The neper
A non-system unit accepted for use · quantity: level, a logarithmic ratio
01 · Definition
One neper is when the amplitude has fallen by a factor of e. Not two, not ten, but 2.718 — because that is how everything decays that decays of itself.
The level of a field quantity is the natural logarithm of the ratio: L = ln(F/F₀) nepers. For a power level one takes half the logarithm, and then one neper still means eight point six eight six decibels.
In the time equal to the resistance times the capacitance, the voltage falls by exactly a factor of e. The engineer says «one tau has passed», the metrologist says «a decay of one neper», and it is the same number. After five time constants less than a per cent is left, so the transient is taken to be over.
02 · Conversion
Enter a level and the sheet turns it into ratios and decibels
The neper is coherent with the SI, so prefixes to it are lawful: the decineper and the centineper turn up in European telecoms practice. The decibel has no such option — it already carries a prefix of its own.
The attenuation of a line is given as a coefficient in nepers per metre: it goes straight into the exponent that describes the wave. The propagation constant is made up of this coefficient and the phase one — nepers and radians side by side.
| Unit | Name | Value | Where it is met |
|---|---|---|---|
| {u} | {name} | {value} | {note} |
03 · Orders of magnitude
the scale is linear in nepers, as it should be04 · Measuring instruments
Where nepers are read off an instrument
From the decay of the reflected pulse one finds the break in the line and its coefficient per unit length. In engineering tables it comes in decibels per hundred metres, in theory — in nepers per metre.
The time constant is read straight off the screen: the time in which the curve falls to 36.8 per cent is one neper of decay. The cursors of today's instruments work it out by themselves.
A vector analyser measures the complex transmission coefficient. Its magnitude is given in decibels, but into the calculation of the propagation constant it enters already in nepers.
The absorption of sound in metal and in living tissue is described by a coefficient in nepers per metre. It is exactly what sets the depth the instrument can look into.
05 · Writing rules
capital N, small p, prefixes allowed
The capital N is because the unit is named after John Napier; the small p finishes the abbreviation of the Latin form of his name. A space goes between the number and the symbol, and the symbol has no plural. Prefixes are allowed: dNp and cNp are legitimate.
A common slip is to lose the half in the definition of a power level. Write the whole logarithm of the power ratio instead of half of it, and the factor for going over to decibels comes out wrongly as 4.343 instead of 8.686.
06 · Neighbouring units
Three units of level describe one and the same ratio and differ only in the base of the logarithm and in the factor. The neper takes the natural logarithm and is therefore at home in equations; the bel and the decibel take the decimal one and are at home in reports.
Reference values are laid down by separate standards. For the attenuation of a line the reference is the amplitude at the input, for a transient it is the initial value, for the skin depth it is the field at the surface of the conductor.
07 · Historical section
Two schools of one scale
Twenty years of reckoning by hand — and tables that turned multiplication into addition. The irony is that Napier's own logarithms were not natural ones: their base is close to the reciprocal of e, and the factor is seven digits.
The telephone men of the continent counted attenuation in nepers: the natural logarithm followed straight from the equations of the line. Across the ocean in the same years the bel was being introduced, and the two scales lived side by side.
Splitting the neper with prefixes is allowed, and in the European telecom norms the tenth part took root. Little by little the decibel pushed it out — simply because in international papers one notation wins.
The decimal logarithm of a power ratio proved handier for reports: convenient round numbers, easy arithmetic in the head. The trades chose the decibel, and the neper stayed in the equations.
The neper lost the paperwork and won the equations
The quarrel of the two scales was settled not by mathematics but by the convenience of the form: in decibels the gain of a stage adds up in round numbers, and the fitter counts them in his head. The neper gives no round numbers — but it is the only one that enters the exponent without a factor.
Catalogue · units of measurement
A data sheet for every quantity
Seven SI base units, twenty-two derived ones with their own names, and non-system units alongside them. Each has its own sheet: definition, dimension, conversion, instruments, rules of writing. In 36 languages.
Non-SI units
open the neper data sheetSI base units
the neper is dimensionless — there are no base units in itSI derived units with names of their own
in terracotta, those whose ratios are counted in nepersSheet language
36 languages. The symbol Np and the formulas are not translated — they are the same in every locale; what is translated is the prose, the local land measures and the writing rules.