micromachines Article
Interrogation Techniques and Interface Circuits for Coil-Coupled Passive Sensors Marco Demori * , Marco Baù, Marco Ferrari
and Vittorio Ferrari
Department of Information Engineering, University of Brescia, Via Branze, 38-25123 Brescia, Italy;
[email protected] (M.B.);
[email protected] (M.F.);
[email protected] (V.F.) * Correspondence:
[email protected]; Tel.: +39-030-371-5897 Received: 17 August 2018; Accepted: 5 September 2018; Published: 9 September 2018
Abstract: Coil-coupled passive sensors can be interrogated without contact, exploiting the magnetic coupling between two coils forming a telemetric proximity link. A primary coil connected to the interface circuit forms the readout unit, while a passive sensor connected to a secondary coil forms the sensor unit. This work is focused on the interrogation of sensor units based on resonance, denoted as resonant sensor units, in which the readout signals are the resonant frequency and, possibly, the quality factor. Specifically, capacitive and electromechanical piezoelectric resonator sensor units are considered. Two interrogation techniques, namely a frequency-domain technique and a time-domain technique, have been analyzed, that are theoretically independent of the coupling between the coils which, in turn, ensure that the sensor readings are not affected by the interrogation distance. However, it is shown that the unavoidable parasitic capacitance in parallel to the readout coil introduces, for both techniques, an undesired dependence of the readings on the interrogation distance. This effect is especially marked for capacitance sensor units. A compensation circuit is innovatively proposed to counteract the effects of the parasitic input capacitance, and advantageously obtain distance-independent readings in real operating conditions. Experimental tests on a coil-coupled capacitance sensor with resonance at 5.45 MHz have shown a deviation within 1.5 kHz, i.e., 300 ppm, for interrogation distances of up to 18 mm. For the same distance range, with a coil-coupled quartz crystal resonator with a mechanical resonant frequency of 4.432 MHz, variations of less than 1.8 Hz, i.e., 0.5 ppm, have been obtained. Keywords: coil-coupled sensor; passive sensor unit; resonant sensor; telemetric sensor; distance-independent contactless interrogation
1. Introduction The ongoing downscaling of modern sensing devices is facing the main challenges of ensuring adequate power supply sources and removing wired connections. The power supply in wireless sensors has been traditionally provided by batteries that, however, have limited lifetime and need periodic recharge/replacement. Moreover, issues related to their degradation and the environmental impact for their disposal need to be considered. As an alternative approach, energy harvesting techniques have gained increasing interest and undergone extensive investigations. Energy is harvested from the surroundings in the form of vibrations, motion, thermal energy, or solar energy, just to name a few. Suitable energy converters have been developed to transform the harvested energy into electrical energy using different principles, like piezoelectric [1,2], electromagnetic [3], thermoelectric [4] or pyroelectric [5,6] effects. Depending on the input source, the converted power can be sufficient to supply, continuously or intermittently, one or more sensing devices, which can transmit the measurement information through a radio frequency
Micromachines 2018, 9, 449; doi:10.3390/mi9090449
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(RF) link to a receiving and supervising unit, thus creating a completely autonomous system without the need for power supply and cabling [7]. Alternatively, solutions based on the radio frequency identification (RFId) technologies can be adopted to implement sensing solutions exploiting electromagnetic coupling or RF fields to energize and transmit measurement information [8,9]. These solutions are typically based on low power configurations relying on a microcontroller to interface passive sensors, such as capacitive or resistive sensors [10]. Implantable sensors for medical analyses and monitoring are important examples where this solution can be advantageously applied [11–13]. Both energy harvesting and RFId systems use active electronics in the sensor unit which, in specific situations, can be a limitation, like in hostile, high-temperature, and chemically-harsh environments, where traditional silicon-based electronics cannot operate. In this context, the use of coil-coupled passive sensors, i.e., devices which do not need active components and integrated circuits to operate, is attractive. This solution exploits the magnetic coupling between a primary and a secondary coil to read passive sensors. The primary coil, along with the reading circuitry, forms the readout unit, which reads the sensor unit composed of the sensor element