Information about Transmission Line
A transmission line is the material medium or structure that forms all or part of a path from one place to another for directing the transmission of energy, such as electromagnetic waves or acoustic waves, as well as electric power transmission.
Components of transmission lines include wires, coaxial cables, dielectric slabs, optical fibers, electric power lines, and waveguides.
A common rule of thumb (justified in the input impedance section) is that the cable or wire should be treated as a transmission line if the length is greater than 1/100 of the wavelength. At this length the phase delay and the interference of any reflections on the line become important and can lead to unpredictable behavior in systems which have not been carefully designed using transmission line theory.
For the purposes of analysis, an electrical transmission line can be modelled as a two-port network (also called a quadrupole network), as follows:
In the simplest case, the network is assumed to be linear (i.e. the complex voltage across either port is proportional to the complex current flowing into it when there are no reflections), and the two ports are assumed to be interchangeable. If the transmission line is uniform along its length, then its behaviour is largely described by a single parameter called the characteristic impedance, symbol Z0. This is the ratio of the complex voltage of a given wave to the complex current of the same wave at any point on the line. Typical values of Z0 are 50 or 75 ohms for a coaxial cable, about 100 ohms for a twisted pair of wires, and about 300 ohms for a common type of untwisted pair used in radio transmission.
When sending power down a transmission line, it is usually desirable that as much power as possible will be absorbed by the load and as little as possible will be reflected back to the source. This can be ensured by making the source and load impedances equal to Z0, in which case the transmission line is said to be matched.
Some of the power that is fed into a transmission line is lost because of its resistance. This effect is called ohmic or resistive loss (see ohmic heating). At high frequencies, another effect called dielectric loss becomes significant, adding to the losses caused by resistance. Dielectric loss is caused when the insulating material inside the transmission line absorbs energy from the alternating electric field and converts it to heat (see dielectric heating).
The total loss of power in a transmission line is often specified in decibels per metre, and usually depends on the frequency of the signal. The manufacturer often supplies a chart showing the loss in dB/m at a range of frequencies. A loss of 3 dB corresponds approximately to a halving of the power.
High-frequency transmission lines can be defined as transmission lines that are designed to carry electromagnetic waves whose wavelengths are shorter than or comparable to the length of the line. Under these conditions, the approximations useful for calculations at lower frequencies are no longer accurate. This often occurs with radio, microwave and optical signals, and with the signals found in high-speed digital circuits.
The Telegrapher's Equations (or just Telegraph Equations) are a pair of linear differential equations which describe the voltage and current on an electrical transmission line with distance and time. They were developed by Oliver Heaviside who created the transmission line model, and are based on Maxwell's Equations.
The transmission line model represents the transmission line as an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:
, 
, 
, and 
may also be functions of frequency. An alternative notation is to use 
, 
, 
and 
to emphasize that the values are derivatives with respect to length.
The line voltage
and the current 
can be expressed in the frequency domain as
When the elements and are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the and elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:
These are wave equations which have plane waves with equal propagation speed in the forward and reverse directions as solutions. The physical significance of this is that electromagnetic waves propagate down transmission lines and in general, there is a reflected component that interferes with the original signal. These equations are fundamental to transmission line theory.
If and are not neglected, the Telegrapher's equations become:
where
and the characteristic impedance is:
The solutions for and are:
The constants
and 
must be determined from boundary conditions. For a voltage pulse 
, starting at 
and moving in the positive 
-direction, then the transmitted pulse 
at position can be obtained by computing the Fourier Transform, 
, of , attenuating each frequency component by 
, advancing its phase by 
, and taking the inverse Fourier Transform. The real and imaginary parts of 
can be computed as
where atan2 is the two-parameter arctangent, and
For small losses and high frequencies, to first order in
and 
one obtains
Noting that an advance in phase by
is equivalent to a time delay by 
, 
can be simply computed as
of a transmission line is the ratio of the amplitude of a single voltage wave to its current wave. Since most transmission lines also have a reflected wave, the characteristic impedance is generally not the impedance that is measured on the line.
