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Capacitor i-v equations

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00:00:01.070
we're going to talk about the equations
00:00:03.500 00:00:03.510 that describe how a capacitor works and
00:00:05.630 00:00:05.640 then I'll give you an example of how
00:00:08.540 00:00:08.550 these equations work so the basic
00:00:11.509 00:00:11.519 equation of a capacitor says the charge
00:00:14.390 00:00:14.400 Q on a capacitor is equal to the
00:00:17.870 00:00:17.880 capacitance value times the voltage
00:00:20.420 00:00:20.430 across the capacitor okay so here's our
00:00:23.450 00:00:23.460 capacitor over here let's say we have a
00:00:25.340 00:00:25.350 voltage on it plus or minus V and it has
00:00:30.890 00:00:30.900 a we say it has a capacitance value of C
00:00:33.979 00:00:33.989 and that's a property of this device
00:00:37.310 00:00:37.320 here and C is equal to just looking at
00:00:40.310 00:00:40.320 the equation over there C is equal to
00:00:42.530 00:00:42.540 the ratio of the charge stored in the
00:00:44.569 00:00:44.579 capacitor divided by the voltage on the
00:00:46.970 00:00:46.980 capacitor so what we mean by stored
00:00:49.729 00:00:49.739 charge is if a current flows into this
00:00:53.600 00:00:53.610 capacitor it can leave some excess
00:00:55.340 00:00:55.350 charge on the top I'll just mark that
00:00:58.010 00:00:58.020 with plus signs and there'll be a
00:00:59.810 00:00:59.820 corresponding set of minus charges on
00:01:03.529 00:01:03.539 the other other plate of the capacitor
00:01:07.030 00:01:07.040 this collection of excess charge will
00:01:10.160 00:01:10.170 will be Q plus and this down here will
00:01:13.609 00:01:13.619 be Q minus and they're going to be the
00:01:15.530 00:01:15.540 same value and what we say here is when
00:01:18.649 00:01:18.659 the capacitors in this state we say it's
00:01:20.330 00:01:20.340 storing this much charge and we'll just
00:01:23.390 00:01:23.400 name one of these numbers here and
00:01:24.950 00:01:24.960 they're going to be the same with
00:01:26.270 00:01:26.280 opposite signs so that's what it means
00:01:28.460 00:01:28.470 for a capacitor to store charge so what
00:01:33.109 00:01:33.119 I want to do now is develop some sort of
00:01:35.090 00:01:35.100 expression that relates the current
00:01:37.550 00:01:37.560 through a capacitor to the voltage so we
00:01:40.249 00:01:40.259 wanted to develop we want to develop an
00:01:42.490 00:01:42.500 IV characteristic so this will
00:01:45.319 00:01:45.329 correspond to sort of like Ohm's law for
00:01:47.120 00:01:47.130 a capacitor what relates the current to
00:01:50.179 00:01:50.189 the voltage so the way I'm going to do
00:01:52.670 00:01:52.680 that is to exercise this equation by
00:01:55.130 00:01:55.140 causing some changes and in particular
00:01:57.649 00:01:57.659 will change the voltage on this
00:02:00.380 00:02:00.390 capacitor and we'll see what happens
00:02:02.690 00:02:02.700 over here so when we say we're going to
00:02:05.149 00:02:05.159 change a voltage that means we're going
00:02:06.620 00:02:06.630 to create something a condition of of DV
00:02:09.619 00:02:09.629 DT
00:02:11.990 00:02:12.000 change in voltage / change in time and
00:02:15.010 00:02:15.020 so I can do that by taking the
00:02:17.420 00:02:17.430 derivative of both sides of this
00:02:19.730 00:02:19.740 equation here I've already done it for
00:02:22.070 00:02:22.080 this side and over here what I'll have
00:02:24.620 00:02:24.630 is DQ DT and I took the derivative of
00:02:30.200 00:02:30.210 both sides just to be sure I treated
