[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.76,0:00:04.17,Default,,0000,0000,0000,,Let's do this example of Kirchhoff's law.
Dialogue: 0,0:00:04.17,0:00:08.22,Default,,0000,0000,0000,,We've got a circuit here that we're going\Nto first define all of our unknowns.
Dialogue: 0,0:00:08.22,0:00:10.70,Default,,0000,0000,0000,,We'd like to find all\Nthe currents in this circuit.
Dialogue: 0,0:00:10.70,0:00:14.17,Default,,0000,0000,0000,,So here is the first current, I1.
Dialogue: 0,0:00:14.17,0:00:17.19,Default,,0000,0000,0000,,Here is the second current, I2.
Dialogue: 0,0:00:17.19,0:00:19.75,Default,,0000,0000,0000,,The third current, I3.
Dialogue: 0,0:00:19.75,0:00:22.22,Default,,0000,0000,0000,,The fourth current, I4.
Dialogue: 0,0:00:22.22,0:00:25.98,Default,,0000,0000,0000,,So the number of unknowns\Nthat we have is N = 4.
Dialogue: 0,0:00:25.98,0:00:30.08,Default,,0000,0000,0000,,I1, I2, I3, and I4.
Dialogue: 0,0:00:30.08,0:00:32.74,Default,,0000,0000,0000,,How did I know which direction\Nto define these currents?
Dialogue: 0,0:00:32.74,0:00:34.46,Default,,0000,0000,0000,,I didn't, I just guessed.
Dialogue: 0,0:00:34.46,0:00:37.65,Default,,0000,0000,0000,,I typically guess left to right and\Ntop to bottom.
Dialogue: 0,0:00:37.65,0:00:41.50,Default,,0000,0000,0000,,Once I've guessed the direction of the\Ncurrent though, I can write the polarity.
Dialogue: 0,0:00:41.50,0:00:46.27,Default,,0000,0000,0000,,Where the plus goes on the tail of\Nthe arrow and the minus goes on the tip.
Dialogue: 0,0:00:46.27,0:00:49.47,Default,,0000,0000,0000,,Now that I have defined\Nthe polarity of each of my arrows.
Dialogue: 0,0:00:49.47,0:00:53.20,Default,,0000,0000,0000,,I need to be able to write four\Nequations for my four unknowns.
Dialogue: 0,0:00:53.20,0:00:55.51,Default,,0000,0000,0000,,Lets begin by writing loop equations.
Dialogue: 0,0:00:55.51,0:00:56.62,Default,,0000,0000,0000,,Here is loop number 1.
Dialogue: 0,0:00:57.70,0:00:59.48,Default,,0000,0000,0000,,And what I am going to\Ndo is start someplace,
Dialogue: 0,0:00:59.48,0:01:01.21,Default,,0000,0000,0000,,here's where I am going\Nto choose to start.
Dialogue: 0,0:01:01.21,0:01:03.66,Default,,0000,0000,0000,,As I move up through the loop,
Dialogue: 0,0:01:03.66,0:01:08.82,Default,,0000,0000,0000,,I'm going to encounter minus,\Nplus, minus, and so on.
Dialogue: 0,0:01:08.82,0:01:12.61,Default,,0000,0000,0000,,So -V1 + I1 R1 + I3 R3,
Dialogue: 0,0:01:12.61,0:01:18.48,Default,,0000,0000,0000,,back to the same place, is equal to 0.
Dialogue: 0,0:01:18.48,0:01:19.62,Default,,0000,0000,0000,,Now lets do another loop.
Dialogue: 0,0:01:19.62,0:01:22.78,Default,,0000,0000,0000,,Each new loop has to pick up\Nsome additional elements.
Dialogue: 0,0:01:22.78,0:01:26.04,Default,,0000,0000,0000,,This loop is going to pick up R4 and V3.
Dialogue: 0,0:01:26.04,0:01:30.80,Default,,0000,0000,0000,,So loop number 2, let's start in the same\Nplace, and I'm going up through I3.
Dialogue: 0,0:01:30.80,0:01:34.89,Default,,0000,0000,0000,,So it's -I3 R3 +
Dialogue: 0,0:01:34.89,0:01:40.67,Default,,0000,0000,0000,,I4 R4- V3 is equal to 0.
