# Interrogation Techniques and Interface Circuits for Coil-Coupled Passive Sensors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Coil-Coupled Passive Sensors

_{1}with inductance L

_{1}and series resistance R

_{1}is magnetically coupled to the secondary coil CL

_{2}with inductance L

_{2}and resistance R

_{2}. The magnetic coupling is accounted for by the mutual inductance M, which depends on the geometry of L

_{1}and L

_{2}and their spatial arrangement. Alternatively, the magnetic coupling can be described through the coupling factor k, which is a nondimensional parameter defined as $k=M/\sqrt{({L}_{1}{L}_{2})}$, resulting in |k|≤1. In the following, the values of L

_{1}, R

_{1}and L

_{2}, R

_{2}will be considered as fixed, while the value of M, and hence k, can change due to variations of the distance or orientation between CL

_{1}and CL

_{2}.

_{2}is connected to the generic impedance Z

_{S}, which models the sensing element. In the following, the relevant cases will be considered where Z

_{S}either forms, with L

_{2}, a second order network with complex conjugate poles, i.e., Z

_{S}is predominantly capacitive, or Z

_{S}itself includes a second order network with complex conjugate poles, i.e., Z

_{S}comprises an LCR network. In both cases, resonance can occur in the secondary circuit where the quantity to be sensed via Z

_{S}influences the resonant frequency and, possibly, the damping. Therefore, the resulting combination will be termed Resonant Sensor Unit (RSU).

_{1}and the RSU.

_{S}and the resulting RSU.

_{S}is a capacitance sensor of value C

_{S}, forming, with L

_{2}, an LC resonant circuit as shown in Figure 2a. The resonant frequency f

_{S}and quality factor Q

_{S}of the RSU are

_{S}is the equivalent impedance of piezoelectric resonant sensors, like QCRs and RPLs. Their electromechanical behavior around resonance can be modelled with the Butterworth–van Dyke (BVD) equivalent lumped-element circuit, as shown in Figure 2b. The BVD circuit is composed of a motional, i.e., mechanical branch, and an electrical branch. The motional branch comprises the series of inductance L

_{r}, capacitance C

_{r}, and resistance R

_{r}, which respectively represent the equivalent mass, compliance, and energy losses of the resonator. The electrical branch is formed by the parallel capacitance C

_{0}, due to the dielectric material of the resonator. Under excitation by a voltage source, the mechanical resonant frequency f

_{r}, i.e., the frequency at which the current in the motional arm is maximum, corresponds to the series resonant frequency of the BVD circuit, i.e., the frequency at which the reactance of the mechanical branch impedance vanishes [26]. Accordingly, f

_{r}and the quality factor Q

_{r}of the electromechanical resonator can be expressed as

_{r}–C

_{r}–R

_{r}and, as a consequence, of f

_{r}and Q

_{r}.

## 3. Analysis of the Interrogation Techniques

#### 3.1. General Considerations

_{1}. The second is a time-domain technique, termed time-gated technique, which considers the free damped response of the RSU measured at the primary coil after that the RSU has been energized.

#### 3.2. k-Independent Techniques Applied to Coil-Coupled Capacitance Sensors

_{1}. From the equivalent circuit of Figure 3b, the impedance Z

_{1}, as a function of ω = 2πf, is

_{R}in series with the primary coil that makes the total impedance Z

_{1}dependent on the coupling factor k. Nevertheless, the resonant frequency f

_{S}and the quality factor Q

_{S}of the RSU, defined in Equation (1), can be obtained from the real part of Z

_{1}[20], given by

_{1}} has a local maximum at the frequency f

_{m}= ω

_{m}/2π, which can be found by equating to zero the derivative of Equation (4) with respect to ω. Interestingly enough, f

_{m}is independent of k, and it can be related to f

_{S}and Q

_{S}only. Then, combining Equations (1) and (4), the following relations hold:

_{m}is the full width at half maximum (FWHM) of Re{Z

_{1}}, around f

_{m}[20]. If Q

_{S}is sufficiently large, then f

_{m}≈ f

_{S}, with a relative deviation |f

_{m}− f

_{S}|/f

_{S}< 100 ppm for Q

_{S}> 50. Equations (4) and (5) demonstrate that from the measurement of f

_{m}and Δf

_{m}in Re{Z

_{1}}, the frequency f

_{S}and quality factor Q

_{S}of the capacitive RSU can be advantageously extracted independently from k. Figure 4 shows sample plots of Re{Z

_{1}} calculated for three different values of k, and illustrates the definition of Δf

_{m}. Consistently with Equation (4), k only affects amplitude.

_{1}is connected to the sinusoidal signal v

_{exc}(t) to excite the RSU through inductive coupling. During the subsequent detection phase, when the switch is in the D position, the excitation signal is disconnected, and CL

_{1}is connected to a readout circuit with a high-impedance input, resulting in a virtually zero current in CL

_{1}.

