RLC Circuit Calculator | Resonance, Impedance & Q Factor
Analyze series and parallel RLC circuits. Computes resonant frequency, impedance at any input frequency, damping ratio, Q factor, bandwidth, and phase angle. Classifies the circuit as overdamped, critically damped, or underdamped and shows the time-domain step response type.
What Is the RLC Circuit Calculator | Resonance, Impedance & Q Factor?
An RLC circuit contains a resistor R, inductor L, and capacitor C. At resonant frequency f₀ the reactive impedances cancel: for a series circuit the total impedance is minimized to R; for parallel it is maximized. The Q factor measures selectivity — higher Q means sharper resonance and narrower bandwidth. The damping ratio ζ classifies the transient response as underdamped, critically damped, or overdamped.
Formula
f₀ = 1/(2π√(LC)) · ζ = (R/2)√(C/L) (series) · Q = (1/R)√(L/C) · |Z| = √(R² + (ωL − 1/ωC)²)
How to Use
- 1
Select Series RLC or Parallel RLC from the toggle at the top.
- 2
Enter the resistance R in ohms (can be 0 for ideal LC circuit).
- 3
Enter the inductance L and choose the unit: H, mH, or μH.
- 4
Enter the capacitance C and choose the unit: F, mF, μF, nF, or pF.
- 5
Enter the operating frequency f in Hz to evaluate impedance at that frequency.
- 6
Click "Analyze Circuit" — result cards show f₀, ω₀, |Z|, phase φ, ζ, Q, and bandwidth.
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Read the damping classification and review the step-by-step working.
Select Series or Parallel circuit topology, then enter R (Ω), L with its unit (H/mH/μH), C with its unit (F/μF/nF/pF), and an operating frequency f. Click Analyze Circuit to compute all parameters.
Example Calculation
Series RLC: R = 10 Ω, L = 10 mH, C = 10 μF. ω₀ = 1/√(0.01 × 10⁻⁵) = 3162 rad/s, f₀ ≈ 503 Hz. Q = (1/10)√(0.01/10⁻⁵) = 10. Bandwidth = 503/10 ≈ 50 Hz. ζ = (10/2)√(10⁻⁵/0.01) = 0.05 → underdamped. At f = 503 Hz: |Z| ≈ R = 10 Ω, phase ≈ 0°.
Understanding RLC Circuit | Resonance, Impedance & Q Factor
Series vs Parallel RLC: key formulas
| Parameter | Series RLC | Parallel RLC |
|---|---|---|
| Resonant freq f₀ | 1 / (2π√(LC)) | 1 / (2π√(LC)) |
| Angular freq ω₀ | 1 / √(LC) | 1 / √(LC) |
| Impedance |Z| at f₀ | Z = R (minimum) | Z = R (maximum) |
| Q factor | Q = (1/R)√(L/C) = ω₀L/R | Q = R√(C/L) = R/(ω₀L) |
| Bandwidth BW | BW = R/(2πL) = f₀/Q | BW = 1/(2πRC) = f₀/Q |
| Damping ratio ζ | ζ = (R/2)√(C/L) | ζ = (1/2R)√(L/C) |
| Phase angle | φ = arctan((ωL−1/ωC)/R) | φ = −arctan(ωC·R − R/ωL) |
| At resonance | I maximized, V across L and C cancel | V maximized, I through L and C cancel |
Damping classification by ζ
| Condition | Classification | Step response behaviour | Typical use |
|---|---|---|---|
| ζ = 0 | Undamped | Sustained oscillations (no R) | Ideal LC oscillator |
| 0 < ζ < 1 | Underdamped | Decaying oscillations, overshoot | Resonant filters, radio tuners |
| ζ = 1 | Critically damped | Fastest non-oscillatory return | Servo systems, shock absorbers |
| ζ > 1 | Overdamped | Slow exponential decay, no overshoot | Power supply filters, stability-critical |
Practical applications of RLC circuits
- ›Radio / TV tuners: Parallel RLC circuits select a station frequency. Adjusting C shifts f₀; high Q gives narrow bandwidth for channel selectivity.
- ›Audio crossover networks: Series and parallel RLC filters split audio into frequency bands for woofers, midranges, and tweeters.
- ›Power factor correction: Capacitors added in parallel to inductive loads form RLC circuits that shift the phase angle toward zero, improving efficiency.
- ›Oscillators: LC tank circuits (ζ ≈ 0) sustain oscillations; used in clock generators, transmitters, and crystal oscillators.
- ›EMI filters: Low-pass RLC filters suppress high-frequency noise on power lines in switched-mode power supplies.
- ›Impedance matching: RLC networks transform source impedance to maximize power transfer, essential in RF amplifier design.
Frequently Asked Questions
What is resonant frequency and why does it matter?
Resonant frequency f₀ = 1/(2π√(LC)) is where the inductive and capacitive reactances are equal and opposite (X_L = X_C). In a series circuit, total impedance is minimized to R alone, maximizing current. In a parallel circuit, impedance is maximized. At resonance, the circuit can transfer energy most efficiently between the inductor and capacitor.
What does Q factor mean in practice?
Q (quality factor) measures how underdamped a resonant circuit is. Q = f₀/BW, so higher Q gives narrower bandwidth and sharper frequency selectivity. A radio tuner needs Q > 100 to separate adjacent stations. A power filter may use Q < 1 to avoid peaking. Q also represents the ratio of energy stored to energy dissipated per cycle.
What is the difference between series and parallel RLC impedance?
In a series circuit, R, L, and C are in series: Z = R + j(ωL − 1/ωC). The magnitude is minimum at resonance (= R). In a parallel circuit, Y = 1/R + j(ωC − 1/ωL); impedance |Z| = 1/|Y| is maximum at resonance (= R). Series RLC is used for bandpass/notch filters; parallel RLC for tank circuits and parallel resonant filters.
How is damping ratio related to circuit behaviour?
The damping ratio ζ determines the transient step response. ζ < 1 (underdamped): oscillatory decay with overshoot — common in resonant filters. ζ = 1 (critically damped): fastest return without oscillation — used in precision instruments. ζ > 1 (overdamped): slow monotonic decay — used where stability matters more than speed.
How do I use unit selectors for L and C?
The calculator accepts inductance in H (henrys), mH (millihenrys, ×10⁻³), or μH (microhenrys, ×10⁻⁶). Capacitance accepts F, mF (×10⁻³), μF (×10⁻⁶), nF (×10⁻⁹), or pF (×10⁻¹²). For example, entering 100 with unit nF means 100 × 10⁻⁹ = 100 nF = 0.1 μF. The calculator converts all values to SI units before computing.
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