Transmission line

A transmission line is a conductor pair — coaxial cable, twisted pair, stripline, waveguide — used to carry high-frequency signals where the wavelength becomes comparable to the physical length of the cable. Below this crossover (roughly λ/10 ≳ cable length), ordinary Kirchhoff's laws apply and the cable is just a wire. Above it, the cable becomes a distributed system: voltage and current vary along its length, reflections bounce off mismatched terminations, and the behaviour is governed by the telegrapher's equations of Heaviside (1876–1892).

The telegrapher's equations come from modelling each infinitesimal length of the cable as a ladder of series inductance L (per unit length), series resistance R, shunt capacitance C, and shunt conductance G. Taking Kirchhoff's laws in the distributed limit produces two coupled PDEs: ∂V/∂x = −L ∂I/∂t − RI and ∂I/∂x = −C ∂V/∂t − GV. In the lossless limit (R = G = 0), these reduce to the wave equation with propagation speed v = 1/√(LC), and signals travel along the cable as forward and backward travelling waves. The ratio of voltage to current in a single travelling wave is the characteristic impedance Z₀ = √(L/C), typically 50 Ω for coax and 75 Ω for video cables.

A mismatch between the line impedance Z₀ and the load impedance Z_L creates reflections: part of the incident signal bounces back to the source with reflection coefficient Γ = (Z_L − Z₀)/(Z_L + Z₀). For Z_L = Z₀, Γ = 0 and all power is absorbed (terminated line, no reflection). For Z_L = 0 (short), Γ = −1 (full-amplitude inverted reflection). For Z_L = ∞ (open), Γ = +1 (full-amplitude in-phase reflection). Reflections cause standing waves, power loss, and — in digital high-speed links — signal-integrity failures that manifest as eye-diagram closure. Every FPGA routing tool, every RF amplifier matching network, every antenna balun, and every multi-gigabit signal trace on a modern circuit board is designed around the telegrapher's equations.