Advertisement

Circuits, Systems, and Signal Processing

, Volume 39, Issue 1, pp 456–473 | Cite as

Frequency Characteristics of Two Topologies Representing Fractional Order Transmission Line Model

  • Stevan M. CvetićaninEmail author
  • Dušan Zorica
  • Milan R. Rapaić
Short Paper

Abstract

Classical telegrapher’s equation is generalized in order to account for the hereditary nature of polarization and magnetization phenomena of the medium by postulating fractional order constitutive relations for capacitive and inductive elements in the elementary circuit, as well as by the two topological modifications of Heaviside’s elementary circuit, referred as series and parallel, differing in the manner in which the effect of charge accumulation effect along the line is taken into consideration. Frequency analysis of generalized telegrapher’s equations is performed, with a particular emphasis on the asymptotic behavior for low and high frequencies. It is found that, like Heaviside’s elementary circuit, parallel topology leads to low-pass frequency characteristics, while the series topology leads to band-pass characteristics. It is also demonstrated that the logarithmic phase characteristics are linear functions of frequency, being suitable for determining some of the fractional differentiation orders in generalized telegrapher’s equations.

Keywords

Generalized telegrapher’s equation Transmission line Fractional order electrical elements Frequency characteristics 

Notes

Compliance with ethical standards

Conflict of interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and publication of this article.

References

  1. 1.
    T.M. Atanackovic, S. Pilipovic, D. Zorica, Diffusion wave equation with two fractional derivatives of different order. J. Phys. A Math. Theor. 40, 5319–5333 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    T.M. Atanackovic, S. Pilipovic, D. Zorica, Time distributed-order diffusion-wave equation. II. Applications of the Laplace and Fourier transformations. Proc. R. Soc. A Math. Phys. Eng. Sci. 465, 1893–1917 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    S.M. Cvetićanin, Frakciono i topološko uopštenje jednačine telegrafičara kao model električnog voda. Ph.D. thesis, Fakultet tehničkih nauka, Univerzitet u Novom Sadu (2017)Google Scholar
  4. 4.
    S.M. Cvetićanin, M.R. Rapaić, D. Zorica, Frequency analysis of generalized time-fractional telegrapher’s equation, in European Conference on Circuit Theory and Design, Catania, Italy (September 4–6, 2017)Google Scholar
  5. 5.
    S.M. Cvetićanin, D. Zorica, M.R. Rapaić, Frekvencijska analiza frakcionog modela električnog voda (ETRAN, Kladovo, Srbija (2017), pp. 5–8Google Scholar
  6. 6.
    S.M. Cvetićanin, D. Zorica, M.R. Rapaić, Generalized time-fractional telegrapher’s equation in transmission line modeling. Nonlinear Dyn. 88, 1453–1472 (2017)CrossRefGoogle Scholar
  7. 7.
    A. Dzieliński, G. Sarwas, D. Sierociuk, Comparison and validation of integer and fractional order ultracapacitor models. Adv. Differ. Equ. 2011(11), 1–15 (2011)MathSciNetzbMATHGoogle Scholar
  8. 8.
    E. Fendzi-Donfack, J.P. Nguenang, L. Nana, Fractional analysis for nonlinear electrical transmission line and nonlinear Schroedinger equations with incomplete sub-equation. Eur. Phys. J. Plus 133, 32 (2018).  https://doi.org/10.1140/epjp/i2018-11851-1 CrossRefGoogle Scholar
  9. 9.
    J.F. Gómez-Aguilar, D. Baleanu, Fractional transmission line with losses. J. Phys. Sci. 69, 539–546 (2014)Google Scholar
  10. 10.
    I.S. Jesus, J.A.T. Machado, Development of fractional order capacitors based on electrolyte processes. Nonlinear Dyn. 56, 45–55 (2009)CrossRefGoogle Scholar
  11. 11.
    M.S. Krishna, S. Das, K. Biswas, B. Goswami, Fabrication of a fractional order capacitor with desired specifications: a study on process identification and characterization. IEEE Trans. Electron Devices 58, 4067–4073 (2011)CrossRefGoogle Scholar
  12. 12.
    J.A.T. Machado, A.M.S.F. Galhano, Fractional order inductive phenomena based on the skin effect. Nonlinear Dyn. 68, 107–115 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    R. Martin, J.J. Quintana, A. Ramos, I. Nuez, Modeling electrochemical double layer capacitor, from classical to fractional impedance, in Electrotechnical conference, MELECON 2008, The 14th IEEE Mediterranean (Ajaccio, Corsica, France, 2008), pp. 61–66Google Scholar
  14. 14.
    D. Mondal, K. Biswas, Packaging of single-component fractional order element. IEEE Trans. Device Mater. Reliab. 13, 73–80 (2013)CrossRefGoogle Scholar
  15. 15.
    Z.B. Popović, B.D. Popović, Introductory Electromagnetics (Prentice Hall, New Jersey, 1999)Google Scholar
  16. 16.
    J.J. Quintana, A. Ramos, I. Nuez, Modeling of an EDLC with fractional transfer functions using Mittag-Leffler equations. Math. Probl. Eng. 2013, 807034–1–7 (2013)Google Scholar
  17. 17.
    A.G. Radwan, M.E. Fouda, Optimization of fractional-order RLC filters. Circuits Syst. Signal Process. 32, 2097–2118 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    A.G. Radwan, K.N. Salama, Fractional-order RC and RL circuits. Circuits Syst. Signal Process. 31, 1901–1915 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    M.R. Rapaić, Z.D. Jeličić, Optimal control of a class of fractional heat diffusion systems. Nonlinear Dyn. 62, 39–51 (2010)MathSciNetCrossRefGoogle Scholar
  20. 20.
    I. Schäfer, K. Krüger, Modelling of coils using fractional derivatives. J. Magn. Magn. Mater. 307, 91–98 (2006)CrossRefGoogle Scholar
  21. 21.
    Y. Shang, W. Fei, H. Yu, A fractional-order RLGC model for terahertz transmission line, in IEEE MTT-S International Microwave Symposium Digest (IMS), (Seattle, WA, 2013), pp. 1–3Google Scholar
  22. 22.
    R. Süsse, A. Domhardt, M. Reinhard, Calculation of electrical circuits with fractional characteristics of construction elements. Forsch Ingenieurwes 69, 230–235 (2005)CrossRefGoogle Scholar
  23. 23.
    C. Yang, H. Yu, Y. Shang, W. Fei, Characterization of CMOS metamaterial transmission line by compact fractional-order equivalent circuit model. IEEE Trans. Electron Devices 62, 3012–3018 (2015)CrossRefGoogle Scholar
  24. 24.
    C. Yang-Yang, S.H. Yu, A compact fractional-order model for terahertz composite right/left handed transmission line, in General Assembly and Scientific Symposium (URSI GASS), 2014 XXXIth URSI, (Beijing, 2014), pp. 1–4Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Power, Electronic and Telecommunication Engineering, Faculty of Technical SciencesUniversity of Novi SadNovi SadSerbia
  2. 2.Mathematical InstituteSerbian Academy of Arts and SciencesBelgradeSerbia
  3. 3.Department of Physics, Faculty of SciencesUniversity of Novi SadNovi SadSerbia
  4. 4.Department of Computing and Control Engineering, Faculty of Technical SciencesUniversity of Novi SadNovi SadSerbia

Personalised recommendations