View source for Electrical length

{{Short description|Parameter characterizing an AC conductor}}
{{distinguish|Antenna effective length}}
{{Antennas|characteristics}}

In [[electrical engineering]], '''electrical length'''  is a dimensionless parameter equal to the physical length of an [[electrical conductor]] such as a cable or wire, divided by the [[wavelength]] of [[alternating current]] at a given [[frequency]] traveling through the conductor.<ref name="ATIS">
{{cite web
  | title = Electrical length
  | work = ATIS Telecom Glossary
  | publisher = Alliance for Telecommunications Industry Solutions website
  | date = 2019
  | url = https://glossary.atis.org/glossary/electrical-length/?char=E&page_number=12&sort=ASC
  | doi = 
  | accessdate = 24 December 2022}} ANSI (American National Standards Institute) accredited definition</ref><ref name="Kaiser">{{cite book
 | last1  = Kaiser
 | first1 = Kenneth L. 
 | title  = Electromagnetic Compatibility Handbook
 | publisher = CRC Press
 | date   = 2004
 | pages  = 3.1–3.2
 | url    = https://books.google.com/books?id=nZzOAsroBIEC&dq=%22electrical+length%22&pg=SA2-PA25
 | doi    = 
 | id     = 
 | isbn   =  9780849320873
 }}</ref><ref name="Weik">{{cite book
 | last1  = Weik
 | first1 = Martin 
 | title  = Communications Standard Dictionary
 | publisher = Springer
 | date   = 2012
 | location = 
 | pages  = 283
 | language = 
 | url    = https://books.google.com/books?id=pVvSBwAAQBAJ&pg=PA283
 | doi    = 
 | id     = 
 | isbn   = 9781461304296
 }}</ref> In other words, it is the length of the conductor measured in wavelengths. It can alternately be expressed as an [[angle]], in [[radian]]s or [[degree (angle)|degree]]s, equal to the [[phase shift]] the alternating current experiences traveling through the conductor.<ref name="ATIS" /><ref name="Weik" />

Electrical length is defined for a conductor operating at a specific frequency or narrow band of frequencies. It is determined by the construction of the cable, so different cables of the same length operating at the same frequency can have different electrical lengths.  A conductor is called ''electrically long''  if it has an electrical length much greater than one; that is it is much longer than the wavelength of the alternating current passing through it, and ''electrically short'' if it is much shorter than a wavelength. '''Electrical lengthening''' and '''electrical shortening''' means adding [[Electrical reactance|reactance]] ([[capacitance]] or [[inductance]]) to an antenna or conductor to increase or decrease the electrical length,<ref name="ATIS" /> usually for the purpose of making it [[resonant]] at a different [[resonant frequency]].

This concept is used throughout  [[electronics]], and particularly in [[radio frequency]] circuit design, [[transmission line]] and [[antenna (radio)|antenna]] theory and design. Electrical length determines when wave effects ([[phase shift]] along conductors) become important in a circuit. Ordinary lumped element [[electric circuit]]s only work well for alternating currents at frequencies for which the circuit is electrically small (electrical length much less than one).  For frequencies high enough that the wavelength approaches the size of the circuit (the electrical length approaches one) the [[lumped element model]] on which circuit theory is based becomes inaccurate, and [[transmission line]] techniques must be used.<ref name="Schmitt">{{cite book
 | last1  = Schmitt
 | first1 = Ron 
 | title  = Electromagnetics Explained: A Handbook for Wireless RF, EMC, and High-Speed Electronics
 | publisher = Newnes
 | date   = 2002
 | url    = https://books.google.com/books?id=MLzPNpJQz9UC&dq=%22electrical+length%22&pg=PA12
 | doi    = 
 | id     = 
 | isbn   =  9780750674034
 }}</ref>{{rp|p.12–14}}

