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Construction of Root Locus
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The lesson covers steps to construct Root Locus

Pawan Bhatla
Currently working for an Engineering Consultancy. I have completed my Instrumentation Engineering from Swami Vivekanand college. Educating i

1. ROOT LOCUS

2. DEFINITION THE LOCUS OF ROOTS OF CHARACTERISTIC EQUATION WHEN GAIN IS VARIED FROM ZERO TO INFINITY IS CALLED ROOT LOCUS. i.e it is the plot of characteristic equation poles of closed loop system Characteristic Equation1G(s).H(s)0

3. Steps for Construction of Root Loci Step 1 Root Locus is symmetrical about the real axis.

4. Steps for Construction of Root Loci Step 2: The root loci starts from an open loop pole and terminates either on open loop zero or at infinity Pole to Zero OR Pole to Infinity

5. Steps for Construction of Root Loci Step 3 : Find the Starting and Ending point of the root loci. G(s)H(s) K(s+5) (s+3) Open loop pole s -3 Open loop zero s -5 Hence, the root locus starts from s--3 and terminates at s-5

6. Steps for Construction of Root Loci Step 3 : Find the Starting and Ending point of the root loci. G(s)H(s)K (s+3) Open loop pole s -3 Open loop zero s 0 Hence, the root locus starts from s--3 and terminates at s INFINITY

7. Steps for Construction of Root Loci Step 4 The angle of asymptotes -(2K + 1)180 ( -0, 1, 2, 3, ) The branches of root locus tend to infinity along a straight line called asymptotes. If P-Z=2 the K = 0, 1 (P-Z) If P-Z 3 then K 0, 1, 2

8. Steps for Construction of Root Loci Example: G(s)H(s) P=3, Z=0 P-Z = 3 1 = (2x0 + 1180 3 Oz = (2x1 + 1180 ,-(2x2 + 1180 1= 60 1800 9,-3000

9. Steps for Construction of Root Loci Step 5: Centroid of Asymptotes The intersection of Asymptotes on real axis. CA= Real part of poles- Real part of zeros P- Z CA= (-3 +-2 +-1 )-(0) CA2

10. Steps for Construction of Root Loci 1. Centroid at s-2 2. Asymptotes starting from the centroid at angles 60, 180 and 300 S -3 S -1 0

11. Steps for Construction of Root Loci ldentification of break away/in point 1. Between 2 poles or 2 zeros 2. Odd numbers of poles and zeros to the right hand side. S -3 s--2 S -1 0

12. Steps for Construction of Root Loci Location of Break away point 5. -(3s2 +12s +11) 0 6. s-1.42 and s-2.5 Substitute the value of s in the equation : K (S+3)(s+2)(s+1) If K is positive, the break away point is valid. In this case break away point is s1.4

13. Steps for Construction of Root Loci Step 7: Intersection with imaginary axis Routh Hurwitz: s3+6s+11s 6 K 0. s3 11 K 60 Auxillary Equation: 6s2+ 66 0 s K-60 0 6 s +3.32j and -3.32j

14. Steps for Construction of Root Loci 1. Centroid at s--2 2. + 3.32j Asymptotes starting from the centroid at angles 60, 180 and 300 Points of intersection with imaginary axis. +3.32j and -3.32j 3. S -3 3.32j

15. RECAP Step 1 Root locus is always symmetrical about its real axis Step 2 : Root locus starts from the pole and terminates on the zero or infinity Step 3: Find Starting and End points Step 4: Angles of Asymptotes Step 5: Centroid of Asymptotes Step 6: Break Away/In point Step 7: Intersection with Imaginary Axis

16. THANK YOU