Michael J. Coleman and Frank Baginski

George Washington University, Washington, DC

Robert R. Romanofsky

Glenn Research Center, Cleveland, Ohio

The Effect of Boundary Support and Refl ector

Dimensions on Infl atable Parabolic Antenna

Performance

NASA/TM—2011-217110

November 2011
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Michael J. Coleman and Frank Baginski

George Washington University, Washington, DC

Robert R. Romanofsky

Glenn Research Center, Cleveland, Ohio

The Effect of Boundary Support and Refl ector

Dimensions on Infl atable Parabolic Antenna

Performance

NASA/TM—2011-217110

November 2011

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

The fi rst two authors were supported by NASA Grants NNX07AR67G and NNX09AH08G.

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Alexandria, VA 22312

Available electronically at http://www.sti.nasa.gov

Trade names and trademarks are used in this report for identifi cation 

only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

Level of Review: This material has been technically reviewed by technical management. 
For parabolic antennas with sufficient surface accuracy, more power can be radiated

with a larger aperture size. This paper explores the performance of antennas of various

size and reflector depth. The particular focus is on a large inflatable elastic antenna reflector that is supported about its perimeter by a set of elastic tendons and is subjected

to a constant hydrostatic pressure. The surface accuracy of the antenna is measured by an

RMS calculation, while the reflector phase error component of the efficiency is determined

by computing the power density at boresight. In the analysis, the calculation of antenna

efficiency is not based on the Ruze Equation. Hence, no assumption regarding the distribution of the reflector surface distortions is presumed. The reflector surface is modeled

as an isotropic elastic membrane using a linear stress–strain constitutive relation. Three

types of antenna reflector construction are considered: one molded to an ideal parabolic

form and two different flat panel design patterns. The flat panel surfaces are constructed

by seaming together panels in a manner that the desired parabolic shape is approximately

attained after pressurization. Numerical solutions of the model problem are calculated under a variety of conditions in order to estimate the accuracy and efficiency of these antenna

systems. In the case of the flat panel constructions, several different cutting patterns are

analyzed in order to determine an optimal cutting strategy.

NASA/TM—2011-217110 1

The Effect of Boundary Support and Reflector Dimensions on 

Inflatable Parabolic Antenna Performance 

Michael J. Coleman and Frank Baginski 

George Washington University 

Washington, DC 20052 

Robert R. Romanofsky 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 
Nomenclature

A Aperture area of the antenna (square meters)

D Diameter of antenna aperture (meters)

Ef Gravitational potential energy due to film (Joules)

Ep Hydrostatic pressure potential energy (Joules)

ET Total energy of the inflatable reflector system (Joules)

F Focal length of antenna (meters)

G Effective antenna gain

GI Ideal antenna gain

g Acceleration due to gravity (meters per square second)

Kt,m Tendon stiffness constant in the mth supporting tendon (Newtons)

S

∗

f Relaxed film strain energy (Joules)

S

∗

t Relaxed tendon strain energy (Joules)

N Number of triangular elements in the upper and lower shells

Nf Number of triangular elements in the lower shell

Nt Number of supporting tendons

p0 Constant differential pressure (Pascals)

W∗

f Relaxed film strain energy density (Joules per square meter)

W∗

t,m Relaxed tendon strain energy density in the mth supporting tendon (Joules per square meter)

E Young’s modulus of film (Pascals)

ν Poisson’s ratio of film material

ΩR Reference configuration

Ω Deformed configuration

α Rim modification factor

γ Tendon length adjustment factor

εm Strain in the mth supporting tendon (meters per meter)

λ Radiation wavelength (meters)

f Radiation frequency (Hertz)

IRkk-dimensional Euclidean space

ˆr Unit direction vector from the antenna to a point in the far field (unit meter)

ˆj Unit direction vector (0, 1, 0) (unit meter)

kˆ Unit direction vector (0, 0, 1) (unit meter)

NASA/TM—2011-217110 2
I. Introduction

The future of communication in near Earth orbit and deep space missions may require large deployable

circular aperture antennas. It is conceivable that the diameter of the aperture may need to reach 10 meters

or more in order to support data transmission rates of hundreds of megabits per second. Similarly, much

larger apertures may be required for futuristic remote sensing and imaging applications.1 A rigid reflector of

such large diameter, however, will not fit aboard a small launch vehicle for transport into orbit. Deployable

space structures are advantageous in deep–space and near Earth orbit missions due to their low construction

expense, low weight and high packing efficiency.2 For these reasons, the use of deployable antenna reflectors

is an attractive option.

Although inflatable antennas have been considered since the 1960s with successful deployment for L and

S band applications,3there is now interest in using such antennas for shorter wavelength bands. There are

three fundamental deployable antenna families, each with its own set of attributes: mesh reflectors with

various deployment mechanisms, shape memory polymer and tensioned membrane shells, and the inflatable membrane reflectors under consideration here. Mesh reflectors dominate the space arena and have an

impressive 20 year plus track record. Nevertheless, there have been a number of deployment anomalies

compromising mission success, including NASA’s Galileo satellite, the Asia Cellular Satellite, the Japanese

VLBI Space Observatory Program, and most recently SkyTerra 1. Shape memory polymer reflectors are a

nascent concept and perhaps at too low a technology readiness level to predict future mission acceptance.

These observations make the case for an alternative antenna technology, namely inflatable membranes, which

promise less complex deployment mechanisms (hence higher reliability), substantially lower aerial density,

and superior packaging efficiency.

It is critical that the reflector have minimal surface distortions and retain a parabolic shape, particularly when operating in frequencies with shorter wavelengths (Ka–Band, for example). One estimate for the

tolerance of the reflector surface accuracy is an RMS value less than 0.5 millimeters.1 A common method

for calculating antenna efficiency directly from RMS involves the Ruze Equation. While the Ruze Equation

simplifies the computation of antenna efficiency, applying it requires the assumption of random reflector

surface distortions having a Gaussian distribution.4 Some earlier research has questioned the accuracy of

the Ruze Equation in analyzing inflatable antenna efficiency. This is specifically a concern since the surface

errors tend to concentrate near the antenna rim.1

NASA/TM—2011-217110 3
The purpose of this paper is to investigate the elastic deformation of inflatable antennas in order to more

carefully analyze the areas of deformation, material stresses, regions of potential wrinkling and the effect of

these characteristics on the overall antenna efficiency. In the model presented, the reflector of an inflatable

antenna is treated as an elastic membrane which deforms in response to external forces. The model employs

a finite element method to directly compute the power gain of a given deformed reflector. A key advantage

of this analysis is that the Ruze Equation is not used and therefore no assumption is made regarding the

distribution of reflector surface distortions.

Earlier work by Greschik5showed that the precision of the membrane shape is sensitive to membrane

properties and wrinkling. Previous work by the authors, explored non–symmetric boundary and gravitational loading and found that asymmetric boundary conditions or high supporting tendon forces can cause

substantial distortion to the reflector surface.6 Reflector construction methods (cutting patterns) were also

explored to help reduce the surface error. Some preliminary findings suggested that modifications to the cutting patterns could help reduce the deflection of the reflector’s rim and thereby improve the surface accuracy.7

This paper extends some earlier results8for the flat panel constructions of small antennas to larger antenna sizes. In addition, two parametric studies focusing on both size and depth of the antenna reflector

are presented. The shape of the deformed membrane is calculated using a model for pneumatic envelopes

developed by the second author and his collaborators.9, 10 The geometry of the discretized model for a

molded reflector is presented in Section III while the geometry of the flat panel constructions are described

in Section IV. Section V introduces RMS calculations that are used to evaluate the antenna shapes obtained.

These calculations are based on a numerical model that approximates the reflector as a faceted surface. No

assumption about the distribution of surface errors is made, nor is the Ruze Equation applied in computation of the antenna efficiency. Finally, some numerical results will be presented in Section VI. The results

include parametric studies that vary the tendon forces, size of the aperture and depth of the antenna reflector.

II. Basic Antenna Design

The inflatable antenna system considered in this study is formed by molding a light–weight material into

parabolic form. Two such parabolic membranes are created and seamed along their rims. The two memNASA/TM—2011-217110 4
branes are held to parabolic shape by an inflation gas between them and are supported by tendons which

are attached to the seamed rim of the membranes on one end and a rigid frame on the other. Typically, one

of the parabolic sheets is coated with 1200  ̊A of aluminum to make the surface a reflective one. The coated

half of the closed surface will serve as the antenna’s reflector. See Figure 1 for a schematic of the inflatable

antenna system.

Figure 1. Parabolic reflector. The lines emanating from F are typical light paths.

For large reflectors (diameters closer to the 10 meter range), the molding process may be infeasible or

too expensive. One may therefore wish to 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. The ability

of such a reflector to achieve the desired parabolic shape will be investigated. For these constructions, the

accuracy of the pressurized reflector will depend on the cutting pattern.

III. Faceted Surface Construction and Mathematical Model

This section presents a mathematical model in which the membrane is approximated by a faceted surface, using constant–strain, plane–stress triangular finite elements. Figure 1 is an image of the actual faceted

surface used for a molded reflector in computations. Although the coated reflector is labeled in Figure 1 for

exposition, the effect of the vaporized aluminum is included on neither the reflective properties nor mechanical properties of the reflector. For these studies, it is assumed that the entire membrane is made of 12.7μm

Kapton. The material parameters will therefore be identical for both the upper and lower paraboloids in

NASA/TM—2011-217110 5

these investigations. The mechanical properties of Kapton are presented in Table 1.
In order to generate the faceted surface, a parabolic mold is created by the parametric equations

x = a + r cos θ; y = a + r sin θ; z = r

2

/(4F), (1)

for 0 ≤ θ ≤ 2π and 0 ≤ r ≤ Rf where Rf is the radius of the reflector aperture and F is the focal length of

the antenna. In Equations (1), a ≥ 0 is a translation that is used to generate a mold for an off–axis reflector.

For axisymmetric reflectors, set a = 0. The paraboloid of Equations (1) is then translated and rotated so

that the rim lies in the plane z = 0 and the 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 a molded reflector, one

takes this initial configuration as the unstrained reference configuration ΩR ⊂ IR3. Simulating the two flat

panel constructions will involve a different unstrained reference configuration ΩR ⊂ IR2(see Section IV).

One of the parametric studies varies the depth of the parabolic reflector. The depth of the reflector is entirely determined by the diameter of the aperture, D = 2Rf and the focal length of the antenna, F. The ratio

F/D is used to express the depth of the antenna. If the reflector size is fixed at a certain aperture diameter,

then the depth is controlled by adjusting F. See Figure 2 for several antenna profiles having D = 2.13 meters.

The supporting tendons are modeled as linearly elastic strings. When the envelope is pressurized, the

entire inflatable reflector system (including the supporting tendons) will seek an equilibrium configuration

of minimum potential energy. In the model, the pressure, tendon stiffness and gravity are treated as parameters 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 gravitational environments.

This model follows an optimization–based solution process which determines the equilibrium position of

the inflated reflector by minimizing the total energy of the discretized system.9 Consider a deformation x

of the reference configuration ΩR ⊂ IRkfor k = 2, 3. The deformed configuration is a surface in IR3, i.e.,

x(ΩR) = Ω ⊂ IR3. For a particular deformed configuration, the energy of the antenna system is calculated

by the functional

ET (x) = Ef (x) + Ep(x) + S

∗

f

(∇x) + S

∗

t

(∇x), (2)

where Ef is the gravitational potential energy of the film, Ep is the hydrostatic pressure potential energy,

NASA/TM—2011-217110 6
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

 z (meters)

 r (meters)

 F / D = 0.2146

 F / D = 0.4291

 F / D = 0.8582

Figure 2. Cross section of inflated antennas for different F/D values and z = r

2/(4F ).

