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Chapt.4 Mechanical Design of MEMS

Chapt.4 Mechanical Design of MEMS


Chapter 4

Mechanical Design of MEMS Gyroscopes

This chapter describes the fundamental mechanical elements in the MEMS implementation of vibratory gyroscopes. Common mechanical structures are presented and analyzed for both linear and torsional gyroscopes, discussing primary vibratory system design issues to realize the gyroscopic dynamical system. Analysis of various ?exure systems is followed by discussions on anisoelasticity and quadrature error due to mechanical imperfections. Finally, damping related issues are addressed, important material properties of silicon are highlighted, and mechanical design considerations to achieve a robust sensing element are discussed.

4.1 Mechanical Structure Designs
Various vibratory MEMS gyroscopes have been reported in the literature based on a wide range of mechanical structures. The common goal of all vibratory gyroscope structures is to realize a drive oscillator that generates and maintains a constant momentum, and a sense-mode accelerometer that measures the sinusoidal Coriolis force. Vast majority of micromachined rate gyroscopes form the drive oscillator and the sense-mode accelerometer out of a mass or a combination of masses suspended by ?exible beams above a substrate. The Coriolis force induced on the masses due to the drive vibration and the angular rate input has to be transferred to the sense-mode accelerometer in the orthogonal direction. Thus, at least one proof mass is required to be common to both the drive oscillator and the sense-mode accelerometer. The primary objective of mechanical structure is to form the coupled orthogonal drive and sense dynamical systems by providing the required degree-of-freedom (DOF) for the masses. Usually both the drive and sense dynamical systems are 1DOF oscillators. Thus, the resulting overall gyroscope can be modeled as a 2-DOF dynamical system as in Figure 4.1, where the modes are coupled by the Coriolis effect.

73

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4 Mechanical Design of MEMS Gyroscopes

The drive-mode oscillator and the sense-mode accelerometer can be based on either linear or torsional motion. In the case of a linear vibratory gyroscope, conservation of linear momentum results in energy transfer from the drive axis to the sense axis, while in a torsional gyroscope conservation of angular momentum results in energy transfer. The following sections outline the basics of both linear and torsional gyroscope designs.

Fig. 4.1 The proof mass, which is free to oscillate in the drive and sense directions, forms the 2-DOF gyroscope system.

4.2 Linear Vibratory Systems
Linear or translational micromachined vibratory gyroscopes are based on sustaining a linear drive oscillation, and detecting a linear sense-mode response to the sinusoidal Coriolis force in the presence of an angular-rate input. Since the induced Coriolis force is orthogonal to the drive-mode vibration, the proof-mass is required to be free to oscillate in two orthogonal linear directions, and desired to be constrained in other vibrational modes. The suspension system design becomes critical in achieving these objectives. The drive and sense axes are determined primarily by the desired angular rate detection axis. For example, a z-Axis gyroscope (Figure 4.2) requires the proof mass to be free to oscillate in the two in-plane orthogonal directions: the drive direction along the x-Axis and the sense direction along the y-Axis. The proof mass that provides the Coriolis coupling is allowed to oscillate along both x-Axis and y-Axis, becoming a 2-DOF system.

4.2 Linear Vibratory Systems

75

Fig. 4.2 Z-Axis gyroscope, the drive direction is along the x-Axis, and the sense direction is along the y-Axis.

An in-plane y-Axis gyroscope (Figure 4.3) proof mass is required to oscillate in one in-plane direction along the x-Axis and one out-of-plane direction along the z-Axis. The drive and sense axes can be interchanged depending on the actuation and detection schemes. An x-Axis gyroscope requires either the drive or the sense direction to be along the y-Axis. Thus, in many cases a y-Axis gyroscope design can be used as x-Axis gyroscope by rotating 90? in plane.

Fig. 4.3 Y-Axis gyroscope, the drive direction is along the x-Axis, and the sense direction is along the z-Axis.

4.2.1 Linear Suspension Systems
The ?exure system that suspends the proof-mass above the substrate usually consists of thin ?exible beams, formed in the same structural layer as the proof-mass. The thin beams have to be oriented to be compliant in both the drive and sense motion directions. Common suspension structures utilized in z-Axis micromachined gyroscopes include crab-leg suspensions (Figure 4.4), serpentine suspensions (Figure 4.5), hairpin suspensions (Figure 4.6), H-type suspensions (Figure 4.7), and U-beam suspensions (Figures 4.9 and 4.11).

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4 Mechanical Design of MEMS Gyroscopes

Fig. 4.4 Crab-leg suspensions.

Fig. 4.5 Serpentine suspensions.

These suspension systems are widely used for both z-Axis gyroscopes with inplane drive and sense modes, and x/y-Axis gyroscopes with one in-plane and one out-of-plane mode. Gyroscopes with an out-of-plane mode are usually fabricated with a thin structural layer to allow out-of-plane de?ections of the beams, while gyroscopes with in-plane drive and sense modes are preferably fabricated with a thicker structural layer to suppress out-of-plane modes. The crab-leg and H-type suspensions are known to provide better symmetry among the drive and sense-modes, allowing to easily locate the drive and sense

4.2 Linear Vibratory Systems

77

Fig. 4.6 Hairpin suspensions.

Fig. 4.7 H-type suspensions.

modes closer. In the crab-leg, serpentine and hairpin suspensions, drive motion results in de?ections also in the sense-mode beams, which often causes undesired energy transfer into the sense-mode. Thus, H-type suspension and especially U-beam suspensions with decoupling frames provide better mode-decoupling, which will be discussed further next.

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4 Mechanical Design of MEMS Gyroscopes

4.2.1.1 Frame Structures Suspension systems similar to crab-leg, serpentine or hairpin suspensions are compliant in two orthogonal directions. The same beams experience de?ections in both modes, resulting in undesired coupling between the drive and sense modes. Since the drive-mode amplitude is orders of magnitude larger than sense-mode, it is often required to isolate the drive motion from the sense motion. It is also desired to limit the de?ection direction of the drive and sense electrodes, so that drive electrodes de?ect only in drive direction, and sense electrodes de?ect only in sense direction. This enhances the precision and stability of the drive actuation and sense detection electrodes.

