WEBVTT 00:00:00.000 --> 00:00:02.420 align:middle line:90% [SQUEAKING] 00:00:02.420 --> 00:00:03.872 align:middle line:90% [RUSTLING] 00:00:03.872 --> 00:00:07.744 align:middle line:90% [CLICKING] 00:00:07.744 --> 00:00:09.680 align:middle line:90% 00:00:09.680 --> 00:00:13.310 align:middle line:84% DAVID PERREAULT: OK, why don't we get started? 00:00:13.310 --> 00:00:16.490 align:middle line:84% So last class, we talked about some things 00:00:16.490 --> 00:00:19.910 align:middle line:84% that happened with rectifier, on the DC side. 00:00:19.910 --> 00:00:23.330 align:middle line:84% Today, I'd like to talk about some measures 00:00:23.330 --> 00:00:26.450 align:middle line:84% that we use when we're looking at the AC side, so for example, 00:00:26.450 --> 00:00:29.040 align:middle line:84% if I had a rectifier connected to the grid. 00:00:29.040 --> 00:00:30.800 align:middle line:84% But what we're going to talk about applies 00:00:30.800 --> 00:00:34.670 align:middle line:84% to all kinds of energy transfers mediated 00:00:34.670 --> 00:00:37.670 align:middle line:90% by time varying waveforms. 00:00:37.670 --> 00:00:39.410 align:middle line:84% Before I get into it, however, I just 00:00:39.410 --> 00:00:42.170 align:middle line:84% wanted to remind you of a few definitions 00:00:42.170 --> 00:00:45.080 align:middle line:84% because we'll come back and use them. 00:00:45.080 --> 00:00:49.190 align:middle line:84% And the first of these is simply the RMS, or the Root Mean 00:00:49.190 --> 00:00:59.120 align:middle line:90% Square, of a signal. 00:00:59.120 --> 00:01:03.830 align:middle line:84% So if I have a signal x of t, the root 00:01:03.830 --> 00:01:14.690 align:middle line:84% mean square is the square root of the mean of x squared of t. 00:01:14.690 --> 00:01:16.683 align:middle line:90% And I'd call that x RMS. 00:01:16.683 --> 00:01:20.090 align:middle line:90% 00:01:20.090 --> 00:01:21.740 align:middle line:84% So there's a variety of reasons why 00:01:21.740 --> 00:01:24.650 align:middle line:84% you might want to calculate the RMS of a signal. 00:01:24.650 --> 00:01:29.670 align:middle line:84% But it's particularly useful in thinking about energy. 00:01:29.670 --> 00:01:30.320 align:middle line:90% Why? 00:01:30.320 --> 00:01:34.940 align:middle line:84% Because suppose I had, for example, a current waveform, i 00:01:34.940 --> 00:01:39.980 align:middle line:90% of t, into a resistor. 00:01:39.980 --> 00:01:43.880 align:middle line:84% And I said, how much power is being transferred 00:01:43.880 --> 00:01:47.150 align:middle line:90% into that resistor on average? 00:01:47.150 --> 00:01:53.330 align:middle line:84% Well, p of t is simply equal to v 00:01:53.330 --> 00:02:02.610 align:middle line:84% times i, which would be equal to i of t squared times R. 00:02:02.610 --> 00:02:07.220 align:middle line:84% And then if I wanted the average power, 00:02:07.220 --> 00:02:14.910 align:middle line:84% I would calculate that as 1 over T, the integral over t of i 00:02:14.910 --> 00:02:22.290 align:middle line:84% squared of t dt times R. So I can take out the R. 00:02:22.290 --> 00:02:28.230 align:middle line:84% And this gives me i RMS squared times R. 00:02:28.230 --> 00:02:31.980 align:middle line:84% So calculating the RMS of a current 00:02:31.980 --> 00:02:36.570 align:middle line:84% means it's the DC quantity-- it's the fixed quantity-- that 00:02:36.570 --> 00:02:38.940 align:middle line:84% will transfer the same amount of power 00:02:38.940 --> 00:02:41.020 align:middle line:90% into a resistor as a current. 00:02:41.020 --> 00:02:45.210 align:middle line:84% Or I could have done this with the voltage, if it's a resistor. 00:02:45.210 --> 00:02:49.630 align:middle line:84% So I don't need to know everything about the waveform. 00:02:49.630 --> 00:02:53.040 align:middle line:84% All I need to know is this RMS, and i RMS 00:02:53.040 --> 00:02:56.850 align:middle line:84% squared times R or v RMS squared divided by R 00:02:56.850 --> 00:02:59.790 align:middle line:84% gives me the power dissipation or resistor. 00:02:59.790 --> 00:03:01.560 align:middle line:84% Now if I don't have joule heating, 00:03:01.560 --> 00:03:04.890 align:middle line:84% if I don't have resistive loss, maybe the RMS 00:03:04.890 --> 00:03:05.940 align:middle line:90% isn't the right quantity. 00:03:05.940 --> 00:03:07.390 align:middle line:90% But very often it is. 00:03:07.390 --> 00:03:12.260 align:middle line:84% So that's why we tend to care about it. 00:03:12.260 --> 00:03:16.220 align:middle line:84% The second thing-- first of all, any questions about the meaning 00:03:16.220 --> 00:03:17.875 align:middle line:90% or use of root mean square? 00:03:17.875 --> 00:03:20.470 align:middle line:90% 00:03:20.470 --> 00:03:23.140 align:middle line:84% The second thing I'd like to remind you of 00:03:23.140 --> 00:03:26.395 align:middle line:90% is the notion of orthogonality. 00:03:26.395 --> 00:03:31.630 align:middle line:90% 00:03:31.630 --> 00:03:34.945 align:middle line:84% Now, I think it's particularly obvious what orthogonality means 00:03:34.945 --> 00:03:36.820 align:middle line:84% when you have a pair of vectors, that they're 00:03:36.820 --> 00:03:38.450 align:middle line:90% perpendicular in some sense. 00:03:38.450 --> 00:03:41.630 align:middle line:84% But what does it mean for continuous time waveforms? 00:03:41.630 --> 00:03:46.840 align:middle line:84% Well, two waveforms, x of t and y of t, 00:03:46.840 --> 00:04:01.060 align:middle line:84% are orthogonal on an interval if the following-- 00:04:01.060 --> 00:04:10.660 align:middle line:84% the integral from a to b of x of t times y of t dt is equal to 0. 00:04:10.660 --> 00:04:14.500 align:middle line:84% This is the natural assumption of the inner product 00:04:14.500 --> 00:04:18.880 align:middle line:84% for discrete sets or vectors or something onto continuous time 00:04:18.880 --> 00:04:20.149 align:middle line:90% waveforms. 00:04:20.149 --> 00:04:23.832 align:middle line:84% So if I take the product of the two, x of t and y of t, 00:04:23.832 --> 00:04:26.165 align:middle line:84% and I integrate them over the interval I'm interested in 00:04:26.165 --> 00:04:29.330 align:middle line:84% and I get 0, that means they're orthogonal. 00:04:29.330 --> 00:04:32.630 align:middle line:84% What that implies, among many other things, 00:04:32.630 --> 00:04:40.280 align:middle line:84% for example, is if I had some element and I had a voltage. 00:04:40.280 --> 00:04:45.920 align:middle line:84% Say the voltage was x of t and the current was y of t. 00:04:45.920 --> 00:04:50.270 align:middle line:84% That would suggest that over that interval of time, 00:04:50.270 --> 00:04:53.120 align:middle line:84% x of t and y of t would conspire to deliver 00:04:53.120 --> 00:04:56.120 align:middle line:90% 0 energy into the element. 00:04:56.120 --> 00:04:59.240 align:middle line:84% So we have a general definition for orthogonality 00:04:59.240 --> 00:05:02.690 align:middle line:84% and it has some implications about voltages and currents 00:05:02.690 --> 00:05:05.270 align:middle line:90% if I use it for that. 00:05:05.270 --> 00:05:08.630 align:middle line:90% Just as a reminder-- 00:05:08.630 --> 00:05:10.600 align:middle line:90% come back to some trigonometry. 00:05:10.600 --> 00:05:20.720 align:middle line:84% If I have the integral over 2 pi of sine of n omega t times 00:05:20.720 --> 00:05:26.540 align:middle line:84% sine of m omega t plus some angle, phi, 00:05:26.540 --> 00:05:35.960 align:middle line:84% d omega t is equal to 0 if n not equal to m, meaning 00:05:35.960 --> 00:05:39.620 align:middle line:84% if I have two sinusoids at different frequencies-- 00:05:39.620 --> 00:05:43.230 align:middle line:84% and here I'm thinking about on some commensurate interval. 00:05:43.230 --> 00:05:46.910 align:middle line:90% So n and m are just different. 00:05:46.910 --> 00:05:48.060 align:middle line:90% They're orthogonal. 00:05:48.060 --> 00:05:49.880 align:middle line:84% If I multiply sine at one frequency 00:05:49.880 --> 00:05:53.870 align:middle line:84% by cosine or sine of at another frequency and I integrate them, 00:05:53.870 --> 00:05:56.940 align:middle line:84% I'm going to get 0, as long as I'm 00:05:56.940 --> 00:06:00.000 align:middle line:90% over a commensurate interval. 00:06:00.000 --> 00:06:04.950 align:middle line:84% So sine at one frequency is orthogonal to sine waves 00:06:04.950 --> 00:06:06.260 align:middle line:90% at any other frequency. 00:06:06.260 --> 00:06:08.130 align:middle line:84% And this is the fundamental trick 00:06:08.130 --> 00:06:10.140 align:middle line:84% that we use to develop Fourier series. 00:06:10.140 --> 00:06:12.510 align:middle line:84% That's how we extract out Fourier coefficients, 00:06:12.510 --> 00:06:17.310 align:middle line:84% is using the orthogonality of different frequencies. 00:06:17.310 --> 00:06:20.670 align:middle line:84% Another thing I'll remind you of is simply 00:06:20.670 --> 00:06:26.310 align:middle line:84% that if I have the integral of 0 over 2 pi of sine 00:06:26.310 --> 00:06:35.460 align:middle line:84% of omega t times cosine of omega t d omega t, I will get 0. 00:06:35.460 --> 00:06:38.700 align:middle line:84% So not only are sinusoids at different frequencies 00:06:38.700 --> 00:06:41.760 align:middle line:84% orthogonal, but sine is orthogonal to cosine even 00:06:41.760 --> 00:06:43.750 align:middle line:90% at the same frequency. 00:06:43.750 --> 00:06:47.180 align:middle line:84% And that's just a straightforward trig identity, 00:06:47.180 --> 00:06:48.450 align:middle line:90% easy to prove. 00:06:48.450 --> 00:06:56.440 align:middle line:84% In general, more generally, we can write that the 1 over 2 pi, 00:06:56.440 --> 00:07:06.640 align:middle line:84% the integral over 0 to 2 pi of sine omega t times sine of omega 00:07:06.640 --> 00:07:17.620 align:middle line:84% t plus some angle phi d omega t is simply equal to 1/2 cosine 00:07:17.620 --> 00:07:18.850 align:middle line:90% of phi. 00:07:18.850 --> 00:07:25.630 align:middle line:84% So if phi is 90 degrees, that becomes a cosine, and I get 0. 00:07:25.630 --> 00:07:28.600 align:middle line:84% If phi is 0 and I have sine squared, 00:07:28.600 --> 00:07:31.570 align:middle line:84% this is really saying that the average value of sine squared 00:07:31.570 --> 00:07:34.100 align:middle line:90% over an interval is 1/2. 00:07:34.100 --> 00:07:37.510 align:middle line:84% So this is just basic trigonometry again, 00:07:37.510 --> 00:07:39.910 align:middle line:84% but it's a handy result to remember. 00:07:39.910 --> 00:07:43.510 align:middle line:84% And we'll see why we care about this shortly. 00:07:43.510 --> 00:07:44.870 align:middle line:90% Any questions about that? 00:07:44.870 --> 00:07:50.884 align:middle line:90% 00:07:50.884 --> 00:07:53.920 align:middle line:90% So we'll come back and use that. 00:07:53.920 --> 00:08:00.970 align:middle line:84% But let's start to think about AC waveforms. 00:08:00.970 --> 00:08:07.210 align:middle line:84% And let's talk about plugging something into the wall. 00:08:07.210 --> 00:08:08.890 align:middle line:90% So here's my wall. 00:08:08.890 --> 00:08:10.525 align:middle line:90% I have Vs sine omega t. 00:08:10.525 --> 00:08:17.660 align:middle line:90% 00:08:17.660 --> 00:08:20.590 align:middle line:84% And if I go back to the source, somewhere in there 00:08:20.590 --> 00:08:22.990 align:middle line:90% I'll have a breaker or a fuse. 00:08:22.990 --> 00:08:25.330 align:middle line:84% And then I'll have some wall wiring, 00:08:25.330 --> 00:08:26.930 align:middle line:90% hopefully very low resistance. 00:08:26.930 --> 00:08:29.170 align:middle line:90% I'll call this R wire. 00:08:29.170 --> 00:08:33.039 align:middle line:84% And then I'll come somewhere eventually to my plug, 00:08:33.039 --> 00:08:36.174 align:middle line:90% and I can plug something in. 00:08:36.174 --> 00:08:38.590 align:middle line:84% And here's my load resistor that I'm going 00:08:38.590 --> 00:08:45.550 align:middle line:90% to plug into the wall, R sub L. 00:08:45.550 --> 00:08:49.300 align:middle line:84% Maybe this is my space heater to keep me nice 00:08:49.300 --> 00:08:54.020 align:middle line:84% and toasty warm in the winter because it's a little bit cold. 00:08:54.020 --> 00:08:58.010 align:middle line:84% So I'm going to go plug this into the wall. 00:08:58.010 --> 00:09:02.960 align:middle line:84% First of all, just as an aside, why do we 00:09:02.960 --> 00:09:04.940 align:middle line:90% have this fuse or breaker here? 00:09:04.940 --> 00:09:06.235 align:middle line:90% What's it designed to do? 00:09:06.235 --> 00:09:09.937 align:middle line:90% 00:09:09.937 --> 00:09:12.522 align:middle line:84% AUDIENCE: To prevent overcurrents. 00:09:12.522 --> 00:09:13.980 align:middle line:84% DAVID PERREAULT: Well, that's true. 00:09:13.980 --> 00:09:15.630 align:middle line:84% But for what reason, specifically? 00:09:15.630 --> 00:09:18.440 align:middle line:84% What's it preventing an overcurrent of? 00:09:18.440 --> 00:09:21.189 align:middle line:84% AUDIENCE: That the source is not spiked consistent. 00:09:21.189 --> 00:09:23.382 align:middle line:90% And if it spikes, it'll break. 00:09:23.382 --> 00:09:24.590 align:middle line:90% DAVID PERREAULT: That's true. 00:09:24.590 --> 00:09:27.500 align:middle line:84% I think those are exactly correct answers. 00:09:27.500 --> 00:09:30.800 align:middle line:84% The thing I was actually looking for is a little bit more subtle. 00:09:30.800 --> 00:09:34.840 align:middle line:84% Why would the-- certainly if the source goes crazy 00:09:34.840 --> 00:09:37.250 align:middle line:84% and gets too big, we would want it to pop out. 00:09:37.250 --> 00:09:39.190 align:middle line:84% But what it's really designed to protect-- 00:09:39.190 --> 00:09:40.060 align:middle line:90% AUDIENCE: Load. 00:09:40.060 --> 00:09:42.310 align:middle line:84% DAVID PERREAULT: Actually, that's a very good thought. 00:09:42.310 --> 00:09:45.225 align:middle line:84% Actually, it turns out not to be the load. 00:09:45.225 --> 00:09:46.600 align:middle line:84% I mean, one thing is if something 00:09:46.600 --> 00:09:48.142 align:middle line:84% goes wrong with the load and the load 00:09:48.142 --> 00:09:51.100 align:middle line:84% draws too much current, you don't really 00:09:51.100 --> 00:09:53.090 align:middle line:84% rely on the breaker or the fuse to protect it. 00:09:53.090 --> 00:09:56.800 align:middle line:90% By then, this thing is toast. 00:09:56.800 --> 00:09:59.600 align:middle line:84% What the breaker is specifically designed to do-- 00:09:59.600 --> 00:10:01.480 align:middle line:84% I mean, if it's the fuse in the equipment, 00:10:01.480 --> 00:10:03.605 align:middle line:84% maybe it's trying to help you prevent the equipment 00:10:03.605 --> 00:10:04.370 align:middle line:90% from burning down. 00:10:04.370 --> 00:10:07.270 align:middle line:84% But the breaker back in your home or in the wall 00:10:07.270 --> 00:10:10.810 align:middle line:84% is designed to protect the wiring because you 00:10:10.810 --> 00:10:12.370 align:middle line:84% could, in principle, plug something 00:10:12.370 --> 00:10:16.090 align:middle line:84% in that would be happy enough, wouldn't die, but would 00:10:16.090 --> 00:10:17.480 align:middle line:90% draw too much current. 00:10:17.480 --> 00:10:20.030 align:middle line:84% And then what happens when your wall wires overheat? 00:10:20.030 --> 00:10:22.720 align:middle line:84% They start a fire and then really bad things happen. 00:10:22.720 --> 00:10:25.040 align:middle line:84% So the fuse of the breaker is really 00:10:25.040 --> 00:10:28.990 align:middle line:84% designed to make sure that the wiring doesn't overheat. 00:10:28.990 --> 00:10:31.513 align:middle line:84% You can't count on that helping you protect your equipment. 00:10:31.513 --> 00:10:33.680 align:middle line:84% Your equipment, you might put your own fuse in there 00:10:33.680 --> 00:10:36.030 align:middle line:84% to help you keep the equipment from burning down. 00:10:36.030 --> 00:10:39.450 align:middle line:90% But it's a good thought. 00:10:39.450 --> 00:10:41.090 align:middle line:90% So it is to protect things. 00:10:41.090 --> 00:10:44.900 align:middle line:84% But really, it's designed to protect the wire. 00:10:44.900 --> 00:10:50.120 align:middle line:84% Now, for most purposes, we can usually 00:10:50.120 --> 00:10:53.330 align:middle line:90% ignore the wire resistance. 00:10:53.330 --> 00:10:56.600 align:middle line:84% So let's for the moment pretend the wire resistance is 0 00:10:56.600 --> 00:10:59.000 align:middle line:84% because if you're burning lots of power in the wire, 00:10:59.000 --> 00:11:01.040 align:middle line:90% you're probably unhappy anyways. 00:11:01.040 --> 00:11:04.170 align:middle line:84% So we'll just ignore that for our calculations. 00:11:04.170 --> 00:11:09.470 align:middle line:84% So what is the power going into my space heater here? 00:11:09.470 --> 00:11:12.320 align:middle line:90% Well, P of t-- 00:11:12.320 --> 00:11:14.880 align:middle line:90% here is V sub L. Here is i. 00:11:14.880 --> 00:11:18.950 align:middle line:90% So P of t would equal-- 00:11:18.950 --> 00:11:20.540 align:middle line:84% as we calculated, it would just be 00:11:20.540 --> 00:11:27.660 align:middle line:84% equal to i squared of t times R, which would be-- 00:11:27.660 --> 00:11:30.180 align:middle line:84% if VL is approximately equal to Vs 00:11:30.180 --> 00:11:32.850 align:middle line:84% because I'm ignoring the wiring resistance, 00:11:32.850 --> 00:11:34.270 align:middle line:90% I could calculate it that way. 00:11:34.270 --> 00:11:36.770 align:middle line:84% And I could say that the average power-- 00:11:36.770 --> 00:11:46.770 align:middle line:84% I'll just call that P or PAv is simply IRMS squared times RL. 00:11:46.770 --> 00:11:51.720 align:middle line:90% So if, for example-- 00:11:51.720 --> 00:11:58.830 align:middle line:90% if VRMS is equal to 115 volts-- 00:11:58.830 --> 00:12:02.700 align:middle line:84% that means a peak sinusoid more like 170 volts-- 00:12:02.700 --> 00:12:16.650 align:middle line:84% and the fuse or the breaker is 15 amps, now this plug here-- 00:12:16.650 --> 00:12:19.590 align:middle line:90% if you're two plugs-- 00:12:19.590 --> 00:12:21.720 align:middle line:84% you have a long one and a short one-- 00:12:21.720 --> 00:12:26.700 align:middle line:84% if your plug looks like this, that's a 15-amp circuit. 00:12:26.700 --> 00:12:30.270 align:middle line:84% So it's intended that you're plugging that back into a 15-amp 00:12:30.270 --> 00:12:30.910 align:middle line:90% line. 00:12:30.910 --> 00:12:33.180 align:middle line:84% If you looked at this one in particular, 00:12:33.180 --> 00:12:37.560 align:middle line:84% it actually has another sideways pin that you could actually 00:12:37.560 --> 00:12:39.390 align:middle line:84% plug something in up to 20 amps, and you'd 00:12:39.390 --> 00:12:41.140 align:middle line:84% have a slightly different plug to do that. 00:12:41.140 --> 00:12:43.037 align:middle line:84% And that means the wall wiring in this case 00:12:43.037 --> 00:12:45.120 align:middle line:84% is a little bit heftier and can carry more current 00:12:45.120 --> 00:12:47.530 align:middle line:84% and the breaker's designed to carry that. 00:12:47.530 --> 00:12:49.530 align:middle line:90% So this is by the by. 00:12:49.530 --> 00:12:52.920 align:middle line:84% But suppose I had a 115-volt RMS, 00:12:52.920 --> 00:12:56.730 align:middle line:84% which is about what it's supposed to be, and 15 amps. 00:12:56.730 --> 00:12:58.230 align:middle line:90% That would give me-- 00:12:58.230 --> 00:13:05.820 align:middle line:84% if I plug this into here would give me about 1.7 kilowatts. 00:13:05.820 --> 00:13:07.830 align:middle line:84% So in fact, if you go to the store and you look, 00:13:07.830 --> 00:13:10.260 align:middle line:84% you'll see a lot of space heaters rated 00:13:10.260 --> 00:13:13.398 align:middle line:84% for just about 1.7 kilowatts or a lot of things you're 00:13:13.398 --> 00:13:14.440 align:middle line:90% going to plug in at home. 00:13:14.440 --> 00:13:14.640 align:middle line:90% Why? 00:13:14.640 --> 00:13:16.515 align:middle line:84% Because you can put them on a 15-amp circuit, 00:13:16.515 --> 00:13:20.084 align:middle line:84% and that's what a lot of people have in their homes. 