This is the accepted version of the journal article: Muñoz Enano, Jonathan; Vélez Rasero, Paris; Su, Lijuan; [et al.]. «On the Sensitivity of Reflective-Mode Phase-Variation Sensors Based on Open-Ended Stepped-Impedance Transmission Lines: Theoretical Analysis and Experimental Validation». IEEE transactions on microwave theory and techniques, Vol. 69, Issue 1 (January 2021), p. 308-324. DOI 10.1109/TMTT.2020.3023728 This version is available at https://ddd.uab.cat/record/258243 under the terms of the license On the Sensitivity of Reflective-Mode Phase- Variation Sensors Based on Open-Ended Stepped- Impedance Transmission Lines: Theoretical Analysis and Experimental Validation Jonathan Muñoz-Enano, Student Member IEEE, Paris Vélez, Member IEEE, Lijuan Su, Member IEEE, Marta Gil, Pau Casacuberta, and Ferran Martín, Fellow IEEE  Abstract— This paper presents an exhaustive study of the within the microware community. Other advantageous sensitivity in reflective-mode phase-variation sensors based on aspects concern the inherently low cost of microwave an open-ended transmission line with a step-impedance generation and detection systems. Moreover, microwave discontinuity. Such discontinuity delimits the sensing region sensors are especially suited for material characterization, (which extends up to the open end of the so-called sensing line), from the transmission line section connected to the input port since microwaves are very sensitive to the properties of the (design line), which is used to enhance the sensitivity. The materials to which they interact (including interaction at theoretical analysis provides the design guidelines to achieve a different scales, i.e., through the near-field or the far-field). sensor with high sensitivity, as compared to the one based on an Probably most planar microwave sensors reported to date ordinary (uniform) line with similar length. In particular, it is are based on frequency variation [1]-[9]. In such sensors, the shown that for sensitivity optimization, the electrical length of working principle is typically the variation of the resonance the design line must be set to 90º (or an odd multiple), whereas either a 90º (or an odd multiple) or a 180º (or an even or odd frequency and quality factor of a resonator-loaded line, multiple) sensing line can be alternatively used in order to caused by the presence of the material under test (MUT). maximize the sensitivity. It is shown that the impedance These sensors are relatively simple, but a wideband contrast, defined as the ratio between the characteristic interrogation signal is needed for measurement. Wideband impedances of the design and sensing line, is a key parameter signals are also required in frequency splitting sensors [10]- for sensitivity enhancement, and it must be as low or as high as [16], a variant of frequency variation sensors. Although possible for the 90º or 180º sensing lines, respectively. For frequency splitting sensors are not truly differential-mode validation purposes, two prototype devices (one with a 90º and the other one with a 180º sensing line) have been designed and sensors [17]-[28], their working principle is based on fabricated following the design guidelines. Such devices have symmetry disruption between the so-called reference (REF) been tested by loading the sensing region with several materials material and the MUT (such materials should be placed on with different dielectric constants. As compared to the ordinary top of two identical resonators symmetrically loading a line) line based sensors, it is found that the maximum sensitivity is [29]. enhanced by a factor of 19.7 and 11.4 in the phase-variation Although frequency variation and frequency splitting sensor based on a 90º and 180º sensing line, respectively. Finally, the sensor concept is generalized to a multi-section step- sensors exhibiting good performance have been reported, the impedance transmission line as a means of further increasing main limitation of these sensors concerns the fact that the the sensitivity, and a prototype device exhibiting 528.7º bandwidth required for measurement is given by the output maximum sensitivity is implemented. dynamic range. Thus, implementing sensors with a high Index Terms–Reflective-mode sensor, microwave sensor, dynamic range is possible at the expense of a higher cost of phase-variation sensor, microstrip, stepped-impedance the associated electronics (based on wideband voltage transmission line. controlled oscillators –VCOs). According to the previous paragraph, it is clear that the I. INTRODUCTION implementation of sensors operating at a single frequency represents a very interesting solution in terms of sensor costs P LANAR microwave sensors have attracted the interest of many researchers in recent years. Their low cost and low profile (including the possibility to implement conformal in a real scenario. Examples of single-frequency sensors can be found in the literature, including coupling modulation sensors on flexible substrates), their compatibility with fully sensors [30]-[40], differential mode sensors [17],[25],[28] planar fabrication processes, and with other sensing and phase-variation sensors [20],[25]. In this paper, we are technologies (e.g., microfluidics, lab-on-a-chip, etc.), their interested in reflective-mode phase-variation sensors, due to potential for wireless connectivity, and their robustness their simplicity (the structure is a simple one-port network) against hostile and harsh conditions, among others, are key and low cost (other reflective mode sensors have been aspects that explain the increasing interest of these sensors recently reported [22],[41],[42]). The specific sensor This work was supported by MINECO-Spain (project TEC2016-75650- Universidad Politécnica de Madrid Young Researchers Support Program R), by Generalitat de Catalunya (project 2017SGR-1159), by Institució (VJIDOCUPM18MGB) for its support. Catalana de Recerca i Estudis Avançats (who awarded Ferran Martín), and J. Muñoz-Enano, P. Vélez, L. Su, P. Casacuberta, and F. Martín are with by FEDER funds. J. Muñoz-Enano acknowledges Secreteraria GEMMA/CIMITEC, Departament d’Enginyeria Electrònica, Universitat d’Universitats i Recerca (Gen. Cat.) and European Social Fund for the FI Autònoma de Barcelona, 08193 Bellaterra, Spain. E-mail: grant. Paris Vélez acknowledges the Juan de la Cierva Program for Ferran.Martin@uab.es. supporting him through Project IJCI-2017-31339. M. Gil acknowledges the M. Gil is with Departamento Ingeniería Audiovisual y Comunicaciones, Universidad Politécnica de Madrid, 28031 Madrid, Spain. structure is an open-ended line with a step-impedance This sensor is devoted to material characterization, and the discontinuity, and the main objective of the paper is to carry working principle is the variation experienced by the phase out an exhaustive sensitivity analysis in order to obtain the of the reflection coefficient, , when a certain material design guidelines for sensitivity optimization. Moreover, it is (material under test –MUT) is placed on top of the sensing also an objective of the paper to experimentally validate the region. Note that, ideally (i.e., by excluding the effects of main conclusions of the analysis. For that purpose, a pair of losses), the modulus of the reflection coefficient should be prototype device sensors are designed, fabricated, and tested  = 1. By considering a uniform and matched open-ended (by considering materials with different dielectric constant). line, i.e., with characteristic impedance identical to the To gain insight on the potential of the approach for sensitivity reference impedance of the port, Z0, the reflection coefficient enhancement, the proposed sensors are compared with is simply [43] phase-variation sensors based on ordinary (uniform) lines.  = 𝑒 −2𝑗𝑠 (1) As it will be shown, the results demonstrate that the sensitivity can substantially be improved by considering a Note that the previous reflection coefficient is the one stepped-impedance transmission line adequately designed corresponding to the structure of Fig. 1 by considering  = 0 (i.e., according to the inferred guidelines). Moreover, a (absence of design line) and Zs = Z0. In such uniform (and procedure to further optimize the sensitivity, by considering matched) line, a variation of the phase of the line, s, a multi-section step-impedance transmission line, is produced by a change in the dielectric constant of the MUT, reported. generates a variation in the phase of the reflection coefficient The work is organized as follows: Section II reports the given by structure and working principle of the proposed sensor, and  = −2𝑠 (2) justifies the need to include a step-impedance discontinuity for sensitivity enhancement. The sensitivity analysis is and, consequently, the sensitivity of the phase of the carried out in Section III. Such section ends with the required reflection coefficient, , with the phase of the line is simply design conditions for sensitivity optimization, derived from 𝑑  the analysis. Section IV is devoted to the experimental = −2 (3) 𝑑 𝑠 validation, and performance comparison with other sensors based on ordinary (uniform) lines. The generalization of the However, the key sensor parameter is the sensitivity of  concept to multiple step-impedance discontinuities is with the dielectric constant of the MUT, MUT, given by considered in Section V, where it is shown that the sensitivity 𝑑  𝑑  𝑑 𝑠 can be enhanced at wish without the need of elongating the 𝑆= 𝑑 𝑀𝑈𝑇 =  𝑑 𝑠 𝑑 𝑀𝑈𝑇 (4) sensing line (a relevant property of the proposed sensors). Moreover, a comparison with other sensors where phase In (2)-(4), the