How it works
In AC analysis, all voltages and currents are represented as phasors: V = V_m∠φ. Impedance Z = R + jX (Ω) where X = ωL for inductors and X = −1/ωC for capacitors. Admittance Y = 1/Z = G + jB. KVL and KCL apply unchanged to phasors: the only difference from DC analysis is that all quantities are complex numbers. Series resonance occurs at ω₀ = 1/√(LC), where |Z| is minimum and equals R. Parallel resonance occurs at the same ω₀ for ideal components, where |Z| is maximum. Quality factor Q = ω₀L/R = 1/(ω₀CR) for series RLC; bandwidth BW = ω₀/Q rad/s or f₀/Q Hz.
Key points to remember
At series resonance the voltage across L and across C individually can be Q times the source voltage — for Q = 10 and Vs = 10 V, VL = VC = 100 V. This voltage magnification is why series resonant circuits must be rated carefully. For parallel resonance, current in each branch can be Q times the supply current. The half-power frequencies are at ω₀ ± BW/2, where power drops to half the resonant value. Power factor = cos φ = R/|Z|; at resonance, PF = 1 and all source power is dissipated in R. Average power P = Vrms·Irms·cosφ; reactive power Q = Vrms·Irms·sinφ; apparent power S = Vrms·Irms.
Exam tip
The examiner always asks you to find the Q factor, bandwidth, and half-power frequencies of a given series RLC circuit — state Q = ω₀L/R, then BW = R/L rad/s, and half-power frequencies as ω₀ ± R/(2L) before plugging in numbers.