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| {{About|the radio-frequency transmission line|the power transmission line|electric power transmission}}
| | This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users. |
| [[File:Transmission line animation.gif|right|thumb|300px|Schematic showing how a wave flows down a lossless transmission line. Red color indicates high [[voltage]], and blue indicates low voltage. Black dots represent [[electron]]s. The line is terminated at an [[impedance matching|impedance-matched]] load resistor (box on right), which fully absorbs the wave.]]
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| [[Image:F-Stecker und Kabel.jpg|thumb|One of the most common types of transmission line, [[coaxial cable]]. ]]
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| In [[Telecommunications engineering|communications]] and [[electronic engineering]], a '''transmission line''' is a specialized cable or other structure designed to carry [[alternating current]] of [[radio frequency]], that is, currents with a [[frequency]] high enough that their [[wave]] nature must be taken into account. Transmission lines are used for purposes such as connecting [[Transmitter|radio transmitters]] and [[Radio receiver|receivers]] with their [[antenna (radio)|antennas]], distributing [[cable television]] signals, [[trunking|trunklines]] routing calls between telephone switching centers, computer network connections, and high speed computer [[data bus]]es.
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| This article covers two-conductor transmission line such as parallel line ([[ladder line]]), [[coaxial cable]], [[stripline]], and [[microstrip]]. Some sources also refer to [[waveguide]], [[dielectric waveguide]], and even [[optical fiber]] as transmission line, however these lines require different analytical techniques and so are not covered by this article; see [[Waveguide (electromagnetism)]].
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| == Overview ==
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| Ordinary electrical cables suffice to carry low frequency [[alternating current]] (AC), such as [[mains power]], which reverses direction 100 to 120 times per second, and [[audio signal]]s. However, they cannot be used to carry currents in the [[radio frequency]] range or higher,<ref name="Jackman">{{cite book
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| | last = Jackman
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| | first = Shawn M.
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| | coauthors = Matt Swartz, Marcus Burton, Thomas W. Head
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| | title = CWDP Certified Wireless Design Professional Official Study Guide: Exam PW0-250
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| | publisher = John Wiley & Sons
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| | year = 2011
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| | location =
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| | pages = Ch. 7
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| | url = http://books.google.com/books?id=AQ8WJGshLBEC&pg=PT300&lpg=PT300&dq=%22what+is+a+transmission+line?%22+%22A+cable's+nature
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| | doi =
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| | id =
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| | isbn = 1118041615}}</ref> which reverse direction millions to billions of times per second, because the energy tends to radiate off the cable as [[radio wave]]s, causing power losses. Radio frequency currents also tend to reflect from discontinuities in the cable such as [[electrical connector|connectors]] and joints, and travel back down the cable toward the source.<ref name="Jackman" /><ref name="Oklobdzija">{{cite book
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| | last = Oklobdzija
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| | first = Vojin G.
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| | coauthors = Ram K. Krishnamurthy
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| | title = High-Performance Energy-Efficient Microprocessor Design
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| | publisher = Springer
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| | year = 2006
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| | location =
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| | pages = 297
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| | url = http://books.google.com/books?id=LmfHof1p3qUC&pg=PA297&dq=%22transmission+line%22+%22uniform
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| | doi =
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| | isbn = 0387340475}}</ref> These reflections act as bottlenecks, preventing the signal power from reaching the destination. Transmission lines use specialized construction, and [[impedance matching]], to carry electromagnetic signals with minimal reflections and power losses. The distinguishing feature of most transmission lines is that they have uniform cross sectional dimensions along their length, giving them a uniform ''[[Electrical impedance|impedance]]'', called the [[characteristic impedance]],<ref name="Oklobdzija" /><ref name="Guru">{{cite book
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| | last = Guru
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| | first = Bhag Singh
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| | coauthors = Hüseyin R. Hızıroğlu
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| | title = Electromagnetic Field Theory Fundamentals, 2nd Ed.
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| | publisher = Cambridge Univ. Press
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| | year = 2004
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| | location =
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| | pages = 422–423
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| | url = http://books.google.com/books?id=qzNdDtZUPXMC&pg=PA422&dq=%22transmission+line%22+uniform
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| | doi =
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| | id =
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| | isbn = 1139451928}}</ref><ref name="Schmitt">{{cite book
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| | last = Schmitt
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| | first = Ron Schmitt
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| | title = Electromagnetics Explained: A Handbook for Wireless/ RF, EMC, and High-Speed Electronics
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| | publisher = Newnes
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| | year = 2002
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| | location =
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| | pages = 153
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| | url = http://books.google.com/books?id=7gJ4RocvEskC&pg=PA153&dq=%22transmission+line%22+uniform
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| | doi =
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| | isbn = 0080505236}}</ref> to prevent reflections. Types of transmission line include parallel line ([[ladder line]], [[twisted pair]]), [[coaxial cable]], [[stripline]], and [[microstrip]].<ref name="Carr">{{cite book
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| | last = Carr
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| | first = Joseph J.