connected to the secondary coil [14–17]. This approach offers the promising advantage of reducing the cost of the passive sensor unit, allowing the production of disposable sensors, such as labels, with a passive sensor connected to the embedded coil [18,19]. In this paper, passive coil-coupled sensor units having a resonant behavior will be considered. The resonant behavior allows extracting the measurement information through the reading of the resonant frequency of the sensor unit [14,20]. This approach is robust because it is unaffected by the disturbances, such as noise and electromagnetic interferences, which typically affect the signal amplitude. Specifically, two kinds of sensors are investigated, as introduced in Section 2, namely, capacitive sensors, which form a resonant LC circuit with the secondary coil, and piezoelectric resonators, such as Quartz Crystal Resonators (QCRs) [21] or ceramic Resonant Piezo Layers (RPLs) [22]. One of the challenges of the contactless readout of passive sensors is to adopt reading techniques independent of the coupling between the primary and secondary coils [20,23]. This, in turn, would ensure that the readings are not affected by the interrogation distance. Two readout techniques, that are virtually independent of the coupling, are presented and discussed in detail in Section 3. In particular, a frequency-domain technique based on impedance measurements [20] and a time-domain technique called time-gated technique [21] are discussed. Both techniques suffer from significant accuracy degradation, due to the unavoidable parasitic capacitance in parallel to the readout coil that introduces a dependence of the readings on the interrogation distance. This undesirable effect is investigated in detail. Section 4 illustrates a compensation circuit that is innovatively proposed to counteract the effects of the parasitic input capacitance and advantageously obtain distance-independent readings in real operating conditions. Section 5 reports a set of experimental results on prototypes that successfully demonstrate the validity of the proposed approach and circuit. 2. Coil-Coupled Passive Sensors A coil-coupled passive sensor is represented in its basic form by the schematic diagram of Figure 1. A primary coil CL1 with inductance L1 and series resistance R1 is magnetically coupled to the secondary coil CL2 with inductance L2 and resistance R2 . The magnetic coupling is accounted for by the mutual inductance M, which depends on the geometry of L1 and L2 and their spatial arrangement. Alternatively, the magnetic coupling can be described through the coupling factor p k, which is a nondimensional parameter defined as k = M/ ( L1 L2 ), resulting in |k|≤1. In the following, the values of L1 , R1 and L2 , R2 will be considered as fixed, while the value of M, and hence k, can change due to variations of the distance or orientation between CL1 and CL2 . CL2 is connected to the generic impedance ZS , which models the sensing element. In the following, the relevant cases will be considered where ZS either forms, with L2 , a second order network with
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complex conjugate poles, i.e., ZS is predominantly capacitive, or ZS itself includes a second order network with complex conjugate poles, i.e., ZS comprises an LCR network. In both cases, resonance Micromachines x FOR PEER circuit REVIEWwhere the quantity to be sensed via ZS influences the resonant 3 of 22 can occur in2018, the9,secondary frequency and, possibly, the damping. Therefore, the resulting combination will be termed Resonant Importantly, Sensor Unit (RSU).for the RSU, the measurement information is carried by the frequency of the readout signal instead its amplitude. The adoption of the principle two Importantly, for theofRSU, the measurement information is resonant carried bymeasuring the frequency of the has readout main advantages with respect to amplitude-based techniques [24,25]. Firstly, the resonant principle signal instead of its amplitude. The adoption of the resonant measuring principle has two main is robust against externaltointerferences or nonidealities that affect the signal amplitude. Secondly, as advantages with respect amplitude-based techniques [24,25]. Firstly, the resonant principle is robust it will be illustrated in the following, the resonant principle, combined with suitable electronic against external interferences or nonidealities that affect the signal amplitude. Secondly, as it will techniques, can that the the readout frequency is made independent of the electronic distance between CL1 be illustrated in ensure the following, resonant principle, combined with suitable techniques, and ensure the RSU. can that the readout frequency is made independent of the distance between CL and the RSU. 1
Figure 1. 1. Equivalent Equivalent circuit circuit of of aa coil-coupled coil-coupled passive passive sensor. Figure sensor.
