An Adaptive Measurement System for the Simultaneous Evaluation of Frequency Shift and Series Resistance of QCM in Liquid

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Article An Adaptive Measurement System for the Simultaneous Evaluation of Frequency Shift and Series Resistance of QCM in Liquid

Ada Fort * , Enza Panzardi , Valerio Vignoli , Marco Tani , Elia Landi , Marco Mugnaini and Pietro Vaccarella

Department of Information Engineering and Mathematical Sciences, University of Siena, 53100 Siena, Italy; [email protected] (E.P.); [email protected] (V.V.); [email protected] (M.T.); [email protected] (E.L.); [email protected] (M.M.); [email protected] (P.V.) * Correspondence: [email protected]

Abstract: In this paper, a novel measurement system based on Quartz Crystal Microbalances is presented. The proposed solution was conceived specifically to overcome the measurement problems related to Quartz Crystal Microbalance (QCM) applications in dielectric liquids where the Q-factor of the resonant system is severely reduced with respect to in-gas applications. The QCM is placed in a Meacham oscillator embedding an amplifier with adjustable gain, an automatic strategy for gain tuning allows for maintaining the oscillator frequency close to the series resonance frequency of the quartz, which is related in a simple way with the physical parameters of interest. The proposed system can be used to monitor simultaneously both the series resonant frequency and the equivalent electromechanical resistance of the quartz. The feasibility and the performance of the proposed method are proven by means of measurements obtained with a prototype based on a 10-MHz AT-cut quartz. Citation: Fort, A.; Panzardi, E.; Vignoli, V.; Tani, M.; Landi, E.; Keywords: QCM sensors; in-liquid measurements; bridge oscillators Mugnaini, M.; Vaccarella, P. An Adaptive Measurement System for the Simultaneous Evaluation of Frequency Shift and Series Resistance of QCM in Liquid. Sensors 2021, 21, 1. Introduction 678. https://doi.org/10.3390/ In-liquid measurements with quartz-based sensors are used in many different ap- s21030678 plication fields, among which are chemical sensing, biosensing and viscosity measurements [1–4]. Received: 29 December 2020 Sensing systems based on quartzes exploit their electromechanical resonant behavior: Accepted: 16 January 2021 their piezoelectric properties provide a transduction of mechanical quantities into electrical Published: 20 January 2021 quantities, whereas the pure elastic behavior of the quartz, which is an almost ideal resonator, is extremely sensitive to the changes of mechanical properties (mass, stiffness Publisher’s Note: MDPI stays neutral and damping). These particular properties have been exploited to obtain many different with regard to jurisdictional claims in sensors: the most traditional ones are Quartz Crystal Microbalances (QCM), operating as published maps and institutional affil- mass sensors with resolutions down to a few nanograms [5–8], although, if used in liquid iations. environments, it is possible to employ resonant measurement systems to characterize liquid viscosity and density. In this contest, the use of resonant structures is widely diffused and spread in different applications. Just to mention a couple, García-Arribas et al. developed a magnetoelastic sensor to sensitively measure the viscosity of fluids, aimed at developing Copyright: © 2021 by the authors. an online and real-time monitoring of the lubricant oil degradation in machinery [9], Licensee MDPI, Basel, Switzerland. and Zang et al. used a self-oscillating tuning fork device to perform fast measurements This article is an open access article of viscosity and density, with the aim of characterizing different phases of liquids in a distributed under the terms and multiphase flow [10]. Moreover, in several applications, the research interest is focused on conditions of the Creative Commons the realization of distributed sensor networks using sensor fusion techniques [11–13]. Attribution (CC BY) license (https:// The basic structure of a quartz-based chemical or biosensor is the QCM, which is creativecommons.org/licenses/by/ usually an AT-cut quartz disk operating in the MHz range provided by two electrodes 4.0/).

Sensors 2021, 21, 678. https://doi.org/10.3390/s21030678 https://www.mdpi.com/journal/sensors Sensors 2021, 21, 678 2 of 23

on the two opposite surfaces. When the quartz vibrates in air, it is possible to assume no damping, and the modal frequencies of the device are set by the disk thickness [14]. In particular, it can be written [15]: s 1 µQ fsN ≈ N (1) 2t ρQ

where fsN represents the N-th modal frequency, N ∈ ℵ and N odd and t is the thickness of the quartz, whereas ρQ and µQ are the quartz density and shear modulus. A mass sensor is obtained by functionalizing one of the quartz surfaces with a material providing the selective adsorption of a target species. If, due to functionalization and to adsorption, a small additional mass (∆m) is deposited on one of the quartz surfaces, by assuming that the mechanical properties of the added layer are the same as those of the quartz, then the effect changes the resonator thickness and, accordingly, shifts the modal frequencies of a quantity (∆ fsN) related to the added mass:

∆t ∆m ∆ fsN = − fsN = − fsN (2) t mQ

where mQ is the quartz mass, and the linear relation stems from the assumption of small thickness variations. Notice that Equation (2), written in a different form, is also known as the Sauerbrey equation [16]. By means of the piezoelectric effect, the mechanical vibrations are converted into an electrical charge or voltage signal. Therefore, a QCM is, in principle, a linear mass sensor providing the frequency as an output quantity. A typical 10-MHz AT-cut QCM provides, theoretically, a response of ∆ fs1(Hz) = −0.8 ∆m (ng) [17]. A complete sensing system can be obtained by inserting the quartz into an electronic circuit providing the frequency metering [1]. Indeed, the measurement problem is far more complex [1,18]. First of all, the electromechanical vibrations can shift frequency not only due to the added mass (desired effect) but, also, due to the unwanted mechanical damping caused by the added layer, which has usually different and worse mechanical properties with respect to the quartz, especially in biosensors, where it consists of a lossy viscoelastic functionalization film and of the adsorbed target species (biological targets). Moreover, and more importantly, differently from AT-cut quartz oscillating in the air, where the surrounding medium can be well-described as an ideal fluid, and the standing elastic wave causing the vibration is confined in the solid disk, for in-liquid sensors, the surrounding medium is better described by a Newtonian fluid. In this case, part of the shear vibration can be transferred to the liquid and part of the elastic energy dissipated. The interaction with the fluid is, in this case, a non-negligible mechanical load (with both dissipative and conservative characteristics) that causes very large frequency shifts (up to hundreds of ppms). Due to this fact, some sensing systems exploit the dependency of the modal frequency of the quartz on the characteristics of the surrounding fluid to obtain viscosity measurements [14,19,20]. As a consequence, QCM-based measurement systems operating in liquid often couple the modal frequency shift assessment with the evaluation of the viscoelastic characteristics of the added layer through the monitoring of other dynamic parameters, such as the dissipation coefficient δ or, equivalently, the Q-factor [21,22]. This could unravel the inertial and dissipative behaviour of the ad-layer and enable distinguishing between the added mass and the surrounding liquid effects. To continue with the issues related to the QCM-based measurements, it must be underlined that the way in which the vibrations are generated and measured can influence their frequency, which can be different from the searched modal frequency: for instance, the front-end circuit loads and influences the electromechanical resonance [1]. In a typical QCM system, the quartz is embedded in the feedback network of an oscillator circuit, and its output frequency is measured by, e.g., a digital frequency meter. Sensors 2021, 21, 678 3 of 23

QCM systems with dissipation monitoring (D-QCM) also provide a measurement of the dissipation factor δ (equivalent to the Q-factor) by inducing transients, e.g., disconnect- ing the feedback loop [23] and evaluating the transient decay duration. This allows for assessing the damping coefﬁcient of the resonant electromechanical system. In this paper, the problems related to measurements based on QCM in in-liquid applications are summarized, and a novel system based on a Meacham oscillator embedding an ampliﬁer with adjustable gain is proposed. An automatic strategy is described that allows for gain tuning and maintaining the oscillator frequency close to the modal resonance of the quartz. From the adjusted gain value, an estimation of the series electromechanical resistance can be obtained, providing also the monitoring of the dissipative behavior of the electromechanical system through a robust estimation technique. This paper is organized as follows: in Section2, the operation of quartzes in liquid is presented, and the main measurement problems are discussed. In Section3, the proposed oscillator topology, the series resistance estimation technique and the gain adjustment strategy are described. In Section4, a prototype system, used to verify the proposed solution, is described, whereas Section5 shows and discusses the experimental results. The conclusions are drawn in Section6.

2. In Liquid Measurements Based on QCM and Oscillators In-liquid quartz resonators can be used either to assess the viscoelastic properties of a liquid or to measure the mass of a target biological species. In the latter case, a functionalization layer is deposited over one of the quartz surfaces, ensuring the selective adsorption of the target. The target species are usually dispersed in a liquid; therefore, in both cases, the pristine or functionalized surface of the quartz is in contact with a liquid medium. A general model for this kind of measurement set-up is represented in Figure1 , Sensors 2021, 21, x FOR PEER REVIEW 4 of 22 where the lumped parameter electrical equivalent circuit (Butterworth-Van Dyke) of a QCM operating at the ﬁrst modal resonance, fs1, is shown [1].

Figure 1. 1.Equivalent Equivalent lumped lumped parameter parameter circuit for circuit a Quartz for Crystal a Quartz Microbalance Crystal (QCM) Microbalance in in-liquid (QCM) in in- liquidapplications. applications.

