Description

Technical field

The present examples mainly refer to a device (e.g., having a micro electrical mechanical system, MEMS device), e.g. for obtaining a variable capacitor, e.g.. for having a variable capacitance (e.g., linearly variable), as well as to a method for making such a device, and applications of the device and of the method (e.g., phase-locked loop, circuits, and so on).

Prior art

A varicap is known, which is based on the physics of semi-conductor. When applying a reversed bias to a p-n junction, a depletion zone is created at the junction between the electrons (n) and the holes (p) locally creating a capacity. A variation of the amplitude modifies the thickness of the depletion zone and consequently the associated capacitance. This component is showing a strong non-linear behavior regarding the v/C characteristic (v being the voltage at the component and C its capacitance).

Generating undesired distortions and complexifying the designs due to the requirement of compensation modules has a strong impact on the specifications (such as reducing the frequency range), the footprint and the cost of the systems such as voltage controlled oscillators (VCO) [1], phase-locked loop (PLL) [2] or parametric amplifiers [1]. As an example, [3] is corresponding to the design required for a FM demodulator of a PLL system to compensate the distortion resulting of the non-linearities of the integrated VCO. It requires a low pass filter, a circuit for producing harmonic distortion from the output of the low pass filter, and a circuit for combining the produced harmonic distortion with the output of the low pass filter, so as to obtain non-distorted FM demodulated signal. The compensation stage would not be necessary if the varactors were linear.

Another recurrent problem is the undesired transfer of electrons through the depletion zone resulting to the creation of a leakage current through the varactor.

Varactors based on micro-electro-mechanical technology have been developed in the intent of overcoming the parasitic current [4], The capacitance is generated by a pair of interdigitated plate electrodes or comb-drive electrodes, one being fixed and the second being able to move. The required displacement is achieved by a set of electrostatic actuators named comb-drive. The reduction of the leakage current allows a reduction of the consumption and the thermal noise of the systems integrating such a component. However, this device is tuning its capacitance by changing its gap size, taking place at the denominator of the capacitance equation, and thus resulting, like the varicap, to a non-linear behavior [5]

US 2013/154754 A describes a method for tuning an oscillator based on microelectromechanical systems. An acoustic resonator, a tuning circuit and an amplifier circuit are arranged in a loop. The method involves determining the oscillation frequency of the oscillator and modifying a capacitance of the circuit based on the oscillation frequency. In addition, the power level of the oscillator can be adjusted based on the modified capacitance. The disadvantage is that it does not describe how linear behavior of the underlying system technology can be realized.

US 2002/079743 A describes a variable capacitor based on MEMS. The variable capacitor is driven by a variety of MEMS switches. However, no features can be derived from this document that allow a linear control of a variable capacitor.

US 2015262758 A describes an arrangement of two variable capacitors formed out of a substrate. Other components, such as transistors, interconnect technology and insulating layers, can be integrated into the substrate itself. Both capacitors are designed in such a way that a respective lower electrode is connected to the substrate and another electrode at a distance from the lower electrode is movable and arranged above the lower electrodes. The upper electrodes are connected by spring elements to so-called anchors, which in turn are connected to the substrate. The two upper electrodes are electrically and mechanically connected to each other by a connecting element. A suitable geometry and topography can influence the spring stiffness and deflection of these spring elements and thus the linearity of the overall system. However, these features do not adversely affect the adjustability of the linearity of the capacitors.

US 2011/0050251 A1 (DE 10 2009 028 924 A1) discloses an electrostatic actuator.

B. Morgan, G. Reza. “Design and simulation of comb-drive actuators incorporating gray-scale technology for tailored actuation characteristics.” Smart Sensors, Actuators, and MEMS II. Vol. 5836. International Society of Optics and Photonics, 2005. S. 468-476 discloses a grey-scale electrostatic actuator.

B. Morgan, G. Reza. “Vertically-shaped tunable MEMS resonators.” Journal of microelectromechanical systems 17.1, 2008. S. 85-92 discloses a comb-like electrostatic actuator.

Electrostatic actuators are disclosed also in (5) and (6).

An electrostatic actuator has electrodes subjected to a biasing voltage, which generates electrostatic forces between the electrodes. The result is the motion of one electrode with respect to the other one. Hence, an electrostatic actuator can be used for actuating a movement, but cannot provide, at its electrodes, a variable capacitance for a circuit, and cannot operate as a varactor.

Bibliography

[1] J. W. M. Rogers, “The Effect of Varactor Nonlinearity on the Phase Noise of Completely Integrated VCOs,” vol. 35, no. 9, p. 8, 2000.

[2] C. Wiegand, C. Hedayat, and U. Hilleringmann, “Non-linear behaviour of charge-pump phase-locked loops,” Advances in Radio Science, vol. 8, no. C. 5, pp. 161-166, 2010.

[3] M. Naito, “COMPENSATION FOR NONLINEAR VARACTOR CHARACTERISTICS,” p. 6.

[4] P. Monajemi and F. Ayazi, “A high-Q low-voltage HARPASS tunable capacitor,” in IEEE MTT-S International Microwave Symposium Digest, 2005., Long Beach, CA, USA, 2005, pp. 749-752.

[5] A. Gupta et al., “MEMS electrostatic actuator device for RF varactor applications,” US9966194B2, 08-May-2018.

[6] H. Conrad et al., “A small-gap electrostatic micro-actuator for large deflections,” Nat Commun, vol. 6, no. 1, pp. 1-7, Dec. 2015.

Summary

The device (e.g. variable capacitor, varactor, etc.) according to examples may have an actuator actuating a displacement between a first and a second electrodes, the actuator being fed by an input signal (e.g. biasing voltage). The actuator may operate a displacement between the first electrode and the second electrode according to a displacement function which maps, for each input signal, a corresponding displacement between the first electrode and the second electrode. The actuator as such may be in general either linear or non-linear, in the sense that the displacement function may be, respectively, linear or non-linear with the input signal.

In some examples, because of some constraints, the shape of at least one electrode is pre-determined, in the sense that the degree of freedom for choosing its shape is limited (in some cases, there is no freedom at all). For example, if one electrode is to be fixed, in the domain of radio frequencies (RF), the shape of the line is influencing its impedance [Josefsson, Lars G., Bengt T. Svensson, and Lars F. Moeschlin. "Impedance matching stripline transition for microwave signals." U.S. Patent No. 4,494,083. 15 Jan. 1985.]. Hence, the shape of one electrode shall be pre-assigned in some cases.

With or without constraints on the shape of the electrodes, with or without linear actuator, the present examples permit to achieve a device with a response linear with the input signal.

In accordance to an aspect, there is disclosed a device (e.g. varactor) comprising: a plurality of electrodes including a first electrode and a second electrode, wherein the first electrode and second electrode are separated from each other by a gap, wherein the plurality of electrodes form a capacitor having a variable capacitance associated to an overlapping area between the first electrode and the second electrode; an actuator configured to actuate, as an effect of an input signal, a displacement between at least the first electrode and the second electrode, so that the displacement obeys to a displacement function which maps input signals onto displacements, wherein the shapes and the relative positions of the plurality of electrodes are chosen so as to obtain an overlapping function which maps displacements onto overlapping areas, and in such a way that the overlapping function verifies, at least for a determined interval of displacements, at least one of the following conditions: the overlapping function increases, or respectively decreases, less than proportionally in correspondence with displacements at which the displacement function increases, or respectively decreases, more than proportionally; the overlapping function increases, or respectively decreases, more than proportionally in correspondence with displacements for which the displacement function increases, or respectively decreases, less than proportionally; the overlapping function is not constant but decreases, or respectively increases, linearly in correspondence with displacements at which the displacement function increases, or respectively decreases, linearly.

According to one aspect, the overlapping function evolves linearly with the inverse function of the displacement function or an approximated version of the inverse function of the displacement function.

In accordance to an aspect, there is disclosed a device (e.g., varactor) comprising: a plurality of electrodes including a first electrode and a second electrode, wherein the first electrode and second electrode are separated from each other by a gap, wherein the plurality of electrodes form a capacitor having a variable capacitance associated to an overlapping area between the first electrode and the second electrode; an actuator configured to actuate, as an effect of an input signal, a displacement between the first electrode and the second electrode, so that the displacement obeys to a displacement function which maps input signals onto displacements, wherein the shapes and the relative positions of the plurality of electrodes are chosen so as to define an overlapping function which maps displacements onto overlapping areas, and in such a way that the overlapping function evolves linearly with the inverse function of the displacement function, or an approximated version of the inverse function of the displacement function, at least for a determined interval of displacements.

According to one aspect, the overlapping function evolves proportionally with the inverse function of the displacement function or an approximated version of the inverse function of the displacement function. According to one aspect, the overlapping function is the inverse function of the displacement function or an approximated version of the inverse function of the displacement function.

According to one aspect, at least one electrode of the plurality of electrodes is a compensation electrode having a shape which, at least along one electrode compensation portion, obeys to a shaping function which maps coordinates on the compensation electrode onto overlapping heights of the compensation electrode.

According to one aspect, the shaping function of at least one compensation electrode is defined, at least along one electrode compensation portion, as the derivative function of the overlapping function or as a function which is proportional to or linear with the derivative function of the overlapping function.

According to one aspect, the plurality of electrodes includes, further to the at least one compensation electrode, at least one further electrode whose shape, at least in one portion, is constant or exceeds the shape of the at least one compensation electrode.

According to one aspect, the at least one further electrode has a shape which is pre-assigned, and the shaping function of the at least one compensation electrode is chosen so as to verify the condition that the overlapping function is linear with, or proportional to, or the same of, the inverse function of the displacement function or an approximated version of the inverse function of the displacement function.

According to one aspect, the displacement function is quadratic, the shaping function of the compensation electrode is, at least along the electrode compensation portion, or linear with or proportional to where x is a coordinate in a width direction.

According to one aspect, the determined interval of displacements includes: a third displacement associated with a third overlapping area; and a second displacement associated with a second overlapping area , wherein the contour of the at least one compensation electrode obeys to the shaping function between the second displacement and the third displacement, wherein a first displacement is defined which is external to the determined interval, wherein the second displacement is interposed between the first displacement and the third displacement, wherein, between the first displacement and the second displacement, the contour of the at least one compensation electrode does not obey to the shaping function, but has an overlapping area which is the integral of the shaping function between the first displacement and the second displacement.

According to one aspect, the first displacement correspond to a null input signal.

According to one aspect, the compensation electrode includes a non- compensation portion with a constant maximum overlapping height, so that along the non-compensation portion the overlapping area is equal to an area obtained by deriving the overlapping function and exceeding the constant maximum overlapping height.

According to one aspect, the determined interval of displacements includes: a third displacement associated with a third overlapping area; and a second displacement associated with a second overlapping area, wherein the contour of the at least one compensation electrode is defined so that: a first displacement is defined which is external to the determined interval, wherein the second displacement is interposed between the first displacement an the third displacement , wherein, at the first displacement, an overlapping area is present which is an offset area which permits a compensation in the determined interval.

According to one aspect, the overlapping function is , where x is a coordinate on the first electrode and second electrode along a width direction of each electrode, d is the displacement, min (ƒ(x),g(δ - x)) is the function minimum and, for a value of x, it provides the minimum value given by the shaping functions f(x) and g(δ - x) respectively associated to the first electrode and the second electrode.

According to one aspect, configured to operate between a second displacement associated to a second capacitance, and a third displacement associated to a third capacitance larger than the second capacitance, wherein the positions between the second displacement and the third displacement correspond to coordinates, in at least one electrode, which are overlapped to the front side of the other electrode.

According to one aspect, the actuator is configured so that the displacement function is linear with the input signal.

According to one aspect, the displacement function is non-linear.

According to one aspect, the first electrode and the second electrode are different from each other but are shaped so that the overlapping area is linear with the displacement.

According to one aspect, at least one of the first electrode and second electrode is a moveable electrode, wherein the actuator is configured to actuate the displacement by at least translating the moveable electrode.

According to one aspect, at least one of the first and second electrodes is a moveable electrode, wherein the actuator is configured to actuate the displacement by at least rotating the moveable electrode, wherein the displacement is an angular displacement.

According to one aspect, at least one of the first and second electrodes is a moveable electrode, wherein the actuator is configured to actuate the displacement by at least roto-translating the at least one moveable electrode.

According to one aspect, the actuator is a piezoelectric actuator.

According to one aspect, the actuator is an electrostatic actuator.

According to one aspect, the actuator is a thermomechanical actuator.

According to one aspect, the actuator is a nano E-drive, NED, actuator.

According to one aspect, the actuator is a balanced nano E-drive, BNED, actuator.

According to one aspect, at least the actuator is a micro mechanical electrical system, MEMS, actuator.

In accordance to an aspect, there is disclosed a device of any of the preceding aspects, wherein the plurality of electrodes includes a third electrode separated from the second electrode by a second gap, wherein the first electrode and the third electrode are electrically connected in parallel to each other. In accordance to an aspect, the actuator may be configured to operate the displacement between the first and second electrodes so that said variable capacitance (C) is controlled by said actuator.

In accordance to an aspect, said variable capacitance (C) may be effect of the displacement (d) between the first and second electrodes.

In accordance to an aspect, at least one of the first electrode and the second electrode may be electrically non-connected to any of the terminals of the actuator.

In accordance to an aspect, the actuator may include an actuator capacitor which is different from said capacitor having variable capacitance (C).

The actuator capacitor may include first and second actuator electrodes which may be cantilevered at a proximal end to a fixed portion of the device, and which may have a moveable distal end, and/or the actuator further including a main body in insulating material which is a deflectable material, the main body being also cantilevered to the fixed portion and attached to the second actuator electrode along the length of the actuator electrode between the proximal end and the distal end, and/or one electrode among the first electrode and the second electrode being fixed to the distal end, so that, by virtue of a voltage between the first and second actuator electrodes, the first and second actuator electrodes may be subjected to an attractive force which causes said displacement between the first electrode and the second electrode.

According to one aspect, the actuator may be a thermomechanical actuator.

According to one aspect, the actuator may be configured to actuate the displacement by thermomechanical excitation.

According to one aspect, the actuator may be thermomechanical bimorph.

In accordance to an aspect, said capacitor with variable capacitance may have capacitance variable between 500 picofarad and 50 nanofarad.

In accordance to an aspect, said gap may be between 100 nm and 1 cm, and the area of each of the first and the second electrodes is between 1 μm ² and 1 cm ² . In accordance to an aspect, there is disclosed a phase-locked loop, PLL, circuit comprising a phase comparator and a voltage-controlled oscillator, VCO, wherein the VCO includes at least one device (e.g., varactor) according to any of the preceding claims as a frequency determining element.

In accordance to an aspect, there is disclosed a use of the device (e.g., varactor) of as above for tuning the output frequency in a phase-locked loop, PLL, circuit.

In accordance to an aspect, there is disclosed a method for manufacturing a device (e.g., varactor) having a capacitor with variable capacitance, the method comprising: preparing a first electrode and a second electrode separated from each other by a gap, wherein the gap is elongated in a gap direction, so that the capacitance is associated to an overlapping area between the first electrode and the second electrode; configuring an actuator so it actuates a displacement between the first electrode and the second electrode, so that the displacement obeys to a displacement function mapping input signals onto displacements, wherein the capacitance is associated to an overlapping function between at least the first and the second electrode, the overlapping function mapping displacements onto overlapping areas between the first electrode and the second electrode, so that the overlapping function observes at least one of the following conditions: the overlapping function increases, or respectively decreases, less than proportionally in correspondence with displacements at which the displacement function increases, or respectively decreases, more than proportionally; the overlapping function increases, or respectively decreases, more than proportionally in correspondence with displacements for which the displacement function increases, or respectively decreases, less than proportionally; the overlapping function is not constant but decreases, or respectively increases, linearly in correspondence with displacements at which the displacement function increases, or respectively decreases, linearly. In accordance to an aspect, there is disclosed a method for manufacturing a device (e.g., varactor) having a capacitor with variable capacitance, the method comprising: preparing a first electrode and a second electrode separated from each other by a gap, wherein the gap is elongated in a gap direction, so that the capacitance is associated to an overlapping area between the first electrode and the second electrode; configuring an actuator so that it actuates a displacement between the first electrode and the second electrode so that the displacement obeys to a displacement function mapping input signals onto displacements, wherein the capacitance is associated to an overlapping function, mapping the displacement onto an overlapping area between the first electrode and the second electrode, so that the overlapping function is linear with, or proportional to, or the same of, the inverse function of the displacement function or an approximated version of the inverse function of the displacement function.

In accordance to an aspect, there is disclosed a method for manufacturing a device (e.g., varactor) having a capacitor with variable capacitance, the capacitor having: a plurality of electrodes including a first electrode and a second electrode, wherein the first electrode and second electrode are separated from each other by a gap, wherein the plurality of electrodes form a capacitor having a variable capacitance associated to an overlapping area between the first electrode and the second electrode; an actuator configured to actuate a displacement between the first electrode and the second electrode, so that the displacement obeys to a displacement function which maps input signals onto displacements, the method including: setting a constraint on an overlapping function to which the overlapping area shall obey, wherein the overlapping function maps displacements onto overlapping areas, wherein the constraint includes the overlapping function to be linear with or proportional to or the same of the inverse function of the displacement function or an approximated version of the inverse function of the displacement function; and defining the shape and the relative positions of at least one of the first and second electrodes by choosing shaping functions for the first and second electrodes which, for at least one interval of the displacements, verify the constraint.

In accordance to an aspect, the approximated version of the inverse function of the displacement function is obtained by covariance matrix adaptation evolution strategy, CMA-ES.

In accordance to an aspect, there is disclosed a non-transitory memory unit storing instructions which, when executed by a processor, cause the processor to perform a method as above.

Drawings

Fig. 1 shows a device (e.g., varactor) according to an example.

Figs. 2 and 3 (respectively divided among Figs. 2(a), 2(b), and 2(c) and Figs. 3(a), 3(b), and 3(c)) show operational stages of the device of Fig. 2 according to different displacements.

Fig. 4 (divided among Figs. 4(a), 4(b), and 4(c)) illustrates mathematical passages for defining the shape of an element of the device of Fig. 1a.

Fig. 5 shows a mathematical relationship for a component of the example of Fig. 1.

Fig. 6 shows a device (e.g., varactor) according to another example.

Fig. 7 (divided among Figs. 7(a), 7(b), and 7(c)) shows operational stages of the device of Fig. 2a according to different displacements.

Fig. 8 (divided among Figs. 8(a) and 8(b)) shows a device according to an example.

Fig. 9 (divided among Figs. 9(a) and 9(b)) shows an example of an actuator.

Fig. 10 shows another example of an actuator.

Fig. 11 shows a device according to another example.

Fig. 12 shows operational stages of the device of Fig. 11 according to different displacements.

Fig. 13 shows a phase-locked loop, PLL, circuit, including a device as in any of Figs. 1-12.

Electrodes

Figs. 1-4 refer to a device 100 (e.g. varactor) according to examples, e.g., operating as a variable capacitor (e.g., linear variable capacitor). The device 100 may have a micromechanical electrical system (MEMS) component (e.g. a MEMS actuator). The device 100 may include an actuator 140 operating a displacement between two electrodes, so as to obtain a capacitor with variable capacitance (e.g., a capacitance controlled by the actuator 140).

The device 100 may include a plurality of electrodes, e.g., a first electrode 110 and a second electrode 120. The first electrode 110 and the second electrode 120 may overlap with each other, but be separated from each other by a gap G (see in particular Fig. 3(c)) which is extended in a gap direction (hence the overlapping is seen in the cap direction, as in Figs. 1 and 2). In correspondence of the gap G, a dielectric material (e.g., void or another dielectric fluid, such as a dielectric liquid), may be present. In several examples, the gap G remains constant even though the electrodes 110, 120 are displaced (e.g. by relative movement) with respect to each other according to a displacement δ. In general terms, the overlapping area A between the electrodes 110 and 120 changes with the displacement δ, as shown in the sequence of Figs. 2(a)-(c) (e.g. causing the capacitance to accordingly vary).

The electrodes 110, 120 form a capacitor 130 having a variable capacitance C associated to the overlapping area A between the first electrode 110 and a second electrode 120 (e.g., as seen in the gap direction). The variable capacitance C may be an effect of a relative displacement δ between the electrodes 110 and 120 (see below). The capacitance (which expresses the ratio between the voltage v _cap and the charge q) is a positive (or non-negative) value (C=v _cap /q≥0, keeping in mind that V _cap is not the biasing voltage that operates the displacement, but the voltage at the different electrodes).

Each of the electrodes 110, 120 is developed along a width direction and a height direction. In the example of Figs. 1-4, the width direction is considered to be parallel to the displacement direction along with a displacement between the electrodes occurs. The height direction may be perpendicular to the width direction. The height direction and the width direction may form a plane (e.g. perpendicular to the gap direction). The electrodes 110 and 120 may therefore have an extension which is parallel to the plane formed by the height direction and the width direction. The thickness of each electrode 110 and 120 may be much smaller (e.g. at least ten times smaller or at least five times smaller) than its height and/or its width. It is to be noted that, in the present document, the “width direction” and the “displacement direction” are understood as being parallel with each other. Notwithstanding, the “width direction” is imagined as being integral with the electrode: if the electrode moves, the width direction moves integrally with it. Each electrode may be understood as having coordinates along one width direction moving (or resting) integrally with it. It will be shown that, in some examples, the shape of at least one electrode may change along the width direction (e.g., different coordinates in the width direction may be associated to different heights in the height direction, hence defining a so-called “shaping function”). To the contrary, the displacement direction is imagined as being fixed, so as to measure the displacement between the electrodes.

