Topology Design And Performance Research Of Fractional-order Capacitor

Since the concept of fractional order capacitor was proposed,the model constructing,approximate circuit design,performance analysis,physical development and circuit application of fractional order capacitor have been researched one after another based on the theory of fractional calculus,and significant research results have been achieved so far.However,for the complexity of the manufacturing process,the fractional capacitor has not been widely used in the actual circuit.Therefore,in order to improve the practical application,this paper focuses on the realization of fractional capacitaor approximate circuit,analyzes and compares the characteristics of different approximate topologies.The quantitative expressions between the iterative equations and the parameters of approximate circuits are given.Several approximate topology circuits with good performances are shown in this paper.The simulation analysis and experimental results suggest that the circuit design methods proposed and the new topologies are simple and effective,and they can be applied to a practical circuit with a certain bandwidth.The following work has been completed in this paper.From the perspective of the rational approximation and the approximation of the calculus operator,an Aitken accelerated convergence iteration method based on the first-order Newton method is proposed,and the specific circuit design and parameter performance analysis of the method are given.At the same time,a detailed circuit design method is given for the commonly used fractional integral operator Oustaloup algorithm in the control field,and the permanent magnet synchronous motor control system is used as an experimental platform to verify the applicability of the method to the integral operator approximation.This paper focuses on fractal structure of the fractional order capacitor approximation,and mainly discusses the circuit complexity and approximation benefit of the fractal topology structures,such as the tree type and the bridge type,etc.The calculation formula of different fractal circuit parameters is derived by means of Matlab simulation.The series improved and parallel improved new structures are proposed to develop the defect in frequency bandwidth of four kinds of Domino ladder topology circuits.The structures achieve the index growth of approximate bandwidth of the fractional capacitor,and the accuracy of 1% in a wide frequency range only by doubling the number of components.And the approximate efficiency is much better than that of the Domino ladder structure.In order to compare the advantages and disadvantages of various topologies,this paper provides different performance indexes,such as the approximate error and the relative complexity,etc,and the comparison results show that the approximate performances of the bridge and Domino improved new topologies are the best.At the end of the paper,the series and parallel performances of fractional order capacitor are analyzed.The order and capacitance of the fractional equivalent capacitors after the series and parallel are different from that of the integer order.The order of the equivalent capacitor varies with the system frequency according to the laws of hyperbolic tangent function.In this paper,the Matlab numerical analysis tool and experimental platform are used to simulate and verify the mathematical relationship among various parameters,the performances of various approximate circuits,and the series and parallel performances of fractional order capacitors.The simulation and experimental results prove the correctness of the theoretical research.