Closed-loop silicon accelerometers

M. Kraft C.P. Lewis T.G.Hesketh

Abstract: The controlsystem structure and behaviour are considered for an analogue and a digital servo-accelerometer having a dynamic range from mg to several g ( lg = 9.81mss2), for use in low-frequency applications. The paper emphasises the derivation of mathematical models, which are presented together with simulated and test results obtained from the implementation of these strategies on a bulk- micromachined silicon sensing element employing capacitivc signal pick-off. The digital device, which is based upon oversampling conversion, proved to have superior !stability compared to the analogue accelerometer. Furthermore, it has the advantages of producing a direct digital output signal and an inherent self-test feature.

List of symbols

a = acceleration, m/s2

A , b = damping factor, kgls C, = top capacitance of the sensing element, F C, = bottom capacitance of the sensing element, F do = distance from the seismic mass at central posi-

dTE = distance from the seismic mass to the top elec-

dBE = distance from the seismic mass to the bottom

D = operating level of idleal relay E = dielectric constant, considered to be E , = 8.854

Fm-' J; = sampling frequency., Hz F,, = electrostatic force on the seismic mass, N Fell = electrostatic force generated by the voltage on

= area of the seismic mass, m2

tion to either electrode, m

trode, m

electrode, m

top electrode, N 0 IEE, 1998 IEE Procec~rlin~s online no. 1998227.5 Paper first receivcd 5th December 1997 and in revised form 19th May 1998 M. Kraft is with the University of California, Berkeley, Electrical Engi- neering and Computer Scicnce, Bcrkeley Sensor and Actuator Center, Cory Hall, Berkelcy, CA 94720-1774, USA C.P. Lewis (deceased) was with, ancl T. G. Hesketh is with the Nonlinear Systems Design Group, School of Engineering, Q-Block Covcntry Univer- sity, Priory St., CVI 5FB, UK

Fc,12 = electrostatic force generated by the voltage on bottom electrode, N

k = suspension spring constant, N/m KF = Ceedback gain, NIV K,,, = pick-off circuit gain, Vim Kp = proportional gain K, = integral gain Kc, = differential gain MZ = seismic mass, kg N n

8

s = Laplace operator V,,,, = {excitation voltage, V V, = bias voltage, V Vr = feedback voltage, V V,,,, = output voltage amplitude of the open-loop

w = angular frequency, s-' x = deflection of scismic mass from the central

x 1 Introduction

= !sinusoidal input describing function = number of samples per half cycle, positive intc-

= lagging angle introduced by sampling action, ,ger

rad

accelerometer, V

position between the electrodes, 111 = magnitude of the input signal to a nonlinearity

Several types of micromachined accelcrometer structure have been reported, using surface and bulk microma- chined devices manufactured principally by either plasma or chemical etching. The latter technique is employed in the construction of the transducers described here which, owing to their physical dinien- sions, tend to use capacitive signal pick-off and clectro- static forces for reset, and are the most frcquently described in the literature.

There are two basic approaches to the production of high performance, micromachined transducers, one is to seek an optimum mechanical design involving a min- imuin of electronics, the other, which is adopted here, is to apply closed-loop control system techniques to a more rudimentary sensing element. The objcct of the worlk described is to show that the choice of an appro- priate control-system strategy can lead to a robust, sta- blc d e s i p suitable for use in low-Frequency applications such as inertial navigation. As an example, a comparison is made of analogue and digital control strategies applied to a silicon, bulk-micromachined sensing element to produce a scrvo-accelerometer.

325

The sensing element consisted of a proof mass sus- pended between two fixed electrodes, which were used as a capacitive half-bridge for signal pick-off [l]. The nominal gap between the mass and a plate electrode was 1 0 ~ with a nominal capacitance of 5pF. For the purposes of this investigation, the sensing element used had the open-loop characteristics shown in Fig. 1, which indicate a bandwidth of 56Hz and hysteresis of 0.84'%.

