DETERMINATION OF THE MINIMAL REFLECTING SURFACE POINTS NUMBER REQUIRED FOR ASSESSMENT OF LARGE-SIZE TRANSFORMABLE ANTENNA PATTERN DEVIATION

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Introduction. Today one of the priority trends in the world satellite construction is producing communication spacecraft (SC) with large-size transformable antennas (LTA) [1; 2]. LTA is a complex technical system on the deployment and on the required configuration of which depend both the quality of the signal provided for sub- scribers in the required coverage zone, and general accu- racy of SC specified performance. The quality of the signal is determined by the orienta- tion accuracy of antenna pattern (AP) of LTA [3] for the target coverage zone, and by the level of LTA gain in this zone. The necessary radio engineering characteristics (REC) of LTA are determined by its practically applied construction configuration designed with the required accuracy. One of the LTA configuration characteristics that influence its REC is the accuracy of the feed and re- flector interposition, greatly dependent on the reflector load-carrying structure - the rod. The next important LTA component (from the point of view of the AP formation) is the reflector [4-6]. The reflector is a rigid framework consisting of a base and spring-type spokes, a form- building structure fastened to the framework, and a net sheet sewn to the form-building structure (fig. 1). To maintain the quality of the signal the form of the net sheet surface of the reflector must be close to the paraboloid of revolution. Because of the large size of LTA for its rather small weight, the structure rigidity is not sufficient. As a result, during the operation at the orbit the structure is subject to temperature and elastic deformations [7; 8] which, in turn, cause deterioration of the antenna REC and make the sig- nal in the coverage zone weaker. It is possible to compensate REC deterioration, also with the help of the stated current AP [9]. Fig. 1. Reflector design Рис. 1. Конструкция рефлектора So the following task can be formulated: to measure the AP and determine the deviations of its axis from the specified values. To assess these AP deviations of SC working at the orbit, two different methods can be used. The first method is based on the REC measurements made by ground SC service stations. This method is suffi- ciently accurate, but it has some drawbacks. For example, it can’t be applied when the spacecraft is being used for its specified purpose. The requirements for ground stations - their number, location and equipment - also increase when this method is applied. The second method is based on deriving REC meas- urements from the configuration and orientation of the antenna reflector. The reflector is imaged as a cloud of monitored checkpoints able to reflect the deviations of the construction’s configuration and orientation from the specified values. To obtain AP measurements using the second method, an antenna configuration control system (ACCS) must be worked out [10-13] for measuring the coordinates of the reflector surface points. The system should have the fol- lowing components: measuring equipment mounted on the spacecraft casing, and control elements (light reflec- tors), on the construction elements. This configuration allows to present the LTA construction in the form of a checkpoint cloud. The initial data required for AP calculation are: - coordinates of the radio-reflecting surface points or general orientation of the given surface (when radio- reflecting surface deformations are negligible); - feed directional pattern; - operation frequency. Thus ACCS measurements complementing AFS char- acteristics specified at the stage of its design and produc- tion form the initial data sufficient for AP setting. There is also a problem of the required number of checkpoints. To control the configuration of “rigid” re- flectors (reflectors having negligible radio-reflecting sur- face deformations), it is enough to monitor several check- points, the number and location of which is chosen to meet the precision requirements when determining the orientation of such reflectors. However, to control the configuration of elastic large-size reflectors functioning as parts of antennas with UHF operation frequencies, it is necessary to assess the reflecting surface deformations, since the deformation data contribute a lot to REC of the antenna. In these cases the required number of monitored checkpoints can reach several thousand. Besides the fac- tors mentioned, the number of checkpoints determined by ACCS depends on: - weight and dimensions of control elements; - structural features of the antenna; - characteristics of calculation algorithms and on-board computer; - characteristics of measuring instruments. This paper shows how the number of checkpoints may influence the precision of REC determination for an antenna with a large-size deformed reflector. A calculation was made to find the AP representing the LTA focal beam electrical axis orientation for different numbers of check- points. AP calculations were made both for the theoreti- cally determined reflector profile (without deformation) and for the deformed reflector profile. The lower KU- band frequency was taken for the antenna operational frequency. Initial data for evaluating the electrical character- istics of antennas. To find the necessary number of points of the reflector surface sufficient for calculating the antenna electrical characteristics with proper accuracy, AP and orientation of the focal beam electrical axis were calculated for the antenna reflector with a 12-meter aper- ture. The calculations were made for different sets of points: 1488, 1250, 1000, 750, 500, 250, 25. In the documentation for software sets used to calcu- late AP, it is advisable to choose the number and orienta- tion of control surfaces that make the distance between two neighboring points not more than l/(5-6) mm, where l is wave length. However, it must be taken into account that the reflec- tor surface is a set of flat trapezoidal facets formed by the frontal network checkpoints. It means that the location of points inside the facets can be determined by the points forming each facet. Thus, the influence of points within each facet on the AP can be expressed through the 4 nodal points of the frontal network. Consequently, the basic data on reflector surface de- formations can be derived from coordinates of the frontal network nodal points, and to obtain a reliable AP it is enough to use for calculation the data on points, the dis- tance between which (for the current implementation of the frontal network) does not meet the above-mentioned requirement. So, for the checkpoints, at their maximum number, the frontal network points with coordinates derived from the reflector finite-element model (FEM) can be taken, in case the model is framed according to the method described in [14]. The deformations were obtained in modelling how the reflector KEM is affected by a model device of reflector configuration finishing adjustment (RCFA). RCFA is a device that allows to adjust the position of the spoke in the plane of the reflector symmetry axis. The main con- trols are the tie-rod devices which tighten the cords attached to the spokes. A more detailed description of RCFA model and im- plementation logic is presented in [15]. The cartogram of distribution of the frontal network checkpoints deviations from the theoretical profile made The deformed configuration was framed artificially by changing the RCFA tie-rods position in FEM. Tie-rod movements are presented in tab. 1. The cartogram of distribution of the frontal network checkpoints deviations from the theoretical profile for the reflector with deformations is presented in fig. 3. The SCD of the frontal network checkpoints from the reflector theoretical profile is 18.1 mm. The theoretical paraboloid coordinate system (TPCS) is a coordinate system associated with the SC casing, in which the theoretical location of the reflecting surface points (theoretical profile) of the reflector is described by the canonical equation of paraboloid of revolution. The reflector coordinate system (RCS) is the coordi- nate system determined in [9]. The coordinates of different numbers of reflecting sur- face points were formed by performing the following se- quence of actions: 1. The coordinates of a large number (1001000) of the reflecting surface points were formed. The coordinates y and z of these points were calculated from the following parametric equations: y = 3u cos(n) + 8.3, for the reflector initial design is shown in fig. 2. Devia- tions are calculated as the shortest distance from frontal z = 3u sin(n), (1) network checkpoints to theoretical profile. The SCD of the frontal network checkpoints from the reflector theo- retical profile is 0.2 mm. where υ - parameter changing from 0.002 to 2 with step of 0.002; ν - parameter changing from 0 to 2π with step of 0.002. 0.8 , мм 6 0.6 4 0.4 2 0.2 м z, Подпись: ZТП, м0 0 -0.2 -2 -0.4 -4 -0.6 -6 y, м 2 4 6 8 10 12 14 -0.8 Fig. 2. Distribution of deviations from TP in the initial design Рис. 2. Распределение отклонений от ТП в проектном состоянии Table 1 Rod movements, mm Rod 7 Rod 8 Rod 9 Rod 10 Rod 11 Rod 12 Rod 1 Rod 2 Rod 3 Rod 4 Rod 5 Rod 6 20 0 20 0 20 0 20 0 20 0 20 0 Note: “+” - is the movement in direction of the reflector folding. 6 4 2 м z, Подпись: ZТП, м0 -2 -4 -6 y, м 2 4 6 8 10 12 14 43.96, мм 32.97 21.98 10.99 0 -10.99 -21.98 -32.97 -43.96 Fig. 3. Distribution of deviations from TP in the deformed state Рис. 3. Распределение отклонений от ТП в деформированном состоянии The x coordinates of the deformed and undeformed profile were calculated by interpolating the frontal net- X ТП X Р X Р_ТП work checkpoints (1488) of the deformed reflector at the points with y and z coordinates, calculated from the expression (1). YТП ZТП = M Р_ТП × YР ZР + YР_ТП , ZР_ТП (3) 2. The coordinates of the reflecting surface points formed in 1 were recalculated in RCS: where M Р_ТП cos(q) -sin(q) 0 = sin(q) cos(q) 0 - transition matrix X Р æ X ТП X Р_ТП ö 0 0 1 ç ÷ YР = M ТП_Р × ç YТП - YР_ТП ÷ , è ø ZР ç ZТП ZР_ТП ÷ (2) from RCS to TPCS; θ - rotation angle of RCS about the axis of OZ TPCS; XТП, YТП, ZТП - coordinates of the point in TPCS; XР, YР, ZР - coordinates of the point in RCS; where M ТП_Р cos(q) sin(q) 0 = -sin(q) cos(q) 0 0 0 1 - transition matrix XР_ТП, YР_ТП, ZР_ТП - coordinates of RCS center location in TPCS. A network of 250 points of radio-reflecting surface in YOZ plane of TPCS is shown in fig. 5. Dashed lines in from TPCS to RCS; θ - rotation angle of RCS about the axis of OZ TPCS equal to 26.5651 degr; XТП, YТП, ZТП - coordinates of the point in TPCS; XР, YР, ZР - coordinates of the point in RCS; XР_ТП, YР_ТП, ZР_ТП - coordinates of RCS center location in TPCS equal to [2296.35 mm, 8731.49, mm 0]. 3. Coordinates of the required number of reflecting surface points were formed. For the purpose a formation of a network of points uniformly distributed in the YOZ RCS plane was carried out: y coordinate of the network points changed from -7 to 7 m; z coordinate of the network points changed from -6 до 6 m; interpoint step relative to both coordinates made h (tab. 2). The coordinates x of the points of the reflector’s deformed and undeformed profiles were determined by interpolation of the points obtained in 2, at the points of the pre-formed network. In the process of interpolation, the network points located outside the radio-reflecting surface were discarded. A network of 250 points of radio- reflecting surface in the YOZ plane of RCS after interpo- lation is shown in fig. 4. 4. The coordinates of the required number of reflect- ing surface points formed in 3 were recalculated in TPCS fig. 5 show the circle that is the generator of a cylinder, the intersection of which with the paraboloid of revolution forms the radio-reflecting surface. Antenna REC calculation. That was made for the following numbers of reflector surface points: 1488, 1250, 1000, 750, 500, 250 and 25. Fig. 6 shows the focal beam pattern at the lower operational band frequency for the antenna reflector theo- retical profile at different profile points number. Fig. 7 shows the focal beam pattern at the lower operational band frequency for the deformed reflector profile of antennas with different profile points number. Fig. 8 shows the focal beam pattern for the