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Distributed-element circuits are designed with the [[distributed-element model]], an alternative to the [[lumped-element model]] in which the passive [[electrical element]]s of [[electrical resistance]], [[capacitance]] and [[inductance]] are assumed to be "lumped" at one point in space in a [[resistor]], [[capacitor]] or [[inductor]], respectively. The distributed-element model is used when this assumption no longer holds, and these properties are considered to be distributed in space. The assumption breaks down when there is significant time for [[electromagnetic wave]]s to travel from one terminal of a component to the other; "significant", in this context, implies enough time for a noticeable [[Phase (waves)|phase]] change. The amount of phase change is dependent on the wave's [[frequency]] (and inversely dependent on [[wavelength]]). A common rule of thumb amongst engineers is to change from the lumped to the distributed model when distances involved are more than one-tenth of a wavelength (a 36° phase change). The lumped model completely fails at one-quarter wavelength (a 90° phase change), with not only the value, but the nature of the component not being as predicted. Due to this dependence on wavelength, the distributed-element model is used mostly at higher frequencies; at low frequencies, distributed-element components are too bulky. Distributed designs are feasible above {{nowrap|300 [[MHz]]}}, and are the technology of choice at [[microwave]] frequencies above {{nowrap|1 GHz}}.<ref>Vendelin ''et al.'', pp. 35–37</ref>
There is no clear-cut demarcation in the frequency at which these models should be used. Although the changeover is usually somewhere in the 100-to-{{nowrap|500 MHz}} range, the technological scale is also significant; miniaturised circuits can use the lumped model at a higher frequency. [[Printed circuit board]]s (PCBs) using [[through-hole technology]] are larger than equivalent designs using [[surface-mount technology]]. [[Hybrid integrated circuit]]s are smaller than PCB technologies, and [[monolithic integrated circuit]]s are smaller than both. [[Integrated circuit]]s can use lumped designs at higher frequencies than printed circuits, and this is done in some [[radio frequency]] integrated circuits. This choice is particularly significant for hand-held devices, because lumped-element designs generally result in a smaller product.<ref>{{
=== Construction with transmission lines ===
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== Advantages and disadvantages ==
Distributed-element circuits are cheap and easy to manufacture in some formats, but take up more space than lumped-element circuits. This is problematic in mobile devices (especially hand-held ones), where space is at a premium. If the operating frequencies are not too high, the designer may miniaturise components rather than switching to distributed elements. However, [[Parasitic element (electrical networks)|parasitic elements]] and resistive losses in lumped components are greater with increasing frequency as a proportion of the nominal value of the lumped-element impedance. In some cases, designers may choose a distributed-element design (even if lumped components are available at that frequency) to benefit from improved [[Q factor|quality]]. Distributed-element designs tend to have greater power-handling capability; with a lumped component, all the energy passed by a circuit is concentrated in a small volume.<ref>{{
== Media ==
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=== Paired conductors ===
Several types of transmission line exist, and any of them can be used to construct distributed-element circuits. The oldest (and still most widely used) is a pair of conductors; its most common form is [[twisted pair]], used for telephone lines and Internet connections. It is not often used for distributed-element circuits because the frequencies used are lower than the point where distributed-element designs become advantageous. However, designers frequently begin with a lumped-element design and convert it to an open-wire distributed-element design. Open wire is a pair of parallel uninsulated conductors used, for instance, for [[telephone line]]s on [[telegraph pole]]s. The designer does not usually intend to implement the circuit in this form; it is an intermediate step in the design process. Distributed-element designs with conductor pairs are limited to a few specialised uses, such as [[Lecher line]]s and the [[twin-lead]] used for [[antenna (radio)|antenna]] [[feed line]]s.<ref>{{
=== Coaxial ===
[[File:Koaxrichtkoppler.jpg|thumb|alt=Photograph|A collection of coaxial [[directional coupler]]s. One has the cover removed, showing its internal structure.]]
