WEBVTT 00:00:00.000 --> 00:00:02.410 align:middle line:90% 00:00:02.410 --> 00:00:03.374 align:middle line:90% [SQUEAKING] 00:00:03.374 --> 00:00:04.338 align:middle line:90% [RUSTLING] 00:00:04.338 --> 00:00:05.302 align:middle line:90% [CLICKING] 00:00:05.302 --> 00:00:09.010 align:middle line:90% 00:00:09.010 --> 00:00:10.510 align:middle line:84% DAVID PERREAULT: So I wanted to keep 00:00:10.510 --> 00:00:14.200 align:middle line:84% talking about DC-to-DC converters today 00:00:14.200 --> 00:00:19.870 align:middle line:84% and keep talking about the impact of ripple and operating 00:00:19.870 --> 00:00:21.580 align:middle line:90% modes of DC-to-DC converters. 00:00:21.580 --> 00:00:24.640 align:middle line:84% So we said-- we did some analysis last time, 00:00:24.640 --> 00:00:27.940 align:middle line:84% and what we found is that under reasonable approximations, 00:00:27.940 --> 00:00:33.670 align:middle line:84% the inductor current waveforms in a PWM DC-to-AC converter 00:00:33.670 --> 00:00:39.260 align:middle line:84% tend to look like some DC value plus some triangular ripple. 00:00:39.260 --> 00:00:42.590 align:middle line:84% And the same thing for the capacitor voltage waveforms. 00:00:42.590 --> 00:00:46.970 align:middle line:84% And we said that we could represent this kind 00:00:46.970 --> 00:00:50.390 align:middle line:84% of waveform-- sort of a triangle wave with an offset-- 00:00:50.390 --> 00:00:56.030 align:middle line:84% as having some DC value plus triangular ripple, 00:00:56.030 --> 00:00:58.040 align:middle line:90% where we could find the-- 00:00:58.040 --> 00:01:03.330 align:middle line:84% we could define the average value and then the peak value. 00:01:03.330 --> 00:01:06.140 align:middle line:84% And because of the properties of a triangle wave, 00:01:06.140 --> 00:01:07.880 align:middle line:90% the peak is right in-- 00:01:07.880 --> 00:01:11.490 align:middle line:84% the average is right in the middle, between the two peaks. 00:01:11.490 --> 00:01:14.390 align:middle line:84% So you swing delta x peak to peak over 00:01:14.390 --> 00:01:17.090 align:middle line:84% to above the average and delta x peak 00:01:17.090 --> 00:01:19.460 align:middle line:84% to peak over to below the average. 00:01:19.460 --> 00:01:24.560 align:middle line:84% And so we can define a ripple ratio for such a waveform, 00:01:24.560 --> 00:01:27.230 align:middle line:84% a notion of a percentage ripple, which 00:01:27.230 --> 00:01:31.550 align:middle line:84% we define in this class as half of the peak-to-peak swing, 00:01:31.550 --> 00:01:36.630 align:middle line:84% or the mean-to-peak swing normalized to the average value. 00:01:36.630 --> 00:01:40.020 align:middle line:84% And one of the nice things about that is we can 00:01:40.020 --> 00:01:42.900 align:middle line:84% then-- if I knew the average value, from our calculations, 00:01:42.900 --> 00:01:44.910 align:middle line:84% I can actually find the peak value 00:01:44.910 --> 00:01:47.345 align:middle line:84% as being the average times 1 plus the ripple ratio. 00:01:47.345 --> 00:01:50.520 align:middle line:90% 00:01:50.520 --> 00:01:54.650 align:middle line:90% If we apply that-- 00:01:54.650 --> 00:02:00.020 align:middle line:84% we also talked about the device stresses in a direct converter, 00:02:00.020 --> 00:02:02.090 align:middle line:84% such as a buck or a boost converter, 00:02:02.090 --> 00:02:07.700 align:middle line:84% and an indirect converter, such as the inverting 00:02:07.700 --> 00:02:09.120 align:middle line:90% buck-boost converter. 00:02:09.120 --> 00:02:10.850 align:middle line:90% And we had written it as-- 00:02:10.850 --> 00:02:14.600 align:middle line:84% for a direct converter, the peak switch voltage, the peak diode 00:02:14.600 --> 00:02:17.420 align:middle line:84% voltage, which is also the peak capacitor voltage, 00:02:17.420 --> 00:02:23.630 align:middle line:84% is this maximum of V1 and V2 times 1 plus the ripple ratio. 00:02:23.630 --> 00:02:28.690 align:middle line:84% This is the average DC value we see, in terms of the DC value. 00:02:28.690 --> 00:02:30.730 align:middle line:84% Plus, the ripple on the capacitor voltage 00:02:30.730 --> 00:02:34.180 align:middle line:84% increases the peak voltage, as we see, a little bit. 00:02:34.180 --> 00:02:39.310 align:middle line:84% And likewise, we can talk about the switch current and the diode 00:02:39.310 --> 00:02:41.710 align:middle line:84% current peak values, which are related 00:02:41.710 --> 00:02:45.120 align:middle line:84% to the maximum of the average input and output currents. 00:02:45.120 --> 00:02:48.340 align:middle line:84% So for a buck converter, the output current 00:02:48.340 --> 00:02:51.260 align:middle line:84% is the bigger one, and the input voltage is the bigger one. 00:02:51.260 --> 00:02:55.120 align:middle line:84% But we see peak stresses and voltage and current 00:02:55.120 --> 00:02:57.820 align:middle line:84% that are related to the maximum of the input and output DC 00:02:57.820 --> 00:03:00.680 align:middle line:90% waveforms, and the ripple. 00:03:00.680 --> 00:03:04.220 align:middle line:84% Then we looked, also, at this indirect converter. 00:03:04.220 --> 00:03:05.810 align:middle line:84% And we found the difference is instead 00:03:05.810 --> 00:03:08.030 align:middle line:84% of seeing the maximum of the input and output, 00:03:08.030 --> 00:03:12.140 align:middle line:84% we see the sum of the input and output currents and voltages. 00:03:12.140 --> 00:03:16.010 align:middle line:84% So this tells us, no matter what, for a given conversion 00:03:16.010 --> 00:03:19.055 align:middle line:84% ratio of V1V2, I1I2, we're always 00:03:19.055 --> 00:03:21.680 align:middle line:84% going to have higher stresses in an indirect converter compared 00:03:21.680 --> 00:03:22.970 align:middle line:90% to a direct converter. 00:03:22.970 --> 00:03:26.060 align:middle line:84% And that makes sense because of how we're processing energy 00:03:26.060 --> 00:03:27.060 align:middle line:90% through the converter. 00:03:27.060 --> 00:03:28.910 align:middle line:84% The indirect converter-- you're basically 00:03:28.910 --> 00:03:31.400 align:middle line:84% storing all of the energy in the middle of the converter. 00:03:31.400 --> 00:03:32.858 align:middle line:84% And in the direct converter, you're 00:03:32.858 --> 00:03:35.720 align:middle line:84% some of it gets to flow right through the converter, 00:03:35.720 --> 00:03:38.270 align:middle line:84% and you can take advantage of that. 00:03:38.270 --> 00:03:41.000 align:middle line:84% So this is just a quick review of the conclusions 00:03:41.000 --> 00:03:42.240 align:middle line:90% we came to last time. 00:03:42.240 --> 00:03:43.915 align:middle line:90% Any questions about this so far? 00:03:43.915 --> 00:03:48.830 align:middle line:90% 00:03:48.830 --> 00:03:52.490 align:middle line:84% The other thing we did was then we took that 00:03:52.490 --> 00:03:56.330 align:middle line:84% and we said, OK, well, I know that my inductor 00:03:56.330 --> 00:03:59.540 align:middle line:84% current, for example, has some triangular ripple on it, 00:03:59.540 --> 00:04:05.180 align:middle line:84% So this IL looks like some average value and some ripple, 00:04:05.180 --> 00:04:07.790 align:middle line:84% And we could calculate-- if I wanted 00:04:07.790 --> 00:04:11.720 align:middle line:84% to limit my peak-to-peak inductor current ripple 00:04:11.720 --> 00:04:15.050 align:middle line:84% below some value, because that's my input ripple current, 00:04:15.050 --> 00:04:18.290 align:middle line:84% or my peak-to-peak output voltage 00:04:18.290 --> 00:04:20.720 align:middle line:84% ripple below a certain value, I can come up 00:04:20.720 --> 00:04:24.360 align:middle line:84% with minimum capacitor and inductor values 00:04:24.360 --> 00:04:29.100 align:middle line:84% to do that that are related to the conversion ratio. 00:04:29.100 --> 00:04:32.790 align:middle line:84% And I expressed it here in terms of the allowed ripple ratio 00:04:32.790 --> 00:04:34.500 align:middle line:90% of the designs. 00:04:34.500 --> 00:04:39.840 align:middle line:84% So now we have a means to start sizing inductors and capacitors, 00:04:39.840 --> 00:04:42.660 align:middle line:84% We can't practically use nearly infinite ones 00:04:42.660 --> 00:04:44.830 align:middle line:84% because they get too big and expensive. 00:04:44.830 --> 00:04:47.430 align:middle line:84% If you have too small a one, you get too much ripple, 00:04:47.430 --> 00:04:48.490 align:middle line:90% and bad things happen. 00:04:48.490 --> 00:04:50.870 align:middle line:84% And so the question is, how do you pick those values? 00:04:50.870 --> 00:04:53.190 align:middle line:84% And one of the things that you use to pick those values 00:04:53.190 --> 00:04:56.550 align:middle line:84% is some of notion of an allowable ripple, which 00:04:56.550 --> 00:04:58.800 align:middle line:84% we could express in terms of peak-to-peak ripple, 00:04:58.800 --> 00:05:02.710 align:middle line:84% or we could express it in terms of ripple ratios. 00:05:02.710 --> 00:05:06.480 align:middle line:84% Typically, in this example, you're 00:05:06.480 --> 00:05:10.080 align:middle line:84% not usually allowed very much ripple ratio on the capacitor 00:05:10.080 --> 00:05:14.850 align:middle line:84% because if you want a DC output voltage, 00:05:14.850 --> 00:05:17.020 align:middle line:84% you can't have 10% ripple in your output voltage. 00:05:17.020 --> 00:05:18.660 align:middle line:90% That would usually be bad. 00:05:18.660 --> 00:05:20.860 align:middle line:84% So maybe that's a percent or something. 00:05:20.860 --> 00:05:22.700 align:middle line:84% It depends on your specifications. 00:05:22.700 --> 00:05:27.410 align:middle line:84% Inductor current ripple-- maybe a bit more is acceptable. 00:05:27.410 --> 00:05:29.920 align:middle line:84% Maybe you could have 25% ripple in that current, 00:05:29.920 --> 00:05:33.790 align:middle line:84% especially if you add on another capacitor to help you filter it 00:05:33.790 --> 00:05:35.080 align:middle line:90% back here. 00:05:35.080 --> 00:05:40.470 align:middle line:90% So there's the notion of sizing. 00:05:40.470 --> 00:05:43.380 align:middle line:84% And we'll talk about-- this is one way you 00:05:43.380 --> 00:05:45.840 align:middle line:84% can start to pick your inductor and capacitor 00:05:45.840 --> 00:05:47.500 align:middle line:90% values based on ripple. 