S

∗

f

is the relaxed strain energy in the membrane and S

∗

t

is the relaxed strain energy in the supporting elastic

tendons.9

III.A. Gravitational Potential

The direction and magnitude of gravity are included as parameters in the 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 gravitational potential energy

can be calculated by

Ef (x) = Z Z

ΩR

wf (x · g) dA

where wf is the film weight density, g is the directed acceleration due to gravity and dA is the surface area

measure over the faceted reference configuration ΩR. This quantity is calculated exactly for each triangular

facet and then summed over the set of triangles.

For the remainder of this paper, fix the parameter g = 9.80665 meters/sec2for the acceleration due to

NASA/TM—2011-217110 7
gravity. The only case for which the gravity vector g will not have the aforementioned magnitude is for a

zero gravity environment.

III.B. Hydrostatic Pressure Potential

For the deformed configuration Ω, let B be the enclosed interior space. Assuming that the differential

pressure is a constant p0, the hydrostatic pressure potential energy is given by

Ep(x) = Z Z Z

B

p0 dV =

Z Z

Ω

p0 k · ˆn dS (3)

where ˆn is the unit outward normal to the deformed surface Ω and dS is surface area measure on Ω. The

divergence theorem was applied to obtain the last integral in Equation (3). The quantity Ep is calculated

exactly for the faceted surface Ω.

III.C. Film Strain Energy

The strain energy in the film is calculated by

S

∗

f

(∇x) = Z Z

ΩR

W∗

f

(∇x) dA

where dA is area measure in ΩR and W∗

f

is the relaxation of Wf (i.e., W∗

f

is the largest convex function that

does not exceed Wf ). The strain energy density is

Wf =

1

2

(S : G)

where G is the Cauchy–Green strain tensor, S is the second Piola–Kirchoff stress resultant tensor, and ‘:’

denotes the tensor inner product. A piecewise constant strain model is applied to the faceted surface. Each

triangle has two principal stresses and two principal strains. The principal strains are the eigenvalues of G

and are denoted δ1 and δ2. The principal stress resultants are the eigenvalues of S and are given by μ1 and

μ2. A thin compliant membrane will not resist compressive stresses but wrinkle instead. The wrinkling is

modeled by using the relaxed strain energy density, W∗

f

. Following the method due to Pipkin,11 the faceted

surface is partitioned into three sets: the set of slack facets, the set of wrinkled facets, and the set of taut

facets. For any given triangular facet, T ∈ ΩR, the state is determined by the principal strains and stresses.

The conditions of Equation (4) describe the possible states: (a) slack, (b)–(c) wrinkled or (d) taut. The

NASA/TM—2011-217110 8
relaxed strain energy density for T is

W∗

f

(T) = hE

2







0 if δ1 < 0 and δ2 < 0; (a)

δ

2

2

if μ1 ≤ 0 and δ2 ≥ 0; (b)

δ

2

1

if μ2 ≤ 0 and δ1 ≥ 0; (c)

δ

2

1+δ

2

2+2νδ1δ2

1−ν2 if μ1 ≥ 0 and μ2 ≥ 0, (d)

(4)

where E and ν are the Young’s modulus and Poisson’s ratio of the film, respectively.9 The thickness of the

material is given by h. See Baginski, et. al.9, 10 for details.

III.D. Tendon Strain Energy

Let Nt be the total number of tendons. The supporting tendons are modeled as linearly elastic strings.

Therefore, the relaxed strain energy density in the mth supporting tendon is

W∗

t,m =







1

2 Kt,mε

2

m if εm ≥ 0;

0 if εm < 0,

(5)

where Kt,m and εm m = 1, . . . , Nt are the tendon stiffness constant and strain of the tendon, respectively.9

The strain in the tendon is calculated by

εm =

Γm − ΓR,m

ΓR,m

(6)

where Γm and ΓR,m 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 of the ΓR,m by

a percentage γ. Support loss in a region can be modeled by setting Kt,m = 0 for desired values of m. The

total energy in the tendons can then be found by calculating

S

∗

t

(∇x) = X

Nt

m=1

Z ΓR,m

0

W∗

t,m(s) ds.

For computational simplicity, the rigid support structure (to which one end of each tendon is attached) is

always in the plane z = 0.

NASA/TM—2011-217110 9
IV. Flat Panel Constructions

Since the molding process is an expensive one,1there is a cost benefit to using flat construction patterns.

Two different construction patterns based on flat sheets of elastic material are considered. One involves flat

gores which seam together like slices of a pie (see Figure 3(a)) and the other involves annular bands which

are seamed to form the lateral surface of a frustum (see Figure 3(b)).

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 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. One can then generate the triangular mesh of the reference configuration ΩR with these vertices. The result is a reference configuration that models a surface having zero

curvature in the circumferential direction within the panels. The vertices of the ribs used to generate the

entire configuration are highlighted in Figure 3(a) as dots over the corresponding vertices. Notice that the

distribution and pattern of the triangular mesh is largely the same as the model for the molded antenna of

Figure 1. The configurations differ in that a gore of Figure 3(a) is initially a developable surface while a

typical “gore” of Figure 1 is initially doubly curved.

(a) Flat gore construction. (b) Flat band construction.

Figure 3. Reference configurations with generating ribs for (a) and generating rings for (b).

The other flat construction pattern consists of concentric bands of material which is a surface whose panels have zero 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 the desired parabolic

shape. These are shown by the rings of dots in Figure 3(b). Rows of triangles are then generated between

the rings to complete the reference configuration. These bands (on an actual reflector) would be sections of

cones and are therefore constructible from flat panels of material.

NASA/TM—2011-217110 10
It is expected that the flat panel constructions should achieve a nearly parabolic shape when the reflector

shell is pressurized. The mesh size of the partition for the flat panels will affect the accuracy of the deformed

reflector. These configurations are analyzed in Section VI.

