NORME INTERNATIONALE CEI IEC INTERNATIONAL STANDARD 60027 3 Troisième édition Third edition 2002 07 Symboles littéraux à utiliser en électrotechnique – Partie 3 Grandeurs logarithmiques et connexes, e[.]
Rapports logarithmiques de grandeurs de champ
Une grandeur dont le carré est proportionnel à une puissance lorsqu'elle agit sur un système linéaire est appelée ici grandeur de champ, de symbole général F.
Courant électrique, tension électrique, champ électrique, pression acoustique, vitesse d'une particule, force.
Pour des grandeurs de champ à variation sinusọdale dans le temps, l'argument du logarithme est le rapport des amplitudes ou des valeurs efficaces.
For non-sinusoidal field quantities, the effective value is determined over a specified appropriate time interval In the case of a periodic quantity, the appropriate time interval is the period.
Pour les rapports logarithmiques de grandeurs de champ, on utilise les logarithmes de deux bases différentes pour exprimer les valeurs numériques Ce sont:
– les logarithmes népériens, symbole ln (ou log e ),
– les logarithmes décimaux, symbole lg (ou log 10 ).
Pour des rapports de grandeurs de champ réelles, F 1 /F 2 , on obtient les expressions générales suivantes du rapport logarithmique, Q ( F ) , exprimé en différentes unités: dB lg
This article discusses licensed materials provided by MECON Limited for internal use in Ranchi and Bangalore It highlights logarithmic quantities, specifically focusing on information-theory metrics such as decision content, which involves a number of mutually exclusive events, and information content, defined as the reciprocal of a probability Additionally, it addresses other logarithmic quantities relevant to the topic.
The set of logarithmic quantities encompasses not only logarithmic values themselves but also includes linear combinations of these quantities, derivatives of logarithmic quantities, and quotients involving logarithmic quantities and other variables, such as the attenuation coefficient.
The logarithm of an argument to any specified base conveys the same information as the argument itself, although logarithms of different bases yield different values and represent distinct quantities In any specific application, it is essential to use logarithms of a single base to define logarithmic quantities Due to the proportionality of logarithms, numerical values can be expressed using various bases and units To prevent ambiguities, it is important to explicitly state the unit after the numerical value in a logarithmic quantity.
NOTE 1 In this part of IEC 60027, complex quantities are denoted by underlining their symbols However, this does not constitute a compulsory rule in applications (see IEC 60027-1).
4 Logarithmic ratios of field quantities and power quantities
4.1 Logarithmic ratios of field quantities
A quantity the square of which is proportional to power when it acts on a linear system is here called a field quantity, general symbol F.
Electric current, voltage, electric field strength, sound pressure, particle speed, and force are field quantities.
For sinusoidal time-varying field quantities, the ratio of the amplitudes or the root-mean- square values is the argument of the logarithm.
For non-sinusoidal field quantities, the root-mean-square (RMS) value is calculated over a specified time interval In the case of periodic quantities, this interval corresponds to the periodic time.
For logarithmic ratios of field quantities, logarithms with two different bases are used for the numerical values These logarithms are:
― natural logarithm, symbol ln (or loge),
― decimal logarithm, symbol lg (or log10).
For real field quantity ratios, F 1/F 2, the following general expressions of a logarithmic ratio,
Q (F), expressed in different units are obtained: dB lg
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Le néper, symbole Np, est ici la valeur de Q ( F ) lorsque F 1 /F 2 = e; et le bel, symbole B, est la valeur de Q ( F ) lorsque F 1 /F 2 = 10 Le décibel, symbole dB, est défini par 1 dB = (1/10) B Par conséquent
1 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)
Le facteur 2 intervient dans la valeur numérique de Q ( F ) exprimée en bels dans la formule (1) pour des raisons historiques qui sont expliquées en 4.2.
Complex notation is frequently employed for field quantities, particularly in telecommunications and acoustics To express logarithmic ratios of complex quantities, only natural logarithms yield easily usable results.
Many mathematical operations and relationships become simpler when the logarithm is natural This is evident from the fact that the natural logarithm of the ratio \$\frac{x_2}{x_1}\$ can be defined using an integral.
2 x x x x x x sans facteur numérique comme avec les autres bases.
C'est pourquoi on emploie les logarithmes népériens dans le système de grandeurs sur lequel le SI est fondé, c'est-à-dire le Système international de grandeurs (ISQ).
In a plenary meeting of ISO Study Committee 12 on quantities, units, symbols, conversion factors, and conversion tables held in Washington D.C in 1973, it was unanimously decided to adopt natural logarithms in the system of quantities underlying the SI, recognizing the neper (Np) as a coherent unit within the SI framework This decision was later reaffirmed by the International Committee for Weights and Measures (CIPM) and the International Organization of Legal Metrology (OIML).
En définissant par convention la grandeur Q ( F ) par un logarithme népérien, soit
Q ( F ) = ln(F 1 /F 2 ) (5) le néper (Np) devient l'unité cohérente, qui peut être remplacée par l'unité un, symbole 1 (voir l'ISO 31-2, 2-9).
It is generally advisable to define a quantity before introducing its corresponding units For historical reasons, this section of IEC 60027 follows the traditional order of presentation.
Pour les applications pratiques, notamment dans les télécommunications et l'acoustique, le sous-multiple décibel (dB) du bel (B), fondé sur les logarithmes décimaux, est d'usage courant.
NOTE 4 En pratique, l'emploi du décibel (dB) l'a emporté sur le plan international depuis que l'UIT a décidé en
In 1968, the use of the decibel became standard practice This approach is somewhat similar to the common use of degrees as a unit of plane angle (… °) instead of the SI coherent unit, the radian (rad).
