Frequency-reconfigurable metasurface circular polarization array antenna based on liquid crystal

Abstract

The utility model discloses a frequency reconfigurable metasurface circular polarization array antenna based on liquid crystal, which comprises an upper Rogers substrate, a metasurface array, a liquid crystal interlayer, a middle Rogers substrate, a metal ground, a lower Rogers substrate and a 180-degree hybrid circuit type feed network which are sequentially connected from top to bottom, and is characterized in that the metasurface array is printed on the lower surface of the upper Rogers substrate; the metasurface array is formed by sequentially rotating four 4 * 4 metasurface units by 90 degrees in the circumferential direction, the liquid crystal interlayer adopts a packaging mode that liquid crystal is injected between two layers of glass, the metal ground is printed on the lower surface of the middle Rogers substrate, four H-shaped air gaps are sequentially and rotationally etched in the metal ground, and the 180-degree hybrid circuit type feed network is printed on the lower surface of the lower Rogers substrate. The metasurface circularly polarized antenna array substantially disclosed by the utility model has the advantages of wide bandwidth, high gain, small size, low profile and reconfigurable frequency.

Description

Technical Field

[0001] The utility model relates to the technical field of antennas, and in particular to a frequency-reconfigurable metasurface circularly polarized array antenna based on liquid crystal. Background Art

[0002] Frequency reconfigurable antennas can operate in different frequency bands, thus adapting to a variety of communication standards and frequency requirements, improving the flexibility and applicability of antennas, and can achieve the optimal performance of antennas in specific frequency bands through frequency reconstruction, so that spectrum resources can be used more effectively and the spectrum efficiency of communication systems can be improved. Circularly polarized antennas have good signal coverage and penetration capabilities in wireless communications, and can effectively overcome problems such as multipath propagation and signal attenuation. The frequency reconfigurable design allows circularly polarized antennas to adjust their operating frequency according to actual environment and application requirements, further optimizing signal coverage and penetration capabilities. The flexibility and applicability of frequency reconfigurable circularly polarized antennas have made them widely used in a variety of application scenarios, including satellite communications, drone communications, mobile communications and other fields, promoting the development and application of communication technology.

[0003] Liquid crystal has been widely used in various display devices and microwave devices due to its flexibility and anisotropic properties as a liquid dielectric. In 2020, Che et al. proposed a low-sidelobe holographic antenna based on liquid crystal, which has the characteristics of large beam scanning range and multi-beam scanning. The antenna has a low sidelobe level of -12.2dB and a large beam scanning range of -57.7° to +57.4°. In addition to using liquid crystal to realize the beam scanning characteristics of the antenna, liquid crystal can also be used to tune the frequency of the antenna. In 2021, Kim et al. used a smaller number of liquid crystals by considering the electric field distribution on the surface of the antenna, and tuned the resonant frequency of the antenna between 3.38GHz and 3.43GHz by changing the dielectric constant of the liquid crystal. In addition, liquid crystal also plays an important role in the terahertz band. In 2021, Garu et al. used the dielectric constant of polymer dispersed liquid crystal to change. When the dielectric constant changed from 2.93 to 2.51, the resonant frequency of the antenna changed from 0.504 THz to 0.540 THz. It can be seen that liquid crystal plays an important role in the beam steering and frequency tuning of the antenna. Utility Model Content

[0004] This paper designs a frequency-reconfigurable metasurface circularly polarized antenna array based on liquid crystal. First, the periodically arranged metasurface antenna with circularly polarized radiation performance is designed by using characteristic mode analysis, and the designed metasurface antenna is arrayed. Then, a parallel sequential rotation feeding network with a hybrid circuit is designed. Finally, a layer of liquid crystal is added between the dielectric substrates, and the relative dielectric constant of the liquid crystal is adjusted by changing the external temperature to achieve the frequency shift of the antenna. Under the regulation of different temperatures, the effective impedance bandwidth and axial ratio bandwidth of the antenna move from 39.0% (4.51~6.70GHz) to 40.3% (4.61~6.94GHz), and the peak gain moves from 11.59dBic at 5.31GHz to 11.71dBic at 5.41GHz.

