Splitter/Combiner Microstrip Sections Loaded With Pairs of Complementary Split Ring Resonators (CSRRs): Modeling and Optimization for Differential Sensing Applications

This paper focuses on the analysis of splitter/ combiner microstrip sections where each branch is loaded with a complementary split ring resonator (CSRR). The distance between CSRRs is high, and hence, their coupling can be neglected. If the structure exhibits perfect symmetry with regard to the axial plane, a single transmission zero (notch) at the fundamental resonance of the CSRR, arises. Conversely, two notches (i.e., frequency splitting) appear if symmetry is disrupted, and their positions are determined not only by the characteristics of the CSRRs but also by the length of the splitter/combiner sections. A model that includes lumped elements (accounting for the CSRR-loaded line sections) and distributed components (corresponding to the transmission lines) is proposed and used to infer the position of the transmission zeros. Frequency splitting is useful for the implementation of differential sensors and comparators based on symmetry disruption. Using the model, the length of the splitter/combiner sections necessary to optimize the sensitivity of the structures as sensing elements is determined. Parameter extraction and comparison with electromagnetic simulations and measurements in several symmetric and asymmetric structures is used to validate the model. Finally, a prototype device sensor/comparator based on the proposed CSRR-loaded splitter/combiner microstrip sections is presented.

variable of interest (measurand) [1]- [18]. Among them, microwave sensors consisting of transmission lines loaded with planar resonators have been the subject of an intensive research activity in the last years [8], [9], [12], [18]. These structures are not exempt from a general drawback of sensors that limits their performance: cross sensitivity. Namely, the electrical variables are not only sensitive to the measurand, but also to other physical quantities. For instance, in resonance-based permittivity sensors, variations in temperature or moisture (environmental factors) may cause unintentional frequency shifts [15], [19], which in turn may produce systematic sensing errors. One solution to partially alleviate the effects of cross sensitivity, particularly those derived from changing environmental conditions, is differential sensing. Differential sensors are robust against variations in the ambient factors since such changes are seen as common-mode perturbations [16]- [18], [20], [21].
Differential sensors are typically implemented by means of two sensing elements, e.g., two loaded transmission lines, one of them acting as [16]. Nevertheless, it is possible to implement sensors scarcely sensitive to environmental factors by means of a single sensing element [22]- [24]. This can be achieved, for instance, by symmetrically loading a transmission line with a planar symmetric resonator [22], [25]- [32]. The sensing principle of these sensors is based on symmetry disruption. As pointed out in [22]- [24], by symmetrically loading a line with a resonator exhibiting symmetry plane of a different electromagnetic nature (at the fundamental resonance) from the symmetry plane of the line (typically a magnetic wall, at least in the microstrip or coplanar waveguide technology), the structure is transparent since coupling is prevented under these conditions. However, by truncating the symmetry, either electric or magnetic coupling (or both) between the line and the resonator may arise, with the result of a frequency notch (transmission zero) at the fundamental resonance frequency. Moreover, the magnitude of the notch (related to the coupling level) is determined by the level of asymmetry. Note, however, that in these coupling-modulated resonance sensors [24], environmental conditions do not produce misalignment between the line and the resonator or any other type of symmetry disruption. Therefore, these sensors are similar to differential sensors in terms of robustness against environmental factors.
Another type of sensors also exhibiting small cross sensitivity to environmental conditions is based on the symmetric loading (or coupling) of a transmission line with a pair of resonant elements [33]- [38]. Such sensors, referred to as frequency-splitting sensors [24], are true differential sensors with two sensing elements (the resonators). In this case, the sensing principle is also related to symmetry, but different than the one for coupling-modulated resonance sensors. In brief, under perfect symmetry, the structure exhibits a single transmission zero (provided the resonant elements are coupled to the line or in contact with it); however, by truncating symmetry, e.g., by loading the resonators with unbalanced dielectric loads, two notches arise, and the frequency difference depends on the level of asymmetry. One limitation of these frequency-splitting sensors may be caused by the possible coupling between resonant elements (unavoidable if such elements are close enough). Such coupling, if present, severely degrades the sensitivity for small perturbations [34]. Interresonator coupling can be prevented by considering two resonant elements separated enough, each one coupled to (or in contact with) a different transmission line in a splitter/combiner configuration as the one proposed in [39] (an alternative is a cascaded configuration, as reported in [40], where the authors consider stepped impedance resonators). However, in this case, the sensitivity is degraded by the length of the lines, and sensing, also related to symmetry disruption and based on the separation between transmission zeros (the output variable), is influenced by the interference between the pair of resonator-loaded lines. In other words, the transmission zeros are not only dictated by the intrinsic resonance frequency of the resonators, but also by the length of the lines, and such transmission zeros occur, in general, at those frequencies where the signals at the end of each loaded line exactly cancel.
