14 - MEEN260 IntroductiontoEngineering Experimentation

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1 MEEN 260 Introduction to Engineering Experimentation Complex Numbers and Fourier Analysis Reza Langari* Department of Mechanical Engineering Texas A&M University rlangari@tamu.edu *Incorporates a number of slides from Brian Rasmussen, TAMU MEEN, Copyright 2008. Complex Numbers Re Im θ j Ae bj a s = + = a b A 2 2 b a A s + = = = = a b s 1 tan A few identities Complex conjugate Euler Identity bj a s = () a s s s 2 Re 2 = = + b s s s 2 Im 2 = = ( ) ( ) sin cos j e j + = 2 cos j j e e + = j e e j j 2 sin =
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2 Calculations Determining the magnitude of a complex number Helpful Hint Examples bj a s + = 2 2 b a s + = j j s + = 2 5 3 () ( ) () () 5 34 1 2 5 3 2 5 3 2 2 2 2 = + + = + = j j s dj c bj a s + + = 2 2 2 2 d c b a s + + = ω j s + = 1 1 () ( ) 2 2 2 2 2 1 1 1 0 1 1 1 + = + + = + = j s Calculations Determining the angle of a complex number Be careful… bj a s + = = a b s 1 tan j s = 1 o 45 1 tan 1 1 tan 1 1 + = = = s III quadrant I quadrant? Re Im Calculations Rules when calculating phase angle Phase measured from the positive real axis Phase conventionally measured bj a s + = = a b s 1 tan π θ < < if a>0 + = a b s 1 tan if a<0, b>0 = a b s 1 tan if a<0, b<0 I & IV quadrants II quadrant III quadrant
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3 Calculations Helpful Hints: Examples: j j s + = 2 5 3 () ( ) o 6 . 85 6 . 26 0 . 59 2 1 tan 3 5 tan 1 1 = = ⎛ − = s dj c bj a s + + = = = c d a b s 1 1 tan tan r denominato numerator j s 5 3 = o o o 0 . 121 180 0 . 59 3 5 tan 1 = = = s Complex functions Functions of a complex variable
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This note was uploaded on 12/26/2010 for the course MEEN 260 taught by Professor Langari during the Fall '08 term at Texas A&M.

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14 - MEEN260 IntroductiontoEngineering Experimentation

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