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Microcontroller-based simulation of a nonlinear resistive-capacitive-inductance shunted Josephson junction model and applications in electromechanical engineering

  • Arnaud Notué Kadjie EMAIL logo , Hyacinthe Tchakounté , Isaac Kemajou and Paul Woafo

Abstract

The equations, modelling a nonlinear resistive-capacitive-inductance shunted Josephson junction (NRCLJJ) subjected to various signal shapes of the electrical current, are simulated experimentally using the Arduino Uno-type microcontroller that takes benefit of its simplicity, lost cost, high precision, ease of implementation, and stability compared to the voltage-controlled oscillators (VCO) circuitry. Real time electrical signals are observed presenting various dynamics. Shapiro steps (SS) from the IV-characteristics are also obtained. These real electrical signals are then used to power an electromechanical pendulum in the second part of this work. Bifurcation diagram shows that the pendulum exhibits periodic and chaotic dynamics.


Corresponding author: Arnaud Notué Kadjie, School of Wood, Water and Natural Resources, Faculty of Agronomy and Agricultural Sciences, Ebolowa Campus, University of Dschang, P. O. Box 786, Ebolowa, Cameroon; and Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes, TWAS Research Unit, Department of Physics, Faculty of Science, University of Yaoundé I, P. O. Box 812, Yaoundé, Cameroon, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix

Let’s consider Eqs. (A1)(A3)

(A1) C d V d t + V R ( V ) + I c sin θ + i s = I ext

(A2) h 2 π e d θ d t = V

(A3) L s d i s d t + R s i s = V

Let’s substitute Eq. (A2) into Eq. (A1):

(A4) h C 2 π e d 2 θ d t 2 + 1 R ( V ) h 2 π e d θ d t + I c sin θ + i s = I ext

The differentiation of Eq. (A4) using Eq. (A3) leads to:

(A5) h C 2 π e d 3 θ d t 3 + 1 R ( V ) h 2 π e d 2 θ d t 2 + I c cos θ + 1 L s h 2 π e d θ d t R s i s = d I ext d t

Finally, if one takes i s in the expression of Eq. (A4) and substitute to Eq. (A5), one obtains:

(A6) h C 2 π e d 3 θ d t 3 + 1 R ( V ) h 2 π e + h C 2 π e R s L s d 2 θ d t 2 + 1 R ( V ) h 2 π e + R s L s h 2 π e L s d θ d t + I c cos θ + R s L s I c sin θ = d I ext d t + R s L s I ext

References

[1] K. Sifeu Takougang, F. Kuiate, R. Kengne, R. Tchitnga, and P. Woafo, “Analysis of a no equilibrium linear resistive-capacitive-inductance shunted junction model, dynamics, synchronization, and application to digital cryptography in its fractional-order form,” Complexity, 2017.10.1155/2017/4107358Search in Google Scholar

[2] S. Kumar Dana, D. Chandra Sengupta, and K. D. Edoh, “Chaotic dynamics in Josephson junction,” IEEE Trans. Circuits Syst. I. Fundam. Theory Appl., vol. 48, pp. 990–996, 2001. https://doi.org/10.1109/81.940189.Search in Google Scholar

[3] C. B. Whan, C. J. Lobb, and M. G. Forester, “Effect of inductance in externally shunted Josephson tunnel junction,” J. Appl. Phys., vol. 77, pp. 382–389, 1995. https://doi.org/10.1063/1.359334.Search in Google Scholar

[4] C. B. Whan and C. J. Lobb, “Complex dynamical behaviors in RCL- shunted Josephson tunnel junction,” Phys. Rev. E, vol. 53, pp. 405–413, 1996. https://doi.org/10.1103/physreve.53.405.Search in Google Scholar PubMed

[5] A. B. Cawthorne, C. B. Whan, and C. J. Lobb, “Complex dynamics of resistively and inductively shunted Josephson junction,” J. Appl. Phys., vol. 84, pp. 1126–1132, 1998. https://doi.org/10.1063/1.368113.Search in Google Scholar

[6] Y. Zhang, C. N. Wang, J. Tang, J. Ma, and G. D. Ren, “Phase coupling synchronization of FHN neurons connected by a Josephson junction,” Sci. China Technol. Sci., vol. 63, pp. 2328–2338, 2020. https://doi.org/10.1007/s11431-019-1547-5.Search in Google Scholar

