This is the accepted version of the journal article: Muñoz-Enano, Jonathan; Vélez, Paris; Gil Barba, Marta; [et al.]. «Frequency- variation sensors for permittivity measurements based on Dumbbell-Shaped defect ground structures (DB-DGS) : Analytical method and sensitivity analysis». IEEE sensors journal, Vol. 22, Issue 10 (May 2022), p. 9378-9386. DOI 10.1109/JSEN.2022.3163470 This version is available at https://ddd.uab.cat/record/258986 under the terms of the license IEEE SENSORS JOURNAL, VOL. XX, NO. XX, MONTH X, XXXX 1 Frequency-Variation Sensors for Permittivity Measurements Based on Dumbbell-Shaped Defect Ground Structures (DB-DGS): Analytical Method and Sensitivity Analysis Jonathan Muñoz-Enano, Student Member, IEEE, Paris Vélez, Senior Member, IEEE, Marta Gil, Member, IEEE, and Ferran Martín, Fellow, IEEE Abstract—It is shown in this paper that a microstrip line loaded with a dumbbell-shaped defect ground structure (DB-DGS) is useful for complex permittivity measurements. The working principle of the sensor is the variation in the notch (resonance) frequency and depth caused by the material under test (MUT), when it is put in contact with the sensitive region of the device, i.e., the capacitive slot. It is demonstrated that the relative sensitivity of the sensor, defined as the variation of the resonance frequency of the DB-DGS with the dielectric constant of the MUT relative to the resonance frequency of the bare structure, does not depend on the geometry of the DB-DGS, provided the substrate is thick enough. The relative sensitivity, the key figure of merit, is dictated by the equivalent dielectric constant of the substrate, and it increases as the substrate permittivity decreases. Using the circuit model of the sensing structure, simple analytical expressions providing the dielectric constant and the loss tangent of the MUT are derived. Such analytical formulas depend on the notch frequency and depth of the sensor with and without MUT in contact with it, i.e., easily measurable quantities. The analysis carried out is corroborated through full-wave electromagnetic simulation and experiments. Index Terms— Microwave sensors, permittivity sensors, defect ground structures (DGS), microstrip technology. I. INTRODUCTION (temperature, humidity, etc.) [37]. Moreover, calibration in P lanar resonant elements have been exhaustively used for symmetry-based sensors is not always required, as far as a sensing purposes using microwaves. The main reasons are reference material with well-known properties is typically used the high sensitivity, low cost, small size, and the inherent for sensing. Highly-sensitive differential sensors, e.g., devices integration potential of planar microwave resonators. This latter able to detect extremely small concentrations of solutes in aspect is fundamental in order to implement low-profile planar diluted solutions, have been reported [9],[13],[15]. Despite sensors, such as many applications demand, including sensors these relevant advantageous aspects of symmetry-based combined with microfluidic technology, of interest for liquid sensors, frequency-variation sensors are still highly demanded characterization and bio-sensing [1]-[15]. since they are inherently simpler, cheaper and smaller, provided Most microwave resonant sensors are implemented by a single resonant element suffices for sensing purposes. loading a transmission line with a planar resonator (the sensing Nevertheless, it should be mentioned that phase-variation element), and the most usual working principle is frequency sensors, operating in reflection, based on a planar resonant variation [2],[4],[6],[10],[16]-[20]. Nevertheless, sensors based element (either semi-lumped or distributed), have been recently on symmetry truncation, including frequency-splitting sensors reported and demonstrated to exhibit very competitive [8],[21]-[25], coupling-modulation sensors [26]-[35], and performance [38],[39]. differential sensors [9],[13]-[15],[36] have also been reported. Several types of planar resonant elements have been used for As compared to those sensors that exploit frequency variation, the implementation of frequency-variation sensors, including sensors based on symmetry disruption are typically more robust split ring resonators (SRRs) [2], complementary split ring against cross-sensitivities related to ambient factors resonators (CSRRs) [4],[6],[10],[18],[19], step impedance This work was supported by MINECO-Spain (project PID2019- J. Muñoz-Enano, P. Vélez, and F. Martín are with GEMMA/CIMITEC, 103904RB-I00), Generalitat de Catalunya (project 2017SGR-1159), Departament d’Enginyeria Electrònica, Universitat Autònoma de Institució Catalana de Recerca i Estudis Avançats (who awarded Ferran Barcelona, 08193 Bellaterra, Spain (e-mail: Ferran.Martin@uab.es). Martín), and by FEDER funds. J. Muñoz-Enano acknowledges M. Gil is with Audiovisual Engineering and Communications Secreteraria d’Universitats i Recerca (Gen. Cat.) and European Social Department, Universidad Politécnica de Madrid, 28031, Madrid, Spain. Fund for the FI grant. 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. 