A New Quartz Monolithic Differential Vibrating Beam ...

A New Quartz Monolithic Differential Vibrating Beam Accelerometer O. Le Traon*, D. Janiaud*, M. Pernice*, S. Masson*, S. Muller*, J-Y. Tridera+ *

Instrumentation & Physics Department, ONERA, Chatillon, France + DGA/LRBA, Vernon, France

Abstract-This paper deals with a new monolithic differential quartz Vibrating Beam Accelerometers (VBA), called VIASC. The concept of this accelerometer includes the three major insulation requirements of the beams in the same quartz monolithic structure: insulation of the beams against thermal stresses, insulation of the beam vibrations and insulation between the two beams. In this paper, the concept of this accelerometer and its manufacturing process by chemical etching are presented. Preliminary results, which confirm the relevance of the VIASC concept, are given and future work on the sensing element is discussed. Two VIASC developments are envisaged: an accurate VIASC version compatible with inertial navigation requirements and a miniaturized version adapted to guidance applications.

I.

INTRODUCTION

Vibrating inertial sensors - i.e. Vibrating Beam Accelerometers (VBA) and Coriolis Vibrating Gyros (CVG) are a very promising new generation of sensors, compatible with micro-machining technologies, and particularly well suited for low cost and miniaturized inertial measurement units for example in stabilization, guidance or inertial navigation applications [1][2][3] (coupled with radioelectric positioning systems, such as GPS or future Galileo systems). For a few years, ONERA has been developing two monolithic quartz sensors (Fig. 1), the “VIA” accelerometer (Vibrating Inertial Accelerometer) [4] and the “VIG” gyro (Vibrating Integrated Gyro) [5]. The use of quartz crystal gives two main advantages: firstly, it allows, through the piezoelectric effect, a very easy excitation and detection of suitable vibrations and secondly, it takes advantage of the highly stable mechanical properties of the quartz itself. Thus, low cost/high performance vibrating sensors are made possible with the use of quartz.

Fig. 1. Quartz Vibrating Inertial Sensors developed at ONERA

The VIA accelerometer accuracy [6] is ~300 µg (measurement range of ±100g) and the expected VIG accuracy [7] is ~10 °/h (measurement range of 1000 °/s). Vibrating Beam Accelerometers (VBA) are particularly attractive for their scale factor accuracy and their frequency output. VBA are based on the change in the resonance frequency of a vibrating beam when subjected to an acceleration, and generally use two vibrating beams in a differential arrangement in order to reduce common parasitic sensitivities e.g. temperature, pressure, ageing. Accurate VBA require three major insulations of the beams: - Insulation against thermal stresses: under temperature variations, thermal stresses, due to differences of materials expansion between the quartz structure and its base, are induced into the structure. The beams must be insulated from these stresses in order to obtain a good thermal behavior of the VBA (weak thermal sensitivity with weak hysteresis); - Insulation of the beams vibrations: to preserve the high quality factors of the quartz beams, and thus to obtain a good output stability of the accelerometer, the vibrating energy losses out of the quartz structure must be minimized; - Insulation between the two beams: a differential VBA uses two beams in a push-pull configuration. So, generally, the two frequencies cross in the measurement range and can lead to a specific problem known as "lock-in". The lock-in phenomena is due to the mechanical coupling between the two beams and is characterized, close to the frequency crossing, by a lock between the two frequencies, with no frequency variations in spite of the applied acceleration. That leads to a blind zone of measurement which degrades the VBA accuracy. Thus, a high insulation between the two beams is required to reduce the "lock-in" phenomena. The VIA is based on two monolithic transducers, each containing a simple beam and mounted in a push-pull configuration. The VIA concept [8] includes all essential insulating requirements for high accuracy VBA, and thanks to the use of two separate transducers to procure a high isolation between the two beams, the lock-in zone of this accelerometer is not discernable. This accelerometer has now been transferred to industry. The VIA accuracy is around 300 µg, all errors combined, for a measurement range of +/-

100 g (in harsh environments and including long term stability). Nevertheless, the VIA configuration is not so well suited to miniaturization and the two separate transducers complicate the assembly process. For these reasons the development of a single chip differential accelerometer has been undertaken. In this paper, an original monolithic differential VBA structure is presented. This new configuration [9], called VIASC (VIA single chip), includes two VIA transducers in a planar push-pull arrangement, linked together by a frame. The efficiency of the beam’s insulation is discussed and the role of the frame around the two transducers as an isolating system is explained. The manufacturing process is described and preliminary experimental results of VIASC prototypes are presented. These first results confirm the relevance of the VIASC concept, since no lock-in zone is observed on the two prototypes. Future work on the VIASC sensing element is discussed and two developments are envisaged: an accurate VIASC version compatible with inertial navigation specifications and a miniaturized one adapted to guidance applications. II.

VIASC CONFIGURATION

Fig. 1 shows the VIASC configuration. This accelerometer is made up of two VIA transducers in a planar pushpull configuration and linked together by an external frame. Each VIA transducer (Fig. 2) includes an active part composed of a proof mass, a simple beam and two articulations. The beam vibrates in the plane of chip and the sensitive axis is quasi perpendicular to the chip. When the transducer is submitted to acceleration, the proof mass induces tensile or compressive stresses into the beam, which modify its resonance frequency. An insulating system (the insulating frame and its two links) connects the active part of each transducer to the external frame. The resonance frequencies F1 and F2 are about 60 kHz and their sensitivity K1 is 15 Hz/g. The output of the accelerometer is the difference between the two transducer beam frequencies F1 and F2, the scale factor of this accelerometer is thus 30 Hz/g.

Fig. 1. Differential configuration of the VIASC: F = F + K ⋅ Γ + K ⋅ Γ2 + F(temp, pressure,...)  1 10 1 2  2 F2 = F20 − K1 ⋅ Γ + K2 ⋅ Γ + F( temp, pressure,...) => F1 − F2 = F10 − F20 + 2.K1.Γ

This differential arrangement reduces common parasitic sensitivities (e.g. temperature, pressure and ageing), and also improves the frequency output linearity with respect to the applied acceleration (the difference of the two beams frequencies removes the acceleration quadratic terms of each beam). This configuration is even more effective the more similar the two transducers. This configuration is particularly well suited to a quartz crystal chemical etching process, since two nominally identical VIA transducers can be obtained in spite of chemical etching facets (this differential configuration takes advantage of the quartz crystal symmetry). This is beneficial for an efficient differential effect.

Fig. 2. VIA transducer.

The VIASC configuration includes the three major insulations of the beams, required for an accurate VBA: A.

Insulation against thermal stresses

The whole VIASC accelerometer is composed of the quartz structure mounted on a metallic case, and two electronic oscillator circuits. Under temperature variations, thermal stresses due to differences in material expansions are induced into the quartz structure. The two beams must be protected from those stresses in order to avoid any undesirable frequency variation against temperature. Thanks to the insulating system of each VIA transducer, and in particular its upper and lower links which allows a free dilatation of the active part, thermal stresses stay located in the external frame of the quartz structure (Fig. 3.a) and stresses in the beams are very weak, around 107 times smaller than in the frame, as shown Fig. 3.b. The efficiency of the insulating system can be precisely evaluated by comparing beam axial stresses due to temperature to those induced under a 1 g acceleration: the obtained sensitivity for each VIA transducer is very weak, about 0,3 µg/°C, thus allowing a good behavior of the sensor under temperature variations, with low hysteresis.

B.

Insulation of the beams vibrations

The bias stability of the sensor depends directly on the frequency stability of each beam, and thus requires a high quality factor Q for the beam vibrations. Indeed, the higher the quality factor, the less sensitive the beam is to the electronic oscillator phase drift. The quartz structure must be designed to obtain high quality factors for the beams. With regard to the general expression of the Q factor (Eq. 1) where W is the stored vibrating energy and ∆W the total dissipated energy per cycle of vibration, the Q inverse can be written as the sum of different terms depending on the different loss sources (Eq. 2). Q = 2π Q −1 =

∑Q

−1 i

W ∆W

(1)

−1 −1 −1 −1 = Qbulk + Q surface + Q TED + Q sup port

(2)

i

- Qbulk corresponds to bulk losses due to the intrinsic viscosity of the material, defects, impurities…Generally, for high quality material like quartz crystal or silicon, these losses are negligible. - Qsurface represents surface losses, e.g. gas damping, surface defects, deposited electrodes. For chemical etched sensors under vacuum, surface losses are also very weak.