00:05:47.500 --> 00:05:49.770 align:middle line:90% There are other factors, too. 00:05:49.770 --> 00:05:52.170 align:middle line:84% We said size, So you can figure the biggest 00:05:52.170 --> 00:05:53.670 align:middle line:84% value you can get before it gets too 00:05:53.670 --> 00:05:55.920 align:middle line:90% big is probably a good thing. 00:05:55.920 --> 00:05:58.440 align:middle line:84% But also, in terms of transient performance-- 00:05:58.440 --> 00:06:00.420 align:middle line:84% the dynamic performance of your converter-- 00:06:00.420 --> 00:06:02.800 align:middle line:84% we'll talk about that aspect a little later. 00:06:02.800 --> 00:06:05.393 align:middle line:84% But that can also come into to how you pick the values. 00:06:05.393 --> 00:06:06.810 align:middle line:84% And the reason I mentioned that is 00:06:06.810 --> 00:06:08.430 align:middle line:84% because in your design project, you're 00:06:08.430 --> 00:06:10.320 align:middle line:84% going to have to pick values, and you're 00:06:10.320 --> 00:06:14.562 align:middle line:84% going to have to think about these different aspects. 00:06:14.562 --> 00:06:21.990 align:middle line:84% The other thing we said was, OK, if I have some minimum values 00:06:21.990 --> 00:06:26.520 align:middle line:84% based on ripple, what does that say about the energy storage 00:06:26.520 --> 00:06:29.208 align:middle line:84% requirement of my inductor and my capacitor? 00:06:29.208 --> 00:06:31.500 align:middle line:84% And the reason we talk about energy storage requirement 00:06:31.500 --> 00:06:33.360 align:middle line:90% is because we ought to-- 00:06:33.360 --> 00:06:37.590 align:middle line:84% it's reasonable to expect that the size of your component 00:06:37.590 --> 00:06:41.080 align:middle line:84% is probably monotonic with its energy storage capability. 00:06:41.080 --> 00:06:44.760 align:middle line:84% It's going to get bigger if you have to store more energy, 00:06:44.760 --> 00:06:45.610 align:middle line:90% usually. 00:06:45.610 --> 00:06:50.520 align:middle line:84% There's a complex trade-off to design actual-- 00:06:50.520 --> 00:06:52.270 align:middle line:90% especially magnetic components. 00:06:52.270 --> 00:06:55.020 align:middle line:84% But nonetheless, it's reasonable to expect that. 00:06:55.020 --> 00:06:58.260 align:middle line:84% And what we can see from this expression 00:06:58.260 --> 00:07:02.250 align:middle line:84% is that if I looked at the energy stored in the capacitor, 00:07:02.250 --> 00:07:07.530 align:middle line:84% it gets bigger if I have smaller ripple ratio. 00:07:07.530 --> 00:07:12.310 align:middle line:84% This factor here tends to get big 00:07:12.310 --> 00:07:15.950 align:middle line:84% if I make ripple ratio very small. 00:07:15.950 --> 00:07:19.370 align:middle line:84% It gets bigger proportional to the power of my converter 00:07:19.370 --> 00:07:21.117 align:middle line:90% needs to handle. 00:07:21.117 --> 00:07:22.200 align:middle line:90% That's kind of reasonable. 00:07:22.200 --> 00:07:24.980 align:middle line:84% If I want a kilowatt converter, it's going to be physically-- 00:07:24.980 --> 00:07:28.930 align:middle line:84% have more energy storage than a 100-watt converter. 00:07:28.930 --> 00:07:31.010 align:middle line:84% It depends on the conversion ratio. 00:07:31.010 --> 00:07:33.100 align:middle line:84% And in this case, for a boost converter, 00:07:33.100 --> 00:07:36.610 align:middle line:84% a bigger conversion ratio is a bigger duty cycle. 00:07:36.610 --> 00:07:39.640 align:middle line:84% So the more voltage conversion I'm doing, 00:07:39.640 --> 00:07:42.220 align:middle line:84% the more energy storage I'm going to need to do it. 00:07:42.220 --> 00:07:45.370 align:middle line:84% And the last thing, which weighs heavily 00:07:45.370 --> 00:07:47.980 align:middle line:84% on a lot of designers' minds, is-- 00:07:47.980 --> 00:07:51.010 align:middle line:84% we expressed it last time in terms of the switching period. 00:07:51.010 --> 00:07:55.540 align:middle line:84% Here, I'm expressing it in terms of the switching frequency. 00:07:55.540 --> 00:08:00.040 align:middle line:84% The energy storage gets smaller if I go to a higher switching 00:08:00.040 --> 00:08:01.060 align:middle line:90% frequency. 00:08:01.060 --> 00:08:05.770 align:middle line:84% So usually, I'd like to switch as fast as I can, commensurate 00:08:05.770 --> 00:08:09.400 align:middle line:84% with other constraints, because my energy storage got smaller. 00:08:09.400 --> 00:08:11.840 align:middle line:84% My components will then hopefully get smaller. 00:08:11.840 --> 00:08:12.860 align:middle line:90% Life is good. 00:08:12.860 --> 00:08:14.590 align:middle line:84% So there's been, always, a lot of effort 00:08:14.590 --> 00:08:17.260 align:middle line:84% to go as high as the frequency you can 00:08:17.260 --> 00:08:19.420 align:middle line:84% until we run into some other limits, 00:08:19.420 --> 00:08:22.270 align:middle line:84% and we'll talk about those limits-- loss being one of them. 00:08:22.270 --> 00:08:24.730 align:middle line:84% And I will say that everything we just 00:08:24.730 --> 00:08:26.980 align:middle line:84% said about the capacitor we could say in the inductor. 00:08:26.980 --> 00:08:30.290 align:middle line:84% And I should note that in this set of lecture notes, 00:08:30.290 --> 00:08:32.960 align:middle line:84% I think I wrote 1 minus D here, and this 00:08:32.960 --> 00:08:37.419 align:middle line:84% should be D, which is correct in the last set of lecture notes. 00:08:37.419 --> 00:08:41.130 align:middle line:84% But nonetheless, the same ideas pertain. 00:08:41.130 --> 00:08:44.410 align:middle line:84% Smaller ripple means larger energy storage requirement. 00:08:44.410 --> 00:08:47.070 align:middle line:84% Larger conversion ratio means larger energy 00:08:47.070 --> 00:08:48.030 align:middle line:90% storage requirement. 00:08:48.030 --> 00:08:51.930 align:middle line:84% And higher frequency means lower energy storage requirements. 00:08:51.930 --> 00:08:54.320 align:middle line:90% Any questions about that? 00:08:54.320 --> 00:08:57.590 align:middle line:90% 00:08:57.590 --> 00:08:58.090 align:middle line:90% OK. 00:08:58.090 --> 00:09:05.380 align:middle line:84% I'd like to keep talking about this notion of conversion 00:09:05.380 --> 00:09:07.750 align:middle line:84% and the ripple you see in your converters 00:09:07.750 --> 00:09:12.580 align:middle line:84% because we haven't actually gotten to all possible cases 00:09:12.580 --> 00:09:13.480 align:middle line:90% yet. 00:09:13.480 --> 00:09:17.980 align:middle line:84% And let's just go think about this boost converter. 00:09:17.980 --> 00:09:22.660 align:middle line:84% And what we'll do is let's think about the inductor current, 00:09:22.660 --> 00:09:26.170 align:middle line:84% and I'll think about the inductor voltage V sub L here. 00:09:26.170 --> 00:09:29.650 align:middle line:84% So if I'm switching my converter, 00:09:29.650 --> 00:09:33.520 align:middle line:84% I'm going to have some switching signal which is driving-- 00:09:33.520 --> 00:09:37.720 align:middle line:84% which is telling when this gate is turning on and off. 00:09:37.720 --> 00:09:41.320 align:middle line:84% So I'll have some switching function. 00:09:41.320 --> 00:09:42.590 align:middle line:90% I'm just doing this. 00:09:42.590 --> 00:09:48.250 align:middle line:84% So it's high for some time, DT, and then it's low for some time, 00:09:48.250 --> 00:09:52.400 align:middle line:84% and then it's high for some time after a period. 00:09:52.400 --> 00:09:56.050 align:middle line:84% And then I can say, OK, well what is my inductor voltage 00:09:56.050 --> 00:09:59.270 align:middle line:90% do during that time? 00:09:59.270 --> 00:10:03.380 align:middle line:84% Well, my inductor voltage is positive at V1 00:10:03.380 --> 00:10:06.470 align:middle line:90% when I turn the switch on. 00:10:06.470 --> 00:10:13.260 align:middle line:84% So it looks like this at value V1. 00:10:13.260 --> 00:10:15.360 align:middle line:84% And then when I turn the switch off, 00:10:15.360 --> 00:10:22.610 align:middle line:84% I have V1 minus V2 across it, and I get some negative voltage. 00:10:22.610 --> 00:10:27.020 align:middle line:84% And what we know is that in periodic steady state, 00:10:27.020 --> 00:10:32.450 align:middle line:90% this area must equal this area. 00:10:32.450 --> 00:10:36.500 align:middle line:84% And this ends up being V1 minus V2. 00:10:36.500 --> 00:10:41.360 align:middle line:84% And if I were to plot the current 00:10:41.360 --> 00:10:48.140 align:middle line:84% in the inductor, what would I see over the same time period? 00:10:48.140 --> 00:10:52.040 align:middle line:90% I have some DC value. 00:10:52.040 --> 00:10:55.830 align:middle line:84% In the inductor current, which is I1. 00:10:55.830 --> 00:11:00.300 align:middle line:84% The DC input current here is the DC current 00:11:00.300 --> 00:11:02.220 align:middle line:90% coming from the source. 00:11:02.220 --> 00:11:08.450 align:middle line:90% And that's my average value. 00:11:08.450 --> 00:11:11.228 align:middle line:84% And what we'll have is in the first part of the cycle, 00:11:11.228 --> 00:11:12.770 align:middle line:84% I store energy in the inductor, and I 00:11:12.770 --> 00:11:14.730 align:middle line:84% get this positive triangular ripple. 00:11:14.730 --> 00:11:19.250 align:middle line:84% And in the second part of the cycle I get negative slope, 00:11:19.250 --> 00:11:23.460 align:middle line:84% and I have my periodic steady state waveform like this. 00:11:23.460 --> 00:11:35.460 align:middle line:84% And this is delta IL, peak to peak. 00:11:35.460 --> 00:11:39.920 align:middle line:84% So that's what our waveforms look like. 00:11:39.920 --> 00:11:43.310 align:middle line:84% So let's think about this in terms of ripple ratio. 00:11:43.310 --> 00:11:46.610 align:middle line:90% OK, so what do I know? 00:11:46.610 --> 00:11:52.490 align:middle line:84% Delta IL, peak to peak is equal-- delta 00:11:52.490 --> 00:12:04.760 align:middle line:84% I is V over L delta T. So V is V1 divided by L times DT. 00:12:04.760 --> 00:12:07.530 align:middle line:84% So this is an expression of how much I rise. 00:12:07.530 --> 00:12:11.630 align:middle line:84% I'm applying V1 for time DT, so the delta I is V1 over L DT. 00:12:11.630 --> 00:12:14.600 align:middle line:90% 00:12:14.600 --> 00:12:18.020 align:middle line:90% What is the input current? 00:12:18.020 --> 00:12:20.720 align:middle line:90% I1. 