Menu
Home Explore People Places Arts History Plants & Animals Science Life & Culture Technology
On this page
Level (logarithmic quantity)
Logarithm of the ratio of two quantities

In science and engineering, a power level and a field level (also called a root-power level) are logarithmic magnitudes 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 (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 (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, telecommunications, acoustics 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.

We don't have any images related to Level (logarithmic quantity) yet.
We don't have any YouTube videos related to Level (logarithmic quantity) yet.
We don't have any PDF documents related to Level (logarithmic quantity) yet.
We don't have any Books related to Level (logarithmic quantity) yet.
We don't have any archived web articles related to Level (logarithmic quantity) yet.

Power level

Level of a power quantity, denoted LP, is defined by

L P = 1 2 log e ( P P 0 )   N p = log 10 ( P P 0 )   B = 10 log 10 ( P P 0 )   d B . {\displaystyle L_{P}={\frac {1}{2}}\log _{\mathrm {e} }\!\left({\frac {P}{P_{0}}}\right)\!~\mathrm {Np} =\log _{10}\!\left({\frac {P}{P_{0}}}\right)\!~\mathrm {B} =10\log _{10}\!\left({\frac {P}{P_{0}}}\right)\!~\mathrm {dB} .}

where

  • P is the power quantity;
  • P0 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 LF, is defined by5

L F = log e ( F F 0 )   N p = 2 log 10 ( F F 0 )   B = 20 log 10 ( F F 0 )   d B . {\displaystyle L_{F}=\log _{\mathrm {e} }\!\left({\frac {F}{F_{0}}}\right)\!~\mathrm {Np} =2\log _{10}\!\left({\frac {F}{F_{0}}}\right)\!~\mathrm {B} =20\log _{10}\!\left({\frac {F}{F_{0}}}\right)\!~\mathrm {dB} .}

where

  • F is the root-power quantity, proportional to the square root of power quantity;
  • F0 is the reference value of F.

If the power quantity P is proportional to F2, and if the reference value of the power quantity, P0, is in the same proportion to F02, the levels LF and LP are equal.

The neper, bel, and decibel (one tenth of a bel) are units of level that are often applied to such quantities as power, intensity, or gain.6 The neper, bel, and decibel are related by7

  • 1 B = ⁠1/2⁠ loge10 Np;
  • 1 dB = 0.1 B = ⁠1/20⁠ loge10 Np.

See also: Decibel § Conversions, and Neper § Units

Standards

Level and its units are defined in ISO 80000-3.

The ISO standard defines each of the quantities power level and field level to be dimensionless, with 1 Np = 1. This is motivated by simplifying the expressions involved, as in systems of natural units.

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 LQ, as8

L Q = log r ( Q Q 0 ) , {\displaystyle L_{Q}=\log _{r}\!\left({\frac {Q}{Q_{0}}}\right)\!,}

where

  • r is the base of the logarithm;
  • Q is the quantity;
  • Q0 is the reference value of Q.

For the level of a root-power quantity, the base of the logarithm is r = e. For the level of a power quantity, the base of the logarithm is r = e2.9

Logarithmic frequency ratio

The logarithmic frequency ratio (also known as 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 10:10

L f = log 2 ( f f 0 )   oct = log 10 ( f f 0 )   dec . {\displaystyle L_{f}=\log _{2}\!\left({\frac {f}{f_{0}}}\right)~{\text{oct}}=\log _{10}\!\left({\frac {f}{f_{0}}}\right)~{\text{dec}}.}

In music theory, the octave is a unit used with logarithm base 2 (called interval).11 A semitone is one twelfth of an octave. A cent is one hundredth of a semitone. In this context, the reference frequency is taken to be C0, four octaves below middle C.12

See also

Notes

References

  1. IEEE/ASTM SI 10 2016, pp. 26–27. - IEEE/ASTM SI 10: American National Standard for Metric Practice, IEEE Standards Association, 2016 ↩

  2. ISO 80000-3 2006. - ISO 80000-3: Quantities and units, vol. Part 3: Space and Time, International Organization for Standardization, 2006 ↩

  3. Carey 2006, pp. 61–75. - Carey, W. M. (2006), "Sound Sources and Levels in the Ocean", IEEE Journal of Oceanic Engineering, 31 (1): 61, Bibcode:2006IJOE...31...61C, doi:10.1109/JOE.2006.872214, S2CID 30674485 https://ui.adsabs.harvard.edu/abs/2006IJOE...31...61C ↩

  4. ISO 80000-8 2007. - ISO 80000-8: Quantities and units, vol. Part 8: Acoustics, International Organization for Standardization, 2007 ↩

  5. D'Amore 2015. - D'Amore, F. (2015), Effect of moisturizer and lubricant on the finger‒surface sliding contact: tribological and dynamical analysis https://www.researchgate.net/publication/316104462 ↩

  6. Taylor 1995. - Taylor, Barry (1995), Guide for the Use of the International System of Units (SI): The Metric System, Diane Publishing Co., p. 28, ISBN 9780788125799 https://books.google.com/books?id=I3HRRrDd1n8C&pg=PA28 ↩

  7. Ainslie, Halvorsen & Robinson 2022. - Ainslie, Michael A.; Halvorsen, Michele B.; Robinson, Stephen P. (January 2022) [2021-11-09]. "A terminology standard for underwater acoustics and the benefits of international standardization". IEEE Journal of Oceanic Engineering. 47 (1). IEEE: 179–200. Bibcode:2022IJOE...47..179A. doi:10.1109/JOE.2021.3085947. eISSN 1558-1691. ISSN 0364-9059. S2CID 243948953. https://doi.org/10.1109%2FJOE.2021.3085947 ↩

  8. ANSI/ASA S1.1 2013, entry 3.01. - ANSI/ASA S1.1: Acoustical Terminology, vol. ANSI/ASA S1.1-2013, Acoustical Society of America, 2013 ↩

  9. Ainslie 2015. - Ainslie, Michael A. (2015), "A Century of Sonar: Planetary Oceanography, Underwater Noise Monitoring, and the Terminology of Underwater Sound", Acoustics Today, 11 (1) http://acousticstoday.org/a-century-of-sonar-planetary-oceanography-underwater-noise-monitoring-and-the-terminology-of-underwater-sound-michael-a-ainslie/ ↩

  10. Ainslie, Halvorsen & Robinson 2022. - Ainslie, Michael A.; Halvorsen, Michele B.; Robinson, Stephen P. (January 2022) [2021-11-09]. "A terminology standard for underwater acoustics and the benefits of international standardization". IEEE Journal of Oceanic Engineering. 47 (1). IEEE: 179–200. Bibcode:2022IJOE...47..179A. doi:10.1109/JOE.2021.3085947. eISSN 1558-1691. ISSN 0364-9059. S2CID 243948953. https://doi.org/10.1109%2FJOE.2021.3085947 ↩

  11. Fletcher 1934, pp. 59–69. - Fletcher, H. (1934), "Loudness, pitch and the timbre of musical tones and their relation to the intensity, the frequency and the overtone structure", Journal of the Acoustical Society of America, 6 (2): 59, Bibcode:1934ASAJ....6...59F, doi:10.1121/1.1915704 https://ui.adsabs.harvard.edu/abs/1934ASAJ....6...59F ↩

  12. ANSI/ASA S1.1 2013. - ANSI/ASA S1.1: Acoustical Terminology, vol. ANSI/ASA S1.1-2013, Acoustical Society of America, 2013 ↩