Vasil G. Angelov, PhD
Department of Mathematics, Faculty of Mine Electromechanics, University of Mining and Geology”St. I.Rilski”, Sofia, Bulgaria

Series: Mathematics Research Developments
BISAC: MAT027000

It is known that transmission lines model various guiding structures such as power lines, wires, cables, printed circuit board traces, buses for carrying digital data in electronic circuits, and microwave circuits and so on. The transmission line theory has received renewed attention because of`important applications in the area of high-speed VLSI interconnect, while it has retained its significance in the area of power transmission.

The behavior of the transmission lines is described by transmission line equations. From a mathematical point of view they represent a hyperbolic system of first order partial differential equations. The transmission lines are terminated by various configurations of linear or nonlinear loads. The Kirchoff’s laws regulate the behavior of these lumped circuits and define boundary conditions. Adding appropriate initial conditions, we can formulate a mixed problem for the transmission line system with unknown functions of the voltage and current of the line.

The main purpose of the present book is to propose a method for solving the mixed problem for transmission line systems reducing it to a neutral equation (or system) on the boundary. Arising nonlinearities in the neutral systems are caused by nonlinear characteristics of the RGCL-loads. In view of the applications we consider mainly periodic and oscillatory problems for lossless transmission lines. We point out, however, that here we propose an extended procedure for reducing the mixed problem for lossless and lossy transmission lines. We introduce also an extension of Heaviside condition and this way we can consider the case of time-varying specific parameters-per-unit length resistance, conductance, inductance and capacitance. We find a solution of the obtained neutral equations by discovering operators whose fixed points in suitable function spaces are periodic or oscillatory solutions of the formulating problems.

Using fixed point theorems for contractive mappings in uniform and metric spaces (proved by the author in the previous papers) we prove existence-uniqueness results for periodic and oscillatory problems. We obtain also successive approximations of the solution with respect to a suitable family of pseudo-metrics and give an estimate of the rate of convergence. Although the question of finding the initial approximation is not trivial. We show that one can begin with a simple harmonic initial approximation. The rate of convergence depends on the parameters of the transmission lines and characteristics of the nonlinear RCL-loads. Our conditions are applicable even in some cases to non-uniform transmission lines. Numerical examples demonstrate the applicability of the main results to design of circuits. It is easy to verify a system of inequalities between basic parameters without examining the proofs of the theorems. (Imprint: Novinka )