design of a cluster-fed multibeam reflector system using

MASTER’S THESIS

2004:190 CIV

TORE LINDGREN

Design of a Cluster-fedMultibeam Reflector SystemUsing Hard Horns as Feeds

MASTER OF SCIENCE PROGRAMME

Department of Computer Science and Electrical EngineeringEISLAB - Embedded Internet Systems Laboratory

2004:190 CIV • ISSN: 1402 - 1617 • ISRN: LTU - EX - - 04/190 - - SE

Design of a Cluster-fed MultibeamReflector System Using Hard Horns

as Feeds

Tore Lindgren

Department of Electromagnetics, Antenna Group

CHALMERS UNIVERSITY OF TECHNOLOGYGoteborg, Sweden 2004-02-05

Copyright c©2004 by Tore LindgrenGoteborg, SwedenPrepared with LATEX

2

Abstract

In this report the performance of the hard horn antennas in a cluster-fed offset reflectorsystem operating in the Ka-band (20-30 GHz) is analyzed. The analysis is done witha program that uses aperture integration to find the far-field of the antenna system.The results for different positions of the feed antenna in the offset reflector systemare compared with results obtained for a rotationally symmetric antenna with the samediameter and a focused feed (BOR1 approximation). The results show a good agreementfor the focused feeds and some differences as the feed is displaced from the focal point.Several improvements of the reflector design are proposed and analyzed.

It is concluded that the BOR1 approximation is a powerful tool when designing acluster-fed offset reflector system. Especially in the initial stage it gives the possibilityto test a large number of different reflector designs. However, there are also degradationin the co- and crosspolar isolation for off-focus feeds that can only be studied by locatingthe feed in its real off-focus position in the actual offset reflector.

3

Preface

This report is a Master thesis at Lulea University of Technology. The examiner isDr. Kalevi Hyyppa at the Department of Computer Science and Electrical Engineering.

The work was done in the Antenna Group at the Department of Electromagneticsat Chalmers University of Technology in Goteborg, Sweden. The project was a part ofOmid Sotoudeh’s research in hard horn antennas for cluster-fed multibeam antennas.All feed patterns needed to calculate the far-field of these reflector antennas was pro-vided by him. The computation of the offset reflector was done with the ARECIBOcode developed by Professor Per-Simon Kildal. I developed my own Matlab programsfor analyzing and plotting the far-field calculated by ARECIBO and also to calculateand analyze the far-field using the BOR1 approximation. The supervisors were Profes-sor Per-Simon Kildal and Omid Sotoudeh.

I would like to thank everyone in the Antenna Group at Chalmers for a very funand instructive stay with the Antenna Group. Special thanks to Omid Sotoudeh andPer-Simon Kildal for great support and feedback during the project and for alwayshaving answers to questions. Others who have contributed with ideas and feedback areDr. Kalevi Hyyppa and Ake Wisten at Lulea University of Technology.

5

Contents

1 Introduction 91.1 Hard Horn Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2 Frequency Reuse and Footprint . . . . . . . . . . . . . . . . . . . . . . 101.3 Weakest Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Theory 132.1 Polarization and Far-field Function . . . . . . . . . . . . . . . . . . . . 132.2 Aperture Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Body of Revolution (BOR) . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Directive Gain and Radiated Power . . . . . . . . . . . . . . . . . . . . 172.5 Phase Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6 Beam Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Description of Reflector System 213.1 Feed Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Initial Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Simulation and Data Analysis 254.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Feed Pattern and Aperture Field . . . . . . . . . . . . . . . . . . . . . 264.3 Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.4 Beam Pointing Direction . . . . . . . . . . . . . . . . . . . . . . . . . . 304.5 Beam Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.6 Initial Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Results 355.1 Analysis of Initial Design . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Increased Focal Length . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.3 Frequency Selective Surface . . . . . . . . . . . . . . . . . . . . . . . . 435.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6 Conclusion 49

Bibliography 51

7

8

A Geometry 53A.1 Offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53A.2 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

B Numerical Values of Reflector Performance 57

Chapter 1

Introduction

In our modern society, Internet has become an important mean of communicating andfinding information. Moore’s law state that the capacity of computers is doubled in 18months, and the amount of data that needs to be transferred between internet usersis naturally increasing with this. The Internet has evolved from being a project foronly military and scientific use to become every man’s property. This developmenthas brought with it a lot of new applications, like for example distance education andapplications for medical use. These often demand access to the Internet in remote placesof the world where the infrastructure is poorly developed. For some years, people haveturned their eyes toward space in search for a solution to this problem.

There are several ways of addressing the problem stated above. In the presentproject, a satellite located in geostationary orbit is considered. This satellite will carrya cluster-fed multibeam antenna system covering a larger area, in this case Europe.The system will operate in the Ka-band (20-30 GHz) using 27.5-30 GHz for uplink and17.7-20.2 GHz for downlink. This concept is not new, but there is a constant need ofimproving the performance of the antenna and make the equipment smaller and morelightweight. It has been proposed to use the hard horn antenna (see Section 1.1) asfeed for both uplink and downlink in order to halve the number of reflectors needed.The hard horn has certain advantages over dominant mode and dual mode horns asfeed in cluster-fed multibeam antennas [1] - [2]. It has higher edge of coverage (EOC)directivity and lower cross polarization than conventional horns. It also gives betterbeam isolation. This study has previously been done in [1], using the BOR1 analysismodel which assumes rotational symmetry (see Section 2.3). The main advantage ofthis model is its simplicity which makes it fast. In [3] it was shown that this modelcan also be used for analysis of non-rotationally symmetric antennas. When used foranalysis of focus-located hard horns in offset reflector antennas, the accuracy was quitegood.

The object of this report is to evaluate the performance of the hard horn antennas ina cluster-fed offset reflector system, where some feed elements are located out of focus.The results are also compared with those obtained using the BOR1 model.

The possibility to use this model in the design process of a cluster-fed offset reflectorsystem is evaluated. Only reflectors with parabolic shape (see Section 2.3) have been

9

10 1. Introduction

considered. This is done by using the ARECIBO code, a reflector program for analysisof multi-reflector antennas, see [4] - [5]. There are many different analysis models foroffset reflectors implemented in ARECIBO. The one used for the present study isaperture integration after Fourier series expansion of the aperture field. This is fastand accurate within main beam and first sidelobes. It is, however, slow compared tothe BOR1 model in which the number of necessary calculations are significantly lower(see Chapter 2).

1.1 Hard Horn Antennas

The name hard horn refers to the surface of the horn and has its origin in acoustics,where different boundary conditions are applied to surfaces that feel soft or hard whentouched. In electromagnetics, a soft surface is defined as a surface where the electricfield density is zero. At the hard surface the electric field density has a maximum. Thehard surface can be realized by longitudinal corrugations filled with dielectric [6]. Hardhorns have a very high efficiency at the designed frequency, the efficiency decrease onlyslowly below this frequency [1]. Figure 1.1 shows a sketch of a hard horn antenna. Thelighter gray areas represents the dielectric.

Figure 1.1: Hard horn antenna ( c© Omid Sotoudeh).

1.2 Frequency Reuse and Footprint

In a multi-beam antenna system it is important that the individual beams do notinterfere with each other. To achieve this it is common to use more than one reflector

1.3 Weakest Point 11

with each reflector operating at different frequencies. Here, four reflectors will be used.Each band (27.5-30 GHz and 17.7-20.2 GHz) is then separated in four smaller bandsthat do not interfere with each other. The footprint (the area on the ground coveredby the antenna system) of this system is illustrated in Figure 1.2. The angular distancebetween two beams is ∆θ = 1. From the figure we can then see that the angulardistance between two beams of the same frequency is 2.

1 1

1

1 1

1 1

4

4 4

4

2 2

2 2

3 3

3 3

Figure 1.2: Frequency reuse plan using four reflectors. This footprint is created byfour antennas of the type shown on the cover of this report. The footprint of the finalsystem will be larger than this and may have another shape.

1.3 Weakest Point

The actual footprint of each beam is not circular, as indicated in Figure 1.2, but hexag-onal to ensure full coverage on the ground. Of interest is then the directivity at theedge of coverage (EOC), also called directivity at the weakest point, Dwp (which is theterm that will be used here). The location of this point is illustrated in Figure 1.3 andcan be found to be at θwp ≈ 0.58 from the center of the main beam. When simulating,the main beam is analyzed for θ ≤ 0.58. The sidelobes are analyzed in the intervalwhere the closest beam of the same frequency is located, this is for 1.5 ≤ θ ≤ 2.57

(including the weakest point behind this beam).

12 1. Introduction

∆θ

θwp

Figure 1.3: The weakest point in the footprint.

Chapter 2

Theory

The purpose of this chapter is to give an overview of the theory used by the simulationprogram ARECIBO and the simplified theory used for comparison and the reason whythe latter theory is faster. The parameters used to describe the performance of theantenna system are defined in Section 2.6. Parts of the derivation of the expressionsin Section 2.2 and 2.3 are left out since they are lengthy and not essential for theunderstanding of the final results of this report. Interested readers are instead referredto [7] for more details.

2.1 Polarization and Far-field Function

The polarization considered for this project is right-hand circular (RHC) polarization.This means that, for a wave propagating along the z-axis, the y-component of theelectric field is delayed by a quarter of a period compared to the x-component. Theco-polar (desired polarization) unit vector is then

coRHC = (x − jy)/√

2 (2.1)

in a cartesian coordinate system (x, y, z), x and y are unit vectors along the x- andy-axes, respectively. This can also be written in a spherical coordinate system (r, θ, ϕ)as

coRHC = e−jφ(θ − jϕ)/√

2, (2.2)

where φ is a real constant and θ and ϕ are unit vectors along the θ- and ϕ-axes,respectively. A small part of the electric field will have the opposite polarization (calledcross-polarization). Here, this means that the x-component of the electric field is de-layed by a quarter of a period compared to the y-component. The cross-polar unitvector is then

xpRHC = (x + jy)/√

2 = ejφ(θ + jϕ)/√

2. (2.3)

This can be compared with linear polarization along the y-axis. The co-and cross-polar unit vectors is here given by

13

14 2. Theory

coy = y = sin ϕθ + cos ϕϕ (2.4)

and

xpy = x = sin ϕθ − cos ϕϕ. (2.5)

The far-field at a point r can generally be written as

E(r) =1

re−jkrG(r), (2.6)

where 1r

is the divergence factor, e−jkr is the phase factor, r is a unit vector alongthe r-axis, and G(r) is the complex far-field function. The far-field function can beexpanded into its co- and cross-polar parts:

G(r) = Gco(r)co + Gxp(r)xp, (2.7)

where Gco and Gxp are the co-and cross-polar parts of the far-field function. Bytaking the scalar product of (2.7) with the complex conjugate of the co- and cross-polar unit vectors (co∗ and xp∗) Gco and Gxp can be found (since co · co∗ = 1 andxp · xp∗ = 1, and co · xp∗ = 0 and xp · co∗ = 0):

Gco(r) = G(r) · co∗(r) (2.8)

Gxp(r) = G(r) · xp∗(r). (2.9)

2.2 Aperture Integration

The simulation program used in this project (ARECIBO) calculates the far-field func-tion using aperture integration. This method gives accurate results for all antennasthat radiated through an opening or aperture. The aperture may be physical as inopen ended waveguides and horn antennas, or virtual as in reflector antennas. In thelatter case the reflector antenna is enclosed by a virtual surface at which the tangentialcomponents of the electric- and the magnetic fields are assumed to be zero, except atthe aperture directly in front of the reflector at which the approximate fields can befound. The equivalent electric current, Ja, and magnetic current, Ma, are introduced.These are related to the electric- and magnetic field at the aperture as

ηJa = na × ηHa = −Ea (2.10)

Ma = Ea × na = ηJa × na, (2.11)

where na is the outgoing normal to the aperture and η is the free space waveimpedance (377Ω). The currents in (2.10) and (2.11) together becomes the Huygen’ssource ([7], Chapter 3) which has the far-field function

2.3 Body of Revolution (BOR) 15

GH(ηJa, na, r) = Ck[ηJa − (ηJa · r) − r − (ηJa × na) × r], (2.12)

where Ck = −jk/(4π) and k is the wave number. The far-field of the whole aperturein the direction r is

GH(ηIa, na, r) = Ck[ηIa − (ηIa · r) − r − (ηIa × na) × r], (2.13)

where ηIa is given by the integral

Ia =

∫∫

A

ηJa(r’)ejk(r’·r)dS =

∫∫

A

Ea(r’)ejk(r’·r)dS, (2.14)

where A is the aperture area and r’ is a coordinate (vector) in the aperture. Theelectric field at the aperture may be found in different ways. Here geometrical optics(GO) is used. GO is an approximation which assumes that all fields in free space prop-agate geometrically along rays that are reflected at a surface by the classical reflectionlaw. For a parabolic surface all reflected rays will be parallel if the rays originate froma feed antenna located at the focal point. If the feed antenna is displaced the GO aper-ture field will change in phase, amplitude, and polarization. The most rapid changewill be in the phase [4]. The effects of this is discussed in more detail in Chapter 4.

2.3 Body of Revolution (BOR)

The ϕ-variation of any far-field function can always be expanded in a Fourier seriessince it is periodic with 2π. We can then write the far-field function as

G(θ, ϕ) = Gθ(θ, ϕ)θ + Gϕ(θ, ϕ)ϕ =∑∞

n=0[An(θ) sin(nϕ) + Bn(θ) cos(nϕ)]θ+∑∞

n=0[Cn(θ) sin(nϕ) − Dn(θ) cos(nϕ)]ϕ.

(2.15)

The minus sign in front of Dn(θ) is chosen for symmetry reasons. If the antenna isrotationally symmetric and excited by a short transverse current source (or an incre-mental current as in aperture antennas) the radiation field will contain only the firstterm of the expansion in (2.15), these are called BOR1 antennas. For an antenna withlinear polarization along the y-axis the far-field function becomes

Gy(θ, ϕ) = GE(θ) sin(ϕ)θ + GH(θ) cos(ϕ)ϕ, (2.16)

where GE(θ) = A1(θ) and GH(θ) = C1(θ) are the complex far-field functions inthe E- and H-planes, respectively. Of interest here, however, is the co- and cross polarfar-field functions obtained when the antenna is excited for RHC-polarization. Thiscan be shown to be the same as the co- and cross polar far-field functions for linear y-polarization in the ϕ = 45 plane ([7], chapter 2.4.2). By using (2.8) and (2.9) togetherwith (2.4) and (2.5) we get the relationships

Gco(θ) = Gco45(θ) =1

2[GE(θ) + GH(θ)] (2.17)

16 2. Theory

Gxp(θ) = Gxp45(θ) =1

2[GE(θ) − GH(θ)]. (2.18)

We can from (2.17) and (2.18) see that the cross polar level depends on the differ-ences of the field in the E- and H-planes. The ideal situation is when these are equaland we get no cross-polarization.