For a lossless transmission line, it can be shown that the impedance measured at a given position
from the load impedance 
is
where
is the wavenumber.
For the special case where
where n is an integer (meaning that the length of the line is a very close to a multiple of half a wavelength), the expression reduces to the load impedance so that 
for all . This includes the case when 
, meaning that the length of the transmission line is less than about 1/100 of the wavelength. The physical significance of this is that the transmission line can be ignored (i.e. treated as a wire) in either case.
Another special case is when the load impedance is equal to the characteristic impedance of the line (i.e. the line is matched), in which case the impedance reduces to the characteristic impedance of the line so that
In calculating
, the wavelength is generally different inside the transmission line to what it would be in free-space and the velocity constant of the material the transmission line is made of needs to be taken into account when doing such a calculation.
Coaxial lines confine the electromagnetic wave to the area inside the cable, between the center conductor and the shield. The transmission of energy in the line occurs totally through the dielectric inside the cable between the conductors. Coaxial lines can therefore be bent and twisted (subject to limits) without negative effects, and they can be strapped to conductive supports without inducing unwanted currents in them.
In radio-frequency applications up to a few gigahertz, the wave propagates in the transverse electric and magnetic mode (TEM), which means that the electric and magnetic fields are both perpendicular to the direction of propagation. However, above a certain frequency called the cutoff frequency, the cable behaves as a waveguide, and propagation switches to either a transverse electric (TE) or a transverse magnetic (TM) mode or a mixture of modes. This effect enables coaxial cables to be used at microwave frequencies, although they are not as efficient as the more expensive, purpose-built waveguides.
The most common use for coaxial cables is for television and other signals with bandwidth of multiple Megahertz. In the middle 20th Century they carried long distance telephone connections.
A microstrip circuit uses a thin flat conductor which is parallel to a ground plane. Microstrip can be made by having a strip of copper on one side of a printed circuit board (PCB) or ceramic substrate while the other side is a continuous ground plane. The width of the strip, the thickness of the insulating layer (PCB or ceramic) and the dielectric constant of the insulating layer determine the characteristic impedance.
A stripline circuit uses a flat strip of metal which is sandwiched between two parallel ground planes. The insulating material of the substrate forms a dielectric. The width of the strip, the thickness of the substrate and the relative permittivity of the substrate determine the characteristic impedance of the strip which is a transmission line.
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History
Mathematical analysis of the behaviour of electrical transmission lines grew out of the work of James Clerk Maxwell, Lord Kelvin and Oliver Heaviside. In 1855 Lord Kelvin formulated a diffusion model of the current in a submarine cable. The model correctly predicted the poor performance of the 1858 trans-Atlantic submarine telegraph cable. In 1885 Heaviside published the first papers that described his analysis of propagation in cables and the modern form of the telegrapher's equations. [1]Transmission line vs wire
In many electric circuits, the length of the wires connecting the components can for the most part be ignored. That is, the voltage on the wire at a given time can be assumed to be the same at all points. However, when the voltage changes in a time interval comparable to the time it takes for the signal to travel down the wire, the length becomes important and the wire must be treated as a transmission line. Stated another way, the length of the wire is important when the signal includes frequency components with corresponding wavelengths comparable to the length of the wire.A common rule of thumb (justified in the input impedance section) is that the cable or wire should be treated as a transmission line if the length is greater than 1/100 of the wavelength. At this length the phase delay and the interference of any reflections on the line become important and can lead to unpredictable behavior in systems which have not been carefully designed using transmission line theory.
The four terminal model
){kind=spacer}
Variations on the schematic electronic symbol for a transmission line.
