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Series RLC resonance: f0, bandwidth, half power, Q

Short answer

Resonance: XL=XC at f0 = 1/(2π√LC). Quality factor Q = ω0L/R = 1/(ω0CR). Half-power frequencies: ω1,2 = ∓R/2L + √[(R/2L)² + 1/LC]. Bandwidth BW = ω2−ω1 = R/L rad/s = ω0/Q.

Explain the phenomenon of resonance in an RLC series circuit. Derive the expression for resonant frequency, bandwidth, half power frequencies and quality factor.

Resonance in a series RLC circuit is the state in which X_L = X_C, so the impedance is purely resistive and the current is in phase with the supply voltage, with the maximum value V/R.

Resonant frequency

Z = R + j(ωL − 1/ωC)
Im(Z) = 0  →  ω0L = 1/(ω0C)
ω0 = 1/√(LC) rad/s,   f0 = 1/(2π√(LC)) Hz

Quality factor

Q is 2π × (maximum energy stored)/(energy lost per cycle), or the voltage magnification at resonance.

Q = ω0L/R = 1/(ω0CR) = (1/R)√(L/C)
V_L = V_C = Q·V at resonance

Half power frequencies

The half power points are where I = I0/√2, i.e. |Z| = √2 R, i.e. |X| = R:

ωL − 1/ωC = ±R
ω2 = R/2L + √[(R/2L)² + 1/LC]
ω1 = −R/2L + √[(R/2L)² + 1/LC]
ω1 ω2 = ω0²

Bandwidth

BW = ω2 − ω1 = R/L rad/s   (R/2πL Hz)
BW = ω0/Q
QuantityExpression
Resonant frequency1/(2π√LC)
Quality factorω0L/R
Half power frequencies∓R/2L + √[(R/2L)² + 1/LC]
BandwidthR/L = ω0/Q

Answer: f0 = 1/(2π√LC), Q = ω0L/R, BW = R/L, ω1,2 = ∓R/2L + √[(R/2L)² + 1/LC].

✓ Answer checked by Claude Opus 5.5 (high effort)

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