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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 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.
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Keywords:
- memristor /
- hysteresis loop /
- memory loss /
- fractional-order derivative
[1] Chua L O 1971 IEEE Trans. Circ. Theory 18 507
[2] Chua L O, Kang S M 1976 Proc IEEE 64 209
[3] Strukov D B, Snider G S, Stewart D R, Williams R S 2008 Nature 453 80
[4] Williams R S 2008 IEEE Spectrum 45 24
[5] Pershin Y V, Di Ventra M 2008 Phys. Rev. B 78 113309
[6] Wang X B, Chen Y R, Xi H, Li H, Dimitrov D 2009 IEEE Electron Device Letters 30 294
[7] Borghetti J, Li Z, Straznicky J, Li X, Ohlberg D A, Wu W, Stewart D R, Williams R S 2009 Proc Natl Acad Sci. USA 106 1699
[8] Borghetti J, Snider G S, Kuekes P J, Yang J J, Stewart D R, Williams R S 2010 Nature 464 873
[9] Thomas A 2013 J. Phys. D: Appl. Phys. 46 093001
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[19] Kim H, Sah M P, Yang C J, Seongik C, Chua L O 2012 IEEE Trans. Circ. Syst. I: Regular Papers 59 2422
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[22] Hilfer R 2000 Applications of fractional calculus in physics (Singapore: World Scientific Publishing Co Pte Ltd) 430
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[24] Cafagna D, Grassi G 2012 Nonlinear Dyn. 70 1185
[25] Sabatier J, Agrawal O P, Machado J A 2007 Advances in Fractional Calculus (Netherlands: Springer) 419
[26] Podlubny I 1999 IEEE Trans. Autom. Control 44 208
[27] Monje C A, Chen Y Q, Vinagre B M, Xue D Y, Feliu V 2010 Fractional-order Systems and Controls: Fundamentals and Applications(London: Springer-Verlag)
[28] Podlubny I 1999 Fractional Differential Equations (SanDiego: Academic Press)
[29] Rossikhin Yu A, Shitikova M V 2010 Applied Mechanics Reviews 63 010801
[30] Wang Z H, Hu H Y 2009 Sci. China Ser. G 39 1495 (in Chinese) [王在华, 胡海岩 2009 中国科学 G 辑 39 1495]
[31] Biolek D, Biolek Z, Biolková V 2014 Electron Lett. 50 74
[32] Biolek Z, Biolek D, Biolková V 2012 IEEE Trans. Circ. Syst. II: Exp.Briefs 59 607
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[1] Chua L O 1971 IEEE Trans. Circ. Theory 18 507
[2] Chua L O, Kang S M 1976 Proc IEEE 64 209
[3] Strukov D B, Snider G S, Stewart D R, Williams R S 2008 Nature 453 80
[4] Williams R S 2008 IEEE Spectrum 45 24
[5] Pershin Y V, Di Ventra M 2008 Phys. Rev. B 78 113309
[6] Wang X B, Chen Y R, Xi H, Li H, Dimitrov D 2009 IEEE Electron Device Letters 30 294
[7] Borghetti J, Li Z, Straznicky J, Li X, Ohlberg D A, Wu W, Stewart D R, Williams R S 2009 Proc Natl Acad Sci. USA 106 1699
[8] Borghetti J, Snider G S, Kuekes P J, Yang J J, Stewart D R, Williams R S 2010 Nature 464 873
[9] Thomas A 2013 J. Phys. D: Appl. Phys. 46 093001
[10] Hu X F, Duan S K, Wang L D, Liao X F 2011 Sci. China: Informationis 41 500 (in Chinese) [胡小方, 段书凯, 王丽丹, 廖晓峰 2011 中国科学: 41 500]
[11] Cai K P, Wang R, Zhou J 2010 Electronic Components and Materials 29 78 (in Chinese) [蔡坤鹏, 王睿, 周济 2010 电子元件与材料 29 78]
[12] Bao B C, Liu Z, Xu J P 2010 Acta Phys. Sin. 59 3785 (in Chinese) [包伯成, 刘中, 许建平 2010 59 3785]
[13] Bao B C, Hu W, Xu J P, Liu Z, Zou L 2011 Acta Phys. Sin. 60 120502 (in Chinese) [包伯成, 胡文, 许建平, 刘中, 邹凌 2011 60 120502]
[14] Yuan F, Wang G Y, Wang X Y 2015 Chin. Phys. B 24 060506
[15] Strukov D B, Williams R S 2009 Appl. Phys. A 94 515
[16] Shkabko A, Aguirre M H, Marozau I, Lippert T, Weidenkaff A 2009 Appl. Phys. Lett. 95 152109
[17] Muthuswamy B, Chua L O 2010 Int. J. Bifurcation and Chaos 20 1567
[18] Adhikari S P, Sah M P, Kim H, Chua L O 2013 IEEE Trans. Circ. Syst. I: Regular Papers 60 3008
[19] Kim H, Sah M P, Yang C J, Seongik C, Chua L O 2012 IEEE Trans. Circ. Syst. I: Regular Papers 59 2422
[20] Corinto F, Ascoli A 2012 IEEE Trans. Circuits Syst. I: Regular Papers 59 2713
[21] Xu M Y, Tan W C 2002 Sci. China Ser. A 32 673 (in Chinese) [徐明瑜, 谭文长 2002 中国科学 A辑 32 673]
[22] Hilfer R 2000 Applications of fractional calculus in physics (Singapore: World Scientific Publishing Co Pte Ltd) 430
[23] Fouda M E, Radwan A G 2013 J. Frac. Calcu. Appl. 4 1
[24] Cafagna D, Grassi G 2012 Nonlinear Dyn. 70 1185
[25] Sabatier J, Agrawal O P, Machado J A 2007 Advances in Fractional Calculus (Netherlands: Springer) 419
[26] Podlubny I 1999 IEEE Trans. Autom. Control 44 208
[27] Monje C A, Chen Y Q, Vinagre B M, Xue D Y, Feliu V 2010 Fractional-order Systems and Controls: Fundamentals and Applications(London: Springer-Verlag)
[28] Podlubny I 1999 Fractional Differential Equations (SanDiego: Academic Press)
[29] Rossikhin Yu A, Shitikova M V 2010 Applied Mechanics Reviews 63 010801
[30] Wang Z H, Hu H Y 2009 Sci. China Ser. G 39 1495 (in Chinese) [王在华, 胡海岩 2009 中国科学 G 辑 39 1495]
[31] Biolek D, Biolek Z, Biolková V 2014 Electron Lett. 50 74
[32] Biolek Z, Biolek D, Biolková V 2012 IEEE Trans. Circ. Syst. II: Exp.Briefs 59 607
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