Admittance

The admittance from the Latin admittere , in German "to accept", is a term from electrical engineering , equivalent to complex conductance . It describes the ratio of sinusoidal alternating current flowing through a linear consumer (component, line, etc.) to the alternating voltage applied to it . In certain contexts, the term is also much broader. ${\displaystyle {\underline {Y}}}$

Significance in alternating current theory

For sinusoidal processes, the mathematical representation using complex-valued quantities is advantageous. These are marked here in the equations by an underscore, the imaginary unit by the letter . ${\displaystyle \mathrm {j} }$

The admittance is the reciprocal of the impedance : ${\displaystyle {\underline {Y}}}$ ${\displaystyle {\underline {Z}}}$

${\displaystyle {\underline {Y}}={{\underline {Z}}^{-1}}={\frac {1}{\underline {Z}}}}$

It is made up of the real part which is referred to as the conductance or conductance :

${\displaystyle G=\operatorname {Re} \,{\underline {Y}}=\operatorname {Re} \,{\frac {1}{\underline {Z}}}}$

and from the imaginary part called susceptance or susceptance :

${\displaystyle B=\operatorname {Im} \,{\underline {Y}}=\operatorname {Im} \,{\frac {1}{\underline {Z}}}}$

The amount of admittance is as admittance  referred. All of these terms are standardized in this way. ${\displaystyle Y}$

${\displaystyle {\underline {Y}}=G+\mathrm {j} B}$

${\displaystyle Y=|\,{\underline {Y}}|={\sqrt {G^{2}+B^{2}}}}$

With the complex impedance of active resistance (Resistance)  and reactance (reactance)  results to ${\displaystyle {\underline {Z}}=R+\mathrm {j} X}$ ${\displaystyle R}$ ${\displaystyle X}$ ${\displaystyle {\underline {Y}}}$

${\displaystyle {\underline {Y}}={\frac {1}{R+\mathrm {j} X}}={\frac {R}{R^{2}+X^{2}}}-\mathrm {j} {\frac {X}{R^{2}+X^{2}}}}$

and thus

${\displaystyle G={\frac {R}{R^{2}+X^{2}}}\quad {\text{und}}\quad B={\frac {-X}{R^{2}+X^{2}}}}$

It can be seen from this that the conductance is generally something different from the reciprocal effective resistance and the susceptance is  something different from the reciprocal reactance . The term conductance is also translated as conductance when it comes to an ohmic consumer . Since the alternating variables depend on the frequency, they also depend on the frequency. ${\displaystyle G}$ ${\displaystyle 1/R}$ ${\displaystyle B}$ ${\displaystyle 1/X}$ ${\displaystyle X}$ ${\displaystyle Y,\ G{\text{ und }}B}$

In exponential form one can write with the phase shift angle between voltage and current strength or their zero phase angles and : ${\displaystyle \varphi _{ui}}$ ${\displaystyle \varphi _{u}}$ ${\displaystyle \varphi _{i}}$

${\displaystyle {\underline {Z}}=Z\ \mathrm {e} ^{\mathrm {j} \varphi _{ui}}=Z\ \mathrm {e} ^{\mathrm {j} (\varphi _{u}-\varphi _{i})}}$

${\displaystyle {\underline {Y}}={\frac {1}{Z}}\ \mathrm {e} ^{-\mathrm {j} \varphi _{ui}}=Y\ \mathrm {e} ^{\mathrm {j} (\varphi _{i}-\varphi _{u})}}$

and, if one applies Euler's formula , ${\displaystyle \mathrm {e} ^{\mathrm {j} x}=\cos x+\mathrm {j} \sin x}$

${\displaystyle B=Y\sin(-\varphi _{ui})=Y\sin(\varphi _{i}-\varphi _{u})}$

${\displaystyle G=Y\cos \varphi _{ui}}$

The unit of measurement in the SI system of units for all specified types of leading values ​​is Siemens with the S as the unit symbol .

Special case

For a lossless ideal capacitor with the capacitance , the information on the resistances applies to sinusoidal alternating voltage with the angular frequency ${\displaystyle C}$ ${\displaystyle \omega }$

${\displaystyle {\underline {Z}}={\frac {1}{\mathrm {j} \omega C}}\ ;\quad R=0\ ;\quad X=-{\frac {1}{\omega C}}}$

This applies to the guiding values

${\displaystyle {\underline {Y}}=\mathrm {j} \omega C;\quad G=0\ ;\quad B=-{\frac {1}{X}}=\omega C\ ;}$

Insulation resistance and dielectric losses of the capacitor are recorded as the effective conductance . In most practical cases remains at least a hundred times greater than ; then remain unchanged within the framework of this approximation and and . ${\displaystyle G>0}$ ${\displaystyle \omega C}$ ${\displaystyle G}$ ${\displaystyle X}$ ${\displaystyle {\underline {Y}}}$ ${\displaystyle R\ll {\frac {1}{\omega C}}}$

Extended meaning

In the theory of linear electrical networks , the ratio of a current to a voltage is also referred to as admittance if they are not measured on the same component. Typical examples are the short-circuit core admittance and the transmission admittance in four-pole theory . Finally, the admittance matrix is ​​also introduced there.

On the other hand, the ratio of a non-sinusoidal current to a non-sinusoidal voltage is also referred to as admittance if the current and voltage are calculated using an operator calculation , e.g. B. the Laplace transformation , in the so-called image area and in this way forms their relationship as an "admittance operator". Such an admittance then does not have the imaginary frequency as a variable, but the complex frequency . The outer shape of such a broken rational function is called within the network synthesis as admittance function . ${\displaystyle \mathrm {j} \omega }$ ${\displaystyle s}$

literature

• Karl Küpfmüller, Wolfgang Mathis and Albrecht Reibiger: Theoretical electrical engineering: An introduction . 18th edition. Springer, 2008, ISBN 978-3-540-78589-7 .
• Wilfried Weißgerber: Electrical engineering for engineers 2 . 8th edition. Vieweg + Teubner, 2013, ISBN 978-3-8348-1031-1 .

Individual evidence

1. ↑ DIN 1304-1, symbols , 1994
2. ↑ DIN 5483-3 time-dependent quantities, complex representation of sinusoidal time-dependent quantities , 1994
3. ↑ ^{a b} DIN 40110-1, alternating current quantities ; Two-wire circuits , 1994
4. ↑ ^{a b} IEC 60050-131, see DKE German Commission for Electrical, Electronic and Information Technologies in DIN and VDE: International Electrotechnical Dictionary
5. ↑ EN ISO 80000-1 , Sizes and units - Part 1: General , 2013
6. ↑ Erwin Böhmer, Dietmar Ehrhardt, Wolfgang Oberschelp: Elements of applied electronics. Vieweg + Teubner, 16th edition 2010, p. 42
7. ↑ Klaus Lunze : Theory of alternating current circuits . 8th edition. Verlag Technik, Berlin 1991, ISBN 3-341-00984-1 .
8. ↑ Gerhard Wunsch : Elements of network synthesis . Verlag Technik, Berlin 1969, DNB  458706396 .

Web links

Wiktionary: Admittance  - explanations of meanings, word origins, synonyms, translations