Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

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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. using nodal analysis.
Example of nodal ac circuit analysis 

We first convert the circuit to the frequency domain:

Hence, the frequency domain equivalent circuit is as drawn in Figure.
Frequency domain equivalent of the circuit in Figure 

Applying KCL at node 1,

or
 .....(1.1)
At node 2,
 
But I_x = V₁/-j2.5. 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 V₁ and V₂ in the circuit of Figure.
Example of supernode ac circuit analysis 

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