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Micro-cantilever sensors for mass detection using resonance frequency have attracted considerable attention over the last decade in the field of gas sensing. For such a sensing system, an oscillator circuit loop is conventionally used to actuate the micro-cantilever, and trace the frequency shifts. In this paper, gas experiments are introduced to investigate the mechanical resonance frequency shifts of the micro-cantilever within the circuit loop(mechanical resonance frequency, MRF) and resonating frequency shifts of the electric signal in the oscillator circuit (system working frequency, SWF). A silicon beam with a piezoelectric zinc oxide layer is employed in the experiment, and a Self-Actuating-Detecting (SAD) circuit loop is built to drive the micro-cantilever and to follow the frequency shifts. The differences between the two resonating frequencies and their shifts are discussed and analyzed, and a coefficient

Micromachined resonant devices are attracting increasing interest in the field of chemical sensor applications, due to some of their interesting properties such as sensitivity, compactness and low energy consumption [

In the study of Zhou _{0} shifts accordingly [_{r} of the circuit. The details of the SAD circuit loop will be introduced in Section 2.1.

Most of the actual studies follow the same pattern as presented previously [_{0}, should be proportional to the mass loading Δ_{r} in this study, is measured instead. The MRF shift and SWF shift are not actually identical. Li

An investigation is performed in this paper to demonstrate the existence of a coefficient _{r} = _{0}. The experimental investigation consists of an open-loop test, where the MRF shift Δ_{0} is measured, and a close-loop test, where the SWF shift Δ_{r} is recorded. The origin of this coefficient is also discussed. The results indicate that the conventional usage of SWF shift as a measurement of gas concentration is not appropriate, as the SWF shift is not identical to MRF shift. Instead, SWF shift measured should be revised by a coefficient to approach the MRF shift. The further discussion in this work also proves that this finding can be applied to all the sensing system using an actuating-detecting oscillator loop.

With a presumption of small deformation, an electrical model of a piezoelectric micro-cantilever is developed [

The mechanical function analogy of the electrical components is presented in _{effect}_{0}_{1}, _{2} are proportionality constants:

Therefore, the electrical components in the model can be determined by the mechanical properties of the beam, among which _{0}. This means the micro-cantilever does not have a good capability of selecting the resonance frequency.

Therefore, for ameliorating the frequency selection performance of the beam, a frequency selection network module is designed. The constant capacitor

_{0}. The output signal of the frequency selection network is amplified and then adjusted by the following phase compensator. An amplitude limiting module is used to control the amplitude of the signal. Finally, as in reality the output of frequency selection network cannot be reduced to zero at frequencies far away from _{0}, a band-pass filter is needed, to reduce the secondary mode oscillation of the beam.

Suppose the gain of the frequency selection network is _{a}_{b}

Generally, the amplitude condition in

_{0} ± 100 Hz, the frequency selection network also shows a good linearity. After mass loading, mechanical resonance frequency decreases, and _{a}_{0}, to _{a}

The sensing system is mounted in a glass canister, with a thin glass slide as top window. Drops of liquid ethanol were jetted by a transfer pipette through this window to provide defined vapor concentrations to the canister. Suppose the volume of the canister is _{vessel}_{analyte}

For the experiments, we calculate the saturated concentration of ethanol vapor, known the ambient temperature and pressure of atmosphere. Then, an appropriate volume _{analyte}

An Open-loop test records the variation of mechanical resonance frequency _{0} of the micro-cantilever. The micro-cantilever used in this work is a probe from a commercial Atomic Force Microscopy (AFM) system supplied by the Veeco^{®} Company. It consists of a micro-machined silicon beam with a piezoelectric zinc oxide layer. The MRF of the AFM cantilever is 50,000 Hz, according to the specificationz and is measured at 66,640 Hz. Polyethyleneoxide (PEO) was chosen as the sensitive material, for its good sensitivity and selectivity to ethanol vapor [

The open-loop test instruments system is illustrated by _{0} can be detected. The Q-factor of the coated probe is near 130 at ambient conditions of standard atmospheric pressure and 293 K temperature.

In a close-loop experiment, the system working frequency _{r} of the SAD circuit loop was detected. Schematic diagram of the test can is illustrated in _{0}, the output of the frequency selection network is equal to 0. Secondly, connect the frequency selection network with the feedback module, and adjust the phase compensator, to meet the phase condition in the Barkhausen conditions.

A pair of experimental datapoints can be observed in _{r} and the MRF shift Δ_{0} what we exploit. It is easy to notice that in this result, the initial frequency, which corresponds to a vapor concentration of 0, is different. Such an offset of the initial frequency is caused the by the offset of the initial phase. During the adjustment of the phase compensator before a close-loop test, an error of operation is inevitable as the signal superposition is depending on subjective judgment. Therefore, an offset of initial phase was introduced, noted as _{offset}_{offset}

As _{offset}_{0} in an open-loop adjustment, from

When the system works in a close-loop mode, we also have:

Developing _{0}, using a Taylor development:

As the linearity shown in

From

It is seen from _{offset}_{offset}

It is the ratio of the slope of two experimental curves that represents the coefficient

Both experimental result and simulation result demonstrate the existence of a coefficient _{r} = _{0}. The value of _{real} =_{simulate}

We are going to explore this a little bit further. As a frequency offset _{offset}_{r} to
_{0} to

After mass loading, at the new resonance frequency

From the discussion of

A deduction follows the routine of

For the first term of the denominator, use

For the second term of the denominator, develop

Notice the linearity of _{b}

Substitute the denominator of

Using

Thus:

From

The literature expression shows that _{offset}_{0}, the slope of the phase-frequency curve _{a}_{b}

To calculate the coefficient _{0}, whereas at the point far away from _{0}, the slopes of the two curves deviate. The deviation is likely to be a result of non-ideal properties of the micro-cantilever. As for the feedback module, both two curves show a good linearity, and an approximate slope.

Based on literature result in _{real}_{simulate}

Gas experiments have been performed on a MEMS gas sensing system. The system is based on a silicon beam with a piezoelectric zinc oxide layer, and a SAD circuit loop is built to actuate the cantilever and detect its frequency shifts. The experimental results confirm, as predicted by another paper, the existence of a coefficient

As analyzed in this paper, the mass loading of the micro-cantilever modifies its MRF as well as the phase-frequency curves of the SAD circuit loop, and therefore the SWF shifts to meet the phase condition of

The authors would like to acknowledge the support by National 863 Project of China and Science Fund of China Post Doctoral Scientists.

Electrical model of piezoelectric micro-cantilever.

Impedance of the piezoelectric micro-cantilever.

Frequency selection network module.

Schematic illustration of a SAD circuit.

Simulation of phase-frequency curves of the SAD circuit loop, with AB and A’B’ the linear interval. Where _{0} = 65,670 Hz and Δ_{0} = 200 Hz.

Scanning electron micrographs of the micro-cantilever: (a) Before the deposition of PEO sensitive material. (b) After the deposition of PEO sensitive material.

Schematic drawing of the experimental setup of the open-loop tests [

Typical raw data of the frequency shift of a SAD circuit loop and the internal micro-cantilever. To improve the visibility of the data points, the data are sampled with a time step of 40 seconds. The abscissa is time and each unit represents 40 seconds.

Frequency shift in response to the injection of ethanol.

Experimental phase-frequency curves in compare with simulation results at MRF _{0} = 65,670 Hz and SWF _{r} = 65,708 Hz. (a) Simulation results for SAD[