**MSC2010** 82D55

### Mathematical modeling of voltage harmonicsfor current-voltage characteristics with singularities

#### N. D. Kuzmichev^{1}, M. A. Vasyutin^{2}, E. A. Lapshina^{3}, D. A. Shilkin^{4}

Annotation | In the paper mathematical modeling of the voltage harmonics occurring in superconductor is performed. It is assumed that the superconductoris in the resistive state when an alternating harmonic current passes through them. In such the I-V characteristics of a superconductor are nonlinear and often have singularities caused by instability. For this reason, the I-V characteristics of superconductors obtained for alternating and direct currents are very different. Analytical expressions for the voltage amplitude of the first harmonic for wide films of type-II superconductors with singularitiesof jump-type and break-type in the current dependence of the I-V characteristics are obtained. The dependences of the amplitudes of the 1st, 2nd, 3rd and the 5th voltage harmonics on the value of the constant current and on the amplitude of the alternating current are calculated numerically. It is shown that the positions of the current dependencies’ singularities determine important characteristics of the superconductor. The experimental I-V characteristic of the first harmonic of YBCO single crystal is considered. Basing on the mathematical modeling performed, it is shown that at currents above the critical one, the single crystal overheats because of poor heat transfer. The resistance of a single crystal depends on the temperature and increases in proportion to the current; the I-V characteristic has a parabolic relationship. |
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Keywords | current-voltage characteristic of superconductors, singularities of the current-voltage characteristic, amplitude voltage of harmonic, current-voltage characteristic of the voltage harmonics, critical current. |

^{1}**Nikolay D. Kuzmichev**, Professor of Department of Computer Science and CAD-technology Ruzaevsky engineering Institute (branch) National Research Mordovia State University (68 Bolshevistskaya Str., Saransk 430005, Republic of Mordovia, Russia), Dr. Sci. (Physics and Mathematics), ORCID: http://orcid.org/0000-0001-6707-4950, kuzmichevnd@yandex.ru.^{2}**Mikhael A. Vasyutin**, Docent of Department of Computer Science and CAD-technology Ruzaevsky engineering Institute (branch) National Research Mordovia State University (68 Bolshevistskaya Str., Saransk 430005, Republic of Mordovia, Russia), ORCID: http://orcid.org/0000-0002-4856-7407, vasyutinm@mail.ru.^{3}**Elena A. Lapshina**, Docent of Department of Computer Science and CAD-technology Ruzaevsky engineering Institute (branch) National Research Mordovia State University (68 Bolshevistskaya Str., Saransk 430005, Republic of Mordovia, Russia), ORCID: http://orcid.org/0000-0002-8828-273X, e.lapshina2010@yandex.ru.^{4}**Dmitri A. Shilkin**, Engineer of Department of Computer Science and CAD-technology Ruzaevsky engineering Institute (branch) National Research Mordovia State University (68 Bolshevistskaya Str., Saransk 430005, Republic of Mordovia, Russia), ORCID: http://orcid.org/0000-0001-9061-3174, dwi8hi@outlook.com.

**Citation**: N. D. Kuzmichev, M. A. Vasyutin, E. A. Lapshina, D. A. Shilkin, "[Mathematical modeling of voltage harmonicsfor current-voltage characteristics with singularities]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,19:4 (2017) 68–78 (In Russian)

**DOI** 10.15507/2079-6900.19.201704.68-78