Distortionless Lossy Transmission Lines Terminated by in Series Connected Rcl-loads

The paper deals with a lossy transmission line terminated at both ends by non-linear RCL elements. The mixed problem for the hyperbolic system, describing the transmission line, to an initial value problem for a neutral equation is reduced. Sufficient conditions for the existence and uniqueness of periodic regimes are formulated. The proof is based on the finding out of suitable operator whose fixed point is a periodic solution of the neutral equation. The method has a good rate of convergence of the successive approximations even for high frequencies.


Introduction
The principal importance of transmission lines investigations has been discussed in many papers (cf.for instance [1][2][3][4][5][6][7][8]). In a previous paper [9] we have investigated lossless transmission lines terminated by in series connected RCL-loads.In [10] we have considered a lossy transmission line terminated by a resistive load with exponential V-I characteristic.In [11] we have considered periodic regimes for lossy transmission lines terminated by parallel connected RCL-loads.Here we investigate lossy transmission lines terminated at both ends by in series connected RCL-loads but in contrast of [11] the capacitive element has a nonlinear V-C characteristic.Unlike of the usually accepted approach (cf.for instance [12,13]) we consider first order hyperbolic system instead of the Telegrapher's equation derived from it.First we reduce the mixed problem for the hyperbolic system to an initial value problem for neutral system of equations on the boundary [14].Extending ideas from [15][16][17] we introduce operators whose fixed points are periodic solutions of the neutral system.Our treatment is based on the fixed point method (cf.[18]).All derivation are performed under assumption The last condition is known as Heaviside one and it implies that the waves propagate without distortion.
We would like to mention the advantages of our method in comparison of the other used ones: lumped element method, finite element method and finitedifference time-domain method (cf.for instance [19][20][21]).
If we use numerical methods we have to keep one and same accuracy.But here we consider nonlinearities of polynomial and transcendental type (for exponential ones cf.[10]).For such "bad" nonlinearities (cf.[2]) there are examples showing that if we want to keep the same accuracy it should be reduced step thousands of times.
Here we obtain (even though approximate) an analytical solution for voltage and current beginning with simple initial approximations.We proceed from the system: where L, C, R and G are prescribed specific parameters of the line and 0   is its length.Here the current   , i x t and voltage   , u x t are unknown functions.The initial conditions for the foregoing system (1) are prescribed functions . The boundary conditions can be derived from the loads and sources at the ends of the line (cf.Figure 1).In view of the Kirchoff's voltage law for the voltages between the nodes a, b and d for x = 0 we obtain: where is the source voltage.The voltage is: .
To calculate the voltage of the condenser we proceed from the relation (assuming To calculate the voltage of the inductor we proceed from .
Therefore the first boundary condition is: Here . , .R C and are characteristics (in general, nonlinear functions) of RCL-loads at right end.

. L
Now we are able to formulate the initial-boundary (mixed) value problem for the hyperbolic transmission line system of equations: to find a solution , , , u x t i x t of the hyperbolic system (1) for and the boundary conditions where


. and (.) are prescribed functions.So the system (1) and conditions ( 6) -( 8) form a mixed problem for a lossy transmission line equations.

Reducing the Mixed Problem for the Transmission Line System to an Initial Value Problem for a Nonlinear Neutral System
First we present system (1) in matrix form: where 1 where 0 1 , 1 0 form we solve the characteristic equation: Denote the matrix formed by eigenvectors by  and its inverse one-by , , Replacing in Equation ( 10) we obtain Since H -1 is a constant matrix we have: After multiplication from the left by H we obtain We consider distortionless lossy transmission lines that means the following Heaviside condition is fulfilled: Then HBH -1 can be simplified and the last system becomes: The new initial conditions we obtain from conditions (6) and system (12): Further on we set Then system ( 14) can be written in the form: The initial conditions (15), ( 16) remain the same ones: From system (12) and denotation (17) we obtain and then We assume that the unknown functions are  .An integration along the charate- t T  and then Equations ( 22) and (23) become Now replace expressions (20) into the first boundary condition and obtain: (21) into the second boundary condition we obtain the following equation Then we put t T t   and change the variables in the integrals: To solve the above equations with respect to the derivatives d ( ) d W t t and d ( ) d J t t we have to divide the above equations into and respectively.
0 ( , We have to ensure a strict positive lower bound for .We can find an interval This can be done if the polynomials have suitable properties (cf. 4. Numerical example).
We also briefly recall: are constants and The derivatives of the functions ( ) ( ) The explicit form of the inverse function for 2  h is: We need the derivative given below (see Equation ( 26)) and the following estimates: For the I-V characteristics we assume that they are po Now we are able to formulate a periodic problem for th lynomial functions The initial functions can be obtained from the initial conditions (7) by shifting along the characteristics of the initial functions [9][10][11]).

Main Results
Here we formulate conditions for the existence-uniqueness of a periodic solution of neutral functional differential system ( 27), (28) (for definition of neutral equation see [15]).First we define a suitable operator generating by the right-hand sides of the Equations ( 27), (28).We find a 0 periodic solution of Equations ( 27), (28) on the interval coinciding with prescribed 0 riodic function on   0,T and then one can continue it periodically because our system is autonomous one.
By 0 we mean the space consisting of all measurable essentially bounded 0 T periodic functions whose derivatives are also essentially bounded and Introduce the sets: , .

B B 
are T periodic functions.

t i T T T T T T t T T B W J t I W J s s t T I W J s s T I W J s s t t T T T T
It is easy to see that changing the integration order one obtains

T T T T t T T T T T T I W J s s t T T I W J s s s I W J s s
The periodic problem ( 27), (28) has a solution Here the constant 0   will be prescribed below.The sets u X and i X turn out into metric spaces with metrics
Proof: Define the operator ( , ): X X  by the above formulas.In what follows we show Using inequality (24) and proceeding as in [11] we obtain (for sufficiently large ):

T T t u T T T T t t T T T T T T R L T B W J t U W J s s U W J s s e U W J s s t e e J T W J e
( 1 ) For the second component of the operator we obtain In what follows we show that B is contractive operator.We need the following preliminary inequalities

T T T T t T T T T T t t T T T T t T B W J t B W J t t T I W J s I W J s s I W J s s I W J s s T I W J s s I W J s s t T e
e e e e W J J W , , , .
Finally we have to obtain an estimate for .It is easy to prove the following inequalities: , , , .
For the derivative of the second component of B we obtain The above inequalities imply We choose Then should be chosen smaller than 1.

Numerical Example
We choose resistive elements with the following V-I characteristics i.e.
and consequently 1 31 Taking into account Most often the initial approximation is chosen to be simple functions, namely: Then we have ), ) and therefore (recall 1 2 2 we can disregard this terms and obtain

Lemma 2 .
If the assumptions (IN) and (E) are fulfilled and  Therefore B is contractive operator and has a unique fixed point in M (cf.[18]).It is a unique periodic solution of system (26), (27).