Exploring the Boundaries of MEMS Gyroscope Scale Factor Temperature Stability via Distributed On-Chip Strain Measurements

JOURNAL OF MICROELECTROMECHANICAL SYSTEMS 1 Exploring the Boundaries of MEMS Gyroscope Scale Factor Temperature Stability via Distributed On-Chip Strain Measurements Mehran Hosseini-Pishrobat , Member, IEEE, Ahmet Arif Aslan, and Erdinc Tatar , Member, IEEE Abstract—We report a combined temperature-stress compensation method that achieves 31 ppm peak-to-peak and less than 1 ppm Allan deviation scale factor (SF) temperature stability for an amplitude-modulated (AM) MEMS gyroscope. The gyroscope operates in the degenerate n=2 wineglass mode shapes of a double-ring structure (within 1.6 mm radius) at 57.6 kHz drive mode resonance and 60–85 Hz frequency split. Sixteen 45-degreeapart capacitive stress sensors (each 480 ×386 µm2) encircle the main ring, eight inside and eight outside. Using an on-PCB heater, we perform full temperature cycling tests in the 25–90 ◦C range while applying a ±40◦/s dithered rate input. The symmetric arrangement of the stress sensors picks up the nonuniform, non-repeatable thermal stresses that develop in the substrate. These stresses stem from the coefficient of thermal expansion (CTE) mismatches along different MEMS packaging and PCB layers. Experiments with two different dies demonstrate the effect of die-attach. Linear stress and temperature compensation performs 8–40×better than temperature-only compensation. This improvement highlights distinct advantages of stress measurement in terms of capturing 1) differential stress sensitivities of the drive/sense modes due to fabrication imperfections, 2) gap variations, and 3) hysteresis effects that dynamically form during temperature cycles. Through analytical modeling, we delineate the contributions of these factors to the SF drift and quantify their impact on the thermal variations of stiffness and capacitive gains. [2025-0155] Index Terms—Calibration, MEMS gyroscope, scale factor stability, stress sensing. I. INTRODUCTION TEMPERATURE stability poses a fundamental challenge to the development of high-performance MEMS gyroscopes demanded by emerging application areas such as autonomous navigation, industrial robotics, and the Internet of Things (IoT) [1]. In that respect, the stability of the scale factor (SF) in dynamic temperature environments is a critical performance metric to maintain reliable rate measurements Received 22 August 2025; revised 7 October 2025; accepted 22 October 2025. This work was supported by the European Union’s European Research Council (ERC) Starting Grant under the grant agreement 101116162-0-driftERC-2023-STG. Views and opinions expressed are, however, those of the authors only and do not necessarily reflect those of the European Union or ERC. Subject Editor H. Chang. (Mehran Hosseini-Pishrobat and Ahmet Arif Aslan contributed equally to this work.) (Corresponding author: Mehran Hosseini-Pishrobat.) Mehran Hosseini-Pishrobat and Ahmet Arif Aslan are with the Department of Electrical and Electronics Engineering, Bilkent University, 06800 Ankara, T¨ urkiye (e-mail: [email protected]; [email protected]). Erdinc Tatar is with the Department of Electrical and Electronics Engineering and the National Nanotechnology Research Center (UNAM), Bilkent University, 06800 Ankara, T¨ urkiye (e-mail: [email protected]). Digital Object Identifier 10.1109/JMEMS.2025.3626121 over long operation periods. For example, it is well-established that inertial navigation requires 10 ppm SF accuracy [2]. On-chip stress sensing has shown great potential for bias stability, substantially improving upon temperature-only calibration (e.g., by 3-4×) through enriching the fitting dataset with thermal stresses that temperature measurements cannot capture [3],[4]. As the primary motivation, we demonstrate that the benefits of stress sensing extend to SF stability as well, exemplified by 8–40×improvement that combined temperature-stress calibration achieves over standard temperature calibration. A. Related Works Improving the SF stability has been a central research problem in inertial MEMS, and various compensation methods have been proposed in the literature. In a broad categorization, the