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A micromachined vibrating rate gyroscope with independent beams for the drive and detection modes
Sensors and Actuators 80 Ž2000. 170–178 www.elsevier.nlrlocatersna
A micromachined vibrating rate gyroscope with independent beams for the drive and detection modes Yoichi Mochida ) , Masaya Tamura, Kuniki Ohwada Yokohama Technical Center, Murata Manufacturing, 1-18-1, Hakusan, Midori-ku, Yokohama 226-0006, Japan Received 13 April 1999; accepted 19 July 1999
Abstract We investigated the mechanical coupling between the drive and detection modes of micromachined vibrating rate gyroscopes, and designed and fabricated a gyroscope with a new structure to reduce the coupling. The coupling of the oscillator was measured using a new measurement system with a two-dimensional laser displacement meter. The oscillations were found to have an elliptical motion due to the coupling. When the frequency mismatch was reduced by means of electrostatic frequency tuning with a DC bias voltage, the coupling increased. The new structure, which has independent beams for the drive and detection modes, exhibited weaker coupling; and the resolution was 0.078rs at a band width of 10 Hz. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Gyroscope; Micromachining; Mechanical coupling; Laser displacement meter
1. Introduction Recently, a number of micromachined vibrating rate gyroscopes, which use separate oscillation modes for drive and detection, have been developed w1–12x. In these gyroscopes, the sensitivity improves as the frequency mismatch between the two modes decreases. One way to reduce the mismatch is to employ electrostatic frequency tuning with a DC bias voltage w5,7,8,10x. However, as the mismatch decreases, the mechanical coupling between the two modes makes operation more and more unstable. Previous studies have failed to show how coupling affects a micromachined gyroscope. Conventionally, a microscope is used to measure the coupling of a vibrating gyroscope with an oscillator consisting of a silicon cantilever w9x, but this method is difficult to use when an extremely small oscillator is fabricated using silicon micromachining. This paper reports on the measurement of the coupling using a two-dimensional laser displacement meter. In order to investigate the influence of coupling on a micromachined gyroscope, we measured the mechanical character-
istics of fabricated oscillators when the frequency mismatch was tuned with a DC bias voltage. To reduce the coupling, we designed and fabricated a micromachined gyroscope with independent beams for the drive and detection modes, and then measured its characteristics.
2. Principle
2.1. Principle of Õibrating gyroscope A micromachined vibrating rate gyroscope is generally composed of a mass and beams that oscillate. The mass readily oscillates in two orthogonal directions. A simple model of a vibrating gyroscope is shown in Fig. 1. When the mass is driven along the x-direction by an external force Ž F Ž t . s F0 sin v t ., and it is subjected to a constant angular rate Ž V ., a Coriolis force Ž Fc s 2 m V x˙ . is induced in the y-direction. In this case, the equation of motion is mx¨ q c x x˙ q k x x s Fo sin v t ,
Ž 1.
my¨ q c y y˙ y 2 m V x˙ q k y y s 0,
Ž 2.
)
Corresponding author. Tel: q81-45-939-7150; fax: q81-45-939-7156; E-mail: [email protected]
0924-4247r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 Ž 9 9 . 0 0 2 6 3 - 0
Y. Mochida et al.r Sensors and Actuators 80 (2000) 170–178
171
fabrication error. Consequently, the energy of the drive oscillation leaks in the y-direction, resulting in the mechanical coupling between the oscillations in the two directions. This coupling is assumed to depend on the amplitude and velocity of the drive oscillation w13x. When both types of coupling are present, the equation of motion is mx¨ q Ž c x y c c . x˙ q c c y˙ q Ž k x y k c . x q k c y s Fo sin v t , Ž 5. my¨ q Ž c y y c c . y˙ q Ž cc y 2 m V . x˙ q Ž k y y k c . y q k c x s 0,
where k c and c c are the coupling coefficients for the amplitude and velocity, respectively. When there is little velocity coupling and V s 0, the equation of motion is
Fig. 1. Simple model of vibrating gyroscope.
where m is the mass; c x and c y are the resistances due to viscosity, and k x and k y are the spring constants, for the x- and y-directions, respectively. When the mass is driven such that it oscillates at the resonance frequency, the oscillatory motion derived from Eqs. Ž1. and Ž2. is x Ž t . s yx m cos v x t , yŽ t. s
Ž 3.
2 m Vv x x m ky
1
(½
1 y Ž v xrv y .
=sin Ž v x t y w . ,
2 2
5 q Ž v rv . x
2
Fo Q x kx
w s tany1 ls
vx vy
Qx s
mx¨ q c x x˙ q Ž k x y k c . x s Fo sin v t ,
Ž 7.
my¨ q c y y˙ q k y y q k c x s 0,
Ž 8.
where c c s 0, x 4 y, k x ,k y 4 k c , and V s 0. When the mass is driven so that it oscillates at the resonance frequency, the oscillatory motion derived from Eqs. Ž7. and Ž8. is given by x Ž t . s yx m cos v x t ,
y
Ž 4.
in which xm s
Ž 6.
,
yŽ t. sy
kc xm ky
1
(½
1 y Ž v xrv y .
=cos Ž v x t y w . .
2 2
5 q Ž v rQ v . x
y
2
y
Ž 10 .
When w f 0 or 1808, the mass oscillates along a linear path at an angle to the x-axis, as shown in Fig. 2a.
l Q y Ž 1 y l2 .
Ž 9.
