A
band-pass filter can be constructed using any elements that can resonate. Filters using stubs can clearly be made band-pass; numerous other structures are possible and some are presented below. An important parameter when discussing band-pass filters is the fractional bandwidth. This is defined as the ratio of the bandwidth to the geometric centre frequency. The inverse of this quantity is called the
Q-factor,
Q. If ω1 and ω2 are the frequencies of the passband edges, then: :bandwidth \Delta\omega=\omega_2-\omega_1\,, :geometric centre frequency \omega_0=\sqrt{\omega_1\omega_2} and :Q=\frac{\omega_0}{\Delta\omega}
Capacitive gap filter The capacitive gap structure consists of sections of line about λ/2 in length which act as resonators and are coupled "end-on" by gaps in the transmission line. It is particularly suitable for planar formats, is easily implemented with printed circuit technology and has the advantage of taking up no more space than a plain transmission line would. The limitation of this topology is that performance (particularly
insertion loss) deteriorates with increasing fractional bandwidth, and acceptable results are not obtained with a
Q less than about 5. A further difficulty with producing low-
Q designs is that the gap width is required to be smaller for wider fractional bandwidths. The minimum width of gaps, like the minimum width of
tracks, is limited by the resolution of the printing technology.
Parallel-coupled lines filter Parallel-coupled lines is another popular topology for printed boards, for which open-circuit lines are the simplest to implement since the manufacturing consists of nothing more than the printed track. The design consists of a row of parallel λ/2 resonators, but coupling over only λ/4 to each of the neighbouring resonators, so forming a staggered line as shown in figure 9. Wider fractional bandwidths are possible with this filter than with the capacitive gap filter, but a similar problem arises on printed boards as dielectric loss reduces the
Q. Lower-
Q lines require tighter coupling and smaller gaps between them which is limited by the accuracy of the printing process. One solution to this problem is to print the track on multiple layers with adjacent lines overlapping but not in contact because they are on different layers. In this way, the lines can be coupled across their width, which results in much stronger coupling than when they are edge-to-edge, and a larger gap becomes possible for the same performance. For other (non-printed) technologies, short-circuit lines may be preferred since the short-circuit provides a mechanical attachment point for the line and
Q-reducing dielectric insulators are not required for mechanical support. Other than for mechanical and assembly reasons, there is little preference for open-circuit over short-circuit coupled lines. Both structures can realize the same range of filter implementations with the same electrical performance. Both types of parallel-coupled filters, in theory, do not have spurious passbands at twice the centre frequency as seen in many other filter topologies (e.g., stubs). However, suppression of this spurious passband requires perfect tuning of the coupled lines which is not realized in practice, so there is inevitably some residual spurious passband at this frequency. The hairpin filter is another structure that uses parallel-coupled lines. In this case, each pair of parallel-coupled lines is connected to the next pair by a short link. The "U" shapes so formed give rise to the name
hairpin filter. In some designs the link can be longer, giving a wide hairpin with λ/4 impedance transformer action between sections. The angled bends seen in figure 10 are common to stripline designs and represent a compromise between a sharp right angle, which produces a large discontinuity, and a smooth bend, which takes up more board area which can be severely limited in some products. Such bends are often seen in long stubs where they could not otherwise be fitted into the space available. The lumped-element equivalent circuit of this kind of discontinuity is similar to a stepped-impedance discontinuity.
Interdigital filter Interdigital filters are another form of coupled-line filter. Each section of line is about λ/4 in length and is terminated in a short-circuit at one end only, the other end being left open-circuit. The end which is short-circuited alternates on each line section. This topology is straightforward to implement in planar technologies, but also particularly lends itself to a mechanical assembly of lines fixed inside a metal case. The lines can be either circular rods or rectangular bars, and interfacing to a coaxial format line is easy. As with the parallel-coupled line filter, the advantage of a mechanical arrangement that does not require insulators for support is that dielectric losses are eliminated. The spacing requirement between lines is not as stringent as in the parallel line structure; as such, higher fractional bandwidths can be achieved, and
Q values as low as 1.4 are possible. The comb-line filter is similar to the interdigital filter in that it lends itself to mechanical assembly in a metal case without dielectric support. In the case of the comb-line, all the lines are short-circuited at the same end rather than alternate ends. The other ends are terminated in capacitors to ground, and the design is consequently classified as semi-lumped. The chief advantage of this design is that the upper stopband can be made very wide, that is, free of spurious passbands at all frequencies of interest.
Stub band-pass filters As mentioned above, stubs lend themselves to band-pass designs. General forms of these are similar to stub low-pass filters except that the main line is no longer a narrow
high impedance line. Designers have many different topologies of stub filters to choose from, some of which produce identical responses. An example stub filter is shown in figure 12; it consists of a row of λ/4 short-circuit stubs coupled together by λ/4 impedance transformers. The stubs in the body of the filter are double paralleled stubs while the stubs on the end sections are only singles, an arrangement that has
impedance matching advantages. The impedance transformers have the effect of transforming the row of shunt anti-resonators into a ladder of series resonators and shunt anti-resonators. A filter with similar properties can be constructed with λ/4 open-circuit stubs placed in series with the line and coupled together with λ/4 impedance transformers, although this structure is not possible in planar technologies. Yet another structure available is λ/2 open-circuit stubs across the line coupled with λ/4 impedance transformers. This topology has both low-pass and band-pass characteristics. Because it will pass DC, it is possible to transmit biasing voltages to active components without the need for blocking capacitors. Also, since short-circuit links are not required, no assembly operations other than the board printing are required when implemented as stripline. The disadvantages are : (i) the filter will take up more board real estate than the corresponding λ/4 stub filter, since the stubs are all twice as long; : (ii) the first spurious passband is at 2ω0, as opposed to 3ω0 for the λ/4 stub filter. Konishi describes a wideband 12 GHz band-pass filter, which uses 60° butterfly stubs and also has a low-pass response (short-circuit stubs are required to prevent such a response). As is often the case with distributed-element filters, the bandform into which the filter is classified largely depends on which bands are desired and which are considered to be spurious. ==High-pass filters==