Dialogue: 0,0:01:40.67,0:01:42.49,Default,,0000,0000,0000,,Now let's do loop number 3.
Dialogue: 0,0:01:42.49,0:01:46.35,Default,,0000,0000,0000,,I'm going to choose this loop, although\NI certainly could also have done this
Dialogue: 0,0:01:46.35,0:01:49.61,Default,,0000,0000,0000,,outside loop,\Npicking up all of the additional elements.
Dialogue: 0,0:01:49.61,0:01:53.76,Default,,0000,0000,0000,,So loop number 3 is going to be,\Nlet's start right here.
Dialogue: 0,0:01:53.76,0:01:58.62,Default,,0000,0000,0000,,Let's go up through I4- I4 R4
Dialogue: 0,0:01:58.62,0:02:03.14,Default,,0000,0000,0000,,+ I2 R2 + V2 is equal to 0.
Dialogue: 0,0:02:03.14,0:02:05.77,Default,,0000,0000,0000,,Well, there's three equations for\Nmy four unknowns, so
Dialogue: 0,0:02:05.77,0:02:07.70,Default,,0000,0000,0000,,I need one additional equation.
Dialogue: 0,0:02:07.70,0:02:09.34,Default,,0000,0000,0000,,Let's transfer over and
Dialogue: 0,0:02:09.34,0:02:13.06,Default,,0000,0000,0000,,let's do a node equation now because\NI've done all my possible loops.
Dialogue: 0,0:02:13.06,0:02:14.96,Default,,0000,0000,0000,,I've picked up every element.
Dialogue: 0,0:02:14.96,0:02:16.57,Default,,0000,0000,0000,,Well, here's a node.
Dialogue: 0,0:02:16.57,0:02:17.61,Default,,0000,0000,0000,,That's an ordinary node.
Dialogue: 0,0:02:17.61,0:02:20.00,Default,,0000,0000,0000,,That's not going to do me any\Ngood in my equations, and
Dialogue: 0,0:02:20.00,0:02:22.07,Default,,0000,0000,0000,,neither is this ordinary node.
Dialogue: 0,0:02:22.07,0:02:25.14,Default,,0000,0000,0000,,Here's an extraordinary node,\Nhowever, this big red node,
Dialogue: 0,0:02:25.14,0:02:27.78,Default,,0000,0000,0000,,and that's where I'm going\Nto do my current equation.
Dialogue: 0,0:02:27.78,0:02:31.42,Default,,0000,0000,0000,,So what I'm going to do for\Nequation number four is I'm going to say,
Dialogue: 0,0:02:31.42,0:02:35.43,Default,,0000,0000,0000,,all the currents coming in are equal\Nto all the currents going out.
Dialogue: 0,0:02:35.43,0:02:42.100,Default,,0000,0000,0000,,Well, the current coming in as I1 and\Nthe currents going out are I2, I3, and I4.
Dialogue: 0,0:02:42.100,0:02:46.10,Default,,0000,0000,0000,,So here are my four equations for\Nfour unknowns.
Dialogue: 0,0:02:46.10,0:02:49.10,Default,,0000,0000,0000,,Now, let us convert these\Nto matrix equations.
Dialogue: 0,0:02:49.10,0:02:52.42,Default,,0000,0000,0000,,So what I'm going to do, because I\Nhave four equations and four unknowns.
Dialogue: 0,0:02:52.42,0:02:57.54,Default,,0000,0000,0000,,I'm going to have a four by four\Nmatrix equation where my unknowns I1,
Dialogue: 0,0:02:57.54,0:03:02.37,Default,,0000,0000,0000,,I2, I3 and I4 are in this vector.
Dialogue: 0,0:03:03.64,0:03:05.85,Default,,0000,0000,0000,,My matrix is gonna be right here, and
Dialogue: 0,0:03:05.85,0:03:08.43,Default,,0000,0000,0000,,of course my constants\Nare gonna be on the other side.
Dialogue: 0,0:03:08.43,0:03:13.29,Default,,0000,0000,0000,,For my convenience, I'm going to also\Nwrite my unknowns across the top so
Dialogue: 0,0:03:13.29,0:03:16.10,Default,,0000,0000,0000,,that I can keep track of which\Ncolumn I'm putting things in.
