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Optimization of microwave power transmission from solar power satellites

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O p tim iz a tio n o f m icrow ave p o w er tra n s m iss io n fro m so la r p o w e r
sa te llite s
Potter, Seth D., Ph.D.
New York University, 1993
Copyright ©1993 by Potter, Seth D . All rights reserved.
UMI
300 N. ZeebRd.
Ann Aibor, MI 48106
Optimization of Microwave Power Transmission
from
Solar Power Satellites
by
Seth D. Potter
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Applied Science
New York University
May, 1993
© Seth D. Potter
All Rights Reserved 1993
Ill
To my mother, Joyce Potter,
and my brothers, Robert and Warren Potter,
and to the memory of my father, Harold Potter.
Acknowledgments
I gratefully acknowledge the help of my advisor, Professor Gabriel Miller, as
well as my dissertation committee members, Professor Martin Hoffert, who
introduced me to the SPS concept; Professor Daniel Speyer; and Dr. William
Grossmann.
I would also like to thank Dr. Murali Kadiramangalam for his
collaborative efforts; Dr. Andrew Hall Cutler for helping to bring my work to the
attention of the space engineering community; and Dr. Daniel Karron, with whom I
reached several major academic milestones.
-v-
Table of Contents
Dedication Page...........................................................................................................iv
Acknowledgments....................................................................................................... v
List of Figures............................................................................................................. vii
List of Tables............................................................................................................... ix
List of Abbreviations and Acronyms..........................................................................x
Chapter 1: Introduction to the Solar Power Satellite Concept....................................1
Chapter 2: Frequency Selection.................................................................................. 6
Chapter 3: Transmitting Antenna Geometry...............................................................27
Chapter 4: Tapered Microwave Beams....................................................................... 39
Chapter 5: Discussion and Conclusions...................................................................... 71
Appendix: Integration of theBeam Intensity Function............................................... 76
Bibliography.................................................................................................................92
Dissertation completed on Thursday, April 8,1993.
-vi-
List of Figures
Figure 2-1. Geometry of a Horizontal Transmitter and Receiver............................. 20
Figure 2-2. Microwave Beam Intensity for a Geostationary SPS with a Square
Antenna............................................................................................................21
Figure 2-3. Exclusion Zones for an SPS with a Square Antenna for 1 to 11
GHz.................................................................................................................. 22
Figure 2-4. Exclusion Zones for an SPS with a Square Antenna for 1 to 100
GHz.................................................................................................................. 23
Figure 2-5. Exclusion Zones for an SPS with a Square Antenna as a Function
of Safety Threshold......................................................................................... 24
Figure 2-6. Rain Attenuation of Microwaves.............................................................25
Figure 2-7. Atmospheric Zenith Attenuation of Microwaves................................... 26
Figure 3-1. Microwave Beam Intensity for an SPS with a Circular Antenna...........35
Figure 3-2. Exclusion Zones for an SPS with a Circular Antenna for 1 to 11
GHz.................................................................................................................. 36
Figure 3-3. Exclusion Zones for an SPS with a Circular Antenna for 1 to 100
GHz.................................................................................................................. 37
Figure 3-4. Exclusion Zones for an SPS with a Circular Antenna as a Function
of Safety Threshold......................................................................................... 38
Figure 4-1. Intensity of Tapered Microwave Beams at a Circular Transmitting
Antenna............................................................................................................62
Figure 4-2. Received Intensity of Tapered Microwave Beams from a Circular
Transmitting Antenna.......................................................................................63
Figure 4-3. Comparison of Sidelobes for the n = 2 and "Ideal" Beam Tapers..........64
Figure 4-4. Exclusion Zones for Tapered Beams for a Safety Threshold of
0.01 mW/cm2 ...................................................................................................65
Figure 4-5. Exclusion Zones for Tapered Beams for a Safety Threshold of 0.1
mW/cm2........................................................................................................... 66
Figure 4-6. Exclusion Zones for Tapered Beams for a Safety Threshold of 1.0
mW/cm2........................................................................................................... 67
Figure 4-7. Exclusion Zones for Tapered Beams for a Safety Threshold of 5.0
mW/cm2........................................................................................................... 68
Figure 4-8. Exclusion Zones for Tapered Beams for a Safety Threshold of
10.0 mW/cm2...................................................................................................69
Figure 4-9. Main Lobe Radius for Tapered Beams....................................................70
List of Tables
Table 4-1. Characteristics of Tapered Beams............................................................59
Table 4-2. Dependence of Exclusion Zone Radius on Safety Threshold and
Frequency......................................................................................................... 60
Table 4-3. Total Relative Areas for Tapered Beams at 9.8 GHz.............................. 61
-ix-
List of Abbreviations and Acronyms
°C = degrees Celsius
cm = centimeters
CO2 = carbon dioxide
DOE = Department of Energy
EMC = electromagnetic compatibility
EMI = electromagnetic interference
GHz = gigahertz
GW = gigawatts
hr = hour
km = kilometers
kW = kilowatts
m = meters
MW = megawatts
mW = milliwatts
NASA = National Aeronautics and Space Administration
NYU = New York University
SPS = Solar Power Satellite
W = watts
-x-
Chapter 1:
Introduction to the Solar Power Satellite Concept
In recent years, there has been a growing concern about the environment and
global warming among both scientists and the general public. Studies undertaken at
New York University (Hoffert, et al., 1991) and elsewhere indicate that the increase in
man-made carbon dioxide levels in the Earth's atmosphere since the industrial
revolution may cause the Earth's temperature to rise. The Earth has warmed by about
0.3°C to 0.6°C over the last century (Houghton, et al., 1990, page xxviii), although it
is not yet known if this is due to anthropogenic CO2 emissions. Energy is required to
produce wealth, and a growing world population will need increasing amounts of
energy if it is to improve its standard of living. This will lead to an increase in CO2
emissions if this energy is produced through fossil fuel combustion. However, studies
at NYU (Hoffert, et al., 1991) and at Britain's University of East Anglia (Wigley,
1990) indicate that if global warming is to be avoided, fossil fuel use must immediately
be reduced by about two-thirds, and must then continue to decrease over the next
century. More realistic (yet still rather difficult to achieve) scenarios, in which fossil
fuel energy production increases, but at a rate less than that of previous years, may lead
to a global warming of perhaps one or two degrees Celsius per century.
The possibility of global warming has led to an interest in non-fossil fuel energy
sources. Concerns about safety and nuclear waste, as well as security are major
constraints on fission power. In addition, the supply of fissionable material is limited.
This limitation can be circumvented by the use o f breeder reactors (Waltar and
Reynolds, 1981), which produce more fissionable material than they use. This would
require a tremendous commerce in fissionable material, with its associated security
risks. In addition, a breeder reactor program similar to the French may already have to
be in place globally to avoid a depletion of the world’s supply of uranium-235, unless
both burner and breeder reactors are improved (International Institute for Applied
Systems Analysis, 1981, pages 55-56). Fusion is often suggested as a safer, lowwaste form of nuclear power. However, Furth (1990) predicts that a commercially
feasible fusion reactor will not come on line until perhaps the year 2020, and could be
the largest worldwide research collaboration ever attempted.
The problems of fossil fuel energy production, as well as those of other sources
of power make solar energy an attractive alternative, since it requires no fuel. The
intensity of sunlight in near-Earth space is 1353 watts per square meter (Duffie and
Beckman, 1980, page 4). However, much less energy is available at the Earth's
surface. The day-night cycle and the oblique angle of the sun’s rays over most of the
Earth's surface, even at midday, are major limiting factors. This can be summarized by
recalling that for a sphere (such as the Earth), the ratio of cross sectional area to total
surface area is 1/4. Thus, the sunlight available at the Earth's surface, averaged over
latitude and time, is, at most, 1/4 of that which is available in space. Furthermore,
some of the sunlight incident on the top of the atmosphere is absorbed or reflected back
into space by air and clouds. Therefore, there is typically about eight times as much
solar energy available in near-Earth space than there is at the Earth's surface (O'Neill,
1989, page 175). The limits on terrestrial solar energy led Czech-American engineer
Dr. Peter Glaser to propose in 1968 (Science, Vol. 162) that solar energy be harnessed
by means of devices known as a Solar Power Satellites (SPS's). An SPS is a large
solar collector placed in geostationary orbit 35,786 kilometers above the equator. The
solar energy is converted to electricity and the electricity is then converted to
microwaves, which are beamed to the Earth's surface. There, the microwaves are
received and converted back into electricity by a device called a rectenna (or rectifying
antenna). The electricity is then converted to a form compatible with terrestrial power
grids and distributed to users. Such a system produces no greenhouse gases or nuclear
waste. Locating the solar collectors in space allows them to receive solar energy almost
continuously. Unlike terrestrial solar collectors, the rectennas would receive power day
and night, in clouds and in sunlight The elements of the system would be huge, with a
single SPS measuring perhaps 5 x 10 kilometers, having a mass of 34,000 to 51,000
tonnes, and beaming up to 5 gigawatts of power to a rectenna that is typically a 10 x 13
kilometer ellipse. These figures are the result of a NASA/US Department of Energy
study completed in 1980 (see The Final Proceedings o f the Solar Power Satellite
Program Review). Since then, smaller, lighter designs have been proposed.
An important aspect of SPS design is the transmission of energy from the
satellites to the Earth. The NASA/US DOE reference design utilized microwave power
transmission at a frequency of 2.45 GHz. Frequencies of a few gigahertz can pass
through the atmosphere relatively unimpeded. The 2.45 GHz frequency, which can
pass through rain and clouds, has been set aside for industrial and scientific use, and is
the frequency used in microwave ovens. It is therefore the most well-understood
frequency, with equipment such as magnetron tubes readily available. The use of this
frequency constrained the SPS reference design, since the laws of diffraction cause a
microwave beam to spread when it is transmitted from geostationary orbit. Since a
transmitting antenna one kilometer wide results in a main beam lobe several kilometers
wide, it is not economical to build an SPS that supplies less than several gigawatts.
This is the reason for the large size of the NASA/US DOE reference design. Recently,
other frequencies have come under consideration. Although the atmosphere becomes
less transparent as the frequency is increased, there are atmospheric "windows" at
higher frequencies, notably 35 and 94 GHz. Higher frequencies would allow for
smaller transmitting antennas, smaller rectennas, and thus, smaller, more feasible, SPS
units. Rainfall may attenuate some of the higher frequencies, so this may complicate an
SPS-based power system. A provision for redirecting the microwave beam to a
different rectenna location may be needed for rainy days. In addition to microwave
beams, lasers have been suggested for power transmission at infrared or optical
frequencies. A laser beam would spread out much less than a microwave beam. At the
Earth's surface, the energy would be converted to electricity by photovoltaic cells.
However, laser beams can be blocked by clouds, and may be hazardous to birds,
aircraft, or anything else in their path.
The financial and environmental costs of launching large amounts of material
into space for SPS construction prevented the large-scale continuation of SPS research
after the NASA/US DOE study. Since then, studies have been done on the use of lunar
materials for SPS construction (Space Studies Institute, 1985). Launching material
from the Moon requires less than one-twentieth the amount of energy required to launch
the same amount of material from the Earth (Maryniak, 1991). Material from the Moon
could be propelled into space by an electromagnetic launcher and then processed at a
solar-powered facility.
As we have seen, the spreading of the power beam is responsible for the large
size of the SPS. The size, in turn, may cause a need for a lunar infrastructure. Since
building such an infrastructure could take more time than the global warming problem
may allow, it is worth investigating whether or not the beam divergence can be
minimized without concentrating it to dangerous levels. A narrower beam may make
-4-
smaller, lower power SPS's feasible, enabling them to be launched from the Earth. In
addition, the sidelobes of the beam contain microwave energy that is at too low a level
to collect economically, but may still be hazardous to surrounding populations. Thus,
the need to fence off large areas of land in the vicinity of the rectennas is another cost of
building SPS's. Such areas of land are known as exclusion zones. Minimizing their
size will make SPS’s more feasible, since rectennas may need to be placed near
populated areas to avoid transmission losses. Subsequent chapters will thus deal with
power beaming issues, and the effect of power transmission on the SPS system design.
