Analysis and Passive Synthesis of Immittance for Fractional-Order Two-Element-Kind Circuit

  • Guishu Liang
  • Jiawei HaoEmail author


Fractional circuits have attracted an extensive attention of scholars and researchers for their superior performance and potential applications. The passive realization of the fractional-order immittance function plays an important role in fractional circuits theory, which is useful in fractional circuits design and modeling. This paper deals with the analysis and passive synthesis of fractional two-element-kind network. Firstly, the time-domain response of fractional two-element-kind network is analyzed based on its immittance function expressions, and the response shows oscillation only in fractional \( L_{\beta } C_{\alpha } \) circuits. Then, necessary and sufficient conditions to realize the fractional-order immittance functions by a passive network with only two kinds of elements are obtained in view of impedance scaling. A procedure is also proposed to realize such immittance functions using two-element-kind network. Finally, three examples are given to illustrate the proposed method.


Fractional circuits Fractional-order impedance Network synthesis Analysis Impedance scaling 



This research was supported in part by Natural Science Foundation of Beijing Municipality under Grant No. 3192039 and the Natural Science Foundation of Hebei Province under Grant No. E2018502121.


  1. 1.
    A. Adhikary, S. Sen, K. Biswas, Practical realization of tunable fractional order parallel resonator and fractional order filters. IEEE Trans. Circuits Syst. I Regul. Pap. 63(8), 1142–1151 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Adhikary, M. Khanra, J. Pal, K. Biswas, Realization of fractional order elements. Inae Lett. 2(2), 41–47 (2017)CrossRefGoogle Scholar
  3. 3.
    A. Adhikary, S. Choudhary, S. Sen, Optimal design for realizing a grounded fractional order inductor using gic. IEEE Trans. Circuits Syst. I Regul. Pap. PP(99), 1–11 (2018)MathSciNetGoogle Scholar
  4. 4.
    N. Bertrand, J. Sabatier, O. Briat et al., Fractional non-linear modelling of ultracapacitors. Commun. Nonlinear Sci. Numer. Simul. 15(5), 1327–1337 (2010)CrossRefGoogle Scholar
  5. 5.
    R. Caponetto, G. Dongola, L. Fortuna, I. Petráš, Factional Order System-Modeling and Control Applications (World Scientific Publishing, Singapore, 2010)CrossRefGoogle Scholar
  6. 6.
    L.J. Diao, X.F. Zhang, D.Y. Chen, Fractional-order multiple RL α C β circuit. Acta Phys. Sin. 63(3), 38401 (2014)Google Scholar
  7. 7.
    K. Diethelm, The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type. Lect. Notes Math. Springer Verlag 2004(9), 1333–1341 (2010)zbMATHGoogle Scholar
  8. 8.
    T. Dimeas, G. Tsirimokou, C. Psychalinos, A.S.C. Elwakil, Realization of fractional-order capacitor and inductor emulators using current feedback Operational Amplifiers, in International Symposium on Nonlinear Theory and ITS Applications (2015)Google Scholar
  9. 9.
    J.S. Duan, Z. Wang, S.Z. Fu, The zeros of the solutions of the fractional oscillation equation. Fract. Calc. Appl. Anal. 17(1), 10–22 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    A.M.A. El-Sayed, H.M. Nour, Fractional parallel RLC circuit. Alex. J. Math. 3, 11–23 (2012)Google Scholar
  11. 11.
    A.M. Elshurafa, M.M. Almadhoun, K.N. Salama, H.N. Alshareef, Microscale electrostatic fractional capacitors using reduced graphene oxide percolated polymer composites. Appl. Phys. Lett. 102(23), 232901 (2013)CrossRefGoogle Scholar
  12. 12.
    A.S. Elwakil, Fractional-order circuits and systems: an emerging interdisciplinary research area. IEEE Circuits Syst. Mag. 10(4), 40–50 (2010)CrossRefGoogle Scholar
  13. 13.
    A. Elwakil, B. Maundy, L. Fortuna, G. Chen, “Guest editorial fractional-order circuits and systems. IEEE J. Emerg. Sel. Topics Circuits Syst. 3(3), 297–300 (2013)CrossRefGoogle Scholar
  14. 14.
    T.J. Freeborn, B. Maundy, A.S. Elwakil, Fractional-order models of supercapacitors, batteries and fuel cells: a survey. Mater. Renew. Sustain. Energy 4(3), 1–7 (2015)CrossRefzbMATHGoogle Scholar
  15. 15.
    F. Gomez, J. Rosales, M. Guia, RLC electrical circuit of non-integer order. Cent. Eur. J. Phys. 11(10), 1361–1365 (2013)Google Scholar
  16. 16.
