a State Key Laboratory of Millimeter Waves, School of Information Science and Engineering, Southeast University, Nanjing 210096, China
b State Key Laboratory of Space-Ground Integrated Information Technology, Beijing Institute of Satellite Information Engineering, Beijing 100095, China
c Research Institute of Millimeter-Wave and Terahertz Technology (RIMMATT), Nanjing 211111, China
This paper presents a systematic investigation and demonstration of a K-band circularly polarized liquid-crystal-based phased array (LCPA), including the design, over-the-air (OTA) in-array calibration, and experimental validation. The LCPA contains 16 phase-shifting radiating channels, each consisting of a circularly polarized stacked patch antenna and a liquid-crystal-based phase shifter (LCPS) based on a loaded differential line structure. Thanks to its slow-wave properties, the LCPS exhibits a maximum phase-shifting range of more than 360° with a figure of merit of 78.3(° )·dB−1 based on a liquid crystal layer with a thickness of only 5 μm. Furthermore, an automatic OTA calibration based on a state ergodic method is proposed, which enables the extraction of the phase–voltage curve of every individual LCPA channel. The proposed LCPA is manufactured and characterized with a total profile of only 1.76 mm, experimentally demonstrating a scanned circularly polarized beam from −40° to +40° with a measured peak gain of 12.5 dBic and a scanning loss of less than 2.5 dB. The bandwidth of the LCPA, which satisfies the requirements of port reflection (|S11|) < −15 dB, an axial ratio (AR) < 3 dB, beam squinting < 3°, and a gain variation < 2.2 dB, spans from 25.5 to 26.0 GHz. The total efficiency is about 34%, which represents a new state of the art. The use of the demonstrated low-profile LCPA to support circularly polarized scanning beams, along with the systematic design and calibration methodology, holds potential promise for a variety of millimeter-wave applications.
With the commercialization of 5G communications [1] and the development of the Internet of Things [2], planar antenna arrays supporting electronically scanned beams are indispensable components that are widely employed in wireless systems for a variety of applications such as communications, radars, sensing, and imaging [3]. Conventional approaches to realize electronically scanned arrays rely on the usage of a large number of beamforming circuits/chips with high cost and high complexity [4], [5]. Thus, alternative new array technologies that can accomplish such functionalities have been actively sought over the past few decades. These emerging methods either utilize reconfigurable components, such as varactor diodes [6], [7], positive-intrinsic-negative (PIN) diodes [8], and micro-electro-mechanical systems (MEMS) switches [9], [10], or exploit phase-change materials, such as ferroelectric materials [11], barium strontium titanate (BST) films [12], and liquid crystals (LCs) [13], to achieve the tunable phase delays required for dynamically controllable beamforming. Among these methods, LCs are of particular interest due to their low cost and maturity for mass production, which have already been demonstrated in the display industry [14], [15], [16]. An LC is a state of matter partway between a crystalline solid and an amorphous liquid; one of the most common phases of an LC is the nematic phase [17]. Nematic LCs inherently exhibit uniaxial electromagnetic responses and own an electrical tunable permittivity. As a result, they have attracted a tremendous amount of interest in the field of microwave engineering, in applications such as phase shifters [18], [19], [20], filters [21], [22], polarization agile antennas [23], and multi-mode antennas [24]. In particular, LC-based beam-steering antennas [13], [16], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36] have become a promising technological venue for a number of applications and have thus been widely studied. Compared with beam-steering antennas based on existing reconfigurable components [6], [7], [8], [9], [10], LC-based beam-steering antennas have the advantages of a lower cost, better integration, and/or continuous tunable phases. Moreover, in contrast to other phase-change materials [11], [12] used for beam-scanning antennas, LC materials allow for a wider operational frequency range from microwaves to terahertz, while requiring lower bias voltages. The previously reported beam-steering antennas based on LCs can be further divided into LC-based lens antennas [25], [26], LC-based leaky wave antennas [16], [27], LC-based reflectarrays [28], [29], [30], and LC-based phased arrays (LCPAs) [13], [31], [32], [33], [34], [35], [36]. Among these, LCPAs are relatively more promising candidates for potential applications due to their low profile and ease of integration.
Over the past few years, various types of LCPAs have been investigated, with a typical architecture containing multiple channels, each including a radiating element and an LC-based phase shifter (LCPS). Different types of radiators have been used for the array element, such as magneto-electric dipoles [31], patches [13], [33], dielectric resonator antennas [34], and Yagi-Uda antennas [35]. An LCPS usually adopts LCs as the substrate material and provides tunable phases with a resonant or non-resonant structure; a commonly used structure is the LC-loaded inverted microstrip line (IMSL) [13], [19], [20], [33], [34], [35]. As described in Ref. [31], it demonstrates a Ka-band 1 × 4 LCPA constructed with planar magnetoelectric dipoles and reflection-type LCPSs based on a 127 μm-thick LC layer. This LCPA achieves a maximum gain of 6.7 dBi with a small scanning coverage from −22° to +23°. Another reported K-band 2 × 2 LCPA adopts a single-layer patch as the array element and a 127 μm-thick LC layer as the substrate of an IMSL LCPS [13]. Since the two-dimensional (2D) array arrangement is realized by using miniaturized radiators and phase shifters, the LCPA suffers from a relatively small scanning range of less than ±23° and a gain of only 5.9 dBi. As described in Ref. [35], it proposes a 1 × 4 endfire LCPA based on Yagi-Uda elements. The thickness of the LC layer of the LCPS is reduced to 20 μm, while the beam can be scanned from −40° to +40°. However, only a maximum gain of 4.9 dBi is achieved, due to the thin LC layer. This structure is further modified by combining the LCPSs with a Butler matrix [37], which enhances the scanning range and gain to ±60° and 5.8 dBi at the expense of using a more complicated circuitry.
In addition to depending on the LCPA’s structural design, the performance of an LCPA strongly relies on the fabrication capability, especially the LC filling technology [14]. In general, a fully closed cavity between the substrates is required to contain the LC molecules; this can be realized by hollowing out a portion of the printed circuit boards (PCBs) [31], removing parts from the metallic plates [33], or separating two layers of glass with surrounding spacers [13]. The thickness and filling accuracy of the LC layer will directly affect the channel uniformity and switching time of the LCPA. It can be seen from the previously reported LCPA designs that several shortcomings still need to be overcome. First, it is desirable to extend the radiation polarization of LCPAs—which has been limited to linear polarization—into circular polarization for applications such as satellite communications and radar imaging. Second, almost all existing LCPAs have a small number of array elements (i.e., no more than four), resulting in the disadvantages of a very limited gain and a low beam resolution. Third, the majority of the LCPAs reported thus far possess an LC layer thickness greater than 20 μm; however, thinner LC layers are called for in order to reduce the beam-switching time and material cost.
