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{{Short description|Ratio of active power to apparent power}}
{{For|the firearms cartridge ranking system|Power factor (shooting sports)}}
[[File:Power Factor General Case.svg|thumb|Schematic showing how power factor is calculated]]

In [[electrical engineering]], the '''power factor''' of an [[AC power]] system is defined as the [[ratio]] of the ''[[real power]]'' absorbed by the [[electrical load|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|RMS]] current and voltage. 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 may be greater than the real power, so more current flows in the circuit than would be required to transfer real power alone. A power factor magnitude of less than one indicates the voltage and current are not in phase, reducing the average [[Product (mathematics)|product]] of the two. A negative power factor occurs when the device (which is normally the load) 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 higher 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 where there is 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 (such as [[induction motor]]s) can be corrected with a passive network of [[capacitor]]s or [[inductor]]s. Non-linear loads, such as [[rectifier]]s, 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 [[electrical substation|substation]], spread out over a distribution system, or built into power-consuming equipment.

==General case==
==General case==
[[File:Power Factor General Case.svg|thumb|Schematic showing how power factor is calculated]]
[[File:Power Factor General Case.svg|thumb|Schematic showing how power factor is calculated]]

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'{{Short description|Ratio of active power to apparent power}} {{For|the firearms cartridge ranking system|Power factor (shooting sports)}} [[File:Power Factor General Case.svg|thumb|Schematic showing how power factor is calculated]] In [[electrical engineering]], the '''power factor''' of an [[AC power]] system is defined as the [[ratio]] of the ''[[real power]]'' absorbed by the [[electrical load|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|RMS]] current and voltage. 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 may be greater than the real power, so more current flows in the circuit than would be required to transfer real power alone. A power factor magnitude of less than one indicates the voltage and current are not in phase, reducing the average [[Product (mathematics)|product]] of the two. A negative power factor occurs when the device (which is normally the load) 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 higher 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 where there is 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 (such as [[induction motor]]s) can be corrected with a passive network of [[capacitor]]s or [[inductor]]s. Non-linear loads, such as [[rectifier]]s, 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 [[electrical substation|substation]], spread out over a distribution system, or built into power-consuming equipment. ==General case== [[File:Power Factor General Case.svg|thumb|Schematic showing how power factor is calculated]] The general expression for power factor is given by :<math> \mbox{power factor} = P/P_a </math> :<math> P_a = I_{rms} V_{rms} </math> where <math>P</math> is the real power measured by an ideal [[wattmeter]], <math>I_{rms}</math> is the rms current measured by an ideal [[ammeter]], and <math>V_{rms}</math> is the rms voltage measured by an ideal [[voltmeter]]. Apparent power, <math>P_a</math>, is the product of the rms current and the rms voltage. If the load is sourcing power back toward the generator, then <math>P</math> and <math> \mbox{power factor} </math> will be negative. ===Periodic waveforms=== If the waveforms are periodic with the same period which is much shorter than the averaging time of the physical meters, then the power factor can be computed by the following :<math> \mbox{power factor} = P/P_a </math> :<math> P_a = I_{rms} V_{rms} </math> :<math> P =\frac 1 T \int_{t'}^{t'+T} i(t)v(t) dt </math> :<math> I_{rms}^2 =\frac 1 T \int_{t'}^{t'+T} {i(t)}^2 dt </math> :<math> V_{rms}^2 =\frac 1 T \int_{t'}^{t'+T} {v(t)}^2 dt </math> where <math>i(t)</math> is the instantaneous current, <math>v(t)</math> is the instantaneous voltage, <math>t'</math> is an arbitrary starting time, and <math>T</math> is the period of the waveforms. ===Nonperiodic 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 <math>T</math> is the averaging time of the meters instead of the waveform period. {{-}} == Linear time-invariant 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}}.]] [[Linear time-invariant system|Linear time-invariant circuits]] (referred to simply as ''linear circuits'' for the rest of this article), for example, circuits consisting of combinations of resistors, inductors and capacitors have 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, heating elements) 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. In the [[electric power grid]], reactive loads cause a continuous ''ebb and flow'' of nonproductive power. 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}}</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 mathematically identical 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=http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf |archive-date=2022-10-09 |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 with respect to 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 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 ''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 so as 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 all the way 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 a number of [[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 supplied 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. == Non-linear loads == Examples of non-linear loads on a power system are rectifiers (such as used in a power supply), and arc discharge devices such as [[fluorescent lamp]]s, electric [[welding]] machines, or [[arc furnace]]s. Because current in these systems is interrupted by a switching action, the current contains frequency components that are multiples of the power system frequency. ''Distortion power factor'' is a measure of how much the harmonic distortion of a load current decreases the average power transferred to the load. [[File:Power factor 75 2.png|right|thumb|upright=1.36|Sinusoidal voltage and non-sinusoidal current give a distortion power factor of 0.75 for this computer power supply load.]] === Non-sinusoidal components === In linear circuits having only sinusoidal currents and voltages of one frequency, the power factor arises only from the difference in phase between the current and voltage. This is ''displacement power factor''.<ref name="FuchsMasoum2015">{{cite book|author1=Ewald Fuchs|author2=Mohammad A. S. Masoum|title=Power Quality in Power Systems and Electrical Machines|url=https://books.google.com/books?id=wuGcBAAAQBAJ&pg=PA432|date=14 July 2015|publisher=Elsevier Science|isbn=978-0-12-800988-8|pages=432–|quote=The DPF it the cosine of the angle between these two quantities}}</ref> Non-linear loads change the shape of the current waveform from a [[sine wave]] to some other form. Non-linear loads create [[harmonic]] currents in addition to the original (fundamental frequency) AC current. This is of importance in practical power systems that contain [[non-linear]] loads such as [[rectifiers]], some forms of electric lighting, [[electric arc furnace]]s, welding equipment, [[Switched-mode power supply|switched-mode power supplies]], variable speed drives and other devices. Filters consisting of linear capacitors and inductors can prevent harmonic currents from entering the supplying system. To measure the real power or reactive power, a [[wattmeter]] designed to work properly with non-sinusoidal currents must be used. === Distortion power factor === The '''distortion power factor''' is the distortion component associated with the harmonic voltages and currents present in the system. :<math> \begin{align} \mbox{distortion power factor} & = \frac{ I_1}{I_{rms}} \\ & = \frac{I_1} {\sqrt{I_1^2+I_2^2+I_3^2+I_4^2+\cdots}} \\ & = \frac{1} { \sqrt{1+ \frac{I_2^2+I_3^2+I_4^2+\cdots}{I_1^2}}} \\ & = \frac{1} {\sqrt{ 1+THD_i^2}} \\ \end{align} </math> <math>\mbox{THD}_i</math> is the [[total harmonic distortion]] of the load current. :<math>THD_i = \frac{\sqrt{\displaystyle\sum_{h=2}^\infty I_h^2}} {I_1}= \frac{\sqrt{I_2^2+I_3^2+I_4^2+\cdots}} {I_1}</math> <math>I_1</math> is the fundamental component of the current and <math>I_{\mbox{rms}}</math> is the total current – both are [[root mean square]]-values (distortion power factor can also be used to describe individual order harmonics, using the corresponding current in place of total current). This definition with respect to total harmonic distortion assumes that the voltage stays undistorted (sinusoidal, without harmonics). This simplification is often a good approximation for stiff voltage sources (not being affected by changes in load downstream in the distribution network). Total harmonic distortion of typical generators from current distortion in the network is on the order of 1–2%, which can have larger scale implications but can be ignored in common practice.<ref>{{Citation |url=http://ecmweb.com/power-quality/effects-harmonics-power-systems |title=Effects of Harmonics on Power Systems |first=C. |last=Sankaran |year=1999 |publisher=Electro-Test |quote=...and voltage-time relationship deviates from the pure sine function. The distortion at the point of generation is very small (about 1% to 2%), but nonetheless it exists.}}</ref> The result when multiplied with the displacement power factor (DPF) is the overall, true power factor or just power factor (PF): :<math>\mbox{PF} = \frac{\cos{\varphi}} {\sqrt{ 1+THD_i^2}}</math> === Distortion in three-phase networks === In practice, the local effects of distortion current on devices in a [[Three-phase electric power|three-phase distribution network]] rely on the magnitude of certain order harmonics rather than the total harmonic distortion. For example, the triplen, or zero-sequence, harmonics (3rd, 9th, 15th, etc.) have the property of being in-phase when compared line-to-line. In a [[delta-wye transformer]], these harmonics can result in circulating currents in the delta windings and result in greater [[Joule heating|resistive heating]]. In a wye-configuration of a transformer, triplen harmonics will not create these currents, but they will result in a non-zero current in the [[Ground and neutral|neutral wire]]. This could overload the neutral wire in some cases and create error in kilowatt-hour metering systems and billing revenue.<ref>{{Citation | chapter-url = http://www.pge.com/includes/docs/pdfs/mybusiness/customerservice/energystatus/powerquality/harmonics.pdf | title = Power System Harmonics | publisher = Pacific Gas and Electric | chapter = Single-phase load harmonics vs. three-phase load harmonics | chapter-format = [[PDF]]}}</ref><ref>{{Citation | chapter-url = http://energylogix.ca/harmonics_and_ieee.pdf | title = Harmonics and IEEE 519 | publisher = EnergyLogix Solutions | chapter = Harmonic Effects | place = [[Canada|CA]] | chapter-format = [[PDF]]}}</ref> The presence of current harmonics in a transformer also result in larger [[eddy currents]] in the magnetic core of the transformer. Eddy current losses generally increase as the square of the frequency, lowering the transformer's efficiency, dissipating additional heat, and reducing its service life.<ref>{{Citation |url=http://ecmweb.com/power-quality/effects-harmonics-power-systems |title=Effects of Harmonics on Power Systems |first=C. |last=Sankaran |year=1999 |publisher = Electro-Test |section=Transformers}}</ref> Negative-sequence harmonics (5th, 11th, 17th, etc.) combine 120 degrees out of phase, similarly to the fundamental harmonic but in a reversed sequence. In generators and motors, these currents produce magnetic fields which oppose the rotation of the shaft and sometimes result in damaging mechanical vibrations.<ref>{{Citation |url=http://ecmweb.com/power-quality/effects-harmonics-power-systems |title=Effects of Harmonics on Power Systems |first=C. |last=Sankaran |year=1999 |publisher=Electro-Test |section=Motors |quote=The interaction between the positive and negative sequence magnetic fields and currents produces torsional oscillations of the motor shaft. These oscillations result in shaft vibrations.}}</ref> === Switched-mode power supplies === {{Main|switched-mode power supply#Power factor}} A particularly important class of non-linear loads is the millions of personal computers that typically incorporate [[Switched-mode power supply|switched-mode power supplies]] (SMPS) with rated output power ranging from a few watts to more than 1&nbsp;kW. Historically, these very-low-cost power supplies incorporated a simple full-wave rectifier that conducted only when the [[Mains electricity|mains]] instantaneous voltage exceeded the voltage on the input capacitors. This leads to very high [[peak-to-average ratio|ratios of peak-to-average]] input current, which also lead to a low distortion power factor and potentially serious phase and neutral loading concerns. A typical switched-mode power supply first converts the AC mains to a DC bus by means of a [[bridge rectifier]]. The output voltage is then derived from this DC bus. The problem with this is that the [[rectifier]] is a non-linear device, so the input current is highly non-linear. That means that the input current has energy at [[harmonic]]s of the frequency of the voltage. This presents a problem for power companies, because they cannot compensate for the harmonic current by adding simple capacitors or inductors, as they could for the reactive power drawn by a linear load. Many jurisdictions are beginning to require power factor correction for all power supplies above a certain power level. Regulatory agencies such as the [[European Union|EU]] have set harmonic limits as a method of improving power factor. Declining component cost has hastened implementation of two different methods. To comply with current EU standard EN61000-3-2, all switched-mode power supplies with output power more than 75&nbsp;W must at least include passive power factor correction. [[80 Plus]] power supply certification requires a power factor of 0.9 or more.<ref>{{Citation | url = http://www.80plus.org/ | publisher = 80 Plus | title = Certified Power Supplies and Manufacturers | section = What is an 80 PLUS certified power supply?}}</ref> === Power factor correction (PFC) in non-linear loads === ==== Passive PFC ==== The simplest way to control the [[Harmonics (electrical power)|harmonic]] current is to use a [[electronic filter|filter]] that passes current only at [[utility frequency|line frequency]] (50 or 60&nbsp;Hz). The filter consists of capacitors or inductors and makes a non-linear device look more like a [[linear]] load. An example of passive PFC is a [[valley-fill circuit]]. A disadvantage of passive PFC is that it requires larger inductors or capacitors than an equivalent power active PFC circuit.<ref>{{Citation |url=http://www.nuvation.com/corporate/news/newsletter/fall2006/powersupply.html |publisher=Nuvation |date=Fall 2006 |title=Power Supply Design Principles: Techniques and Solutions, Part 3 |newspaper=Newsletter |first=Ben |last=Schramm |url-status=dead |archive-url=https://web.archive.org/web/20070309134617/http://www.nuvation.com/corporate/news/newsletter/fall2006/powersupply.html |archive-date=2007-03-09 }}</ref><ref>{{Citation | newspaper = Xplore | title = Quasi-active power factor correction with a variable inductive filter: theory, design and practice | volume = 18 | issue = 1 | pages = 248–255 | publisher = IEEE| doi = 10.1109/TPEL.2002.807135 | bibcode = 2003ITPE...18..248W | year = 2003 | last1 = Wolfle | first1 = W.H. | last2 = Hurley | first2 = W.G. }}</ref><ref>{{Citation |publisher=Nuigalway |type=project |url=http://www.nuigalway.ie/power_electronics/projects/quasi_active.html |place=[[Ireland|IE]] |title=Power electronics |contribution=Quasi-active Power Factor Correction: The Role of Variable Inductance |last1=Wölfle |first1=W. H. |last2=Hurley |first2=W. G.}}</ref> Also, in practice, passive PFC is often less effective at improving the power factor.<ref name="effi">{{Citation | url = http://www.xbitlabs.com/articles/coolers/display/atx-psu5_3.html | title = ATX Power Supply Units Roundup | publisher = xBit labs | quote = The power factor is the measure of reactive power. It is the ratio of active power to the total of active and reactive power. It is about 0.65 with an ordinary PSU, but PSUs with active PFC have a power factor of 0.97–0.99. […] hardware reviewers sometimes make no difference between the power factor and the efficiency factor. Although both these terms describe the effectiveness of a power supply, it is a gross mistake to confuse them. […] There is a very small effect from passive PFC – the power factor grows only from 0.65 to 0.7–0.75. | url-status = dead | archive-url = https://web.archive.org/web/20081120040707/http://www.xbitlabs.com/articles/coolers/display/atx-psu5_3.html | archive-date = 2008-11-20 }}</ref><ref>{{Citation|date=Mar 16, 2006 |publisher=Find articles |url=http://findarticles.com/p/articles/mi_m0EIN/is_2006_March_16/ai_n26797888 |archive-url=https://web.archive.org/web/20090901140721/http://findarticles.com/p/articles/mi_m0EIN/is_2006_March_16/ai_n26797888/ |url-status=dead |archive-date=September 1, 2009 |title=The Active PFC Market is Expected to Grow at an Annually Rate of 12.3% Till 2011 |quote=Higher-powered products are also likely to use active PFC, since it would be the most cost effective way to bring products into compliance with the EN standard. }}</ref><ref>{{Citation | url = http://www.techarp.com/showarticle.aspx?artno=81&pgno=1 | publisher = TECHarp | title = Power Factor Correction | quote = Passive PFC […] the power factor is low at 60–80%. […] Active PFC ... a power factor of up to 95%}}</ref><ref>{{Citation | publisher = Silverstone Technology | url = http://www.silverstonetek.com/tech/wh_pfc.php?area= | title = Why we need PFC in PSU | quote = Normally, the power factor value of electronic device without power factor correction is approximately 0.5. […] Passive PFC […] 70~80% […] Active PFC […] 90~99.9% | url-status = dead | archive-url = https://web.archive.org/web/20081222085515/http://www.silverstonetek.com/tech/wh_pfc.php?area= | archive-date = 2008-12-22 }}</ref><ref>{{Citation | publisher = Electronic products | newspaper = Taiyo | url = http://www2.electronicproducts.com/PFC_options_for_power_supplies-article-taiyo-mar2004-html.aspx | title = PFC options for power supplies | first = Tom | last = Brooks | date = Mar 2004 | quote = The disadvantages of passive PFC techniques are that they typically yield a power factor of only 0.60 to 0.70 […] Dual-stage active PFC technology [yields] a power factor typically greater than 0.98 | url-status = dead | archive-url = https://web.archive.org/web/20081202100831/http://www2.electronicproducts.com/PFC_options_for_power_supplies-article-taiyo-mar2004-html.aspx | archive-date = 2008-12-02 }}</ref> ==== Active PFC ==== [[File:Active pfc PSU packaging.png|right|thumb|Specifications taken from the packaging of a 610 W [[Power supply unit (computer)|PC power supply]] showing active PFC rating]] Active PFC is the use of [[power electronics]] to change the waveform of current drawn by a load to improve the power factor.