Step response worksheet with answers

Free printable Step response worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy

A first-order system G(s) = 5/(10s + 1) receives a step input of size 1 at t = 0, starting from rest. How long does the output take to reach 99 % of its final value? Find t.

G(s) = 5/(10s + 1)U = 199 %

t = s

2.Easy

An uncharged capacitor C = 10 µF is charged from a 10 V step source through a resistor R = 10 kΩ. How long after closing the switch does the capacitor voltage reach 75 % of its final value? Find t in ms.

R = 10 kΩC = 10 µFV = 10 V75 %

t = ms

3.Medium

A system has the transfer function G(s) = 33/(s² + 2s + 33) and receives a unit step input. The poles are s = σ ± jω_d. Find the real part σ.

G(s) = 33/(s² + 2s + 33)

σ =

4.Medium

A system has the transfer function G(s) = 297/(s² + 8s + 99) and receives a unit step input. Find the natural frequency ω_n in rad/s.

G(s) = 297/(s² + 8s + 99)

ω_n = rad/s

5.Hard

A mass m = 10 kg hangs on a spring k = 5000 N/m with a viscous damper c = 45 N·s/m (m·x″ + c·x′ + k·x = F). At rest in equilibrium, a constant force F = 100 N is suddenly applied. Find the largest displacement x_max in mm.

m = 10 kgk = 5000 N/mc = 45 N·s/mF = 100 N

x_max = mm

6.Hard

A unity negative feedback loop has G(s) = 225/(s(s + 10)). Find the 2 % settling time t_s of the closed-loop step response, using t_s ≈ 4/(ζωₙ).

G(s) = 225/(s(s + 10))

t_s = s

Answer key

1. t = 46.05 s
2. t = 138.6 ms
3. σ = −1
4. ω_n = 9.95 rad/s
5. x_max = 34.56 mm
6. t_s = 0.8 s

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