00:13:20.084 --> 00:13:23.020 align:middle line:90% So all right, that's good. 00:13:23.020 --> 00:13:25.005 align:middle line:90% I can stay nice and warm. 00:13:25.005 --> 00:13:28.150 align:middle line:90% 00:13:28.150 --> 00:13:30.500 align:middle line:84% Suppose I wanted to plug in a different load. 00:13:30.500 --> 00:13:35.440 align:middle line:84% So suppose I come back, and here's Vs sine omega t. 00:13:35.440 --> 00:13:38.330 align:middle line:90% 00:13:38.330 --> 00:13:41.160 align:middle line:90% And here's my fuse and my wire. 00:13:41.160 --> 00:13:43.870 align:middle line:90% 00:13:43.870 --> 00:13:46.060 align:middle line:90% And now I'll come and plug-- 00:13:46.060 --> 00:13:51.940 align:middle line:84% suppose I plug in an inductor into the wall here, L. 00:13:51.940 --> 00:13:53.410 align:middle line:90% It's not that cold these days. 00:13:53.410 --> 00:13:55.990 align:middle line:84% Maybe I just want to, while I'm working, 00:13:55.990 --> 00:13:59.210 align:middle line:84% bathe in the luxuriant magnetic fields from my inductor. 00:13:59.210 --> 00:14:02.950 align:middle line:84% So I plug an inductor into the wall. 00:14:02.950 --> 00:14:11.260 align:middle line:90% Well, again, ignoring our wire-- 00:14:11.260 --> 00:14:15.340 align:middle line:84% so I'm assuming that this voltage is 00:14:15.340 --> 00:14:19.000 align:middle line:90% the same as the source voltage. 00:14:19.000 --> 00:14:20.220 align:middle line:90% Then what would I get? 00:14:20.220 --> 00:14:25.970 align:middle line:84% Well, I would get i of t would be what? 00:14:25.970 --> 00:14:28.090 align:middle line:90% Well, V is equal to Ldi/dt. 00:14:28.090 --> 00:14:36.540 align:middle line:84% So this should be 1 over L, the integral of Vs sine of omega t 00:14:36.540 --> 00:14:37.040 align:middle line:90% dt. 00:14:37.040 --> 00:14:39.630 align:middle line:90% 00:14:39.630 --> 00:14:43.110 align:middle line:84% Well, this should be equal to minus V sub s 00:14:43.110 --> 00:14:47.030 align:middle line:90% over omega L cosine of omega t. 00:14:47.030 --> 00:14:49.590 align:middle line:90% 00:14:49.590 --> 00:14:54.450 align:middle line:84% All right, so that's what I get for a result for my current 00:14:54.450 --> 00:14:57.990 align:middle line:84% in steady state into my inductor. 00:14:57.990 --> 00:15:00.990 align:middle line:84% What is the power that's being transferred 00:15:00.990 --> 00:15:03.740 align:middle line:90% into the inductor on average? 00:15:03.740 --> 00:15:05.170 align:middle line:90% Why don't we calculate that? 00:15:05.170 --> 00:15:13.720 align:middle line:84% We said PAv is equal to 1 over 2 pi, the integral 00:15:13.720 --> 00:15:18.910 align:middle line:84% from 0 to 2 pi of the voltage, which is V sine omega 00:15:18.910 --> 00:15:33.240 align:middle line:84% t, times the current minus Vs over omega L cosine omega t d 00:15:33.240 --> 00:15:35.070 align:middle line:90% omega t. 00:15:35.070 --> 00:15:40.680 align:middle line:84% I just did my average in angle instead of in time. 00:15:40.680 --> 00:15:42.900 align:middle line:84% Well, what does this work out to be? 00:15:42.900 --> 00:15:54.150 align:middle line:84% Well, this is minus Vs squared over 2 pi omega L 00:15:54.150 --> 00:16:00.710 align:middle line:84% times the integral over 2 pi of sine omega t cosine omega t. 00:16:00.710 --> 00:16:07.410 align:middle line:90% 00:16:07.410 --> 00:16:10.002 align:middle line:84% Can anybody tell me what that is? 00:16:10.002 --> 00:16:10.860 align:middle line:90% AUDIENCE: 0. 00:16:10.860 --> 00:16:11.830 align:middle line:90% DAVID PERREAULT: 0. 00:16:11.830 --> 00:16:13.240 align:middle line:90% Why is it 0? 00:16:13.240 --> 00:16:17.890 align:middle line:84% Because the voltage is a sine, and the current is a cosine. 00:16:17.890 --> 00:16:23.410 align:middle line:84% And we just decided that sines and cosines were orthogonal. 00:16:23.410 --> 00:16:33.240 align:middle line:84% So I'm going to draw no average power into my inductor. 00:16:33.240 --> 00:16:37.610 align:middle line:84% That kind of makes sense because the inductor is ideally 00:16:37.610 --> 00:16:38.980 align:middle line:90% a lossless element. 00:16:38.980 --> 00:16:42.660 align:middle line:84% So it shouldn't be able to absorb average power. 00:16:42.660 --> 00:16:46.760 align:middle line:84% What I am doing, however, I'm not absorbing power on average, 00:16:46.760 --> 00:16:48.963 align:middle line:90% but I do have current. 00:16:48.963 --> 00:16:50.630 align:middle line:84% There's current going into the inductor, 00:16:50.630 --> 00:16:52.130 align:middle line:84% and there's voltage on the inductor. 00:16:52.130 --> 00:16:54.830 align:middle line:84% So I do have some energy transfer instantaneously. 00:16:54.830 --> 00:16:59.505 align:middle line:84% Or what I have is slosh of energy from the wall 00:16:59.505 --> 00:17:01.130 align:middle line:84% into the inductor and from the inductor 00:17:01.130 --> 00:17:05.569 align:middle line:84% back into the wall but no loss as long as long 00:17:05.569 --> 00:17:08.990 align:middle line:90% as I can ignore our wire. 00:17:08.990 --> 00:17:12.260 align:middle line:84% That's what's known as reactive power transfer 00:17:12.260 --> 00:17:17.314 align:middle line:84% because it's a reactive component. 00:17:17.314 --> 00:17:20.030 align:middle line:90% So that makes sense. 00:17:20.030 --> 00:17:24.650 align:middle line:84% But notice that while I'm drawing no average power, 00:17:24.650 --> 00:17:27.540 align:middle line:90% I still have current. 00:17:27.540 --> 00:17:31.650 align:middle line:90% And that means I am still-- 00:17:31.650 --> 00:17:33.960 align:middle line:84% I have some current here, so I am still 00:17:33.960 --> 00:17:37.950 align:middle line:84% dissipating some energy in that wiring resistance. 00:17:37.950 --> 00:17:41.490 align:middle line:84% And I'm still limited in the current 00:17:41.490 --> 00:17:45.420 align:middle line:84% I can draw before the breaker goes 00:17:45.420 --> 00:17:47.610 align:middle line:84% or before the fuse pops because the fuse is 00:17:47.610 --> 00:17:49.710 align:middle line:90% designed to essentially pop-- 00:17:49.710 --> 00:17:53.910 align:middle line:84% or the breaker is designed to pop based on a long-term RMS 00:17:53.910 --> 00:17:55.050 align:middle line:90% current. 00:17:55.050 --> 00:17:57.070 align:middle line:90% So what does that look like? 00:17:57.070 --> 00:17:59.250 align:middle line:84% Well, if I could figure out the RMS current, 00:17:59.250 --> 00:18:04.483 align:middle line:84% IRMS here is equal to the peak value-- it's a sinusoid. 00:18:04.483 --> 00:18:06.400 align:middle line:84% It's the peak value over the square root of 2. 00:18:06.400 --> 00:18:12.320 align:middle line:84% So it would be Vs over the square root of 2 omega L. 00:18:12.320 --> 00:18:20.780 align:middle line:84% So if I again have omega is equal to 377 radians per second 00:18:20.780 --> 00:18:25.310 align:middle line:84% in the United States-- that's 60 hertz, essentially-- 00:18:25.310 --> 00:18:36.740 align:middle line:84% L of about 20 millihenries gives IRMS of about 15 amps 00:18:36.740 --> 00:18:38.810 align:middle line:90% at 115 volts AC. 00:18:38.810 --> 00:18:45.110 align:middle line:90% 00:18:45.110 --> 00:18:46.190 align:middle line:90% So what does that mean? 00:18:46.190 --> 00:18:49.100 align:middle line:84% If I go out and get myself a 21-millihenry inductor and I 00:18:49.100 --> 00:18:51.710 align:middle line:90% plug it in, life's good. 00:18:51.710 --> 00:18:53.270 align:middle line:90% I enjoy the magnetic fields. 00:18:53.270 --> 00:18:56.810 align:middle line:84% If I instead brought my 19-millihenry inductor 00:18:56.810 --> 00:18:59.480 align:middle line:84% and plugged it in, well, I'm going to lose the breaker, 00:18:59.480 --> 00:19:02.120 align:middle line:84% and I'm going to be sitting in the dark. 00:19:02.120 --> 00:19:07.490 align:middle line:84% So while the inductor doesn't consume average power, 00:19:07.490 --> 00:19:11.690 align:middle line:84% it does draw a mean square current, 00:19:11.690 --> 00:19:17.310 align:middle line:84% and it does degrade the ability of me to utilize my source. 00:19:17.310 --> 00:19:19.340 align:middle line:90% So I have two outlets here. 00:19:19.340 --> 00:19:22.173 align:middle line:84% Suppose I'm putting in my 20-millihenry inductor. 00:19:22.173 --> 00:19:26.720 align:middle line:90% 00:19:26.720 --> 00:19:28.470 align:middle line:90% And I can do that. 00:19:28.470 --> 00:19:29.750 align:middle line:90% Life's good. 00:19:29.750 --> 00:19:30.720 align:middle line:90% Nothing's going wrong. 00:19:30.720 --> 00:19:32.430 align:middle line:90% I haven't exceeded the breaker. 00:19:32.430 --> 00:19:35.240 align:middle line:84% But now if I want to plug in my space heater 00:19:35.240 --> 00:19:37.010 align:middle line:84% as well because I'm feeling cold, 00:19:37.010 --> 00:19:42.300 align:middle line:84% I'm out of luck because I'm already drawing 15 amps RMS. 00:19:42.300 --> 00:19:47.160 align:middle line:84% I draw any more current, the breaker goes. 00:19:47.160 --> 00:19:52.120 align:middle line:84% So plugging in the inductor hasn't used any average energy, 00:19:52.120 --> 00:19:58.160 align:middle line:84% but it has degraded my ability to draw power from the source. 00:19:58.160 --> 00:20:01.700 align:middle line:84% I've utilized the capability of the source to deliver me energy, 00:20:01.700 --> 00:20:04.450 align:middle line:84% even though I haven't used the energy. 00:20:04.450 --> 00:20:08.780 align:middle line:90% Any questions about that? 00:20:08.780 --> 00:20:11.900 align:middle line:84% Now, as you might imagine, the utility hates that. 00:20:11.900 --> 00:20:13.970 align:middle line:90% Their job is to sell you energy. 00:20:13.970 --> 00:20:19.430 align:middle line:84% And you're using up the capacity without using up the energy. 00:20:19.430 --> 00:20:21.350 align:middle line:84% So what people have done is they've 00:20:21.350 --> 00:20:25.800 align:middle line:84% come up with a measure of energy utilization. 00:20:25.800 --> 00:20:30.500 align:middle line:84% How well are you utilizing the capability of the source 00:20:30.500 --> 00:20:32.390 align:middle line:90% to deliver you energy? 00:20:32.390 --> 00:20:39.425 align:middle line:84% And this measure is known as power factor. 00:20:39.425 --> 00:20:50.580 align:middle line:84% So power factor is defined as follows. 00:20:50.580 --> 00:20:56.120 align:middle line:84% It's the average power you're drawing divided by the RMS 00:20:56.120 --> 00:20:59.425 align:middle line:90% voltage times the RMS current. 00:20:59.425 --> 00:21:02.660 align:middle line:90% 00:21:02.660 --> 00:21:04.420 align:middle line:84% Now, when I talk about power factor, 00:21:04.420 --> 00:21:06.970 align:middle line:84% we mean the power factor at a port. 00:21:06.970 --> 00:21:08.523 align:middle line:90% So I have some pair of wires. 