# Root Locus

• It is the graphical representation of the roots of the characteristic equation.

Rules for Construction of Root Locus :

• The root locus always starts (K = 0) from the open loop poles & ends (K= ) on either finite open-loop zeros or infinity.
• The root locus is always symmetrical with respect to the real axis. The number of separate branches (N) of root locus are N = P if P > Z or N = Z if Z > P.
• The section of root locus lies on real axis, if the sum of total number of open loop poles & zeros is odd.
• Break away point: It is a point calculated when the root locus lies between two poles & is measured by dK/ds = 0.
• Angle of asymptotes: It is calculated as given
where k = 0, 1, ..... (P â€“ Z).
• Centroid: The asymptotes meet the real axis at centroid, given by

• Intersection points with imaginary axis: The value of K (forward path gain) & the point at which the root locus branch crosses the imaginary axis is calculated by applying Routh test to the characteristic equation. The roots at the intersection point are imaginary.
• Angle of departure: It is calculated when we have complex poles & is
where  are sum of all angles subtended by remaining poles & by zeros, respectively. Angle of departure is the tangent to the root locus at the complex pole.
• Angle of arrival: It is calculated when we have complex zeros & is calculated as

where are sum of all angles subtended by remaining zeros & by poles, respectively. The angle of arrival is the tangent to the root locus at the complex zero.

Fig.