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Supermesh Analysis

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.
supermesh 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,
supermesh analysis circuit
Fig.1 Supermesh analysis electric circuit
we set i2 = -5 A and write equation for the other mesh in the usual ways; that is,
supermesh analysis circuit
CASE 2 - When a current source exists between two meshes : Consider the circuit in Figure.(2a), for example.
supermesh analysis circuit
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,
A supermesh results 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
supermesh analysis circuit
We apply KCL to a node in the branch where the two meshes intersect. Applying KCL to node 0 in Figure.(2a) gives
supermesh analysis circuit
Solving Equations.(2) and (3), we get
supermesh analysis circuit
Note the following properties of a supermesh :
  1. The current source in the supermesh provides the constraint equation necessary to solve for the mesh currents.
  2. A supermesh has no current of its own.
  3. 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 i1 to i4 using mesh analysis.
supermesh analysis circuit
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,
supermesh analysis circuit
For the independent current source, we apply KCL to node P :
supermesh analysis circuit
For the dependent current source, we apply KCL to node Q :
supermesh analysis circuit
But Io = - i4 , thus,
supermesh analysis circuit
Applying KVL in mesh 4,
supermesh analysis circuit
From (1.1) to (1.4),
supermesh analysis circuit

Untuk Bahasa Indonesia baca Analisis Supermesh.
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