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Bogazici University Dept. Of ME. Laplace Transforms Very useful in the analysis and design of LTI systems. Operations of differentiation and integration are converted to algebraic equations of s. We can solve differential equations using Laplace transforms. Definition: For a time function F(s) is: Note that the time is integrated out. s has unit 1/Time

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Bogazici University Dept. Of ME. For the transform to exist, the integral should be finite. It means that: The inverse transform is given by : This equation is rarely used. L -1 is usually determined using Partial Fraction Expansion (PFE).

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Bogazici University Dept. Of ME. Example: Step function :

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Bogazici University Dept. Of ME. A TABLE OF LAPLACE TRANSFORMS

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Bogazici University Dept. Of ME. Properties Linearity : Transform of a derivative: Repeated application of the rule above :

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Bogazici University Dept. Of ME. Final Value Theorem Use the property of the derivative: States that the steady state behavior of f(t) is the same as the behavior of F(s) around s=0. Very useful result

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Bogazici University Dept. Of ME. Other Properties Integration : Frequency shift : Time shift :

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