## 2 Materials and Methods
### Laboratory Tests--The Proposed Model for EDLC
To simplify the formulation of the solution and the search for parameters from measurements, the non-linear part of capacity (\(3K_{V}\)) is redefined as \(K_{P}\), that is, \(K_{P}=3K_{V}\):  
\[K_{P}u_{C}^{2}(t)\frac{du_{C}(t)}{dt}+C_{0}\frac{du_{C}(t)}{dt}+\frac{u_{C}(t )}{R_{P}}=-Ig_{0} \tag{10}\]  
From the point of view of measurements, \(u_{C}(t)\) can be easily obtained considering R\({}_{S}\) (and R\({}_{L}\) in some cases, depending on measurement nodes):  
\[u_{C}(t)=u(t)+R_{s}Ig_{0} \tag{11}\]  
From Equation (9), it can be considered that in the non-linear capacitor, the user sees at each voltage level an apparent capacitance C\({}_{\text{ap}}\) given by  
\[C_{ap}=C_{0}+3K_{V}u_{C}^{2}(t)=C_{0}+K_{P}u_{C}^{2}(t) \tag{12}\]  
In this paper, to solve the above IVP, we are going to use the Differential Transformation Method (DTM) proposed in [35]. DTM constitutes an important numerical technique used to solve linear and non-linear initial value problems in the analysis of electric circuits [35]. The DT method is an iterative procedure to obtain the analytical solutions of the Taylor series of differential equations, which presents an important advantage in that it can  
Figure 9: Equivalent circuit proposed for PBLM-EDLC. greatly reduce the size of the computational work while still accurately providing the series solution with a fast convergence rate [36]. Moreover, this technique can be directly applied to non-linear ordinary differential equations without requiring linearization, discretization, or perturbation. For this reason, this technique has been of interest to many researchers, where the DT method has been employed to obtain the approximate analytical solution of first, second, and third-order linear differential equations [37]. Furthermore, in [38; 39; 40], the DT method is used to solve differential equations of the Lane-Emden type. Next, we recall the notion of differential transformation (DT) and some of its main properties. **Definition 1**. : _Assume that f(t) is an infinitely differentiable function. DT of f(t) is expressed as_  
\[F(k)=\frac{1}{k!}\left[\frac{d^{k}f(t)}{dt^{k}}\right]_{t=0}. \tag{13}\]  
**Remark 1**. : _Inverse DT of \(F(k)\) is defined as_  
\[f(t)=\sum_{k=0}^{\infty}F(k)t^{k}, \tag{14}\]  
By substituting Equation (13) in Equation (14), we obtain  
Figure 8:0  
Note that the above equation shows that DT originates from the Maclaurin series expansion. Now, we present theorems 1-4 that collect some essential properties of DT [36]. **Theorem 1**. : _If \(f(t)=\lambda g(t)+\mu h(t)\), then \(F(k)=\lambda G(k)+\mu H(k)\), where \(\lambda\) and \(\mu\) are real constants._  
**Theorem 2**. : _If \(f(t)=g(t)h(t)\), then \(F(k)=\sum_{l=0}^{k}G(l)H(k-l)\)._  
**Theorem 3**. : _If \(f(t)=t^{n}\) with \(n\in N\), then \(F(k)=\delta(k-n)= \begin{cases}1&k=n\\ 0&k\
eq n\end{cases} \)._  
**Theorem 4**. : _If \(f(t)=\frac{d^{n}g(t)}{dt^{n}}\), then \(F(k)=(k+1)(k+2)\ldots(k+n)G(k+n)\)._  
Having briefly laid the foundations of DTM, we are able to solve our IVP. Thus, taking DT in Equation (10) gives  
\[K_{P}\sum_{l=0}^{k}P(l)D(k-l)+C_{0}(k+1)U_{c}(k+1)+\frac{1}{R_{P}}U_{c}(k)=-Ig _{0}\delta(k), \tag{16}\]  
where \(P(l)=\sum_{r=0}^{l}U_{c}(r)u_{c}(l-r),D(k-l)=(k-l+1)U_{c}(k-l+1)\), and  
\[u_{c}(0)=u_{0}. \tag{17}\]By substituting Equation (17) into Equation (16) and using the recurrence relation, we have  
\[ \begin{split} U_{c}(1)=-\frac{lg_{0}+\frac{1}{k_{P}}UL_{c}(0)}{k_{P} UL_{c}(0)^{2}+C_{0}};\,U_{c}(2)=-\frac{\left(2K_{P}UL_{c}(0)UL_{c}(1)+\frac{1}{k_{P} }\right)UL_{c}(1)}{2\left(K_{P}UL_{c}(0)^{2}+C_{0}\right)},\\ U_{c}(3)=-\frac{\left(6K_{P}UL_{c}(0)UL_{c}(1)+\frac{1}{k_{P}} \right)u_{c}(2)+K_{P}UL_{c}(1)^{3}}{3\left(K_{P}UL_{c}(0)^{2}+C_{0}\right)}, \end{split}  \tag{18}\]  
and so on. Finally,  
\[ \begin{split} u_{c}(t)\approx\sum_{k=0}^{3}U_{c}(k)t^{k}=& U_{c}(0)+U_{c}(1)t+U_{c}(2)t^{2}+U_{c}(3)t^{3}.\end{split}  \tag{19}\]