Quadrature Error Research Articles

A Phase Self-Correction Method for Bias Temperature Drift Suppression of MEMS Gyroscopes

Phase error of the demodulation clock in the Coriolis vibratory gyroscope system allows the quadrature errors to leak into the sense channel and causes significant bias and temperature drift at the rate output. A phase self-correction method to suppress the temperature drift of the bias in gyroscopes is proposed. Through sweeping the demodulation clock phase and simultaneously monitoring the mechanical quadrature error output in gyroscopes, the optimal demodulation clock phase with minimum relatively phase shift is determined. Thus the bias influenced by the temperature and surroundings can be calibrated on-chip at start-up or when the environment changes drastically without the requirement of the complicated instruments. The proposed approach is validated by a silicon MEMS gyroscope with the natural frequency of 2.8[Formula: see text]kHz, which shows nearly 22 times improvement in the temperature sensitivity of the system bias, from 550[Formula: see text]mdeg/s/∘C down to 24.7[Formula: see text]mdeg/s/∘C.

PLL Position and Speed Observer With Integrated Current Observer for Sensorless PMSM Drives

In this article, a phase-locked loop (PLL) rotor position and speed observer, including an integrated decoupled stator current observer, is proposed for wide-speed-range sensorless control of permanent magnet synchronous motor (PMSM) drives. This new PLL scheme consists of two interconnected observers: 1) a second-order PLL position observer using the direct current error as correction term and 2) a third-order extended Luenberger speed observer, based on the mechanical model of the drive, using the quadrature current error as the correction term. The estimated speed is just an unrefined reference for the position integrator, and it is further corrected by the PLL. The current observer is a subsystem integrated as a PLL phase detector. Key relations between the current errors and the position and speed errors are proven, and a practical two-step observer design method is presented. A method for real-time identification of the stator resistance at low speeds and of the permanent magnet flux at high speeds is proposed, using the current errors in the current reference frame. Experimental results for a 0.5-kW surface-mounted PMSM drive confirm the effectiveness, excellent dynamic performance, and robustness of the new PLL observer. Stable motor and generator operation at 15 r/min (0.01 p.u.) with ramp and step speed reversals, high-speed operation, and stator resistance adaptation is shown.

GPU‐accelerated simulation of polydisperse multiphase flows using dual‐quadrature‐based moment methods

AbstractPolydisperse multiphase flows can be simulated by coupling a population balance model to the multi‐fluid model. For the first time, this simulation is carried out using the dual‐quadrature method of generalized moments (DuQMoGeM) and its direct version to solve the population balance model. The main disadvantage of these methods is the high computational cost of the embedded cubature. Herein, this challenge was addressed by parallelization on graphics processing units, which resulted in a significant acceleration of the simulations, with speedups that can be larger than 1000. Numerical simulations considering simultaneous particles breakage and aggregation were conducted with direct DuQMoGeM‐FC and DQMoM‐FC, whose results were different due to the existence of quadrature error in the latter.

A 3.2-GHz Quadrature Error Corrector for DRAM Transmitters, Using Replica Serializers and Pulse-Shrinking Delay Lines

For DRAM transmitters, in this letter, we propose a quadrature error corrector (QEC), in which clock signal phase errors are corrected using two replicas of the 4:1 serializer of the output stage. Pulse shrinking is used to compare and equalize the outputs of these two replica serializers. To validate the effectiveness of the QEC, a prototype quarter-rate transmitter including our QEC was fabricated in 65-nm CMOS. Adopting our QEC, the experimental results show that the output phase errors of the transmitter are reduced to a residual error of 0.8 ps and the output eye width and height are improved by 84% and 61%, respectively, at a data rate of 12.8 Gb/s.

Rational averaged gauss quadrature rules

It is important to be able to estimate the quadrature error in Gauss rules. Several approaches have been developed, including the evaluation of associated Gauss-Kronrod rules (if they exist), or the associated averaged Gauss and generalized averaged Gauss rules. Integrals with certain integrands can be approximated more accurately by rational Gauss rules than by Gauss rules. This paper introduces associated rational averaged Gauss rules and rational generalized averaged Gauss rules, which can be used to estimate the error in rational Gauss rules. Also rational Gauss-Kronrod rules are discussed. Computed examples illustrate the accuracy of the error estimates determined by these quadrature rules.

