errors predicted in pyramidal horn antena.pdf

8/14/2019 errors predicted in pyramidal horn antena.pdf

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IEEE TRBSSACTIONS O S AKTENNAS AliD PROPAGATION, VOL. AP-21, KO 1, JAKUARI 1973

Errors in the Predicted Gain of Pyramidal HornsEDWARD V. JULL

Abstract-The concepts of the geometrical heory of difhactionare use d o derive th e on-axis gain of two-dimensional E planesectoral orns.Geometrical optics and single noninteraction)diffraction by the aperture edg es yield essentially the Kirchhoffresult-a monotonic gain versus wavelength curve. Reflection ofdiffracted fields from the horn interior and double diffraction atthe aperture add an oscillation to this curve which is not signifi-cantly alter ed by further diffraction for moderat e o large horns.Including these esul ts approximately in Schelkunoffs equationfor the pyramidal horn explains the gain variations observed nmicrowave gain stand ards and provides an error estimate in theirpredicted gain.

I. IKTROUUCTION

A SERIES of opt,imum pyra.mida1 horns, designed an dcalibrated by Slayt.on [l], are widely used as micro-wave gain tandards. Slayton report.ed [[small, houghdefinite, periodic wiggles in his measured gain versuswavelengt,h curves about. th e monotonic curve predictedby Schelkunoffs gain expression [ a ] . Lat>er measurementson horns based on Slaytons designs have confirmed theseobservations [3], [4]. The object. here is to account. fort.he ga.in oscillations, a t least. qualit.atively, by employingth e c0ncept.s of th e geomet.rica1 heory of dif fraction [>An earlier attempt. to do this [SI was incomplete.

It is impossible to analyze he pymmidal horn rigor-ously. Any modal field repre sen tat Jim s at. once approxi-mate as the adjoining horn walls do not constitute a pairof orthogonal surfaces. This alone may not. cause seriouserror, but. if ray diffraction is used, an d t,hat. app ear s tobe th e most promising approach, all but) th e ra ys a t th ecent,er of th e ape rtu re are obliquely incident on the edgesand, aft er reflect.ion in ,he horn, he structure of thediffracted field is exceedingly complex. To avoid t.hisimpa.sse we observe t ha t Schelkunoffs equation accurat.elyrepresents t he monotonic gain component., and th at i n itt.he effect s of diff raction by t,he E- an d H-plane edges oft.he horn are separable [ 2 ] . In t.he t,erminology of th e

geometrical t.heory of diffraction, Schelkunoffs expressionincludes the geometrical opt.ics field of th e horn an d thesingly diffract.ed fields from the aperture edges. It, omit,smultiple diffract,ion an d diffracted fields eflect,ed fromthe horn nt,erior. To a first approximation, diffractionby the E- an d H-pla,ne edges of the horn can be onsideredseparakly . &o, as excitat>ion of th e apert.ure edgesparallel to th e incident elect,ric field is by rays first dif-

Manuscript received Ma y 16, 19i2 ; revised August 14, 1972.The uthor was with the Radio nd Electrical Engineering

Division, Nati onal Research Council of Canada , Ottawa, Ont.,Canada. He 1s now with the Depart ment . of Electrical Engineering,

University of British Columbia, Vancouver 8, B.C., Canada.

fract ed at th e horn-waveguide junction, t.hese diffract.ed

fields ar e weaker th an those of t he edges parallel to t.heincident magnetic field. Consequently di f f r ad o n by onlythe lat ter is considered in detail here.

We are t,hen led to consider t.he on-a.xis gain of a tx o-dimensional E-plane sectoral horn. This simpler st.ructureyields to a ray diffract.ion analysis; previous studies havesuccessfully predicted th e radiation pattern [7] and reftion coefficient [SI of long horns. Th e analysis revedouble diffraction in the horn produces gain oscillat,ionssimila,r to t>hose observed in pyramidal horns of compa-rable dimensions. Rhen t.his double diffmction by hH-plane edges is included in an approx imak way i

Schelkunoffs gain formula, t,here is some improvement inagreement wit.11 experiment. at . high frequencies an d t.hegain oscillations are partially a.ccounted for.

