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Neper

The neper (symbol: Np) is a dimensionless logarithmic unit used to express the level difference between two similar quantities, particularly field quantities such as amplitudes, pressures, or voltages in fields like acoustics, electronics, and telecommunications; it is defined such that one neper corresponds to a natural logarithm ratio of e (approximately 2.718) for the field quantity or e^2 for the corresponding power quantity.^[1] Named after the Scottish mathematician John Napier (1550–1617), who invented logarithms, the neper provides a coherent measure based on the natural logarithm, distinguishing it from base-10 logarithmic units.^[2] Although not an official SI unit, it is accepted for use with the International System of Units (SI) and can be combined with SI prefixes, such as the millineper (mNp).^[3] Introduced in the early 20th century alongside the bel, the neper emerged from efforts to standardize measurements of signal attenuation and gain in telephony; in 1929, the European International Advisory Committee on Long Distance Telephony recommended both the bel (using common logarithms) and the neper as alternative units for expressing transmission losses.^[4] The neper's adoption reflects ongoing metrological debates, with the Comité International des Poids et Mesures (CIPM) and the Conférence Générale des Poids et Mesures (CGPM) considering it as a potential coherent SI derived unit in the late 1990s and early 2000s due to its alignment with the SI's use of natural logarithms in equations like those for the radian.^[3] Despite this, it remains a supplementary unit, often used in contexts requiring precise logarithmic scaling for root-power quantities. In practice, the neper quantifies phenomena like acoustic absorption, where attenuation is expressed in nepers per unit length (e.g., Np/m), or electrical signal damping, with a level difference \Delta L_F = \ln(F_1 / F_2) Np for field quantities F_1 and F_2.^[1] It relates to the more common decibel (dB) by the approximation 1 Np ≈ 8.686 dB, since for field quantities the bel is defined as $2 \log_{10}(ratio) B and 1 Np = ln(ratio) corresponds to approximately 0.8686 B (with 1 B = 10 dB).^[1] This unit's utility lies in its direct compatibility with exponential decay models in physics and engineering, ensuring accurate representation of ratios without dimensional inconsistencies.^[5]

Introduction

Definition

The neper (symbol: Np) is a dimensionless unit used to express ratios between two similar physical quantities, particularly field quantities such as amplitude or voltage and power quantities in signals and waves.^[6] It serves as a logarithmic measure for relative levels, enabling the quantification of changes in magnitude without inherent dimensions, as the unit represents a pure ratio.^[7] Defined using the natural logarithm with base e ≈ 2.71828, the neper is particularly apt for describing phenomena involving exponential decay or growth, such as signal propagation in media. A change of 1 Np corresponds to a ratio of e:1 for field quantities or e²:1 for power quantities, providing a natural scale for such processes.^[8]^[6] Although not an SI unit itself, the neper is accepted for use with the International System of Units (SI) alongside other logarithmic units, as specified in ISO 80000.^[9] It plays a key role in measuring gain, loss, or attenuation in physical systems, offering a coherent framework for theoretical and practical analyses in relevant fields.^[7]

Etymology

The term "neper" derives from the Latinized form "Joannes Neper" of the name of John Napier (1550–1617), the Scottish mathematician credited with inventing logarithms.^[10]^[11] Napier's seminal publication, Mirifici Logarithmorum Canonis Descriptio in 1614, introduced logarithmic tables that simplified complex calculations in astronomy and navigation, establishing the groundwork for logarithmic measurement scales used in various scientific fields.^[12] This work focused on what became known as Napierian logarithms, precursors to the natural logarithm based on the mathematical constant e.^[13] The neper unit honors Napier's foundational role in logarithm development, as it employs the natural (or Napierian) logarithm to quantify amplitude ratios in signal transmission and attenuation.^[14] The name was first proposed in the mid-1920s during efforts to standardize transmission units in telephony, with the term appearing in technical literature by 1928.^[15]^[16]

Mathematical Definition

Field Quantities

In the context of field quantities, such as voltage, current, or sound pressure, the neper measures the logarithmic ratio between two amplitudes x_1 and x_2. The level difference L in nepers is defined by the formula

