% So when does this occur? 00:33:02.730 --> 00:33:06.620 align:middle line:84% This occurs when you have conductors 00:33:06.620 --> 00:33:09.770 align:middle line:84% that are large compared to a skin depth 00:33:09.770 --> 00:33:12.560 align:middle line:84% or on the order of a skin depth and when 00:33:12.560 --> 00:33:15.680 align:middle line:84% you have many layers of winding such that fields from one 00:33:15.680 --> 00:33:17.610 align:middle line:90% winding are hitting the other. 00:33:17.610 --> 00:33:20.825 align:middle line:84% If you're designing something that's carrying mostly DC, eh, 00:33:20.825 --> 00:33:21.700 align:middle line:90% don't worry about it. 00:33:21.700 --> 00:33:24.120 align:middle line:84% If you're designing a really high-frequency transformer, 00:33:24.120 --> 00:33:25.828 align:middle line:84% you might want to start thinking about it 00:33:25.828 --> 00:33:28.950 align:middle line:90% because it can be a big effect. 00:33:28.950 --> 00:33:32.890 align:middle line:84% And in fact, what does that say you should do? 00:33:32.890 --> 00:33:36.300 align:middle line:84% Typically, if I said, OK, because of skin effect, 00:33:36.300 --> 00:33:41.260 align:middle line:84% I wanted my wire to be on the order of a skin 00:33:41.260 --> 00:33:45.290 align:middle line:84% depth or thicker, or at least my strand diameter for my litz, 00:33:45.290 --> 00:33:49.090 align:middle line:84% if you include proximity effect, very often 00:33:49.090 --> 00:33:51.610 align:middle line:84% you want your strand diameter or your wire diameter 00:33:51.610 --> 00:33:54.780 align:middle line:90% to be maybe 0.3 skin depths. 00:33:54.780 --> 00:33:59.200 align:middle line:84% Or you get into this weird situation-- 00:33:59.200 --> 00:34:01.390 align:middle line:84% and so sometimes that forces you to use 00:34:01.390 --> 00:34:08.830 align:middle line:84% really fine wire or really small strands, even smaller 00:34:08.830 --> 00:34:09.580 align:middle line:90% than a skin depth. 00:34:09.580 --> 00:34:12.480 align:middle line:84% So think 0.3 skin depths might be what you'd 00:34:12.480 --> 00:34:15.571 align:middle line:90% get if you had a lot of layers. 00:34:15.571 --> 00:34:17.750 align:middle line:84% So that's just a warning because you can 00:34:17.750 --> 00:34:19.050 align:middle line:90% get into this weird situation-- 00:34:19.050 --> 00:34:21.170 align:middle line:84% I've been there-- where you think, oh, my conduction 00:34:21.170 --> 00:34:21.900 align:middle line:90% loss is too high. 00:34:21.900 --> 00:34:22.608 align:middle line:90% I'm measuring it. 00:34:22.608 --> 00:34:24.300 align:middle line:90% I've got too much resistance. 00:34:24.300 --> 00:34:26.310 align:middle line:84% So what's my natural inclination? 00:34:26.310 --> 00:34:27.260 align:middle line:90% Put in more copper. 00:34:27.260 --> 00:34:28.880 align:middle line:90% Use fatter wire. 00:34:28.880 --> 00:34:33.290 align:middle line:84% And you put in fatter wire, and the resistance goes up. 00:34:33.290 --> 00:34:35.630 align:middle line:84% And the reason the resistance goes up 00:34:35.630 --> 00:34:40.514 align:middle line:84% is because this loss, as you make this conductor wider, 00:34:40.514 --> 00:34:42.139 align:middle line:84% there's more room for the eddy currents 00:34:42.139 --> 00:34:44.159 align:middle line:84% to circulate because of proximity effect. 00:34:44.159 --> 00:34:47.520 align:middle line:84% If you make the current conductor thinner, 00:34:47.520 --> 00:34:50.840 align:middle line:84% this current going this way and this current coming back kind 00:34:50.840 --> 00:34:54.679 align:middle line:84% of overlap and cancel, and you get less proximity effect loss. 00:34:54.679 --> 00:34:56.480 align:middle line:84% So sometimes it can be better to use 00:34:56.480 --> 00:35:00.060 align:middle line:84% thinner wire instead of thicker wire to get less loss in net. 00:35:00.060 --> 00:35:03.030 align:middle line:84% So that's just a heads up about that. 00:35:03.030 --> 00:35:06.560 align:middle line:90% Any questions? 