Electrical method to measure the dynamic behaviour and the quadrature error of a MEMS gyroscope sensor

Sensors and Actuators A 134 (2007) 88–97

Electrical method to measure the dynamic behaviour and the quadrature error of a MEMS gyroscope sensor Alfredo Cigada ∗ , Elisabetta Leo, Marcello Vanali Mechanical Engineering Department, Politecnico di Milano, 20158 Milan, Italy Received 1 March 2006; received in revised form 20 October 2006; accepted 1 November 2006 Available online 11 December 2006

Abstract The electronics on board of a vibrating MEMS sensor is able to compensate only for small changes in the mechanical characteristics of the device. Thus, it is of great importance to find an easy and fast way to evaluate the mechanical parameters, such as the resonance frequency and the damping coefficient, of a MEMS sensor before putting the device on the market [A. Cigada, E. Leo, M. Vanali, Optical and electrical methods to measure the dynamic behaviour of a MEMS gyroscope sensor, Proceedings of IMECE2005 ASME International Mechanical Engineering Congress and Exposition 2005, Orlando, Florida, USA [14]]. Moreover, for a vibrating MEMS gyroscope also the quadrature error, i.e. [Y.Y. Bao, C.L. Yung, Modelling and compensation of quadrature error for silicon MEMS microgyroscope], has to be kept below a given threshold in order to be able to acurately measure low angular speeds. Verification tests are usually carried out at the end of the production process, i.e. when the package is complete. Thus, only electrical measurements are possible. In the present paper, a reliable approach to determine mechanical parameters as well as the quadrature error of the MEMS device through electrical measurements is proposed and results are compared to more traditional optical measurements. © 2006 Elsevier B.V. All rights reserved. Keywords: Dynamic behaviour; Quadrature error; MEMS gyroscope sensor

1. Introduction The dynamic characterisation of a MEMS vibrating device is of fundamental importance both during the design phase and during the final verification phase. During design phase, testing of the dynamic behaviour of a MEMS vibrating device is mainly used to set up and validate numerical models of the device. Such models are then used to evaluate the effects of design changes on the performances of the device without the need producing a prototype. At this stage, the main target of the testing campaign is to fully characterize the dynamic behaviour of the MEMS device in any possible working condition (e.g. at different working pressures). Thus, the testing usually requires lots of time and has to be as flexible as possible. During the final verification phase, it is necessary to verify that the mechanical properties of the MEMS device are within

∗

Corresponding author. Tel.: +39 02 2399 8487; fax: +39 02 2399 8492. E-mail addresses: [email protected] (A. Cigada), [email protected] (E. Leo), [email protected] (M. Vanali). 0924-4247/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2006.11.006

the accepted tolerance. To keep prices low, it is of fundamental importance to test all produced MEMS devices in the shortest time possible. Not all working conditions have to be tested since the device is in its final configuration (determined during the design phase as the optimal one from the performance point of view). The adopted testing method has therefore to allow high execution speed together with a high degree of accuracy to be able to discriminate between correctly working pieces and pieces that have to be withdrawn. Many testing techniques to characterise the dynamic behaviour of MEMS devices have already been described in the literature. Such techniques can be roughly divided into two main groups, one based on direct measurements of the device movements, mainly using a laser Doppler vibrometer [5,6], and the other based on indirect measurements of the device movements through electrical measures [7]. The excitation of the device may be of different types. The sinusoidal (or slowly changing sweep sine) excitation is the one that allows to test the device in (almost) steady-state conditions thus permitting to easily determine both the linear and nonlinear behaviour of the structure. Moreover, such excitation can be applied with both direct and indirect measurement techniques.

