Question

The output of a standard second-order system for a unit step input is given as $y\left(t\right)=1-\frac{2}{\sqrt{3}}{e}^{-t}\cos (\sqrt{3}t-\frac{\pi }{6}).$The transfer function of the system is

• A. $\frac{2}{(s+2)(s+\sqrt{3})}$
• B. $\frac{1}{s^2+2s+1}$
• C. $\frac{3}{s^2+2s+3}$
• D. $\frac{4}{s^2+2s+4}$

Answer 1

The correct option is D $\frac{4}{s^2+2s+4}$

The output of a second order system for unit step input
$y\left(t\right)=1-\frac{e^{-\xi {\omega }_{n}t}}{\sqrt{1-{\xi }^2}}\sin ({\omega }_{d}t-\theta )....\left(i\right)$

From given data $y\left(t\right)=1-\frac{2}{\sqrt{3}}{e}^{-t}\cos (\sqrt{3}t-\frac{\pi }{6})$

$\therefore \alpha =\xi {\omega }_n=1$

$and{\omega }_d=\sqrt{3}$

${\omega }_n=\sqrt{{\alpha }^2+{\omega }_d^2}=2$

$\xi =\frac{1}{2}$

Therefore, the transfer function
$T\left(s\right)=\frac{{\omega }_n^2}{s^2+2\xi {\omega }_{n}s+{\omega }_n^2}=\frac{4}{s^2+2s+4}$