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Capacitors in AC Circuits with Phasors _ Doc Physics
WEBVTT Kind: captions Language: en
00:00:00.030 hey if I hook up a capacitor to a 00:00:03.74000:00:03.750 battery let's say I've got this D 00:00:05.69000:00:05.700 battery here and this sides positive and 00:00:09.16900:00:09.179 this side is negative and here I have a 00:00:10.66900:00:10.679 parallel plate capacitor and my 00:00:13.07000:00:13.080 intention is to put a wire between here 00:00:16.03900:00:16.049 and then a wire between here and as soon 00:00:19.49000:00:19.500 as I connect that see as soon as I 00:00:22.37000:00:22.380 connect that the voltage on the 00:00:25.73000:00:25.740 capacitor will gradually increase like 00:00:27.80000:00:27.810 this voltage will go like this and 00:00:30.83000:00:30.840 approach some some maximum level 00:00:33.31900:00:33.329 probably the voltage of the battery if 00:00:35.63000:00:35.640 this is a function of time but if I'm 00:00:37.70000:00:37.710 going to also want to know what's 00:00:40.13000:00:40.140 happening to the current see the current 00:00:43.27000:00:43.280 immediately will be huge 00:00:45.47000:00:45.480 they'll immediately be a very large 00:00:47.66000:00:47.670 current as soon as I connect that 00:00:50.38900:00:50.399 there'll be a huge current and then the 00:00:52.63900:00:52.649 current will taper off and approach zero 00:00:56.51000:00:56.520 where these guys are acting very 00:00:59.18000:00:59.190 differently right the current in this 00:01:02.45000:01:02.460 circuit is going to be falling to zero 00:01:04.93900:01:04.949 and the voltage will be rising up and 00:01:06.85900:01:06.869 and that's because we can intuitively 00:01:08.39000:01:08.400 understand this because as soon as I 00:01:10.73000:01:10.740 make this connection here there is no 00:01:14.00000:01:14.010 force against the electrons initially 00:01:17.33000:01:17.340 going this direction there's no force 00:01:19.64000:01:19.650 initially against the current going that 00:01:21.17000:01:21.180 direction but as the capacitor becomes 00:01:22.76000:01:22.770 more and more charged than the voltage 00:01:24.53000:01:24.540 difference between the capacitor and the 00:01:26.21000:01:26.220 battery is approaching zero because well 00:01:28.58000:01:28.590 the voltage of the and the voltage of 00:01:31.24900:01:31.259 the capacitor is approaching the voltage 00:01:32.63000:01:32.640 of the battery so we want to think now 00:01:35.66000:01:35.670 about what happens if instead we connect 00:01:38.48000:01:38.490 a capacitor to an alternating source I'm 00:01:43.39900:01:43.409 going to connect a capacitor to an AC 00:01:45.71000:01:45.720 source and I want you to think about 00:01:48.13900:01:48.149 what this means 00:01:49.19000:01:49.200 see if I connect it to a DC source then 00:01:54.05000:01:54.060 the capacitor actually looks like a 00:01:55.76000:01:55.770 break in the circuit it will just 00:01:57.05000:01:57.060 prevent any current from flowing but if 00:02:00.28900:02:00.299 I connect it to a very fast AC source 00:02:02.33000:02:02.340 it's almost as if the capacitor isn't 00:02:04.13000:02:04.140 there because as soon as I begin to 00:02:05.56900:02:05.579 charge the capacitor the current from 00:02:07.67000:02:07.680 the source changes directions and then 00:02:10.10000:02:10.110 discharges it and can charge it and 00:02:11.96000:02:11.970 discharge it and charge and discharge it 00:02:13.76000:02:13.770 so it's just as if there were a wire 00:02:15.38000:02:15.390 right here so let me summarize that 00:02:17.30000:02:17.310 little observation I will say at oh and 00:02:20.54000:02:20.550 of course I'm going to use Omega to 00:02:23.93000:02:23.940 symbolize how fast I'm sloshing right 00:02:26.54000:02:26.550 here this is the frequency the angular 00:02:29.45000:02:29.460 frequency of my of my AC power source at 00:02:33.68000:02:33.690 low Omega capacitors