Admittance

Admittance Y, measured in siemens, is defined as the inverse of impedance Z, measured in ohms:

$Y \equiv \frac{1}{Z}$

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:

$Y = G + j B,$

where

Y is the admittance (siemens);
G is the conductance (siemens);
B is the susceptance (siemens); and
j² = −1, the imaginary unit.

The dynamic effects of the material's susceptance relate to the universal dielectric response, the power law scaling of a system's admittance with frequency under alternating current conditions.

Inversion

The impedance, Z, is composed of real and imaginary parts, $Z = R + j X,$ where

R is the resistance (ohms); and
X is the reactance (ohms).

$Y = Z^{- 1} = \frac{1}{R + j X} = (\frac{1}{R^{2} + X^{2}}) (R - j X)$

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:

$Y = G + j B,$

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

$\begin{matrix} G & = R e (Y) = \frac{R}{R^{2} + X^{2}}, \\ B & = I m (Y) = - \frac{X}{R^{2} + X^{2}} . \end{matrix}$

The magnitude and phase of the admittance are given by:

$\begin{matrix} | Y | & = \sqrt{G^{2} + B^{2}} = \frac{1}{\sqrt{R^{2} + X^{2}}} \\ ∠ Y & = \arctan (\frac{B}{G}) = \arctan (- \frac{X}{R}), \end{matrix}$

where

G is the conductance, measured in siemens; and
B is the susceptance, also measured in siemens.

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.

One more chapter is free with a Lably account.

Inversion

Shunt admittance in electrical power systems modeling

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