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.

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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

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

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

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

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

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

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

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

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

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

Vector Representation of Complex Number 11 Lecturer : IN SOKVAN

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

Vector Representation of Complex Number General 13 Lecturer : IN SOKVAN

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

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

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

Defining Root Locus 17 Lecturer : IN SOKVAN

Defining Root Locus 18 Lecturer : IN SOKVAN

Root Locus Defining Root Locus 19 Lecturer : IN SOKVAN

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

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

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

Magnitude Condition Angle condition Properties of Root Locus 23 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? 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

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

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

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

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

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

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?