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Calculate resonant frequency, Q factor, bandwidth, and cutoff frequencies for RLC series circuits.
RLC Series Circuit
Impedance vs Frequency — Series RLC (normalized)
An RLC series circuit contains a resistor (R), inductor (L), and capacitor (C) in series. At the resonant frequency f₀ = 1/(2π√(LC)), the inductive reactance XL equals the capacitive reactance XC, and they cancel each other. The impedance drops to its minimum value R, and the circuit passes maximum current.
The quality factor Q = (1/R)×√(L/C) measures how sharp the resonance peak is. A high Q means a narrow bandwidth and strong frequency selectivity — the circuit strongly amplifies signals near f₀ while rejecting others. A low Q gives a broad, gentle response useful for wideband applications.
The bandwidth BW = f₀/Q defines the range of frequencies where the circuit response is within 3 dB of the peak. The lower and upper cutoff frequencies f₁ and f₂ mark the edges of this passband. RLC circuits are fundamental building blocks in radio receivers, audio crossovers, and power factor correction.
Resonant Frequency (f₀)
f₀ = 1 / (2π√(LC))Quality Factor (Q)
Q = (1/R) × √(L/C)Bandwidth (BW)
BW = f₀ / QRLC circuit formulas (complete)
Resonant frequency: f₀ = 1 / (2π√LC)
Quality factor: Q = (1/R) × √(L/C) [series RLC]
Q = R × √(C/L) [parallel RLC]
Bandwidth: BW = f₀ / Q = R / (2πL)
Characteristic impedance: Z₀ = √(L/C)
Damping ratio: ζ = 1/(2Q)
Underdamped (ζ<1): oscillates. Critically damped (ζ=1): fastest no-overshoot. Overdamped (ζ>1): slow.Q needed = f₀/BW = 1,000,000/10,000 = 100. L=250µH → C=101pF, R=√(L/C)/Q = 15.8Ω
Variable capacitor 60–160pF covers 890kHz–1.45MHz (AM band).
L=4.7mH, C=0.6µF: f₀=3.01kHz, Z₀=88.5Ω → R=8Ω speaker: Q=11. BW = 3000/11 = 273Hz.
Use zobel network for flatter response.
L=10µH, C=47µF: f₀=7.3kHz (well below switching freq) → attenuation at 100kHz = 40×log(100k/7.3k) = 49dB
Q = 0.5 for critically-damped response: R_damp = √(L/C)/2 = 0.23Ω
Design tip: Q > 10 gives sharp resonance but sensitive to component tolerance. Critically damped (Q = 0.5, ζ = 1): fastest settling with no overshoot — ideal for filters. In SMPS output filters: add ESR of capacitor (0.01–0.1Ω) — it naturally damps the circuit. Series resonance: impedance minimum. Parallel resonance: impedance maximum.
Did you know? RLC circuits underpin wireless communication. Every antenna has a natural resonant frequency determined by equivalent L and C values. A sharp Q factor (high selectivity) lets a receiver pick one station from thousands broadcasting simultaneously.