This document discusses the root locus technique for analyzing control systems. It describes the 7 step procedure for constructing a root locus plot: (1) locate poles and zeros, (2) determine the real axis path, (3) find asymptote angles, (4) identify breakaway points, (5) calculate departure and arrival angles, (6) find imaginary axis intersections, (7) sketch the root locus using test points. The root locus technique allows observing how closed-loop poles move in the s-plane as the system gain varies, helping to achieve desired performance.

1. Root locus techniques
content
 Root locus
 Step by step procedure
Root Locus Techniques
Kongunadunadu College of Engineering and Technology Depar tment of EEE

2. ROOT LOCUS
• The root locus techniques was introduced by W.R.Evans in 1948 for
the analysis of control systems.
• The root locus techniques is a powerful tool for adjusting the
location of closed loop poles to achieve the desired system
performance by varying one or more system parameters.
Kongunadunadu College of Engineering and Technology Depar tment of EEE
Root Locus Techniques

3. • path taken by the roots of characteristics equation when
open loop gain K is varied from 0 to α are called root loci or root
Locus.
Step 1: Location of poles and zeros
 Draw the real and imaginary axis on an ordinary graph sheet and
choose same scales both on real and imaginary axis.
 The poles are marked by cross “X” and zeros are marked by “0”.
 The number of root locus branches is equal to number of poles of
open loop transfer function.
 The origin of a root locus is at a pole and the end is at zero.
 Let, n = number of poles
m= number of finite zeros
Now, m root locus branches ends at finite zeros. The remaining n-m
root locus branches will end at zeros at infinity.
Kongunadunadu College of Engineering and Technology Depar tment of EEE
Root Locus Techniques

4. Step 2: Root locus on real axis
 In order to determine the part of root locus on real axis, take a
test
point on real axis.
 If the total number of poles and zeros on the real axis to the right
of this test point is odd number, then the test point lies on the root
locus.
 If it is even then the test point does not lie on the root locus.
Step 3: Angles of asymptotes
• If n is the number of poles and m is the number of finite zeros,
then n-m root locus branches will terminate at zeros at infinity.
• These n-m root locus branches will go along an asymptotic path
and meets the asymptotes at infinity.
• Hence number of asymptotes is equal to number of root locus
branches going to infinity.Kongunadunadu College of Engineering and Technology Depar tment of EEE
Root Locus Techniques

5. Step 4: Breakaway and break-in points
 The breakaway or break in points either lie on real axis or
exists as complex conjugate pairs.
 If there is a root locus on real axis between 2 poles then there
exist a break away point.
 If there is a root locus on real axis between 2 zeros then there
exist a breakaway point.
 If there is a root locus on real axis between poles and zeros
then there may be or may not be breakaway or break in point.
Kongunadunadu College of Engineering and Technology Depar tment of EEE
Root Locus Techniques

6. Kongunadunadu College of Engineering and Technology Depar tment of EEE
Root Locus Techniques

7. Step 5: angle of departure and angle of arrival
Kongunadunadu College of Engineering and Technology Depar tment of EEE
Root Locus Techniques

8. Kongunadunadu College of Engineering and Technology Depar tment of EEE
Root Locus Techniques

9. Kongunadunadu College of Engineering and Technology Depar tment of EEE
Root Locus Techniques

10. Step 6: Point of intersection of root locus with imaginary axis
The point where the root loci intersects the imaginary axis can be
found by following methods.
1. By Routh hurwitz method
2. By trial and error approach.
Step 7: Test points and root locus
 Choose a test point. Using protractor roughly estimate the angles
of vectors drawn to this point and adjust the point to satisfy angle
criterion. Repeat the procedure for few more test points.
 Sketch the root locus from the knowledge of typical sketches and
the Information obtained in steps 1 through 6.
Kongunadunadu College of Engineering and Technology Depar tment of EEE
Root Locus Techniques

11. Reference:
• A.Nagoor Kani, “ Control Systems”, RBA Publications, June
2012.
Kongunadunadu College of Engineering and Technology Depar tment of EEE
Root Locus Techniques