Differential-Mode to Common-Mode Conversion Detector Based on Rat-Race Hybrid Couplers: Analysis and Application to Differential Sensors and Comparators

A microwave device able to detect differential-mode to common-mode conversion (or vice versa) in a four-port balanced circuit is proposed. For mode conversion detection, the balanced circuit should be fed by a differential-mode signal, generated by means of a rat-race balun from a single-ended signal. By connecting the pair of output ports of the balanced circuit under test to one of the two pairs of isolated ports of a second rat-race coupler, conversion to the common-mode (if it exists) can be detected and recorded in the $\Sigma $ -port. Since the signal level at the output (single-ended) port depends on mode conversion efficiency, the complete two-port structure, including the pair of rat-race couplers plus the balanced circuit in between, can be used as a comparator or differential microwave sensor based on mode conversion. In such a sensor, the working principle is symmetry disruption (caused, e.g., by an asymmetric dielectric load), and the output variable is the transmission coefficient, related to the level of asymmetry. A detailed analysis, considering an arbitrary unbalanced four-port test structure (to account for symmetry disruption), is carried out. Then, such analysis is particularized to the case of a pair of unbalanced and uncoupled matched lines, and the conditions for sensitivity optimization are obtained. Finally, a differential sensor and comparator based on a pair of balanced (and meandered) matched lines, designed according to the guidelines for sensitivity optimization, is fabricated and validated.


I. INTRODUCTION
T HE interest for differential-mode (or balanced) microwave circuits has increased in recent years due to their high immunity to electromagnetic interference, noise, and crosstalk (at least as compared with single-ended circuits) [1], [2]. A critical aspect in balanced circuits is mode conversion, typically caused by an imperfect balance and responsible for the generation of (undesired) commonmode noise, which in turn may affect device performance. For that reason, many efforts have been dedicated to the design of common-mode filters [2], [3]- [22], and to the design of balanced circuits with inherent common-mode rejection [23]- [57]. Nevertheless, mode conversion in balanced circuits should be avoided, or minimized as much as possible, by ensuring accurate fabrication processes that prevent symmetry disruption (imbalances).
Balanced structures have been also applied to the design of differential sensors [58]- [62]. Such sensors offer a high level of robustness against cross sensitivities related to environmental conditions (e.g., temperature, moisture, and so on), as far as ambient factors are seen as a common-mode stimulus. Typically, these sensors consist of a pair of mirror structures (independent sensors), sensitive to the variable to be sensed (measurand). Sensing principle is symmetry truncation caused by a difference in the measurand in both sensing parts, one for the reference (REF) measurand (well known), and the other one for the measurand under test (MUT). The output variable is usually inferred from the difference between a certain magnitude (sensitive to the measurand) recorded in the REF and MUT sensing parts. It should be mentioned that other similar sensors, which are also based on symmetry disruption but cannot be considered true differential sensors, have been recently reported [63]- [77].
A useful parameter to detect symmetry imbalances caused by differences between the REF and MUT measurand in differential sensors is the cross-mode transmission coefficient. This parameter, sensitive to mode conversion, has been used as an output variable in differential sensors based on pairs of balanced transmission lines loaded with resonant elements (the sensing element) [78]- [80]. In [79] and [80], such sensors were applied to the determination of solute concentration in very diluted solutions, by adding fluidic channels on top of the resonant elements. Very competitive resolution and sensitivity, as compared with the reported literature to date [81]- [85], was achieved. However, in the sensors reported in [79] and [80], four-port devices, the output variable (cross-mode transmission coefficient) was obtained from the measurement of the mixed-mode S-parameters [1], [2], [86], [87], which involves either a four-port vector network analyzer, or multiple two-port measurements of single-ended S-parameters with the corresponding conversion to mixed-mode S-parameters.
In this article, we propose a simple strategy to detect mode conversion in balanced structures (potentially subjected to symmetry imbalances), which is then applied to the implementation of differential sensors. This strategy consists of adding a pair of rat-race couplers conveniently connected to the pair of input and output ports of the balanced sensing structure, resulting in a simple two-port device. The main advantage is the simplicity in the measurement of the output variable, the single-ended transmission coefficient of the whole two-port structure, intimately related to the cross-mode transmission coefficient of the balanced sensing part. A detailed analysis of the proposed system, which will provide important hints for sensitivity optimization, is also carried out. This is a relevant contribution of this article. Moreover, such analysis is validated from full-wave electromagnetic simulations and then used for the design of a simple differential sensor and comparator, based on a pair of meandered lines (intended to demonstrate/validate the proposed sensing strategy, including sensitivity optimization).
