Laplace Transform Properties

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LAPLACE TRANSFORMS.

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Presentation transcript:

Laplace Transform Properties EE3511: Automatic Control Systems Laplace Transform Properties EE3511-L3 Prince Sattam Bin Abdulaziz University 1

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

Definition of Laplace Transform EE3511-L3 Prince Sattam Bin Abdulaziz University 3

Linear Properties of Laplace Transform Special Cases: Multiplication by constant Addition of two functions EE3511-L3 Prince Sattam Bin Abdulaziz University 4

Multiplication by Exponential EE3511-L3 Prince Sattam Bin Abdulaziz University 5

Multiplication by Exponential Examples EE3511-L3 Prince Sattam Bin Abdulaziz University 6

Multiplication by time EE3511-L3 Prince Sattam Bin Abdulaziz University 7 7

Properties Covered so far Linear Property of Laplace Transform Multiplication by Exponential Multiplication by time EE3511-L3 Prince Sattam Bin Abdulaziz University 8

Laplace Transform of Derivative EE3511-L3 Prince Sattam Bin Abdulaziz University 9

Laplace Transform of Derivative Example EE3511-L3 Prince Sattam Bin Abdulaziz University 10

Laplace Transform of Integrals EE3511-L3 Prince Sattam Bin Abdulaziz University 11

Laplace Transform of Functions with Delay f(t) f(t-d)u(t-d) d EE3511-L3 Prince Sattam Bin Abdulaziz University 12

Prince Sattam Bin Abdulaziz University Time delay g(t) G(s) f(t) F(s) EE3511-L3 Prince Sattam Bin Abdulaziz University 13

Laplace Transform of Functions with Delay Example 1 1 2 EE3511-L3 Prince Sattam Bin Abdulaziz University 14

Properties of Laplace Transform Slope =A L EE3511-L3 Prince Sattam Bin Abdulaziz University 15

Properties of Laplace Transform Slope =A _ _ Slope =A A L L L Slope =A = L EE3511-L3 Prince Sattam Bin Abdulaziz University 16

Properties of Laplace Transform These are essential in solving differential equations EE3511-L3 Prince Sattam Bin Abdulaziz University 17

Summary of LT Properties EE3511-L3 Prince Sattam Bin Abdulaziz University 18

Salman bin Abdulaziz University impulse function EE3511_L3 Salman bin Abdulaziz University 19

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

Salman bin Abdulaziz University impulse function EE3511_L3 Salman bin Abdulaziz University 21

Inverse Laplace Transform EE3511: Automatic Control Systems Inverse Laplace Transform EE3511_L3 Salman bin Abdulaziz University

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

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

Definition of Inverse Laplace Transform EE3511_L3 Salman bin Abdulaziz University

Inverse Laplace Transform EE3511_L3 Salman bin Abdulaziz University

Proper / Strictly Proper F(s) is strictly proper  F(s) is proper /─ EE3511_L3 Salman bin Abdulaziz University

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

Notation Poles and Zeros EE3511_L3 Salman bin Abdulaziz University

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

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

Partial Fraction Expansion Three Special Cases are considered Distinct pole Repeated poles Complex poles EE3511_L3 Salman bin Abdulaziz University

Partial Fraction Expansion EE3511_L3 Salman bin Abdulaziz University

Partial Fraction Expansion EE3511_L3 Salman bin Abdulaziz University

Salman bin Abdulaziz University Example EE3511_L3 Salman bin Abdulaziz University

Alternative Way of Obtaining Ai EE3511_L3 Salman bin Abdulaziz University

Salman bin Abdulaziz University Repeated poles EE3511_L3 Salman bin Abdulaziz University 37

Salman bin Abdulaziz University Repeated poles EE3511_L3 Salman bin Abdulaziz University 38

Salman bin Abdulaziz University Repeated poles EE3511_L3 Salman bin Abdulaziz University 39

Salman bin Abdulaziz University Repeated poles EE3511_L3 Salman bin Abdulaziz University 40

Salman bin Abdulaziz University Common Error EE3511_L3 Salman bin Abdulaziz University 41

Salman bin Abdulaziz University Complex Poles EE3511_L3 Salman bin Abdulaziz University 42

Salman bin Abdulaziz University Complex Poles EE3511_L3 Salman bin Abdulaziz University 43

What do we do if F(s) is not strictly proper EE3511_L3 Salman bin Abdulaziz University 44

What do we do if F(s) is not strictly proper EE3511_L3 Salman bin Abdulaziz University 45

Salman bin Abdulaziz University Example − − − EE3511_L3 Salman bin Abdulaziz University 46

Salman bin Abdulaziz University Example EE3511_L3 Salman bin Abdulaziz University 47