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A voltage with 5% fifth-harmonic content is applied to a series RC circuit. What is the fifth-harmonic content (percentage) in the circuit current?

Difficulty: Medium

Correct Answer: More than 5%

Explanation:


Introduction / Context:
Harmonic distortion in voltage sources translates differently into current depending on the load impedance versus frequency. In a series RC circuit, impedance magnitude decreases with frequency, so higher-frequency components produce comparatively larger currents than lower-frequency components, other things equal.


Given Data / Assumptions:

  • Applied voltage contains a fundamental and a fifth-harmonic of 5% amplitude relative to the fundamental.
  • Load is a series RC at steady state (linear, time-invariant).
  • No resonance conditions or nonlinearity; purely frequency-dependent impedance.


Concept / Approach:
Current magnitude for a frequency component at angular frequency ωk is |I(ωk)| = |V(ωk)| / |Z(ωk)|. For a series RC, |Z(ω)| = sqrt(R^2 + (1/(ωC))^2). As frequency increases, the capacitive reactance 1/(ωC) decreases, reducing |Z| and increasing current. Hence, the fifth harmonic (5ω) experiences smaller impedance than the fundamental (ω), so its current proportion increases beyond the voltage’s 5%.


Step-by-Step Solution:

Given voltage fifth-harmonic ratio = 5% of fundamental.For current: I1 = V1 / |Z(ω)|, I5 = V5 / |Z(5ω)|.Since |Z(5ω)| < |Z(ω)|, the ratio I5 / I1 > V5 / V1.Therefore, current fifth-harmonic content > 5%.


Verification / Alternative check:

Compute example with R and C: the impedance drop at higher harmonics always favors more current for RC loads.


Why Other Options Are Wrong:

5% or less than 5%: would be true for RL loads (increasing impedance with frequency) but not RC.Equal to or more than 5%: overly vague; the correct insight is strictly more than 5% in an RC series.Approximately zero: not applicable.


Common Pitfalls:

Assuming all loads behave like resistors (frequency-neutral). Reactive impedances bias harmonic currents.


Final Answer:

More than 5%
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