primary line constants
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The primary line constants are parameters that describe the characteristics of conductive transmission lines, such as pairs of
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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 The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a cir ...
, 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
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to a copper line, whereas the primary constants have no meaning for a waveguide. The constants are conductor resistance and inductance, and insulator capacitance and conductance, which are by convention given the symbols ''R'', ''L'', ''C'', and ''G'' respectively. The constants are enumerated in terms of per unit length. The circuit representation of these elements requires a
distributed-element model : ''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 ...
and consequently
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 ...
must be used to analyse the circuit. The analysis yields a system of two first order, simultaneous linear
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s which may be combined to derive the secondary constants of
characteristic impedance The characteristic impedance or surge impedance (usually written Z0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in ...
and
propagation constant The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a cir ...
. A number of special cases have particularly simple solutions and important practical applications. Low loss cable requires only ''L'' and ''C'' to be included in the analysis, useful for short lengths of cable. Low frequency applications, such as
twisted pair Twisted pair cabling is a type of wiring used for communications in which two conductors of a single circuit are twisted together for the purposes of improving electromagnetic compatibility. Compared to a single conductor or an untwisted ba ...
telephone lines, are dominated by ''R'' and ''C'' only. High frequency applications, such as RF co-axial cable, are dominated by ''L'' and ''C''. Lines loaded to prevent distortion need all four elements in the analysis, but have a simple, elegant solution.


The constants

There are four primary line constants, but in some circumstances some of them are small enough to be ignored and the analysis can be simplified. These four, and their symbols and units are as follows: ''R'' and ''L'' are elements in series with the line (because they are properties of the conductor) and ''C'' and ''G'' are elements shunting the line (because they are properties of the
dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
material between the conductors). ''G'' represents leakage current through the dielectric and in most cables is very small. The word loop is used to emphasise that the resistance and inductance of both conductors must be taken into account. For instance, if a line consists of two identical wires that have a resistance of 25 mΩ/m each, the ''loop'' resistance is double that, 50 mΩ/m. Because the values of the constants are quite small, it is common for manufacturers to quote them per kilometre rather than per metre; in the English-speaking world "per mile" can also be used. The word "constant" can be misleading. It means that they are material constants; but they may vary with frequency. In particular, ''R'' is heavily influenced by the
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. Furthermore, while ''G'' has virtually no effect at
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, it can cause noticeable losses at high frequency with many of the
dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
materials used in cables due to a high
loss tangent Dielectric loss quantifies a dielectric material's inherent dissipation of electromagnetic energy (e.g. heat). It can be parameterized in terms of either the loss angle ''δ'' or the corresponding loss tangent tan ''δ''. Both refer to the ...
. Avoiding the losses caused by ''G'' is the reason many cables designed for use at UHF are air-insulated or foam-insulated (which makes them virtually air-insulated). The actual meaning of constant in this context is that the parameter is constant with ''distance''. That is the line is assumed to be homogenous lengthwise. This condition is true for the vast majority of transmission lines in use today.


Typical values for some common cables

:† Manufacturers commonly omit a value for inductance in their data sheets. Some of these values are estimated from the figures for capacitance and characteristic impedance by \scriptstyle ^2=L/C.


