With a presumption of small deformation, an electrical model of a piezoelectric micro-cantilever is developed [3].

The mechanical function analogy of the electrical components is presented in Equation (1). Where m_effect is the effective mass of the micro-cantilever, k is the spring constant, c₀ is the static capacitance of the piezoelectric layer, ξ is the damping coefficient. The k₁, k₂ are proportionality constants: (1) { C 1 = k 1 k 2 L 1 = m effect kk 1 k 2 R 1 = 2 ξ km effect kk 1 k 2 C 0 = c 0 − k 1 k 2

Therefore, the electrical components in the model can be determined by the mechanical properties of the beam, among which C1 and C0 can be seen as a constant, R1 and L1 will vary with the frequencies. Furthermore, the impedance of the micro-cantilever can be determined, as shown in Figure 2. As the impedance of C0 is much smaller than that of C1-R1-L1 branch, the impedance of the micro-cantilever is rather determined by that of C0, and has only a local minimum at the MRF f₀. 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 C0 in the electrical model was compensated by an external tunable capacitor. In an ideal situation, the output of the network should perform as a band-pass filter, having maximum amplitude at the mechanical resonance frequency (see Figure 3).

Figure 4 shows the block diagram of the SAD circuit loop. The micro-cantilever is integrated in the module of frequency selection network, which constitutes the frequency determining unit of the oscillator circuit, and serves like a band-pass filter with a central frequency equal to f₀. 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 f₀, 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, and its phase-frequency curve is φ_a, the rest of the SAD circuit loop is considered as a feedback module, with a gain of F and phase-frequency curve φ_b. When the SAD circuit loop working in a close-loop mode, the Barkhausen conditions should be met [15]: (2) | A ( f r ) F ( f r ) | = 1 φ a ( f r ) + φ b ( f r ) = 2 π

Generally, the amplitude condition in (2) is not a crucial one, as the non-linearity in the feedback module makes the amplitude of the signal always satisfy the first equation. Therefore, the working frequency is rather determined by the second Equation of (2), the phase condition.

Figure 5 illustrates the phase-frequency curves for frequency selection network as well as the feedback module. As seen from the figure, the feedback module shows a good linearity in its phase-frequency curves. And in a neighborhood of the mechanical resonance frequency, at least in the interval of f₀ ± 100 Hz, the frequency selection network also shows a good linearity. After mass loading, mechanical resonance frequency decreases, and φ_a (f) simply has a parallel shift of Δf₀, to φ_a (f).

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 V_vessel, and liquid ethanol with a volume V_analyte is jetted into the canister. When volatilized completely, the concentration of ethanol vapor follows Equations (3) and (4) where M is the molecular weight of the analyte, and P is the ambient atmospheric pressure: (3) C { mg / m 3 } = ρ analyte V analyte V vessel (4) C { ppm } = C { mg / m 3 } M 22.4 ⋅ 273 273 + T ⋅ P 1011325

For the experiments, we calculate the saturated concentration of ethanol vapor, known the ambient temperature and pressure of atmosphere. Then, an appropriate volume V_analyte is chosen, to make sure that with six liquid injections, the concentration of the ethanol vapor in the canister increases at a constant pace and is finally below the saturated level. The open-loop test and the close-loop one are performed in the same canister, with the same ambient conditions. The transfer pipette has a precision of measurement of 0.3%, which ensures the coherence of the two tests, and the comparability of the results.

An Open-loop test records the variation of mechanical resonance frequency f₀ 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 [16]. An aqueous solution of PEO was jetted on top of the cantilever. After the evaporation of the solvent, the tip of the cantilever was covered with a layer of sensitive material in the form of granule (Figure 6), and the MRF of the cantilever decreases to 65,670 Hz accordingly.

The open-loop test instruments system is illustrated by Figure 7. A signal generator provides an actuating signal, and the laser beam was focused on the free-moving end of the coated cantilever. The laser beam deflection signal was captured by a photodetector, and the amplitude of the oscillation was read out from the screen. By sweeping the actuating signal near the mechanical resonance frequency, f₀ 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 f_r of the SAD circuit loop was detected. Schematic diagram of the test can is illustrated in Figure 4. The resolving power of the tests is approximately 0.5 Hz. To ensure that the SAD circuit loop works in its closed-loop mode, the following two adjustments should be made. Firstly, the tunable capacitor in the frequency selection network should be adjusted to compensate C0, that is to say, at a frequency f ≠ f₀, 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 Figure 8. The liquid ethanol was jetted every 30 minutes in both of the tests. The set of triangle symbols represents the mechanical resonance frequency in the open-loop test. The step-form points record the working frequency of the SAD circuit loop as a function of time. As the volatilization takes place, the working frequency of the circuit loop decreases and at the end of the volatilization, the working frequency stabilizes gradually. As mentioned at the beginning, it is the coefficient α between the SWF shift Δf_r and the MRF shift Δf₀ 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 φ_offset. The frequency offset f_offset is thereby created. The frequency offset is 38 Hz in this experiment.

