This is the accepted version of the journal article: Herrojo, Cristian; Vélez, Paris; Muñoz-Enano, Jonathan; [et al.]. «Highly sensitive defect detectors and comparators exploiting port imbalance in rat-race couplers loaded with step-impedance open-ended transmission lines». IEEE sensors journal, Vol. 21, Issue 23 (December 2021), p. 26731-26745. DOI 10.1109/JSEN.2021.3118088 This version is available at https://ddd.uab.cat/record/258424 under the terms of the license IEEE SENSORS JOURNAL, VOL. XX, NO. XX, MONTH X, XXXX 1 Highly Sensitive Defect Detectors and Comparators Exploiting Port Imbalance in Rat- Race Couplers Loaded With Step-Impedance Open-Ended Transmission Lines Cristian Herrojo, Member, IEEE, Paris Vélez, Senior Member, IEEE, Jonathan Muñoz-Enano, Student Member, IEEE, Lijuan Su, Member, IEEE, Pau Casacuberta, Marta Gil, and Ferran Martín, Fellow, IEEE Abstract— Defect detectors and comparators, able to detect tiny differences between a reference (REF) sample and a material under test (MUT), are presented in this paper. In one version, the comparator is implemented by means of a rat-race hybrid coupler and a single step-impedance open-ended transmission line connected to one of the ports of the coupler. The comparator exploits the imbalance generated in one of the two pairs of isolated ports of the coupler when the MUT sample differs from the REF sample. The other pair of isolated ports is used for comparator feeding (-port) and for recording the output signal (-port), and feeding is achieved by means of a harmonic signal tuned to the operating frequency of the coupler. The resulting device is therefore a two-port structure, and the considered output variable is the magnitude of the transmission coefficient, an easily measurable quantity. By virtue of the step-impedance transmission line, used as MUT f0 the sensitive element of the comparator, the device exhibits very high sensitivity to variations in the dielectric constant between the REF and MUT samples, this being the input variable. A prototype device example, with a maximum sensitivity of 0.72 is reported, and applied to the detection of defects in the REF sample, a dielectric slab, generated by drilling holes of different densities across it. In the second prototype, a true differential-mode sensor, two identical sensing elements (step impedance lines) are connected to one of the pairs of isolated ports of the coupler, and roughly perfect balance is obtained when the MUT is identical to the reference sample. The maximum sensitivity in this case is 1.20, and the device discriminates also small perturbations generated in the REF material. The main relevant aspect of the reported prototypes, phase-variation sensing devices by nature, is the phase-to-magnitude conversion achieved by means of the hybrid coupler. This transforms the reflective-mode phase-variation sensing element to a two-port transmission- mode device where the output variable is the magnitude of the transmission coefficient, an easily measurable quantity. Index Terms— Comparator, defect detector, dielectric constant sensor, microwave sensor, microstrip technology, phase-variation sensor, rat-race coupler, step-impedance transmission line. I. INTRODUCTION comparator functionality is based on detecting variations in the M dielectric constant (or effective dielectric constant) of the MUT ICROWAVE COMPARATORS are sensing devices focused on with regard to the one of the REF sample. Such microwave identifying potential differences between a well-known comparators can therefore be implemented by means of sample (usually designated as reference –REF– sample) and the differential, or quasi-differential, dielectric constant sensors, so-called sample, or material, under test (MUT) using where the input signal is the differential dielectric constant microwave signals. Typically, although not exclusively, the between the REF and the MUT sample. Such differential-mode This work was supported by MICINN-Spain (project PID2019- PRICIT Excellence Program for University Professoriat (Ref. 103904RB-I00), by Generalitat de Catalunya (project 2017SGR-1159), M190020074B) by Institució Catalana de Recerca i Estudis Avançats (who awarded C. Herrojo, P. Vélez, J. Muñoz-Enano, L. Su, P. Casacuberta and F. Ferran Martín), and by ERDF. J. Muñoz-Enano acknowledges Martín are with GEMMA/CIMITEC, Departament d’Enginyeria Secreteraria d’Universitats i Recerca (Gen. Cat.) and European Social Electrònica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Fund for the FI grant. Paris Vélez and Lijuan Su acknowledge the Juan Spain. E-mail: Ferran.Martin@uab.es. de la Cierva Program for supporting them through Projects IJCI-2017- M. Gil is with Departamento Ingeniería Audiovisual y Comunicaciones, 31339 and IJC2019-040786-I, respectively. Marta Gil Barba Universidad Politécnica de Madrid, 28031 Madrid, Spain. aknowledges the Polytechnic University of Madrid and the Administration of the Community of Madrid for the support within the V XXXX-XXXX © XXXX IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 8 IEEE SENSORS JOURNAL, VOL. XX, NO. XX, MONTH X, XXXX devices are robust against cross sensitivities, e.g., caused by sensitive reflective-mode phase-variation sensors are reported. changes in environmental factors (temperature or humidity), or By using a rat-race hybrid coupler with the step-impedance noise, and are of special interest for real-time monitoring open-ended sensing lines connected to two of the isolated ports, potential changes in the MUT. However, differential or quasi- the phase information is converted to magnitude information. differential sensors require two sensing elements and, The resulting overall devices are two-port structures where the consequently, such sensors typically exhibit larger dimensions output variable is the magnitude of the transmission coefficient, than their single-ended counterparts. Nevertheless, there are whereas the input variable is the differential dielectric constant applications where the cited advantages of differential sensing between the REF and the MUT samples. With the proposed are not needed. In these cases, comparators based on (smaller) configuration, any imbalance between the reflection single-ended sensors suffice, but, obviously, at the cost of coefficients seen from the isolated ports of the coupler, caused performing the measurements in a two-step process (first, the by a difference between the REF and the MUT samples, is response of the REF sample is inferred, and then the one of the detected by a finite transmission coefficient (ideally null under MUT is obtained). perfect balance). The use of the imbalance in the rat-race There are various approaches for the implementation of coupler combined with the use of impedance-contrast step- dielectric constant microwave sensors/comparators, including impedance sensing arms is the main novel aspect of the paper. frequency variation [1]-[20], coupling modulation [21], phase A detailed sensitivity analysis is carried out, and the effects of variation [22]-[32], frequency splitting [33]-[43], and, losses are also discussed. obviously, a combination of the previous approaches. The paper is organized as follows. Section II presents the Frequency splitting sensors are, by nature, quasi-differential proposed comparators as well as their working principle. A devices where two resonators are used for sensing. In such sensitivity analysis is reported, and validated through sensors, the comparator functionality is based on the splitting in simulation, in Section III. Section IV is focused on the frequency of the (typically) notched response, caused by experimental validation. Two prototype comparators, designed symmetry disruption (e.g., generated by differences between according to the guidelines of Section III, have been fabricated, the REF and MUT samples). Nevertheless, frequency-splitting and their functionality is demonstrated by comparing the REF sensors implemented by means of bandpass structures have also sample with several defected samples, inferred from the REF been reported [39]. Comparators implemented by means of sample by drilling arrays of holes of different densities, and frequency-splitting sensors, applied to the detection of defects several MUTs. A comparison to other comparators is the in samples, have been reported [37],[42],[43]. The other sensor subject of Section V. Finally, the main conclusions are types can be implemented, in general, as single-ended or as highlighted in Section VI. differential-mode structures. For example, differential-mode phase-variation sensors useful as comparators have been II. THE PROPOSED COMPARATOR AND WORKING PRINCIPLE presented in [24],[44]. Other differential-mode sensors devoted The proposed comparators consist of a reflective-mode to material characterization are reported in [45]-[54]. differential phase-variation dielectric constant sensor In comparators, the key parameter is the resolution, since it connected to one of the pairs of the isolated ports of a rat-race determines the capacity of the device to discriminate small hybrid coupler, as depicted in the schematic of Fig. 1. The differences between the REF and the MUT