effects of MUT on  through the variation of variation is the operating principle is carried out in this Zs have been omitted. In general, such effects are not section. Finally, the main conclusions are highlighted in negligible, and should be accounted for by adding an Section VI. additional term in (4), see Appendix A. However, for the conditions that optimize the sensitivity, to be discussed later, II. THE PROPOSED REFLECTIVE-MODE PHASE-VARIATION such term is identically null, as demonstrated in Appendix A. SENSOR AND WORKING PRINCIPLE Consequently, we adopt expression (4) in the present study, The typical schematic/topology of the proposed reflective- as far as the main objective is the implementation of sensors mode phase-variation sensor is depicted in Fig. 1. It is a one- with optimum sensitivity. port network consisting of an open-ended transmission line The last derivative on the right-hand side member of (4), with a step-impedance discontinuity. The sensitive region, dS/dMUT, depends on the length of the line, its transverse indicated by a dashed rectangle in Fig 1, is delimited by the geometry, and also on the dielectric constant of the substrate, step discontinuity and the open end of the line. For obvious r. Obviously, the longer the line, the higher the sensitivity, reasons, such line, with characteristic impedance, Zs, and as it is well known [25], but this strategy for sensitivity electrical length at the design frequency, s, is designated as enhancement goes against size reduction. The transverse sensing line. The line section comprised between the step- geometry does not have a strong influence on the phase impedance discontinuity and the input port (with variation, since the effective dielectric constant of the line characteristic impedance Z and electrical length ), has a soft dependence on the transverse geometry of the line. fundamental for sensitivity optimization, is identified as Finally, low-dielectric constant substrates favor sensitivity design line in this paper. optimization, but there is a limit concerning the sensitivity improvement achievable with this strategy since r > 1 (moreover, sometimes the substrate is dictated by external constraints or specifications). Thus, in summary, the last derivative in (4) cannot be significantly engineered in order to substantially improve its contribution to the overall sensitivity. Nevertheless, the effects of such term in the total sensitivity will be analyzed at the end of the next section. Concerning the first term, d/ds, for a uniform and matched line, the value is given by equation (3). However, Fig. 1. Typical schematic/topology of the proposed reflective-mode phase- variation sensor based on a step-impedance discontinuity. by introducing further degrees of freedom (i.e., a step- impedance discontinuity, as depicted in Fig. 1), it is possible For the trivial case with Zs = Z0 (matched sensing line), the to engineer the line and substantially improve the first term derivative of the sensitivity is null, as expected since the on the right-hand side member of (4), as it will be sensitivity is constant (and given by expression 3). However, demonstrated in this paper. The design parameters include if Zs  Z0, the values of s that provide extreme values to the the characteristic impedances of both the sensing and the sensitivity (those that null the numerator of 9) satisfy either design transmission line sections, as well as their electrical s = n, or s = (2n + 1)/2, with n = 0, 1, 2,... The lengths. The idea of optimizing the sensitivity in these corresponding sensitivities are found to be reflective-mode phase-variation sensors by means of 𝑍0 2 impedance contrast is inspired by the well-known effect of 𝑆𝑠 = −2 = − ̅̅̅ for s = n (10a) 𝑍𝑠 𝑍𝑠 size reduction in half-wavelength or quarter-wavelength 𝑍𝑠 stepped-impedance resonators [44], which can be attributed 𝑆𝑠 = −2 ̅̅̅𝑠 = −2𝑍 for s = (2n + 1)/2 (10b) 𝑍0 to a further variation of the phase of the signal along the where ̅̅̅ 𝑍𝑠 is the normalized impedance of the sensing line. whole element. In particular, stepped-impedance shunt stubs According to expressions (10), for sensitivity optimization, (SISS) with extreme impedance contrast can be considered the normalized impedance of the sensing line must be high if to be semi-lumped resonators, as discussed in [45]. the electrical length satisfies s = (2n + 1)/2, and it must be III. SENSITIVITY ANALYSIS low for s = n. Note that the limits are dictated by the maximum and minimum achievable values of the normalized The main purpose of this section is to analyze in detail the impedance. For instance, if the impedance span is delimited first term of the right-hand side member of (4), d/ds, in by Zs = 25  (minimum value) and Zs = 150  (maximum order to determine the optimum design parameters (Z, Zs,  value), and Z0 = 50 , the upper limit of the magnitude of the and s) for sensitivity optimization. However, we will also sensitivity is Ss,max = 6 [achieved if Zs = 150 , i.e., ̅̅̅ 𝑍𝑠 = obtain analytically the last derivative in (4), by considering a 3, and s = (2n + 1)/2]. Figure 2 depicts a plot of the semi-infinite MUT. This is important since the transverse sensitivity (expression 8) as a function of ̅̅̅𝑍𝑠 and s, where geometry of the line has influence on this term, but it also the above-cited guidelines for sensitivity optimization are determines the characteristic impedance of the sensing line, confirmed. It can also be concluded that for ̅̅̅ 𝑍𝑠 > 1, the thereby influencing the first term. Let us first carry out the sensitivity is maximized and minimized for s = (2n + 1)/2 sensitivity analysis corresponding to the first term of (4), and s = n, respectively, whereas for ̅̅̅ 𝑍𝑠 < 1, s = n d/ds, by considering a uniform (not necessarily matched) provides the maximum value of the sensitivity, and the sensing line directly connected to the input port (i.e., without sensitivity is a minimum for s = (2n + 1)/2. the presence of the design line, or the discontinuity), and then the general case of the structure with step-impedance discontinuity (Fig. 1). The last subsection will be devoted to the analysis of the last term in (4). A. Sensitivity Analysis for a Uniform and Mismatched Sensing Line Directly Connected to the Input Port By eliminating the design line (equivalent to consider the general case with  = 0), the impedance seen from the input port and the reflection coefficient referred to Z0 (the reference impedance of the port), are given by [43] 𝑍𝑖𝑛 = −𝑗𝑍𝑠 cot 𝑠 (5) and +𝑍 +𝑗𝑍 cot  Fig. 2. Plot of the sensitivity of the phase of the reflection coefficient with  = −𝑍0+𝑗𝑍𝑠 cot 𝑠 (6) the phase of the sensing line, for the case of the sensing line directly 0 𝑠 𝑠 connected to the input port. Note that the sensitivity takes negative values, but its magnitude is maximized, or minimized, according to the line respectively. The phase of the reflection coefficient is thus impedance and electrical length combinations indicated in the text. 𝑍𝑠 cot 𝑠  = 2arctan ( ) (7) 𝑍0 B. Sensitivity Analysis for an Open-Ended Line with a Step-Impedance Discontinuity (General Case) and the sensitivity of  with s, designated as Ss, is found to be In this subsection, the main objective is to explore the 𝑑  possibility of further enhancing the sensitivity by adding 2 𝑆𝑠 = = − 𝑍0 𝑍 (8) degrees of freedom to the phase-variation sensor. For that 𝑑 𝑠 sin2 𝑠 + 𝑠 cos 2𝑠 𝑍𝑠 𝑍0 purpose, the structure of Fig. 1, with a step-impedance Let us now calculate the values of s that maximize, or discontinuity achieved by cascading a transmission line minimize, the sensitivity. For that purpose, the derivative of section between the sensing line and the input port, is (8) is calculated. The result can be expressed as considered. In this case, the impedance seen from the input port is 𝑍 𝑍 𝑑𝑆 4( 0 − 𝑠 ) sin 𝑠 cos 𝑠 𝑍𝑠 𝑍0 𝑠 = (9) 𝑗𝑍(𝑍 tan −𝑍𝑠 cot 𝑠 ) 𝑑 𝑠 𝑍 𝑍 ( 0 sin2 𝑠 + 𝑠 cos 2 𝑠 ) 2 𝑍𝑖𝑛 = (11) 𝑍𝑠 𝑍0 𝑍+𝑍𝑠 cot 𝑠 tan  whereas the reflection coefficient is found to be In view of expressions (19), it follows that if Z < Z0, the −𝑍0 (𝑍+𝑍𝑠 tan  cot 𝑠 )+𝑗𝑍(𝑍tan −𝑍𝑠 cot 𝑠 ) sensitivity is larger for 2, and it is larger for 1 if Z > Z0. Let  = +𝑍 (12) us now calculate the derivative of expressions (19) with s, 0 (𝑍+𝑍𝑠 tan  cot 𝑠 )+𝑗𝑍(𝑍 tan −𝑍𝑠 cot 𝑠 ) in order to calculate the value of s that maximizes, or and the phase of the reflection coefficient is minimizes, these values. The following values are obtained: 𝑍(𝑍𝑠 cot 𝑠 −𝑍tan )  = 2arctan (𝑍 ) (13) 𝑑𝑆 | 𝑠  4𝑍 2 𝑍𝑠 (𝑍 2 −𝑍𝑠2 ) sin 𝑠 cos 𝑠 0 (𝑍+𝑍𝑠 tan  cot 𝑠 ) 1 = 2 (20a) 𝑑 𝑠 𝑍0 (𝑍 2 sin2 𝑠 +𝑍𝑠2 cos 2 𝑠 ) Let us now evaluate the sensitivity as defined in (8). After some simple calculation, the following result is obtained: 𝑑𝑆 | 𝑠  2 (𝑍 2 −𝑍𝑠2 ) sin 𝑠 cos 𝑠 = 4𝑍0 𝑍𝑠 2 (20b) −2𝑍 2 𝑍𝑠 𝑍0 (1+tan2 ) 𝑑 𝑠 (𝑍 2 sin2 𝑠 +𝑍𝑠2 cos 2 𝑠 ) 𝑆𝑠 = 2 2 (14) 𝑍02 (𝑍 sin 𝑠 +𝑍𝑠 tan  cos 𝑠 ) +𝑍 2 (𝑍 tan  sin 𝑠 −𝑍𝑠 cos 𝑠 ) and for both equations, the derivative is null if s = n, or s Note that if  = 0, equation (14) simplifies to expression (8), = (2n + 1)/2. as expected. Concerning (20a), if s = (2n + 1)/2, then 1 = n, Since the main purpose of the analysis is to find the according to (18). Conversely, if s = n, then it follows that conditions for sensitivity optimization by considering the 1 = (2n + 1)/2. The sensitivity in the former case is addition of the design line, it is pertinent to calculate the −2𝑍𝑠 𝑆𝑠 = ̅̅̅𝑠 = −2𝑍 (21) derivative of Ss with the electrical length of this line, . By 𝑍0 forcing the resulting expression to be zero, the value of  and the following value results for s = n, and 1 = (2n + providing the maximum value of the sensitivity (provided the 1)/2: other design values, i.e., s, Zs, and Z, are