Small Offset Parabolic Reflector Antenna Design and Analysis

Wojciech J. Krzysztofik # #Wroclaw University of Technology, Faculty of Electronics Engineering, Institute of Telecommunications, Teleinformatics &

Acoustics, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland, wojciech.krzysztofik@pwr.wroc.pl

Abstract— Complete radiation pattern of a focus-fed offset para-bolic reflector antenna is presented. Forward radiation and wide-angle radiation is calculated using the geometrical theory of diffraction. Effects of the offset angle and the feed pattern asymmetry on the cross-polar isolation are presented. Design criteria of an offset reflector satisfying the WARC-DBS re-quirements for receiving antenna, and for a given feed radiation pattern are presented. It is also shown how to determine the antenna noise temperature experimentally, using direct satellite signal.

I. INTRODUCTION The performance of the front-fed parabolic arrangement is

impaired by the unavoidable blocking of the aperture by feed and supporting structures. However, by using an offset para-bolic (not including the apex) the feed can be kept clear of the reflected beam, since it do not block the aperture. Another its important advantages over its symmetric counterpart are mainly the scanning capability and the reduction of interaction between the primary feed and the reflector. These advantages are paid by the higher level of cross-polar radiation for linear polarization and the beam shift for circular polarization [1, 2, 3].

Increasing interest in the use of the reflector antennas for Direct Broadcast Satellite (DBS) ground station (home) recep-tion, has emphasized the need to study closely its microwave and geometrical (mechanical) properties.

II. BASIC EQUATIONS FOR COMPUTING ANTENNA PATTERN One of the important problem is the knowledge of the rela-

tionship between the reflector shape, the field distribution of the primary feed, and the antenna radiation pattern.

Consider an offset parabolic reflector excited at its focus whose geometry configuration is shown in Figure 1. The ba-sic parameters of reflector are shown as the focal length (F) of the parent parabolic, the offset angle (γo), the half of the angle ( ∗θ ) subtended by the reflector as viewed from the focal point, an aperture diameter d (diameter of the projected circular aperture of the main reflector), the ratio (F/D) of focal length to diameter of parent (symmetrical) reflector (so called shape coefficient of the reflector). The physical contour of the re-flector is elliptical but its projection into the x´ y plane pro-duces a true circle.

To calculate the radiation patterns of reflector antennas we need analytical expressions for the feed pattern and the fields

across the aperture of the antenna for a given reflector feed configuration. Then a procedure for computing the antenna far-field patterns from the aperture distribution is required.

Fig. 1. Geometry of an offset parabolic reflector

The far-field radiation arising from a known tangential electric field distribution in an infinite plane can be determined ex-actly. In dealing with the offset reflector, the infinite surface is chosen as the x´ y plane and the electric fields outside of the projected aperture region are assumed negligible. An electro-magnetic field distribution is introduced around the boundary of the projected aperture to satisfy the continuity criterion, and thus the predicted radiation fields satisfy the radiation condi-tions in the forward hemisphere. The neglect of the electric fields outside of the projected-aperture region is acceptable providing that the dimensions of the aperture are large relative to the electrical wavelength (d/λ>10).

The tangential electric field distribution within the pro-jected-aperture region is determined by use of the physical optics approximation.

That is, the electric field rE reflected from the offset re-flector is obtained from

ininr EEE −⋅⋅= 1)1(2 (1)

where iE is the incident electric field (is taken as the primary-feed radiation) at the reflector, and n1 is the surface unit nor-mal.

When the phase centre of primary-feed is located in the general vicinity of the reflector geometric focus, the incident fields at the reflector can be expressed in the form,

ρρφφθθ /)](exp[)11( 1 −⋅+⋅= RjkAAEi (2)

where Aθ, Aφ are the primary-feed principal plane radiation patterns; R1 is a phase-compensation term that accounts for small offset in the primary-feed location, and

)cossinsincoscos1/(2 00 φθθθθρ −+= F is the distance between a point on reflector surface and the reflector geomet-ric focus.

The physical-optics aperture-field method was selected in preference to the surface-current method since it leads to the more simple mathematical expressions, and offers a signifi-cant reduction in the required computational effort [2]. For an incident field polarized in the y-direction, reference polarization (called co-polar) coE and orthogonal polarization (cross-polar) crE are defined, respectively, by

ΦΦΨΨ ⋅Φ+⋅Φ= 1cos1sin EEE co (3)

ΦΦΨΨ ⋅Φ−⋅Φ= 1sin1cos EEE cr (4) and total E-field is given as

crco EEE += (5)

The model employs a conventional spherical co-ordinate system (R, Ψ, Φ) with origin at the centre of the reflector pro-jected-aperture (Fig. 1), and makes use of spatial Fourier transform formulation for the radiation from an infinite sur-face. The normalized radiation patterns of the offset antenna may be expressed directly in terms of a linear co-polar field component Eco and a cross-polar component Ecr in the matrix form as [1]

⎥⎦

⎤⎢⎣

⎡ΦΨΦΨ

⎥⎦

⎤⎢⎣

⎡

Φ+ΦΦΦ−Ψ+=⎥

⎦

⎤⎢⎣

⎡),(),(

2cos12sin2sin2cos1

)0,0(22cos1

22

22

cr

co

cocr

co

FF

tttt

FEE

(6) where t = tan Ψ/2, and the functions Fco, Fcr are the spatial Fourier transforms of the co-polar and cross-polar components of the tangential electric field in the projected aperture plane. The Fourier transforms are the transverse components of the vector

∫ ∫ Ψ−=ΦΨπθ

φθθρε2

0

*

0

2]exp[),( ddnsinjkRsiF ii (7)

where]sinsinsincos)cossincoscos[(sin 00 φθθθφθθρ Φ−Φ+=R

is the distance from general point in the projected-aperture plane to a far-field point, and the tangential electric field com-ponents εi distributed in the projected-aperture are expressed directly trough the radiation characteristics of the primary-feed (Aθ, Aφ) by the matrix expression as

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡

φ

θ

εε

AA

abba

Kcr

co , (8)

where FFRjkK 2/)]2(exp[ 1 −−= ; φθθ sin)cos(cos 0 +=a ;

)coscos1(cossinsin 00 θθφθθ +−=b ; θφφθ cos)cos(sin 01 ztR Δ+−Δ≈ ; zt ΔΔ , are the small

transverse and axial offsets in the primary-feed location, re-spectively; 0φ is the angle (measured to the x axis in the xy plane) denotes the plane of the offset.

To predict the overall antenna radiation pattern it is neces-

sary to specify the primary-feed directivity characteristics ),( ΦΘ AA and to compute the two-dimensional integrals of the

Fourier transforms ),( ΦΨiF . The feed is the element that connects the source and/or receiver of electromagnetic radia-tion by way of a transmission line, such as waveguide, to the antenna. Number of antennas for illuminating reflectors, e.g. pyramidal horn and conical horn antennas (smooth-walled and corrugated), slots, dipoles, etc. are used.

More typically, aperture antennas are fed by conical corru-gated horn radiators. Horn patterns can often be approximated by Gauss error functions or by the )(cos θN function (Fig. 2).

Fig. 2. Example of measured and approximated patterns of primary fed

So that, for a horn pattern linearly polarized in the y-

direction , we can write

ρρρρφφθθ

φφθθ

φθ

/)exp()11(

/)exp()1cos1)(sin(cos)(

ikAA

ikAg N

−⋅⋅+⋅=

=−⋅⋅+⋅≈ (9)

where N is a constant (ranging from one to perhaps 500) and the )(θNsco pattern is assumed to be zero for θ > θ*.

The horn pattern is assumed to be a function of θ only, in-dependent of φ, and the polarization that of a Huygens source.

The primary parameters that the designer can control are • the degree of offset (through γ0), • the aiming of the feed antenna (i.e., the angle θ0), and • field taper on reflector edge (feed radiation pattern).

H-plane 10.95 GHz

+ 11.85 GHz ◊ 12.4 GHz Δ cos8θ

We investigated the influence of the above parameters on cross polarisation, gain, and sidelobe level, using the geomet-rical diffraction based on presented theory, and adopted in computer code OFFREFAN. Scattering from support struc-tures was not modelled, but for an offset configuration, these are typically negligible.