00:12:20.720 --> 00:12:22.320 align:middle line:90% OK, well, what's I1 going to be? 00:12:22.320 --> 00:12:27.380 align:middle line:84% Well, the DC output current is going 00:12:27.380 --> 00:12:30.210 align:middle line:84% to be V2 divided by whatever this resistor is. 00:12:30.210 --> 00:12:40.810 align:middle line:84% So if I wanted I2, I would have V2 divided by R. But this is I2. 00:12:40.810 --> 00:12:44.590 align:middle line:90% How does I2 relate to I1? 00:12:44.590 --> 00:12:54.880 align:middle line:84% I2 is going to be I1 over 1 minus D. 00:12:54.880 --> 00:12:57.080 align:middle line:90% because I1 is bigger than I2. 00:12:57.080 --> 00:13:01.830 align:middle line:90% That's just-- this factor is I2. 00:13:01.830 --> 00:13:02.830 align:middle line:90% And then this gives me-- 00:13:02.830 --> 00:13:07.840 align:middle line:84% 1 over 1 minus D gives me the conversion to I2. 00:13:07.840 --> 00:13:09.165 align:middle line:90% Any question about that? 00:13:09.165 --> 00:13:11.950 align:middle line:90% 00:13:11.950 --> 00:13:15.460 align:middle line:84% So then if I were to write the ripple ratio, 00:13:15.460 --> 00:13:19.150 align:middle line:84% the ripple ratio in the inductor in this example 00:13:19.150 --> 00:13:27.340 align:middle line:84% is simply equal to delta IL peak to peak over 2 divided by IL. 00:13:27.340 --> 00:13:29.530 align:middle line:90% And this is also equal to IL. 00:13:29.530 --> 00:13:36.480 align:middle line:84% And so this is going to give me V1DT over L. 00:13:36.480 --> 00:13:40.161 align:middle line:90% So this is going to be 2L-- 00:13:40.161 --> 00:13:51.880 align:middle line:90% I sub L is V2 R times 1 minus D. 00:13:51.880 --> 00:14:00.695 align:middle line:84% And I can rewrite this in terms of-- 00:14:00.695 --> 00:14:03.300 align:middle line:90% 00:14:03.300 --> 00:14:07.210 align:middle line:90% I can get rid of V1 and V2 here. 00:14:07.210 --> 00:14:07.920 align:middle line:90% Why? 00:14:07.920 --> 00:14:17.720 align:middle line:84% Because V2 is going to be V1 over 1 minus D. 00:14:17.720 --> 00:14:20.530 align:middle line:90% So I could rewrite this as-- 00:14:20.530 --> 00:14:23.730 align:middle line:90% V1 over 1 minus D is V2. 00:14:23.730 --> 00:14:27.310 align:middle line:90% Or V2 times 1 minus D is v one. 00:14:27.310 --> 00:14:35.930 align:middle line:84% So I could rewrite this as D times 1 minus D quantity squared 00:14:35.930 --> 00:14:39.560 align:middle line:90% TR over 2L. 00:14:39.560 --> 00:14:48.728 align:middle line:90% 00:14:48.728 --> 00:14:52.030 align:middle line:90% And what does this tell me? 00:14:52.030 --> 00:14:55.880 align:middle line:90% Well, this says how-- 00:14:55.880 --> 00:15:00.710 align:middle line:84% essentially, I'm asking, what's this ripple compared 00:15:00.710 --> 00:15:03.620 align:middle line:84% to the average value of my current? 00:15:03.620 --> 00:15:10.100 align:middle line:84% And we can see that the ripple ratio goes up as what? 00:15:10.100 --> 00:15:20.210 align:middle line:84% It goes up when either R goes up or L goes down. 00:15:20.210 --> 00:15:23.890 align:middle line:84% Those are two things that will make the ripple ratio bigger. 00:15:23.890 --> 00:15:27.580 align:middle line:90% And that makes sense. 00:15:27.580 --> 00:15:31.390 align:middle line:84% Making L smaller is going to make the ripple ratio bigger 00:15:31.390 --> 00:15:34.470 align:middle line:84% because I have a higher ripple on the inductor. 00:15:34.470 --> 00:15:38.930 align:middle line:90% Any questions about that? 00:15:38.930 --> 00:15:40.220 align:middle line:90% Yeah. 00:15:40.220 --> 00:15:41.860 align:middle line:84% AUDIENCE: The DC values of the currents 00:15:41.860 --> 00:15:43.950 align:middle line:84% are being-- ignoring losses and that. 00:15:43.950 --> 00:15:46.450 align:middle line:84% DAVID PERREAULT: The DC values of the current ignoring loss? 00:15:46.450 --> 00:15:46.860 align:middle line:90% AUDIENCE: Loss. 00:15:46.860 --> 00:15:48.510 align:middle line:84% DAVID PERREAULT: Yeah, there's no loss. 00:15:48.510 --> 00:15:49.890 align:middle line:90% This is a perfect converter. 00:15:49.890 --> 00:15:52.160 align:middle line:90% Everything's lossless. 00:15:52.160 --> 00:15:54.590 align:middle line:84% You throw in loss, and things happen, 00:15:54.590 --> 00:15:56.950 align:middle line:84% Right now, I'm treating everything as being perfect. 00:15:56.950 --> 00:16:00.560 align:middle line:90% 00:16:00.560 --> 00:16:04.850 align:middle line:84% So let's start to think about the impact of changing 00:16:04.850 --> 00:16:12.400 align:middle line:90% resistance and inductance. 00:16:12.400 --> 00:16:14.580 align:middle line:90% So let me replot this. 00:16:14.580 --> 00:16:16.360 align:middle line:90% So let me plot IL again. 00:16:16.360 --> 00:16:23.350 align:middle line:90% 00:16:23.350 --> 00:16:27.900 align:middle line:84% And we said, OK, maybe I have some average value, 00:16:27.900 --> 00:16:38.624 align:middle line:84% and I have some ripple that does this, and so forth. 00:16:38.624 --> 00:16:41.200 align:middle line:84% And this is at some resistance value. 00:16:41.200 --> 00:16:46.110 align:middle line:84% What would happen if I increase my resistance value? 00:16:46.110 --> 00:16:49.320 align:middle line:84% Well, up here, I see if I increase my resistance value, 00:16:49.320 --> 00:16:50.560 align:middle line:90% I increase the ripple ratio. 00:16:50.560 --> 00:16:53.110 align:middle line:84% But how does it change the IL waveforms? 00:16:53.110 --> 00:16:55.080 align:middle line:90% This is IL. 00:16:55.080 --> 00:16:57.030 align:middle line:84% But if I used a bigger resistance 00:16:57.030 --> 00:17:00.790 align:middle line:84% value, what might happen is it might look like this. 00:17:00.790 --> 00:17:07.470 align:middle line:84% I will have some lower current to supply the same voltage 00:17:07.470 --> 00:17:09.390 align:middle line:90% at the output. 00:17:09.390 --> 00:17:13.560 align:middle line:84% If I doubled my resistance, I need half the current 00:17:13.560 --> 00:17:15.400 align:middle line:84% to get to the same output voltage. 00:17:15.400 --> 00:17:18.690 align:middle line:84% If my output current halves, my input current halves. 00:17:18.690 --> 00:17:21.520 align:middle line:84% And that means my inductor current, on average, halves. 00:17:21.520 --> 00:17:25.619 align:middle line:84% So OK-- I reduce the average inductor current, 00:17:25.619 --> 00:17:28.650 align:middle line:84% but V1 and V2 have stayed the same. 00:17:28.650 --> 00:17:32.550 align:middle line:84% And if V1 and V2 have stayed the same, the slope on the ripple 00:17:32.550 --> 00:17:33.870 align:middle line:90% has stayed the same. 00:17:33.870 --> 00:17:37.320 align:middle line:84% So my inductor current, at a higher resistance, 00:17:37.320 --> 00:17:39.690 align:middle line:90% might look like this. 00:17:39.690 --> 00:17:41.320 align:middle line:90% This is parallel. 00:17:41.320 --> 00:17:44.020 align:middle line:90% This is parallel, and so forth. 00:17:44.020 --> 00:17:45.320 align:middle line:90% And I get this. 00:17:45.320 --> 00:17:47.650 align:middle line:90% So this is IL. 00:17:47.650 --> 00:17:49.305 align:middle line:90% This is IL at R small. 00:17:49.305 --> 00:17:52.600 align:middle line:90% 00:17:52.600 --> 00:17:54.735 align:middle line:90% This is IL at RL medium. 00:17:54.735 --> 00:17:59.350 align:middle line:90% 00:17:59.350 --> 00:18:03.920 align:middle line:84% What would happen if I kept increasing my resistance value? 00:18:03.920 --> 00:18:06.920 align:middle line:84% Well, all we saw is the waveform doesn't-- the waveform shape 00:18:06.920 --> 00:18:07.800 align:middle line:90% just doesn't change. 00:18:07.800 --> 00:18:09.470 align:middle line:90% The average just moves down. 00:18:09.470 --> 00:18:12.320 align:middle line:84% The ripple is the same, but the DC value 00:18:12.320 --> 00:18:15.290 align:middle line:84% is changing because I need more or less average current 00:18:15.290 --> 00:18:18.530 align:middle line:84% to support the output and more or less average power 00:18:18.530 --> 00:18:20.860 align:middle line:90% draw to support the output. 00:18:20.860 --> 00:18:22.990 align:middle line:84% So eventually, what's going to happen 00:18:22.990 --> 00:18:24.820 align:middle line:84% is we're going to get something that-- 00:18:24.820 --> 00:18:27.460 align:middle line:84% we hit another mode of operation. 00:18:27.460 --> 00:18:30.520 align:middle line:84% We're going to get something where it does this. 00:18:30.520 --> 00:18:34.360 align:middle line:84% I'm going to turn my switch on, and then it's 00:18:34.360 --> 00:18:41.880 align:middle line:84% going to come down and then stop. 00:18:41.880 --> 00:18:45.420 align:middle line:84% So here, this is supposed to be the same DT, 00:18:45.420 --> 00:18:49.170 align:middle line:84% and here is the same T. But I'm going 00:18:49.170 --> 00:18:51.750 align:middle line:84% to get this time period where the current comes down to 0. 00:18:51.750 --> 00:18:54.460 align:middle line:84% Now, why am I not drawing the current as going negative? 00:18:54.460 --> 00:18:58.820 align:middle line:90% 00:18:58.820 --> 00:19:03.380 align:middle line:84% I can't really get a negative current through the diode, 00:19:03.380 --> 00:19:07.820 align:middle line:84% So once the inductor current ramps down to zero, 00:19:07.820 --> 00:19:11.750 align:middle line:84% the diode is going to turn off, whereas the diode was conducting 00:19:11.750 --> 00:19:12.860 align:middle line:90% before. 00:19:12.860 --> 00:19:16.430 align:middle line:84% So I could build a converter where this was an active switch 00:19:16.430 --> 00:19:19.620 align:middle line:84% and allowed bidirectional current, and people do that. 00:19:19.620 --> 00:19:21.620 align:middle line:84% But if I'm going to build it with a diode, 00:19:21.620 --> 00:19:24.620 align:middle line:84% I'm going to get this alternative state. 00:19:24.620 --> 00:19:30.530 align:middle line:84% So in this alternative state, what do I have? 00:19:30.530 --> 00:19:32.425 align:middle line:90% Here, I have the switch on. 00:19:32.425 --> 00:19:35.230 align:middle line:90% 00:19:35.230 --> 00:19:39.060 align:middle line:90% Here, I have the diode on. 00:19:39.060 --> 00:19:42.335 align:middle line:84% And in this time period, everything is off. 00:19:42.335 --> 00:19:45.490 align:middle line:90% 00:19:45.490 --> 00:19:48.330 align:middle line:84% So we haven't considered this mode of operation, 00:19:48.330 --> 00:19:50.830 align:middle line:84% We always assumed it was either the switch or the diode that 00:19:50.830 --> 00:19:53.180 align:middle line:90% was conducting. 