Alright, let's formalize our approach to mesh analysis.
When analyzing a circuit using mesh analysis we first of all identify
the meshes.
Once again, a mesh is a closed loop that contains no other loops within it.
We'll next assign variable names for each of the mesh currents.
We'll then write KVL equations around each mesh in terms of the mesh current.
And then at that point, we have a system of equations in terms of our mesh currents
that simply need to be solved, and we've got the problem taken care of.
Let's take a look at this circuit here.
First of all, identify the meshes.
We've got a mesh here on the left, we've got this upper mash, a lower mesh.
And again, this outer loop is not a mesh because in this case it contains three
different meshes within it.
So we identify then or we define three different mesh currents, one for
each of the meshes.
We'll define the current In this mesh to be i1 and
referenced in that direction, the mesh in the other,
mesh current in this mesh we call i2 and the mesh current in this mesh we call i3.
Lets go ahead now then and write the three
Mesh equations in terms of I 1 and I 2 and I 3 starting right here.
Going up this way we cross the voltage source going from minus to plus
therefore it's a voltage increase and again that would be a negative V 0.
Plus the votage drop across R1 is just R1 times I1.
Now coming down across R2, we need to be carefully.
The current through R2
involves two different mis currents because we are going down at this point
The current through here is gonna be the mesh current reference down which is I 1,
minus I 2, which is referenced in the opposite direction.
So we'll have then plus R 2 times I 1 minus I 2.
Now continuing on down, across R5.
The current flowing through R5 is going to be I1- I3.
And those terms must then add to 0.
Let's take a look at the top mesh.
Starting right here and
going up we've got R2 times the current
flowing up which is I2 minus I1.
And again let us just point out that the current going up
I2 minus I1 In this equation is the opposite current that
was flowing down which was I1- I2 and this equation.
All right continuing on around we have in plus
R3 times the current through R3 is simply I2.
Coming back to the left through R4 we'll have plus R4 times the current
flowing in R4, which
is I2- I3.
That brings us back to where we started so the sum of those terms must equal zero.
And finally we do a KVL around this bottom
mesh starting here and we'll have R five times
the current flowing through R five in the direction we're going which is I three.
Minus I1
plus coming across here, plus R4 times
the current flowing left to right in R4.
In terms of the mesh currents is I3 minus i2 and
than finally the current coming down here through that r6 will be r6 times i3
and than some of those currents equal zero.
Up here, lets just go ahead and
complete this by factoring out the mesh currents and combining like terms.
We have, for the top one, the top equation we've got I1 times R1.
Let's see what we've got.
R1 there, there and there.
So, we've R1 plus R2 plus R5.
+ I2 times, we have one I2 term there,
it's got a negative.
R2 + I3 We've got one I3 term here with a negative sign in front of it.
So that would be I + I3 times a negative R5.
And the sum of those then equals, we've got the negative V0 that we
need to bring over to the other side as a positive.
V 0 combining like terms in the second equation we factor out the I 2 I'm
sorry the I 1 here and I've got one
I 1 term, it's got a negative on it so it'd be a negative
R 2 Plus I2 times.
You've got I2 here, here, and here.
All positives will be R2 plus R3 plus R4.
Plus R3 plus R4 and
then our I3 turns There is only one.
It's got a negative, so it'll be a negative R4.
And the sum of those terms have to equal 0.
And finally, the bottom equation here, factoring out the I1s to begin with.
I've got an I1 term here.
Times what is that?
That's R5 isn't it?
So I1's got a negative R5.
Plus I2.
once again there's only one I2 [INAUDIBLE] has a negative R4.
Plus I3.
And there are three I3 terms,
I got R4 plus R5 plus R6,
R4 + R5 + R6.
And the sum of those has to equal 0.
Plug it in, well, of course, with the values of the resistors and the V0,
plug it in to your matrix solver or your solve button on your calculator,
and you've got everything you need to calculate those mesh currents.
Once you know the mesh currents you can calculate any branch voltage or
branch current that you might be interested in.