existing methods can be grouped as follows: 1) Employing Surrogate Variables for Temperature: The drive mode frequency of a quadruple mass gyroscope–thanks to the high quality factor of 1.2 million–was shown to serve as a precise temperature self-sensing mechanism, resulting in a 700 ppm compensated SF error over 25–55 ◦C[5]. In a similar approach for an NEMS-based gyroscope with reported ±1500 ppm SF stability (10–60 ◦C range), the temperature coefficient of frequency (TCf) was shown to be repeatable and thus, a viable temperature indicator [6]. The frequency change ratio to drive force was used in [7] to stabilize the SF, achieving 4 ppm/◦C around room temperature. An ASIC for a mode-matched DRG improved SF temperature stability to 556 ppm from −40–60 ◦C by using the AC driving voltage as a temperature sensor [8]. 2) Modeling-Based Approaches: For a mode split AM gyroscope, SF-temperature sensitivity of 8 ppm/◦C in the −30 to 70 ◦C range was reported in [9] based on a multi-parametric fitting algorithm comprised of drive frequency/amplitude, split, and sense mode’s phase error. The SF was calibrated by injecting a cosine calibration signal to modulate the gyroscope drive frequency, achieving 3526 ppm stability over 0–50 ◦C for force rebalance operation in [10]. The capacitive feedthrough between a butterfly gyroscope’s drive and sense channels was shown to cause phase delay, resulting in a U-shaped SF nonlinearity error in [11]. SF nonlinearity was improved to 20 ppm with a real-time phase compensation method across −40–80 ◦C. The third harmonic of the drive pick signal served as a calibration signal in [12], resulting in an SF 1057-7157 ©2025 IEEE. All rights reserved, including rights for text and data mining, and training of artificial intelligence and similar technologies. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on November 26,2025 at 13:45:54 UTC from IEEE Xplore. Restrictions apply. 2 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS Fig. 1. MEMS gyroscope with on-chip stress sensors. a Gyroscope’s SEM. bn=2 wineglass mode shapes. cWorking principle of stress sensors, the unbalanced bridge amplifies the strain in the orthogonal direction. dArrangement of 16 stress sensors. eComparison of a stress sensor’s response to pure thermal, where there is no response, and mechanical loads, where the sensor responds. temperature sensitivity of 5 ppm/◦C over −30–30 ◦C. This approach was extended to whole-angle operation for an HRG in [13], achieving 0.79 ppm SF nonlinearity with real-time calibration. 3) Ovenization: Ovenization isolates the sensor from the external environment to prevent drift by maintaining a stable temperature through a closed-loop apparatus [14]. An ovenized HRG demonstrated 400 ppm SF stability in −20–60 ◦C range [15]. SF drift sources were identified through analytical modeling, and 10 ppm/◦C SF stability was obtained over −20–40 ◦C in [16] without calibration and fitting. Another approach integrated commercial MEMS IMU’s into an ovenization platform, achieving 50 ppm SF stability across −22–55 ◦C [17]. B. Relevance We introduce and implement a novel concept: stress-based SF calibration, which, to the best of our knowledge, has not been previously investigated. Furthermore, regarding the existing work, we address two significant gaps: 1) full temperature cycling, which reveals residual stresses and ensuing hysteretic effects, is a key aspect that is often overlooked; 2) we take into account thermal stresses that arise from CTE mismatches and affect the stiffness matrix of the gyroscope; these stresses are unobservable from temperature measurements, which, in general, capture only variations of material properties. C. Contribution Our MEMS gyroscope, with the SEM given in Fig. 1a, operates in the n=2 wineglass mode shapes of a double-ring structure (within 1.6 mm-radius) at 57.6 kHz drive frequency, ∼30k quality factor, and ∼60–85 Hz mode split (Fig. 1b). The device was fabricated using a silicon-on-glass (SOG) process ((111)-Si with 35 µm structural thickness) with wafer-level vacuum packaging [18]. Differential drive/sense, frequency tuning, and force-rebalance control are achieved using the 16 pairs of electrodes that