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| | title = Microwave & Wireless Communications Technology
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| | publisher = Newnes
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| | year = 1997
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| | location = USA
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| | pages = 46–47
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| | url = http://books.google.com/books?id=1j1E541LKVoC&pg=PA46&dq=%22parallel+line%22+%22coaxial+cable%22+stripline+waveguide
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| | doi =
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| | id =
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| | isbn = 0750697075}}</ref><ref name="Raisanen">{{cite book
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| | last = Raisanen
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| | first = Antti V.
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| | coauthors = Arto Lehto
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| | title = Radio Engineering for Wireless Communication and Sensor Applications
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| | publisher = Artech House
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| | year = 2003
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| | location =
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| | pages = 35–37
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| | url = http://books.google.com/books?id=m8Dgkvf84xoC&pg=PA35
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| | doi =
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| | isbn = 1580536697}}</ref> The higher the frequency of electromagnetic waves moving through a given cable or medium, the shorter the [[wavelength]] of the waves. Transmission lines become necessary when the length of the cable is longer than a significant fraction of the transmitted frequency's wavelength.
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| At [[microwave]] frequencies and above, power losses in transmission lines become excessive, and [[waveguide]]s are used instead,<ref name="Jackman" /> which function as "pipes" to confine and guide the electromagnetic waves.<ref name="Raisanen" /> Some sources define waveguides as a type of transmission line;<ref name="Raisanen" /> however, this article will not include them. At even higher frequencies, in the [[terahertz]], [[infrared]] and [[light]] range, waveguides in turn become lossy, and [[optics|optical]] methods, (such as lenses and mirrors), are used to guide electromagnetic waves.<ref name="Raisanen" />
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| | :<math forcemathmode="png">E=mc^2</math> |
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| The theory of [[sound wave]] propagation is very similar mathematically to that of electromagnetic waves, so techniques from transmission line theory are also used to build structures to conduct acoustic waves; and these are called [[acoustic transmission line]]s.
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| | :<math forcemathmode="source">E=mc^2</math> --> |
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| ==History== | | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
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| 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 communications cable|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]].<ref>Ernst Weber and Frederik Nebeker, ''The Evolution of Electrical Engineering'', IEEE Press, Piscataway, New Jersey USA, 1994 ISBN 0-7803-1066-7</ref>
| | ==Demos== |
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| ==Applicability== | | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| In many [[electric circuit]]s, 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 [[Harmonic analysis|frequency components]] with corresponding [[wavelength]]s comparable to or less than the length of the wire.
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| A common rule of thumb is that the cable or wire should be treated as a transmission line if the length is greater than 1/10 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.
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| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
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| ==The four terminal model== | | ==Test pages == |
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| [[Image:Transmission line symbols.svg|thumb|Variations on the [[electronic schematic|schematic]] [[electronic symbol]] for a transmission line.]] | | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
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| For the purposes of analysis, an electrical transmission line can be modelled as a [[two-port network]] (also called a quadrupole network), as follows:
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| [[Image:Transmission line 4 port.svg]]
| | ==Bug reporting== |
| | | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
| In the simplest case, the network is assumed to be linear (i.e. the [[complex number|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 Z<sub>0</sub>. 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 Z<sub>0</sub> are 50 or 75 [[Ohm (unit)|ohm]]s 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.
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| 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 load impedance equal to Z<sub>0</sub>, in which case the transmission line is said to be ''[[impedance matching|matched]]''.
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| [[File:TransmissionLineDefinitions.svg|thumb|310px|A transmission line is drawn as two black wires. At a distance ''x'' into the line, there is current ''I(x)'' traveling through each wire, and there is a voltage difference ''V(x)'' between the wires. If the current and voltage come from a single wave (with no reflection), then ''V''(''x'') / ''I''(''x'') = ''Z''<sub>0</sub>, where ''Z''<sub>0</sub> is the ''[[characteristic impedance]]'' of the line.]]
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| 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 transmission line is modeled with a resistance (R) and inductance (L) in series with a capacitance (C) and conductance (G) in parallel. The resistance and conductance contribute to the loss in a transmission line.
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| The total loss of power in a transmission line is often specified in [[decibels]] per [[metre]] (dB/m), 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.
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| High-frequency transmission lines can be defined as those designed to carry electromagnetic waves whose [[wavelength]]s 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 [[light|optical]] signals, [[metal mesh optical filters]], and with the signals found in high-speed [[digital circuit]]s.