The second electrode 120 (here considered a moveable electrode) is here represented as a regular figure, such as a square, or more in general a rectangle, with two sides 120a and 120c parallel to the height direction and two sides 120b and 120d parallel to the width direction (displacement direction). Other shapes may be notwithstanding chosen, even though the present shape is particularly easy to be manufactured. It will be shown that, in the example of Figs. 1-4, the side 120a is here the most advanced border of the electrode 120 in the displacement d. In examples, the most advanced border 120a of the electrode 120 is always overlapping the first electrode 110 (or at least is the element of the electrode 120 which most overlaps with the first electrode 110). The side 120c of the electrode 120 is here the last border of the electrode 120 in the displacement δ . In some examples, the last border 120c is the last element of the electrode 120 which arrives to be overlapped to the first electrode 110 (e.g. at the maximum displacement δ= δ _max as in Fig. 2(b)), and may be the element of the second electrode 120 which least overlaps the first electrode 110.

The first electrode 110 (here considered a fixed electrode) is here represented as a non-regular figure, at least as a figure which has at least one non regular portion. The first electrode 110 may have a last element 110c, which is here represented as a segment parallel to the height direction (and, in this case, also to sides 110a and 110c of the second electrode 110). The last element 110c is to be intended in the sense of the direction of the displacement δ, although in some examples it may play a role analogous (or at least not completely dissimilar) to the role of the most advanced border 120a of the electrode 120: the last border 110c of the first electrode 110 is always overlapping the second electrode 120 (or at least is the element of the first electrode 110 which most overlaps the second electrode 120). In some examples, at the maximum displacement (e.g., in Fig. 2(c)) the last element 110c of the first electrode 110 is overlapped by the last element 120c of the second electrode 120. The first electrode 110 may include a most advanced element 110a, which may be a segment (e.g. in parallel to the height direction, and/or parallel to any of the elements 120a, 120c, and 110c). The most advanced element 110a may be in contact with a fixed portion 150 of the device 100 (e.g. to permit a cantilevered state of the first electrode 110). In some examples, at the maximum displacement (e.g., in Fig. 2(c)) the most advanced element 110a of the first electrode 110 is overlapped by the most advanced element 120a of the second electrode 120. The first electrode 110 may present a side 110b, which may be parallel to the width direction (displacement direction). In some cases, along the displacement d, the side 120b of the second electrode 120 may move to more and more overlap the side 110b of the first electrode 110. In some cases, when the maximum displacement is reached (Fig. 2(c)), the sides 110b and 120b completely overlap (or at least reach the maximum overlapping).

The first electrode 100 may present a compensation portion 112, which may have a non-regular shape and/or a curved shape. The compensation portion 112 may have a shape which is chosen to compensate for the non-linearities of the actuator 140. As will be exhaustively discussed below, the shape of the compensation portion 112 may obey to a shaping function f mapping coordinates in the width direction (displacement direction) into heights of the first electrode 110. In this case, the contour of the compensation portion 112 evolves according to a function f(x)=x ^-1/2 for compensating the fact that the displacement d obeys to a quadratic behavior (e.g. δ(v)=a*v ² +b*v+c with a≠0), but different shapes may be chosen for different behaviors (which are here named as or associated to different displacements functions).

The first electrode 100 may present a non-compensation portion 114, which may be a regular and/or non-curved and/or segment like portion (e.g., parallel to the displacement direction or width direction, and/or at least one of the sides 120b, 120d, and 110b). The non-compensation portion 114 may be more and more overlapped by the side 120d of the second electrode 120, up to the point (Figs. 2(b) and 3(b)) that it is completely covered (e.g., even while the second electrode 120 is not at the maximum displacement).

At least one of the electrodes 110 and 120 may be a moveable electrode (moving electrode), to permit the displacement. For example, the moveable electrode as shown in Figs. 1 and 2 is the electrode 120. The other electrode (e.g. 110) may be a fixed electrode (static electrode), e.g. stably attached to the fixed part 150 of the device 100 (e.g., in correspondence of the side 110a). In other examples, it is the electrode 110 which moves, while the electrode 120 may be fixed. In further examples, both the electrodes 110 and 120 may move with respect to each other (e.g. both moved by the same actuator or by two actuators controlled in parallel to each other).

The first and second electrodes 110, 120 are therefore subjected to the reciprocal displacement d, which is variable, under the effect of the action of the actuator 140. The displacement d may be along the displacement direction (parallel to the width direction).

As can be seen from the sequence (a)-(c) in Figs. 2 and 3, the displacement d may be, in the present case, a translatory displacement, in the sense the displacement is translatory (e.g. the moveable electrode translates with respect to the fixed electrode). The angular relationships between the electrodes 110 and 120 are substantially maintained during the translatory motion.

It will be shown that the concept of “height”, in the sense of the height of the electrode which actually participates to the capacitance, may also not refer to the concept of “extension of the metal constituting the electrode” in the structural sense. More preferably, it is possible to refer to the concept of “overlapping height”, referring to the portion of the height which is actually overlapped by the other electrode. On the other side, portions of metal which do not overlap and will never overlap can be considered as non-being part of the capacitor (and therefore non-being part of the electrode), as they play no role in the definition of the capacitance.

Figs. 6 and 7 show another example of device 200 (e.g. varactor). The device 200 may have a micromechanical electrical system (MEMS) component or device. The device 200 may include an actuator 240 operating a displacement between two electrodes, so as to generate a capacitor with variable capacitance (e.g., a capacitance controlled by the actuator 140).

The device 200 may include a plurality of electrodes, e.g., a first electrode 210 and a second electrode 220. The first electrode 210 and the second electrode 220 may be separated from each other by a gap (not shown), which is extended in a gap direction (e.g. analogously to the device 100). In correspondence of the gap, a dielectric material, (e.g., void or another dielectric fluid, such as a dielectric liquid) may be present. In several examples, the gap remains constant even though the electrodes are displaced with respect to each other.

The first and second electrodes 210, 220 form a capacitor 230 having a variable capacitance C associated to the overlapping area between the first electrode 210 and the second electrode 220 (e.g., as seen in the gap direction).

At least one of the electrodes 210 and 220 may be a moveable electrode. For example, the moveable electrode is here the electrode 220. The electrode 210 may be a fixed electrode (static electrode), e.g. stably attached to a fixed part 250 of the device 200. In other examples, it is the electrode 210 which moves, while the electrode 220 may be fixed. In further examples, both the electrodes 210 and 220 may move with respect to each other.

The first and second electrodes 210, 220 may therefore be subjected to a reciprocal displacement d, which is variable.

As can be seen from the sequence in Fig. 7, the displacement d may be, in the present case, an angular displacement, in the sense the moveable electrode 220 moves angularly (e.g., in a rotatory fashion) with respect to the fixed electrode 210. In other examples both the electrodes may be movable and may be displaced angularly. In further examples, the moveable electrode(s) may be subjected to both a translation and a rotation: in some cases, the translation is before or after the rotation, in other cases at least for some portions of the displacement, a rotation in the translation occurs simultaneous (roto-translation).

In general terms, the features of the device 200 are analogous to those of the device 100 and are here not repeated. The elements of the device 200 are indicated with the same numbers of the analogous elements of the device 100 with the addition of the offset 100. It is here only to be remembered that: - the displacement d is an angular displacement (e.g., measured in degrees), e.g. evolving in anti-clockwise direction in Figs. 6 and 7;

- the width direction (displacement direction) is an angular direction (e.g., measured in degrees);

- the height direction is a radial direction (with reference to Figs. 6 and 7, the height direction is taken in one position corresponding to the center of rotation caused by the actuator; of course, for angle, the radial direction takes a different linear direction);

- the parallel sides 114, 120b and 120d of the device 100 are substituted by concentric arches 214, 210b and 220d in the device 200;

- the sides 210a, 220a, 210c and 220c are in general not parallel;

- the compensation portion 112 is substituted by a compensation portion 214 obeying to a function f which maps, for each angular coordinate in the with direction (displacement direction), a height in the height direction (which is a radial direction).

It is to be noted that the nature of the actuator may impose trajectories which are not necessarily rotatory or translatory but more in general roto-translatory. Therefore, the displacement could take a non-necessarily strictly angular trajectory or strictly linear trajectory. The mutual positions and the shapes of the electrodes can be chosen accordingly. Further, suitable changes of coordinate are known. In some cases, instead of “radial displacements / radial direction”, it is possible to use the more general expression “exiting displacement / exiting direction”.

Compensation portion

As shown in Figs. 1, 2, 6, and 7, at least one electrode (e.g., 110 and 210, also indicated as “compensation electrode”) may have a non-regular shape. For example, the compensation portion 112 or 212 may be present. In correspondence to the compensation portion 112 or 212, the height of the compensation electrode may be reduced with respect to the height of the other electrode. For example, as seen along the displacement direction (e.g., from right to left in Figs. 1 and 2, or in anti-clockwise direction in Figs 6 and 7) the contour of the compensation electrode may evolve according to a descending function (or at least a non-constant function). Hence, following the displacement direction (width direction), the contour of the compensation electrode, in correspondence of the compensation portion 112 or 212, may decrease.

The contour of the compensation electrode in correspondence of the compensation portion 112 or 212 may obey to a particular function, here called “shaping function” and indicated with f(x) (where x refers to a generic coordinate in the width direction). In the present case, it may be f(x)=x ^-1/2 . Figs. 4(a)-(c) show mathematically the evolution of such a shaping function f (different functions may be defined). The shaping function may be, in some examples, a continuous function (in some examples, it may be continuous at least along the compensation portion).

In some cases, the shaping function f is chosen to be the derivative dA/dd of the overlapping function A(δ) (i.e. the function that maps displacements d onto overlapping areas A between the electrodes 110 and 120 or 210 and 220), or at least linear with or proportional to the derivative dA/dd of the overlapping function A(δ). This is because it has been noted that the overlapping function A(δ) may be seen as the integral of the shaping function f (or at least as a value linear with or proportional with the integral of the shaping function f). (The overlapping function is in general a non-negative function, as the overlapping area is always positive or, in case of non-overlapping, is zero).

The shaping function f(x) of the compensation portion 112 or 212 of the compensation electrode 110 or 210 may be chosen so that, while the most advanced element 120a or 220a of the non-compensation electrode 120 overlaps the compensation portion 112 or 212, the overlapping area A increases in such a way that it compensates for the non-linearity of the movement caused by the actuator 140 or 240.

Notably, the compensation electrode 110 or 210 may present a noncompensation portion 114 or 214, which may be regular (e.g., parallel, or concentric, to the displacement direction): here, the height of the compensation electrode may be the same or higher than the height of the other electrode 120 or 220.

