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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 ''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 ''Y'' simply because it is next to ''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 is defined as
:<math>Y \equiv \frac{1}{Z} \,</math>

where 
:''Y'' is the admittance, measured in [[Siemens (unit)|siemens]]
:''Z'' is the [[Electrical impedance|impedance]], measured in [[Ohm (unit)|ohms]]

[[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>Y = G + j B \,</math>

where 
*<math>Y</math> is the admittance, measured in siemens.
*<math>G</math> is the [[Electrical conductance|conductance]], measured in siemens.
*<math>B</math> is the [[susceptance]], measured in siemens.
*<math>j^2 = -1</math>

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, ''Z'', is composed of real and imaginary parts,
:<math>Z = R + jX \,</math>
where
*''R'' is the [[Electrical resistance|resistance]], measured in ohms
*''X'' is the [[Electrical reactance|reactance]], measured in ohms

:<math>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, ''G''), and an [[imaginary number|imaginary]] part (the susceptance, ''B''), thus:

:<math>Y = G + jB \,\!</math>

where ''G'' (conductance) and ''B'' (susceptance) are given by:

:<math>\begin{align}
  G &= \Re(Y) =  \frac{R}{R^2 + X^2} \\
  B &= \Im(Y) = -\frac{X}{R^2 + X^2}
\end{align}</math>

The magnitude and phase of the admittance are given by:

:<math>\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 
*''G'' is the [[Electrical conductance|conductance]], measured in [[Siemens (unit)|siemens]]
*''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 Gc and B, respectively.
[[File:Transformer Model.png|center|frame]]
<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>

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 80 kilometers, this capacitance can be ignored and shunt components are not necessary in the model. Lines between 80 and about 250 kilometers, generally considered to be in the medium-line category, contain a shunt admittance governed by

<math>Y=yl=j\omega Cl</math>

where

* ''Y'' = total shunt admittance
* ''y'' = shunt admittance per unit length
* ''l'' = length of line
* ''C'' = capacitance of line

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

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

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

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