surround the outer ring. Taking advantage of the circular structure of the device, we have equipped the gyroscope with 16 capacitive stress sensors (each 480×386 µm2) located 45◦apart–eight outside and eight inside the outer ring (refer to Fig. 1c-d). Each of these stress sensors measures local radial strains at the substrate level. As the original contribution, we leverage these measurements to achieve 31 ppm SF stability–across ∼60 ◦C variations from room temperature–via combined stress-temperature calibration. The preliminary results of this paper, for a different device with fewer stress sensors, are reported in the conference paper [19]. D. Organization Section II outlines the principles of stress sensing and its relevance to SF stability. Section III provides an analytical examination of SF and its various contributing factors. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on November 26,2025 at 13:45:54 UTC from IEEE Xplore. Restrictions apply. HOSSEINI-PISHROBAT et al.: EXPLORING THE BOUNDARIES OF MEMS GYROSCOPE SF TEMPERATURE STABILITY 3 Section IV explains the experimental setup and presents the key results on stress-based SF calibrations. Section Vgives further experimental results with an emphasis on the analytical modeling angle. Section VI discusses the current challenges and future directions of the research. Finally, Section VII draws the concluding remarks. II. STRESS SENSING A. Stress Sensors Fig. 1-c explains the working principle of our stress sensors. The unbalanced bridge-type hinge architecture converts the ∆x displacement (exerted via anchors) in the orthogonal direction as ∆y, amplified by a mechanical gain, ∆y/∆x=1/tan(γ) (see [20] for the derivation). This gain, which is about 6.5 in our case, depends on the particular design of the bridge structure. A fundamental time-scale separation exists between the gyroscope’s vibration mode (microseconds) and stresses developing at the substrate due to environmental temperature fluctuations (minutes or hours) [21]. The resonance frequency of the stress sensors is 304 kHz, which is far from the 57.6 kHz frequency of the wineglass modes. To read the capacitance change across the two anchored sides of stress sensors, 10 kHz voltages Vmod+and Vmod−are used for AC modulation. Hence, the stress sensors operate in the DC regime of the mechanical domain. The outputs of the 16 stress sensors are multiplexed in a way that each sensor is sampled and held for about 4.5 s before moving to the others in sequence. Each stress sensor, in effect, acts as a capacitive strain gauge that picks up the radial strain at its location. As shown by the finite element method (FEM) simulations in Fig. 1-e, the stress sensors are sensitive only to mechanical loads and not to unconstrained thermal expansions. Therefore, what we measure by the stress sensors reflects CTE mismatches across the MEMS packaging and PCB and not the pure thermal expansion. We determined the resolution of the stress sensors to be 5.3×10−3µ-Strain/√Hz. This resolution is sufficient for our analysis as the range of thermal strains–consequential to SF compensation–we encounter is around 5–10 µ-Strain. B. Why Stress? Two mechanisms predominantly govern the response of a MEMS gyroscope to environmental temperature [21]: 1) changes in thermophysical properties of silicon, namely, ∼60 ppm/◦C drop in its Young’s modulus [22] and a polynomial growth in its CTE [23]; 2) thermal stresses. Compared to the former, the latter is a more complex, manifold factor involving thermal interactions of all the device’s components (substrate, die-attach, packaging, and PCB) along with the thermal deformations in the silicon mechanical structure itself. Unlike the softening effect of the first mechanism, the second one perturbs the gyroscope’s stiffness matrix in a more complicated, less predictable manner. There could be competing hardening-softening stress components that affect not only the modal stiffness but also its symmetry; electrostatic gaps could vary nonuniformly across the device, and, notably, residual stresses could build up due to temperature cycles, resulting in hysteresis. Such intricacies are not encoded in temperature alone, and stress sensing is imperative to unraveling their effects on SF stability. To further elucidate this point, let us consider the openand closed-loop scale factors of an AM ring gyroscope operating at n=2 mode: SFOL =2AgXd ∆ωGOL,(1) SFCL =4Agω1XdGCL.