Correct Answer - Option 2 : 1.6 s
Concept:
The characteristic equation is given by:
1 + G(s) H(s) = 0
Also, the standard second-order characteristic equation is:
\({s^2} + 2\zeta {\omega _n}s + \omega _n^2 = 0\)
Settling Time is the time taken by the response to reach ± 2%, tolerance band.
\({e^{ - \xi {\omega _n}{t_s}}} = \pm 5\% \;\left( {or} \right) \pm 2\% \)
\({t_s} \simeq \frac{3}{{\xi {\omega _n}}}\) for a 5% tolerance band.
\({t_s} \simeq \frac{4}{{\xi {\omega _n}}}\) for 2% tolerance band
Calculation:
Characteristic equation:
s2 + 5s + 25 = 0
By comparing this with the standard second-order equation, we get:
\(\omega _n^2 = 25 \Rightarrow {\omega _n} = 5\)
\(2\zeta {\omega _n} = 5\)
\(\Rightarrow \zeta = \frac{5}{{2 \times 5}} = 0.5\)
Settling time is, therefore:
\({t_s} = \frac{4}{{0.5 \times 5}} = 1.6\;sec\)
