RADIO SCIENCE Journal of Research NBS/USNC- URSI Vo!' 68D, No. 2, February 1964

Capacitor Type Biconical Antennas

Janis Ga1ejs

Contribution from Applied Research Laboratory, Sylvania Electronic Systems, a Division of Sylvania Electric Products Inc ., 40 Sylvan Road, Waltham, Mass.

(Received June 24,1963; revised August 12,1963)

In this paper the bieonical antenna analysis performed by C. T. Tai is extended to top-loaded and half-buried antenna structures whieh may have spherical cores of lossy dielectric. After computing the terminal admittance of the antenna, its input admittance is obtained by transmi ssion line considerations. The numerical calculations emphasize antennas of dimensions which are small relative to the wavelength. \Vide angle antennas of solid cones exhibit smaller radiation power factors (or products of available bandwidth and efficiency) than top-loaded a ntennas of small angle. The hemispherical antenna, located above a perfec tly conducting ground plane, exhibits a larger radiation power factor than the corresponding half-buri ed antenna of the same total volume.

1. Introduction 2. Antenna Admittance

Small cn,pacitor type antennas have been discussed in liLemture by Wheeler [1947] and more recently by Galejs [1962]. This work has esLablished relations between antenna impedance and such parameters as anLenna volume or dielectric loading. A small caviLy-backed annular slot has been compared with a top-loaded electric dipole by Galejs and Thompson [1962]. For a given antenna volume the Lop-loaded electric dipole exhibits a larger radiation power facLor p (or a larger product of bandwidth and efficiency). The cavity-backed annular slot may be approached by depressing the base of Lbe top-loaded dipole below tbe ground level, which may be expected to gradually decrease p of the dipole. However, lhe raLe of this decrease has noL been established and Lhe analysis of a top-loaded dipole will be ralher involved in a cylindrical geometry.

The input admittance Y i of a biconical antenna of figure 1 is related to the effective terminating admittance Y t of the biconical transmission line [Schelkunoff, 1943] as

The effecL of antenna burial may be considered more readily with biconical antennas. The theory of such antennas has been developed by Schelkunoff [1943] and Tai [1949]. The application of this theory to the analysis of a top-loaded biconical anten na, and to a half-buried antenna of figures 1 to 3, is straightforward. It is possible to examine configurations with small 00 and OJ exceeding about 40°, or configurations with 00 = 01> 40°. This anal­ysis is tractable only if the boundaries between the regions follow a constant coordinate surface, which precludes the examination of intermediate burial depths. Still the configuration of figure 1 affords an interesting comparison with the cavity backed slot of Galejs and Thompson [1962] . The geometries of figures 1 to 3 will be considered by the same analy is technique, for several angles OJ and var-lolls dielectric parameters of the antenna.

165

where

Y =~ K1Y t cos k1a+j sin kja (1) t Kl cos lCla+ jK1Y t sin lela

T? Zi l 00 .n.l= -:;;:- og cot 2" (2)

(3)

(4)

The terminal admittance of the biconical antenna of figure 1 can be derived following the variational

TOP LOADING CAP

a = co

FIGURE 1. The half-bw·ied biconical antennCt

REGIO N 2

~o ' '2

FIGURE 2. The symmetrical biconical antenna.

method of Tai [J 949]. The Lerminal admittance Y e depends on integrals involving the electric field Eu in the antenna aperture r = a , fJ1 < fJ< 7r/2. The fi eld E u(a, fJ) = E a( fJ) is expressed as a series of Legendre functions. The coefficients of this series are determined from the stationarity property of the Y e expression. Using a two term approximation to E a (fJ), Y e is of the form

(5)

The term. Y w is the zero order admittance which is due to the aperture field

E a(fJ) = - ~lo . sm fJ

(6)

REGION 2

~o' '2

o = 00

F1 GURE 3. The top-loaded bironical antenna above an infinite ground plane.

