SPACE SOLAR POWER REVIEW Volume 3, Number 4, 1982 ■■
SPACE SOLAR POWER REVIEW Published under the auspices of the SUNSA T Energy Council Editor-in-Chief Dr. John W. Freeman Space Solar Power Research Program Rice University, P.O. Box 1892 Houston, TX 77001, USA Associate Editors Dr. Eleanor A. Blakely Lawrence Berkeley Laboratory Colonel Gerald P. Carr Bovay Engineers, Inc. Dr. M. Claverie Centre National de la Recherche Scientifique Dr. David Criswell California Space Institute Mr. Leonard David PRC Energy Analysis Company Mr. Hubert P. Davis Eagle Engineering Professor Alex J. Dossier Rice University Mr. Gerald W. Driggers L-5 Society Mr. Arthur M. Dula Attorney; Houston, Texas Professor Arthur A. Few Rice University Mr. I.V. Franklin British Aerospace, Dynamics Group Dr. Owen K. Garriott National Aeronautics and Space Administration Professor Norman E. Gary University of California, Davis Dr. Peter E. Glaser Arthur D. Little Inc. Professor Chad Gordon Rice University Dean William E. Gordon Rice University Dr. Arthur Kantrowitz Dartmouth College Mr. Richard L. Kline Grumman Aerospace Corporation Dr. Harold Liemohn Boeing Aerospace Company Dr. James W. Moyer Southern California Edison Company Professor Gerard K. O'Neill Princeton University Dr. Eckehard F. Schmidt AEG—Telefunken Dr. Klaus Schroeder Rockwell International Professor George L. Siscoe University of California, Los Angeles Professor Harlan J. Smith University of Texas Mr. Gordon R. Woodcock Boeing Aerospace Company Dr. John Zinn Los Alamos Scientific Laboratories Editorial Assistant: Edith R. Mahone Editorial Office: John W. Freeman, Editor-in-Chief, Space Solar Power Research Program, Rice University, P.O. Box 1892, Houston, TX 77001, USA.
0191 -9067/82/040253-O2$O3.00/0 Copyright ® 1983 SUNSAT Energy Council TRIBUTE TO FREDERICK H. OSBORN, JR., EXECUTIVE SECRETARY, SUNSAT ENERGY COUNCIL Frederick H. Osborn, Jr., died on August 14, 1982, at the age of 67. He had served with distinction as the Executive Secretary of the Council since 1980. He was hard working, enthusiastic, and effective in administrating the Council's affairs. His voluntary service for the SUNSAT Energy Council during its formative period are deserving of appreciation by all who, in the future, will be the beneficiaries of his actions. Born in Detroit, the son of General Frederick H. Osborn and Margaret (Schieffe- lin) Osborn, he was educated at Princeton University and Trinity College, Cambridge, England. He was an officer of the Penn Surgical Manufacturing Company, and later a Director of the SmithKline Foundation which supports basic research and grants for the advancement of science. After his retirement, Frederick Osborn devoted himself to advancing human activities in space. He served as the Treasurer of
the L-5 Society, and was a leader in conservation and environmental protection groups. He supported national and international efforts to protect the environment and recognized that the evolution of humanity beyond the surface of the Earth would have a most significant impact on life on Earth. Frederick Osborn was an early supporter of the concept of energy from space for use on Earth, and through his activities on behalf of the SUNSAT Energy Council, became well-known to those who had an interest in this major option to meet future global energy demands. His invaluable services on behalf of the Council demonstrated his concern with issues that go beyond ephemeral everyday activities—with the fate of humanity and with the environment in which future generations will have to live. He steadfastly pursued a course which lesser men would have abandoned as too futuristic and with limited potential for short-term success. His wide-ranging interests, enthusiasm, and energy were of great benefit to the Council during its formative stages. He organized the office in Cold Spring, New York, and was instrumental in seeking ties with environmental groups that should be interested in energy from space as an environmentally benign approach to power generation. He took the necessary actions to have the Council recognized as a nongovernmental organization by the United Nations and to be affiliated with the International Astronautical Federation. He took it upon himself to tour the world to discuss energy from space with interested individuals in many nations. His efforts resulted in the foundation of the India chapter of the Council, and in a better appreciation by Third World nations that energy from space will benefit them. On a personal level, Fred communicated a sense of inner happiness and joy to all with whom he came in contact. The friendships he formed with so many people in his quiet, unassuming, and selfless way and the enthusiasm he transmitted have been a source of inspiration to all of us who have had the honor to know him. Frederick H. Osborn's dedication to improving the human condition through energy from space will be an example to all those who shared with him the conviction that humanity is on the threshold of expanding into the heavens beyond. His contribution to humanity's future is the most lasting memorial that any righteous man can hope for. Peter E. Glaser President SUNSAT Energy Council Fred Osborn, an elected member of the International Institute of Space Law and a distinguished graduate of Princeton University, dedicated many years of his life and his funds and services to the advancement of the peaceful use of outer space for the benefit of all mankind. As his friend during his entire lifetime, I hope that this unselfish dedication will be an example to all of us on Earth, who follow the same goal for the peaceful use of outer space for the benefit of all mankind. Hon. Edward R. Finch, Jr.
