### 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) / (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**,… connected in series is:

_{3}**Z _{S} = Z_{1} + Z_{1} + Z_{1} +…**

The total admittance **Y _{P}** of admittances

**Y**,

_{1}**Y**,

_{2}**Y**,… connected in parallel is:

_{3}**Y _{P} = Y_{1} + Y_{1} + Y_{1} +…**

In summary:

– use impedances when operating on series circuits,

– use admittances when operating on parallel circuits.

NOTATION |
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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. |
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BCEfGhIjLPQ |
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] |
QRStVWXYZfw |
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] |

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