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International System of Units

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 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).

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.

Constant	Conventional (defined) value (CIPM, 1988)	Empirical value (in SI units) (CODATA, 2010Template:CODATA2010)
Josephson constant	KJ–90 = 483 597.9 GHz/V	KJ = 483 597.870(11) GHz/V
von Klitzing constant	RK–90 = 25 812.807 Ω	RK = 25 812.807 4434(84) 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, KJ–90.
• The conventional ohm, Ω90, is the electrical resistance measured against a w:quantum Hall effect standard using the defined value of the von Klitzing constant, RK–90.
• Other conventional electrical units are defined by the normal physical relationships, as in the conversion table below.

Unit	Definition	SI equivalent (CODATA 2006)Template:Update after
conventional volt	see above	V90 = (KJ–90/KJ) V = [1 + 1.9(2.5)×10−8] V
conventional ohm	see above	Ω90 = (RK/RK–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

The International System of Units consists of a set of base units, a set of derived units, some of which have special names and a set decimal-based multipliers that are denoted as prefixes. The term "SI Units" includes all three categories, but the term "coherent SI units" includes only base units and coherent derived units.^[1]

Derived units are formed by powers, products or quotients of the base units and are unlimited in number;^[2] Derived units are associated with derived quantities, for example velocity is a quantity that is derived from the base quantities of time and distance which, in SI, has the dimensions metres per second (symbol m/s). The dimensions of derived units can be expressed in terms of the dimensions of the base units.

Some derived units have special names, for example the unit of force is the newton. Coherent units (such as those in SI) are derived units that contain no numerical factor other than 1: in the example above, one newton is the force required to accelerate a mass of one kilogram by one metre per second squared. Since the SI units of mass and acceleration are kg and m⋅s⁻² respectively and F ∝ m × a, the units of force (and hence of newtons) is formed by multiplication to give kg⋅m⋅s⁻². Since the newton is part of a coherent set of units, the constant of proportionality is 1.

Named units derived from SI base units^[2]

Name	Symbol	Quantity	Expresed in terms of other SI units	Expresed in terms of SI base units
radian	rad	angle	1	m/m
steradian	sr	solid angle	1	m2/m2
hertz	Hz	frequency		s−1
newton	N	force, weight		kg⋅m⋅s−2
pascal	Pa	pressure, stress	N/m2	kg⋅m−1⋅s−2
joule	J	energy, work, heat	N⋅m	kg⋅m2⋅s−2
watt	W	power, radiant flux	J/s	kg⋅m2⋅s−3
coulomb	C	electric charge or quantity of electricity		s⋅A
volt	V	voltage, electrical potential difference, electromotive force	W/A	kg⋅m2⋅s−3⋅A−1
farad	F	electric capacitance	C/V	kg−1⋅m−2⋅s4⋅A2
ohm	Ω	electric resistance, impedance, reactance	V/A	kg⋅m2⋅s−3⋅A−2
siemens	S	electrical conductance	A/V	kg−1⋅m−2⋅s3⋅A2
weber	Wb	magnetic flux	V⋅s	kg⋅m2⋅s−2⋅A−1
tesla	T	magnetic field strength	Wb/m2	kg⋅s−2⋅A−1
henry	H	inductance	Wb/A	kg⋅m2⋅s−2⋅A−2
degree Celsius	°C	temperature relative to 273.15 K		K
lumen	lm	luminous flux	cd⋅sr	cd
lux	lx	illuminance	lm/m2	m−2⋅cd
becquerel	Bq	radioactivity (decays per unit time)		s−1
gray	Gy	absorbed dose (of ionizing radiation)	J/kg	m2⋅s−2
sievert	Sv	equivalent dose (of ionizing radiation)	J/kg	m2⋅s−2
katal	kat	catalytic activity		s−1⋅mol
Notes 1. The radian and steradian, once given special status, are now considered dimensionless derived units.[2] 2. The ordering of this table is such that any derived unit is based only on base units or derived units that preceed it in the table.