For the purposes of analysis, an electrical transmission line can be modelled as a two-port network (also called a quadrupole network), as follows:
){kind=spacer}
In the simplest case, the network is assumed to be linear (i.e. the complex voltage across either port is proportional to the complex current flowing into it when there are no reflections), and the two ports are assumed to be interchangeable. If the transmission line is uniform along its length, then its behaviour is largely described by a single parameter called the characteristic impedance, symbol Z0. This is the ratio of the complex voltage of a given wave to the complex current of the same wave at any point on the line. Typical values of Z0 are 50 or 75 ohms for a coaxial cable, about 100 ohms for a twisted pair of wires, and about 300 ohms for a common type of untwisted pair used in radio transmission.
When sending power down a transmission line, it is usually desirable that as much power as possible will be absorbed by the load and as little as possible will be reflected back to the source. This can be ensured by making the source and load impedances equal to Z0, in which case the transmission line is said to be matched.
Some of the power that is fed into a transmission line is lost because of its resistance. This effect is called ohmic or resistive loss (see ohmic heating). At high frequencies, another effect called dielectric loss becomes significant, adding to the losses caused by resistance. Dielectric loss is caused when the insulating material inside the transmission line absorbs energy from the alternating electric field and converts it to heat (see dielectric heating).
The total loss of power in a transmission line is often specified in decibels per metre, and usually depends on the frequency of the signal. The manufacturer often supplies a chart showing the loss in dB/m at a range of frequencies. A loss of 3 dB corresponds approximately to a halving of the power.
High-frequency transmission lines can be defined as transmission lines that are designed to carry electromagnetic waves whose wavelengths are shorter than or comparable to the length of the line. Under these conditions, the approximations useful for calculations at lower frequencies are no longer accurate. This often occurs with radio, microwave and optical signals, and with the signals found in high-speed digital circuits.
Telegrapher's equations
The Telegrapher's Equations (or just Telegraph Equations) are a pair of linear differential equations which describe the voltage and current on an electrical transmission line with distance and time. They were developed by Oliver Heaviside who created the transmission line model, and are based on Maxwell's Equations.
The transmission line model represents the transmission line as an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:
- The distributed resistance of the conductors is represented by a series resistor (expressed in ohms per unit length).
- The distributed inductance (due to the magnetic field around the wires, self-inductance, etc.) is represented by a series inductor (henries per unit length).
- The capacitance between the two conductors is represented by a shunt capacitor C (farads per unit length).
- The conductance of the dielectric material separating the two conductors is represented by a conductance G shunted between the signal wire and the return wire (siemens per unit length).
, , , and may also be functions of frequency. An alternative notation is to use , , and to emphasize that the values are derivatives with respect to length.
The line voltage
and the current can be expressed in the frequency domain as
When the elements
and are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the and elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:
These are wave equations which have plane waves with equal propagation speed in the forward and reverse directions as solutions. The physical significance of this is that electromagnetic waves propagate down transmission lines and in general, there is a reflected component that interferes with the original signal. These equations are fundamental to transmission line theory.
If
and are not neglected, the Telegrapher's equations become:
where
and the characteristic impedance is:
The solutions for
and are:
The constants
and must be determined from boundary conditions. For a voltage pulse , starting at and moving in the positive -direction, then the transmitted pulse at position can be obtained by computing the Fourier Transform, , of , attenuating each frequency component by , advancing its phase by , and taking the inverse Fourier Transform. The real and imaginary parts of can be computed as
where atan2 is the two-parameter arctangent, and
For small losses and high frequencies, to first order in
and one obtains
Noting that an advance in phase by
is equivalent to a time delay by , can be simply computed as
Input impedance of a transmission line
The characteristic impedance of a transmission line is the ratio of the amplitude of a single voltage wave to its current wave. Since most transmission lines also have a reflected wave, the characteristic impedance is generally not the impedance that is measured on the line.
For a lossless transmission line, it can be shown that the impedance measured at a given position
from the load impedance is
where
is the wavenumber.
