## 3 Results and Discussions
### Energy Harvesting Study  
#### 3.3.1 Piezoelectric Harvested Current Model  
The goal of modeling is to express the current harvested by a piezoelectric material in temporal domain. Modeling the effect of piezoelectric materials is written in the form of the following two fundamental equations. The piezoelectric constitutive equations of direct and converse effects are given below [14]:  
\[S=s^{E}\times T+d_{t}\times E \tag{1}\]  
\[D=d_{t}\times T+\varepsilon^{T}\times E \tag{2}\]  
where \(S\), \(T\), \(D\), \(E\), \(\varepsilon^{T}\), \(d\) and \(s^{E}\) are, respectively, the strain, the stress, the electric displacement, the electric field, the dielectric permittivity, the piezoelectric coefficient and the compliance.  
It is possible to express the \(T\) stress as a function of strain \(S\) using Equation (1), given by:  
\[T=\frac{S-d_{t}\times E}{s^{E}} \tag{3}\]  
Figure 7: Piezoelectric coefficient as a function of time after ionization process charged in the silicone oil at room temperature.  
By substituting the expression of Equation (3) into Equation (2), the electrical displacement can be expressed as follows:  
\[D=d_{t}\times\frac{S-d_{t}\times E}{s^{E}}+\varepsilon^{T}\times E \tag{4}\]  
It is known that the expression of current harvested and electrical displacement is:  
\[I=A\times\frac{\partial D}{\partial t} \tag{5}\]  
A is the metalized surface of the polymer. Substituting Equations (4) into (5) gives:  
\[I=A\times\left(\varepsilon^{T}-\frac{d_{t}^{2}}{s^{E}}\right)\frac{\partial E }{\partial t}+A\times\frac{d_{t}}{s^{E}}\frac{\partial S}{\partial t} \tag{6}\]  
\(\frac{\partial E}{\partial t}\) and \(\frac{\partial S}{\partial t}\) are the time variation of the electrical field and strain as a function of time variation, respectively.  
The electrical field output of the polymer across the load resistance is defined by the term:  
\[E_{R}=-\frac{R\times I}{h} \tag{7}\]  
where \(R\) the electrical resistance and \(h\) is the thickness of the polymer film.  
The relation between \(Y\) material Young's modulus and the compliance is \(Y=\frac{1}{s_{11}^{2}}\), so substituting Equation (7) into (6) gives:  
\[I(t)=-A\Big{(}\varepsilon^{T}-Yd_{t}^{2}\Big{)}\frac{R}{h}\times\frac{ \partial I}{\partial t}+A\times Yd_{t}\times\frac{\partial S}{\partial t} \tag{8}\]  
Transforming Equation (8) into frequency domain, we can obtain the current I as:  
\[I(p)=\frac{A\times Yd_{t}\times p}{1+A\big{(}\varepsilon^{T}-Yd_{t}^{2}\big{)} \frac{R}{h}\times p}\;S(p) \tag{9}\]  
Applying the transformation of inverse Laplace to Equation (9), we obtain the current harvested expression:  
\[l^{-1}\{I(p)\}=I(t)=\frac{hYd_{t}}{\big{(}\varepsilon^{T}-Yd_{t}^{2}\big{)}R} \left(\gamma(t)-\frac{h}{A\big{(}\varepsilon^{T}-Yd_{t}^{2}\big{)}R}exp\Bigg{(} -t\frac{h}{A\big{(}\varepsilon^{T}-Yd_{t}^{2}\big{)}R}\Bigg{)}\right)\,S(t) \tag{10}\]  
with the governing equation of mechanical motion being:  
\[S(t)=S_{M}\sin(2\pi f_{M}\times t) \tag{11}\]  
where \(S_{M}\) and \(f_{M}\) are maximum transverse strain and mechanical excitation frequency, respectively.  
Substituting Equation (11) into (10), the current expression as a function of time and mechanical cycle excitation is expressed in the form:  
\[I(t)=\frac{hYd_{t}S_{M}}{\big{(}\varepsilon^{T}-Yd_{t}^{2}\big{)}R}\left( \gamma(t)-\frac{h}{A\big{(}\varepsilon^{T}-Yd_{t}^{2}\big{)}R}exp\Bigg{(}-t \frac{h}{A\big{(}\varepsilon^{T}-Yd_{t}^{2}\big{)}R}\Bigg{)}\right)\sin(2\pi f _{m}\times t) \tag{12}\]  
This harvested current depends on the level of mechanical strain, excitation frequency, material dimensions, electrical load and material proprieties. To validate this principle, an experimental study was performed. In order to predict output performances for charged porous films, the physical parameters of Equation (12) are presented in Table 2.