## 2 DB-DGS Microstrip Sensor and Sensitivity Analysis  
The typical topology of the DB-DGS based microstrip permittivity sensor is depicted in Figure 1, where the circuit model is included. Such model was validated in previous papers [65; 89] by comparing the circuit response given by the extracted parameters with the electromagnetic simulation, where it was concluded that the agreement is very good. The sensitivity of any sensor is defined as the derivative of the output variable with respect to the input variable. In frequency-variation permittivity sensors, there are typically two input variables or measurands, which are the dielectric constant of the MUT, \(\varepsilon_{MUT}\), and the loss tangent tan \(\delta_{MUT}\). The pair of measurands may also be the real and the imaginary parts of the relative complex permittivity (the real part is actually the dielectric constant). Nevertheless, the relevant sensitivity, which is the one used for comparison purposes, is the one providing the variation of the resonance frequency of the sensing element,\(f_{0}\), with \(\varepsilon_{MUT}\). It is obvious that the effects of \(\varepsilon_{MUT}\) on the resonance frequency are stronger in high-frequency sensors. Therefore, for the sake of comparison, the relative sensitivity is defined as follows:  
\[S=\frac{1}{f_{0}}\frac{df_{0}}{d\varepsilon_{MUT}}=\frac{1}{f_{0}}\frac{df_{0} }{d\mathcal{C}^{\prime}_{MUT}}\frac{d\mathcal{C}^{\prime}_{MUT}}{d\varepsilon _{MUT}} \tag{1}\]  
and is considered to be the more realistic and significant performance parameter [89]. The resonance frequency is given by the following:  
\[f_{0}=\frac{1}{2\pi\sqrt{LC^{\prime}_{MUT}}}\, \tag{2}\]  
where \(L\) and \(\mathcal{C}^{\prime}_{MUT}\) are the inductance and capacitance, respectively, of the DB-DGS resonator loaded with the MUT (Actually, the MUT modifies the capacitance and leaves the inductance unaltered). In order to obtain the last derivative of the right-hand side member in (1), which is \(d\mathcal{C}^{\prime}_{MUT}/d\varepsilon_{MUT}\), let us assume that the capacitance of the DB-DGS resonator can be expressed as the parallel combination of the capacitances present at both halves of the resonant element, i.e., the upper half, \(C_{U,MUT}\), and lower half \(C_{L}\). We have the following. \[\mathcal{C}^{\prime}_{MUT}=C_{U,MUT}+C_{L} \tag{3}\]  
The validity of (3) is subjected to the fact that there is a magnetic or quasi-magnetic wall in the plane of the resonator. Indeed, if both the MUT and the substrate are considered to be semi-infinite in the vertical direction, then the electric field lines in the MUT and substrate are mirror images and there is a perfect magnetic wall in that plane (it is assumed that the thickness of the metallization is negligible). The semi-infinite approximation requires that the MUT and substrate are thick enough so that the electric field intensity in the MUT/air and substrate/air interfaces can be neglected. In principle, the MUT can be chosen in order to satisfy such requirement (for instance, if the MUT is a liquid sample, a container or a fluidic channel with sufficient height can be designed). However, the thickness of the substrate might be dictated by other requirements and cannot always be considered to be semi-infinite. Thus, if the substrate has a finite thickness and the electric field intensity in the substrate/air interface is not negligible, it is apparent that the electric field lines will be refracted in that interface due to the discontinuity of the dielectric constant. Consequently and rigorously speaking, the magnetic wall in the resonator plane will no longer exist. However, it is reasonable to assume that if the ratio between the capacitive slot width, \(S\), and the substrate thickness, \(h\), is significantly smaller than one, which is a situation found in most cases, the electric field lines in the slot region are roughly tangential. Thus, a quasi-magnetic wall in that plane is expected in that case and, consequently, Equation (3) can be considered valid.