The proposed sensor, working principle, and modeling

Figure 1 depicts the topology of the proposed sensor. The sensing elements are two identical (OSRRs grounded by one of their extremes, and connected to the isolated ports of a rat-race hybrid coupler by the other termination. The resulting structure is a two-port device, where the input signal is injected to the Δ-port of the coupler and the output signal is collected at the isolated port (Σ-port). In practice, in order to avoid interfering the functionality of the rat-race coupler, the sensing regions on top of the OSRRs must be separated from the isolated ports of the rat-race. Consequently, transmission line sections between the sensing elements (OSRRs) and the isolated ports of the coupler are needed, as depicted in Fig. 1. If the sensor is devoted to the differential measurement of liquids properties, then fluidic channels, not included in Fig. 1, should be added on top of the OSRR sensitive regions.

Fig. 1. Topology of the proposed reflective-mode differential sensor.

Let us call ρ₃ and ρ₄ to the reflection coefficients seen from ports 3 and 4 of the coupler (isolated ports), see Fig. 1. The transmission coefficient between the input and the output port of the structure, at the operating frequency, f ₀, of the coupler, is given by

(1)$$S_{12} = -\displaystyle{\,j \over 2}\;\lpar {\rho_3 -\rho_4} \rpar \comma \;$$

i.e., it is proportional to the differential reflection coefficient between both sensing loads. If we consider that the characteristic impedance of the transmission line sections between the OSRRs and the isolated ports of the coupler is identical to the reference impedance of the ports (i.e., Z ₀ = 50 Ω), such lines do not modify the magnitude of the reflection coefficients ρ_REF and ρ_MUT. The presence of such identical lines merely introduces a phase factor in the reflection coefficients given by 2ϕ, where ϕ is the electrical length of the line sections at f ₀. Thus, the transmission coefficient (1) can be expressed as

(2)$$S_{12} = -\displaystyle{\,j \over 2}{\rm e}^{{-}j2\phi}\;\lpar {\rho_{REF} -\rho_{MUT}} \rpar \comma \;$$

where ρ_REF and ρ_MUT is the reflection coefficient seen from the input termination of the OSRR corresponding to the REF and MUT samples, respectively.

An accurate circuit model of the OSRRs is depicted in Fig. 2(a). In such model, L, C_u, and G_u account for the inductance, capacitance, and conductance, respectively, of the bare (i.e., unloaded) resonator. The presence of the REF and MUT samples on top of their respective OSRRs is taken into account by means of the capacitances C_REF and C_MUT, and by means of the conductances G_REF and G_MUT. In [Reference Muñoz-Enano, Vélez, Mata-Contreras, Gil, Dubuc, Grenier and Martín48], the capacitors C ₁ and C ₂ were not considered. These capacitances have a small effect on the transmission coefficient, as it was demonstrated in [Reference Muñoz-Enano, Vélez, Mata-Contreras, Gil, Dubuc, Grenier and Martín48]. However, for an accurate prediction of the reflection coefficient, these capacitors should be included, as reported in [Reference Durán-Sindreu, Vélez, Aznar, Sisó, Bonache and Martín49]. To validate the model, we have considered an unloaded OSRR with the topology and substrate parameters indicated in Fig. 2. The frequency response (transmission and reflection coefficients), inferred by electromagnetic simulation using Keysight Momentum, is depicted in Fig. 3. Using this response, we have extracted the parameters of the circuit model according to the method indicated in [Reference Martín20] (see the values in the caption of Fig. 3). The response inferred from circuit simulation, using the schematic simulator of the Keysight ADS software package, is also depicted in Fig. 2, where it can be appreciated the good agreement with the electromagnetic simulation up to frequencies beyond the OSRR resonance.

Fig. 2. Circuit model of the pair of sensing loads (loaded OSRRs) (a) and specific topology used to validate the circuit model (b). Dimensions are (in mm): r_ext = 2.1, d = c = 0.2, s = 0.8, w = 0.9, h = 4.4, l_line = 30 and w_line = 3.4. Dimensions of the ground plane window are (in mm) l_w = 8.2 and h_w = 10.8. The considered substrate is the Rogers RO4003C with thickness h = 1.524 mm, dielectric constant ɛ_r = 3.55 and loss tangent tanδ = 0.0022.

