Frequency Tuning of Work Modes in Z-Axis Dual-Mass Silicon

Frequency tuning of work modes in the silicon vibratory gyroscope is studied by the theoretical, numerical, and experimental methods in this paper. First, the schematic structure and simplified kinematics model of the gyroscope were presented for deducing the natural frequencies. Then, the width and length of support beams were optimized to tune work frequencies at their designed value. Besides, the frequency difference was experimentally tested andmanually tuned by varying the voltage applied on the tuning capacitors. The test on a prototype showed that the difference could be localized between −55.8Hz and 160.2Hz when the tuning voltage limit is 20V. Finally, a frequency control loop was developed to automatically tune the sense frequency toward the drive frequency. Both the theoretical analysis and numeric simulation show that the difference is stabilized at 0.8Hz when no Coriolis force or quadrature coupling force is applied. It is proved that the frequency difference is successfully tuned bymodifying the size of support beams before fabrication as well as the voltage applied on the tuning capacitors after fabrication.The automatic tuning loop, used to match the work modes, is beneficial to enhance the performance of the gyroscope as well as its resistance to environment disturbances.


Introduction
Silicon microgyroscopes have a lot of merits, such as low power consumption, small volume and weight, high shock resistivity, and suitability for batch fabrication, but their performance is lower than conventional mechanical, rotating wheel, fiber-optical, and ring laser gyroscopes [1,2].In order to broaden silicon microgyroscopes' application, a lot of work has been done on improving their performance, such as modification and optimization of the gyroscope structure, the fabrication process, the quality of vacuum packaging, and the detect circuit as well as the control system.Among these methods, the research on improvement of the structure and the circuit and system is the most economical.
The dual-mass gyroscopes are now popular for their effective rejection of common-mode disturbance, so many institutions [3][4][5][6][7] have done much research on them.It is known that the natural frequencies of work modes are key factors for performance improvement in silicon gyroscopes such as the bandwidth and the sensitivity [1,8].However, due to the fabrication imperfection, it is difficult to control the work frequencies at their designed value, causing the performance to be unpredictable.Therefore, some methods for postfabrication modification of the work frequencies were studied, such as the selective deposition of polysilicon [9], the localized thermal stressing effect [10], and the negative electrostatic stiffness effect [7,8,[11][12][13].From the three methods, the electrostatic tuning method is popular for low cost, little damage to the gyroscope structure, and easy adaptability in different gyroscopes.
The electrostatic tuning method can be realized by controlling the tuning voltage manually or automatically.The automatic control can be realized based on the frequency response characteristic of the mass-spring resonator.When the resonator vibrates at the natural frequency, the vibration amplitude achieves its maximum value and the phase delay is 90 degree.Thus, [12] designed the automatic control loop using the phase characteristic but the controlled frequency difference is influenced by the quadrature coupling.The method by seeking the maximum amplitude is studied by [13], but when the input angular rate is not constant, the controlled frequency is not stable.Theoretically and numerically, the frequency control algorithm presented in this paper is less influenced by the quadrature coupling and the varied input angular rate, especially when they are small.
In this paper, the methods for frequency localization of the work modes are detailed from two aspects: structure optimization and the postfabrication control.In Section 2, based on the structure of a dual-mass silicon microgyroscope, the simplified kinematics model is firstly built to deduce the natural frequencies for the in-phase and antiphase vibrations.Then, in Section 3, a finite element model of the gyroscope is created using Ansys to analyze the relationship between the work frequencies and the size of support beams.Besides, the capacitors of parallel plates are designed in sense frames for frequency tuning by means of the negative electrostatic stiffness effect.The frequency sweeping experiments on a fabricated gyroscope prototype prove that frequency tuning of the work modes is achieved by adjusting the size of support beams as well as by manually controlling the voltage applied on the tuning capacitors.Finally, in Section 4, the automatic frequency tuning of work modes is realized by a closed frequency-control loop.

