Question

The open loop transfer function of a system is $G\left(s\right)=\frac{s^2+6s+10}{s^2+2s+2}.$ The angles of arrival of its root loci are

• A. $\pm \frac{\pi }{4}$
• B. $\pm \frac{\pi }{3}$
• C. $\pm \frac{\pi }{2}$
• D. $\pm \frac{5\pi }{6}$

Answer 1

The correct option is A $\pm \frac{\pi }{4}$

$\varphi ={180}^o-\angle GH$
$G\left(s\right)=\frac{(s+3+i)(s+3-i)}{(s+1+i)(s+1-i)}$
$G(-3+i)=\frac{[-3+i+3+i]}{[-3-i+1+i][-3+i+1-i]}$
$=\frac{\left[2i\right]}{[-2+2i][-2]}$
$=\frac{\left[i\right]}{[2-2i]}$
$\angle G(-3+i)={90}^o-[-ta{n}^{-1}\frac{2}{2}]$
$={90}^o+{45}^o$
$={135}^o$
${\varphi }_a={180}^o-{135}^o={45}^o=\frac{\pi }{4}$
$\therefore Otherangle,{\varphi }_d=-\frac{\pi }{4}$
$\therefore Anglesofarrival=\pm \frac{\pi }{4}$