# A. Time Domain Approach

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*consist of kth harmonics,

_{0}*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.