Mesh Analysis

Mesh analysis (or the mesh current method) is a method that is used to solve planar circuits for the currents (and indirectly the voltages) at any place in the circuit. Planar circuits are circuits that can be drawn on a plane surface with no wires crossing each other. A more general technique, called loop analysis (with the corresponding network variables called loop currents) can be applied to any circuit, planar or not. Mesh analysis and loop analysis both make use of Kirchhoff’s voltage law to arrive at a set of equations guaranteed to be solvable if the circuit has a solution. Mesh analysis is usually easier to use when the circuit is planar, compared to loop analysis.

Determine the currents in the example using mesh analysis.

We first define mesh currents I1 and I2 matching those currents defined in the figure.

Wring the KVL equation for the two meshes gives

mesh1 = 12 - 2I1 - (2+j2)(I1-I2) = 0

mesh2 = -(2+j2)(I1-I2) - (-j4+4+j6)I2 = 0

The equation in matrix form are 

Solving for the mesh currents gives

The third phasor current I3 is related to the mesh currents by

Use these steps to sytematically apply Mesh Analysis when solving a circuit problem:

1. Find all meshes and label them. Show mesh currents going clockwise.
2. Write down everything you know by inspection. Current sources give you information about
   mesh currents. This means that you will not have to generate as many KVL equations.
3. If there are dependent sources in the problem, express their values in terms of mesh currents.
   (There may be enough information to express the dependent source as a constant, which is even better.)
4. Write a KVL equation in each mesh where the mesh current is unknown.
   You may have to temporarily remove current sources to visualize where the loops are.
5. After step 4 you should have N equations if you started with N unknowns. Using Algebra
   or Linear Algebra you can now solve for all the mesh currents.
6. If you know all the mesh currents for the circuit, then anything else can be found (e.g. voltages,
   power, current through an element, etc.)

Learnings:

Mesh Analysis is a step-by-step approach to solving circuits.
It is based on Kirchoff's Voltage Law.

With Mesh Analysis the mesh currents of the circuit are the unknowns in the equations.Solving Mesh analysis in AC is just the same as in DC.