# Power factor

The power factor of an AC electric power system is defined as the ratio of the real power to the apparent power, and is a number between 0 and 1. Real power is the capacity of the circuit for performing work in a particular time. Apparent power is the product of the current and voltage of the circuit. Due to energy stored in the load and returned to the source, or due to a non-linear load that distorts the wave shape of the current drawn from the source, the apparent power can be greater than the real power. Low-power-factor loads increase losses in a power distribution system and result in increased energy costs.

## Explanation

Instantaneous and average power calculated from AC voltage and current with a unity power factor (Template:Phisymbol=0, cosTemplate:Phisymbol=1)
Instantaneous and average power calculated from AC voltage and current with a zero power factor (Template:Phisymbol=90, cosTemplate:Phisymbol=0)
Instantaneous and average power calculated from AC voltage and current with a lagging power factor (Template:Phisymbol=45, cosTemplate:Phisymbol=0.71)

In a purely resistive AC circuit, voltage and current waveforms are in step (or in phase), changing polarity at the same instant in each cycle. Where reactive loads are present, such as with capacitors or inductors, energy storage in the loads result in a time difference between the current and voltage waveforms. This stored energy returns to the source and is not available to do work at the load. A circuit with a low power factor will have thus higher currents to transfer at a given quantity of power than a circuit with a high power factor.

Circuits containing purely resistive heating elements (filament lamps, strip heaters, cooking stoves, etc.) have a power factor of 1.0. Circuits containing inductive or capacitive elements ( lamp ballasts, motors, etc.) often have a power factor below 1.0. For example, in electric lighting circuits, normal power factor ballasts (NPF) typically have a value of (0.4) - (0.6). Ballasts with a power factor greater than (0.9) are considered high power factor ballasts (HPF).

The significance of power factor lies in the fact that utility companies supply customers with volt-amperes, but bill them for watts. Power factors below 1.0 require a utility to generate more than the minimum volt-amperes necessary to supply the real power (watts). This increases generation and transmission costs. Good power factor is considered to be greater than 85%. Utilities may charge additional costs to customers who have a power factor below some limit.

AC power flow has the three components: real power (P), measured in watts (W); apparent power (S), measured in volt-amperes (VA); and reactive power (Q), measured in reactive volt-amperes (VAr).

The power factor is defined as:

$\frac{P}{S}$.

In the case of a perfectly sinusoidal waveform, P, Q and S can be expressed as vectors that form a vector triangle such that:

$S^2\,\! = {P^2\,\!} + {Q^2\,\!}$

If φ is the phase angle between the current and voltage, then the power factor is equal to $\left|\cos\phi\right|$, and:

$P = S \left|\cos\phi\right|$

By definition, the power factor is a dimensionless number between 0 and 1. When power factor is equal to 0, the energy flow is entirely reactive, and stored energy in the load returns to the source on each cycle. When the power factor is 1, all the energy supplied by the source is consumed by the load. Power factors are usually stated as "leading" or "lagging" to show the sign of the phase angle.

If a purely resistive load is connected to a power supply, current and voltage will change polarity in step, the power factor will be unity (1), and the electrical energy flows in a single direction across the network in each cycle. Inductive loads such as transformers and motors (any type of wound coil) generate reactive power with current waveform lagging the voltage. Capacitive loads such as capacitor banks or buried cable generate reactive power with current phase leading the voltage. Both types of loads will absorb energy during part of the AC cycle, which is stored in the device's magnetic or electric field, only to return this energy back to the source during the rest of the cycle.

For example, to get 1 kW of real power if the power factor is unity, 1 kVA of apparent power needs to be transferred (1 kW ÷ 1 = 1 kVA). At low values of power factor, more apparent power needs to be transferred to get the same real power. To get 1 kW of real power at 0.2 power factor 5 kVA of apparent power needs to be transferred (1 kW ÷ 0.2 = 5 kVA).

It is often possible to adjust the power factor of a system to very near unity. This practice is known as power factor correction and is achieved by switching in or out banks of inductors or capacitors. For example the inductive effect of motor loads may be offset by locally connected capacitors.

Energy losses in transmission lines increase with increasing current. Where a load has a power factor lower than 1, more current is required to deliver the same amount of useful energy. Power companies therefore require that industrial and commercial customers maintain the power factors of their respective loads within specified limits or be subject to additional charges. Engineers are often interested in the power factor of a load as one of the factors that affect the efficiency of power transmission.

## Non-sinusoidal components

In circuits having only sinusoidal currents and voltages, the power factor effect arises only from the difference in phase between the current and voltage. This is narrowly known as "displacement power factor". The concept can be generalized to a total, distortion, or true power factor where the apparent power includes all harmonic components. This is of importance in practical power systems which contain non-linear loads such as rectifiers, some forms of electric lighting, electric arc furnaces, welding equipment, switched-mode power supplies and other devices.

A particularly important example is the millions of personal computers that typically incorporate switched-mode power supplies (SMPS) with rated output power ranging from 150W to 500W. Historically, these very low cost power supplies incorporated a simple full wave rectifier that conducted only when the mains instantaneous voltage exceeded the voltage on the input capacitors. This leads to very high ratios of peak to average input current, which also lead to a low distortion power factor and potentially serious phase and neutral loading concerns.

Regulatory agencies such as the EC have set harmonic limits as a method of improving power factor. Declining component cost has hastened acceptance and implementation of two different methods. Normally, this is done by either adding a series inductor (so-called passive PFC) or the addition of a boost converter that forces a sinusoidal input (so-called active PFC). For example, SMPS with passive PFC can achieve power factor of about 0.7...0.75, SMPS with active PFC -- up to 0.99, while SMPS without any power factor correction has power factor of about 0.55...0.65 only.

To comply with current EU standard EN61000-3-2 all switched-mode power supplies with output power more than 75W must include at least passive PFC.

A typical multimeter will give incorrect results when attempting to measure the AC current drawn by a non-sinusoidal load and then calculate the power factor. A true RMS multimeter must be used to measure the actual RMS currents and voltages (and therefore apparent power). To measure the real power or reactive power, a wattmeter designed to properly work with non-sinusoidal currents must be used.

## Mnemonics

English-language power engineering students are advised to remember: "ELI the ICE man" or "ELI on ICE"- the voltage E leads the current I in an inductor L, the current leads the voltage in a capacitor C.

Or even shorter: CIVIL - in a Capacitor the I (current) leads Voltage, Voltage leads I (current) in an inductor L.