WEBVTT 00:00:00.000 --> 00:00:02.475 align:middle line:90% [SQUEAKING] 00:00:02.475 --> 00:00:04.950 align:middle line:90% [RUSTLING] 00:00:04.950 --> 00:00:07.425 align:middle line:90% [CLICKING] 00:00:07.425 --> 00:00:09.035 align:middle line:90% 00:00:09.035 --> 00:00:10.660 align:middle line:84% DAVID PERREAULT: So just to remind you, 00:00:10.660 --> 00:00:15.460 align:middle line:84% from the last lecture, we talked about the notion, the difference 00:00:15.460 --> 00:00:21.290 align:middle line:84% between a linear power supply and a switching power supply. 00:00:21.290 --> 00:00:23.752 align:middle line:84% And we're talking about, in this case, DC to DC conversion, 00:00:23.752 --> 00:00:25.210 align:middle line:84% but the same thing is true actually 00:00:25.210 --> 00:00:27.135 align:middle line:84% of RF amplifiers and other areas where 00:00:27.135 --> 00:00:28.510 align:middle line:84% you can have switching amplifiers 00:00:28.510 --> 00:00:30.020 align:middle line:90% and linear amplifiers. 00:00:30.020 --> 00:00:32.229 align:middle line:84% And we said the notion of a linear amplifier 00:00:32.229 --> 00:00:37.360 align:middle line:84% was simply that I have some input source, V in. 00:00:37.360 --> 00:00:41.290 align:middle line:90% I'll use a transistor. 00:00:41.290 --> 00:00:43.700 align:middle line:90% And here's my load resistance. 00:00:43.700 --> 00:00:46.060 align:middle line:90% So here's V out. 00:00:46.060 --> 00:00:48.753 align:middle line:84% And what we'll do is we'll have a reference voltage. 00:00:48.753 --> 00:00:51.860 align:middle line:90% 00:00:51.860 --> 00:00:53.990 align:middle line:84% And we'll feed back the output voltage 00:00:53.990 --> 00:00:56.900 align:middle line:84% and control the base or the gate of our transistor 00:00:56.900 --> 00:01:00.260 align:middle line:90% so that we get some drop-- 00:01:00.260 --> 00:01:02.490 align:middle line:90% I'll call it Vd here-- 00:01:02.490 --> 00:01:11.450 align:middle line:84% so that this essentially looks like some variable resistor 00:01:11.450 --> 00:01:14.720 align:middle line:84% that we do a resistor divider, and we get our output voltage 00:01:14.720 --> 00:01:16.700 align:middle line:90% from our input voltage. 00:01:16.700 --> 00:01:19.100 align:middle line:84% And we said the real problem with this 00:01:19.100 --> 00:01:23.360 align:middle line:84% was, OK, I have my voltage drop, Vd, 00:01:23.360 --> 00:01:28.820 align:middle line:84% and whatever current is coming from the source-- 00:01:28.820 --> 00:01:31.650 align:middle line:84% and that's the same thing as the load current. 00:01:31.650 --> 00:01:33.440 align:middle line:84% It's coming out of the source, coming 00:01:33.440 --> 00:01:36.090 align:middle line:84% through this past transistor, and going to the load. 00:01:36.090 --> 00:01:39.200 align:middle line:84% And so if I ask what the power dissipation was, 00:01:39.200 --> 00:01:42.380 align:middle line:84% the instantaneous power dissipation in this transistor 00:01:42.380 --> 00:01:48.560 align:middle line:90% is equal to Vd times IL. 00:01:48.560 --> 00:01:52.430 align:middle line:84% If I said, What's the average power dissipation? 00:01:52.430 --> 00:01:55.820 align:middle line:84% it's the average of that, and it's not equal to 0. 00:01:55.820 --> 00:01:58.100 align:middle line:84% So essentially, if I have a variable resistor, 00:01:58.100 --> 00:02:01.160 align:middle line:84% I'm dumping energy into that variable resistor. 00:02:01.160 --> 00:02:03.560 align:middle line:84% And especially when I have large conversion ratios, 00:02:03.560 --> 00:02:05.850 align:middle line:90% that gets really bad. 00:02:05.850 --> 00:02:09.900 align:middle line:84% So what we said is, OK, we can be way more clever than that, 00:02:09.900 --> 00:02:12.940 align:middle line:84% and we can go build a switching converter. 00:02:12.940 --> 00:02:15.630 align:middle line:90% And the idea would be this. 00:02:15.630 --> 00:02:17.610 align:middle line:90% We'd have V in. 00:02:17.610 --> 00:02:21.030 align:middle line:84% We'll have a switch, and this will be built out 00:02:21.030 --> 00:02:23.700 align:middle line:90% of semiconductor switches-- 00:02:23.700 --> 00:02:28.680 align:middle line:84% q of t is equal to 1, q of t is equal to 0. 00:02:28.680 --> 00:02:33.180 align:middle line:84% And then we will take this output, 00:02:33.180 --> 00:02:36.420 align:middle line:84% and we'll run it through some lossless LC filter, 00:02:36.420 --> 00:02:38.150 align:middle line:84% and then I'll have my output here. 00:02:38.150 --> 00:02:40.710 align:middle line:90% 00:02:40.710 --> 00:02:49.590 align:middle line:84% And the notion is by chopping this back and forth like this-- 00:02:49.590 --> 00:02:52.140 align:middle line:84% so I have some switching function, q of t 00:02:52.140 --> 00:02:57.120 align:middle line:84% is equal to 1 and 0, and I'll control this with some switching 00:02:57.120 --> 00:02:59.640 align:middle line:90% period and some duty ratio. 00:02:59.640 --> 00:03:01.200 align:middle line:90% That's a pulse width modulation. 00:03:01.200 --> 00:03:02.070 align:middle line:90% This is q of t. 00:03:02.070 --> 00:03:04.960 align:middle line:90% 00:03:04.960 --> 00:03:07.680 align:middle line:84% Then I can have this voltage that I'll 00:03:07.680 --> 00:03:11.440 align:middle line:84% call Vx, which is sort of a copy of that. 00:03:11.440 --> 00:03:14.350 align:middle line:90% When q of t is 1 Vx is V in. 00:03:14.350 --> 00:03:16.630 align:middle line:90% When q of t is 0, Vx is 0. 00:03:16.630 --> 00:03:19.770 align:middle line:84% And I get a pulsed waveform that looks like this. 00:03:19.770 --> 00:03:22.290 align:middle line:90% This is Vx. 00:03:22.290 --> 00:03:24.875 align:middle line:90% And so forth. 00:03:24.875 --> 00:03:28.760 align:middle line:90% 00:03:28.760 --> 00:03:35.640 align:middle line:84% And when I filter Vx, the idea is this acts as a low pass 00:03:35.640 --> 00:03:37.410 align:middle line:84% filter, and it basically extracts 00:03:37.410 --> 00:03:39.130 align:middle line:90% the DC component of that. 00:03:39.130 --> 00:03:45.510 align:middle line:84% And so then I get an output voltage, V out, 00:03:45.510 --> 00:03:48.980 align:middle line:84% that is basically approximately equal to-- 00:03:48.980 --> 00:03:52.330 align:middle line:90% 00:03:52.330 --> 00:03:57.170 align:middle line:84% V out is approximately equal to d times V in. 00:03:57.170 --> 00:04:00.160 align:middle line:84% So by controlling timing, by controlling 00:04:00.160 --> 00:04:02.860 align:middle line:84% the width of the pulse and the cycle, 00:04:02.860 --> 00:04:05.840 align:middle line:84% I control the average voltage at the output. 00:04:05.840 --> 00:04:09.940 align:middle line:84% And the beauty is, as we argued, switches are ideally lossless, 00:04:09.940 --> 00:04:13.610 align:middle line:84% inductors are ideally lossless, capacitors are ideally lossless. 00:04:13.610 --> 00:04:19.890 align:middle line:84% So the efficiency of this whole box can in theory approach 100%. 00:04:19.890 --> 00:04:23.740 align:middle line:84% So that's the fundamental concept. 00:04:23.740 --> 00:04:25.320 align:middle line:90% We're going to have switching. 00:04:25.320 --> 00:04:27.180 align:middle line:84% We're going to have energy storage. 00:04:27.180 --> 00:04:30.060 align:middle line:84% And we're going to take energy from an input to an output. 00:04:30.060 --> 00:04:32.550 align:middle line:84% And we're going to operate cyclically. 00:04:32.550 --> 00:04:35.340 align:middle line:90% Now, things are a lot more-- 00:04:35.340 --> 00:04:37.540 align:middle line:84% there's a lot more to it than just that. 00:04:37.540 --> 00:04:41.140 align:middle line:84% There's many design considerations in all of this. 00:04:41.140 --> 00:04:43.510 align:middle line:84% First of all, how do you design these inductors? 00:04:43.510 --> 00:04:45.010 align:middle line:90% How do you pick a topology? 00:04:45.010 --> 00:04:45.940 align:middle line:90% What can it do? 00:04:45.940 --> 00:04:46.913 align:middle line:90% How do you control it? 00:04:46.913 --> 00:04:47.830 align:middle line:90% What are the dynamics? 00:04:47.830 --> 00:04:49.247 align:middle line:84% All these kinds of questions we're 00:04:49.247 --> 00:04:51.660 align:middle line:90% going to answer in this term. 00:04:51.660 --> 00:04:53.670 align:middle line:90% But that's the basic concept. 00:04:53.670 --> 00:04:55.710 align:middle line:84% So before I get on to the material today, 00:04:55.710 --> 00:04:58.460 align:middle line:84% are there any questions just about the basic idea here? 00:04:58.460 --> 00:05:02.420 align:middle line:90% 00:05:02.420 --> 00:05:07.790 align:middle line:84% OK, so what I'd like to start doing today 00:05:07.790 --> 00:05:11.000 align:middle line:84% is giving us some of the tools to analyze 00:05:11.000 --> 00:05:12.175 align:middle line:90% these kinds of systems. 00:05:12.175 --> 00:05:14.750 align:middle line:90% 00:05:14.750 --> 00:05:16.310 align:middle line:84% And what we're going to see is we're 00:05:16.310 --> 00:05:19.070 align:middle line:84% going to do a lot of our analysis in the time domain 00:05:19.070 --> 00:05:22.340 align:middle line:84% because it turns out that our circuits tend 00:05:22.340 --> 00:05:24.620 align:middle line:84% to be either nonlinear or time varying, 00:05:24.620 --> 00:05:27.920 align:middle line:84% depending upon how you want to look at it. 00:05:27.920 --> 00:05:30.890 align:middle line:84% And time domain analysis is a really good way 00:05:30.890 --> 00:05:32.600 align:middle line:90% to look at these things. 00:05:32.600 --> 00:05:35.930 align:middle line:84% And what we would like is some tools 00:05:35.930 --> 00:05:37.560 align:middle line:90% to figure out what's going on. 00:05:37.560 --> 00:05:39.410 align:middle line:84% I'm going to introduce you to two concepts 00:05:39.410 --> 00:05:42.290 align:middle line:84% today that we're going to come back to again and again. 00:05:42.290 --> 00:05:44.210 align:middle line:84% And the notion is that the first thing 00:05:44.210 --> 00:05:49.580 align:middle line:84% I'd like to talk about relates to the switches themselves, 00:05:49.580 --> 00:05:52.220 align:middle line:84% I haven't said how we're going to implement 00:05:52.220 --> 00:05:54.380 align:middle line:84% this single-pull double-throw switch. 00:05:54.380 --> 00:05:57.470 align:middle line:84% And what the switching configuration looks like 00:05:57.470 --> 00:05:59.400 align:middle line:90% varies from circuit to circuit. 00:05:59.400 --> 00:06:01.730 align:middle line:84% But usually, we're going to implement those 00:06:01.730 --> 00:06:04.120 align:middle line:90% with semiconductor switches. 