L = \ln\left(\frac{x_1}{x_2}\right) \, \mathrm{Np},

where \ln denotes the natural logarithm and \mathrm{Np} is the symbol for neper.^[6] This formulation arises because field quantities typically scale linearly with the square root of the corresponding power, and the natural logarithm directly captures the exponential relationships prevalent in physical phenomena like wave propagation and signal attenuation.^[6] By using the base-e logarithm, the neper aligns with the mathematical properties of exponential growth and decay, providing a dimensionally consistent unit for amplitude ratios.^[6] A key property of the neper for field quantities is that a change of 1 Np corresponds to a ratio x_1 / x_2 = [e](/page/E!) \approx 2.718, meaning the amplitude increases or decreases by a factor of approximately 2.718.^[6] This additivity in nepers facilitates the analysis of cascaded systems, such as successive attenuators in a signal chain, where total level differences are the sum of individual neper values since the logarithm of a product equals the sum of logarithms. For example, if a signal's amplitude doubles (x_1 / x_2 = 2), the level change is \ln(2) \approx 0.693 Np.^[6]

Power Quantities

In the context of power quantities, such as intensity or electrical power, the level difference L in nepers is defined as L = \frac{1}{2} \ln \left( \frac{p_1}{p_2} \right) Np, where p_1 and p_2 represent the two power quantities being compared.^[1]^[17] This formulation expresses the logarithmic ratio using the natural logarithm, scaled by a factor of \frac{1}{2} to maintain dimensional consistency with field-based measurements. The inclusion of the \frac{1}{2} factor arises from the quadratic relationship between power and field quantities, where power p is proportional to the square of a field quantity x (i.e., p \propto x^2). Consequently, \ln \left( \frac{p_1}{p_2} \right) = 2 \ln \left( \frac{x_1}{x_2} \right), and dividing by 2 ensures that the neper level for power aligns directly with the neper level for the corresponding field ratio.^[17]^[1] A change of 1 Np in power level corresponds to a power ratio \frac{p_1}{p_2} = e^2 \approx 7.389, which facilitates the additive combination of neper levels when power gains or losses occur in series, such as in cascaded systems.^[17] For example, if power increases by a factor of 4, the level change is \frac{1}{2} \ln(4) = \ln(2) \approx 0.693 Np, which matches the neper level for a field quantity doubling.^[1]

Comparisons with Other Units

Bel and Decibel

The bel (symbol: B) is a dimensionless unit for expressing ratios of power levels, defined using the common logarithm (base 10) as B = \log_{10} \left( \frac{P_1}{P_2} \right), where P_1 and P_2 are the two power quantities being compared. For field quantities such as voltage, current, or amplitude—where power is proportional to the square of the field—a change of one bel corresponds to B = 2 \log_{10} \left( \frac{x_1}{x_2} \right), reflecting the quadratic relationship between field and power. The bel provides a logarithmic scale suited to decimal arithmetic, facilitating the representation of wide-ranging ratios in engineering contexts.^[15] The decibel (symbol: dB), introduced as a more practical subunit, equals one-tenth of a bel and is defined for power ratios as \mathrm{dB} = 10 \log_{10} \left( \frac{P_1}{P_2} \right), allowing finer granularity in measurements of small changes. For field quantities, the decibel formula adjusts to \mathrm{dB} = 20 \log_{10} \left( \frac{x_1}{x_2} \right), ensuring consistency with power-based definitions. This base-10 structure enhances intuitive handling of powers-of-ten variations, making it prevalent in fields like audio engineering and electronics for quantifying gains, losses, and signal levels.^[15] In contrast to the neper, which relies on the natural logarithm (base e) for seamless integration with exponential decay in differential equations and physical models, the bel and decibel prioritize base-10 logarithms to align with decimal scaling and perceptual scales in human-engineered systems. The bel originated in 1923 at Bell Telephone Laboratories, named in honor of Alexander Graham Bell to standardize transmission loss measurements, with the decibel formalized shortly thereafter for widespread adoption in telephony and acoustics.^[4]

Conversion Factors

The conversion between nepers (Np) and decibels (dB) arises from the difference in logarithmic bases used in their definitions: the neper employs the natural logarithm (base e), while the decibel uses the common logarithm (base 10). This relationship is derived from the change of base formula for logarithms, ensuring consistent measurement of ratios for both field and power quantities.^[1]^[8] For field quantities, such as voltage or amplitude ratios, the neper is defined as \ln(x_1 / x_2) Np, while the decibel is $20 \log_{10}(x_1 / x_2) dB. The conversion follows as:

\ln\left(\frac{x_1}{x_2}\right) = \frac{\ln 10}{20} \times 20 \log_{10}\left(\frac{x_1}{x_2}\right),

yielding $1 Np = \frac{20}{\ln 10} dB. Since \ln 10 \approx 2.302585, this approximates to $1 Np \approx 8.685889638 dB. The reciprocal is $1 dB \approx \frac{\ln 10}{20} Np \approx 0.115129255 Np.^[1]^[8] For power quantities, the neper is defined as \frac{1}{2} \ln(p_1 / p_0) Np to maintain consistency with field quantities (since power ratios relate to the square of field ratios), while the decibel is $10 \log_{10}(p_1 / p_0) dB. The derivation parallels the field case:

\frac{1}{2} \ln\left(\frac{p_1}{p_0}\right) = \frac{\ln 10}{20} \times 10 \log_{10}\left(\frac{p_1}{p_0}\right),

resulting in the same factor: $1 Np = \frac{20}{\ln 10} \approx 8.685889638 dB, or $1 dB \approx 0.115129255 Np. This equivalence avoids ambiguity in telecommunications applications, where direct use of \ln(p_1 / p_0) Np for power is prohibited as it would yield approximately 4.343 dB per Np.^[1]^[8] These conversions are exact through the logarithmic base change and apply universally to ratio measurements, though nepers are less commonly used in practice due to their non-decimal scaling compared to the base-10 convenience of decibels.^[8]

Applications

Telecommunications and Electronics

In electronics, the neper quantifies voltage or current ratios, particularly for expressing attenuation and gain in amplifiers, filters, and transmission lines. For instance, the attenuation constant α, measured in nepers per meter (Np/m), describes signal loss per unit length in cables and waveguides.^[18] This unit arises naturally from the exponential decay of signals, where the amplitude ratio is given by e^{-αl} for a length l.^[19] A key application appears in the telegrapher's equations, which model transmission lines in communication systems. The propagation constant γ = α + jβ incorporates α in nepers per meter to represent attenuation due to conductor resistance and dielectric losses, while β denotes the phase constant in radians per meter.^[18] This formulation aids in designing coaxial cables and microstrip lines for telecommunications, ensuring efficient signal propagation over distances.^[20] The neper's advantages stem from its basis in the natural logarithm, making it additive for cascaded networks. When signals pass through multiple stages, such as successive amplifier or filter sections, the total attenuation in nepers is the sum of individual attenuations, simplifying analysis of complex systems.^[21] Additionally, it aligns with differential equations in circuit analysis; for example, in an RC circuit, the decay rate of voltage or current is expressed as the neper frequency α in nepers per second, where α = 1/(RC), facilitating solutions to transient responses.^[22] In RF engineering, nepers describe insertion loss—the reduction in signal power through a device—and return loss—the reflected power at an interface—though decibels are more commonly used for practical measurements and specifications.^[21] This preference for decibels in telecommunications standards arises from their convenience in logarithmic scales, but nepers remain valuable for theoretical computations involving complex propagation.^[19]

Acoustics

In acoustics, the neper serves as a unit for expressing ratios of sound pressure levels, which are field quantities, or sound intensity levels, which are power quantities, particularly in contexts involving exponential decay during wave propagation.^[21] For instance, the attenuation of sound through materials or in enclosed spaces, such as rooms, is quantified using the neper to describe the reduction in pressure or intensity, where the pressure amplitude follows p(x) = p(0) e^{-\alpha x} and \alpha is the attenuation coefficient in nepers per meter (Np/m).^[23] A key application involves reverberation time calculations, where the neper measures the air absorption coefficient in formulas like the modified Sabine equation T = \frac{0.161 V}{A + 4 m V}, with m in Np/m representing viscous and thermal losses in air that contribute to the exponential decay of sound energy.^[24] Similarly, absorption coefficients for materials are sometimes expressed in nepers to model energy dissipation in theoretical analyses of room acoustics or barriers.^[23] The neper is favored in theoretical acoustics due to its basis in the natural logarithm, which naturally aligns with the exponential solutions of the wave equation governing sound propagation, such as in derivations of attenuation from viscosity and thermal conduction.^[23] For pressure ratios, the level is defined as L = \ln(p_1 / p_2) Np, providing a dimensionless measure that simplifies mathematical modeling of acoustic phenomena.^[21] In specialized fields like underwater acoustics, nepers quantify transmission loss over distance, incorporating absorption terms in propagation models where total loss includes spherical spreading plus frequency-dependent attenuation, often expressed as \alpha in Np/m for molecular relaxation and friction effects.^[25] Likewise, in aeroacoustics, nepers describe atmospheric absorption during sound propagation, capturing dissipation rates in Np/m for outdoor noise analysis from aircraft or wind sources.^[23]

Optics

In optics and radiometry, the neper is used to express optical depth or attenuation, defined as the natural logarithm of the ratio of incident to transmitted radiant power through a material, τ = -ln(I/I_0), where I is the transmitted intensity and I_0 the incident. This measures the attenuation due to absorption and scattering in media like the atmosphere or biological tissues, with the absorption coefficient often in Np/m. In some contexts, nepers express power ratios directly without the 1/2 factor conventional for field-to-power relations.^[26]^[1]

