ANALYSIS OF MULTIREFLECTOR ANTENNA CLUSTERSBY SPECTRAL METHODSV.E.Boria, M.Baquero, M.Ferrando

Departamento de Comunicaciones, U.P.V.Camino de Vera s/n 46071, Valencia (SPAIN)Telephone: 34 6 3877307 Fax: 34 6 3877309

E-mail: Boriaedcom.upv.es

ABSTRACT.

This paper analyzes multireflector antennas making use of spectral domain techniques. The behaviourof the multireflector antenna is determined by means of a transference function that relates the planewave spectrum of an incident signal on this antenna to the plane wave spectrum reflected by thestructure. Multireflector antenna clusters that synthesize specific radiation patterns have also beenundertaken.

The paper allows us to generalize the identification of every reflecting object through a transferencefunction that relates the incident spectrum to the reflected one. This will permit us to analyzereflection problems with multiple structures.

INTRODUCTION.

In order to obtain the transference function that characterizes the multireflector antenna, we havestudied the reflected fields produced by parabolic and hiperbolic surfaces that integrate themultireflector structure.

Reflector antennas have usually been analyzed by geometrical optics based on ray tracing. In thispaper physical optics approximation has been used for characterizing reflector elements. It has beenfound the induced current distribution that an incident spectrum produces on the reflector surface.Using this current we get the reflected field and their plane wave spectrum.

Once the behaviour of a multireflector antenna has been defined by an equivalent transference functionwe have shifted feeding sources from system focus and we have studied radiation patterns producedby several sources placed around the focus of a Cassegrain antenna.

MULTIREFLECTOR TRANSFERENCE FUNCTION.

The electromagnetic conduct of a multireflector antenna is defined by a transference function thatrelates the incident plane wave spectrum with the reflected one.

Ew,(K..Kv,) Eu K,v

R El\\ Ex(K.Ay) lEN(.K)Ewr(.K

o.\ (K,K,) - Input Spectrwm (planewava)RI E.., (K..K,) - Output Spectrum (plane wave).

H .,al - Reflectou lYanaereuce FwictslouEm (K5,K,) 1.a - Multireflctor rasfdereaca Function.

Fig. I- Multireflector antenina. Fig. 2. - Multireflector transferenice function.

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The transference function of this Cassegrain multireflector antenna is obtained combining transferencefunctions of parabolic and hyperbolic reflectors. The formulas employed with spectral analysis, whichmultireflector description requires, are summarized in the following picture.

Fig. 3.- Spectral formulas employed with multireflector antetnna analysis.

PARABOLIC REFLECTOR TRANSFERENCE FUNCTION.

First of all a parabolic reflector antenna has been characterized by a transference function relating theinput plane wave spectrum to the output one. A complete diagram of a single reflector analysis isshown in figure 4. Spectral responses of an -offset parabolic reflector with a source placed at focusand outside of focus are presented in figure 5.

j~ ~~~^- -

-II so

-1 IN I i

It

\2 I IZt t: 111 8l{:1l

-ob'''''-ssKw4-r-b e.- .............1...................

-@00-*0 -00 -40 -20 0 20 & *0(

E, (K..Ky)

C,, (K,)

Fig. 4.- Diagram employed in reflectoranalysis.

Fig. 5.- Parabolic reflector radiation patternts.

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Plane Wave Spectrum: =ffp(k(tky) e7jE'txdkdkyCylindrical Wave Spectrum: 1 = fcj(kQz)Hi4)(Kpp) ein*edkzckSpherical Wave Spectrum: *j=e hg'(kr) P."(cosO) es

Cyl. Spec. - Pt. Spec. Conversion: p(kpcosP kspnj)= k r j"cA,,(k4)e-1

Sph. Spec. - P. Spec. Conversion: p(ksinacosp,ksinasinp)mp(ksnccosf3ksic-siP)=-1--j r r.''-'l(COsOK irk m-n- f1iM

1 e~Cm0o(a)F2(2n +1)

Pi. Spec. - P. Spec. Translation: p'(k,,k ) (k. ky) c Izo

Cyl. Spec. - Cyl. Spec. Translation: c,, = 3 ±c,j(kp ')(- )hY jki HV,R(kp'IyoI)j=1 ng-e 2IioI2 e.

Sph. Spec. - Sph. Spec. Translation: E.EE "im (-1)R (j)v P-n (2v + 1)

*a(m,n I -m,v Ip) h0)(kzv)

- S- w w - - W -- = - i - g -- --1 "

8-

EouT (K..Ky)

Next, we offer some results about the evolution through several planes of the amplitude and phaseof electromagnetic fields -reflected by a parabolic antenna. The results have been got propagating thereflected plane wave spectrum. In figures 6 and 7 it can be observed the electromagnetic fieldproduced by an offset parabolic reflector (D=20X, f/D=0.5) with a source that has a 10 dB fall atreflector corner and placed at focus.

Vstd. PloW AmplItudisI ,I. - r

10

1I9

-I

-10

0 2 4 * * 10 12 14 1I

Y (wav lPgth)

wI

I4

16 20

D!e fle Phm

y (Wv lngt)

Fig. 6.- Electric field amplitude. Fig. 7.- Electric field phase.

In figures 8 and 9 it is presented the aniplitude and phase of the former electromagnetic field on twoplanes placed 2X and 22X from reflector surface. We can see that the amplitude is more uniform atlarger distances and we, can see that constant phase planes indicate the advance direction of thereflected field.

