Single and Multi Mesh Analysis

Contents

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Key learnings:

Mesh Definition: A mesh is the smallest loop in a circuit without any smaller loops inside it.
Mesh Analysis: Mesh analysis, based on Kirchhoff’s Voltage Law, helps find voltage, current, or power in planar circuits.
Single Mesh Circuits: In single mesh circuits, Ohm’s law is used directly to find current or voltage.
Multi Mesh Circuits: Multi mesh circuits have multiple meshes, making the analysis more complex.
Steps for Mesh Analysis: Identify if the circuit is planar, count the meshes, label currents, write KVL equations, and solve for the currents.

The word mesh means a smallest loop which is closed one and formed by using circuit components. The mesh must not have any other loop inside it.

Mesh Analysis finds the voltage, current or power in a chosen element by writing a voltage equation around each mesh. The method uses Kirchhoff Voltage Law and applies only to planar circuits. A planar circuit can be drawn on a flat surface so that no branch crosses over or under another branch.

Single Mesh

A closed circuit with only one mesh is a single-mesh circuit.

In that circuit the current or voltage across an element follows from Ohm’s law. If some circuit elements are in parallel, combine them first so the network still has a single mesh.

Multi Mesh

A multi mesh circuit has more than one mesh. Extra meshes mean extra KVL equations, so the algebra is heavier than in a single-mesh circuit.

The video below works through one multi-mesh example:

Steps for Mesh Analysis

The mesh-analysis steps are as follows:

Frst we have to determine whether the circuit is planar or non planar. If it is a non planar circuit, we have to perform other methods of analysis such as nodal analysis.
Then we have to count the number of meshes. The number of equations to be solved is same as the number of meshes.
Then we label each of the mesh currents according to the convenience.
We write KVL equation for each of the meshes. If the element lies between two meshes then we calculate the total current flowing through the element by considering two meshes. If the direction of two mesh currents is same then summation of currents is taken as the total current flowing through the element and if the direction is opposite then the difference of mesh currents is taken. In second case the current in the mesh under consideration is taken as the greatest among all the meshes currents and the procedure is followed.

For mesh ABH, the KVL is

For mesh BCF, the KVL is

For mesh CDEF, the KVL is

For mesh BFG, the KVL is

For mesh BGH, the KVL is

Organize the equation according to the mesh currents.
Solve the mesh equations for i₁, i₂, i₃, i₄, and i₅.
If any dependent source is there in the circuit or any unknown other than mesh currents, express that source in the suitable mesh currents.

Disadvantages of Mesh Analysis

Mesh analysis is only useful for planar circuits.
If the network is large, the number of meshes and corresponding equations increases, making the method inconvenient for complex circuits.