surface accuracy analysis and mathematical …
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SURFACE ACCURACY ANALYSIS AND MATHEMATICAL
MODELING OF DEPLOYABLE LARGE APERTURE
ELASTIC ANTENNA REFLECTORS
By Michael J. Coleman
Bachelor of Arts in MathematicsBoston University, May 2004
A Dissertation submitted to
The Faculty ofThe Columbian College of Arts and Sciences
of The George Washington Universityin partial fulfillment of the requirementsfor the degree of Doctor of Philosophy
August 31, 2010
Dissertation directed by
Frank E. BaginskiProfessor of Mathematics
The Columbian College of Arts and Sciences of The George Washington University certifies that
Michael J. Coleman has passed the Final Examination for the degree of Doctor of Philosophy as of
May 14, 2010. This is the final and approved form of the dissertation.
SURFACE ACCURACY ANALYSIS AND MATHEMATICAL MODELING OFDEPLOYABLE LARGE APERTURE ELASTIC ANTENNA REFLECTORS
Michael J. Coleman
Dissertation Research Committee:
Frank E. Baginski, Professor of Mathematics
Dissertation Director
Katharine F. Gurski, Assistant Professor of Mathematics, Howard University
Dissertation Committee Member
Xiaofeng Ren, Associate Professor of Mathematics
Dissertation Committee Member
ii
Acknowledgements
The research presented in this dissertation has been supported by the National Aeronautics and
Space Administration– Grants NNX07AR67G and NNX09AH08G. I would like to begin by thanking
our technical contact at NASA’s Glenn Research Center, Dr. Robert R. Romanofsky. This work would
not have been possible without his attention and guidance on this project throughout my research
time at GW and NASA/GRC in Cleveland. His colleague at NASA, Dr. Kevin M. Lambert also gave
me very useful guidance on antenna theory.
I am particularly grateful for the undivided attention of my advisor, Frank Baginski throughout
my years of study and research. His constant support and advice have been an integral part in
the development of my mathematical, presentation and writing skills. Frank gave many hours (at
times we even met on weekends) to review the countless pages and hours of presentation material
that I have created throughout my graduate career. He is a certainly a devoted advisor and cares
very much for the work that we do. He also makes the best crab dip!
Many thanks to all others who agreed to serve on my dissertation committee: Robbie Robinson,
Katie Gurski, Xiaofeng Ren and Magda Musielak. I appreciate their help and support for my defense
as well as the support they’ve offered me over my years in the graduate program.
Thank you also to my close friends whom I’ve known throughout my time at GWU and in the
DC region in general. I would especially like to thank Tyler White, Joe Herning and Dzung Trac
who helped me set up my computer, Apollo. Most of the results obtained in this dissertation were
computed on Apollo and so I am very thankful for their assistance.
Finally, I would not have made it to this stage without the unceasing love and support of my
family: Mom, Dad, Meaghan, Grandma and Grandpa, “Grandma and the Kitty” (rest in peace),
Jimmy and Lauren, Ray and Michelle, Diane and Kenny, and all my aunts and uncles. Special
thanks to Mom and Dad for your encouragement in both the best and worst of times. Mom, I hope
that this document can well complement that first calculator of mine!
George Washington University Michael J. Coleman
May 14, 2010
iv
Abstract
SURFACE ACCURACY ANALYSIS AND MATHEMATICAL MODELING OFDEPLOYABLE LARGE APERTURE ELASTIC ANTENNA REFLECTORS
One class of deployable large aperture antenna consists of thin light–weight parabolic reflec-
tors. A reflector of this type is a deployable structure that consists of an inflatable elastic membrane
that is supported about its perimeter by a set of elastic tendons and is subjected to a constant hy-
drostatic pressure. A design may not hold the parabolic shape to within a desired tolerance due
to an elastic deformation of the surface, particularly near the rim. We can compute the equilib-
rium configuration of the reflector system using an optimization–based solution procedure that
calculates the total system energy and determines a configuration of minimum energy. Analysis
of the equilibrium configuration reveals the behavior of the reflector shape under various loading
conditions. The pressure, film strain energy, tendon strain energy, and gravitational energy are all
considered in this analysis. The surface accuracy of the antenna reflector is measured by an RMS
calculation while the reflector phase error component of the efficiency is determined by computing
the power density at boresight. Our error computation methods are tailored for the faceted surface
of our model and they are more accurate for this particular problem than the commonly applied
Ruze Equation.
Previous analytical work on parabolic antennas focused on axisymmetric geometries and loads.
Symmetric equilibria are not assumed in our analysis. In addition, this dissertation contains two
principle original findings: (1) the typical supporting tendon system tends to flatten a parabolic
reflector near its edge. We find that surface accuracy can be significantly improved by fixing
the edge of the inflated reflector to a rigid structure; (2) for large membranes assembled from
flat sheets of thin material, we demonstrate that the surface accuracy of the resulting inflated
membrane reflector can be improved by altering the cutting pattern of the flat components.
Our findings demonstrate that the proper choice of design parameters can increase the perfor-
mance of inflatable antennas, opening up new antenna applications where higher resolution and
greater sensitivity are desired. These include space applications involving high data rates and high
v
bandwidths, such as lunar surface wireless local networks and orbiting relay satellites. A light–
weight inflatable antenna is also an ideal component in aerostat, airship and free balloon systems
that supports communication, surveillance and remote sensing applications.
vi
Contents
Acknowledgements iv
Abstract v
List of Figures viii
List of Tables ix
Chapter 1 Introduction 1
1.1 Parabolic Reflectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Shell and Membrane Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Chapter 2 Inflatable Elastic Reflector Model 14
2.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Flat Panel Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Chapter 3 Parabolic Reflector Efficiency 30
3.1 Electromagnetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Reflectivity Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Antenna Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Geometric Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Chapter 4 Demonstration Cases 64
4.1 A Test Case: The Mylar Balloon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 Analysis of Molded Reflectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
vii
Chapter 5 Parametric Studies 76
5.1 Parametric Study for ǫRMS versus Ng and p0 . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Parametric Study for ǫRMS versus α and γ . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 Tendon Supported vs. Fixed Boundary Reflectors . . . . . . . . . . . . . . . . . . . . . 82
Chapter 6 Conclusions and Future Research 85
References 90
Appendix A Surface Metrology Data 94
viii
List of Figures
1.1 Lenticular deployable reflectors and their support structures. . . . . . . . . . . . . . . 2
1.2 Cross sectional diagram of an axisymmetric parabolic reflector. . . . . . . . . . . . . . 6
1.3 Cross sectional diagram of an off–axis parabolic reflector. . . . . . . . . . . . . . . . . 7
2.1 Inflatable parabolic reflector antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Faceted surface for molded antennas. Dots highlight the reflector rim. . . . . . . . . 17
2.3 Deformation of a material body M under a deformation x. . . . . . . . . . . . . . . . . 19
2.4 Stress components τ2i at point p in material M . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Flat gore construction. Generating ribs indicated by dots. . . . . . . . . . . . . . . . . 28
2.6 Four gores of the reference configuration for the flat gore construction. . . . . . . . . 29
2.7 Flat ring construction. Generating rings indicated by dots. . . . . . . . . . . . . . . . . 29
3.1 Charge with velocity v flowing through a differential cross section, da. . . . . . . . 34
3.2 A contour C attached to two surfaces passing through different current. . . . . . . . 38
3.3 Boundary conditions for Maxwell’s Equations. . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Cross sectional diagram of a paraboloid and its geometry. . . . . . . . . . . . . . . . . 46
3.5 Wave front shape in the near and far field of an AUT. . . . . . . . . . . . . . . . . . . . 49
3.6 Geometry of rays extending to a point P in the far field. . . . . . . . . . . . . . . . . . 50
3.7 Far field patterns for an ideal axisymmetric parabolic reflector antenna. . . . . . . . 58
3.8 Deformed triangle T n with centroid (xn, yn, zn). The vertical displacement, ∆z, is
labeled and the pathlength for this triangle is ℓn = fn+ an. . . . . . . . . . . . . . . . 60
4.1 Cross section of the Mylar Balloon with generating curve z(x). . . . . . . . . . . . . 65
4.2 Generating curve of modeled mylar balloon. . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Fully inflated mylar balloon (actual and modeled). . . . . . . . . . . . . . . . . . . . . 67
ix
4.4 Reflector Distortion Pattern for High and Low Tendon Tension. . . . . . . . . . . . . 71
4.5 Effect of tendon force distribution for Case 3 of Table ??. . . . . . . . . . . . . . . . . 72
4.6 Effect of tendon force distribution for Case 4 of Table ??. . . . . . . . . . . . . . . . . 73
4.7 Shell separation for small off–axis antenna reflector. . . . . . . . . . . . . . . . . . . . 74
5.1 Aperture radiation phase plane at 40 GHz with Ng = 16. . . . . . . . . . . . . . . . . . 78
5.2 Phase distributions for two values of p0 at 40 GHz. . . . . . . . . . . . . . . . . . . . . 79
5.3 Parabolic cross section for a flat ring configuration with α= 0.05. . . . . . . . . . . . 80
5.4 Effect of rim modification factor on ǫRMS for various tendon loading. . . . . . . . . . . 81
5.5 Comparison of error for a fixed rim and a tendon–supported reflector. . . . . . . . . 83
A.1 Surface metrology plot of axisymmetric reflector for φ = 90. . . . . . . . . . . . . . . 96
A.2 Surface metrology plot of axisymmetric reflector for φ = 76. . . . . . . . . . . . . . . 97
A.3 Surface metrology plot of off–axis reflector; exhibits substantial rim deflection. . . 97
x
List of Tables
3.1 Charge and mass of the three fundamental charged particles. . . . . . . . . . . . . . 31
3.2 Common IEEE Radiation bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Comparison of predicted versus actual errors for various grid sizes. . . . . . . . . . . 63
4.1 Geometric results of mylar balloon test for two p0 values. . . . . . . . . . . . . . . . . 66
4.2 Mechanical properties of Kapton, reflector dimensions and grid size. . . . . . . . . . 67
4.3 Comparison of energy components for various antenna sizes. . . . . . . . . . . . . . 68
4.4 Gravitational effects on large reflectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5 Summary of RMS values for various physical testing conditions. . . . . . . . . . . . . 72
4.6 Surface accuracy of off–axis reflectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1 Grid parameters and dimensions of the reflector being modeled. . . . . . . . . . . . . 76
5.2 Comparison of ǫRMS(Ng , p0) values in millimeters. . . . . . . . . . . . . . . . . . . . . . 77
5.3 Comparison of ǫRMS(α,γ) values (mm) for various γ. . . . . . . . . . . . . . . . . . . . 81
5.4 Collection of ǫRMS(p0,g) values (mm) for a fixed boundary reflector. . . . . . . . . . 82
5.5 Characterization of fixed rim inflatable antennas; p0 = 12.5 Pa. . . . . . . . . . . . . 84
5.6 Characterization (for comparison) at 80 GHz. . . . . . . . . . . . . . . . . . . . . . . . . 84
A.1 Least squares paraboloid parameters for surface metrology data. . . . . . . . . . . . . 95
xi
Chapter 1
Introduction
An antenna is a device that concentrates a radio signal in a particular direction. There are many
varieties of antennas that have been used throughout the history of communications; see [14] and
[33]. One can choose an antenna that will radiate its signal in all directions in order to reach as
many locations as possible. This is called an isotropic radiator and may be the preferred antenna
type of a radio station, for example. Other types of antennas can direct radiation in a certain
direction in order to concentrate the signal’s power to a particular destination. While the signal is
only received in a particular direction, it will have greater strength than a signal that is radiated
isotropically.
Reflector antennas are particularly effective for the purpose of concentrating a signal over long
distances. Antennas of this type typically use a parabolic reflector dish to direct the signal in a
particular direction. In the interest of long distance radio communication, this antenna is among
the most widely used [33, Pg. 422]. The common notion of a “satellite dish” is exactly the type
of antenna which falls under this category and will be the basis of the antenna theory that is
contained in this dissertation.
Future space exploration and near earth remote sensing missions will require large data trans-
fer rates– on the order of hundreds of megabits per second [27]. There is also interest in missions
at and beyond Earth’s orbit and so information must be transmitted over long distances. In order
to support such large amounts of data, an antenna that can receive and transmit a large amount
of radiation must be employed. If the distance between the mission and Earth grows to the order
1
(a) Supporting torus [37] (b) L’Garde’s lenticular deployable antenna [36]
Figure 1.1: Lenticular deployable reflectors and their support structures.
of astronomical units, there will be additional burdens on the antenna to radiate enough power to
compensate for a large loss of the signal over such a large distance.
The power that radiates from the antenna increases with the size of the reflector [33, Pg.
433]. Therefore, the need for such large data transfer rates leads to the necessity for aperture
antennas which have large parabolic reflectors. The possibility of 10 meter diameter reflectors has
been mentioned [27]. Unfortunately, the limited volume of a launch vehicle or spacecraft places
restriction on the size of a rigid antenna reflector. To allow larger reflectors to serve these missions,
a deployable antenna is desirable. Such an antenna has a reflector which is made of a light–weight
material, can be collapsed or folded for compact stowing, and is deployable at an appropriate time.
Deployable space structures have a history that extends back to around 1960. The background
on space structures that is presented here is documented by R. Freeland, G. Bilyeu and M. Mikulas
in [14] and [15]. Goodyear developed some of the first versions of inflatable antennas; one was
known as a search radar antenna. This antenna was primarily supported by metallic trusses that
unfolded like an accordion to expand the antenna surface and support it. Goodyear also developed
a Radar Calibration Sphere consisting of a large number of flat hexagonal membrane pieces. The
sphere was inflated to a diameter of 6 meters and was metalized for reflectivity.
The lenticular inflatable parabolic reflector is a design much nearer to the one we will consider.
It consists of an inflatable structure formed by two opposite facing paraboloids seamed together
and supported about the periphery by an inflatable toroidal shaped membrane; see Figure 1.1(a).
2
The diameter of the reflector for this particular antenna is on the order of 10 meters. An antenna
of this type was developed by L’Garde, Inc. and selected for NASA’s IN–STEP Inflatable Antenna
Experiment in 1996 [15]. It includes long supporting beams which attach the reflector to the
satellite; see Figure 1.1(b). The experiment was able to verify that these large inflatable antennas:
• could be built at low costs;
• have high mechanical packaging efficiency;
• have low weight;
• have high deployment reliability;
• have surface precision to within a few millimeters RMS.
In 2000, astromesh reflectors were successfully deployed for operation at 14 and 30 GHz [34].
There is now interest in operating large deployable antennas at higher radiation frequencies which
will require greater surface accuracy. Distortions to the parabolic reflector will perturb the signal
of the antenna and cause a loss of the signal’s power to the observer. Any such distortion is
more conspicuous at higher radiation frequencies, as we will show in Chapter 3. An estimate for
the tolerance of the reflector surface accuracy (for the current performance target of inflatable
antennas) is an RMS value of no more than 0.5 mm [27]. Maintenance of the parabolic shape is
therefore critical for successful operation of the antenna. Initial findings suggest that the edges of
the inflated antenna reflectors we are considering are distorted to a flattened ring causing larger
RMS values and an overall reduction of transmitted power.
Investigation of the reflector shape and response of the reflector to internal and external forces
is the central topic of this dissertation. We will model the inflatable antenna reflector so that the
configuration can be calculated based on: physical parameters that quantify the forces acting on
the entire system, mechanical properties of the materials used to construct the reflector, and design
parameters that adjust the lengths and sizes of select apparatus within the system. It is ultimately
our goal to identify those parameters that can be used to improve the efficiency of the antenna
reflector. From this, we can draw some conclusions regarding the support structure, the design
of the current inflatable antenna prototypes and possible construction patterns for the reflector
membrane that may increase performance.
3
The geometric properties of parabolic reflectors will be discussed in Section 1.1. There we focus
on the geometry of rigid paraboloids as reflecting surfaces and neglect the elastic properties which
we include later. Section 1.2 will provide an overview of the literature that addresses membrane
and elastic reflector problems. Background of the mathematical theory required for development
of the model and surface accuracy calculations that we perform in subsequent chapters is given in
Section 1.3.
In Chapter 2, we outline a model for the shape of the deployable antenna system that we
study. This model is based on one that was developed for the analysis of high altitude large
scientific research balloons by F. Baginski and his collaborators; see [2] – [4]. The model uses a
piecewise linear surface of triangular facets to approximate the reflector surface. The model also
includes analysis of large scale deformations and wrinkling which was developed by Allen C. Pipkin
in [26]. In Section 2.2, specific reference configurations will be introduced which feature flat
panel constructions that approximate a paraboloid. The benefits and drawbacks of these differing
reference configurations will also be discussed in Chapter 2.
In Chapter 3, methods of measuring the efficiency and accuracy of a given reflector configura-
tion will be discussed. This will entail a discussion of electromagnetic theory which is necessary for
understanding how antennas reflect radiation. Some direct computations for the surface accuracy
based solely on the geometry of the reflector configuration are also considered. These calculations
depend on both the displacement and orientation of the surface’s triangular facets.
Results of our model are presented in Chapters 4 and 5. The first of these chapters includes the
results of the model applied to a mylar balloon. Mylar balloons are commonly found in party shops
or grocery stores and are comprised of two flat circular membranes which are seamed together and
then inflated; more details regarding these balloons are in Chapter 4. This test case will compare
the results of the model with known analytical results for the shape of a mylar balloon. There
are also tests involving molded parabolic reflectors. Reflectors of the molded variety are actually
cast on mandrels so that the unstrained state of the reflector material is parabolic. The results of
these experiments show that high tendon forces cause deflection of the reflector surface near the
rim. Furthermore, we show that non–symmetric loading at the rim on both symmetric and off–axis
antennas causes high surface inaccuracy.
In Chapter 5, we will present more recent results of the model’s predictions for flat panel
4
construction methods. We perform several parametric studies involving parameters that adjust the
cutting patterns of the panels. Modifications to the material patterns near the rim are introduced
to combat the rim deflection that is described in Chapter 4. These changes improve the surface
accuracy of the deployed antennas. Changes to the inflation pressure also help improve surface
accuracy but excessive pressure causes surface distortions to appear.
In Appendix A, we present surface metrology data on two inflatable antennas that we observed
at NASA’s Glenn Research Center in 2008. The data sets are for a 2.13 meter inflatable astromesh
reflector and a 0.3 meter off–axis reflector. The data sets were taken by a Leica LR200 laser
scanner which allows us to plot the surface points and see the configuration of the reflectors. We
also calculate best fit paraboloids to this data and then determine the reflectors’ surface error.
1.1 Parabolic Reflectors
A parabolic reflector takes advantage of the geometric properties of a paraboloid. If one places an
electrical transmitter at the focal point of the paraboloid, then the rays of radiation will be reflected
in rays that are parallel to the symmetric axis of the antenna’s coordinate frame. This is the feature
that causes the signal to be amplified in the direction that the antenna’s reflector “points”. In
Figure 1.2 we see the characteristics of a typical antenna with an axisymmetric parabolic reflector.
