Compensation of Parasitic Effects for a Silicon Tuning ... - IEEE Xplore

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IEEE SENSORS JOURNAL, VOL. 6, NO. 3, JUNE 2006

Compensation of Parasitic Effects for a Silicon Tuning Fork Gyroscope Stefan Günthner, Markus Egretzberger, Andreas Kugi, Member, IEEE, Konrad Kapser, Bernhard Hartmann, Ulrich Schmid, and Helmut Seidel, Member, IEEE

Abstract—This paper refers to a silicon micromachined tuning fork gyroscope, which is driven via two piezoelectric thin film actuators. The device responds to an external angular rate by a torsional motion about its sensitive axis due to the Coriolis effect. The shear stress in the upper torsional stem, which is proportional to the angular rate, is detected via a piezoresistive readout structure. In addition to the wanted signal corresponding to the angular rate, there are unwanted contributions from the drive motion, e.g., from mechanical unbalances and from asymmetries of the piezoelectric excitation induced by fabrication tolerances. These effects, which disturb the sensor signal with varying contributions in amplitude and phase, have already been examined for capacitive surface micromachined sensors. In this paper, they are identified for a piezoelectrically driven, bulk-micromachined gyro and compared to results of FEM simulations. System-level simulations are performed and show possibilities to compensate the main parasitic effects. Results of eliminating the mechanical unbalance by femtosecond laser trimming are presented and compared with the simulations.

Fig. 1. SEM picture of a tuning fork gyroscope on the wafer level, including sketch of piezoelectric actuators and drive-control unit.

Index Terms—Gyroscopes, laser ablation, micromachining, piezoresistive devices, silicon.

I. INTRODUCTION ICROMACHINED gyroscopes can be used in many fields, such as aerospace, medical, industrial, and automotive applications. An overview on such devices and their applications can be found in [1]–[3]. In recent years, much effort has been invested concerning the working principle and the fabrication of the silicon tuning fork gyroscope examined in this work, see [4]–[7]. Highly reliable and accurate output signals over the full range of environmental conditions and over its projected lifetime are a key requirement for this device which is mainly intended for safety relevant automotive applications. It is, therefore, of utmost importance to identify all parasitic effects contributing to the sensor signal and compensate them as far as possible in order to extract the pure angular rate equivalent.

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Manuscript received March 9, 2005; revised June 24, 2005. This work was supported by the German Federal Ministry of Education and Research (BMBF), within the project Gyrosil. The associate editor coordinating the review of this paper and approving it for publication was Dr. Subhas Mukhopadhyay. S. Günthner, K. Kapser, and B. Hartmann are with the Continental Teves AG & Co. oHG, MEMS Design and Technology, 60488 Frankfurt, Germany (e-mail: [email protected]; [email protected]; [email protected]). M. Egretzberger and A. Kugi are with the Chair of System Theory and Automatic Control, Saarland University, 66123 Saarbrücken, Germany (e-mail: [email protected]; [email protected]). U. Schmid and H. Seidel are with the Chair of Micromechanics, Microfluidics/Microactuators, Saarland University, 66123 Saarbrücken, Germany (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/JSEN.2006.874451

Fig. 2. FEM-simulations. (Left) Drive and (right) sense modes of the silicon tuning fork gyroscope. Principle of excitation of the sense mode by the Coriolis effect.

In contrast to gyroscopes fabricated by silicon surface micromachining processes [8], [9], the tuning fork sensor examined in this work is implemented by bulk micromachining technology, which was also used for the sensors described in [10], [11]. The sensor element consists of two parallel plates (tines) which are mechanically connected by two KOH-etched pyramid trunks. Two cantilever beams link both pyramid trunks to the surrounding silicon substrate, thus forming the stem of the tuning fork, as can be seen in Fig. 1. The system is excited in the tuning fork mode which is characterized by the anti-phase motion of the tines at the corresponding resonance frequency—the drive frequency . The second mode of interest, characterized by a torsional motion of the tuning fork about its stem (see Fig. 2), exceeds the is the sense mode, whose resonance frequency . Two piezoelectric thin film actuators on drive frequency by top of the upper tine (see Fig. 1) are stimulated by a voltage and excite the drive motion. The fabrication of the aluminum nitride (AlN) thin film is reported in detail in [12]. The drive oscillation is monitored via a piezoresistive Wheatstone bridge.

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GÜNTHNER et al.: COMPENSATION OF PARASITIC EFFECTS

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Fig. 3. (a) Maximum shear strain " on top of the upper torsion stem in the sense mode and readout principle (x and y directions are h110i). (b) Distribution of the electric potential in the readout structure in the sense case.