<ref>{{Citation | publisher = Fairchild Semiconductor | year = 2004 | type = application note | number = 42047 | title = Power Factor Correction (PFC) Basics | url = http://www.fairchildsemi.com/an/AN/AN-42047.pdf | access-date = 2009-11-29 | archive-url = https://web.archive.org/web/20140611063712/http://www.fairchildsemi.com/an/AN/AN-42047.pdf | archive-date = 2014-06-11 | url-status = dead }}</ref> Some types of the active PFC are [[Buck converter|buck]], [[Boost converter|boost]], [[Buck-boost converter|buck-boost]] and [[synchronous condenser]]. Active power factor correction can be single-stage or multi-stage. In the case of a switched-mode power supply, a [[boost converter]] is inserted between the bridge rectifier and the main input capacitors. The boost converter attempts to maintain a constant voltage at its output while drawing a current that is always in phase with and at the same frequency as the line voltage. Another switched-mode converter inside the power supply produces the desired output voltage from the DC bus. This approach requires additional semiconductor switches and control electronics but permits cheaper and smaller passive components. It is frequently used in practice. For a three-phase SMPS, the [[Vienna rectifier]] configuration may be used to substantially improve the power factor. [[Switched-mode power supply|SMPSs]] with passive PFC can achieve power factor of about 0.7–0.75, SMPSs with active PFC, up to 0.99 power factor, while a SMPS without any power factor correction have a power factor of only about 0.55–0.65.<ref>{{Citation |last1=Sugawara |first1=I. |last2=Suzuki |first2=Y. |last3=Takeuchi |first3=A. |last4=Teshima |first4=T. |contribution=Experimental studies on active and passive PFC circuits |title=INTELEC 97, 19th International Telecommunications Energy Conference |date=19–23 Oct 1997 |pages=571–78 |doi=10.1109/INTLEC.1997.646051|isbn=978-0-7803-3996-5 |s2cid=109885369 }}</ref> Due to their very wide input voltage range, many power supplies with active PFC can automatically adjust to operate on AC power from about 100&nbsp;V (Japan) to 240&nbsp;V (Europe). That feature is particularly welcome in power supplies for laptops. ==== Dynamic PFC ==== Dynamic power factor correction (DPFC), sometimes referred to as real-time power factor correction, is used for electrical stabilization in cases of rapid load changes (e.g. at large manufacturing sites). DPFC is useful when standard power factor correction would cause over or under correction.<ref>{{Cite conference|last1=Chavez |first1=C. |last2=Houdek |first2=J. A. |title=Dynamic Harmonic Mitigation and power factor correction |publisher=IEEE |book-title= EPQU'07 |conference=9th International Conference Electrical Power Quality and Utilisation: October 9–11, 2007, Barcelona, Spain |pages=1–5 |doi=10.1109/EPQU.2007.4424144 |isbn=978-84-690-9441-9 }}</ref> DPFC uses semiconductor switches, typically [[thyristor]]s, to quickly connect and disconnect capacitors or inductors to improve power factor. == Importance in distribution systems == [[File:Condensor bank 150kV - 75MVAR.jpg|thumb|upright|75 MVAr capacitor bank in a 150 kV substation]] 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. For example, if the load power factor were as low as 0.7, the apparent power would be 1.4 times the real power used by the load. Line current in the circuit would also be 1.4 times the current required at 1.0 power factor, so the losses in the circuit would be doubled (since they are proportional to the square of the current). Alternatively, all components of the system such as generators, conductors, transformers, and switchgear would be increased in size (and cost) to carry the extra current. When the power factor is close to unity, for the same kVA rating of the transformer more load current can be supplied.<ref>{{cite web|url=https://www.electricalclassroom.com/power-factor/|title=Power Factor – Importance, Calculation and Correction techniques|date=23 November 2018}}</ref> Utilities typically charge additional costs to commercial customers who have a power factor below some limit, which is typically 0.9 to 0.95. Engineers are often interested in the power factor of a load as one of the factors that affect the efficiency of power transmission. With the rising cost of energy and concerns over the efficient delivery of power, active PFC has become more common in consumer electronics.<ref>{{Citation | publisher = ON Semiconductor | year = 2007 | title = Power Factor Correction Handbook | url = http://www.onsemi.com/pub_link/Collateral/HBD853-D.PDF }}</ref> Current [[Energy Star]] guidelines for computers<ref>{{Citation | place = US | url = http://www.energystar.gov/ia/partners/prod_development/revisions/downloads/computer/Version5.0_Computer_Spec.pdf | publisher = Energy Star | title = Program Requirements for Computers | edition = Version 5.0}}</ref> call for a power factor of ≥ 0.9 at 100% of rated output in the [[Power supply unit (computer)|PC's power supply]]. According to a white paper authored by Intel and the [[United States Environmental Protection Agency|U.S. Environmental Protection Agency]], PCs with internal power supplies will require the use of active power factor correction to meet the ENERGY STAR 5.0 Program Requirements for Computers.<ref>{{Citation |last1=Bolioli |first1=T. |last2=Duggirala |first2=M. |last3=Haines |first3=E. |last4=Kolappan |first4=R. |last5=Wong |first5=H. |year=2009 |publisher=Energy Star |title=Version 5.0 System Implementation |type=white paper |url=http://www.energystar.gov/ia/partners/product_specs/program_reqs/Computers_Intel_Whitepaper_Spec5.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.energystar.gov/ia/partners/product_specs/program_reqs/Computers_Intel_Whitepaper_Spec5.pdf |archive-date=2022-10-09 |url-status=live}}</ref> In Europe, [[IEC EN 61000-3-2|EN 61000-3-2]] requires power factor correction be incorporated into consumer products. Small customers, such as households, are not usually charged for reactive power and so power factor metering equipment for such customers will not be installed. == Measurement techniques == The power factor in a single-phase circuit (or balanced three-phase circuit) can be measured with the wattmeter-ammeter-voltmeter method, where the power in watts is divided by the product of measured voltage and current. The power factor of a balanced [[Polyphase system|polyphase circuit]] is the same as that of any phase. The power factor of an unbalanced polyphase circuit is not uniquely defined. A direct reading power factor meter can be made with a [[moving coil meter]] of the electrodynamic type, carrying two perpendicular coils on the moving part of the instrument. The field of the instrument is energized by the circuit current flow. The two moving coils, A and B, are connected in parallel with the circuit load. One coil, A, will be connected through a resistor and the second coil, B, through an inductor, so that the current in coil B is delayed with respect to current in A. At unity power factor, the current in A is in phase with the circuit current, and coil A provides maximum torque, driving the instrument pointer toward the 1.0 mark on the scale. At zero power factor, the current in coil B is in phase with circuit current, and coil B provides torque to drive the pointer towards 0. At intermediate values of power factor, the torques provided by the two coils add and the pointer takes up intermediate positions.<ref>{{Citation |first1=Donald G. |last1=Fink |author1-link=Donald G. Fink |first2=H. Wayne |last2=Beaty |title=Standard Handbook for Electrical Engineers |edition=11 |publisher=McGraw-Hill |place=New York |year=1978 |isbn=978-0-07-020974-9 |page=3‐29 paragraph 80}}</ref> Another electromechanical instrument is the polarized-vane type.<ref>{{Citation |title=Manual of Electric Instruments Construction and Operating Principles |id=GET-1087A |publisher=General Electric, Meter and Instrument Department |place=Schenectady, New York |year=1949 |pages=66–68}}</ref> In this instrument a stationary field coil produces a rotating magnetic field, just like a polyphase motor. The field coils are connected either directly to polyphase voltage sources or to a phase-shifting reactor if a single-phase application. A second stationary field coil, perpendicular to the voltage coils, carries a current proportional to current in one phase of the circuit. The moving system of the instrument consists of two vanes that are magnetized by the current coil. In operation, the moving vanes take up a physical angle equivalent to the electrical angle between the voltage source and the current source. This type of instrument can be made to register for currents in both directions, giving a four-quadrant display of power factor or phase angle. Digital instruments exist that directly measure the time lag between voltage and current waveforms. Low-cost instruments of this type measure the peak of the waveforms. More sophisticated versions measure the peak of the fundamental harmonic only, thus giving a more accurate reading for phase angle on distorted waveforms. Calculating power factor from voltage and current phases is only accurate if both waveforms are sinusoidal.<ref name=ni_white_paper>{{cite web|url=http://www.ni.com/white-paper/4278/en/|title=The Fundamentals of FFT-Based Signal Analysis and Measurement in LabVIEW and LabWindows/CVI|work=National Instruments Corporation|access-date=6 November 2017}}</ref> Power Quality Analyzers, often referred to as Power Analyzers, make a digital recording of the voltage and current waveform (typically either one phase or three phase) and accurately calculate true power (watts), apparent power (VA) power factor, AC voltage, AC current, DC voltage, DC current, frequency, IEC61000-3-2/3-12 Harmonic measurement, IEC61000-3-3/3-11 flicker measurement, individual phase voltages in delta applications where there is no neutral line, total harmonic distortion, phase and amplitude of individual voltage or current harmonics, etc.<ref name=Yokogawa_WT3000E>{{cite web|url=http://www.yokogawa.co.jp/ftp/dist/ks/catalog/en/BUWT3000E-01EN_020.pdf|title=WT3000E Series Precision Power Analyzers|work=Yokogawa Corporation|access-date=6 November 2017|archive-url=https://web.archive.org/web/20171107112155/http://www.yokogawa.co.jp/ftp/dist/ks/catalog/en/BUWT3000E-01EN_020.pdf|archive-date=7 November 2017|url-status=dead}}</ref><ref name=Fluke_1760>{{cite web|url=https://cdn.testequity.com/documents/pdf/1760-ds.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://cdn.testequity.com/documents/pdf/1760-ds.pdf |archive-date=2022-10-09 |url-status=live|title=Fluke 1760 Three-Phase Power Quality Recorder |work=Fluke Corporation|access-date=6 November 2017}}</ref> == 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 I leads the voltage E in a capacitor C. Another common mnemonic is CIVIL – in a capacitor (C) the current (I) leads voltage (V), voltage (V) leads current (I) in an inductor (L). == References == {{Reflist}} == External links == * {{Citation | url = http://www.ece.utexas.edu/~grady/POWERFAC.pdf | title = Harmonics and how they relate to power factor | publisher = U Texas }}. {{Electricity delivery}} {{Authority control}} [[Category:Electrical parameters]] [[Category:AC power]] [[Category:Electrical engineering]] [[Category:Engineering ratios]]'
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'==General case== [[File:Power Factor General Case.svg|thumb|Schematic showing how power factor is calculated]] The general expression for power factor is given by :<math> \mbox{power factor} = P/P_a </math> :<math> P_a = I_{rms} V_{rms} </math> where <math>P</math> is the real power measured by an ideal [[wattmeter]], <math>I_{rms}</math> is the rms current measured by an ideal [[ammeter]], and <math>V_{rms}</math> is the rms voltage measured by an ideal [[voltmeter]]. Apparent power, <math>P_a</math>, is the product of the rms current and the rms voltage. If the load is sourcing power back toward the generator, then <math>P</math> and <math> \mbox{power factor} </math> will be negative. ===Periodic waveforms=== If the waveforms are periodic with the same period which is much shorter than the averaging time of the physical meters, then the power factor can be computed by the following :<math> \mbox{power factor} = P/P_a </math> :<math> P_a = I_{rms} V_{rms} </math> :<math> P =\frac 1 T \int_{t'}^{t'+T} i(t)v(t) dt </math> :<math> I_{rms}^2 =\frac 1 T \int_{t'}^{t'+T} {i(t)}^2 dt </math> :<math> V_{rms}^2 =\frac 1 T \int_{t'}^{t'+T} {v(t)}^2 dt </math> where <math>i(t)</math> is the instantaneous current, <math>v(t)</math> is the instantaneous voltage, <math>t'</math> is an arbitrary starting time, and <math>T</math> is the period of the waveforms. ===Nonperiodic 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 <math>T</math> is the averaging time of the meters instead of the waveform period. {{-}} == Linear time-invariant 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}}.]] [[Linear time-invariant system|Linear time-invariant circuits]] (referred to simply as ''linear circuits'' for the rest of this article), for example, circuits consisting of combinations of resistors, inductors and capacitors have 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, heating elements) 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. In the [[electric power grid]], reactive loads cause a continuous ''ebb and flow'' of nonproductive power. 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}}</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 mathematically identical 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=http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf |archive-date=2022-10-09 |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 with respect to 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 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 ''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 so as 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 all the way 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 a number of [[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 supplied 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. == Non-linear loads == Examples of non-linear loads on a power system are rectifiers (such as used in a power supply), and arc discharge devices such as [[fluorescent lamp]]s, electric [[welding]] machines, or [[arc furnace]]s. Because current in these systems is interrupted by a switching action, the current contains frequency components that are multiples of the power system frequency. ''Distortion power factor'' is a measure of how much the harmonic distortion of a load current decreases the average power transferred to the load. [[File:Power factor 75 2.png|right|thumb|upright=1.36|Sinusoidal voltage and non-sinusoidal current give a distortion power factor of 0.75 for this computer power supply load.]] === Non-sinusoidal components === In linear circuits having only sinusoidal currents and voltages of one frequency, the power factor arises only from the difference in phase between the current and voltage. This is ''displacement power factor''.<ref name="FuchsMasoum2015">{{cite book|author1=Ewald Fuchs|author2=Mohammad A. S. Masoum|title=Power Quality in Power Systems and Electrical Machines|url=https://books.google.com/books?id=wuGcBAAAQBAJ&pg=PA432|date=14 July 2015|publisher=Elsevier Science|isbn=978-0-12-800988-8|pages=432–|quote=The DPF it the cosine of the angle between these two quantities}}</ref> Non-linear loads change the shape of the current waveform from a [[sine wave]] to some other form. Non-linear loads create [[harmonic]] currents in addition to the original (fundamental frequency) AC current. This is of importance in practical power systems that contain [[non-linear]] loads such as [[rectifiers]], some forms of electric lighting, [[electric arc furnace]]s, welding equipment, [[Switched-mode power supply|switched-mode power supplies]], variable speed drives and other devices. Filters consisting of linear capacitors and inductors can prevent harmonic currents from entering the supplying system. To measure the real power or reactive power, a [[wattmeter]] designed to work properly with non-sinusoidal currents must be used. === Distortion power factor === The '''distortion power factor''' is the distortion component associated with the harmonic voltages and currents present in the system. :<math> \begin{align} \mbox{distortion power factor} & = \frac{ I_1}{I_{rms}} \\ & = \frac{I_1} {\sqrt{I_1^2+I_2^2+I_3^2+I_4^2+\cdots}} \\ & = \frac{1} { \sqrt{1+ \frac{I_2^2+I_3^2+I_4^2+\cdots}{I_1^2}}} \\ & = \frac{1} {\sqrt{ 1+THD_i^2}} \\ \end{align} </math> <math>\mbox{THD}_i</math> is the [[total harmonic distortion]] of the load current. :<math>THD_i = \frac{\sqrt{\displaystyle\sum_{h=2}^\infty I_h^2}} {I_1}= \frac{\sqrt{I_2^2+I_3^2+I_4^2+\cdots}} {I_1}</math> <math>I_1</math> is the fundamental component of the current and <math>I_{\mbox{rms}}</math> is the total current – both are [[root mean square]]-values (distortion power factor can also be used to describe individual order harmonics, using the corresponding current in place of total current). This definition with respect to total harmonic distortion assumes that the voltage stays undistorted (sinusoidal, without harmonics). This simplification is often a good approximation for stiff voltage sources (not being affected by changes in load downstream in the distribution network). Total harmonic distortion of typical generators from current distortion in the network is on the order of 1–2%, which can have larger scale implications but can be ignored in common practice.<ref>{{Citation |url=http://ecmweb.com/power-quality/effects-harmonics-power-systems |title=Effects of Harmonics on Power Systems |first=C. |last=Sankaran |year=1999 |publisher=Electro-Test |quote=...and voltage-time relationship deviates from the pure sine function. The distortion at the point of generation is very small (about 1% to 2%), but nonetheless it exists.}}</ref> The result when multiplied with the displacement power factor (DPF) is the overall, true power factor or just power factor (PF): :<math>\mbox{PF} = \frac{\cos{\varphi}} {\sqrt{ 1+THD_i^2}}</math> === Distortion in three-phase networks === In practice, the local effects of distortion current on devices in a [[Three-phase electric power|three-phase distribution network]] rely on the magnitude of certain order harmonics rather than the total harmonic distortion. For example, the triplen, or zero-sequence, harmonics (3rd, 9th, 15th, etc.) have the property of being in-phase when compared line-to-line. In a [[delta-wye transformer]], these harmonics can result in circulating currents in the delta windings and result in greater [[Joule heating|resistive heating]]. In a wye-configuration of a transformer, triplen harmonics will not create these currents, but they will result in a non-zero current in the [[Ground and neutral|neutral wire]]. This could overload the neutral wire in some cases and create error in kilowatt-hour metering systems and billing revenue.