In the above expressions

N _S~(kla) ,- S,(k1a)

M _R~(k2a) k-R k (k2a)

S,(x) = ·lXJ '+1/2( X)

(10)

(11)

(12)

(13)

The prime denotes differentiation with respect to the argument of the respective function. PI' (fJ ) is a Legendre function of the first kind of order fJ., I n(x) is a Bessel function of order 11, H~2) (x) is the Hankel function of the second kind of order 11, and k is restric ted to odd integers. The Legendre func­tion L , (fJ) is defined following Schelkunoff [1943] as

L~(fJ ) = t[P,(fJ) - P ,(7r - fJ ) ] (14) The second term in (5) provides a correction to and Y eo. It follows that (15)

(7)

(9)

166

where 1J is such that L ,(fJ ) = 0 for fJ = fJo and (7r - fJp).

L ,(fJ) defined by (14) is an odd func!;ion of cos fJ, while (15) is an even function.

The symbols I ;i of (7) to (9) involve integrals of Legendre functions . Thus

(16)

fJ1 is selected such that P m(fJ 1) = P m (7r - fJ1) = 0 for m equal to an odd integer. Then

(if k~m),

(17)

For 00< < 1, L v(8) of (14) or (15) is approximated following Schelkunoff [1943] by

L v(O) = P v(O) - oQ,(O) (19)

where (20)

with the integrals I kk , I km , and I mm defined by (16), (17), and (18) respectively. After substituting (15) in (22) I~m is obtained as

I~m= v2+v~:2-m [>m-l (~)p, (~} (2H)

[ ')J-l 0= log io (21 ) Substituting (15) in (23) it follow t hat

11 is an integer or zero and wh ere 0,(0) is a Legendre functioll of the second kind of order v. Then I vm is given by

t 7f /2

I v//!= L ,.(8)P I//(0) sin OdO 61

(22)

where v~m. Th e integral I " is obtain ed as

(23)

Substit uting' (19), and using the approximation of Pv (O) and Q,(O) for the vicinity of 0= 0 and 71" it follows that

I 2 { 2 ?

" = 2n + l 1- 0 2n+ l+0-

(24)

where terms proportion al to higher powers of 0 and proportional to O~ have been neglected.

For 80= 01 the terms of (7) and (8) which are pro­pOl'tional to L ,(81) are equal to zero. The part of the summation over v in (9) which contains integrals of the odd functions L~(O) is reduced to a single t erm with 1,m= I mm and I vv= 2I mm. The integrals in ~v of (9) which involve tbe even functions L~(O) are denoted by superscripts as I~m and I :v . For 00=0] (7) to (9) fLre changed into

167

(29)

Th e characteristic values V= Vt which result in L~ (OI) = 0 and also the derivatives OL:(OI ) /OV for V= Vj are ob tained by llum erically computing L :(OJ a a function of v. The zeros Vi can be also obtained by in terpola tion from the zeros of P 2n (0) and Q2 n ,](0) ) where n is an integer . T

The input admitta nce Y j of a symm etrical biconi­cal antenna of figure 2 is related to thE' tenninal H.c!mittance Y t by (1) . Yt is given b:v (5) together wlth (7) to (9) 01' (25) to (27). The fields are sym­metrical wi t h respect to the 8= 71"/2 plane and onlv the odd functions L~(O) should be considered. Fo"r 80< < 81, v= k+ o where k is an odd integer. For 80= 0], the ~, of (27) in the (3m expression is equal to zero . H owever, the range of int egration of the in­tegrals I kk , I kfl" I mm, and I ,m is doubled, which doubles their values relative to t hose indicn,ted in (16), (17), (18), and (22). For 00 = 01 and 0'1= 0 this terminal admittance is the same as Y t in (68) of Gn.lejs [1962].

The input admittance Y i of 3 biconical antenna a.bove an infinite ground plane in figure 3 is t wo tImes Yi of the corresponding symmetrical biconical a ntenna in figure 2.

