# Analytical Approach in the Development of RF MEMS Switches
## 2 Analytical Approach to the Development of RF MEMS Switches of Electrostatic Type
### Study of the Dynamic Characteristics  
In the electrostatic drive with parallel plates, the device is triggered sharply, which makes some uncertainty. The tensile forces cannot balance the electrostatic forces for a long time and the switch closes as soon as the threshold voltage is reached. The membrane type capacitive microstructure can be modeled as an elastic beam, which is affected by the electrostatic force and the damping force on the film (see Figure 8), which is described by Equation (17):  
\[m\frac{\partial^{2}x}{\partial t^{2}}+b\frac{\partial x}{\partial t}+kx=F_{e} \tag{17}\]  
At low pressures, the damping term in (17) can be neglected and the system is inertia-dominated.  
Figure 8: One-dimensional model of the integrated RF MEMS switch.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(E\), GPa & \(v\) & \(\alpha\), \(10^{-6}(^{\circ}\mathbf{C}^{-1})\) & \(K\), W/m\(\cdot\)K & \(\rho\), \(\Omega\)\(\cdot\)m & \(Density\), kg\(\cdot\)m\({}^{3}\) \\ \hline
57.5 & 0.3175 & 16.7 & 196.55 & \(6.32\times 10^{-8}\) & \(4\times 10^{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: The material properties of the sandwich structure  
Figure 7: View of the elastic suspension element.  
The electrostatic force acting on the membrane is determined by Equation (18):  
\[F_{e}=\frac{1}{2}\epsilon_{0}A\frac{V^{2}}{x^{2}} \tag{18}\]  
where \(A\)--the plate area; \(d\)--the distance between the plates; \(V\)--the applied voltage. The distance \(x\) at different voltages is determined from the balance of forces \(F_{m}=F_{e}\), using Equation (19):  
\[x^{3}-dx^{2}+\frac{\epsilon_{0}AV^{2}}{2k}=0 \tag{19}\]  
By solving Equation (19) in inverse form, it is possible to obtain Equation (20) to determine the value of the threshold voltage:  
\[V(x)=\sqrt{\frac{2k}{\epsilon_{0}A}x^{2}(d-x)} \tag{20}\]  
At a distance of \(x=x_{pull-down}=2d/3\), the \(V(x)\) reaches its maximum value \(V_{pull-down}\) and the membrane falls on the fixed down electrode. \(V_{pull-down}\) determined by Equation (21):  
\[V_{pull-down}=\sqrt{\frac{8k}{27\epsilon_{0}A}d^{3}} \tag{21}\]  
For the MEMS switch to stay in the down-state position the electrostatic force must be larger than the mechanical restoring force \(F_{m}=k(g_{0}-g)\) and this is achieved for a hold-down voltage. The hold-down voltage is determined by the following Equation (22):  
\[V_{hold-down}= \sqrt{\frac{2k}{\epsilon\epsilon_{0}A}(d-x)(x+\frac{t_{d}}{ \epsilon_{r}})^{2}} \tag{22}\]  
The voltage at which the membrane is raised by the elastic forces is determined by Equation (23):  
\[V_{return}= \sqrt{\frac{2t_{d}^{2}kd}{\epsilon_{0}\epsilon_{r}A}} \tag{23}\]  
The switching time of RF MEMS switches with electrostatic activation mechanism strongly depends on the applied voltage. In most cases, this voltage is 1.3-1.4 of the pull-down voltage in order to obtain a small switching time. A very high switching voltage leads to an increase in electrostatic force, which in turn adversely affects reliability.  
The resonant frequency of any system modelled as spring is given by Equation (24):  
\[\omega= \sqrt{k/m} \tag{24}\]  
The frequency of operation of the integrated RF MEMS switch can be defined using Equation (25):  
\[f=\frac{1}{2\pi}\sqrt{\frac{k}{m}} \tag{25}\]  
The switching time in down state of the integrated RF MEMS switch can be determined using Equation (26):  
\[t_{pull-down}=3.67\frac{V_{pull-down}}{V_{s}}\sqrt{\frac{m}{k}} \tag{26}\]Analytical Equation (27) for the determining the switching time in initial state can be derived by using the energy method (\(V_{s}=0\)) [73].  
\[t_{pull\text{-}up}=\frac{1}{4f} \tag{27}\]