### Formulas

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**lagging, the uncorrected and corrected reactive power demands,

_{2}**Q**and

_{1}**Q**, are:

_{2}**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 (tanf_{1} – tanf_{2})**

The uncorrected and corrected apparent power demands, **S _{1}** and

**S**, are related by:

_{2}**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**and frequency

_{line}**f**, the leading reactive power demand

**Q**and the leading reactive line current

_{Cstar}**I**are:

_{line}**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**and frequency

_{line}**f**, the leading reactive power demand

**Q**and the leading reactive line current

_{Cdelta}**I**are:

_{line}**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**on a three phase system of frequency

_{line}**f**, the voltage drop

**V**across each line capacitor and the total leading reactive power demand

_{drop}**Q**of the set of three line capacitors are:

_{Cseries}**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**and not the voltage drop

_{line}**V**:

_{drop}**S _{rating} = Ö3V_{line}I_{line}**

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. | ||||||

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] |