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| ==Telegrapher's equations==
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| {{Main|Telegrapher's equations}}
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| {{See also|Reflections on copper lines}}
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| The '''telegrapher's equations''' (or just '''telegraph equations''') are a pair of linear differential equations which describe the [[voltage]] and [[Electric current|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]].
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| [[Image:Transmission line element.svg|thumb|right|250px|Schematic representation of the elementary component of a transmission line.]]
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| 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:
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| * The distributed resistance <math>R</math> of the conductors is represented by a series resistor (expressed in ohms per unit length). | |
| * The distributed inductance <math>L</math> (due to the [[magnetic field]] around the wires, [[self-inductance]], etc.) is represented by a series [[inductor]] ([[henry (unit)|henries]] per unit length).
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| * The capacitance <math>C</math> between the two conductors is represented by a [[Shunt (electrical)|shunt]] [[capacitor]] C ([[farad]]s per unit length).
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| * The [[Electric conductance|conductance]] <math>G</math> of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire ([[Siemens (unit)|siemens]] per unit length). | |
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| The model consists of an ''infinite series'' of the elements shown in the figure, and that the values of the components are specified ''per unit length'' so the picture of the component can be misleading. <math>R</math>, <math>L</math>, <math>C</math>, and <math>G</math> may also be functions of frequency. An alternative notation is to use <math>R'</math>, <math>L'</math>, <math>C'</math> and <math>G'</math> to emphasize that the values are derivatives with respect to length. These quantities can also be known as the [[primary line constants]] to distinguish from the secondary line constants derived from them, these being the [[propagation constant]], [[attenuation constant]] and [[phase constant]].
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| The line voltage <math>V(x)</math> and the current <math>I(x)</math> can be expressed in the frequency domain as
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| :<math>\frac{\partial V(x)}{\partial x} = -(R + j \omega L)I(x)</math>
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| :<math>\frac{\partial I(x)}{\partial x} = -(G + j \omega C)V(x).</math>
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| When the elements <math>R</math> and <math>G</math> are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the <math>L</math> and <math>C</math> elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:
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| :<math>\frac{\partial^2V(x)}{\partial x^2}+ \omega^2 LC\cdot V(x)=0</math>
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| :<math>\frac{\partial^2I(x)}{\partial x^2} + \omega^2 LC\cdot I(x)=0.</math>
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| These are [[wave equation]]s which have [[plane wave]]s 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.
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| If <math>R</math> and <math>G</math> are not neglected, the Telegrapher's equations become: | |
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| :<math>\frac{\partial^2V(x)}{\partial x^2} = \gamma^2 V(x)</math>
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| :<math>\frac{\partial^2I(x)}{\partial x^2} = \gamma^2 I(x)</math>
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| where
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| :<math>\gamma = \sqrt{(R + j \omega L)(G + j \omega C)}</math>
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| and the characteristic impedance is:
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| :<math>Z_0 = \sqrt{\frac{R + j \omega L}{G + j \omega C}}.</math>
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| The solutions for <math>V(x)</math> and <math>I(x)</math> are:
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| :<math>V(x) = V^+ e^{-\gamma x} + V^- e^{\gamma x} \,</math>
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| :<math>I(x) = \frac{1}{Z_0}(V^+ e^{-\gamma x} - V^- e^{\gamma x}). \,</math>
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| The constants <math>V^\pm</math> and <math>I^\pm</math> must be determined from boundary conditions. For a voltage pulse <math>V_{\mathrm{in}}(t) \,</math>, starting at <math>x=0</math> and moving in the positive <math>x</math>-direction, then the transmitted pulse <math>V_{\mathrm{out}}(x,t) \,</math> at position <math>x</math> can be obtained by computing the Fourier Transform, <math>\tilde{V}(\omega)</math>, of <math>V_{\mathrm{in}}(t) \,</math>, attenuating each frequency component by <math>e^{\mathrm{-Re}(\gamma) x} \,</math>, advancing its phase by <math>\mathrm{-Im}(\gamma)x \,</math>, and taking the inverse Fourier Transform. The real and imaginary parts of <math>\gamma</math> can be computed as
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| :<math>\mathrm{Re}(\gamma) = (a^2 + b^2)^{1/4} \cos(\mathrm{atan2}(b,a)/2) \,</math>
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| :<math>\mathrm{Im}(\gamma) = (a^2 + b^2)^{1/4} \sin(\mathrm{atan2}(b,a)/2) \,</math>
