Electric Circuit Based on Mesh (Loop) Current Method

Let consider a simple dc network as shown in figure below to find the currents via various branches by using Mesh (Loop) current technique.

By applying the KVL around mesh (loop)-1 :( note in mesh-1, I₁ is termed as local current and other mesh currents I₂ & I₃ are termed as foreign currents.)

By applying the KVL around mesh (loop)-2 :( likewise in mesh-2, I₂ is local current and I₁ and I₃ are called foreign currents)

By applying KVL around mesh (loop)-3:

In general, we can write for ith mesh (for i = 1, 2,..... N)

Note:

Usually, R_ij = R_ji (true only for linear bilateral circuits)   
I_i → the unknown mesh currents for the network.

Summarize:

Step-1: Draw the circuit on a flat surface with no conductor crossovers.
Step-2: Label the mesh currents (Ii) cautiously in a clockwise direction.
Step-3: Write the mesh equations by examining the circuit (No. of independent mesh (loop) equations = no. of branches (b) - no. of principle nodes (n) + 1).

Note:

To examination, a resistive network having voltage and current sources by using ‘mesh’ equations technique the following steps are necessary to note:
 
• When possible, transform current source to voltage source.

• Or else, state the voltage across the current source and write the mesh equations as when these source voltages were identified. Augment the set of equations with one equation for each current source stating a known mesh current or difference among two mesh currents.

• Mesh analysis is valid only for circuits which can be drawn in a 2-dimensional plane in such a manner that no element crosses over the other.