(2) Here, Agis the angular gain, Xdand ω1are the drive mode’s displacement amplitude and resonance frequency, respectively, and ∆ω=2π∆fis the mode split, which is assumed to be significantly larger than the half bandwidth. GOL and GCL are the total gains of the final output read-out including capacitive and electronics components, that is, G=Capacitive Gain × Electronics Gain. We also note that closedor open-loop refers to whether the sense mode is force-rebalanced or not, respectively. Based on the above discussion, we observe from (1) and (2) that stress affects the SF stability through 1) Stiffness: mode split and drive frequency for the openand closed-loop cases, respectively; 2) Gap variations: capacitive gains, which, in turn, affect the controlled drive amplitude. Capturing these effects is the key advantage that stress sensing offers for SF calibration. We will discuss this point in more detail in the next section. III. ANALYTICAL MODELING We examine the SF stability more closely, leveraging the analytical model presented in the recent paper [21]. This model builds upon the time-scale separation between the environmental temperature and the gyroscope’s vibration (also discussed in Section II-A), which is illustrated by Fig. 2. As a salient outcome, the model distinguishes contributions of temperature sensitivity of material properties, thermal stresses, and electrostatics to the gyroscope’s stiffness. Importantly, the model differentiates between thermal effects that are endogenous and exogenous to the gyroscope’s vibrating structure. The latter is modeled by the displacements ∆ithat the thermal expansion/contraction of the anchored internal region exerts on Portion#i’s innermost beam. Here, we have divided the gyroscope into eight 45◦portions, where Portion#icovers the angular span [45◦(i−1),45◦i] for i=1 : 8. This partitioning helps us better model the vibrating structure’s response to nonuniform external stresses in the presence of fabrication imperfections. Following Fig. 2, the gyroscope’s stiffness matrix is given by K=E(T) E(T0)Knom +E(T) E(T0) εT εT0 KT+E(T) E(T0) 8 X i=1 ∆i ∆0 K(i) S −Ke−KDuffing,(3) which comprises the following components: •Nominal stiffness, Knom:This is the mechanical stiffness of the wineglass modes at a reference temperature T0; as temperature varies, Knom is scaled by Young’s modulus. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on November 26,2025 at 13:45:54 UTC from IEEE Xplore. Restrictions apply. 4 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS Fig. 2. Superposition principle for temperature effects [21].The model is based on the separation of the gyroscope’s fast and environmental temperature’s slow time-scales. The ratio E(T)/E(T0) accounts for the variations of Young’s modulus. •Thermal stiffness, KT:This is the stiffness developed in the gyroscope’s moving structure due to thermal deformations in the structure itself, excluding the anchored internal region. KTis scaled by both Young’s modulus and thermal strain εTat the silicon structural layer. •Stress stiffness, K(i) S:This is the stiffness induced in Portion#idue to the beam displacement ∆i,i=1 : 8. This component complements KTin the sense that it captures the effects of the anchored internal region. •Electrostatic Keand Duffing KDuffing stiffness: The softening impact of the electrostatic potential represented by the linear stiffness Keand the nonlinear, amplitudedependent stiffness KDuffing arising from cubic Duffing effects. Denoting the electrostatic gap by g0,Keand KDuffing admit the proportionalities Ke∝ 1 g3 0 and KDuffing ∝ 1 g5 0 ,(4) which point out the source of their thermal sensitivity. In view of Equations (1) and (2), we also need to examine drive amplitude and read-out gains. Drive amplitude. Although a large drive amplitude is desirable to improve SNR, it brings about nonlinear electrostatic effects. Mathematically, the read-out voltage of the differential drive-pick electrodes is given by VDP =∆CDP C0 .HV ≈β1Xd+β2X3 d, β1∝ 1 g2 0 , β2∝ 1 g4 0 .