Primary line constants
Encyclopedia
The primary line constants are parameters that describe the characteristics of copper
(or other conductive material) transmission line
s in terms of the physical electrical properties of the line. The primary line constants are only relevant to copper 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.
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
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 as there is some variation with frequency. In particular, R is heavily influenced by the skin effect
. Furthermore, while G has virtually no effect at audio frequency
, it can cause noticeable losses at high frequency with many of the dielectric
materials used in cables due to a high loss tangent
. 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.
The line constants cannot be simply represented as lumped elements
in a circuit; they must be described as distributed elements
. 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 mesh
es in the circuit is increased. To give a true representation of the circuit, the elements must be made infinitesimally
small so that each element is distributed along the line as shown in figure 1. The infinitesimal elements in an infinitesimal distance
are given by;
It is convenient for the purposes of analysis to roll up these elements into general series impedance
, Z, and shunt admittance
, Y elements such that;
and,
Analysis of this network (figure 2) will yield the secondary line constants: the propagation constant
,
, (whose real and imaginary parts
are the attenuation constant,
, and phase change constant, 
, respectively) and the characteristic impedance
,
, which also, in general, will have real, 
, and imaginary, 
, 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 all usually vary quite strongly with frequency, even if the frequency dependence of the primary constants is ignored. This is because the reactances in the circuit (
and 
) introduce a dependence on 
. It is possible to choose specific values of the primary constants that result in 
and 
being constant (the Heaviside condition
) but even in this case there is still
which is directly proportional to 
. As with the primary constants, the meaning is that the secondary constants do not vary with distance along the line, not that they are independent of frequency.
, 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 initial length of the line, the remainder of the line can be replaced by 
as its equivalent circuit. This is so because the remainder of the line is still infinitely long. Considering just the first section of the equivalent circuit of the line (this is an L-network consisting of one element each of 
and 
) the remainder can be replaced by 
. This results in the network shown in figure 3, which can be analysed for 
using the usual network analysis
theorems,
which re-arranges to,
Taking limits of both sides
and since the line was assumed to be homogenous lengthwise,
further down the line, that is, after one infinitesimal section, is given by a standard voltage divider calculation. The remainder of the line to the right, as in the characteristic impedance calculation, is replaced with 
,
Each infinitesimal section will multiply the voltage drop by the same factor. After
sections the voltage ratio will be,
At a distance
along the line, the number of sections is 
so that,
In the limit as
,
The second order term
will disappear in the limit, so we can write without loss of accuracy,
and comparing with the mathematical identity,
yields,
From the definition of propagation constant
,
Hence,
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,
which means that L can also be ignored. In those circumstances the secondary constants become,
The attenuation of this cable type increases with frequency which causes distortion of waveforms. Not so obviously, the variation of
with frequency also causes a distortion of a type called dispersion
. To avoid dispersion the requirement is that
is directly proportional to 
. However, it is actually proportional to 
and dispersion results. 
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
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.
and 
. 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;
by doing which eliminates attenuation and dispersion distortions and ensures that
is constant and resistive. The secondary constants are here related to the primary constants by;
Copper
Copper is a chemical element with the symbol Cu and atomic number 29. It is a ductile metal with very high thermal and electrical conductivity. Pure copper is soft and malleable; an exposed surface has a reddish-orange tarnish...
(or other conductive material) transmission line
Transmission line
In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...
s in terms of the physical electrical properties of the line. The primary line constants are only relevant to copper 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
Waveguide
A waveguide is a structure which guides waves, such as electromagnetic waves or sound waves. There are different types of waveguides for each type of wave...
to a copper line, whereas the primary constants have no meaning for a waveguide.
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;| Name | Symbol | Units | Unit symbol |
|---|---|---|---|
| loop resistance Electrical resistance The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical... |
R | ohm Ohm The ohm is the SI unit of electrical resistance, named after German physicist Georg Simon Ohm.- Definition :The ohm is defined as a resistance between two points of a conductor when a constant potential difference of 1 volt, applied to these points, produces in the conductor a current of 1 ampere,... s per metre Metre The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology... |
Ω/m |
| loop inductance Inductance In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit... |
L | henries per metre | H/m |
| insulator capacitance Capacitance In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor... |
C | farad Farad The farad is the SI unit of capacitance. The unit is named after the English physicist Michael Faraday.- Definition :A farad is the charge in coulombs which a capacitor will accept for the potential across it to change 1 volt. A coulomb is 1 ampere second... s per metre |
F/m |
| insulator conductance | G | siemens per metre | S/m |
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
Dielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...