The drive control unit is placed on top of the upper tine, as shown schematically in Fig. 1, and is sensitive to longitudinal stress in the direction (i.e., parallel to the stem). When the inertial system of the sensor is rotated about an axis parallel to the beams, a force perpendicular to the velocity of the tines and perpendicular to the rotational axis acts on both tines, due to the Coriolis effect, thus exciting the sense mode. The sense oscillation causes torsion of the stems and, therefore, strain on their surfaces [see Fig. 3(a)]. The induced stress is measured by piezoresistive elements. The tuning fork gyroscope is designed to have its excitation and sensing modes and at 50 and 50.2 kHz, respectively. In particular, there are no modes below 10 kHz and no modes within a range of 15 kHz around the drive and sense frequencies. This provides excellent insensitivity against external shock and vibrations, which is a very critical demand for automotive applications. II. PIEZORESISTIVE READOUT The theory of the piezoresistive effect in cubic crystal materials, especially in silicon, is described in detail in [13], [14]. The piezoresistivity tensor relates the change of the resistivity to the mechanical stress . With the electric field tensor strength , the current vector and the resistivity tensor the linear constitutive equation for piezoresistive materials reads as1 (1) with , , , , 1, 2, 3. Thereby, the piezoresistivity tensor satisfies the symmetry conditions (2) For crystals with cubic symmetry, there only exist three independent material parameters. Usually, these parameters are defined with respect to a coordinate system aligned with the crystal directions, namely for for for

and (3)

1Einstein’s convention for sums by summing over, up, and down double-repeated indices is used to shorten the formulas.

Fig. 4. Geometry and placement of the sensor element, progress of the shear stress , and crystal orientation of the torsional stem.

All other coefficients are zero, with , , , 1, 2, 3. The sensor element is oriented in the direction of the axis. As the [110] crystal direction of the silicon wafer is aligned with the axis of the torsional stem (see Fig. 4), we have to perform a transformation, i.e., a rotation of 45 about the axis. With the rotation matrix

(4)

the coefficients of the piezoresistivity tensor transform to (5) The readout structure has to be designed in such a way to gain an output voltage proportional to the shear stress of the torsional stem. An unconventional choice for the geometry of this readout structure is depicted in Fig. 4 (see, e.g., [12]). By as shown in Fig. 4, the current vector applying a voltage has only one nonvanishing component . Thus, utilizing the of transformed piezoresistivity tensor of (5) the component the electric field strength according to (1) reads as

(6) Assuming symmetric current density in the direction, we can integrate the electric field in the direction to get the output voltage

(7)

The best sensitivity for the chosen geometry of the piezoresistive readout structure depicted in Fig. 4 is obtained by using n-doped silicon. The nonvanishing coefficients of the piezoresistivity tensor for an n-doped {100} silicon wafer are shown in Table I. A torsion of the stem of 1 rad about the axis results on the surface of the stem of apin a maximum shear stress Pa. Using the material parameters given proximately

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TABLE I MATERIAL PARAMETERS FOR N-DOPED SINGLE CRYSTAL SILICON FOR T = 300 K AND A DOPING CONCENTRATION OF 10 cm [14]

in Table I and taking into account a decrease of these values for cm the used higher doping concentration of about (see [13]), the piezoresistive readout structure yields an output of approximately 75 V. This result is in fairly voltage good agreement with experimental values that yield a sensitivity V rad. of approximately 60 Fig. 3(b) shows the finite element calculation for the electric potential of the piezoresistive readout structure obtained from the software package “Analyzer” by CoventorWare which validates the analytical calculations above. The relation between the electric field strength and the electric potential is given by

(8)

III. PARASITIC EFFECTS In the calculations discussed in the previous chapter, all parasitic effects have been neglected. In reality, a readout signal may be generated even in the absence of an angular rate. As will be shown, different mechanisms are able to make contributions to the measured voltage in various phase positions with respect to the wanted sensor signal. The following three mechanisms can be identified: • mechanical unbalance; • actuation unbalance; • drive motion parasitics. For capacitive micromachined gyroscopes the sources of the parasitic signals have been identified and solutions for their compensations have been presented, cf. [15] and [16] (where the actuation unbalance is called direct motion coupling). In this paper, we focus on a bulk micromachined tuning fork sensor, which is stimulated by piezoelectric forces. A. Mechanical Unbalance This effect delivers the largest contribution to the above mentioned parasitic mechanisms. It originates from a mechanical asymmetry between the upper and the lower tine of the tuning fork, most prominently from a lateral offset between the two. This offset derives from fabrication tolerances induced during the production process. Taking into account a lateral offset of 1 m in the direction, FEM-simulations show that the drive mode is at the same resonance frequency and has the same mode shape as in the perfectly balanced case, but a closer look at Fig. 5 reveals that the torsional stem is tilted about the axis by 54.6 rad, when the drive amplitude is 1 m and Hz. Scanning laser vibrometers can be used most