This paper is focused on the analysis of frequency-splitting sensors based on splitter/combiner microstrip sections loaded with complementary split ring resonators (CSRRs), first presented in [39]. The main aim is to optimize the sensitivity to unbalanced loads or, more specifically, to obtain the necessary conditions (length of the splitter/combiner sections) to achieve such optimization. The analysis, based on a mixed distributed/lumped model of the considered structures, is presented in Section II. This section, supported by the two appendixes, constitutes the main contribution of this paper, as compared with [39]. It is clearly pointed out that one of the transmission zeros may be given by the resonance of the CSRRs, if a suitable electrical length between the position of the CSRR and the T-junctions is chosen. Validation of the model, including sensitivity optimization, is discussed in Section III, where it is clearly pointed out that sensitivity is optimized if such electrical length is selected. In Section IV, a prototype device acting as a differential sensor and comparator, i.e., a device able to detect differences between a sample under test (SUT) and a reference sample and useful for the differential measurements of dielectric constant, is presented (this includes the measurement of the dielectric constant of a known substrate-to demonstrate the potential of the approach-and the estimation of the effective dielectric constant of a sample with defects). Finally, the main conclusions are highlighted in Section V.

II. TOPOLOGY, CIRCUIT MODEL, AND ANALYSIS
The typical topology (including relevant dimensions) of the considered power splitter/combiner microstrip structure with CSRRs etched in the ground plane is shown in Fig. 1(a). Each branch consists of a 50-line loaded with a CSRR, etched in the ground plane. To match the structure to the 50-ports, impedance inverters implemented by means of 35.35-quarter wavelength transmission line sections are cascaded between the ports and the T-junctions. The circuit schematic, with the distributed and lumped elements, is shown in Fig. 1(b). An asymmetric structure is considered as general case, but asymmetry refers to the dimensions of the CSRR, rather than the transmission line sections. The lumped elements account for the CSRR-loaded microstrip line sections. Thus, L u (L l ) and C u (C l ) model the inductance and capacitance of the microstrip line, respectively, above the CSRR in the upper (lower) parallel branch, and the resonators (CSRRs) are accounted for by the tanks L Cu -C Cu (upper CSRR) and L Cl -C Cl (lower CSRR) [41]. The distributed elements account for the transmission line sections which are not located on top of the CSRRs. The line impedance Z i and the electrical length θ i (with i = 1, 2) define such line sections.
