## 4 Validation  
The objective of this section is to validate the sensitivity analysis of Section 3, including the procedure to infer the equivalent dielectric constant of the substrate as is detailed in Section 2. For that purpose, several DB-DGS geometries and substrates (with different values of the thickness and dielectric constant) are considered. Table 1 indicates such geometry and substrate parameters for the six considered structures (designated as sensors A, B, C, D, E and F). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Sensor**} & \(f_{0,air}\) & \(W_{s}\) & \multirow{2}{*}{\(\varepsilon_{r}\)} & \multirow{2}{*}{\(h\) (mm)} & \multirow{2}{*}{\(S\) (mm)} & \multirow{2}{*}{\(I\) (mm)} & \(W_{a}\) & \multirow{2}{*}{\(I_{a}\) (mm)} \\  & (GHz) & & (mm) & & & & & & (mm) \\ \hline A & 3.204 & 3.91 & 2.20 & 1.270 & 0.300 & 21 & 3.7 & 3.7 \\ B & 3.226 & 0.562 & 3.55 & 0.254 & 0.300 & 20 & 3.5 & 3.5 \\ C & 3.22 & 1.13 & 3.55 & 0.508 & 0.600 & 21.6 & 3.5 & 3.5 \\ D & 3.24 & 0.368 & 6.15 & 0.254 & 0.200 & 20 & 2.3 & 2.3 \\ E & 3.28 & 1.184 & 10.2 & 1.270 & 0.200 & 13.5 & 1.7 & 1.7 \\ F & 3.29 & 1.184 & 10.2 & 1.270 & 0.300 & 15 & 1.6 & 1.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometry and substrate parameters for the considered sensors. First, the responses of the bare sensors and the responses of the sensors loaded with different semi-infinite MUTs, which are obtained from full-wave electromagnetic simulation (using _Keysight Momentum_), are obtained. Since the interest in this paper is to validate the sensitivity analysis based on the concept of the equivalent dielectric constant of the substrate, the simulations have been carried out by considering that the loss tangent of both the substrate and MUTs are identical. The specific considered value is the one of the _Rogers RO3010_ commercial microwave substrate, i.e., \(\tan\delta=0.0035\). The metal layers with a thickness of 35 \(\upmu\)m and the conductivity of Cu are the considered parameters. The results of these electromagnetic simulations are depicted in Figure 4.  
For the six sensing structures, we have first obtained the equivalent dielectric constant of the substrate (indicated in Figure 4, as insets). For that purpose, we have used Equation (7), where the considered reference MUT is the one with dielectric constant  
Figure 4: Responses of the sensors for different dielectric constants of the material under test (MUT) (indicated). (**a**) Sensor A; (**b**) sensor B; (**c**) sensor C; (**d**) sensor D; (**e**) sensor E; (**f**) sensor F.  
\(\varepsilon_{MUT}\) = 3.55 in all cases. Introducing this value in (7) and the resonance frequencies, \(f_{0,air}\) and \(f_{0,MUT}\), which are inferred from the simulations, the equivalent dielectric constant of the substrates, \(\varepsilon_{r,\alpha q}\), is obtained. The six resulting values (one for each sensing structure) are depicted in Figure 3b for comparison purposes. The excellent agreement with the results calculated from the numerical method (briefly explained before and reported in [90]) validates the approach to deduce the equivalent dielectric constant from the simulated or measured responses. As expected, the equivalent dielectric constants for sensors B and C are identical, since the dielectric constant of the substrate is \(\varepsilon_{r}\) = 3.55 in both cases and the ratio between the slot width and the substrate thickness is also identical (i.e., \(S/h\) = 1.181). Next, we have obtained the shift of the resonance frequency for each sensor as a function of the dielectric constant of the MUT. The results are depicted in Figure 5a. Figure 5b depicts the relative sensitivity derived according to Equation (1), where the derivative \(df_{0}/d\varepsilon_{MUT}\) has been numerically obtained from the data points of Figure 5a. It follows from Figure 5b that the relative sensitivity, \(S\), is roughly indistinguishable in those sensors with identical value of the equivalent dielectric constant (sensors B and C).