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| where [[atan2]] is the two-parameter arctangent, and
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| :<math>a \equiv \omega^2 LC \left[ \left( \frac{R}{\omega L} \right) \left( \frac{G}{\omega C} \right) - 1 \right] </math>
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| :<math>b \equiv \omega^2 LC \left( \frac{R}{\omega L} + \frac{G}{\omega C} \right). </math>
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| For small losses and high frequencies, to first order in <math>R / \omega L</math> and <math>G / \omega C</math> one obtains
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| :<math>\mathrm{Re}(\gamma) \approx \frac{\sqrt{LC}}{2} \left( \frac{R}{L} + \frac{G}{C} \right) \,</math>
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| :<math>\mathrm{Im}(\gamma) \approx \omega \sqrt{LC}. \, </math>
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| Noting that an advance in phase by <math>- \omega \delta</math> is equivalent to a time delay by <math>\delta</math>, <math>V_{out}(t)</math> can be simply computed as
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| :<math>V_{\mathrm{out}}(x,t) \approx V_{\mathrm{in}}(t - \sqrt{LC}x) e^{- \frac{\sqrt{LC}}{2} \left( \frac{R}{L} + \frac{G}{C} \right) x }. \,</math>
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| ==Input impedance of transmission line==
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| [[File:SmithChartLineLength.svg|thumb|right|350px|Looking towards a load through a length {{math|''l''}} of lossless transmission line, the impedance changes as {{math|''l''}} increases, following the blue circle on this [[Smith chart|impedance Smith chart]]. (This impedance is characterized by its [[reflection coefficient]] {{math|''V<sub>reflected</sub>'' / ''V<sub>incident</sub>''}}.) The blue circle, centered within the chart, is sometimes called an ''SWR circle'' (short for ''constant [[standing wave ratio]]'').]]
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| The [[characteristic impedance]] {{math|''Z''<sub>0</sub>}} 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.
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| The impedance measured at a given distance, {{math|''l''}}, from the load impedance {{math|''Z<sub>L</sub>''}} may be expressed as,
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| :<math>Z_{in}\left(l\right)=\frac{V(l)}{I(l)}=Z_0 \frac{1 + \Gamma_L e^{-2 \gamma l}}{1 - \Gamma_L e^{-2 \gamma l}}</math>,
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| where {{math|''γ''}} is the propagation constant and <math>\Gamma_L=\left(Z_L - Z_0\right)/\left(Z_L + Z_0\right)</math> is the voltage [[reflection coefficient]] at the load end of the transmission line. Alternatively, the above formula can be rearranged to express the input impedance in terms of the load impedance rather than the load voltage reflection coefficient:
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| :<math>Z_{in}\left(l\right)=Z_0 \frac{Z_L + Z_0 \tanh\left(\gamma l\right)}{Z_0 + Z_L\tanh\left(\gamma l\right)}</math>. | |
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| ===Input impedance of lossless transmission line===
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| For a lossless transmission line, the propagation constant is purely imaginary, {{math|''γ''{{=}}''jβ''}}, so the above formulas can be rewritten as,
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| :<math>
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| Z_\mathrm{in} (l)=Z_0 \frac{Z_L + jZ_0\tan(\beta l)}{Z_0 + jZ_L\tan(\beta l)}
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| </math>
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| where <math>\beta=\frac{2\pi}{\lambda}</math> is the [[wavenumber]].
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| In calculating {{math|''β''}}, 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.
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| ===Special cases of lossless transmission lines===
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| ====Half wave length====
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| For the special case where <math>\beta l= n\pi</math> where n is an integer (meaning that the length of the line is a multiple of half a wavelength), the expression reduces to the load impedance so that
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| :
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| <math>Z_\mathrm{in}=Z_L \,</math>
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| for all {{math|''n''}}. This includes the case when {{math|''n''{{=}}0}}, meaning that the length of the transmission line is negligibly small compared to the wavelength. The physical significance of this is that the transmission line can be ignored (i.e. treated as a wire) in either case.
| |
| | |
| ====Quarter wave length====
| |
| {{Main|quarter wave impedance transformer}}
| |
| For the case where the length of the line is one quarter wavelength long, or an odd multiple of a quarter wavelength long, the input impedance becomes
| |
| :<math>
| |
| Z_\mathrm{in}=\frac{{Z_0}^2}{Z_L}. \,
| |
| </math>
| |
| | |
| ====Matched load====
| |
| 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
| |
| :<math>
| |
| Z_\mathrm{in}=Z_L=Z_0 \,
| |
| </math>
| |
| for all <math>l</math> and all <math>\lambda</math>.
| |
| | |
| ====Short====
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| [[File:Transmission line animation open short.gif|thumb|right|300px|Standing waves on a transmission line with an open-circuit load (top), and a short-circuit load (bottom). Colors represent voltages, and black dots represent electrons.]]
