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used with stubs in parallel on both sides of the main line. The resulting filter looks rather similar to the stepped impedance filter of figure 5, but has been designed on completely different principles. A difficulty with using stubs this wide is that the point at which they are connected to the main line is ill-defined. A stub that is narrow in comparison to λ can be taken as being connected on its centre-line and calculations based on that assumption will accurately predict filter response. For a wide stub, however, calculations that assume the side branch is connected at a definite point on the main line leads to inaccuracies as this is no longer a good model of the transmission pattern. One solution to this difficulty is to use radial stubs instead of linear stubs. A pair of radial stubs in parallel (one on either side of the main line) is called a butterfly stub (see figure 7(b)). A group of three radial stubs in parallel, which can be achieved at the end of a line, is called a clover-leaf stub.
730:
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769:
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1104:-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.
1224:
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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
828:
of figure 6. Where space allows, the stubs may be set on alternate sides of the main line as shown in figure 7(a). The purpose of this is to prevent coupling between adjacent stubs which detracts from the filter performance by altering the frequency response. However, a structure with all the stubs on the same side is still a valid design. If the stub is required to be a very low impedance line, the stub may be inconveniently wide. In these cases, a possible solution is to connect two narrower stubs in parallel. That is, each stub position has a stub on
676:. Stubs can also be used in conjunction with impedance transformers to build more complex filters and, as would be expected from their resonant nature, are most useful in band-pass applications. While open-circuit stubs are easier to manufacture in planar technologies, they have the drawback that the termination deviates significantly from an ideal open circuit (see figure 4(b)), often leading to a preference for short-circuit stubs (one can always be used in place of the other by adding or subtracting λ/4 to or from the length).
325:. The exact point at which distributed-element modelling becomes necessary depends on the particular design under consideration. A common rule of thumb is to apply distributed-element modelling when component dimensions are larger than 0.1λ. The increasing miniaturisation of electronics has meant that circuit designs are becoming ever smaller compared to λ. The frequencies beyond which a distributed-element approach to filter design becomes necessary are becoming ever higher as a result of these advances. On the other hand,
761:. In such cases, each element of the filter is λ/4 in length (where λ is the wavelength of the main-line signal to be blocked from transmission into the DC source) and the high-impedance sections of the line are made as narrow as the manufacturing technology will allow in order to maximise the inductance. Additional sections may be added as required for the performance of the filter just as they would for the lumped-element counterpart. As well as the planar form shown, this structure is particularly well suited for
284:
68:
satellite orbit, there is a problem getting the signal the last few feet from the dish to the point where it will be used inside the property. The difficulty is that the signal is brought inside the property by a cable (called a downlead) and the high satellite signal frequencies are greatly attenuated when in a cable rather than free space. The purpose of the block converter is to convert the satellite signal to a much lower frequency band that can be handled by the downlead and the user's
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elsewhere. Implementing a true open circuit in planar technology is not feasible because of the dielectric effect of the substrate which will always ensure that the equivalent circuit contains a shunt capacitance. Despite this, open circuits are often used in planar formats in preference to short circuits because they are easier to implement. Numerous element types can be classified as coupled lines and a selection of the more common ones is shown in the figures.
391:. A major paper on the subject was published by Mason and Sykes in 1937. Mason had filed a patent much earlier, in 1927, and that patent may contain the first published electrical design which moves away from a lumped element analysis. Mason and Sykes' work was focused on the formats of coaxial cable and balanced pairs of wires – the planar technologies were not yet in use. Much development was carried out during the war years driven by the filtering needs of
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by George
Matthaei, and also including Leo Young mentioned above, in a landmark book which still today serves as a reference for circuit designers. The hairpin filter was first described in 1972. By the 1970s, most of the filter topologies in common use today had been described. More recent research has concentrated on new or variant mathematical classes of the filters, such as pseudo-
800:. However, beyond the resonant frequency of the highest frequency resonator, the stopband rejection starts to deteriorate as the resonators are moving towards open-circuit. For this reason, filters built to this design often have an additional single stepped-impedance capacitor as the final element of the filter. This also ensures good rejection at high frequency.
371:, although some structures are more suitable for some implementations than others. The open wire implementations, for instance, of a number of structures are shown in the second column of figure 3 and open wire equivalents can be found for most other stripline structures. Planar transmission lines are also used in
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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. Examples of such stubs can be seen on the bias inputs to several components in the photograph at the top of the article.
