MEMS oscillator

A piezoresistive MEMS oscillator comprises a resonator body, first and second drive electrodes located adjacent the resonator body for providing an actuation signal; and at least a first sense electrode connected to a respective anchor point. The voltages at the electrodes are controlled and/or processed such that the feedthrough AC current from one drive electrode to the sense electrode is at least partially offset by the feedthrough AC current from the other drive electrode to the sense electrode.

This application claims the priority under 35 U.S.C. §119 of European patent application no. 10172068.8, filed on Aug. 5, 2010, the contents of which are incorporated by reference herein.

FIELD OF THE INVENTION

This invention relates to MEMS oscillators, for example for generating clock signals or reference frequency signals.

BACKGROUND OF THE INVENTION

Electronic components inside flat objects, such as tags, bank cards or ID cards require a low profile of typically less than 0.5 mm. It is anticipated that flat, low-cost, and low power real time clocks (RTCs) and frequency reference oscillators (RFOs) are required residing inside these flat objects for future applications in the e-security, pharmaceutical, and food industry. An oscillator comprises a resonator and a feedback amplifier circuit, which are connected in a closed feedback loop. State-of-the-art RTCs make use of quartz crystal resonators with a height profile that typically exceeds the allowed sub-mm specification needed for their incorporation into flat products like cards, tags, and sheets of (value) paper. The main reason for this is because the packaging technology being used to encapsulate the quartz crystal does not lend itself well to miniaturization.

Furthermore, quartz resonators cannot be integrated easily on a Si chip. Therefore, the integration of a complete oscillator consisting of the crystal and amplifier cannot be realized on a single chip and further prohibits the miniaturization of RTCs and RFOs. In contrast, a MEMS resonator can be processed and packaged using surface micro-machining techniques and can be integrated with the amplifier circuit to form a very small form-factor oscillator.

Surface micro-machining is a technique whereby freestanding and moveable structures are made on top of a substrate using thin film deposition and etching techniques. In this way, both the resonator and its package can be processed on top of e.g. a Si wafer. The packaged resonator has a height of only several thin films measuring about 10 μm in total thickness. Furthermore, surface micro-machining allows for the definition of many thousands of packaged resonators onto a single wafer without making use of expensive assembly steps. The production cost associated with micro-machining decreases when the area occupied by a single device decreases. In this way, miniaturization of the resonator also has cost advantages. For quartz resonators, the production cost increases when the size of the resonator decreases as a result of the assembly-like production process that is being used.

MEMS-resonator based oscillators thus allow for low profile and low cost clocks and oscillators. However, they do not necessarily consume little power. Piezoresistive MEMS resonators require a body DC bias-current & a DC polarization voltage for their electrode. The body of a piezoresistive resonator is fed a DC-current. By applying an AC-signal to an attached but isolated electrode (‘gate’) the resistivity of the body is modulated, so a signal-voltage develops. This takes place in a narrow frequency-region for proper operation.

Parasitic capacitances resulting from the physical layout cause undesired conduction paths through the structure, for example a feedthrough path from the drive electrodes to the sense electrodes. These limit the performance of the oscillator and increase the power consumption.

Low power consumption and high performance are often conflicting requirements: to obtain amplitude selectivity that exceeds the signal transfer caused by the inherent capacitive feedthrough path, more bias and/or polarization are required. This increases power consumption. Power consumption can be lowered and/or amplitude selectivity can be increased if the capacitive feedthrough path can be eliminated or at least be reduced.

SUMMARY OF THE INVENTION

According to the invention, there is provided a piezoresistive MEMS oscillator comprising:

a resonator body which comprises a resonator mass part, a piezoresistive sensor part, and at least one anchor point;

first and second drive electrodes located adjacent the resonator body for providing an actuation signal;

at least a first sense electrode connected to a respective anchor point; and

control circuitry for controlling the voltages applied to the electrodes and for processing the signals measured at the first sense electrode to derive the oscillator output signal,

wherein the control circuitry is adapted to control the voltages applied to the electrodes and process the signals measured at the sense electrode such that the feedthrough AC current from one drive electrode to the sense electrode is at least partially offset by the feedthrough AC current from the other drive electrode to the sense electrode.

This oscillator design uses the signals applied or measured to provide offsetting of the effect of parasitic capacitances in the resonator. In particular, the voltage coupling caused by the capacitance between the two drive electrodes and the sense electrode is balanced. This balancing means that feedthrough currents through these parasitic capacitances at least partially cancel each other out.

