Complex Numbers as Phasors in AC Analysis Evaluate the statement: “A complex number represents a phasor quantity.”

Introduction / Context:
Phasors offer a powerful way to analyze linear AC circuits by representing sinusoidal signals as rotating vectors (complex numbers) at a fixed angular frequency. This abstraction converts differential equations into algebraic ones in the frequency domain, enabling straightforward calculations of voltages, currents, and impedances.

Given Data / Assumptions:

• Steady-state sinusoidal signals at angular frequency ω.
• Linear time-invariant circuits where phasor representation is valid.
• Complex numbers used in rectangular a + j b or polar r ∠ θ form.

Concept / Approach:

A phasor is the complex representation of a sinusoid’s amplitude and phase: x(t) = Re{X e^{j ω t}}, where the complex constant X encodes magnitude and phase. As such, every phasor corresponds to a complex number, and manipulations of phasors follow complex arithmetic rules. Impedances of L and C become j ω L and 1/(j ω C), respectively, fitting naturally into the complex framework.

Step-by-Step Solution:

Verification / Alternative check:

Solving an RLC series circuit via phasors yields I = V / (R + j ω L + 1/(j ω C)). Time-domain solution reconstructed by i(t) = Re{I e^{j ω t}} matches the sinusoidal steady-state solution, confirming the equivalence.

Why Other Options Are Wrong:

A magnitude-1 restriction is unnecessary; phasors carry arbitrary magnitude. Rectangular or polar are both valid; the form does not limit phasor validity. Phasors are inherently complex; insisting on real-only would lose phase information.

Common Pitfalls:

Mixing instantaneous values with phasors (phasors omit the explicit time factor e^{j ω t}); forgetting RMS vs. peak conventions when comparing magnitudes.

Final Answer:

True