== 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  = 6–11
 | 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  = 16–18 
 | 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 360° in a wavelength) or [[radian]]s (with 2π 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.12–14}}   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.12–14}}<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> 2.9979×10<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> 8.854×10<sup>−12</sup> F/m (farads per metre) and a [[Vacuum permeability|magnetic permeability]] of <math>\mu_\text{0} =</math> 1.257×10<sup>−6</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 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" /> 
:<math>v_p = \kappa c</math> 
where <math>\kappa</math> (kappa) is a dimensionless number between 0 and 1 called the ''[[velocity factor]]'' (VF), 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, 20–22 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>\kappa</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>&nbsp;&nbsp;  
}}
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>\kappa = {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>\;\;\kappa = {1 \over \sqrt{\epsilon_\text{eff}}}\;</math>  (no magnetic materials)&nbsp;&nbsp;&nbsp;&nbsp; 
}}
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 kappa: <math>\lambda = v_\text{p}/f = \kappa c/f = \kappa\lambda_\text{0}</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 \kappa \lambda_\text{0}} = {lf \over \kappa c}\;</math>
}}

== Transmission lines ==

{| class="wikitable floatright"  
|+  
! Type of line
!
! Velocity<br/>factor κ<ref name="Keller2">{{cite book | last1  = Keller| first1 = Reto B. | title  = Design for Electromagnetic Compatibility-In a Nutshell| publisher = Springer International| date   = 2022| pages  = 39| url    = https://books.google.com/books?id=62CdEAAAQBAJ&pg=PA39| isbn   = 9783031141867}}</ref>
! Velocity of signal<br/>in cm per ns
|-
| [[Ladder line|Parallel line]],<br />air dielectric    ||  [[File:Ladder line.png|70px]]  || .95  || 29
|-
| Parallel line,<br />polyethylene dielectric ([[Twin lead]])    || [[File:Electronics Technician - Volume 7 - Figure 3-10.jpg|70px]]   || .85  || 28
|-
| [[Coaxial cable]],<br />polyethylene dielectric    || [[File:Electronics Technician - Volume 7 - Figure 3-14.jpg|70px]]   || .66  || 20
|-
| [[Twisted pair]], CAT-5    || [[File:Electronics Technician - Volume 7 - Figure 3-11.jpg|70px]]     || .64  || 19
|-
| [[Stripline]]        ||    || .50  || 15
|-
| [[Microstrip]]        || [[File:Microstrip scheme.svg|70px]]    || .50  || 15
|}

Ordinary electrical cable suffices to carry alternating current when the cable is ''electrically short''; the electrical length of the cable is small compared to one, that is when the physical length of the cable is small compared to a wavelength, say <math>l < \lambda/10</math>.<ref name="Keller">{{cite web
  | last = Keller
  | first = Reto
  | title = Chapter 5: Transmission lines
  | work = Electromagnetic compatibility knowledge base
  | publisher = Academy of EMC website
  | date = 2018
  | url = https://www.academyofemc.com/emc-knowledge-base
  | format = 
  | doi = 
  | accessdate = 24 December 2022}}</ref>

As frequency gets high enough that the length of the cable becomes a significant fraction of a wavelength, <math>l > \lambda/10</math>, ordinary wires and cables become poor conductors of AC.<ref name="Schmitt" />{{rp|p.12–14}}  Impedance discontinuities at the source, load, connectors and switches begin to reflect the electromagnetic current waves back toward the source, creating bottlenecks so not all the power reaches the load.  Ordinary wires act as antennas, radiating the power into space as radio waves, and in radio receivers can also pick up [[radio frequency interference]] (RFI).

To mitigate these problems, at these frequencies [[transmission line]] is used instead.  A transmission line is a specialized cable designed for carrying electric current of [[radio frequency]].  The distinguishing feature of a transmission line is that it is constructed to have a constant [[characteristic impedance]] along its length and through connectors and switches, to prevent reflections.  This also means AC current travels at a constant phase velocity along its length, while in ordinary cable phase velocity may vary.  The velocity factor <math>\kappa</math> depends on the details of construction, and is different for each type of transmission line.  However the approximate velocity factor for the major types of transmission lines is given in the table.

Electrical length is widely used with a graphical aid called the [[Smith chart]] to solve transmission line calculations.  A Smith chart has a scale around the circumference of the circular chart graduated in wavelengths and degrees, which represents the electrical length of the transmission line from the point of measurement to the source or load.