[0005] The present invention is achieved through the following technical solutions:

[0006] Firstly, the characteristic mode is used to analyze the characteristic value, characteristic angle, characteristic current and corresponding far-field radiation pattern of the antenna, and a metasurface circularly polarized antenna element is designed by using the method of magnetic field coupling excitation. In order to further expand the effective bandwidth and gain, a parallel-type directional power-dividing feeding network designed by a 180° hybrid circuit is used to form an array of the circularly polarized antenna element, and a mixed nematic phase liquid crystal interlayer is added to it. The liquid crystal dielectric constant of the liquid crystal is changed by external temperature, and finally the frequency reconfigurable function is realized. Under the regulation of different temperatures, the effective impedance bandwidth and axial ratio bandwidth of the antenna move from 39.0% (4.51~6.70GHz) to 40.3% (4.61~6.94GHz), and the peak gain moves from 11.59dBic at 5.31GHz to 11.71dBic at 5.41GHz.

[0007] A frequency-reconfigurable metasurface circularly polarized array antenna based on liquid crystal comprises an upper Rogers substrate, a metasurface array, a liquid crystal interlayer, a middle Rogers substrate, a metal ground, a lower Rogers substrate and a 180° hybrid circuit feeding network connected in sequence from top to bottom, wherein the metasurface array is printed on the lower surface of the upper Rogers substrate, the metasurface array is formed by four 4×4 metasurface units rotated 90° in sequence along the circumferential direction, the liquid crystal interlayer is composed of an upper and lower layer of glass and a liquid crystal package injected between the two layers of glass, the metal ground is printed on the lower surface of the middle Rogers substrate, four H-shaped air gaps are etched on the metal ground rotated 90° in sequence along the circumferential direction, and the 180° hybrid circuit feeding network is printed on the lower surface of the lower Rogers substrate.

[0008] Furthermore, the materials of the upper Rogers substrate, the middle Rogers substrate, and the lower Rogers substrate are all Rogers RO4003 plates, with a dielectric constant of 3.55, a loss tangent value of 0.0027, a length of 76 mm, a width of 76 mm, and thicknesses of 0.2 mm, 2.5 mm, and 0.508 mm, respectively.

[0009] Furthermore, the 4×4 super surface unit is arranged in a 4×4 periodic manner with a single copper sheet as an element. The single copper sheet is a rectangular copper sheet with two small rectangles cut off at the diagonals. The rectangular copper sheet is 7.2 mm wide and 9.3 mm long. The two cut diagonal small rectangles are 4 mm wide and 2.6 mm long. 16 single copper sheets are arranged in a 4×4 periodic manner with an interval of 1.15 mm from each other to form a 4×4 super surface unit.

[0010] Furthermore, the liquid crystal is a mixed nematic phase liquid crystal, which is composed of materials 4′-Pentyl-4-cyano-biphenyl, 4′-Alkyl-4-isothiocyanato-phenylcyclohexanes and 4"-Alkyl-2,5′fluoro-phenyl-4alkyl-biphenyl-tolanes. When the temperature changes from -20°C to 50°C, the dielectric constant of the liquid crystal can be changed from ε / / =3.1 and ε ⊥ =2.5, and the liquid crystal thickness is 0.7mm.

[0011] Furthermore, the H-shaped air gap is composed of two parallel gaps with a length of 5 mm and a width of 0.8 mm, which are connected vertically at the center by a long gap with a length of 20 mm and a width of 1.8 mm. The H-shaped air gap is located at the center directly below the 4×4 metasurface unit and is placed perpendicular to the metasurface unit.

