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)13G₂1G₃+−9G₁C(s)

The signal R splits into two parallel blocks G₂ = 13 and G₃ = 1. 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 = 13G₃ = 1signs: + G₂, − G₃
T =
2.Easy
R(s)+3s + 2G(s)C(s)−

A plant G(s) = 3/(s + 2) is placed in a unity negative feedback loop. A step of size 1 is applied at R. Find the steady-state output css.

G(s) = 3/(s + 2)unity negative feedbackR = 1
css =
3.Medium
R(s)+Ks(s + 15)G(s)C(s)−

The plant G(s) = K/(s(s + 15)) 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(s + 15))unity negative feedback
K =
4.Medium
R(s)+6s + 10G(s)C(s)2H−

A plant G(s) = 6/(s + 10) is controlled in a negative feedback loop whose sensor has the gain H = 2. Find the time constant τ of the closed loop.

G(s) = 6/(s + 10)H = 2
τ = s
5.Hard
R(s)+11s + 4G₁+3s + 8G₂C(s)4H₁−0.25H₂−

Reduce the block diagram with G₁ = 11/(s + 4), G₂ = 3/(s + 8), 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 = 11/(s + 4)G₂ = 3/(s + 8)H₁ = 4H₂ = 0.25
T(0) =
6.Hard
R(s)+6s(s + 2)G(s)C(s)2H−

The plant G(s) = 6/(s(s + 2)) is in a negative feedback loop with sensor gain H = 2. The closed-loop poles are complex. Find their imaginary part ωd (the positive value), in rad/s.

G(s) = 6/(s(s + 2))H = 2
ωd = rad/s

Answer key

  1. 1. T = 108
  2. 2. css = 0.6
  3. 3. K = 56.25
  4. 4. τ = 0.04545 s
  5. 5. T(0) = 0.3739
  6. 6. ωd = 3.317 rad/s

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