The invention can be applied to piezoresistive oscillators generally. In one example, the resonator body comprises an elongate structure having a pair of parallel connection arms between first and second end regions, wherein the connection arms each have an anchor point;

the first and second drive electrodes are located adjacent the first and second end regions, respectively, for providing actuation signals;

a second sense electrode is provided, with the first and second sense electrodes connected to the anchor points of the first and second connection arms, respectively.

This defines a double-beam dog-bone shaped resonator.

In another example, the oscillator comprises comprising a ring having at least two piezoresistive portions defined at inner and/or outer portions of the ring between anchor points, with a sense electrode coupled to each anchor point, wherein at least first and second drive electrodes are located adjacent the inner and/or outer rims of the ring, for providing actuation signals; and at least one anchor point is provided with a sense electrode connected thereto.

This ring can have at least two piezoresistive portions with a sense electrode coupled to each anchor point, wherein at least first and second drive electrodes are located adjacent the piezoresistive portions, for providing actuation signals. Four anchor points can be provided, each having a sense electrode connected thereto.

In one example, the control circuitry is adapted to apply voltages of equal magnitude and opposite polarity to the drive electrodes with respect to a reference voltage. These opposite voltages thus result in a cancelled voltage offset at the anchor location.

The control circuitry can be adapted to apply voltages with opposite polarity to the drive electrodes with respect to a reference voltage, weighted in magnitude with respect to the capacitance between the respective drive electrode and the sense electrode. This provides a way to offset the effect of the parasitic capacitances even if they have different magnitude.

The control circuitry can be adapted to provide a DC current flow through the piezoresistive spring member (this can be achieved by applying voltages to the sense electrodes or using a current source to provide a current flow between the sense electrodes) and to measure a signal at the sense electrode (the measured signal can be a voltage or a current flow depending on the drive signals applied).

This provides voltage-actuated sensing or current-actuated sensing to derive the resonator output signal.

In another arrangement, the control circuitry is adapted to provide a current flow between two sense electrodes and to measure a signal difference between two sense electrodes, such as a voltage difference.

For example, this can provide current-biased sensing to derive a resonator differential output voltage signal.

The control circuitry can be adapted to apply equal voltages to the drive electrodes and to measure a differential signal between the two sense electrodes. This provides a differential sensing method, again to provide cancellation of the effects of parasitic capacitances, but while still enabling equal voltage profiles to be applied to the drive electrodes.

The differential sensing method can be combined with the opposite drive signal approach.

The invention also provides a method of controlling a piezoresistive oscillator, comprising:

applying voltages to drive electrodes located adjacent a resonator body which comprises a resonator mass part, a piezoresistive sensor part, and at least one anchor point;

processing the signal at a sense electrode which is connected to an anchor point of the resonator body,

wherein the voltages applied to the drive electrodes and the processing of the signal measured at the sense electrode are selected such that the feedthrough AC current from one drive electrode to the sense electrode is at least partially offset by the feedthrough AC current from the other drive electrode to the sense electrode.

DETAILED DESCRIPTION OF EMBODIMENTS

FIG. 1shows a known piezoresistive MEMS resonator, which is described in more detail in WO 2004/053431.

The resonator1shown inFIG. 1comprises a substrate10which is a silicon wafer. Alternatively, substrate10may be a gallium arsenide wafer or it may comprise any other semiconducting, metal or dielectric material. For resonators1designed for operation at frequencies above 10 MHz it is advantageous to use a substrate10comprising a dielectric such as, e. g., glass, because this reduces the loss of electromagnetic energy dissipated in the substrate.

The resonator1further comprises an electrically conductive resonator element20having two parallel connecting elements20a,20b. The resonator extends in a longitudinal direction having a length l, for operation in bulk mode. It is attached to the substrate10via support elements21and22which are connected to anchor elements23and24, respectively. The anchor elements23and24are affixed to the substrate10as is shown inFIG. 2C. The resonator element20and the support elements21and22are free from the substrate10except for the connection via the anchor elements23and24.

The resonator may be manufactured using a technique well known in the field of micro electromechanical systems (MEMS). In short, the substrate10is first provided with an oxide layer11on top of which a silicon layer12is deposited, shown inFIG. 2A.

