In a general RLC circuit, when the capacitive reactance equals the inductive reactance (X_C = X_L), what fundamental operating condition is satisfied?

Introduction:
Resonance in RLC circuits is a cornerstone concept: it occurs when the inductive and capacitive reactances cancel each other in magnitude (X_L = X_C). Recognizing this condition helps predict current, voltage distribution, and power factor behavior in both series and parallel topologies.

Given Data / Assumptions:

• Sinusoidal steady state, linear and ideal components.
• RLC network in either series or parallel configuration.
• Condition under study: X_L = X_C.

Concept / Approach:
At resonance, the net reactive component of impedance is zero in series circuits (Z is purely resistive), or the net reactive component of admittance is zero in parallel circuits (Y is purely conductive). This is the unifying definition of resonance and is directly tied to X_L = X_C.

Step-by-Step Solution:
Write X_L = ωL and X_C = 1 / (ωC)Set X_L = X_C ⇒ ωL = 1 / (ωC)Solve for resonance radian frequency: ω_0 = 1 / sqrt(L * C)Thus f_0 = ω_0 / (2 * π) = 1 / (2 * π * sqrt(L * C))At f_0, the reactive effects cancel, defining resonance.

Verification / Alternative check:
Series RLC: Z = R + j(X_L − X_C) ⇒ at resonance Z = R (minimum magnitude), current is maximized for a given applied voltage. Parallel RLC: input admittance Y has zero imaginary part at resonance, leading to minimum source current. In both cases, the defining condition is X_L = X_C.

Why Other Options Are Wrong:

• The circuit draws maximum current in every topology: True for series only, false for parallel (it is minimum at resonance).
• The applied voltage is zero: Not implied by resonance.
• The circuit draws minimum current in every topology: True for parallel only, false for series.
• The power factor is always zero: At resonance in series, PF = 1; not zero.

Common Pitfalls:

• Overgeneralizing series behavior to parallel circuits (or vice versa).
• Confusing the condition X_L = X_C with purely reactive behavior; in series it yields purely resistive input.

Final Answer:
The circuit is at resonance