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 as there is some variation with frequency. In particular, R is heavily influenced by the skin effect
Skin effect
Skin effect is the tendency of an alternating electric current to distribute itself within a conductor with the current density being largest near the surface of the conductor, decreasing at greater depths. In other words, the electric current flows mainly at the "skin" of the conductor, at an...
. Furthermore, while G has virtually no effect at audio frequency
Audio frequency
An audio frequency or audible frequency is characterized as a periodic vibration whose frequency is audible to the average human...
, it can cause noticeable losses at high frequency with many of the dielectric
Dielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...
materials used in cables due to a high loss tangent
Loss tangent
The loss tangent is a parameter of a dielectric material that quantifies its inherent dissipation of electromagnetic energy. The term refers to the tangent of the angle in a complex plane between the resistive component of an electromagnetic field and its reactive component.-Electromagnetic...
. 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
| Designation | Cable form | Application | R | L | G | C | Z0 |
|---|---|---|---|---|---|---|---|
| Ω/km | μH/km | nS/km | nF/km | Ω | |||
| CAT5 | Twisted pair Twisted pair Twisted pair cabling is a type of wiring in which two conductors are twisted together for the purposes of canceling out electromagnetic interference from external sources; for instance, electromagnetic radiation from unshielded twisted pair cables, and crosstalk between neighboring pairs... |
Data transmission Data transmission Data transmission, digital transmission, or digital communications is the physical transfer of data over a point-to-point or point-to-multipoint communication channel. Examples of such channels are copper wires, optical fibres, wireless communication channels, and storage media... |
176 | 490 | <2 | 49 | 100 |
| CAT5e | Twisted pair | Data transmission | 176 | <2 | 100 | ||
| CW1308 | Twisted pair | Telephony Telephony In telecommunications, telephony encompasses the general use of equipment to provide communication over distances, specifically by connecting telephones to each other.... |
98 | <20 | |||
| RG59 | Coaxial Coaxial In geometry, coaxial means that two or more forms share a common axis; it is the three-dimensional linear analogue of concentric.Coaxial cable, as a common example, has a wire conductor in the centre a circumferential outer conductor and an insulating medium called the dielectric separating... |
Video | 36 | 430 | 69 | 75 | |
| RG59 | Coaxial (foam dielectric) |
Video Video Video is the technology of electronically capturing, recording, processing, storing, transmitting, and reconstructing a sequence of still images representing scenes in motion.- History :... |
20.4 | 303 | 54 | 75 | |
| RG58 | Coaxial | Radio frequency Radio frequency Radio frequency is a rate of oscillation in the range of about 3 kHz to 300 GHz, which corresponds to the frequency of radio waves, and the alternating currents which carry radio signals... |
48 | 253 | <0.01 | 101 | 50 |
| Low loss | Coaxial (Foam dielectric) |
Radio frequency transmitter feed |
2.86 | 188 | 75 | 50 | |
| DIN VDE 0816 | Star quad | Telephony (trunk lines) |
31.8 | <0.1 | 35 |
Circuit representation
Lumped element model
The lumped element model simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions...
in a circuit; they must be described as distributed elements
Distributed element model
In electrical engineering, the distributed element model or transmission line model of electrical circuits assumes that the attributes of the circuit are distributed continuously throughout the material of the circuit...
. 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 mesh
Mesh analysis
Mesh analysis is a method that is used to solve planar circuits for the currents at any place in the circuit. Planar circuits are circuits that can be drawn on a plane surface with no wires crossing each other. A more general technique, called loop analysis can be applied to any circuit, planar...
es in the circuit is increased. To give a true representation of the circuit, the elements must be made infinitesimally
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
small so that each element is distributed along the line as shown in figure 1. The infinitesimal elements in an infinitesimal distance
are given by; It is convenient for the purposes of analysis to roll up these elements into general series impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
, Z, and shunt admittance
Admittance
In electrical engineering, the admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the inverse of the impedance . The SI unit of admittance is the siemens...