History and Standardization

Origins

The neper unit was first proposed in the 1920s as a logarithmic measure based on the natural logarithm, serving as an alternative to the base-10 bel unit during the rapid expansion of telephony and radio communications. This development occurred amid increasing reliance on logarithmic scales to quantify signal attenuation and power ratios in transmission lines, where traditional linear units proved inadequate for handling the wide dynamic range of electrical signals. Engineers recognized that natural logarithms, with base e, offered mathematical advantages in differential equations and exponential decay models common in circuit analysis.^[15] The unit emerged primarily from collaborative efforts at Bell Telephone Laboratories in the United States, alongside contributions from international bodies such as the Comité Consultatif International des Communications Téléphoniques à Grande Distance (C.C.I.) and the British Post Office Engineering Department. Bell Labs researchers advocated for a neper-based unit to simplify calculations in transmission theory, particularly for repeater spacing and impedance matching in long-distance circuits. These discussions built on earlier 1923 Bell System introductions of transmission units and gained momentum through 1924 meetings of the International Advisory Committee on Long Distance Telephony, where the neper was debated as a standardized option for global compatibility.^[15]^[27] Initial adoption of the neper appeared in technical literature between 1924 and 1925, particularly for expressing attenuation in long-distance communication lines, as documented in papers from the American Institute of Electrical Engineers (A.I.E.E.) and Electrical Communication. For instance, W.H. Martin's 1924 A.I.E.E. article and R.V.L. Hartley's July 1924 publication highlighted its use in measuring power losses at speech frequencies around 800 Hz, aiding designs for cable and repeater systems. The name "neper" was selected to honor John Napier, the 17th-century Scottish mathematician who invented logarithms, thereby distinguishing it from base-10 units like the bel and underscoring its roots in natural logarithmic principles.^[27]

Adoption in Standards

The neper (Np) was officially defined in the international standard ISO 80000-3:2006 as the coherent dimensionless unit for logarithmic quantities based on the natural logarithm, where 1 Np = 1 corresponds to the value of the natural logarithm of the ratio of two quantities. This specification establishes the neper as a special name for expressing ratios in field and power quantities without introducing additional dimensions. The definition has been maintained and referenced in subsequent parts of the ISO 80000 series, ensuring consistency in international nomenclature for quantities and units. The neper has been accepted by the International Committee for Weights and Measures (CIPM) for use alongside the International System of Units (SI) since the publication of the 9th edition of the SI Brochure in 2019, though it is explicitly not classified as an SI base or derived unit.^[7] This acceptance by the CIPM, under the auspices of the International Bureau of Weights and Measures (BIPM), allows the neper to be employed in scientific and technical contexts compatible with SI without altering the system's coherence. The status remains unchanged in the brochure's latest revision (version 3.02, August 2025).^[28] In electrotechnical standards, submultiples of the neper are defined to accommodate smaller-scale measurements, such as the decineper (dNp), which equals 0.1 Np.^[29] This unit facilitates practical expression of minor attenuations or gains in fields like telecommunications. As of 2025, the neper sees limited practical adoption, primarily due to the widespread preference for the decibel in international standards and applications, but it is retained for theoretical analyses and high-precision engineering where natural logarithmic ratios provide conceptual advantages. No significant updates to its standardization have occurred since the 2019 SI Brochure.

