*Article* **Multibeam Reflectarrays in Ka-Band for Efficient Antenna Farms Onboard Broadband Communication Satellites †**

**Daniel Martinez-de-Rioja 1,\* , Eduardo Martinez-de-Rioja 2 , Yolanda Rodriguez-Vaqueiro 3 , Jose A. Encinar 1 and Antonio Pino 3**

**Abstract:** Broadband communication satellites in Ka-band commonly use four reflector antennas to generate a multispot coverage. In this paper, four different multibeam antenna farms are proposed to generate the complete multispot coverage using only two multibeam reflectarrays, making it possible to halve the number of required antennas onboard the satellite. The proposed solutions include flat and curved reflectarrays with single or dual band operation, the operating principles of which have been experimentally validated. The designed multibeam reflectarrays for each antenna farm have been analyzed to evaluate their agreement with the antenna requirements for real satellite scenarios in Ka-band. The results show that the proposed configurations have the potential to reduce the number of antennas and feed-chains onboard the satellite, from four reflectors to two reflectarrays, enabling a significant reduction in cost, mass, and volume of the payload, which provides a considerable benefit for satellite operators.

**Keywords:** reflectarray antennas; multibeam antennas; dual band reflectarrays; communication satellites; Ka-band

#### **1. Introduction**

In the past years, the continental contoured beams traditionally used for broadcast satellite applications in Ku-band are being replaced by cellular coverages to provide broadband services typically in Ka-band [1]. The cellular coverages are formed by between 50 and 100 slightly overlapping spot beams, generated with a frequency and polarization reuse scheme of four colors, where each color is associated to a unique combination of frequency and polarization [2]. Thus, the four-color scheme requires splitting the available user spectrum into two different frequency sub-bands and two orthogonal circular polarizations (CP) [2,3]. The generation of four-color coverages makes it possible to spatially isolate the spots generated with same frequency and polarization, so the interference between spots is reduced and the throughput of the users can be increased without modifying the operational bandwidth of the system. The high-gain beams generated by the satellite antennas produce circular spots on the Earth's surface. The directions of the beams are adjusted to produce a triangular lattice of circular spots. Therefore, the service area can be split into hexagonal cells, each of which is covered by a circular beam [2]. Figure 1 shows an example of a four-color coverage based on hexagonal cells. The diameter of the spots is defined depending on the capacity need of the system. Typically, the spots cover a circular area of around 250–300 km in diameter, which corresponds to a beamwidth of about 0.5–0.65◦ for the beams produced by the satellite antennas [3].

**Citation:** Martinez-de-Rioja, D.; Martinez-de-Rioja, E.;

Rodriguez-Vaqueiro, Y.; Encinar, J.A.; Pino, A. Multibeam Reflectarrays in Ka-Band for Efficient Antenna Farms Onboard Broadband Communication Satellites . *Sensors* **2021**, *21*, 207. https://doi.org/10.3390/s21010207

Received: 25 November 2020 Accepted: 28 December 2020 Published: 31 December 2020

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**Copyright:** © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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**Figure 1.** Example of a four-color multispot coverage. (**a**) Hexagonal service cells working in four different colors. (**b**) Hexagonal cells covered by circular beams generated in two orthogonal polarizations (P1, P2) and two frequencies (F1, F2). (**c**) Example of a European four-color coverage that would be generated by a geostationary satellite.

The generation of the multispot coverage from the satellite is typically accomplished by using four multi-fed single offset reflectors operating by a single feed per beam (SFPB) architecture [3,4]. Reflectors offer a reliable and simple operation; however, they cannot generate the beams with an angular separation between adjacent beams as small as required for this application, since overlapping feeds would be required. Thus, four multi-fed reflectors are usually employed onboard the satellite, where each reflector produces all the beams in a specific frequency and polarization, which are spatially isolated from each other. In this way, each reflector generates the beams associated to one color in a lattice of non-contiguous spots (the spots are separated by gaps that will be covered by the beams generated by the other reflectors). Therefore, the interlaced beams produced by the four reflectors form the final four-color cellular coverage of contiguous spots.

Due to the severe constraints in weight and volume of satellites, the use of four reflectors can be seen as a suboptimal antenna farm. Different multibeam antenna solutions have been proposed lately to reduce the number of antennas onboard the satellite [5]. The use of lenses [6] or array antennas [7] reduces the radiation efficiency and increases the complexity of the antenna architecture, while the highly oversized reflector proposed in [8] comprises a prohibitive stowage volume. In this paper, four different multibeam antenna solutions are proposed based on reflectarray antennas [9], following the study introduced in [10]. Each multibeam reflectarray is intended to generate half the required multispot coverage, making it possible to reduce the number of antennas onboard the geostationary satellite from four reflectors to two reflectarrays. The antenna specifications commonly required in real scenarios (described in Section 2) will be used to analyze the performance of the proposed reflectarrays. In Section 3, a 1.8 m flat reflectarray and a 1.8 m parabolic reflectarray will be proposed to generate a four-color coverage only for the transmission link with a single antenna aperture. Then, a dual-reflectarray system and a 1.8 m parabolic reflectarray will be shown in Sections 4 and 5, respectively, to generate half the required spots (two colors) simultaneously in transmission (Tx) and reception (Rx). The main characteristics of the four reflectarrays will be compared in Section 6, proving the great potential of reflectarrays for multibeam satellite applications and their capability to halve the number of antennas and feed-chains required onboard the satellite to generate a four-color coverage.

#### **2. Mission Scenario and Requirements of the Antenna System**

The main specifications of the cellular coverage, shown in Table 1, have been established from those of current multibeam satellites in Ka-band [2,3,11,12] and the Rec. ITU-R S.672-4 [13]. The coverage must be formed by around 100 spots in a triangular lattice,

generated with a four-color reuse scheme based on two different frequencies and two orthogonal CP. The four-color coverage must be simultaneously generated at Tx and Rx user frequencies in Ka-band, which comprise from 19.2 to 20.2 GHz for Tx and from 29.0 to 30.0 GHz for Rx. Thus, both Tx and Rx frequency bands must be divided into two sub-bands to provide the four-color coverage with two orthogonal CP simultaneously in Tx and Rx. Figure 2 shows a schematic representation of a multispot coverage with a four-color reuse scheme of two frequencies (F1, F2) and two polarizations (P1, P2) together with the operating scheme of the current multi-fed reflectors used onboard the satellite, where each reflector generates a lattice of non-contiguous spots in a single color, simultaneously in Tx and Rx (the interlaced beams produced by the four reflectors form the final four-color coverage of contiguous spots). The diameter of the spots is set to 0.65◦ to cover a circular area of around 300 km on the Earth. The angular separation between adjacent spots from center to center to form the appropriate triangular lattice of spots must be 0.56◦ (computed as sin(60◦ )·0.65◦ ). The minimum end of coverage (EOC) gain of the beams is set to 45 dBi, the minimum single-entry carrier over interference ratio (C/I) at 20 dB, and the minimum co-polar over cross-polar discrimination (XPD) is also fixed to 20 dB.

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**Table 1.** Antenna system requirements.

**Figure 2.** (**a**) Four-color multispot coverage required simultaneously in Tx and Rx. (**b**) Operating principle of current reflectors used onboard the satellite, operating simultaneously in Tx and Rx.

The proposed antenna configurations operate by an improved SFPB configuration, taking advantage of the ability of reflectarrays to generate independent beams at different frequencies [14] or polarizations [15] with a single feed. The proposed reflectarrays will be illuminated by a cluster of 27 feeds defined with a triangular lattice, in order to provide the required multibeam coverage with a triangular grid of spots. The feeds have been modelled considering the 54 mm Ka-band feed-chain reported in [4], thus the separation between adjacent feeds has been set to 55 mm. The radiation pattern of the feeds has been

modelled in the simulations by an ideal cosq (*θ*) distribution. Figure 3 shows the proposed cluster of 27 feeds using the local coordinate system (xf , yf ), the origin of which matches the center of the aperture of the central feed (C3 in Figure 3). The symmetry of each antenna configuration will be used to reduce the number of feeds considered in simulations (the lines of feeds A, B, C, plotted with thicker lines, or the array of 3x5 feeds plotted with solid lines in Figure 3).

*θ*

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**Figure 3.** Cluster of 27 feeds defined to illuminate the different antenna farms.

#### **3. Antenna Farm Based on Two Single-Band Reflectarrays**

The first strategy introduced in [10] to generate a complete four-color multispot coverage simultaneously in Tx and Rx is based on the design of a reflectarray antenna to generate four spaced beams per feed in two different operating frequencies and two orthogonal polarizations (i.e., four spaced beams in four different colors). In this way, the limitation of conventional reflectors to provide such closely spaced beams is overcome by the generation of four adjacent beams per feed. Thus, a reflectarray illuminated by the proposed cluster of 27 feeds would generate a complete four-color coverage of 108 spots only for a single band (Tx or Rx). To provide multispot coverage both in Tx and Rx, two single-band reflectarrays designed with the same technique to produce four adjacent beams per feed would be required onboard the satellite (one for Tx and the other for Rx). The operating scheme of the proposed solution is shown in Figure 4 for a flat reflectarray to operate in the Tx band in Ka-band.

This solution requires a reflectarray with independent operation at close frequencies (within the same band for Tx or Rx in Ka-band) and also independent operation in orthogonal polarizations, which increases the complexity of the reflectarray cells and the design technique. The controlled application of the beam squint effect with frequency has been used to reduce the design complexity of the reflectarray antenna, as shown in [16]. The method to generate four spaced beams in four different colors per feed has been experimentally validated in [17] for a 43 cm reflectarray antenna. The prototype operates in linear polarization (LP), but the same design technique can be applied to produce the beams in CP by an appropriate selection of the reflectarray cells. Figure 5 shows the 43 cm prototype in the anechoic chamber and the measured radiation patterns of the four spaced beams in four different colors, according to the normalized angular coordinates u = sin*θ*·cos*ϕ*, v = sin*θ*·sin*ϕ*.

**Figure 4.** Operating principle of the proposed single-band reflectarray. *θ φ θ φ*

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− − − **Figure 5.** Reflectarray prototype to generate four spaced beams per feed [17]. (**a**) Picture of the prototype, (**b**) measured pattern contours at −1, −3, and −4 dB of the four beams generated by the same feed.

− − − Two different approaches have been evaluated to produce a multispot coverage following the design technique developed in [17]: first, using a flat reflectarray as proposed in [10], and second, using a reflectarray with a parabolic surface. The two reflectarrays are expected to generate a four-color coverage for Tx in Ka-band using the feed cluster shown in Figure 3. Due to the symmetry of the antenna system, the simulations have considered the lines of feeds A, B, and C in Figure 3 (plotted with thicker lines in Figure 3), since there will be minimal differences between the beams generated by the lines of feeds B, C and D, E. The two reflectarrays have a diameter of 1.8 m and operate at 19.45 and 19.95 GHz in orthogonal polarizations. The simulations have considered ideal reflectarray elements to provide the required phase-shift in each color without phase errors or ohmic losses. The preliminary results will be used to evaluate the strengths and weaknesses of the design method. The conclusions reached in this study can also be applied to the associated reflectarrays used for Rx, since the design method would be identical.

#### *3.1. Flat Transmit Reflectarray to Generate Four Spaced Beams per Feed in Ka-Band*

A flat reflectarray has been proposed to generate four spaced beams per feed according to the design technique shown in [17]. The reflectarray has a diameter of 1.8 m and it is formed by 44125 reflectarray cells arranged in a 239 × 235 lattice, the period of which has

been set to 7.5 mm to avoid grating lobes. As a result of the design method, four different phase distributions are implemented in the reflectarray surface (one for each combination of frequency and polarization). Figure 6 shows the required phase distributions at 19.45 and 19.95 GHz for one CP (there are no appreciable differences with the phase distributions for the orthogonal CP). The large size of the reflectarray and its flat configuration are the reason why the phase distributions depicted in Figure 6 show a large number of 360◦ cycles. The abrupt variations in the phase distributions limit the antenna performance and reduce the operational bandwidth.

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**Figure 6.** Required phase distributions of the 1.8 m flat reflectarray at the (**a**) lower and (**b**) upper operating frequencies for the same CP.

The results of the conducted simulations when the 1.8 m reflectarray is illuminated by the 16 feeds placed in the lines A, B, and C in Figure 3 are shown in Figure 7, which presents the pattern contours of the beams for a 46 dBi gain. The simulated pattern contours show a coverage of 64 beams generated in a four-color reuse scheme. The distribution of beams meets the required triangular lattice of spots, as well as the separation and diameter of the circular spots (0.56◦ and 0.65◦ , respectively). **2021**, , x FOR PEER REVIEW 7 of 18

**Figure 7.** Simulated 46 dBi pattern contours of the 64 beams generated by 16 feeds in four different colors with a 1.8 m flat reflectarray.

The main cuts of the radiation patterns have been evaluated to estimate the interferences between beams generated in the same color. Figure 8 shows the horizontal cut of the radiation patterns in the plane v = 0 for the beams generated at both frequencies. Due to the monofocal antenna design, the extreme beams of the coverage show some aberrations that increase the C/I. Since the side-lobe levels of the beams can be reduced by slightly increasing the antenna aperture and reducing the edge illumination, this drawback can be

easily overcome (space reflectors in Ka-band usually have a diameter of 2.3 m). Thus, the diameter of 1.8 m has been maintained for the rest of the reflectarray antennas proposed in this paper, expecting similar C/I levels. The use of ideal reflectarray cells prevents an accurate characterization of the cross-polar radiation, which is mainly produced by the reflectarray elements and the offset antenna configuration. Previous reflectarray prototypes with offset configurations have provided XPD above 25 dB [17,18].

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**Figure 8.** Cut of the simulated radiation patterns in v = 0 for the beams generated at the (**a**) lower and (**b**) upper frequency in the same circular polarization (CP).

The main constraint of this strategy is the operational bandwidth of the reflectarray, limited by the abrupt phase variations shown in Figure 6 and the independent operation at two close frequencies within the same link (Tx or Rx) in Ka-band.

#### *3.2. Parabolic Transmit Reflectarray to Generate Four Spaced Beams per Feed in Ka-Band*

The design of large and flat reflectarrays results in inappropriate phase distributions (with fast phase variations) that limit the antenna performance. The previous design of a multibeam reflectarray to generate four spaced beams per feed has been improved to fit a 1.8 m parabolic reflectarray. In this configuration, the parabolic surface shapes the beams as a conventional reflector, while the reflectarray elements only introduce a small phase correction to slightly deviate the beams produced in orthogonal polarizations and/or at different operating frequencies.

The four required phase distributions that must be implemented in the parabolic reflectarray have been computed and Figure 9 shows the required phase distributions at 19.45 and 19.95 GHz for one CP (the phase distributions for the orthogonal CP have a similar behavior). The phase distributions in Figure 9 show a great improvement with respect to the equivalent distributions for a flat reflectarray (see Figure 6), presenting smooth phase variations and a reduced number of 360◦ cycles. The 1.8 m parabolic reflectarray antenna has been simulated considering the same cluster of feeds as in the previous design, producing a four-color coverage of 64 beams similar to that shown in Figure 7. There are no appreciable differences between the simulated radiation patterns of both flat and parabolic reflectarrays (a comparison between the antenna performances of flat and parabolic reflectarrays to produce four beams per feed can be seen in detail in [19]).

The results obtained for this (flat or parabolic) antenna farm show that a single reflectarray with 27 feeds would produce a four-color coverage of 108 beams only in the Tx or Rx link of the Ka-band, so two reflectarrays would be needed to generate a complete multispot coverage both in Tx and Rx. As a result, the number of antennas would be halved

with respect to the conventional four-reflector configuration, and the same would happen with the number of feed-chains (since each feed produces four beams in Tx or Rx).

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**Figure 9.** Required phase distributions of the 1.8 m parabolic reflectarray at the (**a**) lower and (**b**) upper operating frequencies for the same CP.

Moreover, this solution makes it possible to reduce the complexity of the feed-chains, since each reflectarray would be illuminated by single-band feeds, instead of the current dual-band feed-chains [4]. The use of independent reflectarrays to operate in Tx and Rx can be exploited to reduce the size of the Rx antenna, due to its higher operating frequency. However, the main limitation is related to the independent operation at two close frequencies. Although the parabolic surface has overcome the drawback associated with the required phase distributions, further research is needed on the reflectarray cell that should be used in a real scenario to achieve dual-band operation at near frequencies. The cell proposed in [20] could be scaled to operate at two close frequencies within the Tx band in Ka-band in dual CP. Once the reflectarray cell is defined, intense optimizations should be applied to provide an independent operation in both frequencies with a stable performance in band.

#### **4. Antenna Farm Based on Two Dual-Band Dual Reflectarray Configurations**

The constraints arising from the design of reflectarrays with independent operation at two close frequencies (within the same band for Tx or Rx) can be overcome by implementing the dual-frequency operation in two separate frequency bands. Then, the reflectarray must generate two spaced beams per feed in orthogonal CP simultaneously at Tx and Rx frequencies in Ka-band. In this way, the reflectarray is expected to generate all the beams in two different colors simultaneously in Tx and Rx. A second dual-band reflectarray with slightly different operating frequencies would generate the second half of the coverage, and the interlaced coverages of the two reflectarrays would form the complete four-color coverage in Tx and Rx. However, the design of a reflectarray with independent operation in dual CP simultaneously at two frequency bands is a complex task.

In this strategy, the reflectarray has been defined with a dual-antenna configuration based on a flat subreflector reflectarray and a main parabolic reflectarray. The use of a dual configuration makes it possible to simplify the implementation of the dual-band operation in dual CP thanks to the larger number of degrees of freedom provided by the use of two reflectarrays. In this way, the feeds will operate in dual-LP, which implies a lesser complexity of the feed-chain than in the case of dual-CP operation (there is no need for a polarizer in the feed-chain). Then, the subreflectarray will be designed to deviate (without focusing) the dual-LP beams radiated by the feeds by ±0.28◦ (in opposite directions for each LP) simultaneously in Tx and Rx. The main reflectarray will be designed to convert the incident dual-LP field into dual-CP, at the same time as its parabolic surface focuses high-gain beams. Figure 10a shows the operating principle of the dual reflectarray antenna to generate two spaced beams in left-handed and right-handed CP (LHCP and RHCP, respectively) from a single feed operating in horizontal and vertical LP (H and V,

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respectively). Figure 10b shows the operating scheme of the proposed antenna to generate half the required multispot coverage (two colors) simultaneously in Tx and Rx.

**Figure 10.** Dual reflectarray system proposed to operate simultaneously in Tx and Rx. (**a**) Operating principle of the dual-configuration, (**b**) functional scheme of the proposed antenna on the satellite.

The operating principles behind this solution have been experimentally validated separately. The design of a dual-band reflectarray with independent operation in dual LP at two separated frequencies can be reached in a direct way, as proposed in [21], by means of reflectarray cells based on orthogonal sets of parallel dipoles for each band. A polarizing reflectarray to convert dual LP into dual CP with broadband operation from 20 to 30 GHz (covering both Tx and Rx bands) has been demonstrated in [22].

The dual reflectarray designed to produce multiple spot beams in Ka-band has been defined on the basis of a Cassegrain system. The flat subreflectarray has a diameter of 0.65 m and it is formed by 11,497 cells disposed in a 117 × 125 grid, considering the same type of reflectarray cells used in [23] (based on two orthogonal set of parallel dipoles) with a period of 5.3 mm. The main parabolic reflectarray has a diameter of 1.8 m, formed by 62,654 cells disposed in a 286 × 279 grid. It has been designed using the polarizing cells shown in [22] (based on three coplanar parallel dipoles) with a period of 5 mm. In contrast to the complex phase distributions computed for the previous flat 1.8 m reflectarray (see Figure 6), the flat subreflectarray only deviates the orthogonal LP beams radiated by the feeds in opposite directions, so the required phase distributions present a smooth phase variation at both frequencies. Figure 11 shows the phase distributions implemented in the subreflectarray at 19.7 GHz and 29.5 GHz in one LP; note that the phase distributions in the orthogonal LP show the opposite phase variation at each frequency to deviate the orthogonal LP beam in the opposite direction. Then, the main parabolic reflectarray only converts the dual-LP incident field into dual-CP, so their cells have been designed to introduce a phase difference of 90◦ between the two orthogonal components of each incident LP [22].

The design of the subreflectarray and the main parabolic reflectarray has been carried out separately. The radiation patterns of the designed subreflectarray (without the main reflectarray) have been simulated and the results have been compared with those obtained considering ideal reflectarray cells (providing the required phase distributions without any phase errors and zero dielectric losses). As an example, Figure 12a shows the comparison between the ideal and realized patterns in the azimuth plane (ϕ = 90◦ ) for the subreflectarray in H polarization at 19.7 and 29.5 GHz. Note that the patterns in Figure 12a show a wide main lobe since the subreflectarray does not focus the beam radiated by the feed (the parabolic surface of the main reflectarray will focus the beam). In the same way, the simulated radiation patterns of the designed polarizing main reflectarray (without the

subreflectarray, so that the main reflectarray is illuminated from the virtual focus of the dual system) have been computed and compared with the simulations performed with ideal cells. Figure 12b shows the comparison between the ideal and realized patterns in the azimuth plane at 19.7 and 29.5 GHz for the polarizing main reflectarray in RHCP (obtained from the H polarization radiated by the feed).

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**Figure 11.** Required phase distributions of the 0.65 m flat subreflectarray at the (**a**) lower and (**b**) upper operating frequencies for the same linear polarization (LP). **2021**, , x FOR PEER REVIEW 11 of 18

**Figure 12.** Simulated radiation patterns at 19.7 and 29.5 GHz in the azimuth plane of the ideal and realized antenna components. (**a**) Subreflectarray for H polarization, (**b**) main reflectarray for right-handed circular polarizations (RHCP).

The patterns in Figure 12 show a satisfactory agreement between the designed and ideal performances of both reflectarrays. The dual system configuration results in relatively large incidence angles from the feed on the subreflectarray, which complicates the design of the elements of the subreflectarray. As a result, the simulated patterns of the designed subreflectarray show a slightly lower gain and higher cross-polar levels than in the ideal case (see Figure 12a). On the other hand, the simulated results of the designed main reflectarray show an excellent performance, which is practically identical to the ideal behavior (see Figure 12b).