V. Surface Accuracy and Efficiency Analysis

For both the molded and flat construction models, one can vary parameters that control the loading

forces to study the effect on the surface accuracy of the reflector. The techniques of this research provide

the analytical means to assist the design engineer in the selection of appropriate materials, cutting patterns,

and support structures that can reduce surface distortions. The accuracy of the deformed reflector can be

evaluated by considering the global RMS surface errors as determined by both vertical displacement of the

reflector and path length error.

V.A. RMS Calculations

It is necessary to first establish some notation for the discretized reflector. Denote the centroid of Tn ∈ ΩR

by pn. The corresponding triangle in Ω is Tn = x(Tn). Let the centroid of Tn be denoted by xˆn = (xn, yn, zn).

φ

φ

Aperture

∆zn

Focal Point

4Fz = x

2 + y2

nn fn

an

F

Figure 4. Deformed triangle Tn with centroid (xn, yn, zn). The vertical displacement, ∆z, is labeled and the pathlength

for this triangle is ln = fn + an.

A reflector in an ideal position should have each triangle’s centroid nearly satisfying the generating

equation 4F zn = x

2

n + y

2

n

. The centroids will likely not satisfy the generating curve exactly since it was the

vertices of the triangles that were positioned on the parabolic form. For a fine mesh, however, this error is

NASA/TM—2011-217110 11
negligible.8 The Euclidean measure of RMS, εeuc, is defined by

ε

2

euc(Ω) = Z Z

A



∆z

2 da

| A |

=

X

Nf

n=1

(∆z)

2

n

|Tn|(nˆn · k)

| A |

(7)

where ∆z is the vertical displacement between the deformed reflector and an ideal paraboloid, | A | is the

area of the circular aperture, A, da is Lebesgue area measure and |Tn|(nˆ · k) is the area of Tn projected onto

the aperture.5

Another RMS calculation includes the pathlength error of a radiation ray 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 all rays of radiation. Having denoted the total path length of a ray striking the

centroid of Tn by ln, one gets

ε

2

rms(Ω) = Z Z

A



∆l

2

2

da

| A |

=

X

Nf

n=1

( ln − 2F)

2

|Tn|(nˆn · k)

4| A |

(8)

where ∆l = ln − 2F is the pathlength error of a ray reflected at a given point on the reflector surface and

F is the focal distance of the reflector.5

V.B. Antenna Gain and Efficiency Calculations

It is desirable to consider the antenna’s efficiency by computing the far zone electric and magnetic fields that

are reflected by the antenna’s reflector surface. The analysis applied here closely follows the computation

methods in Chapter 8 of Stutzman and Thiele.12 In the following equations, the permeability and permittivity of free space are μ0 = 4π × 10−7 N / Amp2

and ε0 = 8.854 × 10−12 coul2 / N m2, respectively.13

This model uses a simple dipole for the feed having current I0 = 200 Amps oriented along the y–axis with

dipole length ∆y = 1 cm. In this case, the incident magnetic field on the reflector is

Hinc = −i

I0β∆y

4πr′

e

−iβr′

(ˆr′ ×ˆj),

where ˆr

′

is the unit direction vector from the focus to a point on the reflector, β

2 = ω2

ε0μ0 and i =

√

−1.

Also, ω = 2πf is the angular frequency associated with f. The far zone electric and magnetic fields can be

determined from the magnetic vector potential A by

E = −iωμ0



A − (ˆr · A)ˆr



and H = (ˆr × E)/η,

NASA/TM—2011-217110 12
where η

2 = μ0/ε0 is the impedance of free space. The power per steradian in any given direction from the

antenna (in spherical coordinates) can be obtained from the reflected electric field E and magnetic field H

by the Poynting Vector in the relation

U(θ, φ) = r

2

(E × H∗) · ˆr.

The function U is the radiation intensity in a given polar direction (θ, φ) where the antenna is positioned

with its vertex at the origin. This intensity function is independent of the distance from the antenna since

E × H∗ ∼ r

−2

.

A parabolic reflector should yield a radiation pattern U which achieves its maximum in the direction the

antenna is pointed (for this case θ = φ = 0). Denote by Um and U

p

m the maximum radiation intensity for the

deformed faceted surface configuration and an ideal paraboloid of the same dimensions, respectively. Then,

the component of the efficiency due to reflector surface distortions is

es = Um/Up

m. (9)

The results in this paper include efficiency computations that incorporate the trade off between the spill–over

and under–illumination efficiency calculations. The maximum efficiency of any parabolic reflector can be no

more than 82% or the value in Equation (9).12 The effective efficiency e

∗

is defined as

e

∗ = 0.82es.

Besides this tradeoff and surface distortions, no other efficiency losses are considered. The ideal and effective

antenna gain are found by,

GI = 4πA/λ2and G = e

∗GI , (10)

respectively. Note that λ = c/f where c is the speed of light. Unless otherwise noted, the data presented for

gain and efficiency are calculated for an RF value of f = 40 GHz.

VI. Numerical Results

This section consists of results which model reflectors having axisymmetric reference configurations. The

material properties of Kapton are used to serve as the reflector material and the dimensions of an actual

prototype are applied. Table 1 contains parameter values that are used in the following numerical experiNASA/TM—2011-217110 13
ments. In certain instances, the dimensions of Table 1 are scaled in order to study the effect of the size and

depth of a reflector on antenna performance.

Table 1. Mechanical properties of Kapton, reflector dimensions and grid specifications.

Property Variable Value

Kapton Thickness h 12.7 microns

Kapton Mass Density ρ 1.42 g/m3

Kapton Young’s Modulus E 2.59 GPa

Kapton Poisson’s Ratio ν 0.34

Reference Tendon Length ΓR,m 6.00 cm

Reflector Diameter D 2.13 meters

Reflector Focal Length F 0.914 meters

Total Number of Facets N 20,736

Number of Reflector Facets Nf 10,368

Number of Supporting Tendons Nt 72

Previous numerical experiments6, 8 demonstrated that the elastic reflector easily deforms when the force

of the supporting tendons is larger than approximately 0.5 N. For example, a force of 2.304 N corresponded

to a 2 to 3 degree deflection near the rim of the reflector and a vertical displacement of approximately 4 mm

near the vertex.6 This sort of deformation can cause the antenna to suffer a loss of power gain, particularly

at higher radio frequencies.