Chapter 2:
Frequency Selection
Introduction
If solar power satellites (SPS) are to provide a significant portion of the world's
energy needs, a safe and efficient method of transmitting the power to the Earth must be
employed. The NASA/US Department of Energy SPS reference design (1980) utilized
microwave power transmission at a frequency of 2.45 gigahertz. As seen in Chapter 1,
this frequency passes through rain and air with little attenuation, and has been set aside
for industrial and scientific use. If frequencies much lower were used, the transmitting
antenna and/or the rectenna would become unpractically large, due to the diffraction of
the beam. In addition, lower frequencies would interfere with communication. The use
of the 2.45 GHz frequency constrained the SPS reference design, since the beam would
be perhaps 10 kilometers wide at the Earth's surface. Since it is uneconomical to use
this much land to receive small amounts of power, the reference SPS was designed to
supply 5 gigawatts. Recently, higher frequencies have come under consideration,
particularly at the atmospheric "windows" of 35 and 94 GHz. Technology for these
frequencies has therefore been under study in recent papers on power beaming (e.g.,
Koert, et al., 1991).
The width of the main beam lobe is directly proportional to the wavelength (and
thus, inversely proportional to the frequency), so the area occupied by a rectenna
(assuming it is approximately the size of the main lobe, where most of the energy is
concentrated) varies as the inverse of the square of the frequency. It is therefore
tempting to use higher frequencies to save land, even at the cost of some beam
attenuation. The presence of sidelobes may make it necessary to fence off an area
which extends beyond the rectenna in order to protect people from levels of microwave
radiation which are not economical to collect and rectify, but which may pose a hazard
to human health. It is easy to extrapolate that since the first minimum moves inward
with increasing frequency, subsequent maxima may do so as well. This suggests that
the exclusion zone size might decrease with increasing frequency. This makes the use
of higher frequencies appear more desirable. In this chapter, the effect of frequency
increase on exclusion zone size is investigated. For consistency, this assessment
involves microwave power transmission from a geostationary SPS with a square
transmitting antenna array beaming power to a rectenna at the equator, unless otherwise
noted.
Microwave Beam Exclusion Zones
As the frequency of the microwave beam increases, the main lobe becomes
narrower and the peak power level becomes higher. Furthermore, a similar process
affects the secondary maxima. Each peak becomes higher and moves closer to the
center as the frequency is increased. An increase in frequency will cause a potentially
dangerous sidelobe to move inward, but an outer sidelobe may increase in intensity,
undoing this effect. If human beings are to be protected from the effects of the
secondary maxima, we must calculate which of these processes will predominate. In
addition, a standard for microwave exposure must be agreed upon. Standards range
from 0.01 mW/cm2 (Eastern Europe [Lehman and Canough, 1991; and Vondrak,
1987]) to 1 mW/cm2 (Canada [Glaser, 1986]) to 10 mW/cm2 (US and Western Europe
[Glaser, 1986]). The US and Western European standard is based mainly on thermal
biological effects (Glaser, 1986), while the Eastern European standard was established
to avoid neurological effects (Vondrak, 1987). Since there is not yet a general
agreement on the existence of, and safety threshold for, nonthermal effects, more
research must be done in this area. The recommendation of a microwave safety
standard is beyond the scope of this chapter, so we will consider several different
standards in order to show the relationship between the standard and the exclusion zone
size.
To calculate the exclusion zone size, we must first consider the diffractionlimited intensity of a microwave beam. The transmitter under consideration is a square
phased array of side Dt = 1 km, at geostationary altitude h = 35,786 km, with N X N
isotropically radiating elements uniformly spaced less than one-half wavelength apart
Thus, N > 2DtA, where X is wavelength, and N » 20,000 at a frequency of 2.45 GHz.
The microwave beam intensity at the rectenna, normalized to the boresight peak I0 is
expressible as the product of the x and y distributions. Two-dimensional intensity is
given by (Hoffert, et al., 1989):
2
2
A
. ny
m -f—
>
(Equation 2-la)
l Nsin5N .
Since X and y are small compared to N, we make the approximation that:
-8-
Thus:
/s
sin
I(x ,y ) =
TCl
ny
sin2
TCI
2
.
(Equation 2-lb)
KJL
2
where
I = ---------------- - -------- *------(A.h)/(2Dt cos0o cos 0XO )
(Equation 2-2a)
V = --------------- —-------- *------(A,h)/(2Dt cos0Q cos 0yO )
(Equation 2-2b)
and
x, y = distances in from the center of the diffraction pattern at the Earth's surface in the
x and y directions.
-9-
The angles refer to Figure 2-1. These equations were derived for Earth-to-space
microwave power transmission (left side of Figure 2-1 - from Hoffert, et al., 1989),
but they apply equally well to space-to-Earth microwave power transmission (right side
o f Figure 2-1 - from Kadiramangalam, 1990).
Since we are considering a
geostationary SPS beaming power to a rectenna at the equator, we can let 0O - ®xo ~
0yo = 0. Thus,
X = -----
(Equation 2-3a)
y = -— ^----Xh/(2Dt )
(Equation 2-3b)
\h/(2D t )
By integrating Equation 2-lb in two dimensions from -oo to oo, it is seen that the peak
beam intensity is given by:
2
p t f2D t 1
° = T" lThJ
where
(Equation 2-4)
is the total transmitted power. Beam intensity at the rectenna is plotted in
Figure 2-2 for a frequency of 2.45 GHz, and a total transmitted power of 5 GW. Note
that the main lobe extends for roughly 4400 meters on either side of the peak. This
peak reaches an intensity of 26 mW/cm2. These results are similar to the NASA/US
DOE SPS reference design (1980), which used a 10 X 13 kilometer rectenna, and had a
peak power of only 23 mW/cm2. However, the reference design beamed power to a
higher latitude, with a slightly smaller transmitting antenna, so the power was "smeared
out."
To find the approximate exclusion zone around the rectenna, the troughs
between the maxima must be "filled in," and the maxima themselves "connected
together." This can be done by finding the upper bound of the one-dimensional
intensity, which occurs when the sine in the numerator of Equation 2 -lb is set equal to
1. Therefore, in one dimension:
Substituting Equations 2-3a and 2-4 gives
Now if we assume I(x) to be the safety threshold Is, we can rearrange the above
equation to obtain the exclusion zone size xs, which is given by:
(Equation 2-5)
Note that xs is independent of frequency. Thus, for a given total power level, and a
given safety standard for microwaves, the size of the exclusion zone does not depend
on frequency. There is therefore no advantage in going to higher frequencies as far as
exclusion zone boundaries are concerned.
As a check, an independent computer calculation of exclusion zone size was
performed, using Equation 2-la in one dimension. Safety thresholds of 0.1, 1, and 10
mW/cm2 were examined for two different frequency ranges. The distance from the
center to the first minimum was also plotted. Figures 2-3 and 2-4 show the results of
these calculations. The zigzags in Figure 2-3 are due to the fact that as the frequency is
increased, a maximum which exceeds the safety threshold will move slowly inward
until the next outer maximum suddenly exceeds the threshold. As the frequency is
further increased, the distance between successive peaks decreases, dampening the
zigzags until the peaks themselves form the diffraction pattern for all practical purposes.
This is the case for the higher frequencies seen in Figure 2-4; it is also the situation
anticipated by Equation 2-5.
From Figure 2-4, it can be seen that the exclusion zone size is much greater than
the size of the main lobe (which is assumed to be the approximate size of an economical
rectenna) for all but the lowest frequencies. This is especially true for the 0.1 mW/cm2
threshold. This may not be a serious problem if concentrations of microwaves beyond
those acceptable for people are considered acceptable for plant and animal life. Small
rectennas could then be located in unpopulated areas, though transmission losses then
become a problem. However, these results suggest that the Eastern European safety
standard (stricter than the standards shown in Figures 2-3 and 2-4) may be unrealistic
for large amounts of transmitted power.
The relationship between exclusion zone size and safety threshold for 2.45 GHz
and 35 GHz is shown in Figure 2-5. The curve for 35 GHz is near the limiting case for
higher frequencies and has the shape expected from Equation 2-5. The curve for 2.45
GHz resembles one side of a diffraction pattern laying on its side with the troughs filled
in. As the frequency is increased, the plot approaches the shape of the plot for 35 GHz.
There is no noticeable change in the plot beyond 35 GHz.
The above analysis considers a total transmitted power of 5 GW for all
frequencies. However, for higher frequencies, the peak intensities may reach levels
that are unsafe for the environment. For example, the 26 mW/cm2 peak which occurs
at 2.45 GHz scales to 5320 mW/cm2 at 35 GHz and 38,400 mW/cm2 at 94 GHz.
These levels are many times the solar constant, and may be environmentally unsafe.
However, if the total power is decreased, the peak will remain low. This suggests that
smaller SPS units could be designed around the higher frequencies. This might be a
useful way to supply power to small, remote villages, or even to squeeze a rectenna into
a crowded urban area. In addition, the smaller SPS's may be easier to finance.
However, as a national or global power system, there would be no net savings in land,
and there would be added complexity, since many more SPS units and rectennas would
be needed at higher frequencies. Furthermore, atmospheric and rain attenuation may
make the higher frequencies less desirable than lower frequencies.
Rain Attenuation of Microwave Beams
The attenuation of a microwave beam due to rainfall is dependent upon
frequency, rainfall rate, and path length through the medium. The attenuation per unit
path length is given by (Johnson and Jasik, 1984, page 45-8):
Y(R) = aRb
(Equation 2-6)
where
R = surface-point rain rate in mm/hr;
a, b = constants;
and y is in dB/km.
To find the attenuation, A, in dB, y(R) must be multiplied by the equivalent path length,
Le(R), in rain at rain rate R. Thus,
A = Le(R) y(R)
(Equation 2-7)
The value of Le depends on the thunderstorm ratio; that is, the fraction of mean annual
rainfall accumulation from R > 25 mm/hr. Johnson and Jasik (1984, page 45-10) gives
the following formula for effective path length, based on data from western Europe and
eastern North America:
Lei(R,0) = [0.00741R0-766 + (0.232 - 0.00018R) sin 0 ]'1
(Equation 2-8)
where 0 is the elevation angle (this is the complement of 0O in Figure 2-1). For a
rectenna at the equator directly below the SPS, 0 = 90°. Johnson and Jasik (1984,
page 45-10, Figure 45-6) also shows a family of curves for effective path length based
on data from Japan. The curves show an effective path length, called Le2» for several
thunderstorm ratios, as a function of elevation angle. The curves for the various
thunderstorm ratios converge at Le2 = 5 km when the elevation angle reaches 90°.
Johnson and Jasik suggests using L e2 for frequencies less than 10 GHz and Le = 0.75
L e2 + 0.25 Lei for frequencies greater than 10 GHz.
Since there is a slight
discontinuity at 10 GHz, a function, based on the error function, was used that spread
-14-
the step over approximately 8 to 12 GHz. The constants a and b were obtained from
Brookner (1977, page 39, Figure 29).
It must be kept in mind that although Equations 2-6 through 2-8 provide a
straightforward means of calculating attenuation in dB, this is a logarithmic result.
Such a result is more meaningful in radar and communications theory than it is for
energy analysis. For example, a 6 dB loss may not be too significant in transmitting
information, but it does represent a loss of 75%, which is quite significant in designing
a power system. Therefore, the attenuation in dB was converted into a loss factor, and
then into a percentage of transmitted power reaching the ground from an SPS. For
example, a loss of 20 dB translates into a loss factor of 100, which yields only 1% of
the transmitted power reaching the ground. The results for various rainfall rates are
shown in Figure 2-6. A rainfall rate of 5 mm/hr is a typical temperate zone rate, while
25 mm/hr represents a thunderstorm. It is seen that even in a 1 mm/hr drizzle, the 94
GHz window is blocked. At 5 mm/hr, the 35 GHz window is about 70% blocked.
Frequencies below about 10 to 15 GHz must be used if power is to be transmitted
through rainfall. Assuming that the sidelobe and peak power safety and environmental
issues raised earlier are solved, the higher frequencies can be used either in areas where
there is little rainfall, or for applications where power can be stored and need not be
supplied continuously. However, this statement is somewhat optimistic, since there is
attenuation even in clear air.
Clear Air Tropospheric Attenuation of Microwave Beams
The choice of microwave frequency in SPS applications is often made on the
basis of the existence of atmospheric windows. The troposphere has H2 O and O 2
absorption resonances at 22.2 GHz and 60 GHz, respectively (Brookner, 1977, page
39, Figure 28a). Between these resonances lies the 35 GHz transmission window.