[0012] Furthermore, the 180° hybrid circuit feeding network includes a hybrid circuit with a characteristic impedance Z 0 =50Ω, the total length is 6λg / 4. The signal is input from the A end of the hybrid circuit and passes through the characteristic impedance Z 0 , electrical length λg / 4 to end B, through characteristic impedance Z 0 , electrical length λg×3 / 4 to C end, and output from B and C ends with equal amplitude and reverse direction, and C end phase lags 180° behind B end. The electrical length of port 3 relative to port 2 and port 5 relative to port 4 are respectively λg / 4 longer, thus achieving the required 0°, 90°, 180°, and 270° phase differences from port 2 to port 5. The characteristic impedance of the microstrip line required for each port is 50Ω, and a small section of characteristic impedance Z is matched at the four ports. 3= 45Ω microstrip line. Port 2 and port 3 are connected in parallel at point E and become 30.9Ω (452 / 50 = 40.5Ω, 502 / 40.5 = 61.8Ω, 61.8 / 2 = 30.9Ω). The impedance is Z 2 =33Ω impedance transformation section to point B becomes 35.3Ω (332 / 30.9=35.3Ω). 35.3Ω is exactly the characteristic impedance required by the hybrid circuit. Similarly, the characteristic impedance of point A is also 35.3Ω, and a section of λg / 4 is connected in series, and the characteristic impedance is Z 1 =42Ω (422 / 35.3=50Ω) impedance transformation section; port 4 and port 5 are connected in parallel at the D end, and port 1 is connected to the A end.

[0013] Compared with the prior art, the present invention has the following technical effects:

[0014] Under the control of different temperatures, the effective impedance bandwidth and axial ratio bandwidth of the frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal move from 39.0% (4.51-6.70GHz) to 40.3% (4.61-6.94GHz), and the peak gain moves from 11.59dBic at 5.31GHz to 11.71dBic at 5.41GHz. The frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal has the advantages of wide bandwidth, high gain, small size and low profile, and frequency reconfigurability. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 A three-dimensional diagram of a frequency-reconfigurable metasurface circularly polarized array antenna based on liquid crystal according to an embodiment of the present invention;

[0016] Figure 2 A front view of a frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal according to an embodiment of the present invention;

[0017] Figure 3 A plan view of a supersurface array of a frequency-reconfigurable supersurface circularly polarized array antenna based on liquid crystal according to an embodiment of the present invention;

[0018] Figure 4 A metal ground plane diagram of a frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal according to an embodiment of the present invention;

[0019] Figure 5 This is a back view of a 180° hybrid circuit feeding network of a liquid crystal-based frequency reconfigurable metasurface circularly polarized array antenna according to an embodiment of the present invention.

[0020] Figure 6 The frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal according to an embodiment of the present invention is |S 11| Atlas;

[0021] Figure 7 An axial ratio spectrum of a frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal according to an embodiment of the present invention;

[0022] Figure 8 A gain spectrum of a frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal according to an embodiment of the present invention;

[0023] Fig. 9 Schematic diagram of the liquid crystal-based frequency reconfigurable metasurface circularly polarized array antenna according to an embodiment of the present invention.

[0024] In the figure: 1-upper Rogers substrate; 2-metasurface array; 3-glass; 4-liquid crystal interlayer; 5-middle Rogers substrate; 6-metal ground; 7-lower Rogers substrate; 8-180° hybrid circuit feeding network; 9-4×4 metasurface unit; 10-H-type air gap. DETAILED DESCRIPTION

[0025] The technical solution of the present invention is further described below in conjunction with the accompanying drawings and specific embodiments:

[0026] like Figure 1 As shown, the utility model provides a frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal, including an upper Rogers substrate 1, a metasurface array 2, a liquid crystal interlayer 3-4-3, a middle Rogers substrate 5, a metal ground 6, a lower Rogers substrate 7, and a 180° hybrid circuit feeding network 8 connected in sequence from top to bottom, and its stereoscopic diagram is shown as follows Figure 1 The materials of the upper Rogers substrate 1, the middle Rogers substrate 5, and the lower Rogers substrate 7 are all Rogers RO4003 plates, with a dielectric constant of 3.55, a loss tangent of 0.0027, and a length of d 6 =76mm, width is d 6 =76mm, thickness is h 3 =0.2mm,h 6 =2.5mm,h 7 =0.508mm. The metasurface array, 180° hybrid circuit feed network, and metal ground are all copper films with a thickness of 0.036mm. The liquid crystal interlayer 3-4-3 encapsulates the liquid crystal 4 through two layers of glass 3, and the thickness of the two layers of glass 3 is h 5=0.2mm, the liquid crystal 4 is a mixed nematic phase liquid crystal, composed of 4′-Pentyl-4-cyano-biphenyl, 4′-Alkyl-4-isothiocyanato-phenylcyclohexa nes and 4"-Alkyl-2,5′fluoro-phenyl-4alkyl-biphenyl-tolanes, and when the temperature changes from -20°C to 50°C, the dielectric constant of the liquid crystal 4 can be changed in ε / / =3.1 and ε ⊥ =2.5, the thickness of the liquid crystal is d 4 =0.7mm. Its front view is as follows Figure 2 shown.

