Conventional electrical unit

A conventional electrical unit (or conventional unit where there is no risk of ambiguity) is a unit of measurement in the field of electricity which is based on the so-called "conventional values" of the Josephson constant and the von Klitzing constant agreed by the International Committee for Weights and Measures (CIPM) in 1988. These units are very similar in scale to their corresponding SI units, but are not identical because of their different definition. They are distinguished from the corresponding SI units by setting the symbol in italic typeface and adding a subscript "90" – e.g., the conventional volt has the symbol V₉₀ – as they came into international use on 1 January 1990.

This system was developed to increase the precision of measurements: The Josephson and von Klitzing constants can be realized with great precision, repeatability and ease. The conventional electrical units have achieved acceptance as an international standard and are commonly used outside of the physics community in both engineering and industry.

The conventional electrical units are "quasi-natural" in the sense that they are completely and exactly defined in terms of fundamental physical constants. They are the first set of measurement units to be defined in this way, and as such, represent a significant step towards using "natural" fundamental physics for practical measurement purposes. However, the conventional electrical units are unlike other systems of natural units in that some physical constants are not set to unity but rather set to fixed numerical values that are very close (but not precisely the same) to those in the common SI system of units.

Four significant steps were taken in the last half century to increase the precision and utility of measurement units. In 1967 the Thirteenth General Conference on Weights and Measures (CGPM) defined the second of atomic time in the International System of Units as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom. In 1983, the seventeenth CGPM redefined the metre in terms of the second and the speed of light, thus fixing the speed of light at exactly 299,792,458 m/s. And in 1990, the eighteenth CGPM adopted conventional values for the Josephson constant and the von Klitzing constant, fixing the conventional Josephson constant at exactly 483,597.9 ×10⁹ Hz/"V", and the conventional von Klitzing constant at exactly 25 812.807 "Ω" (again, these volts and ohms are not precisely the same as the SI definitions but very nearly equivalent).

Definition

Conventional electrical units are based on defined values of the Josephson constant and the von Klitzing constant, which allow practical measurements of electromotive force and electrical resistance respectively. ^[1]

Constant	Conventional (defined) value (CIPM, 1988)	Empirical value (in SI units) (CODATA, 2014^[1])
Josephson constant	K_J–90 = 483597.9GHz/V	K_J = 483597.8525(30)GHz/V
von Klitzing constant	R_K–90 = 25812.807Ω	R_K = 25812.8074555(59)Ω

The conventional volt, V₉₀, is the electromotive force (or electric potential difference) measured against a Josephson effect standard using the defined value of the Josephson constant, K_J–90. See Josephson voltage standard.
The conventional ohm, Ω₉₀, is the electrical resistance measured against a quantum Hall effect standard using the defined value of the von Klitzing constant, R_K–90.
Other conventional electrical units are defined by the normal physical relationships, as in the conversion table below.

Conversion to SI units

Unit	Definition	SI equivalent (CODATA 2006)
conventional volt	see above	V₉₀ = (K_J–90/K_J) V = [1 + 1.9(2.5)×10⁻⁸] V
conventional ohm	see above	Ω₉₀ = (R_K/R_K–90) Ω = [1 + 2.159(68)×10⁻⁸] Ω
conventional ampere	A₉₀ = V₉₀/Ω₉₀	A₉₀ = [1 − 0.3(2.5)×10⁻⁸] A
conventional coulomb	C₉₀ = A₉₀ s = s V₉₀/Ω₉₀	C₉₀ = [1 − 0.3(2.5)×10⁻⁸] C
conventional watt	W₉₀ = A₉₀V₉₀ = V₉₀²/Ω₉₀	W₉₀ = [1 + 1.6(5.0)×10⁻⁸] W
conventional farad	F₉₀ = C₉₀/V₉₀ = s/Ω₉₀	F₉₀ = [1 − 2.159(68)×10⁻⁸] F
conventional henry	H₉₀ = Ω₉₀ s	H₉₀ = [1 + 2.159(68)×10⁻⁸] H

Comparison with natural units

See also: Natural units

Conventional electrical units can be thought of as a scaled version of a system of natural units defined as

c=e=\hbar =1\

having consequence:

{\frac {1}{4\pi \epsilon _{0}}}={\frac {\mu _{0}}{4\pi }}=\alpha \

This is a more general (or less specific) version of either the particle physics "natural units" or the quantum chromodynamical system of units but that no unit mass is fixed. Like n.u. or QCD units, with conventional electrical units any observed variation over space or time in the value of the fine-structure constant, α, is attributed to variation in the Coulomb constant or vacuum permittivity or, since the speed of light, c, is fixed, as a variation in the vacuum permeability.