More in detail, considering an AT-cut quartz with electric permittivity 𝜖 =𝜖, shear modulus 𝜇 =𝑐, density 𝜌, piezoelectric coefficient 𝑑, electrode area 𝐴, viscosity coefficient 𝜂 and mass mQ, the quartz is represented by the series of 𝐶 = 8 , 𝐿 = and 𝑅 = and by the electrical branch consisting only of the electrical capacitance, which, in air, is 𝐶 = . C0 can take a slightly different value in-liquid, due to a fringe electric field in the surrounding medium. The presence of the additional layer (functionalization and adsorbed target species) Δ with mass m is modeled while adding in the series an inductance 𝐿 = . Finally, the effect of the Newtonian fluid with a density 𝜌 and viscous coefficient 𝜂is represented by a series of a resistor and a nonideal inductors with a resulting impedance [24]:

1+𝑗 𝑍 =𝑡 𝜋𝑓 𝜌 𝜂 (3) 𝜇𝐴𝑑

Usually, since the quartz operates in a small frequency range close to fs1, in Equation (3), it can be assumed that f = fs1. Therefore, we get:

𝐿 = and 𝑅 =2 𝜋𝑓𝐿 (4)

Defining 𝐿,𝐶 𝑎𝑛𝑑 𝑅 , as shown in Figure 1, the impedance of the circuit shown in Figure 1 presents a series resonance 𝑓 = corresponding to the first mechanical modal frequency fs1 and a parallel one 𝑓 = . () The two resonances are very close, because C0 is always much larger than 𝐶. It can be seen that the series resonance depends only on the mechanical property of the quartz, whereas the parallel resonance depends also on C0; therefore, it is influenced by the parasitic components due to, for instance, the read-out circuit. In detail, the series frequency changes due to the presence of an additional mass (and this is the working principle of a microbalance), since the inductance LMass contributes to

Sensors 2021, 21, 678 4 of 23

The equivalent circuit is the parallel of the purely electrical branch (C0) and of the electromechanical branch, which is a series resonant circuit, representing with inductances the inertial properties of the mechanical system, with capacitance its elastic properties and, with resistances, viscous losses. Each component of the mechanical system (quartz, additional layer and nonideal liquid) is represented in the electrical domain by the appropriate equivalent components. In the circuit shown in Figure1, the functionalization layer is modeled as an additional rigid layer (mass load), but if needed, the model can be refined simply accounting also for its viscoelastic behavior, introducing a further resistor describing the viscous losses. The Newtonian fluid is represented by an inertial load and by a dissipative element. More in detail, considering an AT-cut quartz with electric permittivity eQ = e22, shear modulus µQ = c66, density ρQ, piezoelectric coefficient d26, electrode area Ae, viscosity d2 µ A m C = 26 Q e coefficient ηQ and mass Q, the quartz is represented by the series of Q1 8 π2t , ρ t3 η L = Q and R = Q and by the electrical branch consisting only of the Q1 2 2 Q1 µq C 8µQ Aed26 Q1 eQ Ae electrical capacitance, which, in air, is C0 = t . C0 can take a slightly different value in-liquid, due to a fringe electric field in the surrounding medium. The presence of the additional layer (functionalization and adsorbed target species) ∆m t2 with mass ∆m is modeled while adding in the series an inductance LMass = 2 2 2 . 8µQ Ae d26 Finally, the effect of the Newtonian fluid with a density ρL and viscous coefficient ηL is represented by a series of a resistor and a nonideal inductors with a resulting impedance [24]: 1 + j p Z = t2 π f ρ η (3) L 2 2 L L µq Aed26

Usually, since the quartz operates in a small frequency range close to fs1, in Equa- tion (3), it can be assumed that f = fs1. Therefore, we get:

t2 r ρ η L = L L and R = 2 π f L (4) L 2 2 L s1 L µq Aed26 4π fs1

Defining Lm, Cm and Rm, as shown in Figure1, the impedance of the circuit shown in 1 Figure1 presents a series resonance fs = √ corresponding to the first mechanical 2π LmCm√ 1 √ Cm+C0 modal frequency fs1 and a parallel one fsp = 2π . Lm(CmC0) The two resonances are very close, because C0 is always much larger than Cm. It can be seen that the series resonance depends only on the mechanical property of the quartz, whereas the parallel resonance depends also on C0; therefore, it is influenced by the parasitic components due to, for instance, the read-out circuit. In detail, the series frequency changes due to the presence of an additional mass (and this is the working principle of a microbalance), since the inductance LMass contributes to the overall inductance, but, even in the absence of an added mass, the presence of a Newtonian liquid causes a modal frequency shift, due to LL. The shift in this case can be described by the equation derived by Kanazawa and Gordon [25]: 3 r fs LL 2 ηLρL ∆ fs ≈ − ≈ − fs (5) 2 LQ1 πρQµQ However, only under the assumption of LL 1. Note that this equation holds only LQ1 for very small changes of Lm, which is not the case when passing from air to water. It is well-known that, when placed in an oscillator circuit, the quartz vibrates at a frequency comprised between the series and the parallel resonance frequencies, and the actual oscillation frequency within this range depends on the circuit topology and components. Sensors 2021, 21, x FOR PEER REVIEW 5 of 22

the overall inductance, but, even in the absence of an added mass, the presence of a New- tonian liquid causes a modal frequency shift, due to 𝐿. The shift in this case can be described by the equation derived by Kanazawa and Gordon [25]:

𝑓 𝐿 𝜂𝜌 Δ𝑓 ≈− ≈−𝑓 (5) 2 𝐿 𝜋𝜌𝜇

However, only under the assumption of ≪1. Note that this equation holds only for very small changes of 𝐿, which is not the case when passing from air to water. It is well-known that, when placed in an oscillator circuit, the quartz vibrates at a frequency comprised between the series and the parallel resonance frequencies, and the actual oscillation frequency within this range depends on the circuit topology and components. From the above discussion, it can be concluded that the mechanical properties of the resonator are related by known relationships to the series resonance frequency. Therefore, to obtain information about the mechanical properties either of the fluid or of the added mass exploiting a QCM-based oscillator, the oscillator should operate at a frequency as Sensors 2021, 21, 678 close as possible to the series resonance of the quartz in any5 of 23conditions; otherwise, the observed frequency shifts cannot be easily interpreted and related to the variations of the mechanical quantities of interest. From the above discussion, it can be concluded that the mechanical properties of the resonatorTo clarify are related the by above known relationships discussion, to the the series differe resonancent frequency. impedances Therefore, of the same 10-MHz AT- cutto quartz obtain information in air, water about the and mechanical in a solution properties of either water of the and ﬂuid or20% of the glycerol added are shown in Figure 2. mass exploiting a QCM-based oscillator, the oscillator should operate at a frequency as Theclose plots as possible are obtained to the series using resonance values of the quartzfor the in anydifferent conditions; components otherwise, the of the equivalent circuit in Figureobserved frequency1 assessed shifts cannoteither be from easily interpretedthe theoreti and relatedcal model to the variations presented of the in this section or using mechanical quantities of interest. experimentalTo clarify the data above [17]. discussion, As expected, the different impedances beyond of shifting the same 10-MHz the series AT-cut frequency about 2 kHz, thequartz effect in of air, pure water andwater in a solutionis to dramatically of water and 20% re glycerolduce arethe shown Q-factor in Figure due2. to the dissipative effect The plots are obtained using values for the different components of the equivalent circuit representedin Figure1 assessed by 𝑅 either, which from the takes theoretical quite model a large presented value in this (mor sectione orthan using ten times the resistance in air).experimental This reduction data [17]. leads As expected, to a beyondless stable shifting beha the seriesvior frequency and opens about 2many kHz, problems in the design the effect of pure water is to dramatically reduce the Q-factor due to the dissipative effect of oscillators.represented by RInL, whichthis respect, takes quite it a large must value be (more remembered than ten times that the resistance these inkinds of quartzes are used to measureair). This reduction very leadssmall to amasses. less stable behaviorThe typical and opens needed many problems mass in resolution the design is in the order of a few of oscillators. In this respect, it must be remembered that these kinds of quartzes are used nanograms;to measure very therefore, small masses. the The freq typicaluency needed relative mass resolution resolution is in the order has of to a few be smaller than 100 ppb. nanograms; therefore, the frequency relative resolution has to be smaller than 100 ppb.

FigureFigure 2. 2. ImpedanceImpedance (real and(real imaginary and imaginary parts in the upper parts plots in and the magnitudes upper and plots phases and in the magnitudes and phases in lower plots) of an AT-cut 10-MHz quartz in air (solid lines), in water (dashed lines) and in a solution theof lower water and plots) 20% glycerol of an (dottedAT-cut lines). 10-MHz The circuit quartz parameters in used air to(solid obtain thelines), plots arein derivedwater (dashed lines) and in a from the experimental data.

Figure3a shows the impedances of the 10-MHz quartz in water and in a 20% glycerol– water solution, with and without an additional parasitic capacitance of 2 pF in parallel with C0, whereas Figure3b illustrates the effects of an added mass of 10 µg. It can be seen that parasitic capacitances do not affect the series resonance position. Sensors 2021, 21, x FOR PEER REVIEW 6 of 22

solution of water and 20% glycerol (dotted lines). The circuit parameters used to obtain the plots are derived from the experimental data.

Figure 3a shows the impedances of the 10-MHz quartz in water and in a 20% glycerol–water solution, with and without an additional parasitic capacitance of 2 pF in par- 0 μ Sensors 2021, 21, 678 allel with C , whereas Figure 3b illustrates the effects of an added mass of 10 g. It can6 of be 23 seen that parasitic capacitances do not affect the series resonance position.