In some cases, the non-compensation portion 114 or 214 of the compensation electrode 110 or 210 may be so that the height of the electrode 110 or 210 in correspondence of the non-compensation portion 114 or 214 is the same or larger than the maximum height of the other electrode 120 or 220. While in Figs. 1 -7 the compensation portions 112 and 212 are shown having a one particular shape (with f(x)=x ^-1/2 ) and one particular position, the contour of the electrodes is not limited to these examples.

At first, the compensation portion 112 or 212 could be present, for example, in the lower part of the first electrode 110 or 210 (e.g., the electrode 110 could be reversed according to a 180° rotation, or, put in another way, the elements 112 and 114 could be exchanged with the element 110b).

Analogously, the compensation portion 112 or 212 could in principle be present on the second electrode 120 (at least when operating between the state 2 in Fig. 2(b) and state 3 of Fig. 3(c)). With reference to Figs. 1 and 2, if the electrode 110 were the moveable electrode and the electrode 120 were the fixed electrode, the same shapes and mutual positions would lead to have an analogous behavior.

It is also not strictly necessary that the compensation portion 112 or 212 of the first electrode 110 or 210 overlaps the most advanced element 120a or 220a of the second electrode 210 or 220: a similar result may be achieved if the second electrode 120 or 220 moves away from the first electrode 110 or 210, even though the contour is not necessarily the same.

Moreover, more than one compensation portion 112 or 212 may be defined, and also at different sides of the an electrode.

Moreover, multiple or different compensation portions may be present in different electrodes, and may cooperate in performing the compensation. With reference to Figs. 1 and 2, a compensation effect is also obtained if, together with the compensation portion 112 in the top part of the first electrode 110, a second compensation portion could be present in correspondence with the side 110b (hence, a curve would substitute the side 110b, at least for the portion at the same widths of the compensation portion 112). Of course, in that case, the curvature of each compensation portion would be different from the curvature of the compensation portion 112 o Figs. 1 and 2.

It will be understood that, in order to achieve the compensation of the non- linearities of the actuator 140 or 240, it is in general necessary to shape and position the electrodes 110 and 120 or 210 and 220 in such a way that the overlapping area A evolves according to a law (“overlapping function A(δ)”) which compensates for the non-linearity of the actuator. It will be shown, for example, that if the displacement function evolves, in regard to the input signal (e.g. biasing voltage), more than proportionally (or less than proportionally, respectively), then the overlapping function may be chosen to evolve less than proportionally (or more than proportionally, respectively). For example, if the displacement function evolves, in regard to the input signal (e.g. biasing voltage), convexly (or concavely, respectively), then the overlapping function may be chosen to evolve, in regard to the input signal (e.g. biasing voltage), concavely (or convexly, respectively). In addition or alternatively, the displacement function may be chosen to be linear with, proportional to, or the same of the inverse function of the overlapping function. The overlapping function may be chosen to be linear with, proportional to, or the same of the inverse function of the displacement function. The overlapping function may be a bijective function.

It is noted that, besides all the other possibilities, the contour of at least one of the electrodes could be pre-determined (or at least having not full degree of freedom). Hence, the compensation portion may also not be uniquely used to compensate linearities of the actuator, but also for compensating pre-assigned shapes of one electrode (in this case, the actuator can also be linear).

The overlapping function may be chosen in such a way that the trajectory of the displacement (e.g. as implied by the nature of the actuator) is taken into account.

It will be shown that the compensation portion 114 or 214 may correspond to the interval of displacements for which it is possible to obtain the linear behavior.

Actuator

The device 100 or 200 (e.g. varactor) may include an actuator 140 or 240. The actuator 140 or 240 may be subjected to an input signal (e.g., biasing voltage v). The actuator 140 or 240 may have a structure having a proximal end 140a or 240a which is attached (e.g. cantilevered) to a fixed part of the device 100 or 200, and a distal end 140b or 240b which is moveable (see, in particular, Figs. 1 and 6). The movable electrode 120 or 220 may be attached to, and move together with, the distal end 140b or 240b. The actuator 140 or 240 may cause the motion (e.g., translating motion, rotating motion, rototranslatory motion, etc.) of the moveable electrode 120 or 220 under the effect of the input signal (e.g., biasing voltage). As such, for the actuator 140 or 240 is not important whether the moveable electrode is the compensation electrode or another electrode.

Fig. 9 (subdivided between Figs. 9(a) and 9(b)) shows an example of the structure of an actuator (e.g., 140 or 240), even though other examples are possible.

An input signal (e.g. biasing voltage v) may be provided by an input signal generator (e.g., biasing voltage generator) 310. The input signal generator 310 may be a part of the actuator 140 or 240 or may be an external element.

Fig. 9 also shows a possible structure 300 of the actuator 140 or 240 (different examples are possible for embodying the actuator 140 or 240). The structure 300 is a structure to be deflected (deflectable structure) according to the input signal (e.g. biasing voltage v) provided to the actuator 140 or 240, hence actuating the displacement d between the electrodes 110 and 120 (or 210 and 220).

The structure 300 may be a microstructure. The structure 300 may be cantilevered to a fixed element. The structure 300 may present a proximal end 300a (e.g. corresponding to the proximal end 140a or 240a), which may be constrained to be fixed (cantilevered) to a fixed part 150 or 250 of the device 100 or 200. The structure 300 may present a distal end 300b (e.g. corresponding to the distal end 140b or 240b), which may be movable. For example, the moveable electrode (e.g. 110 or 210) may be fixed to the distal end 300b, and be displaced of d with respect to the fixed electrode (e.g. 120 or 220) in function of the input signal.

In examples, the structure 300 may comprise a layer stack of two or more layers 301 and 302 (e.g. made of different materials or having different physical characteristics). In some examples, a lateral strain may be generated in at least one of the layers 301 and 302, as an effect of a biasing voltage (input signal) applied to the two layers. This may cause a deflection of the structure 300. The structure 300 (or at least one of the layers 301 and 302) may be a membrane. The structure 300 (or at least one of the layers 301 and 302) may be a beam. The structure 300 (or at least one of the layers 301 and 302) may be a plate. Different layers 301 and 302 may be configured differently.

In general terms, the actuator 140 or 240 may deflect by virtue of an effect of the biasing voltage v (or more in general the input signal). In several examples, for v=0 the actuator may be in straight status (e.g. like in Figs. 1 , 2(a), 6, and 7(a)), while for v>0 the actuator may curve (Figs. 2(b), 2(c), 7(b), and 7(c)). In several examples, the higher the voltage, the higher the curvature.

In Fig. 9(a), the structure 300 is not deflected. This can be position of Fig. 1 or 2a or in state 1 of Fig. 2 or in state 1 of Fig. 7 In Fig. 9(b), the structure 300 is deflected (e.g. as in states 2 and 3 of Fig. 2 and in states 2 and 3 of Fig. 7): while the proximal end 300a remains in place, the distal end 300b is moved according to the displacement δ, causing an analogous displacement δ of the moveable electrode 140 or 240.

The actuator 140 or 240 may be a thermomechanical actuator (thermoresistive actuator). For example, the actuator 140 or 240 may actuate the deflection (and the displacement δ, in turn) by thermomechanical excitation. In examples, the actuator may be thermomechanical bimorph: the two layers 301 and 302 may be constituted of different materials. The different materials may have different coefficients of linear expansion. The layers 301 and 302 may be firmly connected to one another. When the structure 300 is heated, a lateral strain may result, and hence a lateral force of different intensities in both layers. Due to this, the structure 300 is bent, hence producing the displacement δ. The input signal may be a voltage v which heats the layers 301 and 302 (hence obtaining an integrated electrothermal micro heating = usage of the resistive power, e.g. by Joule effect). In general, a thermomechanical actuator is non-linear: in general the displacement δ does not evolve linearly with the input signal (e.g. biasing voltage). The heat power may be of the type P=v ² /R, with R being the resistance of the resistor obtained from the layers 301 and 302, and v a voltage operating as input signal. Hence, when in the present document reference is made to a non-linear actuator, the non-linear actuator may be a thermomechanical actuator (but other examples are in general possible).

In alternative, the actuator 140 or 240 may be a piezoelectric actuator. For example, the actuator 140 or 240 may actuate the deflection (and the displacement δ, in turn) by piezoelectric and/or electrostrictive action. In examples, the actuator may be electroactive monomorph, multimorph (e.g. bimorph) using the transversal effect. Here, a lateral strain or force may be generated within at least one layer 301 by an electrostatic field and by using an electroactive material. This material strain can be actively changed using the electric voltage or the electric field. As a result of this, the structure 300 is bent, hence generating the displacement δ. In general, a piezoelectric actuator can be linear: the displacement δ may evolve linearly with the input signal (e.g. biasing voltage). Hence, when in the present document reference is made to a linear actuator, the linear actuator may be a piezoelectric actuator (but other examples are in general possible).

In alternative, the actuator 140 or 240 may be embodied by a piezomagnetic actuator. For example, the actuator 140 or 240 may actuate the deflection (and the displacement δ, in turn) by piezomagnetic and/or magnetostrictive excitation. The structure 300 may be magnetoactive monomorph or multimorph (e.g. bimorph). Here, a magnetic field and the usage of a magnetoactive material generate a lateral strain within at least one layer 201 . As a result, the microstructure is bent. The piezomagnetic actuator may be a linear actuator. Hence, when in the present document reference is made to a linear actuator, the linear actuator may be a piezomagnetic actuator.

In alternative, the actuator 140 or 240 may be a Nano-E-Drive (NED). The NED actuator may exhibit a quadratic behavior when considering its displacement d being a function of applied voltage V (e.g. δ ~ V ² ). Hence, when in the present document reference is made to a non-linear actuator, the non-linear actuator may be a NED actuator (but other examples are in general possible).

An example of NED actuator is provided by US 9,676,607 B2, which is here incorporated by reference. An example is provided by Fig. 10. The actuator 140 or 240 may have the structure 500, which may include a cantilevered actuator capacitor 530. The capacitor 530 is not to be confused with the capacitors 130 and 230 discussed above, but is to be understood as a capacitor which actuates the displacement δ on the basis of an input signal, which is here the capacitor voltage v, e.g. provided by a generator 510 (which may be either internal to the actuator or external). The capacitor 530 may be formed by actuator electrodes 518 and 520. The actuator electrodes 518 and 520 may be cantilevered, e.g. at its proximal end 500a (e.g. corresponding to 140a or 240a) to a fixed portion 150 or 250 of the device 100 or 200. A distal end 500b (e.g. corresponding to 140b or 240b) may be moveable. One electrode (e.g., the moveable electrode 120 or 220) may be fixed to the distal end 500b, for example. The structure 500 may include a main body 564, in material which is deflectable, and which may be electrically insulant. The main body 564 may also be cantilevered to the fixed part 150 or 250, and may be attached to the actuator electrode 520 along the length of the actuator electrode 520 between the proximal end 500a and the distal end 500b.

Each actuator electrode 518 and 520 may be formed as a repetitive sequence of segments 522, each segment 522 being arcuate or roof-like shaped. For example, even if the electrode mainly extends along the height direction, the actuator electrodes 518 and 520 are, at each segment 522, slanted or arcuate with respect to the height direction. The gap 532 between the actuator electrodes 518 and 520 may also be locally slanted or arcuate. In correspondence with points of local change of direction of the actuator electrodes 518 and 520, spacers 554 (in insulating material) may be present. Each spacer 554 may be physically in contact with both the actuator electrodes 518 and 520.

By virtue of the action of the capacitor voltage v, the actuator electrodes 518 and 520 are subject to an attractive force. The attractive force would in principle cause a relative motion between the actuator electrodes, which would tend to reduce their relative distance. However, by virtue of the particular mechanical structure (slanted or arcuate shape, presence of the spacers 554, gap conformation, and presence of the main body 564), the structure 500 displaces along the displacement direction.

It is noted that the main body 564 influences the bending, mainly due to the increase of the bending stiffness associated to it. In other word, if the main body 564 is thick, then the beam is stiffer and the beam is bending less. Finally, the main body 664 may be absent in some examples.

Other types of NED actuators may be used.

Another kind of actuator may be the Balanced Nano-E-Drive (BNED), using the in-plane Nano-E-Drive technology [6] Such an actuator in general takes advantage of the possibilities of versatile design allowed by its fabrication process. The BNED may be a beam-like actuator that has the possibility, in specific design condition, to exhibit a linear behavior of its bending, as a function of the applied voltage. The actuator includes a series of cells. The integration of a moving conductive element, localized at the point of the actuator that move linearly with the voltage, will allow the creation of a linear variable capacitor. In this configuration, the capacitance is modified by changing area of the overlapping electrodes, resulting to the linear relation between the displacement and the capacitance. The final component has consequently a linear relation between the applied voltage and the capacitance.

In general terms, the actuator may be seen as mapping, through a displacement function δ(v), an interval in the input signal (e.g. [0, V _max ]) onto an interval of the displacement δ. The displacement function δ(v) may be continuous in the interval, and may be bijective (e.g., strictly increasing or strictly decreasing). The displacement function δ(v) may be non-linear (e.g. more than proportional vs less than proportional; convex vs concave; quadratic or having another non-linear behavior) or linear, e.g. according to the nature of the actuator. It will be shown, for example, that the non-linearities may be compensated, e.g. by choosing intelligently the shapes and/or the relative positions between the electrodes.

It is to be noted that the biasing voltage v (or more in general the input signal) is in general independent from the voltage V _cap at the capacitor (e.g., between the electrodes 110 and 120 or 210 and 220). It may be conceived that at maximum one of the electrodes of the capacitor is electrically connected to one of the terminals feeding the biasing voltage v to the actuator (e.g., one of the terminals connected to the layers 301 , 302, 518, 520), as both may be connected to a common mass, for example. However, in general at least one of the electrodes of the capacitor is electrically non-connected to any of the terminals feeding the biasing voltage v to the actuator (e.g., one of the terminals connected to the layers 301 , 302, 518, 520), and/or is electrically independent thereof.

Displacement

Reference is now made to Figs. 2 and 3, and in particular to the sequence of (a), (b), and(c):

Figs. 2(a) and 3(a), state 1 : the input signal (e.g. biasing voltage) may be v=0 (e.g. as in Fig. 9(a) or 10); the displacement δ may be defined as 0; the capacitance is C(0)=C _min (notably, some overlapping area A _min may be present, see below); here, the most advanced border 120a or 220a of the non- compensation electrode 120 or 220 appears to encounter (if seen along the gap direction, e.g. according to the view of Fig. 2) the non- compensation portion 114 or 214 of the compensation electrode 110 or 210 (said in another way, the projection of the most advanced border 120a or 220a onto the compensation electrode 110 or 210 intersects the non-compensation portion 114 or 214 when seen along the gap direction);

Figs. 2(b) and 3(b), state 2: the input signal (e.g. biasing voltage) is v=vo>0 (in other examples it could be v ₀ <0); the displacement δ is δ=δ ₀ >0; the capacitance is C(δ ₀ )=C ₀ >C _min ; here, the most advanced border 120a or 220a (as seen in the displacement direction) of the non-compensation electrode 120 or 220 appears to encounter (if seen along the gap direction) the vertex 113 or 213, defined between the compensation portion 112 or 212 and the non-compensation portion 114 or 214 of the compensation electrode 110 or 210 (said in another way, the vertex 113 or 213 is in the projection of the most advanced border 120a or 220a onto the compensation electrode 110 or 210 when seen along the gap direction);

Figs. 3(c) and 3(c), state 3: the input signal (e.g. biasing voltage) is v=v _max >v ₀ >0 (in other examples it could be v=v _max <v ₀ <0); the displacement δ is δ(v)=δ _max >δ ₀ ; the capacitance is C(δ _max )—C _max -C _min ; here, the most advanced border 120a or 220a of the non- compensation electrode 120 or 220 arrives at the maximum value (as seen in displacement direction, it may be the most advanced border 120a or 220a of the compensation electrode 110 or 210) and/or abuts onto the fixed portion 152 or 252 of the device 100 or 200; the non-compensation electrode 120 or 220 may appear, when seen along the gap direction, to completely cover the compensation electrode 110 or 210.

In some examples, the device may be configured to operate between: δ ₀ (second displacement), associated to C ₀ (first capacitance); and δ _max (third displacement), associated to C _max (third capacitance) With C _max > C ₀ .

It has been understood that, by operating between δ ₀ and δ _max , it is possible to obtain a linear dependency of the capacitance C (obtained between C ₀ and C _max ) on the input signal v (if between vo and vmax), e.g. also in the cases in which the displacement δ follows a non-linear dependency with the input signal (e.g., quadratic, such as δ=a*v ² +b*v+c, with a≠0). The displacement interval between between δ ₀ and δ _max , is indicated with L in Fig. 4(c).

It is noted that, in examples, the displacement interval between δ ₀ and δ _max , is associated to the extension of the compensation portion 112 or 212 along the width direction: for example, in Fig. 4(c), the width of the interval L is δ _max δ ₀ . This is not a coincidence, as the contour of the compensation portion 112 or 212 is shaped so as to obtain the compensation for those displacements.

Displacement function in regard to the input signal

The actuator 140 or 240 may move the at least one movable electrode 120 or 220 according to a displacement function δ(v) mapping input signals v (input signals) onto displacements δ. The displacement function δ(v) can be linear or non- linear with the input signal v, in particular according to the type of actuator 140 or 240 which is used. In some cases, the translation and/or rotation may be linear with the input signal v, for example, because the d displacement δ is linear with the input signal v.

The displacement function δ(v) may be in general bijective: each value v of the input signal is mapped onto one and only one single displacement δ, while each value of the displacement δ may be understood as being (at least mathematically) mapped onto one and only one input signal. Notably, the displacement function δ(v) is an invertible function, i.e. a function which can be inverted: there exists an inverse function which univocally maps the displacement values δ onto values v of the input signal. Where the displacement function δ(v) is quadratic (e.g. δ(v)=a ^* v ² +b ^* v+c, with a≠0), it is only defined in a restricted interval (e.g., between v=0 and v=v _max >0 or between v=0 and v=v _max <0) where it is bijective. More in general, if the displacement function δ(v) is non-bijective but there may be defined a particular interval [v ₁ , v ₂ ] in which the displacement function δ(v) is bijective, then it is possible to use that bijective function in that particular interval. Invertible functions are in general either strictly increasing or strictly decreasing.

The displacement δ(v) may have a linear relation with the input signal v: for example, δ(v) = a*v + b, with v being the biasing voltage (or more in general the input signal), a and b being constant values (a different from zero). Hence, the displacement δ may be of the type Δδ~Δv (“~” indicates proportionality, i.e. the increment of δ is proportional to the increment of v). This may be the case, for example of a linear BNED and of a piezoelectric actuator. The displacement function δ(v) may be a continuous function.

In other examples, the displacement function δ(v) may be non-linear in respect to the input signal (e.g. biasing voltage). For example, it may be δ(v) = a*v ² +b*v+c, with a≠0 constant value, and b and c constant values (in some cases, it may be δ~v ² ). The displacement δ may evolve more than proportionally (e.g. as in δ(v) = a*v ² with a≠0), but in some cases, it is possible to have a displacement function δ(v) which is less than proportional.

With or without a linear displacement function δ(v), it is possible to obtain that the device 100 or 200 has a capacitance C that evolves linearly with the input signal v. The capacitance C may be the capacitance between the electrodes 110 and 120 (or 210 and 220), and is in general not the capacitance of the capacitor 530 (Fig. 5) or of a capacitor at the actuator 140 (or 240).

In general terms, a particular interval [v ₀ , v _max ] of input signals may be bijectively mapped onto the interval L =[δ ₀ , δ _max ] by the displacement function.

It is to be noted that the word “displacement” may also indicate a displacement which is zero. Notably, the origin of the displacement axis (displacement direction) may be anywhere, and a value δ=0 may therefore be defined. Therefore, even δ=0 may be understood as being a displacement (null displacement). Basically, the displacement may be understood as a coordinate value in the displacement axis Definition of a function describing the dependency of the capacitance on the input signal

Some considerations are here provided on the function C(v) which maps the input signals v (e.g. biasing voltages) onto capacitances C. Thanks to the techniques on which the present examples rely, the function C(v) can be advantageously rendered linear.

The function C(v) may be seen as a composite function C(A(δ(v))), obtained from the composition of the following functions: the displacement function δ(v), mapping values v of the input signals onto values d of the displacement (which is in general consequence of the nature of the actuator); the overlapping function A(δ), mapping values δ of the displacement onto overlapping areas A between the first and second electrodes 110 and 120 (or 210 and 220); the relationship C(A) between overlapping areas A and capacitances C.

In many applications (e.g. PLL applications), it is advantageous that the function C(v) be linear (e.g. Δδ~Δv, in the sense that increments Δδ of δ are proportional with increments Δv of v, which can be expressed with C(v)=a*v+b, with a≠0). This is in general guaranteed if all the function δ(v), A(δ), and C(A) are linear. However, this is not always possible. In some cases, the displacement function δ(v) is not linear (e.g., it may be quadratic, e.g. δ=a*v ² ), e.g. by virtue of the nature of the actuator 140 or 240. In other cases, the shape of one of the electrodes is pre- assigned and cannot be modified, e.g. causing a non-linear overlapping function A(δ).

However, it has been understood that, even in the presence of one non- linear relationship, it is possible notwithstanding to compensate the non-linearity by introducing further non-linearities.

