Phasor

In physics and engineering, a phasor (a portmanteau of phase vector), is a complex number representing a sinusoidal function whose amplitude (A), angular frequency (ω), and initial phase (θ) are time-invariant. 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 that encapsulates the frequency and time dependence. The complex constant, which encapsulates amplitude and phase dependence, is known as phasor, complex amplitude, and (in older texts) sinor or even complexor.    (Eq.1)Since this must hold for all t {displaystyle t,} , specifically: t − π 2 ω {displaystyle t-{frac {pi }{2omega }},} , it follows that    (Eq.2) In physics and engineering, a phasor (a portmanteau of phase vector), is a complex number representing a sinusoidal function whose amplitude (A), angular frequency (ω), and initial phase (θ) are time-invariant. 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 that encapsulates the frequency and time dependence. The complex constant, which encapsulates amplitude and phase dependence, is known as phasor, complex amplitude, and (in older texts) sinor or even complexor. A common situation in electrical networks is the existence of multiple sinusoids all with the same frequency, but different amplitudes and phases. The only difference in their analytic representations is the complex amplitude (phasor). A linear combination of such functions can be factored into the product of a linear combination of phasors (known as phasor arithmetic) 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. Glossing over some mathematical details, the phasor transform can also be seen as a particular case of the Laplace transform, which additionally 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. Euler's formula indicates that sinusoids can be represented mathematically as the sum of two complex-valued functions: or as the real part of one of the functions: The function A ⋅ e i ( ω t + θ ) {displaystyle Acdot e^{i(omega t+ heta )}} is called the analytic representation of A ⋅ cos ⁡ ( ω t + θ ) {displaystyle Acdot cos(omega t+ heta )} . Figure 2 depicts it as a rotating vector in a complex plane. It is sometimes convenient to refer to the entire function as a phasor, as we do in the next section. But the term phasor usually implies just the static vector A e i θ {displaystyle Ae^{i heta }} . An even more compact representation of a phasor is the angle notation: A ∠ θ {displaystyle Aangle heta } . See also vector notation. Multiplication of the phasor   A e i θ e i ω t {displaystyle Ae^{i heta }e^{iomega t},} by a complex constant,   B e i ϕ {displaystyle Be^{iphi },}  , produces another phasor. That means its only effect is to change the amplitude and phase of the underlying sinusoid: In electronics, B e i ϕ {displaystyle Be^{iphi },}   would represent an impedance, which is independent of time. In particular it is not the shorthand notation for another phasor. Multiplying a phasor current by an impedance produces a phasor voltage. But the product of two phasors (or squaring a phasor) would represent the product of two sinusoids, which is a non-linear operation that produces new frequency components. Phasor notation can only represent systems with one frequency, such as a linear system stimulated by a sinusoid.