Fig. 5. Drive mode for displacement of the upper tine in the y direction. The shading indicates the absolute value (in micrometers) of the modal displacement vector for a maximum tine deflection of 1 m.

favorably for studying the vibrational modes of MEMS structures (see, e.g., [17]). By a resonant stimulation of the sense oscillation, the torsional amplitude of the beams can be correlated to the measured readout signal. Furthermore, the ratio between readout signal and applied angular rate is known from experiments with a turntable. Thus, the correlation between an applied angular rate and the torsion of the beam turns out to s rad under standard conditions (drive amplitude be 180 Hz). Thus, a readout signal equivalent to 1 m, an angular rate of abqout 9800 s is generated by the assumed offset of 1 m, which is within the typical fabrication tolerances. has its origin in the deBecause this signal component formation of the drive mode shape, it is in phase or in anti-phase with respect with the tine amplitude, i.e., phase shifted by to the excitation voltage . Measurements show that this unbalance signal can go up to the equivalent of 20000 s under realistic fabrication conditions. Other asymmetries within the sensor have been examined by FEM simulations, but none of them contributes in a similar order of magnitude to the measured signal. B. Actuation Unbalance Another cause for a pseudosense signal is an asymmetric stimulation of the sensor in the drive frequency by the two piezoelectric thin film actuators deposited on the upper tine. This effect can originate either from asymmetries in the electric stimulation or from a -directed misalignment of the two actuators on the tine. Both mechanisms lead to an effective angular momentum about the central axis of the sensor. Thus, the sense oscillation is stimulated directly at the drive frequency, which is off its resonance. Therefore, its phase lags behind the drive voltage by [18]

(9)

if and the sense quality factor , which is a reasonable assumption in our case. Analytical calculations with the software tool “Architect” from CoventorWare help to illustrate this effect. This tool is based on the idea that the whole device can be partitioned into rigid and flexible sub elements [19]. A systematic derivation of an analytical model of


Fig. 6. (a) Analytic simulation of the influence on the sense signal S , if the excitation force F is displaced by 1 m from the symmetry axis. (b) Decompo), mechanical unbalance sition of the signal S into angular rate response (S ), actuator asymmetry (S ), and drive motion (S ). (S

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Fig. 7. (Cosine) Scaled in phase and (sine) out of phase signal components due to an asymmetric excitation of the tuning fork.

IV. EXPERIMENTAL VERIFICATION the tuning fork, which is considered as a coupled flexible structure, can be found in [7]. Doing so, the sensor element is described by a set of two beams representing the torsional stem, two beams for the tines and a rigid mass for the two pyramid trunks. For simplicity, the piezoelectric actuators are replaced by which is applied to the tip of the upper a mechanical force tine orthogonal to the top surface. In reality the piezoelectric actuators induce a bending moment in the vicinity of the suspension of the tine about the axis (compare Fig. 1). In this model, the unbalance is simulated by moving the point of force application out of the center along the direction. For each micron lateral displacement from the center, a sense-mode signal corresponding to 0.69 s is induced with a phase shift of eior 179.87 depending on the sign of the displacether ment [see Fig. 6(a)]. This is only a first order approximation of the sensor response, but the effect is measurable, as will be discussed later. Unfortunately, it is difficult to distinguish the signal caused by the actuation unbalance from the pure sense signal, since they are in phase: The sense signal is stimulated out of resonance by the Coriolis force which is proportional to the velocity of the is orthogtuning fork tines. Furthermore, the tine velocity to the tine position . Since the drive motion is onal excited in resonance, the phase shift between the tine position and the drive voltage equals -90 , too. Thus, the actual compared to sense signal is phase-shifted altogether by and, thus, is in (anti)phase with respect to the actuation unbalance signal. C. Drive Motion Parasitics Apart from the two effects mentioned so far, there are other mechanisms which generate parasitic signals. Readout signals can be measured even without creating any torsional motion of the stem. In [20], the so-called drive motion parasitic effect, that causes signals in phase with the drive motion, has been examined. This effect turned out to be rather small and can, therefore, be neglected in the following.

Fig. 7 shows a series of measurements without external angular rate. Here the sensor is excited by asymmetric electric stimulation of the two piezoelectric thin film plates. An actuation unbalance of, e.g., 100% means that only the right-hand actuator is stimulated. The drive resonance frequency of the sensor is found by tuning the frequency of the drive voltage, until the phase-shift between the drive control signal and the driving voltage is exactly 90 . The piezoresistive readout signal is routed to a phase-sensitive lock-in amplifier, that determines the amplitude of the sense signal and the phase shift with respect to the drive voltage source. The asymmetric way of stimulation implies that the drive mode is excited ineffectively with different piezoelectric forces resulting in different amplitudes. In order to make the measurements comparable among each other, the readout signals are divided by the drive amplitude. Next, the signal is multiplied either by the cosine or by the sine of the phase arising between drive voltage and readout voltage. Furthermore, the sensor signals are split into a contribution being in phase and one being orthogonal to the drive oscillation. Both components are plotted versus the level of actuation unbalance. Whereas the in phase (cosine) part stays constant and, therefore, is proportional to the drive amplitude, the out of phase (sine) component depends directly on the degree of actuation asymmetry. The zero-crossing of the orthogonal component at the actuation asymmetry of 17% indicates an inherent asymmetry in the excitation mode of as this particular sensor. Interpreting the sum and as the vector diagram in Fig. 6(b) explains clearly the different contributions to the measured signal, namely the following: • the mechanical unbalance signal and the drive motion parasitics signal with amplitudes proportional to the drive motion and in phase with the drive oscillation; • the actuation unbalance signal proportional to the asymmetry of excitation and orthogonal to the drive oscillation.

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V. ELIMINATION OF THE PARASITIC EFFECTS We have now identified the most important and crucial parasitic effects, leading to a false interpretation of the sense signal (cf. [20]). Since the influence of the drive motion effect was shown to be rather small compared to the contribution of the mechanical and actuation unbalance, it will be neglected in the following investigations. In order to eliminate the two unbalances which are independent of each other, two different mechanisms have to be applied. They are called the following: • mechanical balancing; • actuation balancing. A. Simulation of Mechanical Balancing As already suggested in [20], a postprocess trimming procedure may be necessary in order to improve the mechanical balancing, e.g., by adding or removing mass from the sensor at appropriate locations on the tine. The highest impact on the moment of inertia of the sensor is achieved, if mass is added or removed as far away as possible from the rotational axis of the tine, as a consequence of Steiner’s Theorem

Fig. 8. AC simulations for the rotation r of the upper torsional stem about the x axis starting with (0) the unbalanced sensor and resulting in (1) a mechanically trimmed sensor.

(10) Here, is the distance of the balancing mass from the rotational axis. It is too complicated to calculate the exact mass necessary for balancing the drive motion analytically. Therefore, balancing has to be done in a quasiheuristic way, i.e., certain units of mass have to be removed successively. The goal of the , as defined in trimming procedure is a vanishing signal Fig. 6(b). Furthermore, the mass units must be reduced when approaching the balanced state in order to avoid over-balancing effects. This procedure can be simulated with the “Architect” model, which has already been described above. The mechanical unbalance is implemented in the same way as in the FEM model by displacing the upper semi-sensor against the lower one in the direction by a distance . After that, a mass cuboid with constant base and variable height is fixed to the free tip of the upper tine. It has to be placed as far as possible from the rotational center at the correct side of the tine, i.e., at the edge direcwith negative coordinate, if the displacement is in tion and vice-versa. Finally, a series of transient and small-signal AC-simulations is performed, beginning with a height of and increasing incrementally. The drive-motion is again stimulated by applying a force to the top tine as mentioned above. The AC-simulations yield the drive and sense frequencies and the amplitude of the rotation of the upper torsional stem about the axis. Fig. 8 shows a choice of simulations for different heights , given as fractions of the optimal height . The is dB, where dB rad , and for unit of the resonance amplitude of at the drive frequency is almost zero. The oscillation of is caused by the lateral displacement. Nevertheless, its amplitude can be expressed as a multiple of the response to an angular rate of 1 s via the known correlation between the torsional angle and the angular rate of 5.84 s . As can be seen in Fig. 8, the change in affects nrad the resonance frequencies, too. Therefore, the unbalance signal, , has to be referred to the which is inversely proportional to

Fig. 9. (a) Change of 9 induced by the reduction of the mechanical unbalance S . (b) (Black and gray solid lines) Simulations and (crosses and dotted lines) measurements of the phase running.

original value of . The phase shift between the excitation voltage , or rather the excitation force and the sense signal , is derived from the time delay between the excitation force and the rotation of the stem at the drive frequency. For illustrating this effect in Fig. 9(a), we use the variable , which is a standardized value of the excitation force . For these purposes, transient simulations are performed neglecting start-up effects (see Fig. 10). Next, the phase is plotted versus the corresponding values of the signal given in s, see the black solid line in Fig. 9(b). If the mechanical unbalance dominates, the phase is almost 90 . The more the unbalance signal is suppressed, the more the phase deviates from its original value. In absence of any actuation unbalance, one would not expect this simulated phase running. However, not only the lateral displacement of the upper tine and of the actuation leads to actuation unbalance, but also adding mass asymmetrically to the system. In our case it is smaller than 1 s, but nevertheless it causes a change in the phase. For illustration of this effect another series of simulations with an increased actuation unbalance has been performed. Displacement of the excitation force out of the center of the upper tine leads to an additional actuation unbalance, as shown above. The gray solid line in Fig. 9(b) represents


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Fig. 10. Calculation of the phase shift between r and F in transient simulations by determination of the time delay. Fig. 12. Flow diagram of the three stages of the balancing procedure for a completely unbalanced tuning fork sensor.

Fig. 13. Response of an ideal sensor due to an external angular rate (t). Left: Envelopes of the fast oscillating stem rotation r . Right: Demodulated, filtered . and amplified output and quadrature signals S and S Fig. 11. SEM picture of ablation by a femtosecond laser from the upper tine of a Silicon tuning fork for testing purposes (by courtesy of Laser Zentrum Hannover e.V.).

a second trimming process simulation loop, for a sensor with the same mechanical unbalance but including an additional lateral displacement of the driving force by 30 m. For high values of the mechanical unbalance the two simulations almost coincide, but for small phase shifts, the first simulation cycle shows almost negligible unbalance, whereas the second simulation cycle ends up in a nonzero remnant unbalance. B. Laser Trimming In practice, the moment of inertia is tuned by a femtosecond laser. Laser pulses can remove defined amounts of mass at desired places from the top surface of the sensor. Fig. 11 shows a set of ablation cavities with different geometries and depths at different locations for testing purposes. The depth can be increased by raising the number of pulses, whereas the rectangular geometries can be chosen rather freely, in this case having edge lengths of 25, 50, and 100 m, respectively. In [21] and [22], the working principle and the configuration of the setup for trimming diced sensors and sensors on wafer level are presented. Fig. 9(b) shows the trimming procedure of a particular sensor. The different points of measurement after a certain number of laser pulses are marked with crosses. For each step a measurement controls the readout signal as well as the phase between the signal and the drive-control signal, which is in phase with the drive motion. The measured readout signal is related to a of 200 Hz and converted from the output voltage into the

unit s in the same way as the simulated one. The error bars represent roughly estimated inaccuracies of the measurements and in the determination of the unbalance of in the phase of 20%. The black dashed line in Fig. 9(b) interpolates the measured values. Qualitatively, these values are in very good agreement with the simulations. Starting from large unbalances, the phase running can be explained when approaching smaller unbalance signals. Obviously, the measured remaining actuation unbalance is by several orders of magnitude smaller than the mechanical one, but it still is not negligible. Although it is not known a priori how much mass has to be removed, as pointed out in the previous section, fully automatic trimming is feasible by smart online verification, cf. the left hand and central columns in Fig. 12. After determining the value of the unbalance signal, a small piece of mass is removed from a position in the outer region of the tine on the correct side. The resulting reduction of the unbalance signal is measured, allowing the prediction of the amount of mass which has to be removed in order to balance the sensor. Removing somewhat less than the required mass and controlling the correlation between ablated material and predicted balancing effect, one can successively approach a balanced state. Fine tuning of the balancing procedure is achieved by shifting the ablation position toward the center of the tine. C. Simulation of Actuation Balancing The “Architect” model introduced above can be expanded by a demodulation circuit and a low pass filter, which are necessary for signal processing [6]. Fig. 13 shows the response of

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Fig. 15. Transient simulations of the output signal S and quadrature signal in =s for the unbalanced sensor (index 0), the mechanically balanced S (index “mech”) and the completely balanced sensor (index “mech&act”).