The present theory will consider two specific cases for ZS and the resulting RSU. The present theory will consider two specific cases for ZS and the resulting RSU. In the first case, ZS is a capacitance sensor of value CS , forming, with L2 , an LC resonant circuit as In the first case, ZS is a capacitance sensor of value CS, forming, with L2, an LC resonant circuit as shown in Figure 2a. The resonant frequency f S and quality factor QS of the RSU are shown in Figure 2a. The resonant frequency fS and quality factor QS of the RSU are s 1 1 L2 f S == √ ;; QS == (1) (1) .. R2 CS 2π L2 CS the second secondcase, case,ZZ theequivalent equivalent impedance piezoelectric resonant sensors, QCRs In the impedance of of piezoelectric resonant sensors, like like QCRs and S Sisisthe and RPLs. Their electromechanical around can resonance can with be the modelled with the RPLs. Their electromechanical behavior behavior around resonance be modelled Butterworth–van Butterworth–van Dyke lumped-element (BVD) equivalentcircuit, lumped-element in Figure The BVD Dyke (BVD) equivalent as shown incircuit, Figure as 2b.shown The BVD circuit 2b. is composed circuit is composed of a motional, i.e.,and mechanical branch, andThe an electrical The motional of a motional, i.e., mechanical branch, an electrical branch. motional branch. branch comprises the branchofcomprises of inductance Lr, capacitance Cr, and resistancerepresent Rr, which series inductancethe Lr ,series capacitance Cr , and resistance Rr , which respectively therespectively equivalent represent the equivalent mass, losses compliance, and energy losses of thebranch resonator. The electrical branch mass, compliance, and energy of the resonator. The electrical is formed by the parallel is formed by capacitance C0, due to the dielectric material of the Under capacitance C0the , dueparallel to the dielectric material of the resonator. Under excitation byresonator. a voltage source, r excitation by a voltage source, the mechanical resonant frequency f , i.e., the frequency at which the mechanical resonant frequency f r , i.e., the frequency at which the current in the motional armthe is current in the motional to arm maximum, corresponds tothe theBVD series resonant frequency of the BVD maximum, corresponds theisseries resonant frequency of circuit, i.e., the frequency at which circuit, i.e., the at which theimpedance reactance of the mechanical branch impedance [26]. the reactance of frequency the mechanical branch vanishes [26]. Accordingly, f r and thevanishes quality factor Accordingly, f and the quality factor Q of the electromechanical resonator can be expressed as r r Qr of the electromechanical resonator can be expressed as
1 1 f r == √ ;; Qr == Rr 2π Lr Cr
r
Lr . . Cr
(2) (2)
Typically, when when electromechanical electromechanical piezoelectric piezoelectric resonators resonators are are used used as as sensors, sensors, the the measurand measurand Typically, r–Cr–Rr and, as a quantity generates variations of the parameters of the motional branch L quantity generates variations of the parameters of the motional branch Lr –Cr –Rr and, as a consequence, consequence, of f and Q . of fr and Qr. r
r
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(a)
(b)
Figure2.2.Equivalent Equivalentcircuits circuitsofofthe the two considered cases a coil-coupled resonant sensor Figure two considered cases forfor a coil-coupled resonant sensor unitunit (RSU): S (RSU): (a) capacitance sensor C ; (b) electromechanical piezoelectric resonator represented with its (a) capacitance sensor CS ; (b) electromechanical piezoelectric resonator represented with its equivalent equivalent Butterworth–van Dyke (BVD) model. Butterworth–van Dyke (BVD) model.
3.3.Analysis Techniques Analysisof ofthe theInterrogation Interrogation Techniques 3.1. 3.1.General GeneralConsiderations Considerations Specific are required requiredtotoextract extractinformation information from RSU through Specificinterrogation interrogation techniques techniques are from thethe RSU through electronic measurements measurements at the primary coil-coupled, i.e., i.e., electronic primary coil, coil, exploiting exploitingthe theadvantage advantageof of coil-coupled, contactless,operation. operation. contactless, Onemajor majorissue issue to to consider consider is the MM and coupling factor One the dependence dependenceofofthe themutual mutualinductance inductance and coupling factor thecoils coilson ongeometrical geometrical parameters, parameters, such alignment, and relative orientation. k kofofthe suchasastheir theirdistance, distance, alignment, and relative orientation. Techniquesthat thatare are influenced influenced by by the require keeping such Techniques the value value of of M, M,or orequivalently equivalentlyk, k,would would require keeping such geometrical parameters fixed and constant [27,28]. On the other hand, in most practical applications, geometrical parameters fixed and constant [27,28]. On the other hand, in most practical applications, keepingthe thedistance distanceand and the the alignment alignment between Therefore, as aas a keeping betweencoils coilsfixed fixedisisunpractical/unfeasible. unpractical/unfeasible. Therefore, key requirement for out-of-the-lab use of coil-coupled