**3.2.4 Practical RC model used for experimentations**

In practice, the open circuit line model will be valid only if the entire electrolyte is between the cell plates. In practice this is usually not true. For our batteries, about one half of the electrolyte volume was beside the plates. In such case there is an additional transversal transport of ions, with still longer time constants. This could be accommodated by an additional RC cell connected at the output of the line.

Fig. 11. Transmission line with additional RC cell

Satisfactory preliminary results for model validation were obtained with a much simplified network, with an experimental fitting of the component values (Fig 11).

Structural Design of a Dynamic Model of the Battery for State of Charge Estimation 139

The general form of the charge density depends essentially on the doping profile of the junction. In the ideal case (constant doping "Na and Nd") , we can easily deduce the electric field form E(x) and the potential V(x) by application of equations of electrostatics (Sari-Ari et al.,2005). In addition, the overall electrical neutrality of the junction imposes the relation:

with Wn and Wp corresponding to the limit of DLA on sides N and P respectively (Fig. 13). It may be demonstrated that according to the Boltzmann relationship, the corresponding

> 

where ni represents the intrinsic carrier concentration. On another hand, note that the width

The PN junction out of equilibrium when a potential difference V is applied across the junction. According to the orientation in figure 14, the polarization will therefore directly reduce the height of the potential barrier which becomes (V0-V) resulting in a decrease in the

*n e i*

<sup>0</sup> <sup>2</sup> ln , *a d T T N N kT VU U*

*NW NW an d <sup>p</sup>* (27)

(28)

Fig. 13. Representation of a PN junction at thermodynamic equilibrium

potential barrier (diffusion potential of the junction) is given by:

of the DLA may be related to the potential barrier (Mathieu H, 1987).

Fig. 14. Representation of a PN junction out of equilibrium thermodynamics

thickness of the DLA. (Fig. 14)

We may consider that c and c1 (c<<c1) account for the transmission line impedance, while Cx, Rx (Cx in the order of C1 ) accounts for external electrolyte storage.

Fig. 12a. Diffusion/storage model -1-

Provided that CD << C1 connection as a parallel RC cell should not modify drastically the resulting impedance. This model was introduced in order to separate "short term" and "long term" overvoltage variations in the experimental investigation.

Fig. 12b. Diffusion/Storage model -2-
