## 4 Applications
### Differential Permittivity Sensors and Comparators  
The last reported application of this paper concerns frequency splitting sensors for dielectric measurements. The idea in these sensors, which are based on a transmission line loaded with a pair of resonant elements in a symmetric configuration, is to sense by inspection of the resonance frequencies (number and shift). If symmetry is preserved, a single notch in the transmission coefficient appears, as discussed in Section 3.3 in reference to CPWs loaded with SRRs. However, if symmetry is disrupted, frequency splitting occurs, with the result of two notches. Since symmetry can be perturbed by several means (by linear or by angular displacements, by asymmetric dielectric loading, _etc._), different physical variables can be measured. Thus, this type of frequency-splitting structures can be used as displacement sensors [35], or as sensors/comparators for dielectric characterization, among others. In the latter case, the one of interest in this section, a pair of dielectric samples is loaded onto the structure (interestingly, one of them can be a reference and well-known sample). Using this approach, a differential permittivity sensor may be designed. Alternatively, a comparison (differential by definition) between the two samples may be readily performed; if these loads are identical, a single notch appears, whereas two separated resonances are expected if they are different (note that in a comparator only the number of resonance frequencies is of interest). These dielectric comparators can be implemented by means of SRR-loaded CPW transmission lines. However, the proof-of-concept comparator illustrated in this section, firstly reported in [34], is based on a pair of shunt-connected stepped-impedance resonators (SIRs) loaded to a CPW or microstrip transmission line. Like in CPWs loaded with pairs of SRRs, there is magnetic coupling between the SIRs loaded in pairs [34]. Indeed, the expression providing the resonance frequencies is identical to that reported in Section 3.3 (Equation (2)). Nevertheless, there are some differences between the two topologies; one is relative to the position of the resonance frequency in the presence of inter-resonator coupling (above or below the intrinsic resonance frequency of the resonant element) for the symmetric configuration; another is in regard to the fact that under certain conditions a single notch is possible for asymmetric structures in SRR-loaded lines, but not in SIR-loaded lines. These details are treated in depth  
Figure 23: Angular velocity measurements for an arbitrary initial time \(t_{0}\). From [47]; reprinted with permission from IEEE 2014.  
in [26,34,51]. The interest here is thus to present a proof-of-concept comparator for dielectric measurements. It was implemented by loading a conductor-backed CPW (CB-CPW) with two identical SIRs. The structure was measured through on-wafer microwave probes without any dielectric load, and with a piece of glass (\(\varepsilon_{r}=4.6\)) on top of one of the SIRs (see Figure 24). The appearance of a lower transmission zero in the latter case caused by the symmetry perturbation validates experimentally the sensing principle indicating that different dielectric loads are present on top of the two SIRs. It is worth mentioning that dielectric microwave sensors for analysis of biological cells and organic tissue or for the determination of component concentration/composition in aqueous (or other liquid) solutions by means of microfluidics, mostly based on frequency shift (_i.e._, not based on symmetry properties), have been proposed recently [18,21,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69  
excited. However, when symmetry is perturbed, the resonator is effectively coupled to the line and a notch in the transmission coefficient appears. On the other hand, in frequency-splitting structures, where the line is loaded with two identical resonators, the resonance frequency is split into two frequencies when symmetry is disrupted.