Simple Steps Nodal and Supernode for AC Circuit

nodal and supernode ac circuit

The basic of nodal analysis of ac circuit steady-state condition is Kirchhoff’s current law. Nodal and supernode for ac circuit are no different from the dc circuits, so we won’t have big difficulty here.

Nodal and Supernode for AC Circuit

Since KCL is valid for phasors, we can analyze the ac circuit by nodal analysis.

Learning Kichhoff’s laws for AC circuit will lead us to:

  1. Node and supernode for ac circuit
  2. Mesh and supermesh for ac circuit
  3. Superposition for ac circuit
  4. Source transformation for ac circuit
  5. Thevenin and Norton for ac circuit

Nodal Analysis of AC Circuit

You will not find the basic theory of the nodal analysis here. We will only cover the step-by-step analyzing the ac circuit using this technique.

If you have not learned about its basic principle, make sure to read Nodal Analysis first.

The following nodal analysis ac circuit examples illustrate this.

Find the ix in the circuit of Figure.(1) using nodal analysis.

Nodal and Supernode for AC Circuit
Figure 1. Example of nodal ac circuit analysis

Solution :
We first convert the circuit to the frequency domain :

nodal analysis ac circuit

Hence, the frequency domain equivalent circuit is as drawn in Figure.(2).

nodal analysis ac circuit
Figure 2. Frequency domain equivalent of the circuit in Figure.(1)

Applying KCL at node 1,

nodal analysis ac circuit

or

nodal analysis ac circuit
(1.1)

At node 2,

nodal analysis ac circuit

But Ix = V1/-j2.5. Substituting this gives

nodal analysis ac circuit

By simplifying, we obtain

nodal analysis ac circuit
(1.2)

Equations.(1.1) and (1.2) can be rewritten in matrix form to

nodal analysis ac circuit

We get the determinants as

nodal analysis ac circuit

The current Ix is given by

nodal analysis ac circuit

Transforming this to the time domain,

nodal analysis ac circuit

Supernode Analysis of AC Circuit

Next, we will analyze an ac circuit containing a supernode in the circuit. Like before, we will not cover the basic principle of supernode used in ac circuit.

You need to read Supernode Analysis first in order to understand the following example.

Compute V1 and V2 in the circuit of Figure.(3).

nodal analysis ac circuit
Figure 3. Example of supernode ac circuit analysis

Solution :
Nodes 1 and 2 form a supernode as drawn in Figure.(4).

nodal analysis ac circuit
Figure 4. Supernode in the circuit

Applying KCL at the supernode gives

nodal analysis ac circuit

or

nodal analysis ac circuit
(2.1)

But a voltage source is connected between nodes 1 and 2, so that

nodal analysis ac circuit
(2.2)

Substituting Equations.(2.2) to (2.1) results

nodal analysis ac circuit

From Equation.(2.2),

nodal analysis ac circuit