phase and vibration analysis for a cmos-mems gyroscope

©Freund Publishing House Ltd International Journal of Nonlinear Sciences and Numerical Simulation 3, 319-324, 2002

Phase and Vibration Analysis for a CMOS-MEMS Gyroscope Huikai Xie*, Gary K. Fedder

Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA; E-mail: [email protected].

Zhiyu Pan, Wilhelm Frey

Research & Technology Center of North America, Robert Bosch Corporation, Palo Alto, CA, USA

Abstract

Phase is an important issue in a vibratory-rate gyroscope for Coriolis signal demodulation and off-axis motion compensation. This paper uses a simplified three-dimensional comb model to analyze the motion coupling and the phase relation between the Coriolis signal and coupled signals for a lateral-axis CMOS-MEMS gyroscope. It is found that the off-axis motion is highly nonlinear and dominates the coupled motions in the comb-drive actuation. The phase relation was verified experimentally.

Key words: Gyroscope, CMOS-MEMS, phase, motion coupling, quadrature

1. Introduction

Gyroscopes are devices that are used to measure rotation rate. Recently, markets in automotive and consumer electronic products are demanding low-cost gyroscopes for driving safety and comfort, and motion stability control. Inspired by the successful commercialization of MEMS accelerometers, MEMS gyroscopes are widely believed to be a solution for medium performance requirements and have drawn extensive attention [1-6]. Bosch developed commercial micromachined gyroscopes a few years ago [4], though still too expensive for many applications. Analog Devices, Inc. is planning to put their integrated MEMS gyroscopes [5] into markets very soon.

Most of MEMS gyroscopes are vibratory gyroscopes that are based on the Coriolis

effect. When a structure is vibrating in a rotating reference frame, a Coriolis acceleration arises and is proportional to the rotation rate Ω and the vibration velocity V, i.e.,

ac=2QxV (1)

Coriolis acceleration is very small. For instance, the displacement due to the Coriolis acceleration in a typical vibratory gyroscope is only about 10"6 per °/s of the excitation vibration amplitude. Therefore the Coriolis motion will be overwhelmed by even a small fraction of the excitation vibration coupled into it. Fortunately, the Coriolis motion has a different phase compared to the coupled motions, which provides opportunities to maximize the Coriolis signal by tuning the phase and/or compensate the coupled motions by properly applying anti-phase forces.

In prior work, it was found that the

•Currently at Department of Electrical & Computer F"c:ne.ering, University of Florida, Gainesville, FL 32611, USA

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Coriolis signal had a phase shift of 12 compared to the mixed signal in a deep reactive-ion-etch (DRIE) CMOS-MEMS gyroscope [6]. This paper is focused on the analysis of motion coupling and phase relationship in that gyroscope through a simplified three-dimensional comb model and using Matlab/Simulink [7]. Experimental results are compared to the simulation results.

2. DRIE CMOS-MEMS Gyroscope

A vibratory gyroscope has been fabricated by the authors using a DRIE CMOS-MEMS process [8], The gyroscope employs a lateral-axis comb-finger actuation and vertical-axis comb-finger sensing, as shown in Fig. 1(a). The design, fabrication and characterization of this device were reported in [6].

Fig. 1(b) shows the simplified 2-D model of the gyroscope. It consists of a frame, a proof mass and a network of suspensions.

sensed as a Coriolis accelerometer in the z-direction (sense mode). Thus, according to Eq. (1), the device is a y-axis gyroscope.

3. Phase Relationship

The mechanical structure can be considered as a second-order system along each axis. The phase delay between the acting force or acceleration and the resultant displacement is given by

φχ = atan ω! ω.

Qx[l-(a>,coxY)^ (2)

where Χ = χ, y or ζ, ω is the operating frequency, Qx and ωχ are the Q-factor and resonant frequency in the X direction, respectively.

Due to defects and/or variations from fabrication processing, imperfections such as asymmetry and anisoelasticity in the structure are generated. Consequently, cross-coupled motions are produced. For example, the

drive mode

sense mode

x-axis drive

x-axis sense x/z-axis actuator

rollers frame W w

Iflgj x-axis f|l|l 'j | | jg^Jp l ' f " ; | ö l i . n |[||Π~~spring

(a) (b) Figure 1: SEM of the y-axis DRIE gyroscope, (a) SEM. (b) 2-D model.

The structure is a two-fold, orthogonal primary vibration is along the x-direction, but spring-mass system where the whole vibration may be also observed in the z-structure vibrates in the x-direction (drive direction. This cross-coupled motion is mode) while the inner mass displacement is commonly referred as "quadrature" motion.

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Both drive and sense springs are made of beams, and therefore the stiffness in any direction is a finite value. The stiffnesses along the primary axes are denoted as kd,x, kd,y

and kd,z for the drive mode, and ksx, ksy and ksz for the sense mode. As we will see in the next section, a x-axis electrostatic comb drive generates forces along all three axes. Thus, the finite stiffness of the drive mode in the z-

Figure 2: Phase relation of various signals in a gyroscope.

axis results in a z-axis vibration that is mixed with the Coriolis motion. This vibration is directly coupled into the sense mode, resulting in a z-axis direct-coupled motion zz

on the proof mass. In the following, the phase relationships among the excitation vibration JC, zz and the Coriolis motion ac are analyzed.

1) Excitation vibration versus drive voltage Normally, the device operates at the

resonant frequency of the drive mode, i.e., the x-axis displacement has a 90° phase delay with respect to the actuation force. For electrostatic actuation, a large d.c. offset voltage is often used to minimize the second harmonic term of the generated force, which is then in-phase with the a.c. voltage, Vac. It often happens especially in open loop operation that the operating frequency is not exactly the resonance frequency. In this case, there will be a phase-offset α from 90° between Vac and JC, as shown in Fig. 2.

2) Coriolis versus quadrature The quadrature motion of the drive frame,

which is in-phase with JC, generates a coupled

quadrature motion zq on the proof mass. Eq. (1) indicates that there is a 90° phase shift between JC and the Coriolis acceleration ac

and thus between zq and zc, too. Therefore, it is possible to suppress zq by locking the phase of the demodulation. 3) Coriolis versus direct-coupled motion

Typically, kd,y and kd,z are much larger than kd.x, i.e., the resonant frequencies in the y- and z- directions are much higher than that in the x-direction (which is normally the operating frequency). Therefore, the generated z-axis acceleration az is in-phase with Vac• The resultant zz of the proof mass has a phase delay φ with respect to az

according to Eq. (2). Similarly, there is a phase delay between ac and zc, as shown in Fig. 2.

The phase difference between zz and zc is a , which is typically equal to zero. Thus, the compensation of this direct-coupled signal cannot be performed with phase-lock during demodulation and has to be implemented in the drive mode by using a cancellation technique to null the z-axis component of drive force.

4) Coriolis versus combined motion coupling Both the quadrature and direct-coupled

motions act on the proof mass through a rigid frame, while the Coriolis force applies directly to the proof mass. Therefore, zq and Zz are actually mixed as a single motion with a compound amplitude and phase. This mixed motion is defined as the combined coupling motion and denoted as zz,q in Fig. 2. The phase difference between zc and zz,q depends on the amplitude ratio of zq to zz. The phase difference is (90°-a) if |zj>> \zz\ and α if

k l « ki-ln order to find the phase difference

between zc and zz,q, the phases corresponding to the maximum output of the coupling signal and the highest sensitivity to the rotation

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must be measured. The motion coupling is analyzed next through a simplified 3-D comb-drive model. The phase relation will be explored experimentally in a later section.

4. Comb-Drive 3-D Motion Analysis

Fig. 3 shows the comb finger model with presumed lateral offset and vertical misalignment. For this first-order analysis, parallel-plate approximation is used and fringing effects are ignored. The capacitance of the comb finger can be expressed as

C(xy,z) ',z)=e0{h-z0+zi V

x0 +x x0 +x

g+y0+y g-y0-yj (3)

The capacitance gradients, dC/dx, dC/dy and dC/dz, are readily derived from Eq. (3). The electrostatic force per comb finger set is given by

x 2 dx (4)

where X = x, y, or z, and V is the applied voltage. Then, the equations of motion can be written as

X = -(p>x/Qx)X-c»2xX + N-Fx/ m (5)

where m is the mass of the movable structure. The above equations of motion are applied to the drive mode. For this actuation analysis, we assume the sense mode is passive, i.e., the rotation rate is zero, only inertial forces act on the central proof mass. Similarly, the equations of motion of the sense mode can be expressed as

= ~(p>xs /Qxs )*s ~ + , (6)

where Xs is the vector of x, y, and ζ displacements of the proof mass and X is the vector of displacements of the drive frame.

Eqs. (3) to (6) can be easily implemented in Matlab/Simulink. The lateral and vertical offsets are set to 0.2 μηι and 0.2 μιη,

respectively and other parameter values are based on the DRIE CMOS-MEMS gyroscope shown in Fig. 1 [6].