- 2 L ' ' ' ' ' ' ' ' ' ' -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0 4 0.6 0.8 1 0

accele;ation,g

frequency,Hz b

Fig. 1 (lower diagram) of the open loop accelerometer

Transfer characteristic (upper diagram) and frequency response

In this application of closed-loop control, both the analogue and digital transducer designs employed elec- trostatically generated reset forces by applying the feed- back voltage to the same electrodes as used for signal pick-off. Consequently, there was a need for some form of multiplexing between the pick-off and reset signals.

A fundamental difficulty which may arise with this method of reset is the nonlinear relationship between the feedback voltage and the force on the mass. Several methods can be used to overcome this problem, the most common are: superimposing two electrostatic forces on the mass to maintain a negative, linear feed- back relationship; this results in a closed-loop, ana- logue sensor, [2-51, and embedding the sensing element in a sigma-delta modulator structure resulting in an inherently digital transducer [6-131. The application of both approaches is considered with regard to: signal pick-off, reset of the seismic mass, the control-system format, the mathematical model and its simulation and the sensor stability and performance.

2 Signal pick-off

Capacitive pick-off is in common use [14], however, in micromachined devices the change in capacitance may be only a few femto farads, particularly in a closed- loop structure in which the pick-off signal is effectively the error signal. A major problem with this type of pick-off is the stray capacitance existing between each plate and the mass resulting from the construction, which effectively produces a parallel passive bridge. Since the stray capacitors may be far larger than the active capacitors, unbalance in the stray bridge may produce offset signals equal to or greater than the transducer range. This results in control problems (unless the offset is nulled by an external signal) since the seismic mass will take up a position, under control action, away from the central plane.

326

2.7 Analogue transducer The signal pick-off arrangement, Fig. 2, consists of a charge amplifier (to reduce sensitivity to cable capaci- tance to ground), a buffer amplifier, a phase-sensitive demodulator and a 300Hz low-pass filter. The signal pick-off employs a double sideband suppressed carrier (DSC); the excitation voltage to the transducer outer electrodes being two equal, but anti-phase, signals at the carrier frequency. The resultant signal from the charge amplifier is an amplitude-modulated DSC sig- nal, which requires synchronous demodulation at the carrier frequency. Consequently, the charge amplifier must have an overall bandwidth, centred on the carrier frequency, of at least twice the transducer bandwidth. In practice, an excitation frequency of IOOkHz was used with a typical servo-accelerometer bandwidth of less than 300Hz. The bandwidth of the overall signal pick-off should be at least one magnitude higher than the bandwidth of the sensing element.

t

comparator J' "ex, s

Fig. 2 Signal pick-off arrangement of the analogue transducer

2.2 Dig ita I transducer In this case, the pick-off arrangement operates in the time rather than the frequency domain, as is the case with the analogue device. This is accomplished by a charge-discharge sampling technique, in which voltage steps of opposite polarity are applied instantaneously to the upper and lower electrodes. The charge induced on the central electrode provides a measure of the imbalance in capacitance. The resulting current charges the feedback capacitor of an operational amplifier, the output voltage of which is then stored in a sample and hold device; this completes the sensing cycle. All charge is then removed from the capacitors prior to the start of the next cycle.

A

. --e--l timing diagram

master- Q3 - 62 Y

v 3-

Fig. 3 Timing diagram: switches are low active

Signal pick-off arrangement of the digital transduce?

The pick-off arrangement, Fig. 3, requires nine ana- logue switches, the timing diagram of which is also shown. The number of switches required and the digital control unit contributes considerably to the complexity of the electronic circuit realisation. However, the per- formance requirements of the circuit, with respect to linearity and accuracy, are low since the error signal,

IEE Proc -Circuits Devices Syst , Vol 145, No 5, October I998

under closed-loop control, is very small and, owing to the sigma-delta modulator control structure employed here, the most important information of the error sig- nal is its polarity.

3 Reset mechanism

3. I Analogue transducer As is common in these transducers, electrostatic meth- ods of reset are employed; two approaches are theoret- ically possible. In the firsl, a free electric charge is maintained on the seismic mass, however, owing to the difficulties of holding this charge most commercially available transducers do riot use this technique [SI. Alternatively, an electrostatic force may be applied to an earthed seismic mass, but, in order to provide a lin- ear feedback characteristic, it is necessary to superim- pose two electrostatic forces on the mass, produced by feedback and bias voltages on the outer electrodes. The resultant feedback force, neglecting the force compo- nents introduced by the voltages required for signal pick-off, is

where A , is the active area (of an electrode, d, and dBE are the distances between thLe seismic mass and top and bottom electrodes, respectively (ideally dTE = d5E = do). V’is the feedback voltage, V, the bias voltage and ‘x’ the displacement of the seilsmic mass from the central position between the electro’des.