_{1}(t) of the readout circuit during the detection phase D can be derived by taking the inverse Laplace transform of the corresponding voltage V

_{1}(s), where s is the complex frequency. Since the RSU forms a second order LCR network, the voltage v

_{1}(t) is expected to be a damped sinusoid with frequency f

_{d}and a decay time τ

_{d}from which the resonant frequency f

_{S}and the quality factor Q

_{S}of the RSU can be inferred.

_{1}(t) depends on the initial conditions at t = 0 of all the reactive elements, namely C

_{S}, L

_{1}, L

_{2}, and M. The effect of the initial conditions on v

_{1}(t) for t > 0 is to globally affect only its starting amplitude, while the complex frequencies of the network, that define f

_{d}and τ

_{d}, are unaltered. Therefore, without losing any generality, the single initial condition V

_{CS0}defined as the voltage across C

_{S}at t = 0 can be considered, neglecting the remaining ones. As an equivalent alternative that does not change the consequences of the present treatment, V

_{CS0}can also be seen as an effective initial condition.

_{1}(s) is

_{1}(t) can be calculated:

_{1}(t) is a damped sinusoid with damped frequency f

_{d}and decay time τ

_{d}that are related to f

_{S}and Q

_{S}of the RSU as

_{S}is sufficiently large, it results in f

_{d}≈ f

_{S}, with a relative deviation |f

_{d}− f

_{S}|/f

_{S}< 50 ppm for Q

_{S}> 50. Notably, the coupling factor k only acts as an amplitude factor on v

_{1}(t) without influencing either f

_{d}or τ

_{d}. Figure 6 reports sample plots of v

_{1}(t) calculated for three different values of k.

_{S}and quality factor Q

_{S}of the capacitive RSU, independently of k.

#### 3.3. k-Independent Techniques Applied to Coil-Coupled Electromechanical Piezoelectric Resonators

_{1}measured at the primary coil can be expressed as

_{1}depends on the coupling factor k. Nevertheless, also in this case, the frequency f

_{r}can be extracted from the frequency of the maximum of the real part of Z

_{1}.

_{r}= 2πf

_{r}, the impedance of the motional arm Z

_{r}= R

_{r}+ jωL

_{r}+ 1/(jωC

_{r}) has a magnitude typically much smaller than that of the impedance of C

_{0}, i.e., |Z

_{r}| << 1/ωC

_{0}. Then, the presence of C

_{0}can be neglected, resulting in the simplified equivalent circuit of Figure 7a. Accordingly, Re{Z

_{1}} around ω

_{r}has the following approximated expression:

_{1}} has a maximum at the frequency f

_{m_r}given by

_{r2}, f

_{m_r}≈ f

_{r2}with a deviation |f

_{m_r}− f

_{r2}|/f

_{r2}< 100 ppm for Q

_{r2}> 50. In addition, assuming that L

_{2}<< L

_{r}, the frequency f

_{r2}approximates f

_{r}and, hence, f

_{m_r}≈ f

_{r}holds. Similarly, if R

_{2}<< R

_{r}, Q

_{r2}approaches Q

_{r}. Importantly, again, the coupling factor k acts only as an amplitude factor that advantageously does not affect either the frequency or the quality factor of the resonance.

_{r}, the impedance magnitude of C

_{0}is smaller than the impedance magnitude of Z

_{r}, which then can be neglected, obtaining the equivalent circuit of Figure 7b. Consequently, the following approximated expression of Re{Z

_{1}} results:

_{1}} now has a maximum at the frequency f

_{m_el}:

_{1}} has two peaks: the first is related to the mechanical resonance f

_{r}, the second to the electrical resonance f

_{el}. With the previous assumptions on the values of L

_{r}and L

_{2}, and considering that, typically, C

_{r}<< C

_{0}, then it follows that f

_{el}>> f

_{r}.

_{m_r}and f

_{m_el}derived respectively from Equations (11) and (13), and the frequency of the maxima derived numerically from Re{Z

_{1}} in Equation (9) as a function of L

_{2}. The following values of the BVD model of a 4.432 MHz AT-cut QCR have been used: C

_{0}= 5.72 pF, R

_{r}= 10.09 Ω, L

_{r}= 77.98 mH, and C

_{r}= 16.54 fF. For CL

_{1}and CL

_{2}, the values of the electrical parameters are L

_{1}= 8.5 µH, R

_{1}= 5 Ω, and R

_{2}= 5 Ω.

_{2}up to 10 µH, the values of f

_{m_r}predicted from Equation (11) are within 3 ppm with respect to the numerical solutions from Equation (9). Additionally, for the same range of variation of L

_{2}, a remarkable agreement is obtained between f

_{m_el}predicted from Equation (13) and the numerical solution.

_{1}(t) at CL

_{1}during the detection phase, after the RSU has been energized in the excitation phase, is the sum of two damped sinusoids: one at frequency f

_{d_r}with exponential decaying time τ

_{r}, and one at frequency f

_{d_el}with exponential decaying time τ

_{el}.

_{d_r}is due to the mechanical response of the resonator, while the one at f

_{d_el}is due to the electrical response of L

_{2}that interacts with the electrical capacitance C

_{0}. In addition, for suitable values of L

_{2}and R

_{2}, and considering the typical values of the equivalent parameters of the BVD model of a QCR, the decaying time τ

_{r}is orders of magnitude larger than τ

_{el}. Thus, the damped sinusoid at frequency f

_{d_el}decays to zero much faster than the damped sinusoid at frequency f

_{d_r}. Hence, the former can be neglected in the expression of v

_{1}(t), which results in

_{r}and θ

_{r}are functions of both the initial conditions at the beginning of the detection phase (t = 0), and the electrical and mechanical parameters of the system. The last term represents the contribution of the initial current i

_{L}

_{1}(0) in the primary inductor. From Equation (14), it can be seen that k acts only as a scaling factor for the amplitude of v

_{1}, without affecting the sensor response parameters f

_{d_r}and τ

_{r}. From a simplified analysis that considers the undamped system with R

_{2}= 0 and R

_{r}= 0, under the hypothesis that (ωC

_{0})

^{−1}>> ωL

_{2}at the frequency f

_{r}and that Q

_{r}is large, it has been obtained that the frequency f

_{d_r}can be approximated with the following relation:

_{d_r}depends on the ratio between L

_{2}and L

_{r}. Nevertheless, if L

_{2}<< L

_{r}the frequency f

_{d_r}tends to the resonant frequency f

_{r}of the electromechanical resonator. A numerical analysis that allows the calculation of the parameters f

_{d_r}and τ

_{r}of the complete system, is also reported in [21]. The results can be directly compared with Figure 8, the values of the parameters of the BVD model used in the numerical analysis being the same. Also in that case, good agreement between the values of f

_{d_r}predicted from Equation (15) and the numerical results have been obtained, with a maximum deviation within 3 ppm for L

_{2}up to 10 µH.

#### 3.4. Effect of Parasitic Capacitance at the Primary Coil on Coil-Coupled Capacitance Sensors

_{P}that appears in parallel to L

_{1}. The parasitic capacitance C

_{P}is mainly composed of the parasitic capacitance of the inductor L

_{1}, the capacitance of the connections, and the input capacitance of the electronic interface.

_{P}is now evaluated, firstly, considering the case of the RSU with the capacitance sensor, extending the treatment of Section 3.2.

_{P}≠ 0, Equation (16) no longer allows extraction of f

_{S}and Q

_{S}independently from the coupling factor k, which now is in the expression of Z

_{1P}and affects Re{Z

_{1P}}, not only as a scaling factor. In particular, it has been shown by a numerical analysis of Equation (16) that Re{Z

_{1P}} has two maxima, corresponding, respectively, to a primary resonance near f

_{S}and a secondary resonance near ${f}_{\mathrm{P}}=\text{}1/\text{}\left(2\mathsf{\pi}\sqrt{{L}_{1}{C}_{\mathrm{P}}}\right)$. Both the frequencies of the maxima and the trend of Re{Z

_{1P}} are influenced by the coupling factor k [23].

_{1P}(t) at the primary coil in the detection phase can be obtained from the circuit of Figure 10b. Adopting the same approach as for the case of C

_{P}= 0, it will be assumed that all the reactive elements, except the capacitor C

_{S}, have zero initial conditions at t = 0. Consequently, the voltage V

_{1P}(s) can be expressed in the Laplace domain as

_{CS0}is the voltage across C

_{S}at t = 0. From Equation (17), it can be seen that k, besides acting as a scaling factor, also features in the coefficient of fourth degree in the polynomial D(s). Consequently, it is expected that the complex frequencies are dependent on k. Taking the inverse Laplace transform of Equation (17), it results that the expression of v

_{1P}(t) is composed of the sum of two damped sinusoids as

_{1}and A

_{2}are amplitude coefficients and θ

_{1}and θ

_{2}are phase angles that depend on the parameters of the circuit and the initial conditions. The frequencies f

_{d1}and f

_{d2}and the decay times, τ

_{d1}and τ

_{d2}are obtained by the complex conjugate solutions p

_{1,2}= 1/τ

_{d1}± j2πf

_{d1}and p

_{3,4}= 1/τ

_{d2}± j2πf

_{d2}of D(s) = 0.

_{1,2}and p

_{3,4}, it can be demonstrated that f

_{d1}is close to f

_{P}, while f

_{d2}is close to f

_{S}, but both f

_{d1}and f

_{d2}are dependent on k. For R

_{2}sufficiently smaller than R

_{1}, a decay time τ

_{d2}larger than τ

_{d1}can be obtained. In this condition, in v

_{1P}(t) the damped sinusoid at f

_{d1}falls off more rapidly than that at f

_{d2}, and it becomes negligible as time elapses. Importantly, since f

_{d2}depends on k, the distance-independent operation of the case C

_{P}= 0 is now lost.

_{P}, on both the proposed techniques, is investigated by numerical analysis. For the RSU and CL

_{1}, the following sample values, which represent real conditions well, have been considered: L

_{2}= 8 µH, C

_{S}= 100 pF, R

_{2}= 3 Ω, L

_{1}= L

_{2}, and R

_{1}= 10 Ω. For the impedance technique, the frequency f

_{SP}has been calculated from the expression of Re{Z

_{1P}}, adopting the definitions in Equation (5). For the time-gated technique, f

_{SP}has been calculated from f

_{d2}and τ

_{d2}, derived from the numerical solution of D(s) = 0, adopting the definitions in Equation (8).