2 shunt stubs [40], and other related resonant elements. In this to include the effects of losses in the MUT, and it is related to paper, a frequency-variation sensor useful for complex its loss tangent. Obviously, in the bare sensor, CMUT and GMUT permittivity measurements, based on a microstrip line loaded are null. Finally, l and Z0 are the electrical length and with a dumbbell-shaped defect ground structure (DB-DGS), is characteristic impedance, respectively, of the microstrip line,  presented. The DB-DGS [41] element is a slot resonator, and l being the phase constant and the physical length, transversally etched in the ground plane of the microstrip line, respectively, of the line. with a resonance frequency very sensitive to the permittivity of the surrounding material. The purpose of the paper is twofold. A first objective is to demonstrate that the relative sensitivity, a figure of merit in frequency-variation sensors, only depends on the so-called equivalent dielectric constant of the substrate, to be defined later. The second objective is to obtain analytical expressions useful to determine the material parameters of the material under test (MUT), i.e. the dielectric constant and the loss tangent, from the measurement of the transmission coefficient of the sensor, the DB-DGS-loaded line. The results of the analysis will be validated from full-wave electromagnetic simulations and through the measurement of the complex (a) (b) dielectric constant of several materials, including solid samples Fig. 1. Typical topology (a) and circuit model (b) of the proposed DB- and deionized (DI) water. It should be mentioned that in [42], a DGS-based permittivity sensor. Dimensions are given in mm. Ground differential-mode sensor based on a pair of DB-DGS-loaded plane in grey. lines was reported. Nevertheless, in that sensor the output The model has been validated by first extracting the circuit variable was the magnitude of the cross-mode transmission parameters in the structure without the presence of MUT. For coefficient. that purpose, we have considered the dimensions indicated in The work is organized as follows: Section II presents the the caption of Fig. 1, and the parameters of the Rogers 4003C proposed sensor and the lumped element equivalent circuit, substrate with dielectric constant r = 3.55, thickness necessary for the analysis. The validity of the model is h = 1.524 mm and loss factor tan = 0.0022. With these demonstrated in this section, by comparing electromagnetic substrate parameters, the electrical length of the line and the simulations of the response of the designed structure with characteristic impedance are l = 147º (at the notch frequency circuit simulations with extracted parameters. Section III presents the sensitivity analysis, validated by considering f0) and Z0 = 50 , respectively. The parameters describing the several case examples. The analytical expressions providing the bare DB-DGS have been extracted from the full-wave dielectric constant and loss tangent of the MUT are derived in electromagnetic simulation response (depicted in Fig. 2 and Section IV. The validity of these expressions is demonstrated at obtained by means of ANSYS HFSS). For that purpose, the simulation level and experimentally, by considering different resonance frequency, f0, and the susceptance slope at f0 have structures (with different dielectric constant and loss factor), in been used. This provides the reactive elements L and C, Section V. A comparative analysis is carried out in Section VI. whereas G has been determined by curve fitting the response (it Finally, Section VII highlights the main conclusions. is dictated by the notch magnitude). The extracted parameters are indicated in Table I. Using these parameters, we have II. PROPOSED DB-DGS BASED SENSOR AND CIRCUIT obtained the circuit response of the structure (using Keysight MODEL ADS), also depicted in Fig. 2. The agreement is very good, pointing out the validity of the model. The typical topology of the proposed sensor and the circuit Nevertheless, we have also obtained the electromagnetically model are depicted in Fig. 1 (the justification of such model is simulated response by considering the DB-DGS loaded with a reported in a recent paper by some of the authors [43]). The semi-infinite MUT, with dielectric constant and loss factor sensor consists of a microstrip line with a DB-DGS resonator MUT = 4.4 and tanMUT = 0.02, respectively. The results are also transversally etched in the ground plane. The DB-DGS is depicted in Fig. 2. We have then extracted the parameters by modeled by the resonant tank formed by the capacitance C (the leaving L unaltered, since this parameter is not affected by the slot capacitance) and by the inductance L (related to the return presence of the MUT. The values of CMUT and GMUT that are current path in the ground plane and dictated by the perimeter necessary to fit the electromagnetic simulation are indicated in of the resonator). The conductance G accounts for substrate Table I (note that CMUT and GMUT have been actually obtained losses, i.e., it is