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Generalized modeling and character analyzing of composite fractional-order memristors in series connection

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Abstract

Memristor is a type of memory device representing the relation between charge q and flux \(\varphi \). Recently, memristor, as well as other memristive elements and their corresponding fractional elements, have become very attractive in many applications, due to their unique behavior which cannot be obtained by using conventional elements. However, there are few studies about composite behaviors or characteristics of multiple memristive elements connected in series or parallel, especially for the fractional-order memristors. This paper focuses on analyzing the composite characteristics of two fractional-order memristors connected in series as well as concerning the window function. Under the applied sinusoidal current source, two charge-controlled memristors connected in series are adopted to theoretically demonstrate the variation of memristance and v–i curves. The two fractional order \(\alpha \) and \(\beta \) are interpreted as two phase offset which makes the analysis much convenient. The obtained formulas of instantaneous memristance of composite memristor are derived and some influential factors are analyzed in detail. The further analyzing demonstrates that the composite memristor circuits behave as a new memristor with higher complexity and some typical phenomena are found especially for the pinched hysteresis loops in v–i plane.

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References

  1. Chua, L.: Memristor—the missing circuit element. IEEE Trans. Circuit Theory 18(5), 507 (1971). https://doi.org/10.1109/tct.1971.1083337

    Article  Google Scholar 

  2. Strukov, D.B., Snider, G.S., Stewart, D.R., Williams, R.S.: The missing memristor found. Nature 453(7191), 80 (2008). https://doi.org/10.1038/nature06932

    Article  Google Scholar 

  3. Biolek, D., Biolek, Z., Biolkova, V.: Pinched hysteretic loops of ideal memristors, memcapacitors and meminductors must be ’self-crossing’. Electron. Lett. 47(25), 1385 (2011). https://doi.org/10.1049/el.2011.2913

    Article  Google Scholar 

  4. Pershin, Y.V., Di Ventra, M.: Memory effects in complex materials and nanoscale systems. Adv. Phys. 60(2), 145 (2011). https://doi.org/10.1080/00018732.2010.544961

    Article  Google Scholar 

  5. Di Ventra, M., Pershin, Y.V., Chua, L.O.: Circuit elements with memory: memristors, memcapacitors, and meminductors. Proc. IEEE 97(10), 1717 (2009). https://doi.org/10.1109/JPROC.2009.2021077

    Article  Google Scholar 

  6. Ielmini, D.: Brain-inspired computing with resistive switching memory (RRAM): devices, synapses and neural networks. Microelectron. Eng. 190, 44 (2018). https://doi.org/10.1016/j.mee.2018.01.009

    Article  Google Scholar 

  7. Xia, L., Li, B., Tang, T., Gu, P., Chen, P.Y., Yu, S., Cao, Y., Wang, Y., Xie, Y., Yang, H.: MNSIM: simulation platform for memristor-based neuromorphic computing system. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 37(5), 1009 (2017). https://doi.org/10.1109/TCAD.2017.2729466

    Article  Google Scholar 

  8. Maan, A.K., Jayadevi, D.A., James, A.P.: A survey of memristive threshold logic circuits. IEEE Trans. Neural Netw. Learn. Syst. 28(8), 1734 (2017). https://doi.org/10.1109/tnnls.2016.2547842

    Article  MathSciNet  Google Scholar 

  9. Maan, A.K., Kumar, D.S., Sugathan, S., James, A.P.: Memristive threshold logic circuit design of fast moving object detection. IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 23(10), 2337 (2015). https://doi.org/10.1109/tvlsi.2014.2359801

    Article  Google Scholar 

  10. Dongale, T., Bagade, A., Mohite, S., Rananavare, A., Orlowski, M., Kamat, R., Rajpure, K.: Bipolar resistive switching with coexistence of mem-elements in the spray deposited \(\text{ CoFe }_{2}\text{ O }_{4}\) thin film. J. Mater. Sci. Mater. Electron. 29(4), 3231 (2017). https://doi.org/10.1007/s10854-017-8258-7