(a) (b)

FigureFigure 3.3. ImpedanceImpedance (magnitude(magnitude upperupper plotplot andand phasephase lowerlower plot)plot) ofof anan AT-cutAT-cut 1010 MHzMHz quartzquartz inin waterwater andand inin aa solutionsolution of water and 20% glycerol (a) with (dashed lines) and without (solid lines) a parasitic capacitance and (b) with (dashed of water and 20% glycerol (a) with (dashed lines) and without (solid lines) a parasitic capacitance and (b) with (dashed lines) or without (solid lines) a 10 μg rigid mass load. lines) or without (solid lines) a 10 µg rigid mass load. 3. Oscillator Topology and Working Principle 3. Oscillator Topology and Working Principle In QCM-based oscillators for in-liquid applications, the design should, usually, sat- In QCM-based oscillators for in-liquid applications, the design should, usually, satisfy isfy the following requirements: the following requirements: • The oscillator should oscillate at a frequency providing an estimation of the series • The oscillator should oscillate at a frequency providing an estimation of the series resonance of the quartz. resonance of the quartz. • One electrode of the quartz should be grounded in order to minimize the parasitic • One electrode of the quartz should be grounded in order to minimize the parasitic ef- effects due to the fluid in contact with the QCM surface and to simplify the develop- fects due to the ﬂuid in contact with the QCM surface and to simplify the development ment of the measurement chamber. of the measurement chamber. • Due to the large reduction of the Q-factor of the quartz, the oscillator circuit must be • Due to the large reduction of the Q-factor of the quartz, the oscillator circuit must be characterized by a high stability. characterized by a high stability. For these reasons, we selected as the oscillator topology the Meacham bridge, shown For these reasons, we selected as the oscillator topology the Meacham bridge, shown in Figure 4, which is characterized by a high stability [26], allows for the use of a grounded in Figure4, which is characterized by a high stability [ 26], allows for the use of a grounded quartz and oscillates at a zero-phase frequency, 𝑓 . The oscillation frequency can be used quartz and oscillates at a zero-phase frequency, f0◦0°. The oscillation frequency can be used 𝑅 asas anan estimationestimation ofof thethe seriesseries resonanceresonance ofof thethe quartz,quartz, as required,required, even if, withwith thethe Rm differentdifferent fromfrom 00 ΩΩ,, thethe seriesseries resonanceresonance frequencyfrequency isis closeclose toto butbut doesdoes notnot coincidecoincide withwith thisthis frequency.frequency. In this conﬁguration, we have a loop gain: ! ZQ R2 βAv = βn − βp Av = Av − (6) ZQ + R f R2 + R1

where ZQ is the equivalent impedance of the quartz (impedance of the network shown in Figure1); Av is the open loop gain of the differential ampliﬁer and β, βp and βn are the overall, the positive and the negative feedback networks gains, respectively. The oscillation conditions are obviously those that grant the existence of a couple of purely imaginary poles of the closed loop gain, i.e., the Barkhausen conditions, requiring the existence of a frequency where the loop gain has a unitary module and a phase equal to −180◦. Ideally, the open loop gain is assumed to be a real number not contributing to the phase of the loop gain. In this case, the oscillator operates at the frequency, fo, where the Sensors 2021, 21, 678 7 of 23

impedance of the quartz is purely real; therefore, at the oscillation frequency, it can be written: ZQ( fo) βn( fo) = (7) R f + ZQ( fo)

where ZQ( fo) ∈ <. Note that, to satisfy the phase Barkhausen condition, it must be granted that βp( fo) > βn( fo); therefore, in any working condition, the following inequality should be satisﬁed: R R f 1 < (8) R2 ZQ( fo) Sensors 2021, 21, x FOR PEER REVIEW 7 of 22 Therefore, in the Meacham oscillator, at the oscillation frequency, the quartz operates as a pure resistive component; therefore, fo = f0◦ , and, if Rm is sufﬁciently small, we have ZQ( fo) ≈ Rm and fo = f0◦ ≈ fs.

V+ + V Av o R2 V- -

Gnd Rf Zq

Gnd

Figure 4. Meacham oscillator topology. Figure 4. Meacham oscillator topology. The inequality in (8) is particularly critical for QCMs operating in-liquid. In fact, these devicesIn this present configuration, a largely variable we value have of aR loopm due gain: to the contributions of the surrounding liquids, which can have different viscosities. Besides the phase condition (Equation (7)), the marginal𝑍 stability of𝑅 the system is granted by selecting the value𝛽𝐴 =(𝛽 of Avto−𝛽 also)𝐴 satisfy = 𝐴 the Barkhausen− module condition, (6) 𝑍 +𝑅 𝑅 +𝑅 which is: ! R2 ZQ( fo) where 𝑍 is the equivalentAv impedance− of the quartz= (impedance1 of the network (9) shown in R2 + R1 ZQ( fo) + R f Figure 1); 𝐴 is the open loop gain of the differential amplifier and 𝛽, 𝛽 and 𝛽 are the overall,Note the that, positive if Equation and (9) the is satisﬁed, negative given feedba the valueck networks of Av and those gains, of therespectively. resistances The oscilla- R , R and R , it is possible to derive the value of Z ( f ) ≈ R . tion1 conditions2 f are obviously those that grantQ o the existencem of a couple of purely imagi- Unfortunately, this equality cannot be perfectly met in the reality, and for a QCM, this naryproblem poles is particularlyof the closed critical, loop because gain, thei.e., value the Barkhausen of ZQ( fo) varies conditions, during operations. requiring the existence of a frequencyIn any case, where the loop the gain loop module gain should has a be unitary as close tomodule 1 as possible and buta phase larger equal than 1; to −180°. otherwise,Ideally, the the oscillator open behaves loop gain as a stableis assumed system, andto be its outputa real isnumber not persistent. not contributing to the These considerations clarify how real oscillator circuits never behave as pure marginally phase of the loop gain. In this case, the oscillator operates at the frequency, 𝑓, where the stable systems; on the contrary, they can be modeled in the linear regime with poles with impedance of the quartz is purely real; therefore, at the oscillation frequency, it can be written:

𝑍(𝑓) 𝛽(𝑓) = (7) 𝑅 +𝑍(𝑓)

where 𝑍(𝑓)∈ℜ. Note that, to satisfy the phase Barkhausen condition, it must be granted that 𝛽(𝑓) >𝛽(𝑓); therefore, in any working condition, the following inequality should be satisfied:

𝑅 𝑅 < (8) 𝑅 𝑍(𝑓) Therefore, in the Meacham oscillator, at the oscillation frequency, the quartz operates as a pure resistive component; therefore, 𝑓 =𝑓°, and, if 𝑅 is sufficiently small, we have 𝑍(𝑓)≈𝑅 and 𝑓 =𝑓° ≈𝑓. The inequality in (8) is particularly critical for QCMs operating in-liquid. In fact, these devices present a largely variable value of 𝑅 due to the contributions of the surrounding liquids, which can have different viscosities. Besides the phase condition (Equation (7)), the marginal stability of the system is granted by selecting the value of 𝐴 to also satisfy the Barkhausen module condition, which is:

Sensors 2021, 21, 678 8 of 23

the real part larger than zero, and they always also operate in nonlinear regions. As a consequence, the imaginary part of the unstable poles, which sets the frequency of the oscillator output, does not coincide with the frequency of the marginally stable poles. To deepen this discussion, the overall gain of the ampliﬁer with feedback, A f , can be written in the Laplace domain in the following way:

Av(s) N(s) A f (s) = = (10) 1 + β(s)Av(s) s2 2ζ P(s) 2 − ω s + 1 ωN N s2 2ζ where N(s) and P(s) 2 − ω s + 1 are polynomials in s. ωN N By describing the gain of the ampliﬁer in Equation (10), the denominator factor, responsible for the oscillating behavior, is written in the Bode form. Therefore, the couple of unstable poles responsible for the oscillation are: q 2 p1,2 = ζωN ± jωN 1 − ζ (11)

where ζ and ωN are positive real numbers, and α = ζωN is the positive real part of the poles. Consequently, the response to the initial condition of the system contains a nonvanishing contribution due to this couple of poles that can be written as: q ζωt 2 V0unst(t) = Ae cos ωN 1 − ζ t + φ (12)

where A and φ are the parameters related to the initial conditions. Therefore, the oscillator operates at the frequency fo, deﬁned as follows:

ω q f = N 1 − ζ2 (13) o 2 π

If Equation (9) is perfectly satisﬁed, and if Rm is sufﬁciently small, then the poles are marginally stable, and ζ is equal to zero; therefore, we can write:

ωN = 2π fS (14)

In this case, the oscillator frequency estimates the series resonance frequency as desired. On the other hand, if the loop gain module at the frequency selected by the phase condition is larger than one, then the frequency of the oscillator is different from the searched resonance frequency, and we can approximate the difference, e f , between the two frequencies as follows: q 2 e f = fs − fo ≈ fs 1 − 1 − ζ (15)

It is interesting relating the deviation e f , which is a measurement error, to the magnitude of the loop gain module. To this end, we propose a simplified analysis considering Av as a real number and the value of Rm small. To obtain the relationship between the oscillator characteristics, i.e., how much the loop gain is larger than 1 and the measurement error, it is convenient to write the denominator of the amplifier gain, A f , defined in Equation (10), with the following equation: ! s2 2ζ ( ) − + = ( ) + ( ) = ( + ( )) ( ) P s 2 s 1 Dβ s Nβ s Av 1 Avβ s Dβ s (16) ωN ωN

where Nβ(s) and Dβ(s) are the numerator and the denominator of β, respectively. At the frequency satisfying the phase condition, which is approximately fs, β is a real number; Sensors 2021, 21, 678 9 of 23

therefore, Dβ( fo) + Nβ( fo)Av is either a real number and its imaginary part is zero and or a pure imaginary number with a null real part.

s2 2ζ ( ) = − + = | + ( )| ∈ < α s 2 s 1 and Γ 1 Avβ fo . (17) ωN ωN

Hence, in the ﬁrst case, Nβ( fo) and Dβ( fo) ∈ <, we have:

|Re(P( fo))Re (α( fo)) − Im(P( fo))Im(α( fo))| = |1 + β( fo)Av| Dβ( fo) (18)

While, in the second case, Nβ( fo) and Dβ( fo) ∈ =, we have:

|Re(P( fo))Im (α( fo)) + Im(P( fo))Re(α( fo))| = |1 + β( fo)Av| Dβ( fo) (19)

At the frequency fs, we have Re(α( fs)) = 0 and Im(α( fs)) = 2ζ. So, in the ﬁrst case, we derive: Rm + R f (R2 + R1) RmR2 + R1Rm + R f R2 + R f R1 + AvRmR1 − AvR f R2 2ζ ≈ Γ = (20) |Im(P( fo))| Im(P( fo))

While, in the second case: Rm + R f (R2 + R1) 2ζ ≈ Γ (21) Re(P( fo))

The last equations describe accurately the behavior of the oscillator for a small value of Rm, whereas, for larger values of Rm, the oscillator frequency deviates from fo in a different manner. In any case, Equations (20) and (21) show that the larger the loop gain is with respect to 1, the larger the coefficient ζ and the deviation between the oscillator output frequency and the quartz series resonance frequency. Moreover, and more importantly, this error is not independent from the measurement conditions, and it cannot be compensated, since it varies with Rm. Concluding the deviation of the oscillator frequency from the measurand is a function of Av and of the characteristics of the fluid. This discussion clarifies that the operating frequency of the oscillator deviates from the desired one (in this case, the series resonance of the quartz) if the loop gain module is much larger than 1 and that the deviation depends on the working conditions of the QCM. To mitigate these problems, we propose a Meacham oscillator for AT-cut 10-MHz quartzes based on a large band differential amplifier with variable gain (VGA) VCA842, by Texas Instruments (Dallas, TX, USA). With this solution, it is possible to adaptively change the open loop differential gain, Av, in Equation (9) to maintain the loop gain as close to 1 as possible, irrespective of the working conditions of the quartz.

3.1. Effect of the Phase of the Amplifier The oscillator working frequency corresponds exactly to the zero-phase frequency of the quartz only in the case of an ideal amplifier; in reality, due to the behavior of a real amplifier, the measured frequency deviates both from the series resonance frequency and from the zero-phase frequency, and the effect of the amplifier nonidealities grows, increasing the acoustic losses. ◦ In particular, the assumption of Av having a 0 phase around 10 MHz (typical working frequency for QCM) accepted until now is critical; this means that the amplifier has to be chosen with extreme care, because the phase lag of the amplifier is compensated by the phase of the feedback network gain, bringing the quartz to operate far from the series resonance. Sensors 2021, 21, x FOR PEER REVIEW 10 of 22

In particular, the assumption of 𝐴 having a 0° phase around 10 MHz (typical working frequency for QCM) accepted until now is critical; this means that the amplifier has to be chosen with extreme care, because the phase lag of the amplifier is compensated by the Sensors 2021, 21, 678 phase of the feedback network gain, bringing the quartz to operate far from the10 series of 23 resonance. The selected amplifier phase is smaller than 6° at 10 MHz for all the possible gain values, granting satisfactory operations. In this subsection, the effect of the amplifier The selected amplifier phase is smaller than 6◦ at 10 MHz for all the possible gain phase lag is evaluated. values, granting satisfactory operations. In this subsection, the effect of the amplifier phase In Figure 5, the oscillator is considered to operate at the optimum gain magnitude so lag is evaluated. that Equation (9) is satisfied, replacing Av with |Av(fo)|. In this figure, the oscillator work- In Figure5, the oscillator is considered to operate at the optimum gain magnitude ing frequency (cyan line) is compared to the quartz series resonance frequency (black so that Equation (9) is satisfied, replacing Av with |Av(f o)|. In this figure, the oscillator dashed line) and to the zero-phase frequency (green line) in the case of R2/R1 = 10 and Rf = working frequency (cyan line) is compared to the quartz series resonance frequency (black 200 Ω and taking into a dominant pole the transfer function for the amplifier gain. The dashed line) and to the zero-phase frequency (green line) in the case of R2/R1 = 10 and comparison points out that the deviation is acceptable also when the fluid is characterized Rf = 200 Ω and taking into a dominant pole the transfer function for the amplifier gain. The bycomparison large losses. points In fact, out it that can the be deviation seen that isthe acceptable effect introduced also when by the the fluid amplifier is characterized phase lag (cyanby large curve losses. in Figure In fact, 5) itis can the beon seene providing that the an effect overestimation introduced by of the𝑓0° amplifier, but since phase 𝑓0° itself lag is(cyan an underestimation curve in Figure 5of) is𝑓 the, the one small providing amplifier an phase overestimation lag partially of fcompensates0◦ , but since fthe0◦ itself error inducedis an underestimation by the large acoustic of fs, the losses small in amplifierthe liquid phase and provides lag partially a working compensates frequency the errorof the oscillatorinduced bycloser the to large the acousticseries resonance losses in thefrequency. liquid and Obviously, provides if a the working phase frequency lag becomes of large,the oscillator the error closer becomes to the very series high, resonance as can frequency. be seen in Obviously, Figure 5, ifwhere the phase the magenta lag becomes line showslarge, the the effect error of becomes a large veryamplifier high, phase as can lag. be seen in Figure5, where the magenta line shows the effect of a large amplifier phase lag. frequency (Hz) frequency

(a) (b) β 𝑅 β β FigureFigure 5. 5. (a()a ) βasas a afunction function of of the the freque frequencyncy at at different different values values of of Rm:: upper plotplot β module andand lowerlower plotplotphase phase of ofβ .(. (bb)) ComparisonComparison between between the the series series resonance resonance of of an an AT-cut AT-cut 10MHz quartz operating inin aa NewtonianNewtonian fluidfluid asas aa functionfunction of of the the series resistance, 𝑓 (which would be the oscillator frequency in the case of an ideal amplifier) and 𝑓 , which is the oscil- series resistance, f°◦ (which would be the oscillator frequency in the case of an ideal amplifier) and f , which is the oscillator 0 Ω o lator frequency accounting for the real amplifier behavior, considering R2/R1 = 10 and 𝑅 = 200 . frequency accounting for the real amplifier behavior, considering R2/R1 = 10 and R f = 200 Ω.

Nonetheless,Nonetheless, the the effect effect of of even even a small a small phase phase lag lag becomes becomes very very important important when when op- eratingoperating with with a gain a gain that that does does not not satisfy satisfy Equa Equationtion (9). (9). This This can can be be seen extending throughthrough simulations,simulations, the the results results obtained obtained from from the theoreticaltheoretical analysisanalysis presentedpresented inin thethe previous previous subsectionsubsection concerning concerning the the frequency frequency error error ef due to the operation withwith largelarge gainsgainsof of the the amplifier,amplifier, taking taking into into account account also also the the frequency frequency behavior of the amplifier.amplifier. AsAs anan example,example, FigureFigure 66 showsshows the the frequency error defineddefined inin EquationEquation (15)(15) forfor anan AT-cutAT-cut 10-MHz 10-MHz quartz, quartz, consideringconsidering the circuitcircuit componentscomponents reported reported in thein the legend, legend, as a as function a function the amplifier the amplifier open openloop loop gain gain (direct (direct current current (DC)value) (DC)valu normalizede) normalized with respect with respect to the valueto theA valuevopt (DC 𝐴 value), (DC value),which which is the valueis the ofvalue the gainof the needed gain needed to satisfy to satisfy Equation Equation (9). In this(9). In case, this even case, a even small a ◦ smallphase phase lag of lag−5.68 of −5.68°introduced introduced by the by selected the selected amplifier amplifier causes causes very large very errors.large errors.

Sensors 2021, 21, x FOR PEER REVIEW 11 of 22

3.2. Evaluation of the Series Resistance

If the open loop amplifier gain 𝐴𝑣 = 𝐴𝑣𝑜𝑝𝑡, which is the value that verifies Equation (9), is known, then the value of 𝑅𝑚 can be evaluated by inverting the same equation, under the assumption of small losses and ideal amplifier behavior, obtaining:

𝑅𝑅 1 − 𝑅 𝑅 +𝑅 𝐴 𝑅 = (22) 1 𝑅 1+ − 𝐴 𝑅 +𝑅

where, due to the low loss assumption, we consider 𝑍(𝑓) =𝑅. Indeed, as diffusely discussed in the previous subsections, the assumptions used to derive Equation (22) are not fully satisfied, and Equation (22) provides only an estimation of 𝑅𝑚with a certain degree of uncertainty. To evaluate the quality of this estimation, simulations were performed, taking into account the behavior of the real amplifier in terms of the frequency response and considering also large values of 𝑅𝑚. The simulation results showed that the accuracy of the 𝑅𝑚 estimation based on Equation (22) is really very high (the error is always smaller than few Ωs). As an example, Figure 7 shows the error between the “true” value of 𝑅𝑚 used for the simulations of the quartz behavior and the value obtained from Equa- tion (22), considering also the amplifier nonideality (as in Figure 6), i.e., accounting for the Sensors 2021, 21, 678 11 of 23 amplifier phase lag.

