Analytical Solutions for Wave Process in Electrical Line with Capacitive, Inductive and Active Load

Abstract

The methodology for obtaining exact analytical solutions for the waveform regimes in the electric lines with loads with concentrated parameters of various types is justified. As a mathematical model the partial differential equations (telegraph equations) are used and the coefficients of these equations are presented in system of relative units. The system of equations is changed by introducing two new unknown functions, which are functions of voltage and current in the line. The mathematical relations are transformed into equations with one independent function. The new variables represent the Riemann invariants. The boundary conditions for these two functions are described by an algebraic relation and an integral- differential equation. The last equation represents the Cauchy problem for which the solutions have been determined. It was found that solutions for variables (voltage, current and Riemann invariants) are discontinuous functions. A new formulation for the boundary conditions has been proposed. It is proposed and described the process for obtaining the exact solution for the capacitance, inductive and resistive loads. The proposed method is applicable for determining voltage and current as time functions at any point of the power line. Analytical solutions can serve as test solutions for verifying precision of numerical solutions.