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| {{main|stub (electronics)#Short circuited stub|l1=stub}}
| |
| For the case of a shorted load (i.e. <math>Z_L=0</math>), the input impedance is purely imaginary and a periodic function of position and wavelength (frequency)
| |
| | |
| :<math>
| |
| Z_\mathrm{in} (l)=j Z_0 \tan(\beta l). \,
| |
| </math>
| |
| | |
| ====Open====
| |
| {{main|stub (electronics)#Open_circuited_stub|l1=stub}}
| |
| For the case of an open load (i.e. <math>Z_L=\infty</math>), the input impedance is once again imaginary and periodic
| |
| | |
| :<math>
| |
| Z_\mathrm{in} (l)=-j Z_0 \cot(\beta l). \,
| |
| </math>
| |
| | |
| ===Stepped transmission line===
| |
| [[Image:Segments.jpg|thumb|left|A simple example of stepped transmission line consisting of three segments.]]A stepped transmission line<ref>{{cite web|url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WJX-4W2122T-1&_user=5755111&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000000150&_version=1&_urlVersion=0&_userid=5755111&md5=fe79f204b33cf7eb6d03cb89ff250c91 |title=Journal of Magnetic Resonance - Impedance matching with an adjustable segmented transmission line |publisher=ScienceDirect.com |date= |accessdate=2013-06-15}}</ref> is used for broad range [[impedance matching]]. It can be considered as multiple transmission line segments connected in series, with the characteristic impedance of each individual element to be Z<sub>0,i</sub>. The input impedance can be obtained from the successive application of the chain relation
| |
| | |
| :<math>
| |
| Z_\mathrm{i+1}=Z_\mathrm{0,i} \frac{Z_i + jZ_\mathrm{0,i}\tan(\beta_i l_i)}{Z_\mathrm{0,i} + jZ_i\tan(\beta_i l_i)}
| |
| </math>
| |
| | |
| where <math>\beta_i</math> is the wave number of the ''i''th transmission line segment and l<sub>i</sub> is the length of this segment, and Z<sub>i</sub> is the front-end impedance that loads the ''i''th segment. [[Image:PolarSmith.jpg|thumb|right|The impedance transformation circle along a transmission line whose characteristic impedance Z<sub>0,i</sub> is smaller than that of the input cable Z<sub>0</sub>. And as a result, the impedance curve is off-centered towards the -x axis. Conversely, if Z<sub>0,i</sub> > Z<sub>0</sub>, the impedance curve should be off-centered towards the +x axis.]]Because the characteristic impedance of each transmission line segment Z<sub>0,i</sub> is often different from that of the input cable Z<sub>0</sub>, the impedance transformation circle is off centered along the x axis of the [[Smith Chart]] whose impedance representation is usually normalized against Z<sub>0</sub>.
| |
| | |
| ==Practical types==<!-- This section is linked from [[Wikipedia:Proposed mergers]] -->
| |
| | |
| ===Coaxial cable===
| |
| | |
| {{Main|coaxial cable}}
| |
| | |
| Coaxial lines confine virtually all of the electromagnetic wave to the area inside the cable. 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) only, which means that the electric and magnetic fields are both perpendicular to the direction of propagation (the electric field is radial, and the magnetic field is circumferential). However, at frequencies for which the wavelength (in the dielectric) is significantly shorter than the circumference of the cable, transverse electric (TE) and transverse magnetic (TM) [[waveguide]] modes can also propagate. When more than one mode can exist, bends and other irregularities in the cable geometry can cause power to be transferred from one mode to another.
| |
| | |
| 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.
| |
| | |
| [[Image:Solec Kujawski longwave antenna feeder.jpg|thumb|right|A type of transmission line called a ''cage line'', used for high power, low frequency applications. It functions similarly to a large coaxial cable. This example is the antenna [[feedline]] for a [[longwave]] radio transmitter in [[Poland]], which operates at a frequency of 225 kHz and a power of 1200 kW.]]
| |
| | |
| ===Microstrip===
| |
| | |
| {{Main|microstrip}}
| |
| | |
| A microstrip circuit uses a thin flat conductor which is [[Parallel (geometry)|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.
| |
| Microstrip is an open structure whereas coaxial cable is a closed structure.
| |
| | |
| ===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===
| |
| {{Main|Balanced line}}
| |
| A balanced line is a transmission line consisting of two conductors of the same type, and equal impedance to ground and other circuits. There are many formats of balanced lines, amongst the most common are twisted pair, star quad and twin-lead.
