A 7PPM, 6◦ /HR FREQUENCY-OUTPUT MEMS GYROSCOPE Igor I. Izyumin1 , Mitchell H. Kline1 , Yu-Ching Yeh1 , Burak Eminoglu1 , Chae Hyuck Ahn2 , Vu A. Hong2 , Yushi Yang2 , Eldwin J. Ng2 , Thomas W. Kenny2 , and Bernhard E. Boser1 1 University of California, Berkeley, California, USA 2 Stanford University, Stanford, California, USA ABSTRACT
LISSAJOUS FM OPERATION
We report the first frequency-output MEMS gyroscope to achieve < 7 ppm scale factor accuracy and < 6◦/hr bias stability with a 3.24 mm2 transducer. By implementing continuous-time mode reversal in an FM gyro, the rate signal is modulated away from DC, making the system insensitive to the resonant frequency of the transducer. The scale factor is almost entirely ratiometric, depending primarily on the mechanical angular gain factor of the transducer and the accuracy of the timing reference. Scale factor sensitivity to variations in quality factor, electro-mechanical coupling coefficients, and circuit drift is significantly reduced compared to conventional open-loop and force-rebalance operating modes.
Background The proposed operating mode employs a mode mismatched but otherwise symmetric transducer. Both transducer axes are driven into oscillation at their natural frequency using sustaining oscillator loops. Because the axis frequencies are not equal, the proof mass follows a Lissajous trajectory. Figure 1 shows the block diagram of the gyroscope system, as well as expressions for the X and Y axis oscillation frequencies (for a Z-axis gyro) resulting from the analysis of the system differential equations in the complex baseband using the method of averaging [3, 4] . It can be seen that applied angular rate sinusoidally modulates the frequency of oscillation of both gyro axes. The depth of modulation is equal to the angular rate scaled by the angular gain factor αz of the gyroscope, and the modulation period is determined by the frequency difference ∆ f . The principle of operation may be understood by first considering the behavior of the circular orbit gyroscope as a function of the phase shift ∆φ between the X and Y oscillations. When ∆φ = ±π/2, the proof mass follows a circular orbit at the natural frequency of the transducer in an inertial frame of reference. Applied rotation causes the frame of the transducer to rotate relative to the orbit, changing the observed orbit frequency. When ∆φ = 0 or ∆φ = ±π, the proof mass oscillates along a straight line, and the oscillation frequency is insensitive to angular rate. In the Lissajous FM (LFM) mode, ∆φ is not a static value, but is instead a periodic ramp between −π to +π with a period 1/∆ f ; as a result, the rate sensitivity of the circular orbit FM gyroscope is modulated sinusoidally at ∆ f . Modulating the rate sensitivity causes the rate signal to move from DC to the frequency of modulation; this is iden-
INTRODUCTION Achieving ppm-level scale factor stability in conventional amplitude-based MEMS gyroscopes has been a difficult challenge. As a typical example, [1] employs a sophisticated self-calibration technique to achieve an rms scale factor accuracy of 350 ppm over a 10 ◦ C temperature range. Any sensor measures its input relative to a reference. For a rate gyro, the input has units of angular frequency; therefore, its reference must also be a frequency. However, amplitude-based rate gyros measure a force, either directly (in force-rebalance mode) or indirectly, by measuring sense axis displacement. The reference is constructed implicitly from a number of transducer and readout circuit variables, which include the effective mass, stiffness, various coupling factors (in turn determined by bias voltages), and readout circuit parameters, such as amplifier gains. It is very difficult to measure or control all of these parameters to the required level of accuracy, thus limiting the achievable scale factor stability. Previously reported circular-orbit FM gyroscopes [2] potentially improve scale factor stability by sensing rate directly, without intermediate voltage-to-force conversions and associated errors. However, they are unable to distinguish rate from transducer resonant frequency drift, resulting in very poor bias stability (1550 ◦/hr in [2]). The bias drift is dominated by the temperature dependence of the resonant frequency on the Young’s modulus of silicon, about 30 ppm/K. For a 30 kHz device, this translates to a temperature coefficient of 324 ◦/s/K, implying that 1 µK accurate temperature regulation is needed for a 10 ◦ /hr gyroscope. A second structure with opposite sensitivity reduces the coefficient to about 10 ppb/K, or 400 ◦/hr/K, but orders of magnitude improvement is still needed in order to reduce this error to < 10 ◦/hr. This work presents an FM-based gyroscope operating mode that is insensitive to the resonant frequency of the transducer, while maintaining high scale factor stability.