assumed that metallic losses have negligible by subtracting C and G, respectively to the extracted effect. The presence of a dielectric material (MUT) in contact parameters). The circuit response obtained with these with the DB-DGS is taken into account by including two parameters is compared to the electromagnetic response in Fig. additional elements, CMUT and GMUT. The former accounts for 2, and, again, good agreement is obtained. According to these the enhancement of the DB-DGS capacitance due to the results, it is clear that the presence of the MUT can be taken into presence of the MUT, and it is related to the dielectric constant account by including CMUT and GMUT in the circuit model. of the MUT, MUT. The conductance GMUT is introduced in order 3 interface). If the semi-infinite substrate approximation is not 200 applicable, the most usual case, then r,eq in (3) must be used, 0 150 -2 -4 100 and r,eq < r. Concerning the MUT, it should be semi-infinite in |S11| (dB) 50 the vertical direction for the validity of expression (3) [10]. In S11 (º) -6 0 -8 -50 practice, it suffices to consider a finite MUT sample extending -100 -10 EM Simulation -150 EM Simulation beyond the region occupied by the DB-DGS perimeter, with thickness enough to consider that the electric field lines Schematic Schematic -12 -200 2.0 2.4 2.8 3.2 3.6 4.0 2.0 2.4 2.8 3.2 3.6 4.0 Frequency (GHz) Frequency (GHz) generated by the DB-DGS do not reach the MUT/air interface. Introducing (2) in (1) gives 0 180 1 𝑆̅ = − -5 150 (4) 2(𝑟,𝑒𝑞 + 𝑀𝑈𝑇 ) |S21| (dB) -10 120 S21 (º) 90 -15 60 and from this simple expression, it follows that the relative -20 EM Simulation 30 EM Simulation Schematic sensitivity does not explicitly depend on the geometry of the Schematic -25 2.0 2.4 2.8 3.2 3.6 4.0 0 2.0 2.4 2.8 3.2 3.6 4.0 DB-DGS resonator (and line). Indeed, for thick enough (semi- Frequency (GHz) Frequency (GHz) infinite) substrates, the relative sensitivity does neither depend Fig. 2. Response of the considered DB-DGS based sensor without (red on the thickness of the substrate (since r,eq = r in this case). curve) and with (blue curve) MUT. Since the dielectric constant of the MUT is not a design TABLE I parameter, the main conclusion of this simple analysis is that PARAMETER EXTRACTION the relative sensitivity in DB-DGS-based microstrip L (nH) C (pF) G (mS) CMUT (pF) GMUT (mS) permittivity sensors increases as the dielectric constant of the Bare 2.50 0.73 0.71 ---- ---- substrate decreases (this is true regardless of the thickness of With MUT 2.50 0.73 0.71 0.74 0.40 the substrate). However, for high dielectric constant MUTs, e.g., liquids, the relative sensitivity is dominated by the value III. SENSITIVITY ANALYSIS of such dielectric constant (since MUT >> r,eq for typical In frequency-variation permittivity sensors, a figure of merit substrate materials, with dielectric constants not exceeding a is the so-called relative sensitivity, defined as follows: dozen). In order to validate the previous conclusion in regard to the 1 𝑑𝑓0 relative sensitivity, we have considered the sensor of Fig. 1 𝑆̅ = (1) 𝑓0 𝑑 𝑀𝑈𝑇 (sensor A), with r = 3.55, h = 1.524 mm and tan = 0.0022, and where the output variable, f0, is the frequency of the resonant we have obtained the relative sensitivity by introducing in (1) element (the notch frequency in the considered DB-DGS the resonance frequencies inferred from electromagnetic structure), and the input variable is the dielectric constant of the simulation, as the dielectric constant of the MUT increases. The MUT, MUT. Normalization to f0 is due to the fact that working input dynamic range is comprised between MUT = 1, the value at high frequencies intrinsically provides higher excursion of corresponding to air (bare DB-DGS), and MUT = 10.2. The the resonance frequency (for a given variation of the dielectric results are depicted in Fig. 3, where the curve corresponding to constant of the MUT sample). Thus, for comparison purposes, the analytical expression (4) is also included for comparison the relative sensitivity, as defined in (1), is the key purposes. As it can be appreciated, very good agreement (representative) sensor parameter. The resonance frequency is between the theoretical curve and the data points inferred from given by the simulated values is obtained. As indicated in the figure, the 1 equivalent dielectric constant of the substrate has been found to 𝑓𝑜 = (2) be r,eq = 2.15. Such value has been inferred by considering the 2√𝐿𝐶′𝑀𝑈𝑇 response of the bare sensor and the response of the sensor where loaded with any MUT with known dielectric constant. Note that 𝑟,𝑒𝑞 + 𝑀𝑈𝑇 the ratio of the squared resonance frequencies is inversely 𝐶′𝑀𝑈𝑇 = 𝐶 + 𝐶𝑀𝑈𝑇 = 𝐶 (3) 𝑟,𝑒𝑞 + 1 proportional to the ratio of capacitances, and, using (3), r,eq can be easily isolated [44]. where r,eq is the equivalent dielectric constant of the substrate. Nevertheless, we have considered three additional sensing Such dielectric constant is defined as the dielectric constant of structures. In one of them (sensor B), the bare DB-DGS a hypothetical semi-infinite substrate providing the same resonates at