Nonlinear Membrane Circuit Loaded on a Lossy Transmission Line without the Heaviside’s Condition


  •   Vasil Angelov


The paper deals with transmission lines terminated by a nonlinear circuit describing a simplified model of membrane. This means that all elements of the membrane circuit are nonlinear ones as follows: in series connected LR-loads parallel to C-load. Using the Kirchhoff’s laws we formulate boundary conditions. For lossy transmission lines systems with the Heaviside’s condition, the mixed problem is considered in previous papers. The main goal of the present paper is to investigate the same problem for lossy transmission lines without the Heaviside’s condition. We reduce the existence of solution of the more complicated mixed problem for such a system to the existence of fixed point of an operator acting on a suitable function space. Then by ensuring the existence of this fixed point we obtain conditions for existence of a generalized solution of the mixed problem. The obtained conditions are easily verifiable. We demonstrate the advantages of our method by a numerical example.

Keywords: Fixed Point Theorem, Heaviside Condition, Lossy Transmission Line, Mixed Problem for Hyperbolic System, Neutral Equation


V. G. Angelov, A Method for Analysis of Transmission Lines Terminated by Nonlinear Loads, Nova Science, New York, 2014.

V. G. Angelov, “Lossy transmission lines with Josephson junction − continuous generalized solutions,” Communication in Applied Analysis, vol. 20, pp. 91-106, 2016.

K. S. Cole, and R. F. Baker, “Longitudinal Impedance of the Squid Giant Axon,” J. Gen. Physiology, vol. 24, p. 771, 1941.

K. S. Cole, and H. T. Curtis, “Electric Impedance of the Squid Giant Axon During Activity, “J. Gen. Physiology, vol. 22, p. 649, 1939.

A. Scott, Active and Nonlinear Wave propagation in Electronics, Wiley-Interscience, New York, London, Sydney, Toronto, 1970.

M. Amin, P. P. Dey and H. Badkoobehi, “A Complete Electrical Equivalent Circuit Model for Biological Cell”, Recent Advances in Systems, Communications & Computers, Selected Papers from the WSEAS Conferences in Hangzhou, China, April 6-8, 2008, pp. 343-348.

J. Stenarson, T. N. T. Do, H. Zhao, P. Sobis, A. Y. Tang, K. Yhland and J. Stake, “Sensitivity Analysis of TRL Calibration in Waveguide Integrated Membrane Circuits,” IEEE Transactions on Terahertz Science and Technology, vol. 5, no. 3, pp. 558-565, August 2013.

P. H. Siegel, R. P. Smith, M. C. Graidis and S. C. Martin, “2.5-THz GaAs Monolithic Membrane-Diode Mixer,” IEEE Trans. Microw. Theory Techn., vol. 47, no. 5, pp. 596-604, May 1996.

H. Zhao, P. Sobis, J. Hanning, J. T. Bryllert, A.-Y. Tang abd J. Stake, “Development of a 557 GHz GaAs Monolithic Membrane-Diode Mixer,” 24th Int. Conf. on Indium Phosphide and Related Mater., 2012.

H. Zhao, A.-Y. Tang, P. Sobis, T. Bryllert, K. Yhland, J. Stenarson and J. Stake, “Submillimeter Wave S-parameter Characterization of Integrated Membrane Circuits,” IEEE Microw. Wireless Compon. Lett., vol. 21, no. 2, pp. 110-112, Feb. 2011.

S. Hirata, A. Ono, M. Kurosawa, M. Ohyama and H. Chiba, “Migration-Proof Membrane Circuits for Fine-Pitch Connector,” Fujikura Technical Review, pp. 25-29, 2007.

L. Georgiev, “On the continuous generalized solutions of the lossless transmission lines system with Josephson junction,” Results in Nonlinear Analysis, vol. 1, pp. 30-45, 2018.

L. Georgiev and V. G. Аngelov, “On a problem of self-oscillations in nonlinear systems,” Proceedings of National Conference “Automatics and Informatics’ 98, pp. 66-69, Sofia, 1998.

D. D. Parashkevova, “Delay equations with a control parameter arising in flotation processes,” International Journal of Engineering Sciences & Research Technology, vol. 7, no. 12, pp. 57-63, 2018.

E. Omorogiuwa, W. Ikonwa, “Distance relay protection improvement of Alaoji-Afam 330 kV transmission line,” European Journal of Engineering Research and Science, vol.3, no.9, pp. 40-49, 2018

J. Stenarson, Than Ngoc Thi Do, Huan Zhao Ternehall, P. Sobis, A. Y. Tang, K. Yhland and J. Stake, “Sensitivity analysis of TRL calibration in waveguide integrated membrane circuits,” IEEE Microwave and Wireless Components Letters, vol. 21, no. 2, pp. 110– 112, 2011.

V. G. Аngelov, Fixed Points in Uniform Spaces and Applications, Cluj University Press, Cluj-Napoca, 2009.


Download data is not yet available.


How to Cite
Angelov, V. 2019. Nonlinear Membrane Circuit Loaded on a Lossy Transmission Line without the Heaviside’s Condition. European Journal of Engineering Research and Science. 4, 10 (Oct. 2019), 190-197. DOI: