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

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## Abstract

**:**

## 1. Introduction

_{r}and dielectric loss tangent tan δ

_{ε}) of the most popular textile fabrics [7,8,9,10,11,12]. The used methods are quite different—resonance 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 (ε

_{r}~1.2–2.0), which is quite comfortable for antenna applications (the minimal dielectric constant for the reinforced substrate is typically ε

_{r}~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) [1,13,14,15] and very rarely to their dielectric parameters.

_{xx}≠ ε

_{yy}≠ ε

_{zz}; tan δ

_{ε_xx}≠ tan δ

_{ε_yy}≠ tan δ

_{ε_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 (ε

_{par}= ε

_{xx}or ε

_{yy}≠ ε

_{perp}= ε

_{zz}; and tan δ

_{ε_par}≠ tan δ

_{ε_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 ε

_{par}and ε

_{perp}.

_{r_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 ε

_{r_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 ε

_{r_Denim}~1.59–1.67 [12,20,21,22]. In this case, the extracted dielectric constant should be close to the perpendicular one, ε

_{perp_Denim}. When the applied method is the popular coaxial dielectric probe (DAK, Dielectric Assessment Kit), the obtained parameters are typically ε

_{r_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, ε

_{par_Denim}. Finally, the extracted dielectric constants from the microstrip ring resonator or other planar methods are typically ε

_{r_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 ε

_{eq}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 ε

_{par_Denim}> ε

_{eq_Denim}> ε

_{perp_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 ε

_{eq}for three textile fabrics by a ring-resonator method always are smaller than the corresponding values measured by the DAK method (ε

_{par}). 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 ε

_{perp}(small changes for the patches resonances are observed), ε

_{eq}(suitable for the microstrip feeding lines, transformers, steps, filters, etc.) or ε

_{par}(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.

_{b}(R

_{b}is the radius of an imaginary cylinder to which the antenna is bent) or bending angle θ

_{b}= L (or W)/R

_{b}, 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 (ε

_{r}= 1.3) with thickness h

_{S}= 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 h

_{S}= 2 mm are chosen to be ε

_{r_Denim}= 1.6 and tan δ

_{ε_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 TM

_{10}mode in the rectangular patch antenna should continuously increase with increasing of the bending radius R

_{b}—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.

_{11}and mutual coupling [38] in cylindrically conformal patch antennas on anisotropic substrates.

## 2. Numerical and Experimental Methods and Materials

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

_{par}; tan δ

_{ε_par}and ε

_{perp}; tan δ

_{ε_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 TE

_{0mn}modes (m = 1, 2, 3, …; n = 1, 2, 3, …) in the cylinder marked as R1 or symmetrical TM

_{0m0}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 d

_{S}is chosen to coincide with the resonator diameters d

_{S}~ 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

_{1,2eq}and equivalent wall conductivity σ

_{1,2eq}of both resonators, which considerably increases the accuracy of the next measurements. The second step includes measurements of the resonance characteristics (f

_{ε}and Q

_{ε}) 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 ε

_{par}and dielectric loss tangent tan δ

_{εpar}in resonator R1 and determination of the perpendicular dielectric constant ε

_{perp}and dielectric loss tangent tan δ

_{εperp}in resonator R2. The measurement uncertainty has been evaluated as relatively small [27]: 1–1.5% for ε

_{par}, 3–5% for ε

_{perp}, 5–7% for tan δ

_{εpar}and 10–15% for tan δ

_{εperp}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].

#### 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

_{eq}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_eq}is the isotropic equivalent dielectric loss tangent of the resultant fabrics, tan δ

_{ε}

_{1}and tan δ

_{ε}

_{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 δ

_{ε}

_{1}+ tan δ

_{ε}

_{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 δ

_{ε}

_{1}≅ 0.005) mixed with air (ε

_{2}= 1.0; tan δ

_{ε}

_{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.

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

^{®}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 f

_{r}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) TE

_{011}mode with resonance frequencies in the interval 19.3–21 GHz (E field along 0x); (b) TE

_{101}mode; 21.9–23.5 GHz (E field along 0y); (c) TM

_{010}mode; 11.6–12.2 GHz (E field along 0z); (d) TE

_{111}mode; 17.6–18.7 GHz (E field in plane 0xy). 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 ΔA

_{ε}, ΔA

_{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:

_{r}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 ε

_{eq}and tan δ

_{εeq}, a coincidence should be reached (typically <1%) between both simulated pairs f

_{r}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_xx}versus the ratio ε

_{thread}/ε

_{air}presented in Figure 5 (a). The parameter ΔA

_{ε_xx}increases with the ratio ε

_{thread}/ε

_{air}increasing, but in different ways. When the cylinders are orientated along 0x (as the electric field E

_{TE}of the exited mode), ΔA

_{ε_xx}has large positive values (~8% for ε

_{thread}= 3.4). Contrariwise, when the cylinders are orientated along 0z (E

_{TM}), ΔA

_{ε_xx}has negative values (~−4.5%). These values strictly correspond to the relative volume portions of the treads orientated along 0x (E

_{TE}) and 0z (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 0y (perpendicularly to E

_{TE}and E

_{TM}, the parameter ΔA

_{ε_xx}is close to 0 (i.e., the sample behaves as almost isotropic one).

_{ε}of the 2D-woven sample is close to this one of the pure cylinders along 0x—Figure 5b. However, when the portion of the threads with orientation along 0z 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 ε

_{thread}= 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

_{10}mode; Figure 8b for TM

_{01}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 S

_{21}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.

_{b}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

_{s}of the slices is chosen to be w

_{s}= 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 h

_{s}and length l

_{s}= W

_{s}. 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.

#### 2.5. Materials Used in the Research

_{ε}from 4.3 to 10.3) by the two-resonator method. The measured results for the pairs of parameters ε

_{par}/tan δ

_{εpar}and ε

_{perp}/tan δ

_{εperp}, as well as for the uniaxial anisotropy ΔA

_{ε}/ΔA

_{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, ΔA

_{ε}< 1%. The last two groups have representatives of relatively flexible reinforced substrates and soft artificial ceramics. Their anisotropy ΔA

_{ε}varies in a big interval—8.2–24.5%.

## 3. Results and Discussion

#### 3.1. Pure Bending Effect

_{bent}/f

_{flat}between the resonance frequencies for the lowest-order TM

_{10}mode for bent and flat rectangular resonators on pure isotropic substrate versus the curvature angle α

_{C}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 R

_{b}and the introduced curvature angle α

_{C}(e.g., α

_{C}= 4° corresponds to R

_{b}= 28.7 mm; α

_{C}= 8°—R

_{b}= 14.3 mm; α

_{C}= 12°—R

_{b}= 9.6 mm, etc.).

_{C}> 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 Figure 9 and Figure 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 L

_{E}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 TM

_{10}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 α

_{C}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.

_{C}< 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 L

_{E}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.

_{bent}/f

_{flat}of the lowest-order TM

_{10}mode in bent and flat rectangular resonators on several isotropic substrates versus the bending radius R

_{b}. 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 (h

_{s}= 0.9 mm), Figure 13c. Good results are obtained by Ro3003 substrate (h

_{s}= 0.52 mm), Figure 13a; however, at small bending radii, this soft substrate undergoes technological stretching and f

_{bent}slightly decreases. The harder substrate PC (h

_{s}= 0.5 mm) shows better stability at low R

_{b}. The results for the soft PTFE substrate (h

_{s}= 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 (h

_{s}= 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 R

_{b}.

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

_{flat_aniso}/f

_{flat_iso}between the resonance frequencies of modes TM

_{10}and TM

_{01}for flat rectangular resonators on anisotropic (ΔA

_{ε}~25%) and isotropic substrates versus the substrate thickness h

_{s}. 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 ε

_{par}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].

_{bent_aniso/}f

_{bent_iso}is shown in Figure 15a between the resonance frequencies of mode TM

_{10}for L-/W-bent rectangular resonators versus the curvature angle α

_{C}. The substrate anisotropy ΔA

_{ε}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.

_{bent_aniso/}f

_{flat_aniso}between the resonance frequencies of mode TM

_{10}for L-/W-bent rectangular resonators on anisotropic substrates versus the curvature angle α

_{C}. 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 ΔA

_{ε}, which is impossible for pure isotropic substrates. We also investigate the influence of the substrate thickness h

_{s}on corresponding ratio f

_{bent_aniso/}f

_{flat_aniso}. Figure 14b presents curves for L- and W-bent resonators at curvature angle α

_{C}= 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 α

_{C}and not investigated in detail.

_{bent}/f

_{flat}of the TM

_{10}mode in bent and flat rectangular resonators versus the bending radius R

_{b}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.

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

_{aniso}/f

_{iso}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 f

_{bent_iso}/f

_{flat_iso}between the resonance frequencies of the lowest-order mode in the same planar resonance structures (bent and flat) on isotropic substrates (Figure 18). Now, this ratio is a measure of the pure effect of bending.

