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'''Electrical elastance''' is the [[multiplicative inverse|inverse]] of [[capacitance]]. The [[SI]] unit of elastance is the inverse [farad]] (F<sup>-1</sup>). |
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{{About|the quantity in electronics|the inverse of compliance in physiology|Compliance (physiology)}} |
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{{Unreferenced|date=July 2008}} |
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'''Electrical elastance''' is the inverse of [[capacitance]]. The [[SI]] unit is the reciprocal [[farad]]. Although the term ''[[daraf]]'' (farad spelled backwards) is sometimes used, this is not approved by SI. Electrical [[RLC circuit|inductor-capacitor-resistor circuit]]s follow differential equations that can be interpreted as representing mass-spring-damper systems. If voltage is taken to be force and current velocity, elastance corresponds to the elastic constant of a [[spring (device)|spring]]. |
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The term ''elastance'' was coined by [[Oliver Heaviside]] through the analogy of a capacitor as a spring. It is not widely used by electrical and electronic engineers. The value of [[capacitor]]s is invariably specified in units of capacitance rather than inverse capacitance. However, it is used in theoretical work in network analysis and has some niche applications at [[microwave]] frequencies. |
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==See also== |
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* [[harmonic oscillator]] |
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* [[SI electromagnetism units]] |
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The term is used for analogous quantities in some other energy domains. It maps to [[stiffness]] in the mechanical domain, and is the inverse of [[Compliance (physiology)|compliance]] in the fluid flow domain, especially in [[physiology]]. It is also the name of the generalised quantitiy in [[bond graph]] analysis and other schemes analysing systems across multiple domains. |
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==Usage== |
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The definition of capacitance (''C'') is the charge (''Q'') stored per unit voltage (''V''). |
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:<math> C = {Q \over V} \ , </math> |
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Elastance (''S'') is the [[multiplicative inverse|inverse]] of capacitance, thus, |
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:<math> S = {V \over Q} \ . </math><ref>Camara, p. 16-11</ref> |
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Expressing the values of [[capacitor]]s as elastance is not done much by practical electrical engineers, although it is sometimes convenient for capacitors in series. The total elastance is simply the sum of the individual elastances in that case. However, it is used by network theorists in their analysis. One advantage is that an increase in elastance increases [[electrical impedance|impedance]]. This is in the same direction as the other two basic passive [[Electrical element|elements]], [[resistance]] and [[inductance]]. An example of the use of elastance can be found in the work of [[Wilhelm Cauer]]. On his path to founding [[network synthesis]] he formed the [[mesh analysis|loop matrix]] '''A''', |
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:<math>\mathbf{A}= s^2 \mathbf{L} + s \mathbf{R} + \mathbf{S} = s \mathbf{Z}</math> |
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where '''L''', '''R''', '''S''' and '''Z''' are the network loop matrices of inductance, resistance and elastance respectively and ''s'' is [[complex frequency]]. This expression would be significantly more complicated if Cauer had tried to use a matrix of capacitances instead of elastances. The use of elastance here is merely for mathematical convenience, in much the same way as mathematicians use [[radian]]s rather than the more common units for angles.<ref>Cauer, Mathis & Pauli, p. 4<br/>The symbols in Cauer's expression have been modified for consistency within this article and with modern practice</ref> |
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Elastance is also used in [[microwave engineering]]. In this field [[varactor diode]]s are used as a voltage variable capacitor in [[frequency multiplier]]s and variable [[electronic filter|filters]]. These diodes store a charge in their [[p-n junction|junction]] when [[reverse bias]]ed which is the source of the capacitor effect. The slope of the voltage-stored charge curve is called ''differential elastance'' in this field.<ref>Miles, Harrison & Lippens, pp. 29-30</ref> |
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==Units== |
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The [[SI]] unit of elastance is the reciprocal [[farad]] (F<sup>-1</sup>). The term ''[[daraf]]'' is sometimes used for this unit, but it is not approved by SI and its use is discouraged.<ref>{{multiref|Michell, p. 168|Mills, p. 17}}</ref> The term is formed by writing ''farad'' backwards, in much the same way as the unit ''[[mho]]'' (unit of conductance, also not approved by SI) is formed by writing ''[[ohm]]'' backwards.<ref>Klein, p. 466</ref> |