It is to be noted that, in some examples, δ(v), A(δ), and C(A) are continuous functions, each mapping an interval at the domain onto an interval at the codomain. Hence, it may be that all the values within an interval of voltages [v ₀ , V _max ] is mapped onto values of an interval of capacitances [C ₀ , C _max ]. C(A) may be in general a strictly increasing function. If δ(v) is strictly increasing (or strictly decreasing, respectively), A(δ) may be strictly increasing (or strictly decreasing, respectively) as well, but if δ(v) is convex (or concave, respectively), then A(δ) may be concave (or convex, respectively).

The capacitance C(A) in function of the overlapping area

In general terms, a capacitor has a capacitance C which is dependent on the overlapping area A and the gap G according to a proportional law:

(“~“ indicates proportionality in this document). With gap G constant, the capacitance C evolves linearly and proportionally with the overlapping area A between the first and second electrodes 110 and 120 (or 210 and 220).

Overlapping function A(δ)

The overlapping area A evolves in function of the (translatory or rotatory or more in general roto-translatory) displacement δ between the first and second electrodes 110 and 120 (or 210 and 220). Hence, an overlapping function A(δ), mapping displacements onto overlapping areas, is defined. For example, in Fig. 2(a), a small displacement causes a small overlapping area and a small capacitance. In contrast, in Fig. 2(c), a large displacement causes a large overlapping area and a large capacitance. In general, even though the overlapping area A is function of the displacement δ (overlapping function A(δ)), this relationship is not necessarily linear. The relationship between A and δ follows the shape of the electrodes 110 and 120 (or 210 and 220), and may also be non-linear. It will be shown that, in some examples, in correspondence to the portion 112 (compensation portion), the overlapping function A(δ) is chosen as being non-linear.

Displacement function δ(n)

As explained above, the displacement function δ(v) may be linear or non- linear, e.g. according to the nature of the actuator 140 or 240.

Dependency of the capacitance on the input signal

The results above are now summarized. From A(δ) and (with G constant), it follows that the capacitance C(δ) can be expressed as function of the displacement (i.e. C=C(δ)), even though it can be linear or non-linear with the displacement. A linear relationship between the displacement d and the overlapping area A (and capacitance C) may be obtained, for example, if the contour of the electrodes 110 and 120 (or 210 and 220) is regular.

It could be imagined that a linear dependency of the overlapping area A on the displacement δ could be somehow advantageous. Further, a regular shape of the electrodes 110 and 120 (or 210 and 220) could be in principle understood as preferable, as it could be imagined that regular electrodes could be easier to be manufactured.

Notwithstanding, it has been understood that such a linear dependency of the overlapping area A (and capacitance C in turn) on the displacement d is not always preferable:

At first, if the actuator 140 (or 240) is non-linear (i.e. the displacement function δ(v) is non-linear), the linearity of the overlapping function A(δ) will cause a non-linear dependency of the capacitance C on the input signal v, which is in general unwanted.

Further, in some cases the shape of one electrode is pre-assigned, and therefore it is not possible to guarantee a linear dependency of the capacitance C on the displacement δ. In the latter case, even if the displacement function δ(v) is linear, the final dependency C(v) will be nonlinear.

However, with the techniques here discussed, it is possible to arrive at rendering the relationship C(v) linear. For example, it is possible to obtain an interval of capacitances [C ₀ , C _max ] mapped linearly from an interval of input values [v ₀ , V _max ] (e.g., through the displacement interval L =[δ ₀ , δ _max ]).

Compensation of non-linear actuator

It has been understood that, even with the displacement function δ(v) being non-linear, it is notwithstanding possible to obtain a linear relationship between the input signal v and the capacitance C, when the overlapping function A(δ) is chosen to be linear with the inverse function of the displacement function δ(v). Hence, even though the displacement function δ(v) is and remains non-linear, its non-linearity is compensated by the non-linearity of the overlapping function A(δ). For example, if the displacement function is δ(v)=v ² (in an interval for which δ(v) is bijective, e.g. between v=0 and v=v _max >0), we may simply define the overlapping function A(δ) as being (with a≠0 constant), since the square root function is the inverse function of the quadratic function. The result will be

Hence, it is chosen to have an overlapping area A (which is not linear with the displacement d) evolving in such a way that it compensates for the non-linearity of the displacement function δ(v), hence implying a linear dependency of the overlapping area with respect to the input signal.

With reference to Figs. 2, 3, and 7, between the state 2 (Figs. 2(b), 3(b), 7(b)) and the state 3 (Figs. 2(c), 3(c), 7(c)), the device 100 or 200 operates in a linear regime.

More in general, if the displacement function δ(v) evolves more than proportionally (or less than proportionally, respectively), then the overlapping function A(δ) may be chosen to evolve less than proportionally (or more than proportionally, respectively). For example, if the displacement function δ(v) evolves convexly (or concavely, respectively), then the overlapping function A(δ) may be chosen to evolve concavely (or convexly, respectively). In addition or alternatively, the displacement function δ(v) may be chosen to be linear with, proportional to, or the same of the inverse function of the overlapping function A(δ).

Compensation of non-homogenous shapes of the electrodes

It has been understood that the same approach may be used in case of one electrode having a shape which does not permit a linear dependency of the capacitance on the input signal v, even in case of the displacement function δ(v) being linear.

For example, the shape of one first electrode (say 120, or 220) may be pre- assigned, without giving the possibility of modifying it (or without giving enough degree of freedom). However, it is possible to counter shape the second electrode (say 110, or 210), in such a way that the overlapping function A(δ) compensates for the pre-assigned shape of the first electrode.

The proceeding follows the same theory to be applied for the case of the compensation of non-linear actuator: the overlapping function A(δ) is simply constrained to be linear with the inverse function of the displacement function δ(v). This will permit to determine the non-pre-assigned shape of the second electrode. Notably, this compensation operates with both linear actuator and non- linear actuator and is, therefore, particularly general.

Shape of the electrodes

It is now possible to exampling how to actually define the shape of at least one of the electrodes 110 and 120 (or 210 and 220) starting from the overlapping function A(δ), after that the overlapping function A(δ) has been constrained to be linear with the inverse function of the displacement function δ(v).

The contour of an electrode may be associated, at least for a particular portion 112 or 212, to a particular shaping function (f(x) for the electrode 110 or 210 and/or g(x) for the electrode 120 or 220; hereinabove reference is mainly made to f(x) for simplicity). The shaping function f(x) may be understood as a function f, mapping coordinates x (taken along the width direction) onto heights h (taken along a height direction perpendicular to the gap direction and the displacement direction or width direction). Hence, heights h=f(x) may be associated to coordinates x in the width direction.

An example is provided by Figs. 1-4(c) showing that, in a portion 112 (referred to as “compensation portion”) a particular contour is chosen to have heights h obeying to a particular shaping function f(x). The compensation portion 112 of the electrode 110 in Figs. 1 and 2 follows the shaping function f(x).

In this case the particular contour is chosen only for the electrode 110, while the electrode 120 has a regular contour. However, in other cases, both the electrode(s) 110 and 120 may have contours which follow particular functions.

It has been understood that the particular contour chosen for the electrode(s) has an effect on the overlapping areas A, and, in return, on the overlapping function A(δ). With reference to Fig. 2(a)-(c), while the electrode 120 is moving from state 1 (Fig. 2(a)) towards state 2 (Fig. 2(b)), the overlapping function A(δ) is linear with the displacement d. However, when the electrode 120 is between the state 2 (Fig. 2(b)) and the state 3 (Fig. 2(c)), the overlapping function A(δ) is not linear anymore with the displacement d, but evolves less than proportionally (e.g. convexly).

Accordingly, it is possible to compensate for the non-linearity of the displacement function δ(v): if we know that the displacement function δ(v) evolves more than proportionally, then we will manufacture the contour of the electrode 110 to be descending along the displacement, hence causing the overlapping area A (and the capacitance C) to evolve less than proportionally, thus compensating for the displacement function δ(v), and vice versa. More in particular, the shaping function f(x) of the electrode 110 (and/or the shaping function g(x) of the electrode 120) may be chosen so as to cause the overlapping function A(δ) to be equal, proportional or linear with the inverse function of the displacement function δ(v).

It has been noted that the overlapping function A(δ) may be seen as being linear with the integral value of the shaping function f(x). This because, in the compensation portion 112, the overlapping area A is the area under the electrode 110. Hence, if the shaping function f(x) is associated to the contour of the electrode, the overlapping area is associated to the integral of the shaping function.

Analogously, the shaping function f(x) may be understood as being bound to (e.g. linear with, proportional to, or equal to) the derivative of the overlapping function A(δ).

These considerations permit to find a technique for choosing the shaping function f(x): the shaping function f(x) may be chosen to be equal, proportional or linear with the derivative of the overlapping function A(δ) (after that the overlapping function A(δ) has been constrained to be linear with the inverse function of the displacement function δ(v)). In the above-mentioned case of δ(v)=v ² , and A (σ) = it may be concluded that and

Hence, the contour of the electrode may be chosen to follow a shaping function f(x) linear with the function x ^-1/2 (i.e. the derivative of the function square root).

Mathematical discussion

Let us first take into consideration an actuator exhibiting a quadratic behavior when considering its displacement function δ(v), i.e. the function of the applied biasing voltage v (e.g. δ ~ v ² ), or more in general the input signal. In this context, designing a specific shape of one or both of the electrodes would permit to compensate the non-linearity. Here, we are developing one approach permitting to obtain the electrode design. As already explained, the displacement of the actuator, and consequently one electrode above the second, is evolving in v ² .

In order to compensate this behavior, it would be of interest to get an overlapping area A, directly defining the value of the capacitance C, evolving with a square root function of the displacement. That statement can be written as follow: with f(x) being the shape of one electrode, x the coordinate in the width direction (parallel to the displacement direction) and α, the parameter permitting to generate the overlapping area A ₀ associated to the capacitance C ₀ for the displacement δ ₀ (state 1, Figs. 2(a), 3(a), and 7(a)) The derivation of this equation permits to obtain an expression of f:

Fig. 4 presents the steps permitting to set up the design from the shaping function f. A priori, as shown in Fig. 4(a), such a shaping function f tends to infinite for x=0. In order to contain the electrode height, it is possible to replace the first section of the curve by an equivalent surface added at the base of the electrode (see passages 1 and 2a between Figs. 4(a) and 4(b) and 4(c)). This surface A _min (state 2, Figs. 2(b), 3(b), and 7(b)) will be associated to the value of C _min , thus imposing the value of the ratio between a fixed C _Max , corresponding to a full overlapping electrodes and C _min , depending of the area of the modified surface. In that manner, a small C _min , requiring a large electrode, will allow a large ratio C _Max / C _min of capacitance for the device 100 or 200. In opposition, a section at a lower f(x), giving a low electrode height, of the curve will be linked to a larger C _min and consequently a lower tuning ratio.

