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

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. using nodal analysis.
Example of nodal ac circuit analysis Example of nodal ac circuit analysis 

We first convert the circuit to the frequency domain:
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
Hence, the frequency domain equivalent circuit is as drawn in Figure.
Frequency domain equivalent of the circuit in Figure Frequency domain equivalent of the circuit in Figure 

Applying KCL at node 1,
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
or
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE) .....(1.1)
At node 2,
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE) 
But Ix = V1/-j2.5. Substituting this gives
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
By simplifying, we obtain
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE) ..... (1.2) 
Equations.(1.1) and (1.2) can be rewritten in matrix form to
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
We get the determinants as
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
The current Ix is given by
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
Transforming this to the time domain,
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)

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 Example of supernode ac circuit analysis 

Nodes 1 and 2 form a supernode as drawn in Figure.
Supernode in the circuit Supernode in the circuit 

Applying KCL at the supernode gives
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
or
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)..... (2.1) 
But a voltage source is connected between nodes 1 and 2, so that
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)..... (2.2) 
Substituting Equations.(2.2) to (2.1) results
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)
From Equation.(2.2),
Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE)

The document Node & Supernode for AC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Network Theory (Electric Circuits).
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