Methods for closed loop operation of capacitive accelerometers

Abstract

A method for closed loop operation of a capacitive accelerometer includes applying a first drive signal V.sub.1 to a first fixed capacitive electrode and a second drive signal V.sub.2 to a second fixed capacitive electrode the first and second drive signals each having a periodic waveform varying in amplitude between zero and a maximum value V.sub.ref and sensing a displacement of the proof mass and applying pulse width modulation to the first and second drive signals with a constant frequency f.sub.mod and a variable mark/space ratio. The method also includes applying a voltage offset V.sub.ref/2 to the proof mass and applying the pulse width modulation such that the first and second drive signals have a waveform that varies so that when either one of the first and second drive signals is at V.sub.ref or zero the other drive voltage is at V.sub.ref/2.

Claims

1. A method for closed loop operation of a capacitive accelerometer, the capacitive accelerometer comprising: a proof mass moveable along a sensing axis in response to an applied acceleration; and first and second fixed capacitive electrodes arranged symmetrically either side of the proof mass along the sensing axis with a gap defined between each of the first and second fixed capacitive electrodes and the proof mass under zero applied acceleration; the method comprising: applying a first drive signal V.sub.1 to the first fixed capacitive electrode and a second drive signal V.sub.2 to the second fixed capacitive electrode, the first and second drive signals each having a periodic waveform varying in amplitude between zero and a maximum value V.sub.ref; and sensing a displacement of the proof mass and applying pulse width modulation to the first and second drive signals with a constant frequency f.sub.mod and a variable mark/space ratio so as to provide a net electrostatic restoring force on the proof mass for balancing the inertial force of the applied acceleration and maintaining the proof mass at a null position; applying a voltage offset V.sub.ref/2 to the proof mass; and applying the pulse width modulation such that the first and second drive signals have a waveform that varies so that when either one of the first and second drive signals is at V.sub.ref or zero the other drive voltage is at V.sub.ref/2.

2. The method of claim 1, wherein the first and second drive signals have a waveform that steps between V.sub.ref/2 and V.sub.ref in a first half cycle and steps between V.sub.ref/2 and zero in a second half cycle.

3. The method of claim 1, wherein the first and second drive signals have a waveform that comprises a single square wave pulse in the first half cycle and a single square wave pulse in the second half cycle that is inverted about V.sub.ref/2 relative to the single square wave pulse in the first half cycle.

4. The method of claim 1, wherein applying a voltage offset V.sub.ref/2 to the proof mass comprises referencing the proof mass to an electrical connection midway between the first and second fixed capacitive electrodes.

5. The method of claim 1, further comprising: sensing a displacement of the proof mass by sampling an output signal at the proof mass; and adding a compensation signal to the output signal, the compensation signal having the same pulse width modulation with a constant frequency f.sub.mod, and the compensation signal being in anti-phase with the output signal.

6. The method of claim 5, further comprising: applying the compensation signal across a compensation capacitor having a capacitance substantially matched to the capacitance of the gap between each of the first and second fixed capacitive electrodes and the proof mass under zero applied acceleration.

7. The method of claim 1, further comprising: outputting a signal indicative of the applied acceleration.

8. The method of claim 1, wherein the capacitive accelerometer comprises a silicon MEMS structure and/or wherein the proof mass is substantially planar.

9. The method of claim 1, wherein the proof mass is mounted to a fixed substrate by flexible support legs so as to be linearly moveable in a plane along the sensing axis in response to an applied acceleration, and wherein the first and second fixed capacitive electrodes are formed in the fixed substrate in the same plane.

10. The method of claim 1, wherein: the proof mass comprises first and second sets of moveable capacitive electrode fingers extending from the proof mass, substantially perpendicular to the sensing axis and spaced apart along the sensing axis; and the first and second fixed capacitive electrodes comprise, respectively, first and second sets of fixed capacitive electrode fingers extending substantially perpendicular to the sensing axis and spaced apart along the sensing axis; and the first set of fixed capacitive electrode fingers is arranged to interdigitate with the first set of moveable capacitive electrode fingers with a first offset in one direction along the sensing axis from a median line between adjacent fixed capacitive electrode fingers, and the second set of fixed capacitive electrode fingers is arranged to interdigitate with the second set of moveable capacitive electrode fingers with a second offset in the opposite direction along the sensing axis from a median line between adjacent fixed capacitive electrode fingers.

