Distributed-element circuit

Distributed-element circuits are electrical circuits composed of lengths of transmission lines or other distributed components. These circuits perform the same functions as conventional circuits composed of passive components, such as capacitors, inductors, and transformers. They are used mostly at microwave frequencies, where conventional components are difficult to implement.
Conventional circuits consist of individual components manufactured separately then connected together with a conducting medium. Distributed-element circuits are built by forming the medium itself into specific patterns. A major advantage of distributed-element circuits is that they can be produced cheaply as a printed circuit board for consumer products, such as satellite television. They are also made in coaxial and waveguide formats for applications such as radar, satellite communication, and microwave links.
A phenomenon commonly used in distributed-element circuits is that a length of transmission line can be made to behave as a resonator. Distributed-element components which do this include stubs, coupled lines, and cascaded lines. Circuits built from these components include filters, power dividers, directional couplers, and circulators.
Distributed-element circuits were studied during the 1920s and 1930s but did not become important until World War II, when they were used in radar. After the war their use was limited to military, space, and broadcasting infrastructure, but improvements in materials science in the field soon led to broader applications. They can now be found in domestic products such as satellite dishes and mobile phones.
File:Lumped-distributed comparison.png|thumb|upright=2|A low-pass filter as conventional discrete components connected on a printed circuit board, and as a distributed-element design printed on the board itself

Circuit modelling

Distributed-element circuits are designed with the distributed-element model, an alternative to the lumped-element model in which the passive electrical elements 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 waves to travel from one terminal of a component to the other; "significant", in this context, implies enough time for a noticeable phase change. The amount of phase change is dependent on the wave's frequency. 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. The lumped model completely fails at one-quarter wavelength, 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, and are the technology of choice at microwave frequencies above.
There is no clear-cut demarcation in the frequency at which these models should be used. Although the changeover is usually somewhere in the range, the technological scale is also significant; miniaturised circuits can use the lumped model at a higher frequency. Printed circuit boards using through-hole technology are larger than equivalent designs using surface-mount technology. Hybrid integrated circuits are smaller than PCB technologies, and monolithic integrated circuits are smaller than both. Integrated circuits 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.

Construction with transmission lines

The overwhelming majority of distributed-element circuits are composed of lengths of transmission line, a particularly simple form to model. The cross-sectional dimensions of the line are unvarying along its length, and are small compared to the signal wavelength; thus, only distribution along the length of the line need be considered. Such an element of a distributed circuit is entirely characterised by its length and characteristic impedance. A further simplification occurs in commensurate line circuits, where all the elements are the same length. With commensurate circuits, a lumped circuit design prototype consisting of capacitors and inductors can be directly converted into a distributed circuit with a one-to-one correspondence between the elements of each circuit.
Commensurate line circuits are important because a design theory for producing them exists; no general theory exists for circuits consisting of arbitrary lengths of transmission line. Although an arbitrary shape can be analysed with Maxwell's equations to determine its behaviour, finding useful structures is a matter of trial and error or guesswork.
An important difference between distributed-element circuits and lumped-element circuits is that the frequency response of a distributed circuit periodically repeats as shown in the Chebyshev filter example; the equivalent lumped circuit does not. This is a result of the transfer function of lumped forms being a rational function of complex frequency; distributed forms are an irrational function. Another difference is that cascade-connected lengths of line introduce a fixed delay at all frequencies. There is no equivalent in lumped circuits for a fixed delay, although an approximation could be constructed for a limited frequency range.

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, 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 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 to benefit from improved 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.

Media

Paired conductors

Several types of transmission line exist, and any of them can be used to construct distributed-element circuits. The oldest 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 lines on telegraph poles. 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 lines and the twin-lead used for antenna feed lines.

Coaxial

, 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 connectors at the circuit ports for interconnection purposes.

Planar

The majority of modern distributed-element circuits use planar transmission lines, especially those in mass-produced consumer items. There are several forms of planar line, but the kind known as microstrip is the most common. It can be manufactured by the same process as printed circuit boards and hence is cheap to make. It also lends itself to integration with lumped circuits on the same board. Other forms of printed planar lines include stripline, finline and many variations. Planar lines can also be used in monolithic microwave integrated circuits, where they are integral to the device chip.

Waveguide

Many distributed-element designs can be directly implemented in waveguide. However, there is an additional complication with waveguides in that multiple modes are possible. These sometimes exist simultaneously, and this situation has no analogy in conducting lines. Waveguides have the advantages of lower loss and higher quality resonators over conducting lines, but their relative expense and bulk means that microstrip is often preferred. Waveguide mostly finds uses in high-end products, such as high-power military radars and the upper microwave bands. Waveguide becomes bulkier with lower frequency, which militates against its use on the lower bands.

Mechanical

In a few specialist applications, such as the mechanical filters in high-end radio transmitters, 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, 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 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.