Oscillator with Distributed Nonlinear Structure on a Segment of Lossy Transmission Line

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1. Introduction

The present paper is devoted to investigation of self-oscillators with distributed amplifying structure of tunnel diode type realized on a segment of lossy transmission line. The transmission line is terminated by nonlinear reactive elements. Such problems and their applications (for instance to RF-circuits, PCB-s problems and so on) are usually considered by means of various methods (slowly varying in time and space amplitudes and phases, numerical methods and so on, cf. [1] - [14] ). We have developed (cf. [15] ) a general approach for investigation of lossy transmission lines terminated by nonlinear loads without Heaviside condition. From mathematical point of view in [15] , we consider just linear hyperbolic systems. In [16] and [17] , we have considered a Josephson superconductive transmission line system with sine type nonlinearities. Our main purpose here is to consider lossy transmission line with polynomial nonlinear distributed structure that leads to a nonlinear hyperbolic system. We extend Abolinya- Myshkis method (cf. reference of [16] ) to attack the nonlinear boundary value problem and propose a new general approach to reduce the mixed problem for such nonlinear systems to an operator form in suitable function spaces. The arising nonlinearity is of polynomial type in view of distributed tunnel diode element. The nonlinear characteristics of the reactive elements generate nonlinear boundary conditions. We prove the existence of an approximated solution of the mixed problem and show a way to reach this solution by successive approximations.

We proceed from the circuit shown on Figure 1, where and are nonlinear reactive elements. We consider that a particular case is a nonlinear capacitance, while is a nonlinear inductance. In a similar way, it can be treated more complicated circuits (cf. [15] ).

A lossy transmission line with distributed nonlinear resistive element can be prescribed by the following first order nonlinear hyperbolic system of partial differential equations (cf. [1] - [14] ):

(1)

where and are the unknown voltage and current, while L, C, R and G are inductance, capacitance, resistance and conductance per unit length; is itslength; and is a prescribed polynomial of arbitrary order with intervalof negative resistance (in the applications most often of third order). For the above

Figure 1. Lossy transmission line with distributed nonlinear resistive element with an interval of negative differential resistance in the characteristic.

system (1), one can formulate the following initial-boundary (or briefly mixed) problem: to find the unknown functions and in such that the following initial and boundary conditions are satisfied

(2)

(3)

where and are prescribed initial functions the current and voltage at the initial instant; are characteristics of the reactive elements.

Rewrite the system (1) in the form

(4)

2. Transformation of the Partial Differential System

First we present the system (4) in matrix form:

.

Introducing denotations

we have

. (5)

To transform the matrix in diagonal form we solve the characteristic equation Its roots are,. The eigen- vectors are,. We form the matrix by eigen-vectors. Then and .

Introduce new variables, where. Therefore

and (6)

Substituting in Equation (5) we obtain

or

(7)

But

,

Then introducing denotations we obtain from Equation (7)

(8)

Introduce again new variables

(9)

and then the system (8) reduces to

The new transformation formulas are

(10)

The new initial conditions we obtain from Equations (2), (6) and (9) for:

The new boundary conditions we obtain from Equations (3):

(11)

In order to solve the last equations with respect to the derivatives we consider the properties of nonlinear capacitive and inductive elements. For the capacitive element (cf. [15] ) we have, where are constants and. If, then has strictly positive lower bound.

Indeed (cf. [15] ),.

To obtain we make

Assumption (C).

If we choose it follows and for and therefore

.

Besides

.

The inductive element has I-L characteristic of polynomial type.

To solve the second equation (11) with respect to we make

Assumptions (L),.

In view of we obtain

We present the above relations in an integral form under

Assumptions (CC),

3. Operator Formulation of the Mixed Problem for the Transmission Line System

Now we are able to formulate the mixed problem with respect to the unknown functions: to find satisfying the system and initial and boundary conditions

(12)

In what follows we give an operator representation of the above mixed problem (12).

Recall that and and. The ordinary differential equations (Cauchy problem) for the characteristics of the hyperbolic system are

for each (13)

for each (14)

The functions and are continuous ones. This im- plies that for every there is a unique (to the left from) solu- tion for;, and respectively for;. Denote by the smallest value of such that the solution of Equation (13) still belongs to and respective-

ly the solution of Equation (14) by. If then or and respectively if then or. In our case

;

Remark 1. We notice that. It is easy to see that

Introduce the sets:

.

Prior to present problem (12) in operator form we introduce

and

or

So we assign to the above mixed problem the following system of operator equations (cf. [16] , [17] ):

4. Existence Theorem

In order to obtain a contractive operator we consider the mixed problem (12) on the subset. We introduce the sets

and

where and μ are positive constants chosen below. It is easy to verify that turns out into a complete metric space with respect to the metric

,

where

,

.

Now we define an operator by the formulas

Remark 2. Assumption (C) and Assumptions (L) in view of Equations (10) imply

;.

Theorem 1. Let the following conditions be fulfilled:

1) Assumption (C), Assumptions (L), Assumption (CC) and, for as are sufficiently small while is sufficiently large;

2);

3);

4);

5).

Then there exists a unique solution of the problem (12).

Proof: We establish that the operator B maps the set into itself.

First we notice that and are continuous functions. We show

,.

Indeed, for sufficiently small and in view of and we have

Then for the first component we have

In view of

and

for sufficiently small for the second component we obtain:

Now we show that B is a contractive operator.

Indeed, for the first component we obtain:

Similarly for the second component we obtain

Therefore

and the operator B has a unique fixed point which is a solution of the mixed problem above formulated in the set.

Theorem 1 is thus proved.

Remark 3. We point out that for every there is a unique solution in. The sequence is not necessary convergent when. To find a convergent subsequence we proceed as in [17] . Extending the solution on we can choose a convergent subsequence. The first approximation can be chosen, for instance, as a solution of the linearized system (12).

5. Conclusion Remarks

1) We note that the interval is not sufficiently small.

2) We show a simple verification of all inequalities of the main theorem for soft nonlinearity (cf. [1] ). Consider a lossy transmission line (cf. [1] - [15] ) satisfying the Heaviside condition with specific parameters:

;

;;

;

;

.

Let us choose a polynomial with interval of negative differential resistance, and. Then; . The pn-junction capacity is , while the pn-junction potential. For and the minimal value of is.

We choose such that.

Then the inequalities from Remark 3 and two of inequalities from Theorem 1 become

;

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