The far-field function is calculated with the aperture integration formulas (2.13) and(2.14). For a BOR1 antenna with RHC-polarization these calculations are simplifiedto ([7], chapter 6.5)

Gco(θ) = −2Ck˜Eco(θ) cos2(θ/2) (2.19)

Gxp(θ) = −2Ck˜Exp(θ) cos2(θ/2), (2.20)

where

˜Eco = 2π

∫ d/2

0

Eco(ρ′)J0(kρ′ sin θ)ρ′dρ′ (2.21)

˜Exp = 2π

∫ d/2

0

Exp(ρ′)J2(kρ′ sin θ)ρ′dρ′, (2.22)

Eco and Exp are the co- and cross polar electric field at the aperture, d is the diameterof the aperture, ρ′ is the radial coordinate of a point in the aperture, and J0 and J2 arethe zeroth and second order Bessel functions, respectively. The aperture field can becalculated either from fields given in the E- and H-planes (used for the feed antenna)or, if the aperture is the one of a reflector, from the far-field of the feed. In the lattercase, the feed is assumed to be located at the focal point of a rotationally symmetric,paraboloidal reflector. This reflector is defined by

z(ρ) = −F +ρ2

4F, (2.23)

where ρ is the radial coordinate of a point in the aperture (of the reflector), z(ρ) isa z-coordinate of a point on the reflector (see Figure 2.1), F is the focal length of thereflector, and θf is the angle from the symmetry axis to the point on the reflector withthe coordinate ρ (see Figure 2.1). An alternative description giving identical shape is

r(θf ) =F

cos2(θf/2)(2.24)

with r(θf ) defined in Figure 2.1. The relationship between ρ and θf is then

ρ(θf ) = r(θf ) sin(θf ) = 2F tan(θf/2). (2.25)

The aperture field of the reflector and the far field of the feed is then related by [7]

Eco(ρ) = − 1

Fcos2(θf/2)Gco(θf )e

−j2kF (2.26)

2.4 Directive Gain and Radiated Power 17

and

Exp(ρ) = − 1

Fcos2(θf/2)Gxp(θf )e

−j2kF . (2.27)

As a conclusion we see that with the BOR1 theory the whole far-field of the antennacan be described by only two equations. Either with the co-polar pattern in the E- andH- plane, Equation (2.16), or with the co- and cross-polar pattern in the ϕ = 45 planeexcept for a phase factor in the cross-polar part, Equation (2.17) - (2.18). The far-fieldfunction is found by solving (2.19) - (2.22) numerically. By comparing these expressionswith the general aperture integration theory in (2.13) - (2.14) we can see that by usingthe BOR1 the far-field function can be calculated by solving a simple integral insteadof a double as in the general theory. This will of course reduce the computation timesignificantly. The BOR1 theory will be used for the feed antennas and as a referencefor the simulations of the whole reflector.

θf

ρ

z(ρ), r(θf)

z

^ ρ

^ r

focal point

F

Figure 2.1: Geometry of a rotationally symmetric paraboloid.

2.4 Directive Gain and Radiated Power

The directive gain (Dco and Dxp) will be used to present the result from the simulation.It is defined as the normalization of the far-field function to that of an isotropic co-polarradiator with the same total radiated power and the unit is dBi (decibels relative toisotropic level). The definition can thus be written as

Dco(θ, ϕ) = 10 lg

( |Gco(θ, ϕ)|2|GISO|2

)= 10 lg

(4π

|Gco(θ, ϕ)|2P

). (2.28)

18 2. Theory

where P is the total radiated power. The directive gain of the cross-polar field isfound in the same way. In the general case, P is found by integrating the square of theamplitude of the far-field function over the far-field sphere:

P =

∫∫

4π

(|Gco(θ, ϕ)|2 + |Gxp(θ, ϕ)|2

)sin θdθdϕ. (2.29)

Of interest here is the directive gain of the reflector antenna. P is in this case thetotal radiated power from the feed antenna. Since the feed antenna is assumed to be ofBOR1 type its far-field does not depend on ϕ, the ϕ-integral will therefore be constant(= 2π). The power integral can then be reduced to

P = 2π

∫ π

0

(|Gco,feed(θ)|2 + |Gxp,feed(θ)|2

)sin θdθ (2.30)

It is required that the directive gain is at least 39.5 dBi in the whole beam.

2.5 Phase Center

When specifying the coordinates of the feed antennas it is the coordinates of the phasecenter that is specified. The phase center is the point which makes the phase of thecomplex co-polar far-field function constant. In practice, this point can seldom be de-termined. Instead it is usually defined as the point which minimizes the phase variationof the co-polar far-field function over a given solid angle of interest. For a hard hornantenna the phase variation due to movement of the phase center is small ([3]) and doesnot have to be taken into account. For convenience the phase center is assumed to bein the middle of the aperture of the horn.

2.6 Beam Isolation

There are several measures of the performance of an antenna system. In the case of amulti-beam antenna for satellite use where the footprint on the ground is divided intodifferent cells it is important that the neighboring beams doesn’t interfere with eachother. The cross-polar level in the beam must also be low. The parameters that arecompared between the different reflector geometries and feed positions are explainedbelow. These parameters are called beam isolation. Due to the large uncertainty aboutthe sidelobe level and the cross-polar level a worst-case approach is adopted [1].

The co-polar beam isolation is here defined as the maximum co-polar sidelobe levelin the neighboring beam over the minimum co-polar level in the main beam. Thisshould be at least 25 dB.

BIco =Gco(wp)

Gco,max(beam 1 → beam 2)(2.31)

2.6 Beam Isolation 19

where Gco(wp) is the co-polar level in the weakest point in beam 1 (same as Dwp).The cross-polar beam isolation, which should be at least 27 dB, is in a similar waydefined as

BIxp =Gco(wp)

Gxp,max(beam 1 → beam 2). (2.32)

The cross-polar decoupling is here defined as the highest cross-polar level in themain beam over the minimum co-polar level in the main beam. This parameter shouldalso be at least 27 dB.

XPD =Gco(wp)

Gxp,max(beam)(2.33)

The directivity in the weakest point, the co- and cross-polar beam isolation, andthe cross-polar decoupling are shown in Figure 2.2.

20 2. Theory

0 0.5 1 1.5 2 2.5 3−5

0

5

10

15

20

25

30

35

40

45

50

θ / °

Dire

ctiv

e ga

in /

dBi

Dwp

XPD BI

co

BIxp

Figure 2.2: The parameters discussed in Section 2.6 for a typical far-field pattern (boththe co- and cross-polar far-field is plotted in this figure). The vertical line at 0.58 isthe end of the beam. The vertical lines at 1.5 and 2.57 marks the sidelobe intervalwhere the beam interferes with a neighboring beam. The far-field is calculated usingthe BOR1 model.

Chapter 3

Description of Reflector System

Figure 3.1 shows a 3D sketch of a parabolic offset reflector. The dot in the lower leftcorner is the focal point. The focal length is here 1.204 m, the diameter of the apertureis 1.321 m and the offset is 0.835 m (from the center of the reflector). This figure alsoillustrates the coordinate system used by ARECIBO in which the origin is in the centerof the aperture behind the reflector.

−1.5−1

−0.50

−1

0

1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x / my / m

z / m

origin

Figure 3.1: 3D sketch of an offset parabolic reflector.

21

22 3. Description of Reflector System

3.1 Feed Cluster

The feed cluster will consist of a number of horn antennas placed in a hexagonal ge-ometry shown in Figure 3.2. The feed cluster is located on a plane perpendicular toa vector created by the focal point and the center of the reflector (as seen from thefeed cluster). Each individual feed element points toward the center of the reflector.To reduce the number of simulations only three different positions of the feed antennawere studied. These were at 0, 85, and 170 mm from the focal point at an angle of 30

to the negative x-axis in Figure 3.2. This is the worst case according to my experienceand [8].

x−axis

y−axis

Figure 3.2: Geometry of the feed cluster.

