Circuits, Systems, and Signal Processing

, Volume 32, Issue 5, pp 2097–2118 | Cite as

Optimization of Fractional-Order RLC Filters

Article

Abstract

This paper introduces some generalized fundamentals for fractional-order RL β C α circuits as well as a gradient-based optimization technique in the frequency domain. One of the main advantages of the fractional-order design is that it increases the flexibility and degrees of freedom by means of the fractional parameters, which provide new fundamentals and can be used for better interpretation or best fit matching with experimental results. An analysis of the real and imaginary components, the magnitude and phase responses, and the sensitivity must be performed to obtain an optimal design. Also new fundamentals, which do not exist in conventional RLC circuits, are introduced. Using the gradient-based optimization technique with the extra degrees of freedom, several inverse problems in filter design are introduced. The concepts introduced in this paper have been verified by analytical, numerical, and PSpice simulations with different examples, showing a perfect matching.

Keywords

Fractional calculus Fractional filters Optimization RLC circuit Sensitivity analysis Fractional-order elements 

References

  1. 1.
    K. Biswas, S. Sen, P. Dutta, Modelling of a capacitive probe in a polarizable medium. Sens. Actuators Phys. 120(1), 115–122 (2005) CrossRefGoogle Scholar
  2. 2.
    G. Carlson, C. Halijak, Approximation of fractional capacitors (1/s)^{1/n} by a regular Newton process. IEEE Trans. Circuit Theory 11(2), 210–213 (1964) CrossRefGoogle Scholar
  3. 3.
    K. Diethelm, N.J. Ford, A.D. Freed, Y.Y. Luchko, Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Eng. 194(6), 743–773 (2005) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    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
  5. 5.
    A.S. Elwakil, B. Maundy, Extracting the Cole-Cole impedance model parameters without direct impedance measurement. Electron. Lett. 46(20), 1367–1368 (2010) CrossRefGoogle Scholar
  6. 6.
    M. Faryad, Q.A. Naqvi, Fractional rectangular waveguide. Prog. Electromagn. Res. 75, 384–396 (2007) CrossRefGoogle Scholar
  7. 7.
    N.J. Ford, A.C. Simpson, The numerical solution of fractional differential equations: speed versus accuracy. Numer. Algorithms 26(4), 333–346 (2001) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    M.E. Fouda, A.G. Radwan, On the fractional-order memristor model. J. Fract. Calc. Appl. 4(1), 1–7 (2013) Google Scholar
  9. 9.
    T.C. Haba, G.L. Loum, J.T. Zoueu, G. Albart, Use of a component with fractional impedance in the realization of an analogical regulator of order ½. J. Appl. Sci. 8(1), 59–67 (2008) CrossRefGoogle Scholar
  10. 10.
    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(2), 364–373 (2007) CrossRefGoogle Scholar
  11. 11.
    T.C. Haba, G. Ablart, T. Camps, F. Olivie, Influence of the electrical parameters on the input impedance of a fractal structure realised on silicon. Chaos Solitons Fractals 24(2), 479–490 (2005) CrossRefGoogle Scholar
  12. 12.
    I.S. Jesus, J.A. Machado, J.B. Cunha, M.F. Silva, Fractional order electrical impedance of fruits and vegetables, in Proceedings of the 25th IASTED International Conference on Modeling Identification and Control (2006), pp. 489–494 Google Scholar
  13. 13.
    I.S. Jesus, J.A. Machado, Development of fractional order capacitors based on electrolyte processes. Nonlinear Dyn. 56(1), 45–55 (2009) MATHCrossRefGoogle Scholar
  14. 14.
    B.T. Krishna, Studies on fractional order differentiators and integrators: a survey. Signal Process. 91(3), 386–426 (2011) MATHCrossRefGoogle Scholar
  15. 15.
    B.T. Krishna, K.V.V.S. Reddy, Active and Passive Realization of Fractance Device of Order 1/2. Act. Passive Electron. Compon. 2008 (2008) Google Scholar
  16. 16.
    H. Li, M. Wu, X. Wang, Fractional-moment capital asset pricing model. Chaos Solitons Fractals 42(1), 412–421 (2009) MATHCrossRefGoogle Scholar
  17. 17.
    R.L. Magin, Fractional calculus in bioengineering. Begell House, Connecticut (2006) Google Scholar
  18. 18.
    R.L. Magin, Fractional calculus in bioengineering, part 3. Crit. Rev. Biomed. Eng. 32(3–4), 195–377 (2004) CrossRefGoogle Scholar
  19. 19.
