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Electrical length
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== Definition == [[Image:Wavelength for sine wave.PNG|right|250px]] Electrical length is defined for conductors carrying [[alternating current]] (AC) at a single frequency or narrow band of frequencies. An alternating [[electric current]] of a single frequency <math>f</math> is an oscillating [[sine wave]] which repeats with a [[frequency|period]] of <math>T=1/f</math>.<ref name="Paul">{{cite book | last1 = Paul | first1 = Clayton R. | title = Transmission Lines in Digital and Analog Electronic Systems | publisher = Wiley | date = 2011 | location = | pages = 6β11 | language = | url = https://books.google.com/books?id=NDa9Vl6WBdMC&pg=PA6 | doi = | id = | isbn = 9781118058244 }}</ref> This current flows through a given conductor such as a wire or cable at a particular [[phase velocity]] <math>v_p</math>. It takes time for later portions of the wave to reach a given point on the conductor so the spatial distribution of current and voltage along the conductor at any time is a moving [[sine wave]]. After a time equal to the period <math>T</math> a complete cycle of the wave has passed a given point and the wave repeats; during this time a point of constant [[phase (waves)|phase]] on the wave has traveled a distance of :<math>\lambda = v_p T = v_p/f</math> so <math>\lambda</math> (Greek [[lambda]]) is the [[wavelength]] of the wave along the conductor, the distance between successive crests of the wave. The ''electrical length'' <math>G</math> of a conductor with a physical length of <math>l</math> at a given frequency <math>f</math> is the number of wavelengths or fractions of a wavelength of the wave along the conductor; in other words the conductor's length measured in wavelengths<ref name="Drollinger">{{cite book | last1 = Drollinger | first1 = Francis J. | title = Ground Radio Communications Specialist: Vol. 7 - Auxiliary circuits and systems | publisher = US Air Force Technical Training School | date = 1980 | pages = 16β18 | url = https://books.google.com/books?id=Bl1erNzgoncC&dq=%22electrical+length%22&pg=PA17 | doi = | id = | isbn = }}</ref><ref name="ATIS" /><ref name="Kaiser" /> {{Equation box 1 |indent = |cellpadding = 0 |border = 1 |border colour = black |background colour = transparent |equation = <math>\quad \text{Electrical length}\, G = {lf\over v_p} = {l \over \lambda} = {\text{Physical length} \over \text{Wavelength}} \quad</math> }} The [[phase velocity]] <math>v_p</math> at which electrical signals travel along a transmission line or other cable depends on the construction of the line. Therefore, the wavelength <math>\lambda</math> corresponding to a given frequency varies in different types of lines, thus at a given frequency different conductors of the same physical length can have different electrical lengths. === Phase shift definition === In [[radio frequency]] applications, when a delay is introduced due to a conductor, it is often the [[phase shift]] <math>\phi</math>, the difference in [[phase (waves)|phase]] of the sinusoidal wave between the two ends of the conductor, that is of importance.<ref name="Paul" /> The length of a [[sinusoidal]] wave is commonly expressed as an angle, in units of [[Degree (angle)|degree]]s (with 360Β° in a wavelength) or [[radian]]s (with 2Ο radians in a wavelength). So alternately the electrical length can be expressed as an [[angle]] which is the [[phase shift]] of the wave between the ends of the conductor<ref name="ATIS" /><ref name="Weik" /><ref name="Paul" /> :<math>\phi = 360^\circ{l \over \lambda} \, \text{degrees}</math> :<math>\quad = 2\pi{l \over \lambda} \, \text{radians}</math> === Significance === The electrical length of a conductor determines when wave effects (phase shift along the conductor) are important.<ref name="Schmitt" />{{rp|p.12β14}} If the electrical length <math>G</math> is much less than one, that is the physical length of a conductor is much shorter than the wavelength, say less than one tenth of the wavelength (<math>l < \lambda/10</math>) it is called ''electrically short''. In this case the voltage and current are approximately constant along the conductor, so it acts as a simple connector which transfers alternating current with negligible phase shift. In [[circuit theory]] the connecting wires between components are usually assumed to be electrically short, so the [[lumped element]] [[circuit theory|circuit model]] is only valid for alternating current when the circuit is ''electrically small'', much smaller than a wavelength.