SI, The International System of Units

SI, Systиme Internationale d’Unitйs, the international measure­ment unit system, is not a completely new system. It is based on an earlier metric system and is coming more and more into world-wide use. The SI system is now systematically con­structed to cover in practice all scientific, technical and daily re­quirements and is subject to international agreement. This means that it is now possible to apply the SI system uniformly throughout the world. A measurement system which is suitable for all technical and scientific purposes has to fulfil many re­quirements. Some of the basic requirements which SI satisfies are consistency, consequential applicability, coherence, the convenient expression of multiples and sub-multiples over a wide range of numerical values and accuracy.

Consistency means that each unit shall represent one, and only one, quantity.

Consequential applicability means that each quantity shall be measured in one, and only one, unit.

Coherence means that all units for every quantity shall be com­patible so as to eliminate the need for arbitrary conversion fac­tors in calculations involving related quantities.

Convenient expression of multiples and sub-multiples means the convenient multiplication of units to enable the use of prac­tical numerical values within a particular application.

Accuracy means that the base units shall be precisely derived and defined. Six of the seven base units are thus determined from distinct precisely defined physical phenomena, the sev­enth, the kilogram, is determined by one standard body which is held in Paris.

In 1971 the Council of Ministers of the EEC ratified a Directive, 71/354/EEC, on units which committed all member states to amend legislation to authorise SI units within 18 months of that date and to implement all provisions of the Directive within a fur­ther five years. An amending Directive, 76/770/EEC, legislates the obligations. The Units of Measurement Directives place non-SI units into four chapters A to D.

Chapter A prescribes those units which are for permanent use and member states are obliged to authorise them in their laws by 21 April, 1978.

Chapter B contains a list of all units which member states have undertaken to cease to authorise in their laws with effect from

31 December, 1977.

Chapter C contains a list of units which member states have un­dertaken to cease to authorise in their laws with effect from 31 December, 1979.

Chapter D covers remaining units and some other units and will be reviewed before 31 December, 1979.

The formal content of the SI is determined and authorised by the General Conference of Weights and Measures (CGPM) and, for more detailed descriptions of the System reference should be made to BS 3763 and SI — The International System of Units published by HMSO. However, the basic advice to in­dustry on the use of SI is now contained in IS01000, BS5555.

The SI system includes three classes of units: 1 ) base units 2) supplementary units 3) derived units.

Brief history of unit systems

Although often called the metric system, the SI system essen­tially has more basic units and overcomes problems encoun­tered during the development of a consistent system of units to serve all science and engineering functions. The metric system was introduced at the beginning of the nineteenth century based on the unit of length being the metre. The unit of mass followed and together with the unit of time formed the basic metric system. In 1873 the British Association for the Advance­ment of Science agreed on the use of the centimetre, gramme and second as the basic units for scientific work (the CGS sys­tem) but engineering within the United Kingdom had been well established using the British units of feet, pounds and seconds (the FPS system). As electrical experiments took place it was the metric system that became the basis for units peculiar to the electrical sciences and many basic electrical units were added to the CGS system.

An international authority on the metric system was established in 1875 with the Bureaux International des Poids et Mesures at Sиvres defining the units of length, mass and time as the metre, kilogramme and second respectively (the MKS system) as these units were more convenient than the CGS system. The basic units of length, mass and time are insufficient to cover electrical units and consequently units employing the perme­ability and permittivity of free space became necessary. Confu­sion also arose because of the links with the CGS system with, in particular, the use of p (the ratio of the circumference of a cir­cle to its diameter) appearing in equations which were usually not associated with circles.

To overcome the complications the International Electro­technical Commission (IEC) rationalized the units in 1950 and adopted as a fourth basic unit the unit of electrical current, the ampere (the MKSA system). This had been suggested about a half century earlier by an Italian professor called Giorgi and this system of units was consequently named the Giorgi System. Although this system of units covered electrical engineering it did not cover all branches of science and consequently the Conference Gйnйrale des Poids et Measures (CGPM) in 1954 agreed a rationalized and coherent system of units which be­came in 1960 the SI system.

