Note

Magnitude and Angle Conditions

A point $s_0$ is on the root-locus of a system with open-loop transfer function $L(s)$ if it satisfies the following magnitude and angle conditions:

|k| \,|L(s_0)| =1 \\ \\ \angle k + \angle L(s_0) = 180 \degree \pm 360 \degree \\ \\ \text{or} \quad \angle k + \angle L(s_0) = -180 \degree \pm 360 \degree

If a point $s_0$ is known to belong to the root locus, the same equations can be used to calculate the corresponding value of $k$

|k| = \frac{1}{|L(s_0)|}

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