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How to Solve Supermesh Circuits with Current Sources

Supermesh analysis is a technique for solving electric circuits that have current sources between two or more meshes. A mesh is a loop that does not contain any other loops within it. A current source is an element that provides a constant current regardless of the voltage across it.

Supermesh analysis can simplify the process of applying Kirchhoff's voltage law (KVL) to the circuit, by reducing the number of equations and variables. The basic steps of supermesh analysis are:

Identify all the meshes and current sources in the circuit.

Assign a mesh current to each mesh, except for the ones that have a current source between them.

Create a supermesh by enclosing the meshes that have a current source between them. The supermesh does not include the current source or any elements in series with it.

Apply KVL to each mesh and supermesh, except for the ones that have a current source between them.

Use the current source equation to relate the mesh currents in the supermesh.

Solve the system of equations for the unknown mesh currents.

Use the mesh currents to find the voltages and currents of any element in the circuit.

To illustrate this technique, let us consider an example circuit with three meshes and two current sources, as shown below:

We can apply supermesh analysis to this circuit as follows:

We identify three meshes (A, B, and C) and two current sources (I1 and I2) in the circuit.

We assign a mesh current to each mesh, except for mesh B, which has a current source I1 between it and mesh A. We label the mesh currents as IA, IB, and IC, as shown below:

We create a supermesh by enclosing meshes A and B, excluding the current source I1 and the resistor R2 in series with it. The supermesh is shown below:

We apply KVL to mesh C and the supermesh, as follows:

Mesh C: V3 + R4(IC - IA) + R5(IC - IB) = 0

Supermesh: V1 + R1IA + R3(IA - IC) + V2 + R6(IB - IC) = 0

We use the current source equation to relate IA and IB in the supermesh: IA - IB = I1

We solve the system of three equations and three unknowns for IA, IB, and IC. We can use any method of our choice, such as substitution, elimination, or matrix inversion. The solution is:

IA = 0.5 A

IB = -0.5 A

IC = 0.25 A

We use the mesh currents to find the voltages and currents of any element in the circuit. For example, to find the voltage across R2, we use Ohm's law: VR2 = R2(IA - IB) = 10(0.5 - (-0.5)) = 10 V

This is how we can use supermesh analysis to solve circuits with current sources. Supermesh analysis can be applied to any planar circuit that has one or more current sources between two or more meshes. It can help us reduce the complexity of solving circuits by mesh analysis. 061ffe29dd