sensors, robust measurement techniques are key requirement for out-of-the-lab use of coil-coupled sensors, robust measurement techniques are demanded that are independent of k. demanded that are independent of k. In the following, two innovative techniques are illustrated to perform k-independent readout of In the following, two innovative techniques are illustrated to perform k-independent readout of RSUs of both capacitance and electromechanical piezoelectric resonator types. In particular, the first RSUs of both capacitance and electromechanical piezoelectric resonator types. In particular, the first is a frequency-domain technique which relies on the measurement of the reflected impedance at CL1. isThe a frequency-domain technique which relies on the measurement of the reflected impedance at second is a time-domain technique, termed time-gated technique, which considers the free CL is of a time-domain technique, termed time-gated technique, which the free 1 . The second damped response the RSU measured at the primary coil after that the RSU has beenconsiders energized. 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
3.2. k-Independent Techniques Applied to Coil-Coupled Capacitance Sensors Figure 3a shows the block diagram of the readout technique based on impedance
Figure 3a shows thethe block diagram of the readoutintechnique based analyzer on impedance measurements, measurements, where readout system consists an impedance connected to the where the readout system consists in an impedance analyzer connected to the primary CL . From primary coil CL1. From the equivalent circuit of Figure 3b, the impedance Z1, as a functioncoil of ω = 12πf, the equivalent circuit of Figure 3b, the impedance Z , as a function of ω = 2πf, is 1 is Z1 = R1 + jωL1 + ZR = R1 + jωL1 + ω 2 k2 L1 L2 Micromachines 2018, 9, x FOR PEER REVIEW
=
+
+
(a)
=
+
+
1 R2 + jωL2 +
1 jωCS
.
.
(3) 5 of 22
(3)
(b)
3. (a) Block diagram of the interrogation based on impedance measurement Figure 3. Figure (a) Block diagram of the interrogation system system based on impedance measurement fromfrom the the primary coil; (b) equivalent circuit for the calculation primary coil; (b) equivalent circuit for the calculation of Z1 . of Z1.
It can be seen from Equation (3) that the effect of the coupling with the RSU results in a reflected impedance ZR in series with the primary coil that makes the total impedance Z1 dependent on the coupling factor k. Nevertheless, the resonant frequency fS and the quality factor QS of the RSU, defined in Equation (1), can be obtained from the real part of Z1 [20], given by
primary coil; (b) equivalent circuit for the calculation of Z1.
It can be seen from Equation (3) that the effect of the coupling with the RSU results in a reflected impedance ZR in series with the primary coil that makes the total impedance Z1 dependent on the 9, 449 of 22 coupling Micromachines factor k. 2018, Nevertheless, the resonant frequency fS and the quality factor QS 5of the RSU, defined in Equation (1), can be obtained from the real part of Z1 [20], given by It can be seen from Equation (3) that the effect of the coupling with the RSU results in a reflected impedance ZR in series with the primary coil that makes the total impedance Z1 dependent on the Re{ }( ) = + . coupling factor k. Nevertheless, the resonant frequency f S and the quality factor QS of the RSU, defined in Equation (1), can be obtained from the real part of Z1 [20], given by
(4)
Re{Z1} has a local maximum at the frequency fm = ωm/2π, R2 which can be found by equating to (4) Re{ Z1 }(ω ) = R1 + ω 2 k2 L1 L2 2 . zero the derivative of Equation (4) with respect to ω. RInterestingly 2 + ωL − 1 enough, fm is independent of k, 2 2 ωCS and it can be related to fS and QS only. Then, combining Equations (1) and (4), the following relations Re{Z1 } has a local maximum at the frequency f m = ω m /2π, which can be found by equating hold: 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 Then, combining | Q (S only. ; Equations ≈ ∆ (1) , and (4), the following { }) = relations hold: 2QS f f m = f |max(Re{ Z1 }) = q (5) f S ; QS ≈ S , ∆ fm 4Q2S − 2
(5)
where Δfm is the full width at half maximum (FWHM) of Re{Z1}, around fm [20]. If QS is sufficiently where the full width at deviation half maximum Re{Zppm If Q large, then fm ≈∆ffSm, is with a relative |fm(FWHM) − fS|/fS of < 100 for fQmS [20]. > 50. Equations (4) and (5) 1 }, around S is sufficiently large, then f ≈ f , with a relative deviation |f − f |/f < 100 ppm for Q > 50. Equations (4) and m m ΔfSm inS Re{Z1}, the frequency S S demonstrate that from the measurement of fm and fS and quality factor (5) demonstrate that from the measurement of f m and ∆f m in Re{Z1 }, the frequency f S and quality QS of the capacitive RSU can be advantageously extracted independently from k. Figure 4 shows factor QS of the capacitive RSU can be advantageously extracted independently from k. Figure 4 shows sample plots of plots Re{Zof1}Re{Z calculated for three different values of k, and illustrates the definition of Δfm. sample 1 } calculated for three different values of k, and illustrates the definition of ∆f m . Consistently with Equation (4), k(4), only affects Consistently with Equation k only affectsamplitude. amplitude.