Fig. 14. AC simulations for z and r starting with the unbalanced sensor (index 0) and resulting firstly in the mechanically balanced sensor (index “mech”) and, second, in a sensor without actuation unbalance (index “mech&act”). The gray lines represent a sweep through the balancing torque.

an ideal tuning fork gyroscope exposed to a time-dependent angular rate , which has a frequency of 25 Hz, a maximum am. The drive-mode is plitude of 1 s and is zero at time forced to oscillate with constant amplitude of 1 m. The variable is the deflection of the center of the free tip of the upper tine. For the complete mechanical system, steady state condi. As can be seen in Fig. 13, the stem tions are assumed at follows the external angular velocity and reaches a rotation maximum of 4.85 nrad, thus differing by approximately 20% from the experimentally determined value. After the stem rotaon one hand tion is demodulated with the tine velocity on the other hand, both reand with the tine deflection sulting signals are passed through a low-pass filter with a cut-off frequency of 100 Hz. Finally, the filtered signals are amplified and the by corresponding factors and yield the output signal represented by the dashed and so-called quadrature signal reproduces the angular rate dotted lines in Fig. 13. Whereas with a time-delay resulting from the filter characteristics very well, the quadrature signal is almost vanishing. In the following consideration, the sensor is again assumed to be mechanically m . Furthermore, an actuation unbalunbalanced ance corresponding to a sense signal of about s is assumed. In Fig. 14, the solid lines represent the tine deflection and the stem rotation in the unbalanced case calculated by an AC-simulation. In a first step, the mechanical unbalance can be compensated as discussed above. The dashed curves in Fig. 14 show the frequency shift of both eigen-modes and the reduction of unbalance motion of at the drive frequency. The remaining actuation unbalance can be compensated by applying suitable different voltages to the two thin film actuators, generating a torque about the central axis of the sensor. This torque must have the same phase and frequency as the excitation force . Sweeping through a range of amplitudes of the torque, one gets the sequence of gray lines. There exists an optimum value for the torque, such that the effect of the actuation unbalance is minimized. Note that the resonance frequencies are not changed in this case, as can be seen in Fig. 14. The right hand

column in Fig. 12 shows the procedure for the actuation balancing described above. In order to see the balancing effects on and transient simulations have to be the output signals performed. The three cases of a completely unbalanced sensor, a sensor balanced only mechanically and a totally balanced sensor are depicted in Fig. 15. Provided that the drive oscillation (1- m amplitude) is in steady state, both outgoing signals tend to an almost constant value after a run-in phase. The initial state yields a quadrature signal of more than 11 000 s, which is in good agreement with the FEM results, but also the sense signal has s. After reaching mechanical balan offset of more than ance, not only the quadrature signal is almost zero, but also the bias of the sense signal is decreased and its sign is changed. is explainable, because the signals , , The reduction of , , in the arrow-diagrams, Fig. 6(b) and Fig. 9(a) are not exactly orthogonal to each other, cf. (9). Therefore, even cause contributions small phase shifts of big signals as to the output signal . Consequently, as is reduced, the also decreases. Finally, balancing of the actuation offset of and leaves leads to compensation of the remaining bias of the quadrature signal almost unchanged. VI. CONCLUSION For a silicon tuning fork gyroscope with piezoelectric actuation and piezoresistive, readout different mechanisms have been identified, yielding an understanding of the composition of the measured parasitic piezoresistive readout signals which are present even in the absence of an external angular rate. Particularly, the measured nonorthogonal and nonzero phase angle of the readout signal can be explained well. Different possibilities exist for handling these parasitic effects. The signals in phase with the drive motion are primarily suppressed by the PLL-based electronic circuitry, but even a small phase shift in the signal path leads to an unwanted contribution to the wanted sense signal output. of the large signal In order to minimize the mechanical unbalance, production tolerances should be kept as small as possible, especially with respect to the lateral offset between the upper and the lower tine. Nevertheless, a postprocess trimming procedure has to be performed. Removing mass from the sensor at appropriate locations on the tine by pulses from a femtosecond-laser can be used to effectively minimize the mechanical unbalance.


