Characteristic impedance

The characteristic impedance Z₀ of a transmission line is the ratio of voltage to current in a single travelling wave moving along the line. For a lossless line with series inductance L and shunt capacitance C per unit length, Z₀ = √(L/C). For a coaxial cable with inner radius a, outer radius b, and dielectric permittivity ε_r, the detailed calculation gives Z₀ = (60/√ε_r) ln(b/a) Ω — entirely geometric.

Terminating a transmission line in its characteristic impedance — Z_load = Z₀ — eliminates reflections: all the incident power is absorbed, and the line looks like an infinite length of cable to the source. Any other termination creates reflected waves, leading to standing-wave patterns and (for high-power applications) potentially destructive voltage maxima along the line. Impedance matching is therefore the central engineering concern in every RF, high-speed digital, or high-power coaxial design.

Five canonical values cover most practical work. 50 Ω (RF coaxial) is the industry standard for test equipment, RF amplifiers, antennas, and connectorised measurements — it's the value that minimises the combined power-handling and loss in standard air-dielectric coax geometry. 75 Ω (video and broadband) is used in cable TV, satellite TV, and some broadband feedlines — it minimises attenuation for long runs. 100 Ω differential (twisted pair) is used in Ethernet, USB, PCI Express differential pairs. 110 Ω is the USB 2.0 differential standard. 377 Ω = √(μ₀/ε₀) is the characteristic impedance of free space, the ratio of E to B magnitudes in a vacuum-propagating electromagnetic wave, and the impedance of a half-wave antenna in free air (modulo geometry factors). The same √(L/C) formula unifies all of them — transmission line or free space, the square-root-of-inductance-over-capacitance is the impedance a wave propagates against.