This calculator converts percent overshoot into damping ratio for a standard second-order response. It also estimates peak time, rise time, settling time, peak value, and plots the step response.
Calculator Input
Formula Used
Percent Overshoot:
PO(%) = 100 × exp[ -(ζπ) / √(1 - ζ²) ]
Overshoot Ratio:
Mp = PO / 100
Inverse Formula for Damping Ratio:
ζ = -ln(Mp) / √(π² + [ln(Mp)]²)
Damped Natural Frequency:
ωd = ωn × √(1 - ζ²)
Peak Time:
tp = π / ωd
Rise Time Approximation:
tr = (π - arccos(ζ)) / ωd
Settling Time Approximation:
Ts = -ln(δ × √(1 - ζ²)) / (ζωn)
where δ = 0.02 for 2% band
and δ = 0.05 for 5% band
Valid for standard second-order underdamped step-response overshoot relationships.
How to Use This Calculator
- Enter the measured percent overshoot from your response curve.
- Enter the natural frequency to unlock time metrics and graphing.
- Set the final value for the response plot scale.
- Select either a 2% or 5% settling band.
- Leave graph time blank for automatic sizing, or enter your own.
- Choose the graph sample count and preferred decimal precision.
- Press Calculate Now to view the result above the form.
- Use the export buttons to save the report as CSV or PDF.
Example Data Table
| Percent Overshoot (%) | Overshoot Ratio (Mp) | Approx. Damping Ratio (ζ) | Response Character |
|---|---|---|---|
| 5 | 0.05 | 0.6901 | Light overshoot, well damped |
| 10 | 0.10 | 0.5912 | Moderately damped |
| 20 | 0.20 | 0.4559 | Typical underdamped |
| 30 | 0.30 | 0.3579 | More oscillatory response |
| 40 | 0.40 | 0.2800 | Low damping, strong overshoot |
| 50 | 0.50 | 0.2155 | Very oscillatory response |
FAQs
1) What does percent overshoot mean?
Percent overshoot measures how far a response rises above its final steady value after a step input. It is commonly used to describe how oscillatory and aggressive a second-order system behaves.
2) Can I use 0% overshoot in this calculator?
Yes. A 0% overshoot case is treated as the critical damping limit in this tool, giving a damping ratio of 1. That represents a non-oscillatory boundary response.
3) Is this formula valid for every physical system?
No. It is intended for standard second-order step responses where percent overshoot follows the classical underdamped relationship. Higher-order, nonlinear, delayed, or strongly coupled systems may not match this model well.
4) Why do I need to enter natural frequency?
Natural frequency is not needed to compute damping ratio from overshoot alone. It is included so the calculator can estimate time-domain quantities and draw a meaningful response graph.
5) What happens when overshoot is 100%?
An overshoot of 100% corresponds to a damping ratio near zero. That means the system is essentially undamped and can oscillate strongly around the final value.
6) Can I export the calculated results?
Yes. After calculation, you can download a results CSV, a graph-data CSV, and a PDF report directly from the buttons shown in the result section.
7) Which settling-time equation is used here?
The calculator uses an envelope-based underdamped settling-time approximation with either a 2% or 5% tolerance band. It is a practical engineering estimate rather than an exact all-system solution.
8) Which units should I use?
Enter overshoot as a percentage, natural frequency in radians per second, and time in seconds. The final value can be any positive output scale used by your response plot.