This article is organized as follows. The proposed mode conversion detector and working principle are presented in Section II. The analysis focused on the determination of the transmission coefficient (the key parameter) for the general case of an arbitrary four-port test structure (to account for possible symmetry imbalances, the case of actual interest) is carried out in Section III. Then, Section IV considers the particular case of a pair of uncoupled matched lines, and the conditions for sensitivity optimization (i.e., enhancement of the transmission coefficient with the phase difference of the two lines), are discussed. In Section V, the conclusions of the previous analysis are applied to the design of a differential sensor/comparator, which is experimentally validated in Section VI. Such validation includes a defect detector, a sensor to determine the dielectric constant of low-loss samples, and a sensor to determine the volume fraction of liquid solutions. Finally, the main conclusions of the work are highlighted in Section VII.

II. PROPOSED MODE CONVERSION DETECTOR AND WORKING PRINCIPLE
The proposed mode conversion detector consists of a pair of rat-race hybrid couplers connected to the four-port (arbitrary) circuit understudy, as shown in Fig. 1. Such four-port circuit, not necessarily reciprocal, can be either described by the single-ended S-parameters or by the mixed-mode S-parameters. According to the port designation ( Fig. 1), the single-ended S-parameter matrix is whereas the mixed-mode S-parameter matrix can be expressed as a combination of four ordered two matrices as follows: where the mixed-mode S-parameters are related to the singlemode S-parameters according to well-known transformations (see [2]), that is, Note that the pair of composite differential-and commonmode ports, necessary to derive the above transformations, are considered to be the ports A-B and A -B (driven as a pair). Thus, for perfect symmetry conditions, with regard to the indicated bisection plane O-O , the cross-mode (or mode conversion) matrices, S cd and S dc , are null. By contrast, in the use of the mode conversion detector as a differential sensor or comparator (Section V), a symmetry imbalance will be deliberately generated. In particular, in such application, the sensitive parts of the structure (symmetrically located  with regard to the O-O plane) are loaded with different  samples [the reference sample, REF, and the sample under  test (SUT)]. The result of such imbalance is mode conversion, and, consequently, S cd = 0 and S dc = 0.
The input port of the two-port mode conversion detector is the -port of the first coupler (port 2), whereas the output port is the -port of the second coupler (port 1 ). The ports in both couplers are differentiated by means of a superindex prime (') in the second coupler. For a matched isolated port (-port) in coupler 1 (port 1), such coupler acts as a balun, providing out of phase signals at ports 3 and 4 (i.e., a pure differential signal in the differential port 3-4, or A-B). Similarly, by terminating the -port in the second coupler (port 2 ) with a matched load, the -port (port 1 ) only detects common-mode signals, if they are present, at the composite port 3 -4 (or A -B ). Therefore, the structure is sensitive to differential-mode to common-mode conversion, potentially caused by symmetry imbalances, either undesired or deliberated. In the former case, the structure can be used to detect fabrication inaccuracies in balanced circuits. In the second case (deliberate symmetry truncation), the device is useful as a differential sensor (as it has been pointed out), the case of interest in this article.
According to the previous words, it is expected that the transmission coefficient of the complete two-port structure (including the pair of couplers and the four-port network in between) is related to the cross-mode S-parameters of the four-port network, in turn, correlated with the level of asymmetry of such network. The calculation of such transmission coefficient is carried out in Section III, where the general case of ports 1 and 2 terminated with arbitrary loads is considered.

