Transmission Lines Loaded With Pairs of Stepped Impedance Resonators: Modeling and Application to Differential Permittivity Measurements

Differential techniques are widely used in communication and sensor systems, as these techniques have been shown to improve the performance. This paper shows how differential sensing of permittivity can be conducted in a simple way. For that purpose, a microstrip line loaded with a pair of stepped-impedance resonators is used in two different resonator connections: parallel and cascade. Each resonator is individually perturbed dielectrically so that: 1) when the two individual permittivities are identical, the structure exhibits a single resonance frequency and 2) when the permittivities are different, resonance frequency splitting occurs, giving rise to two resonances (all these resonances are seen in the form of transmission zeroes). The two sensing approaches are successfully validated through electromagnetic simulations and experiments. By virtue of a differential measurement, robustness against changing ambient factors that may produce sensor miscalibration is expected.

Abstract-Differential techniques are widely used in communication and sensor systems, as these techniques have been shown to improve the performance. This paper shows how differential sensing of permittivity can be conducted in a simple way. For that purpose, a microstrip line loaded with a pair of stepped-impedance resonators is used in two different resonator connections: parallel and cascade. Each resonator is individually perturbed dielectrically so that: 1) when the two individual permittivities are identical, the structure exhibits a single resonance frequency and 2) when the permittivities are different, resonance frequency splitting occurs, giving rise to two resonances (all these resonances are seen in the form of transmission zeroes). The two sensing approaches are successfully validated through electromagnetic simulations and experiments. By virtue of a differential measurement, robustness against changing ambient factors that may produce sensor miscalibration is expected.

I. INTRODUCTION
P ERMITTIVITY sensors are prominent electromagnetic sensors whose development began in the mid-twentieth century [1], [2]. These sensors are highly versatile, as the measurand (the physical quantity of interest) can be either the permittivity or another physical variable directly related to it.
Today, there are multiple technologies available to implement permittivity sensors. Nevertheless, the focus here is restricted to sensors consisting of passive planar transmission lines and/or resonators. In the last few years, many planar resonator-based permittivity sensors, where frequency shift is typically the electrical sensing variable, have been reported. These permittivity sensors are aimed at many topics, including material characterization [3]- [5] (this is indeed the context of the sensors proposed in this paper), analysis of organic tissue [6], [7], microfluidics [8]- [11], biosensing [12]- [16], or environmental factors [17]. Other planar permittivity sensors implemented by means of artificial transmission lines have also been proposed [18]- [20].
A general drawback of resonance-based sensors is that, as permittivity depends on environmental conditions (e.g., temperature), the resonance frequency can be unintentionally shifted by spurious effects [2], [17]. As is well known, ideal sensors are designed to be linear or linear to some simple mathematical function with the measurand. However, sensors may be sensitive to other physical quantities, an effect designated as cross-sensitivity that entails measurement errors (miscalibration). Common cross-sensitivities are those derived from ambient factors; in sensor design, environmental stability is usually a key point and cannot be ignored [21], [22]. Regarding permittivity sensors, it can be argued that environmental drifts should be defined in terms of sensitivity rather than cross sensitivities. Nevertheless, if ambient factors are uncontrolled, the drifts should be indeed categorized into cross sensitivities. Additionally, the resonance frequency is generally dependent not only on the permittivity of interest, but also on the permittivity of the substrate(s) necessary to support the resonant element [3]- [17]; this dependence is clearly a cross sensitivity.
To prevent or reduce systematic errors due to cross sensitivities as much as possible, several strategies can be utilized, such as compensation techniques or the use of environmentally stable materials. Another typical solution to deal with changing environmental factors is through differential measurements (the one considered in this paper). Differential measurements are robust against variable ambient conditions, provided that these conditions perturb in the form of a common-mode stimulus. Generally, two sensors are used to construct a differential sensor, both of them being subjected to the same external factors [18]- [22]; an ideal sensor should be capable of conducting measurements regardless of the surrounding ambient factors.
Depending on the application, differential measurements can be used with sensing or comparison purposes. A differential sensor measures the difference in permittivity (or one of the two permittivities, provided that the other one is known), whereas a comparator (differential by definition) can be used to inquire about whether two permittivities are the same or not. Permittivity comparators may be of interest, for instance, for the detection of defects or alteration of a sample under test (SUT) by comparing it with a well-known reference.
A differential permittivity sensor based on amplitude levels by means of an interferometer setup was reported in [18]- [20]. Naqui et al. [23] presented a differential sensor/comparator consisting of a transmission line loaded with a pair of identical shunt-connected stepped-impedance resonators (SIRs) connected to the same junction. Instead of duplicating and combining sensors in pairs to construct a differential sensor, pairs of resonators connected to a single signal interface (a single transmission line) were employed. Because of the SIR semilumped behavior, the size was compact, and electric energy was concentrated in a small region allowing for small SUTs and for low cross sensitivities (if present) related to substrate(s). Due to the SIR high Q-factor, high-frequency discrimination was expected. The sensing principle, experimentally validated in the vicinity of 60 GHz, is based on resonance frequency splitting derived from symmetry perturbation [23]- [25]; there is one or two transmission zeroes depending on whether the two permittivities are the same or not, respectively. However, the interresonator coupling was seen to degrade considerably the sensitivity and discrimination for small differential inputs. This paper is aimed at expanding the preliminary research work conducted in [23], with a view to performing realtime differential permittivity measurements. Further analysis for the topology proposed in [23] is carried out, and an alternative topology based on a transmission line loaded with a cascade connection of SIRs to face with the drawback of the original topology is proposed. Equivalent circuit models are proposed and validated, from which the theoretical sensitivity is inferred. Experimental data applying material permittivity perturbation is reported to validate the potentiality of the proposed approaches. As will be proved, the new proposed structure exhibits significant benefits against the original one when the individual permittivities attain close values, higher sensitivity, higher discrimination, and narrower bandwidth of operation.
This paper is organized as follows. Section II is devoted to the two considered physical sensing topologies and their modeling by equivalent circuits. In Section III, the capabilities of these structures to operate as a differential sensor or a comparator are investigated by analyzing the sensitivity and the discrimination, respectively. Section IV reports experimental verifications of the sensing approaches. Finally, the main conclusions are drawn in Section V.

