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| Percent Overshoot (%) | Overshoot Ratio (Mp) | Damping Ratio (ζ) | Notes |
|---|---|---|---|
| 5 | 0.05 | 0.6901 | Low overshoot, strong damping. |
| 10 | 0.10 | 0.5912 | Common design target in many loops. |
| 20 | 0.20 | 0.4559 | Faster response, more ringing. |
| 30 | 0.30 | 0.3579 | Noticeable oscillation and peak behavior. |
| 50 | 0.50 | 0.2155 | High overshoot, lightly damped system. |
Values are rounded; your results depend on the chosen precision.
For a standard underdamped second‑order step response, the maximum overshoot ratio is:
If you enter percent overshoot PO, then Mp = PO/100. Solving the overshoot equation for ζ gives:
The calculator also reports ln(Mp) and the quality factor Q = 1/(2ζ) as helpful design indicators.
This tool converts step-response overshoot into damping ratio ζ for a classic second-order underdamped system. It accepts percent overshoot, overshoot ratio Mp, or peak and final values. In many control loops, ζ is used to balance speed and ringing.
Percent overshoot (PO) measures how far the first peak rises above the steady value. For a unit step reaching 1.00, a peak of 1.10 means 10% overshoot. Larger PO usually implies smaller ζ and more oscillation.
The calculator first turns PO into Mp using Mp = PO/100. If you provide peak and final, it computes Mp = (ypeak − yfinal)/|yfinal|. Example: ypeak 1.24 and yfinal 1.00 gives Mp = 0.24 and PO = 24%.
Typical mappings are useful checks: PO 5% → ζ ≈ 0.69, PO 10% → ζ ≈ 0.59, PO 20% → ζ ≈ 0.46, PO 30% → ζ ≈ 0.36, PO 50% → ζ ≈ 0.22. Values near ζ = 0.70 feel “well damped,” while ζ below 0.30 rings strongly.
The quality factor Q is reported as Q = 1/(2ζ). With ζ = 0.59, Q ≈ 0.85. With ζ = 0.22, Q ≈ 2.27, which often corresponds to narrow-band resonance and a sharp peak response.
Use the first maximum after the step as ypeak. Estimate yfinal from the settled region, not the last sample. Noise can inflate peaks; averaging or light filtering helps. Ensure the final value is nonzero so Mp remains meaningful.
Once ζ is known, you can pair it with natural frequency ωn to estimate transient metrics. A common approximation is 2% settling time ts ≈ 4/(ζωn). If your design targets PO under 10%, aim for ζ around 0.6 and then adjust ωn for the desired speed. For example, ζ=0.7 corresponds to about 4.6% overshoot, while ζ=0.5 corresponds to about 16.3%. If your trace shows multiple modes, higher‑order dynamics dominate and ζ becomes a single-number descriptor. Use consistent step size and sampling, and record temperature and load so comparisons stay fair across runs too.
Mp is the fractional overshoot above the final value. Mp=0 means no overshoot, and Mp≥1 means the peak is at least double the final value. The standard second‑order overshoot model assumes 0 Enter a percent between 0 and 100, exclusive. Values near 0% imply ζ close to 1, while larger overshoot implies smaller ζ. For very high overshoot, your response may be dominated by higher‑order dynamics. Use the first maximum after the step as ypeak. Estimate yfinal from the settled region, not from a single noisy sample. If the signal oscillates, average a short window where it has clearly stabilized. Not directly. Those responses have essentially no overshoot, so the overshoot-based formula is not informative. If PO is near zero, treat ζ as “high,” and use rise time and settling behavior instead. No. Settling time depends on both ζ and the natural frequency ωn. A common estimate is ts≈4/(ζωn) for a 2% criterion, but you still need ωn from your model or data. The peak‑final method uses Mp=(ypeak−yfinal)/|yfinal|, so sign is handled by the absolute final value. Make sure ypeak is the first overshoot peak relative to that same steady value.What percent overshoot range should I enter?
How do I measure peak and final values?
Can this be used for overdamped or critically damped systems?
Does damping ratio alone tell me the settling time?
What if my final value is negative?
Use overshoot measurements to estimate damping ratio ζ for a classic second‑order step response. This helps compare tunings, validate test runs, and communicate stability tradeoffs clearly. It is popular in servo, process, and vibration tuning workflows today too.
Percent overshoot is (ypeak − yfinal) / |yfinal| × 100%. The overshoot ratio is Mp = PO/100. The model assumes the first peak is the maximum and the final value represents steady state. For a unit step, yfinal is typically 1.0.
For an underdamped second‑order response, Mp = exp(−ζπ/√(1−ζ²)). Solving gives ζ = −ln(Mp)/√(π² + ln(Mp)²). Because ln(Mp) is negative, ζ becomes positive when 0<Mp<1. The calculator applies the same equation whether you enter PO, Mp, or peak and final values.
Small overshoot means stronger damping. Example values: PO 5% → ζ≈0.690, 10% → ζ≈0.591, 20% → ζ≈0.456, 30% → ζ≈0.358, and 50% → ζ≈0.216. These numbers are useful for quick sanity checks. Plot ζ versus PO to see how small overshoot changes move damping.
Many control designs target ζ from 0.5 to 0.8 to limit overshoot while staying responsive. Lower ζ increases ringing and resonance risk, while higher ζ reduces overshoot but can slow rise time when natural frequency stays fixed.
Measure yfinal from the settled portion of the signal, not immediately after the step. Use consistent step size, sampling rate, and filtering. If noise creates false peaks, smooth lightly or detect the first true maximum. For mechanical systems, repeat three runs and average PO to reduce random variation.
The calculator also outputs ln(Mp) and quality factor Q = 1/(2ζ). Higher Q indicates weaker damping and a sharper resonance. For example, ζ=0.5 gives Q=1.0, while ζ=0.25 gives Q=2.0.
If your system has strong delay, saturation, multiple modes, or nonlinear behavior, ζ from overshoot is an effective estimate only. Use it for comparisons, then confirm with frequency or time‑domain identification when accuracy matters. Large transport delay can create overshoot that is not explained by a single ζ, so treat the output as a diagnostic starting point.
It works best when one underdamped second‑order mode dominates. Higher‑order systems can be approximated, but ζ becomes an effective value rather than a full model parameter.
Zero overshoot implies Mp≈0, which is outside the logarithm range. It often means high damping, overdamped behavior, or nonlinear limiting. Use other identification approaches instead.
Mp<1 keeps ln(Mp) negative and produces a positive ζ. Mp≥1 usually indicates a wrong final value, unstable behavior, or a measurement window that missed the true steady state.
Average a short window after the signal settles. Avoid early samples during drift or oscillation. A stable yfinal makes Mp and ζ far more repeatable across runs.
No. Settling time also depends on natural frequency. ζ describes decay relative to oscillation, but you need frequency information to convert that behavior into seconds.
Use CSV to compare many tests and plot trends. Use PDF for a fixed report snapshot showing inputs, computed ζ, and notes that support traceability.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.