III. GENERAL ANALYSIS
The working principle of the proposed mode conversion detector has been explained by considering the isolated ports of the couplers (ports 1 and 2 ) terminated with matched loads. However, it does not mean that this termination is the optimum one in order to obtain a maximum variation of the modulus of the transmission coefficient of the structure as symmetry imbalances increase (sensitivity optimization). Therefore, the following analysis, devoted to obtaining the transmission coefficient of the mode conversion detector, is carried out by considering arbitrary loads in ports 1 and 2 , described by reflection coefficients ρ and ρ , respectively (see Fig. 1). Let us call f 0 the operating frequency of the device, at which the length of the ring couplers is exactly 1.5λ (λ being the guided wavelength), and let Z 0 be the reference impedance of the ports. With the port designation of Fig. 1, the S-parameter matrix of the couplers (at f 0 ) is [88] In the present analysis, we will deal with the normalized amplitudes of the voltage waves incident to (a i ) or reflected from (b i ) the ports, where the subindex i identifies the port. The variable of interest in this article, the transmission coefficient of the whole structure, is given by where, again, the super index prime is used to distinguish between the normalized amplitudes of the incident and reflected waves in both couplers. According to (4), b 1 is where the normalized amplitude of the incident voltages at ports 3 and 4 can be expressed as a function of the elements of the single-ended S-parameter matrix of the four-port network, that is, Note that (7) results from the following (trivial) correspondence between normalized amplitudes of the incident and reflected voltages waves of the couplers (right side members) and the four-port network (left side members) The normalized amplitudes of the reflected voltage waves (referred to the couplers) that appear in the right-hand side members in (7) are given by with a 1 = ρb 1 and a 2 = ρ b 2 . Note also that a 1 = 0 in (9), since port 1 should be matched for the calculation of the transmission coefficient (see expression 5). By introducing the right-hand side terms of (9) in (7), and the resulting expressions in (6), the normalized amplitude of the output voltage wave at port 1 is found to be where the correspondence between mixed-mode and singleended S-parameters of the four-port network (expressions 3) has been used.
In order to obtain the transmission coefficient of the whole structure, according to (5), the first and third terms of the second member in (10) must be written as a function of a 2 . For that purpose, the first step is to express b 1 and b 2 , given by in terms of the same variables that appear in the second member of (10). The procedure is very similar to the one detailed before to obtain (10), and for that reason, it is not repeated. The obtained results are found to be where, again, (3) has been used. From (12), b 1 can be expressed in terms of a 2 and b 2 , that is, By introducing (14) in (13), b 2 can be isolated and expressed in terms of a 2 as follows: where M , introduced to simplify the notation, depends only on the mixed-mode S-parameters of the four-port network and on the reflection coefficients of the loads present at ports 1 and 2 , that is, From (14), b 1 can be expressed in terms of a 2 as where Finally, by introducing (15) and (17) in (10), b 1 can be expressed as proportional to a 2 , and the transmission coefficient, given by (5), is found to be Inspection of (19) reveals that for perfectly balanced fourport networks (i.e., with S cd = S dc = 0), S 1'2 = 0, as expected (note that M = 0 for balanced structures). A detailed analysis of (19) indicates that for unbalanced four-port networks, S 1'2 can be also null. However, this occurs for very specific combinations of port loading (i.e., ρ and ρ ) and mixedmode S-parameters of the four-port network [89]. Thus, when symmetry is truncated, in general, S 1'2 is not zero. Hence, the structure can be used as a detector able to determine the level of asymmetry in a four-port network (mode conversion detector). Interestingly, for matched terminations at ports 1 and 2 (ρ = ρ = 0), the transmission coefficient of the mode conversion detector is found to be (20) which coincides with the cross-mode transmission coefficient of the four-port network (except the sign). Thus, the proposed structure provides a straightforward way to obtain the crossmode transmission coefficient in four-port networks. Note, however, that M and M depend on various elements of the cross-mode matrices. Therefore, it is not a priori clear that the preferred option for sensitivity optimization is to terminate ports 1 and 2 with matched loads. To determine the convenient terminations at those ports (for sensitivity enhancement), the mixed-mode (or single-ended) S-parameters of the four-port network must be known. Nevertheless, from the point of view of fabrication simplicity and cost, terminating the ports with open circuits (ρ = ρ = 1) is equivalent to eliminate one port of each coupler, thereby avoiding the connection of surface mount devices, as required for matched terminations or for arbitrary loads, or vias (corresponding to short-circuit terminations, i.e., ρ = ρ = −1).