    R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag–Leffler Functions (Related Topics and Applications, Springer, Berlin Heidelberg, 2014)zbMATHGoogle Scholar
  17. 17.
    R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order[J]. Mathematics. 49(2), 277–290 (2008)Google Scholar
  18. 18.
    E.A. Guillemin, Synthesis of Passive Networks; Theory and Methods Appropriate to the Realization and Approximation (Wiley, New York, 1957)Google Scholar
  19. 19.
    T.T. Hartley, R.J. Veillette, J.L. Adams et al., Energy storage and loss in fractional-order circuit elements. Circuits Devices Syst. IET 9(3), 227–235 (2015)CrossRefGoogle Scholar
  20. 20.
    C.-C. Hua, D. Liu, X.-P. Guan, Necessary and sufficient stability criteria for a class of fractional-order delayed systems. IEEE Trans. Circuits Syst. II Express Briefs 61(1), 59–63 (2013)CrossRefGoogle Scholar
  21. 21.
    A. Jakubowska, J. Walczak, Analysis of the Transient State in a Series Circuit of the Class RL β C α (Birkhauser Boston Inc., Cambridge, 2016)zbMATHGoogle Scholar
  22. 22.
    A. Jakubowska, J. Walczak, A. Jakubowska, Resonance in series fractional order RL β C α circuit. Przeglad Elektrotechniczny r 90(4), 210–213 (2014)zbMATHGoogle Scholar
  23. 23.
    A. Jakubowska-Ciszek, J. Walczak, Analysis of the transient state in a parallel circuit of the class RLβCα. Appl. Math. Comput. 319, 287–300 (2018)Google Scholar
  24. 24.
    J. Jerabek, R. Sotner, J. Dvorak et al., Reconfigurable fractional-order filter with electronically controllable slope of attenuation, pole frequency and type of approximation. J. Circuits Syst. Comput. 26(10), 1750157 (2017)CrossRefGoogle Scholar
  25. 25.
    T. Kaczorek, K. Rogowski, Fractional Linear Systems and Electrical Circuits (Springer, Bialystok, 2015), pp. 49–80zbMATHGoogle Scholar
  26. 26.
    C. Li, A. Chen, J. Ye, Numerical approaches to fractional calculus and fractional ordinary differential equation. J. Comput. Phys. 230(9), 3352–3368 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    G. Liang, C. Liu, Positive-real property of passive fractional circuits in W-domain. Int. J. Circuit Theory Appl. 46, 893–910 (2018). CrossRefGoogle Scholar
  28. 28.
    G. Liang, L. Ma, Sensitivity analysis of networks with fractional elements. Circuits Syst. Signal Process. 36(10), 4227–4241 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    G. Liang, L. Ma, Multivariate theory-based passivity criteria for linear fractional networks. Int. J. Circuit Theory Appl. 46(7), 1358–1371 (2018)CrossRefGoogle Scholar
  30. 30.
    G. Liang, S. Gao, Y. Wang, Y. Zang, X. Liu, Fractional transmission line model of oil-immersed transformer windings considering the frequency-dependent parameters. IET Gener. Transm. Distrib. 11(5), 1154–1161 (2017)CrossRefGoogle Scholar
  31. 31.
    G. Liang, Y. Jing, C. Liu, L. Ma, Passive synthesis of a class of fractional immittance function based on multivariable theory. J Circuits Syst. Comput. 27(05), 1850074 (2018). CrossRefGoogle Scholar
  32. 32.
    X. Liu, Xiang Cui, Lei Qi et al., Wide-band modeling of cables based on the fractional order differential theory. Adv. Mater. Res. 860–863(4), 2292–2295 (2014)Google Scholar
  33. 33.
    X. Liu, C. Ti, G. Liang, Wide-band modelling and transient analysis of the multi-conductor transmission lines system considering the frequency-dependent parameters based on the fractional calculus theory. IET Gener. Transm. Distrib. 10(13), 3374–3384 (2016)CrossRefGoogle Scholar
  34. 34.
    L. Ma, G. Liang, Characteristics and applications of fractional LC circuits. Sci. Technol. Eng. 17 (2017)Google Scholar
  35. 35.
    J.T. Machado, And I say to myself: what a fractional world! J Fract. Calc. Appl. Anal. 14(4), 635–654 (2011)zbMATHGoogle Scholar
  36. 36.
    R. Magin, M.D. Ortigueira, I. Podlubny, J. Trujillo, On the fractional signals and systems. Signal Process. 91(3), 350–371 (2011)CrossRefzbMATHGoogle Scholar
  37. 37.
    M.D. Ortigueira, An introduction to the fractional continuous time linear systems: the 21st century systems. IEEE Circuits Syst. Mag. 8(3), 19–26 (2008)CrossRefGoogle Scholar
  38. 38.
    I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (Springer, New York, NY, USA, 2011)CrossRefzbMATHGoogle Scholar
  39. 39.
