## 2 Materials and Methods  
### Circuit Topology and Theory Analysis  
MEMS parallel switches can be integrated in coplanar waveguides (CPW) or microstrip lines. In this paper, 50 \(\Omega\) CPW is used to making the circuit model analysis. CPW is generally defined by center strip width W and gap width G. The CPW transmission line has a dielectric layer of thickness \(t_{d}\), and the dielectric constant is \(\varepsilon_{r}\). The length of the beam is \(L\); the width is \(w\), and the thickness is \(t\). A MEMS capacitive parallel switch top view is shown in Figure 1.  
A common MEMS parallel switch beam is connected between the transmission line and ground, and the switch anchor is connected to the CPW ground. Figure 2 shows the equivalent model of the switch in operation, wherein the beam mainly produces inductance, and the beam with the transmission line and the middle of the dielectric produces the capacitor. When the DC voltage is applied between the MEMS beam and the CPW transmission line, the resulting electrostatic force causes the beam to bend downward and switches to the down-state. When the DC bias voltage is not applied, the beam returns to the initial state (up-state) by the elastic restoring force. Figure 3 shows the circuit topology of the typical MEMS switch. The wave port of the characteristic impedance is \(Z_{0}\). \(Z_{b1}\) (\(\alpha\), \(\beta l\)) denotes the characteristic impedance of the transmission line between the wave port and the edge of the MEMS beam; \(\alpha\) is the transmission attenuation constant, and \(\beta l\) denotes electric length of the transmission line. The capacitive ratio \(C_{r}\) and the actuation voltage \(V_{p}\) of the MEMS switch can be calculated from Formula (1) and (2). Figure 1: MEMS capacitive parallel switch top view. Figure 2: Model of a MEMS switch in a working state. \[C_{r}\ =\ \frac{C_{d}}{C_{u}}\ =\ \frac{\frac{e_{0}e_{r}A}{t_{d}}}{\frac{e_{0}A}{g+ \frac{t_{d}}{t_{d}}}+C_{f}} \tag{1}\]  
\[V_{p}\ =\ \sqrt{\frac{8 kg^{3}}{27\varepsilon_{0}A}} \tag{2}\]  
where parameters of \(C_{d}\), \(C_{u}\), \(\varepsilon_{0}\), \(\varepsilon_{r}\), \(t_{d}\), \(S_{r}\), \(A\) and \(k\) are the down-state capacitance of the MEMS switch, the up-state capacitance of the MEMS switch, vacuum dielectric constant, relative dielectric constant of the dielectric layer, the thickness of the dielectric layer, the gap between the beam and the electrodes, the edge capacitance of the up-state capacitance, the cross-sectional area of the electrode and the spring constant of the beam, respectively. When \(C_{f}\) is not taken into account, the Formula (1) becomes Equation (3). From equation (3), we can use MATLAB R2015a (MathWorks, Natick, MA, USA) to obtain Figure 4, which is about the relationship between \(g\), \(t_{d}\) and \(C_{r}\). \[C_{r}\ =\ \frac{C_{d}}{C_{u}}\ =\ \frac{\frac{e_{0}e_{r}A}{t_{d}}}{\frac{e_{0}A}{g+ \frac{t_{d}}{t_{d}}}}\ =\ \frac{g\varepsilon_{r}}{t_{d}}+1 \tag{3}\]  
As a result, the pull-in voltage \(V_{p}\) can be simplified from (3)-(5). \[g\ =\ \frac{(C_{r}-1)t_{d}}{\varepsilon_{r}} \tag{4}\]  
\[V_{p}\ =\ \sqrt{\frac{8k(C_{r}-1)^{3}{t_{d}}^{3}}{27\varepsilon_{0}\varepsilon_{ r}{}^{3}A}} \tag{5}\]  
From Equation (3) and Figure 4, we may deduce that the capacitive ratio \(C_{r}\) rises as \(g\) (the gap between the beam and the electrodes) rises and falls as \(t_{d}\) (the thickness of the dielectric layer) rises. However, due to the pinhole problems in the dielectric layer, we can not deposit a Si\({}_{3}\)N\({}_{4}\) layer less than 0.1 \(\upmu\)m, and the dielectric layer should be able to sustain the excitation voltage without being pierced. Therefore, it is difficult to achieve low pull-in voltage and a larger capacitance ratio for traditional MEMS switches simultaneously.