[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.46,0:00:05.65,Default,,0000,0000,0000,,Okay, this interesting question came up\Nin class and so let's take a look at it.
Dialogue: 0,0:00:05.65,0:00:10.53,Default,,0000,0000,0000,,We want to be able to find all\Nof the currents in the circuit,
Dialogue: 0,0:00:10.53,0:00:11.52,Default,,0000,0000,0000,,I1, 2, 3, 4, 5, and 6.
Dialogue: 0,0:00:11.52,0:00:16.48,Default,,0000,0000,0000,,So our total number of unknowns, N = 6.
Dialogue: 0,0:00:16.48,0:00:20.26,Default,,0000,0000,0000,,The first thing I'm going to do is\Ndefine the polarity of all of these,
Dialogue: 0,0:00:20.26,0:00:22.05,Default,,0000,0000,0000,,plus on the tail minus on the tip.
Dialogue: 0,0:00:22.05,0:00:28.67,Default,,0000,0000,0000,,Plus to minus, plus to minus, and plus\Nto minus, plus to minus, plus to minus.
Dialogue: 0,0:00:29.69,0:00:32.71,Default,,0000,0000,0000,,Then the next thing I'll do\Nis color all of my nodes so
Dialogue: 0,0:00:32.71,0:00:34.98,Default,,0000,0000,0000,,that we can kinda more\Nclearly see the circuit.
Dialogue: 0,0:00:34.98,0:00:39.68,Default,,0000,0000,0000,,There's one node right there,\Nhere's another node here.
Dialogue: 0,0:00:42.100,0:00:46.68,Default,,0000,0000,0000,,Another node right there.
Dialogue: 0,0:00:46.68,0:00:51.65,Default,,0000,0000,0000,,And there are two ordinary nodes here\Nwe're not going to end up using them.
Dialogue: 0,0:00:51.65,0:00:55.21,Default,,0000,0000,0000,,And we could use this node right here,\Nexcept the other nodes are already going
Dialogue: 0,0:00:55.21,0:00:57.30,Default,,0000,0000,0000,,to have taken care of it by\Nthe time we get to that point.
Dialogue: 0,0:00:57.30,0:01:00.34,Default,,0000,0000,0000,,So let's start by drawing\Nour loop equations,
Dialogue: 0,0:01:00.34,0:01:02.93,Default,,0000,0000,0000,,Kirchhoff's loops, loop equations.
Dialogue: 0,0:01:02.93,0:01:06.06,Default,,0000,0000,0000,,This is loop number one,\Nlet's start right here.
Dialogue: 0,0:01:06.06,0:01:10.94,Default,,0000,0000,0000,,We're going to the minus part of I2,
Dialogue: 0,0:01:10.94,0:01:15.68,Default,,0000,0000,0000,,so it's -I2(9 ohms) + I4(6
Dialogue: 0,0:01:15.68,0:01:22.10,Default,,0000,0000,0000,,ohms) + I6(6 ohms) + 6 volts = 0.
Dialogue: 0,0:01:22.10,0:01:24.08,Default,,0000,0000,0000,,Now let's do this loop,\Nlet's call that loop number 2.
Dialogue: 0,0:01:25.57,0:01:30.13,Default,,0000,0000,0000,,As we're moving through here,
Dialogue: 0,0:01:30.13,0:01:35.20,Default,,0000,0000,0000,,we can see we're going to -I6(6
Dialogue: 0,0:01:35.20,0:01:43.86,Default,,0000,0000,0000,,ohms) + I5(6 ohms)- I1(9\Nohms) + 6 volts = 0.
Dialogue: 0,0:01:43.86,0:01:47.84,Default,,0000,0000,0000,,Now we need to do one more loop,\Nwe can either do a loop all the way around
Dialogue: 0,0:01:47.84,0:01:50.69,Default,,0000,0000,0000,,the outside, or\Nwe can do a small loop right here.
Dialogue: 0,0:01:50.69,0:01:55.35,Default,,0000,0000,0000,,Let me just do that small loop just for\Nfun, so
Dialogue: 0,0:01:55.35,0:02:00.00,Default,,0000,0000,0000,,we'll start there, this is loop number 3,
Dialogue: 0,0:02:00.00,0:02:08.18,Default,,0000,0000,0000,,-I4(6 ohms)- I3(9 ohms),\Nand -I5(6 ohms) back to 0.
Dialogue: 0,0:02:08.18,0:02:11.15,Default,,0000,0000,0000,,Okay that's three loop equations and\Nthey have covered
Dialogue: 0,0:02:11.15,0:02:15.06,Default,,0000,0000,0000,,all of the elements in my circuit, every\Nsingle element has been touched by a loop.
Dialogue: 0,0:02:15.06,0:02:18.26,Default,,0000,0000,0000,,So if I do any more loop equations,\NI will have redundant loops.
Dialogue: 0,0:02:18.26,0:02:20.28,Default,,0000,0000,0000,,Now, it's time for\Nme to do node equations.
Dialogue: 0,0:02:20.28,0:02:23.46,Default,,0000,0000,0000,,So let's do this node equation right here,
Dialogue: 0,0:02:23.46,0:02:27.13,Default,,0000,0000,0000,,that would be equation number 4,\Nthat is the V2 node.