11. FAR FIELDS F MAGNETIC INE SOURCECONDUCTIKGALF-PLAKE

By using the exact. solution for t ,he far field of a mag-netic line source near a conducting half plane, singularitieon shadow boundaries occurring in a. direct application ofth e geometrical t.heory of diffraction a re avoided. I nFig. 1 (a) the far field at r.,Oof t he source at. ra,Oo n ist,ion is

H,inc(r,O) = ;)*exp ( - j a>HO@)k p )

where k = ~ P Xs the frecspace propagation consta.nt.If t,his source is parallel t,o th e edge of a conducting half-plane, here is a reflected fa.r field in 0 < 0 < ngiven by ( 1 ) with 0 + eo replacing 0 - eo, and ne < 2n a diffract.ed far field

where

diffraction have also been described [9], bu t because th e axis1 Gain calculations of conical horns by t.he geometrical theory of

caust,ic of the diffracted rays of a circular ape rtu re, i t is impossible

merical results which were presented seem to be identical withto calculate the on-axis gain from the expressions given. The nu-

values obtained much earlier [lo]by the Kirchhoff method.


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6 IEEE TRANS.4ePIOKS ONAYTENXAS AND PROPdGATIOW, AKUARY Icji.3

C)

g. 1. (a) Coordinates for magnetic l i e source near a conductinghalf-plane. @) Two-dimensional E-plane sectoral horn formed byconduct.ing sector with magnetic line source a t apex. R ay path sof on-axis geometrical optics and noninteract.ion aperture diffractedfields are shown. (c) Ray path of upper edge diffracted fieldsreflected in forward direction from horn interior.

with

r0 ,a) = - exp jk rocosII

+ r / 4 )

- - exp ( jk rocos (II)2

nd

When t.his line source is on the conducting half-planeat ro = from the edge, th e source field

produces a diff rac ted field

( 7 )

111. ON-AXISFIELDS F TWO-DIMENSIONALSECTORAL ORN

,4. ,VoniuteractionFields

A simple model of a two-dimensional E-phne sectoralhor n is a magnetic line source a.t th e apex of a scct.or offinite length I formed by two half-planes int,ersecting atan angle 240, as in Fig. l ( b ) . I n addit.ion to t,he sourcefield of (6) an d t he diffracted field of t he upper edge (7) ,there s a d iff rac ted field from ,he lower edge identicalin the forward direction 0 = T - Co because all fields aresymm etri cal abou t the horn axis. Neglect,ing int.rract,ionbetween the edges and walls of t he horn, the on-a,xis fa.rfield is

H I = H Z i ( r l r +,,) + 2 H Z d ( r , n- +,,)

where

yielding, Jvithout further approximation, (9) with =b/ 2XZ cos 6 ) is. For sma.11 40 th e t.wo results a rc esscntiallythe same, so th e Iiirchhoff solution contains gcometricaloptics and noninteraction singly diffracted fields of t,heapert,ure edges. Whereas \tilth the Iiirchhoff method,

further improvement s not possible, th e concepts ofgeometrical diffraction theory [ 5 ] may now be used toinclude in ka ct io n between the edges and \ d l s of thehorn.

B . R e f i c t i o n sf r o m Horn Inter ior

Th e most significant interact,ion will arise from singlydiffracted fields which reenter the horn and are reflectedin the forward direction, as shown in Fig. 1 (c). The pathof edge diffracted r ays reflected from t.hc horn nteriorma.g be traced by loca,ting th e positions of t,he edgeimages. This was done by Yu et ai. [7], who a.lso deter-


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JCLL:S I S PREDICTEDAIN OF PYRAMIDALORNS

mined t.he angular regions in which these image sourcescontribute. Each edge has ? I images, where 117 is the largest,integer less t,ha.n ~ i 2 4 ~ ) .he mt.11 image source con-tri but es to the on-axis far fields if 0 = s - 2 m + l )is positive. I t can be shown from the geometry of Fig. 1 (e)that t.his must. be the direct.ion for singly diffracted raysfrom the edge which proceed in the forward directionafter JH reflections in the horn nterior. The distance alongthis rag path from the edge to the apert ure plane is

m

s = 22 sin 40 sin 2i40).