Dialogue: 0,0:03:16.10,0:03:20.57,Default,,0000,0000,0000,,And I'm going to indicate which of\Nthe equations make up each of my rows.
Dialogue: 0,0:03:20.57,0:03:24.09,Default,,0000,0000,0000,,Now what I'm going to do,\Nis I'm going to look at my first row and
Dialogue: 0,0:03:24.09,0:03:28.28,Default,,0000,0000,0000,,I'm going to see what's multiplied by I1,\Nwhich is R1.
Dialogue: 0,0:03:28.28,0:03:29.95,Default,,0000,0000,0000,,What's multiplied by I2?
Dialogue: 0,0:03:29.95,0:03:30.64,Default,,0000,0000,0000,,0.
Dialogue: 0,0:03:30.64,0:03:32.99,Default,,0000,0000,0000,,What's multiplied by I3?
Dialogue: 0,0:03:32.99,0:03:33.74,Default,,0000,0000,0000,,R3.
Dialogue: 0,0:03:33.74,0:03:35.34,Default,,0000,0000,0000,,What's multiplied by R4?
Dialogue: 0,0:03:35.34,0:03:36.06,Default,,0000,0000,0000,,0.
Dialogue: 0,0:03:36.06,0:03:40.97,Default,,0000,0000,0000,,And then what would be on\Nthe other side would be plus V1.
Dialogue: 0,0:03:40.97,0:03:42.42,Default,,0000,0000,0000,,Now let's do equation two.
Dialogue: 0,0:03:42.42,0:03:43.96,Default,,0000,0000,0000,,What's multiplied by I1?
Dialogue: 0,0:03:43.96,0:03:44.58,Default,,0000,0000,0000,,0.
Dialogue: 0,0:03:44.58,0:03:45.26,Default,,0000,0000,0000,,I2?
Dialogue: 0,0:03:45.26,0:03:46.09,Default,,0000,0000,0000,,0.
Dialogue: 0,0:03:46.09,0:03:48.79,Default,,0000,0000,0000,,13 and I4?
Dialogue: 0,0:03:48.79,0:03:51.59,Default,,0000,0000,0000,,And then this needs to come\Nover to the other side.
Dialogue: 0,0:03:52.92,0:03:54.07,Default,,0000,0000,0000,,Now let's do this equation.
Dialogue: 0,0:03:54.07,0:03:55.48,Default,,0000,0000,0000,,What's multiplied by I1?
Dialogue: 0,0:03:55.48,0:03:56.62,Default,,0000,0000,0000,,0.
Dialogue: 0,0:03:56.62,0:04:00.03,Default,,0000,0000,0000,,I2, I3, and I4?
Dialogue: 0,0:04:00.03,0:04:04.56,Default,,0000,0000,0000,,And then my constant coming over\Nto this side is a minus V2.
Dialogue: 0,0:04:04.56,0:04:05.98,Default,,0000,0000,0000,,Now let's do our last equation.
Dialogue: 0,0:04:05.98,0:04:09.47,Default,,0000,0000,0000,,I'm going to bring this over here so
Dialogue: 0,0:04:09.47,0:04:15.05,Default,,0000,0000,0000,,I'll actually be programming\N-I1 + I2 + I3 + I4.
Dialogue: 0,0:04:15.05,0:04:20.51,Default,,0000,0000,0000,,So a -1 here, 1, 1, 1, and 0,\Nthere's my matrix equation.
Dialogue: 0,0:04:20.51,0:04:24.59,Default,,0000,0000,0000,,This equation is A x = b.
Dialogue: 0,0:04:24.59,0:04:28.69,Default,,0000,0000,0000,,What I can do is solve this equation any\Nof three ways check out your appendix.
Dialogue: 0,0:04:28.69,0:04:29.67,Default,,0000,0000,0000,,You can solve it out.
Dialogue: 0,0:04:29.67,0:04:32.09,Default,,0000,0000,0000,,You can solve it by hand,\Nthat's a lot of work.
Dialogue: 0,0:04:32.09,0:04:34.72,Default,,0000,0000,0000,,You can put it in your calculator,\Nplease practice that.
Dialogue: 0,0:04:34.72,0:04:36.15,Default,,0000,0000,0000,,And you can also put it in Matlab.