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)+20G₁6G₂C(s)0.05H−

In the block diagram the forward path has the gains G₁ = 20 and G₂ = 6 in series, and the negative feedback path has the gain H = 0.05. Find the overall gain T = C/R.

G₁ = 20G₂ = 6H = 0.05
T =
2.Easy
R(s)15G₂8G₃++9G₁C(s)

The signal R splits into two parallel blocks G₂ = 15 and G₃ = 8. Their outputs meet at a summing junction with the signs shown (+ and +), and the sum passes through G₁ = 9. Find the overall gain T = C/R.

G₁ = 9G₂ = 15G₃ = 8signs: + G₂, + G₃
T =
3.Medium
R(s)+10K+3s + 3G(s)C(s)4H₁−−

The plant G(s) = 3/(s + 3) has an inner (minor) negative feedback loop with gain H₁ = 4. 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) = 3/(s + 3)H₁ = 4K = 10
T(0) =
4.Medium
R(s)+Ks(s + 6)G(s)C(s)−

The plant G(s) = K/(s(s + 6)) is in a unity negative feedback loop. Find the gain K for which the closed loop is critically damped (two equal real poles).

G(s) = K/(s(s + 6))unity negative feedback
K =
5.Hard
R(s)+23controller+D(s)+6s + 3plantC(s)−

A proportional controller K = 23 drives the plant G(s) = 6/(s + 3) in a unity negative feedback loop. A disturbance D enters at the plant input (it is added to the controller output). With R = 0, a disturbance step of size 5 is applied. Find the steady-state output css.

K = 23G(s) = 6/(s + 3)D = 5R = 0
css =
6.Hard
R(s)+7s + 1G₁+7s + 5G₂C(s)1H₁−0.1H₂−

Reduce the block diagram with G₁ = 7/(s + 1), G₂ = 7/(s + 5), an inner negative feedback H₁ = 1 around G₂ and an outer negative feedback H₂ = 0.1. Find the DC gain T(0) of C(s)/R(s).

G₁ = 7/(s + 1)G₂ = 7/(s + 5)H₁ = 1H₂ = 0.1
T(0) =

Answer key

  1. 1. T = 17.14
  2. 2. T = 207
  3. 3. T(0) = 0.6667
  4. 4. K = 9
  5. 5. css = 0.2128
  6. 6. T(0) = 2.899

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