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The term ''daraf'' was coined by [[Arthur E. Kennelly]]. He used it from at least 1920.<ref>{{multiref|Kennelly & Kurokawa, p. 41|Blake, p. 29|Jerrard, p. 33}}</ref> |
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==History== |
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The terms ''elastance'' and ''elastivity'' were coined by [[Oliver Heaviside]] in 1886.<ref>Howe, p. 60</ref> Heaviside coined a great many of the terms used in [[circuit analysis]] today, such as [[electrical impedance|impedance]], [[inductance]], [[admittance]], [[electrical conductance|conductance]] and so on. Heaviside's terminology followed the model of [[resistance]] and [[resistivity]] with the ''-ance'' ending used for [[extensive properties]] and the ''-ivity'' ending used for [[intensive properties]]. The extensive properties are used in circuit analysis (they are the "values" of components) and the intensive properties are used in [[field (physics)|field analysis]]. Heaviside's nomenclature was designed to highlight the connection between corresponding quantities in field and circuit.<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 put it, |
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{{blockquote|Permittivity gives rise to permittance, and elastivity to elastance.<ref>Heaviside, p. 28</ref>|Oliver Heaviside}} |
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Here, ''permittance'' is Heaviside's term for capacitance. He did not like any term that suggested that a [[capacitor]] was a container for holding charge. The terms current in his time for a capacitor were ''condenser'' (suggesting that the "electric fluid" could be condensed out) and ''leyden''<ref>Heaviside, p. 268</ref> after the [[Leyden jar]], an early form of capacitor, also suggesting some sort of storage. Heaviside preferred the analogy of a mechanical spring under compression, hence his preference for terms that suggested a property of a spring. This preference was a result of Heaviside following [[James Clerk Maxwell]]'s view of electric current, or at least, Heaviside's interpretation of it. In this view, electric current is a flow caused by the [[electromotive force]] and is the analogue of [[velocity]] caused by a mechanical [[force]]. At the capacitor, this current causes a "[[displacement current|displacement]]" whose rate of change is equal to the current. The displacement is viewed as an electric [[deformation (mechanics)#Strain|strain]], like a mechanical strain in a compressed spring. The existence of a flow of physical charge is denied, as is the build up of charge on the capacitor plates. This is replaced with the concept of [[divergence]] of the displacement field at the plates, which is numerically equal to the charge collected on the plates in the charge flow view.<ref>Yavetz, pp. 150-151</ref> |
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For a period in the nineteenth and early-twentieth centuries, some authors followed Heaviside in the use of ''elastance'' and ''elastivity''.<ref>See, for instance, Peek, p. 215, writing in 1915</ref> Today, the reciprocal quantities ''capacitance'' and ''permittivity'' are almost universally preferred by electrical engineers. However, elastance does still see some usage by theoretical writers. A further consideration in Heaviside's choice of these terms was a wish to distinguish them from mechanical terms. Thus ''elastivity'' rather than ''[[elasticity (physics)|elasticity]]''. This avoids having to write ''electrical elasticity'' to disambiguate it from ''mechanical elasticity''.<ref>Howe, p. 60</ref> |
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Heaviside carefully chose his terms to be unique to [[electromagnetism]], most especially avoiding commonality with [[mechanics]]. Ironically, many of his terms have subsequently been borrowed back into mechanics and other domains in order to name analogous properties. For instance, it is now necessary to distinguish ''[[electrical impedance]]'' from ''[[mechanical impedance]]'' in some contexts.<ref>van der Tweel & Verburg, pp. 16-20</ref> ''Elastance'' has also been borrowed back into mechanics for the analogous quantity by some authors, but often ''[[stiffness]]'' is the preferred term instead. However, ''elastance'' is widely used for the analogous property in the domain of [[fluid dynamics]], especially in the fields of [[biomedicine]] and [[physiology]].<ref>see for instance Enderle & Bronzino, pp. 197-201, especially equation 4.72</ref> |
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==Mechanical analogy== |
