Linearity optimization using dithering in a 1 st order thermal ΣΔ modulator O. Leman, F. Mailly, L. Latorre and P. Nouet LIRMM, University of Montpellier, UMR CNRS 5506, Montpellier, France - contact: mailly@lirmm.fr Abstract - This work investigates the design of first-order thermal ΣΔ modulators for calorimetric sensor applications. Considering the static resolution of the modulator, a set of equations is proposed for the calculation of system parameters. This work was achieved in the context of a convective accelerometer, and this particular sensor is used here as a case- study. We show that the modulator parameters can be optimized so that the sensor noise offers tuned dithering effect that provides deadbands elimination and good modulator linearity. I. INTRODUCTION Following the “more than Moore” direction, interest for silicon embedded sensors is continually growing. MEMS sensors have found applications in many fields: automotive, mobile devices or entertainment to mention few. In most cases, the signal conditioning of the sensor is a key issue, from which depends the system performance in terms of resolution, linearity, bandwidth and power consumption. This paper deals with the design of a Sigma-Delta (ΣΔ) modulator which has been widely considered in sensing applications due to its intrinsic ability to achieve high SNR with low power requirements while providing a digital output [1]. When dealing with thermal sensors, an interesting approach consists in implementing the loop filter of the modulator in the thermal domain [2]. The low-pass filter inherent to the thermal phenomena is then replacing the traditional integrating function and a 1 st order Σ-Δ modulator is easily obtained. A major drawback of such low order modulator is the linearity error, even for small input variations, due to deadbands. In this paper, we present an original and pragmatic approach to easily and comprehensively tune the clock frequency of a 1 st order thermal Σ-Δ modulator. In the first section, we present the device under study and its associated model. The modulator architecture is then presented and modeled (section III and IV). After a semi-empirical analysis of dithering and its effect on linearity (section V), a simple formulation is established to easily calculate the relationship between full scale and clock frequency. II. SENSOR DESIGN AND MODELING Fig. 1 is a picture of a convective accelerometer and its electrical schematic. This sensor was built with a standard CMOS technology. The three suspended bridges are composed of back-end layers of the CMOS process and they were released with a post-process of anisotropic bulk etching. The hermetic sensor package is filled with atmospheric air. Each suspended bridge is 5 μm thick and features a polysilicon resistor, for heating (R H ) and temperature sensing (R d1 and R d2 ) purposes. The value of these polysilicon resistors as a function of temperature T (in K) expresses: R = R 0 ⋅(1+TCR⋅(T-T 0 )) (Ω) (1) Where TCR = 9⋅10 -4 K -1 and T 0 = 300K. Sensor R ref2 R ref1 R H R d2 R d1 V dd V out Heater R H Detectors R d1 , R d2 x 500μm Sensor R ref2 R ref1 R H R d2 R d1 V dd V out Sensor R ref2 R ref1 R H R d2 R d1 V dd V out Heater R H Detectors R d1 , R d2 x 500μm Fig. 1: Micrograph of a thermal convective accelerometer and electrical schematic, heater dissipation is 21mW The central bridge locally heats the air by Joule’s dissipation and heating expands the air thus creating a gradient of air density. Acceleration along the sensing axis x thus induces buoyancy force in the hot air bubble which creates free convection. This produces a heat transfer that heats one detector and cools the other one. Differential Wheatstone bridge output V out is thus proportional to acceleration along the x-axis. More details about sensor operating principles and dimensions can be found in [3]. Fig. 2 shows the small-signal, differential model of the sensor. Convection produces a differential heat transfer ΔP D between detectors (S p =2.67μW/g), and this phenomenon is characterized with a τ=2.5ms time constant. Detector bridges thermal parameters are R th =25800K/W and τ D ≈1.87ms. Wheatstone bridge sensitivity is S wheat =1mV/K with V dd =5V. Each resistor generates a noise, 4⋅k b ⋅T⋅R, and thus the sensor output noise has a spectral density η i = 32.8nV/√Hz. S Wheat + + s S p ⋅ + τ 1 a (g) ΔP D (W) ΔT D ’ (K) s R D th ⋅ + τ 1 V out Wheatstone bridge convection suspended temperature detectors (V) 4⋅k b ⋅T⋅R noise: η i 4⋅k b ⋅T⋅R noise: η i S Wheat + + + + s S p ⋅ + τ 1 a (g) ΔP D (W) ΔT D ’ (K) s R D th ⋅ + τ 1 V out Wheatstone bridge convection convection suspended temperature detectors suspended temperature detectors (V) 4⋅k b ⋅T⋅R noise: η i 4⋅k b ⋅T⋅R noise: η i Fig. 2: Thermal convective accelerometer model