Dans les calculs théoriques, l'emploi du néper (Np) pour exprimer l'amplitude et celui du radian
(rad) pour exprimer la phase rộsultent de faỗon naturelle de l'application des logarithmes népériens à la notation complexe Considérons par exemple le rapport de deux grandeurs de champ complexes F 1 et F 2
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Here the neper, symbol Np, is the value of Q (F) when F 1 /F 2 = e; and the bel, symbol B is the value of Q (F) when F 1/F 2 = 10 The decibel, symbol dB, is 1 dB = (1/10) B Hence
1 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)
The factor 2 in the numerical value of Q (F) expressed in bels in equation (1) has historical reasons and is explained in 4.2.
Complex notation is commonly employed in fields such as telecommunications and acoustics Utilizing the natural logarithm for logarithmic ratios of complex quantities simplifies many mathematical relations and operations This is evident from the definition of the natural logarithmic function of the ratio \( \frac{x_2}{x_1} \) as an integral.
2 x x x x x x without 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 International System of Quantities (ISQ).
In 1973, during a plenary meeting of ISO/TC 12 in Washington D.C., it was unanimously agreed to incorporate the natural logarithm into the SI system of quantities, recognizing the neper (Np) as coherent with SI units This decision was made with the participation of the Chairman and Secretary of IEC/TC 25 and was subsequently adopted by the Comité.
International des Poids et Mesures (CIPM), and the Organisation Internationale de Métrologie Légale (OIML).
With the quantity Q (F) defined by convention with the natural logarithm, i.e.
Q (F) = ln(F 1/F 2) (5) neper (Np) becomes the coherent unit, which can be replaced with one, symbol 1 (see ISO
In IEC 60027, it is customary to present the traditional order of quantity definitions before introducing the corresponding units, despite the general guideline suggesting that definitions should precede unit introductions.
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.
Since the ITU's decision in 1968, the decibel (dB) has become the internationally accepted unit of measurement This trend mirrors the common use of degrees (°) in practice, despite the coherent SI unit for plane angle being the radian (rad).
In theoretical calculations, the neper (Np) for amplitude and the radian (rad) for phase angle emerge from complex notation and natural logarithms For instance, when examining the ratio of two complex quantities, F₁ and F₂, these units play a crucial role in understanding their relationship.
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Avec les tensions U 1 0e j π /2 V et U 2 > j π /3 V on obtient rad 0,524 j Np 303 2 rad 6 j Np 10) (ln ) e ln(10 V e 3
Rapports logarithmiques de grandeurs de puissance
A quantity that is proportional to a power is referred to as a power quantity, symbolized by P In many instances, energy-related quantities are also regarded as power quantities within this context.
Puissance active, puissance réactive et puissance apparente en électrotechnique, puissance acoustique et puissance électromagnétique, ainsi que les densités de puissance correspondantes.
Power quantities are related to field quantities, and both natural and decimal logarithms are utilized for power measurements Consequently, we derive the general expressions for the logarithmic ratio Q(P) of two active powers.
P 1 et P 2 , exprimé en différentes unités: dB lg 10 B lg Np
Le néper (Np) est ici la valeur de Q ( P ) lorsque P 1 /P 2 = e 2 ; et le bel (B) est la valeur de Q ( P ) lorsque P 1 /P 2 = 10 Le décibel (dB) est défini par 1 dB = (1/10) B Par conséquent
1 (ln e 2 ) Np = (lg e 2 ) B = 10 (lg e 2 ) dB ≈ 8,685 889 dB (9)
Ce sont les mêmes facteurs de conversion que ceux obtenus en 4.1, formules (2) à (4).
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With the voltages U 1 0e j π /2 V and U 2 > j π /3 V we obtain rad 0,524 j Np 303 2 rad 6 j Np 10) (ln ) e ln(10 V e 3
4.2 Logarithmic ratios of power quantities
A quantity that is proportional to power is called a power quantity, general symbol P In many cases also energy-related quantities are labelled as power quantities in this context.
Active power, reactive power, and apparent power in electrical technology, acoustic and electromagnetic power, and corresponding power densities.
Power quantities are connected to field quantities, leading to the use of natural and decimal logarithms for their numerical values This results in general expressions for the logarithmic ratio of two active powers, \( P_1 \) and \( P_2 \), denoted as \( Q(P) \), which can be expressed in different units such as dB, lg, and Np.
Here the neper (Np) is the value of Q ( P ) when P 1 /P 2 = e 2 ; and the bel (B) is the value of Q ( P ) when P 1/P 2 = 10 The decibel (dB) is 1 dB = (1/10) B Hence
1 (ln e 2 ) Np = (lg e 2 ) B = 10 (lg e 2 ) dB ≈ 8,685 889 dB (9)
These are the same conversion factors as those obtained in sub-clause 4.1, equations (2) to
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En définissant par convention la grandeur Q ( P ) par un logarithme népérien, soit
Q ( P ) = (1/2) ln(P 1 /P 2 ) (12) le néper (Np) devient l'unité cohérente, qui peut être remplacée par l'unité un, symbole 1 (voir l'ISO 31-2, 2-10).
D'après la définition d'une grandeur de champ, soit
(15) Dans le cas général la relation entre Q ( P ) et Q ( F ) dépend de k 1 /k 2
Dans le cas particulier ó k 1 = k 2 on a Q ( P ) = Q ( F ).
Cela explique la présence du facteur 1/2 dans la formule (12), des facteurs 2 et 20 dans les valeurs numériques de la formule (1) et du facteur 1/2 dans celles de la formule (8).
En électrotechnique le rapport k 1 /k 2 est souvent un rapport d'impédances ou d'admittances.
When comparing logarithmic ratios of field magnitudes, it can lead to misleading conclusions or lack significance without proper information regarding impedances or admittances.
Considérons les puissances complexes S 1 et S 2 respectivement à l'entrée (1) et à la sortie (2) d'une ligne de transmission. i i i i i i i i i i i i i I I Z I Z
Z i = U i /I i est l'impédance; et ∗ indique le complexe conjugué.
L'exposant de transfert de la puissance complexe Γ S , avec ses parties réelle et imaginaire A S et B S , respectivement, est donc:
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With the quantity Q ( P ) defined by convention with the natural logarithm, i.e.
Q ( P ) = (1/2) ln(P 1/P 2) (12) neper (Np) becomes the coherent unit, which can be replaced with one, symbol 1 (see ISO
Following the definition of a field quantity, let
In the general case, the relation between Q ( P ) and Q ( F ) depends on k 1/k 2.
In the special case when k 1 = k 2 then Q ( P ) = Q ( F ).
This explains why the factor 1/2 appears in equation (12) and the factors 2, 20, and 1/2 appear in the numerical values in the equations (1) and (8), respectively.
In electrical technology, the ratio of impedance or admittance, represented as \$k_1/k_2\$, is crucial for accurate analysis Without sufficient information about these ratios, comparing logarithmic values of field quantities can lead to misleading or meaningless conclusions.
Consider the complex powers S 1 and S 2 at the input (1) and output (2), respectively, of a transmission line. i i i i i i i i i i i i i I I Z I Z
Thus, the transfer exponent for complex power Γ S with its real and imaginary parts A S and B S , respectively, becomes:
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L'exposant de transfert en tension et l'affaiblissement en tension sont respectivement:
L'exposant de transfert en courant et l'affaiblissement en courant sont respectivement:
Z et aussi Γ S = A U = A I si, et seulement si
Niveaux
A level, denoted by L, represents the logarithmic ratio of two field quantities or two power quantities, where the denominator is a reference quantity of the same nature as the numerator.
Les niveaux complexes ne sont pas habituels Les niveaux sont donc généralement exprimés en décibels.
La différence entre deux niveaux déterminés par rapport à la même grandeur de référence est indépendante de la valeur de celle-ci.
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The transfer exponent for voltage and the voltage attenuation, respectively, are:
The transfer exponent for electric current and the electric current attenuation, respectively, are:
Thus it is obtained that
Z and that Г S = A U = A I if, and only if
A level, denoted as L, represents the logarithmic ratio of two field or power quantities, with the denominator being a reference quantity of the same type as the numerator.
Complex levels are not customary Therefore, levels are generally given in decibels.
The difference of two levels determined with the same reference quantity is independent of the value of the reference quantity.
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Pour une différence de niveaux de puissance on obtient dB lg 10 dB lg
L P P quelle que soit la valeur de P ref
Informations supplémentaires relatives aux rapports logarithmiques de
de champ et de grandeurs de puissance
According to the fundamental principles of dimensional algebra, it is incorrect to add a name or unit symbol to convey specific information about the nature of the measured quantity or the measurement context (see ISO 31-0).
However, such additions are still commonly used for levels in telecommunications and acoustics They are also typical for weighting scales in acoustics It is important to associate this additional information with the quantity rather than the unit.
The reference values for sound levels and weighting scales in acoustics should be indicated as follows, with the final example showing both a reference value and an A-weighting scale.
L P (re 1 mW) = 7 dB ou L P /1 mW = 7 dB
L E (re 1 àV/m) = 5 Np ou L E /1 àV/m = 5 Np
L p (re 20 àPa) = 15 dB ou L p /20 à Pa = 15 dB
L A (re 20 àPa) = 60 dB ou L A = 60 dB
If the numerical value of the reference quantity after "re" is equal to 1, it can be omitted, for example, when it appears in parentheses or after a slash in the subscript.
L P (re mW) = 7 dB ou L P /mW = 7 dB.
In practice, the short form is often used, which includes a space between the unit symbol and additional information, such as a reference value or a weighting scale.
Lorsqu'on utilise la forme courte de la notation, il convient d'éviter l'omission de la valeur numérique 1 entre les parenthèses en vue d'éviter la confusion avec l'échelle de pondération A.
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For a power level difference we obtain dB lg 10 dB lg
L P P independent of the value of P ref.
4.4 Additional information on logarithmic ratios of field quantities and power quantities
According to fundamental principles of quantity calculus, attaching a unit name or symbol to convey specific information about a quantity or measurement context is incorrect (ISO 31-0, 3.2.1) Despite this, such attachments are frequently utilized in telecommunications and acoustics, particularly for weighting scales It is essential that any supplementary information is represented by the quantity itself, rather than the unit.
Reference values for acoustic levels and weighting scales should be clearly specified, with the final example illustrating both a reference value and an A-weighting scale.
L P (re 1 mW) = 7 dB or L P/1 mW = 7 dB
L E (re 1 àV/m) = 5 Np or L E/1 àV/m = 5 Np
L p (re 20 àPa) = 15 dB or L p/20 àPa = 15 dB
L A(re 20 àPa) = 60 dB or L A = 60 dB
When the numerical value of the reference quantity following "re" in brackets or after the solidus in the subscript is 1, it can be omitted For instance, L P (re mW) = 7 dB can also be expressed as L P/mW = 7 dB.
