cross coupling in a 5 horn monopulse tracking system

7/27/2019 Cross Coupling in a 5 Horn Monopulse Tracking System

1/7

IEEE TRANSACTIONS ON ANTENNAS AKD PROPAQATION, VOL. AP-20, NO . 4, JULY 1972is a noticeable improvement in the

ern structure a t scan angles near en&e. An isotropict array is used for comparison with the da ta of Loal . [SI and the Hansen-Woodyard condition; however,

e opt,imization of a practical array hould include mutualcts Khich may introduce differences from the

Phase optimization is useful for electronically scanneds when varying the phase with fixed amplitude. Fore case of gain optimization, the improvement is most

near endfire scan.ACKKOWLEDGMEKT

The authors wish to thank Dr. S. AI. Sanzgiri for his

REFEREKCESA. I . Uzkov,.A; approach to the problem of optimum directiveantenna deslgn, Dokl . A M . Nauk SSSR, vol. 53, pp. 35-38,June 1946.E. N. Gilbert and S. P. Morgan, Optimum design of direct.ive

antenna rrays ubject to random ariations, Bell Syst.Tech. J.,vol. 34, pp. 637-663, May 1955.[3] A. Bloch, R . G. Medhurst, and S. D. Pool, A new approachElec. Eng ., vol. 100, pp. 303-314, Sept. 1953.to he design of superdiiective erial &Grays, Pr ic ; Znst.[4] M. Uzsoky and L. Solyrnar, Theory of superdirective ineararrays, Ada Phys., vol. 6, pp. 1.85-204, 1956.151 E. I. Krupitsky, On the maxlmumdirectivity of antennasconsist.1ng of &Creteadiatom, Sov. Phys. Dokl., vol. 7,

[SI Y . T. L o , S. W. Lee, and Q. H. Lee, Optimization of di-no. 3, pp. 257-259, Sept. 1962.rectivity and signal-to-nolse ratlo of an rbi trary ntennaarray, Proc. ZEEE, vol. 54, pp. 1033-1045, Aug. 1966.[ i ] . K. Butler and H. Unz, Beam efficiency and gain optimiza-tion of antenna arrays with nonuniform spacings, Radio Sa..,

[ti] 8. M Sanzgiri and J. I(. But.ler, Constrained optimization ofvol. 2, no. 7, pp. 711-720, July 1967. .the performance ndices of arbitrary array antennas, I E E ETrans. Antmnas Propagat. , vol. AP-19, pp. 493-498, July1971.[9] D. K. Cheng and P. D. Raymond, Jr. , Optimisation of arraydirectiv1t.y by phase adjustments, Electron. Lett., vol. 7, no.18. DD. 562-554. &Dt . 1971.[101T.&.&aaty and J. Bram, Nonlinear YaUlmatics. New York:Ill] R. L. Fox, Optimiration MeUlods for Engineering Design.1121 W. W. Hansen and J. R. Woodvard.Anew Drincide n

McGraw-Hill, 1964, pp. 53-174.Reading, Mass.: Addison-Wesley, 1971, pp. 51-58.. . directional antenna design, P T O C . ~ I R E ,o.. 26, pb. 332-345,Mar. 1938.

113) R. E. Collin and F. J. Zucker, Antenna Theory, pt. I. NewYork: McGraw-Hill, 1969, pp. 156-157.

Cross Coupling in a Five Horn MonopulseTracking System

WILLIAM M . BRIDGE

Abstracf-A model is developed which explains the observedof secondary pattern c ross t a l k in a five horn circularlyion and loss of track for s ac i e n t ly depolarizing targets as

as polarization dependent uncertainties in the angular locationoff-axis targets. Thecross coupling is characterizedby twoin the mono-system. This characterization is assume d o be validof whether the observed econdarypattern effects

result of mutual couplings within the feed structure itselfcaused bycross-polarized backscatter from the paraboloidalA ridge and nonridge polarization terminology ishe error hornsa five horn circularly polarized system ar e heavily ridge loadedinear polarization and not the other.

