sensors Review Planar Phase-Variation Microwave Sensors for Material Characterization: A Review and Comparison of Various Approaches Jonathan Muñoz-Enano 1, * , Jan Coromina 1 , Paris Vélez 1 , Lijuan Su 1 , Marta Gil 2 , Pau Casacuberta 1 and Ferran Martín 1 1 CIMITEC, Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain; Jan.Coromina@uab.cat (J.C.); Paris.Velez@uab.cat (P.V.); Lijuan.Su@uab.cat (L.S.); pau.casacubertao@e-campus.uab.cat (P.C.); Ferran.Martin@uab.cat (F.M.) 2 Departamento Ingeniería Audiovisual y Comunicaciones, Universidad Politécnica de Madrid, 28031 Madrid, Spain; marta.gil.barba@upm.es * Correspondence: Jonatan.Munoz@uab.cat Abstract: Planar phase-variation microwave sensors have attracted increasing interest in recent years since they combine the advantages of planar technology (including low cost, low profile, and sensor integration with the associated circuitry for post-processing and communication purposes, among others) and the possibility of operation at a single frequency (thereby reducing the costs of the associated electronics). This paper reviews and compares three different strategies for sensitivity improvement in such phase-variation sensors (devoted to material characterization). The considered   approaches include line elongation (through meandering), dispersion engineering (by considering Citation: Muñoz-Enano, J.; slow-wave artificial transmission lines), and reflective-mode sensors based on step-impedance open- Coromina, J.; Vélez, P.; Su, L.; Gil, M.; ended lines. It is shown that unprecedented sensitivities compatible with small sensing regions are Casacuberta, P.; Martín, F. Planar achievable with the latter approach. Phase-Variation Microwave Sensors for Material Characterization: A Keywords: microwave sensors; stepped-impedance transmission lines; slow-wave transmission Review and Comparison of Various lines; meander lines; dielectric constant sensor; phase-variation sensors Approaches. Sensors 2021, 21, 1542. https://doi.org/10.3390/s21041542 Academic Editor: 1. Introduction Guillermo Villanueva Planar microwave sensors are very good candidates for material characterization and analysis. The reason is the high sensitivity of microwaves to the properties of the materials Received: 5 January 2021 Accepted: 18 February 2021 to which they interact, combined with the advantageous aspects of planar technology Published: 23 February 2021 (including low cost and profile, compatibility with other technologies, such as microfluidics, full integration with the associated electronics, or the possibility of implementing the Publisher’s Note: MDPI stays neutral sensors in conformal, flexible and organic/compostable substrates, among others). with regard to jurisdictional claims in Sensor cost, design simplicity, robustness, and performance (mainly, although not published maps and institutional affil- exclusively, the sensitivity) are the main aspects that determine the interest and quality iations. of a sensing device. As mentioned before, planar sensors are typically cheap, at least as compared to bulk sensors (e.g., cavity sensors). However, the real cost of a sensor is not merely determined by the sensitive part of the device, but also by the associated electronics needed for sensor feeding, post-processing and, eventually, communication purposes. Copyright: © 2021 by the authors. Probably, the most extended approach for sensing using microwaves is frequency vari- Licensee MDPI, Basel, Switzerland. ation [1–24], where, typically, a planar resonant element combined with a transmission This article is an open access article line constitutes the sensing (microwave) part of the sensor. Frequency-variation sensors distributed under the terms and have been applied to the determination of the complex dielectric constant of solids and conditions of the Creative Commons liquids [3,5–7,10,17–24], from the measurement of the resonance frequency and magnitude Attribution (CC BY) license (https:// variation caused by the so-called material under test (MUT). Permittivity sensors based creativecommons.org/licenses/by/ on frequency splitting have been also recently reported [25–35]. These sensors consist 4.0/). of a transmission line structure loaded with a pair of identical resonant elements, and Sensors 2021, 21, 1542. https://doi.org/10.3390/s21041542 https://www.mdpi.com/journal/sensors Sensors 2021, 21, 1542 2 of 27 sensing is based on