Abstract
The present chapter is devoted to the application of the symmetry properties in resonator-loaded transmission lines described in Chap. 4 to microwave sensors.
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Notes
- 1.
Wireless sensors may be involved in many applications. For instance, industrial processes are driven and controlled by sensor-equipped devices, often autonomous and wirelessly interconnected [2].
- 2.
For example, environmental factors may degrade the performance of displacement and alignment sensors based on transmission lines loaded with split-ring resonators where the operation principle is based on the shift of the resonance frequency [3]. On the other hand, since the resonance frequency in resonator-based sensors changes as the permittivity of the host medium changes in response to variations in environmental parameters, sensing of environmental conditions is indeed possible [20].
- 3.
Permittivity sensors based on a single resonator are also possible. Spatial sensors based on pairs of resonators can also be designed and, in fact, have been already reported in the literature [22]. This thesis, however, does not consider these sensors.
- 4.
Note that the resonator must be coupled to the line, but not connected to it. For instance, a folded stepped-impedance resonator (FSIR) may be used, but not a stepped-impedance shunt-stub (SISS).
- 5.
Though temperature can change the geometry by expansion of the materials, it is assumed that temperature does not change symmetry since microwave substrates exhibit a coefficient of thermal expansion (CTE) that allows for excellent dimensional stabilities.
- 6.
Note that although in most practical circuits and applications losses are undesirable, we take advantage of them to use the notch magnitude as a sensing electrical variable (if losses were absent, the notch magnitude would be always infinite). Losses, however, should be as low as possible to enhance the dynamic range of attenuation.
- 7.
Similarly, the notch depth is expected to exhibit low cross-sensitivities to fabrication tolerances in the nominal values of the dimensions (Sect. 6.5.2). By contrast, the resonance frequency of resonator-loaded lines is generally very sensitive to printing tolerances. For example, in ELC-loaded lines, cross-sensitivities affecting the resonance frequency and the notch magnitude can be analytically evaluated by (4.1) and (4.12), respectively.
- 8.
In these designs, the main motivation is to obtain a frequency-independent notch with the displacement that enables the sensors to operate at a single fixed frequency. Otherwise, a frequency sweeping is necessary (for notch frequency tracking), which increases the complexity of the feeding and readout systems.
- 9.
For the sake of simplicity, the working principle is explained by symmetric/asymmetric distributions. To be rigorous, the principle should be actually described in terms of the absolute perturbations applied to each resonator. This means that a single notch can be still present for asymmetric inclusions that affect equitably the two resonators.
- 10.
Besides the frequency, the depth of the resonance can also be a sensing magnitude.
- 11.
The length of the SRR- and FSIR-based CPWs are different, 3 and 2 cm, respectively. The contribution of the slot mode generation to the frequency response is therefore not comparable in the two structures. In fact, in the FSIR-based CPW the length is reduced to mitigate the generation of the slot mode. To further decrease the slot mode generation, vias and backside strips could have been used.
- 12.
As proposed in [6], the direction can actually be measured, but the dynamic range of the physical spatial quantity drops to a half. Additionally, the horizontal axis origin of the transfer curve does not correspond to the case with transparent propagation, that is, a transmission notch is present.
- 13.
Exclusively, the acronym SRR in this subsection refers also to a single split ring resonator.
- 14.
The direction sensing resonators are placed on one of the sides of the displacement sensing resonators toward miniaturization. In the event that coupling between these two resonators affects sensor performance, a solution is simply to load them on different and sufficiently distant longitudinal locations of the CPW, evidently increasing the sensor size.
- 15.
For \({\Delta x=+0.3}\) mm the measured notch is slightly below \(-3\) dB because for this sample the y-axis position sensing resonators are somewhat misaligned due to the fabrication process (we experienced some difficulties to align the top CPW and the bottom SRRs in the in-house fabrication). On the other hand, this is an indicator of the high sensitivity of the approach to detect misalignments.
- 16.
Inter-notch interference limits the performance (e.g. the input dynamic range) of the proof-of-concept sensor. Better performance is expected using folded SIRs (FSIRs) since further increase in the ratio between the second and first resonances can be achieved.
- 17.
The notch amplitude is chosen as the electrical variable to measure the angular displacement. Additionally, or alternatively, other electrical variables (e.g. phase or frequency) may be used, but the characterization with the amplitude allows for implementing the simple angular velocity sensor system that is presented in the next section.
- 18.
The smaller the distance between the CPW and the resonator, the higher their mutual magnetic coupling. However, for tiny distances, the increase in the dynamic range may be at the expense of some insertion loss for 0\(^\circ \), caused by a strong influence of the resonator to the line parameters (inductance and capacitance) that produces impedance mismatch at the 50-\(\Omega \) port interfaces. The distance is set to 1.27 mm, providing a good balance between the insertion loss at 0\(^\circ \) and the dynamic range.
- 19.
The Teflon slab is used to avoid close proximity between the resonator and the metallic shaft. Otherwise, the electrical performance of the sensor may change.
- 20.
This is absolutely true when the microstrip structure is unloaded. When the microstrip structure is asymmetrically loaded with the ELC resonator, mode conversion is generated, and the single-ended signal at the combiner output is a superposition of common- and differential-mode signals.
- 21.
The lines are uncoupled and the even- and odd-mode characteristic impedances are 50 \(\Omega \).
- 22.
The electrical path in the circular microstrip lines is longer than that for the circularly-tapered CPW of the previous subsection.
- 23.
Although the resonance frequency is sensitive to the displacement, for velocity sensing purposes, a single fixed frequency can also attain a transfer function (in notch magnitude) with a well-defined period necessary to measure the velocity.
- 24.
Optionally, the envelope signal can be in addition converted into a digital rectangular signal (by an electronic comparator) to ease the signal processing of the readout system.
- 25.
The case with \(k_m = 0\) cannot be implemented with a pair of SISSs sharing the same junction plane to the line, but this situation describes the case with isolated SISSs.
- 26.
When \(k_m \ne 0\), the lower resonance frequency, \(f_1\), is more sensitive to \({+|\Delta C_{s2}|}\) than the upper resonance frequency, \(f_2\). Complementarily, \(f_2\) is more sensitive to \({-|\Delta C_{s2}|}\). In these sensitive regions, the resonances tend to follow the resonance frequency of the perturbed uncoupled SISS (\(f_2\) for \(k_m=0\)).
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Naqui, J. (2016). Application of Symmetry Properties to Microwave Sensors. In: Symmetry Properties in Transmission Lines Loaded with Electrically Small Resonators. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-24566-9_6
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