optimum paraboloid aerial and feed design

4
Optimum paraboloid aerial and feed design D. Herbison-Evans, M.A., D.Phil. Synopsis A computer study was made of the spillover, crosspolarisation and illumination efficiency of paraboloids of various focal-length/paraboloid-diameter (F/D) ratios, fed by rectangular, square and circular wave- guides in their lowest-order modes. The feed radiation patterns used were those derived by simple diffraction theory from the incident waveguide-mode field distribution. The approximations made in this derivation give unnaturally low spillovers for very small aperture feeds, and thus favour short FjD systems, which would use such feeds. The feed dimensions were optimised to give a maximum figure of merit (gain/ total noise temperature) for each combination of feed, receiver noise temperature and FID ratio. Some of the optimum horn dimensions and resulting illumination tapers, excess aerial noise temperatures, gain efficiencies and figures of merit produced by the computations are presented. The results show, among other things, that the figure of merit of a low-F/D-ratio system is about 0-8dB less than that of a large- F/D-mtio system. They also show that, if the system has a low receiver temperature, the feed dimensions should be larger than conventional design procedures indicate, to give up to 3dB more edge-illumination taper. List of symbols a = isplane halfwidth of rectangular feed aperture, radius of circular-feed aperture or halfwidth of square-feed aperture b = Hplane halfwidth of rectangular-feed aperture D = diameter of paraboloid reflector 2T e , E4 = components of electric field radiated by feed in spherical co-ordinates with origin at feed , components of electric field radiated by feed in rectangular co-ordinates with origin at feed focal length of paraboloid reflector gain of aerial relative to isotropic radiator Bessel function of first kind and order one with E F = F = G = J,W = k M argument x 2TT/A /G\ /7T 2 Z) 2 \ lOlogf —J — lOlog ( > 2 _ j = figure of merit relative to dish with 100% efficiency and under- graded receiver temperature ^oo = power radiated per unit solid angle by feed along its axis P, = total power radiated by feed T = total equivalent noise temperature at aerial term- inals T R equivalent noise temperature /due to receiver and aerial main beam > T s = equivalent noise temperature of feed spillover past paraboloid y = 2TT\guide wavelength F = reflection coefficient at waveguide mouth rjj = aperture illumination taper efficiency rj p = efficiency factor due to crosspolarisation loss rj s = efficiency factor due to spillover loss 6 = component of spherical co-ordinates about feed A = free-space wavelength p = distance from point on paraboloid to focus <f> = component of spherical co-ordinates about feed <j> = cone semiangle subtended by paraboloid at its focus 1 Introduction One of the simplest methods of obtaining aerial gains in excess of 30dB is the horn-fed paraboloid of revolu- tion. The efficiency of this aerial depends on the focal-length/ aperture-diameter (F/D) radio of the paraboloid and the noise temperature of the receiver, as well as the design and dimensions of its feed. Paper 5428 E, first received 29th June and in revised form 11th August 1967. Crown copyright Dr. Herbison-Evans was formerly with the Ministry of Technology Signals Research & Development Establishment, Christchurch, Hants., England, and is now with the School of Physics, University of New South Wales, N.S.W., Australia. ' PROC. IEE, Vol. 115, No. 1, JANUARY 1968 Previous general design procedures for optimising para- boloids have assumed either a long focal length 1 - 2 or approxi- mations to the feed radiation pattern which lend themselves to integration. 3 - 4 The optimum design of the feed-paraboloid combination for a maximum signal-power/noise-power ratio depends on the particular application. For radar, scatter and line-of- sight links and passive-satellite communication links, the received signal is proportional to the square of the aerial gain, so that the aerial may be designed for maximum G 2 / total-noise-power ratio, where G is the gain above isotropic. For active-satellite communication ground systems, where the downlink is more critical than the uplink, as in the present generation of systems, the signal is proportional to the gain, and the aerial should give maximum G/total-noise-power ratio, i.e. figure of merit. The total noise power can be conveniently expressed as an equivalent noise temperature and split into the sum of components due to the receiver, aerial mainbeam, diffraction sidelobes, defects in the paraboloidal surface, feed-support blockage and spillover by the feed outside the cone subtended by the dish at the feed. This last component is important in this study, as it is the one which is 'traded' for gain by altering the feed dimensions. Assuming the noises due to each com- ponent are relatively incoherent, their powers can simply be summed. In this study, all but the spillover will be lumped together and referred to as the 'receiver temperature'. The optimum feed-paraboloid configuration depends on this parameter. 2 Method In this study, the effects of paraboloid F/D ratio and receiver noise temperature on the optimum design were in- vestigated with more realistic approximations to the feed radiation patterns than previous investigations 3 * 4 using digital computers. The study was directed to the design of ground-terminal aerials for an active-satellite communication system. Thus the figure of merit or gain/total-noise temperature ratio was maximised with the aerial presumed to be directed at low elevation (~ 10°), when satellites tend to be at their maximum range and their signals weakest. Three feeds which lend themselves to circular polarisation were investigated: rectangular, square and circular waveguides in their fundamental modes. The radiation patterns of the feeds were taken to be those derived by scalar diffraction theory from the transverse-field distribution of the modes. 5 For the rectangular and square feeds, 2 Y cos (ka sin d cos <f>) cos 6 } ,->/,„ „:„ a „«„ jv 2 /2ka sin 6 cos <f>\ sin (kb sin 9 sin <f>) kb sin 6 sin (j> sin <f> 0) 87