resonance splitting caused by an asymmetric dielectric loading be- tween the reference (REF) resonator and the MUT resonator. Similar to differential-mode sensors [35–46], frequency-splitting sensors are robust against cross sensitivities caused by environmental factors (e.g., changes in temperature or humidity). This is the main advantage of frequency-splitting over frequency-variation sensors. However, both sensor types need wideband interrogation signals for measurement purposes, at least covering the output dynamic range, and this represents a penalty in terms of the costs related to the associated electronics. Note that, despite the fact that vector network analyzers (VNAs) are typically used for demonstration purposes, the interrogation signal sweeping in a real scenario (where portable devices are usually a need) should be generated by means of wideband voltage-controlled oscillators (VCOs). This represents a challenging issue since a single VCO may not suffice to cover the required bandwidth of the output signal. Therefore, this may represent also a burden in terms of sensor costs. To alleviate the previous cost limitations, single-frequency sensors (requiring a single- tone harmonic signal for sensing) represent a good solution. Coupling modulation sen- sors [47–60] belong to this category. In such sensors, the sensing principle is based on modulation of the coupling between a resonator (or a set of resonators) and a transmission line, caused by a relative displacement between both elements, or by an asymmetric di- electric load in the resonant element. The typical output variable in such sensors is the magnitude of the transmission coefficient at the operating frequency, which can be easily converted to a voltage magnitude by means of an envelope detector [53,59]. The main drawback of such sensors concerns the fact that magnitude measurements exhibit a limited robustness against the effects of interference and noise, at least as compared to frequency or phase measurements. According to the previous words, single-frequency phase-variation sensors, the subject of this review paper, may represent a good tradeoff between overall costs and robustness against noise and interference effects. The main objective of this review paper is to point out, discuss and compare different sensing strategies devoted to performance optimization (specifically sensitivity enhancement, the typical challenging aspect) in phase-variation microwave sensors [36–38,45,46,61–64]. Even though examples of planar phase-variation microwave sensors focused on sensing spatial variables have been reported [65–67], the sensors to be discussed in the present paper are mainly devoted to material characterization. Nevertheless, an example of a rotation sensor based on phase variation is also included in the paper (other sensing strategies different than phase variation are not discussed in the paper). The working principle in such sensors is the phase variation in the sensing element, a transmission line section (either conventional or artificial), caused by variations in the effective dielectric constant of the MUT. Specifically, the considered strategies for sensitivity optimization through the measurement of the phase are (i) line elongation through meandering, (ii) phase velocity reduction by means of slow-wave transmission lines, and (iii) impedance contrast in reflective-mode step-impedance open-ended lines. A phase-variation differential sensor based on highly dispersive composite right/left handed (CRLH lines), where the phase information is transformed to magnitude information, is also included in the paper. Organization of the paper is as follows: In Section 2, the considered sensing approaches for sensitivity optimization are justified. Sections 3–5 are devoted to phase-variation sen- sors based on meander lines, slow-wave transmission lines, and reflective-mode step- impedance lines, respectively. Other phase-variation sensors, including a prototype de- voted to measure angular displacements and a differential phase-variation sensor with phase-to-magnitude conversion, are reported in Section 6. A comparative analysis is carried out in Section 7. Finally, the main concluding remarks are highlighted in Section 8. 