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)+8G₁5G₂C(s)0.1H−

In the block diagram the forward path has the gains G₁ = 8 and G₂ = 5 in series, and the negative feedback path has the gain H = 0.1. Find the overall gain T = C/R.

G₁ = 8G₂ = 5H = 0.1
T =
2.Easy
R(s)+2s + 9G(s)C(s)−

A plant G(s) = 2/(s + 9) is placed in a unity negative feedback loop. Find the time constant τ of the closed loop.

G(s) = 2/(s + 9)unity negative feedback
τ = s
3.Medium
R(s)+7K+9s + 7G(s)C(s)4H₁−−

The plant G(s) = 9/(s + 7) has an inner (minor) negative feedback loop with gain H₁ = 4. A controller gain K = 7 and an outer unity negative feedback loop close the system. Find the closed-loop DC gain T(0).

G(s) = 9/(s + 7)H₁ = 4K = 7
T(0) =
4.Medium
R(s)+7K+10s + 4G(s)C(s)0.5H₁−−

The plant G(s) = 10/(s + 4) has an inner (minor) negative feedback loop with gain H₁ = 0.5. A controller gain K = 7 and an outer unity negative feedback loop close the system. Find the closed-loop pole p.

G(s) = 10/(s + 4)H₁ = 0.5K = 7
p =
5.Hard
R(s)+40s(s + 9)G(s)C(s)2H−

The plant G(s) = 40/(s(s + 9)) is in a negative feedback loop with sensor gain H = 2. The closed-loop poles are complex. Find their imaginary part ωd (the positive value), in rad/s.

G(s) = 40/(s(s + 9))H = 2
ωd = rad/s
6.Hard
R(s)+20s + 1G₁+7s + 2G₂C(s)2H₁−0.1H₂−

Reduce the block diagram with G₁ = 20/(s + 1), G₂ = 7/(s + 2), an inner negative feedback H₁ = 2 around G₂ and an outer negative feedback H₂ = 0.1. Find the DC gain T(0) of C(s)/R(s).

G₁ = 20/(s + 1)G₂ = 7/(s + 2)H₁ = 2H₂ = 0.1
T(0) =

Answer key

  1. 1. T = 8
  2. 2. τ = 0.09091 s
  3. 3. T(0) = 0.5943
  4. 4. p = −79
  5. 5. ωd = 7.73 rad/s
  6. 6. T(0) = 4.667

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