DescrizioneM-Derived 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 m-Derived 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: ${\displaystyle \omega _{\infty }={\frac {\omega _{c}}{\sqrt {1-m^{2}}}}}$ For ${\displaystyle 0<\omega <\omega _{c}\,\!}$ the transmission is lossless; ${\displaystyle \gamma =\alpha +i\beta =0+i{\frac {1}{2}}\cos ^{-1}\left(1-{\frac {2m^{2}}{\left({\frac {\omega _{c}}{\omega }}\right)^{2}-\left({\frac {\omega _{c}}{\omega _{\infty }}}\right)^{2}}}\right)}$ For ${\displaystyle \omega _{c}<\omega <\omega _{\infty }}$ the transmission parameters are; ${\displaystyle \gamma =\alpha +i\beta ={\frac {1}{2}}\cosh ^{-1}\left({\frac {2m^{2}}{\left({\frac {\omega _{c}}{\omega }}\right)^{2}-\left({\frac {\omega _{c}}{\omega _{\infty }}}\right)^{2}}}-1\right)+i{\frac {\pi }{2}}}$ For ${\displaystyle \omega _{\infty }<\omega <\infty }$ the transmission parameters are; ${\displaystyle \gamma =\alpha +i\beta ={\frac {1}{2}}\cosh ^{-1}\left(1-{\frac {2m^{2}}{\left({\frac {\omega _{c}}{\omega }}\right)^{2}-\left({\frac {\omega _{c}}{\omega _{\infty }}}\right)^{2}}}\right)+i0}$
Data	25 gennaio 2009
Fonte	Opera propria
Autore	Inductiveload
Licenza (Riusare questo file)	Public domainPublic domainfalsefalse Io, detentore del copyright su quest'opera, la rilascio nel pubblico dominio. Questa norma si applica in tutto il mondo. In alcuni paesi questo potrebbe non essere legalmente possibile. In tal caso: Garantisco a chiunque il diritto di utilizzare quest'opera per qualsiasi scopo, senza alcuna condizione, a meno che tali condizioni siano richieste dalla legge.

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]