2. Towards Sensitivity Optimization in Phase-Variation Microwave Sensors The typical performance parameters in microwave sensors are the sensitivity, the resolution, the accuracy, the input and output dynamic ranges, the linearity, and the repeatability. Although such parameters are used as indicators of the quality of a sen- Sensors 2021, 21, 1542 3 of 27 sor, other aspects also determine the preferred sensor approach for a certain application. Such aspects are sensor cost, size, design complexity, and robustness (understood as the capacity of the sensor to mitigate the pernicious effects of electromagnetic interference, noise, cross-sensitivities, and harsh environments). Phase-variation microwave sensors are good candidates for sensing, since this type of sensors exhibits a good combination of the previous characteristics and attributes. In many applications, a high sensitivity and resolution is required. For example, highly sensitive sensors are needed in several industrial processes, including the heavy and automotive industry (e.g., defect detection in samples, material composition measurements, motion control), or the food and related industry (e.g., wine fermentation processes), among others. Other applications related to health and bio-sensing (e.g., monitoring the electrolyte content or the glucose level in blood or urine) also require high sensitivity. Thus, sensors able to detect small changes in material composition are highly demanded. Such changes typically manifest through a variation in the dielectric constant (or, more generally, the complex permittivity) of the so-called material, or sample, under test (MUT), a measurable quantity using microwave technology. The phase velocity of an electromagnetic signal propagating in a certain medium depends on the dielectric constant (or relative permittivity), ε, of that medium, according to [68]: c vp = √ (1) ε where c is the speed of light in vacuum. Consequently, a variation in ε can be detected through the variation in the phase of the wave measured at a certain point, provided the phase depends on the phase velocity. However, most phase-variation sensors are implemented by means of planar structures, specifically open transmission lines, such as microstrip lines, coplanar waveguide (CPW) transmission lines, etc., see Figure 1. In such lines, the velocity of the propagating waves (also called phase velocity) is given by: c vp = √ (2) εe f f where εeff is the so-called effective dielectric constant. The effective dielectric constant depends on the dielectric constant of the substrate, εr , and the dielectric constant of the medium surrounding the line. Such medium is typically air, at least when the lines act as distributed circuit elements. However, in transmission-line phase-variation sensors, the line (the sensing element) should be covered by the MUT. Thus, the dielectric constant of the MUT, εMUT , determines the value of the effective dielectric constant, which is also a function of the transverse geometry of the line (the specific value of εeff is given by the distribution of the electric field lines between the substrate and the MUT on top of the line). The specific formula is: ε r + ε MUT εe f f = (3) 2 for CPW transmission lines, and: ε r + ε MUT ε r − ε MUT εe f f = + F (4) 2 2 for microstrip lines. In (4), F is a geometry factor given by:  − 1 h 2 F= 1 + 12 (5a) Ws for Ws /h ≥ 1, or by  − 1  2 h 2 Ws F= 1 + 12 + 0.04 1 − (5b) Ws h 𝐹 = (1 + 12 ) (5a) 𝑊𝑠 for Ws/h ≥ 1, or by 1 ℎ −2 𝑊𝑠 2 (5b) Sensors 2021, 21, 1542 𝐹 = (1 + 12 ) + 0.04 (1 − ) 4 of 27 𝑊𝑠 ℎ for Ws/h < 1, where h and Ws are the substrate thickness and the width of the sensing line, respectively, and it is assumed that t << h, where t is the thickness of the metallic layer. for Ws /h < 1, where h and Ws are the substrate thickness and the width of the sensing line, In Equations (3) and (4), it is assumed that the MUT is semi-infinite in the vertical respectively, and it is assumed that t << h, where t is the thickness of the metallic layer. direction. The validity of Equation (3) is also examined in the case of a CPW transmission In Equations (3) and (4), it is assumed that the MUT is semi-infinite in the vertical line with a semi-infinite substrate. If such conditions are not fulfilled, (3) and (4) no longer direction. The validity of Equation (3) is also examined in the case of a CPW transmission provide the effective dielectric constants for the CPW and microstrip lines, respectively. line with a semi-infinite substrate. If such conditions are not fulfilled, (3) and (4) no longer Nevertheless, the