Nodal Analysis with Supernode

From our previous discussion of Nodal Analysis we have seen, how voltage sources affect nodal analysis. We have also seen how a voltage source makes it easier for us to calculate the node voltages when connected with a reference node. But things get complicated when a voltage source cannot be referenced i.e. it comes in between two non-reference node. This voltage source along with two non-reference nodes forms a supernode. 

In summary, When a voltage source comes in between two non-reference node then these two non-reference nodes and the voltage source form a supernode and we take this supernode as a single node and apply KCL and KVL to solve the circuit.

To solve a problem of supernode follow these steps:

01.   Mark a reference node such that a supernode can�t be formed. Try to avoid supernode at first hand. If it�s not possible then at least make a voltage source referenced. 

02.   Then mark other non-referenced nodes as you do in normal nodal analysis.

03.   Next, mark the supernode with a dotted circle to remind you that it�s a supernode.

04.   Now apply KCL at the supernode. 

05.   At the end apply KVL at the supernode loop to find the node voltages in supernode.

Let us explain this procedure with an example:

First of all we have marked a reference node V0. Then we marked all other nodes as we do normally for nodal analysis.

Now we have given a dotted circle to remind us that this is a supernode along with V1 and V2.

Now we apply KCL at the circuit:

2 = (V1 � 0) / 2 + (V2 � 0) / 4 +7

8 = 2 V1 + V2 +28 (multiplying both sides by 4)

2 V1 + V2 = -20 ������������������. (a)

Now we apply KVL at the supernode loop:

-V1 - 2 +V2 = 0

V2 = V1 + 2 ��������������������. (b)

Putting this value of V2 in equation (a):

2 V1 + (V1 + 2) = -20

3 V1 = -22

V1 = -22/3 V

Now from (b): V2 = -22/3 +2 = -16/3V

Note that the 10 Ohm resistor connected across the supernode does not make any difference in the calculations as it is connected across the supernode.