Mesh Current Analysis

1.

For each mesh, label a current flowing in a loop around the mesh except:

(a)

If a current source is common to only one mesh, label that current with the current of the current source (e.g., current i_S1 below).

2.

For each mesh except for meshes containing current sources write KVL in the form

$$\sum v_{k} = 0$$

where v_k is written in terms of mesh currents (e.g., $V_{R_4} = (i_C - i_B) \times R_4$ below).

The mesh equation for mesh C below is

R₅ i_C + R₄ (i_C - i_B) + v_S3 = 0

and rearrange with all the unknowns grouped on the left hand side:

-R₄ i_B + (R₄ + R₅) i_C = -v_S3

(a)

For meshes of the form of 1(a) there is no need to write KVL since the mesh current is already known.

(b)

For two meshes contain the same current source take a path around the outside of the two meshes avoiding the current source - this is called a super mesh. For the supermesh containing i_S2 below the supermesh equation is

R₁ i_A + R₂ (i_A - i_S1) + R₃ (i_B - i_S1) + R₄ (i_B - i_C) = 0

and rearrange with all the unknowns grouped on the left hand side:

i_A (R₁ + R₂) + i_B (R₃ + R₄) - i_C R₄ = i_S1 (R₂ + R₃)

Then relate to two unknown currents in the supermesh to the current supplied by the current source:

i_B - i_A = i_S2

.

3.

Solve the equations.