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 s18

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 18. Find the time constant τ.

y(∞) = 20y(3) = 18
τ = s
2.Easy
+−5 VR = 4.7 kΩC = 1 µFvC​

An uncharged capacitor C = 1 µF is charged from a 5 V step source through a resistor R = 4.7 kΩ. Find the capacitor voltage vC at t = 2.35 ms.

R = 4.7 kΩC = 1 µFV = 5 Vt = 2.35 ms
vC = V
3.Medium
t (s)y(t)1

A system has the transfer function G(s) = 60/(s² + 11s + 60) and receives a unit step input. Find the natural frequency ωn in rad/s.

G(s) = 60/(s² + 11s + 60)
ωn = rad/s
4.Medium
t (s)y(t)0.5

A system has the transfer function G(s) = 28.5/(s² + 6s + 57) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 28.5/(s² + 6s + 57)
σ =
5.Hard
k = 200 N/mc = 36 N·s/mm = 10 kgF = 50 Nx(t)

A mass m = 10 kg hangs on a spring k = 200 N/m with a viscous damper c = 36 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 = 10 kgk = 200 N/mc = 36 N·s/mF = 50 N
tp = s
6.Hard
R(s)+Ks(s + 8)G(s)C(s)−

A unity negative feedback loop has G(s) = K/(s(s + 8)). Find the gain K that gives a step response with 2 % overshoot.

G(s) = K/(s(s + 8))%OS = 2 %
K =

Answer key

  1. 1. τ = 1.303 s
  2. 2. vC = 1.967 V
  3. 3. ωn = 7.746 rad/s
  4. 4. σ = −3
  5. 5. tp = 0.7674 s
  6. 6. K = 26.32

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