phasor

A phasor is a complex number used to represent the amplitude and phase of a sinusoidal signal with a fixed frequency. In engineering, phasor methods express a time-domain sinusoid as a rotating vector in the complex plane, with length equal to the signal’s magnitude and angle equal to its phase relative to a reference.

If x(t) = X_m cos(ωt + φ), its phasor is X = X_m ∠ φ or, using RMS magnitude, X_rms ∠ φ. Using Euler’s formula, x(t) = Re{X e^{jωt}}. This representation factors out the time dependence, allowing steady-state analysis to use algebraic methods rather than differential equations.

Phasor arithmetic simplifies the analysis of linear time-invariant AC circuits. For a circuit element, voltage and current are related by an impedance Z such that V = I Z, where Z is a complex number. In AC circuits, Z can include resistive and reactive parts, for example Z = R + jX_L − jX_C. Multiplication by j corresponds to a 90-degree phase shift, which is naturally handled in the complex plane.

Power and phasors: complex power S = V I*, where V and I are phasors (typically RMS). The real part P = Re(S) is active power, the imaginary part Q = Im(S) is reactive power, and the magnitude |S| is apparent power. Phasor diagrams illustrate the relationships between voltages and currents.

Limitations: the phasor method assumes sinusoidal excitation at a constant frequency and linear time-invariant behavior, and it does not capture transients or non-sinusoidal components, for which time-domain or transform methods are used.

History: the phasor concept emerged with the development of complex impedance and AC analysis in the late 19th and early 20th centuries, aided by Euler’s formula and widespread adoption in electrical engineering.