# Performance size

In different contexts, especially of electrical engineering and acoustics (eg. As audio level , voltage gain , shielding effectiveness ) are physical quantities not stated directly, but only as a ratio to a second variable or fixed size of the same type. Preferably, the two sizes the ratio of which is specified, in each case by performance quantities or performance root quantities .

If the ratios extend over several powers of ten , it makes sense to specify them as a logarithmic quantity .

## Performance size

A performance quantity is a quantity that is proportional to a performance . ${\ displaystyle P}$

Examples: electrical power , electromagnetic and acoustic power and associated power densities

In this context, energy quantities, i.e. quantities that are related to an energy , are also referred to as power quantities .

Examples: electrical energy , electromagnetic and acoustic energy and associated energy densities ( sound power , sound intensity , sound energy density )

## Achievement root size

A performance root quantity is a quantity whose square is proportional to a performance quantity. Power root quantities were previously referred to as field quantities . ${\ displaystyle F}$

Examples: electrical voltage , electrical current strength , electrical and magnetic field strength , electrical and magnetic flux density , sound pressure , sound velocity

Power root quantities are usually effective values ; for a sine- shaped alternating quantity their can amplitude , complex amplitude or its complex RMS may be used. ${\ displaystyle {\ hat {F}}}$ ${\ displaystyle {\ underline {\ hat {F}}}}$${\ displaystyle {\ underline {F}}}$

## Logarithmic ratios

Determinations
{\ displaystyle {\ begin {aligned} {\ text {Mit}} F ^ {2} & \ sim P \ Leftrightarrow {\ frac {F_ {1} ^ {2}} {F_ {2} ^ {2}} } = {\ frac {P_ {1}} {P_ {2}}} \\ Q _ {(F)} & = \ ln {\ frac {F_ {1}} {F_ {2}}} \, \ mathrm {Np} = 2 \ lg {\ frac {F_ {1}} {F_ {2}}} \, \ mathrm {B} = 20 \ lg {\ frac {F_ {1}} {F_ {2}}} \, \ mathrm {dB} \\ Q _ {(P)} & = {\ frac {1} {2}} \ ln {\ frac {P_ {1}} {P_ {2}}} \, \ mathrm { Np} = \ lg {\ frac {P_ {1}} {P_ {2}}} \, \ mathrm {B} = 10 \ lg {\ frac {P_ {1}} {P_ {2}}} \, \ mathrm {dB} \ end {aligned}}}
Logarithmic ratio with field sizes${\ displaystyle Q _ {(F)}}$

Logarithmic ratio with performance quantities ${\ displaystyle Q _ {(P)}}$

Example for the gain of a two-port with the real voltages at the output and at the input:${\ displaystyle Q_ {U}}$
${\ displaystyle U_ {2}}$${\ displaystyle U_ {1}}$
${\ displaystyle Q_ {U} = \ left (\ ln {\ frac {U_ {2}} {U_ {1}}} \ right) \, \ mathrm {Np} = \ left (\ lg {\ frac {U_ {2} ^ {2}} {U_ {1} ^ {2}}} \ right) \, \ mathrm {B} = 20 \, \ left (\ lg {\ frac {U_ {2}} {U_ { 1}}} \ right) \, \ mathrm {dB}}$
or with the complex sizes :${\ displaystyle {\ underline {U}} _ {2} = | U_ {2} | \ cdot \ mathrm {e ^ {j \ varphi _ {2}}} {\ text {and}} {\ underline {U }} _ {1} = | U_ {1} | \ cdot \ mathrm {e ^ {j \ varphi _ {1}}}}$
${\ displaystyle {\ underline {Q}} _ {U} = \ left (\ ln {\ frac {| U_ {2} |} {| U_ {1} |}} \ right) \, \ mathrm {Np} + \ mathrm {j} (\ varphi _ {2} - \ varphi _ {1}) \, \ mathrm {rad}}$

## literature

• Horst Clausert, Gunther Wiesemann, Volker Hinrichsen , Jürgen Stenzel: Basic areas of electrical engineering. Volume 2: Alternating currents, three-phase currents, cables, applications of Fourier, Laplace and Z transformation. 11th, corrected edition. Oldenbourg, Munich et al. 2011, ISBN 978-3-486-59719-6 .

## Individual evidence

1. a b c DIN 5493: 2013-10: Logarithmic quantities and units
2. a b c DIN EN 60027-3: 2007-11: Symbols for electrical engineering - Part 3: Logarithmic and related quantities and their units