Question

The number of times the Nyquist plot of
$G\left(s\right)=\frac{s-1}{s+1}$
will encircle the origin clockwise is

• A. 1

Answer 1

The correct option is A 1

$G\left(s\right)=\frac{s-1}{s+1}G\left(j\omega \right)=\frac{j\omega -1}{j\omega +1}$
$=1\angle {180}^o-2ta{n}^{-1}\omega$
on varying $\omega =0^{+}\to +\infty$
Now on varying $\omega =0^{-}\to -\infty$
$G\left(j\omega \right){|}_{\omega =0}=1\angle {180}^o$
$G\left(j\omega \right){|}_{\omega =0^{-}}=1\angle {180}^o$
$G\left(j\omega \right){|}_{\omega =1}=1\angle {90}^o$
$G\left(j\omega \right){|}_{\omega =-1}=1\angle {270}^o$
$G\left(j\omega \right){|}_{\omega =\infty }=1\angle 0^o$
$G\left(j\omega \right){|}_{\omega =-\infty }=1\angle {360}^o$
On plotting,


$\therefore$No.of times the Nyquist plot encircle the origin clockwise = 1.