Electric Flld Ampotude

0~Plene at 2 w.11

-c- - - - - Plon t 22w.I.-10- @ --------.,............ .................... ..,_,....

-20 ...... ......... ........................ .............*

1 \

o> *1ees@-*- -*--Xs ............ ........ o......

_i ***, 4,.r

-.0 ................_ ............ ...... ...... ANIt

-15S -10 -5 O 5 10 1!

x (wv length)

Fig. 8.- Electric field amplitude.

O.tfle. ield Phose200

150 . . .........

1t (wol.

-2001-15 -10 -5 0 5 ¶0 1

(wove lngtoh)

Fig. 9.- Electricfield phase.

Finally, we have studied an offset parabolic reflector with source shifted from parabolic surface focusin order to study this well known effect. It permit us to point the radiation pattern of the parabolic

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IK

.-Il

-1 ai

3

5

reflector in another direction. Figures 10 and 11 show the evolution of the electromagnetic fieldproduced by an offset parabolic antenna (D=20X1 f/D=0.5) with a feeding source that has a 10 dBfall at reflector corner and shifted 2X from focus.

I.1

Electric Flld AMlDltudls

y (wave length)

Ic

x

Eletic Field phase

y (a0" legh)

Fig. 10.- Electricfield amplitude. Fig. 11.- Electric field phase.

HYPERBOLIC REFLECTOR TRANSFERENCE FUNCTION.

We have also characterized an hSyperbolic tantenna, which is the secondary reflector in Cassegrainsystems. The method we have employed is based on physical optics and we have followed the samesteps than with parabolic reflectors. In figures 12 and 13 we have the, amplitude and phase of theelectromagnetic field reflected by an offset hyperbolic structure (D=5X, c=5X and e=2) with asource placed at focus.

2

I

Electrie Feld Amplltude

I

y (wov lngth)

Fig. 12.- Electric field amplitude.

y (wove lMgt)

Fig. 13.- Electric field phase.

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MULTIREFLECTOR ANTENNA CLUSTER ANALYSIS.

After analyzing parabolic and hyperbolic reflectors, we can combine their transference functions andstudy a Cassegrain system integrated by such reflecting surfaces. In figures 14 and 15 we can see theelectromagnetic field reflected by a Cassegrain antenna, integrated by an hyperbolic reflector whosereflected field has been shown in the former section, and by a parabolic reflector. The source hasbeen placed at hyperbolic surface focus.

Electric Reid Amplitude NO*c Rd Phies

5 6 10 12 14

y (-e iefgth)

II1

6 a 10 12 14

Y (.v length)

Fig. 15.- Electric field phase.

A general method for analyzing this multireflector system has been employed, and this method allowsus to place sources outside of Cassegrain focus. A diagram of this method is shown in next figure.

E LY = J-Input Spectrum ( cylindrical or spherical waves).WJ CCaster J-Element Weight

-CI wSpectrum Conversion to plane waves.

E-IN -Multireflctor Input Spectrum (plane waves)

HmR = Multireflector Tlansference Function.

EOU- = Multireflector Output Spectrum ( plane waves).

Fig. 16.- Multireflector antenna cluster analysis by spectral methods.

We have a cluster of sources with generic spectral content. Each source spectrum, which can be aplane, cylindrical or spherical one, must be weighted by its concerned weight. After that it is

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II

Fig. 14.- Electricfield amplitude.

represented in terms of an input plane wave spectrum. Next, employing the multireflectoritrans-ferencefunction, we get the output plane .wave sptctrum reflected by the Cassegrain antenna,Thb outputspectrum is directly related to its radiation pattern.

Finally we have studied a Cassegrain cluster configuration integrated by three sources with suitableweights. In figure 18 we can see the radiation patterns produced by each source.

I

t.4. EM0.KeN E.(Kx.K1)

S%

Ccnqr.hn Relector Patrns

C . ./ ./... ....

*2 . .. .... ........

Oa ---iov#-e.rtIuY3ffi- --------

1c ..~-..4......,..z.4h /Uso\....-35 -7 .......~~~~~~~-rl...1...O W ou

40 ~ ~ fX t .

-s0 -60 -40 .120 0 20 40 50 50

IN (do.rs)

Fig. 17.- Cassegrain multireflector antennawith source cluster.

Fig. 18.- Radiation patterns of a Cassegrainantenmna.

CONCLUSION.

In this paper we have outlined a method for designing feeding systems based on the characterizationof a multireflector antenna in terms of spectral contents.of incident and reflected signals. We havegot transference functions in spectral domain for parabolic, hyperbolic and Cassegrain reflectorsmaking use of physical optics.

This paper also. establishes the basis for studying the reflection produced by several stru@ures usingtransference functions, that relate the spectral content of the incident signal and the spedtrum of thereflected signal. This will allow us to treat with problems in open systems with multiple structures,that introduce electromagnetic interaction.

ACKNOWLEDGEMENTS .TS

This work has been granted by Comisi6n Interministerial de Ciencia y Tecnologfa (SPAIN) withinProject TIC 92-0919.

REFERENCES.

[11 R.F. Harrington, "Time-Harmonic Electromagnetic Fields", McGraw-Hill Book Company Inc.,New York, 1961.

[2J S. Stein, "Addition Theorems for spherical wave functions", Quart. AppI. Math, Vol 19, No 1,pp 15-24, 1961.

[31 M. Baquero, Tesis Doctoral."Transformaciones Espectrales y Aplicaciones a s(ntesis deyondas,medida de antenas y difracci6n", U.P.V., Valencia, 1994.

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