Parallel RLC impedance vs frequency: which statement is correct about the total impedance of a parallel RLC circuit?

Introduction / Context:The frequency response of a parallel RLC network is central to filter and resonance behavior. Unlike a series resonant circuit (which has minimum impedance at resonance), a parallel resonant circuit exhibits maximum impedance at resonance, often called antiresonance. Recognizing this contrast prevents common design mistakes.

Given Data / Assumptions:

• Ideal R, L, and C elements connected in parallel.
• Sinusoidal steady state; linear components.
• Resonant frequency f_0 where reactive branch currents cancel in phasor sense.

Concept / Approach:

• Admittance Y_total = 1/R + 1/jX_L + jX_C (in compact form), with X_L = 2 * π * f * L and X_C = 1 / (2 * π * f * C).
• At resonance, susceptances of L and C cancel (imaginary parts sum to zero), minimizing |Y_total| and thus maximizing |Z_total| = 1 / |Y_total|.

Step-by-Step Reasoning:

Verification / Alternative check:

Why Other Options Are Wrong:

• Always increases or always decreases with frequency: False; |Z| varies non-monotonically and peaks at resonance.
• Sum of R, X_L, X_C: Impedances in parallel do not add arithmetically; that is a misuse of series rules.
• None of the above: Incorrect because 'maximum at the resonant frequency' is correct.

Common Pitfalls:

• Confusing series and parallel resonance behaviors.
• Adding magnitudes instead of using complex admittances for parallel combinations.

Final Answer: