# Analytical Approach in the Development of RF MEMS Switches
## 2 Analytical Approach to the Development of RF MEMS Switches of Electrostatic Type
### Coefficient of Stiffness of Elastic Suspension Elements  
Various variants of elastic elements of membrane fastening are known, which are used to reduce the coefficient of elasticity. The most frequently used variants of elastic elements of the membrane suspension are shown in Figure 5.  
The resulting value of the stiffness coefficient \(k\) of the elastic elements fastening and membrane is determined by Equation (10):  
\[k=k^{{}^{\prime}}+k^{{}^{\prime}} \tag{10}\]  
where \(k^{{}^{\prime}}\)--the effective stiffness coefficient of the elastic suspension; \(k^{{}^{\prime}}\)--the coefficient of residual stresses in elastic suspension defined by Equation (11):  
\[k^{{}^{\prime}}=4\left[8\sigma(1-\
u)w\left(\frac{t}{l}\right)\right] \tag{11}\]  
where \(\sigma\)--the tensile strength; \(\
u\) - the Poisson's ratio; \(w,t,l\)--the width, the thickness, the length of elastic suspension. The effective stiffness coefficient \(k^{{}^{\prime}}\) depends on the type of elastic suspension fastening:  
(a) Fixed-fixed flexure [69]:  
\[k^{{}^{\prime}}=4 Ew\left(\frac{t}{l}\right)^{3} \tag{12}\]  
Figure 5: Various support beams used to reduce the spring constant. (**a**) Fixed–fixed flexures. (**b**) Crab-leg flexures. (**c**) Folded flexures. (**d**) Serpentine flexures. 2. Crab-leg flexure [70]: \[k^{{}^{\prime}}=\frac{4 Ew\bigg{(}\frac{t}{l}\bigg{)}^{3}}{1+\frac{l_{s}}{l_{c}} \bigg{[}\bigg{(}\frac{l_{s}}{l_{c}}\bigg{)}^{2}+12\bigg{(}\frac{1+v}{1+(\frac{v }{l})^{2}}\bigg{)}\bigg{]}}\approx 4 Ew\bigg{(}\frac{t}{l_{s}}\bigg{)}^{3}\] (13) for \(l_{s}\gg l_{c}\). 3. Folded flexure [71]: \[k^{{}^{\prime}}\approx 2 Ew\bigg{(}\frac{t}{l}\bigg{)}^{3}\] (14) for very stiff truss. 4. Serpentine flexure [71]: \[k^{{}^{\prime}}\approx\frac{48 GJ}{l_{a}^{2}\bigg{(}\frac{GL}{E_{L}}l_{a}+l_{b} \bigg{)}n^{3}}\] (15) for \(n\gg\frac{3l_{b}}{\frac{3l_{b}}{\frac{3l_{b}}{\frac{3l_{b}  
The available design of an integrated RF MEMS switch contains substrate with a coplanar transmission line located on it, consisting of two grounding lines and a transmission line. Membrane is made in the form of a plate with perforation, located above the dielectric film with air gap between them. The dielectric film is deposited on the surface of the fixed down electrode, which integrated in the transmission line. Membrane is fixed on the anchor areas (1 level) by means of elastic suspensions consisting of elastic beams in the form of a meander. The fixed upper electrode is made in the form of a plate with perforation used to pull-up the membrane. Movable comb structures and two side electrodes are made in the form of comb structures designed to attract the movable comb structures in the case of action on the construction switch of the external acceleration in the negative or positive direction of the X, Y, Z axes, vibrations or stiction of the membrane in the down or up position. The switch operates as a variable capacitor with two states depending on the position of the membrane. In the neutral position of the membrane, the capacitance between the membrane and the transmission line is small and the signal flows freely to the output of the transmission line. When a control voltage is applied between the membrane and the fixed down electrode, the charge redistribution occurs, which leads to the appearance of an electrostatic force between them, independent of the polarity of the applied voltage. The electrostatic force causes the membrane to fall to a stationary one and since the structure is bent, tensile forces appear in it seeking to return it to its original position. When the applied control voltage reaches the threshold value, the tensile forces cease to balance the electrostatic forces and the membrane drops sharply to the fixed down electrode. In the down position of the membrane the capacitance value increases sharply and the signal coming to the input of the transmission line shunts to the ground lines. Materials and topological dimensions of the elements of the integrated RF MEMS switch design are presented in Tables 1 and 2.