m-derived filter

For the low-pass half section shown, the cut-off frequency of the m-type is the same as the k-type and is given by

${\displaystyle \omega _{c}={\frac {1}{\sqrt {LC}}}.}$

The pole of attenuation occurs at;

${\displaystyle \omega _{\infty }={\frac {\omega _{c}}{\sqrt {1-m^{2}}}}.}$

From this it is clear that smaller values of m will produce ${\displaystyle \omega _{\infty }}$ closer to the cut-off frequency ${\displaystyle \omega _{c}\,\!}$ and hence will have a sharper cut-off. Despite this cut-off, it also brings the unwanted stop band response of the m-type closer to the cut-off frequency, making it more difficult for this to be filtered with subsequent sections. The value of m chosen is usually a compromise between these conflicting requirements. There is also a practical limit to how small m can be made due to the inherent resistance of the inductors. This has the effect of causing the pole of attenuation to be less deep (that is, it is no longer a genuinely infinite pole) and the slope of cut-off to be less steep. This effect becomes more marked as ${\displaystyle \omega _{\infty }}$ is brought closer to ${\displaystyle \omega _{c}\,\!}$, and there ceases to be any improvement in response with an m of about 0.2 or less.[11][12][13]

m-derived prototype shunt low-pass filter Z_iTm image impedance for various values of m. Values below cut-off frequency only shown for clarity.

The following expressions for image impedances are all referenced to the low-pass prototype section. They are scaled to the nominal impedance R₀ = 1, and the frequencies in those expressions are all scaled to the cut-off frequency ω_c = 1.

The image impedances of the series section are given by[14]

${\displaystyle Z_{iT}={\sqrt {1-\omega ^{2}}}}$

and is the same as that of the constant k section

${\displaystyle Z_{i\Pi m}={\frac {1-\left(\omega /\omega _{\infty }\right)^{2}}{\sqrt {1-\omega ^{2}}}}.}$

The image impedances of the shunt section are given by[11]

${\displaystyle Z_{i\Pi }={\frac {1}{\sqrt {1-\omega ^{2}}}}}$

and is the same as that of the constant k section

${\displaystyle Z_{iTm}={\frac {\sqrt {1-\omega ^{2}}}{1-\left(\omega /\omega _{\infty }\right)^{2}}}}$

As with the k-type section, the image impedance of the m-type low-pass section is purely real below the cut-off frequency and purely imaginary above it. From the chart it can be seen that in the passband the closest impedance match to a constant pure resistance termination occurs at approximately m = 0.6.[14]

m-Derived low-pass filter transfer function for a single half-section

For an m-derived section in general the transmission parameters for a half-section are given by[14]

${\displaystyle \gamma =\sinh ^{-1}{\frac {mZ}{\sqrt {k^{2}+(1-m^{2})Z^{2}}}}}$

and for n half-sections

${\displaystyle \gamma _{n}=n\gamma \,\!}$

For the particular example of the low-pass L section, the transmission parameters solve differently in three frequency bands.[14]

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

${\displaystyle \gamma =\alpha +i\beta ={\frac {1}{2}}\cosh ^{-1}\left(1-{\frac {2m^{2}}{\left({\frac {\omega _{c}}{\omega }}\right)^{2}-\left({\frac {\omega _{c}}{\omega _{\infty }}}\right)^{2}}}\right)+i0}$

The plots shown of image impedance, attenuation and phase change are the plots of a low-pass prototype filter section. The prototype has a cut-off frequency of ω_c = 1 rad/s and a nominal impedance R₀ = 1 Ω. This is produced by a filter half-section where L = 1 henry and C = 1 farad. This prototype can be impedance scaled and frequency scaled to the desired values. The low-pass prototype can also be transformed into high-pass, band-pass or band-stop types by application of suitable frequency transformations.[15]