• Georgia Tsirimokou
  • Costas Psychalinos
  • Ahmed Elwakil
Part of the SpringerBriefs in Electrical and Computer Engineering book series (BRIEFSELECTRIC)


Fractional calculus is three centuries old as the conventional calculus and consist a super set of integer-order calculus, which has the potential to accomplish what integer-order calculus cannot. Its origins dating back to a correspondence from 1695 between Leibnitz and L’Hôpital, with L’Hôpital inquiring about Leibnitz notation for the n-th derivative of a function d n y/dx n , i.e. what would be the result if n = 1/2. The reply from Leibnitz, “It will lead to a paradox, a paradox from which one day useful consequences will be drawn, because there are no useless paradoxes”, was the motivation for fractional calculus to be born. Fractional calculus does not mean the calculus of fractions, nor does it mean a fraction of any calculus differentiation, integration or calculus of variations. The fractional calculus is a name of theory of integrations and derivatives of arbitrary order, which unify and generalize the notation of integer-order differentiation and n-fold integration. The beauty of this subject is that fractional derivatives and integrals translate better the reality of nature! This feature turns it into an efficient tool, offering the capability of having available a language of nature, which can be used to talk with.


Fractional Derivative Fractional Calculus Continue Fraction Expansion Electronic Tuning Current Feedback Operational Amplifier 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Ortigueira, M.: An introduction to the fractional continuous-time linear systems: the 21st century systems. IEEE Circuits Syst. Mag. 8(3), 19–26 (2008)CrossRefGoogle Scholar
  2. 2.
    Podlubny, I., Petras, I., Vinagre, B., O’Leary, P., Dorcak, L.: Analogue realizations of fractional-order controllers. Nonlinear Dyn. 29(1), 281–296 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Haba, T., Loum, G., Zoueu, J., Ablart, G.: Use of a component with frac-tional impedance in the realization of an analogical regulator of order ½. J. Appl. Sci. 8(1), 59–67 (2008)CrossRefGoogle Scholar
  4. 4.
    Tenreiro-Machado, J.A., Jesus, I.S., Galhano, A., Cunha, J.B.: Fractional order electromagnetics. Signal Process. 86(10), 2637–2644 (2006)CrossRefzbMATHGoogle Scholar
  5. 5.
    Miguel, F., Lima, M., Tenreiro-Machado, J.A., Crisostomo, M.: Experimental signal analysis of robot impacts in a fractional calculus perspective. J. Adv. Comput. Intell. Intell. Informatics. 11(9), 1079–1085 (2007)CrossRefGoogle Scholar
  6. 6.
    Ferreira, N.F., Duarte, F., Lima, M., Marcos, M., Machado, J.T.: Application of fractional calculus in the dynamical analysis and control of mechanical manipulators. Fractional Calculus Appl. Anal. 11(1), 91–113 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Caputo, M.: Linearmodels of dissipation whose Q is almost frequency independent, part II. Geophys. J. Int. 13(5), 529–539 (1967)CrossRefGoogle Scholar
  8. 8.
    Khovanskii, A.N.: The Application of Continued Fractions and Their Generalizations to Problems in Approximation Theory. Noordhoff, Groningen (1963)Google Scholar
  9. 9.
    Chen, Y., Vinagre, B., Podlubny, I.: Continued fraction expansion approaches to discretizing fractional order derivatives – an expository review. Nonlinear Dyn. 38(1), 155–170 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, Y., Moore, K.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuits Syst. I. 49(3), 363–367 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Krishna, B., Reddy, K.: Active and passive realizations of fractance device of order 1/2. Act. Passive Electron. Compon. 2008, Article ID 369421, 5 pages (2008)Google Scholar
  12. 12.
    Abdelliche, F., Charef, A.: R-peak detection Using a complex fractional wavelet. In: Proceedings of the IEEE International Conference on Electrical and Electronics Engineering, pp. 267–270 (2009)Google Scholar
  13. 13.