For the special case where
where n is an integer (meaning that the length of the line is a very close to a multiple of half a wavelength), the expression reduces to the load impedance so that for all . This includes the case when , meaning that the length of the transmission line is less than about 1/100 of the wavelength. The physical significance of this is that the transmission line can be ignored (i.e. treated as a wire) in either case.
Another special case is when the load impedance is equal to the characteristic impedance of the line (i.e. the line is matched), in which case the impedance reduces to the characteristic impedance of the line so that
In calculating
, the wavelength is generally different inside the transmission line to what it would be in free-space and the velocity constant of the material the transmission line is made of needs to be taken into account when doing such a calculation.
Practical types of electrical transmission line
Coaxial cable
Coaxial lines confine the electromagnetic wave to the area inside the cable, between the center conductor and the shield. The transmission of energy in the line occurs totally through the dielectric inside the cable between the conductors. Coaxial lines can therefore be bent and twisted (subject to limits) without negative effects, and they can be strapped to conductive supports without inducing unwanted currents in them.
In radio-frequency applications up to a few gigahertz, the wave propagates in the transverse electric and magnetic mode (TEM), which means that the electric and magnetic fields are both perpendicular to the direction of propagation. However, above a certain frequency called the cutoff frequency, the cable behaves as a waveguide, and propagation switches to either a transverse electric (TE) or a transverse magnetic (TM) mode or a mixture of modes. This effect enables coaxial cables to be used at microwave frequencies, although they are not as efficient as the more expensive, purpose-built waveguides.
The most common use for coaxial cables is for television and other signals with bandwidth of multiple Megahertz. In the middle 20th Century they carried long distance telephone connections.
Microstrip
A microstrip circuit uses a thin flat conductor which is parallel to a ground plane. Microstrip can be made by having a strip of copper on one side of a printed circuit board (PCB) or ceramic substrate while the other side is a continuous ground plane. The width of the strip, the thickness of the insulating layer (PCB or ceramic) and the dielectric constant of the insulating layer determine the characteristic impedance.
Stripline
- Main article : Stripline
A stripline circuit uses a flat strip of metal which is sandwiched between two parallel ground planes. The insulating material of the substrate forms a dielectric. The width of the strip, the thickness of the substrate and the relative permittivity of the substrate determine the characteristic impedance of the strip which is a transmission line.
Balanced lines
Lecher lines
General applications of transmission lines
Transferring signals from one point to another
Electrical transmission lines are very widely used to transmit high frequency signals over long or short distances with minimum power loss. One familiar example is the down lead from a TV or radio aerial to the receiver.Pulse generation
Transmission lines are also used as pulse generators. By charging the transmission line and then discharging it into a resistive load, a rectangular pulse equal in length to twice the electrical length of the line can be obtained, although with half the voltage. A Blumlein transmission line is a related pulse forming device that overcomes this limitation. These are sometimes used as the pulsed energy sources for radar transmitters and other devices.Stub filters
If a short-circuited or open-circuited transmission line is wired in parallel with a line used to transfer signals from point A to point B, then it will function as a filter. The method for making stubs is similar to the method for using Lecher lines for crude frequency measurement, but it is 'working backwards'. One method recommended in the RSGB's radiocommunication handbook is to take an open-circuited length of transmission line wired in parallel with the feeder delivering signals from an aerial. By cutting the free end of the transmission line, a minimum in the strength of the signal observed at a receiver can be found. At this stage the stub filter will reject this frequency and the odd harmonics, but if the free end of the stub is shorted then the stub will become a filter rejecting the even harmonics.Acoustic transmission lines
See also
- Heaviside condition
- Smith chart a graphical method to solve transmission line equations
- Transverse electromagnetic wave
- Longitudinal electromagnetic wave
- Electric power transmission
- Radio Frequency Power Transmission
- Standing wave
References
Part of this article was derived from Federal Standard 1037C.1. ^ Ernst Weber and Frederik Nebeker, The Evolution of Electrical Engineering, IEEE Press, Piscataway, New Jersey USA, 1994 ISBN 0-7803-1066-7
- Steinmetz, Charles Proteus, "The Natural Period of a Transmission Line and the Frequency of lightning Discharge Therefrom". The Electrical world. August 27 1898. Pg. 203 - 205.
