## 3 Simulation, Results, and Discussion
### Complex Permittivity Extraction  
The least squares method [32] was used to determine the complex permittivity (\(\varepsilon^{\prime}x+je^{\prime\prime}x\)) for different concentrations of ethanol in water. The complex permittivity functions are related to the change in the resonant frequency and the Q factor. The change in the resonant frequency and the Q factor is described in terms of the complex permittivity of the liquid sample, using the linear equations [32, 41] translated into matrix M (Equation (1)). For a mathematical analysis allowing the determination of the matrix coefficients, the reference values of the complex permittivity for different concentrations were extracted from [43]. These values are given in tables and illustrated by figures in Appendix A, so as to provide a better understanding of water-ethanol mixture concentrations for important frequency ranges. \[\left[ \begin{array}{c}\Delta f\\ \Delta Q\end{array} \right]=\left[ \begin{array}{cc}m_{11}&m_{12}\\ m_{21}&m_{22}\end{array} \right]\left[ \begin{array}{c}\Delta\varepsilon^{ \prime}\\ \Delta\varepsilon^{\prime\prime}\end{array} \right] \tag{1}\]  
where:  
\[\Delta f=\Delta f_{sample}-\Delta f_{reference} \tag{2}\]  
Figure 13: Simulated results: (**a**) S\({}_{21}\) response and (**b**) resonant frequency and Q factor for different mixture water concentrations, with r = 5 mm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Tube Radius (mm)** & \begin{tabular}{c} **Resonant Frequency** \\ **0\% (MHz)** \\ \end{tabular} & \begin{tabular}{c} **Resonant Frequency** \\ **100\% (MHz)** \\ \end{tabular} & \
\begin{tabular}{c} **Afr** \\ **(MHz)** \\ \end{tabular} &
\begin{tabular}{c} **Maximum Q Factor** \\ **(\%)** \\ \end{tabular} \\ \hline
0.75 & 2403.7 & 2284.3 & 119.4 & 88.4 \\
1.5 & 2303.6 & 2047.8 & 255.8 & 53.2 \\
2.5 & 2061.1 & 1819.6 & 241.5 & 44.1 \\
3.75 & 1990.7 & 1596.3 & 394.5 & 27.9 \\
5 & 1758.6 & 1376.9 & 381.7 & 23.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Sensitivity for different tube radius. \[\Delta Q=\Delta Q_{sample}-\Delta Q_{reference} \tag{3}\]  
\[\Delta\varepsilon=\Delta\varepsilon_{sample}-\Delta\varepsilon_{reference} \tag{4}\]  
where \(m_{11}\), \(m_{12}\), \(m_{21}\), and \(m_{22}\) are unknown coefficients, \(\Delta\varepsilon\) is the complex permittivity, \(\Delta f\) is the resonant frequency, and \(\Delta Q\) is the Q factor. The reference value for analysis purposes is taken at 50% water concentration. The shifts in the values related to the variation in the complex permittivity, the resonant frequency, and the Q factor for different samples of the water-ethanol mixture are defined by the matrices \(X\), \(Y_{1}\), and \(Y_{2}\), respectively, as follows:  
\[X=\left[ \begin{array}{cc}\Delta\varepsilon^{\prime}_{0\%}&\Delta \varepsilon^{\prime\prime}_{0\%}\\ \Delta\varepsilon^{\prime}_{10\%}&\Delta\varepsilon^{\prime}_{10\%}\\ \Delta\varepsilon^{\prime}_{20\%}&\Delta\varepsilon^{\prime}_{20\%}\\ \Delta\varepsilon^{\prime}_{30\%}&\Delta\varepsilon^{\prime}_{30\%}\\ \Delta\varepsilon^{\prime}_{40\%}&\Delta\varepsilon^{\prime}_{40\%}\\ \Delta\varepsilon^{\prime}_{50\%}&\Delta\varepsilon^{\prime}_{50\%}\\ \Delta\varepsilon^{\prime}_{60\%}&\Delta\varepsilon^{\prime}_{60\%}\\ \Delta\varepsilon^{\prime}_{70\%}&\Delta\varepsilon^{\prime}_{70\%}\\ \Delta\varepsilon^{\prime}_{80\%}&\Delta\varepsilon^{\prime}_{80\%}\\ \Delta\varepsilon^{\prime}_{90\%}&\Delta\varepsilon^{\prime}_{90\%}\\ \Delta\varepsilon^{\prime}_{100\%}&\Delta\varepsilon^{\prime}_{100\%}\\ \end{array} \right]=\left[ \begin{array}{cc}-34.2259&-5.2866\\ -26.6591&-1.9484\\ -20.3808&-0.651\\ -12.1428&0.4194\\ -7.7491&-0.3072\\ 0&0\\ 5.5436&-0.1577\\ 16.0126&-0.526\\ 21.5426&-2.4037\\ 28.4697&-5.009\\ 33.4518&-6.732\\ \end{array} \right],\ Y_{1}=\left[ \begin{array}{c}\Delta f_{0\%}\\ \Delta f_{10\%}\\ \Delta f_{20\%}\\ \Delta f_{30\%}\\ \Delta f_{30\%}\\ \Delta f_{30\%}\\ \Delta f_{90\%}\\ \Delta f_{90\%}\\ \Delta f_{100\%}\\ \end{array} \right]=\left[