Another urgent challenge to be solved is the development of a reliable calibration method for LCPAs. Unlike phased arrays that are based on digital phase shifters with discretized phase shifting, LCPSs provide continuously tunable phases by rotating the LC molecules with different bias voltages. Importantly, the relationship between the transmission phase and the bias voltage—that is, the phase-voltage curve—needs to be experimentally determined for accurate beamforming. Due to this unknown phase-voltage relationship, conventional calibration methods are no longer suitable, including the rotating-element electric-field vector (REV) method [38], [39], phase-toggling method [40], or synthetic array calibration [41]. Moreover, inaccuracy of the geometrical dimensions and fluctuation of the material properties of the LCPSs can lead to performance variation, including in both the initial phase and the line shape of the phase-voltage curve, particularly for LCPAs based on an ultrathin LC layer. Thus, all the channels of the LCPA must be individually calibrated by determining the distinct phase-voltage curve for each channel.
Although several calibration methods for LC-based antennas have been proposed, they have significant limitations. One approach is to measure the phase-voltage curve of every employed LCPS without being connected to the radiating element [36]. This method is cumbersome and is only suitable for small-scale LCPAs; thus, it cannot be applied to millimeter-wave LCPAs with seamlessly connected radiating elements and LCPSs. Another method is to measure a separate LCPS and use its phase-voltage curve for all the channels in the LCPA for beamforming [35]. However, this method ignores the differences between different channels. To date, there is still a lack of an over-the-air (OTA) in-array calibration method for extracting the characteristics of each channel in an LCPA; nevertheless, this information is indispensable and critical, especially for LCPAs with a large array size and a thin LC layer (e.g., when the thickness of the LC layer is less than 10 μm).
In this paper, a systematic design, OTA calibration, and experimental validation of a circularly polarized (CP) phased array operating in the K-band based on 5 μm-thick ultrathin LCs are proposed. The LCPA contains 16 phase-shifting radiating channels, each consisting of a CP stacked patch antenna and a novel LCPS. Based on a differential line structure with periodic metallic bar loadings, the proposed LCPSs offer a phase-shifting range of more than 360° and a figure of merit (FoM) of 78.3°·dB −1 with an LC layer only 5 μm thick. Moreover, a novel OTA calibration strategy based on the state ergodic method (SEM) is proposed and implemented to automatically extract the phase-voltage curves of each channel in the LCPA. Finally, by applying a phase assignment strategy to the extracted phase-voltage curves, high-performance beam steering can be achieved, which is experimentally demonstrated with a scanned CP beam from −40° to +40° within a frequency range from 25.5 to 26.0 GHz, with a maximum gain of 12.5 dBic and a total efficiency of about 34%. The proposed generic methodology paves the way for the design and characterization of LCPAs for communications and sensing; it can also be applied to different forms of beam-scanning arrays based on LCs or even other types of phase-change materials. The advantages of the proposed LCPA include its high efficiency, low profile, and faster response time. The current one-dimensional (1D) beam-scanning ability can be useful in vehicular communication, assembly line monitoring, geological detection, and other applications. By expanding the array scale and extending the arrays to 2D arrays, high-gain and 2D beam-scanning properties can be achieved with a low manufacturing cost, holding promise for a broad range of millimeter-wave applications such as satellite communications, radars, and imaging.
2. Design of the LCPA
The schematic diagram of the LCPA sub-system is displayed in Fig. 1. The essence of the LCPA’s beam-scanning capability relies on the tunable phase shifts of each channel when different voltages are applied to the nematic LC molecules in the LCPSs. The entire LCPA sub-system includes the LCPSs, CP array elements, feeding network, and control board. The control board is used to adjust the LC states in each LCPS by providing desirable voltages according to the commands issued from a personal computer (PC). The designs of all the modules are described individually below.
2.1. Design of the LCPSs
As critical components in achieving beamforming for the LCPA, the design of the LCPSs is of great importance in realizing a full 360° phase coverage with a low insertion loss. Considering the limited available channel space and fabrication technology, the currently widely used IMSLs suffer from a small tunable phase coverage and high insertion loss [13], [19], [33], [34], [35]. To address this issue, a periodically loaded differential line (LDL) is employed, which enables an ultrathin LC thickness of less than 5 μm and a high FoM of the phase shifter.
2.1.1. Structure and modeling of the LCPSs
The configuration of the proposed LCPSs is depicted in Figs. 2 (a)-(d). In the top view of the LCPS shown in Fig. 2 (a), two microstrip lines are printed on the top layer of substrate 1 (TLY-5 with a thickness of 0.25 mm) in order to connect to the end launch-connectors for the scattering parameter characterization. The microstrip lines are coupled to the embedded microstrip lines, which are printed on the bottom layer of the top glass panel, through a slot etched out on the ground plane printed on the bottom layer of substrate 1. As shown in the bottom view of the LCPS ( Fig. 2 (b)), the main components of the LCPS include two baluns, two LDL transition sections, and an LDL phase-shifting section. The detailed structure of the differential line transition section with non-uniform loading is illustrated in the enlarged view ( Fig. 2 (c)), where the floating metallic bars printed on the top layer of the bottom glass panel are aligned perpendicularly to the differential line. For impedance matching, the lengths of the metallic bars are gradually reduced from the center to two ends. Indium-tin oxide (ITO) lines are connected to the differential line and metallic bars, respectively, for biasing. When a bias voltage ( Vbias) is applied to the LCPS, the LCs in the overlapped area between the differential line and the metallic bars will exhibit a tunable permittivity value, whereas the properties of the LCs in the other regions will remain almost unchanged. The LCs in these two different types of regions are denoted as the effective LC and constant LC, respectively (see side view in Fig. 2 (d)). The parameters of the LC material are the relative permittivity parallel to the long axis of the LC moleculars ε// ≈ 3.84, tanδ// ≈ 0.007, the relative permittivity orthogonal to the long axis of the LC moleculars ε⊥ ≈ 2.66, and tanδ⊥ ≈ 0.015, where tanδ// and tanδ⊥ are the loss tangent values parallel and orthogonal to the long axis of the LC moleculars.
