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# Neper

[nee-per, ney-]
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 logarithmic 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 $x_1$ and $x_2$ 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. One kind of ratio may be converted into the other. Considering that wave power is proportional to the square of the amplitude, 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


begin{align} L & = 10 lg frac{x_1^2}{x_2^2} & mathrm{dB} & = 10 lg {left(frac{x_1}{x_2}right)}^2 & mathrm{dB} & = 20 lg frac{x_1}{x_2} & mathrm{dB} & = ln frac{x_1}{x_2} & mathrm{Np}. end{align}

Like the decibel, the neper is a dimensionless unit. The ITU recognizes both units.

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