2.1.2. Resonator Method

A resonator method consists of using a resonating structure, such as a ring or an antenna, to derive the dielectric properties from the S-parameters with a conversion technique [33]. For that purpose, a resonant microstrip patch antenna was designed with the permittivity value measured in the previous stage (Section 2.1.1.).

This method was selected due to its simplicity, it is low-cost and low profile, and it could be easily reproduced in any research laboratory. In addition, microstrip patch antennas are intrinsically a narrow bandwidth system, which is a beneficial characteristic for sensing [11,13], because the bandwidth acts as a probe within this method. Other resonator structures, such as rings have been widely used for material characterization [34]. On the other hand, microstrip antennas are a popular solution for long-range communications, allowing them to integrate remotely the feature of passive sensing.

Furthermore, as only the antenna is under specific conditions, it would avoid damaging any expensive piece of equipment making it an ideal option for any environmental test campaign.

After prototyping the antenna and measuring its insertion losses (S11) the actual value of εr was derived based on the shift of the resonant frequency [10]. This method has good accuracy due to the narrow bandwidth nature of microstrip patch antennas, where a small variation on fr is easier to recognise and to measure than for other antenna structures.

Dimensions of a microstrip patch antenna are calculated using following Equations (2)–(4) [13,35,36]

$$\mathcal{W}\_{\rm ant} = (\mathsf{c} / (2 \mathfrak{f}\_{\rm r})) \times (\mathsf{v} / (2 / (\varepsilon\_{\rm r} + 1))) \text{ and } \mathcal{L}\_{\rm ant} = (\mathsf{c} / (2 \mathfrak{f}\_{\rm r} \times (\mathsf{v} / (\varepsilon\_{\rm r, eff}))) - 2 \,\mathsf{AL} \tag{2}$$

where Want is the width of the radiation patch, c is the speed of light in vacuum. Lant is the physical length, <sup>ε</sup>r,eff is the e ffective permittivity of the substrate and ΔL is the additional line length because of fringing fields, which could be calculated from

$$\varepsilon\_{\rm r,eff} = \left( (\varepsilon\_{\rm r} + 1)/2 \right) + \left( (\varepsilon\_{\rm r} - 1)/2 \right) + \left( 1 + (12S\_{\rm ubsh})/W\_{\rm ant} \right)^{-1/2} \tag{3}$$

$$
\Delta\text{L} = 0.412\text{S}\_{\text{ubsh}} \times ((\varepsilon\_{\text{r,eff}} + 0.3)(\varepsilon\_{\text{r,eff}} - 0.258)) \times ((\text{W}\_{\text{anti}}\text{/S}\_{\text{ubsh}} + 0.264)(\text{W}\_{\text{anti}}\text{/S}\_{\text{ubsh}} + 0.8)) \tag{4}
$$

where Subsh is the thickness of the fabric substrate.

As illustrated above, the resonant frequency of a microstrip antenna is sensitive to dielectric constant variations and according to Equations (2)–(4), if εr increases the resonant frequency of the antenna decreases. The theory of this research relays on previous studies [4,37], showing that the dielectric constant of materials increases with temperature. It was shown that the dielectric constant of a piece of Terylene film increased by 0.04 for a 20 ◦C temperature increase, from 20 ◦C to 40 ◦C.