roughly the same frequency, but the geometry is contribution to the capacitance of the DB-DGS resonator [44]. different. In the other one (sensor C), the resonance frequency Thus, the equivalent dielectric constant coincides with the of the DB-DGS is higher. In sensors B and C, the substrate is dielectric constant of the substrate (r,eq = r), provided the identical to the one of sensor A. By contrast, in the last sensing substrate is thick enough, so that it can be considered to be device (sensor D), a thicker substrate is considered (i.e., h = 2 semi-infinite in the vertical direction (i.e., with the field lines mm), whereas the geometry of the DB-DGS is identical to the generated in the slot resonator not reaching the opposite one of sensor A. The corresponding layouts of the DB-DGSs 4 for sensors B and C are depicted in Fig. 4, where the dimensions IV. DETERMINATION OF THE DIELECTRIC CONSTANT AND are indicated. The relative sensitivities for these three additional LOSS TANGENT OF THE MUT sensing structures, inferred from (1), are also included in Fig. 3. In this section, two analytical expressions that provide the As it can be appreciated, the resulting data points for sensors B, dielectric constant, MUT, and the loss tangent, tanMUT, of the C and D are very similar, and very similar to those of sensor A. MUT, from the notch frequency and magnitude of the measured The reason is that the equivalent dielectric constants are similar sensor response, are inferred. The hypothesis for the validity of for the four sensors (the theoretical relative sensitivities, as the following analysis is that the thickness of the MUT is given by 4, are not included in Fig. 3 for sensors B, C and D, enough to avoid the presence of electric field lines in the but the values of the equivalent dielectric constant are indicated MUT/air interface (similar to the hypothesis adopted in Section in the caption). Note that in sensors A and C, the equivalent III). dielectric constant is roughly the same because in these sensors, the ratio between the slot width (s = 0.5 mm) and substrate A. Determination of the dielectric constant thickness (h = 1.524 mm) is identical, and the equivalent The dielectric constant of the MUT can be inferred from the dielectric constant is determined by the ratio s/h, as discussed resonance frequencies of the bare sensor, f0,b, and sensor loaded in [44]. with the MUT, f0,MUT. Such resonance frequencies are given by With this study, based on full-wave electromagnetic (2), with f0,b = (1/2)(LC)1/2, as it is inferred from (2) if MUT = simulation, the reported sensitivity analysis, pointing out that 1 (bare DB-DGS). Therefore, these frequencies are related by the geometry of the DB-DGS and substrate thickness do not significantly affect the relative sensitivity, provided s/h << 1, is 2 𝑓0,𝑏 𝑟,𝑒𝑞 + 𝑀𝑈𝑇 = (5) validated. By contrast, in sensors similar to the one considered 2 𝑓0,𝑀𝑈𝑇 𝑟,𝑒𝑞 + 1 in this paper, but based on complementary split ring resonators From the previous expression, the dielectric constant of the (CSRRs) [10], the relative sensitivity depends on the geometry MUT is found to be of the resonant element and substrate thickness. Moreover, in 2 CSRR-based sensors, the resonance frequency depends on the 𝑓0,𝑏 𝑀𝑈𝑇 = (𝑟,𝑒𝑞 + 1) − 𝑟,𝑒𝑞 (6) coupling capacitance between the line and the CSRR, a 2 𝑓0,𝑀𝑈𝑇 parameter not affected by the MUT. Consequently, the relative sensitivity in CSRR-based microstrip sensors is not as good as Hence, from the knowledge of the equivalent dielectric constant the one achievable with DB-DGSs. of the substrate, and the measured notch frequencies of the unloaded and loaded sensor, the dielectric constant of the MUT -0.04 can be obtained. -0.06 B. Determination of the loss tangent -0.08 The notch magnitude is related to the level of losses of the whole device, including the MUT. Thus, it is expected that the S -0.10 loss tangent of the MUT can be inferred from the measured -0.12 Theoretical (r, eq = 2.15) notch level. If the characteristic impedance of the transmission -0.14 Sensor A Sensor B line sections is Z0 = 50 , the lines introduce a phase shift in Sensor C Sensor D the transmission coefficient, but do not have any influence on -0.16 its magnitude, given by 1 2 3 4 5 6 7 8 9 10 MUT 1 Fig. 3. Dependence of the relative sensitivity with MUT for different |𝑆21 | = | | (7) sensing structures. The dielectric constant of the substrate is r = 3.55 in 1 1+ all cases. The equivalent dielectric constant ( r,eq) for the Sensor A, B, C 2𝑍0 𝑌𝑀𝑈𝑇 and D is 2.15, 2.48, 2.08 and 2.71, respectively. where YMUT is the admittance of the DB-DGS loaded with the MUT sample, i.e., 20,𝑀𝑈𝑇 𝑌𝑀𝑈𝑇 = 𝐺′𝑀𝑈𝑇 + 𝑗𝐶′𝑀𝑈𝑇 (1 − ) (8) 2 where G’MUT = G + GMUT,  = 2f is the angular frequency, and 0,MUT = 2f0,MUT. Expression (8) is also valid for the bare sensor, with GMUT = 0 (i.e., G’MUT = G), CMUT = 0 (that is, C’MUT = C), and 0,MUT replaced with 0,b, the resonance (angular) frequency of the bare structure. For the sensor loaded with the MUT sample, the notch (a) (b) magnitude is given by the transmission coefficient evaluated at Fig. 4. Layouts of the DB-DGS