 
Summary: Homogenized Maxwell's Equations;
a Model for Varistor Ceramics
Bj˜ orn Birnir
Niklas Wellander
Department of Mathematics
University of California
Santa Barbara, CA 93106
USA
Abstract
A model for a semiconducting ceramic material used in devices to pro
tect electrical equipment against overvoltages is presented. The fine struc
ture in the material induces highly oscillating coefficients in the Maxwell
equation. Maxwell equations are homogenized to obtain a coupled system.
The fine scales in the model yield a local problem coupled with the homoge
nized Maxwell's equations. In the electrostatic case upper and lower bounds
are obtained for the effective conductivity in the varistor. These two bounds
are associated with two types of failures in varistor ceramics. The upper
bound corresponds two thermal heating and the puncture failure due to lo
calization of strong currents. The lower bound corresponds to fracturing of
the varistor, due to charge build up at the grain boundaries resulting in stress