0191 -9067/82/040255-26$03.00/0 Copyright 1983 SUNSAT Energy Council GRATING LOBE CHARACTERISTICS AND ASSOCIATED IMPACTS UPON THE SOLAR POWER SATELLITE MICROWAVE SYSTEM E. M. KERWIN and G. D. ARNDT NASA Johnson Space Center Houston, Texas 77058, USA Abstract — The object of this investigation is to determine the grating lobe characteristics for the SPS (solar power satellite) microwave power beam. Using techniques of Fourier array decomposition, equations, and numerical results are given for the complete SPS microwave antenna pattern (main beam plus grating lobes). In order to ensure the grating lobe peaks are below the environmental guideline of 0.01 mW/cm2, constraints are defined for the mechanical alignment (tilt) of the SPS antenna and the satellite attitude control system. Two conditions, phase control to the subarray level and to the power module (tube) level, are analyzed as to their effects upon grating lobes. 1. INTRODUCTION The high-power microwave beam from an SPS is electronically steered and phase controlled through the use of an active, retrodirective phasing system. The phase control system will precisely focus the main beam onto a 10 km diameter ground receiving/rectifying antenna (rectenna), maintaining approximately 88% power transfer efficiency from the satellite to the ground. Grating lobes, or replicas of the main beam, will also be formed on the ground at intervals determined by the physical separation of the phase control centers on the transmit antenna. The purpose of this paper is to determine the grating lobe characteristics (separation distances, amplitudes, alignment tolerances, etc.) for two antenna configurations: phase controlling to the antenna subarray level as originally proposed in the reference SPS system and phase controlling to the tube level. This latter configuration increases by an order of magnitude the number of elements to be individually phase controlled. Because of environmental considerations, the grating lobe characteristics can be translated into constraints upon mechanical alignment of the antenna and its subarrays and upon the satellite attitude control system. In particular, the grating lobe peaks must be below 0.01 mW/cm2 for the microwave radiation levels. 2, PHASE CONTROL SYSTEM The SPS antenna is an active retrodirective array using an uplink pilot beam signal to provide precise phase conjugation. The 1 km diameter antenna is mechanically divided into 10.4328 by 10.4328 m subarrays, numbering 7200. A 10 dB Gaussian
antenna illumination taper has been initially selected for maximizing power transfer to the rectenna while minimizing sidelobe levels. Consequently, the number of klystron tubes per subarray varies from 36 at the center to 4 at the edge of the antenna. Each subarray or each power module (that portion of a subarray fed by a single tube) has its own RF (radio frequency) receiver which processes the uplink pilot beam signal and inserts the conjugate of the uplink phase delay onto the input to the tube. The outputs from the subarrays or power modules combine to form a single coherent beam at the center of the ground rectenna where the pilot beam transmitter is located. The uplink pilot beam signal has a pseudo-noise code added to a square wave subcarrier operating at four times the code clock rate. The subcarrier biphase modulates the pilot carrier signal of 2450 MHz, thereby providing a suppressed carrier spectrum. This modulation allows sufficient RF filtering at the receivers in the antenna to isolate the pilot signal from the unwanted downlink power beam which also operates at 2450 MHz. The PN code is used to protect against intentional or unintentional jamming and to allow satellite discrimination between multiple uplink pilot beam signals. Within the antenna receivers, the PN code is despread and the carrier reconstructed, thereby allowing phase conjugation at the same frequency as the downlink power signal. The conjugated phase is routed to each power tube which has a phase control loop to compensate for phase delay variations between tubes and to maintain the output signal in phase with the conjugated pilot signal. The antenna also has a constant reference phase distribution scheme which delivers the same reference to each receiver. The reference distribution system is referred to as the master slave returnable timing system (MSRTS). This system has a four-level distribution tree with returnable timing to compensate for path-length variations within the distribution cables. The original SPS phase control configuration has individual phasing to the subarray level, implying that all power tubes within a particular subarray are adjusted to the same conjugated phase value. Phase control has since been extended down to the tube (or power module) level due to antenna mechanical alignment constraints imposed by the grating lobe characteristics and because of improvements in power transmission efficiency. 3. GRATING LOBE ANALYSIS FOR PHASE CONTROL TO SUBARRAY LEVEL Before proceeding to the detailed analysis of the magnitude and locations of the SPS grating lobes, a short discussion as to the reason for the occurrence of grating lobes is in order. Each subarray can be treated as a single-point phase source with an electromagnetic amplitude Eq, spaced meters apart. Consider a linear array of individual subarrays as shown in Fig. 1. The transmit signal will have a systematic change in phase with the angle of departure of the wavefront according to the expression where Sx = subarray or element spacing, 6 = transmission angle with respect to antenna boresight. The total output from a one-dimensional array is
A grating lobe occurs at each value of 0S, as given in Eq. 3, which is dependent upon the distance between phase control centers. The size of the subarray or power module determines the grating lobe location, and also, as will be shown later, the antenna mechanical pointing requirements. The grating lobes of interest, i.e., the ones incident upon the earth, occur at small values of 0^. In this spatial region, the periodic lobes are regularly spaced at angular intervals of X/Sx. In the discussion which follows, a simplified planer geometry is used which applies most accurately for equatorial rectenna locations. Asymmetries in the field pattern resulting from the curvature of the Earth and latitude of actual rectenna sites will be neglected. The amplitude dependence of the SPS grating lobes can be determined by first considering the signal Ea from a point source at location (Xa, Ya) in the satellite antenna at a height Zo above the ground as shown in Fig. 2. The far-field electric field intensity at a point on the ground (Xa, Y„, Zo) will lag the transmit phase by (w/c)R (where R is the path length, co is 2tt times 2450 MHz, and c is the velocity of light) and can be expressed as The transmission path length R to the field point on the ground is given by which can be approximated using the binomial expansion by