samples. The resulting device is a two-port structure, where the input signal resolution is given by the minimum detectable value of the is injected to the -port (port 1 in Fig. 1), whereas the output input variable, and it is intimately related to the sensitivity at variable is recorded in the -port (port 2 in Fig. 1). With this small perturbations. Thus, high-resolution comparators should configuration, the differential phase of the reflection be necessarily implemented by means of highly-sensitive coefficients seen from the isolated ports of the coupler (the sensors. Sensitivity enhancement combined with small sized usual output variable in reflective-mode differential phase- sensors is probably the most challenging issue for microwave variation sensors) is converted to magnitude information. The engineers working on sensor’s research and development. In output variable in the proposed sensor/comparator is thus the this regard, it has been recently demonstrated that a kind of magnitude of the transmission coefficient, which can be phase-variation sensors, operating in reflective mode, exhibit measured by means of a vector network analyzer (VNA). This unprecedented sensitivities with a limited size of the sensing approach has been carried out in this paper, as far as the main area, provided they are adequately designed [26]-[29]. These purpose is to validate these comparators by means of proof-of- sensors consist of either a low-impedance half-wavelength or a concept demonstrators at laboratory level. Nevertheless, it is high-impedance quarter-wavelength open-ended sensing line possible to replace the VNA with a microwave oscillator cascaded to a set of high/low impedance quarter-wavelength transmission line sections (impedance inverters). The connected to the -port and tuned to the operating frequency of impedance contrast of the inverters generates a multiplicative the rat-race coupler, f0, and an amplitude detector connected to effect on the sensitivity of these step-impedance transmission the -port. With this strategy, reported previously in various line based sensors, and sensitivities as high as 528.7º in sensors sensors proposed by the authors [21],[55]-[56], the output devoted to the determination of the dielectric constant of the variable is the voltage amplitude of the harmonic signal MUT (the input variable) have been obtained [26]. generated in the output port of the structure, related to signal In this paper, microwave comparators based on these highly- imbalances in the isolated ports of the coupler. 8 IEEE SENSORS JOURNAL, VOL. XX, NO. XX, MONTH X, XXXX III. SENSITIVITY ANALYSIS (1) Under the lossless approximation, the transmission coefficient between the input and the output port of the circuit of Fig. 1, S21, can be expressed in terms of the phases of the (3) reflection coefficients seen from the isolated ports, REF and ρREF MUT, as 1 1 𝑆21 = − (𝑅𝐸𝐹 − 𝑀𝑈𝑇 ) = − (𝑒 𝑗𝑅𝐸𝐹 − 𝑒 𝑗𝑀𝑈𝑇 ) (1) 2 2 (2) ρMUT The magnitude of the transmission coefficient can be easily (4) calculated, i.e., 1 Z, ϕ |𝑆21 | = √2 (1 − cos  ) (2) 2 Zs, MUT /ϕs, MUT where  = REF − MUT is the differential phase of the reflection coefficients. Note that for identical REF and MUT Material Under Test samples S21= 0 (since  = 0), whereas the output variable is a maximum (S21= 1) for a combination of REF and MUT Fig. 1. Schematic of the proposed sensor/comparator with indication of samples providing out-of-phase reflection coefficients (with the ports and the main relevant variables. The dashed rectangles  = , or REF = − MUT). indicate the sensing regions for the REF and MUT samples. The sensitivity, or derivative of the output variable, S21, with the input variable, i.e., the differential dielectric constant, Let us designate by REF and MUT the reflection coefficients  =REF − MUT, can be expressed as seen from port 3 and 4, respectively, of the coupler (where the REF and MUT step-impedance sensing lines are connected), 𝑑|𝑆21 | 𝑑|𝑆21 | 𝑑 𝑑𝑠 𝑑 𝑑𝑍𝑠 𝑆= = {  +  } (3) see Fig. 1. By neglecting ohmic, dielectric and radiation losses, 𝑑 𝑑 𝑑𝑠 𝑑 𝑑𝑍𝑠 𝑑 such reflections coefficients satisfy REF=MUT= 1, regardless of the loads present in the REF and MUT sensing with s = s,REF − s,MUT, s,REF and s,MUT being the electrical lines. Thus, under this approximation, any potential imbalance lengths of the sensing line sections for the REF and MUT lines, between the isolated ports should be entirely due to a difference respectively, and Zs = Zs,REF − Zs,MUT, where Zs,REF and Zs,MUT in the phase of the reflection coefficients, intimately related to are the characteristic impedances of those sections. The first changes in the MUT as compared to the REF sample. Since term of the right-hand side member in (3) is such phase imbalance between the isolated ports of the coupler 𝑑|𝑆21 | sin  provides a finite output signal in the -port dependent on it (as = (4) 𝑑 it will be shown in the next section), it follows that the device 2√2 (1 − cos  ) is able to detect differences between the REF and the MUT samples. Note that the sensitive part of the device is the pair of The derivatives 𝑑 /𝑑𝑠 and 𝑑 /𝑑𝑍𝑠 depend on the open-ended sensing lines, with an electrical length (or phase) electrical length of the sensing line. However, it was that depends on the dielectric constant of the material (REF or demonstrated in [26] that for sensitivity optimization in step- MUT material) on top of it. The cascaded 90º line sections with impedance open-ended sensing line based sensors in the alternating high and low impedance generate a multiplicative vicinity of the REF dielectric constant (the one of the REF effect on the phase of the reflection coefficient of the lines, as sample), the electrical length of the sensing line should not be reported in [26], thereby enhancing the sensitivity. Note that in arbitrary. Particularly, such electrical length (with the REF the schematic of Fig. 1, a single 90º line (with impedance Z and material on top of the sensing line) should be either s = 90º (or phase  = 90º at f0) cascaded to the open-ended sensing line is odd multiples) or s = 180º (or even or odd multiples). considered, in accordance with the prototypes of this work. Moreover, for sensitivity optimization with the phase of the However, further 90º stages, with alternating high and low sensing line set to s = 90º, the characteristic impedance of such impedance, can be introduced, as reported, e.g., in [26], in order line must be high, whereas the impedance of the cascaded 90º to further boost up the sensitivity. line sections (if they are present) must alternately be low and According to these words, it is clear that the comparator high. Additionally, the impedance of the 90º line section functionality relies in the capability of the device to measure the adjacent to the sensing line must be low. By contrast, for differential dielectric constant between the REF and MUT sensitivity optimization with the phase of the sensing line set to samples. Obviously, in order to implement a comparator with s = 180º, the impedance of such line must be low, whereas the high discrimination capability, or resolution, it is necessary to impedance of the adjacent 90º line sections must be high, and, achieve high sensitivity in the limit of small perturbations. This if there are further sections, their impedance value must be justifies the need to cascade the high/low impedance 90º lines alternately low and high. between the sensing line and the isolated ports of the coupler. It was also demonstrated in [26] that for the phases that 8 IEEE SENSORS JOURNAL, VOL. XX, NO. XX, MONTH X, XXXX optimize the sensitivity, s = 90º or s = 180º, the derivative the line does not reach the MUT-air interface. 𝑑 /𝑑𝑍𝑠 in the limit of small perturbations is null. Thus, the In the limit of small perturbations (with  → 0, and, second summand of the right-hand side member in (3) can be consequently,  → 0), the contribution to the sensitivity neglected in that limit. It should be mentioned that for given by (4) is simplified to comparator functionality, the main interest is the optimization 𝑑|𝑆21 | 1 of the sensitivity for small imbalances, or for small variations = (9) 𝑑 2 of the dielectric constant of the MUT with regard to the one of the REF material. The reason is that sensitivity optimization in where the L’Hôpital rule has been applied to expression (4) in the vicinity of the REF dielectric constant improves the order to solve the indeterminacy in the considered limit. resolution of the comparator. Therefore, using (5), (6) and (9), the overall sensitivity in the In order to calculate the sensitivity for low perturbations, the vicinity of the dielectric constant of the REF material (low- two cases corresponding to the two optimum phases of the perturbation) is found to be open-ended sensing line should be analyzed separately. Thus, 𝑑|𝑆21 | 𝑍̅ 2  𝑆= =− (1 − 𝐹) (10a) for s = 180º, the sensitivity of the differential phase of the 𝑑 𝑍𝑠 4𝑒𝑓𝑓 ̅̅̅ reflection coefficients with the differential phase of the lines is for the sensors based on 180º sensing lines, and [26] 𝑑|𝑆21 | 𝑍𝑠  ̅̅̅ 𝑑 −2𝑍 2 𝑍̅ 2 𝑆= =− 2 (1 − 𝐹) (10b) = = −2 (5a) 𝑑 𝑍̅ 8𝑒𝑓𝑓 𝑑𝑠 𝑍0 𝑍𝑠 ̅̅̅ 𝑍𝑠 for the sensors implemented by means of 90º sensing lines. whereas for s = 90º, the following result was obtained Let us next validate the previous sensitivity analysis at 𝑑 −2𝑍0 𝑍𝑠 ̅̅̅𝑠 𝑍 schematic level, where losses, as well as other potential non- = = −2 2 (5b) ideality effects, are ignored. For that purpose, let us consider 𝑑𝑠 𝑍 2 𝑍̅ that the frequency of operation is set to f0 = 2 GHz. The where Z0 is the reference impedance of the ports, Zs is the considered substrate has the characteristics of the RO4003C nominal impedance of the sensing lines, i.e., Zs = Zs,REF, and Z substrate with thickness h = 1.52 mm and dielectric constant is the characteristic impedance of the 90º line section cascaded r = 3.55. Nevertheless, for the purpose of validation of the to the sensing line (a single 90º line section, plus the sensing previous analysis, let us consider ideal components, except the line section, is considered in this paper). In addition, the open-ended sensing line section, subjected to the presence of a normalized impedances ̅̅̅ 𝑍𝑠 = 𝑍𝑠 /𝑍0 and 𝑍̅ = 𝑍/𝑍0 have been MUT on top of it. Thus, the impedance of the line sections of used in (5). From (5), it is clear that for sensitivity optimization, the coupler (with a total ring length of 1.5, or an electrical the impedances must satisfy Zs > Z0 > Z for s = 90º, and Zs < Z0 length of 540º) is 70.71 Ω, i.e., the necessary value to obtain a < Z for s = 180º, as anticipated before. Finally, the sensitivity −3dB hybrid coupler by considering a port impedance of of the differential phase of the sensing lines with the differential Z0 = 50 Ω. Concerning the loads seen from the isolated ports of dielectric constant is [26] the coupler, namely, the step-impedance lines, we have 𝑑𝑠 𝑠,𝑅𝐸𝐹 considered sensing lines with phase and impedance of s = 90º = (1 − 𝐹) (6) 𝑑 4𝑒𝑓𝑓 and Zs = 131 Ω, respectively, when they are covered by a semi- where sensor implementation in microstrip technology has been infinite material with dielectric constant REF = 3.55 considered. In (6), eff is the effective dielectric constant of the (corresponding to one of the low-loss microwave substrates MUT open-ended sensing line, i.e. [57], available in our laboratory), the REF dielectric constant. 𝑟 + 𝑀𝑈𝑇 𝑟 − 𝑀𝑈𝑇 Between the open-ended sensing lines and the respective ports 𝑒𝑓𝑓 = + 𝐹 (7) of the coupler, a low-impedance 90º line section with Z = 29 Ω, 2 2 is inserted. Such line sections (one for each isolated port of the where r is the substrate dielectric constant, and F is a geometry coupler) are considered to be ideal. However, for the sensing factor given by 1 lines, it is necessary to consider “real components”, subjected ℎ −2 to the presence of a material (REF or MUT sample) on top of 𝐹 = (1 + 12 ) (8a) 𝑊𝑠 it. Indeed, the Keysight ADS schematic simulator provides a for Ws/h ≥ 1, or by component consisting of a transmission line section including 1 ℎ −2 𝑊𝑠 2 the presence of a dielectric layer on top of it, in our case the 𝐹 = (1 + 12 ) + 0.04 (1 − ) (8b) REF and the MUT samples. With the considered substrate and 𝑊𝑠 ℎ with a semi-infinite material of REF = 3.55 on top of the sensing for Ws/h < 1, where h and Ws are the substrate thickness and the line, the physical length and width of such line, providing the width of the MUT sensing line, respectively, and it is assumed above-cited phase and impedance values, are 19.89 mm and that t << h, where t is the thickness of the metallic layer. Note 0.20 mm, respectively (actually, the impedance Zs was set to the that the validity of (7) is subjected to the semi-infinite MUT value providing the minimum achievable width with the approximation. That is, the MUT must be thick enough in the available fabrication technology). vertical direction, so that the electromagnetic field generated by 8 IEEE SENSORS JOURNAL, VOL. XX, NO. XX, MONTH X, XXXX Z0, п/2 ρREF (1) Z0, 3п/2 Z0, п/2 (2) ρMUT Z0, п/2 Z, ϕ MUT (εr, tanδ, h) ws , l s Fig. 2. Schematic used for the simulations devoted to the validation of the sensitivity analysis. Fig. 4. Photograph of the sensor/comparator based on a single sensing 1.0 1.0 line. The sensing region is indicated by a dashed rectangle. Dimensions Theoretical sensitivity Ideal circuit sensitivity are (in mm): ws = 0.20, ls = 19.89, wd = 9.07, ld = 21.05, c = 1.88, r = 21.15. 0.8 the simulated data points (the results are also included in the 0.6 figure). The absolute value of the sensitivity in the limit of small |S21|f = f perturbations, i.e., for  = 0 (or REF = MUT), inferred from the |S| 0.5 simulated data, is found to be 0.722 whereas the one predicted 0 0.4 by the theory, by means of (10b), is 0.727, that is, in very good agreement with the simulated value. Therefore, with this study, 0.2 based on schematic simulations, the sensitivity analysis is validated. It is important to mention that there are two values of 0.0 0.0 the dielectric constant of the MUT which provides the same 1 2 3 4 5 6 7 8 9 10 value of the magnitude of the transmission coefficient. In order MUT to discriminate the correct value of the dielectric constant, the phase of the transmission coefficient must be obtained. The Fig. 3. Sensor/comparator response corresponding to the schematic of Fig. 2, inferred from circuit simulation, and sensitivity. different sign of the phase at both sides of REF, as it can be easily inferred from (1), allows us to distinguish whether MUT Once the simulation conditions have been established, the < REF or MUT > REF. next step is to obtain the sensor/comparator response, i.e., the Note that the sensitivity is dimensionless since it involves magnitude of the transmission coefficient at f0 as a function of dimensionless input and output variables. It should also be the differential dielectric constant. Nevertheless, a mentioned that in a real sensor, the effects of losses, including simplification can be made, since the reflection coefficient of (potential) radiation losses, might alter the impedance seen the REF step-impedance line does not change. Indeed, such from the isolated ports of the coupler, when the corresponding reflection coefficient is REF = 1, as explained by the fact that sensing line is covered by the REF material. This may result in the impedance seen from that port is infinite. The reason is that signal imbalance when the MUT coincides with the REF the pair of cascaded 90º lines translate the termination sample, provided one of the isolated ports is left opened (as in impedance (an open-circuit) to the coupler port. Thus, if the Fig. 2), thereby generating sensitivity degradation. frequency is kept to f0, no changes are expected in the Nevertheless, this aspect, related to the effects of losses, will be sensor/comparator response if the port (3) of the coupler is left discussed in the next section and in Appendix A. opened. With the above-mentioned simplification, the schematic of IV. SENSOR/COMPARATOR DESIGN, FABRICATION, AND the simulated structure is the one depicted in Fig. 2. Figure 3 EXPERIMENTAL VALIDATION depicts the dependence ofS21with MUT (at f0) obtained by means of the schematic simulator of Keysight ADS. From this We have designed and fabricated two prototype curve, the sensitivity has been inferred by simple derivation of sensor/comparators. One of them corresponds to the schematic of Fig. 2, and is used to check the validity of performing 8 IEEE SENSORS JOURNAL, VOL. XX, NO. XX, MONTH X, XXXX 1.0 1.0 1.0 AIR RO3010 0.8 0.8 PLA 0.6 0.6 |S21|f = f |S21| |S| 0.4 FR4 0.5 Theoretical sensitivity 0 0.4 Sensitivity obtained from EM simulation 0.2 Sensitivity obtained from EM simulation (fine analysis) FR4 0.2 Experimental data points 0.0 RO3010 RO4003C PLA AIR Measurement RO4003C f0 0.0 0.0 EM Simulation -0.2 1 2 3 4 5 6 7 8 9 10 1.5 2.0 2.5 MUT Frequency (GHz) Fig. 5. Frequency response of the sensor comparator of Fig. 4 for Fig. 7. Sensor/comparator response and sensitivity corresponding to different MUT samples placed in the sensing region. The dielectric the device of Fig. 4. The experimental data points are included in the constants of the MUTs are: for RO4003C, the REF sample, figure. MUT = REF = 3.55, for RO3010, MUT = 10.2, for FR4, MUT = 4.4, for PLA, MUT = 2, and for air, MUT = 1. different values of the dielectric constant of the MUT, corresponding to dielectric samples (uncladded microwave 1.0 substrates in most cases) available in our laboratory. Actually, the figure includes the measured responses, inferred by means of the Agilent N5221A vector network analyzer, as well as the responses obtained by means of electromagnetic simulation, using the ANSYS HFSS electromagnetic solver. The thickness of the samples is 3 mm, reasonably thick to consider them semi- |S11| 0.9 infinite in the vertical direction. The potential effects of air gap between the MUT and the sensing region have been minimized by adding pressure on the MUT sample against the sensor substrate. It should also be mentioned that small lateral shifts in the MUT are not relevant as far as the transverse dimensions of the MUT extend significantly beyond the limits of the sensing Including losses line. According to these comments, as far as the semi-infinite Excluding losses approximation in the MUT is satisfied in both the vertical and 0.8 1.0 1.5 2.0 2.5 3.0 transverse dimensions, the size of the sample is irrelevant. Frequency (GHz) As it can be appreciated in Fig. 5, for an MUT identical to the REF sample (RO4003C with MUT = REF = 3.55), the Fig. 6. Simulated reflection coefficient of the sensing line of Fig. 4 frequency response exhibits a transmission zero (partly (together with the cascaded 90º line section) by including and by obscured by losses) at f0 = 2 GHz, indicative of a good balance excluding ohmic and dielectric losses. between the loads seen from the isolated ports of the coupler comparisons in a two-step process, as indicated before, with the (ports 3 and 4). Despite the fact that in port (3), there is a “real” advantage of a simpler structure, where one of the step- open circuit, the virtual open circuit at port (4) when the MUT impedance lines, the one devoted to the REF sample, is replaced is the REF sample is quite satisfactory. To gain insight on this with an open circuit. The second prototype is a true differential aspect, we have inferred the simulated reflection coefficient of sensor/comparator where both sensing lines are included in the the sensing line (including the cascaded 90º line section), with design. In this second prototype, the sensing lines are low- the REF material on top of the sensing region, see Fig. 6. impedance half-wavelength open-ended lines. Actually, we have performed two different simulations, i.e., in A. Prototype sensor/comparator with a single sensing one case by excluding dielectric and ohmic losses, and in the line (Sensor A) other case by including them. The simulation by excluding ohmic and dielectric losses reveals that radiation losses are very The structure corresponding to the schematic of Fig. 2 was small. The modulus of the reflection coefficient at f0 is close to fabricated by means of the LPKF H100 drilling machine on the unity (0.86), when the ohmic and dielectric losses are included, low-loss microwave substrate indicated in the previous section but the effect of losses cannot be considered to be negligible. (with thickness h = 1.52 mm, dielectric constant r = 3.55 and Nevertheless, the effects of losses on the sensitivity analysis loss tangent tan = 0.0021). The photograph of the device is carried out before are, apparently, irrelevant. To gain insight on depicted in Fig. 4. Figure 5 depicts the frequency responses for this, Fig. 7 depicts the sensor/comparator response, i.e., the 8 IEEE SENSORS JOURNAL, VOL. XX, NO. XX, MONTH X, XXXX magnitude of the transmission coefficient at f0 as a function of D1 D2 D3 D4 ref the dielectric constant of the MUT. The figure includes simulated values (inferred by means of ANSYS HFSS), as well as the measured results directly translated from Fig. 5. The figure includes also the absolute value of the sensitivity, inferred from the simulated data points, where it can be appreciated that it exhibits a maximum at MUT = REF = 3.55, Fig. 8. Defected samples obtained by drilling holes of different densities and the value of the maximum sensitivity is Smax = 0.715, in to the REF sample. very good agreement with the prediction of the theory (0.727). 1.0 Moreover, the agreement between the theoretical sensitivity and the values inferred from the simulated data points, in the considered range of MUT variation, is very good, as revealed by 0.8 Fig. 7. Nevertheless, a fine analysis of the sensitivity in the vicinity of the reference value of the dielectric constant of the MUT, 0.6 |S21| reveals that the sensitivity exhibits a different behaviour at this limit. For that purpose, we have inferred the magnitude of the 0.4 transmission coefficient for varying values of MUT in the range ref D1 3.4−3.6, and the resulting sensitivity is also depicted in Fig. 7. Clearly, there is a minimum at MUT = REF with a continuously 0.2 D2 D3 varying slope. Consequently, the sensitivity should be null at this value of the dielectric constant of the MUT, the reference D4 f0 0.0 value. This effect is clearly related to losses, since, if losses are 1.75 2.00 2.25 absent, there is a minimum at MUT, but the absolute value of the Frequency (GHz) derivative is finite (not null) and should coincide with the Fig. 9. Frequency response of the sensor/comparator of Fig. 4 for the theoretical value, as indeed occurs, provided the derivative, i.e., different MUT samples of Fig. 8. the sensitivity, is inferred by considering a more sparse density of data points around the reference dielectric constant. the structure by eliminating the sensing line devoted to the REF According to these results, the theoretical expression of the material, with the caution of the alteration of the sensitivity in sensitivity inferred by neglecting losses is not able to predict the the limit of small perturbations, which may prevent from the null value of it for MUT = REF when losses cannot be neglected. detection of tiny defects in samples (as compared to a reference Appendix A includes the analysis when losses in the structure sample). Nevertheless, real-time comparisons are not possible (sensing line) are present. From this analysis, it is concluded with this approach, as anticipated before. In order to that when losses are present, the sensitivity in the limit when demonstrate the potential of the sensor/comparator to MUT = REF is null, but the deviation from the theoretical value discriminate changes in the MUT with regard to the REF predicted in the analysis of the previous section is restricted to sample, we have drilled arrays of holes of different densities a narrow range of values in the vicinity of REF. Moreover, in across the REF samples, as depicted in Fig. 8. The measured frequency responses are shown in Fig. 9. The responses shift the limit when losses are null, the dip in the sensitivity upwards as the density of holes increases because this decreases disappears and the theoretical prediction given by (3), with (4), the “effective” dielectric constant of the samples. The (5) and (6), is satisfied. sensor/comparator is able to detect the presence of holes in In summary, when the MUT is the REF sample, the sample D1, the one with the smaller density of holes. However, measured transmission coefficient at f0 is not exactly null. This the response is identical to the one of sample D2, and the reason is explained by the imperfect balance, due to the fact that one is that the rows of holes at the sides of the central one (on top of the ports of the coupler is left open, whereas in the other port of the sensing line) are not under the influence of the electric the losses (though small) prevent from the presence of a true field generated by the line, and, therefore, these samples are open-circuit seen from that port. This effect drastically reduces undistinguishable. Note, however, that for samples D 3 and D4, the sensitivity as compared to the (lossless) theoretical value, the rows of holes are closer and the response varies. From the but only in a very narrow region in the vicinity of REF. This sensor/comparator transfer function of Fig. 7, an effective may have impact on the resolution, or capability to detect dielectric constant can be inferred for those drilled samples, but extremely small variations of the dielectric constant as such effective dielectric constant does not have a physical compared to the REF value. Nevertheless, the sensitivity when meaning other than the necessary value of the dielectric the dielectric constant of the MUT is slightly different than REF, constant of a hypothetical MUT to obtain the same magnitude coincides with the lossless prediction to a very good of the transmission coefficient. Nevertheless, the comparator approximation, and the reason is that losses at f0 progressively functionality is demonstrated, and the device is able to detect decrease when MUT separates from REF. small imbalances between the REF and the MUT samples. The results reported so far confirm that it is licit to simplify 8 IEEE SENSORS JOURNAL, VOL. XX, NO. XX, MONTH X, XXXX 2.5 Theoretical sensitivity 1.0 Sensitivity obtained from EM simulation Experimental data points 2.0 AIR 0.8 PLA RO3010 1.5 0.6 |S21|f = f0 r |S| c FR4 1.0 0.4 0.5 0.2 RO4003C wd ld 0.0 0.0 1 2 3 4 5 6 7 8 9 10 MUT Fig. 12. Sensor/comparator response and sensitivity corresponding to the device of Fig. 10. The experimental data points are included. ls 1.0 Material Under Test ws 0.9 Fig. 10. Photograph of the sensor/comparator based on two sensing lines. The sensing regions, including the one devoted to the MUT and 0.8 the one devoted to the REF sample, are indicated by dashed rectangles. |S11| Dimensions are (in mm): ws = 9.07, ls = 39.81, wd = 0.20, ld = 24.65, c = 1.88, r = 21.15. 0.7 1.0 f0 Measurement 0.8 EM simulation RO3010 0.6 0.6 0.4 Including losses FR4 Excluding losses 0.2 0.5 RO4003C 1.0 1.5 2.0 2.5 3.0 0.0 1.0 Frequency (GHz) |S21| 0.8 Fig. 13. Simulated reflection coefficient of the sensing line of Fig. 10 (together with the cascaded 90º line section) by including and by 0.6 AIR excluding ohmic and dielectric losses. 