Damping Ratio Calculator
Damping ratio of an RLC circuit, the electrical analog of a mass–spring–damper: ζ = (R/2)·√(C/L).
Inputs
Drag the handle (or focus it and press ↑/↓) to reorder. Scientific notation such as 2.2e-6 is accepted. Pick any unit; conversion is automatic.
Results
Enter values and press Calculate. Your answer will appear here.
Step response of the circuit
Normalised response to a sudden step input. Below ζ = 1 the output overshoots and rings; at ζ = 1 it rises as fast as possible without overshoot; above 1 it creeps up slowly. The amber dot marks the first peak. The dashed curve is critical damping.
What is being calculated
Common examples
| Case | ζ | Response |
|---|
Illustrative inputs, not manufacturer data.
Effect of R for L = 10 mH, C = 1 µF
| R | ζ | Response | Overshoot |
|---|
Damping classes
| ζ | Behaviour |
|---|---|
| ζ = 0 | Undamped: oscillates forever |
| 0 < ζ < 1 | Underdamped: overshoots and rings, then settles |
| ζ = 1 | Critically damped: fastest rise with no overshoot |
| ζ > 1 | Overdamped: no overshoot, slower response |
Unit conversions
| Quantity | Conversion |
|---|---|
| Resistance | 1 kΩ = 1,000 Ω · 1 MΩ = 10⁶ Ω · 1 mΩ = 10⁻³ Ω |
| Inductance | 1 mH = 10⁻³ H · 1 µH = 10⁻⁶ H · 1 nH = 10⁻⁹ H |
| Capacitance | 1 mF = 10⁻³ F · 1 µF = 10⁻⁶ F · 1 nF = 10⁻⁹ F · 1 pF = 10⁻¹² F |
| Frequency | f = ω / 2π · 1 kHz = 1,000 Hz |
Formula
- ζ: damping ratio (dimensionless)
- R: series resistance (Ω)
- L: inductance (H)
- C: capacitance (F)
Example: R = 100 Ω, L = 10 mH, C = 1 µF. √(C/L) = √(10⁻⁶ ÷ 10⁻²) = 0.01, so ζ = (100 ÷ 2) × 0.01 = 0.5. This is underdamped, with ω0 = 1/√(LC) = 10,000 rad/s (f0 ≈ 1.59 kHz) and an overshoot of e−πζ/√(1−ζ²) ≈ 16.3 %.
Equivalent forms: ζ = R / (2√(L/C)) = R / Rcrit, where Rcrit = 2√(L/C) gives ζ = 1, and Q = 1/(2ζ). For a parallel RLC circuit (an Advanced option) the formula is ζ = (1/2R)·√(L/C). The mechanical analog is ζ = c / (2√(k·m)), with R ↔ damping coefficient c, L ↔ mass m and 1/C ↔ spring constant k. The overshoot and settling-time figures assume the standard second-order step response and the 2 % settling time is approximate.
References
- Wikipedia, Damping ratio: definition, overshoot and settling time
- Wikipedia, RLC circuit: series and parallel damping factors
- OpenStax University Physics Vol. 2, RLC Series Circuits: underdamped, critically damped and overdamped cases
How to Use This Tool
- Pick a preset, or enter the resistance R and its unit.
- Enter the inductance L and capacitance C with their units.
- In Advanced, switch between a series and a parallel RLC circuit.
- Click Calculate.
- Read ζ and the response type in the highlighted result, then check the step-response chart.
What This Tool Offers
- Basic and Advanced modes on one calculation engine
- Resistance, inductance and capacitance unit conversion
- Underdamped, critically damped and overdamped classification
- Natural frequency, Q, critical resistance, overshoot and settling time (Advanced)
- Step-response chart and labelled series RLC diagram
- Reorderable inputs, presets and a mobile-friendly layout
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| ζ | Damping ratio | dimensionless |
| R | Resistance | Ω |
| C | Capacitance | F |
| L | Inductance | H |
What it means
The damping ratio ζ for a series RLC circuit is given by ζ = R/2 · √(C/L). It determines the transient response: ζ < 1 yields an underdamped response (oscillatory), ζ = 1 is critically damped (fastest without oscillation), and ζ > 1 is overdamped (slow, no oscillation). Damping ratio is a key parameter in control systems and filter design. It affects overshoot, settling time, and peak time. Example: For R=10Ω, L=0.1H, C=100µF, ζ = (10/2) * √(100e-6/0.1) = 5 * √0.001 = 5 * 0.03162 = 0.158. This is underdamped, so the circuit will ring when excited by a step input. To achieve critical damping (ζ=1), the resistance would need to be about 2·√(L/C) = 2·√(0.1/100e-6) = 2·√1000 ≈ 63.2Ω.
Worked example
Damping Ratio – Practical Example
Real‑World| Parameter | Value |
|---|---|
| R | 100 Ω |
| L | 10 mH = 0.01 H |
| C | 1 µF = 1×10⁻⁶ F |
| Formula | ζ = (R/2)·√(C/L) |
Common mistakes
Watch unit consistency and the assumptions behind the formula; misapplying it outside its valid conditions is the most frequent error.Applications
Damping ratio (electrical analogue) ζ = R/2·√(C/L) for a series RLC circuit determines the transient response: underdamped (ringing), critically damped, or overdamped. Engineers use it to design circuits with desired transient behaviour, to avoid excessive overshoot, and to ensure fast settling. This parameter is fundamental to circuit and system dynamics.
- Filter and resonance circuit design
- Control system transient response shaping
- Oscillator and timing circuit design
- Power supply transient analysis
- Educational understanding of damping
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