Angle criterion for root locus

The root locus passes through a point $s_1$ only if $s_1$ satisfies the angle criterion:

$$\angle G(s_1)H(s_1) = \sum \angle(s_1 - z_j) - \sum \angle(s_1 - p_i) = \pm(2q+1)\,180^\circ$$

with $q = 0, 1, 2, \dots$ To apply it, draw vectors from all open-loop zeros and poles to $s_1$. Then take the sum of the zero angles minus the sum of the pole angles. If the result is an odd multiple of $180^\circ$, the point lies on the root locus for $K > 0$; otherwise it does not. Example: for $G = K/[s(s+2)]$, the point $-1 + j1$ gives $-(135^\circ + 45^\circ) = -180^\circ$, so it is on the locus.