Structure
As shown in Figure 1(a), the gyroscope is comprised of two symmetric single-mass gyroscopes coupled with some links, which are coupling U-beams here [6].Each single-mass gyroscope contains a drive frame, a proof mass, and a sense frame, serving as the energy source, the energy transferring unit, and the output unit, respectively.The specific arrangement of multiple frames makes the sense frames move only in the sense direction (-axis), maximally reducing the disturbance from drive frames, such as the quadrature force [14,15].
When subject to a constant out-of-plane input angular rate Ω in , the stable vibration amplitude of sense frames is approximated as shown in where   is the vibration amplitude of drive frames,   is the quality factor of sense mode, and  / is the natural frequency of drive/sense mode.It is easily found that the mechanical sensitivity  =   /Ω in is larger when the frequency difference between   and   is decreased.However, the mechanical working bandwidth BW = 0.54|  −   | decreases.Thus, the size of support beams is optimized in the structure design course and the frequency tuning electrodes are designed in the sense frames, so as to obtain an optimal frequency difference between work modes as well as large sense amplitude.
In Figure 1(b), the simplified kinematics model [16], neglecting the damping effect, is developed for analyzing the motion of the dual-mass gyroscope in the drive or sense direction. 1 and  2 are the effective system stiffness between each vibrating mass and the substrate,   is the coupling stiffness between the two vibrating masses, and  is the effective vibrating mass.The kinematics analysis for the dualmass system is displayed in [17] from which the natural frequencies are derived as shown in the following: The frequency in (3) corresponds to the in-phase vibration, namely, the two masses translating in the same direction, which is considered as a disturbing mode for the dual-mass gyroscope function.Similarly, the frequency in ( 4) is for the antiphase vibration.In the gyroscope, the work modes are the opposite vibrations in the drive and sense direction, so the work frequencies can be obtained from (4).

Frequency Localization
3.1.Tuning Effect of Support Beams.In the design of microsensors, optimization of geometry dimensions is essential for improving structure's performance [18][19][20].Here, the width and length of supporting beams are optimized for the designed work frequency.It is seen that in Figure 1(a) all the links among the frames, anchors, and proof mass are made up of U-beams.A U-beam can be considered as the serial combination of two cantilevers, so its stiffness is simplified [21] as where  is the elastic modulus of silicon and  u , ℎ u , and  u are the width, the depth, and the length of the U-beam, respectively.The dual-mass gyroscope is symmetric so (3) and ( 4) can be simplified through  1 =  2 .Then, it is easily noticed that  ip and  op both increase with  1 and  2 .Besides, their difference increases when   grows larger.Therefore, the work frequency can be increased through increasing the width or decreasing the length of support beams, which is verified by the following finite element analysis.
In Figure 2, the equivalent finite element model of the gyroscope structure is developed using Ansys to analyze the influence of support beams' size on work modes' frequencies.The silicon material properties and the mesh element are listed in Table 1.During simulation, the length and width of the drive U-beams and the sense U-beams are changed separately.The simulated work frequencies, namely, the  op in drive and sense directions, are fitted as shown in Figure 3.In Figures 3(a) and 3(b), it is easily noticed that the frequency of sense mode varies little when the size of drive U-beams changes.Similarly, in Figures 3(c) and 3(d), the frequency of drive mode changes little when the size of the sense U-beams changes.Thus, the separate tuning effect of support beams is realized, which eases the frequency localization of the two work modes.Besides, increasing the width of the sense Ubeams or decreasing their length raises the natural frequency of sense mode.The same tuning trend occurs to the drive mode frequency when the drive U-beams change in the same way.The tuning effect, obtained by the finite element analysis, matches the theoretical expectation from ( 4) and (5).

