Level (logarithmic Quantity)
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science and engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, a power level and a field level (also called a root-power level) are logarithmic measures of certain quantities referenced to a standard reference value of the same type. * A ''power level'' is a logarithmic quantity used to measure power, power density or sometimes energy, with commonly used unit
decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...
(dB). * A ''field level'' (or ''root-power level'') is a logarithmic quantity used to measure quantities of which the square is typically proportional to power (for instance, the square of Voltage is proportional to Power by the inverse of the conductor's Resistance), etc., with commonly used units neper (Np) or
decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...
(dB). The type of level and choice of units indicate the scaling of the logarithm of the ratio between the quantity and its reference value, though a logarithm may be considered to be a dimensionless quantity. The reference values for each type of quantity are often specified by international standards. Power and field levels are used in
electronic engineering Electronics engineering is a sub-discipline of electrical engineering which emerged in the early 20th century and is distinguished by the additional use of active components such as semiconductor devices to amplify and control electric current ...
,
telecommunications Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that fe ...
,
acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
and related disciplines. Power levels are used for signal power, noise power, sound power, sound exposure, etc. Field levels are used for voltage, current,
sound pressure Sound pressure or acoustic pressure is the local pressure deviation from the ambient (average or equilibrium) atmospheric pressure, caused by a sound wave. In air, sound pressure can be measured using a microphone, and in water with a hydrophone ...
.


Power level

Level of a ''power'' quantity, denoted ''L''''P'', is defined by :L_P = \frac \log_\!\left(\frac\right)\!~\mathrm = \log_\!\left(\frac\right)\!~\mathrm = 10 \log_\!\left(\frac\right)\!~\mathrm. where *''P'' is the power quantity; *''P''0 is the reference value of ''P''.


Field (or root-power) level

The level of a ''root-power'' quantity (also known as a ''field'' quantity), denoted ''L''''F'', is defined by :L_F = \log_\!\left(\frac\right)\!~\mathrm = 2 \log_\!\left(\frac\right)\!~\mathrm = 20 \log_\!\left(\frac\right)\!~\mathrm. where *''F'' is the root-power quantity, proportional to the square root of power quantity; *''F''0 is the reference value of ''F''. If the power quantity ''P'' is proportional to ''F''2, and if the reference value of the power quantity, ''P''0, is in the same proportion to ''F''02, the levels ''L''''F'' and ''L''''P'' are equal. The neper, bel, and
decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...
(one tenth of a bel) are units of level that are often applied to such quantities as power, intensity, or gain. The neper, bel, and decibel are related by *; *.


Standards

Level and its units are defined in
ISO 80000-3 ISO 80000 or IEC 80000 is an international standard introducing the International System of Quantities (ISQ). It was developed and promulgated jointly by the International Organization for Standardization (ISO) and the International Electrotech ...
. The ISO standard defines each of the quantities power level and field level to be dimensionless, with . This is motivated by simplifying the expressions involved, as in systems of
natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
.


Related quantities


Logarithmic ratio quantity

Power and field quantities are part of a larger class, logarithmic ratio quantities. ANSI/ASA S1.1-2013 defines a class of quantities it calls ''levels''. It defines a level of a quantity ''Q'', denoted ''L''''Q'', as :L_Q = \log_r\!\left(\frac\right)\!, where *''r'' is the base of the logarithm; *''Q'' is the quantity; *''Q''0 is the reference value of ''Q''. For the level of a root-power quantity, the base of the logarithm is . For the level of a power quantity, the base of the logarithm is .


Logarithmic frequency ratio

The ''logarithmic frequency ratio'' (also "frequency level") of two frequencies is the logarithm of their ratio, and may be expressed using the unit ''octave'' (symbol: oct) corresponding to the ratio 2 or the unit ''decade'' (symbol: dec) corresponding to the ratio 10. In
electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
, the
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
(oct) is used as a unit with logarithm base 2, and the
decade A decade () is a period of ten years. Decades may describe any ten-year period, such as those of a person's life, or refer to specific groupings of calendar years. Usage Any period of ten years is a "decade". For example, the statement that "du ...
(dec) is used as a unit with logarithm base 10: : L_f = \log_2 \!\left( \frac \right) ~\text = \log_ \!\left( \frac \right) ~\text. In
music theory Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...
, the
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
is a unit used with logarithm base 2 (called '' interval''). A
semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...
is one twelfth of an octave. A cent is one hundredth of a semitone. In this context, the reference frequency is taken to be C, four octaves below middle C.


See also

* *
Power, root-power, and field quantities A power quantity is a power or a quantity directly proportional to power, e.g., energy density, acoustic intensity, and luminous intensity. Energy quantities may also be labelled as power quantities in this context. A root-power quantity is a quan ...
*
Logarithmic scale A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
*
Sound level (disambiguation) Sound level refers to various logarithmic measurements of audible vibrations and may refer to: *Sound exposure level, measure of the sound exposure of a sound relative to a reference value *Sound power level, measure of the rate at which sound en ...
*
Leveling (tapered floating point) In computing, tapered floating point (TFP) is a format similar to floating point, but with variable-sized entries for the significand and exponent instead of the fixed-length entries found in normal floating-point formats. In addition to this, ...
*
Level-index arithmetic The level-index (LI) representation of numbers, and its algorithms for arithmetic operations, were introduced by Charles Clenshaw and Frank Olver in 1984. The symmetric form of the LI system and its arithmetic operations were presented by Clensh ...
(LI) and symmetric level-index arithmetic (SLI)


Notes


References

* * * * * * * * * *

(22 pages) * {{citation , date=2022 , orig-date=2017 , title=ISO 18405:2017 Underwater acoustics – Terminology , url=https://www.iso.org/standard/62406.html , publisher=
International Organization for Standardization The International Organization for Standardization (ISO ) is an international standard development organization composed of representatives from the national standards organizations of member countries. Membership requirements are given in Ar ...
, access-date=2022-12-20 Mathematical terminology Logarithmic scales of measurement