II. TOPOLOGIES AND EQUIVALENT CIRCUIT MODELS
This section analyzes transmission lines loaded with a pair of shunt-connected SIRs in two different topologies; one with the SIRs loaded at the same junction and the other one by cascading the SIRs. Two SIRs are used to perform a real-time (i.e., through a single measurement) differential measurement. Furthermore, the two SIRs are identical, with the purpose of rejecting common-mode stimulus as much as possible. Equivalent circuit models including independent and arbitrary capacitive variations in the SIRs are presented and validated. Without loss of generality, we use microstrip implementation. Losses are disregarded throughout this work, eligible approximation when dealing with low-loss substrates and SUTs. Nevertheless, a method to estimate the loss factor is included in the last section.

A. Microstrip Line Loaded With a Pair of Parallel SIRs
The SIR in Fig. 1(a), in shunt connection to a microstrip line section, is a semilumped resonator [26]. Provided that the two sections are electrically small, the behavior of a shunt-connected SIR may be approximated to that of a shunt-connected lumped LC resonator. The inductance/capacitance of the SIR is essentially determined by the inductance/capacitance of the high/low impedance section. Consequently, the circuit model depicted in Fig. 1(b) may accurately describe a microstrip line loaded with a single SIR. The inductance and capacitance of the main host transmission line are L and C, respectively, whereas the SIR inductance and capacitance are L s and C s , respectively. Additionally, for sensing purposes, the model considers a capacitive perturbation, C s , applied to the SIR. According to the circuit model of Fig. 1(b), a transmission zero (or notch) appears at the angular resonance frequency given by As a case study, a single SIR-loaded line with dimensions W = 1.83 mm, l = 6 mm, l 1 = l 2 = 2.6 mm, w 1 = 5.5 mm, and w 2 = 250 μm and substrate Rogers RO4003C with relative permittivity ε r = 3.38, thickness h = 812.8 μm, and loss tangent tanδ = 0.0021 was considered in [23]. The model was validated by circuit simulation of extracted parameters, inferred following the systematic procedure reported in [27], which is based on a mapping of the scattering parameters obtained from full-wave electromagnetic simulation, adapted to the considered circuit. The resulting circuit parameters, characteristic impedances, and electrical lengths were found to be Z 0 = 50 , Z 1 = 23 , and Z 2 = 120 ; electrical lengths at f 0 = 3.83 GHz were βl 0 = 45°, βl 1 = 21°, and βl 2 = 18°; and the circuit elements were L = 1.81 nH, C = 0.57 pF, L s = 2.45 nH, and C s = 0.70 pF ( C s = 0).
To verify the modeling of an alteration in the resonator capacitance, a shape perturbation was produced in [23]. Namely, the length of the wide section of the reference resonator was enlarged (+| l 1 |) or shortened (−| l 1 |), with l 1 = ±0.5 mm = ±0.19l 1 , to increase (+| C s |) or decrease (−| C s |) its associated capacitance, respectively. The circuit values of these perturbations (±| C s |), with C s = ±0.11 pF = ±0.15C s , were obtained from (1) and the transmission zero frequencies were inferred from electromagnetic simulations. The transmission coefficient of these structures confirmed that the circuit simulations were in good accordance with those inferred from electromagnetic solvers in [23].
As expected from (1), any capacitive perturbation may be sensed by monitoring the change in the resonance frequency. However, a real-time differential measurement of two capacitive perturbations cannot be performed by loading a transmission line with a single SIR. Therefore, transmission lines loaded with pairs of SIRs are necessary. Fig. 2(a) shows a microstrip line loaded with a pair of identical shunt-connected SIRs placed at the same junction, and on opposite sides of the line (parallel configuration). In this topology, as was already considered in [23], it is assumed that both resonators are individually and simultaneously capacitively perturbed by C si (i = 1, 2). The lumped element equivalent circuit model that was proposed in [23] is shown in Fig. 2(b). The SIRs are close together, and magnetic coupling between them is modeled, such a coupling being negative since the currents in the mirrored SIRs flow in opposite directions (i.e., the currents are antiparallel at the junction of the SIRs). This coupling is accounted for by the mutual inductance −|M|, the magnetic coupling coefficient being [26] From the equivalent T-circuit model of a two-port network consisting of a pair of coupled inductors, the circuit model shown in Fig. 2(c) is inferred. If the capacitive perturbations are identical ( C s1 = C s2 = C s ), the simpler circuit depicted in Fig. 2(d) results.
In balanced (i.e., identical) perturbations, according to the circuit of Fig. 2(d), it can be readily noticed that the structure exhibits a single transmission zero at In the presence of coupling, the transmission zero simply shifts upward. The resonance frequency in (3) is denoted by ω e , in agreement to the nomenclature used in [26] to emphasize the presence of an open circuit, under even mode excitation, at the symmetry plane of two identical coupled resonators.
In the case with different perturbations ( C s1 = C s2 ), the symmetry in the shunt branch of the circuit model is disrupted. Although the resulting circuit model is not as simple as with balanced perturbations, the resonance condition can be easily obtained by setting the shunt branch impedance to zero. This gives the resonance frequencies of a circuit network composed of two magnetically coupled resonators, given by [26] where ω l and ω u denote the lower and upper resonance frequencies, respectively. Therefore, a jump discontinuity arises due to unbalanced perturbations, giving rise to two split resonance frequencies. The most relevant aspect is the fact that, because of coupling, the two resonances depend on the two perturbations, i.e., ω l,u = f ( C s1 , C s2 ). Accordingly, each resonance frequency cannot be independently tuned by its corresponding perturbation. In other words, when one of the resonators is perturbed, the two resonance frequencies change (the more similar the perturbations, the higher the cross-dependence). It turns out that a parallel combination of shunt-connected SIRs can be viewed as a dual-mode trisection (wide-narrow-wide) SIR [28]- [32], where symmetry disruption is necessary to invoke the dual-mode behavior [33]. Note that multimode resonators are usually utilized to reduce the size of circuits, but in this paper, a dual resonant behavior is used to conduct a differential measurement.
