Diffraction by a multilayered dielectric-coated spherewith an azimuthal slotS. J. Towaij, M.Sc, and Prof. M. A. K. Hamid, Ph.D.

Indexing terms: Electromagnetic-wave diffraction, Antenna radiation patterns, Antenna theory

Abstract

The wall air-gap concept for low-loss lines and reactive-wall waveguides is employed in the exterior problemof a dielectric-coated sphere with an aximuthal slot. The results for the gain and radiation pattern for aspecific case are based on a boundary-value solution, and are optimised numerically to establish the concept.It is shown that, for a spherical antenna of electrical radius koa = 5, coated with a dielectric layer (of thick-ness and relative permittivity equal to 0-254A and 16, respectively), which encloses a wall airgap (of0-00303A thickness), the increase in gain over the uncoated antenna is 5- 17dB, while the shift in the mainlobe is 12-5°.

List of symbols/ = region number = /, //, / / / . . .

e,- = relative permittivity of region iv = r.m.s. voltage exciting the antenna

r, 9, (f> = spherical co-ordinates[Mi = permeability of region iHQ = permeability of free spacekj = 2TT/A,- wavenumber of region /kQ = 2TT/A0 wavenumber of free spaceA,- = wavelength in region iAo = wavelength in free space7] i = intrinsic impedance of region i7)0 = 120TT = intrinsic impedance of free spaceE; = 9 component of the electric field in region iH, = <£ component of the magnetic field in region ia) = angular frequency

An,Bn,Cn,Dn,andFn = nth expansion coefficients of the fieldsHljjcr) = Riccati-Hankel function of /th kind and order nH'n(kr) = derivative of Riccati-Hankel function with

respect to the complete argumentPn(cos0) = Legendre function of the 1st kind and order nPcicosd) = derivative of Legendre function with respect to 9

dlfd2 = thickness of 1st and 2nd dielectric layers, respec-tively, radians

S = average Poynting vectorPt = total radiated power

1 IntroductionThe radiation characteristics of a slot in a perfectly

conducting sphere are well known,1"3 and may be variedsignificantly by varying the electrical radius of the sphere andthe position4 or shape of the slot.5 More recently, it has beenshown that the radiation intensity can be enhanced with anoverdense plasma coating, provided that the operating fre-quency is well below the plasma frequency.6 Although aplasma coating may be present (e.g. blackout phenomenon),this method of enhancing the radiation field may introducepractical difficulties because of the availability of sources. Analternative approach is to design concentric dielectric shellsaround the sphere, the shells having the appropriate thicknessand relative permittivity. By analogy with transmission-linetransformers, the radiation characteristics of the antenna willobviously be very frequency-sensitive.

The object of this investigation is to formulate theboundary-value problem for an azimuthal slot in a multi-layered dielectric-coated sphere, and to show that, for a 2-layer coating, the optimum behaviour corresponds to an airgap for the first layer. The concept of an airgap was recentlyproposed by Barlow7 and Oliner8 for the interior problems of

Paper 6478 E, first received 16th October 1970 and in revised form15th April 1971Mr. Towaij and Prof. Hamid are with the Antenna Laboratory,Department of Electrical Engineering, University of Manitoba,Winnipeg, Man., Canada

PROC. IEE, Vol. 118, No. 9, SEPTEMBER 1971

coaxial cables and reactive-wall waveguides for low-losstransmission. The effect of this airgap is not always favour-able,9 and is, in fact, very sensitive to the frequency and gapdimension. The possibility of a strong field concentration inthe dielectric material, in the presence of a small airgap, wasanalysed recently by Mohsen et a/.,10 who also suggested theextension of the airgap concept for the interior problem toantennas.

Although the advantages of an exterior airgap are es-tablished in this paper for a 2-layer coating, the results may begeneralised by the same method, using more layers, with thepossibility of introducing interlayer airgaps. The analogousproblem of optimising the scattering pattern of multilayereddielectric-coated spheres,11"14 wedges,15-16 cylinders,17

cones,18 and spheroids, using airgaps, is presently underinvestigation.

