How to use the Transfer Function Step Response Tool
- 1. Enter G(s)Numerator and denominator as polynomials in s.
- 2. Pick open or closed loopClosed loop uses unity feedback G/(1 + G).
- 3. See the responsePlot, final value, overshoot, rise and settling times.
Second-order step response formulas
A standard second-order system is ωn² / (s² + 2ζωn s + ωn²). For 0 < ζ < 1 (underdamped), with ωd = ωn√(1 − ζ²):
- Peak overshoot Mp = e−ζπ/√(1−ζ²)
- Peak time tp = π / ωd
- Settling time (2%) ts ≈ 4 / (ζωn)
- Final value from the final value theorem: y(∞) = G(0) for a stable system
Worked example: G(s) = 25 / (s² + 6s + 25). ωn = 5 rad/s and ζ = 6/10 = 0.6, so ωd = 4 rad/s. Mp = e−0.6π/0.8 = 9.48%, tp = π/4 = 0.785 s and ts ≈ 4/3 = 1.33 s. The simulated 2% settling time is 1.19 s; the 4/(ζωn) rule is an envelope estimate.
Closed loop and exam questions
For unity feedback, T(s) = G(s) / (1 + G(s)). With G(s) = 4 / (s(s + 2)), T(s) = 4 / (s² + 2s + 4), so ωn = 2 and ζ = 0.5, giving Mp = 16.3% and tp = 1.81 s.
- Find ωn, ζ, Mp, tp and ts of a given second-order system.
- Find K so that the overshoot is 10%.
- Find the steady-state value using the final value theorem.
- Classify the system as underdamped, critically damped or overdamped.
Higher-order systems are simulated directly, so you can see how a third pole or a zero changes the textbook second-order numbers.
Frequently asked questions
Which rise time does the tool show?
The 10% to 90% rise time, which works for every system type. Some books use 0% to 100% for underdamped systems: tr = (π − cos⁻¹ζ)/ωd.
Why does the settling time differ from 4/(ζωn)?
4/(ζωn) uses the exponential envelope, so it is an upper estimate. The tool measures when the simulated response actually stays within 2%.
What if the system is unstable?
The tool still plots the response so you can see it grow or oscillate, and it reports that there is no final value.
Is this tool free?
Yes. You enter your name and WhatsApp number once to see your result, and all other Uncodemy tools then open without the form.