The equation for the voltage as a function of time along a transmission line with a [[impedance matching|matched load]], so there is no reflected power, is
:<math>v(x, t) = V_\text{p} \cos(\omega t - \beta x)</math>
where
:<math>V_\text{p}</math> is the peak voltage along the line
:<math>\omega = 2\pi f = 2\pi/T</math> is the [[angular frequency]] of the alternating current in radians per second 
:<math>\beta = 2\pi/\lambda</math> is the [[wavenumber]], equal to the number of radians of the wave in one meter
:<math>x</math> is the distance along the line
:<math>t</math> is time
In a matched transmission line, the current is in phase with the voltage, and their ratio is the [[characteristic impedance]] <math>Z_\text{0}</math> of the line
:<math>i(x, t) = {v(x, t) \over Z_\text{0}} = {V_\text{p} \over Z_\text{0}} \cos(\omega t - \beta x) = {V_\text{p} \over Z_\text{0}} \cos \omega(t - x/\kappa c)</math>

== Antennas ==
[[File:Dipole antenna standing waves animation 1-10fps.gif|thumb|upright=1.2|A half-wave [[dipole antenna]] showing the standing waves of voltage ''(red)'' and current ''(blue)'' on the antenna.  The antenna is resonant at the frequency at which the electrical length is approximately equal to <math>\lambda/2 = c/2f</math>]]

An important class of radio [[antenna (radio)|antenna]] is the ''thin element antenna'' in which the radiating elements are conductive wires or rods.  These include [[monopole antenna]]s and [[dipole antenna]]s, as well as antennas based on them such as the [[whip antenna]], [[T antenna]], [[mast radiator]], [[Yagi antenna|Yagi]], [[log periodic antenna|log periodic]], and [[turnstile antenna]]s.  These are [[resonant]] antennas, in which the radio frequency electric currents travel back and forth in the antenna conductors, reflecting from the ends.

If the antenna rods are not too thick (have a large enough length to diameter ratio), the current along them is close to a sine wave, so the concept of electrical length also applies to these.<ref name="Weik" />   The current is in the form of two oppositely directed sinusoidal traveling waves which reflect from the ends, which interfere to form [[standing wave]]s.   The electrical length of an antenna, like a transmission line, is its length in wavelengths of the current on the antenna at the operating frequency.<ref name="ATIS" /><ref name="Radioman3&2">{{cite book
 | title  = Radioman 3 & 2, US Navy Training Course NAVPERS 10228-E
 | publisher = Bureau of Naval Personnel, US Navy
 | date   = 1967
 | pages  = 131
 | url    = https://books.google.com/books?id=v1xjCRjU6pAC&dq=%22electrical+length%22&pg=PA131
 | doi    = 
 | id     = 
 | isbn   = 
 }}</ref><ref name="Singh">{{cite book
 | last1  = Singh
 | first1 = Yaduvir 
 | title  = Electro Magnetic Field Theory
 | publisher = Dorling Kindersley
 | date   = 2011
 | location = 
 | pages  = 451
 | url    = https://books.google.com/books?id=0-PfbT49tJMC&q=%22electrical+length%22&pg=PA451
 | doi    = 
 | id     = 
 | isbn   = 9788131760611
 }}</ref><ref name="Schmitt" />{{rp|p.91–104}}  An antenna's [[resonant frequency]], [[radiation pattern]], and driving point [[input impedance|impedance]] depend not on its physical length but on its electrical length.<ref name="Griffith">{{cite book
 | last1  = Griffith
 | first1 = B. Whitfield 
 | title  = Radio-electronic Transmission Fundamentals
 | publisher = Noble Publishing
 | series = 
 | volume = 
 | edition = 
 | date   = 2000
 | location = 
 | pages  = 335–337
 | language = 
 | url    = https://books.google.com/books?id=m5DIroWLw2EC&pg=PA335
 | doi    = 
 | id     = 
 | isbn   = 9781884932137
 }}</ref>  A thin antenna element is resonant at frequencies at which the standing current wave has a node (zero) at the ends (and in monopoles an [[antinode]] (maximum) at the ground plane).  A [[dipole antenna]] is resonant at frequencies at which its electrical length is a half wavelength (<math>\lambda/2, \phi = 180^\circ \;\text{or}\; \pi \;\text{radians}</math>)<ref name="Radioman3&2" /> or a multiple of it.    A [[monopole antenna]] is resonant at frequencies at which its electrical length is a quarter wavelength (<math>\lambda/4, \phi = 90^\circ \;\text{or}\; \pi/2 \;\text{radians}</math>) or a multiple of it.