| |
| | |
| ====Twisted pair====
| |
| {{Main|Twisted pair}}
| |
| Twisted pairs are commonly used for terrestrial [[telephone]] communications. In such cables, many pairs are grouped together in a single cable, from two to several thousand.<ref>Syed V. Ahamed, Victor B. Lawrence, ''Design and engineering of intelligent communication systems'', pp.130-131, Springer, 1997 ISBN 0-7923-9870-X.</ref> The format is also used for data network distribution inside buildings, but the cable is more expensive because the transmission line parameters are tightly controlled.
| |
| | |
| ====Star quad====
| |
| Star quad is a four-conductor cable in which all four conductors are twisted together around the cable axis. It is sometimes used for two circuits, such as [[4-wire]] telephony and other telecommunications applications. In this configuration each pair uses two non-adjacent conductors. Other times it is used for a single, balanced circuit, such as audio applications and [[2-wire]] telephony. In this configuration two non-adjacent conductors are terminated together at both ends of the cable, and the other two conductors are also terminated together.
| |
| | |
| Interference picked up by the cable arrives as a virtually perfect common mode signal, which is easily removed by coupling transformers. Because the conductors are always the same distance from each other, cross talk is reduced relative to cables with two separate twisted pairs.
| |
| | |
| The combined benefits of twisting, differential signalling, and quadrupole pattern give outstanding noise immunity, especially advantageous for low signal level applications such as long microphone cables, even when installed very close to a power cable. The disadvantage is that star quad, in combining two conductors, typically has double the capacitance of similar two-conductor twisted and shielded audio cable. High capacitance causes increasing distortion and greater loss of high frequencies as distance increases.<ref>{{cite book|last=Lampen|first=Stephen H.|title=Audio/Video Cable Installer's Pocket Guide|year=2002|publisher=McGraw-Hill|isbn=0071386211|pages=32, 110, 112}}</ref><ref>{{cite book|last=Rayburn|first=Ray|title=Eargle's The Microphone Book: From Mono to Stereo to Surround - A Guide to Microphone Design and Application|edition=3|year=2011|publisher=Focal Press|isbn=0240820754|pages=164–166}}</ref>
| |
| | |
| ==== Twin-lead ====
| |
| {{Main|Twin-lead}}
| |
| Twin-lead consists of a pair of conductors held apart by a continuous insulator.
| |
| | |
| ====Lecher lines====
| |
| {{Main|Lecher lines}}
| |
| Lecher lines are a form of parallel conductor that can be used at [[Ultra high frequency|UHF]] for creating resonant circuits. They are a convenient practical format that fills the gap between [[Lumped element model|lumped]] components (used at [[High frequency|HF]]/[[VHF]]) and [[Resonant cavity|resonant cavities]] (used at [[Ultra high frequency|UHF]]/[[Super high frequency|SHF]]).
| |
| | |
| ===Single-wire line===
| |
| [[Unbalanced line]]s were formerly much used for telegraph transmission, but this form of communication has now fallen into disuse. Cables are similar to twisted pair in that many cores are bundled into the same cable but only one conductor is provided per circuit and there is no twisting. All the circuits on the same route use a common path for the return current (earth return). There is a power transmission version of [[single-wire earth return]] in use in many locations.
| |
| | |
| ==General applications==
| |
| | |
| ===Signal transfer===
| |
| | |
| 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 [[Antenna (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 power]] sources for [[radar]] [[transmitters]] and other devices.
| |
| | |
| ===Stub filters===
| |
| {{see also|Distributed element filter#Stub band-pass 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 [[Radio Society of Great Britain|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==
| |
| {{Main|Acoustic transmission line}}
| |
| | |
| An acoustic transmission line is the [[Acoustics|acoustic]] [[analogy|analog]] of the electrical transmission line, typically thought of as a rigid-walled tube that is long and thin relative to the wavelength of sound present in it.
| |
| | |
| ==Solutions of the telegrapher's equations as circuit components==
| |
| {{cleanup|section|reason=Poor style|date=June 2012}}
| |
| | |
| [[Image:Unbalanced Transmission Line Equivalent Sub Circuit.jpg|right|thumb|300px|Equivalent circuit of a transmission line described by the Telegrapher's equations.]]
| |
| | |
| [[Image:Balanced Transmission Line Equivalent Circuit.jpg|right|thumb|300px|Solutions of the Telegrapher's Equations as Components in the Equivalent Circuit of a Balanced Transmission Line Two-Port Implementation.]]
