Nonlinear Membrane Circuit Loaded on a Lossy Transmission Line without the Heaviside Condition

DOI: http://dx.doi.org/10.24018/ejers.2019.4.10.1583 190  Abstract—The paper deals with transmission lines terminated by a nonlinear circuit describing a simplified model of membrane. This means that all elements of the membrane circuit are nonlinear ones as follows: in series connected LRloads parallel to C-load. Using the Kirchhoff’s laws, we formulate boundary conditions. For lossy transmission lines systems with the Heaviside’s condition, the mixed problem is considered in previous papers. The main goal of the present paper is to investigate the same problem for lossy transmission lines without the Heaviside’s condition. We reduce the existence of solution of the mixed problem for such a system to the existence of fixed point of an operator acting on a suitable function space. Then by ensuring the existence of this fixed point we obtain conditions for existence of a generalized solution of the mixed problem. The obtained conditions are easily verifiable. We demonstrate the advantages of our method by a numerical example.


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Abstract-The paper deals with transmission lines terminated by a nonlinear circuit describing a simplified model of membrane.This means that all elements of the membrane circuit are nonlinear ones as follows: in series connected LRloads parallel to C-load.Using the Kirchhoff's laws, we formulate boundary conditions.For lossy transmission lines systems with the Heaviside's condition, the mixed problem is considered in previous papers.The main goal of the present paper is to investigate the same problem for lossy transmission lines without the Heaviside's condition.We reduce the existence of solution of the mixed problem for such a system to the existence of fixed point of an operator acting on a suitable function space.Then by ensuring the existence of this fixed point we obtain conditions for existence of a generalized solution of the mixed problem.The obtained conditions are easily verifiable.We demonstrate the advantages of our method by a numerical example.

Index Terms-Fixed Point Theorem, Heaviside Condition, Lossy Transmission Line, Mixed Problem for Hyperbolic
System, Neutral Equation.

I. INTRODUCTION
The primary purpose of the present paper is to investigate a lossy transmission line terminated by a membrane circuit (cf.Fig. 1).We have considered lossy transmission lines terminated by such circuits in previous papers (cf.References in [1]).However, these considerations are under the Heaviside's condition which implies existenceuniqueness result.Here we study the same problem without the Heaviside's condition and show that the uniqueness of solution fails on the whole interval of time (cf.[2]).The problem of modelling of membrane circuits is stated still in the papers [3], [4] and the monograph [5], and is of interest by now days.We note some papers in this area [6]- [11] and other applications [12]- [16] to nonlinear problems attacked by the fixed point method [17].
The paper consists of six sections.Section 1 is an introduction where the mixed problem for lossy transmission line (hyperbolic) system is formulated and the type of nonlinearity of the circuit elements is described.The boundary conditions by means of the Kirchhoff's law are obtained.Section 2 contains a transformation of the transmission line system in a diagonal form.A new form of the initial and boundary conditions with respect to the new variables is derived.The considerations are made under the basic assumption 0 , that is, the Heaviside's condition is not satisfied.In Section 3 some preliminary formulations are given and the mixed problem in a suitable operator form is presented.Section 4 contains the main result.Under easily verified conditions the existence of a solution of the operator equation is proved.We apply the fixed point method to establish existence of a solution on the domain ]

0
, where  is the length of the transmission line, − the speed of propagation of the waves.The specific parameters of the line are: L − per unit-length inductance, Cper unit-length capacitance, Rper unit length resistance, Gper unit length conductance.On the set ] , the uniqueness of solution fails (cf.[2)].The conditions of the main theorem are easily verifiable.Section 5 contains a numerical example demonstrating the applicability of our method to real cases.In the Conclusion (Section 6) we show of how to obtain a sequence of successive approximations to the solution.
We proceed from the lossy transmission line system 0 ) , ( The boundary condition is already derived (cf.Fig. 1) in [1] for 0 Here we consider a nonlinear capacitive element with For the derivatives we have For the I-L characteristics we take polynomial type

II. TRANSFORMATION OF TRANSMISSION LINE SYSTEM INTO
A DIAGONAL FORM First we present (1) in the form: We assume that the Heaviside's condition is not satisfied, that is, and then (2) becomes: We simplify (3) So we obtain the following mixed problem (MP): to solve the system

III. OPERATOR PRESENTATION OF THE MIXED PROBLEM
First we consider Cauchy problem for the characteristics (cf.[2]): The [1/ (.) ] ( , ) So we assign to the above mixed problem the following system of operator equations As a matter of facts we look for a generalized solution.We call a generalized solution of the mixed problem (MP) the solution of the operator equations (7).
Introduce the function sets and μ are positive constants chosen below.The inequalities are satisfied almost everywhere.It is easy to verify that the set turns out into a complete metric space with respect to the metric Let the following conditions be fulfilled: , where 00 w and 00 j are sufficiently small positive constants.
For the second component we get )) , ( For the third component we obtain Thus the fixed point of the operator B is a solution of the generalized problem.
The Theorem 1 is thus proved.
The solution can be extended on the whole interval ] , 0 [  proceeding as in [2], but this solution is not unique one.

V. NUMERICAL EXAMPLE
Here we check all inequalities guaranteeing the existence result: For a transmission line with specific parameter We choose resistive elements with the following       In this way we have a solution in an analytical type.Usually the initial functions can be chosen the solution of the linear problem or even constant functions.

Fig. 1 .
Fig. 1.Lossy transmission line without the Heaviside's condition terminated by a nonlinear membrane circuit.
differential equation problems in electrodynamics are solved by means of finite difference time domain method.Our fixed point method allows us to obtain by successive approximations by the formulas /