Complex Numbers as Phasors in AC Analysis Evaluate the statement: “A complex number represents a phasor quantity.”
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ATrue
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BFalse
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CTrue only if magnitude is 1
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DTrue only in rectangular form
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EFalse; phasors require real numbers only
Answer
Correct Answer: True
Explanation
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:
Represent a sinusoid x(t) = X_m cos(ω t + φ).Define the phasor X = X_m / √2 ∠ φ (for RMS) or X = X_m ∠ φ (for peak).In rectangular form, X = a + j b with a = X cos φ and b = X sin φ.Circuit laws (KCL/KVL) apply to phasors using complex addition and multiplication with impedances.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