# Power factor

In AC circuits, the power factor is the ratio of the real power that is used to do work and the apparent power that is supplied to the circuit.

The power factor can get values in the range from 0 to 1.

When all the power is reactive power with no real power (usually inductive load) - the power factor is 0.

When all the power is real power with no reactive power (resistive load) - the power factor is 1.

### Power factor definition

The power factor is equal to the real or true power P in watts (W) divided by the apparent power |S| in volt-ampere (VA):

PF = P(W) / |S(VA)|

PF - power factor.

P   - real power in watts (W).

|S|   - apparent power - the magnitude of the complex power in volt⋅amps (VA).

### Power factor calculations

For sinusuidal current, the power factor PF is equal to the absolute value of the cosine of the apparent power phase angle φ (which is also is impedance phase angle):

PF = |cos φ|

PF is the power factor.

φ   is the apprent power phase angle.

The real power P in watts (W) is equal to the apparent power |S| in volt-ampere (VA) times the power factor PF:

P(W) = |S(VA)| × PF = |S(VA)| × |cos φ|

When the circuit has a resistive impedance load, the real power P is equal to the apparent power |S| and the power factor PF is equal to 1:

PF(resistive load) = P / |S| = 1

The reactive power Q in volt-amps reactive (VAR) is equal to the apparent power |S| in volt-ampere (VA) times the sine of the phase angle φ:

Q(VAR) = |S(VA)| × |sin φ|

Single phase circuit calculation from real power meter reading P in kilowatts (kW), voltage V in volts (V) and current I in amps (A):

PF = |cos φ| = 1000 × P(kW) / (V(V) × I(A))

Three phase circuit calculation from real power meter reading P in kilowatts (kW), line to line voltage VL-L in volts (V) and current I in amps (A):

PF = |cos φ| = 1000 × P(kW) / (3 × VL-L(V) × I(A))

Three phase circuit calculation from real power meter reading P in kilowatts (kW), line to line neutral VL-N in volts (V) and current I in amps (A):

PF = |cos φ| = 1000 × P(kW) / (3 × VL-N(V) × I(A))

### Power factor correction

Power factor correction is an adjustment of the electrical circuit in order to change the power factor near 1.

Power factor near 1 will reduce the reactive power in the circuit and most of the power in the circuit will be real power. This will also reduce power lines losses.

The power factor correction is usually done by adding capacitors to the load circuit, when the circuit has inductive components, like an electric motor.

#### Power factor correction calculation

The apparent power |S| in volt-amps (VA) is equal to the voltage V in volts (V) times the current I in amps (A):

|S(VA)| = V(V) × I(A)

The reactive power Q in volt-amps reactive (VAR) is equal to the square root of the square of the apparent power |S| in volt-ampere (VA) minus the square of the real power P in watts (W) (pythagorean theorem):

Q(VAR) = √(|S(VA)|2 - P(W)2)

Qc (kVAR) = Q(kVAR) - Qcorrected (kVAR)

The reactive power Q in volt-amps reactive (VAR) is equal to the square of voltage V in volts (V) divided by the reactance Xc:

Qc (VAR) = V(V)2 / Xc = V(V)2 / (1 / (2πf(Hz)×C(F))) = 2πf(Hz)×C(F)×V(V)2

So the power factor correction capacitor in Farad (F) that should be added to the circuit in parallel is equal to the reactive power Q in volt-amps reactive (VAR) divided by 2π times the frequency f in Hertz (Hz) times the squared voltage V in volts (V):

C(F) = Qc (VAR) / (2πf(Hz)·V(V)2)

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