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)82 s4.8

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 8. At t = 2 s it reads 4.8. Find the time constant τ.

y(∞) = 8y(2) = 4.8
τ = s
2.Easy
+−24 VR = 100 kΩC = 100 µFvC​

An uncharged capacitor C = 100 µF is charged from a 24 V step source through a resistor R = 100 kΩ. Find the 10–90 % rise time tr of the capacitor voltage, in ms.

R = 100 kΩC = 100 µFV = 24 V
tr = ms
3.Medium
t (s)y(t)0.5tp​ = ?

A system has the transfer function G(s) = 18/(s² + 4s + 36) and receives a unit step input. Find the peak time tp of the step response.

G(s) = 18/(s² + 4s + 36)
tp = s
4.Medium
t (s)y(t)1

A system has the transfer function G(s) = 47/(s² + 3s + 47) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 47/(s² + 3s + 47)
σ =
5.Hard
R(s)+235s(s + 10)G(s)C(s)−

A unity negative feedback loop has G(s) = 235/(s(s + 10)). Find the 2 % settling time ts of the closed-loop step response, using ts ≈ 4/(ζωₙ).

G(s) = 235/(s(s + 10))
ts = s
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 = 2.183 s
  2. 2. tr = 21972 ms
  3. 3. tp = 0.5554 s
  4. 4. σ = −1.5
  5. 5. ts = 0.8 s
  6. 6. ωn = 1.948 rad/s

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