Fig. 4.8 Lumped model of the drive frame implementation with U-beam suspensions.

Fig. 4.9 Drive frame implementation with U-beam suspensions, minimizing the component of the actual drive motion along the sense detection axis.

4.2 Linear Vibratory Systems

79

To decouple the drive motion and sense motion, it is common to implement a frame structure that nests the proof-mass. Two basic approaches in frame implementation are using a drive frame as in Figure 4.9, or a sense frame as in Figure 4.11. In the drive frame implementation, the proof mass is nested inside a frame that is constrained to move only in the drive direction. This approach assures that the drive motion is very well aligned with the designed drive axis, and minimizes the component of the actual drive motion along the sense detection axis. It also provides improved side stability and minimal parasitic sense-direction forces in the drive actuators.

Fig. 4.10 Lumped model of the sense frame implementation with U-beam suspensions.

Fig. 4.11 Sense frame implementation with U-beam suspensions, minimizing the undesired capacitance change in the sense electrodes due to the drive motion.

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4 Mechanical Design of MEMS Gyroscopes

The sense frame implementation is based on nesting the proof mass inside a frame that is constrained to move only in the sense direction. The sense electrodes are attached to the frame, and relative motion in the sense electrodes along the drive direction is prevented. This approach minimizes the undesired capacitance change in the sense electrodes due to the drive motion. More sophisticated frame structures that provide the advantages of both drive and sense frame implementations are also possible. For example, the z-axis gyroscope in [94] by Bosch utilizes a drive frame around the proof-mass which oscillates only in the drive direction, and a sense frame inside the proof-mass which is ?xed in the drive direction and oscillates with the proof-mass in the sense direction (Figure 4.12). With this double frame structure, the drive oscillations are very well aligned with the drive axis, parasitic components of the drive forces in the sense direction are suppressed, and the motion of the sense-electrodes in the drive direction is eliminated.

Fig. 4.12 A double-frame implementation example based on [94], which combines the advantages of drive and sense frames.

4.2 Linear Vibratory Systems

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4.2.1.2 Anti-Phase Devices Gyroscopes are inherently sensitive to external inertial inputs such as ambient vibrations and shock. Many applications require gyroscopes to function under a certain degree of vibration environment. Given the small amplitudes of the Coriolis response, the response to external accelerations and vibrations could easily disrupt the rate measurements. Anti-phase systems, also known as tuning fork gyroscopes (TFG), aim to cancel common-mode inputs. In the tuning fork architecture, two identical masses are driven in opposite directions (anti-phase), which causes the Coriolis forces induced on the two masses to be in opposite directions also (Figure 4.13). When the sensemode response of the two masses are detected in a differential mode, their response to Coriolis forces are added, but their common-mode response in the same direction are canceled out. Thus, common-mode rejection is achieved while the rate signal is preserved.

Fig. 4.13 Anti-phase tuning fork gyroscopes (TFG), which provide common-mode rejection by utilizing two anti-phase vibrating masses in a differential mode.

To achieve an anti-phase oscillation in the drive-mode, the two masses are coupled with a coupling spring as in Figures 4.13 and 4.14. The coupling spring results in a 2-DOF drive dynamical system, with an in-phase and an anti-phase mode. The device is operated at the anti-phase drive frequency, which excites the masses in opposite directions as desired.

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4 Mechanical Design of MEMS Gyroscopes

Fig. 4.14 An example anti-phase tuning fork gyroscope system, which consists of two drive-frame gyroscopes coupled in the drive mode by a spring.

4.2 Linear Vibratory Systems

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4.2.2 Linear Flexure Elements
In linear micromachined gyroscopes, the suspension systems are usually designed to be compliant along the desired motion direction, and stiff in other directions. Most suspension systems utilize narrow beams as the primary ?exure elements, aligning the narrow dimension of the beam normal to the motion axis (Figure 4.15).

Fig. 4.15 The ?xed-guided end beam under translational de?ection.

In purely translational modes, the boundary conditions of the beams that connect the components of the gyroscopes are most commonly the ?xed-guided end con?guration (Figure 4.15), in which the moving end of the beam remains parallel to the ?xed end. Many complete gyroscope suspension systems can be modeled as a combination of ?xed-guided end beams. If we de?ne the length of a beam L as the x-axis dimension, width w as the y-axis dimension, and the thickness t as the z-axis dimension, the area moments of inertia of the beam in the y and z directions become Iy = Iz = 1 3 tw 12 (4.1)

1 3 wt (4.2) 12 For a single ?xed-guided beam (Figure 4.15), the translational stiffness for motion in the orthogonal direction to the beam axis is given by [106] ky,z = 1 3EIy,z 2 ( L )3 2 (4.3)

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4 Mechanical Design of MEMS Gyroscopes

where E is the Young’s Modulus. Thus, the stiffness values of the ?xed-guided beam along the three principle axes become kx = E ky = E kz = E wt L (4.4) (4.5)

tw3 L3

kz ky

wt 3 (4.6) L3 It should be noticed that the ratio of the stiffness values along the z and y axes is t = ( w )2 . Thus, large thickness is a key factor in suppressing the out-of-plane de-

L ?ections. The ratio of the axial stiffness to the y-axis stiffness is kx = ( w )2 . This ratio ky could be quite large depending on the beam design, providing excellent suppression of orthogonal motion. Even though theoretical expressions of the beam elements could be a practical guide in design, ?nite element analysis (FEA) simulations are essential in estimation of the ?exure characteristics. For example, let us analyze a ?xed-guided beam with the dimensions L = 500?m, w = 10?m, and t = 100?m. Assuming an elastic modulus of E = 130 GPa, FEA results indicate the y-axis reaction force for a 1?m purely y-axis de?ection to be 142?N, yielding ky = 142N/m. However, the reaction force for 10?m de?ection increases to 1980?N, resulting in ky = 198N/m. This illustrates the non-linearity of the beam for increased de?ections. FEA results on the ?xed-guided beam also reveals one of the major limitations of this suspension type. The x-axis reaction force for a 1?m y-axis de?ection is 410?N. For 10?m de?ection, the x-axis reaction force increases to over 40400?N. This extraordinarily large force is due to the fact that the beam starts to be axially loaded as the lateral force increases. Thus, single ?xed-guided beams should not be used for large de?ection ?exures such as the drive-direction suspensions.

