Impedance

Impedance is a complex quantity that describes how a circuit element or network opposes the flow of alternating current. In sinusoidal steady state, impedance Z relates the phasor voltage V to the phasor current I by V = ZI. It combines resistance, which dissipates energy, with reactance, which stores energy temporarily. As a complex number, Z can be written Z = R + jX, where R is the real part (resistance) and X is the imaginary part (reactance). The magnitude |Z| represents the overall opposition to current, and the phase angle φ = arg(Z) is the phase difference between voltage and current.

Reactance X arises from energy storage elements: inductors and capacitors. X_L = ωL is positive, and X_C = -1/(ωC) is negative. Special cases include Z_R = R for a pure resistor, Z_C = 1/(jωC) = -j/(ωC) for a capacitor, and Z_L = jωL for an inductor.

In AC analysis, Ohm’s law generalizes to V = IZ. Complex power S = VI*, with real power P = I^2R = VI cos φ and reactive power Q = VI sin φ; the apparent power is |S| = VI. The reciprocal of impedance is admittance Y = 1/Z.

Impedance also plays a central role in transmission lines, where the characteristic impedance Z0 = sqrt(L/C) governs propagation and reflections; impedance matching aims to minimize reflections and maximize power transfer.

Impedance can be measured as a function of frequency with LCR meters and impedance spectroscopy, which characterizes how Z(ω) behaves in a given system. The concept of impedance extends beyond electronics to other domains such as acoustics and mechanics, where it similarly quantifies opposition to oscillatory motion.