Note that to predict the transmission zero frequencies through the schematic of Fig. 1(b), the input and output transmission line sections can be neglected (such sections do not have influence on the position of the notches). The two-port network then contains two parallel branches. Thus, for analysis purposes, it is convenient to deal with the admittance matrix. Let us now center on the two-port network of Fig. 2, where the output port is terminated with the reference impedance Z 0 . A transmission zero (i.e., total reflection) results if the current at the output port is zero (I 2 = 0) and I 1 = 0. From the admittance matrix equation, considering the indicated load at port 2, the following results are obtained: and from these equations, I 2 can be isolated, that is, Therefore, Y 21 = 0 with the denominator of (2) different than zero, or Y 11 = ∞ with Y 21 = ∞ are sufficient conditions to obtain a transmission zero. It follows from reciprocity that Y 12 = 0, or Y 22 = ∞ is also sufficient conditions to obtain total reflection. Indeed, since the considered structure is symmetric with regard to the midplane between the input and output ports, the transmission zero frequencies should simply satisfy Y 21 = Y 12 = 0, and/or Y 11 = Y 22 = ∞. Let us distinguish the antidiagonal elements of the admittance matrices of the upper and lower branches of the structure of Fig. 1 by the subindexes u and l. Thus, Y 21 is given by and Y 21,u and Y 21,l can be determined by first obtaining the ABCD matrix of each branch. This is given by the matrix product of the matrices corresponding to the three cascaded two-port networks, that is, the pair of transmission line sections with characteristic impedance Z 1 , and the sandwiched lumped two-port network. From the ABCD matrices for each branch, the elements of the right-hand side in (3) are given by Y 21,u = −1/B u and Y 21,l = −1/B l , where B u and B l are the B elements of the ABCD matrix for the upper and lower branches, respectively [42]. Thus, the transmission zeros related to Y 21 = Y 12 = 0 are given by The general solution of (4) is not simple. However, it is possible to obtain the pair of transmission zeros numerically. Also, we can obtain the influence of the element values on the position of such transmission zeros. Particularly, the effects of the asymmetry produced by the CSRRs can be studied. If two identical CSRRs are considered, (4) gives a unique solution (transmission zero) with angular frequency given by where L Cu = L Cl = L r , C Cu = C Cl = C r , and C u = C l = C. Note that (6) is the frequency that shorts to ground the reactance of the identical shunt branches of the lumped two-port T-networks of Fig. 1(b), as expected.
Except for the symmetric case, where the single transmission zero is simply given by the characteristics of the resonators and their coupling to the line, the two transmission zeros of the general case are the consequence of an interfering phenomenon between the parallel CSRR-loaded line sections.
Let us now consider the alternative situation providing transmission zeros, that is, In this case, we can analyze each branch independently. The reason is that Y 11,u = ∞ and/or Y 11,l = ∞ (where the subindexes u and l have been defined before) suffices to guarantee that Y 11 = Y 11,u + Y 11,l = ∞. Therefore, let us calculate, e.g., Y 11,u . This parameter can be inferred from the elements of the ABCD matrix as Y 11,u = D u /B u [42]. Even though B u has been calculated before, this element can be simplified by designating by Y u the admittance of the shunt branch (formed by C u , L cu , and C cu ). We can proceed similarly in order to calculate D u . Once B u and D u have been inferred, Y 11,u can be expressed as a function of Y u , i.e., (7), as shown at the bottom of the next page. Inspection of (7) reveals that Y u = ∞ (corresponding to a short circuit) may provide Y 11,u = ∞. However, this is not a sufficient condition. To verify this, let us calculate Y 11,u in the limit when Y u → ∞. The result is as follows: and this result is in general finite, unless the following condition is satisfied: In (9), ω 0 is the frequency where Y u = ∞. From (9), the following result is inferred: Equation (10) gives the electrical length at ω 0 that is necessary to obtain Y 11,u = ∞ and, hence, Y 11 = ∞. However, note that in view of (2), Y 11 = ∞ does not guarantee I 2 = 0, as required to obtain a transmission zero. We can, however, express (2), for the upper branch, in terms of the ABCD matrix as follows: With the condition (9), B u = 0 and D = 1 (see Appendix I). Therefore, it is demonstrated that Y u = ∞ and (9) are sufficient conditions to obtain a transmission zero.
The physical interpretation of this transmission zero is very clear. The solution of θ 1 provided by (9) corresponds to the electrical length of the line necessary to translate the shunt branch to the input or output port. Note that if L u = 0, such electrical length is simply θ 1 = π, as expected. Thus, if the frequency that nulls the reactance of the shunt branch satisfies (9), a short is present in the input and output ports of the upper branch, hence providing a transmission zero to the whole structure. Note that such transmission zero frequency does not depend on the characteristics of the other (lower) branch, and hence, it is not associated with an interfering phenomenon, contrary to the other transmission zero (assuming asymmetry) which is still related to the destructive interference of the two branches.
It is interesting to mention that since condition (9) translates to the input and output ports the shunt reactance, such condition should also be derived by forcing the electrical length of the whole upper branch to be 2π (by excluding the shunt reactance and by considering this branch as the unit cell of a periodic structure). This is demonstrated in Appendix II.

