CH2.2(a)_Complex_Number_(A)_annotated

CH2.2(a)_Complex_Number_(A)_annotated - Chapter 2 The...

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Unformatted text preview: Chapter 2 The Laplace Transform Objective: Mathematical preparation for the study of next chapters ontents Contents: 2.1 Introduction 2 Complex Numbers and Harmonic Motion 2.2 Complex Numbers and Harmonic Motion 2.3 Laplace Transformation 2.4 Inverse Laplace Transformation 2.5 Solving Linear Differential Eqs. by Laplace Transform ZG2005 1 2.2 (a) Complex Numbers Why need complex number ? Quadratic equation: 2 = + + c bx ax a ac b b x 2 4 2 2 , 1 − ± − = Roots: a ac b b x ac b 2 4 4 2 2 , 1 2 − ± − = > − If Two real roots a b x ac b 2 4 2 , 1 2 − = = − Two roots i r x x a b ac b x ac b 1 2 4 1 4 2 2 , 1 2 − ± = − − ± − = < − wo complex conjugate roots! ZG2005 2 Two complex conjugate roots! efinition of Complex Number Definition of Complex Number real part imaginary part jy x z + = complex number imaginary operator : 1 − = j eg. z=5+j3 or z=8-j10 Note: In reality, x may mean one component of z, jy may be the quadrature component relative to x omplex plane Im z • complex plane (graphical representation of complex number) Note: a complex number is correspondent to a point or directed line complex plane o Re |z| θ y x in complex plane • magnitude and phase angle magnitude of z: 2 2 y x z + = (easy to calculate) ZG2005 3 phase angle of z: x y 1 tan − = θ (be careful, see next page!) ow to Calculate Phase Angle (1) How to Calculate Phase Angle (1) Remember: phase angle is measured from + Re-axis and counterclockwise is positive direction Im (y) z | Im (y) z o Re (x)...
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This note was uploaded on 12/08/2008 for the course ME 3504 taught by Professor Tschang during the Fall '08 term at Virginia Tech.

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CH2.2(a)_Complex_Number_(A)_annotated - Chapter 2 The...

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