Beyond them lies the 94 GHz window. 'Hie "traditional" frequency of 2.45 GHz lies in
a window in the sense that tropospheric loss is very little for frequencies below about
15 GHz. Data for two-way transit of the entire atmosphere for a 90° elevation angle
were obtained from Brookner (1977, page 39, Figure 28a). The attenuation was
divided by 2, since we are dealing with one-way transit of the atmosphere, and the data
then converted from dB to percentage of transmitted power reaching the ground, and
shown in Figure 2-7. Note that the atmosphere is essentially opaque at the 60 GHz O2
absorption resonance.
The 22.2 GHz H2 O absorption resonance is much less
significant, and, in fact, presents less of a loss than the 94 GHz "window;" however,
22.2 GHz is still not an ideal choice of frequency.
For frequencies of 15 GHz or less, the atmospheric absorption is 2% or less.
At 2.45 GHz, it is only 0.8%. At 35 GHz, it is approximately 5%. At 94 GHz, it is
approximately 18%. These figures are for zenith attenuation, the situation that exists
for an SPS that is directly overhead, at the equator. For higher latitudes, say 41° (i.e.,
New York), the beam passes through the atmosphere at an oblique angle, and must
therefore travel through about 1.46 times as much atmosphere as it would at the
equator. In this case, the percentage absorption for the above three frequencies
becomes 1.2%, 7.2%, and 25% for 2.45, 35, and 94 GHz, respectively.
It is
instructive to compare the amount of microwave energy absorbed by the atmosphere
with the solar constant, which is 1353 W/m2, or 135.3 mW/cm2 for near-Earth space
(Duffie and Beckman, 1980, page 4). As seen in Chapter 1, the ratio of the Earth's
cross-sectional area to its total surface area is 1/4; therefore, the solar constant must be
divided by 4 to yield the solar flux incident on the top of the Earth's atmosphere for a
typical latitude, averaged over the day-night cycle. This flux is 33.83 mW/cm2. (It is
necessary to average the solar constant over the day-night cycle in order to make a valid
comparison with microwave beams since an SPS "shines" microwaves day and night.)
If the peak intensity of a microwave beam is held to the 23 mW/cm2 value of the
NASA/US DOE reference design, then the peak amount absorbed at the above three
frequencies at 41° latitude is 0.28,1.7, and 5.8 mW/cm2, respectively. This last figure
is about 17% of the average solar constant. Although the environmental significance of
this may not be any worse than the thermal pollution from a large conventional power
plant, maintaining the peak beam intensity of 23 mW/cm2, while increasing the
frequency to 94 GHz means deploying many small SPS units instead of fewer, larger
ones. Thus, the same amount of land is being utilized as with 2.45 GHz, but the
thermal pollution is 21 times greater. Furthermore, if the 5 GW power level of the
reference design is squeezed into a smaller area through the use of 35 or 94 GHz, then
the 23 mW/cm2 peak scales to 4700 mW/cm2 (for 35 GHz) and 34,000 mW/cm2 (for
94 GHz). The actual amount of microwave energy absorbed by the atmosphere at these
frequencies, for 41° latitude, is 340 mW/cm2 and 8500 mW/cm2 for 35 and 94 GHz,
respectively.
These figures are 10 and 250 times the average solar constant,
respectively. It is likely that this much atmospheric heating, although rather localized,
will damage the environment, especially if many such SPS units are deployed. Since
frequencies just above 2.45 GHz experience little atmospheric (or rain) attenuation, the
above analysis is repeated here for 10 GHz. At that frequency, only 1.1% of the
energy is absorbed by the atmosphere for an elevation angle of 90°. This translates to
1.6% absorption for a latitude of 41°. The 23 mW/cm2 peak scales to 383 mW/cm2 if
the 5 GW total power is maintained. The peak absorption is 383 mW/cm2 x 1.6%, or
6.1 mW/cm2, or 18% of the average solar constant. This is roughly the same as for the
94 GHz, 23 mW/cm2 case; however, unlike that case, there would be a 17-fold savings
in land area compared with the 2.45 GHz case. The atmospheric absorption of the 10
GHz case is 1.6% instead of 1.2%, while using only l/17th of the land area as
compared with the 2.45 GHz case. Therefore, a frequency of approximately 10 to 15
GHz may represent the best compromise between land use, transmission efficiency,
and avoidance of thermal pollution. However, the sidelobes may still prove hazardous
to surrounding populations, unless they can be minimized through the use of circular
transmitting apertures, and/or beam tapering.
Conclusions
To a first-order approximation, exclusion zone size is independent of frequency
for square transmitting antennas, which implies that there is no savings in land area
when frequency is increased, although the rectenna size becomes smaller. If the
frequency is increased beyond 10 to 15 GHz, then rain attenuation becomes significant
At 35 GHz, rain attenuation is significant in a typical temperate zone rainfall of 5
mm/hr, and at 94 GHz, rain attenuation is significant even in a drizzle of 1 mm/hr. At
frequencies above 15 GHz, significant heating of the atmosphere will occur. To avoid
atmospheric heating at higher frequencies, the total transmitted power per SPS must be
decreased. Since more SPS units will be needed to supply the same amount of power
as the 2.45 GHz case, there will be no net savings in rectenna area. However there will
be more atmospheric absorption than for the 2.45 GHz case, and, in fact, the total
amount of energy absorbed by the atmosphere for many small high frequency SPS
units will be the same as for a few large high frequency SPS units.
In conclusion, 2.45 GHz so far appears to be the preferred frequency for spaceto-Earth power transmission. If the sidelobes can be minimized, then the general
puipose SPS microwave frequency should be increased from 2.45 GHz to roughly 10
to 15 GHz (the exact frequency to be determined by electromagnetic
interference/compatibility requirements).
The non-thermal health effects, and the thermal and non-thermal environmental
effects of microwaves should be further studied.
Higher frequencies, such as 35 and 94 GHz, should not be considered for
general-purpose SPS power transmission; however, these frequencies may have a role
to play in small SPS units designed to deliver power to remote areas, or to crowded
locations. This is especially true for areas with little rainfall, or for situations where
energy storage is feasible. In addition, higher frequencies may have a role in space-tospace power transmission, as well as for terrestrial power transmission (e.g., across
valleys, or to unmanned aircraft).
xo
xo
Ae
Ae
Ae
y
FIGURE 2-1. Geometry of a horizontal transmitter and receiver during a flyby. The
left side (from Hoffert, et al., 1989) shows the geometry for Earth-to-satellite power
transmission, while the right side (from Kadiramangalam, 1990) shows satellite-toEarth (SPS) power transmission. The spherical coordinates r, 0 and <J) are related to
the Cartesian coordinates measured from the center of the transmitter by x’ —
r sin0 cos4> “ rsin0x, y' = r-sin0-sin<J> = r-sin0y and h = rcosO, where 0X =
arcsin(x'/r) and 0y = arcsin(y'/r) are the angles between the x' = 0 and y1= 0 planes
and the range vector r measured in the planes containing r. Subscript zero denotes the
boresight direction, and A 0X, A 0y are the angular deviations in the rx and ry planes
from boresight.
-20-
Intensity (mW/cmA2)
30
20
10
0
-20
-10
0
10
20
Distance from Center (km)
FIGURE 2-2. Microwave beam intensity at the Earth's surface for a geostationary SPS
beaming 5 gigawatts of power from a square antenna with 1000 meter sides to the
equator at a frequency of 2.45 GHz.
30
0.1 mW/cmA2 zone
1 mW/cmA2zone
10 mW/cmA2 zone
Q --
1
FIGURE 2-3.
*
.
1
.
1
.
I .
3
5
7
Frequency (GHz)
I .
9
11
Microwave beam exclusion zones at the Earth's surface for a
geostationary SPS beaming 5 gigawatts of power from a square antenna with 1000
meter sides to the equator for frequencies of 1 to 11 GHz. Exclusion zones for 0.1, 1,
and 10 mW/cm2 safety thresholds are shown, as well as the location of the first
diffraction minimum.
30
0.1 mW/cmA2 zone
<3
■4— »
G
0>
U
<2
<D
O
G
34
•4(7
—
C /3
20
10
1 mW/cmA2 zone
faUjliyW IM W M W
r
First min
. n/
......................................................
0
FIGURE 2-4.
0
20
10 mW/cmA2 zone
■ k lU
40
60
80
Frequency (GHz)
100
Microwave beam exclusion zones at the Earth's surface for a
geostationary SPS beaming 5 gigawatts of power from a square antenna with 1000
meter sides to the equator for frequencies of 1 to 100 GHz. Exclusion zones for 0.1,1,
and 10 mW/cm2 safety thresholds are shown, as well as the location of the first
diffraction minimum.
24
Vh
&
a
16
CD
u
s
«g
CD
o
GHz
8
s
4-»
CO
2.45 GHz
0
0
FIGURE 2-5.
2
4
6
8
Safety Threshold (mW/cmA2)
10
Microwave beam exclusion zones at the Earth's surface for a
geostationary SPS beaming 5 gigawatts of power from a square antenna with 1000
meter sides to the equator are shown as a function of safety threshold for 2.45 and 35
GHz.
-24-
100
1 mm/hr
5 mm/hr
10 mm/hr
25 mm/hr
50 mm/hr
100
Frequency (GHz)
FIGURE 2-6. Rain-induced zenith attenuation of microwaves for various rainfall rates.
Based on data from Brookner (1977), and Johnson and Jasik (1984)
-25-
0
20
40
60
80
100
Frequency (GHz)
FIGURE 2-7. Atmospheric zenith attenuation of microwaves. Based on data from
Brookner (1977).
-26-
Chapter 3:
Transmitting Antenna Geometry
Introduction
In order to maximize the safety and efficiency of microwave power transmission
from a solar power satellite to the Earth, the shape of the satellite's transmitting antenna
must be optimized. The previous chapter, as well as previous work at New York
University (Potter and Kadiramangalam, 1991; and Hoffert, et al., 1989) considered a
square transmitting antenna. In particular, Hoffert, et al. (1989) dealt with Earth-tospace power transmission using square antennas as a means of supplying power to
satellites in low Earth orbit In such a situation, rectenna sizes are small, and there is no
population near the rectenna that may be exposed to beam sidelobes. Thus, transmitting
antenna geometry is not a major issue. For both square and circular antennas, the width
of the main beam lobe, and thus, the width of an economical rectenna, varies as the
inverse of the frequency. In the previous chapter, it was shown that, to a good
approximation, the size of the exclusion zone needed to protect people from microwave
exposure is independent of frequency for a square transmitting antenna. Furthermore,
rain and atmospheric attenuation become significant for frequencies above 10 to 15
GHz. This was considered to be the maximum usable frequency range. Since many
SPS studies have considered circular transmitting antennas (e.g., US Department of
Energy, 1980), it is instructive to examine such antennas in a manner analogous to
Chapter 2.
Square Transmitting Antennas
It was shown that for a square phased array of side Dt = 1 km, at geostationary
altitude h = 35,786 km, with N X N isotropically radiating elements uniformly spaced
less than one-half wavelength apart beaming power to a rectenna at the equator, the
microwave beam intensity at the rectenna, with boresight peak Iq, is expressible as the
product of the x and y distributions, as shown in Equations 2-la and 2 -lb. By
integrating the latter equation in two dimensions from -oo to oo, it was seen that
2
2
(Equation 3-1)
where P{ is the total transmitted power and At = Dt2 = transmitting antenna area. If
Equation 2-lb is integrated in two dimensions over the area of the main lobe and
divided by the integral over all (two-dimensional) space, it is seen that 81.5% of the
energy of the main lobe is in the central maximum. The remaining 18.5% of the energy
is spread out in the form of sidelobes. These sidelobes represent energy that is spread
too thinly to be economically rectifiable, but, as seen in Chapter 2, may be a hazard to
surrounding populations. For Pt = 5 GW, this wasted energy is 925 MW. It is
instructive to investigate if these figures can be improved upon with the use of a circular
transmitting antenna. In addition, the independence of exclusion zone width with
frequency discussed in Chapter 2 is a somewhat counterintuitive result. If the exclusion
zone size decreases significantly with frequency for a circular antenna, then it may pay
to increase the frequency beyond 10 to 15 GHz, despite the atmospheric and rain
attenuation that occurs at higher frequencies.
-28-
Circular Transmitting Antennas
The intensity of a microwave beam transmitted from a circular antenna array is
given by (Bom and Wolf, 1980, page 396; and Johnson and Jasik, 1984, page 20-22):
2
(Equation 3-2)
where Jj is the first order Bessel function of the first kind, and r is the dimensionless
distance from the center of the diffraction pattern and is analogous to X and y given by
Equations 2-3a and 2-3b. Thus,
(Equation 3-3)
and r = radial distance from the center of the diffraction pattern at the earth's surface.