Some earlier results7suggest that the antenna performance can be improved by increasing the number

of gores in the flat panel construction. This initial work also seemed to suggest there was a preferred setting

for the differential pressure, p0. Too low a pressure allowed wrinkling in the deformed reflector while too

high a pressure caused excessive bulging of the reflector membrane.

Two studies are presented here which focus on the behavior of the reflector’s edge. The first entails a

modification to the flat band construction pattern of Section IV. The other consists of fixing the rim of

the reflector. Both experiments were conceived in an attempt to mitigate deflection of the reflector near

the rim.7 Parametric studies conducted for an antenna of diameter D = 2.13 meters will be presented in

Sections VI.B and VI.C. An analysis of a large diameter antenna (D = 8.52 meters) is also presented.

VI.A. Reflector Size Variation

This section contains an examination of the relationship between the reflector’s size and the surface accuracy

of the antenna when other significant parameters are held constant. Two case studies are performed; one

NASA/TM—2011-217110 14
with a tendon foreshortening of γ = 0.03 and the other with γ = 0.06. For each case study, the size of the

reflector is increased while holding the ratio F/D = 0.4291. The reflector sizes in Table 2 are obtained by

scaling the reflector of Table 1 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 the model for these parameters and the surface accuracy data is reported in Table 2.

Table 2. Comparison of efficiency for various antenna sizes and supporting conditions.

Reflector Dimensions Case I: γ = 0.03 Case II: γ = 0.06

D (m) F (m) εrms (mm) e

∗ Gain dB εrms (mm) e∗ Gain dB

2.130 0.914 1.4156 80.84 % 58.09 4.2595 74.91 % 57.75

5.325 2.285 4.7162 76.65 % 65.81 2.5938 80.31 % 66.02

8.520 3.656 15.5799 50.77 % 68.11 14.5017 54.82 % 68.44

11.715 5.027 31.0087 39.64 % 69.80 28.1322 43.64 % 70.22

14.910 6.398 50.8096 29.91 % 70.67 52.2116 33.49 % 71.16

By comparing the results of Cases I and II in Table 2, one can see that the size of the reflector having

greatest efficiency for Case I is different from that of Case II. This is not surprising since the mass of the

reflector is increased by the square of the scaling factor. A greater supporting force in the tendons is likely

necessary to counter the larger mass. Specific guidelines would depend on the particular application and the

surface accuracy tolerances.

For Case II, the antenna power gain is included. One can still attain higher levels of power gain for

larger diameter antennas despite the reduction of efficiency. From Equation (10), it can be found that the

antenna in case (D, γ) = (14.91, 0.06) performs like an antenna operating at 100% efficiency with diameter

8.63 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.

VI.B. Parametric Study of Edge Forces and Reflector Shape Modification

Since one serious problem with an inflatable antenna is the deflection of the reflector near the rim, the following parametric study that involves modifying the reflector rim shape is conducted. The band construction

is well–suited for this experiment since the rim of the reflector is contained in one section of the cutting

pattern. The slope of this outer most band is modified by decreasing the radius of the outer most ring of

vertices (situated at the reflector’s rim) as shown in Figure 5.

NASA/TM—2011-217110 15
−1 −0.5 0 0.5 1

−0.2

0

0.2

0.4

0.6

 r (meters)

 z (meters)

α=0.00

α=0.05

Figure 5. Generating curve for a flat ring configuration with α = 0.05.

of 0.1Rf where Rf =

D

2

is the radius of the aperture. A shape modification factor α ∈ [0, 1] is introduced to

adjust the outer most band. This factor determines the radial distance, d, from the vertex of the paraboloid

to the outer most ring of vertices. In particular, d = (1 − α)Rf . Note that the reference configuration is

unchanged for α = 0. The result is a reference configuration modeling a paraboloid for the inner nine bands

with a tapered adjustment for the tenth band.

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. A set of tests was performed to determine the reflector’s shape accuracy as a function of the rim

radius modification factor, α. The differential pressure is set to p0 = 12.5 Pa and the acceleration due to

gravity is set at g = −gk. Table 3 contains εrms values as calculated by Equation (8) for various values of

the parameters α and γ.

(a) D = 2.13 meters (b) D = 8.52 meters

0.01 0.02 0.03 0.04 0

1

2

3

4

5

α

ε RMS (mm)

 γ = 0.02

 γ = 0.03

 γ = 0.04

0.01 0.02 0.03 0.04 0

5

10

15

20

25

α

ε RMS (mm)

 p0 = 5.0 Pa

 p0 = 12.5 Pa

Figure 6. Effect of rim modification factor on εrms for a small(a) and large(b) inflatable antenna.

The band construction in this parametric study consists of 10 bands with the seams being set at intervals

NASA/TM—2011-217110 16
For each value of γ, the strain energy in the tendons is determined by using Equation (5) where Kt,m =

250 N for all Nt tendons. In Figure 6(a), εrms is plotted 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 6(a) that εrms can be reduced by selecting α

appropriately. For fixed γ, one can find a parameter value α = α

∗

γ

such that

εrms(α

∗

γ

, γ) = min

α ∈ A

εrms(α, γ) (11)

where A = { α | 0 ≤ α ≤ 4.4 }. Results concerning these local minima are summarized in Table 3. Columns

four through six contain the data for the flat band reflector that achieves the minimal εrms value for each

γ. The gain is calculated as in Equation (10). The second and third columns of Table 3 contain the εrms

attained by the molded reflector construction and the unmodified flat band construction (α = 0), for comparison. The reduction in εrms (seventh column) compares the values εrms(0, γ) and εrms(α

∗

γ

, γ).

