Angle of Departure

Q. The open loop transfer function of a unity gain negative feedback control system is given by

$G\left(s\right)=\frac{s^2+4s+8}{s(s+2)(s+8)}$

The angle $\theta$, at which the root locus approaches the zeros of the system, satisfies

1. $|\theta |=\pi -ta{n}^{-1}\left(\frac{1}{4}\right)$
2. $|\theta |=\frac{3\pi }{4}-ta{n}^{-1}\left(\frac{1}{3}\right)$
3. $|\theta |=\frac{\pi }{2}-ta{n}^{-1}\left(\frac{1}{4}\right)$
4. $|\theta |=\frac{\pi }{4}-ta{n}^{-1}\left(\frac{1}{3}\right)$

Q. The feedback configuration and the pole-zero location of $G\left(s\right)=\frac{s^2-2s+2}{s^2+2s+2}$ are shown below. The root locus for Negative values of $K$, i.e for $-\infty <K<0$ , has breakaway/break-in points and angle of departure at pole $P$ (with respect to the positive real axis) equal to

1. $\pm \sqrt{2}$ and $0^{\circ }$
2. $\pm \sqrt{2}$ and ${45}^{\circ }$
3. $\pm \sqrt{3}$ and $0^{\circ }$
4. $\pm \sqrt{3}$ and ${45}^{\circ }$

Q. Which of the following points is NOT on the root locus of a system with the open-loop transfer function
$G\left(s\right)H\left(s\right)=\frac{K}{s(s+1)(s+3)}?$

1. $s=-j\sqrt{3}$
2. $s=-1.5$
3. $s=-3$
4. $s=-\infty$