[[Coaxial cable|Coaxial line]], a centre conductor surrounded by an insulated shielding conductor, is widely used for interconnecting units of microwave equipment and for longer-distance transmissions. Although coaxial distributed-element devices were commonly manufactured during the second half of the 20th century, they have been replaced in many applications by planar forms due to cost and size considerations. Air-[[dielectric]] coaxial line is used for low-loss and high-power applications. Distributed-element circuits in other media still commonly transition to [[coaxial connector]]s at the circuit [[Port (circuit theory)|ports]] for interconnection purposes.<ref>{{multiref|Natarajan, pp. 11–12|}}</ref>
=== Planar ===
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=== Mechanical ===
In a few specialist applications, such as the [[mechanical filter]]s in high-end radio transmitters (marine, military, amateur radio), electronic circuits can be implemented as mechanical components; this is done largely because of the high quality of the mechanical resonators. They are used in the [[radio frequency]] band (below microwave frequencies), where waveguides might otherwise be used. Mechanical circuits can also be implemented, in whole or in part, as distributed-element circuits. The frequency at which the transition to distributed-element design becomes feasible (or necessary) is much lower with mechanical circuits. This is because the speed at which signals travel through mechanical media is much lower than the speed of electrical signals.<ref>{{
== Circuit components ==
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{{main|Stub (electronics)}}
A stub is a short length of line that branches to the side of a main line. The end of the stub is often left open- or short-circuited, but may also be terminated with a lumped component. A stub can be used on its own (for instance, for [[impedance matching]]), or several of them can be used together in a more complex circuit such as a filter. A stub can be designed as the equivalent of a lumped capacitor, inductor, or resonator.<ref>{{
[[File:Microstrip Low Pass Bowtie Stub Filter.jpg|thumb|alt=Five butterfly-shaped stubs in a filter|Butterfly stub filter]]
Departures from constructing with uniform transmission lines in distributed-element circuits are rare. One such departure that is widely used is the radial stub, which is shaped like a [[circular sector|sector of a circle]]. They are often used in pairs, one on either side of the main transmission line. Such pairs are called butterfly or bowtie stubs.<ref>Edwards & Steer, pp. 347–348</ref>
=== Coupled lines ===
Coupled lines are two transmission lines between which there is some electromagnetic [[coupling (physics)|coupling]]. The coupling can be direct or indirect. In indirect coupling, the two lines are run closely together for a distance with no screening between them. The strength of the coupling depends on the distance between the lines and the cross-section presented to the other line. In direct coupling, branch lines directly connect the two main lines together at intervals.<ref>{{
Coupled lines are a common method of constructing [[power dividers and directional couplers]]. Another property of coupled lines is that they act as a pair of coupled [[resonator]]s. This property is used in many distributed-element filters.<ref>Bhat & Koul, pp. 10, 602, 622</ref>
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=== Helical resonator ===
{{main|Helical resonator}}
A helical resonator is a [[helix]] of wire in a cavity; one end is unconnected, and the other is bonded to the cavity wall. Although they are superficially similar to lumped inductors, helical resonators are distributed-element components and are used in the [[VHF]] and lower [[UHF]] bands.<ref>{{
=== Fractals ===
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=== Taper ===
A taper is a transmission line with a gradual change in cross-section. It can be considered the limiting case of the stepped impedance structure with an infinite number of steps.<ref>Zhurbenko, p. 310</ref> Tapers are a simple way of joining two transmission lines of different characteristic impedances. Using tapers greatly reduces the mismatch effects that a direct join would cause. If the change in cross-section is not too great, no other matching circuitry may be needed.<ref>Garg ''et al.'', pp. 180–181</ref> Tapers can provide [[Planar transmission line#Transitions|transitions]] between lines in different media, especially different forms of planar media.<ref>{{