Fig. 3.a Normalized thermal stresses into the accelerometer.

- Q TED is the thermoelastic dissipation which arises from the coupling of the stress-strain state to heat flow into the material, and is generally an important loss mechanism for beam in flexion[10][11]: temperature gradients occur between tensile and compressive fibres of the vibrating beam, generating a heat transfer and inducing a relaxation. This phenomenon is highly dependant on the beam frequency, the thickness (in the plane of motion) and also on the thermal diffusivity, and is well described by the Zener equation: QTED =

Fig. 3.b Thermal stresses with an observation scale divided by 107.(the white areas are out of scale). Fig.3. Thermal stresses into the accelerometer: Under temperature variations, stresses, due to the differences in materials expansions are induced into the quartz structure. Thanks to the two links of the VIA insulating system, the stresses into the beams are very weak, 107 smaller than those in the external frame.

F02 + F 2 F0F α .T.E ρ.C

2

.

with F0 =

π λ ⋅ 2 ρ .C.e2

(3)

where ρ is the density (kg/m3), C the heat capacity (J/kgK), α the thermal expansion (in the direction of motion), T the absolute temperature (K), E the equivalent Young’s modulus (N/m2), F the beam frequency (Hz), λ the thermal conductivity (W/mK), and e the beam thickness (m). F0 represents the frequency transition between isothermal (the temperature is in equilibrium during the vibration strain) and adiabatic (no heat flow during the vibration strain) vibrations, and the maximal dissipation is obtained for F = F0. For the typical beam of the VIASC accelerometer (F ~ 60 kHz , e ~ 68 µm), F >> F0, QTED can be thus written (Eq. 4):

QTED =

2 (ρ ⋅ C )2 .F ⋅ e 2 π α 2 .λ .T.E

(4)

For quartz data, Eq.4 gives a theoretical QTED of the VIASC beams around 24 000. It would be possible to obtain higher QTED by increasing, for example, the beam thickness, but the choice of these beam dimensions is the result of a compromise between all the sensor parameters (beam frequency, K1 sensitivity, quality factor, measurement range, first resonance frequency of the structure). Nevertheless, Zener theory, based on a simple isotropic bending beam with a constant curvature, does not fit so well with experimental results. Complex finite element modelling [12], taking into account the thermopiezoelectric coupling, the real geometry of the beam (Fig. 4), the anisotropy of the quartz crystal and also the effect of the thermal conductivity of gold electrode patterns deposited on the beam, allows more precise quantification of the beam thermoelastic damping. This specific modelling [13] gives a value of 17 000, in perfect accordance with experimental measurements. - Qsupport fits with the energy loss out of the quartz structure. Indeed, the quartz structure has to be fixed on a support, and it is important to reduce the transmitted energy into the support, likely to be dissipated by the viscosity of its material and thus reducing the Q factor of the beams. Eq. 5 gives the expression of Qsupport [14], which is directly proportional to the ratio of the energy into the resonator wresonator to that into the support wsupport. w resonator 1 tan(δ ) w sup port η sup port ⋅ ω s tan(δ ) = E sup port

Fig. 4. Double-ended tuning fork: In a perfect tuning fork, dynamic torques and forces are balanced and vibrating energy stays thus located into the tuning fork. The VIASC chemical etching process does not fit with a balanced tuning fork: the obtained beams sections will induce a torque, transmitting vibrating energy in the rest of the quartz structure.

Unfortunately, the specific quartz chemical etching process of the VIASC can not produce a balancing tuning fork (Fig. 4). It is the reason why the VIA concept is based on a simple beam including an insulating system to reduce the energy losses out of the quartz structure. This insulating system is based on the principle of a filtering suspension (Fig. 5) and thanks to its flexibility in regard to the main dynamic excitations of the beam, the energy losses out of the transducer are very weak.

