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:
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.
|Figure 1. 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.(2).
|Figure 2. Frequency domain equivalent of the circuit in Figure.(1)|
Applying KCL at node 1,
At node 2,
But Ix = V1/-j2.5. Substituting this gives
By simplifying, we obtain
Equations.(1.1) and (1.2) can be rewritten in matrix form to
We get the determinants as
The current Ix 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.(3).
|Figure 3. Example of supernode ac circuit analysis|
Nodes 1 and 2 form a supernode as drawn in Figure.(4).
|Figure 4. Supernode in the circuit|
Applying KCL at the supernode gives
But a voltage source is connected between nodes 1 and 2, so that
Substituting Equations.(2.2) to (2.1) results