[0027] The lower surface of the upper Rogers substrate 1 is printed with a metasurface array 2; the metasurface array 2 is formed by four 4×4 metasurface units 9 rotated 90° in sequence along the circumferential direction, and the 4×4 metasurface units 9 are arranged in a 4×4 periodic manner with a single copper pattern as an element, and the single copper pattern is a shape in which two small rectangles are cut off at the diagonals of a rectangular copper sheet, and the width of the rectangular copper sheet is d 4 =7.2mm, length 9.3mm, the two diagonal rectangles cut off are 4mm wide and d long 3 = 2.6mm, 16 individual copper patterns are spaced d apart 5 =1.15mm for 4×4 periodic arrangement to form a 4×4 metasurface unit 9. The metasurface array plane diagram is as follows Figure 3 shown.

[0028] The metal ground 6 is printed on the lower surface of the middle Rogers substrate. Four H-shaped gaps 10 are etched on the metal ground 6 in a 90° rotation in the circumferential direction. The H-shaped air gap 10 consists of two lengths of l 2 =5mm, width is l 3 = 0.8mm parallel gap, through a 20mm long and l wide 4 =1.8mm long gap vertically connected to the center, the H-shaped air gap 10 is located at the center directly below the 4×4 super surface unit 9 and is placed vertically to the super surface unit 9. Its metal ground plane diagram is shown as follows Figure 4 shown.

[0029] The back of the lower Rogers substrate 7 is printed with a 180° hybrid circuit feeding network 8, with a characteristic impedance Z 0 =50Ω, the total length is 6λg / 4. Port 1 to A end is connected by a wire with a width of n 1 =1.1mm and n 2 =1.45mm two-segment microstrip line, the signal is input from the A end of the hybrid circuit, and passes through the characteristic impedance Z 0, electrical length λg / 4 to end B, through characteristic impedance Z 0 , electrical length λg×3 / 4 to C end, and output from B and C ends in opposite directions with equal amplitude, and C end phase lags 180° behind B end. The electrical length of port 3 relative to port 2 and port 5 relative to port 4 are respectively λg / 4 longer, thus achieving the required 0°, 90°, 180°, and 270° phase differences from port 2 to port 5. Port 2 and port 4 are each connected by a section of width n 4 =1.55mm microstrip line to reduce the electrical length. The characteristic impedance of the microstrip line required for each port is 50Ω, and a small section with characteristic impedance Z is matched at the four ports. 3 =45Ω width is n 4 =1.55mm microstrip line. Port 2 and port 3 are connected in parallel at point E and become 30.9Ω (452 / 50=40.5Ω, 502 / 40.5=61.8Ω, 61.8 / 2=30.9Ω). The impedance is Z 2 =33Ω impedance transformation segment becomes 35.3Ω (332 / 30.9=35.3Ω) at point B, and the impedance between BE and CD is Z 2 =33Ω microstrip line, width n 3 =5.18mm. 35.3Ω is exactly the characteristic impedance required by the hybrid circuit. Similarly, the characteristic impedance of point A is also 35.3Ω, and a section of λg / 4 is connected in series, and the characteristic impedance is Z 1 =42Ω(422 / 35.3=50Ω) impedance transformation section. The back view of its 180° hybrid circuit feeding network is as follows: Figure 5 shown.

[0030] The relevant parameters of the liquid crystal-based frequency reconfigurable metasurface circularly polarized array antenna are shown in Table 1.

