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# Bel

For the Mesopotamian deity, see Bel (god)

A bel (symbol B) is a unit of measure of ratios of power levels, i.e., relative power levels. It is mostly used in telecommunication, electronics and acoustics. Invented by engineers of the Bell Telephone Laboratory, it was originally called the transmission unit or TU, but was renamed in 1923 or 1924 in honour of the laboratory's founder and telecommunications pioneer Alexander Graham Bell.

The bel is a logarithmic measure. The number of bels for a given ratio of power levels is calculated by taking the logarithm, to the base 10, of the ratio. Therefore, one bel corresponds to a ratio of 10:1. Mathematically, the number of bels is calculated as B = log10(P1/P2) where P1 and P2 are power levels. The neper is a similar unit which uses the natural logarithm.

The bel is too large for everyday use, so the decibel (dB), equal to 0.1 B, is more commonly used. One decibel is equivalent to a ratio of about 1.259:1. It is defined as 10 log10(P1/P2) where P1 and P2 are the powers.

The decibel is a dimensionless "unit". The decibel is not an SI unit, although the CIPM has recommended its inclusion in the SI system.

 Table of contents 1 Uses 2 Typical abbreviations 3 Common misconception: +3 dB means "times two" 4 References

## Uses

### Optics

In an optical link, if a known amount of optical power, in
dBm (decibel.milliwatts), is launched into a fibre, and the losses, in dB (decibels), of each component (e.g. connectors, splices, and lengths of fibre) are known, the overall link loss may be quickly calculated by simple addition and subtraction of decibel quantities.

### Acoustics

The decibel is often used in acoustics to quantify sound levels relative to some 0 dB reference. The reference may be defined as a sound pressure level (SPL), commonly 20 micropascal (20 μPa). To avoid confusion with other decibel measures, the term dB(SPL) is used for this. The reference can also be defined as the sound intensity at the threshold of human hearing, which is conventionally taken to be one picowatt per square metre (1 pW/m²), roughly the sound of a mosquito flying 10 feet (3 m) away.

The reason for using the decibel is that the ear is capable of hearing a very large range of sound pressures. The ratio of the sound pressure that causes permanent damage from short exposure to the limit that (undamaged) ears can hear is more than a million. Because the power in a sound wave is proportional to the square of the pressure, the ratio of the maximum power to the minimum power is more than one trillion. To deal with such a range, logarithmic units are useful: the log of a trillion is 12, so this ratio represents a difference of 120 dB.

Psychologists have found that our perception of loudness is roughly logarithmic, see Weber-Fechner Law. In other words, you have to multiply the sound intensity by the same factor to have the same increase in loudness. This is why the numbers around the volume control dial on a typical audio amplifier are related not to the absolute power amplification, but to its logarithm.

Various frequency weightings are used for acoustical measurements to approximate the changes in sensitivity of the ear to different frequencies at different levels. These include the dB(A), dB(B), and dB(C) weightings.

Sounds above 85 dB are considered harmful, while 120 dB is unsafe and 150 dB causes physical damage to the human body. Windows break at about 163 dB. Jet airplanes are about 133 dBA at 33 m, or 100 dBA at 170m. Eardrums pop at 190 to 198 dB. Shock waves and sonic booms are about 200 dB at 330 m. Sounds around 200 dB can cause death to humans and are generated near bomb explosions (e.g. 23 kg of TNT detonated 3 m away). The space shuttle is around 215 dB (or about 175 dBA at 17m). Nuclear bombs are 240 dB to 258 dB (distance unknown). Even louder are earthquakes, tornados, hurricanes and volcanoes.

### Electronics

The decibel is used rather than arithmetic ratios or percentages because when certain types of circuits, such as amplifiers and attenuators, are connected in series, expressions of power level in decibels may be arithmetically added and subtracted.

In radio electronics, the decibel is used to describe the ratio between two measurements of electrical power. It can also be combined with a suffix to create an absolute unit of electrical power. For example, it can be combined with "m" for "milliwatt" to produce the "dBm". 0 dBm is one milliwatt, and 1dBm is one decibel greater than 0 dBm, or about 1.259 mW.

Although decibels were originally used for power ratios, they are nowadays commonly used in electronics to describe voltage or current ratios. In a constant resistive load, power is proportional to the square of the voltage or current in the circuit. Therefore, the decibel ratio of two voltages V1 and V2 can be defined as 20 log10(V1/V2), and similarly for current ratios. Thus, for example, a factor of 2.0 in voltage is equivalent to 6.02 dB (not 3.01 dB!).

This practice is fully consistent with power-based decibels, provided the circuit resistance remains constant. However, voltage-based decibels are frequently used to express such quantities as the voltage gain of an amplifier, where the two voltages are measured in different circuits which may have very different resistances. For example, a unity-gain buffer amplifier with a high input resistance and a low output resistance may be said to have a "voltage gain of 0 dB", even though it is actually providing a considerable power gain when driving a low-resistance load. Although this is pedantically deplorable, it is actually a very common practice and seems likely to persist.

### Telecommunications

In telecommunications, decibels are commonly used to measure signal-to-noise ratios.

### Seismology

Earthquakes are measured on the Richter scale, which is expressed in bels. (The units in this case are always assumed, rather than explicit.)

## Typical abbreviations

Absolute measurements:
Relative measurements:

## Common misconception: +3 dB means "times two"

Not exactly. As stated above, decibels are defined so that +10 dB means "ten times the power". From this, we calculate that +3 dB actually multiplies the power by 103/10. This is a ratio of 1.9953 or about 0.25% different from the "times 2" ratio that is sometimes assumed. A level difference of +6 dB is 3.9811, about 0.5% different from 4.

To contrive a more serious example, consider converting a large decibel figure into its linear ratio, for example 120 dB. This is correctly calculated as a ratio of 1012 or one trillion. But if we use the assumption that 3 dB means "times 2", we would calculate a ratio of 2120/3 = 240 = 1.0995 × 1012, for a 10% error.

## References

• Martin, W. H., "DeciBel – The New Name for the Transmission Unit", Bell System Technical Journal, January 1929.