_{SP}− f

_{S})/f

_{S}as a function of the coupling factor k for three different values of C

_{P}/C

_{S}. For the considered values of the parameters, C

_{P}ranges from 1 pF to 10 pF. As it can be observed, (f

_{SP}− f

_{S})/f

_{S}deviates from zero, corresponding to C

_{P}= 0. The deviation increases for increasing k of an amount that augments with C

_{P}/C

_{S}. Noticeably, both the techniques are equally affected by the inaccuracies introduced by C

_{P}, in terms of the dependence of the readout frequency on k. These results demonstrate that C

_{P}prevents accurate distance-independent measurements from being obtained.

#### 3.5. Effect of Parasitic Capacitance at the Primary Coil on Coil-Coupled Electromechanical Piezoelectric Resonators

_{P}can be evaluated by using the same numerical approach as discussed in Section 3.3. The resonant frequency f

_{rP}can be obtained from numerical analysis of the equivalent circuit in Figure 12a for the frequency-domain technique based on impedance Z

_{1P}, while the equivalent circuit of Figure 12b must be considered for the time-gated technique to determine V

_{1P}(s).

_{0}has been considered high enough to be neglected. For the time-gated technique, C

_{P}is expected to give rise to an additional damped sinusoid in v

_{1P}(t), with a damped frequency related to C

_{P}resonating with L

_{1}. However, the numerical simulations have demonstrated that this sinusoid fades out more quickly than the damped sinusoid, due to the QCR response.

_{rP}− f

_{r})/f

_{r}as a function of k for three different increasing values of the ratio C

_{P}/C

_{r}, is reported in Figure 13. For the considered values of the parameters, C

_{P}ranges from 1.65 pF to 99.2 pF. The baseline, i.e., the dotted curve corresponding to C

_{P}= 0, is at −54.5 ppm because of L

_{2}, that slightly affects f

_{r2}and, hence, f

_{rP}, according to Equation (11). As it can be observed, f

_{rP}has a maximum variation of less than 4 ppm with respect to the baseline. Remarkably, also in this case, the same behaviour with respect to C

_{P}and k is predicted for the two techniques.

_{rP}on k can be ascribed to the fact that the inductive component in the RSU is dominated by L

_{r}. In fact, L

_{r}is three orders of magnitude larger than L

_{2}, and it is not involved in the coupling between the primary coil and the RSU. This result shows that with coil-coupled electromechanical resonators, such as QCRs, the proposed techniques remain practically independent from the coupling factor k, despite a not-negligible C

_{P}.

## 4. Interrogation Techniques and Interface Circuits

#### 4.1. Interrogation System Based on the Impedance-Measurement Technique with Parasitic Capacitance Compensation

_{1}is connected to the impedance analyzer. The total parasitic capacitance C

_{P}accounts for the contributions given by the parasitic capacitances of CL

_{1}, the connections and the equivalent capacitance of the input of the impedance analyzer, represented in Figure 14 with C

_{1}, C

_{L}, and C

_{I}, respectively.

_{1}, it is possible to cancel the effects of C

_{P}. The proposed compensation circuit, described in Section 4.3, behaves as an equivalent negative capacitance −C

_{C}. The ideal condition, where C

_{P}is not present, i.e., Z

_{1P}= Z

_{1}, can be thus obtained when C

_{C}= C

_{P}. In the compensated condition, Equation (5) again applies, and k-independent measurements of the resonant frequency and quality factor can be obtained by considering the maximum of the real part of the measured impedance.

#### 4.2. Interrogation System Based on the Time-gated Technique with Parasitic Capacitance Compensation

_{g}(t), alternatively connects the primary coil to the excitation signal v

_{exc}(t) and to the high-input impedance readout amplifier A

_{G}during the excitation and detection phases, respectively. The noninverting amplifier A

_{G}, with gain G, is based on a high-bandwidth operational amplifier. A frequency meter connected to the output of A

_{G}allows measurement of the frequency of the damped sinusoidal signal v

_{O}(t).

_{P}accounts for the contributions of the parasitic capacitances of the primary coil, the connections, the analog switch SW, and the equivalent input capacitance of the amplifier A

_{G}, represented in Figure 15 with C

_{1}, C

_{L}, C

_{SW}, and C

_{I}, respectively.

_{C}can be introduced to cancel C

_{P}. In the compensated condition, the frequency and decay time of the damped sinusoidal voltage v

_{O}(t) return to be unaffected from the coupling factor k. In this condition, Equation (8) can be used to extract the resonant frequency and quality factor of the RSU from the measured resonant frequency and decay time of v

_{O}(t).

