# American Institute of Mathematical Sciences

March  2006, 6(2): 257-272. doi: 10.3934/dcdsb.2006.6.257

## Homogenized Maxwell's equations; A model for ceramic varistors

 1 University Of California, Santa Barbara, Ca 93106 2 Swedish Defence Research Agency, FOI, Microwave Technology, P.O. Box 1165, SE-581 11, Linköping, Sweden

Received  February 2005 Revised  September 2005 Published  December 2005

Varistor ceramics are very heterogeneous nonlinear conductors, used in devices to protect electrical equipment against voltage surges in power lines. The fine structure in the material induces highly oscillating coefficients in the elliptic electrostatic equation as well as in the Maxwell equations. We suggest how the properties of ceramic varistors can be simulated by solving the homogenized problems, i.e. the corresponding homogenized elliptic problem and the homogenized Maxwell equations. The fine scales in the model yield local equations coupled with the global homogenized equations. Lower and upper bounds are also given for the overall electric conductivity of varistor ceramics. These two bounds are associated with two types of failures in varistor ceramics. The upper bound corresponds to thermal heating and the puncture failure due to localization of strong currents. The lower bound corresponds to fracturing of the varistor, due to charge build up at the grain boundaries resulting in stress caused by the piezoelectric property of the varistor.
Citation: Björn Birnir, Niklas Wellander. Homogenized Maxwell's equations; A model for ceramic varistors. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 257-272. doi: 10.3934/dcdsb.2006.6.257
 [1] Matthias Eller. Stability of the anisotropic Maxwell equations with a conductivity term. Evolution Equations & Control Theory, 2019, 8 (2) : 343-357. doi: 10.3934/eect.2019018 [2] S. S. Krigman. Exact boundary controllability of Maxwell's equations with weak conductivity in the heterogeneous medium inside a general domain. Conference Publications, 2007, 2007 (Special) : 590-601. doi: 10.3934/proc.2007.2007.590 [3] W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431 [4] Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems & Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49 [5] Jiann-Sheng Jiang, Chi-Kun Lin, Chi-Hua Liu. Homogenization of the Maxwell's system for conducting media. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 91-107. doi: 10.3934/dcdsb.2008.10.91 [6] Tommi Brander, Joonas Ilmavirta, Manas Kar. Superconductive and insulating inclusions for linear and non-linear conductivity equations. Inverse Problems & Imaging, 2018, 12 (1) : 91-123. doi: 10.3934/ipi.2018004 [7] Zhong Tan, Qiuju Xu, Huaqiao Wang. Global existence and convergence rates for the compressible magnetohydrodynamic equations without heat conductivity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5083-5105. doi: 10.3934/dcds.2015.35.5083 [8] Y. Efendiev, B. Popov. On homogenization of nonlinear hyperbolic equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 295-309. doi: 10.3934/cpaa.2005.4.295 [9] Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201 [10] Jishan Fan, Fucai Li, Gen Nakamura. A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1757-1766. doi: 10.3934/dcdsb.2018079 [11] Xiangdi Huang, Zhouping Xin. On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4477-4493. doi: 10.3934/dcds.2016.36.4477 [12] D. Sanchez. Boundary layer on a high-conductivity domain. Communications on Pure & Applied Analysis, 2002, 1 (4) : 547-564. doi: 10.3934/cpaa.2002.1.547 [13] Ville Kolehmainen, Matti Lassas, Petri Ola, Samuli Siltanen. Recovering boundary shape and conductivity in electrical impedance tomography. Inverse Problems & Imaging, 2013, 7 (1) : 217-242. doi: 10.3934/ipi.2013.7.217 [14] M. Zuhair Nashed, Alexandru Tamasan. Structural stability in a minimization problem and applications to conductivity imaging. Inverse Problems & Imaging, 2011, 5 (1) : 219-236. doi: 10.3934/ipi.2011.5.219 [15] Victor Isakov. On uniqueness in the inverse conductivity problem with local data. Inverse Problems & Imaging, 2007, 1 (1) : 95-105. doi: 10.3934/ipi.2007.1.95 [16] Luca Rondi. On the regularization of the inverse conductivity problem with discontinuous conductivities. Inverse Problems & Imaging, 2008, 2 (3) : 397-409. doi: 10.3934/ipi.2008.2.397 [17] María Teresa González Montesinos, Francisco Ortegón Gallego. The evolution thermistor problem with degenerate thermal conductivity. Communications on Pure & Applied Analysis, 2002, 1 (3) : 313-325. doi: 10.3934/cpaa.2002.1.313 [18] María Teresa González Montesinos, Francisco Ortegón Gallego. The thermistor problem with degenerate thermal conductivity and metallic conduction. Conference Publications, 2007, 2007 (Special) : 446-455. doi: 10.3934/proc.2007.2007.446 [19] Thierry Colin, Boniface Nkonga. Multiscale numerical method for nonlinear Maxwell equations. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 631-658. doi: 10.3934/dcdsb.2005.5.631 [20] M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473

2018 Impact Factor: 1.008