T $3\mathrm{.}2.$ Consider a unity feedback system with the following open$-$loop transfer functions. Plot the root locus for each system for positive values of $K.$ For each case, find the range of values of $K,$ if it exists, such that the closed loop system is over$-$damped.

$($a$)\frac{Ks}{s^2+s+4\mathrm{.}25}$

$($b$)\frac{K(s+1)}{s^2+s+4\mathrm{.}25}$

$($c$)\frac{K(s^2+9)}{s(s^2+16)}$

Answer: $($a$)K3\mathrm{.}12,($b$)K5\mathrm{.}12,($c$)$ Always under$-$damped

AERO $401$