Here, Dt is the diameter of the circular transmitting antenna. Equation 3-2 is a twodimensional intensity, since, for a transmitter directly overhead, the diffraction pattern
is azimuthally symmetric.
By integrating Equation 3-2, it is seen that
2
(Equation 3-4)
Here, Dt and At are, respectively, the diameter and area of the circular transmitting
antenna, and At = 7tDt2/4. By comparing Equations 3-1 and 3-4, it is seen that, for
square and circular transmitting antennas of equal areas, equal peak beam intensities
will result. Thus, to facilitate a comparison of antenna geometries, calculations will be
-29-
done for a consistent value of At, specifically, 1 square kilometer (used in Chapter 2).
Therefore, Dt = 1000 meters for the square antenna and 1128.38 meters for the circular
antenna. The beam intensity at the earth's surface for a circular antenna of this size
transmitting 5 GW of power at a frequency of 2.45 GHz is plotted in Figure 3-1. The
peak beam intensity is 26 mW/cm2. In order to investigate the dependence of exclusion
zone size on frequency for a circular transmitting antenna, an approximation to the
Bessel function can be used. For large u (Kreyszig, 1983, page 177),
Jl(u) * - — --_[cos(u) - sin(u)]
(Equation 3-5)
Thus:
Beam intensity
■pir} "
t1' 2cos(u) sin(u)]
(Equation 3-6)
In order to eliminate the oscillating term in square brackets on the right side of Equation
3-6, its upper bound will be used. Since the upper bound is 2, Equation 3-6 becomes:
(Equation 3-7)
The dimensionless argument of Equation 3-6 is given by:
(Equation 3-8)
Thus,
(Equation 3-9)
-30-
Substituting Equations 3-4 and 3-8 into Equation 3-9 gives:
I(r) - 2 , P | , ^ h
7t3 r3 D t
(Equation 3-10)
If c is the speed o f light and f is the frequency, then X, = c/f. I(r) can be set equal to the
microwave safety threshold Is, and r can be set equal to the exclusion zone radius rs. If
these substitutions are made in Equation 3-10, and the equation is rearranged, then the
approximate radius of an exclusion zone for a circular transmitting antenna is given by:
1/3
1
it
(Equation 3-11)
Thus, for a circular transmitting antenna, the radius of the exclusion zone rs is
proportional to the frequency to the minus one-third power.
The exclusion zone plots shown in Figures 3-2, 3-3, and 3-4 were done using
Equation 3-2; however, Equation 3-11 yields values at or near the upper bounds of the
curves for Figures 3-2 and 3-3, and provides a reasonable estimate for Figure 3-4. By
comparing Figures 3-2,3-3, and 3-4 with Figures 2-3, 2-4, and 2-5, it can be seen that
at a given frequency and microwave safety threshold, the circular transmitting antenna
yields a smaller exclusion zone than the square transmitting antenna. Furthermore,
exclusion zone size does depend on frequency for a circular antenna, though the
dependence is not strong. Thus, circular antennas represent an improvement over
square antennas. With the use of a circular antenna, it would be advantageous to
increase the frequency from 2.45 GHz to approximately 10 GHz. The frequency
should not be increased significantly more than this, because the rain attenuation and
atmospheric heating problems that affect the square antenna system would have the
same effect on the circular antenna system. The exclusion zone decrease would not be
worth the price paid in attenuation and heating. For example, for At = 1 km2, Pt = 5
GW, and Is= 1 mW/cm2, the square antenna exclusion zone half-width is 7118 m for
all frequencies; its area is thus (2x7118 m)2 or 203 km2. The exclusion zone area for
a circular antenna with the same parameters is given by 7trs2, where rs is given by
Equation 3-11. This area is 79 km2 at 2.45 GHz, 31 km2 at 9.8 GHz, 13 km2 at 35
GHz, and 7 km2 at 94 GHz. The latter two frequencies are subject to significant rain
attenuation, as well as some atmospheric heating, as seen in Chapter 2. This is true
regardless of antenna geometry, since square and circular transmitting antennas of equal
areas, wavelengths, and total transmitted powers have the same peak beam intensity
(Equations 3-1 and 3-4).
By comparing the integral of the circular antenna pattern over the main lobe with
the integral over all two-dimensional space, it is seen that 83.8% of the energy is
concentrated in the main lobe (Bom and Wolf, 1980, page 398). Since this figure is
81.5% for the square aperture, this represents another advantage of using a circular
antenna. This 2.3% difference may not seem like much, but it amounts to 115 MW if
the total transmitted power is 5 GW. One hundred fifteen MW of additional energy
distributed among the sidelobes is what makes the square aperture exclusion zones so
much larger than those of the circular aperture. Furthermore, the size of the main lobe
can be found by letting X = y = 2 in Equations 2-3a and 2-3b, and r = 2.440 (Bom
and Wolf, 1980, page 397) in Equation 3-3, and solving for the dimensionalized
distance in each case. This results in main lobe sizes of:
x=y =
X.h , ,
for the square aperture
VAt
(Equation 3-12a)
and
r = 1.081-^===_ for the circular apeiture.
(Equation 3-12„)
Since x and y are the half-widths of a square lobe and r is the radius of a circular lobe,
the areas of the main lobe can be found as follows:
Jt2h2
Area of square lobe = 4xv = 4 - ^ -
(Equation 3-13a)
.
X2h2
Area of circular lobe = tct2 = 3.671 ^
(Equation 3-13b)
It is thus seen that for a given At, the circular apertiure places [(83.8% - 81.5%) /
81.5%] x 100% or 2.8% more power into a main lobe that is 8.2% smaller in area,
while maintaining the same peak beam intensity. Because of this and the exclusion
zone size issue discussed earlier, the use of circular transmitting antennas is more
economical than the use of square transmitting antennas.
Conclusions
For square transmitting antennas, exclusion zone width is independent of
frequency. For circular transmitting antennas, exclusion zone radius is proportional to
frequency to the minus one-third power. The width of the main lobe is inversely
proportional to frequency for both geometries (see Equations 3-12a and 3-12b). At a
given frequency, the circular aperture allows for somewhat smaller rectennas, much
smaller exclusion zones, and somewhat higher amounts of power in the main lobe than
the square aperture. Furthermore, it does so while maintaining the same peak beam
-33-
intensity. It is therefore recommended that future work on solar power satellites
concentrate on circular apertures. In order to take advantage of the decrease of
exclusion zone and main lobe size with frequency, the frequency should be increased
from 2.45 GHz to roughly 10 GHz. Rain attenuation and atmospheric heating continue
to be problems above that frequency. Further attempts to concentrate more power in the
main lobe and reduce power to the sidelobes should therefore concentrate on beam
tapering.
-34-
30
5.
20
I
•&^
C0
§
10
M
0
-20
-10
0
10
20
Distance from Center (km)
FIGURE 3-1. Microwave beam intensity at the Earth's surface for a geostationary SPS
beaming 5 gigawatts of power from a 1128 meter diameter circular antenna to the
equator at a frequency of 2.45 GHz.
-35-
0.1 mW/cmA2 zone
<5
4 —»
<D
U
e
8
First min
«s
<L>
o
a
cd
4 —»
1 mW/cmA2 zone
4
CO
Q
0 mW/cmA2 zone
0
Frequency (GHz)
FIGURE 3-2.
Microwave beam exclusion zones at the Earth's surface for a
geostationary SPS beaming 5 gigawatts of power from a 1128 meter diameter circular
antenna to the equator for frequencies of 1 to 11 GHz. Exclusion zones for 0.1,1, and
10 mW/cm2 safety thresholds are shown, as well as the location of the first diffraction
minimum.
-36-
0.1
&
g
u
8
£
<U
O
c
•»—»
Al
•C
i—
Q
First
4
1 mW/cmA2zone
10 mW/cmA2 zone
0
>jes
100
Frequency (GHz)
FIGURE 3-3.
Microwave beam exclusion zones at the Earth's surface for a
geostationary SPS beaming 5 gigawatts of power from a 1128 meter diameter circular
antenna to the equator for frequencies of 1 to 100 GHz. Exclusion zones for 0.1, 1,
and 10 mW/cm2 safety thresholds are shown, as well as the location of the first
diffraction minimum.
12
10
u
g
8
u
a
6
c§
<D
O
4
4§
—*
Q
2.45 GHz
2
0
0
FIGURE 3-4.
35 GHz
4
2
6
8
Safety Threshold (mW/cmA2)
10
Microwave beam exclusion zones at the Earth's surface for a
geostationary SPS beaming 5 gigawatts of power from a 1128 meter diameter circular
antenna to the equator are shown as a function of safety threshold for 2.45 and 35
GHz.
-38-
Chapter 4:
Tapered Microwave Beams
Introduction
We have seen that the physics of beaming power from an SPS to a rectenna on
the Earth is a major constraint on the design of the SPS. The spreading of the beam
due to diffraction necessitates the transmission of a large amount of power in order for
the SPS to be economical. In previous chapters, as well as in previous work at New
York University (Potter and Kadiramangalam, 1991; Potter, 1992a; and Hoffert, et al.,
1989), the use of square and circular transmitting antennas beaming power at
frequencies of 1 to 100 GHz was investigated. It was concluded that frequencies
below about 10 to 15 GHz should be used for best penetration of rain and air, and that
circular antennas have lower sidelobe levels than square antennas, allowing for more
efficient power transmission, and smaller exclusion zones in the vicinity of the
rectenna. This analysis was carried out for transmitted power levels that are constant
across the face of the transmitting antenna. It was seen that the sidelobes of the beam
were a source of two difficulties: wasted power (since the power in the sidelobes is too
spread out to rectify economically); and "wasted" land (since the first few sidelobes
may nevertheless have enough power to present a danger to people [though perhaps not
crops], thereby necessitating large exclusion zones around the rectenna). Although the
choice of a circular antenna provides some sidelobe reduction compared to a square
antenna, even further reductions are possible if the amplitude of the transmitted beam is
varied, or tapered, across the face of the transmitting antenna. Circular apertures using
-39-
tapered beams will be considered as a means of concentrating more power in the main
lobe with less power in the sidelobes.
Choice of a Tapered Beam
As the amplitude taper of a transmitted microwave beam is increased, the
sidelobe power density decreases, the percent of the power in the main lobe increases,
and the main lobe becomes broader. The former two tendencies are desirable for SPS
use. The latter tendency is desirable to some extent, since a lower peak beam intensity
is environmentally safer. However, if the rectenna is approximately the same size as
the main lobe, a larger rectenna will be necessary for increased tapers. This may be
acceptable if a substantial reduction in the size of the exclusion zone is achieved.
However, an excessively large rectenna may be expensive, disruptive to the
environment, and may even extend past the exclusion boundary, thus partially defeating
the purpose of beam tapering.
Skolnik (1980, page 233) presents a family of beam tapers which illustrates
these tradeoffs, and may allow for improvements over untapered beam transmission. If
p is the radial distance from the center of a circular transmitting antenna of radius p0,
then the aperture distributions to be considered are of the form [1 - (p/po)2] ^ ' 1), where
n = 1, 2, 3 ,.... Note that n = 1 refers to an untapered beam. This yields a radiation
intensity pattern of the form:
2
-40-
where Jn(u) is the n-th order Bessel function of the first kind and u is a nondimensionalized distance from the center of the radiation pattern at the rectenna.
Appropriate normalization constants for these intensities can be found by integrating
these expressions over all two-dimensional space and setting the result equal to Pt, the
total power transmitted. It will be assumed that the total power transmitted is equal to
the total power received. The power intensity at the transmitting antenna is thus:
Hn(p) =
4nPt T
4 p 2l (n' 1)
^ 2
1 ' dJ2]
(Equation 4-la)
where Dt = 2p0 = diameter of transmitting antenna. Thus, At = area of transmitting
antenna = 7tp02 = 7tDt2/4. The average intensity at the transmitting antenna is thus
<Hav> = Pt/At = 4Pt/(7tDt2). The normalized intensity at the transmitting antenna is
thus:
\2
<Hav>
= n 1
(n-1)
(Equation 4-lb)
-
iPo J J
For n > 1, the intensity is thus 0 at the edge of the transmitting antenna (p = p0), and is
at its peak at the center of the antenna (p = 0). This peak is proportional to n. This
may set a limit on the degree of tapering allowable, since excess power at the center
may overheat the antenna.
-41-
The dimensionalized intensity at the rectenna can also be found by integrating
over all two dimensional space and setting the result equal to Pt (assuming no
atmospheric losses). It is given by:
2n-l
° n2
In(r) = I,
i* ■» J-
{ * t)
(t J
(Equation 4-2)
.
The variables used here are defined as follows:
7tPt f2D t
I0 = peak intensity of untapered beam at rectenna = — •{----16 [ Xh
X = wavelength
h = altitude (typically geostationary, 35,786 km)
Pt, Dt are the same as before
r = non-dimensionalized distance from center of beam pattern at rectenna = r
2Dt
Xh
r = dimensionalized distance from center of beam pattern at rectenna.
These definitions are consistent with the variables used for square and circular antennas
in the previous chapters.
The peak intensity of a tapered beam can be found by letting r approach 0 in
Equation 4-2, giving In(0) = IQ(2n-l)/n2. For n large, In(0) = l l j n . Thus, the
peak beam intensity decreases with increasing n. In order to conserve energy, the
beam must therefore become broader.
-42-
The case where n = 1 represents an untapered beam. For this case, Equation 4lb becomes Hn(p) = <Hav> = Pt/At for all p. Equation 4-2 becomes:
which is identical to the untapered beam intensity given in Chapter 3 (Equation 3-2).
Suddath (1980) has examined the family of diffraction patterns given in
Equation 4-2 and has concluded that a "good" electric field pattern (in terms of sidelobe
minimization) can be obtained by combining the n = 1 case and the n = 4 case. The
field is of the form:
24 4! J4(u)
U4
If this expression is squared (to yield a function proportional to power instead of
electric field) and properly scaled, then the beam pattern can be expressed by:
(Equation 4-3)
where the variables are defined above. The subscript "ideal" should not be taken too
A
98
literally. The beam intensity at the center of the rectenna (r = 0) is ^ I0 = 0.757 I0.
-43-
3
This is nearly identical for the peak beam intensity for the n = 2 case, which is ^ I0.
Suddath derives the transmitted power intensity function which yields this beam
pattern. It is given in non-dimensionalized units by:
If this is dimensionalized, it becomes:
(Equation 4-4)
Hideal (P)
Equation 4-la (with n = 1, 2, 3, and 4) and Equation 4-4 are plotted in Figure 4-1 for
Pt = 5 gigawatts. Equations 4-2 and 4-3 are plotted in Figure 4-2 for these same cases
and power level, for a geostationary SPS beaming power to a rectenna at the equator.
The axis calibrations without brackets in Figure 4-2 refer to the 2.45 GHz frequency,
while the calibrations in brackets refer to the 9.8 GHz (= 4 x 2.45 GHz) frequency.
Note that distance scales as the inverse of frequency. This follows from the definition
of r since a fourfold decrease in X requires a fourfold decrease in r to maintain the same
value of r . In addition, intensity scales as the square of the frequency. This follows
from the definition of I0 given above. The 2.45 GHz frequency is taken from the
NASA/US Department of Energy reference design (1980). The 9.8 GHz frequency is
used to show the effect of frequency scaling on the beam pattern and sidelobe levels.
The author and M. Kadiramangalam have determined that 9.8 GHz is a frequency
worth considering for SPS use. It allows for a simple fourfold scaling (compared to
2.45 GHz), without crossing into a frequency regime that is vulnerable to rain and air
attenuation.
The intensity pattern at the transmitting antenna is unaffected by
frequency, since X is not present in Equations 4-la and 4-4. The tapers for various
-44-
values of n, as well as the "ideal" taper will be further examined, and their suitability
for SPS microwave power transmission will be assessed.
Maximization of Power in the Main Beam Lobe
A desirable beam taper should allow as much of the total power to fall within
the main lobe as possible. The fraction of power in the main lobe of the beam can be
calculated by integrating Equation 4-2 or 4-3 over the area of the main lobe and dividing
by the integral over all two-dimensional space. The main lobe includes all of the area in
a circle whose center is at the peak of the beam and whose edge is at the first zero of the
diffraction pattern. For the n = 1 (untapered) case, a formula derived by Bom and
Wolf (1980, page 398), which they attribute to Lord Rayleigh, can be used. The
fraction of power at non-dimensional radius u = t t /2
Fi(u) = l - J 02( u ) - J i2(u)
is given by:
(Equation 4-5a)
The fraction of power in the main lobe can be found by setting u equal to ao, the
position of the first zero of Ji. Since Ji(ao) is, by definition, 0, this yields:
(Equation 4-5b)
Fl(ao) = 1 - Jo2(ao)
-45-
The value of ao is 3.832, giving Fi = 0.838. Thus, 83.8% of the power is in the main
lobe. The author has derived a more general version of Equations 4-5a and 4-5b,
which can be applied to the tapered beams under consideration here. For n = 1, 2, 3,
..., the fraction of power that falls within non-dimensional radius u is given by:
j- —
F n(u) = 1 -
-j [Jn_i2 (u)+Jn2 (u)]
(Equation 4-6a)
The derivation of this equation is shown in the Appendix. As before, the fraction of
power in the main lobe can be found by setting u equal to ao, the position of the first
zero of J„, giving:
Fn(ao)= l
-
\~-----( -
g
iJ - ' 1 (a°~ \
(Equation4-6b)
The results are summarized in Table 4-1. Finding the fraction of power in the main
lobe of the "ideal" case is slightly more difficult, since the first minimum is not at zero
intensity. The location of the first minimum was calculated numerically, using
Mathematica. It is found at rcr/2 = 6.286, or r = 4. At that point, the intensity is just
under 2x1 O'5 of its peak value. Thus, this point can be considered to be the radius of
the main lobe. By comparing the numerical integral over the main lobe with the
analytical integral (see Appendix) over all two-dimensional space, Fideal = 0.9851 is
obtained. Thus, an untapered beam puts 83.8% of the energy into the main lobe, while
the various tapered beams under consideration place between 98 and nearly 100% of
the energy into the main lobe. It is instructive to compute the size of a rectenna for the
-46-
tapered beams needed to capture the same amount of energy that is in the main lobe of
an untapered beam; that is, 83.8%. Since SPS literature usually considers higher
percentages captured than this, the 96% capture area is also considered here. By using
Equation 4-6a, with Fn(u) = 0.838 (or 0.96) and solving for u, the radius of the portion
of the diffraction pattern that contains 83.8% (or 96%) of the energy can be found for
the tapered beams (see Table 4-1). For n = 2, the 96% capture area is roughly the same
size as the 83.8% capture area for n = 1, while for n > 2, the 96% capture area grows
somewhat larger. For tapers with n > 3, the 83.8% capture area is larger than it is for
the untapered case. In addition, although the edge of the main lobe is further from the
center for the "ideal" case than it is for n = 2, if the "ideal" intensity distribution is
integrated out to the distance of the first minimum of the n = 2 case, it is seen that
98.4% of the energy is contained therein. Since integrating out to the first minimum of
the "ideal" case increases this figure only slightly (to 98.5%), the "ideal" and n = 2
cases have nearly identical main lobes (see also Figure 4-2). The cases for n = 2 and 3,
and the "ideal" case put more energy into the area of the untapered main lobe than the
untapered beam does. To avoid excessive broadening of the main lobe, as well as
excessive heating of the transmitting antenna, larger tapers (for, say, n > 2) may not be
desirable. An examination of the sidelobes will, however, be necessary for a more
thorough evaluation of the beam tapers under consideration.
Sidelobe Levels and Exclusion Zones
To protect surrounding populations from possible hazards of microwave
exposure, a safe distance from the center of the beam must be maintained. The region
within this distance is known as the exclusion zone. For a given Pt and safety
threshold, the exclusion zone radius depends on frequency and choice of beam taper.
-47-
To estimate the exclusion zone radius, the approximate upper bound of the microwave
beam intensity will be considered. This, in effect, connects the sidelobe peaks
together. To find the upper bound of the intensity, the following approximations to the
Bessel functions will be used for large u (Kreyszig, 1983, page 177):
J2k(u) ~
(_l)k
(cos u + sin u)
(Equation 4-7a)
V 7CU
(.nk+l
J 2 k+l(u) =
V 7tU
(cos u - sin u)
(Equation 4-7b)
where k is an integer. Squaring either of these equations gives an equation of the form:
Jn2 (u) * —
Jiu
nu
(cos u ± sin u)2
(Equation 4-8)
(cos2 u + sin2 u ± 2 sin u cos u)
= — (1 ± 2 sin u cos u)
7t U
where n is an integer, and the ± sign refers to even (+) or odd (-) n. Dividing through
by u2n gives:
2
j~ ~
| «—
j- (1 ± 2 sin u cos u)
(Equation 4-9)
-48-
To find the upper bound of this function, the oscillating term in parentheses on the right
side of Equation 4-9 must be replaced by its upper bound, which is 2. Thus,
■j Jn(u) 2 u"
2
7tu2n+l
Applying this approximation to Equation 4-2, and substituting for Iq and r gives:
2
(7tr)2n+1
(Equation 4-10)
where r is the dimensionalized distance from the center of the radiation pattern. If c is
the velocity of light and f is the frequency, then X = c/f. In addition, In(r) can be set
equal to the microwave safety threshold Is, and r can be set equal to the exclusion zone
radius rs. These substitutions can be made in Equation 4-10, and the equation
rearranged to give the approximate exclusion zone radius for case n:
ZI1*1
^- l/(2n+l)
{ f p, |
(2n-l) {(n-1)!)
f Dt
*
r° “
It [ l s
,
(Equation 4-1 la)
For n = 1 (untapered beam), this becomes:
(Equation 4-1 lb)
which was seen in Chapter 3 (Equation 3-11). For the general case, Equation 4-1 la
shows that rs depends on Is to the power of -l/(2n+l) and frequency to the power of
(l-2n)/(l+2n). This is summarized in Table 4-2. Two additional advantages of beam
tapering are thus illustrated. As the taper is increased, changing the safety threshold
has less effect on the exclusion zone radius, making strict thresholds more feasible. An
increased taper also allows for a more rapid contraction of exclusion zone radius with
increasing frequency. Since main lobe radius varies as the inverse of frequency, an
increased taper will allow exclusion zone radius to vary in a similar manner. Thus,
both land and rectenna cost can be minimized. (The possibility that higher frequencies
will involve rectennas with a higher cost per unit area has not been considered.)
The above estimates of exclusion zones apply if the boundary is far from the
main lobe. This occurs when the safety threshold is very small compared with the peak
beam intensity, i.e., for small n, high frequency, and/or low threshold. For thresholds
that are somewhat larger, the exclusion zone boundary is near, or perhaps even within,
the main lobe. In addition, the "ideal” case does not lend itself to as simple an
exclusion zone estimate. The exclusion zone radii were therefore computed by direct
numerical comparison of the intensities given by Equations 4-2 and 4-3 with various
safety thresholds. A total power of 5 GW was considered, as in the NASA/US DOE
reference design (1980). For simplicity, power beamed from a 1 km2 circular antenna
to a rectenna at the equator was considered. As discussed in Chapter 2, standards for
microwave exposure range from 0.01 mW/cm2 (Eastern Europe) to 1 mW/cm2
(Canada) to 10 mW/cm2 (US and Western Europe). Because of this disagreement over
microwave safety standards, the cases considered here will span this entire range. The
US and Western European standard should be considered a "best case" scenario in that
exclusion zones will have to be at least as large as this case requires, and may have to
-50-
be much larger. Standards much stricter than that of Eastern Europe may be moot for
SPS design considerations, since many Americans are already exposed to microwaves
at a level of 0.001 mW/cm2 due to radar, television signals, etc. (Glaser, 1986).
However, since people living near a rectenna will be exposed to microwaves
continuously, a standard as strict as that for Eastern Europe may be desirable for
residential areas. It is also likely that several different types of standards may be
needed, e.g., residential, commercial, and industrial. Thus, the results presented here
can serve as a zoning law study.