After some subsequent additions the system now has seven basic units as follows:

Length metre (m)

Mass kilogramme (kg)

Time second (s)

Electric current Ampere (A)

Light candela (cd)

Temperature Kelvin (K)

Substance mole (mol)

In addition the following supplementary units are in use:

Plane angle radian (rad)

Solid angle steradian (sr)

The supplementary units are both ratios and therefore have no basic units.

The SI system is based around the seven basic units and the two supplementary units together with derived units for the more commonly used quantities and a series of prefixes used for the formation of multiples and submultiples.

Actual temperatures are normally ex-pressed in Celsius units (°C) and temperature differences in Kelvin units (K).

The Council of Ministers of the European Economic Commu­nity (EEC) ratified Directive 71/354/EEC in 1971 calling for all member states to amend legislation to authorize SI units and this was followed in 1976 by Directive 76/770/EEC to legalize the obligations. British Standard BS 5555 gives further details of SI units and their use.

Method of expressing symbols and numbers

The following rules apply to symbols for units

The symbol should be lower case unless the unit is derived from a proper name in which case the symbol should be up­per case (or the first letter upper case if more than one let­ter), for example, metre — m, Ampere — A, Hertz — Hz

The symbol should not contain a final full stop, for example, m not m.

The symbol should remain unaltered in the plural, for exam­ple, m not ms

If multiple symbols are required they should be separated by a space if confusion can occur, for example, kg m/s2 not kgm/s2. If a pair of units are each represented by a single letter they are not separated if the absence of a space is not likely to cause confusion, for example Nm

The symbol should be reduced to its simplest expression, for example, W not J/s, Js1, kg m2/s3 or kg m2 S’3

Where more than one symbol is required and division is in­volved use a solidus or superscript, do not use more than one solidus, for example, use m/s2 or m s2’ do not use m/s/s.

Large numbers should be written with the digits grouped in threes and with a or used to denote the decimal place, for example, 12 345 678, 0.000 012, 0,012 3. Use 10 000 not

10,0 to denote ten thousand, for example. One exception to this rule is that numbers with only four digits and without a deci­mal point do not normally have the space after the first digit, for example 1234 not 1 234.

Alternatively very large or very small numbers can be repre­sented in exponential form, that is a number multiplied by a fac­tor in the form of 10 to a positive or negative power. For example 100 is equivalent to 102 and 0.001 is equivalent to 10’3. There­fore 1 234 000 can be expressed in the form 1.234 x 106 and 0.0123 can be expressed as 1.23 x 10’2.

There is a further group of specialised units which are primarily for use within astronomy and physics.

Multiples of SI units

The prefixes in Table 22.1 are used to form names and symbols of multiples and subdivisions of the SI units. The symbol of a prefix is considered to be combined with the unit symbol for the base unit, supplementary unit or derived unit to which it is di­rectly attached, forming with it a symbol for a new unit which can be provided with a positive or negative exponent and which can

Factor by which the unit is multiplied	Prefix	Example
	Name	Symbol
1024	Yotta	Y
1021	Zetta	Z
1018	Exa	E
1015	Peta	P
1012	Tera	T	1 terrajoule = 1TJ
109	Giga	G	1 gigawatt = 1GM
106	Mega	M	1 megavolt= 1 MV
103	Kilo	K	1 kilometre = 1 km
102	Hecto	H	1 hectogram — 1 hg
101	Deca	Da	1 decalumen — 1 dalm
10-’	Deci	D	1 decimetre — 1 dm
10‘2	Centi	C	1 centimetre = 1 cm
10’3	Milli	M	1 milligram = 1 mg

Factor by which the unit is multiplied	Prefix	Example
10’6	Micro		1 microgram = 1 m
10-9	Nano	N	1 nanohenry = 1 nH
10-’2	Pico	P	1 picofarad = 1 pF
1015	Femto		1 femtometre = 1 fm
1018	Atto	A	1 attosecond = 1 as

Table 22.1 Multiples of SI units

Be combined with other unit symbols to form symbols for com­pound units.

Whenever possible units should be multiples or submultiples of

3 — hecto, deca, deci and centi therefore should not normally be used.

The prefix symbol should appear immediately before the basic symbol, for example mAfor milliampere. It should be noted that mm3 means (0.001 m)3 not 0.001 m3 and that mm1 means (10-3 m)’1 not 10’3 nrv1. The use of dk should be avoided as this may cause confusion, for example, dkg is decagramme not deci killogramme.

