# Electronics/Mesh Analysis

## MeshesEdit

A 'mesh' (also called a loop) is simply a path through a circuit that starts and ends at the same place. For the purpose of mesh analysis, a mesh is a loop that does not enclose other loops.

## Mesh AnalysisEdit

Similar to nodal analysis, mesh analysis is a formalized procedure based on KVL equations. A caveat: mesh analysis can only be used on 'planar' circuits (i.e. there are no crossed, but unconnected, wires in the circuit diagram.)

Steps:

1. Draw circuit in planar form (if possible.)
2. Identify meshes and name mesh currents. Mesh currents should be in the clockwise direction. The current in a branch shared by two meshes is the difference of the two mesh currents.
3. Write a KVL equation in terms of mesh currents for each mesh.
4. Solve the resulting system of equations.

## Complication in Mesh AnalysisEdit

1. Dependent Voltage Sources

Solution: Same procedure, but write the dependency variable in terms of mesh currents.

2. Independent Current Sources

Solution: If current source is not on a shared branch, then we have been given one of the mesh currents! If it is on a shared branch, then use a 'super-mesh' that encircles the problem branch. To make up for the mesh equation you lose by doing this, use the mesh current relationship implied by the current source (i.e. $I_2 - I_1=4 mA$).

3. Dependent Current Sources

Solution: Same procedure as for an independent current source, but with an extra step to eliminate the dependency variable. Write the dependency variable in terms of mesh currents.

## ExampleEdit

Given the Circuit below, find the currents $I_1$, $I_2$.

The circuit has 2 loops indicated on the diagram. Using KVL we get:
Loop1: $0=9-1000I_1-3000(I_1-I_2)$
Loop2: $0=3000(I_1-I_2)-2000I_2-2000I_2$
Simplifying we get the simultaneous equations:
$0=9-4000I_1+3000I_2$
$0=0+3000I_1-7000I_2$
solving to get:
$I_1=3.32mA$
$I_2=1.42mA$