The exclusion zone radii for n = 1 through 4, as well as the "ideal" case, were
computed numerically for frequencies of 1 through 15 GHz. Results for safety
thresholds of 0.01, 0.1, 1, 5, and 10 mW/cm2 are shown in Figures 4-4 through 4-8,
respectively. (The 5 mW/cm2 standard is the U.S. limit on microwave oven leakage
[Vondrak, 1987].) It is seen in Figures 4-4 and 4-5 that for strict safety thresholds,
tapered beams result in a significant decrease in exclusion zone radius, compared to
untapered beams. Interestingly enough, the "ideal" taper does not allow for as much of
an exclusion zone decrease as the other tapers under consideration in Figure 4-4. The
reason for this can be seen in Figure 4-3, which compares sidelobes for the n = 2 and
"ideal" cases. Although these two cases have very similar main lobes, their sidelobe
structures are quite different. The n = 2 case has a large sidelobe at the approximate
point where the "ideal" case has its first minimum, making it appear as though the
"ideal" case has nearly no sidelobes. However, subsequent sidelobes are smaller for
the n = 2 case. Figure 4-5 shows that at 0.1 mW/cm2, all of the tapers are roughly
equally preferable to the untapered (n = 1) case. For the 1.0 mW/cm2 case (Figure 46), beam tapering is advantageous at the higher frequencies (especially 7 to 15 GHz),
but not the lower frequencies (below about 3 GHz); furthermore, all of the tapers
-51-
considered have roughly equal exclusion zone radii for a given frequency. In Figures
4-7 and 4-8 (5 and 10 mW/cm2, respectively), it is seen that beam tapering may actually
be a disadvantage, at least at some of the lower frequencies. These latter two figures
represent exclusion zone boundaries that are within the main lobe (except for the n = 1
case at the higher frequencies), since tapering the beam broadens the main lobe (see
Figure 4-9). The n = 2 case may represent the best overall taper, since, compared with
the other tapers, it causes the least increase in peak intensity at the transmitting antenna
(Figure 4-1), while allowing for some reduction in peak beam intensity at the rectenna
(Figure 4-2 and Table 4-1), without unduly broadening the rectenna (Figure 4-2 and
Table 4-1). In addition, it provides roughly the same exclusion zone reduction as the
"higher" tapers (Figures 4-4,4-5, and 4-6). The "ideal" taper is quite similar to the n =
2 case, but its peak power level at the transmitting antenna is somewhat higher.
However, since the aperture illumination function for these two cases is different
(Figure 4-1), a heat transfer analysis of the transmitting antenna may be needed to
provide a more complete comparison.
Effect of Beam Tapering on the SPS System
The size of the exclusion zone of an untapered beam is not very sensitive to
frequency (Equation 4-1 lb and Chapter 3). In addition, the peak beam intensity at the
rectenna varies as the square of the frequency. An m-fold increase in frequency will
allow for an m2-fold decrease in the area of a given part of the beam pattern (such as the
main lobe or 83.8% capture region), but only an m ^ 'fo ld decrease in exclusion zone
area. If the m2 increase in intensity is compensated for by an m2 decrease in Pt, then
m2 times as many SPS units will be needed. There will be no net change in the total
-52-
area of the either the main lobes or the exclusion zones. However, the use of a tapered
transmission beam may make frequency scaling more feasible.
It was seen earlier that the peak intensity at the transmitting antenna is
proportional to n, while the peak beam intensity at the rectenna is proportional to (2nl)/n2. Furthermore, for an m-fold increase in frequency, the peak beam intensity at the
rectenna is proportional to m2, while the peak intensity at the transmitting antenna is
unchanged. A goal can be set to hold the peak beam intensity at the rectenna to the
same level as in the untapered 2.45 GHz case, while not letting the peak intensity at the
transmitting antenna increase beyond that case. This can be accomplished by increasing
the frequency by a factor of m, using the n-th taper, and decreasing Pt by a factor of N,
where N is the factor of increase in the total number of SPS units needed. Since the
peak beam intensity at the rectenna is proportional to
m2(2n-l)
N n2
N can be found by setting this expression equal to 1. Thus,
N =
m2(2n-l)
n2
The increase in total main lobe area for the system, compared to the untapered 2.45
GHz case, can be found by multiplying N by the square of the relative radius of the
main lobe from Table 4-1 and then dividing by m2. Since the m2 dependence drops
out, frequency scaling will not affect the total main lobe area or total 83.8% or 96%
capture area of the system; it will, however, affect the area of each individual main lobe
-53-
(as per Equations 4 -lla and 4-1 lb), as well as the number of SPS units, and the power
level of each. Thus,
Relative total area of main lobes =
n
X (relative radius of individual main lobes
for case n)2* Similarly,
Relative total area for a given percent power capture =
X (relative radius of
individual capture regions for case n) •
The change in total exclusion zone area will depend on the value of Is, as well as other
parameters in Equations 4 - lla and 4-1 lb. In addition, since these equations do not
apply to regions near the main lobe, the change in total exclusion zone area must be
computed numerically. The frequency will be increased from 2.45 GHz to 9.8 GHz,
as mentioned earlier, so that m = 4. The effect of this frequency increase, combined
with beam tapering, is shown in Table 4-3. Note that there is no net change in total
main lobe area cm- exclusion zone area if the frequency is increased without tapering the
beam. Beam tapering is actually disadvantageous if the effect on the main lobe alone is
considered; N times as many SPS units, each with 1/N as much power will have a
greater total main lobe area for a tapered beam than for an untapered beam. This is due
to the broadening of a tapered beam. The relative areas of the individual main lobes
increase with increasing n at a rate slightly faster than (2n-l)/n2 decreases. Thus, the
relative total main lobe areas (compared to the untapered 2.45 GHz) will increase
slowly with increasing n. As seen in Table 4-3, the relative total 83.8% capture areas
are all about 54% smaller for a tapered beam, compared to an untapered beam, while
the relative total 96% capture areas are about 24% smaller. Note that the "ideal" case
has a somewhat greater capture area than the other tapered cases shown. In addition,
by comparing the total main lobe areas (which contain at least 98% of the power for the
-54-
tapered cases) to the 96% capture area, it is seen that the power at the edge of the main
lobe is rather dilute, and a large increase in rectenna size would be needed to capture
very little additional power. The relative radii shown in Table 4-1 tell a similar story.
In addition, if the tapered beam patterns are integrated over the area of the main lobe of
the untapered pattern, it is seen that the n = 2, n = 3, and "ideal" cases actually put more
power into the untapered main lobe area than the untapered beam does, and the n = 4
case places nearly as much. Thus, the broadening of the main lobe does not appear to
be a problem. For strict safety thresholds, the total area occupied by the exclusion
zones decreases substantially if the beam is tapered. For a safety threshold of 0.01
mW/cm2, the maximum reduction of total exclusion zone area occurs in the n = 4 case.
Although the total main lobe area of the system increases by 72%, this is of little
consequence, since the total 83.8% and 96% capture areas are, respectively, 54%
smaller and 24% smaller than the total untapered main lobe area. For the 0.1 mW/cm2
case, there is no advantage in going to tapers higher than n = 2 (except for a slight
decrease in the 96% capture area at some of the other tapers). As the safety threshold is
increased, the exclusion boundary falls within the main lobe; as the main lobe broadens
with increasing taper, the exclusion zone area increases. These cases are denoted by
asterisks in Table 4-3. Note also that for a safety threshold of 1.0 mW/cm2, the total
exclusion zone area for tapered beams is larger than for the untapered case. For the 5.0
and 10.0 mW/cm2 cases, the total relative areas of the exclusion zones for the tapers
considered range from 1.12 to 1.18 (compared to the total exclusion zone area for the
untapered case). Thus, beam tapering is actually disadvantageous for high safety
thresholds, at least from the point of view of total exclusion zone area. An additional
constraint on beam tapering is the effect on the peak intensity at the transmitting
antenna. From Equation 4-la, it can be seen that this is proportional to nPt. Since Pt is
-55-
divided by N in the re-scaled cases, the peak intensity at the transmitting antenna varies
as n/N (assuming Dt is held constant).
If it is desirable to hold this intensity at or
below its original value, then tapers greater than n = 5 should be avoided. Note that for
the "ideal" case, the peak beam intensity is proportional to
m2 28
N 37
As before, N can be found by setting this expression equal to 1. Thus,
m228
N =
37
42 28
= 3 7
=
12. 11
The peak intensity at the transmitting antenna will thus be greatly reduced in this case
(compared to the 2.45 GHz untapered case). Although the relative area of the main
lobe is given as 2.04 in Table 4-3, in practice, it can be considered to be roughly the
same as that for the n = 2 case. Furthermore, the "ideal" case has a smaller total
exclusion zone area than the n = 2 case for the 0.01 mW/cm2 threshold, but not for the
higher thresholds shown in Table 4-3. Thus, the choice of taper will, to some extent,
depend on safety requirements. Note that Dt is assumed to remain constant at 1128
meters (area = 1 km2) in the re-scaled system. If the reference system has a solar
collector area on the order of 50 km2, then the re-scaled system will have an area of
about 50/N km2, or as little as 3 km2. While it may seem impractical to retain the 1 km2
transmitting antenna size for so little (relatively speaking) collector area, recent work on
thin film technology (Landis and Cull, 1991) may allow for a collector surface that is
the same size as the transmitting antenna. Alternatively, a smaller transmitting antenna
can be used with a microwave lens (Space Studies Institute, 1985, pages 86-88) to
increase its effective size.
-56-
Conclusions
The design of an SPS system involves choosing an appropriate frequency and
aperture illumination function for power beaming. In order to make these choices,
environmental standards for peak beam intensity and safety standards for human
microwave exposure must be established. Strict human exposure limits, such as 0.01
to 0.1 mW/cm2 may be desirable, since people living near a rectenna site (which may
have to be near consumers if transmission line losses are to be kept to a minimum) will
be exposed to microwaves continuously.
If peak beam intensities of a few hundred milliwatts per square centimeter are
considered safe for the atmosphere, then a fourfold increase in frequency over the
NASA/US DOE reference design (from 2.45 to 9.8 GHz) will allow for a reduction in
rectenna and exclusion zone area while avoiding atmospheric and rain attenuation. A
moderate amount of beam tapering (the n = 2, n = 3, or "ideal" cases) will allow for a
further reduction in exclusion zone area, as well as the rectenna size needed to capture
the amount of power in the main lobe of the untapered beam. Capturing 96% of the
power for moderate tapers will require only a slight increase in rectenna size over the
untapered main lobe. In order to avoid excessive heating of the transmitting antenna,
while keeping the rectenna size to a minimum, the n = 2 case is recommended for SPS
systems in which high peak beam intensities can be accepted. If microwaves as intense
as 10 mW/cm2 can be tolerated by humans on a continuous basis, then beam tapering is
less strongly recommended.
If peak beam intensities of only a few tens of milliwatts per square centimeter
are considered safe for the atmosphere, and increased peak intensities at the transmitting
antenna are unacceptable, then it will be necessary to employ a larger number of smaller
SPS's. As seen in Table 4-3, increasing the frequency without tapering the beam
allows for no change in the total rectenna or exclusion zone area compared to the low
frequency case, although the individual SPS units can be scaled down, making the
system easier to finance.
To decrease these areas, the n = 4 beam taper is
recommended, especially for very strict (0.01 mW/cm2) microwave exposure safety
thresholds. If capturing the entire main lobe is desired, or if a less strict safety
threshold (e.g., 0.1 mW/cm2) is established, then the n = 2 or 3 cases may be more
feasible. Cases for n > 5 have larger exclusion zones (compared to the n = 4 case), and
cases for n £ 6 have higher peak intensities at the transmitting antenna (compared to the
2.45 GHz untapered case); therefore, these tapers are not recommended.
-58-
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rs dependence
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1
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TABLE 4-2.
Approximate dependence of exclusion zone radius (rs) on safety
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-60-
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falls within the 83.8% capture region is the untapered (n = 1) 1.0 mW/cm2 threshold case.
capture area and the 1.0 mW/cm2 exclusion zone are similar in size. The only case shown in which the exclusion zone
0.833
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0.756
0.770
0.782
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Total relative areas for N times as many SPS units at 9.8 G H z,
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20000
n-3
10000
Antenna radius
-5 6 4 in
n -2
n - 1 (untapered)
0
100
200
300
400
500
Distance from Center of Transmitting Antenna (meters)
FIGURE 4-1. Microwave intensity at a 1 km2 (1128 m diameter) circular transmitting
antenna for various aperture illumination functions for a total power level of 5 GW.
-62-
[480]
n - 1 (untapered)
CM
n m 2,
"ideal"
[320]
n- 4
[160]
0
2000
4000
[1000]
[500]
6000
[1500]
8000
[2000]
Distance from Center of Rectenna (meters)
FIGURE 4-2. Microwave beam intensity at the Earth's surface for a geostationary SPS
beaming 5 GW of power from a 1 km2 circular antenna to the equator. The intensities
for a variety of tapered beams are shown. Axis calibrations without brackets refer to
the 2.45 GHz frequency, while those in brackets refer to the 9.8 GHz frequency. Note
that the n = 2 and "ideal" cases have similar main lobe intensities.
