Angle Criterion

Now if the zeros of the open-loop system $G_o(s)$ are at $z_1\dots z_m$, and the poles are at $p_1\dots p_n$, then we can write $G_o(s)$ as

$$G_o(s)=\frac{\left(s-z_1\right)\left(s-z_2\right)\dots \left(s-z_m\right)}{\left(s-p_1\right)\left(s-p_2\right)\dots \left(s-p_n\right)}$$

and $G(s)$ is

$$G(s)=K\frac{\left(s-z_1\right)\left(s-z_2\right)\dots \left(s-z_m\right)}{\left(s-p_1\right)\left(s-p_2\right)\dots \left(s-p_n\right)}$$

It follows that

$$\begin{array}{rl}\arg G(s)=\arg G_o(s)=& \arg \left(s-z_1\right)+\arg \left(s-z_2\right)+\cdots \\ & -\arg \left(s-p_1\right)-\arg \left(s-p_2\right)-\cdots \end{array}$$

Using the angle condition ($arg{G}_o(s)=-180^o\pm n360^o,n=0,1,2,...$), this leads to the angle criterion: $\sum$ angles subtended from zeros $-\sum$ angles subtended from poles $=-180^{\circ }\pm k.360^{\circ }$ with $k=0,1,2,3,\dots$.