WEBVTT 00:00:00.186 --> 00:00:04.730 I'd like to go through this problem that we started in class. 00:00:04.730 --> 00:00:08.390 This is an interesting Kirchhoff’s law problem because it has a supernode 00:00:08.390 --> 00:00:09.660 here in the middle. 00:00:09.660 --> 00:00:13.140 A supernode is any time that you have a current source or a voltage source, 00:00:13.140 --> 00:00:16.850 whether it's dependent or independent, that's in between two nodes. 00:00:16.850 --> 00:00:21.100 And it doesn't have anything else, no resistors in between these two nodes. 00:00:21.100 --> 00:00:22.800 We have to handle that in a special way. 00:00:23.840 --> 00:00:26.610 Let's say that we have been given our two sources, our voltage source and 00:00:26.610 --> 00:00:30.470 our current source, and all of the resistors that are in this circuit. 00:00:30.470 --> 00:00:33.792 Then we can use this analysis technique where we know that we have our sources, 00:00:33.792 --> 00:00:37.710 in fact, we also know our current source, and our resistors. 00:00:37.710 --> 00:00:43.920 And we can use Kirchhoff’s law in order to determine the currents, so let's begin. 00:00:43.920 --> 00:00:49.176 The first thing that I'm going to do is this 00:00:49.176 --> 00:00:55.164 loop right here, -Vs + I1R1 + I2R2 = 0. 00:00:55.164 --> 00:00:58.850 Now let's do this loop, in this case, 00:00:58.850 --> 00:01:04.209 I'm going to put pluses and minuses on my currents. 00:01:04.209 --> 00:01:10.985 This would be -I2R2- Vx, 00:01:10.985 --> 00:01:18.031 which is 3I2 + I3R3 = 0. 00:01:18.031 --> 00:01:21.180 I can't do any more loops because if I did this loop, 00:01:21.180 --> 00:01:26.210 it would have to go through a current source, and that's not allowed. 00:01:26.210 --> 00:01:29.230 So the next thing I'm going to do is a node. 00:01:29.230 --> 00:01:31.450 I'll show you the easy node, and 00:01:31.450 --> 00:01:35.904 then we're going to actually do a supernode, so here is one node. 00:01:37.859 --> 00:01:41.100 And that's actually the easy node to do in this case. 00:01:41.100 --> 00:01:45.260 If I did the pink node, I would have all of the currents coming in 00:01:45.260 --> 00:01:49.700 equal all of the currents going out, so what's the current here? 00:01:49.700 --> 00:01:53.255 The current and the branch is always the same, so this is I1. 00:01:53.255 --> 00:01:56.280 I1 is the same current in that whole branch. 00:01:56.280 --> 00:02:02.153 So the currents that are coming in are I2 + I3. 00:02:02.153 --> 00:02:06.856 And the currents that are going out are I1 + Is, 00:02:06.856 --> 00:02:09.850 that's if I did the pink node. 00:02:11.935 --> 00:02:17.234 As an alternative, I could have done this supernode right here. 00:02:17.234 --> 00:02:21.000 Now, you'll notice that I actually have two nodes. 00:02:21.000 --> 00:02:25.841 But in between them is just a voltage source that has nothing else, 00:02:25.841 --> 00:02:27.401 no other resistors. 00:02:27.401 --> 00:02:30.977 So if I wanted to be able to do the orange node, for instance, 00:02:30.977 --> 00:02:35.840 I would have the current coming in, I1, and the current going to out here. 00:02:35.840 --> 00:02:37.440 But what current do I have there? 00:02:37.440 --> 00:02:38.790 I don't know. 00:02:38.790 --> 00:02:41.060 If I tried to do this node, I'd have the current going out, 00:02:41.060 --> 00:02:43.270 the current coming in, and what current is there? 00:02:43.270 --> 00:02:47.380 I don't know, so there are two ways of handling the supernode. 