Chau, KT

Li, F

Li, J

Wang, Z

Zhang, Z

HKU Scholars Hub

TitleAnalysis of chaos in Josephson junctions with externalmagnetic field for high-precision voltage measurement inElectric VehiclesAuthor(s) Zhang, Z; Chau, KT; Wang, Z; Li, F; Li, JCitation IEEE Transactions on Applied Superconductivity, 2012, vol. 22,no. 3, p. pp. 4904704:1-4Issued Date 2012URL http://hdl.handle.net/10722/163992Rights Creative Commons: Attribution 3.0 Hong Kong LicenseIEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 22, NO. 3, JUNE 2012 4904704Analysis of Chaos in Josephson Junctions WithExternal Magnetic Field for High-Precision VoltageMeasurement in Electric VehiclesZhen Zhang, K. T. Chau, Senior Member, IEEE, Zheng Wang, Member, IEEE, Fuhua Li, and Jiangui LiAbstract—This paper deals with analysis and control of chaosfor high-precision voltage measurement in electric vehicles (EVs)based on Josephson voltage standards. The key is to take intoaccount the effect of external magnetic field in EV environmenton Josephson junctions. Based on a modified equivalent circuitmodel to characterize the Josephson junction, it can be identifiedthat chaos is elicited by the external magnetic field. Consequently,a backstepping controller is designed to suppress the chaos, andhence improve the voltage measurement. Both mathematicalderivation and numerical simulation are provided to validate theoccurrence of chaos and the feasibility of the control scheme.Index Terms—Backstepping control, chaos, electric vehicles,Josephson junction.I. INTRODUCTIONW ITH ever increasing concern on clean and en-ergy-efficient transportation, the development ofelectric vehicles (EVs), including land vehicles and marinevehicles, has attracted wide attention [1], such as the energymanagement of EVs [2] and the high temperature supercon-ductor (HTS) application to EVs [3]. However, the energymanagement of EVs has a fundamental problem, namelythe voltage measurement device for batteries is not accurateenough. For example, the nickel-metal hydride cell varies from1.25 V to 1.2 V over the 80% change in state-of-charge (SOC),indicating that a resolution of 0.1% SOC needs an accuratemeasurement of 62.5 at 1.2 V. Thus, a high-precisionvoltage measurement device is highly desirable for EVs.Recently, the Josephson voltage standards have been de-veloped for high-precision voltage measurement [4], [5]. Thisvoltage measurement technique is developed based on a seriesarray of thousands of superconductor-insulator-normal- con-ductor-insulator-superconductor (SINIS) Josephson junctions.However, it exhibits complex nonlinear behavior [6] such aschaos may occur at certain external currents. So, various con-trol schemes are proposed to prevent the Josephson junctionsfrom chaos [7]. Nevertheless, most studies on the JosephsonManuscript received September 12, 2011; accepted December 19, 2011. Dateof publication December 21, 2011; date of current version May 24, 2012. Thiswork was supported in part by a grant of HKU Small Project Funding (ProjectCode 200907176028), and a grant of National Natural Science Foundation ofChina (51007008).Z. Zhang, K. T. Chau, F. Li, and J. Li are with the Department of Electricaland Electronic Engineering, The University of Hong Kong, Hong Kong, China(e-mail: zhzhang@eee.hku.hk).Z. Wang is with the School of Electrical Engineering, Southeast University,Nanjing, China.Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TASC.2011.2181144Fig. 1. Potential application in EVs.junctions only deal with the influence of external current, whileignoring the effect of external magnetic field which is inevitablein EV environment.The purpose of this paper is to analyze the nonlinear behaviorof Josephson junctions in the presence of varying external mag-netic field, and to propose a control scheme to stabilize chaosfor high-precision voltage measurement. Firstly, a modifiedJosephson junction model will be proposed and analyzed inSection II. Secondly, a backstepping control scheme will beproposed and analyzed in Section III. Finally, conclusion andfurther discussion will be given in Section IV.II. MODIFIED JOSEPHSON JUNCTION MODELFig. 