00:06:04.120 --> 00:06:07.390 align:middle line:84% Now, you can imagine semiconductor switches 00:06:07.390 --> 00:06:08.480 align:middle line:90% that are fully controlled. 00:06:08.480 --> 00:06:10.330 align:middle line:84% I tell them when to turn on and off. 00:06:10.330 --> 00:06:13.970 align:middle line:84% But more often than not, the switch-- 00:06:13.970 --> 00:06:16.630 align:middle line:84% some aspects of your switch may be 00:06:16.630 --> 00:06:18.410 align:middle line:84% either semicontrolled or uncontrolled. 00:06:18.410 --> 00:06:22.090 align:middle line:84% That is, external circuit events will dictate when 00:06:22.090 --> 00:06:23.890 align:middle line:90% the device turns on and off. 00:06:23.890 --> 00:06:26.290 align:middle line:84% And so one of the things you'll need 00:06:26.290 --> 00:06:28.810 align:middle line:84% to do if you're going to analyze some circuit 00:06:28.810 --> 00:06:31.780 align:middle line:84% is figure out when do these uncontrolled devices 00:06:31.780 --> 00:06:35.380 align:middle line:84% turn on and turn off as part of figuring out what, for example, 00:06:35.380 --> 00:06:40.420 align:middle line:84% some circuit topology will do or can do, for example. 00:06:40.420 --> 00:06:46.010 align:middle line:84% So let me give you the basic approach that we use to do that. 00:06:46.010 --> 00:06:48.530 align:middle line:84% And I'm going to start with this. 00:06:48.530 --> 00:06:49.960 align:middle line:84% Some people might think-- this is 00:06:49.960 --> 00:06:52.370 align:middle line:84% known as a-- operated this way, it's known as a buck converter. 00:06:52.370 --> 00:06:54.078 align:middle line:84% It's a relatively simple power converter. 00:06:54.078 --> 00:06:55.990 align:middle line:84% I'm going to go to perhaps the simplest power 00:06:55.990 --> 00:06:59.300 align:middle line:90% converter you can think of. 00:06:59.300 --> 00:07:01.650 align:middle line:84% And it just simply looks like this. 00:07:01.650 --> 00:07:04.940 align:middle line:84% And this is just by way of illustration. 00:07:04.940 --> 00:07:13.130 align:middle line:84% So suppose I have Vs sine omega t, some AC source. 00:07:13.130 --> 00:07:16.470 align:middle line:84% And I'm going to put a diode here, 00:07:16.470 --> 00:07:20.610 align:middle line:84% which is my canonical, completely uncontrolled switch. 00:07:20.610 --> 00:07:24.590 align:middle line:84% A diode turns on and off when the external circuit 00:07:24.590 --> 00:07:26.190 align:middle line:90% tells it to turn on and off. 00:07:26.190 --> 00:07:28.830 align:middle line:84% We can have also devices that are semicontrolled. 00:07:28.830 --> 00:07:31.130 align:middle line:84% Maybe we only turn them-- tell them when to turn on. 00:07:31.130 --> 00:07:33.830 align:middle line:84% Or maybe we only control when they're turning on and off 00:07:33.830 --> 00:07:34.970 align:middle line:90% under some conditions. 00:07:34.970 --> 00:07:37.850 align:middle line:84% And we'll see that in much more detail. 00:07:37.850 --> 00:07:40.900 align:middle line:90% But suppose I have this circuit. 00:07:40.900 --> 00:07:43.555 align:middle line:90% 00:07:43.555 --> 00:07:46.570 align:middle line:90% And maybe I have some-- 00:07:46.570 --> 00:07:49.870 align:middle line:84% the interesting thing I'm interested in getting here 00:07:49.870 --> 00:07:54.592 align:middle line:90% is V out. 00:07:54.592 --> 00:07:59.170 align:middle line:84% And maybe this is just some value R. 00:07:59.170 --> 00:08:04.000 align:middle line:84% And we have some current that's going to come here. 00:08:04.000 --> 00:08:06.335 align:middle line:90% And let me call this ID. 00:08:06.335 --> 00:08:10.030 align:middle line:90% 00:08:10.030 --> 00:08:14.110 align:middle line:84% And we will also have some voltage VD. 00:08:14.110 --> 00:08:20.250 align:middle line:90% 00:08:20.250 --> 00:08:25.230 align:middle line:84% Now, let's talk about diodes for a second. 00:08:25.230 --> 00:08:26.980 align:middle line:84% People who have had semiconductor classes, 00:08:26.980 --> 00:08:29.730 align:middle line:84% you know a diode has an exponential characteristic. 00:08:29.730 --> 00:08:32.580 align:middle line:84% But for basic analysis, if we're going 00:08:32.580 --> 00:08:36.039 align:middle line:84% to treat things as switches, we can really simplify that. 00:08:36.039 --> 00:08:40.169 align:middle line:84% So we often think about what we would call an ideal diode. 00:08:40.169 --> 00:08:42.030 align:middle line:90% What is an ideal diode do? 00:08:42.030 --> 00:08:52.980 align:middle line:84% An ideal diode has the following IV characteristic. 00:08:52.980 --> 00:08:57.630 align:middle line:90% OK, so here's ID and VD. 00:08:57.630 --> 00:09:05.250 align:middle line:84% And I might have a situation where the diode is on. 00:09:05.250 --> 00:09:08.495 align:middle line:90% So this is when the diode is on. 00:09:08.495 --> 00:09:12.670 align:middle line:90% 00:09:12.670 --> 00:09:16.810 align:middle line:84% That is, it has no forward drop, but it can carry as much current 00:09:16.810 --> 00:09:19.390 align:middle line:90% as you want. 00:09:19.390 --> 00:09:24.540 align:middle line:84% I might also have a case where I have the diode being off. 00:09:24.540 --> 00:09:27.170 align:middle line:84% And so here is the off state of the diode. 00:09:27.170 --> 00:09:32.580 align:middle line:90% 00:09:32.580 --> 00:09:40.520 align:middle line:84% So if I'm off, basically I'm thinking of VD and ID like this. 00:09:40.520 --> 00:09:43.930 align:middle line:90% Here's VD and ID. 00:09:43.930 --> 00:09:46.540 align:middle line:90% 00:09:46.540 --> 00:09:50.250 align:middle line:84% VD is whatever it wants to be as long as it's negative. 00:09:50.250 --> 00:09:56.500 align:middle line:84% And ID is 0 because this is an open circuit. 00:09:56.500 --> 00:10:00.960 align:middle line:84% And I should just be clear I'm talking about an ideal diode 00:10:00.960 --> 00:10:01.985 align:middle line:90% looking like this. 00:10:01.985 --> 00:10:05.850 align:middle line:90% 00:10:05.850 --> 00:10:17.200 align:middle line:84% And if I'm on, what I'm assuming is that essentially VD is 0, 00:10:17.200 --> 00:10:19.870 align:middle line:90% and I have some positive ID. 00:10:19.870 --> 00:10:22.660 align:middle line:90% 00:10:22.660 --> 00:10:25.070 align:middle line:84% So essentially, in treating it like a switch, 00:10:25.070 --> 00:10:28.210 align:middle line:84% I'm either treating it like a perfectly closed switch, which 00:10:28.210 --> 00:10:31.630 align:middle line:84% happens in this part of the IV plane, 00:10:31.630 --> 00:10:34.240 align:middle line:84% or I think of it as a perfectly open switch, in which case 00:10:34.240 --> 00:10:37.220 align:middle line:84% it happens in this part of the IV plane. 00:10:37.220 --> 00:10:40.450 align:middle line:84% So that's how I will think of it as a diode. 00:10:40.450 --> 00:10:44.360 align:middle line:84% And yes, a real diode would have some exponential characteristic. 00:10:44.360 --> 00:10:48.440 align:middle line:84% But for analysis, it's much simpler to treat it this way 00:10:48.440 --> 00:10:50.430 align:middle line:84% and then add in nonidealities later. 00:10:50.430 --> 00:10:53.930 align:middle line:90% 00:10:53.930 --> 00:10:58.040 align:middle line:84% So a diode is an uncontrolled switch. 00:10:58.040 --> 00:11:00.380 align:middle line:90% What controls its switching? 00:11:00.380 --> 00:11:05.100 align:middle line:84% Well, the right way to think about this is based on what 00:11:05.100 --> 00:11:07.980 align:middle line:90% a diode cannot do. 00:11:07.980 --> 00:11:13.230 align:middle line:84% Notice that there is no characteristic down here 00:11:13.230 --> 00:11:14.560 align:middle line:90% with ID negative. 00:11:14.560 --> 00:11:28.763 align:middle line:84% So a diode cannot support a negative current and no positive 00:11:28.763 --> 00:11:29.263 align:middle line:90% voltage. 00:11:29.263 --> 00:11:33.060 align:middle line:90% 00:11:33.060 --> 00:11:34.770 align:middle line:90% This should be ID and VD. 00:11:34.770 --> 00:11:39.150 align:middle line:84% So it cannot have a positive voltage, because it'll turn on. 00:11:39.150 --> 00:11:42.840 align:middle line:84% It cannot have a negative current, because it'll turn off. 00:11:42.840 --> 00:11:47.250 align:middle line:84% And so in figuring out in our circuit what a diode is doing, 00:11:47.250 --> 00:11:50.220 align:middle line:84% we often figure it out based on what it can't 00:11:50.220 --> 00:11:52.340 align:middle line:90% do rather than what it can do. 00:11:52.340 --> 00:11:57.200 align:middle line:84% And I'll get to what I mean by that in a moment. 00:11:57.200 --> 00:12:01.560 align:middle line:84% So let's think about this circuit. 00:12:01.560 --> 00:12:03.280 align:middle line:90% How will I analyze it? 00:12:03.280 --> 00:12:07.610 align:middle line:84% Suppose I have a circuit like this, for example, 00:12:07.610 --> 00:12:09.110 align:middle line:90% with one or more diodes in it. 00:12:09.110 --> 00:12:10.382 align:middle line:90% And this circuit is trivial. 00:12:10.382 --> 00:12:11.090 align:middle line:90% It has one diode. 00:12:11.090 --> 00:12:13.610 align:middle line:84% But you can actually find power converter circuits 00:12:13.610 --> 00:12:15.660 align:middle line:84% for high power that have 48 diodes in them. 00:12:15.660 --> 00:12:18.380 align:middle line:84% And so then part of your job is they're turning on and off, 00:12:18.380 --> 00:12:20.255 align:middle line:84% and you've got to figure out what's going on. 00:12:20.255 --> 00:12:22.690 align:middle line:90% 00:12:22.690 --> 00:12:24.470 align:middle line:90% What would I do here? 00:12:24.470 --> 00:12:26.560 align:middle line:84% Well, what we would tend to use is 00:12:26.560 --> 00:12:30.340 align:middle line:84% something known as the method of assumed states. 00:12:30.340 --> 00:12:37.420 align:middle line:84% And here's the Method of Assumed States, or MAS, as I call it. 00:12:37.420 --> 00:12:41.025 align:middle line:84% One, assume a state for each switch. 00:12:41.025 --> 00:12:52.630 align:middle line:90% 00:12:52.630 --> 00:12:54.760 align:middle line:90% So some switches you know. 00:12:54.760 --> 00:12:56.260 align:middle line:84% If you fully control the switch, you 00:12:56.260 --> 00:12:57.310 align:middle line:84% know when it's on and off because you're 00:12:57.310 --> 00:12:58.250 align:middle line:90% telling it to turn on and off. 00:12:58.250 --> 00:13:00.550 align:middle line:84% But if you don't know, if you have diodes, for example, 00:13:00.550 --> 00:13:03.190 align:middle line:90% in your circuit, guess. 