functionality of the sensor is preserved, despite the fact that such provide the effective dielectric constants for the CPW and microstrip lines, respectively. approximations are not satisfied. Namely, a change in MUT generates a variation in eff, Nevertheless, the functionality of the sensor is preserved, despite the fact that such approx- which in turn modifies the phase velocity of the transmission line. The phase of the line imations are not satisfied. Namely, a change in εMUT generates a variation in εeff , which in (or electrical length) depends on the angular frequency, 0, the line length l, and the phase turn modifies the phase velocity of the transmission line. The phase of the line (or electrical velocity, length) dependsvp, on according to: frequency, ω , the line length l, and the phase velocity, v , the angular 0 p according to: 0 𝑙 0 𝑙 ω0  l = ω l= √𝑒𝑓𝑓 (6) = 𝑣𝑝0 ε e𝑐f f p φ= (6) vp c Thus, sensing MUT, or any other variable related to it, e.g., material composition, Thus, solute sensing content inεMUT , or any a liquid other variable solution, relatedfrom etc., is possible to it,the e.g., material composition, measurement of the phase of solute content in a liquid solution, etc., is the line, such as Equation (6) suggests. possible from the measurement of the phase of the line, such as Equation (6) suggests. (a) (b) (c) (d) Figure 1. Perspective Figure view of 1. aPerspective microstripview lineof(a) and CPWline a microstrip (b) (a) surrounded and CPW (b)by surrounded air, and cross-sectional view of the by air, and cross-sectional microstrip line (c) andview CPWof(d) covered by a MUT. The electric field lines are indicated. the microstrip line (c) and CPW (d) covered by a MUT. The electric field lines are indicated. The phase of theofline, The phase φ, is not the line, , isnecessarily the output not necessarily variable the output in phase-variation variable in phase-variation sensors. For example, there are sensors where the phase information is transformed to sensors. For example, there are sensors where the phase information is transformed to magnitude information [38,45,46]. Nevertheless, it is clear that for sensitivity optimization, magnitude information [38,45,46]. Nevertheless, it is clear that for sensitivity it is necessary to achieve the maximum possible variation of the phase of the line with the optimization, it is necessary to achieve the maximum possible variation of the phase of dielectric constant of the MUT. The sensitivity in phase-variation sensors where the output the line with the dielectric constant of the MUT. The sensitivity in phase-variation sensors variable is the phase (those of the foremost interest in the present study) is thus defined as: where the output variable is the phase (those of the foremost interest in the present study) is thus defined as: dφ S= (7) dε MUT 𝑑 𝑆= (7) 𝑑 𝑀𝑈𝑇 In ordinary lines, the following result is obtained: In ordinary lines, the following result is obtained: dφ dφ dε e f f ω l dε e f f S= = · = √0 · (8) dε MUT dε e f f dε MUT 2c ε e f f dε MUT Using Equation (3), the sensitivity for a CPW line is found to be: ω0 l S= √ √ (9) 2 2c ε r + ε MUT Sensors 2021, 21, 1542 5 of 27 whereas for microstrip lines, the sensitivity is inferred by introducing Equation (4) in (8), i.e.,: ω0 l S= √ q (10) 2 2c ε r 1+ F 2 + ε MUT 1−1 F (1− F ) It is interesting to mention that for a CPW sensing line with thick enough MUT and substrate, S does not depend on the transverse geometry, contrarily to microstrip lines (note the dependence of S on F in (10)). In both cases, reducing the dielectric constant of the substrate, εr , boosts up the sensitivity. However, the dependence of the sensitivity with εr is through the square root, and hence, it is limited. Moreover, aspects such as the compatibility with the associated sensor electronics, or cost, may dictate the sensor substrate. Thus, εr cannot be considered a true design parameter, useful for sensitivity optimization. Inspection of (9) and (10) reveals that the sensitivity is proportional to the product ω 0 l. Therefore, it is clear that, for sensitivity optimization, sensor operation at high frequencies and/or the use of long sensing lines are good strategies for sensitivity optimization. However, operating at high frequencies is not exempt of certain complexity in the generation of the harmonic signal required for sensing. Therefore, the typical approach