1 shows the potential application of Josephson voltagestandards for the energy management in EVs in which there arevarious electric devices emitting electromagnetic radiation. Inorder to assess the effect of magnetic field on the dynamic per-formance of Josephson junctions, a modified resistive-capaci-tive-inductive shunted junction (RCLSJ) model is proposed asshown in Fig. 2. Compared with the traditional one, the pro-posed model takes into account the external magnetic field inthe surroundings.A. Mathematical ModelAs depicted in Fig. 2, the Josephson junction is shunted toa capacitor and a resistor as well as an inductor anda resistor that are connected in series, where denotes the1051-8223/$26.00 © 2011 IEEE4904704 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 22, NO. 3, JUNE 2012Fig. 2. Modified Josephson junction model.phase difference of the superconducting order parameter acrossthe junction. Usually, represents the junction resistance, andis the junction capacitance.From the modified model, a set of circuit equations can bederived as given by [8], [9]:(1)(2)(3)(4)where represents the effect of the external magnetic field,is the voltage across the Josephson junction, is the Planck’sconstant, is the electron charge, is the critical current, isthe current in the branch, and is the external currentwhich is composed of the DC component and the AC com-ponent [9], [10]. The nonlinear is represented bya piecewise linear resistance as defined by:(5)where denotes the gap junction voltage, is the normalstate resistance when , and is the subgap resis-tance when .By substituting (2) and (3) into (1) and replacing by ,it yields a standard 2nd-order differential equation:(6)where is the so-called damping factorof the junction, is the so-called nat-ural frequency of the junction, and.Selecting the state variables as , and ,and then taking the derivatives with respect to , the systemequation can be transformed as:(7)(8)(9)where andare the capacitive and inductance constants respectively,Fig. 3. Phase portrait of versus with .is the junction conductance,and .When is very small, namely , it yields:(10)By using (10), (8) can be further simplified as given by:(11)B. Nonlinear AnalysisWhen applying the Josephson voltage standards to EVs, theJosephson junctions take the definite advantage of high-preci-sion voltage measurement but their stability is inevitably af-fected by external magnetic fields emitted by electric motordrives and vehicular wireless communication. In order to assessthe nonlinear dynamics of the modified RCLSJ model undervarying external magnetic field, the magnetic field magnitudeis naturally chosen as the bifurcation parameter.Due to the aforementioned nonlinear behavior, the systemexhibits complex oscillations including periodic, multi-peri-odic and chaotic oscillations along with the increasing externalmagnetic field. The Runge-Kutta method is used to numer-ically simulate the modified model. Based on the availabledata of Josephson junctions [11], it can deduce ,, , when andwhen .It should be noted that the damping factor of the adoptedJosephson functions is [11]. When , the junc-tions are over-damped; otherwise, they are under-damped ones.So, the type of Josephson system for this EV application is basedon the programmable Josephson voltage standards.Since this paper aims to investigate the effect of external mag-netic field, the effect of is ignored so that and. Also, the time-varying pattern of magnetic field vari-ation is taken as . When the magnetic field mag-nitude is selected as , the phase portrait of versusis simulated as depicted in Fig. 3. It can be seen that the systemexhibits a periodic solution, namely the state variables oscillatewith a constant period. By increasing the value of , multi-pe-riodic oscillations can be observed as shown in Figs. 4 and 5.When is further increased to , the system exhibitsZHANG et al.: ANALYSIS OF CHAOS IN JOSEPHSON JUNCTIONS WITH EXTERNAL MAGNETIC FIELD 4904704Fig. 4. Phase portrait of versus with .Fig. 5. Phase portrait of versus with .chaos as characterized by the highly fluctuating time waveformin Fig. 6 and very messy phase portrait in Fig. 7.In order to prove the existence of chaos in the modifiedRCLSJ model mathematically, the maximum Lyapunov expo-nent is computed as given by:(12)where is the evolution time, is the map describing thetime evolution of , is the -th base vector of the -dimen-sion state space at the -th step, and is the Lyapunov exponentwhich indicates the separating rate of close trajectories of a dy-namical system. Consequently, the maximum value of canbe calculated, namely when . The pos-itive value of mathematically confirms that chaos occursin the RCLSJ model when . Similarly, chaos that occursat other values of can readily be proved.III. CHAOS CONTROLDiffering from those backstepping control design for linearsystems [12], the backstepping control scheme is modified andextended to nonlinear systems, with emphasis on suppressingchaotic oscillations elicited by external magnetic field inJosephson junctions.The design procedure for the proposed backstepping con-troller is described by the following 4 steps:Fig. 6. Time waveform of with .Fig. 7. Phase portrait of versus with .• Step I: By applying the control signal to the state equa-tion involving the bifurcation parameter which governs theoccurrence of chaotic oscillations, (11) can be rewritten as:(13)• Step II: By taking as the desired value as well asdefining , and ,a new virtual model is obtained as given by:(14)(15)(16)where and are constant coefficients.