Characterising nematic liquid crystals using a circular patch resonator

ABSTRACT Reconfigurable microwave material is a promising candidate for designing and manufacturing tunable microwave components. Nematic liquid crystals (NLC) are such materials since their permittivity can be tuned by an external electric field. However, many NLC mixtures were not properly characterised at higher frequency bands due to requiring a complex measurement setup. In this work, a novel method using circular patch resonator (CPR) is developed to measure the dielectric constant and loss tangent of NLCs at microwave frequencies. In addition to using the cavity model for the preliminary design and analysing the fringing effect for a better accuracy, full-wave simulations are employed to confirm the final design and aid the characteristic analysis. Three prototypes were fabricated and measured to reduce uncertainty from manufacturing defects. To avoid the possible damage when higher voltage is required for a large range tuning, a coupling mechanism is proposed between the microstrip line and coplanar waveguides (CPWs) to replace connection through vias. A high accuracy with an uncertainty of 0.02 for relative permittivity estimate has been demonstrated with experiment verification, approximate 80% improvement than other typical methods. The simple design and PCB-based manufacturing techniques can be widely employed to characterise the properties of newly-developed LC mixtures. Graphical Abstract


Introduction
Microwave-based technologies play an important role in today's society, which utilise frequencies in the millimetrewave (mmWave) regime so as to obtain the necessary large bandwidth and satisfy the demand of higher data rates.In some communication systems, such as 5G and low-orbit satellite systems, ground mobile terminals might access the communication infrastructure via beam-steering antenna arrays to increase the coverage and reduce the latency [1,2].
To implement such devices with the function of beam steering, different methods employing various materials and techniques have been used, such as semiconductor [3], RF Micro-electromechanical systems (MEMS) [4], ferroelectrics [5] and liquid crystals (LCs) [6].However, for systems with a large number of beams, it is not practical to be steered by the bulky mechanical systems when the ground terminals are moving.Hence, smart RF components with compact and low insertion loss properties are demanded to adjust the beams for more efficient use of spectrum resources.To design and manufacture such smart RF components, LC shows a promising characteristic.Based on their unique birefringence property, LC materials are initially utilised mainly for the optical applications (e.g.displays, lenses, etc.) [7,8].In the last two decades, following the evolution of microwave techniques, novel nematic liquid crystal (NLC) has drawn CONTACT Yongwei Zhang david.y.zhang@ntu.edu.cnSupplemental data for this article can be accessed online at https://doi.org/10.1080/02678292.2023.2200741.
significant attention because of its large anisotropy and low loss characteristic at higher frequency band.In addition, compared to the conventional ferroelectric materials, the main advantages of LC material are lower voltages required for tunability and being moderately low cost.Therefore, NLCs are promising materials for reconfigurable millimetre-wave applications [9,10].Several types of applications based on NLC have been demonstrated in the literature, such as phase shifters [11][12][13], tunable reflectarray antennas [14], dielectric waveguides [15,16], and steerable phased arrays [17][18][19].
Some studies have been carried out to determine the properties of LC materials in the literature.Early study for characterising the properties was presented in the 1950s, which applying a magnetic field to align the orientation of LC for a fast proof-of-concept in the lab [20].Nowadays, with the development of microwave devices, several technologies covering a large frequency range have been proposed to estimate the characteristics of LCs with the help of an electric biasing, which can be classified as a broadband method and resonator-based method.Broadband methods can determine the dielectric properties over a broad frequency range while the measurements of permittivity usually are not accurate enough.Thus, they are very useful for applications in higher frequency bands (e.g.above 30 GHz for optical components).Some methods, such as temperature-controlled coaxial transmission line [21] and a covered microstrip line [22,23], have been proposed to obtain a rough estimate of LC properties.
Compared with the broadband methods, resonatorbased techniques can estimate the parameters with a higher accuracy but only at single or some discrete frequencies, and are primarily applied in low microwave frequency range.Meanwhile, one of advantages is that the resonate frequencies and other properties can be accurately analysed by means of a cavity model.Several methods such as using a split-cylinder ring [24,25], a patch resonator [26], a circular patch resonator (CPR) [27], and an inductive coupled ring resonator [28], have been reported in the literature.In [24], it can yield �0:22 uncertainty on permittivity measurement for QYPD-036 material by using a split-cylinder ring.In contrast to a microstrip line resonator [22], a CPR with a cavity can tune a larger volume of LC, which potentially leading to a better orientation alignment of the molecules, accordingly a better accuracy is expected.In [27], a CPR was also used to determine the permittivity but a complex experimental setup is needed.In addition, compared with [27], this work provided a numerical approximation that allows a better prediction of the effective circular radius, and achieves a higher accuracy for dielectric constant determination.
In this work, a low-cost printed circuit board (PCB)based CPR with high resolution is designed, fabricated, and tested to determine the dielectric constant and loss tangent of the newly developed off-the-shelf NLCs.A cavity model is developed to investigate the relationship between the physical dimension and resonant frequency of the CPR.The parameters of the cavity model are verified by a full-wave simulator.Finally, one CPR is designed and three samples are fabricated and measured.The experimental results from vector network analyser (VNA) had a good agreement with those from the fullwave simulation, as well as the cavity model.It should be noted that the proposed CPR has an operating frequency lower than 10 GHz based on the following considerations: i) there are many wireless applications operating in sub-10 GHz frequencies including 5G and low-orbit satellite communication system; ii) the manufacturing cost is relatively low, meanwhile it is easier to analyse the measurement uncertainty; iii) permittivity ε keeps reasonably constant over this frequency range, while it has relatively large variation in other higher millimetre wave frequency bands.
The main contributions of this investigation are as follows: • The proposed design is solely based on a PCB technology without involving a complex process as in the existing methods, hence it is applicable in most scenarios where accurate characterisation are required for high-tunability devices at a low cost.• Compared to other methods such as in [27], this study considers the fringing effect of circular patch, therefore yields a better accuracy.The error of the permittivity in the proposed method is less than 0.02, which was typically over 0.1 based on the previous methods.• To calculate ε r with respect to the resonant frequency, a closed-form expression has been derived to determine the effective radius a e by decoupling the permittivity ε r where a e and ε r have a high correlation.
• To avoid possible damage to the instrument caused by external high bias voltages required for a large range of tuning, this study utilizes a coupling mechanism, which is beneficial for applications that require a higher bias voltage as it is isolated for DC between the microstrip line and two CPWs for feed.
The paper is organised as follows: Section 2 introduces the cavity model of a CPR and presents a closed-form expression with respect to the resonant frequency and the permittivity of the LC material; Section 3 first describes the design and fabrication of the CPR, then the experimental results and the uncertainty analysis are provided, finally the uncertainty of the proposed CPR are compared to those by other typical methods in the literature.The conclusion is presented in Section 4.

