# Negative Gain Root Locus

Consider a negative gain root locus for L (s) = k (s-1)/[s(s^2 + 4s + 8)] with a scale of 20mm = 1 unit.

i. Graphically, find the gain required to give dominant closed loop poles with zeta = 0.5.

ii. For the above value of gain, find natural frequency and hence sketch the expected reference to output step response, assuming that the system is low-passing and dominantly second order.

iii. Add a sketch of the positive gain root locus to the diagram.

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#### Solution Preview

Please see the attached file.

Consider the positive gain root locus for shown on the last page of the examination paper with a scale of 20mm=1unit.

i) Graphically, find the gain required to give dominant closed loop poles with .

ii) For the above value of gain, find ...

#### Solution Summary

Solutions are shown with full calculations including formulas C(s)/R(s) = L(s)/(1-L(s)) = [k(s-1)]/[s^3 + 4s^2 + (8-k)s + k] and Laplace transform pairs. Sketched graph for (ii) is also included.