Pre-distorted sinewave-driven parallel-plate electrostatic actuator for harmonic displacement

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J. Micromech. Microeng. 15 (2005) S103–S108 doi:10.1088/0960-1317/15/7/015

Pre-distorted sinewave-driven

parallel-plate electrostatic actuator for

harmonic displacement

G de Graaf

1

, L Mol

1

, L A Rocha

1

, E Cretu

2

and R F Wolffenbuttel

1

1_{Delft University of Technology, Faculty of EEMCS, Department for MicroElectronics,}

Mekelweg 4, NL-2628 CD Delft, The Netherlands

2_{Melexis, Transportstr. 1, B-3980 Tessenderlo, Belgium}

E-mail:G.deGraaf@ewi.tudelft.nl

Received 10 December 2004, in final form 14 April 2005 Published 20 June 2005

Online atstacks.iop.org/JMM/15/S103

Abstract

Harmonic displacement of a parallel-plate electrostatic actuator up to 50% of the static pull-in displacement has been achieved despite the non-linear voltage-to-displacement function using a driving voltage with a

pre-distorted waveform. The microstructure is fabricated in an epi-poly process and the circuit is implemented in a CMOS process and designed for operation of the MEMS in a frequency range up to 1 kHz. The pre-distorted waveform is synthesized using 16 samples per period with 16 non-uniformly spaced quantization levels, using a ladder with accurately scaled resistors. Harmonic actuation has been demonstrated with 34 dB reduction of

second-order distortion compared to systems with sinusoidal actuation. The residual second harmonic content in the harmonic displacement is typically –42 dB.

1. Introduction

Electrostatic actuators are generally classified into two types: lateral comb drive actuators with a motion along the electrode area and parallel-plate actuators with a motion perpendicular to the electrode area. The lateral comb drive actuator generates a force proportional to the applied voltage squared and the number of movable electrodes. This force is, in a first approximation, independent of the associated momentary lateral displacement of the spring-suspended movable part. In the parallel-plate actuator the force is also proportional to the voltage squared, however, for properly designed structures of given lateral dimensions larger electrostatic forces are generated. The force depends on the voltage squared, divided by the parallel-plate spacing squared (i.e. the electrostatic field squared) and thus increases with displacement at a particular voltage level, whereas the restoring force of the beam is, in a first approximation, linear with deflection. Therefore, the drawback of the parallel-plate actuator is the unstable system that results in the case of a deflection beyond a critical value. The pull-in voltage, Vpi, is defined as the voltage that is required

to obtain this critical deflection [1]. This property limits the

dynamic range of displacement for low-frequency movements (quasistatic motion) to 1/3 of the nominal gap width. It must be noted that in this paper the structure is not operating in resonance mode, in which case harmonic displacement can be achieved at the mechanical resonance frequency of the structure [2].

The relevance of the non-linear voltage–displacement relation and the limited operating range depends on the application. They are not important in bi-stable mechanical devices, such as switches and deflectable micromirrors. The static pull-in does limit the dynamic range in servo-controlled accelerometers [3], but the squared voltage–displacement relation can be corrected for in the readout circuits.

An extension of the servo-operated accelerometer is the mechanical spectrum analyser using a dedicated accelerometer with a common-mode time-varying actuation voltage [4]. In the case of harmonic common-mode actuation, y(t) =

y0sin(ω1t ), the presence of the actuation force in the feedback

path, in combination with the non-linear dependence of the electrostatic force on the voltage, causes a periodically varying gain with frequency ω2 = 2ω1. To eliminate the

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signal, a differential accelerometer configuration is used, with the two accelerometers actuated in quadrature. The output signals, after passing an averaging filter, are proportional to the modulus of the spectral component at 2ω1 in the

mechanical vibration spectrum. By varying the frequency

ω1 of the electrical actuation, the mechanical vibration

spectrum is scanned. The dynamic range advantage of gap-varying electrostatic parallel-plate actuation is essential in this application. Since the performance critically depends on the ability to force the microstructure into a harmonic motion of well-defined frequency, parallel-plate actuation is to be used with a pre-distorted excitation voltage waveform to anticipate both the squared dependence of the force on the excitation voltage and the increasing electric field with plate displacement.

