## 4 Applications
### Angular Displacement and Velocity Sensors  
Resonance-based angular displacement and velocity sensors have been reported in [40,41] and [47] in CPW and microstrip technology, respectively. In all the cases, a circular-shaped ELC resonator and line are used since by this means, linearity is optimized. For the CPW-based sensor, the line is non-uniform along the propagation direction. However, matching is preserved provided the central strip and slot widths have the appropriate dimensions. With regard to the microstrip-based sensor, the transmission line (that behaves like a transducer element) is actually composed of a circular-shaped divider/combiner, as depicted in Figure 21. Let us consider that a circular ELC resonator (attached to the rotating object) is symmetrically placed above a symmetric pair of microstrip lines propagating a common-mode signal. Under these conditions, the symmetry plane (_i.e._, the midplane) behaves as a magnetic wall. As proven in [47], and also according to the discussion of Subsection 2.1, the structure is transparent to signal transmission if the ELC electric wall is aligned with the symmetry plane of the structure. Contrarily, a transmission zero arises if the alignment is with the ELC magnetic wall. In the former situation, the fundamental ELC resonance cannot be excited. By contrast, the resonator is strongly coupled (magnetically) to the lines if the magnetic walls are aligned. Such behavior, explained through the equivalent circuit models of Subsection 3.2, suggests that rotations with a 90\({}^{\circ}\) dynamic range can be measured from the common-mode transmission coefficient. By feeding the structure with a single-ended signal, the divider acts as a single-ended to common-mode signal transition. Conversely, the combiner converts the common-mode signal into a single-ended signal. The circular-shaped lines exhibit 50-\(\Omega\) even-mode characteristic impedance. The combiner/divider is implemented with 35.35-\(\Omega\) impedance inverters (quarter-wavelength lines at 1.45 GHz) to achieve matching to 50-\(\Omega\) reference ports. The circular 50-\(\Omega\) line pair (which is the active part of the transducer-like element and hence responsible for resonator coupling) must be arranged face-to-face with the ELC external ring in order to enhance electromagnetic coupling. Furthermore, the line pair as well as the resonator was circular-shaped to enhance linearity, as mentioned above. Figure 20: Simulated and measured differential-mode return loss \(|S_{dd11}|\) and insertion loss \(|S_{dd21}|\), and common-mode insertion loss \(|S_{cc21}|\) for the structure of Figure 19b with and without the presence of the DS-CSRRs. From [27]; reprinted with permission from IEEE 2012. The displacement measurement was performed using an experimental set-up similar to that reported in [41]. The ELC substrate was attached to a step motor through a Teflon slab with 3.5-mm thickness and \(\varepsilon_{r}=2.08\). The air gap between the ELC and the microstrip lines was set to 0.254 mm. As shown in Figure 22, the rotation angle can be sensed from the notch magnitude (and frequency as well), the dependence being reasonably linear. Let us now consider that the ELC is continuously rotating. To obtain the angular velocity, \(\omega_{r}\), a harmonic signal tuned at a fixed frequency (e.g., the notch frequency corresponding to \(\theta=10^{\circ}\), \(f_{0}=1.308\) GHz) is injected to the input port, whereas an isolator plus an envelope detector are cascaded to the output port. For two different angular velocities, the envelope signals at the output (which are dependent on the transmission coefficient) are captured by an oscilloscope (Figure 23), and the velocities are derived from the time difference between consecutive transmission peaks, \(T_{m}=T_{r}/2=1/2f_{r}\) (\(\omega_{r}=2\pi f_{r}\)). By configuring the step motor with \(f_{r}=1\) Hz and 50 Hz (_i.e._ nominal values), the measured  
Figure 21: Angular displacement sensor. (**a**) Layout; and (**b**) photograph.