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)3G₂3G₃++9G₁C(s)

The signal R splits into two parallel blocks G₂ = 3 and G₃ = 3. 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 = 3G₃ = 3signs: + G₂, + G₃
T =
2.Easy
R(s)+12G₁10G₂C(s)0.25H−

In the block diagram the forward path has the gains G₁ = 12 and G₂ = 10 in series, and the negative feedback path has the gain H = 0.25. Find the overall gain T = C/R.

G₁ = 12G₂ = 10H = 0.25
T =
3.Medium
R(s)+K(s + 1)(s + 8)G(s)C(s)−

The plant G(s) = K/((s + 1)(s + 8)) is in a unity negative feedback loop. Find the gain K for which the closed loop is critically damped (two equal real poles).

G(s) = K/((s + 1)(s + 8))unity negative feedback
K =
4.Medium
R(s)+44s(s + 6)G(s)C(s)−

A unity negative feedback loop has G(s) = 44/(s(s + 6)). The reference is a ramp r(t) = 10·t. Find the steady-state error ess.

G(s) = 44/(s(s + 6))r(t) = 10·t
ess =
5.Hard
R(s)+70s(s + 9)G(s)C(s)1H−

The plant G(s) = 70/(s(s + 9)) 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) = 70/(s(s + 9))H = 1
ωd = rad/s
6.Hard
R(s)+Ks(s + 1)(s + 5)G(s)C(s)−

The plant G(s) = K/(s(s + 1)(s + 5)) is in a unity negative feedback loop. Use the Routh–Hurwitz criterion to find the largest gain Kmax for which the closed loop is stable.

G(s) = K/(s(s + 1)(s + 5))unity negative feedback
Kmax =

Answer key

  1. 1. T = 54
  2. 2. T = 3.871
  3. 3. K = 12.25
  4. 4. ess = 1.364
  5. 5. ωd = 7.053 rad/s
  6. 6. Kmax = 30

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