Highly-linear wide-range voltage-controlled delay element with body bias technique

A highly-linear wide-range voltage-controlled delay element in 55 nm CMOS technology at 1.2 V supply is presented. Unlike traditional voltage-controlled delay unit, the proposed delay unit utilizes two transistors as auxiliary current path and employs body bias technique to widen applicable range of control voltage. At the same time, it guarantees correct output logic voltage and high linearity. Moreover, a simple model is proposed to get a better understanding of circuit characteristics. The unit delay ranges from 110ps to 236 ps. When sweeping control voltage from 0 to 1.2 V at 200 MHz (TT corner 27 °C), calculated gain is 14.2ps/mv and delay quadrature error (DQE) is 6.6%. While operating at 200 MHz, the maximum simulated power consumption is 15.09μW.

Asymptotic convergence of the angular discretization error in the scalar flux computed from the particle transport equation with the method of discrete ordinates

The asymptotic convergence of the angular discretization error in the scalar flux solution of the particle transport equation computed with the Discrete Ordinates (SN) method with increasing quadrature order is examined. Five selected angular quadrature types are considered: Level Symmetric (LS), Legendre-Chebyshev Quadrangular (LCQ), Legendre-Chebyshev Triangular(LCT), Quadruple Range (QR) and Quadruple Range Spence-type (QRS) quadrature sets. We relate the SN angular discretization error to the quadrature error, and split the total flux into the uncollided flux and the fully collided flux, and then we verify the uncollided scalar flux error and the fully collided scalar flux error separately. After developing the theoretical basis for the relationship between true solution regularity and quadrature rule error, we employ a two-dimensional problem to verify the observed and theoretical convergence orders for the region-averaged uncollided and fully collided scalar flux errors. Numerical results show that the angular discretization errors in the region-averaged scalar flux obtained by different quadrature types converge with different rates, that are commensurate with the regularity order of the exact angular flux within the quadrature-designed integration interval. The angular discretization error in the uncollided region-averaged scalar flux obtained by LC class quadratures converges linearly, and the error obtained by QR class quadrature sets converges quadratically. The angular discretization error in the fully collided region-averaged scalar flux converges linearly for LC class quadratures, and faster-than-second order for QR class quadrature sets.

Sensor Design Migration: The Case of a VRG

Vibrating ring gyroscopes (VRG) have been shown to achieve a high scale-factor with low non-linearity while some structural modification can help increase their shock resistance in space applications. With some tuning, their quadrature error can be nearly eliminated, thus, minimizing device sensitivity to external factors while keeping a good alignment between the drive and sense-mode frequencies. VRG design is optimized for high scale-factor, low non-linearity, and high shock-resistance while satisfying process related constraints. When the device wafer is processed independent of the capping wafers, the process imposes topology constraints on the device trenches and forces anchors and other stator features to be on the outside of the device. In this paper, we have shown how a VRG that has been designed with a radially outwards topology, having its anchor in the center as the innermost component, can be reconfigured into a VRG with a radially inwards topology with its anchors at the rim as the outermost component. This design transformation can be construed as a VRG design migration from a three-wafer process to a single-wafer process. The process-driven reconfiguration comes at the cost of a lower scale-factor but with no impact on the operating range of the original design. The paper further proposes the use of directional masses to achieve better inertial modal control of the VRG. The resulting radially inwards VRG has been fabricated in the standard SOIMUMPS process, and preliminary experimental results have shown the functionality of the device under atmospheric pressure.