,
,
mvx cx
, Qy s
mvy cy
,
where v x and v y are the resonant frequencies, and Q x and Q y are the resonance quality factors, for the x- and y-directions, respectively; and x m is the amplitude at resonance for the x-direction. If the drive amplitude is constant, then the amplitude of the resulting oscillatory motion is proportional to the angular rate Ž V .. Furthermore, the sensitivity of the gyroscope Ž< yrV <. is higher when the frequency mismatch Ž< D v < s < v x y v y <. is small. 2.2. Mechanical coupling The mass and beams of a micromachined vibrating rate gyroscope are not usually completely symmetrical due to
Fig. 2. Oscillatory motion of mass: Ža. amplitude coupling for w f 0 or 1808; Žb. velocity coupling for w f 0 or 1808.
Y. Mochida et al.r Sensors and Actuators 80 (2000) 170–178
172
yŽ t. s
cc v x x m ky
1
(½
1 y Ž v xrv y .
2 2
5 q Ž v rQ v .
=sin Ž v x t y w . .
x
y
2
y
Ž 14 .
When w f 0 or 1808, the oscillation of the mass is an elliptical motion, as shown in Fig. 2b. These results show that, in a micromachined gyroscope, the type of mechanical coupling can be determined from the oscillatory motion exhibited by the mass of a fabricated oscillator.
3. Design of gyroscopes 3.1. Gyroscope with simple structure Fig. 3. Structure of type 1: Ža. bird’s eye view; Žb. cross-section view ŽA–AX ..
On the other hand, when there is little amplitude coupling and V s 0, the equation of motion is mx¨ q Ž c x y c c . x˙ q k x x s Fo sin v t ,
Ž 11 .
my¨ q c y y˙ q c c x˙ q k y y s 0,
Ž 12 .
where k c s 0, x˙ 4 y, ˙ c x , c y 4 cc , and V s 0. When the mass is driven so that it oscillates at the resonance frequency, the oscillatory motion derived from Eqs. Ž11. and Ž12. is x Ž t . s yx m cos v x t ,
Ž 13 .
We have investigated a micromachined vibrating rate gyroscope with the simple structure Žtype 1. w4x shown in Fig. 3, in which the mass is supported by four beams. The oscillator is 400 mm = 1200 mm = 10 mm in size. It is driven by means of comb electrodes, and there are two electrodes under the mass: one to detect capacitance, and other to tune the frequency. When AC and DC voltages are applied to the comb electrodes, the mass oscillates along the x-direction Ždrive mode.. An externally induced rotation about the z-axis Žangular rate: V . produces a deflection in the y-direction due to the Coriolis force. This deflection is detected as a change in the capacitance between the mass and the detection electrode. If the frequency mismatch Ž D f . between the drive and detection modes is small, the detection mode enhances the deflection. The shapes of the mass-and-beam structure at the
Fig. 4. Shape of beam-and-mass structure of type 1 during deflection.
Y. Mochida et al.r Sensors and Actuators 80 (2000) 170–178
173
Fig. 7. Fabrication process.
four beams supporting the mass are bent in the x-direction in the drive mode, and in the y-direction in the detection mode. In this structure, coupling between the two modes can easily arise if the beams are not completely symmetrical as a result of fabrication error during silicon micromachining. X Fig. 5. Structure of type 2: Ža. top view; Žb-1. cross-section view ŽA–A .; X Žb-2. cross-section view ŽB–B ..
point of peak deflection obtained by an FEM analysis are shown in Fig. 4 for the drive and detection modes. The
3.2. New design to reduce coupling To reduce the coupling, the new structure Žtype 2. shown in Fig. 5 was designed. It features independent pairs of beams for the drive and detection modes. The mass of the oscillator is connected to a frame supported by four
Fig. 6. Shape of beam-and-mass structure of type 2 during deflection.
174
Y. Mochida et al.r Sensors and Actuators 80 (2000) 170–178
Fig. 8. SEM images of Ža. type 1 Žsimple structure. and Žb. type 2 Žnew design. gyroscope. Fig. 10. Resonant frequency vs. DC bias voltage: Ža. type 1; Žb. type 2.
beams that are anchored to the substrate. The oscillator is 800 mm = 1200 mm = 50 mm in size. The inner beams are 20 mm wide and 10 mm high, and the outer beams are 5 mm wide and 50 mm high. It is driven by comb electrodes, and there are two electrodes under the mass, just as in type 1; so the method of detecting the angular rate is the same as for type 1. The shape of the mass-andbeam structure at the point of peak deflection obtained by
an FEM analysis is shown in Fig. 6 for the drive and detection modes. In the drive mode, the outer beams are bent in the x-direction, but there is little deflection of the inner beams. In the detection mode, the inner beams are bent in the y-direction, but there is little deflection of the outer beams. These results indicate that the outer and inner
Fig. 9. Measurement system.
Y. Mochida et al.r Sensors and Actuators 80 (2000) 170–178
175
beams are rigid with respect to the y- and x-directions, respectively. This means that the two pair of beams move independently in the two modes. In the type 2 structure, the coupling between the modes seems to be very weak because drive mode oscillations have little effect on the beams used in the detection mode.
4. Fabrication The fabrication process for type 2 is shown in Fig. 7. It is the same as for type 1 except for step Ža.. Ža. A SOI Žsilicon-on-insulator. substrate with a 200-mm thick base silicon layer, a 4-mm thick SiO 2 insulator layer, and a 50-mm thick top silicon layer is used. To make the inner beams, which move in the z-direction, about 40 mm of part of the top silicon layer is etched off by reactive ion etching ŽRIE.. For a type 1 device, a different SOI substrate with a 10-mm thick top silicon layer is used; and the top silicon layer is not etched in this step. Žb-1. To make the gap between the two electrodes and the silicon mass, a glass substrate is etched down about 2 mm with HF.
