**A. Sample’s Permittivity Analysis**

It is required to know the effective permittivity and tangent loss of samples at the operating frequency range. Therefore, it is essential to determine the complex dielectric permittivity of the LUTs (Liquid Under Test) used during measurements. The dispersive permittivity characteristics of the samples in complex form can be expressed using Eq. 3:

$${\epsilon }_{r}\left(f\right)={\epsilon }_{r}^{\text{'}}\left(f\right)-{\epsilon }_{r}^{\text{'}\text{'}}\left(f\right)$$

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In Eq. 3, \({\epsilon }_{r}^{\text{'}}\) is real permittivity, \({\epsilon }_{r}^{\text{'}\text{'}}\) is imaginary permittivity while \(f\) is the frequency that defines the dispersive characteristics over the operating range of frequencies (20–35 GHz). The instantaneous permittivity over the operating frequency range is measured because materials’ permittivity exhibit a frequency dispersive phenomenon at high frequencies.19 The electromagnetic response of microwaves with water is considered a good starting point for developing a sensor because water is major content in alcohol.24 In this study, we chose ethanol as the LUT that is divided into 6 values of concentration ranging from 0 to 90% obtained by adding 0, 17.5 ml, 22.5 ml, 35 ml, 40 ml and 45 ml ethanol into 50 ml, 32.5 ml, 27.5 ml, 15 ml, 10 ml and 5 ml of water, respectively. According to extensive research, water has a higher permittivity than alcohol.25,26 The permittivity of the LUTs was measured using the Keysight 85070E Dielectric Probe Kit, which can be expressed in terms of the fitted Debye parameters. The Debye model is defined by Eq. 4:

$$\epsilon ={\epsilon }_{\infty }+\frac{{\epsilon }_{s}-{\epsilon }_{\infty }}{1+j\omega \tau }$$

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where \({\epsilon }_{\infty }\) is the dielectric constant at high frequencies, \({\epsilon }_{s}\) is the dielectric constant at low frequencies and \(\tau\) is relaxation time constant. The fitted parameters of the Debye model for each sample used in measurements are given in Table II.

Table II. Composition of liquid samples

Ethanol (%) | 90 | 80 | 70 | 45 | 35 | 0 |

\({\epsilon }_{\infty }\) | 1.44 | 1.5 | 1.62 | 1.85 | 2.29 | 7.73 |

\({\epsilon }_{s}\) | 10.85 | 24.2 | 44.05 | 61.51 | 71.17 | 78.36 |

\(\tau\) (ps) | 22.5 | 33.86 | 47.8 | 32.85 | 24.45 | 8.92 |

**B. Simulation analysis of ethanol with different concentrations**

During the simulations, the samples (LUTs) were modelled by loading the permittivity values obtained in Section 3A. As illustrated in Fig. 5, the LUT was loaded across the sensor's surface, completely covering the resonator structure. Each sample's resonant frequency was determined by substituting its dielectric properties. The transmission parameter, e.g. *S**21*, is typically taken into account to determine the resonant frequency. This parameter corresponds to the signal's negative peak. Notably, our design had a negative peak in *S**11* rather than *S**21*, which is consistent with previously described sensors with *S**11* as the sensing parameter.12,23,27,28 Fig. 6 illustrates the simulated resonant frequencies for various ethanol concentrations. The resonant frequencies of samples at concentrations of 90%, 80%, 70%, 45.5%, 35.5%, and 0% are 27.06 GHz, 26.98 GHz, 26.9 GHz, 26.68 GHz, 26.58 GHz, and 26.34 GHz, respectively.

It is demonstrated that as the ethanol concentration increases, the resonant frequency moves toward higher frequencies. It is worth noting that the difference between the first and second data points (0 and 35% ethanol) is more significant than the difference between the other intervals, owing to the more significant difference in their permittivity levels as defined by the Debye parameters in Table II. This introduces a small amount of non-linearity into the intervals, as the difference in resonant frequencies between samples containing 70 and 80 per cent ethanol and samples containing 80 and 90% ethanol is only 0.08 GHz.

Additionally, the linear regression analysis was used to determine the sensor's resonance shift in response to changes in ethanol concentration. This application seeks to establish a relationship between the input (ethanol concentration) and the output (resonant frequency). After establishing a functional relationship between these two variables using training data on obtained results, the relationship is validated using the same linear regression method on data that was not used during training. Thus, the validated model can be used to forecast the output when the input data is unknown.29 The output of regression analysis applied to the resonance shift is shown in Fig. 6, where *R**2* is the coefficient of determination. The coefficient of determination indicates the degree to which the model fits accurately. It has a value between 0 and 1 and indicates the number of data points, which lie on the regression line. The obtained *R**2* value of 0.9966 indicates that a significant proportion of data points lie on the regression line, implying that unknown outputs can be predicted with reasonable accuracy.

