Solve the circuit with the nodal analysis and determine .

**Solution**

1) Identify all nodes in the circuit. Call the number of nodes .

There are five nodes in the circuit:

Therefore

2) Select a reference node

The best option is the node in the bottom because it is connected to both voltage sources.

3) Assign a variable for each node whose voltage is unknown.

There are four nodes beside the reference node:

Node III and Node IV are connected to the reference node through voltage sources. Therefore, their node voltages can be determined by the voltage sources.

and .

is the current of . The Ohm’s law can be used to write in terms of the node voltages. Thus, . Substituting , (Eq. 1).

Therefore, .

4) Write down KCL equations.

We only need to write a KCL equation for Node I and Node II:

Node I: . Substituting and known variables,

(Eq. 2).

Node II: . .

Substituting in Eq. 2:

.

Now, we need to determine the required quantities. Eq. 1 implies that

.

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