Q support = with

(5)

where ηsupport is the support material viscosity (N.s.m-2), ωs (rd/s) the excitation pulsation of the support (corresponding to the beam pulsation) and Esupport (N/m2) the Young modulus of the support material. Generally, tan(δ) is around 0.1, and only 1% of energy lost into the support will give a Qsupport equal to 1000, and thus will limit the resulting overall total Q to this low value. In order to preserve the high intrinsic quality factor of the beams (QTED = 17 000), the energy loss out of the quartz structure must to be minimized (with QTED around 17 000, the Qsupport has to be higher than 1 000 000 to not deteriorate the QTED more than 2%). This is the reason why VBAs generally use a tuning fork as the sensitive element [15][16], instead of a simple beam in order to take advantage of its decoupling structure.

Fig.5. Principle of a filtering suspension: The beam solicits the filtering suspension with an alternative force F (frequency ω0~60 kHz). Due to the lower resonance frequency of the filtering suspension (ωF~3000 Hz), the transmitted force into the support Fs is attenuated by the ratio (ω0/ωF)2. In terms of energy, the energy ratio (W0/Wsupport) is equal to [(K0/K).(ω0/ωF)4]. Generally the beam stiffness K0 is greater than the support (glue) stiffness K thus Qsupport>> (ω0/ωF)4 ~160 000, which is highly compatible with the intrinsic QTED ~17 000 of the beam.

The efficiency of the insulating system is illustrated in Fig. 6 which shows the strain energy in the structure when the left beam vibrates at its resonance frequency. The strain energy in the mounting areas is very weak, around 106 times weaker than in the beam. The calculated Qsupport is about 107 (for tan(δ)=0,1), which is highly compatible with the intrinsic beam QTED of 17 000.

Fig. 6.a Normalized strain energy into the structure when the left beam is excited at its resonance frequency. The maximum strain energy is located in the built-in ends of the beam.

Fig. 6.b Strain energy with an observation scale divided by 106. The strain energy in the mounting areas is very weak, 106 weaker than in the beam.

The VIASC concept procures two main advantages in comparison with the tuning fork configuration: firstly, because of the use of a simple beam, for the same acceleration sensitivity K1, the VIASC needs a proof mass two times smaller than for a tuning fork configuration. Secondly, the efficiency of the insulating system, which only depends on the ratio between the beam frequency and suspension frequency, does not require the very strong matching necessary for a tuning fork. This is an obvious advantage, especially at the scale of micro-systems.

Fig. 7 Lock-in phenomenon: close to the crossing frequency, an important non-linearity of the frequency-acceleration relation appears and each beam frequency signal is disturbed by the other beam. This creates a blind zone where the two beams are locked together.

This noise increase leads to a lock-in of the two oscillator circuits: the two beams vibrate at their average frequency, with no frequency variations in spite of the applied acceleration. This creates a “blind zone” of measurement which degrades the VBA accuracy. In monolithic differential VBA, the lock-in phenomena is generally due to the mechanical coupling between the two beams [17], which prevents the crossing of the beam frequencies and leads to a forbidden zone. The forbidden zone does not correspond directly to the locking-zone due to the oscillator electronics coupling, but measures the coupling level between the two beams inducing the lock-in. So, generally, the forbidden zone gives a good order of magnitude of the lock-in zone. A high insulation between the two beams is thus required to reduce the “lock-in” phenomena, and the frame around the VIA transducers constitutes an efficient insulating system, as explained by the equivalent mass/spring system of Fig.8.

C.

Insulation between the two beams In a differential VBA like the VIASC, two vibrating beams in a push-pull configuration are used in order to reduce common parasitic sensitivities e.g. temperature, pressure and aging. The differential configuration is more effective, the more similar the two beams (i.e. the frequencies are very close). Generally the two frequencies cross in the measurement range and can lead to a specific problem known as “lock-in”. The “lock-in” phenomenon is described in Fig. 7 and is characterized (close to the crossing frequency) by firstly a non-linearity of the frequency-acceleration relation, and secondly a drastic noise increase (each beam signal is disrupted by a noise signal at the frequency of the other beam).

Fig. 8 Equivalent mass/springs system of the VIASC accelerometer VIASC .

The fundamental equations of the dynamics applied to the masses m and M produce the 4 pulsations of the system and the expression (assuming that k/(K1+K2)