2 Formulation for slot in a coated sphereConsider a perfectly conducting, multilayered dielec-

tric-coated sphere of radius a, with the centre located at theorigin, as shown in Fig. 1. The dielectric layers are assumed

in

xFig. 1Schematic diagram of coated sphere with azimuthal slot

to be uniform, and their relative permittivities are denoted by€,-,/= 1, 2, 3 . . . N, where N is the total number of layers.The sphere is excited as an antenna by a driving e.m.f. v, whichacts in the 9 direction across a circumferential narrow slot at9 = 9Q. TO facilitate the formulation, and without losing anygenerality, we let N = 2, while 90 will be restricted later to

1209

7r/2 (i.e. the centre-fed case). Our analysis is also restricted tothe TM case, where the field components everywhere areindependent of <f>, because of the rotational symmetry.

The field component E$ in the slot is hence given by

(i)

while the space outside the sphere is subdivided into threeregions, as follows:

(a) region I (a < r < b), fi = fx0, e = eu k = ku rj = rj{

(b) region II (b < r < c), yt, = yb0, e = e2, k = k2, t] = rj2

(c) region III (c < r), JU, = /x0, € = eQ, k = k0, rj = r]0

where k and 77 denote the wavenumber and intrinsic im-pedance of the ith region, respectively, and k0 is the free-space wavenumber 2TT/A.

The only nonvanishing field components in these threeregions are E$ and H^, which will be denoted by E and Hsubscripted by the proper region number. The relationshipsfor these components are as follows:

(a) region I,

{A

Hi=- 2,' n = 1

na<?Xkin)}P

(b) region II,

11 = - S

(2)

(3)

Dnfi™Xk2r)}Pfcos6) (4)

I (5)

(c) region III,

. . . (6)

(7)

where

and the prime notation denotes differentiation with respect to8 for the Legendre functions of the first kind, and with respectto the total argument for the Riccati-Hankel function.19'23

The coefficients of expansion in eqns. 2 and 7 may beevaluated by imposing the boundary condition that thetangential components of the fields are continuous across theinterfaces r = a, b and c. Since EY is zero at r = a, except atthe slot where 6 = 7T[2, we have

«„ =

(9)

Multiplying eqn. 9 by P^(cos 6) sin 6, and integrating over therange of 6, we obtain, using the orthogonality property of theLegendre functions,

no) =(10)

a 2n(n+ 1) "v '

• y-{«-i>2Ci -«)w!/{„ _ i/2)!}2 n odd

0 n even

where 6Q has been set equal to TT/2, which corresponds to thecentre-fed case. From eqn. 10, and the relationships resultingfrom the remaining conditions at r = b, c, we obtain thefollowing matrix equation:

'A,

Ky] OD

where

uu = ftfP'(kxa), ux2 = fifP'ik^), uX3 — uX4 = ul5 = 0

"24 — 72"/I \^2P)* "25 — "> ̂ 31 == •"n

W35 = M41 = M42 = 0, W43 = 7J2H^ ' \k2c)

M44 = rJ2^n2)Xk2c), M45 = ^o#n2)'(^Oc)' M51 = M52 = 0

M53 = &^'(k2c), M54 = 6®-\k2c), u55 — fi(2\k0c)

—

may

Solution for expansion coefficientsThe five unknown expansion coefficients in eqn. 11

be evaluated by Cramer's rule.20 The final results are

A

where

13

44

A = wxw2 +

WX = MUM22 - M12M21

W2 = "55(^/33^/44 — "34^43)

^3 = «11«32 -

(12)

w5 = u2Xu32 —

An = h ("12^2 + "32^4)

A12 = -/l(«31«2 + "21^4)

ZAJ3 = 1 j W$\M44M45 "45^54/

A14 = I\WS(MAZU55 ~ U45u53i)

A15 = —4Ix7]xr]2

For the special case of thin dielectric layers, i.e.

b — a<aorb (13a)

c - b <borc (136)

the Riccati-Hankel functions of the first or second kind maybe expanded in a Taylor series of the form

fin{kxb) = fin{kxa) + dxe\t2B'n{kxa) + . . . (14a)

A'JJctb) = fi'n(kxa) + dxe\l2fi'n'{kxa) + . . . (146)

dx = k0(b-a) (14c)

and similarly for Hn(k2c) and H'n(k2c), by replacing kx, b, a,dx and et by k2, c, b, d2 and e2, respectively. The approximatevalues for wx, w2, vt>3 and w4, resulting from the first two termsin the Taylor-series expansion, are given by

w, = 2/ (15a)

n(n+ I)1

( 1 5 c )

(I5d)

1210

d2 = k0(c-b)which lead to the well known solution for the uncoatedsphere, i.e., when a — b = c.

PROC. IEE, Vol. 118, No. 9, SEPTEMBER 1971

4 Radiation patternThe expressions for the electric and magnetic fields in

region III can now be obtained on inserting the expansioncoefficients Fn from eqn. 12 into eqns. 16 and 7. Using theasymptotic value of the Riccati-Hankel function and itsderivatives for large argument, corresponding to the far-field

izone, i.e.

(16a)

(166)

we obtain the following expressions:

e—jkor oo

r n=\ •

. . . . (17)

4-

. . . . (18)

which are related by the intrinsic impedance r)0 of free spaceas expected.

The average Poynting vector may be expressed as

S = i Re E X H*

2 n + l= g

r2 hx ^n(n + 1) \K\while the total radiated power is given by

J

2TT TT

t = \ \ S.Jo Jo

rr2 sino Jo

r"x {P^cos d)}2 sin Odd

Jo

( 2 0 )

where r is a unit vector in the radial direction.

5 GainThe gain function of a test antenna in a given direction

g(6) is defined as the ratio of the power per unit solid anglein the desired direction p{6) to the power per unit solid angleof a lossless isotropic radiator with the same input power, i.e.

(21)

while the gain G is the maximum value of the gain function21

and p(9) = r2\S\.To compare different cases of dielectric loading, the

radiation and input conductances, denoted by ar and ah

respectively, are needed to normalise the power pattern p(0)for a constant input power P,. The radiation conductance isgiven by

ar = PJ\v\2 (22)

where v is the applied voltage, while the input conductance isfound from the ratio of the current to voltage at 6 = TT/2.

The 6 component of the current density at the sphere sur-face (r = a) is given by 21

± (23)

The total current crossing the surface 6 = constant is hencegiven by

- 2 " /.a7(0) = / e r sin 0 drdd = lira sin (0)#!

Jo JoPROC. IEE, Vol. 118, No. 9, SEPTEMBER 1971

(24)

which may be evaluated for the equatorial plane 6 = TX\1. Inparticular, the input conductance at 8 = 7r/2 is

or, = real {/(TT/2)M

= real [- JJ {(25)

where v has been set equal to unity. Hence, the input poweris given by

p. = Pt = real {/*(T7/2)} = at . . . . (26)

where all antenna and dielectric losses have been neglected.Hence, the power pattern of the coated sphere may benormalised with respect to the uncoated sphere by multiplyingby the normalisation factor F, given by

F = aiOl(a,E^ (27)

Here, CT/0 and EQ are the input conductance and electric field£111(0 = TT/2), respectively, for the uncoated sphere.

6 ResultsA plot of the radiation pattern for the uncoated sphere

is shown in Fig. 2, and is based on eqns. 15 and 17, with

I 3ao

is

/I

;/

- /111

11

• 11

1

• " A\\

• 1

/ \

1 1 . 1 I 1 I 1 . 1

0 10 20 30 40 50 60 70 80 909, deg

Fig. 2Radiation pattern of uncoated sphere for koa = 5

a = b — c. This is the identical result to that given previouslyby Harrington1 and others.2'3

For the case of a fixed sphere of size koa and two dielectriccoatings, there are four variables whose effect on the gain andradiation patterns of the antenna is of interest. These areeu e2, dx and d2.

The optimum value of ex for a large range of values of e2,c/i and d2 has proved to be unity. The effects on the gain andmain lobe due to simultaneous variations in dx and d2, for aparticular value of e2, are illustrated in Fig. 3 and Table I.

Table 1BEHAVIOUR OF DIELECTRIC-COATED ANTENNA WITHAND WITHOUT AN AIRGAP

0-0-0-0-0-0-0-0-

0580000019023800522009200167000510009200

d

0-0-0-0-1-1-1-1-

2

67890134

Relative main-lobe maximum

With airgap

Amplitude

5-03445-63575-95036-17906-33006-32306-32006-2155

6 location

deg

16-0012 0013-00130013 0013-0013-0013-00

Without airgap

Amplitude

3-01035-61253-75133-86894-13784-19254-30874-3508

e

deg

16-6012-0016-2018 0016-0016-00160016-00

e2 = 7-0, koa = 5 01211

These results indicate that the optimum values of dy and d2

for this case are 0-092 and 1 -Orad, respectively. The variationsin the radiation pattern near the optimum value of dx andfixed d2 are illustrated in Fig. 4, while the variations near theoptimum value of d2 for fixed d{ are illustrated in Fig. 5. Theimprovements in the gain and peak of the main lobe over theuncoated case are shown in Fig. 6 for fixed dx and variable d2.

4

3

2

1

I1

2

3-

1

1•1 /

1 />

•

/

/

/

1

i\i i

/ \

.-. / 1

*

\—1iii

—

0O4 0 0 8 012 016

Fig. 3

Variation in gain for various thicknesses of airgap as function ofdielectric thickness

d2 = 0-4d2 = 0-6dz = 0-9d2 = 1 0

0 10 20 30 40 50 60 70 80 90G.deg

Fig. 4Radiation pattern for coated case, layer thickness variable and airgapthickness fixed

El = 63 = 1E2 = 7

rf, = 0092d\ = 0-98d2 = 0-99d2 = 1 0 0

1212

The final optimisation of the gain, with du d2 and e2 varyingsimultaneously, leads to the optimum values dx = 0 019,d2 = 1-593 and e2 = 16 for koa = 5, as given in Table 2.The corresponding optimisation with dx = 0 and e2 = 16leads to the optimum value d2= 1-125, as shown in Fig. 7.

6-

10 ' 20 ' 30 ' 40 ' 50 ' 60 ' 70 ' 80 " 900,deg

Fig. 5Radiation pattern of coated case, airgap variable and layer thicknessfixed

ei = e3 = 1E2 = 7d 2 = 1000rfi = 0092di = 0086

_ . _ . d\ = 0105

- 5

1-4

Fig. 6Maximum gain and field intensity relative to uncoated case asfunction of dielectric thickness, with airgap kept constant

si = ej = 1, 62 = 7di = 0092relative magnitude

- gain

PROC. IEE, Vol. 118, No. 9, SEPTEMBER 1971

Finally, the validity of the approximation in eqn. 14 istested for koa = 5, with et ->• 1, e2 — 1 and d\ and d2 variablebut equal to each other, as shown in Fig. 8. The resulting

Table 2BEHAVIOUR OF ANTENNA IN VICINITY OF OPTIMUM

PARAMETERS

15-00015-60015-90015-99016-0001600116010160201606016-46016-860

dx(opti-mum)

00190001900019000190001900-019000190001900 01900019000190

, * .(opti-mum)

1-5931-5931-5931-5931-5931-5931-5931-5931-5931-5931-593

Improve-ment in gain

dB

1-542661-412591-302361-467215-174944-761592-235241-806911-472841-297971-26206

Relative main-lobe maximum

Magnitude

4-577794-509754-452884-538196-954566-631365015164-719204-541144-450634-43226

e

deg14-8015 00160020-008-60

10-0012-0012-8214-0014-8014-80

02 04 O6 O 8 K5 V2 T4

Fig. 7Maximum relative field intensity as function of dielectric-layerthickness, in the absence ofairgap

*->Q.