00:13:03.190 --> 00:13:07.480 align:middle line:84% Assume it's on or off at a given point in time. 00:13:07.480 --> 00:13:09.987 align:middle line:84% What you would then do is say, OK, take that diode 00:13:09.987 --> 00:13:12.070 align:middle line:84% and replace it either with a short circuit if I've 00:13:12.070 --> 00:13:14.530 align:middle line:84% assumed it's on or an open circuit if I've 00:13:14.530 --> 00:13:18.080 align:middle line:90% assumed it's off. 00:13:18.080 --> 00:13:20.370 align:middle line:84% And now I've replaced things with shortened opens, 00:13:20.370 --> 00:13:22.510 align:middle line:90% so now my circuit's simpler. 00:13:22.510 --> 00:13:25.540 align:middle line:84% So I've assumed a state is substituted in. 00:13:25.540 --> 00:13:31.810 align:middle line:84% Then I analyze the V's and the I's in the circuit. 00:13:31.810 --> 00:13:37.240 align:middle line:84% Now, if what I have is switches, inductors, capacitors, 00:13:37.240 --> 00:13:39.610 align:middle line:84% resistors, that kind of thing, then that just 00:13:39.610 --> 00:13:42.880 align:middle line:84% becomes a linear circuit problem that's straightforward to solve. 00:13:42.880 --> 00:13:45.730 align:middle line:84% So analyzing the V's and I's, in that case, 00:13:45.730 --> 00:13:48.430 align:middle line:84% is pretty easy, whereas if I want exponentials in there, then 00:13:48.430 --> 00:13:51.020 align:middle line:90% it gets a little ugly. 00:13:51.020 --> 00:13:53.960 align:middle line:84% The next thing I do, and this is the key thing, 00:13:53.960 --> 00:13:58.183 align:middle line:84% check if switch conditions are violated. 00:13:58.183 --> 00:14:05.420 align:middle line:90% 00:14:05.420 --> 00:14:09.740 align:middle line:84% So what I'm really going to do is I'm going to go say, oh, 00:14:09.740 --> 00:14:14.330 align:middle line:84% I've assumed my switch is on, so I've replaced it 00:14:14.330 --> 00:14:15.890 align:middle line:90% with a short circuit. 00:14:15.890 --> 00:14:18.660 align:middle line:84% Did the diode current end up being negative? 00:14:18.660 --> 00:14:21.080 align:middle line:84% If it did, I know it can't do that because diodes 00:14:21.080 --> 00:14:22.690 align:middle line:90% can't do that. 00:14:22.690 --> 00:14:26.170 align:middle line:84% If I've assumed it's off, the question I have to ask myself 00:14:26.170 --> 00:14:29.175 align:middle line:84% is, did the diode voltage end up being positive? 00:14:29.175 --> 00:14:31.550 align:middle line:84% So what I'm going to do is I'm going to assume something. 00:14:31.550 --> 00:14:34.210 align:middle line:84% I'm going to assume it's a shorter and open and then check 00:14:34.210 --> 00:14:37.745 align:middle line:84% if the other condition is violated. 00:14:37.745 --> 00:14:39.370 align:middle line:84% And the reason I have to do it that way 00:14:39.370 --> 00:14:41.385 align:middle line:84% is because if I have assumed it's a short, 00:14:41.385 --> 00:14:43.510 align:middle line:84% then I can't check its voltage because I've already 00:14:43.510 --> 00:14:44.690 align:middle line:90% assumed its voltage is 0. 00:14:44.690 --> 00:14:47.390 align:middle line:84% I've got to check the other terminal condition. 00:14:47.390 --> 00:14:50.960 align:middle line:90% 00:14:50.960 --> 00:14:59.240 align:middle line:84% All right, four, if not, we're happy, keep going in time. 00:14:59.240 --> 00:15:03.110 align:middle line:90% 00:15:03.110 --> 00:15:10.900 align:middle line:84% If violated, I have to make a new set of assumptions 00:15:10.900 --> 00:15:12.800 align:middle line:90% and come up here. 00:15:12.800 --> 00:15:14.780 align:middle line:90% So guess again. 00:15:14.780 --> 00:15:17.690 align:middle line:84% What I might do is if I found a device condition violated, 00:15:17.690 --> 00:15:22.010 align:middle line:84% maybe I'll switch the condition state of that, 00:15:22.010 --> 00:15:25.010 align:middle line:84% assume state of that diode, and go back and do it over again. 00:15:25.010 --> 00:15:29.548 align:middle line:84% So I can start to guess at a given point in time 00:15:29.548 --> 00:15:31.340 align:middle line:84% what the state of the uncontrolled switches 00:15:31.340 --> 00:15:34.442 align:middle line:84% is and then check if I'm wrong and keep 00:15:34.442 --> 00:15:36.400 align:middle line:84% iterating that until I figure it out I'm right. 00:15:36.400 --> 00:15:39.252 align:middle line:90% 00:15:39.252 --> 00:15:46.776 align:middle line:84% So let me just illustrate that for a very simple condition. 00:15:46.776 --> 00:15:51.050 align:middle line:84% In this particular circuit, I have two possibilities. 00:15:51.050 --> 00:15:52.235 align:middle line:90% If the diode is on-- 00:15:52.235 --> 00:15:57.320 align:middle line:90% 00:15:57.320 --> 00:15:59.630 align:middle line:84% and this is why the world's simplest circuit-- 00:15:59.630 --> 00:16:07.440 align:middle line:84% if I assume the diodes on, then I've got V out here 00:16:07.440 --> 00:16:09.690 align:middle line:84% is just equal to the input voltage. 00:16:09.690 --> 00:16:14.010 align:middle line:84% And I ought to look at this current 00:16:14.010 --> 00:16:17.640 align:middle line:84% to check whether I'm making an error. 00:16:17.640 --> 00:16:26.990 align:middle line:84% And if I've assumed it's off, then 00:16:26.990 --> 00:16:35.760 align:middle line:84% what I would need to look at, I would have V out like this, 00:16:35.760 --> 00:16:42.677 align:middle line:84% and I would need to look at the diode voltage 00:16:42.677 --> 00:16:47.200 align:middle line:90% to see if it's gone positive. 00:16:47.200 --> 00:16:48.780 align:middle line:90% So let's just apply this. 00:16:48.780 --> 00:16:51.000 align:middle line:84% And yes, I grant you, you've long since 00:16:51.000 --> 00:16:54.060 align:middle line:84% figured out what the circuit's doing in your head, 00:16:54.060 --> 00:16:56.310 align:middle line:84% but it's not so simple if you have 00:16:56.310 --> 00:16:59.340 align:middle line:84% a whole pile of uncontrolled devices or a more complicated 00:16:59.340 --> 00:17:01.320 align:middle line:90% circuit, as we'll see. 00:17:01.320 --> 00:17:02.950 align:middle line:84% So let's just look at this example. 00:17:02.950 --> 00:17:03.900 align:middle line:90% So here we go. 00:17:03.900 --> 00:17:08.760 align:middle line:84% I've got my input source Vs sine omega t, 00:17:08.760 --> 00:17:11.740 align:middle line:90% so it looks like some sine wave. 00:17:11.740 --> 00:17:13.720 align:middle line:90% Here's omega t. 00:17:13.720 --> 00:17:17.560 align:middle line:84% And I want to know, what, for example, is 00:17:17.560 --> 00:17:21.220 align:middle line:90% the output voltage doing there? 00:17:21.220 --> 00:17:24.369 align:middle line:84% Well, let's assume-- let me just start at t 00:17:24.369 --> 00:17:29.460 align:middle line:90% equals 0 plus, like right here. 00:17:29.460 --> 00:17:31.460 align:middle line:90% Or omega t equal to 0 plus. 00:17:31.460 --> 00:17:39.570 align:middle line:84% Well, if I assumed the diode was off, I'd have this picture here. 00:17:39.570 --> 00:17:43.390 align:middle line:84% If I assume the diode is off, I'd have this picture here. 00:17:43.390 --> 00:17:45.530 align:middle line:84% But if this is open, the diode current 00:17:45.530 --> 00:17:47.980 align:middle line:84% is 0, so there's no current through the resistor, 00:17:47.980 --> 00:17:49.560 align:middle line:90% so there's no voltage here-- 00:17:49.560 --> 00:17:54.330 align:middle line:84% so in that case, VD would equal Vs sine omega t, right? 00:17:54.330 --> 00:17:58.290 align:middle line:84% And if I just told you that Vs sine omega t is positive, 00:17:58.290 --> 00:18:00.900 align:middle line:84% well, that would mean that the diode voltage is positive. 00:18:00.900 --> 00:18:04.930 align:middle line:84% And hence, this violates this assumption. 00:18:04.930 --> 00:18:06.270 align:middle line:90% And so it can't be true. 00:18:06.270 --> 00:18:11.140 align:middle line:84% This cannot be the state if Vs sine omega t is positive. 00:18:11.140 --> 00:18:17.950 align:middle line:84% So what that would tell me is that I must 00:18:17.950 --> 00:18:22.550 align:middle line:90% have at least in this range-- 00:18:22.550 --> 00:18:27.080 align:middle line:84% the diode has to be on for at least this range. 00:18:27.080 --> 00:18:30.860 align:middle line:84% And so I would get V out looking exactly like VD 00:18:30.860 --> 00:18:34.310 align:middle line:84% because if I look at the top assumed circuit, 00:18:34.310 --> 00:18:37.820 align:middle line:84% Vs sine omega t is applied directly across V out. 00:18:37.820 --> 00:18:41.810 align:middle line:90% And it looks like this. 00:18:41.810 --> 00:18:45.120 align:middle line:90% Here's V out. 00:18:45.120 --> 00:18:48.270 align:middle line:84% And what would the diode current look like, ID? 00:18:48.270 --> 00:18:51.695 align:middle line:84% Well, ID is just Vs over R, so it would do something like this. 00:18:51.695 --> 00:18:56.480 align:middle line:90% 00:18:56.480 --> 00:19:01.750 align:middle line:84% Well, what happens when I cross this boundary? 00:19:01.750 --> 00:19:04.630 align:middle line:84% If I continue to assume I was in the same state 00:19:04.630 --> 00:19:13.210 align:middle line:84% where the diode is on, then I would have ID here 00:19:13.210 --> 00:19:16.570 align:middle line:90% go negative right here. 00:19:16.570 --> 00:19:18.320 align:middle line:90% But we know it can't do that. 00:19:18.320 --> 00:19:19.400 align:middle line:90% So what do I know? 00:19:19.400 --> 00:19:22.750 align:middle line:84% I know my assumption that the diode was on must be off. 00:19:22.750 --> 00:19:29.350 align:middle line:84% And so in this part of the cycle, the diode must be off. 00:19:29.350 --> 00:19:30.670 align:middle line:90% So pretty simple. 00:19:30.670 --> 00:19:37.360 align:middle line:84% If the diode is off, then ID is 0, and VD is 0, and V out is 0. 00:19:37.360 --> 00:19:39.010 align:middle line:90% And this is the waveform I get. 00:19:39.010 --> 00:19:40.710 align:middle line:90% And it would look like this. 00:19:40.710 --> 00:19:44.305 align:middle line:90% So we've analyzed the circuit. 00:19:44.305 --> 00:19:46.360 align:middle line:84% It took a lot of time to do it, which 00:19:46.360 --> 00:19:49.450 align:middle line:90% is just by way of illustration. 00:19:49.450 --> 00:19:51.340 align:middle line:84% But that's the notion of assumed states. 00:19:51.340 --> 00:19:53.110 align:middle line:90% I assume a state for the device. 00:19:53.110 --> 00:19:54.250 align:middle line:90% I calculate along. 00:19:54.250 --> 00:19:56.740 align:middle line:84% And then I see if I'm wrong at any point about what 00:19:56.740 --> 00:19:59.970 align:middle line:90% the diode can and can't do. 00:19:59.970 --> 00:20:01.800 align:middle line:84% Let me stop there and just ask if there's 00:20:01.800 --> 00:20:03.470 align:middle line:90% any questions about the concept. 00:20:03.470 --> 00:20:07.480 align:middle line:90% 00:20:07.480 --> 00:20:08.343 align:middle line:90% Yeah. 00:20:08.343 --> 00:20:09.760 align:middle line:84% AUDIENCE: Are you always-- are you 00:20:09.760 --> 00:20:12.040 align:middle line:84% going to get a unique answer when you do this? 00:20:12.040 --> 00:20:13.540 align:middle line:84% DAVID PERREAULT: Very good question. 00:20:13.540 --> 00:20:15.670 align:middle line:84% Are you going to get a unique answer? 00:20:15.670 --> 00:20:21.173 align:middle line:84% And the answer to that is so long as-- 00:20:21.173 --> 00:20:24.535 align:middle line:90% 00:20:24.535 --> 00:20:25.910 align:middle line:84% it gets more complicated when you 00:20:25.910 --> 00:20:28.370 align:middle line:84% have controlled and uncontrolled devices. 00:20:28.370 --> 00:20:34.070 align:middle line:84% But if I know an initial state, it will do a unique thing. 00:20:34.070 --> 00:20:36.482 align:middle line:84% I could give you something where I didn't tell you 00:20:36.482 --> 00:20:37.940 align:middle line:84% the state variables in the circuit, 00:20:37.940 --> 00:20:41.520 align:middle line:84% and then you wouldn't know what the state of the devices were. 00:20:41.520 --> 00:20:44.270 align:middle line:90% But generally, yes. 00:20:44.270 --> 00:20:46.490 align:middle line:84% And I should say, by the way, that this notion 00:20:46.490 --> 00:20:50.030 align:middle line:84% of making some assumption and calculating, there's 00:20:50.030 --> 00:20:52.760 align:middle line:84% a lot of circuit simulators-- fast simulators, say, 00:20:52.760 --> 00:20:56.130 align:middle line:84% switching circuits-- that's exactly how they operate. 00:20:56.130 --> 00:20:58.310 align:middle line:84% And they go through, and then they change states, 00:20:58.310 --> 00:20:59.270 align:middle line:90% and they keep going. 00:20:59.270 --> 00:21:03.600 align:middle line:84% So you can do this, and you will get an answer. 00:21:03.600 --> 00:21:06.270 align:middle line:84% Those who are old timers in the field, like myself, 00:21:06.270 --> 00:21:08.040 align:middle line:84% at one point or another have written 00:21:08.040 --> 00:21:10.990 align:middle line:84% a simulator that works that way to simulate some circuit. 00:21:10.990 --> 00:21:14.010 align:middle line:84% These days, these circuit simulators, the general purpose 00:21:14.010 --> 00:21:17.070 align:middle line:84% ones are so good that you'd never bother writing your own. 00:21:17.070 --> 00:21:18.750 align:middle line:84% But nonetheless, it turns out to be very 00:21:18.750 --> 00:21:20.290 align:middle line:90% useful to analyze a circuit. 00:21:20.290 --> 00:21:22.890 align:middle line:84% So yes, generally you'll get a unique solution, 00:21:22.890 --> 00:21:24.840 align:middle line:84% assuming you know all the state variables. 00:21:24.840 --> 00:21:27.450 align:middle line:84% And you can think of that as if you built the thing, 00:21:27.450 --> 00:21:29.220 align:middle line:90% it had to do something, right? 00:21:29.220 --> 00:21:32.540 align:middle line:90% And it can only do one thing. 00:21:32.540 --> 00:21:34.940 align:middle line:84% So that's the method of assumed states, 00:21:34.940 --> 00:21:40.125 align:middle line:84% and we'll come back to this with much more complicated circuits. 00:21:40.125 --> 00:21:41.750 align:middle line:84% But I wanted to give you the notion of, 00:21:41.750 --> 00:21:45.130 align:middle line:84% when you have an uncontrolled device, what do you do? 00:21:45.130 --> 00:21:50.690 align:middle line:84% There is another topic I'd like to cover today 00:21:50.690 --> 00:21:55.040 align:middle line:84% that relates to how we generally analyze power converter 00:21:55.040 --> 00:21:55.920 align:middle line:90% circuits. 00:21:55.920 --> 00:22:00.487 align:middle line:84% And we said, well, I'm going to do something-- 00:22:00.487 --> 00:22:02.570 align:middle line:84% and there's all kinds of power converter circuits, 00:22:02.570 --> 00:22:04.237 align:middle line:84% but generally, I'm going to do something 00:22:04.237 --> 00:22:07.070 align:middle line:84% like this, where typically I'm going to operate something 00:22:07.070 --> 00:22:09.810 align:middle line:90% periodically. 00:22:09.810 --> 00:22:14.310 align:middle line:84% And then I want to analyze what its behavior is. 00:22:14.310 --> 00:22:19.530 align:middle line:84% And because we're operating things periodically, 00:22:19.530 --> 00:22:26.200 align:middle line:84% we can often end up with an operating regime 00:22:26.200 --> 00:22:30.280 align:middle line:84% that we will call periodic steady state. 00:22:30.280 --> 00:22:36.640 align:middle line:84% Or we abbreviate Periodic Steady State as PSS. 00:22:36.640 --> 00:22:38.560 align:middle line:84% Now, the notion of periodic steady state 00:22:38.560 --> 00:22:42.100 align:middle line:84% is a little bit like the notion of sinusoidal steady state, 00:22:42.100 --> 00:22:44.380 align:middle line:84% which you would have studied in linear circuits, 00:22:44.380 --> 00:22:47.020 align:middle line:84% except that we don't typically have all our waveforms being 00:22:47.020 --> 00:22:49.990 align:middle line:84% sinusoidal, because we have some kind of nonlinear circuit going 00:22:49.990 --> 00:22:52.180 align:middle line:90% on. 00:22:52.180 --> 00:22:53.990 align:middle line:90% But it's a related concept. 00:22:53.990 --> 00:22:55.240 align:middle line:90% And the idea is this. 00:22:55.240 --> 00:22:56.750 align:middle line:90% We're operating cyclically. 00:22:56.750 --> 00:22:59.830 align:middle line:84% Eventually what we expect is for every cycle of the circuit 00:22:59.830 --> 00:23:01.210 align:middle line:90% to do the same thing. 00:23:01.210 --> 00:23:04.790 align:middle line:84% So periodic steady state operation 00:23:04.790 --> 00:23:07.640 align:middle line:90% is something like this. 00:23:07.640 --> 00:23:10.640 align:middle line:84% So suppose I have some operating period T-- 00:23:10.640 --> 00:23:15.590 align:middle line:90% 2T, 3T, et cetera. 00:23:15.590 --> 00:23:18.050 align:middle line:84% I would be in periodic steady state operation 00:23:18.050 --> 00:23:21.120 align:middle line:84% if every operating cycle was some carbon copy. 00:23:21.120 --> 00:23:23.510 align:middle line:84% So if I looked at all the waveforms, 00:23:23.510 --> 00:23:25.880 align:middle line:84% maybe it would do-- if it did something in this cycle, 00:23:25.880 --> 00:23:28.800 align:middle line:84% it would do the precise same thing in the next cycle. 00:23:28.800 --> 00:23:31.350 align:middle line:84% The third cycle would be a carbon copy of that. 00:23:31.350 --> 00:23:33.950 align:middle line:84% That's periodic steady state operation. 00:23:33.950 --> 00:23:38.090 align:middle line:84% And what I mean is generally about all the waveforms 00:23:38.090 --> 00:23:39.110 align:middle line:90% in the circuit-- 00:23:39.110 --> 00:23:43.490 align:middle line:84% what I'm saying is this, if I just shifted it back a cycle, 00:23:43.490 --> 00:23:46.520 align:middle line:90% it's just repeating. 00:23:46.520 --> 00:23:48.780 align:middle line:84% Do we have to check all the waveforms in the circuit? 00:23:48.780 --> 00:23:51.530 align:middle line:84% Well, if we check the state variables, 00:23:51.530 --> 00:23:53.655 align:middle line:84% then we sort of know what the rest of the variables 00:23:53.655 --> 00:23:54.947 align:middle line:90% in the circuit are going to do. 00:23:54.947 --> 00:23:57.270 align:middle line:84% So I only really have to think about state variables. 00:23:57.270 --> 00:23:59.337 align:middle line:84% And generally, we could often just say look 00:23:59.337 --> 00:24:00.420 align:middle line:90% at the things of interest. 00:24:00.420 --> 00:24:04.460 align:middle line:84% And maybe if it takes this waveform 00:24:04.460 --> 00:24:06.870 align:middle line:84% x, whatever it is, a voltage or current in the circuit-- 00:24:06.870 --> 00:24:09.200 align:middle line:84% takes on some value at the beginning of the cycle 00:24:09.200 --> 00:24:12.360 align:middle line:84% and it returns to the exact same value at the end of the cycle, 00:24:12.360 --> 00:24:15.163 align:middle line:84% then probably it's also periodic steady state operation. 00:24:15.163 --> 00:24:16.580 align:middle line:84% So sometimes we don't really think 00:24:16.580 --> 00:24:18.288 align:middle line:84% about looking at every point, but we just 00:24:18.288 --> 00:24:21.260 align:middle line:84% look at the beginning and the end of each cycle. 00:24:21.260 --> 00:24:23.180 align:middle line:84% So that's periodic steady state, and we're 00:24:23.180 --> 00:24:25.250 align:middle line:84% going to see that it's really useful to think 00:24:25.250 --> 00:24:27.780 align:middle line:84% about circuits in periodic steady state operation. 00:24:27.780 --> 00:24:28.850 align:middle line:90% They don't have to be. 00:24:28.850 --> 00:24:30.530 align:middle line:84% Even a cyclically operated circuit 00:24:30.530 --> 00:24:32.630 align:middle line:84% may not be in periodic steady state. 00:24:32.630 --> 00:24:35.455 align:middle line:84% You could imagine some operating regime 00:24:35.455 --> 00:24:37.080 align:middle line:84% where you're going through a transient. 00:24:37.080 --> 00:24:41.210 align:middle line:84% We'll see one-- or maybe in this cycle it did this, in this cycle 00:24:41.210 --> 00:24:44.100 align:middle line:84% did this, in this cycle it did this. 00:24:44.100 --> 00:24:47.250 align:middle line:84% So we're switching things cyclically, 00:24:47.250 --> 00:24:49.890 align:middle line:84% but the waveforms are not repeating. 00:24:49.890 --> 00:24:53.176 align:middle line:84% And this would be clearly not PSS. 00:24:53.176 --> 00:24:56.250 align:middle line:90% 00:24:56.250 --> 00:25:00.790 align:middle line:84% So we often want to think about what will our circuits do 00:25:00.790 --> 00:25:02.590 align:middle line:84% in a steady state, when they're doing 00:25:02.590 --> 00:25:04.710 align:middle line:90% the same thing every cycle? 00:25:04.710 --> 00:25:06.700 align:middle line:84% We'll also think about these conditions. 00:25:06.700 --> 00:25:10.140 align:middle line:84% But we often like to start with this condition. 00:25:10.140 --> 00:25:11.500 align:middle line:90% Why is that? 00:25:11.500 --> 00:25:14.040 align:middle line:84% Because we can say a lot of things 00:25:14.040 --> 00:25:16.950 align:middle line:84% about what a circuit will do once 00:25:16.950 --> 00:25:21.020 align:middle line:84% it reaches periodic steady state. 