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

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 =} 
                                                                                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.                     

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.

 

 

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

Ʃɸ_Z  -  Ʃɸ_P = 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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