11. A capacitive accelerometer comprising: a proof mass moveable along a sensing axis in response to an applied acceleration; first and second fixed capacitive electrodes arranged symmetrically either side of the proof mass along the sensing axis with a gap defined between each of the first and second fixed capacitive electrodes and the proof mass under zero applied acceleration; a pulse width modulation signal generator arranged to apply a first drive signal V.sub.1 to the first fixed capacitive electrode and a second drive signal V.sub.2 to the second fixed capacitive electrode, the first and second drive signals each having a periodic waveform varying in amplitude between zero and a maximum value V.sub.ref; and a closed loop circuit arranged to detect a signal resulting from displacement of the proof mass and control the pulse width modulation signal generator to apply the first and second drive signals at a constant frequency f.sub.mod with a variable mark/space ratio so as to provide a net electrostatic restoring force on the proof mass for balancing the inertial force of the applied acceleration and maintaining the proof mass at a null position; characterised in that: a voltage offset V.sub.ref/2 is applied to the proof mass; and the first and second drive signals have a waveform that varies so that when either one of the first and second drive signals is at V.sub.ref or zero the other drive voltage is at V.sub.ref/2.

12. The capacitive accelerometer of claim 11, wherein the proof mass is referenced to an electrical connection midway between the first and second fixed capacitive electrodes.

13. The capacitive accelerometer of claim 11, further comprising: a pre-amplifier arranged to sample an output signal at the proof mass; wherein the pulse width modulation signal generator is arranged to input a compensation signal to the pre-amplifier for the pre-amplifier to add to the output signal, the compensation signal having the same pulse width modulation with a constant frequency f.sub.mod, and the compensation signal being in anti-phase with the output signal.

14. The capacitive accelerometer of claim 13, further comprising: a compensation capacitor connected between the pulse width modulation signal generator and the pre-amplifier, the compensation capacitor having a capacitance substantially matched to the capacitance of the gap between each of the first and second fixed capacitive electrodes and the proof mass under zero applied acceleration.

15. The capacitive accelerometer of claim 11, comprising a silicon MEMS structure.

16. The capacitive accelerometer of claim 11, wherein the proof mass is mounted to a fixed substrate by flexible support legs so as to be linearly moveable in a plane along the sensing axis in response to an applied acceleration, and wherein the first and second fixed capacitive electrodes are formed in the fixed substrate in the same plane.

17. The capacitive accelerometer of claim 11, wherein the proof mass is substantially planar.

18. The capacitive accelerometer of claim 11, wherein the proof mass comprises first and second sets of moveable capacitive electrode fingers extending from the proof mass, substantially perpendicular to the sensing axis and spaced apart along the sensing axis; wherein the first and second fixed capacitive electrodes comprise, respectively, first and second sets of fixed capacitive electrode fingers extending substantially perpendicular to the sensing axis and spaced apart along the sensing axis; and wherein the first set of fixed capacitive electrode fingers is arranged to interdigitate with the first set of moveable capacitive electrode fingers with a first offset in one direction along the sensing axis from a median line between adjacent fixed capacitive electrode fingers, and the second set of fixed capacitive electrode fingers is arranged to interdigitate with the second set of moveable capacitive electrode fingers with a second offset in the opposite direction along the sensing axis from a median line between adjacent fixed capacitive electrode fingers.

19. The capacitive accelerometer of claim 18, wherein the proof mass is mounted to a fixed substrate by flexible support legs so as to be linearly moveable in a plane along the sensing axis in response to an applied acceleration, and wherein the first and second fixed capacitive electrodes are formed in the fixed substrate in the same plane.

