## 2 DB-DGS Microstrip Sensor and Sensitivity Analysis
Nevertheless, the limits of validity of the quasi-magnetic wall approximation will be discussed in the next section. Assuming that the capacitance of the slot is separable in two parts, as given by (3), let us next calculate the relative sensitivity (1). For that purpose, it is necessary to express \(C^{\prime}_{MUT}\) as a function of \(\varepsilon_{MUT}\). If the MUT and the substrate are both semi-infinite, the following result is obtained [21; 48]:  
\[C^{\prime}_{MUT}=C^{\prime}_{air}\frac{\varepsilon_{r}+\varepsilon_{MUT}}{ \varepsilon_{r}+1} \tag{4}\]  
where \(C^{\prime}_{air}\) is the capacitance of the bare resonator, i.e., surrounded by air. The capacitance is given by the following:  
\[C^{\prime}_{air}=C_{U,air}+C_{L} \tag{5}\]  
where \(\varepsilon_{r}\) is the substrate dielectric constant. In (5), \(C_{U,air}\) is the contribution of the upper semi-space (air) to the total capacitance. By comparing (5) and (3), it is clear that the presence of any MUT on top of the resonator does not alter the contribution to the capacitance associated to the substrate, \(C_{L}\), which is an assumption valid under the quasi-magnetic wall approximation. However, (4) is not valid if the semi-infinite substrate approximation does not hold. Under these conditions and assuming that the quasi-magnetic wall approximation still holds, it follows that the capacitance of the resonator loaded with the MUT, \(C^{\prime}_{MUT}\), is given by an expression formally identical to (4):  
\[C^{\prime}_{MUT}=C^{\prime}_{air}\frac{\varepsilon_{r,a}+\varepsilon_{MUT}}{ \varepsilon_{r,a}+1} \tag{6}\]  
Figure 1: Typical topology (**a**) and equivalent circuit model (**b**) of a DB-DGS-loaded microstrip line. Relevant dimensions are indicated. The strip of the line is indicated in black color, whereas the ground plane where the DB-DGS resonator is etched, is depicted in grey color. In (**b**), the resonant tank accounts for the DB-DGS resonator, whereas the line sections of characteristic impedance \(Z_{0}\) and electrical length \(kd\), describe the pair of lines between the ports (P1 and P2) and the position where the resonator is etched (\(k\) is the phase constant and \(d\) is the length of these line sections). where the single change concerns the dielectric constant of the substrate, which is replaced with an equivalent dielectric constant, \(\epsilon_{r,eq}\), defined as the dielectric constant of a semi-infinite substrate providing an identical contribution to the capacitance, namely, \(C_{L}\). The equivalent dielectric constant can be calculated by isolating it from (6) and by considering a MUT with well-known dielectric constant \(\varepsilon_{MUT}\). This provides the following:  
\[\varepsilon_{r,eq}=\frac{\frac{C^{\prime}_{MUT}}{C\varepsilon_{ air}}-\varepsilon_{MUT}}{1-\frac{C^{\prime}_{MUT}}{C^{\prime}_{MUT}}}=\frac{f_{0, air}^{2}-\varepsilon_{MUT}f_{0,MUT}^{2}}{f_{0,MUT}^{2}-f_{0,air}^{2}} \tag{7}\]  
where \(f_{0,air}\) and \(f_{0,MUT}\) are the resonance frequencies of the bare resonator and resonator loaded with the MUT, respectively, i.e., the easily measurable quantities. Once the equivalent dielectric constant is found, the dielectric constant of an unknown MUT is obtained by isolating it from (6). The following is provided. \[\varepsilon_{MUT}=(\varepsilon_{r,eq}+1)\frac{f_{0,air}^{2}}{f_{0,MUT}^{2}}- \varepsilon_{r,eq} \tag{8}\]  
Using (2) and (6), the relative sensitivity (1) is found to be as follows:  
\[S=-\frac{1}{2}\cdot\frac{1}{\varepsilon_{r,eq}+\varepsilon_{MUT}} \tag{9}\]  
and it depends solely on the dielectric constant of the MUT. Moreover, it increases by decreasing the dielectric constant and thickness of the substrate, since this has the effect of reducing the equivalent dielectric constant.