<ref>{{Citation | chapter-url = http://www.pge.com/includes/docs/pdfs/mybusiness/customerservice/energystatus/powerquality/harmonics.pdf | title = Power System Harmonics | publisher = Pacific Gas and Electric | chapter = Single-phase load harmonics vs. three-phase load harmonics | chapter-format = [[PDF]]}}</ref><ref>{{Citation | chapter-url = http://energylogix.ca/harmonics_and_ieee.pdf | title = Harmonics and IEEE 519 | publisher = EnergyLogix Solutions | chapter = Harmonic Effects | place = [[Canada|CA]] | chapter-format = [[PDF]]}}</ref> The presence of current harmonics in a transformer also result in larger [[eddy currents]] in the magnetic core of the transformer. Eddy current losses generally increase as the square of the frequency, lowering the transformer's efficiency, dissipating additional heat, and reducing its service life.<ref>{{Citation |url=http://ecmweb.com/power-quality/effects-harmonics-power-systems |title=Effects of Harmonics on Power Systems |first=C. |last=Sankaran |year=1999 |publisher = Electro-Test |section=Transformers}}</ref> Negative-sequence harmonics (5th, 11th, 17th, etc.) combine 120 degrees out of phase, similarly to the fundamental harmonic but in a reversed sequence. In generators and motors, these currents produce magnetic fields which oppose the rotation of the shaft and sometimes result in damaging mechanical vibrations.<ref>{{Citation |url=http://ecmweb.com/power-quality/effects-harmonics-power-systems |title=Effects of Harmonics on Power Systems |first=C. |last=Sankaran |year=1999 |publisher=Electro-Test |section=Motors |quote=The interaction between the positive and negative sequence magnetic fields and currents produces torsional oscillations of the motor shaft. These oscillations result in shaft vibrations.}}</ref> === Switched-mode power supplies === {{Main|switched-mode power supply#Power factor}} A particularly important class of non-linear loads is the millions of personal computers that typically incorporate [[Switched-mode power supply|switched-mode power supplies]] (SMPS) with rated output power ranging from a few watts to more than 1&nbsp;kW. Historically, these very-low-cost power supplies incorporated a simple full-wave rectifier that conducted only when the [[Mains electricity|mains]] instantaneous voltage exceeded the voltage on the input capacitors. This leads to very high [[peak-to-average ratio|ratios of peak-to-average]] input current, which also lead to a low distortion power factor and potentially serious phase and neutral loading concerns. A typical switched-mode power supply first converts the AC mains to a DC bus by means of a [[bridge rectifier]]. The output voltage is then derived from this DC bus. The problem with this is that the [[rectifier]] is a non-linear device, so the input current is highly non-linear. That means that the input current has energy at [[harmonic]]s of the frequency of the voltage. This presents a problem for power companies, because they cannot compensate for the harmonic current by adding simple capacitors or inductors, as they could for the reactive power drawn by a linear load. Many jurisdictions are beginning to require power factor correction for all power supplies above a certain power level. Regulatory agencies such as the [[European Union|EU]] have set harmonic limits as a method of improving power factor. Declining component cost has hastened implementation of two different methods. To comply with current EU standard EN61000-3-2, all switched-mode power supplies with output power more than 75&nbsp;W must at least include passive power factor correction. [[80 Plus]] power supply certification requires a power factor of 0.9 or more.<ref>{{Citation | url = http://www.80plus.org/ | publisher = 80 Plus | title = Certified Power Supplies and Manufacturers | section = What is an 80 PLUS certified power supply?}}</ref> === Power factor correction (PFC) in non-linear loads === ==== Passive PFC ==== The simplest way to control the [[Harmonics (electrical power)|harmonic]] current is to use a [[electronic filter|filter]] that passes current only at [[utility frequency|line frequency]] (50 or 60&nbsp;Hz). The filter consists of capacitors or inductors and makes a non-linear device look more like a [[linear]] load. An example of passive PFC is a [[valley-fill circuit]]. A disadvantage of passive PFC is that it requires larger inductors or capacitors than an equivalent power active PFC circuit.<ref>{{Citation |url=http://www.nuvation.com/corporate/news/newsletter/fall2006/powersupply.html |publisher=Nuvation |date=Fall 2006 |title=Power Supply Design Principles: Techniques and Solutions, Part 3 |newspaper=Newsletter |first=Ben |last=Schramm |url-status=dead |archive-url=https://web.archive.org/web/20070309134617/http://www.nuvation.com/corporate/news/newsletter/fall2006/powersupply.html |archive-date=2007-03-09 }}</ref><ref>{{Citation | newspaper = Xplore | title = Quasi-active power factor correction with a variable inductive filter: theory, design and practice | volume = 18 | issue = 1 | pages = 248–255 | publisher = IEEE| doi = 10.1109/TPEL.2002.807135 | bibcode = 2003ITPE...18..248W | year = 2003 | last1 = Wolfle | first1 = W.H. | last2 = Hurley | first2 = W.G. }}</ref><ref>{{Citation |publisher=Nuigalway |type=project |url=http://www.nuigalway.ie/power_electronics/projects/quasi_active.html |place=[[Ireland|IE]] |title=Power electronics |contribution=Quasi-active Power Factor Correction: The Role of Variable Inductance |last1=Wölfle |first1=W. H. |last2=Hurley |first2=W. G.}}</ref> Also, in practice, passive PFC is often less effective at improving the power factor.<ref name="effi">{{Citation | url = http://www.xbitlabs.com/articles/coolers/display/atx-psu5_3.html | title = ATX Power Supply Units Roundup | publisher = xBit labs | quote = The power factor is the measure of reactive power. It is the ratio of active power to the total of active and reactive power. It is about 0.65 with an ordinary PSU, but PSUs with active PFC have a power factor of 0.97–0.99. […] hardware reviewers sometimes make no difference between the power factor and the efficiency factor. Although both these terms describe the effectiveness of a power supply, it is a gross mistake to confuse them. […] There is a very small effect from passive PFC – the power factor grows only from 0.65 to 0.7–0.75. | url-status = dead | archive-url = https://web.archive.org/web/20081120040707/http://www.xbitlabs.com/articles/coolers/display/atx-psu5_3.html | archive-date = 2008-11-20 }}</ref><ref>{{Citation|date=Mar 16, 2006 |publisher=Find articles |url=http://findarticles.com/p/articles/mi_m0EIN/is_2006_March_16/ai_n26797888 |archive-url=https://web.archive.org/web/20090901140721/http://findarticles.com/p/articles/mi_m0EIN/is_2006_March_16/ai_n26797888/ |url-status=dead |archive-date=September 1, 2009 |title=The Active PFC Market is Expected to Grow at an Annually Rate of 12.3% Till 2011 |quote=Higher-powered products are also likely to use active PFC, since it would be the most cost effective way to bring products into compliance with the EN standard. }}</ref><ref>{{Citation | url = http://www.techarp.com/showarticle.aspx?artno=81&pgno=1 | publisher = TECHarp | title = Power Factor Correction | quote = Passive PFC […] the power factor is low at 60–80%. […] Active PFC ... a power factor of up to 95%}}</ref><ref>{{Citation | publisher = Silverstone Technology | url = http://www.silverstonetek.com/tech/wh_pfc.php?area= | title = Why we need PFC in PSU | quote = Normally, the power factor value of electronic device without power factor correction is approximately 0.5. […] Passive PFC […] 70~80% […] Active PFC […] 90~99.9% | url-status = dead | archive-url = https://web.archive.org/web/20081222085515/http://www.silverstonetek.com/tech/wh_pfc.php?area= | archive-date = 2008-12-22 }}</ref><ref>{{Citation | publisher = Electronic products | newspaper = Taiyo | url = http://www2.electronicproducts.com/PFC_options_for_power_supplies-article-taiyo-mar2004-html.aspx | title = PFC options for power supplies | first = Tom | last = Brooks | date = Mar 2004 | quote = The disadvantages of passive PFC techniques are that they typically yield a power factor of only 0.60 to 0.70 […] Dual-stage active PFC technology [yields] a power factor typically greater than 0.98 | url-status = dead | archive-url = https://web.archive.org/web/20081202100831/http://www2.electronicproducts.com/PFC_options_for_power_supplies-article-taiyo-mar2004-html.aspx | archive-date = 2008-12-02 }}</ref> ==== Active PFC ==== [[File:Active pfc PSU packaging.png|right|thumb|Specifications taken from the packaging of a 610 W [[Power supply unit (computer)|PC power supply]] showing active PFC rating]] Active PFC is the use of [[power electronics]] to change the waveform of current drawn by a load to improve the power factor.<ref>{{Citation | publisher = Fairchild Semiconductor | year = 2004 | type = application note | number = 42047 | title = Power Factor Correction (PFC) Basics | url = http://www.fairchildsemi.com/an/AN/AN-42047.pdf | access-date = 2009-11-29 | archive-url = https://web.archive.org/web/20140611063712/http://www.fairchildsemi.com/an/AN/AN-42047.pdf | archive-date = 2014-06-11 | url-status = dead }}</ref> Some types of the active PFC are [[Buck converter|buck]], [[Boost converter|boost]], [[Buck-boost converter|buck-boost]] and [[synchronous condenser]]. Active power factor correction can be single-stage or multi-stage. In the case of a switched-mode power supply, a [[boost converter]] is inserted between the bridge rectifier and the main input capacitors. The boost converter attempts to maintain a constant voltage at its output while drawing a current that is always in phase with and at the same frequency as the line voltage. Another switched-mode converter inside the power supply produces the desired output voltage from the DC bus. This approach requires additional semiconductor switches and control electronics but permits cheaper and smaller passive components. It is frequently used in practice. For a three-phase SMPS, the [[Vienna rectifier]] configuration may be used to substantially improve the power factor. [[Switched-mode power supply|SMPSs]] with passive PFC can achieve power factor of about 0.7–0.75, SMPSs with active PFC, up to 0.99 power factor, while a SMPS without any power factor correction have a power factor of only about 0.55–0.65.<ref>{{Citation |last1=Sugawara |first1=I. |last2=Suzuki |first2=Y. |last3=Takeuchi |first3=A. |last4=Teshima |first4=T. |contribution=Experimental studies on active and passive PFC circuits |title=INTELEC 97, 19th International Telecommunications Energy Conference |date=19–23 Oct 1997 |pages=571–78 |doi=10.1109/INTLEC.1997.646051|isbn=978-0-7803-3996-5 |s2cid=109885369 }}</ref> Due to their very wide input voltage range, many power supplies with active PFC can automatically adjust to operate on AC power from about 100&nbsp;V (Japan) to 240&nbsp;V (Europe). That feature is particularly welcome in power supplies for laptops. ==== Dynamic PFC ==== Dynamic power factor correction (DPFC), sometimes referred to as real-time power factor correction, is used for electrical stabilization in cases of rapid load changes (e.g. at large manufacturing sites). DPFC is useful when standard power factor correction would cause over or under correction.<ref>{{Cite conference|last1=Chavez |first1=C. |last2=Houdek |first2=J. A. |title=Dynamic Harmonic Mitigation and power factor correction |publisher=IEEE |book-title= EPQU'07 |conference=9th International Conference Electrical Power Quality and Utilisation: October 9–11, 2007, Barcelona, Spain |pages=1–5 |doi=10.1109/EPQU.2007.4424144 |isbn=978-84-690-9441-9 }}</ref> DPFC uses semiconductor switches, typically [[thyristor]]s, to quickly connect and disconnect capacitors or inductors to improve power factor. == Importance in distribution systems == [[File:Condensor bank 150kV - 75MVAR.jpg|thumb|upright|75 MVAr capacitor bank in a 150 kV substation]] 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. For example, if the load power factor were as low as 0.7, the apparent power would be 1.4 times the real power used by the load. Line current in the circuit would also be 1.4 times the current required at 1.0 power factor, so the losses in the circuit would be doubled (since they are proportional to the square of the current). Alternatively, all components of the system such as generators, conductors, transformers, and switchgear would be increased in size (and cost) to carry the extra current. When the power factor is close to unity, for the same kVA rating of the transformer more load current can be supplied.<ref>{{cite web|url=https://www.electricalclassroom.com/power-factor/|title=Power Factor – Importance, Calculation and Correction techniques|date=23 November 2018}}</ref> Utilities typically charge additional costs to commercial customers who have a power factor below some limit, which is typically 0.9 to 0.95. Engineers are often interested in the power factor of a load as one of the factors that affect the efficiency of power transmission. With the rising cost of energy and concerns over the efficient delivery of power, active PFC has become more common in consumer electronics.<ref>{{Citation | publisher = ON Semiconductor | year = 2007 | title = Power Factor Correction Handbook | url = http://www.onsemi.com/pub_link/Collateral/HBD853-D.PDF }}</ref> Current [[Energy Star]] guidelines for computers<ref>{{Citation | place = US | url = http://www.energystar.gov/ia/partners/prod_development/revisions/downloads/computer/Version5.0_Computer_Spec.pdf | publisher = Energy Star | title = Program Requirements for Computers | edition = Version 5.0}}</ref> call for a power factor of ≥ 0.9 at 100% of rated output in the [[Power supply unit (computer)|PC's power supply]]. According to a white paper authored by Intel and the [[United States Environmental Protection Agency|U.S. Environmental Protection Agency]], PCs with internal power supplies will require the use of active power factor correction to meet the ENERGY STAR 5.0 Program Requirements for Computers.<ref>{{Citation |last1=Bolioli |first1=T. |last2=Duggirala |first2=M. |last3=Haines |first3=E. |last4=Kolappan |first4=R. |last5=Wong |first5=H. |year=2009 |publisher=Energy Star |title=Version 5.0 System Implementation |type=white paper |url=http://www.energystar.gov/ia/partners/product_specs/program_reqs/Computers_Intel_Whitepaper_Spec5.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.energystar.gov/ia/partners/product_specs/program_reqs/Computers_Intel_Whitepaper_Spec5.pdf |archive-date=2022-10-09 |url-status=live}}</ref> In Europe, [[IEC EN 61000-3-2|EN 61000-3-2]] requires power factor correction be incorporated into consumer products. Small customers, such as households, are not usually charged for reactive power and so power factor metering equipment for such customers will not be installed. == Measurement techniques == The power factor in a single-phase circuit (or balanced three-phase circuit) can be measured with the wattmeter-ammeter-voltmeter method, where the power in watts is divided by the product of measured voltage and current. The power factor of a balanced [[Polyphase system|polyphase circuit]] is the same as that of any phase. The power factor of an unbalanced polyphase circuit is not uniquely defined. A direct reading power factor meter can be made with a [[moving coil meter]] of the electrodynamic type, carrying two perpendicular coils on the moving part of the instrument. The field of the instrument is energized by the circuit current flow. The two moving coils, A and B, are connected in parallel with the circuit load. One coil, A, will be connected through a resistor and the second coil, B, through an inductor, so that the current in coil B is delayed with respect to current in A. At unity power factor, the current in A is in phase with the circuit current, and coil A provides maximum torque, driving the instrument pointer toward the 1.0 mark on the scale. At zero power factor, the current in coil B is in phase with circuit current, and coil B provides torque to drive the pointer towards 0. At intermediate values of power factor, the torques provided by the two coils add and the pointer takes up intermediate positions.<ref>{{Citation |first1=Donald G. |last1=Fink |author1-link=Donald G. Fink |first2=H. Wayne |last2=Beaty |title=Standard Handbook for Electrical Engineers |edition=11 |publisher=McGraw-Hill |place=New York |year=1978 |isbn=978-0-07-020974-9 |page=3‐29 paragraph 80}}</ref> Another electromechanical instrument is the polarized-vane type.<ref>{{Citation |title=Manual of Electric Instruments Construction and Operating Principles |id=GET-1087A |publisher=General Electric, Meter and Instrument Department |place=Schenectady, New York |year=1949 |pages=66–68}}</ref> In this instrument a stationary field coil produces a rotating magnetic field, just like a polyphase motor. The field coils are connected either directly to polyphase voltage sources or to a phase-shifting reactor if a single-phase application. A second stationary field coil, perpendicular to the voltage coils, carries a current proportional to current in one phase of the circuit. The moving system of the instrument consists of two vanes that are magnetized by the current coil. In operation, the moving vanes take up a physical angle equivalent to the electrical angle between the voltage source and the current source. This type of instrument can be made to register for currents in both directions, giving a four-quadrant display of power factor or phase angle. Digital instruments exist that directly measure the time lag between voltage and current waveforms. Low-cost instruments of this type measure the peak of the waveforms. More sophisticated versions measure the peak of the fundamental harmonic only, thus giving a more accurate reading for phase angle on distorted waveforms. Calculating power factor from voltage and current phases is only accurate if both waveforms are sinusoidal.<ref name=ni_white_paper>{{cite web|url=http://www.ni.com/white-paper/4278/en/|title=The Fundamentals of FFT-Based Signal Analysis and Measurement in LabVIEW and LabWindows/CVI|work=National Instruments Corporation|access-date=6 November 2017}}</ref> Power Quality Analyzers, often referred to as Power Analyzers, make a digital recording of the voltage and current waveform (typically either one phase or three phase) and accurately calculate true power (watts), apparent power (VA) power factor, AC voltage, AC current, DC voltage, DC current, frequency, IEC61000-3-2/3-12 Harmonic measurement, IEC61000-3-3/3-11 flicker measurement, individual phase voltages in delta applications where there is no neutral line, total harmonic distortion, phase and amplitude of individual voltage or current harmonics, etc.<ref name=Yokogawa_WT3000E>{{cite web|url=http://www.yokogawa.co.jp/ftp/dist/ks/catalog/en/BUWT3000E-01EN_020.pdf|title=WT3000E Series Precision Power Analyzers|work=Yokogawa Corporation|access-date=6 November 2017|archive-url=https://web.archive.org/web/20171107112155/http://www.yokogawa.co.jp/ftp/dist/ks/catalog/en/BUWT3000E-01EN_020.pdf|archive-date=7 November 2017|url-status=dead}}</ref><ref name=Fluke_1760>{{cite web|url=https://cdn.testequity.com/documents/pdf/1760-ds.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://cdn.testequity.com/documents/pdf/1760-ds.pdf |archive-date=2022-10-09 |url-status=live|title=Fluke 1760 Three-Phase Power Quality Recorder |work=Fluke Corporation|access-date=6 November 2017}}</ref> == 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 I leads the voltage E in a capacitor C. Another common mnemonic is CIVIL – in a capacitor (C) the current (I) leads voltage (V), voltage (V) leads current (I) in an inductor (L). == References == {{Reflist}} == External links == * {{Citation | url = http://www.ece.utexas.edu/~grady/POWERFAC.pdf | title = Harmonics and how they relate to power factor | publisher = U Texas }}. {{Electricity delivery}} {{Authority control}} [[Category:Electrical parameters]] [[Category:AC power]] [[Category:Electrical engineering]] [[Category:Engineering ratios]]'
