WEBVTT 00:00:00.000 --> 00:00:00.954 align:middle line:90% 00:00:00.954 --> 00:00:02.385 align:middle line:90% [SQUEAKING] 00:00:02.385 --> 00:00:04.293 align:middle line:90% [RUSTLING] 00:00:04.293 --> 00:00:06.678 align:middle line:90% [CLICKING] 00:00:06.678 --> 00:00:09.237 align:middle line:90% 00:00:09.237 --> 00:00:11.070 align:middle line:84% DAVID PERREAULT: OK, why don't we get going? 00:00:11.070 --> 00:00:14.260 align:middle line:90% 00:00:14.260 --> 00:00:18.040 align:middle line:84% So just as a brief reminder about where 00:00:18.040 --> 00:00:20.100 align:middle line:84% we started last class, we started 00:00:20.100 --> 00:00:24.550 align:middle line:84% to introduce some tools to think about switching circuits 00:00:24.550 --> 00:00:30.490 align:middle line:84% and, in particular, techniques for time domain 00:00:30.490 --> 00:00:32.890 align:middle line:90% analysis of such circuits. 00:00:32.890 --> 00:00:34.390 align:middle line:84% So the first thing we said was we 00:00:34.390 --> 00:00:37.180 align:middle line:84% said if we had diodes in a circuit, 00:00:37.180 --> 00:00:39.760 align:middle line:84% we can't really tell them when to turn on and off. 00:00:39.760 --> 00:00:42.490 align:middle line:84% They turn on and off based on what's going on in the circuit. 00:00:42.490 --> 00:00:45.550 align:middle line:84% So we came up with this notion of the method of assumed states 00:00:45.550 --> 00:00:47.958 align:middle line:84% to instantaneously figure out what's 00:00:47.958 --> 00:00:50.000 align:middle line:84% going on with an individual diode of the circuit. 00:00:50.000 --> 00:00:52.900 align:middle line:84% And essentially, we take a guess, maybe an educated guess, 00:00:52.900 --> 00:00:55.690 align:middle line:84% and set up the diode states to be on or off 00:00:55.690 --> 00:00:58.370 align:middle line:84% and replace them by open or short circuits. 00:00:58.370 --> 00:01:01.280 align:middle line:84% Then we calculate the voltages and currents in the circuits. 00:01:01.280 --> 00:01:03.770 align:middle line:84% Then we check if the switch conditions are violated. 00:01:03.770 --> 00:01:07.520 align:middle line:84% That is, we say, Does a diode have a forward positive voltage 00:01:07.520 --> 00:01:09.930 align:middle line:84% when I think it's off? which we know it can't. 00:01:09.930 --> 00:01:13.490 align:middle line:84% Or does it have a negative current when it's on? which we 00:01:13.490 --> 00:01:14.498 align:middle line:90% know it can't? 00:01:14.498 --> 00:01:16.790 align:middle line:84% And if we found out that one of those switch conditions 00:01:16.790 --> 00:01:19.470 align:middle line:84% violated, it's time to make a new assumption. 00:01:19.470 --> 00:01:23.450 align:middle line:84% Otherwise, we were happy with the state, and we keep going. 00:01:23.450 --> 00:01:26.240 align:middle line:84% And we work this through time to figure out 00:01:26.240 --> 00:01:28.730 align:middle line:84% what's happening as devices are turning on and off 00:01:28.730 --> 00:01:31.840 align:middle line:90% over some switching cycle. 00:01:31.840 --> 00:01:35.590 align:middle line:84% The other idea we came up with last class 00:01:35.590 --> 00:01:38.860 align:middle line:84% was this notion of periodic steady state, which 00:01:38.860 --> 00:01:42.010 align:middle line:84% is like sinusoidal steady state, except we don't 00:01:42.010 --> 00:01:43.280 align:middle line:90% have sinusoidal waveforms. 00:01:43.280 --> 00:01:47.380 align:middle line:84% It's basically, in each cycle that we're operating, 00:01:47.380 --> 00:01:49.750 align:middle line:84% the waveforms just are repeats of one another. 00:01:49.750 --> 00:01:54.100 align:middle line:84% It's settled down to this perfectly repetitive operation. 00:01:54.100 --> 00:01:57.880 align:middle line:84% And that's often-- not all power circuits operate that way, 00:01:57.880 --> 00:02:00.250 align:middle line:90% but mostly they do. 00:02:00.250 --> 00:02:02.710 align:middle line:84% And one of the advantages of thinking 00:02:02.710 --> 00:02:04.158 align:middle line:84% about that mode of operation when 00:02:04.158 --> 00:02:05.950 align:middle line:84% it's running in steady state is because you 00:02:05.950 --> 00:02:09.013 align:middle line:84% can start to say a lot about the circuit waveforms. 00:02:09.013 --> 00:02:11.680 align:middle line:84% And that can help you figure out what the circuit is going to do 00:02:11.680 --> 00:02:14.330 align:middle line:84% or what it's capable of in periodic steady state. 00:02:14.330 --> 00:02:16.660 align:middle line:84% So, for example, what we found-- if we're 00:02:16.660 --> 00:02:23.590 align:middle line:84% in periodic steady state, suppose x is a voltage. 00:02:23.590 --> 00:02:28.030 align:middle line:84% We know that the average rate of change of this voltage 00:02:28.030 --> 00:02:30.140 align:middle line:84% is 0 because it's just going up and down. 00:02:30.140 --> 00:02:31.895 align:middle line:84% And that means if it was a capacitor, 00:02:31.895 --> 00:02:33.770 align:middle line:84% because the average rate of change of voltage 00:02:33.770 --> 00:02:36.590 align:middle line:84% is 0, the average current through a capacitor 00:02:36.590 --> 00:02:38.490 align:middle line:90% in periodic steady state 0. 00:02:38.490 --> 00:02:41.990 align:middle line:84% Likewise, the average voltage across an inductor 00:02:41.990 --> 00:02:43.890 align:middle line:90% in periodic steady state is 0. 00:02:43.890 --> 00:02:48.500 align:middle line:84% So we can start to identify certain characteristics 00:02:48.500 --> 00:02:51.840 align:middle line:84% of circuit operation when it's in periodic steady state. 00:02:51.840 --> 00:02:53.832 align:middle line:84% And we'll elaborate on this further 00:02:53.832 --> 00:02:55.790 align:middle line:84% in some upcoming lectures when we start talking 00:02:55.790 --> 00:02:57.620 align:middle line:90% about converter analysis. 00:02:57.620 --> 00:03:00.770 align:middle line:84% So that's just a lightning review 00:03:00.770 --> 00:03:02.520 align:middle line:84% of what we talked about last class. 00:03:02.520 --> 00:03:04.180 align:middle line:90% Any questions before we move on? 00:03:04.180 --> 00:03:08.520 align:middle line:90% 00:03:08.520 --> 00:03:12.600 align:middle line:84% I'd like to pick up from where we were last class. 00:03:12.600 --> 00:03:15.570 align:middle line:84% As examples of device circuits with diodes in them, 00:03:15.570 --> 00:03:17.160 align:middle line:84% we started talking about rectifiers, 00:03:17.160 --> 00:03:21.100 align:middle line:84% and we introduced this notion of a half wave rectifier. 00:03:21.100 --> 00:03:24.540 align:middle line:84% And I'd like to elaborate on that a little bit. 00:03:24.540 --> 00:03:29.580 align:middle line:84% So here's the circuit we introduced. 00:03:29.580 --> 00:03:35.895 align:middle line:84% We had a sinusoidal voltage source, Vs sine omega t. 00:03:35.895 --> 00:03:38.470 align:middle line:90% 00:03:38.470 --> 00:03:42.840 align:middle line:84% We had a first diode that I'll call D1. 00:03:42.840 --> 00:03:48.330 align:middle line:84% And maybe I'll call this VD1, and I'll call this I1. 00:03:48.330 --> 00:03:49.775 align:middle line:90% I have a second diode. 00:03:49.775 --> 00:03:53.700 align:middle line:90% 00:03:53.700 --> 00:04:00.210 align:middle line:90% We'll call this Vx and I2. 00:04:00.210 --> 00:04:06.570 align:middle line:84% And then we have an output filter and an actual output 00:04:06.570 --> 00:04:10.920 align:middle line:90% that we'll call, say, V sub d. 00:04:10.920 --> 00:04:16.380 align:middle line:84% And so maybe we will think about this voltage 00:04:16.380 --> 00:04:21.390 align:middle line:84% as being something we studied what it was doing, Vx. 00:04:21.390 --> 00:04:26.820 align:middle line:84% And maybe we're interested in I2, 00:04:26.820 --> 00:04:34.640 align:middle line:84% the current through this diode, and perhaps I1, the current 00:04:34.640 --> 00:04:37.160 align:middle line:90% through this diode. 00:04:37.160 --> 00:04:39.610 align:middle line:84% And what did those waveforms look like? 00:04:39.610 --> 00:04:42.830 align:middle line:90% 00:04:42.830 --> 00:04:46.710 align:middle line:84% Well, we did our method of assumed states analysis. 00:04:46.710 --> 00:04:51.515 align:middle line:84% We said, OK, here's my voltage waveform, my sinusoidal voltage 00:04:51.515 --> 00:04:52.015 align:middle line:90% waveform. 00:04:52.015 --> 00:04:56.630 align:middle line:90% 00:04:56.630 --> 00:05:01.100 align:middle line:84% So I'm going to plot this versus electrical angle, omega t. 00:05:01.100 --> 00:05:04.040 align:middle line:90% So this is Vs sine omega t. 00:05:04.040 --> 00:05:08.430 align:middle line:84% And what we found in our analysis is this. 00:05:08.430 --> 00:05:14.720 align:middle line:84% We said once the voltage waveform crosses 0 positive 00:05:14.720 --> 00:05:18.680 align:middle line:84% here, D1 must be on, and D2 must be off, 00:05:18.680 --> 00:05:20.810 align:middle line:84% because otherwise we would violate 00:05:20.810 --> 00:05:22.290 align:middle line:90% the method of assumed states. 