View source for Elastance

{{Short description|Inverse of electrical capacitance}}
{{Good article}}
{{For|the physiological term|Compliance (physiology)}}
{{For|the physical term|Elasticity (physics)}}
'''Electrical elastance''' is the [[multiplicative inverse|reciprocal]] of [[capacitance]]. The [[SI]] unit of elastance is the inverse [[farad]] (F<sup>−1</sup>). The concept is not widely used by electrical and electronic engineers, as the value of [[capacitor]]s is typically specified in units of capacitance rather than inverse capacitance. However, elastance is used in theoretical work in network analysis and has some niche applications, particularly at [[microwave]] frequencies.

The term ''elastance'' was coined by [[Oliver Heaviside]] through the analogy of a capacitor to a spring. The term is also used for analogous quantities in other energy domains. In the mechanical domain, it corresponds to [[stiffness]], and it is the inverse of [[Compliance (physiology)|compliance]] in the fluid flow domain, especially in [[physiology]]. It is also the name of the generalized quantity in [[bond-graph]] analysis and other schemes that analyze systems across multiple domains.

==Usage==
The definition of capacitance (''C'') is the charge (''Q'') stored per unit voltage (''V'').

<math display="block"> C = {Q \over V} </math>

Elastance (''S'') is the [[multiplicative inverse|reciprocal]] of capacitance, thus,<ref>Camara, p. 16-11</ref>

<math display="block"> S = {V \over Q} \ . </math>

Expressing the values of [[capacitor]]s as elastance is not commonly done by practical electrical engineers, but can be convenient for [[capacitors in series]] since their total elastance is simply the sum of their individual elastances. However, elastance is sometimes used by network theorists in their analyses. One advantage of using elastance is that an increase in elastance results in an increase in [[electrical impedance|impedance]], aligning with the behavior of the other two basic passive [[electrical element|elements]], [[electrical resistance|resistance]] and [[inductance]]. An example of the use of elastance can be found in the 1926 doctoral thesis of [[Wilhelm Cauer]]. On his path to founding [[network synthesis]], he developed the [[mesh analysis|loop matrix]] '''A''':

<math display="block"> \mathbf{A} = s^2 \mathbf{L} + s \mathbf{R} + \mathbf{S} = s \mathbf{Z} </math>

where '''L''', '''R''', '''S''', and '''Z''' are the network loop matrices of inductance, resistance, elastance, and impedance, respectively, and ''s'' is [[complex frequency]]. This expression would be significantly more complicated if Cauer had used a matrix of capacitances instead of elastances. The use of elastance here is primarily for mathematical convenience, similar to how mathematicians use [[radian]]s rather than more common units for angles.<ref>Cauer, Mathis & Pauli, p.4. The symbols in Cauer's expression have been modified for consistency within this article and with modern practice.</ref>

Elastance is also applied in [[microwave engineering]]. In this field, [[varactor diode]]s are used as voltage-variable capacitors in devices such as [[frequency multiplier]]s, [[parametric amplifier]]s, and variable [[electronic filter|filters]]. These diodes store charge in their [[p-n junction|junction]] when [[reverse bias]]ed, which generates the capacitor effect. The slope of the voltage-stored charge curve in this context is referred to as ''differential elastance.''<ref>Miles, Harrison & Lippens, pp. 29–30</ref>

== Units ==
The [[SI]] unit of elastance is the reciprocal of the [[farad]] (F<sup>−1</sup>). The term ''daraf'' is sometimes used for this unit, but it is not approved by the SI and its use is discouraged.<ref>{{multiref|Michell, p.168|Mills, p.17}}</ref> The term ''daraf'' is formed by reversing the word ''farad,'' in much the same way as the unit ''[[mho]]'' (a unit of conductance, also not approved by the SI) is formed by writing ''[[ohm]]'' backwards.<ref>Klein, p.466</ref>

The term ''daraf'' was coined by [[Arthur E. Kennelly]], who used it as early as 1920.<ref>{{multiref|Kennelly & Kurokawa, p.41|Blake, p.29|Jerrard, p.33}}</ref>

