本文選題：分數(shù)階微積分 + 分數(shù)階電容　； 參考：《山東大學》2017年碩士論文

【摘要】：近四十年來,分數(shù)階微積分在理工商醫(yī)等領域得到了廣泛的應用。相對于整數(shù)階微積分,分數(shù)階微積分的引入體現(xiàn)出了一定的優(yōu)越性。特別是在電路領域,人們也逐漸認識到,分數(shù)階微積分能更好地描述電路中的一些特有現(xiàn)象。而且在電路設計以及實際應用中分數(shù)階電路相對于整數(shù)階電路有更大的靈活性并包含更多的特征參數(shù)。但是分數(shù)階電路原理尚不完善,亟待補充發(fā)展。本文對分數(shù)階電路原理進行了系統(tǒng)分析并經過相關的實驗仿真得到了一些獨到的結果,對分數(shù)階電路原理體系進行了完善,并對分數(shù)階等效電路進行了研究。提出了一種兼顧時頻域的綜合分析方法,并將其應用到了電解電容分數(shù)階特性的證明與研究當中。首先,作為分數(shù)階電路中的最重要的元件,本文給出了分數(shù)階電容幾種不同的實現(xiàn)方法,并通過實驗仿真得到了相對更加精確的RC電路趨近方法。提出了分數(shù)階電容的普遍性,給出了其模型,并將這個模型應用到了 0.5階電容的RC電路趨近中去。然后,本文分析了分數(shù)階等效電路的原理及應用,對分數(shù)階RC、RL、RLC電路進行了分析,包括阻抗、幅值、相位以及分數(shù)階電路特有的純虛數(shù)阻抗。與整數(shù)階電路做了對比,并通過實驗仿真分析了分數(shù)階階次對分數(shù)階電路阻抗的影響,以及對純虛數(shù)阻抗的影響,得到了若干獨特的特征。然后本文給出了兩種分數(shù)階等效電路的應用,包括生物力學以及鋰離子電池的等效電路模型,并提出了其相對于整數(shù)階等效電路的優(yōu)越性。最后,本文給出了三種不同的分數(shù)階電路的分析方法,時域分析方法、頻域分析方法和一種更適合實際應用或模型分析的時頻域綜合分析方法,并分析了整數(shù)階電路和分數(shù)階電路在時域以及頻域中的不同。其中本文提出的綜合分析方法是一種獨到的分析方法,這種方法既考慮了時域擬合的精度,又有效保留了實際系統(tǒng)的頻率特性,可以有效的應用到實際電路以及多種灰箱模型的分析當中。本文將這種方法應用到了電解電容分數(shù)階特性的驗證與分析當中去,證明了電解電容的分數(shù)階特性,得到了電解電容在不同頻率下的階次,得到了電解電容的一個變階模型,并對得到的大量實驗結果作了分析,得到了很多獨特的結論。
[Abstract]:In recent 40 years, fractional calculus has been widely used in the fields of science, industry, medicine and so on. Compared with integral calculus, the introduction of fractional calculus shows some advantages. Especially in the field of circuits, people have come to realize that fractional calculus can better describe some special phenomena in circuits. In addition, fractional order circuits are more flexible and contain more characteristic parameters than integer order circuits in circuit design and practical application. However, the principle of fractional order circuit is not perfect and needs to be developed urgently. In this paper, the principle of fractional order circuit is systematically analyzed, and some original results are obtained through relevant experimental simulation. The principle system of fractional order circuit is improved, and the fractional equivalent circuit is studied. A comprehensive analysis method in time-frequency domain is proposed, and it is applied to prove and study the fractional order characteristic of electrolytic capacitance. Firstly, as the most important component in fractional order circuits, several different methods of realizing fractional order capacitors are given, and a more accurate approach to RC circuits is obtained by experimental simulation. In this paper, the universality of fractional capacitance is proposed and its model is given, and the model is applied to the approach of RC circuit with 0.5 order capacitance. Then, the principle and application of fractional order equivalent circuit are analyzed, including impedance, amplitude, phase and pure imaginary impedance of fractional order circuit. The effects of fractional order on fractional order impedance and pure imaginary impedance are compared with integer order circuit, and some unique characteristics are obtained. Then, two kinds of fractional equivalent circuits are presented, including biomechanics and equivalent circuit models of lithium ion batteries, and their advantages over integer order equivalent circuits are presented. Finally, three different analysis methods of fractional order circuits, time-domain analysis, frequency-domain analysis and a time-frequency integrated analysis method, which are more suitable for practical application or model analysis, are presented. The difference between integer order circuit and fractional order circuit in time domain and frequency domain is analyzed. The synthetic analysis method proposed in this paper is a unique analysis method, which not only considers the precision of time domain fitting, but also effectively preserves the frequency characteristics of the actual system. It can be effectively applied to the analysis of practical circuits and various grey box models. This method is applied to the verification and analysis of the fractional order characteristics of electrolytic capacitors. The fractional order characteristics of electrolytic capacitors are proved. The order of electrolytic capacitors at different frequencies and a variable order model of electrolytic capacitors are obtained. A large number of experimental results are analyzed and many unique conclusions are obtained.
【學位授予單位】：山東大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：TN70

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本文編號：1856399