Question

A system with the transfer function $\frac{Y\left(s\right)}{X\left(s\right)}=\frac{s}{s+p}$ has an output $y\left(t\right)=\cos (2t-\frac{\pi }{3})$ for the input signal $x\left(t\right)=p\cos (2t-\frac{\pi }{2}).$ Then, the system parameter ${}^{\prime }{p}^{\prime }$ is

• A. $\sqrt{3}$
• B. $\frac{2}{\sqrt{3}}$
• C. $1$
• D. $\frac{\sqrt{3}}{2}$

Answer 1

The correct option is B $\frac{2}{\sqrt{3}}$

Phase difference between input and output,
$\varphi =-\frac{\pi }{3}-(-\frac{\pi }{2})=\frac{\pi }{6}={30}^{\circ }$
$and\omega =2rad/s$
From the transfer function
$\varphi ={90}^{\circ }-{\tan }^{-1}\frac{\omega }{p}$
${90}^{\circ }-{\tan }^{-1}\frac{2}{p}={30}^{\circ }$ ${\tan }^{-1}\frac{2}{p}={60}^{\circ }$
$\frac{2}{p}=\tan {60}^{\circ }=\sqrt{3}$
$p=\frac{2}{\sqrt{3}}$