# Neper (auxiliary unit)

Physical unit
Unit name Neper
Unit symbol ${\ displaystyle \ mathrm {Np}}$
Physical quantity (s) Levels and dimensions
Formula symbol ${\ displaystyle L}$${\ displaystyle; Q}$
dimension ${\ displaystyle {\ mathsf {1}}}$
In SI units ${\ displaystyle 1}$
Named after John Napier

The Neper ( unit symbol Np) is an auxiliary unit of measurement named after the Scot John Napier (1550–1617, Latinized: Neper) for marking levels and measurements (for the meaning of both terms, see the article Logarithmic size ). It is used, among other things, in electrical engineering and acoustics .

## Definition, conversions

The Neper is used to identify the natural logarithm of the ratio of two service root sizes (formerly called "field sizes"). With the mathematical symbol and the real quantities and one writes: ${\ displaystyle \ ln}$${\ displaystyle F_ {1}}$${\ displaystyle F_ {2}}$

${\ displaystyle L = \ ln {\ frac {F_ {1}} {F_ {2}}} \, \ mathrm {Np}}$

The following applies:

${\ displaystyle L = 1 \, \ mathrm {Np} \ quad {\ text {if}} \ quad {\ frac {F_ {1}} {F_ {2}}} = \ mathrm {e} \ quad}$(= Base of the natural logarithm)

The Neper is considered by the BIPM for use with the International System of Units (SI) and by the International Telecommunication Union as a unit coherent with the SI. If the logarithmic quantities are defined by agreement using the natural logarithm, "the neper becomes the coherent unit that can be replaced by one, unit symbol 1" (in Section 4.1 and in Section 0.5):

${\ displaystyle 1 \; \ mathrm {Np} = 1}$

The unit Neper is coherent with the SI, but has not yet been accepted by the CGPM as an SI unit.

An indication in Neper can be based on the relationship

${\ displaystyle \ ln {\ frac {F_ {1}} {F_ {2}}} \, \ mathrm {Np} = 20 \; \ lg {\ frac {F_ {1}} {F_ {2}}} \, \ mathrm {dB}}$

convert to a specification in decibels (dB), where

${\ displaystyle 1 \, \ mathrm {Np} = {\ frac {20} {\ ln 10}} \, \ mathrm {dB} \ approx 8 {,} 686 \, \ mathrm {dB} \ ;; \ quad 1 \, \ mathrm {dB} = {\ frac {\ ln 10} {20}} \, \ mathrm {Np} \ approx 0 {,} 1151 \, \ mathrm {Np}}$

## application

In the case of complex service root quantities (field quantities), for example , the ratio of these complex quantities can also be elegantly treated with the natural logarithm. B. separate a complex damping factor into the damping measure (in neper) and phase shift angle (in radians ): ${\ displaystyle {\ underline {F}} _ {1} = | {\ underline {F}} _ {1} | \ cdot \ mathrm {e} ^ {\ mathrm {j} \ varphi _ {1}}}$

${\ displaystyle \ ln {\ frac {{\ underline {F}} _ {1}} {{\ underline {F}} _ {2}}} = \ ln \ left | {\ frac {{\ underline {F }} _ {1}} {{\ underline {F}} _ {2}}} \ right | + \ mathrm {j} (\ varphi _ {1} - \ varphi _ {2})}$

Such a calculation is not possible with the decibel without including factors. Since the neper has the value 1 as a unit like the radian, the units can simply be omitted from calculations.

In practice, for historical reasons, among other things, the Neper is used more for ratios of performance root sizes than for ratios of capacity sizes. When applied to power quantities that are proportional to the square of the power root quantities, i.e. where : ${\ displaystyle P}$${\ displaystyle P_ {1} / P_ {2} = (F_ {1} / F_ ​​{2}) ^ {2}}$

${\ displaystyle L = \ ln {\ frac {F_ {1}} {F_ {2}}} = \ ln {\ sqrt {\ frac {F_ {1} ^ {2}} {F_ {2} ^ {2 }}}} = {\ frac {1} {2}} \ ln {\ frac {P_ {1}} {P_ {2}}}}$

Since the 1970s, the Neper has been used for voltage levels and power levels , e.g. B. in communications technology , used less and less. Instead, the decibel is mainly used (dB, dBu, dBV, dBm, dBW).

## Individual evidence

1. a b c d DIN EN ISO 80000-3: 2013-08 Sizes and units - Part 3: Space and time
2. a b c d DIN EN 60027-3: 2007-11 Symbols for electrical engineering - Part 3: Logarithmic and related quantities and their units .
3. DIN 5493: 2013-10 Logarithmic sizes and units.
4. DIN EN ISO 80000-1: 2013-08 Sizes and units - Part 1: General , chap. 6.5.6