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)23 s0.8

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 2. At t = 3 s it reads 0.8. Find the time constant τ.

y(∞) = 2y(3) = 0.8
τ = s
2.Easy
t (s)y(t)101 s8

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 10. At t = 1 s it reads 8. Find the time constant τ.

y(∞) = 10y(1) = 8
τ = s
3.Medium
t (s)y(t)2?

A system has the transfer function G(s) = 190/(s² + 6s + 95) and receives a unit step input. Find the percent overshoot %OS of the step response.

G(s) = 190/(s² + 6s + 95)
%OS = %
4.Medium
t (s)y(t)1?

A system has the transfer function G(s) = 29/(s² + 5s + 29) and receives a unit step input. Find the percent overshoot %OS of the step response.

G(s) = 29/(s² + 5s + 29)
%OS = %
5.Hard
t (s)y(t)140 %

A second-order system without zeros must have a step response with 40 % overshoot. Find the damping ratio ζ.

%OS = 40 %
ζ =
6.Hard
t (s)y(t)1tp​ = 2 s10 %

The step response of a second-order system without zeros shows 10 % overshoot, and the first peak occurs at t = 2 s. Find the natural frequency ωn.

%OS = 10 %tp = 2 s
ωn = rad/s

Answer key

  1. 1. τ = 5.873 s
  2. 2. τ = 0.6213 s
  3. 3. %OS = 36.19 %
  4. 4. %OS = 19.27 %
  5. 5. ζ = 0.28
  6. 6. ωn = 1.948 rad/s

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