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)50.5 s2

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 5. At t = 0.5 s it reads 2. Find the time constant τ.

y(∞) = 5y(0.5) = 2
τ = s
2.Easy
+−24 VR = 22 kΩC = 1 µFvC​

An uncharged capacitor C = 1 µF is charged from a 24 V step source through a resistor R = 22 kΩ. How long after closing the switch does the capacitor voltage reach 90 % of its final value? Find t in ms.

R = 22 kΩC = 1 µFV = 24 V90 %
t = ms
3.Medium
t (s)y(t)3ts​ = ?

A system has the transfer function G(s) = 24/(s² + s + 8) and receives a unit step input. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 24/(s² + s + 8)
ts = s
4.Medium
t (s)y(t)1

A system has the transfer function G(s) = 42/(s² + 4s + 42) and receives a unit step input. Find the natural frequency ωn in rad/s.

G(s) = 42/(s² + 4s + 42)
ωn = rad/s
5.Hard
k = 200 N/mc = 38 N·s/mm = 20 kgF = 100 Nx(t)

A mass m = 20 kg hangs on a spring k = 200 N/m with a viscous damper c = 38 N·s/m (m·x″ + c·x′ + k·x = F). At rest in equilibrium, a constant force F = 100 N is suddenly applied. Find the damping ratio ζ.

m = 20 kgk = 200 N/mc = 38 N·s/mF = 100 N
ζ =
6.Hard
R(s)+Ks(s + 5)G(s)C(s)−

A unity negative feedback loop has G(s) = K/(s(s + 5)). Find the gain K that gives a step response with 10 % overshoot.

G(s) = K/(s(s + 5))%OS = 10 %
K =

Answer key

  1. 1. τ = 0.9788 s
  2. 2. t = 50.66 ms
  3. 3. ts = 8 s
  4. 4. ωn = 6.481 rad/s
  5. 5. ζ = 0.3004
  6. 6. K = 17.88

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