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Figure 1: Block diagram of Lissajous FM gyroscope and expressions for X and Y axis instantaneous frequencies.
fx =h27629hHz fy =h2773bhHz Qh>h5btbbb Proofhmass:h93μg
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Figure 2: (a) layout of quad-mass MEMS transducer; (b) finite-element simulation of primary mode shape; (c) SEM photograph of fabricated structure. tical to the chopper stabilization technique commonly used in precision amplifiers. As a result, the gyroscope is made insensitive to slow variations in the natural frequency of the transducer, as might result from temperature fluctuations and other drift sources. Furthermore, from the equations in fig. 1, it may be seen that the scale factor of the gyroscope depends only on the angular gain factor αz (a stable, dimensionless parameter set by the transducer geometry) and the velocity amplitude ratios vxa /vya and vya /vxa . If the demodulated X and Y channel outputs are summed, the sensitivity to velocity amplitude mismatch is greatly reduced due to the reciprocal summation of the two amplitude ratios. For example, a relatively large mismatch of 0.5% between the X and Y axis velocities results in a scale factor error of only 12 ppm. Due to the periodic mode reversal inherent to LFM operation, summation of the X and Y outputs cancels reciprocal anisodamping errors, potentially improving bias stability. As in the ordinary AM mode, quadrature errors are rejected by phase-sensitive demodulation. Signal processing Angular rate measurement in the LFM operating mode requires high-resolution measurements of the frequencies of the X and Y axes. The FM demodulator must have sufficient bandwidth to not attenuate the tone at the split frequency. In order to recover the angular rate, the measured frequencies of the X and Y axes are integrated and then differenced, producing a phase ramp ∆φ . A digital phase detector is used to provide an initial condition for the integrators. Applying a sine operation to ∆φ recovers the carrier; synchronous demodulation is then used to demodulate the angular rate signal to the baseband. The X and Y signals are then summed and filtered to produce the final rate output. The block diagram in fig. 1 illustrates this process. Discussion LFM operation presents several significant advantages over other operating modes. Unlike open-loop and forcerebalance operating modes, LFM gyros can achieve ppmlevel scale factor stability without any additional calibration layers. Because the output signal is a frequency, rather
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Figure 3: Sustaining oscillator differential half-circuit.
than an analog voltage or a current, a very high dynamic range can be accommodated without range switching. Unlike the circular-orbit FM mode, the LFM operating mode is not sensitive to natural frequency drift and does not impair bias stability. Because the required ∆ f is small, fullysymmetric transducers such as the quad-mass gyro can be employed, allowing the gyroscope to benefit from the high quality factor, vibration rejection, and other advantages inherent to such structures. The choice of the split frequency ∆ f incurs some design trade-offs. The split frequency needs to be high enough to avoid injection locking and slow drift components near DC and accommodate the required gyroscope bandwidth. Unlike amplitude-based gyros, there is no trade-off with scale factor stability, allowing a significantly smaller split frequency to be used. Furthermore, because the natural frequency of each mode can be easily observed, active electrostatic tuning can be used to precisely control the split frequency, eliminating fabrication mismatch and drift. Since the noise contribution of the readout circuits depends on the frequency split, reducing the split frequency decreases circuit power quadratically. LFM can thus enable low-power operation while maintaining low ARW and high scale factor stability.
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Figure 4: Time-domain scale factor measurement. A one-hour moving average was applied to the combined curve to reduce short-term ARW-related noise and show the long-term trend.
EXPERIMENTAL CHARACTERIZATION
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Figure 6: Scale factor Allan deviation.
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The LFM operating mode was characterized using a quad-mass structure similar to [5]. The layout, primary eigenmodes, design parameters, and measured resonant frequencies of the MEMS transducer are shown in fig. 2. The transducer was fabricated using the epi-seal vacuum encapsulation process described in [6]; fig. 2c shows an SEM micrograph of the fabricated structure. The structure uses differential parallel-plate actuation and sensing using 8 pairs of differential electrodes. The spring anchors are designed to minimize transfer of packaging stress to the springs. The transducer used for testing had an intrinsic ∆ f of 101 Hz (arising from fabrication mismatch); no electrostatic tuning was used. The sustaining oscillators were implemented as discrete Pierce oscillators with opamp-based active integrators; the schematic of the differential half-circuit of a single oscillator channel is shown on fig. 3. FM demodulation was performed by a pair of low-power, high-resolution frequency-to-digital converters implemented on a custom ASIC [7]. The remaining demodulation operations were performed using digital signal processing. The gyroscope bias and scale factor stability were characterized with a 50-hour test. The test consisted of repeated +90, 0, and −90◦ /s angular rate measurements, which allowed bias and scale factor stability to be measured simultaneously. Testing was performed at room temperature with no active temperature regulation. Figure 4 shows the measured scale factor error drift over time. Due to velocity amplitude mismatch caused by circuit imperfections and drift, individual X and Y axis signals exhibited a 1.9% initial scale factor error and additional drift of 0.45% during the test; this is a typical performance level for a conventional AM gyro with no calibration circuits. Combining the X and Y signals almost entirely eliminates this error source: fig. 4 shows no significant drift trend for the summed output. The Allan variance of the scale factor reaches a minimum of 6.7 ppm at τ = 15400s
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tation in deep-submicron, low-voltage CMOS processes and integration with sensor processors and SoCs. 60 Hz interference tone
ACKNOWLEDGEMENTS The authors would like to thank TSMC and InvenSense for providing IC fabrication. This material is based upon work supported by the Defense Advanced Research Projects Agency (DARPA) under Contract No. W31P4Q12-1-0001.