The silicon layer12is covered by a photosensitive resist, not shown, which is patterned by, e.g. lithography. The patterned resist is then developed yielding the surface areas of the resonator element20, the support elements21and22, the anchor elements23and24, and the actuator30shown inFIG. 1covered by the resist while the remaining part of the surface is free from resist. The surface partly covered by the resist is then subjected to etching which selectively removes those parts of the silicon layer12that are not covered by the resist. The result of the etching is shown inFIG. 2B. Subsequently, the oxide layer11which is exposed due to the previous etching is etched in a second etching step. This etching step removes all exposed parts of oxide layer11and, moreover, some of the oxide adjacent to these parts. As a result of the second etching step, the central parts of silicon layer12inFIG. 2Care free from the substrate. They form the resonator element20. At the same etching step, the oxide layer11under the support elements21and22is also removed such that the resonator element20is attached to the substrate10only via the anchor elements23and24.

The resonator element20has two outer ends205in the longitudinal direction. These can be larger than the combined width of the connecting beams20a,20band spacing19to define wider resonator masses at the ends, and thereby define a so-called dog-bone shape.

Each of the outer ends205is faced by a respective electrode of the electrically conductive actuator30, and is spaced from the electrode by an actuation gap g. The actuators can be considered to be gate terminals, in that the control input is applied to these terminals. The actuator30is able to receive an actuation potential difference VINwith respect to the resonator element20for elastically deforming the resonator element20, using the electrostatic force. The actuation potential difference is a function of the input signal applied to the resonator1. In addition to the input signal the actuation potential difference may typically further contain a DC component. The elastic deformation comprises a change of the length l by an amount dl shown inFIG. 1.

The resonator element20is part of an oscillator circuit which is able to conduct an electrical current through the resonator element20. The resonator element20is electrically connected to the positive or negative pole of a DC voltage source VDCvia an auxiliary resistor27, the anchor element24and the support element22. The anchor24can be considered to be a drain terminal in that the DC voltage bias is applied to this terminal to drive a bias current through the device. The resonator element20is further connected to ground via the support element21and the anchor element23. The anchor23can be considered to be a source terminal in that the bias current is collected at this terminal. Therefore, the resonator element20is able to conduct an electrical current I. It constitutes a resistor with an ohmic resistance R which causes a voltage drop V when the resonator element20conducts the electrical current I.

The resonator element20constitutes a resistor with an ohmic resistance R which is a function of the change dl of the length l because the resonator element20comprises a central part19with open space. The resonator element20comprises the two mutually parallel beams20a,20beach of which is affixed to a respective support element21and22. The two beams are connected with each other at the two outer ends by elements205. The central part19has been created during the lithography step and the etching step described above. It prevents the current from flowing from the support element22to the support element21in a straight line. The current has to follow the conductive path formed by the resonator element20. This conductive path extends in the longitudinal direction.

The circuit is able to produce an output signal which is a function of the change dl of the length I and which is a function of the resistance R. To this end the circuit comprises a measurement point28which is electrically connected to the circuit. It is situated between the auxiliary resistor27and the anchor element24, and in operation it produces an electrical output signal which is the electrical potential difference Vout between the measurement point28and the reference point29which is connected to ground.

In an alternative embodiment, not shown, the auxiliary resistor27is not situated between the voltage source and the anchor element24, but instead between the anchor element23and ground. In this case the measurement point28is situated between the auxiliary resistor27and the anchor element23.

In yet another embodiment, not shown either, the DC voltage source VDCand the auxiliary resistor27are omitted. The anchor element24is connected to the positive pole of a current source and the anchor element23is connected to the negative pole of the current source. The measurement point28is situated between the positive pole of the current source and the anchor element24, and the reference point29is situated between the anchor element23and the negative pole of the current source. Thus, a voltage is measured for a constant current, or else a current portion is measured for a constant total voltage.

The output signal is again a function of the change dl of the length l as will be understood by those skilled in the art. Thus, sensing using current bias or voltage bias can be employed.

The resulting mechanical resonance is in-plane of the drawing and is symmetrical. As mentioned above, the left and right parts of the resonator can be enlarged to define masses of relatively larger stiffness than the intermediate beams, so the compression and expansion that cause the resonator-vibration occurs in the beams.

Since the mechanical vibration is symmetrical, the centre of the structure between the drain and source24,23remains mechanically fixed as well.

The so called actuation-gap on each side is located between the gate electrodes30and the resonator mass and is in the order of a few hundred nm.