00:35:06.560 --> 00:35:08.868 align:middle line:84% Don't worry about it if you're carrying mostly DC. 00:35:08.868 --> 00:35:09.660 align:middle line:90% It's not a problem. 00:35:09.660 --> 00:35:10.700 align:middle line:84% It's just something for you to know 00:35:10.700 --> 00:35:12.920 align:middle line:84% about when you start to think about designing really 00:35:12.920 --> 00:35:15.480 align:middle line:90% high-frequency things. 00:35:15.480 --> 00:35:18.175 align:middle line:84% AUDIENCE: [INAUDIBLE] at 56 [INAUDIBLE]? 00:35:18.175 --> 00:35:20.550 align:middle line:84% DAVID PERREAULT: Well, OK, so the skin depth at 60 hertz, 00:35:20.550 --> 00:35:22.350 align:middle line:84% and this depends again on temperature, 00:35:22.350 --> 00:35:24.990 align:middle line:84% is on the order of 3/4 of a centimeter. 00:35:24.990 --> 00:35:27.180 align:middle line:84% So if you were talking a really mondo-- 00:35:27.180 --> 00:35:30.133 align:middle line:84% like, you wanted to carry 10,000 amps, yeah, well, 00:35:30.133 --> 00:35:32.550 align:middle line:84% then you're going to have to strand and insulate your wire 00:35:32.550 --> 00:35:33.610 align:middle line:90% and litz wire. 00:35:33.610 --> 00:35:37.980 align:middle line:84% And so if you look at things like steam turbine generators, 00:35:37.980 --> 00:35:39.420 align:middle line:90% they do use litz. 00:35:39.420 --> 00:35:41.610 align:middle line:90% Their bundles are like this. 00:35:41.610 --> 00:35:44.680 align:middle line:84% But it's big, compressed chunks of copper. 00:35:44.680 --> 00:35:47.520 align:middle line:84% So this applies at any frequency if you're 00:35:47.520 --> 00:35:48.730 align:middle line:90% carrying enough current. 00:35:48.730 --> 00:35:51.120 align:middle line:84% If I'm building a 60 hertz transformer that's 00:35:51.120 --> 00:35:53.120 align:middle line:84% carrying an amp, my conductors are naturally 00:35:53.120 --> 00:35:54.870 align:middle line:84% going to be much less than the skin depth, 00:35:54.870 --> 00:35:56.328 align:middle line:84% and I wouldn't even think about it. 00:35:56.328 --> 00:35:59.240 align:middle line:90% 00:35:59.240 --> 00:36:02.573 align:middle line:90% OK, so that's winding loss. 00:36:02.573 --> 00:36:04.490 align:middle line:84% And these are just some things to think about. 00:36:04.490 --> 00:36:06.032 align:middle line:84% Watch out for skin effect, and really 00:36:06.032 --> 00:36:07.970 align:middle line:84% watch out for proximity effect if you're 00:36:07.970 --> 00:36:10.190 align:middle line:90% carrying high AC currents. 00:36:10.190 --> 00:36:13.880 align:middle line:84% The other piece, of course, of any inductor or any transformer 00:36:13.880 --> 00:36:14.470 align:middle line:90% is the core. 00:36:14.470 --> 00:36:17.110 align:middle line:90% 00:36:17.110 --> 00:36:23.590 align:middle line:84% And ideally, a core material is lossless. 00:36:23.590 --> 00:36:28.470 align:middle line:84% We have always idealized, if you will, 00:36:28.470 --> 00:36:31.440 align:middle line:84% our BH characteristic of our core 00:36:31.440 --> 00:36:33.886 align:middle line:90% to look something like this. 00:36:33.886 --> 00:36:41.280 align:middle line:84% So I might have plotted B versus H to do something 00:36:41.280 --> 00:36:44.940 align:middle line:90% like, I have some ideal-- 00:36:44.940 --> 00:36:46.710 align:middle line:90% this is mu of my core. 00:36:46.710 --> 00:36:49.050 align:middle line:84% And if I get above saturation, it 00:36:49.050 --> 00:36:51.640 align:middle line:84% starts to do something like this, and it goes back to mu 0. 00:36:51.640 --> 00:36:53.730 align:middle line:90% So that's my idealization. 