# SE 207: Modeling and Simulation Introduction to Laplace Transform

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SE 207: Modeling and Simulation Introduction to Laplace Transform
Dr. Samir Al-Amer Term 072

Why do we use them We use transforms to transform the problem into a one that is easier to solve then use the inverse transform to obtain the solution to the original problem

Laplace Transform L L-1 t is a real variable s is complex variable
f(t) is a real function Time Domain s is complex variable F(s) is a complex valued function Frequency Domain L-1 Inverse Laplace Transform

Use of Laplace Transform in solving ODE
Differential Equation Algebraic Equation Laplace Transform Solution of the Algebraic Equation Solution of the Differential Equation Inverse Laplace transform

Definition of Laplace Transform
Sufficient conditions for existence of the Laplace transform

Examples of functions of exponential order

Example unit step

Example Shifted Step

Integration by parts

Example Ramp

Example Exponential Function

Example sine Function

Example cosine Function

Example Rectangle Pulse

Properties of Laplace Transform Multiplication by a constant

Properties of Laplace Transform Multiplication by exponential

Properties of Laplace Transform Examples Multiplication by exponential

Useful Identities

Example sin Function

Example cosine Function
Laplace Transform Inverse Laplace Transform

Properties of Laplace Transform Multiplication by time

Properties of Laplace Transform

Properties of Laplace Transform Integration

Properties of Laplace Transform Delay

Properties of Laplace Transform
Slope =A L

Properties of Laplace Transform4
Slope =A _ _ Slope =A A L L L Slope =A = L

Summary

SE 207: Modeling and Simulation Lesson 3: Inverse Laplace Transform
Dr. Samir Al-Amer Term 072

Properties of Laplace Transform

Solving Linear ODE using Laplace Transform

Inverse Laplace Transform

Notation

Notation

Notation

Examples

Partial Fraction Expansion

Partial Fraction Expansion

Partial Fraction Expansion

Example

Example

Alternative Way of Obtaining Ai

Repeated poles

Repeated poles

Repeated poles

Repeated poles

Common Error

Complex Poles

Complex Poles

What do we do if F(s) is not strictly proper

Solving for the Response

Final value theorem

Final value theorem

Step function A

impulse function

impulse function

Initial Value& Final Value Theorems

Initial Value Theorem

Final Value Theorems

SE 207: Modeling and Simulation Lesson 4: Additional properties of Laplace transform and solution of ODE Dr. Samir Al-Amer Term 072

Outlines What to do if we have proper function? Time delay
Inversion of some irrational functions Examples

Step function A

impulse function

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

impulse function

Sample property of impulse function

Time delay g(t) G(s) f(t) F(s)

What do we do if F(s) is not strictly proper

What do we do if F(s) is not strictly proper

Example − − −

Example

Solving for the Response