fixed) is expected to be found. However, in order to simplify the calculation, −2𝑍 2 𝑍̅2 𝑆𝑠 = = −2 ̅̅̅ (22) we have inferred the derivative with respect to tan. The 𝑍0 𝑍𝑠 𝑍𝑠 following result has been found: It is clear in view of (21) and (22) that for Z > Z0 > Zs, (22) is 𝑑𝑆𝑠 2 (𝑍02 𝑍𝐴 2 2 +𝑍 2 𝑍𝐵 )𝑍 𝑍𝑠 𝑍0 tan −𝑍 2 𝑍𝑠𝑍0 (1+tan2 )(𝑍02 𝑍𝐴 𝑍𝑠 cos 𝑠 +𝑍 2 𝑍𝐵 𝑍 sin 𝑠 ) a maximum and (21) is a minimum. = −4 𝑑 tan  2 +𝑍 2 𝑍 2 ) (𝑍02 𝑍𝐴 𝐵 2 Concerning (20b), if s = (2n + 1)/2, then 2 = (2n + (15) 1)/2, and 2 = n if s = n. The sensitivity in the former where we have used ZA and ZB, defined as case is −2𝑍0 𝑍𝑠 ̅̅̅𝑠 𝑍 𝑍𝐴 = 𝑍 sin 𝑠 + 𝑍𝑠 tan  cos 𝑠 (16a) 𝑆𝑠 = 𝑍2 = −2 𝑍̅ 2 (23) 𝑍𝐵 = 𝑍 tan  sin 𝑠 − 𝑍𝑠 cos 𝑠 (16b) For 2 = n and s = n, the sensitivity is found to be: −2𝑍0 1 for simplification purposes. The zero (or zeros) in (15) are 𝑆𝑠 = = −2 ̅̅̅ (24) 𝑍𝑠 𝑍𝑠 given by those values of tan that null the numerator. Thus, rearranging the numerator and forcing it to be zero, the Inspection of (23) and (24) reveals that for Z < Z0 < Zs, (23) following second order equation (tan  being the unknown) is a maximum and (24) is a minimum. results: According to the previous analysis, it can be concluded that for sensitivity optimization, there are two optimum 𝑍𝑍𝑠 sin 𝑠 cos 𝑠 tan2  + (𝑍 2 sin2 𝑠 − 𝑍𝑠2 cos 2 𝑠 ) tan  − combinations regarding the electrical lengths of the sensing 𝑍𝑍𝑠 sin 𝑠 cos 𝑠 = 0 (17) and design lines (plus the corresponding multiples): and the two solutions ( = 1 and  = 2) are  Case A: s = (2n + 1)/2 and  = (2n + 1)/2. In this 𝑍𝑠 cos 𝑠 case, the line impedances should satisfy Z < Z0 < Zs, and tan 1 = (18a) the value of the sensitivity is given by (23), 𝑍 sin 𝑠 corresponding to a maximum. Moreover, the lower the 𝑍 sin 𝑠 tan 2 = − (18b) impedance contrast, defined as the ratio between the 𝑍𝑠 cos 𝑠 characteristic impedances of the design and sensing Thus, for 1 and 2 given by (18), the sensitivity (calculated line, the higher the sensitivity. using 14), should be a local (or absolute) maximum or  Case B: s = n and  = (2n + 1)/2. In this case, the minimum. The specific values are: line impedances should satisfy Z > Z0 > Zs, and the value 2𝑍 2 𝑍𝑠 1 of the sensitivity is given by (22), also corresponding to 𝑆𝑠 | =− (19a) a maximum. A high impedance contrast enhances the 1 𝑍0 𝑍 2 sin2 𝑠 +𝑍𝑠2 cos 2 𝑠 sensitivity, in this case. 1 𝑆𝑠 | = −2𝑍0 𝑍𝑠 (19b) It is interesting to compare the values of the maximum 2 𝑍 2 sin2 𝑠 +𝑍𝑠2 cos 2 𝑠 sensitivities, given by (22) and (23) for the s = n and s = Nevertheless, in order to be sure that the convenient value of (2n + 1)/2 sensing lines, respectively, with the  for sensitivity optimization is either 1 or 2, it should be corresponding values without design line, given by (10a) and verified that Ss does not exhibit poles (zeros in the (10b). The ratio for s = (2n + 1)/2 is 1/𝑍̅ 2 , whereas it is 𝑍̅ 2 denominator). Inspection of (14) reveals that the for s = n. According to these values, the sensitivity can be denominator cannot be zero, unless both squared terms are substantially enhanced by including the so-called design line simultaneously null, and this is not possible. with the adequate impedance value (low for s = (2n + 1)/2, and high for s = n). The reason is that the dependence of C. Sensitivity of the Phase of the Sensing Line with the such ratios with the normalized impedance of the design line Dielectric Constant of the MUT is squared. For the calculation of the last derivative in (4), dS/dMUT, Figure 3 plots the dependence of the sensitivity Ss with s let us first express the phase of the sensing line in terms of and , calculated by means of expression (14), for two cases: the effective dielectric constant of the line, eff, i.e., (i) Z < Z0 < Zs, and (ii) Z > Z0 > Zs. The figure confirms that 𝑙 𝑠 = 𝑠 √𝑒𝑓𝑓 𝑐 (25) the sensitivity is optimum (maximum) for the phase combinations predicted by the analysis. It is also interesting In (25), ls is the length of the sensing line,  is the angular to mention that for any value of s, there are two values of  frequency, and c is the speed of light in vacuum. By in the interval [0-](designated as 1 and 2 in expression considering a semi-infinite MUT (in the vertical direction) 18), where the sensitivity is either a maximum or a minimum. on top of the sensing line, the effective dielectric constant can The positions of the maxima in the -s plane are indicated be expressed as [43] in Fig. 3 (blue regions). Note that these positions converge to 𝑟+𝑀𝑈𝑇 𝑟−𝑀𝑈𝑇 the canonical values if the electrical length of the sensing line 𝑒𝑓𝑓 = 2 + 2 𝐹 (26) is either s = n or s = (2n + 1)/2. where r is the dielectric constant of the substrate and F is a The maximum value of the magnitude of the sensitivity for geometry factor given by the two cases considered in Fig. 3, given by equation (22) for −1/2 ℎ Z = 150  and Zs = 25 , and by equation (23) for Z = 25  𝐹 = (1 + 12 ) (27a) 𝑊𝑠 and Zs = 150 , are found to be Ss= 36 and Ss= 24, respectively. With these high values of the sensitivity of the for Ws/h ≥ 1, or by phase of the reflection coefficient with the phase of the −1/2 ℎ 𝑊𝑠 2 sensing line, it is expected that the overall sensitivity, given 𝐹 = (1 + 12 ) + 0.04 (1 − ) (27b) 𝑊𝑠 ℎ by expression (4), can be significantly optimized. Interestingly, these high sensitivity values have been for Ws/h < 1. In (27), h and Ws are the substrate thickness and achieved with perfectly implementable transmission lines the width of the sensing line, respectively, and it is assumed (i.e., the considered high, 150 Ω, and low, 25 Ω, that t << h, where t is the thickness of the metallic layer. In characteristic impedances correspond to implementable line practice, if the MUT is thick enough, so that the widths, at least in microstrip technology and line electromagnetic field generated by the line does not reach the implementation in most commercial microwave substrates). MUT-air interface, the MUT can be considered to be semi- 𝑆𝑠 < 𝑆𝑠 | infinite, and equation (26) can be used for analysis purposes. 𝑧𝑠=𝑍0 The last term in (4) can be expressed as 𝑑 𝑠 𝑑 𝑠 𝑑𝑒𝑓𝑓 𝑙𝑠 = = (1 − 𝐹) (28) 𝑑 𝑀𝑈𝑇 𝑑 𝑒𝑓𝑓 𝑑 𝑀𝑈𝑇 4𝑐√𝑒𝑓𝑓 and introducing (26) in (28), after some simple algebra one obtains 𝑑 𝑠 𝑙𝑠 1 = (29) 𝑑 𝑀𝑈𝑇 2√2𝑐 √ 1+𝐹 1 𝑟 (1−𝐹)2+𝑀𝑈𝑇 1−𝐹 It is apparent by inspection of (29) that operating at high frequency and/or choosing a long sensing line favors sensitivity (as it was anticipated before, and it is well known). (a) For sensitivity optimization, it is also convenient to select the substrate with the smallest possible dielectric constant. The 𝑆𝑠 < 𝑆𝑠 | 𝑧𝑠=𝑍0 effects of the transverse geometry of the line are included in F. From (27), it follows that F decreases by increasing h or by decreasing Ws. A decrease in F favors the sensitivity, as it can be easily deduced from (29). Therefore, for sensitivity optimization it is convenient to deal with high impedance sensing lines, as far as for the implementation of these lines, either narrow line widths and/or or thick substrates are needed. According to these comments, the preferred option for the implementation of the sensors is to choose the line impedances satisfying Z < Z0 < Zs, rather than Z > Z0 > Zs. However, the final decision should be taken by considering the maximum and minimum achievable line impedances (b) (determined by the substrate in use and by the available Fig. 3. Plot of the sensitivity of the phase of the reflection coefficient with fabrication technology), and evaluating the whole sensitivity the phase of the sensing line, as a function of the electrical lengths of the using (4), (29), and either (22) or (23) depending on each sensing and design lines, for two cases: (a) Z = 150  and Zs = 25 ; (b) case. Z = 25  and Zs = 150 . The reference impedance of the port is Z0 = 50 . In this case, the sensitivity is presented in level charts, rather than three- It should also be mentioned that if the thickness of the MUT dimensional views, for better comprehension. does not obey the semi-infinite requirement criterion, expression (29) cannot be considered to be valid, and thereby high sensitivity criterion of 10(a)]. Finally, the last two it cannot be used to predict the overall sensitivity of the sensors are simply a 180º and 90º uniform and matched (50- sensor. Nevertheless, the conclusions of the previous Ω) lines. These four sensors are also depicted in Fig. 4, where paragraph, on the effects of the substrate characteristics the inclusion of the 50-Ω access lines can be appreciated. The (dielectric constant) and transverse geometry of the line on lengths of these access lines have been chosen in order to sensitivity, do hold. achieve sensors with comparable lengths, i.e, for those with identical sensing lines. The sensing regions are indicated by IV. EXPERIMENTAL VALIDATION the dashed rectangles. For the six fabricated sensors, we have placed on top of the Experimental validation of the theory of the preceding sensing areas several dielectric slabs with different dielectric section is carried out by considering two prototype sensors constants. Such slabs are uncoated microwave substrates based on a step-impedance discontinuity. In one case, the line with well-known dielectric constant. Their thickness is impedances and electrical lengths satisfy Z < Z0 < Zs, s = (2n roughly 3 mm (achieved by stacking up two 1.5-mm slabs). + 1)/2 and  = (2n + 1)/2 (case A of the previous Section). With this thickness, the MUT can be considered to be semi- For the second prototype, the transmission line sections are infinite in the vertical direction, as it has been inferred from chosen following the guidelines corresponding to case B of independent simulations (not shown), indicating that the Section