for sensors B (a) and C (b). Dimensions  = 0,MUT, that is are given in mm. Ground plane in grey. 5 2𝑍0 𝐺′𝑀𝑈𝑇 |𝑆21 |0,𝑀𝑈𝑇 = (9a) 2𝑍0 𝐺′𝑀𝑈𝑇 + 1 Similarly, the notch magnitude for the bare sensor is 2𝑍0 𝐺 |𝑆21 |0,𝑏 = (9b) 2𝑍0 𝐺 + 1 and combining (9a) and (9b), the conductance contribution of the MUT is found to be 1 |𝑆21 |0,𝑀𝑈𝑇 |𝑆21 |0,𝑏 𝐺𝑀𝑈𝑇 = 𝐺′𝑀𝑈𝑇 − 𝐺 = { − } (10) 2𝑍𝑜 1 − 21 0,𝑀𝑈𝑇 1 − |𝑆21 |0,𝑏 |𝑆 | The loss tangent of the MUT is given by (a) (b) 𝐺𝑀𝑈𝑇 Fig. 5. Sketch showing the different contributions to the capacitance and tan𝑀𝑈𝑇 = (11) conductance of the bare (a) and loaded (b) DB-DGS resonator. (𝐶′𝑀𝑈𝑇 − 𝐶𝑠𝑢𝑏𝑠 )0,𝑀𝑈𝑇 where Csubs is the substrate contribution to the capacitance of V. VALIDATION the DB-DGS resonator (and, consequently, C’MUT  Csubs is the To validate the previous analysis, different MUT samples contribution of the MUT to the capacitance of the DB-DGS). with well-known electromagnetic parameters (dielectric Csubs is given by [10] constant and loss tangent) have been considered. In particular, 𝑟 we have used three solid samples and DI water. The solid 𝐶𝑠𝑢𝑏𝑠 = 𝐶 (12) 𝑟,𝑒𝑞 + 1 samples are uncladded microwave substrates (Rogers RO4003C and RO3010) and FR4. Actually, three 1.5-mm thick Introducing (12) and (3) in (11), the following expression is slabs have been stacked (and glued) in order to achieve the obtained necessary MUT thickness to guarantee the validity of the semi- 𝐺𝑀𝑈𝑇 (𝑟,𝑒𝑞 + 1) infinite MUT approximation. Concerning DI water, a container tan𝑀𝑈𝑇 = (13) 𝐶 𝑀𝑈𝑇 0,𝑀𝑈𝑇 was fabricated by means of a 3D printer (model Ultimaker 3 Extended) using Polylactic-Acid (PLA) material, and then it Finally, introducing (10) in (13), the loss tangent is found to be was attached to the substrate, on top of the DB-DGS resonator. 𝑟,𝑒𝑞 + 1 |𝑆21 |0,𝑀𝑈𝑇 |𝑆21 |0,𝑏 Such container is high enough to ensure that the height of the tan𝑀𝑈𝑇 = { − } (14) liquid (DI water) suffices to consider it semi-infinite in the 2𝑍𝑜 𝐶 𝑀𝑈𝑇 0,𝑀𝑈𝑇 1 − |𝑆21 |0,𝑀𝑈𝑇 1 − |𝑆21 |0,𝑏 vertical direction, as required. For the measurement with DI and it can be inferred from the measured responses of the bare water, the substrate was protected with a dry film of clear sensor and sensor loaded with the MUT (provided all the polyester (with an estimated dielectric constant of 3.5 and parameters preceding the brackets in (14) are either known or thickness of 50-m), in order to avoid substrate absorption. The previously inferred, e.g., MUT). effects of such thin film are negligible given the high dielectric At this point, it should be mentioned that the capacitance constant of DI water. Figure 6 depicts the photograph of the appearing in the denominator in (11), i.e., the contribution of bare sensor (sensor A) and the sensor equipped with the liquid the MUT to the capacitance of the DB-DGS, C’MUT  Csubs, is container. The sensor was fabricated with the LPKF H100 not CMUT. The latter is simply the difference between the total drilling machine. capacitance of the loaded DB-DGS, C’MUT and the capacitance of the unloaded DB-DGS, C, which includes the contribution of air, Cair. In other words, CMUT = C’MUT  C  C’MUT  Csubs, since C = Csubs + Cair. By contrast, the whole conductance of the loaded DB-DGS is G’MUT = G + GMUT, as indicated before, where G is the conductance of the bare DB-DGS, which coincides with the conductance of the substrate (G = Gsubs), provided the conductance of air is null, and GMUT is the contribution to the conductance of the MUT. Figure 5 shows the different contributions to the capacitance and conductance of the bare and loaded DB-DGS. (a) (b) Fig. 6. Photograph of the fabricated sensor (a), and sensor including the liquid container (b). The simulated (using ANSYS HFSS) and measured (by means of the Keysight N5221A vector network analyzer) frequency 6 responses of the bare sensor, and sensor loaded with the based on resonant cavities [45], bulky and expensive as different MUT samples, are depicted in Fig. 7, where good compared to the proposed method. Nevertheless, let us mention agreement is obtained in all the cases. Nevertheless, note that in that, in order to minimize the well-known effects of the air gap, FR4, the non-negligible (but acceptable) frequency shift we have pressed the MUT samples against the substrate by between measurement and simulation is attributed to the well- means of screws. known tolerances (uncertainties) in the dielectric constant of such material. The nominal dielectric constant and loss tangent VI. COMPARISON TO OTHER APPROACHES for the considered MUTs are indicated in Table II. Application A meaningful comparison between microwave permittivity of expression (6) to the simulated and measured responses, with sensors based on frequency variation and reported in the f0,b and f0,MUT easily identifiable from the results, provides the literature is not simple. In particular, comparing the relative MUT dielectric constants also indicated in the table and sensitivity, the main figure of merit in these sensors, is not identified as simulated (sim.) and experimental (exp.). absent of certain difficulty, since this parameter is not constant Similarly, using (14), with the notch depths inferred from the (it depends on the dielectric constant of the MUT), and it varies simulated and the measured responses, the loss tangents of the (decreases) with the substrate dielectric constant. In many MUTs are obtained (the results are also included in Table II). comparison tables reported in the literature, the relative average sensitivity is the considered parameter subject of comparison. 