Criteria for a point to lie on root locus

A point $s_1$ in the s-plane lies on the root locus only if it is a root of the characteristic equation $1 + KG(s)H(s) = 0$ for some $K \ge 0$. Writing $KG(s_1)H(s_1) = -1 = 1\angle \pm 180^\circ$ gives two criteria.

1. Angle criterion (decides whether the point is on the locus)

$$\angle G(s_1)H(s_1) = \sum_{j} \angle(s_1 - z_j) - \sum_{i} \angle(s_1 - p_i) = \pm(2q+1)\,180^\circ$$

• Draw vectors from every open-loop pole and zero to $s_1$ and measure their angles from the positive real axis.
• If (sum of zero angles) − (sum of pole angles) is an odd multiple of $180^\circ$, the point is on the locus. Otherwise it is not.
• $K$ does not appear in this condition, so it alone fixes the shape of the locus.

2. Magnitude criterion (gives K at that point)

$$|K\,G(s_1)H(s_1)| = 1 \;\Rightarrow\; K = \frac{\prod_i |s_1 - p_i|}{\prod_j |s_1 - z_j|}$$

That is, K is the product of the distances from the poles divided by the product of the distances from the zeros (1 if there are no zeros). This is used only after the angle criterion is satisfied.

Example

$G(s) = \dfrac{K}{s(s+4)}$, test point $s_1 = -2 + j2$:

• $\angle s_1 = 135^\circ$ and $\angle(s_1 + 4) = 45^\circ$.
• Sum $= -(135^\circ + 45^\circ) = -180^\circ$, so the point is on the locus.
• $K = |s_1|\,|s_1 + 4| = 2.83 \times 2.83 = 8$.

Related real-axis rule

A real-axis point is on the locus if the number of real poles plus zeros to its right is odd. This rule follows directly from the angle criterion.