# Analytical Approach in the Development of RF MEMS Switches
## 2 Analytical Approach to the Development of RF MEMS Switches of Electrostatic Type
### Mathematical Model of the RF MEMS Switch with Capacitive Switching Principle  
The applied bias voltage is generally separated with respect to the RF signal. One way or another, the potential difference is applied between the membrane connected to the ground lines of the coplanar waveguide and the signal line. Under these conditions, an electrostatic force acts on the design of the switch, which is balanced by the elastic force, depending on the stiffness coefficient \(k\) of the elastic suspension elements. In theory, the balance exists until the membrane drops \(1/3\) of the initial air gap. The membrane then collapses onto the pull-down electrode and a lower voltage value is required to hold it in this position. The dynamics of the switch are also affected by the presence of a medium (usually air or nitrogen during sealing to eliminate the influence of humidity), which introduces its own friction and causes damping and change in the switching speed [63; 64; 65]. Currently, there are several models that take into account the detailed effect of damping, including the presence of holes in the membrane [1; 66; 67; 68]. In addition, the damping changes the natural oscillation frequency of the membrane. Another contribution to the movement of the membrane is made by the contact forces of the membrane with respect to the plane of the coplanar waveguide. They are due to the interaction between the two surfaces and the local redistribution of charges. So we should take into account the impact of the forces Van der Waals forces with the effect of the approximation [63; 65]. Both last components are important in the case when the membrane is close to the fixed electrode or is in contact with the dielectric film. However, a phenomenological approach is usually applied, which takes into account the influence of specific most important parameters useful to describe the required mechanical and electrical response. Equation (5) of the balance of forces acting on the design of the switch has the form:  
\[m\frac{\partial^{2}z}{\partial t^{2}}=\Sigma Forces \tag{5}\]  
where \(z\)--the displacement of the membrane; \(m=\rho At\)--the mass of the membrane determined by the material density \(\rho\), area \(A\) and thickness \(t\). Taking into account the given physical quantities, Equation (5) can be rewritten in Equation (6) in the following form:  
\[m\dot{z}=F_{e}+F_{s}+F_{m}+F_{d}+F_{c} \tag{6}\]  
where \(F_{e}=\frac{1}{2}\frac{\partial C}{\partial z}V^{2}\)--the electrostatic force as a consequence of the applied potential difference with the capacitance change along the direction of motion \(z\); \(F_{m}=-k\bigg{[}z-(t_{d}+g)\bigg{]}\)--the mechanical force due to the rigidity of the elastic suspension elements, the opposite direction of the electrostatic force \(F_{e}\); \(F_{s}=-k_{s}\bigg{[}z-(t_{d}+g)\bigg{]}^{3}\)--the nonlinear tensile force of the elastic suspension element [1]; \(F_{d}=-\alpha\dot{z}\)--the damping force due to the action of the medium, which depends on the speed of the membrane \(\dot{z}\) and the damping parameter \(\alpha\), which in turn is associated with the geometric parameters of the membrane and the viscosity of the medium; \(F_{c}\)--contact forces, which can be divided into Van der Waals and surface forces; the first acts as an attractive and the second as a repulsive, with a possible equilibrium position at a given distance from the fixed electrode [63]. The total capacity according to Figure 3 is calculated using Equation (7):  
\[C(z)=\frac{\epsilon_{0}\epsilon_{r}A}{t_{d}+\epsilon_{r}(z-t_{d})},z\in[t_{d}, t_{d}+g] \tag{7}\]  
where \(\epsilon_{0}=8.85\times 10^{-12}\) F\(\cdot\)m\({}^{-1}\)--the dielectric constant of vacuum; \(\epsilon_{r}\)--the relative permittivity of the dielectric material film.