This paper presents a novel way of controlling the bouncing phenomenon commonly present in the Radio Frequency Microelectromechanical Systems (RF MEMS) switches using a double-electrode configuration. The paper discusses modeling bouncing using both lumped parameter and beam models. The simulations of bouncing and its control are discussed. Comparison between the new proposed method and other available control techniques is also made. The Galerkin method is applied on the beam model accounting for the nonlinear electrostatic force, squeeze film damping, and surface contact effect. The results indicate that it is possible to reduce bouncing and hence beam degradation, by the use of double electrodes.
Microelectromechanical systems (MEMS) switches have been widely seen as a potential replacement for complementary metal-oxide semiconductor (CMOS) based switches due to their lower leakage current, higher reliability, and better resistance to radiation and hazards [
Input pulse shaping has been also proposed to minimize bouncing. Wong and Lai [
On the other hand, Do et al. [
In previous works, delayed feedback controllers have been proposed to stabilize MEMS resonators and make them more immune to dynamic disturbances [
This paper presents a new method to control bouncing of switches by implementing a double-electrode configuration for actuation and control. Using a lumped parameter and continuous beam models, the new method will be demonstrated by simulations based on several practical case studies.
The modeling of the double-electrode beam is done using both lumped parameter and beam models (Figures
Schematic of a double-electrode beam configuration.
Equivalent lumped mass model representation.
The equation of a motion of a switch sandwiched between two electrodes can be written as [
Note that subscript
For the beam model, the Euler Bernoulli equation is solved using the Galerkin method to obtain a reduced order model [
To obtain the reduced order model, the following beam equation is assumed:
Using the Galerkin method, the displacement is assumed to be of form
For the free mode (when the beam is not in contact with the substrate), the following boundary conditions are used:
The system was modeled both with the case of a constant damping coefficient and in the case of a nonlinear squeeze film damping [
The nonlinear equation governing the pressure variation
Solving the Reynolds equation and assuming the pressure across the width (
Partial electrode configuration.
The forcing term for the double-electrode system is given in (
The nondimensional equation of motion then becomes
To avoid a complex Taylor expansion of the terms in the denominator which may require up to 20 terms, we multiply both sides by the common denominator,
Simulations were run for both the lumped parameter model and the beam model. The first target of the simulation is to obtain the bouncing dynamics of the switch using (
To simulate bouncing, the damping term and the stiffness term are varied when the beam comes into contact with the substrate. The parameters used to model the beam are taken from [
The parameters used in the lumped parameter model based on the case study done in [
Parameter | Value |
---|---|
Natural frequency free ( |
2.1 × 104 [Hz] |
Damping free ( |
2 × 10−2 |
Gap ( |
3.8 × 10−6 [m] |
Maximum distance to travel | 2.6 × 10−6 [m] |
Constant |
1.9 × 10−11 [F⋅m/kg] |
Effective spring constant | 42 [N/m] |
Natural frequency during contact ( |
6.1 × 104 [Hz] |
Damping ratio during contact ( |
2 × 10−1 |
Figure
Uncontrolled bouncing and velocity characteristics.
Tuning to obtain the best possible result (i.e., lowest landing velocity and landing time) is a tedious process. However this process can be made simpler by discussing the effect of some key parameters that affect the switch dynamics. These include the actuation voltage, the control voltage, the pulse width (duration of application of the control voltage), and pulse position (start of the control voltage). Also, it is important to point out that most of these parameters are influenced by each other. For example, variation in the pulse width will not allow for any noticeable change in the landing time and landing velocity if the application of the control pulse occurs after the first contact between the tip and the substrate.
The values of actuation, control voltages, and the pulse width are given in Table
Parameters used for studying pulse position using the lumped model.
Actuation voltage | Control voltage | Pulse width |
---|---|---|
30 V | 20 V | 14 |
From Figures
Landing time versus pulse position.
Landing velocity versus pulse position.
Actuation voltage and control voltage changes are also studied. The parameters used in the simulation are given in Table
The parameters used for studying the actuation voltage.
Pulse position | Control voltage | Pulse width |
---|---|---|
17 |
20 V | 14 |
Figures
Landing time versus actuation voltage.
Landing velocity versus actuation voltage.
However, further reduction in velocity can be achieved for negligible change to landing times by applying the control voltage pulse in a noncontinuous fashion (as in Figure
Noncontinuous application of actuation and control signal.
