MIT 2 003 - Complex Exponentials (5 pages)

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Complex Exponentials



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Complex Exponentials

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Pages:
5
School:
Massachusetts Institute of Technology
Course:
2 003 - Modeling Dynamics and Control I

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2 003 Fall 2003 Complex Exponentials Complex Numbers Complex numbers have both real and imaginary components A complex number r may be expressed in Cartesian or Polar forms r a jb cartesian r e polar The following relationships convert from cartesian to polar forms Magnitude r a2 b2 Angle tan 1 tan 1 b a b a a 0 a 0 Complex numbers can be plotted on the complex plane in either Cartesian or Polar forms Fig 1 Figure 1 Complex plane plots Cartesian and Polar forms Euler s Identity Euler s Identity states that ej cos j sin 1 2 003 Fall 2003 Complex Exponentials This can be shown by taking the series expansion of sin cos and e 3 5 7 3 5 7 2 4 6 cos 1 2 4 6 2 3 4 5 j j ej 1 j 2 3 4 5 sin Combining cos j sin 1 j 3 4 5 2 j j 2 3 4 5 ej Complex Exponentials Consider the case where becomes a function of time increasing at a constant rate t t then r t becomes r t ej t Plotting r t on the complex plane traces out a circle with a constant radius 1 Fig 2 Plotting the real and imaginary components of r t vs time Fig 3 we see that the real component is Re r t cos t while the imaginary component is Im r t sin t Consider the variable r t which is de ned as follows r t est where s is a complex number s j 2 2 003 Fall 2003 Complex Exponentials t Figure 2 Complex plane plots r t ej t t t 0 Re r t cos t t 0 Im r t sin t Figure 3 Real and imaginary components of r t vs time What path does r t trace out in the complex plane Consider r t est e j t e t ej t One can look at this as a time varying magnitude e t multiplying a point rotating on the unit circle at frequency via the function ej t Plotting just the magnitude of ej t vs time shows that there are three distinct regions Fig 4 1 0 where the magnitude grows without bounds This condition is unstable 2 0 where the magnitude remains constant This condition is 3 2 003 Fall 2003 Complex Exponentials called marginally stable since the magnitude does not grow without bound but does not converge to zero 3 0 where the magnitude converges to zero



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