00:05:22.290 --> 00:05:26.720 align:middle line:84% If I assume D2 was on, then D1 would have a positive voltage 00:05:26.720 --> 00:05:28.640 align:middle line:84% across, and I know it can't be true. 00:05:28.640 --> 00:05:35.810 align:middle line:84% So what we found is that in this time period, I have D1 on, 00:05:35.810 --> 00:05:41.300 align:middle line:84% and in this time period I have D2 on. 00:05:41.300 --> 00:05:46.640 align:middle line:84% So that meant that our voltage Vx looked like this. 00:05:46.640 --> 00:05:49.925 align:middle line:84% In the positive part of the cycle, when D1 is on, 00:05:49.925 --> 00:05:53.450 align:middle line:84% it basically just tracks the input voltage. 00:05:53.450 --> 00:06:00.530 align:middle line:84% And in the negative half of the cycle, I get 0 because D2 is on. 00:06:00.530 --> 00:06:03.120 align:middle line:90% And this just repeats. 00:06:03.120 --> 00:06:05.700 align:middle line:90% 00:06:05.700 --> 00:06:09.050 align:middle line:84% And then we went further to say, OK, 00:06:09.050 --> 00:06:11.580 align:middle line:90% what is the output voltage Vd? 00:06:11.580 --> 00:06:14.030 align:middle line:84% And what we did was by basically saying 00:06:14.030 --> 00:06:16.160 align:middle line:84% the average voltage across the inductor 00:06:16.160 --> 00:06:19.130 align:middle line:84% has to be 0 in periodic steady state, 00:06:19.130 --> 00:06:23.010 align:middle line:84% we know that the average voltage of Vx is the average voltage Vd. 00:06:23.010 --> 00:06:25.400 align:middle line:84% If I assume that inductor is really big 00:06:25.400 --> 00:06:28.640 align:middle line:84% and the ripple current is really small, 00:06:28.640 --> 00:06:31.910 align:middle line:84% the output voltage is roughly constant, and the value 00:06:31.910 --> 00:06:34.520 align:middle line:90% it's going to take on-- 00:06:34.520 --> 00:06:37.880 align:middle line:84% by analyzing that, we said the average value of Vd 00:06:37.880 --> 00:06:40.850 align:middle line:84% is simply going to be equal to Vs over pi. 00:06:40.850 --> 00:06:44.180 align:middle line:84% Now, what's happening practically 00:06:44.180 --> 00:06:57.020 align:middle line:84% is let me imagine that this current ID is almost constant, 00:06:57.020 --> 00:06:59.023 align:middle line:84% This is a really big filter inductor. 00:06:59.023 --> 00:07:00.440 align:middle line:84% And we often do that when we start 00:07:00.440 --> 00:07:01.607 align:middle line:90% to analyze power converters. 00:07:01.607 --> 00:07:04.065 align:middle line:84% Let's start by assuming we get great filtering, 00:07:04.065 --> 00:07:05.690 align:middle line:84% and then we can figure out what happens 00:07:05.690 --> 00:07:07.020 align:middle line:90% when we back off from that. 00:07:07.020 --> 00:07:10.380 align:middle line:84% So let's assume that this current is almost constant. 00:07:10.380 --> 00:07:16.240 align:middle line:84% So maybe I would plot ID looking something like this. 00:07:16.240 --> 00:07:20.700 align:middle line:90% 00:07:20.700 --> 00:07:23.200 align:middle line:84% Maybe in reality it has some ripple in it. 00:07:23.200 --> 00:07:24.460 align:middle line:90% And what happens? 00:07:24.460 --> 00:07:27.810 align:middle line:84% Well, in the first part of the cycle, D1 is on. 00:07:27.810 --> 00:07:31.420 align:middle line:84% So I get a current, I1, that does this. 00:07:31.420 --> 00:07:35.970 align:middle line:84% In the first part of-- in the first half of the cycle, sorry-- 00:07:35.970 --> 00:07:41.460 align:middle line:84% In the first half of the cycle, this is my input current. 00:07:41.460 --> 00:07:42.220 align:middle line:90% So this is I1. 00:07:42.220 --> 00:07:44.820 align:middle line:90% 00:07:44.820 --> 00:07:47.070 align:middle line:84% So the diode 1 carries the current. 00:07:47.070 --> 00:07:50.100 align:middle line:84% In the second part of the cycle, diode 2 00:07:50.100 --> 00:07:54.020 align:middle line:90% carries the output current. 00:07:54.020 --> 00:07:57.790 align:middle line:90% So here's like this. 00:07:57.790 --> 00:08:00.630 align:middle line:84% And so they just alternate carrying 00:08:00.630 --> 00:08:03.810 align:middle line:90% current for half the cycle. 00:08:03.810 --> 00:08:06.270 align:middle line:84% First diode 1 carries all the current, then diode 2, 00:08:06.270 --> 00:08:08.070 align:middle line:90% then diode 1, and so forth. 00:08:08.070 --> 00:08:10.770 align:middle line:84% And that's the conclusion we came to about what's 00:08:10.770 --> 00:08:12.100 align:middle line:90% happening with the circuit. 00:08:12.100 --> 00:08:15.960 align:middle line:84% So basically, in the first half of the-- first half 00:08:15.960 --> 00:08:19.410 align:middle line:84% of the cycle, D1 carries the current. 00:08:19.410 --> 00:08:21.740 align:middle line:84% And then immediately the current switches 00:08:21.740 --> 00:08:23.490 align:middle line:84% to diode 2 in the second half of the cycle 00:08:23.490 --> 00:08:25.660 align:middle line:84% and just keeps flopping back and forth, 00:08:25.660 --> 00:08:26.990 align:middle line:90% commutating back and forth. 00:08:26.990 --> 00:08:32.030 align:middle line:90% 00:08:32.030 --> 00:08:37.669 align:middle line:84% Let's now start to elaborate on that a little bit. 00:08:37.669 --> 00:08:42.200 align:middle line:84% And I'd like to consider a specific case. 00:08:42.200 --> 00:08:44.660 align:middle line:84% I'd like to expand this circuit that we're 00:08:44.660 --> 00:08:46.840 align:middle line:90% considering a little bit. 00:08:46.840 --> 00:08:50.590 align:middle line:84% And what I would like to do is add a little bit of inductance 00:08:50.590 --> 00:08:52.550 align:middle line:90% on the AC side. 00:08:52.550 --> 00:08:55.750 align:middle line:84% So instead of having a perfect sinusoidal voltage 00:08:55.750 --> 00:08:58.420 align:middle line:90% source like this-- 00:08:58.420 --> 00:09:00.285 align:middle line:90% here's Vs sine omega t again-- 00:09:00.285 --> 00:09:03.920 align:middle line:90% 00:09:03.920 --> 00:09:09.670 align:middle line:84% I am going to include an inductor 00:09:09.670 --> 00:09:12.570 align:middle line:90% that I'm going to call LC. 00:09:12.570 --> 00:09:18.510 align:middle line:84% And then I'm going to connect this to my rectifier 00:09:18.510 --> 00:09:19.570 align:middle line:90% that I had before. 00:09:19.570 --> 00:09:22.740 align:middle line:90% So here is again D1. 00:09:22.740 --> 00:09:27.090 align:middle line:90% Here's again D2. 00:09:27.090 --> 00:09:28.940 align:middle line:90% And here's my output filter. 00:09:28.940 --> 00:09:32.750 align:middle line:90% 00:09:32.750 --> 00:09:36.160 align:middle line:84% And I have the same kind of waveforms I had before. 00:09:36.160 --> 00:09:47.970 align:middle line:90% I have Vx, I2. 00:09:47.970 --> 00:09:52.740 align:middle line:84% I have I1, which is now this current, I1. 00:09:52.740 --> 00:09:54.940 align:middle line:84% And then I have my output voltage. 00:09:54.940 --> 00:09:56.230 align:middle line:90% I have ID and Vd. 00:09:56.230 --> 00:10:01.160 align:middle line:90% 00:10:01.160 --> 00:10:05.000 align:middle line:84% So why would I go to the effort of introducing 00:10:05.000 --> 00:10:09.410 align:middle line:84% this particular variation on the theme? 00:10:09.410 --> 00:10:12.890 align:middle line:84% And the answer is because this often happens. 00:10:12.890 --> 00:10:14.270 align:middle line:90% Where do we see-- 00:10:14.270 --> 00:10:17.810 align:middle line:84% from here on what I have is a rectifier. 00:10:17.810 --> 00:10:21.770 align:middle line:84% We saw that basically I took a perfect AC voltage, 00:10:21.770 --> 00:10:24.150 align:middle line:84% and then I turned it into something with a DC component, 00:10:24.150 --> 00:10:26.930 align:middle line:84% and I filtered it, and I extracted the DC. 00:10:26.930 --> 00:10:29.960 align:middle line:84% Well, we do that all the time in all kinds of things. 00:10:29.960 --> 00:10:32.513 align:middle line:84% If you're plugging your laptop adapter into the wall, 00:10:32.513 --> 00:10:34.430 align:middle line:84% at the front end of that thing is a rectifier, 00:10:34.430 --> 00:10:38.280 align:middle line:90% which is converting AC to DC. 00:10:38.280 --> 00:10:41.670 align:middle line:84% All kinds of other applications have that. 00:10:41.670 --> 00:10:45.000 align:middle line:84% So if you have-- in fact, it's true for the wall. 00:10:45.000 --> 00:10:47.790 align:middle line:84% It's also true if I build-- what we'll see later in class, if I 00:10:47.790 --> 00:10:50.820 align:middle line:84% build a high-frequency, resonant DC-to-DC converter, what I do 00:10:50.820 --> 00:10:52.650 align:middle line:84% is I create really high frequency AC 00:10:52.650 --> 00:10:55.260 align:middle line:84% signals like a megahertz, run them through a transformer, 00:10:55.260 --> 00:10:59.040 align:middle line:84% and then rectify them in some kind of rectifier like that. 00:10:59.040 --> 00:11:03.250 align:middle line:84% It doesn't have to be that exact rectifier, but you do it. 00:11:03.250 --> 00:11:08.410 align:middle line:84% In both of those cases, you may have inductance on the AC side. 00:11:08.410 --> 00:11:08.910 align:middle line:90% Well, why? 00:11:08.910 --> 00:11:11.670 align:middle line:84% If I'm plugging into the wall, I've got all the wall wiring 00:11:11.670 --> 00:11:12.540 align:middle line:90% going back. 00:11:12.540 --> 00:11:14.610 align:middle line:84% And then somewhere back behind there 00:11:14.610 --> 00:11:16.290 align:middle line:84% is a transformer, which has what's 00:11:16.290 --> 00:11:17.950 align:middle line:90% known as leakage inductance. 