where Ro is the distance from the center of the satellite antenna to the field point on the ground. The third term (X„ + Y^)/2R0 in Eq. 5 can be neglected because the pilot beam phase conjugating signal has the effect of eliminating this spatial variation. Thus, the two-term approximation is adequate for an SPS array. In the case of a circular symmetric antenna, the path length can be written in polar coordinates as When considering a complete antenna aperture, the composite electric field at a point on the ground is obtained by integrating the point source contributions oyer the surface of the antenna. If the elecric field taper across the antenna is given by Ea(Xa, Ya) and the ground field point is in the Fraunhafer, or far-field, region of the one kilometer antenna, the resulting field at the ground can be written
This expression, which is the Fourier transform integral of the taper function Ea(Xa, K„) across the antenna aperture, gives the corresponding electromagnetic field on the ground. The taper and ground pattern are related as a Fourier transform pair. It is assumed that the antenna aperture is nearly equiphase, and the propagated wave is linearly polarized. Convolution and Multiplication One method for predicting the electric field produced by the SPS antenna is to break the aperture distribution into simpler components and use the convolution principle to get the composite pattern. The actual SPS array may be thought of as the product of an infinite two-dimensional array of point sources located at the center of each subarray or power module [Fig. 3(a)] and a circular antenna function [Fig. 3(b)], This antenna function is equal in diameter to the actual array and can be shaped according to the illumination taper. Two tapers will later be considered: a uniform illumination and a 10 dB Gaussian taper. From the convolution principle, the ground pattern of the product (truncated, tapered) array will be the convolution of the two individual ground patterns and can be written A multiplication process in the antenna implies convolution of the ground patterns as shown in Figs. 3(a), (b), and (c). Since each point source in the phase array represents a single subarray, the subarray aperture function denoted by f3, must be multiplied with each source [Fig. 3(d)], Convolution in the antenna associates each point (phase center) of the array with a complete subarray function, resulting in an array of subarrays. When two functions are convolved in the antenna, the composite ground pattern (Fourier transform) is the product of the two separate ground patterns, and can be written where f3 is the subarray function and./'Ii2 is the circular array of point sources. The composite SPS antenna pattern (main beam plus grating lobes) is obtained by combining Eqs. 8 and 9, i.e., multiplying the subarray pattern by the array or grating lobe pattern, and is written The composite main beam plus graing lobe pattern is given in Fig. 3(e). The grating lobes are actually replicas of the main beam multiplied by the subarray pattern. It will be shown that the grating lobes normally occur at nulls of the subarray pattern and hence are greatly attenuated. The impact of grating lobes on the SPS
system is a requirement to maintain a close mechanical antenna alignment in order to keep these lobes in the deep nulls of the subarray beam. Each of the three antenna functions will now be determined and the results combined, using Eq. 10, to obtain the composite pattern. Infinite Array of Point Sources (fi) Each subarray is treated as a point source or impulse separated by distances Sx and Sy. Since it is assumed that the array of impulses extends infinitely in all directions, the antenna electric field distribution is given by The Dirac delta notation, 8(Xa - mSx)8(Ya - nSY) expresses an impulse occurring whenever Xa = mSx and Ya = nSY. It can be shown that the multiplication of two delta functions representing the SPS phase centers may be expressed as a infinite series of harmonic cosine functions: Because of the orthogonality property of the cosine terms, the functions exist only when The electric field at a ground grid point is found by taking the Fourier transform (Eq. 7) of the antenna field distribution (Eq. 11) and is given by which, by using the identity and Eq. 12, allows the electric field to be expressed as
where m2 = meter squared units introduced by the double integration of the unit impulses. Thus, the infinite series of point sources on the antenna transforms into an infinite array of impulses or grating lobes in the far field. These far-field impulses exist only at values of KXg = m2n/Sx and KYg = nlir/Sy. Using the definitions associated with Eq. 7, grating lobes occur whenever For ground locations on the Xg and Yg axes, these expressions for grating lobe locations reduce to the one-dimensional form, sin0 = nK/Sx, given in Eq. 3. Using SPS reference system parameters, Sx = 5y = 10.4328 m, X - 0.1225 m, Ro = 36,000 km, in Eq. 15, the grating lobes occur on-axis with angular spacing of 0 = 0.673° which corresponds to a ground spacing of 423 km. One way to intuitively predict the off-axis lobes is to recognize that the array of point sources acts like a diffraction grating at many polar angles, since many sets of parallel grating lines are implicit in the array. Referring back to Fig. 1, it is seen that a line of evenly-spaced points, properly phased to transmit in the direction 0 = 0°, will also add in phase when 0 = sin'^nX/^j.). Extrapolating to the two-dimensional SPS array of point sources as shown in Fig. 4, a rotation of the antenna axes by 45° allows the phase centers to be intersected by sets of parallel lines with slope SY/SX = 1. Grating lobes occur along the 45° ground axis orthogonal to these parallel lines, at increments determined by the spacing between the lines. Since the new lines are more closely spaced, = Sx/X^, the increment between grating lobes is larger, 0^145< = sin-1(nX/545=). The 45° grating lobes are spaced at intervals of 599 km (423 x V2), fitting into a rectangular grid pattern. The grating lobe pattern for phase control at the subarray level is given by Eq. 14 and shown in Fig. 5. The off-axis lobes occur at linear combinations of the axial grating lobe locations. The amplitudes of the off-axis grating lobes are determined by the subarray characteristics as will be discused later. Circular Antenna Function (f2) The second antenna illumination to consider is the circular antenna of radius RT with an electric field distribution Ea<rn>. The electric field at the ground, as determined by Eqs. 6 and 7, is given by where is symmetric with respect to </>a. The Fourier transform is changed to the Fourier Bessel transform by using the Bessel function identity
and integrating between <f>a = 0 and <t>„ = 2tt. Periodic exponentials cause each term in the infinite series to integrate to zero except for The resultant equation is written Fourier Bessel Transform
and the antenna electric field is zero for ra>RT. Although the Fourier Bessel transform is analogous to the Fourier transform in its treatment of antenna and ground fields, the integral in Eq. 17 cannot be solved analytically for many aperture field distributions. Let us consider two field distributions: a uniform illumination (which can readily be solved) and a 10 dB Gaussian taper as used in the SPS reference system. The aperture distribution is a critical parameter when considering rectenna collection efficiency and sidelobe levels, but is a minor factor in determining grating lobe peaks and antenna attitude constraints. To illustrate this point, consider first a uniform antenna distribution, by letting Ea^ be a constant in Eq. 17 to provide an electric field at the ground This uniform antenna pattern has a relatively narrow mainbeam and low sidelobe peaks as shown in Fig. 6. The power density on the ground is proportional to the electric field squared and may be written
where PT is the total transmit power, PAen(0, Zo> is power density at rectenna boresight on the ground. Using the SPS reference system parameters (/?0 = 36,000 km, X = 0.1225 m, Rt = 500 m, PT = 6.5 x 109 W), the power density at rectenna boresight is calculated to be 26 mW/cm2 for a perfect antenna with no errors. This 26 mW/cm2 is also the peak amplitude of each grating lobe as shown in Fig. 3(c) prior to its multiplication with the subarray pattern. The total power received at the ground as a function of rectenna radius is found by integrating Eq. 19 with respect to and rg. The rectenna power collection efficiency for a uniform illumination satellite antenna may be written where P<jeiivered= power delivered to the ground rectenna and has been evaluated for the SPS system to be 83.8% with a 10 km diameter rectenna. Now consider a 10 dB Gaussian taper for the antenna aperture distribution given by
where TdB is the truncated taper in decibels, RT is the maximum antenna radius, and ra = + Ya • The electric field at the ground can be written by use of Eq. 7 in the form where XI + Y* « Rf. This Fourier transform integral cannot in general be solved in closed form; however, for the special condition at rectenna boresight with Xg = Ya = 0, 0 = 0°; and the field contributions from each subarray adding in-phase, the power density for the 10 dB Gaussian taper is determined to be
antenna. For the SPS reference system parameters, the power density at rectenna boresight is calculated to be 23.6 mW/cm2 for a perfect antenna. The rectenna collection efficiency for the 10 dB Gaussian taper is from simultation results, 95.18% for a 10 km diameter rectenna. Summarizing, the 10 dB taper is very important in terms of collection efficiency (95.18% versus 83.8% for uniform illumination), but has limited impact upon main beam and grating lobe peaks (23.6 mW/cm2 versus 26 mW/cm2 for uniform illumination). This 10% difference in giant lobe peaks is small in comparison to the effects of antenna tilt which will be shown to have orders of magnitude impact upon grating lobe intensities. Rectangular Subarray Pattern (/3) The third antenna function f3 is the pattern from a single subarray of dimensions Lx and LY. Each subarray has a uniform field distribution across its surface as given by This subarray power density pattern is contour plotted in Fig. 7, using the SPS parameters of Lx = LY = 10.4328 m. The subarray's X and Y axes are projected onto the ground pattern. Peak magnitudes on the .Y-axis fall (due to the denominator, Kx^Lx/2) approximately at a \/X„ rate, while the off-axis peaks decrease at \/XJY„. Thus, off-axis magnitudes are given by the dB sum of the two on-axis dB components. For example, the overall pattern is down 13 dB at X„ = 635 km which corresponds to -26 dB at Xo, Ya = 635 km. Since the composite grating lobes are directly proportional to the subarray pattern as given in Eq. 10, the off-axis grating lobes will be several orders of magnitude lower than those on the X and Y axes. Using the SPS reference parameters in Eq. 25, the subarray pattern null lines occur at integral multiples of 423 km from the rectenna boresight. The condition for subarray nulls from Eq. 25 is identical to the condition for grating lobe occurrence given in Eq. 15.
where n = 0, ± 1, ±2 . . .this assumes that the separation between phase centers, Sx, is equal to the subarray dimension, Lx, with little or no mechanical spacings between subarrays. Composite SPS Pattern The three antenna functions can be combined using the convolutionmultiplication techniques developed earlier. The significance of the individual functions may be summarized as follows: the infinite array of point impulses takes into account the discontinuous phasing from subarray to subarray; the circular antenna function provides circular symmetry for the one-kilometer array and an illumination taper to improve rectenna collection efficiency; and the subarray pattern gives the transmission characteristics of the fundamental unit in the array. The composite Gaussian transmit antenna distribution can be written using Eqs. 10, 11, 21, and 24. is the pilot beam conjugated phase inserted at the center of each subarray (refer tc Eq. 5). The composite SPS ground electric field can be written in terms of the convolution integral using Eqs. 10, 14, 22, and 25. The variables Xm.n and Ym,n are used to specify distances from each grating lobe m,n. The composite antenna pattern for a Guassiar illumination taper is
where KXm , ^vmn is proportional to the X and Y distances measured from grating lobe m,n; where X? + K? Rf- The main beam patterns for the 10 dB Gaussian taper and for uniform illumination are approximately the same when considering only grating lobe effects. The ground electric field pattern can be greatly simplified if uniform illumination is used in the transmit antenna. Accordingly, the pattern for uniform illumination may be written
Equation 29 can be readily solved by computer simulations to generate the complete SPS beam pattern with phase control at the subarray level as shown in Fig. 8. Centered at each grating lobe position is a replica of the main beam pattern multiplied by the subarray pattern. 4. GRATING LOBE ANALYSIS FOR POWER MODULE PHASING The original SPS reference system documented in the October 1978 DOE/NASA report had phase control to the 10.4328 by 10.4328 m subarray level. As a result of further system studies, the present reference plan recommends that phase conjugation be performed at each of the 101,552 power modules rather than the 7220 subarrays. The
advantage of phase control at the power module or tube level is a reduction in the antenna mechanical alignment (attitude control) and subarray alignment (antenna flatness) requirements (or a reduction in scattered microwave power if the same tilt requirements are maintained). There is also a reduction in the effects of distributed phase errors within the subarrays. The disadvantage is the increased cost due to the 94,000 additional pilot beam receivers and phase conjugation electronics. The present satellite antenna has 10 types of power modules as shown in Fig. 9 to provide a 10-step approximation to the 10 dB Gaussian illumination taper. The power modules range in size from 1.7388 x 1.7388 m at the center of the antenna to 5.2164 x 5.2164 m at the outer edge, where each mechanical subarray has four tubes or power modules. Since the power modules are now to be separately phased, each of the 10 types produces a grating lobe pattern determined by its dimensions, Sx and SY, as given by Eq. 15. The power modules are small in area compared to the subarrays and, hence, produce greater separation between the grating lobes. Some of the power module dimensions are integral multiples of smaller modules, e .g., the type-8 power module has dimensions 2.6028 by 5.2164 m which is one-half the size of the type-10 module at the edge of the antenna. Thus, grating lobes associated with different power module types may coincide. In the X direction, the first grating lobe of the type-8 module coincides with the second grating lobe of the type-10 modules. Equations similar to Eq. 28 for phase control to the subarray level could be written for each of the 10 types of power modules. These 10 individual patterns combine to form a composite pattern. Since the total transmitted power is divided between the module
types whose grating lobes generally do not coincide, the combined lobes will have considerably less power than those associated with the subarrays. The overall grating lobe pattern for phase control to the power module level is shown in Fig. 10 and should be compared to Fig. 5 with phase control to the subarray level. The result is a reduction in both the amplitude and quantity of grating lobes incident upon the Earth. 5. ANTENNA TILT The actual SPS microwave system will have electrical and mechanical errors and component failures which distort the ground microwave beam pattern. An error budget has been developed through computer simulations to allow 2% tube failures, 0.1 dB amplitude error across each power module, 10° RMS phase error for the phase control system, random subarray misalignments of 3 arc-min due to surface warpage, and antenna tilt of 1 arc-min due to attitude control system errors. These errors scatter energy from the mainbeam into the sidelobes, and, consequently, reduce the power transmission efficiency between the antenna and the ground rectenna from 95.18% (fora perfect antenna) to 88%. The antenna tilt error has its most dramatic effect on the grating lobes, which repeat the mainbeam before being greatly attenuated by subarray pattern nulls. A tilt in the antenna plane causes the subarray pattern (Fig. 7) to shift across the Earth in the direction of the tilt. The grating lobes do not move because of the pilot beam retrodirec- tive phasing technique. As a result, the lobe peaks will no longer coincide with nulls in the subarray pattern as shown in Fig. 11. The magnitude of the antenna tilt and the polar tilt direction determine which grating lobes will be enhanced and to what extent their magnitudes will increase. Even a small tilt can greatly increase the magnitude of the grating lobes because of the steepness of the subarray pattern near its nulls. The condition for an antenna tilted along its X axis is shown in Fig. 12. A tilt of 3 arc-min in the 0° or A axis direction raises the peak power density to the first X axis lobe by a factor of 330, from 0.000421 to 0.139 m W/cm2. It is interesting to note the differences in the subarray pattern off-axis as compared to
on-axis, as shown in Fig. 12. The 45° off-axis subarray pattern nulls are much deeper and wider; hence, antenna tilts produce much lower grating lobe peaks for lobes off the main X and Y axes. Subarray Pattern and Grating Lobe Shifts with Antenna Tilt A small antenna tilt of magnitude 0T in a polar direction </>r changes the subarray aperture attitude with the result that all portions of the subarray pattern are shifted across the ground about the new subarray boresight. The boresight location is shifted from the rectenna origin by an amount rT, to a new ground location at (XT, YT, Zo). The entire subarray pattern shifts correspondingly along the shift vector, rT (see Fig. 13, inset). where |fT|, 0T is the magnitude and direction of subarray pattern shift due to antenna tilt and XT, YT is the distance of subarray pattern shift in the X and Y directions: For a tilt of 3 arc-min in the 0T = 0° direction, the subarray pattern is shifted 31.4 km along the X axis. For the grating lobes, the effects of antenna tilt are somewhat complicated by the pilot beam phasing system. The pilot signal corrects for path length differences from the center of each subarray (or power module) to the pilot beam transmitter at the rectenna boresight. Before tilt, the grating lobes occur at angles, 0gl, where Sxsin0gl = n\ as given in Eq. 3. After tilt the path length correction