NOTE 5 In practice, the following short form with a space between the unit symbol and the additional information giving for example reference value or weighting scale is often used:
When the short form notation is used, the omitting of the numerical equal to 1 in the parenthesis should be avoided in order to avoid confusion with the weighting scale A.
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NOTE 6 Il convient cependant d'éviter les formes suivantes, parce que toute information supplémentaire doit être portée par la grandeur et non par la valeur numérique ou l'unité.
In diagrams, tables, and measuring devices, it is advisable to present numerical values as the ratio of the quantity being measured to the unit in which it is expressed.
5 Grandeurs logarithmiques de la théorie de l'information
Dans la théorie de l'information, on utilise les logarithmes de trois bases différentes pour exprimer les valeurs numériques Ces logarithmes sont :
– les logarithmes binaires, symbole lb (ou log 2 ),
– les logarithmes népériens, symbole ln ou (log e ),
– les logarithmes décimaux, symbole lg ou (log 10 ).
En théorie de l'information, on obtient les expressions générales suivantes d'une grandeur logarithmique, Q, exprimée en différentes unités:
In information theory, the quantity \( Q \) can be expressed in different units depending on the base of the logarithm used Specifically, \( Q = (lb \, x) \) represents the logarithm base 2, denoted as \( Sh \) for Shannon; \( Q = (ln \, x) \) corresponds to the natural unit of information, symbolized as \( nat \), when \( x = e \); and \( Q = (lg \, x) \) refers to the Hartley unit, symbolized as \( Hart \), when \( x = 10 \).
1 Sh = (lb 2) Sh = (ln 2) nat = (lg 2) Hart ≈ 0,693 147 nat ≈ 0,301 030 Hart (17)
1 nat = (ln e) nat = (lg e) Hart = (lb e) Sh ≈ 0,434 294 Hart ≈ 1,442 695 Sh (18)
1 Hart = (lg 10) Hart = (lb 10) Sh = (ln 10) nat ≈ 3,321 928 Sh ≈ 2,302 585 nat (19)
Complex notation is not utilized in information theory, and neither the system of units on which the SI is based nor the SI itself has any relation to information theory Due to technical reasons, binary representation is widely used in information technologies, which is why binary logarithms are conventionally employed instead of natural logarithms in the equations defining the units used in information theory It is also common in information theory to operate with unspecified base logarithms, denoted simply as log, when specific values of the quantities are not required (see ISO 31-11, 11-8.4).
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NOTE 6 The following forms should, however, be avoided because all additional information shall be carried by the quantity and not by the quantity value or unit:
It is advisable to represent numerical values in diagrams, tables, and measuring instruments as the ratio of the quantity being measured to the corresponding unit of measurement.
In information theory, logarithms with three different bases are used for the numerical values.
– binary logarithm, symbol lb (or log2),
– natural logarithm, symbol ln (or loge),
– decimal logarithm, symbol lg (or log10).
In information theory the following general expressions of a logarithmic quantity, Q, expressed in different units are obtained:
In the equation \( Q = (lb \, x) \, Sh = (ln \, x) \, nat = (lg \, x) \, Hart \), where \( x \) is a real number, the symbols represent different units of information The Shannon unit, denoted as \( Sh \), corresponds to the value of \( Q \) when \( x = 2 \) The natural unit of information, represented by \( nat \), is the value of \( Q \) when \( x = e \) Lastly, the Hartley unit, indicated by \( Hart \), is the value of \( Q \) when \( x = 10 \).
1 Sh = (lb 2) Sh = (ln 2) nat = (lg 2) Hart ≈ 0,693 147 nat ≈ 0,301 030 Hart (17)
1 nat = (ln e) nat = (lg e) Hart = (lb e) Sh ≈ 0,434 294 Hart ≈ 1,442 695 Sh (18)
1 Hart = (lg 10) Hart = (lb 10) Sh = (ln 10) nat ≈ 3,321 928 Sh ≈ 2,302 585 nat (19)
Complex notation is not used in information theory Neither the International System of
The International System of Quantities (ISQ) and the SI units do not influence information theory In information technology, binary representation is prevalent, leading to the conventional use of binary logarithms instead of natural logarithms in defining quantities in information theory Additionally, it is common practice in this field to utilize logarithms with an unspecified base, denoted as log, when specific quantity values are not required.
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En définissant par convention la grandeur Q par un logarithme binaire, soit
Q = lb x (20) le shannon (Sh) devient l'unité cohérente, qui peut être remplacée par l'unité un, symbole 1.
Pour un événement de probabilité p = 1/3, la quantité d'information I est
Généralités
Il y a d'autres grandeurs logarithmiques que les rapports logarithmiques de grandeurs de champ ou de grandeurs de puissance et que les grandeurs logarithmiques de la théorie de l'information.
Intervalle logarithmique de fréquence, densité optique, pH.
Units such as neper (Np), bel (B), and the submultiple decibel (dB) should not be used when there is no relationship between the quantity in question and either a field quantity or a power quantity.
On ne doit utiliser les unités shannon (Sh), unité naturelle d'information (nat) et hartley (Hart) que dans la théorie de l'information.
Logarithmic ratios of field quantities
A quantity the square of which is proportional to power when it acts on a linear system is here called a field quantity, general symbol F.
Electric current, voltage, electric field strength, sound pressure, particle speed, and force are field quantities.
For sinusoidal time-varying field quantities, the ratio of the amplitudes or the root-mean- square values is the argument of the logarithm.
For non-sinusoidal field quantities, the root-mean-square (RMS) value is calculated over a specified time interval In the case of a periodic quantity, this interval corresponds to the periodic time.
For logarithmic ratios of field quantities, logarithms with two different bases are used for the numerical values These logarithms are:
― natural logarithm, symbol ln (or loge),
― decimal logarithm, symbol lg (or log10).
For real field quantity ratios, F 1/F 2, the following general expressions of a logarithmic ratio,
Q (F), expressed in different units are obtained: dB lg
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Le néper, symbole Np, est ici la valeur de Q ( F ) lorsque F 1 /F 2 = e; et le bel, symbole B, est la valeur de Q ( F ) lorsque F 1 /F 2 = 10 Le décibel, symbole dB, est défini par 1 dB = (1/10) B Par conséquent
1 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)
Le facteur 2 intervient dans la valeur numérique de Q ( F ) exprimée en bels dans la formule (1) pour des raisons historiques qui sont expliquées en 4.2.
Complex notation is frequently employed for field quantities, particularly in telecommunications and acoustics To express logarithmic ratios of complex quantities, only natural logarithms yield easily usable results.
Many mathematical operations and relationships become simpler when the logarithm is natural This is evident from the fact that the natural logarithm of the ratio \( \frac{x_2}{x_1} \) can be defined using an integral.
2 x x x x x x sans facteur numérique comme avec les autres bases.
C'est pourquoi on emploie les logarithmes népériens dans le système de grandeurs sur lequel le SI est fondé, c'est-à-dire le Système international de grandeurs (ISQ).
In 1973, during a plenary meeting of ISO Study Committee 12 in Washington D.C., it was unanimously decided to adopt natural logarithms in the system of quantities underlying the International System of Units (SI), recognizing the neper (Np) as a coherent unit within the SI framework This decision was later reaffirmed by the International Committee for Weights and Measures (CIPM) and the International Organization of Legal Metrology (OIML).
En définissant par convention la grandeur Q ( F ) par un logarithme népérien, soit
Q ( F ) = ln(F 1 /F 2 ) (5) le néper (Np) devient l'unité cohérente, qui peut être remplacée par l'unité un, symbole 1 (voir l'ISO 31-2, 2-9).
It is generally advisable to define a quantity before introducing its corresponding units For historical reasons, this section of IEC 60027 follows the traditional order of presentation.
Pour les applications pratiques, notamment dans les télécommunications et l'acoustique, le sous-multiple décibel (dB) du bel (B), fondé sur les logarithmes décimaux, est d'usage courant.
NOTE 4 En pratique, l'emploi du décibel (dB) l'a emporté sur le plan international depuis que l'UIT a décidé en
In 1968, the use of the decibel became standard practice This approach is somewhat similar to the common use of degrees as a unit of plane angle (… °) instead of the SI coherent unit, the radian (rad).
Dans les calculs théoriques, l'emploi du néper (Np) pour exprimer l'amplitude et celui du radian
(rad) pour exprimer la phase rộsultent de faỗon naturelle de l'application des logarithmes népériens à la notation complexe Considérons par exemple le rapport de deux grandeurs de champ complexes F 1 et F 2
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Here the neper, symbol Np, is the value of Q (F) when F 1 /F 2 = e; and the bel, symbol B is the value of Q (F) when F 1/F 2 = 10 The decibel, symbol dB, is 1 dB = (1/10) B Hence
1 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)
The factor 2 in the numerical value of Q (F) expressed in bels in equation (1) has historical reasons and is explained in 4.2.
Complex notation is commonly utilized in fields such as telecommunications and acoustics The natural logarithm is particularly advantageous for taking logarithms of complex-quantity ratios, as it simplifies many mathematical relations and operations This is evident from the definition of the natural logarithmic function of the ratio \( \frac{x_2}{x_1} \) as an integral.
2 x x x x x x without 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 International System of Quantities (ISQ).
In 1973, during a plenary meeting of ISO/TC 12 in Washington D.C., it was unanimously agreed to incorporate the natural logarithm into the SI system of quantities, recognizing the neper (Np) as coherent with SI units This decision was made with the participation of the Chairman and Secretary of IEC/TC 25 and was subsequently adopted by the Comité.
International des Poids et Mesures (CIPM), and the Organisation Internationale de Métrologie Légale (OIML).
With the quantity Q (F) defined by convention with the natural logarithm, i.e.
Q (F) = ln(F 1/F 2) (5) neper (Np) becomes the coherent unit, which can be replaced with one, symbol 1 (see ISO
In IEC 60027, it is customary to present the traditional order of quantity definitions before introducing the corresponding units, despite the general guideline suggesting that definitions should precede unit introductions.
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.
Since 1968, the international standard for measuring sound has favored the decibel (dB), as established by the ITU This trend mirrors the common use of degrees (°) in practice, rather than the SI unit radian (rad), for measuring plane angles.
In theoretical calculations, the neper (Np) for amplitude and the radian (rad) for phase angle emerge from complex notation and natural logarithms For instance, when examining the ratio of two complex quantities, F₁ and F₂, these units play a crucial role in understanding their relationship.
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Avec les tensions U 1 0e j π /2 V et U 2 > j π /3 V on obtient rad 0,524 j Np 303 2 rad 6 j Np 10) (ln ) e ln(10 V e 3
4.2 Rapports logarithmiques de grandeurs de puissance
A quantity proportional to a power is referred to as a power quantity, symbolized by P In many instances, energy-related quantities are also regarded within this context as power quantities.
Puissance active, puissance réactive et puissance apparente en électrotechnique, puissance acoustique et puissance électromagnétique, ainsi que les densités de puissance correspondantes.
Power quantities are related to field quantities, and both natural and decimal logarithms are utilized for power measurements Consequently, we derive the general expressions for the logarithmic ratio Q(P) of two active powers.
P 1 et P 2 , exprimé en différentes unités: dB lg 10 B lg Np
Le néper (Np) est ici la valeur de Q ( P ) lorsque P 1 /P 2 = e 2 ; et le bel (B) est la valeur de Q ( P ) lorsque P 1 /P 2 = 10 Le décibel (dB) est défini par 1 dB = (1/10) B Par conséquent
1 (ln e 2 ) Np = (lg e 2 ) B = 10 (lg e 2 ) dB ≈ 8,685 889 dB (9)
Ce sont les mêmes facteurs de conversion que ceux obtenus en 4.1, formules (2) à (4).
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With the voltages U 1 0e j π /2 V and U 2 > j π /3 V we obtain rad 0,524 j Np 303 2 rad 6 j Np 10) (ln ) e ln(10 V e 3
Logarithmic ratios of power quantities
A quantity that is proportional to power is called a power quantity, general symbol P In many cases also energy-related quantities are labelled as power quantities in this context.
Active power, reactive power, and apparent power in electrical technology, acoustic and electromagnetic power, and corresponding power densities.
Power quantities are connected to field quantities, leading to the use of natural and decimal logarithms for their numerical values This results in general expressions for the logarithmic ratio of two active powers, \( P_1 \) and \( P_2 \), denoted as \( Q(P) \), which can be expressed in different units as \( dB \), \( \text{lg} \), \( 10 \), \( B \), \( \text{lg} \), and \( Np \).
Here the neper (Np) is the value of Q ( P ) when P 1 /P 2 = e 2 ; and the bel (B) is the value of Q ( P ) when P 1/P 2 = 10 The decibel (dB) is 1 dB = (1/10) B Hence
1 (ln e 2 ) Np = (lg e 2 ) B = 10 (lg e 2 ) dB ≈ 8,685 889 dB (9)
These are the same conversion factors as those obtained in sub-clause 4.1, equations (2) to
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En définissant par convention la grandeur Q ( P ) par un logarithme népérien, soit
Q ( P ) = (1/2) ln(P 1 /P 2 ) (12) le néper (Np) devient l'unité cohérente, qui peut être remplacée par l'unité un, symbole 1 (voir l'ISO 31-2, 2-10).
D'après la définition d'une grandeur de champ, soit
(15) Dans le cas général la relation entre Q ( P ) et Q ( F ) dépend de k 1 /k 2
Dans le cas particulier ó k 1 = k 2 on a Q ( P ) = Q ( F ).
Cela explique la présence du facteur 1/2 dans la formule (12), des facteurs 2 et 20 dans les valeurs numériques de la formule (1) et du facteur 1/2 dans celles de la formule (8).
En électrotechnique le rapport k 1 /k 2 est souvent un rapport d'impédances ou d'admittances.
When comparing logarithmic ratios of field magnitudes, it can lead to misleading conclusions or lack significance without proper information regarding impedances or admittances.
Considérons les puissances complexes S 1 et S 2 respectivement à l'entrée (1) et à la sortie (2) d'une ligne de transmission. i i i i i i i i i i i i i I I Z I Z
Z i = U i /I i est l'impédance; et ∗ indique le complexe conjugué.
L'exposant de transfert de la puissance complexe Γ S , avec ses parties réelle et imaginaire A S et B S , respectivement, est donc:
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With the quantity Q ( P ) defined by convention with the natural logarithm, i.e.
Q ( P ) = (1/2) ln(P 1/P 2) (12) neper (Np) becomes the coherent unit, which can be replaced with one, symbol 1 (see ISO
Following the definition of a field quantity, let
In the general case, the relation between Q ( P ) and Q ( F ) depends on k 1/k 2.
In the special case when k 1 = k 2 then Q ( P ) = Q ( F ).
This explains why the factor 1/2 appears in equation (12) and the factors 2, 20, and 1/2 appear in the numerical values in the equations (1) and (8), respectively.
In electrical technology, the ratio of impedance or admittance, denoted as \( k_1/k_2 \), is crucial Without sufficient information about these values, comparing logarithmic ratios of field quantities can lead to misleading or meaningless conclusions.
Consider the complex powers S 1 and S 2 at the input (1) and output (2), respectively, of a transmission line. i i i i i i i i i i i i i I I Z I Z
Thus, the transfer exponent for complex power Γ S with its real and imaginary parts A S and B S , respectively, becomes:
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L'exposant de transfert en tension et l'affaiblissement en tension sont respectivement:
L'exposant de transfert en courant et l'affaiblissement en courant sont respectivement:
Z et aussi Γ S = A U = A I si, et seulement si
A level, denoted by L, represents the logarithmic ratio of two field quantities or two power quantities, where the denominator is a reference quantity of the same nature as the numerator.
Les niveaux complexes ne sont pas habituels Les niveaux sont donc généralement exprimés en décibels.
La différence entre deux niveaux déterminés par rapport à la même grandeur de référence est indépendante de la valeur de celle-ci.
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The transfer exponent for voltage and the voltage attenuation, respectively, are:
The transfer exponent for electric current and the electric current attenuation, respectively, are:
Thus it is obtained that
Z and that Г S = A U = A I if, and only if
Levels
A level, denoted as L, represents the logarithmic ratio of two field or power quantities, with the denominator being a reference quantity of the same type as the numerator.
Complex levels are not customary Therefore, levels are generally given in decibels.
The difference of two levels determined with the same reference quantity is independent of the value of the reference quantity.
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Pour une différence de niveaux de puissance on obtient dB lg 10 dB lg
L P P quelle que soit la valeur de P ref
4.4 Informations supplémentaires relatives aux rapports logarithmiques de grandeurs de champ et de grandeurs de puissance
According to the fundamental principles of dimensional algebra, it is incorrect to add a name or unit symbol to convey specific information about the nature of the considered quantity or the measurement context (see ISO 31-0).
However, such additions are still commonly used for levels in telecommunications and acoustics They are also typical for weighting scales in acoustics It is important to associate this additional information with the quantity rather than the unit.
The reference values for sound levels and weighting scales in acoustics should be indicated as follows, with the final example showing both a reference value and an A-weighting scale.
L P (re 1 mW) = 7 dB ou L P /1 mW = 7 dB
L E (re 1 àV/m) = 5 Np ou L E /1 àV/m = 5 Np
L p (re 20 àPa) = 15 dB ou L p /20 à Pa = 15 dB
L A (re 20 àPa) = 60 dB ou L A = 60 dB
If the numerical value of the reference quantity after "o" is equal to 1, it can be omitted, for example, when it appears in parentheses or after the slash in the subscript.
L P (re mW) = 7 dB ou L P /mW = 7 dB.
In practice, the short form is often used, which includes a space between the unit symbol and additional information, such as a reference value or a weighting scale.
Lorsqu'on utilise la forme courte de la notation, il convient d'éviter l'omission de la valeur numérique 1 entre les parenthèses en vue d'éviter la confusion avec l'échelle de pondération A.
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For a power level difference we obtain dB lg 10 dB lg
L P P independent of the value of P ref.
Additional information on logarithmic ratios of field quantities and power quantities
According to fundamental principles of quantity calculus, attaching a unit name or symbol to convey specific information about a measurement context is incorrect (refer to ISO 31-0, 3.2.1) Despite this, such attachments are frequently utilized in telecommunications and acoustics, particularly for weighting scales It is essential that any supplementary information is represented by the quantity itself, rather than the unit.
Reference values for acoustic levels and weighting scales should be clearly specified, with the final example illustrating both a reference value and an A-weighting scale.
L P (re 1 mW) = 7 dB or L P/1 mW = 7 dB
L E (re 1 àV/m) = 5 Np or L E/1 àV/m = 5 Np
L p (re 20 àPa) = 15 dB or L p/20 àPa = 15 dB
L A(re 20 àPa) = 60 dB or L A = 60 dB
When the numerical value of the reference quantity following "re" in brackets or after the solidus in the subscript is 1, it can be omitted For instance, L P (re mW) can be expressed simply as L P/mW = 7 dB.
NOTE 5 In practice, the following short form with a space between the unit symbol and the additional information giving for example reference value or weighting scale is often used:
When the short form notation is used, the omitting of the numerical equal to 1 in the parenthesis should be avoided in order to avoid confusion with the weighting scale A.
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NOTE 6 Il convient cependant d'éviter les formes suivantes, parce que toute information supplémentaire doit être portée par la grandeur et non par la valeur numérique ou l'unité.
In diagrams, tables, and measuring devices, it is advisable to present the numerical value as the ratio of the quantity being measured to the unit in which it is expressed.
5 Grandeurs logarithmiques de la théorie de l'information
Dans la théorie de l'information, on utilise les logarithmes de trois bases différentes pour exprimer les valeurs numériques Ces logarithmes sont :
– les logarithmes binaires, symbole lb (ou log 2 ),
– les logarithmes népériens, symbole ln ou (log e ),
– les logarithmes décimaux, symbole lg ou (log 10 ).
En théorie de l'information, on obtient les expressions générales suivantes d'une grandeur logarithmique, Q, exprimée en différentes unités:
In information theory, the quantity \( Q \) is defined as \( Q = (lb \, x) \) for logarithm base 2, \( Sh = (ln \, x) \) for natural logarithm, and \( lg \, x \) for logarithm base 10 The Shannon unit, denoted as \( Sh \), represents the value of \( Q \) when \( x = 2 \) The natural unit of information, symbolized as \( nat \), corresponds to the value of \( Q \) when \( x = e \) Lastly, the Hartley unit, indicated as \( Hart \), is the value of \( Q \) when \( x = 10 \).
1 Sh = (lb 2) Sh = (ln 2) nat = (lg 2) Hart ≈ 0,693 147 nat ≈ 0,301 030 Hart (17)
1 nat = (ln e) nat = (lg e) Hart = (lb e) Sh ≈ 0,434 294 Hart ≈ 1,442 695 Sh (18)
1 Hart = (lg 10) Hart = (lb 10) Sh = (ln 10) nat ≈ 3,321 928 Sh ≈ 2,302 585 nat (19)
Complex notation is not utilized in information theory, and neither the system of units on which the SI is based nor the SI itself has any relation to information theory Due to technical reasons, binary representation is widely used in information technologies, which is why binary logarithms are conventionally employed instead of natural logarithms in the equations defining the units used in information theory It is also common in information theory to operate with unspecified base logarithms, denoted simply as log, when specific values of the quantities are not required (see ISO 31-11, 11-8.4).
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NOTE 6 The following forms should, however, be avoided because all additional information shall be carried by the quantity and not by the quantity value or unit:
It is advisable to represent numerical values in diagrams, tables, and measuring instruments as the ratio of the quantity being measured to the corresponding unit of measurement.
In information theory, logarithms with three different bases are used for the numerical values.
– binary logarithm, symbol lb (or log2),
– natural logarithm, symbol ln (or loge),
– decimal logarithm, symbol lg (or log10).
