Overshoot Calculator Formula

Understand the math behind the overshoot calculator. Each variable explained with a worked example.

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Formulas Used

Percent Overshoot

overshoot_pct = 100 * exp(-pi * zeta / sqrt(1 - pow(zeta, 2)))

Peak Time

peak_time = pi / wd

Damped Frequency

damped_freq = wd

Variables

VariableDescriptionDefault
zetaDamping Ratio (zeta)0.3
wnNatural Frequency (omega_n)(rad/s)10
wdDerived 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.

zeta = 0.3wn = 10
  1. 01omega_d = 10 × sqrt(1 - 0.09) = 10 × 0.9539 = 9.539 rad/s
  2. 02%OS = 100 × exp(-pi × 0.3 / 0.9539)
  3. 03%OS = 100 × exp(-0.9874) = 100 × 0.3730 = 37.30%
  4. 04Peak time = pi / 9.539 = 0.3293 s

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