A. Cigada et al. / Sensors and Actuators A 134 (2007) 88–97

The drawback is that testing time is usually very long thus unacceptable in the verification phase. Another type of excitation is the random (white noise) one. This excitation introduces the same mean energy content in the considered frequency range thus allowing to significantly reduce testing time. However, the control on the excitation amplitudes is lost thus making it impossible to apply some classical electrical measurement techniques. One last excitation type that can be easily applied is the step one. This type of excitation allows to identify structural parameters in one single fast measurement. However, the energy introduced into the system is very limited. Thus, high resolution identification techniques are required. Knowing the dynamic response of the MEMS device is the first step to determine its modal parameters [1,3], i.e. its resonance frequency fi and the corresponding quality factor Qi . This can either be done in the frequency domain or in the time domain using well known identification techniques [3,4,6]. The aim of this work is to develop a fast, efficient and reliable measurement method that allows to identify the modal parameters of a MEMS tuning fork gyroscope [2] both in the design phase, e.g. for statistical analyses, and in the verification phase, when gyros have already been installed in their final package. It should be noted that, although applied to the gyroscope shown in Fig. 1, the developed technique could be applied to any vibrating MEMS device (Fig. 2). MEMS tuning fork gyroscopes usually operate at low pressure in order to reduce the damping thus providing higher output sensitivity [8]. This means that devices are covered by a sealing cap that makes it impossible to use direct measurement techniques to assess the structure’s dynamic response. Thus, only indirect, i.e. electrical, measurement techniques can be applied. In this paper, optical measurements carried out in a vacuum chamber before sealing the MEMS device with a laser Doppler vibrometer, are used as a reference for the proposed electrical measurement technique to determine its systematic bias error and the related uncertainty. The electrical measurement technique is based on the measurement of the ground currents flowing in different part of the device [7] when excited through a step excitation, the only type of excitation that fulfils the test duration requirements of ready-to-distribute devices. The paper is articulated as follows: at first a brief description of the structure under test is provided; then, a FEA model of the structure is developed thus showing that, in the frequency range of interest, a very simple one degree of freedom (1dof) lumped parameter model correctly describes the dynamic response of the device; hence, the electrical measurement technique is described in great detail; finally, the described measurement technique is

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Fig. 1. (a) SEM photograph and (b) schematic drawing of the four-mass gyro: (1 and 2) square-shaped masses; (3 and 4) C-shaped masses.

applied to the considered MEMS gyroscope showing that it is fully equivalent to a more accurate and long-lasting optical measurement technique. 2. Test structure and FEM model The MEMS device used to validate the proposed electric method for the determination of the dynamic behaviour is shown in Fig. 1a. The gyroscope, produced by STMicroelectronics [11], has a symmetrical structure (see Fig. 1b) with the two halves vibrating in counterphase. This is done to cancel out internal inertia forces, to filter out linear acceleration acting on the MEMS sensor and to double the sensitivity of the device to out-of-plane rotations. Two C-shaped masses mc are elastically connected to each other and to the ground (the substrate) through folded beams and their motion along x-direction (drive-axis) is generated through comb-drive actuators. Two square-shaped

Fig. 2. Scheme of the gyro’s operation principle.

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Fig. 4. Third vibration mode determined through the FE model.

Fig. 3. FE model of the mechanical microstructure.

masses mr are linked to the C-shaped masses through four-folded beams (sensing springs) that allow a relative motion along the y-direction (sense-axis) but not along the drive direction. As already said, the actuation is generated by comb-drives that force the C-shaped masses to vibrate along drive direction at a frequency equal to ΩDRIVE . When the MEMS gyroscope experiences an angular rate Ω around the out-of-plane direction (z-axis), the vibrating masses are subject to Coriolis force (proportional to Coriolis acceleration, see Eq. (1)) that acts along sense direction. |aCoriolis | = 2Ωv

(1)

v being the speed of the moving masses along drive direction. The square-shaped masses, that are free to move along the yaxis, start to vibrate also along this direction [10]. Measuring this motion through linear capacitors (capacitance variation), it is possible to determine the angular rate around the out-of-plane direction. Fig. 3 summarizes the gyroscope’s working principle. In order to have high sensitivity, it is important to force the Cshaped and square-shaped masses along drive direction at their first natural resonance frequency. In fact, this type of excitation guaranties the highest medium speeds of the vibrating masses along drive direction, thus the highest Coriolis accelerations (for a given out-of-plane angular rate), thus the highest Corolis forces, thus the widest movements along sense direction. To determine the first natural resonance frequencies both a simplified calculation, based on the estimation of the suspending beams stiffness and on the assumption of a decoupling of the vibration modes along x- and y-directions, and a FEM analysis