0.4 PLA considered impedance values are Zs = 23 Ω and Z = 157 Ω. 0.2 Obviously, the impedance and phase of the open-ended sensing 0.0 line section is with the REF material (the same as in the 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 previous prototype) on top of it. The sensor has been Frequency (GHz) implemented on the same substrate used for the prototype of the Fig. 11. Frequency response of the sensor comparator of Fig. 10 for preceding subsection, and the photograph of the fabricated different MUT samples placed in the sensing region, and the REF sample located on top of the REF line. The dielectric constants of the device is depicted in Fig. 10, where dimensions are indicated. MUTs are: for RO4003C, the REF sample, MUT = REF = 3.55, for The frequency responses inferred by loading the REF line RO3010, MUT = 10.2, for FR4, MUT = 4.4, for PLA, MUT = 2, and for air, with the REF material and the sensing line with different MUTs MUT = 1. are depicted in Fig. 11, whereas Fig. 12 depicts the sensor/comparator response (transfer function), as well as the B. Prototype sensor/comparator with two sensing lines sensitivity (calculated from the simulated data points). It can be (Sensor B) seen that the sensitivity differs significantly from the theoretical (lossless) value, also included in the figure. This difference is The second prototype is a true real-time differential due to the effects of losses, not accounted for in the theoretical sensor/comparator consisting of the rat-race hybrid coupler and sensitivity analysis. Figure 13 depicts the simulated reflection two step-impedance sensing lines connected to the isolated coefficient of the sensing line (including the cascaded 90º line ports. In this case, we have opted to implement the step- section), with the REF material on top of the sensing region. impedance lines by means of a low-impedance 180º open-ended Analogously to Fig. 6, the figure depicts the reflection sensing line cascaded to a high impedance 90º line. The 8 IEEE SENSORS JOURNAL, VOL. XX, NO. XX, MONTH X, XXXX 0.6 ref f0 even for the sample with a sparser hole density. D1 D2 V. COMPARISON TO OTHER PHASE-VARIATION SENSORS D3 D4 The main relevant advantage of the proposed 0.4 sensor/comparator is the design simplicity combined with the fact that a differential measurement is implemented in a two- |S21| port device structure. An interesting, and competitive, approach was presented in [22], where a differential-mode dielectric 0.2 constant sensor based on phase-variation and implemented by means of composite right/left-handed (CRLH) lines was reported. Such sensor operates in transmission, and the two-port configuration was achieved by feeding the differential port of 0.0 1.75 2.00 2.25 the sensor with imbalanced signals generated by means of a balun, whilst the differential output port was connected to an Frequency (GHz) identical balun in opposite configuration. The very high Fig. 14. Frequency response of the sensor/comparator of Fig. 10 for the different MUT samples of Fig. 8. sensitivity of such sensor is due to the high dispersive behaviour of the CRLH lines [58]. In the sensors presented in this paper coefficients by excluding and by including dielectric and ohmic and that of [22], the phase information is converted to losses. These results reveal that losses are significant (the magnitude information. In [44], a two-port differential-mode modulus of the reflection coefficient at f0 is 0.57) and that a phase-variation dielectric constant sensor operating in significant portion of losses is due to radiation. However, since transmission was also reported. Phase-to-magnitude conversion there is a quasi-perfect balance when the loads of the sensing lines are identical (the REF sample), the resulting transmission was achieved in this case by means of a pair of rat-race hybrid coefficient at f0 is null to a very good approximation, as Fig. 11 couplers, and sensitivity was optimized by using meandered reveals. Indeed, the loss level of 𝑅𝐸𝐹 at f0 for the sensing line lines. The main advantage of the sensors of Figs. 4 and 10, as compared to the sensors in [22],[44], all of them differential and loaded with the REF material can reasonably explain the difference in the sensitivity, at least in the limit of small based on a two-port configuration, is the fact that in the sensors perturbations. That is, if we assume that for MUTs with of this work the sensitivity can be enhanced without the need to dielectric constants close to the one of the REF material the loss increase the size of the sensing area (either a 90º or a 180º open- level at f0 does not vary significantly (i.e., 𝑀𝑈𝑇 𝑅𝐸𝐹 ), ended line section). For that purpose it suffices to add further expression (1) can be rewritten as quarter-wavelength transmission line sections with alternating high/low impedance. Obviously, this increases the overall size 1 1 of the sensor, but not the sensing region. By contrast, in the 𝑆21 = − (𝑅𝐸𝐹 − 𝑀𝑈𝑇 ) ≈ − |𝑅𝐸𝐹 |(𝑒 𝑗𝑅𝐸𝐹 − 𝑒 𝑗𝑀𝑈𝑇 ) (11) 2 2 sensors of [22],[44], the sensitivity is proportional to the length Under this approximation, the sensitivity analysis carried out in of the lines. Moreover, the sensors of this study are simply Section III is valid, with the exception of a multiplicative factor based on ordinary lines, thereby being robust against fabrication given by  𝑅𝐸𝐹 . Consequently, by correcting the lossless related tolerances or other detuning effects. Concerning the sensitivity (S = 2.01, see Fig. 12) taking into account the effects that additional quarter-wavelength line sections may value of𝑅𝐸𝐹 = 0.57, the resulting sensitivity in the limit of produce on the sensor sensitivity through losses, it is expected that by considering low-loss microwave substrates, the case of small perturbations should be reasonably predicted, as it the proposed sensors, such effects do not have significant actually occurs. Note thatS 𝑅𝐸𝐹 = 1.17, whereas the influence on the sensor, provided the sensitivities are not sensitivity resulting from the simulated data points, obviously extremely high. However, for highly sensitive sensors, losses by including losses, is 1.18, i.e., in very good agreement. degrade somehow the sensitivity as compared to the theoretical Losses tend to degrade the sensitivity, as it is clearly shown in prediction by excluding losses (expressions 3 and 10). This this subsection. However, it does not mean that the aspect is analysed in further detail in the Appendix, where a sensor/comparator cannot be useful. Indeed, the results of method for the calculation of the sensitivity by considering the Fig. 11 and 12 show the potential of the approach to determine effects of losses is pointed out, and two additional illustrative the dielectric constant of the MUT from the measurement of the examples of sensors (of type A) with enhanced sensitivity are magnitude of the transmission coefficient. As comparator, by reported. using two identical sensing arms, real-time differential In [25], differential sensors with very high sensitivity were measurements can be carried out by means of the device of reported, but at the expense of very long (meandered) lines, and Fig. 10. Moreover, the quasi-perfect balance, is beneficial in hence large size of the sensing region. Moreover, these sensors terms of resolution. The comparator functionality of the device are four-port structures. The sensors reported in [24] are also of Fig. 10 has been demonstrated by obtaining the frequency very interesting, as far as highly dispersive electro-inductive responses corresponding to the defected REF samples of Fig. 8. wave (EIW) artificial transmission lines are used for sensing. The results are depicted in Fig. 14, and reveal the capability of Such sensors exhibit very good sensitivity but are implemented the device to discriminate the differences with the REF sample, 8 IEEE SENSORS JOURNAL, VOL. XX, NO. XX, MONTH X, XXXX TABLE I respectively. COMPARISON OF VARIOUS PHASE-VARIATION DIELECTRIC Note that there are reflective-mode phase-variation sensors CONSTANT SENSORS exhibiting an excellent figure of merit [26],[29]. The reason is Ref. Mode* Size Max. FoM that in such sensors, further high/low impedance quarter- (2) Sensitivity (º/2) wavelength line sections were included. The main purpose of [22] TRANS-DIFF --- 600 dB --- this paper, rather than achieving an excellent sensitivity, has [23] TRANS-SINGLE --- 54.8º --- been to implement differential-mode structures based on two- [25] TRANS-DIFF 12.90 415.6º 32.2 port configurations, since it eases the measurements. [24] TRANS-DIFF 0.075 25.3 dB --- [44] TRANS-DIFF 0.020 17.6 dB --- Nevertheless, the reported sensitivities and FoM are good, and [30] TRANS-SINGLE 0.030 7.7º 257 these parameters can be further enhanced by merely cascading [32] TRANS-SINGLE 0.040 20.0º 500 additional quarter-wavelength line sections between the open- [60] TRANS-SINGLE 0.050 14.24º 284.8 ended sensing line and the isolated ports of the coupler. [26] REFLECT-SINGLE 0.025 528.7º 21148 Note also that the phase-variation sensors presented in this [28] REFLECT-SINGLE 0.100 45.5º 455 work are differential-mode structures operating in reflection. [29] REFLECT-SINGLE 0.025 101.3º 4052 There are other reflective-mode differential sensors reported in [52] REFLECT-DIFF 0.003 1.6 dB/% --- the literature [52], based on a topology similar to the sensor of [A] REFLECT-DIFF 0.005 0.72/85.5º 16512 Fig. 10. The sensing elements in this case are electrically small [B] REFLECT-DIFF 0.062 1.17/230.6º 3671 resonators, specifically open split ring resonators (OSRRs), but *TRANS-DIFF: transmission-mode differential; TRANS-SINGLE: the design procedure was not carried out on the basis of the transmission-mode single-ended; REFLECT-SINGLE: reflective- phase imbalance between the reflection coefficients seen from mode single-ended; REFLECT-DIFF: reflective-mode differential. the isolated ports of the coupler. Indeed, the sensor of [52] was devoted to the characterization of liquids samples, particularly as four-port devices, and EIW lines are very sensitive to the mixtures of isopropanol in deionized (DI) water, and for that effects of detuning and tolerances, since these artificial lines reason that sensor was equipped with fluidic channels in the exhibit very narrow pass bands [59]. Note that the sensitivity in sensing regions. This sensor is included in Table I, but in this the sensors in [24] is given in dB. The reason is that the case the sensitivity is given in terms of dB/%, since the input considered output variable is the cross-mode transmission variable is the volume fraction of isopropanol in DI water. Due coefficient, proportional to the difference in the transmission to the high losses in liquid samples, port imbalance in the sensor coefficients of the sensing EIW lines, intimately related to of [52] is caused by changes in both the phase and the phase imbalance. Alternately, the sensitivity in such sensors can magnitude of the reflection coefficient seen from the ports. be expressed in linear form, as carried out in the present paper. Nevertheless, the main dominant mechanism contributing to the Other differential sensors based on SRRs or CSRRs also use the output signal in this sensor is phase imbalance and, for this, cross-mode transmission coefficient as output variable reason, the device is also included in Table I. Moreover, it has [41],[47],[49], and exhibit good sensitivity, but such sensors been shown in the previous section that the effects of losses need also four-port measurements. (despite the fact that the MUT are low-loss materials) cannot be As it has been reported in previous papers, the combination neglected in the reported sensors. of sensitivity and size of the sensing region in reflective-mode Recently, single-ended phase-variation sensors operating in phase-variation sensors implemented by means of step- transmission and based on slow-wave transmission lines have impedance transmission lines is unique. The FoM, defined as been reported [30],[32]. Such sensors can be easily the ratio between the maximum sensitivity and the size of the implemented as differential mode structures, and exhibit very sensing region expressed in terms of the squared guided reasonable FoM (especially the device in [32]), but their design wavelength has unprecedented values in such sensors [26]-[29], is significantly more complex than the one of the sensors of this as it can be seen in Table I (where several phase-variation work. sensors are compared). Note that the FoM is given only for Let us mention that at the expense of a more complex those sensors where the output variable is the phase of either design, it is possible to replace the 90º or 180º open-ended the transmission or the reflection coefficient. Nevertheless, the sensing lines of the reported sensors with artificial lines (CRLH overall sensitivity of the sensors reported in this paper is lines, slow-wave line, etc.). By this means, it is potentially entirely determined by the sensitivity of the step-impedance possible to further enhance the sensitivity, yet keeping the size open-ended sensing line (the rat-race coupler mainly converts of the sensing region unaltered (or even reduced), due to the the phase to magnitude information, but does not contribute to higher dispersion of artificial lines, as compared to ordinary boost up the sensitivity). These phase sensitivities are 85.5º and lines. Nevertheless, this aspect requires further investigation 230.6º for the sensors of Fig. 4 and 10, respectively, providing and is left for future works. figures of merit of 16512º/2 and 3671º/2, and included in the Finally, to end this comparative analysis, let us emphasize table for comparison purposes. The maximum sensitivities that that the reported sensors operate at a single frequency, in result by including the rat-race coupler and considering the comparison to frequency-variation sensors, and this reduces the magnitude of the transmission coefficient as output variable are cost of the associated electronics in a real scenario, since |S| = 0.72 and |S| = 1.17 for the sensors of Fig. 4 and 10, wideband voltage controlled oscillators (VCOs) are not needed 8 IEEE SENSORS JOURNAL, VOL. XX, NO. XX, MONTH X, XXXX in order to generate the interrogation signals of the sensors. APPENDIX A Let us consider sensor A (Fig. 4) by including the effects of VI. CONCLUSIONS losses in the step-impedance sensing line. Such losses can be In conclusion, microwave comparators based on reflective- accounted by means of a reflection coefficient seen from port 4 mode differential phase-variation dielectric constant sensors of the coupler satisfying𝑀𝑈𝑇 < 1. By contrast, the reflection have been reported in this paper. The devices are two-port coefficient seen from port 3 satisfies𝑅𝐸𝐹 = 1, provided this structures consisting of a rat-race hybrid coupler with two of the port is left opened in sensor A. Under these conditions, the isolated ports connected to the sensing elements, open-ended transmission coefficient (1) should be rewritten as step-impedance transmission lines. The sensitivity of the 1 1 sensors is high by virtue of the step-impedance configuration of 𝑆21 = − (𝑅𝐸𝐹 − 𝑀𝑈𝑇 ) ≈ − (∓1 − |𝑀𝑈𝑇 |𝑒 𝑗𝑀𝑈𝑇 ) (A1) the sensing lines, with a multiplicative effect caused by the step- 2 2 impedance discontinuities. Two prototype sensor/comparators The negative or positive sign of 𝑅𝐸𝐹 depends on whether the have been reported. In one case (sensor A), the sensing line total number of high/low 90º line sections cascaded between the devoted to the reference (REF) sample has been eliminated, in sensing line and the rat-race coupler, N, is even or odd, order to reduce the overall dimensions of the structure. This is respectively. Note that for the device of Fig. 4, with a single justified because with the specific design, the impedance seen section, i.e., N = 1, the sign should be positive, but for N even, from the input port of the step-impedance REF sensing line is the open-circuit at the extreme of the sensing line is seen as a (ideally) an open-circuit. A comparison measurement with this short-circuit from port 4 of the coupler. Thus, for balancing device must be performed in a two-step process, where first the purposes, ports 3 should also see a short-circuit, and thereby a REF sample is measured, and then the response of the MUT is negative sign in (A1) is required. inferred. The second prototype (sensor B) is a real-time A variation in the dielectric constant of the MUT modifies comparator with both sensing lines present (for the REF and the phase and the characteristic impedance of the sensing line, MUT samples). Apart from this difference, the open-ended and this in turn affects the phase of the reflection coefficient, sensing line in sensor A is a high-impedance 90º line, whereas but also magnitude. Thus, the sensitivity by including the in sensor B, the sensitive element is a low-impedance 180º line. effects of losses in the sensing line cannot be merely expressed This combination of phase (either 90º or 180º) and impedance as (3). An additional term, that includes the influence of (either high or low) for the sensing lines obeys the design 𝑀𝑈𝑇 is needed. Thus, the sensitivity should be expressed as guidelines, inferred from the reported sensitivity analysis. Moreover, the 90º line section cascaded between the sensing 𝑑|𝑆21 | 𝑑|𝑆21 | 𝑑 𝑀𝑈𝑇 𝑑 𝑠,𝑀𝑈𝑇 𝑑 𝑀𝑈𝑇 𝑑𝑍𝑠,𝑀𝑈𝑇 line and the isolated ports of the coupler has been designed with 𝑆= = {  +  } 𝑑𝑀𝑈𝑇 𝑑𝑀𝑈𝑇 𝑑 𝑠,𝑀𝑈𝑇 𝑑 𝑀𝑈𝑇 𝑑𝑍𝑠,𝑀𝑈𝑇 𝑑 𝑀𝑈𝑇 the convenient impedance value (high or low), so as to boost up 𝑑|𝑆21 | 𝑑|𝑀𝑈𝑇 | the sensitivity, also in coherence with the design guidelines. +  (A2) The sensitivities of the reported sensor/comparators are 0.72 𝑑|𝑀𝑈𝑇 | 𝑑 𝑀𝑈𝑇 and 1.17, dimensionless as far as the output variable is the Note, however, that the terms within the brackets () are magnitude of the transmission coefficient, the input variable identical to those of (3), since REF, s,REF, and Zs,REF are being the differential dielectric constant between the REF and MUT samples. Nevertheless, these sensitivities can be constant. Concerning the last term in (A2), for values of MUT expressed in terms of the equivalent phase sensitivity, in order sufficiently distant from REF (moderate and large to ease the comparison with other phase variation sensors (see perturbations), the losses at f0 are negligible, and therefore the Table I). The comparator functionality of both prototypes has sensitivity is expected to be correctly predicted by expression been demonstrated by considering