What settling time criterion does this use?

The 2% criterion — the time for the step response to remain within ±2% of the final value. This yields Ts ≈ 4/(ζ·ωn) for underdamped systems, the most common formula in undergraduate control texts.

Does it work for overdamped systems?

The exact formulas for rise time and peak time assume an underdamped system (0 < ζ < 1). For ζ ≥ 1 the tool switches to the exact pole-based expressions and notes that peak time is undefined (no overshoot).

What is the damped natural frequency?

ωd = ωn · √(1 − ζ²). It is the actual oscillation frequency of the underdamped step response and is always less than ωn.

Why does peak time become undefined for ζ ≥ 1?

Peak time (Tp = π / ωd) only exists when the system oscillates. Critically damped and overdamped systems approach steady state monotonically with no overshoot, so there is no first peak.

Can I use Hz instead of rad/s?

Enter ωn in rad/s. If you know frequency in Hz, multiply by 2π first (e.g., 10 Hz → 62.83 rad/s). The results will then be in seconds.

Control System Settling Time Calculator

Name: Control System Settling Time Calculator
Availability: InStock
Author: Nham Vu

Enter damping ratio and natural frequency to instantly compute settling time, rise time, and peak time for a second-order control system.

System Parameters

Enter damping ratio and natural frequency to compute the time-domain response.

Damping Ratio (ζ)

Underdamped: 0 < ζ < 1 · Critically damped: ζ = 1 · Overdamped: ζ > 1

0.01 (underdamped) 3.00 (overdamped)

Natural Frequency (ω_n) — rad/s

Undamped natural frequency in radians per second.

Quick presets

System Type

—

Settling Time (Ts)

—

2% criterion

Rise Time (Tr)

—

10% → 90%

Peak Time (Tp)

—

Time to first peak

Peak Overshoot

—

% above final value

Frequency Details

Natural Frequency (ω_n) —

Damped Frequency (ω_d) —

Bandwidth (approx.) —

Characteristic Poles —

Step Response Preview

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Summary

Enter damping ratio and natural frequency to instantly compute settling time, rise time, and peak time for a second-order control system.

How it works

Enter the damping ratio (ζ) — values between 0 and 1 give underdamped response; exactly 1 is critically damped; above 1 is overdamped.
Enter the undamped natural frequency (ωn) in radians per second.
The calculator computes the damped natural frequency ωd = ωn√(1 − ζ²) for underdamped systems.
Settling time is estimated as Ts ≈ 4/(ζ·ωn) using the 2% criterion (standard for most control textbooks).
Rise time and peak time are computed from the damped frequency and phase angle.
Percent overshoot is derived directly from the damping ratio: %OS = e^(−πζ/√(1−ζ²)) × 100.

Use cases

Verify PID controller tuning against settling time specifications.
Quickly check whether a chosen damping ratio meets a transient response requirement.
Explore the trade-off between speed (low ζ) and overshoot during design.
Validate hand calculations when studying for controls exams.
Size feedback gains for electromechanical servo systems.
Compare critically damped and underdamped responses side by side.

Frequently Asked Questions

Last updated: 2026-07-22 · Reviewed by Nham Vu