Fig. 3. Response of the OSRR of Fig. 2. (a) Reflection coefficient; (b) transmission coefficient. The extracted parameters of the bare OSRR are C_u = 0.424 pF, L = 9.228 nH, G_u = 0.124 mS, C ₁ = 0.429 pF, C ₂ = 0.625 pF, Z ₀ = 50 Ω and ßl = 118.7°.

The results of Fig. 2 validate the proposed model. However, such a model is complex for analysis purposes. The interest is to carry out an analysis useful to determine the effects of the OSRR circuit parameters (mainly L and C_u) on device sensitivity, the fundamental performance parameter of any sensor. For this main reason, we will neglect the capacitances C ₁ and C ₂ in the following analysis.

Analysis

The subject of the analysis is thus the grounded OSRR (Fig. 1). Note that by grounding the OSRR, the effect of C ₂ is indeed eradicated. As for C ₁, though this capacitance is necessary to accurately predicting the behavior of the OSRR, we will neglect it in order to ease the analysis, as mentioned in the preceding paragraph; however, the main conclusions inferred from the analysis will be verified by numerical calculation and simulation of the full circuit model.

We would like to mention that the analytical calculation of ρ_REF and ρ_MUT (and hence S ₁₂, the output variable) from the circuit model of the OSRRs by excluding C ₁ (or even by including it) is feasible. However, in order to conclude what is the convenient strategy for sensitivity optimization, the resulting expressions are cumbersome. Thus, we have opted to proceed following a different approach. Such approach consists of calculating the real and the imaginary parts of the input impedance of the OSRR at the operating frequency, and then separately consider the effects of varying the OSRR conductance, G = G_u + G_REF, maintaining the value of the capacitance C = C_u + C_REF, and vice versa. It should be clarified at this point that by variation in either G or C we actually refer to the difference in the conductance or capacitance between the REF and SUT samples, i.e., ΔG = G_REF − G_MUT and ΔC = C_REF − C_MUT. The first case obeys, e.g., to a variation of electrolyte content (solute) in a liquid such as DI water (solvent and REF sample), where the main effect is the modification of the conductivity of the solution (MUT) and hence the conductance. At small electrolyte concentrations, the dielectric constant of the solution does not experience significant variations by varying the solute content (this justifies the invariability of C in this case). The second case (variation in C maintaining G unaltered) is more difficult to identify with a real scenario, despite the fact that in many cases low-loss samples are considered, where the conductance G can be neglected. Certainly, in most studies of differential sensing of material properties or composition, both the dielectric constant and the loss factor of the MUT, as compared to the REF sample, simultaneously vary, and, consequently, C and G do too. However, if the strategy for sensitivity optimization coincides for the two considered cases, we can extrapolate it for the general case of varying C and G.

Let us consider that the output variable is the modulus of the transmission coefficient, proportional to |ρ _REF − ρ_MUT|. The sensitivity with ΔG or ΔC is defined as

(3a)$$S_{\Delta G} = \displaystyle{1 \over 2}\;\;\displaystyle{{\vert {\rho_{REF} -\rho_{MUT}} \vert } \over \Delta G}\comma \;$$

(3b)$$S_{\Delta C} = \displaystyle{1 \over 2}\;\;\displaystyle{{\vert {\rho_{REF} -\rho_{MUT}} \vert } \over \Delta C}\comma \;$$

and to enhance the sensitivity, it is necessary to achieve the maxim possible variation in the reflection coefficients between the REF and MUT samples, for a given value of the differential conductance ΔG or capacitance ΔC. This is equivalent to maximize the variation in both the real and the imaginary parts of the impedance. Let us then consider both cited cases separately.