III, i.e., Z > Z0 > Zs, s = n and  = (2n + 1)/2. phase of the reflection coefficient does not significantly vary Moreover, for comparison purposes, a set of four sensors for thicker MUT slabs. Figure 5 depicts the variation of the without the presence of the design line (i.e., without step- phase of the reflection coefficient experienced by the impedance discontinuity), and lengths comparable to those different sensors for the different dielectric loads (indicated of the step-impedance discontinuity based sensors, are also in the caption of Fig. 5). The measurement of the reflection designed and fabricated. All sensors are fabricated on the coefficient has been inferred by means of the Keysight Rogers RO4003C substrate with dielectric constant r = 3.55, 85072A vector network analyzer. It can be seen that the thickness h = 1.524 mm and loss factor tan= 0.0022 (the sensors with step-impedance discontinuity exhibit stronger prototypes are fabricated by means of the LPKF H100 variation of the phase of the reflection coefficient with the drilling machine), and the frequency of operation is set to dielectric constant of the material under test. By contrast, f = 2 GHz. sensors of Fig. 4(e) and Fig. 4(f), based on a uniform matched For case A, the line impedances are set to Z = 25Ω and line, exhibit the softer dependence. Thus, these results point Zs = 150 Ω (with Z0 = 50 Ω). With these values, the width of out the potential of step-impedance lines for sensitivity the lines, inferred from the transmission line calculator enhancement, provided the electrical length and included in Keysight ADS, have been found to be Ws = characteristic impedance of the line sections is adequately 0.242 mm and W = 9.1 mm, where W is the width of the selected. design line. The electrical lengths of both the design and sensing line are set to  = s = 90º, in order to implement the sensor with the minimum physical length. Note that the sensitivity can be improved by elongating the sensing line (considering an electrical length equal to an odd multiple of 90º). However, the objective in this paper is to demonstrate the potential for sensitivity improvement with the proposed strategy (step-impedance discontinuity), leading to high sensitivity phase-variation sensors compatible with small sizes. The calculated physical lengths of the design and sensing lines are l = 18.3 mm and ls = 23.65 mm, respectively (note that these lengths are unequal since the phase velocities of the design and sensing lines are not identical). The photograph of the designed sensor is depicted in Fig. 4(a), where it can be appreciated that a 50-Ω access line (with length of 10 mm) has been added for connector soldering (the Fig. 4. Photographs of the fabricated reflective-mode phase-variation width of this 50-Ω line is 3.42 mm). sensors. (a) Sensor with step-impedance discontinuity and Z < Z0 < Zs; (b) The second sensor (case B) has been designed with the sensor with step-impedance discontinuity and Z > Z0 > Zs; (c) sensor with uniform mismatched sensing line and Zs > Z0; (d) sensor with uniform following values of characteristic impedances and electrical mismatched sensing line and Zs < Z0; (e) sensor based on a 90º uniform 50- lengths: Z = 150Ω, Zs = 25 Ω,  = 90º, and s = 180º. Line Ω sensing line and (f) sensor based on a 180º uniform 50-Ω sensing line. widths and lengths are in this case W = 0.242 mm, Ws = Figure 5 also includes the simulation results, inferred by 9.1 mm, l = 20.7 mm and ls = 42.1 mm. The photograph of means of the Ansys HFSS commercial software. The this prototype sensor is shown in Fig. 4(b). agreement with the measured data is very good. In order to Finally, an additional set of four sensors, based on uniform validate the sensitivity analysis carried out in Section III, we lines, has been designed according to the following have included further simulated data points by considering considerations. One sensor is implemented by means of a additional hypothetical materials with different dielectric s = 90º sensing line with high characteristic impedance constants in the range between MUT = 1 to MUT = 10.5. From (Zs = 150 Ω), in order to achieve good sensitivity [see 10(b)]. the simulated data points, we have obtained the sensitivity, Another sensor consists of a low-impedance (Zs = 25 Ω) also included in Fig. 5. The sensitivity in the limit when sensing line with electrical length s = 180º [according to the MUT = MUT  REF  0 (REF being the dielectric constant for this reason the sensitive is a maximum for MUT = 1. of vacuum, i.e., REF = 1) is indicated in Fig. 5 for each Nevertheless, highly sensitive dielectric constant sensor. We have calculated the sensitivity for MUT = 1 by measurements in the vicinity of a different reference value means of expression (4), where the last term is given by (29), are possible by simply forcing the sensing lines to exhibit the and the first term (Ss) is obtained by means of (3), (10a), required electrical length (90º or 180º) at the operating (10b), (22) or (23), depending on the specific sensor. These frequency and the required value of Zs when they are loaded theoretical sensitivities, designated as Sth, are also indicated with the reference sample. For instance, by considering the in Fig. 5. As it can be seen, the agreement with the Rogers RO4003C with dielectric constant 3.55 as reference sensitivities inferred from the simulated data points, S, is sample, it is expected that the phase of the reflection very good, thereby validating the sensitivity analysis coefficient for the different considered MUT can be clearly developed in the preceding section. distinguished (nevertheless, further designs and validations It is remarkable that the sensitivity for small perturbations are out of the scope of this paper). is significantly enhanced by considering the step-impedance discontinuity. Note that for the sensor of Fig. 4(a), the V. GENERALIZATION OF THE CONCEPT AND DISCUSSION sensitivity is roughly 4 times larger than the one of the sensor It has been theoretically demonstrated and experimentally of Fig. 4(c), with identical sensing line, and 19.7 times larger validated in this paper that a step-impedance discontinuity is than the one of the sensor of Fig. 4(e). On the other hand, the useful to increase the sensitivity in these reflective-mode sensor of Fig. 4(b) exhibits a sensitivity better than the one phase-variation sensors. Moreover, it has been shown that of the sensors of Figs. 4(d) and 4(f) by a factor of 9 and 11.4, the electrical length of the design line must be set to  = (2n respectively. + 1)/2, whereas either a s = (2n + 1)/2 or a s = n 0 sensing line is needed for that purpose (in each case, with the convenient impedance contrast, as discussed in Section III). -30 In this section, it is shown that the sensitivity of the phase of -60 the reflection coefficient with the phase of the sensing line (and thereby the overall sensitivity) for small perturbations  (º) -90 -120 can be further improved by considering a multi-step- impedance transmission line, as the one shown in Fig. 6(a). -150 It is a generalization of the structure of Fig. 1, based on a -180 cascade of high and low impedance transmission line -210 sections of identical electrical length, i.e.,  = (2n + 1)/2, 0 S = 9.4º ; Sth = 9.4º followed by the sensing line (with electrical length s and characteristic impedance Zs). The characteristic impedances -50 S = 19.4º ; Sth = 19.7º of the high or low impedance transmission line sections are S = 24.3º ; Sth = 24.7º (a) -100 S = 45.9º ; Sth = 46.5º (b) designated by Zi,, where i = 1,2,…N denotes the specific S (º) (c) (d) section, and N is the total number of Sections (e.g., N = 1 for (e) -150 (f) the structure of Fig. 1). S = 185.4º ; Sth = 186.1º According to Fig. 6, the impedance seen from the -200 EM Simulation discontinuity in contact with the sensing line, looking at the S = 220.8º ; Sth = 222.1º Measured -250 open end, is called Zin,s, and it is given by expression (5). The 1 2 3 4 5 6 7 8 9 10 11 impedance seen from any other discontinuity looking at the MUT open end of the structure, Zin,i, can be expressed in terms of Fig. 5. Measured and simulated phase of the reflection coefficient for the sensors of Fig. 4, and simulated sensitivity. (a) Sensor with step-impedance the impedance seen from the previous discontinuity as discontinuity and Z < Z0 < Zs; (b) sensor with step-impedance discontinuity 𝑍𝑖2 and Z > Z0 > Zs; (c) sensor with uniform mismatched sensing line and Zs > 𝑍𝑖𝑛,𝑖 = (30) 𝑍𝑖𝑛,𝑖−1 Z0; (d) sensor with uniform mismatched sensing line and Zs < Z0; (e) sensor based on a 90º uniform 50-Ω sensing line and (f), sensor based on a 180º uniform 50-Ω sensing line. The measured dielectric loads are 3-mm slabs of from which it follows (using mathematical induction) that uncoated PLA (ɛMUT = 3), Rogers RO4003C (ɛMUT = 3.55), FR4 (ɛMUT = 4.4) the impedance seen from the input port can be expressed as and Rogers RO3010 (ɛMUT = 10.2) substrates. The sensitivities in the limit of 𝑁 {𝑍𝑖2 (−1)𝑁 (−1)𝑖+𝑁 small perturbations are given in absolute value. 