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((short description|Logarithmic unit for ratios of measurements of physical field and power quantities))
((other uses))
((refimprove|date=December 2018))
[[File:John Napier.jpg|thumb|[[John Napier]] after whom the unit is named]]

The '''neper''' (symbol: '''Np''') is a [[logarithmic unit]] for [[ratio]]s of measurements of physical [[Power, root-power, and field quantities|field and power quantities]], such as [[gain (electronics)|gain]] and loss of electronic signals. The unit's name is derived from the name of [[John Napier]], the inventor of logarithms. As is the case for the [[decibel]] and [[Bel (unit)|bel]], the neper is a unit defined in the [[international standard]] [[ISO 80000]]. It is not part of the [[International System of Units]] (SI), but is accepted for use alongside the SI.<ref name="BIPM">((Cite book|url=https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf/2d2b50bf-f2b4-9661-f402-5f9d66e4b507 |archive-url=https://ghostarchive.org/archive/20221009/https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf/2d2b50bf-f2b4-9661-f402-5f9d66e4b507 |archive-date=2022-10-09 |url-status=live|title=The International System of Units (SI).|publisher=[[International Bureau of Weights and Measures]]|year=2019|edition=9|pages=145–146))</ref>

==Definition==
Like the [[decibel]], the neper is a unit in a [[logarithmic scale]]. While the bel uses the decadic (base-10) [[logarithm]] to compute ratios, the neper uses the [[natural logarithm]], based on [[Euler's number]] (((nowrap|[[e (mathematical constant)|''e'']] ≈ 2.71828))). The level of a ratio of two signal amplitudes or [[root-power quantity|root-power quantities]], with the unit neper, is given by<ref name=IEC> ((cite tech report |title=Letter symbols to be used in electrical technology – Part 3: Logarithmic and related quantities, and their units |number=[[IEC 60027-3]]:2002|type=International standard |institution= International Electrotechnical Commission |date=2002-07-19))</ref>
:<math>
L = \ln\frac{x_1}{x_2}\mathrm{~Np}, 
</math>

where <math>x_1</math> and <math>x_2</math> are the signal amplitudes, and ((math|ln)) is the natural logarithm. The level  of a ratio of two [[power quantity|power quantities]], with the unit neper, is given by<ref name=IEC/>
:<math>
L = \frac{1}{2} \ln\frac{p_1}{p_2}\mathrm{~Np}, 
</math>

where <math>p_1</math> and <math>p_2</math> are the signal powers.

In the [[International System of Quantities]], the neper is defined as ((nowrap|1=1 Np = 1)).<ref>((Cite journal |last=Thor |first=A J |date=1994-01-01 |title=New International Standards for Quantities and Units |url=https://iopscience.iop.org/article/10.1088/0026-1394/30/5/010 |journal=Metrologia |volume=30 |issue=5 |pages=517–522 |doi=10.1088/0026-1394/30/5/010 |issn=0026-1394))</ref>

==Units==
((main|Level (logarithmic quantity)))

The neper is defined in terms of ratios of field quantities — also called ''root-power quantities'' — (for example, [[voltage]] or [[Current (electricity)|current]] amplitudes in electrical circuits, or [[sound pressure|pressure]] in [[acoustics]]), whereas the decibel was originally defined in terms of [[power (physics)|power]] ratios.  A power ratio 10 log ''r'' dB is equivalent to a field-quantity ratio 20 log ''r'' dB, since power in a linear system is proportional to the square ([[Joule's first law|Joule's laws]]) of the amplitude.  Hence the decibel and the neper have a fixed ratio to each other:<ref>((cite journal |date=January 2022 |orig-date=2021-11-09 |title=A terminology standard for underwater acoustics and the benefits of international standardization. |url=https://ieeexplore.ieee.org/document/9607022 |journal=[[IEEE Journal of Oceanic Engineering]] |publisher=[[IEEE]] |volume=47 |issue=1 |pages=179-200 [Appendix B Decibel: Past, Present, and Future – Section D] |doi=10.1109/JOE.2021.3085947 |issn=0364-9059 |eissn=1558-1691 |s2cid=243948953 |access-date=2022-12-20 |doi-access=free |author-last1=Ainslie |author-first1=Michael A. |author-last2=Halvorsen |author-first2=Michele B. |author-last3=Robinson |author-first3=Stephen P.)) [https://web.archive.org/web/20221220112042/https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9607022] (22 pages)</ref>
:<math> 1\ \text{Np} = 20 \log_{10} e\ \text{dB} \approx \text{8.685889638 dB} </math>
and
:<math>
1\ \mathrm{dB} = \frac{1}{20} \ln(10)\ \mathrm{Np} \approx \text{0.115129255 Np} .</math>

