Effects of mutual coupling in conical hornarrays

Prof. P.J.B. Clarricoats, D.Sc.(Eng-). F.Eng., F.I.E.E., F.I.E.E.E., F.lnst.P.,F.C.G.I., S.M. Tun, B.Sc, and C.G. Parini, B.Sc, Ph.D., M.I.E.E.

Indexing terms: Antennas, Radiation, Waveguides

Abstract: The paper considers the copolar and crosspolar radiation patterns of a finite array of conical hornelements in which full account is taken of the effects of mutual coupling. Mutual coupling between, respectively,the TE, , - and TE21-, and the TE, , - and TM01-modes has a marked effect on copolar pattern symmetry and onthe level of crosspolarisation. Contour plots are given for 'clean-beam' excitation of the arrays, and the influenceof horn aperture size is shown to be significant.

1 Introduction

This paper is concerned with the influence of mutual coup-ling on the radiation pattern of an array feed comprisingconical horn elements. The initial stimulus for this investi-gation lay in discussion in 1975 between one of the authors(Prof. Clarricoats) and the European Space Agency. Atthat time a requirement was developing for a Europeanspacecraft with a multiple beam antenna with an arrayfeed in which the feed elements would be used to excitemore than one beam. Subsequently, a study of an offsetdual reflector antenna was undertaken at Queen MaryCollege by Bird, Boomars and Clarricoats [1]. A fullaccount of this work was given in the paper by Bird andBoomars [2], while Bird [3] has described a detailed studyof the array feed used in the antenna which comprised cir-cular waveguides in a ground plane. The elements wereassumed to be excited by the dominant TE^-mode, andmutual coupling between TEU-modes in adjacent ele-ments was evaluated.

The present study extends this earlier investigation byallowing for mutual coupling between any of the modes ofthe horns whether propagating or evanescent. An experi-mental study of the array is in progress at Queen MaryCollege, but confidence in our results has been providedthrough the opportunity to compare our theoretical pre-dictions with experimental patterns obtained at ERATechnology Ltd. [4, 5]. There, an independent study ofarray feeds and their associated beam-forming networks isin progress.

Preliminary reports of the study herein have been madeat Conferences [6, 7, 8], and the continuing relevance ofthe work is highlighted by the use of an array of dual-mode circular horns used in both the receive and transmitantennas of the Intelsat VI spacecraft. A companion paper[9] describes the secondary patterns of both single-offsetand dual-offset reflector antennas incorporating the arraysdescribed.

2 Theory

The elementary theory of arrays separates the array factorand element factor in the expression for the array radiationpattern. However, mutual coupling between the elementcauses the element factor to change and become dependenton the element spacing and its precise location within thearray. We shall show how this disturbs the symmetry ofthe copolar radiation pattern and markedly degrades the

Paper 3097H (Ell), first received 11th November 1983 and in revised form 9thFebruary 1984

The authors are with the Department of Electrical & Electronic Engineering, QueenMary College, University of London, Mile End Road, London El 4NS, England

crosspolar radiation pattern, a feature which is crucial indual-polarisation satellite systems. In the aforementionedstudy at Queen Mary College, Bird [3] investigated thecharacteristics of a seven-element circular waveguide arrayembedded in an infinite ground plane. He utilised amethod originally proposed by Hockham in order toevaluate the mutual coupling between the array elementsfor the specific case of TE-TE coupling. We have extendedthis study in the following ways: first, we account for coup-ling between any of the modes which can exist within thehorns, including those modes which are reflected in thehorns and those which are evanescent in the horn aper-tures. Secondly, the software provides for arrays of any sizelikely to be encountered in planned spacecraft technology.

Fig. 1, which is taken from Reference 3, shows identical

Fig. 1 Planar array with the radiation field point P and the source pointP' in the horn aperture

horn elements in an infinite perfectly conducting groundplane. In practice, the elements will be lodged in a finiteground plane, but the common use of parasitic elements atthe periphery of the array minimises the effect of the infin-ite ground-plane assumption. The present theory also givesuseful insight into the behaviour of feed clusters where noground plane is present. Comparison with unpublishedresults of other workers has shown that levels are in closeaccord with our predictions, although the positions ofcrosspolar peaks are slightly shifted. We shall now providean outline of the theory, further details of which are to befound in Reference 3.