unde- formed profile (left) and for the deformed profile (right) at 25 points. Tab. 3 presents calculations of antenna focal beam DG at the upper KU-band frequency for the deformed reflec- tor profile and for that without deformation. Analysis of calculation results. Fig. 6 and 8 of the focal beam pattern indicate that with the lower number of reflector’s monitored points, the beam pattern is distorted; the effect is present even at 500 checkpoints, and the beam pattern gets completely distorted at 25 checkpoints. Table 2 Network step h, m Reflecting surface points number 1250 1000 750 500 250 Theoretical profile 0.31775 0.355 0.4098 0.499986 0.7052 Deformed profile 0.317 0.3552 0.40954 0.5 0.7055 6 4 2 Подпись: ZР, м0 -2 -4 -6 -8 -6 -4 -2 0 2 4 6 8 YР, м Fig. 4. Network of 250 points in the YOZ plane of RCS after interpolation Рис. 4. Сетка из 250 точек в плоскости YOZ СКР после интерполяции 6 4 2 Подпись: ZТП, м0 -2 -4 -6 2 4 6 8 10 12 14 16 YТП, м Fig. 5. Network of 250 points in the YOZ plane of TPCS Рис. 5. Сетка из 250 точек в плоскости YOZ СКТП 0.40 Подпись: ElevationinDegrees0.20 -0.00 -0.20 -0.40 Number of points 1488 -20 -10 -6 -3 -20 -30 Подпись: SATSOFT-0.40 -0.20 -0.00 0.20 0.40 Azimuth in Degrees 0.40 Подпись: ElevationinDegrees0.20 -0.00 -0.20 -0.40 Number of points 1250 Подпись: SATSOFT-0.40 -0.20 -0.00 0.20 0.40 Azimuth in Degrees 0.40 Подпись: ElevationinDegrees0.20 -0.00 -0.20 -0.40 0.40 Подпись: ElevationinDegrees0.20 -0.00 -0.20 -0.40 Number of points 1000 Подпись: SATSOFT-0.40 -0.20 -0.00 0.20 0.40 Azimuth in Degrees Подпись: SATSOFTNumber of points 500 -30 -20 -10 -3 60.4 3 -20 -30 -0.40 -0.20 -0.00 0.20 0.40 Azimuth in Degrees 0.40 Подпись: ElevationinDegrees0.20 -0.00 -0.20 -0.40 0.40 Подпись: ElevationinDegrees0.20 -0.00 -0.20 -0.40 Number of points 750 -2 -30 0 -10 -3 60.6 3 -6 -30 Подпись: SATSOFT-0.40 -0.20 -0.00 0.20 0.40 Azimuth in Degrees Подпись: SATSOFTNumber of points 250 -0.40 -0.20 -0.00 0.20 0.40 Azimuth in Degrees Fig. 6. Focal beam pattern at the lower operational band frequency with different numbers of the reflector theoretical profile points Рис. 6. Контур фокального луча на нижней частоте рабочего диапазона при различном количестве точек теоретического профиля рефлектора Fig. 7 and 8 showing the focal beam pattern for a de- formed reflector indicate that the given reflector deforma- tion causes a deep distortion of the beam. They also con- firm that with a decrease in the number of reflector’s monitored points the contour of the beam is distorted. For the deformed reflector the beam contour is completely blurred at 25 checkpoints. The results of directive gain of antenna focal beam calculation presented in tab. 2 confirm the deterioration of focal beam characteristics at the upper operational band frequency with the decrease in the number of reflector’s monitored points. The minimum of 250 monitored reflec- tor checkpoints is acceptable; at that the calculated dete- rioration of the focal beam DG at the upper operational band frequency is 2.21 %. That number of the reflector surface monitored points makes it possible to assess the antenna focal beam electrical characteristics with a per- missible error. Conclusion. The problem of choosing the monitored points number of the reflecting surface considered in this paper is relevant for the problem of design of LTA con- figuration control system at the orbit. Such system allows to determine the AP when the orientation and form of the LTA components is stated. The proposed approach to determination of the monitored points number, as well as the results of its application for a specific operating antenna are aimed at perspective SC projects and at formulating the technical specification requirements for making SC antenna-feeder systems. Подпись: SATSOFTПодпись: SATSOFTNumber of points 1488 Number of points 1250 1.50 -20 -6 -30 -30 -30 -20 -10 -10 -3 -30 1.50 -20 -3 -30 -30 -30 -20 -10 -3 -6 -10 -30 1.00 -6 -10 -6 -20 1.00 -6 -20 -30 -30 -30 -10 -3 -20 -10 -30 -10 -20 -10 -20 -10 -6 -10 -10 -10 -6 -20 -3 0.50 -10 -10 Подпись: ElevationinDegrees45.51 0.50 -30 -10 -3 -30 -10 -6 -10 -6 -3 -10 -10 Подпись: ElevationinDegrees30 -10 -10 -10 -3 -10 -10 -6 -30 -10 0.00 -10 -10 -3 -30 0.00 45.76 -10 0 -6 -0.50 -3 -10 -6 -10 -20 -10 -10 -20 -30 -30 -0.50 -3 -20 -6 -10 -10 -10 -6 -10 -20 -30 -30 -30 -1.00 -30 -1.00 -10 -30 -6 -10 -10 -20 -10 -6 -30 -30 -6 -30 -30 -20 -10 -1.50 -30 -30 -30 -30 -30 -30 -20 -10 -1.50 -30 -10 -10 -20 -30 -30 -30 -10 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 Azimuth in Degrees Number of points 1000 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 Подпись: SATSOFTAzimuth in Degrees -6 -10 Подпись: SATSOFT-30 Number of points 750 -20 -30 -30 -30 -30 -20 -6 1.50 -3 -6 -20 1.50 -30 -30 -30 -30 -6 -10 -10 -10 1.00 -6 -20 -30 -20 -10 -30 -20 -10 -3 -10 -10 -10 -10 -6 -30 -10 -20 -30 -6 1.00 -30 -20 -10 -6 -10 -10 -6 -30 -10 -2 -20 -30 0.50 -30 Подпись: ElevationinDegrees-30 -30 -10 -10 -6 -10 -3 -10 -30 0.50 30 Подпись: ElevationinDegrees-30 -30 -10 -3 -10 -6 -3 -10 0.00 -10 45.98 -10 -3 0.00 0 46.37 -10 -10 -30 -3 -6 -10 -10 -30 -30 -10 -6 -10 -6 -10 -0.50 -10 -20 -30 -0.50 -30 -30 -30 -10 -20 30 -6 -10 -20 -10 -20 -3 -6 -1.00 -10 -6 -30 -1.00 -10 -10 -1.50 -30 -30 -10 -30 -30 -3 -30 -10 -30 -6 -30 -30 -20 -30 -10 -30 -30 -20 -30 -1.50 -30 -30 -30 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 Azimuth in Degrees Number of points 500 -20 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 Azimuth in Degrees Подпись: SATSOFTПодпись: SATSOFTNumber of points 250 1.50 -10 -30 -30 -30 -30 -6 -10 -30 1.50 -20 -30 -30 -30 -20 -6 -10 -30 -20 -30 -20 -30 -30 -3 -30 -6 -20 -20 -10 -30 -2 -30 -6 -20 -20 -6 -20 1.00 -30 -3 -20 -10 -6 -10 -10 -20 -6 -30 -10 -30 1.00 -10 -10 -20 -30 -30 -20 -6 20 -10 -10 -6 0.50 -20 0.50 -30 -10 -10 -6 -10 Подпись: ElevationinDegrees-6 -10 -10 -3 -10 -10 -3 -10 -30 -30 -20 -6 -10 -3 44.95 -10 -10 0.00 -10 45.87 -10 0.00 -10 -0.50 -20 -6 -20 -10 -10 -6 -10 -10 -30 -30 -20 -30 -20 Подпись: ElevationinDegrees-0.50 0 -20 30 -20 -6 -10 -10 -6 -10 -6 -20 -20 -30 -20 -1.00 -10 -1.00 -10 -30 -30 -30 -6 -10 -30 -30 -20 -10 -6 -20 -30 -20 -20 -10 -6 -10 -20 -30 -1.50 -30 -30 -1.50 -30 -20 -10 -30 -20 -20 -30 -30 -20 -30 -20 -30 -20 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 Azimuth in Degrees -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 Azimuth in Degrees Fig. 7. Focal beam pattern at the lower operational band frequency with different numbers of the deformed reflector profile points Рис. 7. Контур фокального луча на нижней частоте рабочего диапазона при различном количестве точек деформированного профиля рефлектора Подпись: SATSOFTПодпись: SATSOFTNumber of points 25 Number of points 25 -20 -20 -6 -20 -30 -10 -10 6 -20 1.50 10 -20 -10 - 1.00 -20 -10 -10 -6 -3 38.98 -10 -10 -6 -10 -1 -10 0.50 -10 -10 Подпись: ElevationinDegrees10 -20 -6 Подпись: ElevationinDegrees-10 0.00 -20 -20 -10 -10 -0.50 -20 -20 - -20 -1.00 30 -20 - -10 -10 -1.50 -20 - -20 -10 -30 -20 -10 -20 -10 -30 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 Azimuth in Degrees -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 Azimuth in Degrees Fig. 8. Focal beam pattern at the lower operational band frequency with 25 points for both reflector profiles Рис. 8. Контур фокального луча на нижней частоте рабочего диапазона при 25 точках для двух профилей рефлектора Table 3 Directive gain of the antenna focal beam at the upper operational band frequency for different reflector profiles Number of points DG, percent Reflector profile without deformation Deformed reflector profile 1488 100 74.77 1250 99.83 75.15 1000 99.75 75.52 750 99.62 76.17 500 99.26 75.33 250 97.79 73.83 25 64.03 56.80

About the authors

G. I. Kalabegashvili

JSC “Academician M. F. Reshetnev” Information Satellite Systems”

Email: Kalabegashvili89@yandex.ru
52, Lenin Str., Zheleznogorsk, Krasnoyarsk region, 662972, Russian Federation

E. V. Bikeev

JSC “Academician M. F. Reshetnev” Information Satellite Systems”

52, Lenin Str., Zheleznogorsk, Krasnoyarsk region, 662972, Russian Federation

M. G. Mathylenko

JSC “Academician M. F. Reshetnev” Information Satellite Systems”

52, Lenin Str., Zheleznogorsk, Krasnoyarsk region, 662972, Russian Federation

References

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