Neper

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 (e ≈ 2.71828). The value of a ratio in nepers is given by

${\displaystyle L_{\rm {Np}}=\ln {\frac {x_{1}}{x_{2}}}=\ln x_{1}-\ln x_{2}.}$

where ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ are the values of interest (amplitudes), and 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.

In the International System of Quantities, the neper is defined as 1 Np = 1.[2]

The neper is defined in terms of ratios of field quantities — also called root-power quantities — (for example, voltage or current amplitudes in electrical circuits, or pressure in acoustics), whereas the decibel was originally defined in terms of 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 laws) of the amplitude. Hence the decibel and the neper have a fixed ratio to each other:

${\displaystyle 1\ {\text{Np}}=20\log _{10}e\ {\text{dB}}\approx {\text{8.685889638 dB}}\,}$

and

${\displaystyle 1\ \mathrm {dB} ={\frac {1}{20}}\ln(10)\ \mathrm {Np} \approx {\text{0.115129255 Np}}.\,}$

The (voltage) level ratio is

${\displaystyle {\begin{aligned}L&=10\log _{10}{\frac {x_{1}^{2}}{x_{2}^{2}}}&{\text{dB}}\\&=10\log _{10}{\left({\frac {x_{1}}{x_{2}}}\right)}^{2}&{\text{dB}}\\&=20\log _{10}{\frac {x_{1}}{x_{2}}}&{\text{dB}}\\&=\ln {\frac {x_{1}}{x_{2}}}&{\text{Np}}.\\\end{aligned}}}$

Like the decibel, the neper is a dimensionless unit. The International Telecommunication Union (ITU) recognizes both units. Only the neper is coherent with the SI.[3]

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. The linear approximation for small percentage differences,

${\displaystyle (1+\delta )(1+\epsilon )=1+\delta +\epsilon +\delta \epsilon \approx 1+\delta +\epsilon ,}$

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, and applicable to any magnitude change, by defining the following centineper quantity for any change ${\displaystyle \delta }$

${\displaystyle D_{\delta }=100\ln {\frac {(1+\delta )-1}{1}}=100\ln \delta }$

• Nat (unit)

• Tuffentsammer, 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. Retrieved 2016-12-18.

• What's a neper?
• Conversion of level gain and loss: neper, decibel, and bel
• Calculating transmission line losses