To demonstrate the validity and usefulness of the models in Fig. 2, in [23], three different pairs of SIRs loading a microstrip line were considered, as can be seen in Fig. 3(a): a symmetric pair, an asymmetric pair derived by increasing l 1 in one of the SIRs (+| C s2 |), and another asymmetric pair derived by decreasing l 1 (−| C s2 |). These structures were already fabricated (using a drilling machine) and measured in [23], but some fabrication-related tolerances and uncertainties were observed. For this reason, we have fabricated them again (using a photoetching process) to fit better the measurements with the simulations. As the structures considered are built up from those of single SIR loaded line, the only circuit parameter that needs to be obtained is the mutual inductance. This parameter has been obtained by curve fitting the circuit simulation to the electromagnetic simulation in the case of the host line loaded with symmetric SIRs. Analytically, (3) can be used alternatively. The extracted value is M = −0.31 nH (k m = −0.13). As shown in Fig. 3(b), the circuit simulations are consistent with both electromagnetic simulations and measurements, verifying that loss omission is a reasonable approximation using low-loss substrates.
It should be noted from Fig. 3 that the resonance frequency at f u resembles that at f e , while an additional narrowband lower resonance frequency at f l appears. When symmetry (with regard to the line axis) is broken, the bandwidth at f l is narrower than that at f u . The smaller the difference in the perturbations, the narrower the lower notch. As a result, the discrimination to detect small differences between the perturbations is expected to be degraded by losses.

B. Microstrip Line Loaded With a Pair of Cascaded SIRs
This section deals with an alternative topology presented in this paper to perform differential capacitive measurements in such a way that interresonator coupling is prevented. The proposed topology is illustrated in Fig. 4(a), and consists of a transmission line loaded with a cascade connection of two identical SIRs that are spaced apart by a transmission line section of length l s . If the SIRs are loaded on the same side of the line, as is considered, the resonators may be coupled not only magnetically, but also electrically. However, the resonators are sufficiently separated so that we may assume that the total coupling is negligible. The proposed circuit model is shown in Fig. 4(b). Regardless of the length of the transmission line section between resonators l s , the transmission zero frequencies are given by where ω l and ω u denote again the lower and upper resonance frequencies, respectively. According to the previous expressions, the two resonances can be shifted independently like using single SIRs, even for unbalanced perturbations.
Clearly, the resonance frequency splitting phenomenon, which emerges from unbalanced perturbations, is of different nature in the two configurations under study. In cascaded SIRs, splitting occurs as a mere result of frequency shifting, whereas in parallel SIR, there is a combination of frequency shifting and interresonator coupling. Let us now assume the particular case where the in-between transmission line section is half-wavelength long (l s = λ/2, where λ is the guided wavelength). As the input impedance of a load seen through a λ/2 line is unchanged, the circuit in Fig. 4(b) is equivalent to that shown in Fig. 4(c). Note that the SIRs are virtually connected at the same junction so that the structure behaves as if the SIRs were physically located at the same junction (due to periodicity, this holds at integer multiples of λ/2). Indeed, the latter circuit is the ideal one, which we would like to implement. The problems to implement this idealized circuit are: 1) in parallel SIRs, there is interresonator coupling (Section II-B) and 2) in cascaded SIRs, l s = λ/2 is satisfied at one frequency only (this condition is designed to be satisfied at the resonance frequency of the unperturbed resonators). However, the latter issue is not dramatic, since the resonance frequencies in (5) do not depend on the interresonator distance (as long as interresonator coupling can be ignored). Therefore, the resonance frequencies of the model in Fig. 4(c) always coincide with those of the model in Fig. 4(b).
The last simplification makes the assumption that the perturbations are identical (ω l = ω u = ω 0 ). In this circumstance, the circuit in Fig. 4(c) derives to the one in Fig. 4(d), which is formally identical to that with parallel SIRs depicted in Fig. 2(d).
To demonstrate the usefulness of the cascaded approach, we have considered the same three previous scenarios in line with the parallel connection [see Fig. 5(a)]. Accordingly, the same circuit elements have been used to validate the equivalent circuit models. As can be seen in Fig. 5(b), the agreement between the circuit/TL simulations, the electromagnetic simulations, and measurements is quite good. It deserves mentioning that a small ripple can be noticed from measurements in the symmetric structure, owing to the fact that the in-between transmission line is not exactly half-wavelength at the corresponding resonance frequency. Importantly, the notch frequencies are the same as those using single SIRs. Another relevant result is that the bandwidth of both notches is not narrow by nature, in contrast to what occurs at the lower resonance frequency employing parallel SIRs.