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'@@ -1,12 +1,2 @@ -{{Short description|Ratio of active power to apparent power}} -{{For|the firearms cartridge ranking system|Power factor (shooting sports)}} -[[File:Power Factor General Case.svg|thumb|Schematic showing how power factor is calculated]] - -In [[electrical engineering]], the '''power factor''' of an [[AC power]] system is defined as the [[ratio]] of the ''[[real power]]'' absorbed by the [[electrical load|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|RMS]] current and voltage. 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 may be greater than the real power, so more current flows in the circuit than would be required to transfer real power alone. A power factor magnitude of less than one indicates the voltage and current are not in phase, reducing the average [[Product (mathematics)|product]] of the two. A negative power factor occurs when the device (which is normally the load) 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 higher 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 where there is 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 (such as [[induction motor]]s) can be corrected with a passive network of [[capacitor]]s or [[inductor]]s. Non-linear loads, such as [[rectifier]]s, 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 [[electrical substation|substation]], spread out over a distribution system, or built into power-consuming equipment. - ==General case== [[File:Power Factor General Case.svg|thumb|Schematic showing how power factor is calculated]] '
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[ 0 => '{{Short description|Ratio of active power to apparent power}}', 1 => '{{For|the firearms cartridge ranking system|Power factor (shooting sports)}}', 2 => '[[File:Power Factor General Case.svg|thumb|Schematic showing how power factor is calculated]]', 3 => '', 4 => 'In [[electrical engineering]], the '''power factor''' of an [[AC power]] system is defined as the [[ratio]] of the ''[[real power]]'' absorbed by the [[electrical load|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|RMS]] current and voltage. 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 may be greater than the real power, so more current flows in the circuit than would be required to transfer real power alone. A power factor magnitude of less than one indicates the voltage and current are not in phase, reducing the average [[Product (mathematics)|product]] of the two. A negative power factor occurs when the device (which is normally the load) generates real power, which then flows back towards the source.', 5 => '', 6 => '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 higher 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 where there is a low power factor.', 7 => '', 8 => ''''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 (such as [[induction motor]]s) can be corrected with a passive network of [[capacitor]]s or [[inductor]]s. Non-linear loads, such as [[rectifier]]s, 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 [[electrical substation|substation]], spread out over a distribution system, or built into power-consuming equipment.', 9 => '' ]
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'<div class="mw-parser-output"><div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#General_case"><span class="tocnumber">1</span> <span class="toctext">General case</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Periodic_waveforms"><span class="tocnumber">1.1</span> <span class="toctext">Periodic waveforms</span></a></li> <li class="toclevel-2 tocsection-3"><a href="#Nonperiodic_waveforms"><span class="tocnumber">1.2</span> <span class="toctext">Nonperiodic waveforms</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-4"><a href="#Linear_time-invariant_circuits"><span class="tocnumber">2</span> <span class="toctext">Linear time-invariant circuits</span></a> <ul> <li class="toclevel-2 tocsection-5"><a href="#Definition_and_calculation"><span class="tocnumber">2.1</span> <span class="toctext">Definition and calculation</span></a> <ul> <li class="toclevel-3 tocsection-6"><a href="#Power_triangle"><span class="tocnumber">2.1.1</span> <span class="toctext">Power triangle</span></a></li> <li class="toclevel-3 tocsection-7"><a href="#Lagging,_leading_and_unity_power_factors"><span class="tocnumber">2.1.2</span> <span class="toctext">Lagging, leading and unity power factors</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-8"><a href="#Power_factor_correction_of_linear_loads"><span class="tocnumber">2.2</span> <span class="toctext">Power factor correction of linear loads</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-9"><a href="#Non-linear_loads"><span class="tocnumber">3</span> <span class="toctext">Non-linear loads</span></a> <ul> <li class="toclevel-2 tocsection-10"><a href="#Non-sinusoidal_components"><span class="tocnumber">3.1</span> <span class="toctext">Non-sinusoidal components</span></a></li> <li class="toclevel-2 tocsection-11"><a href="#Distortion_power_factor"><span class="tocnumber">3.2</span> <span class="toctext">Distortion power factor</span></a></li> <li class="toclevel-2 tocsection-12"><a href="#Distortion_in_three-phase_networks"><span class="tocnumber">3.3</span> <span class="toctext">Distortion in three-phase networks</span></a></li> <li class="toclevel-2 tocsection-13"><a href="#Switched-mode_power_supplies"><span class="tocnumber">3.4</span> <span class="toctext">Switched-mode power supplies</span></a></li> <li class="toclevel-2 tocsection-14"><a href="#Power_factor_correction_(PFC)_in_non-linear_loads"><span class="tocnumber">3.5</span> <span class="toctext">Power factor correction (PFC) in non-linear loads</span></a> <ul> <li class="toclevel-3 tocsection-15"><a href="#Passive_PFC"><span class="tocnumber">3.5.1</span> <span class="toctext">Passive PFC</span></a></li> <li class="toclevel-3 tocsection-16"><a href="#Active_PFC"><span class="tocnumber">3.5.2</span> <span class="toctext">Active PFC</span></a></li> <li class="toclevel-3 tocsection-17"><a href="#Dynamic_PFC"><span class="tocnumber">3.5.3</span> <span class="toctext">Dynamic PFC</span></a></li> </ul> </li> </ul> </li> <li class="toclevel-1 tocsection-18"><a href="#Importance_in_distribution_systems"><span class="tocnumber">4</span> <span class="toctext">Importance in distribution systems</span></a></li> <li class="toclevel-1 tocsection-19"><a href="#Measurement_techniques"><span class="tocnumber">5</span> <span class="toctext">Measurement techniques</span></a></li> <li class="toclevel-1 tocsection-20"><a href="#Mnemonics"><span class="tocnumber">6</span> <span class="toctext">Mnemonics</span></a></li> <li class="toclevel-1 tocsection-21"><a href="#References"><span class="tocnumber">7</span> <span class="toctext">References</span></a></li> <li class="toclevel-1 tocsection-22"><a href="#External_links"><span class="tocnumber">8</span> <span class="toctext">External links</span></a></li> </ul> </div> <h2><span class="mw-headline" id="General_case">General case</span></h2> <div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="/wiki/File:Power_Factor_General_Case.svg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Power_Factor_General_Case.svg/220px-Power_Factor_General_Case.svg.png" decoding="async" width="220" height="88" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Power_Factor_General_Case.svg/330px-Power_Factor_General_Case.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Power_Factor_General_Case.svg/440px-Power_Factor_General_Case.svg.png 2x" data-file-width="1716" data-file-height="683" /></a> <div class="thumbcaption"><div class="magnify"><a href="/wiki/File:Power_Factor_General_Case.svg" class="internal" title="Enlarge"></a></div>Schematic showing how power factor is calculated</div></div></div> <p>The general expression for power factor is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{power factor}}=P/P_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>power factor</mtext> </mstyle> </mrow> <mo>=</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{power factor}}=P/P_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/209ece3ed294f7118509e4838c74bebb070f8cb6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:21.144ex; height:2.843ex;" alt="{\displaystyle {\mbox{power factor}}=P/P_{a}}"/></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{a}=I_{rms}V_{rms}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>m</mi> <mi>s</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>m</mi> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{a}=I_{rms}V_{rms}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6e1550d52ce6a5f3e4125838807d0a1d9ba414b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:14.446ex; height:2.509ex;" alt="{\displaystyle P_{a}=I_{rms}V_{rms}}"/></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="P"/></span> is the real power measured by an ideal <a href="/wiki/Wattmeter" title="Wattmeter">wattmeter</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{rms}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>m</mi> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{rms}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e397d267d8fb23b0fef967675b3169bca119b18b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:4.211ex; height:2.509ex;" alt="{\displaystyle I_{rms}}"/></span> is the rms current measured by an ideal <a href="/wiki/Ammeter" title="Ammeter">ammeter</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{rms}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>m</mi> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{rms}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/819c79c8ec51e6393519ec3185731db558edd5d3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:4.543ex; height:2.509ex;" alt="{\displaystyle V_{rms}}"/></span> is the rms voltage measured by an ideal <a href="/wiki/Voltmeter" title="Voltmeter">voltmeter</a>. Apparent power, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5830e67f78d703f1bc6ff6d691691cba661ef48" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.594ex; height:2.509ex;" alt="P_{a}"/></span>, is the product of the rms current and the rms voltage. </p><p>If the load is sourcing power back toward the generator, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="P"/></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{power factor}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>power factor</mtext> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{power factor}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cad23a9f50f7a9e3c6635bc32fb6513ffb247e38" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:12.543ex; height:2.509ex;" alt="{\displaystyle {\mbox{power factor}}}"/></span> will be negative. </p> <h3><span class="mw-headline" id="Periodic_waveforms">Periodic waveforms</span></h3> <p>If the waveforms are periodic with the same period which is much shorter than the averaging time of the physical meters, then the power factor can be computed by the following </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{power factor}}=P/P_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>power factor</mtext> </mstyle> </mrow> <mo>=</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{power factor}}=P/P_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/209ece3ed294f7118509e4838c74bebb070f8cb6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:21.144ex; height:2.843ex;" alt="{\displaystyle {\mbox{power factor}}=P/P_{a}}"/></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{a}=I_{rms}V_{rms}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>m</mi> <mi>s</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>m</mi> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{a}=I_{rms}V_{rms}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6e1550d52ce6a5f3e4125838807d0a1d9ba414b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:14.446ex; height:2.509ex;" alt="{\displaystyle P_{a}=I_{rms}V_{rms}}"/></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P={\frac {1}{T}}\int _{t'}^{t'+T}i(t)v(t)dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>&#x222B;<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>t</mi> <mo>&#x2032;</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>t</mi> <mo>&#x2032;</mo> </msup> <mo>+</mo> <mi>T</mi> </mrow> </msubsup> <mi>i</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\frac {1}{T}}\int _{t'}^{t'+T}i(t)v(t)dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/910e09dcbfa65fd8b90f5a129beeb39058420193" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:23.631ex; height:6.509ex;" alt="{\displaystyle P={\frac {1}{T}}\int _{t&#039;}^{t&#039;+T}i(t)v(t)dt}"/></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{rms}^{2}={\frac {1}{T}}\int _{t'}^{t'+T}{i(t)}^{2}dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>m</mi> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>&#x222B;<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>t</mi> <mo>&#x2032;</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>t</mi> <mo>&#x2032;</mo> </msup> <mo>+</mo> <mi>T</mi> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{rms}^{2}={\frac {1}{T}}\int _{t'}^{t'+T}{i(t)}^{2}dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15b471bc6c74ec4cb5349011c2792a48d57611eb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:23.374ex; height:6.509ex;" alt="{\displaystyle I_{rms}^{2}={\frac {1}{T}}\int _{t&#039;}^{t&#039;+T}{i(t)}^{2}dt}"/></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{rms}^{2}={\frac {1}{T}}\int _{t'}^{t'+T}{v(t)}^{2}dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>m</mi> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>&#x222B;<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>t</mi> <mo>&#x2032;</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>t</mi> <mo>&#x2032;</mo> </msup> <mo>+</mo> <mi>T</mi> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{rms}^{2}={\frac {1}{T}}\int _{t'}^{t'+T}{v(t)}^{2}dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84e9b65a0023ffcc54a3fe835f4f839cae56148d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:24.031ex; height:6.509ex;" alt="{\displaystyle V_{rms}^{2}={\frac {1}{T}}\int _{t&#039;}^{t&#039;+T}{v(t)}^{2}dt}"/></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06931e7bcffd32010f83c3d5dbef2d9bcbdcb670" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:3.451ex; height:2.843ex;" alt="i(t)"/></span> is the instantaneous current, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/243a0bf98a12f48552ba6a70302122d81b237b3d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:3.777ex; height:2.843ex;" alt="v(t)"/></span> is the instantaneous voltage, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mo>&#x2032;</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a69b623f18f6b111645f0ec200b3271729fa99af" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.524ex; height:2.509ex;" alt="t&#039;"/></span> is an arbitrary starting time, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="T"/></span> is the period of the waveforms. </p> <h3><span class="mw-headline" id="Nonperiodic_waveforms">Nonperiodic waveforms</span></h3> <p>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 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="T"/></span> is the averaging time of the meters instead of the waveform period. </p> <div style="clear:both;"></div> <h2><span class="mw-headline" id="Linear_time-invariant_circuits">Linear time-invariant circuits</span></h2> <div class="thumb tright"><div class="thumbinner" style="width:302px;"><a href="/wiki/File:Power_factor_0.svg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Power_factor_0.svg/300px-Power_factor_0.svg.png" decoding="async" width="300" height="247" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Power_factor_0.svg/450px-Power_factor_0.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/Power_factor_0.svg/600px-Power_factor_0.svg.png 2x" data-file-width="619" data-file-height="510" /></a> <div class="thumbcaption"><div class="magnify"><a href="/wiki/File:Power_factor_0.svg" class="internal" title="Enlarge"></a></div>Power flow calculated from AC voltage and current entering a load having a zero power factor (<span class="texhtml mvar" style="font-style:italic;">ϕ</span>&#160;=&#160;90°, cos(<span class="texhtml mvar" style="font-style:italic;">ϕ</span>)&#160;=&#160;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 <i>average</i> power flow (light blue line) of zero.</div></div></div> <div class="thumb tright"><div class="thumbinner" style="width:302px;"><a href="/wiki/File:Power_factor_0.7.svg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Power_factor_0.7.svg/300px-Power_factor_0.7.svg.png" decoding="async" width="300" height="251" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Power_factor_0.7.svg/450px-Power_factor_0.7.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Power_factor_0.7.svg/600px-Power_factor_0.7.svg.png 2x" data-file-width="669" data-file-height="559" /></a> <div class="thumbcaption"><div class="magnify"><a href="/wiki/File:Power_factor_0.7.svg" class="internal" title="Enlarge"></a></div>Instantaneous and average power calculated from AC voltage and current for a load with a lagging power factor (<span class="texhtml mvar" style="font-style:italic;">ϕ</span>&#160;=&#160;45°, cos(<span class="texhtml mvar" style="font-style:italic;">ϕ</span>)&#160;≈&#160;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 <span class="texhtml mvar" style="font-style:italic;">ϕ</span>.</div></div></div> <p><a href="/wiki/Linear_time-invariant_system" title="Linear time-invariant system">Linear time-invariant circuits</a> (referred to simply as <i>linear circuits</i> for the rest of this article), for example, circuits consisting of combinations of resistors, inductors and capacitors have a sinusoidal response to the sinusoidal line voltage.