Verification and validation of large eddy simulation with Nek5000 for cold leg mixing benchmark

Large eddy simulation (LES) of an experimental benchmark investigating buoyant mixing among two fluids is developed for verification and validation. Flow physics involve a completely closed system where after a valve is opened, two initially stationary fluids with different densities are allowed to mix. The Boussinesq approximation and low-Mach equations are coded into spectral element code Nek5000 to predict the buoyancy forces, and miscibility is accounted for using the advection-diffusion equation for concentration. Due to computational costs of higher order solutions, current analysis consists of initial flow development after the valve is opened. Minor differences between the models and Schmidt number are found within the turbulent spectrum and time integrated quantities. Moreover, solutions for higher polynomial orders are compared to quantify error due to spatial discretization. Error estimates using a posteriori spectral derivation of truncation and quadrature error are also calculated. Validation with particle image velocimetry (PIV) reveals good prediction of both average and fluctuating components of the primary flow within the cold leg but less accurate results in the downcomer. It is concluded that instabilities due to resolving more of the interface using the high molecular Schmidt number is essential for accurately predicting this type of flow.

On the determination of the transfer coefficients in the heat and mass transfer equations, based on the Boltzmann equation

At present, as a rule, the determination of the transfer characteristics of viscosity, diffusion, heat conduction and other characteristics in macro-equations of aerohydromechanics is carried out experimentally or on the basis of the kinetic theory of gases - the Boltzmann equation. The right “collisional” part of the Boltzmann equation contains multiple improper integrals with an oscillating integrand function, the values of which are used in determining transfer coefficients. This article proposes numerical algorithms suitable for calculating integrals of this type. They are based on the use of the Gauss - Christoffel quadrature formula that applies orthogonal polynomials. The oscillation nature of the integrand requires control of the quadrature error. In this case, the split nodes of the integration interval are distributed on the basis of minimizing the error of approximation of the integrand by orthogonal polynomials. Such an approach, in contrast to others, allows determining the values of integrals with oscillating integrand functions more qualitatively and efficiently and does not contain an “iterative” procedure for increasing the split nodes of the integration segment to achieve a given accuracy. This article is devoted to the problem of constructing quadrature formulas for calculating improper integrals, which are used in determining the transfer coefficients, based on the Boltzmann equation.

A Mixed Eulerian–Lagrangian Spectral Element Method for Nonlinear Wave Interaction with Fixed Structures

We present a high-order nodal spectral element method for the two-dimensional simulation of nonlinear water waves. The model is based on the mixed Eulerian–Lagrangian (MEL) method. Wave interaction with fixed truncated structures is handled using unstructured meshes consisting of high-order iso-parametric quadrilateral/triangular elements to represent the body surfaces as well as the free surface elevation. A numerical eigenvalue analysis highlights that using a thin top layer of quadrilateral elements circumvents the general instability problem associated with the use of asymmetric mesh topology. We demonstrate how to obtain a robust MEL scheme for highly nonlinear waves using an efficient combination of (i) global $$L^2$$ projection without quadrature errors, (ii) mild modal filtering and (iii) a combination of local and global re-meshing techniques. Numerical experiments for strongly nonlinear waves are presented. The experiments demonstrate that the spectral element model provides excellent accuracy in prediction of nonlinear and dispersive wave propagation. The model is also shown to accurately capture the interaction between solitary waves and fixed submerged and surface-piercing bodies. The wave motion and the wave-induced loads compare well to experimental and computational results from the literature.

A posteriori error control for the finite cell method

AbstractThe paper presents some concepts of the finite cell method and discusses a posteriori error control for this approach. The focus is on the application of the dual weighted residual approach (DWR), which enables the control of the error with respect to a user‐defined quantity of interest. Since both the discretization error and the quadrature error are estimated, the application of the DWR approach provides an adaptive strategy which equilibrates the error contributions resulting from discretization and quadrature. The strategy consists in refining either the finite cell mesh or its associated quadrature mesh. Numerical experiments confirm the performance of the error control and the adaptive scheme for a non‐linear problem in 2D.

A 37.5–45 GHz Superharmonic-Coupled QVCO With Tunable Phase Accuracy in 28 nm CMOS

This paper presents a new superharmonic coupling structure for millimeter-wave (mmW) quadrature voltage-controlled oscillators (QVCOs) with high phase accuracy and zero voltage headroom. The proposed coupling network can adjust the quadrature error with almost no phase noise penalty. A tail impedance theory for superharmonically coupled oscillators is introduced to analyze the locking mechanism, phase noise, phase accuracy, and phase tunability of the proposed superharmonic QVCO. For verification, a 40 GHz QVCO is fabricated in a 28 nm bulk CMOS process, which exhibits a phase noise of −94.32 dBc/Hz at 1 MHz offset and 0.18° quadrature error with 8.4 mW power consumption and 18.4% frequency tuning range.

