1 SATELLITE TRACKING THROUGH THE ANALYSIS OF RADIATION PATTERNS David Olivera Mezquita Abstract This paper describes the process of tracking the trajectory of a satellite by analyzing the radiation pattern of the antenna that links with the satellite. Starting from an initial position of the antenna, and performing two consecutive measurements of the received power, we are able to obtain the phase angle relative to the position of maximum power, this way we could orient the antenna properly. Index Terms position, satellite, radiation pattern, power. I. Introduction The orbit described by the satellite is not very accurate because it is very difficult for the pitcher to put the satellite in the desired orbit position. That is why during the first hours of operation of a satellite, the communication has always been a problem. For this reason, it is necessary to create an algorithm that allows us to orient the antenna toward the satellite as much autonomously and accurately as possible. This is especially important during the first hours of life of a satellite because it is one of the most critical moment. Using radiation patterns, analyzing the power received from the satellite will can ensure that the antenna points correctly to the satellite regardless of its movement. To verify that the results are correct the simulation program Satellite Tool Kit (STK) will be used, which also provides the basis to simulate the communication with the satellite. A. Objectives This system is based on the study of the power received from the satellite ground station, so that the antenna will be oriented to obtain maximum power whenever possible. Unlike other similar systems that are based on a single measurement of power, the advantadge of this system is that it will perform two measurements. Moreover, we will only accept an optimal communication, if the antenna is guided so that it can receive more than 80-85% of the power emitted by the satellite. II. STK The STK (Satellite Tool Kit), is a powerful simulation and analysis of interactions between the satellites, launch vehicles, missiles, aircraft and land vehicles. The capabilities of the core of STK include the generation of position data and attitude. This tool will allow us modeling a system in which the orbit is defined to describe the satellite of our study and the ground station where we can place the linking antenna. Figure 1. Azimuth and elevation of an antenna The program will provide the real values of both azimuth and elevation the linking antenna should have to achieve the optimal communication of the satellite, as illustrated in Figure 1. That is, it gives us the real path that will be followed by the satellite, as well as the approximate values of azimuth and elevation, which represent the prediction or estimation of the path the satellite should follow, as especially during the first hours of operation it is difficult to know which is its actual path. This allows us to simulate the power the antenna will receive at the ground station, and that way predict in which direction we should move the antenna to track the satellite properly. III. Radiation pattern The radiation pattern or radiation diagram of an antenna is defined as a mathematical function or graphical representation of the properties of the antenna according to their spatial coordinates. An antenna does not radiate in the same way in all space direction, but according to its geometry, size or form of excitation it is capable of directing the energy in specific spatial directions. The radiation pattern is a graphical representation of the directional properties of the radiation from an antenna in the space. Some of the properties included are the radiation intensity and field strength or power flux density. This one is in our case the property of interest. The Figure illustrates a typical radiation pattern. The typical radiation pattern of an antenna is of the form: P R = sin(θ) P Max θ (1)
Figure 3. Graphical Representation Figure. Radiation Pattern whose graphical representation can be seen in Figure3. This graph can clearly see the primary and secondary lobes of the antenna. The graph clearly shows that the maximum power value is obtained for θ = 0, that means that the maximum is obtained when the main lobe of the antenna is perfectly oriented towards the satellite. In this graph we can see how is the form of the side lobes and which are the values, or pahse angle, for which the received power is zero. The difference of the antenna bandwidth between the different values of θ that give an amplitude of 0.5 is if 3dB. According to this expression of the radiation pattern in terms of bandwidth would be as follows: P R sin(k θ) = P Max k θ () where k is a constant that is defined by the bandwidth of the antenna. IV. Process Description Before presenting a description of the process undertaken to solve our problem, we define the coordinate system so that we will use the parameters that define the orbit of the satellite chosen for the study. The coordinate system chosen is usually used when working with antennas. In this cases, we usually use the spherical coordinate system, since this system allows us to define a direction in the space. Figure 4 illustrates this coordinate system. A. Radiation pattern equation The first thing to do is to define the formula of the radiation pattern of the antenna. This formula should be one of the input data, because of the fact that in a real case, the antenna and its characteristics are known data. But in our case we will define it from other typical parameters such as bandwidth or the diameter of the antenna. This will leave the following equation. and calculate the value of k, depending on the beamwidth of the antenna from the following equation: sin ( ) k B k B = 0.5 (3) where B is the beamwidth of the antenna in radians. The beamwidth of the antenna as estimated from the following expression: θ = 1 f GHz D Where f GHz, is the frequency of the bandwidth expressed in GHz and D is the antenna diameter in meters (m). This makes the beam width will provide, theta, in degrees, for which the receiver power is 50% of the maximum power, ie the 3dB beamwidth. Being thus defined the equation of power density for our study as follows: P R P Max = B. Calculation of phase angle (4) ( ) sin (4, 774 π θ) (5) 4, 774 π θ After defining the expression of the power density, we turn to the calculation of the phase angle for two values of power receive antenna. These values correspond to power two consecutive measurements taken with the antenna, but in our case would be necessary to simulate since we have no real data. To carry out this, use the mentioned STK. This program will define a typical satellite, which in our case it is a sunsynchronous satellite with an altitude of 600km, and put the link station in Madrid. The program provides a file
3 Figure 4. Coordinates Figure 5. Lobe with the azimuth and elevation values the antenna must have. These values represent the real path described by the satellite. Moreover, varying slightly the trajectory of the satellite itself, slightly changing some of the parameters that define its orbita to get a file with values of azimuth and elevation slightly different, corresponding to the estimate made by the observer at the station on the possible satellite track. This allows us to give an initial orientation to the antenna. Once we have the initial position of the antenna, which is given by the estimate file of the trajectory; and the starting position it should have, that we will get from the file with the values of the real path, we calculate the phase angle the antenna has. In order to solve this task, we turn to spherical trigonometry, namely the BesselAngles by the following equation: cos(θ) = cos(90 φ 1 ) cos(90 φ )+ + sin(90 φ 1 ) sin(90 φ ) cos(λ 1 λ ) where θ is the phase angle we want to find, φ 1 is the initial elevation value we place on the antenna, φ elevation is the real value the antenna should have, λ 1 is the initial azimuth value we place on the antenna, and λ azimuth is the actual value the antenna should have. Putting the antenna in the same position and taking data from the actual position we get another value of phase angle, which when it is introduced into equation 5 we get the two values corresponding to the power. Once we have the two power values with the corresponding angles, we calculate the slope of the line defining the two power values with the angle. This will tell us where in the lobe Figure5 we are and therefore where we should orient the antenna. The chosen time interval between data collection is 10 seconds. This time is large enough to test our system. The slope of a line is calculated as follows: (6) m = P (7) θ Once we have the value of the slope of the line, we derive the equation ecuacion 5 compared to θ as follows: [ 0, 0089θ sin (14, 998θ) 0, 1334θ 3 ] dp dθ = cos (14, 998θ) sin (14, 998θ) θ 5 (8) Since we know that the derivative of a function at a point represents the value of the slope of the tangent to the function at that point, we can calculate what is the average phase angle we have and in which angle and direction we should move the antenna. For this purpose, matching equations 7 and 8, and solving for the value of θ, we get the exact angle that we should move the antenna to point correctly to the satellite. Solving the equation ecuacion5, is undoubtedly the most complex part because it is a a very important equation because of the fact that it is not possible to solve by algebraic methods and thus it will require the use of numerical methods. C. Calculate the new position Once we have calculated the phase angle of the antenna, we have to calculate the new position the antenna must have with the phase angle correction. Analyzing the values of elevation of the satellite estimated the trajectory, we will see that it is a very gradual increase, with little variation. Because of this we can deduce that the actual movement of the satellite elevation is not going to make a big change regarding the prediction. For this reason we decided to set the elevation value of the antenna and vary the azimuth value depending on the phase angle calculated above.