#### 4.3. Parasitic Capacitance Compensation Circuit

_{C}operating as a negative impedance converter (NIC) to produce an effective negative capacitance −C

_{C}. The voltage V

_{1}across CL

_{1}is applied across the reference capacitor C

_{A}, thanks to the virtual short circuit at the input of A

_{C}. The current I

_{CA}through C

_{A}is then amplified with gain –R

_{C2}/R

_{C1}, resulting in the current I

_{1}= −jωC

_{A}V

_{1}(R

_{C2}/R

_{C1}). The equivalent input impedance Z

_{Eq}= V

_{1}/I

_{1}is, therefore,

_{A}and R

_{C1}as fixed, and making R

_{C2}variable, the compensation circuit acts as an adjustable negative capacitance, given by

_{P}.

## 5. Experimental Results and Discussion

#### 5.1. Impedance Measurements with Coil-Coupled Capacitance Sensor and QCR

_{C}.

_{2}= 8.51 µH, R

_{2}= 3.2 Ω, and a reference capacitor C

_{S}= 100 pF. According to Equation (1), the resulting resonant frequency and quality factor are f

_{S}= 5.45 MHz and Q

_{S}= 91, respectively. A PCB square planar spiral coil has also been used for the primary coil, with L

_{1}= 8.5 µH and R

_{1}= 5 Ω. A fixed capacitor C

_{F}= 22 pF is connected in parallel to the primary coil, in order to set the parasitic capacitance and test the effectiveness of the compensation circuit.

_{1P}versus frequency has been measured at varying interrogation distance d, and hence the coupling factor k, for different values of the compensation capacitance C

_{C}. The results are shown in Figure 18.

_{mP}where the maximum of Re{Z

_{1P}} near f

_{S}occurs as a function of d, for different values of the compensation capacitance C

_{C}. A monotonic decrease of k is expected by increasing d [29]. It can be observed that by increasing Cc, the expected undesired effect of the parasitic capacitances described in Section 3.3 decreases. With C

_{C}= 27 pF, the value of f

_{mP}becomes independent of d over the considered interrogation range of 16 mm, with a residual deviation of f

_{mP}within 1 kHz, i.e., less than 200 ppm. The obtained value of Cc = 27 pF, slightly higher than the capacitor C

_{F}= 22 pF, is ascribed to the presence of an extra capacitance of about 5 pF that concurs to form C

_{P}. The results clearly demonstrate the effectiveness of the compensation technique and circuit.

_{mP}approaches the unaffected value of f

_{m}, discussed in Section 3.2, over the considered interrogation distance range. Then, for the considered RSU with a Qs = 91, a relative deviation |f

_{mP}– f

_{S}|/f

_{S}as low as 30 ppm is obtained from Equation (5).

_{r}= 4.432 MHz has been connected to CL

_{2}. The parameters of the BVD equivalent circuit around fr of the adopted QCR are C

_{0}= 5.72 pF, R

_{r}= 10.09 Ω, L

_{r}= 77.98 mH, and Cr = 16.54 fF. The numerical analysis, discussed in Section 3.4, proves that parasitic capacitances in the order of tens of picofarads introduce negligible dependence of the measured resonant frequency on k. For this reason, the compensation circuit is not connected to the primary coil. Figure 20a shows the real part of the impedance Z

_{1P}, measured in the frequency range around f

_{r}for different values of the interrogation distance d. As it can be observed, while the magnitude of the maximum of Re{Z

_{1P}} decreases by increasing d, the frequency f

_{rP}, where the maximum occurs, shows residual variations as low as 1 Hz, i.e., less than 0.3 ppm, in the explored range of d, as shown in Figure 20b. This confirms the predicted independence of f

_{rP}from d, and thus from k.

#### 5.2. Time-Gated Measurements with Coil-Coupled Capacitance Sensor and QCR

_{exc}(t) and v

_{g}(t) are generated by two Agilent 3320A waveform generators (Agilent Technologies, Santa Clara, CA, USA). A tailored circuit comprising the analog switch SW (MAX393, Maxim Integrated, San Jose, CA, USA), the parasitic capacitance compensation circuit, and the readout amplifier A

_{G}(OPA656, Texas Instruments, Dallas, TX, USA), has been developed. The readout output signal v

_{O}(t) has been connected to a high-resolution frequency meter Philips PM6680 (Philips International, Eindhoven, The Netherlands). The frequency meter is configured to perform measurements in a time window of duration T

_{M}, starting after a delay time T

_{D}from the beginning of the detection phase. The delay time T

_{D}is used to skip the initial ringing in v

_{O}(t) [18,21]. The voltage v

_{O}(t) measured during detection phase, and the times T

_{D}and T

_{M}, are shown in Figure 22.

_{2}= 8.51 µH, R

_{2}= 3.2 Ω, and a capacitive sensor with C

_{S}= 100 pF, resulting in a resonant frequency f

_{S}= 5.45 MHz. The same PCB spiral coil described in Section 5.1, with L

_{1}= 8.5 µH and R

_{1}= 5 Ω, has been used as CL

_{1}. The frequency of the excitation signal v

_{exc}(t) is set close to f

_{S}to improve the transferred signal level.