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[[Mechanical-electrical analogies]] are formed by comparing the mathematical description of the two systems. Quantities that appear in the same place in equations of the same form are called ''analogs''. There are two main reasons for forming such analogies. The first is to allow electrical phenomena to be explained in terms of the more familiar mechanical systems. For instance, an electrical [[RLC circuit|inductor-capacitor-resistor circuit]] has [[differential equation]]s of the same form as a mechanical mass-spring-damper system. In such cases the electrical domain is converted to the mechanical domain. The second, and more important, reason is to allow a system ontaining both mechanical and electrical parts to be analysed as a unified whole. This is of great benefit in the fields of [[mechatronics]] and [[robotics]]. In such cases the mechanical domain is most often converted to the electrical domain because [[Network analysis (electrical circuits)|network analysis]] in the electrical domain is highly developed.<ref>Busch-Vishniac, pp. 17-18</ref> |
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===The Maxwellian analogy=== |
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In the analogy developed by Maxwell, now known as the [[impedance analogy]], [[voltage]] is made analogous to [[force]]. The voltage of a source of electric power is still called [[electromotive force]] for this reason. Current is analogous to [[velocity]]. The [[time derivative]] of distance (displacement) is equal to velocity and the time derivative of momentum is equal to force. Quantities in other energy domains that are in this same differential relationship are called respectively ''generalised displacement'', ''generalised velocity'', ''generalised momentum'', and ''generalised force''. In the electrical domain, it can be seen that the generalised displacement is charge, explaining the Maxwellians' use of this term.<ref>Gupta, p. 18</ref> |
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Since elastance is the ratio of voltage over charge, then it follows that the analog of elastance in another energy domain is the ratio of a generalised force over a generalised displacement. Thus, an elastance can be defined in any energy domain. ''Elastance'' is used as the name of the generalised quantity in the formal analysis of systems with multiple energy domains, such as is done with [[bond graph]]s.<ref> Vieil, p. 47</ref> |
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{| class="wikitable" |
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|+Definition of elastance in different energy domains<ref>{{multiref|Busch-Vishniac, pp. 18-19|Regtien, p. 21|Borutzsky, p. 27}}</ref> |
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|- |
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!Energy domain||Generalised force||Generalised displacement||Name for elastance |
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|- |
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|Electrical||Voltage||Charge||Elastance |
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|- |
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|Mechanical (translational)||Force||Displacement||Stiffness/elastance<ref>Horowitz, p. 29</ref> |
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|- |
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|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> |
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|- |
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|Fluid||Pressure||Volume||Elastance |
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|- |
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|Thermal||Temperature difference||[[Entropy]]||Warming factor<ref>Fuchs, p. 149</ref> |
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|- |
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|Magnetic||[[Magnetomotive force]] (mmf)||[[Magnetic flux]]||[[Permeance]]<ref>Karapetoff, p. 9</ref> |
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|- |
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|Chemical||[[Chemical potential]]||[[Mole (unit)|Molar amount]]||Inverse chemical capacitance<ref>Hillert, pp. 120-121</ref> |
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|} |
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===Other analogies=== |
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Maxwell's analogy is not the only way that analogies can be constructed between mechanical and electrical systems. There are any number of ways to do this. One very common system is the [[mobility analogy]]. In this analogy force maps to current instead of voltage. Electrical impedance no longer maps to mechanical impedance, and likewise, electrical elastance no longer maps to mechanical elastance.<ref>Busch-Vishniac, p. 20</ref> |
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==References== |
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{{reflist|23em}} |
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==Bibliography== |
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* Blake, F. C., [http://ieeexplore.ieee.org/document/6593059/ "On electrostatic transformers and coupling coefficients"], ''Journal of the American Institute of Electrical Engineers'', vol. 40, iss. 1, pp. 23–[https://books.google.co.uk/books?id=Xo9MAAAAYAAJ&q=%22He+has+called+the+reciprocal+of+the+farad%22 29], January 1921 |
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* Borutzky, Wolfgang, ''Bond Graph Methodology, ''Springer, 2009 ISBN 1848828829. |
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* Busch-Vishniac, Ilene J., ''Electromechanical Sensors and Actuators'', Springer Science & Business Media, 1999 ISBN 038798495X. |
