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)12G₂1G₃+−9G₁C(s)

The signal R splits into two parallel blocks G₂ = 12 and G₃ = 1. 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 = 12G₃ = 1signs: + G₂, − G₃
T =
2.Easy
R(s)6G₂4G₃++6G₁C(s)

The signal R splits into two parallel blocks G₂ = 6 and G₃ = 4. Their outputs meet at a summing junction with the signs shown (+ and +), and the sum passes through G₁ = 6. Find the overall gain T = C/R.

G₁ = 6G₂ = 6G₃ = 4signs: + G₂, + G₃
T =
3.Medium
R(s)+15K+5s + 4G(s)C(s)1H₁−−

The plant G(s) = 5/(s + 4) has an inner (minor) negative feedback loop with gain H₁ = 1. A controller gain K = 15 and an outer unity negative feedback loop close the system. Find the closed-loop DC gain T(0).

G(s) = 5/(s + 4)H₁ = 1K = 15
T(0) =
4.Medium
R(s)+40s(s + 4)G(s)C(s)−

A unity negative feedback loop has G(s) = 40/(s(s + 4)). The reference is a ramp r(t) = 5·t. Find the steady-state error ess.

G(s) = 40/(s(s + 4))r(t) = 5·t
ess =
5.Hard
R(s)+16controller+D(s)+10s + 3plantC(s)−

A proportional controller K = 16 drives the plant G(s) = 10/(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 = 16G(s) = 10/(s + 3)D = 5R = 0
css =
6.Hard
R(s)+10s + 4G₁+8s + 3G₂C(s)4H₁−0.25H₂−

Reduce the block diagram with G₁ = 10/(s + 4), G₂ = 8/(s + 3), an inner negative feedback H₁ = 4 around G₂ and an outer negative feedback H₂ = 0.25. Find the DC gain T(0) of C(s)/R(s).

G₁ = 10/(s + 4)G₂ = 8/(s + 3)H₁ = 4H₂ = 0.25
T(0) =

Answer key

  1. 1. T = 99
  2. 2. T = 60
  3. 3. T(0) = 0.8929
  4. 4. ess = 0.5
  5. 5. css = 0.3067
  6. 6. T(0) = 0.5

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