Power factor

In electrical engineering, the power factor of an AC power system is defined as the ratio of the real power absorbed by the load to the apparent power flowing in the circuit. Real power is the average of the instantaneous product of voltage and current and represents the capacity of the electricity for performing work. Apparent power is the product of root mean square current and voltage. Apparent power is often higher than real power because energy is cyclically accumulated in the load and returned to the source or because a non-linear load distorts the wave shape of the current. Where apparent power exceeds real power, more current is flowing in the circuit than would be required to transfer real power. Where the power factor magnitude is less than one, the voltage and current are not in phase, which reduces the average product of the two. A negative power factor occurs when the device generates real power, which then flows back towards the source.
In an electric power system, a load with a low power factor draws more current than a load with a high power factor for the same amount of useful power transferred. The larger currents increase the energy lost in the distribution system and require larger wires and other equipment. Because of the costs of larger equipment and wasted energy, electrical utilities will usually charge a higher cost to industrial or commercial customers with a low power factor.
Power-factor correction increases the power factor of a load, improving efficiency for the distribution system to which it is attached. Linear loads with a low power factor can be corrected with a passive network of capacitors or inductors. Non-linear loads, such as rectifiers, distort the current drawn from the system. In such cases, active or passive power factor correction may be used to counteract the distortion and raise the power factor. The devices for correction of the power factor may be at a central substation, spread out over a distribution system, or built into power-consuming equipment.

General case

The general expression for power factor is given by
where is the real power measured by an ideal wattmeter, is the rms current measured by an ideal ammeter, and is the rms voltage measured by an ideal voltmeter. Apparent power,, is the product of the rms current and the rms voltage.
If the load is sourcing power back toward the generator, then and will be negative.
If the waveforms are periodic with the same fundamental period, then the power factor can be computed as follows:
where is the instantaneous current, is the instantaneous voltage, is an arbitrary starting time, and is the period of the waveforms.
If the waveforms are not periodic and the physical meters have the same averaging time, then the equations for the periodic case can be used with the exception that is the averaging time of the meters instead of the waveform period.

Linear circuits

In a linear circuit, consisting of combinations of resistors, inductors, and capacitors, current flow has a sinusoidal response to the sinusoidal line voltage. A linear load does not change the shape of the input waveform but may change the relative timing between voltage and current, due to its inductance or capacitance.
In a purely resistive AC circuit, voltage and current waveforms are in step, changing polarity at the same instant in each cycle. All the power entering the load is consumed.
Where reactive loads are present, such as with capacitors or inductors, energy storage in the loads results in a phase difference between the current and voltage waveforms. During each cycle of the AC voltage, extra energy, in addition to any energy consumed in the load, is temporarily stored in the load in electric or magnetic fields then returned to the power grid a fraction of the period later.
Electrical circuits containing predominantly resistive loads have a power factor of almost 1, but circuits containing inductive or capacitive loads can have a power factor well below 1.
A circuit with a low power factor will use a greater amount of current to transfer a given quantity of real power than a circuit with a high power factor thus causing increased losses due to resistive heating in power lines, and requiring the use of higher-rated conductors and transformers.

Definition and calculation

has two components:

Real power or active power , expressed in watts
Reactive power, usually expressed in reactive volt-amperes

Together, they form the complex power expressed as volt-amperes. The magnitude of the complex power is the apparent power, also expressed in volt-amperes.
The VA and var are non-SI units dimensionally similar to the watt but are used in engineering practice instead of the watt to state what quantity is being expressed. The SI explicitly disallows using units for this purpose or as the only source of information about a physical quantity as used.
The power factor is defined as the ratio of real power to apparent power. As power is transferred along a transmission line, it does not consist purely of real power that can do work once transferred to the load, but rather consists of a combination of real and reactive power, called apparent power. The power factor describes the amount of real power transmitted along a transmission line relative to the total apparent power flowing in the line.
The power factor can also be computed as the cosine of the angle θ by which the current waveform lags or leads the voltage waveform.

Power triangle

One can relate the various components of AC power by using the power triangle in vector space. Real power extends horizontally in the real axis and reactive power extends in the direction of the imaginary axis. Complex power represents a combination of both real and reactive power, and therefore can be calculated by using the vector sum of these two components. We can conclude that the mathematical relationship between these components is:
As the angle θ increases with fixed total apparent power, current and voltage are further out of phase with each other. Real power decreases, and reactive power increases.

Lagging, leading and unity power factors

Power factor is described as leading if the current waveform is advanced in phase concerning voltage, or lagging when the current waveform is behind the voltage waveform. A lagging power factor signifies that the load is inductive, as the load will consume reactive power. The reactive component is positive as reactive power travels through the circuit and is consumed by the inductive load. A leading power factor signifies that the load is capacitive, as the load supplies reactive power, and therefore the reactive component is negative as reactive power is being supplied to the circuit.
If θ is the phase angle between the current and voltage, then the power factor is equal to the cosine of the angle, :
Since the units are consistent, the power factor is by definition a dimensionless number between -1 and 1. When the 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, referred to as the unity power factor, 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. Capacitive loads are leading, and inductive loads are lagging.
If a purely resistive load is connected to a power supply, current and voltage will change polarity in step, the power factor will be 1, and the electrical energy flows in a single direction across the network in each cycle. Inductive loads such as induction motors consume reactive power with the current waveform lagging the voltage. Capacitive loads such as capacitor banks or buried cables generate reactive power with the 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. 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. This apparent power must be produced and transmitted to the load and is subject to losses in the production and transmission processes.
Electrical loads consuming alternating current power consume both real power and reactive power. The vector sum of real and reactive power is the complex power, and its magnitude is the apparent power. The presence of reactive power causes the real power to be less than the apparent power, and so, the electric load has a power factor of less than 1.
A negative power factor can result from returning active power to the source, such as in the case of a building fitted with solar panels when surplus power is fed back into the supply.

Power factor correction of linear loads

A high power factor is generally desirable in a power delivery system to reduce losses and improve voltage regulation at the load. Compensating elements near an electrical load will reduce the apparent power demand on the supply system. Power factor correction may be applied by an electric power transmission utility to improve the stability and efficiency of the network. Individual electrical customers who are charged by their utility for low power factor may install correction equipment to increase their power factor to reduce costs.
Power factor correction brings the power factor of an AC power circuit closer to 1 by supplying or absorbing reactive power, adding capacitors or inductors that act to cancel the inductive or capacitive effects of the load, respectively. In the case of offsetting the inductive effect of motor loads, capacitors can be locally connected. These capacitors help to generate reactive power to meet the demand of the inductive loads. This will keep that reactive power from having to flow from the utility generator to the load. In the electricity industry, inductors are said to consume reactive power, and capacitors are said to supply it, even though reactive power is just energy moving back and forth on each AC cycle.
The reactive elements in power factor correction devices can create voltage fluctuations and harmonic noise when switched on or off. They will supply or sink reactive power regardless of whether there is a corresponding load operating nearby, increasing the system's no-load losses. In the worst case, reactive elements can interact with the system and with each other to create resonant conditions, resulting in system instability and severe overvoltage fluctuations. As such, reactive elements cannot simply be applied without engineering analysis.
File:Condensatorenbatterij.jpg|right|thumb|1. Reactive power control relay; 2. Network connection points; 3. Slow-blow fuses; 4. Inrush-limiting contactors; 5. Capacitors ; 6. Transformer
An automatic power factor correction unit consists of some capacitors that are switched by means of contactors. These contactors are controlled by a regulator that measures power factor in an electrical network. Depending on the load and power factor of the network, the power factor controller will switch the necessary blocks of capacitors in steps to make sure the power factor stays above a selected value.
In place of a set of switched capacitors, an unloaded synchronous motor can supply reactive power. The reactive power drawn by the synchronous motor is a function of its field excitation. It is referred to as a synchronous condenser. It is started and connected to the electrical network. It operates at a leading power factor and puts vars onto the network as required to support a system's voltage or to maintain the system power factor at a specified level.
The synchronous condenser's installation and operation are identical to those of large electric motors. Its principal advantage is the ease with which the amount of correction can be adjusted; it behaves like a variable capacitor. Unlike with capacitors, the amount of reactive power furnished is proportional to voltage, not the square of voltage; this improves voltage stability on large networks. Synchronous condensers are often used in connection with high-voltage direct-current transmission projects or in large industrial plants such as steel mills.
For power factor correction of high-voltage power systems or large, fluctuating industrial loads, power electronic devices such as the static VAR compensator or STATCOM are increasingly used. These systems are able to compensate sudden changes of power factor much more rapidly than contactor-switched capacitor banks and, being solid-state, require less maintenance than synchronous condensers.