# Susceptance

In electrical engineering, susceptance (B) is the imaginary part of admittance, where the real part is conductance. The inverse of admittance is impedance, where the imaginary part is reactance and the real part is resistance. In SI units, susceptance is measured in siemens. Oliver Heaviside first defined this property in June 1887.[1]

## Formula

The general equation defining admittance is given by

${\displaystyle Y=G+jB\,}$

where

Y is the admittance, measured in siemens.
G is the conductance, measured in siemens.
j is the imaginary unit, and
B is the susceptance, measured in siemens.

The admittance (Y) is the inverse of the impedance (Z)

${\displaystyle Y={\frac {1}{Z}}={\frac {1}{R+jX}}=\left({\frac {R}{R^{2}+X^{2}}}\right)+j\left({\frac {-X}{R^{2}+X^{2}}}\right)\,}$

or

${\displaystyle B=Im(Y)=\left({\frac {-X}{R^{2}+X^{2}}}\right)={\frac {-X}{|Z|^{2}}}}$

where

${\displaystyle Z=R+jX\,}$
Z is the impedance, measured in ohms
R is the resistance, measured in ohms
X is the reactance, measured in ohms.

Note: The susceptance is the imaginary part of the admittance.

The magnitude of admittance is given by:

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

## Relationship to Reactance

Reactance is defined as the imaginary part of Electrical impedance, and is analogous but not generally equal to the inverse of the susceptance.
However, for purely-reactive impedances (which are purely-susceptant admittances), the susceptance is equal to negative the inverse of the reactance.
In mathematical notation:

${\displaystyle G=0\iff R=0\iff B=-1/X}$

Note the negation which is not present in the relationship between Electrical resistance and the analogue of conductance G, which = ${\displaystyle \Re (Y)}$.

${\displaystyle B=0\iff X=0\iff G=1/R}$

The negation in one but not the other can be thought of as coming from the sign laws of sine and cosine, given the fact that conductance-analogue/resistance are the real parts and susceptance/reactance are the imaginary parts.

## References

1. ^ Heaviside, Oliver (1892). Electrical Papers, Volume 2. London and New York: Macmillan and Co.