(5) Here, ∆CDP is the capacitive change, C0is the transimpedance amplifier capacitor, and HV is the structural high voltage. The linear β1and nonlinear β2gains are susceptible to temperature/stress-induced variations due to their dependency on the electrostatic gap. In that respect, it is important to note that amplitude control loops take VDP as their feedback variable. Hence, Xdcan still vary with temperature despite VDP remaining constant. Read-out gains. The openand closed-loop gains are given by GOL :=Sense output (V) Sense displacement (m) ∝ 1 g2 0 ,(6) GCL :=Rebalance voltage (V) Rebalance force per mass (N/kg) ∝g2 0,(7) respectively. Similar to the linear coefficient β1in Equation (5), the gap dependencies in the above equations arise from ∂C/∂Xterms. In light of the above analysis, we can summarize the dependency of SF on temperature T, stress σ, and gap g0 as SFOL ∝ Xd(g2 0)GOL 1 g2 0 ∆ωT, σ, 1 g3 0 ,1 g5 0,(8) SFCL ∝ω1T, σ, 1 g3 0 ,1 g5 0Xd(g2 0)GCL(g2 0).(9) The following remarks are in order: Remark 1: Electrostatic gap itself varies because of temperature and stress, i.e., g0=g0(T, σ). Remark 2: In the open-loop case, GOL balances the gap dependency of Xdwhile in the closed-loop, GCL amplifies this dependency. IV. RESULTS I: STRESS-BASED COMPENSATION A. Experimental Setup Fig. 3presents our test setup, along with the zoomed image of the MEMS die, the temperature profile, the applied rate profile, and a schematic of the daughter board. A 44-pin ceramic leadless chip carrier (LCC) housing the MEMS die This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on November 26,2025 at 13:45:54 UTC from IEEE Xplore. Restrictions apply. HOSSEINI-PISHROBAT et al.: EXPLORING THE BOUNDARIES OF MEMS GYROSCOPE SF TEMPERATURE STABILITY 5 Fig. 3. Test setup. a Photo of the test setup on the rate table. bTemperature profiles. cZoomed photo of the MEMS gyroscope. dThe applied rate profile. eOverview of our on-PCB heating method; the heater on the top PCB layer heats the sensor. Fig. 4. Test results. a Drive mode’s frequency variation with respect to temperature; the TCf’s are calculated as the least-squares slopes, lower TCf implies higher stress. bSample measured strains versus temperature, soft epoxy exhibits lower stress but with more hysteresis. is soldered to a daughter board. We designed a PCB heater on the top layer of the daughter board (under the LCC), and a proportional to absolute temperature (PTAT) sensor and gyroscope drive and sense preamplifiers are located on the bottom layer of the daughter board. The daughter board plugs into the main board with the stress sensor readouts and multiplexers, and gyroscope AC signal drivers. The gyroscope control loops (drive amplitude control, PLL, sense quadrature control, and force-rebalance), and the stress sensor multiplexer controls are implemented by a digital lock-in amplifier (Zurich Instruments, HF2LI). This configuration enables heating of the gyroscope independent of most of the electronics, and sensitivity tests without expensive and complicated setups with large ovens. We cycled the temperature twice at a slow rate of ±0.6 ◦C/min with 30 min soaks at the end points to minimize temperature gradients (Fig. 3-b). The temperature control is open-loop, where we linearly ramp the heater power (that is, voltage squared). A relatively large Aluminum lid on LCC helps to achieve an isothermal package. We note that our measurements effectively represent the bulk temperature of the MEMS package due to the heat dissipation along the heater-to-device path [14] and the placement of the temperature sensor under the PCB. Utilizing the same type of PTAT measurements, a fully analytical model provided accurate predictions of frequency variation [21]. Therefore, these measurements offer a reasonable approximation of the gyroscope’s temperature. We applied a dithered rate of ±40 deg/s for 4.5 s in each direction and recorded the temperature, sensitivity, and frequency data at 4.5 s sampling time. Because of the multiplexing of all 16 stress sensors, the output of only one sensor is recorded at each sampling instant before moving to the next one in sequence. B. Compensation Results We report compensation results for two different devices: 1) Device#1: fDrive =57.634 kHz, ∆f=85.7 Hz ( fSense > fDrive), mounted to the package with silicone-based soft epoxy having the properties E≈6.92 MPa and This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on November 26,2025 at 13:45:54 UTC from IEEE Xplore. Restrictions apply. 