Elastance
Electrical elastance is the
The term elastance was coined by
Usage
The definition of capacitance (C) is the charge (Q) stored per unit voltage (V).
Elastance (S) is the reciprocal of capacitance, thus,[1]
Expressing the values of
where L, R, S and Z are the network loop matrices of inductance, resistance, elastance and impedance respectively and s is
Elastance is also used in
Units
The
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
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
For a period in the nineteenth and early-twentieth centuries, some authors followed Heaviside in the use of elastance and elastivity.[14] 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, he chose elastivity rather than elasticity. This avoids having to write electrical elasticity to disambiguate it from mechanical elasticity.[15]
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.[16] 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.[17]
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 analogues. 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 containing 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.[18]
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 the term displacement.[19]
Since elastance is the ratio of voltage over charge, then it follows that the analogue 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.[20]
Energy domain | Generalised force | Generalised displacement | Name for elastance |
---|---|---|---|
Electrical |
Voltage | Charge | Elastance |
Mechanical (translational) | Force | Displacement | Stiffness/elastance[22] |
Mechanical (rotational) |
Torque | Angle | Rotational stiffness/elastance Moment of stiffness/elastance Torsional stiffness/elastance[23] |
Fluid | Pressure | Volume | Elastance |
Thermal | Temperature difference | Entropy | Warming factor[24] |
Magnetic | Magnetomotive force (mmf) | Magnetic flux | Permeance[25] |
Chemical | Chemical potential | Molar amount | Inverse chemical capacitance[26] |
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.[27]
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
- ^ Howe, p.60
- ^ Heaviside, p.268
- ^ Yavetz, pp.150–151
- ^ 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
- Borutzky, 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, no. 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.
- 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, vol. 70, no. 420, pp. 54–61, December 1931.
- Jerrard, H. G., A Dictionary of Scientific Units, Springer, 2013 ISBN 9401705712.
- Kennelly, Arthur 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.