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)+4G₁1G₂C(s)0.05H−

In the block diagram the forward path has the gains G₁ = 4 and G₂ = 1 in series, and the negative feedback path has the gain H = 0.05. Find the overall gain T = C/R.

G₁ = 4G₂ = 1H = 0.05
T =
2.Easy
R(s)5G₂10G₃+−8G₁C(s)

The signal R splits into two parallel blocks G₂ = 5 and G₃ = 10. Their outputs meet at a summing junction with the signs shown (+ and −), and the sum passes through G₁ = 8. Find the overall gain T = C/R.

G₁ = 8G₂ = 5G₃ = 10signs: + G₂, − G₃
T =
3.Medium
R(s)+5K+1s + 2G(s)C(s)1H₁−−

The plant G(s) = 1/(s + 2) has an inner (minor) negative feedback loop with gain H₁ = 1. A controller gain K = 5 and an outer unity negative feedback loop close the system. Find the closed-loop DC gain T(0).

G(s) = 1/(s + 2)H₁ = 1K = 5
T(0) =
4.Medium
R(s)+18K+1s + 5G(s)C(s)2H₁−−

The plant G(s) = 1/(s + 5) has an inner (minor) negative feedback loop with gain H₁ = 2. A controller gain K = 18 and an outer unity negative feedback loop close the system. Find the closed-loop pole p.

G(s) = 1/(s + 5)H₁ = 2K = 18
p =
5.Hard
R(s)+7controller+D(s)+6s + 9plantC(s)−

A proportional controller K = 7 drives the plant G(s) = 6/(s + 9) in a unity negative feedback loop. A disturbance D enters at the plant input (it is added to the controller output). A reference step of size 2 and a disturbance step of size 10 are applied together. Find the steady-state output css.

K = 7G(s) = 6/(s + 9)D = 10R = 2
css =
6.Hard
R(s)+40s(s + 7)G(s)C(s)2H−

The plant G(s) = 40/(s(s + 7)) is in a negative feedback loop with sensor gain H = 2. The closed-loop poles are complex. Find their real part σ.

G(s) = 40/(s(s + 7))H = 2
σ =

Answer key

  1. 1. T = 3.333
  2. 2. T = −40
  3. 3. T(0) = 0.625
  4. 4. p = −25
  5. 5. css = 2.824
  6. 6. σ = −3.5

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