Editing Characteristic impedance

{{Short description|Property of an electrical circuit}}
{{About|impedance in electrical circuits|impedance of electromagnetic waves|Wave impedance|characteristic acoustic impedance|Acoustic impedance}}
[[File:TransmissionLineDefinitions.svg|thumb|upright=1.4|A [[transmission line]] drawn as two black wires. At a distance ''x'' into the line, there is current [[phasor]] ''I''(''x'') traveling through each wire, and there is a [[voltage difference]] phasor ''V''(''x'') between the wires (bottom voltage minus top voltage). If <math>Z_0</math> is the '''characteristic impedance''' of the line, then <math>V(x) / I(x) = Z_0</math> for a wave moving rightward, or <math>V(x) / I(x) = -Z_0</math> for a wave moving leftward.]]
[[File:Transmission line schematic.svg|thumb|upright=1.35|Schematic representation of a [[electrical circuit|circuit]] where a source is coupled to a [[electrical load|load]] with a [[transmission line]] having characteristic impedance <math>Z_0</math>]]

The '''characteristic impedance''' or '''surge impedance''' (usually written Z<sub>0</sub>) of a uniform [[transmission line]] is the ratio of the amplitudes of [[voltage]] and [[Electric current|current]] of a wave travelling in one direction along the line in the absence of [[Reflections of signals on conducting lines|reflections]] in the other direction.  Equivalently, it can be defined as the [[input impedance]] of a transmission line when its length is infinite. Characteristic impedance is determined by the geometry and materials of the transmission line and, for a uniform line, is not dependent on its length. The [[SI]] unit of characteristic impedance is the [[Ohm (unit)|ohm]].

The characteristic impedance of a lossless transmission line is purely [[Real number|real]], with no [[Electrical reactance|reactive]] component (see [[Characteristic impedance#Lossless line|below]]). Energy supplied by a source at one end of such a line is transmitted through the line without being [[Dissipation|dissipated]] in the line itself. A transmission line of finite length (lossless or lossy) that is terminated at one end with an [[Electrical impedance|impedance]] equal to the characteristic impedance appears to the source like an infinitely long transmission line and produces no reflections.

== 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.

== Derivation ==

=== Using the telegrapher's equation ===
{{main|telegrapher's equation}}
[[File:Transmission line element.svg|thumb|Consider one section of the transmission line for the derivation of the characteristic impedance. The voltage on the left would be <math>\ V\ </math> and on the right side would be <math>\ V + \operatorname{d} V ~.</math> This figure is to be used for both the derivation methods.]]
The differential equations describing the dependence of the [[voltage]] and [[Electric current|current]] on time and space are linear, so that a linear combination of solutions is again a solution. This means that we can consider solutions with a time dependence <math>\ e^{j \omega t}</math>. Doing so allows to factor out the time dependence, leaving an ordinary differential equation for the coefficients, which will be [[phasor]]s, dependent on position (space) only. Moreover, the parameters can be generalized to be frequency-dependent.<ref name="Miano">{{cite book | last=Miano | first=Giovanni | last2=Maffucci | first2=Antonio | title=Transmission Lines and Lumped Circuits | publisher=Academic Press | publication-place=San Diego | date=2001 | isbn=0-12-189710-9 | pages=130-135}}</ref><ref>{{cite book | last=Mooijweer | first=H. | title=Microwave Techniques | publisher=Macmillan Education UK | publication-place=London | date=1971 | isbn=978-1-349-01067-7 | doi=10.1007/978-1-349-01065-3 | pages=74-79}}</ref>