Total error compensation of non-ideal signal parameters for Moiré encoders

Non-ideal encoder signal parameters generally include amplitude imbalance error, quadrature phase error, dc-offset error and harmonic distortion. These non-ideal signal parameters are the main cause of nonlinear interpolation errors for Moiré encoders. In this paper, a total error compensation strategy for correcting these non-ideal signal parameters is presented. The compensation strategy consists of a passive compensation stage and an active compensation stage, which are realized through hardware and software means, respectively. In the passive compensation stage, an optimized scanning grating with specified micro-pattern is employed to suppress the harmonic distortion. In the active compensation stage, an adaptive compensation algorithm is proposed to compensate the remained amplitude imbalance error, quadrature phase error and dc-offset error. In the experiment, a Moiré encoder using the optimized scanning grating is developed, and a digital circuit is designed by implementing the adaptive compensation algorithm into a field-programmable gate array (FPGA). The effectiveness of the total error compensation strategy has been validated by experimental results.

A Compact Digitally-Assisted Merged LNA Vector Modulator Using Coupled Resonators for Integrated Beamforming Transceivers

This paper presents a compact differential quadrature generation scheme using transformer coupled resonators for use in scalable fifth-generation (5G) phased-array architectures. A merged low-noise amplifier vector modulator (LNA-VM) architecture is introduced where quadrature generation, conventionally implemented using an explicit quadrature hybrid, is now seamlessly integrated with the load of an LNA gain stage facilitating a compact design. The outputs of a low- $Q$ coupled-resonator load in the LNA is used for quadrature generation, and Cartesian combining is implemented by phase-invariant programmable gain amplifiers (PGAs) with constant input capacitance and output conductance across phase settings—this ensures low rms gain and phase errors. A 25.1–27.6-GHz phased array channel employing the proposed scheme is implemented in 65-nm CMOS. The merged LNA-VM is digitally tunable across 2.5 GHz (25.1–27.6 GHz) and achieves rms gain and phase errors (measured at band edges) better than 0.42 dB (0.64 dB) and 4.5° (6.9°), respectively, for a bandwidth of 500 MHz (700 MHz). It occupies a composite area of only 0.123 mm2 while consuming 30.5 mW from a 1-V supply. To achieve quadrature locking at the desired frequency within the LNA-VM’s tuning range, an on-chip mixer-based quadrature error detection and calibration scheme is presented. The proposed scheme is demonstrated to be robust in the face of gain and layout-induced mismatches which can be significant at millimeter-wave (mm-wave) frequencies.

Effect of circuit phase delay on bias stability of MEMS gyroscope under force rebalance detection and self-compensation method

Bias stability is an important performance indicator for MEMS gyroscopes. In this paper, a gyroscope zero rate output (ZRO) model under force-to-rebalance (FTR) closed-loop detection is presented, and the effect of circuit phase delay and various noises on the ZRO is analyzed. Based on the fact that the two feedback forces in the FTR system are insensitive to the phase delay of the sense mode, while they are sensitive to the phase delay of the drive mode, a method for quickly calculating the circuit phase delay is proposed, and an all-pass filter is used to realize one-time automatic compensation for the phase delay in the drive mode. The control system is implemented with an FPGA. The experimental results show that this method can be used to accurately calculate the circuit phase delay and that phase compensation can be used to effectively reduce the effect of quadrature error on ZRO. This technique provides bias instability and angle random walk performances of 0.338° h−1 and 0.061 ° (√h)−1, respectively. In addition, the sensitivity of ZRO to temperature at 0 °C to 70 °C has reached 0.001°/s/°C.

Dual weighted residual error estimation for the finite cell method

Abstract The paper presents a goal-oriented error control based on the dual weighted residual method (DWR) for the finite cell method (FCM), which is characterized by an enclosing domain covering the domain of the problem. The error identity derived by the DWR method allows for a combined treatment of the discretization and quadrature error introduced by the FCM. We present an adaptive strategy with the aim to balance these two error contributions. Its performance is demonstrated for several two-dimensional examples.