For a maximal height h ₀ of the electrode, it is possible to evaluate the width wo, corresponding to the minimum displacement associated to the minimum voltage V ₀ and the minimum capacitance C ₀ required to reach the linear regime of the varactor: and w _e of the extra area to add at the base of the electrode in order to compensate the loss of surface imposed by h ₀ :

In the same manner, the parameter a can be evaluated in order to fully define the function f(x) : A demonstration of the linearity of the capacitance, being a function of the overlapping area, is presented in Fig. 5.

The present approach permits to compensate any bijective non-linearity. It may as well be associated to the shape of one or both electrodes.

A more general discussion is now presented. In order to compensate the non-linearity, the displacement/voltage behavior should be fitted with a function k. In the case of k being a bijection, the reversed function k ^-1 should then be evaluated.

Resolving the equation: by choosing in the right way the function f and g and shaping the electrodes based on those functions permits to linearize the system over the domain [0 ; δ] . In this equation, min(a,b) is the function minimum. For any value of x, it will report the minimum value given by the functions f(x) and g(δ - x) This applies, for example, when the two electrodes are shaped by a positive function (the height of the electrode) following a common width axis.

Considering the defined function g and k, a discretization of those functions over a given number of points should permit to numerically approximate their value over the entire displacement range authorized by the actuators. For that perspective, the equation given above permit to define an optimization problem over the discreet values y _i =f(x _i ), of the function f. A possible optimization algorithm to achieve that task is Covariance Matrix Adaptation Evolution Strategy (CMA-ES).

In the case of a moving electrode combining a translation and a rotation, the problem can be transposed to a polar coordinate system.

Overlapping areas

By virtue of the discussions above, it may be possible to resume the areas of the electrodes for the devices 100 and 200 (different devices may have different shapes).

It has already been explained that, by virtue of the shapes of the electrodes, the linear behavior of the device 100 or 200 is obtained at least between: the displacement δ=δ ₀ (status 2, Figs. 2(b), 3(b), 7(b)); and the displacement δ=δ _max (status 3, Figs. 2(c), 3(c), 7(c)).

The displacements δ ₀ and δ _max form the interval L=[ δ ₀ , δ _max ] in which the capacitance C evolves linearly with the input signal v.

At status 2 (Figs. 2(b), 3(b), 7(b)), the overlapping area A ₀ is given by the area δ ₀ *h ₀ of the rectangle having basis δ ₀ and height h ₀ , wherein ho is the height of the electrodes 110 and 120 (i.e., height of the sides 120a and 110c). The height ho is also shown in Fig. 4(b). From Fig. 2(b), it can also be noted that do is also the length of the non-compensation portion 114. As explained above, the height ho may be chosen on the basis of dimensional considerations.

At status 3 (Figs. 2(c), 3(c), 7(c), Fig. 4(c)), the overlapping area Amax is given by two contributions: the overlapping area A ₀ = δ _0* h ₀ , corresponding to the non- compensation portion 114; and the area below the compensation portion 112, which the integral value of the shaping function f(x)=1/x ^1/2 between δ ₀ and δ _max .

Between status 2 and status 3, the overlapping function A(δ) is linear with the displacement function δ(v), and the capacitance C(v) evolves linearly with the input signal.

As explained above, the compensation may rely on the fact that both the overlapping areas A ₀ and A _max include the area A _min (see Figs. 4(b) and 4(c)). The area A _min is the overlapping area at the displacement δ=0 (status 1 , Figs. 2(a), 3(a), 7(a)). As explained above and with reference to Figs. 4(b) and 4(c) and in particular the passage 2a between Figs. 4(b) and 4(c), area A _min is the portion, exceeding h ₀ , of the integral of the shaping function f for displacements δ between 0 and δ ₀ : this portion A ₀ is hence put behind δ ₀ , e.g. as a rectangle with height h ₀ and base δ ₀ . As explained above, the length of δ ₀ is w _e , e.g. calculated with formulas above. The parameter α shall be taken into consideration.

Accordingly, at least for v ₀ <v<v _max (i.e. between the statuses 2 and 3), a linear behavior is obtained (e.g. in the interval L between δ ₀ and δ _max ).

Overlapping heights

It is noted that, if an electrode has a portion which will never overlap, for any admitted displacement and for any admitted input signal, with any part of the other electrode, then this never-overlapping portion is useless and will never participate to the capacitance. With reference to Fig. 2, if the height of the second electrode 120 where higher than the height of the first electrode 110 (e.g., if the length of sides 120a and 120c exceeded the height of the side 110c), then there would be defined a useless, exceeding portion of the second electrode 120 which would have no effect at all to the obtained capacitance. Hence, for any height of the second electrode 120 exceeding the height of the first electrode, the capacitance would always be the same for the same displacement.

Analogously, the compensation portion 112 is positioned so that, for at least some displacements (e.g., within the interval L), the compensation area 112 is progressively overlapped by the most advanced element 120a of the second electrode 120. Should the compensation portion 112 be placed, say, at a height over the height of the second electrode 120, then the portion 112 would not have any influence (and would not be a compensation portion at all).

This permits to conclude that the “height” as indicated for the electrode is not necessarily the height of the complete electrode, but is the “overlapping height”, i.e. the portion of the height which is actually overlapped, for a particular displacement, by the other electrode.

In general, terms, the particular compensation can be obtained not only by one single compensation portion and for electrodes aligned at one side (as in Figs. 1 and 2), but also by the particular trajectory taken by the moveable electrode with respect to the fixed electrode (or in any case by the relative trajectory in the relative movement).

For example, more in general than in the examples 100 and 200, the shaping functions f and g may be understood as mapping coordinates onto “overlapping heights”, and not only onto “heights of the electrodes”.

However, in examples such as in Fig. 1 , in which the first electrode 110 has at least one side 110b extending in the width direction (displacement direction) aligned with at least one side of the second electrode 210, and the maximum height of the first electrode 110 is the same of the height of the second electrode 120, the concept of “overlapping height” coincides with the concept of “height of the electrode”.

In general terms, however, portions of metals which do not overlap for any displacement can be also considered as not being part of the electrodes and not being part of the capacitor. This because they do not participate in any way to the definition of the capacitance.

General approach

In some cases, it is possible to have a more general approach. Even without defining the overlapping function A(δ) as being linear with the inverse function of the displacement function δ(v), it is notwithstanding possible to at least alleviate the effects of the non-linearity of the displacement function δ(v).

For example, at least in correspondence with the compensation portion 112 or 212, it is possible to shape the compensation electrode 110 or 210 according to a shaping function f(x) so that: the overlapping function A(δ) increases, or respectively decreases, less than proportionally in correspondence with displacements at which the displacement function δ(v) increases, or respectively decreases, more than proportionally; the overlapping function A(δ) increases, or respectively decreases, more than proportionally in correspondence with displacements for which the displacement function δ(v) increases, or respectively decreases, less than proportionally; the overlapping function A(δ) is not constant but decreases, or respectively increases, linearly in correspondence with displacements at which the displacement function δ(v) increases, or respectively decreases, linearly.

In some examples, if the displacement function δ(v) is concave, the overlapping function A(δ) is convex, and vice versa.

Three electrodes or more

Fig. 8 (divided among Figs. 8(a), 8b), and 8(c), analogous to Figs. 3(a), 1 c(b), and 1c(c), respectively) shows another example of device 400 which is analogous to the device 100, with the difference that here, besides the first and second electrodes 110 and 120, a third electrode 410 is present. The third electrode 410 may be separated from the second electrode 120 by a second gap G2. In some cases, the moveable electrode is the second electrode 120, while the first and third electrodes 110 and 410 are fixed. In alternative, the fixed electrode is the second electrode 120, while the first and third electrodes 110 and 410 are moveable (e.g., one single actuator 300 moves both the first and third electrodes 110 and 410).

In some cases, the first electrode 110 and the third electrode 410 may be electrically in parallel with each other. Here, the capacitance C is obtained as C = C ₁₂ + C ₂₃ , i.e. as the sum of the capacitance C ₁₂ between the first electrode 110 and the second electrode 120 and the capacitance C ₂₃ between the second electrode 120 and the third electrode 410.

Analogously, it is possible to implement the same three-electrode technique to the example of Fig. 6, i.e. in case angular displacement.

It is also possible to make use of more that three electrodes.

Manufacturing method

A method for manufacturing a capacitor with variable capacitance (e.g. the device 100, 200, or 400) may comprise the following steps: separating a first electrode (e.g., 110, 210) and a second electrode (e.g., 120, 220) from each other by a gap, wherein the gap is elongated in a gap direction, so that the capacitance is associated to an overlapping area between the first electrode and the second electrode (e.g., 120, 220); and configuring an actuator so as to move at least one of the first electrode (e.g., 110, 210) and the second electrode (e.g., 120, 220) to actuate a displacement between the first electrode (e.g., 110, 210) and the second electrode (e.g., 120, 220) to obtain, in a displacement direction, according to a displacement function (e.g. δ(v)), mapping input signals [e.g., v] onto displacements [e.g., δ],

A moveable electrode (e.g., 110, 210) among the first and second electrodes may be shaped according to a first shaping function [e.g. f], mapping positions of the first electrode (e.g., 110, 210), in the displacement direction, onto widths of the first electrode (e.g., 110, 210) in a width direction.

The capacitance may be associated to an overlapping function, mapping the displacement [e.g., δ] onto an overlapping area between the first electrode (e.g., 110, 210) and the second electrode (e.g., 120, 220), so that the overlapping function is linear with the inverse function of the displacement function [e.g. δ(v)~v ² ].

More in general, it is possible to define the shaping function such a way that the overlapping function verifies at least one of the following conditions: the overlapping function increases, or respectively decreases, less than proportionally in correspondence with displacements at which the displacement function δ(v) increases, or respectively decreases, more than proportionally; the overlapping function increases, or respectively decreases, more than proportionally in correspondence with displacements for which the displacement function δ(v) increases, or respectively decreases, less than proportionally; the overlapping function is not constant but decreases, or respectively increases, linearly in correspondence with displacements at which the displacement function δ(v) increases, or respectively decreases, linearly.

In general terms, it is possible to: set a constraint on an overlapping function to which the overlapping area shall obey, wherein the overlapping function maps displacements onto overlapping areas, wherein the constraint is that the overlapping function is linear with the inverse function of the displacement function δ(v); and define the shape of the first and second electrodes by choosing shaping functions for the first and second electrodes which, for at least one portion of the first electrode and at least one portion of the second electrode, verify the constraint. For example, setting the constraint on the overlapping function (e.g. when expressed as may include evaluating a formula such as: with k ^-1 (δ) linear with (or proportional to, or the same of) the inverse function of the (linear or non-linear) displacement function δ(v).

Once such a constraint is defined, a family of shaping functions (f(x) for electrode 110 or 210, g(x) for electrode 120 or 220) is a candidate function. The contour of at least one of the electrodes 110 and 120 (or 210 and 220) will therefore be defined (e.g. in the compensation portion 112 or 212) among the family of candidate functions that verify the constraint.

