**Author details**

Ekaterina Kurbatova<sup>1</sup> \*, Pavel Kurbatov<sup>1</sup> and Mikhail Sysoev<sup>2</sup>

1 National Research University "Moscow Power Engineering Institute", Moscow, Russia

2 Bauman Moscow State Technical University (National Research University), Moscow, Russia

\*Address all correspondence to: kurbatovaep@mail.ru

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Bulk High-Temperature Superconductors: Simulation of Electromagnetic Properties DOI: http://dx.doi.org/10.5772/intechopen.92452*

## **References**

Using the proposed models, we simulated the interaction of HTS bulk with external magnetic field after zero-field and field cooling. Distribution of the magnetic field sources inside a superconductor and magnetic field strength near it were obtained and analyzed. As shown in simulation results, the combination of two sources allows to widely regulate the properties of HTS and expand the possibilities

For verification of the proposed method, we compare the results of simulation with the experimental measurements of the levitation force. Comparison shows the good agreement between calculated and measured results, which confirms the possibility to expand and clarify the approximation models describing the electro-

\*, Pavel Kurbatov<sup>1</sup> and Mikhail Sysoev<sup>2</sup>

1 National Research University "Moscow Power Engineering Institute", Moscow,

2 Bauman Moscow State Technical University (National Research University),

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: kurbatovaep@mail.ru

provided the original work is properly cited.

of conventional methods, including the modeling of the Meissner effect.

magnetic properties of bulk high-temperature superconductor.

*On the Properties of Novel Superconductors*

**Author details**

Moscow, Russia

Russia

**50**

Ekaterina Kurbatova<sup>1</sup>

[1] Werfel FN, Floegel-Delor U, Rothfeld R, Riedel T, Goebel B, Wippich D, et al. Superconductor bearings, flywheels and transportation. Superconductor Science and Technology. 2012;**25**:014007. DOI: 10.1088/0953-2048/25/1/014007

[2] Krabbes G, Fuchs G, Canders WR, May H, Palka R. High Temperature Superconductor Bulk Materials: Fundamentals - Processing - Properties Control - Application Aspects. Wiley: Weinheim; 2006. p. 311. DOI: 10.1002/ 3527608044

[3] Coombs T. Bulk high temperature superconductor (HTS) materials. In: Melhem Z, editor. High Temperature Superconductors (HTS) for Energy Applications. Cambridge: Woodhead Publishing; 2012. pp. 101-139

[4] Kurbatova E. Comparative analysis of the specific characteristics of the magnetic bearings with HTS elements. IEEE Transactions on Applied Superconductivity. 2018;**28**:5207704. DOI: 10.1109/TASC.2018.2824347

[5] Naseh M, Heydari H. Analytical method for levitation force calculation of radial HTS magnetic bearings. IET Electric Power Applications. 2017;**11**: 369-377. DOI: 10.1049/ietepa.2016.0070

[6] Kim SB, Ikegami T, Fujii Y, Takahashi M, Onodera H. Development of the noncontact rotating system using combined ring-shaped HTS bulks and permanent magnets. IEEE Transactions on Applied Superconductivity. 2014;**24**: 6800105. DOI: 10.1109/TASC.2013. 2284021

[7] Arsenio AJ, Roque M, Cardeira C, Costa Branco PJ, Melicio R. Prototype of a zero-field-cooled YBCO bearing with continuous ring permanent magnets. IEEE Transactions on Applied

Superconductivity. 2018;**28**:5207407. DOI: 10.1109/tasc.2018.2817279

[8] Mukoyama S et al. Development of superconducting magnetic bearing for 300 kW flywheel energy storage system. IEEE Transactions on Applied Superconductivity. 2017;**27**:3600804. DOI: 10.1109/ TASC.2017.2652327

[9] Wang J, Wang S. High Temperature Superconducting Magnetic Levitation. Berlin, Boston: De Gruyter; 2017. p. 388. DOI: 10.1515/9783110538434

[10] Floegel-Delor U, Schirrmeister P, Riedel T, Koenig R, Kantarba V, Werfel FN. Bulk superconductor levitation devices: Advances in and prospects for development. IEEE Transactions on Applied Superconductivity. 2018;**28**:3601605. DOI: 10.1109/tasc.2018.2809467

[11] Sotelo G, Dias D, Andrade R, Stephan R. Tests on a superconductor linear magnetic bearing of a full-scale MagLev vehicle. IEEE Transactions on Applied Superconductivity. 2011;**21**: 1464-1468. DOI: 10.1109/ TASC.2010.2086034

[12] Kurbatov P, Kurbatova E, Dergachev P, Kulayev Y. Simulation of the body motion in a tube with the linear HTS suspension. In: 2018 IEEE 18th International Power Electronics and Motion Control Conference (PEMC-2018); 26–30 August 2018; Budapest, Hungary. New York: IEEE; 2018. pp. 611-616. DOI: 10.1109/ EPEPEMC.2018.8521935

[13] Kovalev K, Koneev S, Poltavec V, Gawalek W. Magnetically levitated high-speed carriages on the basis of bulk HTS elements. In: Proceedings of 8th International Symposium on Maglev Suspension Technology (ISMST'8), 26–28 September 2005; Dresden, Germany. 2005. p. 51

[14] Deng Z et al. A high-temperature superconducting Maglev ring test line developed in Chengdu, China. IEEE Transactions on Applied Superconductivity. 2016;**26**:3602408. DOI: 10.1109/TASC.2016.2555921

[15] Bean C. Magnetization of high-field superconductors. Reviews of Modern Physics. 1964;**36**:31-39. DOI: 10.1103/ RevModPhys.36.31

[16] Kim Y, Heampstead C, Strnad A. Critical persistent currents in hard superconductors. Physical Review Letters. 1962;**9**:306-309. DOI: 10.1103/ PhysRevLett.9.306

[17] Ruiz-Alonso D, Coombs T, Campbell A. Numerical analysis of hightemperature superconductors with the critical state model. IEEE Transactions on Applied Superconductivity. 2004;**14**: 2053-2063. DOI: 10.1109/TASC.2004. 838316

[18] Chun Y et al. Finite element analysis of magnetic field in high temperature bulk superconductor. IEEE Transactions on Applied Superconductivity. 2001;**11**: 2000-2003. DOI: 10.1109/77.920246

[19] Bean C. Response of high temperature superconductors to a step in magnetic field. In: Superconductivity and Applications (Proceedings of the Third Annual Conference on Superconductivity and Applications): 19–21 September 1989; New York. New York: Springer Science; 1989. pp. 767-772

[20] Sinder M, Meerovich V, Sokolovsky V, Vajda I. Penetration of magnetic field into high-temperature superconductors. IEEE Transactions on Applied Superconductivity. 1999;**9**: 4661-4665. DOI: 10.1109/77.819335

[21] Kim Y, Hempstead C, Strand A. Flux-flow resistance in type-II superconductors. Physical Review. 1965; **139**:A1163-A1172. DOI: 10.1103/ PhysRev.139.A1163

[22] Vinen W, Warren A. Flux flow resistivity in type I superconductors: II. Theoretical discussion. Proceedings of the Physical Society. 1967;**91**:409-421. DOI: 10.1088/0370-1328/91/2/319

[29] Anderson P. Theory of flux creep in hard superconductors. Physical Review Letters. 1962;**9**:309-311. DOI: 10.1103/

*DOI: http://dx.doi.org/10.5772/intechopen.92452*

*Bulk High-Temperature Superconductors: Simulation of Electromagnetic Properties*

[30] Klimenko E, Imenitov A, Shavkin S, Volkov P. Resistance–current curves of high pinning superconductors. Journal of Experimental and Theoretical Physics. 2005;**100**:50-65. DOI: 10.1134/

[31] Kulaev Y et al. Modeling of

electrophysical properties of bulk hightemperature superconductors in calculations of magnetic systems. Russian Electrical Engineering. 2015;**86**: 213-219. DOI: 10.3103/S10683712150

[32] Stratton J. Electromagnetic Theory. New York: McGraw-Hill; 1941. p. 631

[33] Kulaev Y, Kurbatov P, Kurbatova E. Construction of combined models of the properties of bulk high-temperature superconducting materials. Russian Electrical Engineering. 2017;**88**:465-470. DOI: 10.3103/s1068371217070094