00:21:08.523 --> 00:21:11.650 align:middle line:90% 00:21:11.650 --> 00:21:14.530 align:middle line:84% And I look at this port, this interface's two-wire pair, 00:21:14.530 --> 00:21:19.210 align:middle line:84% and I'm talking about some voltage v and some current i 00:21:19.210 --> 00:21:19.910 align:middle line:90% here. 00:21:19.910 --> 00:21:22.690 align:middle line:84% So it's basically relating to the power that's 00:21:22.690 --> 00:21:24.390 align:middle line:90% going across this dashed line. 00:21:24.390 --> 00:21:27.675 align:middle line:90% 00:21:27.675 --> 00:21:30.730 align:middle line:84% And why do I choose this measure? 00:21:30.730 --> 00:21:32.930 align:middle line:90% Well, what would be-- 00:21:32.930 --> 00:21:49.300 align:middle line:84% If I had a resistor, the average power is equal to VRMS IRMS. 00:21:49.300 --> 00:21:53.900 align:middle line:90% So the power factor is one. 00:21:53.900 --> 00:21:58.880 align:middle line:84% So I basically best utilize that source as I can. 00:21:58.880 --> 00:22:01.420 align:middle line:84% I'm drawing energy out of the source, 00:22:01.420 --> 00:22:02.990 align:middle line:90% and it's all going to me. 00:22:02.990 --> 00:22:06.520 align:middle line:84% All the RMS current is contributing to energy transfer 00:22:06.520 --> 00:22:11.890 align:middle line:84% from the source to the load or across this boundary. 00:22:11.890 --> 00:22:19.060 align:middle line:84% If I had an inductor L or a capacitor C, for that matter, 00:22:19.060 --> 00:22:21.850 align:middle line:84% I'd have average power is equal to 0, 00:22:21.850 --> 00:22:24.910 align:middle line:84% and I'd have a power factor of 0. 00:22:24.910 --> 00:22:28.660 align:middle line:84% That means I'm utilizing up the capability of the source 00:22:28.660 --> 00:22:32.770 align:middle line:84% by drawing RMS current but delivering no energy. 00:22:32.770 --> 00:22:37.510 align:middle line:84% So the power factor has a magnitude that's less than 1. 00:22:37.510 --> 00:22:40.120 align:middle line:84% The best case from the utility's perspective, 00:22:40.120 --> 00:22:42.430 align:middle line:84% if your power factor is 1, that means 00:22:42.430 --> 00:22:45.600 align:middle line:84% all the current I'm drawing is going to energy transfer. 00:22:45.600 --> 00:22:47.570 align:middle line:84% The power factor can go down to 0, 00:22:47.570 --> 00:22:49.760 align:middle line:84% which means I'm drawing current, but I'm not 00:22:49.760 --> 00:22:52.940 align:middle line:90% drawing any power on average. 00:22:52.940 --> 00:22:58.310 align:middle line:84% Any questions about that general idea? 00:22:58.310 --> 00:22:58.810 align:middle line:90% Yeah. 00:22:58.810 --> 00:23:00.560 align:middle line:84% AUDIENCE: If you have a power factor of 1, 00:23:00.560 --> 00:23:02.690 align:middle line:84% then wouldn't all the electricity we burned up 00:23:02.690 --> 00:23:05.718 align:middle line:90% in the transmission lines? 00:23:05.718 --> 00:23:08.760 align:middle line:84% DAVID PERREAULT: Power factor of 1-- 00:23:08.760 --> 00:23:12.750 align:middle line:84% OK, first of all, the transmission line-- 00:23:12.750 --> 00:23:17.180 align:middle line:84% when I talk about power factor, I'm talking about it at a place. 00:23:17.180 --> 00:23:20.730 align:middle line:84% So I'm talking about the voltage here and the current here. 00:23:20.730 --> 00:23:24.210 align:middle line:84% So power factor doesn't concern itself 00:23:24.210 --> 00:23:28.740 align:middle line:84% with what energy is being dissipated elsewhere. 00:23:28.740 --> 00:23:37.360 align:middle line:84% All it's saying is that the current I'm using here 00:23:37.360 --> 00:23:40.720 align:middle line:84% is doing the best possible job of transferring energy 00:23:40.720 --> 00:23:45.010 align:middle line:84% across this boundary without wasting any of it. 00:23:45.010 --> 00:23:48.940 align:middle line:84% That's not to say that there is no dissipation back over here 00:23:48.940 --> 00:23:50.170 align:middle line:90% somewhere. 00:23:50.170 --> 00:23:52.930 align:middle line:84% And with a resistive load, a power factor of 1 00:23:52.930 --> 00:23:58.620 align:middle line:84% means you're doing as little badness as you can. 00:23:58.620 --> 00:24:02.060 align:middle line:84% In other words, in order to draw energy in this example, 00:24:02.060 --> 00:24:04.360 align:middle line:84% yeah, I am dissipating something in the wire; 00:24:04.360 --> 00:24:07.470 align:middle line:84% there is some loss for the utility that they're paying for. 00:24:07.470 --> 00:24:12.290 align:middle line:84% But I'm minimizing the amount that's 00:24:12.290 --> 00:24:14.090 align:middle line:84% getting burned in the wiring resistance 00:24:14.090 --> 00:24:16.400 align:middle line:90% to transfer energy to my load. 00:24:16.400 --> 00:24:19.650 align:middle line:84% Whereas in the inductive case, I was doing the opposite thing. 00:24:19.650 --> 00:24:24.440 align:middle line:84% I was sort of not taking any energy 00:24:24.440 --> 00:24:26.720 align:middle line:84% but still causing them to have to pay 00:24:26.720 --> 00:24:29.390 align:middle line:84% for the loss in the transmission line. 00:24:29.390 --> 00:24:30.300 align:middle line:90% Does that make sense? 00:24:30.300 --> 00:24:31.415 align:middle line:90% AUDIENCE: Thank you. 00:24:31.415 --> 00:24:35.380 align:middle line:84% DAVID PERREAULT: Other questions or comments? 00:24:35.380 --> 00:24:36.160 align:middle line:90% Yeah. 00:24:36.160 --> 00:24:39.730 align:middle line:84% AUDIENCE: So when we say the average power out of port, 00:24:39.730 --> 00:24:41.890 align:middle line:84% is the idea then that it would be-- 00:24:41.890 --> 00:24:45.600 align:middle line:84% if you had some resistive element at that port-- 00:24:45.600 --> 00:24:46.850 align:middle line:90% DAVID PERREAULT: Say it again. 00:24:46.850 --> 00:24:49.000 align:middle line:84% AUDIENCE: If we say average power out of port, 00:24:49.000 --> 00:24:52.045 align:middle line:84% then it says if we were to have some resistive element 00:24:52.045 --> 00:24:53.920 align:middle line:84% at that port, that's what it would be, right? 00:24:53.920 --> 00:24:54.795 align:middle line:90% DAVID PERREAULT: Yes. 00:24:54.795 --> 00:24:57.100 align:middle line:84% If I had some resistive element on the-- yeah, 00:24:57.100 --> 00:24:59.720 align:middle line:84% the question is, what is the power at the port? 00:24:59.720 --> 00:25:02.170 align:middle line:84% It's the power that's going across this dashed line 00:25:02.170 --> 00:25:04.060 align:middle line:90% on average. 00:25:04.060 --> 00:25:07.100 align:middle line:84% So if I have a resistor over here, that's where it's going. 00:25:07.100 --> 00:25:10.290 align:middle line:90% Yes 00:25:10.290 --> 00:25:14.277 align:middle line:84% OK, so we have this notion of power factor. 00:25:14.277 --> 00:25:16.110 align:middle line:84% And by the way, power factor can technically 00:25:16.110 --> 00:25:18.810 align:middle line:90% be between 1 and minus 1. 00:25:18.810 --> 00:25:21.060 align:middle line:84% Minus 1-- a negative number would just 00:25:21.060 --> 00:25:23.305 align:middle line:84% suggest that actual power is flowing that way. 00:25:23.305 --> 00:25:26.442 align:middle line:90% 00:25:26.442 --> 00:25:29.220 align:middle line:84% But the utility is quite concerned with power factor 00:25:29.220 --> 00:25:32.320 align:middle line:84% and not to be confused with actual power dissipation. 00:25:32.320 --> 00:25:38.970 align:middle line:84% I should say, by the way, that this average power is sometimes 00:25:38.970 --> 00:25:43.340 align:middle line:84% known as a, quote, unquote, "real power." 00:25:43.340 --> 00:25:46.670 align:middle line:90% 00:25:46.670 --> 00:25:50.480 align:middle line:84% So if you talk, hear somebody talk about real power, 00:25:50.480 --> 00:25:52.170 align:middle line:84% that's what they're talking about. 00:25:52.170 --> 00:26:01.250 align:middle line:84% VRMS IRMS is what's known as apparent power, which 00:26:01.250 --> 00:26:03.455 align:middle line:84% is greater than or equal to the real power. 00:26:03.455 --> 00:26:07.558 align:middle line:90% 00:26:07.558 --> 00:26:11.250 align:middle line:84% And this is just saying, what effective power would I 00:26:11.250 --> 00:26:14.310 align:middle line:84% be drawing considering that I have that much current and that 00:26:14.310 --> 00:26:18.065 align:middle line:84% much voltage if it was going into a resistive load? 00:26:18.065 --> 00:26:23.250 align:middle line:90% 00:26:23.250 --> 00:26:28.410 align:middle line:84% So let's expand our thoughts a little bit because I've 00:26:28.410 --> 00:26:31.440 align:middle line:84% been so far talking about resistors and inductors 00:26:31.440 --> 00:26:33.990 align:middle line:84% and capacitors, which have the characteristic that they all 00:26:33.990 --> 00:26:36.480 align:middle line:90% draw sinusoidal waveforms. 00:26:36.480 --> 00:26:38.700 align:middle line:84% So nothing wrong with anything we've done so far, 00:26:38.700 --> 00:26:40.210 align:middle line:84% but the world is bigger than that. 00:26:40.210 --> 00:26:42.900 align:middle line:84% And let's come back and think about, 00:26:42.900 --> 00:26:47.520 align:middle line:84% for example, the rectifier we had from last time. 00:26:47.520 --> 00:26:51.900 align:middle line:84% So last time, we said, OK, I'm going to hook up 00:26:51.900 --> 00:26:53.480 align:middle line:90% to some AC voltage source. 00:26:53.480 --> 00:26:58.000 align:middle line:90% 00:26:58.000 --> 00:26:59.710 align:middle line:84% So I have a sinusoidal voltage source, 00:26:59.710 --> 00:27:03.400 align:middle line:84% and then that's coming into some rectifier. 00:27:03.400 --> 00:27:09.580 align:middle line:84% And I'm getting some output current, which is ideally DC. 00:27:09.580 --> 00:27:14.440 align:middle line:84% And maybe I would say, OK, let's talk about this current 00:27:14.440 --> 00:27:18.520 align:middle line:84% drawn from the rectifier, i of t. 00:27:18.520 --> 00:27:20.470 align:middle line:90% So what does that look like? 00:27:20.470 --> 00:27:26.050 align:middle line:84% In the example we had last time, if I have a sinusoidal voltage 00:27:26.050 --> 00:27:38.040 align:middle line:84% source, for a power converter circuit, 00:27:38.040 --> 00:27:41.190 align:middle line:84% the best-case scenario is it looks like a resistor 00:27:41.190 --> 00:27:43.330 align:middle line:84% because that's when you get power factor of 1. 