| |
| | |
| The solutions of the telegrapher's equations can be inserted directly into a circuit as components. The circuit in the top figure implements the solutions of the telegrapher's equations.<ref>{{Citation | last = McCammon | first = Roy | title = SPICE Simulation of Transmission Lines by the Telegrapher's Method | url=http://i.cmpnet.com/rfdesignline/2010/06/C0580Pt1edited.pdf | accessdate = 22 Oct 2010 }}</ref>
| |
| | |
| The bottom circuit is derived from the top circuit by source transformations.<ref>{{cite book
| |
| |author=William H. Hayt|title=Engineering Circuit Analysis
| |
| |edition=second
| |
| |publisher=McGraw-Hill
| |
| |location=New York, NY|year=1971|isbn=070273820}}, pp. 73-77</ref> It also implements the solutions of the telegrapher's equations.
| |
| | |
| The solution of the telegrapher's equations can be expressed as an ABCD type '' [[Two-port network]]'' with the following defining equations<ref>
| |
| {{cite book
| |
| |author=John J. Karakash|title=Transmission Lines and Filter Networks
| |
| |edition=First
| |
| |publisher=Macmillan
| |
| |location=New York, NY|year=1950}}, p. 44
| |
| </ref>
| |
| | |
| :<math>V_1 = V_2 \cosh ( \gamma x) + I_2 Z \sinh (\gamma x) \,</math>
| |
| :<math>I_1 = V_2 \frac{1}{Z} \sinh (\gamma x) + I_2 \cosh(\gamma x). \,</math>
| |
| | |
| :The symbols: <math>E_s, E_L, I_s, I_L, l \,</math> in the source book have been replaced by the symbols :<math> V_1, V_2, I_1, I_2, x \,</math> in the preceding two equations.
| |
| | |
| The ABCD type two-port gives <math>V_1 \, </math> and <math>I_1 \, </math> as functions of <math>V_2 \, </math> and <math>I_2 \, </math>. Both of the circuits above, when solved for <math>V_1 \, </math> and <math>I_1 \, </math> as functions of <math>V_2 \, </math> and <math>I_2 \, </math> yield exactly the same equations.
| |
| | |
| In the bottom circuit, all voltages except the port voltages are with respect to ground and the differential amplifiers have unshown connections to ground. An example of a transmission line modeled by this circuit would be a balanced transmission line such as a telephone line. The impedances Z(s), the voltage dependent current sources (VDCSs) and the difference amplifiers (the triangle with the number "1") account for the interaction of the transmission line with the external circuit. The T(s) blocks account for delay, attenuation, dispersion and whatever happens to the signal in transit. One of the T(s) blocks carries the ''forward wave'' and the other carries the ''backward wave''. The circuit, as depicted, is fully symmetric, although it is not drawn that way. The circuit depicted is equivalent to a transmission line connected from <math>V_1 \, </math> to <math>V_2 \, </math> in the sense that <math>V_1 \, </math>, <math>V_2 \, </math>, <math>I_1 \, </math> and <math>I_2 \, </math> would be same whether this circuit or an actual transmission line was connected between <math>V_1 \, </math> and <math>V_2 \, </math>. There is no implication that there are actually amplifiers inside the transmission line.
| |
| | |
| Every two-wire or balanced transmission line has an implicit (or in some cases explicit) third wire which may be called shield, sheath, common, Earth or ground. So every two-wire balanced transmission line has two modes which are nominally called the differential and common modes. The circuit shown on the bottom only models the differential mode.
| |
| | |
| In the top circuit, the voltage doublers, the difference amplifiers and impedances Z(s) account for the interaction of the transmission line with the external circuit. This circuit, as depicted, is also fully symmetric, and also not drawn that way. This circuit is a useful equivalent for an unbalanced transmission line like a coaxial cable or a micro strip line.
| |
| | |
| These are not the only possible equivalent circuits.
| |
| | |
| ==See also==
| |
| {{Div col||25em}}
| |
| * [[Distributed element model]]
| |
| * [[Electric power transmission]]
| |
| * [[Heaviside condition]]
| |
| * [[Longitudinal wave|Longitudinal electromagnetic wave]]
| |
| * [[Lumped components]]
| |
| * [[Propagation velocity]]
| |
| * [[Radio frequency power transmission]]
| |
| * [[Smith chart]], a graphical method to solve transmission line equations
| |
| * [[Standing wave]]
| |
| * [[Time domain reflectometer]]
| |
| * [[Transverse wave|Transverse electromagnetic wave]]
| |
| {{Div col end}}
| |
| | |
| ==References==
| |
| ''Part of this article was derived from [[Federal Standard 1037C]].''
| |
| {{Reflist|30em}}
| |
| | |
| *{{Citation
| |
| |last= Steinmetz
| |
| |first= Charles Proteus
| |
| |authorlink= Charles Proteus Steinmetz
| |
| |title= The Natural Period of a Transmission Line and the Frequency of lightning Discharge Therefrom
| |
| |journal=The Electrical World
| |
| |date= August 27, 1898
| |
| |pages= 203–205
| |
| |issn=
| |
| |doi=}}
| |
| *{{Citation
| |
| |title= Electromagnetism
| |
| |edition= 2nd
| |
| |last=Grant
| |
| |first= I. S.
