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)105 s6

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 10. At t = 5 s it reads 6. Find the time constant τ.

y(∞) = 10y(5) = 6
τ = s
2.Easy
+−24 VR = 22 kΩC = 47 µFvC​

An uncharged capacitor C = 47 µF is charged from a 24 V step source through a resistor R = 22 kΩ. Find the 10–90 % rise time tr of the capacitor voltage, in ms.

R = 22 kΩC = 47 µFV = 24 V
tr = ms
3.Medium
t (s)y(t)2tp​ = ?

A system has the transfer function G(s) = 156/(s² + 9s + 78) and receives a unit step input. Find the peak time tp of the step response.

G(s) = 156/(s² + 9s + 78)
tp = s
4.Medium
t (s)y(t)1ts​ = ?

A system has the transfer function G(s) = 60/(s² + 6s + 60) and receives a unit step input. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 60/(s² + 6s + 60)
ts = s
5.Hard
R(s)+245s(s + 6)G(s)C(s)−

A unity negative feedback loop has G(s) = 245/(s(s + 6)). Find the 2 % settling time ts of the closed-loop step response, using ts ≈ 4/(ζωₙ).

G(s) = 245/(s(s + 6))
ts = s
6.Hard
t (s)y(t)12 %

A second-order system without zeros must have a step response with 2 % overshoot. Find the damping ratio ζ.

%OS = 2 %
ζ =

Answer key

  1. 1. τ = 5.457 s
  2. 2. tr = 2272 ms
  3. 3. tp = 0.4134 s
  4. 4. ts = 1.333 s
  5. 5. ts = 1.333 s
  6. 6. ζ = 0.7797

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