Consider a [[steady-state]] problem such that the voltage and current can be written as:
<math display="block">\begin{align}
v(x,t) &= V(x) e^{j \omega t}\\[.5ex]
i(x,t) &= I(x) e^{j \omega t}
\end{align}
</math>
Take the positive direction for <math> V </math> and <math> I </math> in the loop to be clockwise. Substitution in the telegraph equations and factoring out the time dependence <math>\ e^{j \omega t}</math> now gives:
<math display="block">\begin{align}
\frac{\mathrm{d}V}{\mathrm{d}x} &= -\left( R + j \omega L \right) I = -Z I,\\[.5ex]
\frac{\mathrm{d}I}{\mathrm{d}x} &= -\left( G + j\ \omega C \right) V = -Y V,
\end{align}</math>
with impedance <math>Z</math> and [[admittance]] <math>Y</math>. Derivation and substitution of these two [[first-order differential equation]]s results in two uncoupled second-order differential equations:
<math display="block">\begin{align} 
\frac{\mathrm{d}^2 V}{\mathrm{d}x^2} &= k^2 V,\\[.5ex] 
\frac{\mathrm{d}^2 I}{\mathrm{d}x^2} &= k^2 I,
\end{align}</math>
with <math> k^2 = Z Y = ( R + j \omega L)(G + j \omega C )= (\alpha + j \beta)^2 </math> and <math> k = \alpha + j \beta </math> called the [[propagation constant]]. 

The solution to these type of equations can be written as:
<math display="block">\begin{align} 
V(x) &= A e^{-k x} + B e^{k x}\\[.5ex] 
I(x) &= A_{1} e^{-k x} + B_{1} e^{k x}
\end{align}</math>
with <math>A</math>, <math>A_1</math>, <math>B</math> and <math>B_1</math> the [[constant of integration|constants of integration]]. Substituting these constants in the first-order system gives:
<math display="block">\begin{align} 
A_1 &= A \frac{k}{R+ j \omega L}\\[.5ex] 
B_1 &= -B \frac{k}{R + j \omega L}
\end{align}</math>
where
<math display="block">
\frac{A}{A_1} = -\frac{B}{B_1} = \frac{R + j \omega L}{k} = \sqrt{\frac{R+ j\omega L}{G + j \omega C}} = \sqrt{\frac{Z}{Y}} = Z_{0}.
</math>
It can be seen that the constant <math>Z_0,</math> defined in the above equations has the dimensions of impedance (ratio of voltage to current) and is a function of primary constants of the line and operating frequency. It is called the ''characteristic impedance'' of the transmission line.<ref name=":1"/> 

The general solution of the telegrapher's equations can now be written as:
<math display="block">\begin{align}
v(x,t) &= V(x) e^{j \omega t} = A e^{-\alpha x}e^{j(\omega t - \beta x)} +  B e^{\alpha x}e^{j(\omega t + \beta x)} \\[.5ex]
i(x,t) &= I(x) e^{j \omega t} = \frac{A}{Z_0} e^{-\alpha x}e^{j(\omega t - \beta x)} -  \frac{B}{Z_0} e^{\alpha x}e^{j(\omega t + \beta x)} 
\end{align}</math>
Both the solution for the voltage and the current can be regarded as a superposition of two travelling waves in the <math>x_{+}</math> and <math>x_{-}</math> directions.

For typical transmission lines, that are carefully built from wire with low loss resistance <math>\ R\ </math> and small insulation leakage conductance <math>\ G\ ;</math> further, used for high frequencies, the inductive reactance <math>\ \omega L\ </math> and the capacitive admittance <math>\ \omega C\ </math> will both be large. In those cases, the [[phase constant]] and characteristic impedance are typically very close to being real numbers:
<math display="block">\begin{align}
\beta & \approx \omega \sqrt{L C} \\[.5ex]
Z_{0} & \approx \sqrt{\frac{L}{C}} 
\end{align}</math>
Manufacturers make commercial cables to approximate this condition very closely over a wide range of frequencies.