Figure 4. Real part of Z as a function of frequency from Equation (4) for three different values of k.
Figure 4. Real part of Z1 as a 1function of frequency from Equation (4) for three different values of k. The operating principle of the time-gated technique is shown in Figure 5a [21]. It comprises two subsequent alternating phases, namely, excitation and detection phases. During the excitation phase, when the switch is in the E position, CL1 is connected to the sinusoidal signal vexc (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 CL1 is connected to a readout circuit with a high-impedance input, resulting in a virtually zero current in CL1 . The input voltage v1 (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 v1 (t) is expected to be a
when the switch is in the E position, CL1 is connected to the sinusoidal signal vexc(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 CL1 is connected to a readout circuit with a high-impedance input, resulting in a virtually zero current in CL1. Micromachines 2018, 9, 449 v1(t) of the readout circuit during the detection phase D can be derived6 by of 22 The input voltage taking the inverse Laplace transform of the corresponding voltage V1(s), where s is the complex frequency. Since the RSU forms a second order LCR network, the voltage v1(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 damped sinusoid with frequency fd and a decay time τd from which the resonant frequency fS and the the quality factor QS of the RSU can be inferred. quality factor QS of the RSU can be inferred. Generally, assuming that the detection phase D starts at t = 0, the readout voltage v1 (t) depends Generally, assuming that the detection phase D starts at t = 0, the readout voltage v1(t) depends on the initial conditions at t = 0 of all the reactive elements, namely CS , L1 , L2 , and M. The effect on the initial conditions at t = 0 of all the reactive elements, namely CS, L1, L2, and M. The effect of the of the initial conditions on v (t) for t > 0 is to globally affect only its starting amplitude, while the initial conditions on v1(t) for t >1 0 is to globally affect only its starting amplitude, while the complex complex frequencies of the network, that define f d and τ d , are unaltered. Therefore, without losing any d, are unaltered. Therefore, without losing any frequencies of the network, that define fd and τ generality, the single initial condition V CS0 defined as the voltage across CS at t = 0 can be considered, generality, the single initial condition VCS0 defined as the voltage across CS at t = 0 can be considered, neglecting the remaining ones. As an equivalent alternative that does not change the consequences of neglecting the remaining ones. As an equivalent alternative that does not change the consequences the present treatment, V can also be seen as an effective initial condition. CS0 can also be seen as an effective initial condition. of the present treatment, VCS0 As a result, the equivalent circuit of Figure 5b representing the time-gated configuration during As a result, the equivalent circuit of Figure 5b representing the time-gated configuration during the detection phase in the Laplace domain can be considered, and the expression of V 1 (s) is the detection phase in the Laplace domain can be considered, and the expression of V1(s) is s s L1 . . VCS0 V1 ((s))==k (6) (6) L2 s 2 + s R2 + 1 L2
L2 CS
The corresponding time expression v1v(t)(t) can bebe calculated: The corresponding time expression can calculated: 1
s
=k v1((t)) =
L1 L2
s
1 4Q2s − τt atan ]. ]. d cos [2π f − VCS0 e cos[2π d t − atan 2π f d τd 4Q2s − 1
(7) (7)
The signal (t)(t) is isa adamped decay time τd that are The signalv1v dampedsinusoid sinusoidwith withdamped dampedfrequency frequencyfd f and 1 d and decay time τ d that are related to f S and QS of the RSU as related to f S and QS of the RSU as
=
fd = fS
(a)
s
1−
1−
= Q.S 1; ; τ = . d π fS 4QS 2
(8)
(8)
(b)
Figure 5. 5.(a)(a) Block diagram of of thethe time-gated technique; (b)(b) equivalent circuit of of thethe time-gated Figure Block diagram time-gated technique; equivalent circuit time-gated technique during detection phase. technique during thethe detection phase.
