# Computation Taking Account of Hysteresis Effects Based on Fixed-Point Reluctance

## Fixed-Point Reluctance

The magnetic circuit model established from the geometric structure of the power transformer can be used to calculate and analyze non-linear electromagnetic problems, considering the coupled electric circuits. The fixed-point technique can be introduced in the non-linear magnetic circuit model to present the fixed-point reluctance, which can indicate the continuity of magnetic reluctance when hysteresis effects are involved.

The magnetic reluctance is defined traditionally as follows:

where *L* is the equivalent length of the magnetic circuit, *ะค* is the magnetic flux, S is the cross-sectional area of the iron core, and *H* and *B* are the magnetic field intensity and flux density, respectively. The non-linearity of the iron core can be represented by the single-valued function *H* = H(B).

However, the magnetic reluctance in Equation (6-62) is not continuous, due to the non-monotonic function when hysteresis effects are considered in the numerical computation. That may lead to the unstable convergence in computing the non-linear magnetic field.

The widely used fixed point technique in solving non-linear systems can be introduced in the non-linear magnetic circuit model:

Figure 6.14 Magnetic circuit model of a single-phase three limb transformer

where *F* is the magnetomotive force and *M* is a non-linear quantity independent of the magnetic flux.

The magnetic circuit model of a single-phase three-limb transformer is presented in Figure 6.14, and the general non-linear system equations can be obtained in Equations (6-64) to (6-66).

The fixed-point harmonic-balanced equation can be obtained by substituting Equation (6-63) into Equation (6-64), based on the harmonic balance method:

where l_{2n+1} is a unit matrix, whose order depends on the truncated harmonic number *n* in the computation.

Finally, the magnetic circuits, coupled with the electric circuits, are presented by the following equations:

where *C* is the coupling matrix and *Z* is the impedance matrix.