Admittance

Formulas

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) / (R² + X²) = R / (R² + X²) – jX / (R² + X²) = G – jB
G = R / (R² + X²) = R / |Z|²
B = X / (R² + X²) = X / |Z|²

Using the polar form of impedance Z:

Y = 1 / |Z|Ðf = |Z| ^-1Ð-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) / (G² + B²) = G / (G² + B²) + jB / (G² + B²) = R + jX
R = G / (G² + B²) = G / |Y|²
X = B / (G² + B²) = B / |Y|²

Using the polar form of admittance Y:

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

The total impedance Z_S of impedances Z₁, Z₂, Z₃,… connected in series is:

Z_S = Z₁ + Z₁ + Z₁ +…

The total admittance Y_P of admittances Y₁, Y₂, Y₃,… connected in parallel is:

Y_P = Y₁ + Y₁ + Y₁ +…

In summary:
– use impedances when operating on series circuits,
– use admittances when operating on parallel circuits.