4.1.2 The tanh Shape Function

Although this type, contrarily to the $\textsf {arctan}$ shape function, allows the analytic calculation of the subcycle parameters $k$ and $P_\mathrm{off}$, it provides a less accurate fit, especially for low voltages [UGH+99]. The implementation of this function, as introduced by Jiang et al. [JZJ+97], is

$$P = k \cdot P_\mathrm{Sat}\cdot \textsf{tanh}\Bigl(w \cdot (E \pm E_c)\Bigr) + P_{ \mathrm{off}}.$$ (4.5)

The resulting equation system reads as

$$P_\mathrm{old} = k \cdot P_\mathrm{Sat}\cdot \textsf{tanh}\Bigl(w \cdot (E_\mathrm{old} \pm E_c)\Bigr) + P_{ \mathrm{off}}$$

$$P_\mathrm{turn} = k \cdot P_\mathrm{Sat}\cdot \textsf{tanh}\Bigl(w \cdot (E_\mathrm{turn} \pm E_c)\Bigr) + P_{ \mathrm{off}}.$$ (4.6)

For the calculation of the subcycle two different sets of equations are obtained, depending on whether an initial polarization cycle is calculated or not. In the case of an initial cycle the turning point has the values

$$P_\mathrm{turn}= P_\mathrm{Sat}$$

$$E_\mathrm{turn}= \infty.$$ (4.7)

Consequently, the parameters for the locus curves are

$$k = \frac{P_\mathrm{old} - P_\mathrm{Sat}}{ P_\mathrm{Sat}\cdot(\textsf{tanh}(w \cdot (E_\mathrm{old} - E_c) - 1))}$$ (4.8)

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$$P_\mathrm{off} = P_\mathrm{Sat}\cdot( k-1)$$ (4.9)

if the electric field increases, and

$$k = \frac{P_\mathrm{old} + P_\mathrm{Sat}}{ P_\mathrm{Sat}\cdot(\textsf{tanh}(w \cdot (E_\mathrm{old} + E_c)+1))}$$ (4.10)

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$$P_\mathrm{off} = P_\mathrm{Sat}\cdot( k-1)$$ (4.11)

if the electric field decreases. In general the parameters for the subcycles are calculated as

$$k = \frac{P_\mathrm{old} - P_\mathrm{turn}} { P_\mathrm{Sat}\cdot(\te... ... (E_\mathrm{old} \pm E_c) - \textsf{tanh}(w \cdot (E_\mathrm{turn} \pm E_c))))}$$ (4.12)

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$$P_\mathrm{off} = P_\mathrm{turn} - k \cdot P_\mathrm{Sat}\cdot \textsf{tanh}(w \cdot (E_\mathrm{turn} \pm E_c)).$$ (4.13)

The signs depend on whether the electric field strength increases or decreases.