PhysRevLett.9.309

1.1866198

40070

**53**

[23] Prigozhin L. Analysis of criticalstate problems in type-II superconductivity. IEEE Transactions on Applied Superconductivity. 1997;**7**: 3866-3873. DOI: 10.1109/77.659440

[24] Rhyner J. Magnetic properties and AC-losses of superconductors with power-law current-voltage characteristics. Physica C. 1993;**212**: 292-300. DOI: 10.1016/0921-4534(93) 90592-E

[25] Zou S, Zermeño V, Grilli F. Influence of parameters on the simulation of HTS bulks magnetized by pulsed field magnetization. IEEE Transactions on Applied Superconductivity. 2016;**26**:4702405. DOI: 10.1109/TASC.2016.2535379

[26] Douine B, Bonnard C, Sirois F, Berger K, Kameni A, Lévêque J. Determination of *JC* and *n*-value of HTS pellets by measurement and simulation of magnetic field penetration. IEEE Transactions on Applied Superconductivity. 2015;**25**:8001008. DOI: 10.1109/TASC.2015.2409201

[27] Yokono T, Hasegawa K, Kamitani A. Magnetic shielding analysis of high-Tc superconducting plates by power law, flux-flow, and flux-creep models. IEEE Transactions on Applied Superconductivity. 2003;**13**:1672-1675. DOI: 10.1109/TASC.2003.812860

[28] Ikuno S, Kamitani A. Shielding current density analysis of axisymmetric HTS by element-free galerkin method. IEEE Transactions on Applied Superconductivity. 2005;**15**:3688-3691. DOI: 10.1109/TASC.2005.849393

*Bulk High-Temperature Superconductors: Simulation of Electromagnetic Properties DOI: http://dx.doi.org/10.5772/intechopen.92452*

[29] Anderson P. Theory of flux creep in hard superconductors. Physical Review Letters. 1962;**9**:309-311. DOI: 10.1103/ PhysRevLett.9.309

[14] Deng Z et al. A high-temperature superconducting Maglev ring test line developed in Chengdu, China. IEEE

*On the Properties of Novel Superconductors*

**139**:A1163-A1172. DOI: 10.1103/

[22] Vinen W, Warren A. Flux flow resistivity in type I superconductors: II. Theoretical discussion. Proceedings of the Physical Society. 1967;**91**:409-421. DOI: 10.1088/0370-1328/91/2/319

[23] Prigozhin L. Analysis of critical-

superconductivity. IEEE Transactions on Applied Superconductivity. 1997;**7**: 3866-3873. DOI: 10.1109/77.659440

[24] Rhyner J. Magnetic properties and AC-losses of superconductors with

characteristics. Physica C. 1993;**212**: 292-300. DOI: 10.1016/0921-4534(93)

[25] Zou S, Zermeño V, Grilli F. Influence of parameters on the

Transactions on Applied

Transactions on Applied

Transactions on Applied

simulation of HTS bulks magnetized by pulsed field magnetization. IEEE

Superconductivity. 2016;**26**:4702405. DOI: 10.1109/TASC.2016.2535379

[26] Douine B, Bonnard C, Sirois F, Berger K, Kameni A, Lévêque J.

Determination of *JC* and *n*-value of HTS pellets by measurement and simulation of magnetic field penetration. IEEE

Superconductivity. 2015;**25**:8001008. DOI: 10.1109/TASC.2015.2409201

[27] Yokono T, Hasegawa K, Kamitani A. Magnetic shielding analysis of high-Tc superconducting plates by power law, flux-flow, and flux-creep models. IEEE

Superconductivity. 2003;**13**:1672-1675. DOI: 10.1109/TASC.2003.812860

Superconductivity. 2005;**15**:3688-3691. DOI: 10.1109/TASC.2005.849393

[28] Ikuno S, Kamitani A. Shielding current density analysis of axisymmetric HTS by element-free galerkin method.

IEEE Transactions on Applied

state problems in type-II

power-law current-voltage

90592-E

PhysRev.139.A1163

Superconductivity. 2016;**26**:3602408. DOI: 10.1109/TASC.2016.2555921

[15] Bean C. Magnetization of high-field superconductors. Reviews of Modern Physics. 1964;**36**:31-39. DOI: 10.1103/

[16] Kim Y, Heampstead C, Strnad A. Critical persistent currents in hard superconductors. Physical Review Letters. 1962;**9**:306-309. DOI: 10.1103/

[17] Ruiz-Alonso D, Coombs T,

[19] Bean C. Response of high

Third Annual Conference on

York: Springer Science; 1989.

