Supermensh Analysis: In mesh analysis, when a current source is present between two mesh, a supermesh analysis has to be performed. Mesh analysis stands to be one of the most universal methods of solving electrical circuits or networks. It is used to determine currents flowing in the individual mesh by forming a system of linear equations with help of Kirchhoff’s Voltage Law.
In an electrical circuit mesh is the most fundamental or elementary form of a loop that cannot be further divided into other loops.
Illustration on Super Mesh Analysis
Let us consider an electrical circuit as shown below for the illustration of supermesh analysis.
The circuit above consists of three mesh as indicated in the figure itself. It is clear from the figure that a current source IS comes between mesh 1 and 3. So in such cases, we have to apply the supermesh analysis.
For super mesh analysis, the current source comes between mesh 1 and 3. Mesh 1 and 3 are considered to be the supermesh.
For solving the above circuit using mesh analysis, we have three unknown currents, and accordingly, we should form three equations and in addition apply the concept of supermesh analysis.
In the case of supermesh analysis, the three equations can be formed in the following ways.
First, for the mesh analysis formulate the equation of the mesh which is not a supermesh. Such as in the above circuit analysis, mesh 2 does not fall under the supermesh and we can easily write KVL (Kirchhoff’s Voltage Law) for this mesh.
So, the KVL equation for mesh 2 is
Secondly, formulate the current equation from the current source present between two meshes. Such as in above the equation can be formed from the current source IS present between the mesh 1 and 3.
The equation will be
Thirdly, write the KVL equation for the supermesh. During the circuit analysis, write the KVL equation for supermesh and do not consider the branch of the circuit which contains the current source. Such as in the above circuit, write the KVL equation for super mesh 1 and 3 and while writing the equation does not consider the branch containing the current source IS.
The KVL equation for the super mesh will be
Finally, the system of linear equations can be solved to find the unknown mesh currents.
What if the current source is present in the perimeter of any individual mesh?
As shown in the circuit below if the current source is present in the perimeter of any individual mesh then we do not apply the supermesh analysis. Supermesh analysis is to be performed only when the current source is present between two individual meshes.
So in the case of the current source present in the perimeter, the mesh current of that particular mesh will be equal to the value of the current source.
Such as in the above circuit the mesh current of mesh 1 i.e. I1 = IS.
And other equations for mesh 2 and mesh 3 can be formed using KVL.
Examples of Supermesh Analysis
Let us solve a problem that involves supermesh analysis.
Consider a given circuit where mesh analysis has to be performed.
Here, a current source of 5 Amps is present between mesh I and III. So we have to perform supermesh analysis.
First, the KVL equation for mesh II is
Now, the current of the common boundary of meshes I and III is given by
Then, applying KVL for supermesh I and II we have
Finally, solving equations (i), (ii), and (iii) we have