_{dP}of the damped sinusoid v

_{O}(t) during the detection phase, measured at varying d for different values of the compensation capacitance C

_{C}. A delay time T

_{D}= 2 µs and a measurement time T

_{M}= 6 µs have been chosen for all the measurements. As it can be observed, for the case of compensation of C

_{P}, the dependence of f

_{dP}on d is much reduced with respect to the cases with no or partial compensation. With C

_{C}≈ 48 pF, f

_{dP}has residual variations within 1.5 kHz, i.e., less than 300 ppm, across the explored interrogation range of about 17.6 mm.

_{dP}approaches the unaffected value of f

_{d}discussed in Section 3.2. Then, for the considered RSU with Qs = 91, a relative deviation |f

_{dP}− f

_{S}|/f

_{S}as low as 15 ppm is obtained from Equation (8).

_{rP}from k for coil-coupled QCR. The frequency f

_{rP}of the damped sinusoid v

_{O}(t) has been measured with varying the interrogation distance d.

_{O}(t) at the beginning of the detection phase for three different interrogation distances d. As it can be observed, the magnitude of v

_{O}(t) decreases with the increasing d, i.e., with decreasing k, while, as expected, the frequency f

_{rP}is unaffected, as shown in Figure 24b. A residual variation of about 1.8 Hz, i.e., less than 0.5 ppm, has been obtained over the explored interrogation distance range of about 17.8 mm. In summary, the experimental results with coil-coupled QCRs show that the total parasitic capacitance C

_{P}estimated in about 48 pF, causes a negligible variation of the measured frequency f