Some important terminology for parabolic antenna reflectors is listed here.
• Feed: an apparatus located at the focal point of the parabolic reflector which transmits or
receives a signal.
• Focal plane: a plane orthogonal to the symmetric axis of the paraboloid passing through the
focal point of the antenna.
• Focal length: a geometric property of the antenna which is the distance from the vertex of
the paraboloid along the symmetric axis to the focal point.
• Aperture: an imaginary projection of the reflector onto the focal plane. For the antennas of
interest here, the aperture is always circular.
• F/D value: the ratio of the focal length to the aperture’s diameter.
5
Since the aperture is always circular for a parabolic reflector, these antennas are a subset of
antennas known as circular aperture antennas. The geometry is entirely determined by the focal
length and the aperture diameter. In a system of cylindrical coordinates (r,θ , z), an axisymmetric
reflector is obtained by generating the surface of revolution from the curve
4Fz = r2 (1.1)
for 0≤ r ≤ D2
where F is the focal length of the antenna and D is the aperture diameter.
D
Circular
F
ParabolicReflector
Aperture
Focal Pointn
n
Figure 1.2: Cross sectional diagram of an axisymmetric parabolic reflector.
Another type of parabolic antenna that we will consider is an off–axis reflector. The optical
properties of an off–axis reflector are identical to those of an axisymmetric reflector. The only
difference lies in the construction of the reflector surface which can be visualized by taking a
cylindrical cut of a paraboloid (not an axisymmetric cut). A key advantage of the off–axis reflector
is that the feed and its supporting apparatus do not block the reflected radiation. The paraboloid
from which the off–axis reflector is taken is called the parent paraboloid and the focal length of the
antenna is the focal length of this paraboloid. See Figure 1.3 for a visualization of this construction
method.
1.2 Shell and Membrane Theory
The elasticity of inflatable antenna reflectors is an important component of the mathematical
model that we present in Chapter 2. Elastic shells of various shapes had been studied long be-
6
D
Reflector
F
Parent Paraboloid
Circular Aperture
Focal Point
Figure 1.3: Cross sectional diagram of an off–axis parabolic reflector.
fore the 1960s when deployable structures were first conceived. An early analysis of a membrane
problem is Dr. I. Hencky’s 1915 paper Über den Spannungszustand in kreisrunden Platten mit
verschwindender Biegungssteifigkeit (On the Stress State in Circular Plates With Vanishing Bending
Stiffness) [18]. Hencky’s paper discusses (in cylindrical coordinates) an isotropic circular mem-
brane that is fixed at the rim and exposed to a symmetric lateral (z directed) loading, p. He
includes bending rigidity in his analysis and models the axisymmetric vertical displacement of this
membrane, ζ by
Eh2
12(1− ν2)∆2ζ − hr−1B(ζ, F) = p (1.2)
2∆2F + Er−1B(ζ,ζ) = 0 (1.3)
where E,ν and h are the Young’s Modulus, Poisson’s Ratio and thickness of the plate, respectively,
∆ is the Laplacian in polar coordinates and B(u, v) = (ur vr)r . The boundary condition ζ(a) = 0
must be satisfied where a is the radius of the circular membrane. Equations (1.2) and (1.3) are
equivalent to the Von Kármán Equations for a lateral load on a plate [7, Pg. 285]. The antenna
problem that we consider, however, involves a uniform pressure load on the membrane surface
rather than a lateral load and has outward force around the rim for boundary support rather than
a fixed boundary.
W. Fichter recognized the absence of the radial component of the loading in Hencky’s problem
and developed results for the problem with a true uniform pressure load on the membrane in
7
[13]. Fichter showed that the solutions for lateral load problem of Equations (1.2) and (1.3), and
uniform pressure load are quite similar for small loading but diverge slowly when the load grows
larger. At higher loading, the lateral displacement of the membrane reveals a slightly more spher-
ical shape when under uniform pressure and a more uniform distribution of the stress resultants.
Since the surface accuracy of antenna reflectors is important in application, more recent work
has included results on the causes, magnitudes and distribution of deformations on membranes.
Among these studies are gravitational affects on singly–curved membranes having the shape of a
parabolic cylinder [22]. The existence of wrinkle–free solutions to circular membranes problems
has also been explored in [5] and is of importance to deployable antennas; in order to be effective,
the deployed reflector should not have wrinkled areas.
In more recent antenna specific work, Greschik explores the sensitivity of reflector membranes
to certain environmental factors and physical assumptions in [16]. He analyzes the possible errors
of reflector surface accuracy computations when uniformity of the reflector material is assumed,
extreme temperatures are neglected, or regions of wrinkling are ignored. It was found, for ex-
ample, that RMS values could be miscalculated by 1 mm for shallow reflectors (F/D = 4) at a
pressure of 125 Pa when wrinkling is ignored.
In their 1997 paper, Marker and Jenkins reviewed the typical “W” curve error pattern for a
parabolic antenna membrane which is a result following from Hencky’s analysis [21]. They found
that modifications of the boundary position could reduce the maximum deviation of a deformed
reflector up to 58%. We also investigate, in this project, the possibility of reducing parabolic
surface distortions using geometric modifications for the specific antenna reflectors of interest.
The articles cited here focus primarily on axisymmetric membranes and axisymmetric loadings.
While the initial configurations for our model are axisymmetric, the model is not constrained to
compute axisymmetric equilibrium configurations. For this reason, we are able to apply non–
symmetric loading in our analysis. Membrane wrinkling is also included in the model and is a
critical feature to consider since an antenna reflector should be as free of wrinkling as possible.
Our results add to the current knowledge of reflector deformation. We find that distortion
of the reflector membrane supported about the boundary by elastic tendons does not exhibit the
typical “W” profile error pattern. Rather, there is deflection of the membrane near the rim and
substantial (order of 3 mm) vertical displacement at the vertex. We combat these distortions by
8
finding construction patterns for the membrane (adjustable by a certain set of geometric parame-
ters) which will deform to a surface that is sufficient to serve as an effective antenna reflector at
high radio frequency.
1.3 Mathematical Preliminaries
In this section, we will introduce some of the background topics that are used throughout this
dissertation. These few notes also serve as a convenient reference for the reader. First, we present
some notational conventions.
1.3.1 Notational Conventions
The applications covered in the following chapters draw on the theories of elasticity, electromag-
netism and antennas. The standard notations used for these subjects differ and occasionally over-
lap. We will adhere to standard notation as often as possible and clarify which conventions we
adopt in order to remain consistent with notation throughout this dissertation. Unless otherwise
noted, we follow these conventions:
• Lower case italic roman letters (a, b, x , y): scalar quantities or scalar functions. The context
will be clear.
• Lower case bold roman letters (a, b, x, y): vectors in R2 or R3 or vector valued functions
(also referred to as first order tensors).
• The set of vectors i, j,k is the standard basis of R3.
• Any vector displaying a hat (n, for example) has unit length. We will suppress the hat
notation for the standard basis vectors of R3.
• Upper case bold roman letters (A, B, X, Y): second order tensors (for elasticity theory), vector
fields or phasors (for electromagnetic theory). Whether a notation of this type represents a
tensor, vector field or phasor, will be clear in context. Tensors are introduced in Section 1.3.3
and phasors in Section 3.1.2.
• Greek letters of all cases and types will be defined as they are used.
9
1.3.2 Vector Analysis
The development of Maxwell’s Equations and a solution are included to give the reader a com-
plete picture of the reflector’s role in the antenna’s operation. The following vector relations are
important for that section.
The triple product rule is convenient and relates the cross product of three vector fields. For
any three vectors a, b and c,
a× (b× c) = b(a · c)− c(a · b). (1.4)
In addition to the usual definitions of divergence and curl, the following relations are worth con-
sidering especially in this framework. These formulations provide a more intuitive definition of
these two differential operators [23, Sections 1–5 and 1–6]. In each definition, we have the vector
field F ∈ C1(R3), the point x ∈ R3 and a closed ball Vδ centered at x with radius δ.
Definition 1.3.1. The divergence of a vector field, F is the limit of its surface integral per unit volume
as the volume enclosed by the surface goes to zero. That is,
∇ · F(x) = limδ→0
1
Vδ
∫∫
∂ Vδ
⊂⊏⊃ F · n dS.
Definition 1.3.2. The curl of a vector field F is the limit of the ratio of the integral of its cross product
with the outward drawn normal, over a closed surface, to the volume enclosed by the surface as the
volume goes to zero. That is,
∇× F(x) = limδ→0
1
Vδ
∫∫
∂ Vδ
⊂⊏⊃ n× F dS.
We will take advantage of familiar corollaries of Stoke’s Theorem as they are often presented
in Vector Calculus. The version of Stoke’s Theorem specific to the curl of a vector field as well as
the Divergence Theorem are of particular interest here.
Theorem 1.3.1. (Divergence Theorem) If F is a C1 vector field and B ⊂ R3 is a volume with boundary
∂ B, then∫∫∫
B
∇ · F dV =
∫∫
∂ B
⊂⊏⊃ F · n dS (1.5)
10
where n is the unit outward normal of ∂ B, and dV and dS are Lebesgue volume and surface area
measures, respectively [29, Theorem 10.51].
Theorem 1.3.2. (Stoke’s Theorem) If F is a C1 vector field and Σ ⊂ R3 is a surface then
∫∫
Σ
(∇× F) · n dS =
∮
∂Σ
(F · t) ds (1.6)
where n is the unit outward normal of Σ, t is the unit tangent along the boundary ∂Σ, and dS and
ds are Lebesgue surface area and arc length measure, respectively [29, Theorem 10.50].
The following vector calculus identities are useful so we present them for reference. If F and
G are differentiable vector fields and F a differentiable scalar field, then
∇× (∇F) = 0 (1.7)
∇ · (∇× F) = 0 (1.8)
∇×∇× F = ∇(∇ · F)−∇2F (1.9)
∇ · (F×G) = G · (∇× F)− F · (∇×G) (1.10)
1.3.3 Tensor Algebra
In order to fully treat elasticity, we must introduce a little tensor theory. Unlike force, a vector is
not a sufficient device to express the notion of stress in a material. Tensors are also necessary to
formulate expressions for the deformation of a material body. The relationship between the stress
and the strain in a material play a critical role in the model we use to determine the configuration
of the reflector surface. We follow the introduction given by Antman in [1, Section 11.1] since the
notation is consistent with that of our presentation of the model in Chapter 2.
Let e1,e2,e3 be a basis for R3. We can express a vector (first order tensor) by
v =
3∑
i=1
viei ∈ R3
where v= (v1, v2, v3).
Definition 1.3.3. A second order tensor A is a transformation of R3 onto itself.
11
We can represent the second order tensor A as a matrix. The tensor A is evaluated at a v ∈ R3
and denoted by A · v ∈ R3. A product of two tensors A and B is denoted by A · B and satisfies the
relationship
(A ·B) · v = A · (B · v)
for all v ∈ R3. We will encounter the notion of a dyadic product of two vectors in our model. If
a,b ∈ R3, then the dyadic product a⊗ b is a second order tensor that satisfies
(a⊗ b) · v = (b · v)a
for all v ∈ R3. In the case of the first order tensors a = (a1, a2, a3) and b = (b1, b2, b3), we can
represent the dyadic product in matrix form by
a⊗ b =
a1 b1 a1 b2 a1 b3
a2 b1 a2 b2 a2 b3
a3 b1 a3 b2 a3 b3
.
Notice that the matrix representing the dyadic tensor a⊗a would be symmetric. It is also the case
that any symmetric tensor can be decomposed into a dyadic product of vectors.
Definition 1.3.4. The spectral representation of a symmetric tensor A is
A =
3∑
i=1
λi ei ⊗ ei (1.11)
where ei is an orthonormal basis of eigenvectors that correspond to the eigenvalues of the tensor A
[1, Section 11.1].
If we consider two second order tensors Ci j and Dkl , then the product of these two would be
the fourth order tensor
Ei jkl = Ci j ⊗Dkl .
Writing the entire tensor out is not possible since a four dimensional array would be required to
12
satisfy all the subscripts. For example, fixing i = 1 and l = 2 would leave the portion of the tensor
E1 jk2 = C1 j ⊗Dk2 =
C11D12 C12D12 C13D12
C11D22 C12D22 C13D22
C11D32 C12D32 C13D32
.
A tensor inner product “:” is defined in [1, Section 11.1] for dyads. Since the tensors we use
in our model can be expressed as dyads, this definition will be sufficient.
Definition 1.3.5. The tensor inner product “:” for two tensors of the form a⊗b and c⊗d is given by
(a⊗ b) : (c⊗ d) = (a · c)(b · d).
Antman also defines several usual calculus operators which involve tensors [1, Section 11.2].
We present those which we use later.
Definition 1.3.6. The gradient of a vector field F ∈ C1(R3) is a second order tensor of the form
∇F =∂ F
∂ x⊗ i+
∂ F
∂ y⊗ j+
∂ F
∂ z⊗ k.
Definition 1.3.7. The divergence of a second order tensor, T= t1⊗ i+ t2⊗ j+ t3⊗ k, is
∇ · T = ∂ t1
∂ x+∂ t2
∂ y+∂ t3
∂ z
where ti ∈ C1(R3).
13
Chapter 2
Inflatable Elastic Reflector Model
Inflatable antennas are constructed in several different sizes and configurations. Although reflector
diameters on the order of 10 meters or more are conceivable, we have only investigated prototype
antennas with diameters ranging from 0.3 to 2.13 meters. An ideal pressure range for the reflector
assembly is 10 – 15 Pa, although we also observed the envelope at higher pressures of 25 – 30 Pa
to investigate the predicted membrane shape [28].
Inflatable antennas are constructed by molding light–weight material into parabolic form to
achieve the desired geometry for the reflector. Two such parabolic membranes are created and
seamed along their rims. The resulting envelope is pressurized and resembles a clam shell. The
entire clam shell is supported by tendons which are attached to the rim on one end and a rigid
frame on the other. Prototypes that have been observed are made of the material CP1 which is
space qualified, and highly transparent. Typically, one of the parabolic sheets is coated with 1200
Å of aluminum to make the surface a reflective one. The coated half of the clam shell will serve as
the antenna’s reflector. See Figure 2.1 for a schematic of the inflatable antenna system.
The prototypes investigated at NASA’s Glenn Research Center demonstrate some deflection
from parabolic form near the rim of the reflector which subsequently causes a noticeable loss of
power transmitted by the antenna. The distribution of the surface distortions across the reflector
surface is important to understand if we wish to identify what is ultimately responsible for the
reduced antenna performance. These are types of issues that we address in studying the response
of the system to different supporting mechanisms.
14
For large reflectors (diameters closer to the 10 meter range), the molding process may be in-
feasible or too expensive. We therefore consider models for a reflector constructed from flat panels
of material that are cut and seamed in a manner to approximate the parabolic reflector surface.
We will investigate the ability of such a reflector to achieve the desired parabolic shape. For these
constructions, the accuracy of the pressurized reflector will depend on the cutting pattern.
FGravity Vector &
Polar Direction Angle
φ
−kg
Electrical Feed
Transparent Canopy
Rigid Support Structure
Reflector Rim
Aluminum Coated Reflector
Figure 2.1: Inflatable parabolic reflector antenna.
2.1 Mathematical Model
In our mathematical model, we approximate the membrane by a faceted surface, using constant–
strain, plane–stress triangular finite elements. Figure 2.2(a) is an image of the actual faceted
surface that we use for a molded reflector in computations. Although the coated reflector is labeled
in Figure 2.1 (for exposition), we do not actually incorporate the effect of the vaporized aluminum
in our model. The entire membrane is modeled with the material Kapton for which the mechanical
properties are known; see Table 4.2 in Chapter 4. The material parameters will therefore be
identical for both the upper and lower paraboloids in our investigations.
In order to generate the faceted surface, a parabolic mold is created by the parametric equa-
tions
x = a+ r cosθ ; y = a+ r sinθ ; z = r2/(4F), (2.1)
15
for 0≤ θ ≤ 2π and 0≤ r ≤ R f where R f is the radius of the reflector aperture. In Equations (2.1),
a ≥ 0 is a translation that is used to generate a mold for an off–axis reflector. For axisymmetric
reflectors, we take a = 0. The paraboloid of Equations (2.1) is then translated and rotated so that
the rim lies in the plane z = 0 and vertex lies on the z−axis. A symmetric paraboloid is formed
over the plane z = 0 to generate the canopy and produce a clam shell mold. The triangles of
the faceted surface are formed by connecting vertices that initially lie on this mold. To simulate
the molding process of the reflector, we take this initial configuration as the unstrained reference
configuration ΩR; see Figure 2.2(a) for an axisymmetric model. Figure 2.2(b) shows this reference
configuration for off–axis model. Simulating the two flat panel constructions will involve a slightly
different geometry for ΩR to be described in Section 2.2.
The supporting rubber rings are modeled as linearly elastic strings which are hereafter referred
to as supporting tendons. When the envelope is pressurized and the support tendon loads are ap-
plied, the entire inflatable reflector system will seek an equilibrium configuration of minimum
potential energy. In the model, the pressure, tendon stiffness and gravity are treated as parame-
ters which can be modified for parametric studies. Evaluating the shape for various pressure levels
and orientations is of interest as the reflector is likely to operate in different gravity environments.
We follow an optimization–based solution process which determines the equilibrium position
of the inflated reflector by minimizing the total energy of the discretized system [2]. Consider a
deformed configuration of the inflatable reflector system given by Ω = x(ΩR) where
x : ΩR −→ Ω ⊂ R3
is a function that moves each vertex of the unstrained reference configuration to the deformed
configuration. The reference configuration ΩR may be either a subset of R2 or R3 depending on
whether the surface construction is molded or is comprised of developable surfaces which can be
laid out in a plane. Since the faceted surface is ultimately piecewise linear, we require x ∈ D where
D is the completion of C1(ΩR;R3) with respect to the ||x||1,4 norm.
For a particular configuration of the faceted reflector x(ΩR)⊂ R3, the total energy of the entire
16
(a) Axisymmetric construction
(b) Off–axis construction
Figure 2.2: Faceted surface for molded antennas. Dots highlight the reflector rim.
inflatable reflector system is calculated by the functional
ET (x) = E f (x) + Ep(x) + S∗f (∇x) + S∗t (x) (2.2)
where E f (x) is the gravitational potential energy of the reflector film, Ep(x) is the potential energy
of the inflation gas, S∗f (∇x) is the relaxed strain energy in the membrane, and S∗t (x) is the relaxed
strain energy in the supporting tendons.
The following sections include the details of the computations for each of the energy compo-
nents. Following will be the reference configurations for the flat panel designs.
2.1.1 Gravitational and Hydrostatic Pressure Potentials
We include the direction and magnitude of gravity as parameters in our model since the antennas in
question may be operating in a variety of extraterrestrial environments. Given that these antennas
are tested on the ground and used in space applications, one may wish to explore the effect of
zero gravity, a 1g environment and the gravitational environments of other planets. The total
17
gravitational potential energy is calculated by
E f (x) =
∫∫
ΩR
w f (x · g) dA (2.3)
where w f is the film weight density, g is the directed acceleration due to gravity and dA is the
area measure over the faceted reference configuration ΩR. In our model, we compute this quantity
exactly for each triangular facet and then sum over the set of triangles.