for sensitivity enhancement in phase-variation sensors based on transmission lines is line meandering. By meandering, elongated lines with reasonable shape factors for the sensing region, delimited by the region to which the meandered line is circumscribed, are obtained. An example of a phase-variation sensor devoted to dielectric constant measurements, operating in differential mode, and based on a pair of meandered lines is reported in [37], and will be the subject of Section 3. A different strategy for sensitivity optimization in phase-variation sensors exploits dispersion engineering in artificial transmission lines [69–71]. The idea behind this ap- proach is to reduce the phase velocity of the (artificial) sensing line. For that purpose, composite right/left-handed (CRLH) lines, first reported in [72], electro-inductive wave transmission lines [69,73], and slow-wave transmission lines [69,74–76] are useful. In an ordinary line, the phase velocity can only be reduced by increasing the dielectric constant of the substrate, εr . However, the net effect of increasing εr is a reduction in the sensitivity, as it can be appreciated in Equations (9) and (10). Let us clarify this aspect. Note that from Equation (6), the sensitivity can be alternatively expressed as: dφ dφ dv p ω l dv p S= = · = − 02 · (11) dε MUT dv p dε MUT v p dε MUT and, apparently, a reduction of vp increases the sensitivity. However, in an ordinary transmission line, the last term in Equation (11) can be expressed as: dv p dv p dε e f f v3p dε e f f = · =− 2· (12) dε MUT dε e f f dε MUT 2c dε MUT and it is obvious that, by introducing Equation (12) into Equation (11), the sensitivity, in terms of vp , is found to be: dφ dφ dv p ω0 lv p dε e f f S= = · = · (13) dε MUT dv p dε MUT 2c2 dε MUT and it increases with vp , i.e., it decreases with εr . However, in artificial lines, the phase velocity can be reduced by different approaches, and this opens the possibility to enhance the sensitivity based on the reduction of the phase velocity. The last phase-variation sensors studied in this review paper are one-port reflective- mode sensors based on either half- or a quarter-wavelength open-ended transmission line resonators. In such sensors the output variable is the phase of the reflection coefficient, φρ . As it will be shown, the sensitivity to the dielectric constant of the MUT can be enhanced Sensors 2021, 21, 1542 6 of 27 by appropriately choosing the characteristic impedance of the half- or quarter-wavelength open-ended sensing lines. However, as it has been demonstrated [62], the sensitivity can be unprecedentedly enhanced by cascading high/low impedance quarter-wavelength line sections between the sensing line and the input port of the structure. There is a multiplicative effect in the sensitivity, controlled by the impedance contrast of the quarter- wavelength line sections. The main benefit of these sensors, based on a step-impedance transmission line configuration, is the fact that the sensitivity can be enhanced without the need to increase the dimensions of the sensing region. Therefore, sensors with huge sensitivity compatible with small sensing areas can be designed and fabricated. Indeed, a figure of merit in phase-variation sensors is the ratio between the maximum sensitivity, S, and the sensing area, A, expressed in terms of the squared wavelength, i.e.,: S FoM = (14) A As will be shown in Section 5, these reflective-mode phase-variation sensors exhibit a very high FoM, and are therefore excellent candidates for applications requiring high sensitivity and resolution, as well as small MUT samples. 3. Phase-Variation Sensors Based on Meandered Lines As mentioned before, sensitivity optimization in phase-variation sensors implemented with sensitive regions based on meandered lines relies on the elongated length of the sensi- tive lines. As an example of the potential of meandered lines for sensitivity optimization, let us consider a differential sensor implemented with two identical meandered lines [37]. The sensing principle is the variation of the electrical length of the sensing line, as compared to the reference line, when such lines are loaded with the MUT and REF samples, respec- tively. The input and output variables of the sensor are the differential dielectric constant, ∆ε = εREF − εMUT , and the differential phase of the lines, ∆φ = φREF − φMUT , respectively, where εREF and