6 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS CTE ≈140 ppm/◦C (United Adhesives, Thermobond 3508 [24]). 2) Device#2: fDrive =57.408 kHz, ∆f=61.3 Hz (fSense >fDrive), mounted to the package with silverfilled hard epoxy having the properties E≈198 MPa, CTE ≈31,158 ppm/◦C before, after the glass transition temperature of Tg≈80 ◦C (Epoxy Technology, EPOTEK H20E [25]). Sample test data are plotted in Fig. 4: drive frequency variations versus temperature (Fig. 4-a) and measured strain samples comparing the two devices (Fig. 4-b). The hard epoxy results in a more uniform but larger stress with a smaller hysteresis. Strain flattening towards the higher temperatures in Device#2 is due to the glass transition of the hard epoxy. The lower TCf of Device#2 compared to Device#1 also indicates higher stress in Device#2 (Fig. 4a). Based on −60 ppm/◦C dependency of Young’s modulus, we expect to see −30 ppm/◦C TCf without any stresses. The device topology and stresses push the TCf in the positive direction, and hence, a TCf closer to 0 implies higher stress [21]. The compensation results for Devices#1 and 2 are given in Figs. 5and 6, respectively. By compensation, we refer to least-squares curve fitting performed based on specific choices of (T.σ)-basis functions such as linear, inverse, and secondorder polynomial. We present the residuals for various fitting methods and the Allan deviation to observe any drifts in these residuals. Fig. 7summarizes the test results for both devices. We report the best SF residual (Fig. 7-a) and the improvement of the linear stress (σ) and linear temperature and stress (T, σ) compensation over the linear temperature (T) compensation for all the tests (Fig. 7-b). Linear (T, σ)-compensation for Device#1 achieves 26×(at the worst case) lower residue than linear T-compensation. Linear σ-compensation closely parallels the (T, σ)-compensation, and only in the worst case, there is a 20% performance difference. We obtain the most stable SF with (1/T,1/σ), and second-order (T, σ) compensations down to 31 ppm and 90 ppm peak-to-peak (p-p) for the openand closed-loop, respectively. Since the open-loop SF is a function of 1/∆f, (1/T,1/σ) efficiently captures SF. Second-order temperature and stress calibration can be conceived as an expanded polynomial version of this type of compensation. 1) Temperature Versus Stress: Based on Fig. 7-b, the performance of (T, σ)- and σ-compensations are closely comparable, pointing out the marginal contribution of temperature. Hence, the bulk of vital information necessary for the calibration is embedded in the stress. The open-loop SF is primarily a function of frequency split (∆f), and the gyroscope drive and sense frequencies vary similarly with temperature to the first order, thanks to the ring symmetry. On the contrary, the modes may respond differently to the stress due to anisotropic fabrication imperfections and nonuniform gap variations. Additionally, due to the viscoelasticity of the die-attach, the stress evolution over temperature cycles is predisposed to hysteresis. This argument also extends to the closed-loop case, where the impact of gap changes is more pronounced, and stress effects are pertinent only to the drive mode. Linear temperature compensation cannot effectively suppress SF drifts as it lacks these essential factors. Even after adding the square term T2, we observe a limited improvement with temperature compensation over the uncompensated SF as shown in Fig. 5-a and Fig. 6-a. The relatively lower improvement ratios for the hard epoxy in Fig. 7can be attributed to its more uniform stresses with less hysteresis (see Fig. 4-b). We further point out that the stress data successfully capture the hysteresis despite a 16-fold sampling time compared to temperature. Hence, any delays induced by multiplexing are negligible compared to the time constants of thermal stresses. The stress changes between the sampling instants are smooth and slowly varying. Therefore, their temporal evolution can be recovered via interpolating recorded samples, which explains the success of least-squares fitting. 