RLC Circuit Calculator

Calculate resonance frequency, inductance, capacitance, quality factor Q, bandwidth, and resistance for series or parallel RLC circuits.

RLC Circuit Formulas: Resonance Frequency: f₀ = 1 / (2π√LC) ω₀ = 2πf₀ = 1/√LC Where: • f₀ = resonance frequency (hertz, Hz) • ω₀ = angular resonance frequency (rad/s) • L = inductance (henries, H) • C = capacitance (farads, F) Quality Factor Q: Series: Q = (1/R)√(L/C) = ω₀L/R = 1/(ω₀RC) Parallel: Q = R√(C/L) = R/(ω₀L) = ω₀RC Bandwidth: BW = f₀/Q Series: BW = R/(2πL) Parallel: BW = 1/(2πRC) Damping Ratio: ζ = 1/(2Q) Series: ζ = (R/2)√(C/L) Damping Conditions: • ζ < 1 (Q > 0.5): Underdamped (oscillates) • ζ = 1 (Q = 0.5): Critically damped • ζ > 1 (Q < 0.5): Overdamped Characteristic Impedance: Z₀ = √(L/C) Impedance at Resonance: Series: Z = R (minimum) Parallel: Z = L/(RC) (maximum) -3dB Frequencies: f₁ = f₀ - BW/2 (lower) f₂ = f₀ + BW/2 (upper)
Example 1 (Radio Tuner - Resonance): L = 10 µH, C = 100 pF f₀ = 1/(2π√(10×10⁻⁶ × 100×10⁻¹²)) f₀ = 1/(2π√(10⁻¹⁵)) f₀ = 1/(2π × 10⁻⁷·⁵) = 5.03 MHz Z₀ = √(10×10⁻⁶/100×10⁻¹²) = 316.2 Ω Example 2 (Bandpass Filter - Quality Factor): L = 100 mH, C = 1 µF, R = 100 Ω (series) f₀ = 1/(2π√(0.1 × 10⁻⁶)) = 503.3 Hz Z₀ = √(0.1/10⁻⁶) = 316.2 Ω Q = 316.2/100 = 3.16 BW = 503.3/3.16 = 159.2 Hz ζ = 1/(2×3.16) = 0.158 (underdamped) Example 3 (Design Filter for Specific Bandwidth): Needed: f₀ = 1 kHz, BW = 100 Hz Choose: L = 10 mH Q = f₀/BW = 1000/100 = 10 C = 1/(L×(2πf₀)²) C = 1/(0.01×(2π×1000)²) = 2.53 µF R = (1/Q)√(L/C) = (1/10)√(10m/2.53µ) R = 0.1 × 62.9 = 6.29 Ω Example 4 (LC Tank Oscillator): L = 470 µH, C = 100 pF f₀ = 1/(2π√(470×10⁻⁶ × 100×10⁻¹²)) f₀ = 732.4 kHz (AM radio band) For Q = 50: R = √(L/C)/Q Z₀ = √(470µ/100p) = 2168 Ω R = 2168/50 = 43.4 Ω (series) Example 5 (Audio Crossover): Tweeter crossover at 3 kHz, L = 1 mH C = 1/(L×(2πf₀)²) C = 1/(0.001×(2π×3000)²) = 2.81 µF For Q = 0.707 (Butterworth): R = √(L/C)/Q = √(1m/2.81µ)/0.707 R = 18.9/0.707 = 26.7 Ω Example 6 (Parallel RLC Tank): L = 100 µH, C = 1 nF, R = 10 kΩ (parallel) f₀ = 1/(2π√(100µ×1n)) = 1.59 MHz Q = R√(C/L) = 10k√(1n/100µ) Q = 10k × 0.0001 = 1 (low Q) BW = 1/(2πRC) = 1/(2π×10k×1n) BW = 15.9 kHz (very broad)

What is an RLC circuit?

An RLC circuit contains a resistor (R), inductor (L), and capacitor (C) in series or parallel. These circuits exhibit resonance at a specific frequency where inductive and capacitive reactances cancel, allowing maximum current flow (series) or minimum current (parallel).

What is the resonance frequency formula?

The resonance frequency is f₀ = 1/(2π√LC), where L is inductance in henries and C is capacitance in farads. At this frequency, XL = XC, so the circuit behaves purely resistively. For L=100mH, C=100µF: f₀ = 159 Hz.

What is the quality factor Q?

Quality factor Q measures how "sharp" the resonance is. Q = f₀/BW = (1/R)√(L/C) for series RLC. High Q (>10) means narrow bandwidth and sharp resonance; low Q (<1) means broad, damped response. Q also equals energy stored / energy dissipated per cycle.

What is bandwidth in RLC circuits?

Bandwidth (BW) is the frequency range where power is at least 50% of maximum (the -3dB points). BW = f₀/Q = R/(2πL) for series circuits. For f₀=1kHz, Q=10: BW = 100 Hz, meaning the circuit responds strongly from ~950Hz to ~1050Hz.

What are the three damping conditions?

Underdamped (ζ<1, Q>0.5): oscillates before settling. Critically damped (ζ=1, Q=0.5): fastest settling without overshoot. Overdamped (ζ>1, Q<0.5): slow settling without oscillation. Damping ratio ζ = R/(2)√(C/L) for series RLC.

What is the difference between series and parallel RLC?

Series RLC: maximum current at resonance, impedance minimum (Z=R), used in filters and tuners. Parallel RLC: minimum current at resonance, impedance maximum, used in oscillators and tank circuits. Formulas differ but f₀ = 1/(2π√LC) is same for both.

What are practical applications of RLC circuits?

Radio tuning (selecting stations), filters (audio equalizers, crossovers), impedance matching, oscillators (signal generators), power factor correction, induction heating, wireless charging, metal detectors, and LC tanks in RF transmitters/receivers.

How do I design a bandpass filter using RLC?

Choose center frequency f₀, then pick L or C (typically 1µH-1mH for L). Calculate C from f₀ = 1/(2π√LC). Determine Q from desired bandwidth: Q = f₀/BW. Finally, calculate R = (1/Q)√(L/C). For f₀=10kHz, BW=1kHz, L=1mH: C=253nF, Q=10, R=20Ω.

What is impedance at resonance?

At resonance (f = f₀), XL = XC so they cancel. Series RLC: Z = R (minimum). Parallel RLC: Z = L/(RC) (maximum). Away from resonance, impedance increases for series, decreases for parallel. This selectivity makes RLC circuits useful for filtering.

How does component tolerance affect resonance?

Resonance frequency depends on √LC. If L and C each have ±10% tolerance, f₀ can vary ±14%. Use tight tolerance components (±1-5%) for precision filters. Temperature, aging, and parasitics also affect actual resonance frequency in real circuits.

What is the phase angle in RLC circuits?

Phase angle φ between voltage and current: tan(φ) = (XL - XC)/R. At f < f₀: capacitive (current leads). At f = f₀: φ = 0° (in phase). At f > f₀: inductive (current lags). Phase goes from +90° to -90° through resonance.

Can RLC circuits oscillate on their own?

Yes, if energy loss (resistance) is compensated by an active component (transistor, op-amp). LC tank circuits oscillate at f₀ but decay due to R. Add amplification to sustain oscillation (Colpitts, Hartley oscillators). Q determines oscillator stability and phase noise.

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