Nyquist formulas

The contour is traversed clockwise, encloses $Z$ zeros and $P$ poles of $F(s)$, and passes through none of them. $N_0$ = net number of clockwise encirclements of the ORIGIN by the image of $F(s)$. A counterclockwise encirclement counts as $-1$.

For a type-$r$ loop with low-frequency gain constant $K_r$. The image angle runs from $+r\cdot 90^{\circ }$ (at $\omega =0^{-}$) to $-r\cdot 90^{\circ }$ (at $\omega =0^{+}$), with radius $\to \infty$ and always clockwise. If $K_r<0$ (for example, when an open-loop RHP pole makes the low-frequency gain negative), the arc is rotated by $180^{\circ }$ but still runs clockwise.

Convention: $N$ = net number of CLOCKWISE encirclements of $-1+j0$ by the Nyquist plot of $G(s)H(s)$ as $s$ goes clockwise around the Nyquist contour. Each counterclockwise encirclement counts $-1$. $P$ = number of open-loop poles (poles of $GH$) in the open RHP; $j\omega$-axis poles are left out by the indentation. $Z$ = number of closed-loop poles in the RHP. Valid only if the plot does not pass through $-1+j0$.

The closed loop is stable if and only if the Nyquist plot encircles $-1+j0$ exactly $P$ times COUNTERCLOCKWISE. For $P=0$: stable if and only if $-1+j0$ is not encircled at all ($N=0$).

For $GH=K{G}_1{H}_1$ with $K>0$, the plot of $K{G}_1{H}_1$ encircles $-1$ exactly as the plot of $G_1{H}_1$ encircles $-1/K$. Sketch $G_1{H}_1$ once, then slide the critical point along the negative real axis to get the range of $K$. For $K<0$ the critical point $-1/K$ lies on the positive real axis instead.

For a positive gain, $r$ poles at the origin, $n$ poles and $m$ zeros in total, and every other pole and zero in the LHP. As $\omega \to 0^{+}$ the magnitude tends to $\infty$ for $r\ge 1$ and to $GH(0)>0$ for $r=0$. As $\omega \to \infty$ it tends to $0$ when $n>m$. Each RHP pole shifts the starting angle by $-180^{\circ }$ and each RHP zero by $+180^{\circ }$. The end angle is unchanged.

Holds for $a,b>0$ with negative feedback. The plot crosses the negative real axis once. The closed loop is stable for $0<K<ab(a+b)$ and oscillates at $\sqrt{ab}$ rad/s when $K=K_{\max }$. The low-frequency vertical asymptote is $\text{Re}=-K(a+b)/(ab{)}^2$.

The frequency at which the polar plot cuts the negative real axis (the angle is taken modulo $360^{\circ }$).

Here the plot crosses the negative real axis at $-a$ ($a>0$). GM is the factor by which the loop gain can be multiplied before that crossing reaches $-1$. If the plot never reaches $-180^{\circ }$ at a finite frequency (no phase crossover), GM $=\infty$.

${\omega }_{gc}$ is where the plot cuts the unit circle. Measure the angle as a negative number between $0^{\circ }$ and $-360^{\circ }$: $\angle GH=-140^{\circ }$ gives PM $=40^{\circ }$. PM is the extra phase lag at ${\omega }_{gc}$ that would swing that point onto $-1$. If $|GH|<1$ at every frequency, there is no gain crossover and PM is taken as $\infty$.

A delay $e^{-sT}$ has $|e^{-j\omega T}|=1$ and phase $-\omega T$; on its own it traces the unit circle clockwise. It rotates every point of the plot clockwise by $\omega T$ without changing its magnitude, so at ${\omega }_{gc}$ it uses up phase margin. Valid when the delay-free closed loop is stable with PM $>0$ and the plot cuts the unit circle only once.

Unity negative feedback with $\zeta >0$, so the closed loop is $\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$. PM depends on $\zeta$ alone. Example: $\zeta =0.5$ gives PM $\approx 51.8^{\circ }$. As a rough check only, PM in degrees is close to $100\zeta$ for $0<\zeta \le 0.6$ (within about $3.5^{\circ }$). Use the exact formula for a numerical answer.