## 5 Conclusions  
In conclusion, it has been demonstrated in this paper that the plane of a slot capacitance etched on a finite substrate (uncoated on the opposite side of the slot) behaves as a magnetic wall to a very good approximation, at least under reasonable conditions (substrate thickness and dielectric constant, as well as slot width) found in most practical situations. Under the magnetic wall approximation, the slot capacitance can be considered to be composed by the parallel combination of the capacitances at both sides of the slot, while both are independent of one another. By using this parallel association of capacitances as well as the concept of equivalent dielectric constant of the substrate, an analytical expression providing the dependence of the resonance frequency with the dielectric constant of the material under test (MUT) in dumbbell defect-ground-structure (DB-DGS) based microstrip permittivity sensors has been obtained. Such equivalent dielectric constant of the substrate (defined as the necessary dielectric constant of a semi-infinite substrate providing the same contribution to the capacitance of the slot) is a fundamental parameter that, together with the dielectric constant of the MUT, determines the relative sensitivity, which is the key figure of merit of permittivity sensors based on frequency variation. Specifically, a very simple analytical expression of the relative sensitivity that does not explicitly depend on the geometry of the DB-DGS has been inferred. It depends solely on the dielectric constant of the MUT and on the equivalent dielectric constant of the substrate. Therefore, any combination of DB-DGS geometry, substrate thickness and substrate dielectric constant providing the same equivalent dielectric constant of the substrate is expected to provide the same relative sensitivity. It is also concluded that if the substrate can be considered to be semi-infinite in the vertical direction, the equivalent dielectric constant coincides with the dielectric constant of the substrate and the relative sensitivity neither depends on the DB-DGS geometry nor on the substrate thickness.  
Conceptualization and idea [PERSON]-[PERSON] and F.M. ([PERSON]); theoretical analysis, [PERSON]-E., [PERSON] and F.M. ([PERSON]); parametric analysis [PERSON] and F.M. ([PERSON]); validation, [PERSON]-E. and L.S.; writing--original draft preparation, [PERSON] ([PERSON]); writing--review and editing, [PERSON] ([PERSON]); supervision, [PERSON] and F.M. ([PERSON]). All authors have read and agreed to the published version of the manuscript.  
This work was supported by MICINN-Spain (project PID2019-103904 RB-I00), Generalitat de Catalunya (project 2017 SGR-1159), ICREA (who awarded [PERSON]), and by FEDER funds. [PERSON] acknowledges the Secretaria d'Universitats i Recerca (Gen. Cat.) and European Social Fund for the FI grant. [PERSON] acknowledges the Juan de la Cierva Program for supporting her through Project IJC2019-040786-L  
Not applicable.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript or in the decision to publish the results.