In physics and engineering, a phasor (a portmanteau of phase vector) 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, 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, and (in older texts) sinor or even complexor. 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) 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. 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. The originator of the phasor transform was Charles Proteus Steinmetz working at General Electric in the late 19th century. 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. 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. However, the Laplace transform is mathematically more difficult to apply and the effort may be unjustified if only steady state analysis is required.
Notation
Phasor notation (also known as angle notation) is a mathematical notation used in electronics engineering and electrical engineering. A vector whose polar coordinates are magnitude A {\displaystyle A} and angle θ {\displaystyle \theta } is written A ∠ θ . {\displaystyle A\angle \theta .} 1 ∠ θ {\displaystyle 1\angle \theta } can represent either the vector ( cos θ , sin θ ) {\displaystyle (\cos \theta ,\,\sin \theta )} or the complex number cos θ + i sin θ = e i θ {\displaystyle \cos \theta +i\sin \theta =e^{i\theta }} , according to Euler's formula with i 2 = − 1 {\displaystyle i^{2}=-1} , both of which have magnitudes of 1. The angle may be stated in degrees with an implied conversion from degrees to radians. For example 1 ∠ 90 {\displaystyle 1\angle 90} would be assumed to be 1 ∠ 90 ∘ , {\displaystyle 1\angle 90^{\circ },} which is the vector ( 0 , 1 ) {\displaystyle (0,\,1)} or the number e i π / 2 = i . {\displaystyle e^{i\pi /2}=i.}
Multiplication and division of complex numbers become straight forward through the phasor notation. Given the vectors v 1 = A 1 ∠ θ 1 {\displaystyle v_{1}=A_{1}\angle \theta _{1}} and v 2 = A 2 ∠ θ 2 {\displaystyle v_{2}=A_{2}\angle \theta _{2}} , the following is true:
v 1 ⋅ v 2 = A 1 ⋅ A 2 ∠ ( θ 1 + θ 2 ) {\displaystyle v_{1}\cdot v_{2}=A_{1}\cdot A_{2}\angle (\theta _{1}+\theta _{2})} ,
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