The magnetic-field components at a distance R from theorigin in a plane transverse to the z-direction are related tothe aperture electric field through the identity:

Ht(R) =

dS'Et{R',O)G(\R-R'I) (1)

IEE PROCEEDINGS, Vol. 131, Pt. H, No. 3, JUNE 1984 165

where \R-R'\ = y/(x - x')2 + (y - y')2 + z2, G(R) =exp (— jk0 R)/R is the free-space Green's function and Vf isthe transverse gradient operator. k0 = co/c is the wave pro-pagation constant in free space. As Et vanishes on theground plane, the infinite integral reduces to a sum of inte-grals over the N horn apertures D, (i = 1,2,..., N).

In horn / the tangential fields are given approximatelyby the M mode expansions:

(2)

x z (3)

where hm is the transverse magnetic field of mode m andZm is the wave impedance of mode m, which is given by:

La L mm=l

M

) Z,m\_dm

JVmZ - b(n)ejymZ

e-JVmZ + b(i)eJ

kolym TE-modesyjko sr TM-modes (4)

Co is the intrinsic wave impedance of free space and er isthe relative dielectric constant of the material filling thehorn, although, in practice, this value will normally beunity. ym = fim = jocm is the propagation constant of modem. Mode m is associated with the mth highest transversewave number, or eigenvalue, km, when the wave numbersof the TE- and TM-modes are arranged in order ofascending magnitude.

When the field point lies on the plane z = 0, eqn. 1becomes an integral equation in the field componentstangential to the array aperture. This equation is a con-straint upon all possible trial solutions to the aperturefields. One possible set of trial solutions are the fields ofthe orthogonal modes inside the waveguides. We havefound that the inclusion of 16 pairs of modes is sufficientto achieve very high accuracy at coupling levels as low as- 4 0 dB.

By using the Galerkin procedure, as in Reference 3, A(J]

and Bl£ are obtained from the following matrix equation:

ttx tnn n V /

where A{J] = a{n

j) + b{j] and B^j = a{£ — b(£.Appendix 7 gives complete expressions for C^

(elements of the admittance matrix) supplementing thosegiven in Reference 3 for Cj^, the diagonal elements of thematrix.

The crosscoupling integrals of Appendix 7 involve tripleproducts of Bessel functions or their derivatives; however,they may be evaluated by contour integration in a similarmanner to that described in Reference 3 provided thatterms Jq + p(k0RijO)) are rephrased as the sum of Hankelfunctions of the first and second kind.

For conical horn elements, account must be taken ofhigher-order modes which are excited in the aperture butwhich are cut off before reaching the throat. These reflec-ted modes not only alter the aperture field directly, theyalso cause further mutual coupling, which must beincluded in an accurate representation. Fig. 2 shows, sche-matically, coupling between modes in two horns of thearray. Generally, nearest-neighbour coupling is strongest,but our analysis and software allow for coupling extendingbeyond nearest neighbours. The strength of the coupling isqualitatively indicated by the number of lines forming the'bond'; loops signify propagation part of the way into ahorn of a mode which is cut off in the horn throat. Thesemodes have an associated phase factor which can be mod-elled for small flare angles by the addition of two terms.One corresponds to the phase change at the cutoff point,

+ 7i/2 for TE-modes and — n/2 for TM-modes; the otherto the integrated phase change along the horn (D, where

0 =2Xm(j3 - tan jg)

tan a(6)

horn 2

horn 1(TEn excited)

Fig. 2 Mode coupling between two adjacent elements in a conical hornarray

—»— weak* strongp very strong

and

j9 = c o s ^ka

Xm = characteristic root of modea = radius of horn aperturea = horn flare anglek = 2n/X

Once the aperture field is established, the radiation patternis obtained using the Fourier transform model, which isknown to be accurate for an embedded array.

3 Results

3.1 Coupling between a pair of hornsInsight into the physics of mutual coupling is provided byconsidering first coupling between two horns as in Figs. 3Aand B. The choice of horn parameters corresponds tothose used in an array for which experimental data isavailable. Fig. 3A shows the scattering coefficient of coup-ling between TE^-modes in two horns as a function ofhorn radius with the apertures nearly touching. The resultsare obtained by including all significant propagating and

-20

-25

-30

-35

"-A5

-50

-55

-80

E-planeE-plane(single mode)H-plane

0.3 0.4 0.5 0.6 0.7radius, wavelengths

0.8

Fig. 3A E-plane scattering coefficient for TExl-mode coupling betweentwo adjacent horns as a function of radius of aperture aElement spacing = 2.1a

166 1EE PROCEEDINGS, Vol. 131, Pt. H, No. 3, JUNE 1984

evanescent modes in the computation. For comparison,results are also shown when higher modes are not

horn depends quite critically on the horn radius in wave-lengths.

0.30 035 0.40 0.45 0.50 0.55 0.60 0.65 0.70radius, wavelengths

Fig. 3B E-plane scattering coefficients for T £ u - to TEl{-, TM0l- andTE2l-mode, couplings between two adjacent horns as functions of radius ofapertureElement spacing = 2.\a

included. For the most part, neglect of higher modes leadsto an optimistic level for the mutual coupling. For the£-plane case, coupling falls monotonically until the hornsupports the TMn-mode, when a further increase in hornradius increases the coupling. In the //-plane case, thetransition occurs when the TE12-mode is supported. Fig.3B enables comparison to be made between coupling inthe £-plane between T E n — TE2 1 , TE1X — TM0 1 andT E U — T E n combinations. Evidently, for horn sizes usedin practice, i.e. with radius about 0.5A, coupling to theTE2i-mode is 20 dB stronger than coupling to theTE, ,-mode and 8 dB stronger relative to the TM01-mode.For the //-plane configuration, the level of coupling to theTE2,-m°cle is similar to that for the £-plane, although thefield orientation is rotated by 45°; for the TM01-mode thecoupling is zero as a consequence of the relative field con-figurations. In Reference 6, we have compared the aboveresults with those obtained by Bird using an asymptoticapproximation for the coupling coefficients. There it isshown that the agreement lies within about 1 dB for theT E n — TE21 case provided the horn radius is less thanabout 0.6A; for larger values the agreement is poor, withthe exact method predicting much stronger coupling.

3.2 Radiation patterns of array feedsTo establish a basis for comparison of the radiation pat-terns of various array feeds, we first show, in Fig. 4, thecopolar and crosspolar contours due to radiation from anisolated horn immersed in a ground plane and excited inthe TEn-mode. The results are obtained by taking theFourier transform of the electric field over the array aper-ture. The method has been compared with other exactmethods by, for example, Hombach [10], where it hasbeen shown to be very accurate. Neglect of the phase cur-vature in the aperture of a 5° horn has been justified. Thechoice of horn radius is identical to that of the horns usedin arrays constructed by colleagues at ERA TechnologyLtd., thus allowing comparison to be made between pre-dicted and measured results. As can be seen from Fig. 4b,the maximum level of crosspolarisation is very low, a con-sequence of a deliberate choice of horn radius. Elsewhere[11] it is shown that the crosspolarisation of a circular

-30

-45* 45*

Fig. 4 Predicted linearly polarised radiation fields from a single isolatedconical hornRadius of aperture a = 0.51/Flare angle = 5

copolarcrosspolar

Contours on 0 and <p grid. Scale depicts angular displacement in any <j> directionfrom boresight

Figs. 5 and 6 show, respectively, the predicted and mea-sured copolar and crosspolar radiation contours for aseven-element array feed with the central element excitedby the TE^-mode, and with orthogonal orientations of theelectric vector, a and b. The measured patterns are fromReference 4. Generally, the agreement between the resultsof Figs. 5 and 6 is good. The principal differences inpattern shape are attributed to diffraction at the edge ofthe ground plane in the experimental array. The differencebetween the two polarisation directions stems from therotational asymmetry of the array and the effect this hason mutual coupling.

To interpret the results of Fig. 5, we recall from Fig. 4that the crosspolarisation of the isolated element only risesabove —40 dB in the 45° planes at observation angles wellremoved from boresight. By contrast, the crosspolarisationof the immersed element rises above — 22 dB quite close tothe boresight, which is due almost entirely to the effects ofTE1X — TE2 1 and T E n - T M 0 1 mutual coupling. BothTE21- and TM01-modes are inherently rich in cross-polarisation; thus, their presence strongly disturbs thecrosspolar pattern symmetry. Many designers have sur-rounded the seven-element cluster with rings of identicalparasitic elements in order to reduce mutual couplingbetween subarrays and to reduce edge effects.

Fig. 7 shows the influence on crosspolarisation of suchelements, which are assumed to contain perfectly matchedloads. On comparison with Fig. 5a we see that the cross-polar lobes are slightly increased and their position hasshifted. Thus, an assumption sometimes made that onlynearest neighbours contribute to the mutual couplingprocess, is clearly not entirely valid.

