RLC Circuit

Resonance, impedance, and transient response in AC circuits

An RLC circuit consists of a resistor R, inductor L, and capacitor C in series with an AC voltage source V₀sin(ωt). The impedance Z determines the current amplitude I₀ = V₀/Z. At resonance (ω₀ = 1/√(LC)), the inductive reactance XL = ωL exactly cancels the capacitive reactance XC = 1/(ωC), leaving Z = R (minimum impedance, maximum current). The phase angle φ = arctan((XL - XC)/R) indicates whether current leads or lags voltage. The quality factor Q = ω₀L/R = 1/(R√(C/L)) measures the sharpness of the resonance peak. In transient response (no driving source), the circuit exhibits damped oscillations. The damping ratio ζ = R/(2√(L/C)) determines the behavior: underdamped (ζ < 1, oscillating decay), critically damped (ζ = 1, fastest non-oscillating decay), or overdamped (ζ > 1, slow exponential decay).