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:

- Node and supernode for ac circuit
- Mesh and supermesh for ac circuit
- Superposition for ac circuit
- Source transformation for ac circuit
- 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 i_{x} in the circuit of Figure.(1) using nodal analysis.

Figure 1. Example of nodal ac circuit analysis |

__Solution :__

We first convert the circuit to the frequency domain :

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

Figure 2. Frequency domain equivalent of the circuit in Figure.(1) |

Applying KCL at node 1,

or

(1.1) |

At node 2,

But **I**_{x} = **V**_{1}/-*j*2.5. Substituting this gives

By simplifying, we obtain

(1.2) |

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

We get the determinants as

The current **I**_{x} is given by

Transforming this to the time domain,

## 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 **V**_{1} and **V**_{2} in the circuit of Figure.(3).

Figure 3. Example of supernode ac circuit analysis |

*Solution :*

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

Figure 4. Supernode in the circuit |

Applying KCL at the supernode gives

or

(2.1) |

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

(2.2) |

Substituting Equations.(2.2) to (2.1) results

From Equation.(2.2),

Grate