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 ID: 935348
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Slide1
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 ElectricityControl Engineering
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Lecturer : IN SOKVAN
Slide2Lesson Objectives 01
Aware of Control System Problem 2Lecturer : IN SOKVAN
Slide3The Prerequisites To Root Locus Technique
Understanding the control system problemsVector representation of complex numbers3
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Slide4The Control System Problem
General Feedback Control SystemOpen Loop Transfer FunctionAlso called, Loop GainClosed-loop Transfer Function
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Slide5The Control System Problem
What is the control system problem?Difficult to obtain the polesPoles location varied with the gainLet’s find the closed-loop transfer function
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Slide6OL poles are roots of:
Forward transfer function denominatorFeedback transfer function denominatorCL poles are roots of: Combinations of numerator and denominator of forward and feedback transfer functionsPoles depend on gain, KComparison Between Open Loop and Closed Loop SystemsTherefore, with CL system, the poles are not easily obtained and change with the value of K
The Control System Problem
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Slide7Given:
Open Loop SystemPoles: 0, -2 and -4Closed-loop SystemPoles: We have to factor and also depends on K. Root locus technique help to find poles!
The Control System Problem - Example
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Slide8Aware
of the Vector Representation of Complex Number 8Lecturer : IN SOKVANLesson Objectives 02
Slide9What is complex number?
Complex number is a vectorVector has magnitude and directionTherefore, we can also represent complex number s as:
Vector
Representation of Complex Number
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Slide10But, if s is a variable in a function, how to represent the complex number. For example,
Replacing s,Another complex numberGraphically,
Vector Representation of Complex Number
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Slide11Vector
Representation of Complex Number
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Slide12Same 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
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Slide13Vector
Representation of Complex Number
General
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Slide14What is F(s)?
Vector
Representation of Complex Number
Example
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Slide15define the root locus
15Lecturer : IN SOKVANLesson Objectives 03
Slide16What 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
Slide17Defining Root Locus
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Slide18Defining Root Locus
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Slide19Root Locus
Defining Root Locus19
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Slide20Root Locus
Representation of the paths of closed-loop poles as the gain is variedWhat can we learn from this graphic?0<K<25The system is over-dampedK=25The system is critically dampedK>25The system is under-dampedThe system is stableDefining Root Locus
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Slide21Determine the properties of root locus
21Lecturer : IN SOKVANLesson Objectives 04
Slide22How do we get poles?
The value “-1” is a complex number
Properties
of
Root Locus
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Slide23Magnitude Condition
Angle condition
Properties
of
Root Locus
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Slide24Given a unity feedback system forward transfer function
Is point -3+0j is on a root locus?If the point is on the root locus, find the value of K?Steps:Determine zeros and poles of the forward transfer functionDetermine angles from zeros and poles to the interested pointDetermine the length of vector from zeros and poles to the interested pointAdd all angles. If it is equal to multiple of 180, then the point is on the root locus.Determine K using length of zero and pole vectors
Properties of Root Locus - Example
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Slide25Determine zeros and poles of the forward transfer function
Using quadratic equation,
Properties
of
Root Locus - Example
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Slide262. Determine angles from zeros and poles to the interested point
Properties
of
Root Locus - Example
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Slide273. Determine the length of vector from zeros and poles to the interested point
Properties
of
Root Locus - Example
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Slide284. 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
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Slide295. Determine K using length of zero and pole vectors
Properties
of Root Locus - Example
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Slide30Home work
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Given a unity feedback system forward transfer functionIs point -3+0j is on a root locus?If the point is on the root locus, find the value of K?