1175:
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
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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
603:
of the line, which introduces a discontinuity in the transmission characteristics. This is done in planar technologies by a change in the width of the transmission line. Figure 4(a) shows a step up in impedance (narrower lines have higher impedance). A step down in impedance would be the mirror image
67:
and is intended to be attached to a satellite television receiving dish antenna. It is called a block converter because it converts a large number of satellite channels as a block with no attempt to extract to a particular channel. Even though the transmission has travelled 22,000 miles from the
1167:
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
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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
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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
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of the line. A drawback of this topology is that additional transverse resonant modes are possible along the λ/2 length of line formed by the two stubs together. For a choke design, the requirement is simply to make the capacitance as large as possible, for which the maximum stub width of λ/4 may be
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A more complex example of stepped impedance design is presented in figure 6. Again, narrow lines are used to implement inductors and wide lines correspond to capacitors, but in this case, the lumped-element counterpart has resonators connected in shunt across the main line. This topology can be used
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is similar to a stub, in that it requires a distributed-element model to represent it, but is actually built using lumped elements. They are built in a non-planar format and consist of a coil of wire, on a former and core, and connected only at one end. The device is usually in a shielded can with a
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Early stripline directly coupled resonator filters were end-coupled, but the length was reduced and the compactness successively increased with the introduction of parallel-coupled line filters, interdigital filters, and comb-line filters. Much of this work was published by the group at
Stanford led
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of printed planar technologies greatly simplified the manufacture of many microwave components including filters, and microwave integrated circuits then became possible. It is not known when planar transmission lines originated, but experiments using them were recorded as early as 1936. The inventor
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of about 1.3. Some of Young's procedures in that paper were empirical, but later, exact solutions were published. Young's paper specifically addresses directly coupled cavity resonators, but the procedure can equally be applied to other directly coupled resonator types, such as those found in modern
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and are inextricably mixed together. The filter design is usually concerned only with inductance and capacitance, but because of this mixing of elements they cannot be treated as separate "lumped" capacitors and inductors. There is no precise frequency above which distributed element filters must be
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are difficult, if not impossible, to implement with distributed elements. The usual design approach is to start with a band-pass design, but make the upper stopband occur at a frequency that is so high as to be of no interest. Such filters are described as pseudo-high-pass and the upper stopband is
1199:
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
1074:
Stripline parallel-coupled lines filter. This filter is commonly printed at an angle as shown to minimize the board space taken up, although this is not an essential feature of the design. It is also common for the end element or the overlapping halves of the two end elements to be a narrower width
827:
Another common low-pass design technique is to implement the shunt capacitors as stubs with the resonant frequency set above the operating frequency so that the stub impedance is capacitive in the passband. This implementation has a lumped-element counterpart of a general form similar to the filter
712:
applications with the resonant frequency well outside the band of interest. Figures 3(d) and 3(e) show coupled line structures which are both useful in band-pass filters. The structures of figures 3(c) and 3(e) have equivalent circuits involving stubs placed in series with the line. Such a topology
1139:
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
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Some common structures are shown in figures 3 and 4, along with their lumped-element counterparts. These lumped-element approximations are not to be taken as equivalent circuits but rather as a guide to the behaviour of the distributed elements over a certain frequency range. Figures 3(a) and 3(b)
1234:
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
1203:
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
1195:
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
627:
of the corresponding lumped element filter. This correspondence is not exact since distributed-element circuits cannot be rational and is the root reason for the divergence of lumped element and distributed-element behaviour. Impedance transformers are also used in hybrid mixtures of lumped and
89:
signals respectively and the device can be switched between these two. Many filter structures can be seen in the circuit: there are two examples of band-pass parallel-coupled lines filters which are there to restrict the incoming signal to the band of interest. The relatively large width of the
699:
Coupled lines (figures 3(c-e)) can also be used as filter elements; like stubs, they can act as resonators and likewise be terminated short-circuit or open-circuit. Coupled lines tend to be preferred in planar technologies, where they are easy to implement, whereas stubs tend to be preferred
578:
of their networks. These links were also used by other industries with large, fixed networks, notably television broadcasters. Such applications were part of large capital investment programs. However, mass-production manufacturing made the technology cheap enough to incorporate in domestic
425:
connected elements. This was not possible to implement in planar technologies and was often inconvenient in other technologies. This problem was solved by K. Kuroda who used impedance transformers to eliminate the series elements. He published a set of transformations known as
665:. Over a narrow range of frequencies, a stub can be used as a capacitor or an inductor (its impedance is determined by its length) but over a wide band it behaves as a resonator. Short-circuit, nominally quarter-wavelength stubs (figure 3(a)) behave as shunt
1200:
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.
72:. Frequencies depend on satellite system and geographical region, but this particular device converts a block of frequencies in the band 10.7 GHz to 11.8 GHz. The output going to the downlead is in the band 950 MHz to 1950 MHz. The two
1080:
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
752:
design. The filter consists of alternating sections of high-impedance and low-impedance lines which correspond to the series inductors and shunt capacitors in the lumped-element implementation. Low-pass filters are commonly used to feed
1248:
described as a vestigial stopband. Even structures that seem to have an "obvious" high-pass topology, such as the capacitive gap filter of figure 8, turn out to be band-pass when their behaviour for very short wavelengths is considered.
485:
Some simple planar filter structures are shown in the first column. The second column shows the open-wire equivalent circuit for these structures. The third column is a semi-lumped element approximation where the elements marked
1176:
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.
414:. Commensurate lines are networks in which all the elements are the same length (or in some cases multiples of the unit length), although they may differ in other dimensions to give different characteristic impedances.
1117:
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
704:
show a short-circuit and open-circuit stub, respectively. When the stub length is λ/4, these behave, respectively, as anti-resonators and resonators and are therefore useful, respectively, as elements in band-pass and
266:
used by telephone companies and other organisations with large fixed-communication networks, such as television broadcasters. Nowadays the technology can be found in several mass-produced consumer items, such as the
76:
at the bottom of the device are for connection to downleads. Two are provided on this particular model (block converters can have any number of outputs from one upwards) so that two televisions or a television and
604:
of figure 4(a). The discontinuity can be represented approximately as a series inductor, or more exactly, as a low-pass T circuit as shown in figure 4(a). Multiple discontinuities are often coupled together with
848:
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
433:
Following the war, one important research avenue was trying to increase the design bandwidth of wide-band filters. The approach used at the time (and still in use today) was to start with a lumped element
974:
917:
1021:
262:
had long before been developed but these new military systems operated at microwave frequencies and new filter designs were required. When the war ended, the technology found applications in the
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planar technologies and illustrated in this article. The capacitive gap filter (figure 8) and the parallel-coupled lines filter (figure 9) are examples of directly coupled resonators.