Fig. 12. Lisssajou figures of lateral and vertical displacement of type 2: Ža. DC biass 2 V; D f sy2.3%; Žb. DC biass6 V; D f sy0.8%; Žc. DC biass 7 V; D f sy0.1%.
Žb-2. After the deposition of Au and Cr layers, the two electrodes are patterned by wet etching. Žc. The SOI and glass substrates are bonded together by anodic bonding. Žd. The base silicon and insulator layers are removed by wet etching with TMAH and HF, respectively.
Fig. 11. Lisssajou figures of lateral and vertical displacement of type 1: Ža. DC biass 4 V; D f sy2.2%; Žb. DC biass 5 V; D f sy1.2%; Žc. DC biass6.2 V; D f s 0.3%.
Fig. 13. Amplitude ratio vs. frequency mismatch.
Y. Mochida et al.r Sensors and Actuators 80 (2000) 170–178
176
Fig. 14. Damping force and velocity of misaligned oscillator.
Že. Finally, the top silicon layer on the glass substrate is etched by RIE to form the oscillator. SEM images of fabricated oscillators are shown in Fig. 8.
5. Experimental results
To measure the frequency response of the oscillation amplitude, the output voltages of the laser displacement meter for the lateral and vertical directions are fed back into the network analyzer. Then, the resonant frequency of each mode is determined from the peak of the corresponding frequency response curve. Fig. 10 shows the results of measurements of the resonant frequencies of the drive and detection modes when a DC bias voltage was applied between the tuning electrode and the mass of the oscillator. For both types 1 and 2 devices, as the bias voltage increased, the resonant frequency of the detection mode dropped, while that of the drive mode remained virtually constant. As a result, the frequency mismatch Ž D f . between the two modes decreased, and was nearly zero at a bias of about 6 or 7 V. 5.3. Measurements of the coupling
5.1. Measurement system Resonant frequencies and oscillation amplitudes were measured with a two-dimensional laser displacement meter. The measurement systems are shown in Fig. 9. The laser displacement meter, which employs two laser for a single measurement, can measure both lateral and vertical movement at the same time. DC and AC voltages are applied to the electrodes of the oscillator in a vacuum chamber, which reduces the air damping of the oscillator; the pressure in the chamber is below 100 Pa. A network analyzer is used to measure resonant frequencies, as shown in Fig. 9a; and a function generator and a oscilloscope are used to measure oscillation amplitudes, as shown in Fig. 9b. 5.2. Frequency measurements for tuning with a DC bias Õoltage The oscillator is excited by applying a reference voltage signal from the network analyzer between either the detection or the drive electrode and the mass of the oscillator.
The oscillation amplitudes of oscillators driven at the resonant frequency of the drive mode by AC and DC bias voltages from the function generator were also measured. Combining the output voltages of the displacement meter for the lateral and vertical directions on an oscilloscope produces a Lissajou figure, which shows the motion of the oscillator in the x–y plain. Figs. 11 and 12 show the results for types 1 and 2, respectively. It is clear that the oscillation is an elliptical motion due to coupling between the two modes. The ratio of the amplitude in the y-direction to that in the x-direction was calculated. Fig. 13 shows the amplitude ratio as a function of frequency mismatch. As the frequency mismatch decreased, the ratio increased because the coupling became stronger. The coupling of type 2 was found to be much smaller than that of type 1. Since the characteristics of the oscillator were measured in a vacuum, the Q factor Ž Q y . of the oscillator for the detection mode was above 1000. Fig. 11 shows that, when Q y is 1000, for frequency mismatches Ž D f . of y2.2, y1.2 and 0.3%, the calculated phase lag angle Ž w ., which
Fig. 15. Block diagram of signal processing setup.
Y. Mochida et al.r Sensors and Actuators 80 (2000) 170–178
was used in Eqs. Ž10. and Ž14., is 1.3, 2.4 and 170.58, respectively. As mentioned above, the oscillation of the mass is an elliptical motion when there is velocity coupling and w f 0 or 1808. So, it was concluded that the elliptical motion of the fabricated oscillator was due to velocity coupling. A damping force on the oscillatory velocity in the x-direction leaks to the y-direction when the mass and the substrate are not completely parallel to each other, as shown in Fig. 14. It seems that this misalignment of the oscillator occurs when there is an imbalance in the beams or when the anodic bonding produces residual stress, both of which arise from fabrication error. In type 2, they have little effect because the beams for the drive mode are rigid with respect to the y-direction, and the beams for the detection mode, which support the mass, are separated from the substrate. As the result, the coupling of type 2 should be much smaller than that of type 1.
177
Fig. 17. Noise vs. frequency mismatch.
A block diagram of the signal processing setup for the gyroscope is shown in Fig. 15. The change in capacitance between the detection electrode and the mass of the oscillator is detected with an FET, which functions as a C–V converter w2x. The output signals of types 1 and 2 gyroscopes were measured during frequency tuning with a DC bias voltage. The criteria used to compare the oscillator characteristics were the normalized sensitivity and normalized noise, which are the sensitivity and noise divided by the amplification of the signal processing, respectively. The resolution was calculated for the noise equivalent rate. The normalized sensitivity, normalized noise, and resolution are shown in Figs. 16–18 as a function of the frequency mismatch. As the frequency mismatch decreased, the normalized sensitivity of type 2 increased more than that of type 1, while the increase in the normalized noise
was roughly the same. As a result, the resolution of type 2 was better than that of type 1. One reason for the better resolution of type 2 is the weaker coupling. The resolution of type 2 was 0.078rs at a band width of 10 Hz. Reducing the mismatch by tuning the frequency with a DC bias voltage increased not only the sensitivity, but also the noise. This result suggests that noise, which arises from instability in the oscillations, increases as the coupling becomes stronger. So, the resolution cannot be improved very much when the frequency mismatch is reduced. A micromachined gyroscope with a structure similar to that of type 1 has been reported w10x. The resolution was about 0.18rs when the frequency was tuned with a DC bias voltage. Since different devices use different types of signal processing, it is difficult to compare gyroscope characteristics. However, the better resolution of type 2 indicates that an oscillator structure in which there are independent beams for the drive and the detection modes
Fig. 16. Sensitivity vs. frequency mismatch.
Fig. 18. Resolution vs. frequency mismatch.
5.4. Gyroscope characteristics
178
Y. Mochida et al.r Sensors and Actuators 80 (2000) 170–178
provides better gyroscope characteristics. The other gyroscope using frequency tuning with a DC bias voltage employed the same beams for drive and detection and oscillated in a rotational mode, so the frequency mismatch could only be reduced to 2% w7x. In our study, the frequency of type 2 was tuned so that the mismatch was only 1% because the coupling was small. So, it seems that the structure of type 2 makes it easy to reduce the frequency mismatch by means of frequency tuning with a DC bias voltage. This is why the sensitivity of the gyroscope could be improved so much.
6. Conclusion Two types of micromachined gyroscopes — one with a simple structure, and one with independent beams for the drive and detection modes — were fabricated; and their oscillation characteristic were observed with a two-dimensional laser displacement meter. The oscillations were found to be an elliptical motion caused by coupling, which resulted from a damping force affecting the oscillatory velocity. A device with independent pairs of beams for the two modes exhibited weaker coupling than one with a simple structure. The normalized sensitivity, normalized noise, and resolution were measured as a function of the frequency mismatch. The gyroscope with independent pairs of beams exhibited better resolution: 0.078rs at a band width of 10 Hz. One reason for the better resolution is the weaker coupling.
Acknowledgements This work was performed under the management of the Micromachine Center as the ISTF ŽIndustrial Science and Technology Frontier Program., ‘‘Research and Development of Micromachine Technology’’, of MITI supported by NEDO ŽNew Energy and Industrial Technology Development Organization..
References w1x J.Bernstein, S. Cho, A.T. Kourepenis, P. Maciel, M. Weinberg, A micromachined comb-drive tuning fork rate gyroscope, Proc. MEMS ‘93, Fort Lauderdale, FL, 1993, pp. 143–148.
w2x K. Maenaka, T. Shiozawa, Silicon rate sensor using anisotropic etching technology, Proc. Transducer ‘93, Yokohama, Japan, 1993, pp. 642–645. w3x J. Soderkvist, Micromachied gyroscope, Proc. Transducer ‘93, ¨ Yokohama, Japan, 1993, pp. 638–641. w4x K. Tanaka, Y. Mochida, S. Sugimoto, K. Moriya, T. Hasegawa, K. Atsuchi, K. Ohwada, Micromachined vibrating gyroscope, Proc. MEMS ‘95, Amsterdam, The Netherlands, 1995, pp. 278–281. w5x M. Yamashita, K. Minami, M. Esashi, A silicon micromachined resonant angular rate sensor using electrostatic excitation and capacitive detection, Tech. Dig. 14th Sensor Symp., Tokyo, Japan, 1996, pp. 39–42. w6x F. Paoletti, M.-A. Gretillat, N.F. de Rooij, A silicon micromachined vibrating gyroscope with piezoresistive detection and electromagnetic excitation, Proc. MEMS ‘96, Atlanta, GA, 1996, pp. 162–167. w7x T. Juneau, A.P. Pisano, Micromachined dual input axis angular rate sensor, Proc. Solid-State Sensor and Actuator Workshop ‘96, Hilton Head, SC, 1996, pp. 299–302. w8x W.A. Clark, R.T. Howe, Surface micromachined z-axis vibratory rate gyroscope, Proc. Solid-State Sensor and Actuator Workshop ‘96, Hilton Head, SC, 1996, pp. 283–287. w9x K. Maenaka, T. Fujita, Y. Konishi, M. Maeda, Analysis of a Highly Sensitive Silicon Gyroscope with Cantilever Beam as Vibrating Mass, Sensors and Actuators A 54 Ž1996. 568–573. w10x Y. Oh, B. Lee, S. Baek, H. Kim, J. Kim, S. Kang, C. Song, A Surface-Micromachined Tunable Vibratory Gyroscope, Proc. MEMS ‘97, Nagoya, Japan, 1997, pp. 272–277. w11x W. Geiger, B. Folkmer, U. Sobe, H. Sandmaier, W. Lang, New designs of micromachined vibrating rate gyroscope with decoupled oscillation modes, Transducer ‘97, Chicago, IL, 1997, pp. 1129– 1132. w12x M. Lutz, W. Golderer, J. Gerstenmeier, J. Marek, B. Maihofer, S. ¨ Mahler, H. Munzel, U. Bischof, A precision yaw rate sensor in ¨ silicon micromachining, Transducer ‘97, Chicago, IL, 1997, pp. 847–850. w13x J. Soderkvist, An equivalent circuit description of two coupled ¨ vibrations, J. Acoust. Soc. Am. 90 Ž2. Ž1991. 693–699. Yoichi Mochida received his BE and MS degrees in Applied Physics from Waseda University, Tokyo, Japan, in 1984 and 1986, respectively. He worked at Hitachi Metals from 1986 to 1991. Then he joined Murata Manufacturing, since 1991. His current interests include microsensors and micromachining technologies. Masaya Tamura received his BE degree in Chemical Engineering from Meiji University, Kanagawa, Japan, in 1992. He has been working at Murata Manufacturing, since 1992, in the field of microsensors. Kuniki Ohwada received his BS degree in Physics and PhD degree in Electronic Engineering from Tohoku University, Sendai, Japan, in 1967 and 1984. From 1967 to 1990, he worked at the Electrical Communication Laboratories of Nippon Telegraph, where he was engaged in research on thin film capacitors, CMOS devices and GaAs integrated circuits. He joined Murata Manufacturing in 1990, where he began research on silicon micromachining development. He is now a general manager of a development department in this area.
