Long transmission lines (over roughly 250 km) require a distributed-parameter model because series impedance and shunt admittance are distributed continuously along the line rather than lumped, giving rise to hyperbolic-function relationships between sending-end and receiving-end voltage and current expressed via ABCD two-port parameters. The propagation constant (decomposed into an attenuation constant and a phase constant) and the characteristic (surge) impedance govern how voltage and current vary with distance, and an equivalent-pi circuit with modified series impedance and shunt admittance reproduces this behavior exactly for circuit-based analysis. For the idealized lossless line, these hyperbolic relations reduce to trigonometric ones, enabling closed-form treatment of wavelength, surge-impedance loading, and the steady-state power-angle (stability) limit — core concepts in power-system transmission-line theory within electrical power engineering.
Long transmission lines (over roughly 250 km) require a distributed-parameter model because series impedance and shunt admittance are distributed continuously along the line rather than lumped, givin…