1204:
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
180:
theory; many distributed-element components are made of short lengths of transmission line. In the distributed view of circuits, the elements are distributed along the length of
94:
filters below and to the right of the central metal oblong) reflect the wide bandwidth the filter is required to pass. There are also numerous examples of stub filters supplying
845:
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.
713:
is straightforward to implement in open-wire circuits but not with a planar technology. These two structures are therefore useful for implementing an equivalent series element.
684:
hole in the top for adjusting the core. It will often look physically very similar to the lumped LC resonators used for a similar purpose. They are most useful in the upper
820:
Various forms of stubs, respectively, doubled stubs in parallel, radial stub, butterfly stub (paralleled radial stubs), clover-leaf stub (triple paralleled radial stubs).
1089:
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.
421:
The difficulty with
Richards' transformation from the point of view of building practical filters was that the resulting distributed-element design invariably included
98:
to transistors and other devices, the filter being required to prevent the signal from travelling towards the power source. The rows of holes in some tracks, called
85:
would normally be fitted to the circular hole in the centre of the board, the two probes protruding into this space are for receiving horizontally and vertically
612:. These impedance transformers can be just a short (often λ/4) length of transmission line. These composite structures can implement any of the filter families (
228:
332:
The most noticeable difference in behaviour between a distributed-element filter and its lumped-element approximation is that the former will have multiple
1100:
For other (non-printed) technologies, short-circuit lines may be preferred since the short-circuit provides a mechanical attachment point for the line and
1196:
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.
498:
respectively. The fourth column shows a lumped-element approximation making the further assumption that the impedance transformers are λ/4 transformers.
2321:
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418:
allows a lumped element design to be taken "as is" and transformed directly into a distributed-element design using a very simple transform equation.
164:, which considers each element to be "lumped together" at one place. That model is conceptually simple, but it becomes increasingly unreliable as the
340:
passband, because transmission-line transfer characteristics repeat at harmonic intervals. These spurious passbands are undesirable in most cases.
792:. However, calculating component values for these structures is an involved process and has led to designers often choosing to implement them as
160:
to pass, but to block others. Conventional filters are constructed from inductors and capacitors, and the circuits so built are described by the
438:
and through various transformations arrive at the desired filter in a distributed-element form. This approach appeared to be stuck at a minimum
430:
in 1955, but his work was written in
Japanese and it was several years before his ideas were incorporated into the English-language literature.
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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
204:
of many signals into one channel. Distributed-element filters may be constructed to have any of the bandforms possible with lumped elements (
531:
of printed stripline, however, is known; this was Robert M. Barrett who published the idea in 1951. This caught on rapidly, and
Barrett's
17:
41:
A circuit featuring many of the filter structures described in this article. The operating frequency of the filters is around 11
925:
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1208:(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;
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formats (that is, formats where conductors consist of flat strips) are popular because they can be implemented using established
112:
The PCB inside a 20GHz
Agilent N9344C spectrum analyser showing various microstrip distributed-element filter technology elements
708:. Figure 3(c) shows a short-circuited line coupled to the main line. This also behaves as a resonator, but is commonly used in
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to a wavefront travelling down the line, and these reactances can be chosen by design to serve as approximations for lumped
227:
There are many component forms used to construct distributed-element filters, but all have the common property of causing a
321:
is a significant fraction of the wavelength of the operating frequency, and it becomes difficult to use the conventional
1125:. In some designs the link can be longer, giving a wide hairpin with λ/4 impedance transformer action between sections.
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instead, which perform well and are much easier to calculate. The purpose of incorporating resonators is to improve the
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were needed before filters could be advanced beyond wartime designs. One of these was the commensurate line theory of
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2367:"The use of coaxial and balanced transmission lines in filters and wide band transformers for high radio frequencies"
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A microstrip hairpin filter followed by a low pass stub filter on a PCB in a 20GHz
Agilent N9344C spectrum analyser
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more space than a plain transmission line would. The limitation of this topology is that performance (particularly
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structure dimensions are usually comparable to λ in all frequency bands and require the distributed-element model.
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A microstrip low pass filter implemented with bowtie stubs inside a 20 GHz
Agilent N9344C spectrum analyser
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implementations with alternating discs of metal and insulator being threaded on to the central conductor.
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Distributed-element filters are used in many of the same applications as lumped element filters, such as
559:, while still using the same basic topologies, or with alternative implementation technologies such as
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For clarity of presentation, the diagrams in this article are drawn with the components implemented in
2165:
A History of
Engineering and Science in the Bell System: Volume 5: Communications Sciences (1925–1980)
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lumped-element prototype with the stepped impedance filter shown in figure 5. This is also called a
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1047:) deteriorates with increasing fractional bandwidth, and acceptable results are not obtained with a
757:(DC) bias to active components. Filters intended for this application are sometimes referred to as
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Stepped-impedance low-pass filter formed from alternate high and low impedance sections of line
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or buried stripline techniques (with suitable adjustments to dimensions) and can be adapted to
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2352:"Microwave resonators and filters for wireless communication: theory, design, and application"
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672:, and an open-circuit nominally quarter-wavelength stub (figure 3(b)) behaves as a series LC
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216:, which is usually only approximated. All filter classes used in lumped element designs (
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represents a strap through the board making connection with the ground plane underneath.
