Circuits, Systems, and Signal Processing

, Volume 31, Issue 6, pp 1901–1915 | Cite as

Fractional-Order RC and RL Circuits

  • A. G. RadwanEmail author
  • K. N. Salama


This paper is a step forward to generalize the fundamentals of the conventional RC and RL circuits in fractional-order sense. The effect of fractional orders is the key factor for extra freedom, more flexibility, and novelty. The conditions for RC and RL circuits to act as pure imaginary impedances are derived, which are unrealizable in the conventional case. In addition, the sensitivity analyses of the magnitude and phase response with respect to all parameters showing the locations of these critical values are discussed. A qualitative revision for the fractional RC and RL circuits in the frequency domain is provided. Numerical and PSpice simulations are included to validate this study


Fractional calculus Frequency domain analysis Fractional-order circuit Sensitivity analysis Fractional-order filter 


  1. 1.
    A. Abbisso, R. Caponetto, L. Fortuna, D. Porto, Non-integer-order integration by using neural networks, in Proceedings of International Symposium on Circuits and Systems, vol. 38 (2001), pp. 688–691 Google Scholar
  2. 2.
    K. Biswas, S. Sen, P. Dutta, Realization of a constant phase element and its performance study in a differentiator circuits. IEEE Trans. Circuits Syst. II 53, 802–806 (2006) CrossRefGoogle Scholar
  3. 3.
    G.W. Bohannan, S.K. Hurst, L. Springler, Electrical component with fractional order impedance. Utility Patent Application, US2006/0267595, 11/372,232, 10 March 2006 Google Scholar
  4. 4.
    G. Carlson, C. Halijak, Approximation of fractional capacitors (1/s)1/n by a regular Newton process. IEEE Trans. Circuits Syst. 11, 210–213 (1964) Google Scholar
  5. 5.
    T.C. Doehring, A.H. Freed, E.O. Carew, I. Vesely, Fractional order viscoelasticity of the aortic valve: an alternative to QLV. J. Biomech. Eng. 127(4), 700–708 (2005) CrossRefGoogle Scholar
  6. 6.
    A. Dzielinski, Stability of discrete fractional order state-space systems. J. Vib. Control 14, 1543–1556 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    M. Faryad, Q.A. Naqvi, Fractional rectangular waveguide. Prog. Electromagn. Res. 75, 384–396 (2007) CrossRefGoogle Scholar
  8. 8.
    T.C. Haba, G.L. Loum, G. Ablart, An analytical expression for the input impedance of a fractal tree obtained by a microelectronical process and experimental measurements of its non-integral dimension. Chaos Solitons Fractals 33, 364–373 (2007) CrossRefGoogle Scholar
  9. 9.
    S. Jesus, T.J.A. Machado, B.J. Cunha, Fractional electrical impedances in botanical elements. J. Vib. Control 14(9–10), 1389–1402 (2008) CrossRefzbMATHGoogle Scholar
  10. 10.
    B.T. Krishna, Studies on fractional order differentiators and integrators: a survey. Signal Process. 91(3), 386–426 (2011) CrossRefzbMATHGoogle Scholar
  11. 11.
    B.T. Krishna, K.V.V.S. Reddy, Active and passive realization of fractance device of order 1/2. J. Active Passive Electron. Compon. 1–5 (2008). doi: 10.1155/2008/369421
  12. 12.
    A. Lahiri, T.K. Rawat, Noise analysis of single stage fractional-order low-pass filter using stochastic and fractional calculus. ECTI Trans. Electr. Eng. Electron. Commun. 7(2), 136–143 (2009) Google Scholar
  13. 13.
    H. Li, Y. Luo, Y.Q. Chen, A fractional-order proportional and derivative (FOPD) motion controller: tuning rule and experiments. IEEE Trans. Control Syst. Technol. 18(2), 516–520 (2010) CrossRefGoogle Scholar
  14. 14.
    R.L. Magin, Fractional Calculus in Bioengineering (Begell House, New York, 2006) Google Scholar
  15. 15.
    R. Martin, J.J. Quintara, A. Ramos, L. De La Nuez, Modeling electrochemical double layer capacitor, from classical to fractional impedance. Journal of Computational and Nonlinear Dynamics 3(2), 6 (2008) CrossRefGoogle Scholar
  16. 16.