Variation in G, constant C

Let us first consider that the capacitance associated with the REF and MUT samples is identical and that the operating (angular) frequency is ω ₀ = ω_REF = (LC)^−1/2. The impedance of the OSRR is

(4)$$Z = \displaystyle{{1-{L}{C}{\omega}^2 + {LG\omega}j} \over {G + j\omega C}}.$$

At ω = ω ₀, the impedance can be expressed (separating the real and imaginary parts) as

(5)$$Z_{\omega_0} = \displaystyle{G \over {G^2 + \lpar C/L\rpar }} + \displaystyle{{{L}{G}^2{\omega}_0 } \over {G^2 + \lpar C/L\rpar }}j.$$

Let us now analyze the effects of varying G on the real and the imaginary parts of the above impedance. For that purpose, the derivative with respect to G is obtained, i.e.,

(6a)$$\displaystyle{{\;{\partial\rm Re}\lcub Z_{\omega_0}\rcub } \over {\;\partial G}} = \displaystyle{{-G^2 + \lpar C/L\rpar } \over {{\lpar {G^2 + \lpar C/L\rpar } \rpar }^2}}\comma \;$$

(6b)$$\displaystyle{{\;\partial{\rm Im}\lcub Z_{\omega_0 }\rcub } \over {\;\partial G}} = \displaystyle{{2CG\omega_0} \over {{\lpar {G^2 + \lpar C/L\rpar } \rpar }^2}}.$$

Inspection of 6(a) reveals that the real part of the impedance exhibits a maximum for G = (C/L)^1/2, where the value of the real part of the impedance at the operating frequency is ½[(L/C)^1/2]. For small values of G as compared to (C/L)^1/2, the usual situation is moderate and low-loss materials (the case of interest in this paper), the derivative increases with L/C. This means that increasing the ratio L/C (the product is fixed by the value of ω ₀) has the effect of enhancing the variation of the real part of the impedance of the OSRR when G varies, and this favors sensitivity. Concerning the imaginary part (6b), it is a monotonically increasing function with G, with${\rm \;Im}\lcub Z_{\omega_0 }\rcub = 0$ for G = 0, and ${\rm \;Im}\lcub Z_{\omega_0 }\rcub = L\omega_0$ for G → ∞. Nevertheless, the range of G of interest is G << (C/L)^1/2, where the derivative increases with the ratio L/C. Therefore, from this simple analysis, it is clear that in order to enhance the sensitivity of the output variable with G (considering C constant), it is necessary to increase the ratio L/C as much as possible.

Variation in C, constant G

In this case, the impedance of the OSRR at ω = ω ₀ is

(7)$$Z_{\omega_0 } = \displaystyle{G \over {G^2 + \omega_0^2 C^2}} + \displaystyle{{{L}{G}^2\omega_0 -C\omega_0 + {L}{C}^2\omega_0^3 } \over {G^2 + \omega_0^2 C^2}}j.$$

Note that, in (7), C is the variable, and only for C = C_u + C_REF ≡ C ₀, the term LCω ₀² = 1. Also, note that if ω ₀ is set to a certain value, the real part of the impedance does not depend on the ratio L/C ₀. Such a real part is a monotonically decreasing function with C, where ${\rm \;Re}\lcub Z_{\omega_0 }\rcub = 1/G$ for C = 0, ${\rm \;Re}\lcub Z_{\omega_0 }\rcub = 0$ for C → ∞, and ${\rm \;Re}\lcub Z_{\omega_0 }\rcub = G/\lpar {G^2 + \lpar C_0/L\rpar } \rpar \;$ for C = C ₀. Let us now analyze the effects of varying C in the vicinity of C ₀ on the imaginary part of the impedance. The derivative with C is

(8)$$\displaystyle{{\;\partial{\rm Im}\lcub Z_{\omega_0 }\rcub } \over {\;\partial C}} = \displaystyle{{-G^2 + C^2\omega_0^2 \;} \over {{\lpar {G^2 + \omega_0^2 C^2} \rpar }^2}}\;\omega_0\comma \;$$

and the value at C = C ₀, the point of interest, is found to be

(9)$$\left. {\displaystyle{{\;\partial{\rm Im}\lcub Z_{\omega_0 }\rcub } \over {\;\partial C}}} \right\vert _{C = C_0} = \displaystyle{{-G^2 + \lpar C_0/L\rpar \;} \over {{\lpar {G^2 + \lpar C_0/L\rpar } \rpar }^2}}\;\omega_0.$$

For G << (C/L)^1/2, the derivative is positive and it increases with the ratio L/C ₀. Thus, we obtain the same conclusion as in case A. Increasing the ratio L/C ₀ produces a larger variation of the imaginary part of the impedance by varying C, larger variation in the reflection coefficient, and, therefore, in the sensitivity.