III. MODEL VALIDATION AND OPTIMIZATION FOR SENSING PURPOSES
Model validation has been carried out by comparing lossless electromagnetic simulations of different structures with circuit simulations. To this end, it has been necessary to extract the circuit elements describing the transmission line sections loaded with CSRRs following the procedure described in [43]. For that purpose, we have first independently simulated the considered CSRR-loaded microstrip sections.
We have first considered the symmetric CSRR-loaded splitter/combiner section as shown in Fig. 1(a). The frequency response (magnitude of the transmission coefficient) inferred from electromagnetic simulation, using the Keysight Momentum commercial software, is shown in Fig. 3(a).  − ω 2 L 2 has been increased, and in the other one, it has been decreased. The responses (electromagnetic and circuit simulations) are shown in Fig. 3(b) and (c), where the corresponding sets of extracted parameters are indicated (see caption). In all the cases, there is a very good agreement between the electromagnetic and circuit simulations, pointing out the validity of the model. Fig. 3 also includes the measured responses, inferred from the Agilent N5221A vector network analyzer [see the photograph of the experimental setup in Fig. 3(d)].
We have carried out further electromagnetic simulations with different values of W l . The pairs of transmission zeros as a function of W l /W l are shown in Fig. 4(a). As W l tends to be zero, corresponding to the symmetric structure, the separation between the transmission zeros decreases. However, it can be appreciated in Fig. 4(a) that both transmission zeros do not converge (a sudden jump occurs when the structure is symmetric, with only one transmission zero, as anticipated earlier).
In the structure of Fig. 1(a), giving the responses of Fig. 3 and the pairs of transmission zeros of Fig. 4(a) for different values of W l , the electrical length of the transmission line sections between the T-junctions and the position of the CSRRs, θ 1 , does not satisfy (10). Particularly, in Fig. 1(a), θ 1 < θ 1,∞ , where θ 1,∞ is the phase that satisfies (10). We have repeated the electromagnetic simulations of the structures considered in Fig. 4(a), but by considering θ 1 > θ 1,∞ . The pairs of transmission zeros that result by varying W l are shown in Fig. 4(b). The behavior is very similar to the one observed  in Fig. 4(a). However, the single transmission zero for the symmetric structure belongs now to the opposite curve.
Finally, we have considered the case with θ 1 = θ 1,∞ [ Fig. 4(c)]. In this case, the pair of transmission zeros merge when the structure is symmetric, and the two curves cross. This is an expected result since it was demonstrated in Section II that when condition (10) is satisfied, one of the transmission zeros is given by the frequency that nulls the reactance of the upper shunt branch, regardless of the dimensions of the CSRR present at the other (lower) branch. Concerning the frequency response, a typical characteristic when θ 1 = θ 1,∞ is the similarity between the two notches (depth and width) for asymmetric structures, as it can be appreciated in Fig. 5.
If we use the CSRR-loaded splitter/combiner structures as sensors or comparators based on frequency splitting, where the sensitivity is defined as the variation of the frequency difference between the two transmission zeros ( f z1 and f z2 ) with the variable that generates the asymmetry (typically a difference in dielectric constant between two samples), it follows that the optimum structure in terms of sensitivity is the one satisfying (10), i.e., the one giving the transmission zeros of Fig. 4(c). In Fig. 4, the asymmetries are caused by varying the dimensions of one of the CSRRs, but this behavior (the dependence of the transmission zero curves with θ 1 ) does not depend on the cause of the asymmetry, and it is general for unbalanced loads. Thus, without the loss of generality, we can define the sensitivity as where f z = f z1 − f z2 . The sensitivity for the three considered cases is shown in Fig. 6, where it can be appreciated that the sensitivity for small unbalanced perturbations is clearly optimized when θ 1 = θ 1,∞ (note that if the optimum electrical length, 0.84π in our case, is not considered, but it is very close to this value, the sensitivity is expected to approach the optimum value for practical unbalanced loads).

