Traveling Waves: Lossless Lines

Power lines are modeled as single-phase, two-wire lossless transmission lines characterized by series inductance and shunt capacitance. Each line section has loop inductance (L H/m) and line-to-line capacitance (C F/m), creating a series inductance LΔx and shunt capacitance CΔx over distance Δx. This model captures voltage and current behavior using Kirchhoff's laws.

Kirchhoff's laws generate partial differential equations relating voltage, current, position, and time. Laplace transforms convert these partial differential equations into ordinary differential equations, assuming zero initial conditions. This simplification yields linear, second-order homogeneous differential equations with solutions describing forward and backward traveling waves.

Wave velocity depends on the inductance and capacitance per unit length of the transmission line. The relationship between these parameters governs how fast forward and backward traveling waves propagate along the line. This velocity is fundamental to understanding transient behavior and wave interactions on lossless lines.

Forward waves travel in the positive x-direction toward the receiving end, while backward waves move in the negative x-direction toward the sending end. Both are described by solutions incorporating time shifts and exponential functions. The characteristic impedance relates forward and backward currents to their respective voltages, determining how these waves interact.

Characteristic impedance is a function of inductance and capacitance per unit length, derived by equating coefficients in the second-order differential equation solutions. It defines the relationship between forward and backward voltage and current waves. Understanding characteristic impedance is essential for analyzing wave propagation and applying boundary conditions lossless lines.

Solutions are substituted back into the second-order differential equation, and coefficients of exponential functions on both sides are equated. This process yields forward and backward currents expressed in terms of forward and backward voltages and the characteristic impedance. These relationships fully define the traveling wave behavior on the transmission line.

Voltage and current vary with both position (x) and time (t) along the transmission line. As the line section length Δx approaches zero, partial differential equations emerge to capture these simultaneous dependencies. This mathematical framework is essential for accurately modeling transient wave propagation on lossless transmission lines.