1. Introduction
Microelectromechanical systems (MEMS) variable capacitors using parallel plates are becoming increasingly popular in radio frequency (RF) applications due to their high-quality factor, fast response time, and straightforward design. Reconfigurable RF front-end modules can be made easy by microelectromechanical system (MEMS) changeable capacitors, which can help them fulfill space limitations by requiring fewer components and meeting the requirements of modern multi-band applications. Moreover, MEMS components have reduced power consumption, greater quality factor, and lower insertion loss when compared to solid-state equivalents such as Schottky varactors and p-n junction diodes [
1]. Applications for reconfiguring MEMS variable capacitors include tunable impedance matching or antenna frequency tuning to maximize antenna efficiency in various environments and reconfigurable filters to increase performance selectivity while requiring fewer components overall [
2]. RF filters [
3,
4], phase-locked loop circuits (PLLs) [
5], voltage-controlled oscillators (VCOs) [
6], and other applications are based on MEMS varactors.
Parallel-plate MEMS variable capacitors, which are usually based on attractive electrostatic actuation, show a nonlinear capacitance-to-voltage (C-V) response despite their relatively high-quality factors [
7,
8]. Because capacitance is dependent on distance, a decrease in plate distance results in an increase in capacitance. The capacitance increases as the actuation voltage increases because the plate travels faster in the direction of the decreasing gap. Such nonlinearities can cause loop bandwidth changes in phase-locked loops (PLLs) and nonconstant phase noises in voltage-controlled oscillators (VCOs), according to several studies. [
9].
Different actuation devices are used for actuation purposes in RF MEMS varactors. The most commonly used actuators are electrostatic actuators due to low power consumption and fast response time.
The pull-in point is an important parameter of the parallel plate RF MEMS varactor. After applying the voltage, an electrostatic force is produced. Due to the produced force, the top plate moves in a downward direction. When the top plate covers one-third of the distance, electrostatic force overcomes the mechanical force. This is the instability point of the parallel plate RF MEMS varactor at which the top plate snaps down and no capacitance tuning occurs after this point.
Several techniques, such as utilizing variable gap spacing variations and introducing nonlinear mechanical components, have been put forth to increase the linearity of MEMS varactors [
10]. One more method, which is detailed in [
11,
12], utilizes a levering structure to change the electrostatic varactor’s closing-gap motion into a rising gap movement [
13], producing a capacitance–voltage connection that is more linear. However, this method shows pull-in limitations. A segmented plate linear capacitor with a noteworthy 99.7% LF was introduced in [
14,
15,
16,
17,
18] but the tuning ratio was limited to 68% tuning ratio. A variable capacitor with a high C-V response and 50% tunability was presented in [
19]. An optimized MEMS variable capacitor was proposed to obtain the high linear capacitance–voltage response; however, this method also shows pull-in limitations [
20]. To preserve improved linearity, the C-V spring stiffening technique is used to increase the varactor’s rigidity as the voltage approaches the pull-in point [
21]. However, this method makes it challenging to determine the control voltage range and capacitance value. One of the techniques of torsion spring structure is utilized to obtain a high-capacitance tuning ratio and linear capacitance–voltage response, but it shows pull-in instability after one-third of the distance [
22]. Recently one of the techniques with repulsive actuation was proposed with high linearity and tunability, but this method works within 1–100 volts, which is very high and needs to be addressed [
23].
The tuning ratio of these variable capacitors was limited due to the pull-in point because the capacitance value of these capacitors changes dramatically when the actuation voltage approaches the pull-in point. In the literature, several attempts have been made to improve the linearity factor of capacitance–voltage response but the tuning ratio was limited due to pull-in instability. It was found difficult to obtain a high tuning ratio with high linearity due to pull-in instability. Some of the techniques made it possible to overcome the pull-in instability and nonlinear capacitance–voltage response, but these devices work at very high actuation voltage [
23]. It was challenging to attain a significant capacitance tuning ratio with the high linear capacitance–voltage response. Therefore, during the past ten years, there has been a great deal of interest in the development of MEMS variable capacitors that offer both high linearity factor and a large capacitance tuning ratio.
To overcome these issues, this study proposed a novel design of an RF MEMS varactor with planar and non-planar structures of the electrostatic actuator to improve the performance of the RF MEMS variable capacitor. The proposed study presents several notable contributions to the research. One of the notable contributions is to avoid the risk of pull-in instability by implementing the non-planar structure of the electrostatic torsion actuator. This is the major limitation of the traditional MEMS varactor. Due to pull-in instability, the tuning ratio of the MEMS varactor is limited and affects the performance of the MEMS varactor. Another noteworthy contribution is to obtain the linear capacitance-voltage response by introducing the novel design of the RF MEMS varactor.
A novel linear RF MEMS varactor design is proposed to obtain a high capacitance tuning ratio and high linearity. An electrostatic actuator is used to connect the actuator part to the capacitor plate by utilizing an s-shaped beam. It is the first demonstration of the s-shaped beam to be used as a connecting link between two parts. Two modes of electrostatic actuators are discussed to obtain a high tuning ratio with high linearity. The proposed design requires low actuation voltage in the planar structure of the electrostatic actuator due to the low spring constant. Parametric optimization of the proposed design is performed by utilizing ANSYS APDL analysis and high-frequency structural simulator (HFSS) analysis to obtain a high tuning ratio RF MEMS varactor with low actuation voltage. In the next part of the research paper, the RF MEMS varactor with an electrostatic actuator having a non-planar structure is implemented to obtain the linear RF MEMS varactor with high tuning ratio by avoiding the risk of instability.
2. Design Methodology
The traditional parallel MEMS varactor exhibits nonlinear behavior due to the rise in capacitance and decreases in the air gap between them when an actuation voltage is applied. The capacitance increases as the actuation voltage increases because the plate travels faster in the direction of the decreasing gap. Thus, a nonlinear capacitance–voltage response results from a more steeply increasing rate of capacitance increment with an increase in voltage. This is the drawback of the traditional parallel MEMS varactor.
There is a linear capacitance–voltage response in the proposed design. The capacitance reduces linearly with applied voltage while the distance between the plates extends as the voltage rises. The traditional decreasing-gap movement of the electrostatic actuator is changed into a rising-gap movement by utilizing an s-shaped beam, which allows for the capacitance to decrease as the actuation voltage increases. Linear response of capacitance–voltage is attained by adjusting the rates at which the capacitor plate rises as the actuation voltage increases and the rate at which the capacitance decreases as the plate rises. The linearity response of the conventional and proposed MEMS variable capacitor is shown in
Figure 1.
In the proposed design, the actuator is utilized to move the capacitor plate in the rising direction linearly with applied voltage. An electrostatic torsion actuator is presented, which contains two torsion springs. An electrostatic torsion actuator with its planar and non-planar form is discussed in this design to obtain a high tuning ratio by avoiding the pull-in point.
The length of the upper electrode and bottom plate is
and
, respectively, while w is the width of the electrode. A cross-sectional view of the electrostatic torsion actuator is shown in
Figure 2b. For the planar mode of the electrostatic actuator, the length of the upper electrode is equal to the bottom plate; however, for the non-planar type of electrostatic actuator, unequal lengths of the top electrode and bottom plate are presented.
The proposed design is composed of two segments: the actuator part and the capacitor part. The electrostatic actuator is joined to the capacitor part by introducing the s-shaped beam. This is the first demonstration of an s-shaped beam to be utilized to connect the actuator part and capacitor part. The planned RF-MEMS variable capacitor is shown in
Figure 2a.
After applying the voltage in the electrostatic actuator, electric force is formed between the plates causing the top plate to move. The s-shaped beam is employed here to drive the capacitor plate in the rising gap direction. So, the upstate capacitance will be minimal due to the rising movement of the capacitor plate. Consequently, the tuning ratio will be maximum.
Figure 3a,b shows the simplified view of the RF MEMS variable capacitor before and after applying the voltage.
The electrostatic torque is made in the actuator when voltage is applied across them. At equilibrium, electrostatic torque becomes equal to the mechanical torque [
24].
where
is the spring constant,
w is the width of the plate in the actuator part,
is the pull-in angle, and
is the maximum pull-in angle.
By expanding the left side of Equation (1) into series form:
The force becomes stronger with an increase in voltage. There will be a point when electrostatic force becomes strong enough to overcome the mechanical force, so plates come in contact with each other at this point. This is the instability between two positions and is called the pull-in point and can be described by using the following equation [
24]:
There is a beam force in the s-shape beam due to which the central capacitor plate moves in the upward direction.
The rising movement of the capacitor plate can be expressed as follows:
where
indicates the length of the s-shaped beam and
is the angle at the pull-in.
It is mainly dependent on the length of the s-shaped beam and the pull-in angle of the actuator plate. Capacitor plate displacement will be maximum for maximum downward travel of the actuator plate and having the higher pull-in angle.
Pull-in angle
can be derived from Equation (2):
The capacitance of the proposed varactor at the initial position is shown as follows:
where
A is the area of the central plate, and the initial gap is represented by
d.
As there is an increase in distance between the capacitor plates by
d1 by applying the voltage, the up capacitance can be calculated as follows:
where
is the increased gap between the plates after applying the voltage which can be shown from Equation (4).
