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)y(∞) = ?

A first-order system G(s) = 15/(s + 5) receives a step input of size 2 at t = 0, starting from rest. Find the final value y(∞) of the output.

G(s) = 15/(s + 5)U = 2
y(∞) =
2.Easy
+−9 VR = 100 kΩC = 100 µFvC​

An uncharged capacitor C = 100 µF is charged from a 9 V step source through a resistor R = 100 kΩ. How long after closing the switch does the capacitor voltage reach 90 % of its final value? Find t in ms.

R = 100 kΩC = 100 µFV = 9 V90 %
t = ms
3.Medium
t (s)y(t)1

A system has the transfer function G(s) = 76/(s² + 9s + 76) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 76/(s² + 9s + 76)
σ =
4.Medium
t (s)y(t)2tp​ = ?

A system has the transfer function G(s) = 100/(s² + 11s + 50) and receives a unit step input. Find the peak time tp of the step response.

G(s) = 100/(s² + 11s + 50)
tp = s
5.Hard
k = 500 N/mc = 9 N·s/mm = 4 kgF = 50 Nx(t)

A mass m = 4 kg hangs on a spring k = 500 N/m with a viscous damper c = 9 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 = 4 kgk = 500 N/mc = 9 N·s/mF = 50 N
ζ =
6.Hard
R(s)+Ks(s + 12)G(s)C(s)−

A unity negative feedback loop has G(s) = K/(s(s + 12)). Find the gain K that gives a step response with 10 % overshoot.

G(s) = K/(s(s + 12))%OS = 10 %
K =

Answer key

  1. 1. y(∞) = 6
  2. 2. t = 23026 ms
  3. 3. σ = −4.5
  4. 4. tp = 0.7069 s
  5. 5. ζ = 0.1006
  6. 6. K = 103

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