**C. Experimental analysis of ethanol with different concentrations**

During the measurements, coaxial cables from the N5234B PNA-L Vector Network Analyzer (VNA) were connected to the sensor's transmission lines via 2.4 mm reusable SMA connectors, as shown in Fig. 7. Before performing the measurements, the VNA was calibrated in three stages in the desired frequency range (20–35 GHz): open circuit, short circuit, and 50 loads. The same six liquid samples listed in Table II were prepared with ethanol concentrations ranging from 0–90% to avoid unnecessary complexity. The dropper was used to insert liquid over the sensor’s surface covering the entire resonator structure. The shift in resonance caused by adding alcohol to water is depicted in Fig. 8. Understandably, as the concentration of ethanol increases, the resonant frequency increases that is consistent with the simulated results. The resonant frequencies for samples containing 90% ethanol and 0% ethanol are 27.05 GHz and 25.55 GHz, respectively, while the resonant frequencies for samples containing 80%, 45%, 35%, and 0% ethanol are 26.68 GHz, 26.6 GHz, 26.3 GHz, and 26.15 GHz, respectively. When applying regression analysis on the data points, measured results show a coefficient of determination of *R**2* = 0.978, which is slightly less than the simulated result. The obtained coefficient of determination is still a significant number implying that the higher percentage of data points in measurements lie on the regression line.

The electric energy stored at the resonance frequency must equal the magnetic field stored in the resonating structure. The presence of an external field influences the net electric and magnetic fields, consequently causing a disturbance in the resonance. Eq. 5 can be used to get insight into the variation in dielectric properties of the external disturbances by associating permittivity and permeability of the external medium with the resonant frequency’s perturbation.30 Perturbation is the phenomenon of realising change in quality factor and/or shift in resonant frequency.31

$$\frac{{\varDelta f}_{r}}{{f}_{r}}=\frac{{\int }_{0}^{v}\left(\varDelta \epsilon {{E}_{0}E}_{1}+\varDelta \mu {{H}_{0}H}_{1}\right)dv}{{\int }_{0}^{v}\left({\epsilon }_{0}{\left|{E}_{0}\right|}^{2}+{\left|{H}_{0}\right|}^{2}{\mu }_{0}\right)dv}$$

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where\(v\) represents the volume of external medium that is the volume of a sample that interacts with the EM fields of the structure; \({E}_{0}\) and \({H}_{0}\) are electric field distributions and magnetic field distributions without external distributions, respectively; \({\epsilon }_{0}\) and \({\mu }_{0}\) are permittivity and permeability of free space, respectively; The change in the resonance (quality factor and/or shift) is represented by \(\varDelta f\); change in net permittivity and permeability with an external medium is represented by \(\varDelta \epsilon\) and \(\varDelta \mu\), respectively; and \({E}_{1}\) represents the external fields’ electric field distribution and \({H}_{1}\) represents external fields’ magnetic field distribution.

As stated in Section 2, the external medium’s permittivity adds to the effective capacitance (\({C}_{eff}\)) from FTL’s one side to the other side and its permeability adds to the induced current in the resonant element. Thus, it is reaffirmed that the permittivity of the introduced medium strongly influences the resonator structure on the top layer. Since the original fields have incorporated the aqueous solutions used in this work, it cannot be expected that the resonator’s internal field patterns will be approximately equal to the resonator’s field patterns without samples. It is because the interval field patterns show complicated field distributions after the placement of the sample. Even though the accurate prediction of the relationship is not possible, resonance shift can be realised versus ethanol’s concentration by a change in their permittivity values. From Eq. 6, the relative perturbation in the reflection coefficient can be approximated as 32:

$$\frac{\varDelta {f}_{r}}{{f}_{r}}\approx \frac{-\left|\varDelta \epsilon \right|h}{2\left|\epsilon \right|L}$$

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where \(\varDelta \epsilon\) is the difference in the permittivity of two consecutive ethanol concentrations as listed in Table II, \({f}_{r}\) is the reference resonant frequency, \(\varDelta {f}_{r}\) is the difference between reference and next resonant frequency, \(h\) is the height of the sample’s drop introduced on the sensor’s surface, while \(L\) is the length of the sample’s drop.

According to the measured results, the average \(\frac{\varDelta {f}_{r}}{{f}_{r}}\) for 10% change in ethanol’s concentration is determined as 0.178. The average length and its height of drop introduced on the sensor’s surface covering resonator structure entirely was 8 mm and 1 mm, respectively. Thus, the calculated average \(\frac{\varDelta {f}_{r}}{{f}_{r}}\) using Eq. 6 per 10% change in ethanol’s concentration, 0.146 is comparatively in agreement with measured results, which validates the application of the proposed geometry.