3.2.1 Turning Point

It is assumed that the whole device is polarized into negative saturation with the applied electric field smaller than $-E_\mathrm{max}$. Then the electric field is increased up to a first operating point A with a respective field strength $E_A$. The dipoles will start to turn according to the distribution of the transition fields. According to (3.3) the overall polarization reads as

$$P = P_\mathrm{Sat} \cdot \int_{-\infty}^{E_A}\int_{0}^{E_\mathrm{max}} F(E_u,E_d)\cdot dE_d \cdot dE_u +$$

$$+ (-P_\mathrm{Sat}) \cdot (1 - \int_{-\infty}^{E_A}\int_{0}^{E_\mathrm{max}} F(E_u,E_d)\cdot dE_d \cdot dE_u)$$ (3.4)

The state of the dipoles is plotted in the Preisach-Everett (PE) diagram Fig. 3.5. The light/dark red areas represent the dipoles with positive/negative polarization.

Figure 3.5: Preisach-Everett diagram, increasing electric field

Figure 3.6: Hysteresis loop, increasing electric field

For actual simulations the hysteresis is not calculated by integration of the distribution function, but by an analytic function that matches the measurements for a particular material. In brief, not the distribution function is evaluated, but its integral. Fig. 3.6 shows the related first locus curve.

Next the direction is changed again and the electric field is decreased to $E_B < E_A$. It has to be considered that no dipoles with already negative orientation can be taken into account for transition, so the switching criterion has to be applied to the remaining dipoles only. The resulting Preisach-Everett diagram for the next branch of the hysteresis is shown in Fig. 3.7.

Figure 3.7: Preisach-Everett diagram, turning point

Figure 3.8: Resulting hysteresis loop for a single turning point

The polarization will decrease according to the distribution until finally all of the dipoles are oriented in the same direction again and the hysteresis loop closes. Fig. 3.8 shows this loop and, in order to allow a comparison, the saturation loop is plotted as well. These loops and all the following ones were calculated simulating a capacitor.