Phasor

A phasor is a complex number that represents a sinusoidal steady-state signal of known frequency. The real sinusoid V(t) = V₀ cos(ωt + φ) is rewritten as the real part of V₀ e^(j(ωt+φ)) = (V₀ e^(jφ)) · e^(jωt). The time-varying e^(jωt) factor is the same for every signal in the circuit and can be factored out; what remains, Ṽ = V₀ e^(jφ), is the phasor: a single complex number carrying both the amplitude V₀ and the phase φ. Two signals of the same frequency are fully characterised by their phasors.

The trick works because the operator d/dt in the time domain becomes multiplication by jω in the phasor domain (d/dt(e^(jωt)) = jω e^(jωt)). So differential equations governing RLC circuits — L dI/dt, C dV/dt, etc. — become algebraic equations in complex impedances. A resistor R becomes impedance R. An inductor L becomes impedance jωL. A capacitor C becomes impedance 1/(jωC) = −j/(ωC). Kirchhoff's laws hold unchanged in the phasor domain, and solving a network reduces to complex algebra: Ṽ = ĨZ, V_divider = V_in · Z₂/(Z₁+Z₂), etc.

Phasors turn the whole analysis of AC circuits into what amounts to DC-style algebra over the complex numbers. The cost is that only linear, single-frequency (or superposed-single-frequency) problems can be handled: non-linear components (diodes, transistors in saturation) or transient responses require returning to the time domain. But for the vast majority of power-frequency, audio, and RF design, phasor analysis is the default working framework. Introduced by Charles Steinmetz in the 1890s at General Electric, building on Heaviside's complex-impedance work, phasors made large-scale AC power systems practically designable for the first time.

Definition

History