00:19:53.180 --> 00:19:57.620 align:middle line:84% Now I'm telling you, hey, if I make R big-- 00:19:57.620 --> 00:20:01.250 align:middle line:90% 00:20:01.250 --> 00:20:02.610 align:middle line:90% and how big is big? 00:20:02.610 --> 00:20:03.630 align:middle line:90% We'll get into that. 00:20:03.630 --> 00:20:06.497 align:middle line:84% But if I make RL big enough, the average current 00:20:06.497 --> 00:20:08.330 align:middle line:84% goes down to the fact that my current ripple 00:20:08.330 --> 00:20:10.310 align:middle line:84% is going to make me hit 0 current, 00:20:10.310 --> 00:20:15.120 align:middle line:84% and I'm going to get this new funny state. 00:20:15.120 --> 00:20:18.470 align:middle line:84% Does that make sense to everybody? 00:20:18.470 --> 00:20:21.756 align:middle line:84% So this is what happens with variations, with resistance. 00:20:21.756 --> 00:20:24.980 align:middle line:90% 00:20:24.980 --> 00:20:27.710 align:middle line:84% What happens with variations in inductance? 00:20:27.710 --> 00:20:32.390 align:middle line:90% Well, slightly different. 00:20:32.390 --> 00:20:36.140 align:middle line:90% Maybe what I have is this. 00:20:36.140 --> 00:20:39.620 align:middle line:84% I will have, again, some value, IL. 00:20:39.620 --> 00:20:44.890 align:middle line:90% 00:20:44.890 --> 00:20:46.860 align:middle line:84% Actually, maybe make it a little smaller. 00:20:46.860 --> 00:20:50.090 align:middle line:84% I have some average value IL that I'm generating. 00:20:50.090 --> 00:20:53.620 align:middle line:84% And if I have some big inductance-- 00:20:53.620 --> 00:20:56.700 align:middle line:84% maybe my ripple is small, so it does this. 00:20:56.700 --> 00:21:00.080 align:middle line:90% 00:21:00.080 --> 00:21:06.565 align:middle line:90% So this is IL at L big. 00:21:06.565 --> 00:21:07.940 align:middle line:84% I have a big inductance in there. 00:21:07.940 --> 00:21:10.160 align:middle line:90% The ripple is relatively small. 00:21:10.160 --> 00:21:12.500 align:middle line:90% If I got a smaller inductor-- 00:21:12.500 --> 00:21:15.170 align:middle line:84% I turned off my converter; I pull out one inductor; 00:21:15.170 --> 00:21:18.240 align:middle line:84% I put in a new inductor that's half the value. 00:21:18.240 --> 00:21:19.290 align:middle line:90% What would happen? 00:21:19.290 --> 00:21:21.860 align:middle line:84% Well, the ripple would increase, And I would 00:21:21.860 --> 00:21:25.050 align:middle line:90% get something that did this. 00:21:25.050 --> 00:21:29.940 align:middle line:84% The delta IL peak to peak is inversely proportional to L. So 00:21:29.940 --> 00:21:32.055 align:middle line:90% my ripple current would-- 00:21:32.055 --> 00:21:33.180 align:middle line:90% so it's a terrible drawing. 00:21:33.180 --> 00:21:34.565 align:middle line:84% My ripple current would increase. 00:21:34.565 --> 00:21:37.230 align:middle line:90% 00:21:37.230 --> 00:21:39.390 align:middle line:84% If I turn the converter off and then 00:21:39.390 --> 00:21:42.240 align:middle line:84% I put in even a smaller inductor, what would happen? 00:21:42.240 --> 00:21:43.990 align:middle line:84% If I get a really small inductor-- 00:21:43.990 --> 00:21:46.390 align:middle line:84% I pull out the smallest inductor I can find in my kit, 00:21:46.390 --> 00:21:50.600 align:middle line:84% maybe I get a waveform that does this. 00:21:50.600 --> 00:21:57.040 align:middle line:84% It starts at 0, comes down, hits 0, and then stays at 0. 00:21:57.040 --> 00:22:05.590 align:middle line:84% So I'm going to get, again, a case where I have the diode on. 00:22:05.590 --> 00:22:07.900 align:middle line:90% And then I have-- 00:22:07.900 --> 00:22:10.360 align:middle line:90% I'm sorry, I have the switch on. 00:22:10.360 --> 00:22:15.430 align:middle line:84% In this case-- and this is only pertaining to this bottom 00:22:15.430 --> 00:22:16.580 align:middle line:90% picture-- 00:22:16.580 --> 00:22:19.160 align:middle line:90% the diode's on. 00:22:19.160 --> 00:22:21.360 align:middle line:84% And then I get a time period where both are off. 00:22:21.360 --> 00:22:24.520 align:middle line:90% 00:22:24.520 --> 00:22:28.140 align:middle line:84% And this is a different mode of operation. 00:22:28.140 --> 00:22:30.870 align:middle line:90% 00:22:30.870 --> 00:22:32.090 align:middle line:90% Any questions about that? 00:22:32.090 --> 00:22:38.790 align:middle line:90% 00:22:38.790 --> 00:22:44.370 align:middle line:84% This mode of operation is known as discontinuous conduction 00:22:44.370 --> 00:22:44.870 align:middle line:90% mode. 00:22:44.870 --> 00:22:55.910 align:middle line:90% 00:22:55.910 --> 00:22:59.750 align:middle line:90% It's often abbreviated, DCM. 00:22:59.750 --> 00:23:01.880 align:middle line:84% Now, when we say, discontinuous, there's 00:23:01.880 --> 00:23:04.820 align:middle line:84% nothing discontinuous about the inductor current, obviously. 00:23:04.820 --> 00:23:07.910 align:middle line:84% The inductor current is always a continuous waveform. 00:23:07.910 --> 00:23:10.940 align:middle line:84% But what it's what's meant by that is it's 00:23:10.940 --> 00:23:13.790 align:middle line:84% not continuously flowing positive and negative. 00:23:13.790 --> 00:23:18.140 align:middle line:84% It has this third state, where both devices are off. 00:23:18.140 --> 00:23:21.500 align:middle line:84% And keep in mind that that's all the possible cases because I 00:23:21.500 --> 00:23:24.470 align:middle line:84% cannot have a case where both the switch and the diode are 00:23:24.470 --> 00:23:26.160 align:middle line:90% on and the voltage is positive. 00:23:26.160 --> 00:23:27.150 align:middle line:90% That can't exist. 00:23:27.150 --> 00:23:33.780 align:middle line:84% So we exercise all the things that can happen. 00:23:33.780 --> 00:23:35.170 align:middle line:90% Any questions about that? 00:23:35.170 --> 00:23:38.720 align:middle line:90% 00:23:38.720 --> 00:23:41.330 align:middle line:90% So when does this happen? 00:23:41.330 --> 00:23:45.170 align:middle line:90% Well, the ripple ratio-- 00:23:45.170 --> 00:23:48.660 align:middle line:84% if I were right at the edge of discontinuous conduction, 00:23:48.660 --> 00:23:52.220 align:middle line:84% you could imagine a situation where 00:23:52.220 --> 00:23:58.000 align:middle line:84% I pick some intermediate value of resistance, 00:23:58.000 --> 00:24:03.240 align:middle line:84% for example, in this case, where it does exactly this. 00:24:03.240 --> 00:24:07.730 align:middle line:90% 00:24:07.730 --> 00:24:11.660 align:middle line:84% Oops, I didn't draw that very well. 00:24:11.660 --> 00:24:17.660 align:middle line:84% I'm sitting right at the edge, where, yes, my waveforms 00:24:17.660 --> 00:24:18.570 align:middle line:90% are continuous. 00:24:18.570 --> 00:24:22.950 align:middle line:84% My diode never turns off, but only barely so. 00:24:22.950 --> 00:24:26.570 align:middle line:84% In that case, the peak-to-peak ripple 00:24:26.570 --> 00:24:29.870 align:middle line:84% is exactly twice the average current. 00:24:29.870 --> 00:24:37.940 align:middle line:84% Or this case is ripple ratio equal to 1. 00:24:37.940 --> 00:24:42.730 align:middle line:84% It's rippling up from 0 to twice the average and back down to 0. 00:24:42.730 --> 00:24:43.920 align:middle line:90% It's still a triangle wave. 00:24:43.920 --> 00:24:47.430 align:middle line:84% It's not this funny clipped triangle wave. 00:24:47.430 --> 00:24:49.310 align:middle line:90% It's still a pure triangle wave. 00:24:49.310 --> 00:24:52.740 align:middle line:84% And that's exactly the ripple ratio equal to 1 case. 00:24:52.740 --> 00:24:54.830 align:middle line:84% Does that make sense to everybody? 00:24:54.830 --> 00:24:57.740 align:middle line:90% So what do I get from that? 00:24:57.740 --> 00:25:03.490 align:middle line:84% Well, we can come back and just calculate what happens. 00:25:03.490 --> 00:25:06.980 align:middle line:84% I have my expression for ripple ratio is equal to 1, 00:25:06.980 --> 00:25:08.870 align:middle line:90% so I can write that again here. 00:25:08.870 --> 00:25:18.050 align:middle line:84% I get R sub L is equal to D times 1 minus D quantity squared 00:25:18.050 --> 00:25:19.370 align:middle line:90% TR over 2L. 00:25:19.370 --> 00:25:23.420 align:middle line:90% 00:25:23.420 --> 00:25:27.850 align:middle line:84% and that's equal to 1 at the edge of DCM. 00:25:27.850 --> 00:25:31.970 align:middle line:90% 00:25:31.970 --> 00:25:35.680 align:middle line:84% That's where I transition between the two modes. 00:25:35.680 --> 00:25:43.990 align:middle line:84% So from this, we can say that DCM happens for R is greater 00:25:43.990 --> 00:26:03.210 align:middle line:84% than 2L over D times 1 minus D quantity squared T or for L is 00:26:03.210 --> 00:26:13.565 align:middle line:84% less than D times 1 minus D squared T R over 2. 00:26:13.565 --> 00:26:18.010 align:middle line:90% 00:26:18.010 --> 00:26:25.280 align:middle line:84% This value of L is sometimes called the critical inductance 00:26:25.280 --> 00:26:26.450 align:middle line:90% in the literature. 00:26:26.450 --> 00:26:28.200 align:middle line:84% There's nothing very critical about it. 00:26:28.200 --> 00:26:30.020 align:middle line:84% But all it's saying is if I go out and pick 00:26:30.020 --> 00:26:32.150 align:middle line:84% a big inductor, bigger than this, 00:26:32.150 --> 00:26:33.950 align:middle line:84% for the specified operating conditions, 00:26:33.950 --> 00:26:36.480 align:middle line:84% I'm not going to run into discontinuous conduction. 00:26:36.480 --> 00:26:39.158 align:middle line:90% 00:26:39.158 --> 00:26:43.820 align:middle line:84% But understand, in a lot of cases, for some operating mode, 00:26:43.820 --> 00:26:46.470 align:middle line:84% you're going to have to run into discontinuous conduction. 00:26:46.470 --> 00:26:47.120 align:middle line:90% Why? 00:26:47.120 --> 00:26:50.920 align:middle line:90% Because think about it this way. 00:26:50.920 --> 00:26:54.130 align:middle line:84% I have to run for different load resistances, 00:26:54.130 --> 00:26:56.980 align:middle line:84% Mostly, in a power converter, you build the converter, 00:26:56.980 --> 00:26:58.480 align:middle line:84% and then people hook loads up to it, 00:26:58.480 --> 00:27:00.480 align:middle line:84% and you don't know what load they hook up to it. 00:27:00.480 --> 00:27:01.960 align:middle line:84% There's some maximum power rating, 00:27:01.960 --> 00:27:04.652 align:middle line:84% but maybe they want to put very little load on it sometimes, 00:27:04.652 --> 00:27:06.610 align:middle line:84% Maybe sometimes your microprocessor is running. 