Tapers can be used to match a transmission line to an antenna. In some designs, such as the [[horn antenna]] and [[Vivaldi antenna]], the taper is itself the antenna. Horn antennae, like other tapers, are often linear, but the best match is obtained with an exponential curve. The Vivaldi antenna is a flat (slot) version of the exponential taper.<ref>{{
=== Distributed resistance ===
Resistive elements are generally not useful in a distributed-element circuit. However, distributed resistors may be used in [[attenuator (electronics)|attenuator]]s and line [[electrical termination|terminations]]. In planar media they can be implemented as a meandering line of high-resistance material, or as a deposited patch of [[thin-film]] or [[thick-film]] material.<ref>{{
== Circuit blocks ==
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{{main|Distributed-element filter}}
[[File:Microstrip Hairpin Filter And Low Pass Stub Filter.jpg|thumb|alt=See caption|upright=1.3|Microstrip [[band-pass]] hairpin filter (left), followed by a [[low-pass]] stub filter]]
Filters are a large percentage of circuits constructed with distributed elements. A wide range of structures are used for constructing them, including stubs, coupled lines and cascaded lines. Variations include interdigital filters, combline filters and hairpin filters. More-recent developments include [[fractal]] filters.<ref>Cohen, p. 220</ref> Many filters are constructed in conjunction with [[dielectric resonator]]s.<ref>{{
As with lumped-element filters, the more elements used, the closer the filter comes to an [[brickwall filter|ideal response]]; the structure can become quite complex.<ref>Harrell, p. 150</ref> For simple, narrow-band requirements, a single resonator may suffice (such as a stub or [[spurline filter]]).<ref>Awang, p. 296</ref>
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A directional coupler which splits power equally between the output and coupled ports (a {{nowrap|3 [[decibel|dB]]}} coupler) is called a ''hybrid''.<ref>Maloratsky (2004), p. 117</ref> Although "hybrid" originally referred to a [[hybrid transformer]] (a lumped device used in telephones), it now has a broader meaning. A widely used distributed-element hybrid which does not use coupled lines is the ''hybrid ring'' or [[rat-race coupler]]. Each of its four ports is connected to a ring of transmission line at a different point. Waves travel in opposite directions around the ring, setting up [[standing wave]]s. At some points on the ring, destructive [[wave interference|interference]] results in a null; no power will leave a port set at that point. At other points, constructive interference maximises the power transferred.<ref>Chang & Hsieh, pp. 197–198</ref>
Another use for a hybrid coupler is to produce the sum and difference of two signals. In the illustration, two input signals are fed into the ports marked 1 and 2. The sum of the two signals appears at the port marked Σ, and the difference at the port marked Δ.<ref>Ghione & Pirola, pp. 172–173</ref> In addition to their uses as couplers and power dividers, directional couplers can be used in [[balanced mixer]]s, [[frequency discriminator]]s, [[Attenuator (electronics)|attenuator]]s, [[phase shifter]]s, and [[antenna array]] [[antenna feed|feed]] networks.<ref>{{
=== Circulators ===
[[File:Ferritzirkulator1.jpg|thumb|upright|alt=Square, grey, three-port device with an identifying sticker|A coaxial ferrite circulator operating at {{nowrap|1 GHz}}]]
{{main|Circulator}}
A circulator is usually a three- or four-port device in which power entering one port is transferred to the next port in rotation, as if round a circle. Power can flow in only one direction around the circle (clockwise or anticlockwise), and no power is transferred to any of the other ports. Most distributed-element circulators are based on [[Ferrite (magnet)|ferrite]] materials.<ref>{{
An unusual application of a circulator is in a [[reflection amplifier]], where the [[negative resistance]] of a [[Gunn diode]] is used to reflect back more power than it received. The circulator is used to direct the input and output power flows to separate ports.<ref>Roer, pp. 255–256</ref>
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Distributed-element modelling was first used in electrical network analysis by [[Oliver Heaviside]]<ref>Heaviside (1925)</ref> in 1881. Heaviside used it to find a correct description of the behaviour of signals on the [[transatlantic telegraph cable]]. Transmission of early transatlantic telegraph had been difficult and slow due to [[dispersion (optics)|dispersion]], an effect which was not well understood at the time. Heaviside's analysis, now known as the [[telegrapher's equations]], identified the problem and suggested<ref>Heaviside (1887), p. 81</ref> [[loading coil|methods for overcoming it]]. It remains the standard analysis of transmission lines.<ref>Brittain, p. 39</ref>
