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)50t = ?80 %

A first-order system G(s) = 5/(0.5s + 1) receives a step input of size 10 at t = 0, starting from rest. How long does the output take to reach 80 % of its final value? Find t.

G(s) = 5/(0.5s + 1)U = 1080 %
t = s
2.Easy
t (s)y(t)101 s7

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 10. At t = 1 s it reads 7. Find the time constant τ.

y(∞) = 10y(1) = 7
τ = s
3.Medium
t (s)y(t)3

A system has the transfer function G(s) = 108/(s² + 8s + 36) and receives a unit step input. Find the damping ratio ζ.

G(s) = 108/(s² + 8s + 36)
ζ =
4.Medium
t (s)y(t)1

A system has the transfer function G(s) = 7/(s² + 2s + 7) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 7/(s² + 2s + 7)
σ =
5.Hard
t (s)y(t)14.3 %

A second-order system without zeros must have a step response with 4.3 % overshoot. Find the damping ratio ζ.

%OS = 4.3 %
ζ =
6.Hard
R(s)+45s(s + 8)G(s)C(s)−

A unity negative feedback loop has G(s) = 45/(s(s + 8)). Find the 2 % settling time ts of the closed-loop step response, using ts ≈ 4/(ζωₙ).

G(s) = 45/(s(s + 8))
ts = s

Answer key

  1. 1. t = 0.8047 s
  2. 2. τ = 0.8306 s
  3. 3. ζ = 0.6667
  4. 4. σ = −1
  5. 5. ζ = 0.7077
  6. 6. ts = 1 s

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