_{par}> ε

_{perp}). 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

_{par}and ε

_{perp}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 ε

_{par}> ε

_{perp}) 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 ΔA

_{ε}and the actual bending radius R

_{b}. 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.

_{b}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.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**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.

**Figure 2.**Minimal and maximal bounds for the normalized resultant equivalent dielectric constant ε

_{eq}/ε

_{1}(

**a**) and normalized equivalent dielectric loss tangent tan δ

_{ε_eq}/tan δ

_{ε}

_{_1}(

**b**) of two mixed dielectrics with isotropic parameters ε

_{1}/tan δ

_{ε}

_{1}and ε

_{2}/tan δ

_{ε}

_{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).

**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: 0x, 0y, 0z and in the plane 0xy in the Ku-band (modes: TE

_{011}(

**a**); TE

_{101}(

**b**); TM

_{010}(

**c**); TE

_{111}(

**d**).

**Figure 5.**Dielectric constant anisotropy of artificial textile samples versus the ratio between the dielectric constant of the threads and air ε

_{thread}/ε

_{air}: (

**a**) for ordered straight cylinders, orientated along axes 0x, 0y or 0z; (

**b**) for 2D, 2½D and 3D woven fabrics in Ku-band (see Figure 6) (Arrows: E fields of the used TE and TM modes).

**Figure 6.**(

**a**) Woven threads of corresponding dimensions; three types of woven fabrics: (

**b**) 2D woven; straight threads along 0x, 0y; (

**c**) 2.5D woven; straight threads along 0y, wavy threads along 0x; (

**d**) 3D woven; wavy threads along 0x, 0y (all threads are with equal dielectric constant ε

_{thread}= 3.4).

**Figure 7.**Simulated E-field pattern in microstrip resonators: (

**a**,

**b**) TM

_{10}and TM

_{01}in a flat resonator; (

**c**,

**d**) TM

_{10}and TM

_{01}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.

**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 L

_{s}= 42 and width W

_{s}= 34 mm (last two cases with bending radius R

_{b}= 14.3 and 9.6 mm).

**Figure 11.**Numerical dependencies of the ratio between the resonance frequencies f

_{bent}/f

_{flat}of the lowest-order TM

_{10}mode for bent and flat rectangular resonators on isotropic substrate versus (

**a**) the curvature angle α

_{C}between the substrate slices and (

**b**) substrate thickness h

_{s}. The isotropic dielectric constant is chosen to be 3.0, but its concrete value has negligible influence. Positive and negative curvature angles are used.

**Figure 12.**(

**a**) Definition of the relation between the curvature angle α

_{C}and bending radius R

_{b}; dashed line: middle line in the resonator substrate, where the effective electrical length of the resonator is formed; (

**b**) resonance structures on positive (+α

_{C}) and negative (−α

_{C}) bent substrate (the bending radius R

_{b}is always determined to the side of the resonator layout).

**Figure 13.**Experimental dependencies of the ratio between the resonance frequencies f

_{bent}/f

_{flat}of the lowest-order TM

_{10}mode for bent and flat rectangular resonators on several isotropic substrates versus the bending radius R

_{b}: (

**a**) Ro3003; (

**b**) PC; (

**c**) commercial silicone elastomer; (

**d**) PTFE and Polyguide

^{®}Polyflon (http://www.polyflon.com; dielectric parameters 2.05/0.00045).

**Figure 14.**(

**a**) Numerical dependencies of the ratio f

_{flat_aniso}/f

_{flat_iso}between the resonance frequencies of modes TM

_{10}and TM

_{01}for flat rectangular resonators on anisotropic and isotropic substrate versus the substrate thickness h

_{s}. (

**b**) Numerical dependencies of the ratio f

_{bent_aniso/}f

_{flat_aniso}of mode TM

_{10}for L-/W-bent and flat rectangular resonators on anisotropic substrates versus the substrate thickness h

_{s}.

**Figure 15.**(

**a**) Numerical dependencies of the ratio f

_{bent_aniso}/f

_{bent_iso}between the resonance frequencies of mode TM

_{10}in L/W-bent resonators on anisotropic and isotropic substrates (h

_{s}= 0.52) versus the curvature angle α

_{C}; (

**b**) Numerical dependencies of the ratio f

_{bent_aniso/}f

_{flat_aniso}of mode TM

_{10}in L-/W-bent and flat rectangular resonators on anisotropic substrates versus the curvature angle α

_{C}.

