Nyquist contour mapping of 300/[(s+3)(s+1)(s+2)]

Nyquist contour

The Nyquist contour is a closed path in the $s$-plane that encloses the entire right half plane. It runs up the whole $j\omega$ axis from $-j\infty$ to $+j\infty$ and returns along a semicircle of infinite radius, traversed clockwise. If $G(s)H(s)$ has poles on the $j\omega$ axis, the contour goes around them by small semicircles of radius $\epsilon\to0$ so they are not on the path.

        jw
        |\  +j inf
        | \
        |  \  R -> inf
   -----+---)----- sigma
        |  /  (clockwise)
        | /
        |/  -j inf

Mapping this contour through $G(s)H(s)$ gives the Nyquist plot; by the principle of the argument, $N=Z-P$ (clockwise encirclements of $-1$), so $Z=P+N$.

Mapping $G(s)=\dfrac{300}{(s+1)(s+2)(s+3)}$

Poles at $-1,-2,-3$: $P=0$, none on the $j\omega$ axis.

$$G(j\omega)=\frac{300}{(6-6\omega^2)+j(11\omega-\omega^3)}$$

Section I ($\omega=0\to+\infty$):

• Imaginary-axis crossing: $6-6\omega^2=0\Rightarrow\omega=1$, $G=\dfrac{300}{j10}=-j30$.
• Real-axis crossing: $11\omega-\omega^3=0\Rightarrow\omega=\sqrt{11}=3.317$, $G=\dfrac{300}{6-66}=-5$.

Section II (infinite semicircle, $s=Re^{j\theta}$): $G\approx\dfrac{300}{R^3}e^{-j3\theta}\to0$; maps to the origin.

Section III ($\omega=-\infty\to0$): mirror image of section I about the real axis.

              Im
              |   ___ w<0
         _____|__/   \
 --o--o--+----+-------o--- Re
  -5  -1 \____|__    / 50
    w=3.317   |  \__/ w>0
              | -j30 (w=1)

Nyquist criterion

The plot crosses the negative real axis at $-5$, to the left of $-1$. Following the curve, the point $-1+j0$ is encircled twice in the clockwise direction: $N=2$.

$$Z=P+N=0+2=2$$

Two closed-loop poles lie in the right half plane, so the closed-loop system is unstable. (Check: $s^3+6s^2+11s+306=0$ has roots $-8.74$ and $1.37\pm j5.75$.) GM $=1/5=0.2$ ($-14$ dB); the gain must be below $300/5=60$ for stability.

Answer: N = 2, P = 0, Z = 2 – the closed-loop system is unstable.