[7] U. Simo Domguia, M. V. Tchakui, H. Simo, and P. Woafo, “Theoretical and experimental study of an electromechanical system Actuated by a brusselator electronic circuit simulator,” J. Vib. Acoust., vol. 27, pp. 25–30, 2017.10.1115/1.4037139Search in Google Scholar

[8] S. E. Lyshevski, Electromechanical Systems and Devices, 6000 Broken Sound Parkway NW, CRC Press, Taylor & Francis Group, 2008.10.1201/9781420069754Search in Google Scholar

[9] C. K. Hu, “Spiking and bursting in Josephson junction,” IEEE Trans. Circuits Syst., II, Exp. Briefs, vol. 53, pp. 1031–1034, 2006.10.1109/TCSII.2006.882183Search in Google Scholar

[10] Y. L. Feng and K. Shen, “Chaos synchronization in RCL-shunted Josephson junctions via a common chaos driving,” Eur. Phys. J. B, vol. 61, pp. 105–110, 2008. https://doi.org/10.1140/epjb/e2008-00037-9.Search in Google Scholar

[11] K. S. Ojo, A. O. Adelakun, and A. A. Oluyinka, “Synchronisation of cyclic coupled Josephson junctions and its microcontroller-based implementation,” Pramana - J. Phys., 2019. https://doi.org/10.1007/s12043-019-1733-3.Search in Google Scholar

[12] Y. L. Feng and K. Shen, “Synchronization of chaos in resistive-capacitive-inductive shunted Josephson junctions,” Chin. Phys. B, vol. 17, pp. 550–556, 2008.10.1088/1674-1056/17/2/033Search in Google Scholar

[13] R. Thepi Siewe, U. Simo Domguia, and P. Woafo, “Microcontroller control/synchronization of the dynamics of van der Pol oscillators submitted to disturbances,” Int. J. Nonlinear Sci. Numer. Simul., vol. 19, pp. 153–163, 2017.10.1515/ijnsns-2017-0025Search in Google Scholar

[14] A. Uçar, K. E. Lonngren, and E.-W. Bai, “Chaos synchronization in RCL-shunted Josephson junction via active control,” Chaos, Solit. Fractals, vol. 31, pp. 105–111, 2007. https://doi.org/10.1016/j.chaos.2005.09.035.Search in Google Scholar

[15] Y. Jun-Juh, H. Cheng-Fang, and L. Jui-Sheng, “Robust synchronization of chaotic behavior in unidirectional coupled RCLSJ models subject to uncertainties,” Nonlinear Anal. R. World Appl., vol. 10, pp. 3091–97, 2009.10.1016/j.nonrwa.2008.10.009Search in Google Scholar

[16] H. Cheng-Fang, L. Jui-Sheng, Y. Jun-Juh, and W. Cheng-Chi, “On the robust chaos synchronization of RCLSJ models,” in IEEE Conf. on Soft Computing in Industrial Applications (SMCia/08), Muroran, Japan, 2008.Search in Google Scholar

[17] A. M. Harb and B. A. Harb, “Controlling chaos in Josephson-junction using nonlinear backstepping controller,” IEEE Trans. Appl. Supercond., vol. 16, pp. 1988–1998, 2006. https://doi.org/10.1109/tasc.2006.881811.Search in Google Scholar

[18] Y. M. Shukrinov, I. R. Rahmonov, K. V. Kulikov, and P. Seidel, “Effects of LC shunting on the Shapiro steps features of Josephson junction,” EPL, vol. 110, 2015, Art no. 47001. https://doi.org/10.1209/0295-5075/110/47001.Search in Google Scholar

[19] E. M. Tekougoum, N. U. Gael, S. Noumbissi, F. H. Bertrand, and P. Woafo, “Effects of carrying capacity and delay on the dynamics of Lotka–Volterra system: mathematical, numerical and microcontroller simulation,” Commun. Nonlinear Sci. Numer. Simulat., vol. 62, pp. 454–461, 2018.10.1016/j.cnsns.2018.01.015Search in Google Scholar

