The open loop transfer function of a unity gain negative feedback system is given as G(s)= 1/s(s+1) The Nyquist contour in the s-plane encloses the entire right half plane and a small neighborhood around the origin in the left half plane, as shown in figure below. The number of encirclements of the point ( -1 + j0) by the Nyquist plot of G(s), corresponding to the Nyquist contour, is denoted as N. Then N equals to

The correct answer is 1.

Step 1: Identify the poles of the open-loop transfer function
The given open-loop transfer function is:
$G\left(s\right)=\frac{1}{s(s+1)}$
The open-loop poles are the roots of the denominator:
$s(s+1)=0\Rightarrow s=0,-1$
So, we have a pole at the origin ($s=0$) and another pole at $s=-1$.

Step 2: Determine the open-loop poles enclosed by the Nyquist contour
Looking at the provided contour in the s-plane:

The contour encloses the entire right half-plane (RHP) and wraps around the origin into the left half-plane (LHP) via a small semi-circle of radius $ϵ\to 0$.
- The pole at $s=0$ lies inside this small neighborhood and is therefore enclosed by the contour.
- The pole at $s=-1$ is outside this small neighborhood and is not enclosed.
Thus, the number of open-loop poles enclosed by the contour is:
$P=1$

Step 3: Determine the closed-loop poles enclosed by the contour
The characteristic equation for the closed-loop system is:
$1+G\left(s\right)=0\Rightarrow 1+\frac{1}{s(s+1)}=0$
Simplifying this gives:
$s^2+s+1=0$
The roots (closed-loop poles) are:
$s=-0.5\pm j\frac{\sqrt{3}}{2}$
These roots lie in the left half-plane (with a real part of $-0.5$), well outside the small neighborhood around the origin. Therefore, no closed-loop poles are enclosed by the contour:
$Z=0$

Step 4: Calculate the number of encirclements (N)
Using the Principle of the Argument (Nyquist stability criterion), the number of counter-clockwise encirclements $N$ of the point $-1+j0$ by the Nyquist plot is given by:
$N=P-Z$
Substituting the values we found:
$N=1-0=1$
Thus, the number of encirclements $N$ of the point $-1+j0$ is equal to 1.