00:27:43.330 --> 00:27:46.600 align:middle line:84% Unfortunately, a lot of power converters don't look like that. 00:27:46.600 --> 00:27:50.970 align:middle line:84% And in fact, if we took the half wave rectifier from last time 00:27:50.970 --> 00:27:53.100 align:middle line:84% and I said, What did the line current look like? 00:27:53.100 --> 00:27:54.270 align:middle line:90% it would look like this. 00:27:54.270 --> 00:27:55.560 align:middle line:90% It would look like-- 00:27:55.560 --> 00:27:57.280 align:middle line:84% in positive half of the line cycle, 00:27:57.280 --> 00:28:00.000 align:middle line:84% it would give me some current Id. 00:28:00.000 --> 00:28:04.080 align:middle line:84% And in the second half of the line cycle, it would give me 0. 00:28:04.080 --> 00:28:06.000 align:middle line:90% So I'd be drawing a square wave. 00:28:06.000 --> 00:28:09.278 align:middle line:84% And in fact, in that half wave rectifier waveform, 00:28:09.278 --> 00:28:11.070 align:middle line:84% it's actually a square wave with an offset, 00:28:11.070 --> 00:28:12.690 align:middle line:90% particularly horrible. 00:28:12.690 --> 00:28:15.210 align:middle line:90% And here's omega t. 00:28:15.210 --> 00:28:16.650 align:middle line:90% This is pi. 00:28:16.650 --> 00:28:19.500 align:middle line:90% And this is 2 pi. 00:28:19.500 --> 00:28:23.550 align:middle line:84% So the point is, when we start introducing power conversion, 00:28:23.550 --> 00:28:26.550 align:middle line:84% we can also often have nonlinear loads, 00:28:26.550 --> 00:28:28.440 align:middle line:84% meaning that even when I'm driving it 00:28:28.440 --> 00:28:31.950 align:middle line:84% with a sinusoidal voltage, I don't get a sinusoidal current. 00:28:31.950 --> 00:28:38.273 align:middle line:84% So let's start to think about how might I represent that case. 00:28:38.273 --> 00:28:41.920 align:middle line:90% 00:28:41.920 --> 00:28:44.870 align:middle line:84% And this is particularly-- when I did this half wave rectifier, 00:28:44.870 --> 00:28:48.770 align:middle line:84% it was a half wave rectifier with an inductive filter, which 00:28:48.770 --> 00:28:51.120 align:middle line:84% draws that kind of square, wavy waveform. 00:28:51.120 --> 00:28:55.520 align:middle line:84% Well, let's think about how I might represent 00:28:55.520 --> 00:28:57.040 align:middle line:90% this current i of t. 00:28:57.040 --> 00:29:01.040 align:middle line:90% 00:29:01.040 --> 00:29:07.430 align:middle line:84% One way to think about that is in periodic steady state, 00:29:07.430 --> 00:29:10.520 align:middle line:90% I get a periodic current, right? 00:29:10.520 --> 00:29:16.100 align:middle line:84% So in PSS, what I'm talking about right now, 00:29:16.100 --> 00:29:19.460 align:middle line:84% I can represent i of t with a Fourier series. 00:29:19.460 --> 00:29:23.510 align:middle line:84% So i of t, we might write this as the summation from n 00:29:23.510 --> 00:29:34.220 align:middle line:84% equals 0 to infinity of I sub n sine of n omega t plus phi sub 00:29:34.220 --> 00:29:35.390 align:middle line:90% n. 00:29:35.390 --> 00:29:39.090 align:middle line:84% This is one way I could write a Fourier series. 00:29:39.090 --> 00:29:41.140 align:middle line:84% We can find the Fourier coefficients. 00:29:41.140 --> 00:29:44.820 align:middle line:84% And just to be clear, I'm defining phi 0 00:29:44.820 --> 00:29:49.710 align:middle line:84% to be equal to pi over 2 or 90 degrees so that I 00:29:49.710 --> 00:29:51.790 align:middle line:90% sub 0 is just the DC component. 00:29:51.790 --> 00:29:57.120 align:middle line:90% 00:29:57.120 --> 00:30:04.110 align:middle line:84% OK, well, I can represent this pink waveform or any other AC 00:30:04.110 --> 00:30:07.890 align:middle line:84% waveform that's not sinusoidal as long as it's periodic 00:30:07.890 --> 00:30:09.580 align:middle line:90% with some Fourier series. 00:30:09.580 --> 00:30:15.380 align:middle line:84% You can relatively easily show what the RMS of that waveform 00:30:15.380 --> 00:30:15.880 align:middle line:90% is. 00:30:15.880 --> 00:30:18.213 align:middle line:84% If I have-- if I break it down with this Fourier series, 00:30:18.213 --> 00:30:20.410 align:middle line:90% we could write RMS as follows. 00:30:20.410 --> 00:30:29.610 align:middle line:84% IRMS is simply equal to the square root of I0 squared plus 00:30:29.610 --> 00:30:38.160 align:middle line:84% 1/2 I1 squared plus 1/2 half In squared and so forth. 00:30:38.160 --> 00:30:43.210 align:middle line:84% So it becomes the sum of the squares. 00:30:43.210 --> 00:30:47.160 align:middle line:84% The halves are because of the mean square of the individual 00:30:47.160 --> 00:30:51.090 align:middle line:84% components, and I0 is constant, so it doesn't have 1/2. 00:30:51.090 --> 00:30:52.840 align:middle line:90% How do you find this? 00:30:52.840 --> 00:30:55.320 align:middle line:84% Well, you could just apply the definition 00:30:55.320 --> 00:30:58.600 align:middle line:84% of RMS with this waveform and then use orthogonality, 00:30:58.600 --> 00:30:59.445 align:middle line:90% and it'll pop out. 00:30:59.445 --> 00:31:03.750 align:middle line:90% 00:31:03.750 --> 00:31:09.930 align:middle line:84% So fine, I can express my RMS current. 00:31:09.930 --> 00:31:13.100 align:middle line:84% How would I write the power associated 00:31:13.100 --> 00:31:14.910 align:middle line:90% with that waveform given this? 00:31:14.910 --> 00:31:17.810 align:middle line:84% Well, what I could do is I could say this. 00:31:17.810 --> 00:31:25.070 align:middle line:84% The average power, PAv, I could write as v of t times 00:31:25.070 --> 00:31:27.330 align:middle line:84% the average of v of t times i of t. 00:31:27.330 --> 00:31:34.060 align:middle line:84% So this would be 1 over 2 pi integral from 0 to 2 pi of Vs 00:31:34.060 --> 00:31:44.120 align:middle line:84% sine omega t times the summation n equals 0 00:31:44.120 --> 00:31:52.805 align:middle line:84% to infinity I sub n sine of n omega t plus phi sub n d omega 00:31:52.805 --> 00:31:53.305 align:middle line:90% t. 00:31:53.305 --> 00:31:56.540 align:middle line:90% 00:31:56.540 --> 00:32:00.590 align:middle line:84% So I've got this integral of a product. 00:32:00.590 --> 00:32:06.800 align:middle line:84% I can interchange my summation and integration here. 00:32:06.800 --> 00:32:08.600 align:middle line:90% So what would this give me? 00:32:08.600 --> 00:32:18.320 align:middle line:84% This would give me 1 over 2 pi Vs I sub 1-- 00:32:18.320 --> 00:32:24.500 align:middle line:84% I'm sorry, 1 over 2 pi V sub s, sorry-- 00:32:24.500 --> 00:32:34.030 align:middle line:84% summation n equals 0 to infinity of I 00:32:34.030 --> 00:32:45.715 align:middle line:84% sub n sine of omega t sine of n omega t d omega t. 00:32:45.715 --> 00:32:49.515 align:middle line:84% Does that makes sense to everybody? 00:32:49.515 --> 00:32:52.090 align:middle line:84% AUDIENCE: Shouldn't there be an integral as well? 00:32:52.090 --> 00:32:54.298 align:middle line:84% DAVID PERREAULT: There should be an integral in here, 00:32:54.298 --> 00:32:59.700 align:middle line:90% shouldn't there/ Thank you. 00:32:59.700 --> 00:33:03.630 align:middle line:84% So all I've done is I've pulled out the summation. 00:33:03.630 --> 00:33:05.020 align:middle line:90% But look at this. 00:33:05.020 --> 00:33:07.080 align:middle line:90% What would you tell me-- 00:33:07.080 --> 00:33:09.980 align:middle line:84% and this should be sine omega t plus phi sub n. 00:33:09.980 --> 00:33:15.390 align:middle line:90% 00:33:15.390 --> 00:33:17.810 align:middle line:84% What would you tell me about this integral here? 00:33:17.810 --> 00:33:21.600 align:middle line:90% 00:33:21.600 --> 00:33:27.596 align:middle line:84% Well, if n is not equal to 1, what is this integral? 00:33:27.596 --> 00:33:28.290 align:middle line:90% AUDIENCE: 0. 00:33:28.290 --> 00:33:30.690 align:middle line:84% DAVID PERREAULT: 0 because sinusoids 00:33:30.690 --> 00:33:33.280 align:middle line:84% at different frequencies are orthogonal. 00:33:33.280 --> 00:33:37.560 align:middle line:84% So I can throw every other term but the n equal 1 term away. 00:33:37.560 --> 00:33:39.303 align:middle line:84% And so what I'm going to get is-- 00:33:39.303 --> 00:33:41.950 align:middle line:90% 00:33:41.950 --> 00:33:44.270 align:middle line:84% and I can put this 1 over 2 pi here. 00:33:44.270 --> 00:33:49.180 align:middle line:84% And if we come back to my original equation here, 00:33:49.180 --> 00:33:51.470 align:middle line:90% I have that 1 over 2 pi here. 00:33:51.470 --> 00:34:02.190 align:middle line:84% So what I get is Vs times I1 over 2 cosine of phi 1. 00:34:02.190 --> 00:34:05.540 align:middle line:90% 00:34:05.540 --> 00:34:11.469 align:middle line:84% So the nice thing is that everything drops out. 00:34:11.469 --> 00:34:13.350 align:middle line:90% So what does this tell me? 00:34:13.350 --> 00:34:18.330 align:middle line:84% This says, if I have this distorted waveform, 00:34:18.330 --> 00:34:20.850 align:middle line:84% it's no longer a sinusoid, this pink current 00:34:20.850 --> 00:34:24.360 align:middle line:84% that has a whole bunch of harmonic junk in it. 00:34:24.360 --> 00:34:29.159 align:middle line:84% The only part of that that contributes to average power, 00:34:29.159 --> 00:34:34.810 align:middle line:84% to real power, is the fundamental of the current 00:34:34.810 --> 00:34:37.060 align:middle line:84% because every other current component 00:34:37.060 --> 00:34:40.510 align:middle line:84% at all other frequencies, DC and all harmonics, 00:34:40.510 --> 00:34:42.224 align:middle line:90% is orthogonal to the voltage. 00:34:42.224 --> 00:34:44.989 align:middle line:90% 00:34:44.989 --> 00:34:48.710 align:middle line:84% Moreover, it's not just the fundamental, 00:34:48.710 --> 00:34:52.550 align:middle line:84% but it depends upon the cosine of that fundamental. 00:34:52.550 --> 00:34:55.280 align:middle line:84% In other words, it's only the component of the fundamental 00:34:55.280 --> 00:34:58.340 align:middle line:84% that's in phase with the voltage. 00:34:58.340 --> 00:35:01.770 align:middle line:84% If phi 1 was 0, cosine of phi 1 would be 1. 