1. Properties of Root Locus Lesson Objectives : After finish this lesson, students will be able to  be aware of the control system problem  be aware of the vector representation of complex number  define the root locus  determine the properties of root locus NPIC Faculty of Electricity Control Engineering 1 Lecturer : IN SOKVAN

2. Lesson Objectives 01  Aware of Control System Problem 2 Lecturer : IN SOKVAN

3. The Prerequisites To Root Locus Technique 1.Understanding the control system problems 2.Vector representation of complex numbers 3 Lecturer : IN SOKVAN

4. The Control System Problem General Feedback Control System Open Loop Transfer Function Also called, Loop Gain Closed-loop Transfer Function 4 Lecturer : IN SOKVAN

5. The Control System Problem What is the control system problem? Difficult to obtain the poles Poles location varied with the gain Let’s find the closed-loop transfer function 5 Lecturer : IN SOKVAN

6. OL poles are roots of: 1.Forward transfer function denominator 2.Feedback transfer function denominator CL poles are roots of: 1.Combinations of numerator and denominator of forward and feedback transfer functions 2.Poles depend on gain, K Comparison Between Open Loop and Closed Loop Systems Therefore, with CL system, the poles are not easily obtained and change with the value of K The Control System Problem 6 Lecturer : IN SOKVAN

7. Given:Open Loop System Poles: 0, -2 and -4 Closed-loop System Poles: We have to factor and also depends on K. Root locus technique help to find poles! The Control System Problem - Example 7 Lecturer : IN SOKVAN

8.  Aware of the Vector Representation of Complex Number 8 Lecturer : IN SOKVAN Lesson Objectives 02

9. What is complex number? Complex number is a vector Vector has magnitude and direction Therefore, we can also represent complex number s as: Vector Representation of Complex Number 9 Lecturer : IN SOKVAN

10. But, if s is a variable in a function, how to represent the complex number. For example, Replacing s, Another complex number Graphically, Vector Representation of Complex Number 10 Lecturer : IN SOKVAN

11. Vector Representation of Complex Number 11 Lecturer : IN SOKVAN

12. Same Vector is a complex number and can be represented by a vector drawn from the zero of the function to the point s. Vector Representation of Complex Number Same Vector 12 Lecturer : IN SOKVAN

13. Vector Representation of Complex Number General 13 Lecturer : IN SOKVAN

14. What is F(s)? Vector Representation of Complex Number Example 14 Lecturer : IN SOKVAN

15.  define the root locus 15 Lecturer : IN SOKVAN Lesson Objectives 03

16. What is Root Locus? The root locus is the path of the roots of the characteristic equation shown out in the s-plane as a system parameter is changed. 16

17. Defining Root Locus 17 Lecturer : IN SOKVAN

18. Defining Root Locus 18 Lecturer : IN SOKVAN

19. Root Locus Defining Root Locus 19 Lecturer : IN SOKVAN

20. Root Locus Representation of the paths of closed- loop poles as the gain is varied What can we learn from this graphic? 0<K<25 The system is over-damped K=25 The system is critically damped K>25 The system is under-damped The system is stable Defining Root Locus 20 Lecturer : IN SOKVAN

21.  Determine the properties of root locus 21 Lecturer : IN SOKVAN Lesson Objectives 04

22. How do we get poles? The value “-1” is a complex number Properties of Root Locus 22 Lecturer : IN SOKVAN

23. Magnitude Condition Angle condition Properties of Root Locus 23 Lecturer : IN SOKVAN

24. Given a unity feedback system forward transfer function 1.Is point -3+0j is on a root locus? 2.If the point is on the root locus, find the value of K? Steps: 1.Determine zeros and poles of the forward transfer function 2.Determine angles from zeros and poles to the interested point 3.Determine the length of vector from zeros and poles to the interested point 4.Add all angles. If it is equal to multiple of 180, then the point is on the root locus. 5.Determine K using length of zero and pole vectors Properties of Root Locus - Example 24 Lecturer : IN SOKVAN

25. 1.Determine zeros and poles of the forward transfer function Using quadratic equation, Properties of Root Locus - Example 25 Lecturer : IN SOKVAN

26. 2.Determine angles from zeros and poles to the interested point Properties of Root Locus - Example 26 Lecturer : IN SOKVAN

27. 3.Determine the length of vector from zeros and poles to the interested point Properties of Root Locus - Example 27 Lecturer : IN SOKVAN

28. 4. Add all angles. If it is equal to multiple of 180, then the point is on the root locus. Therefore, the point -3+0j is a point on the root locus Properties of Root Locus - Example 28 Lecturer : IN SOKVAN

29. 5. Determine K using length of zero and pole vectors Properties of Root Locus - Example 29 Lecturer : IN SOKVAN

30. Home work 30 Lecturer : IN SOKVAN Given a unity feedback system forward transfer function 1.Is point -3+0j is on a root locus? 2.If the point is on the root locus, find the value of K?