A case of special interest is the one corresponding to a pair of matched and uncoupled lines, to be discussed in Section IV, and then applied to the implementation of a differential sensor and comparator.

IV. ANALYSIS RESULTS FOR MATCHED AND UNCOUPLED LINES AND SENSITIVITY OPTIMIZATION
For a four-port network consisting of a pair of matched and uncoupled lines (Fig. 2), the mixed-mode S-parameters that explicitly appear in (19) are where φ A and φ B are the electrical lengths of the lines between ports A-A (line A) and B-B (line B), respectively. The other term needed for the determination of S 1'2 is M, given by and the transmission coefficient (19) is found to be Note that expression (23) has been obtained by neglecting losses. This approximation is justified as far as the considered substrate (for device implementation) is a low-loss microwave substrate and provided the imbalances (if they are present) are caused by dielectric loading of the lines (the REF and SUT samples in lines A and B, respectively) with lowloss materials. Nevertheless, the generalization of (23) by considering the lines loaded with lossy materials (e.g., liquids) is carried out in the Appendix.
Let us now designate the first and second term of the right-hand side member of (23) as P and Q, respectively. The modulus of (23), the variable of interest for the use of the considered structure for sensing purposes, is given by the modulus of P, that is, times the modulus of Q. This later quantity cannot be expressed in a straightforward form. However, for small asymmetries (imbalances), corresponding to similar (but not identical) values of the electrical lengths of both lines (φ A ≈ φ B ), the modulus of the transmission coefficient can be approximated by The case of small imbalances is justified since the detection of tiny differences between the phases of both lines (e.g., caused by slightly different sample loading in the lines) is of interest for highly sensitive detectors and sensors, as it will be discussed in Section IV.
In the view of (25), it follows that for small asymmetries, termination of any of the isolated ports (1 or 2 ) with a matched load (i.e., ρ · ρ = 0) is a sufficient condition to obtain a proportional dependence of the modulus of the transmission coefficient with the phase difference. Thus, by considering the phase difference, φ = φ A − φ B , as the input variable, the sensitivity for small perturbations is constant and given by For ρ · ρ = 0, the sensitivity (as defined earlier) for small perturbations is not constant, but it can be either enhanced or degraded, depending on the phase of the lines (φ A ≈ φ B ).
Let us now consider two canonical cases: 1) ρ · ρ = 1 (corresponding, e.g., to ρ = ρ = ±1) and 2) ρ · ρ = −1 (corresponding, e.g., to ρ = 1 and ρ = −1, or to ρ = −1 and ρ = 1). In the former case,  uncoupled lines is roughly a half wavelength (or a multiple of this length). In this case, the sensitivity in the limit of small asymmetries is S = 1. Conversely, for ρ · ρ = −1, Thus, if one port is left open and the other one is shortcircuited to ground, the optimum line length for sensitivity optimization (with S = 1, as well) is an odd multiple of a quarter wavelength.
It should be mentioned that such canonical cases can be also obtained by means of purely reactive loads. For instance, case 1) results by considering, e.g., ρ = j (pure inductance) and ρ = − j (pure capacitance). Nevertheless, the most interesting case from a practical viewpoint is case 1) with ρ = ρ = 1, since neither a specific load nor a via is required in the isolated ports (1 and 2 ) of the couplers.
For the validation of the previous sensitivity analysis, we have represented the exact value of the term Q [in brackets in expression (23)] (Fig. 3). Such a term can be rewritten as It can be seen in Fig. 3 that when φ A ≈ φ B , Q approaches 0 or 2 for the phase conditions indicated earlier, depending on the specific case (ρ · ρ = 1 or ρ · ρ = −1).
For comparison purposes, the transmission coefficient and the sensitivity that results when ρ · ρ = 0 is also included.  Fig. 4(b)]. For small values of φ, the sensitivity is optimized for ρ · ρ = 1 and φ A = π (or nπ), or for ρ · ρ = −1 and φ A = π/2 [or (2n + 1)π/2], in agreement to the previous analysis based on the approximate expression (25). Indeed, we have also inferred the transmission coefficient and the sensitivity that results from (25), valid for small perturbations, and we have shown it in Fig. 4. The agreement is progressively better as φ tends to zero, as expected. Fig. 4 also shows the transmission coefficient and sensitivity corresponding to ρ · ρ = 0. In this case, the sensitivity varies with φ, being roughly constant for small perturbations (φ ≈ 0), but it does not depend on the phase of line A.