Transient analyses are conducted to understand the motion coupling and phase relation. Fig. 4 shows Simulink simulation results at different conditions. The primary excitation is along the x-axis. The y off-axis motion induces cross-axis sensitivity and the ζ off-axis motion directly mixes with the Coriolis signal. At the normal operating voltage (Vac = 5 V, Vdc = 18 V), the y-axis and z-axis displacements are highly non-linear, i.e., harmonics are present (Fig. 4(a)), and have only a 45° phase shift from the resonating x-axis motion. This makes the off-axis motion compensation difficult. If the voltage is small, e.g., Vac = 0.5 V, Vdc = 2 V, the y-axis and z-axis displacements are shifted by 90° from the x-axis motion, and the z-axis displacement in the sense mode is

9-y0 g+y0

Xo: overlap; Zo: vertical offset y0: lateral offset h: thickness g: gap

Figure 3: Comb finger model for 3-D motion analysis.

shifted by about 5° from the z-axis displacement in the drive mode.

The non-linearity of the z-axis displacements is caused by the z-axis electrostatic force, which is proportional to the overlap of the comb fingers. If the x-axis displacement is much smaller than the overlap, the non-linearity of the z-axis displacement is small, as the case shown in Fig. 4(b). If the x-axis displacement is comparable to the overlap, high non-linearity of the z-axis force and displacement arises.

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The overlap may be increased to reduce the non-linearity of the z-axis displacements, as shown in Fig. 4(c), where xo - 60 μηι. However, the y-axis displacements show a high non-linear behavior and both y-axis and z-axis displacements are large.

0.0194 0.0195 0.0196 0.0197 0.0198 0.0199 0.02

0.0194 0.0195 0.0196 0.0197 0.0198 0.0199 0.02

0.0194 0.0195 0.0196 0.0197 0.0198 0.0199 0 02

0.0194 0.0195 0.0196 0.0197 0.0198 0.0199 0.02

0.0194 0.0195 0.0196 0.0197 0.0198 0.0199 0.02 Time (second)

Figure 4: Simulink simulation results, (a) High drive voltage, (b) Low drive voltage, (c) High drive voltage with large overlap.

The phase relations of various signals existing in a vibratory gyroscope are presented. The direct-coupled motion is highly non-linear and difficult to compensate. It also dominates the ZRO signal and is difficult to suppress by demodulation since it has the same phase as the Coriolis signal. Therefore, z-axis force cancellation techniques must be included into vibratory gyroscope designs with comb-finger actuation for performance improvement.

-100 0 100 P h a s e i d e a r e e )

Fieure 5: Phase relation between ZRO and

curve reflects the combined coupling motion, while the sensitivity curve is the Coriolis signal. There is a 12° phase lag between the Coriolis signal and the combined coupling signal. It is clear that the direct-coupled motion dominates the quadrature motion in this case.

6. Conclusion

5. Experimental Results

To find out the phase relation between the Coriolis signal and coupling signal, the DRIE CMOS-MEMS gyroscope (Fig. 1) was mounted on a turntable to perform phase sweeping at differenct rotation rates. The measured phase dependences of the zero-rate output (ZRO) and the rotation sensitivity of the gyroscope are plotted in Fig. 5. The ZRO

Acknowledgement

The first two authors thank Mr. John A. Geen of Analog Devices, Inc. for valuable discussions. This work was sponsored by DARPA under the AFRL, Air Force Materiel Command, USAF, under agreement F30602-97-2-0323.

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References

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[2] M.W. Putty, K. Najafi, "A micromachined vibrating ring gyroscope," Tech. Digest. Solid-State Sensor and Actuator Workshop, Hilton Head Island, SC, 13-16 June 1994, p.213-220.

[3] W.A. Clark, R.T. Howe, R. Horowitz, "Surface micromachined Z-axis vibratory rate gyroscope", Tech. Digest. Solid-State Sensor and Actuator Workshop, Hilton Head Island, SC, USA; 3-6 June 1996, p.283-287.

[4] M. Lutz, et al., "A precision Yaw Rate Sensor in Silicon Micromachining", Transducers '97, Chicago, June 16-19, 1997, p.847-850.

[5] J.A. Geen, S.J. Sherman, J.F. Chang and S.R. Lewis, "Single-chip surface-micromachining integrated gyroscope with 50 deg/hour root Allan variance", Tech. Digest. 2002 IEEE International So I id-State Circuits Conference, San Francisco, CA, Feb. 3-7, 2002, pp.426-427.

[6] H. Xie, and G.K. Fedder, "A DRIE CMOS-MEMS Gyroscope", to be presented at IEEE Sensors 2002 Conference, June 12-14, 2002, Orlando, Florida.

[7] Matlab/Simulink, Version 6.0, The MathWorks, Inc., http: // www. mathworks. com.

[8] H. Xie, L. Erdmann, X. Zhu, K. Gabriel and G. Fedder, "Post-CMOS Processing For High- aspect-ratio Integrated Silicon Microstructures", Journal of Microelectromechanical Systems, 11(2) (2002) 93-101.

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phase and vibration analysis for a cmos-mems gyroscope

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