Centroid & Angle of Asymptotes

Introduction :

The conditions for determining the centroid and the angles of asymptotes in the root locus are essential to understanding the behavior of a control system as the gain (proportional gain, K) varies. These conditions help in visualizing the root locus, and they are based on the characteristics of the system's open-loop transfer function (OLTF), G(s)H(s). Here are the conditions for finding the centroid and angles of asymptotes

Centroid of Asymptotes

1. Determine the number of poles (P) and zeros (Z) of the OLTF G(s)H(s). Count all the poles and zeros, including those in the left-half (LHP) and right-half (RHP) of the complex plane. P and Z may be complex conjugate pairs counted as a single pole or zero.
2. Calculate the sum of the real parts of the poles of G(s)H(s) in the LHP:
3. Sum of Real Parts of Poles in LHP = ∑ Real Parts of Poles in LHP
4. Calculate the sum of the real parts of the zeros of G(s)H(s) in the LHP:
5. Sum of Real Parts of Zeros in LHP = ∑ Real Parts of Zeros in LHP
6. Calculate the difference between the sum of real parts of poles and the sum of real parts of zeros:

Centroid = ( Sum of Real Parts of Poles in LHP − Sum of Real Parts of Zeros in LHP) ÷ (P-Z)

The centroid is the point where the asymptotes intersect, and the root locus branches approach this point as the gain K varies.​

Angles of Asymptotes

θa= ( 2 n + 1 ) π/N ​

• θa is the angle of an asymptote.
• N is the number of poles (or zeros) to the left of the point where the asymptote starts.
• n is an integer ranging from 0 to N-1.

Step for find the angles of asymptotes:

1. Count the total number of poles (P) to the left of the centroid. These are the poles of G(s)H(s) in the LHP.
2. If there are any finite zeros to the left of the centroid, count them as well.
3. Use the formula to calculate the angles of asymptotes for all P and Z. These angles represent the directions in which the root locus branches approach the centroid.