Table 3. Comparison of εrms(α, γ) values (mm) for various γ. Aperture diameter is D = 2.13 meters. Reduction in εrms

is based on a comparison of Column 3 and Column 5.

γ

Molded α = 0 Reflector with α = α

∗

γ Reduction

εrms εrms(0, γ) α

∗

γ

εrms(α

∗

γ

, γ) Gain (dB) in εrms

0.01 0.8082 0.5084 0.002 0.4718 58.11 7.2%

0.02 0.2238 0.4369 0.000 0.4369 58.13 —–

0.03 1.0456 1.3537 0.028 0.8993 56.26 33.6%

0.04 1.9960 2.3340 0.032 1.2478 55.95 46.5%

The results in Table 3 advocate for the use of design parameters such as α to help reduce surface

distortions such as those generated by the boundary support forces. Consider, for example, the tendon

foreshortening factor γ = 0.03. The best surface accuracy attained is εrms = 0.8993 mm for the parameters

(α, γ) = (0.028, 0.03). This εrms 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 εrms value that is 14.0% lower than εrms = 1.0456 mm for the molded

reflector.

To investigate larger antennas, the analysis with the rim modification factor α is repeated with D =

8.52 meters. In this case, one finds that the tendon foreshortening does not affect εrms as much as the

differential pressure, p0. Hence, γ = 0.03 is fixed and the larger antenna tested for several different different

pressure levels. In Figure 6(b), note that εrms can be reduced with a particular value of α, as was the

case for the smaller antenna. The usual choice of p0 = 12.5 Pa, however, does not seem to be preferred

in this case. The data of this study suggests that changing the conditions from (α, p0) = (0.014, 12.5)

to (α, p0) = (0.006, 5) will lead to a reduction in εrms by 65.4%. For comparison with the plots of FigNASA/TM—2011-217110 17
ure 6(b), note that the ideal configuration (such as seen in Figure 1) for an antenna of this size would have

εrms = 0.4256 mm. This datum is based on the ideal reflector geometry and is independent of the differential

pressure, p0.

A practical consideration to reduce the rim effect entails reducing the area of the elastic reflector that is

treated for reflectivity. In particular, one can neglect treating the outer edge of the reflector, where surface distortions tend to concentrate. This scenario is modeled by removing the outer most ring of triangular elements

from the computations of Um as defined in Section V.B. Recall the case with (D, p0, α) = (8.52, 12.5, 0.014)

which resulted in εrms = 13.5066 mm and GdB = 66.82. For this case, neglecting the outermost ring of

triangles is equivalent to treating 2.30 square meters less of the surface for reflectivity (the total reflector

surface for this case is 57.86 square meters). Note that this neglected area is only a portion of the outer most

band of the flat band construction. The result obtained yields εrms = 13.3248 mm and GdB = 67.99. Despite

a decrease in the radiating aperture, removal of the distorted reflector edge from consideration allowed the

gain to increase slightly.

In both the case of a large and small antenna, the rim modification factor is effective in reducing εrms.

Other physical factors must be considered to help further the reduction of εrms. Which physical parameters

are most important and to what extent, will require further investigation.

VI.C. Tendon Supported vs. Fixed Boundary Reflectors

Figure 6(a) reveals that for sufficiently large γ (in that case γ > 0.02), εrms(α

∗

γ

, γ) increases as a function

of γ. This suggests that the surface accuracy may inevitably suffer from very strong tendon forces. This

motivates an investigation of the antenna’s εrms when the reflector is not subject to tendon forces at the

boundary. In this section, the antenna reflector is modeled by using the flat band construction, fixing the

boundary to a rigid frame, and setting α = 0. This entails eliminating S

∗

t

from Equation (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 4.

Table 4. Values of εrms(p0, g) (mm) for a fixed boundary reflector with D = 2.13 meters.

Gravity p0 = 5 Pa p0 = 10 Pa p0 = 15 Pa p0 = 20 Pa p0 = 25 Pa

Vector: g εrms(p0, g) εrms(p0, g) εrms(p0, g) εrms(p0, g) εrms(p0, g)

0 0.1923 0.0853 0.2483 0.4454 0.6450

−gk 0.1861 0.0879 0.2546 0.4519 0.6516

−gj 0.1925 0.0856 0.2484 0.4454 0.6450

−g(j + k)/

√

2 0.1969 0.0807 0.2447 0.4420 0.9556

NASA/TM—2011-217110 18
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 ≤ εrms ≤ 0.2546 as presented in Table 4. Table 3 shows that the band construction

achieves its best surface accuracy of εrms = 0.4369 for the case (α, γ) = (0, 0.02). Figure 7 compares the

vertical discrepancy of the deformed reflector, as calculated in Equation (7), for both a fixed rim antenna

and tendon–supported one. Each data point in Figure 7 plots the average vertical facet displacements of the

facets in each band against the outermost radial extent of the band. The bars surrounding the data points

represent the spread of the displacement data for the given 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 g = −gk.

The top set of data in Figure 7 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 since

α = 0.028 for that case.

0 0.2 0.4 0.6 0.8 1 1.2 0

0.3

0.6

0.9

1.2

1.5

1.8

 r (meters)

| ∆ z | (mm)

Tendon Supported

Fixed Boundary

Figure 7. 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 frequencies

greater than 40 GHz. It is clear that the εrms values, despite their growth for larger pressure values, remain

low compared to the εrms values of the study shown in Table 3. Changes to the gravity field have little

affect on the εrms values. The radiation and efficiency data in Table 5 show that the performance of a fixed

boundary antenna may be able to perform well at frequencies above Ka–Band.

NASA/TM—2011-217110 19
Table 5. Characterization of fixed rim inflatable antennas; p0 = 12.5 Pa.

