# The Theory of HB-FEM in Nonlinear Magnetic Fields

## A. Current Source-Driven Fields

If the excitation waveform is a sinusoidal signal, the current source can be considered as a sine wave (Figure 3.4(a) - primary coil is an excitation coil). The resulting magnetic field contains all harmonic components (order components only) when the magnetic field becomes saturated, as shown in Figure 3.4(b).

The magnetizing characteristic of the core can be expressed as a function of the flux density B - that is,

Figure 3.4 Magnetic core for a 2D transformer structure, and its typical B-H cure. (a) Transformer with nonlinear magnetic core. (b) B-H curve and permeability

where the hysteresis characteristic is neglected. The magnetic reluctivity *v* can be written as:

When hysteresis characteristic is concerned as shown in Figure 3.5(a), the expression H(B) can be obtained from a B-H curve data table, or a function of the flux density *B *with Hysteresis term [3], as follows:

However, in the DC-biased case [11,14], the B-H curve with hysteresis characteristics will be illustrated as Figure 3.6(b), and the expression of H(B) can be obtained from a B-H curve data table.

Figure 3.5 B-H curve with hysteresis characteristics (a), and without hysteresis characteristics (b)

Figure 3.6 B-H curve with hysteresis characteristics and DC-biased condition. (a) H-B curve with hysteresis; (b) H-B curve with hysteresis and DC-biased case