## 5 Discussion of Trends from Section 4 against Models Delivering Analytical Results  
Comparison of Trends of Analytical Results for Parallel Plate MEMS Devices with Numerical Results from Section 4  
Here, the results obtained by numerical simulations in Section 4 are qualitatively related to the analytic formula given in Sections 2 and 3 for a parallel plate MEMS actuator. Equation (4) in Section 2 reveals that the pull-in voltage \(V_{\text{pi}}\) is proportional to the root of the spring constant \(\sqrt{k}\) and inversely proportional to the root of the area \(\sqrt{A}\) of plates. Therefore, the \(V_{\text{pi}}\) of a parallel plate qualitatively increases with growing \(k\) and decreasing \(A\).  
The eigenfrequency \(f_{\text{n}}\) (described in Equation (5) of Section 3) for a parallel plate capacitor is proportional to the root of the spring constant \(\sqrt{k}\) and inversely proportional to the root of the mass \(\sqrt{m}\). Therefore, \(f_{\text{n}}\) increases with growing \(k\) and decreasing \(m\). As mentioned before, the actuation time is reversely related to the frequency. Thus, the actuation time qualitatively increases with growing \(m\) and decreasing \(k\).  
Therefore, the simple formula of the parallel plate actuator can provide a preliminary qualitative tendency, even for such a simplified geometry (parallel plate actuation) compared to a cantilever actuator. For quantitative statements, numerical simulations (as performed in Section 4) are required.  
Comparison of Trends of Analytical Results for MEMS Cantilevers with Numerical Results from Section 4  
For a cantilever system, the spring constant can be roughly approximated by \(kc=\frac{2 Eb}{3}{\left(\frac{b}{l}\right)}^{3}\)[50]. Therefore, \(V_{\text{pi}}\) increases with growing \(E\) and \(h\), and it increases with decreasing \(l\). In the literature, the calculated pull-in voltage \(V_{\text{pi}}\) of cantilevers is reported to be proportional to \(\sqrt{\frac{Eb^{3}}{l^{4}}}\)[51], which means that \(V_{\text{pi}}\) rises with growing \(E\) and \(h\) as well as with decreasing \(l\). This trend is in agreement with the results obtained in Section 4. The results show that the primary factors affecting pull-in voltage are length, followed by thickness, and finally [PERSON]'s modulus (showing decreasing influence in this sequence), which is consistent with the simulation results in Section 4.  
Then, based on the eigenfrequency formula of cantilevers from Ref [52], \(f_{\text{n}}\) is proportional to \(\sqrt{\frac{Eb^{2}}{\rho^{2}}}\). Thus, \(f_{\text{n}}\) grows with increasing \(E\) and \(h\) as well as decreasing \(\rho\) and \(l\). The actuation time is inversely proportional to the frequency, and the relation is also in agreement with our numerical simulations obtained in Section 4.  
In many theoretical papers, the parameters in the results have been converted into reduced parameters (dimension-less). In order to enable a comparison with our results from Section 4, we reconverted the parameters, delivering the following: the actuation voltage \(V\) is proportional to \(\sqrt{\frac{2\rho^{2}EI}{\varepsilon b^{4}}}\) and the actuation time \(t_{a}\) is proportional to \(\sqrt{\frac{\varepsilon b^{4}}{EI}}\)[21; 53]. For a cantilever, the area of the cross-section is \(S=bl\), and the area moment of inertia is \(I=\frac{bl^{3}}{12}\). Therefore, the pull-in voltage of the cantilever \(V_{\text{pi}}\) is proportional to \(\sqrt{\frac{Eb^{3}}{l^{4}}}\), and the actuation time \(t_{a}\) is proportional to \(\sqrt{\frac{\rho^{4}}{EI^{2}}}\). In addition, this trend is in agreement with the above-mentioned trends and in agreement with our numerical results from Section 4.