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Phasor

For other uses, see Phasor (disambiguation).
Not to be confused with phaser.
"Complex amplitude" redirects here. For the quantum-mechanical concept, see Complex probability amplitude.
An example of series RLC circuit and respective phasor diagram for a specific 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 voltages and the reference direction for the current.

In physics and engineering, a phasor (a portmanteau of phase vector[1][2]) is a complex number representing a sinusoidal function whose amplitude (A), and initial phase (θ) are time-invariant and whose angular frequency (ω) is fixed. It is related to a more general concept called analytic representation,[3] 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,[4][5] and (in older texts) sinor[6] or even complexor.[6]

A common application is in the steady-state analysis of an electrical network powered by 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[7]: 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 vectors is possible for phasors as well.[6] An important additional feature of the phasor transform is that differentiation and integration of sinusoidal signals (having constant amplitude, period and phase) corresponds to simple algebraic operations on the phasors; the phasor transform thus allows the analysis (calculation) of the AC steady state of RLC circuits by solving simple algebraic equations (albeit with complex coefficients) in the phasor domain instead of solving differential equations (with real coefficients) in the time domain.[8][9][a] The originator of the phasor transform was Charles Proteus Steinmetz working at General Electric in the late 19th century.[10][11] 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.[12]

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.[9][11] However, the Laplace transform is mathematically more difficult to apply and the effort may be unjustified if only steady state analysis is required.[11]

Fig 2. When function is depicted in the complex plane, the vector formed by its imaginary and real parts rotates around the origin. Its magnitude is A, and it completes one cycle every 2π/ω. θ is the angle it forms with the positive real axis at t = 0 (and at t = n 2π/ω for all integer values of n).
  1. ^ Huw Fox; William Bolton (2002). Mathematics for Engineers and Technologists. Butterworth-Heinemann. p. 30. ISBN 978-0-08-051119-1.
  2. ^ Clay Rawlins (2000). Basic AC Circuits (2nd ed.). Newnes. p. 124. ISBN 978-0-08-049398-5.
  3. ^ Bracewell, Ron. The Fourier Transform and Its Applications. McGraw-Hill, 1965. p269
  4. ^ K. S. Suresh Kumar (2008). Electric Circuits and Networks. Pearson Education India. p. 272. ISBN 978-81-317-1390-7.
  5. ^ Kequian Zhang; Dejie Li (2007). Electromagnetic Theory for Microwaves and Optoelectronics (2nd ed.). Springer Science & Business Media. p. 13. ISBN 978-3-540-74296-8.
  6. ^ a b c J. Hindmarsh (1984). Electrical Machines & their Applications (4th ed.). Elsevier. p. 58. ISBN 978-1-4832-9492-6.
  7. ^ a b Gross, Charles A. (2012). Fundamentals of electrical engineering. Thaddeus Adam Roppel. Boca Raton, FL: CRC Press. ISBN 978-1-4398-9807-9. OCLC 863646311.
  8. ^ William J. Eccles (2011). Pragmatic Electrical Engineering: Fundamentals. Morgan & Claypool Publishers. p. 51. ISBN 978-1-60845-668-0.
  9. ^ a b Richard C. Dorf; James A. Svoboda (2010). Introduction to Electric Circuits (8th ed.). John Wiley & Sons. p. 661. ISBN 978-0-470-52157-1.
  10. ^ Allan H. Robbins; Wilhelm Miller (2012). Circuit Analysis: Theory and Practice (5th ed.). Cengage Learning. p. 536. ISBN 978-1-285-40192-8.
  11. ^ a b c Won Y. Yang; Seung C. Lee (2008). Circuit Systems with MATLAB and PSpice. John Wiley & Sons. pp. 256–261. ISBN 978-0-470-82240-1.
  12. ^ Basil Mahon (2017). The Forgotten Genius of Oliver Heaviside (1st ed.). Prometheus Books Learning. p. 230. ISBN 978-1-63388-331-4.


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