For the deformed configuration Ω, let B be the enclosed interior space. We assume that the
inflation gas is not buoyant and that the differential pressure is the constant, p0. Therefore, the
hydrostatic pressure potential energy of the system is given by
Ep(x) =
∫∫∫
B
p0 dV =
∫∫
Ω
⊂⊏⊃ p0k · n dS (2.4)
where n is the unit outward normal and dS is the surface area measure over the membrane. The
divergence theorem was applied to obtain the last equality in Equation (2.4). The energy Ep can
be calculated exactly for the faceted surface Ω.
2.1.2 Film and Tendon Strain Energy
We calculate the total stain energy stored in the film using a strain energy density function. Be-
fore presenting the details of this energy computation, we recall some terminology and results of
elasticity theory.
Deformation and Strain
Strain is a measure of deformation that a material body undergoes when either an internal or ex-
ternal force is applied. This differs from a rigid body displacement in that said deformation entails
a change in the relative positions of internal material points. Figure 2.3 depicts the deformation
of a body.
Let e1,e2,e3 be a basis for R3 and α1,α2,α3 be the corresponding independent coordinates
for this basis. A deformation x of a material M has three independent components and can be
18
expressed generally as
x = x1(α1,α2,α3)e1+ x2(α1,α2,α3)e2+ x3(α1,α2,α3)e3.
Consider two points which are infinitesimally close in the material M , p = (p1, p2, p3) and p+ δp
where δp = (δp1,δp2,δp3).
e2
e3
e1
M
p
p+δp
M x(M)
x(p)
x(p+δp)
x
Figure 2.3: Deformation of a material body M under a deformation x.
A deformation of M will move both points to new locations given by x(p) and x(p+δp). Since
δp is a small displacement, we assume that x(p+δp)−x(p) is relatively small. In other words, we
assume that x is a continuous and differentiable material deformation. We may therefore expand
the function components x i in a Taylor series about the component p j , i.e.,
x i(p j +δp j) = x i(p j) +∂ x i
∂ α jδp j +
1
2
∂ 2 x i
∂ α2j
(δp j)2 + O(δp j)
3.
In the assumption that the difference δp j is infinitesimal, we are justified in neglecting the terms
of order (δp j)2 and higher. Hence,
x i(p j +δp j) = x i(p j) +∂ x i
∂ α jδp j . (2.5)
Equation (2.5) may expressed in the form x i(p j +δp j) = x i(p j) + ei jδp j + ωi jδp j where
ei j =1
2
∂ x i
∂ α j+∂ x j
∂ αi
and ωi j =1
2
∂ x i
∂ α j− ∂ x j
∂ αi
19
are the differential displacements generating strain and rotation, respectively [6, Section 10.2].
Notice that Equation (2.5) represents nine different equations since the indices i and j each run
from 1 to 3. There are nine partial derivatives of the form ∂ x i/∂ α j for this deformation x which
make up the gradient of the deformation. The gradient ∇x is the deformation gradient and is the
second order tensor explicitly given by
∇x =
3∑
i=1
3∑
j=1
∂ x i
∂ α jei ⊗ e j .
Since ∇x is a tensor of second order, it may be expressed as the 3×3 matrix [8, Section 1.8]
∇x
i j =
∂ x i
∂ α j
for 1≤ i, j ≤ 3. If x is a differentiable deformation at a point p, then we can write
x(p+ δp)− x(p) = ∇x(p)δp+ o||δp||.
Taking the transpose and multiplying on the left gives,
||x(p+δp)− x(p)||2 = δpT∇x(p)T∇x(p)δp+ o||δp||2. (2.6)
The Cauchy–Green Strain Tensor is the second order tensor produced in Equation (2.6), namely
C = ∇xT∇x.
Note that C is a symmetric tensor by construction. This tensor is also positive definite at all points
p where x is differentiable and it can used to compute the length of a curve deformed by x [8,
Section 1.8].
Not all deformations involve strain. For example, a deformation which consists simply of
translations and rotations of the material will not impose strain on the material. In fact, C = I if
and only if the deformation is a rigid one; see [8, Theorem 1.8–1]. Since we might expect that
the strain tensor should be zero for rigid deformations, we are lead to consider the material strain
20
tensor [1, Section 12.2]
G = 12(C− I) = 1
2(∇xT∇x− I ).
The tensor G is also referred to as the Cauchy–St. Venant strain tensor [8, Section 1.8] or the
engineering strain tensor. Note that G is a symmetric tensor by its construction and by the symmetry
of C. Also, G= 0 if and only if the deformation x is rigid. The matrix equivalent form of this tensor
has the linearized components of the strain of the material
G =
ǫ1 γ12 γ13
γ21 ǫ2 γ23
γ31 γ32 ǫ3
where we adopt the convention ǫi = γii and observe that γi j = γ ji by the symmetry G.
Stress and Constitutive Relations
At any given point, p in a material, the stress is a ratio of the force acting on an infinitesimal planar
slice of material at p to the area of that planar slice. Namely, we can define a stress component by
τi j(p) = limA→0
f · e j
A
where A is the area of a planar slice of the material at point p which has normal vector ei and f is
the force acting on that planar slice. See Figure 2.4 for a diagram.
e2
e3
e1
Mτ22 = σ2
τ23
τ21
nA = e2
Net stress on A at p: ~τ2(p) =∑
τ2ieiA
p
Figure 2.4: Stress components τ2i at point p in material M .
21
Since there are three independent planes at any point, there are nine scalars to fully describe
the stress in a material at that point. These stress components are elements of the Cauchy stress
tensor
T =
σ1 τ12 τ13
τ21 σ2 τ23
τ31 τ32 σ3
=
~τ1
~τ2
~τ3
where ~τi are the vector stress components as shown in Figure 2.4. It can be shown that T is a
symmetric tensor using equations of equilibrium for the stress and force components. If F is a
force intensity throughout a static body M that occupies volume B, then the stresses in that body
must satisfy the force equilibrium condition,
∫∫∫
B
F dV +
∫∫
∂ B
⊂⊏⊃ T · n dS = 0
where dS is Lebesgue surface area measure [6, Section 9.3]. Applying the Divergence Theorem to
the second integral, we obtain
∫∫∫
B
F dV +
∫∫∫
B
∇ · T dV = 0.
Since this equation must hold for any volume, B, we obtain the equilibrium condition
F + ∇ · T = 0 (2.7)
where the divergence of the second order tensor T can be obtained by the identity presented in
Definition 1.3.7. We must also have a balance of the moments about the origin at all points in the
static body. We have∫∫∫
B
r× F dV +
∫∫
∂ B
⊂⊏⊃ r× T · n dS = 0 (2.8)
where T ·n is the vector component of the stress tensor in the direction of the normal to the surface
∂ B and r is a position vector in the volume B. Applying the Divergence Theorem to the second
22
integral in Equation (2.8) gives
∫∫∫
B
r× F dV +
∫∫∫
B
∇ · r× T
dV = 0.
Again, since this equation must hold for an arbitrary volume, we have
r× F + ∇ · r× T
= 0. (2.9)
If we consider the i, j and k components of Equation (2.9), we obtain
r2F3− r3F2 = ∇ · (r2~τ3− r3~τ2) = r2(∇ · ~τ3)− r3(∇ · ~τ2) +τ32−τ23 ;
r3F1− r1F3 = ∇ · (r3~τ1− r1~τ3) = r3(∇ · ~τ1)− r1(∇ · ~τ3) +τ13−τ31 ;
r1F2− r2F1 = ∇ · (r1~τ2− r2~τ1) = r1(∇ · ~τ2)− r2(∇ · ~τ1) +τ21−τ12 .
Applying the corresponding components of Equation (2.7), we obtain τi j−τ ji = 0 for i 6= j. Thus,
T is symmetric.
The Cauchy stress tensor contains the stress components depending on the coordinate frame
of the deformed configuration. In our model, we calculate the strain energy over the reference
configuration. We will use the second Piola–Kirchoff stress tensor since the stress components in
that tensor are in terms of coordinates of the reference configuration. The second Piola–Kirchoff
stress tensor is
S = J(∇x)−1T(∇x)−T
where J = det∇x [8, Section 2.6]. One can show that S is a symmetric tensor by using the
symmetry of T and noting that
ST = J
(∇x)−1T(∇x)−TT= J(∇x)−1TT (∇x)−T .
The strain that an elastic material undergoes is directly related to the stress in the material. The
stress and strain tensors that were defined are related by the generalized Hooke’s Law. In particu-
lar,
τi j = ki1γ1 j + ki2γ2 j + ki3γ3 j + ki4γ4 j + ki5γ5 j + ki6γ6 j . (2.10)
23
This linear system may be inverted and solved for the strain components in terms of the stresses.
For a homogeneous material, small elastic deformations are governed by the more general tensor
relation
Si j = Ki jklGkl
where Ki jkl is the fourth order tensor of elastic moduli. This tensor will depend on the nature of the
material and its mechanical properties. Since we know that G and S are both symmetric tensors,
we have the conditions K jikl = Ki jkl = Ki jlk. Thus, there are only 36 independent elements among
the 81 elements in K. These independent elements are the constants ki j in Equation (2.10).
Film Strain Energy
In this section, we derive the total energy contained in the strained film, S∗f (∇x). Since the reflector
material has thickness on the order of microns (10−6 meters), we assume a state of plane stress
in the membrane and a uniform material thickness, h, throughout. A derivation of the stress and
strain tensors gives similar results to those seen above for the three dimensional case; see [6,
Chapters 1 and 2]. In particular, symmetry of the strain and stress tensors still holds and they take
the form
G =
δ1 0
0 δ2
and S =
µ1 0
0 µ2
when expressed in terms of the principal directions.
The strain energy in the film is calculated using the relaxed strain energy density W ∗f by the
equation
S∗f (x) =∫∫
ΩR
W ∗f (x) dA
where dA is the area measure of ΩR. The strain energy density function of a membrane is
Wf =12
G : S (2.11)
where “:” is the tensor inner product as defined in Definition 1.3.5. Equation (2.11) is due to
Koiter and is derived by Ciarlet in [9, Pg. 548]. The full shell equations of Koiter include a
bending energy component which is O(h3). We neglect the bending energy for our problem since
24
the reflector membrane has thickness 10−6 meters. The strain energy of Equation (2.11) is O(h).
We assume that the material is linearly elastic and isotropic. An elastic body is one which
regains its original configuration when the forces acting on it are removed. A material is said to be
isotropic if its mechanical properties related to stress at a point, are identical in all directions [6,
Section 3.1]. In this case, the strain and stress tensors are related by
S =hE
1− ν2
G+ νcof(G)T
, (2.12)
where cof(G) is the 2×2 cofactor matrix of G, E is the Young’s Modulus of the film and ν is the
Poisson’s Ratio of the film. Note that G and S will have the same principal axes (eigenvectors)
since they are related by a linear stress–strain constitutive relation for an isotropic film [2]. Also,
G and S are symmetric tensors so the Spectral Representation Theorem implies that
G = δ1n1⊗ n1+δ2n2⊗ n2
S = µ1n1⊗ n1+µ2n2⊗ n2
where n1 and n2 are orthonormal vectors which we take to be the principal directions.
Since each triangle in the faceted surface is assumed to have constant stress within, each tri-
angle will have its own principal directions, principal strains and stress resultants. For a triangular
facet T ∈ ΩR, the principal strains are δ1 and δ2 and the principal stresses are µ1 and µ2. Applying
the spectral decompositions of G and S along with the Equation (2.12) to Equation (2.11) gives
Wf (T ) = hEδ2
1 +δ22 + 2νδ1δ2
2(1− ν2).
A thin compliant membrane such as those used for these inflatable antennas, will not resist
compressive stresses but will wrinkle instead. We model this wrinkling with the relaxed strain
energy density, W ∗f . The method applied here is due to Pipkin and involves partitioning the faceted
surface into three sets: the set of slack facets, the set of wrinkled facets, and the set of taut facets
[26]. For any given triangular facet, T ∈ ΩR, the state is determined by the principal strains and
stresses. The conditions of Equation (2.13) describe the possible states: slack (a), wrinkled (b,c)
25
or taut (d). The relaxed strain energy density for triangle T is then calculated by
W ∗f (T ) =
0 if δ1 < 0 and δ2 < 0; (a)
hEδ2
2
2if µ1 ≤ 0 and δ2 ≥ 0; (b)
hEδ2
1
2if µ2 ≤ 0 and δ1 ≥ 0; (c)
hEδ2
1 +δ22 + 2νδ1δ2
2(1− ν2)if µ1 > 0 and µ2 > 0. (d)
(2.13)
See [2] for additional details of the relaxed strain energy for an isotropic film.
Tendon Strain Energy
Let Nt be the total number of tendons. The tendons are modeled as linearly elastic strings. There-
fore, the relaxed strain energy density in the mth supporting tendon is
W ∗t,m =
0 if ǫm < 0;
Kt,mǫ2
m
2if ǫm ≥ 0,
(2.14)
where Kt,m and ǫm are the tendon stiffness constant and strain of the tendon, respectively [2]. The
strain in the tendon is calculated by
ǫm =Γm−Γm,R
Γm,R(2.15)
where Γm and Γm,R are the lengths of the mth tendon in the deformed and reference configurations,
respectively. The initial tension in the supporting tendons can be adjusted by foreshortening all
the lengths, Γm,R by a percentage γ. Support loss in a region can be modeled by setting Kt,m = 0
for desired values of m. We can also simulate non–symmetric loading by modifying the stiffness
26
constants Kt,m as necessary. The total strain energy in the tendons can then be found by calculating
S∗t (x) =Nt∑
m=1
∫ Γm,R
0
W ∗t,m(s) ds
!
.
For computational simplicity, the rigid support structure (to which one end of each tendon is at-
tached) is always in the plane z = 0.
2.2 Flat Panel Constructions
As previously mentioned, inflatable antennas are constructed by molding sheets of material into
parabolic form. These antennas are most easily modeled by taking a reference configuration ΩR
which is comprised of vertices that are set initially on a paraboloid. Figure 2.2(a) shows the initial
configuration for an axisymmetric reflector. While this geometry is ideal to approximate the design
of the antennas, it is not clear what impact the molding process might have on the mechanical
properties of the material. Also, since casting material to a certain shape on a precision mandrel is
an expensive process, constructing an antenna of diameter near to 10 meters may be infeasible. It
is reasonable, therefore, to consider other geometric constructions that involve only flat sheets of
material that are seamed together.
2.2.1 Flat Gore Construction
The reference configuration for the flat gore construction is generated by initially setting vertices
along parabolic arcs extending from the vertex to the rim. The arcs fit the desired parabolic shape
of the ideal reflector. There is one arc of vertices on the boundary of each pair of adjacent gores.
The vertices of the ribs used to generate the entire configuration are highlighted in Figure 2.5 as
dots over the corresponding vertices.
The remaining vertices for the reference configuration are positioned on straight lines between
adjacent pairs of these arcs so that each pie slice is a developable surface (a surface that can be
flattened into a plane without any strain or distortion). We then generate the triangular mesh
of this construction with these vertices. The result is a reference configuration that models a
27
Figure 2.5: Flat gore construction. Generating ribs indicated by dots.
surface having no curvature within the panels in the circumferential direction. Notice that the
distribution and pattern of the triangular mesh is similar to the distribution for the molded antenna
of Figure 2.2(a). The configurations differ in that a gore of Figure 2.5 is initially a developable
surface while a typical “gore” of Figures 2.2(a) and 2.2(b) is initially doubly curved.
The description in the previous paragraph describes the construction of an unstrained spacial
configuration for this reflector. The actual reference configuration, however, would involve cutting
the gores apart and laying them out in the plane as seen in Figure 2.6. Note that the gores are
paired and that each pair is joined by a common section, Si at the vertex to form a “super gore”.
For example, gores Ω1 and Ω2 are both attached to the unshaded region S1. We construct the
gores in this fashion in order to avoid a clustering of narrow triangles around the paraboloid’s
vertex. The arrangement of the triangles in the Si region still allows the vertices to lie on a perfect
paraboloid (as shown in Figure 2.5) such that the triangles are unstrained relative to the reference
configuration of Figure 2.6. If Ng is the total number of gores, then the reference configuration ΩR
is
ΩR =
Ng⋃
i=1
Ωi
∪
Ng/2⋃
j=1
S j
⊂ R2.
2.2.2 Flat Band Construction
The other flat construction pattern consists of concentric bands of material which is a surface whose
panels have no curvature in the radial direction but are curved in the circumferential direction.
The different panels are modeled in the reference configuration by first fixing rings of vertices to
28
Ω1 Ω2 Ω3 Ω4
u
v
Vertex
Separation of twogores begins here
S1 S2
Figure 2.6: Four gores of the reference configuration for the flat gore construction.
the desired parabolic shape. These are shown by the rings of dots in Figure 2.7. Rows of triangles
are then generated between the rings to complete the reference configuration. These bands are
sections of cones and are therefore constructible from flat panels of material.
Figure 2.7: Flat ring construction. Generating rings indicated by dots.
In addition to lower construction costs, there are computational advantages to using these flat
panel constructions in our model. We will develop in Chapter 5, a parameter that adjusts the
geometry of the flat band construction. The parameter can then be used to reduce the surface
error. A geometric modification to these constructions translates only to a different cutting pattern
of the panels and therefore very little change in the actual construction procedure. Modifications
are not reasonable to consider in the molded construction case since there is not necessarily a
precision mandrel available to generate a geometry deemed appropriate.
29
Chapter 3
Parabolic Reflector Efficiency
In this chapter, we will establish some methods of quantifying the performance of a particular
antenna reflector shape. We consider two different classes of measurement. The first technique in-
volves characterizing the antenna. Characterization of an antenna entails analyzing how the reflec-
tor in question transmits radiation in certain directions. The electromagnetic fields that comprise
the signal sent must be computed so that both the magnitude and frequency of the transmitted
signal can be determined. These measurements require an understanding of antenna theory as
well as electromagnetic theory.
The other method of measurement relates purely to the geometric distortions in the sense that
only the displacement of the antenna reflector from a parabolic shape is considered. These errors
are referred to as RMS error measurements. We will present two different types of RMS error,
radiometric and Euclidean.
3.1 Electromagnetic Theory
In order to evaluate the effectiveness of a certain antenna geometry, we must understand how it
reflects radiation. The media by which a signal travels is electromagnetic radiation and thus, some
background in Electromagnetic Theory is necessary. We present some material from this theory
and Maxwell’s system of differential equations that govern the behavior of radiation. Note that
the notation for the complex number i =p−1 is used throughout. The engineering notation in
electromagnetic theory for the imaginary unit is j.
