MESH ANALYSIS

Mesh analysis is used to find the voltages and currents in a circuit. Usually mesh analysis is employed when there are more than one meshes/loops and many voltage sources.

Procedure:

1. Identify the number of meshes in a circuit.
2. Assign the mesh currents. The assignment of mesh currents is arbitrary and consistent.
3. Apply Kirchoff's Voltage Law (KVL) to all meshes and obtain the mesh equations.
4. Solve the equations and obtain the mesh currents.

Mesh Analysis with Current Sources:

1. If the current source is in only one mesh, then write KVL as usual and substitute value from the current source and solve. 
2. If current source is between two meshes, then eliminate the current source and element in series with it (like open circuiting the branch) and form a super mesh.
3. Write KVL and solve the equations. 
4. Write Nodal equation for the node where these elements where eliminated.
5. On solving all the equations mesh currents will be obtained.

Example:

Find the mesh/loop currents in the following circuit.

The circuit has a current source in between two meshes. So we eliminate the current source and the resistor in series with it, to form a super mesh.

KVL for the super mesh,

–8 + 2i1 – 2i3 + 12i2 – 4i3 = 0 or i1 + 6i2 – 3i3 = 4                               (1)

KVL for mesh 3,

8i3 – 2i1 – 4i2 = 0 or –i1 – 2i2 + 4i3 = 0                                                (2)

Kirchoff's Current Law (KCL) at node 0 in Fig. (a),

i1 = 4 + i2                     i1 – i2 = 4

Solving (1) to (3) we obtain the mesh currents,

i1 =  4.632 A, i2 =  631.6 mA, i3 =  1.4736 A

Reference:

Fundamentals of Electric Circuits( Fifth Edition) by Charles K Alexander and Matthew N O Sadiku