## Admitance

An
impedance Z comprising a resistance R in series with a reactance X can be
converted to an admittance Y comprising a conductance G in parallel with a
susceptance B:

Y = Z^{ -1} = 1 / (R + jX) = (R - jX) / (R^{2} + X^{2})
= R / (R^{2} + X^{2}) - jX / (R^{2} + X^{2})
= G - jB

G = R / (R^{2} + X^{2}) = R / |Z|^{2}

B = X / (R^{2} + X^{2}) = X / |Z|^{2}

Using the polar form of impedance Z:

Y = 1 / |Z|Ðf = |Z|^{ -1}Ð-f
= |Y|Ð-f = |Y|cosf - j|Y|sinf

**
Conversely,
an admittance Y comprising a conductance G in parallel with a susceptance B
can be converted to an impedance Z comprising a resistance R in series with
a reactance X:
Z = Y ^{ -1} = 1 / (G - jB) = (G + jB) / (G^{2} + B^{2})
= G / (G^{2} + B^{2}) + jB / (G^{2} + B^{2})
= R + jX
R = G / (G^{2} + B^{2}) = G / |Y|^{2}
X = B / (G^{2} + B^{2}) = B / |Y|^{2}
Using the polar form of admittance Y:
Z = 1 / |Y|Ð-f = |Y|^{ -1}Ðf
= |Z|Ðf
= |Z|cosf
+ j|Z|sinf
**

The total
impedance Z_{S} of impedances Z_{1}, Z_{2}, Z_{3},...
connected in series is:

Z_{S} = Z_{1} + Z_{1} + Z_{1} +...

The total admittance Y_{P} of admittances Y_{1}, Y_{2},
Y_{3},... connected in parallel is:

Y_{P} = Y_{1} + Y_{1} + Y_{1} +...

In summary:

- use impedances when operating on series circuits,

- use admittances when operating on parallel circuits.