sufficientlylarge, large,itit results results in withaa relative relative deviation deviation |f < 50 ppm S is S |/f If If QQ sufficiently in ffdd ≈≈fSf,S ,with |fdd −−fSf|/f ppm forfor S is S 50. Notably, the coupling factor k only acts as an amplitude factor on v (t) without influencing 1 without influencing QS >S 50. Notably, the coupling factor k only acts as an amplitude factor on v1(t) either 6 reports sample plots v1calculated (t) calculated three different values of k. 1(t) either fd for τd.τFigure 6 reports sample plots of vof for for three different values of k. d or d . Figure In summary, Equations (7) and (8) demonstrate that, under the assumptions made, the time-gated technique can also allow extraction of the frequency f S and quality factor QS of the capacitive RSU, independently of k.
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In summary, Equations (7) and (8) demonstrate that, under the assumptions made, the time-gated technique Micromachines 2018, 9, 449 can also allow extraction of the frequency fS and quality factor QS of 7 ofthe 22 capacitive RSU, independently of k.
Figure 6. 6. Voltage Voltage vv11(t) (t) during during the the detection detection phase phase calculated calculated for for three three different different values values of of the the coupling coupling Figure factor k. k. factor
3.3. k-Independent Techniques Applied to Coil-Coupled Electromechanical Piezoelectric Resonators 3.3. k-Independent Techniques Applied to Coil-Coupled Electromechanical Piezoelectric Resonators Considering the technique based on impedance measurements with reference to the equivalent Considering the technique based on impedance measurements with reference to the equivalent circuit of Figure 2b, the impedance Z1 measured at the primary coil can be expressed as circuit of Figure 2b, the impedance Z1 measured at the primary coil can be expressed as
+1 + ω2 k+2 L1 L2 Z1 = R1=+ jωL
R2 + jωL2 +
1
1 jωC0
jωLr +
1 jωCr
. .
+ Rr
(9) (9)
As it can be observed in Equation (9), the impedance Z1 depends on the coupling factor k. As it can be observed in Equation (9), the impedance Z depends on the coupling factor k. Nevertheless, also in this case, the frequency fr can be extracted 1from the frequency of the maximum Nevertheless, also in this case, the frequency f r can be extracted from the frequency of the maximum of the real part of Z1. of the real part of Z1 . Close to the angular frequency ωr = 2πfr, the impedance of the motional arm Close to the angular frequency ω = 2πf r , the impedance of the motional arm Zr = Rr + jωLr Zr = Rr + jωLr + 1/(jωCr) has a magnituder typically much smaller than that of the impedance of C0, i.e., + 1/(jωCr ) has a magnitude typically much smaller than that of the impedance of C0 , i.e., |Zr | ω r , the impedance magnitude of C0 is smaller than the impedance magnitude of Zr , which then can be neglected, obtaining the equivalent circuit of Figure 7b. Consequently, the following approximated expression of Re{Z1 } results: Micromachines 2018, 9, x FOR PEER REVIEW
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Re{ Z1 } ≈ R1 + ω 2 k2 L1 L2
Re{
}≈
+
R2 R22 + ωL2 −
2 .
(12)
1 ωC0.
(12)
Also Equation (4), and andititcan canbe beseen seenthat thatRe{Z Re{Z } now Also Equation(12) (12)has hasthe thesame sameform form as as Equation Equation (4), hashas a a 1} 1now maximum at at the frequency maximum the frequencyf m_el fm_el: f m_el
s 1 L 1 √ andand Q=el = , where= f el== = . 2. = f_el p , where 2 R C 2π L C 2 0 4Qel − 2 2 0 2Qel
(a)
(13)(13)
(b)
Figure Block diagramofofthe theinterrogation interrogation system system with Figure 7. 7. (a)(a) Block diagram with equivalent equivalentcircuit circuitofofelectromechanical electromechanical piezoelectric resonator around f ; (b) block diagram of the interrogation system equivalent r piezoelectric resonator around f r ; (b) block diagram of the interrogation system withwith equivalent circuit circuit of electromechanical piezoelectric resonator for f >> f . r of electromechanical piezoelectric resonator for f >> f r .
