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)+9GC(s)0.04H+

An amplifier with gain G = 9 has positive feedback through a path with gain H = 0.04 (the feedback signal is added at the summing junction). Find the closed-loop gain T = C/R.

G = 9H = 0.04positive feedback
T =
2.Easy
R(s)+7G₁10G₂C(s)0.5H−

In the block diagram the forward path has the gains G₁ = 7 and G₂ = 10 in series, and the negative feedback path has the gain H = 0.5. Find the overall gain T = C/R.

G₁ = 7G₂ = 10H = 0.5
T =
3.Medium
R(s)+15s + 7G(s)C(s)0.2H−

A plant G(s) = 15/(s + 7) is controlled in a negative feedback loop whose sensor has the gain H = 0.2. A step of size 2 is applied at R. Find the steady-state output css.

G(s) = 15/(s + 7)H = 0.2R = 2
css =
4.Medium
R(s)+15K+8s + 2G(s)C(s)1H₁−−

The plant G(s) = 8/(s + 2) has an inner (minor) negative feedback loop with gain H₁ = 1. 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 + 2)H₁ = 1K = 15
T(0) =
5.Hard
R(s)+13s + 4G₁+1s + 3G₂C(s)4H₁−1H₂−

Reduce the block diagram with G₁ = 13/(s + 4), G₂ = 1/(s + 3), an inner negative feedback H₁ = 4 around G₂ and an outer negative feedback H₂ = 1. Find the DC gain T(0) of C(s)/R(s).

G₁ = 13/(s + 4)G₂ = 1/(s + 3)H₁ = 4H₂ = 1
T(0) =
6.Hard
R(s)+Ks(s + 7)(s + 8)G(s)C(s)−

The plant G(s) = K/(s(s + 7)(s + 8)) 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 + 7)(s + 8))unity negative feedback
Kmax =

Answer key

  1. 1. T = 14.06
  2. 2. T = 1.944
  3. 3. css = 3
  4. 4. T(0) = 0.9231
  5. 5. T(0) = 0.3171
  6. 6. Kmax = 840

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