# Design and Fabrication of a MEMS Bandpass Filter with Different Center Frequency of 8.5-12 GHz
## 2 Design and Simulation of RF MEMS Filter  
RF MEMS filters are generally composed of a single coplanar waveguide (CPW) line, an actuation electrode, top electrode, air bridge, and resonant ring. These components allow the RF MEMS switch, which is actuated using a specific amount of static electricity, to control a signal by opening or closing. Here, we propose an all-aluminum tunable RF MEMS filter that consists of a substrate, a CPW line, two actuation electrodes, a top electrode, a resonant ring, two fixed anchors, and two air bridges. As illustrated in Figure 1a, quartz glass with a low relative permittivity is used as the substrate material for the filter to reduce its insertion loss. The CPW lines are fixed on the glass substrate. The resonant ring can produce different resonant frequencies and control the distance between the upper and lower electrodes of the switch, which is the most important structure in the RF MEMS switch and is fixed on the CPW lines.  
As shown in Figure 1b, we selected a resonant-ring structure composed of double rings and ground lines to act as the resonant structure to be fixed on the CPW lines, and among them the middle ring is used as an anchor to support the top electrode along with the anchor on the other side. In a double-ended fixed beam RF MEMS switch, the beam is fixed at both ends and is actuated using a voltage that is applied at both ends of the beam. When the maximal voltage (6.48 V) is applied, the displacement reaches its maximum value (1.5 \(\upmu\)m). The operational states of the switch are shown in Figure 2.  
Figure 1: Structure of the RF MEMS filter. (**a**) MEMS filter; (**b**) resonant ring.  
Figure 2: Cross-sectional views of filter operation:(**a**) off state; (**b**) on state.  
The RF MEMS switch in the on-state, where no direct voltage is applied between the top electrode and the actuation electrode, is shown in Figure 2a. The coupling capacitance between the top electrode and the signal line is low, and the RF signal from the input is connected with the output. When the switch is actuated, the top electrode is pulled down by electrostatic force to \((2/3)\)\(g_{0}\), where \(g_{0}\) is the air gap between the top electrode and the actuation electrode. This process achieves direct metal-to-metal contact between the top electrode and the signal line. The RF MEMS switch is turned off, as shown in Figure 2b. In this case, the actuation voltage, \(V_{\text{pull-down}}\), is given by [26]:  
\[V_{\text{pull-down}}=V\left(\frac{2}{3}g_{0}\right)=\sqrt{\frac{8k}{27\epsilon _{0}A}g_{0}{}^{3}}. \tag{1}\]  
here, \(\epsilon_{0}\) is the permittivity of free space and has a value of 8.854 \(\times\) 10\({}^{-12}\) F/m, and \(A\) is the area of the electrodes (on the top). \(k\) is the effective spring constant of the cantilever beams of the top electrode and is given by:  
\[k=\frac{1}{4}Ew(\frac{t}{l}){}^{3}. \tag{2}\]  
The actuation voltage can be defined as the minimum voltage that is required to pull the switch beam of the RF MEMS switch down. One of the design objectives for MEMS devices with actuation capability is to perform using a low actuation voltage, which is dependent on the design of the switch structure. The actuation voltage is dependent on the geometric parameters of the initial gap between the movable structures, the actuation area, and the electrode. The actuation voltage of 6.48 V is obtained in the straight plate type top electrode switch case, as shown in Figure 3.