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 = I_P – jI_Q

The active current I_P and the reactive current I_Q are:

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| = V² / |Z| = |I|²|Z|

P = VI_P = I_P²|Z|² / R = V²R / |Z|² = |I|²R

Q = VI_Q = I_Q²|Z|² / X = V²X / |Z|² = |I|²X

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_Q are:

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| = V²|Y|

P = VI_P = I_P² / G = |I|²G / |Y|² = V²G

Q = VI_Q = I_Q² / B = |I|²B / |Y|² = V²B

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
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]

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