    R.L. Magin, M. Ovadia, Modeling the cardiac tissue electrode interface using fractional calculus. J. Vib. Control 14(9–10), 1431–1442 (2008) MATHCrossRefGoogle Scholar
  20. 20.
    R. Martin, J.J. Quintara, A. Ramos, L. Nuez, Modeling electrochemical double layer capacitor, from classical to fractional impedance, in Proceedings of the 14th IEEE Mediterranean Electrotechnical Conference (2008), pp. 61–66 Google Scholar
  21. 21.
    K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993) MATHGoogle Scholar
  22. 22.
    K. Moaddy, A.G. Radwan, K.N. Salama, S. Momani, I. Hashim, The fractional-order modeling and synchronization of electrically coupled neuron systems. Comput. Math. Appl. 64(10), 3329–3339 (2012) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    M. Nakagawa, K. Sorimachi, Basic characteristics of a fractance device. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E75-A(12), 1814–1819 (1992) Google Scholar
  24. 24.
    K.B. Oldham, J. Spanier, Fractional Calculus (Academic Press, New York, 1974) MATHGoogle Scholar
  25. 25.
    I. Petras, D. Sierociuk, I. Podlubny, Identification of parameters of a half-order system. IEEE Trans. Signal Process. 60(10), 5561–5566 (2012) MathSciNetCrossRefGoogle Scholar
  26. 26.
    I. Petras, Y. Chen, Fractional-order circuit elements with memory, in Proceedings of the 13th International Carpathian Control Conference (2012), pp. 552–558 CrossRefGoogle Scholar
  27. 27.
    I. Petras, Fractional-Order Nonlinear Systems: Modelling, Analysis and Simulation (Springer, Berlin, 2011) CrossRefGoogle Scholar
  28. 28.
    A.G. Radwan, A.S. Elwakil, A.M. Soliman, Fractional-order sinusoidal oscillators: design procedure and practical examples. IEEE Trans. Circuits. Syst., I 55(7), 2051–2063 (2008) MathSciNetCrossRefGoogle Scholar
  29. 29.
    A.G. Radwan, A.M. Soliman, A.S. Elwakil, Fractional-order sinusoidal oscillators: four practical circuit design examples. Int. J. Circuit Theory Appl. 36(4), 473–492 (2008) MATHCrossRefGoogle Scholar
  30. 30.
    A.G. Radwan, A.M. Soliman, A.S. Elwakil, First order filters generalized to the fractional domain. J. Circuits Syst. Comput. 17(1), 55–66 (2008) CrossRefGoogle Scholar
  31. 31.
    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
  32. 32.
    A.G. Radwan, K.N. Salama, Fractional-order RC and RL circuits. Circuits Syst. Signal Process. 31(6), 1901–1915 (2012) MathSciNetCrossRefGoogle Scholar
  33. 33.
    A.G. Radwan, K.N. Salama, Passive and active elements using fractional L β C α circuit. IEEE Trans. Circuits Syst. I 58(10), 2388–2397 (2011) MathSciNetCrossRefGoogle Scholar
  34. 34.
    A.G. Radwan, Stability analysis of the fractional-order RL β C α circuit. J. Fract. Calc. Appl. 3(1), 1–15 (2012) Google Scholar
  35. 35.
    A.G. Radwan, M.H. Bakr, N.K. Nikolova, Transient adjoint sensitivities for discontinuities with Gaussian material distributions. Prog. Electromagn. Res. B 27, 1–19 (2011) CrossRefGoogle Scholar
  36. 36.
    S. Roy, On the realization of a constant-argument immitance or fractional operator. IEEE Trans. Circuit Theory 14(3), 264–274 (1967) CrossRefGoogle Scholar
  37. 37.
    J. Sabatier, O.P. Agrawal, M.J.A. Tenreiro, Advances in Fractional Calculus; Theoretical Developments and Applications in Physics and Engineering (Springer, Berlin, 2007) MATHCrossRefGoogle Scholar
  38. 38.
    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
  39. 39.
    I. Schäfer, K. Krüger, Modelling of lossy coils using fractional derivatives. J. Phys. D, Appl. Phys. 41(4), 045001 (2008) CrossRefGoogle Scholar
  40. 40.
    A. Soltan, A.G. Radwan, A.M. Soliman, Fractional order filter with two fractional elements of dependent orders. J. Microelectron. 7(9), 965–969 (2012) Google Scholar
  41. 41.
    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
  42. 42.
    J. Valsa, Fractional-order electrical components, networks and systems, in Proceedings of the 22nd International Conference Radioelektronika (2012), pp. 1–9 Google Scholar
  43. 43.
    S. Westerlund, L. Ekstam, Capacitor theory. IEEE Trans. Dielectr. Electr. Insul. 1(5), 826–839 (1994) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Engineering Mathematics DepartmentCairo UniversityCairoEgypt
  2. 2.NISC Research CenterNile UniversityCairoEgypt

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