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Laplace Transform Properties
EE3511: Automatic Control Systems Laplace Transform Properties EE3511-L3 Prince Sattam Bin Abdulaziz University 1
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Properties of Laplace Transform
Learning Objectives To be able to state different Laplace transform properties. To be able to apply different properties to simplify calculations of Laplace transform or Inverse Laplace transform. EE3511-L3 Prince Sattam Bin Abdulaziz University 2
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Definition of Laplace Transform
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Linear Properties of Laplace Transform
Special Cases: Multiplication by constant Addition of two functions EE3511-L3 Prince Sattam Bin Abdulaziz University 4
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Multiplication by Exponential
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Multiplication by Exponential Examples
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Multiplication by time
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Properties Covered so far
Linear Property of Laplace Transform Multiplication by Exponential Multiplication by time EE3511-L3 Prince Sattam Bin Abdulaziz University 8
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Laplace Transform of Derivative
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Laplace Transform of Derivative Example
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Laplace Transform of Integrals
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Laplace Transform of Functions with Delay
f(t) f(t-d)u(t-d) d EE3511-L3 Prince Sattam Bin Abdulaziz University 12
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Prince Sattam Bin Abdulaziz University
Time delay g(t) G(s) f(t) F(s) EE3511-L3 Prince Sattam Bin Abdulaziz University 13
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Laplace Transform of Functions with Delay Example
1 1 2 EE3511-L3 Prince Sattam Bin Abdulaziz University 14
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Properties of Laplace Transform
Slope =A L EE3511-L3 Prince Sattam Bin Abdulaziz University 15
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Properties of Laplace Transform
Slope =A _ _ Slope =A A L L L Slope =A = L EE3511-L3 Prince Sattam Bin Abdulaziz University 16
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Properties of Laplace Transform
These are essential in solving differential equations EE3511-L3 Prince Sattam Bin Abdulaziz University 17
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Summary of LT Properties
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Salman bin Abdulaziz University
impulse function EE3511_L3 Salman bin Abdulaziz University 19
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Salman bin Abdulaziz University
impulse function You can consider the unit impulse as the limiting case for a rectangle pulse with unit area as the width of the pulse approaches zero Area=1 EE3511_L3 Salman bin Abdulaziz University 20
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Salman bin Abdulaziz University
impulse function EE3511_L3 Salman bin Abdulaziz University 21
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Inverse Laplace Transform
EE3511: Automatic Control Systems Inverse Laplace Transform EE3511_L3 Salman bin Abdulaziz University
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Inverse Laplace Transform
Outlines Inverse Laplace transform Definitions Partial Fraction Expansion Special Cases Distinct poles Complex poles Repeated poles Examples EE3511_L3 Salman bin Abdulaziz University
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Definition of Inverse Laplace Transform
A real number that is greater than real part of all singularities of F(s) EE3511_L3 Salman bin Abdulaziz University
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Definition of Inverse Laplace Transform
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Inverse Laplace Transform
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Proper / Strictly Proper
F(s) is strictly proper F(s) is proper /─ EE3511_L3 Salman bin Abdulaziz University
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Salman bin Abdulaziz University
Examples Strictly Proper Proper Degree of numerator =0 Degree of denominator =2 Degree of numerator =1 Degree of denominator =1 Degree of numerator =0 Degree of denominator =3 Degree of numerator =2 Degree of denominator =2 EE3511_L3 Salman bin Abdulaziz University
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Notation Poles and Zeros
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Salman bin Abdulaziz University
Examples Zeros Poles -2 -3,-4 3 -0.5 -3 0,0,-1,-2 -1,-1,2±j3 EE3511_L3 Salman bin Abdulaziz University
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Partial Fraction Expansion
Partial Fraction Expansion of F(s) : F(s) is expressed as the sum of simple fraction terms How do we obtain these terms? EE3511_L3 Salman bin Abdulaziz University
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Partial Fraction Expansion
Three Special Cases are considered Distinct pole Repeated poles Complex poles EE3511_L3 Salman bin Abdulaziz University
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Partial Fraction Expansion
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Partial Fraction Expansion
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Example EE3511_L3 Salman bin Abdulaziz University
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Alternative Way of Obtaining Ai
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Salman bin Abdulaziz University
Repeated poles EE3511_L3 Salman bin Abdulaziz University 37
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Salman bin Abdulaziz University
Repeated poles EE3511_L3 Salman bin Abdulaziz University 38
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Salman bin Abdulaziz University
Repeated poles EE3511_L3 Salman bin Abdulaziz University 39
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Repeated poles EE3511_L3 Salman bin Abdulaziz University 40
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Salman bin Abdulaziz University
Common Error EE3511_L3 Salman bin Abdulaziz University 41
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Complex Poles EE3511_L3 Salman bin Abdulaziz University 42
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Salman bin Abdulaziz University
Complex Poles EE3511_L3 Salman bin Abdulaziz University 43
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What do we do if F(s) is not strictly proper
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What do we do if F(s) is not strictly proper
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Salman bin Abdulaziz University
Example − − − EE3511_L3 Salman bin Abdulaziz University 46
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Salman bin Abdulaziz University
Example EE3511_L3 Salman bin Abdulaziz University 47
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