IV. PROTOTYPE DEVICE SENSOR AND COMPARATOR
To demonstrate the potential of the structure with θ 1 = θ 1,∞ as sensor and comparator, we have first loaded it (i.e., the lower CSRR, acting as the active sensor region) with small dielectric slabs with different dielectric constants (the other CSRR is kept unloaded). Specifically, we have cut square-shaped pieces of unmetalized commercial microwave substrates with dielectric constants of 10.2 (Rogers RO3010), 3.55 (Rogers RO4003C), and 2.43 (Arlon CuClad 250). The measured responses are shown in Fig. 7(a), whereas the variation of frequency splitting, f z , with the dielectric constant, exhibiting roughly a linear variation, is shown in Fig. 7(b). This curve can be used to determine the dielectric constant of unknown substrates/samples from the measurement of the resulting frequency splitting. The dielectric constant of the SUT can in principle be arbitrarily small. Nevertheless, as the dielectric constant of the sample approaches unity, the two notches may merge in a single one by the effect of losses of the device (metallic and dielectric). Therefore, the discrimination is limited by this effect. Considering SUTs with large dielectric constant would force us to consider additional samples with known large dielectric constant in order to extend the span of the calibration curve. Apart from that, the curve of Fig. 7(b) reveals a significant variation of f z with the dielectric constant. If we assume that frequency differences (for different samples) of the order of 0.01 GHz can be distinguished (reasonable on account of the peaked responses at the notches), then differences in dielectric constants of the order of 0.35 or even less, can be detected.
As a test example, we have loaded the device with a square slab of uncladded FR4 substrate (with nominal dielectric constant 4.5). The resulting frequency splitting is f z = 0.112 GHz, providing a dielectric constant of 4.56, according to the curve of Fig. 7(b), i.e., in close agreement to the nominal value.
It is worth mentioning that the SUT is assumed to be a dielectric slab larger than the dimensions of the CSRRs. The relative position of the SUT with regard to the CSRR is not relevant as long as the SUT limits are beyond those of the CSRRs. Otherwise, the notch frequency will depend on the relative position between the SUT and the CSRR, situation that must be avoided. In principle, the proposed sensor system is useful for dielectric slabs, not for samples with arbitrary geometry. The reason is that the CSRR slots must be surrounded by the SUT material under consideration. Concerning the potentiality of this approach for the characterization of the dielectric constant of liquids, the system may be useful as long as the experimental setup is able to guarantee sealing. Obviously this needs further work, which is out of the scope of this paper. The main aim of this paper is the determination of the dielectric constant in low-loss dielectric samples.
To demonstrate its use as comparator, we have loaded the structure with two square-shaped pieces of unmetalized Rogers RO3010 substrate (with dielectric constant 10.2), but with defects in one of the samples (the SUT) in the form of a square Fig. 9.
Response of the CSRR-loaded combiner/splitter structure to two unbalanced loads, i.e., a reference sample and the same sample with defects (SUT). The response with both CSRRs loaded with the reference sample (balanced loading) is also included for comparison. array of vertical cylindrical holes with radius 0.2 mm and separated 0.8 mm (see Fig. 8). This reduces the effective dielectric constant. The measured response (Fig. 9) gives two notches, indicative of the difference between the two samples, the SUT and the reference sample (unaltered piece of substrate).
The SUT has also been measured with the other CSRR unloaded [similar to the experiment carried out in Fig. 7]. The resulting frequency splitting, f z = 0.204 GHz, indicates that the effective dielectric constant of the sample is in the vicinity of 7.4, according to the curve in Fig. 7(b). This is a reasonable value on account of the perforated cylindrical holes across the sample.
Another interesting aspect concerns the effects of pressure applied to the SUT, related to the presence of an air gap between the CSRRs and the samples. In our case, we have simply left the samples to rest on top of the CSRRs. By putting pressure, the notch positions certainly change. However, the device will work both as sensor and as comparator as long as the pressure is the same in the SUT and reference sample (comparator), or in the considered SUT as compared with the samples used for calibration (sensor).

