Fractional-Order 2 × n RLC Circuit Network

Abstract

This paper introduces new fundamentals of the 2 Â n RLC circuit network in the fractionalorder domain. First, we derive the three general formulae of the equivalent impedances of the circuit network by using the matrix transform methods and constructing the di®erential equation models in three di®erent cases. Moreover, we systematically study the e®ects of the system parameters on the impendence characteristics in the three di®erent cases. Speci¯cally, the new phenomena and laws are presented by the results of the numerical simulations, which are impossible in the conventional cases. Finally, a comparative sensitivity analysis about the three cases with respect to the fractional orders for the fractional-order circuit network is carried out in detail. Mathematical analyses and numerical simulations are included to validate the study. J CIRCUIT SYST COMP Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/23/15. For personal use only.20th century. 1,2 As a typical example, Andre Geim and Konstantin Novoselov won the 2010 Nobel Prize in Physics for their investigations of the resistance networks of the graphene, which indicates the existence of the planar circuit networks in nature. Therefore, circuit networks cannot merely be considered as a science, but also a technology, which shows their charm. [3][4][5][6][7][8][9][10][11] Over the past few decades, researchers have published some papers on the integer circuit networks with the method of the random walks, 12 the theory of harmonic functions, 13¯r st-passage processes 14 and the lattice Green's functions 15 and so on. However, the past e®orts have been focused mainly on analyzing the resistances 16,17 or capacitances 18,19 of the integer order single-component circuit networks. Little attention has been paid to the impedance of the fractional-order multiple-component circuit networks.Fractional-order mathematical models developed for inductors and capacitors could describe the electrical characteristics more accurately, which allows for higher°e xibility, more freedom, best¯t and better optimization techniques. In other words, the actual inductors and capacitors are fractional-order in nature. 20-23 Especially, Westerlund has creatively created the fractional-order capacitors with di®erent electrolytes. 24,25 Moreover, Tenreiro and Galhano indicated that the fractional-order inductors could be designed based on skin e®ect. 26 If things happen, circuit designers will face the new challenges to the fundamentals and laws due to the fractional-order components.At present, some researchers have also been devoting to the design and realization of the fractional electronic components. 27-31 Furthermore, some researchers have been concentrating on the study of fractional-order circuit theory. 32-38 However, few researchers study the electrical characteristics of 2 Â n circuit networks in fractionalorder sense. Therefore, we focus on the impedance of the fractional-order 2 Â n RLC circuit network, which may lay the groundwork for the furthe...

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