MUT samples consisting on (3). For small perturbations, the derivative 𝑑|𝑀𝑈𝑇 |/𝑑 𝑀𝑈𝑇 is pieces of the REF sample with sparse arrays of holes of expected to be close to zero, because when MUT = REF, |𝑀𝑈𝑇 | different densities. Both devices are able to detect the samples is a minimum. with the smaller density of holes, thanks to the high sensitivity Let us next calculate the derivative 𝑑|𝑆21 |/𝑑 𝑀𝑈𝑇 . The of the reported sensors. The effects of losses have also been modulus of the transmission coefficient is simply analysed. It has been shown that in sensor A losses are low and have not a significant effect on the sensitivity, except for 1 2 |𝑆21 | = √1 ± 2|𝑀𝑈𝑇 | cos 𝑀𝑈𝑇 + |𝑀𝑈𝑇 | (A3) extremely small perturbations. For sensor B, device losses are 2 more significant, and degrade the sensitivity as compared to the and (A3) coincides with (2) if |𝑀𝑈𝑇 | = 1 (lossless case), as one predicted by the theory (where the effects of losses are excluded). Nevertheless, despite the higher level of losses, expected. Thus, using (A3), the first derivative in (A2) is found sensor B exhibits quasi-perfect balanced when the MUT to be sensing line is loaded with the REF sample, and, potentially, 𝑑|𝑆21 | ∓|𝑀𝑈𝑇 | sin 𝑀𝑈𝑇 = (A4) this sensor (with the pair of balanced sensing arms) exhibits 𝑑 𝑀𝑈𝑇 2 better resolution. Nevertheless, both sensors have been able to 2√1 ± 2|𝑀𝑈𝑇 | cos 𝑀𝑈𝑇 + |𝑀𝑈𝑇 | detect the defected samples with the sparser density of defects (holes). Finally, the effects of device losses on the sensitivity If losses are absent (|𝑀𝑈𝑇 | = 1), (A4) is identical to (4), and the have been qualitatively and quantitatively explained (see sensitivity in the limit of small perturbations (MUT = REF) is further details in Appendix A). finite and given by expressions (10), (and, obviously, A4 is 1/2 8 IEEE SENSORS JOURNAL, VOL. XX, NO. XX, MONTH X, XXXX in that limit, in coherence with expression 9). However, if losses 2.0 Theoretical sensitivity are present (|𝑀𝑈𝑇 | < 1), the numerator of (A4) is null in the Sensitivity obtained from EM simulation limit of small perturbations (where MUT = 0 for the case of N Sensitivity obtained from EM simulation (fine analisis) odd and MUT = nπ for the case of N even), whereas the 1.5 denominator is finite in that limit. Therefore, |𝑆21 |/𝑑 𝑀𝑈𝑇 = 0, and the overall sensitivity is null when MUT = REF, as it is |S| verified in the text. This explains the dip in the sensitivity in the 1.0 limit of small perturbations, as reported in the text (see Fig. 7 and related text). The dip is very narrow, and tends to disappear as losses decrease. Indeed, only a fine analysis in the vicinity of 0.5 MUT = REF is able to provide the data points necessary to capture this dip in the sensitivity inferred from the simulated data points, as discussed in the text. Nevertheless, it should be mentioned that as the sensitivity in the limit of small 0.0 perturbations is enhanced, e.g., by including additional 90º line 2 4 6 8 10 sections, the result is a narrowing effects in the sensitivity MUT response. Thus, for sensing structures designed to exhibit Fig. A1. Sensor/comparator sensitivity corresponding to the device of extreme values of the sensitivity in the limit of small Fig. 4 with an added 90º line section in the sensing arm with characteristic impedance Z2 = 70.71 Ω. perturbations (MUT = REF), the prediction of the sensitivity given by the theory may be jeopardized by the effect of losses losses (expression 3) overestimates the maximum sensitivity, (since the interesting part of the sensitivity curve might be due to the effect of losses. Nevertheless, it does not mean that circumscribed to the range of MUT affected by losses). by adding further 90º high/low impedance line sections to the To illustrate the previous aspect, we have considered two sensing line the sensitivity cannot be enhanced. Indeed, Fig. A2 additional structures, analyzed at simulation level. In both demonstrates that the maximum sensitivity (inferred from the cases, the sensors are of type A, and the sensing arms consist of simulated data points) is higher than in Fig. A1, but not as high a high-impedance 90º sensing line, plus two cascaded 90º line as the theory (by excluding losses) predicts. sections with alternating low/high characteristic impedance (N For an accurate prediction of the sensitivity in very high = 2) in order to boost up the sensitivity in the limit of small sensitive sensors, where the effects of losses cannot be perturbations. The difference is that in one case, the additional circumvented, the use of expression (A2) is a due. However, the 90º line (the one in contact with the rat-race coupler) exhibits a evaluation of the last derivative in (A2), i.e., 𝑑|𝑀𝑈𝑇 |/𝑀𝑈𝑇 , high characteristic impedance of Z2 = 157 Ω, whereas in the entirely related to losses, is not simple, since the effects of losses other sensing structure, such impedance is less extreme depend on many factors. Therefore, the a priori determination of (Z2 = 70.71 Ω). Apart from the addition of such high- the sensitivity in highly sensitive sensors, where the effect of losses impedance 90º line sections, the sensors under study are cannot be excluded, is not simple. Nevertheless, given a sensitive identical to the one of Fig. 4. Figure A1 depicts the absolute arm, with well-known substrate parameters and dimensions, it is value of the sensitivity inferred from the simulated data points possible to obtain the dependence of |𝑀𝑈𝑇 | with 𝑀𝑈𝑇 at f0 from (by including the effects of losses) and the one predicted by the electromagnetic simulation by including losses. From such theory by excluding losses, corresponding to the sensor with Z2 dependence, evaluating the term 𝑑|𝑀𝑈𝑇 |/𝑀𝑈𝑇 is possible and, = 70.71 Ω. With this value of Z2, the sensitivity in the limit of consequently, the sensitivity can be calculated. For that purpose, small perturbations (MUT  REF), as predicted by the lossless the other term contributing to the second summand of the theory, is enhanced by a factor of 2, as compared to the one of sensitivity (A2), i.e., 𝑑|𝑆21 |/𝑑|𝑀𝑈𝑇 | , should also be calculated. the sensor of Fig. 4. Such value is also inferred by Using (A3), such derivative is found to be electromagnetic simulation (see Fig. A1). Indeed, except by the presence of the narrow dip, discussed before, the agreement 𝑑|𝑆21 | |𝑀𝑈𝑇 | ± cos 𝑀𝑈𝑇 = (A5) between the theory and the simulated values of the sensitivity 𝑑|𝑀𝑈𝑇 | 2 is excellent, similar to Fig. 7. Note, however, that the sensitivity 2√1 ± 2|𝑀𝑈𝑇 | cos 𝑀𝑈𝑇 + |𝑀𝑈𝑇 | response is narrower, as compared to Fig. 7, as predicted, and due to the enhanced sensitivity in the limit of small and (A5) can be determined from the simulated reflection perturbations. coefficient of the sensing arm, 𝑀𝑈𝑇 = |𝑀𝑈𝑇 |𝑒 𝑗𝑀𝑈𝑇 , for the By contrast, for the sensing structure with Z2 = 157 Ω, the different values of the dielectric constant of the MUT. expected sensitivity in the limit of small perturbations should Nevertheless, it should be emphasized that this method requires be S= 10.3, according to the theoretical prediction by the previous simulation of the reflection coefficient of the excluding losses. However, Fig. A2 reveals that the absolute sensitive arm at f0 as a function of the dielectric constant of the value of the sensitivity inferred from the simulated data points MUT. Fig. A2 also includes the sensitivity by including the does not coincide with the prediction of the theory by excluding effects of losses, i.e., inferred by means of (A2) according to losses, and the reason is the extremely peaked sensitivity the explained procedure, and the result is in good agreement response, as indicated. Thus, for extremely sensitive sensors, with the simulated sensitivity. Therefore, these results point out with very narrow sensitivity responses, the theory by excluding that for moderate values of the sensitivity, the effects of losses can be ignored, and the sensitivity is correctly predicted by the 8 IEEE SENSORS JOURNAL, VOL. XX, NO. XX, MONTH X, XXXX 4 [8] L. Su, J. Mata-Contreras, P. Vélez, A. Fernández-Prieto, and F. Martín, 12 “Analytical method to estimate the complex permittivity of oil Samples,” Sensors, 18(4), paper 984, 2018. 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Mahdi, “Electroinductive waves in chains of complementary metamaterial P. Kar, M. Daneshmand, and K. Shankar, “Effect of phosphonate elements,” Appl. Phys. Lett., vol. 88, Art. no. 083503, Jan. 2006. monolayer adsorbate on the microwave photoresponse of TiO2 nanotube [60] L. Su, J. Muñoz-Enano, P. Vélez, P. Casacuberta, M. Gil, F. Martín, membranes mounted on a planar double ring resonator”, Nanotechnology, “Phase-Variation Microwave Sensor for Permittivity Measurements Based vol. 27, paper 375201, Sep. 2016. on a High-Impedance Half-Wavelength Transmission Line”, IEEE Sensors [40] L. Su, J. Naqui, J. Mata-Contreras, F. Martín, “Cascaded Splitter/Combiner J., vol. 21, no. 9, pp. 10647-10656, May 2021. Microstrip Sections Loaded with Complementary Split Ring Resonators (CSRRs): Modeling, Analysis and Applications”, IEEE MTT-S Int. C. Herrojo was born in Barcelona, Spain, in 1983. Microw. Symp. (IMS’16), San Francisco, May 2016. He received the Telecommunications Technical [41] P. Vélez, L. Su, K. Grenier, J. Mata-Contreras, D. Dubuc, and F. Martín, Engineering degree in electronic systems and "Microwave microfluidic sensor based on a microstrip splitter/combiner Telecommunications Engineering degree from the configuration and split ring resonators (SRR) for dielectric characterization Universitat Autònoma de Barcelona in 2010 and of liquids", IEEE Sensors J., vol. 17, pp. 6589-6598, Oct. 2017. 