###### Abstract
Frequency-variation sensors measure the variation in the notch or peak resonance frequency and eventually the magnitude (or quality factor) of a transmission line loaded with a sensing resonator that is caused by the variation in the permittivity of the so-called material under test (MUT). For sensing in an operational environment, wideband voltage controlled oscillators (VCOs) that are able to cover the entire output frequency dynamic range are needed (thereby increasing the cost). This is the main drawback of these sensors than compared to single-frequency sensors such as phase-variation, coupling modulation or some differential-mode sensors. However, frequency-variation permittivity sensors have been widely investigated with many papers devoted to sensitivity enhancement and they are still the subject of intensive research because the combinations of design simplicity, achievable sensitivity, robustness and required associated electronics in such sensors are good. There are many planar resonators useful for sensing by exploiting frequency variation. Of particular interest are those resonators exhibiting a small electrical size [74; 75; 76], including metallic and slot-based resonators, since the overall size of the sensing region can be minimized by considering such small-sized sensing elements. The examples of electrically small metallic resonators are the step impedance resonator (SIR) [77], the step impedance shunt stub (SISS) [78], the folded SIR [79], the split ring resonator (SRR) [80], the open split ring resonator (OSRR) [81], the spiral resonator (SR) [82] and the electric LC (ELC) resonator [83], among others. The slot (or complementary) counterparts of the previous resonators, i.e., the dumbbell defect ground structure resonator (DB-DGS) [84], the slotted shunt stub [85], the folded DB-DGS [73], the complementary split ring resonator (CSRR) [86], the open complementary split ring resonator (OCSRR) [87] and the magnetic LC (MLC) resonator [88], are also of interest for sensing. Indeed, slotted resonators etched, e.g., in the ground plane of a microstrip line, may represent a better alternative to their metallic counterparts. The reason is that since the MUT should be in contact or in close proximity to the resonant element, such MUT is necessarily located in the backside part of the substrate. By this means, any potential interference of the MUT (including the microfluidic channel and mechanical accessories in sensors devoted to the measurement of liquid properties) with the line and associated electronics can be circumvented. In slotted resonators, the change in the resonance frequency is caused by the variation of the capacitance, in turn produced by the effects of the MUT. It is possible to analytically obtain the expression providing the sensitivity as a function of the dielectric constant of the MUT under some reasonable hypothesis and approximations. Specifically, for that purpose,it is necessary to express the dependence of the resonator's capacitance on the dielectric constant of the MUT and this can be easily achieved if the capacitance of the resonator can be expressed as the parallel combination of the capacitances of the two halves (above and below) of the resonator plane. Such parallel association of capacitances is valid as far as the electric field in the slot region (in the plane of the slot) is tangential or quasi-tangential to it, i.e., if there is a magnetic or quasi-magnetic wall in the slot (or resonator) plane. In this paper, a parametric analysis to infer the validity of this quasi-magnetic wall approximation is carried out and then the results of the analysis are validated by means of electromagnetic simulations in sensing structures consisting in a microstrip line loaded with a DB-DGS resonator. Nevertheless, the sensitivity analysis, which is focused on obtaining the specific expression providing the sensitivity as a function of the dielectric constant of the MUT, is first developed.