Resonant frequency is important because at frequencies at which the antenna is [[resonant]] the input [[electrical impedance|impedance]] it presents to its feedline is purely [[electrical resistance|resistive]].  If the resistance of the antenna is matched to the [[characteristic impedance|characteristic resistance]] of the feedline, it absorbs all the power supplied to it, while at other frequencies it has [[Electrical reactance|reactance]] and reflects some power back down the line toward the transmitter, causing [[standing wave]]s (high [[standing wave ratio|SWR]]) on the feedline.  Since only a portion of the power is radiated this causes inefficiency, and can possibly overheat the line or transmitter.  Therefore, transmitting antennas are usually designed to be resonant at the transmitting frequency; and if they cannot be made the right length they are ''electrically lengthened'' or ''shortened'' to be resonant (see below).

=== End effects ===
[[File:DipoleReductionFactor.jpg|thumb|upright=1.6|Reduction factor of physical length of a resonant dipole from a half-wavelength electrical length as a function of element thickness]] 
A thin-element antenna can be thought of as a transmission line with the conductors separated,<ref name="AF52-19" />   so the near-field electric and magnetic fields extend further into space than in a transmission line, in which the fields are mainly confined to the vicinity of the conductors.  Near the ends of the antenna elements the electric field is not perpendicular to the conductor axis as in a transmission line but spreads out in a fan shape (fringing field).<ref name="Schelkunoff">{{cite book
 | last1  = Schelkunoff
 | first1 = Sergei A.
 | last2  = Friis
 | first2 = Harold T.
 | title  = Antennas: Theory and Practice
 | publisher = John Wiley and Sons
 | date   = 1952
 | pages  = 245
 | url    = https://archive.org/details/antennastheorypr00sche/page/n5/mode/2up
 | doi    = 
 | id     = 
 }}</ref>  As a result, the end sections of the antenna have increased capacitance, storing more charge, so the current waveform departs from a sine wave there, decreasing faster toward the ends.<ref name="Rudge">{{cite book
 | last1  = Rudge
 | first1 = Alan W.
 | last2  = Milne
 | first2 = K.
 | title  = The Handbook of Antenna Design, Vol. 2
 | publisher = IET
 | date   = 1982 
 | pages = 564
 | url    = https://books.google.com/books?id=QjYtNJZmWLEC&dq=%22umbrella+antenna%22&pg=PA588
 | isbn   = 9780906048870
 }}</ref>  When approximated as a sine wave, the current does not quite go to zero at the ends; the [[node (physics)|nodes]] of the current standing wave, instead of being at the ends of the element, occur somewhat beyond the ends.<ref>The effect of this on the antenna is equivalent to the current wave moving along the antenna at a phase velocity <math>v_\text{p}</math> lower than the speed of light <math>c</math>, as in a transmission line. Some sources explain it this way: {{cite book
 | last1  = Carr
 | first1 = Joseph
 | last2  = Hippisley
 | first2 = George
 | title  = Practical Antenna Handbook, 5th Ed.
 | publisher = McGraw-Hill
 | date   = 2012
 | pages  = 105
 | url    = http://hamradio.uz/media/uploads/2018/04/06/practical_antenna_handbook__2012.pdf
 | id     = 
 | isbn   = 9780071639590
 }} and {{cite book
 | last1  = Rudge
 | first1 = Alan W.
 | last2  = Milne
 | first2 = K.
 | title  = The Handbook of Antenna Design, Vol. 2
 | publisher = IET
 | date   = 1982 
 | pages  = 564
 | url    = https://books.google.com/books?id=QjYtNJZmWLEC&pg=PA588&dq=%22umbrella+antenna%22
 | isbn   = 9780906048870
 }}  However, this is a physically misleading description; the phase velocity is not constant along the element.</ref>  Thus the electrical length of the antenna is longer than its physical length.