00:02:47.380 --> 00:02:52.290 The easy way, I think, is to define a current, let's just call it Ix. 00:02:52.290 --> 00:02:54.860 That's the same current everyplace in that branch. 00:02:54.860 --> 00:02:58.540 And then I can actually do both this node and this node. 00:02:58.540 --> 00:03:01.090 Because I've defined this new unknown, 00:03:01.090 --> 00:03:06.370 I now need to find also Ix, so I'm going to need an extra equation. 00:03:06.370 --> 00:03:10.450 So as an alternative to this, I could do the orange node, 00:03:10.450 --> 00:03:13.980 which would be currents coming in, I1 = Ix + I2. 00:03:13.980 --> 00:03:20.456 And the other one would be what's coming in, 00:03:20.456 --> 00:03:24.390 Ix + Is = I3, going out. 00:03:24.390 --> 00:03:29.010 So this is the orange node, and this is the yellow node. 00:03:30.450 --> 00:03:33.310 So I have two different ways I could solve this problem. 00:03:33.310 --> 00:03:38.900 I can either use equations 1 and 2 for my two loops, plus my pink node. 00:03:38.900 --> 00:03:43.250 Or I could use 1 and 2 plus my orange and yellow. 00:03:43.250 --> 00:03:45.290 These, I would turn into a matrix equation. 00:03:45.290 --> 00:03:47.063 Let's do the one with the pink on this side. 00:03:57.661 --> 00:03:59.391 I would be solving for I1, I2, and I3, my variables would be I1, I2, I3. 00:03:59.391 --> 00:04:02.882 I would be solving for these equations, like so. 00:04:02.882 --> 00:04:06.700 And then there would be a constant on this side. 00:04:06.700 --> 00:04:11.255 So it would be Vs coming over to this side. 00:04:11.255 --> 00:04:15.598 I1 times R1, I2 times R2, and 0 times I3. 00:04:15.598 --> 00:04:22.645 Here we would have 0 times I1, -R2, aha, there's a 3R2 there. 00:04:22.645 --> 00:04:25.977 So combine those two, so it's actually, 00:04:27.996 --> 00:04:34.224 The I2 is going to have a -R2 and a -3 right there, 00:04:34.224 --> 00:04:38.690 and then here is R3, any constants? 00:04:38.690 --> 00:04:41.215 No, no constants over on that side. 00:04:41.215 --> 00:04:44.115 Right here I would have, let's get these all on the same side. 00:04:44.115 --> 00:04:46.463 I'm gonna move these two over to this side. 00:04:49.499 --> 00:04:53.330 This was Is, not I3, I had a little trouble keeping track of that one. 00:04:53.330 --> 00:04:57.259 So here's I1, here's -I2, and 00:04:57.259 --> 00:05:02.490 here is -I3, and then over here we have Is. 00:05:02.490 --> 00:05:06.840 So these are the equations that we would solve in order to find our solution. 00:05:08.560 --> 00:05:11.120 Okay, let's go over to this side. 00:05:11.120 --> 00:05:16.180 If I wanted to do this set instead, I would have actually had four unknowns, 00:05:16.180 --> 00:05:20.905 I1, I2, I3, and Ix. 00:05:20.905 --> 00:05:24.715 I1, I2, I3, and Ix, and 00:05:24.715 --> 00:05:30.970 then over here's my vector of constants. 00:05:30.970 --> 00:05:37.370 My equation number 1 would be just the same, R1, R2, 0, 0, and Vs. 00:05:37.370 --> 00:05:43.714 My equation 2 would be the same, 0, (-R2, -3). 00:05:43.714 --> 00:05:46.720 This would be R3, 0, and 0. 00:05:48.760 --> 00:05:52.930 Now, we're replacing the pink with the two orange and yellows. 00:05:52.930 --> 00:05:57.285 So this would be, I'm gonna move this over here. 00:05:57.285 --> 00:06:01.225 0 is equal to minus I1 plus 1 times Ix plus 1 times I2. 00:06:01.225 --> 00:06:04.795 And the yellow one is going to be 1 times Ix. 00:06:05.915 --> 00:06:12.977 Let's move Is over to this side and I3 over to that side, like so.