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)600.3 s?

A first-order system G(s) = 30/(s + 10) receives a step input of size 20 at t = 0, starting from rest. Find the output y(0.3) at t = 0.3 s.

G(s) = 30/(s + 10)U = 20t = 0.3 s
y(0.3) =
2.Easy
+−10 VR = 1 kΩC = 100 µFvC​

An uncharged capacitor C = 100 µF is charged from a 10 V step source through a resistor R = 1 kΩ. How long after closing the switch does the capacitor voltage reach 60 % of its final value? Find t in ms.

R = 1 kΩC = 100 µFV = 10 V60 %
t = ms
3.Medium
t (s)y(t)3

A system has the transfer function G(s) = 222/(s² + 10s + 74) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 222/(s² + 10s + 74)
σ =
4.Medium
t (s)y(t)0.5

A system has the transfer function G(s) = 45/(s² + 8s + 90) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 45/(s² + 8s + 90)
σ =
5.Hard
R(s)+185s(s + 10)G(s)C(s)−

A unity negative feedback loop has G(s) = 185/(s(s + 10)). Find the 2 % settling time ts of the closed-loop step response, using ts ≈ 4/(ζωₙ).

G(s) = 185/(s(s + 10))
ts = s
6.Hard
k = 5000 N/mc = 80 N·s/mm = 8 kgF = 20 Nx(t)

A mass m = 8 kg hangs on a spring k = 5000 N/m with a viscous damper c = 80 N·s/m (m·x″ + c·x′ + k·x = F). At rest in equilibrium, a constant force F = 20 N is suddenly applied. Find the damping ratio ζ.

m = 8 kgk = 5000 N/mc = 80 N·s/mF = 20 N
ζ =

Answer key

  1. 1. y(0.3) = 57.01
  2. 2. t = 91.63 ms
  3. 3. σ = −5
  4. 4. σ = −4
  5. 5. ts = 0.8 s
  6. 6. ζ = 0.2

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