Adding 2 H z d ( r+ sm,emf) to S) , he on-a.xis far fieldbecomes

i= l

HI =exp -jkr)

( k r ) /2{exp ( - j k l cos &) + 2v 2,s - 40)

+ 2 exp (- jk-sm)v[l ,s - 2m + l )40]} . (11)This reflection can occur for all orders of diffract.ed field,as discussed in the next section.

C. M u l t i p l eDi l.adion

Doubly diffracted fields ar e produced by singly dif-fracted rays from the ape rtu re edges which proceed int,he direct.ions e ; = ~ / 2 40, i = 1,2, ---,m. For i = 1t,hese rays traverse the apert,ure and ar e oubly diffracteda t th e opposite edge, (see [S, fig. 01, where i = n + 1 ) .For i > 1 the rays are reflected i - 1 t,imes in the horninterior before being diffracted again at th e ame i ven)or 0pposit.e i odd) edge. The ay-path length betweensingle an d double diffraction is i = 21sin (i J0)nd t,heangle of incidence for double diffraction is ei. Assume the

edges ar e nou- isotropic magnet.ic-line sources H z d r,Bi ) ,defined by (7) . Since th e on-axis fields of t.he tn-o edgesar e equal, their doubly diffracted fields in the B = s - 40direction are

where

Double diffract.ion a t t,he horn waveguide junct,ion is notincluded here because its cont,ribution t.0 t.he on-axis fieldis very small, as discussed in Appendix I.

Doubly diffmcted fields reflected from the interior ofthe horn without int,errupt,ion into the forward directionunder th e condit.ions of Sect.ion III-B may be accountedfor by adding t o (13)

Szr = cxp ( - ks , )

Triply diffracted fields ar e produced by doubly difracted fields from .hr apert,ure edges which proceed inthe directions Bi. Again th e ray-pa.th 1engt.h beh-eendouble and triple diffract ion s d i an d for t.riple diffract.ionth e angle of incidence is Oi a.nd angle of diffraction s -for the on-axis far fields. The riply dif frackd fields oft,he two edges in the forward direct.ion ma?; bc writ tcn

H =2 exp ( - j k r )

(k r ) / ? stwhere

To include trip ly diffracted fields reflected in the for-ward direction, when they oc.cur,

is added to (16).Since v(2,a) O [ k l ( l + cos 0r)]-1/2), k l ( 1+ COS a >

higher order dif fract ed fields will be successively smaller.The numerical results which follow indica,te t,ha t single

an d double diffraction suffices for most, horns.

IV. GAINOF TWO-DIMENSIONAL ORN

The 4 component of incident elect.ric field in the horn is

Power in he incident mode is obtained by ntegrating+Re I? x Bi*. ) where the ast.erisk denotes complexconjugat,e over asurface po < p < in he horn. Thincident power per uni t 1engt.h in the z direction of thhorn is


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JCLL: ERROR3 S PREDICTED G A I S OF PYKAMID.4L HORNS

AiCtul)

Fig. 3. Gain of 10-cm band pyramidal horn. a = 32.41 cm,a = 7.21 cm, b = 24.00 cm, b = 3.40.cm, l a = 42.15 cm,

= 47.45 cm, = li.78 cm. From [ l ] : X measured, caled.From 141: measured. Monotonic curve: from (26). Oscillatingcurve: (26) wiith (30).

where th e Fresnel ntegrals, defined by 5 ) have .hearguments

25)

Both RE and RH have been tabulated [ll].These expressions include a. reliable approximation o

th e geometrical optics field and t.he singly diffra.ct.ed ieldsof t.he apert,ure edges not reflected from t he horn interior.RE contains singly diffracted fields of the edges parallel

to t.he incident magnet.ic field (H-polarized fields) and RHthe corresponding d iff rac ted fields from th e edges parallelto he incident electric field (E-polarized ields). Withinteraction betu-een the edges and x\-alls of the horn E-an d H-polarized fields ar e inseparable, ut implerevision of these expressions based on physical considera-ti om may be made. For t.he H-polarized fields the edge-wall interac tion which affects t.he on-axis fields must occurprimarily i n t,hat portion of th e horn produced by flaringthe waveguide in the E plane only, because of Fermat.sprinciple for diffracted rays [S. Only those fields fromt,he waveguide normally or nearly normally incident. uponthe ape rtu re edges will be diffracted in directions permit-ting reflection or double diffract,ion in to he forwarddirection. Di ffrac ted ray pat.hs of the obliquely incidentrays will lie on a cone with t,Be edge as axis and verylit,tle of these fields will ult,imately ont,ribute to heon-axis radiation intensity. If all relevant. reflection an ddouble diffraction of th e H-polarized horn fields is assumedto occur only in the region x < a / 2 of Fig. 3, then thesefields must be multiplied by