00:27:06.610 --> 00:27:09.080 align:middle line:90% Maybe sometimes it's not. 00:27:09.080 --> 00:27:13.510 align:middle line:84% And so you might have to deal with a very wide range of load 00:27:13.510 --> 00:27:16.050 align:middle line:90% currents, up to some maximum. 00:27:16.050 --> 00:27:18.300 align:middle line:84% Well, if you have to deal with a very high load 00:27:18.300 --> 00:27:21.360 align:middle line:84% resistance or a very low load or a light load, 00:27:21.360 --> 00:27:24.250 align:middle line:84% that means your average is going to come down and down and down. 00:27:24.250 --> 00:27:26.190 align:middle line:84% And the only way to keep it into CCM 00:27:26.190 --> 00:27:29.070 align:middle line:84% would be to use smaller and smaller and smaller 00:27:29.070 --> 00:27:30.843 align:middle line:84% ripple so you never hit that mode. 00:27:30.843 --> 00:27:32.760 align:middle line:84% And the only way to do that is to get a bigger 00:27:32.760 --> 00:27:34.380 align:middle line:90% and bigger and bigger inductor. 00:27:34.380 --> 00:27:35.260 align:middle line:90% But guess what? 00:27:35.260 --> 00:27:36.970 align:middle line:84% You've got to run at high load, too. 00:27:36.970 --> 00:27:42.450 align:middle line:84% And you run into this case where suddenly, if you've done that 00:27:42.450 --> 00:27:44.940 align:middle line:84% by a much bigger inductor, suddenly 00:27:44.940 --> 00:27:48.000 align:middle line:84% you've gotten to the point where you've got a lot of inductor 00:27:48.000 --> 00:27:48.740 align:middle line:90% energy storage. 00:27:48.740 --> 00:27:51.808 align:middle line:90% 00:27:51.808 --> 00:27:54.490 align:middle line:84% If your ripple ratio is small at light load, 00:27:54.490 --> 00:27:58.100 align:middle line:84% it's really tiny at high load, and this factor will kill you. 00:27:58.100 --> 00:28:00.400 align:middle line:84% You just have way too big a component. 00:28:00.400 --> 00:28:02.050 align:middle line:84% So as a practical matter, if you're 00:28:02.050 --> 00:28:04.270 align:middle line:84% going to run over a really wide power range, 00:28:04.270 --> 00:28:06.520 align:middle line:90% you might have to deal with DCM. 00:28:06.520 --> 00:28:09.520 align:middle line:84% So we often design our converters 00:28:09.520 --> 00:28:12.310 align:middle line:84% for continuous conduction mode at full power, 00:28:12.310 --> 00:28:17.880 align:middle line:84% but they might go into DCM if you're at very light load power. 00:28:17.880 --> 00:28:19.270 align:middle line:90% Any questions about that? 00:28:19.270 --> 00:28:25.732 align:middle line:90% 00:28:25.732 --> 00:28:30.960 align:middle line:84% So what can we say about operating in this new switch 00:28:30.960 --> 00:28:33.510 align:middle line:90% state? 00:28:33.510 --> 00:28:37.350 align:middle line:90% Well, let's think about that. 00:28:37.350 --> 00:28:38.850 align:middle line:90% What do our waveforms look like? 00:28:38.850 --> 00:28:42.560 align:middle line:84% Let's recreate our switching waveforms. 00:28:42.560 --> 00:28:48.200 align:middle line:84% We said we were going to have some duty ratio for our switch. 00:28:48.200 --> 00:28:51.010 align:middle line:84% And we were always focusing on the duty ratio of the switch. 00:28:51.010 --> 00:28:53.800 align:middle line:84% And that's useful in CCM because the diode 00:28:53.800 --> 00:28:56.360 align:middle line:84% does exactly the opposite of what the switch does. 00:28:56.360 --> 00:28:58.300 align:middle line:84% But now, if we're going to talk about DCM, 00:28:58.300 --> 00:28:59.390 align:middle line:90% we can't talk about that. 00:28:59.390 --> 00:29:06.650 align:middle line:84% So here's q of T. And q of T is doing something. 00:29:06.650 --> 00:29:10.100 align:middle line:90% So here's T DT. 00:29:10.100 --> 00:29:13.570 align:middle line:90% Here's q of t for the switch. 00:29:13.570 --> 00:29:15.190 align:middle line:90% But I might have-- 00:29:15.190 --> 00:29:19.630 align:middle line:84% then think, OK, maybe what I have is some q sub 00:29:19.630 --> 00:29:30.500 align:middle line:84% D. This is the switching period for the diode, q sub D of T. 00:29:30.500 --> 00:29:33.982 align:middle line:84% The purple waveform tells me when the diode is on because now 00:29:33.982 --> 00:29:36.440 align:middle line:84% I have a time where the switch is on, when the diode is on, 00:29:36.440 --> 00:29:38.970 align:middle line:90% and when neither are on. 00:29:38.970 --> 00:29:42.230 align:middle line:84% And maybe this happens for some time-- 00:29:42.230 --> 00:29:50.180 align:middle line:84% this time period might be D plus D2 times T, so the diode runs 00:29:50.180 --> 00:29:52.850 align:middle line:90% with some duty ratio of D2. 00:29:52.850 --> 00:29:57.800 align:middle line:84% And to be a DCM, D2 is less than 1 00:29:57.800 --> 00:30:03.790 align:middle line:84% minus D. Make sense to everybody? 00:30:03.790 --> 00:30:07.550 align:middle line:84% So-- OK, we have this additional mode. 00:30:07.550 --> 00:30:13.180 align:middle line:84% What does the voltage on the inductor look like? 00:30:13.180 --> 00:30:15.320 align:middle line:84% We had the voltage on the inductor here. 00:30:15.320 --> 00:30:16.470 align:middle line:90% What does he look like? 00:30:16.470 --> 00:30:22.980 align:middle line:90% 00:30:22.980 --> 00:30:25.980 align:middle line:84% Well, in the first part of the cycle, he's at V1. 00:30:25.980 --> 00:30:28.870 align:middle line:90% 00:30:28.870 --> 00:30:30.490 align:middle line:84% Here's the voltage on the inductor. 00:30:30.490 --> 00:30:34.960 align:middle line:84% He started at V1 in the first part of the cycle. 00:30:34.960 --> 00:30:37.690 align:middle line:84% Then when the switch turns off and the diode turns on, 00:30:37.690 --> 00:30:40.720 align:middle line:90% he's at V1 minus V2. 00:30:40.720 --> 00:30:46.340 align:middle line:84% And then he's on for a while, until this second time 00:30:46.340 --> 00:30:49.080 align:middle line:90% period, D plus D2, ends. 00:30:49.080 --> 00:30:54.280 align:middle line:84% What happens if both the switch and the diode are off? 00:30:54.280 --> 00:30:56.680 align:middle line:84% Well, if both the switch and the diode are off, 00:30:56.680 --> 00:30:58.215 align:middle line:84% there's no current in the inductor, 00:30:58.215 --> 00:31:00.970 align:middle line:84% or there's no DI DT in the inductor. 00:31:00.970 --> 00:31:02.590 align:middle line:84% If there's no DI DT in the inductor, 00:31:02.590 --> 00:31:05.030 align:middle line:84% then there can be no voltage across the inductor. 00:31:05.030 --> 00:31:07.870 align:middle line:84% And ideally, this voltage across the inductor 00:31:07.870 --> 00:31:09.977 align:middle line:90% would have to go to 0. 00:31:09.977 --> 00:31:11.560 align:middle line:84% So then I'm going to get a time period 00:31:11.560 --> 00:31:16.180 align:middle line:84% where the inductor voltage is 0 until the end of the cycle. 00:31:16.180 --> 00:31:26.590 align:middle line:84% So this is the ideal waveforms for a discontinuous conduction 00:31:26.590 --> 00:31:31.690 align:middle line:84% mode where I'm plotting V sub L of T. 00:31:31.690 --> 00:31:35.810 align:middle line:84% And what do I have for periodic steady state requirement? 00:31:35.810 --> 00:31:39.260 align:middle line:84% The average voltage across the inductor still has to be 0. 00:31:39.260 --> 00:31:46.080 align:middle line:84% That means that this area now balances against this area. 00:31:46.080 --> 00:31:48.790 align:middle line:84% So the average-- the integral of this over the cycle 00:31:48.790 --> 00:31:51.200 align:middle line:90% still has to be 0. 00:31:51.200 --> 00:31:52.972 align:middle line:84% Does that make sense to everybody? 00:31:52.972 --> 00:31:57.220 align:middle line:90% 00:31:57.220 --> 00:32:01.030 align:middle line:84% And if I just plotted the current to complete everything, 00:32:01.030 --> 00:32:10.220 align:middle line:84% in this case, the current starts at 0 and ramps up. 00:32:10.220 --> 00:32:12.830 align:middle line:84% And the second part of the cycle, it ramps down, 00:32:12.830 --> 00:32:17.810 align:middle line:84% and then it sits at 0 for the rest of the cycle until T. 00:32:17.810 --> 00:32:22.841 align:middle line:90% And so this is I sub L. 00:32:22.841 --> 00:32:24.633 align:middle line:84% So this is what we're going to have in DCM. 00:32:24.633 --> 00:32:29.790 align:middle line:90% 00:32:29.790 --> 00:32:33.300 align:middle line:84% What is the conversion ratio in DCM? 00:32:33.300 --> 00:32:36.110 align:middle line:84% And I say, well, why do I have to ask that question? 00:32:36.110 --> 00:32:40.280 align:middle line:84% Because previously, how did I find the conversion ratio? 00:32:40.280 --> 00:32:42.720 align:middle line:84% The way we got that voltage conversion ratio is-- 00:32:42.720 --> 00:32:46.950 align:middle line:84% what we did was we said that V1 DT is actually equal-- 00:32:46.950 --> 00:32:50.770 align:middle line:84% has got to be equal to V2 minus V1 times 1 minus DT. 00:32:50.770 --> 00:32:52.810 align:middle line:84% The average voltage across the inductor is 0. 00:32:52.810 --> 00:32:55.380 align:middle line:84% But that was based on this waveform. 00:32:55.380 --> 00:32:57.420 align:middle line:84% Our waveforms are now different, so now we've 00:32:57.420 --> 00:32:59.490 align:middle line:90% got to recalculate it. 00:32:59.490 --> 00:33:04.350 align:middle line:84% But we can find the conversion ratio the same way we do it. 00:33:04.350 --> 00:33:08.280 align:middle line:84% We know that the average voltage across the inductor has to be 0. 00:33:08.280 --> 00:33:10.110 align:middle line:90% So let's just calculate that. 00:33:10.110 --> 00:33:18.470 align:middle line:90% In PSS, what do I get? 00:33:18.470 --> 00:33:23.020 align:middle line:84% I get the average voltage across the inductor is equal to 0, 00:33:23.020 --> 00:33:32.880 align:middle line:84% and that's got to be equal to D times V1 plus D2 00:33:32.880 --> 00:33:37.750 align:middle line:84% times V1 minus V2, and that has to equal 0. 00:33:37.750 --> 00:33:40.500 align:middle line:84% So what I get is if I rearrange this equation-- what 00:33:40.500 --> 00:33:47.583 align:middle line:84% I should get is D plus D2 times V1 is equal to D2V2. 