Time Response Analysis Questions and Answers

Answer: D

The damping ratio \(\zeta\) of a second order system decides the nature of the response. For \(\zeta = 0\) the response is undamped, for \(0 < \zeta < 1\) it is underdamped (oscillatory with decaying amplitude), for \(\zeta = 1\) it is critically damped, and for \(\zeta > 1\) the system is overdamped.

In the overdamped case the characteristic roots are real and unequal, and the response creeps to its final value without overshoot. So the damping ratio of an overdamped system is greater than unity, option D.

Answer: A

The characteristic equation of the closed loop system is \(s^2 + 4s + K = 0\), with roots

\[ s = \frac{-4 \pm \sqrt{16 - 4K}}{2} \]

For \(K = 4\) the discriminant is zero, so

\[ s_1 = s_2 = -2 \]

The roots are real and equal (repeated), at the same point \(-2\) on the negative real axis, and the system is critically damped. The roots are not on the imaginary axis, which would need no \(s\) term, and the value \(-6\) in the last option is wrong. The answer is option A.