# Dynamics-Level Design for Discrete- and Continuous-Time ... Sigma-Delta-Modulators for...

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Dynamics-level Design for Discrete- and Continuous-Time Band-passSigma-Delta-Modulators for Micro-machined Accelerometers

Luo, J. Rapisarda, P. Kraft, M.

Abstract High-performance micro-electro-mechanical sys-tems (MEMS) sensors can be implemented by incorporatinga micro-machined capacitive sensing element in a Sigma-Delta-Modulators (M ) force-feedback loop, forming an electro-mechanical M (EM M ). We propose a transfer-function based design methodology to realize discrete- andcontinuous-time low-pass electro-mechanical M systems.The design is performed at the level of the integrated systemconsisting of the electro-mechanical sensor and of the electroniccircuit; we call this the dynamics-level. We also illustratea technique to perform the conversion of the discrete-timedesign to a continuous-time one. The approach is demonstratedthrough an electro-mechanical M design example for a bulkmicro-machined, capacitive accelerometer.

I. INTRODUCTIONMicro-machined accelerometers are widely used in air-

bag systems in vehicles, in vibration measurement, in inertialguidance, and many other applications. Most modern micro-machined sensing elements consist of a proof mass which isdeflected due to the inertial force, resulting in a differentialcapacitive signal which is then amplified. Since the differen-tial changes in the capacitive signal are very small, a high-resolution interface circuit becomes of primary importance indesign. Equally important is the incorporation of the sensingelement in a force-feedback control scheme so as to increasethe linearity, dynamic range and bandwidth of the sensor,and to reduce sensitivity to fabrication imperfections and sus-ceptibility to environmental parameter changes. Compared toanalogue force feedback control strategies, closed-loop force-feedback control schemes based on Sigma-Delta modulators(M in the following) have many advantages [1]; inparticular, they produce a direct digital output which canbe directly processed further by a digital signal processor. Insuch a scheme, the micromachined sensing element is part ofthe control loop to form an electro-mechanical M . Thedesign problem is to determine parameters for the controlloop that guarantee the stability of the overall system and agood performance in terms of signal-to-noise ratio.

Initially, in order to design electro-mechanical M stan-dard approaches developed in the context of purely electronicM analog-digital convertors (ADC) were used, see [2],[3]. However, electro-mechanical M are fundamentallydifferent from purely electronic ones, since integrating themicro-machined sensing element without losing loop stabil-ity and performance introduces additional difficulties in the

J. Luo, P. Rapisarda, H. Ding and M. Kraft are with the Schoolof Electronics and Computer Science, University of Southampton,Southampton, UK. SO17 1BJ.E-mail: luoj@soton.ac.uk, pr3@ecs.soton.ac.uk andmk1@ecs.soton.ac.uk

design. Since there is no exact theoretical analysis methoddue to the non-linearity of the quantizer, and since fab-rication tolerances and environmental conditions introduceuncertainties, designs are often validated through systemlevel simulations. This approach is rather inefficient, sincea large number of different scenarios must be simulated andanalyzed before sufficient confidence can be established toadopt a given design. Moreover, often the optimization ofthe parameters is performed directly on a specific circuitchosen for the implementation, and the resulting designs aretopology-dependent.

Recently in [4], [5], an unconstrained architecture (alsocalled topology in the following) has been proposed forthe case of a MEMS gyroscope. The topology is calledunconstrained since it has the necessary degrees of free-dom to accommodate an arbitrary second order dynamicalmodel. This approach constitutes a major inspiration for theresults presented in this paper; we briefly review its salientfeatures. In [4], [5], the need for system-level simulationsis significantly reduced by the adoption of a filter-leveldesign, of an elegant analytical framework, and of a set ofclear-cut design guidelines. The methodology is developedessentially for a second-order model for the sensing element,and for a specific system topology; this makes the mappingbetween the designed QNTF to the circuit rather straightfor-ward. However, the adoption of higher-order models (oftennecessary to model the complex dynamics of the sensingelement) requires using different topologies, which are notpresented in [5]. Another aspect not discussed explicitlyin [5] is the conversion of the design from discrete- tocontinuous-time.

