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)+6GC(s)0.1H+

An amplifier with gain G = 6 has positive feedback through a path with gain H = 0.1 (the feedback signal is added at the summing junction). Find the closed-loop gain T = C/R.

G = 6H = 0.1positive feedback
T =
2.Easy
R(s)+15G₁4G₂C(s)1H−

In the block diagram the forward path has the gains G₁ = 15 and G₂ = 4 in series, and the negative feedback path has the gain H = 1. Find the overall gain T = C/R.

G₁ = 15G₂ = 4H = 1
T =
3.Medium
R(s)+7(s + 3)(s + 7)(s + 9)G(s)C(s)−

A unity negative feedback loop has G(s) = 7(s + 3)/((s + 7)(s + 9)). A step of size 1 is applied at the reference. Find the steady-state error ess.

G(s) = 7(s + 3)/((s + 7)(s + 9))R = 1
ess =
4.Medium
R(s)+13K+5s + 4G(s)C(s)5H₁−−

The plant G(s) = 5/(s + 4) has an inner (minor) negative feedback loop with gain H₁ = 5. A controller gain K = 13 and an outer unity negative feedback loop close the system. Find the closed-loop DC gain T(0).

G(s) = 5/(s + 4)H₁ = 5K = 13
T(0) =
5.Hard
R(s)+14s + 4G₁+6s + 2G₂C(s)4H₁−0.25H₂−

Reduce the block diagram with G₁ = 14/(s + 4), G₂ = 6/(s + 2), 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 = 14/(s + 4)G₂ = 6/(s + 2)H₁ = 4H₂ = 0.25
T(0) =
6.Hard
R(s)+Kcontroller+D(s)+8s + 4plantC(s)−

A proportional controller K drives the plant G(s) = 8/(s + 4) 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 2.

G(s) = 8/(s + 4)D = 5css ≤ 2
K =

Answer key

  1. 1. T = 15
  2. 2. T = 0.9836
  3. 3. ess = 0.75
  4. 4. T(0) = 0.6915
  5. 5. T(0) = 0.672
  6. 6. K = 2

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