00:25:21.020 --> 00:25:23.515 align:middle line:84% Let's just give an example of that. 00:25:23.515 --> 00:25:26.030 align:middle line:90% 00:25:26.030 --> 00:25:32.980 align:middle line:84% Suppose I have an inductor, and I have some V sub L 00:25:32.980 --> 00:25:34.802 align:middle line:84% voltage across the inductor and I sub 00:25:34.802 --> 00:25:36.580 align:middle line:90% L current into the inductor. 00:25:36.580 --> 00:25:43.570 align:middle line:84% What I know is that V sub L is equal to L dIL/dt 00:25:43.570 --> 00:25:47.530 align:middle line:84% OK, well, suppose I took the average 00:25:47.530 --> 00:25:51.480 align:middle line:84% of each side of this equation, the time average. 00:25:51.480 --> 00:25:53.032 align:middle line:84% And maybe I'll average over all time, 00:25:53.032 --> 00:25:54.490 align:middle line:84% or maybe I'll average over a cycle. 00:25:54.490 --> 00:25:57.000 align:middle line:84% But let's talk about averaging over all time. 00:25:57.000 --> 00:26:01.050 align:middle line:84% Then I could say, OK, the average V sub L is equal to L 00:26:01.050 --> 00:26:08.910 align:middle line:84% times-- is equal to the average of L dIL/dt, which is-- 00:26:08.910 --> 00:26:11.160 align:middle line:84% I could pull this L out of the average, which 00:26:11.160 --> 00:26:14.310 align:middle line:84% is the average of V sub L is equal to L times 00:26:14.310 --> 00:26:15.823 align:middle line:90% the average of dIL/dt. 00:26:15.823 --> 00:26:19.770 align:middle line:90% 00:26:19.770 --> 00:26:21.540 align:middle line:90% So what does that tell me? 00:26:21.540 --> 00:26:24.210 align:middle line:84% Well, if I look at the periodic steady state waveform, 00:26:24.210 --> 00:26:29.040 align:middle line:84% suppose, for example, I said this was some random variable x. 00:26:29.040 --> 00:26:35.750 align:middle line:84% Suppose x was equal to I sub L. What is dIL/dt on average 00:26:35.750 --> 00:26:37.420 align:middle line:90% of this waveform? 00:26:37.420 --> 00:26:40.090 align:middle line:84% Well, the current went up, the current came down, went up, 00:26:40.090 --> 00:26:40.950 align:middle line:90% came down, right? 00:26:40.950 --> 00:26:43.540 align:middle line:84% On average, the current is not going anywhere. 00:26:43.540 --> 00:26:45.890 align:middle line:90% On average, dIL/dt is 0. 00:26:45.890 --> 00:26:48.130 align:middle line:90% I'm just repeating every cycle. 00:26:48.130 --> 00:26:53.740 align:middle line:84% So what I can say is that in this case, 00:26:53.740 --> 00:27:00.360 align:middle line:90% in PSS, this is equal to 0. 00:27:00.360 --> 00:27:03.600 align:middle line:84% If I have an inductor and periodic steady state, 00:27:03.600 --> 00:27:05.790 align:middle line:84% the average voltage across it has 00:27:05.790 --> 00:27:09.940 align:middle line:84% to be 0 because, on average, the current through is not changing. 00:27:09.940 --> 00:27:12.360 align:middle line:84% It might be doing something cyclically, 00:27:12.360 --> 00:27:15.020 align:middle line:84% but it's just sitting there on average. 00:27:15.020 --> 00:27:23.120 align:middle line:84% So what we can say is that for inductors, the average value 00:27:23.120 --> 00:27:28.670 align:middle line:84% of the voltage across an inductor is equal to 0 in PSS. 00:27:28.670 --> 00:27:34.400 align:middle line:84% I could have made the same argument about capacitors. 00:27:34.400 --> 00:27:36.860 align:middle line:84% The average value of a current through a capacitor 00:27:36.860 --> 00:27:41.150 align:middle line:90% is equal to 0 in PSS. 00:27:41.150 --> 00:27:48.590 align:middle line:84% So suddenly I can start having an understanding of requirements 00:27:48.590 --> 00:27:51.650 align:middle line:84% on the circuit operation if I'm sitting there 00:27:51.650 --> 00:27:53.810 align:middle line:84% in periodic steady state operation, which 00:27:53.810 --> 00:27:57.110 align:middle line:84% is very often where we start our analysis. 00:27:57.110 --> 00:27:59.930 align:middle line:84% So if I want to know what some cyclically operating 00:27:59.930 --> 00:28:07.330 align:middle line:84% circuit like this is doing, I might just start there. 00:28:07.330 --> 00:28:10.350 align:middle line:84% And in fact, I did that subtly when I analyzed the circuit. 00:28:10.350 --> 00:28:13.110 align:middle line:84% We said this is a low pass filter. 00:28:13.110 --> 00:28:16.630 align:middle line:84% The output voltage extracts the DC component of Vx. 00:28:16.630 --> 00:28:18.930 align:middle line:84% That's precisely the same thing as saying 00:28:18.930 --> 00:28:22.880 align:middle line:84% the average voltage across the inductor is 0. 00:28:22.880 --> 00:28:27.230 align:middle line:84% So we actually already used that subtly, 00:28:27.230 --> 00:28:30.030 align:middle line:84% but we just recognized that it was a low pass filter. 00:28:30.030 --> 00:28:33.020 align:middle line:84% But it applies to any element in your circuit. 00:28:33.020 --> 00:28:36.470 align:middle line:90% This is very much like saying-- 00:28:36.470 --> 00:28:38.630 align:middle line:84% the average voltage across an inductor as being 0 00:28:38.630 --> 00:28:42.140 align:middle line:84% is like saying at DC, or in steady state when 00:28:42.140 --> 00:28:46.070 align:middle line:84% nothing's changing, an inductor looks like a short. 00:28:46.070 --> 00:28:49.070 align:middle line:84% Or in DC or its steady state, when nothing's changing, 00:28:49.070 --> 00:28:54.560 align:middle line:84% a capacitor looks like an open, has no current on average. 00:28:54.560 --> 00:28:56.930 align:middle line:90% So that's the basic idea. 00:28:56.930 --> 00:28:59.833 align:middle line:84% But what we're going to do going forward 00:28:59.833 --> 00:29:02.000 align:middle line:84% is we're going to start to figure out how to analyze 00:29:02.000 --> 00:29:04.120 align:middle line:90% all kinds of circuits. 00:29:04.120 --> 00:29:07.000 align:middle line:84% And we're going to-- this is a very useful tool 00:29:07.000 --> 00:29:09.730 align:middle line:84% to figure out what they do in the periodic steady state, which 00:29:09.730 --> 00:29:12.640 align:middle line:84% is a very good first step to figure out 00:29:12.640 --> 00:29:16.890 align:middle line:84% what they're going to do in the nonperiodic steady state 00:29:16.890 --> 00:29:17.390 align:middle line:90% condition. 00:29:17.390 --> 00:29:19.710 align:middle line:84% But we're often operating in periodic steady state, 00:29:19.710 --> 00:29:21.120 align:middle line:90% so that's why we look at it. 00:29:21.120 --> 00:29:23.270 align:middle line:84% And I should say, it's not uncommon-- 00:29:23.270 --> 00:29:26.460 align:middle line:90% 00:29:26.460 --> 00:29:28.303 align:middle line:84% as a consultant, for example, there's 00:29:28.303 --> 00:29:29.970 align:middle line:84% some problem with their power converter, 00:29:29.970 --> 00:29:31.830 align:middle line:84% they ship you up their documents. 00:29:31.830 --> 00:29:34.180 align:middle line:84% Here's this circuit, and they tell you nothing else. 00:29:34.180 --> 00:29:35.740 align:middle line:90% And what do you have to do? 00:29:35.740 --> 00:29:40.040 align:middle line:84% The first thing you have to do is figure out, what will it do? 00:29:40.040 --> 00:29:41.157 align:middle line:90% How will it operate? 00:29:41.157 --> 00:29:42.740 align:middle line:84% And the first place to start with that 00:29:42.740 --> 00:29:44.813 align:middle line:84% is, how does it operate in steady state? 00:29:44.813 --> 00:29:47.480 align:middle line:84% And of course, sometimes they'll tell you, well, let me tell you 00:29:47.480 --> 00:29:48.720 align:middle line:90% about how my circuit operates. 00:29:48.720 --> 00:29:50.435 align:middle line:84% But of course, they wouldn't be calling you 00:29:50.435 --> 00:29:51.893 align:middle line:84% if it did what they thought it did. 00:29:51.893 --> 00:29:55.640 align:middle line:84% So very often you have to tell them why. 00:29:55.640 --> 00:29:58.160 align:middle line:84% And tools like this are very good at figuring 00:29:58.160 --> 00:30:03.640 align:middle line:84% out what is it going to do and why is it going to do it. 00:30:03.640 --> 00:30:09.730 align:middle line:84% So let me just give you a super simple example of that 00:30:09.730 --> 00:30:12.460 align:middle line:84% and how it can inform your thinking. 00:30:12.460 --> 00:30:14.610 align:middle line:84% And here's the circuit I'm going to do. 00:30:14.610 --> 00:30:21.880 align:middle line:84% Suppose we came back to this circuit. 00:30:21.880 --> 00:30:25.610 align:middle line:84% And what I got when I looked at that output voltage, 00:30:25.610 --> 00:30:34.000 align:middle line:84% this voltage here is a periodic steady state voltage. 00:30:34.000 --> 00:30:35.650 align:middle line:90% It actually has a DC component. 00:30:35.650 --> 00:30:38.230 align:middle line:84% So that thing is actually the world's simplest rectifier. 00:30:38.230 --> 00:30:41.850 align:middle line:84% It takes an AC voltage and generates an output voltage 00:30:41.850 --> 00:30:43.570 align:middle line:90% that has a DC component. 00:30:43.570 --> 00:30:45.610 align:middle line:90% That's why I would use a switch. 00:30:45.610 --> 00:30:47.830 align:middle line:84% I've now taken AC and turned it into DC. 00:30:47.830 --> 00:30:51.030 align:middle line:84% You can't do that with a linear circuit element, but a diode 00:30:51.030 --> 00:30:52.860 align:middle line:90% you can use to do that. 00:30:52.860 --> 00:30:55.530 align:middle line:84% So I might use this if I had some AC voltage 00:30:55.530 --> 00:30:59.800 align:middle line:84% and I wanted an output with a DC component to it. 00:30:59.800 --> 00:31:04.510 align:middle line:84% Or what I merely might want is a constant DC output. 00:31:04.510 --> 00:31:09.370 align:middle line:84% Well, what did we do last time we wanted a constant DC output? 00:31:09.370 --> 00:31:11.740 align:middle line:84% Last class, when we analyzed our converter, 00:31:11.740 --> 00:31:13.615 align:middle line:84% we kind of generated a pulsing waveform, 00:31:13.615 --> 00:31:14.990 align:middle line:84% and then we threw a filter on it. 00:31:14.990 --> 00:31:17.740 align:middle line:84% Maybe I should just throw an inductive filter on this thing 00:31:17.740 --> 00:31:20.260 align:middle line:84% and I can get now constant output, 00:31:20.260 --> 00:31:25.480 align:middle line:84% which I might want if I'm going to run some DC load with it. 00:31:25.480 --> 00:31:30.430 align:middle line:84% OK, so let's try that just as an example and see where we get. 00:31:30.430 --> 00:31:32.770 align:middle line:84% So here's the circuit I'm going to propose. 00:31:32.770 --> 00:31:35.110 align:middle line:90% And we'll take a look at this. 00:31:35.110 --> 00:31:38.290 align:middle line:90% Here is Vs sine omega t. 00:31:38.290 --> 00:31:42.860 align:middle line:90% 00:31:42.860 --> 00:31:45.920 align:middle line:90% Here's my diode. 