10 20 30 40 50 60 70 80 900,deg

Fig. 8Test of approximation in eqn. 14 for optical airgap of increasingthickness

koa = 5dx=d2 = 0 0d\ = dz — 0-2

- . _ . d\—di = 0-6di = d2= 1 0

PROC. IEE, Vol. 118, No. 9, SEPTEMBER 1971

errors in the peak of the main lobes are 1 • 2 %, 7 % and 17 %,corresponding to dl 2 = 0-2, 0-6 and 1 -Orad, respectively.

7 Discussion and conclusionsExamination of our results shows that, for the special

case of ka = 5, the optimum 2-layered dielectric coatingconsists of a narrow airgap and a relatively thick dielectriclayer of high relative permittivity. Although the computationsare based on the first four terms in the exact series solution,the mode coefficients converge rapidly, and are accurate to atleast four digits. The computation time is considerablydecreased using the method of Shafai et al.22 However, forlarge values of koa and relatively small values of dx and d2,the approximation in eqn. 14 is reasonably accurate, and leadsto considerable saving in computer time.

Our results indicate that there are two significant effectsdue to the dielectric coating of the antenna. The first is that asingle dielectric coating of optimum thickness and dielectricconstant shifts the peak of the main beam towards smallervalues of 9, as shown in Figs. 2 and 5. The second effect isdue to an optimum airgap between the sphere and thedielectric layer, which leads to a significant increase in thelevel of the peak of the main lobe and the main-/side-loberatio, but has little effect on the location of the main beam.Although our results are for the special case of koa = 5 and2-layer coating, the optimum single dielectric layer leads to anincrease of 2-3dB, and a shift of 12-5° in the peak of themain lobe relative to the uncoated antenna. However, in thepresence of an airgap and a dielectric layer, the optimumvalues for the increase and shift in the peak of the main lobeare 5-17dB and 12-5°, respectively. Although the same shiftin the main lobe may be obtained by using a larger uncoatedsphere, the improvement in the gain is significant, and is aconsequence of the airgap. This improvement may be attri-buted to a transformer effect, which is best obtained by thecombination of the airgap and dielectric-layer transformers.Another possibility, which is presently under investigation,is to view the structure as a supergain antenna, with reducedlosses due to the airgap.

Finally, the construction of the antenna may be achievedwith reasonable tolerances on the optimum dimensions andsymmetry in the v.h.f. band, where the airgap ranges fromfew millimetres to several centimetres. In such cases, bothhalves of the sphere are separately coated with commercialfoam material of relative permittivity near 1 -03, to representthe airgap. The outer dielectric layer is then positioned, and acoaxial feed is used to excite the antenna.

8 AcknowledgmentsThe research reported here was supported by National

Research Council of Canada grants A-3326 and A-7240and Defence Research Board of Canada grants 3801-42 and6801-37.

9 References1 HARRINGTON, R. F. : Time-harmonic electromagnetic fields'

(McGraw-Hill, 1961)2 BALADEL, v.: 'Electromagnetic fields' (McGraw-Hill, 1964),

pp. 338-3413 STRATTON, J. A., and CHU, L. J. : 'Steady-state solutions of electro-

magnetic field problems. Pt. 2—Forced oscillations of a conductingsphere', / . Appl. Phys., 1941, 12, pp. 236-240

4 KARR, P. R.: 'Radiation properties of spherical antennas as afunction of the location of the driving force', / . Res. Nat. Bur.Stand., 1951, 46, pp. 422-436

5 MUSHIAKE, Y., and WEBSTER, R. E.: 'Radiation characteristics withpower gain for slots on a sphere', IRE Trans., 1957, AP-5, pp. 47-55

6 LIN, c. c , and CHEN, K. M. : 'Improved radiation from a sphericalantenna by overdense plasma coating', IEEE Trans., 1969, AP-17,pp. 675-678