20. The capacitive accelerometer of claim 19, wherein the proof mass is substantially planar.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) One or more non-limiting examples will now be described with reference to the accompanying drawings, in which:

(2) FIG. 1 schematically illustrates a known electronic control scheme for a closed loop capacitive accelerometer according to the prior art;

(3) FIG. 2 shows the applied voltage waveforms and resultant electrostatic forces for electrodes 1 and 2, for a 50:50 mark:space ratio under a zero applied acceleration condition, according to the prior art;

(4) FIG. 3 shows the applied voltage waveforms and resultant electrostatic forces for electrodes 1 and 2, for a 25:75 mark:space ratio under a positive applied acceleration condition, according to the prior art;

(5) FIG. 4 shows the applied voltage waveforms and resultant electrostatic forces for electrodes 1 and 2, for a 50:50 mark:space ratio under a zero applied acceleration condition, with the proof mass referenced to V.sub.ref/2;

(6) FIG. 5 shows the applied voltage waveforms and resultant electrostatic forces for electrodes 1 and 2, for a 50:50 mark:space ratio under a zero applied acceleration condition, according to examples of the present disclosure;

(7) FIG. 6 shows the applied voltage waveforms and resultant electrostatic forces for electrodes 1 and 2, for a 25:75 mark:space ratio under a positive applied acceleration condition, according to examples of the present disclosure;

(8) FIG. 7 schematically illustrates an electronic control scheme for a closed loop capacitive accelerometer according to examples of the present disclosure;

(9) FIG. 8 shows the pre-amplifier voltage output signals and sampling points, for an open loop accelerometer under positive, zero and negative applied acceleration conditions, according to the prior art;

(10) FIG. 9 shows the pre-amplifier voltage output signals and sampling points, for a closed loop accelerometer under positive, zero and negative applied acceleration conditions, according to the prior art;

(11) FIG. 10 shows the pre-amplifier voltage output signals and sampling points, for an open loop accelerometer under positive, zero and negative applied acceleration conditions, according to examples of the present disclosure;

(12) FIG. 11 schematically illustrates an alternative electronic control scheme for a closed loop capacitive accelerometer according to further examples of the present disclosure;

(13) FIG. 12 shows the pre-amplifier voltage output signals and sampling points, both before and after application of a compensating signal, for an open loop accelerometer under positive applied acceleration conditions, according to examples of the present disclosure;

(14) FIG. 13 shows the pre-amplifier voltage output signals and sampling points, for a closed loop accelerometer under positive, zero and negative applied acceleration conditions, according to further examples of the present disclosure; and

(15) FIG. 14 is a schematic representation of an exemplary electrode arrangement in a capacitive accelerometer.

DETAILED DESCRIPTION

(16) There is generally seen in FIG. 1 a known electronic control scheme for a closed loop capacitive accelerometer according to the prior art exemplified by U.S. Pat. No. 7,267,006, the contents of which are hereby incorporated by reference. A pulse width modulation (PWM) signal generator receives a constant fixed reference voltage V.sub.ref and supplies complementary first and second drive voltages V.sub.1 and V.sub.2 to electrode 1 and electrode 2. Electrodes 1 and 2 typically take the form of first and second sets of fixed capacitive electrode fingers that interdigitate with the moveable capacitive electrode fingers of the proof mass, as is well known in the art.

(17) A pre-amplifier is arranged to sample an output signal at the proof mass. As illustrated in FIG. 1, the pre-amp is configured so that its input from the proof mass is a “virtual earth” where the voltage for the “earth” is 0 V, which is achieved by a DC biasing element such as a resistor. The pre-amp then forces the proof mass DC bias voltage to be the same as the reference voltage V.sub.ref, whereas periodic signals are picked up by the pre-amp in response to movement of the proof mass. These periodic pick-off signals are then demodulated and passed through a loop filter before being fed back to a pulse width modulation (PWM) signal generator in closed loop operation.