look like what and 00:02:43.97000:02:43.980 at high Omega this is very important so 00:02:47.69000:02:47.700 I'm going to ask you to fill those in 00:02:48.95000:02:48.960 you have to think about what it means 00:02:50.71000:02:50.720 capacitors and I don't need to say look 00:02:54.77000:02:54.780 I could say they act like what in a 00:03:02.54000:03:02.550 circuit this is very important to think 00:03:07.97000:03:07.980 about in a DC circuit a capacitor looks 00:03:11.15000:03:11.160 like a break in the circuit and so at 00:03:14.15000:03:14.160 low Omega that's what capacitors look 00:03:16.01000:03:16.020 like it looks as if the circuit is 00:03:17.69000:03:17.700 broken but at high omegas sloshing back 00:03:20.75000:03:20.760 and forth shoo-shoo shoo-shoo the 00:03:22.49000:03:22.500 capacitor doesn't affect the sloshing at 00:03:24.92000:03:24.930 all assuming the capacitor is large 00:03:26.87000:03:26.880 enough right so that it doesn't reach 00:03:28.67000:03:28.680 its maximum voltage very quickly 00:03:30.11000:03:30.120 capacitors that are tiny reach the 00:03:32.09000:03:32.100 maximum voltage very quickly and could 00:03:33.74000:03:33.750 still impede the flow of charge through 00:03:36.05000:03:36.060 this part of the circuit of course there 00:03:37.76000:03:37.770 aren't actually charges going that 00:03:39.80000:03:39.810 direction or that direction it's just 00:03:42.23000:03:42.240 everywhere along here and everywhere 00:03:43.88000:03:43.890 along here so as long as you're outside 00:03:45.32000:03:45.330 the capacitor it looks as if there's 00:03:47.12000:03:47.130 just a wire right there so here we put 00:03:49.64000:03:49.650 wire and there we put break or here you 00:03:52.19000:03:52.200 could put short and here you could put 00:03:53.72000:03:53.730 break a break in a shorter very 00:03:55.10000:03:55.110 different things think about that for a 00:03:56.72000:03:56.730 moment now as we go into this we're 00:03:59.60000:03:59.610 going to have to do a little bit of 00:04:00.68000:04:00.690 thumb we're going to have to do a little 00:04:03.14000:04:03.150 bit of calculus and I think that's 00:04:04.40000:04:04.410 probably very good for us so let's do it 00:04:05.99000:04:06.000 you know that the voltage across a 00:04:08.36000:04:08.370 capacitor is the charge on the capacitor 00:04:10.69900:04:10.709 divided by its capacitance and this 00:04:13.67000:04:13.680 charge is actually the integral of the 00:04:15.80000:04:15.810 current going into the capacitor so as 00:04:18.50000:04:18.510 current goes into a capacitor it fills 00:04:20.51000:04:20.520 up the capacitor and that's what gives 00:04:22.58000:04:22.590 the capacitor charge on its positive 00:04:24.74000:04:24.750 plate so let's write that of again the 00:04:26.65900:04:26.669 voltage on the capacitor 00:04:27.62900:04:27.639 charge / capacitance so that's going to 00:04:30.95900:04:30.969 be well it's going to be the integral 00:04:33.57000:04:33.580 well it'll be one over C times the 00:04:35.39900:04:35.409 integral of current over time and with 00:04:38.49000:04:38.500 my AC generator now I'm going to say 00:04:41.12900:04:41.139 that I have current that does depend on 00:04:44.07000:04:44.080 time and I'm going to say that the 00:04:46.67900:04:46.689 current is well this is current as a 00:04:49.89000:04:49.900 function of time it's going to be IMAX 00:04:51.92900:04:51.939 times the sine of Omega times T that's 00:04:56.52000:04:56.530 my current I plug that in right there 00:04:58.61900:04:58.629 and see what happens it's going to be 00:05:01.49900:05:01.509 some really enjoyable calculus 1 over C 00:05:05.12900:05:05.139 times the integral of IMAX times the 00:05:09.95900:05:09.969 sine of Omega T over time IMAX as a 00:05:15.45000:05:15.460 constant and pulls out and we just have 00:05:17.39900:05:17.409 to do the integral of the sine of Omega 00:05:19.76900:05:19.779 T it gives us the negative cosine