- Electromagnetism 2nd ed., Grant, I.S., and Phillips, W.R., pub John Wiley, ISBN 0-471-92712-0
- Fundamentals Of Applied Electromagnetics 2004 media edition., Ulaby, F.T., pub Prentice Hall, ISBN 0-13-185089-X
- Radiocommunication handbook, page 20, chaper 17, RSGB, ISBN 0-900612-58-4
- Naredo, J.L., A.C. Soudack, and J.R. Marti, Simulation of transients on transmission lines with corona via the method of characteristics. Generation, Transmission and Distribution, IEE Proceedings. Vol. 142.1, Inst. de Investigaciones Electr., Morelos, Jan 1995. ISSN 1350-2360
External articles and further reading
- Annual Dinner of the Institute at the Waldorf-Astoria. Transactions of the American Institute of Electrical Engineers, New York, January 13, 1902. (Honoring of Guglielmo Marconi, January 13 1902)
- Avant! software, Using Transmission Line Equations and Parameters. Star-Hspice Manual, June 2001.
- Cornille, P, On the propagation of inhomogeneous waves. J. Phys. D: Appl. Phys. 23, February 14 1990. (Concept of inhomogeneous waves propagation — Show the importance of the telegrapher's equation with Heaviside's condition.)
- Farlow, S.J., Partial differential equations for scientists and engineers. J. Wiley and Sons, 1982, p. 126. ISBN 0-471-08639-8.
- Han, Hsiu C., Transmission-Line Equations. EE 313 Electromagnetic Fields and Waves.
- Kupershmidt, Boris A., Remarks on random evolutions in Hamiltonian representation. Math-ph/9810020. J. Nonlinear Math. Phys. 5 (1998), no. 4, 383-395.
- Pupin, M., U.S. Patent 1,541,845, Electrical wave transmission.
- Transmission line matching. EIE403: High Frequency Circuit Design. Department of Electronic and Information Engineering, Hong Kong Polytechnic University. (PDF format)
- Wilson, B. (2005, October 19). Telegrapher's Equations. Connexions.
- John Greaton Wöhlbier, ""Fundamental Equation" and "Transforming the Telegrapher's Equations". Modeling and Analysis of a Traveling Wave Under Multitone Excitation.
- Transmission Line Pulse
A transmission medium (plural transmission media) is a material substance (solid, liquid or gas) which can propagate energy waves. For example, the transmission medium for sound received by the ears is usually air, but solids and liquids may also act as transmission media
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A path or pathway is a line representing the course or route of actual, potential or abstract movement (change of location over time).
Path, PATH or pathway may refer to:
In natural and built environments:
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Path, PATH or pathway may refer to:
In natural and built environments:
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In telecommunications, transmission is the forwarding of signal traffic over distances that are too great to be simply connected by a twisted pair wires. Techniques available now may be microwave link, satellite link, coaxial cable or fibre optic cable.
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Electromagnetic (EM) radiation is a self-propagating wave in space with electric and magnetic components. These components oscillate at right angles to each other and to the direction of propagation, and are in phase with each other.
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wave is a mode of energy transfer from one place to another, often with little or no permanent displacement of the particles of the medium (i.e. little or no associated mass transport); instead there are oscillations around almost fixed positions.
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Electric power transmission, a process in the delivery of electricity to consumers, is the bulk transfer of electrical power. Typically, power transmission is between the power plant and a substation near a populated area.