According to these results, a differential sensor and comparator based on a pair of matched and uncoupled lines with the phases set to 2π, and with open-ended loads at ports 1 and 2 , (ρ = ρ = 1) has been designed and fabricated for validation purposes (to be discussed in Section V).

V. SENSOR/COMPARATOR DESIGN AND FABRICATION
The analysis of the previous section has been used to design the proposed sensor/comparator. The operating frequency has been set to f 0 = 2 GHz, and the considered substrate for sensor fabrication is the Rogers RO4003C with dielectric  With this electrical length of the lines, the sensitivity is optimized (see section IV), provided ρ · ρ = 1 (note that the isolated ports of both couplers are opened, and thereby ρ = ρ = 1). It should be mentioned that the characteristic impedance of the line varies slightly in the sensing region, due to the presence of the REF sample. Nevertheless, the conclusions relative to the previous analysis prevail, as will be shown later.
The device has been fabricated by means of an LPKF H100 drilling machine. The simulated and measured responses of the device without any material in the sensing regions, inferred by means of the Keysight Momentum commercial software and through the Agilent N5221A PNA vector network analyzer, respectively, are shown in Fig. 6. The measured transmission coefficient at f 0 is smaller than −60 dB, indicating that the fabricated structure exhibits a good balance between the meandered lines. This is an essential aspect with direct impact on sensor/comparator resolution. The measured response that results by loading both lines with the REF sample indicated before is also included in Fig. 6. The response exhibits an insertion loss of −60 dB at f 0 , pointing out that line balance is maintained by loading the lines with identical samples (to minimize the effects of the air gap, the samples have been attached to the lines by means of screws).
In order to validate the sensitivity analysis of Section IV, we have obtained the simulated transmission coefficient at f 0 by considering the REF line loaded with the REF sample, and the other line loaded with a hypothetical material of identical dimensions and different dielectric constants (the loss tangent being identical). Fig. 7(a) shows the modulus of the transmission coefficient at f 0 , as a function of the variation of the dielectric constant of the SUT, where such variation is expressed as a percentage with regard to the nominal value of the REF sample (with dielectric constant ε r = 10.2). As can be seen in Fig. 7(a), the output variable exhibits a roughly linear variation with the differential dielectric constant, and the average sensitivity with this variable has been found to be S = ∂|S 21 |/∂ %ε r = 0.0052.
From independent simulations, corresponding to line B with the different SUTs, we have inferred the phase of such line for each case, and we have evaluated the expression (23), and the output variable that results from the low-perturbation approximation (expression 25). Note that the phase of line A does not vary and corresponds to the optimum case for sensitivity optimization (with ρ · ρ = 1, as indicated earlier). The agreement between the simulated value of the transmission coefficient and the value inferred from (23) is good. The curve corresponding to the approximate expression (25) is indistinguishable from the one of (23) because the phase difference between the lines is small, even for the larger variation of the dielectric constant (20%). This points out the validity of the considered approximation (25), at least up to moderate variations in the differential dielectric constant.
We have repeated the previous simulations by considering an identical structure but elongating the lines so that the electrical length that results by loading them with the REF sample is 2.5π, corresponding to the worst case in terms of sensitivity (with Q = 0). The variation of the transmission coefficient with the differential dielectric constant for this second case is also included in Fig. 7(a). As can be seen, the variation of the transmission coefficient for small values of the differential dielectric constant is roughly negligible, in agreement with a negligible value of the sensitivity for small perturbations.
In Fig. 7(b), we have shown the dependence of the transmission coefficient with the simulated differential phase for the two considered cases in Fig. 7(a). For the optimum case (electrical length of the lines of 2π), the average slope has been found to be 0.9762, corresponding to an average sensitivity with the differential phase very close to the theoretical value (S = 1).
The previous simulations validate the analysis of Section IV relative to the dependence of the sensitivity with the length of the lines and with the termination of the isolated ports of the couplers. The slight discrepancies between the simulations and the exact expression providing the output variable, |S 21 |, are due to the fact that the lines are not perfectly matched, as discussed earlier.
