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2、art 3: Logarithmic and related quantities, and their unitsICS 01.060; 01.075; 29.020Letter symbols to be used in electrical technology BRITISH STANDARDBS EN 60027-3:2007BS EN 60027-3:2007Licensed Copy: Wang Bin, ISO/EXCHANGE CHINA STANDARDS, 10/07/2008 04:18, Uncontrolled Copy, (c) BSIThis British S
3、tandard was published under the authority of the Standards Policy and Strategy Committee BSI 2008ISBN 978 0 580 60281 8on 30 April 2008Amendments issued since publicationAmd. No. Date CommentsThis publication does not purport to include all the necessary provisions of a contract. Users are responsib
4、le for its correct application.Compliance with a British Standard cannot confer immunity from legal obligations.National forewordThis British Standard is the UK implementation of EN 60027-3:2007. It is identical to IEC 60027-3:2002. It supersedes BS 7998-3:2002 which is withdrawn.The UK participatio
5、n in its preparation was entrusted to Technical Committee SS/7, General metrology, quantities, units and symbols.A list of organizations represented on this committee can be obtained on request to its secretary.EUROPEAN STANDARD EN 60027-3 NORME EUROPENNE EUROPISCHE NORM January 2007 CENELEC Europea
6、n Committee for Electrotechnical Standardization Comit Europen de Normalisation Electrotechnique Europisches Komitee fr Elektrotechnische Normung Central Secretariat: rue de Stassart 35, B - 1050 Brussels 2007 CENELEC - All rights of exploitation in any form and by any means reserved worldwide for C
7、ENELEC members. Ref. No. EN 60027-3:2007 E ICS 01.075; 01.060 Supersedes HD 60027-3:2004English version Letter symbols to be used in electrical technology Part 3: Logarithmic and related quantities, and their units (IEC 60027-3:2002) Symboles littraux utiliser en lectrotechnique Partie 3: Grandeurs
8、logarithmiques et connexes, et leurs units (CEI 60027-3:2002) Formelzeichen fr die Elektrotechnik Teil 3: Logarithmische und verwandte Gren und ihre Einheiten (IEC 60027-3:2002) This European Standard was approved by CENELEC on 2006-12-01. CENELEC members are bound to comply with the CEN/CENELEC Int
9、ernal Regulations which stipulate the conditions for giving this European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such national standards may be obtained on application to the Central Secretariat or to any CENELEC
10、member. This European Standard exists in three official versions (English, French, German). A version in any other language made by translation under the responsibility of a CENELEC member into its own language and notified to the Central Secretariat has the same status as the official versions. CEN
11、ELEC members are the national electrotechnical committees of Austria, Belgium, Cyprus, the Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, the Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Sloveni
12、a, Spain, Sweden, Switzerland and the United Kingdom. Licensed Copy: Wang Bin, ISO/EXCHANGE CHINA STANDARDS, 10/07/2008 04:18, Uncontrolled Copy, (c) BSIForeword The text of the International Standard IEC 60027-3:2002, prepared by IEC TC 25, Quantities and units, and their letter symbols, was approv
13、ed by CENELEC as HD 60027-3:2004 on 2003-12-01. This Harmonization Document was submitted to the formal vote for conversion into a European Standard and was approved by CENELEC as EN 60027-3 on 2006-12-01. This European Standard supersedes HD 60027-3:2004. The following date was fixed: latest date b
14、y which the EN has to be implemented at national level by publication of an identical national standard or by endorsement (dop) 2007-12-01 Annex ZA has been added by CENELEC. _ Endorsement notice The text of the International Standard IEC 60027-3:2002 was approved by CENELEC as a European Standard w
15、ithout any modification. _ EN 60027-3:2007 2 Licensed Copy: Wang Bin, ISO/EXCHANGE CHINA STANDARDS, 10/07/2008 04:18, Uncontrolled Copy, (c) BSICONTENTS1 Scope 42 Normative references. 43 Logarithmic quantities 44 Logarithmic ratios of field quantities and power quantities 54.1 Logarithmic ratios of
16、 field quantities 54.2 Logarithmic ratios of power quantities74.3 Levels94.4 Additional information on logarithmic ratios of field quantities and powerquantities.105 Logarithmic information-theory quantities116 Other logarithmic quantities 126.1 General .126.2 Logarithmic frequency interval .127 Nam
17、es, symbols, and definitions.13Annex ZA (normative) Normative references to international publications with theircorresponding European publications.14EN 60027-3:2007 3 Licensed Copy: Wang Bin, ISO/EXCHANGE CHINA STANDARDS, 10/07/2008 04:18, Uncontrolled Copy, (c) BSILETTER SYMBOLS TO BE USED IN ELE
18、CTRICAL TECHNOLOGY Part 3: Logarithmic and related quantities, and their units1 ScopeThis part of IEC 60027 gives general information about logarithmic and related quantities, andtheir units. Names and symbols for logarithmic quantities are given in other parts ofIEC 60027, mainly in part 2, in the
19、context where they belong.2 Normative referencesThe following referenced documents are indispensable for the application of this document.For dated references, only the edition cited applies. For undated references, the latest editionof the referenced document (including any amendments) applies.IEC
20、60027-2:2000, Letter symbols to be used in electrical technology Part 2: Telecom-munications and electronicsISO 31-0:1992, Quantities and units Part 0: General principlesISO 31-2:1992, Quantities and units Part 2: Periodic and related phenomenaISO 31-7:1992, Quantities and units Part 7: AcousticsISO
21、 31-11:1992, Quantities and units Part 11: Mathematical signs and symbols for use inthe physical sciences and technologyISO/IEC 2382-16:1996, Information technology Vocabulary Part 16: Information theory3 Logarithmic quantitiesLogarithmic quantities are quantities defined by means of logarithmic fun
22、ctions. For adefinition of a logarithmic quantity to be complete, the base of the logarithm must bespecified.Depending on the source of the argument of the logarithm, logarithmic quantities may beclassified as follows:a) logarithmic ratios which are defined by the logarithm of the ratio of two field
23、 quantitiesof the same kind or of two power quantities of the same kind, for example attenuationsand gains in telecommunication, where the argument may be the ratio of two electriccurrents or powers, and levels in acoustics, where the argument may be the ratio of asound pressure or a sound power to
24、a reference quantity of the same kind;EN 60027-3:2007 4 Licensed Copy: Wang Bin, ISO/EXCHANGE CHINA STANDARDS, 10/07/2008 04:18, Uncontrolled Copy, (c) BSIb) logarithmic quantities, in which the argument is given explicitly as a number, forexample logarithmic information-theory quantities, such as d
25、ecision content, where theargument is a number of mutually exclusive events, and information content, where theargument is the reciprocal of a probability;c) other logarithmic quantities.In the set of logarithmic and related quantities there are also included quantities which are alinear combination
26、 of logarithmic quantities, or a derivative of a logarithmic quantity, or aquotient of a logarithmic quantity and another quantity, for example attenuation coefficient.The logarithm to any specified base of an argument gives the same information about thesituation under consideration, as does the ar
27、gument itself. Quantities defined with logarithmsof different bases are proportional to each other, but have different values and thus aredifferent quantities. In a given field of application, only logarithms of one base shall be usedto define logarithmic quantities. Because of the proportionality b
28、etween the logarithms it ispossible to express the numerical values using different bases of the logarithm together withdifferent units. To avoid ambiguities in applications the unit shall be written out explicitly afterthe numerical value in a logarithmic quantity.NOTE 1 In this part of IEC 60027,
29、complex quantities are denoted by underlining their symbols. However, thisdoes not constitute a compulsory rule in applications (see IEC 60027-1).4 Logarithmic ratios of field quantities and power quantities4.1 Logarithmic ratios of field quantitiesA quantity the square of which is proportional to p
30、ower when it acts on a linear system is herecalled a field quantity, general symbol F.EXAMPLESElectric current, voltage, electric field strength, sound pressure, particle speed, andforce are field quantities.For sinusoidal time-varying field quantities, the ratio of the amplitudes or the root-mean-s
31、quare values is the argument of the logarithm.For non-sinusoidal field quantities, the root-mean-square value over an appropriate timeinterval to be specified is used. For a periodic quantity, the appropriate time interval is theperiodic time.For logarithmic ratios of field quantities, logarithms wi
32、th two different bases are used for thenumerical values. These logarithms are: natural logarithm, symbol ln (or loge), decimal logarithm, symbol lg (or log10).For real field quantity ratios, F1/F2, the following general expressions of a logarithmic ratio,Q(F), expressed in different units are obtain
33、ed:dBlg20Blg2Npln212121=FFFFFFQF )(1)EN 60027-3:2007 5 Licensed Copy: Wang Bin, ISO/EXCHANGE CHINA STANDARDS, 10/07/2008 04:18, Uncontrolled Copy, (c) BSIHere the neper, symbol Np, is the value of Q(F)when F1/F2= e; and the bel, symbol B is thevalue of Q(F)when F1/F2= 10 . The decibel, symbol dB, is
