Editing Power factor (section)

== Linear circuits ==

[[File:Power factor 0.svg|right|thumb|upright=1.36|Power flow calculated from AC voltage and current entering a load having a zero power factor ({{mvar|ϕ}}&nbsp;=&nbsp;90°, cos({{mvar|ϕ}})&nbsp;=&nbsp;0). The blue line shows the instantaneous power entering the load: all of the energy received during the first (or third) quarter cycle is returned to the grid during the second (or fourth) quarter cycle, resulting in an ''average'' power flow (light blue line) of zero.]]

[[File:Power factor 0.7.svg|right|thumb|upright=1.36|Instantaneous and average power calculated from AC voltage and current for a load with a lagging power factor ({{mvar|ϕ}}&nbsp;{{=}}&nbsp;45°, cos({{mvar|ϕ}})&nbsp;≈&nbsp;0.71). The blue line (instantaneous power) shows that a portion of the energy received by the load is returned to the grid during the part of the cycle labeled {{mvar|ϕ}}.]]

In a [[Linear time-invariant system|linear circuit]], consisting of combinations of resistors, inductors, and capacitors, current flow has a sinusoidal response to the sinusoidal line voltage.<ref name="Das_2015">{{cite book | title = Power System Harmonics and Passive Filter Design | first = J. C. | last = Das | publisher = Wiley, IEEE Press | year = 2015 | page = 2 | isbn =  978-1-118-86162-2 | quote = To distinguish between linear and nonlinear loads, we may say that linear time-invariant loads are characterized so that an application of a sinusoidal voltage results in a sinusoidal flow of current.}}</ref> A linear load does not change the shape of the input waveform but may change the relative timing (phase) between voltage and current, due to its inductance or capacitance.

In a purely resistive AC circuit, voltage and current waveforms are in step (or [[Phase (waves)|in phase]]), changing polarity at the same instant in each cycle. All the power entering the load is consumed (or dissipated).

Where [[Reactance (electronics)|reactive]] loads are present, such as with [[capacitor]]s or [[inductor]]s, 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 field|electric]] or [[magnetic field]]s then returned to the power grid a fraction of the period later.

Electrical circuits containing predominantly resistive loads ([[incandescent lamps]], devices using [[Heating element|heating elements]] like [[Electric toaster|electric toasters]] and [[Electric oven|ovens]]) have a power factor of almost 1, but circuits containing inductive or capacitive loads (electric motors, [[solenoid]] valves, transformers, [[Electrical ballast|fluorescent lamp ballasts]], and others) 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 [[Joule heating|resistive heating]] in power lines, and requiring the use of higher-rated conductors and transformers.

=== Definition and calculation ===
[[AC power#Instantaneous power, instantaneous active power and instantaneous reactive power in sinusoidal steady-state|AC power]] has two components:
* [[AC power#Active power in sinusoidal steady-state|Real power or active power]] (<math>P</math>) (sometimes called average power<ref>{{Cite book|title=Introductory Circuit Analysis|last=Boylestad|first=Robert|isbn=978-0-13-097417-4|edition=10th|date=2002-03-04|page=857|publisher=Prentice Hall }}</ref>), expressed in [[watt]]s (W)
* [[AC power#Reactive power in sinusoidal steady-state|Reactive power]] (<math>Q</math>), usually expressed in [[volt-ampere reactive|reactive volt-amperes]] (var)<ref>{{cite web |title=SI Units – Electricity and Magnetism |publisher = International Electrotechnical Commission |url=http://www.iec.ch/zone/si/si_elecmag.htm | place = [[Switzerland|CH]] | archive-url = https://web.archive.org/web/20071211234311/http://www.iec.ch/zone/si/si_elecmag.htm#si_epo |archive-date = 2007-12-11 |access-date= 14 June 2013}}</ref>

Together, they form the [[AC power#Complex power in sinusoidal steady-state|complex power]] (<math>S</math>) expressed as [[volt-amperes]] (VA). The magnitude of the complex power is the apparent power (<math>|S|</math>), also expressed in volt-amperes (VA).

