Solve the circuit with nodal analysis and find and .
1) Identify all nodes in the circuit. Call the number of nodes .
There are four nodes in the circuit:
2) Select a reference node
Because of symmetry in the circuit, any node can be chosen as the reference node.
3) Assign a variable for each node whose voltage is unknown.
There are three nodes beside the reference node:
Node I and Node III are connected to each other by a voltage source. Therefore, they form a supernode.
The voltage of Node III can be written in terms of the voltage of Node I. All we need to do is to apply KVL in the loop illustrated below.
A good practice is to avoid assigning a voltage label to Node III and use as its voltage.
There is also another voltage source which connects Node II to the reference node. This voltage source must be used to obtain the node voltage. Since the negative terminal of the voltage source is connected to the node, . You can verify this by applying KVL to the loop between the reference node and Node II.
4) Write down KCL equations.
We only need to write a KCL equation for the supernode:
. Substituting and known variables,
Now, we need to determine the required quantities:
KCL at node 1:
KVL around the loop shown in the figure below: