Distributed-element model

Last updated
Fig.1 Transmission line. The distributed-element model applied to a transmission line. Line model Heaviside.svg
Fig.1 Transmission line. The distributed-element model applied to a transmission line.

In electrical engineering, the distributed-element model or transmission-line model of electrical circuits assumes that the attributes of the circuit (resistance, capacitance, and inductance) are distributed continuously throughout the material of the circuit. This is in contrast to the more common lumped-element model, which assumes that these values are lumped into electrical components that are joined by perfectly conducting wires. In the distributed-element model, each circuit element is infinitesimally small, and the wires 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.

Contents

The distributed model is used where the wavelength 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 lines such as overhead power lines.

Applications

The distributed-element model is more accurate but more complex than the lumped-element model. The use of infinitesimals will often require the application of calculus, whereas circuits analysed by the lumped-element model can be solved with linear algebra. 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. [1]

Transmission lines

Transmission lines 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 lines, 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 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.

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

High-frequency transistors

Fig.2. The base region of a bipolar junction transistor can be modelled as a simplified transmission line. Line model Kelvin.svg
Fig.2. The base region of a bipolar junction transistor can be modelled as a simplified transmission line.

Another example of the use of distributed elements is in the modelling of the base region of a bipolar junction transistor at high frequencies. The analysis of charge carriers crossing the base region is inaccurate when the base region is simply treated as a lumped element. A more successful model is a simplified transmission line model, which includes the base material's distributed bulk resistance and the substrate's distributed capacitance. This model is represented in figure 2.

Resistivity measurements

Fig. 3. Simplified arrangement for measuring the resistivity of a bulk material with surface probes. Resistivity probes.svg
Fig. 3. Simplified arrangement for measuring the resistivity of a bulk material with surface probes.

In many situations, it is desired to measure resistivity of bulk material by applying an electrode array at the surface. Amongst the fields that use this technique are geophysics (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 wafers. [2] 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 material's resistivity 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. [3]

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 coordinates, indeed, in the geophysical application, it may well be that regions of changed resistivity are the very things that it is desired to detect. [4]

Inductor windings

Fig. 4. A possible distributed-element model of an inductor. A more accurate model will also require series resistance elements with the inductance elements. Inductor distributed model.svg
Fig. 4. A possible distributed-element model of an inductor. A more accurate model will also require series resistance elements with the inductance elements.

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 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. [5]

See also

Related Research Articles

<span class="mw-page-title-main">Electrical network</span> Assemblage of connected electrical elements

An electrical network is an interconnection of electrical components or a model of such an interconnection, consisting of electrical elements. An electrical circuit is a network consisting of a closed loop, giving a return path for the current. Thus all circuits are networks, but not all networks are circuits. Linear electrical networks, a special type consisting only of sources, linear lumped elements, and linear distributed elements, have the property that signals are linearly superimposable. They are thus more easily analyzed, using powerful frequency domain methods such as Laplace transforms, to determine DC response, AC response, and transient response.

In electrical engineering, electrical length is a dimensionless parameter equal to the physical length of an electrical conductor such as a cable or wire, divided by the wavelength of alternating current at a given frequency traveling through the conductor. In other words, it is the length of the conductor measured in wavelengths. It can alternately be expressed as an angle, in radians or degrees, equal to the phase shift the alternating current experiences traveling through the conductor.

<span class="mw-page-title-main">Loading coil</span> Inductor in a transmission line

A loading coil or load coil is an inductor that is inserted into an electronic circuit to increase its inductance. The term originated in the 19th century for inductors used to prevent signal distortion in long-distance telegraph transmission cables. The term is also used for inductors in radio antennas, or between the antenna and its feedline, to make an electrically short antenna resonant at its operating frequency.

<span class="mw-page-title-main">Transmission line</span> Cable or other structure for carrying radio waves

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 transmission must be taken into account. This applies especially to radio-frequency engineering because the short wavelengths mean that wave phenomena arise over very short distances. However, the theory of transmission lines was historically developed to explain phenomena on very long telegraph lines, especially submarine telegraph cables.

In electrical engineering, electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks. All electrical networks can be analyzed as multiple electrical elements interconnected by wires. Where the elements roughly correspond to real components, the representation can be in the form of a schematic diagram or circuit diagram. This is called a lumped-element circuit model. In other cases, infinitesimal elements are used to model the network in a distributed-element model.

<span class="mw-page-title-main">Antenna (radio)</span> Electrical device

In radio engineering, an antenna or aerial is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver. In transmission, a radio transmitter supplies an electric current to the antenna's terminals, and the antenna radiates the energy from the current as electromagnetic waves. In reception, an antenna intercepts some of the power of a radio wave in order to produce an electric current at its terminals, that is applied to a receiver to be amplified. Antennas are essential components of all radio equipment.

<span class="mw-page-title-main">Lumped-element model</span> Simplification of a physical system into a network of discrete components

The lumped-element model is a simplified representation of a physical system or circuit that assumes all components are concentrated at a single point and their behavior can be described by idealized mathematical models. The lumped-element model simplifies the system or circuit behavior description into a topology. It is useful in electrical systems, mechanical multibody systems, heat transfer, acoustics, etc. This is in contrast to distributed parameter systems or models in which the behaviour is distributed spatially and cannot be considered as localized into discrete entities.

<span class="mw-page-title-main">Impedance matching</span> Adjusting input/output impedances of an electrical circuit for some purpose

In electrical engineering, impedance matching is the practice of designing or adjusting the input impedance or output impedance of an electrical device for a desired value. Often, the desired value is selected to maximize power transfer or minimize signal reflection. For example, impedance matching typically is used to improve power transfer from a radio transmitter via the interconnecting transmission line to the antenna. Signals on a transmission line will be transmitted without reflections if the transmission line is terminated with a matching impedance.