00:11:17.950 --> 00:11:21.060 align:middle line:84% So generally, you're not looking back 00:11:21.060 --> 00:11:22.732 align:middle line:84% into a perfect voltage source when 00:11:22.732 --> 00:11:23.940 align:middle line:90% you're looking into the wall. 00:11:23.940 --> 00:11:25.410 align:middle line:90% There's some source impedance. 00:11:25.410 --> 00:11:27.720 align:middle line:90% And that's typically inductive. 00:11:27.720 --> 00:11:32.250 align:middle line:84% Likewise, in the high-frequency converter I mentioned, 00:11:32.250 --> 00:11:33.690 align:middle line:90% it has a transformer in it. 00:11:33.690 --> 00:11:36.150 align:middle line:84% And that transformer has some parasitic inductance. 00:11:36.150 --> 00:11:37.750 align:middle line:90% And you're stuck with that. 00:11:37.750 --> 00:11:40.990 align:middle line:84% And so you're stuck with what that does to you. 00:11:40.990 --> 00:11:44.050 align:middle line:84% The last example I'll show you where this kind of thing 00:11:44.050 --> 00:11:46.520 align:middle line:84% turns up is in any kind of generator. 00:11:46.520 --> 00:11:48.740 align:middle line:84% So this is an automotive alternator. 00:11:48.740 --> 00:11:52.150 align:middle line:84% So what it does is it has an electromagnet that essentially 00:11:52.150 --> 00:11:55.300 align:middle line:84% spins inside a set of coils, generates a big voltage. 00:11:55.300 --> 00:11:57.830 align:middle line:84% And then there's a set of rectifiers. 00:11:57.830 --> 00:11:59.740 align:middle line:90% These are the diodes right here. 00:11:59.740 --> 00:12:02.710 align:middle line:90% They turn that AC back into DC. 00:12:02.710 --> 00:12:04.390 align:middle line:84% So it goes mechanical energy input 00:12:04.390 --> 00:12:07.100 align:middle line:84% to AC voltage back to DC voltage. 00:12:07.100 --> 00:12:10.570 align:middle line:84% And that's what charges the battery in your car. 00:12:10.570 --> 00:12:12.250 align:middle line:84% Now, this is a three-phase device 00:12:12.250 --> 00:12:13.750 align:middle line:84% instead of a single phase, and we'll 00:12:13.750 --> 00:12:15.110 align:middle line:90% talk about three-phase later. 00:12:15.110 --> 00:12:17.540 align:middle line:84% But the basic concept is exactly the same. 00:12:17.540 --> 00:12:21.820 align:middle line:84% And the challenge in here is you have all these coils inside, 00:12:21.820 --> 00:12:24.020 align:middle line:84% and, again, there's leakage inductance. 00:12:24.020 --> 00:12:27.490 align:middle line:84% So you're stuck with extra inductance on the AC side 00:12:27.490 --> 00:12:28.670 align:middle line:90% that you don't want. 00:12:28.670 --> 00:12:32.890 align:middle line:84% So in practice, if I'm going to rectify, I often-- 00:12:32.890 --> 00:12:34.360 align:middle line:90% not always, but often-- 00:12:34.360 --> 00:12:36.683 align:middle line:84% there's some inductance here, whether I like it or not. 00:12:36.683 --> 00:12:39.225 align:middle line:84% And so I better know what that inductance does to my circuit. 00:12:39.225 --> 00:12:42.026 align:middle line:90% 00:12:42.026 --> 00:12:47.190 align:middle line:84% So it's important as a practical matter, 00:12:47.190 --> 00:12:52.410 align:middle line:84% but it's also kind of useful to look at from understanding how 00:12:52.410 --> 00:12:54.750 align:middle line:90% to use these analysis tools. 00:12:54.750 --> 00:12:56.165 align:middle line:90% Any questions about that? 00:12:56.165 --> 00:12:59.280 align:middle line:90% 00:12:59.280 --> 00:13:04.530 align:middle line:84% OK, so let's think about how I might model this. 00:13:04.530 --> 00:13:06.030 align:middle line:90% Now, when I think about it-- 00:13:06.030 --> 00:13:09.610 align:middle line:84% maybe I'll call this inductor LD. 00:13:09.610 --> 00:13:15.480 align:middle line:84% Typically, LD is much, much bigger than LC. 00:13:15.480 --> 00:13:18.700 align:middle line:84% This is a parasitic inductance that I'm sort of stuck with. 00:13:18.700 --> 00:13:19.953 align:middle line:90% This is a big filter inductor. 00:13:19.953 --> 00:13:22.620 align:middle line:84% And for the moment, I'm actually going to think about this thing 00:13:22.620 --> 00:13:24.300 align:middle line:90% as being close to infinite. 00:13:24.300 --> 00:13:28.050 align:middle line:84% So let's just remember that this is kind of a smallish inductance 00:13:28.050 --> 00:13:29.290 align:middle line:90% that I'm stuck with. 00:13:29.290 --> 00:13:31.565 align:middle line:84% This is a really big filter inductor usually. 00:13:31.565 --> 00:13:34.225 align:middle line:90% 00:13:34.225 --> 00:13:38.930 align:middle line:84% And what we can do in order to analyze this circuit is maybe 00:13:38.930 --> 00:13:43.660 align:middle line:84% I'll make a simplification just for us going through it. 00:13:43.660 --> 00:13:45.660 align:middle line:84% And I sort of already did it over there 00:13:45.660 --> 00:13:47.322 align:middle line:90% and didn't really mention it. 00:13:47.322 --> 00:13:49.530 align:middle line:84% Maybe what I'll do is I'll come in here and I'll say, 00:13:49.530 --> 00:13:54.570 align:middle line:84% OK, here's my voltage, Vs sine omega t. 00:13:54.570 --> 00:13:57.090 align:middle line:90% 00:13:57.090 --> 00:13:59.970 align:middle line:84% Here's my leakage inductance, or I 00:13:59.970 --> 00:14:03.600 align:middle line:84% call it L sub C for Commutating inductance for reasons 00:14:03.600 --> 00:14:04.660 align:middle line:90% we'll discuss. 00:14:04.660 --> 00:14:07.260 align:middle line:90% Here's my diode D1. 00:14:07.260 --> 00:14:08.520 align:middle line:90% And here's my diode D2. 00:14:08.520 --> 00:14:12.320 align:middle line:90% 00:14:12.320 --> 00:14:15.800 align:middle line:84% And I'm just going to replace this whole second piece 00:14:15.800 --> 00:14:17.605 align:middle line:90% with a constant current ID. 00:14:17.605 --> 00:14:20.850 align:middle line:90% 00:14:20.850 --> 00:14:22.560 align:middle line:84% So I'm going to assume this filter 00:14:22.560 --> 00:14:25.290 align:middle line:84% inductor is so big that I can ignore 00:14:25.290 --> 00:14:28.640 align:middle line:90% the ripple in that current. 00:14:28.640 --> 00:14:31.010 align:middle line:84% So I can think of the load of this thing 00:14:31.010 --> 00:14:32.583 align:middle line:90% as being a current source. 00:14:32.583 --> 00:14:34.250 align:middle line:84% Now, it's a very special current source. 00:14:34.250 --> 00:14:39.170 align:middle line:84% The question is, what is the value of that current? 00:14:39.170 --> 00:14:42.650 align:middle line:84% Well, in reality, what it is, is it's-- 00:14:42.650 --> 00:14:47.083 align:middle line:84% the average value of Vx is the same thing as VD divided by R 00:14:47.083 --> 00:14:48.000 align:middle line:90% would be this current. 00:14:48.000 --> 00:14:52.160 align:middle line:84% So this ID is actually equal to the average value of Vx divided 00:14:52.160 --> 00:14:54.185 align:middle line:90% by R will be that current. 00:14:54.185 --> 00:14:56.310 align:middle line:84% So it's kind of a funny independent current source. 00:14:56.310 --> 00:14:59.010 align:middle line:84% It actually depends upon this average value. 00:14:59.010 --> 00:15:01.860 align:middle line:84% But we're analyzing the periodic steady state case. 00:15:01.860 --> 00:15:03.290 align:middle line:90% It has some value. 00:15:03.290 --> 00:15:06.210 align:middle line:84% Let's just call it constant at that value. 00:15:06.210 --> 00:15:07.625 align:middle line:90% Any questions about that? 00:15:07.625 --> 00:15:10.980 align:middle line:90% 00:15:10.980 --> 00:15:14.780 align:middle line:84% So what do I have in this circuit? 00:15:14.780 --> 00:15:19.340 align:middle line:84% Well, what I want to do is go through and analyze 00:15:19.340 --> 00:15:21.800 align:middle line:84% the behavior of this circuit over time, just the way 00:15:21.800 --> 00:15:24.080 align:middle line:90% we did before. 00:15:24.080 --> 00:15:30.580 align:middle line:84% And if I think about it, I have two diodes. 00:15:30.580 --> 00:15:31.970 align:middle line:90% There's four possible states. 00:15:31.970 --> 00:15:35.170 align:middle line:90% 00:15:35.170 --> 00:15:39.650 align:middle line:84% One possible state is that both diodes are off. 00:15:39.650 --> 00:15:43.570 align:middle line:84% But because I have a current here, both diodes can't be off. 00:15:43.570 --> 00:15:47.770 align:middle line:84% Another possible state would be both diodes on. 00:15:47.770 --> 00:15:51.840 align:middle line:84% A third possible state would be D1 on and D2 off. 00:15:51.840 --> 00:15:55.330 align:middle line:84% And a fourth possible state would be D2 on and D1 off. 00:15:55.330 --> 00:15:57.640 align:middle line:84% That's our universe of possibilities 00:15:57.640 --> 00:15:58.960 align:middle line:90% in analyzing this thing. 00:15:58.960 --> 00:16:02.080 align:middle line:84% So let's go through and analyze and see 00:16:02.080 --> 00:16:04.940 align:middle line:90% what we get as we go along. 00:16:04.940 --> 00:16:08.890 align:middle line:90% OK, so here's the waveforms. 00:16:08.890 --> 00:16:10.615 align:middle line:84% I'm going to use the same color coding. 00:16:10.615 --> 00:16:14.340 align:middle line:90% 00:16:14.340 --> 00:16:15.770 align:middle line:90% Here's my voltage waveform. 