C. Comparison Between the Topologies
To end this section, we compare the transmission zero frequencies given by the two considered configurations, namely, those depicted in Figs. 3(a) and 5(a).
The results for the three considered scenarios with different perturbations [Figs. 3(b) and 5(b), as well as the single SIR] are plotted in Fig. 6 for comparison purposes. From this figure, it is clear that the two concerning topologies manifest a strongly different behavior caused by the presence or absence of interresonator coupling.   5) in the case where a capacitive perturbation is applied to one of the SIRs ( C s1 = 0 and C s2 = 0). The considered circuit parameters of the unperturbed structures are those obtained previously. Additionally, the resonances for the particular perturbed structures are also mapped to the curves. It is especially visible that, when one of the resonators is perturbed ( C s2 = 0), the frequency splitting in parallel SIRs is characterized by a shift in the two resonances. On the contrary, for cascaded SIRs, only the resonance frequency of the perturbed resonator is shifted.
In conclusion, it can be drawn that a cascade connection of SIRs provides a good solution to prevent coupling between them. Moreover, significant advantages arise, particularly, Transmission coefficient magnitude (lossless electromagnetic simulation) of the considered microstrip lines loaded with parallel or cascaded SIRs for C s1 = 0, and (a) C s2 = 0, (b) +| C s2 |, and (c) −| C s2 |. For comparison purposes, the response using a single SIR [23] applying it to C s2 is also plotted. the resonances are independent to each other (enhancing the sensitivity, as will be shown in Section III-C), their bandwidth is moderately wide by nature (improving the discrimination), the spectral separation between the resonance frequencies is not enhanced by coupling (requiring a narrower bandwidth of operation), and unknown perturbations can be physically identified. As will be shown, these advantages hold as long as the perturbations are similar.

III. SENSITIVITY ANALYSIS
Thus far, capacitive perturbations have been produced by changing the physical dimensions of the resonators. Evidently, in practice, the capacitive perturbations in a permittivity sensor must be due to permittivity perturbations of SUTs. We use the term sample in order not to lose generality, meaning that the sample can be a material, liquid, organic tissue, and so forth. Furthermore, the SUT may also refer to a functional layer that enhances the measurand-to-permittivity relationship for sensing purposes (e.g., in environmental sensors based on functional layers, a material highly sensitive to the environmental factor of interest is used).
This section evaluates the sensitivity in differential permittivity measurements, for both capacitive and permittivity perturbations. The study is conducted on the basis of the proposed equivalent circuit models, together with analytical expressions of capacitances, and on electromagnetic simulations.

