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# Admittance

In electrical engineering, **admittance** is a measure of how easily a circuit or device will allow a current to flow. It is defined as the inverse of impedance. The SI unit of admittance is the siemens (symbol S). Oliver Heaviside coined the term *admittance* in December 1887.^{[1]}

Admittance is defined as

- <math>Y \equiv \frac{1}{Z} \,</math>

where

The synonymous unit mho, and the symbol ℧ (an upside-down uppercase omega Ω), are also in common use.

Resistance is a measure of the opposition of a circuit to the flow of a steady current, while impedance takes into account not only the resistance but also dynamic effects (known as reactance). Likewise, admittance is not only a measure of the ease with which a steady current can flow, but also the dynamic effects of the material's susceptance to polarization:

- <math>Y = G + j B \,</math>

where

- <math>Y</math> is the admittance, measured in siemens.
- <math>G</math> is the conductance, measured in siemens.
- <math>B</math> is the susceptance, measured in siemens.
- <math>j^2 = -1</math>

## Conversion from impedance to admittance

*Parts of this article or section rely on the reader's knowledge of the complex impedance representation of capacitors and inductors and on knowledge of the frequency domain representation of signals*.

The impedance, Z, is composed of real and imaginary parts,

- <math>Z = R + jX \,</math>

where

*R*is the resistance, measured in ohms*X*is the reactance, measured in ohms

- <math>Y = Z^{-1}= \frac{1}{R + jX} = \left( \frac{1}{R^2 + X^2} \right) \left(R - jX\right) </math>

Admittance, just like impedance, is a complex number, made up of a real part (the conductance, *G*), and an imaginary part (the susceptance, *B*), thus:

- <math>Y = G + jB \,\!</math>

where *G* (conductance) and *B* (susceptance) are given by:

- <math>\begin{align}

G &= \Re(Y) = \frac{R}{R^2 + X^2} \\ B &= \Im(Y) = -\frac{X}{R^2 + X^2}

\end{align}</math>

The magnitude and phase of the admittance are given by:

- <math>\begin{align}

\left | Y \right | &= \sqrt{G^2 + B^2} = \frac{1}{\sqrt{R^2 + X^2}} \\ \angle Y &= \arctan \left( \frac{B}{G} \right) = \arctan \left( -\frac{X}{R} \right)

\end{align}</math>

where

Note that (as shown above) the signs of reactances become reversed in the admittance domain; i.e. capacitive susceptance is positive and inductive susceptance is negative.

## See also

40x40px | Look up in Wiktionary, the free dictionary.admittance |

## References

**^**Ushida, Jun; Tokushima, Masatoshi; Shirane, Masayuki; Gomyo, Akiko; Yamada, Hirohito (2003). "Immittance matching for multidimensional open-system photonic crystals".*Physical Review B***68**(15). Bibcode:2003PhRvB..68o5115U. arXiv:cond-mat/0306260. doi:10.1103/PhysRevB.68.155115.

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