| Περίληψη: | The aim of this Ph.D. dissertation is to demonstrate the interdisciplinary nature of fractional calculus and the innovative advantages that fractional-order circuits can offer in a wide range of scientific fields. The design of such circuits for diverse filtering, control, and biological/biomedical applications is the core of this work. The main objective is to bridge the gap between theory and practical applications and, consequently, to prove the practical value of this type of circuits in numerous branches of science and technology.
Different types of non-integer order transfer functions are studied, including: (i) fractional-order transfer functions, which are based on fractional-order Laplace operator(s), (ii) power-law transfer functions, which are formed through the exponentiation of the whole initial function to a fractional power, and (iii) double-order transfer functions, derived from the combination of the two previous types. Even though the power-law and double-order function types have been utilized for describing controllers and impedances, it is the first time in the literature that power-law and double-order filters are introduced, and this is an important contribution of this Thesis.
As the exploitation of fractional calculus in the branch of engineering is one of the recently developed research disciplines, there is no commercial production of fundamental elements that could directly build a fractional-order circuit. This condition generates a requirement for development of alternative methods, in order to approximate the fractional behavior and realize the corresponding circuitry. Towards this purpose, a thorough research on the provided approximation techniques, that are able to approach different forms of fractional-order functions, is performed with the aim of presenting and comparing the available tools. Subsequently, an extensive reference on various types of structures, capable of implementing the function derived from the approximation process, is made, including passive and active configurations. The target is to achieve the most effective combination of applied approximation and utilized structure, in order to assure the best option for each case study.
The research pivots on three principal fields: (I) filtering, (II) control systems, and (III) biology/biomedicine. A methodical approach to the mathematical framework of the case under consideration is initially performed, in order to extract the critical characteristics. Then, depending on the type of the application, the specifications and the performance requirements that need to be fulfilled, the proposed procedure for the development of the mathematical functions into the corresponding implementable circuits, is presented step by step. The final stage includes simulation and/or experimental verification of the process and the derived circuitry, supported by suitable equipment that is available in the laboratory.
The theoretical analysis, as well as the approximation processes, for each case study, are performed using the MATLAB software. The performance evaluation of the proposed circuits is based on simulation results obtained using the OrCAD PSpice simulator or the Cadence software and the Design Kit provided by the Austria Mikro Systeme (AMS) CMOS 0.35 μm technology process, or on experimental results obtained using an Anadigm AN231K04 Field-Programmable Analog Array development board.
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