## 2 Problem
In order to determine the voltage change due to the SoC, these variables can be considered constant. In contrast, the SoC is a key determinant of the activity of the redox partner (\(a_{Rel}\) and \(a_{Ox}\), respectively). The activity indicates the concentration corresponding to the behavior of a real mixture. As an approximation, the concentration of the reduced or oxidized species and thus the SoC can be used. The Nernst equation describes the behavior at one electrode at a time, so that the differences of both electrodes accumulate. Using the example of a lithium-ion battery, the nominal voltage is 3.6 V, but a charging cycle of an empty battery starts at 3.4 V and finishes at 4.2 V. The resulting voltage difference due to the SoC is  
\[V_{\text{Lilon,SoC,diff}}=V_{\text{Lilon}}(\text{SoC}=1)-V_{\text{Lilon}}( \text{SoC}=0)=4.2\,\text{V}-3.4\,\text{V}=0.8\,\text{V}. \tag{3}\]  
In both lead-acid batteries and lithium-ion cells, the voltage differences are therefore sufficiently high to enable simple detection of the SoC. In individual variants of lithium-ion batteries, the voltage difference can already be reduced in some areas. Lithium-iron phosphate batteries, for example, show only a slight change in cell voltage in the SoC range between 10% and 90% during both charging and discharging, making it difficult to determine the SoC. When analyzing zinc-air batteries, the change in cell voltage as a function of the SoC is even less significant. Usually, zinc-air batteries use oxygen from the ambient air. The amount of oxygen consumed during discharging or the amount released during charging is relatively small compared to the amount of oxygen in the air, and therefore has only a very small influence on the partial pressure of oxygen in the ambient air. The redox potential of the air electrode therefore shows almost no dependence on the SoC of the cell. Instead, only the zinc anode contributes to the change in open circuit voltage. Applying the Nernst equation to the zinc anode, the activity of the oxidation partner can be assumed to be a constant concentration of KOH solution \(c_{KOH}\). In contrast, the chemical activity of the reduction partner is strongly influenced by the SoC, as it is the ratio \(x_{ZnO}\) of zinc to zinc:  
\[E_{red,ZnAir,anode}=E_{red,ZnAir,anode}^{\Theta}-\frac{RT}{zF}\ln\frac{x_{ZnO }}{c_{KOH}}. \tag{4}\]  
Unfortunately, the resulting impact is not very distinct as the redox potential difference between a fully charged and a nearly empty battery is about  
\[ \begin{array}{rcl}V_{\text{ZnAir,SoC,diff}}&=&E_{red,ZnAir,anode}(x_{ZnO} \to 0)-E_{red,ZnAir,anode}(x_{ZnO}=1)\\ V_{\text{ZnAir,SoC,diff}}&\sim&60\,\text{mV}.\end{array}  \tag{5}\]  
The influence of the SoC is thus an order of magnitude smaller than with traditional technologies. In particular, the temperature and possible aging of the cell and especially of the electrolyte have a stronger effect on the cell voltage than the SoC. It is therefore not possible to determine the SoC from the open-circuit voltage of a zinc-air cell. Figure 4 shows the cell voltage of a zinc-air cell during an active charge and discharge cycle. Both the charging process and the discharging process are performed with a constant current. Even with an active charge current, the change in cell voltage over the charge cycle is so small that neither SoC detection nor effective end-of-charge detection is possible from the cell voltage. In addition, it is problematic that, towards the end of the charging process, an accompanying electrolysis process starts at a similar voltage level, so that overcharging with gassing cannot be detected either. The voltage characteristic during the discharge process is also largely constant. Towards the end of the discharge process, however, there is a voltage drop, therefore deep discharge detection is at least possible with the aid of a conventional voltage threshold.