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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 ZS of impedances Z1, Z2, Z3,… connected in series is:

ZS = Z1 + Z1 + Z1 +…

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

YP = Y1 + Y1 + Y1 +…

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

 NOTATION The symbol font is used for some notation and formulae. If the Greek symbols for alpha beta delta do not appear here [ a b d ] the symbol font needs to be installed for correct display of notation and formulae. B C E f G h I j L P Q susceptance capacitance voltage source frequency conductance h-operator current j-operator inductance active power reactive power [siemens, S] [farads, F] [volts, V] [hertz, Hz] [siemens, S] [1Ð120°] [amps, A] [1Ð90°] [henrys, H] [watts, W] [VAreactive, VArs] Q R S t V W X Y Z f w quality factor resistance apparent power time voltage drop energy reactance admittance impedance phase angle angular frequency [number] [ohms, W] [volt-amps, VA] [seconds, s] [volts, V] [joules, J] [ohms, W] [siemens, S] [ohms, W] [degrees, °] [rad/sec]