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
+−12 VR = 1 kΩC = 22 µFvC​

An uncharged capacitor C = 22 µF is charged from a 12 V step source through a resistor R = 1 kΩ. Find the capacitor voltage vC at t = 5.5 ms.

R = 1 kΩC = 22 µFV = 12 Vt = 5.5 ms
vC = V
2.Easy
t (s)y(t)y(∞) = ?

A first-order system G(s) = 2/(0.8s + 1) receives a step input of size 10 at t = 0, starting from rest. Find the final value y(∞) of the output.

G(s) = 2/(0.8s + 1)U = 10
y(∞) =
3.Medium
t (s)y(t)1ts​ = ?

A system has the transfer function G(s) = 21/(s² + 3s + 21) and receives a unit step input. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 21/(s² + 3s + 21)
ts = s
4.Medium
t (s)y(t)1

A system has the transfer function G(s) = 16/(s² + 2s + 16) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 16/(s² + 2s + 16)
σ =
5.Hard
k = 100 N/mc = 18 N·s/mm = 20 kgF = 25 Nx(t)

A mass m = 20 kg hangs on a spring k = 100 N/m with a viscous damper c = 18 N·s/m (m·x″ + c·x′ + k·x = F). At rest in equilibrium, a constant force F = 25 N is suddenly applied. Find the percent overshoot %OS of the displacement.

m = 20 kgk = 100 N/mc = 18 N·s/mF = 25 N
%OS = %
6.Hard
t (s)y(t)1tp​ = 0.25 s25 %

The step response of a second-order system without zeros shows 25 % overshoot, and the first peak occurs at t = 0.25 s. Find the natural frequency ωn.

%OS = 25 %tp = 0.25 s
ωn = rad/s

Answer key

  1. 1. vC = 2.654 V
  2. 2. y(∞) = 20
  3. 3. ts = 2.667 s
  4. 4. σ = −1
  5. 5. %OS = 52.44 %
  6. 6. ωn = 13.74 rad/s

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