Abs+act― Two second-order bandpass delta-sigma A/D modulators have been implemented in a 0.8 μ BiCMOS process to demonstrate the feasibility of converting a 10.7 MHz radio IF signal to digital form. The circuits, based on switched-capacitor biquads, demonstrated 57 dB SNR in a 200 kHz bandwidth when clocked at 42.8 MHz, dissipating 60 mW from a 5 V supply. The two modulators use different clocking strategies to allow evaluation of a tradeoff between active and passive sensitivites.

Switched-capacitor bandpass delta-sigma A/D modulation at 10.7 MHz

Analog-to-digital converters based on sigma–delta modulators ( $\Sigma \Delta {\mathrm{ M}}$ ) are a popular choice for high resolution conversion from the analog to the digital domain. With relatively small modifications, they can also be used as electromechanical $\Sigma \Delta {\mathrm{ M}}$ (EM- $\Sigma \Delta {\mathrm{ M}}$ ) force feedback interfaces for capacitive micro-electromechanical systems (MEMSs) inertial sensors. Such interfaces are able to combine the benefits of force feedback and analog-to-digital conversion at relatively modest circuit cost. This paper provides a comprehensive review of EM- $\Sigma \Delta {\mathrm{ M}}$ interfaces for capacitive MEMS inertial sensors. The principle and the design methodology of EM- $\Sigma \Delta {\mathrm{ M}}$ interfaces are introduced. A classification of EM- $\Sigma \Delta {\mathrm{ M}}$ accelerometers and gyroscopes is presented, and a detailed analysis of different EM- $\Sigma \Delta {\mathrm{ M}}$ architectures is given. The most representative EM- $\Sigma \Delta {\mathrm{ M}}$ inertial sensors systems are discussed and compared with regard to their performance characteristics. In particular, the properties of various discrete and continuous-time techniques and a system parameter optimization methodology are illustrated through specific examples. Finally, current challenges and future development trends of EM- $\Sigma \Delta {\mathrm{ M}}$ interfaces for inertial sensors have been identified.

/pdf/electromechanical-sigma-delta-modulators-sigma-delta-mathrm-17imq1xsr2.pdf

Electromechanical Sigma–Delta Modulators ( $\Sigma \Delta {\mathrm{ M}}$ ) Force Feedback Interfaces for Capacitive MEMS Inertial Sensors: A Review

This paper endeavors to investigate the output-feedback sliding mode control (SMC) issue of the networked singularly perturbed systems (SPSs) under fast sampling. In order to improve the reliability of network communication, a redundant channel transmission protocol is introduced in the SMC design. Based on the measurement outputs, a sliding function is constructed with the consideration of the transmission protocol. With the aid of some appropriate Lyapunov functions, the sufficient conditions are derived to ensure the mean-square exponentially ultimately boundedness of the sliding mode dynamics and the reachability of the specified sliding surface. Moreover, a convex optimization algorithm is formulated to solve the output-feedback SMC law by searching the available upper bound of the singularly perturbed parameter. Finally, an operational amplifier circuit is exploited to explore the influences from the redundant channel transmission protocol to the output-feedback SMC performance and the estimated $\varepsilon$ -bound.

/pdf/reliable-sliding-mode-control-of-fast-sampling-singularly-3s3mde3amq.pdf

Reliable Sliding Mode Control of Fast Sampling Singularly Perturbed Systems: A Redundant Channel Transmission Protocol Approach

This paper presents a design method of force rebalance control for the sense mode of a micromachined vibratory gyroscope with multiobjective. The control strategy is mainly based on constraining sensitivity margin specifications via numerical optimization approach, which embeds the sense mode in a closed-loop system to possess more robust performance. This paper also provides a quantitative methodology of configuring the system parameters to realize the functional electrostatic force feedback control for the microgyroscope in the case of nonideal coupling and external angular disturbance. Theoretical results predicted by the control loop are shown to be in close agreement with the experimental results using a practical microgyroscope, which indicated a satisfactory performance of the proposed control algorithm. The maximums of the sensitivity function and complementary sensitivity function are measured to be 4.5 and 1 dB, resulting in GM ≥ 7.84 dB and PM ≥ 35°. The gyroscope achieves a scale factor of 7.1 mV/deg/s with nonlinearity of 0.2% which is improved by one order of magnitude compared with that of the open loop. The bandwidth is extended to about 98 Hz from 30 Hz in the open loop. The quadrature error and temperature stability of the scale factor are reduced from - 10.33 to -59.22 dB and from 4858 to 646 ppm/°C with the force rebalance control, respectively.