00:02:32.360 00:02:32.370 both sides of the equation the same so
00:02:35.210 00:02:35.220 let's look at this little expression
00:02:36.560 00:02:36.570 right here this is kind of interesting
00:02:38.300 00:02:38.310 this is change of charge with change of
00:02:42.410 00:02:42.420 time and that's equal to that's what we
00:02:45.380 00:02:45.390 mean by current that is current and the
00:02:51.020 00:02:51.030 symbol for current is I so DQ DT is
00:02:56.870 00:02:56.880 current essentially by definition we
00:03:00.020 00:03:00.030 give it the symbol I and that's going to
00:03:02.090 00:03:02.100 be equal to C DV DT and this is an
00:03:12.740 00:03:12.750 important equation that's basically the
00:03:16.610 00:03:16.620 the IV relationship between current and
00:03:20.570 00:03:20.580 voltage in a capacitor and what it tells
00:03:23.900 00:03:23.910 us that the current is actually
00:03:25.940 00:03:25.950 proportional to and the proportionality
00:03:28.040 00:03:28.050 constant is C the current is
00:03:30.710 00:03:30.720 proportional to the rate of change of
00:03:32.330 00:03:32.340 voltage not to the voltage itself but to
00:03:34.310 00:03:34.320 the rate of change of voltage all right
00:03:38.240 00:03:38.250 now what I want to do is find an
00:03:40.310 00:03:40.320 expression that expresses V in terms of
00:03:43.370 00:03:43.380 I so here here we have I in terms of DV
00:03:46.430 00:03:46.440 DT let's figure out if we can express V
00:03:48.890 00:03:48.900 in terms of some expression containing I
00:03:51.789 00:03:51.799 the way I do that is I need to eliminate
00:03:54.080 00:03:54.090 this derivative here and what I'm going
00:03:56.780 00:03:56.790 to do that by taking the integral of
00:03:59.090 00:03:59.100 this side of the equation and at the
00:04:00.949 00:04:00.959 same time I'll take the integral of the
00:04:02.509 00:04:02.519 other side of the equation to keep
00:04:03.890 00:04:03.900 everything equal so what that looks like
00:04:06.470 00:04:06.480 is the integral of AI
00:04:12.140 00:04:12.150 with respect to time is equal to the
00:04:17.759 00:04:17.769 integral of c dv/dt with respect to time
00:04:25.580 00:04:25.590 dt and on this side i have basically a I
00:04:30.930 00:04:30.940 do something like this and I have the
00:04:33.030 00:04:33.040 integral of DV so this is this looks
00:04:36.240 00:04:36.250 like an antiderivative this is an
00:04:38.040 00:04:38.050 integral acting like an antiderivative
00:04:39.600 00:04:39.610 and what is it what function has a
00:04:43.830 00:04:43.840 derivative of DV and that would be just
00:04:46.680 00:04:46.690 plain V so I can rewrite this side of
00:04:49.500 00:04:49.510 the equation constant C comes out of the
00:04:51.390 00:04:51.400 expression and we end up with V on this
00:04:54.480 00:04:54.490 side just plain V and that equals the
00:04:56.909 00:04:56.919 integral of I DT so we're partway
00:05:06.600 00:05:06.610 through we're developing and what's
00:05:08.190 00:05:08.200 going to be called an integral form of
00:05:09.600 00:05:09.610 the capacitor IV equation what I need to
00:05:13.740 00:05:13.750 look at next is what are the bounds on
00:05:15.900 00:05:15.910 this integral so the bounds on this
00:05:18.120 00:05:18.130 integral are basically minus time equals
00:05:21.360 00:05:21.370 minus infinity to time equals some time
00:05:24.990 00:05:25.000 T which is sort of like the time now and
00:05:27.380 00:05:27.390 that equals capacitance times voltage
00:05:30.420 00:05:30.430 and let me take the C over on the other
00:05:32.880 00:05:32.890 side and actually I'm going to move V
00:05:35.310 00:05:35.320 over here on to the left and then I can