6000 R = 60 m R = 20 5000 m R = 5 m 4000 R = 240 m R = 270 3000 m

2000

1000

-1000 234567 A /A v vopt

FigureFigure 6. 6.The The error erroref definedef defined in Equationin Equation (15) evaluated(15) evaluated for an for AT-cut an AT-cut 10 MHz 10 quartz, MHz consideringquartz, considering Ω ◦ RR22/R/R11 == 1010 andandR Rf =f = 200 200Ω , and, and a phasea phase lag lag of the of the amplifier ampli offier−5.68 of −5.68°.. 3.2. Evaluation of the Series Resistance

If the open loop amplifier gain Av = Avopt, which is the value that verifies Equation (9), is known, then the value of Rm can be evaluated by inverting the same equation, under the assumption of small losses and ideal amplifier behavior, obtaining:

R2R f 1 − R f R1+R2 Avopt Rm = (22) 1 + 1 − R2 Avopt R2+R1

where, due to the low loss assumption, we consider ZQ( fo) = Rm. Indeed, as diffusely discussed in the previous subsections, the assumptions used to derive Equation (22) are not fully satisfied, and Equation (22) provides only an estimation of Rm with a certain degree of uncertainty. To evaluate the quality of this estimation, simulations were performed, taking into account the behavior of the real amplifier in terms of the frequency response and considering also large values of Rm. The simulation results showed that the accuracy of the Rm estimation based on Equation (22) is really very high (the error is always smaller than few Ωs). As an example, Figure7 shows the error between the “true” value of Rm used for the simulations of the quartz behavior and the value obtained from Equation (22), considering also the amplifier nonideality (as in Figure6), i.e., accounting for the amplifier phase lag. Finally, concerning the uncertainty of the estimation related to the uncertainty of the gain estimation, we can write:

∂R R f u(R ) ≈ m u(A ) = u(A ) (23) m ∂A v 2 v v A2 1 + 1 − R2 v Av R1+R2

which, considering operating close to the optimum gain, can be expressed as a function of Rm as follows: 2 RmR1 − R f R2 ( ) = ( ) u Rm 2 u Av (24) R f (R1 + R2) Sensors 2021, 21, x FOR PEER REVIEW 12 of 22

Sensors 2021, 21, 678 12 of 23 mest - R - m R

Figure 7.7.Error Error in in the the estimation estimation of R ofm from𝑅 from Equation Equation (22) for (22) an AT-cut for an 10 AT-cut MHz quartz10 MHz considering quartz consider- Ω Ring2/ RR12/R= 101 = and10 andR f = 𝑅 200 =Ω 200. .

∗ R f R2 Showing that there exists an optimum value of Rm, R = , where the estimation Finally, concerning the uncertainty of the estimationm relatedR1 to the uncertainty of the of R is robust with respect to the estimation errors of the optimum gain. gainm estimation, we can write: As a concluding remark, it is shown that the estimation of the series resistance value is found to be significantly more𝜕𝑅 robust than the one of the series𝑅 resonance frequency, and 𝑢(𝑅) ≈ u(𝐴)= 𝑢(𝐴) as such, Rm is a very useful parameter, which should be taken into account for the correct 𝜕𝐴 1 𝑅 (23) interpretation of measurements. 𝐴 1 + − 𝐴 𝑅 +𝑅 4.which, Measurement considering System operating Architecture close and to the Gain optimum Adjustment gain, Procedure can be expressed as a function of RmThe as proposedfollows: measurement system is composed by the oscillator, an amplitude meter able to monitor the oscillator output amplitude and a control block that provides gain 𝑅 𝑅 −𝑅 𝑅 adjustment with a feedback loop exploiting the measured oscillator output amplitude, as (24) 𝑢(𝑅) = 𝑢(𝐴) shown in Figure8. 𝑅(𝑅 +𝑅) Gain adjustment is provided by a digital controller (the block named “gain ad- ∗ justmentShowing logic” that in Figure there8 exists) that executesan optimum the iterative value of dichotomic Rm, 𝑅 = search, where of the the opti- estimation mum gain value illustrated in Figure8b with the objective of minimizing the quantity of Rm is robust with respect to the estimation errors of the optimum gain. ( ) R2 Re(ZQ fo ) γ = As Av a concluding+ − remark, it is − shown1 in N that steps. the estimation of the series resistance value R2 R1 Re(ZQ( fo))+R f is foundEach to step belasts significantly a time Ts ,more so the robust search than process the concluded one of the in seriesN iterations, resonance a mea- frequency, surementand as such, sample 𝑅𝑚of is the a very oscillator useful frequency, parameter,fo, and whichRm is sh providedould be aftertaken a timeinto equalaccount to for the tcorrectmeas = NT interpretations. For the real-time of measurements. monitoring of adsorption phenomena, tmeas must be less than about 2 s. 4. MeasurementThe search algorithm System is Architecture based on the twoand following Gain Adjustment ideas: Procedure • theThe system proposed works measurement in a predefined system range ofis Rcomposm, withedRm by< R themMAX oscillator,. an amplitude me- • At the start, the gain is set at the maximum value, and this is sufficient for the oscillator ter able to monitor the oscillator output amplitude and a control block that provides gain to also work in the case of Rm = R (minimum Q-factor), with the oscillator adjustment with a feedback loop exploitingmMAX the measured oscillator output amplitude, as behaving as an unstable (or marginally stable) system with Av ≥ Avopt. •shownAt eachin Figure search 8. process step, the amplitude of the oscillator output signal is measured and used to detect the presence of persisting oscillations. The search procedure, described in Figure8b, is based on reducing the gain on a dichotomic basis, checking for the presence of oscillations with an amplitude larger than a predefined threshold related to the noise floor and taking as fo the frequency measured at

Sensors 2021, 21, 678 13 of 23

Sensors 2021, 21, x FOR PEER REVIEW 13 of 22

the last step. The corresponding gain is the estimated Avopt and can be used to assess the value of Rm based on Equation (9).

oscillator Gain adjustment logic Amplitude/Frequency meter

(a)

k=0

Set Av(0)=AvMAX Δ Av(0) = AvMAX-Avmin

NO N > k Avopt=Av(N) STOP YES Measure oscillator output

amplitude (Vo)

Δ Δ Δ Δ Av(k)= Av(k-1)/2 YES NO Av(k)= Av(k-1)/2 Δ Δ Av(k)=Av(k-1) - Av(k) Vo> Vthr Av(k)=Av(k-1) + Av(k) k=k+1 k=k+1

(b)

FigureFigure 8. (a) 8.Schematics(a) Schematics of the of thebuilding building blocks blocks op operationserations of thethe proposed proposed system. system. (b) ( Gainb) Gain adjustment adjustment strategy. strategy.

Using N search steps, the gain can be adjusted with a resolution given by ∆Av = Gain− adjustment is provided by a digital controller (the block named “gain adjust- Av MAX Av MIN , and consequently, using the same procedure adopted to derive the uncertainty ment logic”2N in Figure 8) that executes the iterative dichotomic search of the optimum gain ∂Rm Av MAX−Av MIN as in Equation (24), the resolution of the Rm measurements is ∆Rm = . value illustrated in Figure 8b with the objective of minimizing∂Av the2N quantity 𝛾= Notice that, as usual, accurate measurements need longer measurement times; there- (())) 𝐴 fore, the− settings in terms of − gain 1 in resolution N steps. and measurement time must be selected as a (()) trade-off. Each step lasts a time Ts, so the search process concluded in N iterations, a measure- mentMeasurement sample of System the oscillator Setup frequency, 𝑓, and 𝑅 is provided after a time equal to tmeas = NTs. ForThe the system, real-time described monitoring in principle of adsorption in Figure8, wasphenomena, implemented tmeas in must the laboratory be less than to about 2 s. test the performance of the proposed oscillator. The implemented measurement set-up is describedThe search in Figure algorithm9. In particular, is based on the the AT-Cut two 10-MHzfollowing quartz ideas: (gold electrodes having • a diameter of 0.6 cm) was placed in an ad-hoc-built measurement chamber that allowed the system works in a predefined range of 𝑅, with 𝑅 < 𝑅. • forAt static the start, measurements the gain is in set liquid. at the The maximum measurement value, chamber, and this shown is sufficient in Figure for9b–d, the oscil- has a cylindrical shape with a 49-mm outer diameter and 50-mm height. The chamber is 𝑅 𝑅 composedlator to also of two work main in blocksthe case (see of Figure = 9c,d), the (minimum bottom one Q-factor), is a polyvinyl with chloride the oscillator behaving as an unstable (or marginally stable) system with 𝐴 ≥ 𝐴. • At each search process step, the amplitude of the oscillator output signal is measured and used to detect the presence of persisting oscillations. The search procedure, described in Figure 8b, is based on reducing the gain on a dichotomic basis, checking for the presence of oscillations with an amplitude larger than a predefined threshold related to the noise floor and taking as 𝑓 the frequency measured at the last step. The corresponding gain is the estimated 𝐴 and can be used to assess the value of 𝑅 based on Equation (9).

Sensors 2021, 21, x FOR PEER REVIEW 14 of 22

Using N search steps, the gain can be adjusted with a resolution given by Δ𝐴 = , and consequently, using the same procedure adopted to derive the uncertainty as in Equation (24), the resolution of the 𝑅 measurements is Δ𝑅 = . Notice that, as usual, accurate measurements need longer measurement times; therefore, the settings in terms of gain resolution and measurement time must be selected as a trade-off.