Force Rebalance Controller Synthesis for a Micromachined Vibratory Gyroscope Based on Sensitivity Margin Specifications

This paper describes the development and experimental evaluation of a microelectromechanical system vibratory gyroscope using an optimized double closed-loop control strategy. An automatic gain control self-oscillation interface is used to resonate the gyroscope in the drive mode; the sense mode is controlled by a sixth-order continuous-time and force-feedback band-pass sigma-delta modulator. The parameters of both control loops are optimized by a genetic algorithm (GA). System level simulations show that the settling time of the drive mode self-oscillation is 125 ms, the root mean square displacement of the proof mass is in the sense mode, and the signal-to-noise ratio is 90 dB in a bandwidth of 64 Hz with a 200 °/s angular rate input signal. The system is implemented using symmetrical and fully decoupled silicon on insulator gyroscope operating at atmospheric with the circuit implemented on printed circuit board. The measured power spectral density of the output bitstream shows an obvious band-pass noise shaping and a deep notch at the gyroscope resonant frequency. The measured noise floor is approximately -120 dBV/Hz1/2. In the drive mode, the relative drift of the resonant frequency and amplitude is 3.2 and 10.7 ppm for 1 h measurements, respectively. The settling time, scale factor, zero bias stability, and bandwidth of the gyroscope controlled by the optimized control system are 200 ms, 22.5 mV/°/s, 34 °/h, and 110 Hz, respectively. This is compared with a non-optimized system for which the corresponding values are 300 ms, 17.3 mV/°/s, 58 °/h, and 98 Hz; hence, by GA optimization a considerable performance improvement is achieved.

/pdf/design-and-implementation-of-an-optimized-double-closed-loop-579nqcm8a8.pdf

Design and Implementation of an Optimized Double Closed-Loop Control System for MEMS Vibratory Gyroscope

The present work that exists on predicting the stability of Delta-Sigma modulators is confined to DC input signals and unity quantizer gains. This poses a limitation for numerous Delta-Sigma modulator applications. The proposed research work gives the stability curves for DC, sine, and dual sinusoidal inputs for any value of the quantizer gain. The maximum stable input limits for third-, fourth-, and fifth-order Chebyshev-Type-II-based Delta-Sigma modulators are established using the describing-function method for DC and sinusoidal inputs. Closed-form mathematical expressions for the gains of the quantizer for higher order Delta-Sigma modulators whose inputs are two concurrent sinusoids are derived from first principles. The derived stability curves are shown to agree reasonably well with the simulation results for different types of input signals and amplitudes.

/pdf/nonlinear-stability-analysis-of-higher-order-delta-sigma-1159ziwy4o.pdf

Nonlinear-Stability Analysis of Higher Order $\Delta$ – $\Sigma$ Modulators for DC and Sinusoidal Inputs