Material and methods
As an anisotropic material, the unique feature of nematic LC is that the direction of LC molecules can be reoriented, by means of an external low-frequency electric or magnetic field, which can then lead to different dielectric constants.Anisotropy is the key characteristic attractive in microwave devices.According to the orientation of LC molecules, the permittivity of LC can be simply represented as a tensor vector from the parallel direction ε k to the perpendicular direction ε ?, as shown in Figure 1.
Anisotropy can be defined as the maximal difference of these two extreme values, Δε ¼ ε k À ε ? .The choice of LC material for a microwave device depends on some intrinsic parameters, such as permittivity, loss tangent and Frank elastic constants, etc.Among these parameters, elastic constants are mainly considered in optical applications, and out of scope of this work.Loss tangent is a useful parameter to evaluate the dissipation factor of RF devices, and can be derived from the quality factor.Both the relative permittivity and loss tangent have been examined in this study due to their importance for the design of microwave components.
Taking conventional phase shifter as an example, the maximal phase delay ΔΦ mainly depends on the physical length of shifter l and two extreme dielectric constants ε k and ε ?, can be written as [13]: where f is the operating frequency, c is the speed of light.
Several LC mixtures initially designed for the display devices, such as E7 (Merck KGaA), K15 (5CB), etc., have been used for the microwave components.However, they demonstrate relatively small anisotropy and high loss tangent at mmW frequency band.Recently some highperformance LC materials have been developed to meet the demand of microwave applications, e.g.GT7-29001 provided by Merck KGaK showing a large anisotropy (Δε r > 1 at 19 GHz), but it is not widely available.In addition, some LC mixtures might have a great potential for the microwave applications, but they only provide parameters for the optical devices, and their characteristics at the microwave frequency band need to be thoroughly examined.Thus, to achieve the optimal performance of LCbased microwave devices, it is very important to develop low-cost and easily implemented technique to accurately determine the characterisation of various LC materials.
This study uses a CPR to analyse the characteristics of LC because of its simple structure and high estimation accuracy.A basic model of CPR is shown in Figure 2. In contrast to examining structures with a rectangular patch and transmission line, we used the cylindrical coordinate ðρ; ϕ; zÞ system to investigate CPRs.In addition, only one parameter (radius a of the disc) is needed to determine the orders of electromagnetic modes.For LC-based CPR design, the circular patch is in ðx; yÞ plane, LC material is filled in the middle cavity which is formed between the inverted circular patch and the ground plane.The bulk of LC material in the cavity can be treated as a substrate for characteristic analysis.Meanwhile, to control the direction of LC molecules, the patch and ground plane are also used as electrodes to provide bias voltage.
Several approaches have been adopted to design CPRs, including transmission line model, cavity model and full-wave simulations.In this work, cavity model and full-wave simulation were implemented to design the CPRs and characterise the LC materials used in the devices.Full-wave simulation is more accurate than any other models, however it needs a large number of optimisation iterations and usually gives less  physical insight.Whereas the cavity model is simpler and can indicate good physical insight.The following presents the main principle and key parameters of a cavity model.The results for a CPR design based on full-wave simulation and the cavity model are given and compared in the end.