It is interesting to note that if G ≠ 0, the capacitance that nulls the imaginary part of the impedance at ω ₀ is no longer C ₀, but

(10)$$C = \displaystyle{{1 \pm\sqrt {1-4L^2G^2\omega_0^2} } \over {2L\omega_0^2 }}\comma \;$$

as it is inferred by forcing the numerator of the imaginary part of (7) to be zero. Indeed, there are two solutions for the capacitance, one of them very close to C = C ₀ (provided G is very small), and the other one very close to C = 0. It is also interesting to comment that for the ideal situation of G → 0 (lossless case), it is very simple to justify that the sensitivity increases with the ratio L/C ₀. The reason is as follows: at ω ₀, the impedance seen from the input port of the OSRR is a short-circuit; consequently, the reflection coefficient is (ideally) −1. Since the modulus of the reflection coefficient is always 1 (due to the lack of losses), enhancing the sensitivity requires that the OSRR reactance at ω ₀ changes as much as possible by varying C. For that purpose, a peaked (i.e., narrow) reactance in the vicinity of ω ₀ is needed, and this is achieved if L/C ₀ is large.

Validation

According to the previous analysis, it can be concluded that the OSRR must be chosen with the largest possible ratio between its inductance and its capacitance (although geometrical constraints limit, in practice, such ratio). Nevertheless, the analysis has been carried out by simplifying the circuit model of the OSRR, i.e., by neglecting C ₁ (the effects of C ₂ are null since this capacitance is short-circuited to the ground). Therefore, validation by considering the complete circuit model is necessary. For that purpose, we have considered the circuit parameters of the OSRR topology of Fig. 2, extracted from the electromagnetic simulation of the frequency response, and indicated in the caption of Fig. 3. We have considered two different scenarios. In one case, we have modified the conductance of the MUT OSRR, leaving unaltered the one of the REF OSRR. In the other case, the variation corresponds to the capacitance. The idea in both cases has been to obtain the output variable of the sensor, |S ₁₂|, by considering the ratio L/C ₀ as a parameter. The specific procedure has been the numerical calculation of expression (2) using MATLAB. The considered phase, at f ₀, of the pair of connecting lines is ϕ = 118.7°. The reflection coefficients seen from the input ports of the OSRRs are given by the well-known expressions

(11)$$\rho_{{\rm REF}} = \displaystyle{{Z_{REF}-Z_0} \over {Z_{REF} + Z_0}}\;\;\comma \;\;\;\;\rho_{MUT} = \displaystyle{{Z_{MUT}-Z_0} \over {Z_{MUT} + Z_0}}\comma \;$$

where the impedance of the REF OSRRs, including C ₁, is

(12)$$Z_{REF} = \displaystyle{{1-LC\omega_0^2 + LG\omega_0j} \over {G\lpar {1-LC_1\omega_0^2 } \rpar + j\omega_0\lpar {C + C_1-LCC_1\omega_0^2 } \rpar }}\comma \;$$

with G = G_u + G_REF and C = C_u + C_REF. An identical expression for the impedance of the MUT OSRR is obtained (a simple replacement of the sub-indexes REF with MUT is needed).

Figure 4 depicts the variation of |S ₁₂| as a function ΔG, parametrized by L/C (where L and C must satisfy LC = ω ₀⁻²). It is appreciated from the figure that the variation of the output variable is stronger as L/C increases, as predicted in the previous analysis based on the simplified circuit model. By considering the variation in the capacitance of the MUT sample as compared to the one of the REF sample, the effect of the ratio L/C is exactly the same (Fig. 5). Therefore, these numerical results depicted in Figs 4 and 5 confirm the previous analysis based on an approximate model of the OSRR: increasing the ratio L/C has the effect of optimizing the sensitivity of the sensor. With this concluding remark in mind, we have designed the sensing OSRRs of the reported sensing structure, to be discussed in the next section.

