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| View Original: | Mm'-Type Low-pass Transfer Function (1 Half-section).svg (600x660) | |||
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| Keywords: Mm'-Type Low-pass Transfer Function (1 Half-section).svg 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 <math>\omega_ \infin \frac \omega_c \sqrt 1- mm' 2 </math> For <math>0<\omega<\omega_c\ \ </math> the transmission is lossless; <math>\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_ \infin \right 2 \right </math> For <math>\omega_c<\omega<\omega_\infin</math> the transmission parameters are; <math>\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_ \infin \right 2 - 1 \right + i\frac \pi 2 </math> For <math>\omega_\infin<\omega<\infin</math> the transmission parameters are; <math>\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_ \infin \right 2 \right +i0</math> own Inductiveload 2009-01-25 own Inductiveload 2009-11-26 Mathematica Code <pre><nowiki> winfwc_ m_ wc/ Sqrt1 - m 2 ; gammaw_ wc_ winf_ m_ Piecewise 0 + I 0 5 ArcCos1 - 2 m 2/ wc/w 2 - wc/winf 2 w \LessSlantEqual wc 0 5 ArcCosh2 m 2/ wc/w 2 - wc/winf 2 - 1 + I Pi/2 wc < w < winf 0 5 ArcCosh1 - 2 m 2/ wc/w 2 - wc/winf 2 winf < w p LogLinearPlot Table Regammaw 1 winf1 m m m 0 2 0 8 0 2 w 0 1 10 PlotRange -> -0 1 3 6 PlotPoints -> 100 q LogLinearPlot Table Imgammaw 1 winf1 m m m 0 2 0 8 0 2 w 0 1 10 PlotRange -> -0 1 1 6 PlotPoints -> 100 </nowiki></pre> Mm'-Derived filters | ||||
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