V. CONCLUSION
In conclusion, splitter/combiner microstrip sections loaded with pairs of CSRRs have been studied in detail. These structures are useful as sensors and comparators, and their working principle is based on frequency splitting caused by symmetry disruption on the CSRR pair. Thus, the structures are especially suitable for the measurement of the dielectric constant of samples (sensor functionality) and for the detection of defects or anomalies in an SUT, as compared with a reference sample (comparator functionality). We have proposed a mixed distributed/lumped circuit model of the structures, which has been validated by comparison between the circuit responses obtained from circuit simulation (with extracted parameters) and those inferred from electromagnetic simulation and experiment. From the circuit model, we have obtained the necessary conditions, relative to the electrical length of the transmission line sections of the splitter/combiner, to optimize the sensitivity of the structures as sensing elements and the discrimination as comparators. Finally, we have demonstrated the potential of the reported CSRR-based structures as sensors and comparators by considering different dielectric loads, located in the active region of the sensor (one of the CSRRs), and by obtaining the resulting frequency splitting, and by comparing a defected SUT with a reference (unaltered) sample. In the latter case, the presence of two notches in the frequency response indicates the difference between the two samples. The reported approach solves the limitation of previous sensing structures based on the pairs of CSRRs loading a single line, where coupling between resonators degrades the sensitivity and discrimination. As pointed out in [39], several samples can be sensed/compared simultaneously by cascading several splitter/combiner sections, each one loaded with a different pair of CSRRs. Nevertheless, up to three samples can also be measured with a single splitter/combiner, by adequately locating three pairs of CSRRs in the structure: two of them in the input/output access lines, two of them in the parallel microstrip lines, at a distance given by (10) from the input T-junction, and the remaining two CSRRs also in the parallel microstrip lines at a distance given by (10) from the T-junction adjacent to the output port.
APPENDIX I B u , the denominator of (7), can be expressed as Since the condition (9) is a common factor in the previous equation, it follows that B u = 0 if (9) is satisfied. Actually, from a mathematical viewpoint, the second term in the right-hand side can be null, infinite or finite, depending on the variation of Y u with frequency in the limit when ω → ω 0 . However, for any shunt admittance, Y u , it can be demonstrated that such term is null. To this end, let us calculate this term in the indicated limit, that is, where Z u = 1/Y u is the impedance of the shunt branch, which nulls at ω 0 . Since both the numerator and the denominator of (AI.2) are zero in the indicated limit, it is necessary to apply the L'Hôpital rule. However, for simplicity, let us take the derivatives with θ 1 , rather than with the angular frequency, that is (AI.3), as shown at the top of the next page, where θ 1,0 is the electrical length of the lines at ω 0 , v p is the phase velocity of the lines, l 1 is their physical length, and Z u ' is the derivative of the shunt impedance with θ 1 (proportional to the derivative with ω). Inspection of (AI.3) reveals that the numerator is null since the left-hand term is null (condition 9), whereas the right-hand term is finite. However, the denominator is finite, as corresponds to the derivative of any reactance with frequency at resonance. Therefore, the previous limit is null, and hence, B u = 0. For which concerns D, the numerator of (7), it can be expressed as follows: According to the previous equation, if (9) is satisfied, the second term is null. The third term is also null, unless Y u → ∞. Hence, if Y u is finite, D u = 1, I 2 given by (11) is not null, and therefore, a transmission zero does not occur (as one expects since a transmission zero requires that Y u → ∞).
If Y u → ∞, the third term in (AI.4) is neither null nor infinite, but finite. The reason is that, in this case, the application of the L'Hôpital rule, by considering Z u = 1/Y u , provides a finite value of both the numerator and the denominator. According to these words, it follows that D = 1.

APPENDIX II
By excluding the shunt reactance of the upper branch in the circuit of Fig. 1(b), the elements of the ABCD matrix can be calculated. In particular, the diagonal elements are given by Since the electrical length, ϕ, of the unit cell of a periodic structure in the allowed bands is given by [23] cos ϕ = A + D 2 (AII.2) by forcing ϕ = 2π, it follows that A = 1, which is equivalent to (9). He is currently involved in the subjects related to metamaterials sensors for fluidics detection at LAAS-CNRS through a TECNIOSpring fellowship cofounded by the Mari Curie program. His current research interests include the miniaturization of passive circuits RF/microwave and sensors-based metamaterials.
Dr. Vélez was the recipient of a predoctoral teaching and research fellowship by the Spanish Government from 2011 to 2014. He is a Reviewer for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and for other journals.