[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.19,0:00:04.73,Default,,0000,0000,0000,,I'd like to go through this\Nproblem that we started in class.
Dialogue: 0,0:00:04.73,0:00:08.39,Default,,0000,0000,0000,,This is an interesting Kirchhoff’s law\Nproblem because it has a supernode
Dialogue: 0,0:00:08.39,0:00:09.66,Default,,0000,0000,0000,,here in the middle.
Dialogue: 0,0:00:09.66,0:00:13.14,Default,,0000,0000,0000,,A supernode is any time that you have\Na current source or a voltage source,
Dialogue: 0,0:00:13.14,0:00:16.85,Default,,0000,0000,0000,,whether it's dependent or independent,\Nthat's in between two nodes.
Dialogue: 0,0:00:16.85,0:00:21.10,Default,,0000,0000,0000,,And it doesn't have anything else,\Nno resistors in between these two nodes.
Dialogue: 0,0:00:21.10,0:00:22.80,Default,,0000,0000,0000,,We have to handle that in a special way.
Dialogue: 0,0:00:23.84,0:00:26.61,Default,,0000,0000,0000,,Let's say that we have been given our\Ntwo sources, our voltage source and
Dialogue: 0,0:00:26.61,0:00:30.47,Default,,0000,0000,0000,,our current source, and all of\Nthe resistors that are in this circuit.
Dialogue: 0,0:00:30.47,0:00:33.79,Default,,0000,0000,0000,,Then we can use this analysis technique\Nwhere we know that we have our sources,
Dialogue: 0,0:00:33.79,0:00:37.71,Default,,0000,0000,0000,,in fact, we also know our current source,\Nand our resistors.
Dialogue: 0,0:00:37.71,0:00:43.92,Default,,0000,0000,0000,,And we can use Kirchhoff’s law in order to\Ndetermine the currents, so let's begin.
Dialogue: 0,0:00:43.92,0:00:49.18,Default,,0000,0000,0000,,The first thing that\NI'm going to do is this
Dialogue: 0,0:00:49.18,0:00:55.16,Default,,0000,0000,0000,,loop right here, -Vs + I1R1 + I2R2 = 0.
Dialogue: 0,0:00:55.16,0:00:58.85,Default,,0000,0000,0000,,Now let's do this loop, in this case,
Dialogue: 0,0:00:58.85,0:01:04.21,Default,,0000,0000,0000,,I'm going to put pluses and\Nminuses on my currents.
Dialogue: 0,0:01:04.21,0:01:10.98,Default,,0000,0000,0000,,This would be -I2R2- Vx,
Dialogue: 0,0:01:10.98,0:01:18.03,Default,,0000,0000,0000,,which is 3I2 + I3R3 = 0.
Dialogue: 0,0:01:18.03,0:01:21.18,Default,,0000,0000,0000,,I can't do any more loops\Nbecause if I did this loop,
Dialogue: 0,0:01:21.18,0:01:26.21,Default,,0000,0000,0000,,it would have to go through a current\Nsource, and that's not allowed.
Dialogue: 0,0:01:26.21,0:01:29.23,Default,,0000,0000,0000,,So the next thing I'm\Ngoing to do is a node.
Dialogue: 0,0:01:29.23,0:01:31.45,Default,,0000,0000,0000,,I'll show you the easy node, and
Dialogue: 0,0:01:31.45,0:01:35.90,Default,,0000,0000,0000,,then we're going to actually do\Na supernode, so here is one node.
Dialogue: 0,0:01:37.86,0:01:41.10,Default,,0000,0000,0000,,And that's actually the easy\Nnode to do in this case.
Dialogue: 0,0:01:41.10,0:01:45.26,Default,,0000,0000,0000,,If I did the pink node,\NI would have all of the currents coming in
Dialogue: 0,0:01:45.26,0:01:49.70,Default,,0000,0000,0000,,equal all of the currents going out,\Nso what's the current here?
Dialogue: 0,0:01:49.70,0:01:53.26,Default,,0000,0000,0000,,The current and the branch is\Nalways the same, so this is I1.
Dialogue: 0,0:01:53.26,0:01:56.28,Default,,0000,0000,0000,,I1 is the same current\Nin that whole branch.