It should be pointed out that the sensitivity of the output variable with the differential dielectric constant increases as the length of the line increases. The reason is that the differential phase is proportional to the length l of the lines, according to (28) and, obviously, the effects of the differential dielectric constant on the differential phase are more pronounced as l increases. For this reason, we have designed the sensor with relatively long lines (nevertheless, a tradeoff is necessary in order to avoid an excessively large sensor). In (28), c is the speed of light in vacuum, and ε eff,A and ε eff,B are the effective dielectric constants of line A and B, related to the dielectric constants of the REF and SUT samples, respectively.

A. Comparator Functionality
The functionality of the structure as comparator has been carried out by loading line A, the REF line, with the REF sample, and line B subsequently with eight identical samples,  but with arrays of drilled holes of different densities. The measured transmission coefficient corresponding to the different SUT samples are shown in Fig. 8, where it can be appreciated the significant variation of the transmission coefficient at the operating frequency, f 0 , as the density of holes varies. The comparator is able to detect tiny differences between the REF and SUT samples, as derived from the different transmission coefficient that results when line B is loaded with the REF sample and with the SUT with the smaller density of holes (the photograph of the eight fabricated SUTs is shown in Fig. 9).

B. Dielectric Constant Measurements
We have also loaded line B with different samples with well-known dielectric constant, particularly different types of uncladded dielectric substrates (i.e., FR4 with ε r = 4.6 and RO4003C with ε r = 3.55). The measured responses of the sensor that result by loading line B with such SUT samples and line A with the REF sample are shown in Fig. 10. The transmission coefficients at f 0 are shown in Fig. 11, from which we have obtained the calibration curve, also shown in Fig. 11. From this curve, the dielectric constant of the SUT can be inferred by measuring the corresponding transmission coefficient. Note that such a curve is useful for the dielectric characterization of materials with dielectric constant smaller than the one of the REF sample, with ε r = 10.2. Moreover, it is necessary that the thickness of the SUT is comparable to the one of the REF samples and SUTs used for calibration. Nevertheless, if the REF sample and the SUTs are thick enough, such that the electric field lines generated by the lines do not reach the interface between the sample and  Fig. 11. Transmission coefficient at f 0 corresponding to the SUT samples of Fig. 10. air, the response of the sensor does not depend on sample thickness.
We have fabricated a pair of SUT samples by means of a 3-D printer (Ultimaker 3 Extended). The measured dielectric constants, inferred by means of a commercial resonant cavity (Agilent 85072A), have been found to be 3 for PLA and 7.6 for RS Pro MT-COPPER material, respectively. Such samples have the same thickness to the one of the REF sample and to those of the samples used for sensor calibration. The measured transmission coefficients corresponding to such samples are also included in Figs. 10 and 11, and are in close proximity to the calibration curve, which means that the proposed sensor provides a good estimation of the dielectric constant of the SUT samples. The calibration curve taken from the measured substrates (excluding the 3-D-printed SUT samples) for the determination of the dielectric constant is included in Fig. 11. The resulting correlation factor has been found to be R 2 = 0.9998. The maximum sensitivity, inferred from the calibration curve, has been found to be |S| = 0.097, and the maximum sensitivity, in dB, as derived from the results of Fig. 10 at f 0 is 17.62 dB.
In order to discern if the dielectric constant of the SUT is larger or smaller than the one of the REF sample, let us consider again that the phase variation of line B does not experience a strong change when it is loaded with the SUT sample. Under these conditions, expression (23) can be approximated by (φ A +φ B ) ). (29) For φ A = π and φ A ≈ φ B , the last term in (29) is roughly Q ≈ 2, as discussed earlier, and the transmission coefficient can be expressed as According to (30), the phase of the transmission coefficient is +90 • for φ A < φ B (corresponding to a smaller dielectric constant for the REF sample), and −90 • for φ A > φ B (with a larger dielectric constant for the REF sample). Hence, the phase of the transmission coefficient at f 0 allows us to distinguish whether the dielectric constant of the SUT is larger or smaller than the one of the REF sample if it is needed.