=== As a limiting case of infinite ladder networks ===

==== Intuition ====
{{see also|Iterative impedance|Constant k filters}}

{{Multiple image|
|align=right
|direction=vertical
<!--image 1-->
| image1            = Ladder iterative impedance.svg
| alt1              = Iterative impedance of an infinite ladder of L-circuit sections
| caption1          = Iterative impedance of an infinite ladder of L-circuit sections
| width=300px
<!--image 2-->
| image2            = L-section iterative impedance.svg
| alt2              = Iterative impedance of the equivalent finite L-circuit 
| caption2          = Iterative impedance of the equivalent finite L-circuit 
}}

Consider an infinite [[ladder network]] consisting of a series impedance <math>\ Z\ </math> and a shunt admittance <math>\ Y ~.</math> Let its input impedance be <math>\ Z_\mathrm{IT} ~.</math> If a new pair of impedance <math>\ Z\ </math> and admittance <math>\ Y\ </math> is added in front of the network, its input impedance <math>\ Z_\mathrm{IT}\ </math> remains unchanged since the network is infinite. Thus, it can be reduced to a finite network with one series impedance <math>\ Z\ </math> and two parallel impedances <math>\ 1 / Y\ </math> and <math>\ Z_\text{IT} ~.</math> Its input impedance is given by the expression<ref>{{cite book |title=The Feynman Lectures on Physics|title-link=The Feynman Lectures on Physics |volume=2 |first1=Richard |last1=Feynman |author1-link=Richard Feynman |first2=Robert B. |last2=Leighton |author2-link=Robert B. Leighton |first3=Matthew |last3=Sands |author3-link=Matthew Sands |section=Section 22-6. A ladder network |section-url=https://www.feynmanlectures.caltech.edu/II_22.html#Ch22-S6}} </ref><ref name=lee2004/>

:<math>\ Z_\mathrm{IT} = Z + \left( \frac{\ 1\ }{ Y } \parallel Z_\mathrm{IT} \right)\ </math>

which is also known as its [[iterative impedance]]. Its solution is:

:<math>\ Z_\mathrm{ IT } = {Z \over 2} \pm \sqrt { {Z^2 \over 4} + {Z \over Y} }\ </math>

For a transmission line, it can be seen as a [[Mathematical limit|limiting case]] of an infinite ladder network with [[infinitesimal]] impedance and admittance at a constant ratio.<ref name="feynman">{{cite book|title=The Feynman Lectures on Physics|title-link=The Feynman Lectures on Physics|volume=2|first1=Richard|last1=Feynman|author1-link=Richard Feynman|first2=Robert B.|last2=Leighton|author2-link=Robert B. Leighton|first3=Matthew|last3=Sands|author3-link=Matthew Sands|section=Section 22-7. Filter |section-url=https://www.feynmanlectures.caltech.edu/II_22.html#Ch22-S7 |quote=If we imagine the line as broken up into small lengths Δℓ, each length will look like one section of the L-C ladder with a series inductance ΔL and a shunt capacitance ΔC. We can then use our results for the ladder filter. If we take the limit as Δℓ goes to zero, we have a good description of the transmission line. Notice that as Δℓ is made smaller and smaller, both ΔL and ΔC decrease, but in the same proportion, so that the ratio ΔL/ΔC remains constant. So if we take the limit of Eq. (22.28) as ΔL and ΔC go to zero, we find that the characteristic impedance z0 is a pure resistance whose magnitude is √(ΔL/ΔC). We can also write the ratio ΔL/ΔC as L0/C0, where L0 and C0 are the inductance and capacitance of a unit length of the line; then we have <math>\sqrt{\frac{L_0}{C_0}}</math>}}.</ref><ref name=lee2004>{{cite book |first=Thomas H. |last=Lee |author-link=Thomas H. Lee (electronic engineer) |year=2004 |title=Planar Microwave Engineering: A practical guide to theory, measurement, and circuits |publisher=Cambridge University Press |section=2.5 Driving-point impedance of iterated structure |page=44 }}</ref> Taking the positive root, this equation simplifies to:

:<math>\ Z_\mathrm{IT} = \sqrt{ \frac{\ Z\ }{ Y }\ }\ </math>

==== Derivation ====

Using this insight, many similar derivations exist in several books<ref name="feynman"/><ref name=lee2004/> and are applicable to both lossless and lossy lines.<ref>{{cite book |first=Thomas H. |last=Lee |author-link=Thomas H. Lee (electronic engineer) |year=2004 |title=Planar Microwave Engineering: A practical guide to theory, measurement, and circuits |publisher=Cambridge University Press |section=2.6.2. Characteristic impedance of a lossy transmission line |page=47}}</ref>