    Abdelliche, F., Charef, A., Talbi, M., Benmalek, M.: A fractional wavelet for QRS detection. IEEE Inf. Commun. Technol. 1(2), 1186–1189 (2006)Google Scholar
  14. 14.
    Ferdi, Y., Hebeuval, J., Charef, A., Boucheham, B.: R wave detection using fractional digital differentiation. ITBMRBM. 24(5–6), 273–280 (2003)Google Scholar
  15. 15.
    Goutas, A., Ferdi, Y., Herbeuval, J.P., Boudraa, M., Boucheham, B.: Digital fractional order differentiation-based algorithm for P and T-waves detection and delineation. Int. Arab. J. Inf. Technol. 26(2), 127–132 (2005)Google Scholar
  16. 16.
    Benmalek, M., Charef, A.: Digital fractional order operators for R-wave detection in electrocardiogram signal. IET Signal Proc. 3(5), 381–391 (2009)CrossRefGoogle Scholar
  17. 17.
    Jesus, I.S., Machado, J.A.: Development of fractional order capacitors based on electrolytic process. Nonlinear Dyn. 56(1), 45–55 (2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Haba, T.C., Ablart, G., Camps, T., Olivie, F.: Influence of the electrical parameters on the input impedance of a fractal structure realized on silicon. Chaos, Solitons Fractals. 24(2), 479–490 (2005)CrossRefGoogle Scholar
  19. 19.
    Biswas, K., Sen, S., Dutta, P.: Realization of a constant phase element and its performance study in a differentiator circuit. IEEE Trans. Circuits Syst. II Express Briefs. 53(9), 802–806 (2006)CrossRefGoogle Scholar
  20. 20.
    Mondal, D., Biswas, K.: Performance study of fractional order integrator using single component fractional order elements. IET Circuits Devices Syst. 5(4), 334–342 (2011)CrossRefGoogle Scholar
  21. 21.
    Krishna, M.S., Das, S., Biswas, K., Goswami, B.: Fabrication of a fractional - order capacitor with desired specifications: a study on process identification and characterization. IEEE Trans. Electron Devices. 58(11), 4067–4073 (2011)CrossRefGoogle Scholar
  22. 22.
    Elshurafa, A.M., Almadhoun, M.N., Salama, K.N., Alshareef, H.N.: Microscale electrostatic fractional capacitors using reduced graphene oxide percolated polymer composites. Appl. Phys. Lett. 102(23), 232901–232904 (2013)CrossRefGoogle Scholar
  23. 23.
    Carlson, G.E., Halijak, C.A.: Approximation of fractional capacitors (1/s)(1/n) by a regular Newton process. IEEE Trans. Circuit Theory. 11(2), 210–213 (1964)CrossRefGoogle Scholar
  24. 24.
    Steiglitz, K.: An RC impedance approximation to s-1/2. IEEE Trans. Circuits Syst. 11(1), 160–161 (1964)Google Scholar
  25. 25.
    Roy, S.C.D.: On the realization of a constant-argument immittance or fractional operator. IEEE Trans. Circuits Syst. 14(3), 264–274 (1967)Google Scholar
  26. 26.
    Valsa, J., Vlach, J.: RC models of a constant phase element. Int. J. Circuit Theory Appl. 41(1), 59–67 (2013)Google Scholar
  27. 27.
    Ali, A., Radwan, A., Soliman, A.: Fractional order Butterworth filter: active and passive realizations. IEEE J. Emerging Sel. Top. Circuits Syst. 3(3), 346–354 (2013)CrossRefGoogle Scholar
  28. 28.
    Freeborn, T.J., Maundy, B.J., Elwakil, A.S.: Approximated fractional-order Chebyshev lowpass filters. Math. Probl. Eng. 2015, 1 (2015)CrossRefGoogle Scholar
  29. 29.