<sup id="cite_ref-Das_2015_1-0" class="reference"><a href="#cite_note-Das_2015-1">&#91;1&#93;</a></sup> 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. </p><p>In a purely resistive AC circuit, voltage and current waveforms are in step (or <a href="/wiki/Phase_(waves)" title="Phase (waves)">in phase</a>), changing polarity at the same instant in each cycle. All the power entering the load is consumed (or dissipated). </p><p>Where <a href="/wiki/Reactance_(electronics)" class="mw-redirect" title="Reactance (electronics)">reactive</a> loads are present, such as with <a href="/wiki/Capacitor" title="Capacitor">capacitors</a> or <a href="/wiki/Inductor" title="Inductor">inductors</a>, 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 <a href="/wiki/Electric_field" title="Electric field">electric</a> or <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic fields</a> then returned to the power grid a fraction of the period later. </p><p>Electrical circuits containing predominantly resistive loads (incandescent lamps, heating elements) have a power factor of almost 1, but circuits containing inductive or capacitive loads (electric motors, <a href="/wiki/Solenoid" title="Solenoid">solenoid</a> valves, transformers, <a href="/wiki/Electrical_ballast" title="Electrical ballast">fluorescent lamp ballasts</a>, and others) can have a power factor well below 1. </p><p>In the <a href="/wiki/Electric_power_grid" class="mw-redirect" title="Electric power grid">electric power grid</a>, reactive loads cause a continuous <i>ebb and flow</i> of nonproductive power. 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 <a href="/wiki/Joule_heating" title="Joule heating">resistive heating</a> in power lines, and requiring the use of higher-rated conductors and transformers. </p> <h3><span class="mw-headline" id="Definition_and_calculation">Definition and calculation</span></h3> <p><a href="/wiki/AC_power#Instantaneous_power,_instantaneous_active_power_and_instantaneous_reactive_power_in_sinusoidal_steady-state" title="AC power">AC power</a> has two components: </p> <ul><li><a href="/wiki/AC_power#Active_power_in_sinusoidal_steady-state" title="AC power">Real power or active power</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="P"/></span>) (sometimes called average power<sup id="cite_ref-2" class="reference"><a href="#cite_note-2">&#91;2&#93;</a></sup>), expressed in <a href="/wiki/Watt" title="Watt">watts</a> (W)</li> <li><a href="/wiki/AC_power#Reactive_power_in_sinusoidal_steady-state" title="AC power">Reactive power</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="Q"/></span>), usually expressed in <a href="/wiki/Volt-ampere_reactive" class="mw-redirect" title="Volt-ampere reactive">reactive volt-amperes</a> (var)<sup id="cite_ref-3" class="reference"><a href="#cite_note-3">&#91;3&#93;</a></sup></li></ul> <p>Together, they form the <a href="/wiki/AC_power#Complex_power_in_sinusoidal_steady-state" title="AC power">complex power</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="S"/></span>) expressed as <a href="/wiki/Volt-amperes" class="mw-redirect" title="Volt-amperes">volt-amperes</a> (VA). The magnitude of the complex power is the apparent power (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |S|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |S|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28d901e98a035ff4c0e37fe6dd8e750ece6c1f0b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:2.793ex; height:2.843ex;" alt="|S|"/></span>), also expressed in volt-amperes (VA). </p><p>The VA and var are non-SI units mathematically identical to the watt, but are used in engineering practice instead of the watt to state what <a href="/wiki/Physical_quantity" title="Physical quantity">quantity</a> is being expressed. The <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> explicitly disallows using units for this purpose or as the only source of information about a physical quantity as used.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4">&#91;4&#93;</a></sup> </p><p>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.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5">&#91;5&#93;</a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6">&#91;6&#93;</a></sup> </p><p>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,<sup id="cite_ref-SureshKumar_2013_7-0" class="reference"><a href="#cite_note-SureshKumar_2013-7">&#91;7&#93;</a></sup>. </p> <h4><span class="mw-headline" id="Power_triangle">Power triangle</span></h4> <p><a href="/wiki/File:Power_triangle_diagram.jpg" class="image"><img alt="Power triangle diagram.jpg" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Power_triangle_diagram.jpg/370px-Power_triangle_diagram.jpg" decoding="async" width="370" height="146" data-file-width="442" data-file-height="175" /></a> </p><p>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: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S&amp;=P+jQ\\|S|&amp;={\sqrt {P^{2}+Q^{2}}}\\{\text{pf}}&amp;=\cos {\theta }={\frac {P}{|S|}}=\cos {\left(\arctan {\left({\frac {Q}{P}}\right)}\right)}\\Q&amp;=P\,\tan(\arccos({\text{pf}}))\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>S</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>P</mi> <mo>+</mo> <mi>j</mi> <mi>Q</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>pf</mtext> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03B8;<!-- θ --></mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>P</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>cos</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>arctan</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mi>P</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>Q</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mi>tan</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>pf</mtext> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S&amp;=P+jQ\\|S|&amp;={\sqrt {P^{2}+Q^{2}}}\\{\text{pf}}&amp;=\cos {\theta }={\frac {P}{|S|}}=\cos {\left(\arctan {\left({\frac {Q}{P}}\right)}\right)}\\Q&amp;=P\,\tan(\arccos({\text{pf}}))\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37992d4db007c5931353013d024c857c4dcdaafe" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -8.171ex; width:40.926ex; height:17.509ex;" alt="{\displaystyle {\begin{aligned}S&amp;=P+jQ\\|S|&amp;={\sqrt {P^{2}+Q^{2}}}\\{\text{pf}}&amp;=\cos {\theta }={\frac {P}{|S|}}=\cos {\left(\arctan {\left({\frac {Q}{P}}\right)}\right)}\\Q&amp;=P\,\tan(\arccos({\text{pf}}))\end{aligned}}}"/></span></dd></dl> <p>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. </p> <h4><span id="Lagging.2C_leading_and_unity_power_factors"></span><span class="mw-headline" id="Lagging,_leading_and_unity_power_factors">Lagging, leading and unity power factors</span></h4> <p>Power factor is described as <i>leading</i> if the current waveform is advanced in phase with respect to voltage, or <i>lagging</i> when the current waveform is behind the voltage waveform. A lagging power factor signifies that the load is inductive, as the load will <i>consume</i> reactive power. The reactive component <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="Q"/></span> is positive as reactive power travels through the circuit and is <i>consumed</i> by the inductive load. A leading power factor signifies that the load is capacitive, as the load <i>supplies</i> reactive power, and therefore the reactive component <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="Q"/></span> is negative as reactive power is being supplied to the circuit. </p><p><a href="/wiki/File:Lagging-Leading.jpg" class="image"><img alt="Lagging-Leading.jpg" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Lagging-Leading.jpg/590px-Lagging-Leading.jpg" decoding="async" width="590" height="224" data-file-width="737" data-file-height="280" /></a> </p><p>If θ is the <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase angle</a> between the current and voltage, then the power factor is equal to the <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">cosine</a> of the angle, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mi>&#x03B8;<!-- θ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/611e5c70de1d1cf4ebc3b70d2b5467f45d17a483" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:4.589ex; height:2.176ex;" alt="\cos \theta "/></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |P|=|S|\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mi>&#x03B8;<!-- θ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |P|=|S|\cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d68e3a292318e732659831b37364d0bc2bfe6a1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:13.906ex; height:2.843ex;" alt="|P|=|S|\cos \theta "/></span></dd></dl> <p>Since the units are consistent, the power factor is by definition a <a href="/wiki/Dimensionless_number" class="mw-redirect" title="Dimensionless number">dimensionless number</a> between -1 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, referred to as <i>unity</i> power factor, all the energy supplied by the source is consumed by the load. Power factors are usually stated as <i>leading</i> or <i>lagging</i> to show the sign of the phase angle. Capacitive loads are leading (current leads voltage), and inductive loads are lagging (current lags voltage). </p><p>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. </p><p>For example, to get 1&#160;kW of real power, if the power factor is unity, 1&#160;kVA of apparent power needs to be transferred (1&#160;kW ÷ 1 = 1&#160;kVA). At low values of power factor, more apparent power needs to be transferred to get the same real power. To get 1&#160;kW of real power at 0.2 power factor, 5&#160;kVA of apparent power needs to be transferred (1&#160;kW ÷ 0.2 = 5&#160;kVA). This apparent power must be produced and transmitted to the load and is subject to losses in the production and transmission processes. </p><p>Electrical loads consuming <a href="/wiki/AC_power" title="AC power">alternating current power</a> 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. </p><p>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.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8">&#91;8&#93;</a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9">&#91;9&#93;</a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10">&#91;10&#93;</a></sup> </p> <h3><span class="mw-headline" id="Power_factor_correction_of_linear_loads">Power factor correction of linear loads</span></h3> <div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="/wiki/File:Blindleistungskompensation.svg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Blindleistungskompensation.svg/220px-Blindleistungskompensation.svg.png" decoding="async" width="220" height="118" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Blindleistungskompensation.svg/330px-Blindleistungskompensation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Blindleistungskompensation.svg/440px-Blindleistungskompensation.svg.png 2x" data-file-width="1860" data-file-height="1000" /></a> <div class="thumbcaption"><div class="magnify"><a href="/wiki/File:Blindleistungskompensation.svg" class="internal" title="Enlarge"></a></div>Power factor correction of linear load</div></div></div> <p>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 <a href="/wiki/Electric_power_transmission" title="Electric power transmission">electric power transmission</a> 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 so as to reduce costs. </p><p>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 all the way 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. </p><p>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 <a href="/wiki/Overvoltage" title="Overvoltage">overvoltage</a> fluctuations. As such, reactive elements cannot simply be applied without engineering analysis. </p> <div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="/wiki/File:Condensatorenbatterij.jpg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Condensatorenbatterij.jpg/220px-Condensatorenbatterij.jpg" decoding="async" width="220" height="165" class="thumbimage" data-file-width="2272" data-file-height="1704" /></a> <div class="thumbcaption"><div class="magnify"><a href="/wiki/File:Condensatorenbatterij.jpg" class="internal" title="Enlarge"></a></div>1. <a href="/wiki/Static_VAR_compensator" title="Static VAR compensator">Reactive power control relay</a>; 2. Network connection points; 3. <a href="/wiki/Fuse_(electrical)" title="Fuse (electrical)">Slow-blow fuses</a>; 4. Inrush-limiting <a href="/wiki/Contactor" title="Contactor">contactors</a>; 5. <a href="/wiki/Capacitor" title="Capacitor">Capacitors</a> (single-phase or three-phase units, delta-connection); 6. <a href="/wiki/Transformer" title="Transformer">Transformer</a> (for controls and ventilation fans)</div></div></div> <p>An <b>automatic power factor correction unit</b> consists of a number of <a href="/wiki/Capacitor" title="Capacitor">capacitors</a> that are switched by means of <a href="/wiki/Contactor" title="Contactor">contactors</a>. 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. </p><p>In place of a set of switched <a href="/wiki/Capacitor" title="Capacitor">capacitors</a>, an unloaded <a href="/wiki/Synchronous_motor" title="Synchronous motor">synchronous motor</a> can supply reactive power. The <a href="/wiki/Reactive_power" class="mw-redirect" title="Reactive power">reactive power</a> drawn by the synchronous motor is a function of its field excitation. It is referred to as a <b><a href="/wiki/Synchronous_condenser" title="Synchronous condenser">synchronous condenser</a></b>. It is started and connected to the <a href="/wiki/Electrical_network" title="Electrical network">electrical network</a>. It operates at a leading power factor and puts <a href="/wiki/Volt-ampere_reactive" class="mw-redirect" title="Volt-ampere reactive">vars</a> onto the network as required to support a system's <a href="/wiki/Voltage" title="Voltage">voltage</a> or to maintain the system power factor at a specified level. </p><p>The synchronous condenser's installation and operation are identical to those of large <a href="/wiki/Electric_motor" title="Electric motor">electric motors</a>. 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 supplied is proportional to voltage, not the square of voltage; this improves voltage stability on large networks. Synchronous condensers are often used in connection with <a href="/wiki/High-voltage_direct_current" title="High-voltage direct current">high-voltage direct-current</a> transmission projects or in large industrial plants such as <a href="/wiki/Steel_mill" title="Steel mill">steel mills</a>. </p><p>For power factor correction of high-voltage power systems or large, fluctuating industrial loads, power electronic devices such as the <a href="/wiki/Static_VAR_compensator" title="Static VAR compensator">static VAR compensator</a> or <a href="/wiki/STATCOM" class="mw-redirect" title="STATCOM">STATCOM</a> 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. </p> <h2><span class="mw-headline" id="Non-linear_loads">Non-linear loads</span></h2> <p>Examples of non-linear loads on a power system are rectifiers (such as used in a power supply), and arc discharge devices such as <a href="/wiki/Fluorescent_lamp" title="Fluorescent lamp">fluorescent lamps</a>, electric <a href="/wiki/Welding" title="Welding">welding</a> machines, or <a href="/wiki/Arc_furnace" class="mw-redirect" title="Arc furnace">arc furnaces</a>. Because current in these systems is interrupted by a switching action, the current contains frequency components that are multiples of the power system frequency. <i>Distortion power factor</i> is a measure of how much the harmonic distortion of a load current decreases the average power transferred to the load. </p> <div class="thumb tright"><div class="thumbinner" style="width:302px;"><a href="/wiki/File:Power_factor_75_2.png" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Power_factor_75_2.png/300px-Power_factor_75_2.png" decoding="async" width="300" height="270" class="thumbimage" data-file-width="393" data-file-height="354" /></a> <div class="thumbcaption"><div class="magnify"><a href="/wiki/File:Power_factor_75_2.png" class="internal" title="Enlarge"></a></div>Sinusoidal voltage and non-sinusoidal current give a distortion power factor of 0.75 for this computer power supply load.</div></div></div> <h3><span class="mw-headline" id="Non-sinusoidal_components">Non-sinusoidal components</span></h3> <p>In linear circuits having only sinusoidal currents and voltages of one frequency, the power factor arises only from the difference in phase between the current and voltage. This is <i>displacement power factor</i>.<sup id="cite_ref-FuchsMasoum2015_11-0" class="reference"><a href="#cite_note-FuchsMasoum2015-11">&#91;11&#93;</a></sup> </p><p>Non-linear loads change the shape of the current waveform from a <a href="/wiki/Sine_wave" title="Sine wave">sine wave</a> to some other form. Non-linear loads create <a href="/wiki/Harmonic" title="Harmonic">harmonic</a> currents in addition to the original (fundamental frequency) AC current. This is of importance in practical power systems that contain <a href="/wiki/Non-linear" class="mw-redirect" title="Non-linear">non-linear</a> loads such as <a href="/wiki/Rectifiers" class="mw-redirect" title="Rectifiers">rectifiers</a>, some forms of electric lighting, <a href="/wiki/Electric_arc_furnace" title="Electric arc furnace">electric arc furnaces</a>, welding equipment, <a href="/wiki/Switched-mode_power_supply" title="Switched-mode power supply">switched-mode power supplies</a>, variable speed drives and other devices. Filters consisting of linear capacitors and inductors can prevent harmonic currents from entering the supplying system. </p><p>To measure the real power or reactive power, a <a href="/wiki/Wattmeter" title="Wattmeter">wattmeter</a> designed to work properly with non-sinusoidal currents must be used. </p> <h3><span class="mw-headline" id="Distortion_power_factor">Distortion power factor</span></h3> <p>The <b>distortion power factor</b> is the distortion component associated with the harmonic voltages and currents present in the system. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\mbox{distortion power factor}}&amp;={\frac {I_{1}}{I_{rms}}}\\&amp;={\frac {I_{1}}{\sqrt {I_{1}^{2}+I_{2}^{2}+I_{3}^{2}+I_{4}^{2}+\cdots }}}\\&amp;={\frac {1}{\sqrt {1+{\frac {I_{2}^{2}+I_{3}^{2}+I_{4}^{2}+\cdots }{I_{1}^{2}}}}}}\\&amp;={\frac {1}{\sqrt {1+THD_{i}^{2}}}}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>distortion power factor</mtext> </mstyle> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>m</mi> <mi>s</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msqrt> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>&#x22EF;<!-- ⋯ --></mo> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>&#x22EF;<!