<ref name="Schmitt" />{{rp|p.12β14}}<ref name="Paul" /> When the electrical length approaches or is greater than one, a conductor will have significant [[electrical reactance|reactance]], [[inductance]] or [[capacitance]], depending on its length. So simple circuit theory is inadequate and [[transmission line]] techniques (the [[distributed-element model]]) must be used. === Velocity factor === In a vacuum an [[electromagnetic wave]] ([[radio wave]]) travels at the [[speed of light]] <math>v_p = c = </math> 2.9979Γ10<sup>8</sup> meters per second, and very close to this speed in air, so the ''free space wavelength'' of the wave is <math>\lambda_\text{0} = c/f</math>.<ref name="Paul" /> (in this article free space variables are distinguished by a subscript 0) Thus a physical length <math>l</math> of a radio wave in space or air has an electrical length of :<math>G_\text{0} = {l \over \lambda_\text{0}} = {lf \over c}</math> wavelengths. In the [[Systeme International|SI]] system of units, empty space has a [[Permittivity of Free Space|permittivity]] of <math>\epsilon_\text{0} =</math> 8.854Γ10<sup>β12</sup> F/m (farads per metre) and a [[Vacuum permeability|magnetic permeability]] of <math>\mu_\text{0} =</math> 1.257Γ10<sup>β6</sup> H/m (henries per meter). These universal constants determine the speed of light<ref name="Paul" /><ref name="Rao">{{cite book | last1 = Rao | first1 = R. S. | title = Electromagnetic Waves and Transmission Lines | publisher = PHI Learning | date = 2012 | pages = 445 | url = https://books.google.com/books?id=7gK9XEQ9a9QC&pg=PA445 | doi = | id = | isbn = 9788120345157 }}</ref> :<math>c = {1 \over \sqrt{\epsilon_\text{0}\mu_\text{0}}}</math> [[File:Transmission line equivalent circuit - Lossless.svg|thumb|Equivalent circuit of a lossless transmission line. <math>L</math> and <math>C</math> represent the [[inductance]] and [[capacitance]] per unit length of a small section of line]] In most transmission lines, the series [[electrical resistance|resistance]] of the wires and shunt [[electrical conductance|conductance]] of the insulation is low enough that the line can be approximated as lossless (see diagram). This means the inductance and capacitance per unit length of the line determine the phase velocity. In an electrical cable, for a cycle of the alternating current to move a given distance along the line, it takes time to charge the [[capacitance]] between the conductors, and the rate of change of the current is slowed by the series [[inductance]] of the wires. This determines the phase velocity <math>v_p</math> at which the wave moves along the line. In cables and transmission lines an electrical signal travels at a rate determined by the effective shunt [[capacitance]] <math>C</math> and series [[inductance]] <math>L</math> per unit length of the transmission line :<math>v_p = {1 \over \sqrt{LC} }</math> Some transmission lines consist only of bare metal conductors, if they are far away from other high permittivity materials their signals propagate at very close to the speed of light, <math>c</math>. In most transmission lines the material construction of the line slows the velocity of the signal so it travels at a reduced [[phase velocity]]<ref name="Paul" /> This property of the line is specified by a dimensionless number between 0 and 1 called the ''[[velocity factor]]'' <math>\mathit{VF}</math>: :<math>\mathit{VF} = {v_p \over c}</math> characteristic of the type of line, equal to the ratio of signal velocity in the line to the speed of light.