Fig. 4. Variation of |S ₁₂| with ΔG at f₀ = 2.894 GHz for different values of L/C.

Fig. 5. Variation of |S ₁₂| at f₀ = 2.894 GHz with the capacitance of the MUT as compared to the REF capacitance. REF and MUT conductance are identical and equal to 0.122 mS. Note that when C_REF and C_MUT are equal, the modulus of the cross-mode transmission coefficient is zero (perfect symmetry case).

Sensor design and fabrication

The reflective-mode differential sensor will be applied to the measurement of solute concentrations in solutions of DI water. For this reason, fluidic channels must be added on top of the OSRRs, the sensitive regions. The topology of the sensing OSRRs is indeed the one depicted in Fig. 2(b), where it can be appreciated that the arms of the OSRRs are narrow and substantially separated in order to achieve a small capacitance and a large inductance (the requirement to obtain good sensitivity). However, we have replaced the short-circuits to the ground of the OSRRs with quarter-wavelength open-ended lines (with length l_inv), thereby avoiding the use of vias. The photograph (top view) of the sensor by excluding the fluidic channels and mechanical accessories is depicted in Fig. 6(a), whereas Fig. 6(b) shows a perspective view of the complete differential sensor. The sensor structure, including the rat-race coupler, connecting lines, open-ended lines and access lines, have been fabricated by means of a drilling machine (LPKF-H100 model), and the considered substrate is the one indicated in the caption of Fig. 2. The fluidic channels and the necessary mechanical accessories are identical to those reported in [Reference Muñoz-Enano, Vélez, Mata-Contreras, Gil, Dubuc, Grenier and Martín48], and therefore the details are not given in the present paper, with the exception of channel dimensions, indicated in the caption of Fig. 6 for completeness. We would like to emphasize that, similar to [Reference Muñoz-Enano, Vélez, Mata-Contreras, Gil, Dubuc, Grenier and Martín48], the sensor has been protected by means of a dry film, in order to avoid liquid absorption by the substrate.

Fig. 6. Top view of the sensor topology (a) and perspective view of the whole fabricated sensor (b). OSRR dimensions and substrate parameters are those indicated in the caption of Fig. 2. The other sensor dimensions (in mm) are l_inv = 23.473, w_rat = 1.8474 and r_rat = 23.3. Channel dimensions (in mm) are h_ch = 1.5 mm (height), l_ch = 26 mm (length), and w_ch = 4.6 mm (width).

The resonance frequency of the bare OSRRs is f ₀ = 2.894 GHz. However, by loading the designed OSRRs with DI water, the frequency is reduced to f ₀ = 1.8 GHz (nominal value inferred from electromagnetic simulation). In this study, the REF sample is DI water, and, consequently, the operating frequency of the sensor is 1.8 GHz. This frequency determines the dimensions of the rat-race coupler, as well as the dimensions of the pair of connecting lines and quarter-wavelength open-ended lines (indicated in the caption of Fig. 6).

Experimental results and comparison to other sensors

The functionality of the designed and fabricated sensor has been validated by considering solutions of isopropanol in DI water (DI water is the REF material, as indicated above). Thus, we have injected DI water in the REF channel, whereas in the MUT channel we have subsequently injected solutions of isopropanol in DI water of different concentrations, varying between 0 and 100% (pure isopropanol). The responses of the sensor for the different concentrations are depicted in Fig. 7, where the response corresponding to pure DI water in the MUT channel (indicated as REF) is also included. The measurements have been carried out by means of the Agilent N5221A PNA vector network analyzer. It can be appreciated in the figure that the minimum of |S ₁₂| for the Ref case does not exactly coincide with the nominal resonance frequency of the OSRR (1.8 GHz). This slight variation is due to tolerances in the substrate parameters and geometrical variables. Therefore, we have considered as output variable |S ₁₂| at the frequency of the minimum for the symmetric loading (i.e., DI water in both channels). Such frequency has been found to be 1.77 GHz. The results are depicted in Fig. 8. From these data, we have obtained the calibration curve, also depicted in the figure. Such a curve, with a correlation coefficient of R ² = 0.9987, is

(13)$$Fv\lpar \percnt \rpar = 184.942{\rm e}^{\lpar {S_{21}{\rm \lpar dB}\rpar /5.687} \rpar } + 81.618{\rm e}^{\lpar {S_{21}\lpar {\rm dB}\rpar /45.449} \rpar }-34.793.$$

Fig. 7. Frequency response of the sensing structure for different concentrations of isopropanol in DI water injected in the MUT channel, and pure DI water injected in the REF channel.