System-level simulations reproduce the laser-trimming process very well and give a prospect for completely balancing the sensor by additional compensation of the actuation unbalance. This can be reached by applying suitable different voltages to the two piezoelectric actuators, causing a torque with the same frequency and phase as the driving force about the center axis of the sensor. Electrical compensation of the actuation unbalance can be implemented by applying signals with adequate amplitude in phase and in anti-phase, respectively, to both piezoelectric actuators in superposition to the normal drive excitation. This approach is limited by the maximum available force provided by the thin film actuators and has to be examined further. In summary, the tuning fork gyroscope exhibits several advantages of significant relevance, among these its outstanding behavior under external shock, its CMOS-compatible fabrication process, and a relatively small chipsize (below 15 mm ). The necessity for compensational post processing steps to increase the yield can be found in a similar way for other types of gyroscopes, too. ACKNOWLEDGMENT The authors would like to thank Dr. M. Rose, R. Gottinger, and R. Grünberger from Infineon Technologies AG, as well as M. Frey, for many fruitful discussions and for many years of excellent collaboration. They would also like to thank Dr. F. Nuscheler and M. Kluge from EADS Deutschland GmbH for providing some of the measurements, as well as U. Klug from Laserzentrum Hannover e.V. for providing the instructive pictures about the laser trimming process. REFERENCES [1] N. Yazdi, F. Ayazi, and K. Najafi, “Micromachined inertial sensors,” Proc. IEEE, vol. 86, no. 8, pp. 1640–1659, Aug. 1998. [2] J. Söderkvist, “Micromachined gyroscopes,” Sens. Actuators A, vol. 43, pp. 65–71, 1994. [3] H. Seidel, M. Aikele, M. Rose, and S. Tölg, “Safety relevant microsystems for automotive applications,” in Proc. Microsystems Conf., Düsseldorf, Germany, Mar. 27–29, 2001. [4] S. Sassen, R. Voss, J. Schalk, E. Stenzel, T. Gleissner, R. Grünberger, F. Nuscheler, F. Neubauer, W. Ficker, W. Kupke, and K. Bauer, “Robust and selfstable silicon tuning fork gyroscope with enhanced resolution,” in Proc. Advanced Microsystems for Automotive Applications, Berlin, Germany, 2000, pp. 233–245. [5] R. Voss, K. Bauer, W. Ficker, T. Gleissner, W. Kupke, M. Rose, S. Sassen, J. Schalk, H. Seidel, and E. Stenzel, “Silicon angular rate sensor for automotive applications with piezoelectric drive and piezoresistive readout,” in Proc. Transducers, Chicago, IL, 1997, pp. 879–882. [6] S. Sassen, R. Voss, J. Schalk, E. Stenzel, T. Gleissner, R. Grünberger, F. Neubauer, W. Ficker, W. Kupke, K. Bauer, and M. Rose, “Tuning fork silicon angular rate sensor with enhanced performance for automotive applications,” Sens. Actuators A, vol. 83, pp. 80–84, 2000. [7] A. Kugi, D. Thull, and H. Seidel, “Modeling and optimization of a silicon tuning fork gyroscope,” Proc. Appl. Math. Mech, vol. 4, pp. 59–62, 2004. [8] W. Geiger, J. Merz, T. Fischer, B. Folkmer, H. Sandmaier, and W. Lang, “The silicon angular rate sensor system DAVED,” Sens. Actuators A, vol. 84, pp. 280–284, 2000. [9] S. E. Alper and T. Akin, “A symmetric surface micromachined gyroscope with decoupled oscillation modes,” Sens. Actuators A, vol. 97-98C, pp. 347–358, 2002. [10] H. Kuisma, T. Ryhänen, J. Lahdenperä, E. Punkka, S. Ruotsalainen, T. Sillanpäa, and H. Seppä, “A bulk micromachined silicon angular rate sensor,” in Proc. Transducers, Chicago, IL, 1997, pp. 875–878.

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[11] K. Maenaka and T. Shiozawa, “Silicon rate sensor using anisotropic etching technology,” in Proc. Transducers, Yokohama, Japan, 1993, pp. 642–645. [12] M. Rose, “Entwicklung eines mikromechanischen Drehratensensors aus Silizium,” Ph.D. dissertation, Technical Univ. Berlin, Berlin, Germany, 1998. [13] Y. Kanda, “A graphical representation of the piezoresistance coefficients in silicon,” IEEE Trans. Electron Devices, vol. ED-29, no. 1, pp. 64–70, Jan. 1982. [14] B. Klöck, “Piezoresistive sensors,” in Sensors, A Comprehensive Survey: Vol. 7, Mechanical Sensors, W. GöpelGopel, J. Hesse, and J. Zemel, Eds. New York: Wiley, 1994, pp. 145–172. [15] M. Braxmaier, A. Gaißer, T. Link, A. Schumacher, I. Simon, J. Frech, H. Sandmaier, and W. Lang, “Cross-coupling of the oscillation modes of vibratory gyroscopes,” in Proc. Transducers, Boston, MA, 2003, pp. 167–170. [16] H. Xie and G. K. Fedder, “Integrated microelectromechanical gyroscopes,” J. Aerosp. Eng., vol. 16, pp. 65–75, 2003. [17] R. A. Lawton, M. Abraham, and E. Lawrence, “Characterization of nonplanar motion in MEMS involving scanning laser interferometry,” Proc. SPIE, vol. 3880, pp. 46–50, 1999. [18] H. Kuchling, Taschenbuch der Physik. Leipzig, Germany: Fachbuchverlag Leipzig-Köln, 1995. [19] G. Lorenz, Netzwerksimulation mikromechanischer Systeme Ph.D. dissertation, Univ. Bremen. Aachen, Germany, 1999. [20] S. Günthner, K. Kapser, M. Rose, B. Hartmann, M. Kluge, U. Schmid, and H. Seidel, “Analysis of piezoresistive read-out signals for a silicon tuning fork gyroscope,” in Proc. IEEE Sensors Conf., Vienna, Austria, Oct. 2004, pp. 1411–1414. [21] U. Klug, private communication. [22] Final Report of joint project Gyrosil, funded by German Federal Ministry of Education and Research (BMBF) within program, Mikrosystemtechnik 2000+ Förderungskennzeichen: 16SV1325/0, to be published.

