Monday, March 10, 2008

For Neper as a mythological god, see Neper (mythology), for the lunar crater named Neper, see Neper (crater), and for the Scottish mathematician, phycisist and astronomer, see John Napier.
A neper (Symbol: Np) is a unit of ratio. It is not an SI unit but is accepted for use alongside the SI. It is used to express ratios, such as gain and loss, and relative values. The name is derived from John Napier, the inventor of logarithms.
Like the decibel, it is a unit in a logarithmic scale, the difference being that where the decibel uses base-10 logarithms to compute ratios, the neper uses base e ≈ 2.71828. The value of a ratio in nepers, Np, is given by
$
Np = lnfrac{x_1}{x_2} = ln x_1 - ln x_2 ,
$
.
where x1 and x2 are the values of interest, and ln is the natural logarithm.
The neper is often used to express ratios of voltage and current amplitudes in electrical circuits (or pressure in acoustics), whereas the decibel is used to express power ratios. Taking that into account, we have
$
1 mbox{Np} = frac{20}{ln 10} mbox{dB} = 20 log_{10} e mbox{dB} approx 8{.}685889638 mbox{dB} ,
$

and
$
1 mbox{dB} = frac{ln 10}{20} mbox{Np} = frac{1}{20 log_{10} e} mbox{Np} approx 0{.}115129254 mbox{Np} ,
$
.
The decibel and the neper have a fixed ratio to each other. The (voltage) level is
$
L = 20 lg frac{x_1}{x_2} ,mathrm{dB} = ln frac{x_1}{x_2} ,mathrm{Np} ,
$
.
Like the decibel, the neper is a dimensionless unit. The ITU recognizes both units.