# Planar Microwave Resonant Sensors: A Review and Recent Developments
## 3 Planar Resonant Elements for Sensing
For this kind of application, with more than one input  
Figure 3: Typical topology of some electrically small slot resonators useful for sensing. (**a**) Complementary split-ring resonator (CSRR), (**b**) open complementary split-ring resonator (OCSRR), (**c**) dumbbell-shaped DGS (DB-DGS), (**d**) magnetic-LC (MLC). Figure 2: Typical topology of some electrically small metallic resonators useful for sensing. (**a**) Step impedance resonator (SIR), (**b**) step impedance shunt stub (SISS), (**c**) split-ring resonator (SRR), (**d**) open split-ring resonator (OSRR), (**e**) electric-LC (ELC). variable (or measurand), several sensitivities, including cross-sensitivities, can be defined, and this further prevents from obtaining concluding remarks with regard to the effects of resonator topology on sensitivity and resolution. However, the real part of the complex permittivity (or the dielectric constant) of the MUT mainly affects the resonance frequency of the sensing element (through the effects on the capacitance), whereas the imaginary part (related to the loss tangent) primarily affects the quality factor or the notch magnitude. For a given resonator, the strategy for optimizing such \"natural sensitivities\" (in terms of geometry) typically coincides (e.g., for the OCSRR and dumbbell-DGS, as it was demonstrated in [62, 64], respectively). It is difficult, however, to determine absolutely the optimum resonator (among those depicted in Figure 2) in terms of sensitivity (and resolution) optimization. Nonetheless, to the light of a comparative analysis of the circuit model of the considered resonators, it is possible to infer some helpful hints. As an example, let us compare three slot resonators: the CSRR, the OCSRR, and the DB-DGS. The circuit models of such resonant elements loading a transmission line are depicted in Figure 4.  
Let us assume that the resonant element (the sensing part) is loaded with a dielectric material slab (MUT) of sufficient thickness to ensure that the field lines generated in the slot do not reach the opposite interface, and let us consider that the substrate thickness satisfies also this requirement. If these sensors are intended to work based on frequency variation, the key figure of merit is the relative sensitivity of the resonance frequency, \(f_{0}\), with the dielectric constant of the MUT, \(\varepsilon_{MUT}\), given by  
\[S\ =\ \frac{1}{f_{0}}\frac{df_{0}}{d\varepsilon_{MUT}} \tag{1}\]  
where the resonance frequency is given by  
\[f_{o}\ =\ \frac{1}{2\pi}\Big{(}l\Big{(}C+C_{c}\frac{\varepsilon_{r}+\varepsilon_ {MUT}}{\varepsilon_{r}+1}\Big{)}\Big{)}^{-1/2} \tag{2}\]  
Expression (2) is valid for the CSRR-loaded line, whereas for the OCSRR- and the DB-DGS-loaded line, the previous expression is valid by skipping \(C\), the coupling capacitance between the line, and the CSRR. The unloaded resonators are described by the capacitance \(C_{c}\) and by the inductance \(L\). Notice that if the dielectric constant of the MUT is different than the one of air (i.e., \(\varepsilon_{\rm air}=1\)), the result is  
Figure 4: Circuit models of the CSRR- (**a**), OCSRR- (**b**), and DB-DGS-loaded line (**c**). a variation of the capacitance of the resonator (and hence a shift \(\inf f_{0}\)), as expected. In (1), \(\varepsilon_{r}\) is the dielectric constant of the substrate. Introducing (2) in (1), the relative sensitivity is found to be  
\[S = -\frac{1}{2}\frac{C_{c}}{C_{c}(\varepsilon_{r}+\varepsilon_{MUT})+C( \varepsilon_{r}+1)} \tag{3}\]  
In view of (3), we can conclude that it is convenient to reduce \(C\) and \(\varepsilon_{r}\) as much as possible for sensitivity optimization. Consequently, the DB-DGS and the OCSRR are good candidates for the implementation of high sensitive frequency variation sensors, at least as compared to CSRRs. In [90], it was demonstrated that the relative sensitivity of DB-DGS-loaded lines is better than the one of CSRR-loaded lines.