00:33:47.583 --> 00:33:51.390 align:middle line:90% 00:33:51.390 --> 00:34:02.730 align:middle line:84% Or V2 over V1 is equal to D plus D2 over D2V1, 00:34:02.730 --> 00:34:13.060 align:middle line:84% or D2 plus D over D2, which is also equal to 1 plus D over D2. 00:34:13.060 --> 00:34:15.860 align:middle line:90% 00:34:15.860 --> 00:34:20.760 align:middle line:84% So that's not exactly an answer in terms of it, 00:34:20.760 --> 00:34:23.170 align:middle line:84% because it depends on the duty ratio of the switch. 00:34:23.170 --> 00:34:24.670 align:middle line:84% But then I got to go figure out what 00:34:24.670 --> 00:34:28.560 align:middle line:84% the duty ratio of that diode is going to look like. 00:34:28.560 --> 00:34:32.110 align:middle line:90% But what do I know? 00:34:32.110 --> 00:34:38.010 align:middle line:84% I know that D2 is less than 1 minus D. Whatever 00:34:38.010 --> 00:34:43.550 align:middle line:84% it is, it's less than 1 minus D because this is the duration D2, 00:34:43.550 --> 00:34:44.929 align:middle line:90% or D2T. 00:34:44.929 --> 00:34:45.860 align:middle line:90% Right. 00:34:45.860 --> 00:34:49.580 align:middle line:90% And that's less than 1 minus DT. 00:34:49.580 --> 00:34:53.699 align:middle line:90% So if I took the CCM-- 00:34:53.699 --> 00:34:57.770 align:middle line:90% what is it-- this is DCM. 00:34:57.770 --> 00:35:00.870 align:middle line:90% What is it in CCM? 00:35:00.870 --> 00:35:06.720 align:middle line:84% In CCM, V2 over V1 is equal to 1 over 1 00:35:06.720 --> 00:35:13.260 align:middle line:84% minus D, which I could rewrite as 1 minus D plus D 00:35:13.260 --> 00:35:17.010 align:middle line:90% over 1 minus D. 00:35:17.010 --> 00:35:21.450 align:middle line:84% All I did was add a minus D and a plus D in the numerator. 00:35:21.450 --> 00:35:30.524 align:middle line:84% And then I could rewrite this as 1 plus D over 1 minus D. 00:35:30.524 --> 00:35:34.800 align:middle line:84% So here, we can say that the conversion ratio in CCM 00:35:34.800 --> 00:35:36.190 align:middle line:90% can be rewritten. 00:35:36.190 --> 00:35:40.390 align:middle line:84% It's a less convenient form, but it's 1 plus D over 1 minus D. 00:35:40.390 --> 00:35:42.880 align:middle line:84% So it's clear for any value of D, 00:35:42.880 --> 00:35:46.250 align:middle line:84% the conversion ratio is going to be greater than 1. 00:35:46.250 --> 00:35:51.920 align:middle line:84% Here-- because D2 is less than 1 minus D, V2 over V1 00:35:51.920 --> 00:35:54.332 align:middle line:84% is bigger in discontinuous conduction mode 00:35:54.332 --> 00:35:56.290 align:middle line:84% than it would be in continuous conduction mode. 00:35:56.290 --> 00:35:59.010 align:middle line:90% 00:35:59.010 --> 00:36:04.710 align:middle line:84% So what's going to happen is for a given duty ratio, whatever 00:36:04.710 --> 00:36:07.560 align:middle line:84% I get for my V2, it's going to be 00:36:07.560 --> 00:36:10.320 align:middle line:84% bigger than I would get in continuous conduction mode. 00:36:10.320 --> 00:36:13.265 align:middle line:84% The output voltage is going to get pumped up a bit. 00:36:13.265 --> 00:36:16.205 align:middle line:90% 00:36:16.205 --> 00:36:17.465 align:middle line:90% So I can tell that. 00:36:17.465 --> 00:36:21.080 align:middle line:90% 00:36:21.080 --> 00:36:25.490 align:middle line:90% That's pretty straightforward. 00:36:25.490 --> 00:36:29.030 align:middle line:90% It's a little bit messy. 00:36:29.030 --> 00:36:32.420 align:middle line:84% But we can go figure this out because I have the means 00:36:32.420 --> 00:36:35.300 align:middle line:84% to calculate what D2 is because I 00:36:35.300 --> 00:36:38.600 align:middle line:84% know exactly what these ripple waveforms look like. 00:36:38.600 --> 00:36:40.790 align:middle line:84% If I write the exact expression-- 00:36:40.790 --> 00:36:45.050 align:middle line:84% if I basically eliminate D2 from this expression, 00:36:45.050 --> 00:36:46.380 align:middle line:90% here's what you get. 00:36:46.380 --> 00:36:57.710 align:middle line:84% You get V2 over V1 is equal to a half plus 1/2 the square root 00:36:57.710 --> 00:37:08.540 align:middle line:84% of 1 plus 2D squared RT over 2L-- 00:37:08.540 --> 00:37:11.485 align:middle line:90% or RT over L, for DCM boost. 00:37:11.485 --> 00:37:16.870 align:middle line:90% 00:37:16.870 --> 00:37:23.918 align:middle line:84% So I'm just writing what this is for a boost converter. 00:37:23.918 --> 00:37:26.210 align:middle line:84% It would be different for different kind of converters. 00:37:26.210 --> 00:37:27.610 align:middle line:90% This is just an example. 00:37:27.610 --> 00:37:29.540 align:middle line:90% But what's the difference? 00:37:29.540 --> 00:37:34.180 align:middle line:84% Well, we said, look, if I'm in CCM, 00:37:34.180 --> 00:37:36.830 align:middle line:84% the voltage conversion ratio is just a function of duty ratio. 00:37:36.830 --> 00:37:37.960 align:middle line:84% It doesn't matter the inductance. 00:37:37.960 --> 00:37:39.543 align:middle line:84% It doesn't matter the load resistance. 00:37:39.543 --> 00:37:41.470 align:middle line:90% It's just going to-- 00:37:41.470 --> 00:37:43.690 align:middle line:84% very straightforward relationship. 00:37:43.690 --> 00:37:46.448 align:middle line:84% Suddenly, when I get into DCM, it depends on everything. 00:37:46.448 --> 00:37:47.740 align:middle line:90% It depends on my load resistor. 00:37:47.740 --> 00:37:49.630 align:middle line:84% It depends on my switching period. 00:37:49.630 --> 00:37:51.080 align:middle line:90% It depends on my inductance. 00:37:51.080 --> 00:37:52.387 align:middle line:90% It's a mess. 00:37:52.387 --> 00:37:53.720 align:middle line:90% It doesn't mean you can't do it. 00:37:53.720 --> 00:37:57.380 align:middle line:84% It's just not a simple relationship anymore. 00:37:57.380 --> 00:38:03.590 align:middle line:84% So suddenly, it's a lot more complicated 00:38:03.590 --> 00:38:06.450 align:middle line:84% than it was in continuous conduction mode. 00:38:06.450 --> 00:38:09.530 align:middle line:84% And it also affects the converter's dynamics, 00:38:09.530 --> 00:38:10.180 align:middle line:90% as we'll see. 00:38:10.180 --> 00:38:13.400 align:middle line:90% 00:38:13.400 --> 00:38:14.770 align:middle line:90% Any questions about that? 00:38:14.770 --> 00:38:17.510 align:middle line:90% 00:38:17.510 --> 00:38:19.410 align:middle line:84% But that said, we can't avoid it. 00:38:19.410 --> 00:38:21.200 align:middle line:90% Yeah, go ahead. 00:38:21.200 --> 00:38:24.110 align:middle line:84% AUDIENCE: How do we know that we stay in periodic steady state 00:38:24.110 --> 00:38:27.230 align:middle line:84% when we go through these different modes of operation? 00:38:27.230 --> 00:38:28.920 align:middle line:84% DAVID PERREAULT: That's a good question. 00:38:28.920 --> 00:38:34.200 align:middle line:84% It's a question of, will it reach periodic steady state? 00:38:34.200 --> 00:38:38.970 align:middle line:84% That's the same analysis you go do, You can go do this analysis 00:38:38.970 --> 00:38:43.140 align:middle line:84% and find out whether it'll end up in periodic steady state. 00:38:43.140 --> 00:38:44.740 align:middle line:90% It generally will settle-- 00:38:44.740 --> 00:38:47.380 align:middle line:84% I mean, because you can solve for these things, 00:38:47.380 --> 00:38:50.730 align:middle line:84% the boost converter will end up in periodic steady state 00:38:50.730 --> 00:38:52.800 align:middle line:90% unless-- 00:38:52.800 --> 00:38:56.760 align:middle line:84% suppose I disconnected this load resistor in this case. 00:38:56.760 --> 00:39:01.436 align:middle line:84% That's equivalent to making R equal to infinity. 00:39:01.436 --> 00:39:09.420 align:middle line:84% If I come over here and I make R equal to infinity, V2 over V1 00:39:09.420 --> 00:39:10.860 align:middle line:90% goes to infinity. 00:39:10.860 --> 00:39:15.025 align:middle line:84% That means it's just going to wander off to infinity. 00:39:15.025 --> 00:39:16.650 align:middle line:84% The output will wander off to infinity. 00:39:16.650 --> 00:39:17.760 align:middle line:90% Why would that happen? 00:39:17.760 --> 00:39:22.040 align:middle line:84% Because look, this is IL in DCM, So every cycle, 00:39:22.040 --> 00:39:25.970 align:middle line:84% I take energy out of the input, put it in the inductor, 00:39:25.970 --> 00:39:29.630 align:middle line:84% turn off the switch, and that energy goes to the output. 00:39:29.630 --> 00:39:32.980 align:middle line:84% And I build up some energy on the capacitor. 00:39:32.980 --> 00:39:35.650 align:middle line:84% And the feedback mechanism that gets you to steady state 00:39:35.650 --> 00:39:37.895 align:middle line:84% is the resistor drains some of that energy. 00:39:37.895 --> 00:39:39.520 align:middle line:84% But if you don't ever drain the energy, 00:39:39.520 --> 00:39:41.313 align:middle line:84% this just wanders off to infinity. 00:39:41.313 --> 00:39:42.730 align:middle line:84% I actually had a converter explode 00:39:42.730 --> 00:39:43.855 align:middle line:90% on me once because of that. 00:39:43.855 --> 00:39:48.362 align:middle line:84% The electrolytic capacitor literally blew up in my face. 00:39:48.362 --> 00:39:49.820 align:middle line:84% AUDIENCE: Were you wearing glasses? 00:39:49.820 --> 00:39:50.410 align:middle line:90% DAVID PERREAULT: What's that? 00:39:50.410 --> 00:39:51.770 align:middle line:84% AUDIENCE: Were you wearing glasses? 00:39:51.770 --> 00:39:52.820 align:middle line:84% DAVID PERREAULT: You're damn right I was-- 00:39:52.820 --> 00:39:53.725 align:middle line:90% I can still see. 00:39:53.725 --> 00:39:55.590 align:middle line:90% [LAUGHTER] 00:39:55.590 --> 00:39:59.892 align:middle line:84% I always wear glasses when I'm working on a higher powered 00:39:59.892 --> 00:40:00.600 align:middle line:90% thing these days. 00:40:00.600 --> 00:40:03.230 align:middle line:84% And I should on any converter because literally-- 00:40:03.230 --> 00:40:04.340 align:middle line:90% I was looking down to it. 00:40:04.340 --> 00:40:07.310 align:middle line:84% And I bumped the connection to the load with my elbow, 00:40:07.310 --> 00:40:11.420 align:middle line:84% and the capacitor just literally exploded in my face. 00:40:11.420 --> 00:40:14.520 align:middle line:84% They often don't vent so dramatically. 00:40:14.520 --> 00:40:18.680 align:middle line:84% They're not supposed to, but in this case, it exploded. 00:40:18.680 --> 00:40:21.200 align:middle line:84% So, yes, wearing glasses is a good idea 00:40:21.200 --> 00:40:22.500 align:middle line:90% in these kind of things. 