- Selected Problems and Solutions of GATE Examinations from 2002 to 2012

Solution: Answer is (c) both real and negative. Explanation: Poles are the roots of the characteristic equation of a closed-loop transfer

function at controller gain, Kc = 0. For a second-order system, the characteristic equation is

τ2s2 + 2τξs + 1 + Kc = 0

At Kc = 0, τ2s2 + 2τξs + 1 = 0. Solving the above quadratic equation, we get roots s = {–ξ ± √(ξ2 – 1)}/τ,

which indicates that the roots of the characteristic equation may be real, imaginary, or complex, but having real parts for both the roots are negative.