2. Specification of the electrical signal source

The shape of the excitation voltage required for obtaining a sinusoidal displacement of a moving electrode in a parallel-plate electrostatic field can be found by solving the equation of motion in the absence of an external acceleration for a harmonic displacement [1,2,5] md 2_{y(t )} dt2 + b dy(t) dt + ky(t)= Felect, (1) where y(t) denotes the structure displacement, m represents the movable mass, b is the damping coefficient, k is the spring constant and Felect= C0d0V

2

2(d0−x)2 is the electrostatic force caused

by a voltage V (t) applied to a capacitor with initial value C0

and initial gap d0. Since the excitation force is proportional

to the driving voltage squared, an initial displacement should be applied. Substitution with y(t)= y0(1− β) sin(ωt), with β <1 in the differential equation (1) and solving this equation for V (t) yields:

V (t )=[√2{d0− y0(sin(ωt)− β)}

×mω2_y

0sin(ωt)− bωy0cos(ωt)− ky0(sin(ωt)− β)]/

d0C0.

An analytical solution of equation (1) is generally not always possible in the case of a non-linear damping term b. Figure1

shows this driving voltage and the calculated resulting displacement drawn as solid lines. The dashed lines show the effects of quantization, both in time (number of samples per period) and in amplitude (number of bits) of the excitation voltage on the electrode displacement function. The acceptable quantization errors are specified in the frequency domain, V (ω), in terms of the spurious free dynamic range (SFDR).

The excitation waveform should be such that the associated gain for an external acceleration is a perfect sine function (see figure1). In terms of frequency diagram, for

y(t )= y0sin(ωt), the drive voltage should therefore contain

only a harmonic component at ω. Digital implementation of the signal generator implies quantization both in time and in amplitude. The quantization in time (phase steps) introduces harmonics around the sampling frequency and therefore parasitic harmonics. For eight samples per period the maximum is at the 7th harmonic and is equal to−16.9 dB. Similarly 16 and 32 samples per period yield −23.5 dB

Figure 1. Actuation voltage V (t) (above) and resulting electrode

displacement δ(t) (below), using a properly pre-distorted sinewave.

Figure 2. Simplified block diagram of the pre-distorted sinewave

oscillator.

at the 15th harmonic and −29.8 dB at the 31st harmonic, respectively. The SFDR increases by 6 dB per doubling of the number of samples per period. The quantization of the signal amplitude also causes harmonics. The resolution of the signal

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Figure 3. Structure of the direct digital frequency synthesizer.

should be highest at the peak of the waveform, especially at high sensivities where the device operates near pull-in. The design of the direct digital frequency synthesizer (DDFS) and the use on a typical MEMS structure fabricated in an epi-poly process is discussed in the next section.

3. Design of the DDFS

The DDFS for variable frequency is based on a ramp generator that produces a linear increasing digital code plus a DA converter, as shown schematically in figure 2

[6, 7]. The special aspect in this application is the non-linear DA conversion. The number of output bits of the phase accumulator determines the number of time steps per period. Considering the required amplitude resolution at the peak of the waveform a straightforward implementation of a binary ROM look-up table and DA converter using a uniform

Figure 4. Structure of the MEMS mechanical spectrum analyser.

step size would require a very large amount of circuit area. A generic CMOS process is used to leave the option for on-chip system integration open using IC-compatible MEMS post-processing. Such a single-chip solution does not favour the integration of very large digital circuits. Therefore, the number of quantization steps should be minimized by making the intervals in the waveform amplitude non-uniform, while matching the non-quantized theoretical waveform within specifications in terms of SFDR.