0 However, since the relative sensitivity decreases with the -5 dielectric constant of the MUT, it follows that the relative -10 average sensitivity depends significantly on the considered |S21| (dB) input dynamic range of the dielectric constant of the MUT. -15 Thus, a significant comparison requires that the considered -20 input dynamic ranges are similar. According to these Air RO4003C comments, and because one of the considered MUTs in this -25 FR4 RO3010 Measurement work is DI water, with measured dielectric constant of 78.2 (see DI Water Simulation -30 Table II), we have compared the sensor proposed in this work 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 with other sensors exhibiting an input dynamic range (in the Frequency (GHz) dielectric constant) of the same order. Table III reports such Fig. 7. Measured responses (transmission coefficient) of the bare comparison. In the table, f0,b is the frequency of the bare sensor and sensor loaded with stacks of FR4, Rogers RO3010, and resonator. Sav is the average sensitivity, calculated as the ratio Rogers RO4003C, as well as DI water. between the output dynamic range (or difference between the TABLE II resonance frequency of the bare resonator and the resonance MUT PARAMETERS frequency of the resonator loaded with the MUT exhibiting the RO4003C FR4 RO3010 DI Water maximum dielectric constant) and the input dynamic range MUT (nom.) 3.55 4.40 10.2 80.9 (difference between the maximum and minimum values of tan (nom.) 0.0022 0.020 0.0020 0.042 MUT). Finally, 𝑆𝑎𝑣 ̅ is the relative average sensitivity, calculated MUT (sim.) 3.57 4.44 10.6 79.5 by simply dividing the average sensitivity by the resonance tan (sim.) 0.0030 0.035 0.0022 0.050 ̅ = 𝑆𝑎𝑣 /𝑓0,𝑏 frequency of the bare resonator, i.e., 𝑆𝑎𝑣 MUT (exp.) 3.53 4.05 10.0 78.2 ̅ (nevertheless, note that in the table, 𝑆𝑎𝑣 is expressed as tan (exp.) 0.0017 0.039 0.0022 0.041 percentage). The table includes also the considered input dynamic range relative to the dielectric constant of the MUT. As it can be appreciated from Table II, the reported analytical procedure provides the material parameters of the different TABLE III MUTs to a good approximation, particularly the dielectric COMPARISON OF VARIOUS SENSORS IN TERMS OF SENSITIVITY constant. The dielectric constant of FR4 inferred from the Reference f0,b (GHz) Sav (MHz) ̅ 𝑺𝒂𝒗 (%) Din. Range measured notch frequency is moderately smaller than the [2] 1.9 1.53 0.081 9.0-80.0 nominal value. This is an expected result taking into account [4] 2.0 4.76 0.238 9.0-79.5 [6] 3.5 9.16 0.261 6.5-80.0 the frequency deviation visible in Fig. 7, and attributed to the [8] 0.87 0.79 0.091 27.8-80.8 high dispersion of this parameter from sample to sample in FR4. This work 3.75 38.21 1.019 1.0-78.2 The loss tangent of the MUT cannot be accurately predicted, in part because expression (14) is highly sensitive to variations in the notch depth, which can be caused by diverse factors (e.g., According to the results of Table III, the proposed sensor air gaps between stacked layers, and with the substrate, effects exhibits by far the higher relative average sensitivity. It should of connectors, etc.). Nevertheless, the proposed sensor, using be mentioned that the substrate dielectric constant in the sensors the reported analytical procedure, is able to provide a rough of [2],[8] is r = 10.2, and this explains in part the limited estimation of the loss tangent of the MUT (or order of relative average sensitivity in such sensors. However, in the magnitude). It is well known that the accurate measurement of sensors reported in [4] and [6], the dielectric constant of the the loss tangent in low-loss solid samples requires methods substrate is r = 2.94 and r = 2.33, respectively, i.e., smaller 7 than the one considered in the sensor proposed in this work VII. CONCLUSIONS (r = 3.55). The reason that explains the superior relative In summary, an analytical method to determine the dielectric average sensitivity in the present work is the considered constant and loss tangent of solids and liquids in DB-DGS- resonator, a DB-DGS. With this sensing resonant element, the based frequency-variation permittivity sensors has been changes in the dielectric constant of the MUT directly affect the reported and experimentally validated. The method is based on unique capacitance that determines the resonance frequency, analytical expressions that depend on the notch frequency and the output variable. By contrast, in the sensors presented in magnitude of the response of the bare sensor and sensor loaded [4],[6], where a complementary split ring resonator (CSRR) with the MUT, i.e., easily measurable quantities. The results was