# 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,

${\displaystyle Y}$  is the admittance, measured in siemens.
${\displaystyle G}$  is the conductance, measured in siemens.
${\displaystyle j}$  is the imaginary unit, and
${\displaystyle B}$  is the susceptance, measured in siemens.

The admittance (${\displaystyle Y}$ ) is the inverse of the impedance (${\displaystyle Z}$ )

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

and

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

where

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

Note: The susceptance ${\displaystyle B}$  is the imaginary part of the admittance ${\displaystyle Y}$ .

The magnitude of admittance is given by:

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

## Relation to capacitance

In electronic and semiconductor devices, transient or frequency-dependent current between terminals contains both conduction and displacement components. Conduction current is related to moving charge carriers (electrons, holes, ions, etc.), while displacement current is caused by time-varying electric field. Carrier transport is affected by electric field and by a number of physical phenomena, such as carrier drift and diffusion, trapping, injection, contact-related effects, and impact ionization. As a result, device admittance is frequency-dependent, and the simple electrostatic formula for capacitance, ${\displaystyle C=q/V,}$  is not applicable. A more general definition of capacitance, encompassing electrostatic formula, is:[2]

${\displaystyle C={\frac {\operatorname {Im} (Y(\omega ))}{\omega }},}$

where ${\displaystyle Y(\omega )}$  is the device admittance, and ${\displaystyle \omega }$  is the angular frequency.

## 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 equals ${\displaystyle \Re (Y)}$ .

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

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.

## Applications

High susceptance materials are used in susceptors built into microwavable food packaging for their ability to convert microwave radiation into heat.[3]