30
3.1.1 Radiation, Electric and Magnetic Fields
The fundamental electromagnetic particles are the electron, the proton and the neutron. Each is
attributed with a certain charge and mass that is listed in Table 3.1 [17, Section 26–6]. Charge is
a quantity measured in coulombs and is analogous to the notion of mass in mechanics.
Particle Symbol Charge (coul) Mass (kg)Electron e −1.602× 10−19 9.10910× 10−31
Proton p 0 1.67482× 10−27
Neutron n +1.602× 10−19 1.67252× 10−27
Table 3.1: Charge and mass of the three fundamental charged particles.
Coulomb’s Law for force, F, acting on electric charges (analogous to Newton’s law of gravitation
for masses) is
F =q1q2
4πǫ0
r2− r1
| r2− r1|3
where q1 and q2 are point charges positioned at r1 and r2, respectively. It is important to note
that the force repels the two charges when they have the same sign and attracts them when the
charge signs are opposite. The constant ǫ0 is the permittivity of free space and has an experimental
value of ǫ0 = 8.854 × 10−12 coul2 / N m2 [17, Section 26–4]. Note also that the coefficient in
Coloumb’s Law is1
4πǫ0= 8.9874× 109 N · m2 / coul2.
Electric Fields and Gauss’ Law
Definition 3.1.1. Suppose there are a collection of charges in space which together generate an electric
force field. The electric field is the limit of the ratio of the force on a test charge q to the charge value
as q→ 0, or
E(r) = limq→0
Fq(r)
q
where r is the spatial position of the test charge. The electric field has units of volts per meter; see [17,
Section 27–2].
While Definition 3.1.1 is an intuitive definition, electric fields are rarely measured in this man-
ner due to experimental difficulties. Electric fields are typically computed from the scalar electric
31
potential, V . The electric potential has units of volts. One can find the electrostatic potential
energy at any point r by integrating over the charge density ρ(r′) throughout space [23, Section
2–4]. We have,
V (r) =1
4πǫ0
∫∫∫
R3
1
| r− r′| ρ(r′)dr′.
The electric potential energy is related to the electric field by E = −∇V . Note also that an electric
field generated by n point source charges can be found by the vector sum
E(r) =
n∑
i=1
qi
4πǫ0
r− ri
| r− ri |3. (3.1)
Theorem 3.1.1. Gauss’ Law states that the total electric flux through a closed surface, Σ, is propor-
tional to the total charge enclosed by the surface. In particular,
∫∫
Σ
⊂⊏⊃ E · n dS =n∑
i=1
qi
ǫ0. (3.2)
Furthermore, any charge located outside Σ gives no contribution to the value of the integral on the left
hand side of Equation (3.2); see [23, Section 2–6].
Proof. This law can be shown by considering an arbitrary closed surface enclosing a collection of
n charges. Recalling Equation (3.1), we have
∫∫
Σ
⊂⊏⊃ E · n dS =1
4πǫ0
n∑
i=1
qi
∫∫
Σ
⊂⊏⊃ r− ri
| r− ri |3· n dS
. (3.3)
To handle the integral on the right hand side, we recall Vi(r) = | r− ri |−1 which is the potential
generated by the charge qi . Notice that
∇Vi =r− ri
| r− ri |3and ∇2Vi = 0
for r 6= ri . These two relations can be seen by applying the Laplacian and gradient operators for
spherical coordinates. Recall that [35, Pg. 188]
∇2V =∂ 2V
∂ r2 +2
r
∂ V
∂ r+
cscθ
r2
∂
∂ θ
sinθ∂ V
∂ θ
+csc2 θ
r2
∂ 2V
∂ φ2 .
32
It remains to show that
∫∫
Σ
⊂⊏⊃ ∇Vi · n dS =
4π if qi is inside Σ;
0 if qi is outside Σ.(3.4)
If qi is located outside of the surface Σ, then ∇2Vi = 0 everywhere inside Σ. By Green’s Theorem,
the integral in Equation (3.4) vanishes. This establishes that no charge outside Σ contributes to
the flux of E through Σ.
For qi inside Σ, we take the usual approach to handling singularities in the study of Laplace’s
Equation [32, Section 4.2]. Take a sphere, Σǫ with radius ǫ which is centered at the singularity ri .
Notice that | r− ri |= ǫ on Σǫ and that the normal of Σǫ is directed inward along the vector r− ri .
Hence,∫∫
Σ
⊂⊏⊃ ∇Vi · n dS = −∫∫
Σǫ
⊂⊏⊃ r− ri
| r− ri |3· −(r− ri)
| r− ri |dS.
We reduced this integral to one over the small sphere since the potential function Vi is harmonic
elsewhere within the closed surface S. Changing to spherical coordinates gives,
∫∫
Σ
⊂⊏⊃ ∇Vi · n dS = limǫ→0
∫ π
ǫ
∫ 2π
0
1
ǫ2 ǫ2 sinφ dθ dφ = 4π.
Applying the Divergence Theorem
Equation (1.5)
to Gauss’ Law, we obtain
∫∫∫
B
∇ · E dV =
∫∫∫
B
ρ
ǫ0dV (3.5)
where ∂ B = Σ. Since Equation (3.5) must hold for any simply connected region B, we have
∇ · E = ρ
ǫ0. (3.6)
Current and the Continuity Equation
In the previous section, the electric fields were computed assuming that the charges generating
them were fixed for all time. We now introduce the notion of traveling charge which is given the
33
term current.
Definition 3.1.2. Current is the rate of flow of charge in a conducting medium past a given point. If
q(t) is the net charge transported past a particular point at time t, then current is defined by
I =dq
d t
at said point. The unit of current is the ampere equal to 1 coulomb / second; see [23, Section 7–2].
Assume that a conducting medium is carrying charged particles qi at corresponding velocities
vi . Then the differential current through an element of area da is given by
dI =
n∑
i=1
qiNivi
!
· n da
where the quantity J =∑
i qiNivi is the current density; see Figure 3.1. Current is usually the
result of one class of charge carrier namely, electrons. Hence, we can simplify the expression of
current to
dI = qeNv · n da
where qe = 1.602× 10−19 coul, the charge of one electron. Here, N is the number of electrons in
the element da and v is assumed to be a uniform velocity of the electrons.
e
e
ee
e
e
e
e
da
n
v
Figure 3.1: Charge with velocity v flowing through a differential cross section, da.
34
It follows that the current passing through any surface S′ is obtained immediately by
I =
∫∫
S′J · n dS.
For an arbitrary closed surface ∂ B, we can relate the current passing through the surface to the
charge density in the enclosed region B. Since current is also the time derivative of charge, we
form the equationd
d t
∫∫∫
B
ρ(r, t) dV =dq
d t= −
∫∫
∂ B
⊂⊏⊃ J · n dS,
where the negative sign indicates an increase in charge density for an inward flow of current.
Using the Dominated Convergence Theorem on the left hand side and the Divergence Theorem on
the right, we see that∫∫∫
B
∂ ρ
∂ tdV = −
∫∫∫
B
∇ · J dV
for an arbitrary region B. We therefore obtain the Continuity Equation
∇ · J+ ∂ ρ∂ t
= 0. (3.7)
Magnetic Fields and Biot–Savart Law
The magnetic field is as important to electromagnetic theory as the electric field. We begin with
the magnetic induction field which is defined using the notion of a test charge.
Definition 3.1.3. The magnetic induction B is the vector field which satisfies the relation
F = qv× B
for all velocities v. The unit of magnetic induction is the tesla (T) equal to 1 Weber / m2.
A Weber is related to the fundamental SI units by 1 Weber = 1 Joule / Ampere. Let us also
simultaneously introduce the magnetic intensity field H,
B = µ0H
where µ0 = 4π× 10−7 N / Amp2 is the permeability of free space [33, Pg. 6]. The field H has
35
units of Ampere / meter.
The relation in Definition 3.1.3 is limited to the computation of a magnetic force on a charge.
In order to calculate the magnetic induction at a point, we first derive a new expression that relates
B to current and then apply a law due to Biot and Savart.
Consider a line of current, C , having electrons as the charge carrier. First note that the force
on the element of conducting material d~ℓ from Definition 3.1.3 can be expressed as
dF = NA|d~ℓ|qev× B
where N is the number of electrons per unit volume, A is the cross section of the conducting
material and d~ℓ is the differential direction vector of the current. Since d~ℓ and v are parallel
vectors, we can rewrite this expression as
dF = NA|v|qd~ℓ× B.
Recalling the relations developed for the computation of current, we find that the force on a test
charge as a result of the moving current is given by
F =
∮
C
Id~ℓ× B. (3.8)
The Biot–Savart Law for the force of one loop of current on another is given by
F =µ0
4π
∮
C1
∮
C2
d~ℓ2×
d~ℓ1× (r2− r1)
| r2− r1|3
where C1 and C2 are the two paths of the two current loops [23, Section 8–3].
From the Biot–Savart Law and Equation (3.8), we obtain the magnetic field generated by the
first loop of current, C1 as
B(r) =µ0
4π
∮
C1
I1d~ℓ× (r− r1)
| r− r1|3.
Rather than integrating along C1, we can (equivalently) integrate over a volume B that encloses
36
the contour C1 and define the current density J that has supp(J) = C1. Then,
B(r) =µ0
4π
∫∫∫
B
J(r1)×r− r1
| r− r1|3dV.
Take the divergence of both sides to obtain
∇ ·B(r) = µ0
4π
∫∫∫
B
∇ ·
J(r1)×r− r1
| r− r1|3
dV. (3.9)
We refer to the vector identity in Equation (1.10) while noting that J(r1) is constant. Thus, Equa-
tion (3.9) becomes
∇ ·B(r) = − µ0
4π
∫∫∫
B
J(r1) ·
∇× r− r1
| r− r1|3
dV. (3.10)
The term within the parenthesis of Equation (3.10) is a curl of a gradient. Therefore, by the vector
identity of Equation (1.8), we obtain
∇ ·B = 0. (3.11)
This is a significant result since the Divergence Theorem implies that the magnetic flux through
any closed surface S must be zero [23, Pg. 154]. Hence, no isolated magnetic poles exist.
3.1.2 Maxwell’s Equations
Maxwell’s Equations consist of Equations (3.6) and (3.11) of the previous section as well as Equa-
tions (3.14) and (3.15) of this section. The last two equations can be derived from widely accepted
observations known as Ampere’s Law and Farady’s Law.
The Laws of Ampere and Faraday
Ampere’s Law states that the magnetic field due to a distribution of current satisfies
∮
C
H · d~ℓ =∫∫
Σ
J · n dS
where C is the contour boundary of some surface Σ [23, Section 15–1]. This law, however, fails
for a class of examples involving capacitors as shown in Figure 3.2.
37
I
S1
C
S2
Capacitor Plates
n1
n2
Figure 3.2: A contour C attached to two surfaces passing through different current.
Example 3.1.1. From Figure 3.2, we can apply Ampere’s law to the surface S1 where C is the contour
boundary. We can also use Ampere’s Law with S2 where C is again the contour boundary. Note that
we obtain
I =
∫∫
S1
J · n1 dS =
∮
C
H · d~ℓ =∫∫
S2
J · n2 dS = 0 (3.12)
which is a contradiction.
This contradiction arises because
∫∫
S2
J · n2 dS −∫∫
S1
J · n1 dS 6= 0. (3.13)
We notice that the integral difference in Equation (3.13) is the closed surface integral on S1 ∪ S2.
Notice that n1 is in the direction of the current in Figure 3.2 and so −n1 is needed for the outward
normal. In order to eliminate the contradiction we found in Equation (3.12), we need a vector
field J′ which has the property that
∫∫
S1∪S2
J′ · n dS =
∫∫
S2
J · n2 dS −∫∫
S1
J · n1 dS = 0.
Since S1 ∪ S2 is a closed surface, the Divergence Theorem implies that this condition is equivalent
to finding a vector field with ∇ · J′ = 0. The Continuity Condition in Equation (3.7) and Gauss’
Law in Equation (3.6) suggest that we should select
J′ = J+ ǫ0∂ E
∂ t.
38
Taking the divergence, we indeed obtain
∇ · J′ = ∇ · J+∇ ·
ǫ0∂ E
∂ t
=∂ ρ
∂ t− ǫ0
∂ (ρ/ǫ0)
∂ t= 0.
This major contribution is due Maxwell and yields the modified Ampere’s Law
∮
C
H · d~ℓ =∫∫
Σ
J+ ǫ0∂ E
∂ t
· n dS
which holds for an arbitrary surface Σ. An application of Stoke’s Theorem
Equation (1.6)
on the
right hand side allows us to find
∇×H− ǫ0∂ E
∂ t= J. (3.14)
Farady hypothesized that the electromotive force in a circuit is related to the magnetic flux by
∮
Γ
E · d~ℓ+ d
d t
∫∫
Σ
B · n dS = 0
where Γ is the closed contour boundary of the surface Σ. Using Stoke’s theorem on the left hand
side and using the Dominated Convergence Theorem on the right hand side, we see that
∫∫
Σ
∇× E+∂ B
∂ t
· n dS = 0
for any simple C1 surface Σ [23, Pg. 171]. This implies that the integrand must be the zero vector
which yields the equation
∇× E+∂ B
∂ t= 0. (3.15)
Phasors and Maxwell’s Equations
Before proceeding, we will change the mathematical nature of the vector fields that have been
presented in this chapter. We assume that the radiation waves in the scope of this dissertation
will be monochromatic, i.e., characterized by a single frequency, ω. The time dependence of the
fields is introduced with a factor of ei(ωt+φ) to incorporate the oscillation of the radiation waves of
frequency ω. The angle φ is the phase of the radiation. From this point forward, the fields E and
39
B are therefore understood to represent
E(r, t) =
E(r)eiφeiωt and B(r, t) =
B(r)eiφeiωt ,
respectively. These vector fields with the exponential factor amended are known as phasors and
encompass (in addition to E and B) the fields H and J. We can therefore express the Equa-
tions (3.6), (3.11), (3.14) and (3.15) as
∇ · E = ρ/ǫ0 (3.16)
∇ ·H = 0 (3.17)
∇×H− iωǫ0E = J (3.18)
∇× E+ iωµ0H = 0 (3.19)
where ω is the frequency of the radiation wave and i =p−1. Equations (3.16) – (3.18) are
Maxwell’s Equations for monochromatic electromagnetic waves.
We assume that the region where these waves live is composed of a single media and that the
region ends where a new media is encountered. This can be physically represented by a wall that
interferes with radiation waves, a boundary between oil and water, or any other obstruction that
impedes electromagnetic waves. A boundary is essentially any place where refraction or reflection
occurs. The main boundary of interest for our problem, is the antenna reflector surface. The
boundary conditions that we will use are
n× (H2−H1) = Js and (E2− E1)× n = Ms
where n is the unit normal of the boundary surface pointing into medium 2. The fields Js and Ms
are the electric and magnetic surface currents at the boundary, respectively. See Figure 3.3 for an
illustration.
40
Medium 1
JsMedium 2
Ms
E1
E2
H1
H2
n
Figure 3.3: Boundary conditions for Maxwell’s Equations.
Solution to Maxwell’s Equations
A solution to Maxwell’s Equations can be found by taking advantage of the two decoupled equa-
tions among Equations (3.16) – (3.19). From Equation (3.17), we see that the magnetic field H
has no divergence. By the vector identity in Equation (1.8) there exists a field A such that
H = ∇×A. (3.20)
The field A is referred to as the magnetic vector potential. Substituting Equation (3.20) into Equa-
tion (3.19) gives
∇× (E+ iωµ0A) = 0.
By Equation (1.7), there exists a scalar field Φ such that
E+ iωµ0A = −∇Φ. (3.21)
We refer to the scalar field Φ as the electric scalar potential. From Equation (3.18) and the definition
of A, we have
∇×∇×A− iωǫ0E = J. (3.22)
Using Equations (1.9) and (3.21), Equation (3.22) becomes
∇(∇ ·A)−∇2A− iωǫ0(−iωµA−∇Φ) = J.
Linearity of the gradient then gives
∇2A+ω2ǫ0µ0A−∇(iωǫ0Φ+∇ ·A) = −J.
41
We chose the divergence of A such that the third term on the left hand side is zero [33, Pg. 10].
This leaves the differential equation of A,
∇2A+ω2ǫ0µ0A = −J. (3.23)
Solving this differential equation is equivalent to solving the system of scalar Helmholtz equations
∇2+ β2Aα = −Jα (3.24)
for α= x , y, z and β2 =ω2µ0ǫ0.
Proposition 3.1.1. Suppose Aα satisfies the outgoing radiation condition given by
∂ Aα∂ r
+ iβAα = O
1
r2
,
which models the decay of the radiation as it travels from the current density source. Then the solution
to Equation (3.24) is
Aα(r) =
∫∫∫
R3
Jα(r′)
exp− iβ | r− r′|
4π| r− r′| dr′.
See [25, Pg. 460].
Proof. Define the operator L(u) = ∇2u+ β2u. A fundamental solution of the homogeneous form
of Equation (3.24) is
Ψ(r, r′) =exp− iβ | r− r′|
| r− r′| (3.25)
for r′ 6= r [12, Pg. 314]. Since there is a singularity in Ψ at r, we define the domain
B′ǫ =n
r′ ∈ R3
ǫ ≤ | r− r′| ≤ Ro
and apply Green’s Identity to the functions Ψ and Aα on the domain B′ǫ. We obtain,
∫∫∫
B′ǫ
Ψ L
Aα− Aα L
Ψ
dr′ =∫∫
∂ B′ǫ
Ψ∂ Aα∂ n− Aα
∂Ψ
∂ n
dS (3.26)
where ∂ B′ǫ has two components: a small sphere of radius ǫ centered at r, and a large outer sphere
42
of radius R also centered at r. For the sphere | r− r′| = ǫ, the normal to B′ǫ is directed toward the
center point, r. Hence,
Iǫ =
∫∫
| r−r′|=ǫ
e−iβ | r−r′|
| r− r′|∂ Aα∂ n− Aα
e−iβ | r−r′|iβ | r− r′|+ 1
| r− r′|2!
dS
=
∫ π
0
∫ 2π
0
e−iβǫ
ǫ
∂ Aα∂ n− Aα
e−iβǫiβǫ+ 1
ǫ2
ǫ2 sinφ dθ dφ
=
∫ π
0
∫ 2π
0
ǫe−iβǫ
∂ Aα∂ n− iβAα
− Aαe−iβǫ
sinφ dθ dφ
Since the factors ǫ and e−iβǫ are constant over the sphere | r− r′| = ǫ, they can be moved outside
the integral. Since the sphere | r− r′|= ǫ is centered at r, we obtain
Iǫ = −4πAα(r)e−iβǫ +O(ǫ). (3.27)
On the outer component of the boundary, | r− r′|= R, we have
IR =
∫∫
| r−r′|=R
Ψ∂ Aα∂ n− Aα
∂Ψ
∂ n
dS =
∫∫
| r−r′|=R
Ψ∂ Aα∂ r− Aα
∂Ψ
∂ r
dS
where the second equality was obtained since n = r−r′| r−r′| for the sphere | r − r′| = R. Using the
outgoing radiation condition for both Ψ and Aα, we obtain
IR =
∫∫
| r−r′|=R
Ψ
− iβAα +O
r−2
− Aα
− iβΨ+O
r−2
dS.
