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# Reactive Loads and Power Factor

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### 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) / (R2 + X2) = 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 = IP – jIQ

The active current IP and the reactive current IQ are:

IP = VR / |Z|2 = |I|cosf

IQ = VX / |Z|2 = |I|sinf

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

S = V|I| = V2 / |Z| = |I|2|Z|

P = VIP = IP2|Z|2 / R = V2R / |Z|2 = |I|2R

Q = VIQ = IQ2|Z|2 / X = V2X / |Z|2 = |I|2X

The power factor cosf and reactive factor sinf are:

cosf = IP / |I| = P / S = R / |Z|

sinf = IQ / |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 = IP – jIQ

The active current IP and the reactive current IQ are:

IP = VG = V / R = |I|cosf

IQ = VB = V / X = |I|sinf

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

S = V|I| = |I|2 / |Y| = V2|Y|

P = VIP = IP2 / G = |I|2G / |Y|2 = V2G

Q = VIQ = IQ2 / B = |I|2B / |Y|2 = V2B

The power factor cosf and reactive factor sinf are:

cosf = IP / |I| = P / S = G / |Y|

sinf = IQ / |I| = Q / S = B / |Y|

 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]