-- ⋯ --></mo> </mrow> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <mi>T</mi> <mi>H</mi> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mbox{distortion power factor}}&amp;={\frac {I_{1}}{I_{rms}}}\\&amp;={\frac {I_{1}}{\sqrt {I_{1}^{2}+I_{2}^{2}+I_{3}^{2}+I_{4}^{2}+\cdots }}}\\&amp;={\frac {1}{\sqrt {1+{\frac {I_{2}^{2}+I_{3}^{2}+I_{4}^{2}+\cdots }{I_{1}^{2}}}}}}\\&amp;={\frac {1}{\sqrt {1+THD_{i}^{2}}}}\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027a061193041b5732fec25cf4464f07cdd3a48" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -15.171ex; width:53.142ex; height:31.509ex;" alt="{\displaystyle {\begin{aligned}{\mbox{distortion power factor}}&amp;={\frac {I_{1}}{I_{rms}}}\\&amp;={\frac {I_{1}}{\sqrt {I_{1}^{2}+I_{2}^{2}+I_{3}^{2}+I_{4}^{2}+\cdots }}}\\&amp;={\frac {1}{\sqrt {1+{\frac {I_{2}^{2}+I_{3}^{2}+I_{4}^{2}+\cdots }{I_{1}^{2}}}}}}\\&amp;={\frac {1}{\sqrt {1+THD_{i}^{2}}}}\\\end{aligned}}}"/></span></dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{THD}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>THD</mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{THD}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e47a55ca3b41f55160baacd0b8cbe5dab3dfecc9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:5.996ex; height:2.509ex;" alt="{\mbox{THD}}_{i}"/></span> is the <a href="/wiki/Total_harmonic_distortion" title="Total harmonic distortion">total harmonic distortion</a> of the load current. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle THD_{i}={\frac {\sqrt {\displaystyle \sum _{h=2}^{\infty }I_{h}^{2}}}{I_{1}}}={\frac {\sqrt {I_{2}^{2}+I_{3}^{2}+I_{4}^{2}+\cdots }}{I_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>H</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">&#x221E;<!-- ∞ --></mi> </mrow> </munderover> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </msqrt> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>&#x22EF;<!-- ⋯ --></mo> </msqrt> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle THD_{i}={\frac {\sqrt {\displaystyle \sum _{h=2}^{\infty }I_{h}^{2}}}{I_{1}}}={\frac {\sqrt {I_{2}^{2}+I_{3}^{2}+I_{4}^{2}+\cdots }}{I_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac832d1338f9e67b0db47335901a623fa4045c38" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:43.01ex; height:10.676ex;" alt="{\displaystyle THD_{i}={\frac {\sqrt {\displaystyle \sum _{h=2}^{\infty }I_{h}^{2}}}{I_{1}}}={\frac {\sqrt {I_{2}^{2}+I_{3}^{2}+I_{4}^{2}+\cdots }}{I_{1}}}}"/></span></dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03f18d041b2df30adef07164dbf285878893dedc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.077ex; height:2.509ex;" alt="I_{1}"/></span> is the fundamental component of the current and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mbox{rms}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>rms</mtext> </mstyle> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mbox{rms}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d08cb0931d6168f0beb94a26e4ecbebb5be5de" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:5.019ex; height:2.509ex;" alt="I_{\mbox{rms}}"/></span> is the total current – both are <a href="/wiki/Root_mean_square" title="Root mean square">root mean square</a>-values (distortion power factor can also be used to describe individual order harmonics, using the corresponding current in place of total current). This definition with respect to total harmonic distortion assumes that the voltage stays undistorted (sinusoidal, without harmonics). This simplification is often a good approximation for stiff voltage sources (not being affected by changes in load downstream in the distribution network). Total harmonic distortion of typical generators from current distortion in the network is on the order of 1–2%, which can have larger scale implications but can be ignored in common practice.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12">&#91;12&#93;</a></sup> </p><p>The result when multiplied with the displacement power factor (DPF) is the overall, true power factor or just power factor (PF): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{PF}}={\frac {\cos {\varphi }}{\sqrt {1+THD_{i}^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>PF</mtext> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03C6;<!-- φ --></mi> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <mi>T</mi> <mi>H</mi> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{PF}}={\frac {\cos {\varphi }}{\sqrt {1+THD_{i}^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/598c51e116683db75d9e5bb13fdb40b56ec1334b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -4.671ex; width:20.04ex; height:7.676ex;" alt="{\displaystyle {\mbox{PF}}={\frac {\cos {\varphi }}{\sqrt {1+THD_{i}^{2}}}}}"/></span></dd></dl> <h3><span class="mw-headline" id="Distortion_in_three-phase_networks">Distortion in three-phase networks</span></h3> <p>In practice, the local effects of distortion current on devices in a <a href="/wiki/Three-phase_electric_power" title="Three-phase electric power">three-phase distribution network</a> rely on the magnitude of certain order harmonics rather than the total harmonic distortion. </p><p>For example, the triplen, or zero-sequence, harmonics (3rd, 9th, 15th, etc.) have the property of being in-phase when compared line-to-line. In a <a href="/wiki/Delta-wye_transformer" title="Delta-wye transformer">delta-wye transformer</a>, these harmonics can result in circulating currents in the delta windings and result in greater <a href="/wiki/Joule_heating" title="Joule heating">resistive heating</a>. In a wye-configuration of a transformer, triplen harmonics will not create these currents, but they will result in a non-zero current in the <a href="/wiki/Ground_and_neutral" title="Ground and neutral">neutral wire</a>. This could overload the neutral wire in some cases and create error in kilowatt-hour metering systems and billing revenue.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13">&#91;13&#93;</a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14">&#91;14&#93;</a></sup> The presence of current harmonics in a transformer also result in larger <a href="/wiki/Eddy_currents" class="mw-redirect" title="Eddy currents">eddy currents</a> in the magnetic core of the transformer. Eddy current losses generally increase as the square of the frequency, lowering the transformer's efficiency, dissipating additional heat, and reducing its service life.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15">&#91;15&#93;</a></sup> </p><p>Negative-sequence harmonics (5th, 11th, 17th, etc.) combine 120 degrees out of phase, similarly to the fundamental harmonic but in a reversed sequence. In generators and motors, these currents produce magnetic fields which oppose the rotation of the shaft and sometimes result in damaging mechanical vibrations.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16">&#91;16&#93;</a></sup> </p> <h3><span class="mw-headline" id="Switched-mode_power_supplies">Switched-mode power supplies</span></h3> <style data-mw-deduplicate="TemplateStyles:r1033289096">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Switched-mode_power_supply#Power_factor" title="Switched-mode power supply">switched-mode power supply §&#160;Power factor</a></div> <p>A particularly important class of non-linear loads is the millions of personal computers that typically incorporate <a href="/wiki/Switched-mode_power_supply" title="Switched-mode power supply">switched-mode power supplies</a> (SMPS) with rated output power ranging from a few watts to more than 1&#160;kW. Historically, these very-low-cost power supplies incorporated a simple full-wave rectifier that conducted only when the <a href="/wiki/Mains_electricity" title="Mains electricity">mains</a> instantaneous voltage exceeded the voltage on the input capacitors. This leads to very high <a href="/wiki/Peak-to-average_ratio" class="mw-redirect" title="Peak-to-average ratio">ratios of peak-to-average</a> input current, which also lead to a low distortion power factor and potentially serious phase and neutral loading concerns. </p><p>A typical switched-mode power supply first converts the AC mains to a DC bus by means of a <a href="/wiki/Bridge_rectifier" class="mw-redirect" title="Bridge rectifier">bridge rectifier</a>. The output voltage is then derived from this DC bus. The problem with this is that the <a href="/wiki/Rectifier" title="Rectifier">rectifier</a> is a non-linear device, so the input current is highly non-linear. That means that the input current has energy at <a href="/wiki/Harmonic" title="Harmonic">harmonics</a> of the frequency of the voltage. This presents a problem for power companies, because they cannot compensate for the harmonic current by adding simple capacitors or inductors, as they could for the reactive power drawn by a linear load. Many jurisdictions are beginning to require power factor correction for all power supplies above a certain power level. </p><p>Regulatory agencies such as the <a href="/wiki/European_Union" title="European Union">EU</a> have set harmonic limits as a method of improving power factor. Declining component cost has hastened implementation of two different methods. To comply with current EU standard EN61000-3-2, all switched-mode power supplies with output power more than 75&#160;W must at least include passive power factor correction. <a href="/wiki/80_Plus" title="80 Plus">80 Plus</a> power supply certification requires a power factor of 0.9 or more.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17">&#91;17&#93;</a></sup> </p> <h3><span id="Power_factor_correction_.28PFC.29_in_non-linear_loads"></span><span class="mw-headline" id="Power_factor_correction_(PFC)_in_non-linear_loads">Power factor correction (PFC) in non-linear loads</span></h3> <h4><span class="mw-headline" id="Passive_PFC">Passive PFC</span></h4> <p>The simplest way to control the <a href="/wiki/Harmonics_(electrical_power)" title="Harmonics (electrical power)">harmonic</a> current is to use a <a href="/wiki/Electronic_filter" title="Electronic filter">filter</a> that passes current only at <a href="/wiki/Utility_frequency" title="Utility frequency">line frequency</a> (50 or 60&#160;Hz). The filter consists of capacitors or inductors and makes a non-linear device look more like a <a href="/wiki/Linear" class="mw-redirect" title="Linear">linear</a> load. An example of passive PFC is a <a href="/wiki/Valley-fill_circuit" title="Valley-fill circuit">valley-fill circuit</a>. </p><p>A disadvantage of passive PFC is that it requires larger inductors or capacitors than an equivalent power active PFC circuit.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18">&#91;18&#93;</a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19">&#91;19&#93;</a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20">&#91;20&#93;</a></sup> Also, in practice, passive PFC is often less effective at improving the power factor.<sup id="cite_ref-effi_21-0" class="reference"><a href="#cite_note-effi-21">&#91;21&#93;</a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22">&#91;22&#93;</a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23">&#91;23&#93;</a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24">&#91;24&#93;</a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25">&#91;25&#93;</a></sup> </p> <h4><span class="mw-headline" id="Active_PFC">Active PFC</span></h4> <div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="/wiki/File:Active_pfc_PSU_packaging.png" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Active_pfc_PSU_packaging.png/220px-Active_pfc_PSU_packaging.png" decoding="async" width="220" height="131" class="thumbimage" data-file-width="283" data-file-height="169" /></a> <div class="thumbcaption"><div class="magnify"><a href="/wiki/File:Active_pfc_PSU_packaging.png" class="internal" title="Enlarge"></a></div>Specifications taken from the packaging of a 610 W <a href="/wiki/Power_supply_unit_(computer)" title="Power supply unit (computer)">PC power supply</a> showing active PFC rating</div></div></div> <p>Active PFC is the use of <a href="/wiki/Power_electronics" title="Power electronics">power electronics</a> to change the waveform of current drawn by a load to improve the power factor.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26">&#91;26&#93;</a></sup> Some types of the active PFC are <a href="/wiki/Buck_converter" title="Buck converter">buck</a>, <a href="/wiki/Boost_converter" title="Boost converter">boost</a>, <a href="/wiki/Buck-boost_converter" class="mw-redirect" title="Buck-boost converter">buck-boost</a> and <a href="/wiki/Synchronous_condenser" title="Synchronous condenser">synchronous condenser</a>. Active power factor correction can be single-stage or multi-stage. </p><p>In the case of a switched-mode power supply, a <a href="/wiki/Boost_converter" title="Boost converter">boost converter</a> is inserted between the bridge rectifier and the main input capacitors. The boost converter attempts to maintain a constant voltage at its output while drawing a current that is always in phase with and at the same frequency as the line voltage. Another switched-mode converter inside the power supply produces the desired output voltage from the DC bus. This approach requires additional semiconductor switches and control electronics but permits cheaper and smaller passive components. It is frequently used in practice. </p><p>For a three-phase SMPS, the <a href="/wiki/Vienna_rectifier" title="Vienna rectifier">Vienna rectifier</a> configuration may be used to substantially improve the power factor. </p><p><a href="/wiki/Switched-mode_power_supply" title="Switched-mode power supply">SMPSs</a> with passive PFC can achieve power factor of about 0.7–0.75, SMPSs with active PFC, up to 0.99 power factor, while a SMPS without any power factor correction have a power factor of only about 0.55–0.65.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27">&#91;27&#93;</a></sup> </p><p>Due to their very wide input voltage range, many power supplies with active PFC can automatically adjust to operate on AC power from about 100&#160;V (Japan) to 240&#160;V (Europe). That feature is particularly welcome in power supplies for laptops. </p> <h4><span class="mw-headline" id="Dynamic_PFC">Dynamic PFC</span></h4> <p>Dynamic power factor correction (DPFC), sometimes referred to as real-time power factor correction, is used for electrical stabilization in cases of rapid load changes (e.g. at large manufacturing sites). DPFC is useful when standard power factor correction would cause over or under correction.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28">&#91;28&#93;</a></sup> DPFC uses semiconductor switches, typically <a href="/wiki/Thyristor" title="Thyristor">thyristors</a>, to quickly connect and disconnect capacitors or inductors to improve power factor. </p> <h2><span class="mw-headline" id="Importance_in_distribution_systems">Importance in distribution systems</span></h2> <div class="thumb tright"><div class="thumbinner" style="width:172px;"><a href="/wiki/File:Condensor_bank_150kV_-_75MVAR.jpg" class="image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Condensor_bank_150kV_-_75MVAR.jpg/170px-Condensor_bank_150kV_-_75MVAR.jpg" decoding="async" width="170" height="254" class="thumbimage" data-file-width="926" data-file-height="1382" /></a> <div class="thumbcaption"><div class="magnify"><a href="/wiki/File:Condensor_bank_150kV_-_75MVAR.jpg" class="internal" title="Enlarge"></a></div>75 MVAr capacitor bank in a 150 kV substation</div></div></div> <p>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. For example, if the load power factor were as low as 0.7, the apparent power would be 1.4 times the real power used by the load. Line current in the circuit would also be 1.4 times the current required at 1.0 power factor, so the losses in the circuit would be doubled (since they are proportional to the square of the current). Alternatively, all components of the system such as generators, conductors, transformers, and switchgear would be increased in size (and cost) to carry the extra current. When the power factor is close to unity, for the same kVA rating of the transformer more load current can be supplied.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29">&#91;29&#93;</a></sup> </p><p>Utilities typically charge additional costs to commercial customers who have a power factor below some limit, which is typically 0.9 to 0.95. Engineers are often interested in the power factor of a load as one of the factors that affect the efficiency of power transmission. </p><p>With the rising cost of energy and concerns over the efficient delivery of power, active PFC has become more common in consumer electronics.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30">&#91;30&#93;</a></sup> Current <a href="/wiki/Energy_Star" title="Energy Star">Energy Star</a> guidelines for computers<sup id="cite_ref-31" class="reference"><a href="#cite_note-31">&#91;31&#93;</a></sup> call for a power factor of ≥ 0.9 at 100% of rated output in the <a href="/wiki/Power_supply_unit_(computer)" title="Power supply unit (computer)">PC's power supply</a>. According to a white paper authored by Intel and the <a href="/wiki/United_States_Environmental_Protection_Agency" title="United States Environmental Protection Agency">U.S. Environmental Protection Agency</a>, PCs with internal power supplies will require the use of active power factor correction to meet the ENERGY STAR 5.0 Program Requirements for Computers.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32">&#91;32&#93;</a></sup> </p><p>In Europe, <a href="/wiki/IEC_EN_61000-3-2" class="mw-redirect" title="IEC EN 61000-3-2">EN 61000-3-2</a> requires power factor correction be incorporated into consumer products. </p><p>Small customers, such as households, are not usually charged for reactive power and so power factor metering equipment for such customers will not be installed. </p> <h2><span class="mw-headline" id="Measurement_techniques">Measurement techniques</span></h2> <p>The power factor in a single-phase circuit (or balanced three-phase circuit) can be measured with the wattmeter-ammeter-voltmeter method, where the power in watts is divided by the product of measured voltage and current. The power factor of a balanced <a href="/wiki/Polyphase_system" title="Polyphase system">polyphase circuit</a> is the same as that of any phase. The power factor of an unbalanced polyphase circuit is not uniquely defined. </p><p>A direct reading power factor meter can be made with a <a href="/wiki/Moving_coil_meter" class="mw-redirect" title="Moving coil meter">moving coil meter</a> of the electrodynamic type, carrying two perpendicular coils on the moving part of the instrument. The field of the instrument is energized by the circuit current flow. The two moving coils, A and B, are connected in parallel with the circuit load. One coil, A, will be connected through a resistor and the second coil, B, through an inductor, so that the current in coil B is delayed with respect to current in A. At unity power factor, the current in A is in phase with the circuit current, and coil A provides maximum torque, driving the instrument pointer toward the 1.0 mark on the scale. At zero power factor, the current in coil B is in phase with circuit current, and coil B provides torque to drive the pointer towards 0. At intermediate values of power factor, the torques provided by the two coils add and the pointer takes up intermediate positions.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33">&#91;33&#93;</a></sup> </p><p>Another electromechanical instrument is the polarized-vane type.