Fig. 4.16 Folded beam (Ushaped) suspensions consist of two ?xed-guided beams in series, and eliminate the nonlinearity and axial-loading limitations of single ?xedguided beams.

4.2 Linear Vibratory Systems

85

Folded beam (U-shaped) suspensions eliminate the nonlinearity and axial-loading limitations of single ?xed-guided beams. By connecting two ?xed-guided beams in series in the folded beams (Figure 4.16), the two connection points of the suspension are on the same side, and lateral de?ections do not result in axial loading. Since a folded beam consists of two ?xed-guided beams of stiffness ky in series, the stiffness of a folded beam of length L, width w and thickness h becomes 1 1 1 = + k f olded ky ky k f olded = E (4.7)

tw3 (4.8) 2L3 One limitation of the folded beams is the reduced axial stiffness. The distance between the two beams results in a moment arm under an axial load, and causes bending as in Figure 4.17. This could become a disadvantage in designs that require substantial suppression of axial motion.

Fig. 4.17 The folded beam (U-shaped) suspensions under lateral and axial loading. The compliance of the folded beam under axial loading due to bending could be undesirable.

The axial compliance problem of the folded beams could be solved by providing symmetry in the axial direction. The double-folded suspension beam contains two folded beams symmetrically connected (Figure 4.18). Since it could be modeled as two folded beams in parallel, the stiffness of a double-folded beam is tw3 (4.9) L3 In summary, for small de?ections the folded beam provides half the stiffness of a ?xed-guided beam while the double-folded beam has the same stiffness as a ?xedguided beam. As the de?ections get larger compared to the beam dimensions, the ?xed-guided beam stiffness becomes non-linear, as seen in Figure 4.19. In complete suspension systems, a number of ?exure elements are connected to the proof-mass. The total stiffness in a certain direction could be approximated by the sum of all ?exure stiffness values in that direction. However, this approximation assumes that the compliance of the proof-mass, frame structures, and ?exures kdouble? f olded = 2k f olded = E

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4 Mechanical Design of MEMS Gyroscopes

Fig. 4.18 Double-folded beam suspension, which contains two symmetrically connected folded beams, provide excellent axial stiffness and linearity.

in other directions are negligible. In reality, these factors dramatically reduce the overall stiffness. Thus, modal analyses in FEA software is absolutely necessary for accurate estimation and design of resonant frequencies.

Fig. 4.19 Comparison of the y-axis reaction forces in ?xed-guided, folded, and double-folded beams.

4.3 Torsional Vibratory Systems

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4.3 Torsional Vibratory Systems
Torsional or rotation-based micromachined gyroscopes utilize rotational vibratory motion in their drive and sense modes. The operation principle is based on conservation of primarily angular momentum, instead of linear momentum as in translational vibratory gyroscopes. Torsional gyroscope structures consist of a rotational drive oscillator that generates and maintains a constant angular momentum, and a sense-mode angular accelerometer that measures the sinusoidal Coriolis moment. Thus, similar to linear vibratory gyroscopes, a 2-DOF oscillatory system is formed.

Fig. 4.20 Torsional Z-Axis gyroscope: The drive oscillation is about the x-Axis, and the sense oscillation is about the y-Axis.

In a z-Axis torsional gyroscope, the proof mass rotates about the two in-plane orthogonal directions: the drive direction rotation about the x-Axis and the sense direction rotation about the y-Axis as in Figure 4.20. The Coriolis moment is about the axis cross-product of the input angular rate and the drive angular velocity vectors.

Fig. 4.21 Torsional Y-Axis gyroscope: The drive oscillation is about the z-Axis, and the sense oscillation is about the x-Axis.

An in-plane y-Axis torsional can be implemented with either an in-plane drive (about the z-Axis) and out-of-plane sense (about the x-Axis) con?guration, or an out-of-plane drive (about the x-Axis) and in-plane sense (about the z-Axis) con?guration as in Figure 4.21.

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4 Mechanical Design of MEMS Gyroscopes

Fig. 4.22 Torsional gyroscope by Bosch, with a drivemode about the z-Axis [95]. SEM courtesy of Bosch.

4.3.1 Torsional Suspension Systems
Gimbals are commonly used in torsional gyroscope suspension systems to decouple the drive and sense modes, and to suppress undesired modes. Many suspension system and gimbal con?gurations are possible in torsional vibratory gyroscopes. Similar to linear gyroscope systems, the suspension system that supports the masses and gimbals usually consists of thin ?exible beams, formed in the same structural layer as the proof-mass.

Fig. 4.23 Torsional z-Axis gyroscope with drive gimbal structure. The drive-mode de?ection angle of the gimbal is θd , and the sense-mode de?ection angle of the sensing mass is φ .

An example gimbal system for a z-Axis torsional gyroscope based on [97] was shown in Figure 4.23. In the drive-mode, the outer drive gimbal is excited about the x-axis. In the presence of an angular rate input about z-axis, the sinusoidal Coriolis torque is induced about the y-axis, which causes the sense-mode response of the inner mass (Figure 4.24).

4.3 Torsional Vibratory Systems

89

Fig. 4.24 The drive and sense modes of a typical torsional z-axis gyroscope, similar to [97].

A representative gimbal implementation in y-Axis torsional gyroscopes based on [99] is presented in Figure 4.25. The system consists of an inner gimbal that can de?ect torsionally in-plane about the z-Axis, and an the outer mass attached to the inner gimbal. The drive-mode is in-plane about the z-Axis, and the sensemode is out-of-plane about the x-Axis. In the drive-mode, the inner gimbal and the outer mass oscillate together, and the angular rate input about the y-Axis generates a Coriolis torque about the x-Axis. The outer mass responds to the Coriolis torque by de?ecting torsionally about the x-Axis relative to the drive gimbal. The sensemode response is detected by the out-of-plane electrodes located underneath the outer mass structure.

Fig. 4.25 The gimbal system in a y-axis torsional gyroscope, based on [99].