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* Camara, John A., ''Electrical and Electronics Reference Manual for the Electrical and Computer PE Exam'', Professional Publications, 2010 ISBN 159126166X. |
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* Cauer, E.; Mathis, W.; Pauli, R., "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. [http://www.cs.princeton.edu/courses/archive/fall03/cs323/links/cauer.pdf Retrieved online] 19 September 2008. |
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* Enderle, John; Bronzino, Joseph, ''Introduction to Biomedical Engineering'', Academic Press, 2011 ISBN 0080961215. |
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* Fuchs, Hans U., ''The Dynamics of Heat: A Unified Approach to Thermodynamics and Heat Transfer'', Springer Science & Business Media, 2010 ISBN 1441976043. |
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* Gupta, S. C., ''Thermodynamics'', Pearson Education India, 2005 ISBN 813171795X. |
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* Heaviside, Oliver, ''Electromagnetic Theory: Volume I'', Cosimo, 2007 ISBN 1602062714 (first published 1893). |
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* Hillert, Mats, ''Phase Equilibria, Phase Diagrams and Phase Transformations'', Cambridge University Press, 2007 ISBN 1139465864. |
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* Horowitz, Isaac M., ''Synthesis of Feedback Systems'', Elsevier, 2013 ISBN 1483267709. |
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* 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'', pp. 54-61, iss. 420, vol. 70, December 1931. |
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* Jerrard, H. G., ''A Dictionary of Scientific Units'', Springer, 2013 ISBN 9401705712. |
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* Kennelly, A. 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. 56, no. 1, pp. 3-42, 1921. |
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* Klein, H. Arthur, ''The Science of Measurement: A Historical Survey'', Courier Corporation, 1974 ISBN 0486258394. |
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* Miles, Robert; Harrison, P.; Lippens, D., ''Terahertz Sources and Systems'', Springer, 2012 ISBN 9401008248. |
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* Mills, Jeffrey P., ''Electro-magnetic Interference Reduction in Electronic Systems'', PTR Prentice Hall, 1993 ISBN 0134639022. |
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* Mitchell, John Howard, ''Writing for Professional and Technical Journals'', Wiley, 1968 {{OCLC|853309510}} |
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* Peek, Frank William, ''Dielectric Phenomena in High Voltage Engineering'', Watchmaker Publishing, 1915 (reprint) ISBN 0972659668. |
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* Regtien, Paul P. L., ''Sensors for Mechatronics'', Elsevier, 2012 ISBN 0123944090. |
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* 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. |
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* Tschoegl, Nicholas W., ''The Phenomenological Theory of Linear Viscoelastic Behavior'', Springer, 2012 ISBN 3642736025. |
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* Vieil, Eric, ''Understanding Physics and Physical Chemistry Using Formal Graphs'', CRC Press, 2012 ISBN 1420086138 |
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* Yavetz, Ido, ''From Obscurity to Enigma: The Work of Oliver Heaviside, 1872–1889'', Springer, 2011 ISBN 3034801777. |
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<!--Categories--> |
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[[Category:Electrostatics]] |
[[Category:Electrostatics]] |
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[[Category:Physical quantities]] |
[[Category:Physical quantities]] |
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[[Category:Concepts in physics]] |
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{{physics-stub}} |
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[[Category:Electromagnetism]] |
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[[ca:Elastància (electricitat)]] |
Revision as of 11:10, 18 December 2016
Electrical elastance is the inverse of capacitance. The SI unit of elastance is the inverse [farad]] (F-1).
The term elastance was coined by Oliver Heaviside through the analogy of a capacitor as a spring. It is not widely used by electrical and electronic engineers. The value of capacitors is invariably specified in units of capacitance rather than inverse capacitance. However, it is used in theoretical work in network analysis and has some niche applications at microwave frequencies.
The term is used for analogous quantities in some other energy domains. It maps to stiffness in the mechanical domain, and is the inverse of compliance in the fluid flow domain, especially in physiology. It is also the name of the generalised quantitiy in bond graph analysis and other schemes analysing systems across multiple domains.
Usage
The definition of capacitance (C) is the charge (Q) stored per unit voltage (V).
