Definition of Power Factor

The power factor is the ratio of kW to kVA. The closer the power factor approaches its maximum possible value of 1, the greater the benefit to consumer and supplier. PF = P (kW) / S (kVA) P = Active power S = Apparent power

Definition of power factor

The power factor of a load, which may be a single power-consuming item, or a number of items (for example an entire installation), is given by the ratio of P/S i.e. kW divided by kVA at any given moment.
The value of a power factor will range from 0 to 1.
If currents and voltages are perfectly sinusoidal signals, power factor equals cos φ.
A power factor close to unity means that the reactive energy is small compared with the active energy, while a low value of power factor indicates the opposite condition.

Power vector diagram

• Active power P (in kW)

  - Single phase (1 phase and neutral): P = V I cos φ
  - Single phase (phase to phase): P = U I cos φ
  - Three phase (3 wires or 3 wires + neutral): P = [math]\displaystyle{ \sqrt 3 }[/math] U I cos φ

• Reactive power Q (in kvar)

- Single phase (1 phase and neutral): P = V I sin φ
- Single phase (phase to phase): Q = U I sin φ
- Three phase (3 wires or 3 wires + neutral): P = [math]\displaystyle{ \sqrt 3 }[/math] U I sin φ

• Apparent power S (in kVA)

  - Single phase (1 phase and neutral): S = V I
  - Single phase (phase to phase): S = U I
  - Three phase (3 wires or 3 wires + neutral): P = [math]\displaystyle{ \sqrt 3 }[/math] U I
where:
V = Voltage between phase and neutral
U = Voltage between phases
I = Line current
φ = Phase angle between vectors V and I.
  - For balanced and near-balanced loads on 4-wire systems

Current and voltage vectors, and derivation of the power diagram
The power “vector” diagram is a useful artifice, derived directly from the true rotating vector diagram of currents and voltage, as follows:
The power-system voltages are taken as the reference quantities, and one phase only is considered on the assumption of balanced 3-phase loading.
The reference phase voltage (V) is co-incident with the horizontal axis, and the current (I) of that phase will, for practically all power-system loads, lag the voltage by an angle φ.
The component of I which is in phase with V is the “wattful” component of I and is equal to I cos φ, while VI cos φ equals the active power (in kW) in the circuit, if V is expressed in kV.
The component of I which lags 90 degrees behind V is the wattless component of I and is equal to I sin φ, while VI sinφ equals the reactive power (in kvar) in the circuit, if V is expressed in kV.
If the vector I is multiplied by V, expressed in kV, then VI equals the apparent power (in kVA) for the circuit.
The simple formula is obtained: S² = P² + Q²
The above kW, kvar and kVA values per phase, when multiplied by 3, can therefore conveniently represent the relationships of kVA, kW, kvar and power factor for a total 3-phase load, as shown in Figure L3.

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Fig. L3: Power diagram

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An example of power calculations (see Fig. L4)

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Type of circuit		Apparent power S (kVA)	Active power P (kW)	Reactive power Q (kvar)
Single-phase (phase and neutral)		S = VI	P = VI cos φ	Q = VI sin φ
Single-phase (phase to phase)		S = UI	P = UI cos φ	Q = UI sin φ
Example	5 kW of load	10 kVA	5 kW	8.7 kvar
	cos φ = 0.5
Three phase 3-wires or 3-wires + neutral		S = [math]\displaystyle{ \sqrt 3 }[/math] UI	P = [math]\displaystyle{ \sqrt 3 }[/math] UI cos φ	Q = [math]\displaystyle{ \sqrt 3 }[/math] UI sin φ
Example	Motor Pn = 51 kW	65 kVA	56 kW	33 kvar
	cos φ= 0.86
	ρ= 0.91 (motor efficiency)

Fig. L4: Example in the calculation of active and reactive power

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