00:36:53.730 --> 00:36:59.400 align:middle line:84% But it actually doesn't look like that quite on a real core. 00:36:59.400 --> 00:37:08.306 align:middle line:84% So if I just thought, suppose I built a core, 00:37:08.306 --> 00:37:17.290 align:middle line:84% and it had some area of the core. 00:37:17.290 --> 00:37:21.342 align:middle line:84% It had some mu of the core, area of the core, length of the core. 00:37:21.342 --> 00:37:23.440 align:middle line:84% And I put a winding on it, and let 00:37:23.440 --> 00:37:26.146 align:middle line:84% me imagine that my winding has no resistance. 00:37:26.146 --> 00:37:30.180 align:middle line:90% 00:37:30.180 --> 00:37:33.870 align:middle line:84% What would the net power going into the core 00:37:33.870 --> 00:37:37.810 align:middle line:90% look like or the net energy? 00:37:37.810 --> 00:37:40.820 align:middle line:84% Let's calculate the energy that I'm putting into the core. 00:37:40.820 --> 00:37:46.480 align:middle line:84% So the input energy would be the integral over a period 00:37:46.480 --> 00:37:51.440 align:middle line:90% of v of t times i of t, dt. 00:37:51.440 --> 00:37:54.020 align:middle line:84% It's the voltage applied times the current. 00:37:54.020 --> 00:37:59.540 align:middle line:84% And if there's any loss in this system in the core, 00:37:59.540 --> 00:38:03.730 align:middle line:84% I will have an integral of vi that would be not 0. 00:38:03.730 --> 00:38:07.090 align:middle line:84% Ideal inductor, of course, that ought to be 0. 00:38:07.090 --> 00:38:10.280 align:middle line:90% But if I have loss, it won't be. 00:38:10.280 --> 00:38:14.100 align:middle line:90% And what does this look like? 00:38:14.100 --> 00:38:18.240 align:middle line:84% Well, let me rewrite this in magnetic terms. 00:38:18.240 --> 00:38:20.310 align:middle line:90% What is v? 00:38:20.310 --> 00:38:30.690 align:middle line:84% v is equal to N times the area of the core times dB dt. 00:38:30.690 --> 00:38:33.090 align:middle line:90% Why? 00:38:33.090 --> 00:38:37.520 align:middle line:84% Because v is d lambda dt, and lambda is NVAc so then 00:38:37.520 --> 00:38:41.140 align:middle line:90% I have NAc times dB/dt. 00:38:41.140 --> 00:38:43.090 align:middle line:90% What is the current? 00:38:43.090 --> 00:38:47.350 align:middle line:84% Well, keep in mind, if I integrated the H of the core 00:38:47.350 --> 00:38:50.230 align:middle line:84% around the length, I should have Ni. 00:38:50.230 --> 00:38:53.590 align:middle line:84% So what I get is H in the core times the length of the core 00:38:53.590 --> 00:38:55.960 align:middle line:90% is equal to Ni. 00:38:55.960 --> 00:38:58.030 align:middle line:84% So I ought to be able to write i as being 00:38:58.030 --> 00:39:05.241 align:middle line:90% equal to Hc times lc over N. 00:39:05.241 --> 00:39:07.630 align:middle line:84% So then I could write i of t as being 00:39:07.630 --> 00:39:17.030 align:middle line:84% equal to Hc times lc divided by N, dt. 00:39:17.030 --> 00:39:21.020 align:middle line:84% So let me cancel out the N's here, and let me isolate-- 00:39:21.020 --> 00:39:22.400 align:middle line:90% I could write this as-- 00:39:22.400 --> 00:39:24.390 align:middle line:90% let me take out lc and Ac-- 00:39:24.390 --> 00:39:26.435 align:middle line:90% 00:39:26.435 --> 00:39:28.560 align:middle line:84% the length of the core times the area of the core-- 00:39:28.560 --> 00:39:31.860 align:middle line:84% this is just the volume of the core here-- 00:39:31.860 --> 00:39:37.960 align:middle line:90% times the integral of H dB. 00:39:37.960 --> 00:39:41.600 align:middle line:84% And I'm going over a cycle, so presumably, B is looping. 00:39:41.600 --> 00:39:45.920 align:middle line:84% So this is around a closed cycle. 00:39:45.920 --> 00:39:52.630 align:middle line:84% So this is the energy I lose in going and cycling around. 00:39:52.630 --> 00:39:55.370 align:middle line:84% It's the integral of H dB in the core. 00:39:55.370 --> 00:40:00.680 align:middle line:84% And what we will see is that, in a real core material, 00:40:00.680 --> 00:40:03.980 align:middle line:84% B and H don't quite go up and down proportional to one another 00:40:03.980 --> 00:40:05.150 align:middle line:90% the way this illustrates. 