Negative Electrostatic Stiffness
Effect. Figure 4 is a part of the tuning combs.When the fixed plate is loaded with the voltage V and the moving plate translates only in the  axis, the generated negative electrostatic stiffness along the sense direction is derived [22]: where  is the number of the tuning combs attached to one sense frame, ℎ 1 ,  1 , and  1 are the depth, length, and distance of the comb capacitor, and  is the absolute dielectric constant of the gap.According to (4) and Figure 3, by changing the effective system stiffness between the vibrating mass and the substrate, that is,  1 and  2 , the work frequencies can be tuned towards their expected value.Therefore, the negative electrostatic stiffness effect of the squeeze capacitors in sense frames is used to change  1 and  2 and modify the natural frequency of sense mode.When the tuning capacitors are loaded with V, the frequencies of the in-phase and antiphase translation in the sense direction are derived as seen in where   ,  1 , and   are the effective system mass and stiffness in the sense direction, as explained in Section 2. It is easily seen that natural frequencies of the in-phase and antiphase vibration both decrease when the tuning voltage  increases.

Experiment Measurement.
The dual-mass gyroscope design, with the frequency difference between work modes around 100 Hz, is fabricated using the SOG process [23] and vacuum packaged in ceramic [24].The frequency localization effect of the support beams and the tuning capacitors are verified by the frequency sweeping tests.As seen in Figure 5, the experimental setup is made up of the power supplies, the detect circuit board, an oscilloscope, and the packaged gyroscope.To test the frequency response of the fabricated gyroscope, 2 V AC biased with 5 V DC is applied on the drive/sense electrodes and the detect circuit is connected with one anchor to sense the vibration of the gyroscope.The  oscilloscope is used to display the vibration signal from the circuit.Figure 6 gives the frequency response curves of work modes, obtained by fitting the vibration amplitude (in Volt) displayed on the oscilloscope at every frequency step within the range of interest.It is known that the vibration amplitude changes along with the actuation frequency and the largest amplitude is achieved when the drive frequency is equal to the natural frequency.Thus, the work frequencies of drive mode and sense mode, 4007.9Hz and 4168.1 Hz, are obtained from the two curves.And the frequency difference after fabrication is 160.2Hz, much larger than the designed 100 Hz.As the drive mode and sense mode resonators can be considered as band-pass filters, there exists a relationship between the 3 dB bandwidth and the quality factor, BW 3 dB =  0 /, where  0 and  are the natural frequency and quality factor of each mode.Thus, the quality factor of drive mode and sense mode, around 5000 and 950, is obtained based on Figure 6.
The frequency sweeping methods are also used to obtain the natural frequency of sense mode when varied DC voltage is applied on the tuning electrodes.Figure 7 gives the frequency tuning curves obtained by the negative electrostatic stiffness effect.It is noticed that the experimental tuning effect matches with the theoretical results, calculated by ( 7) using the parameter value in Table 2.As the applied tuning voltage increases, the two natural frequencies in the sense direction (-axis) both decrease but the two frequencies in the drive direction (-axis) are almost constant.Besides, when the tuning voltage increases, the frequency gap between the drive mode and the in-phase vibration along the sense direction (a disturbing mode) is enlarged.
With the aid of the tuning capacitors, the mode distribution of the gyroscope structure is more desirable because the frequencies of work modes are further separated from those of disturbing modes.Besides, the frequency difference between sense mode and drive mode can be adjusted from −55.8 Hz to 160.2 Hz when the tuning voltage limit is 20 V. Thus, the bandwidth BW can be tuned down from 86.5 Hz to 0 Hz according to different application conditions.Meanwhile, the sensitivity can be increased gradually.The experimental matching voltage 18 V is a bit smaller than the theoretical voltage 18.3 V, mainly resulting from the underetching effect during fabrication.
Assuming   = 1m and Ω in = 100 deg/s, the sense amplitude of the gyroscope prototype, calculated by (1), is shown in Figure 8.It is noticed that the amplitude is increased by over 20 times when the difference is decreased from 50 Hz to 0 Hz, verifying that the study about tuning frequency difference is beneficial to increase the sensitivity of the gyroscope.