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Page 1: Optimum paraboloid aerial and feed design

Optimum paraboloid aerial and feed designD. Herbison-Evans, M.A., D.Phil.

Synopsis

A computer study was made of the spillover, crosspolarisation and illumination efficiency of paraboloidsof various focal-length/paraboloid-diameter (F/D) ratios, fed by rectangular, square and circular wave-guides in their lowest-order modes. The feed radiation patterns used were those derived by simple diffractiontheory from the incident waveguide-mode field distribution. The approximations made in this derivationgive unnaturally low spillovers for very small aperture feeds, and thus favour short FjD systems, whichwould use such feeds. The feed dimensions were optimised to give a maximum figure of merit (gain/total noise temperature) for each combination of feed, receiver noise temperature and FID ratio. Someof the optimum horn dimensions and resulting illumination tapers, excess aerial noise temperatures,gain efficiencies and figures of merit produced by the computations are presented. The results show, amongother things, that the figure of merit of a low-F/D-ratio system is about 0-8dB less than that of a large-F/D-mtio system. They also show that, if the system has a low receiver temperature, the feed dimensionsshould be larger than conventional design procedures indicate, to give up to 3dB more edge-illuminationtaper.

List of symbols

a = isplane halfwidth of rectangular feed aperture,radius of circular-feed aperture or halfwidth ofsquare-feed aperture

b = Hplane halfwidth of rectangular-feed apertureD = diameter of paraboloid reflector

2Te, E4 = components of electric field radiated by feed inspherical co-ordinates with origin at feed ,

components of electric field radiated by feed inrectangular co-ordinates with origin at feed

focal length of paraboloid reflectorgain of aerial relative to isotropic radiatorBessel function of first kind and order one with

E F =

F =G =

J,W =

k

M

argument x2TT/A

/G\ /7T2Z)2\lOlogf —J — lOlog ( >2_ j = figure of merit

relative to dish with 100% efficiency and under-graded receiver temperature

^oo = power radiated per unit solid angle by feed alongits axis

P, = total power radiated by feedT = total equivalent noise temperature at aerial term-

inalsTR — equivalent noise temperature /due to receiver and

aerial main beam >Ts = equivalent noise temperature of feed spillover past

paraboloidy = 2TT\guide wavelengthF = reflection coefficient at waveguide mouthrjj = aperture illumination taper efficiencyrjp = efficiency factor due to crosspolarisation lossrjs = efficiency factor due to spillover loss6 = component of spherical co-ordinates about feedA = free-space wavelengthp = distance from point on paraboloid to focus<f> = component of spherical co-ordinates about feed<j> = cone semiangle subtended by paraboloid at its

focus

1 IntroductionOne of the simplest methods of obtaining aerial

gains in excess of 30dB is the horn-fed paraboloid of revolu-tion. The efficiency of this aerial depends on the focal-length/aperture-diameter (F/D) radio of the paraboloid and thenoise temperature of the receiver, as well as the design anddimensions of its feed.

Paper 5428 E, first received 29th June and in revised form 11th August1967. Crown copyrightDr. Herbison-Evans was formerly with the Ministry of TechnologySignals Research & Development Establishment, Christchurch, Hants.,England, and is now with the School of Physics, University of New SouthWales, N.S.W., Australia. '

PROC. IEE, Vol. 115, No. 1, JANUARY 1968

Previous general design procedures for optimising para-boloids have assumed either a long focal length1-2 or approxi-mations to the feed radiation pattern which lend themselvesto integration.3-4

The optimum design of the feed-paraboloid combinationfor a maximum signal-power/noise-power ratio depends onthe particular application. For radar, scatter and line-of-sight links and passive-satellite communication links, thereceived signal is proportional to the square of the aerialgain, so that the aerial may be designed for maximum G2/total-noise-power ratio, where G is the gain above isotropic.For active-satellite communication ground systems, where thedownlink is more critical than the uplink, as in the presentgeneration of systems, the signal is proportional to the gain,and the aerial should give maximum G/total-noise-powerratio, i.e. figure of merit.

