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:
Write branch admittances and sum: Y = G + j(B_C + B_L).Set B_C + B_L = 0 at f_0 → imaginary part vanishes, leaving Y ≈ G (small if R is large).Result: Z_total = 1 / Y peaks at f_0 (maximum impedance at resonance).
Verification / Alternative check:
Plotting |Z| vs f for a parallel RLC shows a peak at f_0; the series RLC shows a dip at f_0. This opposite behavior distinguishes band-stop (parallel resonance) from band-pass (series resonance) characteristics.
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:
is maximum at the resonant frequency
Discussion & Comments