Automatic Tuning Algorithm
The schematic control loop for tuning the natural frequency of sense mode towards the natural frequency of drive mode is displayed in Figure 9.When gyroscope works, the sense mode is driven by the Coriolis and quadrature coupling force.Ω equ is the equivalent angular input of the quadrature coupling force.An additional external signal  ℎ cos  ℎ  is applied as well, where  ℎ and  ℎ are the amplitude and frequency of the signal.The response of the external signal is used as the information to modify the applied tuning voltage so as to change the natural frequency of the sense mode.The tuning algorithm is derived based on the approximate symmetry of frequency response of the second order resonator, as seen in where    and    are the amplitude and phase characteristic, respectively.Similarly,    ± ℎ and    ± ℎ are the characteristic at frequency   ±  ℎ .When the resonator is applied with two respective forces at   ±  ℎ and   =   , their responses after demodulation and filtering are nearly opposite because the two frequencies are symmetric about the natural frequency   .By comparing the two responses, the natural frequency of sense mode and the drive frequency can be in the approximate match condition.The following equations detail the algorithm and processing.

Working Principle.
In the sense mode, the low frequency carrier  ℎ cos  ℎ  is loaded, modulated by sin    with  ℎ ≪   , and then applied on the feedback combs in sense frames.Then, the generated electrostatic force applied on the sense frames is where   is the equivalent transfer factor of feedback combs from the applied voltage to the generated electrostatic force.Thus, the displacement of sense vibration driven by  ℎ is Then, the displacement is amplified by a factor of   through a capacitive interface circuit and demodulated by − sin    and cos  ℎ , the process of which is expressed in where Finally, the control signal, used for tuning the natural frequency of sense mode, is obtained by low-pass filtering the demodulated signal  ℎ2 .Substituting ( 8) into ( 13), the control signal is where and Δ is the frequency difference between the natural frequency   and the drive frequency   .Suppose  ℎ is 2 × 80 rad/s; the relationship between (Δ) and Δ is displayed in Figure 10.As seen in ( 8), the amplitude characteristic as well as the phase characteristic is not strictly symmetric about the natural frequency   , leading to the equilibrium state of approximate matching.Theoretically, the stable frequency difference Δ between drive mode and sense mode is localized at 0.8006 Hz by using this control loop.However, the symmetry of ( 8) can be enhanced when the drive frequency and the natural frequency grow larger.Thus, the difference can be tuned closer to 0 Hz.
Based on Figure 10, the detailed control process is demonstrated when different initial frequency difference appears.It is noticed that when −79.9738 Hz < Δ < 0.8006 Hz, (Δ) < 0; under the control of the closed tuning loop, the  tuning voltage is decreased to enlarge   to approach   .Similarly, when 0.8006 Hz < Δ < 79.9705 Hz, (Δ) > 0 increases the tuning voltage to make   smaller and approach   .However, when Δ < −79.9738Hz, (Δ) > 0; if the tuning voltage is not limited, the natural frequency is further decreased and Δ is away from −79.9738 Hz towards −∞.
Similarly, when Δ > 79.9705 Hz, (Δ) < 0; the natural frequency is increased until it reaches its initial value and Δ is larger than 79.9705 Hz.Thus, the initial frequency difference should be controlled within [−79.9738Hz, 79.9705 Hz] so that the stable difference can converge to 0.8006 Hz.

