The bel
A non-system unit accepted by the SI · quantity: level of a quantity
01 · Definition
The decimal logarithm of the ratio of two powers. One bel means exactly a tenfold difference, two bels a hundredfold: the unit counts zeros.
The bel is dimensionless and always describes a ratio, so on its own it means nothing: one has to say what the quantity is referred to. Zero bels means not the absence of a signal but equality with the reference.
The bel was introduced at Bell Labs to replace the «mile of standard cable», but a step of ten proved too coarse: a person tells apart a change in loudness of about one decibel, and whole bels would describe hearing in four steps. Within a few years practice moved to the tenth part, and today the bel turns up almost only in the wording of standards. There is a case the other way in computing: the noise level of disc drives is sometimes given in bels precisely so that the number looks small.
02 · Conversion
Enter a level or a ratio and the sheet turns one into the other
Every row refers to a power level. For voltage, current and sound pressure the number doubles: one and the same ratio gives twice the level, because power grows as the square.
The chief convenience of a logarithmic scale is that the successive links of a path add up with a plain plus. An amplifier of plus three bels and a cable of minus one leave plus two, and no multiplication is needed anywhere.
| Unit | Name | Value | Where it is met |
|---|---|---|---|
| {u} | {name} | {value} | {note} |
03 · Orders of magnitude
every bel is another decimal order04 · Measuring instruments
What levels are measured with
A needle instrument with a logarithmic scale where zero means not silence but the nominal level. The red zone past zero marks overload: what matters to the engineer is not the loudness itself but the distance from it to the ceiling.
A set of calibrated attenuators switched in steps of 1, 3, 10 and 30 decibels. It is precisely with these that the reference attenuation is set when instruments are verified: bels and decibels here are not an estimate but a measure.
The vertical axis of instruments is nearly always marked logarithmically, because otherwise the weak components merge with zero. One screen covers eight to ten orders at once — that is eighty to a hundred decibels.
In fibre links the budget of a line is counted wholly in decibels: the attenuation of the fibre and the losses at splices and connectors are added, not multiplied. The receiver states its sensitivity in the same decibels, and the margin is found by subtraction.
05 · Writing rules
Capital B — and not the byte
The symbol is formed from a surname and is therefore capital: B, in Russian Б. The word «bel» is written in small letters. The clash with the byte is settled by context and by the prefix: dB is unambiguous, while B alone calls for care in computing texts.
The last entry is incomplete: a level without a stated reference is undefined, and the standard requires naming it either in the text or by a suffix. Hence the family of notations dBm, dBW, dBu, dBFS — each of them fixes its own reference.
06 · Neighbouring units
There are three logarithmic units: the bel with the decimal logarithm, the neper with the natural one, and the decibel as the working part of a bel. All are dimensionless and all require a reference to be stated.
The neper is handier in theory, where the solutions of the equations carry an exponential, the bel in measurement, where orders are more familiar. Going from one to the other needs no reference at all. Simetrium data sheets nearby stand decibel and neper, and also hydrogen exponent — one more dimensionless quantity obtained by taking a logarithm.
07 · Historical section
A unit that was divided by ten straight away
The attenuation of a line was measured in miles of standard cable: as many miles as it takes to get the same decay. The unit depended on the frequency and on the properties of the particular cable, and it made comparing the networks of different countries hopeless.
In place of the mile of cable they brought in the transmission unit, a logarithmic measure independent of the material of the line. The quantity was tied to the decimal logarithm of a power ratio, and it turned out numerically close to the old mile — so the tables did not have to be rewritten.
The unit was renamed the bel, after Alexander Graham Bell, and the working measure became its tenth part, the decibel. Practice chose the part at once: whole bels gave too few gradations for hearing and for the reckoning of a line.
The International Bureau of Weights and Measures lists the bel and the decibel among the units accepted for use with the SI, but requires that the reference quantity always be stated. ISO 80000 puts it more sharply still: without a reference the record is not a level.
A unit with no absolute value of its own
The bel is not realised as a physical quantity: what is realised is a ratio. An attenuation standard is a divider whose factor is worked out from resistances, and its accuracy rests on the accuracy of those resistances alone.
Catalogue · units of measurement
A data sheet for every quantity
Seven SI base units, twenty-two derived ones with their own names, and non-system units alongside them. Each has its own sheet: definition, dimension, conversion, instruments, rules of writing. In 36 languages.
Non-SI units
the bel data sheet is openSI base units
the bel is dimensionless and rests on none of themSI derived units with names of their own
in terracotta, the watt: levels are most often counted from itSheet language
36 languages. The symbol Np and the formulas are not translated — they are the same in every locale; what is translated is the prose, the local land measures and the writing rules.