εMUT are the dielectric constants of the REF and MUT samples, respectively, and φREF and φMUT are the electrical lengths of the REF and MUT line, respectively. In order to express ∆φ in terms of ∆ε, let us first write the differential phase as a function of the effective dielectric constants of the REF line, εeff,REF , and MUT line, εeff,MUT , i.e.,: p  ∆φ = β 0 l ε e f f ,REF − p ε e f f ,MUT (15) where β0 = ω 0 /c is the phase constant in vacuum. These effective dielectric constants are given by (3) or by (4), depending on the considered line technology (microstrip or CPW). Obviously, the sub-index MUT in (3) or (4) should be replaced with the sub-index REF for the corresponding line. Introducing the effective dielectric constants of both lines in (15), the following result is obtained: q  β0 l q ∆φ = √ ε r (1 + F ) + ε REF (1 − F ) − ε r (1 + F ) + ε MUT (1 − F ) (16) 2 where line in implementation in microstrip technology has been considered. Let us now consider that the interest is to measure small variations of the dielectric constant of the MUT in reference to the one of the REF material, with a maximum input dynamic range designated as ±∆εmax . Under these conditions, it follows that ∆ε << εREF , and Equation (16) can be approximated by: β0 l ∆ε(1 − F ) ∆φ = √ p (17) 2 2 ε r (1 + F ) + ε REF (1 − F ) Sensors 2021, 21, 1542 7 of 27 For sensitivity optimization, and taking into account the considered maximum input dynamic range, it is convenient to force ∆φ = ±π when ∆ε = ±∆εmax . Introducing these extreme values in Equation (17) and isolating the length of the lines, one obtains: √ p ε r (1 + F ) + ε REF (1 − F ) l = 2 2π (18) β 0 ∆ε max (1 − F ) and the sensitivity is: π β l (1 − F ) S= = √0 · p (19) ∆ε max 2 2 ε r (1 + F ) + ε REF (1 − F ) with l given by Equation (18). Note that by simply increasing l, the sensitivity can be made as high as desired (with the operating frequency, and consequently β0 , set to a certain value). In view of Equation (18), it is clear that small dielectric constant substrates help to reduce the length of the lines (by decreasing εREF , the same effect is obtained, but this parameter is not a design variable). For the prototype device sensor, reported in [37], a Rogers RO4003C substrate with dielectric constant εr = 3.5, thickness h = 0.8128 mm and loss tangent tanδSUB = 0.0027 was used. A piece of Nelco N4350-13RF substrate, with dielectric constant εREF = 3.55, and dissipation factor tanδREF = 0.0065, was considered to be the REF sample. With such thickness of the REF sample, it can be considered to be semi-infinite in the vertical direction. With these material properties, the width of the REF line necessary to provide a 50 Ω characteristic impedance was found to be Ws = 1.6 mm. The considered input dynamic range measurement was limited to ∆εmax = 0.45. Such dynamic range corresponds to 12.7% variations of the dielectric constant of the REF sample. By evaluating (18), it follows that β0 l = 84.23 rad. This gives a length for the meandered lines of l = 670.3 mm, by considering an operating frequency of f 0 = ω 0 /2π = 6 GHz. Such long line length is the motivation for line meandering. Nevertheless, similar results are achievable by means of straight lines (but in this case with the penalty of an excessive sensor dimension in the direction of the line axis). A photograph of the fabricated bare sensor, with relevant dimensions indicated, is depicted in Figure 2a, whereas Figure 2b shows the sensor loaded with the REF and MUT samples. The differential phase of the structure (inferred from full-wave electromagnetic simulation using Keysight Momentum) as a function of the dielectric constant of the MUT (or sample under test, SUT), by considering variations in the vicinity of εREF = 3.55 with ∆εmax = 0.45, is depicted in Figure 3. In reality, Figure 3 shows the results obtained by considering different values of the loss tangent of the MUT. The fact that the curves are very similar indicates that the loss factor of the MUT does not play an important role on the differential phase. The simulated curves of Figure 3 are in very good agreement with the analytical Equation (17) (Equation (5) in the source paper [37]), also included in the figure. Such agreement validates the previous analysis, and points out that it is not necessary to know in advance the loss tangent of the MUT sample in order to determine its dielectric constant. Note