==History==
The terms ''elastance'' and ''elastivity'' were coined by [[Oliver Heaviside]] in 1886.<ref>Howe, p.60</ref> Heaviside coined many of the terms used in [[circuit analysis]] today, such as [[electrical impedance|impedance]], [[inductance]], [[admittance]], and [[electrical conductance|conductance]]. His terminology followed the model of [[electrical resistance|resistance]] and [[resistivity]], with the ''-ance'' ending used for [[extensive properties]] and the ''-{{not a typo|ivity}}'' ending used for [[intensive properties]]. Extensive properties are used in circuit analysis (they represent the "values" of components), while intensive properties are used in [[field (physics)|field analysis]]. Heaviside's nomenclature was designed to emphasize the connection between corresponding quantities in fields and circuits.<ref>Yavetz, p.236</ref> 

Elastivity is the intensive property of a material, corresponding to the bulk property of a component, elastance. It is the reciprocal of [[permittivity]]. As Heaviside stated, 
{{blockquote|Permittivity gives rise to permittance, and elastivity to elastance.<ref>Heaviside, p.28</ref>|Oliver Heaviside}} 
Here, ''permittance'' is Heaviside's term for capacitance. He rejected any terminology that implied a [[capacitor]] acted as a container for holding charge. He opposed the terms ''capacity'' (capacitance) and ''capacious'' (capacitive) along with their inverses, ''incapacity'' and ''incapacious.''<ref>Howe, p.60</ref> At the time, the capacitor was often referred to as a ''condenser'' (suggesting that the "electric fluid" could be condensed), or as a ''leyden,''<ref>Heaviside, p.268</ref> after the [[Leyden jar]], an early capacitor, both implying storage. Heaviside preferred a mechanical analogy, viewing the capacitor as a compressed spring, which led to his preference for terms suggesting properties of a spring.<ref>Yavetz, pp.150–151</ref> 

Heaviside's views followed [[James Clerk Maxwell]]'s perspective on electric current, or at least Heaviside's interpretation of it. According to this view, electric current is analogous to [[velocity]], driven by the [[electromotive force]], similar to a mechanical [[force]]. At a capacitor, current creates a "[[displacement current|displacement]]" whose rate of change is equivalent to the current. This displacement was seen as an electric [[deformation (mechanics)#Strain|strain]], like mechanical strain in a compressed spring. Heaviside denied the idea of physical charge flow and accumulation on capacitor plates, replacing it with the concept of the [[divergence]] of the displacement field at the plates, which was numerically equal to the charge collected in the flow view.<ref>Yavetz, pp.150–151</ref>

In the late 19th and early 20th centuries, some authors adopted Heaviside's terms ''elastance'' and ''elastivity.''<ref>See, for example, Peek, p.215, writing in 1915</ref> Today, however, the reciprocal terms ''capacitance'' and ''permittivity'' are almost universally preferred by electrical engineers. Despite this, elastance still sees occasional use in theoretical work. One of Heaviside's motivations for choosing these terms was to distinguish them from mechanical terms. Thus, he selected ''elastivity'' rather than ''[[elasticity (physics)|elasticity]]'' to avoid the need to clarify between ''electrical elasticity'' and ''mechanical elasticity.''<ref>Howe, p.60</ref>

Heaviside carefully crafted his terminology to be unique to [[electromagnetism]], specifically avoiding overlaps with [[mechanics]]. Ironically, many of his terms were later borrowed back into mechanics and other domains to describe analogous properties. For example, it is now necessary to differentiate ''[[electrical impedance]]'' from ''[[mechanical impedance]]'' in some contexts.<ref>van der Tweel & Verburg, pp.16–20</ref> ''Elastance'' has also been used by some authors in mechanics to describe the analogous quantity, though ''[[stiffness]]'' is often preferred. However, ''elastance'' is widely used for the analogous property in the domain of [[fluid dynamics]], particularly in fields such as [[biomedicine]] and [[physiology]].<ref>See for example Enderle & Bronzino, pp.197–201, especially equation 4.72</ref>

==Mechanical analogy==
[[Mechanical–electrical analogies]] are established by comparing the mathematical descriptions of mechanical and electrical systems. Quantities that occupy corresponding positions in equations of the same form are referred to as ''analogues.'' There are two main reasons for creating such analogies.

The first reason is to explain electrical phenomena in terms of more familiar mechanical systems. For example, the differential equations governing an electrical [[RLC circuit]] (inductor-capacitor-resistor circuit) are of the same form as those governing a mechanical mass-spring-damper system. In such cases, the electrical domain is translated into the mechanical domain for easier understanding.