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34">&#91;34&#93;</a></sup> In this instrument a stationary field coil produces a rotating magnetic field, just like a polyphase motor. The field coils are connected either directly to polyphase voltage sources or to a phase-shifting reactor if a single-phase application. A second stationary field coil, perpendicular to the voltage coils, carries a current proportional to current in one phase of the circuit. The moving system of the instrument consists of two vanes that are magnetized by the current coil. In operation, the moving vanes take up a physical angle equivalent to the electrical angle between the voltage source and the current source. This type of instrument can be made to register for currents in both directions, giving a four-quadrant display of power factor or phase angle. </p><p>Digital instruments exist that directly measure the time lag between voltage and current waveforms. Low-cost instruments of this type measure the peak of the waveforms. More sophisticated versions measure the peak of the fundamental harmonic only, thus giving a more accurate reading for phase angle on distorted waveforms. Calculating power factor from voltage and current phases is only accurate if both waveforms are sinusoidal.<sup id="cite_ref-ni_white_paper_35-0" class="reference"><a href="#cite_note-ni_white_paper-35">&#91;35&#93;</a></sup> </p><p>Power Quality Analyzers, often referred to as Power Analyzers, make a digital recording of the voltage and current waveform (typically either one phase or three phase) and accurately calculate true power (watts), apparent power (VA) power factor, AC voltage, AC current, DC voltage, DC current, frequency, IEC61000-3-2/3-12 Harmonic measurement, IEC61000-3-3/3-11 flicker measurement, individual phase voltages in delta applications where there is no neutral line, total harmonic distortion, phase and amplitude of individual voltage or current harmonics, etc.<sup id="cite_ref-Yokogawa_WT3000E_36-0" class="reference"><a href="#cite_note-Yokogawa_WT3000E-36">&#91;36&#93;</a></sup><sup id="cite_ref-Fluke_1760_37-0" class="reference"><a href="#cite_note-Fluke_1760-37">&#91;37&#93;</a></sup> </p> <h2><span class="mw-headline" id="Mnemonics">Mnemonics</span></h2> <p>English-language power engineering students are advised to remember: <i>ELI the ICE man</i> or <i>ELI on ICE</i> – the voltage E, leads the current I, in an inductor L. The current I leads the voltage E in a capacitor C. </p><p>Another common mnemonic is CIVIL – in a capacitor (C) the current (I) leads voltage (V), voltage (V) leads current (I) in an inductor (L). </p> <h2><span class="mw-headline" id="References">References</span></h2> <style data-mw-deduplicate="TemplateStyles:r1011085734">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Das_2015-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Das_2015_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1067248974">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFDas2015" class="citation book cs1">Das, J. C. (2015). <i>Power System Harmonics and Passive Filter Design</i>. Wiley, IEEE Press. p.&#160;2. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-1-118-86162-2" title="Special:BookSources/978-1-118-86162-2"><bdi>978-1-118-86162-2</bdi></a>. <q>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.</q></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Power+System+Harmonics+and+Passive+Filter+Design&amp;rft.pages=2&amp;rft.pub=Wiley%2C+IEEE+Press&amp;rft.date=2015&amp;rft.isbn=978-1-118-86162-2&amp;rft.aulast=Das&amp;rft.aufirst=J.+C.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFBoylestad2002" class="citation book cs1">Boylestad, Robert (2002-03-04). <i>Introductory Circuit Analysis</i> (10th&#160;ed.). p.&#160;857. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-13-097417-4" title="Special:BookSources/978-0-13-097417-4"><bdi>978-0-13-097417-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Introductory+Circuit+Analysis&amp;rft.pages=857&amp;rft.edition=10th&amp;rft.date=2002-03-04&amp;rft.isbn=978-0-13-097417-4&amp;rft.aulast=Boylestad&amp;rft.aufirst=Robert&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20071211234311/http://www.iec.ch/zone/si/si_elecmag.htm#si_epo">"SI Units – Electricity and Magnetism"</a>. <a href="/wiki/Switzerland" title="Switzerland">CH</a>: International Electrotechnical Commission. Archived from <a rel="nofollow" class="external text" href="http://www.iec.ch/zone/si/si_elecmag.htm">the original</a> on 2007-12-11<span class="reference-accessdate">. Retrieved <span class="nowrap">14 June</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=unknown&amp;rft.btitle=SI+Units+%E2%80%93+Electricity+and+Magnetism&amp;rft.place=CH&amp;rft.pub=International+Electrotechnical+Commission&amp;rft_id=http%3A%2F%2Fwww.iec.ch%2Fzone%2Fsi%2Fsi_elecmag.htm&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf"><i>The International System of Units (SI) &#91;SI brochure&#93;</i></a> <span class="cs1-format">(PDF)</span>. §&#160;5.3.2 (p.&#160;132, 40 in the <a href="/wiki/PDF" title="PDF">PDF</a> file): <a href="/wiki/BIPM" class="mw-redirect" title="BIPM">BIPM</a>. 2006. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=The+International+System+of+Units+%28SI%29+%5BSI+brochure%5D&amp;rft.place=%C2%A7+5.3.2+%28p.+132%2C+40+in+the+PDF+file%29&amp;rft.pub=BIPM&amp;rft.date=2006&amp;rft_id=http%3A%2F%2Fwww.bipm.org%2Futils%2Fcommon%2Fpdf%2Fsi_brochure_8_en.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location (<a href="/wiki/Category:CS1_maint:_location" title="Category:CS1 maint: location">link</a>)</span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite class="citation cs2"><i>Authoritative Dictionary of Standards Terms</i> (7th&#160;ed.), <a href="/wiki/Institute_of_Electrical_and_Electronics_Engineers" title="Institute of Electrical and Electronics Engineers">IEEE</a>, 2000, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-7381-2601-2" title="Special:BookSources/978-0-7381-2601-2"><bdi>978-0-7381-2601-2</bdi></a>, Std. 100</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Authoritative+Dictionary+of+Standards+Terms&amp;rft.edition=7th&amp;rft.pub=IEEE&amp;rft.date=2000&amp;rft.isbn=978-0-7381-2601-2&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite class="citation cs2"><i>Trial-Use Standard Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions</i>, IEEE, 2000, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-7381-1963-2" title="Special:BookSources/978-0-7381-1963-2"><bdi>978-0-7381-1963-2</bdi></a>, Std. 1459–2000</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Trial-Use+Standard+Definitions+for+the+Measurement+of+Electric+Power+Quantities+Under+Sinusoidal%2C+Nonsinusoidal%2C+Balanced%2C+or+Unbalanced+Conditions&amp;rft.pub=IEEE&amp;rft.date=2000&amp;rft.isbn=978-0-7381-1963-2&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span>. 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 <a rel="nofollow" class="external free" href="http://powerstandards.com/Shymanski/draft.pdf">http://powerstandards.com/Shymanski/draft.pdf</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304071333/http://powerstandards.com/Shymanski/draft.pdf">Archived</a> 2016-03-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-SureshKumar_2013-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-SureshKumar_2013_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFSuresh_Kumar2013" class="citation book cs1">Suresh Kumar, K. S. (2013). <i>Electric Circuit Analysis</i>. Pearson. p.&#160;8.10. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-8-13-179155-4" title="Special:BookSources/978-8-13-179155-4"><bdi>978-8-13-179155-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Electric+Circuit+Analysis&amp;rft.pages=8.10&amp;rft.pub=Pearson&amp;rft.date=2013&amp;rft.isbn=978-8-13-179155-4&amp;rft.aulast=Suresh+Kumar&amp;rft.aufirst=K.+S.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFDuddell1901" class="citation cs2">Duddell, W. (1901), "On the resistance and electromotive forces of the electric arc", <i>Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences</i>, <b>203</b> (359–371): 512–15, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frsta.1904.0022">10.1098/rsta.1904.0022</a></span>, <q>The fact that the solid arc has, at low frequencies, a negative power factor, indicates that the arc is supplying power to the alternator…</q></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+A%3A+Mathematical%2C+Physical+and+Engineering+Sciences&amp;rft.atitle=On+the+resistance+and+electromotive+forces+of+the+electric+arc&amp;rft.volume=203&amp;rft.issue=359%E2%80%93371&amp;rft.pages=512-15&amp;rft.date=1901&amp;rft_id=info%3Adoi%2F10.1098%2Frsta.1904.0022&amp;rft.aulast=Duddell&amp;rft.aufirst=W.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFZhang2006" class="citation cs2">Zhang, S. (July 2006), "Analysis of some measurement issues in bushing power factor tests in the field", <i>IEEE Transactions on Power Delivery</i>, <b>21</b> (3): 1350–56, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftpwrd.2006.874616">10.1109/tpwrd.2006.874616</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:39895367">39895367</a>, <q>…(the measurement) gives both negative power factor and negative resistive current (power loss)</q></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=IEEE+Transactions+on+Power+Delivery&amp;rft.atitle=Analysis+of+some+measurement+issues+in+bushing+power+factor+tests+in+the+field&amp;rft.volume=21&amp;rft.issue=3&amp;rft.pages=1350-56&amp;rft.date=2006-07&amp;rft_id=info%3Adoi%2F10.1109%2Ftpwrd.2006.874616&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A39895367%23id-name%3DS2CID&amp;rft.aulast=Zhang&amp;rft.aufirst=S.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFAlmarshoud2004" class="citation cs2">Almarshoud, A. F.; et&#160;al. (2004), "Performance of Grid-Connected Induction Generator under Naturally Commutated AC Voltage Controller", <i>Electric Power Components and Systems</i>, <b>32</b> (7): 691–700, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F15325000490461064">10.1080/15325000490461064</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:110279940">110279940</a>, <q>Accordingly, the generator will consume active power from the grid, which leads to negative power factor.</q></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Electric+Power+Components+and+Systems&amp;rft.atitle=Performance+of+Grid-Connected+Induction+Generator+under+Naturally+Commutated+AC+Voltage+Controller&amp;rft.volume=32&amp;rft.issue=7&amp;rft.pages=691-700&amp;rft.date=2004&amp;rft_id=info%3Adoi%2F10.1080%2F15325000490461064&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A110279940%23id-name%3DS2CID&amp;rft.aulast=Almarshoud&amp;rft.aufirst=A.+F.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-FuchsMasoum2015-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FuchsMasoum2015_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFEwald_FuchsMohammad_A._S._Masoum2015" class="citation book cs1">Ewald Fuchs; Mohammad A. S. Masoum (14 July 2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wuGcBAAAQBAJ&amp;pg=PA432"><i>Power Quality in Power Systems and Electrical Machines</i></a>. Elsevier Science. pp.&#160;432–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-12-800988-8" title="Special:BookSources/978-0-12-800988-8"><bdi>978-0-12-800988-8</bdi></a>. <q>The DPF it the cosine of the angle between these two quantities</q></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Power+Quality+in+Power+Systems+and+Electrical+Machines&amp;rft.pages=432-&amp;rft.pub=Elsevier+Science&amp;rft.date=2015-07-14&amp;rft.isbn=978-0-12-800988-8&amp;rft.au=Ewald+Fuchs&amp;rft.au=Mohammad+A.+S.+Masoum&amp;rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DwuGcBAAAQBAJ%26pg%3DPA432&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFSankaran1999" class="citation cs2">Sankaran, C. (1999), <a rel="nofollow" class="external text" href="http://ecmweb.com/power-quality/effects-harmonics-power-systems"><i>Effects of Harmonics on Power Systems</i></a>, Electro-Test, <q>...and voltage-time relationship deviates from the pure sine function. The distortion at the point of generation is very small (about 1% to 2%), but nonetheless it exists.</q></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Effects+of+Harmonics+on+Power+Systems&amp;rft.pub=Electro-Test&amp;rft.date=1999&amp;rft.aulast=Sankaran&amp;rft.aufirst=C.&amp;rft_id=http%3A%2F%2Fecmweb.com%2Fpower-quality%2Feffects-harmonics-power-systems&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://www.pge.com/includes/docs/pdfs/mybusiness/customerservice/energystatus/powerquality/harmonics.pdf">"Single-phase load harmonics vs. three-phase load harmonics"</a> <span class="cs1-format">(<a href="/wiki/PDF" title="PDF">PDF</a>)</span>, <i>Power System Harmonics</i>, Pacific Gas and Electric</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.atitle=Single-phase+load+harmonics+vs.+three-phase+load+harmonics&amp;rft.btitle=Power+System+Harmonics&amp;rft.pub=Pacific+Gas+and+Electric&amp;rft_id=http%3A%2F%2Fwww.pge.com%2Fincludes%2Fdocs%2Fpdfs%2Fmybusiness%2Fcustomerservice%2Fenergystatus%2Fpowerquality%2Fharmonics.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://energylogix.ca/harmonics_and_ieee.pdf">"Harmonic Effects"</a> <span class="cs1-format">(<a href="/wiki/PDF" title="PDF">PDF</a>)</span>, <i>Harmonics and IEEE 519</i>, <a href="/wiki/Canada" title="Canada">CA</a>: EnergyLogix Solutions</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.atitle=Harmonic+Effects&amp;rft.btitle=Harmonics+and+IEEE+519&amp;rft.place=CA&amp;rft.pub=EnergyLogix+Solutions&amp;rft_id=http%3A%2F%2Fenergylogix.ca%2Fharmonics_and_ieee.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFSankaran1999" class="citation cs2">Sankaran, C. (1999), "Transformers", <a rel="nofollow" class="external text" href="http://ecmweb.com/power-quality/effects-harmonics-power-systems"><i>Effects of Harmonics on Power Systems</i></a>, Electro-Test</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.atitle=Transformers&amp;rft.btitle=Effects+of+Harmonics+on+Power+Systems&amp;rft.pub=Electro-Test&amp;rft.date=1999&amp;rft.aulast=Sankaran&amp;rft.aufirst=C.&amp;rft_id=http%3A%2F%2Fecmweb.com%2Fpower-quality%2Feffects-harmonics-power-systems&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFSankaran1999" class="citation cs2">Sankaran, C. (1999), "Motors", <a rel="nofollow" class="external text" href="http://ecmweb.com/power-quality/effects-harmonics-power-systems"><i>Effects of Harmonics on Power Systems</i></a>, Electro-Test, <q>The interaction between the positive and negative sequence magnetic fields and currents produces torsional oscillations of the motor shaft. These oscillations result in shaft vibrations.</q></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.atitle=Motors&amp;rft.btitle=Effects+of+Harmonics+on+Power+Systems&amp;rft.pub=Electro-Test&amp;rft.date=1999&amp;rft.aulast=Sankaran&amp;rft.aufirst=C.&amp;rft_id=http%3A%2F%2Fecmweb.com%2Fpower-quality%2Feffects-harmonics-power-systems&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite class="citation cs2">"What is an 80 PLUS certified power supply?", <a rel="nofollow" class="external text" href="http://www.80plus.org/"><i>Certified Power Supplies and Manufacturers</i></a>, 80 Plus</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.atitle=What+is+an+80+PLUS+certified+power+supply%3F&amp;rft.btitle=Certified+Power+Supplies+and+Manufacturers&amp;rft.pub=80+Plus&amp;rft_id=http%3A%2F%2Fwww.80plus.org%2F&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFSchramm2006" class="citation cs2">Schramm, Ben (Fall 2006), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070309134617/http://www.nuvation.com/corporate/news/newsletter/fall2006/powersupply.html">"Power Supply Design Principles: Techniques and Solutions, Part 3"</a>, <i>Newsletter</i>, Nuvation, archived from <a rel="nofollow" class="external text" href="http://www.nuvation.com/corporate/news/newsletter/fall2006/powersupply.html">the original</a> on 2007-03-09</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Newsletter&amp;rft.atitle=Power+Supply+Design+Principles%3A+Techniques+and+Solutions%2C+Part+3&amp;rft.ssn=fall&amp;rft.date=2006&amp;rft.aulast=Schramm&amp;rft.aufirst=Ben&amp;rft_id=http%3A%2F%2Fwww.nuvation.com%2Fcorporate%2Fnews%2Fnewsletter%2Ffall2006%2Fpowersupply.html&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFWolfleHurley2003" class="citation cs2">Wolfle, W.H.; Hurley, W.G. (2003), "Quasi-active power factor correction with a variable inductive filter: theory, design and practice", <i>Xplore</i>, IEEE, vol.&#160;18, no.&#160;1, pp.&#160;248–255, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003ITPE...18..248W">2003ITPE...18..248W</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTPEL.2002.807135">10.1109/TPEL.2002.807135</a></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Xplore&amp;rft.atitle=Quasi-active+power+factor+correction+with+a+variable+inductive+filter%3A+theory%2C+design+and+practice&amp;rft.volume=18&amp;rft.issue=1&amp;rft.pages=248-255&amp;rft.date=2003&amp;rft_id=info%3Adoi%2F10.1109%2FTPEL.2002.807135&amp;rft_id=info%3Abibcode%2F2003ITPE...18..248W&amp;rft.aulast=Wolfle&amp;rft.aufirst=W.H.&amp;rft.au=Hurley%2C+W.G.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFWölfleHurley" class="citation cs2">Wölfle, W. H.; Hurley, W. G., "Quasi-active Power Factor Correction: The Role of Variable Inductance", <a rel="nofollow" class="external text" href="http://www.nuigalway.ie/power_electronics/projects/quasi_active.html"><i>Power electronics</i></a> (project), <a href="/wiki/Ireland" title="Ireland">IE</a>: Nuigalway</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.atitle=Quasi-active+Power+Factor+Correction%3A+The+Role+of+Variable+Inductance&amp;rft.btitle=Power+electronics&amp;rft.place=IE&amp;rft.pub=Nuigalway&amp;rft.aulast=W%C3%B6lfle&amp;rft.aufirst=W.+H.&amp;rft.au=Hurley%2C+W.+G.&amp;rft_id=http%3A%2F%2Fwww.nuigalway.ie%2Fpower_electronics%2Fprojects%2Fquasi_active.html&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-effi-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-effi_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20081120040707/http://www.xbitlabs.com/articles/coolers/display/atx-psu5_3.html"><i>ATX Power Supply Units Roundup</i></a>, xBit labs, archived from <a rel="nofollow" class="external text" href="http://www.xbitlabs.com/articles/coolers/display/atx-psu5_3.html">the original</a> on 2008-11-20, <q>The power factor is the measure of reactive power. It is the ratio of active power to the total of active and reactive power. It is about 0.65 with an ordinary PSU, but PSUs with active PFC have a power factor of 0.97–0.99. […] hardware reviewers sometimes make no difference between the power factor and the efficiency factor. Although both these terms describe the effectiveness of a power supply, it is a gross mistake to confuse them. […] There is a very small effect from passive PFC – the power factor grows only from 0.65 to 0.7–0.75.</q></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=ATX+Power+Supply+Units+Roundup&amp;rft.pub=xBit+labs&amp;rft_id=http%3A%2F%2Fwww.xbitlabs.com%2Farticles%2Fcoolers%2Fdisplay%2Fatx-psu5_3.html&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20090901140721/http://findarticles.com/p/articles/mi_m0EIN/is_2006_March_16/ai_n26797888/"><i>The Active PFC Market is Expected to Grow at an Annually Rate of 12.3% Till 2011</i></a>, Find articles, Mar 16, 2006, archived from <a rel="nofollow" class="external text" href="http://findarticles.com/p/articles/mi_m0EIN/is_2006_March_16/ai_n26797888">the original</a> on September 1, 2009, <q>Higher-powered products are also likely to use active PFC, since it would be the most cost effective way to bring products into compliance with the EN standard.