The electrical length of an antenna element also depends on the length-to-diameter ratio of the conductor.<ref name="Lewis">{{cite book
 | last1  = Lewis
 | first1 = Geoff 
 | title  = Newnes Communications Technology Handbook
 | publisher = Elsevier
 | date   = 2013
 | pages  = 46
 | url    = https://books.google.com/books?id=_7z8BAAAQBAJ&q=%22electrical+length%22+%22velocity+factor%22
 | doi    = 
 | id     = 
 | isbn   =  9781483101026
 }}</ref><ref name="AF52-19">{{cite book
 | title  = US Air Force Manual 52-19: Antenna Systems
 | publisher = US Air Force
 | date   = 1953
 | pages  = 104–105
 | url    = https://books.google.com/books?id=bDC0pGpz5EQC&dq=%22length+to+diameter+ratio%22+%22electrical+length%22&pg=PA105
 | doi    = 
 | id     = 
 | isbn   = 
 }}</ref><ref name="ARRL">{{cite book
 | title  = The A.R.R.L. Antenna Book, 5th Ed.
 | publisher = American Radio Relay League
 | date   = 1949
 | pages  = 27–28
 | language = 
 | url    = https://books.google.com/books?id=2Y1RAAAAMAAJ&dq=%22electrical+length%22&pg=PA27
 | doi    = 
 | id     = 
 | isbn   = 
 }}</ref><ref name="Carr1">{{cite book
 | last1  = Carr
 | first1 = Joseph
 | title  = Antenna Toolkit, 2nd Ed.
 | publisher = Elsevier
 | date   = 2001
 | location = 
 | pages  = 52–54
 | language = 
 | url    = https://books.google.com/books?id=kEbQ3io1q6sC&dq=%22transmission+line%22+%22velocity+factor%22&pg=PA52
 | doi    = 
 | id     = 
 | isbn   =  9780080493886
 }}</ref>  As the ratio of the diameter to wavelength increases, the capacitance increases, so the node occurs farther beyond the end, and the electrical length of the element increases.<ref name="Lewis" /><ref name="ARRL" />  When the elements get too thick, the current waveform becomes significantly different from a sine wave, so the entire concept of electrical length is no longer applicable, and the behavior of the antenna must be calculated by [[electromagnetic simulation]] computer programs like [[Numerical Electromagnetics Code|NEC]].

As with a transmission line, an antenna's electrical length is increased by anything that adds shunt capacitance or series inductance to it, such as the presence of high permittivity dielectric material around it.  In [[microstrip antenna]]s which are fabricated as metal strips on [[printed circuit board]]s, the [[dielectric constant]] of the substrate board increases the electrical length of the antenna.  Proximity to the Earth or a [[ground plane]], a dielectric coating on the conductor, nearby grounded towers, metal structural members, [[guy line]]s and the capacitance of insulators supporting the antenna also increase the electrical length.<ref name="ARRL" />

These factors, called "end effects", cause the electrical length of an antenna element to be somewhat longer than the length of the same wave in free space.  In other words, the physical length of the antenna at resonance will be somewhat shorter than the resonant length in free space (one-half wavelength for a dipole, one-quarter wavelength for a monopole).<ref name="Lewis" /><ref name="ARRL" />  As a rough generalization, for a typical [[dipole antenna]], the physical resonant length is about 5% shorter than the free space resonant length.<ref name="Lewis" /><ref name="ARRL" />

=== Electrical lengthening and shortening ===
In many circumstances for practical reasons it is inconvenient or impossible to use an antenna of resonant length.  An antenna of nonresonant length at the operating frequency can be made resonant by adding a [[electrical reactance|reactance]], a [[capacitance]] or [[inductance]], either in the antenna itself or in a [[matching network]] between the antenna and its [[feedline]].<ref name="ARRL" />  A nonresonant antenna appears at its feedpoint electrically equivalent to a [[electrical resistance|resistance]] in series with a reactance.  Adding an equal but opposite type of reactance in series with the feedline will cancel the antenna's reactance; the combination of the antenna and reactance will act as a series [[resonant circuit]], so at its operating frequency its input impedance will be purely resistive, allowing it to be fed power efficiently at a low [[standing wave ratio|SWR]] without reflections.

In a common application, an antenna which is ''electrically short'', shorter than its fundamental resonant length, a monopole antenna with an electrical length shorter than a quarter-wavelength (<math>\lambda/4</math>), or a dipole antenna shorter than a half-wavelength (<math>\lambda/2</math>) will have [[capacitive reactance]].  Adding an [[inductor]] (coil of wire), called a [[loading coil]], at the feedpoint in series with the antenna, with [[inductive reactance]] equal to the antenna's capacitive reactance at the operating frequency, will cancel the capacitance of the antenna, so the combination of the antenna and coil will be resonant at the operating frequency.  Since adding inductance is equivalent to increasing the electrical length, this technique is called '''electrically lengthening''' the antenna.  This is the usual technique for matching an electrically short transmitting antenna to its feedline, so it can be fed power efficiently.  However, an electrically short antenna that has been loaded in this way still has the same [[radiation pattern]]; it does not radiate as much power, and therefore has lower [[antenna gain|gain]] than a full-sized antenna.