𝑍𝑖𝑛,𝑁 = 𝑍𝑖𝑛,𝑠  ∏ } (31) It can be seen in Fig. 5 that as MUT increases, the magnitude 𝑖=1 of the sensitivity decreases, and such decrement is more or (using 5) pronounced for the sensors exhibiting the larger sensitivity (−1)𝑁 𝑁 {𝑍𝑖2 𝑁 (−1)𝑖+𝑁 for small perturbations, i.e., those including the step- 𝑍𝑖𝑛,𝑁 = (−𝑗𝑍𝑠 )(−1) (cot 𝑠 ) ∏ }(32) 𝑖=1 impedance discontinuity. These sensors exhibit smaller linearity, as compared to those based on uniform matched In (31) and (32), the symbol  denotes the product operator. lines, but they are able to detect tiny variations of the The reflection coefficient seen from the input port is thus dielectric constant as compared to the one of a reference (−1)𝑁 𝑗(−1)𝑁+1 (𝑍𝑠 cot 𝑠 )  ∏ −𝑍0 material by virtue of their high sensitivity. Since the sensing = (−1)𝑁 (33) lines have been designed in order to exhibit either 90º or 180º 𝑗(−1)𝑁+1 (𝑍𝑠 cot 𝑠 )  ∏ +𝑍0 by considering that they are unloaded, it follows that the and the phase of the reflection coefficient is given by reference dielectric constant in our study is the one of air, and (−1)𝑁 (𝑍𝑠 cot 𝑠 ) (−1)𝑁 ∏ expressions (36). Particularly, for case A’, with the product  = 2arctan ( 𝑍0 ) (34) operator in the denominator, the odd-order transmission line sections must exhibit low impedance values, whereas high Note that in (33) and (34) the argument of the product characteristic impedance sections are required for the even operator has been omitted for simplicity. After some simple sections. The opposite conditions apply for case B’, as far as (but tedious) algebra, the sensitivity of  with s, designated the product operator appears in the numerator of (36b). For as Ss in (8), is found to be N even (cases C’ and D’) and i odd, the impedance is negative 2 squared (𝑍𝑖−2 ), whereas it appears as 𝑍𝑖2 for N even and i 𝑆𝑠 = − 𝑍0 (sin 𝑠 ) (−1)𝑁 +1 𝑁 ∏  (𝑍𝑠 )(−1) (cos 𝑠 ) (−1)𝑁 +1 even. Consequently, for case C’, with the product operator in (−1) 𝑁 (−1) 𝑁 −1 + 𝑍  (−1) 𝑁 −1 the numerator of (36c), the odd sections must exhibit low ∏  (𝑍𝑠 ) (cos 𝑠 ) 0 (sin 𝑠 ) (35) characteristic impedance, and the line impedance must be high for the even sections. Finally, it is obvious that for case and it takes a maximum, or a minimum, value, depending on D’ (half-wavelength sensing line), the sections that should the set of impedances Zi, when the electrical length of the exhibit high impedance for sensitivity optimization are those sensing line is either s = (2n + 1)/2 or s = n, as with odd index. anticipated before. In order to evaluate (35) for these specific From the previous analysis, it can be concluded that for values of s, it is necessary to distinguish if the number of sensitivity optimization, regardless of the number of sections sections, N, is even or odd. Consequently, four different N, if the electrical length of the sensing line is s = (2n + cases appear, and, for each case, the sensitivity is found to 1)/2, the impedance of this line must be high, and the be: impedance of the cascaded quarter-wavelength sections must  Case A’: s = (2n + 1)/2 and N odd. alternatively exhibit low and high values (with a low 2𝑍𝑠 𝑍0 impedance value for the section adjacent to the sensing line, 𝑆𝑠 = − (36a) 𝑁 2(−1)𝑖+𝑁 i.e., the one with i = 1). For s = n, the sensing line must be ∏ {𝑍𝑖 } 𝑖=1 a low-impedance line, a high-impedance line is required for the first section (i = 1), a low-impedance line for the second  Case B’: s = n and N odd. section, and so on. Thus, it is clear that a non-uniform 𝑁 2(−1)𝑖+𝑁 stepped-impedance transmission line based on quarter 2 ∏ {𝑍𝑖 } 𝑖=1 wavelength sections, cascaded to an open-ended quarter- or 𝑆𝑠 = − (36b) 𝑍𝑠 𝑍0 half-wavelength line, is a very useful structure for sensitivity  Case C’: s = (2n + 1)/2 and N even. enhancement in phase-variation sensors. Note that, with the 𝑁 reported approach, the sensitivity can be as high as desired, 2(−1)𝑖+𝑁 without the need to increase the length of the sensing line (as 2𝑍𝑠  ∏ {𝑍𝑖 } 𝑆𝑠 = − 𝑖=1 (36c) occurs in typical phase-variation sensors [20],[25]). For that 𝑍0 purpose, it suffices to include the necessary number of  Case D’: s = n and N even. sections (with as much contrast of impedance as possible) to 2𝑍0 obtain the required (high or low) value of the product 𝑆𝑠 = − 𝑁 (36d) 𝑍𝑠  ∏ 2(−1)𝑖+𝑁 {𝑍𝑖 } operator in (36). 𝑖=1 Note that the effects of losses on sensor performance (sensitivity) have not been considered so far. From the Note that for N = 1 (cases A’ and B’), expressions (36a) and analysis developed in detail in Appendix B, it can be (36b) coincide with expressions (23) and (22), respectively, concluded that the sensitivity is not substantially degraded, as expected, since the multi-stepped impedance transmission at least under the low-loss approximation (an approximation line for this case is the one of Fig. 1. valid for many MUTs). It should also be clarified that Inspection of expressions (36) reveals that Zs appears in the although in the Appendix B a method for estimating the loss numerator for s = (2n + 1)/2 and it appears in the tangent of the MUT is suggested (see last part), resonant denominator for s = n. Thus, it can be concluded that for methods are, in general, preferred for that purpose. In the quarter-wavelength (or odd multiple) sensing lines, a high present study, validation of the theoretical analysis is carried characteristic impedance is required for sensitivity out by considering samples with different dielectric optimization (regardless of the number of sections of the constants, and also with different loss tangents. Nevertheless, structure). Conversely, low impedance values are needed in the considered input variable is the dielectric constant of the half-wavelength (or multiple) sensing lines. To infer the MUT. Note, however, that the proposed phase-variation effects of the characteristic impedances of the quarter- sensors can be applied to material analysis and composition, wavelength transmission line sections, Zi (with i = 1,2…N), quality control processes, or to defect detection in samples, on the sensitivity, we should analyze carefully the product since changes in the dielectric constant of the MUT are operator that appears in expressions (36). For N odd (cases involved in these applications. Through a proper design, the A’ and B’), it follows that the characteristic impedance of a proposed sensing strategy may be also useful for the section with odd order (i.e., with i odd) appears as 𝑍𝑖2 , characterization of liquid composition (e.g., determination of whereas for an even-order section the corresponding term in solute content in liquid solutions, etc.). Nevertheless, this is the product is 𝑍𝑖−2. Under these circumstances, the out of the scope of the present paper. requirement of a high or low value of Zi for sensitivity Despite the fact that the reported experimental results of optimization depends on whether the product operator is Section IV (with N = 1) are indicative of the potential of the present either in the numerator or in the denominator in approach for sensitivity enhancement, let us consider a different air gaps in the interval from g = 0 mm to g = 3 mm further example, corresponding to case C’ with N = 2. The (far enough so as to consider that the electromagnetic field impedances are set to Zs = 150 , Z1 = 25  and Z2 = 86.6  generated by the line does not reach the MUT). From the (the operating frequency and substrate characteristics are output variable, , resulting for each air gap distance (Fig. identical to those of the prototypes of Section IV). The 8), the equivalent dielectric constant, MUT,eq, can be impedance Z2 has been chosen in order to obtain Ss = 72, i.e., determined by reading it from Fig. 7 (i.e., using the simulated 3 times the sensitivity of the previous structure with Zs = 150 phase variation of Fig. 8, Fig. 7 provides MUT,eq for each  and Z = 25  (N = 1). Figure 6(b) shows the photograph simulated point). Thus, we can generate a correspondence of the fabricated prototype. between the air gap g, and the equivalent dielectric constant MUT,eq. Such dependence is given in the inset of Fig. 9, where the phase dependence with MUT,eq for the experimental data (squared symbols) and with MUT for the simulations is depicted. The agreement is reasonably good taking into account the difficulty in controlling the air gap distance for small values. In addition, in this experiment, similar to the (a) previous experimental campaigns, two slabs have been stacked up. By screwing the whole MUT against the sensing region, as done in the previous measurements, the presence of a thin air layer between the slabs is minimized. However, this has not been done in the measurements of Fig. 8, because the MUT is not in contact with the sensing region. (b) 0 0 Fig. 6. General schematic/topology of the N-sections reflective-mode phase- -50 -100 variation sensor (a) and photograph of the fabricated device with N = 2 (b). Dimensions (in mm) are ws = 0.242, ls = 23.65, w1 = 9.1, l1 = 18.3, w2 = 1.22 -100 -200  (º) S (º) and l2 = 21.5. -150 -300 The phase variations of the reflection coefficient, in -200 -400 reference to the one of the bare sensor, inferred from full wave electromagnetic simulation by considering MUTs with -250 -500 S = 528.7º ; Sth = 558º dielectric constants varying in the interval 1 to 4, are depicted -300 -600 in Fig. 7. From these simulated data, the sensitivity in the 1.0 1.5 2.0 2.5 3.0 3.5 4.0 MUT low-perturbation limit (MUT  1) is found to be S= 528.7º, by far superior to those achieved in the structures with N = 1 Fig. 7. Simulated phase of the reflection coefficient for different values of of Section IV. As indicated before, with the considered MUT for the sensor of Fig. 6, and simulated sensitivity. The theoretical sensitivity, designated as Sth, is also indicated. The sensitivities in the limit impedance values, the sensitivity of the phase of the of small perturbations are given in absolute value. reflection coefficient with the phase of the sensing line (calculated from 36c) is found to be Ss = 72. The sensitivity 0 of s with MUT was calculated in the previous section (i.e., -20 ds/dMUT = 7.75º), since in one of the prototypes of that -40 section the sensing line is identical to the one considered in -60  (º) the sensor of Fig. 6. From the product (see 4), the total -80 sensitivity is found to be Sth= 558º, in reasonable agreement -100 with the result inferred from full wave simulation. The -120 discrepancy is mainly attributed to the excessive variation of -140  with MUT, which limits the accurate calculation of the -160 -180 sensitivity from the derivative of a set of discrete data points. 