2) Contribution of Each Stress Sensor: To compare stress sensors in terms of their contributions to the calibration, we provide their standardized coefficients in Fig. 8. These normalized coefficients1reflect the change, in terms of standard deviation, in SF per unit increase in the output of a particular stress sensor [26]. As expected, stress sensors with larger strain values and hysteresis have a larger contribution. The standardized coefficient of the temperature is of the same order for both devices. 3) Effect of Drive Amplitude: For Device#1, we have included separate tests with small (Xd=0.5µm) and large (Xd=1µm) drive amplitudes (with respect to the g0=3.6µm gap) to observe any possible effects of drive mode electrostatic nonlinearity. We observe ∼2×lower residues for Xd=1µm in the open-loop and comparable residues in the closed-loop operation. Hence, we conclude that the electrostatic Duffing nonlinearities of the drive mode do not have a major effect on the SF temperature sensitivity. 4) Uncompensated SFs: Device#1’s uncompensated SF variations are 3600 ppm and 8500 ppm for the openand closed-loop, respectively. The sense pick-offelectrodes are differentially positioned inside and outside the ring with a double-sided gap; however, the force-rebalance electrode is single-ended and located only on the inside with a single side gap (the outside electrode is utilized for sense frequency tuning). The single-sided gap is more prone to stress and electrostatic gap variations, resulting in higher closed-loop SF variation for the uncompensated SF. For Device#2, the open-loop uncompensated SF variation is comparable to Device#1 at 3800 ppm. However, in the closedloop, this variation is 26000 ppm, 3×of Device#1’s. The single-ended closed-loop operation, combined with the higher stress and hence higher gap variations, leads to such amplified closed-loop SF variation. We conducted a comparison test with fully differential force-rebalance by skipping the frequency tuning, the results are shown with “Differential electrodes” in Fig. 6-a. The uncompensated closed-loop SF variation reduced to 4900ppm from 26000ppm, proving the effect of the differential drive. However, including stress in the single-sided 1Mathematically, the standardized coefficient of the i-th variable in the regression model Y=PaiXiis given by aiσXi/σY. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on November 26,2025 at 13:45:54 UTC from IEEE Xplore. Restrictions apply. HOSSEINI-PISHROBAT et al.: EXPLORING THE BOUNDARIES OF MEMS GYROSCOPE SF TEMPERATURE STABILITY 7 Fig. 5. Device#1 (soft epoxy) compensation results.a Scale factor residuals with peak-to-peak values. Closedor open-loop refers to whether the sense mode is forcere balanced or not, respectively; Xdis the drive mode’s amplitude, and Poly(T, σ) denotes a second order polynomial in (T, σ). bAllan deviation plots generated for SF residuals normalized by the initial SF; the reported ppm numbers are the final values of each graph. force-rebalance compensation dramatically reduced the peakto-peak SF variation to less than 100 ppm and Allan deviation to 2 ppm. As shown in Fig. 7, higher stress calibration ratios are obtained in the soft epoxy device due to larger hysteresis. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on November 26,2025 at 13:45:54 UTC from IEEE Xplore. Restrictions apply. 8 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS Fig. 6. Device#2 (hard epoxy) compensation results. a Scale factor residuals with peak-to-peak values. bAllan deviation plots with the final attained values. Fig. 7. Summary of compensation results for all the tests. a Best p-p SF residual in ppm. bImprovement achieved by linear(σ) and linear (T, σ)- compensation over linear T-compensation in terms of p-p SF residual. 