From the previousanalysis, analysis,ititcan canbe beconcluded concluded that Re{Z first is is related to to From the previous Re{Z11}}has hastwo twopeaks: peaks:the the first related the mechanical resonance f , the second to the electrical resonance f . With the previous assumptions r el the mechanical resonance f r , the second to the electrical resonance f el . With the previous assumptions r and L2, and considering that, typically, Cr > fr. values onon thethe values ofof LrLand L2 , and considering that, typically, Cr > f r . To validate, numerically, proposedapproximations, approximations,Figure Figures 8a,b reportthe thecomparison comparisonofofthe To validate, numerically, thethe proposed 8a,b report m_r and fm_el derived respectively from Equations (11) and (13), and the frequency of the the values of f values of f m_r and f m_el derived respectively from Equations (11) and (13), and the frequency of the maxima derived numerically from Re{Z1} in Equation (9) as a function of L2. The following values of maxima derived numerically from Re{Z1 } in Equation (9) as a function of L2 . The following values of the BVD model of a 4.432 MHz AT-cut QCR have been used: C0 = 5.72 pF, Rr = 10.09 Ω, Lr = 77.98 mH, the BVD model of a 4.432 MHz AT-cut QCR have been used: C0 = 5.72 pF, Rr = 10.09 Ω, Lr = 77.98 mH, and Cr = 16.54 fF. For CL1 and CL2, the values of the electrical parameters are L1 = 8.5 µH, R1 = 5 Ω, and Cr = 16.54 fF. For CL1 and CL2 , the values of the electrical parameters are L1 = 8.5 µH, R1 = 5 Ω, and R2 = 5 Ω. and R2 = 5 Ω. Figure 8a shows that for L2 up to 10 µH, the values of fm_r predicted from Equation (11) are Figure 8a shows that for L up to 10 µH, the values of f predicted from Equation (11) are within within 3 ppm with respect to2 the numerical solutions fromm_r Equation (9). Additionally, for the same 3 ppm with respect to the numerical solutions from Equation (9). Additionally, for the same range range of variation of L2, a remarkable agreement is obtained between fm_el predicted from of variation of (13) L2 , aand remarkable agreement is obtained between f m_el predicted from Equation (13) and Equation the numerical solution.
the numerical solution.
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Micromachines 2018, 9, 449 Micromachines 2018, 9, x FOR PEER REVIEW
(a) (a)
9 9ofof2222
(b) (b)
Figure 8. (a) (a) Comparison of fof fm_rfm_r derived Re{Z frequenciesaround aroundfrf,fr,in in Figure 8. Comparison (a) Comparison derivedfrom fromthe themaximum maximum of of Re{Z Re{Z111}} for for Figure 8. of from the maximum of for frequencies frequencies around m_r derived r , in Equation (9), and the approximate value from Equation (11) as a function of L ; (b) comparison of f 2 m_el Equation approximate value from Equation (11)asasa afunction functionofofLL22;; (b) (b) comparison comparison of Equation (9),(9), andand thethe approximate value from Equation (11) of ffm_el m_el derived from the maximum of Re{Z f >> f Equation (9), and the approximate value from 1} 1for r,fr,in derived from the maximum of Re{Z } for f >> in Equation (9), and the approximate value derived from the maximum of Re{Z1 } for f >> f r , in Equation (9), and the approximate valuefrom from Equation (13) as of LL2.L. 2. Equation a function Equation (13)(13) as aaasfunction function of of 2
possibility interrogatecoil-coupled coil-coupledelectromechanical electromechanical piezoelectric piezoelectric The possibility to interrogate resonators with the TheThe possibility to to interrogate coil-coupled electromechanical piezoelectricresonators resonatorswith withthe the time-gated technique independently from the coupling has been previously demonstrated [21]. time-gated technique independently from the coupling has been previously demonstrated [21]. time-gated technique independently from the coupling has been previously demonstrated [21]. RSU configuration ofFigure Figure hasbeen been studied in showing [21], showing showing that the open The RSU configuration 9 9has studied in [21], that thecircuit opencircuit circuit TheThe RSU configuration ofofFigure 9 has been studied in [21], that the open voltage voltage v at CL the detection phase, after the RSU has been energized in the excitation 1(t) 1 during voltage v (t) at CL during the detection phase, after the RSU has been energized in the excitation 1 1 v1 (t) at CL1 during the detection phase, after the RSU has been energized in the excitation phase, is the d_r with is the sum of two damped sinusoids:one oneatatfrequency with exponential exponential decaying time τrτ, r, phase, the sum of two damped sinusoids: ffd_r decaying time sumphase, ofistwo damped sinusoids: one at frequency f d_rfrequency with exponential decaying time τ r , and one at and one at frequency f d_el with exponential decaying time τel. and one at ffrequency f decaying time τ d_el with exponential el. frequency with exponential decaying time τ . d_el el
Figure 9. Block diagram of the time-gated technique applied to a coil-coupled electromechanical Figure Blockresonator. diagram of of the the time-gated time-gated technique technique applied applied to a coil-coupled electromechanical Figure 9. Block diagram electromechanical piezoelectric piezoelectric resonator.
