Phasors and Impedance

Inductors and capacitors relate voltage and current through derivatives, so KVL around an RLC loop gives a second-order differential equation. Solvable, and thoroughly tedious to do for every circuit.

Differentiating a sinusoid scales it by ω and shifts it 90°; the frequency is unchanged. So in steady state every voltage and current in a linear circuit is a sinusoid at the source frequency, differing only in amplitude and phase — two numbers each, instead of a function of time.

A phasor encodes those two numbers as a complex number, dropping ωt because it is common to every signal. That step is legitimate only at a single frequency in steady state — the time dependence has not been solved, it has been factored out.

Every element now satisfies V = I·Z — Ohm's law with a complex constant. Impedances combine exactly like resistances: series adds, parallel takes reciprocals, dividers divide. Nodal analysis, mesh analysis and Thévenin all carry over unchanged.

The j is doing real work — it is a 90° rotation, which is precisely the phase shift a reactive element introduces. With exactly 90° between voltage and current, the average power is exactly zero: energy stored and returned rather than consumed.

Z = R + jX separates the dissipative part from the storing part. Resistance converts energy to heat permanently; reactance stores it in a field and returns it twice per cycle. The same distinction reappears as real against reactive power.

Inductive reactance rises with frequency and capacitive reactance falls, so every RC and RL network is a filter whether intended or not. Where the two are equal they cancel exactly, leaving pure resistance — which is resonance.

• Steady state only — the transient after a switch closes is invisible to the method.
• One frequency only — a square wave needs each harmonic solved separately and summed, which is Fourier plus superposition.
• Linear only — a diode produces harmonics that were not in the input.

Laplace generalises it: s = σ + jω covers transient and steady state together, and phasor analysis is the special case with σ = 0. Phasors handle the case an engineer meets most often, which is why they are worth the restriction.