Composite image filter
The researchers involved in this work and their contributions are briefly listed below; The image analysis starts with a calculation of the input and output impedances (the image impedances) and the transfer function of a section in an infinite chain of identical sections.
This can be shown to be equivalent to the performance of a section terminated in its image impedances.
They will usually be terminated with fixed resistances that the filter cannot match perfectly except at one specific frequency.
This mismatch leads to multiple reflections at the filter terminations and at the junctions between sections.
These reflections result in the filter response deviating quite sharply from the theoretical, especially near the cut-off frequency.
[8] The requirement for better matching to the end impedances is one of the main motivations for using composite filters.
These prototypes may be scaled and transformed to the desired frequency bandform (low-pass, high-pass, band-pass or band-stop).
T and Π are the smallest symmetrical sections that can be constructed, as shown in diagrams in the topology chart (below).
The most prominent feature of the m-type is a pole of attenuation just past the cut-off frequency inside the stopband.
The drawback with m-type sections is that they have poor stopband rejection past the pole of attenuation.
They are therefore good for matching in to the filter terminations, in the passband at least, the stopband is another story.
Its chief advantage is that it rather better at matching in to resistive end terminations than the k-type or m-type.
There is thus some leeway in the choice, but Zobel suggests[9] the values m=0.7230 and m′=0.4134 which give a deviation of the impedance of less than 2% over the useful part of the band.
Since mm′=0.3, this section will also have a much faster cut-off than an m-type of m=0.6 which is an alternative for impedance matching.
However, the improvements obtained diminish at each iteration and are not usually worth the increase in complexity.
This section has the advantage of being able to place the pole of attenuation much closer to the cut-off frequency than the Zobel filter, which starts to fail to work properly with very small values of m because of inductor resistance.
Clearly, the Zobel network filter does not have a problem matching to its terminations and this is its main advantage.
The constant resistance means that when used in combination with other image filter sections the same problem of matching arises as with end terminations.
Zobel networks also suffer the disadvantage of using far more components than other equivalent image sections.
The most severe deviation of the response from that predicted occurs in the passband close to cut-off.
Further into the passband the impedance match progressively improves, thus limiting the error.
So while stopband impedance mismatch may be severe, it has only limited effect on the filter response.
Sections at the beginning and end of the filter are often chosen for their impedance match in to the terminations rather than the shape of their frequency response.
[12] The inner sections of the filter are most commonly chosen to be constant k since these produce the greatest stopband attenuation.
However, one or two m-type sections might also be included to improve the rate of fall from pass to stopband.
The lower the value of m, the faster the transition, while at the same time, the stopband attenuation becomes less, increasing the need to use extra k-type sections as well.
Another reason for using m-types in the body of the filter is to place an additional pole of attenuation in the stopband.
In this way the m-type sections serve to give good stopband rejection near to cut-off and the k-type sections give good stopband rejection far from cut-off.
Constant resistance sections may also be required, if the filter is being used on a transmission line, to improve the flatness of the passband response.
All sections can be made to have precisely the same image impedance of a fixed resistance.
