Editing Electrical length (section)

== Transmission lines ==

{| class="wikitable floatright"  
|+  
! Type of line
!
! Velocity<br/>factor <math>\mathit{VF}</math><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>\mathit{VF}</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/\mathit{VF} c)</math>