A micromachined vibrating rate gyroscope with independent beams for the drive and detection modes Yoichi Mochida ) , Masaya Tamura, Kuniki Ohwada Yokohama Technical Center, Murata Manufacturing, 1-18-1, Hakusan, Midori-ku, Yokohama 226-0006, Japan Received 13 April 1999; accepted 19 July 1999
Abstract We investigated the mechanical coupling between the drive and detection modes of micromachined vibrating rate gyroscopes, and designed and fabricated a gyroscope with a new structure to reduce the coupling. The coupling of the oscillator was measured using a new measurement system with a two-dimensional laser displacement meter. The oscillations were found to have an elliptical motion due to the coupling. When the frequency mismatch was reduced by means of electrostatic frequency tuning with a DC bias voltage, the coupling increased. The new structure, which has independent beams for the drive and detection modes, exhibited weaker coupling; and the resolution was 0.078rs at a band width of 10 Hz. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Gyroscope; Micromachining; Mechanical coupling; Laser displacement meter
1. Introduction Recently, a number of micromachined vibrating rate gyroscopes, which use separate oscillation modes for drive and detection, have been developed w1–12x. In these gyroscopes, the sensitivity improves as the frequency mismatch between the two modes decreases. One way to reduce the mismatch is to employ electrostatic frequency tuning with a DC bias voltage w5,7,8,10x. However, as the mismatch decreases, the mechanical coupling between the two modes makes operation more and more unstable. Previous studies have failed to show how coupling affects a micromachined gyroscope. Conventionally, a microscope is used to measure the coupling of a vibrating gyroscope with an oscillator consisting of a silicon cantilever w9x, but this method is difficult to use when an extremely small oscillator is fabricated using silicon micromachining. This paper reports on the measurement of the coupling using a two-dimensional laser displacement meter. In order to investigate the influence of coupling on a micromachined gyroscope, we measured the mechanical character-
istics of fabricated oscillators when the frequency mismatch was tuned with a DC bias voltage. To reduce the coupling, we designed and fabricated a micromachined gyroscope with independent beams for the drive and detection modes, and then measured its characteristics.
2. Principle
2.1. Principle of Õibrating gyroscope A micromachined vibrating rate gyroscope is generally composed of a mass and beams that oscillate. The mass readily oscillates in two orthogonal directions. A simple model of a vibrating gyroscope is shown in Fig. 1. When the mass is driven along the x-direction by an external force Ž F Ž t . s F0 sin v t ., and it is subjected to a constant angular rate Ž V ., a Coriolis force Ž Fc s 2 m V x˙ . is induced in the y-direction. In this case, the equation of motion is mx¨ q c x x˙ q k x x s Fo sin v t ,
Ž 1.
my¨ q c y y˙ y 2 m V x˙ q k y y s 0,
Ž 2.
)
Corresponding author. Tel: q81-45-939-7150; fax: q81-45-939-7156; E-mail: [email protected]
0924-4247r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 Ž 9 9 . 0 0 2 6 3 - 0
Y. Mochida et al.r Sensors and Actuators 80 (2000) 170–178
171
fabrication error. Consequently, the energy of the drive oscillation leaks in the y-direction, resulting in the mechanical coupling between the oscillations in the two directions. This coupling is assumed to depend on the amplitude and velocity of the drive oscillation w13x. When both types of coupling are present, the equation of motion is mx¨ q Ž c x y c c . x˙ q c c y˙ q Ž k x y k c . x q k c y s Fo sin v t , Ž 5. my¨ q Ž c y y c c . y˙ q Ž cc y 2 m V . x˙ q Ž k y y k c . y q k c x s 0,
where k c and c c are the coupling coefficients for the amplitude and velocity, respectively. When there is little velocity coupling and V s 0, the equation of motion is
Fig. 1. Simple model of vibrating gyroscope.
where m is the mass; c x and c y are the resistances due to viscosity, and k x and k y are the spring constants, for the x- and y-directions, respectively. When the mass is driven such that it oscillates at the resonance frequency, the oscillatory motion derived from Eqs. Ž1. and Ž2. is x Ž t . s yx m cos v x t , yŽ t. s
Ž 3.
2 m Vv x x m ky
1
(½
1 y Ž v xrv y .
=sin Ž v x t y w . ,
2 2
5 q Ž v rv . x
2
Fo Q x kx
w s tany1 ls
vx vy
Qx s
mx¨ q c x x˙ q Ž k x y k c . x s Fo sin v t ,
Ž 7.
my¨ q c y y˙ q k y y q k c x s 0,
Ž 8.
where c c s 0, x 4 y, k x ,k y 4 k c , and V s 0. When the mass is driven so that it oscillates at the resonance frequency, the oscillatory motion derived from Eqs. Ž7. and Ž8. is given by x Ž t . s yx m cos v x t ,
y
Ž 4.
in which xm s
Ž 6.
,
yŽ t. sy
kc xm ky
1
(½
1 y Ž v xrv y .
=cos Ž v x t y w . .
2 2
5 q Ž v rQ v . x
y
2
y
Ž 10 .
When w f 0 or 1808, the mass oscillates along a linear path at an angle to the x-axis, as shown in Fig. 2a.
l Q y Ž 1 y l2 .