and 00:05:21.86900:05:21.879 then we have to do the chain rule for 00:05:23.49000:05:23.500 integrals oh man so we get negative IMAX 00:05:30.52900:05:30.539 over C and and then we have cosine of 00:05:36.52900:05:36.539 Omega T and we've also got to note what 00:05:40.80000:05:40.810 the integral of that sucker is and that 00:05:44.15900:05:44.169 would be oh man I think I'm going to get 00:05:46.26000:05:46.270 divided by Omega right here so that's 00:05:50.24900:05:50.259 really lovely and I could write it just 00:05:52.98000:05:52.990 slightly differently I'm going to write 00:05:54.65900:05:54.669 it as oh gosh well this is negative 00:05:57.77900:05:57.789 cosine here but I happen to know that 00:06:00.20900:06:00.219 negative cosine of theta is the sine of 00:06:03.92900:06:03.939 theta this is spelled this is really 00:06:05.61000:06:05.620 lovely sine of theta minus PI by 2 all 00:06:10.74000:06:10.750 right so that's a substitution I'm about 00:06:13.17000:06:13.180 to use and I'm just going to clean 00:06:14.42900:06:14.439 things up a little bit I'm going to say 00:06:16.01900:06:16.029 that this is 1 over Omega times C times 00:06:21.32900:06:21.339 the maximum current possible times the 00:06:25.64900:06:25.659 sine of well what do we have to put we 00:06:29.27900:06:29.289 have to put Omega t minus PI over 2 00:06:35.24900:06:35.259 this equation is a very powerful 00:06:37.11000:06:37.120 equation we call this constant here 00:06:40.82000:06:40.830 phase-shift and thats related to the 00:06:43.29000:06:43.300 term fazer of course and what we're 00:06:46.32000:06:46.330 seeing is that the fazer for voltage is 00:06:50.16000:06:50.170 not facing the same direction as the 00:06:53.10000:06:53.110 fazer for the current so first of all we 00:06:57.69000:06:57.700 can identify some maximum values when 00:07:00.24000:07:00.250 the sine function is at a maximum it 00:07:02.13000:07:02.140 just disappears it becomes 1 so then we 00:07:04.32000:07:04.330 say that V cap is 1 over Omega C times 00:07:08.57000:07:08.580 IMAX on the cap so that would be the 00:07:11.91000:07:11.920 maximum voltage so I can say V Max is 1 00:07:16.56000:07:16.570 over Omega C times IMAX I hope that you 00:07:20.55000:07:20.560 believed that particular observation and 00:07:23.49000:07:23.500 I want to make a substitution also 00:07:25.95000:07:25.960 because I like I mean it's one of my 00:07:28.47000:07:28.480 favorite electrical equations I like the 00:07:30.75000:07:30.760 idea of a resistor where V is I times R 00:07:34.65000:07:34.660 and for a resistor we also find that V 00:07:37.47000:07:37.480 Max is I max times the resistance of the 00:07:41.01000:07:41.020 resistor so what if capacitor is acted 00:07:43.50000:07:43.510 kind of like resistors and and they sort 00:07:46.14000:07:46.150 of do because they are sometimes able to 00:07:49.47000:07:49.480 act like a line in the circuit a short 00:07:52.59000:07:52.600 circuit sometimes they're able to act 00:07:54.42000:07:54.430 like a break in the circuit so that's 00:07:56.25000:07:56.260 sort of a resistance it's not exactly a 00:07:59.31000:07:59.320 resistance we call it a reactance so I 00:08:02.88000:08:02.890 want to say that this is V V is equal to 00:08:06.57000:08:06.580 I times something so I'm going to define 00:08:08.91000:08:08.920 1 over Omega C to be the reactance of a 00:08:13.59000:08:13.600 capacitor and it's abbreviated with the 00:08:15.30000:08:15.310 letter X this is the reactance of the 00:08:17.43000:08:17.440 capacitor and you'll find that it 00:08:18.90000:08:18.910 actually has the same units of 00:08:20.34000:08:20.350 resistance this is radians per second 00:08:22.38000:08:22.390 and that is the capacitance in farad's 00:08:24.45000:08:24.460 and you get units of ohms Wow go figure 00:08:28.20000:08:28.210 try that out it's really beautiful so we 00:08:30.27000:08:30.280 have this analogy of Ohm's law for well 00:08:34.77000:08:34.780 