00:40:05.150 --> 00:40:07.920 align:middle line:84% What you really see is something that does this. 00:40:07.920 --> 00:40:14.830 align:middle line:84% It goes up, and then it comes down with some slight spacing, 00:40:14.830 --> 00:40:17.470 align:middle line:90% like that. 00:40:17.470 --> 00:40:19.960 align:middle line:84% And if I think about the integral of H dB, 00:40:19.960 --> 00:40:25.166 align:middle line:90% that's really this area. 00:40:25.166 --> 00:40:28.750 align:middle line:84% So H and B really form a hysteresis loop, 00:40:28.750 --> 00:40:31.360 align:middle line:84% and the area inside that hysteresis loop 00:40:31.360 --> 00:40:35.950 align:middle line:90% is the energy I lose in a cycle. 00:40:35.950 --> 00:40:37.110 align:middle line:90% Any questions about that? 00:40:37.110 --> 00:40:41.660 align:middle line:90% 00:40:41.660 --> 00:40:48.400 align:middle line:84% This would sort of suggest that I lose so much energy per cycle 00:40:48.400 --> 00:40:51.880 align:middle line:90% for a given B or H swing. 00:40:51.880 --> 00:40:57.410 align:middle line:84% And if I do it twice as fast, I lose twice as much and so forth. 00:40:57.410 --> 00:41:00.210 align:middle line:84% And at low frequencies, that's kind of true. 00:41:00.210 --> 00:41:03.230 align:middle line:84% At low enough frequencies, that's kind of true. 00:41:03.230 --> 00:41:05.800 align:middle line:84% When that's true, they call it hysteresis loss. 00:41:05.800 --> 00:41:10.790 align:middle line:90% 00:41:10.790 --> 00:41:13.550 align:middle line:84% Unfortunately, when you get higher in frequency-- 00:41:13.550 --> 00:41:15.680 align:middle line:84% where does this loss come from, I should say. 00:41:15.680 --> 00:41:20.540 align:middle line:84% It's because you're doing work to move magnetic domains around. 00:41:20.540 --> 00:41:22.302 align:middle line:84% They're mechanically moving around. 00:41:22.302 --> 00:41:24.260 align:middle line:84% And by the way, when the magnetic domains jump, 00:41:24.260 --> 00:41:26.430 align:middle line:84% they introduce little eddy currents in the material. 00:41:26.430 --> 00:41:28.940 align:middle line:90% And so that generates loss. 00:41:28.940 --> 00:41:32.670 align:middle line:84% So you do have a bunch of loss moving the magnetic domains 00:41:32.670 --> 00:41:33.170 align:middle line:90% around. 00:41:33.170 --> 00:41:37.183 align:middle line:84% And hence, that appears as this net loss 00:41:37.183 --> 00:41:38.600 align:middle line:84% that-- even if you ignored winding 00:41:38.600 --> 00:41:41.790 align:middle line:84% loss or some additional loss by swinging AC flux in the core. 00:41:41.790 --> 00:41:46.310 align:middle line:90% 00:41:46.310 --> 00:41:53.130 align:middle line:84% This loss is sometimes called classical hysteresis loss. 00:41:53.130 --> 00:41:55.698 align:middle line:84% Unfortunately, while at low frequencies, 00:41:55.698 --> 00:41:57.990 align:middle line:84% it's sort of proportional to frequency, when you get up 00:41:57.990 --> 00:41:59.365 align:middle line:84% to high frequencies, it turns out 00:41:59.365 --> 00:42:01.140 align:middle line:90% the shape of the loop changes. 00:42:01.140 --> 00:42:05.010 align:middle line:84% So it turns out that actual loss in many magnetic materials 00:42:05.010 --> 00:42:06.850 align:middle line:84% is not proportional to frequency. 00:42:06.850 --> 00:42:09.580 align:middle line:84% It's proportional to something higher than frequency. 00:42:09.580 --> 00:42:11.290 align:middle line:84% It just depends on the magnetic material. 00:42:11.290 --> 00:42:14.020 align:middle line:84% So that's one thing you've got to keep in mind. 00:42:14.020 --> 00:42:15.990 align:middle line:84% So people sometimes talk about hysteresis 00:42:15.990 --> 00:42:19.050 align:middle line:84% loss, but what this hysteresis loop looks 00:42:19.050 --> 00:42:22.840 align:middle line:90% like is frequency-dependent. 