A 200 Ω resistor, a coil with inductive reactance XL = 30 Ω, and a capacitor with unknown reactance are in series across an AC source. The circuit is at resonance. What is the total series impedance at resonance?

Introduction / Context:
At series resonance, the inductive and capacitive reactances cancel each other exactly (XL = Xc). The net reactive part becomes zero, leaving only the series resistance to determine the total impedance.

Given Data / Assumptions:

• R = 200 Ω (series resistor).
• XL = 30 Ω, Xc chosen so that resonance occurs.
• Series connection; ideal components at resonance.

Concept / Approach:

Series impedance is Z = R + j(XL − Xc). At resonance, XL = Xc, hence Z = R + j0 = R. Therefore, the total impedance equals the resistance alone.

Step-by-Step Solution:

Verification / Alternative check:

At resonance, current is maximum for a given source because impedance is minimized to R. Voltage drops across L and C may be large and opposite in phase, but they cancel in the series sum.

Why Other Options Are Wrong:

230 Ω or 170 Ω imply residual reactance (not at resonance). 'Cannot be determined' ignores the standard resonance property.

Common Pitfalls:

Assuming the 30 Ω adds with R; forgetting that L and C cancel reactively at resonance in series circuits.

Final Answer:

is 200 Ω