Dominance of reactance in a series RLC: Across operating frequencies, which quantity determines whether the net reactive behavior is inductive or capacitive in a series RLC circuit?

Introduction:
Series RLC circuits exhibit frequency-dependent reactance. Whether the circuit looks inductive or capacitive depends on the relative magnitudes of the inductive reactance XL and the capacitive reactance XC at the operating frequency. This underlies resonance and filter behavior.

Given Data / Assumptions:

• Series connection of R, L, and C
• Sinusoidal steady-state analysis
• Net reactance X = XL - XC (sign indicates inductive or capacitive)

Concept / Approach:

In a series RLC, the total impedance is Z = R + j(XL - XC). If XL > XC, X is positive (inductive). If XC > XL, X is negative (capacitive). At resonance, XL = XC and the net reactance is zero. Thus, the larger magnitude of the two reactances determines the reactive character away from resonance.

Step-by-Step Solution:

Verification / Alternative check:

Plot XL and XC versus frequency: XL rises linearly with f, while XC falls hyperbolically. The crossover (XL = XC) marks resonance and confirms dominance flips around f0.

Why Other Options Are Wrong:

• Capacitive reactance is always dominant.: False; dominance depends on frequency.
• Inductive reactance is always dominant.: False; at low f, capacitive dominates.
• Resistance is always dominant.: Resistance affects damping, not reactive sign.
• Both reactances cancel at all frequencies.: Cancellation happens only at resonance.

Common Pitfalls:

• Thinking R determines inductive vs capacitive behavior; it does not.
• Forgetting sign conventions when computing X = XL - XC.

Final Answer:

The larger of the two reactances is dominant.