The total noise power can be conveniently expressed as anequivalent noise temperature and split into the sum ofcomponents due to the receiver, aerial mainbeam, diffractionsidelobes, defects in the paraboloidal surface, feed-supportblockage and spillover by the feed outside the cone subtendedby the dish at the feed. This last component is important inthis study, as it is the one which is 'traded' for gain by alteringthe feed dimensions. Assuming the noises due to each com-ponent are relatively incoherent, their powers can simply besummed. In this study, all but the spillover will be lumpedtogether and referred to as the 'receiver temperature'. Theoptimum feed-paraboloid configuration depends on thisparameter.

2 MethodIn this study, the effects of paraboloid F/D ratio and

receiver noise temperature on the optimum design were in-vestigated with more realistic approximations to the feedradiation patterns than previous investigations3*4 usingdigital computers. The study was directed to the design ofground-terminal aerials for an active-satellite communicationsystem. Thus the figure of merit or gain/total-noisetemperature ratio was maximised with the aerial presumed tobe directed at low elevation (~ 10°), when satellites tend to beat their maximum range and their signals weakest.

Three feeds which lend themselves to circular polarisationwere investigated: rectangular, square and circular waveguidesin their fundamental modes. The radiation patterns of thefeeds were taken to be those derived by scalar diffractiontheory from the transverse-field distribution of the modes.5

For the rectangular and square feeds,

2 Y cos (ka sin d cos <f>)cos 6 } ,->/,„ „:„ a „«„ jv 2/2ka sin 6 cos <f>\

sin (kb sin 9 sin <f>)kb sin 6 sin (j>

sin <f> 0)87

Page 2: Optimum paraboloid aerial and feed design

-{©' -f cos0cos (ka sin 6 cos<f>)

~C sin

sin

77

• sin sin <f>)

and for the circular feed,kb sin 0 sin <f> cos . (2)

V 1-841

In both cases, the reflection coefficient appearing in theequations in Reference 5 has been approximated by that dueto the change in refractive index implied by the change inphase velocity. A more exact complex expression for thereflection coefficient is derived in Reference 5, but the extracomplexity of its use was not considered to be justified atthis stage. Thus the reflection coefficient was taken as

k-y(5)

The calculation of the figure of merit of a particular combi-nation of horn, paraboloid F/D ratio and receiver noisetemperature, fell into three parts: spillover, crosspolarisationand illumination efficiency.

The spillover is the proportion of power radiated by thefeed not intercepted by the paraboloid. It defines an efficiencyfactor 7)s:

f \(El + El) sin d dcbdd

— . • • • • « >ii + El) sin n

'o Jo

where O is the semiangular aperture of the paraboloid. Thedenominator should be equal to the power leaving the wave-guide aperture. So as to avoid the wiggly integral involved inits direct evaluation, the analytical formula for the gain wasused to find it indirectly:

n oo

where pTt /•2ir

'/ = (ElJo Jo

G = axial gain of feed over isotropic radiator

and Poo = (El + E2) at 0 = <j> = 0

In our case,6 for the rectangular and square feeds

(7)

(8)

(9)

_ k2ab 8O = —2

77 77

and for the circular feed

2 1 2

• • • (ID

• • • (12)

03)

The crosspolarisation efficiency was found by integrationacross the focal plane of the wanted linear component:

\h[ \ h x sin e dcf>dd(14)

f [ (El + E2) sin 6 d(f>deJo Jo

where Ex — Ee sin </> + E$ cos <f> (15)

E y = EQ C O S <f> — Ef s i n <f> (16)

The aperture-illumination efficiency was found by using

nJo Ji

2* £ vp2 sin

sin(D r2;r

where. . . . (17)

P =IF

1 + cos

and F = focal length of paraboloid

The crosspolarisation was assumed to give rise to a finalcrosspolarisation pattern with most of its radiation near theaxis. Hence it was assumed not to add to the noise tem-perature but only to diminish the gain. The total noisetemperature was taken as

T=TR + Ts(l -rjs)

where 7^ is the receiver temperature. Assuming that onehalf of the spillover is absorbed by a black ground at 300° Kand that the other half sees, the sky at 0°K, 7^ was taken as150°K. The results can be expressed simply for other dishorientations and sky and ground temperatures by linearscaling of TR.