It should be mentioned that the proposed sensor is useful to measure the dielectric constant of the SUT (subjected to the required thickness, i.e., identical to the one of the REF sample, as indicated before). In the considered samples, the loss tangent is small or moderately small so that the low-loss approximation holds (and therefore, the analysis of Section IV can be adopted). The proposed system is not as sensitive to the effects of losses (in the SUT sample) as resonant methods are [90]- [94]. This means that the system is not appropriate for the measurement of the loss tangent of the SUT sample, and for this reason, the measurement of this material parameter is not considered in this article. Nevertheless, it does not mean that differential sensing involving lossy materials, e.g., liquids, is not possible. On the contrary, the proposed sensor is very appropriate to measure the volume fraction of solute in diluted liquid solutions, to be discussed next. Since liquids are lossy materials, line imbalance may be caused not only by variations in the dielectric constant between the REF and SUT sample but also by changes in the loss tangent. For this reason, the generalization of the formulation of Section IV for the lossy case is considered in the Appendix. From the Appendix, it is concluded that the same phase conditions for the lines resulting in the lossless case should be applied for sensitivity optimization.

C. Solute Concentration Measurements in Liquid Solutions
In the last campaign of experiments, we have demonstrated the potential of the sensor for the determination of volume fraction in liquid solutions. In particular, we have considered mixtures of isopropanol in deionized (DI) water. In order to determine small volume fractions of isopropanol, the natural REF liquid should be DI water. Moreover, for sensitivity optimization, the sensor, particularly the length of the meandered sensing lines, should be redimensioned so that the electrical length of the REF line (line A) is φ A ≈ nπ when such line is loaded with the REF liquid. This has provided a total line length of 131.77 mm, corresponding to n = 3, i.e., 46.84 mm longer than the one of the previous device.
For the determination of the volume fraction of different solutions, we have equipped the sensing regions of lines A and B with microfluidic channels. Such channels and the necessary accessories for liquid injection (through syringes) are described in [80]. It should be mentioned that in order to avoid liquid absorption by the substrate, it has been protected by a dry film with thickness 50 μm and dielectric    Fig. 13, where it can be seen that the sensor is able to detect small concentrations of isopropanol in DI water. In particular, a volume fraction as small as 1% of isopropanol can be resolved.
The dependence of the transmission coefficient at f 0 as a function of the isopropanol content is shown in Fig. 14. The calibration curve, useful for the determination of the volume fraction (F v ) of solute in unknown mixtures of isopropanol and DI water, is also shown in Fig. 14. Such a curve, with a correlation factor of R 2 = 0.9992, is  In Table I, a comparison of various fluidic sensors devoted to measuring the volume fraction of isopropanol, methanol, or ethanol is reported. As can be seen, the proposed sensor is able to resolve 1% of volume fraction. This is a relevant result, only improved by the sensor reported in [99], where a solute concentration as small as 0.005% of volume fraction is detected, although with frequency variations smaller than 1 MHz centered around 1.83 GHz.
An advantage of the proposed sensor is that it does not require a wide frequency span for measuring purposes, contrary to most sensors reported in Table I, based on frequency variation or frequency splitting. The sensors reported in [79] and [103] (resonant differential sensors) are based on the measurement of the cross mode transmission coefficient. The output variable is indeed the maximum magnitude of the cross mode transmission coefficient, but it occurs at different frequencies (dictated by the SUT). Therefore, a wideband frequency measurement is also required in this case. Despite the fact that the results of Fig. 13 show the transmission coefficient of the reported sensor in a relatively wide frequency range, the output variable is actually the magnitude of the transmission coefficient at f 0 , the operating frequency. Namely, the sensor operates at a single frequency. This singlefrequency operation also applies to the dielectric constant sensor of the previous subsection.
Sensitivity comparison of the reported sensor with other sensors available in the literature is not easy, as far as most sensors are based on frequency variation or frequency splitting. Nevertheless, it is worth mentioning that, as compared with [79] and [103] (sensors that use similar working principle), the maximum sensitivity of the reported sensor for the measurement of volume fraction is superior (i.e., 11.52 dB/%, as compared with 0.70 dB/% in [79] and 0.62 dB/% in [103]). For dielectric constant measurements, the sensitivity of the sensor of the previous subsection (17.62 dB) is by far superior to those of the sensors reported in [79] (0.71 dB) and [103] (0.63 dB).