Dialogue: 0,0:01:56.28,0:02:02.15,Default,,0000,0000,0000,,So the currents that\Nare coming in are I2 + I3.
Dialogue: 0,0:02:02.15,0:02:06.86,Default,,0000,0000,0000,,And the currents that\Nare going out are I1 + Is,
Dialogue: 0,0:02:06.86,0:02:09.85,Default,,0000,0000,0000,,that's if I did the pink node.
Dialogue: 0,0:02:11.94,0:02:17.23,Default,,0000,0000,0000,,As an alternative, I could have\Ndone this supernode right here.
Dialogue: 0,0:02:17.23,0:02:21.00,Default,,0000,0000,0000,,Now, you'll notice that I\Nactually have two nodes.
Dialogue: 0,0:02:21.00,0:02:25.84,Default,,0000,0000,0000,,But in between them is just a voltage\Nsource that has nothing else,
Dialogue: 0,0:02:25.84,0:02:27.40,Default,,0000,0000,0000,,no other resistors.
Dialogue: 0,0:02:27.40,0:02:30.98,Default,,0000,0000,0000,,So if I wanted to be able to do\Nthe orange node, for instance,
Dialogue: 0,0:02:30.98,0:02:35.84,Default,,0000,0000,0000,,I would have the current coming in,\NI1, and the current going to out here.
Dialogue: 0,0:02:35.84,0:02:37.44,Default,,0000,0000,0000,,But what current do I have there?
Dialogue: 0,0:02:37.44,0:02:38.79,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:02:38.79,0:02:41.06,Default,,0000,0000,0000,,If I tried to do this node,\NI'd have the current going out,
Dialogue: 0,0:02:41.06,0:02:43.27,Default,,0000,0000,0000,,the current coming in,\Nand what current is there?
Dialogue: 0,0:02:43.27,0:02:47.38,Default,,0000,0000,0000,,I don't know, so there are two\Nways of handling the supernode.
Dialogue: 0,0:02:47.38,0:02:52.29,Default,,0000,0000,0000,,The easy way, I think, is to define\Na current, let's just call it Ix.
Dialogue: 0,0:02:52.29,0:02:54.86,Default,,0000,0000,0000,,That's the same current\Neveryplace in that branch.
Dialogue: 0,0:02:54.86,0:02:58.54,Default,,0000,0000,0000,,And then I can actually do\Nboth this node and this node.
Dialogue: 0,0:02:58.54,0:03:01.09,Default,,0000,0000,0000,,Because I've defined this new unknown,
Dialogue: 0,0:03:01.09,0:03:06.37,Default,,0000,0000,0000,,I now need to find also Ix, so\NI'm going to need an extra equation.
Dialogue: 0,0:03:06.37,0:03:10.45,Default,,0000,0000,0000,,So as an alternative to this,\NI could do the orange node,
Dialogue: 0,0:03:10.45,0:03:13.98,Default,,0000,0000,0000,,which would be currents coming in,\NI1 = Ix + I2.
Dialogue: 0,0:03:13.98,0:03:20.46,Default,,0000,0000,0000,,And the other one would\Nbe what's coming in,
Dialogue: 0,0:03:20.46,0:03:24.39,Default,,0000,0000,0000,,Ix + Is = I3, going out.
Dialogue: 0,0:03:24.39,0:03:29.01,Default,,0000,0000,0000,,So this is the orange node,\Nand this is the yellow node.
Dialogue: 0,0:03:30.45,0:03:33.31,Default,,0000,0000,0000,,So I have two different ways\NI could solve this problem.
Dialogue: 0,0:03:33.31,0:03:38.90,Default,,0000,0000,0000,,I can either use equations 1 and 2 for\Nmy two loops, plus my pink node.
Dialogue: 0,0:03:38.90,0:03:43.25,Default,,0000,0000,0000,,Or I could use 1 and\N2 plus my orange and yellow.
Dialogue: 0,0:03:43.25,0:03:45.29,Default,,0000,0000,0000,,These, I would turn\Ninto a matrix equation.
Dialogue: 0,0:03:45.29,0:03:47.06,Default,,0000,0000,0000,,Let's do the one with\Nthe pink on this side.
Dialogue: 0,0:03:57.66,0:03:59.39,Default,,0000,0000,0000,,I would be solving for I1, I2, and\NI3, my variables would be I1, I2, I3.