These signal values represent the application of the control voltage and actuation voltage for the specified time periods. Using a similar application, the landing velocity curve is found to be as shown in Figure
Landing velocity versus actuation voltage for the lumped parameter model for the case of noncontinuous actuation voltage.
It is seen in Figure
Landing times for different actuation voltages for constant control pulse parameters.
From Figure
For lower voltage, after the activation of the control voltage, the actuation voltage is reapplied. Due to the small distance between the beam and the substrate, the actuation voltage is the dominant force and hence reaccelerates the beam rapidly. This can be used to explain the presence of a minimum point in Figure
Control voltage follows a similar pattern to actuation voltage with the exception that increasing the control voltage has an opposite effect to that of the actuation voltage.
By tuning the parameters presented earlier (such as in Table
Displacement and velocity profiles for tuned system using the double-electrode scheme.
To assess the efficiency of the double-electrode control scheme, we compare its performance with two different types of actuation waves in the literature: the dual pulse actuation waveform (DP) and the exponentially increasing dual pulse waveform (EDP) (Figure
(a) Dual pulse actuation waveform (DP). (b) Exponentially increasing dual pulse actuation waveform (EDP).
(a) Modified dual pulse actuation waveform (MDP). (b) Modified exponentially increasing dual pulse actuation waveform (MEDP).
Table
Parameters for the simulated DP and EDP waves.
|
|
|
|
|
|
---|---|---|---|---|---|
DP | 17.4 | 80 | 40 | — | — |
MDP | 17.4 | 85 | 40 | 53.1 | 19.9 |
EDP | 26.4 | 80 | 40 | — | — |
MEDP | 29 | 80 | 40 | 50 | 31.405 |
Using the data in Table
Displacement profile for DP and MDP.
From Figures
Velocity profile for DP and MDP.
For the EDP case in Figures
Displacement profile for EDP and MEDP.
Velocity profile for EDP and MEDP.
Double-electrode model using a continuous beam model is developed to improve the accuracy of the results. As such, the parameters of the model are taken from [
Parameters used for the beam model [
Parameter | Value |
---|---|
Length | 70 [ |
Width | 30 [ |
Thickness | 2 [ |
Gap ( |
1.5 [ |
Max distance ( |
0.75 [ |
Electrode position | 21 [ |
Young’s modulus ( |
207 [GPa] |
Density | 8900 [kg/ |
The bouncing profile obtained using the beam model for actuation voltage 280 V and control voltage 0 V (uncontrolled response) is given in Figure
Uncontrolled bouncing characteristics.
When varying the application time of the pulse, Figures
Landing time versus pulse position.
Landing velocity versus pulse position.
As in the case of lumped parameter system, the beam model shows similar results for landing time and landing velocity. For Figure
Similar to the lumped parameter model, it is possible to tune the system to get reduced overall bouncing and landing velocity in the beam. Tuning is a dynamic process and requires use of the results derived earlier in the form of the models response to different parameters. Table
Parameters used for obtaining tuned system.
Actuation voltage | Control voltage | Pulse width | Pulse position |
---|---|---|---|
300 V | 200 V | 0.13 |
0.36 |
Tuned displacement profile for the beam model.
Table
From Figure
This paper discusses a method that can allow the users to actively tune their devices to obtain the required landing time and velocities up to the limitations allowed by the physics of the system. The dual electrode configuration hence has practical use as it may be easily tuned as compared to other techniques that may require extensive simulations for each switch, as switches even in the same wafer may have wide variations in the parameters. The switch used for the simulations in this paper is a double-electrode switch. Similar to the single-electrode switch, one electrode is used for the actuation of the switch towards the gate. However, an extra electrode (the control electrode) is used to decelerate the beam to control bouncing. The lumped model for the double electrode was simulated and the dependence on the switch landing time and velocity has been modeled. Furthermore, the beam model for the system was also derived to seek more accurate results. Simulations done on the system confirmed that the switch of this form may be used to reduce bouncing and high landing velocities.
Effective mass/kg
Damping coefficient [Ns/m]
Effective stiffness/[N/m]
Permittivity of free space/[F/m]
Area of electrode [m2]
Substrate-beam gap/m
Young’s modulus/Pa
Moment of inertia/m4
Density/[kg/m3]
Beam deflection/m
Mode shape
Modal coordinate
Electrostatic constant =
Damping ratio.
There is no conflict of interests among the authors. This paper is based on this thesis titled “Control of Bouncing in RF MEMS Switches Using Double Electrode” written by the first author.
This work has been supported by KAUST.