In order to clarify the working principle of the LCPS based on the periodically LDL, the equivalent circuit and the dispersion curves of a single unit cell with and without the metallic bar loading are illustrated in Fig. 2 (e). The equivalent circuit of the unloaded differential line, which is a transverse electromagnetic transmission line, is composed of a series inductor ( L0) and a parallel capacitor (C0) that are almost constant and unaffected by the LC states [20], as manifested by the dispersion curves (see the dot-dash lines in Fig. 2 (f)). After loading the metallic bars, an extra shunt inductor (Lbar) is introduced, along with two series-connected shunt capacitors (CLC ) formed in the overlapped regions between the bars and the differential line. The dispersion curves of the unit cell with and without the bar loading obtained by a full-wave eigenmode solver are displayed in Fig. 2 (f). It can be seen that the loaded unit cells exhibit the characteristics of slow-wave structures [42]. When increasing the permittivity value of the LCs, the dispersion curves of the unloaded differential line remain almost unchanged; in contrast, a phase difference of 10.8° occurs in the LDL across a single unit cell at 25.5 GHz. From the vectorial of the electric field in Fig. 2 (g), it can be seen that the fields of the LDL structure are strongly bounded in the overlap areas, revealing that the effective permittivity is mainly controlled by the 5 μm-thick LC layer instead of the glass substrates. Since a thinner LC layer can result in a larger CLC, a wide variation range in ΔCLC can be achieved by modifying the permittivity value of the 5 μm-thick LC material. Correspondingly, the propagation constant β can also be changed to realize the phase-shifting function of the unit cell. In our design, the number of unit cells is selected to be 41 in order to achieve a smaller size and a lower insertion loss, on the premise of reaching a phase-shifting range of 360°.
2.1.2. Simulated and measured results of the LCPSs
In order to verify the performance of the designed LCPSs, a number of prototypes were fabricated and measured. The bottom view and details of an LCPS are displayed in Fig. 3 (a). First, the top PCBs and the bottom LDL located on the two glass substrates were manufactured individually. The LCs were filled in between the two glass panels whose separation distance was controlled by glue spacers at the periphery of the glass panels. Then, the two parts were bonded together. Two southwest end-launch connectors were installed at the two microstrip line ports for testing. In order to adjust the permittivity of the LC material, 300 Hz square-wave pulses were applied to the LCPS through a flexible printed circuit. The peak-to-peak value of the bias voltage ( Vbias) was set within the range of [Vmin, Vmax], where Vmin and Vmax, defined as the minimum and maximum value of Vbias, are set to be 0 V and 23.5 V in our design.
The simulated and measured scattering parameters of the LCPS, which are reported in Figs. 3 (b) and (c), show good agreement. The different states of the LCs were realized by changing the permittivity of the LC material in full-wave simulations and by applying different bias voltages in the experiments. The measured reflection coefficient indicates that the proposed LCPS achieves a good impedance match from 24 to 27 GHz with port reflection (| S11|) < −15 dB. Both the simulated and measured transmission phases of the LCPS are normalized to align with the case of LCs in the initial state—that is, an unbiased state (Vbias = 0 V), at 25.5 GHz. The relationships between the transmission phase and the bias voltage resemble smooth inverted S-shaped curves at all different frequency points from 24.5 to 26.5 GHz. According to the simulations and measurements, the LCPS achieves maximum phase shifts (ΔΦmax) of about 366° and 390°, and maximum insertion losses (ILmax) of around 4 and 5 dB at 25.5 GHz, respectively. The fluctuation of the measured insertion loss is less than 1.3 dB across the frequency band from 24.5 to 26.5 GHz and less than 0.57 dB among different states of the LC material.
To further evaluate the performance of the LCPS, the widely used FoM, which is defined as the ratio of the maximum phase shift to the maximum insertion loss [43], was employed:
The simulated and measured FoMs of the demonstrated LCPS were found to be 91.7 and 78.3(° )·dB−1, respectively. A comparison between this LCPS and other previously reported LCPSs operating at frequencies higher than 10 GHz is provided in Table 1 [13], [18], [19], [20], [31], [33], [35]. It can be seen that the LCPS based on an LDL structure has the highest FoM, while possessing a relatively small footprint and low fabrication complexity. The experimentally achieved full-360° phase coverage of the LCPS, along with its low and stable insertion loss, makes it highly useful for realizing LCPAs with good beam-scanning capabilities.
2.2. Design of the CP element
The configuration of the radiating element of the LCPA is depicted in Fig. 4 (a). It contains four substrates and four metallic layers. Substrate 1 and substrate 2 are made of Taconic TLY-5 with a thickness of 0.51 and 0.25 mm, respectively, between which is a 0.1 mm-thick bonding film made of Rogers RO4450F. Substrate 3 and substrate 4 are glass, both with a thickness of 0.3 mm. Glue is used on the outer edges of the two glass panels, forming a container with a thickness of 5 μm filled by a homogeneous layer of LCs. Between the top PCBs and the glass panels is a 0.025 mm-thick bonding film. Stacked rectangular patches with a pair of diagonal truncations are employed as the CP radiators, which are printed on the top layers of substrate 1 and substrate 2. The solid metallic layer on the bottom layer of substrate 2 serves as the ground plane of the array elements, where an I-shaped slot is etched out for electromagnetic excitation. On the bottom layer of substrate 3, an embedded microstrip line is used to feed the patches. It should be noted that the LCPS is also placed on this layer, in between the two glass panels. More detailed dimension information is presented in the top and bottom views, as shown in Figs. 4 (b) and (c).
The simulated results of the reflection coefficient, axial ratio (AR), and gain of the LC-based CP element are illustrated in Fig. 4 (d). The element achieves a good impedance match, with |S11| < −14 dB from 24 to 27 GHz. At frequencies ranging from 24.95 to 26.06 GHz, the element radiates a right-handed circularly polarized (RHCP) wave with an AR < 3 dB. The maximum gain value reaches 4.86 dBic with a gain variation of less than 0.5 dB.
2.3. Integrated configuration and fabrication of the LCPA
In order to combine the channels of the CP elements and LCPSs into an integrated array, an N-way power divider was designed. The integrated configuration of the LCPA, with an overall footprint of 195 mm × 91.7 mm and a thickness of 1.76 mm, is shown in Fig. 5 (a); it includes a 1-16 stripline power divider, 16 LCPSs, 16 CP radiating elements, and two dummy elements. To facilitate the connection of the control board, the substrate is slightly extended on both the left and right sides. Moreover, a stepped board process is introduced at the input port for converting the stripline into a microstrip line that can be more easily connected to a southwest end-launch connector. The characteristic impedance of the input microstrip line is set to be 50 Ω, with a width of 0.65 mm.