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4 Mechanical Design of MEMS Gyroscopes

4.3.2 Torsional Flexure Elements
4.3.2.1 Out-of-Plane Torsional Hinges Out-of-plane de?ections, which are about the x-axis or y-axis, are most commonly achieved by torsional beams (Figure 4.26) aligned along the de?ection axis. In the purely torsional mode, the boundary conditions of the beam are such that the moving end of the beam remains parallel to the ?xed end, but rotates about the axis normal to the end plane.

Fig. 4.26 Typical torsional beams used for out-of-plane de?ections, about the x or y axes.

Assuming each torsional beam is straight with a uniform cross-section, and the structural material is homogeneous and isotropic; the torsional stiffness of each beam with a length of L can be modeled as SG L

K=

(4.10)

E where G = 2(1?ν) is the shear modulus with the elastic modulus E and Poisson’s ratio ν. For a beam with a rectangular cross-section of width w and thickness t, given that w ≤ t, the cross-sectional coef?cient S can be expressed as [106]

S = tw3

w t4 1 ? 0.21 1? 3 t 12t 4

(4.11)

4.3 Torsional Vibratory Systems

91

4.3.2.2 In-Plane Torsional Flexures In-plane torsional de?ections about the z-axis are usually achieved by a combination of ?xed-guided end beams (Figure 4.27). The beams are con?gured such that the center lines along their lengths intersect at the rotation center of the mass. The boundary conditions of the beams are different from linear gyroscope systems, in that the moving end of the beam does not remain parallel to the ?xed end, and de?ects with an angle equal to the total rotation angle.

Fig. 4.27 Fixed-guided ?exure beams commonly used for in-plane (about the z-Axis) torsional suspensions.

The suspension beams can be located inside or outside the mass. The direction of the moving end de?ection angle, and thus the boundary condition depends on the location of the beams. Let us consider a single ?xed-guided beam, and de?ne the length of a beam L as the y-axis dimension, width w as the y-axis dimension, and the thickness t as the z-axis dimension. The area moment of inertia of the beam in the y direction is Iy = 1 3 tw 12 (4.12)

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4 Mechanical Design of MEMS Gyroscopes

When the beam is located inside the mass as in Figure 4.28 and the moving end is at a distance R from the rotation center, the torsional spring constant of the single beam about the z-axis becomes Kzz = 4 EIy R 3 LN L
2

R ?3 +1 L

(4.13)

Fig. 4.28 Interior beam con?guration for in-plane torsional suspensions.

where N is the number of folds in the beam. When the beam is located outside the mass as in Figure 4.29 and the moving end is at a distance R from the rotation center, the torsional spring constant of a single beam is Kzz = 4 EIy R 3 LN L
2

R +3 +1 L

(4.14)

Fig. 4.29 Exterior beam con?guration for in-plane torsional suspensions.

Derivations of the torsional spring constants are presented in detail in [115]. The interior beam con?guration allows to achieve more compliant suspension systems with compact dimensions. However, large residual stresses in the structural layer could cause excessive curling. Exterior beam con?guration minimizes curling, while consuming more die area. Thus, interior beams could be more suitable for bulk

4.4 Anisoelasticity and Quadrature Error

93

micromachined low-stress devices, and exterior beams more suitable for surface micromachined devices.

4.4 Anisoelasticity and Quadrature Error
Most of the major challenges in vibratory gyroscopes arise because of the fact that the magnitude of the sense-mode response amplitude is extremely small. Let us illustrate common order of magnitudes on an example gyroscope system. In Chapter 2, the sense-mode response amplitude of a matched mode gyroscope system with ωs = ωd was derived as y0 matched = ?z 2Qs x0 mC ms ωs (4.15)

If we consider a mode-matched gyroscope system with the total sense-mode mass equal to the mass that generates the Coriolis force, i.e. ms = mC , the ratio of the sense amplitude to the drive amplitude becomes ?z y0 matched = 2Qs x0 ωs (4.16)

Typical operation frequencies of vibratory gyroscopes are around fs = 10 kHz, yielding ωs = 2π · 10, 000 rad/s. Assuming a sense-mode quality factor Qs = 1000, the sense to drive amplitude ratio for a 1? /s = π/180 rad/s angular rate input is 556 ppm. If the drive-mode amplitude is 10 ?m, the sense-mode amplitude is 5.56 nm. This example illustrates a best case scenario, since it is based on a high-Q and modematched system. If the drive and sense modes are separated with ? f = 100 Hz, the sense response amplitude drops to 0.16 nm. In reality, fabrication imperfections result in non-ideal geometries in the gyroscope structure, which in turn causes the drive oscillation to partially couple into the sense-mode. Even though several cross-coupling mechanisms such as elastic, viscous and electrostatic couplings exist, in most cases the elastic coupling in the suspension elements is the largest in magnitude. Considering the relative magnitudes of the drive and sense oscillations, even extremely small undesired coupling from the drive motion to the sense-mode could completely mask the Coriolis response. To investigate the dynamical effects of cross-coupling, let us start with the ideal system dynamics. The simpli?ed dynamics of an ideal z-axis gyroscope system with the drive-mode along the x-axis and the sense-mode along the y-axis in vector form can be expressed as md 0 0 ms x ¨ c 0 + x y ¨ 0 cy x ˙ k 0 + x y ˙ 0 ky x Fd = y ?2mC ?z x ˙

(4.17)

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4 Mechanical Design of MEMS Gyroscopes

where mC is the portion of the driven proof mass that contributes to the Coriolis force, md is the total drive-mode mass, and ms is the total sense-mode mass. The total stiffness matrix is equal to the sum of the stiffness matrices of each suspension element in the system. Almost all suspension elements in real implementations of vibratory gyroscopes have elastic cross-coupling between their principal axes of elasticity. This phenomenon is called anisoelasticity, and is the primary cause of mechanical quadrature error in gyroscopes. The anisoelastic forces that result in elastic coupling between the x and y axes are modeled through the off-diagonal springs constants kxy and kyx in the system stiffness matrix. When these anisoelastic elements are included in the dynamics, the equations of motion become md 0 0 ms c 0 x ¨ + x y ¨ 0 cy k k x ˙ + x xy y ˙ kyx ky Fd x = y ?2mC ?z x ˙