The feed radiation pattern is assumed with a -10 dB beamwidth of ±θ*, this led to a feed taper (edge illuminations) of -10 dB (exactly: -10.1 dB and -14.6 dB on horizontal and vertical reflector edges, respectively). This was for the case of d = 0.55 m; F/D = 0.326; γ

0 = 420; θ*= 720, and the feed-pattern maximum pointed to bisect the angle subtended by the reflector.

III. NUMERICAL AND EXPERIMENTAL RESULTS Results of numerical calculations, together with experi-

ments, and ITU-R-standards for the radiation patterns are ploted in Fig.3.

a)

b)

Fig. 3. Computed and measured far-field radiation co-polar pattern in E- and H-plane (a) and cross-polar pattern in E-plane (Φ=90o) (b) of the offset-parabolic reflector antenna of diameter d = 0.55 m

Comparison of theoretical end experimental ([4], [5]) data shows a favourable agreement. In both polarisations antenna satisfies requirements of ITU-R WARC-DBS (Geneva 1977).

IV. G/T MEASUREMENT The need to verify RF performance of an Earth Station an-

tenna for satellite telecommunications prior to commencement of operational activities is paramount in order to secure any

potential degradation of service and co-channel/satellite inter-ference [8].

We will discuss procedure of the gain to noise ratio (G/T) (figure of merit) obtained by making use of prevailing televi-sion carriers [6], [7]. Generally a G/T characterisation is a C/N measurement via a calibrated filter having a known noise equivalent bandwidth. This filter can be that of spectrum ana-lyser if the carrier used is pure or has little modulation, or a TV filter, if a TV carrier is used. It is the latter method that we shall describe here as this has been used to characterise the above-mentioned antenna. The configuration of the measured set up contains AUT (Antenna Under Test), LNB ( Low Noise Block) with down converter, and spectrum analyser (Tek-tronix -TEK 2710 ).

The main advantage of this method is that space segment is always available in the form of TV carriers (Fig. 3), entailing routine G/T measurements throughout the operational life of the antenna.

a)

b) Fig. 4. Spectra of TV-signals from satellites ASTRA 1A(a), and EUTELSAT-II (b), received in Wroclaw (Poland) using the offset parabolic reflector an-tenna model OG-66

The procedure would thus be that the AUT measures the TV carrier level with the antenna pointed to the satellite. The power level, says Ps, is recorded. Next the antenna is de-pointed to clear sky and a new power meter readings (Pcs) is noted. The satellite EIRP is assessed by the co-operating sta-tion and the G/T value is obtained as follows:

kBLLLEIRPPPTG Neqrsdatbfcss −++++−−= )(/ (10)

ITU-R ITU-R

ITU-R

where: (Ps-Pcs) [dBm] is the power meter readings ; Lbf, Lat, Lrad are the free space , atmospheric, and radiometer losses, respectively; BNeq is an equivalent noise bandwidth; and k [dBW/kHz] is the Boltzmann´s constant. Antenna noise temperature TA have been estimated theoreti-cally and verified experimentally by means of direct satellite signal measurements.

We have made use of the existing TV carriers within the ASTRA and EUTELSAT II space segments (Fig. 4) to assess the G/T figure of an offset parabolic reflector antenna and compared this to traditional methods.

In Table 1, the measured geometrical, and electrical pa-rameters of WISI Model OG 66 parabolic reflector antenna, are presented.

TABLE 1. ANTENNA GEOMETRICAL AND ELECTRICAL PARAMETERS

Parameter Producer Data Measured Diameter, d [m] 0.55 0.55

Ellipse Main Axis [m] - 0.60 Focal Length, F [m] - 0.589

Aperture Angle, θ* [deg] - 72.36 Offset Angle, γ0 [deg] - 42.37

G [dBi] 35 34.1

Effeciency - 0.71 HPBW [deg] - 3.3 FSLL [dB] - -23 G/T [dB] 13 14.4

In Fig. 5, the calculated radiation patterns of an offset para-

bolic antenna OG-66 are presented.

