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 . 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:
Since the trigonometric functions are orthogonal functions, the harmonic potential Pk (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 Pk is the sum of harmonics on each boundary node i.