Electrical susceptance

In electrical engineering, susceptance (B) is the imaginary part of admittance, where the real part is conductance. The reciprocal 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.

Formula

The general equation defining admittance is given by

$Y=G+jB\,$

where,

Y is the complex admittance, measured in Siemens.

G is the real-valued conductance, measured in Siemens.

j is the imaginary unit (i.e. j² = −1), and

B is the real-valued susceptance, measured in Siemens.

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

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

and

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

where

$Z=R+jX\,$

Z is the complex impedance, measured in Ohms

R is the real-valued resistance, measured in Ohms

X is the real-valued reactance, measured in Ohms.

Note: The susceptance $B$ is the imaginary part of the admittance $Y$.

The magnitude of admittance is given by:

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

And similar formulas transform admittance into impedance, hence susceptance (B) into reactance (X):

$Z={\frac {1}{Y}}={\frac {1}{G+jB}}=\left({\frac {G}{\;G^{2}+B^{2}}}\right)+j\left({\frac {-B\;\;}{\;G^{2}+B^{2}}}\right)\,$

hence

$X=\operatorname {Im} (Z)={\frac {-B\;}{\;G^{2}+B^{2}}}={\frac {-B~\;}{~\;\left|Y\right|^{2}}}=-B\,\left|Z\right|^{2}\,\,$.

Note that reactance and susceptance are only reciprocals in the absence of either resistance or conductance (only if either R = 0 or G = 0).

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, $C={\frac {q}{V}},$ is not applicable. A more general definition of capacitance, encompassing electrostatic formula, is:

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

where $Y$ is the device admittance, evaluated at the angular frequency in question, and $\omega$ is the angular frequency. Note that it is common for electrical components to have slightly reduced capacitances at extreme frequencies, due to slight inductance of conductors used to make capacitors (not just the leads), and permittivity changes in insulating materials with frequency: C is very nearly, but not quite a constant.

Relationship to reactance

Reactance is defined as the imaginary part of electrical impedance, and is analogous but not generally equal to the reciprocal of the susceptance.

However, for purely-reactive impedances (which are purely-susceptive admittances), the susceptance is equal to negative the inverse of the reactance.

In mathematical notation:

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

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

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

The negative sign for one pair, but not the other, can be thought of as coming from the sign laws of sine and cosine, given the fact that the conductance :: resistance analogy involves the real parts and susceptance :: reactance analogy involves the imaginary parts. If the imaginary unit is included, we get

$jB={\frac {1}{jX}}~,$

for the reactance-free (susceptance-free) case.

Applications

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

See also

• Electrical measurements
• SI electromagnetism units