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Circle in the complex plane

  1. Jan 24, 2009 #1

    f95toli

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    I have a set of experimental data that I am trying to fit to an equation; but I have what I believe is a a silly problem (something I would probably have solved in 5 minutes 10 years ago....)
    Anyway, the data set consists of a complex impedance as a function of a normalized frequency (the parameter x below).
    Now, if I plot my data in the complex plane (i.e. Im(f) vs. Re(f) ) I get a nice circle which I can easily fit using a standard algorithm; from this fit I get the coordinates of the centre of the circle (C, a complex number) and its radius (R)

    [itex] |z-C|=R^2 [/itex].

    Now to my problem. The equation I am trying to fit to is

    [itex]\frac{1+a_2xi}{a_1+a_2xi}[/itex]

    which describes a circle in the complex plane when plotted as a function of the parameter x (but the points are not equidistant).
    The goal is to determine [itex]a_1, a_2 [/itex]

    How do I rewrite this equation in a form where I can use my known values (R and C) to determine [itex]a_1, a_2 [/itex]?

    I am really stuck, I realize that the equation looks like a bilinear transformation but I am not even sure if that helps or not(?)
     
  2. jcsd
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