As φ_offset is the phase offset at f = f₀ in an open-loop adjustment, from (2) we have: (5) φ a ( f 0 ) + φ b ( f 0 ) = 2 π + φ offset

When the system works in a close-loop mode, we also have: (6) φ a ( f 0 + f offset ) + φ b ( f 0 + f offset ) = 2 π

Developing (6) at f = f₀, using a Taylor development: φ a ( f 0 ) + ∂ φ a ( f 0 ) ∂ f f offset + 1 2 ! ∂ φ a 2 ( f 0 ) ∂ f 2 f offset 2 + ... + φ b ( f 0 ) + ∂ φ b ( f 0 ) ∂ f f offset + 1 2 ! ∂ φ b 2 ( f 0 ) ∂ f 2 f offset 2 + ... = 2 π

As the linearity shown in Figure 4, in the interval of this development, the high order derivatives are far less than the first order derivative. Thus: (7) φ a ( f 0 ) + ∂ φ a ( f 0 ) ∂ f f offset + φ b ( f 0 ) + ∂ φ b ( f 0 ) ∂ f f offset = 2 π

From (5) and (7), we have: (8) f offset = φ offset ∂ φ a ( f 0 ) ∂ f + ∂ φ b ( f 0 ) ∂ f

It is seen from (8) how φ_offset causes f_offset, the offset of the initial frequency in experimental results. The literature expression of α later on will show that this coefficient will not be affected by the initial frequency offset.

Figure 9 shows the MRF and the SWF as a function of gas concentration. After six sequential ethanol injections of equal quantity, the MRF decreases from 65,670 Hz to 65,455 Hz, and the SWF decreases from 65,708 Hz to 65,498 Hz. A result of simulation of SWF calculated from the MRF measured is also presented for comparison.

It is the ratio of the slope of two experimental curves that represents the coefficient α. The linear expression is obtained by interpolating the curves using the least-square principle. As the close-loop has a higher resolution power, its data shows a better correlation: f 0 :     y = − 36.393 x + 65676 ,   R 2 = 0.9966 f r :     y = − 35.179 x + 65710 ,   R 2 = 0.9999 f r − s i m u l a t e :     y = − 35.089 x + 65710 ,   R 2 = 0.9969

Both experimental result and simulation result demonstrate the existence of a coefficient α which makes Δf_r = α·Δf₀. The value of α is also calculated: α_real = 0.9666, α_simulate = 0.9642.

We are going to explore this a little bit further. As a frequency offset f_offset is observed, we are going to demonstrate it has no influence to α. Suppose that the SWF of the SAD circuit loop decreases from f_r to f r ′, when the MRF decreases form f₀ to f 0 ′. Transform the definition f r − f r ′ = α ( f 0 − f 0 ′ ) to: (9) ( 1 − α ) ( f 0 − f 0 ′ ) = f offset ′ − f offset

After mass loading, at the new resonance frequency f 0 ′, a new phase offset φ offset ′ exists: (10) φ a ′ ( f 0 ′ ) + φ b ( f 0 ′ ) = 2 π + φ offset ′

From the discussion of Figure 4, there is: (11) φ a ( f ) = φ a ′ ( f − Δ f 0 )

A deduction follows the routine of (8), we have: (12) f offset ′ = φ offset ′ ∂ φ a ′ ( f 0 ′ ) ∂ f + ∂ φ b ( f 0 ′ ) ∂ f

For the first term of the denominator, use (11), we have: (13) ∂ φ a ′ ( f 0 ′ ) ∂ f = ∂ φ a ′ ( f 0 − Δ f 0 ) ∂ f = ∂ φ a ( f 0 ) ∂ f

For the second term of the denominator, develop ∂ φ b ( f 0 ) ∂ f using Taylor development at f 0 ′: (14) ∂ φ b ( f 0 ) ∂ f = ∂ φ b ( f 0 ′ ) ∂ f + 1 2 ! ∂ 2 φ b ( f 0 ′ ) ∂ f 2 Δ f 0 + 1 3 ! ∂ 3 φ b ( f 0 ′ ) ∂ f 3 Δ f 0 2 + ...

Notice the linearity of φ_b(f), we have: (15) ∂ φ b ( f 0 ′ ) ∂ f = ∂ φ b ( f 0 ) ∂ f

Substitute the denominator of (12) by (13) and (15), using (8), we have: (16) f offset ′ − f offset = φ offset ′ − φ offset ∂ φ a ( f 0 ) ∂ f + ∂ φ b ( f 0 ) ∂ f

Using (5) and (10), we have: (17) φ offset ′ − φ offset = ∂ φ b ( f 0 ) ∂ f Δ f 0

Thus: (18) f offset ′ − f offset = ∂ φ b ( f 0 ) ∂ f ∂ φ a ( f 0 ) ∂ f + ∂ φ b ( f 0 ) ∂ f Δ f 0

From (18) α can be written as: (19) α = ∂ φ a ( f 0 ) ∂ f ∂ φ a ( f 0 ) ∂ f + ∂ φ b ( f 0 ) ∂ f

The literature expression shows that α is independent with φ_offset. It is the phase-frequency curve of frequency selection network and of feedback module that determine the coefficient. As shown in Figure 5, at the point of f₀, the slope of the phase-frequency curve φ_a (f) is much larger than that of φ_b (f), thereby the coefficient α has a value close to 1.

To calculate the coefficient α from Equation (19), we draw the phase-frequency curves in Figure 10, where the experimental data is presented together with the simulation one. For the frequency selection network, both the results from experiment and simulation show a good linearity at the neighborhood of f₀, whereas at the point far away from f₀, 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 (19), the coefficient α can be determined from the slopes of the phase-frequency curves: α_real = 0.9659, α_simulate = 0.9640. These results confirm our findings in gas experiments. Furthermore, as the coefficient α is related to the phase-frequency curves of the frequency selection network and of the feedback module, it is a typical value for each realization of a SAD circuit loop.