Capacitors and inductors as used in electric circuits are not ideal components with only capacitance or inductance. However, they can be treated, to a very good degree of approximation, as being ideal capacitors and inductors in series with a resistance; this resistance is defined as the equivalent series resistance (ESR). If not otherwise specified, the ESR is always an AC resistance, which means it is measured at specified frequencies, 100 kHz for switched-mode power supply components, 120 Hz for linear power-supply components, and at its self-resonant frequency for general-application components. Additionally, audio components may report a "Q factor", incorporating ESR among other things, at 1000 Hz.

<span class="mw-page-title-main">Resonator</span> Device or system that exhibits resonance

A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. The oscillations in a resonator can be either electromagnetic or mechanical. Resonators are used to either generate waves of specific frequencies or to select specific frequencies from a signal. Musical instruments use acoustic resonators that produce sound waves of specific tones. Another example is quartz crystals used in electronic devices such as radio transmitters and quartz watches to produce oscillations of very precise frequency.

<span class="mw-page-title-main">Electronic circuit</span> Electrical circuit with active components

An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. It is a type of electrical circuit. For a circuit to be referred to as electronic, rather than electrical, generally at least one active component must be present. The combination of components and wires allows various simple and complex operations to be performed: signals can be amplified, computations can be performed, and data can be moved from one place to another.

<span class="mw-page-title-main">Gyrator–capacitor model</span> Model for magnetic circuits

The gyrator–capacitor model - sometimes also the capacitor-permeance model - is a lumped-element model for magnetic circuits, that can be used in place of the more common resistance–reluctance model. The model makes permeance elements analogous to electrical capacitance rather than electrical resistance. Windings are represented as gyrators, interfacing between the electrical circuit and the magnetic model.

<span class="mw-page-title-main">Primary line constants</span> Parameters of transmission lines

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 relevant to transmission lines and are to be contrasted with the secondary line constants, which can be derived from them, and are more generally applicable. The secondary line constants can be used, for instance, to compare the characteristics of a waveguide to a copper line, whereas the primary constants have no meaning for a waveguide.

<span class="mw-page-title-main">Distributed-element filter</span> Type of electronic filter circuit

A distributed-element filter is an electronic filter in which capacitance, inductance, and resistance are not localised in discrete capacitors, inductors, and resistors as they are in conventional filters. Its purpose is to allow a range of signal frequencies to pass, but to block others. Conventional filters are constructed from inductors and capacitors, and the circuits so built are described by the lumped element model, which considers each element to be "lumped together" at one place. That model is conceptually simple, but it becomes increasingly unreliable as the frequency of the signal increases, or equivalently as the wavelength decreases. The distributed-element model applies at all frequencies, and is used in transmission-line 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 conductors 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 used but they are especially associated with the microwave band.

<span class="mw-page-title-main">Waveguide filter</span> Electronic filter that is constructed with waveguide technology

A waveguide filter is an electronic filter constructed with waveguide technology. Waveguides are hollow metal conduits inside which an electromagnetic wave may be transmitted. Filters are devices used to allow signals at some frequencies to pass, while others are rejected. Filters are a basic component of electronic engineering designs and have numerous applications. These include selection of signals and limitation of noise. Waveguide filters are most useful in the microwave band of frequencies, where they are a convenient size and have low loss. Examples of microwave filter use are found in satellite communications, telephone networks, and television broadcasting.

<span class="mw-page-title-main">Mechanical filter</span> Type of signal processing filter

A mechanical filter is a signal processing filter usually used in place of an electronic filter at radio frequencies. Its purpose is the same as that of a normal electronic filter: to pass a range of signal frequencies, but to block others. The filter acts on mechanical vibrations which are the analogue of the electrical signal. At the input and output of the filter, transducers convert the electrical signal into, and then back from, these mechanical vibrations.

<span class="mw-page-title-main">Commensurate line circuit</span>

Commensurate line circuits are electrical circuits composed of transmission lines that are all the same length; commonly one-eighth of a wavelength. Lumped element circuits can be directly converted to distributed-element circuits of this form by the use of Richards' transformation. This transformation has a particularly simple result; inductors are replaced with transmission lines terminated in short-circuits and capacitors are replaced with lines terminated in open-circuits. Commensurate line theory is particularly useful for designing distributed-element filters for use at microwave frequencies.

The impedance analogy is a method of representing a mechanical system by an analogous electrical system. The advantage of doing this is that there is a large body of theory and analysis techniques concerning complex electrical systems, especially in the field of filters. By converting to an electrical representation, these tools in the electrical domain can be directly applied to a mechanical system without modification. A further advantage occurs in electromechanical systems: Converting the mechanical part of such a system into the electrical domain allows the entire system to be analysed as a unified whole.

<span class="mw-page-title-main">Distributed-element circuit</span> Electrical circuits composed of lengths of transmission lines or other distributed components

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.

<span class="mw-page-title-main">Performance and modelling of AC transmission</span>

Performance modelling is the abstraction of a real system into a simplified representation to enable the prediction of performance. The creation of a model can provide insight into how a proposed or actual system will or does work. This can, however, point towards different things to people belonging to different fields of work.

References

  1. Kaiser, p. 3·2.
  2. Lark-Horovitz & Johnson, p. 54.
  3. Sharma, pp. 210–212.
  4. Sharma, p. 211.
  5. Northrop, pp. 141–142.

Bibliography