The bel
When a welder lowers a shield with a shade ten glass over the face, the blinding arc turns into a calm green spark. Such glass transmits about one seven-thousandth of the light, and that fraction is not easy to remember. It is much more convenient to say that the glass attenuates the light by 3.9 bels. The bel simply shows how many times a quantity had to be divided by ten, and so instead of unwieldy fractions you get small numbers that can, moreover, be added to one another.
Strictly speaking, the bel is the decimal logarithm of the ratio of two powers. If one power is ten times the other, the difference between them is one bel; if a hundred times, two bels; and an attenuation by a factor of a thousand is written as minus three bels. The unit is dimensionless and always compares a quantity with a reference, so zero bels means not emptiness at all but equality with the reference. When voltages, currents or sound pressures are compared, one has to remember that power grows in proportion to their square, and double the logarithm. The bel was devised by telephone engineers and named after Alexander Graham Bell, but almost at once in practice people began to use its tenth part, the decibel.
Conversion
Level and ratios — both ways
Enter a number and say what it means: a level in bels, decibels or nepers, a ratio of powers or voltages, optical density, camera exposure steps or star magnitudes. On the right the same ratio will be written out in all the ways at once.
Etymology
A bell, a word and a number
The bel got its name from the surname of the inventor of the telephone, and its old rival the neper from the Latin signature of the Scottish lord who invented logarithms. The word “logarithm” itself is made up of two Greek words. So three concepts without which no communication line is ever calculated came to us from Edinburgh, from the Scottish estate of Merchiston and from ancient Greece.
01 · Definition
How many times divided by ten
Imagine an amplifier whose output power is a thousand times greater than its input. A thousand has three zeros, and so its gain is three bels. And if a communication line leaves only one hundred-thousandth part of the signal, its attenuation is five bels. When the ratio is not a round power of ten, the number of bels is fractional: doubling of power, for example, corresponds to 0.301 bel, because ten to the power 0.301 gives exactly two.
The main convenience of the logarithm is that it turns multiplication into addition. A signal that has passed first along a cable, then through an amplifier and again along a cable was attenuated and amplified several times in a row, and all these factors would have to be multiplied, whereas bels need only be added, not forgetting the signs. One should also remember that power is proportional to the square of voltage, current or sound pressure. So a level calculated from these quantities contains a two, and a voltage that has doubled gives not 0.301 but 0.602 bel.
From a billionth to a billion
Move the slider and see how the long number expressing the ratio folds into a short number of bels. Each rectangle stands for one decimal order: to the right of zero the power grows, to the left it falls.
Stretch, darken, double
In the “Line” experiment choose what the signal travels along and how far. A telephone wire, a cable or a glass fibre take away one bel for every few kilometres, and the level diagram will show where the signal drowns in noise if an amplifier is not placed in time. In the “Glasses” experiment assemble light filters, sunglasses and a welding glass in a beam of light and check that their densities in bels simply add up. And in the “Amplitude” experiment double the voltage, and you will see that the power has grown fourfold.
02 · Glass transparency
From a hundred bels to a hundredth of a bel per kilometre
Even the best optical glass of the mid-twentieth century lost about a hundred bels on every kilometre, so light died out completely in it after a few tens of metres. In 1966 in the English town of Harlow Charles Kao proved that light is absorbed not by the glass itself but by impurities in it, and predicted a glass that would lose only two bels per kilometre. Four years later such a glass was actually made at the American company Corning, nine years after that Japanese engineers brought the losses down to two hundredths of a bel, and communication between continents moved to fibre. The graph plots record losses in bels per kilometre; its scale is logarithmic.
03 · Where the bel stayed whole
Five scales that really count in bels
Communication engineers long ago went over to decibels, but in other trades the decimal logarithm of a ratio took root whole. Photographers and radiologists call it optical density, astronomers count star magnitudes in it, multiplying by two and a half, welders pick a filter shade number by it, and computer makers state directly in bels how noisy their machines are. Choose a scale and move the value, and the conversion will show how many bels it is and by what factor the quantity changes.
04 · The bel, the neper and the mile of cable
Every telephone school counted in its own way
Until the end of the 1920s American telephone engineers expressed line attenuation in miles of standard cable, that is, in the length of a reference copper cable that would weaken speech in exactly the same way. This measure depended on frequency and on the properties of the cable itself, and so in the Bell company it was replaced first by the transmission unit and then by the bel and the decibel. European engineers, who solved the equations of long lines, preferred the natural logarithm and the neper, because in those equations attenuation is expressed by a single exponential without extra factors.
The International Consultative Committee for Telephony recognised both units, and for a long time the reference books of Germany, the Soviet Union and a number of other countries gave cable attenuation in nepers per kilometre, while American and British ones gave it in decibels. Today the decibel has won almost everywhere, and the neper has remained in circuit theory and in textbooks.
The mile of standard cable was defined at a frequency of 800 Hz, to which the ear is especially sensitive when listening to speech. At other frequencies the same mile weakened the signal differently, and it was hard to compare measurements from different laboratories. It was exactly this trouble that the transition to the pure logarithm of the power ratio removed.
05 · Writing rules
06 · Neighbouring units
07 · Historical section
In 1924 William Martin, an engineer of the Bell telephone company, proposed replacing the mile of cable with a “transmission unit” equal to ten decimal logarithms of the power ratio. It nearly coincided with the old mile, so telephone men got used to it without difficulty, and line calculation turned into simple addition. Four years later, in 1928, the units were renamed: the tenfold logarithm became the decibel, and the logarithm without a factor received the name of the bel.
Thus metrology acquired the only pair in which the base unit is hardly ever used while its tenth is known to everyone. The bel proved too large: the ear notices a change of loudness of about one tenth of a bel, and in whole bels hearing would be described by only a few coarse steps. In optics, on the other hand, the decimal logarithm of attenuation stayed whole under the name of optical density, and in the data sheets of noisy equipment it stayed whole under its own name, the bel.
Catalogue of quantities
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