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)10010 s80

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 100. At t = 10 s it reads 80. Find the time constant τ.

y(∞) = 100y(10) = 80
τ = s
2.Easy
t (s)y(t)81 s3.2

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 8. At t = 1 s it reads 3.2. Find the time constant τ.

y(∞) = 8y(1) = 3.2
τ = s
3.Medium
t (s)y(t)0.5ts​ = ?

A system has the transfer function G(s) = 44.5/(s² + 7s + 89) and receives a unit step input. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 44.5/(s² + 7s + 89)
ts = s
4.Medium
t (s)y(t)1tp​ = ?

A system has the transfer function G(s) = 35/(s² + 5s + 35) and receives a unit step input. Find the peak time tp of the step response.

G(s) = 35/(s² + 5s + 35)
tp = s
5.Hard
+−5 VR = 268 ΩL = 500 mHC = 10 µFvC​

A series RLC circuit with R = 268 Ω, L = 500 mH and C = 10 µF is connected to a 5 V step at t = 0. The capacitor is uncharged and no current flows before the step. Find the undamped natural frequency ωn in rad/s.

R = 268 ΩL = 500 mHC = 10 µFV = 5 V
ωn = rad/s
6.Hard
t (s)y(t)1tp​ = 0.5 s30 %

The step response of a second-order system without zeros shows 30 % overshoot, and the first peak occurs at t = 0.5 s. Find the natural frequency ωn.

%OS = 30 %tp = 0.5 s
ωn = rad/s

Answer key

  1. 1. τ = 6.213 s
  2. 2. τ = 1.958 s
  3. 3. ts = 1.143 s
  4. 4. tp = 0.5859 s
  5. 5. ωn = 447.2 rad/s
  6. 6. ωn = 6.729 rad/s

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