can be carried out. The advantage of the FEM analysis is that no assumptions and estimations have to be done. Moreover, it allows to determine adjacent vibration modes, e.g. out-of-plane modes, that could disturb the sensor’s response along sense direction. Thus, a FE model of the MEMS gyroscope was set up (Fig. 3) and its first vibration modes were determined. Table 1 sums up the first vibration modes of the considered gyroscope. As expected, these modes are rigid ones: the first four modes are rigid vibration modes either along x- or y-directions with no coupling between them. Hence, in the frequency range of interest (about 4 kHz), exciting the gyroscope along drive direction, only the third vibration mode of C-shaped and squareshaped masses starts. Fig. 4 graphically shows the third vibration mode. Exciting the square-shaped masses along sense direction, instead, only the firth vibration mode appears (the excitation along sense direction is generated by an angular rate around the out-of plane direction and determines a counter-phase vibration of the inner masses). Exploiting the symmetry of the device, it is possible to schematize the gyroscope through two simple one degree of freedom lumped parameter models [12,13], one describing the dynamics of the C-shaped and square-shaped masses along drive direction and one describing the dynamics of the square-shaped mass along sense direction. These models can be used to identify the modal parameters both in the time domain and in the frequency domain. In this paper, attention will be focused on the drive direction but the same procedure can be applied to the sense direction. 3. Electrical measurement method As already said, the excitation along drive direction is generated by comb-drive actuators. Fig. 5 shows a scheme

Table 1 First vibration modes identified through the FE model 1 2 3 4 5

2657 Hz 3780 Hz 3782 Hz 3793 Hz 12617 Hz

In-phase linear vibration of C-shaped and square-shaped masses along x-axis Counter-phase linear vibration of C-shaped and square-shaped masses along x-axis In-phase linear vibration of square-shaped masses along y-axis Counter-phase linear vibration of square-shaped masses along y-axis In-phase rotational vibration of the C-shaped and square-shaped masses (both x- and y-axis are involved)

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Fig. 5. Scheme of a comb-drive actuator.

of the comb fingers highlighting the stator and rotor parts, the capacitances C1 and C2 and the applied voltages V1 and V2 . A bias voltage VB and a harmonic voltage having VA amplitude and Ωdrive frequency are applied to the stators of the comb-drive. The harmonic voltage is applied in counter-phase to the two-stator parts of the comb-drive: V1 = VB − VA sin(Ωdrive t) V2 = VB + VA sin(Ωdrive t)

(2)

C1 = C0 − C x

(3)

C2 = C0 + C x

C0 being the capacitance when no voltage is applied, x the rotor displacement (hence the mass displacement) along drive direction and C the derivative of the capacitance with respect to the x-coordinate. It is assumed that C is independent of the rotor displacement along drive direction. This assumption is correct if boundary effects are neglected. The excitation force on the C-shaped and square-shaped masses is therefore equal to: 

1 ∂C1 2 1 ∂C2 2 V + V 2 ∂x 1 2 ∂x 2

x = X0 sin(Ωdrive t + ϕ), |X0 | = 

F¯ 0 /ki 2 2 (1 − (Ωdrive /ωi2 ))



Assuming that the capacitance C1 of the left part of the combdrive is equal to the capacitance C2 of the right part when no voltage is applied and that the capacitances on the two sides of the comb-drive vary linearly with the rotor displacement, we obtain:

Fdrive = F1 + F2 = N

Fig. 6. Electrical connections of the mechanical part of the gyroscope: the ROT pad is connected to the mobile parts, the D1 (S1) and the D2 (S2) pads are connected to the stators of comb-drive actuators (parallel plates) and the SD1 and SD2 pads are connected to the stators of the SD comb-drives.



= 2C VA VB sin(Ωdrive t) = F¯ 0 sin(Ωdrive t)

(3)

where N is the number of actuation comb-drives. Applying the same excitation force to the simple one degree of freedom lumped parameter model that describes the dynamics of the MEMS gyroscope along drive direction, the motion of the C-shaped and square-shaped masses can easily be determined:

ϕ = −a tan

,

+ (Ωdrive /ωi Qi ) 

Ωdrive /ωi Qi 2 1 − (Ωdrive /ωi2 )

2

(5)

Thus, the transfer function of the gyroscope along drive direction can be determined if the motion of the C-shaped and square-shaped masses and the excitation force are known. Measuring the applied voltages and the capacitance variation in the comb-drives due to a rotor movement, it is possible to easily determine the excitation force. The motion of the masses instead can either be directly measured through an optical system (e.g. a laser Doppler vibrometer) or through the ground currents flowing though some of the electrical connections of the device. Fig. 6 shows the electrical connections of the gyro’s mechanical part. The moving parts, i.e. the C-shaped and square-shaped masses together with the rotors of both the comb-drives and the sense plates, are electrically connected (ROT pad). Connecting the ROT pad to the ground through a current measuring device, it is therefore possible to measure the ground current i. Neglecting eddy current effects [7], the measured ground current is proportional to the variation of the capacitances and of the applied voltages of both the comb-drives and the parallels plates. If no voltage is applied to the stators of the parallel plates, the ground current i measured at ROT pad due to a motion of C-shaped and square-shaped masses along drive direction is equal to: irot,D = ND

d d d (C1 V1 ) + ND (C2 V2 ) = ND (C1 V1 + C2 V2 ) dt dt dt (6)

