Question

The dominant poles of a servo system are located at $s=(-2\pm j2)$. The damping ratio of the system is

• A. 1
• B. 0.8
• C. 0.707
• D. 0.6

Answer 1

The correct option is C 0.707

$T\left(s\right)=\frac{1}{(s-(-2+j2\left)\right)(s-(-2-j2\left)\right)}$
$=\frac{1}{(s+2-j2)(s+2+j2)}$
$\frac{1}{(s+2{)}^2+4}=\frac{1}{s^2+4s+4+4}$
$=\frac{1}{s^2+4s+8}$
${\omega }_n^2=8$
$\Rightarrow {\omega }_n=\sqrt{8}$
$2\xi {\omega }_n=4$
$\therefore \xi =\frac{4}{2.\sqrt{8}}=\frac{2}{\sqrt{8}}=\frac{1}{\sqrt{2}}=0.707$