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)+7G₁9G₂C(s)1H−

In the block diagram the forward path has the gains G₁ = 7 and G₂ = 9 in series, and the negative feedback path has the gain H = 1. Find the overall gain T = C/R.

G₁ = 7G₂ = 9H = 1
T =
2.Easy
R(s)+15G₁9G₂C(s)1H−

In the block diagram the forward path has the gains G₁ = 15 and G₂ = 9 in series, and the negative feedback path has the gain H = 1. Find the overall gain T = C/R.

G₁ = 15G₂ = 9H = 1
T =
3.Medium
R(s)+60s(s + 8)G(s)C(s)−

A unity negative feedback loop has G(s) = 60/(s(s + 8)). The reference is a ramp r(t) = 10·t. Find the steady-state error ess.

G(s) = 60/(s(s + 8))r(t) = 10·t
ess =
4.Medium
R(s)+3K+4s + 2G(s)C(s)4H₁−−

The plant G(s) = 4/(s + 2) has an inner (minor) negative feedback loop with gain H₁ = 4. A controller gain K = 3 and an outer unity negative feedback loop close the system. Find the closed-loop DC gain T(0).

G(s) = 4/(s + 2)H₁ = 4K = 3
T(0) =
5.Hard
R(s)+Kcontroller+D(s)+9s + 1plantC(s)−

A proportional controller K drives the plant G(s) = 9/(s + 1) 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 4.5.

G(s) = 9/(s + 1)D = 5css ≤ 4.5
K =
6.Hard
R(s)+28s(s + 3)G(s)C(s)0.5H−

The plant G(s) = 28/(s(s + 3)) is in a negative feedback loop with sensor gain H = 0.5. The closed-loop poles are complex. Find their real part σ.

G(s) = 28/(s(s + 3))H = 0.5
σ =

Answer key

  1. 1. T = 0.9844
  2. 2. T = 0.9926
  3. 3. ess = 1.333
  4. 4. T(0) = 0.4
  5. 5. K = 1
  6. 6. σ = −1.5

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