The detailed geometry of different portions of the stripline power divider with a corporate feeding structure is displayed in Fig. 5 (b). There are four stages of cascaded “T”-shaped 1-2 power dividers for realizing an equal power division between the 16 output ports. At the input port, three sections of quarter-wave stripline are employed to improve the impedance match. As shown in Fig. 5 (c), the 1-16 power divider, together with the phase shifters, possesses a return loss of greater than 12.5 dB within the frequency band from 24 to 27 GHz. The overall insertion loss for the case in which the beam scans to broadside—that is, the beam scanning angle θscan equals to 0°—is 4.55 dB at 25.5 GHz, which can still be maintained within 4.38 to 4.64 dB when the beam scans from −40° to +40°.
Fig. 5 (d) illustrates the laminated structure of the proposed LCPA, where the upper PCBs and the lower glass panels are connected by means of a bonding film without any direct metallic connection. Compared with the conventional LC filling method, in which a cavity must be milled in the substrate for LC filling [31], the employed structure allows for the separated fabrication of the CP elements and glass-based LCPSs, thereby improving the manufacturing efficiency and uniformity of the LC filling. The integrated 16-element LCPA was fabricated and fixed on an acrylic frame using nylon screws ( Fig. 5 (e)). The enlarged view at the top right of the figure illustrates the CP radiation elements and that at the bottom right illustrates the LCPS. A flexible printed circuit is used to connect the ITO bias lines for all the channels to the control board.
2.4. Design and implementation of the control board
After the design and fabrication of the LCPA, another critical part of the sub-system is the control circuit, which is responsible for applying the correct bias voltage to each channel. The control board includes a microcontroller unit (MCU) and 16 operational amplifiers (OPAs), which provide bias voltages in the form of square pulses for the 16 channels individually ( Fig. 5 (f)). The power management of the control board includes a source module and a low dropout linear chip. For each channel of the LCPA, the top bias lines are connected in parallel to the control board through the flexible printed circuit. The bias lines on the bottom, however, are combined together for all the channels and connected to the ground.
In terms of functionality, the control board is designed to generate multiple square waves with adjustable voltages of alternating current (AC) source. First, the original digital signals output from the MCU is transformed into analog signals by the built-in digital-to-analog converters (DACs). The appropriate voltage of the analog signals is then obtained using an OPA. Finally, a 300 Hz square wave with a bias voltage in the range of 0-23.5 V is generated by the control circuit and applied to the ITO electrodes of the LCPS via the flexible printed circuit. By establishing serial communication between the MCU and the PC, the state of each LCPA channel can be individually controlled by applying the targeted bias voltage. With the help of the control board, the automatic in-array calibration and phase assignment of the LCPA can be further implemented, as will be described in the next section.
3. In-array calibration of the LCPA
In order to achieve the desired beam-scanning functionality, calibration is indispensable for phased arrays with a large number of channels. Unlike conventional phased arrays based on digital phase shifters, for LCPAs, the relationship between the phase shifts and bias voltage is unknown and differs from channel to channel. Hence, the phase-voltage relationship of each individual channel, including the initial phase and line shape of the phase shift-voltage curve, must be extracted in the array environment. To this end, an SEM is proposed and implemented to achieve automatic LCPA calibration by utilizing the ergodic states of LC molecule deflection without relying on any prior knowledge.
3.1. Theory of the SEM
For the OTA in-array calibration of a conventional phased array, only the channel under test is turned on (i.e., is radiating), while all the other channels are turned off (i.e., are not radiating) [44]. However, for the LCPAs, all the channels are turned on simultaneously. Considering an LCPA of N channels calibrated by a scanning near-field probe placed in front of the array at a distance of about 0.5λ0 to 1.5λ0 ( Fig. 6 (a)), a multipath transmission model can be used to describe the waves received by the probe. As such, the received signal is a superposition of the waves radiated from all the channels, as follows:
where denotes the total signal received by the probe when channel k is being calibrated and applied with a bias voltage of Vi, while the contribution of the fields radiated by each single channel j with a bias voltage of Vi is . Since different bias voltages are applied to each channel when it is under calibration, the voltage set can be denoted as V = {V1, V2, …, Vi, …, Vn}, where the total number of ergodic states is M and Vi ∈ [Vmin, Vmax]. Graphically, and can be described as vectors due to their complex nature. Hence, the purpose becomes to obtain the quantity for i = 1, 2, …, M, and k = 1, 2, …, N, which can be used to retrieve the distinct phase–voltage curves of all the channels.
When extracting of channel k, the bias voltages for all the other channels are set to be unchanged, such as at 0 V, which ensures that the influence of the radiated fields from the other channels are kept the same. Under such circumstances, the summation of the fields radiated from all the other channels except for channel k, namely , can be regarded as nearly a constant, which is defined as the environment signal . Thus, any complex signal directly measured by the probe can be represented by a vectorial superposition of and in the complex plane, as shown in Fig. 6(b). The required corresponding to channel k alone under a bias voltage Vi can be described as follows:
With a nearly unchanged environment signal , according to our experimental observations, the endpoint trajectory of the vector resembles a quasi-ellipse in the complex plane as Vi traverses from Vmin to Vmax. As shown in Fig. 6 (c), the shape of the ellipse is determined by the response of channel k, including both the amplitudes and phases of the radiated field with different bias voltages, while the center of the ellipse is determined by the environment vector . Because the phase shift of each channel is not linearly proportional to the bias voltage, as revealed by the measured results in Section 2.1, the endpoints of would not be uniformly distributed on the ellipse. It should also be noted that, in practice, the endpoints of may not be able to trace out a complete ellipse under circumstances when the phase-shifting range of the channel is smaller than 360°.
