The mesh analysis relies on the prior concept of the Kirchhoff's voltage law,

which is a sum of the voltages around any loop is zero.

And in fact the mesh analysis is a systematic application of

the Kirchhoff's voltage law.

There are three basic steps in applying mesh analysis.

First to define mesh currents, one for each non-inclusive loop.

By non-inclusive, I mean a loop doesn't include any other loops.

So I've got three non-inclusive loops here.

And then define these mesh currents to be I1, I2, and I3.

And I give it a direction of the current flow around that loop.

And then I do a KVL for around each of these loops.

And then I solve for the mesh currents.

In this particular case,

the only unknowns in my equation should be these mesh currents.

So even if I have another variable floating around in there,

I ignore that variable for the time being,

I try to write my equations only in terms of those mesh currents.

Now, let's take a look at coming up with the equations for this particular circuit.

Now in coming up with the equations for

this example, we're going to need to use the KVL.

And in using the KVL we're going to need to be able to do

find the voltage across a resistor like this.

Well, that means that we really want to know what the current

is through this resistor.

Now I'm going to put something off to the side here.

Suppose we have a resistor like this and

the voltage is equal to IR, Ohm's law.

But suppose our current is shown in terms of these branch currents, so we've got

one branch current going this way and then another branch current going this way.

The combination i is equal to I1 which is

going in the same direction as i minus I2 which is going in the opposite direction.

So then this voltage across that resistor is R(I1-I2).