Mesh Analysis

What is Mesh Analysis?

Loop is nothing but any closed path. A loop is said to be a mesh if it is not having any other loop inside it. The mesh analysis is based on Kirchhoff's Voltage Law.

Mesh Analysis Definition

In the Mesh analysis method, we will analyze the currents of meshes in the given electric circuit. Hence, mesh analysis is equal to a mesh current analysis method. Similarly, in the Loop analysis method, we will analyze the currents of loops present in the given electric circuit. Hence, we can call this method the Loop current analysis method or Loop analysis in short. Since the number of mesh equations will always be less than or equal to the number of loop equations, we prefer Mesh analysis to Loop analysis.

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Procedure of Mesh Analysis

We can easily solve network theory problems using the Mesh Analysis method. Here, first, you will get to know that the Mesh analysis is based on which laws & then the method Mesh analysis is generally used to determine which electrical quantities etc. Now, let’s see the steps for solving the given network in the Mesh Analysis method.

• Step 1: Observe the number of meshes in the given circuit and then represent the current in each mesh either in a clockwise or in anti-clockwise direction.
• Step 2: Represent the current flows through each circuit element in terms of the mesh currents.
• Step 3: Write the mesh equation (KVL & Ohm’s law) for every mesh whose mesh current is unknown.
• Step 4: Get the mesh currents by solving the above mesh equations. Using these mesh currents, we can find the response (voltage or current) of any branch in the given circuit.

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Mesh Analysis Solved Problems

Question 1: For the following circuit, find the current through the ammeter by using Mesh Analysis.

Solution:

Step 1: In the given electric circuit, there are 3 meshes. For the time being, let us consider the mesh currents in a clockwise direction.

Step 2: In the following diagram, the current flowing through each circuit element is represented in terms of the mesh currents.

Step 3: We no need to write the mesh equations for meshes 1 and 3. Rather, we can directly write the mesh current values by looking at the branch currents.

i₁ = 2 A and i₃ = 5 A

Mesh equation around Mesh 2 is

-i₂(1)-(i₂-i₃)(1)-(i₂-i₁)(1)=0

=>-i₂(1)-(i₂-5)(1)-(i₂-2)(1)=0

=>3i₂=7

=>i₂=7/3 A

Step 4: The current, I = i₂-2 = 7/3-2 = 1/3 A. So, using the Mesh analysis method, we got the current I value as 1/3 A.

Question 2. Find the current I of the following electric circuit by using the Mesh analysis method.

Solution:

Step 1: In the given electric circuit, there are 2 meshes. For the time being, let us consider the mesh currents in a clockwise direction.

Step 2: In the following diagram, the current flowing through each circuit element is represented in terms of the mesh currents.

Step 3: Mesh equation around Mesh 1 is

20-i₁(10)-(i₁-i₂)10- 5 = 0

=>20i₁-10i₂=15

=>4i₁-2i₂=3 …(1)

Mesh equation around Mesh 2 is

-20 i₂+5-(i₂-i₁)10=0

=>10 i₁-30i₂ = -5

=>2 i₁-6i₂ = -1 …(2)

Step 4: By solving equations (1) and (2), we will get i₁=1 A & i₂=0.5 A.

=>I=i₂=0.5 A. So, using the Mesh analysis method, we got the current I value as 0.5 A.

In this article, we discussed the Mesh analysis method for DC circuits and solved the Mesh analysis problems for DC circuits. Similarly, we can apply the Mesh analysis method for AC circuits and solve the Mesh analysis problems in AC circuits.