Cavity model of a CPR
RF properties of LC materials can be studied by using a CPR with the cavity model, which exploits the relationship between the resonant frequency and dielectric constant, and is a commonly-used method to estimate the permittivity ε r and the loss tangent.As shown in Figure 2, since the substrate height h is much smaller than the wavelength of RF signal λ (h< <λ), the primary electric and magnetic modes that are supported in a CPR are TM z , where z is taken perpendicular to the patch.
Based on the electromagnetic field theory, the fields propagating in the cavity can be derived by using the vector potential method, which satisfies the homogeneous wave formulation in the cylindrical coordinate.In this section, the principle equations are provided to derive the dielectric properties, and more detailed description regarding the cavity model can be found in [29] and Appendix A. The resonant frequency corresponding to the permittivity for the dominant mode TM z mn0 can be written as where f mn0 is the resonant frequency, which is the key parameter required to derive ε r , and can be obtained from scattering parameters of S 11 or S 22 based on VNA measurements or the full-wave simulations.ε r is the relative permittivity of LC material, a is the radius of the inverted circular patch, h is the height of LC in the cavity, υ 0 is the speed of light in free space, χ 0 mn (m ¼ 0; 1; 2; ::: ; n ¼ 1; 2; 3; :::) is the zeros of the derivative of Bessel function J m ðχÞ which is used to determine the order of resonant frequency.Therefore, according to a given TM z mn0 mode, the dominant value of χ 0 mn can be calculated using the Bessel functions.Taking the first four modes of TM z mn0 as an example, we can calculate, in ascending order, χ

Effective radius of a circular patch
In practice, f mn0 predicted from the cavity model based on (2) is usually slightly higher than that from the measured results.The reason is that it does not take into account the fringing effect of the circular patch, which makes the electrical radius a larger than the physical size, as shown in Figure 3(a).To optimise the parameter for the cavity model and improve the estimate accuracy of ε r , we need to compensate the extra length for the radius a.In this work, effective radius a e is considered to include the fringing effect, and a good approximation is made in [30] and can be written as Based on the geometry of the proposed CPR (a ¼ 6:5 mm; h ¼ 0:265 mm), when the range of the permittivity of 1 < ε < 3 is considered, the resonant frequencies corresponding to the physical radius a and the effective radius a e are calculated and plotted in Figure 3(b), respectively.We can observe the frequencies determined from a e by (3) are less than those with the physical radius a, which mainly because a e is greater than a due to the the fringing effect.Meanwhile, the deviation decreases with an increase of ε r and the maximum discrepancy is about 1 GHz when ε r ¼ 1. Adopting the effective radius a e and taking χ 0 11 as an example, the resonant frequency for the dominant mode TM z 110 of (2) can be rewritten as