00:31:45.920 --> 00:31:48.620 align:middle line:84% I'm just going to now put an inductor. 00:31:48.620 --> 00:31:51.370 align:middle line:90% And here's my output. 00:31:51.370 --> 00:31:59.010 align:middle line:84% So I can look at some voltage that I'll call, for example-- 00:31:59.010 --> 00:32:03.410 align:middle line:90% 00:32:03.410 --> 00:32:05.210 align:middle line:90% I'll call this ID. 00:32:05.210 --> 00:32:08.010 align:middle line:90% 00:32:08.010 --> 00:32:10.520 align:middle line:84% Let me now-- I have some distinction 00:32:10.520 --> 00:32:14.880 align:middle line:84% now between the voltage here and the voltage at the output. 00:32:14.880 --> 00:32:15.990 align:middle line:90% But let me do this. 00:32:15.990 --> 00:32:23.010 align:middle line:84% Let me call this voltage Vx, and I'll call this voltage V out. 00:32:23.010 --> 00:32:27.788 align:middle line:90% 00:32:27.788 --> 00:32:33.270 align:middle line:84% And let's think about analyzing this circuit. 00:32:33.270 --> 00:32:38.100 align:middle line:84% Well, my goal was to use this inductor 00:32:38.100 --> 00:32:41.540 align:middle line:84% so that ID, which is also IL here, is smooth. 00:32:41.540 --> 00:32:43.250 align:middle line:84% So maybe the assumption I could make 00:32:43.250 --> 00:32:45.375 align:middle line:84% is let me just start by assuming this diode is just 00:32:45.375 --> 00:32:46.910 align:middle line:90% on all the time. 00:32:46.910 --> 00:32:48.448 align:middle line:90% I'll make that assumption. 00:32:48.448 --> 00:32:50.240 align:middle line:84% We said we're going to analyze the circuit. 00:32:50.240 --> 00:32:52.032 align:middle line:84% We'll make some assumption about the state. 00:32:52.032 --> 00:32:53.520 align:middle line:84% Let me just assume the diode is on. 00:32:53.520 --> 00:32:58.910 align:middle line:84% So in that case, I got Vs sine omega t-- 00:32:58.910 --> 00:33:00.750 align:middle line:84% I'll assume that the diode is always on. 00:33:00.750 --> 00:33:02.330 align:middle line:90% I'll replace it with a short. 00:33:02.330 --> 00:33:05.665 align:middle line:84% I have my inductor, and I have my output. 00:33:05.665 --> 00:33:08.396 align:middle line:90% 00:33:08.396 --> 00:33:10.780 align:middle line:90% And here's my diode current, ID. 00:33:10.780 --> 00:33:14.090 align:middle line:90% 00:33:14.090 --> 00:33:16.710 align:middle line:84% And here's my output voltage, V out. 00:33:16.710 --> 00:33:20.490 align:middle line:90% 00:33:20.490 --> 00:33:23.860 align:middle line:90% Let me analyze this circuit. 00:33:23.860 --> 00:33:27.200 align:middle line:84% So I'll just do KVL around the loop here. 00:33:27.200 --> 00:33:35.240 align:middle line:84% So I've got Vs sine omega t minus VL-- 00:33:35.240 --> 00:33:36.440 align:middle line:90% I'll call that VL-- 00:33:36.440 --> 00:33:40.540 align:middle line:90% minus V out is equal to 0. 00:33:40.540 --> 00:33:43.660 align:middle line:90% 00:33:43.660 --> 00:33:45.368 align:middle line:84% Well, this is a legitimate equation. 00:33:45.368 --> 00:33:47.660 align:middle line:84% I could take the average of each side of this equation, 00:33:47.660 --> 00:33:50.240 align:middle line:84% and it's still a legitimate equation. 00:33:50.240 --> 00:33:53.400 align:middle line:84% So maybe what I'll do is I'll take this KVL equation-- 00:33:53.400 --> 00:33:55.730 align:middle line:84% I'll take the average of the whole thing, 00:33:55.730 --> 00:34:01.500 align:middle line:84% and that gives me the average of each individual component. 00:34:01.500 --> 00:34:05.544 align:middle line:84% What's the average of Vs sine omega t? 00:34:05.544 --> 00:34:06.440 align:middle line:90% AUDIENCE: 0? 00:34:06.440 --> 00:34:08.389 align:middle line:90% DAVID PERREAULT: 0. 00:34:08.389 --> 00:34:09.560 align:middle line:90% That goes away. 00:34:09.560 --> 00:34:11.969 align:middle line:84% What is the average of the inductor voltage? 00:34:11.969 --> 00:34:15.846 align:middle line:84% Well, I just told you that if I'm in periodic steady state 00:34:15.846 --> 00:34:17.929 align:middle line:84% and maybe I'm interested in what this circuit does 00:34:17.929 --> 00:34:20.600 align:middle line:84% in the periodic steady state, this 00:34:20.600 --> 00:34:29.969 align:middle line:84% is 0 in PSS, which just gives me then the average voltage of V 00:34:29.969 --> 00:34:36.719 align:middle line:84% out would be 0 in that case because the other two terms are 00:34:36.719 --> 00:34:39.580 align:middle line:84% 0, and this is the only thing left. 00:34:39.580 --> 00:34:41.330 align:middle line:84% Well, something's funny about that, right? 00:34:41.330 --> 00:34:43.699 align:middle line:84% I just told you we were building a rectifier, 00:34:43.699 --> 00:34:45.525 align:middle line:84% and now I've told you I've got a 0 output 00:34:45.525 --> 00:34:47.900 align:middle line:84% voltage, which isn't exactly what a rectifier is supposed 00:34:47.900 --> 00:34:48.659 align:middle line:90% to do. 00:34:48.659 --> 00:34:50.659 align:middle line:90% Something's funny here. 00:34:50.659 --> 00:34:54.080 align:middle line:84% What's funny is my assumption's stupid. 00:34:54.080 --> 00:34:56.265 align:middle line:84% I assumed that this diode was on all the time. 00:34:56.265 --> 00:34:57.890 align:middle line:84% Well, if the diode was on all the time, 00:34:57.890 --> 00:34:59.515 align:middle line:84% I suppose I could have just replaced it 00:34:59.515 --> 00:35:00.912 align:middle line:90% with a short circuit, right? 00:35:00.912 --> 00:35:05.390 align:middle line:84% But what this voltage being 0 in periodic steady state 00:35:05.390 --> 00:35:07.320 align:middle line:90% told me is I've made a mistake. 00:35:07.320 --> 00:35:09.560 align:middle line:84% I've done something, perhaps not stupid, 00:35:09.560 --> 00:35:12.950 align:middle line:84% but I've gotten over aggressive with my assumption 00:35:12.950 --> 00:35:17.420 align:middle line:84% that this diode stays on all the time. 00:35:17.420 --> 00:35:20.720 align:middle line:84% And in fact, I can tell that because if the average voltage 00:35:20.720 --> 00:35:24.300 align:middle line:84% was 0, and presumably some of the time 00:35:24.300 --> 00:35:26.940 align:middle line:84% this current is positive, so I have a positive voltage-- 00:35:26.940 --> 00:35:30.390 align:middle line:84% and if it's average is going to be 0, then some of the voltage 00:35:30.390 --> 00:35:32.550 align:middle line:84% must be negative some of the time. 00:35:32.550 --> 00:35:34.260 align:middle line:84% And if the voltage across the resistor 00:35:34.260 --> 00:35:36.090 align:middle line:84% is negative some of the time, the current through the resistor 00:35:36.090 --> 00:35:37.590 align:middle line:90% is negative some of the time. 00:35:37.590 --> 00:35:40.560 align:middle line:84% But we already said this diode can never 00:35:40.560 --> 00:35:42.840 align:middle line:90% have a negative current. 00:35:42.840 --> 00:35:47.760 align:middle line:84% So clearly my assumption was bad. 00:35:47.760 --> 00:35:52.390 align:middle line:84% Or when I assumed that this diode was turned on, 00:35:52.390 --> 00:35:53.880 align:middle line:84% that might be true for some time, 00:35:53.880 --> 00:35:56.130 align:middle line:84% but it can't be true all the time. 00:35:56.130 --> 00:36:02.820 align:middle line:84% So my quest to use this inductor to filter the waveforms maybe 00:36:02.820 --> 00:36:06.630 align:middle line:84% can't work out or at least as I've done it so far. 00:36:06.630 --> 00:36:10.620 align:middle line:84% What would I get for waveforms if I did this? 00:36:10.620 --> 00:36:12.630 align:middle line:84% What you would see is something like this. 00:36:12.630 --> 00:36:15.380 align:middle line:90% 00:36:15.380 --> 00:36:18.380 align:middle line:84% With this circuit what I would get is this. 00:36:18.380 --> 00:36:24.040 align:middle line:90% 00:36:24.040 --> 00:36:26.030 align:middle line:90% Here's my Vs sine omega t. 00:36:26.030 --> 00:36:28.580 align:middle line:90% 00:36:28.580 --> 00:36:32.690 align:middle line:84% I know my diode has to be off some of the time. 00:36:32.690 --> 00:36:37.410 align:middle line:84% Well, let me assume that right where Vs sine omega t is 0, 00:36:37.410 --> 00:36:38.200 align:middle line:90% the diode is off. 00:36:38.200 --> 00:36:41.840 align:middle line:90% 00:36:41.840 --> 00:36:44.090 align:middle line:84% All right, so if I assume this diode is off 00:36:44.090 --> 00:36:45.955 align:middle line:84% and I replace this with an open circuit-- 00:36:45.955 --> 00:36:48.980 align:middle line:90% 00:36:48.980 --> 00:36:50.250 align:middle line:90% so I'll just go do that. 00:36:50.250 --> 00:36:51.980 align:middle line:90% Let me just assume-- 00:36:51.980 --> 00:36:53.550 align:middle line:90% I just went through 0. 00:36:53.550 --> 00:36:55.910 align:middle line:90% It's an open circuit. 00:36:55.910 --> 00:36:57.000 align:middle line:90% What do I know? 00:36:57.000 --> 00:36:59.420 align:middle line:84% Well, I know that if there's no current-- 00:36:59.420 --> 00:37:02.150 align:middle line:84% if this is an open, there's no current this way. 00:37:02.150 --> 00:37:05.630 align:middle line:84% That means there's also no dI/dt, Which 00:37:05.630 --> 00:37:08.390 align:middle line:84% means that the voltage across the inductor 00:37:08.390 --> 00:37:10.670 align:middle line:90% is 0 because V is LDI/dt. 00:37:10.670 --> 00:37:13.590 align:middle line:84% Voltage across the resistor is 0 because the current is 0. 00:37:13.590 --> 00:37:17.690 align:middle line:90% And that means that this VD-- 00:37:17.690 --> 00:37:20.480 align:middle line:84% since this is 0 voltage and this is 0 voltage, 00:37:20.480 --> 00:37:24.870 align:middle line:84% VD is the same as Vs sine omega t. 00:37:24.870 --> 00:37:28.550 align:middle line:84% So what I can tell is that, geez, 00:37:28.550 --> 00:37:30.590 align:middle line:90% if the diode was off here-- 00:37:30.590 --> 00:37:33.800 align:middle line:84% and I know it's off some time, so probably maybe up 00:37:33.800 --> 00:37:36.800 align:middle line:84% to some point here the diode is off. 00:37:36.800 --> 00:37:39.500 align:middle line:84% But as soon as I cross 0, that assumption 00:37:39.500 --> 00:37:42.360 align:middle line:84% is bad because VD would no longer be negative. 00:37:42.360 --> 00:37:44.760 align:middle line:90% It would be positive. 00:37:44.760 --> 00:37:50.610 align:middle line:84% So the diode must turn on right at t equals 0-- 00:37:50.610 --> 00:37:52.090 align:middle line:90% or omega t equals 0. 00:37:52.090 --> 00:37:55.180 align:middle line:84% So the first part of what I did was correct. 