Stefan Günthner was born in Eichstaett, Germany, in 1975. He received the diploma degree in Physics in 2000 from the Ludwig-Maximilians University, Munich, Germany. He is currently pursuing the Ph.D. degree at the Chair of Micromechanics, Saarland University, Saarbruecken, Germany. He was with Epcos AG, Munich, until 2003. He is currently engaged in the design and simulation of micromachined gyroscopes at Continental Teves, Frankfurt/Main, Germany.

Markus Egretzberger was born in Tulln/Donau in 1977. He studied Mechatronics at the Johannes Kepler University Linz, Austria, and received the diploma degree in 2004. He is currently pursuing the Ph.D. degree at the Chair of System Theory and Automatic Control, Saarland University, Saarbruecken, Germany. In November 2004, he joined the Chair of System Theory and Automatic Control, Saarland University, where he is currently a Research Assistant. His main research interests are in the modelling and control of MEMS sensors.

Andreas Kugi (M’93) was born in Villach, Austria, in 1967. He received the Dipl.-Ing. degree in electrical engineering from the TU Graz, Graz, Austria, and the Ph.D. (Dr.Techn.) degree in control engineering and the Habilitation in automatic control and control theory from the Johannes Kepler University (JKU), Linz, Austria, in 1992, 1995, and 2000, respectively. In 2002, he was appointed to Full Professor at Saarland University, Saarbruecken, Germany, where he currently holds the Chair of System Theory and Automatic Control.

Konrad Kapser received the diploma degree in physics from the Technical University in Munich, Munich, Germany, in 1991, and the Ph.D. degree in the field of silicon based infrared sensors from the University of Erlangen-Nuernberg, Germany, in 1996. He then joined Continental Teves, Frankfurt, Germany, where he was engaged in several development projects for MEMS devices (low-g sensors and gyroscopes). Currently, he is with Infineon Technologies AG, Munich, where he is an Application Manager for magnetic field sensors.

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Bernhard Hartmann was born in 1964. He studied physics at the Technical University of Munich, Munich, Germany, and received the diploma in 1996. During the last one-and-a-half years, he performed the thesis work at the Lab for the Integration of Sensors and Actuators, Montrèal, QC, Canada. After his studies, he was an Engineer for MEMS inertial sensors and the MEMS Technology Group Leader at Temic mic. GmbH. Since 2002, he has been Head of the MEMS Design Group at Continental Teves, Frankfurt, Germany.

Ulrich Schmid was born in Munich, Germany, in 1972. He started studies in physics at the University of Kassel, Germany, in 1992. He performed his diploma work at the Daimler-Benz AG on the electrical characterization of 6H-SiC microelectronic devices. In 2003, he received the Ph.D. degree from the Technical University of Munich. In 1999, he joined the research laboratories of DaimlerChrysler (now EADS Deutschland GmbH), Munich. Currently, he is a Postdoctorate at the Chair of Micromechanics, Saarland University, Saarbruecken, Germany


Helmut Seidel (M’04) received the Diploma in physics from the Ludwig-Maximilians University, Munich, Germany, in 1980, and the Ph.D. degree in physical chemistry from the Free University of Berlin, Berlin, Germany, in 1986. In 1980, he joined Fraunhofer-Society, Munich, as a Research Scientist focusing on MEMS. From 1986 to 2002, he held various positions at MBB, Daimler-Benz, and Temic. Since 2002, he has been a Professor at Saarland University, Saarbruecken, Germany, holding the Chair for Micromechanics.