7 BARLOW, H. M. : 'Low-loss waveguides'. Conference abstracts ofthe URSI 16th general assembly, Ottawa, 1969, p. 71

8 OLINER, A. A. : 'A new class of reactive wall waveguides for low-lossapplications'. Conference abstracts of the URSI 16th generalassembly, Ottawa, 1969, pp. 72-73

9 BURRELL, R., and DMITREVSKY, s.: 'Approximate cutoff frequencyof dielectric ridge waveguide, with application to travelling-wavemasers', Proc. Inst. Elect. Electron. Engrs., 1968, 56, pp. 2183-2184

10 MOSHEN, A., KASHYAP, S. C, BOERNER, W. M., and HAMID, M. A. K.:'Field distribution in multilayered dielectric-loaded rectangularwaveguides', Proc. IEE, 1970, 117, (4), pp. 709-712

1213

11 WESTON, v. H., and HEMENGER, R. : 'High frequency scattering froma coated sphere', / . Res. Nat. Bur. Stand., 1962, 66D, pp. 613-619

12 KOMATA, A., and MUSHIAKE, Y. : 'Effects of dielectric coatings withnonuniform thickness on the radar cross section of the perfectlyconducting sphere', Rep. Res. Inst. Elect. Commun. Tohoku Univ.,1960, 11, pp. 191-201

13 LEVINE, s., and KERKER, M. : 'Scattering of electromagnetic wavesfrom concentric spheres, when the outer shell has a variablerefraction index'. Proceedings of the interdisciplinary conferenceon electromagnetic scattering, Potsdam, New York, 1962, inKERKER, M. (Ed.): 'Electromagnetic scattering' (Pergamon, 1963)

14 USLENGHI, P. L. E. : 'High frequency backscattering from a coatedsphere', Aha Frequenza, 1965, 34, pp. 1-4

15 HAMID, M. A. K., BOULANGER, R., MOSTOWY, N. J., a n d MOHSEN, A. :'Radiation characteristics of dielectric-loaded horn antennas',Electron. Lett., 1970, 6, pp. 20-21

16 HAMID, M. A. K., and MOHSEN, A.: 'Diffraction by dielectric-loaded

horns and corner reflectors', IEEE Trans., 1969, AP-17, pp. 660-662 •

17 USLENGHI, P. L. E. : 'High-frequency scattering from a coatedcylinder', Can. J. Phys., 1964, 42, pp. 2121-2128

18 HAMID, M. A. K., BOERNER, W. M., BHARTIA, P . , a n d MOHSEN, A . :'Diffraction by dielectric loaded conical horn antennas'. Proceed-ings of the European microwave conference, London, 1969

19 GALOGERO, F. : 'Variable phase approach to potential scattering',in 'Mathematics in science and engineering—Vol. 35' (AcademicPress, 1967), pp. 198-204MILLER, K. s.: 'Engineering mathematics' (Dover, 1956), pp. 13-14SILVER, s.: 'Microwave antenna theory and design' (Dover, 1965)SHAFAI, L., TOWAIJ, s. J., and HAMID, M. A. K. : 'Fast generation ofspherical Bessel functions with complex argument', Electron. Lett.,1970, 6, pp. 612-613ABRAMOWIK, M., and STEGUN, i. A. (Eds.): Handbook of mathe-matical functions' (Dover, 1965), p. 445

202122

23

CorrespondenceRADIATION PATTERNS OF CIRCULARAPERTURES WITH STRUCTURALSHADOWS

Mr. Cornbleet's paper [Proc. IEE, 1970, 117, (8),pp. 1620-1626] is of considerable interest in connection withthe design of radiotelescopes. Mr. Cornbleet has achieved ananalytical treatment of the complicated case when theillumination of the aperture is nonuniform. One of the mainconclusions of the paper is that the degradation of the radia-tion pattern produced by a tripod feed support is more severethan that produced by a quadrupod support. However, thisresult does not appear to be correct.

The radiation patterns with tripod shadowing are given inFigs. 7 and 8 of the paper for uniform- and tapered-apertureilluminations, respectively. These predict that the halfpowerbeamwidth should vary by approximately 50% for a changeof TT/2 in the azimuth angle. A distortion of this magnitude,however, is altogether unreasonable for such a small apertureblockage (9-25%) and, in fact, does not occur in practice.