that for the MUT sample with the highest or lowest dielectric constant (corresponding to ±∆εmax ), the differential phase is found to be approximately ∆φ = φREF − φMUT = ±π. The sensitivity is roughly constant, in agreement with Equation (19), and it is found to be S = d∆φ/d∆ε = −415.6◦ (very close to the theoretical value, i.e., −400◦ ). With such high sensitivity, the sensor is able to detect extremely small differential dielectric constants. Sensors 2021, 21, 1542 8 of 27 Sensors2021, Sensors 2021,21, 21,xxFOR FORPEER PEERREVIEW REVIEW 88ofof27 27 (a) (a) (b) (b) Figure2.2.Photograph Figure Photographofofthe thedifferential-mode differential-modephase phasevariation variationsensor sensorbased basedon on apair pairof ofmean- mean- Figure 2. Photograph of the differential-mode phase variation sensor based on aapair of meandered deredlines, dered lines,without without(a) (a)and andwith with(b) (b)REF REFand andMUT MUT(or (orSUT) SUT)samples sampleson ontop topof ofitit(dimensions (dimensionsare are lines, without (a) and with (b) REF and MUT (or SUT) samples on top of it (dimensions are given in givenininmm). given mm).Reprinted Reprintedwith withpermission permissionfrom from[37]. [37]. mm). Reprinted with permission from [37]. 180 180 120 120 (º )) REF -- SUT (º 60 60 RO4003C SUT RO4003C 00 FR4 FR4 REF -60 -60 Usingeq. Using eq.(5) (5) Ideallines Ideal lines Meanderedlines Meandered lines(tan (tanSUT==0.0022) 0.0022) -120 -120 SUT Meanderedlines Meandered lines(tan (tanSUT==0.02) 0.02) SUT -180 -180 3.1 3.2 3.1 3.2 3.3 3.3 3.4 3.4 3.5 3.5 3.6 3.6 3.7 3.7 3.8 3.8 3.9 3.9 4.0 4.0 SUT SUT Figure Simulated 3.3.Simulated Figure3. Figure differential Simulateddifferential phase differentialphase atf0ff00=== phaseatat 6GHz GHz 66GHz for for for the the the sensor sensor sensor ofof of Figure Figure Figure 2,2,2, asas as afunction function aafunction ofofof the the the dielectric dielectric constant constant of ofthe MUT, the MUT, parameterized parameterized by bythe the loss loss tangent tangent of ofthe the dielectric constant of the MUT, parameterized by the loss tangent of the MUT. The curve given byMUT. MUT. The The curve curve given given by by Equation Equation (17), (17), or or byby Equation Equation (5)(5) in in thethe source source paper paper [37], [37], is is alsoalso included. included. The Equation (17), or by Equation (5) in the source paper [37], is also included. The mean of the meas- The meanmeanof of the the meas- measured ureddifferential ured differential differential phases phasesphases N(with (with (with = 3N N==33independent independent independent measurements) measurements) measurements) for the for for theindicated indicated indicated the materials, materials, as well materials, as as as well well as as the the error error bars bars inferred inferred from from the the standard standard deviation, deviation, are are also also included. included. the error bars inferred from the standard deviation, are also included. Reprinted with permission Reprinted Reprinted with with per- per- mission mission from [37]. from from [37]. [37]. The Thesensor The sensorwas sensor wasexperimentally was experimentallyvalidated experimentally validated validated ininin[37]. [37].For [37]. Forthat For thatpurpose, that purpose,the purpose, thesensing the sensingregion sensing region region was was first loaded first loaded withwith a commercial a commercial (uncladded) (uncladded) microwave microwave was first loaded with a commercial (uncladded) microwave substrate with well-known substrate substrate with with well-known well-known dielectric dielectricconstant, dielectric constant,i.e., constant, i.e.,the i.e., the the FR4 FR4 FR4 substrate substrate substrate with with with MUT εMUT MUT ===4.5. 4.5.The 4.5. Thethicknesses The thicknessesof thicknesses ofofthe theREF the REFand REF and and MUT samples MUTsamples MUT samplesareare 3.3 are3.3 mm 3.3mm and mmand 3.2 and3.2 mm, 3.2mm, respectively mm,respectively respectively(such (such thicknesses (suchthicknesses thicknesseswere were achieved wereachieved achievedby byby stacking stackingtwo stacking twosamples). two samples).With samples). Withthese With thesethicknesses, these thicknesses,the thicknesses, thesamples the samplescan samples canbe can beroughly be roughlyconsidered roughly considered considered semi-infinite. semi-infinite.ItIt semi-infinite. isimportant Itisis important important tototo