IEE PROCEEDINGS, Vol. 131, Pt. H, No. 3, JUNE 1984 167

Next, we turn our attention to so-called clean-beam crosspolarisation. Fig. 8 shows a seven-element clusterexcitation of arrays. Such arrays have been used in space- with a 12-element passive guard ring. The central elementcraft antennas in order to reduce sidelobes and to improve is excited in phase with and +10.5 dB above the six sur-

rounding elements of the inner cluster. The array factorserves to suppress the crosspolarisation peak by about

• 4 5 C

a - 4 5 C

a+45°

-45 "

b

• 4 5 * -45°

b

-34

• 4 5 °

Fig. 5 Predicted linearly polarised radiation fields from an immersed Fig. 6 Linearly polarised radiation fields from an immersed conicalconical element element (measured by ERA Technology Ltd)Element spacing = 2.01a Element spacing = 2.01aFor details of element size and figures see Fig. 4 For details of element size and figures see Fig. 4

copolar copolarcrosspolar crosspolar

a Vertical polarisation a Vertical polarisationb Horizontal polarisation b Horizontal polarisation

168 IEE PROCEEDINGS, Vol. 131, Pt. H, No. 3, JUNE 1984

- A 5 C • 45°

Fig. 7 Predicted linearly polarised radiation fields from an immersedconical element in a 19-element cluster

Element spacing = 2.01aFor details of element size and figures see Fig. 4Vertical polarisation

copolarcrosspolar

/^\% VA--' A'

,-40 -20

-45C.45°

- 4 5 * 0" *45U

IEE PROCEEDINGS, Vol. 131, Pt. H, No. 3, JUNE 1984

Fig. 9 Predicted linearly polarised radiation fields from a clean-beamcluster without mutual coupling

copolarcrosspolar

For details see Fig. 8

5 dB, and there is a corresponding improvement in copolarpattern symmetry. As this array is one of the most impor-tant used in spacecraft antenna design, it is worthwhile tocompare the patterns of Fig. 8 with those of Fig. 9, wherethe effect of mutual coupling has been intentionally sup-pressed from the prediction process. Clearly, mutual coup-ling has a significant influence on the radiation patterncontours, especially crosspolarisation, and it must beincluded in the design process.

From our earlier study in Figs. 3A and B of the depen-dence of crosspolarisation on horn aperture size, it is to beexpected that this parameter will also influence the cross-polarisation of a clean-beam array. Fig. 10 shows that thepeak crosspolar level falls below — 32 dB when the radiusexceeds about 0.52. Higher levels may be encountered inmore complex arrays when element re-use is beingdeployed.

Finally, we turn to an example of a 29-element arraycontaining three seven element cells which, in conjunctionwith a reflector, could be used to generate three spotbeams or, with suitable excitation, an elliptically shapedbeam such as could be required for a European coveragezone. The element dimensions and excitation levels areshown in the inset of Fig. 11 for the elliptical beam case.

Fig. 8 Predicted linearly polarised radiation fields from a clean-beamcluster with mutual couplingElement spacing = 2.0laFor details of element size and figures see Fig. 4

— copolarcrosspolar

Inner ring of elements is excited in amplitude 10.5 dB below the central element.Outer ring elements are terminated in matched loads

169

This Figure shows the copolar and crosspolar contours,and, within the 10 dB copolar contour, the crosspolar levellies close to —40 dB. Furthermore, crosspolarisation iseffectively suppressed by the array factor in the vertical

-20

-25

-30

I "35o

-40

b -45

-400.40 0.45 0.50 0.55

radius of aperture, wavelengths0.60

Fig. 10 Peak crosspolar variation as a function of aperture radius for aclean-beam cluster

For details of dimensions see Fig. 8

- 4 5 ' • 4 5 s

Fig. 11 Predicted linearly polarised radiation fields from three overlap-ping clean-beam clusters

a = 0.55AElement spacing = 1.2/Flare angle = 5°For details of figure see Fig. 4

copolar— crosspolar

Overall excitation levels (vertical polarisation):a = 0 d B c = -8.01 dB/ > = - 1.925 dB d= -14.03 dB

All excitations are in phase, and the outer ring of dummy elements is terminated inmatched loads

plane. However, it should be noted that the above resultsare displayed at 11.2 GHz, which is in the middle of the10.7-11.7 GHz range. Because of the rapid increase incrosspolarisation with decreasing horn aperture size (inwavelengths), deterioration in performance by about7.5 dB occurs at the lower band edge where the elementradius is 0.52A.

4 Conclusions

The effect of mutual coupling in a conical horn array hasbeen analysed quite generally by extending a methoddeveloped originally by Bird. It is shown that coupling tohigher modes, especially between the T E ^ - andTE2i -modes, significantly increases the level of cross-polarisation and causes distortion of the copolar pattern.The effect depends quite critically on the horn aperturesize. For example, in a typical case, a 20% increase inaperture size causes a 15 dB suppression in the maximumlevel of crosspolarisation. However, in practice, large hornsizes may be precluded because of the overall array size,but, were they to be used, other effects would almost cer-tainly dominate over mutual coupling in determiningcrosspolar levels.

