Considering the orthogonal characteristic of trigonometric functions, substituting magnetic vector A, interpolation functions Nj, magnetic reluctivity v and current density J_{s }into (3-7), the following FEM matrix equation can be obtained:

and the compact form can be written as:

? dA

where [S] is the system coefficient matrix, [A] represents —, and [M] can be obtained as:

dt

The equation can be solved using the Newton-Raphson method for nonlinear magnetic fields.

B. Frequency Domain Approach

For a frequency domain, the governing equation can be rewritten as:

In the two-dimensional case, and using Galerkin’s method to discretize the governing equation and (3-11) for the two-dimensional problems, the weighted residual can be obtained as:

The system matrix can be obtained as follows:
and the compact form can be written as:

and [5] is the system coefficient matrix, and [M] matrix can be obtained as:

C. Multi-Frequency Domain Approach Using the Harmonic Balance Method

For a multi-frequency domain or harmonic domain problem, the governing equation for nonlinear magnetic field can be rewritten as:

Where magnetic vector potential A_{k} and current density J_{0} consist of kth harmonics, о is the conductivity of the conductors, the expression will have the form of:

The nonlinear 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 [8, 9]. v (=1/^) is the nonlinear magnetic reluctivity and the Fourier coefficients obtained from Equations (3-20) to (3-22), respectively.

0

If equations (3-17), (3-18) and (3-19) are substituted into (3-16), the HBFEM single element matrix equation can be derived as follows [8,9]:

The coefficients of the matrix D are determined by only the Fourier coefficients in Equations (3-19) to (3-22). The matrix D acts as a reluctivity and is called the Reluctivity Matrix. On the other hand, the matrix N is a constant concerned with harmonic orders, and is called the Harmonic Matrix. The magnetic reluctivity coefficient matrix, D, can be derived as:

and the harmonic coefficient matrix, harmonics number N, can be derived as:

However, the harmonic matrix can include DC, as well as even and order harmonics, depending on the application problems.

If only the fundamental frequency and third harmonics are considered, {A} and {k} can be expressed as:

and:

Finally, the HBFEM system matrix can be obtained as:

and the compact form can be written as:
where [M] is the harmonic matrix:

The time harmonic matrix N can be expressed as:

where only the order harmonic components are considered.