    P. Melchior, B. Orsoni, O. Lavialle, A. Oustaloup, The CRONE toolbox for Matlab: fractional path planning design in robotics. Int. J. Circuit Theory Appl. 36, 473–492 (2008). doi: 10.1002/cta. Laboratoire d’Automatique et de Productique (LAP) 2001. Copyright 2007 Wiley, New York CrossRefGoogle Scholar
  17. 17.
    K. Moaddy, A.G. Radwan, K.N. Salama, S. Momani, I. Hashim, The fractional-order modeling and synchronization of electrically coupled neurons system. J. Comput. Math. Appl. (2012). doi: 10.1016/j.camwa.2012.01.005 zbMATHGoogle Scholar
  18. 18.
    S. Mukhopadhyay, C. Coopmans, Y.Q. Chen, Purely analog fractional-order PI control using discrete fractional capacitors (fractals): synthesis and experiments, in International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE, USA (2009) Google Scholar
  19. 19.
    M. Nakagawa, K. Sorimachi, Basic characteristics of a fractance device. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E75(12), 1814–1819 (1992) Google Scholar
  20. 20.
    F.L. Oustaloup, B. Mathieu, Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47, 25–39 (2000) CrossRefGoogle Scholar
  21. 21.
    A.G. Radwan, Stability analysis of the fractional-order RLC circuit. J. Fract. Calc. Appl. 3 (2012) Google Scholar
  22. 22.
    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
  23. 23.
    A.G. Radwan, A.M. Soliman, A.S. Elwakil, Design equations for fractional-order sinusoidal oscillators: practical circuit examples, in International Conference on Microelectronics (ICM) (2007), pp. 91–94 Google Scholar
  24. 24.
    A.G. Radwan, A.S. Elwakil, A.M. Soliman, Fractional-order sinusoidal oscillators: design procedure and practical examples. IEEE Trans. Circuits Syst. I, Regul. Pap. 55, 2051–2063 (2008) MathSciNetCrossRefGoogle Scholar
  25. 25.
    A.G. Radwan, A.S. Elwakil, A.M. Soliman, On the generalization of second-order filters to fractional-order domain. J. Circuits Syst. Comput. 18(2), 361–386 (2009) CrossRefGoogle Scholar
  26. 26.
    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
  27. 27.
    A.G. Radwan, K. Moddy, S. Momani, Stability and nonstandard finite difference method of the generalized Chua’s circuit. Comput. Math. Appl. 62, 961–970 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    A.G. Radwan, A. Shamim, K.N. Salama, Theory of fractional-order elements based impedance matching networks. IEEE Microw. Wirel. Compon. Lett. 21(3), 120–122 (2011) CrossRefGoogle Scholar
  29. 29.
    S. Roy, On the realization of a constant-argument immittance or fractional operator. IEEE Trans. Circuits Syst. 14, 264–274 (1967) Google Scholar
  30. 30.
    J. Sabatier, O.P. Agrawal, J.A. Tenreiro Machado, Advances in Fractional Calculus; Theoretical Developments and Applications in Physics and Engineering (Springer, Berlin, 2007) zbMATHGoogle Scholar
  31. 31.
    K. Saito, M. Sugi, Simulation of power-law relaxations by analog circuits: fractal distribution of relaxation times and non-integer exponents. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E76(2), 205–209 (1993) Google Scholar
  32. 32.
    A. Shamim, A.G. Radwan, K.N. Salama, Fractional Smith chart theory and application. IEEE Microw. Wirel. Compon. Lett. 21(3), 117–119 (2011) CrossRefGoogle Scholar
  33. 33.
    M. Sugi, Y. Hirano, Y.F. Miura, K. Saito, Simulation of fractal immittance by analog circuits: an approach to the optimized circuits. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E82(8), 1627–1634 (1999) Google Scholar
  34. 34.
    H. Zhu, S. Zhou, Z. He, Chaos synchronization of the fractional order Chen’s system. Chaos Solitons Fractals 41, 2733–2740 (2009) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Engineering Mathematics Department, Faculty of EngineeringCairo UniversityCairoEgypt
  2. 2.Nanoelectronics Integrated Systems CenterNile UniversityCairoEgypt
  3. 3.Electrical Engineering ProgramKing Abdullah University of Science and Technology (KAUST)ThuwalKingdom of Saudi Arabia

Personalised recommendations