FIG. 3shows as plot300the frequency response for the standard drive approach for the circuit ofFIG. 1.

There is a capacitive feedthrough path between the gates30and drain/source24,23.

FIG. 4shows equivalent AC circuits of the resonator inFIG. 1, taking into account feedthrough capacitances Cs1and Cs2. The left circuit is a complete circuit representation and the right circuit is a simplified version.

The total resonator feedthrough-capacitance Cs is built up from contributions of the gap capacitance Cg1, Cg2and the other shown capacitances resulting from the capacitances to and from the imperfectly grounded substrate. For the arrangement where the lower resonator-body connection is grounded, the capacitances Cs1& Cs2between the gate electrodes and the drain terminal24(Vd) are the dominant contributions to the capacitive-feedthrough.

These parasitic capacitances result in additional power consumption and the capacitive feedthrough path also has the effect of reducing the magnitude of the resonant peak of the resonator. The distance between peak-height and background is the amplitude selectivity.

While indeed the height of the resonant peak decreases, the absolute height of the transfer at the peak location remains about the same: it is the ‘base’/‘background’ (the height of the slope from which the resonance-peak arises) that increases due to more capacitive feedthrough.

In the known arrangement ofFIG. 1, both gates carry the same signal (Vin), and they are electrically connected together as shown.

The invention provides designs in which the influence of the capacitive feedthrough can be removed or at least significantly be reduced by using the same resonator-structure in a modified configuration. The feedthrough currents resulting from different parasitic capacitive couplings are caused to cancel each other, at least partially.

A first example of approach of the invention is to make each connection of the two originally interconnected gates available separately and drive them by signals with opposite polarity, both for the DC-polarization and the AC-signals.

For such an arrangement, two identical feedthrough currents but of opposite sign arrive at the resonator output (drain node24). This arrangement can be considered to have balanced driving (in that the capacitive coupling of the drive signals is balanced so that the voltage effect is cancelled) and unbalanced sensing. As a result, the net feedthrough is cancelled/reduced.

The cancellation principle can be explained usingFIG. 4.

InFIG. 4, only the AC components of the capacitive feedthrough currents are considered. vg1and vg2are AC actuation signals applied at the electrodes. The actuation signals have the same frequency ω, but may have different phase. vdis the AC voltage at the drain node. ZLis the impedance of the load, which is the readout circuit of the oscillator. Rbis the resistance of each beam of the connecting element20a,20b. Due to the symmetry of the resonator body, the beams of the connecting elements have the same resistance. In the simplified circuit ZLand Rbare in parallel, resulting in Ztot. Cg1and Cg2are much smaller than Cs1and Cs2, and therefore are ignored in the simplified circuit.

The useful signal that is desired at the drain node is the AC signal corresponding to the change dl in strain, which is highly frequency dependent: it shows a sharp peak at the resonant frequency of the resonator body. At the drain node, besides the actual useful signal coming from the change in strain (not shown inFIG. 4), there is a parasitic voltage vdinduced by feedthrough currents i1and i2coming from the actuation electrodes, through the parasitic feedthrough capacitances Cs1and Cs2. The feedthrough-induced voltage vddoes not contribute to the resonant signal, therefore deteriorates the frequency selectivity of the signal (i.e. the background around the peak in the frequency response curve increases, thus the resonance peak height is reduced). For this voltage vdto be equal to zero, no net current should be able to flow into Ztotand develop a voltage across it. So the following must hold:
jωCs1*(vg1−vd)=jωCs2*(vd−vg2)  [1]

This is based on Kirchhoffs current law applied to the drain-node.

By assuming that Cs1=Cs2=Cs, which is likely true due to the symmetry of the resonator body, and with a condition that vg1=−vg2=vg, Equation [1] can be simplified to the following relationships:
jωCs*(vg1−vd)=jωCs*(vd+vg2)  [2]
(vg−vd)=(vd+vg)  [3]

This shows that for perfect symmetry and actuation signals of opposite signs, the capacitive feedthrough is eliminated (vd=0) and the resonator amplitude selectivity is improved.

Thus, a first implementation of the invention provides the application of equal and opposite AC voltages to the drive electrodes with respect to a reference voltage. In the equations above, the reference voltage is zero—however, this is not necessarily the case, as symmetrical voltages can be provide around a non-zero reference voltage. In practice, improvements of resonance peak height of up to 25 dB have been measured.

