Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T15:13:27.307Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Magneto-Heat Coupling Model for Large Power Transformers

Part of: Parabolic equations and systems Optics, electromagnetic theory - General Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  06 July 2017

Xujing Li ,
Shipeng Mao ,
Kangkang Yang  and
Weiying Zheng
Xujing Li*
Affiliation:
LSEC, NCMIS, Academy of Mathematics and Systems Science, University of Chinese Academy of Sciences, CAS, Beijing, 100190, China
Shipeng Mao*
Affiliation:
LSEC, NCMIS, Academy of Mathematics and Systems Science, University of Chinese Academy of Sciences, CAS, Beijing, 100190, China
Kangkang Yang*
Affiliation:
LSEC, NCMIS, Academy of Mathematics and Systems Science, University of Chinese Academy of Sciences, CAS, Beijing, 100190, China
Weiying Zheng*
Affiliation:
LSEC, NCMIS, Academy of Mathematics and Systems Science, University of Chinese Academy of Sciences, CAS, Beijing, 100190, China
*
*Corresponding author. Email addresses:[email protected] (X. Li), [email protected] (S. Mao), [email protected] (K. Yang), [email protected] (W. Zheng)
*Corresponding author. Email addresses:[email protected] (X. Li), [email protected] (S. Mao), [email protected] (K. Yang), [email protected] (W. Zheng)
*Corresponding author. Email addresses:[email protected] (X. Li), [email protected] (S. Mao), [email protected] (K. Yang), [email protected] (W. Zheng)
*Corresponding author. Email addresses:[email protected] (X. Li), [email protected] (S. Mao), [email protected] (K. Yang), [email protected] (W. Zheng)
Get access

Abstract

This paper studies the magneto-heat coupling model which describes iron loss of conductors and energy exchange between magnetic field and Ohmic heat. The temperature influences Maxwell's equations through the variation of electric conductivity, while electric eddy current density provides the heat equation with Ohmic heat source. It is in this way that Maxwell's equations and the heat equation are coupled together. The system also incorporates the heat exchange between conductors and cooling oil which is poured into and out of the transformer. We propose a weak formulation for the coupling model and establish the well-posedness of the problem. The model is more realistic than the traditional eddy current model in numerical simulations for large power transformers. The theoretical analysis of this paper paves a way for us to design efficient numerical computation of the transformer in the future.

Keywords

Magneto-heat coupling modeleddy current problemMaxwell's equationsradiative boundary condition

