1. Introduction

The main purpose of the present paper is to consider a lossless transmission line loaded by in series connected nonlinear RL-loads parallel to C-load. Such a configuration arises not only in radio frequencies devices but in various geophysical studies as well (cf.[1]). Our goal is to demonstrate the advantages of our method[2] used in analogous problems. So we came up with an approach to solve this set of problems (cf.[3]-[7]).

The primary purpose of the present paper is twofold. First, to formulate the mixed problem for hyperbolic Telegrapher equations corresponding to the nonlinear circuits on Figure 1. The boundary conditions are nonlinear ones in view of the nonlinear characteristics of the loads. Important first step on the base of Kirchoff’s law is to derive the boundary conditions in the form of differential equations on the boundary. This is done in 2.1. Reducing the mixed problem to a neutral system on the boundary is made in 2.2 following the technique from[2],[5] and[6].

Second, to present a method for solving neutral equations with “bad” (non-Lipschitz) nonlinearities. In 2.3 the domains of the nonlinear characteristics are defined. They have to possess strictly positive lower bounds and namely Assumption (C) for capacitive functions and (L) for inductive functions ensure that. In 2.3 the choice of functional spaces and a family of pseudo-metrics is directly related to the operator representation of the periodic problem in 2.4. Let us point out that the extension of Bielecki norm allows to overcome the difficulties caused by nonlinearity of the characteristic functions.

The key role plays Lemma 3. It guaranties that the fixed points of the operator defined in 2.4 are periodic solutions of the neutral system and conversely. The main result is Theorem 1 (cf. 2.5) and its proof consists of two parts. The first one is to show that operator B maps the set into itself. The second one is to show that B is contractive operator. The fixed point of B is the required periodic solution. The numerical example in 2.6 shows that for applications are required only inequalities obtained in the proof of the main theorem.

2. Main Results

2.1. Formulation of the Problem

Let be the length of the transmission line and , where L is per unit length inductance and C – per unit-length capacitance. In accordance of Kirchoff’s voltage-law (Figure 1) we have to add the voltages of the elements and after that to define the current of and finally to add it with the current of . In the real cases parallel to and is connected an input voltage , where is the amplification coefficient. Since we have to add the currents one can replaced by an equivalent current source . We assume that the second end is terminated by the same configuration.

Assume that and are nonlinear elements, that is, and are nonlinear functions. So we have

and then

Figure 1. Lossless transmission line terminated at both ends by in series connected RL-loads parallel to C-load

Since current is connected parallel to RL-elements then Kirchoff’s current-law yields

(1)

Since then can be found as a solution of differential equation

But is unknown too and then from (1) we have

or one more differential equation:

Analogously for the right end we have

Now we are able to formulate the initial-boundary value (mixed) problem for the transmission line equations: to find a solution of the first order partial differential system of hyperbolic type

(2)

for

satisfying the initial conditions

(3)

and the boundary conditions for

(4)

and for

(5)

2.2. Reducing the Mixed Problem to an Initial Value Problem on the Boundary

We proceed from the lossless transmission line equations

(6)

Rewrite system (6) in the form

(7)

Adding and subtracting (6) and (7) in view of we get:

(8)

Let us put

and hence

It follows

(9)

For and we have

Replace them in (4) and (5) we obtain

and

But

We assume that the unknown functions are

and then in view of

solve with respect to the derivatives we reach the system

(10)

So we have obtained a neutral system of differential equations with retarded arguments.

2.3. Estimates of the Arising Nonlinearities and Introducing Metrics

We consider C-V characteristics

,

for

where have a strictly positive lower bounds. Put

..

Then

For the I-V characteristics we assume

Then . For we get

Assumptions (L):

Assumptions (C):

For the V-I characteristics we assume that they are of polynomial type:

We introduce the sets for the unknown functions

where is the set of all continuously differentiable -periodic functions and are positive constants (chosen below) and

Introduce the metrics

The set turns out into a complete metric space with respect to the metric:

2.4. Operator Presentation of the Periodic Problem

Now we formulate the main problem: to find a -periodic solution of the system (10) on the interval coinciding with prescribed -periodic initial functions on the interval respectively:

Remark 1. As in[3] one can shift the initial function of the mixed problem from the interval along the characteristic to the interval

The main difficulty is to define a suitable operator whose fixed points are solutions sought. We define it in the following way: the four-tuple functions

are defined on every interval (for every k = 0,1,2, …, m-1) by the expressions

and are translated to the right initial functions over .

From now on the following assumptions will be fulfilled:

(E):

(IN):

It follows

Lemma 1. If (E) and (IN) are satisfied and

then

are -periodic ones.

Lemma 2. If

then

The proofs can be accomplished as in[5],[6].

The following lemma guaranties that the fixed points of the above defined operator are periodic solutions of the neutral system (10).

Lemma 3. The periodic problem (10) has a solution

that is,

Proof: Let

be a -periodic solution of (10).

Then after integration of the first equation we have (recall that ):

Therefore

Analogously we obtain

that is, is a fixed point of B.

Conversely, let have a fixed point

Then

If we assume in view of one obtains a contradiction.

It follows

Analogously

Consequently

Differentiating the last equalities we conclude that (10) has -periodic solution.

Lemma 3 is thus proved.

2.5. Existence-Uniqueness of Periodic Solution

The main result contains in the following

Theorem 1. Let assumptions (L), (C), (E) and (IN) be fulfilled. Then there exists a unique T₀-periodic solution of (10).

Proof: We show that maps into itself. Indeed

We have

Further on we have

and

and

On the other hand

We have

Then

Finally

and

It remains to obtain the Lipschitz estimates for the right-hand sides of the equations. Omitting calculation of the partial derivatives we obtain:

Then

It follows

Further on we get

It follows

For the third component we obtain:

For the fourth component we have

It follows

For the derivatives we obtain

Consequently

Further on we have

Therefore

Consequently

Further on we have

Then

For the last component of the derivative we obtain

and

On the other hand

Consequently

.

Finally we have

where

Then B has a unique fixed point which is a periodic solution of (10).

Theorem 1 is thus proved.

2.6. Numerical Example

For a transmission line with length Then .

Let us check the propagation of waves with . We have

We choose , then and

.

We choose resistive elements with the following V-I characteristics and inductive elements with Then For one obtains .

Let us take

where . Let us choose and . Then

.

Then the above inequalities for become:

We calculate just since are of order : ; ; ; .

3. Conclusions

We consider transmission lines neglecting the lossies. This makes it possible to find conditions for the existence and uniqueness of periodic regimes. This natural physical fact is confirmed by the mathematical method we apply.

• In order to prove an existence-uniqueness theorem we introduce an operator (unknown in the literature up to now) whose fixed points are periodic solutions of the problem stated. We apply contractive fixed point theorems in metric spaces. By extended Bielecki metrics we overcome the difficulties caused by polynomial and transcendental nonlinearities.

• The numerical example 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 simple inequalities between the basic specific parameter of the lines and loads.

• We show a unified approach for solving problems for analysis of transmission lines terminated various configurations of nonlinear loads. In contrast to various results devoted to numerical methods[8]-[14] we obtain an explicit approximated solution.

• We emphasize on the fact that first we prove not only an existence but and uniqueness of the solution as well. So our successive approximations tend to this solution. All other methods need such a uniqueness result. Unfortunately in most papers the uniqueness is not ensured.