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where ND is the total number of the comb-drive capacitors. Substituting Eqs. (3) and (2) into Eq. (6), we obtain d irot,D = ND (2C0 VB + 2C VA x sin(Ωdrive t)) (7) dt where x is given by Eq. (5). Thus, the ground current measured at ROT pad with no voltage applied to the stators of the parallel plates is equal to irot,D = ND (2C VA X0 Ωdrive sin(2Ωdrive t + ϕ))

(8)

Hence, the harmonic component of the measured ground current having a frequency twice that of the applied voltage is proportional to the motion of the C-shaped and square-shaped masses along drive direction. It is therefore possible to determine the transfer function of the gyroscope along drive direction if an harmonic voltage is applied to the comb-drives. Another possible way of determining the motion of the vibrating masses along drive direction is to measure the ground current at pads SD1 and SD2 that are commonly used to verify the mass displacement and that are electrically linked to the “sense” comb-drive. The approach is similar to the one described above. Applying a constant voltage V¯ 1 to the stators through SD pads, the ground currents iSD1 and iSD2 measured at SD1 and SD2 pads are equal to: d (C1 V¯ 1 ) = −NSD C X0 Ωdrive V¯ 1 cos(Ωdrive t + ϕ) dt = −iSD2 (9)

iSD1 = NSD

where NSD is the number of capacitors of the “sense” comb-drive Subtracting iSD2 from iSD1 (to double the sensitivity) and considering the harmonic component having the same frequency of the applied voltage, the motion of the C-shaped and squareshaped masses along drive direction can be determined: i = iSD1 − iSD2 = NSD = NSD V¯ 1 (−2C x˙ )

d d (C1 V¯ 1 ) − NSD (C2 V¯ 1 ) dt dt (10)

In fact, as shown by Eq. (10), the measured current is proportional to the vibration velocity of the masses. It should be observed that the above equation is truely independent of the applied excitation (harmonic, step or impulsive) if and only if a constant voltage V¯ 1 is applied. The two techniques based on the measurement of the ground currents at pads ROT and SD1–SD2 were used to determine the motion of the suspended masses along drive direction, i.e. to determine its model parameters along x-axis. As already said, this is of fundamental importance to verify the correspondence of the gyroscope to design specification, i.e. to verify that the excitation frequency along x-axis generated by the electronic parts of the MEMS device is able to excite the gyroscope at its resonance [10].

Fig. 7. Scheme of a sense plate.

In order to measure the quadrature error, an harmonic voltage is also applied to the stators of the parallel plates and the resulting motion of the C-shaped and square-shaped masses is determined. In absence of any comb-drive excitation, a motion of the inner masses along sense direction would result in a ground current at ROT pad that would be interpreted as a motion of the C-shaped and square-shaped masses along drive direction. Thus, the voltage applied to the stators of the parallel plates must not produce any motion of the inner masses. The excitation frequency has therefore to be much above any (rigid) resonance frequency along sense direction (seismographic region). Considering that both the rigid resonance frequencies along sense direction are below 4 kHz, an excitation frequency of 90 kHz applied to the stators of the parallel plates should satisfy the above request. Fig. 7 shows a scheme of a sense plate highlighting the stator and rotor parts, the capacitances CS1 and CS2 and the applied voltages VS1 and VS2 . As already said, to measure the quadrature error an harmonic voltage having VAS amplitude and Ωsense frequency is applied to the stators of the parallel plates. Thus, V1S = −VAS sin(Ωsense t) V2S = VAS sin(Ωsense t)

(11)