00:16:15.770 --> 00:16:20.650 align:middle line:90% 00:16:20.650 --> 00:16:25.080 align:middle line:90% And here is my current. 00:16:25.080 --> 00:16:26.470 align:middle line:90% So here's omega t. 00:16:26.470 --> 00:16:31.630 align:middle line:84% And I'm just going to assume my load current rid is constant. 00:16:31.630 --> 00:16:38.470 align:middle line:84% All right, so what did I have at just before t equals 0? 00:16:38.470 --> 00:16:45.400 align:middle line:84% Over here, just before t equals 0, D2 was on. 00:16:45.400 --> 00:16:50.950 align:middle line:84% So let me just assume that coming up to this point here, 00:16:50.950 --> 00:16:52.080 align:middle line:90% D2 is on. 00:16:52.080 --> 00:16:55.145 align:middle line:90% 00:16:55.145 --> 00:16:58.770 align:middle line:84% So when D2 is on, let's see, what do we know? 00:16:58.770 --> 00:17:00.820 align:middle line:90% Vx is 0. 00:17:00.820 --> 00:17:02.010 align:middle line:90% Vx looks like this. 00:17:02.010 --> 00:17:07.470 align:middle line:90% 00:17:07.470 --> 00:17:10.329 align:middle line:84% Since D2 is on, D2 is carrying all the current, 00:17:10.329 --> 00:17:12.950 align:middle line:90% so it looks like this. 00:17:12.950 --> 00:17:16.637 align:middle line:90% Here's I2. 00:17:16.637 --> 00:17:22.290 align:middle line:84% And the question happens, what happens now when 00:17:22.290 --> 00:17:24.690 align:middle line:90% the voltage crosses 0? 00:17:24.690 --> 00:17:28.440 align:middle line:84% OK, well, let's just draw that out again. 00:17:28.440 --> 00:17:32.520 align:middle line:84% So we said, all right, what I'm going 00:17:32.520 --> 00:17:37.665 align:middle line:84% to have is this is my method of assumed states. 00:17:37.665 --> 00:17:41.130 align:middle line:90% 00:17:41.130 --> 00:17:44.440 align:middle line:90% I had this device open. 00:17:44.440 --> 00:17:47.720 align:middle line:90% This device is a short circuit. 00:17:47.720 --> 00:17:51.881 align:middle line:90% And here's my load current here. 00:17:51.881 --> 00:17:56.240 align:middle line:90% And this is VD1. 00:17:56.240 --> 00:17:59.870 align:middle line:90% This is VD2. 00:17:59.870 --> 00:18:01.220 align:middle line:90% And this is Vx. 00:18:01.220 --> 00:18:06.590 align:middle line:90% 00:18:06.590 --> 00:18:09.530 align:middle line:84% And it would help if I included my commutating reactants 00:18:09.530 --> 00:18:10.790 align:middle line:90% in here. 00:18:10.790 --> 00:18:13.830 align:middle line:90% Here we go. 00:18:13.830 --> 00:18:14.880 align:middle line:90% LC. 00:18:14.880 --> 00:18:17.610 align:middle line:84% And I should add in my current, I1. 00:18:17.610 --> 00:18:22.060 align:middle line:90% 00:18:22.060 --> 00:18:29.020 align:middle line:84% OK, so what happens when Vs sine omega t crosses 0 positive? 00:18:29.020 --> 00:18:33.790 align:middle line:84% Well, what happened over there was D1 had to turn on. 00:18:33.790 --> 00:18:38.080 align:middle line:84% And I'm going to argue that D1 still has to turn on. 00:18:38.080 --> 00:18:39.200 align:middle line:90% Why is that? 00:18:39.200 --> 00:18:43.930 align:middle line:84% Because if I assume he's off, there's no current through here. 00:18:43.930 --> 00:18:46.300 align:middle line:84% There's no dI/dt in this inductor, 00:18:46.300 --> 00:18:48.860 align:middle line:84% and hence there's no voltage across that inductor. 00:18:48.860 --> 00:18:50.680 align:middle line:84% So what that means is essentially 00:18:50.680 --> 00:18:55.120 align:middle line:84% Vs sine omega t drops through 0 and then appears as VD1. 00:18:55.120 --> 00:18:59.680 align:middle line:84% So basically, as soon as this voltage crosses 0 positive, 00:18:59.680 --> 00:19:02.660 align:middle line:84% the diode 1 voltage would go positive. 00:19:02.660 --> 00:19:06.910 align:middle line:84% And we know that diode voltages can't be positive. 00:19:06.910 --> 00:19:10.135 align:middle line:84% So basically, as soon as this voltage crosses 0 positive, 00:19:10.135 --> 00:19:11.760 align:middle line:84% this diode is going to have to turn on. 00:19:11.760 --> 00:19:14.692 align:middle line:90% 00:19:14.692 --> 00:19:16.300 align:middle line:90% But what do I know here? 00:19:16.300 --> 00:19:20.170 align:middle line:90% So I can say, OK, he turns on. 00:19:20.170 --> 00:19:23.570 align:middle line:90% So now I'll assume he's a short. 00:19:23.570 --> 00:19:26.120 align:middle line:84% In the other circuit or the earlier version, 00:19:26.120 --> 00:19:29.330 align:middle line:90% D2 immediately turned off. 00:19:29.330 --> 00:19:30.655 align:middle line:90% But can that happen here? 00:19:30.655 --> 00:19:34.150 align:middle line:90% 00:19:34.150 --> 00:19:40.070 align:middle line:84% Keep in mind that if I had I1 of 0 minus is equal to 0-- 00:19:40.070 --> 00:19:43.720 align:middle line:84% because he had no current through him. 00:19:43.720 --> 00:19:46.630 align:middle line:90% This diode was open. 00:19:46.630 --> 00:19:52.105 align:middle line:84% Right after t equals 0 plus, well, he's an inductor. 00:19:52.105 --> 00:19:53.480 align:middle line:84% His state doesn't want to change. 00:19:53.480 --> 00:19:55.397 align:middle line:84% Unless you put an infinite voltage across him, 00:19:55.397 --> 00:19:58.840 align:middle line:84% he's still going to have 0 current. 00:19:58.840 --> 00:20:06.390 align:middle line:84% So that must suggest that I1 of 0 plus is still equal to 0. 00:20:06.390 --> 00:20:09.600 align:middle line:84% If I1 of O plus is equal to 0, and this is Id, 00:20:09.600 --> 00:20:14.500 align:middle line:84% that means I still have current going through this device. 00:20:14.500 --> 00:20:17.290 align:middle line:84% I can't make that go open, because there's 00:20:17.290 --> 00:20:18.770 align:middle line:90% no current going through here. 00:20:18.770 --> 00:20:21.910 align:middle line:84% And so this Id has to go somewhere. 00:20:21.910 --> 00:20:25.390 align:middle line:84% So now what I find is I can-- if I 00:20:25.390 --> 00:20:28.210 align:middle line:84% work through that argument, that just says that both 00:20:28.210 --> 00:20:30.630 align:middle line:90% of these devices have to be on. 00:20:30.630 --> 00:20:38.790 align:middle line:84% So I'm going to have a new state where I have both D1 on 00:20:38.790 --> 00:20:40.055 align:middle line:90% and D2 on. 00:20:40.055 --> 00:20:42.795 align:middle line:90% 00:20:42.795 --> 00:20:51.710 align:middle line:84% And essentially what I have is now this circuit is-- 00:20:51.710 --> 00:20:54.300 align:middle line:84% some of the current from Id is going to go through here. 00:20:54.300 --> 00:20:56.750 align:middle line:84% Some of it's going to come from the source. 00:20:56.750 --> 00:21:01.290 align:middle line:84% And this current I1 is going to change over time. 00:21:01.290 --> 00:21:04.460 align:middle line:84% Essentially, the input voltage, Vs sine omega t, 00:21:04.460 --> 00:21:09.090 align:middle line:84% is being impressed across this inductance. 00:21:09.090 --> 00:21:13.090 align:middle line:90% Any questions about that? 00:21:13.090 --> 00:21:14.290 align:middle line:90% So what does that mean? 00:21:14.290 --> 00:21:16.300 align:middle line:90% Well, I can calculate-- 00:21:16.300 --> 00:21:19.254 align:middle line:90% what I get is dI1/dt-- 00:21:19.254 --> 00:21:22.218 align:middle line:90% 00:21:22.218 --> 00:21:27.510 align:middle line:84% L dI1/dt is equal to Vs sine omega t. 00:21:27.510 --> 00:21:30.740 align:middle line:90% 00:21:30.740 --> 00:21:36.390 align:middle line:84% So that means I ought to be able to calculate I1 of t 00:21:36.390 --> 00:21:50.230 align:middle line:84% is equal to Vs over L, the integral of sine omega t dt. 00:21:50.230 --> 00:21:52.940 align:middle line:84% But what I'm going to do is I'm going to change my variables. 00:21:52.940 --> 00:21:56.050 align:middle line:84% Let me change my variables to omega t, 00:21:56.050 --> 00:21:58.210 align:middle line:84% so I'll put an omega here and omega here. 00:21:58.210 --> 00:22:03.430 align:middle line:84% And I'm integrating from 0 to some time omega t. 00:22:03.430 --> 00:22:09.150 align:middle line:84% Maybe I should go back and change this symbol to theta just 00:22:09.150 --> 00:22:10.780 align:middle line:90% to make it a little clearer. 00:22:10.780 --> 00:22:15.430 align:middle line:84% But that's the integral I'm calculating. 00:22:15.430 --> 00:22:17.145 align:middle line:90% So what do I get? 00:22:17.145 --> 00:22:18.520 align:middle line:84% What I'm going to get here is I'm 00:22:18.520 --> 00:22:30.720 align:middle line:84% going to get Vs over omega L times cosine of omega t minus-- 00:22:30.720 --> 00:22:35.000 align:middle line:84% I'm sorry, cosine of 0 minus cosine of omega t and cosine 00:22:35.000 --> 00:22:36.450 align:middle line:90% of 0 is 1. 00:22:36.450 --> 00:22:40.030 align:middle line:84% So this just becomes 1 minus cosine of omega t. 00:22:40.030 --> 00:22:43.226 align:middle line:90% 00:22:43.226 --> 00:22:49.550 align:middle line:84% So now I know what I1 of t or I1 of omega t looks like. 00:22:49.550 --> 00:22:52.330 align:middle line:84% Does that make sense to everybody? 00:22:52.330 --> 00:22:56.750 align:middle line:84% And essentially, right as he crosses 0 positive, 00:22:56.750 --> 00:23:00.930 align:middle line:84% I have no voltage across this because it goes as a sine wave. 00:23:00.930 --> 00:23:02.663 align:middle line:84% But then the voltage starts to increase. 