Comparing 21) with (24) a.nd taking into account. t,hepreceding considerations, a revised fa.ctor R E to be uin (26) is

exp ( - j k l cos 40) + 2a(l ,n - o) +a

in which (1 + cos ) / 2 appears rom using (10)I = ZE for zv, rat.her than (27) . The final term naccounts for double diffraction of th e H-polarized fields.If diffracted fields are reflect.ed in the forward dirediont.hen

is used where a Z,cy) , 52: and S2r are defined by 4) (and (14 ) respectively.

DiffractcJd fields of the aper tu re edges parallel to th einciden.t electric fieid (E-polarized fields) a re initiallydiffracted at. t,he horn-waveguide junct ion and conquen tly arc weaker tha n t.he corresponding H-polarizedfields. At t.he l e d of approximat.ion considered here,reflection and double diffract,ion of these fields may becmittcd. A pyramidal horngain expre-ssion containing

first-order edge-wall interac tion of only th e H-polarizedfields is 2 6 ) with RH defined by (28) and R E byor (31).

VI. EXPERIMENTALND XUMERICAL ESULT

The preceding expressions were used to calculate t.hegain of three of Slaytons horns [l]. Other horns in thiseries arc scale models of these examples. Slayto n pre-sented experimental results for his horns and other rmultshave since bcen reported [3], [4]. The measured gavalues of Figs. 3-5 ar e for matched horns, 11-hereas thecalculated values nclude dominant mode nlismstch a t

the aprrt.urr? but thc eflection coefficients in Appendixindicate his actor is negligibly small here. Equation(26) u-ith (30) is represented by t.hc oscillating curve forthe 10 cm band horn of 1;ig. 3, while Schelkunoffs expres-sion, (26j with 22) , yields the curve which decreasesmonotonically with wavclength. This horn has the sameE-plane dimensions as in 1:ig. 2 and the gain oscillationproduced by doubly diffra.cted H-polarized fields in hehorn is contained in (30). It does not accurately coincidewith t.he observed gain oscillations for t,his horn, evallowing for experimental error in the mea.surements. Thecauses of this discrepa,ncy are discussed lat.er, but: a t least


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0 IEEE TRAKSACTIOSS O N AXTEliN.48 AND ROPAGATIOK, ANUARY 1973

I

I I \ A3.6 3 8 4 0246 48 5 0 5 2

k C M I

g. 4. Gain of 4.75-cm band pyramidal horn. a = 28.85 cm,

l = 50.84 cm, I = 8.90 cm. from [I], from [4]. Monotonica = 3.49 cm, b = 21.37 cm, b = 1.58 cm, 1~ = 47.50 cm,

curve: from (26). Oscillating curve: (26) with (31).

25 0

24 0 -, , , ;

3

17 1 8 I 9 2 0 2 1 2 23 2 4A CM1

g. 5. Gain of a 1.80-cm band pyramidal horn. a = 15.20 cm,a = 1.58 em, b = 12.47 cm, b = 0.79 cm, 1~ = 34.63 cm,Zg = 36.41 em, I = 7.62. Measured values: X from [l ], from[4]. onotonic curve: from (26). Oscillating curve: (26) with (30).

n approx imat ion to t.he gain oscilla6ion app rars to havcbeen achieved.

Bett.er agreement should appea r a t higher frequrnciesnd this seems to be evident for the 4.75 cm band hornf Fig. 4. Here bhe conditions of section 3 .2 apply and

RE defined by (31) has been used in (26). The singlyi ffract ed fields reflected from t.he horn interior alone canroduce a gain mri at io n of about. f 0 . 0 6 dB a.t X = 4.4 c mn d ar e responsible for most. of t he observed gain oscilla-on, which is of larger magnitude th an would othern-isee expected for a horn with this gain.