Dialogue: 0,0:02:27.13,0:02:30.07,Default,,0000,0000,0000,,What we'll do is say all\Nthe currents coming in,
Dialogue: 0,0:02:30.07,0:02:33.39,Default,,0000,0000,0000,,I4 is equal to all the currents going out,\NI5 + I6.
Dialogue: 0,0:02:34.44,0:02:39.10,Default,,0000,0000,0000,,Now let's do the V3 node that\Nwould be equation number 5,
Dialogue: 0,0:02:39.10,0:02:45.46,Default,,0000,0000,0000,,what's coming in there, looks like I3 is\Ncoming in and going out is I2 and I4.
Dialogue: 0,0:02:45.46,0:02:49.54,Default,,0000,0000,0000,,Our last equation is going to\Nbe this green node, the V4 node.
Dialogue: 0,0:02:49.54,0:02:56.98,Default,,0000,0000,0000,,And what's coming in, I5 and\NI1 are coming in and I3 is going out.
Dialogue: 0,0:02:56.98,0:03:01.22,Default,,0000,0000,0000,,So those were our Kirchhoff's equations,\Nlet's convert them now to matrix form.
Dialogue: 0,0:03:01.22,0:03:07.32,Default,,0000,0000,0000,,I'm going to have a 6 by 6 matrix\Nbecause I'm going to have unknowns,
Dialogue: 0,0:03:07.32,0:03:10.69,Default,,0000,0000,0000,,I1, I2, I3, I4, I5, and I6.
Dialogue: 0,0:03:10.69,0:03:15.88,Default,,0000,0000,0000,,I'm just writing them for convenience\Nacross the top, and equations 1,
Dialogue: 0,0:03:15.88,0:03:21.63,Default,,0000,0000,0000,,2, 3, 4, 5 and 6, that's just written for\Nconvenience as well.
Dialogue: 0,0:03:21.63,0:03:27.99,Default,,0000,0000,0000,,And then I write my unknowns here\Nin this vector, 3, 4, 5 and 6 and
Dialogue: 0,0:03:30.59,0:03:33.73,Default,,0000,0000,0000,,this will be my vector of constants.
Dialogue: 0,0:03:33.73,0:03:39.55,Default,,0000,0000,0000,,Okay, so let's look at our first equation,\NI1 times 0, I2 times -9,
Dialogue: 0,0:03:39.55,0:03:44.30,Default,,0000,0000,0000,,I3 times 0, I4 times 6,\NI5 times 0, I6 times 6 ohms.
Dialogue: 0,0:03:44.30,0:03:51.59,Default,,0000,0000,0000,,And then I need to bring this 6 over\Nhere in order to be in the constants.
Dialogue: 0,0:03:51.59,0:03:56.09,Default,,0000,0000,0000,,My second equation has 9 ohms- 9 ohms,
Dialogue: 0,0:03:56.09,0:04:00.85,Default,,0000,0000,0000,,times I1, nothing times I2 or I3 or I4.
Dialogue: 0,0:04:00.85,0:04:05.60,Default,,0000,0000,0000,,It has 6 ohms times I5, and\Nit has -6 ohms times I6, and
Dialogue: 0,0:04:05.60,0:04:09.71,Default,,0000,0000,0000,,this 6 volt also has to come\Nover to the other side.
Dialogue: 0,0:04:09.71,0:04:14.65,Default,,0000,0000,0000,,Right here we have 0, 0, I3 has -9,
Dialogue: 0,0:04:14.65,0:04:19.86,Default,,0000,0000,0000,,I4 has -6, I5 has -6, and I6 has 0,
Dialogue: 0,0:04:19.86,0:04:24.68,Default,,0000,0000,0000,,and there are no constants on this side.
Dialogue: 0,0:04:24.68,0:04:29.75,Default,,0000,0000,0000,,Now let us do equation number 4,\Nnothing times I1, I2, or I3.
Dialogue: 0,0:04:29.75,0:04:32.28,Default,,0000,0000,0000,,We have to get these on the same\Nside of the equation, so
Dialogue: 0,0:04:32.28,0:04:35.82,Default,,0000,0000,0000,,I'm gonna bring this over here, so\Nthat whole equation will be equal to 0.
Dialogue: 0,0:04:35.82,0:04:40.68,Default,,0000,0000,0000,,So what I will be writing is\N0 is equal to -I4 + I5 + I6,
Dialogue: 0,0:04:40.68,0:04:45.45,Default,,0000,0000,0000,,so there's a -1 here,\Na 1 there and a 1 there.
Dialogue: 0,0:04:45.45,0:04:53.02,Default,,0000,0000,0000,,Similarly, I'm going to bring this\Nover here, 0, 1, -1, 1, 0, 0.
Dialogue: 0,0:04:53.02,0:04:57.52,Default,,0000,0000,0000,,And in this last equation,\Nlet's bring that over here, so
Dialogue: 0,0:04:57.52,0:05:02.57,Default,,0000,0000,0000,,we'll have 1 times I1, 0,\N-1, 0, 1 and 0, equal to 0.
Dialogue: 0,0:05:02.57,0:05:07.97,Default,,0000,0000,0000,,So there's my matrix equation, just\Na minute, I'll put that into MATLAB and
Dialogue: 0,0:05:07.97,0:05:12.98,Default,,0000,0000,0000,,give you a final answer but there's\Nour solution to our matrix problem.