Correct Answer - Option 3 : Settling time

The open-loop transfer function of a unity feedback 2^{nd} order system is,

\(G\left( s \right) = \frac{k}{{s\left( {Js + B} \right)}}\;\)

Closed-loop transfer function:

\(H\left( s \right) = \frac{k}{{s\left( {Js + B} \right) + k}}\)

\(H\left( s \right) = \frac{{k/J}}{{{s^2} + \frac{B}{J}s + \frac{k}{J}}}\)

Now, compare it with the standard equation:

\(H\left( s \right) = \frac{{{\omega _n}}}{{{s^2} + 2\xi {\omega _n}s + \omega _n^2}}\)

Now, settling time

\( = \frac{1}{{\xi {\omega _n}}} = \frac{1}{{\frac{B}{{2J}}}} = \frac{{2J}}{B}\)

Hence only settling time is not affected by system gain k.