The capacitance tuning ratio can be established as follows [
25]:
For the RF MEMS variable capacitor to have the maximum tuning ratio, downstate capacitance should be maximum, and up-capacitance should be minimum. In the proposed design, minimum up capacitance is achieved by increasing the capacitor plate’s height, which is desirable for a maximum tuning ratio.
4. Simulation Results with Non-Planar Electrostatic Actuator
Optimization of the proposed RF MEMS variable capacitor with a high linearity factor and large tuning ratio is presented in
Section 3. Optimized results are obtained with low actuation voltage.
In this section, an RF MEMS varactor using a non-planar electrostatic actuator is presented. For a non-planar structure of the electrostatic actuator, the length of the top electrode and bottom plates are unequal, as shown in
Figure 17. In this case, after applying a voltage to the electrostatic actuator, the top electrode starts moving down due to electrostatic force. The pull-in angle of the actuator plate is mainly dependent on the length of the fixed electrode. Shorter fixed electrode length is used here as compared to the planar structure. So, lower electrode length leads to a maximum pull-in angle, and it allows for the moveable plate to cover the maximum distance in the downward direction. Hence, the actuator plate covers the maximum distance in the downward direction by avoiding the risk of pull-in stability.
ANSYS APDL is utilized to obtain a capacitance–voltage response by using a non-planar structure of an electrostatic torsion actuator.
Figure 18a,b represents the response of capacitor plate displacement and up capacitance at different actuation voltages. Maximum capacitor plate displacement of 2.022 um is achieved at 47.2 volts by avoiding the pull-in point. The tuning ratio of the proposed varactor is 167%, which is higher as compared to the device by utilizing a planar electrostatic actuator. A non-planar electrostatic actuator is presented to obtain the high tuning ratio by avoiding the pull-in point and allowing for the actuator plate to have maximum travel in the downward direction.
Table 3 represents the comparison of the proposed RF MEMS variable capacitor with the literature.
Up-capacitance in the form of pull-in angle can be represented as follows:
After putting the pull-in angle from Equation (5), capacitance dependence on voltage can be found by the following equation:
To determine the level of linearity of the C–V relationship, a linear factor (
LF), which is defined as the linear correlation between the capacitance and the voltage response, is introduced [
26].
where
n is the number of tests and
Ci is the capacitance that correlates with the voltage
Vi. The linearity factor (
LF) is a function of both 0 and 1, and as the curve becomes more straight, it becomes closer to 1. Equation (11) is used to access the linearity factor between capacitance and voltage, which is 99%.
Table 4 represents the comparison of the tuning ratio for the MEMS variable capacitor with the planar and non-planar electrostatic actuators.
5. Conclusions
The proposed study presents the design and modeling of an RF MEMS variable capacitor with an s-shaped beam. This is the first demonstration of an s-shaped beam to be utilized to connect the actuator part and capacitor part. An electrostatic actuator is employed for actuation purposes. Two modes of electrostatic actuators with planar and non-planar structures are proposed to obtain a high tuning ratio for the RF MEMS variable capacitor. Initially, electrical and mechanical analysis of RF MEMS variable capacitor by utilizing the planar structure of the electrostatic actuator is presented at low actuation voltage. The proposed design by introducing the planar structure of the electrostatic actuator shows a 138% tuning ratio at 15.2 V. High linear capacitance–voltage response is achieved for the proposed MEMS variable capacitor. The optimized RF MEMS varactor shows better performance at 5.3 GHz in terms of isolation losses.
For long-term reliability, some of the analyses were carried out in the proposed research design. There are slices in the central plate to avoid stiction issues, which is essential for the switching cycle of the RF MEMS variable capacitor. Optimization of the proposed RF MEMS variable capacitor with a planar electrostatic actuator is performed to obtain high performance with low actuation voltage. Parametric optimization of the proposed design is essential because when the varactor is being fabricated, a parametric investigation of the device is required to study the impact of changing various parameters. Stress analysis is carried out to evaluate the collapsing issues in the device which guarantees that the structure can sustain the applied voltage without collapsing because maximum stress is less than aluminum’s yield strength.
Another approach of the electrostatic actuator with non-planar mode is presented. RF MEMS variable capacitor with a non-planar electrostatic actuator operates at high actuation voltage as compared to the RF MEMS variable capacitor with a planar electrostatic actuator. This is because the RF MEMS varactor with the non-planar structure of the actuator allows for the moveable plate to travel long in the downward direction by avoiding the occurrence of pull-in instability. A high linearity factor of 99% is obtained in this case. A tuning ratio of 167% is achieved by preventing the occurrence of pull-in instability. The proposed RF MEMS variable capacitor design has applications in various tunable devices like VCOs and PLL.