# Parameter Identification Method of a Double-Layer Supercapacitor by Using a Real Voltage Source
## 3 Methods
### Three-Point Method (3 PM)  
The external voltage as a function of time during cell charging will produce a curve like that shown in Figure 3. Three measurements will be taken at sufficiently distant points in time, whose values will be \(\mathrm{P_{1}}=(0,\mathrm{u_{co1}})\), \(\mathrm{P_{2}}=(\mathrm{t_{2}},\mathrm{u_{co2}})\), and \(\mathrm{P_{3}}=(\mathrm{t_{3}},\mathrm{u_{co3}})\). It is assumed that the initial sample will be collected at \(\mathrm{t_{1}}=0\) s. If the value of the voltage source, \(\mathrm{E}\), is added to both members of the function \(\mathrm{t(u_{co})}\) presented in (15), the relationship between the time \(\mathrm{t}\) and the external voltage \(\mathrm{u_{co}}\) is expressed as follows:  
\[\mathrm{k_{3}}\cdot\mathrm{t}+\mathrm{E}= \mathrm{k_{1}}\cdot\log\left(\frac{\mathrm{U_{0}}-\mathrm{E}}{ \mathrm{k_{2}}\cdot(\mathrm{u_{co}}-\mathrm{E})}\right)+\mathrm{U_{0}}+\mathrm{ k_{2}}\cdot(\mathrm{E}-\mathrm{u_{co}}). \tag{16}\]  
The value of \(\mathrm{R}\) can be calculated from the first value of the external voltage, \(\mathrm{u_{co1}}\). At time \(\mathrm{t}=0\), this voltage is given by:  
\[\mathrm{u_{co1}}=\frac{\mathrm{R_{c}}\cdot\mathrm{U_{0}}+\mathrm{R}\cdot \mathrm{E}}{\mathrm{R}+\mathrm{R_{c}}}. \tag{17}\]  
From (17), the value of \(\mathrm{R}\) is directly calculated as follows:  
\[\mathrm{R}= \left(\frac{\mathrm{u_{co1}}-\mathrm{U_{0}}}{\mathrm{E}-\mathrm{u_{ co1}}}\right)\cdot\mathrm{R_{c}}. \tag{18}\]  
Once the value of \(\mathrm{R}\) has been obtained, the value of the constant \(\mathrm{k_{2}}\) can be calculated using Equation (13). The two remaining measurements, corresponding to the points \(\mathrm{P_{2}}\) and \(\mathrm{P_{3}}\), can then be combined into a system of equations, where the value of \(\mathrm{k_{2}}\) is already known:  
\[ \begin{cases}\mathrm{k_{3}}\cdot\mathrm{t_{2}}+\mathrm{E}= \mathrm{k_{1}}\cdot\log\left(\frac{\mathrm{U_{0}}-\mathrm{E}}{\mathrm{k_{2}} \cdot(\mathrm{u_{co2}}-\mathrm{E})}\right)+\mathrm{U_{0}}+\mathrm{k_{2}}\cdot (\mathrm{E}-\mathrm{u_{co2}})\\ \mathrm{k_{3}}\cdot\mathrm{t_{3}}+\mathrm{E}= \mathrm{k_{1}}\cdot\log\left(\frac{\mathrm{U_{0}}-\mathrm{E}}{\mathrm{k_{2} }\cdot(\mathrm{u_{co3}}-\mathrm{E})}\right)+\mathrm{U_{0}}+\mathrm{k_{2}}\cdot (\mathrm{E}-\mathrm{u_{co3}})\end{cases} . \tag{19}\]  
Solving this system of equations yields the values of the constants \(\mathrm{k_{1}}\) and \(\mathrm{k_{3}}\):  
\[\mathrm{k_{1}}=\frac{\mathrm{E}-\mathrm{U_{0}}+\mathrm{k_{2}}\cdot\left(\left( \frac{\mathrm{t_{2}}}{\mathrm{t_{2}}-\mathrm{t_{2}}}\right)\cdot(\mathrm{u_{ co2}}-\mathrm{u_{co3}})+\mathrm{u_{co2}}-\mathrm{E}\right)}{\log\left(\left(\frac{ \mathrm{U_{0}}-\mathrm{E}}{\mathrm{k_{2}}\cdot(\mathrm{u_{co2}}-\mathrm{E})} \cdot\left(\frac{\mathrm{u_{co1}}-\mathrm{E}}{\mathrm{u_{co2}}-\mathrm{E}} \right)^{\frac{\mathrm{t_{2}}}{\mathrm{t_{3}}-\mathrm{t_{2}}}}\right)\right.}. \tag{20}\]  
\[\mathrm{k_{3}}=\frac{\mathrm{k_{1}}\cdot\log\left(\frac{\mathrm{u_{co2}}-\mathrm{ E}}{\mathrm{u_{co3}}-\mathrm{E}}\right)+\mathrm{k_{2}}\cdot(\mathrm{u_{co2}}- \mathrm{u_{co3}})}{\mathrm{t_{3}}-\mathrm{t_{2}}}. \tag{21}\]  
Figure 3: External voltage as a function of time, with three sufficiently spaced samples to be chosen. Once the three constants, k\({}_{1}\), k\({}_{2}\), and k\({}_{3}\), are known, it is straightforward to obtain C\({}_{0}\) and k\({}_{\rm c}\) values from the expressions (12) and (14):  
\[\rm{k_{c}=\frac{1}{2\cdot R_{c}\cdot k_{2}\cdot k_{3}}.} \tag{22}\]  
\[\rm{C_{0}=\frac{k_{1}-E}{R_{c}\cdot k_{2}\cdot k_{3}}.} \tag{23}\]  
The proposed method is straightforward, requiring only three measurements of the function u\({}_{\rm co}\)(t) to obtain the desired parameters. When the three external voltage measurements (u\({}_{\rm co1}\), u\({}_{\rm co2}\), and u\({}_{\rm co3}\)) are acquired, they may have oscillations with respect to the real value due to the errors that occur when using a data acquisition system, e.g., due to the resolution of the A/D converter.