U I. INTRODUCTION1;IEROUS possibilities exist for cross coupling in afive horn monopulse tracking system. For tmheollow-cross coupling wll be defined as the spu-

Manuscript receivedDecember 2, 1971; revised February 17,The author iswith theMITRE Corporation, Bedford, Mas.01730.

rious coupling of energy from one error channel to another.It includes t,he effects of mutual coupling between adjacenthorns, finite isolation between the orthogonal signals ofeach dual polarized horn, and the inherentcross-polarizedbackscat.ter from a parabolic reflector. In a circularlypolarized five horn system, the desired error signals areusua1l;v obtained by aking he difference between thehorizontal and vertical componentsf the circularly polar-ized input signal and hen combining t.hese H and Vdifference signals in a 3 dB quadrature hybrid toproducethe desired error signal output as shown in Figs. 1 and 2.Consequently the feed basically receives H and V signalsand, in this paper, any cross coupling which exists d lbe considered as between the H and V components of thesignal.

The previous definition of cross coupling is to o generalto be very useful in characterizing a particular network.Isolation is more definit,ive and refers to measurementsa t specific ports of .t.he network (notapertures),Theinsertion loss between aaimut,h ridge (AR) and elevationnonridge (EN) ports of Fig. 1 would be defined as ridge-to-nonridge isolation. Crosstalk, on the other hand, refers


2/7

BRIDGE: FIVE HORN YONOPULSE T R A C K I N G SYSTEM 43

5=$-=$1

Fig. 1. Ridge-to-ridge crosstalk, pedest.al-42> target AZ.

to pattern-t,ype measurements associated with the recep-tion of signals at. the apertures. Crosst.alk may be termedas primary or secondary depending on whether or not, asec0nda.q reflector is included. The following discussionswillbe limited to he results obtained from secondarypat.tern crosstalk measurements performed on an 84 dishwith an F / D rat.io of 0.375 and mounted with conven-tional azimuth and elevation axes.-4.Cross-Polarized Crosstalk

Cross-polarized crosstalk refers to asingle axis (azimuthor elevation) and is a measure of the relative response ofthe H and V channels to a cross-polarized input signal.For example, azimuth cross-polarized crosstalk could bemeasured by observing he level of spurious signal presentat port AN (Fig. 1 or 2) when theant,enna is rot.at,edabout a vert.ica1 axis while it is illuminated by a verticallypolarized source located in the far ield.B . Ridge-to-Sonridg e Crosstalk

This erm refers to crosstalk betxeenazimuthandelevation channels of t,he same polarization. For example,ridge-to-nonridge crosstalk could be measured by observ-ing the level of spurious signal present, in the E N channelm-hen the antenna is rotated and illuminated as describedin Section I-A.

I 1 1 J

Fig. 2. Nonridge-to-nonridge crosstalk, pedestal AZ > target A Z

C . Ridge-to-Ridge CrosstalkThis erm refers to crosst.alk between t,he vertically

polarized azimuth channel a,nd the horizontally polarizedelevation channel. Forexample, this crosstalk could bemeasured by noting t,he spurious signal present in theER channel when the antenna is rotated and illuminatedas described in Section I-A.D . LVmridge-to-Konl.idge Crosstalk

This t,erm refers to crosstalk between the horizontallypolarized azimut,h channeland the vertically polarizedelevation channel. For example, t.his crosstalk couldbemeasured by noting the spurious signal in theE N channewhen the antenna is rotated as in Section 1-8,while it iilluminated by a horizont,a,llypolarized source located inthe far field.It should be pointed out that both ridge-to-ridge crosstalk and nonridge-t,o-nonridge crosstalk could also be obt.ained by rotating t.he antenna about a. horizonta.1 axiwhile illuminating it, with an appropriately polarized farfield source.

In a welldesigned eed, the cross-polarized crosstalkshould beat least 20 dB less than the esponse to correctlypolarizedsignals. In addition, it can be easoned fromsymmetryrguments,hat ridge-t.o-nonridge crosstalkshould be independent of angular offset in the principal


3/7

IEEE TRLLNSACTIONS ON ANTENNAS AND PROPAGATION, JULY 1972

12 -0 0 7

Fig.

RIDGE-TO-RIDGERIDGEVERT-RIDGE

2 3 4 5 6 7 8 9TIME IN SEC. (2 M w / S )3. Vertical azimut,hscan.

-NOWE L

xis and hat, he coupled signals should cancel in heput port of the difference hybrid. This sverified by the3, which includes ridge-to-ridge cross-

high econdary patternof this antenna was substant.iated by primary

and isolation measurements, and may not bentative of other five horn systems. The measure-

of cross-polarized crosstalk and ridge-to-nonridgeto isolate the H and V channels. Ron-ever, having

that t,hese crosstalk levels are negligible,e remaining crosst.alk can be evaluat,ed without, removalthese hybrids.