by the pilot signal largely compensates for the tilt, producing new virtual phase sources in a plane very close to the original untilted antenna plane. Referring to Fig. 13, the new condition for grating lobes with a small antenna tilt, 0T, is
The incremental angle, A0gH results from the antenna tilt. Substituting Eqs. 32 and 3 into 31, and using the small angle approximations the incremental changes in grating lobe angle and distance from rectenna boresight can be expressed in the form Grating Lohe Intensities as a Function of Tilt It was previously shown (Eq. 25) that the subarray pattern had a \IXgYa dependence for off-axis locations, and a \/X„ or \/Yg dependence on the X or Y axes. Since the steepest fall off of the subarray pattern is along the </>„ = 45° (also -45°, 135°, etc.) axis and the pattern has its slowest decay along the X and Y axes, the = 0 and <b„ = 45° axes give the maximum and minimum values for the variations in grating lobe intensities as a function of tilt. The grating lobes occur when Xg = nkR0/Sx, from Eq. 15. When the antenna is tilted in the </>r = 0° direction, the magnitude of the subarray pattern at the location of the grating lobe is given by Eq. 35 evaluated at where the second term includes the pattern shift. The argument of the subarray
The resulting grating lobe peak power density is the main beam maximum density (all elements in phase) reduced by the subarray field term squared and can be written using Eq. 23 as For the SPS reference system parameters (1 km antenna, 10 dB Gaussian taper, Lx = 10.43 m, Ro = 36,000 km, Pden(ant) = 23 kW/m2), the peak power density for the first X-axis grating lobe is 0.0158 mW/cm2 for one arc-min antenna tilt and 0.1424 mW/cm2 for 3 arc-min tilt. These peaks are virtually unchanged when power amplifier failures, amplitude and phase errors, and subarray misalignments are included. Summarizing, the on-axis grating lobe intensities increase as the antenna tilt angle squared. On the 45° axis, Xg = Yg, and the normalized subarray pattern can be written for a square subarray as An antenna tilt in the 0r = 45° direction shifts the subarray pattern by Zotan </>T/ V 2 in the X and T directions. Assuming small tilt angles, the grating lobe peak power density along the = 45° axis is Since 0T is very small (on the order of a few arc-minutes or less), fourth power scaling means the 45° axis grating lobes are much lower than the axial lobes. For the SPS reference system, the first 45° grating lobe occurs at Xg = Yg = 423 km with a peak
power density of 0.0000024 mW/cm2 for a one arc-min antenna tilt, rising to 0.0002 mW/cm2 for 3 arc-min tilt. This performance for the X-axis and 45° grating lobes can be anticipated from an examination of the subarray pattern shown in Figs. 7 and 12. Off-axis subarray
pattern nulls are deeper and wider than on-axis nulls, and require greater antenna tilt in favorable directions before the corresponding grating lobes reach higher intensities. 6. ANALYTICAL RESULTS The performances of the grating lobe peaks as a function of phase control to the subarray level and the power module level are shown in Fig. 14. The advantages of phase control at the power module level can be seen from this data, since one arc-minute tilt for subarray phase control and 3 arc-min for power module phasing give similar results. The peaks along the Y-axis represent the worst case condition. Using 0.01 mW/cm2 as a constraint for the maximum power density level, the allowable antenna tilt is approximately one arc-minute for subarray phase control and 6 arc-min for power module phase control. The best case grating lobes, or those with the lowest power density peaks, are also shown in Fig. 14. These lobes along the 45° axis are 20-40 dB or 2-4 orders of magnitude below the axial peaks. Other off-axis lobes in the 26.6°, 63.4°, etc., ground directions are also several orders of magnitude below on-axis peaks. Thus, for assessing environmental effects from grating lobes, only the axial lobes need to be considered. The direction of the antenna tilt is also a critical factor in determining which
grating lobes will be enhanced. For example, an antenna tilt in the X-axis direction, </>r = 0°, will have little or no effect upon F-axis grating lobes. This can be seen by referring to the subarray pattern in Fig. 7; a F-axis grating lobe remains in the null when the pattern is shifted in the X-axis direction. The complete field pattern, main beam plus grating lobes, is summarized in Fig. 15 for a ground cut along the X-axis and the 45° axis. It can readily be seen that the grating lobes and sidelobes are considerably lower along the 45° side. This analysis has assumed minimal mechanical separations between the 10.43 m
subarrays. The effect of finite gaps between the subarrays upon grating lobe peaks is shown in Fig. 16. This degradation (increase in grating lobe peaks) occurs because Lx in Eq. 26. The result indicates a 15 cm gap as originally proposed was unacceptable; however, the 0.635 cm gap for the reference design has minimal impact. 7. CONCLUSION In summary, only .Y-axis and F-axis grating lobes with antenna tilts in those axial directions are important in determining antenna attitude control constraints and antenna flatness requirements for the SPS. Phase control to the power module level is quite beneficial in reducing the grating lobe peaks and/or tolerating greater errors in attitude control. Off-axis grating lobes are 20-40 dB lower than corresponding X and Y axis lobes. Antenna tilt requirements, or attitude control constraints, vary from one arc-minute for phase control at the subarray level to approximately 6 arc-min for power module phase control. Acknowledgements — The authors would like to acknowledge the informative discussions with W.B. Warren of TRW Systems, Houston, Texas, S. Rathjen and W. Lund of the Boeing Aerospace Company, Seattle, Washington, and R.H. Durrett of Marshall Space Flight Center, Huntsville, Alabama. REFERENCES 1. W.C. Lindsey and A.V. Kantak, Solar Power Satellite Baseline Phase Control System Design and Performance Evaluation, Phase 11 Final Report, NAS 9-15237. 2. S. Silver, Microwave Antenna Theory and Design, Radiation Laboratory Series, p. 173, 1949. 3. R. Courant and D. Hilbert, Methods of Mathematical Physics— Vol. I, Interscience, New York, 1953. 4. Department of Energy, SPS Reference System Report, DOE/ER-0023, 1978.