-63-
0.08
[1.28
n -2
ideal
0.04
[0.64
’Ideal" case
1st minimum
Eastern European
safety standard
0.00
4000
[1000]
12000
[3000]
20000
[5000]
Distance from Center of Rectenna (meters)
FIGURE 4-3. Comparison of sidelobes for the n = 2 and "ideal" beam tapers. Axis
calibrations without brackets refer to the 2.45 GHz frequency, while those in brackets
refer to the 9.8 GHz frequency. The first minimum for the "ideal" case does not quite
fall to zero intensity. Furthermore, it occurs in a region where the n = 2 case has its
first sidelobe. Since this sidelobe is rather large, it appears at first glance that the
"ideal" case has virtually no sidelobes. However, subsequent sidelobes decrease faster
in the n = 2 case than in the "ideal" case. Thus, the n = 2 case may be preferable to the
"ideal" case for strict safety thresholds. The Eastern European microwave safety
standard (0.01 mW/cm2) is shown with respect to the 2.45 GHz calibration.
-64-
32000
CO
V -t
L>»
■<
*—
o
B
24000
CO
=3
16000
n - 1 (untapered)
n«i
§
o
N
c
o
'ideal'
8000
CO
n «■2
13
x
W
1
3
5
7
9
11
13
15
Frequency (GHz)
FIGURE 4-4.
Microwave beam exclusion zones at the Earth's surface for a
geostationary SPS beaming 5 GW of power from a 1 km2 circular antenna to the
equator for various beam tapers, with a safety threshold of 0.01 mW/cm2.
-65-
16000
12000
n - 1 (untapered)
Frequency (GHz)
FIGURE 4-5.
Microwave beam exclusion zones at the Earth's surface for a
geostationary SPS beaming 5 GW of power from a 1 km2 circular antenna to the
equator for various beam tapers, with a safety threshold of 0.1 mW/cm2.
-66-
8000
<*)
u
&
<L>
S,
CO
a
T3
c«
04
<L>
C
6000
4000
n - 1 (untapered)
o
(untapered)
N
c
o
•i-H
2000
co
3
i“H
o
X
n«2
'ideal'
ffl
1
3
5
7
9
11
13
15
Frequency (GHz)
FIGURE 4-6.
Microwave beam exclusion zones at the Earth's surface for a
geostationary SPS beaming 5 GW of power from a 1 km2 circular antenna to the
equator for various beam tapers, with a safety threshold of 1.0 mW/cm2.
-67-
4000
<L>
S
CO
S3
<L>
G
O
3000
n -.1 (untapered)
2000
n«1 /
(untapered)
N
c
o
C«
G
O
1000
n -2
'ideal'
W
1
3
5
7
9
11
13
15
Frequency (GHz)
FIGURE 4-7.
Microwave beam exclusion zones at the Earth's surface for a
geostationary SPS beaming 5 GW of power from a 1 km2 circular antenna to the
equator for various beam tapers, with a safety threshold of 5.0 mW/cm2.
3000
<D
£
§
2000
T3
cd
04
n -1 /
(untapered)
'ideal'
O
X
m
i
3
5
7
9
11
13
15
Frequency (GHz)
FIGURE 4-8.
Microwave beam exclusion zones at the Earth's surface for a
geostationary SPS beaming 5 GW of power from a 1 km2 circular antenna to the
equator for various beam tapers, with a safety threshold of 10.0 mW/cm2.
-69-
24000
16000
T3
Cd
n -2
<D
o
h-4
n-3,
"ideal"
8000
n -4
.a
cd
2
n« 1 X
(untapered)
1
3
5
7
9
11
13
15
Frequency (GHz)
FIGURE 4-9. Radius of the main lobe of various tapered microwave beams for a
geostationary SPS beaming power from a 1 km2 circular antenna to the equator. Note
the similarity between n = 3 and "ideal" main lobe radii (as expected from Table 4-1).
Note also the broadening of the main lobe with increasing taper, at any given
frequency.
Chapter 5:
Discussion and Conclusions
Peter Glaser, the originator of the solar power satellite concept, has written:
The concept of the SPS is not tied to any particular technology for
beaming power from space to Earth, the magnitude of power
delivered to Earth, a preferred design, the percentage of terrestrial
or extraterrestrial materials used for its construction, deployment in
specific Earth orbits, on the surface of the moon or on other
planets. The key determinants for the realization of the SPS
concept is the state-of-the-art of applicable technologies at various
stages of their development, economic considerations, and societal
issues including policies, regulatory guidelines, and above all the
perceived value of meeting the growing energy demands of the
global population in the 21st Century. (Glaser, 1991.)
The goal in designing a power beaming system in previous chapters was not to stray
too far from the purpose of Glaser's 1968 paper or the NASA/US DOE reference
design: namely, to supply the world with electricity on a large scale in a manner that
minimizes or eliminates carbon dioxide emissions and nuclear waste, while avoiding
the inherent limits on the availability of non-renewable energy sources. Geostationary
orbit was chosen to minimize or eliminate the need for storage batteries.
The years since the NASA/DOE study have presented new challenges, as well
as new opportunities. While the oil crises of the 1970's prompted an interest in
renewable energy sources due to fears that fossil fuel would "run out" or become
unavailable for political reasons, the environment has been a major concern in recent
years.
The carbon cycle model discussed in Chapter 1 suggests that a global
greenhouse warming may be upon us very soon. The need to implement an SPS
system as quickly as possible was therefore an important motivation for the work in
the previous chapters. When the NASA/DOE study, as well as other early work on
-71-
SPS, yielded designs with masses of tens of thousands of tonnes, frequent launches of
enormous space vehicles needed to be considered. The financial and environmental
costs of this threatened to render the SPS concept unfeasible. The possibility of
mining the Moon thus became increasingly attractive. However, the difficulty of
constructing even a small-scale space infrastructure such as Space Station Freedom,
combined with the relative immediacy of a possible global warming, suggest that
Earth-launchable (i.e., lighter) SPS's be developed in parallel with a lunar mining
system. To some extent, this is already being done in the SPS community. For
example, a Japanese project, called SPS 2000 (Nagatomo and Kiyohiko, 1991), would
consist of a prism-shaped satellite 800 meters long by
100
meters wide, with a
100
meter diameter circular transmitting antenna. It would orbit at an altitude of perhaps
1000 km, and would supply power to remote regions near the equator. Such a system,
if it is to be more than a demonstration project, would require some form of energy
storage, or perhaps a network of satellites and ground stations spaced in such a way as
to maximize the power transmission duty cycle.
In light of the results of previous chapters, as well as recent work on thin film
technology (Landis and Cull, 1991), it is recommended that geostationary Earthlaunchable SPS's be considered.
Since it is likely that the peak intensity of a
microwave beam may have to be limited to fairly low levels (say, 26 mW/cm2, as
discussed in Chapter 4), let us consider the results presented at the end of Chapter 4.
To minimize land use, the n = 4 beam taper was recommended, for use with a 1
square km (1128 m diameter) circular transmitting antenna. The recommended
frequency was 9.8 GHz. In order to maintain the peak beam intensity at 26 mW/cm2,
the total transmitted power was cut by a factor of 7 (from 5 GW to roughly 700 MW),
although 7 times as many SPS units would now be needed. Despite the increase in the
-72-
number of SPS units needed, the total rectenna and exclusion zone areas were
decreased, due to the combined effect of increasing the frequency and tapering the
beam. The 51,000 tonne NASA/DOE reference design can now be scaled down to
roughly 7300 tonnes (or slightly more, since the transmitting antenna is not being
scaled down, and is, in fact, slightly larger than the
1000
meter diameter reference
antenna). In addition, Landis and Cull (1991) report that thin film technology can, in
the near term, allow the construction of SPS's with masses of 0.7 kg/kW (compared to
about 10 kg/kW for the NASA/DOE reference design). This would allow a 7300
tonne SPS to be reduced to 510 tonnes. Landis and Cull also state that thin film
technology with masses of 0.08 kg/kW may be available after the year 2000. This can
further reduce the SPS mass to 58 tonnes. Thus, geostationary SPS's that are light
enough to be lofted from the Earth in one (or perhaps a few) launch(es) may be
feasible if a frequency increase, beam tapering, and thin film technology are
combined. Such a system also allows for reduced rectenna and exclusion zone sizes
(compared to a 2.45 GHz untapered beam), without increasing the peak beam
intensity. The individual steps leading to a large scale world-wide SPS system can
thus be made much smaller. Since more SPS's will therefore be needed, it is likely
that a lunar infrastructure will eventually become desirable.
It is therefore
recommended that work on such an infrastructure proceed in parallel with the
deployment of Earth-launchable geostationary SPS's.
The above-mentioned SPS design is not etched in stone. In particular, it was
stated in previous chapters that there is a wide range of human microwave exposure
standards in effect in different nations, and that the choice of a standard will affect the
SPS design. In addition, it is possible that peak beam intensities of perhaps 50 or 100
mW/cm 2 may prove relatively safe for the environment. This, too, would affect the
-73-
SPS design. The selection of health and environmental safety standards may require
more research by physicians, biophysicists, and environmental scientists. The results
of such research will have to be subjected to a cost/benefit analysis. It is possible that
clear-cut health and environmental limits for microwaves cannot be determined.
Society may then have to make choices about how it wishes to obtain energy, what
types of risks to accept, and even what standard of living it wishes to strive for.
Another constraint on the use of microwaves for beaming power is electromagnetic
compatibility and interference (EMC/EMI). Fiber-optic technology may eventually
reduce or eliminate the need for communications satellites, thereby freeing
geostationary orbit for SPS's, and, in addition, reducing the possibility that power
beaming will interfere with communications. Weighing against this is the increasing
use of cellular telephones, pagers, and other communications technologies, which may
compete with power beaming for the microwave spectrum. Thus, the 9.8 GHz SPS
frequency recommended earlier should be considered approximate. In addition, even
if relatively liberal safety standards (such as 10 mW/cm2) are in effect for human
exposure, EMC/EMI requirements are likely to make strict exclusion zones desirable
in some circumstances. For example, sidelobe reduction may be essential if a
rectenna is to be located within a few kilometers of an airport, or even a frequently
used air route, in order to avoid interference with radar and other navigation
equipment.
Because of all of these unknowns, general results in the form of analytical
formulas were presented in this dissertation (particularly in Chapter 4 and in the
Appendix) in addition to numerical results. The use of dimensionless variables and
normalized (relative) quantities also allows the work described here to be generalized
to other cases. The results presented in this dissertation can therefore be viewed as a
-74-
feasible set of SPS design characteristics in their own right, as well as a "jumping-off
point for further solar power satellite research.
-75-
Appendix:
Integration of the Beam Intensity Function
Chapter 4 introduced the following equation for the intensity of a particular set
of tapered microwave beams:
(Equation 4-2)
where n = 1, 2, 3 , ; Jn is the n-th Bessel function of the first kind; r is the nondimensionalized distance from the center of the beam pattern at the rectenna; and I0 is
the peak intensity of the untapered beam (n = 1 case) at the rectenna. In addition, an
"ideal" intensity pattern, based on Suddath (1980), was discussed, and is given as
follows:
(Equation 4-3)
lideal (^)
The derivation of the fraction of the microwave energy that falls within a given radius
(Equations 4-6a and 4-6b) is shown here.
-76-
For the purpose of this derivation, it is convenient to incorporate the factor of
tc/ 2 into the non-dimensionalized distance r by defining a new non-dimensionalized
radius u = tct/2. In addition, we can define non-dimensionalized intensities
= In/Io
and '{'ideal = lidealAo- Equation 4-2 becomes:
'Fn(u)
=
(Equation A-la)
Equation 4-3 becomes:
lT,
, ,
'J'ideal(u) =
24 4! J4(u)r
J4 (u)V
7 f2 Ji(u) ^. 2^4!
“3 7
|
u
+
^4
...............