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REFERENCES
Average ARW: 0.014 deg/s/rtHz
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Figure 7: In-band rate noise power spectral density (angle random walk). The tone is caused by 60 Hz power line interference coupling to the sustaining circuits. The rate signal is at about 100 Hz; after demodulation the 60 Hz tone appears near 40 Hz. (fig. 6), which is near the specified performance limit of the rate table used for this test. Bias stability reaches a minimum of 5.9 ◦/hr at τ = 3800 s (fig. 5). This represents a 250× improvement over the single-transducer circular orbit FM result in [2]. The ARW in fig. 5 is higher than actual due to noise folding resulting from the test protocol. Figure 7 shows the measured √ rate noise density; the angle random walk is 0.014 ◦/s/ Hz, and is limited by the phase noise of the discrete oscillator circuits. The bandwidth of the gyroscope is 50 Hz; the full-scale rate exceeds ±1000 ◦/s.
CONCLUSIONS This work has presented a frequency-readout operating mode for vibratory gyroscopes that dramatically improves bias stability over the circular-orbit FM mode while demonstrating state-of-the-art scale factor stability. The operating mode can take full advantage of high-Q fully-symmetric transducers to reduce cross-damping errors and power consumption. Unlike the whole-angle operating mode, Lissajous FM operation does not require a complex controller and is well-suited for low-cost low-power consumer-grade applications. In addition to ensuring accurate scale factor, frequency readout permits a large dynamic range to be accommodated without encountering analog dynamic range limitations and without range switching. The mostly-digital implementation of the FM readout circuits is well-suited to implemen-
[1] A. Trusov, I. Prikhodko, D. Rozelle, A. Meyer, and A. Shkel, “1 ppm precision self-calibration of scale factor in MEMS Coriolis vibratory gyroscopes,” in TRANSDUCERS 2013: 17th Int. Conf. Solid-State Sensors, Actuators and Microsystems, Jun. 2013, pp. 2531–2534. [2] M. Kline, Y. Yeh, B. Eminoglu, H. Najar, M. Daneman, D. Horsley, and B. Boser, “Quadrature FM gyroscope,” in 2013 IEEE 26th Int. Conf. MEMS, Jan. 2013, pp. 604–608. [3] M. H. Kline, “Frequency modulated gyroscopes,” Ph.D. dissertation, University of California, Berkeley, Dec. 2013. [4] D. Lynch, “Vibratory gyro analysis by the method of averaging,” in 2nd Saint Petersburg Int. Conf. Gyroscopic Technol. and Navigation I, May 1995, pp. 26– 34. [5] A. Trusov, I. Prikhodko, S. Zotov, A. Schofield, and A. Shkel, “Ultra-high Q silicon gyroscopes with interchangeable rate and whole angle modes of operation,” in 2010 IEEE Sensors, Nov. 2010, pp. 864–867. [6] R. Candler, M. Hopcroft, B. Kim, W.-T. Park, R. Melamud, M. Agarwal, G. Yama, A. Partridge, M. Lutz, and T. Kenny, “Long-term and accelerated life testing of a novel single-wafer vacuum encapsulation for MEMS resonators,” J. Microelectromech. Syst., vol. 15, no. 6, pp. 1446–1456, Dec. 2006. [7] I. Izyumin, M. Kline, Y.-C. Yeh, √ B. Eminoglu, and B. Boser, “A 50 µW, 2.1 mdeg/s/ Hz frequency-todigital converter for frequency-output MEMS gyroscopes,” in ESSCIRC 2014 – 40th European Solid State Circuits Conf., Sep. 2014, pp. 399–402.
CONTACT *I. I. Izyumin;
[email protected]
