## 2 Materials and Methods  
### Electrical Analysis of a DLSC Charged or Discharged with a Voltage Source  
This section will show how to express the different electrical variables of a DLSC when connected to a voltage source. The voltage source will be represented by a constant no load voltage, E, and a series connected resistor, R\({}_{\rm C}\). As mentioned above, the supercapacitor model will be that of a capacitance linearly varying with its internal voltage. All the obtained functions will be valid for charging and discharging with a voltage source or discharging with a constant resistor. In the second case, it is sufficient to cancel the no-load voltage, E = 0 V, and the discharge resistance will be R\({}_{\rm C}\). The internal resistance of the DLSC will be denoted as R and its capacitance will vary with the internal voltage called u. The external voltage of the DLSC is u\({}_{\rm co}\) and will obviously coincide with the terminal voltage of the voltage source (Figure 1). The capacitance of the DLSC shall be modelled as the parallel association of two capacitances, one of constant value, C\({}_{0}\), called \"Initial Capacitance\", which is the capacitance when the DLSC is fully discharged, and the other which is proportional to the internal voltage, C\({}_{1}\)(u). The final capacitance of the cell, C(u), will be the parallel association of the two. \[C(u)=C_{0}+C_{1}(u)=C_{0}+k_{\rm c}\cdot u. \tag{1}\]  
The values of C\({}_{0}\) and k\({}_{\rm c}\) are not specified in the datasheets of the cells provided by the manufacturers but can be expressed as a function of the rated capacitance, C\({}_{N}\), and  
Figure 1: Discharge of the DLSC through a voltage source. the rated voltage, \(U_{N}\). In this way, the capacitance can be expressed as a function of the internal voltage as follows:  
\[C(u)=C_{0}+\frac{C_{N}}{U_{N}}.\left(1-\frac{C_{0}}{C_{N}}\right)\cdot u=C_{N} \cdot k_{0}+\frac{C_{N}}{U_{N}}.(1-k_{0})\cdot u. \tag{2}\]  
The value of \(k_{0}\) (dimensionless), which is the quotient of the initial capacitance and the nominal capacitance, is known as the \"Normalized Initial Capacitance\". In a conventional capacitor, with constant capacitance, it is true that: \(k_{0}=1\), \(k_{c}=0\) and \(C_{N}=C_{0}\). The constant \(k_{0}\) is also not usually specified in the datasheets, but in most families of SCs from different manufacturers, \(k_{0}\) presents values between 0.7 and 0.8. Considering a discharge process, according to the definition of an electric current, this is equal to the time variation of charges experienced by the device, i.e.,  
\[i=-\frac{dq}{dt}=-\frac{dq}{du}\cdot\frac{du}{dt}. \tag{3}\]  
The negative sign indicates that the current represents a discharge. In both conventional SCs and DLSCs, the capacitance establishes the well-known relationship between electric charge and internal voltage:  
\[q=C(u)\cdot u. \tag{4}\]  
Considering the capacitance value in (1), the stored charge can be expressed as a function of the internal voltage and the values of \(C_{0}\) and \(k_{c}\):  
\[q=C_{0}\cdot u+k_{c}\cdot u^{2}. \tag{5}\]  
By deriving the charge \(q\) with respect to the internal voltage:  
\[\frac{dq}{du}=C_{0}+2\cdot k_{c}\cdot u. \tag{6}\]  
If the value of the derivative calculated in (6) is substituted in the expression for the current, the relationship between the current and the internal voltage can be obtained:  
\[i=-\frac{dq}{dt}=-\underbrace{(C_{0}+2\cdot k_{c}\cdot u)}_{Virtual Capacitance} \cdot\frac{du}{dt}=-C_{v}\cdot\frac{du}{dt}. \tag{7}\]  
The \(C_{V}\) value is called \"Virtual Capacitance\" or \"Dynamic Capacitance\".