# Bel (unit)

Physical unit
Unit name Bel
Unit symbol ${\ displaystyle \ mathrm {B}}$
Physical quantity (s) Levels and dimensions
Formula symbol ${\ displaystyle Q, \, L}$(Level), (dimensions)
${\ displaystyle A, \, G}$
dimension ${\ displaystyle {\ mathsf {1}}}$
Named after Alexander Graham Bell

The Bel ( unit symbol  B) is an auxiliary unit of measurement named after Alexander Graham Bell for the identification of the decadic logarithm of the ratio of two quantities of the same kind for levels and measurements . These are u. a. in electrical engineering and in the acoustics applied, such as information on losses and gains , voltages and sound pressure .

As a rule, the decibel (unit symbol dB) is used instead of the bel , i.e. the tenth part of a bel.

Unlike in other European countries, the decibel is a legal unit in Switzerland and Austria .

## Definition of bel and decibel

Conversion: examples
${\ displaystyle Q}$ ${\ displaystyle P_ {1} / P_ {2}}$ ${\ displaystyle F_ {1} / F_ ​​{2}}$
40 dB 10,000 100
20 dB 100 10
10 dB 10 ≈ 3.16
6 dB ≈ 4 ≈ 2
3 dB ≈ 2 ≈ 1.41
1 dB ≈ 1.26 ≈ 1.12
0 dB 1 1
−1 dB ≈ 0.79 ≈ 0.89
−3 dB ≈ 0.5 ≈ 0.71
−6 dB ≈ 0.25 ≈ 0.5
−10 dB 0.1 ≈ 0.32
−20 dB 0.01 0.1
−40 dB 0.0001 0.01

More common than Bel is the decibel, which is formed using the unit prefix "Dezi" (prefix d):

{\ displaystyle {\ begin {aligned} 1 \, \ mathrm {dB} & = {\ frac {1} {10}} \, \ mathrm {B} \\ 10 \, \ mathrm {dB} & = 1 \ , \ mathrm {B} \ end {aligned}}}

The bel (or decibel) is used to identify the level , i.e. the decadic logarithm of the ratio of two similar energy or power quantities and : ${\ displaystyle Q _ {(P)}}$ ${\ displaystyle P_ {1}}$${\ displaystyle P_ {2}}$

${\ displaystyle Q _ {(P)} = \ lg {\ frac {P_ {1}} {P_ {2}}} \, \ mathrm {B} = 10 \, \ lg {\ frac {P_ {1}} {P_ {2}}} \, \ mathrm {dB}}$

If the power ratio is, for example , the result is a level of 2 B (or 20 dB), da . One could also say: What you have to 10 potentiate (the "decadent" logarithm always has the base 10), so that one - as in this example - comes to 100 ?: ${\ displaystyle {P_ {1}} / {P_ {2}} = 100}$${\ displaystyle 10 ^ {2} = 100}$

${\ displaystyle Q _ {(P)} = \ lg {\ frac {100} {1}} \, \ mathrm {B} = 2 \, \ mathrm {B} = 20 \, \ mathrm {dB} \}$

In linear systems, the power quantities are proportional to the squares of the effective values of the acting field quantities (e.g. electrical voltage , sound pressure ). Accordingly, the term "field size" has been replaced in the standardization by the term "service root size". ${\ displaystyle P}$ ${\ displaystyle F}$

${\ displaystyle P \ sim F ^ {2}; \ quad {\ frac {P_ {1}} {P_ {2}}} = {\ frac {F_ {1} ^ {2}} {F_ {2} ^ {2}}} = \ left ({\ frac {F_ {1}} {F_ {2}}} \ right) ^ {2}; \ quad \ lg {\ frac {P_ {1}} {P_ {2 }}} = 2 \ cdot \ lg {\ frac {F_ {1}} {F_ {2}}}.}$

If the performance ratio is to be calculated on the basis of performance root variables, then:

${\ displaystyle Q _ {(F)} = 10 \, \ lg {\ frac {F_ {1} ^ {2}} {F_ {2} ^ {2}}} \, \ mathrm {dB} = 20 \, \ lg {\ frac {F_ {1}} {F_ {2}}} \, \ mathrm {dB}.}$

The logarithmic ratios of the power quantities and the power root quantities differ by a factor of two, see also the conversion table.

## Conversion into the unit Neper

Both decibel and neper are used to identify the logarithms of ratios. They differ by a fixed factor. With the definition

${\ displaystyle Q _ {(F)} = 20 \ lg {\ frac {F_ {1}} {F_ {2}}} \, \ mathrm {dB} = \ ln {\ frac {F_ {1}} {F_ {2}}} \, \ mathrm {Np},}$

where denotes the natural logarithm , and with the conversion valid for every > 0${\ displaystyle \ ln}$${\ displaystyle x}$

${\ displaystyle \ ln x = \ lg x \ cdot \ ln 10}$

is independent of ${\ displaystyle F_ {1} / F_ ​​{2}}$

${\ displaystyle 20 \; \ mathrm {dB} = \ ln 10 \; \ mathrm {Np}; \ quad 1 \; \ mathrm {dB} \ approx 0 {,} 1151 \; \ mathrm {Np}; \ quad 1 \; \ mathrm {Np} \ approx 8 {,} 686 \; \ mathrm {dB}.}$

