### Formulas

#### Resistance and Series Reactance

The impedance **Z** of a reactive load comprising resistance **R** and series reactance **X** is:

**Z = R + jX = |Z|Ðf**

Converting to the equivalent admittance **Y**:

**Y = 1 / Z = 1 / (R + jX) = (R – jX) / (R ^{2} + X^{2}) = R / |Z|^{2} – jX / |Z|^{2}**

When a voltage **V** (taken as reference) is applied across the reactive load **Z**, the current **I** is:

**I = VY = V(R / |Z| ^{2} – jX / |Z|^{2}) = VR / |Z|^{2} – jVX / |Z|^{2} = I_{P} – jI_{Q}**

The active current **I _{P}** and the reactive current

**I**are:

_{Q}**I _{P} = VR / |Z|^{2} = |I|cosf**

**I _{Q} = VX / |Z|^{2} = |I|sinf**

The apparent power **S**, active power **P** and reactive power **Q** are:

**S = V|I| = V ^{2} / |Z| = |I|^{2}|Z|**

**P = VI _{P} = I_{P}^{2}|Z|^{2} / R = V^{2}R / |Z|^{2} = |I|^{2}R**

**Q = VI _{Q} = I_{Q}^{2}|Z|^{2} / X = V^{2}X / |Z|^{2} = |I|^{2}X**

The power factor **cosf** and reactive factor **sinf** are:

**cosf = I _{P} / |I| = P / S = R / |Z|**

sinf = I_{Q} / |I| = Q / S = X / |Z|

#### Resistance and Shunt Reactance

The impedance **Z** of a reactive load comprising resistance **R** and shunt reactance **X** is found from:

**1 / Z = 1 / R + 1 / jX**

Converting to the equivalent admittance **Y** comprising conductance **G** and shunt susceptance **B**:

**Y = 1 / Z = 1 / R – j / X = G – jB = |Y|Ð-f**

When a voltage **V** (taken as reference) is applied across the reactive load **Y**, the current **I** is:

**I = VY = V(G – jB) = VG – jVB = I _{P} – jI_{Q}**

The active current **I _{P}** and the reactive current

**I**are:

_{Q}**I _{P} = VG = V / R = |I|cosf**

**I _{Q} = VB = V / X = |I|sinf**

The apparent power **S**, active power **P** and reactive power **Q** are:

**S = V|I| = |I| ^{2} / |Y| = V^{2}|Y|**

**P = VI _{P} = I_{P}^{2} / G = |I|^{2}G / |Y|^{2} = V^{2}G**

**Q = VI _{Q} = I_{Q}^{2} / B = |I|^{2}B / |Y|^{2} = V^{2}B**

The power factor **cosf** and reactive factor **sinf** are:

**cosf = I _{P} / |I| = P / S = G / |Y|**

**sinf = I _{Q} / |I| = Q / S = B / |Y|**

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] |