00:35:01.770 --> 00:35:05.990 align:middle line:84% So if I decomposed the current, the fundamental current 00:35:05.990 --> 00:35:08.543 align:middle line:84% into something that was in phase with the voltage 00:35:08.543 --> 00:35:10.460 align:middle line:84% and something that was 90 degrees out of phase 00:35:10.460 --> 00:35:11.127 align:middle line:90% of the voltage-- 00:35:11.127 --> 00:35:13.310 align:middle line:84% I can split it up that way-- only 00:35:13.310 --> 00:35:15.350 align:middle line:84% the component that's in phase with the voltage 00:35:15.350 --> 00:35:18.590 align:middle line:90% contributes to energy transfer. 00:35:18.590 --> 00:35:24.020 align:middle line:84% So if I want to transfer energy, the only part 00:35:24.020 --> 00:35:25.910 align:middle line:84% of that pink waveform that's helping me 00:35:25.910 --> 00:35:29.390 align:middle line:84% is the part at the frequency of the sine wave of voltage 00:35:29.390 --> 00:35:33.795 align:middle line:84% and only the piece of that fundamental component 00:35:33.795 --> 00:35:35.170 align:middle line:84% that's in phase with the voltage. 00:35:35.170 --> 00:35:37.960 align:middle line:90% 00:35:37.960 --> 00:35:39.640 align:middle line:90% I could rewrite this-- 00:35:39.640 --> 00:35:43.600 align:middle line:84% Remember that for a sinusoid, the RMS 00:35:43.600 --> 00:35:46.490 align:middle line:84% is 1 over the square root of 2 times the peak value. 00:35:46.490 --> 00:35:58.590 align:middle line:84% So I could have rewritten this as VRMS I1 RMS cosine of phi 1. 00:35:58.590 --> 00:36:01.330 align:middle line:90% 00:36:01.330 --> 00:36:08.780 align:middle line:84% That's just to say that this 1/2 comes from the RMS component. 00:36:08.780 --> 00:36:12.660 align:middle line:90% So I can write it this way too. 00:36:12.660 --> 00:36:17.440 align:middle line:84% So what does this say about power factor? 00:36:17.440 --> 00:36:20.447 align:middle line:84% Well, I can write this average power factor. 00:36:20.447 --> 00:36:22.030 align:middle line:84% So now that I can write it, I can say, 00:36:22.030 --> 00:36:24.250 align:middle line:90% what is the power factor? 00:36:24.250 --> 00:36:26.500 align:middle line:84% And when I'm saying what is the power 00:36:26.500 --> 00:36:28.930 align:middle line:84% factor, in this particular example, 00:36:28.930 --> 00:36:32.770 align:middle line:84% I'm saying, what's the power factor at this port right 00:36:32.770 --> 00:36:34.960 align:middle line:84% here, at the rectifier input port? 00:36:34.960 --> 00:36:38.050 align:middle line:84% Because I have this current and Vs sine omega t. 00:36:38.050 --> 00:36:41.020 align:middle line:90% 00:36:41.020 --> 00:36:49.510 align:middle line:84% The power factor is simply going to be the average power, which 00:36:49.510 --> 00:37:00.430 align:middle line:84% is VRMS times I1 RMS, the RMS of just the fundamental component, 00:37:00.430 --> 00:37:07.510 align:middle line:84% times the cosine of the fundamental phase of the current 00:37:07.510 --> 00:37:11.163 align:middle line:90% divided by VRMS IRMS. 00:37:11.163 --> 00:37:14.440 align:middle line:90% 00:37:14.440 --> 00:37:18.550 align:middle line:84% Or I could cancel out VRMS because they're both just 00:37:18.550 --> 00:37:20.260 align:middle line:90% the RMS of a sine wave. 00:37:20.260 --> 00:37:28.750 align:middle line:84% And this would give us I1 RMS over IRMS 00:37:28.750 --> 00:37:31.140 align:middle line:90% times the cosine of phi 1. 00:37:31.140 --> 00:37:35.476 align:middle line:90% 00:37:35.476 --> 00:37:39.050 align:middle line:90% This decomposition is useful. 00:37:39.050 --> 00:37:43.270 align:middle line:84% This piece here is sometimes called k sub 00:37:43.270 --> 00:37:45.300 align:middle line:90% d, the distortion factor. 00:37:45.300 --> 00:37:51.190 align:middle line:90% 00:37:51.190 --> 00:37:52.970 align:middle line:90% What's it a measure of? 00:37:52.970 --> 00:37:59.140 align:middle line:84% It's saying, look, this is the IRMS of the whole thing. 00:37:59.140 --> 00:38:11.550 align:middle line:84% I1 RMS is simply equal to the square root of 1/2 I1 squared. 00:38:11.550 --> 00:38:16.050 align:middle line:84% So it's saying this is a measure-- 00:38:16.050 --> 00:38:18.540 align:middle line:84% this factor, this distortion factor 00:38:18.540 --> 00:38:23.430 align:middle line:84% is a measure of how much the harmonic content is 00:38:23.430 --> 00:38:25.830 align:middle line:90% degrading my power factor. 00:38:25.830 --> 00:38:28.350 align:middle line:84% I just told you that all the harmonic content doesn't 00:38:28.350 --> 00:38:30.020 align:middle line:90% help you transfer energy. 00:38:30.020 --> 00:38:32.140 align:middle line:84% That means it's degrading your power factor. 00:38:32.140 --> 00:38:35.100 align:middle line:84% So this number kd is clearly some number 00:38:35.100 --> 00:38:39.000 align:middle line:90% of magnitude less than 1. 00:38:39.000 --> 00:38:43.530 align:middle line:84% If it's a pure sinusoid, I1 one of RMS is equal to IRMS. 00:38:43.530 --> 00:38:45.400 align:middle line:90% And so that becomes 1. 00:38:45.400 --> 00:38:49.080 align:middle line:84% But when I start to add in any harmonic components or DC 00:38:49.080 --> 00:38:53.160 align:middle line:84% components, IRMS becomes bigger than I1 of RMS, 00:38:53.160 --> 00:38:54.960 align:middle line:84% and this distortion factor starts 00:38:54.960 --> 00:38:56.345 align:middle line:90% degrading my power factor. 00:38:56.345 --> 00:38:59.800 align:middle line:90% 00:38:59.800 --> 00:39:03.910 align:middle line:84% This here is sometimes called k theta 00:39:03.910 --> 00:39:05.460 align:middle line:90% or the displacement factor. 00:39:05.460 --> 00:39:10.630 align:middle line:90% 00:39:10.630 --> 00:39:14.025 align:middle line:84% And phi 1 is sometimes called the power factor angle. 00:39:14.025 --> 00:39:19.510 align:middle line:90% 00:39:19.510 --> 00:39:21.760 align:middle line:84% This is saying, suppose I had a-- 00:39:21.760 --> 00:39:24.320 align:middle line:84% suppose my distortion factor was just 1-- 00:39:24.320 --> 00:39:27.970 align:middle line:90% there's no distortion. 00:39:27.970 --> 00:39:32.940 align:middle line:84% Well, if my phase angle phi 1 of the current with respect 00:39:32.940 --> 00:39:35.250 align:middle line:84% to the voltage is 0-- they're in phase-- 00:39:35.250 --> 00:39:37.410 align:middle line:90% then this becomes 1. 00:39:37.410 --> 00:39:41.620 align:middle line:84% As the current starts to get out of phase with the voltage, 00:39:41.620 --> 00:39:46.530 align:middle line:84% this cosine of phi 1 has a magnitude less than 1. 00:39:46.530 --> 00:39:48.800 align:middle line:84% So we can hurt our power factor, we 00:39:48.800 --> 00:39:54.248 align:middle line:84% can hurt our ability to utilize the source to transfer energy. 00:39:54.248 --> 00:39:56.040 align:middle line:84% It's not how much energy we're dissipating. 00:39:56.040 --> 00:39:58.530 align:middle line:84% It's how much utilization do we have. 00:39:58.530 --> 00:40:00.670 align:middle line:90% I can damage it in two ways. 00:40:00.670 --> 00:40:03.870 align:middle line:84% I can damage it by drawing a bunch of harmonic current that 00:40:03.870 --> 00:40:10.325 align:middle line:84% doesn't deliver power or by delivering components of current 00:40:10.325 --> 00:40:12.200 align:middle line:84% that are out of phase with the voltage, which 00:40:12.200 --> 00:40:14.160 align:middle line:90% also doesn't deliver power. 00:40:14.160 --> 00:40:18.470 align:middle line:84% So this is kind of a very useful way sometimes 00:40:18.470 --> 00:40:24.670 align:middle line:90% to decompose what's going on. 00:40:24.670 --> 00:40:28.170 align:middle line:84% Because if I really cared about the power factor, 00:40:28.170 --> 00:40:29.710 align:middle line:90% I might need to know that. 00:40:29.710 --> 00:40:31.450 align:middle line:84% And you say, well, why is that true? 00:40:31.450 --> 00:40:34.770 align:middle line:84% Because suppose I connected up a rectifier like this 00:40:34.770 --> 00:40:39.240 align:middle line:84% with drawing the square wave current. 00:40:39.240 --> 00:40:40.830 align:middle line:84% First of all, this would be horrible 00:40:40.830 --> 00:40:43.568 align:middle line:84% because it has a DC value, which a lot of transformers 00:40:43.568 --> 00:40:44.360 align:middle line:90% doesn't let you do. 00:40:44.360 --> 00:40:45.705 align:middle line:90% But suppose I put that aside. 00:40:45.705 --> 00:40:48.330 align:middle line:90% 00:40:48.330 --> 00:40:54.300 align:middle line:84% Within my 15-amp limit, I have a limited amount of power I can 00:40:54.300 --> 00:40:58.740 align:middle line:84% draw with this rectifier just because of the harmonic 00:40:58.740 --> 00:40:59.790 align:middle line:90% currents. 00:40:59.790 --> 00:41:01.860 align:middle line:84% The fundamental of this pink waveform 00:41:01.860 --> 00:41:04.450 align:middle line:84% is actually in phase with the voltage. 00:41:04.450 --> 00:41:07.260 align:middle line:84% So I'm not hurting myself by having 00:41:07.260 --> 00:41:10.560 align:middle line:84% phase shifts like I did with an inductor or capacitor. 00:41:10.560 --> 00:41:13.710 align:middle line:84% But I am hurting myself or limiting the amount of power 00:41:13.710 --> 00:41:17.010 align:middle line:84% I can draw by the fact that there's 00:41:17.010 --> 00:41:20.220 align:middle line:84% a certain amount of nonfundamental component 00:41:20.220 --> 00:41:22.170 align:middle line:90% in the current waveform. 00:41:22.170 --> 00:41:26.740 align:middle line:84% Well, if I got rid of the DC component, for example, 00:41:26.740 --> 00:41:30.700 align:middle line:84% suddenly my RMS would go down and my power transfer 00:41:30.700 --> 00:41:31.630 align:middle line:90% would stay the same. 00:41:31.630 --> 00:41:33.110 align:middle line:90% And I'd be a happier camper. 00:41:33.110 --> 00:41:35.650 align:middle line:84% That might be one thing I could do, for example. 00:41:35.650 --> 00:41:40.490 align:middle line:84% But these kind of calculations can tell you, how are you-- 00:41:40.490 --> 00:41:43.303 align:middle line:84% what should you do to do better utilization of your source? 00:41:43.303 --> 00:41:45.720 align:middle line:84% Whereas the best thing you can do is look like a resistor. 