Frequency
Response.The simulation model in Figure 9 is built using Matlab and Simulink.The sense mode vibration is developed using the -Function module with two inputs; one is the total applied force and the other is the tuned sense frequency by the negative electrostatic stiffness effect.The simulation parameters are shown in Table 3, in which  is approximated according to Figure 7(a).As the initial difference of 160.2 Hz between drive mode and sense mode is out of the range of the desirable difference shown in Figure 10, a tuning voltage  ref is preloaded on the tuning combs to make the frequency difference below 79 Hz.An integrator is used as the controller to tune the voltage applied on the tuning combs.
When the gyroscope works with the Coriolis and quadrature coupling force applied, the response of the frequency control loop, namely, the tuning voltage and the sense frequency, is given in Figure 11.Besides, the integrating parameter   is optimized for fast response and small overshoot.It is noticed that when   increases from 10 to 60, the equilibrium time decreases but the oscillation at the beginning grows to be severe.When   = 30, the control loop can be stabilized within 8 seconds with the final tuning voltage around 18.01 V.
The natural frequency response under different force loads is shown in Figure 12.When no Coriolis force or quadrature force is applied, the stable frequency difference is around 0.8 Hz, matching the theoretical difference.But when a large force is applied on the sense mode, the frequency control signal  ℎ3 will be contaminated, causing different stable difference.It is seen that when only the Coriolis force is applied and Ω in = 100 deg/s, the difference turns to 0.3 Hz.If the quadrature force, whose equivalent angular signal Ω equ = 100 deg/s, is applied as well, the difference is around 1.6 Hz.It is concluded that the difference distributes around 0.8 Hz  when the gyroscope works and it approaches 0.8 Hz when the applied force decreases.Thus, when the applied force is large, a force feedback loop is needed to diminish the influence of the Coriolis and quadrature force on the frequency difference.
Besides, the influence of the quality factor   is investigated, as displayed in Figure 13.It is shown that the frequency response is almost the same when   varies from 200 to 2000, indicating that the frequency control loop is independent on   within the simulated range.However, as   grows larger, the stable sense frequency vibrates in sine wave, caused by the enlarged gyroscope response under the Coriolis and quadrature force.

Gyro Output.
In Figure 9, two forces are applied; one is the useful signal, the Coriolis force, and the other is the disturbing signal, the quadrature coupling force.As seen in the control loop, the gyro output is obtained by demodulating the gyroscope response.If the drive frequency is not equal to the natural frequency, namely, the frequency difference is not 0 Hz, the phase delay of the sense resonator is not 90 degree.Then, during the demodulation by − sin   , the gyro output includes the element influenced by the quadrature signal.Therefore, as seen in Figure 14, the gyro output, obtained by simulating the system in Figure 9, is different when the quadrature coupling is present and absent.The useful gyro output signal is contaminated by the quadrature coupling.Thus, the quadrature coupling should be compensated to decrease the output error when this frequency tuning loop is used.The quadrature compensation is discussed in [25,26], and in the following simulation the quadrature coupling is assumed to be 0.
The response of the gyro output under different input angular rate, Ω in =  win cos(2 win ), is illustrated in Figure 15.As shown in Figure 12, the frequency tuning loop takes nearly 8 seconds to make the natural frequency of sense mode reach its equilibrium state.Besides, it is known that the gyroscope output is dependent on the difference between the natural frequency and the drive frequency.Thus, as displayed in Figure 15(a), during the initial 8 seconds the gyroscope output gradually increases as the natural frequency of sense mode approaches the drive frequency.
When  win ̸ = 0, the maximum output happens when the corresponding drive frequency   ±  win equals the natural frequency.Therefore, when  win is 5 Hz, the maximum output happens at 4012.9 Hz, which explains why a peak appears Effective sense frequency (Hz)