(4.18)

Since the oscillation amplitudes in the sense-mode are orders of magnitude smaller than the drive-mode, the coupling spring kxy in the drive dynamics is negligible. The impact of anisoelasticity is primarily on the sense-mode dynamics due to kyx , which couples the drive-mode displacement into the sense-mode oscillator. The simpli?ed sense-mode equation of motion with anisoelasticity can be expressed as ¨ ˙ ˙ ms y + cy y + ky y = ?2m?z x ? kyx x

(4.19)

The total cross-coupling stiffness kyx in the suspension system is equal to the sum of the kyx values of each suspension beam. In an ideal gyroscope system with identical springs located symmetrically, even if the cross-axis stiffness values of each individual spring are not zero, the off-diagonal cross-axis stiffness values exactly cancel out when added. Thus, the total kxy and kyx become zero, and the total stiffness matrix becomes diagonal. With a diagonal stiffness matrix, system eigenmodes align perfectly with the drive and sense axes. However, fabrication imperfections and variations are inevitable, and exist to a certain degree in every actual gyroscope structure. Non-uniform variations within the die result in slight differences among the suspension elements. Therefore, the off-diagonal terms do not exactly cancel out in real suspension systems, and yield residual off-diagonal terms in the total stiffness matrix. For z-axis gyroscopes, slight variation in the average widths of the suspension beams is the basic cause of suspension asymmetries which ultimately leads to anisoelasticity. The nominal cross-axis coupling values for folded beams (assumL1 ) and crab-leg suspensions are derived in [98] as ing L1 ? L2 and LC

4.4 Anisoelasticity and Quadrature Error

95

Fig. 4.30 Models of folded beams and crab-leg suspensions for calculation of cross-axis coupling stiffness values.

Folded beam: kyx =

3EIC (L1 ? L2 ) 3 LC L1 9EI1 I2 Crab-leg: kyx = L1 L2 (I1 L2 + I2 L1 )

(4.20) (4.21)

where IC and LC are the moment of inertia and the length of the connecting beam in the folded beam, and I1,2 and L1,2 are the moment of inertia and the length of the ?rst and second springs in the folded beam and the crab-leg suspension as shown in Figure 4.30. Cross-axis stiffness of the folded beam nominally becomes zero when the length of the two beams are equal. Even though in reality geometrical mismatches between the two beams of a folded suspension could result in a net cross-axis stiffness, this value is relatively small compared to crab-leg or similar suspensions, especially when the length of the connecting beam is kept to a minimum. The resulting net cross-axis stiffness of each suspension element is canceled out further by symmetrically placing suspension beams, but will not be nulled completely due to slight variations among the suspension elements. In gyroscope systems with out-of-plane operational modes, cross-axis stiffness between the in-plane and out-of-plane directions, i.e. kzx or kzy , becomes critical. The primary factor that results in this type of anisoelasticity is the deviation of the cross-section of the beams from a perfect rectangle. Sidewall tilt in DRIE is known to result in a non-rectangular cross-section, causing the principle axes of elasticity of the suspension beams to deviate from perfectly parallel and orthogonal to the device surface.

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4 Mechanical Design of MEMS Gyroscopes

Fig. 4.31 Sidewall angle in suspension beams, resulting in the cross-axis stiffness kzx or kzy between the in-plane and out-of-plane directions.

If we consider a y-axis gyroscope system with drive direction along the x-axis and sense direction along the z-axis as an example, the cross-axis coupling stiffness term kzx excites the sense-mode. For a single suspension beam with a parallelogram cross section, the cross-axis coupling stiffness due to the sidewall angle θ is [101] t2 θ w2

kzx = kx

(4.22)

Similarly, the kzx values of symmetrically designed beams ideally cancel out in the total stiffness matrix, while non-uniform variations among the beams result in a net cross-axis coupling from the in-plane drive motion to the out-of-plane sensemode. Having explained the fundamental causes of cross-axis stiffness in suspension beams, let us investigate the impact of these off-diagonal stiffness terms on the system dynamics. The two forces that excite the sense-mode oscillator in a z-axis gyroscope system were previously shown to be the Coriolis force FC and the quadrature force FQ FC = ?2mC ?z x ˙ FQ = ?kyx x (4.23) (4.24)

The rate-equivalent quadrature error in the sense-mode response can be found by taking the ratio of the quadrature force FQ to the Coriolis force per unit angular rate (FC /?z ) FQ
FC ?z

?Q =

=

kyx |x0 sin ωd t| 2mC |ωd x0 cos ωd t|

(4.25)

Assuming that the mass that generates the Coriolis force is equal to the drivemode mass, i.e. mC md , the rate equivalent quadrature becomes

4.4 Anisoelasticity and Quadrature Error

97

?Q =

kyx ωd kx 2

(4.26)

This expression illustrates how very small imbalances in the suspension structure can lead to high quadrature error values. For example, if the cross coupling stiffness kyx is only 1% of the drive stiffness in a gyroscope system with ωd = 10kHz, the resulting quadrature error is 18000? /sec. Simply due to the fact that the Coriolis force FC is proportional to the drive velocity x and the quadrature force FQ is proportional to the drive position x, there ˙ is always a 90? phase difference between the Coriolis response and the mechanical quadrature. Since both FC and FQ are excitation forces applied on the sense-mode oscillator, both the relative amplitude and phase of the quadrature and Coriolis response are independent of the sense-mode dynamics (Figure 4.32). This means that the rate-equivalent quadrature remains constant for varying mode mismatch ? f and sense quality factor Qs , even though the actual quadrature signal magnitude varies.

Fig. 4.32 Block diagram of the mechanical quadrature model, showing that both the Coriolis force FC and the quadrature force FQ are simultaneously applied on the sense-mode oscillator. Both the amplitude and phase of the quadrature signal relative to the Coriolis response are independent of the sense-mode dynamics.