Gain oscillations are small for high gain horns [13]nd the 1.8 cm band horn results in Fig. 5 clearly illus-ate t,his reduct.ion in oscillation. While many doublyiffracted fields cont,ribute here m = 8 in ( 3 0 ) ) , theyre weak in high ga.in horns and, as reflection in the for-

ward direction does not occur, th e oscilla.tion in gain ismall. I n this example ohmic losses in th e waveguide feed

.nd horn amount, o about 0.03 d B a t=

2.0 em, accord-

ing to th e calculation in Appendix 11, an d t.his accountsin part for the discrepancy in Fig. 5 . Ohmic losscs amo untto about 0.01 dB and .02 dB in thr orns o f Figs. 3 and 4,respect,ively.

VII CONCLUSIONAND DISCUSSION

I n a t,wo-dimensional E-plane sectoral horn singly dif-fracted fields of t he ape rtu re edges subsequently reflectedfrom th e interior and doubly diffracted produce oscilla-tions in th e on-axis gain versus u-avelengt,h curve. Higherorder multiply diffracted an d reflected fields have ittleeff ect on he gai n of most horns. While the a.nalysis issufficient.ly comp1et.e and accurate or th is two-dimensionalhorn, rat.her arb itr ary assumpt.ions are required to a.pplyi t t o t.he pyramidal horn. The resulting xpression prrdict,sonlF approximately the amplitude, period and phasc ofoscillations observed in the gain versus wavelength curvesof pyramida.1 horns.

Some mprovement may be made hhrough a imilaranalysis of th e two-dimensional H-plane sect,oral horn,leading t.0 a revised RH in 26), but, it is probably moreimportant to account for the differing propagation con-stant. s of modes reflected from the horn nterior. Thisaffect,s t.he period a.nd phase of th e varying gain com-ponent , especially at. longer wavelengths, where the largestdiscrepancies mith experiment appear. The est.imate oft,he region of relevant, edge-wall interaction, which a.ffectsthe amplitude of th e predicted gain oscillat,ions, may alsobr adjusted, but it l ot be easy to properly account forcoupling between ,he E- and H-polarized fields at theaperture.

Clearly an accurate quan titat,i ve description of these

gainvariations n pyramidal horns is dffi cult , if notimpossible, to achieve. 4 new design of antenna gainstandard for microwave frequencies is perhaps a morepractical objective. In th e meantime, errors in he pre-dieted gain of pyramidal horns ma.y be rcduccd by ensur-ing, in th ri r design, that. diffract.ed ields from the apert.ureare not reflected in the forward direction. Equa tion (26)with (30) or (31) accounts approximatrly for the gainvariations, providing a n estimate of their magnitude fora particular horn and? at, thc higher frequencies for somehorns, a more accurate valuc of the gain.

APPENDIX

DIFFRACTION AT HORN-WAVEGUIDE UNCTION

A crude approximat.ion to th e horn-wavcguide junct.ionis the closed apex of a conduct.ing wedge. This was usedby Yu et al. [7], an d may be adequate for pa tt ern ca.lcu-lations, alt.hough t.heir expression for t he diffracted fieldis of doubtful validit,y for these wedge angles [7, eq. 1) 1.These diff racted fields were omitt.ed from th ri r numericalcalculations.

The tangent,ial magnetic and elect.ric do mimnt mode

fields rxcit.ed near the junction by singly diffra.ctrd fields


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JELL: ERROL< IX PREDICTED GAINOF PPHAMIDAL HORNS

from the aperture may be \\-ribten

is th e domina.nt, mode amplit.ude of singly dif fmcted fieldsexcit.ed at. tahe apertu re by (6) , [S, eq. (13) 1. /3 is thedominant mode ampli tude of t,he fields diffract,ed at,p = po, the horn-waveguide unction. Equa ting 3 2 ) and(33) there to the dominant (TEN) mode field producedin t.he waveguide gives

if 40 is not arge. If kl >> 1, a is small and fi

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errors predicted in pyramidal horn antena.pdf