00:23:02.663 --> 00:23:04.330 align:middle line:84% And hence the current starts to increase 00:23:04.330 --> 00:23:06.320 align:middle line:90% as the integral of the sine. 00:23:06.320 --> 00:23:07.960 align:middle line:90% So what's going to happen? 00:23:07.960 --> 00:23:11.985 align:middle line:84% I'm going to get this waveform for I1 one of t. 00:23:11.985 --> 00:23:13.360 align:middle line:84% If I were to plot that over here, 00:23:13.360 --> 00:23:15.170 align:middle line:84% it would look something like this. 00:23:15.170 --> 00:23:17.750 align:middle line:90% It would start off as 0. 00:23:17.750 --> 00:23:22.310 align:middle line:84% And it would come up, and it would keep going up like that. 00:23:22.310 --> 00:23:25.690 align:middle line:90% That make sense to everybody? 00:23:25.690 --> 00:23:27.530 align:middle line:90% And what's I2 of t? 00:23:27.530 --> 00:23:30.670 align:middle line:90% Well, I2 of t-- 00:23:30.670 --> 00:23:35.090 align:middle line:84% I2 is simply equal to Id minus I1. 00:23:35.090 --> 00:23:41.640 align:middle line:84% So basically, I2 started out as Id. 00:23:41.640 --> 00:23:43.650 align:middle line:84% And he does the exact complementary thing, 00:23:43.650 --> 00:23:44.770 align:middle line:90% and he slopes down. 00:23:44.770 --> 00:23:46.770 align:middle line:84% He initially has 0 slope, and he's doing this, 00:23:46.770 --> 00:23:50.431 align:middle line:84% and then he keep going like that. 00:23:50.431 --> 00:23:51.710 align:middle line:90% Any questions about that? 00:23:51.710 --> 00:23:56.190 align:middle line:90% 00:23:56.190 --> 00:23:58.303 align:middle line:84% Well, why did I stop drawing things? 00:23:58.303 --> 00:23:59.970 align:middle line:84% Because keep in mind, we have our method 00:23:59.970 --> 00:24:03.390 align:middle line:90% of assumed states going on. 00:24:03.390 --> 00:24:10.920 align:middle line:84% Once I1 hits Id is exactly when I2 hits 0. 00:24:10.920 --> 00:24:15.240 align:middle line:84% I would say by this calculation that presumably it 00:24:15.240 --> 00:24:17.890 align:middle line:84% would keep going and I2 would keep going negative. 00:24:17.890 --> 00:24:22.200 align:middle line:84% But I2 can't go negative because there's a diode here. 00:24:22.200 --> 00:24:26.310 align:middle line:84% So as soon as I2 hits 0 and tries to go negative, 00:24:26.310 --> 00:24:27.790 align:middle line:90% D2 is going to turn off. 00:24:27.790 --> 00:24:34.350 align:middle line:84% So I'm going to hit some point here, right here, where-- 00:24:34.350 --> 00:24:41.910 align:middle line:84% and I'm going to call this some time u, omega t is equal to u-- 00:24:41.910 --> 00:24:47.070 align:middle line:84% where basically D2 is going to turn off. 00:24:47.070 --> 00:24:49.505 align:middle line:84% And maybe then I'll only have D1 on. 00:24:49.505 --> 00:24:53.000 align:middle line:90% 00:24:53.000 --> 00:24:55.600 align:middle line:90% Does that make sense, everybody? 00:24:55.600 --> 00:24:58.550 align:middle line:84% Because otherwise, he would have a negative current in. 00:24:58.550 --> 00:25:00.410 align:middle line:90% That's my third assumption. 00:25:00.410 --> 00:25:02.890 align:middle line:90% Well, how can I find that time? 00:25:02.890 --> 00:25:06.920 align:middle line:84% I can find that time by a simple integral. 00:25:06.920 --> 00:25:09.240 align:middle line:90% I can say, OK, what do I know? 00:25:09.240 --> 00:25:14.560 align:middle line:90% I know that I1 of t equals Id. 00:25:14.560 --> 00:25:17.260 align:middle line:84% That's when this happens is that. 00:25:17.260 --> 00:25:22.180 align:middle line:84% And that's equal to Vs over omega L times 1 00:25:22.180 --> 00:25:26.380 align:middle line:90% minus cosine of u. 00:25:26.380 --> 00:25:31.870 align:middle line:84% And so I can rearrange this, and I can find cosine of u 00:25:31.870 --> 00:25:34.131 align:middle line:90% is equal to omega-- 00:25:34.131 --> 00:25:37.290 align:middle line:90% 00:25:37.290 --> 00:25:47.870 align:middle line:84% this should be Lc-- omega Lc Id over Vs. I'm sorry, cosine of u 00:25:47.870 --> 00:25:48.743 align:middle line:90% is 1 minus. 00:25:48.743 --> 00:25:56.685 align:middle line:90% 00:25:56.685 --> 00:26:02.520 align:middle line:84% This angle u is what's known as the commutating angle. 00:26:02.520 --> 00:26:09.810 align:middle line:84% And it has units of electrical degrees or electrical radians. 00:26:09.810 --> 00:26:13.300 align:middle line:90% 00:26:13.300 --> 00:26:16.340 align:middle line:84% And why am I saying that's the commutating angle? 00:26:16.340 --> 00:26:19.180 align:middle line:90% Because what's happened here? 00:26:19.180 --> 00:26:22.030 align:middle line:84% If I were to draw what's going to go on now, 00:26:22.030 --> 00:26:26.590 align:middle line:90% I have D1 on, D2 off. 00:26:26.590 --> 00:26:28.790 align:middle line:84% I have a constant current out here 00:26:28.790 --> 00:26:30.760 align:middle line:84% that's now going to go through D1, 00:26:30.760 --> 00:26:35.900 align:middle line:84% and I'm carrying a constant current. 00:26:35.900 --> 00:26:40.520 align:middle line:84% So what happened is in this original circuit break, 00:26:40.520 --> 00:26:41.956 align:middle line:90% I was carrying-- 00:26:41.956 --> 00:26:47.020 align:middle line:90% 00:26:47.020 --> 00:26:49.990 align:middle line:84% diode 2 is carrying the current, then immediately diode 1 00:26:49.990 --> 00:26:52.540 align:middle line:84% switches and carries the current, then diode 2 switches 00:26:52.540 --> 00:26:54.940 align:middle line:84% and carries the current, et cetera. 00:26:54.940 --> 00:26:58.210 align:middle line:84% Once I add inductance here, it doesn't switch instantly, 00:26:58.210 --> 00:27:00.647 align:middle line:84% because it can't, because there's inductance 00:27:00.647 --> 00:27:02.980 align:middle line:84% which doesn't like to have its current change through it 00:27:02.980 --> 00:27:04.000 align:middle line:90% instantly. 00:27:04.000 --> 00:27:08.560 align:middle line:84% So I get some new period in which the current switches 00:27:08.560 --> 00:27:11.590 align:middle line:84% between going through this path and going 00:27:11.590 --> 00:27:15.100 align:middle line:84% through this path, what we would say commutates 00:27:15.100 --> 00:27:17.590 align:middle line:90% from this branch to this branch. 00:27:17.590 --> 00:27:21.940 align:middle line:84% And if I were to do the same of analysis, what I would find is, 00:27:21.940 --> 00:27:25.360 align:middle line:84% guess what, when I hit omega t is equal to pi, 00:27:25.360 --> 00:27:27.200 align:middle line:90% it would do the same thing. 00:27:27.200 --> 00:27:31.240 align:middle line:84% And this current would ramp down like this. 00:27:31.240 --> 00:27:33.370 align:middle line:90% And I2 would ramp-- 00:27:33.370 --> 00:27:35.800 align:middle line:84% I2 wasn't carrying any current in diode 2, 00:27:35.800 --> 00:27:38.140 align:middle line:84% then picks up the current and does this, 00:27:38.140 --> 00:27:39.640 align:middle line:90% and then it switches over. 00:27:39.640 --> 00:27:43.690 align:middle line:84% And this would happen at an angle pi plus u. 00:27:43.690 --> 00:27:49.540 align:middle line:84% So all I get is some finite time in which the current switches 00:27:49.540 --> 00:27:51.430 align:middle line:84% between the two diodes instead of just, 00:27:51.430 --> 00:27:53.640 align:middle line:90% bam, happening instantly. 00:27:53.640 --> 00:27:57.698 align:middle line:90% Any questions about that? 00:27:57.698 --> 00:27:59.240 align:middle line:84% AUDIENCE: So in the previous example, 00:27:59.240 --> 00:28:02.120 align:middle line:84% we just had the filter inductance. 00:28:02.120 --> 00:28:08.205 align:middle line:84% Do we get the same phase shift over the period where D1 is on? 00:28:08.205 --> 00:28:11.130 align:middle line:84% DAVID PERREAULT: So to repeat the question, 00:28:11.130 --> 00:28:14.380 align:middle line:84% the question is, did I get the same phase shift? 00:28:14.380 --> 00:28:15.780 align:middle line:90% And the answer is no. 00:28:15.780 --> 00:28:17.970 align:middle line:84% The difference here isn't the fact 00:28:17.970 --> 00:28:22.740 align:middle line:90% that this is no longer finite. 00:28:22.740 --> 00:28:25.380 align:middle line:84% I actually assumed it was kind of infinite over here too. 00:28:25.380 --> 00:28:28.490 align:middle line:84% The difference is that I have that commutating inductance 00:28:28.490 --> 00:28:31.340 align:middle line:84% in there, and one of the effects of that commutating inductance 00:28:31.340 --> 00:28:34.130 align:middle line:84% is to smear the currents a bit later. 00:28:34.130 --> 00:28:36.780 align:middle line:84% Does that address your question, or did I misunderstand it? 00:28:36.780 --> 00:28:37.240 align:middle line:90% AUDIENCE: No, no. 00:28:37.240 --> 00:28:37.740 align:middle line:90% It was good. 00:28:37.740 --> 00:28:39.320 align:middle line:90% DAVID PERREAULT: OK. 00:28:39.320 --> 00:28:41.600 align:middle line:84% All right, so why am I making a big deal about this? 00:28:41.600 --> 00:28:43.580 align:middle line:90% Why am I even mentioning it? 00:28:43.580 --> 00:28:46.080 align:middle line:84% Yeah, OK, we get another diode state, and it happens. 