we can write it here the analogy of 00:08:36.51000:08:36.520 Ohm's law for capacitors is that voltage 00:08:39.48000:08:39.490 on a capacitor is and sorry I have to 00:08:44.49000:08:44.500 put Max's in here or rms is in here 00:08:46.74000:08:46.750 because it's not an instantaneous 00:08:49.02000:08:49.030 equation at all it's going to be I 00:08:51.99000:08:52.000 max times the reactive capacitance now 00:08:57.32900:08:57.339 my next plan is to show you how this 00:09:00.44900:09:00.459 works we've got we've got no I'm gonna 00:09:04.61900:09:04.629 have to use a different page because 00:09:05.67000:09:05.680 this is really important stuff um I'll 00:09:08.57900:09:08.589 set up my axes and I've got Y and X and 00:09:11.28000:09:11.290 this is where my phasor is going to live 00:09:12.90000:09:12.910 before I go on though I want to remind 00:09:15.30000:09:15.310 you this is what the phasor of a 00:09:16.94900:09:16.959 resistor looks like starts here and if 00:09:19.47000:09:19.480 the voltage is max then the current 00:09:21.26900:09:21.279 through the resistor will be max at that 00:09:22.55900:09:22.569 same time so right now I'm saying what 00:09:24.92900:09:24.939 does the y-component gives the value the 00:09:32.69900:09:32.709 Y component of the phasor at any instant 00:09:34.98000:09:34.990 in time gives the actual value of that 00:09:36.99000:09:37.000 variable so right now I've got no y 00:09:39.15000:09:39.160 component for anything 00:09:40.05000:09:40.060 and now I've got maximum remember this 00:09:42.84000:09:42.850 is this is everything here is rotating 00:09:46.05000:09:46.060 at Omega so that's the frequency of my 00:09:49.88900:09:49.899 power source and as I reach this point 00:09:53.12900:09:53.139 right here I get maximum value for 00:09:55.80000:09:55.810 current and maximum value for voltage 00:09:57.90000:09:57.910 and that's a resistor and then I get 00:10:00.32900:10:00.339 none and then I get negative current and 00:10:04.53000:10:04.540 negative voltage right because you're 00:10:07.23000:10:07.240 pushing the other way it's going the 00:10:08.46000:10:08.470 other way and then I get none again and 00:10:10.76900:10:10.779 so if I draw this graph for a resistor 00:10:15.94900:10:15.959 I'm going to find that the voltage might 00:10:19.23000:10:19.240 do this and the current one we want to 00:10:24.92900:10:24.939 use for current let's use orange for 00:10:26.28000:10:26.290 current the current would do this for 00:10:28.98000:10:28.990 instance where these guys have the same 00:10:34.17000:10:34.180 time dependence and they are we can call 00:10:36.32900:10:36.339 these in phase and they differ only by 00:10:41.81900:10:41.829 this this Ohm's law relationship this is 00:10:45.66000:10:45.670 going to be let's see this is V and this 00:10:49.49900:10:49.509 level is the Max and the orange is I and 00:10:55.63900:10:55.649 this value here is I max and we can 00:11:01.19900:11:01.209 define I max by Ohm's law these 00:11:04.11000:11:04.120 are so I is V over our IMAX is going to 00:11:08.34000:11:08.350 be V Max divided by the resistance of my 00:11:11.43000:11:11.440 resistor it's time to go to capacitors 00:11:14.43000:11:14.440 though so resistors are really really 00:11:16.35000:11:16.360 Pleasant 00:11:16.82900:11:16.839 everybody likes resistors because 00:11:18.18000:11:18.190 they're super simple it's time to go to 00:11:20.76000:11:20.770 capacitors and in a capacitor let me get 00:11:23.55000:11:23.560 you the same idea of a graph in a 00:11:26.40000:11:26.410 capacitor graph if the voltage starts 00:11:29.82000:11:29.830 out well let's see I'm actually going to 00:11:32.55000:11:32.560 say that the voltage on the capacitor is 00:11:34.56000:11:34.570 going to start out I mean it really 00:11:36.93000:11:36.940 doesn't matter where start will just 00:11:37.89000:11:37.900 start you let you go voltage big voltage 00:11:42.54000:11:42.550 small I'm zooming in