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, the von Klitzing constant agreed by the International Committee for Weights and Measures (CIPM) in 1988, as well as Δν_Cs used to define the second. These units are very similar in scale to their corresponding SI units, but are not identical because of the different values used for the constants. 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, and are exactly defined in terms of the universal constants e and h. The conventional electrical units represent a significant step towards using "natural" fundamental physics for practical measurement purposes. They achieved acceptance as an international standard in parallel to the SI system of units and are commonly used outside of the physics community in both engineering and industry. Addition of the constant c would be needed to define units for all dimensions used in physics, as in the SI.

The SI system made the transition to equivalent definitions 29 years later but with values of the constants defined to match the old SI units more precisely. Consequently, the conventional electrical units slightly differ from the corresponding SI units, now with exactly defined ratios.

Historical developmentEdit

Several significant steps have been 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 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom.^[1]
• 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 299792458 m/s.^[2]
• In 1988, the CIPM recommended adoption of conventional values for the Josephson constant as exactly K_J-90 = 483597.9×10⁹ Hz/V^[3] and for the von Klitzing constant as exactly R_K-90 = 25812.807 Ω^[4] as of 1 January 1990.
• In 1991, the eighteenth CGPM noted the conventional values for the Josephson constant and the von Klitzing constant.^[5]
• In 2000, the CIPM approved the use of the quantum Hall effect, with the value of R_K-90 to be used to establish a reference standard of resistance.^[6]
• In 2018, the twenty-sixth CGPM resolved to abrogate the conventional values of the Josephson and von Klitzing constants with the 2019 redefinition of SI base units.^[7]

DefinitionEdit

Conventional electrical units are based on defined values of the caesium-133 hyperfine transition frequency, Josephson constant and the von Klitzing constant, the first two which allow a very precise practical measurement of time and electromotive force, and the last which allows a very precise practical measurement of electrical resistance.^[8]

Constant	Conventional exact value (CIPM, 1988; until 2018)	Empirical value (in SI units) (CODATA, 2014[8])	Exact value (SI units, 2019)
133Cs hyperfine transition frequency	Δν(133Cs)hfs = 9192631770 Hz		Δν(133Cs)hfs = 9192631770 Hz[9]
Josephson constant	KJ-90 = 483597.9 GHz/V[10]	KJ = 483597.8525(30) GHz/V	KJ = 2 × 1.602176634×10−19 C/6.62607015×10−34 J⋅s
von Klitzing constant	RK-90 = 25812.807 Ω[11]	RK = 25812.8074555(59) Ω	RK = 6.62607015×10−34 J⋅s/(1.602176634×10−19 C)2

• 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; that is, by the relation K_J = 483597.9 GHz/V₉₀. 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; that is, by the relation R_K = 25812.807 Ω₉₀.
• Other conventional electrical units are defined by the normal relationships between units paralleling those of SI, as in the conversion table below.

Conversion to SI unitsEdit

Unit	Symbol	Definition	Related to SI	SI value (CODATA 2014)	SI value (2019)
conventional volt	V90	see above	KJ-90/KJ V	1.0000000983(61) V	1.00000010666... V[12]
conventional ohm	Ω90	see above	RK/RK-90 Ω	1.00000001765(23) Ω	1.00000001779... Ω[13]
conventional ampere	A90	V90/Ω90	KJ-90/KJ⋅RK-90/RK A	1.0000000806(61) A	1.00000008887... A[14]
conventional coulomb	C90	s⋅A90 = s⋅V90/Ω90	KJ-90/KJ⋅RK-90/RK C	1.0000000806(61) C	1.00000008887... C[15]
conventional watt	W90	A90V90 = V902/Ω90	(KJ-90/KJ)2  ⋅RK-90/RK W	1.000000179(12) W	1.00000019553... W[16]
conventional farad	F90	C90/V90 = s/Ω90	RK-90/RK F	0.99999998235(23) F	0.99999998220... F[17]
conventional henry	H90	s⋅Ω90	RK/RK-90 H	1.00000001765(23) H	1.00000001779... H[18]

The 2019 redefinition of SI base units defines all these units in a way that fixes the numeric values of K_J, R_K and Δν_Cs exactly, albeit with values of the first two that differ slightly from the conventional values. Consequently, these conventional units all have known exact values in terms of the redefined SI units. Because of this, there is no accuracy benefit from maintaining the conventional values.

Comparison with natural unitsEdit

See also: Natural units

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

${\displaystyle c=e=\hbar =1.}$

This is a more general (or less specific) version of either the particle physics "natural units" or the quantum chromodynamical system of units but without fixing unit mass.