Elastance (S) is the inverse of capacitance, thus,
Expressing the values of capacitors as elastance is not done much by practical electrical engineers, although it is sometimes convenient for capacitors in series. The total elastance is simply the sum of the individual elastances in that case. However, it is used by network theorists in their analysis. One advantage is that an increase in elastance increases impedance. This is in the same direction as the other two basic passive elements, resistance and inductance. An example of the use of elastance can be found in the work of Wilhelm Cauer. On his path to founding network synthesis he formed the loop matrix A,
where L, R, S and Z are the network loop matrices of inductance, resistance and elastance respectively and s is complex frequency. This expression would be significantly more complicated if Cauer had tried to use a matrix of capacitances instead of elastances. The use of elastance here is merely for mathematical convenience, in much the same way as mathematicians use radians rather than the more common units for angles.[2]
Elastance is also used in microwave engineering. In this field varactor diodes are used as a voltage variable capacitor in frequency multipliers and variable filters. These diodes store a charge in their junction when reverse biased which is the source of the capacitor effect. The slope of the voltage-stored charge curve is called differential elastance in this field.[3]
Units
The SI unit of elastance is the reciprocal farad (F-1). The term daraf is sometimes used for this unit, but it is not approved by SI and its use is discouraged.[4] The term is formed by writing farad backwards, in much the same way as the unit mho (unit of conductance, also not approved by SI) is formed by writing ohm backwards.[5]
The term daraf was coined by Arthur E. Kennelly. He used it from at least 1920.[6]
History
The terms elastance and elastivity were coined by Oliver Heaviside in 1886.[7] Heaviside coined a great many of the terms used in circuit analysis today, such as impedance, inductance, admittance, conductance and so on. Heaviside's terminology followed the model of resistance and resistivity with the -ance ending used for extensive properties and the -ivity ending used for intensive properties. The extensive properties are used in circuit analysis (they are the "values" of components) and the intensive properties are used in field analysis. Heaviside's nomenclature was designed to highlight the connection between corresponding quantities in field and circuit.[8] 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 put it,
Permittivity gives rise to permittance, and elastivity to elastance.[9]
— Oliver Heaviside
Here, permittance is Heaviside's term for capacitance. He did not like any term that suggested that a capacitor was a container for holding charge. The terms current in his time for a capacitor were condenser (suggesting that the "electric fluid" could be condensed out) and leyden[10] after the Leyden jar, an early form of capacitor, also suggesting some sort of storage. Heaviside preferred the analogy of a mechanical spring under compression, hence his preference for terms that suggested a property of a spring. This preference was a result of Heaviside following James Clerk Maxwell's view of electric current, or at least, Heaviside's interpretation of it. In this view, electric current is a flow caused by the electromotive force and is the analogue of velocity caused by a mechanical force. At the capacitor, this current causes a "displacement" whose rate of change is equal to the current. The displacement is viewed as an electric strain, like a mechanical strain in a compressed spring. The existence of a flow of physical charge is denied, as is the build up of charge on the capacitor plates. This is replaced with the concept of divergence of the displacement field at the plates, which is numerically equal to the charge collected on the plates in the charge flow view.[11]
For a period in the nineteenth and early-twentieth centuries, some authors followed Heaviside in the use of elastance and elastivity.[12] Today, the reciprocal quantities capacitance and permittivity are almost universally preferred by electrical engineers. However, elastance does still see some usage by theoretical writers. A further consideration in Heaviside's choice of these terms was a wish to distinguish them from mechanical terms. Thus elastivity rather than elasticity. This avoids having to write electrical elasticity to disambiguate it from mechanical elasticity.[13]
Heaviside carefully chose his terms to be unique to electromagnetism, most especially avoiding commonality with mechanics. Ironically, many of his terms have subsequently been borrowed back into mechanics and other domains in order to name analogous properties. For instance, it is now necessary to distinguish electrical impedance from mechanical impedance in some contexts.[14] Elastance has also been borrowed back into mechanics for the analogous quantity by some authors, but often stiffness is the preferred term instead. However, elastance is widely used for the analogous property in the domain of fluid dynamics, especially in the fields of biomedicine and physiology.[15]