00:28:46.080 --> 00:28:48.200 align:middle line:84% And that's true, but it turns out 00:28:48.200 --> 00:28:50.900 align:middle line:84% to have reasonably important consequences 00:28:50.900 --> 00:28:52.940 align:middle line:90% in a lot of designs. 00:28:52.940 --> 00:28:54.270 align:middle line:90% Why is that? 00:28:54.270 --> 00:28:56.360 align:middle line:84% Well, let's think about-- let's come back, 00:28:56.360 --> 00:29:01.040 align:middle line:84% and let's look at this voltage Vx here. 00:29:01.040 --> 00:29:02.970 align:middle line:90% Well, what does Vx do? 00:29:02.970 --> 00:29:07.120 align:middle line:84% What is Vx when both D1 and D2 are on? 00:29:07.120 --> 00:29:16.020 align:middle line:90% 00:29:16.020 --> 00:29:19.560 align:middle line:84% Anybody want to take a crack at that? 00:29:19.560 --> 00:29:20.685 align:middle line:90% AUDIENCE: Wouldn't it be 0? 00:29:20.685 --> 00:29:22.840 align:middle line:90% DAVID PERREAULT: Yes, right. 00:29:22.840 --> 00:29:24.040 align:middle line:90% D2 is on. 00:29:24.040 --> 00:29:25.400 align:middle line:90% This is a short circuit. 00:29:25.400 --> 00:29:27.850 align:middle line:90% V2 better be 0. 00:29:27.850 --> 00:29:33.180 align:middle line:84% So before, I got this nice half sine wave bump. 00:29:33.180 --> 00:29:35.910 align:middle line:84% What's going to happen is between 0-- 00:29:35.910 --> 00:29:39.000 align:middle line:84% for the entire time the current's commutating, 00:29:39.000 --> 00:29:41.820 align:middle line:90% I've got 0 here. 00:29:41.820 --> 00:29:48.280 align:middle line:84% Once it's only D1 one on, then Lc has a constant current in it, 00:29:48.280 --> 00:29:51.170 align:middle line:84% and so it has no voltage drop across it. 00:29:51.170 --> 00:29:56.780 align:middle line:84% And then I get the remainder of this. 00:29:56.780 --> 00:29:58.310 align:middle line:84% And the second half of the cycle, it 00:29:58.310 --> 00:30:00.650 align:middle line:84% does the same thing as last time. 00:30:00.650 --> 00:30:04.040 align:middle line:84% So what happens is once I have this commutating 00:30:04.040 --> 00:30:06.770 align:middle line:84% inductance in there or commutating reactance in there, 00:30:06.770 --> 00:30:12.480 align:middle line:84% I take a bite out of my half sine voltage. 00:30:12.480 --> 00:30:17.340 align:middle line:84% And remember, this is the thing that we're averaging 00:30:17.340 --> 00:30:20.240 align:middle line:90% to get our output voltage. 00:30:20.240 --> 00:30:23.790 align:middle line:84% So now let me say, what is the average voltage? 00:30:23.790 --> 00:30:26.720 align:middle line:84% What's the average of Vx, which is the same thing 00:30:26.720 --> 00:30:29.420 align:middle line:90% as the average of Vd? 00:30:29.420 --> 00:30:31.670 align:middle line:90% Well, I can calculate that. 00:30:31.670 --> 00:30:36.590 align:middle line:84% The average value of Vx with commutating reactance 00:30:36.590 --> 00:30:40.820 align:middle line:84% is going to be equal to 1 over 2 pi. 00:30:40.820 --> 00:30:51.500 align:middle line:84% The integral-- from when?-- from some angle u to some angle pi, 00:30:51.500 --> 00:30:57.800 align:middle line:90% Vs cosine of omega t d omega t. 00:30:57.800 --> 00:31:00.230 align:middle line:84% OK, so what does this end up being? 00:31:00.230 --> 00:31:06.003 align:middle line:84% This ends up being 1 over 2 pi or Vs over 2 pi times-- 00:31:06.003 --> 00:31:11.210 align:middle line:90% 00:31:11.210 --> 00:31:15.760 align:middle line:90% oops that should be sine, sorry. 00:31:15.760 --> 00:31:17.750 align:middle line:90% The voltage waveform is a sign. 00:31:17.750 --> 00:31:26.700 align:middle line:84% This should be equal to times cosine of u minus cosine of pi. 00:31:26.700 --> 00:31:29.740 align:middle line:90% 00:31:29.740 --> 00:31:33.330 align:middle line:84% Well, if I work that out, then what's the cosine of u? 00:31:33.330 --> 00:31:34.970 align:middle line:84% Well, conveniently, I have that here. 00:31:34.970 --> 00:31:41.400 align:middle line:84% That's 1 minus omega Lc Id over Vs. What's cosine of pi? 00:31:41.400 --> 00:31:42.660 align:middle line:90% It's minus 1. 00:31:42.660 --> 00:31:48.110 align:middle line:84% So this becomes Vs over 2 pi times 2 00:31:48.110 --> 00:31:54.800 align:middle line:90% minus omega Lc Id over Vs. 00:31:54.800 --> 00:32:01.140 align:middle line:84% So if I rewrote that, what I can say is the average value of Vx 00:32:01.140 --> 00:32:05.980 align:middle line:84% once I filter it is also Vd, the output of the rectifier. 00:32:05.980 --> 00:32:23.890 align:middle line:84% So Vd then, then becomes Vs over pi times 1 minus 1/2 omega Lc Id 00:32:23.890 --> 00:32:27.610 align:middle line:84% over Vs. Sorry, that's a little bit of a mess. 00:32:27.610 --> 00:32:32.290 align:middle line:84% But it should be basically just this. 00:32:32.290 --> 00:32:40.750 align:middle line:84% By the way, this comes up all the time 00:32:40.750 --> 00:32:42.500 align:middle line:84% in these kind of rectifier applications. 00:32:42.500 --> 00:32:47.593 align:middle line:90% This is a unitless expression. 00:32:47.593 --> 00:32:49.760 align:middle line:84% I'm subtracting it from 1, so it better be unitless. 00:32:49.760 --> 00:32:56.890 align:middle line:84% Omega Lc Id over Vs, or we might write it as Xc Id over s 00:32:56.890 --> 00:32:58.725 align:middle line:84% is known as the reactance factor. 00:32:58.725 --> 00:33:05.222 align:middle line:90% 00:33:05.222 --> 00:33:08.960 align:middle line:84% And what it's-- and actually, let me just rewrite this 00:33:08.960 --> 00:33:09.940 align:middle line:90% because it's a mess. 00:33:09.940 --> 00:33:24.300 align:middle line:90% 00:33:24.300 --> 00:33:27.390 align:middle line:90% What it says is this. 00:33:27.390 --> 00:33:30.360 align:middle line:90% Suppose Lc was 0. 00:33:30.360 --> 00:33:32.460 align:middle line:84% This term would be 0, and I'd just 00:33:32.460 --> 00:33:35.070 align:middle line:90% get Vd is equal to Vs over pi. 00:33:35.070 --> 00:33:38.340 align:middle line:84% So that makes sense because my first analysis 00:33:38.340 --> 00:33:40.475 align:middle line:90% was with Lc equals 0. 00:33:40.475 --> 00:33:44.050 align:middle line:90% 00:33:44.050 --> 00:33:51.880 align:middle line:84% But as any of the load current or the commutating inductance 00:33:51.880 --> 00:33:55.300 align:middle line:84% or the frequency increase, this tends 00:33:55.300 --> 00:33:58.760 align:middle line:90% to degrade my output voltage. 00:33:58.760 --> 00:34:03.380 align:middle line:90% So suppose I was-- 00:34:03.380 --> 00:34:07.350 align:middle line:84% I generate this high-frequency voltage and then I rectify it, 00:34:07.350 --> 00:34:08.570 align:middle line:90% expecting to get-- 00:34:08.570 --> 00:34:12.030 align:middle line:84% in this rectifier, it would be Vs over pi as my output voltage. 00:34:12.030 --> 00:34:14.820 align:middle line:84% And suddenly I do that, and what happens? 00:34:14.820 --> 00:34:18.500 align:middle line:84% Well, I get this droop that's load dependent. 00:34:18.500 --> 00:34:24.199 align:middle line:84% So if I was to plot this, what I would see is this. 00:34:24.199 --> 00:34:25.045 align:middle line:90% I would see-- 00:34:25.045 --> 00:34:28.580 align:middle line:90% 00:34:28.580 --> 00:34:29.883 align:middle line:90% Let me plot this versus-- 00:34:29.883 --> 00:34:31.550 align:middle line:84% I could plot it versus different things, 00:34:31.550 --> 00:34:33.230 align:middle line:90% but let me plot this versus Id. 00:34:33.230 --> 00:34:39.080 align:middle line:84% In my original rectifier, if I plotted Vd versus Id, what 00:34:39.080 --> 00:34:42.810 align:middle line:90% I would just get is Vs over pi. 00:34:42.810 --> 00:34:45.179 align:middle line:84% No matter how much current I draw 00:34:45.179 --> 00:34:48.150 align:middle line:84% from this rectifier over here, I'm going to get Vs over pi 00:34:48.150 --> 00:34:49.170 align:middle line:90% at the output. 00:34:49.170 --> 00:34:51.750 align:middle line:84% So I could kind of reliably generate 00:34:51.750 --> 00:34:55.090 align:middle line:84% that AC voltage, rectify it, and I'd get what I want. 00:34:55.090 --> 00:34:58.170 align:middle line:84% And very typically, I'm trying to get a voltage out 00:34:58.170 --> 00:35:00.340 align:middle line:90% of a system. 00:35:00.340 --> 00:35:04.240 align:middle line:84% What this says is that the voltage that I get, 00:35:04.240 --> 00:35:07.880 align:middle line:84% Vd, actually droops versus current. 00:35:07.880 --> 00:35:10.700 align:middle line:84% And in fact, I'm going to get 0 voltage 00:35:10.700 --> 00:35:16.720 align:middle line:84% once the current gets to be equal to 2 Vs over omega Lc. 00:35:16.720 --> 00:35:21.320 align:middle line:90% 00:35:21.320 --> 00:35:24.350 align:middle line:84% So actually, my actual characteristic 00:35:24.350 --> 00:35:31.370 align:middle line:84% looks like this when I have commutating inductance. 00:35:31.370 --> 00:35:34.110 align:middle line:84% If Lc goes to 0, this point moves off to infinity, 00:35:34.110 --> 00:35:36.650 align:middle line:84% and I get the straight line I was hoping for. 00:35:36.650 --> 00:35:43.130 align:middle line:84% This phenomenon is called load regulation, 00:35:43.130 --> 00:35:47.420 align:middle line:84% meaning that my load current somewhat 00:35:47.420 --> 00:35:50.070 align:middle line:90% determines my output voltage. 