(18) FIG. 2 illustrates that the first and second drive signals have a standard square waveform varying in amplitude between zero and a maximum value V.sub.ref, which is typically 30 V in order to give the force required to achieve a 70 g dynamic range. With zero applied acceleration, when the gaps d between the proof mass electrode and each of the two fixed electrodes are nominally equal, the waveforms consists of a square wave with a 50:50 mark:space ratio. The resultant force, F, for each electrode is given by:

(19) $F=\frac{C{V}^2}{d}$
where C is the gap capacitance and V is the voltage. The mean voltage level for a 50:50 mark:space waveform, as shown by the dashed line, is equal to V.sub.ref/2. The corresponding mean force will therefore similarly be half of the peak value, and is also shown as a dashed line in FIG. 2. As the fixed electrodes are located at opposite sides of the proof mass, the forces act in opposite directions and therefore, the net force acting on the proof mass will be zero. These waveforms are conveniently modulated at a very high frequency (e.g. f.sub.mod˜100 kHz) compared to the mechanical resonant frequency of the proof mass, which is critically damped, and therefore no significant motion occurs at this frequency.

(20) When an acceleration is applied to the accelerometer, the signal generated by the displacement of the proof mass is fed back into the control loop which adjusts the mark:space ratio of the PWM signal. This differentially changes the forces between the fixed electrodes and proof mass to drive the proof mass back to the null position. The net force is given by:

(21) $F=V_{ref}^2\left[\frac{C_1{w}_1}{2d_1}-\frac{C_2{w}_2}{2d_2}\right]$

(22) Where d.sub.1 and d.sub.2 refer to the electrode 1 and 2 capacitor gaps respectively, and w.sub.1 and w.sub.2 are the pulse widths (i.e. voltage is at V.sub.ref) for the two waveforms applied to the fixed electrodes. The waveforms and resultant forces for an exemplary positive acceleration, giving rise to a 25:75 mark:space ratio, are shown in FIG. 3. The reduced pulse width applied to electrode 1 results in a reduced mean voltage and thus a reduced mean force while the mean voltage and mean force increases for electrode 2. The mean net force on the proof mass is given by the differential force between electrodes 1 and 2. The individual mean force produced by each drive signal is non-linear, but the two operating together cancels the non-linearity and produces a net force which varies linearly with respect to the mark:space ratio.

(23) In such a prior art capacitive accelerometer, dielectric charging arises due to the large mean DC voltages between the fixed electrodes and the proof mass. These voltage levels will also vary depending on the applied acceleration level thus changing the charging characteristics. In the control scheme of FIG. 1, applying a voltage level of 30 V for V.sub.ref results in a mean DC offset of 15 V between each of the fixed electrodes and the proof mass which generates large, opposing forces on the proof mass even when 0 g is applied. The mean voltage offsets will vary depending on the applied g level, however, large offsets will typically be present at all times in any practical application.

(24) The Applicants have recognised that such dielectric charging can, however, be substantially eliminated by removing the mean voltage offset between the fixed electrodes and the proof mass. This may be achieved by firstly offsetting the proof mass voltage level to V.sub.ref/2. The voltage waveforms will therefore vary symmetrically around the proof mass voltage level to give a mean zero offset between the proof mass and fixed electrodes. This eliminates the voltage gradients driving the charge migration and thus the problematic scale factor and bias drifts.

(25) However, the Applicants have realised that simply offsetting the proof mass voltage in the known control scheme of FIGS. 1-3 does not allow the necessary force feedback of closed loop operation to be achieved. FIG. 4 shows the effect of simply offsetting the proof mass voltage. It can be seen that, while the mean DC voltage is eliminated, the force, which varies as V.sup.2, is rectified and thus positive and negative pulses produce identical forces of the same polarity. Varying the mark:space ratio does not therefore induce any change in the mean force and thus no force feedback can be applied.

(26) In examples according to the present disclosure, the drive signal waveforms are modified, as shown in FIG. 5, such that the mean forces can be adjusted by applying pulse width modulation. The electrode 1 voltage waveform V.sub.1 steps between V.sub.ref/2 and V.sub.ref in a first half cycle to give a first ‘mark’ with a positive pulse (with respect to the proof mass voltage), and steps between V.sub.ref/2 and zero in a second half cycle to give a second negative pulse (with respect to the proof mass voltage). These pulses are separated by the ‘space’ at a voltage level of V.sub.ref/2. The voltages relative to the proof mass are given on the right hand side of the plots for V.sub.1 and V.sub.2. The overall voltage change of 15 V seen by the proof mass is the same 30 V range that would have been applied in a prior art control scheme, but now there is a mean zero voltage as indicated by the dashed line. When comparing the waveforms of the prior art (FIG. 2) with those of an example of the present disclosure (FIG. 5), for identical timescales, the frequency, f.sub.mod, of the waveform for the present disclosure will be at half that of the prior art. The temporal form of the resultant force, shown in FIG. 5, is at 2×f.sub.mod, due to the rectification of the positive and negative voltage cycles and is essentially identical to that of FIG. 2.

(27) FIG. 6 shows the effect of adjusting the mark:space ratio to 25:75, where the pulse widths of the positive and negative cycles are adjusted identically, allowing the differential force to be adjusted whilst maintaining a zero mean voltage. As before, the electrode 1 voltage waveform V.sub.1 steps between V.sub.ref/2 and V.sub.ref in a first half cycle to give a first ‘mark’ with a positive pulse (with respect to the proof mass voltage), and steps between V.sub.ref/2 and zero in a second half cycle to give a second negative pulse (with respect to the proof mass voltage), albeit the marks are shorter in time. The mark:space ratio of 25:75 produces a differential force between electrodes 1 and 2 and hence a net electrostatic restoring force on the proof mass for balancing the inertial force of the applied acceleration and maintaining the proof mass at a null position. The voltages relative to the proof mass are given on the right hand side of the plots for V.sub.1 and V.sub.2. Again, the resultant force temporal form is identical to that shown in FIG. 3.

(28) In the example seen in FIGS. 5-6, the mean drive force F∝(V.sub.ref/2).sup.2 so this control scheme reduces the drive force by a factor of ×4. This limits the overall g range for a given accelerometer unless a larger voltage range is applied. However, many high precision applications do not require a large g range. The g range may however be increased by increasing the value of V.sub.ref, if required.

(29) FIG. 7 shows a block diagram of an exemplary means to implement the disclosed method for closed loop operation of a capacitive accelerometer, i.e. a force feedback control scheme. A pulse width modulation (PWM) signal generator is supplied with voltages V.sub.ref, V.sub.ref/2 and 0 V and outputs electrode voltages waveforms V.sub.1 and V.sub.2, shown in FIGS. 5 and 6, to electrodes 1 and 2 respectively. The signal from the proof mass is applied to a pre-amplifier which is referenced to V.sub.ref/2 via a DC biasing element (which may e.g. consist of a resistor), which offsets the proof mass reference voltage to the same level. This differs from the prior art where the proof mass is referenced to 0 V, as shown in FIG. 1. The pre-amplifier output is demodulated, and applied to a loop filter which integrates the signal and sets the dynamic response of the system with the output used to control the PWM signal generator to adjust the mark:space ratio. The pre-amplifier, demodulator, loop filter and feedback to the PWM signal generator form a closed loop circuit 10.

