As the excitation frequency increases, how do the impedances of series RC and parallel RC networks qualitatively change?

Introduction / Context:RC networks are fundamental frequency-selective elements. Understanding how their impedances vary with frequency is essential for filter design, coupling/decoupling, and timing applications. This question contrasts series and parallel RC behaviors as frequency rises.

Given Data / Assumptions:

• Capacitive reactance Xc = 1 / (2 * pi * f * C).
• Resistor R is frequency-independent (ideal).
• Comparison is qualitative: increase frequency f while R and C are fixed.

Concept / Approach:As frequency increases, Xc decreases. In a series RC, total impedance magnitude |Z_series| = sqrt(R^2 + Xc^2) falls because the reactive term shrinks. In a parallel RC, the total admittance increases because the capacitive susceptance (1 / Xc) grows with frequency, so the equivalent impedance |Z_parallel| = 1 / |Y| decreases as well.

Step-by-Step Solution:Compute Xc = 1 / (2 * pi * f * C); increasing f → decreasing Xc.Series RC: |Z_series| = sqrt(R^2 + Xc^2) → decreases as Xc decreases.Parallel RC: Y_total = 1 / R + 1 / (jXc); |Y| increases with frequency, so |Z_parallel| = 1 / |Y| decreases.Therefore, both series and parallel RC impedances decrease with increasing frequency.

Verification / Alternative check:Limiting cases: as f → 0, Xc → infinity; series RC behaves like largely resistive with large |Z|, parallel RC tends toward R (capacitor open). As f → infinity, Xc → 0; series RC tends to R (smaller than at low f), parallel RC tends to near 0 (capacitor acts short), confirming decreasing trends.

Why Other Options Are Wrong:

• Series decreases, parallel increases: Parallel does not increase; added capacitive path lowers impedance.
• Series increases, parallel decreases: Opposite of Xc behavior trend.
• Both increase: Contradicts the fact that Xc decreases as frequency rises.

Common Pitfalls:

• Confusing impedance with reactance; while Xc alone decreases, total Z must be evaluated for the configuration.
• Assuming R changes with frequency; ideal R is constant.

Final Answer:both series and parallel RC impedance decrease