0191 -9067/82/040281-19$03.00/0 Copyright ® 1983 SUN SAT Energy Council ANTENNA OPTIMIZATION OF SINGLE BEAM MICROWAVE SYSTEMS FOR THE SOLAR POWER SATELLITE E. M. KERWIN, D. J. JEZEWSKI, and G. D. ARNDT NASA Johnson Space Center Houston, Texas 77058, USA Abstract — A generalized antenna design technique is applied to the unique environmental requirements pertaining to solar power satellite (SPS) systems. Optimal illumination tapers and antenna/rectenna sizings are generated which allow increased transmit powers and lower electricity costs while minimizing sidelobe levels to meet a 0.01 mWTcm2 environmental standard. These optimal tapers also provide other advantages over the 10 dB Gaussian reference system. INTRODUCTION The initial design of the solar power satellite (SPS) microwave power transmission system had a 1-km diameter phased array antenna radiating 6.5 GW (gigawatts) of RF energy to a ground receiving/rectifying system (rectenna). A 10 dB Gaussian illumination taper was chosen for the reference system configuration in order to maximize the rectenna collection efficiency and to minimize sidelobe levels (1). The overall system performance included 5 GW of power delivered to a 10 km diameter rectenna at an electricity cost of 46.8 mills per kWh (1 mill = 0.001 dollar). The first sidelobe peak had a maximum power density of 0.08 mW/cm2. A detailed SPS cost model with seven major categories and 34 subcategories was subsequently developed and provides insight into the relative cost factors (2). Based upon this cost model and an antenna optimization procedure to be described, it is now possible to greatly improve the end-to-end microwave system performance. The purpose of this paper is to describe a general ank..na design optimization method which, when applied to the environmental constraints imposed by the SPS system, generates optimal antenna illumination tapers. The developed tapers allow more power to be transmitted and received at a reduced cost while minimizing the sidelobe levels to meet a 0.01 mW/cmz environmental standard. Other benefits include smaller rectenna sizes and fewer geosynchronous satellites. The paper is structured to present the rationale for the reference Gaussian taper, the cost model used for optimization, microwave theory and equations, optimization procedure, and then the results and system configurations. INITIAL TAPER AND SIZING ANALYSIS The initial SPS sizing with a 1-km transmitting antenna and 5 GW of d.c. output power from the rectenna was based upon Ref. 1
[1] A thermal limitation of 23 kW/m2 in the transmitting antenna; [2] A peak power density of 23 mW/cm2 in the ionosphere; [3] Cost effectiveness (the larger the power system, the more cost effective). The thermal limitation at the antenna is a function of the amount of heat generated by the d.c.-RF power converter tubes and of the effective radiator area. The reference configuration has 70 kW klystron tubes operating at 85% conversion efficiency and cooled by passive heat pipe radiators. From thermal considerations, larger antennas or lower transmit powers are desirable; however, as the antenna size increases, beam focusing is enhanced and the power density in the ionosphere increases. At some threshold power density level, which is dependent on the operating frequency, nonlinear interactions between the ionosphere and the power beam could begin to occur. These nonlinear heating effects are of concern because of possible disruptions produced in low frequency communications and navigation systems by RF interference (RFI) and by multipath effects. Theoretical studies of the ionosphere completed during the early phases of the SPS evaluation program indicated the power density should be limited to 23 mW/cm2 in order to prevent such nonlinear heating effects (3,4). This theoretical value was taken as the SPS design guideline. Based on these two opposing constraints, the reference system was sized to produce 5 GW of rectenna output power with an antenna 1-km in diameter. A Gaussian taper was chosen for the reference design as previous studies had shown this type of illumination function was a good approximation to an optimum aperture distribution (4,5). An evaluation of the relative performance of various Gaussian tapers can be obtained from a comparison of their rectenna collection efficiencies, i.e., the percentage of the transmitted power intercepted by the rectenna. For the SPS concept, only a portion of the main beam will be collected; the sidelobe energy occupies a very large area at minimal power density levels and is not economically feasible to collect. The rectenna collection efficiencies for a number of Gaussian tapers are shown in Fig. 1. Increasing the taper produces a lower main- beam gain, a wider beam, and reduced sidelobes. In summarizing, the 10 dB Gaussian taper was initially selected for the best overall performance (maximum power delivered at a high efficiency) when considering the power density constraints at the transmit array and in the ionosphere. SPS COST MODEL A detailed analysis of subsystem costs and masses for the reference 5 GW satellite with silicon solar cells for photovoltaic conversion is developed in Ref. 2. That cost model, figured in 1977 dollars, is used in this paper in optimizing the antenna design to achieve minimum electricity costs (minimum mills per kWh). Future changes in the absolute costs of the reference system should not have a great impact on the conclusions stated herein since this analysis is based upon the relative costing of various illumination functions. The principal elements in the SPS recurring costs are |1] Satellite hardware, [2] Transportation, [3] Space construction and support, [4] Rectenna, [5] Program management and integration, [6] Cost allowance for mass growth.
Other costing assumptions include a 30-year operating lifetime, a 92% plant factor for 2.45 GHz operation, a 15% rate of return on investment capital, and a 17% increase in net SPS hardware costs to account for mass growth. The cost statements for the individual subsystems may be further divided as shown in Table 1 for the reference satellite. Proportionality factors are used to rescale costs and masses for alternative configurations, e.g., P signifies normalized transmit power; the solar cell blankets have costs and masses proportional to the amount of power transmitted. The nomenclature for each proportionality factor is given in the diagram. The information in Table 1 can be reduced into the following comprehensive SPS cost equation: where Cref is the cost for the 1978 reference design SPS; P is the transmit pow- er/reference, 6.7 GW; A is the antenna area/reference, 0.7854 km2; R is the rectenna area/reference, 78.54 km2; Ma is the power transmission (antenna) mass/13,627 MT; C„o is the total satellite hardware cost, $4947 x 106; Ch is the satellite hardware cost less amortization, $4474 x 106.