(30) There are significant differences between examples of the present disclosure and the prior art control scheme described in U.S. Pat. No. 7,267,006, in terms of the signal detected by the pre-amplifier and its subsequent demodulation. For the previous scheme of FIG. 1, the AC signal detected by the pre-amplifier (in an open loop accelerometer configuration) is shown schematically in FIG. 8. For example, when a positive acceleration is applied, the electrode 1 signal will be larger than the electrode 2 signal due to the differential gap change. FIG. 8 shows the resultant signal for positive, zero and negative applied accelerations. The rising and falling edges of the drive waveform V.sub.1 are used to trigger the signal sampling which takes place after a fixed delay to avoid any signal transients arising from the input waveform. The sampling points are shown by the crosses in FIG. 8. The corresponding points on the drive waveform V.sub.1 are shown in FIGS. 2-3. The delay is conveniently set to equal approximately 5 percent of the pulse duration which, for an exemplary 100 kHz modulation frequency, f.sub.mod, is equivalent to a 0.25 micro-second delay. A limitation of 5%:95%<mark:space<95%:5% is set to ensure that no switching occurs during the sampling period. The acceleration signal is derived from the difference in the levels between Samples A and Samples B as follows:
Acceleration Signal=(Sample A−Sample B)

(31) In closed loop operation this AC signal is used to adjust the mark:space ratio in order to achieve a null at the input to the pre-amplifier. The waveforms shown in FIG. 8 would therefore be modified such that the pre-amplifier output is maintained at zero at all times. The sampling points would however vary in accordance with the rising and falling edges of the drive waveform, which is now subject to pulse width modulation at a variable mark:space ratio. The pre-amplifier output and corresponding sample points for a closed loop configuration are shown in FIG. 9. The corresponding points on the drive waveform V.sub.1 are shown in FIGS. 2-3, with FIG. 2 (50:50) representing zero acceleration and FIG. 3 (25:75) representing positive and negative accelerations.

(32) The corresponding open loop AC signals measured by the pre-amplifier for examples of the present disclosure, for positive, zero and negative applied accelerations, are shown in FIG. 10. It can be seen that, even at zero applied acceleration, a large AC signal exists at f.sub.mod. For this implementation, with the displacement of the proof mass induced by a positive acceleration, the sample A level will give a smaller positive value than that for the sample B. For the next two samples, the sample C level will be a smaller negative value than that for sample D. The acceleration signal is derived by summing the samples as follows:
Acceleration Signal=(Sample A−Sample B)−(Sample C−Sample D)

(33) For a negative acceleration, the polarity of the resultant acceleration signal will be reversed. This process enables the acceleration signal to be extracted despite the presence of the large AC background signal. In other words, the acceleration signal appears as a relatively small perturbation superimposed on the background (zero acceleration) signal. For an open loop accelerometer this provides a direct measurement of the acceleration, however, in closed loop operation this signal is used to adjust the mark:space ratio such that the input acceleration signal is nulled.

(34) The large AC waveform shown in FIG. 10 at f.sub.mod is superimposed on the smaller acceleration-induced signal variation. This large signal input to the pre-amplifier may be problematic as it limits the gain which may be applied in order to avoid saturation. This limitation can be overcome by summing an anti-phased compensation signal input to the pre-amplifier of equal amplitude. This may be implemented as shown schematically in FIG. 11. The capacitive accelerometer is basically the same as already described in relation to FIG. 7, except that in this example a square wave compensation signal (at f.sub.mod) is additionally produced by the PWM signal generator which varies between V.sub.ref/2 and −V.sub.ref/2. This ensures that the timing and amplitude of the compensation signal are accurately synchronised with the drive signal waveforms applied to electrodes 1 and 2. The compensation signal is applied to a fixed “compensation” capacitor C which has a capacitance substantially of equal value to the gap capacitance of electrodes 1 and 2, with the signal from the capacitor C then applied to the pre-amplifier input. The compensation signal received at the input to the pre-amplifier is then substantially of equal amplitude but of opposite phase to the signal input from the proof mass electrode. FIG. 12 shows the effect of the compensation for an exemplary case with a positive applied acceleration. The uncompensated signal shows a large peak to peak variation of which only a small component is due to the applied acceleration. After application of the compensation signal (dashed line), the variation is significantly reduced. For an exactly matched compensation signal, the resultant output signal will contain only components arising due to the applied acceleration.