If Vr < V, and x + 0 then eqn. 1 simplifies to

lim 2 + 0

which is a linear relationship between Fel and Vf

81 I

-8 -30 -20 -10 0 10 20 30

feedback voltage, V Fig. 4 Feedbuck trunsfer churucte,pistic relating Fe, to V ,

In Fig. 4, the relationship between the electrostatic force on the mass and the feedback voltage for x €{-6, -3, 0, 3, 6) p,m is shown for V, = 15V. For the partic- ular case, x = 0, the characteristic is a straight line as predicted in eqn. 2. The nonlinear feedback relation- ship between Vf and Fcl as; the mass departs from the central plane is clearly shown. Consequently, it is desir- able that the feedback force is sufficient to prevent motion beyond 5 1 I . L ~ ovri the transducer operating range.

For displacement appreciably above this value, the characteristics show a change of polarity in the gain, resulting in positive feedback which can cause ‘latch up’; the mass being irreversibly attracted to the nearest

IEE Proc.-Circuits Devices Syst., Vol 145, No 5, October 1998

electrode. The spring suspension acts to stabilise the transducer, but at large feedback voltages the effect is negligible.

........... i.- ....... . jVg=20 v ... ..i ... .-, . i” - ,

3

2 7-

0 z o g -1 c

............ j ...... ..,& ..... ..;. ....... ; ....... ;-. ...............

-5 -4 -3 -2 -1 0 1 2 3 4

deflection, pm (xi0 ) -6

Fig. 51 jbr dij%rent V,

Electrostatic ,force on the muss under shock condition ( v/ = 0)

When the accelerometer is subjected to a shock, owing to the limited bandwidth of the control system, a sudden (change in ‘x’ does not result in a corresponding change in vf and hence in Fe[, thus closed-loop control of the sensor is ineffective. Consequently, V, causes Fe[ to act in the same direction as the deflection, resulting in positive feedback and ‘latch-up’ as shown in Fig. 5 where, ffor simplicity Fel, as in eqn. 1, is plotted as a function of ‘x’ for a range of V5 under the condition Vf = 0; an increase in V, leads to a higher sensitivity to this condition. It should be noted, however, that in normal operation the higher the value of V5 the greater the linear operating range [15]. ‘Latch-up’ can also happen on initial energisation of the transducer, since the mass position is not sensed until the circuit is oper- ational. To recover from this condition, complete removal of the electrical supply from the transducer is required.

3.2 Digital transducer Here, oversampling is employed with one bit D to A conversion in the feedback path. One operating cycle comprkes a sensing and a much longer reset phase, i.e. the pick:-off and feedback signals are separated in time. In the reset phase, only the electrode furthest from the seismic mass is energised (the other being grounded), this ensures a restoring force of the correct polarity. The duration of a voltage pulse is much shorter than the sma,llest time constant of the sensing element, con- sequently, negligible motion of the mass is produced. Thus, there is an insignificant change in the feedback force during one operating cycle, consequently the seis- mic mass is subjected to a bipolar, quantised reset force. The average force on the mass, in a given time period, is determined by the number and polarity of the feedback pulses. A condition can be realised during a voltage pulse in which the mass crosses the central position, this results in a change of polarity of the restoring force; however, this has no consequence, since it is corrected by the next pulse. It is this action which gives riise to a limit cycle essential to normal operation.

The control-system output signal now effectively con- sists of a stream of high-frequency, constant-amplitude clock pulses, the number of which in a given period represents the input-signal amplitude. Thus, there is a

321

linear relationship between the feedback voltage and the force on the mass without the requirement for a bias voltage, consequently, the locking to an electrode, which can occur in the analogue transducer, is no longer possible.