Assuming that the system is symmetric when no voltage is applied, i.e. that the capacitance C1S of the upper part of the parallel-plate is equal to the capacitance C2S of the lower part, we obtain     L+x L+x C1S = ε0 h =A g+y g+y     (12) L+x L+x C2S = ε0 h =A g−y g−y where L is the length of the overlapping sense plates, g the distance between adjacent plates in their initial position and h is their thickness along the out-of-plane direction (z-axis). Approximating the above equations through a Taylor series truncated at the third order, C1S and C2S become

 x2 + c y2 + 2c xy] + 1 [c x3 + 3c x2 y + 3c xy2 + c y3 ] C1S ∼ = C1S0 + cx x + cy y + 21 [cxx yy xy xxy xyy yyy 3! xxx  x2 + c y2 + 2c xy] + 1 [c x3 + 3c x2 y + 3c xy2 + c y3 ] C2S ∼ = C2S0 + cx x + cy y + 21 [cxx yy xy xxy xyy yyy 3! xxx

(13)

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where the term c x (c x ) is the partial derivative of C1S (C2S ) with respect to the x coordinate evaluated in x = 0 and y = 0 (initial condition). Determining the various partial derivatives that appear in Eq. (13) we obtain L 1  L 1 L 1 L C1S = A C2S

g

+

x−

y− 2

xy + 2

y2 + 3

xy2 − 3

y3 4

g g 2g g 3g g  1 L 1 L 2 1 L 2 =A + x − 2 y + 2 xy + 3 y + 3 xy + 4 y3 g g g 2g g 3g g

L

(14)

Due to the fact that the gap g between the sense plates in the initial condition is much higher then the displacement y along sense direction due to the quadrature error [9], the second and third order terms of Eq. (14) can be neglected. Thus, the capacitance of the upper and lower parts of the parallel plates can be rewritten as:

L 1 L C1S = A + x − 2y g g g

(15) L 1 L C2S = A + x + 2y g g g Eqs. (11) and (15) (as done with Eqs. (2) and (3)) can be used to determine the ground current measured at ROT pad due to the movement of the square-shaped masses along sense direction (quadrature error): d d (C1S V1S ) + NS (C2S V2S ) dt dt d = NS (C1S V1S + C2S V2S ) dt

irot,S = NS

(16)

NS being the number of parallel plates. Substituting Eqs. (11) and (15) into Eq. (16) we obtain

d 2L irot,S = NS (17) A 2 yVAS sin(Ωsense t) dt g As already said, the excitation frequency Ωsense along y-axis does not determine any motion of the square-shaped masses along sense direction (seismographic region). Thus, if a motion along sense direction is measured, this is due to the quadrature error and has a frequency equal to Ωdrive . In such case, the above

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equation becomes d 2L YVAS A 2 [cos((Ωsense − Ωdrive )t − ϕS ) irot,S = NS dt g 2 − cos((Ωsense + Ωdrive )t + ϕS )]]

(18)

where ϕS is the phase of the y displacement with respect to the x displacement and Y is the amplitude of displacement along y-direction. As can be noticed, the ground current measured at ROT pad due to the capacitance between the parallel plates has two harmonic components, one at Ωsense − Ωdrive and one at Ωsense + Ωdrive , i.e. these two harmonics are centred around the sense frequency Ωsense . This is true in case of symmetric structures (equal capacitances of the upper and lower part of the parallel plates in the static conditions) and in presence of quadrature error. Otherwise, an additional term having a frequency equal to Ωsense is obtained. Therefore, if the spectrum of the measured ground current at ROT pad shows only two peaks around Ωsense frequency, the structure is symmetric (the capacitances of the upper and lower parallel plates are equal in static conditions) but there is quadrature error; if a third peak having a frequency equal to Ωsense is obtained, the structure is not symmetric. Thus, exciting the MEMS gyroscope through the combdrives along drive direction and through the parallel plates along sense direction, the ground current measured at ROT pad is equal to d d d (C1 V1 ) + ND (C2 V2 ) + NS (C1S V1S ) dt dt dt d + NS (C2S V2S ) = irot,D + irot,S dt

irot = ND

(19)

The two contributions can be easily split due to the fact that they have completely different frequencies: irot,D has a frequency that is equal to 2Ωdrive while irot,S has two (three) harmonic contributions at frequencies Ωsene − Ωdrive and Ωsene + Ωdrive (and Ωsene ). Therefore, through the described methodology both the dynamic response of the device and the quadrature error of just the mechanical part can easily be measured.

Fig. 8. Test bench.