Mechanical analogy
Mechanical-electrical analogies are formed by comparing the mathematical description of the two systems. Quantities that appear in the same place in equations of the same form are called analogs. There are two main reasons for forming such analogies. The first is to allow electrical phenomena to be explained in terms of the more familiar mechanical systems. For instance, an electrical inductor-capacitor-resistor circuit has differential equations of the same form as a mechanical mass-spring-damper system. In such cases the electrical domain is converted to the mechanical domain. The second, and more important, reason is to allow a system ontaining both mechanical and electrical parts to be analysed as a unified whole. This is of great benefit in the fields of mechatronics and robotics. In such cases the mechanical domain is most often converted to the electrical domain because network analysis in the electrical domain is highly developed.[16]
The Maxwellian analogy
In the analogy developed by Maxwell, now known as the impedance analogy, voltage is made analogous to force. The voltage of a source of electric power is still called electromotive force for this reason. Current is analogous to velocity. The time derivative of distance (displacement) is equal to velocity and the time derivative of momentum is equal to force. Quantities in other energy domains that are in this same differential relationship are called respectively generalised displacement, generalised velocity, generalised momentum, and generalised force. In the electrical domain, it can be seen that the generalised displacement is charge, explaining the Maxwellians' use of this term.[17]
Since elastance is the ratio of voltage over charge, then it follows that the analog of elastance in another energy domain is the ratio of a generalised force over a generalised displacement. Thus, an elastance can be defined in any energy domain. Elastance is used as the name of the generalised quantity in the formal analysis of systems with multiple energy domains, such as is done with bond graphs.[18]
Energy domain | Generalised force | Generalised displacement | Name for elastance |
---|---|---|---|
Electrical | Voltage | Charge | Elastance |
Mechanical (translational) | Force | Displacement | Stiffness/elastance[20] |
Mechanical (rotational) | Torque | Angle | Rotational stiffness/elastance Moment of stiffness/elastance Torsional stiffness/elastance[21] |
Fluid | Pressure | Volume | Elastance |
Thermal | Temperature difference | Entropy | Warming factor[22] |
Magnetic | Magnetomotive force (mmf) | Magnetic flux | Permeance[23] |
Chemical | Chemical potential | Molar amount | Inverse chemical capacitance[24] |
Other analogies
Maxwell's analogy is not the only way that analogies can be constructed between mechanical and electrical systems. There are any number of ways to do this. One very common system is the mobility analogy. In this analogy force maps to current instead of voltage. Electrical impedance no longer maps to mechanical impedance, and likewise, electrical elastance no longer maps to mechanical elastance.[25]
References
- ^ Camara, p. 16-11
- ^ Cauer, Mathis & Pauli, p. 4
The symbols in Cauer's expression have been modified for consistency within this article and with modern practice - ^ Miles, Harrison & Lippens, pp. 29-30
- ^
- Michell, p. 168
- Mills, p. 17
- ^ Klein, p. 466
- ^
- Kennelly & Kurokawa, p. 41
- Blake, p. 29
- Jerrard, p. 33
- ^ Howe, p. 60
- ^ Yavetz, p. 236
- ^ Heaviside, p. 28
- ^ Heaviside, p. 268
- ^ Yavetz, pp. 150-151
- ^ See, for instance, Peek, p. 215, writing in 1915
- ^ Howe, p. 60
- ^ van der Tweel & Verburg, pp. 16-20
- ^ see for instance Enderle & Bronzino, pp. 197-201, especially equation 4.72
- ^ Busch-Vishniac, pp. 17-18
- ^ Gupta, p. 18
- ^ Vieil, p. 47
- ^
- Busch-Vishniac, pp. 18-19
- Regtien, p. 21
- Borutzsky, p. 27
- ^ Horowitz, p. 29
- ^
- Vieil, p. 361
- Tschoegl, p. 76
- ^ Fuchs, p. 149
- ^ Karapetoff, p. 9
- ^ Hillert, pp. 120-121
- ^ Busch-Vishniac, p. 20
Bibliography
- Blake, F. C., "On electrostatic transformers and coupling coefficients", Journal of the American Institute of Electrical Engineers, vol. 40, iss. 1, pp. 23–29, 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., "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. Retrieved online 19 September 2008.
- 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., "The nomenclature of the fundamental concepts of electrical engineering", Journal of the Institution of Electrical Engineers, pp. 54-61, iss. 420, vol. 70, December 1931.
- Jerrard, H. G., A Dictionary of Scientific Units, Springer, 2013 ISBN 9401705712.
- Kennelly, A. E.; Kurokawa, K., "Acoustic impedance and its measurement", Proceedings of the American Academy of Arts and Sciences, vol. 56, no. 1, pp. 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, From Obscurity to Enigma: The Work of Oliver Heaviside, 1872–1889, Springer, 2011 ISBN 3034801777.