Neper

(′nē·pər)

(physics) Abbreviated Np. Also known as napier. A unit used for expressing the ratio of two currents, voltages, or analogous quantities; the number of nepers is the natural logarithm of this ratio. A unit used for expressing the ratio of two powers (even when this ratio is not the square of the corresponding current or voltage ratio); the number of nepers is the natural logarithm of the square root of this ratio; to avoid confusion, this usage should be accompanied by a specific statement.

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A unit of attenuation used in transmission-line theory. On a uniform transmission line having waves traveling in only one direction, the magnitudes of voltage E and of current I decrease with distance x traveled, as given
1. $\frac{E}{E_0}=\frac{I}{I_0}=\epsilon^{-\alpha x}$
by Eq. (1), where E₀, I₀, and α are constants. The attenuation in nepers between the points where E₀, and I₀ are measured and where E and I are measured is given by Eq. (2), in which In denotes the natural
2. $\alpha x={\rm ln}\ \frac{E_0}{E}={\rm ln}\ \frac{I_0}{I}$
(or napierian) logarithm. One neper equals 8.686 dB, the decibel being the practical unit of attenuation. See also Decibel; Transmission lines.

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The unit of measurement based on Napierian logarithms that represents the ratio between two values, such as current or voltage.

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napier

[Etymology: I. Nepero = J. Napier] engineering. Symbol Ne. A geometric scale for power ratios, akin to, and close to, the bel but using the natural logarithm and addressing primarily the amplitudes, defined quantitatively for amplitudes A₁ and A₂ as

where P₁ and P₂ are the respective powers. Each unit increase thus represents an increase in the amplitude ratios of e = 2.718 3~, in the power ratio of e² = 7.389 1~. The bel, in contrast, addressing primarily the power ratios, is calculated as

each unit increase representing a ten-fold increase in the power.

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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

$Np = \ln\frac{x_1}{x_2} = \ln x_1 - \ln x_2. \,$

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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Best of the Web: Neper

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Some good "Neper" pages on the web:

Egyptian Mythology
www.pantheon.org

Learn More

	decineper (physics)
	Napier (mechanics)
	skin depth (electromagnetism)

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