# Design of Broadband High-Frequency Multi-Throw RF-MEMS Switches
## 2 Structure and Theoretical Analysis
This study adopts a gradient upper electrode approach to minimize the coupling capacitance value, consequently enhancing the isolation of the RF MEMS switch. The cantilever beam used in this study has the following fundamental characteristics: the beam's bending occurs only in the longitudinal direction without involving lateral deformation, and the beam's deflection is relatively small compared to its length. Based on these characteristics, two fundamental assumptions are proposed: first, under the action of bending moments alone, the beam's cross-section remains planar during the deflection process, known as the pure bending assumption; second, the beam's deflection is much smaller than its length, referred to as the small deformation assumption, which allows us to neglect shear stress relative to normal stress. Based on these assumptions, the mechanical deformation of the switch's top plate can be described using a linear elastic coefficient \(k\). A pivotal metric for the RF MEMS switch is the drive voltage. An excessively high drive voltage complicates its integration with other devices. Their fundamental vertical movement model consists of two parallel substrates regardless of series contact or parallel capacitive types. The electrostatic force exerted on the upper electrode is as follows:  
\[F_{\mathrm{e}}=\frac{1}{2}V^{2}\frac{dc(g_{0})}{dg_{0}}=-\frac{1}{2}\frac{ \epsilon_{0}AV^{2}}{g_{0}^{2}} \tag{2}\]  
Figure 2: Equivalent circuit of the proposed switch. Here, \(\epsilon_{0}\) represents the dielectric constant of vacuum or air, \(A\) signifies the face-to-face area between the upper electrode and the driving electrode, \(V\) denotes the switch's drive voltage, and \(g_{0}\) is the gap between the upper electrode and the driving electrode at the critical equilibrium state. The switch beam can be equivalently regarded as rigid due to minimal deformation, and its mechanical properties can be described by the linear elastic coefficient [26]. The expression for the restoring force experienced by the upper electrode can be formulated as  
\[F_{0}=k(g-g_{0}) \tag{3}\]  
\[k_{1\ldots 7}=\frac{EWT^{3}}{4L^{3}} \tag{4}\]  
\[\frac{1}{k}=\frac{1}{k_{1}}+\ldots+\frac{1}{k_{7}} \tag{5}\]  
Here, \(k\) is the equivalent elasticity coefficient of the upper electrode, g represents the gap between the upper electrode and the driving electrode, and \(E\), \(W\), \(T\), and \(L\) correspond to the Young's modulus, width, thickness, and length of the cantilever, respectively. When the upper electrode is drawn to two-thirds of the distance \(g\) between the upper and lower electrodes, the electrostatic force and restoring force achieve a critical equilibrium state, from which the drive voltage formula can be derived:  
\[V_{pull-down}=V\bigg{(}\frac{2}{3}g\bigg{)}=\sqrt{\frac{8k}{27\epsilon_{0}A} g^{3}} \tag{6}\]  
By examining Equation (6), to design a switch with the desired low drive voltage, we crafted a folded beam with a low \(k\) value. The corresponding switch response time, considering the presence of damping, is [27]  
\[t_{s}\approx\frac{9V_{pull-down}^{2}}{4\omega_{0}QV_{s}^{2}} \tag{7}\]  
where, \(V_{s}\) is generally taken as \(1.3V_{pull-down}\sim 1.4V_{pull-down}\), \(Q\) is the quality factor, and the resonant frequency is \(w_{0}=\sqrt{k/m}\). While optimizing the cantilever beam to achieve lower actuation voltages and response times, the ability of the beam to withstand stress under the optimized geometry should also be considered. Stress on the cantilever beam begins to develop as deformation progresses, and exceeding a certain threshold will lead to beam failure. Equation (8) provides the critical stress value [28]. Figure 1:0  
where \(v\) is the material's Poisson's ratio and \(\sigma_{cr}\) is the critical stress value. The self-actuating voltage can be calculated through the following steps:  
Figure 1:1  
The maximum power it can withstand can be calculated using the following equation:  
Figure 1:2  
where \(Z_{0}\) is the characteristic impedance.