The following table provides a comparison of conventional electrical units with other natural unit systems:

Quantity / Symbol	Planck	Stoney	Schrödinger	Atomic	Electronic	Conventional Electrical Units
speed of light in vacuum $c\,$	$1\,$	$1\,$	$\frac{1}{\alpha} \$	$\frac{1}{\alpha} \$	$1\,$	$299792458\$
Planck's constant $h\,$	$2\pi \,$	${\frac {2\pi }{\alpha }}\$	$2\pi \,$	$2\pi \,$	${\frac {2\pi }{\alpha }}\$	${\frac {4\times 10^{{-18}}}{(25812.807)(483597.9)^{2}}}\$
reduced Planck's constant $\hbar=\frac{h}{2 \pi}$	$1\,$	$\frac{1}{\alpha} \$	$1\,$	$1\,$	$\frac{1}{\alpha} \$	${\frac {2\times 10^{{-18}}}{\pi (25812.807)(483597.9)^{2}}}\$
elementary charge $e\,$	$\sqrt{\alpha} \,$	$1\,$	$1\,$	$1\,$	$1\,$	${\frac {2\times 10^{{-9}}}{(25812.807)(483597.9)}}\$
Josephson constant $K_{J}={\frac {2e}{h}}\,$	$\frac{\sqrt{\alpha}}{\pi} \,$	$\frac{\alpha}{\pi} \,$	$\frac{1}{\pi} \,$	$\frac{1}{\pi} \,$	$\frac{\alpha}{\pi} \,$	$483597.9\times 10^{9}\,$
von Klitzing constant $R_{K}={\frac {h}{e^{2}}}\,$	$\frac{2\pi}{\alpha} \,$	$\frac{2\pi}{\alpha} \,$	$2\pi \,$	$2\pi \,$	$\frac{2\pi}{\alpha} \,$	$25812.807\,$
characteristic impedance of vacuum $Z_{0}=2\alpha R_{K}\,$	$4\pi \,$	$4\pi \,$	$4\pi \alpha \,$	$4\pi \alpha \,$	$4\pi \,$	$2\alpha (25812.807)\,$
electric constant (vacuum permittivity) $\varepsilon _{0}={\frac {1}{Z_{0}c}}\,$	${\frac {1}{4\pi }}\,$	${\frac {1}{4\pi }}\,$	${\frac {1}{4\pi }}\,$	${\frac {1}{4\pi }}\,$	${\frac {1}{4\pi }}\,$	${\frac {1}{2\alpha (25812.807)(299792458)}}\$
magnetic constant (vacuum permeability) $\mu _{0}={\frac {Z_{0}}{c}}\,$	$4\pi \,$	$4\pi \,$	$4\pi \alpha ^{2}\,$	$4\pi \alpha ^{2}\,$	$4\pi \,$	${\frac {2\alpha (25812.807)}{299792458}}\$
Newtonian constant of gravitation $G\,$	$1\,$	$1\,$	$1\,$	$-\,$	$-\,$	$-\,$
electron mass $m_{e}\,$	$-\,$	$-\,$	$-\,$	$1\,$	$1\,$	$-\,$
Hartree energy $E_{h}=\alpha ^{2}m_{e}c^{2}\,$	$-\,$	$-\,$	$-\,$	$1\,$	$\alpha ^{2}\,$	$-\,$
Rydberg constant $R_{\infty }={\frac {E_{h}}{2hc}}\,$	$-\,$	$-\,$	$-\,$	${\frac {\alpha }{4\pi }}\,$	${\frac {\alpha ^{3}}{4\pi }}\,$	$-\,$
caesium ground state hyperfine transition frequency	$-\,$	$-\,$	$-\,$	$-\,$	$-\,$	$9\ 192\ 631\ 770\,$

References

Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006". Reviews of Modern Physics. 80 (2): 633–730. arXiv:0801.0028. Bibcode:2008RvMP...80..633M. doi:10.1103/RevModPhys.80.633.

1 2 Mohr, Peter J.; Newell, David B.; Taylor, Barry N. (2015). "CODATA recommended values of the fundamental physical constants: 2014". Zenodo. arXiv:1507.07956. doi:10.5281/zenodo.22826.

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