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)1005 s40

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 100. At t = 5 s it reads 40. Find the time constant τ.

y(∞) = 100y(5) = 40
τ = s
2.Easy
+−9 VR = 10 kΩC = 47 µFvC​

An uncharged capacitor C = 47 µF is charged from a 9 V step source through a resistor R = 10 kΩ. Find the 10–90 % rise time tr of the capacitor voltage, in ms.

R = 10 kΩC = 47 µFV = 9 V
tr = ms
3.Medium
t (s)y(t)1

A system has the transfer function G(s) = 37/(s² + 3s + 37) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 37/(s² + 3s + 37)
σ =
4.Medium
t (s)y(t)0.5ts​ = ?

A system has the transfer function G(s) = 37.5/(s² + 13s + 75) and receives a unit step input. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 37.5/(s² + 13s + 75)
ts = s
5.Hard
+−5 VR = 57 ΩL = 500 mHC = 100 µFvC​

A series RLC circuit with R = 57 Ω, L = 500 mH and C = 100 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 = 57 ΩL = 500 mHC = 100 µFV = 5 V
ωn = rad/s
6.Hard
t (s)y(t)18 %

A second-order system without zeros must have a step response with 8 % overshoot. Find the damping ratio ζ.

%OS = 8 %
ζ =

Answer key

  1. 1. τ = 9.788 s
  2. 2. tr = 1033 ms
  3. 3. σ = −1.5
  4. 4. ts = 0.6154 s
  5. 5. ωn = 141.4 rad/s
  6. 6. ζ = 0.6266

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