[20] R. Chiu, M. Mora-Gonzalez, and D. Lopez-Mancilla, “Implementation of a chaotic oscillator into a simple microcontroller,” IERI Procedia, vol. 4, pp. 247–252, 2013. https://doi.org/10.1016/j.ieri.2013.11.035.Search in Google Scholar

[21] H. Hamiche, S. Guermah, R. Saddaoui, K. Hannoun, M. Laghrouche, and S. Djennoune, “Analysis and implementation of a novel robust transmission scheme for private digital communications using Arduino Uno board,” Nonlinear Dynam., vol. 81, pp. 1921–1932, 2015. https://doi.org/10.1007/s11071-015-2116-z.Search in Google Scholar

[22] A. Valizadeh, M. R. Kolahchi, and J. P. Straley, “On the origin of fractional Shapiro steps in systems of Josephson junctions with few degrees of freedom,” J. Nonlinear Math. Phys., vol. 15, pp. 407–416, 2008. https://doi.org/10.2991/jnmp.2008.15.s3.39.Search in Google Scholar

[23] J. Akhtar Khan, A. S. Almazyad, and M. Shahabuddin, “Simulation study of noise effect on Shapiro steps in high-Tc Josephson junctions using RCLSJ model,” J. Supercond. Nov. Magnetism, vol. 24, pp. 1649–1651, 2011. https://doi.org/10.1007/s1098-010-1072-6.Search in Google Scholar

[24] A. Notué Kadjie and P. Woafo, “Effects of springs on a pendulum electromechanical energy harvester,” Theor. Appl. Mech. Lett., vol. 4, 2014. https://doi.org/10.1063/2.1406301.Search in Google Scholar

[25] A. Notué Kadjie, I. Kemajou, and P. Woafo, “Control of an electromechanical pendulum subjected to impulse disturbances using the Melnikov theory approach,” J. Mech. Sci. Technol., vol. 32, no. 2, pp. 865–874, 2018. https://doi.org/10.1007/s12206-018-0137-x.Search in Google Scholar

[26] B. Nana, S. B. Yamgoué, R. Tchitnga, and P. Woafo, “Dynamics of a pendulum driven by a DC motor and magnetically controlled,” Chaos, Solit. Fractals, vol. 104, pp. 18–27, 2017. https://doi.org/10.1016/j.chaos.2017.07.027.Search in Google Scholar

[27] F. D. O. Tcheutchoua, “Sieving devices based on nonlinear dynamics of electromechanical systems with rotary electric actuator: theory and experiment,” Ph.D. Thesis, University of Yaoundé I, Yaoundé-Cameroon, 2011.Search in Google Scholar

[28] R. Tsapla Fotsa and P. Woafo, “Chaos in a new bistable rotating electromechanical system,” Chaos, Solit. Fractals, vol. 93, pp. 48–57, 2016. https://doi.org/10.1016/j.chaos.2016.09.025.Search in Google Scholar

[29] A. Notué Kadjie, P. R. Nwagoum Tuwa, and P. Woafo, “An electromechanical pendulum robot arm in action: dynamics and control,” Shock. Vib., vol. 2017, 2017, Art no. 3979384. https://doi.org/10.1155/2017/3979384.Search in Google Scholar

[30] J. B. Mogo and P. Woafo, “Dynamics of a cantilever arm actuated by a nonlinear electrical circuit,” Nonlinear Dynam., vol. 63, pp. 807–818, 2011. https://doi.org/10.1007/s11071-010-9839-7.Search in Google Scholar

[31] J. M. Ottino, F. J. Muzzio, and M. Tjahjadi, “Chaos, symmetry and self-similarity exploiting order and disorder in mixing process,” Science, vol. 257, pp. 754–760, 1992. https://doi.org/10.1126/Science.257.5071.754.Search in Google Scholar PubMed

[32] S. Ye and K. T. Chau, “Chaotization of DC motors for industrial mixing,” IEEE Trans. Ind. Electron., vol. 54, pp. 2024–2032, 2007. https://doi.org/10.1109/TIE.2007.895150.Search in Google Scholar

Received: 2018-05-23
Revised: 2021-04-26
Accepted: 2021-05-12
Published Online: 2021-06-10
Published in Print: 2022-02-26

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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