• Step III: By defining the Lyapunov function as:(17)4904704 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 22, NO. 3, JUNE 2012Fig. 8. Time response of with under chaos control.Fig. 9. Phase portrait of versus with under chaos control.where , it yields:(18)• Step IV: In order to stabilize the system, (18) should be anegative-definite function, namely . Thus, the corre-sponding parameters are chosen as , ,and , and the control signal is designed as:(19)To assess the feasibility of the proposed backstepping con-troller, the dynamic performance of the system is simulated.Namely, when , the controller is initially inactive andthen activated after time . Fig. 8, shows the time re-sponse of which verifies that the controller can successfullystabilize the chaotic oscillation to the stable periodic oscillation.Additionally, the amplitude of this periodic oscillation is greatlysuppressed. The phase portrait of versus is plotted in Fig. 9which also indicates that the system is free from the chaos.Therefore, by using this backstepping controller, theJosephson junctions can avoid the occurrence of chaos evenunder a large magnitude of external magnetic field. It indicatesthat the Josephson voltage standards can be employed forhigh-precision voltage measurement in EVs. Consequently,by providing high-precision voltage feedback of EV batteries,the performance of the EV energy management system can besignificantly improved.Finally, it should be noted that there are other techniques suchas a careful shielding to avoid chaos in the Josephson junctionselicited by an external magnetic field.IV. CONCLUSIONIn this paper, analysis and control of chaos for adoptingJosephson voltage standards in EVs have been presented. Thekey is to propose the modified RCLSJ model to characterize theJosephson junction in the presence of external magnetic field.Both mathematical derivation and numerical simulation con-firms that complex behaviors including chaotic oscillations canbe elicited along with the increase of external magnetic field.Consequently, a backstepping controller has been proposed anddesigned to suppress the chaos occurred at Josephson junctions,and hence improve the accuracy of voltage measurement.Dynamic performance of the system has also been simulated tovalidate the feasibility of the proposed control scheme.Because of the costly Josephson voltage standards and theassociated energy consumption for cryogenic cooling, a precisevoltage measurement based on Zener diodes may be a more eco-nomical way to provide the desired measurement. It is antici-pated that the application of Josephson voltage standards willbe justifiable only for those military or large EVs such as elec-tric tanks or electric vessels.REFERENCES[1] C. C. Chan and K. T. Chau, Modern Electric Vehicle Technology. :Oxford University Press, 2001.[2] K. T. Chau and C. C. Chan, “Emerging energy-efficient technologiesfor hybrid electric vehicles,” Proc. IEEE, vol. 95, no. 4, pp. 821–835,2007.[3] C. Liu, K. T. Chau, J. Zhong, and J. Li, “Design and analysis of aHTS brushless doubly-fed doubly-salient machine,” IEEE Trans. Appl.Supercond., vol. 21, no. 3, pp. 1119–1122, 2011.[4] F. Malétras, P. Gournay, I. A. Robinson, and G. Genevès, “A biassource for dynamic voltage measurements with a programmableJosephson junction array,” IEEE Trans. Instrum. Meas., vol. 56, no.2, pp. 495–499, 2007.[5] C. A. Hamilton, “Josephson voltage standards,” Rev. Sci. Instrum., vol.71, no. 10, pp. 3611–2623, 2000.[6] K. T. Chau and Z. Wang, Chaos in Electric Drive Systems—Analysis,Control and Application. : Wiley-IEEE Press, 2011.[7] S. K. Dana, D. C. Sengupta, and K. D. Edoh, “Chaotic dynamics inJosephson junction,” IEEE Trans. Circuit Syst. I, vol. 48, no. 8, pp.405–413, 2001.[8] K. K. Likharev, Dynamics of Josephson Junctions and Circuits. :Gordon and Breach, 1986.[9] C. B. Whan, C. J. Lobb, and M. G. Forester, “Effect of inductance inexternally shunted Josephson tunnel junction,” J. Appl. Phys., vol. 77,no. 1, pp. 382–389, 1995.[10] A. B. Cawthorne, C. B. Whan, and C. J. Lobb, “Complex dynamics ofresistively and inductively shunted Josephson junction,” J. Appl. Phys.,vol. 84, no. 2, pp. 1126–1132, 1998.[11] A. Uçar, K. E. Lonngren, and E. W. Bai, “Chaos synchronization inRCL-shunted Josephson junction via active control,” Chaos, Solitonsand Fractals, vol. 31, no. 1, pp. 105–111, 2007.[12] F. Ikhouane and M. 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Analysis of chaos in Josephson junctions with external magnetic field for high-precision voltage measurement in Electric Vehicles

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Analysis of chaos in Josephson junctions with external magnetic field for high-precision voltage measurement in Electric Vehicles

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