phasornotation

Phasor notation is a mathematical shorthand used in the analysis of linear, time-invariant systems driven by sinusoidal signals. It represents a sinusoid as a complex number with a magnitude and a phase, allowing operations such as addition, differentiation, and integration to be carried out algebraically rather than via time-domain calculus.

In practical terms, a sinusoid of the form A cos(ωt + φ) is represented by a phasor V = A ∠ φ (sometimes written as V or V̂). The actual time-domain signal is recovered by taking the real part of V e^{jωt}, i.e., v(t) = Re{V e^{jωt}}. When working with RMS amplitudes, the phasor is often taken as the RMS value, which simplifies power calculations.

Phasor notation has useful algebraic properties. Adding two sinusoids of the same frequency corresponds to adding their phasors. Differentiation with respect to time corresponds to multiplication by jω, and integration corresponds to division by jω. These rules make it easier to analyze circuits and systems with impedances and transfer functions, where frequency-dependent behavior can be treated as algebraic operations on phasors.

Conventions include the choice of angle measurement (degrees or radians) and whether the phasor uses peak or RMS magnitude. Phasors are most effective for single-frequency, linear responses and are less applicable to transients or nonlinear systems; for multiple frequencies, the principle of superposition is applied across each frequency component.

Applications of phasor notation are widespread in electrical engineering, including AC circuit analysis, impedance calculation, control systems, and signal processing. See also phasor, complex number, Fourier transform, and impedance.