[0031] Table 1

[0032]

[0033] Within the range of -20 to 50°C, the dielectric constant of the liquid crystal used in the array antenna is adjusted within the range of 2.5 to 3.1. At the same time, the frequency band of the metasurface circularly polarized antenna array will slowly move from low frequency to high frequency. Figure 6 The array antenna is based on different dielectric constants |S 11| Curve, From this figure, it can be seen that in the range of 5 to 7.5 GHz, as the dielectric constant of the liquid crystal changes, the impedance bandwidth of the antenna moves to high frequency in turn. In addition to the obvious change at the center frequency, compared with the liquid crystal dielectric constant of 3.1, the liquid crystal with a dielectric constant of 2.5 can move the frequency of the antenna to high frequency by 0.1 GHz, but the change in the cutoff frequency at low and high frequencies is not obvious enough. When the liquid crystal dielectric constant is 3.1, the impedance bandwidth of the antenna is 60.6% (4.03 to 7.53 GHz), and when the liquid crystal dielectric constant is 2.5, the impedance bandwidth of the antenna is 60.2% (4.06 to 7.56 GHz). Compared with the impedance bandwidth, the frequency shift of the axial ratio bandwidth has a more obvious change.

[0034] Figure 7 It is an antenna axial ratio curve diagram of the array antenna based on different dielectric constants. When the liquid crystal dielectric constant changes from 3.1 to 2.5, the axial ratio bandwidth of the antenna can move from 39.0% (4.51-6.70GHz) to 40.3% (4.61-6.94GHz). At low frequencies, the axial ratio bandwidth shifts right by 0.1GHz, and at high frequencies, the bandwidth shifts right by 0.24GHz. Figure 8 It is an antenna gain curve diagram of the array antenna based on different dielectric constants. The overall gain of the antenna also has an obvious frequency shift phenomenon at 5 to 7 GHz.

[0035] Fig. 9 is the directional pattern of the array antenna based on different dielectric constants, and Figure (a) corresponds to ε LC = 2.5, the directional diagrams at 5.5 GHz and 6 GHz, Figure (b) corresponds to ε LC =3.1, the directional diagrams at 5.2GHz and 5.7GHz show that the main polarization of the circularly polarized antenna is RHCP and the cross polarization is LHCP. LC =2.5, at 5.5GHz, the maximum polarization isolation of left and right rotation on the main lobe on the xoz and yoz planes can reach 28.8dB and 26.3dB respectively, and the 3dB axial ratio beamwidth reaches 60° and 41.2° respectively. At 6.0GHz, the maximum polarization isolation of left and right rotation on the main lobe on the xoz and yoz planes can reach 23.7dB and 29.6dB respectively, and the 3dB axial ratio beamwidth reaches 26.6° and 39.4° respectively. When ε LC=3.1, at 5.2GHz, the maximum polarization isolation of left and right rotation on the main lobe on the xoz and yoz planes can reach 26dB and 22.2dB respectively, and the 3dB axial ratio beamwidth reaches 54.5° and 39.35° respectively. At 5.7GHz, the maximum polarization isolation of left and right rotation on the main lobe on the xoz and yoz planes can reach 18.6dB and 16.7dB respectively, and the 3dB axial ratio beamwidth reaches 57.2° and 25.6° respectively.

[0036] So in summary, as the temperature increases, the effective impedance bandwidth and axial ratio bandwidth of the antenna move from 39.0% (4.51-6.70GHz) to 40.3% (4.61-6.94GHz), and the peak gain moves from 11.59dBic at 5.31GHz to 11.71dBic at 5.41GHz. The simulated directional pattern shows that the radiation pattern of the circularly polarized antenna array has good symmetry and stable directivity. At the same time, it can be seen that when the dielectric constant of the liquid crystal gradually changes from 3.1 to 2.5, the directional pattern from low frequency to high frequency remains basically unchanged, which also confirms the frequency reconfigurable characteristics of the metasurface circularly polarized antenna array. The metasurface circularly polarized array antenna has the advantages of wide bandwidth, high gain, small size and low profile, and frequency reconfigurability.