a little bit I 00:11:44.61000:11:44.620 changed my angular frequency so that you 00:11:46.41000:11:46.420 can study this a little bit better so 00:11:47.97000:11:47.980 this is my voltage and this is all as a 00:11:51.78000:11:51.790 function of time sorry I should have 00:11:53.13000:11:53.140 labeled that up there now the 00:11:55.29000:11:55.300 interesting thing about a capacitor is 00:11:57.00000:11:57.010 that if you have a large voltage across 00:12:00.84000:12:00.850 the capacitor then that's when the 00:12:03.75000:12:03.760 current actually stops so let's go back 00:12:06.30000:12:06.310 to this diagram right here as I'm 00:12:08.13000:12:08.140 putting a really large voltage across 00:12:10.07900:12:10.089 the capacitor that means the capacitor 00:12:11.88000:12:11.890 is fully charged because Q is CV and 00:12:15.69000:12:15.700 when Q reaches a maximum that means the 00:12:17.69900:12:17.709 capacitor is going to be impeding the 00:12:19.68000:12:19.690 flow of charge you won't have any charge 00:12:22.82000:12:22.830 flowing at that time so the current is 00:12:26.28000:12:26.290 zero when the voltage reaches a maximum 00:12:29.82000:12:29.830 and we can also argue that the current 00:12:32.97000:12:32.980 is zero when the voltage reaches a 00:12:35.31000:12:35.320 minimum so we're going to have these 00:12:36.63000:12:36.640 zeros right here I want to also say that 00:12:40.92000:12:40.930 if the if the voltage is decreasing then 00:12:46.89000:12:46.900 you can imagine that will be what are we 00:12:49.59000:12:49.600 doing here this is going to be a graph 00:12:51.00000:12:51.010 of current ultimately and I'm going to 00:12:53.61000:12:53.620 say that the current will be going the 00:12:57.09000:12:57.100 opposite direction as the voltage 00:12:59.19000:12:59.200 decreases so there we've got voltage at 00:13:02.19000:13:02.200 a peak and the current now is going 00:13:04.53000:13:04.540 below zero and then it comes up and goes 00:13:08.94000:13:08.950 above zero and comes down and goes below 00:13:11.61000:13:11.620 zero and comes up and goes above zero so 00:13:15.54000:13:15.550 you've got this very interesting pattern 00:13:18.03000:13:18.040 notice in this case we can look at in 00:13:19.88900:13:19.899 terms of current the current is a 00:13:21.66000:13:21.670 minimum when the voltage across the 00:13:23.87900:13:23.889 capacitor is zero and the current is 00:13:26.87900:13:26.889 zero when the voltage across the 00:13:28.37900:13:28.389 capacitor is very negative 00:13:30.18000:13:30.190 if the voltage across the capacitor is 00:13:32.55000:13:32.560 very negative then you can bet that that 00:13:35.16000:13:35.170 will induce a current to start going the 00:13:37.19900:13:37.209 other direction so it's sloshing Chuchu 00:13:39.50900:13:39.519 Chuchu 00:13:40.13900:13:40.149 but the key fact is that they are out of 00:13:43.17000:13:43.180 phase we've already seen that they're 00:13:45.78000:13:45.790 out of phase I want to remind you that 00:13:47.63900:13:47.649 they're out of phase by this equation 00:13:49.17000:13:49.180 right here it says the voltage on a 00:13:50.75900:13:50.769 capacitor is the current max on the 00:13:54.12000:13:54.130 capacitor with this stuff right here 00:13:56.28000:13:56.290 this reactance but it's out of phase of 00:13:59.40000:13:59.410 the same Omega it's minus PI over 2 or 00:14:02.61000:14:02.620 90 degrees or it's a quarter revolution 00:14:05.40000:14:05.410 out of phase that's what it means to be 00:14:07.68000:14:07.690 a quarter revolution out of phase these 00:14:09.18000:14:09.190 guys are in phase and these guys are a 00:14:11.22000:14:11.230 quarter out of phase so I'll say PI by 00:14:13.88900:14:13.899 two out of phase hi Kira and if I take 00:14:21.15000:14:21.160 this capacitor phase or you see that 00:14:25.19900:14:25.209 initially if I start it out like this I 00:14:28.68000:14:28.690 didn't actually start it out like this 