11. M O D E L DEVELOPMENTThe observed effects of crosstalk can be predicted quite

ly with the a.id of the model shonn in Figs. 1 ands model assumes a ridge-to-ridge coupling coefficient0L 7 ) and a nonridge-t,o-nonridge coupling coefficient

between adjacent horns. Before these c0efficient.sbe evaluated, i t is necessary to determine the phaseants of the system. I n general, the only phase infor-

is the IF phase difference betweenchannel andeither of the twoerrorchannels

- Z) and (AEL - 2 ) . The RF phaseconstantsand +EN can be determined by noting the change info r vertical and horizontal input signalsa boresight source at fixed angular offsets. The IFng of the system BEL, ez, and eAzcan be determined

a constant by noting the asymptotic phase differ-during azimuth and elevation scans. The crosstalk

coefficients 0L y and aL 6 ca.n be obtained fromaxis scans of a linearly polarized boresight sourceshown in Figs. 1 and 2. The following arbitrary con-

shonmFigs. 1 and 2.1) An LC received signal corresponds to the clock-wisetation of electric vector.2 ) A + A Z offset (pedestal A Z > target A Z ) causes

na.1 in the right horn of the diagram and pro-a 90" ( AAZ / Z ) phase.3) -4+EL offset (pedestal EL > target EL) causessignal in hebottomhorn of the diagram and

90" (AEL/Z) phase.4) The inst.antaneous phase of a signal is assumed tozero when t.he electric vector is directed toward the

output porta.nd 180"when directed away from the outputport.5) The direction of the cross-coupled vector is deter-mined by extending the vector representing the signal inthe principal axis about a 90' arc of a circle into the hornwhich receives the cross-coupled energy.6) The feed and its associated components are assumed

to be rotationally ymmetric and ideal with the exceptiont.hat the error orns need not have the ame gain for ridgeand nonridge excitation.

111. DERIVATIONF CROSSTALKQUATIONSThe following general expressions for crosstalk are de-

rived from the assumption that any arbi trary input ignalcanbe expressed in erms of leftrhand and right-handcircular components, and that ach of these can be furtherdivided into H and V components. An LC signal ofmagnitude ( A ) is always assumed present. Without lossin genera.lity a "viewing time" may be chosen when th einstantaneous orientation of this vector is vertical. Theright-circular component is then assumed to haveanymagnitude ( K A ) and an inst,antaneous orientation anglef elative to the LC component, L f = LLC - LRC.The vertical and horizontal components of the left- andright-circular input signals can be expressed as follows:

LC (vertical) A = A LOLC (horizontal) B = j A

RC (vertical) A' = KA L fRC (horizontal) B' = - j K A L \k.

The center reference horn is assumed tohaveperfectcircularity so that it does not respond to the RC com-ponent of the input signal. This assumption is supportedby the act hat t.he measured RC cancellation fromspherical t.argets wason the order of -30 to -35 dB.The reference signal Z of Fig. 1 can now be expressed as

n

Fora positive azimuth offset (pedestal A Z > targetAZ) , the azimuth error signal ( A A Z ) can be expressed

A A Z (RC) = - j ("-") K A L f [ 1 - GL$AR] Leaz(2)

where R and L represent the unequal voltages induced inright and left horns.

The factor G has been introduced to account for thedifference in secondary error pat tern gain between ridgeand nonridge excitation. Using (1 ) the normalized azimuth


4/7

BRIDGE: FIVE HORN MONOPULSE TRACKING SYSTEM 43error,hen becomes and RC component.sit.h an instant.aneous Dhme angl* = 0". Equat,ions ( 3 ) , (4) nd ( 5 ) , (6 ) the; reduce tov5CZL C ) = - R- L ) [ + GL $-4R] L (OAZ - 6,)z 2 2 for +AZ offset (vertical signal, K = 1, = Oo)AAZ v%- RC) = -- R- L )z 2Fora positive azimuth offset the spurious normalized for +EL offset (vert,ical signal, K = 1, = 0")elevation error signal can be expressed as:AEL l 6 -: = j y( B- T ) ~ L [ + E N (eEL- , )] (9AEL fi- L C ) = - R - L )z 2AEL v5- ( R C ) = - R- L )2

The gain factor G, which is principally t.he result, ofdifferent llumination unctions for ridge and nonridgepolarizations, does not appear in the previous expressionsbecause it is assumed that he crosstalk sprincipallycaused by mutual coupling between adjacent error horns.I n any event, the effect of differential gain could be in-cluded in the ma.gnitude of a and 6 .

