Passive Synthesis Of Fractional Order Biquadratic Impedance Functions

With the development of fractional calculus theory,a large number of application examples of fractional-order modeling have emerged in electrical engineering and other fields.Some fractional-order circuits and systems have also emerged.Due to the introduction of fractional-order elements,fractional-order circuits and systems are characterized by different characteristics from integer-order circuits and systems,which poses new challenges to the analysis and synthesis theory of traditional integer-order circuits.The problem of integer-order passive network synthesis has been more maturely developed in the last century,forming a complete set of theories for the definition and criteria of passivity,and and the methods of circuit implementation.The emergence of fractional-order circuits makes us unable to limit the modeling and design of systems to integers.Therefore,it is necessary to discuss the synthesis of fractional-order passive circuits.Due to the complexity of fractional-order expressions compared with traditional integer-order expressions,this paper mainly explores the synthetic methods of fractional-order biquadratic expressions.The study of its implementation problems can provide important discussion for the implementation of more general fractional-order functions.Based on the circuit synthesis theory,this paper analyzes and synthesizes the fractional-order biquadratic impedance functions.In this paper,we first study the positivity criterion of fractional-order biquadratic impedance functions.Then analyze the network composed of simple network and simple connection form from the topological structure,and derive its implementation conditions and implementation steps.It mainly studies the problem that the biquadratic impedance function can be realized by at-most four elements and the implementation of the five elements networks with two fractional reactance elements.In order to classify the network,we introduce the concept of quartets.This paper attempts to obtain a comprehensive process of simple fractional order biquadratic impedance functions using a more systematic and simple method.Secondly,the synthesis of a class of special functions,ie,the fractional-order biquadratic minimum function,is studied.Three comprehensive methods for the fractional-order biquadratic minimum function are introduced and their implementation range are summarized.Finally,based on this,the arbitrary fractional biquadratic function is synthesized.