3.0 2.5 2.0 1.5 1.0 0.5 0.0 Nevertheless, potential non-ideality effects caused by the g (mm) step discontinuities, not considered in the theoretical analysis, may play also a role. Fig. 8. Measured phase of the reflection coefficient for the sensor of Fig. 6, by loading the sensing region with a MUT separated a variable distance g. Due to the lack of available samples with MUT in the high The considered MUT is the uncoated Rogers RO4003C with thickness h = 3 sensitive region (MUT  1), we have carried out an mm and dielectric constant MUT = 3.55. experiment consisting of obtaining the phase of the reflection coefficient for a MUT with dielectric constant MUT = 3.55 (the uncoated Rogers RO4003C substrate used before), but for different vertical distances (air gap, g) with regard to the sensing line. Note that this is equivalent to modify the dielectric constant of a hypothetical semi-infinite MUT in contact with the sensing line. Such equivalent dielectric constant, MUT,eq, can be estimated by simulation. For that purpose we have simulated the structure by considering 0 the magnitude of the transmission coefficient of the resulting two-port device). Nevertheless, the sensitivity enhancement -50 capability of the strategy proposed in this work is by far -100 superior to the one of [28]. In [17] and [27], the reported  (º) -150 3.5 3.0 sensors are also based on phase variation, although the output -200 variable is not the phase, but the magnitude of the MUT, eq 2.5 -250 2.0 transmission coefficient [17], or the magnitude of the cross- 1.5 1.0 EM Simulation mode transmission coefficient [27]. The sensitivity of such -300 Measured 0.0 0.5 1.0 1.5 g (mm) 2.0 2.5 3.0 sensors [17],[27] is high, thanks to the strong dispersion of -350 the considered sensing lines (left handed lines in [17] and 1.0 1.1 1.2 1.3 1.4 1.5 MUT electro-inductive-wave transmission lines in [27]). It is Fig. 9. Measured phase of the reflection coefficient for the sensor of Fig. 6, especially remarkable the sensitivity of the sensor in [17], but by loading the sensing region with a MUT separated a variable distance g. at the expense of a relatively large sensing area (not given in The considered MUT is the uncoated Rogers RO4003C with thickness h = [17]). However, further increasing the sensitivity requires the 3 mm (two stacked slabs) and dielectric constant MUT = 3.55. The equivalent dielectric constant for the measured data is given by g, according to the elongation of the sensing lines in both sensors. Moreover, the correspondence curve shown in the inset. For the simulated data, the robustness of these sensors against fabrication related horizontal axis is the dielectric constant of a hypothetical semi-infinite MUT tolerances is limited since either defect ground structures, in contact with the sensing region. In other words, the simulated curve is the one of Fig. 7, but restricted to a smaller range. vias, or other elements loading the lines are needed. By contrast, the phase-variation sensors reported in this paper Nevertheless, the sensor is able to detect tiny variations of are implemented by a mere cascade of alternating high/low MUT,eq in the vicinity of MUT,eq = 1, corresponding to air gap impedance transmission line sections. A comparison of these separations close to 3 mm. This experiment, emulating low sensors is summarized in Table I. perturbations in the dielectric constant of the MUT in the TABLE I highly sensitive region, validates the structure of Fig. 6 as a COMPARISON OF VARIOUS SENSORS BASED ON PHASE VARIATION sensor exhibiting very high sensitivity with relatively small Maximum Ref. Resolution** Size*** Frequency sensing region. Sensitivity* The results of these section and those of Section IV are [17] 600 dB 0.0017 N.A. 2.3 GHz indicative of the huge potential of these step-impedance [20] 54.85º 0.0911 2.5 cm2 10 GHz transmission line reflective-mode phase-variation sensors for [25] 415.6º 0.0120 105.6 cm2 6 GHz sensitivity enhancement, without the need to consider large [27] 25.33 dB 0.0395 44.55 cm2 4.6 GHz [28] 17.62 dB 0.0560 1.2 cm2 2 GHz sensing areas (or MUTs). Comparison of the reported phase- Proposed variation sensors with other sensors devoted to dielectric design 528.7º 0.0095 15.08 cm2 2 GHz characterization is not easy, since there are few sensors (Fig. 6) available in the literature whose working principle is phase * Units are degrees for sensors where the output variable is a phase, and units variation. In [20],[25], the reported phase-variation sensors are dB for sensors where the output variable is a transmission coefficient. ** are based on long (uniform) sensing lines (meandered in The resolution in the dielectric constant is calculated by assuming that either 5º or 1dB in the output variable (depending on the case) can be [25]), in order to achieve high sensitivity. In [25], the discriminated. sensitivity of the sensor (a differential-mode sensor operating *** The size corresponds to the sensing area. in transmission) is very good (S = 415.6º), but at the expense In summary, the reported sensors exhibit a good of a pair of meandered lines of length 67.03 cm (the reported combination of size (sensing region) and performance sensitivity of the sensor of [20] is worse, i.e., S = 54.85º). (sensitivity), they operate at a single frequency, they are Moreover, the sensing area for the MUT in the sensor of [25] based on a reflective-mode one-port structure, and, most is as large as 105.56 cm2. By contrast, the sensing area of the important, the sensitivity, the key sensor parameter, can be sensor reported in this section (Fig. 6) is 15.08 cm2, and the enhanced at wish, keeping unaltered the region devoted to reported sensitivity is as high as 528.7º (simulated). This the MUT. This later aspect is very relevant and a unique combination of sensitivity and sensing area is very feature of the reported sensors. Moreover, the part of the competitive. Moreover, the sensitivity can be further sensors excluding the sensing region is as simple as a stepped enhanced (maintaining the sensing region unaltered) by impedance transmission line consisting of cascaded quarter- simply considering additional sections, as indicated before wavelength transmission line sections. As it has been shown, (this interesting property is not present in the sensors few sections suffice to achieve very high sensitivity. reported in [20],[25]). It should also be mentioned that the sensing region of the device can actually be made much VI. CONCLUSIONS smaller, since a width of few times the width of the sensing line suffices. In conclusion, a strategy to significantly enhance the There are other sensors available in the literature where the sensitivity of reflective-mode phase-variation sensors based working principle is phase variation, but the output variable on open-ended lines has been reported in this paper. From a is not the phase [17],[27],[28]. In the recent paper [28], it was detailed sensitivity analysis, it is demonstrated that by demonstrated that by combining a pair of mirrored considering a step-impedance discontinuity, with the sensing meandered lines with a double rat-race structure region comprising the space between the discontinuity and conveniently designed, the sensitivity of the differential the open end, the sensitivity can be engineered. It is sensor can be enhanced (in that sensor, the output variable is concluded from such analysis that for sensitivity enhancement, the electrical length of the sensing line must coefficient, , with MUT should be expressed as be either a multiple of 180º, or an odd multiple of 90º, 𝑑  𝑑  𝑑 𝑠 𝑑  𝑑𝑍𝑠 whereas the line present between the discontinuity and the 𝑆= 𝑑 𝑀𝑈𝑇 =  𝑑 𝑠 𝑑 𝑀𝑈𝑇 + 𝑑𝑍𝑠 𝑑 𝑀𝑈𝑇  (A1) input port (design line) should exhibit a phase of 90º (or an odd multiple). Moreover, sensitivity is enhanced by However, by including the last term of the right-hand side impedance contrast. Particularly, high impedance sensing member of (A1), the phase and impedance conditions for line and low impedance design line is a necessary condition sensitivity optimization are roughly the same as those give in for sensitivity optimization in sensors with 90º sensing lines, the body of the text. Specifically, for the structure based only whereas in sensors with 180º sensing lines, the impedance on the sensing line, the sensitivity is optimized when the contrast should be the opposite one. For validation purposes, phase of such line is either s = n (with Zs low), or s = (2n a first set of six prototype sensors has been fabricated. Two + 1)/2 (with Zs high), with n = 0, 1, 2,... Moreover, under of them exhibit a step-impedance discontinuity and have such phase conditions, the derivative d/dZs is null. Hence, been designed following the design guidelines of the the sensitivity given by (4) provides an accurate result as far theoretical analysis. In two sensors, a uniform mismatched as it is calculated for the optimum phase conditions. Indeed, sensing line (with high impedance in one case and low for the different structures considered in this paper, the phase impedance in the other one) has been considered. Finally, the conditions that optimize d/ds are those that null d/dZs. other sensors have been implemented by means of uniform Consequently, the exhaustive study focused on d/ds 90º and 180 