_{rP}over the explored interrogation range.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ferrari, M.; Ferrari, V.; Guizzetti, M.; Marioli, D. An autonomous battery-less sensor module powered by piezoelectric energy harvesting with RF transmission of multiple measurement signals. Smart Mater. Struct.
**2009**, 18, 085023. [Google Scholar] [CrossRef] - Ferrari, M.; Ferrari, V.; Guizzetti, M.; Marioli, D.; Taroni, A. Piezoelectric multifrequency energy converter for power harvesting in autonomous microsystems. Sensor Actuator A Phys.
**2008**, 142, 329–335. [Google Scholar] [CrossRef] - Tan, Y.; Dong, Y.; Wang, X. Review of MEMS electromagnetic vibration energy harvester. J. Microelectromech. Syst.
**2017**, 26, 1–16. [Google Scholar] [CrossRef] - Dalola, S.; Ferrari, M.; Ferrari, V.; Guizzetti, M.; Marioli, D.; Taroni, A. Characterization of thermoelectric modules for powering autonomous sensors. IEEE Trans. Instrum. Meas.
**2009**, 58, 99–107. [Google Scholar] [CrossRef] - Cuadras, A.; Gasulla, M.; Ferrari, V. Thermal energy harvesting through pyroelectricity. Sensor Actuator A Phys.
**2010**, 158, 132–139. [Google Scholar] [CrossRef] - Dalola, S.; Faglia, G.; Comini, E.; Ferroni, M.; Soldano, C.; Zappa, D.; Ferrari, V.; Sberveglieri, G. Planar thermoelectric generator based on metal-oxide nanowires for powering autonomous microsystem. Procedia Eng.
**2012**, 47, 346–349. [Google Scholar] [CrossRef] - Demori, M.; Ferrari, M.; Bonzanini, A.; Poesio, P.; Ferrari, V. Autonomous sensors powered by energy harvesting by von karman vortices in airflow. Sensors
**2017**, 17, 2100. [Google Scholar] [CrossRef] [PubMed] - Sample, A.P.; Yeager, D.J.; Powledge, P.S.; Mamishev, A.V.; Smith, J.R. Design of an RFID-based battery-free programmable sensing platform. IEEE Trans. Instrum. Meas.
**2008**, 57, 2608–2615. [Google Scholar] [CrossRef] - Siddiqui, A.; Mahboob, M.R.; Islam, T. A passive wireless tag with digital readout unit for wide range humidity measurement. IEEE Trans. Instrum. Meas.
**2017**, 66, 1013–1020. [Google Scholar] [CrossRef] - Demori, M.; Baù, M.; Dalola, S.; Ferrari, M.; Ferrari, V. RFID powered system for contactless measurement of a resistive sensor array. In Proceedings of the 2018 IEEE International Instrumentation and Measurement Technology Conference (I2MTC), Houston, TX, USA, 14–17 May 2018. [Google Scholar]
- Chatzandroulis, S.; Tsoukalas, D.; Neukomm, P.A. A miniature pressure system with a capacitive sensor and a passive telemetry link for use in implantable applications. J. Microelectromech. Syst.
**2000**, 9, 18–23. [Google Scholar] [CrossRef] - Rodriguez, S.; Ollmar, S.; Waqar, M.; Rusu, A. A batteryless sensor ASIC for implantable bio-impedance applications. IEEE Trans. Biomed. Circuit Syst.
**2016**, 10, 533–544. [Google Scholar] [CrossRef] [PubMed] - Bhamra, H.; Tsai, J.W.; Huang, Y.W.; Yuan, Q.; Shah, J.V.; Irazoqui, P. A subcubic millimeter wireless implantable intraocular pressure monitor microsystem. IEEE Trans. Biomed. Circuit Syst.
**2017**, 11, 1204–1215. [Google Scholar] [CrossRef] [PubMed] - Nopper, R.; Has, R.; Reindl, L. A wireless sensor readout system—Circuit concept, simulation, and accuracy. IEEE Trans. Instrum. Meas.
**2011**, 60, 2976–2983. [Google Scholar] [CrossRef] - Huang, Q.A.; Dong, L.; Wang, L.F. LC passive wireless sensors toward a wireless sensing platform: Status, prospects, and challenges. J Microelectromech. Syst.
**2016**, 25, 822–840. [Google Scholar] [CrossRef] - Babu, A.; George, B. A linear and high sensitive interfacing scheme for wireless passive LC sensors. IEEE Sens. J.
**2016**, 16, 8608–8616. [Google Scholar] [CrossRef] - Zhang, C.; Wang, L.F.; Huang, J.Q.; Huang, Q.A. An LC-type passive wireless humidity sensor system with portable telemetry unit. J Microelectromech. Syst.
**2015**, 24, 575–581. [Google Scholar] [CrossRef] - Demori, M.; Masud, M.; Baù, M.; Ferrari, M.; Ferrari, V. Passive LC sensor label with distance-independent contactless interrogation. In Proceedings of the 2017 IEEE Sensors Conference, Glasgow, UK, 30 October–1 November 2017. [Google Scholar]
- Wang, X.; Larsson, O.; Platt, D.; Nordlinder, S.; Engquist, I.; Berggren, M.; Crispin, X. An all-printed wireless humidity sensor label. Sensor Actuators B Chem.
**2012**, 166, 556–561. [Google Scholar] [CrossRef] [Green Version] - Nopper, R.; Niekrawietz, R.; Reindl, L. Wireless readout of passive LC sensors. IEEE Trans. Instrum. Meas.
**2010**, 59, 2450–2457. [Google Scholar] [CrossRef] - Baù, M.; Ferrari, M.; Ferrari, V. Analysis and validation of contactless time-gated interrogation technique for quartz resonator sensors. Sensors
**2017**, 17, 1264. [Google Scholar] [CrossRef] [PubMed] - Ferrari, M.; Baù, M.; Tonoli, E.; Ferrari, V. Piezoelectric resonant sensors with contactless interrogation for mass sensitive and acoustic-load detection. Sensors Actuators A Phys.
**2013**, 202, 100–105. [Google Scholar] [CrossRef] - Demori, M.; Baù, M.; Ferrari, M.; Ferrari, V. Electronic technique and circuit topology for accurate distance-independent contactless readout of passive LC sensors. AEU Int. J. Electron. Commun.
**2018**, 92, 82–85. [Google Scholar] [CrossRef] - Morshed, B.I. Dual coil for remote probing of signals using resistive wireless analog passive sensors (rWAPS). In Proceedings of the 2016 United States National Committee of URSI National Radio Science Meeting, Boulder, CO, USA, 21 March 2016. [Google Scholar]
- Yang, B.; Meng, F.; Dong, Y. A coil-coupled sensor for electrolyte solution conductivity measurement. In Proceedings of the 2013 2nd International Conference on Measurement, Information and Control, Harbin, China, 6 March 2014. [Google Scholar]
- Arnau, A.; Ferrari, V.; Soares, D.; Perrot, H. Interface electronic systems for AT-Cut QCM sensors: A comprehensive review. In Piezoelectric Transducers and Applications, 2nd ed.; Springer-Verlag Berlin: Heidelberg, Germay, 2008; pp. 187–203. [Google Scholar]
- DeHennis, A.; Wise, K.D. A double-sided single-chip wireless pressure sensor. In Proceedings of the MEMS 2002 IEEE International Conference, Las Vegas, NV, USA, 21–24 January 2002. [Google Scholar]
- Harpster, T.J.; Hauvespre, S.; Dokmeci, M.R.; Najafi, K. A passive humidity monitoring system for in situ remote wireless testing of micropackages. J Microelectromech. Syst.
**2002**, 11, 61–67. [Google Scholar] [CrossRef] - Jacquemod, G.; Nowak, M.; Colinet, E.; Delorme, N.; Conseil, F. Novel architecture and algorithm for remote interrogation of battery-free sensors. Sensor Actuators A Phys.
**2010**, 160, 125–131. [Google Scholar] [CrossRef]

**Figure 2.**Equivalent circuits of the two considered cases for a coil-coupled resonant sensor unit (RSU): (

**a**) capacitance sensor C

_{S}; (

**b**) electromechanical piezoelectric resonator represented with its equivalent Butterworth–van Dyke (BVD) model.

**Figure 3.**(

**a**) Block diagram of the interrogation system based on impedance measurement from the primary coil; (

**b**) equivalent circuit for the calculation of Z

_{1}.

**Figure 4.**Real part of Z

_{1}as a function of frequency from Equation (4) for three different values of k.

**Figure 5.**(

**a**) Block diagram of the time-gated technique; (

**b**) equivalent circuit of the time-gated technique during the detection phase.

**Figure 6.**Voltage v

_{1}(t) during the detection phase calculated for three different values of the coupling factor k.