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Development of distributed-element filters began in the years before World War II.
2461:, Massachusetts Institute of Technology Radiation Laboratory, Dover Publications, 1965.
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The development of distributed-element filters was spurred on by the military need for
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Planar
Microwave Engineering: A Practical Guide to Theory, Measurement, and Circuits
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A microstrip hairpin PCB filter implemented in an Agilent N9344C spectrum analyser
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Another form of stepped-impedance low-pass filter incorporating shunt resonators
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Matthaei, G. L. "Comb-line band-pass filters of narrow or moderate bandwidth",
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The initial non-military application of distributed-element filters was in the
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soon had fierce commercial competition from rival planar formats, especially
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manufacturing techniques. The structures shown can also be implemented using
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depends on which bands are desired and which are considered to be spurious.
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of the signal being transmitted on the line or a section of line of that
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Design and Realizations of Miniaturized Fractal Microwave and RF Filters
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as they are in conventional filters. Its purpose is to allow a range of
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Microwave Filters, Impedance-Matching Networks, and Coupling Structures
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can be tuned to two different channels at the same time. The receiving
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Distributed-element filters are mostly used at frequencies above the
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Complex Electromagnetic Problems and Numerical Simulation Approaches
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Three Interdigital Coupled Line filters from a spectrum analyser PCB
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Another very common component of distributed-element filters is the
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distributed-element filters (the so-called semi-lumped structures).
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34:
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Stripline-like Transmission Lines for Microwave Integrated Circuits
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Satellite Television: Techniques of Analogue and Digital Television
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with a distributed-element prototype. This prototype was based on
403:, but other laboratories in the US and the UK were also involved.
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resonators (compare to the microstrip example in figure 2, or the
2322:"A History of microwave filter research, design, and development"
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bands whereas stubs are more often applied in the higher UHF and
564:
224:, etc.) can be implemented using a distributed-element approach.
95:
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for matching purposes (not shown in this diagram, see Figure 1).
599:
The simplest structure that can be implemented is a step in the
2048:
Barrett, R. M. and Barnes, M. H. "Microwave printed circuits",
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More stripline elements and their lumped-element counterparts.
102:, are not filtering structures but form part of the enclosure.
2541:"Direct-coupled cavity filters for wide and narrow bandwidths"
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Barrett, R. M. "Etched sheets serve as microwave components",
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format. This does not imply an industry preference, although
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969:{\displaystyle \omega _{0}={\sqrt {\omega _{1}\omega _{2}}}}
1059:, is limited by the resolution of the printing technology.
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less than about 5. A further difficulty with producing low-
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Standard stubs on alternating sides of main line λ/4 apart.
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A parallel-coupled lines filter in microstrip construction
231:
on the transmission line. These discontinuities present a
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Stripline stub filter composed of λ/4 short-circuit stubs
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in modern usage usually refers to the form then known as
466:
and was able to produce designs with bandwidths up to an
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78:
912:{\displaystyle \Delta \omega =\omega _{2}-\omega _{1}\,}
1016:{\displaystyle Q={\frac {\omega _{0}}{\Delta \omega }}}
2150:"Hairpin line/half-wave parallel-coupled-line filters"
2120:"Parallel-coupled transmission-line resonator filters"
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Networks and Devices using Planar Transmission Lines
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Matthaei, George L.; Young, Leo and Jones, E. M. T.
1255:
574:
used by telecommunications companies to provide the
505:
An open-circuit stub in parallel with the main line.
502:
A short-circuit stub in parallel with the main line.
2423:"Microwave and millimeter-wave integrated circuits"
2544:IEEE Transactions: Microwave Theory and Techniques
2427:IEEE Transactions: Microwave Theory and Techniques
2326:IEEE Transactions: Microwave Theory and Techniques
2311:IEEE Transactions: Microwave Theory and Techniques
2154:IEEE Transactions: Microwave Theory and Techniques
1015:
968:
911:
45:(GHz). This circuit is described in the box below.
2386:IRE Transactions: Microwave Theory and Techniques
2124:IRE Transactions: Microwave Theory and Techniques
865:are the frequencies of the passband edges, then:
185:used but they are especially associated with the
2557:
2210:Microstrip Filters for RF/Microwave Applications
1062:
317:). At these frequencies, the physical length of
168:of the signal increases, or equivalently as the
1462:
1460:
458:published a method for designing filters which
2536:, vol. 81, iss. 2, pp. 570-571, February 1987.
2421:Niehenke, E. C.; Pucel, R. A. and Bahl, I. J.
508:A short-circuit line coupled to the main line.
144:of the circuit) are not localised in discrete
2034:Lumped Elements for RF and Microwave Circuits
2534:Journal of the Acoustical Society of America
1752:
1750:
1457:
2226:The Worldwide History of Telecommunications
1790:
1788:
1786:
1784:
817:Similar construction using butterfly stubs.