This first implementation of a balanced drive (but unbalanced sensing) is shown inFIG. 5A, and with a non-zero DC reference voltage.

FIG. 5Ashows the resonator body20. A DC bias-current through the resonator body is realized by connecting a current source to the drain24. As shown, opposite DC voltages +Vg and −Vg are applied to the gate electrodes30as well as opposite AC voltages +vac,inand −vac,in. Here, the DC voltages are referred to the middle point30of the electrodes. The sensing is based on the AC voltage at the drain. As shown, a differential amplifier500provides the differential (balanced) driving,

When a combination of DC and AC voltages are applied at the actuation electrodes, the mechanical vibration is caused by the electrostatic actuation force across each actuation-gap (between electrode & mass) and is for each gap given by:

where A is the electrode area and g is the gap width; ε0is the permittivity of vacuum.

From this equation it is clear that changing both the signs of Vgand vgresults in an identical actuation force as for the original configuration, hence the mechanical vibration mode remains completely unchanged for the modified arrangement.

As a consequence, the described method of balanced resonator-drive combines improved electrical resonator-behaviour with maintained mechanical-behaviour (as desired).

As mentioned above,FIG. 3shows as plot300the frequency response for the standard drive approach for the circuit ofFIG. 1. Plot302shows how the response is modified by using this embodiment of the invention. The height of the resonant peak is increased and the width is reduced, giving better frequency selectivity.

The invention is not restricted to a balanced drive and unbalanced sensing approach as explained above.

A drive scheme with interconnected gate drive electrodes (as in the prior art, which can be considered to be an unbalanced drive scheme) can also be used, if a differential sensing scheme is used. This differential sensing scheme can be considered to be a balanced sensing scheme, in that the effect of capacitive effects at the two anchor points is cancelled.

This approach is shown inFIG. 5B.

In this approach, the drain and source AC signals can be subtracted. The symmetry of the design means that the capacitive coupling to the drain24and source23are almost identical, so that the differential measurement cancels the effect of the capacitive feedthrough path. However, the signal at the drain and source vary oppositely. If the resistance of the resonator20drops, the voltage on the source23will rise and the voltage at the drain will fall as there is then a smaller resistance between them.

This approach involves applying equal voltages +Vg to the drive electrodes and measuring a differential voltage between the sense electrodes. Thus, this approach results in an improved situation as well: the capacitive feedthrough appears as a common-mode AC-signal while the desired signals appear with opposite polarities.

The approach described above for differential drive (balanced drive) can be combined with the approach described above for differential sensing (balanced sensing).

This approach is shown inFIG. 5C.

A DC bias-current through the resonator-body is realized by connecting equal resistors510(or current sources) between the supply rails and the anchor points24,23. Due to the symmetry of the arrangement, the DC voltage at the outer ends205of the resonator body is 0.5 times the supply voltage Vdc. The differential amplifier500provides differential (balanced) sensing, as explained above. The drive electrodes are DC biased with opposite voltages +Vg and −Vg and with AC components of opposite polarity +vac,inand −vac,in. This means the drive electrodes are provided with equal signal voltages of opposite sign at the outer ends205.

Thus,FIG. 5Cshows how to implement both the balanced drive and balanced sensing approach. As explained above, these two approaches may be used individually to provide the first two examples above, or they may be used in combination.

The configuration which is most suited depends on the requirements of the complete oscillator.

For example, the advantage of the conventional drive scheme with balanced sensing configuration is the requirement for only a single polarization polarity.

As also explained above, the sensing can either be voltage- or current-biased. For instance, current sensing can be employed, wherein a fixed voltage is applied to the piezoresistive element, and the current is measured. Whether voltage-biasing or current-sensing is used—or current-biasing in combination with voltage-sensing, depends on the requirements of the complete oscillator. Thus, in addition to the choice for one of the three balancing configurations shown inFIG. 5, the biasing/sensing approach provides further possibilities for adaptation to the requirements. For example, the amplifier stage500inFIG. 5can either be applying voltage-amplification to the drain & source signals (voltage sensing) or can be providing virtual-earths at its input for current sensing (forming a transimpedance amplifier).

To obtain the largest reduction of the unwanted effect of capacitive feedthrough it is required that the provided signals are as accurately matched as possible.

However, even if there is non-perfect matching, the improvement can still be significant: more than 20 dB improvement in amplitude selectivity was measured in an experimental setup based on a resonator that had separated gates, but was otherwise not truly symmetrical.