Measurement System Setup The system, described in principle in Figure 8, was implemented in the laboratory to test the performance of the proposed oscillator. The implemented measurement set-up is described in Figure 9. In particular, the AT-Cut 10-MHz quartz (gold electrodes having a diameter of 0.6 cm) was placed in an ad-hoc-built measurement chamber that allowed for static measurements in liquid. The measurement chamber, shown in Figure 9b–d, has a Sensors 2021, 21, 678 14 of 23 cylindrical shape with a 49-mm outer diameter and 50-mm height. The chamber is composed of two main blocks (see Figure 9c,d), the bottom one is a polyvinyl chloride (PVC) material and hosts a proper designed holder and connector for the quartz. The quartz is sandwiched(PVC) material between and hoststwo conductive a proper designed O-rings. holder The andtop connectorblock (cap) for is the stainless quartz. steel The and presentsquartz a is central sandwiched hole, between under twowhich conductive the quartz O-rings. is placed. The top The block steel (cap) cap is stainless is grounded steel by and presents a central hole, under which the quartz is placed. The steel cap is grounded by means of spring-loaded contacts and connected to the upper quartz electrode. The quartz means of spring-loaded contacts and connected to the upper quartz electrode. The quartz upperupper surface surface acts acts as as the the bottom bottom wall ofof aa measurement measurement well well with with a diameter a diameter of 11.5 of mm11.5 mm andand a height a height of of 2 2mm, mm, guaranteeing guaranteeing thethe liquidliquid deposition deposition on on the the quartz quartz surface. surface. All the All the teststests in liquid in liquid are are performed performed using using thethe volumevolume of of the the liquid liquid such such that that the thicknessthe thickness of the of the q liquid layer is much larger than the extinction depth δ0 = 2ηL [27]; therefore, the liquid liquid layer is much larger than the extinction depth 𝛿 = ωsρ [27]; therefore, the liquid L can be seen as a semi-inﬁnite layer. can be seen as a semi-infinite layer.

Sensors 2021, 21, x FOR PEER REVIEW 15 of 22

(a)

Figure 9. (a) Laboratory set-up for the characterization of the proposed system. (b–d) Measurement chamber: assembled Figure(b) and 9. building (a) Laboratory blocks views set-up (c,d ).for the characterization of the proposed system. (b–d) Measurement chamber: assembled (b) and building blocks views (c,d).

The quartz is connected to the developed Meacham oscillator; the gain of the VGA is set through the 16-bit D/A converter of an acquisition board (NI PCI 6052E, D/A) and a conditioning amplifier. With reference to Figure 4, the Meacham bridge was realized using R1 = 2.4 kΩ, R2 = 24 kΩ and Rf = 200 Ω (which are the same values considered for the all the simulations whose results were presented in the previous sections). The amplitude and the frequency of the oscillator output are digitally measured by a LabView VI executed by a PC after having acquired, with the same acquisition board (16- bit A/D), a frequency downshifted version of the signal. The frequency downshift of the oscillator output is obtained using an ad-hoc developed circuit hosting a mixer and a low- pass filter used to mix the oscillator output with a reference signal the with frequency 𝑓 generated by a signal generator AG33220A (Agilent, Santa Clara, CA, USA). The 𝑓 is automatically selected by exploiting a coarse measurement of 𝑓, performed during the ∗ set-up phase, with 𝐴 = 𝐴, such that 𝑓 = 𝑓 − 𝑓. The frequency 𝑓 is selected to guarantee, during the whole measurement, 𝑓 > 𝑓 and Δ𝑓 = 𝑓 −𝑓 in the kHz range. The mixer output signal is acquired with a suitable sampling frequency for the time windows of the duration tacq < Ts. The gain adjustment algorithm (Figure 8b) is implemented by the same VI controlling the acquisition and performing the amplitude measurements. The same VI assesses also the frequency of the acquired signal at each search process step, using an accurate single tone estimation algorithm. The selected acquisition time is 150 ms, and the optimum gain search procedure consists of a number of steps, N, that can be selected in the range 9–12. The measurement time, tmeas, was fixed at 2 s; therefore, each two seconds, the system provides and saves a measurement of 𝑓 performed with the optimum gain (obtained at the last step of the gain adjustment procedure) and, for comparison sake, a measurement of 𝑓 obtained at the starting step of the gain adjustment procedure (with maximum gain). The TI VCA842 differential amplifier has a gain range 1/10 < Av <10, but, in this application, we consider Avmin = 1 and a linear analog control of the gain value, obtained by means of a voltage input VG. The gain is related to the control voltage by the following equation:

𝐴 = 4.90 𝑉 (𝑉 +1𝑉)

Sensors 2021, 21, 678 15 of 23

The quartz is connected to the developed Meacham oscillator; the gain of the VGA is set through the 16-bit D/A converter of an acquisition board (NI PCI 6052E, D/A) and a conditioning amplifier. With reference to Figure4, the Meacham bridge was realized using R1 = 2.4 kΩ, R2 = 24 kΩ and Rf = 200 Ω (which are the same values considered for the all the simulations whose results were presented in the previous sections). The amplitude and the frequency of the oscillator output are digitally measured by a LabView VI executed by a PC after having acquired, with the same acquisition board (16-bit A/D), a frequency downshifted version of the signal. The frequency downshift of the oscillator output is obtained using an ad-hoc developed circuit hosting a mixer and a low-pass filter used to mix the oscillator output with a reference signal the with frequency fre f generated by a signal generator AG33220A (Agilent, Santa Clara, CA, USA). The fre f is automatically selected by exploiting a coarse measurement of fo, performed during the ∗ set-up phase, with Av = AvMAX, such that fre f = f o − fo f f set. The frequency fo f f set is selected to guarantee, during the whole measurement, fo > fre f and ∆ f = fo − fre f in the kHz range. The mixer output signal is acquired with a suitable sampling frequency for the time windows of the duration tacq < Ts. The gain adjustment algorithm (Figure8b) is implemented by the same VI controlling the acquisition and performing the amplitude measurements. The same VI assesses also the frequency of the acquired signal at each search process step, using an accurate single tone estimation algorithm. The selected acquisition time is 150 ms, and the optimum gain search procedure consists of a number of steps, N, that can be selected in the range 9–12. The measurement time, tmeas, was fixed at 2 s; therefore, each two seconds, the system provides and saves a measurement of fo performed with the optimum gain (obtained at the last step of the gain adjustment procedure) and, for comparison sake, a measurement of fo obtained at the starting step of the gain adjustment procedure (with maximum gain). The TI VCA842 differential amplifier has a gain range 1/10 < Av <10, but, in this application, we consider Avmin = 1 and a linear analog control of the gain value, obtained by means of a voltage input VG. The gain is related to the control voltage by the following equation: −1 Av = 4.90 V (VG + 1V) The search strategy resolution, which is considered as the largest source of uncertainty for the estimation of Avopt, is related to the number of steps that are used to adjust the gain. In the presented case, the number of steps, N, used can be selected in the range 9–12; we have, therefore: Av MAX − Avmin 10 ∆Av = < N = 9, . . . , 12 2N 2N Exploiting the results of Equation (24), and using the circuit component values, we found that the obtainable resolution for the Rm estimation is in the range 2 Ω–4 Ω with N = 9, whereas, with N = 12, the resolution is improved a factor of 8 and lies in the range 0.25 Ω–0.5 Ω. The VI front-panel is shown in Figure 10. Sensors 2021, 21, x FOR PEER REVIEW 16 of 22

The search strategy resolution, which is considered as the largest source of uncertainty for the estimation of Avopt, is related to the number of steps that are used to adjust the gain. In the presented case, the number of steps, N, used can be selected in the range 9–12; we have, therefore:

𝐴 − 𝐴 10 Δ𝐴 = < 𝑁=9,...,12 2 2 Exploiting the results of Equation (24), and using the circuit component values, we found that the obtainable resolution for the Rm estimation is in the range 2 Ω–4 Ω with N = 9, whereas, with N = 12, the resolution is improved a factor of 8 and lies in the range 0.25 SensorsΩ2021–0.5, 21 ,Ω 678. 16 of 23 The VI front-panel is shown in Figure 10.

Figure 10. VI front-panel. Figure 10. VI front-panel.

5. Experimental Results 5. Experimental Results To test the theory discussed in the previous section, fo and Rm were monitored in To test the theorydifferent discusse experimentald in the conditions previous for section, time windows fo and 25-min-long Rm were performing monitored the in optimum different experimentalgain conditions search (Avopt for); simultaneously,time windows the f25-min-longo value obtained pe withoutrforming using thethe gain optimum adjustment procedure at the maximum gain AvMAX was also acquired. Table1 shows the mean gain search (Avopt); simultaneously, the fo value obtained without using the gain adjustment deviation ef between the two frequencies (i.e., e f = fo@Avopt − fo@AvMAX) in the tested procedure at the maximumdifferent conditions. gain AvMAX The was experimental also acquired. measurements Table were 1 shows repeated the at leastmean ﬁve devia- times for each of the considered test cases in the same environmental conditions. tion ef between the two frequencies (i.e.,𝑒 =𝑓@𝐴 − 𝑓@𝐴) in the tested different

conditions. The experimentalTable 1. e f , the measurements estimated Rm,Avmax/A werevopt and repeated standard deviation at least of the five measured times frequency for each (10 min) of in the considered test casesdifferent in working the same conditions. environmental Gly20% is a solution conditions. of glycerol in pure water with a 20% concentration. In particular, the reported results concern tests performed in the air, pure water (ηH2O ef = fo@AvOpt Estimated σfo@AvMAX σfo@AvOpt −4 3 Avmax/Avopt = 8.4 × 10 Pa∙s, ρH2O = 1000 kg/m−) fando@AvMAX a mixtureRm of(Ω )pure water and 10%(10 of min) glycerol(10 (η min)gly = 1.5 Pa∙s, ρgly = 1260 kg/m3Air). 3377 Hz 64 6.66 2.3 Hz 1.3 Hz Pure water 4390 Hz 237 3.71 20 Hz 8 Hz o s Assuming f as a goodGly20% estimation 4300 Hz of f in the 265 case of the 3.35 adjusted 31 gain, Hz we find 27 Hz an error due to the loop gain module inaccuracy of about 3 kHz in the air (ef = 3377 Hz) and a larger error in the liquid;In particular, these results the reported are in results accordance concern tests with performed the simulation in the air, analysis pure water −4 3 reported in Figure 6.(η H2OMoreover,= 8.4 × 10 operatingPa·s, ρH2O at= an 1000 optimum kg/m ) and gain a mixture usually of increases pure water andthe 10%sta- of 3 bility of the oscillator,glycerol especially (ηgly = 1.5 when Pa·s, ρthegly = optimum 1260 kg/m value). is far from the maximum gain Assuming f as a good estimation of f in the case of the adjusted gain, we ﬁnd an error value. o s due to the loop gain module inaccuracy of about 3 kHz in the air (ef = 3377 Hz) and a larger error in the liquid; these results are in accordance with the simulation analysis reported