176:applies at all frequencies, and is used in
2172:An Introduction to Linear Network Analysis
1692:
1690:
1688:
1656:
1654:
1652:
1650:
2482:Radio Frequency Integrated Circuit Design
2350:Makimoto, Mitsuo and Yamashita, Sadahiko
2307:"Theory of direct coupled-cavity filters"
1747:
1211:(ii) the first spurious passband is at 2ω
1179:
908:
810:Low-pass filters constructed from stubs.
744:can be implemented quite directly from a
258:during World War II. Lumped element
2105:The Technician's Radio Receiver Handbook
1781:
1222:
1183:
1158:
1147:
1127:
1108:
1091:
1066:
1030:
1026:
802:
767:
728:
720:
630:
477:
282:
107:
33:
1685:
1647:
1289:Power dividers and directional couplers
651:A transverse half-slit across the line.
583:systems. An emerging application is in
189:band (wavelength less than one metre).
14:
2558:
1143:
443:
313:(Very High Frequency) band (30 to 300
271:(figure 1 shows an example) used with
63:The circuit depicted in figure 1 is a
2479:Rogers, John W. M. and Plett, Calvin
2466:"Resistor-transmission-line circuits"
2207:Hong, Jia-Sheng and Lancaster, M. J.
2177:Ford, Peter John and Saunders, G. A.
1238:
836:
591:operated by mobile phone companies.
496:impedance or admittance transformers
2294:, Cambridge University Press, 2004
2174:, English Universities Press, 1961.
2167:, AT&T Bell Laboratories, 1984.
2085:Bhat, Bharathi and Koul, Shiban K.
1172:values as low as 1.4 are possible.
716:
594:
464:quarter wave impedance transformers
278:
24:
2410:McGraw-Hill 1964 (1980 edition is
2392:, pp. 479–491, November 1962.
2239:Jarry, Pierre and Beneat, Jacques
2160:, pp. 719–728, November 1972.
1004:
876:
25:
2582:
1395:Mason, Warren P., "Wave filter",
788:with poles of attenuation in the
399:. A good deal of this was at the
296:The symbol λ is used to mean the
27:Type of electronic filter circuit
2382:"Interdigital band-pass filters"
2337:Understanding Digital Television
1258:
518:
52:
2459:Microwave transmission circuits
2179:The Rise of the Superconductors
2148:Cristal, E. G. and Frankel, S.
2130:, pp. 223–231, April 1958.
2025:
2009:
2000:
1987:
1974:
1961:
1952:
1939:
1930:
1921:
1918:Hong and Lancaster, pp.130–132.
1912:
1903:
1890:
1877:
1864:
1855:
1846:
1837:
1828:
1819:
1806:
1797:
1772:
1759:
1734:
1721:
1712:
1699:
1676:
1663:
1638:
1629:
1620:
1611:
1598:
1589:
1580:
1571:
1558:
1549:
1540:
1531:
1522:
1505:
1496:
1487:
1478:
1469:
1448:
1439:
1430:
1421:
1412:
1389:
1038:Capacitive gap stripline filter
645:A line coming to an abrupt end.
336:replicas of the lumped-element
2476:, pp. 217–220, Feb. 1948.
2403:, pp. 82–91, August 1963.
2365:Mason, W. P. and Sykes, R. A.
2317:, pp. 340–348, June 1967.
2091:, New Age International, 1989
2074:, Butterworth-Heinemann, 1999
2067:, pp. 114–118, June 1952.
1644:Ford and Saunders, pp.157–159.
1380:
1371:
1358:
1345:
1336:
1327:
1318:
1309:
1300:
608:to produce a filter of higher
511:Coupled short-circuited lines.
212:, etc.) with the exception of
13:
1:
2550:, pp. 162–178, May 1963.
2274:Microwave Integrated Circuits
2257:Microwave Integrated Circuits
2194:The RF and Microwave Handbook
2163:Fagen, M. D. and Millman, S.
2056:, p. 16, September 1951.
1294:
1155:Stripline interdigital filter
1063:Parallel-coupled lines filter
620:, etc.) by approximating the
514:Coupled open-circuited lines.
247:, as required by the filter.
2566:Distributed element circuits
2530:"Warren P. Mason: 1900-1986"
2500:, John Wiley and Sons, 2006
2243:, John Wiley and Sons, 2009
2213:, John Wiley and Sons, 2001
1577:Levy and Cohn, pp.1057–1059.
1436:Makimoto and Yamashita, p.2.
1230:Konishi's 60° butterfly stub
642:An abrupt stepped impedance.
446:below for an explanation of
389:distributed-element circuits
7:
2137:, Edward Arnold Ltd., 1972
1251:
922:geometric centre frequency
456:Stanford Research Institute
406:Some important advances in
397:electronic counter-measures
273:satellite television dishes
256:electronic counter measures
10:
2587:
1958:Hong and Lancaster, p.140.
1803:Hong and Lancaster, p.117.
1586:Cristal and Frankel, 1972.
1342:Golio, pp.1.2–1.3,4.4–4.5.
378:
122:distributed-element filter
18:Distributed element filter
2377:, pp. 275–302, 1937.
1511:Barrett and Barnes, 1951,
1418:Fagen and Millman, p.108.
648:A hole or slit in a line.
174:distributed-element model
65:low-noise block converter
57:Low-noise block converter
2335:Lundström, Lars-Ingemar
1219:for the λ/4 stub filter.
