The topic of this problem is Mesh Analysis.

And we're going to be working with circuits with independent voltage and

current sources.

The problem is to determine V0 at the circuit below.

You can see in the circuit that we have one current source 2 Milli amps and

we have 2 voltage sources a 6 volts source and a 12 volts source.

And V out is measured across a 1 kilo ohms load resistor and

it's on the right hand side of the circuit.

So we are going to use Mesh analysis to solve this problem.

So the first thing that we know is that Mesh analysis

is performed using Kirchhoff's voltage law.

So we're going to use Kirchhoff's voltage law and after we realize that our first

step as far as circuit analysis will be to determine and assign loops so

that we can sum up the voltages around the loops using Kirchhoff's voltage law.

So we're going to assign our loops now, we're going to start with the lower left

hand corner we're going to call this loop one and it'll have loop current I1.

We have another loop at the top of the circuit we're going to

call it loop 2 or mesh 2 and it has current I2 and

our last independent loop is the lower right hand loop and

we'll give it a current I3.

So now that we've assigned our loops and

our loop currents, we now want to go around and

write our equations using Kirchhoff's voltage law for each one of the loops.

So we're going to start in the lower left hand corner first

summing up the voltages around loop one, so

we'll start the lower left hand corner and we'll go around loop 1.

What we're looking for when we're doing mesh analysis is we're looking for

the loop currents.

If we can find the loop or mesh currents, that's really what we're interested in.

Once we find I1, I2, and I3, we could solve for

any other value that we want in our circuit.

So that's really what we're after.

And so we see with loop 1 that we have a voltage drop across a 2 milliamp source,

and 1 kilo Ohm resistor, and a 1 kilo Ohm resistor, and

then a voltage source at the bottom toward the end of our loop.

So we don't know the voltage across the 2 milliamp source and

in fact if we wanted to sum up the voltage around this loop

we'd have to add another variable for like the voltage across the 2 milliamp source.

So we immediately realize that this 2 milliamp source is I sub 1,

because I sub 1 is the only current flowing up through this left most

leg of our circuit, a branch of our circuit.