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)+13G₁6G₂C(s)0.25H−

In the block diagram the forward path has the gains G₁ = 13 and G₂ = 6 in series, and the negative feedback path has the gain H = 0.25. Find the overall gain T = C/R.

G₁ = 13G₂ = 6H = 0.25
T =
2.Easy
R(s)7G₂12G₃+−5G₁C(s)

The signal R splits into two parallel blocks G₂ = 7 and G₃ = 12. Their outputs meet at a summing junction with the signs shown (+ and −), and the sum passes through G₁ = 5. Find the overall gain T = C/R.

G₁ = 5G₂ = 7G₃ = 12signs: + G₂, − G₃
T =
3.Medium
R(s)+8s(s + 9)G(s)C(s)−

A unity negative feedback loop has G(s) = 8/(s(s + 9)). The reference is a ramp r(t) = 1·t. Find the steady-state error ess.

G(s) = 8/(s(s + 9))r(t) = 1·t
ess =
4.Medium
R(s)+15K+8s + 6G(s)C(s)3H₁−−

The plant G(s) = 8/(s + 6) has an inner (minor) negative feedback loop with gain H₁ = 3. 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) = 8/(s + 6)H₁ = 3K = 15
T(0) =
5.Hard
R(s)+62s(s + 10)G(s)C(s)2H−

The plant G(s) = 62/(s(s + 10)) is in a negative feedback loop with sensor gain H = 2. The closed-loop poles are complex. Find their real part σ.

G(s) = 62/(s(s + 10))H = 2
σ =
6.Hard
R(s)+K(s + 4)(s + 6)(s + 7)G(s)C(s)−

The plant G(s) = K/((s + 4)(s + 6)(s + 7)) is in a unity negative feedback loop. At the largest stable gain the loop oscillates. Find the frequency of oscillation ω in rad/s.

G(s) = K/((s + 4)(s + 6)(s + 7))unity negative feedback
ω = rad/s

Answer key

  1. 1. T = 3.805
  2. 2. T = −25
  3. 3. ess = 1.125
  4. 4. T(0) = 0.8
  5. 5. σ = −5
  6. 6. ω = 9.695 rad/s

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