00:41:45.720 --> 00:41:48.291 align:middle line:90% 00:41:48.291 --> 00:41:51.910 align:middle line:84% Any questions about any of this so far? 00:41:51.910 --> 00:41:52.450 align:middle line:90% Yeah. 00:41:52.450 --> 00:41:54.075 align:middle line:84% AUDIENCE: You made a couple of comments 00:41:54.075 --> 00:41:56.560 align:middle line:84% about utilities caring a lot about this. 00:41:56.560 --> 00:41:58.660 align:middle line:84% It seems like all of the stuff that this 00:41:58.660 --> 00:42:00.380 align:middle line:90% involves is on the user's end. 00:42:00.380 --> 00:42:02.290 align:middle line:84% Can the utility do anything about this? 00:42:02.290 --> 00:42:04.360 align:middle line:84% Or is it just they wish we could do better? 00:42:04.360 --> 00:42:08.590 align:middle line:84% DAVID PERREAULT: Yeah, that's a very, very good question. 00:42:08.590 --> 00:42:13.090 align:middle line:84% The answer is if you are an industrial user of power, 00:42:13.090 --> 00:42:16.030 align:middle line:84% if you're running a big factory that's 00:42:16.030 --> 00:42:18.940 align:middle line:84% drawing lots of electricity for whatever your industrial process 00:42:18.940 --> 00:42:21.733 align:middle line:84% is, yes, the utility can come and say, 00:42:21.733 --> 00:42:22.900 align:middle line:90% you've got to clean this up. 00:42:22.900 --> 00:42:25.540 align:middle line:84% Or what they really do is they start charging if your power 00:42:25.540 --> 00:42:26.945 align:middle line:90% factor is not good enough. 00:42:26.945 --> 00:42:28.570 align:middle line:84% And then you go clean it up because you 00:42:28.570 --> 00:42:30.660 align:middle line:90% don't want to pay them. 00:42:30.660 --> 00:42:34.060 align:middle line:90% So that's one way it's limited. 00:42:34.060 --> 00:42:35.630 align:middle line:90% But you're absolutely right. 00:42:35.630 --> 00:42:37.550 align:middle line:84% There are some perverse incentives. 00:42:37.550 --> 00:42:43.430 align:middle line:84% So increasingly, for different kinds of loads, 00:42:43.430 --> 00:42:46.030 align:middle line:84% requirements are being placed on the quality 00:42:46.030 --> 00:42:47.660 align:middle line:90% of the waveforms drawn. 00:42:47.660 --> 00:42:50.690 align:middle line:90% And that's a very complex thing. 00:42:50.690 --> 00:42:53.690 align:middle line:84% In some cases, they actually quote power factor. 00:42:53.690 --> 00:42:56.020 align:middle line:84% In some cases, they quote other measures. 00:42:56.020 --> 00:42:58.133 align:middle line:84% Like, they assume that your phase 00:42:58.133 --> 00:42:59.800 align:middle line:84% angle is going to be happy, but how much 00:42:59.800 --> 00:43:01.400 align:middle line:90% distortion do you put in? 00:43:01.400 --> 00:43:04.450 align:middle line:84% And that becomes-- if you want your Energy Star 00:43:04.450 --> 00:43:06.400 align:middle line:84% rating, for example, on your appliance, 00:43:06.400 --> 00:43:08.750 align:middle line:84% then suddenly you've got to meet that. 00:43:08.750 --> 00:43:11.680 align:middle line:84% Or if you want to sell a computer power supply, 00:43:11.680 --> 00:43:16.640 align:middle line:84% there are certain standards you have to meet. 00:43:16.640 --> 00:43:19.360 align:middle line:84% But those are a little bit haphazard 00:43:19.360 --> 00:43:22.580 align:middle line:84% because people were doing all kinds of nasty stuff 00:43:22.580 --> 00:43:24.830 align:middle line:84% to the grid back in the day and nobody paid attention. 00:43:24.830 --> 00:43:27.038 align:middle line:84% And now they're trying to clean it up after the fact. 00:43:27.038 --> 00:43:30.460 align:middle line:84% So it's mostly if you're a big industrial user of energy 00:43:30.460 --> 00:43:34.060 align:middle line:84% that they're really regulating your power factor. 00:43:34.060 --> 00:43:36.880 align:middle line:84% But increasingly with other kinds of loads, especially 00:43:36.880 --> 00:43:39.610 align:middle line:84% as new kinds of power supplies and things come up, 00:43:39.610 --> 00:43:42.410 align:middle line:84% there are more and more strict regulations. 00:43:42.410 --> 00:43:43.640 align:middle line:90% But excellent question. 00:43:43.640 --> 00:43:44.955 align:middle line:90% Other questions or thoughts? 00:43:44.955 --> 00:43:49.520 align:middle line:90% 00:43:49.520 --> 00:43:52.886 align:middle line:84% OK, I should point out one thing about this breakdown. 00:43:52.886 --> 00:43:56.990 align:middle line:90% 00:43:56.990 --> 00:43:59.570 align:middle line:84% This whole breakdown into distortion factor 00:43:59.570 --> 00:44:03.050 align:middle line:84% and displacement factor worked because I 00:44:03.050 --> 00:44:05.270 align:middle line:84% assumed that the voltage was a sinusoid 00:44:05.270 --> 00:44:07.620 align:middle line:84% and that the current might have harmonics. 00:44:07.620 --> 00:44:10.670 align:middle line:84% I could have done the same kind of breakdown 00:44:10.670 --> 00:44:14.190 align:middle line:84% if the current was a sinusoid and the voltage had harmonics. 00:44:14.190 --> 00:44:17.780 align:middle line:84% In that case, it would just be V1 RMS over VRMS 00:44:17.780 --> 00:44:20.300 align:middle line:84% and then cosine of the phase shift. 00:44:20.300 --> 00:44:23.900 align:middle line:84% If you have both the voltage is not sinusoidal 00:44:23.900 --> 00:44:26.180 align:middle line:84% and the current is not sinusoidal, 00:44:26.180 --> 00:44:29.210 align:middle line:84% you can't break things down this way. 00:44:29.210 --> 00:44:33.440 align:middle line:84% You can, however, still calculate power factor. 00:44:33.440 --> 00:44:35.330 align:middle line:84% And it turns out that the way you 00:44:35.330 --> 00:44:38.245 align:middle line:84% get a power factor of 1, which is the best you can do, 00:44:38.245 --> 00:44:39.620 align:middle line:84% is you make your current waveform 00:44:39.620 --> 00:44:41.287 align:middle line:84% look exactly like your voltage waveform. 00:44:41.287 --> 00:44:43.820 align:middle line:84% It has the same harmonic content and the same phase 00:44:43.820 --> 00:44:45.240 align:middle line:90% shift at every frequency. 00:44:45.240 --> 00:44:48.020 align:middle line:84% And that's the best you can do to transfer energy. 00:44:48.020 --> 00:44:52.040 align:middle line:84% And you can think about it as transfer the least-- 00:44:52.040 --> 00:44:56.430 align:middle line:84% the most energy per unit loss back in the source somewhere 00:44:56.430 --> 00:44:58.370 align:middle line:84% is the way you might think about that. 00:44:58.370 --> 00:45:02.299 align:middle line:90% 00:45:02.299 --> 00:45:07.060 align:middle line:84% Any other questions before we show you some of these things? 00:45:07.060 --> 00:45:07.900 align:middle line:90% Yeah. 00:45:07.900 --> 00:45:10.570 align:middle line:84% AUDIENCE: Is there any intuitive way 00:45:10.570 --> 00:45:14.460 align:middle line:84% to understand where the parent power is 00:45:14.460 --> 00:45:18.820 align:middle line:90% going from harmonic components? 00:45:18.820 --> 00:45:24.408 align:middle line:84% Like, what is actually causing that distortion? 00:45:24.408 --> 00:45:25.950 align:middle line:84% DAVID PERREAULT: Well, what's causing 00:45:25.950 --> 00:45:28.740 align:middle line:84% the distortion has something to do with the circuit. 00:45:28.740 --> 00:45:33.500 align:middle line:84% So this is all posited on I have a voltage waveform 00:45:33.500 --> 00:45:34.750 align:middle line:90% and I have a current waveform. 00:45:34.750 --> 00:45:36.810 align:middle line:84% And suppose the current is distorted. 00:45:36.810 --> 00:45:40.530 align:middle line:84% So what's causing all that has to do with the circuit? 00:45:40.530 --> 00:45:44.150 align:middle line:84% The way you might think about the impact 00:45:44.150 --> 00:45:52.280 align:middle line:84% is very much in thinking about how does the harmonics content-- 00:45:52.280 --> 00:45:54.950 align:middle line:84% this comes back to the fact that sinusoids 00:45:54.950 --> 00:45:57.500 align:middle line:84% at different frequencies are orthogonal. 00:45:57.500 --> 00:46:00.800 align:middle line:84% And so you degrade your energy transport 00:46:00.800 --> 00:46:02.690 align:middle line:90% because of that orthogonality. 00:46:02.690 --> 00:46:06.110 align:middle line:84% And sine and cosine at a given frequency is orthogonal, 00:46:06.110 --> 00:46:07.598 align:middle line:84% and that causes this degradation. 00:46:07.598 --> 00:46:09.140 align:middle line:84% So that's the way I would look at it. 00:46:09.140 --> 00:46:12.060 align:middle line:84% In terms of what's causing that or how do you fix that, 00:46:12.060 --> 00:46:16.360 align:middle line:84% that's more of a circuit thing or a system design issue. 00:46:16.360 --> 00:46:19.530 align:middle line:90% 00:46:19.530 --> 00:46:21.480 align:middle line:90% OK, let's do a demo. 00:46:21.480 --> 00:46:25.480 align:middle line:84% [INAUDIBLE] has very kindly set this up. 00:46:25.480 --> 00:46:27.630 align:middle line:84% And what this is, this is a piece of equipment 00:46:27.630 --> 00:46:33.090 align:middle line:84% that's designed to look at line waveforms and measure power 00:46:33.090 --> 00:46:34.950 align:middle line:84% factor and other things of signals. 00:46:34.950 --> 00:46:41.130 align:middle line:84% So basically, things that get plugged into this power strip, 00:46:41.130 --> 00:46:46.950 align:middle line:84% this box analyzes the voltage and the current waveform 00:46:46.950 --> 00:46:49.810 align:middle line:84% and then computes things like power factor for you. 00:46:49.810 --> 00:46:52.470 align:middle line:84% And I don't know what this particular Yokogawa power 00:46:52.470 --> 00:46:53.770 align:middle line:90% analyzer cost. 00:46:53.770 --> 00:46:55.350 align:middle line:90% I think it was under $10k. 00:46:55.350 --> 00:46:58.525 align:middle line:84% So for you, if you want one at home, you can go buy one. 00:46:58.525 --> 00:47:00.400 align:middle line:84% Actually, you can get one cheaper these days. 00:47:00.400 --> 00:47:03.720 align:middle line:90% But that's about what this cost. 00:47:03.720 --> 00:47:05.880 align:middle line:84% So what you can see here is she has 00:47:05.880 --> 00:47:07.480 align:middle line:90% the voltage waveform running. 00:47:07.480 --> 00:47:09.180 align:middle line:84% It's 110 volts, but there's no current 00:47:09.180 --> 00:47:10.930 align:middle line:84% because she has nothing plugged in. 00:47:10.930 --> 00:47:12.305 align:middle line:84% Now, [INAUDIBLE] is going to plug 00:47:12.305 --> 00:47:15.180 align:middle line:84% in a traditional incandescent light bulb. 00:47:15.180 --> 00:47:17.850 align:middle line:90% Here we go. 00:47:17.850 --> 00:47:23.390 align:middle line:84% Lambda 4 up there, for whatever reason, 00:47:23.390 --> 00:47:26.000 align:middle line:84% is the symbol they're using for power factor. 