A. Analytical Sensitivity in Terms of Capacitance
With a view to performing differential sensing of capacitances in the considered approach based on frequency splitting, the differential input is the difference between the capacitances, that is, The output electrical variable to be monitored is frequency, and the differential output is defined as the difference between the upper and lower resonance frequencies, that is, Obviously, despite the fact that the resonance for balanced perturbations is denoted by f e or f 0 , when a single resonance is monitored, implicitly results f l = f u . The curve that relates the output and input quantities, namely f d (C sd ), is named transfer function, whose slope is the sensitivity. Hence, the sensitivity is defined as the variation in the difference between the resonance frequencies divided by the variation in the difference between the capacitances. Mathematically, the sensitivity can be written as where C si (i = 1, 2) and f l,u stand for individual incremental/decremental changes in capacitance and resonance frequency, respectively. For simplicity, if one of the SIRs is not perturbed and considered to be a reference (so that C s1 = 0), then (8) reduces to In addition, in the cascade configuration, the magnitude of the analytical sensitivity for null perturbation (y-axis) becomes The sensitivity should be as high as possible, and constant values of it are preferred. To gain insight into the sensitivity, instead of analyzing (9), which is cumbersome when the resonance frequencies are governed by (4), numerical solutions will be given. The resonance frequencies determined by (3) and (4) for two different coupling coefficients and by (5) are plotted in Fig. 8(a) assuming C s1 = 0. It is evident that the output is null in the case of a perfect balance of perturbations ( C s1 = C s2 = 0). On the other hand, unbalanced perturbations have been introduced by a capacitive variation in one of the SIRs ( C s1 = 0, C s2 = 0). As expected, when k m = 0, corresponding to the case with parallel SIRs, f u is more sensitive to −| C s2 | than f l , and complementarily, f l is more sensitive to +| C s2 |. Both frequencies tend to the frequencies without coupling as the capacitive perturbation increases. Fig. 8(b) shows the corresponding transfer function from which it is apparent that, as |k m | increases, the frequency splitting, f d , strengthens. It turns out that the sensitivity, shown in Fig. 8(c), worsens as |k m | increases. Indeed, the maximum sensitivity corresponds to the case of (uncoupled) cascaded SIRs, and is given by (10). Specifically, a significant degradation in the sensitivity results in small perturbations. As was stressed earlier, even for loose coupling coefficients, coupling cannot be neglected for similar perturbations. Nevertheless, the parallel configuration can be useful when the perturbations differ from each other significantly. In these situations, coupling plays an insignificant role.
The results in Fig. 8 were already published in [23], where using parallel SIRs with k m = 0 was assumed to be fictitious. Here, k m = 0 is no longer fictitious, as it is implemented employing cascaded SIRs. For small inputs, since the interresonator coupling in parallel SIRs decreases the sensitivity, parallel-connected SIRs are apparently not much appropriate to properly operate as a sensor. In addition, this topology working as a comparator between two capacitively perturbed SIRs is not very promising. As mentioned before, the discrimination for small differential inputs is expected to be rather limited due to the narrowband nature of the lower resonance. Conversely, SIRs in cascade connection are expected to achieve high sensitivities and discriminations.
It should be mentioned that the dynamic range of measurement is not limited to the considered one (| C s2 | ≤ 0.6C s ). This relatively small dynamic range, however, allows for illustrating properly the handicap of the parallel configuration, i.e., low performance under small inputs. Finally, it is also important to highlight that the sensitivity is not constant due to the fact that the transfer function is not linear [this is indeed evident from the simplest form of the sensitivity given by (10)]. Moreover, sensitivity is not a symmetric function with respect to the y-axis, i.e., S (+| C sd |) = S(−| C sd |). Instead, the sensitivity magnitude increases or decreases as the input capacitance is increasingly lower or higher, respectively.

B. Analytical Sensitivity in Terms of Permittivity
In Section III-A, we considered a capacitive input as a generalization. Capacitive variations can be produced by several means, e.g., by changing the physical dimensions of the resonators (i.e., shape perturbation as done in Section II) or by changing dielectric properties (i.e., material perturbation, as it must be in practice). Therefore, in a real sensor the differential input is the difference in the relative permittivity of the SUTs, defined here as It should be noted that the sensor measures the difference between two permittivities. If one of the two permittivities is known, the other permittivity can be inferred. The sensitivity in terms of relative permittivity becomes where ε ri (i = 1, 2) stands for changes in relative permittivity. The partial derivatives indicate that the output differential frequency may be influenced by other physical variables related to cross sensitivities, as will be illustrated further in this section. By assuming a linear dependence of the capacitance with the permittivity (with proportionality constant k), and letting ε r1 = 0 and k m = 0 once again, the sensitivity magnitude becomes To analytically establish the transfer function and sensitivity above, the SIR capacitance was approximated by the parallelplate capacitance of its wide section, namely, where ε 0 is the vacuum permittivity. In (14), the substrate in the vicinity of the wide section is supposed to be replaced with one SUT. The most relevant aspect in this approximation is that the SIR capacitance is linearly dependent on the permittivity. The corresponding capacitance for the reference SIR is C p = 0.53 pF. However, (14) must be considered as a first-order approximation as mentioned above, where l 1 , w 1 h are required for accurate results. Likewise, C s is neither exactly proportional to l 1 ; l 1 = ±0.19l 1 produces C s = ±0.15C s . Finally, to illustrate the robustness of differential measurements against cross sensitivities, let us assume that we compare two identical materials ( C s = C s1 = C s2 ). Using a two-step process with a single SIR, the resulting resonance frequencies are Let us now introduce a spurious change in the capacitance related to a cross sensitivity, denoted by C xi (i = 1, 2). As the effect of cross sensitivities may change over time, one may assume C x1 = C x2 (neither a compensation technique nor recalibration is considered). Hence, in the presence of cross sensitivities, the different drifts shift the resonance frequencies to Therefore, f d = 0 and the readout is going to be wrong. By contrast, performing the proposed real-time differential measurement, using the cascaded configuration, the resulting single resonance frequency is Since the two materials are the same and affected by cross sensitivities at the same time, it is reasonable to assume that C x = C x1 = C x2 , so that Consequently, f d = 0, indicating that there is no difference between the materials. Note, however, that a comparison Fig. 9.
Electric field amplitude at the SIR resonance frequency of the reference structure (single SIR) over the mid-plane between the SIR and the ground plane. The SUT height is set to h −2t where t = 35 μm, and its length and width extend 1 mm beyond the wide section of the SIR, as depicted. The SUT size is 4.6 mm × 7.5 mm × 812.8 μm.
between identical materials is the most favorable case to reject cross sensitivities, this case being theoretically totally immune to them, as indicated by (18). In a differential sensing, some degree of robustness is expected, as different materials may respond in different ways to the same stimulus. Particularly, when we compare f d for different materials (characterized by C s1 = 0, C s2 = C s ), and the same quantity under the presence of a common mode stimulus f d , the difference is found to be That is, the difference is not exactly null if C x = 0, but small (and identical to zero if C s = 0).

