# Useful Equations

Power Factor Correction
Useful Equations:
The following electrical formulas may be used to calculate basic PFC values.

Active power
The amount of input power converted to output power is the active power.
P= √3 × V × I × cos ∅(W)
Formula 1

Reactive power
The reactive power is the power consumed in an AC circuit due to the expansion and collapse of magnetic (inductive) and electrostatic (capacitive) fields.
Q= √3 × V × I × sin  (VAR)
Formula 2

Apparent Power
The apparent power is the power delivered to an electric circuit.
S= √3 x V × I (VA)
Formula 3

Power factor
The power factor of an AC electrical power system is defined as the ratio of the real (active) power to the apparent power.

Active Power (P)             P
Power Factor (PF) = -----------------------------  =  -------
Apparant Power (S)          S
Formula 4

Power Factor Correction
When the AC load is partly capacitive or inductive, the current waveform is out of phase with the voltage. This requires additional AC current to be generated that is not consumed by the load, creating I2R losses in power cables. Capacitors are used to supply reactive energy to inductive loads. Reactive energy must be produced as closely as possible to the loads to prevent unnecessary flow of current in the network. This is known as power factor correction
QC = P x (tan p1 – tan p2) [VAr]
Formula 5

QC: reactive power needed
P:total reactive power
p1:actual angle of cos p actual
p2:target angle of cos p target

Connection and rating of capacitors
The reactive power of the capacitor is a function of its rated voltage and current.
Qc = VC x IC [VAr]
Formula 6

Vc x Vc            Vc2
Qc = ------------------ = -----------
Xc                 Xc
Formula 7

1                         1
Xc = ------------------ = --------------------
w x C              2pie x f x C
Formula 8

f: frequency of network
XC: impedance of capacitor
C:capacitance value

QC = (VC)2 x w x C = (VC)2 x 2pie x f x C
Formula 9

Capacitor in three-phase PFC application
Three-phase PFC applications have two types of capacitor connections: star and delta. Qtot = Qc x 3

Formula 10

VL

Vc = -------

√3

Formula 11

(VL)2

Qtot = 3 x ------------ x w x Cstar

(√3)2

Formula 12

Qtot                      Qtot

Cstar = ---------- x w = -----------------------

(VL)2              (VL)2 x 2pie x f

Formula 12 Vc = VL

Formula 13

From formulas (9) (10) and (13)

Qtot = 3 x (VL)2 x w Cdelta

Qtot                            Qtot

Cdelta = ---------------------  = -------------------------

3 x (VL)2 x w        3 x (VL)2 x 2pie x f

Formula 14

Cstar

Cdelta = ----------------

3

Formula 15

Capacitor output kvar:
From the formula (9), if we find the Qnew with ratio: C will be constant.

Vnew          fnew
Qnew = ------------  x ---------- x Qc
VR            fR
Formula 16

These values are operating conditions:
Qnew: new reactive power
Vnew: new voltage
fnew: new frequency

These values are the values capacitor is designed: QC: rated capacitor reactive power
VC: rated capacitor voltage
fR:rated frequency

Calculation examples
Example 1:
The relationship between active, reactive and real power and cos

In the diagram below, the power triangle shows an initial power factor of 0.70 for a 100 kW (real power) inductive load. The reactive power required by the load is 100 kvar. By installing a 67-kvar capacitor, the apparent power is reduced from 142 to 105 kvar, resulting in a 26% reduction in current. The power
factor is improved to 0.95.Formulas used (1), (2), (3) and (4). Power factor calculations:

Before PFC: 100/142 = 0.70 or 70% After PFC: 100/105 = 0.95 or 95%

Formulas used (1) (2) (3) and (4)

Power factor calculations:
Before PFC 100/142 = 0.7 or 70%
After PFC 100/105 = 0.95 or 95%