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Home / Resources (video trainings, theorems) / Knowledge / Reactive Loads and Power Factor

Formulas

Resistance and Series Reactance

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

Z = R + jX = |Z|รf

Converting to the equivalent admittance Y:

Y = 1 / Z = 1 / (R + jX) = (R โ€“ jX) / (R2 + X2) = R / |Z|2 โ€“ 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 โ€“ jX / |Z|2) = VR / |Z|2 โ€“ jVX / |Z|2 = IP โ€“ 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 โ€“ j / X = G โ€“ jB = |Y|ร-f

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

I = VY = V(G โ€“ jB) = VG โ€“ jVB = IP โ€“ 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]
[1ร120ยฐ]
[amps, A]
[1ร90ยฐ]
[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, ยฐ]
[rad/sec]

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