Fig. 8. Variation of the transmission coefficient measured at 1.77 GHz with the isopropanol content, and calibration curve.

It is worth mentioning that the sensor is able to resolve isopropanol concentrations as small as 2.5%. Such volume fraction resolution is better than the one reported in most works where similar sensors were presented [Reference Ebrahimi, Withayachumnankul, Al-Sarawi and Abbott11, Reference Vélez, Su, Grenier, Mata-Contreras, Dubuc and Martín36, Reference Vélez, Grenier, Mata-Contreras, Dubuc and Martín40, Reference Vélez, Muñoz-Enano, Gil, Mata-Contreras and Martín43, Reference Chretiennot, Dubuc and Grenier50–Reference Kilpijärvi, Halonen, Juuti and Hannu55]. Exceptions are the sensors reported in [Reference Abdolrazzaghi, Daneshmand and Iyer56, Reference Muñoz-Enano, Vélez, Gil, Mata-Contreras and Martín47], with resolutions as small as 0.005 and 1%, respectively. Nevertheless, it should be mentioned that in [Reference Abdolrazzaghi, Daneshmand and Iyer56], the reported resolution corresponds to a frequency variation smaller than 1 MHz (centered around 1.83 GHz), which seems to be difficult to discriminate in practice. The maximum sensor sensitivity, defined as the derivative of |S ₁₂| at 1.77 GHz with respect to the isopropanol concentration, is found to be 1.6 dB/%. Note that such maximum sensitivity (maximum derivative) occurs when the isopropanol content tends to zero, and then it progressively decreases. This maximum sensitivity is better than the one reported in [Reference Vélez, Grenier, Mata-Contreras, Dubuc and Martín40] (0.7 dB/%) and [Reference Vélez, Muñoz-Enano, Gil, Mata-Contreras and Martín43] (0.62 dB/%), but it is worse than the one reported in [Reference Muñoz-Enano, Vélez, Gil, Mata-Contreras and Martín47] (11.52 dB/%). Thus, according to these data, the sensor reported in [Reference Muñoz-Enano, Vélez, Gil, Mata-Contreras and Martín47] is somehow superior in terms of sensitivity and resolution, as compared to the sensor of Fig. 6. In both cases, rat-race couplers are used in order to implement two-port differential sensors operating at a single frequency. In [Reference Muñoz-Enano, Vélez, Gil, Mata-Contreras and Martín47], the sensor operates in transmission, whereas in this work we have reported a reflective-mode sensor. This is the main advantage of this sensor, as compared to the one reported in [Reference Muñoz-Enano, Vélez, Gil, Mata-Contreras and Martín47]. Namely, by operating in reflection mode, the sensing structure of this paper uses only one rat-race hybrid coupler, whereas in the transmission-mode differential sensor of [Reference Muñoz-Enano, Vélez, Gil, Mata-Contreras and Martín47], two couplers are needed in order to obtain the differential transmission coefficient (the output variable) with a simple two-port measurement. Thus, by using a single coupler, the benefits of the proposed approach in terms of size are apparent. Thus, the proposed sensor combines good performance, relatively small electrical size, differential-mode operation (using OSRR sensing loads in reflection), and it is based on simple two-port measurements at a single frequency for sensing purposes. Table 1 summarizes the comparison of the sensor proposed in this work with other sensors devoted to the characterization of ethanol or isopropanol content in aqueous solutions. Concerning the sensitivity, note that it is given in dB/% units in those sensors based on amplitude variation (operating at a single frequency), and in MHz/% for the sensors based on frequency variation (in these sensors the operation frequency is also given).

Table 1. Comparison of various sensors devoted to the determination of alcohol content in aqueous solutions