一个分数阶忆阻器模型及其简单串联电路的特性

忆阻器是具有时间记忆特性的非线性电阻. 经典HP TiO₂忆阻器模型的忆阻值为此前通过忆阻器电流的时间积分, 即记忆没有损失. 而最近研究证实HP TiO₂ 线性忆阻器掺杂层厚度不能等于零或者器件整体厚度, 导致器件的记忆有损失. 基于此发现, 本文首先提出了一个阶数介于0 与1间的分数阶HP TiO₂ 线性忆阻器模型, 研究了当受到周期外激励时, 分数阶导数的阶数对其忆阻值动态范围和输出电压动态幅值的影响规律, 推导出了磁滞旁瓣面积的计算公式. 结果表明, 分数阶导数阶数对磁滞回线的形状及所围成区域面积有重要影响. 特别地, 在外激频率大于1时, 分数阶忆阻器的记忆强度达到最大. 然后讨论了此分数阶忆阻器与电容或电感串联组成的单口网络的伏安特性. 结果表明, 在周期激励驱动时, 随着分数阶导数阶数的变化, 此分数阶忆阻器与电容的串联电路呈现出纯电容电路与忆阻电路的转换, 而它与电感的串联电路则呈现出纯电感电路与忆阻电路的转换.

A fractional-order memristor model and the fingerprint of the simple series circuits including a fractional-order memristor

A memristor is a nonlinear resistor with time memory. The resistance of a classical memristor at a given time is represented by the integration of all the full states before the time instant, a case of ideal memory without any loss. Recent studies show that there is a memory loss of the HP TiO₂ linear model, in which the width of the doped layer of HP TiO₂ model cannot be equal to zero or the whole width of the model. Based on this observation, a fractional-order HP TiO₂ memristor model with the order between 0 and 1 is proposed, and the fingerprint analysis of the new fractional-order model under periodic external excitation is made, thus the formula for calculating the area of hysteresis loop is obtained. It is found that the shape and the area enclosed by the hysteresis loop depend on the order of the fractional-order derivative. Especially, for exciting frequency being bigger than 1, the memory strength of the memristor takes its maximal value when the order is a fractional number, not an integer. Then, the current-voltage characteristics of the simple series one-port circuit composed of the fractional-order memristor and the capacitor, or composed of the fractional-order memristor and the inductor are studied separately. Results demonstrate that at the periodic excitation, the memristor in the series circuits will have capacitive properties or inductive properties as the fractional order changes.