Since only the complex value of can be directly recorded in an experiment, it becomes important to extract the near-constant environment vector that is uncorrelated to the bias voltage Vi applied to the channel under calibration. Mathematically, the problem to be solved is to find the coordinates of the center of the quasi-elliptical trace on the complex plane. To this end, as illustrated in Fig. 6 (d), an ellipse fitting algorithm is employed with the fitted ellipse function written as follows:
where, Re(·) and Im(·) represent the real part and imaginary part of the complex signal, respectively, a and b are the major and minor axes of the ellipse, respectively. Hence, the coordinates of the fitted ellipse center—that is, the environment vector —can be obtained by using an optimization algorithm for the least squares fitting of the ellipse [45], which can make full use of the responses of each channel under calibration in all different LC states. Importantly, such a method could be effective for LCPA channels with almost any kind of phase-shift coverage and phase–voltage relationships.
3.2. The OTA in-array calibration setup and procedure
In order to implement the proposed SEM for the phase-voltage curve extraction of the in-array channels of the LCPA, an automatic calibration system is implemented ( Fig. 7 (a)). The system includes an LCPA along with the control board, a dual-linearly polarized open-waveguide probe, a vector network analyzer (VNA), a robotic arm with a precise displacement machine, and a PC for controlling the above devices. As shown in Fig. 7 (b), port 1 of the VNA is connected to the input port of the LCPA, while port 2 and port 3 are connected to the two input ports of the open-waveguide probe for receiving the two orthogonal linearly polarized wave components emitted from the LCPA channels. The values of S21 and S31 represent the vertically polarized (VP) and horizontally polarized (HP) complex transmission coefficients between the LCPA and probe, respectively. The probe is fixed on the robotic arm, which can be moved in the horizontal direction in front of the LCPA elements. The robotic arm, control board, and VNA are all connected to the PC through serial communication or socket communication, for realizing the automatic procedures of the moving probe, applying bias voltages, and recording data, respectively.
A flowchart of the automatic OTA in-array calibration is shown in Fig. 7 (c). First, the dual-port open-waveguide probe is placed in front of the radiating element of the channel k with k = 1. Second, different bias voltages Vi are applied to the channel k of the LCPA through the control board, and the two corresponding linearly polarized transmission coefficients, and , are recorded, from which the RHCP transmission coefficient, that is, , can be obtained. This step iterates until the data for all different bias voltages of Vi ranging from 0 to 23.5 V, with a step of 0.1 V, are recorded. Third, based on the calculated , the RHCP environment vector of channel k, that is, , can be extracted by the ellipse fitting algorithm. Furthermore, the phase–voltage curves can be calculated utilizing Eq. (3). Then, the probe moves to the next channel to calibrate channel (k + 1). The abovementioned iterative procedure continues until the calibration for all the channels is completed; namely, the phase–voltage dataset with a size of N × M is recorded at every frequency point within the required frequency band.
3.3. Phase-voltage curve extraction of the LCPA channels
Based on the proposed SEM calibration strategy and the automatic calibration procedure, the phase-voltage curves of the 16 channels of the LCPA at 24.5, 25.0, 25.5, 26.0, and 26.5 GHz were obtained, as shown in Fig. 8 (a). To facilitate comparison, all the curves are normalized to the initial phase, with Vbias = 0 V at each frequency. It can be seen that the transmission phases of the LCPA channels as a function of the bias voltage all share a line shape similar to that of the LCPS illustrated in Section 2 —that is, an inverted S-shaped curve—which corroborates the measurement results reported previously in the literature [46], [47]. From 24.5 to 26.5 GHz, the averaged maximum phase shift ΔΦmax among all 16 channels is 258.0°, 286.6°, 288.7°, 306.1°, and 315.2°, respectively, which covers the majority of a full cycle. The difference in ΔΦmax among the 16 channels at the five frequency points is about 50.4°, 41.9°, 43.4°, 47.9°, and 27.8°, respectively. Such a difference in phase coverage among the channels mainly comes from the nonuniform thickness of the LC layer, whose thickness can be controlled within a fabrication error of less than 10% ( Section S1 in Appendix A ). More consistent responses from the LCPA channels can be expected with further improvement of the fabrication accuracy. The normalized transmission magnitudes as a function of the bias voltage for the 16 channels at different frequencies are presented in Fig. 8 (b), exhibiting a variation of less than 1.5 dB under different bias voltages from 24.5 to 25.5 GHz. At higher frequencies, the fluctuation slightly increases to around 2 dB due to the existence of a resonance.
In order to further evaluate the static characteristics of each channel in an unbiased state with Vbias = 0 V, the extracted initial phases and transmission magnitudes of the 16 channels at different frequencies are reported in Figs. 8 (c) and (d), respectively. It can be seen that the initial phases of the 16 channels quasi-monotonically increase as a function of the channel number, with a variation of about 60°. The transmission magnitudes among different channels also show a small fluctuation of less than 2 dB at the central frequency of 25.5 GHz and of about 3 dB at the lower or higher frequency ends.
In summary, response differences can be observed among all the LCPA channels in terms of both transmission magnitude and phase. This phenomenon is mainly attributed to the non-uniform thickness of the LC and the deposited copper layers, which is difficult to completely eliminate due to the extreme thinness of the LC layer. Hence, the proposed automatic OTA calibration method, which can effectively extract the characteristics of each individual channel in the LCPA, becomes particularly important. The transmission phase difference can be effectively extracted by the SEM and separately compensated for, thereby ensuring the accuracy of far-field beam scanning. The fluctuation in the transmission magnitudes is less than 3 dB; that is, the amplitude variation in the radiated fields among all the elements is no greater than 1.5 dB and thus will only have a minor influence on the steerable beams of the LCPA. It should be mentioned that the response of the LCPA channels remains stable even after thousands of switching cycles of repeatedly changing the bias voltage ( Section S2 in Appendix A ).
4. Experimental results
4.1. Calculation of bias voltage distribution
Based on the extracted phase–voltage curves of each channel of the LCPA, the bias voltage distributions of the proposed linear LCPA for achieving steerable directive beams can be obtained. For a beam pointing at the direction θscan, the excitation phase φj required for element Ej can be described as follows:
where k0 is the free-space wave number, d is the spacing between the array elements, φ0 is a redundant reference phase that refers to the initial phase of the first element E1 in the LCPA. According to the tunability principle of the LCPA, φj can be achieved by applying the bias voltage Vj according to the extracted relationship between the phase and voltage for channel j, as described by φj = fj(Vj). The schematic diagram for the phase assignment is shown in Fig. 9 (a), where the phase–voltage curves are represented by black lines and the LC states of each channel are marked as yellow dots. The required bias voltage Vj can be obtained by mapping the corresponding excited phase φj to the phase–voltage curve; that is, . It should be noted that the desired phase distribution for pointing the beam at θscan is not unique. In other words, the reference phase φ0 is an optimizable variable. When testing the far-field pattern of the LCPA with beams steered at different θscan, the value of φ0 is tuned to obtain beams with better performance.