Here, we follow an approach posted by Tim Healy.<ref name=":2">{{cite web |url=http://www.ee.scu.edu/eefac/healy/char.html |title=Characteristic impedance |website= ee.scu.edu |access-date=2018-09-09 |archive-date=2017-05-19 |archive-url=https://web.archive.org/web/20170519040949/http://www.ee.scu.edu/eefac/healy/char.html |url-status=dead }}</ref> The line is modeled by a series of differential segments with differential series elements <math>\ \left( R\ \operatorname{d}x,\ L\ \operatorname{d}x \right)\ </math> and shunt elements <math>\ \left(C\ \operatorname{d}x,\ G\ \operatorname{d}x\ \right)\ </math> (as shown in the figure at the beginning of the article). The characteristic impedance is defined as the ratio of the input voltage to the input current of a semi-infinite length of line. We call this impedance <math>\ Z_0 ~.</math> That is, the impedance looking into the line on the left is <math>\ Z_0 ~.</math> But, of course, if we go down the line one differential length <math>\ \operatorname{d}x\ ,</math> the impedance into the line is still <math>\ Z_0 ~.</math> Hence we can say that the impedance looking into the line on the far left is equal to <math>\ Z_0\ </math> in parallel with <math>\ C\ \operatorname{d}x\ </math> and <math>\ G\ \operatorname{d}x\ ,</math> all of which is in series with <math>\ R\ \operatorname{d}x\ </math> and <math>\ L\ \operatorname{d}x ~.</math> Hence:
<math display="block">\begin{align}
 Z_0 &= (R + j\ \omega L)\ \operatorname{d}x + \frac{ 1 }{\ (G + j \omega C)\ \operatorname{d}x + \frac{1}{\ Z_0\ }\ } \\[1ex]
 Z_0 &= (R + j\ \omega L)\ \operatorname{d}x + \frac{\ Z_0\ }{Z_0\ (G + j \omega C)\ \operatorname{d}x + 1\, } \\[1ex]
 Z_0 + Z_0^2\ (G + j\ \omega C)\ \operatorname{d}x &= (R + j\ \omega L)\ \operatorname{d}x + Z_0\ (G + j\ \omega C)\ \operatorname{d}x\ (R + j\ \omega L)\ \operatorname{d}x + Z_0
\end{align} </math>

The added <math>\ Z_0\ </math> terms cancel, leaving
<math display="block">\ Z_0^2\ (G + j\ \omega C)\ \operatorname{d}x = \left( R + j\ \omega L \right)\ \operatorname{d}x + Z_0\ \left( G + j\ \omega C \right)\ \left( R + j\ \omega L \right)\ \left( \operatorname{d}x \right)^2 </math>

The first-power <math>\ \operatorname{d}x\ </math> terms are the highest remaining order. Dividing out the common factor of <math>\ \operatorname{d}x\ ,</math> and dividing through by the factor <math>\ \left( G + j\ \omega C \right)\ ,</math> we get
<math display="block">\ Z_0^2 = \frac{ \left( R + j\ \omega L \right) }{\ \left( G + j\ \omega C \right)\ } + Z_0\ \left( R + j\ \omega L \right)\ \operatorname{d}x ~.</math>

In comparison to the factors whose <math>\ \operatorname{d}x\ </math> divided out, the last term, which still carries a remaining factor <math>\ \operatorname{d}x\ ,</math> is infinitesimal relative to the other, now finite terms, so we can drop it. That leads to
<math display="block">\ Z_0 = \pm \sqrt{\frac{\ R + j\ \omega L\ }{G + j\ \omega C}\ } ~.</math>

Reversing the sign {{math|±}} applied to the square root has the effect of reversing the direction of the flow of current.