used, the resonance frequency is also dependent on the obtained by considering samples with known dielectric constant coupling capacitance between the line and the resonant element. and loss tangent (including low-loss microwave substrates, FR4 Since such capacitance does not vary with the dielectric and DI water) have demonstrated the accuracy of the method constant of the MUT, it is expected that such CSRR-based for the determination of the dielectric constant. The loss sensors exhibit poorer sensitivity, as compared to those of DB- tangent, related to the notch magnitude, cannot be estimated DGS-based sensors, as it actually occurs. In other words, the with the same level of accuracy, but the method has been coupling capacitance of the CSRR obscures in part the effects demonstrated to provide a rough estimate of the nominal value. Moreover, a sensitivity analysis, that demonstrates that the of the dielectric constant of the MUT on the variation of the relative sensitivity (the main figure of merit in frequency resonance frequency, thereby reducing the sensitivity. variation sensors) does not depend on the geometry of the DB- There are other recently reported sensors based on frequency DGS resonator (for thick enough substrates), has been carried variation that exhibit good performance, especially high out in the paper. It has been found that the sensitivity can be sensitivity [46],[47]. However, such sensors have been applied optimized by decreasing the dielectric constant of the substrate, to MUT samples exhibiting much smaller input dynamic range, and it decreases with the dielectric constant of the MUT. The and for this main reason, they are not included in the proposed DB-DGS frequency-variation permittivity sensor comparative Table III. exhibits higher relative sensitivity as compared to other sensors Concerning the measurement of the loss tangent in low-loss based on slot resonators, such as CSRR-based permittivity solid samples by means of planar sensors, papers [19],[20] sensors. Since microstrip lines with slot resonators etched in the present sensing structures based on CSRRs in both cases. Such ground plane are very interesting permittivity sensors, due to sensors provide not only the dielectric constant of the MUT, but the inherent backside isolation of the ground plane, it can be also the loss tangent, similar to the sensor reported in this work. concluded that DB-DGS-loaded microstrip lines are very However, the accuracy in the determination of the loss tangent promising devices for permittivity sensing. Such devices not with those sensors is comparable to that of the sensor of Fig. 6, only exhibit superior sensitivity than their CSRR-based i.e., it is limited (an inherent characteristic of planar sensors). counterparts, but also, a simple circuit model describes them, Consequently, the loss factor of the MUT can be roughly and the sensors can be easily designed with the optimum estimated with these (planar) sensors, rather than being (relative) sensitivity, since it mainly depends on the substrate accurately predicted. dielectric constant. To end this section, let us mention that for the measurement of complex variables, or for the characterization of complex REFERENCES phenomena, such as corrosion, among others, post processing [1] K. Grenier et al., “Integrated broadband microwave and microfluidic algorithms such as feature fusion, as discussed in the [48], sensor dedicated to bioengineering,” IEEE Trans. Microw. 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Tian, "Multiresonance Chipless RFID Sensor Ferran Martín (M’04-SM’08-F’12) was born in Tag for Metal Defect Characterization Using Principal Component Barakaldo (Vizcaya), Spain in 1965. He received Analysis," IEEE Sensors J., vol. 19, no. 18, pp. 8037-8046, Sep. 2019. the B.S. Degree in Physics from the Universitat [50] N. Hosseini and M. Baghelani, "Selective real-time non-contact multi- Autònoma de Barcelona (UAB) in 1988 and the variable water-alcohol-sugar concentration analysis during fermentation PhD degree in 1992. From 1994 up to 2006 he was process using microwave split-ring resonator based sensor," Sens. Act. A: Associate Professor in Electronics at the Phys., vol. 325, pp, 112695, 2021. Departament d’Enginyeria Electrònica (Universitat Autònoma de Barcelona), and since 2007 he is Full Jonathan Muñoz-Enano was born in Mollet del Professor of Electronics. In recent years, he has Vallès (Barcelona) Barcelona, Spain, in 1994. He been involved in different research activities received the Bachelor's Degree in Electronic including modelling and simulation of electron devices for high Telecommunications Engineering in 2016 and the frequency applications, millimeter wave and THz generation systems, Master's Degree in Telecommunications and the application of electromagnetic bandgaps to microwave and Engineering in 2018, both at the Autonomous millimeter wave circuits. He is now very active in the field of University of Barcelona (UAB). Actually, he is metamaterials and their application to the miniaturization and working in the same university in the elaboration of optimization