分数阶电路及其时频域综合分析方法研究

发布时间：2018-05-07 09:35

本文选题：分数阶微积分 + 分数阶电容　；参考：《山东大学》2017年硕士论文

【摘要】：近四十年来,分数阶微积分在理工商医等领域得到了广泛的应用。相对于整数阶微积分,分数阶微积分的引入体现出了一定的优越性。特别是在电路领域,人们也逐渐认识到,分数阶微积分能更好地描述电路中的一些特有现象。而且在电路设计以及实际应用中分数阶电路相对于整数阶电路有更大的灵活性并包含更多的特征参数。但是分数阶电路原理尚不完善,亟待补充发展。本文对分数阶电路原理进行了系统分析并经过相关的实验仿真得到了一些独到的结果,对分数阶电路原理体系进行了完善,并对分数阶等效电路进行了研究。提出了一种兼顾时频域的综合分析方法,并将其应用到了电解电容分数阶特性的证明与研究当中。首先,作为分数阶电路中的最重要的元件,本文给出了分数阶电容几种不同的实现方法,并通过实验仿真得到了相对更加精确的RC电路趋近方法。提出了分数阶电容的普遍性,给出了其模型,并将这个模型应用到了 0.5阶电容的RC电路趋近中去。然后,本文分析了分数阶等效电路的原理及应用,对分数阶RC、RL、RLC电路进行了分析,包括阻抗、幅值、相位以及分数阶电路特有的纯虚数阻抗。与整数阶电路做了对比,并通过实验仿真分析了分数阶阶次对分数阶电路阻抗的影响,以及对纯虚数阻抗的影响,得到了若干独特的特征。然后本文给出了两种分数阶等效电路的应用,包括生物力学以及锂离子电池的等效电路模型,并提出了其相对于整数阶等效电路的优越性。最后,本文给出了三种不同的分数阶电路的分析方法,时域分析方法、频域分析方法和一种更适合实际应用或模型分析的时频域综合分析方法,并分析了整数阶电路和分数阶电路在时域以及频域中的不同。其中本文提出的综合分析方法是一种独到的分析方法,这种方法既考虑了时域拟合的精度,又有效保留了实际系统的频率特性,可以有效的应用到实际电路以及多种灰箱模型的分析当中。本文将这种方法应用到了电解电容分数阶特性的验证与分析当中去,证明了电解电容的分数阶特性,得到了电解电容在不同频率下的阶次,得到了电解电容的一个变阶模型,并对得到的大量实验结果作了分析,得到了很多独特的结论。
[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.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TN70

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