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. In other words, it is the length of the conductor measured in wavelengths. It can alternately be expressed as an angle, in radians or degrees, equal to the phase shift the alternating current experiences traveling through the conductor.
Definition
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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 period of <math>T=1/f</math>. 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 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
:<math>c = {1 \over \sqrt{\epsilon_\text{0}\mu_\text{0}</math>
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 resistance of the wires and shunt 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
! Velocity of signal<br/>in cm per ns
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| Parallel line,<br />air dielectric || 70px || .95 || 29
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| Parallel line,<br />polyethylene dielectric (Twin lead) || 70px || .85 || 28
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| Coaxial cable,<br />polyethylene dielectric || 70px || .66 || 20
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| Twisted pair, CAT-5 || 70px || .64 || 19
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| Stripline || || .50 || 15
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| Microstrip || 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>.
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. 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>) 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. When approximated as a sine wave, the current does not quite go to zero at the ends; the nodes of the current standing wave, instead of being at the ends of the element, occur somewhat beyond the ends. 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. 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. 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>, the action occurs in the near field. The 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 waves, 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 near-field electric and magnetic fields. Therefore, the approximation of the lumped element model can be used, and electric currents oscillating at these frequencies can be processed by electric circuits consisting of lumped circuit elements such as resistors, capacitors, inductors, transformers, transistors, and integrated circuits 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 lines and waveguide and lumped elements are replaced by resonant stubs, irises, and cavity resonators. Often only a single 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 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 modes propagating is usually large. Since little of the energy is stored in the quasistatic (induction) electric or magnetic fields at the surface boundaries between media (called evanescent fields 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>, permeability <math>\mu</math>, and dispersion. At these frequencies electromagnetic waves are manipulated by optical elements such as lenses, mirrors, prisms, optical filters and diffraction gratings. 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>||farads / meter ||Permittivity per meter of the dielectric in cable
|-
|<math>\epsilon_\text{0}</math>||farads / meter ||Permittivity of free space, a fundamental constant
|-
|<math>\epsilon_\text{eff}</math>||farads / 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 ||Wavelength of radio waves in conductor
|-
|<math>\lambda_\text{0}</math>||meter ||Wavelength of radio waves in free space
|-
|<math>\mu</math>||henries / meter ||Effective magnetic permeability per meter of cable
|-
|<math>\mu_\text{0}</math>||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>||radians or degrees ||Phase shift of current between the ends of the conductor
|-
|<math>\omega</math>||radians / second ||Angular frequency of alternating current <math>= 2\pi/f</math>
|-
|<math>c</math>||meters / second ||Speed of light in vacuum
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|<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 ||Length of the conductor
|-
|<math>L</math>||henries / meter ||Inductance per unit length of the conductor
|-
|<math>T</math>||second ||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
|-
|}