J
(Equation A-lb)
To find the amount of energy enclosed within a given radius "a," let us define Rn(a), a
quantity proportional to this amount of energy. It is given by:
2%
a
Rn(a) —
J 'Fn(u)
u du d 0
(Equation A-2)
0
where 0 is the azimuthal angle about the center of the rectenna. Since 'P n(u) is
azimuthally symmetric,
R„(a) = 2 k j 'F„(u)udu = 2jc
f ^n
u
-77-
udu
(Equation A-3)
Note that if dimensionalized units were used, then R n(°°) could be used to find PtThe fraction of power that falls within radius a is therefore given by:
(Equation A-4)
Fn(a)=fRn(°°)
f^
2 n-l
2re
n
f2 n n! Jn(»)12
u du
u
oo
2 n-l
2 jc
un
n
J
udu
(T
a
f r a 2-
du
oo
J Jn(u)
j
udu
U«
0
I
Jn2 (u) du
U2 n-l
oo
/
Jn2 (u) du
U2 n-l
It is now convenient to define:
/
Sn(a) —
Jn2(u)
du
u2n-l
(Equation A-5)
Thus,
Fn(a) =
(Equation A-6 )
Sn(°°)
It is therefore necessary to integrate Sn(a). This can be done by parts in a manner
analogous to what Bom and Wolf (1980, pages 397-398) did for the untapered (n = 1)
case. Kreyszig (1983, page 177) gives the following relationships between Bessel
functions:
[un J„(u)] = un Jn-i(u)
(Equation A-7a)
(Equation A-7b)
In Equations A-7a and A-7b, n can be any real number. Therefore, n in Equation A7b can be replaced by n-1, yielding:
k
=' ^
(Equation A-8 a)
Multiplying both sides by -u n_1 gives:
Jn(u) = -
(Equation A-8 b)
-79-
Multiplying both sides of Equation A-7a by Jn(u) gives:
Jn(u)
[un J„(u)] = un Jn-i(u) J„(u)
(Equation A-9)
Carrying out the differentiation on the left side of Equation A -9 causes this equation
to become:
Jn(u) [un ^
U) + Jn(u) nu"-^ = un Jn-i(u) Jn(u)
(Equation A - 10)
Distributing Jn(u) among the terms inside the brackets on the left side gives:
un Jn( u ) ^ j ^ + Jn2(u) nun_1 = un Jn-i(u) Jn(u)
(Equation A - 11)
Dividing both sides by un_1 gives:
u Jn(u)
+ n Jn2(u) = u Jn-l(u) Jn(u)
(Equation A -12)
The left-most term can be subtracted from both sides. In addition, we will now
consider only cases where n = 1, 2, 3,... . Thus, n * 0, so all terms can be divided
through by n, giving:
Jn2(u) = iL k y n i") . “ W d W
(Equation A. 13)
-80-
Dividing both sides by u2n_1 gives:
^
(EquationA-14)
Note that the left side of Equation A-14 is the integrand of Sn(a), shown in Equation
A-5. Now, we can replace Jn(u) in the left bracketed term by the right side of
Equation A-8 b. This gives:
Jn2 (u) =
dJn(u) j
U2 n -1
i r j n-i(u)d
nj_ un-l du
(Equation A-15)
Note that both terms in square brackets in Equation A-15 contain a function times its
derivative. This is equal to one-half of the derivative of the square of the function.
Thus,
Substituting the right side of Equation A-16 into Equation A-5 gives:
Sn(a) — ~
*d j /Jn-i(u)>
du
dull u" - 1 ,
_1_
2n
2n
Lo
-81-
(Equation A-17)
The left bracketed term is simply the integral of a derivative. The right bracketed
term can be integrated by parts. This gives:
u=a
S „ (a )= -
1 /Jn-l(u)\ 2
2n
I u"'1J
u=0
1
2n
|Jn (u )f
J
u= a
u= 0
0
(Equation A-18)
Note that the last term on the right is equal to
Sn(a) = [" - l ] s n(a).
Thus, the
terms containing Sn(a) can be combined and the equation divided through by the
coefficient of Sn(a), giving:
iU = a
1
Sn(a)“ ” 4m2
( ^ 2+
(Equation A-19)
u= 0
To evaluate these terms at u = 0, note that the n-th Bessel function of the first kind is
given by (Kreyszig, 1983, page 175):
l„(u) = un£
m
(-l)mu2m
2 2m+n m!
(Equation A-20)
(n+m)!
-82-
Thus:
Un ' l
J
= U2
2
m
(-l)m U2m
2 2m+n m!
(Equation A-21)
(n+m)!
Note that this last expression is 0 at u = 0. In addition, by substituting n-1 for n in
Equation A-20, it can be seen that:
, x
Jn-l(u) =
(-l)m u2*11
2m+„.,
(Equation A-22)
-
m= 0
Thus,
Jn-l(u)
un-l
2
m
/■Jn-ltuK2 _
{ U --1
J -
(-l)mu:2 m
2 2 m+n-l m! (n+m- 1 )!
2
m
(Equation A-23)
(.p m u2m
2 2m +n' 1
(Equation A-24)
m! (n+m-1 )!
-83-
If u =%
0 is substituted into Equation A-24, then all terms in the summation become 0,
except for the m = 0 term, which is:
4 n -l(u > '2
(Equation A-25)
I u»-l J
Equation A-21 (for u = 0) and Equation A-25 can be substituted into Equation A-19,
giving:
1
Sn(a) ” 4n-2
1
_ /Jn.i(aK 2 _ /Jn( a ) f '
( 2 n‘l (n- 1 ) ! ) 2
I
*"’1
J
(Equation A-26)
U " '1 J
Sn(°°) must now be found. Recall, from Chapter 4, that for large u:
Jn2 (u) « ~7vU (1± 2 sin u cos u)
(Equation 4-8)
where the ± sign refers to even (+) or odd (-) n. Taking the upper bound and changing
the variable name from u to a gives:
Jn2 (a) < ^
(Equation A-27)
-84-
Note that the right side of Equation A-27 is independent of n, so this inequality
applies to n-1 as well. Thus:
(a £ r)2 £ :^ J b l
(Equation A-28a)
and
s
(Equation A-28b)
Since n = 1, 2, 3, ... , the right side of these inequalities will always have "a" to a
positive power in the denominator, and hence, the expressions in Equations A-28a and
A-28b go to 0 as"a" becomes infinite. Applying this limit to Equation A-26 gives:
Sn(°°) = 1 ^ 2 7 ------ -------72
[2 n_1 (n-1 )!]
(Equation A-29)
Applying Equations A-26 and A-29 to Equation A - 6 gives:
2
F n (a)= !^= l
-
[Jn-l2 (a)+Jn 2 (a)]
(Equation A-30)
Changing the name of the variable from a to u gives:
2
Fn(u)= 1 -
j"
un” " ^ ‘j [Jn-l2 (u) + Jn2 (u)]
-85-
(Equation A-3la)
which is identical to Equation 4-6a in Chapter 4. This equation was used to compute
the fraction of energy that falls within a specified non-dimensionalized radius u
(specifically, the main lobe radius for various values of n). Using Mathematica, it was
also used to compute the non-dimensionalized radius that encloses a specified fraction
of energy (specifically, 0.838 and 0.96). The fraction of power in the main lobe for
case n can be found by setting u equal to ao, the position of the first zero of Jn, giving:
Fn(ao) = 1 -
{•2 n~ (l~ n - f c 1^ 0)}
(Equation A-3 lb)
which is identical to Equation 4-6b in Chapter 4.
A similar procedure can now be applied to the "ideal" case. We can define a
function proportional to the amount of energy enclosed within radius a as:
2k
/•
a
J 'Fideal(u) u du d 0
u
Rideal(a) =
(Equation A-32)
0
where 0 is the azimuthal angle about the center of the rectenna, and ^ /ideal(u) is given
by Equation A-lb. Since ' P i d e a l ( u ) is azimuthally symmetric,
a
Rideal(a) = 2n J 'Fideal(u) u du
(Equation A-33)
-86-
Since we are interested in Rideal(a)/Rideal(°°)»we can drop the factor of 2k, and define
Sideal(a) as the integral in Equation A-33. Thus,
a
Sideal(a) =
^ideal(u)« du
(Equation A-34)
The fraction of energy enclosed within radius a is therefore given by:
Fideal(a) =
(Equation A-35)
Sideal(°°)
Substituting Equation A -lb into Equation A-34 gives:
; ^ ( a ) = 7 _ J ^ u ) + ( 2 U i m ( u ) + 2 6 4 lb <
a
-J-i
- 37
4J
^ ^ d u
m
a
+
(24 4!
y dxl
a
+
26 4 ! j ’Jl(u^4J4(u)du
(Equation A-36)
We can now define:
a
Q(a) = J ^ W
du
(Equation A-37)
-87-
Substituting Equation A-37 into Equation A-36 and making use of the definition of
Sn(a) in Equation A-5 yields:
Sideal(a) = ^
{4Si(a) +
(24 4!)2s4(a)
+
26 4! Q(a)}
(Equation A-38)
Expressions for Si (a) and S4 (a) can be obtained by using Equation A-26 with
appropriate values of n. Therefore, to find Sideal(a), it is necessary to find Q(a). This
can be done by rewriting Equation A-37 as follows:
a
(Equation A-39)
By applying Equation A-7b, with n = 3, to the expression in parentheses, the
following is obtained:
a
(Equation A-40)
This can be integrated by parts as follows:
=
+
(Equation A-41)
Equation A-7b, with n = 1, can be applied to the derivative inside the integral sign.
Thus:
Jl(u)J 3 (u)
Q(a) =
vr
u=a
•J3(u)J2(u)du
u3 u
(Equation A-42)
u=0
u= a
_
Ji(u)J 3 (u)
u**
■ m m du
uz uz
u= 0
Equation A-7b, with n = 2, can now be applied to the J 3 (u)/u2 term in the integral,
giving:
Ji(u)J 3 (u)
u
Q(a) =
_ _
(Equation A-43)
I
+
u= a
u=a
u= 0
u=0
M h ) M *)
U
u3
To evaluate these terms at u = 0, note that Equation A-20 says that:
Jn(u)
u1
’n
_
^
mi:= 0
( - l ) m u2m
2 2m+n
(Equation A-44)
m! (n+m)!
-89-
When u = 0, all terms in the summation vanish, except for the m = 0 term. Thus:
Jn(u)
un
(Equation A-45)
2 n n!
u= 0
By applying Equation A-45, with n = 1,2, and 3, to Equation A-43, we obtain:
0(a) Jl(a) J 3 <a)
Q (a)-
I R - jO f
a
a3 +
2L a2 J
+
I _L_
+
2 233!
_
i p
_
f
2 L222 LI
(Equation A-46)
By putting this expression, as well as appropriate expressions for Si (a) and S4 (a)
(from Equation A-26), into Equation A-38, the following is obtained:
f
. /fttaK2 M aX2 1
1
L(23 31)2
+
L
U 3J I *3J J
+
a
a3
M_*aX2 +
2
1 a2 J
1 1
2 233!
.
1 - 1 - 1 )
2 (222!)2 J J
(Equation A-47)
This equation is so unwieldy for general "a" that Sideal(a) was found by applying
Mathematica directly to Equation A-34 and finding the integral numerically.
However, Mathematica was not able to do the integral numerically for a = <*>, a
necessary step in finding Fideal(a). Therefore, Sideal(°°) was found by using Equation
-90-
A-47, and letting "a" approach °°. To do this, recall that from Equations 4-7a and 47b (based on Kreyszig, 1983, page 177), for large u, Jn(u) goes as 1/Vu for all n.
Thus, Jn(u) approaches 0 as u approaches
This is true even if Jn(u) is divided by a
positive power of u and/or raised to a positive power. Therefore, all of the terms
containing a in Equation A-47 are zero for a = «*>. Therefore:
(Equation A-48)
By combining this result with Equations A-34 and A-35, it is seen that the fraction of
energy contained within non-dimensionalized radius "a" for the "ideal" case is given
by:
a
Fideal(a)
J
'Fideal(u)
(Equation A-49)
where 'Pideal(u) is given by Equation A-lb. This equation is equivalent to Equation 2
in Chang, et al. (1989), who studied the "ideal" taper as a means of space-to-space
power transmission. Equation A-49 was used with Mathematica in Chapter 4 to find
Fideal(a) numerically for a given "a" (such as the "ideal" main lobe radius), or "a"
numerically for a given energy fraction Fideal(a) (e.g., 0.838 and 0.96).
Bibliography
Bom, Max, and Wolf, Emil, 1980, Principles o f Optics, Sixth Edition, Pergamon
Press, New York.
Brookner, Eli, 1977, Radar Technology, Artech House, Dedham, Massachusetts.
Chang, K., McCleary, J.C., and Pollock, M.A., 1989, "Feasibility Study of 35 GHz
Microwave Power Transmission in Space," Space Power, Volume 8 , Number
3, pages 365-370. (Presented at the International Astronautical Federation
International Conference on Space Power, Cleveland, Ohio, 5-7 June 1989.)
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