== Lossless line ==
The analysis of lossless lines provides an accurate approximation for real transmission lines that simplifies the mathematics considered in modeling transmission lines. A lossless line is defined as a transmission line that has no line resistance and no [[dielectric loss]]. This would imply that the conductors act like perfect conductors and the dielectric acts like a perfect dielectric. For a lossless line, {{math|''R''}} and {{math|''G''}} are both zero, so the equation for characteristic impedance derived above reduces to:
<math display="block">Z_0 = \sqrt{\frac{L}{C}\,}\,.</math>

In particular, <math>Z_0</math> does not depend any more upon the frequency. The above expression is wholly real, since the imaginary term {{mvar|j}} has canceled out, implying that <math>Z_0</math> is purely resistive. For a lossless line terminated in <math>Z_0</math>, there is no loss of current across the line, and so the voltage remains the same along the line. The lossless line model is a useful approximation for many practical cases, such as low-loss transmission lines and transmission lines with high frequency. For both of these cases, {{mvar|R}} and {{mvar|G}} are much smaller than {{math|''ωL''}} and {{math|''ωC''}}, respectively, and can thus be ignored.

The solutions to the long line transmission equations include incident and reflected portions of the voltage and current:
<math display="block">\begin{align}
V &= \frac{V_r + I_r Z_c}{2} e^{\gamma x} + \frac{V_r - I_r Z_c}{2} e^{-\gamma x} \\[1ex]
I &= \frac{V_r/Z_c + I_r}{2} e^{\gamma x} - \frac{V_r/Z_c - I_r}{2} e^{-\gamma x}
\end{align}</math>
When the line is terminated with its characteristic impedance, the reflected portions of these equations are reduced to 0 and the solutions to the voltage and current along the transmission line are wholly incident. Without a reflection of the wave, the load that is being supplied by the line effectively blends into the line making it appear to be an infinite line. In a lossless line this implies that the voltage and current remain the same everywhere along the transmission line. Their magnitudes remain constant along the length of the line and are only rotated by a phase angle.

== Surge impedance loading ==
In [[electric power transmission]], the characteristic impedance of a transmission line is expressed in terms of the '''surge impedance loading''' ('''SIL'''), or natural loading, being the power loading at which [[reactive power]] is neither produced nor absorbed:
<math display="block">\mathit{SIL} = \frac{{V_\mathrm{LL}}^2}{Z_0}</math>
in which <math>V_\mathrm{LL}</math> is the [[Root mean square AC voltage|RMS]] line-to-line [[voltage]] in [[volts]].

Loaded below its SIL, the voltage at the load will be greater than the system voltage. Above it, the load voltage is depressed. The [[Ferranti effect]] describes the voltage gain towards the remote end of a very lightly loaded (or open ended) transmission line. [[Underground cable]]s normally have a very low characteristic impedance, resulting in an SIL that is typically in excess of the thermal limit of the cable.

