Nodal Analysis with complex numbers

Nodal analysis is a method of determining the voltage between nodes in an electrical circuit in terms of the branch currents.

Sample problem #2

Find I

The elements in this circuit are already converted to impedances.

First let’s identify the nodes in this circuit.

The node V2 has already its value which is 8 V because of the supply connected under it.

Solving V1 could help identify the current entering the –j3 ohm capacitor. From the node V1, we can get:

V1 (1/j6 + 1/-j3 + 1/8) – V2 (1/8) = 0

(The reciprocal of the nearby impedances of node V1 times V1 minus the neighbor node of V1which is V2 times the reciprocal of the impedance that connects them.)

We could substitute V2= 8.

V1 (0.125+ j0.1666) – 8 (1/8) = 0

                                                       V1 (0.125 +0.1666) = 1

       V1= 1/ (0.125+ j0.1666)

                                                       V1= 2.881 – j3.840 V

                                                       V1= 4.80 ∟-53.12V

To get I, just use ohms law and divide V1 to the capacitor.

Other problems that involves nodal and mesh analysis requires matrix to solve for the unknown. The more complex the problem, the greater the chance matrix is needed.

Supernode

A super node is contains two nodes, one a non-reference node and another node that may be a second non-reference node or the reference node.

When a super node occurs in a circuit it involves shortening the supply voltage and combining or adding the formulated equation of the two nodes that are connected to the supply voltage. Then applying KVL to the original circuit to get another equation and applying matrix, ohms law or simple calculations to obtain the unknown value in the circuit.