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A wire is a single, usually cylindrical, elongated strand of drawn metal. Wires are used to bear mechanical loads and to carry electricity and telecommunications signals . Standard sizes are determined by various wire gauges.
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Coaxial cable is an electrical cable consisting of a round conducting wire, surrounded by an insulating spacer, surrounded by a cylindrical conducting sheath, usually surrounded by a final insulating layer (jacket).
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A dielectric is a physical model commonly used to describe how an electric field behaves inside a material. It is characterised by how an electric field interacts with an atom. It is possible to approach dielectrics from either a classical interpretation or a quantum one.
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An optical fiber (or fibre) is a glass or plastic fiber designed to guide light along its length. Fiber optics is the overlap of applied science and engineering concerned with such optical fibers.
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A waveguide is a structure which guides waves, such as electromagnetic waves, light, or sound waves. There are different types of waveguide for each type of wave.
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Electromagnetic waveguides
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James Clerk Maxwell
James Clerk Maxwell
Born May 13 1831
Edinburgh, Scotland
Died November 5 1879 (aged 48)
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James Clerk Maxwell
Born May 13 1831
Edinburgh, Scotland
Died November 5 1879 (aged 48)
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William Thomson, 1st Baron Kelvin, OM, GCVO, PC, PRS, FRSE, (26 June 1824 – 17 December 1907) was a British mathematical physicist, engineer, and outstanding leader in the physical sciences of the 19th century.
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Oliver Heaviside
Portrait of Oliver Heaviside (1850-1925) by Frances Hodge
Born May 18 1850
Camden Town, London, England
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Portrait of Oliver Heaviside (1850-1925) by Frances Hodge
Born May 18 1850
Camden Town, London, England
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Submarine cables may be divided into two types:
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- Submarine communications cables
- Submarine power cables
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Telegraphy (from the Greek words (τηλη) = far and (γραφειν) = write) is the long-distance transmission of written messages without physical transport of letters, originally by changing something that could
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electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources, and switches.
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Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms.
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In physics, wavelength is the distance between repeating units of a propagating wave of a given frequency. It is commonly designated by the Greek letter lambda (λ). Examples of wave-like phenonomena are light, water waves, and sound waves.
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In mathematics, a complex number is a number of the form
where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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characteristic impedance or surge impedance of a uniform transmission line, usually written , is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections.
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ohm (symbol: Ω) is the SI unit of electrical impedance or, in the direct current case, electrical resistance, named after Georg Ohm.
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Definition
An ohm is the electrical resistance offered by a current-carrying element that produces a voltage drop of one volt when a..... Click the link for more information.
Coaxial cable is an electrical cable consisting of a round conducting wire, surrounded by an insulating spacer, surrounded by a cylindrical conducting sheath, usually surrounded by a final insulating layer (jacket).
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Impedance matching is the practice of attempting to make the output impedance of a source equal to the input impedance of the load to which it is ultimately connected, usually in order to maximize the power transfer and minimize reflections from the load.
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Joule heating is the process by which the passage of an electric current through a conductor releases heat. It was first studied by James Prescott Joule in 1841. Joule immersed a length of wire in a fixed mass of water and measured the temperature rise due to a known current
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Dielectric heating (also known as electronic heating, RF heating, high-frequency heating) is the phenomenon in which radiowave or microwave electromagnetic radiation heats a dielectric material, especially as caused by dipole rotation.
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The decibel (dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power) relative to a specified or implied reference level.
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1 metre =
SI units
1000 mm 0 cm
US customary / Imperial units
0 ft 0 in
The metre or meter[1](symbol: m) is the fundamental unit of length in the International System of Units (SI).SI units
1000 mm 0 cm
US customary / Imperial units
0 ft 0 in
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In physics, wavelength is the distance between repeating units of a propagating wave of a given frequency. It is commonly designated by the Greek letter lambda (λ). Examples of wave-like phenonomena are light, water waves, and sound waves.
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