Correct Answer - Option 3 : 1/√G

__Concept__:

Standard second-order closed-loop transfer function is given by:

\(\frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{ω _n^2}}{{{s^2} + 2\xi {ω _n}s + ω _n^2}}\) .... (1)

\(\frac{C(s)}{R(s)}=\frac{{G}}{{{s^2} + 2\xi {ω _n}s + ω _n^2}}\) ... (2)

ξ = damping ratio

ωn = undamped natural frequency

G = Gain

From equation (1) and (2),

G = ω_{n}^{2}

∴ \(\omega _n=\sqrt{G}\) .... (3)

Now we have to find the pole of the system, and it will be located at,

\(s=-\zeta \omega_n\pm\omega_n \sqrt{1-\zeta^2}\)

The decaying exponential has a time constant equal to,

\(1=\frac{1}{\zeta \omega_n}\)

∴ \(\zeta =\frac{1}{\omega_n}\) .... (4)

From equation (3) and (4),

\(\large{\zeta=\frac{1}{\sqrt{G}}}\)