Conversely, an antenna longer than resonant length at its operating frequency, such as a monopole longer than a quarter wavelength but shorter than a half wavelength, will have [[inductive reactance]]. This can be cancelled by adding a [[capacitor]] of equal but opposite reactance at the feed point to make the antenna resonant.  This is called '''electrically shortening''' the antenna.

=== Scaling properties of antennas ===
Two antennas that are [[similarity (geometry)|similar]] (scaled copies of each other), fed with different frequencies, will have the same [[radiation resistance]] and [[radiation pattern]] and fed with equal power will radiate the same power density in any direction if they have the same electrical length at the operating frequency; that is, if their lengths are in the same proportion as the wavelengths.<ref name="Levin">{{cite book
 | last1  = Levin
 | first1 = Boris 
 | title  = Wide-Range Antennas
 | publisher = CRC Press
 | date   = 2019
 | pages  = 26 
 | url    = https://books.google.com/books?id=fn-JDwAAQBAJ&dq=antenna+scaling&pg=SA9-PA26
 | doi    = 
 | id     = 
 | isbn   = 9781351043229
 }}</ref><ref name="Schmitt" />{{rp|p.12–14}}
:<math>{l_\text{1} \over l_\text{2}} = {\lambda_\text{1} \over \lambda_\text{2}} = {f_\text{2} \over f_\text{1}}</math>
This means the length of antenna required for a given [[antenna gain]] scales with the wavelength (inversely with the frequency), or equivalently the [[antenna aperture|aperture]] scales with the square of the wavelength.

=== Electrically short antennas ===
An electrically short conductor, much shorter than one wavelength, makes an inefficient radiator of [[electromagnetic wave]]s.  As the length of an antenna is made shorter than its fundamental resonant length (a half-wavelength for a dipole antenna and a quarter-wavelength for a monopole), the [[radiation resistance]] the antenna presents to the feedline decreases with the square of the electrical length, that is the ratio of physical length to wavelength, <math>(l/\lambda)^2</math>.    As a result, other resistances in the antenna, the ohmic resistance of metal antenna elements, the ground system if present, and the loading coil, dissipate an increasing fraction of transmitter power as heat.  A monopole antenna with an electrical length below  .05<math>\lambda</math> or 18° has a radiation resistance of less than one ohm, making it very hard to drive.

A second disadvantage is that since the capacitive reactance of the antenna and inductive reactance of the required loading coil do not decrease,  the [[Q factor]] of the antenna increases; it acts electrically like a high Q [[tuned circuit]].  As a result, the [[bandwidth (signal processing)|bandwidth]] of the antenna decreases with the square of electrical length, reducing the [[data rate]] that can be transmitted.  At [[very low frequency|VLF]] frequencies even the huge toploaded wire antennas that must be used have  bandwidths of only ~10 hertz, limiting the [[data rate]] that can be transmitted.