3.2 Offset

The reflector is offset to avoid blockage from the feed cluster. Figure 3.3 shows thereflector from Figure 3.1 projected on the symmetry plane. The dots in the lower leftcorner represent the feed cluster. The size of the offset depends on the size and geometryof the feed cluster. All beams from the reflector should be clear of all elements in thefeed cluster. To achieve this, both the size of the feed cluster (δf in Figure 3.3) and thedirection of the lowest beam must be taken into account. The beam pointing directiondepends on the distance between the feed and the focal point and also on the distancefrom the focal point to the center of the reflector. This distance should be kept constantwhen the geometry of the feed cluster and reflector is changed. It is equal to the focallength for a BOR1 antenna and is therefore referred to as FBOR1 in this text. We thenhave a relationship between the focal length and the offset (xC in Figure 3.3)

F =1

2

(√F 2

BOR1 − x2C + FBOR1

). (3.1)

The offset distance at which there is no blockage can be found from

3.3 Initial Design 23

x0 = δf cos θC +

(F − x2

0

4F+ δf sin θC

)tan θB. (3.2)

The angles θB and θC are given by

θB = arctan

(δf

FBOR1

)(3.3)

and

θC = arctan

(Dx/2 + x0

F − 14F

(Dx/2 + x0)

). (3.4)

Dx is the diameter of the aperture in x-direction. Equations (3.1) to (3.4) are derivedin Appendix A.

δf

θC

θB

FBOR1

F

xC

x0

xf

Figure 3.3: Geometry of reflector and feed cluster.

3.3 Initial Design

Since several different geometries of the reflector will be tested and this affects the focallength and offset of the reflector, it is convenient to plot these two parameters as a

24 3. Description of Reflector System

1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.500.60

0.70

0.80

0.90

1.00

Dx / m

x C /

m

1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.501.16

1.18

1.20

1.22

1.24

1.26

Dx / m

F /

m

Figure 3.4: Plot of focal length and offset as a function of the diameter of the reflector.The focal length of the corresponding BOR1 antenna is 1.349 m.

function of Dx (the diameter of the reflector). This is done in Figure 3.4, the focallength of the corresponding BOR1 antenna is 1.349 m.

For the initial reflector design, a feed cluster consisting of five feed elements in thesymmetry plane was considered (see Figure 3.2). The parameters used to specify theinitial reflector design are presented in Table 5.1. These are the values used whenplotting Figure 3.1 and 3.3.

Parameter Description Value

D Diameter 1.321 mF Focal length 1.204 mxC Offset 0.835 mDf Diameter of feed antenna 0.048 m

Table 3.1: Parameters used to specify feed and reflector.

Chapter 4

Simulation and Data Analysis

The program used for the simulation (FRTDUAL in the ARECIBO code) uses apertureintegration to numerically find the far-field radiation of the antenna. The reflector isspecified by a number of parameters. These are the geometrical specification of thereflector (diameter in x- and y-direction, offset, and focal length), the position of thephase center of the feed antenna in the coordinate system of the reflector (see Figure3.1), the complex far-field of the feed antenna, and the total radiated power of the feedantenna. The program uses a grid of the type shown in Figure 4.1. The grid used is finerthan this, 49 ρ-values and up to 256 ϕ-values (closest to the edge). The simulationsare done at six different frequencies. These are one for each endpoint of the two bands(17.7 GHz, 20.2 GHz, 27.5 GHz, and 30.0 GHz) and one in the center of each band(18.95 GHz and 28.75 GHz). The feed antenna always points toward the center of thereflector.

The outputs from the program are many. Of interest for this analysis is the ampli-tude and phase of the aperture field of the reflector (in ϕ = 0/ − 180, 45/ − 135,90/ − 90, and 135/ − 45 planes), beam pointing direction, and amplitude of thefar-field (in a coordinate system described in Section 4.1). In this chapter the methodsused to analyze the data will be presented as well as some general effects seen in thesimulations. In the last section the simulation results from ARECIBO are comparedwith the BOR1 theory for validation.

4.1 Coordinate System

The far-field of the reflector is presented in a grid using a coordinate system in whichtwo angles, θx and θy, are directed from the positive z-axis toward the the positive x-and y-axis, respectively. The distance to a point in space is r. This transforms to thespherical coordinate system as

r = r

θ = arccos(

cos θx cos θy

r

)

ϕ = arctan(

tan θy

sin θx

) (4.1)

25

26 4. Simulation and Data Analysis

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x / m

y / m

Figure 4.1: Type of grid used by ARECIBO.

(see appendix A.2 for more information about these relationships). This coordinatesystem is illustrated in Figure 4.2. The angles θx and θy can be found if the angles θand ϕ are given in the spherical coordinate system.

θx = arctan(tan θ cos ϕ)θy = arcsin(− sin θ sin ϕ)

(4.2)

4.2 Feed Pattern and Aperture Field

The aperture fields used when calculating the far-field of the feed are shown in Fig-ure 4.3. The horns are designed to have the most uniform aperture distribution at afrequency of 31.8 GHz and have a diameter of five wavelengths at this frequency.

The far-field of the feed is calculated using the BOR1 approximation from Section2.3. The co- and cross-polar patterns of the far-field function in an arbitrary plane (thefar-field is rotationally symmetric) are shown in Figure 4.4. The vertical line marks theedge of the reflector (at 27.5). The cross-polar level is highest for the low frequenciesbecause the largest differences of the aperture field in the E- and H-planes are at thesefrequencies.

The aperture field of the reflector in the ϕ = 0/ − 180, 45/ − 135, 90/ − 90,

4.2 Feed Pattern and Aperture Field 27

θx θ

y

x y

z

r

Figure 4.2: Coordinate system using θx,θy, and r.

and 135/ − 45 planes is shown in Figure 4.5. The feed is here placed 48 mm fromthe focal point in the negative x-direction. The frequency is 30.0 GHz. For this highfrequency the first sidelobes hit the reflector, this can also be seen in Figure 4.4. Theaperture field in Figure 4.5 is not completely symmetric which is mainly due to the factthat the feed is placed out of focus and also to some extent that the reflector is offset.


0 5 10 15 20 250

0.5

1

ρ / mm

Rel

ativ

e am

pitu

de

17.7 GHz

0 5 10 15 20 250

0.5

1

ρ / mm

Rel

ativ

e am

pitu

de

18.95 GHz

0 5 10 15 20 250

0.5

1

ρ / mm

Rel

ativ

e am

pitu

de

20.2 GHz

0 5 10 15 20 250

0.5

1

ρ / mmR

elat

ive

ampi

tude

27.5 GHz

0 5 10 15 20 250

0.5

1

ρ / mm

Rel

ativ

e am

pitu

de

28.75 GHz

0 5 10 15 20 250

0.5

1

ρ / mm

Rel

ativ

e am

pitu

de

30.0 GHz

E−planeH−plane

Figure 4.3: Aperture field of feed antenna at different frequencies.

4.3 Phase

In the previous section magnitude of the aperture field and the far-field of the feedantenna were discussed. This field is complex and therefore also have a phase whichhave some interesting effects on the far-field of the reflector. Figure 4.6 shows thephase of the aperture field shown in Figure 4.5. The phase variation is large in allplanes except for the 90/ − 90-plane. This is expected since the feed is moved out offocus in the 0/ − 180-plane. It is this phase variation that causes the main beam tobe directed in different directions for different locations of the feed and it can also beused to find this direction.