Having explained the underlying reason behind the constant 90? phase difference between the Coriolis response and quadrature, let us now investigate the phase relations between the drive motion and quadrature error for the two main system types: mode-matched system with ? f = 0 and mode-mismatched system with ? f > 0. It is assumed that in the steady state the drive-mode displacement is of the form x(t) = x0 sin(ωd t + φd )

(4.27)

where φd is the drive-mode position phase relative to the drive AC signal. Since the drive oscillator is usually operated at resonance, in most systems φd = ?90? .

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4 Mechanical Design of MEMS Gyroscopes

For a system with matched drive and sense modes, i.e. ? f = 0, the phases of the Coriolis response and quadrature relative to the drive signal are φCoriolis = φd ? 180? φQuadrature = φd ? 270
?

(4.28) (4.29)

Fig. 4.33 The drive position, drive velocity, Coriolis response and quadrature phase relations for a mode-matched system, when ? f = 0 Hz.

4.4 Anisoelasticity and Quadrature Error

99

For a mismatched system with the sense-mode suf?ciently higher than the drivemode, i.e. ? f > 0, the Coriolis response phase converges to the Coriolis force phase, which is ?90? from drive position, and the quadrature response lags the Coriolis response by 90? φCoriolis = φd ? 90? (4.30) ? (4.31) φQuadrature = φd ? 180

Fig. 4.34 The drive position, drive velocity, Coriolis response and quadrature phase relations for a mode-mismatched system, when ? f = 100 Hz.

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4 Mechanical Design of MEMS Gyroscopes

In a mode-matched system, the proof-mass oscillation trajectory due to mechanical quadrature becomes an ellipse, since the sense-mode quadrature response has a 90? phase difference with the drive position (Figure 4.35).

Fig. 4.35 Coriolis response and quadrature trajectories for a mode-matched system.

When the drive and sense frequencies are mismatched, the oscillation trajectory is already an ellipse without anisoelasticity. For a system with ? f > 0, the quadrature response has a 180? phase difference with the drive position, and the oscillation trajectory due to quadrature becomes a straight line as in Figure 4.36.

4.4.1 Quadrature Compensation
Since there is always a 90? phase difference between the Coriolis response and the mechanical quadrature, the quadrature signal can be separated from the Coriolis signal during amplitude demodulation at the drive frequency. However, the substantially large relative magnitude of the quadrature signal has many implications on the detection electronics: ? First and foremost, the dynamic range of front-end electronics has to be designed to accommodate the large levels of quadrature signal, which could be quite larger than the dynamic range required for the full range of the gyroscope. This could result in lower resolution and signal to noise ratio for the Coriolis signal. ? To be able to discriminate large levels of quadrature signal, the phase accuracy of synchronous demodulation also has to be high. Lower phase accuracy results in more quadrature signal to mix into the rate signal, which could result in large rate bias for high quadrature levels.

4.4 Anisoelasticity and Quadrature Error

101

Fig. 4.36 Coriolis response and quadrature trajectories for a mode-mismatched system. The straight line of oscillation due to quadrature turns into a narrow ellipse in the presence of an angular rate input.

? The stability of quadrature over temperature and over time is extremely important when the quadrature level is high. If the part of the quadrature signal that mixes into the rate signal varies over temperature and time, then the temperature stability and long term stability of the gyroscope deteriorate. Even though the quadrature signal can be electrically nulled, it is desirable to cancel the actual mechanical quadrature motion at the sensing element level. Several approaches could be implemented to compensate for mechanical quadrature: ? Mechanically balancing the sensing element by altering the mechanical system. Post-fabrication trimming by laser ablation can be used to remove mass from proper areas of the proof-mass until mechanical imbalance is eliminated [100]. Although this process can be automated, it is not compatible with wafer-level packaged devices. ? Electrostatically canceling quadrature force via DC bias. This approach requires quadrature compensation electrodes that vary overlap area as a result of the drive motion. The electrostatic force is modulated by the drive motion at the drive frequency, proportional to the drive amplitude. By adjusting the DC bias level, a quadrature cancellation force Fcomp = kyx x0 sin ωd t equal and opposite to the quadrature force is achieved. ? Electrostatically canceling quadrature force by directly applying an AC compensation signal. In this approach, an AC signal in-phase with the drive position is extracted from the drive oscillator, and applied together with a DC bias on an electrode that exerts force in the sense direction. The amplitude of the AC signal and the DC bias are controlled to achieve a quadrature cancellation force Fcomp = kyx x0 sin ωd t that exactly nulls the quadrature force. Since the compen-

102

4 Mechanical Design of MEMS Gyroscopes

sation force does not depend on the drive motion, precise phase control and an AGC loop that regulates the drive amplitude are required. The quadrature error can reach thousands of ? /s signals in actual gyroscope systems. Usually a combination of different measures have to be taken to minimize the yield loss due to excessive quadrature. Tight process control, robust gyroscope design, and electrical/electrostatic quadrature cancellation capabilities have to work together to achieve acceptable quadrature levels over the entire wafer area.

4.5 Damping
Damping is the energy dissipation effect in an oscillatory system. In micromachined vibratory systems such as gyroscopes, many dissipation mechanisms contribute to the total damping. The following is an overview of prominent damping phenomena such as viscous damping and structural damping.

4.5.1 Viscous Damping
In the presence of a gas surrounding the vibratory structure of a gyroscope, the primary damping mechanism in the gyroscope dyanmical system is the viscous effects of the gas con?ned between the proof mass surfaces and the stationary surfaces. The damping of the structural material is usually orders of magnitude lower than the viscous damping except under high-vacuum conditions. Viscous damping in the gyroscope dynamical system is dominated by the internal friction of the gas between the proof-mass and the substrate, and between the combdrive and sense capacitor ?ngers. These viscous damping effects can be captured by using two general damping models: slide ?lm damping and squeeze ?lm damping.