C. Simulated Sensitivity in Terms of Permittivity
It is apparent that the optimum position (i.e., the most sensitive region) where the SUTs must be loaded depends on the technology of implementation. In the microstrip technology, as is well known, most of the electric field is concentrated in the substrate. Therefore, the SUTs should be located between the SIR and the ground plane, like filling a parallel-plate structure. Otherwise, the sensitivity will be decreased. Fig. 9 shows a 2-D cut of the electric field amplitude simulated with C ST Microwave Studio, confirming that most of the electric field confinement lies below the wide section of the SIR (i.e., the electric field is dominated by the capacitance effect of the resonator). Moreover, since the field is roughly uniformly distributed, the wide section behaves approximately like a parallel-plate capacitance.
To characterize the sensitivity in terms of permittivity perturbations, we have considered the same reference structures as in Section II [Figs. 3(a) and 5(a)], and we have introduced a bulk dielectric SUT in one of the SIRs, as shown in Fig. 9. The structures have been simulated with different values of the SUT relative permittivity ε r2 , and the resulting resonance frequencies are plotted in Fig. 10. The resulting sensitivity derived from the results in Fig. 10 is plotted Fig. 10. Resonance frequencies for parallel and cascaded SIRs, obtained by electromagnetic simulation, versus a perturbation in the relative permittivity of an SUT loaded to one of the SIRs. The relative permittivity is perturbed by steps of ±10% so that 1.352 ≤ εr2 ≤ 5.408 (−0.6 ≤ εr2/εr ≤ 0.6) where εr = 3.38. Losses are considered. Fig. 11. Magnitude of the sensitivity derived from Fig. 10 by steps of ε r2 /ε r = ±0.1 (±10%). The sensitivity in the jump discontinuity of the parallel configuration is not calculated. As stated by (13), the sensitivity and the perturbation have the same sign.
in Fig. 11. Clearly, for small perturbations, the sensitivity in cascaded SIRs is superior to that obtained using parallel SIRs.
It can be observed that the results in Fig. 10 are similar to those in Fig. 7. More specifically, the sensitivity to a permittivity perturbation ( ε r2 / ε r ) is a bit smaller than that to a capacitive perturbation ( C s2 / C s ) with the same amount of relative perturbation. This verifies again that the capacitance is not exactly proportional to the permittivity.
The discrimination for small differential inputs ε rd is of relevant importance in comparators. To gain insight into the performance of the considered topologies to operate as comparators, the discrimination from balanced to unbalanced perturbations should be defined as the minimum ε rd necessary to produce a discriminable doubly notched response. It is assumed that two −3-dB notches is sufficient for reasonably accurate discrimination. For the considered topologies, these values are about ±6% and ±2% in the parallel and cascaded configurations. The transmission coefficients for these inputs  are plotted in Fig. 12. Therefore, the cascade configuration exhibits higher discrimination than the parallel one, and it is an appropriate means to enhance the capability of detecting inputs as small as possible (i.e., to enhance the discrimination). As was expected, wideband notches are required. Otherwise losses may mask the notches, as occurs in the lower notch for parallel SIRs when the input differential permittivity is close to zero. In summary, the discrimination is better with the cascaded configuration, where both notches are wide.
On the other hand, in the case of large perturbations and as shown in Fig. 13, the parallel and cascaded configuration have similar resonance frequencies and, thus, similar sensitivities.

IV. EXPERIMENTAL VALIDATION OF THE SENSING APPROACHES AND DISCUSSION
In this section, permittivity perturbations are experimentally validated. To do so, the reference structures in Figs. 3 and 5 are considered again. To apply both positive and negative differential permittivity perturbations, the substrate in the vicinity of one of the SIRs is removed using a drilling machine, creating a cavity. Next, the cavity is either unfilled (−| ε r2 |) or filled with a Rogers RO3010 substrate ε r = 11.2 (+| ε r2 |), and covered with a metallic adhesive tape to act as the ground plane etched from the microstrip structure. Fig. 14 shows different photographs to illustrate the cavities, and their filling and covering. Note that the effects of the air-gap present between the cavity and the SUT can be minimized by putting some pressure to the structure, as we have done in the measurements with the Rogers slab (in [5] Fig. 14. Photographs of the reference structures composed of microstrip lines loaded with pairs of (a) parallel and (b) cascaded SIRs. From left to right: 1) cavity in one of the SIRs; 2) filled cavity with Rogers RO3010 with ε r2 = 11.2; and 3) cavity covered with a metallic tape. (c) Photograph of the experimental setup for measurement, including the Agilent N5221A network analyzer, cables, connectors, and sensing system. a systematic method to eliminate the effects of the air gap is reported).

