Optimization of Fractional-Order RLC Filters
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Optimization of Fractional-Order RLC Filters Ahmed G. Radwan · M.E. Fouda
Received: 11 August 2012 / Revised: 13 March 2013 © Springer Science+Business Media New York 2013
Abstract This paper introduces some generalized fundamentals for fractional-order RLβ Cα 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
1 Introduction The history of fractional calculus dates back to 1695 with the work of scientists such as L’Hospital and Leibniz, but the first logic definitions were proposed by Liouville A.G. Radwan () · M.E. Fouda Engineering Mathematics Department, Cairo University, 12613, Cairo, Egypt e-mail: [email protected] M.E. Fouda e-mail: [email protected] A.G. Radwan NISC Research Center, Nile University, Cairo, Egypt
Circuits Syst Signal Process
in 1834, Riemann in 1847, and Grünwald in 1867 [21, 24]. Fractional calculus can be considered as a super set of integer-order calculus, which has the potential to accomplish what integer-order calculus cannot. The first approximation of the fractionalorder derivative in terms of a complicated system of integer orders was proposed in 1964 [2], but this approximation is good only in a certain band of frequencies. Furthermore, different realizations of the fractional elements were introduced during the last few years [14, 15, 23, 25, 36, 38, 41] using different techniques. The theory of fractional-order elements comes from the frequency-dependent losses in the conventional elements as proved recently in [39, 42, 43]. Moreover, this theory was extended to the memristive elements (memristor, memcapacitor, and meminductor) [8, 26]. Many books and researches during the last three decades have aimed to increase the accessibility of fractional calculus for remodeling most of the existing applications and analyzing new models in basic natural sciences. For example, many papers recently have tried to model the electrical impedance of vegetables and fruits into simple electrical circuit connections using a single fractional capacitor [5, 12]. Generally, the fingerprints of the applied fractional calculus could
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