Editing Phasor (section)

{{short description|Complex number representing a particular sine wave}}
{{other uses}}
{{distinguish|phaser (disambiguation){{!}}phaser}}
{{redirect|Complex amplitude|the quantum-mechanical concept|Complex probability amplitude}}
[[Image:Wykres wektorowy by Zureks.svg|thumb|An example of series [[RLC circuit]] and respective '''phasor diagram''' for a specific {{mvar|ω}}. The arrows in the upper diagram are phasors, drawn in a phasor diagram ([[complex plane]] without axis shown), which must not be confused with the arrows in the lower diagram, which are the reference polarity for the [[voltage]]s and the reference direction for the [[electric current|current]].]]

In [[physics]] and [[engineering]], a '''phasor''' (a [[portmanteau]] of '''phase vector'''<ref name="FoxBolton2002">{{cite book|author1=Huw Fox|author2=William Bolton|title=Mathematics for Engineers and Technologists|url=https://archive.org/details/mathematicsforen00foxh_204|url-access=limited|year=2002|publisher=Butterworth-Heinemann|isbn=978-0-08-051119-1|page=[https://archive.org/details/mathematicsforen00foxh_204/page/n36 30]}}</ref><ref name="Rawlins2000">{{cite book|author=Clay Rawlins|title=Basic AC Circuits|url=https://archive.org/details/basicaccircuits00mscl|url-access=limited|year=2000 |publisher=Newnes|isbn=978-0-08-049398-5|page=[https://archive.org/details/basicaccircuits00mscl/page/n134 124]|edition=2nd}}</ref>) is a [[complex number]] representing a [[sine wave|sinusoidal function]] whose [[amplitude]] {{mvar|A}} and [[Phase (waves)|initial phase]] {{mvar|θ}} are [[time-invariant system|time-invariant]] and whose [[angular frequency]] {{mvar|ω}} is fixed. It is related to a more general concept called [[analytic signal|analytic representation]],<ref name=Bracewell>Bracewell, Ron. ''The Fourier Transform and Its Applications''. McGraw-Hill, 1965. p269</ref> which decomposes a sinusoid into the product of a complex constant and a factor depending on time and frequency.  The complex constant, which depends on amplitude and phase, is known as a '''phasor''', or '''complex amplitude''',<ref name="Kumar2008">{{cite book|author=K. S. Suresh Kumar|title=Electric Circuits and Networks|year=2008|publisher=Pearson Education India|isbn=978-81-317-1390-7|page=272}}</ref><ref name="ZhangLi2007">{{cite book|author1=Kequian Zhang|author2=Dejie Li|title=Electromagnetic Theory for Microwaves and Optoelectronics|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-74296-8|page=13|edition=2nd}}</ref> and (in older texts) '''sinor'''<ref name="Hindmarsh2014"/> or even '''complexor'''.<ref name="Hindmarsh2014">{{cite book|author=J. Hindmarsh|title=Electrical Machines & their Applications|year=1984|edition=4th|publisher=Elsevier|isbn=978-1-4832-9492-6|page=58}}</ref>

A common application is in the steady-state analysis of an [[electrical network]] powered by [[Alternating current|time varying current]] where all signals are assumed to be sinusoidal with a common frequency. Phasor representation allows the analyst to represent the amplitude and phase of the signal using a single complex number. The only difference in their analytic representations is the complex amplitude (phasor). A linear combination of such functions can be represented as a linear combination of phasors (known as '''phasor arithmetic''' or '''phasor algebra<ref name=":02">{{Cite book |last=Gross |first=Charles A. |title=Fundamentals of electrical engineering |date=2012 |publisher=CRC Press |others=Thaddeus Adam Roppel |isbn=978-1-4398-9807-9 |location=Boca Raton, FL |oclc=863646311}}</ref>{{Rp|page=53}}''') and the time/frequency dependent factor that they all have in common.

The origin of the term phasor rightfully suggests that a (diagrammatic) calculus somewhat similar to that possible for [[Euclidean vector|vectors]] is possible for phasors as well.<ref name="Hindmarsh2014"/> An important additional feature of the phasor transform is that [[derivative|differentiation]] and [[integral|integration]] of sinusoidal signals (having constant amplitude, period and phase) corresponds to simple [[algebraic operation]]s on the phasors; the phasor transform thus allows the [[network analysis (electrical circuits)|analysis]] (calculation) of the [[alternating current|AC]] [[steady state (electronics)|steady state]] of [[RLC circuit]]s by solving simple [[algebraic equation]]s (albeit with complex coefficients) in the phasor domain instead of solving [[differential equation]]s (with [[real number|real]] coefficients) in the time domain.<ref name="Eccles2011">{{cite book|author=William J. Eccles|title=Pragmatic Electrical Engineering: Fundamentals|year=2011| publisher=Morgan & Claypool Publishers|isbn=978-1-60845-668-0|page=51}}</ref><ref name="DorfSvoboda2010">{{cite book| author1=Richard C. Dorf|author2=James A. Svoboda|title=Introduction to Electric Circuits|url=https://archive.org/details/introductiontoel00dorf_304|url-access=limited|year=2010|publisher=John Wiley & Sons|isbn=978-0-470-52157-1|page=[https://archive.org/details/introductiontoel00dorf_304/page/n680 661]|edition=8th}}</ref>{{Efn|name="ac-circuits"|Including analysis of the AC circuits.{{r|:02|pp=53}}}} The originator of the phasor transform was [[Charles Proteus Steinmetz]] working at [[General Electric]] in the late 19th century.<ref name="RobbinsMiller2012">{{cite book|author1=Allan H. Robbins|author2=Wilhelm Miller|title=Circuit Analysis: Theory and Practice|year=2012| edition=5th| publisher=Cengage Learning|isbn=978-1-285-40192-8|page=536}}</ref><ref name="YangLee2008"/> He got his inspiration from [[Oliver Heaviside]]. Heaviside's operational calculus was modified so that the variable p becomes jω. The complex number j has simple meaning: phase shift.<ref name="BasilMahon2017">{{cite book|author1=Basil Mahon|title=The Forgotten Genius of Oliver Heaviside |year=2017| edition=1st| publisher=Prometheus Books Learning|isbn=978-1-63388-331-4|page=230}}</ref>

Glossing over some mathematical details, the phasor transform can also be seen as a particular case of the [[Laplace transform]] (limited to a single frequency), which, in contrast to phasor representation, can be used to (simultaneously) derive the [[transient response]] of an RLC circuit.<ref name="DorfSvoboda2010"/><ref name="YangLee2008">{{cite book|author1=Won Y. Yang|author2=Seung C. Lee|title=Circuit Systems with MATLAB and PSpice|year=2008|publisher=John Wiley & Sons|isbn=978-0-470-82240-1|pages=256–261}}</ref> However, the Laplace transform is mathematically more difficult to apply and the effort may be unjustified if only steady state analysis is required.<ref name="YangLee2008"/>

[[File:unfasor.gif|thumb|right|Fig 2. When function <math>A \cdot e^{i(\omega t + \theta)}</math> is depicted in the complex plane, the vector formed by its [[complex number|imaginary and real parts]] rotates around the origin. Its magnitude is ''A'', and it completes one cycle every 2{{pi}}/ω. ''θ'' is the angle it forms with the positive real axis at {{math|1=''t'' = 0}} (and at {{math|1=''t'' = ''n'' 2''π''/''ω''}} for all [[integer]] values of {{mvar|n}}).]]