Some years ago, we carried out a study of the effect of atripod shadowing of about 4% on the performance of the210ft-diameter paraboloid at Parkes, NSW, Australia.Experimental work at wavelengths of 21 and 75 cm usingradioastronomical sources showed negligible asymmetry inthe halfpower width of the main beam, although the sidelobestructure was much more complicated than that of an un-obstructed aperture. The main features of the results for thisparticular aerial were in agreement with a theoretical scalardiffraction analysis of a uniformly illuminated aperture with atripod shadow. This showed that even-numbered sidelobes areeffectively eliminated for azimuth directions normal to atripod-leg shadow (cf> = 0, TT/3, 2TT/3, . . .), and that odd-numbered sidelobes are somewhat increased in amplitude. Inazimuth directions lying along a tripod-leg shadow or itsextension across the aperture (<f> = TT/6, TT/2, 5TT/6, . . .), thereis no significant broadening of the main beam, and thedisturbance to the sidelobe structure is less severe apart froma substantial filling in of the minima. The principal conclusionsof the theoretical analysis were also verified by an opticaldiffraction experiment. It is not to be expected that taperingthe aperture illuminations would alter these general con-clusions.

The difference in the results predicted by Cornbleet appearsto be due to algebraic errors in eqns. 24 and 28 of his paper.If the angle X is TT/3, the first line of eqn. 24 should reduce to

(A)

the diffraction integral for a circular aperture blocked by acentral circular obstruction of normalised radius D. This doesnot happen, however, and re-evaluation of eqn. 2 shows thatthe correct form of eqn. 24 is

g(u,0) = [/(/•){6XJ0(ur) + 2 S J2rt(«r)JD { n = l

( 1 + 2 cos — ) } r d r •\ 3 / n J

. (B)

Furthermore, the Jacobi expansion of exp (jur sin 0') givenin eqn. 25 should be*

exp (jur sin </»') = J0(ur) + 2 S J2/,(w) cos 2n<f>'i

2/ S J2n+i(ur) sin(2«n=0

• • (Q

When this expression is used, it is found that eqn. 28 shouldread

6Xjo(ur)n=\

4/m\ . »—- ) +4y S (-l)nJ2w-

2n+ 1(D)

Both eqns. B and D now reduce to eqn. A for X = TT/3 as theyshould. Also for uniform illumination these equations agreewith those obtained in Reference A.

The general conclusions resulting from a comparison of thetripod obstructed aperture with the quadrupod case (as givenby Cornbleet) may be summarised as follows. The radiationpattern (magnitude) with the tripod shadow exhibits a 6-foldsymmetry in </>, whereas that for the quadrupod shadow has a4-fold symmetry. The extent of the halfpower beamwidthasymmetry and the level of the sidelobes are not significantlydifferent in the two cases.

30th March 1971

School of Electrical EngineeringUniversity of New South WalesSydney, NSW, Australia

C. J. E. PHILLIPS

H. C. MINNETT

1214

Division of RadiophysicsCSIROSydney, NSW, Australia

ReferenceA PHILLIPS, c. J. E. : Ph.D. dissertation, University of Sydney, Sydney,

Australia, 1967

I should like to thank Mr. Phillips and Mr. Minnettfor pointing out the hidden error in eqn. 24, which wasomitted, together with the consequences, from the Corrigenda.I accept that the results now show negligible differences in themainlobes of the patterns.

S. CORNBLEETDepartment of PhysicsUniversity of SurreyGuildford, Surrey, England* This error has also been pointed out in a recent corrigenda [Proc. IEE, 1971,118, (1), p. 78]. Unfortunately, the form of eqn. 28 given there is still incorrect,because of the error in eqn. 24 discussed above. Our conclusion that beam broaden-ing in the two cases is insignificantly different is unchanged

PROC. IEE, Vol. 118, No. 9, SEPTEMBER 1971