The damped sinusoid at fd_r is due to the mechanical response of the resonator, while the one at The damped sinusoid atresponse due themechanical mechanical response ofthe thecapacitance resonator, while while the one one at at d_risisdue The ffd_r response of resonator, the 2 the 0. In addition, fd_el is damped due to thesinusoid electricalat of to Lto that interacts with the electrical C duetotothe the electrical response ofLconsidering L22that thatinteracts interacts withthe the electrical capacitance C . In addition, d_elfor 00. In addition, ffd_el isisdue electrical of with electrical suitable values of L2response and R2, and the typical values of thecapacitance equivalent C parameters of for suitable values of L and R , and considering the typical values of the equivalent parameters of the 2 2 BVD values model of of La2 QCR, decaying time τthe magnitude than τparameters r is orders el. Thus, the for the suitable and Rthe considering typical of values of the larger equivalent of 2, and BVD model of a QCR, the decaying time τ is orders of magnitude larger than τ . Thus, the damped r sinusoid at frequency fd_el decays to zero much than the damped el than τsinusoid the damped BVD model of a QCR, the decaying time τr is orders of faster magnitude larger the el. Thus, at sinusoid at frequency f d_el to zero much than the damped at frequency f d_r frequency fd_r. Hence, the decays formerfd_el can be neglected in the expression ofthan v1(t),sinusoid which results in damped sinusoid at frequency decays to faster zero much faster the damped sinusoid at.
Hence, thefd_r former canthe be former neglected expression of vexpression results 1 (t), which of frequency . Hence, caninbethe neglected in the v1(t), in which results in (14) p ( )≅ − ( ) (0), − τtr cos 2π _ + ∼ (14) v1 ( t ) = k L1 L2 Ar e cos(2π f d_r t + θr ) − δ(t) L1 i L1 (0), (14) ) ≅ phase coefficientscos 2π θ_r are+functions − ( of) both (0), where the amplitude( and Ar and the initial conditions at the the beginning of theand detection phase (t = 0),Arand and mechanical parameters of the where amplitude phasecoefficients coefficients andthe arefunctions functions both initial conditions r electrical where the amplitude and phase Ar and θrθare ofof both thethe initial conditions at system. The last term representsphase the contribution ofthe theelectrical initial current iL1(0) in the primary inductor. at the beginning of the detection (t = 0), and and mechanical parameters of the the the beginning of the detection phase (t = 0), and the electrical and mechanical parameters of FromThe Equation (14),represents it can be seen k acts onlyofasthe a scaling foriL1 the of v1, without system. last term term thethat contribution initial factor current (0)amplitude inthe theprimary primary inductor. system. The last represents the contribution of the initial current iL1(0) in inductor. d_r and τr. From a simplified analysis that considers the affecting the sensor response parameters f From Equation (14), it can be seen that k acts only as a scaling factor for the amplitude of v , without From Equation (14), it can be seen that k acts only as a scaling factor for the amplitude of v11, without undamped systemresponse with R2 =parameters 0 and Rr = 0,f under the hypothesis that (ωC0)−1analysis >> ωL2 at theconsiders frequency the fr affecting the sensor and τ . From a simplified that d_r and τr.r From a simplified analysis that considers the affecting the sensor response parameters fd_r 1 >> ωL at the frequency undamped system system with and R Rrr ==0, 0,under under the the hypothesis hypothesis that that (ωC (ωC00))−1−>> undamped with R R22 == 00 and ωL2 at2 the frequency fr
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f r and that Qr is large, it has been obtained that the frequency f d_r can be approximated with the following relation: 1 L2 f d_r ≈ f r 1 − . (15) 2 Lr It can be observed in Equation (15) that f d_r depends on the ratio between L2 and Lr . Nevertheless, if L2
