BeschreibungM-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; ${\displaystyle \gamma =\alpha +i\beta ={\frac {1}{2}}\cosh ^{-1}\left(1-{\frac {2m^{2}}{\left({\frac {\omega _{c}}{\omega }}\right)^{2}-\left({\frac {\omega _{c}}{\omega _{\infty }}}\right)^{2}}}\right)+i0}$
Datum	25. Januar 2009
Quelle	Eigenes Werk
Urheber	Inductiveload
Genehmigung (Weiternutzung dieser Datei)	Public domainPublic domainfalsefalse Ich, der Urheberrechtsinhaber dieses Werkes, veröffentliche es als gemeinfrei. Dies gilt weltweit. In manchen Staaten könnte dies rechtlich nicht möglich sein. Sofern dies der Fall ist: Ich gewähre jedem das bedingungslose Recht, dieses Werk für jedweden Zweck zu nutzen, es sei denn, Bedingungen sind gesetzlich erforderlich.

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]