To end this section, we would like to mention that in the proposed sensor, the considered sensing mechanism is phase variation in the meandered transmission lines. Alternatively, it is possible to consider a sensing region based on a pair of resonator-loaded lines, as in many other differential sensors [79], [80], [103]- [105] not based on a two-port measurement by means of hybrid couplers. In this case, resonator loading modifies the resonance frequency, and the peak or notch magnitude (depending on the specific configuration). However, in such potential configuration based on a pair of resonator-loaded lines and a pair of rat-race couplers, mode conversion at the operating frequency of the rat-race couplers is expected to occur, provided the sensing resonators are asymmetrically loaded. Therefore, the proposed mode conversion detector can be also applied in this case. Concerning sensor size, it is mainly dictated by the pair of couplers. Size reduction is possible, obviously, by increasing the operational frequency. Alternatively, the ordinary couplers can be replaced by couplers based on slow-wave artificial lines, with a substantially smaller size (see [106] and references therein). All these aspects are left for future works.

VII. CONCLUSION
In conclusion, a differential sensor and comparator, based on a differential-mode to common-mode conversion detector implemented by means of a pair of rat-race hybrid couplers, has been presented in this article. The measuring principle of the device is phase variation in a pair of meandered lines. The output variable in such sensor/comparator is the transmission coefficient at the operating frequency, whereas the input variable can be, e.g., the differential dielectric constant (i.e., the one of the SUT relative to the one of the reference sample), or the volume fraction of solute in a liquid solution. The designed device is also useful to detect tiny differences between the SUT and the reference sample (comparator functionality). The conditions for sensitivity optimization have been obtained analytically, and from such analysis, it has been concluded that the length of the pair of meandered lines (the sensing regions of the device) must be a multiple of a half wavelength, provided the isolated ports of the rat race are left opened (this is a relevant contribution of this article). The results of the analysis have been corroborated through electromagnetic simulation by considering sensing structures with different line length. Then, a prototype device has been fabricated and used in different scenarios, including the detection of defects (emulated by drilling hole arrays of different densities in dielectric samples), the measurement of the dielectric constant of slabs, and the measurement of the volume fraction of solute (isopropanol) in liquid solutions (with DI water as solvent). The proposed approach is useful for the implementation of single-frequency differential sensors with very competitive sensitivity and resolution.

APPENDIX GENERALIZATION OF THE TRANSMISSION COEFFICIENT BY CONSIDERING LOSSES
By considering lines A and B loaded with lossy materials, e.g., liquids, the low-loss approximation used to obtain (23) is no longer valid. However, expression (23) can be easily generalized by simply replacing j φ A = jβ A l and j φ B = jβ B l with γ A l and γ B l, respectively, where γ A = α A + jβ A and γ B = α B + jβ B are the complex propagation constant of lines A and B, respectively. In the limit of small asymmetries (imbalances), i.e., γ A ≈ γ B , P and Q can be expressed as The modulus of the transmission coefficient is, thus, given by Inspection of (A.3)-(A.5) reveals that in order to enhance the sensitivity with either the differential phase (mainly related to the differential dielectric constant) or the differential attenuation constant (mainly dictated by the differential loss tangent), the same phase conditions for the lines resulting for the lossless case (see Section IV), should be applied. Namely, if ρ · ρ = 1, it is convenient to choose φ A ≈ φ B ≈ nπ. By contrast, for ρ · ρ = −1, the phases should satisfy φ A ≈ φ B ≈ (2n + 1)π/2. In both cases, the modulus of Q is optimized and given by Note that the resulting |Q| is smaller than 2 (the lossless value), due to the effect of losses. Note that the optimum sensitivity for small imbalances is also no longer S = 1 (lossless case), but given by by considering the differential phase as an input variable. If the input variable is considered to be the differential attenuation constant, the resulting sensitivity is found to be 1+e −(α A +α B )l . (A.8) Inspection of (A.7) indicates that if losses can be neglected (α A = α B = 0), then the optimum sensitivity satisfies S φ = S = 1, as expected.