    Y.F. Pu, Research on Application of Fractional Calculus to Latest Signal Analysis and Processing (SiChuan University, Chengdu, 2006)Google Scholar
  40. 40.
    A.G. Radwan, Stability analysis of the fractional-order RLC circuit. J. Fract. Calc. Appl. 3(3), 1–15 (2012)Google Scholar
  41. 41.
    A.G. Radwan, Resonance and quality factor of the RLαCα fractional circuit. IEEE J. Emerg. Sel. Top. Circuits Syst. 3(3), 377–385 (2013)CrossRefGoogle Scholar
  42. 42.
    A.G. Radwan, K.N. Salama, Passive and active elements using fractional L β C α circuit. IEEE Trans. Circuits Syst. I Regul. Pap. 58(10), 2388–2397 (2011)MathSciNetCrossRefGoogle Scholar
  43. 43.
    A.G. Radwan, K.N. Salama, Fractional-order RC and RL circuits. Circuits Syst. Signal Process. 31(6), 1901–1915 (2012)MathSciNetCrossRefGoogle Scholar
  44. 44.
    A.G. Radwan, A.M. Soliman, A.S. Elwakil, First-order filters generalized to the fractional domain. J. Circuits Syst. Comput. 17(1), 55–56 (2008)CrossRefGoogle Scholar
  45. 45.
    A.G. Radwan, A.S. Elwakil, A.M. Soliman, Fractional-order sinusoidal oscillators: design procedure and practical examples. IEEE Trans. Circuits Syst. I Reg. Pap. 55(7), 2051–2063 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    A.G. Radwan, A.M. Soliman, A.S. Elwakil, A. Sedeek, On the stability of linear systems with fractional-order elements. Chaos Solitons Fractals 40(5), 2317–2328 (2009). CrossRefzbMATHGoogle Scholar
  47. 47.
    E.J. Routh, W.K. Clifford, C. Sturm et al., Stability of motion (1975)Google Scholar
  48. 48.
    M.S. Sarafraz, M.S. Tavazoei, Realizability of fractional-order impedances by passive electrical networks composed of a fractional capacitor and RLC components. IEEE Trans. Circuits Syst. I Regul. Pap. 62(12), 2829–2835 (2015)MathSciNetCrossRefGoogle Scholar
  49. 49.
    M.S. Sarafraz, M.S. Tavazoei, Passive realization of fractional-order impedances by a fractional element and RLC components: conditions and procedure. IEEE Trans. Circuits Syst. I Regul. Pap. 64(3), 585–595 (2017)CrossRefGoogle Scholar
  50. 50.
    M.S. Semary, A.G. Radwan, H.N. Hassan, Fundamentals of fractional-order LTI circuits and systems: number of poles, stability, time and frequency responses. Int. J. Circuit Theory Appl. 44, 2114–2133 (2016)CrossRefGoogle Scholar
  51. 51.
    X. Shu, Z. Bo, A fractional-order method to reduce the resonant frequency of integer-order wireless power transmission system. Trans. China Electrotech. Soc. 32(18), 83–89 (2017)Google Scholar
  52. 52.
    A. Soltan, A.G. Radwan, A.M. Soliman, CCII based fractional filters of different orders. J. Adv. Res. 5(2), 157–164 (2014)CrossRefGoogle Scholar
  53. 53.
    M.S. Tavazoei, M. Tavakoli-Kakhki, Minimal realizations for some classes of fractional order transfer functions. IEEE J. Emerg. Sel. Top. Circuits Syst. 3, 313–321 (2013)CrossRefGoogle Scholar
  54. 54.
    G. Temes, J. Lapatra, Introduction to Circuit Synthesis and Design (McGraw-Hill, New York, 1977)Google Scholar
  55. 55.
    M.C. Tripathy, D. Mondal, K. Biswas et al., Design and performance study of phase-locked loop using fractional-order loop filter. Int. J. Circuit Theory Appl. 43(6), 776–792 (2015)CrossRefGoogle Scholar
  56. 56.
    G. Tsirimokou, C. Laoudias, C. Psychalinos, 0.5-V fractional-order companding filters. Int. J. Circuit Theory Appl. 43(9), 1105–1126 (2015)CrossRefGoogle Scholar
  57. 57.
    G. Tsirimokou, C. Psychalinos, A.S. Elwakil, K.N. Salama, Electronically tunable fully integrated fractional-order resonator. IEEE Trans. Circuits Syst. II Express Briefs PP(99), 1 (2017)Google Scholar
  58. 58.
    J. Walczak, A. Jakubowska, Resonance in parallel fractional-order reactance circuit, in Proceedings of the XXIII Symposium Electromagnetic Phenomena in Nonlinear Circuits (EPNC), Pilsen (2014)Google Scholar
  59. 59.
    C. Wu, G. Si, Y. Zhang et al., The fractional-order state-space averaging modeling of the buck-boost DC/DC converter in discontinuous conduction mode and the performance analysis. Nonlinear Dyn. 79(1), 689–703 (2015)CrossRefGoogle Scholar

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

Authors and Affiliations

  1. 1.Department of Electric EngineeringNorth China Electric Power UniversityBaodingChina

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