# HBFEM with Fixed-Point Technique

## Introduction

DC flux and AC flux co-exist in a ferromagnetic core when a power transformer works under DC-biased magnetization, which leads to the distortion and asymmetry of hysteresis loops. The loss curve of the iron core under DC-biased excitation is also different from that under sinusoidal excitation [38]. Figure 5.32 depicts two different hysteresis loops under DC-biased and sinusoidal excitations. *B*_{acm} and *B*_{dc} are, respectively, the

Figure 5.31 Harmonic flux distributions *(I _{dc}* = 1.27A,

*U*= 370V). (a) DC flux; (b) Fundamental

_{ac}*(at*= n/2); (c) Second order

*(2at*= n/2); (d) Third order

*(3at*= n/2); (e) Fourth order

*(4at*= n/2); (f) Fifth order (5at = n/2)

Figure 5.32 Hysteresis loops under DC-biased and sinusoidal excitations

magnitudes of the AC component and the DC component of magnetic flux density. The hysteresis loop is distorted and the ferromagnetic core is saturated due to the DC flux density *B _{dc}.* As a result, the magnetization curve under DC-biased excitation is different from the basic one.

The fixed-point technique has been used in finite element analysis of the nonlinear magnetic field under sinusoidal excitation. The optimal convergence strategy is discussed in nonlinear eddy current problems [39, 40]. Two different methods to determine the fixed-point reluctivity are presented in time-stepping FEM, and the convergent performance is discussed in detail [41, 42]. In this section, the fixed-point technique is combined with the HBFEM to calculate the DC-biasing magnetic field by means of the measured magnetization curves of SLC under DC-biased magnetization.