Since the operational bandwidth of the LCPA is relatively narrow, only the phase-voltage curves of all the channels at 25.5 GHz were utilized when calculating the desirable bias voltages for different scanned beams. For verification, a total of 13 pencil beams scanning from −60° to +60° were generated with a step of 10°; their bias voltages were calculated and are listed in Table 2.
4.2. Simulated and measured results of the LCPA
In order to verify the proposed LCPA, a prototype was measured in an anechoic far-field chamber with its RHCP beam steering from −60° to +60° and with a step of 10°, by applying appropriate bias voltage sets to the LCPA ( Fig. 9 (b)). The simulated and measured |S11| of the LCPA with different beam directions are displayed in Fig. 9 (c). It can be observed that the reflection coefficient remains below −10 dB for all cases from 23.6 to 28.0 GHz for the simulated results and from 23.5 to 28.0 GHz for the measured results. The variation in the measured | S11 | due to different bias voltage sets being applied to the LCPA is less than 2 dB, implying that the LCPA can maintain a stable and well-matched input impedance.
The simulated and measured far-field RHCP gain patterns and AR patterns of the LCPA at 25.50, 25.75, and 26.00 GHz are reported in Fig. 10 for cases with different beam-scanning angles. It should be noted that the simulated patterns were obtained based on array synthesis theory [48] and the cascading method of multi-port networks [49], where the vectorial radiated fields and S -parameters of the array elements, LCPSs, and feeding network must be separately simulated and extracted. It can be observed that the simulated and measured patterns agree well with each other at all three frequencies in terms of the beam pointing angle and beam shape. When the scanning angle is gradually increased from broadside to ±60°, the half-power beamwidth of the main beam is broadened from 6° to 8° with a scanning loss of less than 3 dB. At 25.50 GHz, the maximum sidelobe levels (SLLs) of all the patterns are kept below −10 dB. No significant beam shape distortion or abnormally high sidelobes occur, even for beams with a large scanning angle. The measured AR pattern also matches well with the simulated results, indicating that scanning CP beams with AR < 3 dB can be obtained from −40° to +40° within the band from 25.50 to 26.00 GHz. Both the measured results of the AR and the gain pattern confirm the validity of the proposed OTA in-array calibration method and the correctness of the bias voltage sets for beamforming. The far-field patterns were measured repeatedly under different temperatures of 10 and 30 °C, respectively. The corresponding patterns remained almost unchanged, indicating that the LCPA has a stable radiation performance within normal temperature fluctuation.
The gain and AR curves of the beams steering at directions ranging from −40° to +40° as a function of frequency are displayed in Fig. 11 (a) ( Section S3 in Appendix A ). The maximum gain values of the nine beams are about 11.2, 11.8, 12.2, 12.5, and 12.4 dBic at 25.00, 25.25, 25.50, 25.75, and 26.00 GHz, respectively. The 3 dB gain bandwidth covers from 24.25 to 27.00 GHz, corresponding to a relative bandwidth of 8.7%, with a gain fluctuation of less than 2.5 dB for all the beams at all five frequencies. The averaged difference between the simulated and measured gain is about 0.7 dB, which can be attributed to the non-ideal gradient phase excitation due to the imperfect calibration of the channels and the non-uniform amplitude excitation caused by the inherent inconsistencies among different channels. The AR of the generated CP beams is below 3 dB from 25.5 to 26.0 GHz for all nine beams, indicating that the LCPA possesses stable CP radiation as the beam scans from −40° to +40°. When the scanning angle is further increased to ±50° and ±60°, the AR bandwidth of the CP beam becomes narrower and shifts toward lower frequencies. For all the CP beams scanning from −60° to +60°, the AR value remains smaller than 6 dB in the frequency band from 25.2 to 25.6 GHz. The simulated and measured beam directions of the thirteen beams are presented in Fig. 11 (b). It can be seen that the beam direction error is less than 3° at 25.5 GHz within a scanning range between −60° and +60°. Within the frequency band from 25.0 to 26.0 GHz, the simulated and measured maximum beam squinting is within ±3° and ±4°, respectively.
A detailed comparison between the proposed LCPA and its previously reported counterparts is provided in Table 3 [13], [31], [32], [33], [34], [35], [36]. First, compared with existing designs, the proposed LCPA realizes CP radiation characteristics, while the others are linearly polarized. Second, the demonstrated LCPA possesses a wide scanning angle from −40° to +40°. Third, the array contains more elements than the other arrays but still realizes accurate beam scanning, thus offering a higher gain and a finer angular resolution, which is partially made possible by the efficient automatic calibration method.
5. Conclusions
In summary, this work reported the design and experimental validation of an integrated 1×16 K-band CP LCPA, together with its OTA calibration. Each channel of the LCPA includes a diagonally chamfered stacked patch antenna and an LCPS based on an LDL structure with 5 μm-thick ultrathin LCs, which can provide a maximum phase-shifting range of more than 360° and an FoM of 78.3(° )·dB −1. Moreover, a near-field OTA automatic calibration based on the SEM was proposed and implemented to extract the phase-voltage curves of the 16 channels of the LCPA. A prototype of the fabricated LCPA was measured and was shown to provide a scanned CP beam covering from −40° to +40° with a peak gain of 12.5 dBic and a scanning loss of less than 2.5 dB. The bandwidth of the LCPA, which satisfies the requirements of a joint | S11 | < −15 dB, AR < 3 dB, beam squinting < 3°, and gain variation < 2.2 dB, spans from 25.5 to 26.0 GHz. Compared with existing LCPAs, the proposed LCPA has the advantages of a high gain, high efficiency, low profile, and faster response time, making it potentially promising for millimeter-wave communications such as satellite communications and radar systems. The methodology presented here offers a systematic solution for LCPAs, which can be beneficial for the design and characterization of various types of beam-scanning arrays based on LCs and even on other phase-change materials.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (NSFC; 62122019 and 62293492 ), the National Key Research and Development Program of China ( 2019YFB2204704 ), and the Fundamental Research Funds for the Central Universities and the Zhishan Scholar Program of Southeast University (2242022R40038). The authors thank Tianyi Huo for his assistance during the antenna measurements.