4.3 Phase 29

0 20 40 60 80−50

−40

−30

−20

−10

0

θ / °

Rel

ativ

e ga

in /

dB

17.7 GHz

0 20 40 60 80−50

−40

−30

−20

−10

0

θ / °

Rel

ativ

e ga

in /

dB

18.95 GHz

0 20 40 60 80−50

−40

−30

−20

−10

0

θ / °

Rel

ativ

e ga

in /

dB

20.2 GHz

0 20 40 60 80−50

−40

−30

−20

−10

0

θ / °R

elat

ive

gain

/ dB

27.5 GHz

0 20 40 60 80−50

−40

−30

−20

−10

0

θ / °

Rel

ativ

e ga

in /

dB

28.75 GHz

0 20 40 60 80−50

−40

−30

−20

−10

0

θ / °

Rel

ativ

e ga

in /

dB

30.0 GHz

Gco

Gxp

Figure 4.4: Far-field of feed antenna at different frequencies. The vertical line (at 27.5)marks the edge of the reflector.

Figure 4.7 shows the phase of the aperture field when the direction of the beamhas been taken into account (compare with Figure 4.6). The phase in the center of theaperture is set to 0. It is displayed between −180 and 180 which causes the largejumps when the phase difference relative to the center exceeds 180.


−0.5 0 0.5−30

−20

−10

0

10

20

ρ / m

Am

plitu

de /

dB

0°/−180° − plane

−0.5 0 0.5−30

−20

−10

0

10

20

ρ / m

Am

plitu

de /

dB

45°/−135° − plane

−0.5 0 0.5−30

−20

−10

0

10

20

ρ / m

Am

plitu

de /

dB

90°/−90° − plane

−0.5 0 0.5−30

−20

−10

0

10

20

ρ / m

Am

plitu

de /

dB

135°/−45° − plane

co−polarcross−polar

Figure 4.5: Aperture field of the reflector specified in Table 5.1. The feed is placed48 mm from the focal point in the negative x-direction. The frequency is 30.0 GHz.

4.4 Beam Pointing Direction

The far-field of the reflector is calculated in the range −5 ≤ θx ≤ 5 and −5 ≤ θy ≤ 5.The beam isolations and cross-polar decoupling is calculated by examine all values inthe beam (θ ≤ 0.58) and in the interval for the neighboring beam (1.5 ≤ θ ≤ 2.57).To get as accurate values as possible of these parameters it is important to know thebeam pointing direction accurately.

ARECIBO uses the aperture field to calculate the direction of the main beam. Thisis done by finding the direction that maximizes the aperture efficiency. As stated earlier,the phase will change most rapidly when the feed is displaced from the focal point. Thedirection that maximizes the aperture efficiency can therefore be assumed to be thesame as the direction that maximizes the phase efficiency over the aperture. This isthe method used by ARECIBO and it is described in detail in [4]. It has the advantage

4.4 Beam Pointing Direction 31

−0.5 0 0.5

−1500

−1000

−500

ρ / m

Pha

se / °

0°/−180° − plane

−0.5 0 0.5

−1000

−800

−600

−400

−200

0

ρ / m

Pha

se / °

45°/−135° − plane

−0.5 0 0.5

−200

−100

0

100

200

300

ρ / m

Pha

se / °

90°/−90° − plane

−0.5 0 0.5

0

200

400

600

800

1000

ρ / m

Pha

se / °

135°/−45° − plane


Figure 4.6: Phase of the aperture field of the same reflector, feed, and frequency as inFigure 4.5. The beam pointing direction have not been taken into account.

that the main beam direction is already known when the far-field is calculated. Thedisadvantage is that the method assumes that the phase over the aperture is roughlyconstant, which we can see from Figure 4.7 that it is not for the higher frequencies,where it changes with more than 180. This causes a small error in the main beamdirection calculated by ARECIBO. This problem is solved by adjusting the calculateddirection so that two points on the same level, one on each side of the beam, are atthe same distance from the center. The chosen level is in this case 39.5 dBi, the lowestacceptable gain in the beam.


−0.5 0 0.5

−150

−100

−50

0

50

100

150

ρ / m

Pha

se / °

0°/−180° − plane

−0.5 0 0.5

−150

−100

−50

0

50

100

150

ρ / m

Pha

se / °

45°/−135° − plane

−0.5 0 0.5

−150

−100

−50

0

50

100

150

ρ / m

Pha

se / °

90°/−90° − plane

−0.5 0 0.5

−150

−100

−50

0

50

100

150

ρ / m

Pha

se / °

135°/−45° − plane


Figure 4.7: Phase of the aperture field of the same reflector, feed, and frequency as inFigure 4.5 when the beam pointing direction have been taken into account.

4.5 Beam Isolation

The different beam isolations described in Section 2.6 are found by searching the wholemain beam for the minimum value and comparing this with the highest co- and crosspolar value in the sidelobes and the highest cross-polar value in the main beam. Thevalues presented is therefore the worst possible. This value is not always representativefor the whole beam since the spread over different ϕ-planes is in some cases significant.

4.6 Initial Simulation

The first step of the analysis is to compare results from simulations of a rotation-ally symmetric antenna with the feed in the focal point using both BOR1 theory and

4.6 Initial Simulation 33

aperture integration (from the ARECIBO code). The results obtained with the BOR1

theory is compared with the results obtained using ray tracing in order to show thatthe latter is accurate. The reflector used is the corresponding BOR1 reflector used toget the results in Table 5.1. It then has the same diameter (1.321 m), but a focal lengthof 1.349 m. The far-field of the feed antenna was calculated for 1000 different values ofθ.

Figure 4.8 shows the far-field of the reflector obtained from BOR1 theory and fromsimulations using aperture integration. As the figure shows, the two methods gives verysimilar results (the two curves are almost superposed). The largest difference betweenthe two curves are in the cross-polar level and the far-out sidelobes. For the upper band(27.5 - 30.0 GHz) the main beam is flattened in the center and it is possible to see adip here for the highest frequency. This is due to the fact that sidelobes from the feedenter into the reflector. These have a phase-shift of 180 relative to the main beam (seeFigure 4.7) and will therefore interfere destructively with the feeds main beam in thecenter of the beam from the reflector.

Since the results from the BOR1 model and ARECIBO are very similar the con-clusion is that ARECIBO gives accurate results and that the analysis programs workcorrectly. It is also concluded that the resolution of the far-field of the feed antenna issufficient.


−2 0 2

0

10

20

30

40

50

θ / °

Dire

ctiv

e ga

in /

dBi

17.7 GHz

−2 0 2

0

10

20

30

40

50

θ / °

Dire

ctiv

e ga

in /

dBi

18.95 GHz

−2 0 2

0

10

20

30

40

50

θ / °

Dire

ctiv

e ga

in /

dBi

20.2 GHz

−2 0 2

0

10

20

30

40

50

θ / °

Dire

ctiv

e ga

in /

dBi

27.5 GHz

−2 0 2

0

10

20

30

40

50

θ / °

Dire

ctiv

e ga

in /

dBi

28.75 GHz

−2 0 2

0

10

20

30

40

50

θ / °

Dire

ctiv

e ga

in /

dBi

30.0 GHz

BOR1

aperture integration

Figure 4.8: Comparison between far-field obtained by BOR1 theory and aperture inte-gration. The fields are rotationally symmetric and therefore shown in only one plane.

Chapter 5

Results

In the previous chapters the focus has been on design and analysis of the antennasystem. A simulation of a rotationally symmetric reflector with the feed antenna placedat the focal point were made in order to validate the results. In this chapter the reflectoris analyzed with the feed antenna placed out of focus. First, the reflector from Table 5.1is analyzed. Some possible improvements of the design is then presented and analyzedin the subsequent sections. Values of XPD, BIco, and BIxp above 40 dB are consideredto be ”out of range” and are in Figure 5.1, 5.5, and 5.9 assigned the value 40.0 dB.