Electricity cost in mills per kilowatt hour (kWh) may be calculated, using the previously stated costing assumptions, by To get a simpler, more usable form for the cost equation (Eq. 1), a Taylor Series multivariable approximation can be made for deviations from the reference system values for P, A, R, and delivered power D. This allows electricity costs to be written as a linear function depending only upon four microwave parameters. From Table 1, the simplified cost equation can be written
where PD = d.c. power delivered from rectenna, PT = antenna transmit power, RT = antenna radius, Rg = rectenna radius, A, = 9.6 GW mills/kWh, K2 = 22.05 mills/kWh, K3 = 174.4 GW mills/km2 kWh, K4 = 1.31 GW mills/km2kWh. Equation 3 may be used with an optimization procedure to produce cost-efficient microwave systems constrained to preselected environmental limits. ANTENNA THEORY The SPS microwave beam characteristics are found theoretically using the Fourier transform relation between the antenna electric field and the resulting E-field pattern in the Fraunhofer region. The SPS antenna may be considered a continuous, circular aperture with near-uniform phasing, beaming energy to a circular rectenna as shown in Fig. 2. The relationship between the antenna and ground electric fields is the Fourier-Bessel (Hankel) transform and the associated power densities may be stated (6):
The Hankel transform allows the ground microwave pattern to be found for radially symmetric antenna tapers, though a computer technique is usually required to accomplish the integration. Analytical attempts to solve the optimal SPS power transfer problem have so far had only limited success because of the difficulty of analytical integration (7). Expression 4 can be simplified by formulating the antenna taper as an even-powered polynomial series: where u = yRTru (argument of the Jo Bessel function), N„ = number of terms in a series approximation for Jn, where a series expression for the Bessel function has been used which is exact only for This expression is straightforward for a computer evaluation giving the ground microwave power density as a function of parameters AX-AN rH, and RT.
The power received by the SPS rectenna is found by integrating power density over the reception area: Received power is now a function of Ai - Anlc, and RT and Rc. Other micro wave beam characteristics may also be found from Eqs. 5 and 6. The power transmitted by the antenna, PT, is From the standpoint of minimizing electricity cost, it was shown previously (Eq. 3) that mills/kWh cost can be formulated as a simplified function of PG, PT, RT, and RG, with cost penalties added to constrain thermal, ionospheric, and sidelobe power densities within bounds:
By use of the equations above, cost is translated into a definite multivariable function of parameters Rr. RG, and A,—AK: The optimal SPS configuration, operating at the lowest electricity cost and predetermined environmental standards, is then found by performing a multivariable iteration to find the least cost combination of taper shapes, antenna and rectenna dimensions, transmit power, etc. METHOD OF SOLUTION («) Optimization The Davidon-Fletcher-Power (DFP) penalty method is a way of determining the local minimum of a function,/(x), where .r is an n-dimensional parameter vector, subject to vector equality and inequality constraints, g(x) = 0 and h(x) 0 (9). The method finds the minimum of a quadratic function in n one-dimensional search steps which are conjugate with respect to a matrix containing second partial derivative information. The particular DFP computer approach used is an excellent algorithm due to Johnson (10). The DFP method requires computation of (a) a penalty function which is the sum of cost function and adjoined inequality constraints as given by Eq. 10, and (b) the gradient of the penalty function with respect to the n parameters of.r, i.e., Rh RG, and At- ANK. In the SPS problem, since the cost function requires evaluating Bessel functions and a quadrature, the respective gradients are determined numerically using a central difference method. The DFP scheme then iteratively generates the approximate inverse of the Hessian matrix (initially chosen as any positive definite matrix, usually an identity matrix) in n-iterations by the accuracy of one-dimensional searches along the gradient of the penalty function which locate a directional minimum by a cubic fit or golden section approach. (b) Inequality Constraints Inequality constraints are adjoined to the cost function by positive penalty constraints. For example, consider the ionosphere inequality contraint defined in Eq. 10,
constraints on a solution has proved to be practical, efficient, and expedient. The choice of the constant C; prescribes how rigid the inequality constraint is to be adhered. A word of caution to the analyst. It is his responsibility to determined that a feasible solution exists, i.e., that contradictory inequality constraints are not imposed. (c) Converging to the Solution Given a reference value for the parameter vectorx (i.e., for RT, RG, and A, - Ak) and a method for computing the cost function which includes the inequality constraint functions h(x), a gradient vector at the reference point is computed. An accurate one-dimensional search is performed to determine a directional minimum point. Using this information and the values of the gradient vectors in two successive iterations, a correction is computed to the approximate inverse of the Hessian matrix. The parameter vector is updated by using (a) the search parameter or minimum points, (b) the approximate inverse of the Hessian matrix, and (c) the gradient vector. A new iteration is then initiated. The solution is terminated when either [1] the magnitude of the gradient vector is less than a given tolerance, [2] the difference in the function in two successive iterations is less than a given tolerance, or [3] the value of the derivative of the penalty function with respect to the one-dimensional parameter at the origin is less than a given tolerance. These conditions imply that the function cannot be further reduced with the specified constants and constraints. For condition 2, it is sometimes possible to further reduce the cost function by restarting the solution with the inverse of the approximate Hessian matrix reset to the identity matrix. ANALYSIS OF RESULTS (a) Effects of Antenna Thermal Limits The antenna thermal limitation is due to waste heat rejection by the d.c.-to-RF power converter tubes, i.e., klystrons. The present SPS guideline is 23 kW/m2 for the maximum of RF power density that can be radiated. However, subsequent investigations on the referenced system thermal radiators indicated the design was conservative and improvements in the amount of heat rejection may be possible. These improvements would be made by using graphite composite materials with a high emissivity coating for the radiators. The reduction is electricity costs as a function of increasing the antenna thermal limit (To) is shown in Fig. 3. Each point on the curve represents a unique taper optimized at its allowable thermal limit. The data indicate large cost advantages by increasing To above 23 kW/m2. The reason is that more power can be delivered since the ionospheric limit has not yet been reached for these particular configurations. The overall cost reduction per satellite for effect of a 1 -mill per kWh reduction in the electricity rate is approximately $260 million. This reduction would be taken from a total satellite cost of approximately $12.4 billion per system. (b) Effects of Sidelobe Power Density Limits The effects upon electricity costs of increasing the maximum allowable sidelobe
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