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{{Short description|Ease of electrical current flow}}
{{about|electrical engineering}}
In [[electrical engineering]], '''admittance''' is a measure of how easily a circuit or device will allow a current to flow. It is defined as the [[multiplicative inverse|reciprocal]] of [[Electrical impedance|impedance]], analogous to how [[Electrical resistance and conductance|conductance and resistance]] are defined. The [[SI]] unit of admittance is the [[siemens (unit)|siemens]] (symbol S); the older, synonymous unit is [[mho]], and its symbol is ℧ (an upside-down uppercase omega Ω). [[Oliver Heaviside]] coined the term ''admittance'' in December 1887.<ref>{{Cite journal|doi=10.1103/PhysRevB.68.155115|title=Immittance matching for multidimensional open-system photonic crystals|year=2003|last1=Ushida|first1=Jun|last2=Tokushima|first2=Masatoshi|last3=Shirane|first3=Masayuki|last4=Gomyo|first4=Akiko|last5=Yamada|first5=Hirohito|journal=Physical Review B|volume=68|issue=15|page=155115|arxiv = cond-mat/0306260 |bibcode = 2003PhRvB..68o5115U |s2cid=119500762}}</ref> Heaviside used {{mvar|Y}} to represent the magnitude of admittance, but it quickly became the conventional symbol for admittance itself through the publications of [[Charles Proteus Steinmetz]]. Heaviside probably chose {{mvar|Y}} simply because it is next to {{mvar|Z}} in the alphabet, the conventional symbol for impedance.<ref>Ronald R. Kline, ''Steinmetz: Engineer and Socialist'', p. 88, Johns Hopkins University Press, 1992 {{ISBN|0801842980}}.</ref>

Admittance {{mvar|Y}}, measured in [[Siemens (unit)|siemens]], is defined as the inverse of [[Electrical impedance|impedance]] {{mvar|Z}}, measured in [[Ohm (unit)|ohms]]:

<math display="block">Y \equiv \frac{1}{Z}</math>

[[electrical resistance|Resistance]] is a measure of the opposition of a circuit to the flow of a steady current, while impedance takes into account not only the resistance but also dynamic effects (known as [[Electrical reactance|reactance]]). Likewise, admittance is not only a measure of the ease with which a steady current can flow, but also the dynamic effects of the material's susceptance to polarization:

<math display="block">Y = G + j B \,,</math>

where 
* {{mvar|Y}} is the admittance (siemens);
* {{mvar|G}} is the [[Electrical conductance|conductance]] (siemens);
* {{mvar|B}} is the [[susceptance]] (siemens); and
* {{math|1= ''j''{{isup|2}} = −1}}, the [[imaginary unit]].

The dynamic effects of the material's susceptance relate to the [[universal dielectric response]], the power law scaling of a system's admittance with frequency under alternating current conditions.

==Conversion from impedance to admittance==
{{complex Z}}
The impedance, {{mvar|Z}}, is composed of real and imaginary parts,
<math display="block">Z = R + jX \,,</math>
where
* {{mvar|R}} is the [[Electrical resistance|resistance]] (ohms); and
* {{mvar|X}} is the [[Electrical reactance|reactance]] (ohms).

<math display="block">Y = Z^{-1}= \frac{1}{R + jX} = \left( \frac{1}{R^2 + X^2} \right) \left(R - jX\right) </math>

Admittance, just like impedance, is a complex number, made up of a [[real number|real]] part (the conductance, {{mvar|G}}), and an [[imaginary number|imaginary]] part (the susceptance, {{mvar|B}}), thus:

<math display="block">Y = G + jB \,,</math>

where {{mvar|G}} (conductance) and {{mvar|B}} (susceptance) are given by:

<math display="block">\begin{align}
  G &= \mathrm{Re}(Y) =  \frac{R}{R^2 + X^2}\,, \\
  B &= \mathrm{Im}(Y) = -\frac{X}{R^2 + X^2}\,.
\end{align}</math>

The magnitude and phase of the admittance are given by:

<math display="block">\begin{align}
  \left | Y \right | &= \sqrt{G^2 + B^2} = \frac{1}{\sqrt{R^2 + X^2}} \\
            \angle Y &= \arctan \left( \frac{B}{G} \right) = \arctan \left( -\frac{X}{R} \right)\,,
\end{align}</math>

where 
* {{mvar|G}} is the [[Electrical conductance|conductance]], measured in [[Siemens (unit)|siemens]]; and
* {{mvar|B}} is the [[susceptance]], also measured in [[Siemens (unit)|siemens]].

Note that (as shown above) the signs of reactances become reversed in the admittance domain; i.e. capacitive susceptance is positive and inductive susceptance is negative.

== Shunt admittance in electrical power systems modeling ==
In the context of electrical modeling of transformers and transmission lines, shunt components that provide paths of least resistance in certain models are generally specified in terms of their admittance. Each side of most transformer models contains shunt components which model magnetizing current and core losses. These shunt components can be referenced to the primary or secondary side. For simplified transformer analysis, admittance from shunt elements can be neglected. When shunt components have non-negligible effects on system operation, the shunt admittance must be considered. In the diagram below, all shunt admittances are referred to the primary side. The real and imaginary components of the shunt admittance, conductance and susceptance, are represented by {{math|''G''{{sub|c}}}} and {{mvar|B}}, respectively.<ref>{{Cite book|title=Power System Analysis|last1=Grainger|first1=John J.|last2=Stevenson|first2=William D.|publisher=McGraw-Hill|year=1994|location=New York}}</ref>
[[File:Transformer Model.png|center|frame]]

Transmission lines can span hundreds of kilometers, over which the line's capacitance can affect voltage levels. For short length transmission line analysis, which applies to lines shorter than {{convert|80|km|mi}}, this capacitance can be ignored and shunt components are not necessary in the model. Lines from {{convert|80|to about|250|km|mi}}, generally considered to be in the medium-line category, contain a shunt admittance governed by<ref>J. Glover, M. Sarma, and T. Overbye, ''Power System Analysis and Design, Fifth Edition'', Cengage Learning, Connecticut, 2012, {{ISBN|978-1-111-42577-7}}, Chapter 5 ''Transmission Lines: Steady-State Operation''</ref><ref>{{Cite web|title=Equivalent- π Representation of a Long Line|url=http://nptel.ac.in/courses/Webcourse-contents/IIT-KANPUR/power-system/chapter_2/2_7.html|last=Ghosh|first=Arindam|access-date=30 Apr 2018}}</ref>
<math display="block">Y=yl=j\omega Cl\,,</math>
where

* {{mvar|Y}} is the total shunt admittance;
* {{mvar|y}} is the shunt admittance per unit length;
* {{mvar|l}} is the length of the transmission line; and
* {{mvar|C}} is the capacitance of the line.

[[File:Long Transmission Line Model.png|center|frameless|389x389px]]

==See also==
{{Wiktionary|admittance}}
*[[Nodal admittance matrix]]
*[[SI electromagnetism units]]
*[[Immittance]]

==References==
<references />{{Authority control}}

[[Category:Physical quantities]]
[[Category:Electrical resistance and conductance]]