CIRCUIT ANALYSIS AND TECHNIQUES
Node Voltage Method (Special Cases)

Suppose there is a dependent voltage source in the circuit:

Care must be taken in writing the circuit equations, since the dependent source may look like an additional variable. This can be easily handled by expressing the current if in terms of the node voltages:

                                      

then proceed in the usual way.

A complication may also arise when a voltage source is connected between two essential nodes, and is the only connection between these nodes. On the figure below:

if there is an attempt to try to write node-voltage equations at nodes 2 or 3, the current in the dependent voltage source is unknown. The problem is solved by introducing a current i, through the source, and immediately eliminating it =>

At node 2:

...........(7)

At node 3:

.................(8)

When these two equations are combined:

...(9)

Note that equation (9) could have been derived directly by visualising nodes 2 and 3 as a single node and summing currents of the combined node in terms of the node voltages

This is the concept of a supernode. At this stage there has to be ensured that Kirchhoff's current law is obeyed. Using the supernode and summing the currents =>

...........(10)

Equations (9) and (10) are identical. Note that the voltages at nodes 2 and 3 are not the same, and the nodes are not connected, but just visualised as a single node for the convenience of current summation. The concept of a supernode can be used whenever two essential nodes are connected only by a voltage source.