4 System strategy

The basic control system for both types of transducer has a type-I structure to minimise steady-state error. In general, stability will be determined by the sensing-ele- ment dynamics and the compensation circuit; the dynamic characteristics of the pick-off and reset elec- tronics playing little part.

vout

pick- demodulation low- I Off I

“exc Fig .6 Analogue, closed-loop accelerometer

4.1 Analogue transducer The basic structure of the analogue servo-accelerometer is shown in Fig. 6. The PID-block represents a series compensator and ensures a type- 1 characteristic. The integrator is not in the ideal position within the loop and hence sensor offsets, etc. are now important. The application of the bias voltage necessary for a linear operating range is indicated in the feedback path.

digital bitstream

vout

Fig. 7 Digital, closed-loop accelerometer

4.2 Digital transducer In Fig. 7, the general structure of the digital accelero- meter is shown. The distinctive features are: in the for- ward path, the signal-pick-off, sample and hold, and the comparator and, in the monitoring feedback path, the method of producing the quantised force.

5 Mathematical system models

For both transducers, it was essential to derive a math- ematical model of the sensing element and control elec- tronics, so that a systematic approach could be made to the development of a closed-loop device [16]. The accuracy of the models was verified by a comparison of the results of simulation with measurements on proto- types. Simulation was undertaken principally at com- ponent level using the MicroSim Design Centre V.7 117, 181.

328

5. I Analogue transducer In Fig. 8 , a model of the analogue transducer is shown, the important features of which are the nonlinear damping effect, which is a function of the displacement of the seismic mass, and its velocity; the saturation ele- ment represents the physical limit on ‘x’. The spring constant represents the mechanical suspension and the feedback path introduces the nonlinear effects pro- duced by the generation of the electrostatic forces which are a function of ‘x’, V, and Vf The pick-off and demodulator are in the forward path of the trans- ducer. Since the system is second order (ignoring the effects of the filter) compensation is required for satis- factory performance, PID compensation being used for this purpose. C,(x) and C2(x), representing the changes in capacitance of the sensing element with displace- ment, are given by:

spring constant output voltage

Fig. 8 Matliemutical model of the analogue, closed-loop accelerometer

Fig.9 Small signal model of the closed-loop, anulogue accelerometer

In order to apply conventional control system analysis, a linearised small signal model was derived, Fig. 9. This was obtained by making the following assumptions: the damping coefficient, ‘b’, is independent of ‘x’, i.e. the seismic mass is viscously damped [2, 191; the pick-off may be represented by a gain constant; the electrostatic reset force for a constant bias voltage is directly pro- portional to the feedback voltage. The general form of the closed-loop transfer function is:

K~ 2 + K , s + ~ , Kpe,m

ms3+ (b+ K F K p o K,+)s2+(Ic+ICF K p o K,)s+ I<F I<,, I<,

(4) where K,, is the gain of the pick-off circuit, KF the feedback path gain, Kd, Kp, K, the PID controller gain settings, m the seismic mass, k the spring constant of

IEE Proc -Cvcuits Devices Sjirt., Vel. 145, N o 5, October 1998

the suspension system and I;) the damping coefficient. A closed-loop frequency response (Vout/A) (ju) for

typical values of system parameters is shown in Fig. 10; with K!, = 6, K, = 0.6, Kd = 0. No attempt was made to optimise the transducer compensation owing to the problems of 'latch-up' already identified and the realisation that a digital transducer offered a method of overcoming these nonlinear characteristics. Neverthe- less, the transducer frequency response was considered satisfactory within the above constraints.

0

m -10 9 ._ g -20

-30 1 2 3 4

10 10 10 10 10

0 1 2 3 4 10 10 i o 10 10

frequency, radis

Frequency response of the lineavised model,for the closed-loop, Fig. 10 analogue accelerometer

displacement, x digital

D to A Fig. 1 1 Muthematical model of the digital accelerometer

5.2 Digital transducer The same sensing element vias used as in the analogue transducer. The pick-off circuit gain was assumed to be constant, this provided a s ipa l which, after compensa- tion, was applied to a sample and hold and compara- tor; producing a pulse-densil-y-modulated output signal. A one bit D to A converter represents the method of producing a bipolar reset signal necessary for control. The model, Fig. 11, shows t'hat the reset force is a non- linear function of 5 and '2;'. However, as the magni- tude of the feedback volta.ge pulses is constant, the major nonlinear effect is due to the mass deflection.

The model of Fig. 1 1 does not portray the fact that the phases of operation of the pick-off and feedback are separated in time; however, since the pick-off phase is much shorter than that of the reset, this is a valid approximation.

Under control action, the mass is normally restricted to a position very close to the central plane (i.e. within 25% of the gap); this facilitates the derivation of a small signal model as in Fig. 12, and the analytical pre- diction of limit cycle modc:s. In the unforced, type1 transducer, any limit cycle waveform at the input to the serial nonlinear combinatioin must be symmetric. This implies an equal number, n, of sampling pulses per half cycle of the mode of oscillation, referred to as an (n , n)

IEE Proc.-Circuits Devices Syst., Vol. 14.5, No. j, October 1998

mode 1201. To aid limit cycle analysis, the combination of comparator and D to A converter may be repre- sented by an ideal relay transfer characteristic. Conse- quently, the combination of the sample and hold and this nonlinearity may be modelled by a describing func- tion, N(x, e), where x is the magnitude of the assumed sinusoidal signal to the sample and hold and 0 is the sampling angle. The limit cycle modes are predicted [15, 211 by the solution of the characteristic equation:

= 0 (5) where

and D is the operating level of the ideal relay.

diaital output

force _1___

KF Fig. 12 Small signul rnodel of the digital crccelwomeler

A graphical solution was obtained, with K, = 100, K,, = 1, from which it is possible to predict (1 , I ) and (2, 2) limit cycle modes with periods equal to 2 and 4 sam- pling periods, respectively. These are typical of a low- order sigma-delta modulator 1221. A pure integrator, as used in a conventional modulator, would lead to an undesirably high mode of oscillation; in practice, it has been found that KiJKp < 1000 ensures low-order modes.

The predicted magnitudes of the deflection of the seismic mass for the (1, 1) and (2, 2) modes were 0.1 1 pm and 0 . 3 4 ~ , respectively; these are less than 5% of the gap. This agreed with simulation using Simulink made on the full model [15], which justified the original assumption of negligible mass motion.

It is well known 1201 that limit cycling systems have a closed-loop transfer characteristic which is dependent on tlhe feedback element, provided the input signal bandwidth is appreciably less than the limit cycle fre- quency. In this transducer, since the feedback elements are constant, a linear transfer characteristic is therefore predicted with a flat frequency response down to DC. This agrees with measured results on a prototype which are discussed below.

6 IMeasurement results

To obtain a static transfer characteristic, both types of transducer were tested by vertical rotation of 360" in the earth's gravitational field; a dividing head being used for accurate control of the rotation. To overcome the practical limitations of the hest equipment, the closed loop frequency response was obtained by mounting on a vibration table the device undcr test and a reference accelerometer (Sunstrand, Q-Flex 1000) as in Fig. 13, this equipment in turn being mounted on a large flexibly mounted table. The reference accelerome- ter had a -3dB cut-off frequency of 300Hz which was

129

acceptable, since in normal use a low-pass mechanical filter would be used as part of the mounting. This would eliminate any high frequency vibration effects in both types of accelerometer.

Fig. 13 Accelerometer test arrangement for frequency response determi- nation (The Q-Flex, quartz servo-accelerometer, and the micromachined sensing element under test (mounted on an alunlinium block) are shown assembled on the vibrator.)

During initial characterisation of the silicon devices, it was observed that the spring constant varied from 8 to 40Nim. In addition, the position of the seismic mass was difficult to determine accurately, owing to the wide range of parasitic capacitance which dominated the active capacitance by a factor of between 9 and 11 to 1. However, as far as could be determined, the offset of the mass was within +I- 1 . 2 ~ in a gap of 1 0 ~ .

1 .o 0.8

0.6

0.4 - 2 0.2

,a -0.2

0) ._ W O +-

3 -0.4

-0.6

-0.8

-1 .o

....................

................... ~~~~~~~~~~ ......................................

-1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0

acceleration,g

Fig. 14 ters (1) Digital accelerometer; (ii) analogue accelerometer

Transfer characteristic for the analogue and digital accelerome-

6. I Analogue transducer Fig. 14 shows the transfer characteristic for the ana- logue servo-accelerometer. The performance of the device was mainly limited by noise generated by the discrete component implementation of the interface electronics (approximately 5mV peak), nevertheless the linearity and hysteresis errors were approximately 0.4%. The resolution was 7mg and the sensitivity 0.7VI g; the latter can be adjusted by changing the feedback gain. A zero offset is obviously present, this is owing to the offset of the seismic mass caused by manufacturing tolerances.

The closed-loop frequency response obtained with PID settings, as used in Fig. 10, is shown in Fig. 15 (upper trace), indicating a bandwidth equal to the ref- erence accelerometer of 300Hz; this compares with an open loop bandwidth of 56Hz. Bearing in mind the

330

simplifications made for the theoretical analysis, the agreement is considered very good.

The uncontrollable condition of locking to an elec- trode was observed when V, > 20V and moderate acceleration shocks were applied to the device. Stand- ard industrial shock tests [23] could not be performed, since damage to the limited number of prototype sens- ing elements was unacceptable.

m 9 -5 , , >

m U) I I I , , I I I

._

, , , , , -1 0 0

10 1 2

10 l o a

1 2 10 10

frequency, Hz

b Fig. 15 ometers a Analogue accelerometer h Digital accelerometer

Frequency responses of the analogue and digital servo-acceler-

2- 4

E 3

E 1 8 2

0 0 0.5 I .o 1.5 2.0

time,ms Fig. 16 mode jumping from ( I , I ) to (2, 2) modes

Ouput bitstream of the unforced digital accelerometer showing

6.2 Digital accelerometer Fig. 16 shows the output signal of the digital acceler- ometer under unforced conditions and a sampling fre- quency of 10kHz. The limit cycles jump between (1, 1) and (2, 2) modes as predicted.

Conventionally, the bitstream would be processed by a decimation filter [22]. In this work, for simplicity, an analogue low-pass filter was used. The static character- istic, Fig. 14, and the frequency response of Fig. 15, were similar to those of the analogue device. However, the signal processing used to average the digital bit stream automatically removed the zero-offset. As expected, the digital device exhibited superior stability, freedom from non-recoverable saturated conditions being observed.

Unfortunately, with the computing facilities availa- ble, it was completely impractible to obtain a simulated system frequency response at component level using SPICE. Hence no comparison could be made between the characteristics of the mathematical model and the real transducer.

7 Conclusions

The prototype implementations of two control strate- gies for a micromachined, closed-loop accelerometer presented in this paper demonstrate the improvements of the digital approach which are:

IEE Proc.-Circuits Devices Syst., Vol. 145, No. 5, October 1998

~ superior system stability (freedom from a non-recov- erable, saturated condition), suitable for consideration in high integrity applications;

~ a direct digital output signal in the form of a serial, pulse density modulated bitstream; - linearisation of the reset action; ~ an inherent self-test feature (failure of the limit cycle indicates a system fault); ~ adaptivity towards compoinent tolerance [24]; - and fewer analogue circuit components are expected to result in a better cost to performance ratio owing to the smaller die area required. Significant improvement of resolution and sensitivity is to be expected with ASIC implementation of the interface circuit.

Rather than attempt to optimise the design and man- ufacturing tolerances of the sensing element, the appli- cation of closed-loop techniques can provide an economical means of achieving a desired specification. A practical disadvantage of both analogue and digital approaches for the particular sensing element used is the requirement of relatively high operating voltages to provide feedback action, however, sensing elements are commercially available with reduced physical dimen- sions, (which allow sufficient electrostatic reset force to be generated (thereby increasing the dynamic range) with the use of standard supply voltages. However, the principles of operation oul lined above are generally independent of the physical dimensions of the sensing element.

In making this study of thLese accelerometers no cost/ performance considerations were undertaken since the aim was to satisfy a defence requirement; the objective being to determine the technical feasibility of designing such transducers.

8 Acknowledgment

The support of Druck Ltd., UK is gratefully acknowl- edged.

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