00:40:22.500 --> 00:40:25.760 align:middle line:84% I actually did that this morning on our demo converter, too, 00:40:25.760 --> 00:40:27.440 align:middle line:90% but that's not what happened. 00:40:27.440 --> 00:40:32.462 align:middle line:84% But what happened was the MOSFET started breaking down. 00:40:32.462 --> 00:40:33.920 align:middle line:84% In other words, it would just start 00:40:33.920 --> 00:40:37.670 align:middle line:84% dumping energy into the MOSFET, which isn't good for the MOSFET. 00:40:37.670 --> 00:40:39.400 align:middle line:84% So fortunately, I didn't kill the demo 00:40:39.400 --> 00:40:42.160 align:middle line:84% because I was being stupid by taking off the load. 00:40:42.160 --> 00:40:43.840 align:middle line:84% But the point is you don't necessarily 00:40:43.840 --> 00:40:46.132 align:middle line:84% have to get to periodic steady state in all converters. 00:40:46.132 --> 00:40:48.800 align:middle line:84% And this is an example where you can get to really high voltage. 00:40:48.800 --> 00:40:51.070 align:middle line:84% And in fact, if you look at little photo flash 00:40:51.070 --> 00:40:54.700 align:middle line:84% chargers for those little cameras 00:40:54.700 --> 00:40:58.270 align:middle line:84% that you use at weddings, the actual film cameras, 00:40:58.270 --> 00:41:02.920 align:middle line:84% yielding cameras, they often use a boost or a flyback in DCM 00:41:02.920 --> 00:41:05.860 align:middle line:84% to charge the output voltage up from a volt and a half 00:41:05.860 --> 00:41:07.570 align:middle line:90% up to a few 100 volts. 00:41:07.570 --> 00:41:10.060 align:middle line:84% Same trick because you can get a very large conversion 00:41:10.060 --> 00:41:11.180 align:middle line:90% ratios this way. 00:41:11.180 --> 00:41:14.000 align:middle line:90% 00:41:14.000 --> 00:41:18.830 align:middle line:90% So your DCM is harder to model. 00:41:18.830 --> 00:41:23.160 align:middle line:84% How would I think about modeling it? 00:41:23.160 --> 00:41:25.350 align:middle line:84% And I'll let Monsi start to set up 00:41:25.350 --> 00:41:28.950 align:middle line:90% and roll down screens and stuff. 00:41:28.950 --> 00:41:35.070 align:middle line:84% One way I could think about this is the following. 00:41:35.070 --> 00:41:37.305 align:middle line:90% Maybe I would think about-- 00:41:37.305 --> 00:41:40.620 align:middle line:84% if I looked at my converter here, 00:41:40.620 --> 00:41:42.993 align:middle line:84% this is my inductor current waveform. 00:41:42.993 --> 00:41:45.610 align:middle line:90% 00:41:45.610 --> 00:41:48.420 align:middle line:84% What does my diode current waveform look like-- 00:41:48.420 --> 00:41:49.190 align:middle line:90% I sub D? 00:41:49.190 --> 00:41:54.210 align:middle line:90% 00:41:54.210 --> 00:41:59.110 align:middle line:84% I sub D is 0 in the first part of the cycle. 00:41:59.110 --> 00:42:02.050 align:middle line:84% Then it conducts in the second part of the cycle, 00:42:02.050 --> 00:42:06.140 align:middle line:84% until D plus D2, and then it's 0 for the rest of the cycle. 00:42:06.140 --> 00:42:07.690 align:middle line:90% So this is DT. 00:42:07.690 --> 00:42:12.140 align:middle line:84% This is D plus D2, T, and so forth. 00:42:12.140 --> 00:42:18.230 align:middle line:84% So maybe, I could think about modeling my converter 00:42:18.230 --> 00:42:20.480 align:middle line:84% from a control perspective as just 00:42:20.480 --> 00:42:23.510 align:middle line:84% whatever this current ID is into the output, 00:42:23.510 --> 00:42:25.970 align:middle line:84% So maybe I would model it by saying, 00:42:25.970 --> 00:42:33.470 align:middle line:84% OK, let me think of my converter as this current ID, which 00:42:33.470 --> 00:42:34.360 align:middle line:90% looks like this-- 00:42:34.360 --> 00:42:37.800 align:middle line:90% 00:42:37.800 --> 00:42:44.500 align:middle line:84% DT D plus D2 T and T, and just repeats-- 00:42:44.500 --> 00:42:47.620 align:middle line:90% 00:42:47.620 --> 00:42:55.840 align:middle line:84% into my output capacitor and resistor. 00:42:55.840 --> 00:42:58.840 align:middle line:84% So maybe I would start to start to think of my converter 00:42:58.840 --> 00:43:01.990 align:middle line:84% as some sort of funny current source. 00:43:01.990 --> 00:43:05.740 align:middle line:84% And maybe I can relate this current source back 00:43:05.740 --> 00:43:08.170 align:middle line:84% to the component values and just start 00:43:08.170 --> 00:43:10.300 align:middle line:84% thinking about my converter as a current source 00:43:10.300 --> 00:43:11.980 align:middle line:84% because that's really what I've turned it into. 00:43:11.980 --> 00:43:14.272 align:middle line:84% I'm going to be squirting some current into that output 00:43:14.272 --> 00:43:17.470 align:middle line:90% capacitor every cycle. 00:43:17.470 --> 00:43:20.260 align:middle line:90% Well, what does that look like? 00:43:20.260 --> 00:43:26.560 align:middle line:84% What I know here is that this current, I peak-- 00:43:26.560 --> 00:43:38.400 align:middle line:84% I peak is equal to V1DT over L. And this time period-- 00:43:38.400 --> 00:43:42.080 align:middle line:90% 00:43:42.080 --> 00:43:44.780 align:middle line:84% let me call this time period to discharge delta 00:43:44.780 --> 00:43:49.170 align:middle line:90% T, which is just D2 T. 00:43:49.170 --> 00:43:51.440 align:middle line:90% What do I know? 00:43:51.440 --> 00:43:55.080 align:middle line:84% I know that V is equal to L DI DT. 00:43:55.080 --> 00:44:03.790 align:middle line:84% So delta T is equal to D2 T is equal to L 00:44:03.790 --> 00:44:12.220 align:middle line:84% I peak divided by L I peak divided 00:44:12.220 --> 00:44:14.890 align:middle line:90% by V, which is V2 minus V1. 00:44:14.890 --> 00:44:19.504 align:middle line:90% 00:44:19.504 --> 00:44:22.840 align:middle line:90% And I can put in I peak here. 00:44:22.840 --> 00:44:32.860 align:middle line:84% So that's just going to be V1 DT V1DT 00:44:32.860 --> 00:44:38.800 align:middle line:90% over L divided by V2 minus V1. 00:44:38.800 --> 00:44:41.030 align:middle line:90% That make sense to everybody? 00:44:41.030 --> 00:44:43.850 align:middle line:84% And so let me ask, what is the average-- what's 00:44:43.850 --> 00:44:45.350 align:middle line:84% important over the switching cycle 00:44:45.350 --> 00:44:47.450 align:middle line:84% is what's the average current that I'm 00:44:47.450 --> 00:44:49.740 align:middle line:90% sticking into the output? 00:44:49.740 --> 00:44:54.300 align:middle line:84% I should ask, each cycle, how much on average 00:44:54.300 --> 00:44:55.860 align:middle line:90% am I squirting into the output? 00:44:55.860 --> 00:45:00.000 align:middle line:84% Well, I sub D-- the average value of I sub D is basically 00:45:00.000 --> 00:45:05.130 align:middle line:84% the area there, which would be 1/2 the base of that, 00:45:05.130 --> 00:45:12.090 align:middle line:84% which is this delta T, V1D2 over V2 minus V1. 00:45:12.090 --> 00:45:16.360 align:middle line:90% 00:45:16.360 --> 00:45:18.880 align:middle line:84% 1/2 the base times the height, which is IP, 00:45:18.880 --> 00:45:23.810 align:middle line:90% which is V1DT over L. 00:45:23.810 --> 00:45:26.230 align:middle line:84% This is the area under that triangle-- 00:45:26.230 --> 00:45:32.740 align:middle line:84% 1/2 base times height-- times 1 over T. If I rewrite that, then, 00:45:32.740 --> 00:45:48.430 align:middle line:84% what I'm going to get is 1/2 V1 T V1 squared T over 2 V2 minus 00:45:48.430 --> 00:45:51.030 align:middle line:90% V1 L times D squared. 00:45:51.030 --> 00:45:57.940 align:middle line:90% 00:45:57.940 --> 00:45:58.930 align:middle line:90% Let me just see-- 00:45:58.930 --> 00:46:02.020 align:middle line:84% double check to make sure I did my math right here. 00:46:02.020 --> 00:46:03.470 align:middle line:84% It's something like this, anyways. 00:46:03.470 --> 00:46:05.060 align:middle line:90% V1 squared T over 2L. 00:46:05.060 --> 00:46:07.670 align:middle line:90% V2 minus V1 D squared. 00:46:07.670 --> 00:46:08.480 align:middle line:90% OK, yeah. 00:46:08.480 --> 00:46:10.780 align:middle line:90% So really what I have is, now-- 00:46:10.780 --> 00:46:14.980 align:middle line:84% I can think of all the converter leading up to the output cap 00:46:14.980 --> 00:46:18.580 align:middle line:84% as just a current source that's a function of the duty ratio 00:46:18.580 --> 00:46:20.620 align:middle line:90% I control. 00:46:20.620 --> 00:46:23.230 align:middle line:84% So I've turned this thing into a current source that squirts 00:46:23.230 --> 00:46:24.230 align:middle line:90% current into the output. 00:46:24.230 --> 00:46:26.770 align:middle line:84% And I think about it that way and control it that way. 00:46:26.770 --> 00:46:28.385 align:middle line:84% And if I know what my load resistor is 00:46:28.385 --> 00:46:30.010 align:middle line:84% and the voltage I want, I know how much 00:46:30.010 --> 00:46:32.730 align:middle line:84% averaged current I want to stick into it. 00:46:32.730 --> 00:46:36.480 align:middle line:84% So that's how I might model a DCM converter. 00:46:36.480 --> 00:46:38.607 align:middle line:84% It's more complicated, in some sense, 00:46:38.607 --> 00:46:40.440 align:middle line:84% than my continuous conduction mode converter 00:46:40.440 --> 00:46:42.720 align:middle line:84% because it's a function of all this junk. 00:46:42.720 --> 00:46:45.870 align:middle line:84% My switching period and my and my inductor value 00:46:45.870 --> 00:46:48.490 align:middle line:84% and all that stuff matters, where it didn't before. 00:46:48.490 --> 00:46:50.820 align:middle line:84% But I have to live with that because it's this more 00:46:50.820 --> 00:46:53.640 align:middle line:90% complicated operating rule. 00:46:53.640 --> 00:46:54.540 align:middle line:90% Yeah. 00:46:54.540 --> 00:46:57.753 align:middle line:84% AUDIENCE: So does that mean when you go from CCM to DCM, 00:46:57.753 --> 00:46:59.670 align:middle line:84% you're probably going to change the duty cycle 00:46:59.670 --> 00:47:00.920 align:middle line:90% to maintain a target voltage? 00:47:00.920 --> 00:47:02.170 align:middle line:90% DAVID PERREAULT: That's right. 00:47:02.170 --> 00:47:05.175 align:middle line:84% You're going to use a lower duty cycle than you would otherwise 00:47:05.175 --> 00:47:06.300 align:middle line:90% use in the boost converter. 