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4. Experimental setup and results A first set of measurements has been carried out during the design phase, i.e. on just the mechanical part of the MEMS gyroscope. Thus, both optical and electrical measurements could be carried out at the same time. Fig. 8 shows the used test setup. The facility is equipped with a vacuum chamber in order to be able to work at various pressure conditions. The gyroscope is mounted on a specially designed test board that is used to excite the gyroscope both along drive and sense directions with a changing excitation frequency and to measure the ground currents necessary for the proposed measurement methodology. The same board can also be used for the testing of the final package of the gyroscope. The optical measurements are performed using a Polytech laser Doppler vibrometer (LDV), equipped with modules OFV512 and OFV3001. To optically measure the vibrations of such small structures, special care has to be taken in the focusing of the laser spot. To this purpose, a microscope is used. The connections between the gyroscope, the test board and the signal generators and sensors are schematically depicted in Fig. 9. As can be seen, an external signal generator produces • the bias voltage VB and the harmonic voltage VA sin(Ωdrive t) to be applied to the stators of the comb-drives; • the bias voltage V¯ 1 to be applied to the stators of the SD combdrives, when the system’s response is measured through the SD ground currents (Eq. (10)). The ‘Driving/Sensing actuation’ block sums/subtracts the bias and the harmonic voltages and applies the two voltages V1 and V2 to the comb-drives. This block can be bypassed via a switch to be able to apply a step voltage directly to the actuators. The ground currents at the ROT, SD1 and SD2 pads are converted into voltages by two current-to-voltage converters to be able to use standard acquisition systems. The adopted data acquisition system is a PC-based system with built-in anti-aliasing filters.

Fig. 9. Scheme of the connections of the test bench.

Table 2 Tests carried out during the design phase to compare the electrical and optical methodologies Pressure (Pa)

105 14

Method Optical (LDV)

Electrical (rotor current)

× ×

×

Electrical (stator currents) ×

The applied force along drive direction is known once the bias voltage VB and the harmonic voltage VA are known. However, as shown in the experimental results, the test board that allows to apply these voltages to the stators of the comb-drives introduces a constant phase delay in the transfer function that has to be taken into account during the identification of the system response. The tests carried out during the design phase to compare the electrical and optical testing methodologies were performed at two different pressure levels (105 Pa and 14 Pa, see Table 2). All tests were carried out using a stepped sine excitation in the frequency range 3000–5000 Hz (around the resonance frequency) and were repeated for two gyroscopes to assess the variability of modal parameters. The acquired signals were post-elaborated to determine the system’s transfer functions (asterisks in Figs. 10–13). Using the simple one degree of freedom model, i.e. using Eq. (5), the resonance frequency f and quality factor Q were then identified in a least square sense. The numerical transfer functions with the identified model parameters are shown in Figs. 10–13 with the continuous line. It should be observed that the modulus of the transfer functions is given in V/V as no calibration was applied to the acquired voltages. The resonance frequency and the quality factor identified at ambient pressure (105 Pa) with the two measurement methods are shown in Table 3. As can be seen, there’s a very good agreement between the results achieved with the two methodologies.

Fig. 10. Modulus of the transfer function at atmospheric pressure (105 Pa) measured with the LDV.

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Fig. 11. Phase of the transfer function at atmosphericpressure (105 Pa) measured with the LDV.

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Fig. 13. Phase of the transfer function at atmospheric pressure (105 Pa) measured via the rotor ground current. Table 4 Resonance frequency and quality factor identified at a pressure of 14 Pa with the two measurement methods

Resonance frequency [Hz] Quality factor

Fig. 12. Modulus of the transfer function at atmospheric pressure (105 Pa) measured via the rotor ground current.

Optical method (LASER)

Electrical (SD rotor current)

5172 9435

5172 9189

It can therefore be concluded that the electrical method, based on the rotor current or stator currents, can be used instead of the optical (direct) method to identify the modal parameters of the considered MEMS gyroscope. The next step was to measure the dynamic response of the complete package. It should be observed that the working pressure of the fully packaged device is 19 Pa. As already said, only the electrical methodologies can be applied in this case. To significantly reduce the testing time, instead of a stepped sine

Similar good agreements can be obtained if, instead of the rotor current, the stator currents are used to determine the system’s response. Table 4 sums up the resonance frequency and the quality factor identified at a pressure of 14 Pa using the optical method and the electrical method based on the stator currents. As can be seen, also at low pressure levels the identified modal parameters differ for less then 3%, which is comparable with the uncertainty usually considered as acceptable for such kind of measurements. Table 3 Resonance frequency and quality factor identified at ambient pressure (105 Pa) with the two measurement methods Optical (LDV) Resonance frequency [Hz] Quality factor

3926 23

Electrical (rotor current) 3925 23

Fig. 14. Time history of the gyroscope’s dynamic response along drive direction in case of a step excitation.