The analogous expressions for a positive elevation offset(pedestal EL > target EL) are

This condit.ion is particularly useful for evaluating hecoupling coefficients LY L and P L y as shown in Figs.and 2. A similar situation exists for horizont,al signalwhich may also be used for evaluating the coupling co-efficients.

The LC only case is represented by the first expressioof each set. It is intereding to note that in general thmagnitude of the spurious crosstalk relative to the principal error signal is less for LC signaLs than for th e lineaH o r V signal which maximizes the crosstalk. This can beseen from the followingLC ratios for AZ and EL offsetsfor +AZ offset ( L C )

AEL fiz- RC) = j - ( B- T)2AAZ v5z- L C ) = - j - ( B- T)2AAZ d2z RC) = - j - ( B - T )2

(6 )A few special cases are w0rt.h examining in detail. A purevertical signal ca.n be considered as cont,aining equal LC

For G = 1 and ' $ a ~ nd EN negligibly small, the onlyway the LC spurious crosstalk could be as severe as theworst. of the H or 1 crosstalks would be for the ridge andnonridge coupling coefficients to be equal and oppositelyphased (a = a, = 6 + T).If both azimuth and elevation offsets are assumed,a.cerror signal is a combination of a true error signal anda spurious crosstalk ignal. The magnit.ude of t,he spuriouscrosstalk signal dependson the phase constants of thesyst.em as well as the polarization cha.racteristics of theinput signal, and its sign depends on t.he direction of t,hoffset in the otheraxis. Thus for certain offset and polarization conditions, the tracking system could become un-stable and drive in the opposite direction. This stabilityconditiondl be examined only for the case of negligiblesmall '$AR a.nd +EX and G = 1 (equal ridge a.nd nonridgegains).Under hese condit,ions t,he normalized azimuth


5/7

IEEE TRBNSACTIONS ON ANTENNAS AND PROPAGATION, ULY 1972

sin function of the phasee reference signal. Thus

AZ I@@: 2 2s i n e = - ( R - L ) + - ( B - T ) [ ~ s i n 6 - O s i n y

+Kasin ( 6 +e)+ KPsin (y + JE)] (15)L I6 I@= -((B - T ) +-(R-L)[-a.sin6+psiny2: 2 2

= tan-' a cos 6 + p cos ya s i n s + P s i n y ' (17)for 2 values of * differing byone resu1t.s i n a maximum value of crosstalk whileother results% either minimum crosstalk or a maxi-

m vaIue of opposite sense crosstalk. The sense of theis determined by the direction of the off-combined unit offsets in he "left-up" or

mn-right." direction the tracking system may becomefor sufficiently large K. For t.he system to actu-AZ and EL

have the incorrect sign. This meansof t.he smaller crosstalk maxima must ber than t,he true unit error signal.m. X P E R Ih l E h T A LV E R IF IC A T IO N O F A :f OD E L

The model thus derived has been very successful incting the performance of theantenna system for

conditions.The following list gives the measured secondary char-

f this particular five horn antenna system.1) Linear polarizationnull depths referenced 40 dB2 ) Cross-polarized crosstalk referenced to the 23 dB

3) Ridge-t,o-nonridge crosstalk referenced to 30 dB

to t.he sum pattern (sphere).principal error signal.the principal error signal.

4) Ridge-to-ridge crosstalk referenced to he 6 dB5 ) Nonridge-to-nonridge crosstalk referenced 15.5 dB6) Error harnel H-to-V ratio (G), ridge 2.5 dB7) Circularolarizationrosstalksphere) 12 dB.

principal error signal.to the principa.1 error signal.polarizat.ion low.

The derived model characteristics are as follows:Excess azimuth ridge phase = 8".Excess elevation nonridge phase = 5".Azimuth IF phasing = 97".Reference IF phasing = 20".Elevation IF phasing = 0".Ridge-to-ridge voltage coupling coefficient = 4.Ridge-to-ridge coupling phase = 180'.Sonridge-to-nonridge to voltage coupling coe6-cient = A.Sonridge-to-nonridge coupling phase = 255".Secondary error pattern voltage gain differencefor ridge and nonridge excitation = 1.The model --as first evaluated for pure LC returns from

a sphere. Equations 3 ) and (4)ndicate tha t the C cross-talk level should be 12 dB, and thiss precisely the level ofcrosstalk observed. The theoretical crosstalk has also beenevaluated for various linear polarization conditions, usingthe derived phase constants of t he system. The resultsare plotted along -n-it,h the experimental data in Fig. 4and the agreement is very close. The experimental datawas obtained by rotating a linearly polarized horn locatedon a. boresight tower.