Making appropriate cancellations leaves
IR =
∫∫
| r−r′|=R
O
r−2Ψ− Aα
R2 sinφ dθ dφ.
Since Ψ and Aα have the same asymptotic behavior and since we know Ψ from Equation (3.25),
43
we can argue that |Ψ− Aα|= O
r−1. Hence,
IR =
∫∫
| r−r′|=R
O
r−3R2 sinφ dθ dφ = 4πO
R−1. (3.28)
For the left hand side of Equation (3.26), recall that L(Ψ) = 0 in B′ and L(Aα) =−Jα. Then,
∫∫∫
B′ǫ
Ψ L
Aα− Aα L
Ψ
dr′ = −∫∫∫
B′ǫ
Ψ Jα dr′.
Replacing this result along with Iǫ of Equation (3.27) and IR of Equation (3.28) into Equa-
tion (3.26) gives
−∫∫∫
B′ǫ
Ψ(r, r′) Jα(r′) dr′ = −4πAα(r) +O(ǫ) + 4πO
R−1.
Taking the limit ǫ→ 0 and then R→∞ gives the desired result.
From Proposition 3.1.1, we combine the solutions for α= x , y, z to get the vector solution
A =
∫∫∫
R3
J(r′)exp
iβ | r− r′|
4π| r− r′| dr′. (3.29)
For the antenna reflector problem, supp(J) is the reflector surface. Therefore, we reduce the
solution of Equation (3.29) to
A =
∫∫
RJ(r′)
exp
iβ | r− r′|
4π| r− r′| dS′ (3.30)
where R is the reflector surface. Equation (3.30) is an accepted computation for the field A in
antenna theory; see [33, Chapter 8] and [30].
With this solution for the magnetic vector potential, we can use Equation (3.20) to obtain
H = ∇×A.
We then use Equation (3.18) to recover the electric field
E =1
iωǫ0(∇×H− J) =
1
iωǫ0(∇×∇×A− J).
44
Using the vector identity from Equation (1.9), we can transform this equation to
E =1
iωǫ0
∇(∇ ·A)−∇2A− J
.
Recalling the differential equation (3.23), we obtain
E =1
iωǫ0
∇(∇ ·A) + β2A
.
Using the definition of β , we get the solution
E =∇(∇ ·A)
iωǫ0− iωµ0A. (3.31)
3.2 Reflectivity Properties
In order to transmit large quantities of data, higher radiation frequencies are desired. As men-
tioned in Chapter 1, inflatable antennas are already in use and have been successful for L and S
band radiation frequencies [34]. In Table 3.2 we present the common radiation frequencies as
established by the IEEE [19].
Band L S C X Ku K Kaω (GHz) 1 – 2 2 – 4 4 – 8 8 – 12 12 – 18 18 – 27 27 – 40λ (mm) 300 – 150 150 – 75 75 – 37.5 37.5 – 25 25 – 16.7 16.7 – 11.1 11.1 – 7.5
Table 3.2: Common IEEE Radiation bands.
The frequency of the radiation,ω plays an important role in the efficiency of the reflector being
used for an antenna. To understand this, let us first establish the exact geometry of a paraboloid.
Given a focus f= (0,0, F) and a directrix plane z =−F , a paraboloid is the set of all points p ∈ R3
that satisfy
d
f,p
= d
p,p′
where d(x,y) is standard Euclidean distance between x ∈ R3 and y ∈ R3, p = (x , y, z) is an arbi-
trary point on the paraboloid and p′ is the point on the directrix plane closest to p. See Figure 3.4
for an illustration. The parabolic curve shown in Figure 3.4 can be revolved about the symmetric
45
axis (z−axis) to generate a paraboloid. In our application, F is the focal distance of the parabolic
reflector as described in Chapter 1.
p′
f
Aperture Plane: z = F + C
Directrix Plane: z =−F
D/2
p
−D/2
a
r
z
Paraboloid: 4Fz = r2
n
C
Figure 3.4: Cross sectional diagram of a paraboloid and its geometry.
The path that a radiation wave takes from the focus f to aperture point a, is by way of reflection
on the paraboloid at point p. There are two key properties of this path that are determined by the
paraboloid’s geometry. Before listing these properties, we perform some calculations with the
vectors a, f and p.
f− p =
®
−x ,−y, F − x2+ y2
4F
¸
a− p =
®
0,0, F + C − x2+ y2
4F
¸
(3.32)
At point p on the paraboloid, a normal vector is
n = ∇
z − x2+ y2
4F
=
− x
2F,− y
2F, 1·
. (3.33)
46
The two properties of the radiation path are:
1. The radiation from f is reflected at p in a path that is parallel to the symmetric axis. This can
be shown by demonstrating that the angle between n and f − p is the same as the angle
between n and a− p. This is clear from the relation x · y= |x | |y | cosθ and the equality
(a− p) · n|a− p | |n | −
(f− p) · n| f− p | |n | =
1
|n |
(a− p) · n|a− p | −
(f− p) · n| f− p |
= 0. (3.34)
The last equality in Equation (3.34) can be established by noting that
(a− p) · n = |a− p | and (f− p) · n = | f− p |
which are derived from Equations (3.32) and (3.33). It must also be shown that the vectors
(a − p), (f − p) and n are coplanar. This can be checked by using the relations in Equa-
tions (3.32) and (3.33) and then calculating
(a− p)× (f− p) · n= 0.
2. The total distance of the path that any line of radiation takes from f to the aperture is the
constant 2F + C . It is important that the aperture have a height value that exceeds the z
components of all points on the paraboloid. Often, the paraboloid lies entirely below the
focal point, and so we may take C = 0. Using Equation (3.32), it is straightforward to show
that
| f− p | + |a− p | = 2F + C . (3.35)
Since all reflected waves of radiation travel parallel to the z−axis, an observer looking at the
paraboloid from the +z axis (this is referred to as the boresight direction) will see all the reflected
paths of radiation coming directly to him. Equation (3.35) guarantees that said radiation will be
completely in phase, regardless of the where it was reflected on the paraboloid. This holds because
the phase is φ = (ωℓ)/c where c is the speed of light and ℓ is the constant path length traveled.
The total sum of the radiation (signal seen by the observer) is therefore amplified by the constant
phase and direction of travel of the reflected radiation.
47
If the paraboloid is distorted, then some local areas of the reflector may reflect radiation in
a manner that the traveling to the distance is changed by ∆ℓ. The phase error suffered by this
pathlength error is
∆φ =ω∆ℓ
c. (3.36)
If there is a discrepancy in the phase, then some cancellation will occur in the total sum of the
radiation and cause a reduction in the total signal. We see that the phase error is directly propor-
tional to the frequency of the radiation. Hence, phase error increases as the radiation frequency
increases. For this reason, antenna reflectors have lower distortion tolerance limits when operating
at or above Ka–Band versus below Ka–Band.
3.3 Antenna Efficiency
Antenna performance can be quantified by the amount of power that is lost between the feed
(electronic source of a signal) and the receiving probe. A collection of losses, each pertaining to a
different property of the antenna or accompanying apparatus, is calculated in order to ultimately
compute the amalgamated power loss and the antenna’s overall efficiency. For aperture antennas,
there are several factors that contribute to power loss during operation; we list some here.
• Spillover loss is radiation from the electrical feed that is lost beyond the edge of the reflector.
In other words, the reflector simply does not intercept this “outer edge” radiation.
• Taper loss is caused by under–illumination of the reflector near the edge in order to reduce
spillover loss.
• Random surface error loss is caused by Gaussian distributed path length error and can be
calculated by the Ruze Equation [33, Pg. 434].
• Cross polarization loss refers to energy lost due to misalignment of the feed with the re-
ceiving probes.
• Aperture blockage loss is caused when part of the supporting apparatus obstructs radiation
between the reflector and aperture window.
• Feed phase error is non–uniformity of the radiation wave phase caused by the feed output.
48
• Reflector phase error is non–uniformity of the radiation wave phase errors caused by re-
flector surface errors.
The scope of this project concerns only the reflector phase error component and so we will
neglect all other efficiencies. One should note that these other sources of power loss are not neces-
sarily subdued even if they are understood and controlled. For example, the maximum efficiency
that can be accomplished just by the trade off between the taper and spillover losses is 82% for
any parabolic antenna system [33, Pg. 434]. The remainder of Section 3.3 focuses on methods to
compute the phase error for a distorted reflector.
3.3.1 Far Field
The shape of a wave leaving a source, such as an AUT (Antenna Under Test), is spherical. When
the generating current of the AUT is sufficiently far away, the radiation waves appear to be planar
as depicted in Figure 3.5.
AUT Probe
Spherical Waves Nearly Planar Waves
Figure 3.5: Wave front shape in the near and far field of an AUT.
The far field region is where propagation lines of radiation arriving at the point P are suffi-
ciently “parallel” that planar wave structure can be assumed. Equivalently, the observer can make
the judgement that the entire AUT is at a particular distance away and hence contributions of
power at all positions on the aperture are from the same direction.
In order to evaluate the discrepancy between the distances R and r in Figure 3.6, we will need
the Binomial Theorem
(1+ x)n = 1+ nx +n(n− 1)
2!x2+ . . . (3.37)
for |x |< 1 [31, Section 456]. Figure 3.6 shows the parallel approximation that we make for paths
traveling a long distance to the far field. An observer at point P will actually intercept both paths
49
θ
r′
R
r
Symmetric Axis
P
r − R= r ′ cosθ
Figure 3.6: Geometry of rays extending to a point P in the far field.
and so the vectors R and r both pass through point P. We can write the exact relationship between
R, r and r′ as
R2 = (r− r′) · (r− r′).
Expanding this product and using the angle θ between the two vectors r and r′ gives
R2 = r2
1− 2r ′
rcosθ +
r ′
r
2
.
Taking the square root of both sides and applying the Binomial Theorem
Equation (3.37)
with
n= 12
and x = 1− 2r ′r−1 cosθ , we find
R = r − r ′ cosθ +(r ′)2
2r+O(r−2). (3.38)
Equation (3.38) contains the far field approximation
R = r − r · r′ (3.39)
which is exactly the geometric relation between the distances R and r as they are illustrated in
Figure 3.6. The discrepancy between the far field approximation of R in Equation (3.39) and the
exact computation of R in Equation (3.38) can be bounded by placing a restriction on the third
50
term of Equation (3.38). We require that the third and largest neglected term of Equation (3.38)
be no more than λ/16. This corresponds to one–sixteenth of a wave length or 22.5 phase error,
which is accepted in antenna theory [33, Pg. 23]. We therefore require
(r ′)2
2r≤ (D/2)2
2rF=λ
16
where D is the diameter of the reflector and rF is the distance to the far field. Solving for rF
gives a definition of the far field distance in terms of the reflector’s diameter and the radar band
wavelength.
Definition 3.3.1. The far field is said to begin at a distance where the assumption that parallel lines
are converging yields a path length error of not more than λ/16 (corresponding to 22.5 phase error).
This distance is given by
rF =2D2
λ
away from the antenna, where D is the diameter of the aperture and λ is the wavelength of radiation.
In the far field, the reflected E and H fields are orthogonal to one another as well as the
direction of propagation, r. They are related by the Plane Wave Equation
H =r× E
η(3.40)
where η =p
µ0/ǫ0 is the impedance of free space [33, Pg. 26].
The E field in the far field of the AUT is the second term in Equation (3.31) without the radial
component
E = −iωµ0A+ (iωµ0A · r)r. (3.41)
3.3.2 Directivity and Power Gain
Directivity is a function that expresses how much the AUT prefers to radiate power in one direction
versus another. Before generating an equation for directivity, we must determine how power
is calculated. Poynting’s Theorem provides us with a method of computing the power radiated
through a surface as a result of electric and magnetic fields. We take the dot product of Maxwell’s
51
Equations (3.19) and (3.18) with H and E respectively to get
(∇× E) ·H+ iωµ|H |2 = 0 and (∇×H) · E− iωǫ0|E |2 = J · E.
Subtracting these equations gives
(∇× E) ·H− (∇×H) · E+ iωµ|H |2+ iωǫ0|E |2+ J · E = 0.
We apply the vector identity of Equation (1.10) and find
∇ · (E×H∗) + iωµ|H |2 + iωǫ0|E |2+ J · E = 0
where the superscripted star indicates complex conjugation. Integration over a region B enclosed
by the surface S leaves
∫∫
∂ B
⊂⊏⊃ (E×H∗) · n dS +
∫∫∫
B
(iωµ|H |2+ iωǫ0|E |2+ J · E) dV = 0 (3.42)
where the Divergence Theorem has been applied to the first integral. The integrand E × H∗ is
known as the Poynting Vector. Poynting asserts that the surface integral in Equation (3.42) is the
power flowing out of the region B while the volume integral is the power within the region.
Since we are interested in the power obtained far from the antenna, we can take the surface S
in Equation (3.42) to be a sphere of radius R > rF . In this case, the power radiated through any
particular solid angle Θi is
Pi =1
2Re
∫∫
Θi
(E×H∗) · r r2 dΘ
!
where dΘ = sinφ dθdφ. We assume that S is in the far field of the antenna, so the Plane Wave
Equation (3.40) applies. Hence,
Pi =1
2ηRe
∫∫
Θi
E× (r× E∗) dΘ
!
=1
2η
∫∫
Θi
|E |2r2 dΘ. (3.43)
52
The integrand in Equation (3.43) is the radiation intensity for any given direction (θ ,φ), i.e.,
U(θ ,φ) =| rE |22η
. (3.44)
Radiation intensity per unit solid angle has units of watts / steradian. Typically, the maximum
radiation intensity occurs in the boresight direction, or (θ ,φ) = (0,0). The total power radiated
by the antenna is
Pr = max(U)
∫ π
0
∫ 2π
0
U(θ ,φ) sinφ dθ dφ,
and the average power radiated per steradian is given by
U =1
4π
∫ π
0
∫ 2π
0
U(θ ,φ) sinφ dθ dφ =Pr
4π.
It is often convenient to consider the radiation intensity normalized. This allows one to compare
how much radiation the antenna reflects to directions other than boresight. It is important to note
that radiation not radiated in the boresight direction is considered loss since the intended observer
is not at any other angle in the far field.
Definition 3.3.2. The far field pattern of an antenna is a normalized radiation intensity function, F
given by
F(θ ,φ)
2=
U(θ ,φ)
max(U).
Note that F = 1 in the direction of maximum radiation intensity (usually the boresight direction).
Definition 3.3.3. Directivity is the ratio of radiation intensity in a certain direction (θ ,φ) to the
average radiation intensity of the antenna. Hence,
D(θ ,φ) =U(θ ,φ)
U.
Definition 3.3.4. The maximum power one can receive from an antenna is known as the ideal an-
tenna gain and is expressed numerically as
GI =4πA
λ2 (3.45)
53
where A is the area of the aperture [33, Pg. 433].
Equation (3.45) yields a calculation of the antenna gain assuming that the antenna is 100%
efficient. As mentioned at the start of this section, no aperture antenna can achieve an efficiency
of greater than 82%, so we will seek an alternative definition.
Definition 3.3.5. The effective antenna gain is 4π times the ratio of power radiated by the antenna
in a certain direction to the power transmitted by the feed. Mathematically, effective gain is
G(θ ,φ) =4πU(θ ,φ)
P0=
D(θ ,φ)Pr
P0,
where P0 is the power transmitted by the antenna.
The effective antenna gain can also be defined as the ratio of the radiation intensity in a certain
direction to the radiation intensity of the isotropic radiator with power P0. The isotropic radiator
will take the input power, P0, and radiate over all solid angles. This justifies the division by 4π in
the denominator which presents itself in the numerator of Definition 3.3.5. If a discussion of gain,
G and/or directive gain, D ensues without any reference to an angular direction, it is presumed
that
G = max G(θ ,φ) and D = max D(θ ,φ).
Since power gain is a power ratio, it is often expressed in decibels:
GdB = 10 log G.
Example 3.3.1. Suppose we have an antenna which takes in a total power of 100 watts at the
electrical input terminals. Let the antenna have a parabolic reflector with an aperture radius of 0.25
meters and a radiation frequency of 20 GHz (λ = 0.015 meter). The ideal antenna gain is therefore
GI =4πA
λ2 =4π ·π(0.25 meters)2
(0.015 meters)2= 10966 =⇒ GI dB = 40.401 dB.
Remark: The significance of a 0 dB gain measure is that the antenna is behaving like an isotropic
radiator. In particular, the antenna performs “0 times” better than the feed radiating without any
reflector.
54
Definition 3.3.6. The efficiency of an antenna is equal to the ratio of the effective power gain to the
ideal power gain, or
e =G
GI.
In practice, a measurement of gain will reflect all power losses and so the efficiency as defined
in Definition 3.3.6 is the true efficiency of the antenna. In this report, we will be calculating
a theoretical power gain whose value is dependent only on the power input and the reflector’s
surface distortions.
3.3.3 Antenna Characterization
Characterization of an antenna involves calculating the resulting far field magnetic and electric
fields that are reflected by the antenna. The distribution of the signal strength as a function of
angle to the far field is of interest. We present here the computation of the far field pattern for
an ideal paraboloid and then show the method we use to compute the pattern for a deformed
faceted surface. For our computations, we assume that the reflector has a reflectivity constant of
1 and neither absorbs nor refracts any radiation. We also assume that there are no obstructions or
changes in media between the feed and the receiving probe.
Ideal Parabolic Reflector
In this section, we calculate and plot the far field pattern for an ideal axisymmetric parabolic
reflector. From Equation (3.30), we can determine the magnetic vector potential in the far field.
In this case, we integrate over the aperture of the antenna, A . It is sufficient to integrate over
A since the entire radio signal passes through the aperture. We could also integrate the current
density over the reflector surface, R as shown in Equation (3.30). Integration overA , however is
less complicated when we assume a perfect parabolic reflector. We have,
A(r) =
∫∫
AJa(r′)
e−iβ | r−r′|
4π| r− r′| dA′ (3.46)
where Ja is the current density at the aperture, r′ ∈ A is an aperture point and the far field
point is denoted r. We assume, without loss of generality, that the aperture is located in the plane
55
z = 0 and is centered at the origin. From far field approximation of Equation (3.39), we have
| r− r′|= r − r · r′ and | r− r′| ≈ r. Hence, Equation (3.46) can be expressed as
A(r) =e−iβ r
4πr
∫∫
AJa(r′)eiβ r·r′ dA′.
Note that the field A is essentially obtained by a Fourier Transform of the current density field Ja
at the antenna’s aperture. From the boundary conditions of Maxwell’s Equations, we have
Ja = na ×HINCa
where na = k is the outward normal of the aperture and HINCa is the incident magnetic field arriving
at the aperture. A key property of a perfect parabolic reflector is that the reflected fields E and H
are constant in the aperture. We may therefore assume that Ha = H0 j. Then, we have
A =e−iβ r
4πrna × H0 j
∫∫
Aeiβ r·r′ dA′ = −i
H0e−iβ r
4πr
∫∫
Aeiβ r·r′ dA′. (3.47)
The direction to the far field is r= (cosθ sinφ, sinθ sinφ, cosφ). Recalling that r′ is in the circular
aperture, we can write r′ = (r ′ cosθ ′, r ′ sinθ ′, 0). Hence,
r · r′ = r ′ sinφ cos(θ − θ ′).
