In this article, we will discuss mesh analysis (also called the mesh current method or loop current method).

What is Mesh Analysis?

Mesh analysis is one of the simple and powerful techniques used to solve and analyze electrical circuits. The mesh analysis is also known as the mesh current method or loop current method. This is because, in the case of mesh analysis, the circuit equations are expressed in terms of loop currents.

The mesh analysis method gives the solution of a circuit in the form of linear equations. The mesh analysis method is based on KVL (Kirchhoff’s Voltage Law).

In mesh analysis, mesh or loop currents are assumed that do not divide at nodes or junctions as in the case of nodal analysis. But, they complete their paths around the loop/mesh.

What is a Mesh?

A mesh is the smallest loop or closed path that does not contain any other closed path. In figure-1, loops ABCF and CDEF are the two meshes, but ABCDEFA is not a mesh because it consists of two smaller loops inside it.

Steps to Solve Circuits using Mesh Analysis

In the beginning, we will first discuss the mesh analysis to solve circuits that contain only voltage sources and such current sources which are transformable into voltage sources.

Following are the steps applied to solve an electric circuit having voltage sources only using mesh analysis:

Step 1 – Convert all the current sources into voltage sources (by Source Transformation).

Step 2 – Identify all the meshes and assign the mesh currents in any direction, clockwise or anticlockwise.

Step 3 – Write the KVL equation for each mesh and solve all the mesh equations simultaneously to find the mesh current values.

Now, let us understand these steps by considering an example.

Consider an electric circuit as shown in figure-2. We clearly identify two meshes in the circuit. We assume the two mesh currents I₁ and I₂ to be in the clockwise direction (advisable). However, we may also assume different current directions for different meshes.

Now, the KVL equation for mesh-1 is,

`\V_1-I_1 R_1-(I_1-I_2 ) R_2=0`

`\⇒V_1=I_1 R_1+I_1 R_2-I_2 R_2`

`\⇒V_1=I_1 (R_1+R_2 )-I_2 R_2…(1)`

Similarly, the KVL equation for mesh-2 is,

`\-V_2-(I_2-I_1 ) R_2-I_2 R_3=0`

`\⇒V_2=-(I_2-I_1 ) R_2-I_2 R_3`

`\⇒V_2=I_1 R_2-I_2 (R_2+R_3 )…(2)`

By solving equations (1) and (2) simultaneously, we can determine the value of currents I₁ and I₂, and other circuit parameters such as power, energy, etc.

Mesh Analysis with Unaccompanied Current Sources

An unaccompanied current source is one that cannot be converted into a voltage source. To solve such circuits, the steps of mesh analysis explained in the above section are not applicable as such.

The procedure to solve an electric circuit with a current source by mesh analysis method is explained step-by-step below with the help of an example.

Step 1 – Identify the meshes, and assume mesh currents.

Step 2 – Write the KVL equations for each mesh and solve them. But, in the case of the mesh analysis with the current source can have the following two cases:

Case (a) – If a mesh has a current source which is not common to any other mesh. Then, set the mesh current of the mesh having the current source equal to its current source value. Now, we do not need to write the KVL equation for this mesh because the mesh current is already known.

Consider an electric circuit with a current source as shown in figure-3. Here, the current source is subjected to one mesh only and not to any other mesh.

KVL equation for mesh-3 is,

`\I_3=I_S…(3)`

KVL equation for mesh-1 is,

`\V_1-I_1 R_1-(I_1-I_2 ) R_2=0`

`\⇒V_1=I_1 R_1+(I_1-I_2 ) R_2`

`\⇒V_1=I_1 (R_1+R_2 )-I_2 R_2…(4)`

KVL equation for mesh-2 is,

`\-V_2-(I_2-I_1 ) R_2-(I_2+I_3 ) R_3=0`

`\V_2=I_1 R_2-I_2 (R_2+R_3 )-I_3 R_3…(5)`

Solve equations (3), (4), & (5) to find the mesh currents I₁ and I₂.

Case (b) – If the current source is in the common branch, as shown in figure-4. The procedure to solve this circuit is explained below:

• Identify meshes and assign mesh current. In the circuit of figure-4, two meshes are identified and their mesh currents are assumed.
• Form a supermesh (Supermesh is one which is formed by combining two nearby meshes having a common current source). In the given circuit, the supermesh is shown by dotted lines.
• Write the KVL equation for the supermesh. For the given circuit, the KVL equation for the supermesh is,

`\V-I_1 R_1-I_2 R_2-I_2 R_3=0…(6)`

• The current source inside the supermesh gives a KCL equation. For the given circuit, this KCL equation is,

`\I_2=I_1+I_S…(7)`

• Finally, to determine the unknown currents, solve the supermesh equation and KCL equation simultaneously.