## 2 Model Principles
### Dynamic Effects--Electromechanical Transient Regime and Harmonic Relaxation
Consistently with Laplace and Fourier transforms, these fractional operators in the time-domain (as physical systems are ruled by--potentially non-integer--partial differential equations in time-domain) yield a multiplication by \(s^{n}\) and \(\left(j\omega\right)^{n}\) (respectively, in Laplace and Fourier spaces), with \(s\) the Laplace variable, \(\omega\) the angular frequency and \(j\) the unit imaginary number. As an illustration, Figures 3 and 4 show the frequency response and transfer characteristics of the fractional order unit relaxor \(H\) defined in the Laplace domain as:  
\[H(s)=\tfrac{out(s)}{in(s)}\ =\frac{1}{1+s^{n}} \tag{4}\]  
where \(s\) is recalled as the Laplace variable and \(n\) is the fractional order (\(n\in\mathbb{R}\)). \(in(s)\) and \(out(s)\), respectively, denote the input and output signals of the system represented by the transfer function \(H(s)\). These illustrative results have been obtained using the FOMCON (Fractional-order Modeling and Control - version 1.22.0.0, Tallinn University of Technology, Tallinn, Estonia) toolbox for MATLAB  
Figure 3: Frequency response of fractional order unit relaxors (\(n\) is the fractional order). (version R2020a, Mathworks, MA, USA) developed by [PERSON] et al. [34; 35]. Therefore, it can be seen from Figure 3 that the frequency response can be finely tuned both in terms of magnitude response and phase, allowing changing the attenuation slope and bandwidth in a precise way. Phase tuning through the fractional order \(n\) also permits shaping differently the elliptical input-output characteristics and its frequency behavior as shown in Figure 4. Moreover, a remarkable property when considering unipolar input (Figure 4b) is the ability of relating an almost constant shift after a transient stage, while the entire derivative order \(n=1\) yields offsets that significantly change as the frequency increases. Finally, the possibility of finely controlling the system dynamics can also be observed from the step response depicted in Figure 5, showing that low fractional order expands the settling time and thus well relates long-term phenomena. Figure 4: Input (“\(in(s)\)”)-output (“\(out(s)\)”) characteristics of fractional order unit relaxors (\(n\) is the fractional order) at several frequencies (a.u. refers to “arbitrary units”). Figure 5: Step responses of fractional order unit relaxors (\(n\) is the fractional order). (**Left**) linear time scale; (**Right**) logarithmic time scale (a.u. refers to “arbitrary units”; “Input” and “Output”, respectively, refer to the input and output signal of the system represented by the transfer function \(H(s)\) (i.e., \(in(s)\) and \(out(s)\)). Therefore, in order to take into account transient and harmonic effects into the static model and according to Figure 2, the driving voltage \(V\) is changed into an equivalent static model input voltage \(V^{*}\) that is obtained after two filtering processes: one representing the transient effect whose transfer function is denoted \(G_{t}\) and the other harmonic phenomena with an associated transfer function \(G_{d}\). This, therefore, yields in the Laplace domain:  
\[V^{*}(s)=G_{t}(s)G_{d}(s)V(s) \tag{5}\]  
with \(G_{t}(s)\) and \(G_{d}(s)\), respectively, defined as  
\[ \begin{array}{rcl}G_{t}(s)&=&\frac{1}{1+\tau_{t}^{\lambda_{t}}s^{\lambda_{t} }}\\ G_{d}(s)&=&\frac{1}{1+\tau_{d}^{\lambda_{d}}s^{\lambda_{d}}}\end{array}  \tag{6}\]  
with \(\tau_{i}\) and \(\lambda_{i}\) (\(i=t,d\)) the time constant and fractional order associated to each effect, respectively. Obviously, as the transient effect is expected to have a response spanning a wider time period compared to harmonic losses and conversion, one can expect that \(\tau_{d}^{\lambda_{d}}.s^{\lambda_{d}}<\tau_{t}^{\lambda_{t}}.s^{\lambda_{t}}\) for relatively small value of \(s\) and with \(\lambda_{t}<\lambda_{d}\).