View source for Susceptance

{{short description|Imaginary part of electrical admittance}}

In [[electrical engineering]], '''susceptance''' ({{mvar|B}}) is the [[Imaginary unit|imaginary]] part of [[admittance]] ({{math|1=''Y'' = ''G'' + ''jB''}}), where the [[real part]] is [[Electrical conductance|conductance]] ({{mvar|G}}). The [[multiplicative inverse|reciprocal]] of admittance is [[Electrical impedance|impedance]] ({{math|1=''Z'' = ''R'' + ''jX''}}), where the imaginary part is [[Electrical reactance|reactance]] ({{mvar|X}}) and the real part is [[Electrical resistance|resistance]] ({{mvar|R}}). In [[SI]] units, susceptance is measured in [[siemens (unit)|siemens]] (S).

==Origin==
The term was coined by [[Charles Proteus Steinmetz|C.P. Steinmetz]] in a 1894 paper.<ref>
{{cite journal
 |first=C.P. |last=Steinmetz  |author-link=Charles Proteus Steinmetz 
 |date=May 1894
 |title=On the law of hysteresis (part III), and the theory of ferric inductances
 |journal=Transactions of the American Institute of Electrical Engineers
 |volume=11 |pages=570–616
 |doi=10.1109/T-AIEE.1894.4763808 |s2cid=51648079 |url=https://ieeexplore.ieee.org/document/4763808
}}
</ref>

In some sources [[Oliver Heaviside]] is given credit for coining the term,<ref>
{{cite book
 |first=Graydon |last=Wetzer
 |year=2019
 |section=Wayfinding ''re/dicto''
 |pages=295–324
 |editor1=Flynn, Susan 
 |editor2=Mackay, Antonia
 |title=Surveillance, Architecture and Control: Discourses on spatial culture
 |publisher=Springer
 |isbn=978-3030003715
}}
</ref> or with introducing the concept under the name ''permittance''.<ref>For example:<br/>
{{cite book
 |first1=Sverre |last1=Grimnes
 |first2=Orjan G. |last2=Martinsen
 |title=Bioimpedance and Bioelectricity Basics
 |page=499
 |publisher=Academic Press
 |year=2014
 |isbn=978-0124115330
}}
</ref>
This claim is mistaken according to Steinmetz's biographer.<ref>
{{cite book
 |first=Ronald R. |last=Kline
 |year=1992
 |title=Steinmetz: Engineer and Socialist
 |page=88
 |place=Baltimore, MD
 |publisher=Johns Hopkins University Press
 |isbn=0801842980
}}
</ref>
The term ''susceptance'' does not appear anywhere in Heaviside's collected works, and Heaviside used the term ''permittance'' to mean [[capacitance]], not ''susceptance''.<ref>
{{cite book
 |first=Ido |last=Yavetz
 |year=2011
 |title=From Obscurity to Enigma: The work of Oliver Heaviside, 1872–1889
 |publisher=Springer
 |isbn=978-3034801775
 |url=https://books.google.com/books?id=JgXZQTeYby8C
 |via=Google Books
}}
</ref><!-- Be cautious with sources from 21st century that make this claim - there may be some circular copying from Wikipedia going on! -->

==Formula==
The general equation defining admittance is given by
<math display=block>Y = G + j B \,</math>

where

{{plainlist|1=
* {{mvar|Y}} is the complex [[admittance]], measured in [[siemens (unit)|siemens]];
* {{mvar|G}} is the real-valued [[Electrical conductance|conductance]], measured in siemens;
* {{mvar|j}} is the [[imaginary unit]] (i.e. {{math|''j''² {{=}} &minus;1}}); and
* {{mvar|B}} is the real-valued susceptance, measured in siemens.
|indent=1}}

The admittance ({{mvar|Y}}) is the [[Multiplicative inverse|reciprocal]] of the impedance ({{mvar|Z}}), if the impedance is not zero:

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

and

<math display=block>B \equiv \operatorname\mathcal{I_m}\{\, Y \,\}  = \frac{-X\;}{\;R^2+X^2} = \frac{-X~\;}{~\;\left| Z \right|^2\,} ~,</math>

where

{{plainlist|1=
* <math>Z = R + j X \,;</math>
* {{mvar|Z}} is the complex [[electrical impedance|impedance]], measured in [[ohm (unit)|ohms]];
* {{mvar|R}} is the real-valued [[electrical resistance|resistance]], measured in ohms; and
* {{mvar|X}} is the real-valued [[Reactance (electronics)|reactance]], measured in ohms.
|indent=1}}

The susceptance <math>B</math> is the imaginary part of the admittance <math>Y~.</math>

The magnitude of admittance is given by:

<math display=block>\left| Y \right| = \sqrt{G^2 + B^2\;} ~.</math>

And similar formulas transform admittance into impedance, hence susceptance ({{mvar|B}}) into reactance ({{mvar|X}}):

