Oscillations in Lossy Transmission Lines Terminated by in Series Connected Nonlinear RCL-Loads

We formulate conditions for an existence-uniqueness of oscillatory regimes in transmission lines terminated by in series connected nonlinear RGLC-loads. This is achieved by introducing suitable operator whose fixed point is a seeking solution. The method allows obtaining approximated solutions and an estimate of the rate of convergence.


Introduction
In a recent paper [1] we have investigated a transmission line terminated by in series connected RCL-loads and we have reduced the mixed problem for the lossy transmission line equations to an initial value problem for a system of neutral equations on the boundary. Then we proved an existence-uniqueness result of periodic boundary value problem for the neutral system obtained but for one period. Global behaviour of the solution is declining rather than periodically. In contrast to parallel connected RCL-loads (cf. [2]) here the mu ltip lier Fro m the physical point of view this means that the signal vanishes in time. Moreover one can notice the voltage and current in the lossy transmission line are products of a periodic function and an exponential function, that is, , should be oscillat ing functions and vanish exponentially at infinity. This suggest us to state the problem for an existence-uniqueness of a global oscillatory solution on ) , [ ¥ T vanishing exponentially at infinity. As in [1], [2] we assume the Heaviside condition is satisfied, that is, . It imp lies th at the t rans miss ion line is without distortion. Here we o mit the reduction (cf. [1]) of the mixed problem for the hyperbolic system to a neutral system on the boundary and prove by the fixed point method (cf. [3]) an existence-uniqueness of the oscillatory solution for the n eu tral s ystem. Th e main d ifficu lty is g en erated by n o n lin earities o f RLC-lo ads (cf. [4]- [ 7]). W e o b tain successive approximation of the seeking oscillatory solution in an exp licit fo rm and estimate the rate of convergence.
Especially we want to draw attention to a fundamental disadvantage of lot of papers -solving equations without guaranteed uniqueness of the solution. Whatever the method (numerical, appro ximated and so on) to apply without uniqueness is not known to which o f the many solutions are coming.
We mention recent various approaches for an analysis of transmission lines [8]- [15] .We would like to point out that we obtain an appro ximated solution but in explicit form beginning with simple functions.

Mixed Problem for the Lossy Transmission Line System terminated by in Series Connected RCL-Loads
We proceed fro m the lossy transmission line equations system: (2) where L, C, R and G are p rescribed specific parameters of the line and L >0 is its length and ) ( ), (  (3) and analogously for  Recall briefly some results fro m [1]. For the voltages of the condenser p C we proceed fro m the relat ion (assuming   (6) The voltage of the inductor p L is  The mixed problem (1)- (4) can be reduced to an init ial value problem for a nonlinear neutral system. Recall so me transformations fro m [1]: , ( 2 ) , , ( 2 ) , The init ial conditions remain unchanged because ) ( Assume that the unknown functions are (cf. and taking into account the relations obtained after integration along the characteristics ) , ( ( should be oscillatory ones. They should satisfy the following So o mitting some transformat ions given in details in [1] we reach the problem for existence-uniqueness of oscillatory solution ( ) of the fo llo wing system: )( , ( The init ial functions are obtained as in [1]. First we formu late the conditions for the initial functions Assume that one can find an interval This can be done if the polynomial has suitable properties (cf. Nu merical example).

An Existence-Uniqueness of Oscillatory Solution for the Neutral System
Here we introduce an operator representation of the oscillatory problem and by a fixed point theorem in uniform spaces [3] we establish an existence-uniqueness of global oscillatory solution. Now we are able to formu late the main problem: to find a solution of (8) with advanced prescribed zeros on an interval be the set of zeros of the in itial functions, that is, 0 be a strictly increasing sequence of real numbers satisfying the following conditions (C): Re mark 3.3. Let us comment the conformity condition (CC). It could be obtained replacing . We notice that (CC) beco mes a relation between the initial functions ), 0 ( ) ( If the last condition is not satisfied then the ju mp of the derivative at 0 t t = propagates to the right and it falls at some zero point because of . We do not go beyond our function space because the derivative of our functions might have ju mps at Introduce the following family o f pseudo-metrics Further on the follo wing assumptions will be hold: Therefore operator equations (13) become Differentiating the last integral equations we obtain (8). Lemma 3.1 is thus proved. Preliminary assertions: is increasing and Then there exists an unique oscillatory solution of (8), belonging to In order to show that ( ) , ( ), , ( we have to establish the follo wing inequalities: Using Preliminary assertions we have .
For the second component we have  ( 1)  (                                 . .

Numerical Example
Finally we demonstrate of how to apply the above theorem to engineering problems. We collect all inequalities imp lying an existence-uniqueness of oscillatory solution: Then

Conclusions
• We consider transmission lines taking into account the lossies. This means there is attenuation in time of the signals. This natural physical fact is confirmed by the mathematical method we apply. Namely, the transformat ion (we have used to reduce the mixed problem for hyperbolic system to a problem for neutral system on the boundary) contains an exponential function t L R e ) / ( which implies that signals (current and voltage) vanish exponentially. It reminds us that natural global solutions are not periodic ones. That is why we formulate the problem o f existence-uniqueness of an oscillatory solution.
• In order to prove an existence-uniqueness theorem we introduce an operator (unknown in the literature up to now) whose fixed points are oscillatory solution of the problem stated.
• It turns out that the space of oscillat ing functions does not form a metric space but a uniform one. Th is requires applying fixed point theorems of operators acting on uniform spaces.
• We would like to point out that by means of this fixed point method we solve nonlinear equations with various nonlinearities as polynomial, exponential and transcendental ones.
• By virtue of the theorems obtained in this paper we show that attenuating oscillat ing modes are natural for the lossy transmission lines terminated by such configuration of the nonlinear loads.
• The numerical examp le demonstrates a frame of applicability of the theory exposed (for instance to design of circuits) and shows that the method could be applied checking few simp le inequalities between the basic specific parameter of the lines and loads.
• Finally we note that a lot o f papers have been done where numerical (or other) methods are applied without uniqueness is assured. Then it is not clear to which solution is approaching. Our fixed point method guarantees a uniqueness of solution.
• The calculation of the successive approximations and the estimations of some terms (lead ing to their disregarding) simp lify the calculation of the next appro ximations. It is extremely important for any program imp lementing the method.