Construction Rules of Root Locus

Following are the rules which need to be follow step by step for construction of Root Locus .

RULE 1 : Write down the characteristic equation 1 + G(s) H(s) = 0 and find open loop poles and zeros . Plot these open loop poles and zeros in s – plane . Root Locus start from open loop poles ( k=0 ) and ends at open loop zeros or infinite ( k= infinite) .

RULE 2 : Determine the number of branches of Root Locus , N

N = P if P > Z (if number of poles is grater than number of zeros than total number of branches is equal to number of poles.)

N = Z if P < Z (if number of zeros is grater than number of poles than total number of branches is equal to total number of zeros.)

N = P = Z if P = Z

where N = number of branches ; P = number of Poles ; Z = number of Zeros .

RULE 3 : The existence of root locus on a section of real axis is confirmed if sum of open loop poles and zeros to right of that point is odd number.

RULE 4 : Find Break away or Break in Point of root locus . It can be find out by writing the k in term of characteristic equation in s and then differentiating with respect to s and putting equals to 0 and solve for s .

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

dk / ds =0

When two poles are placed adjacent to each other and and there exist root locus in between them then there must be minimum one Break Away Point between them.

When two zeros are placed adjacent to each other and and there exist root locus in between them then there must be minimum one Break In Point between them.

RULE 5 : Find Centroid point . Centroid is point where asymptotes intersect real axis . It is given as .

x = ( ∑ Real Part of pole − ∑ Real Part of Zeroes ) / (P -Z )

RULE 6 : Find the angle of Asymptotes . It can be determined from below given formula .

θ = ( 2 m + 1 ) 180⁰ / ( P -Z ) ; where m = 0 , 1 , 2 , 3 , …. , P- Z – 1

RULE 7 : To determine value of k and point at which root locus intersect imaginary axis we apply Routh Array Criteria in characteristic equation and find the value of k by solving auxiliary equation .

RULE 8 : Angle of departure is to be find out only for complex poles and its conjugate .

ϕ_d = 180⁰ – ( ϕ_p – ϕ_z ) ; where ϕ_d = Angle of departure , ϕ_P = sum of all the angle substituted to that pole by remaining poles , ϕ_Z = sum of all the angle substituted to that pole by remaining zeros.

RULE 9 : Angle of Arrival is to be find out only for complex zeros and its conjugate.

ϕ_a = 180⁰ + ( ϕ_P – ϕ_Z ) ; where ϕ_a = Angle of departure , ϕ_P = sum of all the angle substituted to that zero by remaining poles , ϕ_Z = sum of all the angle substituted to that pole by remaining zeros.

The Root Locus is the Technique to identify the roots of Characteristics Equations Within a Transfer Function. It Follows the Process of Plotting the roots on a graph, Showcasing their Variations across Different Parametric Values. In this article, we will be going through What is Root Locus?, Angle and Magnitude Conditions, Construction Rules of Root Locus, and At last we will solve the Examples.