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)50.4 s?

A first-order system G(s) = 0.25/(s + 0.5) receives a step input of size 10 at t = 0, starting from rest. Find the output y(0.4) at t = 0.4 s.

G(s) = 0.25/(s + 0.5)U = 10t = 0.4 s
y(0.4) =
2.Easy
+−5 VR = 1 kΩC = 1 µFvC​

An uncharged capacitor C = 1 µF is charged from a 5 V step source through a resistor R = 1 kΩ. How long after closing the switch does the capacitor voltage reach 60 % of its final value? Find t in ms.

R = 1 kΩC = 1 µFV = 5 V60 %
t = ms
3.Medium
t (s)y(t)1

A system has the transfer function G(s) = 91/(s² + 8s + 91) and receives a unit step input. Find the natural frequency ωn in rad/s.

G(s) = 91/(s² + 8s + 91)
ωn = rad/s
4.Medium
t (s)y(t)3?

A system has the transfer function G(s) = 12/(s² + s + 4) and receives a unit step input. Find the percent overshoot %OS of the step response.

G(s) = 12/(s² + s + 4)
%OS = %
5.Hard
R(s)+Ks(s + 4)G(s)C(s)−

A unity negative feedback loop has G(s) = K/(s(s + 4)). Find the gain K that gives a step response with 25 % overshoot.

G(s) = K/(s(s + 4))%OS = 25 %
K =
6.Hard
k = 5000 N/mc = 141 N·s/mm = 4 kgF = 50 Nx(t)

A mass m = 4 kg hangs on a spring k = 5000 N/m with a viscous damper c = 141 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 time tp at which the displacement is largest.

m = 4 kgk = 5000 N/mc = 141 N·s/mF = 50 N
tp = s

Answer key

  1. 1. y(0.4) = 0.9063
  2. 2. t = 0.9163 ms
  3. 3. ωn = 9.539 rad/s
  4. 4. %OS = 44.43 %
  5. 5. K = 24.54
  6. 6. tp = 0.1025 s

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