5.1 Analysis of Initial Design

In this section the design from Table 5.1 is analyzed. The results from the simulationsusing the ARECIBO code and the BOR1 model is presented in Figure 5.1. We cansee that the general trend is that the spread of the results due to defocusing effects ofthe feed elements are largest for the higher frequencies. This is because sidelobes fromthe feed antenna hit the reflector in this band, making the pattern more sensitive tochanges in the aperture field. The results obtained using the BOR1 model agrees wellwith the results for the focused feed elements in the low band except for the co-polarbeam isolation. In Figure 5.2 the deviation of the beam pointing direction from thedesired direction is shown. The deviation increases in the same way for all frequenciesas the feed is moved out from the focal point. The reason for this is that the feedcluster is on a plane instead of on a spherical surface. Since the deviation is similar forall frequencies the position of the feed can be adjusted to cancel out this effect.

From Figure 5.1 we can see that the performance is clearly not good enough for thisdesign.

35

36 5. Results

15 20 25 30 3539

40

41

42

43

frequency / GHz

Dw

p / dB

i

15 20 25 30 3520

25

30

35

40

frequency / GHz

XP

D /

dB

15 20 25 30 35

15

20

25

30

frequency / GHz

BI co

/ dB

15 20 25 30 3525

30

35

40

frequency / GHz

BI xp

/ dB

δf = 0 mm

δf = 48 mm

δf = 96 mm

BOR1 model

Required

Figure 5.1: Performance of the reflector for different frequencies and feed positions.The values of δf in this figure is the distance from the focal point and correspond tothe three black colored feed positions in Figure 3.2.

5.1 Analysis of Initial Design 37

δf = 0 mm

δf = 48 mm

δf = 96 mm

17.7 GHz

0.1°

0.2°18.95 GHz

0.1°

0.2°20.2 GHz

0.1°

0.2°

27.5 GHz

0.1°

0.2°28.75 GHz

0.1°

0.2°30.0 GHz

0.1°

0.2°

Figure 5.2: Deviation of beam pointing direction from the desired direction. The valuesof δf in this figure is the distance from the focal point and correspond to the three blackcolored feed positions in Figure 3.2.

38 5. Results

5.2 Increased Focal Length

The performance of the reflector for the defocused feeds can be expected to improve ifthe focal length of the reflector is increased while the diameter is kept constant. Thisis because the reflector will become less curved as the ratio of the focal length to thediameter (F/D-ratio) increases. The phase errors due to the curvature will then bereduced. When increasing the F/D-ratio the directive gain of the feed antennas mustalso be increased. In this case that means larger horn antennas. We can see that thesystem will require more space as the F/D-ratio is increased, it must therefore not beincreased too much.

Since the behavior of the horn antennas changed as the diameter was increasedthese must be redesigned. This means that the aperture field of the redesigned hornantennas will not be identical to the ones shown in Figure 4.4. The far-field of the newhorn antennas is shown in Figure 5.3, the vertical line marks the edge of the reflector.These horn antennas have a diameter of 5 wavelengths at a frequency of 17.7 GHz, orapproximately 85 mm.

In Figure 5.4 a 3D-plot of a reflector with increased focal length is shown togetherwith the reflector analyzed in the previous section. The new reflector have a focallength of 2.307 m and a diameter of 1.28 m, the focal length of the correspondingBOR1 antenna is 2.42 m. It can in this figure be seen that the new reflector have a lesscurved surface than the old reflector with a smaller F/D-ratio.

From Figure 5.5 we can see that especially the co-polar beam isolation has improvedsignificantly from the initial geometry, but is still not good enough. Also the cross-polarbeam isolation has increased and is well above the required value of 27 dB. The cross-polar decoupling has not increased significantly. The deviation of the beam pointingdirection is shown in Figure 5.6. The result is here the same as for the initial reflectordesign.

5.2 Increased Focal Length 39

0 20 40 60 80−50

−40

−30

−20

−10

0

Rel

ativ

e ga

in /

dB

17.7 GHz

0 20 40 60 80−50

−40

−30

−20

−10

0

θ / °R

elat

ive

gain

/ dB

18.95 GHz

0 20 40 60 80−50

−40

−30

−20

−10

0

θ / °

Rel

ativ

e ga

in /

dB

20.2 GHz

0 20 40 60 80−50

−40

−30

−20

−10

0

θ / °

Rel

ativ

e ga

in /

dB27.5 GHz

0 20 40 60 80−50

−40

−30

−20

−10

0

θ / °

Rel

ativ

e ga

in /

dB

28.75 GHz

0 20 40 60 80−50

−40

−30

−20

−10

0

θ / °

Rel

ativ

e ga

in /

dB

30.0 GHz

Gco

Gxp

θ / °

Figure 5.3: Far-field of new feed antenna at different frequencies. The vertical line (at15.1) marks the edge of the reflector.

40 5. Results

−1.5 −1.0 −0.5 0 0.5 1.0 −1.00

1.0

0

0.5

1.0

1.5

2.0

2.5

x / m

y / m

z / m

Figure 5.4: 3D plot of two different offset reflector geometries. The focal lengths are1.204 m and 2.307 m.

5.2 Increased Focal Length 41

15 20 25 30 3539

40

41

42

43

frequency / GHz

Dw

p / dB

i

15 20 25 30 3520

25

30

35

frequency / GHz

XP

D /

dB

15 20 25 30 35

15

20

25

30

frequency / GHz

BI co

/ dB

15 20 25 30 3525

30

35

40

frequency / GHz

BI xp

/ dB

δf = 0 mm

δf = 85 mm

δf = 170 mm

BOR1 model

Required

frequency / GHz

Figure 5.5: Performance of the reflector for different frequencies. The values of δf inthis figure is the distance from the focal point and correspond to the three black coloredfeed positions in Figure 3.2.

42 5. Results

δf = 0 mm

δf = 85 mm

δf = 170 mm

17.7 GHz

0.1°

0.2°18.95 GHz

0.1°

0.2°20.2 GHz

0.1°

0.2°

27.5 GHz

0.1°

0.2°28.75 GHz

0.1°

0.2°30.0 GHz

0.1°

0.2°

Figure 5.6: Deviation of beam pointing direction from the desired direction.The valuesof δf in this figure is the distance from the focal point and correspond to the three blackcolored feed positions in Figure 3.2.

5.3 Frequency Selective Surface 43

5.3 Frequency Selective Surface

The dimensions used in the previous two reflector design was obtained by finding theBOR1 antenna that gives the best performance over the whole frequency range (17.7-30.0 GHz). Naturally, the far-field in the two frequency bands differ significantly, whichwas shown in the previous section. The performance of the reflector can be expectedto increase if it could be possible to optimize the reflector for the two frequency bandsseparately, that is, to use only a part of the reflector for the higher band.

This can be realized by use of a frequency selective surface (FSS). These surfacesare designed to reflect the incoming wave only at certain frequencies. Generally, theyconsists of a periodic structure with certain resonance frequencies, more informationabout this subject can be found in [10]. Here it is assumed that the area coveredwith the FSS reflects perfectly in the lower band (17.7-20.2 GHz) and not at all in thehigh band (27.5-30.0 GHz). This is a crude assumption but is made to simplify thecalculations.

Curves used to find the dimensions of the reflector that gives the optimum perfor-mance are shown in Figure 5.7 and 5.8. In these figures the expected directive gainin the weakest point and beam isolations are shown for the low and high frequencybands, respectively. We can see that the requirement for the directive gain and thecross-polar beam isolation should be easy to meet while the co-polar beam isolationand the cross-polar decoupling for the low band may be more difficult. The dimensionsof the reflector was chosen to give the best possible trade-off between the different pa-rameters. In this case a diameter of 1.28 m was chosen for the low frequency band. Forthe high band only the co-polar beam isolation is below the required value for somereflector diameters. For this band a diameter of 0.825 m was chosen since this gives thehighest possible co-polar beam isolation.