MSC classification

Secondary: 35K55: Nonlinear parabolic equations 35Q60: PDEs in connection with optics and electromagnetic theory 78A25: Electromagnetic theory, general
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 3 , September 2017 , pp. 683 - 711
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ammari, H., Buffa, A., and Nédélec, J., A justification of eddy current model for the Maxwell equations, SIAM J. Appl. Math., 60 (2000), pp. 1805–1823.Google Scholar
[2] Amrouche, C., Bernardi, C., Dauge, M., and Girault, V., Vector potentials in three-dimensional non-smooth domains, Math. Meth. Appl. Sci., 21 (1998), pp. 823–864.3.0.CO;2-B>CrossRefGoogle Scholar
[3] Bachinger, F., Langer, U., Schöberl, J., Numerical Analysis of Nonlinear Multiharmonic Eddy Current Problems, Numer. Math., 100 (2005), 593–616.CrossRefGoogle Scholar
[4] Beck, R., Hiptmair, R., Hoppe, R. and Wohlmuth, B., Residual based a posteriori error estimators for eddy current computation, Math. Model. Numer. Anal., 34 (2000), 159–182.CrossRefGoogle Scholar
[5] Chen, J., Chen, Z., Cui, T. and Zhang, L., An adaptive finite element method for the eddy current model with circuit/field couplings, SIAM J. Sci. Comput., 32 (2010), 1020–1042.CrossRefGoogle Scholar
[6] Cheng, Z., Takahashi, N., and Forghani, B., TEAM Problem 21 Family (V.2009), http://www.compumag.org/jsite/team, approved by the International Compumag Society at Compumag 2009.Google Scholar
[7] Cheng, Z., Takahashi, N., Forghani, B., and et al., Electromagnetic and Thermal Field Modeling and Application in Electrical Engineering, Science Press, Beijing, 2009.Google Scholar
[8] Cheng, Z., Takahashi, N., Forghani, B., Gilbert, G., Zhang, J., Liu, L., Fan, Y., Zhang, X., Du, Y., Wang, J., and Jiao, C., Analysis and measurements of iron loss and flux inside silicon steel laminations, IEEE Trans. Magn., 45 (2009), 1222–1225.CrossRefGoogle Scholar
[9] Costabel, M., Dauge, M., and Nicaise, S., Singularities of eddy current problems, ESAIM: Mathematical Modelling and Numerical Analysis, 37 (2003), 807–831.CrossRefGoogle Scholar
[10] Druet, P.-É., Existence of weak solutions to the time-dependent MHD equations coupled to the heat equation with nonlocal radiation boundary conditions, Nonlinear Analysis: Real World Applications, 10 (2009), 2914–2936.Google Scholar
[11] Hiptmair, R., Analysis of multilevel methods for eddy current problems, Math. Comp., 72 (2002), 1281–1303.CrossRefGoogle Scholar
[12] Jiang, X. and Zheng, W., An efficient eddy current model for nonlinear Maxwell equations with laminated conductors, SIAM J. Appl. Math., 72 (2012), pp. 1021–1040.CrossRefGoogle Scholar
[13] Jiang, X. and Zheng, W., Homogenization of quasi-static Maxwell's equations, Multiscal Modeling and Simulation: A SIAM Interdisciplinary Journal, 12(2014). pp. 152–180.CrossRefGoogle Scholar
[14] Kurose, H., Miyagi, D., Takahashi, N., Uchida, N., Kawanaka, K., 3-D eddy current analysis of induction heating apparatus considering heat emission, heat conduction, and temperatue dependence of magnetic characteristics, IEEE Trans. Magn., 45 (2009), 3, 1847–1850.CrossRefGoogle Scholar
[15] Ledger, P. D., and Zaglmayr, S., hp-Finite element simulation of three-dimensional eddy current problems on multiply connected domains, Computer Methods in Applied Mechanics and Engineering, 199 (2010), 3386–3401.CrossRefGoogle Scholar
[16] Li, P., and Zheng, W., An H-ψ formulation for the three-dimensional eddy current problem in laminated structures, J. Diff. Eqn., 254 (2013), pp. 3476–3500.CrossRefGoogle Scholar
[17] Muramatsu, K., Takahashi, N., Mimula, T., Magneto-thermal-fluid analysis taking account of natural convection using lagrange coordinate system, IEEE Trans. Magn., 35 (1999), 3, 1670–1673.CrossRefGoogle Scholar
[18] Nédélec, J.C. and Wolf, S., Homogenization of the problem of eddy currents in a transformer core, SIAM J. Numer. Anal., 26 (1989), 1407–1424.CrossRefGoogle Scholar
[19] Roubíc˘ek, T., Nonlinear partial differential equations with applications, Birkhäuser Verlag, Basel, 2005.Google Scholar
[20] Yin, H.-M., Existence and regularity of a weak solution toMaxwells equations with a thermal effect, Math. Meth. Appl. Sci., 29 (2006), 1199–1213.CrossRefGoogle Scholar
[21] Zeidler, E., Nonlinear Functional Analysis and its Applications, II/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990.Google Scholar
[22] Zheng, W., Chen, Z., and Wang, L., An adaptive finite element method for the h–ψ formulation of time-dependent eddy current problems, Numer. Math., 103 (2006), 667–689.CrossRefGoogle Scholar