AC maximum power transfer – Complex-conjugate matching Statement: In an AC circuit, power delivered to the load is maximized at the frequency where the load impedance is the complex conjugate of the source (output) impedance. Is this correct?
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ATrue
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BFalse
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CTrue only for purely resistive sources
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DFalse; the load must equal the source impedance exactly (not conjugate)
Answer
Correct Answer: True
Explanation
Introduction / Context:Maximum power transfer is a cornerstone in RF, audio, and power electronics. In AC systems with reactance, matching is not just about magnitudes; the phase (reactive part) must be addressed via conjugation. This question confirms the complex-conjugate matching condition for peak power delivery.
Given Data / Assumptions:
- Linear AC circuit analyzed at a specific frequency using phasors.
- Source (output) impedance Zs = Rs + jXs, load impedance ZL = RL + jXL.
- Goal: maximize average power absorbed by the load.
Concept / Approach:
The average power to the load is maximized when ZL = Zs* (the complex conjugate of Zs). This cancels reactive parts (XL = −Xs) and matches resistive parts (RL = Rs), maximizing the magnitude of load voltage/current product that contributes to real power without reactive circulation.
Step-by-Step Solution:
Let Zs = Rs + jXs and ZL = RL + jXL.Power to the load P = |Vth|^2 * RL / |Zs + ZL|^2 (Thevenin form).To maximize P with respect to RL and XL, set XL = −Xs (reactive cancellation).Differentiate P with respect to RL; optimum occurs at RL = Rs.Thus ZL,opt = Rs − jXs = Zs* (complex conjugate).Verification / Alternative check:
At the match: input seen by source is purely resistive; voltage standing wave ratio is minimized in transmission-line terms, confirming peak power transfer.
Why Other Options Are Wrong:
“False” and “must equal exactly (not conjugate)” ignore reactive cancellation. “True only for resistive sources” is unnecessary; the conjugate form explicitly covers reactive sources.
Common Pitfalls:
Matching magnitudes only; forgetting that reactive parts must cancel to avoid circulating reactive power and suboptimal real power absorption.
Final Answer:
True.