Phasor

In <a href="/facts/Physics/a7aH2y08">physics</a> and <a href="/facts/Engineering/RFVg8H3I">engineering</a>, a phasor (a <a href="/facts/Portmanteau/d124JEbz">portmanteau</a> of phase vector) is a <a href="/facts/Complex_number/2w7mqI7e">complex number</a> representing a <a href="/facts/Sine_wave/mD1CB09M">sinusoidal function</a> whose <a href="/facts/Amplitude/aEckuXJM">amplitude</a> A and <a href="/facts/Phase_(waves)/cpGtuVll">initial phase</a> θ are <a href="/facts/Time-invariant_system/NpWUk0YG">time-invariant</a> and whose <a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a> ω is fixed. It is related to a more general concept called <a href="/facts/Analytic_signal/8CB4Cyul">analytic representation</a>, 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 <a href="/facts/Electrical_network/zt94kTW0">electrical network</a> powered by <a href="/facts/Alternating_current/qxICS3xa">time varying current</a> 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: 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 <a href="/facts/Euclidean_vector/bCLrdksy">vectors</a> is possible for phasors as well. An important additional feature of the phasor transform is that <a href="/facts/Derivative/7Ito70Dh">differentiation</a> and <a href="/facts/Integral/V7lyV997">integration</a> of sinusoidal signals (having constant amplitude, period and phase) corresponds to simple <a href="/facts/Algebraic_operation/wl69LmeN">algebraic operations</a> on the phasors; the phasor transform thus allows the <a href="/facts/Network_analysis_(electrical_circuits)/IjmVqbG8">analysis</a> (calculation) of the <a href="/facts/Alternating_current/qxICS3xa">AC</a> <a href="/facts/Steady_state_(electronics)/U3ylWEtp">steady state</a> of <a href="/facts/RLC_circuit/68f1gLJw">RLC circuits</a> by solving simple <a href="/facts/Algebraic_equation/1QauIGxs">algebraic equations</a> (albeit with complex coefficients) in the phasor domain instead of solving <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a> (with <a href="/facts/Real_number/R02gw5Pb">real</a> coefficients) in the time domain. The originator of the phasor transform was <a href="/facts/Charles_Proteus_Steinmetz/j4TplnwJ">Charles Proteus Steinmetz</a> working at <a href="/facts/General_Electric/bBm5CzVY">General Electric</a> in the late 19th century. He got his inspiration from <a href="/facts/Oliver_Heaviside/ktYBpAt7">Oliver Heaviside</a>. 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 <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transform</a> (limited to a single frequency), which, in contrast to phasor representation, can be used to (simultaneously) derive the <a href="/facts/Transient_response/m4Nr61Px">transient response</a> 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.

Phasor open-in-new

Phasor