Series/parallel resonance criterion: In a linear R–L–C circuit, does resonance occur when the magnitudes of capacitive and inductive reactances are equal (|XC| = |XL|)?

Introduction / Context:
Resonance is a foundational concept in filter design, oscillators, and matching networks. At resonance, reactive effects cancel in a way that produces either a minimum or maximum in impedance, depending on whether the tuned network is series or parallel. The simplest recognition rule uses the equality of reactance magnitudes.

Given Data / Assumptions:

• Sine-wave steady state; linear components.
• Inductive reactance XL = 2 * pi * f * L.
• Capacitive reactance XC = 1 / (2 * pi * f * C).

Concept / Approach:
Setting XL = XC yields 2 * pi * f * L = 1 / (2 * pi * f * C), which solves to f0 = 1 / (2 * pi * sqrt(L * C)). In a series RLC, this makes the net reactance zero so impedance is near R (minimum), maximizing current (band-pass via the resistor). In a parallel RLC, the branch susceptances cancel, producing a large input impedance (a notch when used in series with a load). The criterion |XC| = |XL| is the unifying test for both cases.

Step-by-Step Solution:

Verification / Alternative check:
Impedance/admittance plots show phase crossing zero at f0, with magnitude extremum consistent with the topology.

Why Other Options Are Wrong:
Incorrect: contradicts standard resonance math.
Series-only / parallel-only: the equality holds for both; only the impedance extremum type differs.

Common Pitfalls:
Confusing equality of reactances with equality of impedances; forgetting that resistance alters Q and bandwidth but not the resonance condition itself.

Final Answer:
Correct