Although in the interests of brevity results have beengiven only for linearly polarised excitation of the arrayfeed, we have also studied circularly polarised excitation.Because of the rotational asymmetry of the seven-elementcluster, contour patterns are different, but the levels ofcrosspolarisation are nearly the same.

5 Acknowledgment

The authors gratefully acknowledge discussions with T.S.Bird and R.C. Brown on the subject of this paper. One ofthem (S.M. Tun) thanks Queen Mary College and the Uni-versity of London for scholarships enabling him to under-take the research reported.

6 References

1 BIRD, T.S, BOOMARS, J.L., and CLARRICOATS, P.J.B.:'Multiple-beam dual-offset reflector antennas with array feed', Elec-tron. Lett., 1978,14, pp. 439-441

2 BIRD, T.S., and BOOMARS, J.L.: 'Evaluation of focal fields andradiation characteristics of a dual-offset reflector antenna', IEE Proc.H, Microwaves, Opt. & Antennas, 1980,127, pp. 209-218

3 BIRD, T.S.: 'Mode coupling in a planar circular waveguide array',ibid., 1979,3, pp. 172-180

4 ADATIA, N., CLAYDON, B., and BRAIN, D.: 'Primary feed ele-ments for multiple and contoured beam satellite antennas', Publ.ICAP(York), 1981, pp. 98-103

5 ADATIA, N , BRAIN, D., PHILIPPOU, G., and RINOUS, R.: 'Atheoretical and experimental investigation of a multiple primary feedreflector antenna suitable for contour beam applications'. IEE Conf.Publ. 219, 1983, pp. 6-11

6 CLARRICOATS, P.J.B., PARINI, C.G., and TUN, S.M.: 'Conicalhorn array feed performance'. IEE Conf. Publ. 219, 1983, pp. 195-199

7 CLARRICOATS, P.J.B., TUN, S.M, BROWN, R.C, and PARINI,C.G.: 'An array fed offset-reflector for spacecraft antennas'. Proc.International URSI Symposium, Spain, 1983

8 CLARRICOATS, P.J.B., TUN, S.M., and PARINI, C.G.: 'Primaryand secondary patterns from array feed offset reflector antennas'.European Microwave Conf. Publ. 1983

9 CLARRICOATS, P.J.B., TUN, S.M, and BROWN, R.C: The per-formance of offset reflector antennas with array feeds', IEE Proc. H,Microwaves, Opt. & Antennas, 1984, 131, pp. 172-178

10 HOMBACH, V.: 'Radiation from flanged waveguide'. Proc. Interna-tional URSI Symposium, 1983, pp. 77-80

11 RUDGE, A.W., MILNE, K, OLVER, A.D., and KNIGHT, P.:'Handbook of antenna design' (Peter Peregrinus Ltd, 1982), Vol. 1,pp. 392

170 IEE PROCEEDINGS, Vol. 131, Pt. H, No. 3, JUNE 1984

7 Appendix: Coefficients of the admittancematrix Cmn

7.1 TE'- -> TE'-mode coupling

CL = Nm lK + (xmxf

<oj\ - (D2 J'p{K(D)J'a{Kco)Gmn - Jo (co2 - u2)(co2 - u2 +

7.2 TNI'- -> TM'-mode coupling

rU _ -na3Keryny'my'n

(D3JJKco)JJKco)*-^wwtn

Jo yi-co2(a,2-u2)(tt2-«2r™ ""

7.3 TE'- —> TM'-mode coupling

O)JJK(D)J (Kco)

7.4 TM'- —» TE'-mode coupling

m" N

(x2 — p2) TE-modes

2

TM-modes

dco

£ 0p —

1 p = 0

2 p > 0

ith aperture: 'm' mode with p azimuthal period)th aperture: V mode with g azimuthal period

/c0fl>^j) cos [(p

J cos [(p - fl)0

k0coRJ sin [(p

d sin [(p - g)0

- <An - <A J

= o if ^ = o

i/̂ m = polarisation angle defined relative to the y-direction for TE-modes and x-direction for TM-modes

(jyjj = angle between the line joining the centres of jthand jth apertures and the x-direction from the jthaperture

R.j = distance between the centres of ith and jth aper-tures

a = radius of apertureK = koa

xm = Xma

ym = JP(xm)y'm = J'p(Xm)

"m = XJK

The scattering matrix is given by S = [/ + C]~l[I — C]where C is the admittance matrix.

IEE PROCEEDINGS, Vol. 131, Pt. H, No. 3, JUNE 1984 171