6.2.3 Detection of the Locus Curves

Calculating the locus curves for a new operating point is a sophisticated task. As outlined in Fig. 6.4, one of two possible locus curves has to be chosen at each operating point, depending on the history of the electric field [JZJ+97].

Figure 6.4: Possible locus curves in an operating point

As a consequence of the two-dimensional algorithm, the common starting point $C$ of these two branches will move during the nonlinear iteration. In fact it highly depends on the actual orientation of the electric field at a particular iteration step. Therefore, it cannot be guaranteed that always the same branch is used in the numerical equation system during the iteration process. Regarding the different derivatives of the two functions this will lead to poor convergence and in the worst case to oscillations of the nonlinear iteration.

As this shows, a preselection of the appropriate branch is necessary in order to achieve convergence, especially for the simulation of complex structures. A suitable approach for detecting the direction of the change of the electric field is solving a linearized equation system.

The equation

$$\vec{D}= \epsilon \cdot \vec{E} + f(\vec{E})$$ (6.10)

is transformed by keeping the nonlinear part constant.

$$\vec{D} = \epsilon \cdot \vec{E} + f(\vec{E}_\mathrm{old}) = \epsilon \cdot \vec{E} + \vec{P}_\mathrm{old}$$ (6.11)

With this method an approximation of the electric field at the new operating point is obtained.

The parallel component of the old field vector to this approximation is calculated and the result is interpreted with respect to the orientation of the new field vector as outlined in Fig. 6.5 and Fig. 6.6.

Figure 6.5: Detection of the change of the electric field, electric field decreases.

Figure 6.6: Detection of the change of the electric field, electric field increases.

With this information it is now possible to select the correct branch of the hysteresis curve. The complete scheme is outlined in Fig. 6.7.

Figure 6.7: Modified trivial iteration scheme