## Decibel and Neper, historical development

Although not the bel or decibel, but the neper is the coherent auxiliary unit for logarithmic ratios in relation to the International System of Units SI , the decibel is mainly used in practice. On the one hand there are historical reasons: in the USA until 1923 the auxiliary unit “Mile Standard Cable” (msc) was used as a unit for the attenuation of a telephone connection. This unit corresponds to the attenuation of a certain cable type (“19 gauge”) at a length of one mile and a frequency of 800 Hz and at the same time the mean subjective perceptibility threshold when comparing two volume levels . The latter also applies to the decibel. Therefore, when using the decibel, the numerical values ​​were roughly the same as when using the “Mile Standard Cable” (1 msc = 0.9221 dB). Another reason for the preferred use of the decibel is that it results in easily comprehensible numerical values. So is z. B. the doubling of the power as a power variable a change of about 3 dB and the tenfold a change of 10 dB. In contrast, however, z. B. the doubling of the voltage or the sound pressure as a field size a change of about 6 dB and the tenfold a change of 20 dB.

To avoid a common misunderstanding: A level change is not separate for e.g. B. Determine voltage and power. The same level changes apply. So +6 dB means a doubling of the voltage, which corresponds to a quadrupling of the power.

## Use with other units of measurement, accessories

Like any other unit of measurement, the bel or decibel can be used together with other units of measurement if it is used to describe a quantity in which a level or measure is linked by multiplication or division with another quantity. Examples are the attenuation level of a line in decibels per meter (dB / m) or the related sound power level of an extended sound source in decibels per square meter (dB / m 2 ).

According to the calculation rules that apply to quantities , it is not correct to add additions to a unit in order to convey information about the type of quantity being considered. B. still used in the recommendations of the ITU . Because of the uniqueness and the potential confusion with unit products (. Eg dB · m instead dBm) are as specified in DIN , IEC and ISO - standards always not to link this information with the size and with the unit. The most common examples of dB additions are summarized in the following table:

Unit with suffix (ITU) meaning Notation according to DIN, IEC, ISO
dBu Voltage level with the reference quantity${\ displaystyle {\ sqrt {600 \, \ Omega \ cdot 1 \, \ mathrm {mW}}} \, \ approx 0 {,} 7746 \, \ mathrm {V}}$ ${\ displaystyle L_ {u} (\ mathrm {re \; 0 {,} 775 \, V}) = \ dots \, \ mathrm {dB}}$
dBV Voltage level with the reference value 1 V. ${\ displaystyle L_ {V} (\ mathrm {re \; 1 \, V}) = \ dots \, \ mathrm {dB}}$
dBA A-weighted sound pressure level with the reference quantity${\ displaystyle \ mathrm {\; 20 \, \ mu Pa}}$ ${\ displaystyle L_ {pA} (\ mathrm {re \; 20 \, \ mu Pa}) = \ dots \, \ mathrm {dB}}$
A-weighted sound power level ${\ displaystyle L_ {WA} (\ mathrm {re \; 1 \, pW}) = \ dots \, \ mathrm {dB}}$
dBm Power level with the reference value 1 mW ${\ displaystyle L_ {P} (\ mathrm {re \; 1 \, mW}) = \ dots \, \ mathrm {dB}}$
dBW Power level with the reference value 1 W ${\ displaystyle L_ {P} (\ mathrm {re \; 1 \, W}) = \ dots \, \ mathrm {dB}}$
dBµ Level of the electric field strength with the reference value 1 µV / m ${\ displaystyle L_ {E} (\ mathrm {re \; 1 \, \ mu V / m}) = \ dots \, \ mathrm {dB}}$

In addition, there are a large number of other additives that are used inconsistently in different specialist areas. There are standardized reference values for many levels .

## application

Example of a representation with a linear quantity: Transfer factor of a 2nd order Butterworth filter${\ displaystyle u_ {o} / u_ {i}}$
Example of a representation with logarithmic size: Transfer rate of a Butterworth filter of the 2nd order${\ displaystyle A_ {0}}$

In both representations the vertical axis is divided linearly, the horizontal one logarithmically.

The specification of levels, level differences and dimensions play a role in various specialist areas. Particularly in the acoustic and the sound equipment , the communications technology and high frequency technology as well as in automation technology , the used have sizes often ranges over several orders of magnitude . The specification as a logarithmic ratio often allows a quick and clear interpretation of quantities if certain relationships are to be made just as clear in the area of ​​small values ​​as in the area of ​​large values.

In all of these technical applications, the decadic logarithm is preferred together with the decibel, especially since this representation enables a simple power of ten estimation. The natural logarithm is only preferred in theoretical treatises.

The human sensory impression is roughly logarithmic to the intensity of the physical stimulus ( Weber-Fechner law ). This means that the level of the physical quantity involved corresponds linearly to human perception. This is important for acoustics, for example, where the unit of measurement of the psychoacoustic variable volume , the phon , is defined in decibels through a link with the physical sound pressure level .

## literature

• Jürgen H. Maue, Heinz Hoffmann, Arndt von Lüpke: 0 decibels plus 0 decibels equals 3 decibels . 8th edition. Erich Schmidt Verlag, Berlin 2003, ISBN 3-503-07470-8 (1st edition published in 1975).