Impedance

Impedance Z is the complex generalisation of resistance to AC circuits. At a single driving frequency ω, every two-terminal linear component has an impedance Z = V/I — the complex ratio of the voltage phasor to the current phasor. A resistor has Z = R (purely real, in phase). An inductor has Z = jωL (purely imaginary, current lags voltage by 90°). A capacitor has Z = 1/(jωC) = −j/(ωC) (purely imaginary, current leads voltage by 90°). Combinations follow the familiar series (Z_total = Z₁ + Z₂) and parallel (1/Z_total = 1/Z₁ + 1/Z₂) rules, now with complex arithmetic.

The magnitude |Z| = √(R²+X²) tells you the ratio of voltage-amplitude to current-amplitude; the phase angle arg(Z) = arctan(X/R) tells you how much the current lags the voltage. Only the real part R dissipates real power (P = ½|I|²R, the average over a cycle); the imaginary part X shuttles energy back and forth between the source and the reactive element (inductor or capacitor) without net dissipation. This distinction — between real power (watts) and reactive power (volt-amperes reactive, VAR) — is what "power factor" corrections on industrial grids are about: the reactive part is real current that electrical wiring has to carry even though it does no net work on the load.

Impedance matching — making the source impedance equal to the load impedance — is the condition for maximum power transfer, derived from dP/dR_L = 0 where P is the power dissipated in the load. It is the design rule behind transmission-line termination (50 Ω coax needs a 50 Ω terminator to prevent reflections), antenna matching networks (a quarter-wave matching transformer between a 75 Ω feedline and a 50 Ω amplifier), and audio amplifier output stages. Heaviside introduced the word impedance in 1886 in his cable-theory papers, where the complex ratio of voltage to current in a loaded transmission line first became essential.