00:42:22.840 --> 00:42:26.440 align:middle line:84% The other thing I would say is that for this kind of loss, 00:42:26.440 --> 00:42:29.590 align:middle line:84% you can calculate it as loss per unit volume. 00:42:29.590 --> 00:42:31.600 align:middle line:84% If I told you the shape of the hysteresis loop 00:42:31.600 --> 00:42:34.150 align:middle line:84% and what B you were swinging over, you could say, 00:42:34.150 --> 00:42:39.080 align:middle line:84% this area times the volume of the core times the frequency 00:42:39.080 --> 00:42:42.280 align:middle line:84% would give me my power dissipation 00:42:42.280 --> 00:42:44.510 align:middle line:90% in the core material. 00:42:44.510 --> 00:42:47.880 align:middle line:90% Make sense, everybody? 00:42:47.880 --> 00:42:50.040 align:middle line:84% I should say that this kind of loss, 00:42:50.040 --> 00:42:53.940 align:middle line:84% owing to the magnetic domains, is one form of loss you can get. 00:42:53.940 --> 00:42:56.820 align:middle line:84% There are other forms of loss you can get, however, 00:42:56.820 --> 00:43:00.760 align:middle line:84% in cores that are not so-called hysteresis or related loss. 00:43:00.760 --> 00:43:04.960 align:middle line:84% And one of those is, in fact, eddy current loss in the core. 00:43:04.960 --> 00:43:06.410 align:middle line:90% Think about this. 00:43:06.410 --> 00:43:07.940 align:middle line:90% What is this core doing? 00:43:07.940 --> 00:43:10.180 align:middle line:84% If I put a current in here, I'm putting 00:43:10.180 --> 00:43:18.121 align:middle line:84% a flux and a flux density of B through the core itself. 00:43:18.121 --> 00:43:20.320 align:middle line:84% Well, at low frequencies, especially, we often 00:43:20.320 --> 00:43:22.850 align:middle line:84% use conductive magnetic materials. 00:43:22.850 --> 00:43:23.840 align:middle line:90% We use steel. 00:43:23.840 --> 00:43:25.330 align:middle line:90% Steel is pretty conductive. 00:43:25.330 --> 00:43:30.070 align:middle line:84% And so if I looked at this cross-section 00:43:30.070 --> 00:43:39.320 align:middle line:84% that I drew here in white, what I would get 00:43:39.320 --> 00:43:42.685 align:middle line:84% is I would have some flux going in like this. 00:43:42.685 --> 00:43:45.490 align:middle line:90% 00:43:45.490 --> 00:43:46.550 align:middle line:90% And what would that do? 00:43:46.550 --> 00:43:49.520 align:middle line:84% If I have an AC flux, cB, dt, I'm 00:43:49.520 --> 00:43:55.293 align:middle line:84% going to generate a field inside the conductor that does this. 00:43:55.293 --> 00:43:58.780 align:middle line:90% 00:43:58.780 --> 00:44:02.530 align:middle line:84% And that AC field is then going to generate eddy currents which 00:44:02.530 --> 00:44:07.990 align:middle line:84% try to counteract that and reject the flux in the core. 00:44:07.990 --> 00:44:10.240 align:middle line:84% So in other words, the fact that I'm 00:44:10.240 --> 00:44:12.730 align:middle line:84% carrying AC flux through a conductive material 00:44:12.730 --> 00:44:15.285 align:middle line:84% means I can get eddy currents circulating in the core. 00:44:15.285 --> 00:44:18.290 align:middle line:90% 00:44:18.290 --> 00:44:20.960 align:middle line:90% That's kind of unfortunate. 00:44:20.960 --> 00:44:21.880 align:middle line:90% I'm sorry. 00:44:21.880 --> 00:44:24.460 align:middle line:84% If I put the flux in the core, the eddy currents 00:44:24.460 --> 00:44:26.380 align:middle line:90% are going to circulate this way. 00:44:26.380 --> 00:44:29.530 align:middle line:90% So this is i eddy. 00:44:29.530 --> 00:44:32.900 align:middle line:84% When do I start to care about this eddy current loss? 00:44:32.900 --> 00:44:37.220 align:middle line:84% I care about it when the dimension of the core 00:44:37.220 --> 00:44:39.200 align:middle line:84% is on the order of the skin depth. 00:44:39.200 --> 00:44:42.530 align:middle line:84% And if I get a cross-section that's too big, 00:44:42.530 --> 00:44:43.360 align:middle line:90% two things happen. 