# Logarithmic size

Under logarithmic sizes is understood in the electrical engineering , communication engineering and acoustic sizes as logarithm of two values of the ratio of size to be formed.

If the logarithm of the ratio of an output value to a variable input value is formed, as in the case of amplification (see audio amplifier , amplifier (electrical engineering) or amplification (physics) ), the logarithmic quantity is called a measure . On the other hand , if the reference value is fixed , such as the volume , one speaks of a level .

When using the decadic logarithm, levels are specified in the auxiliary unit of measurement Bel or its tenth part, the decibel (unit symbol dB), when using the natural logarithm in neper (unit symbol Np).

## level

### definition

A level (signal level) is a logarithmic quantity that is defined by the logarithmic ratio of a power root quantity (previously called a field quantity) or a power quantity to a reference value that has the same dimension as the counter quantity. The meter size is used to describe the level in more detail. The usual formula symbol is (for level). ${\ displaystyle L}$

Example:

${\ displaystyle L = \ lg {\ frac {P} {P_ {0}}} \; \ mathrm {B}}$

is the level of the power or the power level in relation to the reference value . Due to the more manageable numerical values, in practical use levels are almost exclusively given in decibels instead of Bel. For the example given it results as follows: ${\ displaystyle P_ {0}}$

${\ displaystyle L = 10 \ cdot \ lg {\ frac {P} {P_ {0}}} \; \ mathrm {dB}}$.

If the difference is formed between two levels with the same reference value, this does not depend on the reference value (see calculation rules for logarithms ). For the example of the difference between two power levels we get:

${\ displaystyle \ Delta L = L_ {2} -L_ {1} = 10 \ cdot \ lg {\ frac {P_ {2}} {P_ {0}}} \; \ mathrm {dB} -10 \ cdot \ lg {\ frac {P_ {1}} {P_ {0}}} \; \ mathrm {dB} = 10 \ cdot \ lg \ left ({\ frac {P_ {2}} {P_ {0}}} \ cdot {\ frac {P_ {0}} {P_ {1}}} \ right) \; \ mathrm {dB}}$
${\ displaystyle \ Delta L = 10 \ cdot \ lg {\ frac {P_ {2}} {P_ {1}}} \; \ mathrm {dB}}$.

Although also specified in decibels, the quantity is not a level, but a measure , as the quantity in the denominator of the logarithmic ratio is not a fixed reference value. Occasionally the outdated and misleading term “relative level” is also used for it. ${\ displaystyle \ Delta L}$ ${\ displaystyle \ Delta L}$

### Levels of field quantities and power quantities

Power root variables or field variables such as electrical voltage or sound pressure are used to describe physical fields . The square of the effective value of such a field size is proportional in a linear system to its energetic state, which is recorded by a power size . In this context, quantities that are related to energy are also referred to as power quantities . Without having to know the exact laws, it follows that the ratio of two power quantities is equal to the ratio of the squares of the associated effective values ​​of the field quantities. For the direct calculation of levels from the ratios of effective values ​​of field sizes, an additional factor 2 results, for example when calculating the voltage level from the effective value of the electrical voltage : ${\ displaystyle L_ {u}}$${\ displaystyle U}$

${\ displaystyle L_ {u} = 10 \ cdot \ lg {\ frac {U_ {1} ^ {2}} {U_ {0} ^ {2}}} \; \ mathrm {dB} = 20 \ cdot \ lg {\ frac {U_ {1}} {U_ {0}}} \; \ mathrm {dB}}$.

For a voltage level of 10 decibels, the voltage must therefore be that the reference value (3.16 times approx) times be. ${\ displaystyle U_ {1}}$${\ displaystyle {\ sqrt {10}}}$${\ displaystyle U_ {0}}$

In physics, signal amplitudes often range over several orders of magnitude : For example, megavolt to nanovolt as the ratio of field sizes and megawatts to picowatts as the ratio of power parameters . Thanks to the logarithm, these quantities can be represented in easy-to-read, mostly two- to three-digit numbers for practical use.

Characteristic curves of amplifiers, filters or other electronic elements and spectra in acoustics can be displayed more easily and clearly, as the diagram covers a high level of dynamics due to the logarithmic display .