00:37:55.180 --> 00:37:56.700 align:middle line:90% The diode turns on. 00:37:56.700 --> 00:38:00.830 align:middle line:84% Vx when the diode is on is exactly 00:38:00.830 --> 00:38:06.330 align:middle line:84% going to follow Vs sine omega t for t greater than 0. 00:38:06.330 --> 00:38:11.640 align:middle line:84% So it's going to follow the same argument. 00:38:11.640 --> 00:38:14.540 align:middle line:84% What's going to happen here, though? 00:38:14.540 --> 00:38:16.350 align:middle line:84% Is the diode going to turn off here? 00:38:16.350 --> 00:38:19.590 align:middle line:84% Well, let's think about what this current would be. 00:38:19.590 --> 00:38:21.990 align:middle line:84% I could analyze, and I could say, look, 00:38:21.990 --> 00:38:28.120 align:middle line:84% at t equals 0 plus, inductor current 0, 00:38:28.120 --> 00:38:29.580 align:middle line:84% and I've replaced-- at that point, 00:38:29.580 --> 00:38:32.710 align:middle line:84% I've replaced this switch with a short circuit. 00:38:32.710 --> 00:38:35.635 align:middle line:90% I can analyze that circuit. 00:38:35.635 --> 00:38:37.243 align:middle line:90% That circuit's just this. 00:38:37.243 --> 00:38:43.190 align:middle line:90% 00:38:43.190 --> 00:38:49.510 align:middle line:84% I can analyze this circuit for the inductor current, 00:38:49.510 --> 00:38:52.730 align:middle line:90% knowing that IL of 0 is 0. 00:38:52.730 --> 00:38:55.370 align:middle line:84% And if I plotted that, what I would get 00:38:55.370 --> 00:38:58.220 align:middle line:84% is something that did, I don't know, something like this. 00:38:58.220 --> 00:39:02.030 align:middle line:84% It would start off very small, and it would get bigger. 00:39:02.030 --> 00:39:03.440 align:middle line:90% And then it would-- 00:39:03.440 --> 00:39:04.940 align:middle line:84% eventually, it would cross through 0 00:39:04.940 --> 00:39:07.430 align:middle line:84% because the voltage would get negative and negative-- more 00:39:07.430 --> 00:39:10.040 align:middle line:90% and more negative. 00:39:10.040 --> 00:39:13.020 align:middle line:84% And then it would keep going, et cetera. 00:39:13.020 --> 00:39:14.220 align:middle line:90% But what do I know? 00:39:14.220 --> 00:39:16.250 align:middle line:84% I know that IL can never go negative 00:39:16.250 --> 00:39:20.240 align:middle line:84% because the diode current would go negative. 00:39:20.240 --> 00:39:26.870 align:middle line:84% So what would happen is if this was ID, 00:39:26.870 --> 00:39:31.170 align:middle line:84% the diode would stay on as long as ID is positive. 00:39:31.170 --> 00:39:34.950 align:middle line:84% And when ID tried to go negative, 00:39:34.950 --> 00:39:38.620 align:middle line:84% this would go negative, and the diode would just turn off, 00:39:38.620 --> 00:39:40.080 align:middle line:90% and ID would go to 0. 00:39:40.080 --> 00:39:41.470 align:middle line:90% It'd do something like this. 00:39:41.470 --> 00:39:45.970 align:middle line:84% Now, what does exactly that solution look like? 00:39:45.970 --> 00:39:47.140 align:middle line:90% We can calculate that. 00:39:47.140 --> 00:39:49.060 align:middle line:84% It depends on the parameter values. 00:39:49.060 --> 00:39:52.870 align:middle line:84% But what we can tell from all this is that at some point, 00:39:52.870 --> 00:39:55.010 align:middle line:90% this diode is going to turn off. 00:39:55.010 --> 00:39:56.730 align:middle line:84% And it's going to limit the current. 00:39:56.730 --> 00:39:59.272 align:middle line:84% And the current is not going to be a smooth, constant current 00:39:59.272 --> 00:40:00.860 align:middle line:84% like I might want if I wanted a purely 00:40:00.860 --> 00:40:02.215 align:middle line:90% DC output current and voltage. 00:40:02.215 --> 00:40:06.930 align:middle line:90% 00:40:06.930 --> 00:40:10.990 align:middle line:84% Just to illustrate that, I've got a simulation 00:40:10.990 --> 00:40:13.000 align:middle line:90% here to show you. 00:40:13.000 --> 00:40:15.400 align:middle line:84% And all I did was go into LTspice, 00:40:15.400 --> 00:40:23.000 align:middle line:84% which I encourage you to all download, and simulate it. 00:40:23.000 --> 00:40:27.220 align:middle line:84% So here we have a sinusoidal voltage source, 00:40:27.220 --> 00:40:29.030 align:middle line:90% an almost ideal diode. 00:40:29.030 --> 00:40:31.240 align:middle line:84% We gave it a small forward drop, but if we gave it 00:40:31.240 --> 00:40:33.050 align:middle line:84% a 0 forward drop, it would be ideal. 00:40:33.050 --> 00:40:36.010 align:middle line:84% We put in a big inductor and a resistor. 00:40:36.010 --> 00:40:43.130 align:middle line:84% And here you can see in red, on the left axis, we have voltage. 00:40:43.130 --> 00:40:47.500 align:middle line:84% It's just a sinusoidal voltage, and the 0 is here. 00:40:47.500 --> 00:40:51.260 align:middle line:84% In the right-hand axis, you can see the diode current. 00:40:51.260 --> 00:40:53.230 align:middle line:84% And what you can see is right when-- 00:40:53.230 --> 00:40:57.820 align:middle line:84% just after the voltage crosses 0 positive, 00:40:57.820 --> 00:40:59.860 align:middle line:84% the diode current starts to ramp up. 00:40:59.860 --> 00:41:01.370 align:middle line:90% It goes up and up and up. 00:41:01.370 --> 00:41:03.430 align:middle line:90% It hits about just under an amp. 00:41:03.430 --> 00:41:09.220 align:middle line:84% But then the voltage gets negative and, boom, turns off. 00:41:09.220 --> 00:41:13.640 align:middle line:84% If I plotted this voltage as well-- 00:41:13.640 --> 00:41:15.680 align:middle line:90% oops, I'm not sure it gave me-- 00:41:15.680 --> 00:41:20.020 align:middle line:84% see, we can see this thing where the diode-- 00:41:20.020 --> 00:41:25.200 align:middle line:84% the voltage here, which I plotted in yellow, 00:41:25.200 --> 00:41:26.830 align:middle line:90% jumps back to 0. 00:41:26.830 --> 00:41:28.760 align:middle line:84% This particular parameter is just 00:41:28.760 --> 00:41:31.290 align:middle line:84% letting go much more of the cycle. 00:41:31.290 --> 00:41:34.230 align:middle line:84% Does that make sense to everybody? 00:41:34.230 --> 00:41:35.435 align:middle line:90% Any questions about that? 00:41:35.435 --> 00:41:39.810 align:middle line:90% 00:41:39.810 --> 00:41:42.330 align:middle line:90% So what would I do to fix this? 00:41:42.330 --> 00:41:46.560 align:middle line:84% The real problem I have is what's going on here? 00:41:46.560 --> 00:41:48.300 align:middle line:90% This voltage went positive. 00:41:48.300 --> 00:41:49.200 align:middle line:90% Diode turns on. 00:41:49.200 --> 00:41:53.070 align:middle line:84% I start shoving current through the inductor. 00:41:53.070 --> 00:41:56.350 align:middle line:84% Voltage goes negative-- what I wanted to do 00:41:56.350 --> 00:42:03.330 align:middle line:84% and what happened over here was that when the voltage went 00:42:03.330 --> 00:42:05.730 align:middle line:84% negative, the diode shut off, and then I 00:42:05.730 --> 00:42:07.300 align:middle line:90% got 0, which is what I wanted. 00:42:07.300 --> 00:42:09.750 align:middle line:84% So I had the positive part of the sine wave 00:42:09.750 --> 00:42:12.180 align:middle line:84% and then 0 for the other half, and I got an average value. 00:42:12.180 --> 00:42:15.120 align:middle line:84% Here, what happens is the inductor 00:42:15.120 --> 00:42:16.590 align:middle line:84% forces the diode to stay on, and I 00:42:16.590 --> 00:42:20.610 align:middle line:84% get this chunk of negative voltage until the diode voltage 00:42:20.610 --> 00:42:21.760 align:middle line:90% averages to 0-- 00:42:21.760 --> 00:42:23.760 align:middle line:84% I'm sorry, until the inductor current averages-- 00:42:23.760 --> 00:42:28.170 align:middle line:84% until the inductor voltage averages to 0, in which case 00:42:28.170 --> 00:42:30.820 align:middle line:90% the circuit switches. 00:42:30.820 --> 00:42:32.830 align:middle line:84% What could I do if I wanted to do 00:42:32.830 --> 00:42:38.990 align:middle line:84% something more like this waveform but smooth the output? 00:42:38.990 --> 00:42:41.030 align:middle line:84% Well, if I got clever, I might say, 00:42:41.030 --> 00:42:48.050 align:middle line:84% hey, why don't I build instead this circuit? 00:42:48.050 --> 00:42:51.770 align:middle line:84% I will build this, but instead of just having D, 00:42:51.770 --> 00:42:54.050 align:middle line:90% let me call this D1. 00:42:54.050 --> 00:42:56.300 align:middle line:90% And I'll call this then ID1. 00:42:56.300 --> 00:42:59.880 align:middle line:90% 00:42:59.880 --> 00:43:04.630 align:middle line:84% My problem is that Vx is going negative. 00:43:04.630 --> 00:43:08.890 align:middle line:84% And then that means that instead of getting this positive bump 00:43:08.890 --> 00:43:11.142 align:middle line:84% and then 0, I'm getting this negative part 00:43:11.142 --> 00:43:13.350 align:middle line:84% because the inductor current is holding the diode on. 00:43:13.350 --> 00:43:16.480 align:middle line:84% Well, maybe I can just get clever, come in, and add 00:43:16.480 --> 00:43:19.810 align:middle line:90% a second diode here, D2. 00:43:19.810 --> 00:43:25.024 align:middle line:90% 00:43:25.024 --> 00:43:29.310 align:middle line:84% This diode D2 can take the inductor current. 00:43:29.310 --> 00:43:34.047 align:middle line:84% So if I have I sub L, now I sub D1 00:43:34.047 --> 00:43:35.630 align:middle line:84% is going to be different than I sub L, 00:43:35.630 --> 00:43:38.150 align:middle line:84% so I guess I'll have to give it a different color. 00:43:38.150 --> 00:43:43.130 align:middle line:84% Let me call this I sub L, if anybody can see that. 00:43:43.130 --> 00:43:47.480 align:middle line:90% So I sub L can now be big. 00:43:47.480 --> 00:43:49.730 align:middle line:84% It doesn't have to go through D1. 00:43:49.730 --> 00:43:51.485 align:middle line:84% Instead, it can go through this diode D2. 00:43:51.485 --> 00:43:55.140 align:middle line:90% 00:43:55.140 --> 00:43:56.950 align:middle line:84% Well, how would I analyze this circuit? 00:43:56.950 --> 00:43:59.420 align:middle line:84% Well, I can do the method of assumed states. 00:43:59.420 --> 00:44:03.870 align:middle line:84% But now I have two switches, which could give me, in theory, 00:44:03.870 --> 00:44:05.470 align:middle line:90% up to four possible states-- 00:44:05.470 --> 00:44:09.030 align:middle line:84% both off; one off, one on; the other off, the other on; 00:44:09.030 --> 00:44:11.010 align:middle line:90% both on. 00:44:11.010 --> 00:44:14.040 align:middle line:84% But if I'm clever, maybe I assume that they're not both 00:44:14.040 --> 00:44:17.400 align:middle line:84% on at the same time, typically, or off at the same time, 00:44:17.400 --> 00:44:18.960 align:middle line:90% hopefully. 