Foundation
A phasor is a complex number that fully describes a sinusoidal signal of known frequency. By packaging amplitude and phase into a single complex value, phasors reduce differential equations in time into algebraic equations in the complex plane — the central mathematical move that makes AC circuit theory and power-system analysis tractable.
Consider a single voltage v(t) = Vm cos(ωt + θ) oscillating at angular frequency ω. The full description in the time domain requires three numbers — amplitude Vm, frequency ω, and phase θ — and any operation on this signal (addition with another sinusoid, integration through a capacitor, differentiation across an inductor) requires manipulating trigonometric identities.
Phasor analysis observes that when ω is fixed across the entire system (as it is in steady state), we only need to track amplitude and phase. We represent the sinusoid as a complex number V = Vm · ejθ, called the phasor. The time-domain signal is recovered by:
The imaginary unit j (preferred over i in electrical engineering, to avoid clash with current symbols) acts as a rotation operator: multiplying a phasor by j rotates it 90° counterclockwise in the complex plane. This geometric interpretation is the key to why inductors and capacitors map naturally to imaginary impedances.
The fundamental property: the real-part operator is linear. For two signals v1(t) and v2(t) at the same frequency with phasors V1 and V2:
Adding signals in time corresponds to adding their phasors in the complex plane. Scaling a signal scales its phasor. Linear operations on time signals become linear operations on phasors.
Differentiating v(t) = Re(V ejωt) with respect to time yields:
Differentiation in time corresponds to multiplication by jω in phasor space. Integration corresponds to division by jω. This is the algebraic miracle: linear differential equations describing circuit dynamics collapse into linear algebraic equations relating phasors.
For a resistor R, Ohm’s law gives v = R·i in time, hence V = R·I in phasor space — a real-valued ratio. For an inductor, v = L·di/dt becomes V = jωL · I, giving a purely imaginary impedance jωL. For a capacitor, i = C·dv/dt becomes V = I / (jωC). Combining these:
The real part R is resistance (dissipates energy); the imaginary part X is reactance (stores energy in fields, oscillating between electric and magnetic forms). The reciprocal Y = G + jB is admittance, with G the conductance and B the susceptance.
The instantaneous power absorbed by an element is p(t) = v(t) · i(t). Substituting phasor expressions and averaging over one cycle yields:
The factor cos(θV − θI) is the power factor. To capture both active and reactive components in a single expression, define complex power:
The real part P is active power (in watts, the energy actually consumed); the imaginary part Q is reactive power (in volt-amperes reactive, the energy oscillating between source and reactive elements). The conjugate on I is the convention that makes lagging currents (inductive loads, the typical case) yield positive Q.
Phasors are the universal language of power-system analysis. Every voltage, current, and impedance in a steady-state OPF model is a phasor, often expressed in per-unit on a common MVA base. The bus admittance matrix Ybus, derived in the next foundation, packages phasor relationships between every pair of buses into a single complex matrix.
Beyond steady-state analysis, phasors generalize to symmetrical components (decomposing three-phase systems into positive, negative, and zero sequences) and to dynamic phasors (capturing slow envelope dynamics of fast-oscillating signals). They are the foundation on which power-flow, OPF, and small-signal stability analysis are built.