| |
| |last2= Phillips
| |
| |first2= W. R.
| |
| |publisher= John Wiley
| |
| |isbn= 0-471-92712-0
| |
| |doi=}}
| |
| *{{Citation
| |
| |title=Fundamentals of Applied Electromagnetics
| |
| |edition= 2004 media
| |
| |last= Ulaby
| |
| |first= F. T.
| |
| |publisher= Prentice Hall
| |
| |isbn= 0-13-185089-X
| |
| |doi= }}
| |
| *{{Citation
| |
| |title=Radio communication handbook
| |
| |year= 1982
| |
| |page= 20
| |
| |chapter= Chapter 17
| |
| |publisher= [[Radio Society of Great Britain]]
| |
| |isbn= 0-900612-58-4
| |
| |doi= }}
| |
| *{{Citation
| |
| |last= Naredo
| |
| |first= J. L.
| |
| |first2= A. C.
| |
| |last2= Soudack
| |
| |first3= J. R.
| |
| |last3= Marti
| |
| |title= Simulation of transients on transmission lines with corona via the method of characteristics
| |
| |journal= IEE Proceedings. Generation, Transmission and Distribution.
| |
| |volume= 142
| |
| |issue= 1
| |
| |publisher= Institution of Electrical Engineers
| |
| |location= Morelos <!-- dubious -->
| |
| |date= Jan 1995
| |
| |issn= 1350-2360
| |
| |doi=}}
| |
| | |
| ==Further reading==
| |
| {{Commons category|Transmission lines}}
| |
| * [http://earlyradiohistory.us/1902wt.htm 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, [http://web.archive.org/web/20050925041320/http://www.ece.cmu.edu/~ee762/hspice-docs/html/hspice_and_qrg/hspice_2001_2-124.html Using Transmission Line Equations and Parameters]. Star-Hspice Manual, June 2001.
| |
| * Cornille, P, [http://www.iop.org/EJ/abstract/0022-3727/23/2/001 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.
| |
| * Kupershmidt, Boris A., [http://arxiv.org/abs/math-ph/9810020 Remarks on random evolutions in Hamiltonian representation]. Math-ph/9810020. J. Nonlinear Math. Phys. 5 (1998), no. 4, 383-395.
| |
| * [[Mihajlo Pupin|Pupin, M.]], {{US patent|1541845}}, ''Electrical wave transmission''.
| |
| * [http://cktse.eie.polyu.edu.hk/eie403/ Transmission line matching]. EIE403: High Frequency Circuit Design. Department of Electronic and Information Engineering, Hong Kong Polytechnic University. ([[Portable Document Format|PDF]] format)
| |
| * Wilson, B. (2005, October 19). ''[http://cnx.rice.edu/content/m1044/latest/ Telegrapher's Equations]''. Connexions.
| |
| * John Greaton Wöhlbier, "''[http://www.wildwestwohlbiers.org/john/files/ms_thesis.pdf "Fundamental Equation''" and "''Transforming the Telegrapher's Equations"]''. Modeling and Analysis of a Traveling Wave Under Multitone Excitation.
| |
| * Agilent Technologies. Educational Resources. ''Wave Propagation along a Transmission Line''. [http://education.tm.agilent.com/index.cgi?CONTENT_ID=6 Edutactional Java Applet].
| |
| * Qian, C., [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WJX-4W2122T-1&_user=5755111&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000000150&_version=1&_urlVersion=0&_userid=5755111&md5=fe79f204b33cf7eb6d03cb89ff250c91 Impedance matching with adjustable segmented transmission line]. J. Mag. Reson. 199 (2009), 104-110.
| |
| | |
| ==External links==
| |
| * [http://www.cvel.clemson.edu/emc/calculators/TL_Calculator/index.html Transmission Line Parameter Calculator]
| |
| * [http://www.amanogawa.com/archive/transmissionB.html Interactive applets on transmission lines]
| |
| * [http://www.eetimes.com/design/microwave-rf-design/4200760/SPICE-Simulation-of-Transmission-Lines-by-the-Telegrapher-s-Method-Part-1-of-3-?Ecosystem=microwave-rf-design SPICE Simulation of Transmission Lines]
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| {{Telecommunications}}
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| {{DEFAULTSORT:Transmission Line}}
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| [[Category:Cables]]
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| [[Category:Telecommunications engineering]]
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| [[Category:Distributed element circuits]]
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