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{{short description|Logarithmic unit for ratios of measurements of physical field and power quantities}}
{{about|unit of measurement|other uses|Neper (disambiguation)}}

The '''neper''' (symbol: Np) is a logarithmic unit for [[ratio]]s of measurements of physical 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, 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>Bureau International des Poids et Mesures (2006), [http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf The International System of Units (SI) Brochure, 8th edition], pp. 127–128.</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 value of a ratio in nepers is given by
:<math>
L_{\rm Np} = \ln\frac{x_1}{x_2} = \ln x_1 - \ln x_2. 
</math>

where <math>x_1</math> and <math>x_2</math> are the values of interest (amplitudes), and {{math|ln}} is the natural logarithm.  When the values are quadratic in the amplitude (e.g. power), they are first linearised by taking the square root before the logarithm is taken, or equivalently the result is halved.{{citation needed|reason=It is clear from references (e.g. BIPM brochure) that this is true for amplitudes.  That there is a convention of referring power-like quantities to field-like quantities with the associated halving to determine a quantity in Np (the reverse of the established convention for dB) is not obvious, and I'm not even convinced that this is the case.  We must not assume from electronic texts that this is so.  We should have an authoritative reference.|date=July 2017}}

In the [[International System of Quantities]], the neper is defined as {{nowrap|1=1 Np = 1}}.<ref>[http://iopscience.iop.org/article/10.1088/0026-1394/30/5/010/pdf Thor, A. J. (1994). New International Standards for Quantities and Units. Metrologia, 30(5), 517.]</ref>

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

The neper is defined in terms of ratios of [[Field, power, and root-power quantities|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:

:<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==
{{refimprove section|date=December 2018}}
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.

Particular to the neper, however, is that the derived unit of ''[[centineper]]'' is asymptotically equal to percentage difference for very small differences – since the derivative of the natural log (at 1) is 1; this is not shared with other logarithmic units, which introduce a scaling factor due to the derivative not being unity. The centineper can thus be used as a linear replacement for percentage differences.{{citation needed|reason=It is unclear that this is of practical benefit, nor that this is an established practice; we need an authoritative reference.|date=December 2018}} The linear approximation for small percentage differences,
:<math>(1+\delta)(1+\epsilon) = 1 + \delta + \epsilon + \delta\epsilon \approx 1 + \delta + \epsilon,</math>
is widely used, particularly in finance—see for example the [[Fisher equation]]. However, it is only approximate, with error increasing for large percentage changes. Measured instead in centinepers, these linear approximations can be replaced with exact equalities{{OR|date=December 2018}}, and applicable to any magnitude change, by defining the following centineper quantity for any change <math>\delta</math>

<math>
D_{\delta} = 100 \ln\frac{(1+\delta) -1}{1} = 100 \ln\delta
</math>

==See also==
*[[Decibel]]
*[[Level (logarithmic quantity)]]
*[[Nat (unit)]]

== Notes and references ==
{{Reflist}}

== Further reading ==
* {{cite journal |author-last=Tuffentsammer |author-first=Karl |title=Das Dezilog, eine Brücke zwischen Logarithmen, Dezibel, Neper und Normzahlen |language=German |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=German |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 ==
*[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]
*[http://www.microwaves101.com/encyclopedia/transmission_loss.cfm Calculating transmission line losses]

{{SI units}}

[[Category:Units of measurement]]
[[Category:Logarithmic scales of measurement]]