º matched lines. In all these sensors, the variation optimization, conducted in Section III and generalized in of the phase of the reflection coefficient (the output variable) Section V, is fully justified. Let us next demonstrate the with the dielectric constant of the material under test (MUT) previous assertions for the structure based simply on the has been inferred by considering different dielectric slabs. sensing line (and considering implementation in microstrip The sensor sensitivity, the key parameter, has been inferred technology). from simulations, since a large number of data points is The two terms of the first product in (A1) are given by (8) required for the accurate determination of such parameter. and (28), but they can be alternatively expressed as: From the results, it is concluded that the proposed strategy, 𝑑  𝑍 2 𝑠 sin−2 𝑠 𝑍0 based on the step-impedance discontinuity, is very useful for =− 2 (A2) 𝑑 𝑠 𝑍 1+( 𝑠 ) cot 2 𝑠 the implementation of highly sensitive phase-variation 𝑍0 sensors. In the last part of the paper, it has been demonstrated 𝑑 𝑠 𝑠 that the sensitivity can further be enhanced by simply = (1 − 𝐹) (A3) 𝑑 𝑀𝑈𝑇 4𝑒𝑓𝑓 cascading further quarter-wavelength transmission line sections with alternating high and low impedance. By this The two terms of the second product are easily inferred from means, the sensitivity can be increased at wish, maintaining (7) and from the expression providing the characteristic the sensing region unaltered. This is an essential and unique impedance of a microstrip line, i.e., property of these phase-variation sensors. The sensitivity 𝑑  2 cot 𝑠 = 2  (A4) achieved in the last designed and fabricated prototype, based 𝑑𝑍𝑠 𝑍 1+( 𝑠 ) cot 2 𝑠 𝑍0 𝑍0 on three quarter-wavelength step-impedance transmission 𝑑𝑍𝑠 𝑑𝑍𝑠 𝑑𝑒𝑓𝑓 𝑍𝑠 line sections (including the sensing line section), is S = = =− (1 − 𝐹) (A5) 𝑑 𝑀𝑈𝑇 𝑑 𝑒𝑓𝑓 𝑑 𝑀𝑈𝑇 4𝑒𝑓𝑓 528.7º. This is 56.2 times larger than the sensitivity achieved in the reflective-mode phase-variation sensor based on an Introducing (A2)-(A5) in (A1), the sensitivity is found to be ordinary and matched line with identical sensing line length. 𝑍𝑠 (1−𝐹) 𝑠 sin−2 𝑠 +cot 𝑠 These sensors may be of special interest in applications 𝑆=− 2𝑍0 𝑒𝑓𝑓  𝑍 2 (A6) related to the measurement of small changes in the dielectric 1+( 𝑠 ) cot 2 𝑠 𝑍0 constant of the MUT, including determination of material Let us now calculate the derivative of (A6) with respect to s, composition of solids and liquids (e.g. concentration of in order to obtain the values of s that maximize the solute in very diluted solutions), detection of defects in sensitivity. After some algebra, one obtains samples (as compared to a reference), sensing and 𝑍 2 𝑍 2 monitoring of industrial processes (e.g., wine fermentation), 𝑑𝑆 𝑠 𝑠 𝑍𝑠 (1−𝐹)sin−2 𝑠 cot 𝑠 {(𝑍0 ) ·cot 𝑠 +𝑠 (𝑍0 ) −𝑠 } etc. These applications are left for future works. In the 𝑑 𝑠 =− 𝑍0 𝑒𝑓𝑓  2 2 (A7) 𝑍 (1+( 𝑠 ) cot 2 𝑠 ) present paper, the main objective has been to demonstrate the 𝑍0 potential of these step-impedance reflective-mode phase- It is apparent that s = (2n + 1)/2 nulls (A7). For such phase variation sensors for sensitivity optimization, on the basis of of the sensing line, the sensitivity (A6) is a detailed sensitivity analysis, validated through simulation 𝑍𝑠 (1−𝐹) and simple experiments. 𝑆=− 2𝑍0 𝑒𝑓𝑓 𝑠 (A8) APPENDIX A This value is identical to the one inferred from (4), using EXACT CALCULATION OF THE SENSITIVITY (10b) and (28), and for such phase value the line impedance Actually, the variation of MUT modifies both the phase, s, Zs must be high in order to obtain high sensitivity. By and the characteristic impedance, Zs, of the sensing line. contrast, for s = n, equation (A7) is not strictly null. There Therefore, the sensitivity of the phase of the reflection is, however, an additional condition that nulls (A7), i.e., 𝑍2 𝑠 tan 𝑠 = 𝑍2 −𝑍 𝑠 2 (A9) 0 𝑠 and the solution of (A9) can be approximated by s = n as where  =  + j, and  and  are the attenuation constant far as Zs < Z0. Thus, it can be concluded that for Zs < Z0 there and the phase constant, respectively, of the sensing line. is a maximum of the sensitivity very close to s = n. After some simple algebra, (B.1) can be expressed as Consequently, it is reasonable to set the phase of the sensing sinh 2𝑙𝑠 −𝑗 sin 2𝑠 line to such value for low impedance sensing lines. For s = 𝑍𝑖𝑛,𝑠 = 𝑍𝑠 (B.2) cosh 2𝑙𝑠 −cos 2𝑠 n, the sensitivity is with s = ls. The impedance seen from the input port of the 𝑍0 (1−𝐹) 𝑆=− 2𝑍𝑠 𝑒𝑓𝑓 𝑠 (A10) whole sensor, Zin,N, is calculated by means of (31), with Zin,s given by (B.2). The reflection coefficient is thus also given by (4), but in this case utilizing (10a) and (28). For (−1)𝑁 sinh 2𝑙𝑠 −𝑗 sin 2𝑠 (𝑍𝑠 )  ∏ −𝑍0 both phases, s = n and s = (2n + 1)/2, equation (A4) is = cosh 2𝑙𝑠 −cos 2𝑠 (B.3) (−1)𝑁 null and, for that reason, the sensitivity can be calculated sinh 2𝑙𝑠 −𝑗 sin 2𝑠 (𝑍𝑠 cosh 2𝑙𝑠 −cos 2𝑠 )  ∏ +𝑍0 using the simplified expression (4). Despite the fact that the previous analysis has been carried where  is the product operator defined in (31) with the out on the basis of the structure of Section III.A, for the argument omitted, for simplicity. At this point, in order to structure of Section III.B, consisting of the sensing line plus calculate the phase of the reflection coefficient by an additional (design) line, a similar behavior occurs. considering losses, it is convenient to distinguish between Namely, for the optimum phase conditions, summarized in even- and odd-order structures. For N even, the reflection Section III.B, the derivative d/dZs is also null, and therefore coefficient can be expressed as the sensitivity can be calculated according to (4). Let us 𝑍 ∏ (sinh 2𝑙 −𝑗 sin 2 )−𝑍 (cosh 2𝑙 −cos 2 ) demonstrate this by calculating such derivative from (13).  = 𝑍𝑠 ∏ (sinh 2𝑙𝑠 −𝑗 sin 2𝑠 )+𝑍0 (cosh 2𝑙𝑠 −cos 2𝑠 ) (B.4) 𝑠 𝑠 𝑠 0 𝑠 𝑠 After some simple calculation, the following result is obtained: and the phase of the reflection coefficient is 𝑍𝑠 ∏ sin 2𝑠 𝑑  2𝑍0 𝑍 2 cot 𝑠 (1+tan2 )  = arctan (𝑍 )+ = 2 2 (A11) 0 (cosh 2𝑙𝑠 −cos 2𝑠 )−𝑍𝑠 ∏ sinh 2𝑙𝑠 𝑑𝑍𝑠 𝑍02 (𝑍 + 𝑍𝑠 tan  cot 𝑠 ) +𝑍 2 (𝑍𝑠 cot 𝑠 −𝑍 tan ) 𝑍𝑠 ∏ sin 2𝑠 and it follows that d/dZs = 0 for  = (2n + 1)/2 and s = + arctan ( ) (B.5) 𝑍0 (cosh 2𝑙𝑠 −cos 2𝑠 )+𝑍𝑠 ∏ sinh 2𝑙𝑠 n or s = (2n + 1)/2 (the optimum phase conditions according to the analysis of Section III.B). This applies also The sensitivity with the phase of the sensing line is therefore to the generalized structure, with an arbitrary number of 𝑆𝑠 = cascaded high/low impedance design lines with i = (2n + 2𝑍𝑠 ∏ cos 2𝑠 (𝑍0 (cosh 2𝑙𝑠 −cos 2𝑠 )−𝑍𝑠 ∏ sinh 2𝑙𝑠 )−2𝑍𝑠 𝑍0 ∏(sin 2𝑠 ) 2 1)/2. 𝐹 2 + (𝑍0 (cosh 2𝑙𝑠 −cos 2𝑠 )−𝑍𝑠 ∏ sinh 2𝑙𝑠 ) Obviously, the calculation of the sensitivity for phases 2 different than those corresponding to the optimum values 2𝑍𝑠 ∏ cos 2𝑠 (𝑍0 (cosh 2𝑙𝑠 −cos 2𝑠 )+𝑍𝑠 ∏ sinh 2𝑙𝑠 )−2𝑍𝑠 𝑍0 ∏(sin 2𝑠 ) +𝐹′ 2 [i.e., i = (2n + 1)/2 and s = n or s = (2n + 1)/2] would (𝑍0 (cosh 2𝑙𝑠 −cos 2𝑠 )+𝑍𝑠 ∏ sinh 2𝑙𝑠 ) require the use of (A1), rather than (4). Nevertheless, the (B.6) paper is entirely focused on sensitivity optimization for small where the factors F and F’ defined as perturbations (representing small variations of the phase of 1 𝐹= 2 (B.7a) the sensing line around the optimum value). For this reason, 1+(𝑋) the analysis of Section III, generalized in Section V, is 1 𝐹′ = (B.7b) justified. Indeed, the good agreement between the theoretical 1+(𝑋′)2 sensitivities for small perturbations, inferred from (4), and have been introduced for simplicity. In (B.7), X and X’ are those calculated from the simulated data when MUT  1 (see the arguments of the arctan in the first and second term of the Figs. 5 and 7) validate the theory. right hand side member of expression (B.5). For N odd, the reflection coefficient is found to be APPENDIX B −𝑍0 (sinh 2𝑙𝑠 −𝑗 sin 2𝑠 )+𝑍𝑠−1 ∏ (cosh 2𝑙𝑠 −cos 2𝑠 ) EFFECTS OF LOSSES ON SENSOR SENSITIVITY = 𝑍0 (sinh 2𝑙𝑠 −𝑗 sin 2𝑠 )+𝑍𝑠−1 ∏ (cosh 2𝑙𝑠 −cos 2𝑠 ) (B.8) Let us evaluate Ss by considering the effects of losses in the MUT (to a first-order approximation, the ohmic losses of and the phase of the reflection coefficient is 𝑍 sin 2 the step-impedance transmission line, as well as the dielectric  = arctan (𝑍−1 ∏ (cosh 2𝑙 0−cos 2𝑠 )+ losses of the substrate, can be neglected, provided the 𝑠 𝑠 𝑠 )−𝑍0 sinh 2𝑙𝑠 considered substrate is a low-loss material). The impedance 𝑍0 sin 2𝑠 + arctan ( ) (B.9) seen from the step-impedance discontinuity adjacent to the 𝑍𝑠−1 ∏ (cosh 2𝑙𝑠 −cos 2𝑠 )+𝑍0 sinh 2𝑙𝑠 sensing line, Zin,s, is no longer given by equation (5), since Taking the derivative of (B.9), the sensitivity is obtained, i.e., the complex propagation constant, , must be considered in 𝑆𝑠 = order to account for the effects of the lossy MUT. Thus, such 2 2𝑍0 cos 2𝑠 (𝑍𝑠−1 ∏ (cosh 2𝑙𝑠 −cos 2𝑠 )−𝑍0 sinh 2𝑙𝑠 )−2𝑍0 𝑍𝑠−1 ∏(sin 2𝑠 ) impedance is given by 𝐹 2 + (𝑍𝑠−1 ∏ (cosh 2𝑙𝑠 −cos 2𝑠 )−𝑍0 sinh 2𝑙𝑠 ) 𝑍𝑖𝑛,𝑠 = 𝑍𝑠 coth 𝑙𝑠 (B.1) 2 2𝑍0 cos 2𝑠 (𝑍−1 −1 𝑠 ∏ (cosh 2𝑙𝑠 −cos 2𝑠 )+𝑍0 sinh 2𝑙𝑠 )−2𝑍0 𝑍𝑠 ∏(sin 2𝑠 ) +𝐹′ 2 (𝑍𝑠−1 ∏ (cosh 2𝑙𝑠 −cos 2𝑠 )+𝑍0 sinh 2𝑙𝑠 ) (B.10) with F and F’ given by (B.7), and X and X’ corresponding to variable of the considered sensors, , for the lossless and the arguments of the arctan of the first and second term of the low-loss approximation cases. For N even, expression (B.5), right hand side member of (B.9), respectively. with sinh(2ls)  2ls and cosh(2ls)  1 + 22ls2, If we now consider the two optimum phases of the sensing provides line for sensitivity optimization, four cases (similar to those 𝑍𝑠 ∏ sin 2𝑠 enumerated in Section V) arise:  = arctan ( 2𝑙𝑠 (𝑍0 𝑙𝑠 −𝑍𝑠 ∏ ) )+ 𝑍0 (1−cos 2𝑠 ){1+ } 𝑍0 (1−cos 2𝑠 )  Case A’: s = (2n + 1)/2 and N odd. Evaluation of (B.10) gives: 