Construction of root locus

Root locus


Root locus is defined as the locus of closed loop poles when system gain is varied from to zero to infinity. It determine the relative stability of system.


Steps to design a root locus



Step 1:- The root locus is symmetrical about real axis


Step 2:- Let 

P = number of open loop poles


 Z  =  number of open loop zeroes


And P > Z then


No. of branches of root locus = P


Number of branches terminating at zeroes is equal to Z


Number branches terminating at infinity P - Z


Step 3:- A point on real axis is said to be on root locus if to right side of this point the sum of open loop poles and zeroes is odd. Root locus is outward from poles and inwards to zeroes

root locus

Step 4:-  Angle of asymptotes

 

The p-z branches terminating at infinity will go along certain straight lines known as asymptotes of root locus.


The no of asymptotes = P – Z

Angle of asymptotes   ɵ = root locus
   
                                                                            where q = 0,1,2,3,……………



Step 5:-  Centroid


It is the intersection of asymptotes on the real axis. It may or may not be part of root locus.

Centroid   =         



Step 6:-  Breakaway points


They are those points where multiple roots of the characteristics equation occur.

1 + G(s).H(s) = 0



Write in terms of K

dk/ds  =  0  will give breakaway point



Whenever there are two adjacently placed poles on the real axis with the section of real axis between them as a part of root locus then there exist a breakaway point between the adjacently placed poles.                     

root locus

Whenever there are two adjacently placed zeroes on the real axis with the section of real axis between them as a part of root locus then there exist a breakinn point between the adjacently placed zeroes.

 

root locus

 

Step 7:-  Intersection of root locus with imaginary axis



The roots of auxilary equation at K = K marginal from Routh Array gives intersection of root locus with imaginary axis.

 

Step 8:-  Angle of departure and angle of arrival


Ø Angle of departure is obtained when complex poles terminate at infinity.
Ø The angle of arrival is obtained at complex zeroes


ɸd  =  1800 + ɸ

ɸA  =  1800 - ɸ


ƩɸZ  ƩɸP = ɸ


 
Angle and magnitude condition.


Ø The angle condition is used for checking whether the certain points lies on root locus or not.

1+G(s).H(s) = 0

G(s).H(s) = -1 + j0

ÐG(s).H(S) =  1800 – tan-1(0/1)   1800

   ≈ ±[2q+1]1800



The angle condition may be stated as “for a point to lie on root locus the angle evaluated at that point should be an odd multiple of ±180



Ø The magnitude condition is used for finding the value of system gain k at any point on the root locus.

1+G(s).H(s) = 0

G(s).H(s) = -1 + j0

|G(s).H(s)| = √((-1²) + 0²)  =  1

|G(s).H(s)| =  1


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