Flat Band Construction, α 0.000 0.000 0.000 0.028

Edge Condition Fixed Rim Fixed Rim Fixed Rim γ = 0.03

Radiation Frequency, f (GHz) 40 60 80 80

Intensity at Boresight (Watts/Str) 1282.71 2884.89 5125.71 3004.45

Antenna Gain (dB) 58.1427 61.6630 64.1591 60.9769

Side Lobe Level – H plane (dB) −17.57 −17.59 −17.56 −17.29

Side Lobe Level – E plane (dB) −24.19 −24.18 −24.18 −21.82

Radiometric RMS (mm) 0.1509 0.1509 0.1509 0.8993

Euclidean RMS (mm) 0.1641 0.1641 0.1641 4.1233

Efficiency, e

∗ 81.91% 81.88% 81.83% 47.96%

At a radio frequency level of 80 GHz, the fixed rim antenna is much more efficient than the tendon

supported one; see Table 5. Note that the tendon supported reflector has an εrms on the order of 1 mm. For

high radio frequency, however, the εrms has far lower tolerance. If large reflector antennas are used for high

frequency applications, one may wish to re–examine the boundary support.

Table 6. Values of εrms(p0, g) (mm) for a reflector with D = 8.520 meters.

Gravitational p0 = 5 Pa p0 = 10 Pa p0 = 15 Pa p0 = 20 Pa p0 = 25 Pa

Environment εrms(p0, g) εrms(p0, g) εrms(p0, g) εrms(p0, g) εrms(p0, g)

g = 0 1.8725 5.0434 8.1638 11.2398 14.2787

g = −gk 1.7677 4.9387 8.0599 11.1368 14.1764

From Figure 6(b), recall that εrms for the larger antenna (D = 8.520 meters) has greater dependence

on p0 than on γ. The rim is fixed for this reflector and the same test is performed as was for the smaller

fixed rim reflector. The data in Table 6 shows that the surface distortions are not confined to sub–millimeter

quantities for the larger antenna. This data does confirm that the pressure term has a substantial effect on

εrms. While a fixed rim may be a good solution for a reflector with a diameter of approximately 2 meters,

the model does not suggest the fixed rim for an antenna with a diameter of nearly 8 meters.

VI.D. Parametric Study of Reflector Depth

The results in this section suggest that the depth of an antenna reflector is an important factor affecting the

antenna’s performance. Since the antenna that is used for most of the experiments has F/D = 0.4291, the

reflector is tested with multiples of this value. First, a base value of F is determined depending on the reflector diameter, D. In the case D = 2.13 meters, the base value Fb = 0.914 meters is obtained. Then, various

F/D ratios are tested by substituting the values {0.25Fb, 0.5Fb, 0.75Fb, . . . , 3.0Fb} for the focal length.

NASA/TM—2011-217110 20
For this experiment, the pressure is set at 12.5 Pa and the tendon foreshortening to γ = 0.03. See

Figure 8. The antenna reflector having D = 2.13 meters, holds an efficiency e

∗ of at least 80.16% for all

the F/D values tested. For the case, D = 5.325 meters, the efficiency drops substantially as the F/D ratio

increases. At F/D = 1.2873, the model predicts a performance efficiency of only e

∗ = 4.08%. An even faster

reduction of the antenna performance is seen for the large antenna of D = 10.650 meters. In that case,

e

∗ = 0.30% is obtained when F/D = 1.2873. Additional data points for antennas with aperture diameters

of 3.195 and 3.728 meters are also plotted. This data shows that the efficiency of mid–size antennas is also

compromised when F/D grows, although less severely.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0.2

0.4

0.6

0.8

1

 F / D

 e*

 D = 2.130 m

 D = 3.195 m

 D = 3.728 m

 D = 5.325 m

 D = 10.650 m

Figure 8. Antenna performance for various F/D values and antenna sizes.

Finally, consider the case with (F/D, D, γ) = (1.2873, 5.325, 0.03) where e

∗ = 4.08% was found. A small

improvement, e

∗ = 7.81%, is achieved when a fixed boundary is applied (see Section VI.C). Upon further

examination of the fixed rim boundary condition for a shallow reflector, one finds that the reflector is displaced by approximately 2.5 cm. For an equivalent tendon supported reflector, the vertex of the reflector was

displaced by approximately 4.2 cm. Since the two parabolic sheets are joined together at a shallow angle for

these antennas, there is tendency for the two sheets to separate after pressurizing; in other words, “bulging”

is found to occur near the rim.

It is expected that the stress resultants will increase in the case of a shallow reflector. The Laplace–Young

Equation asserts that

p0 = σ1k1 + σ2k2,

where σi are the principal stress resultants and ki are the principal curvatures. For a fixed differential pressure p0, the σi must increase as the principle curvatures decrease.

NASA/TM—2011-217110 21
(a) D = 2.13 meters (b) D = 10.65 meters

0.2 0.4 0.6 0.8 1

0

40

80

120

 r / Rf

Average 

σ

+ and 

σ

− (N/m)

 F / D = 0.4291

 F / D = 0.9655

0.2 0.4 0.6 0.8 1

0

40

80

120

 r / Rf

Average 

σ

+ and 

σ

− (N/m)

Figure 9. Antenna performance for various F/D values and antenna sizes.

In Figure 9, the stress resultants for four antenna cases are plotted. Since the stress distribution of the

deformed reflector is nearly radially symmetric in these cases, the principle stress resultants are plotted as a

function of radial distance from the vertex. The solid lines of Figure 9 indicate a deeper reflector while the

dotted lines are for a shallower reflector. For each case, two lines are plotted to show the average maximum

stress (σ

+(r)) and average minimum stress (σ−(r)). The plots indicate that the stress resultants at the

vertex of the reflector are nearly twice the magnitude for the shallow reflector versus the deeper one. This

holds true for both the smaller and larger diameter reflectors. Also note that σ

− nears zero for the larger

shallow reflector. This suggests an increase likelihood of wrinkling near the rim of the reflector, especially

since the maximum stress resultant σ

+ remains high around 90 N/m.

