Mesh Analysis

MESH ANALYSIS

Mesh analysis is a method that is used to solve planar circuits for the current at any place in the circuit. Planar circuits are that can be drawn on a plane surface with no wires crossing each other. It is also called as loop analysis. Generally KVL and KCL are used in deriving the mesh and nodal equations respectively.

Mesh equations are desired from the division of current in these meshes. A loop current is different from branch current. In order to illustrate this difference, consider the circuit shown in Figure 1.19(a).

The given circuit is redrawn with nodes as shown in Figure 1.20(b). Here we can identify two closed loops or meshes represented as mesh 1 (abda) and mesh '2' (bcdb). Here I₁, and I₂are the mesh currents flowing in mesh '1' and mesh '2' respectively,

As the branch 'bd' consisting of resistor, R, is shared between the meshes 1 and 2, the resultant branch current is I,. The magnitude of I, depends on the magnitudes of both the mesh currents I, and I, from the circuit shown in Figure 1.11(b) the current directions I, and I, in R, are opposite to each other. Hence

The possible way of drawing one more closed path through the elements

V₁ → R₁ → R₂ →V₂ and→ V₁ is shown in Figure 1.11(c).

Assigning the direction of mesh current is arbitrary. In order to simplify the procedure for writing the mesh equation, the preferred direction for the flow of current is usually clockwise.

If an element is located on the boundary between two meshes such as R, in figure 1.11(d) the element current is the algebraic sum of the currents flowing through it.

Steps for Solving Mesh Equation

Step 1: Ensure that the circuit has only voltage sources. If there is current source convert it into the voltage source first.

Step 2: Assume the direction of current in advance.

Step 3: Mark the polarities of voltage drops across each element.

Step 4: Assign correct polarity to the voltage source.

Step 5: Write the mesh equation using KVL and equate the algebraic sum of voltage drops to zero in that particular mesh.

Step 6: For the shared branch the algebric sum of mesh currents flowing through it is considered.

Step 7: Solve the mesh equations to find the solution for unknown quantities.

Example 1.14: Write the mesh equations for the circuit  shown below and determine the

Example 1.15: Determine the loop currents of the circuit shown in figure below. Also find the current through 6ῼ resistor.

Since I₂ and I₃ have a negative sign the actual current direction of I₂ is opposite to the assumed clockwise direction.

Hence I₂ = 1 126 A and I₃ = 2 676 A.

Current through 6. Q resistor is 16ῼ = 13 — I2 = 1.55 A.

Example 1.16: Determine the value of R in the circuit shown below when the current is zero in the branch CD.

Assuming current direction to be clockwise.

Applying KVL for

Example 1.17: While loop and matrix equations for the network shown in figure below and determine the loop and branch currents in the network.

Example 1.18: Write the mesh equation for the circuit shown below and determine the current in each loop.

Solution:

From the circuit the mesh 1 consists of outer parameter current is I₁ = 25 A.

Example 1.19: For the given circuit find the voltage across 5 2 resistor using mesh analysis method.

As the branches BC and DE consist of current sources, we can form a super CEBDC and the equation is: