: ''This article is an example from the domain of electrical systems, which is a special case of the more general distributed-parameter systems.'' In electrical engineering, the distributed-element model or transmission-line model of electrical circuits assumes that the attributes of the circuit ( resistance,

capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized ar ...

, and inductance) are distributed continuously throughout the material of the circuit. This is in contrast to the more common

lumped-element model The lumped-element model (also called lumped-parameter model, or lumped-component model) simplifies the description of the behaviour of spatially distributed physical systems, such as electrical circuits, into a topology consisting of discrete e ...

, which assumes that these values are lumped into

electrical component An electronic component is any basic discrete device or physical entity in an electronic system used to affect electrons or their associated fields. Electronic components are mostly industrial products, available in a singular form and are not ...

s that are joined by perfectly conducting wires. In the distributed-element model, each circuit element is infinitesimally small, and the

wire Overhead power cabling. The conductor consists of seven strands of steel (centre, high tensile strength), surrounded by four outer layers of aluminium (high conductivity). Sample diameter 40 mm A wire is a flexible strand of metal. Wire is c ...

s connecting elements are not assumed to be perfect conductors; that is, they have impedance. Unlike the lumped-element model, it assumes nonuniform current along each branch and nonuniform voltage along each wire. The distributed model is used where the

wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...

becomes comparable to the physical dimensions of the circuit, making the lumped model inaccurate. This occurs at high frequencies, where the wavelength is very short, or on low-frequency, but very long,

transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmi ...

s such as

overhead power line An overhead power line is a structure used in electric power transmission and distribution to transmit electrical energy across large distances. It consists of one or more uninsulated electrical cables (commonly multiples of three for three-p ...

Applications

The distributed-element model is more accurate but more complex than the

. The use of infinitesimals will often require the application of

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

whereas circuits analysed by the lumped-element model can be solved with

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...

. The distributed model is consequently usually only applied when accuracy calls for its use. The location of this point is dependent on the accuracy required in a specific application, but essentially, it needs to be used in circuits where the wavelengths of the signals have become comparable to the physical dimensions of the components. An often quoted engineering rule of thumb (not to be taken too literally because there are many exceptions) is that parts larger than one tenth of a wavelength will usually need to be analysed as distributed elements.

Transmission lines

Transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmi ...

s are a common example of the use of the distributed model. Its use is dictated because the length of the line will usually be many wavelengths of the circuit's operating frequency. Even for the low frequencies used on

power transmission line Electric power transmission is the bulk movement of electrical energy from a generating site, such as a power plant, to an electrical substation. The interconnected lines that facilitate this movement form a ''transmission network''. This is d ...

s, one tenth of a wavelength is still only about 500 kilometres at 60 Hz. Transmission lines are usually represented in terms of the

primary line constants The primary line constants are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevan ...

as shown in figure 1. From this model the behaviour of the circuit is described by the secondary line constants which can be calculated from the primary ones. The primary line constants are normally taken to be constant with position along the line leading to a particularly simple analysis and model. However, this is not always the case, variations in physical dimensions along the line will cause variations in the primary constants, that is, they have now to be described as functions of distance. Most often, such a situation represents an unwanted deviation from the ideal, such as a manufacturing error, however, there are a number of components where such longitudinal variations are deliberately introduced as part of the function of the component. A well-known example of this is the

horn antenna A horn antenna or microwave horn is an antenna that consists of a flaring metal waveguide shaped like a horn to direct radio waves in a beam. Horns are widely used as antennas at UHF and microwave frequencies, above 300 MHz. They are ...

. Where

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

s are present on the line, quite short lengths of line can exhibit effects that are simply not predicted by the lumped-element model. A quarter wavelength line, for instance, will

transform Transform may refer to: Arts and entertainment * Transform (scratch), a type of scratch used by turntablists * ''Transform'' (Alva Noto album), 2001 * ''Transform'' (Howard Jones album) or the title song, 2019 * ''Transform'' (Powerman 5000 album ...

the terminating impedance into its dual. This can be a wildly different impedance.

High-frequency transistors

Another example of the use of distributed elements is in the modelling of the base region of a

bipolar junction transistor A bipolar junction transistor (BJT) is a type of transistor that uses both electrons and electron holes as charge carriers. In contrast, a unipolar transistor, such as a field-effect transistor, uses only one kind of charge carrier. A bipola ...

at high frequencies. The analysis of charge carriers crossing the base region is not accurate when the base region is simply treated as a lumped element. A more successful model is a simplified transmission line model which includes distributed bulk resistance of the base material and distributed capacitance to the substrate. This model is represented in figure 2.

Resistivity measurements

In many situations it is desired to measure

resistivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...

of a bulk material by applying an

electrode array An electrode array is a configuration of electrodes used for measuring either an electric current or voltage. Some electrode arrays can operate in a bidirectional fashion, in that they can also be used to provide a stimulating pattern of electric c ...

at the surface. Amongst the fields that use this technique are

geophysics Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' so ...

(because it avoids having to dig into the substrate) and the semiconductor industry (for the similar reason that it is non-intrusive) for testing bulk

silicon wafer In electronics, a wafer (also called a slice or substrate) is a thin slice of semiconductor, such as a crystalline silicon (c-Si), used for the fabrication of integrated circuits and, in photovoltaics, to manufacture solar cells. The wafer serv ...

s. The basic arrangement is shown in figure 3, although normally more electrodes would be used. To form a relationship between the voltage and current measured on the one hand, and the resistivity of the material on the other, it is necessary to apply the distributed-element model by considering the material to be an array of infinitesimal resistor elements. Unlike the transmission line example, the need to apply the distributed-element model arises from the geometry of the setup, and not from any wave propagation considerations. The model used here needs to be truly 3-dimensional (transmission line models are usually described by elements of a one-dimensional line). It is also possible that the resistances of the elements will be functions of the co-ordinates, indeed, in the geophysical application it may well be that regions of changed resistivity are the very things that it is desired to detect.

Inductor windings

Another example where a simple one-dimensional model will not suffice is the windings of an inductor. Coils of wire have capacitance between adjacent turns (and also more remote turns as well, but the effect progressively diminishes). For a single layer solenoid, the distributed capacitance will mostly lie between adjacent turns as shown in figure 4 between turns T₁ and T₂, but for multiple layer windings and more accurate models distributed capacitance to other turns must also be considered. This model is fairly difficult to deal with in simple calculations and for the most part is avoided. The most common approach is to roll up all the distributed capacitance into one lumped element in parallel with the inductance and resistance of the coil. This lumped model works successfully at low frequencies but falls apart at high frequencies where the usual practice is to simply measure (or specify) an overall '' Q'' for the inductor without associating a specific equivalent circuit.Northrop, pp. 141–142.

References

Bibliography

*Kenneth L. Kaiser, ''Electromagnetic compatibility handbook'', CRC Press, 2004 . *Karl Lark-Horovitz, Vivian Annabelle Johnson, ''Methods of experimental physics: Solid state physics'', Academic Press, 1959 . *Robert B. Northrop, ''Introduction to instrumentation and measurements'', CRC Press, 1997 . *P. Vallabh Sharma, ''Environmental and engineering geophysics'', Cambridge University Press, 1997 . {{DEFAULTSORT:Distributed-Element Model Electronic design Electronic circuits Distributed element circuits Conceptual models ja:分布定数回路