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**Development of fractional order capacitors based on electrolyte processes.**
*(English)*
Zbl 1173.78313

Summary: In recent years, significant research in the field of electrochemistry was developed. The performance of electrical devices, depending on the processes of the electrolytes, was described and the physical origin of each parameter was established. However, the influence of the irregularity of the electrodes was not a subject of study and only recently this problem became relevant in the viewpoint of fractional calculus. This paper describes an electrolytic process in the perspective of fractional order capacitors. In this line of thought, are developed several experiments for measuring the electrical impedance of the devices. The results are analyzed through the frequency response, revealing capacitances of fractional order that can constitute an alternative to the classical integer order elements. Fractional order electric circuits are used to model and study the performance of the electrolyte processes.

### MSC:

78A55 | Technical applications of optics and electromagnetic theory |

78-05 | Experimental work for problems pertaining to optics and electromagnetic theory |

28A80 | Fractals |

92E20 | Classical flows, reactions, etc. in chemistry |

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\textit{I. S. Jesus} and \textit{J. A. Tenreiro Machado}, Nonlinear Dyn. 56, No. 1--2, 45--55 (2009; Zbl 1173.78313)

### References:

[1] | Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974) · Zbl 0292.26011 |

[2] | Miller, K.S., Ross, B.: Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) · Zbl 0789.26002 |

[3] | Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993) · Zbl 0818.26003 |

[4] | Debnath, L.: In: 41st IEEE Conference. Fractional Calculus and Its Applications, November 2002 |

[5] | Tenreiro Machado, J.A., Jesus, I.S.: A suggestion from the past?. J. Fract. Calc. Appl. Anal. 7(4), 403–407 (2005) · Zbl 1121.26004 |

[6] | Jesus, I.S., Tenreiro Machado, J.A., Cunha, J.B., Silva, M.F.: Fractional order electrical impedance of fruits and vegetables. In: Proc. 25th IASTED International Conference on Modeling, Identification and Control–MIC06, pp. 489–494, Spain, February 6–8 2006 |

[7] | Nigmatullin, R.R., Alekhin, A.P.: Quasi-fractals: new method of description of a structure of disordered media. In: 2nd IFAC Workshop on Fractional Differentiation and Its Applications–FDA06, Porto, Portugal, 19–21 July 2006 · Zbl 1206.82052 |

[8] | Nigmatullin, R.R., Arbuzov, A.A., Salehli, F., Giz, A., Bayrak, I., Catalgil-Giz, H.: The first experimental confirmation of the fractional kinetics containing the complex-power-law exponents: Dielectric measurements of polymerization reactions. Physica B 388, 418–434 (2007) |

[9] | Samavati, H., Hajimiri, A., Shahani, A.R., Nasserbakht, G.N., Lee, T.H.: Fractal capacitors. IEEE J. Solid-State Circuits 33(12), 2035–2041 (1998) |

[10] | Jonscher, A.K.: Dielectric Relaxation in Solids. Chelsea Dielectric Press, London (1993) |

[11] | Bohannan, G.W.: Analog realization of a fractional order control element. Wavelength Electronics (2002) |

[12] | Bohannan, G.W.: Interpretation of complex permittivity in pure and mixed crystals. Wavelength Electronics (2002) |

[13] | Westerlund, S.: Capacitor theory, IEEE Trans. Dielectr. Electr. Insul. 1(5), 826–839 (1994) |

[14] | Barsoukov, E., Macdonald, J.R.: Impedance Spectroscopy, Theory, Experiment, and Applications. Wiley, New York (2005) |

[15] | Jesus, I.S., Tenreiro Machado, J.A., Cunha, J.B.: Fractional electrical dynamics in fruits and vegetables. In: 2nd IFAC Workshop on Fractional Differentiation and its Applications, Porto, Portugal, 19–21 July 2006 |

[16] | Jesus, I.S., Tenreiro Machado, J.A., Cunha, J.B.: Fractional electrical impedances in botanical elements, J. Vibr. Control (2008, in press) · Zbl 1229.78023 |

[17] | Jesus, I.S., Tenreiro Machado, J.A., Silva, M.F.: Fractional order capacitors. In: Proc. 27th IASTED International Conference on Modeling, Identification and Control, Austria, February 11–13 2008 |

[18] | Clerc, J.P., Tremblay, A.-M.S., Albinet, G., Mitescu, C.D.: A.c. response of a fractal networks. J. Phys. Lett. 45(19), L913–L924 (1984) |

[19] | Kaplan, T., Gray, L.J.: Effect of disorder on a fractal model for the ac response of a rough interface. J. Phys. B 32(11), 7360–7366 (1985) |

[20] | Liu, S.H.: Fractal model for the ac response of a rough interface. J. Phys. 55(5), 529–532 (1985) |

[21] | Kaplan, T., Liu, S.H., Gray, L.J.: Inverse-Cantor-bar model for the ac response of a rough interface. J. Phys. B 34(7), 4870–4873 (1986) |

[22] | Kaplan, T., Gray, L.J., Liu, S.H.: Self-affine fractal model for a metal–electrolyte interface. J. Phys. B 35(10), 5379–5381 (1987) |

[23] | Falconer, K.: Fractal Geometry–Mathematical Foundation and Applications. Wiley, New York (1990) · Zbl 0689.28003 |

[24] | Mehaute, A.L., Fractal Geometries: Theory and Applications, Penton Press, London (1990) |

[25] | Nigmatullin, R.R., Le Mehaute, A.: The geometrical and physical meaning of the fractional integral with complex exponent. Int. J. Sci. Georesources 1(8), 2–9 (2004) |

[26] | Nigmatullin, R.R., Le Mehaute, A.: Is there geometrical/physical meaning of the fractional integral with complex exponent? J. Non-Cryst. Solids 351, 2888–2899 (2005) |

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