Filtered Discrete Ordinates Equations for Radiative Transport

We present and analyze a discrete ordinates ( $$\text {S}_N$$ ) discretization of a filtered radiative transport equation (RTE). Under certain conditions, $$\text {S}_N$$ discretizations of the standard RTE create numeric artifacts, known as “ray-effects”; the goal of the filter is to remove such artifacts. We analyze convergence of the filtered discrete ordinates solution to the solution of the non-filtered RTE, taking into account the effect of the filter as well as the usual quadrature and truncation errors that arise in discretize ordinate methods. We solve the filtered $$\text {S}_N$$ equations numerically with a discontinuous Galerkin spatial discretization and implicit time stepping. The form of the filter enables the resulting linear systems to be solved in an established Krylov framework. We demonstrate, via the simulation of two benchmark problems, the effectiveness of the filtering approach in reducing ray effects. In addition, we also examine efficiency of the method, in particular the balance between improved accuracy and additional cost of including the filter.

A Reconfigurable 28-/37-GHz MMSE-Adaptive Hybrid-Beamforming Receiver for Carrier Aggregation and Multi-Standard MIMO Communication

This paper presents a multi-standard fully connected hybrid beamforming (FC-HBF) receiver that can be reconfigured between two fully connected two-stream multi-input-multi-output (MIMO) modes at 28- or 37-GHz band, and an inter-band carrier-aggregation (CA) mode enabling concurrent single-stream operation at 28 and 37 GHz. A new image-reject (IR) heterodyne beamforming architecture is introduced that facilitates easy reconfiguration. Concurrent dual-band operation of the beamformer front end is achieved by using an inherently wideband Cartesian-combining-based RF-domain complex-weighting architecture, which is implemented using coupled-resonator-based dual-band gain stages, and current-mode dual-band active combiners. The downconversion stage comprises a cascade of complex-quadrature mixing stages that combines Cartesian complex-weighting operation with image rejection. A new quadrature error detection and calibration scheme is also introduced to improve the image rejection of the proposed architecture. A minimum mean-square error (MMSE) beam adaptation technique, introduced for the first time in an RF or hybrid beamformer, breaks the need for individual access to the beamformer inputs needed in a traditional least-mean-square (LMS) scheme and allows both main lobe and null adaptation. A 28-/37-GHz hybrid beamforming receiver with four antenna inputs and two baseband output streams is designed in 65-nm CMOS that demonstrates the above features. In each antenna path, the receiver achieves 33-dB (26.5 dB) peak conversion gain, 2.75-GHz (3.75 GHz) RF bandwidth, and 5.7-dB (8.5 dB) NF at 28 GHz (37 GHz). The entire receiver consumes 310 mW and occupies 2.2 mm2 core area. Extensive characterization results for single-element and several multi-element scenarios are presented including concurrent dual-band receiving (>35-dB image rejection), null-steering (>25-dB peak to null for two elements), and multi-element CA (~50-dB inter-carrier interference rejection).

Spectral Analysis of Quadrature Rules and Fourier Truncation-Based Methods Applied to Shading Integrals.

We propose a theoretical framework, based on the theory of Sobolev spaces, that allows for a comprehensive analysis of quadrature rules for integration over the sphere. We apply this framework to the case of shading integrals in order to predict and analyze the performances of quadrature methods. We show that the spectral distribution of the quadrature error depends not only on the samples set size, distribution and weights, but also on the BRDF and the integrand smoothness. The proposed spectral analysis of quadrature error allows for a better understanding of how the above different factors interact. We also extend our analysis to the case of Fourier truncation-based techniques applied to the shading integral, so as to find the smallest spherical/hemispherical harmonics degree L (truncation) that entails a targeted integration error. This application is very beneficial to global illumination methods such as Precomputed Radiance Transfer and Radiance Caching. Finally, our proposed framework is the first to allow a direct theoretical comparison between quadrature- and truncation-based methods applied to the shading integral. This enables, for example, to determine the spherical harmonics degree L which corresponds to a quadrature-based integration with N samples. Our theoretical findings are validated by a set of rendering experiments.