Question

Consider a system with the transfer function $G\left(s\right)=\frac{s+6}{K{s}^2+s+6}$. Its damping ratio will be $0.5$ when the value of $K$ is

• A. $2/6$
• B. $3$
• C. $1/6$
• D. $6$

Answer 1

The correct option is C $1/6$

$\frac{s+6}{K(s^2+\frac{s}{K}+\frac{6}{K})}$
Comparing with $s^2+2\xi {\omega }_n+{\omega }_n^2$
${\omega }_n=\sqrt{\frac{6}{K}}$
$2\xi {\omega }_n=\frac{1}{K}$
$2\times 0.5\times \sqrt{\frac{6}{K}}=\frac{1}{K}$ $\Rightarrow \frac{6}{K}=\frac{1}{K^2}$
$K=\frac{1}{6}$