4.5.1.1 Slide Film Damping Slide ?lm damping, or lateral damping, occurs when two plates of an area A, separated by a distance y0 , slide parallel to each other (Figure 4.37). At low pressures and when the mean free path of the gas is comparable to the gap, gas rarefaction effects can be modeled by the effective viscosity of the gas ?eff . Assuming a Newtonian gas, the lateral damping coef?cient can be expressed as A (4.32) d where A is the overlap area of the plates, and d is the plate separation. The effective viscosity ?eff is approximated in [109] as cslide = ?eff

4.5 Damping

103

?eff =

? 1 + 2Kn + 0.2Kn 0.788 e?Kn /10

(4.33)

The Knudsen number Kn is the measure of the gas rarefaction effect, which is a function of the gas mean free path λ and the gap d: λ (4.34) d Pressure dependence of the gas viscosity is captured in the mean free path, which can be calculated for air as Kn = Pλ = 5.1 × 10?5 Torr.m (4.35)

Fig. 4.37 Slide-?lm damping between two plates, which occurs when the plates slide parallel to each other.

4.5.1.2 Squeeze Film Damping Squeeze ?lm damping occurs when two parallel plates move toward each other and squeeze the ?uid ?lm in between (Figure 4.38). Squeeze ?lm damping effects are quite complicated, and can exhibit both damping and stiffness effects depending on the compressibility of the ?uid. The effective viscosity ?eff to model gas rarefaction effects in squeeze ?lm damping is given in [110] as ?eff = ? 1 + 9.638Kn 1.159 (4.36)

Solving the linearized Reynolds equation yields one force in-phase with displacement, and one force out-of-phase. The in-phase force due to squeeze-?lm effect is the spring force, and the out-of-phase force is the damping force. The squeeze-?lm damping force Fc and spring force Fk are reported in [111] as

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4 Mechanical Design of MEMS Gyroscopes

64σ Pa A Fc m2 + c2 n2 = ∑ (mn)2 [(m2 + c2 n2 )2 + σ 2 /π 4 ] 6d z π m,n odd 64σ 2 Pa A 1 Fk = ∑ (mn)2 [(m2 + c2 n2 )2 + σ 2 /π 4 ] 8d z π m,n odd

(4.37) (4.38)

where z is the plate de?ection, Pa is the ambient pressure, m and n are odd integers, c = w/l and A = wl for a plate with width w and length l. The squeeze number σ as a function of frequency ω is 12?eff w2 ω Pa d 2

σ=

(4.39)

Fig. 4.38 Squeeze-?lm damping between two plates, which occurs when the plates move towards each other.

Detailed discussions on slide-?lm and squeeze-?lm damping in micromacined structures along with more advanced phenomena such as non-linear effects, kinetic gas models, plate motions that propagate into the ?uid with rapidly diminishing amplitude, and computational ?uid dynamics simulations are presented in [108– 110, 112, 113].

4.5.2 Viscous Anisodamping
Mechanical cross-coupling sources between the two oscillation axes of vibratory gyroscopes is not limited to the elastic coupling due to anisoelasticity. Depending on the design, viscous coupling due to hydrodynamic forces could be a major error mechanism. Hydrodynamic lift, also known as the surfboard effect, is the primary source of anisodamping, which couples the drive motion into the sense-mode [101]. This phenomenon occurs when a plate slides over a viscous medium and the hydrodynamic lift due to the plate velocity generates a force orthogonal to the motion direction.

4.5 Damping

105

The anisodamping forces appear as the off-diagonal damping terms cxy and cyx in the system damping matrix: md 0 0 ms x ¨ c c + x xy y ¨ cyx cy x ˙ k k + x xy y ˙ kyx ky x Fd = y ?2mC ?z x ˙

(4.40)

Similar to anisoelasticity, the coupling in the drive dynamics is negligible, and the impact is primarily on the sense-mode dynamics. The simpli?ed sense-mode equation of motion including the anisodamping effect is ¨ ˙ ˙ ˙ ms y + cy y + ky y = ?2m?z x ? cyx x

(4.41)

It should be noticed that, unlike anisoelasticity, anisodamping is proportional to the drive velocity, and causes a coupling exactly in phase with the Coriolis response. Thus, it is indistinguishable from the rate response of the gyroscope. Even though the bias due to anisodamping could be canceled from the output as on offset, viscous forces are highly temperature dependent, and could contribute greatly to the temperature bias of the gyroscope. Vacuum packaging becomes crucial in minimizing this effect. The design of the gyroscope is also critical. Avoiding the use of parallel plate sense electrodes that move with the drive motion, or providing symmetry about the drive axis alleviate the effects of anisodamping.

4.5.3 Intrinsic Structural Damping
Although viscous damping is the dominating damping mechanism in the presence of a gas in the gyroscope ambient, the total damping in the gyroscope system is a combination of multiple effects. The damping components other than viscous damping start limiting the quality factor as the pressure inside the gyroscope cavity approaches high vacuum. Under vacuum conditions, thermoelastic damping is one of the primary damping mechanisms. Thermoelastic damping is the intrinsic material damping that occurs as a result of thermal energy dissipation due to elastic deformation. In a vibrating beam, alternating tensile and compressive strains across the width cause irreversible heat ?ow, which in turn results in an effective damping due to dissipation of vibrational energy [103]. Thermoelastic damping has been reported to limit the Q factor of vacuum packaged gyroscopes to values from 100,000 to 200,000. Many other factors from anchor losses to die attach methods contribute to the total damping in vibratory gyroscopes. The total quality factor can be expressed as a combination of these effects as [102] 1 1 1 1 1 1 = + + + + Qtotal Qviscous QT ED Qanchor Qelectronics Qother

(4.42)

106

4 Mechanical Design of MEMS Gyroscopes

where QT ED is due to thermoelastic damping, Qelectronics is due to electronics damping which is in the order of 1011 , Qother captures remaining damping effects estimated around 250,000 [102]. Qanchor is due to the anchor losses which could be as low as 10,000 depending on the anchor type and material. Anti-phase devices that locally cancel vibration injection into the anchors provide much higher Qanchor values.

Fig. 4.39 Typical quality factor versus pressure curve, where the structural damping becomes the Q limit as viscous damping diminishes.

Usually viscous damping and other intrinsic damping components are very dif?cult to estimate theoretically for complicated gyroscope systems. It is common to empirically measure the overall Q factor of drive and sense modes by a frequency response or ring-down test. Q factor versus pressure curves as in Figure 4.39 are usually obtained to guide the packaging pressure requirements of gyroscopes. At suf?ciently low pressures, the quality factors usually start converging to a limit value set by the total structural damping, which depends on the speci?cs of the device geometry and anchor structure. To minimize effects of pressure variations, it is desirable to operate the device in the ?at region of the curve.

