Time Divided Architecture for Closed Loop MEMS Capacitive

Accelerometer

Jingqing Huang, Tingting Zhang, Meng Zhao, Lichen Hong,

Yacong Zhang*, Wengao Lu, Zhongjian Chen, Yilong Hao

Key Laboratory of Microelectronic Devices and Circuits

Department of Microelectronics, Peking University, Beijing, 100871, PRC

* Email: zhangyc@pku.edu.cn

Abstract

This paper mainly discusses issues concerning the

architecture of time divided closed loop accelerometer.

For this particular architecture mathematical relationship

between the external acceleration detected by sensor and

the voltage output of the readout circuits is deduced.

Both Matlab/Simulink model and Verilog-A model for

such architecture are established. Simulation results

agree with the mathematical formula. Readout circuits

designed to work under 50kHz with feedback duty cycle

η being 60% are fabricated using 0.35μm HV CMOS

process. Test results show a sensitivity of 1.518V/g.

1. Introduction

Microelectromechanical system (MEMS) capacitive

accelerometers, owing to their advantages such as small

size, low power consumption and compatibility with

CMOS circuits, have gained a lot of attention. In order to

achieve better linearity, higher resolution and wider

signal bandwidth, closed loop architecture is often

applied. For the implementation of closed loop, kinds of

methods have been reported, such as sigma-delta loops

[1], continuous time interface [2], time divided method

[3], and so forth.

The time divided closed loop architecture presented in

this paper is derived from an open loop accelerometer

[4]. A proportional integral derivative controller (PID) is

the only major analog block added.

Formulae deduced in this paper help to determine design

parameter both for sensor and readout circuits, such as

mass of proof mass, power supply voltage, feedback

duty cycle, and so on.

In this paper, first mathematical formulae are deduced.

Second, Matlab/Simulink and Verilog-A models are

established. Finally, test results are shown.

2. Mathematical calculation

Micromachined sensor which detects acceleration

consists of a moving proof mass that is attached to

elastic beams and two fixed plates that serve as

electrodes [5], as shown in Fig. 1.

Poof mass

Top layer

Bottom layer Proof mass

Damping Elasticity

Figure 1. Model of sensor

Theoretically, this kind of sensor in a closed loop system

is applied with four different forces, which are inertial

force, elastic force, viscous force and electrostatic force,

as shown in Fig. 2. Electrostatic force is realized when

the top and bottom plates are bias at voltage of VR and –

VR respectively, and the proof mass is bias at Vf.

Fe Fd Fk Fi

Figure 2 Illustration of forces

For a continuous time accelerometer, the electrostatic

force is effective all the time. Therefore, in such state

when static equilibrium is achieved, the poof mass is

static in the rest position. Under this condition, only two

of the four forces are effective, and thus inertial force is

equal to electrostatic force.

𝑚 ∙ 𝑎 =2 ∙ 𝜀0 ∙ 𝐴 ∙ 𝑉𝑅

𝑑02 ∙ 𝑉𝑓 (1)

where m is the mass of the poof mass, a is the

acceleration, ε0 is the vacuum permittivity, A and d0 are

the area and separation of the capacitor formed by the

each fixed plate and the proof mass.

Accelerometers of time divided closed loop architecture

are working under a main frequency. For each period,

there are two phases, detecting phase and feedback phase.

978-1-4673-2475-5/12/$31.00 ©2012 IEEE

In detecting phase, the system works similarly to an open

loop one as described in [4], and no feedback force is

available. In feedback phase, electrostatic force is

applied to the proof mass though proper biasing.

For time divided architecture shown in Fig. 3, equation

(1) has to be modified, since the feedback electrostatic

force is only partially applied to the proof mass.

sensor CSA S&H

LPFPIDBUF

Figure 3. Block diagram of time divided system

As the whole system is working under a frequency much

high than the signal bandwidth, thus the following

equation is true on the mean sense.

𝑚 ∙ 𝑎 =1

𝑇0∫

2 ∙ 𝜀0 ∙ 𝐴 ∙ 𝑉𝑅𝑑02 ∙ 𝑉𝑓

𝑡0+𝑇0

𝑡0

∙ 𝑑𝑡 (2)

where T0 is the period of the control signal of the

feedback switch.

For each period of T0, there is time span of Tf during

which electrostatic force is applied. Therefore, equation

(2) becomes as follows.

𝑚 ∙ 𝑎 = 𝜂 ∙2 ∙ 𝜀0 ∙ 𝐴 ∙ 𝑉𝑅

𝑑02 ∙ 𝑉𝑓 (3)

𝜂 =𝑇𝑓𝑇0

(4)

3. Matlab/Simulink and Verilog-A modeling

Matlab/Simulink model and Verilog-A model for sensors

in time divided closed loop accelerometers can be

established by modifying the open loop model [6, 7] as

well as the closed loop model [8]. Feedback block which

simulates the behavior of the electrostatic force of the

time divided architecture must be correctly added.