Compliance with ethics guidelines
Xin Yu, Fengshuo Wan, Hongyuan Feng, Shichao Jin, Chong Guo, Yu Wei, Dunge Liu, Yuqian Yang, Longzhu Cai, Zhi Hao Jiang, and Wei Hong declare that they have no conflict of interest and financial conflicts to disclose.
W.Hong, Z.H.Jiang, C.Yu, D.Hou, H.Wang, C.Guo, et al.. The role of millimeter-wave technologies in 5G/6G wireless communications. IEEE J Microwaves, 1 (1) ( 2021), pp. 101-122
[2]
J.Lin, W.Yu, N.Zhang, X.Yang, H.Zhang, W.Zhao. A survey on internet of things: architecture, enabling technologies, security and privacy, and applications. IEEE Internet Things J, 4 (5) ( 2017), pp. 1125-1142
[3]
R.J.Mailloux.Phased array antenna handbook. ( 3rd ed.), Artech House Publishers, Boston ( 2018)
[4]
Y.Yin, B.Ustundag, K.Kibaroglu, M.Sayginer, G.M.Rebeiz.Wideband 23. 5-29.5-GHz phased arrays for multistandard 5G applications and carrier aggregation. IEEE Trans Microwave Theory Tech, 69 (1) ( 2021), pp. 235-247
[5]
J.W.Kim, S.C.Chae, H.W.Jo, T.D.Yeo, J.W.Yu. Wideband circularly polarized phased array antenna system for wide axial ratio scanning. IEEE Trans Antennas Propag, 70 (2) ( 2022), pp. 1523-1528
[6]
S.Xiao, C.Zheng, M.Li, J.Xiong, B.Z.Wang. Varactor-loaded pattern reconfigurable array for wide-angle scanning with low gain fluctuation. IEEE Trans Antennas Propag, 63 (5) ( 2015), pp. 2364-2369
[7]
Y.Ji, L.Ge, J.Wang, Q.Chen, W.Wu, Y.Li. Reconfigurable phased-array antenna using continuously tunable substrate integrated waveguide phase shifter. IEEE Trans Antennas Propag, 67 (11) ( 2019), pp. 6894-6908
[8]
Y.F.Cao, X.Y.Zhang. A wideband beam-steerable slot antenna using artificial magnetic conductors with simple structure. IEEE Trans Antennas Propag, 66 (4) ( 2018), pp. 1685-1694
[9]
K.Topalli, O.A.Civi, S.Demir, S.Koc, T.Akin. A monolithic phased array using 3-bit distributed RF MEMS phase shifters. IEEE Trans Microwave Theory Tech, 56 (2) ( 2008), pp. 270-327
[10]
C.Jung, M.Lee, G.P.Li, F. deFlaviis. Reconfigurable scan-beam single-arm spiral antenna integrated with RF-MEMS switches. IEEE Trans Antennas Propag, 54 (2) ( 2006), pp. 455-463
[11]
A.Nafe, F.A.Ghaffar, M.F.Farooqui, A. Shamim. A ferrite LTCC-based monolithic SIW phased antenna array. IEEE Trans Antennas Propag, 65 (1) ( 2017), pp. 196-205
[12]
K.K.Karnati, Y.Shen, M.E.Trampler, S.Ebadi, P.F.Wahid, X.Gong. A BST-integrated capacitively loaded patch for Ka- and X-band beamsteerable reflectarray antennas in satellite communications. IEEE Trans Antennas Propag, 63 (4) ( 2015), pp. 1324-1333
[13]
O.H.Karabey, A.Gaebler, S.Strunck, R.Jakoby. A 2-D electronically steered phased-array antenna with 2 × 2 elements in LC display technology. IEEE Trans Microwave Theory Tech, 60 (5) ( 2012), pp. 1297-1306
[14]
H.Maune, M.Jost, R.Reese, E.Polat, M.Nickel, R. Jakoby. Microwave liquid crystal technology. Crystals, 8 (9) ( 2018), p. 355
[15]
R.Jakoby, A.Gaebler, C.Weickhmann. Microwave liquid crystal enabling technology for electronically steerable antennas in SATCOM and 5G millimeter-wave systems. Crystals, 10 (6) ( 2020), p. 514
[16]
StevensonR, SazegarM, BilyA, JohnsonM, KundtzN. Metamaterial surface antenna technology:commercialization through diffractive metamaterials and liquid crystal display manufacturing. In:Proceedings of 2016 10th International Congress on Advanced Electromagnetic Materials in Microwaves and Optics (METAMATERIALS); 2016 Sep 19-22; Chania, Greece. Piscataway: IEEE; 2016. p. 349-51.
[17]
D.K.Yang, S.T.Wu.Fundamentals of liquid crystal devices. ( 2nd ed.), John Wiley & Sons, Inc., Hoboken ( 2014)
[18]
F.Goelden, A.Gaebler, M.Goebel, A.Manabe, S.Mueller, R.Jakoby. Tunable liquid crystal phase shifter for microwave frequencies. Electron Lett, 45 (13) ( 2009), pp. 686-687
[19]
S.Bulja, D. Mirshekar-Syahkal. Meander line millimetre-wave liquid crystal based phase shifter. Electron Lett, 46 (11) ( 2010), pp. 769-771
[20]
D.Wang, E.Polat, C.Schuster, H.Tesmer, G.P.Rehder, A.L.C.Serrano, et al.. Fast and miniaturized phase shifter with excellent figure of merit based on liquid crystal and nanowire-filled membrane technologies. IEEE J Microwaves, 2 (1) ( 2022), pp. 174-184
[21]
F.Kamrath, E.Polat, S.Matic, C.Schuster, D.Miek, H.Tesmer, et al.. Bandwidth and center frequency reconfigurable waveguide filter based on liquid crystal technology. IEEE J Microwave, 2 (1) ( 2022), pp. 134-144
[22]
E.Polat, F.Kamrath, S.Matic, H.Tesmer, A.Jiménez-Sáez, D.Wang, et al.. Novel hybrid electric/magnetic bias concept for tunable liquid crystal based filter. IEEE J Microwaves, 2 (3) ( 2022), pp. 490-495
[23]
O.H.Karabey, S.Bildik, S.Bausch, S.Strunck, A.Gaebler, R.Jakoby. Continuously polarization agile antenna by using liquid crystal-based tunable variable delay lines. IEEE Trans Antennas Propag, 61 (1) ( 2013), pp. 70-76
[24]
J.Shu, Y.Zhang. Design of a liquid crystal beam-steerable antenna with characteristic mode analysis. IET Microwaves Antennas Propag, 16 (1) ( 2022), pp. 29-36
[25]
H.Tesmer, R.Reese, E.Polat, M.Nickel, R.Jakoby, H. Maune. Liquid-crystal-based fully dielectric lateral wave beam-steering antenna. IEEE Antennas Wireless Propag Lett, 18 (12) ( 2019), pp. 2577-2581
[26]
R.Reese, M.Jost, E.Polat, H.Tesmer, J.Strobl, C.Schuster, et al.. A millimeter-wave beam-steering lens antenna with reconfigurable aperture using liquid crystal. IEEE Trans Antennas Propa, 67 (8) ( 2019), pp. 5313-5324
[27]
StevensonRA, FotheringhamD, FreemanT, NoelT, MasonT, ShafieS.High-throughput satellite connectivity for the constant contact vehicle. In:Proceedings of 2018 48th European Microwave Conference (EuMC); 2018 Sep 23-27; Madrid, Spain. Piscataway: IEEE; 2018. p. 316-9.