The figure of merit M of the combination was approximatedto by

• • 0 8 )

giving a value relative to a uniformly illuminated aperture ofthe paraboloid diameter with no exess aerial noise. Noaccount was taken of feed blocking, mismatching or othereffects.

In the circular-feed case, the integrations were done ana-lytically, and only single integrations were performed by thecomputer. For the rectangular and square feeds, the doubleintegrations were done by computer. Integrations were doneusing an improved version of the adaptive Simpson pro-cedure6 to a relative accuracy of better than 0 001.

The approximations made in deriving the feed radiationpatterns make them invalid in the range of feed dimensionsused. This was brought out clearly by the spillover integration.This revealed that more energy was radiated into the conesubtended by the paraboloid than was transmitted throughthe waveguide aperture when the aperture transverse dimen-sion dropped below about 0-9A. Thus the results are biasedin favour of low F/D ratios by having unnaturally lowspillovers for small feed apertures.

The optimum horn dimensions were found by a golden-search procedure for each TR and FjD. The golden searchmaximises a function of a variable over a limited range ofthat variable. This it does by examining the function at thegolden points of the range; i.e. at \ (\/5 — 1) and \ (3 — \/5)of the range). Whichever is lower is taken as the boundary of anew range, and the procedure is recursively repeated. Itbrackets the maximum of the function more and moreclosely, until a desired accuracy is attained.

The results for a circular feed of the maximum figure ofmerit are shown in Fig. 1, with the i/plane illuminationsrequired at the edge of the paraboloid relative to that at theapex (including space taper due to the different distancesfrom the focus to the edge and the apex of the paraboloid)in Fig. 2, and the resulting aperture efficiencies and excessnoise temperatures shown in Figs. 3 and 4. Similar curveswere obtained with rectangular and square feeds. Figs. 5,6 and 7 show the optimum feed dimensions for round, rect-angular and square feeds. Fig. 8 shows figure of merit againstTR for FID = 0-4 and 1 -5 for all three feeds.

The scatter on many of the graphs is probably spurious andthe result of the finite accuracy of the integration and searchprocedures.

The asymptotic values obtained for a rectangular feed asFID tends to infinity for 3000° K and 30° K were comparedwith the results of Milne and Raab2 for a single-horn feed,optimising gain alone and gain/excess temperature, respec-tively. The efficiencies and temperatures (allowing for the

PROC. IEE, Vol. 115, No. I, JANUARY 1968

Page 3: Optimum paraboloid aerial and feed design

ffl -2

-4

-5

3000300°K

025 0 5 075F/D

10

Fig. 1Figure of merit against F/D for round feed

1-25 1-5

300 K

0-25 0-5 0-75F/D

1-0 1-25 1-5

Fig. 5Aperture diameter against F/D for round feed

-20

$ -15

a> -10oi

a -5x

30°K

300°K3000°K

025 0-5 075F/D

10 1-25

Fig. 2Hplane edge illumination against F/D for round feed

1-5

30°K300°K

3000°K.

30 K300°K

3000°K

0-25 0-5 0 7 5F/D

1-0 1-25

Fig. 6Aperture width against F/D for rectangular feedO i/planex £plane

1-5

9O

£ 70

60

50

3000°Kand300°K

30°K

0-25 05 .0.-75,, 10' F/D

Fig. 3Gain efficiency against F/D for round feed

125 15

4,3

t1

0) 3000 K

0-25 0-5 0-75F/D

1-0

Fig. 7Aperture width against FjD for square feed

1-25 1-5

30

20

15

10

3000°K

300°K

30°K

-0 fl.0.

0-25 0-5 0-75F/D

10 1-25

Fig. 4Excess temperature against F/D for round feed

PROC. IEE, Vol. 115, No. 1, JANUARY 1968

1-5

- 2

I " 3

oID

cgi- 4

- 53 0 10

F/D=1-5

-0-4

30 100 300TR deg K

1000 3000

Fig. 8Figures of merit comparedy— square, — O — rectangular, . . . • . . . round

89

Page 4: Optimum paraboloid aerial and feed design

different environment models) agree closely, but the optimumhorn dimensions derived here are slightly larger.