𝑍𝑠 ∏ sin 2𝑠 + arctan ( 2𝑙𝑠 (𝑍0 𝑙𝑠 +𝑍𝑠 ∏ ) ) (B.19) 2𝑍 1 1 𝑍0 (1−cos 2𝑠 ){1+ } 𝑍0 (1−cos 2𝑠 ) 𝑆𝑠 = − −1 ∏(cosh0 𝑍𝑠 2𝑙𝑠 +1) { 𝑍0 sinh 2𝑙𝑠 + 𝑍0 sinh 2𝑙𝑠 } 1− 1+ 𝑍−1 𝑠 ∏(cosh 2𝑙𝑠 +1) 𝑍−1 𝑠 ∏(cosh 2𝑙𝑠 +1) (B.11) and such expression can be further approximated by 𝑍𝑠 ∏ 𝑍𝑠 ∏ cot 𝑠 42 𝑙2 For moderate and low-loss materials (i.e., satisfying ls  = 2arctan ( 𝑍0 cot 𝑠 ) − 𝑍0 𝑍 ∏ 2  (1−cos 2𝑠  1+( 𝑠 cot 𝑠 ) 𝑠) << 1), (B.11) can be approximated by 𝑍0 (B.20) 2𝑍0 𝑆𝑠 = − (B.12) 𝑍𝑠−1 ∏(1+2 𝑙𝑠2 ) where the first-order Taylor expansion arctan[A/(1+x)] = arctan[A]  Ax/(1+A2), with x small, and the trigonometric where it has been assumed that sinh(2ls)  2ls and identity sin(2s)/(1-cos2s) = cots have been used. For N cosh(2ls)  1 + 22ls2. odd, following a similar procedure, the phase of the  Case B’: s = n and N odd. In this case, the sensitivity reflection coefficient can be approximated by 𝑍𝑠 𝑍0 (using B.10) is found to be 𝑍𝑠 𝑍0 cot 𝑠 42 𝑙2 2𝑍0 2𝑍0  = 2arctan ( ∏ cot 𝑠 ) − ∏ 𝑍𝑠 𝑍0 2  (1−cos 𝑠2 𝑆𝑠 = + 1+( ∏ cot 𝑠 ) 𝑠) 𝑍𝑠−1 ∏(cosh 2𝑙𝑠 −1)−𝑍0 sinh 2𝑙𝑠 𝑍𝑠−1 ∏(cosh 2𝑙𝑠 −1)+𝑍0 sinh 2𝑙𝑠 (B.13) (B.21) and the result for the low-loss approximation can be As it can be appreciated, if losses are small, the phase of the expressed as reflection coefficient does not significantly change as 𝑍0 𝑍0 2𝑍0 𝑍𝑠−1 ∏ compared to the lossless case (note that, again, the lossy 𝑆𝑠 = 2 𝑍𝑠−1 ∏ 2 𝑙𝑠 −𝑍0 𝑙𝑠 + 2 𝑍𝑠−1 ∏ 2 𝑙𝑠 +𝑍0 𝑙𝑠 = 2 𝑍𝑠 2 𝑙𝑠 ∏ 2 −𝑍02 −2 (B.14) term, ls, appears squared). Moreover, (B.20) and (B.21) coincide with expression (34) in the limit ls  0, as  Case C’: s = (2n + 1)/2 and N even. Evaluation of expected. (B.6) provides Indeed, for low-loss MUTs, it can be concluded from 𝑆𝑠 = − 2𝑍𝑠 ∏ { 1 + 1 } (B.20) and (B.21) that the phase of the reflection coefficient, 𝑍𝑠 ∏ sinh 2𝑙𝑠 𝑍𝑠 ∏ sinh 2𝑙𝑠 𝑍0 (cosh 2𝑙𝑠 +1) 1− 𝑍0 (cosh 2𝑙𝑠 +1) 1+ 𝑍0 (cosh 2𝑙𝑠 +1) , does not depend on the attenuation constant of the sensing (B.15) line,  (in turn related to the loss factor of the MUT). Moreover, it is well known that if losses are small, the phase and the following result is obtained for the low-loss approximation: constant of the sensing line, , is uniquely determined by the 2 𝑍𝑠 ∏ effective dielectric constant of that line [43]. Therefore, the 𝑆𝑠 = − 𝑍0 (1+2 𝑙𝑠2 ) (B.16) output variable of the proposed sensor, , can be used for the determination of the dielectric constant of the MUT,  Case D’: s = n and N even. In this case, (B.6) is despite the presence of losses (provided losses are small). 2𝑍𝑠 ∏ 2𝑍𝑠 ∏ To end this appendix, it is worth-mentioning that from the 𝑆𝑠 = + 𝑍0 (cosh 2𝑙𝑠 −1)−𝑍𝑠 ∏ sinh 2𝑙𝑠 𝑍0 (cosh 2𝑙𝑠 −1)+𝑍𝑠 ∏ sinh 2𝑙𝑠 measurement of the reflection coefficient at the operating (B.17) frequency (including not only the phase, but also the and the result for the low-loss approximation can be modulus), it is possible to infer the attenuation constant of expressed as the sensing line, . It can be isolated from (B.4) or (B.8), 𝑍𝑠 ∏ 𝑍𝑠 ∏ 2𝑍0 𝑍𝑠 ∏ depending on whether N is even or odd, as far as  is inferred 𝑆𝑠 = 2 𝑍0 2 𝑙𝑠 −𝑍𝑠 ∏ + 2 𝑍0 2 𝑙𝑠 +𝑍𝑠 ∏ = 2 𝑍02 2 𝑙𝑠 −𝑍𝑠2 ∏ 2 (B.18) from measurement, and s can be obtained from (34). 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He received the Bachelor's Degree in Electronic Telecommunications B.S. Degree in Physics from the Universitat Engineering in 2016 and the Master's Degree in Autònoma de Barcelona (UAB) in 1988 and the PhD Telecommunications Engineering in 2018, both at the degree in 1992. From 1994 up to 2006 he was Autonomous University of Barcelona (UAB). Associate Professor in Electronics at the Actually, he is working in the same university in the Departament d’Enginyeria Electrònica (Universitat elaboration of his PhD, which is focused on the Autònoma de Barcelona), and since 2007 he is Full development of microwave sensors based on Professor of Electronics. In recent years, he has been metamaterials concepts for the dielectric characterization of materials and involved in different research activities including biosensors. modelling and simulation of electron devices for high frequency applications, millimeter wave and THz generation systems, and the Paris Vélez (S’10–M’14) was born in Barcelona, application of electromagnetic bandgaps to microwave and millimeter wave Spain, in 1982. He received the degree in circuits. He is now very active in the field of metamaterials and their Telecommunications Engineering, specializing in application to the miniaturization and optimization of microwave circuits electronics, the Electronics Engineering degree, and and antennas. Other topics of interest include microwave sensors and RFID the Ph.D. degree in Electrical Engineering from the systems, with special emphasis on the development of high data capacity Universitat Autònoma de Barcelona, Barcelona, in chipless-RFID tags. He is the head of the Microwave Engineering, 2008, 2010, and 2014, respectively. His Ph.D. thesis Metamaterials and Antennas Group (GEMMA Group) at UAB, and director concerned common mode suppression differential of CIMITEC, a research Center on Metamaterials supported by TECNIO microwave circuits based on metamaterial concepts (Generalitat de Catalunya). He has organized several international events and semi-lumped resonators. During the Ph.D., he was related to metamaterials and related topics, including Workshops at the awarded with a pre-doctoral teaching and research fellowship by the Spanish IEEE International Microwave Symposium (years 2005 and 2007) and Government from 2011 to 2014. From 2015-2017, he was involved in the European Microwave Conference (2009, 2015 and 2017), and the Fifth subjects related to metamaterials sensors for fluidics detection and International Congress on Advanced Electromagnetic Materials in characterization at LAAS-CNRS through a TECNIOSpring fellowship Microwaves and Optics (Metamaterials 2011), where he acted as Chair of cofounded by the Marie Curie program. His current research interests the Local Organizing Committee. He has acted as Guest Editor for six include the miniaturization of passive circuits RF/microwave and sensors- Special Issues on metamaterials and sensors in five International Journals. based metamaterials through Juan de la Cierva fellowship. Dr. Vélez is a He has authored and co-authored over 600 technical conference, letter, Reviewer for the IEEE Transactions on Microwave Theory and Techniques journal papers and book chapters, he is co-author of the book on and for other journals. Metamaterials entitled Metamaterials with Negative Parameters: Theory, Design and Microwave Applications (John Wiley & Sons Inc.), author of Lijuan Su was born in Qianjiang (Hubei), China in the book Artificial Transmission Lines for RF and Microwave Applications 1983. She received the B.S. degree in (John Wiley & Sons Inc.), co-editor of the book Balanced Microwave Filters communication engineering and the M.S. degree in (Wiley/IEEE Press) and co-author of the book Time-Domain Signature circuits and systems both from Wuhan University of Barcodes for Chipless-RFID and Sensing Applications (Springer). Ferran Technology, Wuhan, China, in 2005 and 2013 Martín has generated 21 PhDs, has filed several patents on metamaterials respectively, and the Ph.D. degree in electronic and has headed several Development Contracts. engineering from Universitat Autonoma de Prof. Martín is a member of the IEEE Microwave Theory and Techniques Barcelona, Barcelona, Spain, in 2017. From Nov. Society (IEEE MTT-S). He is reviewer of the IEEE Transactions on 2017 to Dec. 2019, she worked as a postdoc Microwave Theory and Techniques and IEEE Microwave and Wireless researcher in Flexible Electronics Research Center, Huazhong University of Components Letters, among many other journals, and he serves as member Science and Technology, Wuhan, China. She is currently a postdoc of the Editorial Board of IET Microwaves, Antennas and Propagation, researcher in CIMITEC, Universitat Autonoma de Barcelona, Spain. Her International Journal of RF and Microwave Computer-Aided Engineering, current research interests focus on the development of novel microwave and Sensors. He is also a member of the Technical Committees of the sensors with improved performance for biosensors, dielectric European Microwave Conference (EuMC) and International Congress on characterization of solids and liquids, defect detection, industrial processes, Advanced Electromagnetic Materials in Microwaves and Optics etc. (Metamaterials). Among his distinctions, Ferran Martín has received the 2006 Duran Farell Prize for Technological Research, he holds the Parc de Marta Gil Barba (S’05-M’09) was born in Recerca UAB – Santander Technology Transfer Chair, and he has been the Valdepeñas, Ciudad Real, Spain, in 1981. She recipient of three ICREA ACADEMIA Awards (calls 2008, 2013 and 2018). received the Physics degree from Universidad de He is Fellow of the IEEE and Fellow of the IET. 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 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 assistant professor in the Universidad Politécnica de Madrid. She has worked in metamaterials, piezoelectric MEMS and microwave passive devices. Her current interests include metamaterials sensors for fluidic detection. Pau Casacuberta was born in Sabadell (Barcelona), Spain, in 1997. Currently he is in his senior year of the Bachelor's Degree in Electronic Telecommunications Engineering and the Bachelor's Degree in Computer Engineering, both at the Universitat Autònoma de Barcelona (UAB). He received the Collaboration fellowship by the Spanish Government in 2019 and developing his Bachelor’s Thesis in highly sensitive microwave sensors based in stepped impedance structures.