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 step is applied to a first-order process at t = 0. The output starts at 0 and settles at 2. At t = 10 s it reads 0.8. Find the time constant τ.

y(∞) = 2y(10) = 0.8

τ = s

2.Easy

A first-order system G(s) = 8/(s + 4) receives a step input of size 5 at t = 0, starting from rest. How long does the output take to reach 75 % of its final value? Find t.

G(s) = 8/(s + 4)U = 575 %

t = s

3.Medium

A system has the transfer function G(s) = 14.5/(s² + 8s + 29) and receives a unit step input. Find the 2 % settling time t_s of the step response, using t_s ≈ 4/(ζωₙ).

G(s) = 14.5/(s² + 8s + 29)

t_s = s

4.Medium

A system has the transfer function G(s) = 27/(s² + 11s + 54) and receives a unit step input. Find the damping ratio ζ.

G(s) = 27/(s² + 11s + 54)

ζ =

5.Hard

A mass m = 1 kg hangs on a spring k = 400 N/m with a viscous damper c = 8 N·s/m (m·x″ + c·x′ + k·x = F). At rest in equilibrium, a constant force F = 10 N is suddenly applied. Find the time t_p at which the displacement is largest.

m = 1 kgk = 400 N/mc = 8 N·s/mF = 10 N

t_p = s

6.Hard

A series RLC circuit with R = 190 Ω, L = 100 mH and C = 1 µF is connected to a 24 V step at t = 0. The capacitor is uncharged and no current flows before the step. Find the time t_p of the first voltage peak, in ms.

R = 190 ΩL = 100 mHC = 1 µFV = 24 V

t_p = ms

Answer key

1. τ = 19.58 s
2. t = 0.3466 s
3. t_s = 1 s
4. ζ = 0.7485
5. t_p = 0.1603 s
6. t_p = 1.042 ms

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