1135:Stripline hairpin filter
601:characteristic impedance
416:Richards' transformation
401:MIT Radiation Laboratory
349:planar transmission line
1927:Jarry and Beneat, p.15.
1274:RF and microwave filter
587:filters for use in the
2470:Proceedings of the IRE
2328:, pp. 1055–1067,
1673:, pp.144–149, 203–207.
1626:Huurdeman, pp.369–371.
1617:Levy and Cohn, p.1065.
1595:Levy and Cohn, p.1063.
1475:Levy and Cohn, p.1057.
1466:Levy and Cohn, p.1056.
1445:Levy and Cohn, p.1055.
1386:Mason and Sykes, 1937.
1231:
1192:
1180:Stub band-pass filters
1164:
1156:
1136:
1114:
1097:
1076:
1039:
1017:
970:
913:
824:
776:
737:
726:
658:
606:impedance transformers
589:cellular base stations
524:
291:
113:
46:
2528:Thurston, Robert N.,
2485:, Artech House, 2003
2320:Levy, R. Cohn, S.B.,
2037:, Artech House, 2003
1696:Bhat and Koul, p.499.
1682:Bhat and Koul, p.539.
1660:Bhat and Koul, p.498.
1398:U.S. patent 1,781,469
1226:
1187:
1162:
1151:
1131:
1112:
1095:
1070:
1034:
1027:Capacitive gap filter
1018:
971:
914:
806:
771:
732:
724:
634:
481:
470:, corresponding to a
387:founded the field of
353:printed circuit board
286:
111:
37:
2571:Microwave technology
2224:Huurdeman, Anton A.
2192:Golio, John Michael
982:
926:
873:
581:satellite television
323:lumped element model
162:lumped element model
2517:, Wiley-IEEE, 2003
2498:History of Wireless
2457:Ragan, G. L. (ed.)
2371:Bell Syst. Tech. J.
2271:Konishi, Yoshihiro
2228:, Wiley-IEEE, 2003
1909:Kneppo, pp.216–221.
1852:Kneppo, pp.212–213.
1756:Kneppo, pp.213–214.
1528:Sarkar, pp.556–559.
1324:Lundström, pp.80–82
1144:Interdigital filter
561:suspended stripline
543:. The generic term
428:Kuroda's identities
2446:, CRC Press, 2000
2277:, CRC Press, 1991
2196:, CRC Press, 2001
2181:, CRC Press, 2005
2006:Konishi, pp.80–82.
1971:, pp.424, 497–518.
1949:, pp.424, 614–632.
1936:Paolo, pp.113–116.
1900:, pp.422, 472–477.
1874:, pp.422, 440–450.
1266:Electronics portal
1232:
1215:, as opposed to 3ω
1193:
1165:
1157:
1137:
1115:
1098:
1077:
1040:
1013:
966:
909:
825:
798:stopband rejection
782:elliptical filters
777:
738:
727:
659:
654:A gap in the line.
525:
373:integrated circuit
319:passive components
292:
233:reactive impedance
196:of radio channel,
158:signal frequencies
114:
47:
2496:Sarkar, Tapan K.
2440:Di Paolo, Franco
2437:2002, pp.846–857.
2397:Microwave Journal
2354:, Springer, 2001
2339:, Elsevier, 2006
2260:, Springer, 1994
2135:Wave Transmission
1333:Connor, pp.13–14.
1315:Benoit, pp.44–51.
1306:Bahl, pp.290–293.
1245:high-pass filters
1239:High-pass filters
1011:
964:
837:Band-pass filters
794:m-derived filters
786:Chebyshev filters
706:band-stop filters
681:helical resonator
625:transfer function
444:Band-pass filters
302:electrical length
178:transmission-line
126:electronic filter
118:
117:
106:
105:
16:(Redirected from
2578:
2464:Richards, P. I.
2436:
2380:Matthaei, G. L.
2332:, issue 9, 1984.
2102:Carr, Joseph J.
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1843:Lee, pp.790–792.
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1825:Lee, pp.792–794.
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1778:Lee, pp.789–790.
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1377:Thurston, p. 570
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1349:
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1279:Waveguide filter
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894:
893:
843:band-pass filter
717:Low-pass filters
595:Basic components
529:
528:The introduction
522:
436:prototype filter
279:General comments
260:analogue filters
92:local oscillator
53:
30:
29:
21:
2586:
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2434:
2288:Lee, Thomas H.
2108:, Newnes, 2001
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1718:Carr, pp.63–64.
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1555:Matthaei, 1963.
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1546:Matthaei, 1962.
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1454:Richards, 1948.
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839:
823:
746:ladder topology
742:low-pass filter
719:
710:low-pass filter
657:
597:
585:superconducting
572:microwave links
527:
517:
385:Warren P. Mason
381:
281:
264:microwave links
172:decreases. The
28:
23:
22:
15:
12:
11:
5:
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2574:
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2511:Sevgi, Levent
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2205:
2190:
2175:
2170:Farago, P. S.
2168:
2161:
2146:
2133:Connor, F. R.
2131:
2116:
2100:
2083:
2070:Benoit, Hervé
2068:
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1951:
1938:
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1911:
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1836:
1834:Kneppo, p.212.