The second, more significant reason is to analyze systems containing both mechanical and electrical components as a unified whole. This approach is especially beneficial in fields like [[mechatronics]] and [[robotics]], where integration of mechanical and electrical elements is common. In these cases, the mechanical domain is often converted into the electrical domain because [[network analysis (electrical circuits)|network analysis]] in the electrical domain is more advanced and highly developed.<ref>Busch-Vishniac, pp.17–18</ref>

===The Maxwellian analogy===
In the analogy developed by Maxwell, now known as the [[impedance analogy]], [[voltage]] is analogous to [[force]]. The term "electromotive force" used for the voltage of an electric power source reflects this analogy. In this framework, [[electric current|current]] is analogous to [[velocity]]. Since the time derivative of displacement (distance) is equal to velocity and the time derivative of momentum equals force, quantities in other energy domains with similar differential relationships are referred to as ''generalized displacement,'' ''generalized velocity,'' ''generalized momentum,'' and ''generalized force.'' In the electrical domain, the generalized displacement is [[electric charge|charge]], which explains the Maxwellians' use of the term ''displacement''.<ref>Gupta, p.18</ref>

Since elastance is defined as the ratio of voltage to charge, its analogue in other energy domains is the ratio of a generalized force to a generalized displacement. Therefore, elastance can be defined in any energy domain. The term ''elastance'' is used in the formal analysis of systems involving multiple energy domains, such as in [[bond graph]]s.<ref>Vieil, p.47</ref>

{| class="wikitable"
|+Definition of elastance in different energy domains<ref>{{multiref|Busch-Vishniac, pp.18–19|Regtien, p.21|Borutzky, p.27}}</ref>
|-
!Energy domain||Generalized force||Generalized displacement||Name for elastance
|-
|[[Electrical science|Electrical]]||Voltage||Charge||Elastance
|-
|[[Mechanics|Mechanical (translational)]]||Force||Displacement||Stiffness/elastance<ref>Horowitz, p.29</ref>
|-
|[[Rotational mechanics|Mechanical (rotational)]]||[[Torque]]||[[Angle]]||Rotational stiffness/elastance<br/>Moment of stiffness/elastance<br/>Torsional stiffness/elastance<ref>{{multiref|Vieil, p.361|Tschoegl, p.76}}</ref>
|-
|[[Fluid dynamics|Fluid]]||Pressure||Volume||Elastance
|-
|[[Thermodynamics|Thermal]]||Temperature difference||[[Entropy]]||Warming factor<ref>Fuchs, p.149</ref>
|-
|[[Magnetostatics|Magnetic]]||[[Magnetomotive force]] (mmf)||[[Magnetic flux]]||[[Permeance]]<ref>Karapetoff, p.9</ref>
|-
|[[Chemistry|Chemical]]||[[Chemical potential]]||[[Mole (unit)|Molar amount]]||Inverse chemical capacitance<ref>Hillert, pp.120–121</ref>
|}

===Other analogies===
Maxwell's analogy is not the only method for constructing analogies between mechanical and electrical systems. There are multiple ways to create such analogies. One commonly used system is the [[mobility analogy]]. In this analogy, force is mapped to current rather than voltage. As a result, electrical impedance no longer corresponds directly to mechanical impedance, and similarly, electrical elastance no longer corresponds to mechanical elastance.<ref>Busch-Vishniac, p.20</ref>

== See also ==
* [[Compliance (physiology)]]
* [[Elasticity (physics)]]