References

[1]
NIST Guide to the SI, Chapter 8
Jan 28, 2016 · These definitions imply that the numerical value of LF when LF is expressed in the unit neper is {LF}Np = ln(F/F0), and that the numerical ...
[2]
Nepers – Knowledge and References - Taylor & Francis
(The unit 'neper” is named after the mathematician John Napier and “decibel” is used to honor Alexander Graham Bell; 1 neper = 8.868 dB.)Missing: origin | Show results with:origin
[3]
[PDF] The unit one, the neper, the bel and the future of the SI - BIPM
The neper would no longer be used to express levels, as stated in the SI brochure, it could now be defined as SI derived unit for natural logarithmic amplitude ...
[4]
Using the Decibel - Part 1: Introduction and underlying concepts - EDN
May 28, 2008 · In the beginning, it was among several units of measurement. Prior to 1923 “gains” and “losses” of telephone circuits were labeled in miles ...<|control11|><|separator|>
[5]
Definitions of the Units Radian, Neper, Bel, and Decibel | NIST
Aug 1, 2001 · Abstract. The definition of coherent derived units in the International System of Units (SI) is reviewed, and the important role of the ...
[6]
[PDF] Guide for the Use of the International System of Units (SI)
Feb 3, 1975 · These units are not SI units, but they have been accepted by the CIPM for use with the SI. For additional information on the neper and bel, see ...
[7]
[PDF] SI Brochure - 9th ed./version 3.02 - BIPM
May 20, 2019 · In this respect, the English text presented here follows the ISO/IEC 80000 series Quantities and units. ... Non-SI units accepted for use with the ...
[8]
None
### Summary of Neper Unit from ITU-R V.574-5
[9]
https://www.iso.org/obp/ui/#iso:std:iso:80000:-3:ed-1:v1:en
[10]
What is a Neper & Conversion to dB Chart - Electronics Notes
The neper is named after John Napier who invented logarithms. Napier was a scot who lived between 1550 and 1617, and his latinised name was Ioannes Neper ...
[11]
neper - Wiktionary, the free dictionary
Etymology. From Joannes Neper, the Latinized name of John Napier, a Scottish ... The unit of α is nepers per meter. It can be easily seen that if a line ...
[12]
Mathematical Treasure: Napier's Logarithms in English
John Napier originally published his theory of logarithms in 1614 under the title, Mirifici Logarithmorum Canonis.
[13]
Napierian Logarithm -- from Wolfram MathWorld
The first definition of the logarithm was constructed by Napier and popularized through his posthumous pamphlet (Napier 1619).
[14]
Neper - Oxford Reference
1 neper = 8.686 decibels. The unit is named after John Napier (1550–1617), the inventor of natural logarithms. From: neper in A Dictionary of Physics ».Missing: origin | Show results with:origin
[15]
[PDF] Bell System Technical Journal January, 1929 Decibel—The Name ...
It was further sug- gested that the naperian unit be called the "neper" and that the fundamental decimal unit be called the "bel," these names being derived ...<|control11|><|separator|>
[16]
NEPER Definition & Meaning - Dictionary.com
Neper definition: the unit used to express the ratio of two amplitudes as a ... Word History and Origins. Origin of neper. First recorded in 1920–25 ...
[17]
None
### Definitions and Formulas for Neper
[18]
2.2: Transmission Line Theory - Engineering LibreTexts
Jun 23, 2023 · with SI units of m − 1 and where α is the attenuation coefficient and has units of nepers per meter ( Np/m ), and β is the phase-change ...Missing: telecommunications | Show results with:telecommunications
[19]
What's a Neper? - Microwave Encyclopedia
The Neper (Np) is similar to the decibel (dB). It's unitless and is used to express ratios. The Neper uses the natural logarithm, and is named after Microwave ...
[20]
Transmission Line Loss - Microwave Encyclopedia
Notice the "natural" units of transmission line loss are Nepers/length. To get loss in dB/length, multiply Nepers by 8.686.
[21]
None
### Summary of Neper Use in Telecommunications (ITU-R V.574-3)
[22]
Complex Frequency in Network Functions - Sanfoundry
The dimension of neper frequency is neper per second. The imaginary part, ωn is the radian frequency or the real frequency. The dimension of radian ...
[23]
[PDF] Chapter 8 – Absorption and Attenuation of Sound
Here α x is specified in Nepers (no units) where the Neper is the natural logarithm of the ratio of pressures as follows. (0) ln. nepers (no units) ( )
[24]
Proposal of a simplified methodology for reverberation time ...
Jan 27, 2023 · ... reverberation time prediction, mainly, as a calculation tool in building acoustics design. ... Neper/m) and A is the sound absorption area ...
[25]
[PDF] Principles of Underwater Sound
What is a neper? natural log ratio. Absorption. Page 22. Which Loss is ... Sound Level = Sound Pressure Level – Transmission Loss. Sound Pressure Level ...
[26]
[PDF] C.C.I.F. Plenary Session: Documents (Paris, 1926)
... Unit of Transmission is discussed and a proposal made that the unit in terms of the Naperian logarithm shall be the Neper and that the unit on the Briggs ...
[27]
SI Brochure - BIPM
SI Brochure: The International System of Units (SI). Note for translators and editors: Details on changes made since May 2019 are available on request to ...Missing: neper | Show results with:neper
[28]
IEC 60050 - Details for IEV number 801-22-04: "neper" - Electropedia
Note 1 – The decineper is one-tenth of a neper. Note 2 – One neper equals 8,686... decibels. fr. neper, m. unité de niveau utilisée pour une grandeur de champ ...