In information theory the following general expressions of a logarithmic quantity, Q, expressed in different units are obtained:
In the equation \( Q = (lb \, x) \, Sh = (ln \, x) \, nat = (lg \, x) \, Hart \), where \( x \) is a real number, the symbols represent different units of information The Shannon unit, denoted as \( Sh \), corresponds to the value of \( Q \) when \( x = 2 \) The natural unit of information, represented by \( nat \), is the value of \( Q \) when \( x = e \) Lastly, the Hartley unit, indicated by \( Hart \), is the value of \( Q \) when \( x = 10 \).
1 Sh = (lb 2) Sh = (ln 2) nat = (lg 2) Hart ≈ 0,693 147 nat ≈ 0,301 030 Hart (17)
1 nat = (ln e) nat = (lg e) Hart = (lb e) Sh ≈ 0,434 294 Hart ≈ 1,442 695 Sh (18)
1 Hart = (lg 10) Hart = (lb 10) Sh = (ln 10) nat ≈ 3,321 928 Sh ≈ 2,302 585 nat (19)
Complex notation is not used in information theory Neither the International System of
The International System of Quantities (ISQ) and the SI units do not influence information theory In information technology, binary representation is prevalent, leading to the conventional use of binary logarithms instead of natural logarithms in defining quantities in information theory Additionally, it is common practice in this field to utilize logarithms with an unspecified base, denoted as log, when specific quantity values are not required.
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En définissant par convention la grandeur Q par un logarithme binaire, soit
Q = lb x (20) le shannon (Sh) devient l'unité cohérente, qui peut être remplacée par l'unité un, symbole 1.
Pour un événement de probabilité p = 1/3, la quantité d'information I est
Il y a d'autres grandeurs logarithmiques que les rapports logarithmiques de grandeurs de champ ou de grandeurs de puissance et que les grandeurs logarithmiques de la théorie de l'information.
Intervalle logarithmique de fréquence, densité optique, pH.
Units such as neper (Np), bel (B), and the submultiple decibel (dB) should not be used unless there is a relationship between the quantity in question and either a field quantity or a power quantity.
On ne doit utiliser les unités shannon (Sh), unité naturelle d'information (nat) et hartley (Hart) que dans la théorie de l'information.
Pour les intervalles logarithmiques de fréquence, on utilise les logarithmes de deux bases différentes pour exprimer les valeurs numériques Ces logarithmes sont :
– les logarithmes binaires, symbole lb (ou log 2 ),
– les logarithmes décimaux, symbole lg ou (log 10 ).
Pour un intervalle logarithmique de fréquence, G, on obtient les expressions générales suivantes exprimées en différentes unités: dec lg oct lb
G f (21) ó f 1 et f 2 ≥ f 1 sont deux fréquences ; l'octave (oct) est ici la valeur de G lorsque l'argument f 2 /f 1 = 2; et la décade (dec) est la valeur de G lorsque f 2 /f 1 = 10.
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With the quantity Q defined by convention with the binary logarithm, i.e.
Q = lb x (20) shannon (Sh) becomes the coherent unit which can be replaced with one, symbol 1.
For an event with probability p = 1/3, the information content I is
General
There are other logarithmic quantities than logarithmic ratios of field or power quantities, and logarithmic information-theory quantities.
Logarithmic frequency interval, optical density, pH.
The units neper (Np) and bel (B), including its sub-multiple decibel (dB), should only be utilized when there is a relationship between the quantity in question and either a field quantity or a power quantity.
The units shannon (Sh), natural unit of information (nat), and hartley (Hart), shall be used only in information theory.
Logarithmic frequency interval
For logarithmic frequency interval, logarithms with two different bases are used for the numerical value These logarithms are:
– binary logarithm, symbol lb (or log2),
– decimal logarithm, symbol lg (or log10).
For logarithmic frequency interval, the following general expressions expressed in different units are obtained: dec lg oct lb
G f (21) where f 1 and f 2≥ f 1 are two frequencies; here the octave (oct) is the value of G when the argument f 2 /f 1 = 2; and the decade (dec) is the value of G when f 2/f 1 = 10.
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1 oct = (lb 2) oct = (lg 2) dec ≈ 0,301 030 dec (22)
1 dec = (lg 10) dec = (lb 10) oct ≈ 3,321 928 oct (23)
NOTE 7 Un sous-multiple de la décade est le savart, 1 savart = 0,001 dec.
En définissant par convention la grandeur G par un logarithme binaire, soit
G = lb(f 2 /f 1 ) (24) ce qui est l'usage courant en acoustique, l'octave, symbole oct, devient l'unité cohérente qui peut être remplacée par l'unité un, symbole 1 (voir l'ISO 31-7, 7-3).
The names, symbols, and definitions of logarithmic quantities and their units in electrotechnology are provided in other sections of IEC 60027, primarily in part 2 Additionally, the names, symbols, and definitions of logarithmic quantities and their units in information theory are outlined in ISO/IEC 2382-16.
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1 oct = (lb 2) oct = (lg 2) dec ≈ 0,301 030 dec (22)
1 dec = (lg 10) dec = (lb 10) oct ≈ 3,321 928 oct (23)
NOTE 7 A sub-multiple of the decade is the savart, 1 savart = 0,001 dec.
With the quantity G defined by convention with the binary logarithm, i.e.
G = lb(f 2/f 1) (24) which is the practice in acoustics, the octave, symbol oct, becomes the coherent unit which can be replaced with one, symbol 1 (see ISO 31-7, 7-3).
Logarithmic quantities, along with their names, symbols, definitions, and units in electrical technology, are detailed in IEC 60027, particularly in part 2 Additionally, ISO/IEC 2382-16 provides similar information for logarithmic quantities in the field of information theory.
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