1827:
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1513:Barrett, 1952,
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1405:1927, issued:
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1045:insertion loss
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811:
755:direct current
750:cascaded lines
718:
715:
670:antiresonators
656:
655:
652:
649:
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643:
639:
596:
593:
516:
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499:
408:network theory
380:
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361:coaxial cables
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2527:
2524:
2523:0-471-43062-5
2520:
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2510:
2507:
2506:0-471-71814-9
2503:
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2491:1-58053-502-X
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2452:0-8493-1835-1
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2416:0-89006-099-1
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2360:3-540-67535-3
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2283:0-8247-8199-6
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2266:0-412-54700-7
2263:
2259:
2258:
2254:Kneppo, Ivan
2253:
2250:
2249:0-470-48781-X
2246:
2242:
2238:
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2234:0-471-20505-2
2231:
2227:
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2219:0-471-38877-7
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2206:
2203:
2202:0-8493-8592-X
2199:
2195:
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2188:
2187:0-7484-0772-3
2184:
2180:
2176:
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2169:
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2159:
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2147:
2144:
2143:0-7131-3278-7
2140:
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2125:
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2115:
2114:0-7506-7319-2
2111:
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2106:
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2098:
2097:81-224-0052-3
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2080:0-340-74108-2
2077:
2073:
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2047:
2044:
2043:1-58053-309-4
2040:
2036:
2035:
2030:
2029:
2018:
2012:
2003:
1997:, pp.605–614.
1996:
1990:
1984:, pp.595–605.
1983:
1977:
1970:
1964:
1955:
1948:
1942:
1933:
1924:
1915:
1906:
1899:
1893:
1887:, pp.585–595.
1886:
1880:
1873:
1867:
1861:Farago, p.69.
1858:
1849:
1840:
1831:
1822:
1816:, pp.373–380.
1815:
1809:
1800:
1794:Sevgi, p.252.
1791:
1789:
1787:
1785:
1775:
1769:, pp.373–374.
1768:
1762:
1753:
1751:
1744:, pp.217–229.
1743:
1737:
1731:, pp.217–218.
1730:
1724:
1715:
1709:, pp.203–207.
1708:
1702:
1693:
1691:
1689:
1679:
1672:
1666:
1657:
1655:
1653:
1651:
1641:
1635:Benoit, p.34.
1632:
1623:
1614:
1607:
1601:
1592:
1583:
1574:
1567:
1561:
1552:
1543:
1534:
1525:
1518:
1508:
1502:Aksun, p.142.
1499:
1490:
1481:
1472:
1463:
1461:
1451:
1442:
1433:
1424:
1415:
1399:
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1186:
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1173:
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1161:
1154:
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1141:
1134:
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1126:
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1111:
1107:
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1103:
1094:
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1087:
1083:
1073:
1069:
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1054:
1050:
1046:
1037:
1033:
1007:
998:
994:
988:
985:
978:
959:
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949:
945:
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934:
930:
921:
903:
899:
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868:
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846:
844:
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731:
723:
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493:
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476:
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469:
465:
461:
457:
453:
449:
445:
442:of five (see
441:
437:
431:
429:
424:
419:
417:
413:
412:Paul Richards
409:
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398:
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305:
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276:
274:
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265:
261:
257:
253:
248:
246:
242:
238:
234:
230:
229:discontinuity
225:
223:
219:
215:
211:
207:
203:
200:of noise and
199:
195:
190:
188:
183:
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159:
155:
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147:
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110:
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97:
93:
88:
84:
80:
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71:
66:
62:
61:
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55:
54:
51:
50:
44:
40:
36:
32:
31:
19:
2553:
2547:
2543:
2533:
2513:
2497:
2481:
2473:
2469:
2458:
2442:
2430:
2426:
2407:
2400:
2396:
2389:
2385:
2374:
2370:
2336:
2329:
2325:
2314:
2310:
2290:
2273:
2256:
2240:
2225:
2209:
2193:
2178:
2171:
2164:
2157:
2153:
2134:
2127:
2123:
2118:Cohn, S. B.
2104:
2087:
2071:
2064:
2060:
2053:
2050:Radio Telev.
2049:
2033:
2031:Bahl, I. J.
2026:Bibliography
2016:
2011:
2002:
1994:
1989:
1981:
1976:
1968:
1963:
1954:
1946:
1941:
1932:
1923:
1914:
1905:
1897:
1892:
1884:
1879:
1871:
1866:
1857:
1848:
1839:
1830:
1821:
1813:
1808:
1799:
1774:
1766:
1761:
1741:
1736:
1728:
1723:
1714:
1706:
1701:
1678:
1670:
1665:
1640:
1631:
1622:
1613:
1605:
1600:
1591:
1582:
1573:
1565:
1560:
1551:
1542:
1533:
1524:
1516:
1507:
1498:
1489:
1484:Young, 1963.
1480:
1471:
1450:
1441:
1432:
1427:Ragan, 1965.
1423:
1414:
1391:
1382:
1373:
1365:
1360:
1352:
1347:
1338:
1329:
1320:
1311:
1302:
1242:
1233:
1227:
1202:
1198:
1194:
1188:
1174:
1169:
1166:
1152:
1138:
1132:
1119:
1116:
1106:
1101:
1099:
1085:
1081:
1078:
1071:
1052:
1048:
1041:
1035:
854:
847:
840:
829:
826:
807:
778:
772:
758:
749:
739:
733:
702:
698:
678:
660:
635:
598:
569:
553:
548:
544:
540:
536:
532:
526:
491:
487:
482:
471:
459:
450:). In 1957,
447:
439:
432:
420:
405:
382:
342:
331:
308:
295:
287:
249:
226:
202:multiplexing
198:bandlimiting
191:
121:
119:
74:F connectors
56:
38:
2061:Electronics
1537:Cohn, 1958.