</q></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=The+Active+PFC+Market+is+Expected+to+Grow+at+an+Annually+Rate+of+12.3%25+Till+2011&amp;rft.pub=Find+articles&amp;rft.date=2006-03-16&amp;rft_id=http%3A%2F%2Ffindarticles.com%2Fp%2Farticles%2Fmi_m0EIN%2Fis_2006_March_16%2Fai_n26797888&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://www.techarp.com/showarticle.aspx?artno=81&amp;pgno=1"><i>Power Factor Correction</i></a>, TECHarp, <q>Passive PFC […] the power factor is low at 60–80%. […] Active PFC ... a power factor of up to 95%</q></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Power+Factor+Correction&amp;rft.pub=TECHarp&amp;rft_id=http%3A%2F%2Fwww.techarp.com%2Fshowarticle.aspx%3Fartno%3D81%26pgno%3D1&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20081222085515/http://www.silverstonetek.com/tech/wh_pfc.php?area="><i>Why we need PFC in PSU</i></a>, Silverstone Technology, archived from <a rel="nofollow" class="external text" href="http://www.silverstonetek.com/tech/wh_pfc.php?area=">the original</a> on 2008-12-22, <q>Normally, the power factor value of electronic device without power factor correction is approximately 0.5. […] Passive PFC […] 70~80% […] Active PFC […] 90~99.9%</q></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Why+we+need+PFC+in+PSU&amp;rft.pub=Silverstone+Technology&amp;rft_id=http%3A%2F%2Fwww.silverstonetek.com%2Ftech%2Fwh_pfc.php%3Farea%3D&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFBrooks2004" class="citation cs2">Brooks, Tom (Mar 2004), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081202100831/http://www2.electronicproducts.com/PFC_options_for_power_supplies-article-taiyo-mar2004-html.aspx">"PFC options for power supplies"</a>, <i>Taiyo</i>, Electronic products, archived from <a rel="nofollow" class="external text" href="http://www2.electronicproducts.com/PFC_options_for_power_supplies-article-taiyo-mar2004-html.aspx">the original</a> on 2008-12-02, <q>The disadvantages of passive PFC techniques are that they typically yield a power factor of only 0.60 to 0.70 […] Dual-stage active PFC technology [yields] a power factor typically greater than 0.98</q></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Taiyo&amp;rft.atitle=PFC+options+for+power+supplies&amp;rft.date=2004-03&amp;rft.aulast=Brooks&amp;rft.aufirst=Tom&amp;rft_id=http%3A%2F%2Fwww2.electronicproducts.com%2FPFC_options_for_power_supplies-article-taiyo-mar2004-html.aspx&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140611063712/http://www.fairchildsemi.com/an/AN/AN-42047.pdf"><i>Power Factor Correction (PFC) Basics</i></a> <span class="cs1-format">(PDF)</span> (application note), Fairchild Semiconductor, 2004, archived from <a rel="nofollow" class="external text" href="http://www.fairchildsemi.com/an/AN/AN-42047.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2014-06-11<span class="reference-accessdate">, retrieved <span class="nowrap">2009-11-29</span></span></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Power+Factor+Correction+%28PFC%29+Basics&amp;rft.pub=Fairchild+Semiconductor&amp;rft.date=2004&amp;rft_id=http%3A%2F%2Fwww.fairchildsemi.com%2Fan%2FAN%2FAN-42047.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite id="CITEREFSugawaraSuzukiTakeuchiTeshima1997" class="citation cs2">Sugawara, I.; Suzuki, Y.; Takeuchi, A.; Teshima, T. 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Retrieved <span class="nowrap">6 November</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=unknown&amp;rft.jtitle=Yokogawa+Corporation&amp;rft.atitle=WT3000E+Series+Precision+Power+Analyzers&amp;rft_id=http%3A%2F%2Fwww.yokogawa.co.jp%2Fftp%2Fdist%2Fks%2Fcatalog%2Fen%2FBUWT3000E-01EN_020.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> <li id="cite_note-Fluke_1760-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-Fluke_1760_37-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://cdn.testequity.com/documents/pdf/1760-ds.pdf">"Fluke 1760 Three-Phase Power Quality Recorder"</a> <span class="cs1-format">(PDF)</span>. <i>Fluke Corporation</i>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/https://cdn.testequity.com/documents/pdf/1760-ds.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09<span class="reference-accessdate">. Retrieved <span class="nowrap">6 November</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=unknown&amp;rft.jtitle=Fluke+Corporation&amp;rft.atitle=Fluke+1760+Three-Phase+Power+Quality+Recorder&amp;rft_id=https%3A%2F%2Fcdn.testequity.com%2Fdocuments%2Fpdf%2F1760-ds.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span></span> </li> </ol></div></div> <h2><span class="mw-headline" id="External_links">External links</span></h2> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1067248974"/><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://www.ece.utexas.edu/~grady/POWERFAC.pdf"><i>Harmonics and how they relate to power factor</i></a> <span class="cs1-format">(PDF)</span>, U Texas</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Harmonics+and+how+they+relate+to+power+factor&amp;rft.pub=U+Texas&amp;rft_id=http%3A%2F%2Fwww.ece.utexas.edu%2F~grady%2FPOWERFAC.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3APower+factor" class="Z3988"></span>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul 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style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"/><style data-mw-deduplicate="TemplateStyles:r1063604349">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output 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href="/wiki/Electricity_delivery" title="Electricity delivery">Electricity delivery</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Automatic_generation_control" title="Automatic generation control">Automatic generation control</a></li> <li><a href="/wiki/Backfeeding" title="Backfeeding">Backfeeding</a></li> <li><a href="/wiki/Base_load" title="Base load">Base load</a></li> <li><a href="/wiki/Demand_factor" title="Demand factor">Demand factor</a></li> <li><a href="/wiki/Droop_speed_control" title="Droop speed control">Droop speed control</a></li> <li><a href="/wiki/Merit_order" title="Merit order">Economic dispatch</a></li> <li><a href="/wiki/Electric_power" title="Electric power">Electric power</a></li> <li><a href="/wiki/Energy_demand_management" title="Energy demand management">Demand management</a></li> <li><a href="/wiki/Energy_return_on_investment" title="Energy return on investment">Energy return on investment</a></li> <li><a href="/wiki/Electrical_fault" title="Electrical fault">Electrical fault</a></li> <li><a href="/wiki/Home_energy_storage" title="Home energy storage">Home energy storage</a></li> <li><a href="/wiki/Grid_energy_storage" title="Grid energy storage">Grid storage</a></li> <li><a href="/wiki/Grid_code" title="Grid code">Grid code</a></li> <li><a href="/wiki/Short_circuit_ratio" title="Short circuit ratio">Grid strength</a></li> <li><a href="/wiki/Load-following_power_plant" title="Load-following power plant">Load-following</a></li> <li><a href="/wiki/Merit_order" title="Merit order">Merit order</a></li> <li><a href="/wiki/Nameplate_capacity" title="Nameplate capacity">Nameplate capacity</a></li> <li><a href="/wiki/Peak_demand" title="Peak demand">Peak demand</a></li> <li><a class="mw-selflink selflink">Power factor</a></li> <li><a href="/wiki/Electric_power_quality" title="Electric power quality">Power quality</a></li> <li><a href="/wiki/Power-flow_study" title="Power-flow study">Power-flow study</a></li> <li><a href="/wiki/Repowering" title="Repowering">Repowering</a></li> <li><a href="/wiki/Utility_frequency" title="Utility frequency">Utility frequency</a></li> <li><a href="/wiki/Variable_renewable_energy" title="Variable renewable energy">Variability</a></li> <li><a href="/wiki/Vehicle-to-grid" title="Vehicle-to-grid">Vehicle-to-grid</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="8" style="width:1px;padding:0 0 0 2px"><div><a href="/wiki/File:Abspannportal.jpg" class="image"><img alt="Abspannportal.jpg" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Abspannportal.jpg/120px-Abspannportal.jpg" decoding="async" width="120" height="90" data-file-width="2048" data-file-height="1536" /></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sources</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7em"><a href="/wiki/Non-renewable_resource" title="Non-renewable resource">Non-renewable</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coal" title="Coal">Coal</a></li> <li><a href="/wiki/Fossil_fuel_power_station" title="Fossil fuel power station">Fossil fuel power station</a></li> <li><a href="/wiki/Natural_gas" title="Natural gas">Natural gas</a></li> <li><a href="/wiki/Petroleum" title="Petroleum">Petroleum</a></li> <li><a href="/wiki/Nuclear_power" title="Nuclear power">Nuclear</a></li> <li><a href="/wiki/Oil_shale" title="Oil shale">Oil shale</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7em"><a href="/wiki/Renewable_energy" title="Renewable energy">Renewable</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Biogas" title="Biogas">Biogas</a></li> <li><a href="/wiki/Biofuel" title="Biofuel">Biofuel</a></li> <li><a href="/wiki/Biomass" title="Biomass">Biomass</a></li> <li><a href="/wiki/Geothermal_power" title="Geothermal power">Geothermal</a></li> <li><a href="/wiki/Hydroelectricity" title="Hydroelectricity">Hydro</a></li> <li><a href="/wiki/Marine_energy" title="Marine energy">Marine</a> <ul><li><a href="/wiki/Marine_current_power" title="Marine current power">Current</a></li> <li><a href="/wiki/Osmotic_power" title="Osmotic power">Osmotic</a></li> <li><a href="/wiki/Ocean_thermal_energy_conversion" title="Ocean thermal energy conversion">Thermal</a></li> <li><a href="/wiki/Tidal_power" title="Tidal power">Tidal</a></li> <li><a href="/wiki/Wave_power" title="Wave power">Wave</a></li></ul></li> <li><a href="/wiki/Solar_power" title="Solar power">Solar</a></li> <li><a href="/wiki/Sustainable_biofuel" title="Sustainable biofuel">Sustainable biofuel</a></li> <li><a href="/wiki/Wind_power" title="Wind power">Wind</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Power_station_technology" title="Category:Power station technology">Generation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/AC_power" title="AC power">AC power</a></li> <li><a href="/wiki/Cogeneration" title="Cogeneration">Cogeneration</a></li> <li><a href="/wiki/Combined_cycle_power_plant" title="Combined cycle power plant">Combined cycle</a></li> <li><a href="/wiki/Cooling_tower" title="Cooling tower">Cooling tower</a></li> <li><a href="/wiki/Induction_generator" title="Induction generator">Induction generator</a></li> <li><a href="/wiki/Micro_combined_heat_and_power" title="Micro combined heat and power">Micro CHP</a></li> <li><a href="/wiki/Microgeneration" title="Microgeneration">Microgeneration</a></li> <li><a href="/wiki/Rankine_cycle" title="Rankine cycle">Rankine cycle</a></li> <li><a href="/wiki/Three-phase_electric_power" title="Three-phase electric power">Three-phase electric power</a></li> <li><a href="/wiki/Virtual_power_plant" title="Virtual power plant">Virtual power plant</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Electric_power_transmission" title="Electric power transmission">Transmission</a><br />and <a href="/wiki/Electric_power_distribution" title="Electric power distribution">distribution</a></div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Demand_response" title="Demand response">Demand response</a></li> <li><a href="/wiki/Distributed_generation" title="Distributed generation">Distributed generation</a></li> <li><a href="/wiki/Dynamic_demand_(electric_power)" title="Dynamic demand (electric power)">Dynamic demand</a></li> <li><a href="/wiki/Electric_power_distribution" title="Electric power distribution">Electric power distribution</a></li> <li><a href="/wiki/Electricity_retailing" title="Electricity retailing">Electricity retailing</a></li> <li><a href="/wiki/Electrical_busbar_system" title="Electrical busbar system">Electrical busbar system</a></li> <li><a href="/wiki/Electric_power_system" title="Electric power system">Electric power system</a></li> <li><a href="/wiki/Electric_power_transmission" title="Electric power transmission">Electric power transmission</a></li> <li><a href="/wiki/Electrical_grid" title="Electrical grid">Electrical grid</a></li> <li><a href="/wiki/Interconnector" title="Interconnector">Electrical interconnector</a></li> <li><a href="/wiki/High-voltage_direct_current" title="High-voltage direct current">High-voltage direct current</a></li> <li><a href="/wiki/High-voltage_shore_connection" title="High-voltage shore connection">High-voltage shore connection</a></li> <li><a href="/wiki/Load_management" title="Load management">Load management</a></li> <li><a href="/wiki/Mains_electricity_by_country" title="Mains electricity by country">Mains electricity by country</a></li> <li><a href="/wiki/Overhead_power_line" title="Overhead power line">Power line</a></li> <li><a href="/wiki/Power_station" title="Power station">Power station</a></li> <li><a href="/wiki/Pumped-storage_hydroelectricity" title="Pumped-storage hydroelectricity">Pumped hydro</a></li> <li><a href="/wiki/Smart_grid" title="Smart grid">Smart grid</a></li> <li><a href="/wiki/Electrical_substation" title="Electrical substation">Substation</a></li> <li><a href="/wiki/Single-wire_earth_return" title="Single-wire earth return">Single-wire earth return</a></li> <li><a href="/wiki/Super_grid" title="Super grid">Super grid</a></li> <li><a href="/wiki/Transformer" title="Transformer">Transformer</a></li> <li><a href="/wiki/Transmission_system_operator" title="Transmission system operator">Transmission system operator</a> (TSO)</li> <li><a href="/wiki/Transmission_tower" title="Transmission tower">Transmission tower</a></li> <li><a href="/wiki/Utility_pole" title="Utility pole">Utility pole</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Failure modes</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Power_outage" title="Power outage">Blackout</a> (<a href="/wiki/Rolling_blackout" title="Rolling blackout">Rolling blackout</a>)</li> <li><a href="/wiki/Brownout_(electricity)" title="Brownout (electricity)">Brownout</a></li> <li><a href="/wiki/Black_start" title="Black start">Black start</a></li> <li><a href="/wiki/Cascading_failure" title="Cascading failure">Cascading failure</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Protective<br />devices</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arc-fault_circuit_interrupter" title="Arc-fault circuit interrupter">Arc-fault circuit interrupter</a></li> <li><a href="/wiki/Circuit_breaker" title="Circuit breaker">Circuit breaker</a></li> <li><a href="/wiki/Earth-leakage_circuit_breaker" title="Earth-leakage circuit breaker">Earth-leakage circuit breaker</a></li> <li><a href="/wiki/Generator_interlock_kit" title="Generator interlock kit">Generator interlock kit</a></li> <li><a href="/wiki/Residual-current_device" title="Residual-current device">Residual-current device</a> (GFI)</li> <li><a href="/wiki/Power_system_protection" title="Power system protection">Power system protection</a></li> <li><a href="/wiki/Protective_relay" title="Protective relay">Protective relay</a></li> <li><a href="/wiki/Numerical_relay" title="Numerical relay">Numerical relay</a></li> <li><a href="/wiki/Sulfur_hexafluoride_circuit_breaker" title="Sulfur hexafluoride circuit breaker">Sulfur hexafluoride circuit breaker</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Economics<br />and policies</div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Availability_factor" title="Availability factor">Availability factor</a></li> <li><a href="/wiki/Capacity_factor" title="Capacity factor">Capacity factor</a></li> <li><a href="/wiki/Carbon_offset" title="Carbon offset">Carbon offset</a></li> <li><a href="/wiki/Cost_of_electricity_by_source" title="Cost of electricity by source">Cost of electricity by source</a></li> <li><a href="/wiki/Environmental_tax" title="Environmental tax">Environmental tax</a></li> <li><a href="/wiki/Energy_subsidy" title="Energy subsidy">Energy subsidies</a></li> <li><a href="/wiki/Feed-in_tariff" title="Feed-in tariff">Feed-in tariff</a></li> <li><a href="/wiki/Fossil_fuel_phase-out" title="Fossil fuel phase-out">Fossil fuel phase-out</a></li> <li><a href="/wiki/Load_factor_(electrical)" title="Load factor (electrical)">Load factor</a></li> <li><a href="/wiki/Net_metering" title="Net metering">Net metering</a></li> <li><a href="/wiki/Pigovian_tax" class="mw-redirect" title="Pigovian tax">Pigovian tax</a></li> <li><a href="/wiki/Renewable_Energy_Certificate_(United_States)" title="Renewable Energy Certificate (United States)">Renewable Energy Certificates</a></li> <li><a href="/wiki/Renewable_Energy_Payments" title="Renewable Energy Payments">Renewable energy payments</a></li> <li><a href="/wiki/Renewable_energy_commercialization" title="Renewable energy commercialization">Renewable energy policy</a></li> <li><a href="/wiki/Spark_spread" title="Spark spread">Spark/Dark/Quark/Bark spread</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Statistics and<br />production</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_electricity_sectors" title="List of electricity sectors">List of electricity sectors</a></li> <li><a href="/wiki/Electric_energy_consumption" title="Electric energy consumption">Electric energy consumption</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <dl><dt>Categories</dt> <dd><a href="/wiki/Category:Electric_power_distribution" title="Category:Electric power distribution">Electric power distribution</a></dd> <dd><a href="/wiki/Category:Electricity_economics" title="Category:Electricity economics">Electricity economics</a></dd> <dd><a href="/wiki/Category:Power_station_technology" title="Category:Power station technology">Power station technology</a></dd> <dt>Portals</dt> <dd><a href="/wiki/Portal:Energy" title="Portal:Energy">Energy</a></dd> <dd><a href="/wiki/Portal:Renewable_energy" title="Portal:Renewable energy">Renewable energy</a></dd></dl> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"/><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1061467846"/></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control</a>: National libraries <a href="https://www.wikidata.org/wiki/Q750454#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" style="vertical-align: text-top" class="noprint" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4167276-8">Germany</a></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="http://uli.nli.org.il/F/?func=find-b&amp;local_base=NLX10&amp;find_code=UID&amp;request=987007535929405171">Israel</a></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/subjects/sh85041886">United States</a></span></li></ul> </div></td></tr></tbody></table></div></div>'
Whether or not the change was made through a Tor exit node (tor_exit_node)
false
Unix timestamp of change (timestamp)
'1672999746'