== Practical examples ==
{| class="wikitable"
|-
! Standard
! Impedance<br>(Ω)
! Tolerance
|- align="center";
| [[Category 5 cable#Characteristics|Category 5]]
| 100
| &nbsp;±5Ω<ref name="drakacom_cat5">{{cite web |title=SuperCat OUTDOOR CAT 5e U/UTP |url=http://communications.draka.com/sites/eu/Datasheets/SuperCat5_24_U_UTP_Install.pdf |archive-url=https://web.archive.org/web/20120316111058/http://communications.draka.com/sites/eu/Datasheets/SuperCat5_24_U_UTP_Install.pdf |archive-date=2012-03-16}}</ref>
|- align="center";
| [[USB#Signaling|USB]]
| &nbsp;90
| ±15%<ref>{{cite web |series=USB in a NutShell |title=Chapter 2 – Hardware |publisher=Beyond Logic.org |url=http://www.beyondlogic.org/usbnutshell/usb2.htm |access-date=2007-08-25}}</ref>
|- align="center";
| [[HDMI]]
| &nbsp;95
| ±15%<ref name=an10798/>
|- align="center";
| [[IEEE 1394]]
| 108
| &nbsp;{{su|b=−2%|p=+3%}}<ref name=ieee1394tdr>{{cite web |title=Evaluation |publisher=materias.fi.uba.ar |url=http://materias.fi.uba.ar/6644/info/reflectometria/avanzado/ieee1394-evalwith-tdr.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://materias.fi.uba.ar/6644/info/reflectometria/avanzado/ieee1394-evalwith-tdr.pdf |archive-date=2022-10-09 |url-status=live |access-date=2019-12-29}}</ref>
|- align="center";
| [[Video Graphics Array#Specifications|VGA]]
| &nbsp;75
| &nbsp;±5%<ref name=vga_klotz>{{cite web |title=VMM5FL |series=pro video data sheets |url=http://www.promusic.cz/soubory/File/Downloads/Data%20sheet/Klotz/Kabely%20pro%20video/VMM5FL__e.pdf |access-date=2016-03-21 |archive-url=https://web.archive.org/web/20160402033004/http://www.promusic.cz/soubory/File/Downloads/Data%20sheet/Klotz/Kabely%20pro%20video/VMM5FL__e.pdf |archive-date=2016-04-02 |url-status=dead }}</ref>
|- align="center";
| [[DisplayPort]]
| 100
| ±20%<ref name=an10798>{{cite web |title=AN10798 DisplayPort PCB layout guidelines |url=https://www.nxp.com/documents/application_note/AN10798.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.nxp.com/documents/application_note/AN10798.pdf |archive-date=2022-10-09 |url-status=live |access-date=2019-12-29}}</ref>
|- align="center";
| [[Digital Visual Interface|DVI]]
| &nbsp;95
| ±15%<ref name=an10798/>
|- align="center";
| [[PCI Express|PCIe]]
| &nbsp;85
| ±15%<ref name=an10798/>
|- align="center";
| [[Overhead power line]]
| &nbsp;400
| Typical{{sfn|Singh|2008|p=212}}
|- align="center";
| [[Undergrounding|Underground power line]]
| &nbsp;40
| Typical{{sfn|Singh|2008|p=212}}
|}

The characteristic impedance of [[coaxial cable]]s (coax) is commonly chosen to be {{nowrap|50 Ω}} for [[radio frequency|RF]] and [[microwave]] applications. Coax for [[video]] applications is usually {{nowrap|75 Ω}} for its lower loss.
{{see also|Nominal impedance#50 Ω and 75 Ω}}

== See also ==
* {{annotated link|Ampère's circuital law}} 
* {{annotated link|Characteristic acoustic impedance}} 
* [[Characteristic admittance]] – Mathematical inverse of the characteristic impedance
* [[Iterative impedance]] – Characteristic impedance is a limiting case of this
* {{annotated link|Maxwell's equations}} 
* {{annotated link|Wave impedance}} 
* {{annotated link|Smith chart}}
* {{annotated link|Space cloth}}

== References ==
{{reflist|25em}}

===Sources===
{{refbegin}}
*{{cite book
 | last = Guile | first = A.E.
 | year = 1977
 | title = Electrical Power Systems
 | isbn = 0-08-021729-X
}}
*{{cite book
 | last = Pozar | first = D.M. | author-link = David M. Pozar
 | date = February 2004
 | title = Microwave Engineering
 | edition = 3rd
 | isbn = 0-471-44878-8
}}
*{{cite book
 | last = Ulaby | first = F.T.
 | year = 2004
 | title = Fundamentals of Applied Electromagnetics
 | edition = media
 | publisher = Prentice Hall
 | isbn = 0-13-185089-X
}}
*{{cite book | first1 = S. N. | last1 = Singh | date = 23 June 2008 | title = Electric Power Generation: Transmission and Distribution | edition = 2 | publisher = PHI Learning Pvt. Ltd. | pages = 212 | isbn = 9788120335608 | oclc = 1223330325 | url = https://books.google.com/books?id=dBtbDYdRsdkC&pg=PA212}}
{{refend}}

== External links ==
{{FS1037C}}

{{Telecommunications}}

{{DEFAULTSORT:Impedance, Characteristic}}
[[Category:Electricity]]
[[Category:Physical quantities]]
[[Category:Distributed element circuits]]
[[Category:Transmission lines]]

[[de:Leitungstheorie#Die allgemeine L.C3.B6sung der Leitungsgleichungen]]