Circuit representation

The line constants cannot be simply represented as
lumped elements 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 ...
in a circuit; they must be described as
distributed elements : ''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 E ...
. For instance "pieces" of the capacitance are in between "pieces" of the resistance. However many pieces the ''R'' and ''C'' are broken into, it can always be argued they should be broken apart further to properly represent the circuit, and after each division the number of
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es in the circuit is increased. This is shown diagramtically in figure 1. To give a true representation of the circuit, the elements must be made
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
ly small so that each element is distributed along the line. The infinitesimal elements in an infinitesimal distance \scriptstyle dx are given by;Connor, pp. 8–10. :dL=\lim_(L\delta x)=Ldx  :dR=\lim_(R\delta x)=Rdx  :dC=\lim_(C\delta x)=Cdx  :dG=\lim_(G\delta x)=Gdx  It is convenient for the purposes of analysis to roll up these elements into general series impedance, ''Z'', and shunt
admittance In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance, analogous to how conductance & resistance are defined. The SI unit of admittance ...
, ''Y'' elements such that; :dZ=(R+i \omega L)dx=Zdx\, ,  and, :dY=(G+i \omega C)dx=Ydx\, . Analysis of this network (figure 2) will yield the secondary line constants: the
propagation constant The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a cir ...
, \scriptstyle \gamma, (whose
real and imaginary parts In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
are the
attenuation constant The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a ci ...
, \scriptstyle \alpha, and phase change constant, \scriptstyle \beta, respectively) and the
characteristic impedance The characteristic impedance or surge impedance (usually written Z0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in ...
, \scriptstyle Z_0, which also, in general, will have real, \scriptstyle R_0, and imaginary, \scriptstyle X_0, parts, making a total of four secondary constants to be derived from the four primary constants. The term constant is even more misleading for the secondary constants as they usually vary quite strongly with frequency, even in an ideal situation where the primary constants do not. This is because the reactances in the circuit (\scriptstyle \omega L and \scriptstyle 1/(\omega C)) introduce a dependence on \scriptstyle \omega. It is possible to choose specific values of the primary constants that result in \scriptstyle \alpha and \scriptstyle Z_0 being independent of \omega (the
Heaviside condition The Heaviside condition, named for Oliver Heaviside (1850–1925), is the condition an electrical transmission line must meet in order for there to be no distortion of a transmitted signal. Also known as the distortionless condition, it can be used ...
) but even in this case, there is still \scriptstyle \beta which is directly proportional to \scriptstyle \omega. As with the primary constants, the meaning of "constant" is that the secondary constants do not vary with distance along the line, not that they are independent of frequency.


Characteristic impedance

The characteristic impedance of a transmission line, \scriptstyle Z_0, is defined as the impedance looking into an infinitely long line. Such a line will never return a reflection since the incident wave will never reach the end to be reflected. When considering a finite length of the line, the remainder of the line can be replaced by \scriptstyle Z_0 as its equivalent circuit. This is so because the remainder of the line is still infinitely long and therefore equivalent to the original line. If the finite segment is very short, then in the equivalent circuit it will be modelled by an L-network consisting of one element of \scriptstyle dZ and one of \scriptstyle dY; the remainder is given by \scriptstyle Z_0. This results in the network shown in figure 3, which can be analysed for \scriptstyle Z_0 using the usual
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theorems, :Z_0 = \delta Z + \frac which re-arranges to, :^2 - Z_0 \delta Z = \frac Taking limits of both sides :\lim_(^2 - Z_0 \delta Z) = ^2 = \frac and since the line was assumed to be homogenous lengthwise, :^2 = \frac


Propagation constant

The ratio of the line input voltage to the voltage a distance \scriptstyle \delta x further down the line (that is, after one section of the equivalent circuit) is given by a standard
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calculation. The remainder of the line to the right, as in the characteristic impedance calculation, is replaced with \scriptstyle Z_0, :\frac = \frac = 1 + \frac + \delta Z \delta Y Each infinitesimal section will multiply the voltage drop by the same factor. After \scriptstyle n sections the voltage ratio will be, :\frac = \left ( 1 + \frac + \delta Z \delta Y \right)^n At a distance \scriptstyle x along the line, the number of sections is \scriptstyle x/\delta x so that, :\frac = \left ( 1 + \frac + \delta Z \delta Y \right)^ In the limit as \scriptstyle \delta x \to 0, :\frac = \lim_ \frac = \lim_ \left ( 1 + \frac + \delta Z \delta Y \right)^ The second order term \scriptstyle \delta Z \delta Y will disappear in the limit, so we can write without loss of accuracy, :\frac = \lim_ \left ( 1 + \frac \right)^ and comparing with the mathematical identity, :e^x \equiv \lim_ (1+1/p)^ yields, :V_\mathrm i = V_x e^ From the definition of
propagation constant The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a cir ...
, :V_\mathrm i = V_x e^\,\! Hence, :\gamma = \frac = \sqrt