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 [[physical quantity|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.<ref>{{cite book|title=The International System of Units (SI) [SI brochure]|url=https://www.bipm.org/documents/20126/41483022/si_brochure_8.pdf |archive-url=https://web.archive.org/web/20220319080426/https://www.bipm.org/documents/20126/41483022/si_brochure_8.pdf |archive-date=2022-03-19 |url-status=live|year=2006|publisher=[[BIPM]]|location=§&nbsp;5.3.2 (p.&nbsp;132, 40 in the [[PDF]] file)}}</ref>

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.<ref>{{Citation | publisher = [[Institute of Electrical and Electronics Engineers|IEEE]] | id = Std. 100 | title = Authoritative Dictionary of Standards Terms | edition = 7th | isbn = 978-0-7381-2601-2| year = 2000 }}</ref><ref>{{Citation | publisher = IEEE | id = Std. 1459–2000 | title = Trial-Use Standard Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions | year = 2000 | isbn = 978-0-7381-1963-2}}.  Note 1, section 3.1.1.1, when defining the quantities for power factor, asserts that real power only flows to the load and can never be negative. As of 2013, one of the authors acknowledged that this note was incorrect, and is being revised for the next edition. See http://powerstandards.com/Shymanski/draft.pdf {{Webarchive|url=https://web.archive.org/web/20160304071333/http://powerstandards.com/Shymanski/draft.pdf |date=2016-03-04 }}</ref>

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.<ref name="SureshKumar_2013">{{cite book | title = Electric Circuit Analysis | first = K. S. | last = Suresh Kumar | publisher = Pearson | year = 2013 | page = 8.10 | isbn = 978-8-13-179155-4}}</ref>

==== Power triangle ====
[[File:Power triangle diagram.jpg|frameless|upright=1.68]]

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 (and its magnitude, apparent 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:

:<math>\begin{align}
  S &= P + jQ \\
  |S| &= \sqrt{P^2 + Q^2} \\
  \text{pf} &= \cos{\theta} = \frac{P}{|S|} = \cos{ \left( \arctan{ \left( \frac{Q}{P} \right) } \right) } \\
  Q &= P \, \tan(\arccos(\text{pf}))
\end{align}</math>

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 <math>Q</math> 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 <math>Q</math> is negative as reactive power is being supplied to the circuit.

[[File:Lagging-Leading.jpg|frameless|upright=2.66]]

If θ is the [[phase (waves)|phase angle]] between the current and voltage, then the power factor is equal to the [[Trigonometric functions|cosine]] of the angle, <math>\cos\theta</math>:
:<math>|P| = |S| \cos\theta</math>

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 (current leads voltage), and inductive loads are lagging (current lags voltage).

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 (any type of wound coil) 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&nbsp;kW of real power, if the power factor is unity, 1&nbsp;kVA of apparent power needs to be transferred (1&nbsp;kW ÷ 1 = 1&nbsp;kVA). At low values of power factor, more apparent power needs to be transferred to get the same real power. To get 1&nbsp;kW of real power at 0.2 power factor, 5&nbsp;kVA of apparent power needs to be transferred (1&nbsp;kW ÷ 0.2 = 5&nbsp;kVA). 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 [[AC power|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 (0 to −1) 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.<ref>{{Citation | title = On the resistance and electromotive forces of the electric arc |first=W. | last = Duddell | journal = Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=203 |issue=359–371 |doi=10.1098/rsta.1904.0022  | pages = 512–15 | year = 1901 | quote = The fact that the solid arc has, at low frequencies, a negative power factor, indicates that the arc is supplying power to the alternator…| doi-access = free }}</ref><ref>{{Citation |title=Analysis of some measurement issues in bushing power factor tests in the field |first=S. |last=Zhang |journal= IEEE Transactions on Power Delivery|volume=21 |issue=3 |pages=1350–56 |date=July 2006 |quote=…(the measurement) gives both negative power factor and negative resistive current (power loss) |doi=10.1109/tpwrd.2006.874616|s2cid=39895367 }}</ref><ref>{{Citation | title = Performance of Grid-Connected Induction Generator under Naturally Commutated AC Voltage Controller |first=A. F. |last=Almarshoud |display-authors=etal |journal=Electric Power Components and Systems |volume=32 |issue=7 |pages=691–700 |year=2004 |quote=Accordingly, the generator will consume active power from the grid, which leads to negative power factor.|doi=10.1080/15325000490461064 |s2cid=110279940 }}</ref>

=== Power factor correction of linear loads ===
[[File:Blindleistungskompensation.svg|thumb|Power factor correction of linear load]]
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. [[Static VAR compensator|Reactive power control relay]]; 2. Network connection points; 3. [[Fuse (electrical)|Slow-blow fuses]]; 4. Inrush-limiting [[contactor]]s; 5. [[Capacitor]]s (single-phase or three-phase units, delta-connection); 6. [[Transformer]] (for controls and ventilation fans) ]]

An ''' automatic power factor correction unit''' consists of some [[capacitor]]s that are switched by means of [[contactor]]s. 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 [[capacitor]]s, 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 [[volt-ampere reactive|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 motor]]s. 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|high-voltage direct-current]] transmission projects or in large industrial plants such as [[steel mill]]s.

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.