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

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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

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Laplace Transform
L Laplace Transform
t is a real 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
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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
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Definition of Laplace Transform

• Sufficient conditions for existence of the
  Laplace transform

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Examples of functions of exponential order
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Exampleunit step
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ExampleShifted Step
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Integration by parts
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ExampleRamp
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ExampleExponential Function
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Examplesine Function
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Examplecosine Function
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ExampleRectangle Pulse
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Properties of Laplace TransformAddition
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Properties of Laplace TransformMultiplication by
a constant
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Properties of Laplace TransformMultiplication by
exponential
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Properties of Laplace TransformExamples
Multiplication by exponential
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Useful Identities
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Examplesin Function
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Examplecosine Function
Laplace Transform
Inverse Laplace Transform
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Properties of Laplace TransformMultiplication by
time
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Properties of Laplace Transform
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Properties of Laplace TransformIntegration
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Properties of Laplace TransformDelay
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Properties of Laplace Transform
Slope A
L
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Properties of Laplace Transform4
Slope A
_
_
Slope A
A L
L
L
Slope A

L
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Summary
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SE 207 Modeling and SimulationLesson 3 Inverse
Laplace Transform

• Dr. Samir Al-Amer
• Term 072

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Properties of Laplace Transform
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Solving Linear ODE using Laplace Transform
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Inverse Laplace Transform
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Notation
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Notation
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Notation
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Examples
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Partial Fraction Expansion
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Partial Fraction Expansion
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Partial Fraction Expansion
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Example
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Example
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Alternative Way of Obtaining Ai
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Repeated poles
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Repeated poles
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Repeated poles
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Repeated poles
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Common Error
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Complex Poles
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Complex Poles
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What do we do if F(s) is not strictly proper
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Solving for the Response
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Final value theorem
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Final value theorem
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Step function
A
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impulse function
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impulse function
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Initial Value Final Value Theorems
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Initial Value Theorem
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Final Value Theorems
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SE 207 Modeling and SimulationLesson 4
Additional properties of Laplace transform and
solution of ODE

• Dr. Samir Al-Amer
• Term 072

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Outlines

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

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Step function
A
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impulse function
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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
Area1
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impulse function
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Sample property of impulse function
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Time delay
g(t) G(s)
f(t) F(s)
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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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Example
- - -
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Example
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Solving for the Response