Ž 9.
,
,
mvx cx
, Qy s
mvy cy
,
where v x and v y are the resonant frequencies, and Q x and Q y are the resonance quality factors, for the x- and y-directions, respectively; and x m is the amplitude at resonance for the x-direction. If the drive amplitude is constant, then the amplitude of the resulting oscillatory motion is proportional to the angular rate Ž V .. Furthermore, the sensitivity of the gyroscope Ž< yrV <. is higher when the frequency mismatch Ž< D v < s < v x y v y <. is small. 2.2. Mechanical coupling The mass and beams of a micromachined vibrating rate gyroscope are not usually completely symmetrical due to
Fig. 2. Oscillatory motion of mass: Ža. amplitude coupling for w f 0 or 1808; Žb. velocity coupling for w f 0 or 1808.
Y. Mochida et al.r Sensors and Actuators 80 (2000) 170–178
172
yŽ t. s
cc v x x m ky
1
(½
1 y Ž v xrv y .
2 2
5 q Ž v rQ v .
=sin Ž v x t y w . .
x
y
2
y
Ž 14 .
When w f 0 or 1808, the oscillation of the mass is an elliptical motion, as shown in Fig. 2b. These results show that, in a micromachined gyroscope, the type of mechanical coupling can be determined from the oscillatory motion exhibited by the mass of a fabricated oscillator.
3. Design of gyroscopes 3.1. Gyroscope with simple structure Fig. 3. Structure of type 1: Ža. bird’s eye view; Žb. cross-section view ŽA–AX ..
On the other hand, when there is little amplitude coupling and V s 0, the equation of motion is mx¨ q Ž c x y c c . x˙ q k x x s Fo sin v t ,
Ž 11 .
my¨ q c y y˙ q c c x˙ q k y y s 0,
Ž 12 .
where k c s 0, x˙ 4 y, ˙ c x , c y 4 cc , and V s 0. When the mass is driven so that it oscillates at the resonance frequency, the oscillatory motion derived from Eqs. Ž11. and Ž12. is x Ž t . s yx m cos v x t ,
Ž 13 .
We have investigated a micromachined vibrating rate gyroscope with the simple structure Žtype 1. w4x shown in Fig. 3, in which the mass is supported by four beams. The oscillator is 400 mm = 1200 mm = 10 mm in size. It is driven by means of comb electrodes, and there are two electrodes under the mass: one to detect capacitance, and other to tune the frequency. When AC and DC voltages are applied to the comb electrodes, the mass oscillates along the x-direction Ždrive mode.. An externally induced rotation about the z-axis Žangular rate: V . produces a deflection in the y-direction due to the Coriolis force. This deflection is detected as a change in the capacitance between the mass and the detection electrode. If the frequency mismatch Ž D f . between the drive and detection modes is small, the detection mode enhances the deflection. The shapes of the mass-and-beam structure at the
Fig. 4. Shape of beam-and-mass structure of type 1 during deflection.
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Fig. 7. Fabrication process.
four beams supporting the mass are bent in the x-direction in the drive mode, and in the y-direction in the detection mode. In this structure, coupling between the two modes can easily arise if the beams are not completely symmetrical as a result of fabrication error during silicon micromachining. X Fig. 5. Structure of type 2: Ža. top view; Žb-1. cross-section view ŽA–A .; X Žb-2. cross-section view ŽB–B ..
point of peak deflection obtained by an FEM analysis are shown in Fig. 4 for the drive and detection modes. The
3.2. New design to reduce coupling To reduce the coupling, the new structure Žtype 2. shown in Fig. 5 was designed. It features independent pairs of beams for the drive and detection modes. The mass of the oscillator is connected to a frame supported by four
Fig. 6. Shape of beam-and-mass structure of type 2 during deflection.
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Fig. 8. SEM images of Ža. type 1 Žsimple structure. and Žb. type 2 Žnew design. gyroscope. Fig. 10. Resonant frequency vs. DC bias voltage: Ža. type 1; Žb. type 2.
beams that are anchored to the substrate. The oscillator is 800 mm = 1200 mm = 50 mm in size. The inner beams are 20 mm wide and 10 mm high, and the outer beams are 5 mm wide and 50 mm high. It is driven by comb electrodes, and there are two electrodes under the mass, just as in type 1; so the method of detecting the angular rate is the same as for type 1. The shape of the mass-andbeam structure at the point of peak deflection obtained by
an FEM analysis is shown in Fig. 6 for the drive and detection modes. In the drive mode, the outer beams are bent in the x-direction, but there is little deflection of the inner beams. In the detection mode, the inner beams are bent in the y-direction, but there is little deflection of the outer beams. These results indicate that the outer and inner
Fig. 9. Measurement system.
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beams are rigid with respect to the y- and x-directions, respectively. This means that the two pair of beams move independently in the two modes. In the type 2 structure, the coupling between the modes seems to be very weak because drive mode oscillations have little effect on the beams used in the detection mode.
4. Fabrication The fabrication process for type 2 is shown in Fig. 7. It is the same as for type 1 except for step Ža.. Ža. A SOI Žsilicon-on-insulator. substrate with a 200-mm thick base silicon layer, a 4-mm thick SiO 2 insulator layer, and a 50-mm thick top silicon layer is used. To make the inner beams, which move in the z-direction, about 40 mm of part of the top silicon layer is etched off by reactive ion etching ŽRIE.. For a type 1 device, a different SOI substrate with a 10-mm thick top silicon layer is used; and the top silicon layer is not etched in this step. Žb-1. To make the gap between the two electrodes and the silicon mass, a glass substrate is etched down about 2 mm with HF.
