**4. Electromechanical models for electrostatically actuated RF-MEMS switches: energy considerations**

Mechanical design plays an important role in the RF behavior of MEMS switches because it couples important parameters such as the required actuation voltage (also called pull-in voltage, *Vpull-in*), actuation time, release time, and the appearance of a bouncing phenomenon after release, which delays a complete release of the switch. The pull-in voltage is commonly calculated assuming a static mechanical behavior in RF-MEMS switches. For compatibility of RF-MEMS with low-voltage CMOS and BiCMOS technologies [14], charge-pump techniques are often used [53]. Nonetheless, a current trend is to decrease the MEMS high *Vpull-in*, because charge pump has some limitations given the ever-reducing voltages in CMOS and BiCMOS [54]. One strategy is to dynamically drive the RF-MEMS switches with an input step voltage waveform, which has shown that can decrease the actuation voltage [55]. However, the induced dynamic mechanical behavior can cause an important bouncing phenomenon after release and deeply affect the switch RF/microwave isolation relevant to the RF/microwave behavior, disabling fast-switching applications.

Another trend to achieve fast-switching is to use actuation voltages beyond pull-in. The accumulated electrostatic energy will generate mechanical energy that will be released in the form of mechanical oscillations (bouncing) of the switch membrane [30–35]. To accurately approach RF-MEMS mechanical design, the dynamic behavior of RF-MEMS switches should be considered rather than only static behavior.

The analysis of the electromechanical exchange of energy in the RF-MEMS is an analytical tool that can provide inside knowledge on the required minimum *Vpull-in*. It also takes into account the rebound after release due to the increased actuation voltages [26]. During the switching process, the mechanical membrane or the cantilever undergoes an important deformation. To capture this influence, nonlinear terms should be used in the mechanical model [35].

**Figure 8** shows the schematic of the one-dimensional (1D) lumped-mass model that, combined with classical Newtonian mechanics, can be used to predict the behavior under applied electrostatic forces of an RF-MEMS switch (either a membrane or a cantilever). If air damping is considered the only non-conservative force, then the equation of the motion for the 1D model shown in **Figure 8** is

$$m\_{q\dot{q}}\ddot{y} + b\_{q'}\dot{y} + k\_{q'}y = F\_o \tag{1}$$

where *V* is the applied actuation voltage, *ε*<sup>0</sup>

linear spring with spring constants *k*<sup>l</sup>

*Vpull*\_*in* <sup>≈</sup> <sup>√</sup>

**reconfigurable uniplanar circuits**

follows:

is the dielectric constant, *gi*

RF-MEMS Switches Designed for High-Performance Uniplanar Microwave and mm-Wave Circuits

between the electrodes, and *A* is the area of the electrodes. In Eq. (2), the first term in brackets is the kinetic energy, the second is the potential energy *Um* stored as the mechanical deforma-

processes in the RF switch fabrication (explained in Section 3) can induce an intrinsic residual stress. This effect produces a nonlinear mechanical behavior which can be modeled as a non-

**Figure 9** shows the evolution of the total energy *E* with respect to the position of the contact point of the mechanical structure along the gap when the switch is released from the actuated position. It can be observed that the total energy *E* evolves due to inertia and damping forces

From Eq. (2), the actuation voltage can be obtained since at the point of instability (or switching) ∂*E*/∂*y* = 0. In case of nonlinear mechanical behavior, the resulting expression of *Vpull-in* is as

> 3 \_\_\_\_ <sup>4</sup> *<sup>ε</sup><sup>o</sup> <sup>A</sup>*(

This section presents electrostatically actuated switch configurations which can easily be integrated in reconfigurable uniplanar circuits. All the considered devices were fabricated using the eight-mask surface micromachining process from FBK explained in Section 3. The structures are composed of a 1.8-μm-thick gold layer and reinforced with a 3.5-μm-thick

Eq. (3) is used in the simulations of the fabricated switches, discussed in the next Section.

**5. Electrostatically actuated RF-MEMS switch configurations for** 

\_\_\_\_\_\_\_\_\_\_\_\_ *gi*

*k*<sup>3</sup> *gi* 2 \_\_\_\_

tion of the structure, and the third is the electrical potential energy *Ue*

**Figure 9.** Simulated evolution of the switch total energy after release from the actuated position.

but is bounded by the potential energy curve (*U* = *Um* + *Ue*

 and *k*<sup>3</sup> .

energy accumulated can modify both the rebounds and the isolation time.

is the initial gap

. The different thermal

), and the change on the potential

http://dx.doi.org/10.5772/intechopen.76445

127

<sup>8</sup> + *k*1) (3)

where *meff* is the effective mass of the mechanical structure, *beff* is the damping coefficient (= √ \_\_\_\_\_\_ *<sup>m</sup>eff <sup>k</sup>eff* /*Q*, where Q is the mechanical quality factor), *keff* = *k*<sup>l</sup> + *k*<sup>3</sup> *y*<sup>2</sup> , where *k*<sup>l</sup> and *k*<sup>3</sup> are the spring constants in the direction of the motion of the mechanical structure, and *Fo* is the summation of all external forces applied (i.e., the electrostatic force).

The study of the energy exchange of the mechanical system not only provides the position of the contact point of the MEMS switch but also provides deep insight to the required *Vpull-in*, the maximum rebound height, and the actual actuation/release times. For the lumped model shown in **Figure 8**, the total energy of the system *E* is expressed as *ε* \_*o A*

$$E = \left(\frac{1}{2}m\_{gt}\dot{y}^2\right) + \left(\frac{1}{2}k\_1y^2 + \frac{1}{2}k\_3y^4\right) + \left(\frac{\varepsilon\_\circ A}{2\left(g\_i^-y\right)^2}V^2\right) \tag{2}$$

**Figure 8.** 1D mechanical lumped-mass model of an RF-MEMS switch.

**Figure 9.** Simulated evolution of the switch total energy after release from the actuated position.

One strategy is to dynamically drive the RF-MEMS switches with an input step voltage waveform, which has shown that can decrease the actuation voltage [55]. However, the induced dynamic mechanical behavior can cause an important bouncing phenomenon after release and deeply affect the switch RF/microwave isolation relevant to the RF/microwave behavior,

Another trend to achieve fast-switching is to use actuation voltages beyond pull-in. The accumulated electrostatic energy will generate mechanical energy that will be released in the form of mechanical oscillations (bouncing) of the switch membrane [30–35]. To accurately approach RF-MEMS mechanical design, the dynamic behavior of RF-MEMS switches should

The analysis of the electromechanical exchange of energy in the RF-MEMS is an analytical tool that can provide inside knowledge on the required minimum *Vpull-in*. It also takes into account the rebound after release due to the increased actuation voltages [26]. During the switching process, the mechanical membrane or the cantilever undergoes an important deformation. To

**Figure 8** shows the schematic of the one-dimensional (1D) lumped-mass model that, combined with classical Newtonian mechanics, can be used to predict the behavior under applied electrostatic forces of an RF-MEMS switch (either a membrane or a cantilever). If air damping is considered the only non-conservative force, then the equation of the motion for the 1D

*mefy*¨ + *bef y*̇+ *kef y* = *Fo* (1)

where *meff* is the effective mass of the mechanical structure, *beff* is the damping coefficient

The study of the energy exchange of the mechanical system not only provides the position of the contact point of the MEMS switch but also provides deep insight to the required *Vpull-in*, the maximum rebound height, and the actual actuation/release times. For the lumped model

<sup>2</sup> *k*<sup>3</sup> *y*<sup>4</sup>

) <sup>+</sup> (

*ε* \_*o A*

2 (*gi* − *y*)

<sup>2</sup> *V*<sup>2</sup>

spring constants in the direction of the motion of the mechanical structure, and *Fo*

, where *k*<sup>l</sup>

and *k*<sup>3</sup>

) (2)

are the

is the

*<sup>m</sup>eff <sup>k</sup>eff* /*Q*, where Q is the mechanical quality factor), *keff* = *k*<sup>l</sup> + *k*<sup>3</sup> *y*<sup>2</sup>

summation of all external forces applied (i.e., the electrostatic force).

shown in **Figure 8**, the total energy of the system *E* is expressed as

) + ( \_1 <sup>2</sup> *<sup>k</sup>*<sup>1</sup> *<sup>y</sup>*<sup>2</sup> <sup>+</sup> \_1

\_1 <sup>2</sup> *meff y*˙<sup>2</sup>

**Figure 8.** 1D mechanical lumped-mass model of an RF-MEMS switch.

capture this influence, nonlinear terms should be used in the mechanical model [35].

disabling fast-switching applications.

126 MEMS Sensors - Design and Application

model shown in **Figure 8** is

*E* = (

(= √ \_\_\_\_\_\_

be considered rather than only static behavior.

where *V* is the applied actuation voltage, *ε*<sup>0</sup> is the dielectric constant, *gi* is the initial gap between the electrodes, and *A* is the area of the electrodes. In Eq. (2), the first term in brackets is the kinetic energy, the second is the potential energy *Um* stored as the mechanical deformation of the structure, and the third is the electrical potential energy *Ue* . The different thermal processes in the RF switch fabrication (explained in Section 3) can induce an intrinsic residual stress. This effect produces a nonlinear mechanical behavior which can be modeled as a nonlinear spring with spring constants *k*<sup>l</sup> and *k*<sup>3</sup> .

**Figure 9** shows the evolution of the total energy *E* with respect to the position of the contact point of the mechanical structure along the gap when the switch is released from the actuated position. It can be observed that the total energy *E* evolves due to inertia and damping forces but is bounded by the potential energy curve (*U* = *Um* + *Ue* ), and the change on the potential energy accumulated can modify both the rebounds and the isolation time.

From Eq. (2), the actuation voltage can be obtained since at the point of instability (or switching) ∂*E*/∂*y* = 0. In case of nonlinear mechanical behavior, the resulting expression of *Vpull-in* is as follows:

## 1000ws: 
$$V\_{pull\\_in} \approx \sqrt{\frac{g\_i^3}{4} \frac{\left(k\_3 g\_i^2 + k\_1\right)}{8}}\tag{3}$$

Eq. (3) is used in the simulations of the fabricated switches, discussed in the next Section.