Ψ

-100

-80

-60

-40

-20

0

20

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

co-polar

cros-polar

E/Emax [dB]

[deg] Fig. 5. Calculated radiation pattern of OG-66 antenna

V. ANTENNA GAIN – LOSSES BILANS An offset parabolic reflector illuminated axially-symmetric,

always produces an orthogonal radiated field component. It means if part of power radiated by primary fed is useless. The phase errors in the aperture field are due to: imprecise pattern of primary-fed, the reflector surface and shape tolerance, shift-ing of primary-fed phase centre with respect to reflector focus, etc.

The real an offset parabolic reflector antenna can be charac-terized with an equivalent power gain as follows:

cfphsaphraphfprDG ηηηηηηη −−−= (11)

where: D is the antenna directive gain, cfphsaphraphfpr ηηηηηηη ,,,,,, −−− are, respectively, all losses

due to: radiation, polarization, phase of primary fed character-istic, random and systematic phase errors in the antenna aper-ture field distribution, feeding line between primary fed and down-converter, and the coating surface protecting the reflec-tor.

The estimated numbers of different losses of designed an-tenna are shown in Table 2.

TABLE 2. ANTENNA LOSSES, GAIN, AND HPBW

Parameter Frequency, GHz 10.95 11.3 11.7

ηr ,dB -0.96 -0.89 -0.92 ηp ,dB -0.11 -0.11 -0.11

ηra-ph ,dB -0.1 -0.1 -0.1 ηsa-ph ,dB -0.18 -0.2 -0.24

ηc ,dB -0.15 -0.15 -0.15 ηtotal ,dB -1.5 -1.45 -1.52

Efficiency 0.708 0.716 0.705 D, dB 36.27 35.55 35.85 G, dB 34.77 34.10 34.33

HPBW, deg 3.08 3.34 3.25

VI. CONCLUSIONS It is verified numerically that aperture field integration can

predict radiation fields of offset parabolic reflectors that are close to the measured results, if the aperture integration sur-face is chosen to cap the reflector. We conclude that as long as the necessary precautions evoked above are achieved, good qualitative results of G/T measurements can be obtained. The accuracy associated with this type of measurement is +/-0.6 dB. Taking into account the losses in the antenna structure, the antenna gain was estimated with good accuracy with respect to measurements.

REFERENCES [1] A.W. Rudge, “Multiple-Beam Antennas: Offset Reflectors with Offset

Feeds”, IEEE Transactions on Antennas & Propagation, vol.AP-23, pp. 317-22, May 1975

[2] H. Matsuura, K. Hongo, “Comparison of Induced Current and Aperture Field Integration for an Offset Parabolic Reflector”, i.b.i.d, vol. AP-35, pp. 101-105, January 1987.

[3] W. Stutzman, M. Terada, “Design of Offset-Parabolic-Reflector An-tennas for Low Cross-Pol and Low Sidelobes”, IEEE Antennas and Propagation Magazine, vol. 35, No. 6, pp. 46-49, December, 1993.

[4] A.W. Rudge, K. Milne, A.D. Olver, P. Knight, eds, The Handbook of Antenna Design, Peter Peregrinus Ltd., vol. 1, London, GB, 1982

[5] A.W. Love, ed., Reflector Antenna, IEEE Press, New York, US,1978 [6] B.J. Kasstan, An Ideal Test Range for Accurate Earth Station Antenna

Measurements, Six International Conference on Antennas & Propaga-tion, ICAP´ 89, Coventry, United Kingdom, Part 1: Antennas, 473-7, 1989

[7] D.J. Bem, Satellite Radiocommunication, (in Polish), Communication & Telecommunication Publications, Warsaw, Poland, 1990

[8] W.J. Krzysztofik, Z. Langowski, W. Papierniak, Design of an Offset Parabolic Reflector Antenna for Direct Reception of TV-Satellite, Scientific Report, Wroclaw University of Technology, Wroclaw, Pol-and,1992

[9] ITU-R, “Final Acts of the World Administrative Conference for Plan-ning of Broadcast Satellite Service in Frequency Band 11.7-12.5 GHz (in Region 1), and 11.7-12.2 GHz (in Regions 2 and 3), Geneve 1977