DescriptionConstant-k Filter Transmission Parameters.svg	English: Plot of the transmission parameters ${\displaystyle \gamma =\alpha +i\beta }$ of a prototype low-pass constant-k filter of a single half-section. ${\displaystyle \gamma =\alpha +i\beta ={\begin{cases}0+\sin ^{-1}{\frac {\omega }{\omega _{c}}},&\omega <\omega _{c}\\\cosh ^{-1}{\frac {\omega }{\omega _{c}}}+i{\frac {\pi }{2}},&\omega >\omega _{c}\end{cases}}}$ α is measured in nepers and β in radians.
Date	23 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

l = 1;
c = 1;

wc = 1/Sqrt[l*c];

gam[w_] := Piecewise[
  {{I*ArcSin[w/wc], w < wc},
   {ArcCosh[w/wc] + I*Pi/2, w \[GreaterSlantEqual] wc}}
  ]

alpha[w_] := Re[gam[w]];
beta[w_] := Im[gam[w]];

LogLinearPlot[
 {alpha[w],
  beta[w]},
 {w, 0.01, 10^3},
 PlotRange -> All,
 PlotPoints -> 1000
 ]