00:44:43.360 --> 00:44:45.027 align:middle line:84% First, I get a lot of eddy current loss, 00:44:45.027 --> 00:44:49.440 align:middle line:84% and that just manifests itself as additional loss. 00:44:49.440 --> 00:44:51.350 align:middle line:84% And by the way, it scales up with area 00:44:51.350 --> 00:44:54.560 align:middle line:84% because I have more room for eddy currents to run around in, 00:44:54.560 --> 00:44:56.880 align:middle line:84% so it's not even lost per unit volume. 00:44:56.880 --> 00:44:59.000 align:middle line:84% It gets worse as the cross-sectional area 00:44:59.000 --> 00:45:00.980 align:middle line:90% gets bigger. 00:45:00.980 --> 00:45:03.290 align:middle line:84% But the other thing is, if it's bad enough, 00:45:03.290 --> 00:45:05.550 align:middle line:84% it sends a reject flux from the center of the core, 00:45:05.550 --> 00:45:06.440 align:middle line:90% and I don't even-- 00:45:06.440 --> 00:45:08.900 align:middle line:84% I might drive up the concentration of flux 00:45:08.900 --> 00:45:12.100 align:middle line:84% at the edges and maybe even saturate it. 00:45:12.100 --> 00:45:15.820 align:middle line:90% So what do people do about this? 00:45:15.820 --> 00:45:18.220 align:middle line:84% If you have high frequency materials like ferrite, 00:45:18.220 --> 00:45:21.070 align:middle line:84% it turns out, they're not very conductive, 00:45:21.070 --> 00:45:24.490 align:middle line:84% so you don't usually worry about eddy currents in your core, 00:45:24.490 --> 00:45:26.610 align:middle line:84% unless you're at really high frequencies. 00:45:26.610 --> 00:45:29.187 align:middle line:84% But if you're at low-frequency materials like steels, 00:45:29.187 --> 00:45:30.520 align:middle line:90% you do have to worry about this. 00:45:30.520 --> 00:45:31.750 align:middle line:90% How do they fix this? 00:45:31.750 --> 00:45:34.890 align:middle line:84% Instead of making one big, giant core cross-sectional area, 00:45:34.890 --> 00:45:36.770 align:middle line:84% they sort of take the same trick. 00:45:36.770 --> 00:45:38.580 align:middle line:84% What they do-- they don't need to litz 00:45:38.580 --> 00:45:41.550 align:middle line:84% transpose it because it's not like they're 00:45:41.550 --> 00:45:42.640 align:middle line:90% carrying current in it. 00:45:42.640 --> 00:45:45.390 align:middle line:84% What they'll do is instead of making a big cross section, 00:45:45.390 --> 00:45:47.730 align:middle line:84% they'll make this out of little sheets 00:45:47.730 --> 00:45:49.590 align:middle line:84% like this that are insulated from one 00:45:49.590 --> 00:45:51.900 align:middle line:90% another called laminations. 00:45:51.900 --> 00:45:57.450 align:middle line:84% So instead of building up a big, fat piece of steel, 00:45:57.450 --> 00:45:59.580 align:middle line:84% they'll build it usually out of stampings 00:45:59.580 --> 00:46:02.150 align:middle line:84% and insulate them and then stack them up. 00:46:02.150 --> 00:46:04.010 align:middle line:84% And the nice thing about this is, instead 00:46:04.010 --> 00:46:06.350 align:middle line:84% of this big eddy current being allowed 00:46:06.350 --> 00:46:09.290 align:middle line:84% to run around this huge loop and cause lots of loss, 00:46:09.290 --> 00:46:12.490 align:middle line:84% it's being forced to circulate in these tiny little areas. 00:46:12.490 --> 00:46:15.297 align:middle line:84% And as long as these laminations are thin 00:46:15.297 --> 00:46:17.630 align:middle line:84% compared to a skin depth, the eddy currents more or less 00:46:17.630 --> 00:46:20.450 align:middle line:84% cancel out, and you don't get much eddy current loss. 00:46:20.450 --> 00:46:22.700 align:middle line:84% So you'll see a lot of 60 hertz transformers 00:46:22.700 --> 00:46:25.200 align:middle line:84% or electric machines are all built out of laminations. 00:46:25.200 --> 00:46:28.490 align:middle line:84% How do they size the skin depth of those laminations-- 00:46:28.490 --> 00:46:30.360 align:middle line:84% how do they size the lamination thickness? 