Determination of permittivity
With the known values of f 110 and a e , ε r can be directly calculated using (4) and can be written as: However, in order to calculate a e using (3), we must know the value of ε r , which means there is a high correlation between a e and ε r in (3), so we must decouple a e from ε r first.In the literature, considering the fact that a> >h for LC-based devices (since h should be small enough to have a quick response time for tuning), and for the majority of microwave LC materials, ε r is usually between 1.6 to 3.5 at sub-10 GHz band.We can fit a e by assuming different values to ε r in the range from 1.6 to 3.5 in (3).With this approach, for the proposed CPR structure, the associated effective radius a e with respect to a is obtained by interpolation method, and can be expressed as

Determination of loss tangent
In addition to permittivity, loss tangent tanδ can be estimated by seeking the quality factor and the relationship between them is approximately given as [29] where Q is the unloaded quality factor when the energy dissipation of conductor is considered and radiation loss is neglected in the microwave resonator.Accurate determination of loss tangent is a very complex task [31] since the loss from various sources is difficult to distinguish.For transmission type resonators such as a CPR, the loaded Q L can be calculated as where f 1 and f 2 are respectively the half-power frequencies according to the resonant frequency f mn0 for a given mode, which can be obtained at 3 dB power points deviated from the resonant frequency.The loss tangent tanδ is initially defined as [32] where ε 00 r is the imaginary part of permittivity, which can be determined by where V c is the volume of the empty cavity and Q 0 is the value of quality factor.When the LC is filled in the cavity, the value of the volume (V s ) remains unchanged.But the value of the quality factor will be reduced (Q L ). ε 0 r is the real part of permittivity, which can be obtained as where f c is the resonant frequency with an empty cavity, and f s is the resonant frequency after the NLC sample is filled.With ( 8)-( 11), the loss tangent can be derived accordingly.
To sum up, according to the above-mentioned equations, ε r and loss tangent tanδ can be determined by a fullwave simulation or VNA measurements.Taking the VNA measurement as an example, f 110 is obtained from the S-parameter (S 11 or S 22 ) measurements with TM 110 as the dominant mode, then we can derive the relative dielectric constant ε r using (5) and the numerical approximation a e defined in (6).Finally, loss tangent tanδ can be estimated by (9) with the corresponding frequencies.

Feasibility analysis of the cavity model
In order to verify the feasibility of the proposed cavity model, we first acquire the resonant frequencies using (4), then the results are compared with those from the full-wave simulation.The main parameters used for this analysis are identical to the fabricated CPR provided in Section 3, e.g. a = 6.5 mm and h = 0.265 mm.Five distinct dielectric constants are assigned separately (the range from 2.22 to 2.98) for verification, we obtained the resonant frequencies based on the cavity model and the full-wave simulation, respectively, as illustrated in Figure 4.In Figure 4, the curves represent the refection coefficient S 11 from the full-wave simulation, and the straight lines are the frequencies calculated by using ( 4) from the cavity model.Taking ε r ¼ 2:22 as an example, the resonant frequency from the lowest value of S 11 using the full-wave simulation is 8.84 GHz, and it is estimated 8.81 GHz based on the cavity model, the discrepancy is only 0.03 GHz, which is close enough to demonstrate the high accuracy on determination of dielectric characteristics through the cavity model.

Experimental results and discussion
To accurately evaluate the characteristics of LC in mmW band, a CPR was designed with the optimised parameters by a full-wave simulation.Three prototypes were fabricated and tested.The experimental results including permittivity and loss tangent for the two types of newly-developed LC materials are provided and verified.