𝑚 ∙ 𝑎 = 𝑚 ∙𝑑2𝑥

𝑑𝑡2+ 𝑏 ∙

𝑑𝑥

𝑑𝑡+ 𝑘 ∙ 𝑥 + 𝐹𝑒𝑙 (5)

𝐹𝑒𝑙 = −𝜀0 ∙ 𝐴

2[(𝑉𝑅 − 𝑉𝑓)

2

(𝑑0 − 𝑥)2−(𝑉𝑅 + 𝑉𝑓)

2

(𝑑0 + 𝑥)2] (6)

The model of sensor shown in Fig. 4 is mainly based on

equations (5) and (6) with modifications adapted to time

divide architecture.

Σ ∫ ∫

Damping

Elasticity

Electrostatic

a x

Biasing of

sensor

ΔC

Figure 4. Model of sensor

The readout circuits, as shown in Fig. 3, consist of

several main blocks, which are charge sensitive amplifier

(CSA), sample and hold circuit (S&H), low pass filter

(LPF), and PID controller. Submodels for blocks can

firstly be established by Matlab/Simulink as if blocks were ideal, which leads to a simple linear model of the

whole circuits, as shown in Fig. 5.

𝐻(𝑠)𝐶𝑆𝐴 =𝑉𝑅𝐶𝑓

(7)

𝐻(𝑠)𝐿𝑃𝐹 =𝐾

𝑎 ∙ 𝑠2 + 𝑏 ∙ 𝑠 + 𝑐 (8)

𝐻(𝑠)𝑃𝐼𝐷 = 𝑃 +𝐼

𝑠+ 𝐷 ∙ 𝑠 (9)

In Matlab/Simulink, CSA is modeled as a linear

amplifier with a constant gain, and LPF and PID are both

modeled as s-domain transfer functions. While in

Verilog-A, realistic amplifiers, resistors, capacitors,

switches and so forth can be added, so that more

complicated features can be accurately modeled.

ΔC CSAS&H

LPF

num(s)/den(s)PID

P+I/s+D·s

Biasing of

sensor

Clock

Figure 5. Model of circuits

Combining both mechanical and electronics models,

simulation of the accelerometer as a whole is feasible.

Simulation results with different values of η are

compared, as shown in Fig. 6. Other conditions being

fixed, the larger the η is, the better the system

approaches to a continuous time interface, while the less

the sensitivity of the accelerometer.

Figure 6. Simulation results of different η

4. Test results

The readout circuits are fabricated using 0.35μm HV

CMOS process. This chip consumes an area of about

10mm2. For testing purposes, a printed circuit board

(PCB) is designed. The chip lies in the center of the PCB.

Photographs of the chip and PCB are shown in Fig. 7.

Figure 7. Photographs of layout and PCB

Test results show that the chip works under a frequency

of 47.31kHz and η is around 60.1%. The waves shown in

Fig. 8 are attained by Agilent DSO-X 2014A.

Figure 8. Waves of top and bottom electrode of sensor

and the control signal of feedback switch

Together with a micromachined sensor, the PCB is used

to test the chip in closed loop. Output voltage responses

to accelerations are measured by Agilent 34401A. The

curve is shown in Fig. 9. The sensitivity of the tested

accelerometer is 1.518V/g.

Figure 9. Test reulsts of accelerometer

5. Summary

Mathematical relationship between the acceleration

detected by sensor and the output voltage of readout

circuits of time divided architecture has been deduced.

Simulations of Matlab/Simulink and Verilog-A both

support the mathematical equation. Readout circuits

designed to work under 50kHz with feedback duty cycle

η being 60% have been fabricated. Test results show a

sensitivity of 1.518V/g.

Acknowledgments

The project is supported by the National High

Technology Research and Development Program of

China (Grant No.2008AA042201).

References

[1] Haluk Külah, Junseok Chae, Navid Yazdi, Khalil

Najafi, IEEE Journal of Solid-State Circuits, Vol. 41,

No. 2, p. 352, February (2006)

[2] Lasse Aaltonen, Kari Halonen, Sensors and Actuators

A 154, p. 46 (2009)

[3] Yuntao Liu, Xiaowei Liu, Liang Yin, Weiping Chen,

Qun Wu, Conference on Nano/Micro Engineered and

Molecular Systems, p. 510 (2009)

[4] Huang Jingqing, Zhang Mingming, Chen Zhongjian,

et al, International Conference of Electron Devices

and Solid-State Circuits, (2011)

[5] Qifang Hu, Chengchen Gao, Yilong Hao, et al, Micro

& Nano Letters, 2011, Vol. 6, Iss. 7, p. 510 (2011)

[6] Li Haojiong, Zhang Mingming, Chen Zhongjian, et

al, International Conference of Electron Devices and

Solid-State Circuits, (2011)

[7] Teodor Lucian Grigorie, MEMSTECH’2008, p.105

(2008).

[8] Teodor Lucian Grigorie, MEMSTECH’2008, p. 115

(2008)