Fig. 12. Lisssajou figures of lateral and vertical displacement of type 2: Ža. DC biass 2 V; D f sy2.3%; Žb. DC biass6 V; D f sy0.8%; Žc. DC biass 7 V; D f sy0.1%.
Žb-2. After the deposition of Au and Cr layers, the two electrodes are patterned by wet etching. Žc. The SOI and glass substrates are bonded together by anodic bonding. Žd. The base silicon and insulator layers are removed by wet etching with TMAH and HF, respectively.
Fig. 11. Lisssajou figures of lateral and vertical displacement of type 1: Ža. DC biass 4 V; D f sy2.2%; Žb. DC biass 5 V; D f sy1.2%; Žc. DC biass6.2 V; D f s 0.3%.
Fig. 13. Amplitude ratio vs. frequency mismatch.
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Fig. 14. Damping force and velocity of misaligned oscillator.
Že. Finally, the top silicon layer on the glass substrate is etched by RIE to form the oscillator. SEM images of fabricated oscillators are shown in Fig. 8.
5. Experimental results
To measure the frequency response of the oscillation amplitude, the output voltages of the laser displacement meter for the lateral and vertical directions are fed back into the network analyzer. Then, the resonant frequency of each mode is determined from the peak of the corresponding frequency response curve. Fig. 10 shows the results of measurements of the resonant frequencies of the drive and detection modes when a DC bias voltage was applied between the tuning electrode and the mass of the oscillator. For both types 1 and 2 devices, as the bias voltage increased, the resonant frequency of the detection mode dropped, while that of the drive mode remained virtually constant. As a result, the frequency mismatch Ž D f . between the two modes decreased, and was nearly zero at a bias of about 6 or 7 V. 5.3. Measurements of the coupling
5.1. Measurement system Resonant frequencies and oscillation amplitudes were measured with a two-dimensional laser displacement meter. The measurement systems are shown in Fig. 9. The laser displacement meter, which employs two laser for a single measurement, can measure both lateral and vertical movement at the same time. DC and AC voltages are applied to the electrodes of the oscillator in a vacuum chamber, which reduces the air damping of the oscillator; the pressure in the chamber is below 100 Pa. A network analyzer is used to measure resonant frequencies, as shown in Fig. 9a; and a function generator and a oscilloscope are used to measure oscillation amplitudes, as shown in Fig. 9b. 5.2. Frequency measurements for tuning with a DC bias Õoltage The oscillator is excited by applying a reference voltage signal from the network analyzer between either the detection or the drive electrode and the mass of the oscillator.
The oscillation amplitudes of oscillators driven at the resonant frequency of the drive mode by AC and DC bias voltages from the function generator were also measured. Combining the output voltages of the displacement meter for the lateral and vertical directions on an oscilloscope produces a Lissajou figure, which shows the motion of the oscillator in the x–y plain. Figs. 11 and 12 show the results for types 1 and 2, respectively. It is clear that the oscillation is an elliptical motion due to coupling between the two modes. The ratio of the amplitude in the y-direction to that in the x-direction was calculated. Fig. 13 shows the amplitude ratio as a function of frequency mismatch. As the frequency mismatch decreased, the ratio increased because the coupling became stronger. The coupling of type 2 was found to be much smaller than that of type 1. Since the characteristics of the oscillator were measured in a vacuum, the Q factor Ž Q y . of the oscillator for the detection mode was above 1000. Fig. 11 shows that, when Q y is 1000, for frequency mismatches Ž D f . of y2.2, y1.2 and 0.3%, the calculated phase lag angle Ž w ., which
Fig. 15. Block diagram of signal processing setup.
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was used in Eqs. Ž10. and Ž14., is 1.3, 2.4 and 170.58, respectively. As mentioned above, the oscillation of the mass is an elliptical motion when there is velocity coupling and w f 0 or 1808. So, it was concluded that the elliptical motion of the fabricated oscillator was due to velocity coupling. A damping force on the oscillatory velocity in the x-direction leaks to the y-direction when the mass and the substrate are not completely parallel to each other, as shown in Fig. 14. It seems that this misalignment of the oscillator occurs when there is an imbalance in the beams or when the anodic bonding produces residual stress, both of which arise from fabrication error. In type 2, they have little effect because the beams for the drive mode are rigid with respect to the y-direction, and the beams for the detection mode, which support the mass, are separated from the substrate. As the result, the coupling of type 2 should be much smaller than that of type 1.
177
Fig. 17. Noise vs. frequency mismatch.
A block diagram of the signal processing setup for the gyroscope is shown in Fig. 15. The change in capacitance between the detection electrode and the mass of the oscillator is detected with an FET, which functions as a C–V converter w2x. The output signals of types 1 and 2 gyroscopes were measured during frequency tuning with a DC bias voltage. The criteria used to compare the oscillator characteristics were the normalized sensitivity and normalized noise, which are the sensitivity and noise divided by the amplification of the signal processing, respectively. The resolution was calculated for the noise equivalent rate. The normalized sensitivity, normalized noise, and resolution are shown in Figs. 16–18 as a function of the frequency mismatch. As the frequency mismatch decreased, the normalized sensitivity of type 2 increased more than that of type 1, while the increase in the normalized noise
was roughly the same. As a result, the resolution of type 2 was better than that of type 1. One reason for the better resolution of type 2 is the weaker coupling. The resolution of type 2 was 0.078rs at a band width of 10 Hz. Reducing the mismatch by tuning the frequency with a DC bias voltage increased not only the sensitivity, but also the noise. This result suggests that noise, which arises from instability in the oscillations, increases as the coupling becomes stronger. So, the resolution cannot be improved very much when the frequency mismatch is reduced. A micromachined gyroscope with a structure similar to that of type 1 has been reported w10x. The resolution was about 0.18rs when the frequency was tuned with a DC bias voltage. Since different devices use different types of signal processing, it is difficult to compare gyroscope characteristics. However, the better resolution of type 2 indicates that an oscillator structure in which there are independent beams for the drive and the detection modes
Fig. 16. Sensitivity vs. frequency mismatch.
