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)+6G₁8G₂C(s)0.5H−

In the block diagram the forward path has the gains G₁ = 6 and G₂ = 8 in series, and the negative feedback path has the gain H = 0.5. Find the overall gain T = C/R.

G₁ = 6G₂ = 8H = 0.5
T =
2.Easy
R(s)+4s + 3G(s)C(s)−

A plant G(s) = 4/(s + 3) is placed in a unity negative feedback loop. Find the time constant τ of the closed loop.

G(s) = 4/(s + 3)unity negative feedback
τ = s
3.Medium
R(s)+K(s + 9)(s + 12)G(s)C(s)−

The plant G(s) = K/((s + 9)(s + 12)) 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 + 9)(s + 12))unity negative feedback
K =
4.Medium
R(s)+4s(s + 6)G(s)C(s)−

A unity negative feedback loop has G(s) = 4/(s(s + 6)). The reference is a ramp r(t) = 5·t. Find the steady-state error ess.

G(s) = 4/(s(s + 6))r(t) = 5·t
ess =
5.Hard
R(s)+74s(s + 4)G(s)C(s)2H−

The plant G(s) = 74/(s(s + 4)) 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) = 74/(s(s + 4))H = 2
ωd = rad/s
6.Hard
R(s)+15s + 3G₁+5s + 7G₂C(s)1H₁−0.1H₂−

Reduce the block diagram with G₁ = 15/(s + 3), G₂ = 5/(s + 7), an inner negative feedback H₁ = 1 around G₂ and an outer negative feedback H₂ = 0.1. Find the DC gain T(0) of C(s)/R(s).

G₁ = 15/(s + 3)G₂ = 5/(s + 7)H₁ = 1H₂ = 0.1
T(0) =

Answer key

  1. 1. T = 1.92
  2. 2. τ = 0.1429 s
  3. 3. K = 2.25
  4. 4. ess = 7.5
  5. 5. ωd = 12 rad/s
  6. 6. T(0) = 1.724

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