Calibration Equation

The value of $K$ for a specific point on the root-locus can be calculated using equation (2.10),

$$K=1/\left|G_o(s)\right|$$

(remembering that our base characteristic is $1+K{G}_o(s)=0$)

With

$$\left|G_o(s)\right|=\frac{\left|s-z_1\right|\left|s-z_2\right|\dots \left|s-z_m\right|}{\left|s-p_1\right|\left|s-p_2\right|\dots \left|s-p_n\right|}=\frac{{\prod }_1^n\text{ distance from }s\text{ to open loop zeros }}{{\prod }_1^m\text{ distance from }s\text{ to open loop poles }}$$

this leads to the calibration equation

$$K=\frac{1}{\left|G_o(s)\right|}=\frac{{\prod }_1^n\text{ distance from }s\text{ to open loop poles }}{{\prod }_1^m\text{ distance from }s\text{ to open loop zeros }}$$