Special cases


Ideal transmission line

An ideal transmission line will have no loss, which implies that the resistive elements are zero. It also results in a purely real (resistive) characteristic impedance. The ideal line cannot be realised in practice, but it is a useful approximation in many circumstances. This is especially true, for instance, when short pieces of line are being used as circuit components such as stubs. A short line has very little loss and this can then be ignored and treated as an ideal line. The secondary constants in these circumstances are; :\gamma = i \omega \sqrt :\alpha = 0\, :\beta = \omega \sqrt :Z_0 = \sqrt \frac


Twisted pair

Typically,
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cable used for audio frequencies or low data rates has line constants dominated by ''R'' and ''C''. The dielectric loss is usually negligible at these frequencies and ''G'' is close to zero. It is also the case that, at a low enough frequency, \scriptstyle R \gg \omega L which means that ''L'' can also be ignored. In those circumstances the secondary constants become, :\gamma \approx \sqrt :\alpha \approx \sqrt \frac :\beta \approx \sqrt \frac :Z_0 \approx \sqrt \frac = \sqrt \frac - i \sqrt \frac The attenuation of this cable type increases with frequency, causing distortion of waveforms. Not so obviously, the variation of \scriptstyle \beta with frequency also causes a distortion of a type called
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. To avoid dispersion the requirement is that \scriptstyle \beta is directly proportional to \scriptstyle \omega. However, it is actually proportional to \scriptstyle \sqrt \omega and dispersion results. \scriptstyle Z_0 also varies with frequency and is also partly reactive; both these features will be the cause of reflections from a resistive line termination. This is another undesirable effect. The
nominal impedance Nominal impedance in electrical engineering and audio engineering refers to the approximate designed impedance of an electrical circuit or device. The term is applied in a number of different fields, most often being encountered in respect of: ...
quoted for this type of cable is, in this case, very nominal, being valid at only one spot frequency, usually quoted at 800 Hz or 1 kHz.


Co-axial cable

Cable operated at a high enough frequency (
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radio frequency or high data rates) will meet the conditions \scriptstyle R \ll \omega L and \scriptstyle G \ll \omega C. This must eventually be the case as the frequency is increased for any cable. Under those conditions ''R'' and ''G'' can both be ignored (except for the purpose of calculating the cable loss) and the secondary constants become; :\gamma \approx i \omega \sqrt :\alpha \approx \frac = \tfrac\left(Z_0 G + \frac\right) \approx \frac :\beta \approx \omega \sqrt :Z_0 \approx \sqrt \frac


Loaded line

Loaded lines are lines designed with deliberately increased inductance. This is done by adding iron or some other magnetic metal to the cable or adding coils. The purpose is to ensure that the line meets the
Heaviside condition The Heaviside condition, named for Oliver Heaviside (1850–1925), is the condition an electrical transmission line must meet in order for there to be no distortion of a transmitted signal. Also known as the distortionless condition, it can be used ...
, which eliminates distortion caused by frequency-dependent attenuation and dispersion, and ensures that \scriptstyle Z_0 is constant and resistive. The secondary constants are here related to the primary constants by; :\gamma = \sqrt + i \omega \sqrt :\alpha = \sqrt :\beta = \omega \sqrt :Z_0 = \sqrt \frac = \sqrt \frac


Velocity

The velocity of propagation is given by, : v = \lambda f . Since, : \omega = 2 \pi f and \beta = \frac then, : v = \frac . In cases where can be taken as, : \beta = \omega \sqrt the velocity of propagation is given by, : v = . The lower the capacitance the higher the velocity. With an air dielectric cable, which is approximated to with low-loss cable, the velocity of propagation is very close to ''c'', the speed of light ''in vacuo''.Connor, pp. 10, 19-20.


Notes


References

*F.R. Connor, ''Wave Transmission'', Edward Arnold Ltd., 1972 . *John Bird, ''Electrical Circuit Theory and Technology'', Newnes, 2007 . *Ian Hickman, ''Analog Electronics'', Newnes, 1999 . *Fred Porges, ''The Design of Electrical Services for Buildings'', Taylor & Francis, 1989 {{ISBN, 0-419-14590-7. Electronic design Cables Telecommunications engineering Distributed element circuits