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)2G₂7G₃+−3G₁C(s)

The signal R splits into two parallel blocks G₂ = 2 and G₃ = 7. Their outputs meet at a summing junction with the signs shown (+ and −), and the sum passes through G₁ = 3. Find the overall gain T = C/R.

G₁ = 3G₂ = 2G₃ = 7signs: + G₂, − G₃
T =
2.Easy
R(s)+8GC(s)0.025H+

An amplifier with gain G = 8 has positive feedback through a path with gain H = 0.025 (the feedback signal is added at the summing junction). Find the closed-loop gain T = C/R.

G = 8H = 0.025positive feedback
T =
3.Medium
R(s)+6K+5s + 3G(s)C(s)3H₁−−

The plant G(s) = 5/(s + 3) has an inner (minor) negative feedback loop with gain H₁ = 3. A controller gain K = 6 and an outer unity negative feedback loop close the system. Find the closed-loop DC gain T(0).

G(s) = 5/(s + 3)H₁ = 3K = 6
T(0) =
4.Medium
R(s)+56s(s + 3)G(s)C(s)−

A unity negative feedback loop has G(s) = 56/(s(s + 3)). The reference is a ramp r(t) = 2·t. Find the steady-state error ess.

G(s) = 56/(s(s + 3))r(t) = 2·t
ess =
5.Hard
R(s)+16s + 3G₁+5s + 6G₂C(s)4H₁−0.1H₂−

Reduce the block diagram with G₁ = 16/(s + 3), G₂ = 5/(s + 6), an inner negative feedback H₁ = 4 around G₂ and an outer negative feedback H₂ = 0.1. Find the DC gain T(0) of C(s)/R(s).

G₁ = 16/(s + 3)G₂ = 5/(s + 6)H₁ = 4H₂ = 0.1
T(0) =
6.Hard
R(s)+16s(s + 2)G(s)C(s)1H−

The plant G(s) = 16/(s(s + 2)) is in a negative feedback loop with sensor gain H = 1. The closed-loop poles are complex. Find their imaginary part ωd (the positive value), in rad/s.

G(s) = 16/(s(s + 2))H = 1
ωd = rad/s

Answer key

  1. 1. T = −15
  2. 2. T = 10
  3. 3. T(0) = 0.625
  4. 4. ess = 0.1071
  5. 5. T(0) = 0.9302
  6. 6. ωd = 3.873 rad/s

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