00:47:26.000 --> 00:47:27.500 align:middle line:84% But what you can see is the current, 00:47:27.500 --> 00:47:30.470 align:middle line:84% which is in purple, is basically sinusoidal 00:47:30.470 --> 00:47:32.370 align:middle line:84% and in phase with our nice voltage. 00:47:32.370 --> 00:47:34.927 align:middle line:84% And so we get a power factor of 1. 00:47:34.927 --> 00:47:37.010 align:middle line:84% And you can see she does a nice job of lighting up 00:47:37.010 --> 00:47:40.160 align:middle line:84% this kind of very Edisonian bulb. 00:47:40.160 --> 00:47:42.510 align:middle line:90% So thanks. 00:47:42.510 --> 00:47:44.310 align:middle line:84% And one thing I didn't point out, 00:47:44.310 --> 00:47:45.930 align:middle line:90% that was about 50 watts there. 00:47:45.930 --> 00:47:49.760 align:middle line:84% So that kind of makes sense for that kind of brightness. 00:47:49.760 --> 00:47:52.430 align:middle line:84% What [INAUDIBLE] is going to plug in right 00:47:52.430 --> 00:47:55.945 align:middle line:90% now is a fluorescent light bulb. 00:47:55.945 --> 00:47:58.480 align:middle line:90% 00:47:58.480 --> 00:48:01.780 align:middle line:84% And there we're actually getting a lot brighter light here. 00:48:01.780 --> 00:48:05.260 align:middle line:84% This is brighter than the incandescent light bulb. 00:48:05.260 --> 00:48:08.240 align:middle line:84% Notice that we're only drawing about 12 watts. 00:48:08.240 --> 00:48:11.348 align:middle line:84% So from an energy consumption, just purely how much power we're 00:48:11.348 --> 00:48:13.390 align:middle line:84% drawing from the grid, this is a way better deal, 00:48:13.390 --> 00:48:15.950 align:middle line:84% even though the light's kind of not as nice. 00:48:15.950 --> 00:48:19.150 align:middle line:84% And if we did an LED bulb, it would be even better. 00:48:19.150 --> 00:48:22.540 align:middle line:84% But look at the current waveform. 00:48:22.540 --> 00:48:27.010 align:middle line:84% That current waveform, the purple, is absolutely horrible. 00:48:27.010 --> 00:48:29.600 align:middle line:84% It's massively dominated by its harmonics. 00:48:29.600 --> 00:48:34.360 align:middle line:84% You can visually see that, yeah, it has a fundamental component, 00:48:34.360 --> 00:48:36.400 align:middle line:90% but it's pretty bad, right? 00:48:36.400 --> 00:48:39.100 align:middle line:84% The reason, this has a diode rectifier in it 00:48:39.100 --> 00:48:40.280 align:middle line:90% at the front end. 00:48:40.280 --> 00:48:42.610 align:middle line:84% But instead of running it with an inductive filter, 00:48:42.610 --> 00:48:44.030 align:middle line:90% they used a capacitive filter. 00:48:44.030 --> 00:48:45.760 align:middle line:84% So you get this kind of big pulse 00:48:45.760 --> 00:48:47.420 align:middle line:90% of current and then nothing. 00:48:47.420 --> 00:48:51.610 align:middle line:84% And then that internal capacitor discharges. 00:48:51.610 --> 00:48:54.080 align:middle line:84% So this leads to terrible power factor. 00:48:54.080 --> 00:48:57.250 align:middle line:84% And you can see that the power factor is only about 0.53. 00:48:57.250 --> 00:49:01.900 align:middle line:84% So what that says is if you use lots of these bulbs, 00:49:01.900 --> 00:49:06.460 align:middle line:84% you're kind of utilizing up the capability of your source 00:49:06.460 --> 00:49:07.610 align:middle line:90% to deliver power. 00:49:07.610 --> 00:49:10.450 align:middle line:84% What that means is while we're not measuring it here, 00:49:10.450 --> 00:49:13.330 align:middle line:84% you're dissipating more power back into the wall wiring 00:49:13.330 --> 00:49:14.740 align:middle line:90% than you need to. 00:49:14.740 --> 00:49:18.190 align:middle line:84% And in fact, if you went and bought 00:49:18.190 --> 00:49:19.710 align:middle line:90% one of the really new LED-- 00:49:19.710 --> 00:49:22.290 align:middle line:90% 00:49:22.290 --> 00:49:27.210 align:middle line:84% one of the really new LED power supplies, 00:49:27.210 --> 00:49:29.340 align:middle line:84% they might have a much better power factor 00:49:29.340 --> 00:49:31.380 align:middle line:84% because they're being incentivized 00:49:31.380 --> 00:49:34.650 align:middle line:84% if you want the Energy Star stamp to do that. 00:49:34.650 --> 00:49:39.600 align:middle line:84% Last example we'll show you is one that 00:49:39.600 --> 00:49:40.900 align:middle line:90% doesn't have a power converter. 00:49:40.900 --> 00:49:41.560 align:middle line:90% This is a fan. 00:49:41.560 --> 00:49:45.285 align:middle line:90% So this is a typical motor load. 00:49:45.285 --> 00:49:50.130 align:middle line:84% And here you see the current is reasonably sinusoidal, 00:49:50.130 --> 00:49:52.650 align:middle line:84% but there's a phase shift between the voltage 00:49:52.650 --> 00:49:53.380 align:middle line:90% and the current. 00:49:53.380 --> 00:49:55.043 align:middle line:84% The current lags the voltage why? 00:49:55.043 --> 00:49:57.210 align:middle line:84% Because it has to magnetize the motor, and the motor 00:49:57.210 --> 00:49:59.500 align:middle line:84% doesn't look like a perfect resistive load. 00:49:59.500 --> 00:50:03.660 align:middle line:84% So we get energy transfer, and there's 00:50:03.660 --> 00:50:07.800 align:middle line:84% a little bit of distortion owing to the machine, 00:50:07.800 --> 00:50:09.090 align:middle line:90% to the electric machine. 00:50:09.090 --> 00:50:11.680 align:middle line:84% But nonetheless, it's pretty close to sinusoidal. 00:50:11.680 --> 00:50:15.820 align:middle line:84% But we only get about 0.7 power factor because of this lag. 00:50:15.820 --> 00:50:17.910 align:middle line:84% So people are asking about, well, 00:50:17.910 --> 00:50:20.100 align:middle line:84% what about if you're a big industrial user? 00:50:20.100 --> 00:50:21.820 align:middle line:84% A lot of big industrial companies 00:50:21.820 --> 00:50:24.075 align:middle line:90% have lots of motor loads. 00:50:24.075 --> 00:50:27.460 align:middle line:90% 00:50:27.460 --> 00:50:30.163 align:middle line:84% They're using the pumps and whatever have you 00:50:30.163 --> 00:50:30.830 align:middle line:90% they're driving. 00:50:30.830 --> 00:50:34.940 align:middle line:84% And that tends to generate inductively loaded power factor. 00:50:34.940 --> 00:50:36.860 align:middle line:84% So this isn't so much the distortion factor. 00:50:36.860 --> 00:50:38.240 align:middle line:90% This is displacement factor. 00:50:38.240 --> 00:50:40.730 align:middle line:84% The interesting thing, you can fix that one. 00:50:40.730 --> 00:50:44.530 align:middle line:84% If I put some capacitor in parallel with the line, 00:50:44.530 --> 00:50:46.910 align:middle line:90% I could source. 00:50:46.910 --> 00:50:49.030 align:middle line:84% I could kind of cancel out the inductive element 00:50:49.030 --> 00:50:52.520 align:middle line:84% and clean up my power factor and make it very close to 1. 00:50:52.520 --> 00:50:54.910 align:middle line:84% So if your problem is phase shift, 00:50:54.910 --> 00:50:57.670 align:middle line:84% you can fix it by adding some reactants to the line 00:50:57.670 --> 00:51:00.100 align:middle line:90% or susceptance to the line. 00:51:00.100 --> 00:51:02.830 align:middle line:84% If your problem is because you've got a crummy power 00:51:02.830 --> 00:51:05.170 align:middle line:84% converter distorting things, that's much harder to fix, 00:51:05.170 --> 00:51:07.540 align:middle line:84% and you've got to fix your power converter typically, 00:51:07.540 --> 00:51:09.940 align:middle line:84% although there are other tricks as well. 00:51:09.940 --> 00:51:13.000 align:middle line:84% So that's about it for today on power factor. 00:51:13.000 --> 00:51:15.285 align:middle line:84% Are there any final questions before we wrap up? 00:51:15.285 --> 00:51:18.350 align:middle line:90% 00:51:18.350 --> 00:51:19.190 align:middle line:90% Yeah. 00:51:19.190 --> 00:51:21.465 align:middle line:84% AUDIENCE: Is the power supply coming from the outlet? 00:51:21.465 --> 00:51:22.590 align:middle line:90% DAVID PERREAULT: I'm sorry? 00:51:22.590 --> 00:51:24.802 align:middle line:84% AUDIENCE: Is the power supply coming from the outlet? 00:51:24.802 --> 00:51:26.177 align:middle line:84% DAVID PERREAULT: Oh, is the power 00:51:26.177 --> 00:51:28.010 align:middle line:84% that she was showing coming from the outlet? 00:51:28.010 --> 00:51:28.585 align:middle line:90% Yes, it was. 00:51:28.585 --> 00:51:31.680 align:middle line:84% AUDIENCE: Is it commonplace to not be beyond 20 volts? 00:51:31.680 --> 00:51:34.290 align:middle line:84% DAVID PERREAULT: Ah, so that's a very good question. 00:51:34.290 --> 00:51:38.150 align:middle line:84% In this particular case, we were running this 00:51:38.150 --> 00:51:40.790 align:middle line:84% through a variac, which lets us modulate 00:51:40.790 --> 00:51:42.120 align:middle line:90% the voltage a little bit. 00:51:42.120 --> 00:51:48.200 align:middle line:84% But the answer is, more broadly, typical range of voltage 00:51:48.200 --> 00:51:51.712 align:middle line:84% is minus 15% to plus 10% of the nominal. 00:51:51.712 --> 00:51:54.170 align:middle line:84% And in fact, if you're going to make a laptop power supply, 00:51:54.170 --> 00:51:56.990 align:middle line:84% you've got to make it go down way low because the line can 00:51:56.990 --> 00:51:58.010 align:middle line:90% be all over the place. 00:51:58.010 --> 00:52:00.450 align:middle line:84% The thing is, that's pretty sinusoidal actually. 00:52:00.450 --> 00:52:02.780 align:middle line:84% You'll often see it often clips at the top 00:52:02.780 --> 00:52:07.580 align:middle line:84% with very old-fashioned peak clipping loads. 00:52:07.580 --> 00:52:10.190 align:middle line:84% So you can get a lot of distortion on your voltage too, 00:52:10.190 --> 00:52:12.000 align:middle line:84% even though this is pretty sinusoidal. 00:52:12.000 --> 00:52:17.270 align:middle line:84% So yeah, it's not as accurate as you might wish it to be. 00:52:17.270 --> 00:52:19.540 align:middle line:90% All right, have a great day. 00:52:19.540 --> 00:52:23.000 align:middle line:90%