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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