**Figure 7.**(

**a**) Block diagram of the interrogation system with equivalent circuit of electromechanical piezoelectric resonator around f

_{r}; (

**b**) block diagram of the interrogation system with equivalent circuit of electromechanical piezoelectric resonator for f >> f

_{r}.

**Figure 8.**(

**a**) Comparison of f

_{m_r}derived from the maximum of Re{Z

_{1}} for frequencies around f

_{r}, in Equation (9), and the approximate value from Equation (11) as a function of L

_{2}; (

**b**) comparison of f

_{m_el}derived from the maximum of Re{Z

_{1}} for f >> f

_{r}, in Equation (9), and the approximate value from Equation (13) as a function of L

_{2}.

**Figure 9.**Block diagram of the time-gated technique applied to a coil-coupled electromechanical piezoelectric resonator.

**Figure 10.**(

**a**) Block diagram of the interrogation system with equivalent circuit of the impedance Z

_{1P}for the technique based on impedance measurements applied to a coil-coupled capacitance sensor; (

**b**) block diagram of the interrogation system with equivalent circuit in the Laplace domain to derive V

_{P1}(s) during the detection phase of the time-gated technique applied to a coil-coupled capacitance sensor.

**Figure 11.**Comparison of the (f

_{SP}− f

_{S})/f

_{S}obtained from the two techniques as a function of k for three different values of the ratio C

_{P}/C

_{S}. The exact value of f

_{S}without the parasitic capacitance, i.e., C

_{P}= 0, is f

_{S}= 5.626977 MHz.

**Figure 12.**(

**a**) Block diagram of the interrogation system with equivalent circuit of the impedance Z

_{1P}for the technique based on impedance measurements applied to an electromechanical piezoelectric resonator; Z

_{Rr}represents the reflected impedance of the RSU with electromechanical piezoelectric resonator. (

**b**) Block diagram of the interrogation system with equivalent circuit in the Laplace domain to derive V

_{1P}(s) during the detection phase of the time-gated technique applied to an electromechanical piezoelectric resonator.

**Figure 13.**Comparison of the relative deviation (f

_{rP}− f

_{r})/f

_{r}obtained from the time-gated technique and the impedance technique as a function of k for three different values of the ratio C

_{P}/C

_{r}.

**Figure 14.**Block diagram of the interrogation system based on impedance measurement technique with parasitic capacitance compensation circuit.

**Figure 15.**Block diagram of the interrogation system based on of time-gated technique with parasitic capacitance compensation circuit.

**Figure 17.**Experimental setup and interrogation system based on impedance-measurement technique with parasitic capacitance compensation.

**Figure 18.**Measured maxima in Re{Z

_{1P}} around f

_{S}for different values of the compensation C

_{C}, varying the distance d between CL

_{1}and the RSU. The frequency of the maxima at f

_{mP}is highlighted with a black circle.

**Figure 19.**Measured frequency f

_{mP}as a function of d for different values of C

_{C}. The no compensation data are extrapolated from experimental values.

**Figure 20.**(

**a**) Real part of Z

_{1P}measured around the mechanical resonant frequency f

_{r}of the quartz crystal resonator (QCR) connected to the primary coil CL

_{1}for different distances d. The frequency of the maxima at f

_{rP}is highlighted with a black circle. (

**b**) Frequency f

_{rP}as a function of d. The error bars report the standard deviations calculated over 5 repeated measurements.

**Figure 22.**Measured output signal v

_{O}(t) during the detection phase. Indications of the adopted delay time T

_{D}and measurement time T

_{M}are reported.

**Figure 23.**Frequency f

_{dP}of the damped sinusoid v

_{1P}(t) measured as a function of the interrogation distance d for different values of the compensation capacitance C

_{C}. A delay time T

_{D}= 2 µs and a measurement time T

_{M}= 6 µs have been set in the measurements.

**Figure 24.**(

**a**) Measured output signal v

_{O}(t) at the beginning of the detection phase for three different interrogation distances d. (

**b**) Frequency f

_{rP}as a function of d measured with a delay time T

_{D}= 5 µs and a measurement time T

_{M}= 10 ms. The error bars report the standard deviations calculated over 30 repeated measurements.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Demori, M.; Baù, M.; Ferrari, M.; Ferrari, V.
Interrogation Techniques and Interface Circuits for Coil-Coupled Passive Sensors. *Micromachines* **2018**, *9*, 449.
https://doi.org/10.3390/mi9090449

**AMA Style**

Demori M, Baù M, Ferrari M, Ferrari V.
Interrogation Techniques and Interface Circuits for Coil-Coupled Passive Sensors. *Micromachines*. 2018; 9(9):449.
https://doi.org/10.3390/mi9090449

**Chicago/Turabian Style**

Demori, Marco, Marco Baù, Marco Ferrari, and Vittorio Ferrari.
2018. "Interrogation Techniques and Interface Circuits for Coil-Coupled Passive Sensors" *Micromachines* 9, no. 9: 449.
https://doi.org/10.3390/mi9090449