00:14:29.85000:14:29.860 but look at this instant right here this 00:14:32.04000:14:32.050 is the instant right here 00:14:33.44900:14:33.459 where the voltage that's my purple is a 00:14:35.81900:14:35.829 oh no sorry this is maximum current oh 00:14:38.81900:14:38.829 that is right there yeah good so I get 00:14:40.65000:14:40.660 maximum current initially before the 00:14:43.13900:14:43.149 capacitor is charged right and there's 00:14:45.90000:14:45.910 no voltage on the capacitor as I start 00:14:47.55000:14:47.560 so I start charging the capacitor 00:14:49.47000:14:49.480 there's a huge current and the voltage 00:14:52.18900:14:52.199 rises up as I'm going to here the 00:14:55.11000:14:55.120 voltage is rising up because I'm 00:14:56.91000:14:56.920 charging my capacitor boom now the 00:15:00.36000:15:00.370 capacitor is fully charged so the 00:15:02.06900:15:02.079 voltage is huge but the current has 00:15:04.11000:15:04.120 dropped to zero because the capacitor is 00:15:05.87900:15:05.889 fully charged and the power supply 00:15:07.65000:15:07.660 starts and no let's go the other 00:15:10.23000:15:10.240 direction so the current actually starts 00:15:12.24000:15:12.250 unloading going away from the capacitor 00:15:14.79000:15:14.800 and back towards the power supply and 00:15:17.04000:15:17.050 that's here right now and now we have no 00:15:20.46000:15:20.470 voltage C the Y component of this phasor 00:15:23.81900:15:23.829 right here is at zero but we have a very 00:15:26.97000:15:26.980 negative current and if we continue 00:15:28.92000:15:28.930 going counterclockwise 00:15:30.99000:15:31.000 right here for instance we've got no 00:15:33.39000:15:33.400 current but we've got a very very 00:15:35.73000:15:35.740 negative voltage and as I continue 00:15:38.19000:15:38.200 spinning this around you see how these 00:15:39.72000:15:39.730 two things interact which one is leading 00:15:41.85000:15:41.860 would you say that the current is 00:15:44.52000:15:44.530 leading the voltage or that the voltage 00:15:47.82000:15:47.830 is leading the current now take that 00:15:50.91000:15:50.920 answer that you got right there the 00:15:52.74000:15:52.750 voltage leading the current and the 00:15:53.88000:15:53.890 current leading the voltage and see what 00:15:56.25000:15:56.260 you think if you look at this which one 00:15:57.93000:15:57.940 of these does it look like is leading 00:15:59.78000:15:59.790 does it look like the voltage is leading 00:16:01.89000:16:01.900 here or the current is leading I mean I 00:16:03.87000:16:03.880 guess if it's erased that direction then 00:16:05.64000:16:05.650 it looks like the voltage is leading but 00:16:07.14000:16:07.150 it's not what we're doing is we're 00:16:08.85000:16:08.860 scrolling across this so we're saying at 00:16:11.37000:16:11.380 this instant right here we've got that 00:16:13.05000:16:13.060 and then oh man see that's going down 00:16:15.21000:16:15.220 and now that's going up which one looks 00:16:18.00000:16:18.010 like it's leading now it looks to me 00:16:20.46000:16:20.470 like the orange is leading because the 00:16:22.62000:16:22.630 orange leaps up and then the purple is 00:16:24.21000:16:24.220 like okay we can go up and then the 00:16:26.01000:16:26.020 orange shoots down the Purple's like 00:16:27.45000:16:27.460 okay we can go down and then the orange 00:16:29.22000:16:29.230 shoots up and purple says okay we can go 00:16:31.32000:16:31.330 up all right so that's why it's 00:16:33.30000:16:33.310 consistent to say that the voltage lags 00:16:36.21000:16:36.220 the current for a capacitor and that 00:16:38.61000:16:38.620 will be our final statement here voltage 00:16:42.82900:16:42.839 lags current by a phase shift of PI over 00:16:50.25000:16:50.260 2 they are 90 degrees out of phase I'm 00:16:54.21000:16:54.220 gonna go get some lunch I think you 00:16:55.74000:16:55.750 should have some
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