The results from the simulation of the offset reflector system is shown for threedifferent positions of the feed antenna. We can see that the directive gain in theweakest point and the cross-polar beam isolation is well above the requirement for allpositions of the feed. The cross-polar decoupling is above the required limit for thehigh band but still remains below the limit for part of the lower band. The co-polarbeam isolation is below the requirement in most cases except for the focused feeds atsome frequencies. It is interesting to note that defocusing the feed elements has almostno effect on the directive gain in the weakest point and the cross-polar decoupling. Thedeviation of the beam pointing direction behaves in the same way as in the two previoussections. This effect is then of no great concern since it is independent of the geometryof the reflector.

Diameter at 20 GHz band 1.28 mDiameter at 30 GHz band 0.825 m

Table 5.1: Effective reflector diameter with FSS.

44 5. Results

1 1.2 1.440.8

41

41.2

41.4

41.6

Diameter / m

Dw

p / D

bi

1 1.2 1.420

25

30

35

40

Diameter / m

XP

D /

Db

1 1.2 1.420

22

24

26

28

30

Diameter / m

BI co

/ D

b

1 1.2 1.433

34

35

36

37

38

39

Diameter / m

BI xp

/ D

b

17.7 GHz18.95 GHz20.2 GHz

Figure 5.7: Performance of a BOR1 reflector at the low frequency band.


0.8 140.5

41

41.5

42

42.5

Diameter / m

Dw

p / D

bi

0.8 125

30

35

40

45

50

Diameter / m

XP

D /

Db

0.8 120

22

24

26

28

30

Diameter / m

BI co

/ D

b

0.8 136

38

40

42

44

46

48

Diameter / m

BI xp

/ D

b

27.5 GHz28.75 GHz30.0 GHz

Figure 5.8: Performance of a BOR1 reflector at the high frequency band.

46 5. Results

15 20 25 30 3539

40

41

42

43

frequency / GHz

Dw

p / dB

i

15 20 25 30 3520

25

30

35

40

frequency / GHz

XP

D /

dB

15 20 25 30 35

15

20

25

30

frequency / GHz

BI co

/ dB

15 20 25 30 3525

30

35

40

frequency / GHz

BI xp

/ dB

δf = 0 mm

δf = 85 mm

δf = 170 mm

BOR1 model

Required

frequency / GHz

Figure 5.9: Performance of the reflector for different frequencies and feed positionswhen the reflector is partially covered with a FSS surface. The values of δf in thisfigure is the distance from the focal point and correspond to the three black coloredfeed positions in Figure 3.2.


δf = 0 mm

δf = 85 mm

δf = 170 mm

17.7 GHz

0.1°

0.2°18.95 GHz

0.1°

0.2°20.2 GHz

0.1°

0.2°

27.5 GHz

0.1°

0.2°28.75 GHz

0.1°

0.2°30.0 GHz

0.1°

0.2°

Figure 5.10: Deviation of beam pointing direction from the desired direction.The valuesof δf in this figure is the distance from the focal point and correspond to the three blackcolored feed positions in Figure 3.2.

48 5. Results

5.4 Optimization

The performance of the reflector system could be expected to increase if the positionof the feed elements is optimized. This can be done in various ways. As stated earlier,the performance of the antenna depends on in which φ-plane the feed is located. Sincethe final geometry of the feed cluster depends on the desired footprint on earth it maybe possible to exclude the feed elements that gives the worst performance. Even if thewhole cluster must be used, it may be possible to increase the performance by movingthis relative the focal point. John Ruze found in [9] that the abberations caused whendefocusing the feed elements are minimized if these are moved from the plane where ithas so far been located slightly closer to the reflector. The optimum distance is givenby

ǫz =ǫ2x

2F, (5.1)

where ǫx is the displacement of the feed element in the x-direction and ǫz is thesearched distance.

Chapter 6

Conclusion

The reflectors presented in Section 5.3 and 5.2 fulfil most of the requirements for theperformance but is still below the required value in some cases. The defocus effectsare of most importance for the co-polar beam isolation. The general trend is howeverthat the variations of the displaced feed elements due to frequency follow the variationof the focused elements. It is therefore concluded that it is possible to use the BOR1

theory extensively in the design of a cluster-fed reflector system.

The main advantage of the BOR1 model is its simplicity, which makes it possibleto in a short time test a large number of different reflectors and feed antennas. This isimportant in the initial stage of the design procedure and when considering to use anFSS surface on the reflector. The disadvantage is that, due to defocusing effects, theBOR1 model is inaccurate if the feed antenna is displaced from the focal point. Theagreement is better when a larger F/D-ratio is used. This is the case even though thesize of the cluster increase due to the higher gain needed for the feed antennas.

There are several ways to further improve the performance of this antenna system.There is ongoing research by Omid Sotoudeh on the hard horn antennas and a consider-ably improvement on these antennas can be expected. This is of great importance sincethe performance of the whole reflector system largely depends on the feed antennas. Inthis report, an FSS surface was considered and gave promising results. It was assumedto reflect perfectly in one band and not at all in the other. This is not the case in reality,and if these surfaces are to be used they should be studied in detail. The reflectors con-sidered in this report have all been parabolic. This is not necessarily the best solution.It may be possible to find a shape that gives a more even performance for the wholecluster, which could improve the performance of the most displaced elements. It shouldbe noted that the results presented in this report are worst case and the design is notoptimized. It should also be mentioned that the chosen specifications are very strict,when we take into consideration that the reference is the co-polar level in the weakestpoint. Therefore, it may be possible to relax on them. Also, after the exact shape ofthe footprint on earth is known, the performance could be improved significantly byoptimizing the geometry of each of the 4 feed clusters. It is worth putting more effortinto this, because the advantages concerning the total payload needed on a satellitewould be significant if the number of reflectors needed for the same coverage is halved.

49

50 6. Conclusion

The hard horn is therefore a candidate for cluster feeds in multi-beam antennas for dualband operation.

Bibliography

[1] O. Sotoudeh, P-S. Kildal, Z. Sipus, C. Mangenot and P. Ingvarson, ”Theoreticalstudy of the hard horn used as feed in multi-beam antennas for dual band operationat 20/30 Ghz,” Proceedings of 2003 IEEE AP-S International Symposium, Vol. 4,Columbus, June 2003.

[2] O. Sotoudeh, P-S. Kildal and P. Ingvarson, ”Comparison of different theoreticalhorn antenna types used as cluster feeds in reflector systems with multiple beams,”JINA Conference 2002, Nice, France, November 2002.

[3] O. Sotoudeh, P-S. Kildal, P. Ingvarson and T. Lindgren, ”Fast and accurate para-metric studies of alternative feeds for offset multibeam antennas by assuming rota-tionally symmetric geometries (BOR1 modelling),” 26th ESA Antenna Technology

Workshop on Satellite Antenna Modelling and Design Tools, p 255-261, Nov 2003.

[4] P-S. Kildal, M. Johansson, T. Hagfors and R. Giovanelli, ”Analysis of a clusterfeed for the Arecibo trireflector system using forward ray tracing and aperture in-tegration,” IEEE Transactions on Antennas and Propagation, Vol. 41, no 8, p1019-1025, Aug 1993.

[5] P-S. Kildal, ”ARECIBO code for Analysis of Multi-Reflector Antennas,” availablefrom the Author, see www.kildal.se.

[6] P-S. Kildal, ”Artificial soft and hard surfaces in electromagnetics,” IEEE Transac-

tions on Antennas and Propagation, Vol. 38, no 10, pp 1537-1544, Oct 1990.