==References==
{{reflist|23em}}

==Bibliography==
* Blake, F. C., [https://ieeexplore.ieee.org/document/6593059/ "On electrostatic transformers and coupling coefficients"], ''Journal of the American Institute of Electrical Engineers'', vol.&nbsp; 40, no.&nbsp;1, pp.&nbsp;23–[https://books.google.com/books?id=Xo9MAAAAYAAJ&q=%22He+has+called+the+reciprocal+of+the+farad%22 29], January 1921
* Borutzky, Wolfgang, ''Bond Graph Methodology, ''Springer, 2009 {{ISBN|1848828829}}.
* Busch-Vishniac, Ilene J., ''Electromechanical Sensors and Actuators'', Springer Science & Business Media, 1999 {{ISBN|038798495X}}.
* Camara, John A., ''Electrical and Electronics Reference Manual for the Electrical and Computer PE Exam'', Professional Publications, 2010 {{ISBN|159126166X}}.
* Cauer, E.; Mathis, W.; Pauli, R., "[http://www.cs.princeton.edu/courses/archive/fall03/cs323/links/cauer.pdf Life and Work of Wilhelm Cauer (1900 – 1945)]", ''Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000)'', Perpignan, June, 2000. 
* Enderle, John; Bronzino, Joseph, ''Introduction to Biomedical Engineering'', Academic Press, 2011 {{ISBN|0080961215}}.
* Fuchs, Hans U., ''The Dynamics of Heat: A Unified Approach to Thermodynamics and Heat Transfer'', Springer Science & Business Media, 2010 {{ISBN|1441976043}}.
* Gupta, S. C., ''Thermodynamics'', Pearson Education India, 2005 {{ISBN|813171795X}}.
* Heaviside, Oliver, ''Electromagnetic Theory: Volume I'', Cosimo, 2007 {{ISBN|1602062714}} (first published 1893).
* Hillert, Mats, ''Phase Equilibria, Phase Diagrams and Phase Transformations'', Cambridge University Press, 2007 {{ISBN|1139465864}}.
* Horowitz, Isaac M., ''Synthesis of Feedback Systems'', Elsevier, 2013 {{ISBN|1483267709}}.
* Howe, G. W. O., [http://digital-library.theiet.org/content/journals/10.1049/jiee-1.1931.0203 "The nomenclature of the fundamental concepts of electrical engineering"], ''Journal of the Institution of Electrical Engineers'', vol.&nbsp; 70, no.&nbsp; 420, pp.&nbsp;54–61, December 1931.
* Jerrard, H. G., ''A Dictionary of Scientific Units'', Springer, 2013 {{ISBN|9401705712}}.
* Kennelly, Arthur E.; Kurokawa, K., "[https://archive.org/stream/proceedingsofam56amer#page/40/mode/2up Acoustic impedance and its measurement"], ''Proceedings of the American Academy of Arts and Sciences'', vol.&nbsp; 56, no.&nbsp; 1, pp.&nbsp;3–42, 1921.
* Klein, H. Arthur, ''The Science of Measurement: A Historical Survey'', Courier Corporation, 1974 {{ISBN|0486258394}}.
* Miles, Robert; Harrison, P.; Lippens, D., ''Terahertz Sources and Systems'', Springer, 2012 {{ISBN|9401008248}}.
* Mills, Jeffrey P., ''Electro-magnetic Interference Reduction in Electronic Systems'', PTR Prentice Hall, 1993 {{ISBN|0134639022}}.
* Mitchell, John Howard, ''Writing for Professional and Technical Journals'', Wiley, 1968 {{OCLC|853309510}}
* Peek, Frank William, ''Dielectric Phenomena in High Voltage Engineering'', Watchmaker Publishing, 1915 (reprint) {{ISBN|0972659668}}.
* Regtien, Paul P. L., ''Sensors for Mechatronics'', Elsevier, 2012 {{ISBN|0123944090}}.
* van der Tweel, L. H.; Verburg, J., "Physical concepts", in  Reneman, Robert S.; Strackee, J., ''Data in Medicine: Collection, Processing and Presentation'', Springer Science & Business Media, 2012 {{ISBN|9400993099}}.
* Tschoegl, Nicholas W., ''The Phenomenological Theory of Linear Viscoelastic Behavior'', Springer, 2012 {{ISBN|3642736025}}.
* Vieil, Eric, ''Understanding Physics and Physical Chemistry Using Formal Graphs'', CRC Press, 2012 {{ISBN|1420086138}}
* Yavetz, Ido, ''[https://books.google.com/books?id=JgXZQTeYby8C From Obscurity to Enigma: The Work of Oliver Heaviside, 1872–1889]'', Springer, 2011 {{ISBN|3034801777}}.

[[Category:Electrostatics]]
[[Category:Physical quantities]]
[[Category:Electromagnetism]]
[[Category:Capacitance]]
[[Category:Electromagnetic quantities]]

[[ca:Elastància (electricitat)]]