**4.1 Comparison to PN junction**

A PN junction is formed of two zones respectively doped N (rich in electrons: donor atoms) and P (rich in holes: acceptor atoms). When both N and P regions are assembled (Fig. 13), the concentration difference between the carriers of the N and P will cause a transitory current flow which tends to equalize the concentration of carriers from one region to another. We observe a diffusion of electrons from the N to the P region, leaving in the N region of ionized atoms constituting fixed positive charges. This process is the same for holes in the P region which diffuse to the N region, leaving behind fixed negative charges. As for electrolytes, it then appears a double layer area (DLA). These charges in turn create an electric field that opposes the diffusion of carriers until an electrical balance is established.

We may consider that c and c1 (c<<c1) account for the transmission line impedance, while

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

A PN junction is formed of two zones respectively doped N (rich in electrons: donor atoms) and P (rich in holes: acceptor atoms). When both N and P regions are assembled (Fig. 13), the concentration difference between the carriers of the N and P will cause a transitory current flow which tends to equalize the concentration of carriers from one region to another. We observe a diffusion of electrons from the N to the P region, leaving in the N region of ionized atoms constituting fixed positive charges. This process is the same for holes in the P region which diffuse to the N region, leaving behind fixed negative charges. As for electrolytes, it then appears a double layer area (DLA). These charges in turn create an electric field that

opposes the diffusion of carriers until an electrical balance is established.

Cx, Rx (Cx in the order of C1 ) accounts for external electrolyte storage.

"long term" overvoltage variations in the experimental investigation.

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

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

**4.1 Comparison to PN junction** 

**4. Activation voltage** 

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

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:

$$\mathcal{N}\_a \mathcal{V} \mathcal{V}\_n = \mathcal{N}\_d \mathcal{V} \mathcal{V}\_p \tag{27}$$

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

$$V\_0 = \mathcal{U}\_T \ln \left(\frac{N\_a N\_d}{n\_l^{\ast 2}}\right), \mathcal{U}\_T = \frac{kT}{e} \tag{28}$$

where ni represents the intrinsic carrier concentration. On another hand, note that the width of the DLA may be related to the potential barrier (Mathieu H, 1987).

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 thickness of the DLA. (Fig. 14)

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

Structural Design of a Dynamic Model of the Battery for State of Charge Estimation 141

for one single PN junction, the static and dynamic modeling of a diode is given by the

0

 

*dt*

It is noted that one can easily model the current through the diode with an equivalent model of stored charge; this approach is valid for one current direction and not referring to the battery charge. We must therefore provide a more complete model that can be used in charge or discharge. This analysis therefore reflects a model with two antiparallel diodes. The static and dynamic modeling of the two antiparallel diodes is given by the current

> () *<sup>S</sup> <sup>a</sup> st a <sup>V</sup> I G V J sh*

The static relation corresponds to the Butler Volmer equation (symmetric case). It is

*Q I GV dQ I I dt*

After an analysis resulting static (and dynamic) and an experimental validation, we get the model of the phenomenon of activation with a parallel non linear capacitance and

> 0 0 0 0 0 0

*<sup>J</sup> V c <sup>V</sup> I G V sh and Q V sh v v v v* (33)

( ) ( ) *a a st a <sup>a</sup>*

*st*

conductance circuit (fig.15) whose expressions are given by the following equations:

 

 

Fig. 15. Activation model : non-linear capacitance and conductance

( ) *st a*

 

*Va st s <sup>v</sup> <sup>Q</sup> I Je*

1

0

(30)

(31)

(32)

0

*v*

*Q t dQ <sup>I</sup>*

with Q representing an amount of stored charge and a time constant τ associated.

expression (simplified symmetric model):

I

I

completed by the charge driven model:

*Va*

*Va*

current expression:

The decrease in potential barrier allows many electrons of the N region and holes from the P region to cross this barrier and appear as carriers in excess on the other side of the DLA. These excess carriers move by diffusion and are consumed by recombination. It is readily seen that the total current across the junction is the sum of the diffusion currents, and that these current may be related to the potential difference V in the form (Mathieu H, 1987):

$$J = J\_S \left( \exp\left(\frac{V}{U\_T}\right) \cdot 1\right) \tag{29}$$

where Js is called the current of saturation.

On the other hand the diffusion current is fully consumed by recombination with time constant , so that the stored charge Q may be expressed as Q = J. This expression will be used for the dynamic model of the diode.