The effect of cross coupling on the angular location ofoff-axis targets is graphicallydepicted i n Fig. 5 . Theapparent, target osition for fixed 3 n1ra.d offsets in azimuthand elevation is plot,ted as a function of orientation for alinearly polarized return ( I LC I = I RC I ) . The boresighttracking 1ocat.ionwas first established for the existingphasing. Then, approximately 3 mrad offsets were takenfrom this particular designat,ion point and data were re-cordedwhile t.he boresight linearly polarized horn wasrotated. The data clea.rly indicate the uncertaint,y in off-axis target location for target. returns having equal LCand RC magnitudes.

The effect of off-axis target locat,ion for ot,her polariza-tion ratios has been investigated theoretically using themodel previously developed. For these analyses a ridge-to-nonridge gain difference of 2.5 dB and ideal phasingwere assumed,( G ~ R 0 = GEN), (eAz - e,) = go", (B E L - ,) = 0".

The results are shown in Fig. 6 for four polarizations( L C ,RC = +LC,R C = L C , RC = 2 L C ) , where the crosscoupling is depicted on t.he asis of normalized error volt-ages for unit.offsets. The shape f the t,heoreticalRC = LCcontours is similar to the measured contours shown inFig. 5. The elliptical nature of the contours is caused bythe gain difference between t.he ridge and nonridge po1a.r-


6/7

BRIDGE: FIVE HORN MONOPULSE TRACKING SYSTEN 441v A Z Ix =TffiORETICAL AMP.Q =THEORETICAL WASE* EL CHANNELGAIN'1.2 rb-=MEASURED DATA

I"' !180 x

167.5

,HIGH1"40 4 9 se D Iu. I W 2 W ZZYn m a VERT ncmu.ORIENTATION OFLINEAR NFUTSIGNAL

Fig. 4. Elevation cross coupling fo r +3 mrad AZ offset.t EL OFFSET IN MRAD

2250-A Z OFFSET + AZ OFFSETIN MRAD

2 3 4v35'- 2 1 Y

-EL OFFSET N MRAOFig. .5. Crosstalk for off axis targets n4t.h I LC = I RC I .- HASE = 90NORMALIZED +EL OFFSET

NORMA LIZED- EL OFFSET+ H A S E ' ZWFig. 6. Theoretical crosstalk for ellipt,ical polarizations.

izations. If this gain difference could be reduced to zeroor compensated by the introduction of pads ( 2 . 5 dB) inthe nonridge error lines, t.he contours mouldcollapse tostraight lines. It should be pointed out t.hat t,he center(R C = 0) as well as the size and shape of the contours iscritically dependent on the IF phasing. The dashed linesat 45"/135O are included in Fig. 6 to indicate the point

NORMALIZED EL OFFSET-EL PHASE = 270"Fig. 7. Composite crosstalk for A2 and EL offset, K = 4.NORMALIZED-AZ OFFSET; 4 -1:z -9 $ NORMALIZED+ A2 OFFSET

NORMALIZED-E L OFFSET ' 3-\

Fig. 8. Theoretical elevat.ion crosstalk for BEL = -30") 0, + 30"K = 3.2.where the magnitude of the spurious crosstalk is equato the principal error signal.

The det.ailed sha.pe of the measured contours in Fig.is not symmetrical and is not in exact agreement, with thetheoretical results of Fig. 6. However, at the ime of thesedat,a, .he reference cha.nne1 pha.singrior to themicrowavcircular combining hybrid had not been adjusted for optimum reference channel circularity. Thismisalignment(approximately 15") combinedwith any IF phase misalignment (BAZ,BZ,OEL) could account for much of the observed differences. The bottom contour of Fig. 5 is somewhat unusual and this is attributed to different, grounreflections for H a.nd V polarizations at boresight elevation angles less than 28 mad.

V. COMPUTERESULTSThe model equations presented in the previous sectio

ha.ve been programmed on a digital computer so that theffect.of perturbing t.he various parameters can be nvestigated.Thisprogram ha.s been used to nvestigate hetheoretica.1 stability of t,he antenna system for idea.1 phasing and a unit offset in the "down-right" direction. Thresults for G = 2, K = 4 (RC signal 12 dB greater thanLC signal) is shown in Fig. 7. Fig. 7 is a composite plotfo r a unit offset, in both azimuth and levation and clearlindica.tes a region of instability for be he en -80" and- 10".

