###### Abstract  
This paper presents a parametric analysis relative to the effects of the dielectric constant of the substrate, substrate thickness and slot width on the edge capacitance of a slot-based resonator. The interest is to find the conditions (ranges of the previously cited parameters) compatible with the presence of a quasi-magnetic wall in the plane of the slot (or plane of the metallization). If such magnetic wall is present (or roughly present), the electric field in the plane of the slot is tangential (or quasi-tangential) to it and the edge capacitance can be considered to be the parallel combination of the capacitances at both sides of the slot. Moreover, variations in one of such capacitances, e.g., caused by a change in the material on top of the slot, or by a modification of the dielectric constant of the substrate do not affect the opposite capacitance. Under the magnetic wall approximation, the capacitance of certain electrically small slot-based resonators can be easily linked to the dielectric constant of the material present on top of it. The consequence is that such resonators can be used as sensing elements in a permittivity sensor and the dielectric constant of the so-called material under test (MUT) can be determined from the measured resonance frequency and a simple analytical expression. In this paper, the results of this parametric analysis are validated by considering several sensing structures based on dumbbell defect ground structure (DB-DGS) resonators of different dimensions. microwave sensor; permittivity sensor; dielectric constant; slot resonator; magnetic wall +
Footnote †: journal: Applied Sciences  
1 CIMTEC, Departament d'Enginyeria Electronica, Universitat Autonoma de Barcelona, 08193 Bellaterra, Spain; [PERSON]ESS] (P.V. ); [PERSON](L.S. ); [PERSON]S] (F.M.) 2 Grupo de Microondas, Universidad de Sevilla, Avenida Reina Mercedes s/n, 41012 Sevilla, Spain;  
[PERSON]_ADDRESS] (J.M. ); [PERSON]_ADDRESS] (F.M.) 2 Grupo de Microondas, Universidad de Sevilla, Avenida Reina Mercedes s/n, 41012 Sevilla, Spain;
and, consequently, resonant methods are good candidates for the implementation of highly sensitive permittivity sensors. There are several strategies or working principles for the implementation of permittivity sensors based on planar resonators, including frequency variation [12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27], frequency splitting [28; 29; 30; 31; 32; 33; 34; 35; 36], phase variation [37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50], coupling modulation [51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61] and differential-mode sensors [62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72] (Nevertheless, some differential sensors are based also on some of the previous working principles, e.g., phase variation [37; 38; 39; 40; 41]). Frequency-splitting, coupling-modulation and differential-mode sensors exploit electromagnetic symmetry properties [73; 74] and exhibit high robustness against cross-sensitivities related to changes in environmental factors, such as temperature or humidity (Nevertheless, most coupling-modulation sensors are devoted to the measurement of spatial variables, rather than material properties). Frequency-variation and phase-variation sensors, by contrast, are less tolerant to the effects of ambient factors, unless these sensors are implemented differentially. However, the design of these sensors is, in general, very simple. Typically, frequency-variation and phase-variation sensors are implemented by means of a transmission line and a loading or coupled planar resonant element and examples of transmission-mode (e.g., [16; 21; 47]) and reflective-mode (e.g., [45; 48]) devices have been reported. Recently, phase-variation sensors with unprecedented sensitivity have been presented [45]. In such sensors, a harmonic interrogation signal suffices for sensing. However, phase measurements, especially in reflective-mode structures, i.e., those exhibiting the highest reported sensitivities [45], are not so simple in a real scenario.