    Article  Google Scholar 

  11. Rajagopal, K., Karthikeyan, A., Srinivasan, A.: Dynamical analysis and FPGA implementation of a chaotic oscillator with fractional-order memristor components. Nonlinear Dyn. 91(3), 1491 (2017). https://doi.org/10.1007/s11071-017-3960-9

    Article  MATH  Google Scholar 

  12. Rziga, F.O., Mbarek, K., Ghedira, S., Besbes, K.: The basic I–V characteristics of memristor model: simulation and analysis. Appl. Phys. A 123(4), 288 (2017). https://doi.org/10.1007/s00339-017-0902-9

    Article  Google Scholar 

  13. Fouda, M.E., Radwan, A.G.: Resistive-less memcapacitor-based relaxation oscillator. Int. J. Circuit Theory Appl. 43(7), 959 (2015). https://doi.org/10.1002/cta.1984

    Article  Google Scholar 

  14. Gomez Aguilar, J.F.: Behavior characteristics of a cap-resistor, memcapacitor, and a memristor from the response obtained of RC and RL electrical circuits described by fractional differential equations. Turk. J. Electr. Eng. Comput. Sci. 24(3), 1421 (2016). https://doi.org/10.3906/elk-1312-49

    Article  Google Scholar 

  15. Fouda, M.E., Radwan, A.G.: Memcapacitor response under step and sinusoidal voltage excitations. Microelectron. J. 45(11), 1372 (2014). https://doi.org/10.1016/j.mejo.2014.08.002

    Article  Google Scholar 

  16. Wang, G.Y., Cai, B.Z., Jin, P.P., Hu, T.L.: Memcapacitor model and its application in a chaotic oscillator. Chin. Phys. B 25(1), 010503 (2016). https://doi.org/10.1088/1674-1056/25/1/010503

    Article  Google Scholar 

  17. Yang, X.J., Machado, J.T., Cattani, C., Gao, F.: On a fractal LC-electric circuit modeled by local fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 47, 200 (2017). https://doi.org/10.1016/j.cnsns.2016.11.017

    Article  Google Scholar 

  18. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent—II. Geophys. J. Int. 13(5), 529 (1967). https://doi.org/10.1111/j.1365-246x.1967.tb02303.x

    Article  Google Scholar 

  19. Sun, H., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. (2018). https://doi.org/10.1016/j.cnsns.2018.04.019

    Article  Google Scholar 

  20. Zhang, H., Wang, X.Y., Lin, X.H.: Stability and control of fractional chaotic complex networks with mixed interval uncertainties. Asian J. Control 19(1), 106 (2017). https://doi.org/10.1002/asjc.1333

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang, G., Zang, S., Wang, X., Yuan, F., Iu, H.H.C.: Memcapacitor model and its application in chaotic oscillator with memristor. Chaos Interdiscip. J. Nonlinear Sci. 27(1), 013110 (2017). https://doi.org/10.1063/1.4973238

    Article  MATH  Google Scholar 

  22. Huang, S., Zhang, R., Chen, D.: Stability of nonlinear fractional-order time varying systems. J. Comput. Nonlinear Dyn. 11(3), 031007 (2016). https://doi.org/10.1115/1.4031587

    Article  Google Scholar 

  23. Chen, L., Cao, J., Wu, R., Machado, J.T., Lopes, A.M., Yang, H.: Stability and synchronization of fractional-order memristive neural networks with multiple delays. Neural Netw. 94, 76 (2017). https://doi.org/10.1016/j.neunet.2017.06.012

    Article  Google Scholar 

  24. Arqub, O.A.: Solutions of time-fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space. Numer. Methods Partial Differ. Equ. 34(5), 1759 (2017). https://doi.org/10.1002/num.22236

    Article  MathSciNet  Google Scholar 

  25. Arqub, O.A.: Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm. Int. J. Numer. Methods Heat Fluid Flow 28(4), 828 (2018). https://doi.org/10.1108/hff-07-2016-0278

    Article  Google Scholar 

  26. Arqub, O.A.: Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions. Comput. Math. Appl. 73(6), 1243 (2017). https://doi.org/10.1016/j.camwa.2016.11.032