5) Limits of Allan Deviation: We perform Allan deviation analysis on the SF residues in Fig. 5-a and Fig. 6-a over the entire duration of the tests (∼9 h), resulting in the plots in Fig. 5-b and Fig. 6-b. Allan deviation reveals the time domain residue characteristics for different compensations and helps us better understand the effectiveness of stress in capturing the long-term drift. While the original and temperature compensation exhibit drift, stress compensation removes such long-term components from the SF residues. The best stress compensation results closely follow the reference pure white noise This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on November 26,2025 at 13:45:54 UTC from IEEE Xplore. Restrictions apply. HOSSEINI-PISHROBAT et al.: EXPLORING THE BOUNDARIES OF MEMS GYROSCOPE SF TEMPERATURE STABILITY 9 Fig. 8. Standardized coefficients. Comparing the contributions of temperature and stress sensors’ outputs in terms of their standardized regression coefficients in (T, σ) fitting; i: inner, o: outer, and numbers indicate the angular position of the stress sensors. −1/2 slope Allan deviation line, achieving 0.6 ppm and 2 ppm stability for the best cases of Device#1 and #2, respectively. These results suggest that we may be limited by the rate table noise and stability (Acutronics AC1120Si). Upon characterization of the rate table, we found that the rate table has a rate-dependent noise, and the sensor white noise increased to 3×of the stationary value while rotating. We obtained the highest rate-to-noise ratio at 40◦/s and ran the tests at this rate input. 6) Comparison With Existing Methods: Table Icompares our best SF with the literature in terms of total value (ppm) and temperature coefficient (ppm/◦C). The latter quantity should be considered judiciously, as the residual SFs do not necessarily follow a linear pattern. Most studies did not perform temperature cycling, a critical aspect that reveals temperature-SF hysteresis and represents practical applicability. The temperature ranges vary between studies. Since we use a heater, our setup cannot go below room temperature. Our result, 31 ppm, is 1.5×better than the closest work (50 ppm) for a similar temperature range. We also note that we collected data at ∼4.5 s sampling time while ramping the temperature at ±0.6◦C/min. In contrast, most studies typically perform measurements after extended soaking periods, which tends to generate less drift. In that respect, the high level of SF stability we achieved further demonstrates the promise of stress calibration. V. RESULTS II: FURTHER ANALYTICAL EXAMINATION In this section, we present further results incorporating analytical modeling to substantiate the conclusions drawn in Section III. A. Frequency Split in Open-Loop SF As a corollary of Remark 1in Section III, frequency split is the prevalent factor in SF stability under the open-loop operation. We investigate this claim by conducting separate experiments using Device#1, where we record both drive and sense frequencies–in sync with the open-loop SF–over TABLE I COMPARISON OF SF STABILITY RESULTS two temperature cycles. We continuously halted the gyroscope operation, obtained the sense frequency with a Phase Locked Loop (PLL), and restarted the gyroscope operation during the temperature sweeps. Fig. 9plots the variation of frequency split ∆fversus the SF and the fitting residues of T-, fd-, and ∆f-compensations. We observe that fdcompensation closely follows and offers only a marginal improvement over T-compensation. This result further confirms that our measured temperatures are accurate indicators of the device’s actual temperature. As expected from the SF-∆f correlation, ∆f-compensation outperforms T-compensation by 5×. However, ∆f-compensation does not entirely compensate for the long-term effects. Two factors explain this observation: 1) Hysteresis. Although we observe a first-order linear relation, accumulating residual effects over temperature cycles spoils this linearity. 2) Gap variations. The g0terms in Xdand GOL in Equation (8) pertain to different angular spans over the drive-pick (at 90◦and 270◦) and sense electrodes (at 135◦and 315◦), respectively. So, Xdand GOL variations may not exactly cancel each other. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. Authorized licensed use limited to: ULAKBIM UASL - Bilkent University. Downloaded on November 26,2025 at 13:45:54 UTC from IEEE Xplore. Restrictions apply.