Finally, the dual antenna configuration is illuminated by the cluster of 27 feeds shown in Figure 3 rotated 90◦ in the xfyf -plane. The simulations of the complete dual reflectarray have considered 15 of the 27 feeds (an array of 3 × 5 feeds, plotted with solid lines in

Figure 3), since the beams produced by the remaining feeds are expected to have similar behavior. The simulated pattern contours of the 30 beams generated by the designed dual reflectarray simultaneously at 19.7 GHz and 29.5 GHz are depicted in Figure 13 at the EOC levels of the beams (between 45 and 46 dBi). The simulated beams match with the required spot distribution, where there is room for the beams that would be generated by a second dual reflectarray operating at slightly different frequencies to form the complete four-color multispot coverage in Tx and Rx. **2021**, , x FOR PEER REVIEW 12 of 18

**Figure 13.** Simulated pattern contours for the 30 beams generated by 15 feeds illuminating the proposed dual reflectarray at 19.7 GHz and 29.5 GHz.

The main cuts of the radiation patterns have revealed similar C/I levels as those of the previous 1.8 m reflectarray. Figure 14 shows the cut of the radiation patterns in the plane u = 0 for the beams produced by the central row of feeds at both frequencies. The simulated cross-polar radiation provides an XPD lower than 20 dB for some beams. The analysis of the results has shown that the increase of the cross-polar component is mainly produced in the subreflectarray, so an optimization procedure should be applied to the dual-band dual-LP cells to reduce the cross-polarization. Moreover, the use of a dual configuration could be used to implement a bifocal technique to improve the shaping of the beams generated in the edge of the coverage [24].

In conclusion, the simulated results of the dual reflectarray configuration prove the capability of the proposed multibeam antenna to generate half the required four-color coverage simultaneously in Tx and Rx. The dual-band operation involving separate frequencies simplifies the design of the reflectarray elements, while the dual configuration simplifies the process of generating two spaced beams in orthogonal CP per feed, splitting it into two straightforward stages (deviation of the LP beams in opposite directions and dual-LP to dual-CP conversion), the principles of which have been experimentally validated previously. The simulated results show that a further optimization is required for the subreflectarray to improve the XPD, although the flexibility provided by the dual configuration could be also used to improve the overall performance of the antenna.

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**Figure 14.** Cut of the simulated radiation patterns in u = 0 for the beams generated at (**a**) 19.7 GHz and (**b**) 29.5 GHz in same CP.

#### **5. Antenna Farm Based on Two Dual-Band Offset Parabolic Reflectarrays**

Bearing in mind the better performance of the previous dual-band dual reflectarray with respect to the single-band reflectarray, the final proposed antenna farm is also based on the use of a dual-band reflectarray to generate all the beams associated to two colors simultaneously in Tx and Rx. In this case, a single offset parabolic reflectarray is proposed instead of the previous dual configuration, reducing the number of components in the antenna configuration.

The generation of two spaced beams in orthogonal CP per feed is now completely managed by the reflectarray elements placed on the parabolic reflectarray surface. The operating principle of the antenna is based on the concept proposed in [25], where a singleband parabolic reflectarray deviates the orthogonal CP beams focused by the parabolic surface in opposite directions by means of the variable rotation technique (VRT) applied to the reflectarray elements. In our solution, a dual-band offset parabolic reflectarray is proposed to focus the beams by its parabolic surface and deviate the orthogonal CP by the application of VRT simultaneously at Tx and Rx frequencies. The dual-band operation by VRT is achieved using the reflectarray cells proposed in [20]. As a result, a single offset parabolic reflectarray illuminated by dual-CP feeds generates two spaced beams in orthogonal CP per feed at Tx and Rx frequencies in Ka-band. The operating scheme of the proposed reflectarray to generate half the required four-color coverage is shown in Figure 15.

**Figure 15.** Operating principle of the proposed single-band parabolic reflectarray.

Recently, the proposed operating principle has been experimentally validated in [26] by a 0.9 m parabolic reflectarray prototype that generates two spaced beams per feed at Tx and Rx frequencies using the reflectarray cell described in [20]. Figure 16 shows the prototype in the anechoic chamber and the 40.6 dBi pattern contours of the two measured beams generated simultaneously at 19.7 GHz and 30 GHz with a single dual-CP feed. **2021**, , x FOR PEER REVIEW 14 of 18 **2021**, , x FOR PEER REVIEW 14 of 18

**Figure 16.** Dual band parabolic reflectarray prototype to generate two spaced beams per feed [26]. (**a**) Picture of the prototype, (**b**) measured 40.6 dBi pattern contours of the two beams generated at 19.7 and 30 GHz by the same feed.

A 1.8 m parabolic reflectarray formed by 62,654 reflectarray cells with a period of 6.5 mm has been designed to deviate the beams in orthogonal CP ± 0.28◦ simultaneously at 19.7 and 29.5 GHz. Since the parabolic surface of the antenna focuses the beams radiated by the feeds, the required phase distributions on the reflectarray to split the orthogonal CP beams present a similar aspect as those of the subreflectarray shown in Section 4. Figure 17 shows the required phase distributions on the parabolic surface at 19.7 GHz and 29.5 GHz in one CP (the phase distributions in the orthogonal CP show the opposite phase variation to deviate the orthogonal CP beams in the opposite direction).

**Figure 17.** Required phase distributions of the 1.8 m parabolic reflectarray at the (**a**) lower and (**b**) upper operating frequencies for the same CP.

As in the first antenna solution shown in Section 3, the proposed dual-band parabolic reflectarray has been illuminated by the cluster of 27 feeds shown in Figure 3, and the simulations have considered three lines of feeds (16 feeds). Thus, the designed parabolic reflectarray is expected to generate 32 beams in orthogonal CP simultaneously in Tx and Rx. The simulated pattern contours of the 32 beams generated by the 1.8 m parabolic reflectarray at 19.7 and 29.5 GHz are depicted in Figure 18, where the contours of the beams at 19.7 and 29.5 GHz correspond to a gain level of 46 and 45 dBi, respectively. In a similar way to the previous dual reflectarray configuration, a second parabolic reflectarray 0.01

0.02

0.03

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operating at slightly different frequencies would produce the second half of the required four-color coverage. **2021**, , x FOR PEER REVIEW 15 of 18

RHCP 19.7 GHz LHCP 29.5 GHz RHCP 29.5 GHz

**Figure 18.** Contours of 32 beams generated simultaneously at 19.7 GHz and 29.5 GHz by the 1.8-m dual-band parabolic reflectarray illuminated by 16 feeds.

Figure 19 shows the horizontal cut in the plane v = 0 of the radiation patterns for the beams generated at 19.7 and 29.5 GHz. The main cuts of the radiation pattern have shown similar values of C/I to those of the previous configurations, with a XPD larger than 20 dB. Due to the large electrical size of the antenna at 29.5 GHz, the beams in Rx show a higher maximum gain than the beams in Tx; however, the shaping of the beams in Tx and Rx can be matched by the optimization procedure applied in [26].

**Figure 19.** Cut of the simulated radiation patterns in v = 0 for the beams generated at (**a**) 19.7 GHz and (**b**) 29.5 GHz in same CP.

In conclusion, the simulated results of the dual-band parabolic reflectarray have demonstrated a very satisfactory performance. The simulations show a suitable distribution of the beams with appropriate values of gain and XPD. As in the previous configurations, the C/I can be improved by a small increase of the antenna size to reduce the edge illumination levels on the reflectarray. Moreover, the reflectarray cell considered in the design makes it possible to implement additional optimization to shape the beams in Rx and to maintain the cross-polar levels in band, as in [26], by a proper adjustment of the dimensions and rotation angle of the elements [27].

#### **6. Conclusions**

In this paper, four novel antenna farms based on reflectarrays for broadband communication satellites in Ka-band have been proposed and evaluated. The exclusive characteristics of reflectarrays, which are able to operate independently at different frequencies or polarizations, have been applied in different ways to improve the SFPB operation compared to conventional reflectors. The impact of using flat or curved surfaces, single or dual-band operation, and single or dual antenna configurations for large reflectarrays in Ka-band has been assessed to achieve the best antenna performance. Table 2 summarizes the main characteristics of the proposed antenna solutions.

**Table 2.** Main characteristics of the proposed antenna solutions.

The preliminary simulations of single band reflectarrays to generate four beams per feed permits operation with a single antenna for Tx and an independent antenna for Rx, which simplifies the simultaneous operation in both bands as well as the design of the feed-chains. However, this solution involves a major difficulty in providing a stable performance in band due to the independent operation at close frequencies, requiring complex reflectarray cells to which intense optimizations must be applied. On the other hand, the two proposed dual-band reflectarrays show promising results. The simulations of the dual reflectarray system would require a further optimization of the subreflectarray to slightly reduce the cross-polar radiation, although the use of a dual configuration with two reflectarrays provides greater flexibility of operation than single aperture systems, which can be used to correct the shaping of the beams in the edge of the coverage. Finally, the dual-band parabolic reflectarray achieves an excellent antenna performance with a simple antenna configuration similar to that of the currently used reflectors. Moreover, the reflectarray elements can be optimized to properly shape the beams in Tx and Rx without the need to shape the antenna surface.

The core technologies of the proposed reflectarrays have been experimentally validated, which confirms the capabilities of reflectarrays to provide novel efficient multibeam antenna farms for broadband communications satellites in Ka-band. All the proposed antenna solutions make it possible to halve the number of antennas and feed-chains required onboard the satellite, from four reflectors to two reflectarray antennas, and from a total of N feed-chains (to produce a coverage of N spots) to N/2 feeds, allowing an important reduction in cost, as well as in mass and volume of the payload, which are severely limited in the satellite.

**Author Contributions:** Conceptualization, J.A.E. and A.P.; methodology, J.A.E. and A.P.; software, D.M.-d.-R., E.M.-d.-R. and Y.R.-V.; validation, D.M.-d.-R., E.M.-d.-R. and J.A.E.; formal analysis, D.M.-d.-R., E.M.-d.-R. and Y.R.-V.; investigation, D.M.-d.-R., E.M.-d.-R., Y.R.-V., J.A.E. and A.P.; data curation, Y.R.-V. and A.P.; writing—original draft preparation, D.M.-d.-R.; writing—review and editing, E.M.-d.-R., Y.R.-V., J.A.E. and A.P.; visualization, D.M.-d.-R., E.M.-d.-R. and Y.R.-V.; supervision, J.A.E. and A.P.; project administration, J.A.E.; funding acquisition, J.A.E. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research has been supported in part by the Spanish Ministry of Economy and Competitiveness under the project TEC2016-75103-C2-1-R, and under the project FJCI-2016-29943; by the European Regional Development Fund (ERDF), and by the European Space Agency (ESA) under contract<PHONE_NUMBER>/16/NL/AF.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

## *Article* **Numerical and Experimental Investigation of the Opposite Influence of Dielectric Anisotropy and Substrate Bending on Planar Radiators and Sensors**

**Plamen I. Dankov 1,\* , Praveen K. Sharma 2 and Navneet Gupta 2**

1 Faculty of Physics, Sofia University "St. Kliment Ohridski", 1164 Sofia, Bulgaria

2 Department of EEE, Birla Institute of Technology and Science (BITS), Pilani 333031, India;

**\*** Correspondence<EMAIL_ADDRESS>Tel.: +359-899-052-097

**Abstract:** The simultaneous influences of the substrate anisotropy and substrate bending are numerically and experimentally investigated in this paper for planar resonators on flexible textile and polymer substrates. The pure bending effect has been examined by the help of well-selected flexible isotropic substrates. The origin of the anisotropy (direction-depended dielectric constant) of the woven textile fabrics has been numerically and then experimentally verified by two authorship methods described in the paper. The effect of the anisotropy has been numerically divided from the effect of bending and for the first time it was shown that both effects have almost comparable but opposite influences on the resonance characteristics of planar resonators. After the selection of several anisotropic textile fabrics, polymers, and flexible reinforced substrates with measured anisotropy, the opposite influence of both effects, anisotropy and bending, has been experimentally demonstrated for rectangular resonators. The separated impacts of the considered effects are numerically investigated for more sophisticated resonance structures—with different types of slots, with defected grounds and in fractal resonators for the first three fractal iterations. The bending effect is stronger for the slotted structures, while the effect of anisotropy predominates in the fractal structures. Finally, useful conclusions are formulated and the needs for future research are discussed considering effects in metamaterial wearable patches and antennas.

**Keywords:** anisotropy; dielectric constant; material characterization; planar resonators; substrate bending; textile fabrics; wearable radiators

#### **1. Introduction**

Recently, many artificial materials known with their traditional applications in the human life can be considered as electrodynamic media due to the propagation of waves through them. Textile fabrics are typical examples. Most of these materials and some of their flexible polymer substitutes have been transformed into a new type of electronic components—antenna/sensor substrates due to their new applications in the wearable communication systems (antennas, sensors, radio-frequency identification or RFID, millimetrewave identification or mmID, etc.) [1–5]. In this role, they look like the commercial reinforced substrates with PCB (printed circuit board) applications (a comparison has been given in [6], chapter IV). From a long time, the PCB designers have required manufacturers of the traditional reinforced substrates to provide up-to-date and reliable information on their dielectric parameters. Nowadays, the situation with the antenna designers of the wearable textile devices is almost the same—they must know the right information for the actual dielectric parameters of these specific materials as substrates. Our observations show that a lot of research papers appeared in the last several years concerning the characterization of the dielectric parameters (dielectric constant *εr* and dielectric loss tangent tan *δε*) of the most popular textile fabrics [7–12]. The used methods are quite different—resonance

**Citation:** Dankov, P.I.; Sharma, P.K.; Gupta, N. Numerical and Experimental Investigation of the Opposite Influence of Dielectric Anisotropy and Substrate Bending on Planar Radiators and Sensors. *Sensors* **2021**, *21*, 16. https://dx.doi.org/10.3390/s21010016

Received: 25 November 2020 Accepted: 18 December 2020 Published: 22 December 2020

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/).

<EMAIL_ADDRESS>(P.K.S<EMAIL_ADDRESS>(N.G.)

and non-resonance—and most of them are implemented in the traditional ISM bands (typically around 2.45 GHz). However, the textile substrates differ from the reinforced substrates consist of natural and/or synthetic fibres (threads, yarns, filaments, etc.) in air and form fibrous structures with a considerably bigger variety of different cross-section views [13,14] in comparison with the simple woven or non-woven reinforced substrates. Thus, depending on the used fibre materials, their density, applied fabrication technology, and selected stitch, they act as porous materials with relatively low permittivity (*εr* ~1.2– 2.0), which is quite comfortable for antenna applications (the minimal dielectric constant for the reinforced substrate is typically *εr* ~3.0). The other differences are that the textile fabrics are more flexible and compressible materials, the thickness and density of which can be easily changed by low mechanical pressure. One property seems common—the existence of an intrinsic planar anisotropy due to the predominant orientation of the fibres. However, the anisotropy the woven/knitted fabrics is mainly related to their mechanical properties (e.g., tensile coefficients) [1,13–15] and very rarely to their dielectric parameters.

Like typical artificial materials, the textile fabrics can be considered as mixtures between two or more dielectrics (reinforced fibre nets with an appropriate filling/air). In such cases, effective-media models have been developed [16], which can predict numerically the resultant isotropic dielectric constant and the dielectric loss tangent of these materials. However, the variety of technologies used for the manufacturing of these fibrous materials and the complex cross sections [5] can provoke a measurable dielectric uni- or biaxial anisotropy—direction-dependent dielectric parameters (*εxx* 6= *εyy* 6= *εzz*; tan *δε\_xx* 6= tan *δε\_yy* 6= tan *δε\_zz*). Our previous research [17] showed that most of the textile fabrics have typical uniaxial anisotropy: different dielectric parameters in parallel and perpendicular directions regarding to the sample surface (*εpar* = *εxx* or *εyy* 6= *εperp* = *εzz*; and tan *δε\_par* 6= tan *δε\_perp*), which is also typical for the wide-spread microwave reinforced substrates [6,18]. Actually, the anisotropy of both types of woven materials is an undesired property, but it should be taken into account in the RF design of different microwave (incl. antenna/sensor) components especially in the mm-wavelength range. Exactly here is the difference. Nowadays, the major manufacturers of reinforced substrate started to share information about the possible anisotropy of some of their commercial products, while the designers of wearable antennas usually completely ignore this property for the textile substrate. We found only a few papers, which comment on the dielectric anisotropy of textile fabrics. The authors of the review paper [1] considered this problem for the textile fabrics without presenting concrete data. A recent paper ([19], Table 6) investigated the influence of the percentage of the normal and in-plane (parallel) components (fibres) in the woven fabrics (at microstructural level) on the resultant dielectric constant, but without to give separate values of *εpar* and *εperp*.

At the same time, we found another interesting fact—many papers, devoted to the dielectric characterization of textile materials, give different results for similar materials. A typical example is the measured dielectric constant of denim textile substrates, *εr\_Denim*. Our survey shows that the used values vary from 1.4 to 2.0 (~35% scatter) [4]. One of the reasons is the possible different types of applied weaving stitch in different cases. However, we additionally encountered a relationship between the measured dielectric constant *εr\_Denim* of denim fabrics on the applied measurement method. Researchers, who derive the dielectric constant from the resonance parameters of standards rectangular flat patches give values *εr\_Denim* ~1.59–1.67 [12,20–22]. In this case, the extracted dielectric constant should be close to the perpendicular one, *εperp\_Denim*. When the applied method is the popular coaxial dielectric probe (DAK, Dielectric Assessment Kit), the obtained parameters are typically *εr\_Denim* ~1.78–1.8 and beyond [23,24]. The free-space method confirms these values 1.75–2 in the frequency range 14–40 GHz [25]. Both considered methods give values close to the parallel one, *εpar\_Denim*. Finally, the extracted dielectric constants from the microstrip ring resonator or other planar methods are typically *εr\_Denim* ~1.69–1.73 [23,24]. In this case, the planar methods extract the equivalent dielectric constant (see the concept developed in [26]). The equivalent dielectric constant *εeq* appears for characterization of

the whole substrate when the real anisotropic structure has been replaced with an isotropic equivalent. That's why, we observe the inequality *εpar\_Denim* > *εeq\_Denim* > *εperp\_Denim*, which is a typical situation for the woven materials, e.g., for the reinforced substrates [18,27]. Very interesting are the obtained results in [23]; they confirm the assumption above because the measured values for the equivalent dielectric constants *εeq* for three textile fabrics by a ring-resonator method always are smaller than the corresponding values measured by the DAK method (*εpar*). Therefore, we can conclude that the anisotropy of the textile fabrics is a natural property, and its existence can explain the behaviour of their permittivity. Our investigations show that the anisotropy of the materials in the antenna project directly influences mainly the matching conditions of the patches and radome transparency [28], while then at the working frequency it indirectly slightly changes the gain, radiation patterns, efficiency and even polarization thought the anisotropic radome. The most common circumstance in the research papers considering wearable radiating components is the observation of small, moderate, and sometimes big differences between the simulated and measured resonance characteristics, explained by the authors with different experimental and simulation conditions. Our opinion is that in the most cases this effect depends on the selected by the authors values of the dielectric constant—close to *εperp* (small changes for the patches resonances are observed), *εeq* (suitable for the microstrip feeding lines, transformers, steps, filters, etc.) or *εpar* (applicable for the coplanar and slotted wearable structures) [26]. If the actual anisotropy of the used substrates is smaller than 2–3% (see below for this parameter), its influence is usually negligible.

The other important issue for the wearable antennas is the bending effect (for conformal patches) [29]. A part of the research papers dealing with the wearable antennas on textile fabrics and polymers usually include additional information for the effect of bending at typical radii, compliant with the human body [30–33]. A measure for the degree of bending is the curvature radius *Rb* (*Rb* is the radius of an imaginary cylinder to which the antenna is bent) or bending angle *θb* = *L* (or *W*)/*Rb* , where *L* and *W* are the length and width of the rectangular patch antenna [34]. Most of the papers simply registered the bending effect on the working frequency and/or frequency bandwidth (usually a decrease of the resonant frequency) and rarely on the gain and radiation pattern. Sometimes, unexpected discrepancies are detected between the simulated and measured results from the bending [32] explained by imperfect measurements. Only a few researchers provide discussions for the nature of the bending effect. When the measurements are well performed, the obtained results are useful for understanding the bending effect. For example, the results obtained in the paper [30] give the information that the thickness of the flexible substrate is important for the degree of the bending influence. For the substrate as a flexible felt (*εr* = 1.3) with thickness *hS* = 0.5–12 mm, the optimal thickness for minimizing the effect of bending over the frequency shift is about 6 mm. Very helpful results for the bending effect on rectangular patch antenna on denim substrate are presented in [35]. The parameters of this material with thickness *hS* = 2 mm are chosen to be *εr\_Denim* = 1.6 and tan *δε\_Denim* = 0.01 at 2.4 GHz. For the first time, the authors definitely show by simulations that the resonance frequency of the lowest-order TM10 mode in the rectangular patch antenna should continuously increase with increasing of the bending radius *Rb*—with a relatively low degree for the width-bent patches and with a higher degree—for the length-bent patches. However, the measurement results slightly differ from the simulations, as relatively big ripples appear in the experimental frequency shifts: ±2.5 MHz for width-bent and ±85 MHz for length-bent patches (compared to the resonance frequency ~2.4 GHz for the flat patches). Nevertheless, the tendency for increasing of the resonance frequency is visible. The authors commented that this behaviour was not expected from simulations. They attribute this discrepancy to other physical properties that the conductive textile was subjected to upon bending that were not correctly replicated in simulations.

In our paper [36], we supposed for the first time that both effects (bending of the flexible substrate and its anisotropy) can simultaneously affect the resonance behaviour of the resonance patches. There we presented some preliminary experimental results for a rectangular resonator with isotropic substrates, but the influence of the substrate anisotropy was not separately investigated. We cannot find other research papers, where the anisotropy and bending are considered in parallel, excepting some calculations of the input impedance [37] and additionally return losses S11 and mutual coupling [38] in cylindrically conformal patch antennas on anisotropic substrates.

In this paper, we continue to investigate more deeply the opposite impacts of the dielectric anisotropy and bending of the substrate on the resonance characteristics of planar radiators. We follow the same strategy in this paper—not to consider fed patches and antennas, but to examine pure resonant structures and to avoid any parasitic influence of the feeding lines. This paper includes new experimental and simulation results for the frequency shift of the modes in planar rectangular resonators and their modifications, which makes possible the separation between the effects of anisotropy and bending and the independent characterization of the degree of these effects. In the Materials and Methods section, two experimental and numerical methods have been used for determination of the uniaxial anisotropy of the textile fabrics. A methodology for accurate measurements of the bending effects on the resonance characteristics of the planar resonator has been described. Then, an efficient procedure is introduced for creating suitable 3D models of planar resonators for separate numerical investigations of the bending and anisotropy and both effects together. Data for the measured anisotropy of several selected for the research flexible anisotropic and isotropic materials are presented. In the Results and Discussions section, very interesting results are obtained and discussed for the separate and simultaneous influence of the anisotropy and bending for materials with different anisotropy and for conformal resonance structures bent at different radii. The results for the influence of the anisotropy and bending on several planar resonators with sophisticated shapes are added—for slotted rectangular patches, fractal structures, and resonators with defected grounds. Finally, the origins of the considered competitive effects on the resonance planar structures are discussed and explained and useful conclusions are offered. A possible future work has been formulated.

#### **2. Numerical and Experimental Methods and Materials**

The aim of this research is to investigate numerically and experimentally the possible competitive influences of the uniaxial anisotropy and bending of textile substrates on the resonance performances of wearable planar radiators. Therefore, in this section, we describe all applied experimental and numerical methods for the determination of the substrate anisotropy and reliable characterization of the bending effect in these radiating structures. The selected materials and their important characteristics for the research have been obtained.

#### *2.1. Two-Resonator Method for Measurement of the Uniaxial Anisotropy of Textile Fabrics*

The considered below method has been proposed in [27] and applied for anisotropy characterization of a variety of materials [19]. In this paper, it has been applied for the determination of the pairs of parameters, *εpar*; tan *δε\_par* and *εperp*; tan *δε\_perp*, of textile fabrics. Figure 1a schematically presents the idea of the used method: a textile disk sample is placed sequentially in two resonators, which are designed to support either symmetrical TE0*mn* modes (*m* = 1, 2, 3, . . . ; *n* = 1, 2, 3, . . . ) in the cylinder marked as R1 or symmetrical TM0*m*0 modes (*m* = 1, 2, 3, . . . ) in the cylinder marked as R2 with mutually perpendicular E fields—parallel to the sample surface in R1 or perpendicular to this surface in R2. The sample is placed in the middle of R1 and on the bottom of R2 ensuring the best conditions for the excited TE or TM modes to be influenced by the sample and these modes to be maximally separated (e.g., the resonators heights to be *H*1 ~ *D*1 and *H*2 < *D*2 and the coupling probes to be orientated to excite only TE modes in R1 or TM modes in R2). The sample diameter *dS* is chosen to coincide with the resonator diameters *dS* ~ *D*1,2. In this case, the extraction of the dielectric parameters can be accurately performed by the analytical model described in [27,39]. In short, the measurement procedure is as follows. First, the

resonance characteristics are measured (resonance frequency *f* 0 and unloaded quality factor *Q*0) of each TE or TM mode under interest in the empty R1 or R2 resonator. This step makes possible a fine determining the equivalent resonator diameters *D*1,2*eq* and equivalent wall conductivity *σ*1,2*eq* of both resonators, which considerably increases the accuracy of the next measurements. The second step includes measurements of the resonance characteristics (*fε* and *Qε*) of the same TE or TM modes (well-identified) in the R1 or R2 resonators with a sample. Finally, the set of obtained data ensures the determination of the parallel dielectric constant *εpar* and dielectric loss tangent tan *δεpar* in resonator R1 and determination of the perpendicular dielectric constant *εperp* and dielectric loss tangent tan *δεperp* in resonator R2. The measurement uncertainty has been evaluated as relatively small [27]: 1–1.5% for *εpar*, 3–5% for *εperp*, 5–7% for tan *δεpar* and 10–15% for tan *δεperp* in the case of 0.5–1.5 mm thick substrates with dielectric constants ~1.3–5 in the Ku band. The main source of the pointed inaccuracy is the uncertainty for the determination of the sample thickness. Another circumstance is the selectivity of the considered method; due to the E-fields orientation the cylinder resonators measure the corresponding "pure" parameters (parallel ones in R1 and perpendicular ones in R2) with selectivity uncertainty less than ±0.3–0.4% for the dielectric constant and less than ±0.5–1.0% for the dielectric loss tangent in a wide range of substrate anisotropy and thickness [39]. *σ ε ε ε δε ε δε ε ε δε δε*

**Figure 1.** Two-resonator method: (**a**) Pair of resonators for measurement of parallel (R1) and perpendicular (R2) dielectric parameters of disk samples in cylindrical TE (R1) and TM-mode (R2) resonators; (**b**) Photography of resonators R1 and R2 with denim textile sample 1; E—electric field.

#### *2.2. Numerical Models for Determination of the Dielectric Constant and Anisotropy of Textile Fabrics as Dielectric Mixtures*

2.2.1. Limits for the Dielectric Parameters of Mixed Textile Threads

There exists a big variety of technologies to mix or blend two or more types of textile threads from different materials and with different mechanical properties, which makes possible to obtain new fabrics with desired specific elasticity moduli and stiffness and to control the stability of these properties. Due to these purposes, textile engineers have developed different models and effective- medium theories for reliable characterization of these structures [13–16,40–42]. Our survey shows that these models with some modifications could be successfully applied also to the electromagnetic properties of the textile fabrics dielectric mixtures, as it has been done in [16].

The simplest models (as the first stage of approximation, if the details of geometry are ignored) give the so-called upper and lower bounds of the resultant dielectric constant and loss tangent on the base of the modified Reuss (iso-strain) and Voigt (iso-stress) bound models (for series or parallel layered mixtures). The Bruggman formula [14] presents relatively accurate approximation for near-to-isotropic materials:

$$\varepsilon\_{\epsilon\eta} = \frac{(\varepsilon\_1 + \mathfrak{u})(\varepsilon\_2 + \mathfrak{u})}{V\_1(\varepsilon\_2 + \mathfrak{u}) + V\_2(\varepsilon\_1 + \mathfrak{u})} - \mathfrak{u}; \quad 0 \le \mathfrak{u} \le \infty, \quad V\_1 + V\_2 = 1 \tag{1}$$

where *εeq* is the scalar isotropic equivalent dielectric constant of the mixture, *ε*1 and *ε*2 are the dielectric constants of the mixed threats, *V*1 and *V*2 are the corresponding normalized volumes, and *u* ⊂ (0; ∞) is a parameter which depends on the method of mixing. Three cases could be derived from this expression depending on the type of mixing: for series mixing *u* = 0 (Reuss bound); for parallel mixing *u* = ∞ (Voigt bound) and for random mixing, *u* = (*ε*1*ε*2) 1/2 (Bruggman curve). All these curves for the normalized dielectric constant are plotted in Figure 2a. *ε ε ε ε ε*

*ε ε δε δε ε δε ε δε* **Figure 2.** Minimal and maximal bounds for the normalized resultant equivalent dielectric constant *εeq*/*ε*1 (**a**) and normalized equivalent dielectric loss tangent tan *δε\_eq*/tan *δε*\_1 (**b**) of two mixed dielectrics with isotropic parameters *ε*1/tan *δε*1 and *ε*2/tan *δε*2 and normalized volumes *V*1 and *V*2 (*V*1 + *V*2 = 1) (insets: of parallel, series and random mixing regarding the direction of the applied E field).

; 0 (tan )(tan ) tan 1 2 δ δ The Equation (1) is a complex one; it can be rewritten also for a direct calculation of the corresponding dielectric loss tangents bounds (modified Bruggman formula); see the dependencies in Figure 2b:

$$\tan \delta\_{\varepsilon\_{\varepsilon} \eq q} = \frac{(\tan \delta\_{\varepsilon 1} + v)(\tan \delta\_{\varepsilon 2} + v)}{V\_1(\tan \delta\_{\varepsilon 2} + v) + V\_2(\tan \delta\_{\varepsilon 1} + v)} - v; \quad 0 \le v \le \infty,\tag{2}$$

*δε δε ε ε δε δε ε δε ε δε* where tan *δε\_eq* is the isotropic equivalent dielectric loss tangent of the resultant fabrics, tan *δε*1 and tan *δε*2 are the dielectric loss tangent of the mixed/blended threats, and *v* ⊂ (0; ∞) is a new parameters; now we have again *v* = 0 for series mixing, *v* = ∞ for parallel mixing, however, *v* = [*ε*1*ε*2 (tan *δε*1 + tan *δε*2)]−1/2 for random mixing. We have selected a concrete synthetic material for the presented examples in Figure 2a,b—Polyester threads (*ε*1 ∼= 3.4; tan *δε*1 ∼= 0.005) mixed with air (*ε*2 = 1.0; tan *δε*2 = 0). However, the predicted anisotropy by Equations (1) and (2) is too large, does not take into account the concrete sizes and shapes of the threads and therefore, the results do not correspond to the realistic textile fabric. The survey of other effective-media analytical expressions for the resultant permittivity in different mixtures, presented in [43], show that they also cannot give the actual anisotropy.

Therefore, in this paper, we accepted another more realistic approach. Most of the textile fabrics can be considered as complex fabrics of cylindrical single or multi-fibre threads (a short survey on the Internet of the free microscopic images of the popular fabrics illustrates well the predominant existence of the cylindrical cross-section shape of the threads). Such an approach is very popular in the mechanical models of the textile fabrics [13–15], but also applicable for characterization and modelling of their dielectric properties [10,19,44]. In this research, a similar approach has been accepted. In the next subsection, we present an effective numerical model for accurate prediction of the real anisotropy of textile fabrics on the base of cylindrical unit cells.

\_

*δε*

2.2.2. Numerical Models for Evaluation of the Dielectric Anisotropy of the Textile Fabrics

The degree of anisotropy can be predicted for artificial textile fabrics by the numerical method introduced in [17]. The idea of this method is to build a unit cell by two or more isotropic cylindrical fibres (threads), to reproduce it in a hosting isotropic substrate (e.g., air) and to put the whole sample in a rectangular resonator, which supports TE and TM modes with exited E fields in three mutually perpendicular directions. The simulations are performed by electromagnetic simulators (HFSS® in this case). Figure 3 illustrates the selected unit cells with three mutually perpendicular cylinders of equal diameter *d*. The concrete unit cell is a prism with sides *a* = *b* = 1.0; *c* = 1.5 mm. They form a rectangular sample with dimensions 9.5 × 8 × 1.5 mm and this sample is placed in the middle of a rectangular box with dimensions 9.5 × 8 × 10 mm. Actually, this box is one-quarter part of a rectangular resonator with dimensions 19 × 16 × 10 mm, which support TE and TM mode in the Ku and K bands depending on the diameters, filling and dielectric constant of the threads. The resonator with a sample is solved in "eigenmode" option of the used HFSS simulator (calculating the resonance frequency *fr* and the unloaded quality factor *Q*), where appropriate symmetrical boundary conditions are accepted at side A and B of the box: "symmetrical E-field" for TE modes and "symmetrical H-field" for TM modes. Thus, the considered resonator with 1.5-mm thick artificial textile sample supports the following mode of interest, illustrated in Figure 4 for a 3D-woven textile sample: (a) TE011 mode with resonance frequencies in the interval 19.3–21 GHz (E field along 0*x*); (b) TE101 mode; 21.9–23.5 GHz (E field along 0*y*); (c) TM010 mode; 11.6–12.2 GHz (E field along 0*z*); (d) TE111 mode; 17.6–18.7 GHz (E field in plane 0*xy*). The considered set of modes makes it possible the extraction of the dielectric constants and dielectric loss tangent of the investigated textile samples in different directions, considered as samples with bi- or uniaxial symmetry as it is shown in Figure 4. The concrete resonator dimensions are chosen relatively small (to facilitate simulations). However, larger dimensions can be selected for lower-frequency ISM bands. The only rule is the size of the unit cell to be smaller than the free-space wavelength to ensure homogenization of the artificial structure at a given frequency. As quantitative measures are used, the parameters ∆*Aε* , ∆*A*tan*δε* for the degree of the dielectric anisotropy of the resulting (equivalent) dielectric constant/loss tangent for bi-/uni-axial anisotropy are calculated by the following expressions: *ε δε*

$$
\Delta A\_{\varepsilon\_{\text{\tiny\text{xx},yy}}} = 2(\varepsilon\_{\text{xx},yy} - \varepsilon\_{zz}) / (\varepsilon\_{\text{xx},yy} + \varepsilon\_{zz}) . \tag{3}
$$

$$
\Delta A\_{\tan \delta \varepsilon\_{\text{aux},yy}} = \mathcal{Z} (\tan \delta\_{\varepsilon\_{\text{aux},yy}} - \tan \delta\_{\varepsilon\_{\text{zz}},zz}) / (\tan \delta\_{\varepsilon\_{\text{xx},yy}} + \tan \delta\_{\varepsilon\_{\text{zz}},zz}).\tag{4}$$

**Figure 3.** (**a**) Unit cell *a* × *b* × *c* with cylindrical threads of diameter *d*; (**b**) constructed artificial sample with repeated unit cells in a hosting isotropic substrate (air); (**c**) equivalent sample in a rectangular box, which is a quarter part of the whole resonator with symmetrical boundary conditions on Side A and B.

**Figure 4.** Artificial sample (3D-woven fabrics) in a rectangular resonator (not shown), which supports different modes with mutually perpendicular E fields (red arrows) along the axes: 0*x*, 0*y*, 0*z* and in the plane 0*xy* in the Ku-band (modes: TE011 (**a**); TE101 (**b**); TM010 (**c**); TE111 (**d**).

*ε δε* Independent extraction of the resultant dielectric constant along all three axes is possible after the replacing of the anisotropic sample under test with an equivalent isotropic sample (as in Figure 3c). The procedure is as follows. After the selection of each unit cell and the construction of the whole artificial 3D sample (see below), placed in the middle of the selected resonator, the resonance frequency *fr* and Q factor of the corresponding mode can be obtained by simulations in "eigenmode option". Then, the anisotropic structure is replaced with an equivalent prism of the same dimensions and by tuning the corresponding isotropic values *εeq* and tan *δεeq*, a coincidence should be reached (typically <1%) between both simulated pairs *fr* and Q for the anisotropic sample and its isotropic equivalent. Thus, the corresponding dielectric parameters of the biaxial anisotropic textile samples can be obtained by using the excited modes in Figure 4a–c, while the parameters for the uniaxial anisotropic textile samples (most of the cases) can be obtained by using the modes in Figure 4c,d.

*ε ε ε ε ε ε ε ε ε* − *ε* The described procedure is effective and enough accurate for preliminary prediction of the anisotropy of different artificial materials. In the paper [17], some preliminary results have been obtained for several artificial woven and knitted tactile fabrics, but the method has been successfully applied for many other materials (incl. 3D printed) [6]. In this paper, we present new results to show how the anisotropy depends on the structure and threads' orientation of the textile fabrics and to establish the origin of this property. Some attempts to find such relations have been done in [19] but without to present quantitative results. In the beginning, three simple structures have been constructed as in paper [17] based on ordered single cylinders built from the unit cell in Figure 3 of diameter 0.5 mm and distance between their axes 1.0 mm. The cylinders are consistently orientated along 0*x*, 0*y* and 0*z* axes. Applying the modes from Figure 4a,c with electric fields orientated along 0*x* or 0*z* axis, the described model makes it possible to determine the dependence of the dielectric constant anisotropy ∆*Aε\_xx* versus the ratio *εthread*/*εair* presented in Figure 5 (a). The parameter ∆*Aε\_xx* increases with the ratio *εthread*/*εair* increasing, but in different ways. When the cylinders are orientated along 0*x* (as the electric field *E*TE of the exited mode), ∆*Aε\_xx* has large positive values (~8% for *εthread* = 3.4). Contrariwise, when the cylinders are orientated along 0*z* (*E*TM), ∆*Aε\_xx* has negative values (~−4.5%). These values strictly correspond to the relative volume portions of the treads orientated along 0*x* (*E*TE) and 0*z* (*E*TM) in these simple cases (detailed geometrical calculations are not performed at this stage of the research). Only when the cylinders are orientated along 0*y* (perpendicularly to *E*TE and *E*TM, the parameter ∆*Aε\_xx* is close to 0 (i.e., the sample behaves as almost isotropic one).

*ε*

*ε*

*ε*

*ε ε* **Figure 5.** Dielectric constant anisotropy of artificial textile samples versus the ratio between the dielectric constant of the threads and air *εthread*/*εair*: (**a**) for ordered straight cylinders, orientated along axes 0*x*, 0*y* or 0*z*; (**b**) for 2D, 2 1 2D and 3D woven fabrics in Ku-band (see Figure 6) (Arrows: E fields of the used TE and TM modes). *ε ε*

*ε ε* **Figure 6.** (**a**) Woven threads of corresponding dimensions; three types of woven fabrics: (**b**) 2D woven; straight threads along 0*x*, 0*y*; (**c**) 2.5D woven; straight threads along 0*y*, wavy threads along 0*x*; (**d**) 3D woven; wavy threads along 0*x*, 0*y* (all threads are with equal dielectric constant *εthread* = 3.4).

Very interesting are the results for the uni-axial anisotropy (obtained by the modes from Figure 4c,d) of constructed three artificial woven fabrics (shown in Figure 6). They are conditionally named 2D, 2.5D and 3D woven samples due to the applied straight and/or wavy threads (see the figure captions of Figure 6). The behaviour of the uni-axial parameter ∆*Aε* of the 2D-woven sample is close to this one of the pure cylinders along 0*x*—Figure 5b. However, when the portion of the threads with orientation along 0*z* axis increases (for 2.5D and especially for 3D-woven samples) applying wavy threads, the anisotropy becomes smaller, from 10% (for 2D woven samples) to 7% (2.5D) and 3.5% (3D) for *εthread* = 3.4. This result shows that the dense woven fabrics have relatively small anisotropy, close to the realistically measured values of 4–6% for most of the textile fabrics. However, their anisotropy exists and can be taken into account in the design of different wearable devices, when the final design accuracy is important and for the higher 5G frequency bands.

#### *2.3. Procedure for Accurate Measurements of Bent Planar Resonators on Textile Fabrics*

− − The accurate measurement of bent wearable structures is not an easy task. There appear strong mechanical changes during the bending—deformations in the substrates; deformations in the metal layout (it should always tightly cover the substrate); the feeding lines can affect the resonance behaviour. Following the strategy in this paper to investigate only pure resonance structures, we apply coaxial probes to excite the lowest-order resonances in the planar structures. Figure 7 represents the simulated E-field pattern of the first two planar modes in flat and bent microstrip resonators. Coaxial probes from electric type (short coaxial pin orientated along the E field) should be put close to the E-maximums. However, in this research, we apply more stable coaxial magnetic loops placed close to the H-maximums of the magnetic field of the corresponding mode (Figure 8a for TM10

mode; Figure 8b for TM01 mode and Figure 8c for both modes). The measurements are performed by a vector network analyzer in the L and S bands in transmission regime. The place and the orientation of the loops are tuned during the measurements until the transmission losses S21 increase more than −40 dB. At these conditions, the resonance frequency practically does not depend on the loop proximity and the measured resonance frequencies are enough accurate.

**Figure 7.** Simulated E-field pattern in microstrip resonators: (**a**,**b**) TM10 and TM01 in a flat resonator; (**c**,**d**) TM10 and TM01 in a bent resonator. Legend: *L*—length; *W*—width.

**Figure 8.** (**a**–**c**) Pair of magnetic coaxial loops placed on the length, width, and diagonal of the planar resonator; (**d**–**f**) length (L)-bent, width (W)-bent and diagonal (D)-bent microstrip resonators. Legend: 1—resonator; 2—substrate; 3—pair of magnetic coaxial probes.

In this research, we apply self-adhesive 0.05-mm thick metal (Al or Cu) folio to form the resonator layout. We start measurements in a flat position of the resonator and then measure the bent resonator with continuously decreasing bending radius. The resonator substrates are bending over a set of smooth metallic cylinders with radii *Rb* from 80 to 12.5 mm. Three types of bending are applied—length-(L), width-(W) and diagonal-bent (D) resonators—see the illustrations in Figure 8. When we bend, special care is taken to ensure that the metallization remains well adhered to the substrate and that it does not detach itself. Therefore, measurements are performed only for decreasing bending radius and not in reverse order. Each of the pointed types of bending is realized with a new fresh resonator folio. In this research, the results are presented for the ratio between the resonance frequencies for the bent and flat resonators.

#### *2.4. Numerical Models for Investigations of Bent Planar Resonators on Anisotropic Substrates*

Most of the modern electromagnetic simulators have options for the introduction of anisotropic materials. However, in the case of conformal planar structures, this is not easy to perform directly, when the substrate has been introduced as a single object and to be sure that the anisotropy is accurately described. Therefore, we chose a geometrical approach. The anisotropic substrate is divided into several equal slices with a form of prisms (with rectangular cross-section view for the flat resonators and with trapezoidal cross-section view for the flat resonators). The slices have equal anisotropic properties as the whole substrate, but the parallel and perpendicular directions used to determine the uniaxial anisotropic dielectric parameters can be controlled now for each slice with the change of the bending radius—as it is sown in Figure 9 for the half of structures. In this research, the concrete width *ws* of the slices is chosen to be *ws* = 2 mm but can be decreased for

thicker substrates or smaller bending radii for better fitting of the cross-section of the bent substrate. The other sizes are height *hs* and length *ls* = *Ws* . The 3D views of flat and bent microstrip resonators on sliced anisotropic substrates are presented in Figure 10. During the bending, we satisfy the rule to keep the resonator dimensions *L* and *W*. However, the ground and the slices may undergo some deformations.

**Figure 9.** Flat and bent microstrip resonators on substrate constructed by sliced prisms, each with own anisotropic properties. Arrows represent the normal direction in each slice in flat and curved substrates.

**Figure 10.** 3D view of flat and bent microstrip resonators of length *L* = 30 and width *W* = 26 mm on sliced substrates with length *Ls* = 42 and width *Ws* = 34 mm (last two cases with bending radius *Rb* = 14.3 and 9.6 mm).

#### *2.5. Materials Used in the Research*

*ε ε δε ε δε ε δε ε ε ε ε δε ε δε ε δε ε ε* Based on the purposes of the paper, several types of materials have been selected. One of the groups consists of several textile and polymer samples with different measured degrees of anisotropy (∆*Aε* from 4.3 to 10.3) by the two-resonator method. The measured results for the pairs of parameters *εpar*/tan *δεpar* and *εperp*/tan *δεperp*, as well as for the uniaxial anisotropy ∆*Aε*/∆*A*tan*δε* are presented in the upper part of Table 1. The other group includes several flexible isotropic substrates, selected for measurement of the pure bending effect. The measured anisotropy of these materials is very small, ∆*Aε* < 1%. The last two groups have representatives of relatively flexible reinforced substrates and soft artificial ceramics. Their anisotropy ∆*Aε* varies in a big interval—8.2–24.5%.

*ε δε εε δε ε δε*

*ε δε εε δε ε δε*

−

−

−

−

−

−

**Table 1.** Measured dielectric parameters and anisotropy of selected materials for this research (averaged values for the frequency interval 6–13 GHz).

#### **3. Results and Discussion**

Three types of results and corresponding discussions are presented in this section. First, numerical and experimental results are presented for the pure bending effect in planar resonators on flexible isotropic and near-to-isotropic substrates (Section 3.1). The next step is to verify with results the assumption that the bending effect and substrate anisotropy have opposite impacts on the wearable radiators and sensors (Section 3.2). Finally, the simultaneous bending and anisotropy influence is investigated for several sophisticated planar resonators with magnetic slots, defected grounds and for Koch fractal resonators (Section 3.3).

#### *3.1. Pure Bending Effect*

As we mentioned in the Introduction, the investigated bending effect in wearable planar patches and devices usually has been masked by other phenomena, not considered in the simulations [32,33]. We try to solve these problems applying experimentally-proven pure flexible isotropic substrates, using pure resonance structures (to minimize the effects of the feeding lines) and follow an accurate measurement procedure described in Section 2.3.

First, Figure 11a presents the dependencies of the ratio *fbent*/*fflat* between the resonance frequencies for the lowest-order TM10 mode for bent and flat rectangular resonators on pure isotropic substrate versus the curvature angle *αC* between the neighbour slices used to construct the substrate. This is a new measure for the bending degree, which is more comfortable in our research. Figure 12 illustrates the relationship between the bending radius *Rb* and the introduced curvature angle *αC* (e.g., *αC* = 4◦ corresponds to *Rb* = 28.7 mm; *αC* = 8◦—*Rb* = 14.3 mm; *αC* = 12◦—*Rb* = 9.6 mm, etc.).

*α*

*α*

*α α*

*α α*

*α*

**Figure 11.** Numerical dependencies of the ratio between the resonance frequencies *fbent*/*fflat* of the lowest-order TM10 mode for bent and flat rectangular resonators on isotropic substrate versus (**a**) the curvature angle *αC* between the substrate slices and (**b**) substrate thickness *hs*. The isotropic dielectric constant is chosen to be 3.0, but its concrete value has negligible influence. Positive and negative curvature angles are used. *α*

*α α α α* **Figure 12.** (**a**) Definition of the relation between the curvature angle *αC* and bending radius *Rb* ; dashed line: middle line in the resonator substrate, where the effective electrical length of the resonator is formed; (**b**) resonance structures on positive (+*αC*) and negative (−*αC*) bent substrate (the bending radius *Rb* is always determined to the side of the resonator layout).

The presented results show the expected fact (mentioned in [35]) that the resonance frequency of the L-bent resonator increases in comparison to the flat case for pure isotropic substrates. The dependence is not exactly linear. At the same time, the effect on the bending is relatively small for W-bent resonators, which is also an expected result. These dependencies correspond to the classical "positive" bending (*αC* > 0). What happens during the bending? The material undergoes mechanical deformations, e.g., stretching at the top (to the resonator) and shrinking at the bottom area (to the ground). In our model, we take into account this effect by changing the cross-section shape of the separate slices from rectangular to trapezoidal (illustrated in Figures 9 and 12a). The narrow side of the trapezoid is orientated to the ground of the resonance structure. Thus, the model confirms the assumption that the electrical length *LE* of the L-bent resonator decreases in comparison to the geometrical length *L* (illustrated with the dashed line in Figure 12a). The standing wave of the lowest order TM10 mode is located exactly along the curvature in the L-bent structures (see Figure 7a,c) and it explains the increase of the resonance frequency when the curvature angle *αC* increase. Contrariwise, during the W-bending the standing wave is located in a perpendicular direction and the influence of the bending is negligible, especially for thin substrates.

Figure 11a presents also the bending effect for the "negative" bending (*αC* < 0). It is just the opposite and this confirms the origin of the bending effect for the wearable structures. Now, the narrow side of the trapezoid of each slice is orientated to the resonator layout of the resonance structure and in this case, the effective electrical length *LE* of the L-bent resonator increases in comparison to the geometrical length *L* and the corresponding resonance frequency decreases. This type of bending is rarely used and not discussed in detail.

Finally, Figure 11b additionally shows the variations of the bending effect in substrates with different thickness. Now, the effect considerable increases for a thickness interval of 0.5–2.5 mm and then saturation appears for L-bent structures (relatively strong increase is observed also for W-bent structures at bigger thicknesses). However, we cannot observe here the existence of an optimal thickness, where the bending effects are minimized as shown in [30].

The next step is to prove experimentally these tendencies. Figure 13 gives a set of measurement results for the ratio *fbent*/*fflat* of the lowest-order TM10 mode in bent and flat rectangular resonators on several isotropic substrates versus the bending radius *Rb* . Three types of dependencies are shown—for L-, W and D-bent resonators. All the results are close to results from the numerical simulations in Figure 11a (D-bent resonators are not simulated). They depend on substrate flexibility and deformations. The best results are got for the well-flexible silicone elastomer (*hs* = 0.9 mm), Figure 13c. Good results are obtained by Ro3003 substrate (*hs* = 0.52 mm), Figure 13a; however, at small bending radii, this soft substrate undergoes technological stretching and *fbent* slightly decreases. The harder substrate PC (*hs* = 0.5 mm) shows better stability at low *Rb* . The results for the soft PTFE substrate (*hs* = 1.0 mm) deviate from the theoretical dependencies due to the poor adhesion properties of this materials to the metal folio. However, the PTFE-like material with the commercial mark Polyguide®Polyflon (*hs* = 1.5 mm) demonstrates better behaviour. In all presented cases, the curves for D-bent substrates (moderate influence) lie between the curves for L-bent (upper curves; stronger influence) and W-bent substrates (lower curves; smaller influence). Thus, we can conclude that the experimental results for the pure bending effect on planar resonators on isotropic substrate fully confirm the numerical simulations, taking into account the possible substrate deformation during the bending on very small radii *Rb* .

**Figure 13.** Experimental dependencies of the ratio between the resonance frequencies *fbent*/*fflat* of the lowest-order TM10 mode for bent and flat rectangular resonators on several isotropic substrates versus the bending radius *Rb* : (**a**) Ro3003; (**b**) PC; (**c**) commercial silicone elastomer; (**d**) PTFE and Polyguide® Polyflon (http://www.polyflon.com; dielectric parameters 2.05/0.00045).

*ε*

*ε*

#### *3.2. Investigation of the Simultaneous Effects of Anisotropy and Bending of Planar Resonators*

The main expected results in the research are included in this section. In the beginning, it is important to evaluate the effect of anisotropy in flat resonators. Figure 14a shows the simulated dependencies of the ratio *fflat\_aniso*/*fflat\_iso* between the resonance frequencies of modes TM10 and TM01 for flat rectangular resonators on anisotropic (∆*Aε* ~25%) and isotropic substrates versus the substrate thickness *hs* . The effect is visibly weak. Only for relatively thick substrates does the resonance frequency shift due to the anisotropy influence with 1–1.5%, which explains why this property is not so popular in the patch antenna design. The explanation is easy—the parallel E fields (to have a noticeable influence of the *εpar* component) appear only close to the edge of such wide planar structure and the relative effect is practically negligible in comparison to the microstrip line [26]. *ε ε*

**Figure 14.** (**a**) Numerical dependencies of the ratio *fflat\_aniso*/*fflat\_iso* between the resonance frequencies of modes TM10 and TM01 for flat rectangular resonators on anisotropic and isotropic substrate versus the substrate thickness *hs*. (**b**) Numerical dependencies of the ratio *fbent\_aniso/fflat\_aniso* of mode TM10 for L-/W-bent and flat rectangular resonators on anisotropic substrates versus the substrate thickness *hs*.

However, we expect a stronger effect when the resonators are bent. To perform deeper research, a set of bent resonators with different curvature angle are simulated by the help of the 3D models shown in Figures 9 and 10. First, the ratio *fbent\_aniso/fbent\_iso* is shown in Figure 15a between the resonance frequencies of mode TM10 for L-/W-bent rectangular resonators versus the curvature angle *αC*. The substrate anisotropy ∆*Aε* is chosen to be small (~3.5%), moderate (~11%) and big (~25%). This ratio is not measurable, but it shows in a pure form the effect of anisotropy in bent resonators. The results give the useful information, obtained for the first time, that this influence is considerably bigger in comparison with the flat case (up to −5% shifts down). One can see from the presented dependencies that the influence of the substrate anisotropy decreases the resonance frequency in comparison to the hypothetical case of an isotropic bent substrate. Therefore, we can conclude that the effect of the anisotropy of the substrate is just opposite to the effect of bending (as it is shown in Figure 11a). This was our preliminary hypothesis, and it can be considered as proven numerically. Therefore, one can expect that both effects can strongly change the behaviour of these dependencies.

*α ε*

*α α* **Figure 15.** (**a**) Numerical dependencies of the ratio *fbent\_aniso*/*fbent\_iso* between the resonance frequencies of mode TM10 in L/W-bent resonators on anisotropic and isotropic substrates (*hs* = 0.52) versus the curvature angle *αC*; (**b**) Numerical dependencies of the ratio *fbent\_aniso/fflat\_aniso* of mode TM10 in L-/W-bent and flat rectangular resonators on anisotropic substrates versus the curvature angle *αC*.

*α ε α α* Figure 15b presents the ratio *fbent\_aniso/fflat\_aniso* between the resonance frequencies of mode TM10 for L-/W-bent rectangular resonators on anisotropic substrates versus the curvature angle *αC*. Now, this ratio is measurable and can be verified experimentally. The new dependencies show that the resonance frequency shift in resonator on realistic (anisotropic) substrates may have as positive, as well as negative signs depending on the actual parameter ∆*Aε* , which is impossible for pure isotropic substrates. We also investigate the influence of the substrate thickness *hs* on corresponding ratio *fbent\_aniso/fflat\_aniso*. Figure 14b presents curves for L- and W-bent resonators at curvature angle *αC* = 12◦ . The results show that the bending effect can compensate the anisotropy influence for thicker substrates to some degree. It is interesting to note that as in [30], we observe the fact that for mediate thicknesses (named "optimal thickness" in [30]) the effect of anisotropy decreases the bending effect; this property probably depends on the curvature angle *αC* and not investigated in detail.

Let's now present some experimental dependencies for bent resonators on anisotropic substrates, selected in Section 2.5. The measurement results for the ratio *fbent*/*fflat* of the TM10 mode in bent and flat rectangular resonators versus the bending radius *Rb* are presented in Figure 16. They differ from the dependencies shown in Figure 13 for isotropic substrates. In anisotropic case, more or less expressed ripples in the resonance shifts is observed in both L- and W-bent resonators below the resonance frequencies of the corresponding flat resonators (as in paper [35]), which is practically impossible for the isotropic case when accurate measurement procedure has been applied. Therefore, all these cases confirm the simultaneous effects of the anisotropy and bending of used substrates. Very typical are the curves for the textile fabrics denim, linen and commercial multilayer GORE-TEX® and for the flexible polymer PDMS with a small degree of stretching. Similar behaviour is observed for three relatively flexible commercial reinforced substrates: Ro4003; NT9338 and soft ceramic Ro3010. However, the course of dependences here is affected also by the non-plastic deformation in these substrates, which does not allow bending at very small radii. Of course, all presented experimental curves cannot be directly compared with the theoretical ones in Figure 15b due to the difficulties to satisfy the perfect measurement conditions especially at small bending radii, but the trends that reveal the impact of the anisotropy together with the bending effect in wearable structure is obvious.

**Figure 16.** Experimental dependencies of the ratio between the resonance frequencies *fbent*/*fflat* of the lowest-order TM10 mode for bent and flat rectangular resonators on several anisotropic substrates versus the bending radius *Rb* : (**a**) Denim; (**b**) Linen; (**c**) commercial textile fabrics GORE-TEX®; (**d**) PDMS; (**e**) NT9338, Ro4003; (**f**) Ro3010.

#### *3.3. Effects of Anisotropy and Bending on More Sophisticated Planar Resonators*

The fact, that the substrate anisotropy visible influences together with the bending the resonance behaviour of such simple structure as the rectangular resonator gives us the idea to verify this influence for more complicated planar resonance structures on anisotropic substrates. In this subsection, several resonance structures with slots, defected grounds and Koch fractal contours are numerically investigated to verify the effects of anisotropy and bending in the L and S bands.

Two types of results are presented in Figures 17 and 20. We again investigate the ratio *faniso*/*fiso* between the resonance frequencies of the lowest-order mode for each structure on anisotropic and isotropic substrate (Figure 17). This ratio is a measure of the pure effect of anisotropy. The second type of result is for the ratio *fbent\_iso*/*fflat\_iso* between the resonance frequencies of the lowest-order mode in the same planar resonance structures (bent and flat) on isotropic substrates (Figure 20). Now, this ratio is a measure of the pure effect of bending.

*ε α α* **Figure 17.** Simulated values of the ratio *faniso*/*f iso* between the resonance frequencies of the lowestorder mode in several planar resonance structures with dimensions 30 × 30 mm on anisotropic and isotropic substrates (*hs* = 0.52; ∆*Aε* ~ 25%) (this ratio gives the pure effect of anisotropy). The shapes of the considered structures are presented in Figures 18 and 19. The first column for each case corresponds to a flat structure, second—bent at *αC* = 8◦ ; third—bent at *αC* = 12◦ ; Solid and hollow points correspond to two mutually perpendicular orientations (V & H) of the structure during the bending (when this is possible).

*α α α* **Figure 18.** Top view of several resonance structures (bent at *αC* = 8◦ ) with dimensions in mm: Case 1—resonator (30 × 30); Case 2—resonator with two slots (V—vertical orientation; H—horizontal orientation); Case 3—resonator with U-shaped slot (V&H); Case 4—resonator with double U-shaped slot (V&H); Case 5—resonator with swastika slot; Case 8—resonator with a defected ground (V&H).

*ε ε*

*α* **Figure 19.** Top view of the first three iterations of Koch fractal contours as a planar resonator (flat first row and bent at *αC* = 8◦—second row) with dimensions in mm: Case 1—iteration 0; Case 6—iteration 1; Case 7—iteration 2. *α α*

*ε ε α α* **Figure 20.** Simulated values of the ratio *fbent\_iso*/*fflat\_iso* between the resonance frequencies of the lowest-order mode in several planar resonance structures with dimensions 30 × 30 mm on isotropic substrates (*hs* = 0.52; ∆*Aε* ~ 25%) (this ratio gives the pure effect of bending). The shapes of the considered structures are presented in Figures 18 and 19. The first column for each case corresponds to a flat structure, second—bent at *αC* = 8◦ ; third—bent at *αC* = 12◦ ; Solid and hollow points correspond to two mutually perpendicular orientations (V & H) of the structure during the bending (when this is possible).

In the beginning, several rectangular patches with magnetic slots have been considered. These structures are usually applied for a widening of the bandwidth of the corresponding planar patches in comparison to the standard planar patch (Case 1). They include several types of slots (see also Figure 18): Case 2—resonator with two slots [45]; Case 3—resonator with U-shaped slot [30,46]; Case 4—resonator with double U-shaped slot [47]; Case 5 resonator with swastika slot [48]. The structures are not optimized; their dimensions are presented in Figure 18 and are compliant with the used grid of the sliced substrates. The results from Figure 17 show that the effect of the anisotropy decreases (4–1%) with adding the listed slots in the resonator layout. These slots are placed relatively far from the edges,

where the parallel E fields exist and the anisotropy cannot change effectively the electrical dimensions of the slotted resonators.

The considered defected-ground resonator [49] is also not strongly influenced by the anisotropy (2–3%); the effect is comparable with the effect in the planar resonator with a standard ground. However, the pure bending effect is strong, especially for horizontallyplaced slots in the defected ground (case 8H; the increase is more than 25–50%; the values are not shown in Figure 20, because are out of the scale).

Actually, only the considered fractal resonators demonstrate relatively big resonance shifting due to the anisotropy of the substrates—Figure 17 (4–6%), while the pure bending effect is even smaller than in the case of the standard planar resonator (iteration i0)—Figure 20. We investigate the first and second iterations (i1, i2) of classical Koch fractal contours [50], performed on flat and bent substrates—Figure 19. The reason for the increased anisotropy influence is that the portions of the parallel E fields increase considerably with the iteration number of the fractal resonators, which provokes stronger resonance frequency shift down (when *εpar* > *εperp*). A similar effect can be expected in most of the metamaterial surfaces used in the wearable flat and bent antennas, which is the objective of our future work.

#### **4. Conclusions**

The main objective of this study has been accomplished—to prove the opposite influences of the effects of anisotropy and bending on the resonance characteristics of flexible wearable structures. The advantage of this paper is that both effects have been separated in the numerical simulations, which makes it possible to evaluate the degree and sign of the resonance frequency shifts of simple rectangular planar resonators on anisotropic and isotropic substrates in flat and bent states. All simulations and the obtained experimental results show that the pure bending effect, performed only by experimentallyverified isotropic substrates, increases the resonance frequency of the bent rectangular resonators in comparison to the flat ones and proves the origin of this effect in a pure form. Contrariwise, the presented numerical analysis shows that the anisotropy (the existence of direction-dependent dielectric constants *εpar* and *εperp* of the textile materials and similar woven substrates) has just an opposite influence—the resonance frequency of the flat or bent rectangular resonators on anisotropic substrates always decreases (when *εpar* > *εperp*) in comparison to the same structures on pure isotropic substrates. The last effect is not directly measurable, but it gives the expected pure effect of the substrate anisotropy, which depends on the degree of anisotropy ∆*Aε* and the actual bending radius *Rb* . The combined effects, anisotropy and bending, lead to a more complicated behaviour of the investigated resonance structures when the bent and flat rectangular resonators are considered—as positive, as well as negative resonance frequency shifts.

Now, these combined effects are fully measurable. Applying well-selected flexible anisotropic substrates (including textile fabrics), the resonance shifts in bent and flat resonance structures are measured in the L and S bands. The obtained dependencies for bending radii *Rb* from 80 up to 10 mm show as increasing (as for the pure bending effect), as well as decreasing of the resonance frequencies (the last phenomenon is theoretically impossible for pure bending effect). Due to the mechanical deformations in the same of the materials during the bending, the obtained dependencies do not fully coincide with the numerical ones, but the tendencies for the opposite influence of the anisotropy and bending are considered as proven. The obtained results explain well the observed dependencies by other authors, even the existence of optimal substrate thicknesses, where the effect of bending (but we add the anisotropy, too), could be minimized. Of course, the last phenomenon depends on the concrete bending radius and anisotropy degree.

Encouraged by the results obtained for such a simple structure as the planar rectangular resonator on wearable substrates, we performed a useful numerical study for the combined effects of anisotropy and bending for more sophisticated structures—planar

resonators with slots and defected grounds and fractal resonators. In some of them, the bending effect predominates (resonators with slots and defected grounds), while in the other structures, e.g., the fractal resonators with increased iteration number, the effect of the anisotropy is stronger than the bending effect. These new results determine also the direction of our future research—to investigate complex metamaterial structures for wearable antennas performed on anisotropic substrates at different bending angles. The proposed experimental and numerical methods by the applied TE/TM modes for reliable determination of the direction-dependent equivalent dielectric parameters of different metasurfaces ensure the preliminary determination of the anisotropy of these structures, while the proposed methods for parallel investigation of the effects of bending and anisotropy—the accurate behaviour of metasurfaces with accurate curvature. This is a new scheme of research concerning such meta structures, which will be developed in our future work.

**Author Contributions:** Conceptualization, P.I.D., P.K.S. and N.G.; methodology, P.I.D.; software and simulations, P.I.D. and P.K.S.; measurements, validation, P.I.D. and P.K.S.; investigation, P.I.D.; resources, P.I.D., P.K.S. and N.G.; writing—original draft preparation P.I.D.; writing—review and editing, P.K.S. and N.G.; visualization, P.I.D.; supervision, P.I.D. and N.G.; project administration, P.I.D. and N.G.; funding acquisition, P.I.D. and N.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Department of Science and Technology, Ministry of Science and Technology, New Delhi, India and the National Science Fund, Ministry of Education and Science, Sofia, Bulgaria under grant number DST INT/BLG/P-01/2019 and KP-06-India-7/2019 under India-Bulgaria Joint Research Projects to the Department of Electrical and Electronics Engineering, Birla Institute of Technology and Science (BITS)—Pilani, Pilani Campus, Rajasthan, India and Faculty of Physics, Sofia University "St. Kliment Ohridski", Sofia, Bulgaria, respectively. The research of anisotropy and methods for its measurement was funded by the National Science Fund, Ministry of Education and Science, Sofia, Bulgaria under grant number DN-07/15 in Faculty of Physics, Sofia University, Sofia, Bulgaria.

**Data Availability Statement:** Data is contained within the article. More detailed data and data presented in this study are available on request from the corresponding author. Part of them could be included in the Final reports to the corresponding funding organizations.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

*Article*