== Regimes of electromagnetics ==
The field of [[electromagnetics]] is the study of [[electric field]]s, [[magnetic field]]s, [[electric charge]], [[electric current]]s and [[electromagnetic wave]]s.  Classic electromagnetism is  based on the solution of [[Maxwell's equations]].  These equations are mathematically difficult to solve in all generality, so approximate methods have been developed that apply to situations in which the electrical length of the apparatus is very short (<math>G \ll 1</math>) or very long (<math>G \gg 1</math>).  Electromagnetics is divided into three regimes or [[fields of study]] depending on the electrical length of the apparatus, that is the physical length <math>l</math> of the apparatus compared to the wavelength <math>\lambda = c/f</math> of the waves:<ref name="Schmitt" />{{rp|p.21}}<ref name="Azadeh">{{cite book
 | last1  = Azadeh
 | first1 = Mohammad 
 | title  = Fiber Optics Engineering
 | publisher = Springer Science and Business Media
 | date   = 2009
 | pages  = 11
 | url    = https://books.google.com/books?id=8yV-Gwa2ByMC&dq=%22circuit+theory%22+microwave++optics+electromagnetics+wavelength&pg=PA11
 | doi    = 
 | id     = 
 | isbn   =  9781441903044
 }}</ref><ref name="Pozar">{{cite book
 | last1  = Pozar
 | first1 = David M. 
 | title  = Microwave Engineering, 4th Ed.
 | publisher = Wiley Global Education
 | date   = 2011
 | location = 
 | pages  = 1–2
 | url    = https://books.google.com/books?id=JegbAAAAQBAJ&dq=%22circuit+theory%22+microwave+optics&pg=PA1
 | doi    = 
 | id     = 
 | isbn   = 9781118213636
 }}</ref><ref name="Karmel">{{cite book
 | last1  = Karmel
 | first1 = Paul R.
 | last2  = Colef
 | first2 = Gabriel D.
 | last3  = Camisa
 | first3 = Raymond L.
 | title  = Introduction to Electromagnetic and Microwave Engineering
 | publisher = John Wiley and Sons
 | date   = 1998
 | location = 
 | pages  = 1–2
 | url    = https://books.google.com/books?id=iruLnH941OEC&dq=%22circuit+theory%22+microwave++optics&pg=PA1
 | doi    = 
 | id     = 
 | isbn   = 9780471177814
 }}</ref>  Completely different apparatus is used to conduct and process electromagnetic waves in these different wavelength ranges
*<math>\lambda \gg l</math> ''[[Circuit theory]]'': When the wavelength of the electrical oscillations is much larger than the physical size of the circuit (<math> G \ll 1</math>), say <math>\lambda > 50l</math>,<ref name="Clark">{{cite book
 | last1  = Clark
 | first1 = Alan Robert 
 | last2  = Fourie
 | first2 = Andre P. C. 
 | title  = Antennas in Practice
 | publisher = Poynting Innovations
 | date   = 2001
 | pages  = 3
 | language = 
 | url    = https://poynting.tech/wp-content/uploads/downloads/reference_materials/Antennas-in-Practice.pdf
 | doi    = 
 | id     = 
 | isbn   = 0620276193
 }}</ref> the action occurs in the [[Near and far field|near field]].   The [[phase (waves)|phase]] of the oscillations and therefore the current and voltage can be approximated as constant along the length of connecting wires.  Also little energy is radiated in the form of [[electromagnetic wave]]s, the power radiated by a conductor as an antenna is proportional to the electrical length squared <math>(l/\lambda)^2 = G^2</math>.   So the electrical energy remains in the wires and components as [[quasistatic approximation|quasistatic]] near-field [[electric field|electric]] and [[magnetic field]]s.  Therefore, the approximation of the [[lumped element model]] can be used, and electric currents  oscillating at these frequencies can be processed by [[electric circuit]]s consisting of lumped [[circuit element]]s such as resistors, capacitors, inductors, transformers, transistors, and [[integrated circuit]]s linked by ordinary wires.  Mathematically Maxwell's equations reduce to [[circuit theory]] ([[Kirchhoff's circuit laws]]).   
*<math>\lambda \approx l</math>, ''[[Distributed-element model]] ([[microwave]] theory)'': When the wavelength of the waves is of the same order of magnitude as the size of the equipment (<math> G \approx 1</math>),  as it is in the [[microwave]] part of the spectrum, full solutions of Maxwell's equations must be used.  At these frequencies, wires are replaced by [[transmission line]]s and [[waveguide]] and lumped elements are replaced by [[resonant stub]]s, irises, and [[microwave cavity|cavity resonators]].  Often only a single [[mode (electromagnetism)|mode]] (wave pattern) is propagating through the apparatus, which simplifies the mathematics.  A modification of circuit theory called the [[distributed-element model]] can often be used, in which extended objects are regarded as electrical circuits with capacitance, inductance and resistance distributed along their length.   A graphical aid called the [[Smith chart]] is often used to analyze transmission lines. 
*<math>\lambda \ll l</math>, ''[[Optics]]'': When the wavelength of the electromagnetic wave is much smaller than the physical size of the equipment that manipulates it (<math> G \gg 1</math>), say <math>\lambda < l/50</math>, most of the path of the waves is in the [[Near and far field|far field]].  In the far field, the electric and magnetic fields cannot be separated but propagate together as an electromagnetic wave.  Unlike in the case of microwaves, unless [[coherent light]] sources like lasers are used, the number of [[mode (electromagnetism)|mode]]s propagating is usually large. Since little of the energy is stored in the [[quasistatic approximation|quasistatic]] (induction) electric or magnetic fields at the surface boundaries between media (called [[evanescent field]]s in optics), the concepts of voltage, current, capacitance, and inductance have little meaning and are not used, and the medium is characterized by its [[index of refraction]] <math>\nu = c/v_\text{p} = \sqrt{\epsilon_\text{r}\mu_\text{r}}</math>, absorption, [[permittivity]] <math>\epsilon</math>, [[Magnetic permeability|permeability]] <math>\mu</math>, and [[Dispersion (optics)|dispersion]].  At these frequencies electromagnetic waves are manipulated by optical elements such as [[lens]]es, mirrors, [[Prism (optics)|prism]]s, [[optical filter]]s and [[diffraction grating]]s.   Maxwell's equations can be approximated by the equations of [[geometrical optics]] or [[physical optics]].
Historically, electric circuit theory and optics developed as separate branches of physics until at the end of the 19th century [[James Clerk Maxwell]]'s electromagnetic theory and [[Heinrich Hertz]]'s discovery that light was electromagnetic waves unified these fields as branches of electromagnetism.