(35) When operated in a closed loop mode and with the compensation signal applied, the pre-amplifier output waveforms shown in FIG. 10 for open loop will be modified as shown in FIG. 13. This enables the gain of the pre-amplifier to be set substantially higher than would be possible in the absence of the compensation signal, thus providing a significant improvement in the measurement resolution and hence in the sensitivity and noise performance of the accelerometer.

(36) The proof mass and fixed capacitive electrodes may have any suitable arrangement in a capacitive accelerometer as generally disclosed herein. For example, the proof mass may be moveable in a pendulous or hinged structure. EP0338688 provides an applicable example of a moveable proof mass electrode formed at the tip of a silicon cantilever and fixed electrodes arranged to oppose the moveable electrode. However, in some preferred examples the proof mass is planar and comprises moveable electrodes fingers that interdigitate with fixed electrode fingers extending from the first and second fixed capacitive electrodes. Such an interdigitated or comb-like electrode structure is well-known in the art, for example as disclosed in any of U.S. Pat. Nos. 6,761,069, 6,631,643, or U.S. Pat. No. 7,267,006, the contents of each of which are hereby incorporated by reference.

(37) An exemplary electrode structure for a capacitive accelerometer 101 is schematically illustrated in FIG. 14, which is similar to that described in U.S. Pat. No. 7,267,006. In this example, the proof mass 102 is substantially planar and mounted to a fixed substrate (not seen) by flexible support legs 114 so as to be linearly moveable in a plane along the sensing axis (as indicated by the double-headed arrow) in response to an applied acceleration. The flexible support legs 114 extend from the body of the proof mass 102 and are fixed at anchor points 116 to the fixed substrate.

(38) First and second fixed capacitive electrodes 104, 106 are formed in the fixed substrate in the same plane. The proof mass 102 comprises first and second sets of moveable capacitive electrode fingers 108 extending from the proof mass 102, substantially perpendicular to the sensing axis and spaced apart along the sensing axis. It may also be seen that the first and second fixed capacitive electrodes 104, 106 comprise, respectively, first and second sets of fixed capacitive electrode fingers 110, 112 extending substantially perpendicular to the sensing axis and spaced apart along the sensing axis. The first set of fixed capacitive electrode fingers 110 is arranged to interdigitate with the first set of moveable capacitive electrode fingers 108a with a first offset in one direction along the sensing axis from a median line m between adjacent fixed capacitive electrode fingers 110, and the second set of fixed capacitive electrode fingers 112 is arranged to interdigitate with the second set of moveable capacitive electrode fingers 108b with a second offset in the opposite direction along the sensing axis from a median line m between adjacent fixed capacitive electrode fingers 112.

(39) The proof mass 102 can move in-plane relative to the fixed electrodes 104, 106 in a direction along the sensing axis in response to an applied acceleration. As the two sets of fixed electrode fingers 110, 112 are offset from the proof mass fingers 108a, 108b in opposite directions, a movement in either direction can be measured. These offsets may be equal in size. The difference in offset for the first set of fixed electrode fingers 110 and the second set of fixed electrode fingers 112 relative to the moveable fingers 108a, 108b causes an attractive force when a drive signal (e.g. voltage waveform) is applied to the first and second sets of fixed electrode fingers 110, 112.

(40) In open loop operation, movement of the proof mass 102 in response to an applied acceleration causes a change in the offset between the proof mass fingers 108a, 108b and the fixed electrode fingers 110, 112. This change can be used to calculate the acceleration, as it will cause a change in differential capacitance. In closed loop operation, the interdigitated electrode fingers do not actually move relative to one another. Applying pulse width modulation (PWM) to the first and second drive signals applied to the first and second fixed capacitive electrodes 104, 106, an electrostatic restoring force acts on the proof mass fingers 108a, 108b so that under acceleration the proof mass 102 does not move from the null position seen in FIG. 14, with the inertial force of the applied acceleration being balanced by a net electrostatic restoring force.

(41) It will be appreciated by those skilled in the art that the disclosure has been illustrated by describing one or more specific examples thereof, but is not limited to these aspects; many variations and modifications are possible, within the scope of the accompanying claims.