Block diagrams worksheet with answers

Free printable Block diagrams 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
R(s)+14GC(s)0.02H+

An amplifier with gain G = 14 has positive feedback through a path with gain H = 0.02 (the feedback signal is added at the summing junction). Find the closed-loop gain T = C/R.

G = 14H = 0.02positive feedback
T =
2.Easy
R(s)+18G₁1G₂C(s)0.25H−

In the block diagram the forward path has the gains G₁ = 18 and G₂ = 1 in series, and the negative feedback path has the gain H = 0.25. Find the overall gain T = C/R.

G₁ = 18G₂ = 1H = 0.25
T =
3.Medium
R(s)+22s(s + 3)G(s)C(s)−

A unity negative feedback loop has G(s) = 22/(s(s + 3)). The reference is a ramp r(t) = 2·t. Find the steady-state error ess.

G(s) = 22/(s(s + 3))r(t) = 2·t
ess =
4.Medium
R(s)+10K+4s + 5G(s)C(s)1H₁−−

The plant G(s) = 4/(s + 5) has an inner (minor) negative feedback loop with gain H₁ = 1. A controller gain K = 10 and an outer unity negative feedback loop close the system. Find the closed-loop DC gain T(0).

G(s) = 4/(s + 5)H₁ = 1K = 10
T(0) =
5.Hard
R(s)+Kcontroller+D(s)+9s + 3plantC(s)−

A proportional controller K drives the plant G(s) = 9/(s + 3) in a unity negative feedback loop. A disturbance step of size 5 enters at the plant input while R = 0. Find the smallest gain K that keeps the steady-state output at or below 0.3.

G(s) = 9/(s + 3)D = 5css ≤ 0.3
K =
6.Hard
R(s)+64s(s + 8)G(s)C(s)1H−

The plant G(s) = 64/(s(s + 8)) is in a negative feedback loop with sensor gain H = 1. The closed-loop poles are complex. Find their imaginary part ωd (the positive value), in rad/s.

G(s) = 64/(s(s + 8))H = 1
ωd = rad/s

Answer key

  1. 1. T = 19.44
  2. 2. T = 3.273
  3. 3. ess = 0.2727
  4. 4. T(0) = 0.8163
  5. 5. K = 16.33
  6. 6. ωd = 6.928 rad/s

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