00:35:50.070 --> 00:35:56.160 align:middle line:84% It's as if I had hoped to create a voltage source, 00:35:56.160 --> 00:35:57.910 align:middle line:84% and I don't quite get a voltage source. 00:35:57.910 --> 00:36:00.540 align:middle line:90% I get a droopy voltage source. 00:36:00.540 --> 00:36:03.580 align:middle line:90% Now, do I care? 00:36:03.580 --> 00:36:07.890 align:middle line:84% Well, if Lc's really small, and for full current, 00:36:07.890 --> 00:36:10.470 align:middle line:84% I only drop 1%, no, I don't care. 00:36:10.470 --> 00:36:14.020 align:middle line:84% If Lc gets too big, however, maybe I have a problem. 00:36:14.020 --> 00:36:17.790 align:middle line:84% In fact, this actually limits the amount of power 00:36:17.790 --> 00:36:19.740 align:middle line:90% I can get out of that source. 00:36:19.740 --> 00:36:23.910 align:middle line:84% Because if you think about it, if I draw too much current, 00:36:23.910 --> 00:36:25.120 align:middle line:90% I get 0 voltage. 00:36:25.120 --> 00:36:29.525 align:middle line:84% So I'm not really getting any power at the output. 00:36:29.525 --> 00:36:31.900 align:middle line:90% And in fact, we once had that. 00:36:31.900 --> 00:36:34.470 align:middle line:84% We designed a really high-frequency resonant power 00:36:34.470 --> 00:36:36.275 align:middle line:84% converter, and we had rated it for, 00:36:36.275 --> 00:36:37.650 align:middle line:84% I don't remember how many it was, 00:36:37.650 --> 00:36:39.280 align:middle line:90% like 500 watts or something. 00:36:39.280 --> 00:36:41.670 align:middle line:84% And what we found out is we couldn't actually 00:36:41.670 --> 00:36:43.960 align:middle line:84% get more than 400 watts out of the thing. 00:36:43.960 --> 00:36:44.820 align:middle line:90% Why? 00:36:44.820 --> 00:36:47.790 align:middle line:84% Because the transformer leakage inductance 00:36:47.790 --> 00:36:50.760 align:middle line:84% was making it droop just enough that it was stealing our ability 00:36:50.760 --> 00:36:53.860 align:middle line:84% to deliver the current we wanted. 00:36:53.860 --> 00:36:56.050 align:middle line:84% Or as we delivered more and more current, 00:36:56.050 --> 00:36:57.050 align:middle line:90% the voltage would droop. 00:36:57.050 --> 00:36:58.675 align:middle line:84% And then I wasn't getting enough power, 00:36:58.675 --> 00:37:01.990 align:middle line:84% so we weren't making our voltage rating anymore. 00:37:01.990 --> 00:37:04.240 align:middle line:84% So this can be a practically important phenomenon, 00:37:04.240 --> 00:37:13.420 align:middle line:84% both for line frequency rectifiers where you're 00:37:13.420 --> 00:37:17.110 align:middle line:84% trying to get power out of the wall or even 00:37:17.110 --> 00:37:21.870 align:middle line:84% a high-frequency megahertz power converter. 00:37:21.870 --> 00:37:24.940 align:middle line:84% Now, first of all, let me stop there 00:37:24.940 --> 00:37:29.850 align:middle line:84% and ask if there's any questions about any of this concept 00:37:29.850 --> 00:37:30.365 align:middle line:90% so far. 00:37:30.365 --> 00:37:35.020 align:middle line:90% 00:37:35.020 --> 00:37:39.161 align:middle line:90% OK, how would I model this? 00:37:39.161 --> 00:37:43.360 align:middle line:84% What would be the DC equivalent circuit model of this thing? 00:37:43.360 --> 00:37:46.810 align:middle line:84% Well, if I thought of-- if I wanted to think about, 00:37:46.810 --> 00:37:55.480 align:middle line:84% for example, everything over here in an average sense, like, 00:37:55.480 --> 00:38:00.420 align:middle line:84% What is the IV characteristic looking back here?-- 00:38:00.420 --> 00:38:07.590 align:middle line:84% well, Vd is also the average of Vx here-- 00:38:07.590 --> 00:38:16.380 align:middle line:84% I might think of this as a voltage source, Vs over pi, 00:38:16.380 --> 00:38:21.930 align:middle line:84% in series with an output resistance. 00:38:21.930 --> 00:38:23.610 align:middle line:84% How big is that output resistance? 00:38:23.610 --> 00:38:31.820 align:middle line:84% It's really Vs over pi divided by 2 Vs over omega Lc. 00:38:31.820 --> 00:38:38.000 align:middle line:84% So what I get is an effective output impedance, 00:38:38.000 --> 00:38:41.090 align:middle line:90% omega Lc over 2 pi. 00:38:41.090 --> 00:38:45.660 align:middle line:90% This has units of resistance. 00:38:45.660 --> 00:38:47.880 align:middle line:90% Omega is reactance. 00:38:47.880 --> 00:38:49.680 align:middle line:90% It has units of ohms. 00:38:49.680 --> 00:38:54.570 align:middle line:84% So omega Lc over 2 pi has the right units to be a resistance. 00:38:54.570 --> 00:38:57.030 align:middle line:84% And then I could connect this thing 00:38:57.030 --> 00:39:02.710 align:middle line:84% to, effectively, my inductive filter or more 00:39:02.710 --> 00:39:06.970 align:middle line:90% simply my load current Id. 00:39:06.970 --> 00:39:15.320 align:middle line:84% So this is not a bad model for how on the AC side-- 00:39:15.320 --> 00:39:17.630 align:middle line:84% looking back from the DC side or looking back 00:39:17.630 --> 00:39:20.000 align:middle line:84% from the side of this filter how this thing 00:39:20.000 --> 00:39:24.040 align:middle line:90% will behave on average. 00:39:24.040 --> 00:39:26.128 align:middle line:90% Why is it drooping? 00:39:26.128 --> 00:39:28.420 align:middle line:84% Why is there this effective output versus-- well, first 00:39:28.420 --> 00:39:31.910 align:middle line:84% of all, this is a DC equivalent model. 00:39:31.910 --> 00:39:34.210 align:middle line:90% There is no loss here. 00:39:34.210 --> 00:39:35.090 align:middle line:90% I think a resistor-- 00:39:35.090 --> 00:39:36.300 align:middle line:90% I think it got lost, right? 00:39:36.300 --> 00:39:39.840 align:middle line:90% I'm not burning power anywhere. 00:39:39.840 --> 00:39:40.940 align:middle line:90% There's no loss. 00:39:40.940 --> 00:39:44.630 align:middle line:84% There's no watts being turned into heat. 00:39:44.630 --> 00:39:49.940 align:middle line:84% What's happening instead is that each time this voltage 00:39:49.940 --> 00:39:53.540 align:middle line:84% swings positive, I've got to take some time 00:39:53.540 --> 00:39:58.246 align:middle line:84% to charge up this inductor to the right current. 00:39:58.246 --> 00:40:01.330 align:middle line:84% And that time eats away from the time 00:40:01.330 --> 00:40:04.270 align:middle line:84% at which I generate my positive pulse, which is rectified 00:40:04.270 --> 00:40:05.600 align:middle line:90% to give me output voltage. 00:40:05.600 --> 00:40:09.370 align:middle line:84% So I'm of eating into my ability to synthesize a DC output 00:40:09.370 --> 00:40:15.730 align:middle line:84% voltage because I have to charge him up. 00:40:15.730 --> 00:40:18.370 align:middle line:84% So if I have a higher frequency, it's 00:40:18.370 --> 00:40:20.260 align:middle line:84% a higher percentage of the overall cycle 00:40:20.260 --> 00:40:22.510 align:middle line:84% because it takes a certain amount of time with voltage 00:40:22.510 --> 00:40:23.360 align:middle line:90% to do this. 00:40:23.360 --> 00:40:26.170 align:middle line:84% Or if I have a bigger inductance, it takes more time. 00:40:26.170 --> 00:40:29.510 align:middle line:84% Or if my voltage is smaller, it takes more time-- 00:40:29.510 --> 00:40:30.890 align:middle line:90% my source voltage is smaller. 00:40:30.890 --> 00:40:36.285 align:middle line:84% So it all comes back to this unitless factor omega Lc 00:40:36.285 --> 00:40:41.055 align:middle line:84% Id over Vs. If that factor gets bigger, the droop gets bigger. 00:40:41.055 --> 00:40:44.160 align:middle line:90% 00:40:44.160 --> 00:40:45.930 align:middle line:90% It's also, by the way-- 00:40:45.930 --> 00:40:47.520 align:middle line:90% if I wanted to figure out-- 00:40:47.520 --> 00:40:49.560 align:middle line:84% imagine I came to this rectifier and I 00:40:49.560 --> 00:40:51.610 align:middle line:90% dropped a short circuit on it. 00:40:51.610 --> 00:40:56.610 align:middle line:84% If I told you take that thing and make the resistance go to 0, 00:40:56.610 --> 00:40:59.200 align:middle line:84% you'd think, well, I'd get infinite current, wouldn't I? 00:40:59.200 --> 00:41:01.893 align:middle line:84% Because I'm basically taking a voltage source 00:41:01.893 --> 00:41:03.810 align:middle line:84% and putting a short circuit on it, and you get 00:41:03.810 --> 00:41:06.160 align:middle line:90% infinite current. 00:41:06.160 --> 00:41:08.990 align:middle line:84% Even if you're commutating reactance is small, 00:41:08.990 --> 00:41:11.580 align:middle line:84% it's what's going to limit-- ultimately, 00:41:11.580 --> 00:41:14.180 align:middle line:84% there's a limit to how much current that will synthesize. 00:41:14.180 --> 00:41:18.370 align:middle line:84% So if you wanted to calculate the short circuit current, 00:41:18.370 --> 00:41:20.045 align:middle line:84% that's how you would figure it out. 00:41:20.045 --> 00:41:22.420 align:middle line:84% You'd have to include-- even if it was a small reactance, 00:41:22.420 --> 00:41:24.650 align:middle line:84% you'd have to include that AC side reactance. 00:41:24.650 --> 00:41:26.500 align:middle line:84% So maybe, in that sense, it's a good thing 00:41:26.500 --> 00:41:28.390 align:middle line:84% if your currents don't go off to infinity. 00:41:28.390 --> 00:41:34.020 align:middle line:84% They only go off to some really big number that you don't like. 00:41:34.020 --> 00:41:35.910 align:middle line:84% But it influences all kinds of things. 