Suppose we have the following transfer function:
$G(s) = K \frac{s+3}{s(s+1)}$
Given the above Nyquist path
I want to sketch Nyquist plot. By using the Nyquist plot rules, I made the following sketch:
According to Nyquist plot, there is counter-clockwise encirclement of $-1$.
$N = P - Z$
where $P$ and $Z$ are the number of open loop and closed loop poles enclosed by the path, respectively. Since there is CCW encirclement of $-1$, then $N = 1$. There is also $1$ open loop poles which is enclosed by the path, so $P = 1$. According to this information, the number of closed loop poles $Z$ is zero. But if you quickly sketch a root locus, we see that this is not true. Can you help me out for finding my mistake?
When using a Nyquist plot using a different contour than the one encircling the right half plane, then the meaning of the variables in the equation $N=P-Z$ changes. Namely, those variables represent the number of poles/zeros that lie inside the used contour.
If you calculate the closed poles for $K=1$, by solving $s^2+2s+3=0$, yields $s=-1\pm\,i\sqrt{2}$, which lies outside the the selected contour.