A. Determining the Dielectric Constant of the SUT
The measured results of the positive/negative perturbations, plotted in Fig. 15, together with those with no perturbation, are in accordance with theory. Even though the cavity dimensions cannot be controlled very accurately with the in-house drilling, these experiments validate the sensing principle under permittivity perturbation. Nevertheless, inspection of Fig. 15 reveals that the difference in notch frequencies is somehow smaller than the results of Fig. 10 for the considered dielectric constant values (1 for the unfilled cavity, and 11.2 for the filled cavity with the considered Rogers material). The reason is that the cavity has been implemented by milling, and we have not completely removed all the substrate material, since the SIR needs some material for mechanical stability. Therefore, the unfilled cavity, including the remaining substrate layer, has an effective dielectric constant larger than 1, and the cavity filled with the Rogers substrate (dielectric constant 11.2) has actually an effective dielectric constant smaller than 11.2. In other words, the measurement provides the effective dielectric constant of the structure below the SIR, including not only the SUT but also the presence of a narrow dielectric layer of relative permittivity of 3.38. Moreover, the thickness of the whole structure, layer on top the cavity plus SUT, is not necessarily the same as the thickness of the substrate. In the simulations that have been carried out to obtain the results of Fig. 10, we have not considered this substrate layer between the SIR metal level and the SUT. In practice, it is very difficult for us to precisely control the thickness of the remaining substrate between the SIR patch and the cavity. For this main reason, such a layer has not been considered in the simulations. However, note that with a more sophisticated fabrication technology (e.g., micromachining), such control would not be a problem.
Nevertheless, we have proposed a method to determine this thickness that subsequently allows us to obtain the dielectric constant of the SUT once the reference permittivity (the one of the substrate, i.e., ε r1 ) and the thickness of the SUT are known. Let us consider that h 1 and h c are the thicknesses of the substrate layer on top of the cavity and the cavity, respectively, so that h 1 + h c = h, the substrate thickness. Moreover let us assume that the SUT corresponds to a material with wellknown dielectric constant ε r2 and thickness h 2 (see Fig. 16). The method is based on the fact that the effective dielectric constant of the composite formed by the substrate layer on the top of the cavity plus the SUT ε eff is related to the respective dielectric constants by From the previous expression, the dielectric constant of the SUT can be isolated  where ε eff can be inferred from the split in frequency of the SUT and the curve of Fig. 10 corresponding to the cascade connection. Thus, if ε r2 and h 2 are known, h 1 can be obtained from (21). For the SUT corresponding to the Rogers substrate with ε r2 = 11.2 and h 2 = 635 μm, and considering that the notch frequencies of Fig. 15(b) provide ε eff = 4.9, according to Fig. 10, the resulting thickness of the layer on top of the cavity is found to be h 1 = 369 μm. To verify the validity of this result, we have considered the curve of Fig. 15(b) corresponding to the unfilled cavity (ε r2 = 1). By introducing the corresponding effective dielectric constant ε eff = 1.69 in (21), and h 1 = 369 μm, the air thickness is found to be h 2 = 371 μm, which is in reasonable agreement with the thickness of the cavity (note, however, that the metallic tape is somehow flexible and hence may reduce the effective value of the cavity thickness).
Once h 1 is known, we can calculate the dielectric constant of another SUT, an Arlon slab with ε r2 = 2.43 and h 2 = 490 μm. The measured transmission coefficient for the cascaded configuration is depicted in Fig. 17. The effective dielectric constant that results from Fig. 10 is ε eff = 2.70, and using (21), the dielectric permittivity is found to be ε r2 = 2.50, very close to the nominal value (2.43).
It is interesting to analyze expression (21) and the dependence of the dielectric constant of the SUT with the different parameters. Particularly, it linearly depends on the thickness of the SUT, h 2 . Therefore, uncertainties in this parameter are directly translated into the dielectric constant. The dependence on the other geometrical parameters h and h 1 can be appreciated in expression (21) as well. The thickness of the substrate h is typically known with good accuracy. On the other hand, the dependence on h 1 is modulated by ε eff , and the effects of the uncertainty with this parameter are minimized in samples with small dielectric constant (and hence small ε eff ). In view of (21), and considering that ε eff is roughly proportional to ε r1 , namely, it follows that ε r2 is proportional to ε r1 [see (21)] and hence the uncertainties in the dielectric constant of the reference material are (roughly) directly translated into the one of the measured materials.
Comparison of the proposed differential sensor with other permittivity sensors based on microstrip technology, such as those sensors based on CSRRs [3]- [5], is not easy. Nevertheless, the fact that the proposed sensors are differential represents advantages in several aspects. As comparators, common-mode stimuli are minimized by means of a differential-mode approach, as discussed before, and small changes between two (apparently identical) samples can be detected. Comparators based on this approach can be of interest to determine defects or abnormalities in a sample, compared with a well-known reference, as well as soft permittivity changes in circuits manufactured on microwave laminates. In a real scenario, two identical cavities must be present, each one below the corresponding SIR patch.
Concerning sensitivity, the capacitance of the SIR is broadside and hence very sensitive to the permittivity, compared to the case of edge capacitances (such as the one of CSRRs). The sensitivity that can be inferred from the sensors reported in [3]- [5] is very reasonable, but not as good as the one reported in this paper [of the order of 0.6 GHz for small perturbations, according to Fig. 10 and expression (13)-note that we have considered the relative permittivity, i.e., dimensionless, in the denominator of (13)]. Note that according to expression (13), such sensitivity can be modulated by means of the SIR dimensions, which provide the resonance frequency for the unperturbed state.

