Second-order system explorer
Explore the standard second-order system H(s) = ωₙ² / (s² + 2ζωₙs + ωₙ²). Drag the damping ratio ζ and natural frequency ωₙ and watch the step response, overshoot, peak time, and settling time change in real time.
Damping ratio and natural frequency
The damping ratio ζ controls the shape: ζ < 1 is underdamped and overshoots, ζ = 1 is critically damped, and ζ > 1 is overdamped. The natural frequency ωₙ sets the speed. Percent overshoot depends only on ζ, while the settling time depends on the product ζωₙ.
Frequently asked questions
- What is a second-order system?
- A standard second-order system is H(s) = ωₙ² / (s² + 2ζωₙs + ωₙ²), where ζ is the damping ratio and ωₙ the natural frequency. A great many control, mechanical, and RLC problems reduce to this canonical form.
- How does the damping ratio affect the response?
- ζ = 0 is undamped (pure oscillation); 0 < ζ < 1 is underdamped, with overshoot and ringing; ζ = 1 is critically damped, the fastest response with no overshoot; and ζ > 1 is overdamped, slow with no overshoot.
- How do you calculate overshoot and settling time?
- For an underdamped system, percent overshoot = 100·e^(−ζπ/√(1−ζ²)), peak time = π/(ωₙ√(1−ζ²)), and the 2% settling time ≈ 4/(ζωₙ). This tool computes these from the simulated response and shows the theory values beside them.
- Is the second-order calculator free?
- Yes, it is completely free and runs in your browser.