Editing Characteristic impedance (section)

== Transmission line model ==
The characteristic impedance <math>Z(\omega)</math> of an infinite transmission line at a given angular frequency <math>\omega</math> is the ratio of the voltage and current of a pure sinusoidal wave of the same frequency travelling along the line. This relation is also the case for finite transmission lines until the wave reaches the end of the line. Generally, a wave is reflected back along the line in the opposite direction. When the reflected wave reaches the source, it is reflected yet again, adding to the transmitted wave and changing the ratio of the voltage and current at the input, causing the voltage-current ratio to no longer equal the characteristic impedance. This new ratio including the reflected energy is called the [[input impedance]] of that particular transmission line and load.

The input impedance of an infinite line is equal to the characteristic impedance since the transmitted wave is never reflected back from the end. Equivalently: ''The characteristic impedance of a line is that impedance which, when terminating an arbitrary length of line at its output, produces an input impedance of equal value''. This is so because there is no reflection on a line terminated in its own characteristic impedance.
[[File:Transmission line element.svg|thumb|[[Circuit diagram|Schematic]] of [[telegrapher's equations|Heaviside's model]] of an [[infinitesimal]] segment of transmission line]]

Applying the transmission line model based on the [[telegrapher's equations]] as derived below,<ref name=":1">{{cite news |url=https://gateece.org/2016/04/16/derivation-of-characteristic-impedance-of-transmission-line/|title=Derivation of Characteristic Impedance of Transmission line |date=2016-04-16 |work=GATE ECE 2018 |access-date=2018-09-09 |language=en-US |archive-url=https://web.archive.org/web/20180909221832/https://gateece.org/2016/04/16/derivation-of-characteristic-impedance-of-transmission-line/ |archive-date=2018-09-09 |url-status=dead |df=dmy-all}}</ref> the general expression for the characteristic impedance of a transmission line is:
<math display="block">Z_0 = \sqrt{ \frac{R + j\omega L}{G + j\omega C}\, }</math>
where
{{unbulleted list | style = padding-left:1.2em
|<math>R</math> is the [[Electrical resistance|resistance]] per unit length, considering the two conductors to be [[in series]],
|<math>L</math> is the [[inductance]] per unit length,
|<math>G</math> is the [[Electrical conductance|conductance]] of the dielectric per unit length,
|<math>C</math> is the [[capacitance]] per unit length,
|<math>j</math> is the [[imaginary unit]], and
|<math>\omega</math> is the [[angular frequency]].
}}
This expression extends to DC by letting <math>\omega</math> tend to 0.

A surge of energy on a finite transmission line will see an impedance of <math>Z_0</math> prior to any reflections returning; hence ''surge impedance'' is an alternative name for ''characteristic impedance''.
Although an infinite line is assumed, since all quantities are per unit length, the “per length” parts of all the units cancel, and the characteristic impedance is independent of the length of the transmission line.

The voltage and current [[Phasor (electronics)|phasor]]s on the line are related by the characteristic impedance as:
<math display="block"> Z_\text{0} = \frac{V_{(+)}}{I_{(+)}} =-\frac{V_{(-)}}{I_{(-)}}</math>
where the subscripts (+) and (&minus;) mark the separate constants for the waves traveling forward (+) and backward (&minus;).  The rightmost expression has a negative sign because the current in the backward wave has the opposite direction to current in the forward wave.