Lesson+11 - 1 Lesson 11 Lesson 11 Operational...

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Operational Calculus (Transforms) Properties of Laplace Transforms Ch 12.5 Lesson 11 2 Lesson 11
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Challenge 10 Lesson 11 Given cosh( ) = (e t + e - t )/2, what is the Laplace transform of cosh( )? 2 2 0 0 0 1 2 1 1 2 1 2 1 2 1 2 1 β β β β β β β β β β s s s s s e s e dt e e L t s t s t s t s ) ) ( cosh ( 3 Could have deduced these two terms from a Laplace table.
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Lesson 11 Laplace transform pair Laplace transform (analysis equation) Inverse Laplace transform (synthesis equation) dt e t x s X st Lesson 11 c j c st ds e s X j t x 2 1 ) ( 4
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It is be assumed that the Laplace transform pairs for common (primitive) signals have been pre-computed and reduced to table entries . The signal transforms are combined and modified to produce the transform of increasingly complex signals. Understanding this process allows Laplace transforms to be applied to complicated signals and systems. This process is defined in terms of Laplace transform properties . Lesson 11 5
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Lesson 11 Examples of Laplace transforms for common signals. 6
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Lesson 11 More Laplace transforms of common signals. 7
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Lesson 11 dt e t t u t t x s Y st 0 0 Property Example: Time Shift Assume x(t) X(s) and y(t)=x(t-t 0 ). Then: Let t-t 0 = dt e u x s Y t s ) ( 0 ) ( 0 0 ) ( s X e dt e u x e st s st Delay Theorem 8
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In-class Problem Lesson 11 Property Example: Frequency Shift Assume x(t) X(s); y(t)=x(t)e s 0 t , what is Y(s)? 9
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Lesson 11 dt e dt t dx s Y st / Property Example: Differentiation Assume x(t) X(s) and y(t)=dx(t)/dt ) ( ) ( ) ( ) ( s X s x s s X dt e t x s e t x s Y st st 0 0 0 0 Integrate by parts This rule provides a mechanism to convert ODEs into an algebraic equation. Note the presence of an IC 10 Negative 1-IC
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Lesson 11 0 2 2 dt e dt t x d s Y st / Assume x(t) X(s); y(t)=dx 2 (t)/dt 2 ) ( ) ( ) ( 0 0 2 x sx s X s s Y and so on. 2 negative ICs 11 From table.
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Lesson 11 ds s dX dt e t tx s Y st ) ( ) ( Property Eample: Multiplication by t (ramping) Assume x(t) X(s); y(t)=tx(t); Example: y(t)= t u(t) U(s) = 1/s Y(s)= - dU(s)/ds = - d(1/s) /ds = -1/s 2 12 (Given in table)
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Lesson 11 ) ( ) ( a s X a s Y 1 Property Example: Scaling Assume x(t) X(s); y(t) = x(at) 13 (Given in tables)
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