A voltage dividing resistor string especially designed for well-defined amplitude levels and two sets of tap switches are used for forming the proper excitation voltages. Only the part for the phase range in between 0◦and 180◦is generated and switching is used to synthesize the entire waveform. The ramp with adjustable increments is generated in the phase accumulator by repetitive addition of a frequency control word (FCW), as shown in figure3. The four most significant bits are used to drive the DA converter to yield the momentary value of the excitation voltage. In the actual circuit the total number of bits m = 20 and a 1 MHz system clock is used to enable the generation of a ramp between 1 Hz and 1 kHz with 1 Hz resolution. The number of output bits of the phase accumulator determines the number of samples per period and thus the magnitude of the parasitic harmonic components. The number of bits for the DA converter is a trade-off between the required accuracy and the size and complexity of the resistive ladder. A design based on 16 samples per period (SFDR= 23.5 dB) is sufficient for the application as a mechanical spectrum analyser and requires only eight accurately scaled resistors. For larger values the mismatch between the resistors in the string is expected to dominate system performance.

For best matching the resistors in the DA converter consist of identical resistors (unit resistors) connected in series and in

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Figure 5. Calculated frequency spectrum of the displacement of the

structure using a DDFS with perfectly sized resistors.

Figure 6. Die-photograph of the DDFS.

parallel, in such a way that the required quantization levels are generated. The ideal values are scaled to relative values and subsequently an optimum network of series/parallel connected resistors has been calculated. Table1shows the results (16/1 + 2/3 indicates 16 unit resistors in series plus two series connected sets of three unit resistors in parallel). The solution is not unique, but yields a deviation from the ideal value smaller than the expected component mismatch.

Prototypes have been realized in the Bosch epipoly process. Basically an 11 µm thick polysilicon layer is patterned and released in a surface-micromachining-like process [8, 9]. The planar MEMS accelerometer structure fabricated is shown in figure4. The springs are shown in the upper-right corner and interdigitated finger electrodes shown in the lower left figure are available for both electrostatic actuation and capacitive detection of displacement. Stoppers shown in the lower right figure are included on either side of the movable mass to limit the lateral displacement range. Table2gives an overview of the specifications of the fabricated device. Figure5shows the calculated displacement spectrum

HF Carrier HF Carrier VH VL MEMS device Cs1 Cs2 Ca1 Ca2 Charge Amplifier MIXER LP Filter From DDFS From DDFS Actuation

Figure 7. Block diagram for actuation and measuring the

displacement.

Table 1. Resistor values for the intended non-linear function.

Scaled value Resistor configuration Actual value Error (%)

16.6818 16/1 + 2/3= 16.666 7 −0.090 16.3283 16/1 + 2/6= 16.333 3 0.031 3.854 73 11/3 + 1/5= 3.866 67 0.310 1.267 89 3/5 + 2/3= 1.266 67 −0.097 0.540 711 1/25 + 3/6= 0.540 00 −0.131 0.265 114 1/12 + 2/11= 0.265 15 0.0154 0.134 629 1/10 + 1/29= 0.134 48 −0.109 0.058 8235 1/17= 0.058 8235 0.000

Table 2. Specifications of the MEMS device.

Parameter Value Mass (m) 4.27 µg Mechanical spring (k) 1.2930 N m−1 Initial gap (d0) 2.25 µm Damping coefficient (b) 1.92× 10−4N s m−1 (linearized around x= d0/3)

Cd0(zero-displacement actuation capacitor) 141 fF Cs0(zero-displacement sensing capacitor) 611 fF

of the voltage-to-displacement of the electrostatically driven structure. The ideal 2nd harmonic distortion at the given sampling rate and using perfectly selected resistors is at −68 dB. Figure6shows the 2.1× 2.6 mm2die of the DDFS fabricated in CMOS.

4. Measurement results

The MEMS presented is basically a coherent detector operating in the electro-mechanical domain. Readout requires the measurement of the displacement of the structure and demodulation in a synchronous detector circuit, as shown in the block diagram in figure7. The displacement is measured capacitively using a 500 mV amplitude sinewave carrier. The frequency (3 MHz) of the carrier is far above the band of the displacement signals and cannot cause electrostatic actuation of the structure. This signal is connected to the mass (the centre electrode) of the device and displacement of the mass causes a differential signal at the inputs of the charge amplifier. The structure is actuated by the DDFS waveform. The displacement of the centre electrode introduces amplitude modulation of the HF carrier signal. This signal is amplified and converted back to the baseband by the mixer followed by a 4 kHz lowpass second-order Sallen–Key filter. Figure8shows the accelerometer chip, the CMOS chip DDFS chip and the CMOS chip with the charge amplifier for the readout in one package. The power consumption of the DDFS chip has been

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Figure 8. Photograph of the packaged system: containing the DDFS

and the readout chip (left) and the microstructure (right).

(a) 200 400 600 0.35 0.3 0.25 0.2 0.15 0.1 0.05 Amplitude 0 (b) 2 4 6 8 10 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 t [ms] Amplitude Displacement t [ms] Displacement

Figure 9. Measured time plot of the displacement when driving the

actuator with a (a) 16 Hz pre-distorted waveform and (b) 400 Hz pre-distorted waveform.

measured as 3 mW at a 1 MHz clock frequency and 0.8 mW for the resistor string at a 5 V power supply voltage. Figures9(a) and (b) show the measured displacement signals with this setup using a 16 Hz and 400 Hz pre-distorted waveform respectively. The 16 phase steps per period are clearly visible in these figures. The distortion at zero crossings of the signal in figure 9(b) is not caused by the slew-rate of the electrical signals, but by the frequency-dependent squeezed-film damping of the structure [10]. Figures10(a) and (b) show the measured spectrum of the displacement at excitation frequencies of 16 Hz and 400 Hz respectively; the measured residual second-order distortion at low frequencies is typically−42 dB. Figures11(a) and (b) show the measured displacement and its spectrum using a 16 Hz sinewave for excitation. The improvement in second harmonic distortion using the pre-distorted sinewave instead of a pure sinewave is about 34 dB, as can be seen by comparing figures 10(b) and11(b). (a) 10 20 30 40 50 60 70 80 90 100 90 80 70 60 50 40 30 20 10 Frequency [Hz] Magnitude [dB] Displacement Frequency [Hz] Magnitude [dB] Displacement (b) 200 400 600 800 1000 1200 1400 1600 1800 2000 70 60 50 40 30 20 10

Figure 10. Measured spectra of the displacement when driving the

actuator with a (a) 16 Hz pre-distorted waveform and (b) 400 Hz pre-distorted waveform. (a) 100 200 300 400 500 600 700 0.2 0.15 0.1 0.05 t [ms] Amplitude Displacement 16Hz sinewave (b) 10 20 40 60 80 80 70 60 50 40 30 20 10 0 Frequency [Hz] Magnitude [dB] Displacement

Figure 11. Measured displacement using a 16 Hz sinewave voltage

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5. Conclusions

A direct digital frequency synthesizer for harmonic electrostatic actuation of parallel-plate MEMS structures has been developed and tested. The frequency of the waveform can be digitally programmed in 1 Hz steps from 1 Hz to 1 kHz. The signal waveform is pre-programmed in the resistor ladder and depends on the required range of travel of the moving plate. The non-linear excitation principle can be used to drive the structure very close to the pull-in point. The accuracy of amplitude of the waveform becomes more critical if the mass moves close to the pull-in point. The improvement in second-order distortion compared to systems using full travel range with sinusoidal actuation is more than 34 dB. The measured residual second-order distortion of a MEMS structure is about −42 dB. Accuracy is now limited by the matching of the resistors in the ladder, but can be improved using for instance dynamic element matching techniques. Further research will be focused on integration of the actuating and readout circuits on a single chip and on applications where a harmonic varying gain is required for signal processing in the mechanical domain, such as mechanical spectrum analysers.

Acknowledgement

The authors acknowledge the DIMES/ICP group of the Delft University of Technology for fabrication of the DDFS chip.

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