It is normal practice to use millimetre (mm) as the unit of length on engineering drawings.

Derived units

These are expressed in terms of base units and/or supplemen­tary units by multiplication and division according to the laws of physics relating the various quantities, see Table 22.2.

Quantity	Name of derived SI unit	Symbol	Expressed in terms of base or supplementary units
Frequency	Hertz	Hz	1 Hz = 1/s
Force	Newton	N	1 N = 1 kg m/s2
Pressure, stress	Pascal	Pa	1 Pa = 1 N/m2
Energy, work, heat	Joule	J	1J = 1Nm
Power	Watt	W	1 W = 1 J/s
Electric charge, quantity of electricity	Coulomb	C	1C = 1 As
Electric potential	Volt	V	1 V = 1 J/C = 1 W/A
Electric capacitance	Farad	F	1 F = 1 C/V
Electric resistance	Ohm		1 = V/A
Electric conductance	Siemens	S	1 S = 1/
Magnetic flux	Weber	Wb	1 Wb =1V s
Magnetic flux density	Tesla	T	1 T = 1 Wb/m2
Inductance	Henry	H	1 H = 1 Wb/A
Luminous flux	Lumen	Im	1 Im = 1 cd sr
Illuminance	Lux	Ix	1lx = 1 lm/m2
Radioactivity	Becquerel	Bq	1 Bq = 1/s
Absorbed dose	Gray	Gy	1 Gy = 1 J/kg

Table 22.2 Some derived SI units having special names

Non SI units. There are certain units not included in SI which cannot, for a variety of reasons, be eliminated, despite the fact that these can, in principle, be expressed in SI units. Some of the non-SI units which may be used together with the SI units and their multiples and are recognised by the CIPM, Comitй In­ternational des Poids et Mesures, are shown in Table 22.3.

Conductance is sometimes used as the reciprocal of resistance for which the unit is the Siemen (S), also referred to as mho.

Quantity	Name of unit	Unit symbol	Definition
	Minute	Min	1 min	— 60 sec
Time	Hour	H	1 h	— 60 min
	Day	D	1 d	= 24 h
	Degree	»	1°	= (p/180) rad
Plane angle	Minute		1	= (1/60)°
	Second		1	= (1/60)
Volume	Litre	I	11 =1 dm3
Mass	Tonne	T	11	= 103 kg
Pressure of fluid	Bar	Bar	1 bar	= 105Pa

Table 22.3 Non-SI units

Checking units in equations

A/L

K

It is sometimes useful to check equations by checking whether the units are consistent on each side of the equals sign. In par­ticular force and mass are often confused, for example, a mass of 1 kg produces a force due to gravity of 1 kg multiplied by the acceleration due to gravity (9.807 m/s2) to give 9.807 N but this is often referred to as 1 kilogramme force (1 kgf) — if the units of kilogramme force are taken as the same as kilogramme then errors will result. Checking the consistency of the units should immediately indicate whether an error is present.

It is normal practice to replace the units with the letters M for mass, L for length and T for time when checking units and this avoids any consideration of multiples and submultiples that are not relevant when checking for consistency. For example if a constant torque is applied to a rotating mass, the time to change the speed by a given amount is given by:

Equ 22.1

Where

D = current density

= conductor resistivity

D = density of conductor

C = specific heat of conductor

The units of D may be A/m2 which can be expressed as A/L2.

The units of are usually expressed in terms of a resistance across the faces of a cube, for example, m2/m or m. is a derived unit equivalent to a voltage divided by a current. In turn voltage is power divided by current, power is work divided by time, work is force multiplied by distance and force is mass mul­tiplied by acceleration. Repeated substitutions then leads to the basic units of resistivity as equivalent to ML3/T3A2. The unit of density is equivalent to M/L3. Specific heat is usually ex-pressed in terms of J/kg Kand therefore after substituting for J the unit of c is equivalent to L/T2K. Substituting in equation

22.3 gives the units as:

ML /T A

Equ 22.4

M/L3 L2/T2K T

Which gives consistent units for the rate of temperature rise.

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