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The standard deviations reported in the table describe the short-time frequency stability provided by the oscillator in different test conditions. Nevertheless, the analysis of repeated measurements showed larger deviations that can be explained by the variations of the influence quantities (e.g., ambient temperature varied approximately in a 5 °C interval during the measurement campaign). The Allan variance was assessed to gain information on the variability/stability of the oscillation frequency over time in different conditions: in air and in water. The standard deviation values of the frequency variations for one hour and one day are: <3 Hz, ≈20 Hz in the air and <20 Hz, ≈50 Hz in the liquid.

Table 1. 𝑒, the estimated Rm, Avmax/Avopt and standard deviation of the measured frequency (10 min) in different working conditions. Gly20% is a solution of glycerol in pure water with a 20% concentration. Sensors 2021, 21, 678 17 of 23 ef = fo@AvOpt − Estimated σfo@AvMAX σfo@AvOpt Avmax/Avopt fo@AvMAX Rm (Ω) (10 min) (10 min) in Figure6. Moreover, operating at an optimum gain usually increases the stability of the oscillator,Air especially when the 3377 optimum Hz value is far from 64 the maximum 6.66 gain value. 2.3 Hz 1.3 Hz PureThe standard water deviations 4390 reported Hz in the table describe 237 the short-time 3.71 frequency sta- 20 Hz 8 Hz bility provided by the oscillator in different test conditions. Nevertheless, the analysis of repeatedGly20% measurements showed 4300 larger Hz deviations that 265 can be explained by 3.35 the variations of 31 Hz 27 Hz the inﬂuence quantities (e.g., ambient temperature varied approximately in a 5 ◦C interval during the measurement campaign). The Allan variance was assessed to gain information on theFigure variability/stability 11 reports of the the oscillation optimum frequency gain over time predicted in different conditions:by simulations in (solid line) in the same air and in water. The standard deviation values of the frequency variations for one hour conditionsand one day are: used <3 Hz, to≈20 derive Hz in the Figures air and <20 Hz,5–7≈ 50as Hz a infu thenction liquid. of the series resistance and the opti- mumFigure gain 11 reports provided the optimum by gainthe predictedadjustment by simulations procedure (solid line) experimentally in the same (markers), showing a conditions used to derive Figures5–7 as a function of the series resistance and the optimum verygain provided good byaccordance. the adjustment procedure experimentally (markers), showing a very good accordance.

4.5

3.5

2.5

2 Experimantal Experimental 1.5 Simulated

1 50 100 150 200 250 300 350 400 450 500 R , estimated R ( ) m m FigureFigure 11. 11.Optimum Optimum gain predicted gain bypredicted the simulations by (solidthe simulations line) in the same (solid conditions line) used in the same conditions used to deriveto derive Figures Figures5–7 5–7as a functionas a function of the “true” of series the resistance.“true” seri Optimumes resistance. gain provided Optimum the gain provided the adjust- adjustment procedure experimentally (markers) as a function of the estimated Rm. ment procedure experimentally (markers) as a function of the estimated Rm. Figure 12 reports the results obtained in a 13-h measurement performed in the air with the quartz placed in the chamber and shows the performance of the system in terms of the short timeFigure stability. 12 It isreports worth noticing the thatresults the constraint obtained exerted in by thea 13-h placement measurement of the performed in the air quartz in a chamber for in-liquid applications causes a perturbation of the resonant system withaccompanied the quartz by a modest placed reduction in the of the chamber Q-factor. Therefore, and sh theows short-term the performance stability of the system in terms ofof thethe measurements short time in stability. the air results It are is somewhat worth worsenotici thanng the that ones the obtainable constraint for exerted by the placement in-gas applications where the QCM ﬁxing is obtained by suspending the quartz with small ofsprings, the whichquartz cause in a a negligible chamber mechanical for in-liquid load. applications causes a perturbation of the resonant systemFigure accompanied 13 reports the results by obtained a modest in a 10-h reduction measurement performedof the Q-factor. in a solution Therefore, the short-term sta- of pure water and 10% of glycerol. The effect of evaporation of the water during the bilitymeasurement, of the accompanied measurements by an increase in ofthe the solutionair results viscosity, are can somewhat be seen both from worse than the ones obtainable forthe decreasingin-gas ofapplications the frequency shift where with time the and fromQCM the increasingfixing is of theobtained optimum gain. by suspending the quartz with The results obtained in the different conditions are compared in Figure 14, showing smallthat the springs, effect of the gainwhich setting cause is relevant a negligible particularly when mechanical considering ﬂuidsload. with a large difference in viscosity.

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(a) (b) ) ( m R (a) (b)

Figure 12. Measurement results obtained in the air in approximately 13 h:m (a) oscillation frequency fo at the maxi- R mum(𝐴) and optimum gain (𝐴) and (b) frequency deviation 𝑒 =𝑓@𝐴 −𝑓@𝐴. (c) Estimated 𝐴 and (d) estimated Rm. N = 10.

Figure 13 reports the results obtained in a 10-h measurement performed in a solution (c) of pure water and 10% of glycerol. The effect of evaporation of(d )the water during the meas-

Figure 12.Figure Measurement 12. Measurement resultsurement, resultsobtained obtained accompanied in the inair the in airapproximately by in an approximately increase 13 of h: 13 the(a h:) oscillation (soa)lution oscillation viscosity, frequency frequency can fo fato beat the theseen maxi- maximum both from the mum(𝐴(AvMAX) and) and optimum optimum gaindecreasing gain (𝐴AvOpt) ofandand the ( (bb )frequency frequency deviation shiftdeviation withe f time=𝑒 =𝑓fo@ andA@𝐴vOpt from− fo−𝑓 @theAvMAX @𝐴increasing.(c). Estimated (c) ofEstimated theA optimumvOpt 𝐴and gain. and (d) estimated(d) estimated Rm. RNm =. N10=. 10.

Figure 13 reports the results obtained -4180in a 10-h measurement performed in a solution of pure water and 10% of glycerol. The effect-4200 of evaporation of the water during the measurement, accompanied by an increase of the-4220 solution viscosity, can be seen both from the decreasing of the frequency shift with time-4240 and from the increasing of the optimum gain. -4260 -4280 -4300 -4180 0 100 200 300 400 500 600 Time (min) -4200 (a) -4220 (b) -4240 3.08 -4260 -4280 3.06 -4300 0 100 200 300 400 500 600 vOpt

A Time (min) 3.04 (a) (b)

3.02 0 100 200 300 400 500 600 3.08 Time (min) (c) (d) 3.06 FigureFigure 13. Measurement 13. Measurement results results obtained obtained in water in water and and 10% 10% glycerol glycerol during during approximately approximately 1010 h: h: ( a(a)) oscillation oscillation frequency frequency vOpt

A fo at thef at maximum the maximum (𝐴(A) and) optimumand optimum gains gains 𝐴A and and(b) frequency(b) frequency deviation deviation 𝑒e=𝑓=@𝐴f @A −𝑓−@𝐴f @A . (c.() Esti-c) 3.04 o vMAX vOpt f o vOpt o vMAX 𝐴 m. matedEstimated andAvOpt (d) andestimated (d) estimated R N =R 12m..N = 12.

3.02 0 100 200 300 400 500 600 Time (min) (c) (d)

Figure 13. Measurement results obtained in water and 10% glycerol during approximately 10 h: (a) oscillation frequency fo at the maximum (𝐴) and optimum gains 𝐴 and (b) frequency deviation 𝑒 =𝑓@𝐴 −𝑓@𝐴. (c) Esti- mated 𝐴 and (d) estimated Rm. N = 12.

