Definition check for series resonance: In a series R–L–C circuit, which condition defines resonance?

Introduction / Context:Series resonance is a key operating point where a series R–L–C presents minimum impedance and maximum current. Recognizing the proper equality that marks this condition is essential for setting center frequency and predicting bandwidth and Q.

Given Data / Assumptions:

• Series connection of R, L, and C driven by a sinusoidal source.
• Inductive reactance XL = 2 * pi * f * L; capacitive reactance XC = 1 / (2 * pi * f * C).
• No exotic elements; standard notation (XT sometimes denotes total reactance).

Concept / Approach:In a series path, the total reactance is XT = XL − XC. Resonance occurs when the reactive terms cancel, i.e., XL = XC, making XT = 0. The loop impedance collapses to approximately R, maximizing current and producing a band-pass profile when the output is taken across R. Alternatives like XC = XC (tautology), XL = XB (undefined symbol), or XT = R (mixing reactance with resistance) do not define resonance.

Step-by-Step Solution:

Verification / Alternative check:At resonance, current is maximal and the phase angle between source voltage and current is zero (power factor unity), confirming XT = 0 and XL = XC.

Why Other Options Are Wrong:XC = XC: identity, gives no frequency condition. XL = XB: symbol XB not defined; meaningless here. XT = R: mixing reactive and resistive quantities; resonance requires XT = 0, not equality with R.

Common Pitfalls:Forgetting that series and parallel resonance share the XL = XC magnitude equality, but differ in whether impedance is minimized or maximized.

Final Answer:XL = XC