In this paper we address both these issues. We distinguishthree phases in the design of M , namely:

dynamics-level design: designing the QNTF and thesignal transfer function at the system-level, i.e. with thesensing element integrated in the loop filter;

topology-level design: synthesizing the loop filter into adesired circuit topology;

circuit-level design: circuit behavior and layout design.

We concentrate on dynamics-level design, taking as start-ing point the work illustrated in [6]; standard techniquesfor design at the topology- and circuit-level for purelyelectronic M (see for example [7], [8], [9], [10], [11])can be used for the other two phases. This approach entailstwo advantages. Firstly there is the decoupling of design(performed in the first phase) and implementation (performedin the last two phases and aimed at solving issues specific to

Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems MTNS 2010 59 July, 2010 Budapest, Hungary

ISBN 978-963-311-370-7 735

the application). Secondly, in the design phase it is easy toaccommodate higher-order models for the sensing elementgiven that the computations are transfer-function based.

At the dynamics level, we design the QNTF of a discrete-time M under constraints that specifically model theintegration of the sensing element in the system, as in theapproach of [5], [12]. After converting the QNTF into anopen-loop filter, the splitting of the sensing element and theelectronic part of this filter is carried out. In this phase weonly use transfer function manipulations, without restrictingourselves to an a priori specified topology.

In this paper we also consider the problem of how toconvert a discrete-time design to an equivalent continuous-time one at the dynamics-level. A direct conversion does notpreserve the separation of the mechanical and the electronicpart; consequently, we introduce a mixed-feedback topologyincluding a signal- and a compensation path in order toaccommodate the dynamics of the sensing element, andthen convert separately the resulting transfer functions tocontinuous-time through standard techniques.

This paper is organized as follows. In section II, a system-level analysis is carried out for the discrete-time case toestablish a relationship between the quantization noise trans-fer function and the open-loop system transfer function. Weproceed to illustrate our design procedure and to exemplifyit with a fourth-order example. In section III, the conversionfrom a discrete-time electro-mechanical M design to anequivalent continuous-time system is discussed. In the finalsection of the paper, conclusions are drawn and some currentand future research work is outlined.

II. TRANSFER-FUNCTION-BASEDDISCRETE-TIME DESIGN

In this section we present a design procedure for single-loop EM-M systems, which are often chosen for elec-tromechanical applications (see [2]) due to their high toler-ance to parameters variations.

A. System analysis of single-loop electro-mechanical M

A simplified single-feedback electro-mechanical M isshown in Fig. 1, including the sensing element with transferfunction Gs(s), the electronic interface and filter circuit partwith transfer function Hc(s), the feedback-force block kfband the quantizer.

Fig. 1. Block diagram of electro-mechanical M system

This simplified model neglects several factors affectingthe dynamics of a real-world electro-mechanical system, forexample nonlinearities arising from non-ideal components,

from additional electrostatic force components, and fromthe feedback voltage to electro-static force conversion; andnon-constant damping and spring coefficients. However, inthe simulations used for validating the design techniqueproposed in this paper we do consider several of these non-ideal effects. It is also important to note that although thethe sensor element Gs in Fig. 1 is described by a second-order transfer function, our methodology can be used withoutmodifications for higher-order models.

In the second-order case the dynamics of the sensingelement are described in continuous-time as:

Gs(s) =1

ms2 + bs+ k, (1)

where m is the proof mass (measured in Kg), b is thedamping coefficient (measured in N/m/s), and k is thestiffness constant (measured in N/m). As shown in [5],assuming a zeroth order DAC, the transfer function (1) hasa discrete-time domain equivalent

Gs(z) = ksz bs

(z as)(z as), (2)

for suitable values of the constants as (computed from thecontinuous-time poles of (1) via the usual continuous-to-discrete-time mapping formula), ks, and bs. Of particularrelevance for the following is the presence in (2) of the zerobs, which is absent in the continuous-time model.

Quasi-linear models have proved to be effective in dealingwith the nonlinearity of M in many successfully Mapplications, see [13], [1], [5]. In these models the nonlinear-ity of the one-bit quantizer is replaced with a gain kq and aquantisation error eQ (see section III of [14]). Consequently,a linearized model for the discrete-time QNTF, denoted withQNTF(z), can be written as:

QNTF (z) =1

1Gs(z)Hc(z)k

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