Design and fabrication
Figure 5 illustrates the structure of the proposed CPR, LC is contained in a closed cavity between the circular patch on the top and the ground plane on the bottom.As shown in Figure 5(a), on top layer three holes are In contrast to a conventional CPR design, this work proposed a coupling mechanism for microwave transmission, to avoid the possible damage to the instrument caused by a higher bias voltage.Instead of a direct contact through the vias, we used copper patches to couple the RF signal between the transmission line of the resonator and the CPWs for feeding, as shown in Figure 5 (c).More detailed analysis on the coupling mechanism is given in Section 3.2.
Figure 6 shows the key fabrication processes of the CPR.The top board consists of the primary elements, including an inverted circular patch, coupling patches, CPWs, feeding lines, and filling holes, CPWs are designed with a 50-Ω characteristic impedance, as shown in 6(a).The board after the deposition and curing treatments with Polyimide (PI, AL 1254) on the patch surface is presented in Figure 6(b), the thickness of PI is about 100 nm and the curing process at 230 � C took 20 minutes.The microscopic grooves are realised by rubbing the PI surface with velvet in the same direction, shown in Figure 6(c).Figure 6(d) illustrates the device under test (DUT) by a vector network analyser (VNA, N9918A of Keysight).

Experiments
In this section, experiments are carried out to estimate the dielectric characteristics of new developed LC materials.The procedure to determine the dielectric properties of the NLCs is summarised into three steps: Step 1, the reflection coefficients of DUTs are acquired by using the VNA, to determine the resonant frequencies.Step 2, the values of dielectric constant ε r are calculated using ( 5) by the cavity model.Finally, ε r obtained from Step 2 was further verified with the full-wave simulation.
As mentioned earlier, in order to support higher bias voltages for achieving the full anisotropy potential, a coupling mechanism is proposed for feeding the resonator in this work.Thus we first analyse the impact of the coupling mechanism, in comparison to conventional vias arrangements.The relativity permittivity of air (where ε r ¼ 1 without the presence of the LC material) is used as a testing benchmark to evaluate the accuracy of the method.The scattering parameter S 11 obtained from measurement and the full-wave simulation is shown in Figure 7.It depicted that the coupling mechanism pushes the resonant frequency slightly towards higher frequency compared with via arrangement.When the coupling mechanism is employed, we observe that the resonant frequency from measurement (12.48 GHz) is very close to that from simulation (12.45 GHz) (see Table 1 for details).
Two types of LC mixtures, JC-M-LC-E7 from JCOPTIX and QYPD-470-10-N001 from Qingdao QY Liquid Crystal Co., are used for testing.They are commercially available and both have been developed primarily for optical applications.Their dielectric properties at the optical spectrum are summarised in Table 2 but short of description on characteristics at microwave frequencies, and the molecule structure of the intermediate compounds for typical E7 mixtures is depicted in Figure 8.They can be potentially employed to design and manufacture microwave or mmWave components.Thus this study investigates their dielectric characteristics (dielectric constant and loss tangent) at the microwave frequency bands.It is pointed out that for JC-M-LC-E7, this is the first time to examine its properties in microwave frequencies as far as we understand.
To determine the anisotropy and loss tangent of the two types of LCs (JC-M-LC-E7 and QYPD-470-10-N001), the experiments were performed by applying external bias voltages.To reduce the random errors, three devices dubbed as DUT1, DUT2, and DUT3, were fabricated, as shown in Figure 9.The LC material was filled into the cavity through the filling holes, and a 1 KHz sine wave voltage is applied with a AC power source.By adjusting the voltage step by step, a largest phase shift was observed when the bias voltage was reaching 32 V (the peak to peak voltage), where the long axis of the LC molecules was expected to be approximately in parallel to electric field of the propagated waves.
When JC-M-LC-E7 was filled in the cavity of DUT1, Figure 10 demonstrates the scattering parameter S 11 obtained from the VNA and full-wave simulation, respectively.We can see there is a good agreement on the resonant frequency between simulation and experiments, whether 0 or 32 V voltage was applied.In Figure 11, S 11 for JM-M-LC-E7 from experiments using three DUTs was presented.As expected, three DUTs have the approximate performance.The similar results for QYPD-470-10-N001 were also observed, as shown in Figure 12.Using the frequencies acquired from measurements, the dielectric constant and loss tangent for both types of LC mixtures were calculated based on the cavity model, and the results were listed in       Tables 3 and 4, respectively.The term unbiased and biased in these tables mean that 0 and 32 Volts are applied, respectively.

