DescriptionMm'-Type Low-pass Transfer Function (1 Half-section).svg	English: A plot (attenuation in nepers and phase in degrees) of the transfer function of one half-section of an mm'-type low-pass filter (the prototype L-section model) with response as follows: The transmission parameters solve differently in three frequency bands. The pole of attenuation (where the attenuation is infinite) is found at: ${\displaystyle \omega _{\infty }={\frac {\omega _{c}}{\sqrt {1-(mm')^{2}}}}}$ For ${\displaystyle 0<\omega <\omega _{c}\,\!}$ the transmission is lossless; ${\displaystyle \gamma =\alpha +i\beta =0+i{\frac {1}{2}}\cos ^{-1}\left(1-{\frac {2(mm')^{2}}{\left({\frac {\omega _{c}}{\omega }}\right)^{2}-\left({\frac {\omega _{c}}{\omega _{\infty }}}\right)^{2}}}\right)}$ For ${\displaystyle \omega _{c}<\omega <\omega _{\infty }}$ the transmission parameters are; ${\displaystyle \gamma =\alpha +i\beta ={\frac {1}{2}}\cosh ^{-1}\left({\frac {2(mm')^{2}}{\left({\frac {\omega _{c}}{\omega }}\right)^{2}-\left({\frac {\omega _{c}}{\omega _{\infty }}}\right)^{2}}}-1\right)+i{\frac {\pi }{2}}}$ For ${\displaystyle \omega _{\infty }<\omega <\infty }$ the transmission parameters are; ${\displaystyle \gamma =\alpha +i\beta ={\frac {1}{2}}\cosh ^{-1}\left(1-{\frac {2(mm')^{2}}{\left({\frac {\omega _{c}}{\omega }}\right)^{2}-\left({\frac {\omega _{c}}{\omega _{\infty }}}\right)^{2}}}\right)+i0}$
Date	26 November 2009
Source	Own work
Author	Inductiveload
Permission (Reusing this file)	Public domainPublic domainfalsefalse I, the copyright holder of this work, release this work into the public domain. This applies worldwide. In some countries this may not be legally possible; if so: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

Mathematica Code

winf[wc_, m_] = wc/(Sqrt[1 - m^2]);

gamma[w_, wc_, winf_, m_] = Piecewise[
  {
   {0 + I* 0.5 * ArcCos[1 - 2 m^2/((wc/w)^2 - (wc/winf)^2)],
    w \[LessSlantEqual] wc},
   {0.5 * ArcCosh[2 m^2/((wc/w)^2 - (wc/winf)^2) - 1] + I*Pi/2,
    wc < w <= winf},
   {0.5 * ArcCosh[1 - 2 m^2/((wc/w)^2 - (wc/winf)^2)],
    winf < w}
   }
  ]

p = LogLinearPlot[
  {Table[
    Re[gamma[w, 1, winf[1, m], m]],
    {m, 0.2, 0.8, 0.2}]},
  {w, 0.1, 10},
  PlotRange -> {-0.1, 3.6},
  PlotPoints -> 100]

q = LogLinearPlot[
  {Table[
    Im[gamma[w, 1, winf[1, m], m]],
    {m, 0.2, 0.8, 0.2}]},
  {w, 0.1, 10},
  PlotRange -> {-0.1, 1.6},
  PlotPoints -> 100]