VII. Conclusions

Large aperture reflectors are able to radiate more energy than smaller reflectors. However, the results of

this paper show that larger reflectors typically are not able to attain sub–millimeter RMS values. For this

reason, large antennas exhibit a trade off between the ideal gain that could be achieved with a large aperture

and the effective antenna gain which is likely reduced due to surface distortions.

Flat construction patterns, in certain cases, may lead to lower fabrication costs and simplification of the

construction method. Furthermore, the rim modification factor can be used to reduce εrms. In the case

of a large size antenna, additional factors must also be considered. These results suggest that the surface

accuracy has greater dependence on the differential pressure than on the rim modification parameter when

NASA/TM—2011-217110 22
the reflector diameter is of order 10 meters. This is in contrast to a smaller antenna, where the supporting

tendons have a greater effect on surface accuracy. One result, however, suggested the possibility of a slight

gain increase by not treating the outer edge of the paraboloid for reflectivity.

The support tendons are the principle factor that distorts the 2.13 meter reflector from the desired

parabolic shape. A fixed reflector boundary leads to a substantial reduction of the RMS values. Variations

in the direction and strength of gravity as well as the internal pressure do not cause overall RMS values

greater than 1 mm. Unfortunately, elimination of the supporting tendons may lead to deployment problems

with a different support structure. Also, the model for a larger antenna does not perform as well with a fixed

boundary. The RMS values do not remain below 1 mm and it is clear that an increase in the differential

pressure could cause a deformation that negates the benefits of a fixed rim.

Many of the results presented here point to key design parameters that can be tuned to increase the

performance of inflatable parabolic reflectors. Further analysis on large reflectors will need to be carried out

in order to characterize their behavior for different operating environments. However, the analytical methods

developed here show promise that with additional analysis a more complete picture of large aperture reflector

behavior can be attained.

References

1 Romanofsky, Robert R., Kevin Lambert, Brian W. Welch and Irene Bibyk. “The Potential for Gossamer Deployable Antenna

Systems in Ka–Band Exploration and Science Communications Architectures.” Twelfth Ka and Broadband Communications

Conference. Napoli, Italy. September 26–27 2006.

2 Freeland, R. E., G. Bilyeu, G.R. Veal and M.M. Mikulas. “Inflatable Deployable Structures Technology Summary.” International Astronautical Federation, Paper IAF–98–I.5.01, September – October 1998.

3 Thomson, Mark W. “Astromesh Deployable Reflectors for Ku and Ka Band Commercial Satellites.” AIAA 2002–2032.

4 Ruze, John. “Physical Limitations on Antennas.” MIT Research Laboratory of Electronics. Technical Report 248, October

1952.

5 Greschik, G. et al. “Sensitivity Study of Precision Pressurized Membrane Reflector Deformations.” AIAA Journal, Volume

39, No. 2, February 2001.

6 Coleman, Michael J. and Frank E. Baginski. “Modeling the Shape of Deployable Large Aperture Antennas in Non–symmetric

Loading Environments.” AIAA Student Region MA–I Conference. April 2009.

7 Coleman, Michael J. and Frank E. Baginski. “A Shape Deformation Study of Large Aperture Inflatable Elastic Parabolic

Antenna Reflectors.” Eleventh AIAA Gosammer Systems Forum. Orlando, Florida. AIAA 2010 – 2501 (2010).

NASA/TM—2011-217110 23
8 Coleman, Michael J. Surface Accuracy Analysis and Mathematical Modeling of Deployable Large Aperture Elastic Antenna

Reflectors. Dissertation George Washington University, 2010. Ann Arbor, Michigan: ProQuest UMI, 2010. 108138709.

9 Baginski, Frank E., Michael C. Barg and William Collier. “Existence Theorems For Tendon–Reinforced Thin Wrinkled

Membranes Subjected to a Hydrostatic Pressure Load.” Mathematics and Mechanics of Solids 13: 532 – 570, 2008.

10 Baginski, Frank E. and Willi W. Schur. “Structural Analysis of Pneumatic Envelopes: Variational Formulation and

Optimization–Based Solution Process.” AIAA Journal, Volume 41, No. 2, February 2003.

11 Pipkin, Allen C. “Relaxed Energy Densities for Large Deformations of Membranes.” IMA Journal of Applied Mathematics,

No. 52, Pages 297 – 308, 1994.

12 Stutzman, Warren L. and Gary A. Thiele. Antenna Theory and Design. John Wiley and Sons: New York, 1981.

13 Halliday, David and Robert Resnick. Physics Part II. New York: John Wiley and Sons, 1960.

NASA/TM—2011-217110 24
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The Effect of Boundary Support and Reflector Dimensions on Inflatable Parabolic Antenna 

Performance 

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13. SUPPLEMENTARY NOTES

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14. ABSTRACT

For parabolic antennas with sufficient surface accuracy, more power can be radiated with a larger aperture size. This paper explores the 

performance of antennas of various size and reflector depth. The particular focus is on a large inflatable elastic antenna reflector that is 

supported about its perimeter by a set of elastic tendons and is subjected to a constant hydrostatic pressure. The surface accuracy of the 

antenna is measured by an RMS calculation, while the reflector phase error component of the efficiency is determined by computing the 

power density at boresight. In the analysis, the calculation of antenna efficiency is not based on the Ruze Equation. Hence, no assumption 

regarding the distribution of the reflector surface distortions is presumed. The reflector surface is modeled as an isotropic elastic membrane 

using a linear stress-strain constitutive relation. Three types of antenna reflector construction are considered: one molded to an ideal 

parabolic form and two different flat panel design patterns. The flat panel surfaces are constructed by seaming together panels in a manner 

that the desired parabolic shape is approximately attained after pressurization. Numerical solutions of the model problem are calculated 

under a variety of conditions in order to estimate the accuracy and efficiency of these antenna systems. In the case of the flat panel 

constructions, several different cutting patterns are analyzed in order to determine an optimal cutting strategy. 

15. SUBJECT TERMS

Microwave antennas 

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