34、 1 dB = (1/10) B. Hence1 Np = (ln e) Np = 2 (lg e) B = 20 (lg e) dB 8,685 889 dB (2)1 B = 2 (lg 10 ) B = 10 dB = (ln 10 ) Np 1,151 292 Np (3)1 dB = 101B = 101(ln 10 ) Np 0,115 129 2 Np (4)The factor 2 in the numerical value of Q(F)expressed in bels in equation (1) has historicalreasons and is explai
35、ned in 4.2.Complex notation is frequently used for field quantities, for example in telecommunicationsand acoustics. The taking of logarithms of complex-quantity ratios is usefully done only withthe natural logarithm. Many other mathematical relations and operations also become simplerif, and only i
36、f, the natural logarithm is used. This is seen from the fact that the naturallogarithmic function of x2/x1be defined as the integral=21dln12xxxxxxwithout any numerical factors as with other bases.That is why natural logarithms are used in the system of quantities on which the SI is based,i.e. the In
37、ternational System of Quantities (ISQ).NOTE 2 At a plenary meeting of ISO/TC 12, Quantities, units, symbols, conversion factors and conversion tablesin Washington D.C., US, in 1973 with participation of the Chairman and the Secretary of IEC/TC 25 it wasunanimously decided to use the natural logarith
38、m in the system of quantities on which the SI is based, i.e. toconsider the neper, symbol Np as coherent with the SI. This decision has later been adopted by the ComitInternational des Poids et Mesures (CIPM), and the Organisation Internationale de Mtrologie Lgale (OIML).With the quantity Q(F)define
39、d by convention with the natural logarithm, i.e.Q(F)= ln(F1/F2)(5)neper (Np) becomes the coherent unit, which can be replaced with one, symbol 1 (see ISO31-2, 2-9).NOTE 3 In general, a quantity definition should be given before the introduction of the corresponding units. Forhistorical reasons, howe
40、ver, the traditional order of presentation is followed in this part of IEC 60027.For practical applications mainly in telecommunication and acoustics, the sub-multiple decibel(dB) of the bel (B) based on decimal logarithms is in common usage.NOTE 4 In practice, the use of decibel (dB) has prevailed
41、internationally since the ITU decided in 1968 to useonly the decibel. This is in some respects similar to the fact that the unit degree () is commonly used in practiceinstead of the coherent SI unit radian (rad) for plane angle.In theoretical calculations, neper (Np) for the amplitude together with
42、radian (rad) for thephase angle, result naturally from complex notation and natural logarithms. Consider forexample a ratio of two complex quantities F1and F2.EN 60027-3:2007 6 Licensed Copy: Wang Bin, ISO/EXCHANGE CHINA STANDARDS, 10/07/2008 04:18, Uncontrolled Copy, (c) BSI)j(21j2j1212121eee=FFFFF
43、F(6)j(lnln212121( +=FFFFQF )(7)EXAMPLEWith the voltages Ve30/2j1=U and Ve3/3j2=U we obtainrad0,524jNp3032rad6jNp10)(ln)eln(10Ve3Ve30lnln6j3jj21+=,/2UUUQ4.2 Logarithmic ratios of power quantitiesA quantity that is proportional to power is called a power quantity, general symbol P. In manycases also e
44、nergy-related quantities are labelled as power quantities in this context.EXAMPLESActive power, reactive power, and apparent power in electrical technology, acoustic andelectromagnetic power, and corresponding power densities.Since power quantities are related to field quantities, natural logarithms
45、 and decimallogarithms are used also for the numerical values of power quantities. Hence, the followinggeneral expressions of a logarithmic ratio of two active powers P1and P2, Q(P), expressed indifferent units are obtained:dBlg10BlgNpln21212121=PPPPPPQP)(8)Here the neper (Np) is the value of Q(P)wh
46、en P1/P2= e2; and the bel (B) is the value of Q(P)when P1/P2= 10. The decibel (dB) is 1 dB = (1/10) B. Hence1 Np = 21(ln e2) Np = (lg e2) B = 10 (lg e2) dB 8,685 889 dB (9)1 B = (lg 10) B = 10 dB = 21(ln 10) Np 1,151 292 Np (10)1 dB = 101B = 201(ln 10) Np 0,115 129 2 Np (11)These are the same conver
47、sion factors as those obtained in sub-clause 4.1, equations (2) to(4).EN 60027-3:2007 7 Licensed Copy: Wang Bin, ISO/EXCHANGE CHINA STANDARDS, 10/07/2008 04:18, Uncontrolled Copy, (c) BSIWith the quantity Q(P)defined by convention with the natural logarithm, i.e.Q(P)= (1/2) ln(P1/P2)(12)neper (Np) b
48、ecomes the coherent unit, which can be replaced with one, symbol 1 (see ISO31-2, 2-10).Following the definition of a field quantity, letP1= k1F12(13)P2= k2F22(14)Therefore21212122221121ln21ln21lnln21ln21kkQkkFFFkFkPPQFP+=+=)()(15)In the general case, the relation between Q(P)and Q(F)depends on k1/k2
49、.In the special case when k1= k2then Q(P)= Q(F).This explains why the factor 1/2 appears in equation (12) and the factors 2, 20, and 1/2appear in the numerical values in the equations (1) and (8), respectively.In electrical technology, the ratio k1/k2is often an impedance or admittance ratio. Therefore,the comparison of values of logarithmic ratios involving field quantities without adequateinformation about the impedance