Uncertainty analysis
Uncertainty may exist in every step of the investigation.There were many potential sources of errors, such as material stability, manufacturing tolerance, optimal position for biasing, etc.Two key steps have been taken to reduce the uncertainty: (1) Coupling mechanism is used to make connection between the transmission line where the CPR is connected to and the CPW, which can obtain a more stable and accurate measurement than the case where a direct connection exists through a vias, especially for higher external voltages.(2) More than one samples have been made for cross checking and reducing the uncertainty.Three devices have been fabricated to make reliable test for the proposed method.The resonant frequencies and the corresponding dielectric constant from the measurement and simulation are summarised in Table 5, and a slight discrepancy among these devices is observed.It is shown that the error for anisotropy determination is within a scope of 0.02.
To clearly demonstrate the performance of the proposed CPR, uncertainty comparison between the proposed method and other similar methods in the literature is listed in Table 6.As shown in Table 6, the uncertainty for the permittivity determination proposed by this work is about 0.02, which is much lower than   that based on other conventional techniques, a significant improvement is observed.

Discussion
This work shows that the resonator-based technique can achieve a high accuracy in determining permittivity and loss tangent of NLC with a low cost.However, there are still some problems to be addressed in the future work.For example, although we have made some analysis on the effect of coupling mechanism, quantitative analysis such as numerical model, propagation characteristics and impedance mismatching, should be investigated more thoroughly.The alignment procedure is critical to establish and control the relationship between the orientation of the LC molecules and polarisation of the waves propagating through the material [33].However, we can not evaluate the performance of alignment procedure by some simple methods.In addition, one of the concerns in practical applications is the response time of LC. which depends primarily on the thickness of the LC cells.The response time is not the main focus of this study and would be investigated in future studies.

Conclusions
A technique using a CPR to determine the dielectric constant and loss tangent of NLC materials has been developed and validated at a moderately low cost.Numerical methods, including the cavity model and the full-wave simulation, are used to assess the CPR performance with the optimised design parameters, aiming for a high-level determination accuracy for material characterisation.Three prototypes are manufactured with a PCB-based process and measured to reduce the random errors.The results demonstrate a good agreement between the simulation and experimental measurements.The uncertainty for determining anisotropy of the NLC materials based on the proposed CPR is below 0.02, which is significantly lower than that reported in the literature.The proposed method can be used to design highperformance microwave devices at microwave and mmWave frequency bands.

Figure 1 .
Figure 1.(Colour online) Orientation of LC molecules under different biasing voltages.

Figure 2 .
Figure 2. (Colour online) Geometry of a CPR with parameters of circular radius a, substrate height h and dielectric constant ε r in a cylindrical coordinate (ρ; ϕ; z) system.

Figure 3 .
Figure 3. (Colour online) Fringing effect and resonant frequencies corresponding to the circular radius a and effective radius a e vs. different dielectric constants.

Figure 4 .Figure 5 .
Figure 4. (Colour online) Comparison of the resonant frequencies from the cavity model and a full-wave simulation vs. different values for dielectric constant of the LC material.

Figure 6 .
Figure 6.(Colour online) Key fabrication steps for the CPR: (a) main board including the circular patch, (b) depositing and curing treatment of PI on the patch surface, (c) microscopic grooves on the PI surface, and (d) a CPR under test using a VNA.

Figure 7 .
Figure 7. (Colour online) Scattering parameters of the empty CPR device with the coupling mechanism and via connection without LC filled in the cavity (ε r ¼ 1).

Figure 9 .
Figure 9. (Colour online) Three CPR devices manufactured for material characterization.

Figure 8 .
Figure 8. Molecule structures for typical E7 intermediate compounds, and associated empirical formulas.

Figure 10 .
Figure 10.(Colour online) Scattering parameter S 11 for JC-M-LC-E7 from the full-wave simulation and experiment using DUT1 at 0 and 32 V, respectively.

Table 1 .
Resonant frequency and calculated permittivity for air from the CPR with the coupling mechanism and vias connection (where ε r ¼ 1 is the norm).

Table 2 .
Comparison of dielectric properties of available new NLCs and classical mixture E7 (Merck) for display applications.

Table 3 .
Resonant frequency, calculated dielectric constant and loss tangent for JC-M-LC-E7 using three DUTs.

Table 5 .
Uncertainty analysis based on resonant frequency and interpolation from full-wave simulations when the cavity is empty.

Table 6 .
Uncertainty comparison of the proposed work to that of methods found in the literature.