The program has also been used to show the effect oIF phasing fo r a unit azimuth offset with G = E, K = 3.2(RC 10 dB > L C ) . This dat.a is shown in Fig. 8 and indi


7/7

42 IEEE TRANSACTIONS ON H T E N N A S AND PROPAGATION, VOL. AP-20, NO. 4, m Y 972s Dhe variability in crosstalk with F phasing. Perhaps

is the marked difference intarget elevation for the same incoming polariza-iP. This points up the necessity for fairly st.rin-if the effect of crosstalk is to be re-

The cross-coupling effect of varying thedifferential4 EN was negligible and this parameter as

~ A Rhould be optimized for best error channeltha.n minimization f crosstalk.VI. CONCLUSIONS

A model has been developed for the secondary patt.ernf a five horn monopulse tracking system. The

was adequate for predicting the system perform-incoming olarizat,ions. The most severe

of crosstalk is to cause angle tracking to becomefor sufficiently large cross polarized returns. The

alk also results n serious pola.rizationdependentn the angular ocation of off-axis targets.latt.er effect is critically dependent on the system

phasing but could be removed by computer processingLC and

RC returns was measured. In order to accomplish this theabsolute nsertionphase of each receiver channelmusteit-herbe set to a prescribed value or measured as part ofpremission calibrations.

The most severecrosstalk in the measured antennasyst,em was the ridge-to-ridge variety. It can account formuch of the observed 2.5-dB loss in secondary patternridge polarimtion difference gainf the effectsf reradiatedenergyare considered. The analysis s notparticularlyhelpful in the design of an improved feed s tructure butdoes point up the need for extensive secondary patiernmeasurements to characterize the finaldevice. Futuremodifications and new feed designs should attempt oequalize the ridge and nonridge secondary pattern differ-ence gains, by reducing the crosstalk levels and possiblyby the insertion of fixed pads in the proper channels. Ascheme of intentionally introducing additional cross cou-pling of the proper magnitude and phase to cancel theundesired cross coupling has also been suggested.

A C K N O ~ E D G M E N TThe author would like to thank Dr. J. Ruze of M.I.T.Lincoln Laboratory, for valuable criticism and iscussions,and J. Pearlman, who programmed the equations.

The Behavior of Electromagnetic Fields at EdgesJOSEF MEIXlUER

Abstract-The behavior of an electromagnetic field n the neigh-of the common edge of angular dielectric or conductingis determined from the condition that the energy densitybe integrableoverany ~ t eomain the o-callededge

n). Two cases are treated in detail. 1) A region consistinga conducting wedge and tw o different dielectric wedges with a

edge. 2) A regionconsisting of tw o differentdielectricwith a common edge. It is also shown that near such edges,

ill exhibit the same behaviorthe electromagnetic ield.

I. INTRODUCTIOKN THE SOLUTION of diffraction problems, it isfound that at sharp edges of the diffracting obst.acleinfinite.

work was supported by Air Force Cambridge Research Cent.erManuscript. received January 21, 1971 ; evised January 2 4 , 1972.Contract, AF-19(122)-42.The aut.hor was with Courant Institute f Mathematical Science,from the Institutef Theoretical Physics, Rhine-W&falian Institute of Technology,Germany.

The order of this singularity is , however, subject to theso-called edge condition [l], [a]. The edge conditionstates th at the electromagnetic energy density must beintegrable over anyfinite domain evenif this domain con-tains singularities of t.he electromagnetic field. In otherwords, t.he electromagnetic energy in any finite domainmust be finite. In the case of a perfectly conducting sur-face n-ith an edge, one concludes from this condition thatnear the edge the singular components of the electric andmagnetic field vectors are of the order p- l I2 , where p is thedistance from the edge, while the component,sof the elec-tric and magnetic field strengths parallel to the edge areah-ays finite.In this paper we generalize the preceding result to findt.hebehavior of the field vectorsnear the edges of di-elect.ric and perfectly conducting bodies. We restrict our-selves to the neighborhood of points for which there is awell-defined tangent along the edge. We may then con-sider the edge as locally straight. Hence it suftices to con-sider a space filled with wedges of homogeneous materia1wititha common straight edge. We shall treat in det,ail

cross coupling in a 5 horn monopulse tracking system

Documents