4.6 Material Properties of Silicon

107

4.6 Material Properties of Silicon
Single crystal silicon is one of the most common materials used in inertial sensors. Since the cubic nature of the single crystal silicon lattice results in orthotropic material properties, it requires special attention in design and modeling. The following table summarizes the important material properties of single crystal silicon and polysilicon for reference.
Table 4.1 Material properties of Silicon Material Property Young’s modulus Poisson ratio Density Single Crystal Si [100] [111] 131 GPa 190 GPa 0.28 0.26 2330 kg/m3 Polysilicon 161 GPa 0.23

When suspension beams with arbitrary angles are designed in a gyroscope system, anisotropy of Young’s modulus has to be taken into account. For example, in a (100) silicon wafer, elastic modulus is 131 GPa parallel to the ?at and 169 GPa at a 45? angle to the ?at. Variation of Young’s modulus in the <100> plane for single crystal silicon is presented in Figure 4.40 [96].

Fig. 4.40 Young’s modulus vs. direction in the <100> plane for Silicon and Germanium. Reprinted with permission from [96]. Copyright 1965, American Institute of Physics.

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4 Mechanical Design of MEMS Gyroscopes

It should be noted that silicon is an excellent structural material since it exhibits no plastic deformation or creep below 500 ? C. Since it is impervious to fatigue, it can withstand millions of cycles without failure, which is essential for vibratory gyroscopes.

4.7 Design for Robustness
The micromachined gyroscope sensing element is a highly complex dynamic mechanical system. In Chapter 2 we illustrated that the dynamic response of the gyroscope is very sensitive to variations in system parameters that shift the drive or sense resonant frequencies. Both structural variations and environmental effects are known to result in quite large variations in the resonant frequencies. Even though mode-mismatching is a common practice to reduce sensitivity to variations by operating away from the sense resonant frequency, many other factors and failure modes have to be considered to achieve robustness. Some of the most important design aspects are summarized below.

4.7.1 Yield
Numerous types of fabrication process variations have direct effect on the device performance. Variations in structural thickness, critical dimensions such as beam widths and capacitive gaps, sidewall angles and cross-section pro?les result in differences in mechanical and electrical characteristics from die to die on a wafer, from wafer to wafer, and from lot to lot. The sensing element design must meet speci?cations such as scale factor and quadrature error with a given process window to maximize yield. Many critical features can often be determined with a top-down approach. For example, the scalefactor compensation capability of the electronics determines the acceptable range of capacitive gap variations and relative variations of the drive and sense resonant frequencies. Then in turn, acceptable relative tolerance on the suspension beam widths is estimated. With a given the critical dimension control capability of the fabrication process, the minimum beam widths allowed in the gyroscope design are determined. Especially in low-cost and high-volume applications, design for yield is of utmost importance. Corner analysis and Monte Carlo methods are widely used to estimate the impact of the combination of the variation in many fabrication parameters. In corner analysis, every combination of maximum and minimum values of each variable is simulated. In Monte Carlo simulations, each variable is randomly sampled and combined based on their statistical distribution, and the resulting statistical distribution of the system parameters is obtained.

4.7 Design for Robustness

109

4.7.2 Vibration Immunity
Gyroscopes are typically complicated vibratory systems with many high-Q resonant modes. Severe vibration environments in many applications could easily excite an undesired mode that could lead to a large bias error, saturation and even catastrophic failure. In the gyroscope design cycle, it is crucial to de?ne the vibration spectrum in the target application. Thorough FEA simulations and design optimization have to be performed to locate the operational modes and parasitic modes of the system away from the high-energy region of the vibration spectrum. FEA simulations also allow to estimate the de?ections in the gyroscope structure under acceleration. Simulation of de?ections under acceleration reveal many important design aspects, such as maximum de?ection points to guide the shock-stop design, the level of acceleration that cause contact, and capacitance changes due to acceleration. Common-mode rejection methods, such as anti-phase devices previously presented in this chapter, help alleviate the adverse effects of acceleration and vibration.

4.7.3 Shock Resistance
In micromachined gyroscopes, the most common failure mechanism due to shock is fracture. Crystalline silicon is a purely brittle material, which means that it deforms elastically until the maximum stress level reaches the yield strength. Even though the yield strength of Silicon is 7 GPa [107], a very wide range of yield strength values have been measured in the literature. To minimize the probability of fracture at a given shock level, a practical design criteria is to keep the maximum von-Mises stress lower than 1 GPa. Almost all FEA software packages have the capability to model an acceleration on the structure and calculate the von-Mises stress distribution. In gyroscopes, the highest stress concentration points are usually suspension beam connections. In drive suspension beams, the superposition of the stresses due to the drive motion together with the shock stress has to be taken into consideration.

4.7.4 Temperature Effects
Stability of the gyroscope scale factor and bias over temperature are among the most challenging performance speci?cations. Temperature affects the electro-mechanical system parameters in many ways. Due to thermal expansion, the device geometry and the characteristics of the actuation and detection electrodes are altered. Thermally induced stresses due to expansion also affect the suspension stiffness values. Stress relief mechanisms and central anchors help minimize the thermal stresses.

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4 Mechanical Design of MEMS Gyroscopes

Avoiding bi-morph effects by the use of materials with matched thermal expansion coef?cients is crucial in process design. FEA simulations are essential in identifying and controlling thermal effects in a particular design. Temperature also directly affects the Young’s modulus of the structural material. It should also be noted that the viscous damping is also temperature dependent, which could have a drastic in?uence on scale factor and bias in devices with closely matched modes.

4.8 Summary
In this chapter, we presented fundamental mechanical design aspects of linear and torsional micromachined vibratory gyroscopes. Basic gyroscope dynamical system structures and their corresponding ?exure systems were covered. The root cause and system-level implications of anisoelasticity and quadrature error were explained. Damping in vibratory systems, material properties of silicon, and critical design aspects for a robust sensing element were discussed.


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