Overshoot Calculator Formula
Understand the math behind the overshoot calculator. Each variable explained with a worked example.
Formulas Used
Percent Overshoot
overshoot_pct = 100 * exp(-pi * zeta / sqrt(1 - pow(zeta, 2)))Peak Time
peak_time = pi / wdDamped Frequency
damped_freq = wdVariables
| Variable | Description | Default |
|---|---|---|
zeta | Damping Ratio (zeta) | 0.3 |
wn | Natural Frequency (omega_n)(rad/s) | 10 |
wd | Derived value= wn * sqrt(1 - pow(zeta, 2)) | calculated |
How It Works
Second-Order Overshoot
Overshoot is the amount by which the step response exceeds the final steady-state value, expressed as a percentage. It depends only on the damping ratio.
Formula
%OS = 100 × exp(-pi × zeta / sqrt(1 - zeta²))
Peak time: tp = pi / omega_d, where omega_d = omega_n × sqrt(1 - zeta²) is the damped natural frequency.
For zeta = 0 (undamped), overshoot is 100%. For zeta >= 1 (critically damped or overdamped), there is no overshoot.
Worked Example
A system with zeta = 0.3, omega_n = 10 rad/s.
- 01omega_d = 10 × sqrt(1 - 0.09) = 10 × 0.9539 = 9.539 rad/s
- 02%OS = 100 × exp(-pi × 0.3 / 0.9539)
- 03%OS = 100 × exp(-0.9874) = 100 × 0.3730 = 37.30%
- 04Peak time = pi / 9.539 = 0.3293 s
Frequently Asked Questions
What damping ratio gives 5% overshoot?
Setting %OS = 5 and solving: zeta = -ln(0.05) / sqrt(pi² + ln²(0.05)) = 2.996 / sqrt(9.870 + 8.976) = 2.996 / 4.340 = 0.690. So zeta ≈ 0.69 gives about 5% overshoot.
Can overshoot be eliminated?
Yes, by making zeta >= 1 (critically or over-damped). However, this makes the system slower. In practice, designers often accept 5-10% overshoot for faster response (zeta ≈ 0.6-0.8).
Is overshoot always bad?
Not always. Some systems (like position servos) may tolerate small overshoot if it means faster response. But for processes like temperature control or chemical dosing, any overshoot can be harmful or dangerous.
Ready to run the numbers?
Open Overshoot Calculator