A prefix may be added to a unit to produce a multiple of the original unit. All multiples are integer powers of ten, and beyond a hundred(th) all are integer powers of a thousand. For example, kilo- denotes a multiple of a thousand and milli- denotes a multiple of a thousandth; hence there are one thousand millimetres to the metre and one thousand metres to the kilometre. The prefixes are never combined, and multiples of the kilogram are named as if the gram was the base unit. Thus a millionth of a metre is a micrometre, not a millimillimetre, and a millionth of a kilogram is a milligram, not a microkilogram.^[3]

Template:SI-Prefixes

Although, in theory, SI can be used for any physical measurement, it is recognised that some non-SI units still appear in the scientific, technical and commercial literature, and will continue to be used for many years to come. In addition, certain other units are so deeply embedded in the history and culture of the human race that they will continue to be used for the foreseeable future. The CIPM has catalogued a number of such non-SI units accepted for use with SI and published them in the SI Brochure thereby ensuring that their use is consistent across the globe. These units have been grouped as follows:^{[4][Note 1]}

• Non-SI units accepted for use with the SI (Table 6):

Certain units of time, angles and legacy non-SI metric units have a long history of consistent use. Most of mankind has used the day and its non-decimal subdivisions as a basis of time and, unlike the foot or the pound, these were the same regardless of where it was being measured. The radian, being 1/2π of a revolution has mathematical niceties, but it is cumbersome for navigation, and, as with time, the units used in navigation have a large degree of consistency around the world. The tonne, litre and hectare were adopted by the CGPM in 1879 and have been retained as units that may be used alongside SI units, having been given unique symbols. The catalogued units are

minute, hour, day, degree of arc, minute of arc, second of arc, hectare, litre and tonne

• Non-SI units whose values in SI units must be obtained experimentally (Table 7).

Physicists often use units of measure that are based on natural phenomena, particularly when the quantities associated with these phenomena are many orders of magnitude greater than or less than the equivalent SI unit. The most common ones have been catalogued in the SI brochure together with consistent symbols and accepted values, but with the caveat that their physical values need to be measured.^{[Note 2]} :

electronvolt, dalton/unified atomic mass unit, astronomical unit, speed of light, Planck constant and electron mass

• Other non-SI units (Table 8):

A number of non-SI units that had never been formally sanctioned by the CGPM have continued to be used across the globe in many spheres including health care and navigation. As with the units of measure in Tables 6 and 7, these have been catalogued by the CIPM in the SI brochure to ensure consistent usage, but with the recommendation that authors who use them should define them wherever they are used.

bar, millimetre of mercury, ångström, nautical mile, barn, knot, neper and [deci]bel

• Non-SI units associated with the CGS and the CGS-Gaussian system of units (Table 9)

The SI manual also catalogues a number of legacy units of measure that are used in specific fields such as geodesy and geophysics or are found in the literature, particularly in classical and relativistic electrodynamics where they have certain advantages: The units that are catalolgued are:

erg, dyne, poise, stokes, stilb, phot, gal, maxwell, gauss and œrsted.

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

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

having consequence:

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

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	Neoclassical
speed of light in vacuum ${\displaystyle c\,}$	${\displaystyle 1\,}$	${\displaystyle 1\,}$	${\displaystyle {\frac {1}{\alpha }}\ }$	${\displaystyle {\frac {1}{\alpha }}\ }$	${\displaystyle 1\,}$	${\displaystyle 299792458\ }$	${\displaystyle 10^{8}\ }$
Planck's constant ${\displaystyle h\,}$	${\displaystyle 2\pi \,}$	${\displaystyle {\frac {2\pi }{\alpha }}\ }$	${\displaystyle 2\pi \,}$	${\displaystyle 2\pi \,}$	${\displaystyle {\frac {2\pi }{\alpha }}\ }$	${\displaystyle {\frac {4\times 10^{-18}}{(25812.807)(483597.9)^{2}}}\ }$	${\displaystyle 2\pi \times 10^{-34}\,}$
reduced Planck's constant ${\displaystyle \hbar ={\frac {h}{2\pi }}}$	${\displaystyle 1\,}$	${\displaystyle {\frac {1}{\alpha }}\ }$	${\displaystyle 1\,}$	${\displaystyle 1\,}$	${\displaystyle {\frac {1}{\alpha }}\ }$	${\displaystyle {\frac {2\times 10^{-18}}{\pi (25812.807)(483597.9)^{2}}}\ }$	${\displaystyle 10^{-34}\,}$
elementary charge ${\displaystyle e\,}$	${\displaystyle {\sqrt {\alpha }}\,}$	${\displaystyle 1\,}$	${\displaystyle 1\,}$	${\displaystyle 1\,}$	${\displaystyle 1\,}$	${\displaystyle {\frac {2\times 10^{-9}}{(25812.807)(483597.9)}}\ }$	${\displaystyle 10^{-18}\,}$
Josephson constant ${\displaystyle K_{J}={\frac {2e}{h}}\,}$	${\displaystyle {\frac {\sqrt {\alpha }}{\pi }}\,}$	${\displaystyle {\frac {\alpha }{\pi }}\,}$	${\displaystyle {\frac {1}{\pi }}\,}$	${\displaystyle {\frac {1}{\pi }}\,}$	${\displaystyle {\frac {\alpha }{\pi }}\,}$	${\displaystyle 483597.9\times 10^{9}\,}$	${\displaystyle {\frac {10^{16}}{\pi }}\,}$
von Klitzing constant ${\displaystyle R_{K}={\frac {h}{e^{2}}}\,}$	${\displaystyle {\frac {2\pi }{\alpha }}\,}$	${\displaystyle {\frac {2\pi }{\alpha }}\,}$	${\displaystyle 2\pi \,}$	${\displaystyle 2\pi \,}$	${\displaystyle {\frac {2\pi }{\alpha }}\,}$	${\displaystyle 25812.807\,}$	${\displaystyle 200\pi \,}$
characteristic impedance of vacuum ${\displaystyle Z_{0}=2\alpha R_{K}\,}$	${\displaystyle 4\pi \,}$	${\displaystyle 4\pi \,}$	${\displaystyle 4\pi \alpha \,}$	${\displaystyle 4\pi \alpha \,}$	${\displaystyle 4\pi \,}$	${\displaystyle 2\alpha (25812.807)\,}$	${\displaystyle 400\pi \alpha \,}$
electric constant (vacuum permittivity) ${\displaystyle \varepsilon _{0}={\frac {1}{Z_{0}c}}\,}$	${\displaystyle {\frac {1}{4\pi }}\,}$	${\displaystyle {\frac {1}{4\pi }}\,}$	${\displaystyle {\frac {1}{4\pi }}\,}$	${\displaystyle {\frac {1}{4\pi }}\,}$	${\displaystyle {\frac {1}{4\pi }}\,}$	${\displaystyle {\frac {1}{2\alpha (25812.807)(299792458)}}\ }$	${\displaystyle {\frac {10^{-10}}{4\pi \alpha }}\,}$
magnetic constant (vacuum permeability) ${\displaystyle \mu _{0}={\frac {Z_{0}}{c}}\,}$	${\displaystyle 4\pi \,}$	${\displaystyle 4\pi \,}$	${\displaystyle 4\pi \alpha ^{2}\,}$	${\displaystyle 4\pi \alpha ^{2}\,}$	${\displaystyle 4\pi \,}$	${\displaystyle {\frac {2\alpha (25812.807)}{299792458}}\ }$	${\displaystyle 4\pi \alpha \times 10^{-6}\,}$
Newtonian constant of gravitation ${\displaystyle G\,}$	${\displaystyle 1\,}$	${\displaystyle 1\,}$	${\displaystyle 1\,}$	${\displaystyle -\,}$	${\displaystyle -\,}$	${\displaystyle -\,}$	${\displaystyle -\,}$
electron mass ${\displaystyle m_{e}\,}$	${\displaystyle -\,}$	${\displaystyle -\,}$	${\displaystyle -\,}$	${\displaystyle 1\,}$	${\displaystyle 1\,}$	${\displaystyle -\,}$	${\displaystyle -\,}$
Hartree energy ${\displaystyle E_{h}=\alpha ^{2}m_{e}c^{2}\,}$	${\displaystyle -\,}$	${\displaystyle -\,}$	${\displaystyle -\,}$	${\displaystyle 1\,}$	${\displaystyle \alpha ^{2}\,}$	${\displaystyle -\,}$	${\displaystyle -\,}$
Rydberg constant ${\displaystyle R_{\infty }={\frac {E_{h}}{2hc}}\,}$	${\displaystyle -\,}$	${\displaystyle -\,}$	${\displaystyle -\,}$	${\displaystyle {\frac {\alpha }{4\pi }}\,}$	${\displaystyle {\frac {\alpha ^{3}}{4\pi }}\,}$	${\displaystyle -\,}$	${\displaystyle -\,}$
caesium ground state hyperfine transition frequency	${\displaystyle -\,}$	${\displaystyle -\,}$	${\displaystyle -\,}$	${\displaystyle -\,}$	${\displaystyle -\,}$	${\displaystyle 9\ 192\ 631\ 770\,}$	${\displaystyle 2\ 808\ 349\ 000\,}$

• Centimetre–gram–second system of units

• Template:CODATA2006

Template:Systems of measurement
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