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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= 749.png) 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
    754.png where k = 0, 1, ..... (P – Z).
  • Centroid: The asymptotes meet the real axis at centroid, given by
    759.png 
  • 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 764.png
    where 769.png 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
    774.png
    where 779.pngare 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. 

784.png 

Fig.

 





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