The (voltage) level ratio is
:<math>
\begin{align}
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{align}
</math>

Like the decibel, the neper is a dimensionless unit. The [[International Telecommunication Union]] (ITU) recognizes both units. Only the neper is [[coherence (units of measurement)|coherent]] with the SI.<ref>[[ISO 80000]]-3:2007 §0.5</ref>

==Applications==
The neper is a natural linear unit of [[relative 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) and '''centineper''' (1 cNp = 0.01 neper) are also used.<ref>((cite book |title=Glossary of Telecommunication Terms |date=1980 |publisher=General Services Administration, Federal Supply Service |page=73 |url=https://books.google.com/books?id=3cQ65jxbaTAC&pg=PA73 |language=en))</ref> The centineper for root-power quantities corresponds to a [[log point]] or log percentage, see ''((section link|Relative change and difference#Logarithmic scale))''.<ref name="Blythe2020">((cite journal |last1=Karjus |first1=Andres |last2=Blythe |first2=Richard A. |last3=Kirby |first3=Simon |last4=Smith |first4=Kenny |title=Quantifying the dynamics of topical fluctuations in language |journal=Language Dynamics and Change |date=10 February 2020 |volume=10 |issue=1 |pages=86–125 |doi=10.1163/22105832-01001200 |s2cid=46928080 |url=https://brill.com/view/journals/ldc/10/1/article-p86_5.xml|doi-access=free |arxiv=1806.00699 ))</ref>

== See also ==
*[[Nat (unit)]]
*[[Nepers per metre]]

== References ==
((Reflist))

== Works ==
[[File:Napier, John – Mirifici logarithmorum canonis constructio, 1825 – BEIC 6555650.jpg|thumb|''Mirifici logarithmorum canonis constructi''o, 1825]]
* ((Cite book|title=Mirifici logarithmorum canonis constructio|volume=|publisher=Librairie scientifique Hermann et C.ie|location=Paris|year=1825|language=fr|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=6555650))

== Further reading ==
* ((cite journal |author-last=Tuffentsammer |author-first=Karl |title=Das Dezilog, eine Brücke zwischen Logarithmen, Dezibel, Neper und Normzahlen |language=de |trans-title=The decilog, a bridge between logarithms, decibel, neper and preferred numbers |journal=VDI-Zeitschrift |volume=98 |date=1956 |pages=267–274))
* ((cite book |title=Logarithmen, Normzahlen, Dezibel, Neper, Phon - natürlich verwandt! |language=de |trans-title=Logarithms, preferred numbers, decibel, neper, phon - naturally related! |author-first=Eugen |author-last=Paulin |date=2007-09-01 |url=http://www.rechenschieber.org/Normzahlen.pdf |access-date=2016-12-18 |url-status=live |archive-url=https://web.archive.org/web/20161218223050/http://www.rechenschieber.org/Normzahlen.pdf |archive-date=2016-12-18))

==External links ==
*[https://web.archive.org/web/20140716201523/http://www.microwaves101.com/ENCYCLOPEDIA/neper.cfm What's a neper?]
*[http://www.sengpielaudio.com/calculator-neper.htm Conversion of level gain and loss: neper, decibel, and bel]
*[https://web.archive.org/web/20140429141423/http://www.microwaves101.com/encyclopedia/transmission_loss.cfm Calculating transmission line losses]

((SI units))

[[Category:Units of level]]

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