[0037] The above shows and describes the basic principles, main features and advantages of the utility model. Those skilled in the art should understand that the utility model is not limited by the above specific embodiments. The above specific embodiments and the description in the specification are only for further illustrating the principles of the utility model. Without departing from the spirit and scope of the utility model, the utility model may have various changes and improvements, which fall within the scope of the utility model to be protected. The scope of protection claimed by the utility model is defined by the claims and their equivalents.

Claims

1. Frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal, It is characterized in that It includes an upper Rogers substrate, a metasurface array, a liquid crystal interlayer, a middle Rogers substrate, a metal ground, a lower Rogers substrate and a 180° hybrid circuit feeding network connected in sequence from top to bottom, the metasurface array is printed on the lower surface of the upper Rogers substrate, the metasurface array is formed by four 4×4 metasurface units rotated 90° in a circumferential direction, the liquid crystal interlayer is composed of an upper and lower layer of glass and a liquid crystal package injected between the two layers of glass, the metal ground is printed on the lower surface of the middle Rogers substrate, four H-shaped air gaps are etched on the metal ground rotated 90° in a circumferential direction, and the 180° hybrid circuit feeding network is printed on the lower surface of the lower Rogers substrate.

2. The frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal according to claim 1, It is characterized in that The upper Rogers substrate, the middle Rogers substrate, and the lower Rogers substrate are all Rogers RO4003 plates with a dielectric constant of 3.55, a loss tangent value of 0.0027, a length of 76 mm, a width of 76 mm, and thicknesses of 0.2 mm, 2.5 mm, and 0.508 mm, respectively.

3. The frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal according to claim 1, It is characterized in that The 4×4 super surface unit is arranged in a 4×4 periodic manner with a single copper sheet as an element, and the single copper sheet is a rectangular copper sheet with two small rectangles cut off at the diagonals.

4. The frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal according to claim 3, It is characterized in that The rectangular copper sheet is 7.2 mm wide and 9.3 mm long, the cut small rectangle is 4 mm wide and 2.6 mm long, and 16 individual copper sheets are spaced 1.15 mm apart from each other in a 4×4 periodic arrangement.

5. The frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal according to claim 1, It is characterized in that The thickness of the upper and lower layers of glass is 0.2 mm. The liquid crystal is a mixed nematic phase liquid crystal composed of 4′-Pentyl-4-cyano-biphenyl, 4′-Alkyl-4-isothiocyanato-phenylcy clohexanes and 4"-Alkyl-2,5′fluoro-phenyl-4alkyl-biphenyl-tolanes. When the temperature changes from -20°C to 50°C, the dielectric constant of the liquid crystal changes from ε / / =3.1 and ε ⊥ =2.5, and the liquid crystal thickness is 0.7 mm.

6. The frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal according to claim 1, It is characterized in that The H-shaped air gap is formed by connecting two parallel gaps with a length of 5 mm and a width of 0.8 mm and a long gap with a length of 20 mm and a width of 1.8 mm. The H-shaped air gap is located at the center directly below the 4×4 metasurface unit and is placed perpendicular to the metasurface unit.

7. The frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal according to claim 1, It is characterized in that The 180° hybrid circuit feeding network is a four-port parallel power division hybrid feeding network with phase difference, including ports 1, 2, 3, 4, 5 and terminals A, B, C, D, and E. The signal is input from the terminal A and passes through the characteristic impedance Z 0 , electrical length λg / 4 to end B, through characteristic impedance Z 0 , electrical length λg×3 / 4 to C terminal, with equal amplitude and reverse output from B and C terminals, and the phase of C terminal lags behind B terminal by 180°; the electrical length of port 3 relative to port 2 and port 5 relative to port 4 are respectively longer by λg / 4, realizing the required phase difference of 0°, 90°, 180°, and 270° from port 2 to port 5; port 2 and port 3 are connected in parallel at E terminal, port 4 and port 5 are connected in parallel at D terminal, and port 1 is connected to A terminal.

8. The frequency reconfigurable metasurface circularly polarized array antenna based on liquid crystal according to claim 1, It is characterized in that The metasurface array, metal ground, and 180° hybrid circuit feeding network are all made of copper films with a thickness of 0.036 mm.

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