## 3 Mathematical Model of the Proposed MEMS Gyroscope
### Air Damping Model for MEMS Gyroscope  
In resonant MEMS gyroscopes, the accurate estimation of the viscous air damping is crucial since it affects the overall quality factor and hence sensitivity. In the proposed gyroscope design, both the slide and squeeze film air-damping factors result in the energy dissipation.  
In the drive axis, the slide film air damping is the sole energy-loss mechanism since the comb-drive fingers slide parallel to each other with an application of actuation voltage in the \(x\)-direction. The movement of the sense frame and drive mass in the \(y\)-axis results in the squeeze film air damping [20]. With the displacement of the sense mass in the \(y\)-axis, the air between the oscillating and stationary parallel plates is compressed. The same damping effect occurs in the interdigitated drive and tuning comb fingers. The damping coefficients in the \(x\)-axis and \(y\)-axis, for the gyroscope design, due to slide and squeeze film damping can be obtained as:  
\[c_{x(slide~{}film)}=\frac{2N_{d}~{}\mu_{eff}~{}x_{0d}~{}t}{y_{0d}} \tag{11}\]  
\[c_{y(squeeze~{}film)}=\mu_{eff}~{}t^{3}(b_{d}+b_{t}+b_{s}) \tag{12}\]  
where \(b_{d}=\frac{2N_{t}x_{0d}}{y_{0d}^{3}}\), \(b_{t}=\frac{2N_{t}x_{0s}}{y_{0d}^{3}}\), \(b_{s}=N_{s}x_{0s}\Big{(}\frac{1}{d_{1}^{3}}+\frac{1}{d_{2}^{3}}\Big{)}\), \(\mu_{eff}\) is the effective viscosity of air, \(N_{d}\), \(N_{t}\) and \(N_{s}\) are respectively the total number of comb-drive fingers attached to the drive mass, electrostatic tuning-comb fingers and sensing parallel plates both attached to the sense frame, \(x_{0d}\), \(x_{0d}\) and \(x_{0s}\) are the corresponding overlap lengths, \(y_{0d}\) and \(y_{0t}\) are the gap between the comb-drive fingers and tuning-comb fingers, respectively, while \(d_{1}\) and \(d_{2}\) are the smaller and larger gap sizes of the sensing parallel plates, respectively. The effective air viscosity at given temperature and pressure in the case of slide film damping can be expressed as [21]:  
\[\mu_{eff}=\frac{\mu}{1+2K_{n}+0.2K_{n}0.788e^{-K_{n}/10}} \tag{13}\]  
while the effective air viscosity in squeeze film damping is given as [22]:  
\[\mu_{eff}=\frac{\mu}{1+9.638K_{n}^{1.1}} \tag{14}\]  
where \(K_{n}\) is the Knudsen number and \(\mu\) is the air viscosity at room conditions. The Knudsen number is an estimate of the gas rarefaction effect and it can be defined as;  
\[K_{n}=\frac{\lambda}{d} \tag{15}\]where \(\lambda\) is the mean free air path at a fixed temperature and \(d\) is the air gap thickness in the drive combs, tuning combs and sensing parallel plates, respectively equal to \(y_{0d}\), \(y_{0t}\) and \(d_{1}\). The \(\lambda\) value can be estimated as [23]:  
\[\lambda=\frac{\mu}{P}\sqrt{\frac{\pi K_{B}T}{2m_{air}}} \tag{16}\]  
where \(m_{air}\) is the mass of air, \(P\) and \(T\) are the air pressure and temperature, respectively, and \(K_{B}\) is the Boltzmann constant.  
Figure 3 shows the effect of temperature and pressure variations on the sense Quality Factor value for the proposed MEMS gyroscope, equal to \(Q_{s}=\frac{(m_{d}+ms)\omega_{s}}{c_{y}}\). The variation of the sense Quality Factor can be considered as negligible when the ambient pressure is fixed at 101 kPa and the operating temperature increases in the range of \(-40\)\({}^{\circ}\)C to 100 \({}^{\circ}\)C (Figure 3a), while for a fixed temperature at 25 \({}^{\circ}\)C the sense Quality Factor drastically decreases as the operating pressure increases (Figure 3b).