1493:Levy, 1967.
1407:11 November
1355:, pp.17–18.
614:Butterworth
218:Butterworth
194:selectivity
130:capacitance
70:set-top box
2560:Categories
2548:vol.MTT-11
2539:Young, L.
2390:vol.MTT-10
2315:vol.MTT-15
2158:vol.MTT-20
1295:References
1228:Figure 13.
1189:Figure 12.
1153:Figure 11.
1133:Figure 10.
869:bandwidth
830:both sides
780:to design
688:and lower
541:microstrip
369:waveguides
365:twin leads
357:microstrip
298:wavelength
269:converters
245:resonators
241:capacitors
182:conductors
170:wavelength
146:capacitors
138:resistance
134:inductance
100:via fences
2305:Levy, R.
2128:vol.MTT-6
2015:Matthaei
1993:Matthaei
1980:Matthaei
1967:Matthaei
1945:Matthaei
1896:Matthaei
1883:Matthaei
1870:Matthaei
1812:Matthaei
1765:Matthaei
1740:Matthaei
1727:Matthaei
1705:Matthaei
1669:Matthaei
1604:Niehenke
1564:Matthaei
1515:Niehenke
1401:, filed:
1351:Matthaei
1072:Figure 9.
1036:Figure 8.
1008:ω
1005:Δ
995:ω
956:ω
946:ω
931:ω
900:ω
896:−
887:ω
880:ω
877:Δ
808:Figure 7.
773:Figure 6.
734:Figure 5.
674:resonator
636:Figure 4.
618:Chebyshev
545:stripline
533:stripline
483:Figure 3.
452:Leo Young
375:designs.
345:stripline
338:prototype
288:Figure 2.
237:inductors
222:Chebyshev
214:high-pass
210:band-pass
187:microwave
166:frequency
154:resistors
150:inductors
128:in which
87:polarized
43:gigahertz
39:Figure 1.
2019:, p.541.
1608:, p.847.
1519:, p.846.
1368:, p.129.
1284:Spurline
1252:See also
1243:Genuine
1084:. Lower-
851:Q-factor
790:stopband
622:rational
576:backbone
557:elliptic
549:triplate
537:triplate
334:passband
206:low-pass
142:elements
2435:3 March
1568:, 1964.
1403:25 June
1364:Rogers
1121:hairpin
763:coaxial
696:bands.
565:finline
460:started
379:History
327:antenna
96:DC bias
2521:
2504:
2489:
2474:vol.36
2450:
2433:, Iss.
2431:vol.50
2414:
2375:vol.16
2358:
2343:
2330:vol.32
2298:
2281:
2264:
2247:
2232:
2217:
2200:
2185:
2141:
2112:
2095:
2078:
2065:vol.25
2054:vol.46
2041:
2017:et al.
1995:et al.
1982:et al.
1969:et al.
1947:et al.
1898:et al.
1885:et al.
1872:et al.
1814:et al.
1767:et al.
1742:et al.
1729:et al.
1707:et al.
1671:et al.
1606:et al.
1566:et al.
1517:et al.
1366:et al.
1353:et al.
1123:filter
1057:tracks
857:. If ω
759:chokes
468:octave
423:series
152:, and
136:, and
124:is an
2401:vol.6
1409:1930.
861:and ω
610:order
393:radar
252:radar
140:(the
2519:ISBN
2502:ISBN
2487:ISBN
2448:ISBN
2412:ISBN
2356:ISBN
2341:ISBN
2296:ISBN
2279:ISBN
2262:ISBN
2245:ISBN
2230:ISBN
2215:ISBN
2198:ISBN
2183:ISBN
2139:ISBN
2110:ISBN
2093:ISBN
2076:ISBN
2039:ISBN
663:stub
563:and
539:and
494:are
395:and
367:and
254:and
83:horn
2429:, '
976:and
784:or
694:SHF
690:UHF
686:VHF
490:or
454:at
315:MHz
311:VHF
243:or
79:VCR
2562::
2546:,
2532:,
2472:,
2468:,
2425:,
2418:).
2399:,
2388:,
2384:,
2373:,
2369:,
2324:,
2313:,
2309:,
2156:,
2152:,
2126:,
2122:,
2063:,
2052:,
1783:^
1749:^
1687:^
1649:^
1459:^
853:,
841:A
740:A
679:A
667:LC
616:,
567:.
551:.
363:,
275:.
239:,
220:,
208:,
148:,
132:,
120:A
2525:.
2508:.
2493:.
2454:.
2362:.
2347:.
2302:.
2285:.
2268:.
2251:.
2236:.
2221:.
2204:.
2189:.
2145:.
2099:.
2082:.
2045:.
1217:0
1213:0
1170:Q
1102:Q
1086:Q
1082:Q
1053:Q
1049:Q
999:0
989:=
986:Q
960:2
950:1
940:=
935:0
919:,
904:1
891:2
883:=
863:2
859:1
855:Q
492:J
488:K
472:Q
448:Q
440:Q
304:.
20:)
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