Circuits and Systems, 2011, 2, 297-310 doi:10.4236/cs.2011.24042 Published Online October 2011 (http://www.scirp.org/journal/cs) Copyright © 2011 SciRes. CS Distortionless Lossy Transmission Lines Terminated by in Series Connected RCL-Loads Vasil G. Angelov 1 , Marin Hristov 2 1 Department of Mathematics, University of Mining and Geology, Sofia, Bulgaria 2 Department of Microelectronics, Technical University of Sofia, Sofia, Bulgaria E-mail: angelov@mgu.bg, mhristov@ecad.tu-sofia.bg Received July 29, 2011; revised August 26, 2011; accepted September 4, 2011 Abstract 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. Keywords: Lossy Transmission Line, RCL-Nonlinear Loads, Fixed Point, Periodic Solution 1. Introduction The principal importance of transmission lines investi- gations has been discussed in many papers (cf. for instance [1-8]). In a previous paper [9] we have inves- tigated 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 cons- idered periodic regimes for lossy transmission lines term- inated by parallel connected RCL-loads. Here we inve- stigate 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 in- stead 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 equ- ations on the boundary [14]. Extending ideas from [15- 17] we introduce operators whose fixed points are peri- odic solutions of the neutral system. Our treatment is based on the fixed point method (cf. [18]). All derivation are performed under assumption / / R L G C  ( ). 0, 0 R G   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 finite- difference time-domain method (cf. for instance [19-21]). 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:       , , , 0 uxt ixt C Gu , xt t x        (1)       , , , 0 ixt uxt L Ri t x   , xt                  2 , , : , 0, 0, xt xt R xt       ,           0 0 ,0 , ,0 , 0, ux u x ix i x x     (2) where L, C, R and G are prescribed specific parameters of the line and 0   is its length. Here the current   , ixt and voltage   , uxt are unknown functions. The initial conditions for the foregoing system (1) are prescribed functions     i x 0 , u x 0 . The boundary con- ditions can be derived from the loads and sources at the ends of the line (cf. Figure 1).