Mm'-type filter

Derivation

The m-derived process starts with a k-type filter midsection and transforms it into a m-derived filter with a different transfer function but retaining the same image impedance and passband. Two different results are obtained depending on whether the process began with a T-section or a Π-section. From a T-section, the series Z and shunt Y are multiplied by an arbitrary parameter m (0 < m < 1). An additional impedance is then inserted in series with Y whose value is that which restores the original image impedance. The half-sections resulting from splitting the T-section, however, will show a different image impedance at the split, ZiΠm. Two such half-sections cascaded with the ZiT impedances facing, will form a Π-section with image impedance ZiΠm. The m-derived process can now be applied to this new section, but with a new parameter m'. The series Z and shunt Y are multiplied by m' and an additional admittance is inserted in parallel with the series elements to bring the image impedance back to its original value of ZiΠm. Again, the half sections will have a different image impedance at the split ports and this is designated ZiTmm'.

The dual realisation of this filter is obtained in a completely analogous way by first transforming a mid-shunt k-type Π-section, forming the resulting mid-series m-type T-section and then transforming using m', resulting in a new Π flavour of Zimm', ZiΠmm', which is the dual of ZiTmm'.

The m-derivation transformation can, in principle, be applied ad-infinitum and produce mm'm''-types etc. There is, however, no practical benefit in doing this. The improvements obtained diminish at each iteration and are not worth the increase in complexity. Note that applying the m-derived transformation twice to a T-section (or Π-section) will merely result in an m-type filter with a different value of m. The transformation must be applied alternately to T-sections and Π-sections for a completely new form of filter to be obtained.

Operating frequency

For a low-pass prototype, the cut-off frequency is given by the same expression as for the m-type and the k-type;

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

The pole of attenuation occurs at;

${\displaystyle {\omega }_{\mathrm{\infty }}=\frac{{\omega }_c}{\sqrt{1-(m{m}^{\prime }{)}^2}}.}$

Image impedance

See also Image impedance#Derivation

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.

"T" port image impedance

The image impedance looking into a "T" topology port of a shunt derived section is given by,

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

For comparison;

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

"Π" port image impedance

The image impedance looking into a "Π" topology port of a series derived section is given by,

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

For comparison;

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

Optimisation

Note that ${\displaystyle {\omega }_{\mathrm{\infty }}}$ can be adjusted independently of m by adjusting m'. It is therefore possible to adjust the impedance characteristic and frequency response characteristic independently. However, for optimum impedance match it is necessary to adjust both parameters solely for maximally flat image resistance in the passband. The term resistance is used because the image impedance is purely real in the passband between cut-off frequencies and purely imaginary outside the passband. It is not possible to obtain an exact impedance match across the entire band. With two degrees of freedom it is only possible to match the impedance exactly at two spot frequencies. It has been determined empirically that a good match is made with the values;

${\displaystyle m=0.7230}$

${\displaystyle m^{\prime }=0.4134}$

This amounts to setting the match to be exact at frequencies 0.8062 and 0.9487 rad/s for the prototype filter and the impedance departs less than 2% from nominal from 0 to 0.96 rad/s, that is, nearly all of the passband.

The transfer function of an mm'-type is the same as an m-type with m set to the product mm' and in this case mm'=0.3. Where an m-type section is used for impedance matching the optimum value of m is m=0.6. Steepness of cut-off increases with decreasing m so an mm'-type section has this as an incidental advantage over an m-type in this application.

Transmission parameters

See also Image impedance#Transfer function

The operating frequencies, transmission parameters and transfer function are identical to those for the m-type and details can be found in that article if the parameter m is replaced by the product mm'. Only the image impedance is different in the mm'-type filter in terms of black box behaviour.

Prototype transformations

The plots shown of image impedance and attenuation 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=1Ω. 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.

Cascading sections

As with all image filters, it is required to match each section into a section of identical image impedance if the theoretical filter response is to be achieved. This is a particular difficulty for the end sections of the filter which are often working into resistive terminations which cannot be matched exactly by an image impedance. Hence the use of the mm'-type as end sections for the filter because of its nearly flat impedance with frequency characteristic in the passband. However, it is not desirable to use it throughout the entire filter. The workhorse of image filters are the k-type sections and these will usually be required somewhere in the filter for good rejection in the stopband well away from cut-off and also because they are the simplest topology and lowest component count. Unfortunately, neither side of a mm'-type can match into a k-type. The solution is to form a composite section from a mm'-type half-section and a m-type section which will match each other on one side if m has the same value for both half-sections. This can, for instance, produce a composite T-section with ZiTmm' facing the termination and ZiT facing the rest of the filter. The T-section will be matched internally with ZiTm. This has the additional incidental advantage of producing two poles of attenuation in the stopband at different frequencies. This is a consequence of m and mm' necessarily being different values.