Modeling and homogenization of processes in cylindrical layered composites like capacitors and transformers
Gennadiy V. Sandrakov, Mykola Kyrychek · Bulletin of Taras Shevchenko National University of Kyiv Series Physics and Mathematics · 2025
Capacitors and transformers are widely used devices. There is a trend towards minutearization of such devices. Thus, a natural problem of modeling thermal and electromagnetic processes in such devices arises. Understanding the models of such processes allows, for example, to optimize or make cheaper of these devices. But, the geometry of such devices is very specific. For example, a regular capacitor is a long strip with an insulator rolled into a cylinder. Direct modeling of processes in such a device is complicated, since there are very many layers, which leads to equations with highly oscillating coefficients. On the other hand, methods of homogenization theory can be applied to such equations and equations with constant coefficients can be obtained, the solutions of which approximate the solutions of the equations under consideration. The solution of equations with constant coefficients is much simpler both by numerical and analytical methods. As a first step in this direction, one dynamic equation will be considered here, which corresponds to the modeling of wave and thermal processes. Thus, the problem of modeling wave and thermal processes in a cylinder consisting of a large number of alternating layers with different characteristics is considered. Using the asymptotic methods of the homogenization theory, this problem is approximately reduced to modeling processes in a homogeneous (homogenized) cylinder. Asymptotic expansions and homogenized problems are presented, the solutions of which determine the approximate asymptotic of solutions to the problem under consideration. Formulas for calculating the characteristics of the homogeneous cylinder and accuracy estimates of the approximations are given. The presented accuracy estimates of such approximations are essential for numerical calculations and computer modeling of the problems with guaranteed accuracy. In addition, the given approximations are undoubtedly useful for understanding wave and thermal processes in devices similar to capacitors and transformers. This approach can be considered as a basis for considering more general electromagnetic and thermal processes, which are modeled by Maxwell's equations coupled with the heat equation.