TransmissionLine

A transmission line is a specialized electrical network designed to carry electrical energy or signals from one location to another. It is typically a pair of conductors or a single conductor with a return path, whose length is comparable to or greater than the wavelength of the signals it transmits. Unlike lumped-element circuits, a transmission line must be treated as a distributed system with parameters that vary along its length.

The behavior is described by the telegrapher's equations, using per-unit-length resistance R, inductance L, conductance G, and capacitance C. In the frequency domain, the equations are dV/dx = - (R + jωL) I and dI/dx = - (G + jωC) V. From these, the characteristic impedance Z0 = sqrt((R + jωL)/(G + jωC)) and the propagation constant γ = α + jβ follow. In a lossless line (R and G negligible), Z0 = sqrt(L/C) and β = ω sqrt(LC). The phase velocity is v = ω/β ≈ 1/ sqrt(LC).

Impedance matching is critical to minimize reflections. When the load impedance ZL differs from Z0, part of the incident wave is reflected, described by the reflection coefficient Γ = (ZL - Z0)/(ZL + Z0). Proper terminations and impedance control ensure efficient energy transfer and reduce standing waves.

Common electrical transmission lines include coaxial cables, twisted-pair cables, and microstrip lines, as well as waveguides used at microwave frequencies. Optical transmission lines include optical fibers that guide light via total internal reflection.

Applications span radio frequency and microwave communications, digital interconnects, and power transmission. Design considerations include loss, dispersion, radiative and conductor losses, impedance control, crosstalk, and physical constraints.