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 о 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.