**4.2 Comparison of PN junction and electrochemical interface**

From the analysis of PN junction diodes, following similarities can be cited in relation to electrochemical interfaces (Coupan and al., 2010):


However, significant differences may be identified:


Based on method for modeling the PN junction, and the comparison seen above, we propose to analyze and model the phenomenon of activation in a lead-acid battery.

### **4.3 Phenomenon of non-linear activation**

The activation phenomenon is characterized by an accumulation of reactants at the space charge region. This electrokinetic phenomenon obeys to the Butler-Voltmer: exponential variation of current versus voltage, for direct and reverse polarization (Sokirko Artjom et al., 1995). Dynamical behavior can be introduced using a "charge driven model", familiar for PN junctions, connected to an excess carrier charge Q stored in the activation phenomenon. Bidirectional conduction can be accommodated using two antiparallel diodes. Based on the

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):

> exp - 1 *T*

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

From the analysis of PN junction diodes, following similarities can be cited in relation to

The electrical neutrality is preserved outside an area of "double layer" formed at the

 The voltage drop located in the double layer zone is connected to limit concentrations of carriers by an exponential law (according to the Nernst's equation in electrochemistry,

 For the PN junction, it is the concentration ratio that leads to predominant diffusion current for the minority carriers by diffusion. For lead acid battery, it is the mobility

 There is no recombination of the carriers in the battery. As a result, in constant current operation the stored charge builds up linearly with time, instead of reaching a limit

 The diffusion length is in fact the distance between electrodes, resulting in very long time constants (time constants even longer if one takes into account the migration of

 within the overall "double-layer", additional "activation layers" build up in the presence of current, corresponding to the accumulation of active carries close to the reaction

Based on method for modeling the PN junction, and the comparison seen above, we propose

The activation phenomenon is characterized by an accumulation of reactants at the space charge region. This electrokinetic phenomenon obeys to the Butler-Voltmer: exponential variation of current versus voltage, for direct and reverse polarization (Sokirko Artjom et al., 1995). Dynamical behavior can be introduced using a "charge driven model", familiar for PN junctions, connected to an excess carrier charge Q stored in the activation phenomenon. Bidirectional conduction can be accommodated using two antiparallel diodes. Based on the

to analyze and model the phenomenon of activation in a lead-acid battery.

ions move almost exclusively by diffusion

(29)

*<sup>V</sup> J Js <sup>U</sup>* 

where Js is called the current of saturation.

used for the dynamic model of the diode.

interface electrode / electrolyte.

ration that explains that <sup>2</sup> *SO*<sup>4</sup>

ions from outside the plates).

**4.3 Phenomenon of non-linear activation** 

interface.

electrochemical interfaces (Coupan and al., 2010):

the Boltzmann law for semi-conductors). However, significant differences may be identified:

value proportional to the recombination time.

**4.2 Comparison of PN junction and electrochemical interface** 

In the neutral zone, conduction is predominantly by diffusion.

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

$$V\_a \left[ \bigvee\_{t=0}^{\omega} \begin{array}{c} \mathbf{I} \\ \bigvee \\ \mathbf{I} \end{array} \right] \quad \left| I\_{st} = J\_s \begin{pmatrix} \left(\frac{V\_{\underline{\alpha}}}{\nu\_0}\right)\_{-1} \\ e^{\left(\frac{V\_{\underline{\alpha}}}{\nu\_0}\right)\_{-1}} \end{pmatrix} - \frac{\underline{Q}}{\tau\_0} \right. \tag{30}$$

$$I = \frac{\underline{Q}(t)}{\tau} + \frac{d\underline{Q}}{dt}$$

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

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 expression (simplified symmetric model):

$$V\_a \stackrel{\bullet}{\longmapsto} \left\| \sum\_{t=1}^{\bullet} \sum\_{t=1}^{\bullet} \right\| \qquad \qquad I\_{st} = G(V\_a) = I\_s \operatorname{ sh}\left(\frac{V\_a}{v\_0}\right) \tag{31}$$

The static relation corresponds to the Butler Volmer equation (symmetric case). It is completed by the charge driven model:

$$\begin{cases} Q = \tau I\_{st} = \tau \mathbf{G}(V\_a) \\ I = I\_{st} + \frac{dQ}{dt} \end{cases} \tag{32}$$

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

$$I\_{st} = G(V\_a) = \frac{J\_0}{\nu\_0} s h \left(\frac{V\_a}{\nu\_0}\right) \quad \text{and} \quad \underbrace{Q(V\_a)}\_{\text{I}^1} = \underbrace{c\_0}\_{\text{I}^2} s h \left(\frac{V\_a}{\nu\_0}\right) \tag{33}$$

$$\underbrace{I}\_{\text{I}^1} = \underbrace{I}\_{\text{II}^2} \tag{34}$$

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

Structural Design of a Dynamic Model of the Battery for State of Charge Estimation 143

Fig. 17. Experimental analysis and simulation of activation phenomena

Fig. 18. Highlight of the diffusion and activation process