, Y elements such that;
and,Analysis of this network (figure 2) will yield the secondary line constants: the propagation constant
Propagation constant
The propagation constant of an electromagnetic wave is a measure of the change undergone by the amplitude of the wave as it propagates in a given direction. The quantity being measured can be the voltage or current in a circuit or a field vector such as electric field strength or flux density...
,
, (whose real and imaginary partsComplex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
are the attenuation constant,
, and phase change constant, , respectively) and the characteristic impedanceCharacteristic impedance
The characteristic impedance or surge impedance of a uniform transmission line, usually written Z_0, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. The SI unit of characteristic impedance is the ohm...
,
, which also, in general, will have real, , and imaginary, , 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 all usually vary quite strongly with frequency, even if the frequency dependence of the primary constants is ignored. This is because the reactances in the circuit ( and ) introduce a dependence on . It is possible to choose specific values of the primary constants that result in and being constant (the Heaviside conditionHeaviside condition
The Heaviside condition, due to Oliver Heaviside , is the condition an electrical transmission line must meet in order for there to be no distortion of a transmitted signal...
) but even in this case there is still
which is directly proportional to . As with the primary constants, the meaning 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,, 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 initial length of the line, the remainder of the line can be replaced by as its equivalent circuit. This is so because the remainder of the line is still infinitely long. Considering just the first section of the equivalent circuit of the line (this is an L-network consisting of one element each of and ) the remainder can be replaced by . This results in the network shown in figure 3, which can be analysed for using the usual network analysisNetwork analysis (electrical circuits)
A network, in the context of electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, every component in the network. There are a number of different techniques for achieving this...
theorems,
which re-arranges to,
Taking limits of both sides
and since the line was assumed to be homogenous lengthwise,
Propagation constant
The ratio of the line input voltage to the voltage a distance further down the line, that is, after one infinitesimal section, is given by a standard voltage divider calculation. The remainder of the line to the right, as in the characteristic impedance calculation, is replaced with ,Each infinitesimal section will multiply the voltage drop by the same factor. After
sections the voltage ratio will be,At a distance
along the line, the number of sections is so that,In the limit as
,The second order term
will disappear in the limit, so we can write without loss of accuracy,and comparing with the mathematical identity,
yields,
From the definition of propagation constant
Propagation constant
The propagation constant of an electromagnetic wave is a measure of the change undergone by the amplitude of the wave as it propagates in a given direction. The quantity being measured can be the voltage or current in a circuit or a field vector such as electric field strength or flux density...
,
Hence,
Special cases
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;Twisted pair
Typically, twisted pairTwisted pair
Twisted pair cabling is a type of wiring in which two conductors are twisted together for the purposes of canceling out electromagnetic interference from external sources; for instance, electromagnetic radiation from unshielded twisted pair cables, and crosstalk between neighboring pairs...
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,
which means that L can also be ignored. In those circumstances the secondary constants become,The attenuation of this cable type increases with frequency which causes distortion of waveforms. Not so obviously, the variation of
with frequency also causes a distortion of a type called dispersionDispersion relation
In physics and electrical engineering, dispersion most often refers to frequency-dependent effects in wave propagation. Note, however, that there are several other uses of the word "dispersion" in the physical sciences....
. To avoid dispersion the requirement is that
is directly proportional to . However, it is actually proportional to and dispersion results. 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 impedanceNominal impedance
Nominal impedance in electrical engineering and audio engineering refers to the approximate designed impedance of an electrical circuit or device...
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 (VHF radio frequency or high data rates) will meet the conditions and . 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;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 conditionHeaviside condition
The Heaviside condition, due to Oliver Heaviside , is the condition an electrical transmission line must meet in order for there to be no distortion of a transmitted signal...
by doing which eliminates attenuation and dispersion distortions and ensures that
is constant and resistive. The secondary constants are here related to the primary constants by;