Dialogue: 0,0:03:59.39,0:04:02.88,Default,,0000,0000,0000,,I would be solving for\Nthese equations, like so.
Dialogue: 0,0:04:02.88,0:04:06.70,Default,,0000,0000,0000,,And then there would be\Na constant on this side.
Dialogue: 0,0:04:06.70,0:04:11.26,Default,,0000,0000,0000,,So it would be Vs coming\Nover to this side.
Dialogue: 0,0:04:11.26,0:04:15.60,Default,,0000,0000,0000,,I1 times R1, I2 times R2, and 0 times I3.
Dialogue: 0,0:04:15.60,0:04:22.64,Default,,0000,0000,0000,,Here we would have 0 times I1,\N-R2, aha, there's a 3R2 there.
Dialogue: 0,0:04:22.64,0:04:25.98,Default,,0000,0000,0000,,So combine those two, so it's actually,
Dialogue: 0,0:04:27.100,0:04:34.22,Default,,0000,0000,0000,,The I2 is going to have a -R2 and\Na -3 right there,
Dialogue: 0,0:04:34.22,0:04:38.69,Default,,0000,0000,0000,,and then here is R3, any constants?
Dialogue: 0,0:04:38.69,0:04:41.22,Default,,0000,0000,0000,,No, no constants over on that side.
Dialogue: 0,0:04:41.22,0:04:44.12,Default,,0000,0000,0000,,Right here I would have,\Nlet's get these all on the same side.
Dialogue: 0,0:04:44.12,0:04:46.46,Default,,0000,0000,0000,,I'm gonna move these\Ntwo over to this side.
Dialogue: 0,0:04:49.50,0:04:53.33,Default,,0000,0000,0000,,This was Is, not I3, I had a little\Ntrouble keeping track of that one.
Dialogue: 0,0:04:53.33,0:04:57.26,Default,,0000,0000,0000,,So here's I1, here's -I2, and
Dialogue: 0,0:04:57.26,0:05:02.49,Default,,0000,0000,0000,,here is -I3, and\Nthen over here we have Is.
Dialogue: 0,0:05:02.49,0:05:06.84,Default,,0000,0000,0000,,So these are the equations that we would\Nsolve in order to find our solution.
Dialogue: 0,0:05:08.56,0:05:11.12,Default,,0000,0000,0000,,Okay, let's go over to this side.
Dialogue: 0,0:05:11.12,0:05:16.18,Default,,0000,0000,0000,,If I wanted to do this set instead,\NI would have actually had four unknowns,
Dialogue: 0,0:05:16.18,0:05:20.90,Default,,0000,0000,0000,,I1, I2, I3, and Ix.
Dialogue: 0,0:05:20.90,0:05:24.72,Default,,0000,0000,0000,,I1, I2, I3, and Ix, and
Dialogue: 0,0:05:24.72,0:05:30.97,Default,,0000,0000,0000,,then over here's my vector of constants.
Dialogue: 0,0:05:30.97,0:05:37.37,Default,,0000,0000,0000,,My equation number 1 would be just\Nthe same, R1, R2, 0, 0, and Vs.
Dialogue: 0,0:05:37.37,0:05:43.71,Default,,0000,0000,0000,,My equation 2 would be the same,\N0, (-R2, -3).
Dialogue: 0,0:05:43.71,0:05:46.72,Default,,0000,0000,0000,,This would be R3, 0, and 0.
Dialogue: 0,0:05:48.76,0:05:52.93,Default,,0000,0000,0000,,Now, we're replacing the pink\Nwith the two orange and yellows.
Dialogue: 0,0:05:52.93,0:05:57.28,Default,,0000,0000,0000,,So this would be,\NI'm gonna move this over here.
Dialogue: 0,0:05:57.28,0:06:01.22,Default,,0000,0000,0000,,0 is equal to minus I1 plus\N1 times Ix plus 1 times I2.
Dialogue: 0,0:06:01.22,0:06:04.80,Default,,0000,0000,0000,,And the yellow one is\Ngoing to be 1 times Ix.
Dialogue: 0,0:06:05.92,0:06:12.98,Default,,0000,0000,0000,,Let's move Is over to this side and\NI3 over to that side, like so.