# 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 dispersive nature of dielectrics can be easily included in circuit models through the Fourier transform technique, which permits to work in the frequency domain; finally Inverse Fast Fourier Transform (IFFT) techniques can be employed to restore the time-domain behavior of the quantities under investigation. A possible very useful application for non Debye dielectric models is in the framework of the Partial Elements Equivalent Circuit (PEEC) method, an integral equation based method that permits performance of electromagnetic (EM) simulations of complex structures introducing a circuit representation of the EM phenomena. The PEEC method is very useful in Electromagnetic Compatibility (EMC) and Signal integrity (SI) areas, allowing prediction and solving of electromagnetic interference (EMI) issues. Dielectric materials are a fundamental part of modern electric and electronic devices and for this reason it is important to accurately include their presence; this can be done in the PEEC method through the concept of excess capacitance [53]. In PEEC models is feasible to include in the excess capacitance also the fractional relative permittivity of Equation (70), in order to provide a more accurate representation of the EM phenomena that affect the structures under observation. It is reported here the example of an RC series circuit fed by a Gaussian pulse voltage, with the capacitor supposed to consist of flat and parallel plates, filled by a dielectric characterized by a relative permittivity generally described by Equation (70); all the four dielectric laws presented are examined here. \begin{table}
\begin{tabular}{c c c} _Model_ & \(\alpha\) & \(\beta\) \\ _[PERSON]_ & 0 & 0 \\ _[PERSON]_—_[PERSON]_ & 0.4 & 0 \\ _[PERSON]_—_[PERSON]_ & 0 & 0.1 \\ _[PERSON]_—_[PERSON]_ & 0.4 & 0.1 \\ \end{tabular}
\end{table} 
Table 1: Dielectric models. The ordinary differential equation, describing an RC series circuit through the electric charge \(q(t)\) in the time-domain, is the following:  
\[R\frac{d}{dt}q(t)+\frac{q(t)}{C(t)}=V(t)=V_{0}f(t), \tag{72}\]  
where \(R\) is the resistance value of the RC circuit and \(V(t)=V_{0}f(t)\) is the forcing function which represents the voltage source applied to the circuit; the capacitance \(C(t)\) is a time-dependent function because of the dispersive nature of the dielectric inside the capacitor. In the frequency domain, the capacitor can be described by its complex capacitance  
\[C(\omega)=\varepsilon_{0}\varepsilon_{r}(\omega)\frac{S}{d}=C_{0}\varepsilon _{r}(\omega), \tag{73}\]  
where \(S\) is the surface of the plates and \(d\) is the distance between them. Therefore, its time-domain counterpart is of the form  
\[C(t)=C_{0}\varepsilon_{r}(t) \tag{74}\]  
where  
\[\varepsilon_{r}(t)=\varepsilon_{\infty}\Bigg{(}1+\frac{\kappa}{(1+(\tau\frac {d}{dt})^{(1-\alpha)})^{(1-\beta)}}\Bigg{)} \tag{75}\]  
is the time-domain fractional dielectric law described in Equation (71). We now define the parameter: \(\tau_{c}=RC_{0}\) and introduce the dimensionless variable \(\eta=\frac{t}{\tau_{c}}\). Hence, the forcing function \(f(t)\) is also a \(f(\tau_{c}\eta)=f^{*}(\eta)\) and the variable function \(q(t)\) is also expressible as a \(q(\tau_{c}\eta)\).