Mesh Analysis in Electrical Circuits

Introduction

• Mesh analysis is a method used to analyze electrical networks.
• Main goal: To obtain power delivered or absorbed by electrical elements.
• Requires knowledge of voltage and current in the network.

What is a Mesh?

• A mesh is a loop in a circuit that does not contain any inner loops.
• Definition of a loop: a path where the first and last nodes are the same.
• Identifying meshes is crucial for performing mesh analysis.

Steps to Perform Mesh Analysis

1. Identify the Total Number of Meshes

   • Count the total number of loops without inner loops in the network.

2. Assign Mesh Currents

   • Each mesh will have its own mesh current flowing around its perimeter.
   • Direction can be clockwise or anti-clockwise; clockwise is preferred for convenience.

3. Develop KVL Equations for Each Mesh

   • Apply Kirchhoff’s Voltage Law (KVL) to each mesh to create equations.

4. Solve KVL Equations for Mesh Currents

   • Solve equations to find the values of the mesh currents assigned in step 2.

Important Points Regarding Mesh Analysis

• Planar Networks: Mesh analysis is only applicable for planar networks, where no branches cross each other.

• Direction of Mesh Currents: Choosing clockwise is often easier and aligns with common circuit configurations.

• Number of Equations: The number of equations required is equal to the number of meshes. Formula:

  [ M = B - N + 1 ]

  where M = number of meshes, B = number of branches, N = number of nodes.

Example Problem

1. Identify Total Number of Meshes:

   • Example circuit has 2 meshes. The outer loop cannot be considered a mesh due to inner loops.

2. Assign Mesh Currents:

   • Assign ( i_1 ) to first mesh and ( i_2 ) to second mesh, both in the clockwise direction.

3. Develop KVL Equations:

   • For mesh 1:
     • KVL: ( 10 - 5i_1 - 5(i_1 - i_2) = 0 )
     • Resulting equation: ( 2i_1 - i_2 = 2 )
   • For mesh 2:
     • KVL: ( -5(i_2 - i_1) - 10i_2 = 0 )
     • Resulting equation: ( i_1 - 3i_2 = 0 )

4. Solve KVL Equations:

   • Use methods such as substitution or elimination to solve for ( i_1 ) and ( i_2 ).
   • Example result: ( i_2 = \frac{2}{5} ) A.

5. Calculate Power:

   • Power loss in the 10 ohm resistor:
     • ( P = i_2^2 \times R = \left( \frac{2}{5} \right)^2 \times 10 = 1.6 ) W.

Conclusion

• Mesh analysis is a systematic approach to solving electrical networks that simplifies calculations of current and power.
• Important to remember how to set up KVL equations correctly and to identify mesh currents appropriately.