Guide to Theory and Problems of Basic AC/DC Circuit Analysis by John O’Malley

In AC circuit analysis, voltage and current phasors are utilized alongside resistances and reactances, similar to how voltages and currents are used with resistances in DC circuit analysis. The original AC circuit, known as a time-domain circuit, is converted into a phasor-domain circuit. In this transformation, sinusoidal voltages and currents are represented by phasors, and inductances and capacitances are replaced by reactances.

Basic AC/DC Circuit Theory, Analysis and Problems

Resistances remain unchanged. The phasor-domain circuit is the one that is actually analyzed. It has the advantage that resistances and reactances share the same ohm unit, allowing them to be combined in a manner similar to how resistances are combined in DC circuit analysis.

Analyzing a phasor-domain circuit doesn't require calculus, only complex algebra. Additionally, all the concepts from DC circuit analysis for finding voltages and currents are applicable to phasor-domain circuits, with the key difference being the use of complex numbers instead of real numbers.

Phasor-domain Circuit Elements

To transform a time-domain circuit into a phasor-domain circuit, we need to establish the relationships between the voltage and current phasors for resistors, inductors, and capacitors. Let's start, firstly, by considering obtaining this relation for a resistor of R ohms. For a current i = Im sin (ωt+θ), the voltage across the resistor is r = RImsin (ωt+θ).

The corresponding phasors are: I = Im / √2 θ A and V = RIm / √2 θ V

Dividing the voltage equation by the current equation produces a relation between the voltage and current phasors:

V/I = (ImR / √2) θ / (Im / √2) θ = R

This result indicates that the resistance R of a resistor connects the resistor's voltage and current phasors in the same manner as it connects the resistor's voltage and current.

Figure 1 - Phasor-domain circuit

Because of this similarity, the relation V/I = R can be represented in a phasor-domain circuit in the same way that V/I = R is represented in the original time-domain circuit. Figure 1 shows this.

Due to this similarity, the relationship with V/I = R can be represented in a phasor-domain circuit just as it is in the original time-domain circuit. This is illustrated in Figure 1.

Theory and Problems – Basic Circuit Analysis  by John O’Malley

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