Applying mesh analysis to circuits containing current sources (dependent or independent) may appear complicated. Before learning this method, make sure you have read the mesh analysis.

## Mesh Analysis with Current Sources

But to be honest, the current sources will reduce the number of equations, makes it easier to solve. Let us take a look on two possibilities.

**CASE 1**- When a current source exist only in one mesh : Consider in Figure.(1) for example,

Fig.1 Supermesh analysis electric circuit |

we set

*i*= -5 A and write equation for the other mesh in the usual ways; that is,_{2}(1) |

**CASE 2**- When a current source exists between two meshes : Consider the circuit in Figure.(2a), for example.

Figure 2. Supermesh with intersect |

We create a

*supermesh*by excluding the current source and any elements connected in series with it, as shown in Figure.(2b). Thus,Asupermeshresults when two meshes have a (dependent or independent) current source in common.

As shown in Figure.(2b), we create a supermesh as the periphery of the two meshes and treat it differently. (If a circuit has two or more supermeshes that intersect, they should be combined to form a larger supermesh). Why treat supermesh differently? Because mesh analysis applies KVL - which requires that we know the voltage across each branch - and we do not know the voltage across the current source in advance. However, a supermesh must satisfy KVL like any other mesh. Therefore, applying KVL to the supermesh in Figure.(2b) gives

(2) |

We apply KCL to a node in the branch where the two meshes intersect. Applying KCL to node 0 in Figure.(2a) gives

(3) |

Solving Equations.(2) and (3), we get

(4) |

Note the following properties of a supermesh :

- The current source in the supermesh provides the constraint equation necessary to solve for the mesh currents.
- A supermesh has no current of its own.
- A supermesh requires the application of both KVL and KCL.

## Supermesh Analysis Examples

For better understanding, let us review the example below :**1. For the circuit in Figure.(3), find**

*i*to_{1}*i*using mesh analysis._{4}Figure 3 |

__Solution :__
Note that meshes 1 and 2 form a supermesh since they have an independent current source in common. Also, meshes 2 and 3 form another supermesh because they have dependent current source in common. The two supermeshes intersect and form a larger supermesh as shown. Applying KVL to the larger supermesh,

(1.1) |

For the independent current source, we apply KCL to node P :

(1.2) |

For the dependent current source, we apply KCL to node Q :

But

*I*, thus,_{o}= - i_{4}(1.3) |

Applying KVL in mesh 4,

(1.4) |

From (1.1) to (1.4),

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