Thus, even partially offsetting the capacitive coupling effect will give improved performance.

Compensation of any Cs1vs. Cs2unbalance is possible by adjusting the AC signal levels, so that the voltages are weighted with respect to the capacitance between the respective drive electrode and the sense electrode. In this case, only the AC component is weighted. The AC signal levels should be adjusted such that Eq. [5] below is satisfied.

This approach simply follows from equation [1] with vd=0:
Cs1*vg1=−Cs2*vg2[5]

The invention has been described in detail with reference to a particular design of resonator. However, other resonator designs are possible. For example, a central resonator mass may have opposite anchored spring members on each side.

The resonator described above has bulk mode vibration, but the invention can be applied to flexural mode resonators as well.

In WO 2010/044058, a type of flexural mode piezoresistive resonator is described. The resonator has a shape of a ring, or a symmetrical polygon frame, such as a square frame. The structure has an in-plane mode shape in which segments of the structure alternatively bend inward and outward, while the width of the structure substantially remains unchanged. Anchors are positioned at the four quasi-nodal points of the mode shape, i.e. at the boundaries between the segments. An example of a ring-shaped resonator is given inFIG. 6. The flexural vibration mode shape is given inFIG. 7.

FIG. 6shows the basic known resonator configuration from WO2010/044058, but also shows how the actuation electrodes are modified to implement the same conceptual cancellation of capacitive feedthrough.

FIG. 7shows a finite element simulation showing the flexural ellipse mode shape of the ring-shaped resonator. The gray-scale coding indicates strain in the X (left picture) and Y (right picture) directions. During vibration, the inner and outer rims of consecutive segments contract and expand alternately.

At one or more locations on the structure, the Si material is locally doped (n- or p-type) to enable the piezoresistive effect. The piezoresistive regions are connected together and connected to the outside world via the anchors by highly doped regions (low resistance), or regions coated with metal layers. The connections are made in such a way that during vibration, the piezoresistive signal, being the change in resistance as a function of strain in the doped Si material, at different piezoresistive regions can be added up. During operation, a current is sent via the anchors, through the different piezoresistive regions to collect the piezoresistive signal. This signal is proportional to the deformation of the structure during its vibration. The vibration is excited by applying a combination of AC and DC voltages on the electrodes.

The left and right images ofFIG. 7show grey-scale maps of strain in the X, and Y directions, respectively. The light grey means expansion, and the dark grey-contraction. One can easily recognize that at every segment of the ring, the outer rim and the inner rim have opposite strain signs and the strain sign alternates from one segment to the next. For instance, the outer rim of the upper segment expands while the inner rim of the same segment contracts. At the same time, the outer rim of the next segment on the right side contracts while its inner rim expands, and so on.

To collect the piezoresistive signals due to strain induced in the structure, four regions600are locally doped with a suitable doping concentration (normally it is a relatively low concentration compared to the connection regions), as seen inFIG. 6. These regions, called the piezoresistive regions, should exhibit good piezoresistive effect. The piezoresistive regions are located at every segment of the ring and alternatively at the inner and outer rim, where the magnitudes of strain are maximum. The arrangement of the piezoresistive regions should be such that during vibration, they undergo strain with the same sign. The areas610in between the piezoresistive regions are doped with high concentration to make the material low ohmic. These low ohmic regions also extend to the anchors615and from the anchors to the outside world. The low ohmic regions610serve as the electrical connections between the piezoresistive regions and from these regions to the outside world.

These low ohmic regions therefore have no significant contribution to the piezoresistive signal. The rest of the ring area (such as area620) should be left undoped, thus having very high resistance.

During operation, a sense current is sent through two opposite anchors and out through the other two opposite anchors. In this configuration, the four piezoresistive regions are connected in parallel and the sign of change in signals in all resistors are the same, thus the signals can be added up. The total piezoresistive signal change can be used as the output signal of the resonator, just as in the resonator described inFIG. 1. The “current-in” anchors can be considered the drain, and the “current-out” anchors can be considered the source.

Using the approach of the current invention, the ring-shaped resonator of WO 2010/044058 can be used in a different way, in order to cancel the effect of the feedthrough capacitance. In order to form the balanced drive configuration of the ring-shaped resonator, four actuation electrodes6301to6304are positioned at the outer rim of the ring structure, facing the four segments. This is shown inFIG. 6.

