## Power factor correct.

**
If
an inductive load with an active power demand P has an uncorrected power
factor of cosf _{1}
lagging, and is required to have a corrected power factor of cosf_{2}
lagging, the uncorrected and corrected reactive power demands, Q_{1}
and Q_{2}, are:**

Q_{1} = P tanf_{1}

Q_{2} = P tanf_{2}

where tanf_{n} = (1 / cos^{2}f_{n} - 1)^{½}

The leading
(capacitive) reactive power demand Q_{C} which must be connected
across the load is:

Q_{C} = Q_{1} - Q_{2} = P (tan**f _{1}
- tanf_{2})**

**
The uncorrected and corrected
apparent power demands, S _{1} and S_{2}, are related by:**

S_{1}cosf_{1} = P = S_{2}cosf_{2}

Comparing corrected and uncorrected load currents and apparent power demands:

I_{2} / I_{1} = S_{2} / S_{1} = cosf_{1} / cosf_{2}

**
If the load is required to have a
corrected power factor of unity, Q _{2} is zero and:
Q_{C} = Q_{1} = P tanf_{1}
I_{2} / I_{1} = S_{2} / S_{1} = cosf_{1}
= P / S_{1} **

**
Shunt Capacitors
For star-connected shunt capacitors each of capacitance C _{star} on
a three phase system of line voltage V_{line} and frequency f, the
leading reactive power demand Q_{Cstar} and the leading reactive
line current I_{line} are:
Q_{Cstar} = V_{line}^{2} / X_{Cstar} = 2pfC_{star}V_{line}^{2}
I_{line} = Q_{Cstar} /
Ö3V_{line}
= V_{line} /
Ö3X_{Cstar}
C_{star} = Q_{Cstar} / 2pfV_{line}^{2}
**

**
For
delta-connected shunt capacitors each of capacitance C _{delta} on a
three phase system of line voltage V_{line} and frequency f, the
leading reactive power demand Q_{Cdelta} and the leading reactive
line current I_{line} are:**

Q_{Cdelta} = 3V_{line}^{2} / X_{Cdelta} = 6pfC_{delta}V_{line}^{2}

I_{line} = Q_{Cdelta} / Ö3V_{line} = Ö3V_{line} / X_{Cdelta}

C_{delta} = Q_{Cdelta} / 6pfV_{line}^{2}

Note that
for the same leading reactive power Q_{C}:

X_{Cdelta} = 3X_{Cstar}

C_{delta} = C_{star} / 3

**
Series Capacitors
For series line capacitors each of capacitance C _{series} carrying
line current I_{line} on a three phase system of frequency f, the
voltage drop V_{drop} across each line capacitor and the total
leading reactive power demand Q_{Cseries} of the set of three line
capacitors are:
V_{drop} = I_{line}X_{Cseries} = I_{line} /
2pfC_{series}
Q_{Cseries} = 3V_{drop}^{2} / X_{Cseries} =
3V_{drop}I_{line} = 3I_{line}^{2}X_{Cseries}
= 3I_{line}^{2} / 2pfC_{series}
C_{series} = 3I_{line}^{2} / 2pfQ_{Cseries}
**

Note that the apparent power rating S_{rating}
of the set of three series line capacitors is based on the line
voltage V_{line} and not the voltage drop V_{drop}:

S_{rating} =
Ö3V_{line}I_{line}