phasors

Phasors are a mathematical tool used in electrical engineering to simplify the analysis of linear, time-invariant systems driven by sinusoidal sources. A phasor represents the instantaneous sinusoidal quantity by its amplitude and phase as a complex number, with a fixed angular frequency ω. If a voltage is v(t) = Vm cos(ωt + φ), its phasor is V = Vm ∠ φ (or Vrms ∠ φ depending on convention).

In the complex plane, a phasor can be written as V = Re{V} + j Im{V}. The time-domain signal is recovered by v(t) = Re{V e^{jωt}} = |V| cos(ωt + ∠V). Polar and rectangular forms are used interchangeably.

Operations: Multiplication by an impedance Z relates voltage and current: V = I Z. Components have fixed impedances: R is real, L contributes jωL, C contributes 1/(jωC). Addition of phasors corresponds to vector addition, i.e., Kirchhoff's laws apply in the phasor domain. Phasor analysis converts differential equations into algebraic ones, facilitating steady-state AC analysis.

Limitations: phasor methods assume a sinusoidal source at a single frequency and linear, time-invariant behavior; they do not describe transients or non-sinusoidal signals directly. For non-steady or broadband signals, Fourier or other time-frequency methods are used, often working with phasors at each frequency component.