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The results obtained in the different conditions are compared in Figure 14, showing that the effect of the gain setting is relevant particularly when considering fluids with a

Sensors 2021, 21, 678 large difference in viscosity. 19 of 23

0 -3200 Gly10% f - f @A o oAIR vMAX Gly10% f - f @A -3400 -1000 o oAIR vOpt Air f - f @A Gly10% -2000 o oAIR vMAX -3600 Air f - f @A Air o oAIR vOpt -3800 Water Water f - f @A -3000 o oAIR vOpt Water f - f @A -4000 -4000 o oAIR vMAX -4200 -5000 -4400 -6000 -4600

-7000 -4800

-8000 -5000 0 5 10 15 20 25 0 5 10 15 20 25 time (min) Time (min) (a) (b)

(c)

Figure 14. MeasurementFigure 14. Measurement results resultsobtained obtained in the in theair air(N (N= 12),= 12), water water (NN= = 10)10) and and 10% 10% glycerol glycerol (N = 12)(N during= 12) during 25 min: 25 min: (a) oscillation frequency(a) oscillation shifts, frequency fo − foAIR shifts,, wherefo − foAIR foAIR, where is thef oAIRoscillatoris the oscillator frequency frequency in air in and air andat optimum at optimum gain; gain; dashed dashed lines: shifts ( ) at maximumlines: (𝐴 shifts) atand maximum continuousAvMAX lines:and continuousshifts at optimum lines: shifts gain at optimum 𝐴 gain. (b)A FrequencyvOpt .(b) Frequency deviation deviation 𝑒 =𝑓@𝐴 − e = fo@AvOpt − fo@AvMAX.(c) Estimated Rm. 𝑓@𝐴. (cf) Estimated Rm.

Finally, Figure 15 shows a dynamical test, fo, was monitored starting from pure water Finally,(140 µL) andFigure adding, 15 shows with a micropipette, a dynamical subsequent test, fo, doseswas ofmonitored 13 µL or 26 startingµL of a solution from pure water (140 μofL) water and adding, with 30% glycerol,with a micropipette, obtaining solutions subsequent with glycerol doses concentrations of 13 μL ofor 0%, 26 4.7%,μL of a solution 6.5%, 8.1%, 9.5%, 10.7% and 11.8%. The transients induced by the perturbation due to the of waterdrop with cast can30% be glycerol, seen. The glycerol obtaining solution solution is droppeds with manually glycerol with aconcentrations micropipette, and of 0%, 4.7%, 6.5%, 8.1%,the added 9.5%, solution 10.7% has and a different 11.8%. density The transients and temperature induced with respect by the to perturbation the one in the due to the drop castwell, can so, whenbe seen. the drop The fallsglycerol (even ifsolution we tried is to dropped perform this manually operation with as smoothly a micropipette, as and the addedpossible), solution a ﬂuid dynamicalhas a different transient density is expected and before temperature the solution diffuseswith respect and reaches to the a one in the steady state. Due to the manual deposition, the transients obtained with the individual well, so,drop when depositions the drop are somewhat falls (even different. if we tried to perform this operation as smoothly as possible), a fluid dynamical transient is expected before the solution diffuses and reaches a steady state. Due to the manual deposition, the transients obtained with the individual drop depositions are somewhat different. These measurements show that the gain successfully adapts itself to the different measurement conditions, compensating as needed for the change of Rm during measurements.

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(a)

(b)

(c)

FigureFigure 15. QCM15. QCM monitoring monitoring starting starting from pure from water pure and water adding, an withd adding, a micropipette, with a subsequentmicropipette, doses subsequent of 13 µL or 26 µL of adoses solution of 13 with μL 30% or glycerol,26 μL of obtaining a solution solutions with 30% with glycerol, glycerol concentrations obtaining solutions of 0%, 4.7%, with 6.5%, glycerol 8.1%,9.5%, concen- 10.7% and 11.8%.trations (a) Oscillation of 0%, 4.7%, frequency 6.5%, shifts, 8.1%,fo 9.5%,− foAIR 10.7%, where andfoAIR 11.8%.is the oscillator (a) Oscillation frequency frequency in the air andshifts, at the fo − optimum foAIR, gain; dashedwhere lines: foAIR shifts is the at maximumoscillator( frequencyAvMAX) and in continuous the air and lines: at shiftsthe optimum at the optimum gain; gain dashedAvOpt lines:, N shifts= 12. ( bat) Frequency deviationmaximume f = (𝐴fo@AvOpt) and− fo continuous@AvMAX.(c) lines: Estimated shiftsRm at. the optimum gain 𝐴, N = 12. (b) Fre- quency deviation 𝑒 =𝑓@𝐴 −𝑓@𝐴. (c) Estimated Rm. These measurements show that the gain successfully adapts itself to the different measurement conditions, compensating as needed for the change of Rm during measurements. The relationship betweenThe relationship the glycerol between conc theentration, glycerol concentration, the measured the frequency measured frequencyshifts shifts and the estimated resistanceand the estimated was evaluated resistance and was reported evaluated in and Figure reported 16. in It Figurecan be 16 seen. It can that, be seen that, as expected by it beingas expected derived by from it being the derived theory from and thesimulations theory and in simulations the previous in the section, previous section, the effect of using thethe maximum effect of using gain the of maximum the ampl gainifier of is the not ampliﬁer a simple is offset not a simple of the offset frequency of the frequency shifts, but it causes anshifts, error but depending it causes an erroron the depending working on conditions, the working and conditions, as such, and it ascan such, be it can be hardly compensated. Using the proposed system allows both for reducing the frequency shift error and for adding an accurate measurement of the series resistance, which can be used to assess the viscosity of the interacting liquid.

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Sensors 2021, 21, x FOR PEER REVIEW 21 of 22 hardly compensated. Using the proposed system allows both for reducing the frequency shift error and for adding an accurate measurement of the series resistance, which can be used to assess the viscosity of the interacting liquid.

0 (a)

-500 f - f @ A o owater vMAX f - f @ A o owater vopt -1000 024681012 glycerin concentration (%) 280

260

(b) 240 024681012 glycerin concentration (%)

− Figure 16. Figure(a) Oscillation 16. (a) frequency Oscillation shifts, frequencyfo fowater shifts,(where fo −f owaterfowater is(where the oscillator fowater is frequency the oscillator in pure frequency water) as in a function pure of the glycerinwater) concentration. as a function (b) Estimated of the glycerinRm as a functionconcentration. of the glycerin (b) Estimated concentration, Rm as aN function= 12. of the glycerin concentration, N = 12. 6. Conclusions 6. Conclusions This paper showed the feasibility of a novel measurement system based on Quartz This paperCrystal showed Microbalances, the feasibility on a of Meacham a novel oscillator measurement embedding system an amplifierbased on with Quartz adjustable gain and on an automatic strategy for gain tuning. It was shown that maintaining the Crystal Microbalances on a Meacham oscillator embedding an amplifier with adjustable oscillator frequency close to the series resonance frequency of the quartz needs ad-hoc gain and on ansolutions automatic when strategy the quartz for operatesgain tuning. in-liquid, It was where shown the Q-factorthat maintaining is severely the reduced. oscillator frequencyThe system close uses to the optimumseries reso gainnance value frequency to derive theof the value quartz of the needs quartz ad-hoc impedance at solutions whenf 0 ◦the, which quartz is aoperates good estimation in-liquid, of where the series the resistance Q-factor ofis severely the quartz. reduced. The simultaneous The system uses themonitoring optimum of thegain frequency value to shift deri andve ofthe the value series resistanceof the quartz allows impedance for gaining informationat f0°, which is a goodon estimation the mechanical of the load series type andresistance distinguishing of the quartz. between The the simultaneous viscous loading moni- and inertial toring of the frequencyloading of shift the QCM.and of The the proposedseries resistance system wasallows tested for andgaining verified information using a laboratory on the mechanicalset-up, load buttype it can and be distinguishing realized with low-cost between components the viscous exploiting loading a microcontroller and inertial for the implementation of the gain adjustment algorithm and for frequency downshifted signal loading of the QCM. The proposed system was tested and verified using a laboratory set- acquisition and processing, whereas the reference signal can be easily generated with a up, but it can Directbe realized Digital with Synthesis low-cost integrated components circuit (DDS exploiting IC). a microcontroller for the implementation of the gain adjustment algorithm and for frequency downshifted signal acquisition andAuthor processing, Contributions: whereasConceptualization, the reference A.F., signal V.V. andcan M.M.; be easily methodology, generated A.F., with V.V. and a M.M.; Direct Digital validation,Synthesis E.L.; inte formalgrated analysis, circuit A.F., (DDS P.V. IC). and E.P.; investigation, P.V., E.L. and E.P.; data curation, P.V., A.F. and M.T., writing—original draft preparation, A.F. and writing—review and editing, E.P., M.M. and V.V. All authors have read and agreed to the published version of the manuscript. Author Contributions: Conceptualization, A.F., V.V. and M.M.; methodology, A.F., V.V. and M.M.; validation, E.L.;Funding: formal analysis,This paper A. receivedF., P.V. and no external E.P.; investigation, funding. P.V., E.L. and E.P.; data curation, P.V., A.F. and M.T.,Institutional writing—origin Review Boardal draft Statement: preparation,Not applicable.A.F. and writing—review and editing, E.P., M.M. and V.V. All authors have read and agreed to the published version of the manuscript. Informed Consent Statement: Not applicable. Funding: This paper received no external funding. Data Availability Statement: Data sharing not applicable. Institutional Review Board Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. Informed Consent Statement: Not applicable. Data Availability Statement: Data sharing not applicable. Conflicts of Interest: The authors declare no conflicts of interest.

References 1. Alassi, A.; Benammar, M.; Brett, D. Quartz crystal microbalance electronic interfacing systems: A review. Sensors 2017, 17, 2799, doi:10.3390/s17122799. 2. Nomura, T.; Zherlitsyn, S.; Kohama, Y.; Wosnitza, J. Viscosity measurements in pulsed magnetic fields by using a quartz-crystal microbalance. Rev. Sci. Instrum. 2019, 90, 065101, doi:10.1063/1.5098451.

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