Logarithmic unit for ratios of measurements of physical field and power quantities
John Napier after whom the unit is namedTheneper (symbol:Np ) is alogarithmic unit forratios of measurements of physicalfield and power quantities , such asgain and loss of electronic signals. The unit's name is derived from the name ofJohn Napier , the inventor of logarithms. As is the case for thedecibel andbel , the neper is a unit defined in theinternational standard ISO 80000 . It is not part of theInternational System of Units (SI), but is accepted for use alongside the SI.[ 1]
Like thedecibel , the neper is a unit in alogarithmic scale . While the bel uses the decadic (base-10)logarithm to compute ratios, the neper uses thenatural logarithm , based onEuler's number (e root-power quantities , with the unit neper, is given by[ 2]
L = ln x 1 x 2 N p , {\displaystyle L=\ln {\frac {x_{1}}{x_{2}}}\mathrm {~Np} ,} wherex 1 {\displaystyle x_{1}} x 2 {\displaystyle x_{2}} ln is the natural logarithm. The level of a ratio of twopower quantities , with the unit neper, is given by[ 2]
L = 1 2 ln p 1 p 2 N p , {\displaystyle L={\frac {1}{2}}\ln {\frac {p_{1}}{p_{2}}}\mathrm {~Np} ,} wherep 1 {\displaystyle p_{1}} p 2 {\displaystyle p_{2}}
In theInternational System of Quantities , the neper is defined as1 Np = 1 .[ 3]
The neper is defined in terms of ratios of field quantities — also calledroot-power quantities — (for example,voltage orcurrent amplitudes in electrical circuits, orpressure inacoustics ), whereas the decibel was originally defined in terms ofpower ratios. A power ratio 10 logr dB is equivalent to a field-quantity ratio 20 logr dB, since power in a linear system is proportional to the square (Joule's laws ) of the amplitude. Hence the decibel and the neper have a fixed ratio to each other:[ 4]
1 Np = 20 log 10 e dB ≈ 8.685889638 dB {\displaystyle 1\ {\text{Np}}=20\log _{10}e\ {\text{dB}}\approx {\text{8.685889638 dB}}} and
1 d B = 1 20 ln ( 10 ) N p ≈ 0.115129255 Np . {\displaystyle 1\ \mathrm {dB} ={\frac {1}{20}}\ln(10)\ \mathrm {Np} \approx {\text{0.115129255 Np}}.} The (voltage) level ratio is
L = 10 log 10 x 1 2 x 2 2 dB = 10 log 10 ( x 1 x 2 ) 2 dB = 20 log 10 x 1 x 2 dB = ln x 1 x 2 Np . {\displaystyle {\begin{aligned}L&=10\log _{10}{\frac {x_{1}^{2}}{x_{2}^{2}}}&{\text{dB}}\\&=10\log _{10}{\left({\frac {x_{1}}{x_{2}}}\right)}^{2}&{\text{dB}}\\&=20\log _{10}{\frac {x_{1}}{x_{2}}}&{\text{dB}}\\&=\ln {\frac {x_{1}}{x_{2}}}&{\text{Np}}.\\\end{aligned}}} Like the decibel, the neper is a dimensionless unit. TheInternational Telecommunication Union (ITU) recognizes both units. Only the neper iscoherent with the SI.[ 5]
The neper is a natural linear unit ofrelative difference , meaning in nepers (logarithmic units) relative differences add rather than multiply. This property is shared with logarithmic units in other bases, such as the bel.
The derived units decineper (1 dNp = 0.1 neper) andcentineper (1 cNp = 0.01 neper) are also used.[ 6] log point or log percentage, seeRelative change and difference § Logarithmic scale [ 7]
^ The International System of Units (SI) International Bureau of Weights and Measures . 2019. pp. 145– 146.Archived from the original on 2022-10-09.^a b Letter symbols to be used in electrical technology – Part 3: Logarithmic and related quantities, and their units (International standard). International Electrotechnical Commission. 2002-07-19.IEC 60027-3 :2002.^ Thor, A J (1994-01-01)."New International Standards for Quantities and Units" Metrologia .30 (5):517– 522.doi :10.1088/0026-1394/30/5/010 .ISSN 0026-1394 . ^ 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 [Appendix B Decibel: Past, Present, and Future – Section D].Bibcode :2022IJOE...47..179A .doi :10.1109/JOE.2021.3085947 eISSN 1558-1691 .ISSN 0364-9059 .S2CID 243948953 . [1] (22 pages)^ ISO 80000 -3:2007 §0.5^ Glossary of Telecommunication Terms ^ Karjus, Andres; Blythe, Richard A.; Kirby, Simon; Smith, Kenny (10 February 2020)."Quantifying the dynamics of topical fluctuations in language" .Language Dynamics and Change .10 (1):86– 125.arXiv :1806.00699 doi :10.1163/22105832-01001200 S2CID 46928080 . Mirifici logarithmorum canonis constructi o, 1825Tuffentsammer, Karl (1956). "Das Dezilog, eine Brücke zwischen Logarithmen, Dezibel, Neper und Normzahlen" [The decilog, a bridge between logarithms, decibel, neper and preferred numbers].VDI-Zeitschrift (in German).98 :267– 274. Paulin, Eugen (2007-09-01).Logarithmen, Normzahlen, Dezibel, Neper, Phon - natürlich verwandt! Logarithms, preferred numbers, decibel, neper, phon - naturally related! ](PDF) (in German).Archived (PDF) from the original on 2016-12-18. Retrieved2016-12-18 .