00:05:39.600 00:05:39.610 write this one over C this is the normal
00:05:43.140 00:05:43.150 looking version of this equation I DT
00:05:47.400 00:05:47.410 and minus infinity to time T T time Big
00:05:53.520 00:05:53.530 T is time right now and what this says
00:05:57.210 00:05:57.220 it says that the voltage on a capacitor
00:05:58.760 00:05:58.770 has something to do with the summation
00:06:01.469 00:06:01.479 or the integral of the current over its
00:06:04.020 00:06:04.030 entire life all the way back to T equals
00:06:06.600 00:06:06.610 minus infinity and this is not so
00:06:09.960 00:06:09.970 convenient what we're going to do
00:06:11.610 00:06:11.620 instead is we're going to pick a time
00:06:13.020 00:06:13.030 let we'll call it
00:06:14.990 00:06:15.000 we'll pick a time called T equals zero
00:06:17.360 00:06:17.370 and we'll say that the voltage on the
00:06:19.460 00:06:19.470 capacitor was equal to let's say V
00:06:23.210 00:06:23.220 naught with some value and that what
00:06:27.050 00:06:27.060 we'll do is we're going to change the
00:06:28.730 00:06:28.740 limit on our integral here from minus
00:06:31.280 00:06:31.290 infinity to time T equals zero and then
00:06:34.220 00:06:34.230 then we'll use the integral from instead
00:06:36.800 00:06:36.810 zero to the time we're interested in so
00:06:40.100 00:06:40.110 that equation looks like this we're just
00:06:41.540 00:06:41.550 going to change the limits on the on the
00:06:45.740 00:06:45.750 integral
00:06:54.300 00:06:54.310 and we have the integral now but we have
00:06:56.439 00:06:56.449 to actually account for all the time
00:06:58.180 00:06:58.190 before T equals zero and what we do
00:07:01.180 00:07:01.190 there is we just basically add V not
00:07:04.510 00:07:04.520 whatever V not is that's the starting
00:07:06.610 00:07:06.620 point at time equals zero and then the
00:07:08.920 00:07:08.930 integral takes us from time zero until
00:07:11.499 00:07:11.509 time now
00:07:14.490 00:07:14.500 this is the integral form of the
00:07:16.480 00:07:16.490 capacitor equation and I want to
00:07:18.220 00:07:18.230 actually make one more little change
00:07:25.380 00:07:25.390 this is the current at V of as a
00:07:28.660 00:07:28.670 function of big T what we would really
00:07:31.900 00:07:31.910 want to write here is we want to write V
00:07:34.120 00:07:34.130 of a little T just this is just
00:07:36.070 00:07:36.080 stylistically this is what we like this
00:07:38.230 00:07:38.240 equation to look like and so I want my
00:07:41.380 00:07:41.390 the limits on my integral to be 0 to
00:07:44.680 00:07:44.690 little T and now I need to sort of make
00:07:46.630 00:07:46.640 up a new replacement for this T that's
00:07:50.530 00:07:50.540 inside here I can call it something else
00:07:52.090 00:07:52.100 I can call it I of will call it I'll
00:07:56.800 00:07:56.810 call it tau this is basically just a
00:07:59.860 00:07:59.870 little fake variable d tau plus V naught
00:08:06.130 00:08:06.140 and this is now we finally have it this
00:08:09.580 00:08:09.590 is the integral form of the capacitor
00:08:13.030 00:08:13.040 equation
00:08:14.650 00:08:14.660 we have the other form of the equation
00:08:16.720 00:08:16.730 that goes with this which was I equals
00:08:19.660 00:08:19.670 big c dv/dt so there's the two forms of
00:08:30.580 00:08:30.590 the capacitor equation now I want to do
00:08:34.150 00:08:34.160 an example with this one here
00:08:36.190 00:08:36.200 just to see how how this it works when
00:08:38.200 00:08:38.210 we have a capacitor circuit

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