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)502 s15

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 50. At t = 2 s it reads 15. Find the time constant τ.

y(∞) = 50y(2) = 15
τ = s
2.Easy
+−5 VR = 2.2 kΩC = 2.2 µFvC​

An uncharged capacitor C = 2.2 µF is charged from a 5 V step source through a resistor R = 2.2 kΩ. How long after closing the switch does the capacitor voltage reach 90 % of its final value? Find t in ms.

R = 2.2 kΩC = 2.2 µFV = 5 V90 %
t = ms
3.Medium
t (s)y(t)0.5

A system has the transfer function G(s) = 6/(s² + s + 12) and receives a unit step input. Find the damping ratio ζ.

G(s) = 6/(s² + s + 12)
ζ =
4.Medium
t (s)y(t)2

A system has the transfer function G(s) = 8/(s² + s + 4) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 8/(s² + s + 4)
σ =
5.Hard
R(s)+Ks(s + 10)G(s)C(s)−

A unity negative feedback loop has G(s) = K/(s(s + 10)). Find the gain K that gives a step response with 15 % overshoot.

G(s) = K/(s(s + 10))%OS = 15 %
K =
6.Hard
R(s)+250s(s + 5)G(s)C(s)−

A unity negative feedback loop has G(s) = 250/(s(s + 5)). Find the 2 % settling time ts of the closed-loop step response, using ts ≈ 4/(ζωₙ).

G(s) = 250/(s(s + 5))
ts = s

Answer key

  1. 1. τ = 5.607 s
  2. 2. t = 11.14 ms
  3. 3. ζ = 0.1443
  4. 4. σ = −0.5
  5. 5. K = 93.56
  6. 6. ts = 1.6 s

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