# 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
### Theory  
In order to meet the actual application requirements, the filter design needs to meet practical parameter specifications, such as cutoff frequency, center frequency, insertion loss, and stopband rejection. Here is a brief introduction to these parameter indicators: [27]  
Center frequency \((f_{0})\) refers to the intermediate frequency of the filter passband. Generally, for bandpass and bandstop filters, the specific calculation formula is:  
\[f_{0}=\sqrt{f_{1}\times f_{2}}. \tag{3}\]  
here, \((f_{1})\) and \((f_{2})\) represent the frequencies corresponding to 3 dB points on both sides of the passband.  
Figure 3: Displacement versus actuation voltage.  
Insertion loss: When the signal passes through the filtered two-port network, it produces a certain energy loss, which can be expressed as the ratio of the output power of the filter to the input power. The specific calculation formula is:  
\[IL=-10 log\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}=-10 log\Big{(}1-\left|\Gamma_{ \mathrm{in}}\right|^{2}\Big{)}. \tag{4}\]  
here, \(P_{\mathrm{in}}\) is the input power, \(P_{\mathrm{out}}\) is the output power, and \(\left|\Gamma_{\mathrm{in}}\right|\) refers to the reflection coefficient at the input end. In general, the dielectric substrate and the high-temperature superconducting film with low-loss corners can greatly reduce the insertion loss and improve the sensitivity of the filter.  
Stopband rejection (SR) is a technical indicator used to indicate the anti-interference ability of a filter. The wider the stopband, the better the anti-interference ability. In general, a step impedance resonator is used, the zero point is generated at the parasitic passband, and different coupling structures are designed to widen the stop band to improve the anti-interference ability of the filter.  
The groove structure on the middle signal line can be equivalent to a Pi type network, which is composed of an equivalent series capacitor, C1, and two parallel capacitors, C2. The wires on both sides of the CPW can be equivalent to a series inductor, Lc, and a shunt capacitor, Cc. The annular groove structure etched on the coplanar waveguide ground can be equivalent to the parallel inductance, La, due to the magnetic coupling to the signal line. Ignoring the resistance of the MEMS switch, there is a capacitance, Ca1, between the MEMS switch beam and the signal line, and the beam itself has an inductance, La1, so it can be equivalent to a circuit form in which two sets of capacitance and inductance are connected in series. The specific equivalent circuit diagram is shown in Figure 4.  
As shown in Figure 5, Lc and Lb can be equivalent to an admittance converter, and a hybrid circuit of Ca1 and La1 can be equivalent to a resonant circuit. By combining the capacitor inductance, it can be simplified to the equivalent circuit diagram shown in Figure 5. Because the La1 is small, the inductance La1 can be ignored, so the series circuits Ca1 and La1 can be directly represented by Ca1, while Co = Cc + Ca - C2, Cp = Ca + Cb + Cc - C2.  
Figure 4: Equivalent circuit of the filter.  
Figure 5: Simplified equivalent circuit.  
In Figure 4, the admittance inverters can be calculated by the following equation [24]:  
\[J_{01}=\sqrt{\frac{Y_{0}\omega_{0}C_{a}BW}{g_{0}g_{1}\omega_{1}}}=\left.\frac{1}{ \omega L_{p}}\right|_{\omega=\omega_{0}} \tag{5}\]  
\[J_{12}=\frac{\omega_{0}C_{a}BW}{\omega_{1}}\sqrt{\frac{1}{g_{0}g_{1}}}=\left. \omega C_{1}\right|_{\omega=\omega_{0}} \tag{6}\]