Let's do this example of Kirchhoff's law.
We've got a circuit here that we're going to first define all of our unknowns.
We'd like to find all the currents in this circuit.
So here is the first current, I1.
Here is the second current, I2.
The third current, I3.
The fourth current, I4.
So the number of unknowns that we have is N = 4.
I1, I2, I3, and I4.
How did I know which direction to define these currents?
I didn't, I just guessed.
I typically guess left to right and top to bottom.
Once I've guessed the direction of the current though, I can write the polarity.
Where the plus goes on the tail of the arrow and the minus goes on the tip.
Now that I have defined the polarity of each of my arrows.
I need to be able to write four equations for my four unknowns.
Lets begin by writing loop equations.
Here is loop number 1.
And what I am going to do is start someplace,
here's where I am going to choose to start.
As I move up through the loop,
I'm going to encounter minus, plus, minus, and so on.
So -V1 + I1 R1 + I3 R3,
back to the same place, is equal to 0.
Now lets do another loop.
Each new loop has to pick up some additional elements.
This loop is going to pick up R4 and V3.
So loop number 2, let's start in the same place, and I'm going up through I3.
So it's -I3 R3 +
I4 R4- V3 is equal to 0.
Now let's do loop number 3.
I'm going to choose this loop, although I certainly could also have done this
outside loop, picking up all of the additional elements.
So loop number 3 is going to be, let's start right here.
Let's go up through I4- I4 R4
+ I2 R2 + V2 is equal to 0.
Well, there's three equations for my four unknowns, so
I need one additional equation.
Let's transfer over and
let's do a node equation now because I've done all my possible loops.
I've picked up every element.
Well, here's a node.
That's an ordinary node.
That's not going to do me any good in my equations, and
neither is this ordinary node.
Here's an extraordinary node, however, this big red node,
and that's where I'm going to do my current equation.
So what I'm going to do for equation number four is I'm going to say,
all the currents coming in are equal to all the currents going out.
Well, the current coming in as I1 and the currents going out are I2, I3, and I4.
So here are my four equations for four unknowns.
Now, let us convert these to matrix equations.
So what I'm going to do, because I have four equations and four unknowns.
I'm going to have a four by four matrix equation where my unknowns I1,
I2, I3 and I4 are in this vector.
My matrix is gonna be right here, and
of course my constants are gonna be on the other side.
For my convenience, I'm going to also write my unknowns across the top so
that I can keep track of which column I'm putting things in.
And I'm going to indicate which of the equations make up each of my rows.
Now what I'm going to do, is I'm going to look at my first row and
I'm going to see what's multiplied by I1, which is R1.
What's multiplied by I2?
0.
What's multiplied by I3?
R3.
What's multiplied by R4?
0.
And then what would be on the other side would be plus V1.
Now let's do equation two.
What's multiplied by I1?
0.
I2?
0.
13 and I4?
And then this needs to come over to the other side.
Now let's do this equation.
What's multiplied by I1?
0.
I2, I3, and I4?
And then my constant coming over to this side is a minus V2.
Now let's do our last equation.
I'm going to bring this over here so
I'll actually be programming -I1 + I2 + I3 + I4.
So a -1 here, 1, 1, 1, and 0, there's my matrix equation.
This equation is A x = b.
What I can do is solve this equation any of three ways check out your appendix.
You can solve it out.
You can solve it by hand, that's a lot of work.
You can put it in your calculator, please practice that.
And you can also put it in Matlab.