00:41:35.910 --> 00:41:38.820 align:middle line:84% I mentioned we had a problem with the high-frequency resonant 00:41:38.820 --> 00:41:41.370 align:middle line:84% converter we built. Or if you were thinking, 00:41:41.370 --> 00:41:43.380 align:middle line:84% I'm going to rectify the line voltage to get 00:41:43.380 --> 00:41:45.610 align:middle line:84% a voltage I know out, and I'm getting droop, 00:41:45.610 --> 00:41:47.020 align:middle line:90% I probably don't like that. 00:41:47.020 --> 00:41:51.450 align:middle line:84% That shows up in a big way in these automotive alternators. 00:41:51.450 --> 00:41:55.830 align:middle line:84% It turns out that the reactance is really, really very big. 00:41:55.830 --> 00:41:59.130 align:middle line:84% So the output of this thing is supposed to charge basically 00:41:59.130 --> 00:42:02.940 align:middle line:90% a 14-volt automotive battery. 00:42:02.940 --> 00:42:05.400 align:middle line:84% How big do you think the AC voltage, 00:42:05.400 --> 00:42:09.330 align:middle line:84% the effective AC voltage is to do that, this Vs? 00:42:09.330 --> 00:42:13.100 align:middle line:90% 00:42:13.100 --> 00:42:17.500 align:middle line:84% And whereas you might think you wanted to synthesize a rectified 00:42:17.500 --> 00:42:19.460 align:middle line:84% voltage of like 14 volts, it doesn't. 00:42:19.460 --> 00:42:22.360 align:middle line:84% It's more like 70 or 80 volts when 00:42:22.360 --> 00:42:24.970 align:middle line:90% it's under full current load. 00:42:24.970 --> 00:42:29.170 align:middle line:84% So really, you have something where that alternator is really 00:42:29.170 --> 00:42:33.280 align:middle line:84% generating a really big voltage with a really steep droop to it. 00:42:33.280 --> 00:42:37.570 align:middle line:84% And so it actually almost looks closer to a current source 00:42:37.570 --> 00:42:40.590 align:middle line:84% or somewhere between a voltage source and a current source. 00:42:40.590 --> 00:42:42.730 align:middle line:84% And in fact, that can cause trouble 00:42:42.730 --> 00:42:45.370 align:middle line:84% in an automotive application because there's 00:42:45.370 --> 00:42:46.960 align:middle line:90% something called a load dump. 00:42:46.960 --> 00:42:48.580 align:middle line:84% Imagine you're charging your battery 00:42:48.580 --> 00:42:49.850 align:middle line:90% with lots and lots of current. 00:42:49.850 --> 00:42:52.780 align:middle line:84% So what they do is they crank up that voltage Vs. 00:42:52.780 --> 00:42:56.880 align:middle line:84% That voltage Vs is controlled by an electromagnet which is then 00:42:56.880 --> 00:42:59.700 align:middle line:84% regulated to get the right voltage to give you 00:42:59.700 --> 00:43:01.030 align:middle line:90% the charging you want. 00:43:01.030 --> 00:43:03.480 align:middle line:84% But if your load suddenly goes away, 00:43:03.480 --> 00:43:05.520 align:middle line:84% that's way too much voltage, and you get 00:43:05.520 --> 00:43:08.080 align:middle line:90% this big spike in DC voltage. 00:43:08.080 --> 00:43:08.730 align:middle line:90% Why? 00:43:08.730 --> 00:43:11.850 align:middle line:84% Because what happened is you were operating down here, 00:43:11.850 --> 00:43:14.460 align:middle line:84% and suddenly the current went away, and you went over here. 00:43:14.460 --> 00:43:16.095 align:middle line:84% And now you've got this big voltage. 00:43:16.095 --> 00:43:16.720 align:middle line:90% And guess what. 00:43:16.720 --> 00:43:18.120 align:middle line:84% That big voltage is being slapped 00:43:18.120 --> 00:43:19.810 align:middle line:90% across all your electronics. 00:43:19.810 --> 00:43:21.450 align:middle line:84% So all the electronics in your car, 00:43:21.450 --> 00:43:24.150 align:middle line:84% despite the fact that they're designed for 14 volts, 00:43:24.150 --> 00:43:27.150 align:middle line:84% all those modules, every single damn thing in your car has to be 00:43:27.150 --> 00:43:30.210 align:middle line:84% rated for 80 volts-- it's kind of crazy-- 00:43:30.210 --> 00:43:32.820 align:middle line:84% all because of this crummy alternator 00:43:32.820 --> 00:43:36.390 align:middle line:84% that has massive amounts of load regulation in it. 00:43:36.390 --> 00:43:39.790 align:middle line:84% So there's all kinds of examples. 00:43:39.790 --> 00:43:41.373 align:middle line:84% And like I said, I showed it to you 00:43:41.373 --> 00:43:42.790 align:middle line:84% in the simple half-wave rectifier. 00:43:42.790 --> 00:43:43.770 align:middle line:90% This is an even-- 00:43:43.770 --> 00:43:47.200 align:middle line:84% this is a three-phase rectifier that the math for 00:43:47.200 --> 00:43:48.790 align:middle line:90% is all about the same. 00:43:48.790 --> 00:43:51.690 align:middle line:90% It does the same kind of thing. 00:43:51.690 --> 00:43:54.360 align:middle line:84% You can also get this in other contexts 00:43:54.360 --> 00:44:00.480 align:middle line:84% where instead of series inductance, 00:44:00.480 --> 00:44:05.490 align:middle line:84% shunt capacitance, such as owing to diode capacitance, 00:44:05.490 --> 00:44:08.430 align:middle line:84% can also mess with your ability to rectify voltage and get 00:44:08.430 --> 00:44:11.340 align:middle line:90% energy out of an AC system. 00:44:11.340 --> 00:44:14.140 align:middle line:90% So that's just-- 00:44:14.140 --> 00:44:18.400 align:middle line:84% I wanted to show you some of the effects of how 00:44:18.400 --> 00:44:21.190 align:middle line:84% we think about analyzing a rectifier 00:44:21.190 --> 00:44:24.970 align:middle line:84% and using the method of assumed states and some of the things 00:44:24.970 --> 00:44:26.547 align:middle line:84% that happen in practical rectifiers. 00:44:26.547 --> 00:44:28.630 align:middle line:84% And we're going to come back to this because we're 00:44:28.630 --> 00:44:32.530 align:middle line:84% going to look at those as parts of different kinds of power 00:44:32.530 --> 00:44:33.530 align:middle line:90% converters. 00:44:33.530 --> 00:44:38.844 align:middle line:84% But perhaps I'll pause there and ask if there's any questions. 00:44:38.844 --> 00:44:39.658 align:middle line:90% Yeah. 00:44:39.658 --> 00:44:41.200 align:middle line:84% AUDIENCE: Do you know if it does this 00:44:41.200 --> 00:44:46.910 align:middle line:84% by adding a capacitor in parallel after the inductance? 00:44:46.910 --> 00:44:49.030 align:middle line:84% DAVID PERREAULT: Oh, could you play some game 00:44:49.030 --> 00:44:51.560 align:middle line:84% to reduce the effect of the source impedance? 00:44:51.560 --> 00:44:53.590 align:middle line:84% AUDIENCE: Yeah, like the [INAUDIBLE] 00:44:53.590 --> 00:44:55.125 align:middle line:90% for the diode current. 00:44:55.125 --> 00:44:57.580 align:middle line:84% DAVID PERREAULT: Yeah, you could try. 00:44:57.580 --> 00:45:00.820 align:middle line:90% That would tend to be a very-- 00:45:00.820 --> 00:45:02.690 align:middle line:84% I mean, it depends on the application. 00:45:02.690 --> 00:45:05.843 align:middle line:84% The challenge with that is it tends to-- 00:45:05.843 --> 00:45:09.013 align:middle line:90% 00:45:09.013 --> 00:45:10.430 align:middle line:84% in that kind of application, you'd 00:45:10.430 --> 00:45:12.200 align:middle line:84% need a really big capacitor because it's 00:45:12.200 --> 00:45:13.650 align:middle line:90% relatively low frequency. 00:45:13.650 --> 00:45:16.560 align:middle line:84% So yes, you could play some games like that. 00:45:16.560 --> 00:45:19.760 align:middle line:84% There are other ways you can fix the problem, too. 00:45:19.760 --> 00:45:22.850 align:middle line:84% So the answer is yes, that's a very clever idea. 00:45:22.850 --> 00:45:27.150 align:middle line:84% And you can apply that, but it's very application dependent. 00:45:27.150 --> 00:45:29.930 align:middle line:84% So, yeah, there's ways to fix the problem, yes. 00:45:29.930 --> 00:45:32.812 align:middle line:84% Or you could just design a better machine that 00:45:32.812 --> 00:45:34.020 align:middle line:90% doesn't have so much leakage. 00:45:34.020 --> 00:45:35.630 align:middle line:84% That would be another way to do it. 00:45:35.630 --> 00:45:38.642 align:middle line:90% Other questions? 00:45:38.642 --> 00:45:41.050 align:middle line:84% OK, so that's all I wanted to say for today. 00:45:41.050 --> 00:45:43.390 align:middle line:84% What we're going to do next time is 00:45:43.390 --> 00:45:47.380 align:middle line:84% we've looked at this rectifier from the DC side, 00:45:47.380 --> 00:45:50.380 align:middle line:84% like what is the characteristics when I take AC energy 00:45:50.380 --> 00:45:51.880 align:middle line:90% and turn it to DC. 00:45:51.880 --> 00:45:56.810 align:middle line:84% The thing we haven't looked at is what about this poor AC grid? 00:45:56.810 --> 00:45:58.510 align:middle line:84% And what happens when I start pulling 00:45:58.510 --> 00:46:01.400 align:middle line:84% weird currents through it like from some rectifier? 00:46:01.400 --> 00:46:04.190 align:middle line:84% And that turns out to be quite important for energy efficiency. 00:46:04.190 --> 00:46:07.130 align:middle line:84% And so we'll pick up that topic next class. 00:46:07.130 --> 00:46:09.450 align:middle line:90% So have a great day. 00:46:09.450 --> 00:46:15.000 align:middle line:90%