Alternatively, the electrodes6301to6304can be positioned facing the inner rim of the segments. The electrodes should be paired such that any two opposite electrodes are connected to each other. The two pairs of electrode are fed with AC voltages of opposite signs but DC polarization voltages of the same sign, as shown.

For example, as shown inFIG. 6, the top-bottom electrodes are fed with +vg+Vg, while the left-right electrodes are fed with −vg+Vg. According to Eq [4], the force that exerts on the top-bottom segments has opposite polarity compared to that exerting on the left-right segments. These anti-phase forces are required to drive the ring in the ellipse mode shape. On the other hand, the use of vgin opposite signs forms the balanced drive configuration, thus the feedthrough current is cancelled out. An advantage of this configuration over the balanced drive configuration of the dog-bone resonator inFIG. 1is that only one DC polarization is needed, which makes the circuit design much easier, for a given limited voltage supply budget.

The ring-shaped resonator can also be arranged to form the balanced sensing configuration. This is shown in the example ofFIG. 8. The left and the right segments810are lowly doped at the outer and inner rim, respectively, to form two piezoresistive regions. The top and bottom regions are not doped, so that there are only two piezoelectric regions. The two segments have the same polarity in strain during a vibration cycle. However, because the piezoresistive regions are located in the opposite sides of the segments, they should result in signals with opposite polarity during a vibration cycle, which is +vac-outand −vac-out. These differential signals are fed into a differential amplifier. In this way, the useful signal is amplified while the common mode feedthrough voltages are cancelled out in the same amplifier.

Yet another example of balanced sensing configuration is given inFIG. 9. In this example, four piezoresistive regions910are positioned at the same rim of the ring, e.g. at the outer rim. As a consequence of the alternating bending directions of the ring segments, the four piezoresistive regions have alternating resistance changes during a vibration cycle. For example, during a cycle, resistances of the top and bottom segments increase while resistances of the left and right segments decrease. In other words, the ring forms a Wheatstone bridge configuration. When a constant DC voltage Vdcis applied to an anchor, e.g. the top-left anchor, and the opposite anchor (e.g. bottom-right) is grounded, the other two anchors provide output signals of opposite polarity +vac-outand −vac-out, which are fed into the differential amplifier, as in the previous example. The feedthough cancellation is also done in the same way.

Any combinations of balanced drive and balanced sensing (e.g.FIG. 6withFIG. 8orFIG. 6withFIG. 9) are possible. An example of such a combination is given inFIG. 10, which shows a balanced drive—balanced sensing configuration for the ring-shaped resonator.

For the example of a dogbone resonator, the length may be 40 um, and the width of each beam around 3 um. The dimensions of a ring-shaped resonator may be an outer diameter of around 70 um, an inner diameter of around 48 um.

Vg is in the order of a few V up to a few tens of V; vg is in the order of mV up to a few hundreds of mV.

The actuation force has the same frequency ω as the drive AC voltage. By using a combination of Vdrive=Vg+vg, in which vg=vg0*cos ωt, and with a condition that Vg>>vg0, e.g Vgin the order of a few V to a few tens of V, while vgis in the order of mV to a few tens of mV. An equation for the actuation force is:

F=⁢∂C∂x⁢Vdrive22=⁢∂C∂x⁢Vg2+2⁢Vg⁢vg⁢⁢0⁢cos⁢⁢ω⁢⁢t+vg⁢⁢02⁢cos2⁢ω⁢⁢t2=⁢ɛ0⁢Ag2⁢Vg2+2⁢Vg⁢vg⁢⁢0⁢cos⁢⁢ω⁢⁢t+vg⁢⁢02⁢cos2⁢ω⁢⁢t2
in which C is the capacitance across the actuation gap, x is the displacement of the ends of the resonator. In the above equation, the last term (vg02*cos2ωt), after a trigonometric change, is a function of double frequency 2ω. However, since the condition is that Vg>>vg0, and this term has the factor of vg0^2, it is considered very small and ignored. The first term (Vg^2) is a DC component, therefore has no function in driving the vibration, can also be ignored. The rest is the middle term, which is a function of frequency ω, is actually used to drive the vibration. This leads to equation [4] above.

In the balanced drive version, equal and opposite voltages are applied to the drive electrodes with respect to a reference value. For example, with +Vg+vacand −(Vg+vgac) the reference is zero. With +Vg+vacand +Vg−vac, the reference voltage is Vg.