<ref name="Carr2">{{cite book |last1=Carr |first1=Joseph J. |title=Microwave & Wireless Communications Technology |publisher=Newnes |date=1997 |pages=51 |url=https://books.google.com/books?id=1j1E541LKVoC&pg=PA51 |isbn=0750697075}}</ref><ref name="Amlaner">{{cite conference |last=Amlaner |first=Charles J. Jr. |title=The design of antennas for use in radio telemetry |book-title=A Handbook on Biotelemetry and Radio Tracking: Proceedings of an International Conference on Telemetry and Radio Tracking in Biology and Medicine, Oxford, 20β22 March 1979 |pages=260 |publisher=Elsevier |date=March 1979 |url=https://books.google.com/books?id=IXXYBAAAQBAJ&pg=PA260 |accessdate=23 November 2013}}</ref><ref name="Drollinger" /> Most transmission lines contain a [[dielectric]] material (insulator) filling some or all of the space in between the conductors. The [[permittivity]] <math>\epsilon</math> or ''dielectric constant'' of that material increases the distributed capacitance <math>C</math> in the cable, which reduces the velocity factor below unity. If there is a material with high [[Permeability (electromagnetism)|magnetic permeability]] (<math>\mu</math>) in the line such as steel or [[Ferrite (magnet)|ferrite]] which increases the distributed inductance <math>L</math>, it can also reduce <math>\mathit{VF}</math>, but this is almost never the case. If all the space around the transmission line conductors containing the near fields was filled with a material of permittivity <math>\epsilon</math> and permeability <math>\mu</math>, the phase velocity on the line would be<ref name="Paul" /> {{Equation box 1 |indent =: |cellpadding = 0 |border = 2 |border colour = black |background colour = transparent |equation = <math>\;\;v_p = {1 \over \sqrt{\epsilon\mu}}\;</math> }} The effective permittivity <math>\epsilon</math> and permeability <math>\mu</math> per unit length of the line are frequently given as dimensionless constants; [[relative permittivity]]: <math>\epsilon_\text{r}</math> and [[magnetic permeability|relative permeability]]: <math>\mu_\text{r}</math> equal to the ratio of these parameters compared to the universal constants <math>\epsilon_\text{0}</math> and <math>\mu_\text{0}</math> :<math>\epsilon_\text{r} = {\epsilon \over \epsilon_\text{0}} \qquad \mu_\text{r} = {\mu \over \mu_\text{0}}</math> so the phase velocity is :<math>v_\text{p} = {1 \over \sqrt{\epsilon\mu}} = {1 \over \sqrt{\epsilon_\text{0}\epsilon_\text{r}\mu_\text{0}\mu_\text{r}}} = c{1 \over \sqrt{\epsilon_\text{r}\mu_\text{r}}}</math> So the velocity factor of the line is :<math>\mathit{VF} = {v_p \over c} = {1 \over \sqrt{\epsilon_\text{r}\mu_\text{r}}}</math> In many lines, for example [[twin lead]], only a fraction of the space surrounding the line containing the fields is occupied by a solid dielectric. With only part of the electromagnetic field effected by the dielectric, there is less reduction of the wave velocity. In this case an ''effective permittivity'' <math>\epsilon_\text{eff}</math> can be calculated which if it filled all the space around the line would give the same phase velocity. This is computed as a weighted average of the relative permittivity of free space, unity, and that of the dielectric: <math display="block">\epsilon_\text{eff}= (1-F) + F \epsilon_\text{r}</math> where the ''fill factor'' {{math|''F''}} expresses the effective proportion of space around the line occupied by dielectric. In most transmission lines there are no materials with high magnetic permeability, so <math>\mu = \mu_\text{0}</math> and <math>\mu_\text{r} = 1</math> and so {{Equation box 1 |indent =: |cellpadding = 0 |border = 2 |border colour = black |background colour = transparent |equation = <math>\;\;\mathit{VF} = {1 \over \sqrt{\epsilon_\text{eff}}}\;</math> (no magnetic materials) }} Since the electromagnetic waves travel slower in the line than in free space, the wavelength of the wave in the transmission line <math>\lambda</math> is shorter than the free space wavelength by the factor VF: <math>\lambda = v_\text{p}/f = \mathit{VF}(c/f) = \lambda_\text{0}\mathit{VF}</math>. Therefore, more wavelengths fit in a transmission line of a given length <math>l</math> than in the same length of wave in free space, so the electrical length of a transmission line is longer than the electrical length of a wave of the same frequency in free space<ref name="Paul" /> {{Equation box 1 |indent =: |cellpadding = 0 |border = 2 |border colour = black |background colour = transparent |equation = <math>\;G = {l \over \lambda} = {l \over \lambda_\text{0}\mathit{VF}} = {lf \over c\mathit{VF}}\;</math> }}
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