impedances

Impedances are the complex opposition that a circuit presents to alternating current signals. They extend the idea of resistance to AC by incorporating both magnitude and phase, so voltage and current need not be in step. In phasor form Ohm’s law becomes V = Z I, where Z is the complex impedance Z = R + jX. The real part R is resistance, and the imaginary part X is reactance, which captures energy storage in magnetic or electric fields. Reactance is frequency dependent: inductive reactance X_L = ωL and capacitive reactance X_C = 1/(ωC), with X = X_L − X_C. The unit of impedance is the ohm (Ω).

The magnitude and phase of impedance are |Z| = sqrt(R^2 + X^2) and φ = arctan(X/R). If φ is positive, the impedance is inductive; if negative, capacitive. This framework enables analysis of networks across frequencies.

In networks, impedances add differently depending on configuration. For components in series, Z_total = Σ Z_i. For parallel, 1/Z_total = Σ 1/Z_i. Impedance matching, where the load is the complex conjugate of the source impedance, maximizes power transfer. In transmission-line contexts, the characteristic impedance Z0 and the reflection coefficient describe how mismatches reflect signals.

Impedance is measured with instruments such as impedance analyzers or vector network analyzers and is widely used in audio electronics, RF design, power systems, and sensor technologies. The concept also has mechanical and acoustical analogues, where impedance relates force to velocity in a given medium.