Integrating over the circular aperture, Equation (3.47) becomes,
A = −iH0e−iβ r
4πr
∫ Ra
0
∫ 2π
0
eiβ r ′ sinφ cos(θ−θ ′) dθ ′!
r ′ dr ′
where Ra is the radius of the aperture. The θ ′ and r ′ integrations are performed using the two
Bessel Function relations [33, Appendix F .3]
J0(x) =1
2π
∫ 2π
0
ei x cosα dα and xJ1(x) =
∫
xJ0(x) d x ,
respectively.
56
Then we obtain the magnetic vector potential
A = −iH0Rae−iβ r
2rβ sinφJ1(βRa sinφ)
The electric field in the antenna’s far field is then recovered by Equation (3.41). The details have
been worked out in Stutzman and Thiele [33, Chapter 8]. The radiation integral of Equation (3.46)
is quite arduous to compute, in general. Only in rare cases (such as a perfect paraboloid), is it even
possible to find closed form solutions; it is common to use numerical techniques for reflectors with
variable dimensions or distortions [30]. We will ultimately obtain a far field pattern of the form
F(θ ,φ) =|E |
max |E | =2J1(βRa sinφ)
βRa sinφ(3.48)
which is equivalent to the result we would obtain using Definition 3.3.2. For the ideal parabolic
reflector, F(θ ,φ) is axisymmetric since there is no θ dependence in Equation (3.48). The far field
pattern is plotted in a logarithm plot with units of decibels as in Figure 3.7. In particular, we plot
10 log(F) versus φ.
Remarks:
1. As long as the boresight direction is that for which U is maximum, the pattern F will always
have a maximum of 0 dB at φ = 0.
2. The far field pattern, F , is plotted in this manner so that we can observe how many “dB
down” the antenna radiates for other directions (φ 6= 0).
3. More negative dB levels for non–boresight directions indicate that a lower portion of the
total radiated power is sent in the non–boresight direction. Hence, the antenna performs
more optimally.
4. It is common to measure the dB level of the highest peak after boresight in the F pattern.
This peak is known as the side lobe and it is typically the direction where the most power is
radiated other than boresight. For Figure 3.7, the side lobe level is −17.6 dB.
57
−15 −10 −5 0 5 10 150
500
1000
−15 −10 −5 0 5 10 15
−40
−30
−20
−10
0
Angle φ to the Far Field, Degrees
Far
Fiel
dPa
tter
nF(φ)
dB
Rad
iati
onIn
tens
ity
U(φ)
Wat
ts/St
r
Figure 3.7: Far field patterns for an ideal axisymmetric parabolic reflector antenna.
Faceted Surface
The deformed reflector in our model is a faceted surface. In this section we give an overview of
the method used to calculate the far field pattern for such a surface. The far field pattern that we
find can then be used to calculate the efficiency and gain of the antenna as shown in Section 3.3.2.
Let us first establish some notation for the discretized reflector. For each triangle Tn ∈ ΩR, we
denote the centroid of Tn by pn. The deformed reflector has the configuration Ω which contains the
triangles Tn = x(Tn) ∈ Ω. We denote the centroid of the deformed triangle by x(pn) = (xn, yn, zn).
See Figure 3.8 for an illustration.
We model the antenna feed with a simple dipole having current I0 = 200 Amps oriented along
the y–axis with dipole length ∆y = 2 cm. In this case, the incident magnetic field to Tn is
HINCn = −i
I0β∆y
4πr ′ne−iβ r r′n× j
, (3.49)
58
where r′n is the vector from the focal point to x(pn). The radiation integral of Equation (3.30)
is calculated over the deformed faceted surface for a selected set of directions to the far field,
(θ f ,φ f ).
For our results, we consider only antenna diameters less than 20 meters. Also, the radiation
wavelength, λ is always greater than 0.003 meters. With these conditions, the least possible
distance to satisfy the far field approximation, by Definition 3.3.2, is 2.6×105 meters. Choosing
all the sample far field points to be at a distance of 107 meters is sufficient for all of our cases. We
then calculate
A(θ f ,φ f ) =exp(−iβ107)
4π× 107
Nt/2∑
n=1
Jn exp(iβ r f · r′n) |Tn|
where r f is the unit direction vector corresponding to the direction (θ f ,φ f ). Also, Jn is the current
density on Tn as determined by facet’s normal vector nn and the boundary condition
Jn = 2nn×HINCn .
We then use Equation (3.41) to calculate E in the far field.
3.4 Geometric Error Analysis
In addition to the efficiency computations presented in Section 3.3, we also seek some surface ac-
curacy measurements that relate purely to the geometry of the deformed reflector and not on the
reflected radiation. While the efficiency computations based on the antennas reflective properties
are certainly important for understanding the operation of the antenna, the following measure-
ments will help identify regions where shape deformation is more pronounced.
3.4.1 RMS Calculation Methods
For both the molded and flat construction models, we can vary parameters that control the loading
forces to determine what effect these parameters have on the surface accuracy of the reflector.
Our results will assist in developing a cutting pattern, support structure or other constructions
that may reduce the surface distortions. The geometric configuration of the deformed reflector
can be evaluated by considering the global RMS surface errors as determined by both vertical
59
φφ
Aperture
∆zn
Focal Point
4Fz = x2+ y2
nn fn
an
F
Figure 3.8: Deformed triangle Tn with centroid (xn, yn, zn). The vertical displacement, ∆z, islabeled and the pathlength for this triangle is ℓn = fn+ an.
displacement of the reflector and path length error.
A reflector in an ideal position should have each triangle’s centroid nearly satisfying the gen-
erating equation 4Fzn = x2n + y2
n . The centroids will likely not satisfy the generating curve exactly
since it was the vertices of the triangles that were positioned using the parabolic form. For a fine
mesh, however, this error is negligible as will be demonstrated in Section 3.4.2. The Euclidean
measure of RMS, ǫEUC, is obtained by the integral [16]
ǫ2EUC(Ω) =
∫∫
A
∆z2 dA
|A | (3.50)
where ∆z is the vertical displacement between the deformed reflector and an ideal paraboloid,
|A | is the area of the circular aperture, and dA is Lebesgue area measure. We calculate ǫEUC by
considering all of the reflector facets and then computing Equation (3.50) discretely by
ǫ2EUC(Ω) =
1
|A |N f∑
n=1
∆z2
n |Tn|(nn · k) (3.51)
where |Tn|(nn · k) is the projected area of Tn onto the aperture.
Another RMS calculation includes the pathlength error of a radiation wave traveling from the
focal point to the focal plane via triangle Tn. For an ideal paraboloid, this distance should be twice
the focal length of the parabolic reflector for each triangle (recall Section 3.2). Having denoted
60
the total path length of a ray striking the centroid of triangle Tn by ℓn, we have
ǫ2RMS(Ω) =
∫∫
A
∆ℓ
2
2 dA
|A | (3.52)
where ∆ℓ = ℓn − 2F is the pathlength error of a wave reflected at a given point on the reflector
surface and F is the focal distance of the reflector [16]. As we did for the Euclidean RMS, we also
compute the integral of Equation (3.52) discretely by
ǫ2RMS(Ω) =
1
4|A |N f∑
n=1
ℓn− 2F2 |Tn|(nn · k). (3.53)
3.4.2 Numerical Accuracy of ǫRMS
The order of error in the surface accuracy computations depends on the discretization of the tri-
angular mesh we choose. The two primary controlling factors for the mesh size are the number of
rings of triangles, Nr and the number of gores, Ng .
In Equation (3.50), we see that the surface accuracy is determined by the vertical discrepancy
of the facet’s centroid from a paraboloid. An ideal configuration for our model is one whose vertices
lie exactly on the parabolic mold. This configuration, however, will not have facet centroids that lie
on the actual paraboloid and so the ideal configuration would not have ǫEUC = 0. To understand the
error in these computations, we calculate ǫEUC(ΩR)where ΩR is the molded reference configuration.
Let Tn ∈ ΩR be a typical facet with vertices denoted by v j = (vxj , v y
j , vzj ) for j = 1,2,3. Then,
the centroid can be defined based on these three vertices as
pn =
v x1 + v x
2 + v x3
3,
v y1 + v y
2 + v y3
3,
vz1 + vz
2 + vz3
3
. (3.54)
The vertical discrepancy, ∆zn between pn and the ideal paraboloid 4Fz = x2+ y2 is
∆zn = pzn−(px
n)2+ (p
yn )
2
4F
where pxn ,py
n and pzn are the coordinates of pn. Using Equation (3.54), we can write the vertical
61
discrepancy as a function of the vertices of Tn. We have,
∆zn =vz
1 + vz2 + vz
3
3− 1
4F
v x1 + v x
2 + v x3
3
2
+
v y1 + v y
2 + v y3
3
2
.
After expanding the trinomials and performing some algebra, we can obtain
∆zn =1
18F
(v x1 − v x
3 )(vx1 − v x
2 ) + (vx2 − v x
3 )2+ (v y
1 − v y3 )(v
y1 − v y
2 ) + (vy2 − v y
3 )2
.
Now define vi ∈ R2 to be the projection of the point vi to the plane z = 0. Also, define P(Tn) to
the projection of the triangle Tn to the plane z = 0. Then we can write
∆zn =1
18F
(v1− v3) · (v1− v2) + |v2− v3|2
.
The definition of dot product leaves
∆zn =1
18F
|v1− v3| |v1− v2| cosθ1+ |v2− v3|2
where θ1 is the interior angle of P(Tn) at vertex 1. Let hn be the length of the longest edge of
P(Tn). Then, we can bound ∆zn by
∆zn
≤
1+ | cosθn,1|
h2n
18F≤ h2
n
9F.
From Equation (3.50), we obtain
ǫ2EUC(ΩR) =
1
|A |N f∑
n=1
∆z2
n | Tn |(nn · k) ≤1
|A |N f∑
n=1
h4n
81F2 | Tn |(nn · k).
Let H = maxTn ∈ΩR
hn. Then, we can reduce this further to
ǫ2EUC(ΩR) ≤
H4
81F2|A |N f∑
n=1
| Tn |(nn · k).
Recall that | Tn |(nn · k) is the area of P(Tn). Therefore, the sum over all reflector facets will cancel
62
with the aperture area |A |. This leaves
ǫEUC(ΩR) ≤H2
9F.
We now compute a reasonable value for H with the knowledge that the triangles P(Tn) that
are adjacent to the vertex are always the largest by construction of the grid. The longest side of
one of these triangles is
H = max
R f
Nr,
R f
Nr
p2
1− cos
2π/Ng
where Nr is the number of rings of triangles on the reflector and Ng is the number of gores. It is
straightforward to show thatp
2
1− cos
2π/Ng
< 1
whenever Ng > 5. The grid parameters for all our experiments use a value of Ng ≥ 7. Therefore,
we can fix the quantity
H =R f
Nr.
Notice that the error bound no longer depends on Ng as long we keep Ng > 5. The bound on
ǫEUC(ΩR) becomes
ǫEUC(ΩR) ≤R2
f
9FN2r
=D2
36FN2r
(3.55)
where D, R f and F are the diameter and radius and focal length of the reflector aperture, respec-
tively. We compare some of the model’s computations for error on the reference configuration ΩR
to these bounds. The results in Chapters 4 and 5 typically use Nr = 36.
NrReflector Size Eqn. (3.55) Model CalculationsD (m) F (m) ǫEUC (mm) ǫEUC (mm) ǫRMS (mm)
36 2.130 0.914 0.1064 0.0805 0.066536 10.650 4.570 0.5320 0.4027 0.332544 10.650 4.570 0.3561 —– —–
Table 3.3: Comparison of predicted versus actual errors for various grid sizes.
63
Chapter 4
Demonstration Cases
The results we present in this chapter demonstrate the behavior of the reflector surface under
various conditions. In Section 4.2, we model the antenna reflector using the molded reference
configuration. Some of the results of the parametric studies shown here will therefore demonstrate
predictions that this model gives for the surfaces that are already constructed and tested. To assess
the actual accuracy of the reflector and the degree to which it may be useful in operation, we
apply the analysis developed in Chapter 3. In Section 4.1, a mylar balloon is also considered as
a test case of the model. The construction methods for the faceted surface that we presented in
Chapter 2 can be easily adapted to model the construction of such a balloon.
4.1 A Test Case: The Mylar Balloon
The design of the triangular mesh and the reference configuration pattern are appropriate for
modeling the shape of a mylar balloon. We make some predictions for the geometry of an inflated
mylar balloon and compare them to an idealized mylar balloon as discussed in [24]. The mylar
balloon is constructed by seaming together two flat disks of material and then inflating the region
in between. The equilibrium configuration for this balloon will exhibit wrinkling which is large
relative to the balloon’s dimensions. In Figure 4.1, we show the generating curve of an idealized
inflated mylar balloon whose reference configuration has radius a.
We label the semi–major radius of the inflated mylar balloon RM and the depth τM . Assuming
that the mylar does not stretch, the arclength along the profile curve must be the same as the
64
z(x)
xRM
z
North Pole
South Pole
τM
Figure 4.1: Cross section of the Mylar Balloon with generating curve z(x).
radius of the uninflated circular membrane, i.e.,
∫ RM
0
p
1+ z′(x) d x = a (4.1)
where a is the radius of the uninflated membrane. We can also construct an equation for the
volume of the balloon using the fact that it is a surface of revolution. Implementing the shell
method gives
V =
∫ RM
0
4πxz(x) d x . (4.2)
Using Equations (4.1) and (4.2), one can apply Calculus of Variations and obtain the correspond-
ing Euler–Lagrange Equations. From that, the profile shape is ultimately found to be the elliptic
integral
z(x) =
∫ RM
x
t2
p
R4M − t4
d t,
where 0 ≤ x ≤ RM [24]. Relationships between the semi–major radius, depth and reference
configuration radius can be obtained in terms of gamma functions. In [24], Mladenov and Oprea
derive the ratios for these geometric properties. In Table 4.1, we compare Oprea’s inelastic analytic
results for the ratios RM/a, τM/a and τM/(2RM ) to those obtained by our elastic model.
It is interesting to note that there is good agreement between our results that include wrin-
kling and the idealized problem of Oprea that neglects elastic effects. In our model, we have
a = 1.065 meters and we can obtain the generating curve as shown in Figure 4.2. Though this
65
RM/a τM/a τM/(2RM )
Oprea [24] 0.7627 0.9139 0.5991Model for p0 = 125 Pa 0.7720 0.9273 0.6009Model for p0 = 500 Pa 0.7756 0.9506 0.6128
Table 4.1: Geometric results of mylar balloon test for two p0 values.
curve is smooth and convex, in reality there are many wrinkled regions in the mylar balloon. For
certain cross sectional cuts of the mylar balloon, the profile shape will locally collapse.
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
x−axis
z−ax
is
Figure 4.2: Generating curve of modeled mylar balloon.
The stress resultants for an axisymmetric balloon surface are derived in Section 5.3 of [20]
and are given by
σθ =p0r2
2
2− r2
r1
and σφ =p0r2
2
where φ is the radial direction and θ is circumferential direction. A mylar balloon is assumed to
have zero circumferential stress, σθ . This is not, however, physically plausible in a doubly curved
surface with p0 > 0. Also, the curvatures are related by
k1 = 2k2 and K =8
9H2
where k1 and k2 are the principal curvatures and K and H are the Gaussian and mean curvatures,
respectively [24]. For this reason, the mylar balloon is an example of a Weingarten Surface. Actual
mylar balloons exhibit “crimpling” of the inflated surface which is responsible for the large folds
that can be seen around the rim of the balloon in Figure 4.3.
66
(a) A real mylar balloon [38]. (b) Top view of model. (c) Side view of model.
Figure 4.3: Fully inflated mylar balloon (actual and modeled).
4.2 Analysis of Molded Reflectors
In this section, we investigate the response of the reflector shape and its surface accuracy to
changes in the boundary support tendons, gravity and symmetry of the loading conditions. These
were discussed in [10] using a coarser mesh.
We use the material properties of Kapton to serve as the reflector material and apply the di-
mensions of an actual prototype. Table 4.2 contains the parameter values of Kapton that are used
in our numerical experiments as well as the dimensions of the reflector geometry.
Property Variable ValueKapton Thickness h 12.7 micronsKapton Mass Density ρ 1419.98 kg/m3
Kapton Young’s Modulus E 25.9 ×108 N/m2
Kapton Poisson’s Ratio ν 0.34Reference Tendon Length Γm,R 0.06 metersReflector Diameter D 2.13 metersReflector Focal Length F 0.914 metersTriangular Facets N 20,736Reflector Facets N f 10,368Supporting Tendons Nt 72
Table 4.2: Mechanical properties of Kapton, reflector dimensions and grid size.
For the remainder of this dissertation, we fix the parameter g = 9.80665 meters/sec2 for the
acceleration due to gravity. In each case, the gravity vector g will be defined with a direction. The
only case for which g will not have the aforementioned magnitude is for a 0g environment.
67
The surface accuracy results will include the RMS measurement, the antenna power gain
and the reflector efficiency as they were defined in Chapter 3. We use the radiometric RMS,
ǫRMS as defined (3.52). Expressions for the antenna gain and efficiency can be found in Defini-
tions 3.3.5 and 3.3.6. We will denote efficiency by e.
The tendon foreshortening parameter, γ is important in many of the experiments of Chap-
ters 4 and 5; it was briefly mentioned in Section 2.1.2. The reference length of the tendons are
foreshortened by the percentage γ where 0 ≤ γ < 1. This parameter enables us to vary the force
on the reflector’s initial configuration.
4.2.1 Reflector Size Variation
The effects on a larger reflector are of concern since a large radiating aperture is desired for long
range missions. In particular, we are interested in the relationship between the reflector’s size and
the surface accuracy of the antenna when other significant parameters are held constant.
We perform two case studies; one with a tendon foreshortening of γ= 0.03 and the other with
γ = 0.06. For each case study, we increase the size of the reflector while holding the ratio F/D
constant. The reflector sizes in Table 4.3 are obtained by taking the reflector dimensions of Ta-
ble 4.2 and increasing the dimensions by factors of 1.0, 2.5, 4.0, 5.5 and 7.0. The tendon lengths
are not changed by the size factor. Gravity is held at g = −gk for these tests and p0 = 10 Pa.
The equilibrium configuration is determined by our model for these parameters and we report the
surface accuracy data in Table 4.3.