The current approaches on predicting the stability of delta-sigma (Δ-Σ) modulators are mostly confined to dc inputs. This poses limitations as practical applications of Δ-Σ modulators involve a wide range of signals other than dc. In this paper, a quasi-linear model for Δ-Σ modulators with nonlinear feedback control analysis is presented, which accurately predicts the stability of higher-order single-loop 1-bit Δ-Σ modulators for various types of input signals, such as single sinusoids, dual sinusoids, multiple sinusoids, and Gaussian. Theoretical values are shown to match closely with simulation results. The results of this paper would significantly speed up the design and evaluation of higher-order single-loop 1-bit Δ-Σ modulators for various applications, including those that may require multiple-sinusoidal inputs or any general input composed of a finite number of sinusoidal components, circumventing the need to perform detailed time-consuming simulations to quantify stability limits. By using the proposed method, the difference between the predicted and the actual stable amplitude limits results in an error of less than 1 dB in the in-band signal-to-noise ratio for the third-order and higher-order Δ-Σ modulators for single-sinusoidal inputs. For single-sinusoidal, dual-sinusoidal, multiple-sinusoidal, and Gaussian inputs, the error is less than 2 dB for the fifth-order modulator and reduces to less than 1 dB for the sixth-order and higher-order Δ-Σ modulators.

/pdf/nonlinear-model-based-approach-for-accurate-stability-277c4id0rj.pdf

Nonlinear Model-Based Approach for Accurate Stability Prediction of One-Bit Higher-Order Delta–Sigma Modulators

In this paper, the stability boundaries of higher order 1-bit bandpass (BP) Σ-Δ modulators for single- and dual-sinusoidal inputs are determined using quasi-linear modeling together with the describing-function method. Closed-form mathematical expressions are derived for the stability curves for different quantizer gain values. The stability prediction for the maximum stable input limits for eighth- and tenth-order (i.e., effective fourth- and fifth-order) inverse-Chebyshev- and Butterworth-based BP Σ-Δ modulators for different stopband attenuation and bandwidths are established. The theoretical results are shown to be in good agreement with the simulation results for a variety of input amplitudes, bandwidths, and modulator orders. The proposed mathematical models of the quantizer will immensely help speed up the design of higher order BP Σ-Δ modulators.

Stability Analysis of Bandpass Sigma–Delta Modulators for Single and Dual Tone Sinusoidal Inputs

The paper presents simulation results from investigating the behaviour of multistage (MASH) oversampled bandpass sigma-delta (Σ-Δ) modulators for use in analogue to digital converters for high frequency narrowband applications such as the signals out of the intermediate frequency (IF) section of a superheterodyne radio receiver. The bandpass configurations under consideration have in their loop filter a cascade of second-order resonator structures in order to achieve acceptable noise shaping. The quantisation noise in each stage is suppressed by feeding the error of each section into the input of the following stages. It is demonstrated that the triple effective-first-order bandpass MASH structure has significantly better performance compared with the effective-second-order effective-first-order bandpass MASH structure.

MASH structures for bandpass sigma-delta modulators

This paper proposes a novel algorithm that can be integrated with various design and evaluation tools, to more accurately and rapidly predict stability in multibit delta-sigma (Δ-Σ) modulators. Analytical expressions using the nonlinear gains from the concept of modified nonlinearity in control theory are incorporated into the mathematical model of multibit Δ-Σ modulators to predict the stable amplitude limits for sinusoidal input signals. The nonlinear gains lead to a set of equations, which can numerically estimate the quantizer gain as a function of the input sinusoidal signal amplitude. This method is shown to accurately predict the stable amplitude limits of sinusoids for second-sixth-order three- and five-level midtread quantizer-based Δ-Σ modulators. The algorithm is simple to apply and can be extended to midrise quantizers or to any number of quantizer levels. The only required input parameters for this algorithm are the number of quantizer levels and the coefficients of the noise transfer function.

/pdf/nonlinear-stability-prediction-of-multibit-delta-sigma-48t21ym30i.pdf

M. Al-Janabi

Papers

Nonlinear-Stability Analysis of Higher Order $\Delta$ – $\Sigma$ Modulators for DC and Sinusoidal Inputs

Nonlinear Model-Based Approach for Accurate Stability Prediction of One-Bit Higher-Order Delta–Sigma Modulators

Stability Analysis of Bandpass Sigma–Delta Modulators for Single and Dual Tone Sinusoidal Inputs

MASH structures for bandpass sigma-delta modulators

Nonlinear Stability Prediction of Multibit Delta–Sigma Modulators for Sinusoidal Inputs