    Article  MathSciNet  Google Scholar 

  27. Abdelouahab, M.S., Lozi, R., Chua, L.: Memfractance: a mathematical paradigm for circuit elements with memory. Int. J. Bifurc. Chaos 24(9), 1430023 (2014). https://doi.org/10.1142/s0218127414300237

    Article  MATH  Google Scholar 

  28. Guo, Z., Si, G., Diao, L., Jia, L., Zhang, Y.: Generalized modeling of the fractional-order memcapacitor and its character analysis. Commun. Nonlinear Sci. Numer. Simul. 59, 177 (2018). https://doi.org/10.1016/j.cnsns.2017.11.007

    Article  MathSciNet  Google Scholar 

  29. Machado, J.T., Galhano, A.M.: Generalized two-port elements. Commun. Nonlinear Sci. Numer. Simul. 42, 451 (2017). https://doi.org/10.1016/j.cnsns.2016.05.030

    Article  Google Scholar 

  30. Machado, J.T.: Fractional generalization of memristor and higher order elements. Commun. Nonlinear Sci. Numer. Simul. 18(2), 264 (2013). https://doi.org/10.1016/j.cnsns.2012.07.014

    Article  MathSciNet  MATH  Google Scholar 

  31. Si, G., Diao, L., Zhu, J., Lei, Y., Babajide, O., Zhang, Y.: Modeling and character analyzing of current-controlled memristors with fractional kinetic transport. Commun. Nonlinear Sci. Numer. Simul. 48, 224 (2017). https://doi.org/10.1016/j.cnsns.2016.12.030

    Article  Google Scholar 

  32. Fouda, M.E., Radwan, A.G.: Fractional-order memristor response under DC and periodic signals. Circuits Syst. Signal Process. 34(3), 961 (2014). https://doi.org/10.1007/s00034-014-9886-2

    Article  Google Scholar 

  33. Pu, Y.F., Yuan, X., Yu, B.: Analog circuit implementation of fractional-order memristor: arbitrary-order lattice scaling fracmemristor. IEEE Trans. Circuits Syst. I Regul. Pap. (2018). https://doi.org/10.1109/tcsi.2018.2789907

  34. Qiyan, Z., Dongsheng, Y., Yan, L., Mengke, C.: Composite behaviors of dual meminductor circuits. Chin. Phys. B 24(11), 180 (2015). https://doi.org/10.1088/1674-1056/24/11/110701

    Article  Google Scholar 

  35. Wang, X.Y., Iu, H.H.C., Wang, G.Y., Liu, W.: Study on time domain characteristics of memristive RLC series circuits. Circuits Syst. Signal Process. 35(11), 4129 (2016). https://doi.org/10.1007/s00034-016-0250-6

    Article  MATH  Google Scholar 

  36. Zhi, Z., Dongsheng, Y., Xiaoyuan, W.: Investigation on the dynamic behaviors of the coupled memcapacitor-based circuits. Chin. Phys. B 26(12), 179 (2017). https://doi.org/10.1088/1674-1056/26/12/120701

    Article  Google Scholar 

  37. Tavazoei, M.S.: A note on fractional-order derivatives of periodic functions. Automatica 46(5), 945 (2010). https://doi.org/10.1016/j.automatica.2010.02.023

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by Shaanxi Natural Science Foundation (Grant No: 2018JM5095), the Fundamental Research Funds for the Central Universities and the China Scholarship Council Fund (Grant No: 201806285022).

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Authors and Affiliations

  1. State Key Laboratory of Electrical Insulation and Power Equipment, Shaanxi Key Laboratory of Smart Grid, School of Electrical Engineering, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi Province, China

    Zhang Guo, Gangquan Si, Xiang Xu, Kai Qu & Shuang Li

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  1. Zhang Guo
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Correspondence to Gangquan Si.

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Guo, Z., Si, G., Xu, X. et al. Generalized modeling and character analyzing of composite fractional-order memristors in series connection. Nonlinear Dyn 95, 101–115 (2019). https://doi.org/10.1007/s11071-018-4553-y

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