4 Figure 6. Figure 7. Results Once we have this new azimuth and elevation position we should perform a process similar to the one mentioned above, that is, make two new power measurements with corresponding phase angle to recalculate the new average and phase angle so it would be the 3rd position of the antenna. This process would be necessary for each of the points of the path, so you always get optimal communication. When calculating the new azimuth value the antenna should have, it could be the case, due to the shape of the function illustrated in Figure 6, you could find one solution, two solutions or even any solution. In case you do not find any solution, we will take the closest value to the optimal solution. In case you find two solutions, we will choose the solution that provides a more optimal power values, as one of the solutions will have very far from the optimal power values. This error occurs when the change in elevation angle is greater than the value of the phase angle and therefore, there is not value of azimuth which satisfies equation 7. V. Analysis of results In our study we calculated the first 9 positions the antenna must have, making eighteen power measurements. No more positions were calculated from the antenna, as with the 9 we have already calculated, we have obtained enough results to validate our system. In view of the obtained results, we will always achieve our objective of having the transmission powers higher of 80-85%. That is why the communication can be considered optimal. However, we find some results that despite being within the limits are not very positive results. The movement of the antenna is not linear at all. However, we find some results that even though the fact that they are within the limits, they are not very positive results. The main reason why the movement of the antenna is not a linear movement, and sometimes phase angles above degrees, and even reaching nearly 3 degrees, is due to the high time interval between each data collection. Therefore, in order to verify the validity of our system obtaining even more accurate data, we repeat the process with a much smaller time lag between data collection: seconds. With seconds time interval, the results are significantly better obtaining phase angles not greater than 0.4 degrees. Figure 7 shows the results obtained for azimuth and elevation for a time interval between data collection for seconds compared to the values of real azimuth and elevation values. As we can see, the antenna performs very linear movement and the values it gets are close to real values. VI. Conclusion The most used methods used until now, consisted on reorienting the antenna continously always looking for achieving the maximum power value. In our case the system is slightly different from the systems used before. The idea is to keep the antenna fixed on a point to make two measurements of power in order to appreciate more accurately the movement you are describing by the satellite. This system has clear advantages over conventional systems. The first has to do with atmospheric attenuation, a factor that has to be taken into account for these types of communications. In our case, during these two measures in a very small interval of time and being the satellite to a relatively large distance; atmospheric attenuation can be considered constant and it is not an important factor. Another problem that arises in this type of communication is the noise that may contain the signal. If you only take into account one single measurement of the power and we are orienting the antenna due to that measure; the noise signal may contain considerable error and it could cause some errors if not treated properly by some kind of filter. In our case, since we perform two consecutive measurements in very small time intervals and then make the difference between these two signals, the mentioned noisy signal is removed.
5 Moreover, in view of the results, we conclude that the interval between data collection, significantly affects the results. As we reduce the time interval between data collection, the quality of the connection increases. For time intervals of seconds we see that we get optimal communication, and in the case of having another type of antenna with other conditions that require higher quality, could well reduce the time interval between collection of data values to less than 1 second. For these two reasons, we consider this system as a breakthrough. References [1] Capderou, Michel. (005). SATELLITES ORBITS AND MISSIONS. Jean-Jacques Dordain, ESA, Director General. [] Constantine A. Balanis. (005). ANTENNA THEORY ANALYSIS AND DESIG. Wiley-Intersciences. [3] Wiley J. Larson and James R. Wertz, (1999). SPACE MISSION ANALYSIS AND DESIGN, Third Edition. Douglas Kirkpatrick, United States Air Force Academy Donna Klungle, Microcosm, Inc. [4] Fundamentos de Antenas. CONSIDERACIONES GENERALES SOBRE ANTENAS [5] RECOMENDACIONES DE LAS CARACTERÍSTI- CAS TÉCNICAS DE LAS ESTACIONES TERRENAS DE HISPASAT (CTETH) [6] SPACECRAFT ANTENNA PATTERN CONTROL SYSTEM. Milton Berkowitz, King of Prussia [7] Master of Science in Electronics. DESIGN OF GROUND STATION ANTENNA FOR A DOUBLE CUBESAT STUDENT PROJECT. Mireia Oliver Miranda [8] RADIATION PATTERN ANALYSIS OF A FOUR- ELEMENT LINEAR ARRAY. John J. Lemmon