B. Loss Tangent Estimation
Even though this paper is focused on low-loss substrates and SUTs (as mentioned before), let us discuss a procedure to estimate the loss tangent. It is based on the depth of the notch (similar to [4] and [5]), and for this reason a structure with a single SIR is preferred (i.e., nondifferential). Otherwise, the presence of closed notches (as results in cases with small differential permittivities) may obscure the results. The initial assumption is that the substrate material below the SIRs is completely removed and replaced with the SUT. Let us consider two causes of losses, i.e., metallic losses, mainly associated to the narrow inductive strip of the SIR, and dielectric losses, related to the SUT. The model of the SIRloaded line with losses included is the one of Fig. 1(b), but including a resistance, R M , in series with the inductance L s , plus a resistance, R D , parallel connected to the capacitance C s . The impedance of this shunt branch is (23) and this expression can be approximated by where it has been assumed that R 2 D C 2 s ω 2 1, corresponding to SUTs with moderate or low-loss levels (note that in an ideal lossless SUT, R D = ∞). At the notch frequency, the reactive part of Z in nulls, and the resulting resistance can be expressed as where Q D is the SUT quality factor. Such resistance is related to the magnitude of the transmission coefficient at the notch frequency by and therefore it can be inferred from the measured frequency response. On the other hand, If R M is known, (25) gives Q D and hence tanδ can be inferred from (27). To determine the contribution of the metallic losses R M , a possible procedure is to consider a reference SUT (or substrate) with a well-known tanδ (and hence Q D ). From (25), R M can be isolated, and used subsequently for the determination of the loss tangent of the SUT (it is assumed that R M does not vary with the SUT).
The previous procedure cannot be directly applied in general to our proof-of-concept cascaded SIR-based sensor since it is based on two SIRs, and, moreover, there is a remaining substrate layer on top of the cavity. However, for the considered SUT, the Arlon slab with nominal permittivity and thickness ε r2 = 2.43 and h 2 = 490 μm, the frequency response shows quite uncoupled notches and, therefore, we can modify the procedure and at least make an estimation of the tanδ. By introducing the SUT into the cavity, expression (25) can be rewritten as where the subindex 1 and 2 refer to the layer on top of the cavity and SUT, respectively. By identifying the last two terms with the loss contribution of the composite (C eq ωQ eq ) −1 , it follows that: This expression depends on the ratio C 2 /C 1 , which can be determined from the thickness of the layer and SUT and from the dielectric constants. Using (25) with C s = C eq and Q D = Q eq and with R LOSS inferred from (26), where S 21 is obtained from the depth of the second notch in Fig. 17, we can determine Q eq , which is found to be Q eq = 366. Since Q 1 is known (Q 1 = tan δ) −1 = 476, we can isolate Q 2 from (29), and the resulting value is found to be Q 2 = 353, which gives tanδ = 0.0028 (in reasonable agreement with the nominal value of 0.0020). Note that we cannot expect an accurate value of the loss factor with the considered in-house measurement system, necessarily affected by additional losses such as those derived from the connectors, soldering, metallic adhesive tape, etc. Nevertheless, a procedure to estimate the tanδ of the SUT has been reported.

V. CONCLUSION
This paper analyzes two simple strategies to conduct differential measurements of permittivity. These strategies are based on loading pairs of SIRs, in either parallel or cascade connection, to a microstrip line. It has been shown that, although the most canonical symmetric configuration to perform a differential measurement is the one where the SIRs are parallel-connected, such a structure exhibits lower performance for small differential permittivities due to interresonator coupling, and hence the cascade connection is preferred in these situations. The differential technique is simple, as a transmission line is driven with a single-ended RF/microwave signal, and the SIRs are simply simultaneously and individually loaded with dielectric materials. The sensing principle is based on resonance frequency splitting, so that when the permittivities of the two dielectric materials are identical the sensor exhibits a single resonance frequency, whereas two resonances are present if such permittivities are different. The sensing principle has been experimentally validated by both shape and material perturbations. This main focus in this paper has been on the differential measurement. A method to determine the dielectric constant of a certain SUT, provided the one of the reference is known, has been proposed. Finally, despite that the analysis of the present work has been carried out by excluding losses, a method to estimate the loss tangent of the SUT has been also reported. As is well known, changing environmental factors may change the permittivity and, therefore, miscalibration. By virtue of a differential measurement, sensing and comparison with some degree of immunity to these external factors are expected.