ece_350_lecture_5_pt1_sp10_v5

ece_350_lecture_5_pt1_sp10_v5 - ECE 350 Lecture 5, Part 1...

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ECE 350 D. van Alphen 1 ECE 350 – Lecture 5, Part 1 Lecture Overview • The Transform Concept • Laplace Transforms • Laplace Transform Properties – Time Shift or Delay – Frequency Shift – Time Differentiation
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ECE 350 D. van Alphen 2 Introduction: The Transform Concept Hard Problem Easy Problem For Laplace Transforms (one application) Given Domain Different Domain Time Domain (or the t domain) Complex Frequency Domain (or the s domain) Differential Equation Problem Algebra Problem
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ECE 350 D. van Alphen 3 The Transform Concept, continued Strategy: Given an “ugly” problem in the time-domain, – transfer to another domain – work the easy problem in the other domain – transfer back to the time domain (with the answer) One transform you already know: Phasor Transforms – Used to find steady-state system output, given sinusoidal inputs – Find transformed circuit; transform sinusoidal inputs to complex numbers – Differential equation problem becomes complex arithmetic problem
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ECE 350 D. van Alphen 4 (Bilateral) Laplace Transform Definitions Defn : The (bilateral) Laplace Transform , X(s), of the function x(t), is The integral given in (1) does not necessarily converge for all values of s. The values of s for which the integral does converge make up the Region of Convergence (ROC) in the complex s-plane. Defn : The Inverse Laplace Transform , x(t), of the function X(s), is The integral in (2) implies integration in the complex s-plane, which we will avoid by doing table look-ups and other “tricks.” (Constant c is chosen to force convergence.) (1) ds e ) s ( X j 2 1 ) t ( x st j c j c π = + (2) dt e ) t ( x ) s ( X st =
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ECE 350 D. van Alphen 5 Laplace Transforms, continued Notation: X(s) = L [x(t)] and x(t) = L -1 [X(s)], or x(t) X(s) Linearity Property : If x 1 (t) X 1 (s), and x 2 (t) X 2 (s), then ax 1 (t) + bx 2 (t) ______ + ______
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ECE 350 D. van Alphen 6
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ece_350_lecture_5_pt1_sp10_v5 - ECE 350 Lecture 5, Part 1...

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