00:46:30.360 --> 00:46:32.910 align:middle line:84% They ask what the skin depth in the steel is, 00:46:32.910 --> 00:46:36.880 align:middle line:84% and they make the lamination sort of thin compared to that. 00:46:36.880 --> 00:46:37.910 align:middle line:90% Any questions? 00:46:37.910 --> 00:46:40.670 align:middle line:90% 00:46:40.670 --> 00:46:45.380 align:middle line:84% AUDIENCE: What characterizes [INAUDIBLE] core 00:46:45.380 --> 00:46:49.280 align:middle line:90% as being higher [INAUDIBLE]? 00:46:49.280 --> 00:46:52.530 align:middle line:84% Is it related to permeability, or is it just a property? 00:46:52.530 --> 00:46:56.940 align:middle line:84% DAVID PERREAULT: Well, so what's the-- yeah, it comes down to, 00:46:56.940 --> 00:46:58.585 align:middle line:90% what's the skin depth? 00:46:58.585 --> 00:47:00.710 align:middle line:84% So if you have high permeability of the skin depth, 00:47:00.710 --> 00:47:02.375 align:middle line:90% it gets small. 00:47:02.375 --> 00:47:05.120 align:middle line:84% So you just kind of plunk this in and find out, 00:47:05.120 --> 00:47:07.170 align:middle line:84% what is the dimension that I would care about? 00:47:07.170 --> 00:47:10.180 align:middle line:90% 00:47:10.180 --> 00:47:17.170 align:middle line:84% I should say, I can fix eddy currents just by laminating. 00:47:17.170 --> 00:47:21.580 align:middle line:84% So then I'm left with this hysteresis kind of behavior. 00:47:21.580 --> 00:47:25.120 align:middle line:84% But I said that hysteresis, in reality, frequency-dependent. 00:47:25.120 --> 00:47:28.750 align:middle line:84% And the truth of the matter is, it's not 00:47:28.750 --> 00:47:30.850 align:middle line:90% proportional to frequency. 00:47:30.850 --> 00:47:32.090 align:middle line:90% How do people treat it? 00:47:32.090 --> 00:47:33.580 align:middle line:84% They treat it rather empirically. 00:47:33.580 --> 00:47:36.430 align:middle line:84% What they really do is they plot loss 00:47:36.430 --> 00:47:39.880 align:middle line:84% per unit volume versus flux density 00:47:39.880 --> 00:47:41.300 align:middle line:90% at different frequencies. 00:47:41.300 --> 00:47:43.330 align:middle line:84% And you might get curves like this. 00:47:43.330 --> 00:47:46.060 align:middle line:84% And you can see that the loss goes up some. 00:47:46.060 --> 00:47:47.300 align:middle line:90% This is a log log plot. 00:47:47.300 --> 00:47:52.190 align:middle line:84% So it comes up, goes up at some power of the flux density, 00:47:52.190 --> 00:47:55.100 align:middle line:84% and it also increases somehow in frequency. 00:47:55.100 --> 00:48:00.890 align:middle line:84% And they model it, and they say, OK, the power per unit volume-- 00:48:00.890 --> 00:48:08.840 align:middle line:84% P sub V is power per unit volume is equal to some constant K 00:48:08.840 --> 00:48:17.590 align:middle line:84% times some frequency to a first power and some AC flux density 00:48:17.590 --> 00:48:18.277 align:middle line:90% amplitude. 00:48:18.277 --> 00:48:20.860 align:middle line:84% And this is usually calculated-- this is only usually measured 00:48:20.860 --> 00:48:25.300 align:middle line:84% for sinusoids, unfortunately-- so some other power, beta. 00:48:25.300 --> 00:48:28.760 align:middle line:84% And basically, what they do is they'll collect data like that. 00:48:28.760 --> 00:48:31.510 align:middle line:84% They'll figure out the core loss from the kind of measurements 00:48:31.510 --> 00:48:32.260 align:middle line:90% we're going to do. 00:48:32.260 --> 00:48:33.310 align:middle line:90% I'll show you right now. 00:48:33.310 --> 00:48:35.770 align:middle line:84% And they'll say, K alpha and beta, 00:48:35.770 --> 00:48:37.040 align:middle line:90% or they'll give you a plot. 00:48:37.040 --> 00:48:38.740 align:middle line:84% And you look at this and say, OK, this 00:48:38.740 --> 00:48:40.070 align:middle line:90% is a high-frequency material. 