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Laplace Transform Properties

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Presentation on theme: "Laplace Transform Properties"— Presentation transcript:

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

2 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

20 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

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

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

23 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

24 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

25 Definition of Inverse Laplace Transform
EE3511_L3 Salman bin Abdulaziz University

26 Inverse Laplace Transform
EE3511_L3 Salman bin Abdulaziz University

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

28 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

29 Notation Poles and Zeros
EE3511_L3 Salman bin Abdulaziz University

30 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

31 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

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

33 Partial Fraction Expansion
EE3511_L3 Salman bin Abdulaziz University

34 Partial Fraction Expansion
EE3511_L3 Salman bin Abdulaziz University

35 Salman bin Abdulaziz University
Example EE3511_L3 Salman bin Abdulaziz University

36 Alternative Way of Obtaining Ai
EE3511_L3 Salman bin Abdulaziz University

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

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

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

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

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

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

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

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

45 What do we do if F(s) is not strictly proper
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46 Salman bin Abdulaziz University
Example − − − EE3511_L3 Salman bin Abdulaziz University 46

47 Salman bin Abdulaziz University
Example EE3511_L3 Salman bin Abdulaziz University 47

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