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**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.