00:48:40.070 --> 00:48:41.830 align:middle line:84% I'm going to put 10 millitesla through it, 00:48:41.830 --> 00:48:44.110 align:middle line:84% and I'm going to do it at 5 megahertz, 00:48:44.110 --> 00:48:47.440 align:middle line:84% and I have 500 milliwatts per centimeter cubed, 00:48:47.440 --> 00:48:49.150 align:middle line:90% and that's how you calculate it. 00:48:49.150 --> 00:48:51.820 align:middle line:90% So it's very, very empirical. 00:48:51.820 --> 00:48:54.790 align:middle line:84% Unlike winding loss, where you can write lots of equations 00:48:54.790 --> 00:48:58.390 align:middle line:84% and predict it very beautifully, core loss 00:48:58.390 --> 00:49:01.280 align:middle line:84% is very much determined by measurement. 00:49:01.280 --> 00:49:03.640 align:middle line:84% It's very hard to link the microscopic behavior 00:49:03.640 --> 00:49:09.060 align:middle line:84% to the macroscopic behavior except in limited circumstances. 00:49:09.060 --> 00:49:11.070 align:middle line:84% Now, I'm almost out of time here, 00:49:11.070 --> 00:49:13.200 align:middle line:90% but what we wanted to show you-- 00:49:13.200 --> 00:49:16.680 align:middle line:84% and maybe I will hand this over to Monsi. 00:49:16.680 --> 00:49:18.430 align:middle line:84% PROFESSOR 2: So we have our inductor here, 00:49:18.430 --> 00:49:22.010 align:middle line:84% and I'm plotting the current through the inductor that's 00:49:22.010 --> 00:49:25.020 align:middle line:84% proportional to H, and it's on the x-axis. 00:49:25.020 --> 00:49:27.530 align:middle line:84% And we are also plotting the voltage through the inductor, 00:49:27.530 --> 00:49:30.420 align:middle line:84% and it's integrating it, and it's on the y-axis. 00:49:30.420 --> 00:49:33.110 align:middle line:84% So we are trying to get the BH core for the core 00:49:33.110 --> 00:49:35.810 align:middle line:84% And I'll slowly increase the current that's being 00:49:35.810 --> 00:49:37.110 align:middle line:90% pushed through the inductor. 00:49:37.110 --> 00:49:39.070 align:middle line:90% So let's see. 00:49:39.070 --> 00:49:43.830 align:middle line:90% 00:49:43.830 --> 00:49:44.330 align:middle line:90% Yeah. 00:49:44.330 --> 00:49:54.690 align:middle line:90% 00:49:54.690 --> 00:49:57.540 align:middle line:84% So this is where we can see it's slightly reaching 00:49:57.540 --> 00:49:59.130 align:middle line:90% the saturation of the core. 00:49:59.130 --> 00:50:00.730 align:middle line:90% And this is the BH core. 00:50:00.730 --> 00:50:03.687 align:middle line:84% Do you want to add something to that? 00:50:03.687 --> 00:50:04.520 align:middle line:90% DAVID PERREAULT: No. 00:50:04.520 --> 00:50:07.250 align:middle line:84% So there you can see, just based on the earlier calculation we 00:50:07.250 --> 00:50:11.900 align:middle line:84% did, there's this hysteresis loop between B and H. 00:50:11.900 --> 00:50:14.940 align:middle line:84% You would see that it changed shape versus frequency. 00:50:14.940 --> 00:50:18.500 align:middle line:84% And from this, that's what a real core material does look 00:50:18.500 --> 00:50:20.760 align:middle line:84% like, even though we pretend it doesn't do that. 00:50:20.760 --> 00:50:22.760 align:middle line:84% So all you've got to do is be careful. 00:50:22.760 --> 00:50:27.030 align:middle line:84% If you're going to have a lot of AC flux density, 00:50:27.030 --> 00:50:29.620 align:middle line:84% figure out your loss per unit volume and use it. 00:50:29.620 --> 00:50:31.950 align:middle line:84% So that's my very hand-wavy introduction 00:50:31.950 --> 00:50:33.520 align:middle line:90% to loss in magnetics. 00:50:33.520 --> 00:50:35.040 align:middle line:84% I think it'll be enough to get you 00:50:35.040 --> 00:50:38.040 align:middle line:84% through to designing some pretty reasonable inductors 00:50:38.040 --> 00:50:39.120 align:middle line:90% and transformers. 00:50:39.120 --> 00:50:42.760 align:middle line:84% And we will pick a new topic up next class. 00:50:42.760 --> 00:50:44.690 align:middle line:90% So have a great day. 00:50:44.690 --> 00:50:54.000 align:middle line:90%