Reflector Dimensions Case I: γ= 0.03 Case II: γ= 0.06D (m) F (m) ǫRMS (mm) e (40 GHz) ǫRMS (mm) e (40 GHz) Gain dB2.130 0.914 1.4156 98.59 % 4.2595 91.35 % 58.615.325 2.285 4.7162 93.47 % 2.5938 97.94 % 66.888.520 3.656 15.5799 61.91 % 14.5017 66.85 % 69.30
11.715 5.027 31.0087 48.34 % 28.1322 53.22 % 71.0714.910 6.398 50.8096 36.48 % 52.2116 40.84 % 72.02
Table 4.3: Comparison of energy components for various antenna sizes.
By comparing the results of Cases I and II in Table 4.3, we see that the size of the reflector
having greatest efficiency for Case I is different from that of Case II. This suggests that the force
68
in the tendons is dependent on the size of the antenna and should be considered when choosing
a certain diameter for an antenna reflector. Additional analysis is needed to assess the benefits or
drawbacks of certain parameter values which can be combined to help improve a large antenna
reflector’s surface accuracy.
For Case II, we include the antenna power gain. We see that one can still attain higher levels of
power gain for larger diameter antennas despite the reduction of efficiency. From Equation (3.45),
we can find that the antenna in case (D,γ) = (14.910,0.06) performs like an antenna operating
at 100% efficiency with diameter 9.53 meters. While larger aperture antennas are desired for
increased power transmission, higher accuracy will be required to make use of the full size of the
reflector at hand.
Gravitational Variation for Large Reflectors
Earlier results reported in [10] and [11] suggested that changes in the gravitational environment
have lesser effects on the surface accuracy of the reflector than the supporting tendons. These
conclusions were based on results from our model for antennas of diameter up to 2.13 meters. In
this section we will demonstrate that larger reflectors are somewhat more sensitive to changes in
the gravitational environment. The pressure is set to p0 = 12.5 Pa for this experiment.
−g/gCase I: D = 2.13 meters Case II: D = 14.91 metersǫRMS (mm) e (40 GHz) ǫRMS (mm) e (40 GHz) Gain dB
k 1.4156 98.59 % 50.8096 36.48% 71.53( j+ k )/
p2 1.3568 98.51 % 44.9216 47.29% 72.66
j 1.6830 98.35 % 41.6859 46.23% 72.560 1.6802 98.41 % 40.8349 48.31% 72.75
Table 4.4: Gravitational effects on large reflectors.
Case I of Table 4.4 shows that the 2.13 meter axisymmetric reflector is not particularly sensitive
to a rotation in the gravity vector. The RMS values have a range of less than 0.3 mm. In [10],
we showed that a smaller size 0.3 meter off–axis reflector had even less sensitivity to changes in
the gravitational environment. In Case II of Table 4.4, however, we see that the larger reflector
is affected particularly when the gravity is in the direction −k. The ǫRMS ranges from 40.83 mm
(for a 0g environment) to 50.80 mm (for a 1g environment). Larger weight and minimal vertical
69
support are likely the key factors for this observation. The affect of gravity on the deformation
and overall displacement of the entire reflector should be considered when working with antenna
reflectors on the order of 10 meters.
4.2.2 Boundary Support Variation
Initial findings for the inflatable antennas that we are investigating suggest that deflection near the
rim is a cause of efficiency loss. In this section, we investigate the reflector surface accuracy under
the impact of both low and high boundary support tension. In [10], we found that if the tension
was sufficiently high, there can be significant deflection near the rim. Similarly, we found that the
reflector displaces vertically near the vertex of the reflector. These effects can lead to severe loss
of the radiation reflected by the antenna.
Symmetric Boundary Forces
Assume that each tendon has the same stiffness constant Kt,m and has the same foreshortening
factor, γ. Then, the force applied to the antenna is uniform around the entire rim. We also as-
sume gravity is g = −gk so that totally symmetric loading applies. In this case, the computed
equilibrium configurations of the reflector are axisymmetric although the solver does not assume
an axisymmetric geometry. In Figure 4.4, we plot the radial and vertical components of the facet
displacements as a function of radial distance from the symmetric axis. It is clear that the dis-
placements are smaller and hence less severe for the lower parameter value of γ = 0.02. Surface
accuracy values of these results are presented in Table 4.5.
Most of our results do not contain a “W” shaped error curve. The W shape has a maximum
vertical error displacement at about 3/4R f , where R f is the radius of the reflector. See [21] for an
example where a circular pressurized membrane with a fixed edge is studied. In Figure 4.4, both
vertical error plots are monotonic and reveal maximum displacement at the rim of the reflector.
This is significant because a substantial vertical displacement of the reflector’s vertex indicates a
general vertical shift of the reflector’s position with respect to the feed. A loss of power gain is
expected since the optical properties of the paraboloid and its focus are disturbed.
70
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
Profile of Facet Displacements (mm)
Rad
ial D
isp
0 0.2 0.4 0.6 0.8 1
0
1
2
3
Radial Distance from Vertex (meters)
Ver
tical
Dis
p
γ = 0.02γ = 0.04
Radial Distance from Vertex (meters)
Profile of Facet Displacements in millimeters
Rad
ial
Dis
plac
emen
tVe
rtic
alD
ispl
acem
ent
γ= 0.02
γ= 0.04
Figure 4.4: Reflector Distortion Pattern for High and Low Tendon Tension.
Non–symmetric Boundary Forces
In practice, the reflector may be subject to non–symmetric loading forces. The asymmetry can be
modeled by setting the tendon stiffness constants Kt,m in W ∗t,m of Equation (2.14) to
Kt,m = K t,m
1+ coskπm
Nt
(4.3)
where K t,m = 250 N is the average tendon stiffness and k is a positive integer. For k = 4, the
tendon force is maximum at the ±i positions on the rim and 0 in the ±j positions. Hence, the
entire reflector responds by straining more in the x direction and less in the y direction. As shown
in Figure 4.5, this results in a large variation in the separation angles between the upper and lower
shells of the clam shell. Also note that the variation in this angle corresponds to the peaks and
troughs of Kt,m in Equation (4.3).
71
0 50 100 150 200 250 300 350
55
60
65
70
75
80
85
90
95
100
105
Angular Position on Reflector Rim (Degrees)
Ang
leB
etw
een
Upp
er&
Low
erSh
ells
(Deg
rees
)
Shell Separation for Deformed Antenna Reflector
Deformed Configurationof Case 3 in Table 4.5
Molded ReferenceConfiguration
Figure 4.5: Effect of tendon force distribution for Case 3 of Table 4.5.
Figure 4.6 shows the shell separation angles for a tendon stiffness distribution with a sudden
jump at the direction +i. The first and seventy–second tendons (lying next to each other and on
either side of +i) exert a maximum and minimum force, respectively. This jump generates a non–
symmetric distribution of force along the rim of the reflector.
Case Figure No. Stiffness, Kt,m (N) Foreshortening ǫRMS (mm)1 4.4 Kt,m = 250 γ= 0.02 0.55132 4.4 Kt,m = 250 γ= 0.04 2.31903 4.5 Eqn (4.3) with k = 4 γ= 0.03 4.25924 4.6 Eqn (4.3) with k = 3 γ= 0.03 6.0464
Table 4.5: Summary of RMS values for various physical testing conditions.
72
0 50 100 150 200 250 300 350
60
70
80
90
100
110
120
Angular Position on Reflector Rim (Degrees)
Ang
leB
etw
een
Upp
er&
Low
erSh
ells
(Deg
rees
)
Shell Separation for Deformed Antenna Reflector
Deformed Configurationof Case 4 in Table 4.5
Molded ReferenceConfiguration
Figure 4.6: Effect of tendon force distribution for Case 4 of Table 4.5.
4.2.3 Off–axis Reflector
The off–axis antenna we consider here is substantially smaller in size than the axisymmetric re-
flector considered in Section 4.2.1. The diameter for the off–axis antenna is 0.3048 meters with
a focal distance of 0.1524 meters. While this antenna holds the advantage of not having the feed
and its mounting device overshadow the reflector, it does not hold its parabolic shape as well as
its axisymmetric cousin. This is most likely due to the fact that, unlike the axisymmetric case, a
non–constant edge force needs to be applied. The pressure is set to p0 = 12.5 Pa and the tendon
foreshortening is γ= 0.03 for this experiment.
In Figure 4.7, we present the lower and upper shell separation angle around the rim of the
antenna for two off–axis antennas. A summary of the dimensions for these two antenna reflectors
and the surface accuracy analysis is presented in Table 4.6. Recall that the off–axis antenna is not a
symmetric shape on its own (refer back to Section 1.1), therefore we see an ideal separation angle
73
0 50 100 150 200 250 300 350
5
10
15
20
25
30
35
40
45
50
Angular Position on Reflector Rim (Degrees)
Ang
leB
etw
een
Upp
er&
Low
erSh
ells
(Deg
rees
)
Shell Separation for Deformed Antenna Reflector
Deformed Configurationof Case 1 in Table 4.6
Molded ReferenceConfiguration
Figure 4.7: Shell separation for small off–axis antenna reflector.
which is periodic around the reflector rim.
In Table 4.6, we see that the ǫRMS is much larger for Case 1 than for Case 2. Since the size
of the reflector is far larger for Case 2, but the translation parameter, a
see Equation (2.1)
is
left unchanged, the overall shape is close to axisymmetric. Hence, we find a much lower ǫRMS as
we were able to attain in Case 1 of Table 4.5, for example. Case 3 of Table 4.6 uses a translation
parameter which was increased proportionally with the diameter and focal lengths. A much higher
ǫRMS value is obtained in Case 3, yet the efficiency remains at 93.43%. The reflector was deformed
in a manner that caused severe deflection near the rim. This deflection is primarily responsible for
the ǫRMS calculation of 28.5148 since the ǫRMS calculation restricted to the last ring of facets is 120
mm. The remainder of the antenna was moved to a position where its pathlength error over the
majority of the surface was actually quite low. The median contribution to the RMS for the interior
rings was only 3.18 mm.
74
CaseReflector Dimensions Surface Accuracy
D (meters) F (meters) a (meters) ǫRMS (mm) e (40 GHz) Gain (dB)1 0.3048 0.1524 0.1778 8.0649 84.53% 41.392 3.0480 1.5240 0.1778 0.4223 99.40% 62.093 3.0480 1.5240 1.7780 28.5148 93.43% 61.82
Table 4.6: Surface accuracy of off–axis reflectors.
There was no design involved to obtain the fairly efficient antenna of Case 3 in Table 4.6. Our
model would not advocate that a deformation for an antenna of this type will generally deform
to an efficient reflector. Generally, these tests show that the two parabolic sheets are easily forced
together as the separation angle is dramatically reduced around the entire antenna. The off–axis
antenna class will need substantial modifications to reduce the deflection of the rim. This will
likely entail a correct computation of ideal boundary conditions.
75
Chapter 5
Parametric Studies
We present here some results which model an (initially) axisymmetric reflector constructed from
flat panels of material. The results presented in this chapter focus on experiments involving varia-
tions in the tendon supports, the internal pressure and changes to the cutting patterns for the flat
panels. The properties of the triangular grid for the faceted surface will vary slightly depending
on which construction is used. While the faceted surface models an antenna with the same dimen-
sions in each case, the triangular grid will change slightly for the gore construction and require
a change in the number of supporting tendons. The grid specifications are provided in Table 5.1.
The mechanical properties used for the membrane surface are again that of Kapton and can be
found in Table 4.2.
5.1 Parametric Study for ǫRMS versus Ng and p0
This parametric study involves the flat gore construction that was outlined in Section 2.2. We
investigate changes in the surface accuracy when the internal pressure is increased and when the
Number of . . . VariableFlat Gore Construction All Other
14 Gores 16 Gores 18 Gores ConstructionsTriangular Facets N 17,528 20,032 22,536 20,736Reflector Facets N f 8,764 10,016 11,268 10,368Supporting Tendons Nt 70 80 75 72
Table 5.1: Grid parameters and dimensions of the reflector being modeled.
76
number of gores is changed. We foreshorten the tendons by 1% (γ = 0.01) and set the gravita-
tional acceleration to g = −gk where the unit vector k = ⟨0,0,1⟩ is normal to the plane of the
aperture. Recall that g = 9.80665 meters/sec2. As shown in Table 5.2, we test the reflector for the
differential pressures of p0 = 10 Pa, p0 = 12.5 Pa, p0 = 25 Pa and p0 = 50 Pa and the number of
gores set to 14, 16 and 18. We use our model to compute the equilibrium configurations and the
corresponding surface accuracy values, ǫRMS, which are presented in Table 5.2. If we consider a
fixed pressure of 25 Pa, the ǫRMS value is decreased by 57.7% to 0.7590 mm when we increase the
number of gores from 14 to 18. This suggests that an increase in the number of gores can help to
reduce surface distortions. We can also reduce ǫRMS by fixing the number of gores and increasing
the differential pressure to 25 Pa. For a grid with 18 gores, ǫRMS is reduced by 47.8% to a value of
0.7590 mm by increasing p0 from 10 to 25 Pa. The data in Table 5.2 suggests that for a fixed Ng ,
there is a p0 that minimizes ǫRMS. For higher differential pressure, we would expect deformation
of the surface in the normal direction in addition to high film stresses. In the case of p0 = 50 Pa,
the deformations become too large and cause a decrease in surface accuracy.
Ng ǫRMS(ΩR)p0 = 10.0 Pa p0 = 12.5 Pa p0 = 25.0 Pa p0 = 50.0 PaǫRMS(Ng , p0) ǫRMS(Ng , p0) ǫRMS(Ng , p0) ǫRMS(Ng , p0)
14 4.6484 2.7030 2.5675 1.7947 1.565116 3.6082 2.0223 1.9020 1.2119 1.524918 2.8808 1.4564 1.3342 0.7590 1.9245
Table 5.2: Comparison of ǫRMS(Ng , p0) values in millimeters.
We use the deformed reflector’s facet normal vectors to predict the path of a sample light ray
from the feed to the aperture; see Figure 3.8. The phase plane in Figure 5.1 is an image of the
paths’ terminal points on the aperture where the shading indicates the phase of the radiation at
the aperture for that path. For the reflector dimensions we use here (see Table 4.2), the expected
phase of the radiation at the aperture is
φ =ωℓ
cmod 360 = 284.89
where the frequency is ω = 40 GHz, the ideal traveling distance of a radiation wave from the focal
point to the aperture is ℓ= 2F and c is the speed of light.
77
−0.9 −0.6 −0.3 0 0.3 0.6 0.9
−0.9
−0.6
−0.3
0
0.3
0.6
0.9
x (meters)
y (m
eter
s)
AUT Aperture Phase Diagram −− p0 = 10 Pa
0
60
120
180
240
300
360
x (meters)
y(m
eter
s)
AUT Aperture Phase Diagram – p0 = 10 Pa
Figure 5.1: Aperture radiation phase plane at 40 GHz with Ng = 16.
Figure 5.1 demonstrates how the radiation pattern at the aperture is disturbed by the cutting
pattern of the reference configuration. Variation in the phase at the aperture is clear along the
lines where the seams of the panels are located. Figure 5.2 shows plots of the phase distributions
at the aperture based on the sampling of the 10,016 light rays (one for each reflector facet) for
two different values of p0. Each data point plots the number of light rays having a phase within
2 of the corresponding phase value. The peak of the distribution is closer to the expected phase
value of 284.89 for the higher pressure value of 25 Pa. Higher pressure can help reduce the
phase deviation, but one must be careful to consider the loss of surface accuracy for much greater
(relatively) values p0; recall Table 5.2.
78
180 200 220 240 260 280 300 3200
500
1000
1500
2000
2500
3000
Phase (degrees)
Num
ber
of
Ligh
t R
ays
Aperture Radiation Phase Distributions
p0 = 10 Pa
p0 = 25 Pap0 = 25 Pa
p0 = 10 Pa
Phase (Degrees)
AUT Aperture Radiation Phase Distributions
Num
ber
ofLi
ght
Ray
s
Figure 5.2: Phase distributions for two values of p0 at 40 GHz.
5.2 Parametric Study for ǫRMS versus α and γ
Since one serious problem with this inflatable antenna design is the deflection of the reflector
near the rim, we conduct the following parametric study that involves modifying the reflector rim
shape. The band construction is well–suited for this experiment since the rim of the reflector is
contained in one section of the cutting pattern. We modify the slope of this outer–most band by
decreasing the radius of the outer–most ring of vertices (situated at the reflector’s rim) as shown
in Figure 5.3.
The band construction we consider for this parametric study consists of 10 bands with the
seams being set at intervals of 0.1R f where R f =D2
is the radius of the aperture. We introduce
a shape modication factor α ∈ [0,1] to adjust this outer–most band. This factor determines the
79
distance, d, from the symmetric axis of the paraboloid to the ring of vertices at the reflector’s rim;
see Figure 5.3. In particular, d = (1 − α)R f . Note that the nominal reference configuration is
α = 0. The result is a reference configuration modeling a paraboloid for the inner 9 bands with a
tapered adjustment for the tenth.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
Parabolic Cross Sections
Sym
met
ric A
xis
Radial Axis (meters)
R f
d
Sym
met
ric
Axi
s
Radial Axis (meters)
−1 − 0.8 − 0.6 − 0.4 − 0.2 0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
Figure 5.3: Parabolic cross section for a flat ring configuration with α= 0.05.
The geometric adjustment to the rim of the reference configuration was introduced to explore
means of adjusting the construction patterns in order to help reduce the deflection due to the
tension in the supporting tendons. We perform a set of tests to determine the reflector’s shape
accuracy as a function of the rim radius modification factor, α. The differential pressure is fixed to
p0 = 12.5 Pa and the acceleration due to gravity is set at g = −gk. Table 5.3 contains ǫRMS values
as calculated by Equation (3.52) for various values of the parameters α and γ.
For each value of γ, the strain energy in the tendons is determined by using Equation (2.14)
where Kt,m = 250 N for all Nt tendons. In Figure 5.4, we plot ǫRMS as a function of α for 0 ≤ α ≤0.044 and for γ= 0.01,0.02,0.03 and 0.04. One can see in Figure 5.4 that the ǫRMS can be reduced
by adjusting the parameter α. For a fixed value of γ, we can find a parameter value α = α∗γ such
that
ǫRMS(α∗γ,γ) = min
α∈AǫRMS(α,γ) (5.1)
where A is the set of α values for which we computed reflector configurations. The results of
these local minima are summarized in Table 5.3. The third column contains the data for the flat
band reflector that achieves the minimal ǫRMS value for each γ. The gain is calculated as in Defi-
nition 3.3.5. In the second column of Table 5.3, we report the ǫRMS attained by both the molded
80
0 0.01 0.02 0.03 0.04 0.050
0.5
1
1.5
2
2.5
3
3.5
4
γ = 0.01γ = 0.02γ = 0.03γ = 0.04
ǫR
MS(α
,γ)
Rim Modification Factor, α.
Figure 5.4: Effect of rim modification factor on ǫRMS for various tendon loading.
reflector construction and the unmodified flat band construction (α = 0), for comparison. The
reduction in ǫRMS (fourth column) compares the values ǫRMS(0,γ) and ǫRMS(α∗γ,γ).