[28]
W.Zhang, Y.Li, Z.Zhang. A reconfigurable reflectarray antenna with an 8 μm-thick layer of liquid crystal. IEEE Trans Antennas Propag, 70 (4) ( 2022), pp. 2770-2778
[29]
X.Li, Y.Wan, J.Liu, D.Jiang, T.Bai, K.Zhu, et al.. Broadband electronically scanned reflectarray antenna with liquid crystals. IEEE Antennas Wireless Propag Lett, 20 (3) ( 2021), pp. 396-400
[30]
G.Perez-Palomino, P.Baine, R.Dickie, M.Bain, J.A.Encinar, R.Cahill, et al.. Design and experimental validation of liquid crystal-based reconfigurable reflectarray elements with improved bandwidth in F-band. IEEE Trans Antennas Propag, 61 (4) ( 2013), pp. 1704-1713
[31]
X.Y.Li, D.Jiang, J.Liu, M.S.Tong. A Ka-band multilayer beaming-scanning antenna using liquid crystals. IEEE Antennas Wireless Propag Lett, 21 (1) ( 2022), pp. 44-48
[32]
J.Shu, G.Xu, H.Peng, J.Mao. An electrically steerable parasitic array radiator in package based on liquid crystal. IEEE Antennas Wireless Propag Lett, 18 (11) ( 2019), pp. 2365-2369
[33]
S.Ma, S.Q.Zhang, L.Q.Ma, F.Y.Meng, D.Erni, L.Zhu, et al.. Compact planar array antenna with electrically beam steering from backfire to endfire based on liquid crystal. IET Microwaves Antennas Propag, 12 (7) ( 2018), pp. 1140-1146
[34]
O.H.Karabey, A.Mehmood, M.Ayluctarhan, H.Braun, M.Letz, R. Jakoby. Liquid crystal based phased array antenna with improved beam scanning capability. Electron Lett, 50 (6) ( 2014), pp. 426-428
[35]
D.Wang, E.Polat, H.Tesmer, R.Jakoby, H.Maune. A compact and fast 1 × 4 continuously steerable endfire phased-array antenna based on liquid crystal. IEEE Antennas Wireless Propag Lett, 20 (10) ( 2021), pp. 1859-1862
[36]
A.Panahi, L.Yeung, M.Hedayati, Y.E.Wang. Sub-6 GHz high FOM liquid crystal phase shifter for phased array antenna. IEEE J Microwaves, 2 (2) ( 2022), pp. 316-325
[37]
D.Wang, E.Polat, H.Tesmer, H.Maune, R.Jakoby. Switched and steered beam end-fire antenna array fed by wideband via-less butler matrix and tunable phase shifters based on liquid crystal technology. IEEE Trans Antennas Propag, 70 (7) ( 2022), pp. 5383-5392
[38]
S.Mano, T.Katagi. A method for measuring amplitude and phase of each radiating element of a phased array antenna. Electron Commun Jpn, 65 (5) ( 1982), pp. 58-64
[39]
T.Takahashi, Y.Konishi, S.Makino, H.Ohmine, H.Nakaguro. Fast measurement technique for phased array calibration. IEEE Trans Antennas Propag, 56 (7) ( 2008), pp. 1888-1899
[40]
K.M.Lee, R.S.Chu, S.C.Liu. A built-in performance-monitoring/fault isolation and correction (PM/FIC) system for active phased-array antennas. IEEE Trans Antennas Propag, 41 (11) ( 1993), pp. 1530-1540
[41]
W.P.M. N.Keizer. Fast and accurate array calibration using a synthetic array approach. IEEE Trans Antennas Propag, 59 (11) ( 2011), pp. 4115-4122
[42]
A.Kianinejad, Z.N.Chen, C.W.Qiu. Design and modeling of spoof surface plasmon modes-based microwave slow-wave transmission line. IEEE Trans Microwave Theory Tech, 63 (6) ( 2015), pp. 1817-1825
[43]
S.Gevorgian. Ferroelectrics in microwave devices, circuits and systems: physics, modeling, fabrication and measurements. Springer, Berlin ( 2009)
[44]
H.Gao, W.Wang, Y.Wu, Y.Liu, G.F.Pedersen, W.Fan.Experimental comparison of on-off and all-on calibration modes for beam-steering performance of mmWave phased array antenna-in-package. IEEE Trans Instrum Meas, 70 ( 2021), p. 8002509
[45]
A.Fitzgibbon, M.Pilu, R.B.Fisher. Direct least square fitting of ellipses. IEEE Trans Pattern Anal Mach Intell, 21 (5) ( 1999), pp. 476-480
[46]
T.W.Kim, J.S.Park, S.O.Park. A theoretical model for resonant frequency and radiation pattern on rectangular microstrip patch antenna on liquid crystal substrate. IEEE Trans Antennas Propag, 66 (9) ( 2018), pp. 4533-4540
[47]
S.Bulja, D.Mirshekar-Syahkal, R.James, E.Day, F.A.Fernández. Measurement of dielectric properties of nematic liquid crystals at millimeter wavelength. IEEE Trans Microwave Theory Tech, 58 (12) ( 2010), pp. 3493-3501
R.J.Cameron, M.K.Chandra, R.M.Raafat. Microwave filters for communication systems: fundamentals, design, and applications. John Wiley & Sons, Inc., Hoboken ( 2018)
PDF
(6588KB)
