# Nonlinear Medium Description

Nonlinear phenomena in EM fields are caused by nonlinear materials, which are normally field strength-dependent. Therefore, when the time-periodic quasi-static EM field is applied to the nonlinear material, the electromagnetic properties of the material will be functions of the EM field. They will also be time-dependent. Simple nonlinear materials and their harmonic expressions are defined as follows:

## A. Magnetic Medium:

The magnetic reluctivity v corresponding to *B(t)* can be expressed as:

where flux density B(t) is time dependent. This can be expressed by the B-H curve including the hysteresis characteristic [11]. *v* (=1/^) is the nonlinear magnetic reluctivity and the Fourier coefficients obtained from Equations (38) to (40), respectively.

## B. Electric Medium:

The electrical conductivity ะพ related *E(t)* can be expressed as:

where *a* is the field strength-dependent conductivity, and the Fourier coefficients are obtained from:

# Boundary Conditions

Since the trigonometric functions are orthogonal functions, the harmonic potential *P _{k }*(degrees of freedom) on the boundary satisfies Dirichlet and Neumann boundary conditions. The frequency-domain representation, or spectrum on each boundary node, can then be expressed as follows:

## A. Dirichlet Boundary Condition:

## B. Neumann Boundary Condition:

where the potential *P _{k}* is the sum of harmonics on each boundary node i.