00:44:18.960 --> 00:44:21.300 align:middle line:84% So what would I do if I analyze this circuit? 00:44:21.300 --> 00:44:24.180 align:middle line:84% And let me just assume the inductor is pretty big so 00:44:24.180 --> 00:44:27.545 align:middle line:84% that IL is going to be approximately constant. 00:44:27.545 --> 00:44:30.320 align:middle line:90% What would I get in that case? 00:44:30.320 --> 00:44:32.870 align:middle line:90% I get something like this. 00:44:32.870 --> 00:44:34.990 align:middle line:90% Here's my sinusoidal voltage. 00:44:34.990 --> 00:44:38.230 align:middle line:90% 00:44:38.230 --> 00:44:41.810 align:middle line:84% If my inductor current is positive-- 00:44:41.810 --> 00:44:44.330 align:middle line:90% suppose D2 was on-- 00:44:44.330 --> 00:44:51.710 align:middle line:84% as soon as Vs sine omega t crosses 0 positive, if D2 is on, 00:44:51.710 --> 00:44:55.110 align:middle line:84% this voltage would then get to be positive. 00:44:55.110 --> 00:44:58.880 align:middle line:84% So I know as soon as I cross 0 positive, 00:44:58.880 --> 00:45:04.650 align:middle line:84% D1 is going to be on in this range. 00:45:04.650 --> 00:45:08.350 align:middle line:84% And I'm going to get a voltage Vx that looks like this. 00:45:08.350 --> 00:45:13.400 align:middle line:90% It's going to follow this. 00:45:13.400 --> 00:45:16.700 align:middle line:84% Now, what happens when x goes negative? 00:45:16.700 --> 00:45:22.470 align:middle line:84% If I assume that D1 one is on and Vs sine omega 00:45:22.470 --> 00:45:27.690 align:middle line:84% t becomes negative and D2 is off, 00:45:27.690 --> 00:45:29.436 align:middle line:90% I'd have this configuration. 00:45:29.436 --> 00:45:34.020 align:middle line:90% 00:45:34.020 --> 00:45:37.770 align:middle line:84% I'm assuming he's on, and I'm assuming he's off. 00:45:37.770 --> 00:45:42.930 align:middle line:84% And my inductor current is doing something here. 00:45:42.930 --> 00:45:46.410 align:middle line:90% This diode 2 voltage-- 00:45:46.410 --> 00:45:52.260 align:middle line:84% if Vs sine omega t becomes negative, 00:45:52.260 --> 00:45:55.990 align:middle line:84% that means this diode voltage would become positive. 00:45:55.990 --> 00:45:59.170 align:middle line:84% Can I have a positive diode voltage? 00:45:59.170 --> 00:45:59.830 align:middle line:90% No. 00:45:59.830 --> 00:46:02.770 align:middle line:84% That means that this diode must turn 00:46:02.770 --> 00:46:05.690 align:middle line:90% on when this voltage goes to 0. 00:46:05.690 --> 00:46:09.520 align:middle line:84% So at the minimum, I know that D2 two has to be on here. 00:46:09.520 --> 00:46:12.370 align:middle line:84% Now, if I assume that both D1 and D2 are on here, 00:46:12.370 --> 00:46:13.670 align:middle line:90% I'd be shorting out the source. 00:46:13.670 --> 00:46:14.545 align:middle line:90% So that can't happen. 00:46:14.545 --> 00:46:19.030 align:middle line:84% So what happens is that when I cross this point, 00:46:19.030 --> 00:46:22.510 align:middle line:84% this diode turns on, and this diode turns off. 00:46:22.510 --> 00:46:25.860 align:middle line:84% And what I will get is when the bottom diode is on, 00:46:25.860 --> 00:46:28.710 align:middle line:90% Vx is going to be 0. 00:46:28.710 --> 00:46:31.040 align:middle line:84% And I'm going to get something that looks like this. 00:46:31.040 --> 00:46:34.030 align:middle line:90% 00:46:34.030 --> 00:46:36.440 align:middle line:84% And I get the waveform I had before. 00:46:36.440 --> 00:46:38.000 align:middle line:90% But now what happens? 00:46:38.000 --> 00:46:45.460 align:middle line:84% I'm applying this yellow waveform, Vx, to this big LR 00:46:45.460 --> 00:46:46.770 align:middle line:90% filter. 00:46:46.770 --> 00:46:52.250 align:middle line:84% Well, if the L is really big, it extracts this, 00:46:52.250 --> 00:46:56.930 align:middle line:84% and I get V out is equal to-- is approximately 00:46:56.930 --> 00:46:58.720 align:middle line:90% equal to the average of Vx. 00:46:58.720 --> 00:47:00.880 align:middle line:90% So the average of V out-- 00:47:00.880 --> 00:47:07.580 align:middle line:90% 00:47:07.580 --> 00:47:10.000 align:middle line:84% well, what's the average of voltage Vx? 00:47:10.000 --> 00:47:12.560 align:middle line:84% Well, I would say that the average value of Vx 00:47:12.560 --> 00:47:22.550 align:middle line:84% is simply 1 over 2 pi, the integral from 0 to pi of Vs 00:47:22.550 --> 00:47:27.680 align:middle line:84% sine omega t, d omega t, because I did everything 00:47:27.680 --> 00:47:31.040 align:middle line:84% in electrical angle here instead of time. 00:47:31.040 --> 00:47:32.630 align:middle line:84% If I take this integral, this just 00:47:32.630 --> 00:47:35.930 align:middle line:84% ends up being Vs. This means being cosine of 0 00:47:35.930 --> 00:47:37.230 align:middle line:90% minus cosine of pi. 00:47:37.230 --> 00:47:40.860 align:middle line:84% This just ends up being Vs over pi. 00:47:40.860 --> 00:47:44.360 align:middle line:84% So what I can say is the average value of Vx 00:47:44.360 --> 00:47:50.040 align:middle line:84% here is just going to be Vs over pi. 00:47:50.040 --> 00:47:54.270 align:middle line:84% And because the average value across the inductor is 0, 00:47:54.270 --> 00:47:57.590 align:middle line:84% that means V out has to be approximately Vs over pi. 00:47:57.590 --> 00:48:03.060 align:middle line:84% So basically, V out is going to be basically Vs over pi. 00:48:03.060 --> 00:48:06.750 align:middle line:84% And now I've got a DC output current and a DC output voltage, 00:48:06.750 --> 00:48:10.880 align:middle line:84% which is what I wanted out of this circuit. 00:48:10.880 --> 00:48:14.910 align:middle line:84% Let me just show you that really quickly. 00:48:14.910 --> 00:48:19.060 align:middle line:84% If I put down this simulation, I'll put up my other simulation. 00:48:19.060 --> 00:48:24.450 align:middle line:84% All we've done here is I've added a second diode. 00:48:24.450 --> 00:48:26.900 align:middle line:84% This is what's known as a half wave rectifier. 00:48:26.900 --> 00:48:28.030 align:middle line:90% Let me show you two things. 00:48:28.030 --> 00:48:30.840 align:middle line:84% First of all, you can see Vs sine omega t here 00:48:30.840 --> 00:48:33.790 align:middle line:84% is this red, which is on the left axis. 00:48:33.790 --> 00:48:35.450 align:middle line:90% I'm sorry, this is-- 00:48:35.450 --> 00:48:36.660 align:middle line:90% this is V sub x. 00:48:36.660 --> 00:48:39.150 align:middle line:84% I'm sorry, this is my yellow waveform over here. 00:48:39.150 --> 00:48:41.790 align:middle line:84% But look at the inductor current. 00:48:41.790 --> 00:48:43.890 align:middle line:84% What do you think is happening here? 00:48:43.890 --> 00:48:46.330 align:middle line:84% This is the inductor current building up over time. 00:48:46.330 --> 00:48:50.290 align:middle line:84% The blue waveform is I sub L starting at some time equals 0. 00:48:50.290 --> 00:48:52.590 align:middle line:90% It's not constant. 00:48:52.590 --> 00:48:55.735 align:middle line:84% If I look at this, would you say that this blue waveform-- 00:48:55.735 --> 00:48:57.360 align:middle line:84% you can see all these cycles going on-- 00:48:57.360 --> 00:49:00.700 align:middle line:84% is the blue waveform in periodic steady state? 00:49:00.700 --> 00:49:02.990 align:middle line:90% No, it's ramping up over time. 00:49:02.990 --> 00:49:06.340 align:middle line:84% So anything I might say about steady state operation 00:49:06.340 --> 00:49:08.380 align:middle line:84% is not true down here right after I've 00:49:08.380 --> 00:49:10.430 align:middle line:84% started this circuit up and it's coming up. 00:49:10.430 --> 00:49:12.620 align:middle line:90% Real circuits have dynamics. 00:49:12.620 --> 00:49:13.940 align:middle line:90% It's doing something dynamic. 00:49:13.940 --> 00:49:17.890 align:middle line:84% But the interesting thing is, eventually, I get way over here, 00:49:17.890 --> 00:49:19.330 align:middle line:90% and look at this. 00:49:19.330 --> 00:49:21.220 align:middle line:84% If I come over here-- and let me just zoom in 00:49:21.220 --> 00:49:22.660 align:middle line:84% on this right hand portion of the circuit 00:49:22.660 --> 00:49:23.910 align:middle line:90% after I've waited for a while. 00:49:23.910 --> 00:49:26.540 align:middle line:90% 00:49:26.540 --> 00:49:28.760 align:middle line:84% Everything's repeating every cycle. 00:49:28.760 --> 00:49:31.630 align:middle line:84% That's periodic steady state operation. 00:49:31.630 --> 00:49:35.460 align:middle line:84% And the interesting thing is, if I look at this-- 00:49:35.460 --> 00:49:38.330 align:middle line:90% 00:49:38.330 --> 00:49:40.710 align:middle line:90% if I look at-- 00:49:40.710 --> 00:49:44.430 align:middle line:84% this inductor current is almost constant. 00:49:44.430 --> 00:49:49.530 align:middle line:84% Vx is exactly this half wave rectified waveform. 00:49:49.530 --> 00:49:55.590 align:middle line:84% If I looked at the voltage here, and I will now look at it, 00:49:55.590 --> 00:50:00.490 align:middle line:90% it is almost constant. 00:50:00.490 --> 00:50:06.130 align:middle line:84% And it's just about 1/3 or 1 over pi times the peak value 00:50:06.130 --> 00:50:07.530 align:middle line:90% of the sine wave. 00:50:07.530 --> 00:50:10.840 align:middle line:84% So we've been able to use the fact 00:50:10.840 --> 00:50:13.835 align:middle line:84% that I understand the average value across the inductor 00:50:13.835 --> 00:50:16.210 align:middle line:84% to tell us what the output voltage is this thing is going 00:50:16.210 --> 00:50:18.340 align:middle line:90% to be in periodic steady state. 00:50:18.340 --> 00:50:22.810 align:middle line:84% So to conclude, what we've got is method of assumed states 00:50:22.810 --> 00:50:25.780 align:middle line:84% to figure out when our diodes are turning on and off. 00:50:25.780 --> 00:50:27.887 align:middle line:84% And periodic steady state operation, 00:50:27.887 --> 00:50:29.470 align:middle line:84% we can use some of those relationships 00:50:29.470 --> 00:50:31.780 align:middle line:84% to start figuring out what voltages and currents will 00:50:31.780 --> 00:50:33.850 align:middle line:90% get in my circuit. 00:50:33.850 --> 00:50:34.967 align:middle line:90% That's all for today. 00:50:34.967 --> 00:50:36.550 align:middle line:84% I'm like a minute over, but if there's 00:50:36.550 --> 00:50:38.870 align:middle line:84% any last-minute question, I'll happily take it. 00:50:38.870 --> 00:50:41.550 align:middle line:90% 00:50:41.550 --> 00:50:42.670 align:middle line:90% OK, have a great day. 00:50:42.670 --> 00:50:44.720 align:middle line:90% I'll see you next class. 00:50:44.720 --> 00:50:48.000 align:middle line:90%