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Fig. 15. Time history of the gyroscope’s dynamic response along drive direction in case of an impulsive excitation.

excitation, both a step and an impulsive excitation were tested (Figs. 14 and 15). For analysis purposes, the impulse responses have been divided in two groups, those corresponding to positive steps (rising edge) and those corresponding to negative steps (falling edge). This is done because the working conditions of the gyroscope are different in the two cases: the rising step is obtained by applying a constant bias voltage to all combdrives and this may change the system mechanical response. The Hilbert transform is then used to determine both the resonance frequency (phase of the transform) and the envelope of the function (modulus of the transform). Figs. 16 and 17 show an example of the identification of modal parameters through the Hilbert transform. Table 5 sums up the obtained results in terms of identified resonance frequency and quality factor together with the spread of the population (95% confidence interval) derived by the analysis of the various considered steps (separating jumps up from jumps down). While the resonance frequency has only a very

Fig. 16. Time history of phase of the Hilbert transform.

Fig. 17. Time history of the modulus of the Hilbert transform. Table 5 Resonance frequency and quality factor identified considering both the rising step and the falling step

Resonance frequency [Hz] Quality factor

Step up

Step down

4429 ± 0.2 3805 ± 80

4429 ± 0.6 4238 ± 83

small dispersion and the estimated mean value is equal for step up and step down excitations, the estimated mean value (as well as dispersion) of the quality factor is significantly different in the two cases. As already said, this is due to the different working conditions of the gyroscope. The proposed electrical measurement methodology is then used to determine the quadrature error. Referring to Eqs. (2) and (11), a sinusoidal excitation along drive direction and along sense direction were applied, the first having a frequency equal to the corresponding resonance frequency (around 4 kHz) and the second having a frequency much higher than the corresponding resonance frequency (90 kHz) to avoid any motion of the inner masses. As already said, two peaks in the transfer function at Ωsene − Ωdrive and Ωsene + Ωdrive indicate that there is quadrature error and their amplitude is proportional to the amplitude of this error. Fig. 18 shows the auto spectrum of the measured voltage signal proportional to the ground current. The two peaks due to the quadrature error are clearly visible. As already said,

Fig. 18. Auto spectrum of the ground current (converted into a voltage signal) measured during the quadrature error tests.

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for the comb-drives. As expected, the considered device shows up a quadrature error whose amplitude varies with the excitation force amplitude and with the working pressure. Acknowledgements

Fig. 19. Amplitude of the quadrature error of the sensor for two different working pressure values as a function of the harmonic voltage amplitude along x-axis.

The authors gratefully acknowledge STMicroelectronics staff of Cornaredo plant that have helped during the various research phases. In particular, Ing. Guido Spinola and Ing. Fabiano Frigoli are greatly acknowledsged for their precious advices and help. References

instead, the peak at 90 kHz is due to the non-symmetric layout of the gyroscope, i.e. to differences in the capacitances of the upper and of the lower parallel plates. The same test has been performed for different values of the harmonic voltage amplitude applied to the comb-drives (VA ), i.e. for different excitation forces, and for two pressure values (14 and 75 Pa). The results are shown in Fig. 19. As expected, for the same applied excitation force, the quadrature error is higher when the pressure level is lower. Moreover, the relation between amplitude of quadrature error and harmonic voltage amplitude is almost linear at 75 Pa while it reaches (almost linearly) a maximum level at 14 Pa. This means that, at 14 Pa, there is no linear relation between system response and applied force. 5. Conclusions The proposed electrical methodology based on the measurement of the ground current was used both to identify the main mechanical parameters (resonance frequency and quality factor) of the gyroscope and its quadrature error. Two different approaches of this methodology have been presented: a first one based on the measurement of the ground current at ROT pad (flowing through the mobile parts) and a second one based on the measurement of the ground current at SD1 and SD2 pads (flowing through the comb- drives stators). Both approaches allow to use a step excitation thus significantly reducing the time required for the identification of the modal parameters. To check the validity of the obtained results, the identified modal parameters are compared to those obtained through the widely used direct optical method. A very good agreement has been found. The electrical method is then used to measure the quadrature error by applying a high frequency excitation for the parallel plates and a low frequency excitation (tuned at the resonance)

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