In a parallel RLC circuit operating exactly at its resonant frequency, how are the inductive reactance (X_L) and capacitive reactance (X_C) related?

Introduction / Context:Resonance is a fundamental phenomenon in AC circuits, filters, and RF design. In a parallel RLC network, resonance yields desirable properties like maximized impedance magnitude and minimal reactive current at the resonant frequency. Recognizing the condition that defines resonance is vital for tuning circuits and understanding bandwidth and quality factor (Q).

Given Data / Assumptions:

• Parallel connection of ideal R, L, and C.
• Operation at the exact resonant frequency f_0.
• Focus on the relationship between X_L and X_C.

Concept / Approach:At resonance in either series or parallel RLC, the magnitudes of the reactive impedances are equal and opposite: X_L = 2 * pi * f_0 * L and X_C = 1 / (2 * pi * f_0 * C). The resonance condition is X_L = X_C in magnitude. In a parallel circuit, this equality causes the reactive branch admittances to cancel, leaving a predominantly resistive input admittance and a high equivalent impedance, which is exploited in tank circuits and oscillators.

Step-by-Step Solution:Write X_L = 2 * pi * f * L; X_C = 1 / (2 * pi * f * C).Set resonance condition: X_L = X_C (magnitudes equal).At f = f_0, reactive effects cancel, so net susceptance is near zero.Therefore, choose “inductive reactance is equal to the capacitive reactance.”

Verification / Alternative check:For a given L and C, solve f_0 = 1 / (2 * pi * sqrt(L * C)). Substituting f_0 back into X_L and X_C yields equal magnitudes, confirming the condition.

Why Other Options Are Wrong:A and B contradict the resonance condition; D introduces resistance into the equality, which is not the defining relation for resonance; “None” is wrong because the equality is the standard rule.

Common Pitfalls:Confusing series with parallel resonance effects on impedance magnitude; forgetting that equality refers to magnitudes, while the signs are opposite (inductive positive, capacitive negative susceptance).

Final Answer:inductive reactance is equal to the capacitive reactance