<math display=block>Z = \frac{1}{Y} = \frac{1}{G + j B} = \left( \frac{G}{\;G^2+B^2} \right) + j \left( \frac{-B\;\;}{\;G^2+B^2} \right) ~.</math>

hence

<math display=block>X \equiv \operatorname\mathcal{I_m}\{\, Z \,\} = \frac{\,-B\;~}{\;G^2+B^2} = \frac{\,-B~\;}{~\;\left| Y \right|^2\,} ~.</math>

The reactance and susceptance are only reciprocals in the absence of either resistance or conductance (only if either {{math|''R'' {{=}} 0}} or {{math|''G'' {{=}} 0}}, either of which implies the other, as long as {{math|''Z'' ≠ 0}}, or equivalently as long as {{math|''Y'' ≠ 0}}).

== Relation to capacitance ==
In electronic and semiconductor devices, transient or frequency-dependent current between terminals contains both conduction and displacement components. Conduction current is related to moving charge carriers (electrons, holes, ions, etc.), while displacement current is caused by time-varying electric field. Carrier transport is affected by electric field and by a number of physical phenomena, such as carrier drift and diffusion, trapping, injection, contact-related effects, and impact ionization. As a result, device [[admittance]] is frequency-dependent, and the simple electrostatic formula for capacitance, <math>C = \frac{q}{V}~,</math> is not applicable.

A more general definition of capacitance, encompassing electrostatic formula, is:<ref name="LauxCapacitance">{{cite journal |last=Laux |first=S.E. |date=Oct 1985 |title=Techniques for small-signal analysis of semiconductor devices |journal=IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems |volume=4 |issue=4 |pages=472–481 |doi=10.1109/TCAD.1985.1270145 |s2cid=13058472}}</ref>

<math display=block>C = \frac{~\operatorname\mathcal{I_m}\{\, Y \,\}~}{\omega} = \frac{B}{~\omega~} ~ ,</math>

where <math>Y</math> is the device admittance, and <math>B</math> is the susceptance, both evaluated at the angular frequency in question, and <math>\omega</math> is that angular frequency. It is common for electrical components to have slightly reduced capacitances at extreme frequencies, due to slight inductance of the internal conductors used to make capacitors (not just the leads), and [[permittivity]] changes in insulating materials with frequency: {{mvar|C}} is ''very nearly'', but ''not quite'' a constant.

==Relationship to reactance==
[[Electrical reactance|Reactance]] is defined as the imaginary part of [[electrical impedance]], and is ''analogous'' to but not generally equal to the negative reciprocal of the susceptance – that is their reciprocals are equal and opposite only in the special case where the real parts vanish (either zero resistance or zero conductance). In the special case of entirely zero admittance or exactly zero impedance, the relations are encumbered by infinities.

However, for purely-reactive impedances (which are purely-susceptive admittances), the susceptance is equal to the negative [[reciprocal (mathematics)|reciprocal]] of the [[Electrical reactance|reactance]], except when either is zero.

In mathematical notation:

:<math display=block>\forall ~ Z \ne 0 ~ \Leftrightarrow ~ Y \ne 0 \quad \Longrightarrow \quad G = 0 \Leftrightarrow R = 0 \quad \iff \quad B = -\frac{1}{\, X \,} ~.</math>

The minus sign is not present in the relationship between [[electrical resistance]] and the analogue of conductance <math>~ G \equiv \operatorname\mathcal{R_e}\{\, Y \,\} ~,</math> but otherwise a similar relation holds for the special case of reactance-free impedance (or susceptance-free admittance):
:<math display=block>\forall ~ Z \ne 0 ~ \Leftrightarrow ~ Y \ne 0 \quad \Longrightarrow \quad B = 0 \Leftrightarrow X = 0 \quad \iff \quad G = +\frac{1}{\, R \,}</math>

If the imaginary unit is included, we get

:<math display=block> jB = \frac{1}{\,jX\,} ~,</math>

for the resistance-free case since,

:<math display=block> \frac{1}{\, j \,} = -j ~.</math>

==Applications==
High susceptance materials are used in [[susceptor|susceptors]] built into microwavable food packaging for their ability to convert [[microwave|microwave radiation]] into heat.<ref>{{cite journal |last1=Labuza |first1=T. |last2=Meister |first2=J. |title=An alternate method for measuring the heating potential of microwave susceptor films |journal=Journal of International Microwave Power and Electromagnetic Energy |volume=27 |issue=4 |pages=205–208 |year=1992 |doi=10.1080/08327823.1992.11688192 |bibcode=1992JMPEE..27..205L |url=http://www.jmpee.org/JMPEE_PDFs/27-4_bl/JMPEE-Vol27-Pg205-Labuza.pdf  |access-date=23 Sep 2011}}</ref>

==See also==
*[[Electrical measurements]]
*[[SI electromagnetism units]]

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

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[[Category:Physical quantities]]
[[Category:Electrical engineering]]