indicate indicate indicate that that MUT thatMUT (orthe εMUT (or (orcorresponding the the corresponding corresponding differential differential differential dielec- dielec- dielectric   constant ∆ε) is out of the considered input dynamic tric constant ) is out of the considered input dynamic range. Nevertheless, the measured tric constant ) is out of the considered input dynamic range. range. Nevertheless,Nevertheless, the measuredthe measured differentialdifferential phase,   phase, , is a ∆φ, is periodic a periodic function function with   with ∆ε differential phase, , is a periodic function with  (actually,  varies linearly with , , (actually,   (actually, varies ∆φ varies linearly linearly with with but ∆ε, measured butany any but any measured measured valueof value out ofvalue outof ofthe of ∆φrange the out of range the range [,+] [,+] [−π,+π] isisexpressed expressed byisits by expressed its equivalent equivalent by its value value equivalent withinthat value thatrange). within range).Thus, that range). Thus,ititisispossible possibleto Thus, tocheck it checkthe is possible thevalidity validityof to ofthecheck the theapproach approachfrom validity fromthe of the themeas- meas- within approach from the phasemeasured differential limitationisisphase (the by limitation is dictatedby < Z0 > Zs, and s = 180°; (c) sensor with uniform mismatched s = 90° sensing line and and Z1 > Z0 > Zs , and φs = 180◦ ; (c) sensor with uniform mismatched φs = 90◦ sensing line and Zs > Z0 ; (d) sensor with uniform mismatched φs = 180◦ sensing line and Zs < Z0 ; (e) sensor based on a 90◦ uniform 50 Ω sensing line and (f) sensor based on a 180◦ uniform 50 Ω sensing line. The considered high and low impedances are 150 Ω and 25 Ω, respectively. The dimensions are given in the pictures (in mm), and the sensing regions are indicated by dashed rectangles. Reprinted with permission from [62]. The sensitivity can be further enhanced by merely cascading additional 90◦ line sections to sensors A and B of Figure 7. In [62], a sensor with N = 2 quarter wavelength (90◦ ) stages, based on a 90◦ sensing line, and impedances set to Zs = 150 Ω, Z1 = 25 Ω and Z2 = 86.6 Ω, was also reported. The achieved sensitivity for small perturbations (i.e., in the limit were εMUT = 1) was as high as 528.7◦ , a very competitive value. With such sensitivity, the FoM, defined as the ratio between the maximum sensitivity and the area of the sensing region expressed in terms of the squared guided wavelength, is found to be FoM = 21148◦ /λ2 . However, linearity is poor in this sensor, specifically devoted to measure Sensors 2021, 21, x FOR PEER REVIEW 17 of 27 Sensors 2021, 21, 1542 17 of 27 Zs > Z0; (d) sensor with uniform mismatched s = 180° sensing line and Zs < Z0; (e) sensor based on a 90° uniform 50 Ω sensing line and (f) sensor based on a 180° uniform 50 Ω sensing line. The con- sidered high and low impedances are 150 Ω and 25 Ω, respectively. The dimensions are given in tiny variations in the dielectric constant in the vicinity of the reference dielectric constant, the pictures (in mm), and the sensing regions are indicated by dashed rectangles. Reprinted with i.e., the one of air. For this reason, experimental data concerning the MUT samples of permission from [62]. Figure 10 were not considered for this sensor. 0 -30 -60  (º) -90 -120 -150 -180 -210 S = 9.4º; Sth = 9.4º 0 -50 S = 19.4º; Sth = 19.7º S = 24.3º; Sth = 24.7º (a) S = 45.9º; Sth = 46.5º (b) -100 S (º) (c) (d) (e) -150 (f) S = 185.4º; Sth = 186.1º -200 EM Simulation S = 220.8º; Sth = 222.1º Measured -250 1 2 3 4 5 6 7 8 9 10 11 MUT Figure10. Figure 10. Measured Measured andand simulated simulated phase phase of ofthe thereflection reflectioncoefficient coefficientfor forthe thesensors sensorsofofFigure Figure9, 9, and simulated sensitivity. (a) Sensor with step-impedance discontinuity and Z < Z0 < Zs; (b) sensor and simulated sensitivity. (a) Sensor with step-impedance discontinuity and Z < Z0 < Zs ; (b) sensor with step-impedance discontinuity and Z > Z0 > Zs; (c) sensor with uniform mismatched sensing with step-impedance discontinuity and Z > Z0 > Zs ; (c) sensor with uniform mismatched sensing line and Zs > Z0; (d) sensor with uniform mismatched sensing line and Zs < Z0; (e) sensor based on line a 90°and Zs > Z50 uniform 0; Ω (d)sensing sensor line withand uniform mismatched (f), sensor based onsensing line and50ZΩ a 180o uniform Z0 ; (e) line. s