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)205 s8

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 20. At t = 5 s it reads 8. Find the time constant τ.

y(∞) = 20y(5) = 8
τ = s
2.Easy
t (s)y(t)250.175 s?

A first-order system G(s) = 20/(s + 4) receives a step input of size 5 at t = 0, starting from rest. Find the output y(0.175) at t = 0.175 s.

G(s) = 20/(s + 4)U = 5t = 0.175 s
y(0.175) =
3.Medium
t (s)y(t)1

A system has the transfer function G(s) = 92/(s² + 11s + 92) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 92/(s² + 11s + 92)
σ =
4.Medium
t (s)y(t)1

A system has the transfer function G(s) = 94/(s² + 14s + 94) and receives a unit step input. Find the natural frequency ωn in rad/s.

G(s) = 94/(s² + 14s + 94)
ωn = rad/s
5.Hard
t (s)y(t)1tp​ = 1.5 s4.3 %

The step response of a second-order system without zeros shows 4.3 % overshoot, and the first peak occurs at t = 1.5 s. Find the natural frequency ωn.

%OS = 4.3 %tp = 1.5 s
ωn = rad/s
6.Hard
+−5 VR = 17 ΩL = 20 mHC = 47 µFvC​

A series RLC circuit with R = 17 Ω, L = 20 mH and C = 47 µF is connected to a 5 V step at t = 0. The capacitor is uncharged and no current flows before the step. Find the peak capacitor voltage vC,max.

R = 17 ΩL = 20 mHC = 47 µFV = 5 V
vC,max = V

Answer key

  1. 1. τ = 9.788 s
  2. 2. y(0.175) = 12.59
  3. 3. σ = −5.5
  4. 4. ωn = 9.695 rad/s
  5. 5. ωn = 2.964 rad/s
  6. 6. vC,max = 6.208 V

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