[20] Sinder M, Meerovich V,

pp. 767-772

**52**

Campbell A. Numerical analysis of hightemperature superconductors with the critical state model. IEEE Transactions on Applied Superconductivity. 2004;**14**: 2053-2063. DOI: 10.1109/TASC.2004.

[18] Chun Y et al. Finite element analysis of magnetic field in high temperature bulk superconductor. IEEE Transactions on Applied Superconductivity. 2001;**11**: 2000-2003. DOI: 10.1109/77.920246

temperature superconductors to a step in magnetic field. In: Superconductivity and Applications (Proceedings of the

Superconductivity and Applications): 19–21 September 1989; New York. New

Sokolovsky V, Vajda I. Penetration of magnetic field into high-temperature superconductors. IEEE Transactions on Applied Superconductivity. 1999;**9**: 4661-4665. DOI: 10.1109/77.819335

[21] Kim Y, Hempstead C, Strand A. Flux-flow resistance in type-II

superconductors. Physical Review. 1965;

Transactions on Applied

RevModPhys.36.31

PhysRevLett.9.306

838316

[30] Klimenko E, Imenitov A, Shavkin S, Volkov P. Resistance–current curves of high pinning superconductors. Journal of Experimental and Theoretical Physics. 2005;**100**:50-65. DOI: 10.1134/ 1.1866198

[31] Kulaev Y et al. Modeling of electrophysical properties of bulk hightemperature superconductors in calculations of magnetic systems. Russian Electrical Engineering. 2015;**86**: 213-219. DOI: 10.3103/S10683712150 40070

[32] Stratton J. Electromagnetic Theory. New York: McGraw-Hill; 1941. p. 631

[33] Kulaev Y, Kurbatov P, Kurbatova E. Construction of combined models of the properties of bulk high-temperature superconducting materials. Russian Electrical Engineering. 2017;**88**:465-470. DOI: 10.3103/s1068371217070094

**Chapter 4**

**Abstract**

like effect for high-Tc SCs.

**1. Introduction**

**55**

**1.1 A historical note**

the Tc of the Bi-based superconductors

Generalized BCS Equations:

of Ba2Sr2CaCu2O8

A Review and a Detailed Study

*Gulshan Prakash Malik and Vijaya Shankar Varma*

of the Superconducting Features

High-Tc superconductors (SCs) are most widely studied via the multiband approach (MBA) based on the work of Suhl et al. and the Nambu-Eliashberg-McMillan extension of the BCS theory. Complementing MBA and presented in a recent monograph is an approach based on the generalized BCS equations

(GBCSEs), which too has been applied to a significant number of SCs. GBCSEs are obtained via a Bethe-Salpeter equation and the Matsubara technique. One of the key features of this approach is the characterization of a composite SC by Cooper pairs with different binding energies—each of which is identified with a gap (Δ) of the SC—depending on whether pairing is due to phonon exchanges with one, two, or more ion species. Another feature is the incorporation of chemical potential in the GBCSEs, which enables one to calculate the critical current density j0 of an SC using the same parameters that determine its Tc and Δ. Following a review of the concepts of this approach, given herein, for the first time, is a detailed explanation of the multitude of reported empirical values of {Tc, Δ, j0} of Bi2Sr2CaCu2O8. Also discussed are the currently topical issues of s-superconductivity and the isotope-

**Keywords:** high-Tc superconductors, multiband approach, GBCSE-based approach, superconducting features of Bi2Sr2CaCu2O8, s-superconductivity, isotope-like effect for high-Tc superconductors, suggestions for further increasing

The phenomenon of superconductivity was first observed by Kamerlingh Onnes [1] in 1911 while studying the electrical resistivity of metallic mercury as a function of temperature when cooled to liquid helium temperatures. He noticed that at the critical temperature Tc = 4.2 K (269°C), the resistance vanished abruptly. This was an exciting discovery because it suggested that such materials could have immense practical applications. Since cooling a material to very low temperatures is a tedious and expensive process, Onne's discovery triggered a

## **Chapter 4**