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

Quantity		Other systems					Conventional electrical units
Name	Symbol	Planck	Stoney	Schrödinger	Hartree	Electronic
Speed of light in vacuum	${\displaystyle c}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle {\frac {1}{\alpha }}}$	${\displaystyle {\frac {1}{\alpha }}}$	${\displaystyle 1}$	${\displaystyle 299792458}$
Planck's constant	${\displaystyle h}$	${\displaystyle 2\pi }$	${\displaystyle {\frac {2\pi }{\alpha }}}$	${\displaystyle 2\pi }$	${\displaystyle 2\pi }$	${\displaystyle {\frac {2\pi }{\alpha }}}$	${\displaystyle {\frac {4\times 10^{-18}}{(25812.807)(483597.9)^{2}}}}$
Reduced Planck's constant	${\displaystyle \hbar ={\frac {h}{2\pi }}}$	${\displaystyle 1}$	${\displaystyle {\frac {1}{\alpha }}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle {\frac {1}{\alpha }}}$	${\displaystyle {\frac {2\times 10^{-18}}{\pi (25812.807)(483597.9)^{2}}}}$
Elementary charge	${\displaystyle e}$	${\displaystyle {\sqrt {\alpha }}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle {\frac {2\times 10^{-9}}{(25812.807)(483597.9)}}}$
Josephson constant	${\displaystyle K_{\text{J}}={\frac {2e}{h}}}$	${\displaystyle {\frac {\sqrt {\alpha }}{\pi }}}$	${\displaystyle {\frac {\alpha }{\pi }}}$	${\displaystyle {\frac {1}{\pi }}}$	${\displaystyle {\frac {1}{\pi }}}$	${\displaystyle {\frac {\alpha }{\pi }}}$	${\displaystyle 483597.9\times 10^{9}}$
von Klitzing constant	${\displaystyle R_{\text{K}}={\frac {h}{e^{2}}}}$	${\displaystyle {\frac {2\pi }{\alpha }}}$	${\displaystyle {\frac {2\pi }{\alpha }}}$	${\displaystyle 2\pi }$	${\displaystyle 2\pi }$	${\displaystyle {\frac {2\pi }{\alpha }}}$	${\displaystyle 25812.807}$
Characteristic impedance of vacuum	${\displaystyle Z_{0}=2\alpha R_{\text{K}}}$	${\displaystyle 4\pi }$	${\displaystyle 4\pi }$	${\displaystyle 4\pi \alpha }$	${\displaystyle 4\pi \alpha }$	${\displaystyle 4\pi }$	${\displaystyle 2\alpha (25812.807)}$
Electric constant (vacuum permittivity)	${\displaystyle \varepsilon _{0}={\frac {1}{Z_{0}c}}}$	${\displaystyle {\frac {1}{4\pi }}}$	${\displaystyle {\frac {1}{4\pi }}\,}$	${\displaystyle {\frac {1}{4\pi }}\,}$	${\displaystyle {\frac {1}{4\pi }}\,}$	${\displaystyle {\frac {1}{4\pi }}\,}$	${\displaystyle {\frac {1}{2\alpha (25812.807)(299792458)}}\ }$
Magnetic constant (vacuum permeability)	${\displaystyle \mu _{0}={\frac {Z_{0}}{c}}}$	${\displaystyle 4\pi }$	${\displaystyle 4\pi }$	${\displaystyle 4\pi \alpha ^{2}}$	${\displaystyle 4\pi \alpha ^{2}}$	${\displaystyle 4\pi }$	${\displaystyle {\frac {2\alpha (25812.807)}{299792458}}}$
Newtonian constant of gravitation	${\displaystyle G}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle -}$	${\displaystyle -}$	${\displaystyle -}$
Electron mass	${\displaystyle m_{\text{e}}}$	${\displaystyle -}$	${\displaystyle -}$	${\displaystyle -}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle -}$
Hartree energy	${\displaystyle E_{\text{h}}=\alpha ^{2}m_{\text{e}}c^{2}}$	${\displaystyle -}$	${\displaystyle -}$	${\displaystyle -}$	${\displaystyle 1}$	${\displaystyle \alpha ^{2}}$	${\displaystyle -}$
Rydberg constant	${\displaystyle R_{\infty }={\frac {E_{\text{h}}}{2hc}}}$	${\displaystyle -}$	${\displaystyle -}$	${\displaystyle -}$	${\displaystyle {\frac {\alpha }{4\pi }}}$	${\displaystyle {\frac {\alpha ^{3}}{4\pi }}}$	${\displaystyle -}$
Caesium hyperfine transition frequency	${\displaystyle \Delta \nu _{\text{Cs}}}$	${\displaystyle -}$	${\displaystyle -}$	${\displaystyle -}$	${\displaystyle -}$	${\displaystyle -}$	${\displaystyle 9\ 192\ 631\ 770}$

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