Fig. 18. Resolution vs. frequency mismatch.
5.4. Gyroscope characteristics
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provides better gyroscope characteristics. The other gyroscope using frequency tuning with a DC bias voltage employed the same beams for drive and detection and oscillated in a rotational mode, so the frequency mismatch could only be reduced to 2% w7x. In our study, the frequency of type 2 was tuned so that the mismatch was only 1% because the coupling was small. So, it seems that the structure of type 2 makes it easy to reduce the frequency mismatch by means of frequency tuning with a DC bias voltage. This is why the sensitivity of the gyroscope could be improved so much.
6. Conclusion Two types of micromachined gyroscopes — one with a simple structure, and one with independent beams for the drive and detection modes — were fabricated; and their oscillation characteristic were observed with a two-dimensional laser displacement meter. The oscillations were found to be an elliptical motion caused by coupling, which resulted from a damping force affecting the oscillatory velocity. A device with independent pairs of beams for the two modes exhibited weaker coupling than one with a simple structure. The normalized sensitivity, normalized noise, and resolution were measured as a function of the frequency mismatch. The gyroscope with independent pairs of beams exhibited better resolution: 0.078rs at a band width of 10 Hz. One reason for the better resolution is the weaker coupling.
Acknowledgements This work was performed under the management of the Micromachine Center as the ISTF ŽIndustrial Science and Technology Frontier Program., ‘‘Research and Development of Micromachine Technology’’, of MITI supported by NEDO ŽNew Energy and Industrial Technology Development Organization..
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w2x K. Maenaka, T. Shiozawa, Silicon rate sensor using anisotropic etching technology, Proc. Transducer ‘93, Yokohama, Japan, 1993, pp. 642–645. w3x J. Soderkvist, Micromachied gyroscope, Proc. Transducer ‘93, ¨ Yokohama, Japan, 1993, pp. 638–641. w4x K. Tanaka, Y. Mochida, S. Sugimoto, K. Moriya, T. Hasegawa, K. Atsuchi, K. Ohwada, Micromachined vibrating gyroscope, Proc. MEMS ‘95, Amsterdam, The Netherlands, 1995, pp. 278–281. w5x M. Yamashita, K. Minami, M. Esashi, A silicon micromachined resonant angular rate sensor using electrostatic excitation and capacitive detection, Tech. Dig. 14th Sensor Symp., Tokyo, Japan, 1996, pp. 39–42. w6x F. Paoletti, M.-A. Gretillat, N.F. de Rooij, A silicon micromachined vibrating gyroscope with piezoresistive detection and electromagnetic excitation, Proc. MEMS ‘96, Atlanta, GA, 1996, pp. 162–167. w7x T. Juneau, A.P. Pisano, Micromachined dual input axis angular rate sensor, Proc. Solid-State Sensor and Actuator Workshop ‘96, Hilton Head, SC, 1996, pp. 299–302. w8x W.A. Clark, R.T. Howe, Surface micromachined z-axis vibratory rate gyroscope, Proc. Solid-State Sensor and Actuator Workshop ‘96, Hilton Head, SC, 1996, pp. 283–287. w9x K. Maenaka, T. Fujita, Y. Konishi, M. Maeda, Analysis of a Highly Sensitive Silicon Gyroscope with Cantilever Beam as Vibrating Mass, Sensors and Actuators A 54 Ž1996. 568–573. w10x Y. Oh, B. Lee, S. Baek, H. Kim, J. Kim, S. Kang, C. Song, A Surface-Micromachined Tunable Vibratory Gyroscope, Proc. MEMS ‘97, Nagoya, Japan, 1997, pp. 272–277. w11x W. Geiger, B. Folkmer, U. Sobe, H. Sandmaier, W. Lang, New designs of micromachined vibrating rate gyroscope with decoupled oscillation modes, Transducer ‘97, Chicago, IL, 1997, pp. 1129– 1132. w12x M. Lutz, W. Golderer, J. Gerstenmeier, J. Marek, B. Maihofer, S. ¨ Mahler, H. Munzel, U. Bischof, A precision yaw rate sensor in ¨ silicon micromachining, Transducer ‘97, Chicago, IL, 1997, pp. 847–850. w13x J. Soderkvist, An equivalent circuit description of two coupled ¨ vibrations, J. Acoust. Soc. Am. 90 Ž2. Ž1991. 693–699. Yoichi Mochida received his BE and MS degrees in Applied Physics from Waseda University, Tokyo, Japan, in 1984 and 1986, respectively. He worked at Hitachi Metals from 1986 to 1991. Then he joined Murata Manufacturing, since 1991. His current interests include microsensors and micromachining technologies. Masaya Tamura received his BE degree in Chemical Engineering from Meiji University, Kanagawa, Japan, in 1992. He has been working at Murata Manufacturing, since 1992, in the field of microsensors. Kuniki Ohwada received his BS degree in Physics and PhD degree in Electronic Engineering from Tohoku University, Sendai, Japan, in 1967 and 1984. From 1967 to 1990, he worked at the Electrical Communication Laboratories of Nippon Telegraph, where he was engaged in research on thin film capacitors, CMOS devices and GaAs integrated circuits. He joined Murata Manufacturing in 1990, where he began research on silicon micromachining development. He is now a general manager of a development department in this area.