γMolded α= 0 Reflector with α= α∗γ ReductionǫRMS ǫRMS α∗γ ǫRMS(α
∗γ,γ) Gain (dB) in ǫRMS
0.01 0.8082 0.5084 0.002 0.4718 58.97 7.2%0.02 0.2238 0.4369 0.000 0.4369 58.99 —–0.03 1.0456 1.3537 0.028 0.8993 57.12 33.6%0.04 1.9960 2.3340 0.032 1.2478 56.81 46.5%
Table 5.3: Comparison of ǫRMS(α,γ) values (mm) for various γ.
Consider, for example, the tendon foreshortening factor γ = 0.03. The best surface accuracy
attained is ǫRMS = 0.8993 mm for the parameters (α,γ) = (0.028,0.03). This ǫRMS value is 33.6%
less than that of the case (α,γ) = (0,0.03). The case for (α,γ) = (0.028,0.03) also has an ǫRMS
81
value that is 14.0% lower than ǫRMS = 1.0456 mm for the molded reflector. These results advo-
cate for the use of design parameters such as α to help reduce surface distortions such as those
generated by the boundary support forces.
5.3 Tendon Supported vs. Fixed Boundary Reflectors
From Figure 5.4, we see that for sufficiently large γ (in our case γ > 0.02), ǫRMS(α∗γ,γ) increases
as a function of γ. This suggests that the surface accuracy may inevitably suffer for very strong
tendon forces. This motivates an investigation of the antenna’s ǫRMS when the reflector is not sub-
ject to tendon forces at the boundary. In this section, we model the antenna reflector using the flat
band construction with the boundary fixed to a rigid frame and α = 0. This entails eliminating S∗t
from Equation (2.2) and fixing the vertices of the antenna’s rim. Equilibrium shapes are calculated
for the differential pressures p0 = 5, 10, 15, 20, and 25 Pa. The shapes are also tested in three
different gravitational fields as listed in Table 5.4.
Gravitational p0 = 5 Pa p0 = 10 Pa p0 = 15 Pa p0 = 20 PaEnvironment ǫRMS(p0,g) ǫRMS(p0,g) ǫRMS(p0,g) ǫRMS(p0,g)g= 0 0.1923 0.0853 0.2483 0.4454g=−gk 0.1861 0.0879 0.2546 0.4519g=−gj 0.1925 0.0856 0.2484 0.4454g=−g(j+ k)/
p2 0.1969 0.0807 0.2447 0.4420
Table 5.4: Collection of ǫRMS(p0,g) values (mm) for a fixed boundary reflector.
Generally, the shape of the reflector is far more accurate for these experiments than for tests
involving the supporting tendons. For the pressure values of 10 and 15 Pa, the fixed boundary
reflector models have surface accuracy range 0.0853≤ ǫRMS ≤ 0.2546 as presented in Table 5.4. In
Table 5.3, we see that the band construction achieves its best surface accuracy of ǫRMS = 0.4369 for
the case (α,γ) = (0,0.02). Figure 5.5 compares the vertical discrepancy of the deformed reflector
to an actual paraboloid for both a fixed rim antenna and tendon–supported one. Each group of
data points in Figure 5.5 is the minimum, average and maximum vertical facet displacement for
all the facets in that band. With exception of the boundary support and configuration type, all
conditions for the two tests are identical. The internal pressure is held at 12.5 Pa and gravity is set
82
to g= −gk.
The top set of data in Figure 5.5 is for a tendon supported antenna modeled with the band
configuration and (α,γ) = (0.028,0.03). The lower set of data is for a fixed boundary reflector
modeled with the band configuration. The tenth band of the tendon–supported reflector has a
large variance in displacement due to α= 0.028 for that case.
0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
1
1.5
2
Tendon Supported
Fixed Boundary
Radial Distance from Vertex (meters)
Vert
ical
Dis
plac
emen
tof
Cen
troi
d(m
illim
eter
s)
Distribution of Facet Centroid Displacements
Figure 5.5: Comparison of error for a fixed rim and a tendon–supported reflector.
The low RMS values for the fixed boundary reflector suggest good performance at radio fre-
quencies greater than 40 GHz. We see that the ǫRMS values, despite their growth for larger pressure
values, remain low compared to the ǫRMS values of the studies shown in Tables 5.2 and 5.3. Even
changes to the gravity field have little affect on the ǫRMS values that the experiments in this sec-
tion attain. The radiation and efficiency data in Table 5.5 show that the performance of a fixed
boundary antenna may be able to perform well at frequencies above Ka–Band.
83
Radiation Frequency (GHz) 40 60 80Radiation Wavelength (mm) 7.50 5.00 3.75Intensity at Boresight (Watts/Str) 1282.71 2884.89 5125.71Antenna Gain (dB) 59.0046 62.5246 65.0208Side Lobe Level – H plane (dB) −17.57 −17.59 −17.56Side Lobe Level – E plane (dB) −24.19 −24.18 −24.18Radiometric RMS (mm) 0.1609 0.1609 0.1609Euclidean RMS (mm) 0.1707 0.1707 0.1707Efficiency 99.89% 99.85% 99.79%
Table 5.5: Characterization of fixed rim inflatable antennas; p0 = 12.5 Pa.
At a radio frequency level of 80 GHz, we can see how much more efficient the fixed rim an-
tenna is compared with the tendon supported one; see Table 5.6. Note that the tendon supported
reflector has ǫRMS < 1 mm. For high radio frequency, however, the ǫRMS has far lower tolerance.
If these large reflector antennas are used at these frequencies, one may wish to re–examine the
current tendon support boundary condition.
Tendon Supported Fixed Rim
Flat Band Construction: α 0.028 0.000Tendon Foreshortening: γ 0.03 —–Intensity at Boresight (Watts/Str) 3004.45 5125.71Antenna Gain (dB) 62.7009 65.0208Side Lobe Level – H plane (dB) −17.29 −17.56Side Lobe Level – E plane (dB) −21.82 −24.18Radiometric RMS (mm) 0.8993 0.1609Euclidean RMS (mm) 4.1233 0.1707Efficiency 58.49% 99.79%
Table 5.6: Characterization (for comparison) at 80 GHz.
84
Chapter 6
Conclusions and Future Research
Inflatable antenna reflectors have been successfully deployed and operated in radio frequencies
below Ka Band in past missions. There is interest in improving the surface accuracy of elastic
deployable antennas to meet the demands of future data transmission in both long and short
range space missions. Deployable antennas of the type discussed throughout this dissertation are
generally sensitive to either one or many of the supporting boundary conditions. The shape of the
deformed membrane varies when the parameters or boundary conditions are adjusted.
In Chapter 4, we considered a number of demonstration cases. For antenna applications, the
geometry is typically a paraboloid or an off–axis section of a parabaloid. To assess validity of our
model of a pressurized membrane formed by sealing together two opposite facing surfaces, we
considered the idealized problem of an inelastic mylar balloon in Section 4.1. A mylar balloon is
not molded, but constructed by sealing together the edges of two disks of equal diameter. The
equilibrium equations for the ideal mylar balloon are set in the deformed configuration, and yield
an axisymmetric smooth surface free of wrinkling. The generating curve can be computed explicitly
using elliptic functions. We find that the height to diameter ratios for the ideal mylar balloon
and our numerical solutions are in good agreement. Furthermore, our model is able to capture
wrinkling, an effect that must occur since a flat surface (like a disk) cannot be mapped smoothly
onto a doubly curved surface.
For an inflatable antenna, we fix the F/D ratio and vary the diameter of the reflector. Our
results find that the support tendon force is a key parameter that significantly influences the
85
efficiency of the reflector. If this support force is fixed, then the antenna’s RMS increases as
the reflector diameter increases. One particular example is an antenna that has the parameters
(D,γ) = (14.91 meters, 0.06) and has the same gain as an ideal D = 9.53 meter antenna. A large
antenna may be desirable, but greater surface accuracy is needed to obtain the corresponding ra-
diating aperture size. In the case of a smaller antenna, a more careful analysis was considered. It
was concluded that tendon forces that are too high deform the reflector in a manner that vertically
displaces the vertex of the reflector. This sort of deformation shifts the position of the reflector
with respect to the feed. In future work, we plan to carry out a dimensional analysis so that we
can better isolate the key parameters for additional trade studies.
Gravitational effects are noticeable for large (10 meter range) antennas but not for smaller
ones. It is particularly clear that large antennas experience less deformation in zero gravity en-
vironments or when they are positioned with the aperture plane containing the gravity vector.
Gravitational forces that are normal to the aperture plane can displace a large reflector away from
the feed. Hence, the surface accuracy of a large reflector is better when the rim is oriented verti-
cally.
We also probed the effects of varying the boundary support system. In our results, asymmetric
loading of the symmetric antenna system can dramatically influence the radiometric RMS. While
we only considered a sinusoidal variation of Kt,m with period (2π)/(kπ/Nt) = 2Nt/k and k = 3,4,
tendon failure (i.e., Kt,m = 0) in a number of contiguous locations could be catastrophic, rendering
the reflector inoperable. This strengthens the argument for attaching the reflector edge directly to
a rigid structure and eliminating the tendons altogether. However, if an off–axis antenna system is
desirable, then the edge forces will need to be applied in an appropriate non–symmetric manner
and attaching the reflector edge to a rigid structure could be problematic. A backup system of
tendons could eliminate some of the tendon support concerns.
In Chapter 5, we discovered that reflectors constructed from flat panels of elastic material can
be pressurized to nearly parabolic shape. This removes the necessity of the molding process (which
is expensive) and also opens the possibility of altering the construction pattern. Either geometric
or physical parameters can be analyzed to determine what affect they may have on the reflector’s
surface distortions. It was shown that the parameters can be implemented to improve antenna
performance.
86
In Section 5.1, we considered parametric studies where we varied the differential pressure
p0 and the number of gores Ng . As one might expect, increasing the number of gores leads to
an increase in surface accuracy. Similarly, we found that the differential pressure can be used to
reduce surface distortions. If the pressure is too low, then it is impossible for the antenna to hold
the parabolic shape. However, if the pressure is too high, surface distortion increases. For a given
set of design criteria, there should be an optimal value of p0.
We also explored how the radiation pattern at the aperture is perturbed by the cutting pattern
of the reference configuration. In our investigations, we found that most distortion was concen-
trated along the gore seams, but these effects could be mitigated by increasing the pressure. The
gore structure induces the wrinkling to take place along the seams. However, these distortions
could also be reduced by altering the cutting pattern. For example, one might curve the bound-
aries of Ωi in Figure 2.6. This approach has been followed in the fabrication of high altitude large
scientific balloons and spherical pressure vessels.
In Section 5.2, we investigated the effects of varying the tendon foreshortening γ and the rim
modification factor α. The results in Section 5.2 advocate for the use of design parameters such
as α to help reduce surface distortions such as those generated by boundary support forces. Our
parametric studies showed that surface accuracy degrades with very strong applied tendon forces.
This suggested an investigation of a parabolic reflector that eliminated the tendons altogether. For
this reason, we attached the edges of the paraboloid to a rigid structure. In general, we found
that the shape of the reflector for this boundary support is far more accurate than the shapes
supported by tendons. Moreover, the low RMS values for the fixed boundary reflector suggest
good performance at radio frequencies greater than 40 GHz. The results in Section 5.3 show the
performance of a fixed boundary antenna should perform well at Ka Band. Moreover, for high
radio frequency applications (beyond Ka Band), ǫRMS had a far lower value than the 1 mm value
found for the tendon supported system.
We altered the α for the outer band, but there is no reason why similar parameters α1, . . . ,αk
cannot be introduced for other bands. Certainly, this is worth exploring if it will further reduce
surface distortions. In Appendix A, we present metrology data that was collected in June 2008 at
NASA’s Glenn Research Center.
Our general approach to compute the reflector’s configuration can be applied to a wide vari-
87
ety of construction types and large range of parameter values. Combined with the more accurate
surface error computations (compared to the Ruze Equation), these methods should be able to pre-
dict the performance capabilities of larger antenna reflectors subject to a host of different factors.
This model has shown that properly selected design parameters and construction types can help
increase the performance of inflatable antennas which will open new applications where greater
surface accuracy and reliability are required.
Directions for Future Research
There are a number of projects that naturally follow from this research. There is interest in devel-
oping large aperture antennas that are deployable and operable for sustained periods in potentially
extreme temperatures or harsh environments. Many avenues of research can be pursued to explore
the shape retention of inflatable antennas even in controlled environments.
If it is desirable to use an off–axis system, then one needs to derive the correct tendon loads
that will yield the appropriate parabolic geometry. If the reference geometry of the reflector is
molded, calculation of the boundary conditions should be straightforward. However, if we are
considering a large reflector of flat panels, then the cutting pattern may also need to be adjusted
in order to achieve the appropriate inflated geometry.
For our preliminary investigations, it was sufficient to use an isotropic membrane model. How-
ever, it may be desirable to exact more precise control over certain regions of the inflated mem-
brane. This could be done using orthotropic materials and so it would be desirable to develop an
orthotropic material model in order to carry out the necessary trade studies.
In the future, it would be useful to carry out a dimensional analysis of the equilibrium equations
in order to isolate key parameters to better understand how the inflated membrane structure scales
as the aperture diameter increases. Once this is done, one could carry out trade studies involving
the deployed to packaged volume ratio, a critical parameter in space applications.
Because the physical models that were made available to us had an F/D ratio of 0.4291, we
restricted our attention to this class of parabolic reflectors. However, it would be revealing to study
the radiometric RMS, antenna power gain, and reflector efficiency as a function of Kt,m,α,γ, p0
for different F/D ratios. It is not too difficult to imagine that a pressurized large aperture deep
88
parabolic antenna in a 1g gravity field would be less susceptible to gravity effects than a shallow
parabolic antenna of the same size. How might the design engineer trade the reduced mass of
the shallow antenna for the more robust, albeit heavier, deep antenna? A shallow antenna would
require higher support tendon tension than a comparably sized deep antenna, since more of the
membrane tension will be transferred across the interface between the opposite facing paraboloids
in a deep antenna. How significant is this, if the antenna is to operate in zero-gravity? Additional
trade studies would be able to provide partial answers to these questions.
The inflatable antenna system has a number of features that make it very appealing for ap-
plications where surface accuracy, efficiency, low mass, ease of deployment, and high packaging
efficiency are important. It is not a simple task to find a solution to such a multi–parameter opti-
mization problem, let alone the optimal one. However, our findings as supported by our analytical
modeling and numerical simulations suggest that these questions are tractable and can be probed
with modest computing power and an appropriate mathematical model that captures the physics
of the problem with sufficient fidelity.
89
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93
Appendix A
Surface Metrology Data
Some surface metrology data on inflatable antennas was collected at NASA’s Glenn Research Center
in June 2008. The data was obtained using a high precision Leica LR200 Laser Scanner. This
scanner can measure the position of a point on a surface with a precision of 20 microns at a
distance of up to 48 meters away. For our experiments, the surface of the antennas were measured
at a distance of not more than 10 meters away. The scanner defines a particular point in space as
the origin and data is measured based on that reference point in a Cartesian coordinate system.
The units of the metrology data are inches, but we convert to meters in our presentation here.
Since the coordinate frame of the laser scanner may not put the antenna near the origin, we
will need to rotate and translate the metrology data so that it corresponds to a paraboloid with
a vertex at the origin and directrix plane perpendicular to the z–axis, as in Equation (2.1). The
original data points (xn, yn, zn) can be transformed to (xn, yn, zn) = T (xn, yn, zn), where T is a
matrix of translations and rotations. The translation parameters are x0, y0 and z0; the rotation
parameters are θ0 (a rotation about the z–axis) and φ0 (a rotation about the x–axis).
For the given metrology data set, we aim to determine a “least squares fit” paraboloid. In other
words, we seek the set of parameters (x0, y0, z0, F0,θ0,φ0) that minimizes
Γ(x0, y0, z0, F0,θ0,φ0) =
N∑
n=1
x2n+ y2
n
4F0− zn
2!1/2
where N is the number of data points and F0 is the focal length. A MATLAB code was written to
solve this minimization problem and it utilizes the MATLAB function lsqnonlin.
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For several data sets, we present the results of the least squares fit paraboloid in Table A.1 and
show some corresponding scatter plots in Figures A.1, A.2 and A.3.
Case NManufacturer’s Properties Results of lsqnonlinF (meters) R (meters) F0 (m) R0 (m) Γ (mm)
1 32,830 0.914 1.065 0.9143 1.0841 1.00512 83,466 0.914 1.065 0.9161 1.0805 1.32033 9,353 0.152 0.152 170.17 0.1670 8.65094 7,077 0.152 0.152 0.1445 0.1501 2.4556
Table A.1: Least squares paraboloid parameters for surface metrology data.
The following are the key physical conditions that were present during each of the data sampling
cases in Table A.1.
Case 1. The antenna reflector was situated with the gravity polar angle φ = π/2. The differen-
tial pressure was held at p0 = 39.75 Pa (0.159 inAq). The scanner was set to measure
the position of surface points with a rectangular discretization of 1.02 cm × 1.02 cm
over the surface.
Case 2. The antenna was leaning so that φ = 76. We set p0 = 15 Pa (0.06 inAq) and the
discretization of the surface points was 0.63 cm × 0.63 cm.
Case 3. Small off–axis antenna which is positioned upside down (φ = π). Differential pressure
was held within the range 13.25 to 14.25 Pa (0.053 to 0.057 inAq). The discretization
of the surface points for this case was 0.25 cm × 0.25 cm. Severe deflection at the rim
results in a least squares fit paraboloid which is far from the actual intended geometric
dimensions; see Table A.1 and note Figure A.3.
Case 4. Small off–axis antenna with φ = π/2 and p0 = 14.75 Pa (0.059 inAq). The dis-
cretization is 0.25 cm× 0.25 cm. Several tendons are removed such that are only 7
of the total 32 are supporting. These 7 tendons are distributed so that the reflector
is supported in three principal directions. While these tendons and the pressure offer
physical support for the reflector, substantial wrinkling is present. The least squares
fit paraboloid for this case is more representative of the intended geometry than in
Case 3. This is due to less deflection at the rim and a resulting shape which is globally
nearer to the desired geometry, despite wrinkling.
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In Figures A.1 and A.2, gravity is oriented in the −x direction. In Figure A.3, gravity is oriented
in the −z direction. The units are meters and the axes are equally scaled within each figure.
−1.0−0.5
00.5
1.03.23.8
−1.0
−0.5
0
0.5
1.0
y − axis
z − axis
x −
axi
s
Figure A.1: Surface metrology plot of axisymmetric reflector for φ = 90.
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−1.0−0.5
00.5
1.03.2 3.8
−1.0
−0.5
0
0.5
1.0
y − axisz − axis
x −
axi
s
Figure A.2: Surface metrology plot of axisymmetric reflector for φ = 76.
00.1
0.22.1 2.2 2.3 2.4
−1.00−0.98−0.96
x − axisy − axis
z −
axi
s
Figure A.3: Surface metrology plot of off–axis reflector; exhibits substantial rim deflection.
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