# About the Use of Generalized Forms of Derivatives in the Study of Electromagnetic Problems
## 5 Permittivity Models and Fractional Derivatives
### RC Circuit Tests Including Fractionary Dielectrics
The last positions permit the writing of Equation (72) as  
\[R\frac{d}{dt}q(\tau_{c}\eta)+\frac{q(\tau_{c}\eta)}{C(t)}=V_{0}f^{*}(\eta). \tag{76}\]  
By considering that \(dt=\tau_{c}d\eta\) and the Equation (74), we write Equation (76) as  
\[\frac{R}{\tau_{c}}\frac{d}{d\eta}q(\tau_{c}\eta)+\frac{q(\tau_{c}\eta)}{C_{0} \varepsilon_{r}(t)}=V_{0}f^{*}(\eta) \tag{77}\]  
which, using the relation \(\tau_{c}=RC_{0}\) and expressing that \(\varepsilon_{r}(t)=\varepsilon_{r}(\tau_{c}\eta)\), becomes  
\[\frac{1}{C_{0}}\frac{d}{d\eta}q(\tau_{c}\eta)+\frac{q(\tau_{c}\eta)}{C_{0} \varepsilon_{r}(\tau_{c}\eta)}=V_{0}f^{*}(\eta). \tag{78}\]  
By introducing the parameter \(q_{0}=C_{0}V_{0}\), which has the dimensions of the electric charge, we can write  
\[\frac{d}{d\eta}q(\tau_{c}\eta)+\frac{q(\tau_{c}\eta)}{\varepsilon_{r}(\tau_{ c}\eta)}=q_{0}f^{*}(\eta) \tag{79}\]  
and finally we can obtain the differential equation for the normalized charge \(\tilde{q}(\tau_{c}\eta)\) in the normalized variable \(\eta\)  
\[\frac{d}{d\eta}\tilde{q}(\tau_{c}\eta)+\frac{\tilde{q}(\tau_{c}\eta)}{ \varepsilon_{r}(\tau_{c}\eta)}=f^{*}(\eta), \tag{80}\]  
where the relative permittivity can be expressed as  
\[R\frac{d}{dt}q(t)+\frac{q(t)}{C(t)}=V(t)=V_{0}f(t), \tag{72}\]0A solution of the Equation (80) can be found first by expressing it in the frequency domain, where \(\frac{d}{d\eta}=jk\), \(k\) being the dimensionless pulsation  
\[R\frac{d}{dt}q(t)+\frac{q(t)}{C(t)}=V(t)=V_{0}f(t), \tag{72}\]1  
where \(\tilde{Q}(k)\) is the Fourier transform with respect to the variable \(\eta\) of the normalized charge \(\tilde{q}(\tau_{c}\eta)\). Furthermore, it is necessary to express the fractional dielectric permittivity law in the dimensionless frequency domain as  
\[R\frac{d}{dt}q(t)+\frac{q(t)}{C(t)}=V(t)=V_{0}f(t), \tag{72}\]2  
Finally, in the frequency domain, we have the following final expression for the normalized charge  
\[R\frac{d}{dt}q(t)+\frac{q(t)}{C(t)}=V(t)=V_{0}f(t), \tag{72}\]3  
where \(F^{*}(k)\) is the Fourier transform of \(f^{*}(\eta)\). We considered a forcing function \(f^{*}(\eta)=e^{-(\eta-10)^{2}}\), represented in Figure 14, and the data for the test in \[R\frac{d}{dt}q(t)+\frac{q(t)}{C(t)}=V(t)=V_{0}f(t), \tag{72}\]4  
The function \(\tilde{q}^{*}(\eta)\) is restored by the IFFT algorithm, which is an approximation of the integral below. \[R\frac{d}{dt}q(t)+\frac{q(t)}{C(t)}=V(t)=V_{0}f(t), \tag{72}\]5 \begin{table}
\begin{tabular}{l c} \hline \hline \(\tau\) & 153 ps \\ \(\tau_{c}\) & 88.54 ps \\ \(\kappa\) & 3 \\ \(\varepsilon_{\infty}\) & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Data for the test. Figure 14: Normalized source voltage. The results for the time-domain behavior of the normalized charge using the IFFT technique are depicted in Figure 15. It is clear, from Figure 15, that considering different frequency domain dielectric laws for the capacitor changes necessarily also the time-domain behavior of the overall circuit. In order to validate the IFFT technique, in Figure 16 is reported a comparison with the approximated integral computation, for the Havriliak-Negami model, showing a very good agreement between the two techniques. Moreover, it is possible to observe in Figure 17 the normalized charge behavior for a dielectric described by the Havriliak-Negami model, considering three different values of the parameter \(\kappa\). The main advantage of the IFFT technique is its simplicity and ease of coding. On the other hand, its inclusion in a 3D time-stepping electromagnetic solver is not straightforward since it requires an a priori knowledge of the source term in the integral (15) which is not the case when the fractionary dielectric is part of a more complex 3D model. In this case, recursive schemes are recommended which are based on the knowledge of a rational approximation of the fractional permittivity. Figure 16: Time-domain comparison for the normalized charge considering the Havriliak–Negami dielectric.