of microwave circuits and antennas. Other topics of interest his PhD, which is focused on the development of include microwave sensors and RFID systems, with special emphasis microwave sensors based on metamaterials concepts for the dielectric on the development of high data capacity chipless-RFID tags. He is the characterization of materials and biosensors. head of the Microwave Engineering, Metamaterials and Antennas Group (GEMMA Group) at UAB, and director of CIMITEC, a research Center on Metamaterials supported by TECNIO (Generalitat de Catalunya). He Paris Vélez (S’10–M’14–SM’21) was born in has organized several international events related to metamaterials and Barcelona, Spain, in 1982. He received the degree related topics, including Workshops at the IEEE International Microwave in Telecommunications Engineering, specializing in Symposium (years 2005 and 2007) and European Microwave electronics, the Electronics Engineering degree, Conference (2009, 2015 and 2017), and the Fifth International Congress and the Ph.D. degree in Electrical Engineering from on Advanced Electromagnetic Materials in Microwaves and Optics the Universitat Autònoma de Barcelona, Barcelona, (Metamaterials 2011), where he acted as Chair of the Local Organizing in 2008, 2010, and 2014, respectively. His Ph.D. Committee. He has acted as Guest Editor for six Special Issues on thesis concerned common mode suppression metamaterials and sensors in five International Journals. He has differential microwave circuits based on authored and co-authored over 650 technical conference, letter, journal metamaterial concepts and semi-lumped papers and book chapters, he is co-author of the book on Metamaterials resonators. During the Ph.D., he was awarded with entitled Metamaterials with Negative Parameters: Theory, Design and a pre-doctoral teaching and research fellowship by the Spanish Microwave Applications (John Wiley & Sons Inc.), author of the book Government from 2011 to 2014. From 2015-2017, he was involved in Artificial Transmission Lines for RF and Microwave Applications (John the subjects related to metamaterials sensors for fluidics detection and Wiley & Sons Inc.), co-editor of the book Balanced Microwave Filters characterization at LAAS-CNRS through a TECNIOSpring fellowship (Wiley/IEEE Press) and co-author of the book Time-Domain Signature cofounded by the Marie Curie program. From 2018 to 2021 he has Barcodes for Chipless-RFID and Sensing Applications (Springer). worked in miniaturization of passive circuits RF/microwave and sensors- Ferran Martín has generated 21 PhDs, has filed several patents on based metamaterials through Juan de la Cierva fellowship. His current metamaterials and has headed several Development Contracts. research interests include the miniaturization of passive circuits Prof. Martín is a member of the IEEE Microwave Theory and RF/microwave and sensors-based metamaterials. Dr. Vélez is a Techniques Society (IEEE MTT-S). He is reviewer of the IEEE Reviewer for the IEEE TRANSACTIONS ON MICROWAVE THEORY Transactions on Microwave Theory and Techniques and IEEE AND TECHNIQUES and for other journals. Microwave and Wireless Components Letters, among many other journals, and he serves as member of the Editorial Board of IET Marta Gil (S’05-M’09) was born in Valdepeñas, Microwaves, Antennas and Propagation, International Journal of RF and Ciudad Real, Spain, in 1981. She received the Microwave Computer-Aided Engineering, and Sensors. He is also a Physics degree from Universidad de Granada, member of the Technical Committees of the European Microwave Spain, in 2005, and the Ph.D. degree in electronic Conference (EuMC) and International Congress on Advanced engineering from the Universitat Autònoma de Electromagnetic Materials in Microwaves and Optics (Metamaterials). Barcelona, Barcelona, Spain, in 2009. She studied Among his distinctions, Ferran Martín has received the 2006 Duran one year with the Friedrich Schiller Universität Jena, Farell Prize for Technological Research, he holds the Parc de Recerca Jena, Germany. During her PhD Thesis she was UAB – Santander Technology Transfer Chair, and he has been the holder of a METAMORPHOSE NoE grant and recipient of three ICREA ACADEMIA Awards (calls 2008, 2013 and National Research Fellowship from the FPU 2018). He is Fellow of the IEEE and Fellow of the IET. Program of the Education and Science Spanish Ministry. As a postdoctoral researcher, she was awarded with a Juan de la Cierva fellowship working in the Universidad de Castilla-La Mancha. She was postdoctoral researcher in the Institut für Mikrowellentechnik und Photonik in Technische Universität Darmstadt and in the Carlos III University of Madrid. She is currently associate professor in the Universidad Politécnica de Madrid within the Excellence Program for University Professoriat of the V Regional Plan for Scientific Research and Technological Innovation of the Administration of the Community of Madrid. She has worked in metamaterials, piezoelectric MEMS and microwave passive devices. Her current interests include metamaterials sensors for fluidic detection.