what is neper and decibal give relation between them
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Answer:
Before dealing with various types of filters, it will be desirable to know the unites in which the attenuation of a network is measured. These unites are (i) the decibel (dB) and (ii) the neper. The origin of the decible (dB) can be traced back to the telephony industry. Our ears respond to sound intensity in a non-linear logartithmic fashion. Since many of the amplifiers wear used to supply an audio-output, the use of a lognarithmic power output scale was deemed reasonable and convenient. The basic unit of power gain (the ratio of power output and the power input to a network) was termed the bel in honour of A.G. Bell. The number to bels by which the power output P_{out} exceeds the power input P_{in} is defined as
Number of bels =log_{10}\dfrac{p_{out}}{P_{in}}
For example, a power gain of 100 would be equivalent to log_{10} 100 i.e. 2 bels. The bel proved to be too large for most work, so one tenth of a bel the decible (dB), was adopted. In decibels power gain of 100 is equivalent to 10 log_{10} 100=10*2=20 dB.
In short, to convert a power expressed in bels into decibels, we merely multiply the number of bels by 10. Symbolically
Number dB = 10*number of bels
Number of dB 10 log_{10}\dfrac{P_{out}}{P_{in}}
In practice, it is much easier and covenient to measure voltage than to measure power levels. Therefore, a relationship between the voltage gain and the power gain in decibels was developed. It is derived as below
Number of dB = 10 log_{10}\dfrac{V^2_{out}/R_L}{V^2_{in}/R_{in}}=10 log_{10}\left[\dfrac{V_{out}}{V_{in}}\right]^2 \dfrac{R_{in}}{R_L}
=20 log_{10}\dfrac{V_{out}}{V_{in}}
if the power P_{out} and P{in} are associated with equal impedances.
In a similar fashion, the current gain in in decibles can be related to the power gain in decibels and
Number of dB =20 log_{10}\dfrac{I_{out}}{V_{in}}
The nepar is fundamentally a unit of current or voltage ratio.
The attenuation in nepers =log_e\dfrac{I_{out}}{I_{in}}
or =log_e\dfrac{V_{out}}{V_{in}}
The attenuation dB =20 log_{10}\dfrac{I_{out}}{I_{in}}=\dfrac{20 log_e\dfrac{I_{out}}{I_{in}}}{log_{10}e} =8.686 log_e\dfrac{I_{out}}{I_{in}}
Thus attenuation in dB = 8.686*attenuation in nepers.
It should be noted that the two units are defined on alogarithmic base, the decibel being being expressed in logarithms to the base 10 and the neper being expressed in logarithms to the base 10 and the neper being expressed to the base ‘e’. This is since their greatest field of application is in sound transmission, the loudness of which is a logarithmic function of the sound energy.
Example 1. The output and input voltage of a filter network are 10 mV and 20 mV respectively. Determine the attenuation in dB and nepers.
Solution: Input voltage, V_{in}=20 mV
Output voltage, V_{out}=10 mV
Attenuation in dB =20 log_{10}\dfrac{V_{out}}{V_{in}}=20 log_{10}\dfrac{10}{20}
=-6 dB Ans.
Attenuation in nepers \dfrac{Attenuation\enspace in\enspace dB}{8.686}=\dfrac{-6}{8.686}
= -0.6908 nepers Ans.
The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. As is the case for the decibel and bel, the neper is a unit defined in the international standard ISO 80000. It is not part of the International System of Units (SI), but is accepted for use alongside the SI.
Like the decibel, the neper is a unit in a logarithmic scale. While the bel uses the decadic (base-10) logarithm to compute ratios, the neper uses the natural logarithm, based on Euler's number (e ≈ 2.71828). The level a ratio of two signal amplitudes or root-power quantities.
The natural logarithm of the ratio of two amplitudes is measured in nepers. Show that one neper = 8.68 dB. The neper (symbol Np) expresses the ratio of two field quantities such as voltage or current, the square of which is proportional to power by the natural logarithm of this ratio. The value of a power ratio in nepers is one-half of the natural logarithm of the power ratio.
dB↔Np 1 Np = 8.6860000036933 dB.
The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms.
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