Series R–L–C resonance behavior: How does the total impedance of a series RLC circuit vary at the resonant frequency compared to off-resonance?

Introduction / Context:
Resonance in a series RLC circuit is a foundational AC topic. At resonance, the inductive reactance and capacitive reactance cancel, leaving only the resistive part. Understanding this helps predict current peaks and bandwidth in filters and tuners.

Given Data / Assumptions:

• Series connection of R, L, and C.
• Sinusoidal steady state.
• Resonant frequency f_0 where X_L = X_C.

Concept / Approach:
Total impedance Z for series RLC is Z = R + j(X_L − X_C). At resonance, X_L = X_C, so Z = R. Since R is the smallest possible magnitude of Z (reactances cancel out), the impedance magnitude is minimal and the current is maximal for a given source voltage.

Step-by-Step Solution:
Write reactances: X_L = 2πfL, X_C = 1/(2πfC).At f = f_0, X_L − X_C = 0.Therefore Z = R (purely real), |Z| = R.Away from resonance, |Z| = sqrt(R^2 + (X_L − X_C)^2) > R.Hence |Z| is minimum at resonance.

Verification / Alternative check:
Examine current I = V/|Z|. The current peaks at resonance—empirical confirmation of minimal impedance.

Why Other Options Are Wrong:
(b) and (c) impose monotonic behavior with frequency that is not true; impedance decreases toward resonance and increases away. (d) The maximum occurs far from resonance for large reactance differences, not at resonance.

Common Pitfalls:
Confusing series with parallel resonance (where behavior differs) and mixing up current/impedance relationships.

Final Answer:
It is minimum at the resonant frequency.