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
t (s)y(t)203 s12

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 20. At t = 3 s it reads 12. Find the time constant τ.

y(∞) = 20y(3) = 12
τ = s
2.Easy
t (s)y(t)y(∞) = ?

A first-order system G(s) = 2/(0.8s + 1) receives a step input of size 1 at t = 0, starting from rest. Find the final value y(∞) of the output.

G(s) = 2/(0.8s + 1)U = 1
y(∞) =
3.Medium
t (s)y(t)0.5tp​ = ?

A system has the transfer function G(s) = 15.5/(s² + 5s + 31) and receives a unit step input. Find the peak time tp of the step response.

G(s) = 15.5/(s² + 5s + 31)
tp = s
4.Medium
t (s)y(t)1

A system has the transfer function G(s) = 15/(s² + 5s + 15) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 15/(s² + 5s + 15)
σ =
5.Hard
+−5 VR = 41 ΩL = 500 mHC = 47 µFvC​

A series RLC circuit with R = 41 Ω, L = 500 mH and C = 47 µF is connected to a 5 V step at t = 0. The capacitor is uncharged and no current flows before the step. Find the time tp of the first voltage peak, in ms.

R = 41 ΩL = 500 mHC = 47 µFV = 5 V
tp = ms
6.Hard
k = 2000 N/mc = 240 N·s/mm = 20 kgF = 50 Nx(t)

A mass m = 20 kg hangs on a spring k = 2000 N/m with a viscous damper c = 240 N·s/m (m·x″ + c·x′ + k·x = F). At rest in equilibrium, a constant force F = 50 N is suddenly applied. Find the damping ratio ζ.

m = 20 kgk = 2000 N/mc = 240 N·s/mF = 50 N
ζ =

Answer key

  1. 1. τ = 3.274 s
  2. 2. y(∞) = 2
  3. 3. tp = 0.6315 s
  4. 4. σ = −2.5
  5. 5. tp = 15.54 ms
  6. 6. ζ = 0.6

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