Bel

For the Mesopotamian deity, see Bel (mythology); for the Celtic deity, see Belenus

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 power 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 Richter scale uses numbers expressed in bels as well, though this is implied by definition rather than expressly stated.

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

1.1 Optics
1.2 Acoustics
1.3 Electronics
1.4 Telecommunications
1.5 Seismology

2 Typical abbreviations
3 Common misconceptions

3.6 +3 dB means "two times" the power

4 References

4.7 External links

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

See Bel (Acoustics)

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.

dBm or dBmW - dB(1 mW@600 Ω) - in analogue audio, power measurement relative to 1 milliwatt into a 600-ohm load
dBW - same as dBm, with reference level of 1 watt

Electric voltage:

dBu or dBv - dB(0.775 V) - (usually RMS) voltage amplitude referenced to 0.775 volts, not related to any impedance. dBu is preferable, since dBv is easily confused with dBV.
dBV - dB(1 V) - (usually RMS) voltage amplitude of an audio signal in a wire, relative to 1 volt, not related to any impedance

Radio power:

dBm - dB(mV/m²) - millivolts per square metre - signal strength of a radio signal
dBμ or dBu - dB(μV/m²) - microvolts per square metre - strength of a radio signal
dBf - dB(fW) - femto watts - amount of power required to drive a radio receiver
dBW - dB(W) - watts - amount of power transmitted by a low-power radio station
dBk - dB(kW) - kilowatts - amount of power transmitted by a broadcast radio station

Acoustics:

dB SPL - (Sound Pressure Level) relative to 20 micropascals (μPa), the quietest sound a human can hear. Roughly the sound of a mosquito flying 3 meters away.

Relative measurements:

dB(A), dB(B), or dB(C) - different frequency weightings used to approximate the human ear's response to sound. Also written dB_A, etc.
dBd - dB(dipole) - effective radiated power compared to a dipole antenna
dBi - dB(isotropic) - effective radiated power compared to an imaginary isotropic antenna
dBFS or dBfs - dB(full scale) - amplitude of a signal (usually audio) compared to the maximum which a device can handle before clipping occurs. In digital systems, 0 dBFS would equal the highest level (number) the processor is capable of representing.
dBr - simply a relative difference to something else, which is made apparent in context. Difference of a filter's response to nominal levels, for instance.

Common misconceptions

+3 dB means "two times" the power

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 10^3/10. This is a power ratio of 1.9953 or about 0.25% different from the "times 2" power 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 power ratio of 2^120/3 = 2⁴⁰ = 1.0995 ÃÂÃÂ 10¹², for a 10% error.

References

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