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 = ωn2
∴ \(\omega _n=\sqrt{G}\) .... (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),
\(\large{\zeta=\frac{1}{\sqrt{G}}}\)