    Freeborn, T.J., Elwakil, A.S., Maundy, B.J.: Approximating fractional-order inverse Chebyshev lowpass filters. Circuits Syst. Signal Process. 35(6), 1973–1982 (2016)CrossRefGoogle Scholar
  30. 30.
    Maundy, B.J., Elwakil, A.S., Gift, S.: On the realization of multiphase oscillators using fractional-order allpass filters. Circuits Systems and Signal Processing. 31(1), 3–17 (2012)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Radwan, A., Soliman, A., Elwakil, A.: First-order filters generalized to the fractional domain. J. Circuits Syst. Comput. 17(1), 55–66 (2008)CrossRefGoogle Scholar
  32. 32.
    Radwan, A., Elwakil, A., Soliman, A.: On the generalization of second-order filters to the fractional-order domain. J. Circuits Syst. Comput. 18(2), 361–386 (2009)CrossRefGoogle Scholar
  33. 33.
    Radwan, A., Salama, K.: Fractional-order RC and RL circuits. Circuits Systems and Signal Processing. 31, 1901–1915 (2012)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Tripathy, M., Mondal, D., Biswas, K., Sen, S.: Experimental studies on realization of fractional inductors and fractional-order bandpass filters. International Journal of Circuit Theory and Applications. 43(9), 1183–1196 (2015)CrossRefGoogle Scholar
  35. 35.
    Psychalinos, C., Pal, K., Vlassis, S.: A floating generalized impedance converter with current feedback amplifiers. Int. J. Electron. Commun. 62, 81–85 (2008)CrossRefGoogle Scholar
  36. 36.
    Chen, Y.Q., Moore, K.L.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuits Syst. I, Fundam Theory Appl. 49(3), 363–367 (2002)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Tseng, C.C., Lee, S.L.: Design of fractional order digital differentiator using radial basis function. IEEE Trans. Circuits Syst. I, Regul. Pap. 57(7), 1708–1718 (2010)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Das, S., Pan, I.: Fractional-order signal processing: introductory concepts and applications. Springer Briefs in Applied Sciences and Technology, Chapter, vol. 2, pp. 13–30 (2012)Google Scholar
  39. 39.
    Freeborn, T.J., Maundy, B.J., Elwakil, A.S.: Field programmable analogue array implementation of fractional step filters. IET Circuits Devices Syst. 4(6), 514–524 (2010)CrossRefGoogle Scholar
  40. 40.
    Maundy, B.J., Elwakil, A.S., Freeborn, T.J.: On the practical realization of higher-order filters with fractional stepping. Signal Process. 91(3), 484–491 (2011)CrossRefzbMATHGoogle Scholar
  41. 41.
    Ahmadi, P., Maundy, B.J., Elwakil, A.S., Belostotski, L.: High-quality factor asymmetric-slope band-pass filters: a fractionalorder capacitor approach. IET Circuits Devices and Systems. 6(3), 187–197 (2012)CrossRefGoogle Scholar
  42. 42.
    Soltan, A., Radwan, A., Soliman, A.: Fractional-order filter with two fractional elements of dependant orders. Microelectron. J. 43(11), 818–827 (2012)CrossRefGoogle Scholar
  43. 43.
    Freeborn, T.J., Maundy, B.J., Elwakil, A.S.: Towards the realization of fractional step filters. In: Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS), pp. 1037–1040 (2010)Google Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Georgia Tsirimokou
    • 1
  • Costas Psychalinos
    • 1
  • Ahmed Elwakil
    • 2
    • 3
  1. 1.Physics Department Electronics LaboratoryUniversity of PatrasRio PatrasGreece
  2. 2.Department of Electrical and Computer EngineeringUniversity of SharjahSharjahUnited Arab Emirates
  3. 3.Nanoelectronics Integrated Systems Center (NISC)Nile UniversityCairoEgypt

Personalised recommendations