[[Warren P. Mason]] was the first to investigate the possibility of distributed-element circuits, and filed a patent<ref>Mason (1930)</ref> in 1927 for a coaxial filter designed by this method. Mason and Sykes published the definitive paper on the method in 1937. Mason was also the first to suggest a distributed-element acoustic filter in his 1927 doctoral thesis, and a distributed-element mechanical filter in a patent<ref>Mason (1961)</ref> filed in 1941. Mason's work was concerned with the coaxial form and other conducting wires, although much of it could also be adapted for waveguide. The acoustic work had come first, and Mason's colleagues in the [[Bell Labs]] radio department asked him to assist with coaxial and waveguide filters.<ref>{{
Before [[World War II]], there was little demand for distributed-element circuits; the frequencies used for radio transmissions were lower than the point at which distributed elements became advantageous. Lower frequencies had a greater range, a primary consideration for [[Broadcasting|broadcast]] purposes. These frequencies require long antennae for efficient operation, and this led to work on higher-frequency systems. A key breakthrough was the 1940 introduction of the [[cavity magnetron]] which operated in the microwave band and resulted in radar equipment small enough to install in aircraft.<ref>Borden, p. 3</ref> A surge in distributed-element filter development followed, filters being an essential component of radars. The signal loss in coaxial components led to the first widespread use of waveguide, extending the filter technology from the coaxial ___domain into the waveguide ___domain.<ref>Levy & Cohn, p. 1055</ref>
The wartime work was mostly unpublished until after the war for security reasons, which made it difficult to ascertain who was responsible for each development. An important centre for this research was the [[MIT Radiation Laboratory]] (Rad Lab), but work was also done elsewhere in the US and Britain. The Rad Lab work was published<ref>Fano & Lawson (1948)</ref> by Fano and Lawson.<ref>Levy & Cohn, p. 1055</ref> Another wartime development was the hybrid ring. This work was carried out at [[Bell Labs]], and was published<ref>Tyrrell (1947)</ref> after the war by W. A. Tyrrell. Tyrrell describes hybrid rings implemented in waveguide, and analyses them in terms of the well-known waveguide [[magic tee]]. Other researchers<ref>{{
George Matthaei led a research group at [[Stanford Research Institute]] which included [[Leo C. Young|Leo Young]] and was responsible for many filter designs. Matthaei first described the interdigital filter<ref>Matthaei (1962)</ref> and the combline filter.<ref>Matthaei (1963)</ref> The group's work was published<ref>Matthaei ''et al.'' (1964)</ref> in a landmark 1964 book covering the state of distributed-element circuit design at that time, which remained a major reference work for many years.<ref>Levy and Cohn, pp. 1057–1059</ref>
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Planar formats began to be used with the invention of [[stripline]] by [[Robert M. Barrett]]. Although stripline was another wartime invention, its details were not published<ref>Barrett & Barnes (1951)</ref> until 1951. [[Microstrip]], invented in 1952,<ref>Grieg and Englemann (1952)</ref> became a commercial rival of stripline; however, planar formats did not start to become widely used in microwave applications until better dielectric materials became available for the substrates in the 1960s.<ref>Bhat & Koul, p. 3</ref> Another structure which had to wait for better materials was the dielectric resonator. Its advantages (compact size and high quality) were first pointed out<ref>Richtmeyer (1939)</ref> by R. D. Richtmeyer in 1939, but materials with good temperature stability were not developed until the 1970s. Dielectric resonator filters are now common in waveguide and transmission line filters.<ref>Makimoto & Yamashita, pp. 1–2</ref>
Important theoretical developments included [[Paul I. Richards]]' [[commensurate line theory]], which was published<ref>Richards (1948)</ref> in 1948, and [[Kuroda's identities]], a set of [[Transformation (function)|transforms]] which overcame some practical limitations of Richards theory, published<ref>{{multiref|First English publication:
== References ==
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