DescriptionMm'-Derived Low-pass Image Impedance.svg	English: A graph comparing the image impedances of prototype low-pass mm'-derived filter, m-derived filter and constant-k filter in the region immediately below ω=ωc=1. The parameters of the mm'-derived filter are m=0.7230, m'=0.4124 (mm'=0.3). The parameter of the m-derived filter is m=0.6. These filters both have the optimum impedance matching for their types. ${\displaystyle Z_{iTmm'}={\frac {{\sqrt {1-\omega ^{2}}}\left(1-(1-m^{2})\omega ^{2}\right)}{1-\left(\omega /\omega _{\infty }\right)^{2}}}}$ where ${\displaystyle \omega _{\infty }={\frac {\omega _{c}}{\sqrt {1-(mm')^{2}}}}.}$ ${\displaystyle Z_{iTm}={\frac {\sqrt {1-\omega ^{2}}}{1-\left(\omega /\omega _{\infty }\right)^{2}}}}$ where ${\displaystyle \omega _{\infty }={\frac {\omega _{c}}{\sqrt {1-m^{2}}}}.}$ ${\displaystyle Z_{iT}={\sqrt {1-\omega ^{2}}}}$
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

wc = 1;

(*mm'-derived*)
m1 = 0.7230;
m2 = 0.4134;
mm = m1*m2;

winfmm = wc/Sqrt[1 - mm^2];
zitmm[w_] := (Sqrt[1 - w^2]*(1 - (1 - m1^2)*w^2))/(1 - (w/winfmm)^2);

(*m-derived*)
m = 0.6;

winfm = wc/Sqrt[1 - m^2];
zitm[w_] := Sqrt[1 - w^2]/(1 - (w/winfm)^2);

(*constant-k*)
zit[w_] := Sqrt[1 - w^2]


Plot[
 {zitmm[w],
  zitm[w],
  zit[w]},
 {w, 0.6, 1},
 PlotRange -> All
 ]