[7] P-S. Kildal, Foundations of Antennas, a Unified Approach, Antenna textbook withMathcad handbook, Studentlitteratur, Lund, 2000.

[8] C. Sletten, Reflector and Lens Antennas, Analysis Using a Personal Computer,Artech House Inc., Norwood, 1988.

[9] J. Ruze, ”Lateral-feed displacement in a paraboloid,” IEEE transactions on Anten-

nas and Propagation, Vol. 13, no 5, pp 660-665, sept 1965.

[10] B. Munk, Frequency Selective Surfaces, John Wiley & Sons, Inc., 2000.

[11] L. Rade and B. Westergren, Mathematics Handbook for Science and Engineering.Studentlitteratur, Lund, 1998.

51

52 BIBLIOGRAPHY

Appendix A

Geometry

A.1 Offset

δf

θC

θB

FBOR1

F

xC

x0

xf

zC

zF

Figure A.1: Geometry of reflector and feed cluster.

The focal length of the offset reflector (F in Figure A.1) is related to the offset(xC), and the focal length of the corresponding BOR1 antenna (FBOR1, constant) bythe Pythagorean relation.

53

54 A. Geometry

z2C + x2

C = F 2BOR1, (A.1)

which can be written

(F − x2

C

4F

)2

+ x2C = F 2

BOR1. (A.2)

By rearranging and taking the square root of (A.2) we get

F − x2C

4F=

√F 2

BOR1 − x2C . (A.3)

This can be written in the form of a second degree equation. We have

F 2 − F√

F 2BOR1 − x2

C − x2C

4= 0. (A.4)

This equation has the solution

F =1

2

(√F 2

BOR1 − x2C ± FBOR1

). (A.5)

We are only interested in the positive solution since the negative represents a pointbehind the reflector.

The offset, x0, for which the blockage can be assumed to be neglected depends onthe distance xF , the distance between the highest located feed and the reflector, zF ,and the angle θB. From trigonometry we can see that this relationship is

x0 = xF + zF tan θB, (A.6)

where

xF = δf + cos θC (A.7)

and

zF = F − x20

4F+ δf sin θC . (A.8)

This gives

x0 = δf + cos θC +

(F − x2

0

4F+ δf sin θC

)tan θB. (A.9)

The angle θC is given by

θC = arctan

(xC

zC

), (A.10)

where

A.2 Coordinate System 55

xC =Dx

2+ x0 (A.11)

and

zC = F − x2C

4F= F − 1

4F

(Dx

2+ x0

)2

. (A.12)

Dx is the diameter of the reflector in x-direction. The relationship for zC is foundfrom the parabolic equation. We have

θC = arctan

(Dx

2+ x0

F − 14F

(Dx

2+ x0

)2

). (A.13)

The angle θB is

θB = arctan

(δf

FBOR1

). (A.14)

The offset, x0, is found by using (A.14) and iterating (A.5), (A.9), and (A.13).

A.2 Coordinate System

Consider a point, p, located on the z-axis at a distance r from the origin. This canbe rotated an angle θy around the x-axis and an angle θx around the y-axis. The newpoint, p′ will then given by

p′xp′yp′z

= Ry(θx)Rx(θy)

00r

. (A.15)

The rotation matrices can be found in a mathematical handbook ([11]) to be

Rx(θy) =

1 0 00 cos θy sin θy

0 − sin θy cos θy

(A.16)

and

Ry(θx) =

cos θx 0 sin θx

0 1 0− sin θx 0 cos θx

. (A.17)

The rotation around the x-axis (A.16) is in the negative (left-hand) direction whichresults in opposite signs on the sine-terms in (A.16) compared to what is found in [11].Inserting (A.16) and (A.17) in (A.15) and multiplying gives

56 A. Geometry

p′xp′yp′z

=

r sin θx cos θy

r sin θy

r cos θx cos θy

. (A.18)

The relationship between cartesian and spherical coordinates is

r =√

x2 + y2 + z2

θ = arccos

(z√

x2+y2+z2

)

ϕ = arctan(

yx

). (A.19)

The point p′ can then be described in the spherical coordinate system as

r = rθ = arccos (cos θx cos θy)

ϕ = arctan(

tan θy

sin θx

) . (A.20)

The angles θx and θy can be found if the angles θ and ϕ are given in the sphericalcoordinate system. To find this relationship, consider the point p′ above. Its locationin cartesian coordinates can be calculated with

x = r sin θ cos ϕy = r sin θ sin ϕz = r cos θ

(A.21)

or

x = r sin θx cos θy

y = −r sin θy

z = r cos θx cos θy

. (A.22)

Using (A.21) and (A.22) we get

sin θ cos ϕ = sin θx cos θy (a)sin θ sin ϕ = − sin θy (b)cos θ = cos θx cos θy (c)

. (A.23)

θy can be found directly from (A.23 b). To find θx we divide (A.23 a) with (A.23c). The final result is then

θx = arctan(tan θ cos ϕ)θy = arcsin(− sin θ sin ϕ)

. (A.24)

Appendix B

Numerical Values of ReflectorPerformance

In this section the numerical values used in Figure 5.1, 5.5, and 5.9 are presented.

57

58 B. Numerical Values of Reflector Performance

Frequency/GHz 17.7 18.95 20.2 27.5 28.75 30.0Dwp/dBiRequired 39.5 39.5 39.5 39.5 39.5 39.5BOR1 41.4 41.4 41.3 42.4 42.7 42.00 mm 41.4 41.4 41.4 42.4 42.7 42.048 mm 41.2 41.2 41.0 41.3 41.4 41.696 mm 41.1 41.0 40.9 40.0 39.6 39.2

XPD/dBRequired 27.0 27.0 27.0 27.0 27.0 27.0BOR1 30.1 28.2 26.6 27.5 30.3 33.60 mm 29.8 28.0 26.6 25.9 27.8 29.548 mm 29.1 27.6 25.9 23.3 24.6 26.496 mm 28.3 26.6 25.2 23.2 24.2 23.8

BIco/dBRequired 25.0 25.0 25.0 25.0 25.0 25.0BOR1 26.2 28.9 30.1 28.8 29.3 29.50 mm 23.7 25.4 26.8 28.9 29.4 29.548 mm 21.3 21.5 22.1 22.7 24.4 25.496 mm 13.8 15.1 16.4 14.9 14.4 13.1

BIxp/dBRequired 27.0 27.0 27.0 27.0 27.0 27.0BOR1 39.0 37.8 36.8 40.0 39.9 37.80 mm 36.3 37.4 36.4 40.0 39.6 37.648 mm 32.1 34.9 33.9 35.8 34.1 32.396 mm 29.5 29.9 29.2 29.2 27.6 25.9

Table B.1: Results for initial design (see Figure 5.1).

59




BIxp/dBRequired 27.0 27.0 27.0 27.0 27.0 27.0BOR1 37.1 36.0 35.2 >40 39.8 37.90 mm 37.6 36.2 35.3 >40 39.9 38.285 mm 35.4 35.7 34.8 39.4 38.6 37.0170 mm 33.5 35.0 34.0 37.4 36.6 34.4

Table B.2: Results for reflector with FBOR1 = 2.42 m (see Figure 5.5).

60 B. Numerical Values of Reflector Performance




BIxp/dBRequired 27.0 27.0 27.0 27.0 27.0 27.0BOR1 37.1 36.0 35.2 >40 >40 40.00 mm 37.6 36.2 35.3 >40 >40 39.885 mm 35.4 35.7 34.8 >40 40.0 38.4170 mm 33.5 35.0 34.0 >40 39.7 37.0

Table B.3: Results for reflector with FBOR1 = 2.42 m having a FFS surface (see Figure5.9).

design of a cluster-fed multibeam reflector system using

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