In the present examples, k ^-1 may be the inverse function of a fitting function. It may be obtained using the CMA-ES procedure. The function k ^-1 may be provided by points, and may be understood as representing an approximated version of a function representing the inverse function of the (linear or non-linear) displacement function δ(v) (or to a function linear with or proportional to the inverse function of the displacement function δ(v)).

At least some of the steps may be performed by a processor executing instructions stored in a storage unit. For example dimensioning steps (such as implementing using the equation or finding out the most preferable shapes f and g, etc.) may be performed by the processor.

Linearity, proportionality

In examples above, reference is often made to linear relationships. In general terms, linearity between two generic variables x and y is obtained when a relationship is obtained such that y=a*x+b, with a≠0 constant and b constant. In these cases, Δx~Δy, in the sense that increments in x are mapped onto proportional increments of y. In some cases, the b linear relationships may be also proportional relationships, e.g. with y=a*x, with a≠0.

For example, it is possible to obtain that, at least in correspondence with the compensation portion 112 or 212, C=a*v+b (with generic b and a≠0), even if the displacement function δ(v) is not linear with the input signal c and/or the overlapping function A(δ) is not linear with the displacement δ.

With reference to the sequences of Figs. 2, 3, 7 and 8, in correspondence with the compensation portion 112 or 212 (e.g. for displacements between δ ₀ and δ _max ), it may be understood that, despite the deflection of the structure 300 of the actuator being more than proportional (and the displacement δ increasing more than proportionally), the overlapping area A increases less than proportionally with respect to the displacement δ, hence compensating the more than proportional behavior of the actuator.

Linear Actuator with homogenous shapes of the electrodes

Figs. 11 and 12 show examples of a device 600 (e.g. varactor) having a linear actuator 640 (e.g. a BNED actuator or a piezoelectric actuator), actuating a displacement δ between two electrodes 610 and 620 (one of them can be movable, the other one being fixed, according to the possibilities discussed above). The two electrodes 610 and 620 may have the same area (or at least the broader of the two electrodes may be configured to completely overlap the smaller of the two electrodes). Elements of the two electrodes 610 and 620 of the device 600, if analogous to elements of the electrodes 110 and 120 of the device 100, are indicated with the same number + 500 (i.e. by substituting the first “1” with a “6”).

The displacement δ may be a translator/ displacement or an angular displacement or more in general rototranslatory (in Figs. 11 and 12 an angular displacement is shown). Here, the displacement function δ(v) is linear, hence δ=a+b*v, with a≠0 constant and b constant. The input signal v may be between a minimum V _min (e.g., v _min <0) and a maximum v _max (e.g. v _max >0, e.g., v _min = -v _max ). For v= v _min , the displacement may be δ _min <0; for v=0, the displacement may be 0, and for v=v _max , the displacement may be δ _max >0.

At state 1 (Fig. 12(a)) the overlapping area A may be A=0. At state 2 (Fig. 12(b)) the overlapping area A may be A=A _electr /2, where A _electr is the area of each of electrodes 610 and 620. At state 3 (Fig. 12(c)) the overlapping area A may be A=A _electr .

The two electrodes 610 and 620 may have a regular shape: for example, in respect to the device 100, the so-called compensation portion 112 has a regular surface (e.g. concentric or parallel to the side 610b). Hence, contrary to the devices 100 and 200, there is no a portion of missing area. This is because the shapes of the electrodes 610 and 620 area already regular, and the actuator 640 is also linear: hence, the compensation portion is banal. This is the same to say that the shaping function f and g for the first and second electrodes 610 and 620 are of the type inconstant or g=constant.

Hence, in this case the overlapping function A(δ) decreases, or respectively increases, linearly in correspondence with displacements at which the displacement function δ(v) increases, or respectively decreases, linearly.

Units

It is also to be noted that, where it is here affirmed that the overlapping function A(δ) may be the inverse function of the displacement function δ(v) or an approximated version of the inverse function of the displacement function δ(v), this is valid apart a unit conversion coefficient. This is because:

- the overlapping function A(δ) maps displacements onto areas; while

- the displacement function δ(v) may map voltages onto displacements.

Hence, A(δ) and δ(v) are the inverse function with respect to each other if a conversion coefficient is taken into account.

Some values

Here, some constructional values are provided for a device 100, 200, or 600 according to the present examples.

The area of each electrode may be between 1 μm ² (where μm refers to micrometers) and 1 m ² (or 1 cm in some cases, e.g. in some examples of microsystem technology).

In particular, the height may be between 1 μm and 1 m (or 1 cm in some cases, e.g. in some examples of microsystem technology).

The width (in the width direction or displacement direction) may be between 1μm and 1m (or 1 cm in some cases, e.g. in some examples of microsystem technology).

The gap may be between 100nm and 1cm.

The maximum displacement δ _max may be 1m (or 1 cm in some cases, e.g. in some examples of microsystem technology).

The obtained capacitances C ₀ and C _max (in the linear interval L between δ ₀ and δ _max ) may be 500 picofarad and 50 nanofarad. The maximum biasing voltage (e.g., v _max , associated to δ _max ) applied to the actuator may be 300V when the actuator is an actuator of type Nano-E-Drive.

The numbers provided above may be varied e.g. by ±10%.

Applications

In order to address the correct value of capacitance to the correct driving voltage, the system needs to be actuated in the quasi-static regime, i.e. at a frequency often chosen at 1/10 ^th of the resonant frequency of the system. However, in the case of a capacitance variation associated to a cosines function, the linear varactor can be used at its resonant frequency. In that case, the amplitude of the excitation of the varactor should be properly set up, in order to generate mechanical oscillations that are not larger in amplitude than the oscillation linked to the applied potential V _Max in quasi static.

PLL

A phased-locked loop, PLL, circuit 800 is shown in Fig. 13. The PLL circuit 800 may generate a PLL output signal 804 whose phase is related with a PLL input signal 802. The PLL circuit 800 may comprise a phase comparator 810, and a voltage-controlled oscillator (VCO) 830. A filter 820 (e.g. a loop filter, e.g. a low-pass filter) may be optionally interposed between the phase comparator 810 and the voltage-controlled oscillator 830. The output signal 804 may be provided, as feedback, to the phase comparator 810, through the feedback line 834. In some cases, along the feedback line 834, a frequency divider (not shown) is also present.

The PLL input signal 802 may have a first frequency f ₈₀₂ , while the PLL output signal 804 may have a second frequency feo4, which is intended to be related to the first frequency f ₈₀₂ (e.g., it may be intended to have f ₈₀₄ =f ₈₀₂ , or having a particular relationship thereto, e.g. f ₈₀₄ =a*f ₈₀₂ , with a≠0 constant value, and/or it may be intended to put the output signal 804 in phase with the input signal 802). At the phase comparator 810, the first frequency f ₈₀₂ (and/or the phase of the input signal 802) may be compared to the second frequency f ₈₀₄ (and/or the phase of the output signal 804), so that the signal 812 provides information on the frequency error (or phase error) impairing the output signal 804 with respect to the input signal 802. The signal 812 (or its low-passed filtered version 822) may be input to the VCO 830, e.g. as a control signal. The VCO 830 may generate the output signal 804 at the second frequency f ₈₀₄ , which therefore keeps into account the error information contained in the signal 812.

The VCO 830 may include at least one device as any of the examples above (e.g. a device 100, 200 or 600) to generate the output signal 804 at the second frequency fso4. For example, the device 100, 200 or 600 may be used as having the function of a varactor in the PLL circuit 800 or, in any case as a device controlling the capacitance which tunes the second frequency f ₈₀₄ of the output signal 804. Notably, the signal 822 may be used as input signal v of the actuator 140 or 240 (see above). In general terms, the VCO 830 may use the device 100, 200 or 600 as the frequency determining element for the output signal 804.

Contrary to examples of the prior art, the VCO 830 may avoid the use of at least one of a low pass filter at the output side of the VCO 830, a circuit for producing harmonic distortion from the output of the low pass filter, and a circuit for combining the produced harmonic distortion with the output of the low pass filter. Hence, a non- distorted output signal 804 may be obtained by reducing the number of components of the VCO 830 and of the PLL circuit 800.

Aspects

Some aspects of the present disclosure refer to a technology, of a linear variable capacitor a. Based on the deflectable element b. The deflectable element can be based on NED technology of a one-sided clamped c.The deflectable elements are based on an electrostatic, piezoelectric, thermomechanical principle of operation. At the movable end of the deflectable element, a first electrode is connected to the deflectable element, which thus follows the movement of the deflectable element. d. In addition, a second electrode at a distance from the first electrode, which cannot be moved.

Some aspects of the present disclosure refer to a method describing the operation of a linear varactor. The capacitance of the capacitor is changed as a result of the movement of the first electrode relative to the second electrode. Some aspects of the present disclosure refer to a description associated to the shape of the electrodes and the process of designing those electrode in agreement with the nonlinearity to compensate.

Other implementations

The implementation in hardware or in software may be performed using a digital storage medium, for example cloud storage, a floppy disk, a DVD, a Blue- Ray, a CD, a ROM, a PROM, an EPROM, an EEPROM ora FLASH memory, having electronically readable control signals stored thereon, which cooperate (or are capable of cooperating) with a programmable computer system such that the respective method is performed. Therefore, the digital storage medium may be computer readable.

Some examples according to the invention comprise a data carrier having electronically readable control signals, which are capable of cooperating with a programmable computer system, such that one of the methods described herein is performed.

Generally, examples of the present invention may be implemented as a computer program product with a program code, the program code being operative for performing one of the methods when the computer program product runs on a computer. The program code may for example be stored on a machine-readable carrier.

Other examples comprise the computer program for performing one of the methods described herein, stored on a machine-readable carrier. In other words, an examples of the inventive method is, therefore, a computer program having a program code for performing one of the methods described herein, when the computer program runs on a computer.

A further examples of the inventive methods is, therefore, a data carrier (or a digital storage medium, or a computer-readable medium) comprising, recorded thereon, the computer program for performing one of the methods described herein. A further examples of the inventive method is, therefore, a data stream or a sequence of signals representing the computer program for performing one of the methods described herein. The data stream or the sequence of signals may for example be configured to be transferred via a data communication connection, for example via the Internet. A further examples comprises a processing means, for example a computer, or a programmable logic device, configured to or adapted to perform one of the methods described herein. A further examples comprises a computer having installed thereon the computer program for performing one of the methods described herein. In some examples, a programmable logic device (for example a field programmable gate array) may be used to perform some or all of the functionalities of the methods described herein. In some examples, a field programmable gate array may cooperate with a microprocessor in order to perform one of the methods described herein. Generally, the methods are preferably performed by any hardware apparatus.

The above described examples are merely illustrative for the principles of the present invention. It is understood that modifications and variations of the arrangements and the details described herein will be apparent to others skilled in the art. It is the intent, therefore, to be limited only by the scope of the impending patent claims and not by the specific details presented by way of description and explanation of the examples herein.