### Calculating with levels

Since the calculation rules for logarithms apply to level calculations, z. B. Multiplication of the physical quantities in additions. The output level of amplifier or attenuation elements connected in series (e.g. cables or plug connections) can be obtained by simply adding the input level with the individual logarithmic amplification or attenuation values.

For performance variables such as energy, intensity and power applies: As lg lg 1 and 10 = 2 ≈ 0.3, you can remember a rule of thumb:

+10 dB means tenfold, +3 dB means doubling, −10 dB means one tenth, −3 dB means half.

Other values ​​can be estimated from this, e.g. B. +16 dB = (+ 10 + 3 + 3) dB, so: original value × 10 × 2 × 2; +16 dB is 40 times that.

The following rule of thumb applies to field quantities such as linear sound field quantities , electrical voltage and current strength :

+20 dB corresponds to a tenfold increase, +6 dB means a doubling, −20 dB a tenth, −6 dB a halving.

Other values ​​can be estimated from this; z. B. for an attenuation of −26 dB related to 1 volt: −20 dB corresponds to one tenth; this results in: 0.1 volt = 100 mV; a further −6 dB (corresponding to halving) related to this 100 mV thus result in 50 mV.

### application

Level indications are widespread , especially in acoustics . Applications can also be found in high-frequency technology as part of communications technology , sound technology (see audio level ) and automation technology . For special applications with voltages in electrical engineering, see voltage level .

Filters for frequency weighting are predominantly used when indicating the level of audible sounds . These filters are intended to produce a measurement result that matches the actual volume impression better than the unevaluated information. According to all ISO standards , a frequency weighting must be indicated by an index on the level size. In contrast to this, the following notations are often used to indicate the use of the different weighting filters.

• dB A , dB (A), "dBA"
• dB B , dB (B), "dBB"
• dB C , dB (C), "dBC"

## Dimensions

A logarithmized ratio of two field quantities or two power quantities is formed as a measure , which is used to describe the properties of a system considered as a two-port , for example an amplifier. As a rule, the word "-Maß" is used as the ending of a compound word that describes the size in more detail.

Examples of such logarithmic measures are:

for performance sizes : sound reduction index ${\ displaystyle R}$

${\ displaystyle R = 10 \ lg {\ frac {I_ {0}} {I}} \; \ mathrm {dB}}$
(transmitted sound intensity , incident sound intensity ),${\ displaystyle I}$${\ displaystyle I_ {0}}$

for field sizes: voltage damping measure ${\ displaystyle A_ {U}}$

${\ displaystyle A_ {U} = 20 \ lg \ left | {\ frac {U_ {1}} {U_ {2}}} \ right | \; \ mathrm {dB} = \ ln \ left | {\ frac { U_ {1}} {U_ {2}}} \ right | \; \ mathrm {Np}}$
(Input voltage , output voltage ).${\ displaystyle U_ {1}}$${\ displaystyle U_ {2}}$

The advantages and calculation rules for levels also apply to dimensions.

## Classification of logarithmic quantities

The logarithmic quantities are quantities that are defined with the aid of logarithmic functions. They are divided according to the origin of the argument of the logarithm as follows.

Logarithmic ratios
Logarithmic ratios are defined by the ratio of two power quantities or two field quantities. This includes levels and dimensions. The units of measurement neper and bel or decibel may only be used for their identification .
Logarithmic quantities of information theory
These logarithmic quantities are defined by the fact that their argument is given from the outset as a number, as they are used, for example, in information theory. The units of measurement Shannon , Hartley and nat may only be used for their identification .
Other logarithmic quantities
Other specially defined logarithmic quantities are e.g. B.

## literature

• Jürgen H. Maue, Heinz Hoffmann, Arndt von Lüpke: 0 decibels plus 0 decibels equals 3 decibels . Erich Schmidt Verlag, Berlin 2003, ISBN 3-503-07470-8
• Frank Gustrau: High frequency technology: Basics of mobile communication technology. 2nd Edition. Carl Hanser Verlag, Munich 2013, ISBN 978-3-446-43245-1 .
• Hermann Weidenfeller: Basics of communication technology . Springer Fachmedien, Wiesbaden 2002, ISBN 3-519-06265-8 .