Results and Discussion
The frequency tuning method of the dual-mass gyroscope is investigated in this paper from two aspects; one is to modify the size of the drive and sense U-beams, respectively, before fabrication; the other is to change the voltage applied on the tuning capacitors after fabrication.The frequency tuning rate of the support beams and the tuning capacitors are summarized in Table 4. Width/length tuning rate is the frequency change when the width/length of support beams in drive or sense mode varies by 1 m.Similarly, the capacitor tuning rate is the frequency change when the applied tuning voltage varies by 1 V.It is indicated that the width-tuning rate is much larger than the length-tuning rate, but the variation range of the width is smaller by two orders of magnitude than that of the length in the gyroscope structure.
In general, frequency localization of the work modes is implemented independently for the -axis dual-mass silicon microgyroscope by tuning the size of the support beams or the voltage applied on the tuning capacitors.
The frequency difference between work modes is set at 100 Hz in the structure design procedure but the tested difference of the gyroscope sample is 160.2Hz.By using the negative electrostatic stiffness of the squeeze capacitors in the sense frames, the difference can be tuned from 160.2 Hz down to −55.8 Hz when the tuning voltage limit is 20 V. Thus, the bandwidth BW can be tuned down from 86.5 Hz to 0 Hz.Besides, the sensitivity can be increased by over 20 times when the difference is decreased from 50 Hz to 0 Hz.Furthermore, according to the frequency characteristic of the second order resonator, the automatic frequency control loop is successfully designed to match the frequencies of the drive and sense modes.According to the theoretical and simulation analysis, the frequency difference can be stabilized at 0.8 Hz when no Coriolis force or quadrature force is applied, as seen in Figures 10 and 12.When a large force is applied on the sense mode, the difference is varied; when Ω in = 100 deg/s and Ω equ = 100 deg/s, the difference is 1.6 Hz, as displayed in Figure 12; it is concluded that an additional force feedback loop is needed to suppress the disturbance when a large force is applied.Besides, quadrature compensation is needed as the gyro output is influenced by the quadrature signal, caused by the nonzero frequency difference.It is also proved that the frequency control loop can work properly under varied angular input, as shown in Figure 15.The dynamic 3 dB bandwidth, 0.836 Hz, is calculated when  win varies along with time in a slope of 0.1 Hz/s.Therefore, the electrostatic frequency tuning method is able to modify the frequency difference after fabrication, which eases the frequency accuracy required in the fabrication process.Besides, the automatic frequency control loop is able to tune the natural frequency towards the drive frequency when the gyroscope works with the Coriolis and quadrature force applied.When the applied force is small, the difference is almost not influenced by the force.On the other hand, when the force is large, a force feedback is required to suppress the disturbance so as to keep the difference unvaried when  dynamic input is applied.

Conclusion
This paper systematically demonstrates the frequency localization of a dual-mass gyroscope from two aspects; one is to optimize the support beams' size and the other is to take advantage of the negative electrostatic stiffness effect of the squeeze capacitors.As the fabrication imperfection influences the frequency difference between the work modes randomly, the experiment results show that the tested difference is 160.2Hz much larger than the designed 100 Hz.However, the frequency difference can be localized between −55.8 Hz and 160.2 Hz by manually applying a proper voltage under 20 V on the tuning capacitors.Besides, the automatic frequency control loop is designed to tune the natural frequency of sense mode towards the natural frequency of drive mode.This method makes the frequency difference resistive to the environment disturbances, such as the variation of the ambient temperature and work frequencies.The simulated results indicate the difference can be stabilized at 0.8 Hz within 8 seconds when no Coriolis force or quadrature coupling force is applied.Besides, the control loop still works when the input angular rate varies dynamically; Ω in =  win cos(2 win ).Therefore, the study made in this paper ensures that the frequency difference can be controlled for performance improvement as well as disturbance robustness in silicon vibratory microgyroscopes.

Figure 2 :
Figure 2: Equivalent finite element model of the dual-mass gyroscope structure.

Figure 3 :
Figure 3: The dependence of work frequencies on the size of U-beams: (a) the length of drive U-beams, (b) the width of drive U-beams, (c) the length of sense U-beams, and (d) the width of sense U-beams.

Figure 4 :
Figure 4: Part of the tuning parallel capacitors.

Figure 5 :
Figure 5: Experimental setup for frequency sweeping test.

Figure 8 :
Figure 8: Dependence of sense amplitude on frequency difference.

Figure 9 :Figure 10 :
Figure 9: Schematic control loop for automatic frequency tuning of the sense frequency.

Figure 11 :
Figure 11: Response of the tuning voltage  and the sense frequency   when   varies.

Figure 12 :
Figure 12: Sense frequency under different applied force.

Figure 13 :Figure 14 :
Figure 13: Response of the sense frequency   when   varies.

A0 1 1 Figure 15 :
Figure 15: Responses of gyro output when the input angular rate Ω in =  win cos(2 win ) is applied.

Table 1 :
Parameters for FE simulation.

Table 2 :
Designed structure parameters for gyroscope.

Table 3 :
Simulation parameters for frequency control loop.

Table 4 :
Frequency tuning rate for drive mode and sense mode.