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# Derivation of SIBCs for Nonlinear and Nonhomogeneous Problems

DOI link for Derivation of SIBCs for Nonlinear and Nonhomogeneous Problems

Derivation of SIBCs for Nonlinear and Nonhomogeneous Problems book

# Derivation of SIBCs for Nonlinear and Nonhomogeneous Problems

DOI link for Derivation of SIBCs for Nonlinear and Nonhomogeneous Problems

Derivation of SIBCs for Nonlinear and Nonhomogeneous Problems book

## ABSTRACT

In previous chapters, we assumed the properties of the conducting medium were constant. In practice, however, there are problems in which these assumptions are not valid and the governing Maxwell’s equations for the electromagnetic ﬁeld distribution in conductors must be supplemented by additional equation(s) describing material properties. For example, the magnetic ﬂux and magnetic induction in ferromagnetic materials are related through a nonlinear function. When currents ﬂowing through the conductors are high enough, Joule heating starts to play a role and the dependence between conductivity and temperature cannot be neglected. In the latter case the heat transfer equation must be considered together with Maxwell’s equations. Objects consisting of several conductive layers or conductors coated with dielectrics belong to yet another class of problems where assumption of constant material properties is invalid. Intuitively it is clear that the SIBCs considered in Chapters 2 through 4

cannot be used directly in modeling nonlinear problems. However, the general application area of the surface impedance concept is not restricted to homogeneous conductors with constant material properties. The perturbation approach developed in Chapter 2 for derivation of SIBCs consists of two major stages:

1. Approximation of the original three-dimensional equations of the electromagnetic ﬁeld distribution inside the conductor by onedimensional equations in the direction normal to the conductor’s surface

2. Derivation of the boundary conditions from analytical solution(s) of the one-dimensional equation(s) of diffusion of the magnetic ﬁeld inside the conductor

Stage 1 can be performed if the electromagnetic ﬁeld in the skin layer varies in the normal direction much faster than in the tangential directions so that tangential derivatives are much smaller than normal derivatives. But this is exactly the main condition of applicability of the surface impedance concept

to the problem under consideration. This means that the conditions of applicability of the surface impedance concept do not restrict the transformations performed at stage 2. In other words, even if the equations considered at stage 2 do not provide SIBCs in the classical single frequency form or do not allow exact analytical solution at all, it does not mean that the surface impedance concept cannot be applied. The objective of this chapter is to demonstrate how the perturbation

approach developed in the previous chapters can be applied to derive boundary conditions for nonlinear and nonhomogeneous problems. It is expected that these SIBCs will be more complex than the classical conditions for homogeneous conductors. Thus we will restrict ourselves to low-order approximation to reduce the number of transformations. High-order terms can be derived following the same approach.