mm'-type filters, also called double-m-derived filters, are a type of electronic filter designed using the image method. They were patented by Otto Zobel in 1932.[1] Like the m-type filter from which it is derived, the mm'-type filter type was intended to provide an improved impedance match into the filter termination impedances and originally arose in connection with telephone frequency division multiplexing. The filter has a similar transfer function to the m-type, having the same advantage of rapid cut-off, but the input impedance remains much more nearly constant if suitable parameters are chosen. In fact, the cut-off performance is better for the mm'-type if like-for-like impedance matching are compared rather than like-for-like transfer function. It also has the same drawback of a rising response in the stopband as the m-type. However, its main disadvantage is its much increased complexity which is the chief reason its use never became widespread. It was only ever intended to be used as the end sections of composite filters, the rest of the filter being made up of other sections such as k-type and m-type sections.
An incidental advantage of the mm'-type is that it has two independent parameters (m and m') that the designer can adjust. This allows for two different design criteria to be independently optimised.
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The mm'-type filter was an extension of Zobel's previous m-type filter, which itself grew out of George Campbell's k-type design. Zobel's m-type is arrived at by applying the m-derivation process (see m-derived filter) to the k-type filter. Completely analogously, the mm'-type is arrived at by applying the m-derived process to the m-type filter. The value of m used in the second transformation is designated m to distinguish it from m, hence the naming of the filter mm'-type. However, this filter is not a member of the class of filters, general mn-type image filters, which are a generalisation of m-type filters. Rather, it is a double application of the m-derived process and for those filters the arbitrary parameters are usually designated m1, m2, m3 etc., rather than m, m', m'' as here.
The importance of the filter lies in its impedance properties. Some of the impedance terms and section terms used in image design theory are pictured in the diagram below. As always, image theory defines quantities in terms of an infinite cascade of two-port sections, and in the case of the filters being discussed, an infinite ladder network of L-sections.
The sections of the hypothetical infinite filter are made up of series impedance elements of 2Z and shunt admittance elements of 2Y. The factor of two is introduced since it is normal to work in terms of half-sections where it disappears. The image impedance of the input and output port of a section, ''Z''<sub>i1</sub> and ''Z''<sub>i2</sub>, will generally not be the same. However, for a mid-series section (that is, a section from halfway through a series element to halfway through the next series element) will have the same image impedance on both ports due to symmetry. This image impedance is designated ''Z''<sub>iT</sub> due to the "T" topology of a mid-series section. Likewise, the image impedance of a mid-shunt section is designated ''Z''<sub>iΠ</sub> due to the "Π" topology. Half of such a "T" or "Π" section is (unsurprisingly) called a half-section. The image impedances of the half section are dissimilar on the input and output ports but are equal to the mid-series ''Z''<sub>iT</sub> on the side presenting the series element and the mid-shunt ''Z''<sub>iΠ</sub> on the side presenting the shunt element.
A mid-series derived section (that is, a series m-type filter) has precisely the same image impedance, ''Z''<sub>iT</sub>, as a k-type mid-series "T" filter. However, the image impedance of a half-section of such a filter (on the shunt side) is not the same and is designated ''Z''<sub>iΠm</sub>. Similarly, the series element side of a shunt m-derived filter half-section is designated ''Z''<sub>iTm</sub>.
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 . 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, ''Z''<sub>iΠm</sub>. Two such half-sections cascaded with the ''Z''<sub>iT</sub> impedances facing, will form a Π-section with image impedance ''Z''<sub>iΠm</sub>. 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 ''Z''<sub>iΠm</sub>. Again, the half sections will have a different image impedance at the split ports and this is designated ''Z''<sub>iTmm'</sub>.
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 ''Z''<sub>imm'</sub>, ''Z''<sub>iΠmm'</sub>, which is the dual of ''Z''<sub>iTmm'</sub>.
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.
For a low-pass prototype, the cut-off frequency is given by the same expression as for the m-type and the k-type;
| \omega | ||||
|
The pole of attenuation occurs at;
| \omega | ||||
|
See also Image impedance#DerivationThe 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 ωc = 1.
The image impedance looking into a "T" topology port of a shunt derived section is given by,
ZiT=
| \sqrt{1-\omega2 | |
| \left(1-(1-m |
2)\omega2
| 2} | |
| \right)}{1-\left(\omega/\omega | |
| infin\right) |
For comparison;
ZiT=\sqrt{1-\omega2}
ZiT=
| \sqrt{1-\omega2 | |
The image impedance looking into a "Π" topology port of a series derived section is given by,
Zi\Pi=
| |||||||
| \sqrt{1-\omega2 |
\left(1-(1-m2)\omega2\right)}
For comparison;
Zi\Pi=
| 1 | |
| \sqrt{1-\omega2 |
Zi\Pi=
| |||||||
| \sqrt{1-\omega2 |
Note that
\omegainfin
m=0.7230
m'=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.