00:47:06.300 --> 00:47:07.860 align:middle line:90% Yes. 00:47:07.860 --> 00:47:10.470 align:middle line:84% That's right, because the conversion ratio goes up. 00:47:10.470 --> 00:47:12.960 align:middle line:84% And so your controller-- and we'll talk about this later-- 00:47:12.960 --> 00:47:15.850 align:middle line:84% your controller usually has to manage that boundary. 00:47:15.850 --> 00:47:19.440 align:middle line:84% You have to have a controller that works for either case. 00:47:19.440 --> 00:47:21.660 align:middle line:90% Let's just take a look at this. 00:47:21.660 --> 00:47:24.540 align:middle line:84% So Monsi has set up an actual boost 00:47:24.540 --> 00:47:26.530 align:middle line:90% converter that's doing this. 00:47:26.530 --> 00:47:29.770 align:middle line:84% And let's take a look at the waveforms. 00:47:29.770 --> 00:47:32.770 align:middle line:90% In yellow, you can see-- 00:47:32.770 --> 00:47:35.470 align:middle line:90% 00:47:35.470 --> 00:47:38.680 align:middle line:84% you can essentially see the switch voltage waveform, 00:47:38.680 --> 00:47:41.830 align:middle line:84% which is essentially this waveform moved up 00:47:41.830 --> 00:47:44.060 align:middle line:90% where this is 0. 00:47:44.060 --> 00:47:46.570 align:middle line:84% So in the first part of the cycle, 00:47:46.570 --> 00:47:51.710 align:middle line:84% you get 0 in the yellow wave-- from the second part 00:47:51.710 --> 00:47:53.730 align:middle line:90% of the cycle, it's at V2. 00:47:53.730 --> 00:47:55.250 align:middle line:84% And then you get this time period 00:47:55.250 --> 00:47:58.850 align:middle line:90% where you see this voltage ring. 00:47:58.850 --> 00:48:00.180 align:middle line:90% What the heck is that? 00:48:00.180 --> 00:48:03.260 align:middle line:84% Well, I said we had this time period 00:48:03.260 --> 00:48:06.680 align:middle line:84% where both the switch and the diode were open, 00:48:06.680 --> 00:48:10.800 align:middle line:84% and so that this voltage here ought to be exactly at V1. 00:48:10.800 --> 00:48:16.370 align:middle line:84% But what really happens in the real world is that-- guess what. 00:48:16.370 --> 00:48:21.600 align:middle line:84% There's some switch capacitance and some diode capacitance. 00:48:21.600 --> 00:48:27.380 align:middle line:84% And so what happens is this voltage, instead of going to 0, 00:48:27.380 --> 00:48:31.100 align:middle line:84% rings with those capacitance-- the inductor ring 00:48:31.100 --> 00:48:33.060 align:middle line:90% with the parasitic capacitances. 00:48:33.060 --> 00:48:34.320 align:middle line:90% People don't like that. 00:48:34.320 --> 00:48:36.770 align:middle line:84% That's unpleasant, It causes noise and stuff. 00:48:36.770 --> 00:48:38.610 align:middle line:84% But you can't do anything about it. 00:48:38.610 --> 00:48:40.350 align:middle line:84% If you're going to have both devices on. 00:48:40.350 --> 00:48:41.930 align:middle line:84% In some cases, actually, people actually 00:48:41.930 --> 00:48:43.722 align:middle line:84% even put in another switch to short it out. 00:48:43.722 --> 00:48:45.920 align:middle line:90% But generally, you don't-- 00:48:45.920 --> 00:48:48.295 align:middle line:84% generally, you just let it ring, and that's life. 00:48:48.295 --> 00:48:52.070 align:middle line:90% 00:48:52.070 --> 00:48:55.200 align:middle line:84% But you can now see that you have the rising inductor 00:48:55.200 --> 00:48:57.120 align:middle line:84% current, the falling inductor current at 0. 00:48:57.120 --> 00:49:02.811 align:middle line:84% It's exactly doing what this yellow waveform is doing here. 00:49:02.811 --> 00:49:04.100 align:middle line:90% Any questions about that? 00:49:04.100 --> 00:49:08.660 align:middle line:90% 00:49:08.660 --> 00:49:09.160 align:middle line:90% Yeah. 00:49:09.160 --> 00:49:10.660 align:middle line:84% AUDIENCE: Can you mention the source 00:49:10.660 --> 00:49:12.638 align:middle line:90% that caused that ring again? 00:49:12.638 --> 00:49:14.513 align:middle line:84% DAVID PERREAULT: It's basically the inductor. 00:49:14.513 --> 00:49:17.290 align:middle line:90% 00:49:17.290 --> 00:49:22.050 align:middle line:84% resonating with parasitic capacitance at this node. 00:49:22.050 --> 00:49:23.370 align:middle line:90% Yeah. 00:49:23.370 --> 00:49:24.635 align:middle line:90% It's an unfortunate thing. 00:49:24.635 --> 00:49:25.510 align:middle line:90% People don't like it. 00:49:25.510 --> 00:49:26.370 align:middle line:90% It causes noise. 00:49:26.370 --> 00:49:27.650 align:middle line:90% It's a little bit annoying. 00:49:27.650 --> 00:49:28.430 align:middle line:90% That's life. 00:49:28.430 --> 00:49:31.980 align:middle line:90% 00:49:31.980 --> 00:49:35.850 align:middle line:84% So I said-- sometimes you have to live with DCM operation, 00:49:35.850 --> 00:49:39.570 align:middle line:84% especially if you operate over really wide load ranges 00:49:39.570 --> 00:49:44.220 align:middle line:84% or you want a finite value for your inductor sizes. 00:49:44.220 --> 00:49:46.690 align:middle line:84% Sometimes you actually want DCM operation. 00:49:46.690 --> 00:49:50.020 align:middle line:84% People often design their converters for DCM operation. 00:49:50.020 --> 00:49:51.930 align:middle line:90% Why is that? 00:49:51.930 --> 00:49:56.700 align:middle line:90% Well, suppose I thought of-- 00:49:56.700 --> 00:50:00.080 align:middle line:90% I had a reference current. 00:50:00.080 --> 00:50:02.960 align:middle line:84% And in this case, since I'm talking about a boost converter, 00:50:02.960 --> 00:50:05.570 align:middle line:84% let me think-- suppose I had a reference current for current 00:50:05.570 --> 00:50:08.420 align:middle line:84% to draw to deliver the power I wanted from the output, 00:50:08.420 --> 00:50:11.330 align:middle line:90% from the input. 00:50:11.330 --> 00:50:13.550 align:middle line:84% If I have a small ripple converter 00:50:13.550 --> 00:50:18.170 align:middle line:84% and I say, OK, ideally speaking, to support the load, 00:50:18.170 --> 00:50:21.017 align:middle line:84% I had this low current reference and then 00:50:21.017 --> 00:50:23.100 align:middle line:84% I stepped it up for this higher current reference. 00:50:23.100 --> 00:50:26.510 align:middle line:84% So this is I ref for the reference current-- 00:50:26.510 --> 00:50:29.150 align:middle line:90% for the inductor waveform, 00:50:29.150 --> 00:50:34.495 align:middle line:84% If I have a small ripple converter 00:50:34.495 --> 00:50:38.110 align:middle line:84% and it's rippling like this and then I step the reference-- 00:50:38.110 --> 00:50:39.660 align:middle line:84% I want much more current to deliver 00:50:39.660 --> 00:50:41.770 align:middle line:90% more power, what has to happen? 00:50:41.770 --> 00:50:45.565 align:middle line:84% Well, the inductor current is going to slew up, 00:50:45.565 --> 00:50:46.940 align:middle line:84% and eventually, it can get there, 00:50:46.940 --> 00:50:49.320 align:middle line:90% and I can deliver this again. 00:50:49.320 --> 00:50:52.040 align:middle line:84% But you notice that there's this time 00:50:52.040 --> 00:50:54.770 align:middle line:84% period where the inductor current takes a lot of time 00:50:54.770 --> 00:50:55.320 align:middle line:90% to change. 00:50:55.320 --> 00:50:57.750 align:middle line:84% If the ripple is small, the inductor value is big, 00:50:57.750 --> 00:51:00.500 align:middle line:84% it takes a lot of time to put the voltage across the inductor 00:51:00.500 --> 00:51:02.152 align:middle line:90% to slew the current in it. 00:51:02.152 --> 00:51:05.000 align:middle line:84% And that means for all this time period, 00:51:05.000 --> 00:51:07.190 align:middle line:84% you're not delivering the right output power, 00:51:07.190 --> 00:51:10.970 align:middle line:84% or you're at least not drawing the right input power. 00:51:10.970 --> 00:51:15.270 align:middle line:84% So this converter-- its ability to change its operating point 00:51:15.270 --> 00:51:18.340 align:middle line:90% is limited by the small ripple. 00:51:18.340 --> 00:51:21.370 align:middle line:90% Suppose I had a DCM converter. 00:51:21.370 --> 00:51:24.600 align:middle line:84% Well, maybe I have something like this, 00:51:24.600 --> 00:51:26.100 align:middle line:90% and I-- again, I step my ripple. 00:51:26.100 --> 00:51:29.070 align:middle line:84% But maybe in a DCM converter, my inductor current 00:51:29.070 --> 00:51:29.930 align:middle line:90% looks like this. 00:51:29.930 --> 00:51:34.370 align:middle line:90% 00:51:34.370 --> 00:51:36.030 align:middle line:90% And the next cycle, I want more? 00:51:36.030 --> 00:51:36.560 align:middle line:90% Fine. 00:51:36.560 --> 00:51:37.060 align:middle line:90% Boom. 00:51:37.060 --> 00:51:39.810 align:middle line:90% 00:51:39.810 --> 00:51:43.000 align:middle line:84% I can deliver that almost instantly. 00:51:43.000 --> 00:51:46.350 align:middle line:84% In other words, the high ripple means 00:51:46.350 --> 00:51:49.620 align:middle line:84% that I can change the average current in the inductor much 00:51:49.620 --> 00:51:52.320 align:middle line:84% faster, maybe even within a cycle. 00:51:52.320 --> 00:51:55.320 align:middle line:84% So there's-- on the one hand, we often want small ripple 00:51:55.320 --> 00:51:58.140 align:middle line:84% because we want nice waveforms and not a lot of ripple and we 00:51:58.140 --> 00:51:59.340 align:middle line:90% got to filter it. 00:51:59.340 --> 00:52:03.110 align:middle line:84% On the other hand, the ability to change the energy storage 00:52:03.110 --> 00:52:04.860 align:middle line:84% and the energy processing of the converter 00:52:04.860 --> 00:52:07.920 align:middle line:84% quickly might take you towards using DCMs. 00:52:07.920 --> 00:52:11.080 align:middle line:84% But for a very high transient response converter, 00:52:11.080 --> 00:52:13.287 align:middle line:84% having high ripple and possibly DCM operation 00:52:13.287 --> 00:52:14.120 align:middle line:90% can be a good thing. 00:52:14.120 --> 00:52:17.358 align:middle line:90% 00:52:17.358 --> 00:52:19.460 align:middle line:84% Any last questions before we wrap up? 00:52:19.460 --> 00:52:22.570 align:middle line:90% 00:52:22.570 --> 00:52:23.380 align:middle line:90% OK. 00:52:23.380 --> 00:52:25.750 align:middle line:84% Next class, we're going to start talking about, how do 00:52:25.750 --> 00:52:28.490 align:middle line:84% I actually do things like design our inductors? 00:52:28.490 --> 00:52:30.510 align:middle line:90% Have a great day. 00:52:30.510 --> 00:52:34.000 align:middle line:90%