2012, respectively and the PhD degree in [42] A. Ebrahimi, G. Beziuk, J. Scott, and K. Ghorbani, “Microwave Electronics Engineering from the same university Differential Frequency Splitting Sensor Using Magnetic-LC Resonators”, in 2018. His research interests include Sensors, vol. 20, p. 1066, 2020. RF/microwave devices, Chipless-RFID and RFID [43] A. Ebrahimi, J. Scott, and K. Ghorbani, “Differential Sensors Using technology, and Metamaterials. Microstrip Lines Loaded With Two Split-Ring Resonators”, IEEE Sens. J., vol. 18, pp. 5786–5793, 2018. [44] J. Muñoz-Enano, P. Vélez, M. Gil, J. Mata-Contreras, and F. Martín, “Differential-mode to common-mode conversion detector based on rat- race couplers: analysis and application to microwave sensors and comparators,” IEEE Trans. Microw. Theory Techn., vol. 68, pp. 1312- 1325, Apr. 2020. [45] J. Muñoz-Enano, P. Vélez, M. Gil, and F. Martín, “Planar microwave resonant sensors: a review and recent developments”, Appl. Sci., vol. 10, p. 2615 (29 pages), 2020; doi:10.3390/app10072615. 8 IEEE SENSORS JOURNAL, VOL. XX, NO. XX, MONTH X, XXXX Paris Vélez (S’10–M’14–SM’21) was born in postdoctoral researcher, she was awarded with a Juan de la Cierva Barcelona, Spain, in 1982. He received the degree fellowship working in the Universidad de Castilla-La Mancha. She was in Telecommunications Engineering, specializing in postdoctoral researcher in the Institut für Mikrowellentechnik und electronics, the Electronics Engineering degree, and Photonik in Technische Universität Darmstadt and in the Carlos III the Ph.D. degree in Electrical Engineering from the University of Madrid. She is currently associate professor in the Universitat Autònoma de Barcelona, Barcelona, in Universidad Politécnica de Madrid within the Excellence Program for 2008, 2010, and 2014, respectively. His Ph.D. thesis University Professoriat of the V Regional Plan for Scientific Research concerned common mode suppression differential and Technological Innovation of the Administration of the Community of microwave circuits based on metamaterial concepts Madrid. She has worked in metamaterials, piezoelectric MEMS and and semi-lumped resonators. During the Ph.D., he microwave passive devices. Her current interests include metamaterials was awarded with a pre-doctoral teaching and research fellowship by sensors for fluidic detection the Spanish Government from 2011 to 2014. From 2015-2017, he was involved in the subjects related to metamaterials sensors for fluidics Ferran Martín (M’04-SM’08-F’12) was born in detection and characterization at LAAS-CNRS through a TECNIOSpring Barakaldo (Vizcaya), Spain in 1965. He received fellowship cofounded by the Marie Curie program. From 2018 to 2020 the B.S. Degree in Physics from the Universitat he has worked in miniaturization of passive circuits RF/microwave and Autònoma de Barcelona (UAB) in 1988 and the sensors-based metamaterials through Juan de la Cierva fellowship. His PhD degree in 1992. From 1994 up to 2006 he was current research interests include the miniaturization of passive circuits Associate Professor in Electronics at the RF/microwave and sensors-based metamaterials. Dr. Vélez is a Departament d’Enginyeria Electrònica (Universitat Reviewer for the IEEE Transactions on Microwave Theory and Autònoma de Barcelona), and since 2007 he is Full Techniques and for other journals. Professor of Electronics. In recent years, he has been involved in different research activities Jonathan Muñoz-Enano was born in Mollet del including modelling and simulation of electron devices for high Vallès (Barcelona), Spain, in 1994. He received the frequency applications, millimeter wave and THz generation systems, Bachelor's Degree in Electronic and the application of electromagnetic bandgaps to microwave and Telecommunications Engineering in 2016 and the millimeter wave circuits. He is now very active in the field of Master's Degree in Telecommunications metamaterials and their application to the miniaturization and Engineering in 2018, both at the Autonomous optimization of microwave circuits and antennas. Other topics of interest University of Barcelona (UAB). Actually, he is include microwave sensors and RFID systems, with special emphasis working in the same university in the elaboration of on the development of high data capacity chipless-RFID tags. He is the his PhD, which is focused on the development of head of the Microwave Engineering, Metamaterials and Antennas Group microwave sensors based on metamaterials concepts for the dielectric (GEMMA Group) at UAB, and director of CIMITEC, a research Center characterization of materials and biosensors. on Metamaterials supported by TECNIO (Generalitat de Catalunya). He has organized several international events related to metamaterials and Lijuan Su was born in Qianjiang (Hubei), China in related topics, including Workshops at the IEEE International Microwave 1983. She received the B.S. degree in Symposium (years 2005 and 2007) and European Microwave communication engineering and the M.S. degree in Conference (2009, 2015 and 2017), and the Fifth International Congress circuits and systems both from Wuhan University of on Advanced Electromagnetic Materials in Microwaves and Optics Technology, Wuhan, China, in 2005 and 2013 (Metamaterials 2011), where he acted as Chair of the Local Organizing respectively, and the Ph.D. degree in electronic Committee. He has acted as Guest Editor for six Special Issues on engineering from Universitat Autonoma de metamaterials and sensors in five International Journals. He has Barcelona, Barcelona, Spain, in 2017. From Nov. authored and co-authored over 650 technical conference, letter, journal 2017 to Dec. 2019, she worked as a postdoc papers and book chapters, he is co-author of the book on Metamaterials researcher in Flexible Electronics Research Center, Huazhong entitled Metamaterials with Negative Parameters: Theory, Design and University of Science and Technology, Wuhan, China. She is currently Microwave Applications (John Wiley & Sons Inc.), author of the book a postdoc researcher in CIMITEC, Universitat Autonoma de Barcelona, Artificial Transmission Lines for RF and Microwave Applications (John Spain. Her current research interests focus on the development of novel Wiley & Sons Inc.), co-editor of the book Balanced Microwave Filters microwave sensors with improved performance for biosensors, dielectric (Wiley/IEEE Press) and co-author of the book Time-Domain Signature characterization of solids and liquids, defect detection, industrial Barcodes for Chipless-RFID and Sensing Applications (Springer). processes, etc. Ferran Martín has generated 21 PhDs, has filed several patents on metamaterials and has headed several Development Contracts. Pau Casacuberta was born in Sabadell Prof. Martín is a member of the IEEE Microwave Theory and Techniques (Barcelona), Spain, in 1997. He received the Society (IEEE MTT-S). He is reviewer of the IEEE Transactions on Bachelor's Degree in Electronic Microwave Theory and Techniques and IEEE Microwave and Wireless Telecommunications Engineering and the Components Letters, among many other journals, and he serves as Bachelor's Degree in Computer Engineering in member of the Editorial Board of IET Microwaves, Antennas and 2020, both from the Universitat Autònoma de Propagation, International Journal of RF and Microwave Computer- Barcelona (UAB). Currently, he is in his junior year Aided Engineering, and Sensors. He is also a member of the Technical of the Master's Degree in Telecommunications Committees of the European Microwave Conference (EuMC) and Engineering in the UAB. He received the International Congress on Advanced Electromagnetic Materials in Collaboration fellowship by the Spanish Government in 2019 for Microwaves and Optics (Metamaterials). Among his distinctions, Ferran developing his Bachelor’s Thesis in highly sensitive microwave sensors Martín has received the 2006 Duran Farell Prize for Technological based in stepped impedance structures. Research, he holds the Parc de Recerca UAB – Santander Technology Transfer Chair, and he has been the recipient of three ICREA Marta Gil Marta Gil (S’05-M’09) was born in ACADEMIA Awards (calls 2008, 2013 and 2018). He is Fellow of the Valdepeñas, Ciudad Real, Spain, in 1981. She IEEE and Fellow of the IET. received the Physics degree from Universidad de Granada, Spain, in 2005, and the Ph.D. degree in electronic engineering from the Universitat Autònoma de Barcelona, Barcelona, Spain, in 2009. She studied one year with the Friedrich Schiller Universität Jena, Jena, Germany. During her PhD Thesis she was holder of a METAMORPHOSE NoE grant and National Research Fellowship from the FPU Program of the Education and Science Spanish Ministry. As a