Research On Properties And Synthesis Method Of Fractional Circuits

As the powerful extension of integer-order calculus,fractional calculus can describe complex systems,processes and phenomena more concisely and accurately,and that is why fractional calculus has drawn much attention of researchers all over the world.With the vigorous development of fractional calculus,there emerge a large number of applications of fractional calculus in electrical engineering and other fileds,which will certainly cause the emergence of massive novel fractional order circuits and systems.On one hand,conventional circuit and system theory could be extended,and more choices,better circuit performance and more design freedoms can be provided to the users;on the other hand,due to the introduction of the fractional order parameters,fractional circuits exhibit many distinct characteristics and hence constitute a new challenge for the analysis and synthesis methods in conventional theory.In recent decades,scholars have conducted a lot of researches on fractional circuits and systems,and many fruitful achievements have been achieved.However,because of the complexity of fractional calculus,there still remain many unsolved problems,for example,researches on characteristics of fractional order elements which are the base of the fractional circuits and systems need to be further studied;likely,the fabrication and realization methods of fractional order elements should be developed and improved;and the fractional-order circuit synthesis which is the key to fractional modeling of devices and equipment is just at its begining.Hence,it is of great significance to conduct the researches on properties of fractional elements and fractional-order circuit synthesis,which will certainly promote the progress and improvement of circuit theory of fractional order,solve the key problems in fractional modeling,and push forward the steps of applciations of fractional calculus in electrical engineering.Based on the above statements and aimed at fractional circuits and systems,this thesis mainly studies the properties and synthesis methods of fractional-order circuits.By the aids of the conventional circuit theory and considering the properties of fractional calculus,many important conclusions and theorem are obtianed.The main results of this paper are listed as follows:1.According to the definition of reciprocity,the conditions of reciprocity and anti-reciprocity are derived for fractional elements which are in current-controlled(or voltage-controlled)and hybrid expressions.The fractional-order state space representations of two-terminal nonlinear fractional higher-oder are given and the a-? plane are segmented into 14 regions to study their passivity by the aids of the definition of available energy.Besides,the passivity criteria of linear fractional elements are discussed based on the dissipative matrices,and the main difference form the conventional passivity creiteria are explained;In the light of the fact that the passivity of fractional order circuits is no longer equivalent to the Positive-Real properties of the immittance functions,a multivariable passive criterion for fractional order circuits is proposed.2.Based on the difference form of Tellegent's theorem,we have derived the frequency domain and time domain sensitivity formulae for two-terminal and multiport fractional elements.With the help of generalized duality,definition of inter-reciprocity and Tellegent's theorem,the hybrid-transmission presentation for multiport elements and the method of getting the adjoint elements based on the hybrid-transmission presentation are proposed.The sensitivity formulae based on the hybrid-transmission presentation is also derived.3.The relationship between fractional series/parallel LC circuits with traditional series/parallel RC and RL circuits is explored,and the similarities and differences are emphasized under the condition that the sum of the orders of fractional inductor and capacitor is equal 1.Besides,the applications and sensitivity analysis are also given.Moreover,four circuits topologies of a kind of AD844-Based convertor for fractional elements are proposed,which can be used to realize the convertion of fractional elements.4.The multivariable synthesis method for fractional order circuits is proposed and the syntheis problem of a kind of fractional order driving point immittance functions is solved.Simulations examples and fractional modeling of supercapacitors are given to validate the proposed method.