A point that arose in the calculations was that no optimumfeed produced more than 2 % of crosspolarised energy. Thisfactor depends on the effective electric/magnetic dipolemoment ratio at the feed aperture, and this is sensitive to theform taken for the reflection coefficient, since the mismatchat the feed aperture augments the electric field but diminishesthe magnetic field there.5

3 ResultsThe following general conclusions can be made from

the results:

(a) All three feeds give similarly shaped curves of figure ofmerit against FID or TR. However, the curves for rect-angular and circular feeds are identical to within OldB andshow an advantage over a square feed of 0-2-0-4dB. Forcircular polarisation, the square and circular feeds can beused directly, but a rectangular feed has to be synthetisedfrom a square one, with internal fins at the edges of theaperture to constrict it for each linear-polarisation com-ponent separately. However, the difference in performancebetween the optimum circular and rectangular feeds isprobably less than the excess conduction losses that this finloading would entail for a rectangular feed, and so a conicalhorn is the indicated feed for circular polarisation.8

(6) The usual feed-design procedures specify the size of thefeed by the illumination occurring in the direction of the edgeof the paraboloid relative to that in the direction of the apex.The present work shows that, for most values of F/D and forall three types of feed, reducing the receiver temperaturefrom 3000 to 300° K requires an increase in the optimum feeddimension to give an extra — 0 • 7 dB edge illumination in the//plane. Reducing the receiver temperature from 3000 to30°K requires an extra -3dB.(c) Despite approximations leading to a bias in favour oflow FID ratios, the results show an advantage in gain ofabout 0-8dB for large over small F/D ratios for all receivertemperatures. This advantage would also hold for Cassegrainor Gregorian systems with a short-focus main dish but highmagnification, as the spillover, crosspolarisation and apertureefficiency will be those apertaining to the long-focus equivalentparaboloid.7 However, for these this advantage must bebalance against extra losses due to blockage by the sub-reflector and diffraction effects due to its finite size.(d) For all three feeds and over most FJD ratios, there islittle change in the design for the optimum C/Tratio betweenreceiver temperatures of 3000 and 300° K. Small changes inaerial temperature are swamped by the large receiver tem-peratures, and the resulting designs are virtually for maximumG alone. Thus the conclusions about optimum horn type and

dimensions, and paraboloid FjD ratios are also true forapplications requiring maximum G2\T for this range ofreceiver temperature. The similar excess aerial noise tem-peratures in this range for rectangular and circular feedsindicates a spillover of about 15%. However, with a 30° Kreceiver temperature, the edge illumination should betapered by a further 2-4dB. The rectangular and circularfeeds give excess aerial noise temperatures of 18 and 16°K,respectively. These are nearly constant with respect to F\Dand indicate a spillover for the optimum designs of approxi-mately 12 and 10-5%, respectively.

4 ConclusionsThe graphs of figure of merit against F\D enable a

system designer who is intent on using a horn-fed paraboloidto choose an F\D ratio for the paraboloid he must use. Hecan trade the losses in the necessary waveguide run to a frontfeed, or the subreflector blocking and truncation losses of aCassegrain feed, for the merit loss as the F\D ratio of theparaboloid is reduced. The curves presented here guide thechoice of feed and give the optimum feed-aperture dimen-sions. A small point is that the designer must ensure that thereis room for the length of the flare of the feed from normalwaveguide dimensions to the required feed-aperture dimen-sions, giving less than a A/8 phase error across the feedaperture.

He still has many other problems in the design of the aerialfor the system. It is hoped that the results presented here canenable him to concentrate properly upon them.

5 AcknowledgmentsThanks are due to the Mathematical Services staff

of SRDE and AAEE for their tolerance and understanding,and to K. Milne for illuminating discussions. Acknowledg-ment is due to the Controller of HM Stationery Office forpermission to publish this paper.

6 References1 CROMPTON, j . w.: 'On the optimum illumination taper for the

objective of a microwave aerial', Proc. IEE, 1954, 101, Pt. Ill,pp. 371-382

2 MILNE, K., and RAAB, A. R.: 'Optimum illumination tapers for four-horn and five-horn monopulse aerial systems' in 'Design andconstruction of large steerable aerials', IEE Conf. Rep. Ser. 21, 1966,pp.12-16

3 SILVER, s.: 'Microwave antenna theory and design' (McGraw-Hill,1949), p. 425

4 LIVINGSTON, M. L.: 'The effect of antenna characteristics on antennanoise temperature and system s.n.r.', IRE Trans., 1961, SET-7,p. 71

5 FRADIN, A. z.: 'Microwave antennas' (Pergamon, 1961), p. 1476 KUNCIR, G. F.: Algorithm 103, Commun. ACM, 1962, 5, p. 3477 HANNAN, P. w.: 'Microwave antennas derived from the Cassegrain

telescope', IRE Trans., 1961, AP-9, pp. 140-1538 REINTJES, J. F., and COATE, G. T. : 'Principles of radar' (McGraw-

Hill, 1952, 3rd edn.), p. 955

90 PROC. IEE, Vol. 115, No. I, JANUARY 1968