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)+4G₁3G₂C(s)1H−

In the block diagram the forward path has the gains G₁ = 4 and G₂ = 3 in series, and the negative feedback path has the gain H = 1. Find the overall gain T = C/R.

G₁ = 4G₂ = 3H = 1
T =
3.Medium
R(s)+42s(s + 7)G(s)C(s)−

A unity negative feedback loop has G(s) = 42/(s(s + 7)). The reference is a ramp r(t) = 1·t. Find the steady-state error ess.

G(s) = 42/(s(s + 7))r(t) = 1·t
ess =
4.Medium
R(s)+21s + 7G(s)C(s)0.5H−

A plant G(s) = 21/(s + 7) is controlled in a negative feedback loop whose sensor has the gain H = 0.5. A step of size 5 is applied at R. Find the steady-state output css.

G(s) = 21/(s + 7)H = 0.5R = 5
css =
5.Hard
R(s)+26s(s + 8)G(s)C(s)2H−

The plant G(s) = 26/(s(s + 8)) 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) = 26/(s(s + 8))H = 2
ωd = rad/s
6.Hard
R(s)+18s + 3G₁+5s + 7G₂C(s)2H₁−0.2H₂−

Reduce the block diagram with G₁ = 18/(s + 3), G₂ = 5/(s + 7), an inner negative feedback H₁ = 2 around G₂ and an outer negative feedback H₂ = 0.2. Find the DC gain T(0) of C(s)/R(s).

G₁ = 18/(s + 3)G₂ = 5/(s + 7)H₁ = 2H₂ = 0.2
T(0) =

Answer key

  1. 1. T = −15
  2. 2. T = 0.9231
  3. 3. ess = 0.1667
  4. 4. css = 6
  5. 5. ωd = 6 rad/s
  6. 6. T(0) = 1.304

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