== Definition of variables ==
{| class="wikitable"
!Symbol ||Unit ||Definition
|-
|<math>\beta</math>||meter<sup>−1</sup>||[[Wavenumber]] of wave in conductor <math>= 2\pi/\lambda</math>
|-
|<math>\epsilon</math>||[[farad]]s / meter ||[[Permittivity]] per meter of the dielectric in cable
|-
|<math>\epsilon_\text{0}</math>||[[farad]]s / meter ||[[Permittivity of free space]], a fundamental constant
|-
|<math>\epsilon_\text{eff}</math>||[[farad]]s / meter ||Effective relative [[permittivity]] per meter of cable
|-
|<math>\epsilon_\text{r}</math>||none ||[[Relative permittivity]] of the dielectric in cable
|-
|<math>\kappa</math>||none ||[[Velocity factor]] of current in conductor <math>= v_p/c</math>
|-
|<math>\lambda</math>||[[meter (unit)|meter]] ||[[Wavelength]] of radio waves in conductor
|-
|<math>\lambda_\text{0}</math>||[[meter (unit)|meter]] ||[[Wavelength]] of radio waves in free space
|-
|<math>\mu</math>||[[Henry (unit)|henries]] / meter ||Effective [[magnetic permeability]] per meter of cable 
|-
|<math>\mu_\text{0}</math>||[[Henry (unit)|henries]] / meter ||[[Permeability of free space]], a fundamental constant 
|-
|<math>\mu_\text{r}</math>||none ||[[Relative permeability]] of dielectric in cable
|-
|<math>\nu</math>||none ||[[Index of refraction]] of dielectric material 
|-
|<math>\pi</math>||none ||Constant = 3.14159
|-
|<math>\phi</math>||[[radian]]s or [[degree (angle)|degree]]s ||[[Phase shift]] of current between the ends of the conductor
|-
|<math>\omega</math>||[[radian]]s / second ||[[Angular frequency]] of alternating current <math>= 2\pi/f</math>
|-
|<math>c</math>||meters / second ||[[Speed of light]] in vacuum
|-
|<math>C</math>||farads / meter ||Shunt [[capacitance]] per unit length of the conductor
|-
|<math>f</math>||hertz ||[[Frequency]] of radio waves
|-
|<math>F</math>||none ||Fill factor of a transmission line, the fraction of space filled with dielectric
|-
|<math>G</math>||none ||Electrical length of conductor
|-
|<math>G_\text{0}</math>||none ||Electrical length of electromagnetic wave of length l in free space
|-
|<math>l</math>||[[meter (unit)|meter]] ||Length of the conductor
|-
|<math>L</math>||[[henry (unit)|henries]] / meter ||[[Inductance]] per unit length of the conductor
|-
|<math>T</math>||second ||[[Period (physics)|Period]] of radio waves 
|-
|<math>t</math>||second ||time 
|-
|<math>v_p</math>||meters / second ||[[phase velocity]] of current in conductor
|-
|<math>x</math>||meter ||distance along conductor 
|-
|}

== References ==
{{reflist}}

{{DEFAULTSORT:Electrical Length}}
[[Category:Telecommunication theory]]
[[Category:Antennas]]