**Figure 16.**Experimental dependencies of the ratio between the resonance frequencies f

_{bent}/f

_{flat}of the lowest-order TM

_{10}mode for bent and flat rectangular resonators on several anisotropic substrates versus the bending radius R

_{b}: (

**a**) Denim; (

**b**) Linen; (

**c**) commercial textile fabrics GORE-TEX

^{®}; (

**d**) PDMS; (

**e**) NT9338, Ro4003; (

**f**) Ro3010.

**Figure 17.**Simulated values of the ratio f

_{aniso}/f

_{iso}between the resonance frequencies of the lowest-order mode in several planar resonance structures with dimensions 30 × 30 mm on anisotropic and isotropic substrates (h

_{s}= 0.52; ΔA

_{ε}~ 25%) (this ratio gives the pure effect of anisotropy). The shapes of the considered structures are presented in Figure 19 and Figure 20. The first column for each case corresponds to a flat structure, second—bent at α

_{C}= 8°; third—bent at α

_{C}= 12°; Solid and hollow points correspond to two mutually perpendicular orientations (V & H) of the structure during the bending (when this is possible).

**Figure 18.**Simulated values of the ratio f

_{bent_iso}/f

_{flat_iso}between the resonance frequencies of the lowest-order mode in several planar resonance structures with dimensions 30 × 30 mm on isotropic substrates (h

_{s}= 0.52; ΔA

_{ε}~ 25%) (this ratio gives the pure effect of bending). The shapes of the considered structures are presented in Figure 19 and Figure 20. The first column for each case corresponds to a flat structure, second—bent at α

_{C}= 8°; third—bent at α

_{C}= 12°; Solid and hollow points correspond to two mutually perpendicular orientations (V & H) of the structure during the bending (when this is possible).

**Figure 19.**Top view of several resonance structures (bent at α

_{C}= 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).

**Figure 20.**Top view of the first three iterations of Koch fractal contours as a planar resonator (flat—first row and bent at α

_{C}= 8°—second row) with dimensions in mm: Case 1—iteration 0; Case 6—iteration 1; Case 7—iteration 2.

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

Material | h_{s}, mm | ε_{par}/tan δ_{ε}_{_par} | ε_{εperp}/tan δ_{ε}_{_perp} | ΔA_{ε}/ΔA_{tan}_{δε}, % |
---|---|---|---|---|

Textile and polymer samples | ||||

Denim | 0.90 | 1.74/0.048 | 1.61/0.030 | 7.8/38 |

Linen | 0.65 | 1.65/0.043 | 1.58/0.044 | 4.3/−2.3 |

Waterproof fabric with breathability GORE-TEX^{®} | 0.20 | 1.53/0.0057 | 1.38/0.0043 | 10.3/28 |

Polydimethylsiloxane (PDMS) | 0.70 | 2.73/0.022 | 2.57/0.019 | 6.00/15 |

Flexible isotropic and near-to-isotropic samples | ||||

Polytetrafluoroethylene (PTFE) | 0.45 | 2.05/0.00027 | 2.04/0.00026 | 0.49/3.8 |

Polycarbonate (PC) | 0.50 | 2.77/0.0056 | 2.76/0.0055 | 0.36/1.8 |

Silicone elastomer | 0.90 | 2.21/0.0010 | 2.19/0.0008 | 0.91/22 |

Ro3003 | 0.51 | 3.00/0.0012 | 2.97/0.0013 | 1.0/−8 |

Relatively flexible anisotropic reinforces substrates | ||||

Ro4003 | 0.21 | 3.67/0.0037 | 3.38/0.0028 | 8.2/28 |

NT9338 | 0.52 | 4.02/0.005 | 3.14/0.0025 | 24.6/67 |

Relatively flexible anisotropic soft ceramics | ||||

Ro3010 | 0.645 | 11.74/0.0025 | 10.13/0.0038 | 14.7/−41 |

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**MDPI and ACS Style**

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://doi.org/10.3390/s21010016

**AMA Style**

Dankov PI, Sharma PK, Gupta N.
Numerical and Experimental Investigation of the Opposite Influence of Dielectric Anisotropy and Substrate Bending on Planar Radiators and Sensors. *Sensors*. 2021; 21(1):16.
https://doi.org/10.3390/s21010016

**Chicago/Turabian Style**

Dankov, Plamen I., Praveen K. Sharma, and Navneet Gupta.
2021. "Numerical and Experimental Investigation of the Opposite Influence of Dielectric Anisotropy and Substrate Bending on Planar Radiators and Sensors" *Sensors* 21, no. 1: 16.
https://doi.org/10.3390/s21010016