3. Elementary Design Considerations

The half-buried antenna structure of figure 1 is compared in this note with an antenna which has the same above-the-ground stru cture (fig. 3) (called antennas 1 and 3). It can also be shown by elemen­tary considerations that antenna 1 is superior to antenna 3. However, antenna 3 becomes the better one, if its volume is increased to be equal to the total volume of ant enna 1.

o. 03 ,-------,----rr--,--~n_____,,,__,___-_____,--rl

0 .025 Re Yti

---

~ 0. 01

~ ~ a

'"

- 0 .005

91 ~ 39.2°

90 = 10- 3 RAD

'/'0 = 2.8

TAN 81 = 0

- 0.01 "--_----" __ ---'-L-_--'---__ --'------'---_'---_--'

0.5 1.5 2.0 2.5

NORMALIZED RAD IUS - k20

FrGUHE 4. 'Terminal admittance of a half-bU1'ied cap-loaded biconical antenna (case 1 a) loss-less dielectric.

Antennas 1 and 3 are compared for constant volt­age V across the aperture. These antennas have approximately the same magnetic dipole moment. The radiated power P and the radiation conductance Gr= P r/V 2 are therefore approximately constant for antennas 1 and 3. Geometrical considerations show that the antenna capacities are related as

(3 0)

The radiation power factors P= Gr/wO are rela ted for Gr= constant as PI!P3=03/01. Applying the in­equality (30) gives

(31)

The loss conductance GL of the antenna due to di­electric losses may be defined from PL= fJ f E 2dv = (fJ /e)OiV2, where Oi is the capacity of the antenna associated with the fields inside the dielectric (the fringing fields are excluded from the definition of Ot; hence, Oi< O). It follows that GL = fJ Ot/e. With Gr = cons tan t , effl /eff3 = GL3/GLl == 0;3/ 0il. The ine­quality (30) applies also to Oi. Hence

(32)

The antenna structure 4 is obtained by increasing the linear dimensions of antenna 3 by 21/3"" 1.26 . (The volume of antenna 4 is the same as the total volume of antenna 1.) Hence, 04 = 21 / 303 , P4 = 2P3, GL4= 21/3GL3, G74= 24/3G73 , and eff4=2 eff3 .

0.03,---------n-......,.,,--------,:,-------.rTT-------,m

o :!: :::;

0 .025

0.02

0.0 15

I 0 . 0 1 w U Z ~ ~ 0. 005 a

'"

91 ~ 39 . 2°

90, ' 10- 3 RAD

'/'0 = 2. 8

TAN 81 ~ 0.01

o ~--~---+-------_.r---

-0.005

0. 5 1.5

NORMALIZED RADI US - k :,o

2.5

FrG U HE 5. 'Terminal admittance of a half-buried cap-loaded biconical antenna (case 1 a) loss y dielectric.

Comparison of antennas 1 and 4 yields

p4/2 < PI < P4

eff4/2 < eff 1 < eff4.

(33)

(34)

For a given volume the above-ground antenna 4 is superior to the half-buried antenna 1. Th e cal­culations or measurements which specify the input admittance

are sufficient for specifying the ratios between radia­tion power factors P or efficiencies eff of the antenna types under consideration. It follows from the preceding development that

P4 2P3 201

PI= P:=C;

eff4 2 eff3 2GL 1

Off1 = eff1 = GL3 ·

4. Discussion of Numerical Results

(36)

(37)

Numerical results will be presented for half­buried antennas of figure 1 and above-the-ground antennas of figure 3, which will be designated in the subsequent discussion as cases 1 and 3, respec­tively. The configurations involving top loading spherical caps (80« 81) and solid wide angle cones

168

0.025 r---.----,-----.----.,.----,---, l -2 I x 10-3 I x 10 ro __ -,---,--.-~-.~--,---,

w U Z

0. 02

0.015

~ ~ 0.01 Cl «

0.005

91 = 39.2"

'/'0 = 2.8 TAN °1 = 0

1. 5 2. 0 NORMALI ZED RADIUS koo

2.5

FIG URE 6. Terminal admillanl'e of a half-buried wide angle biconical antenna (case lb).

(00 = 01) are labeled by subscripts a and b respecLively. The zero-order Lerminal admittance Y IO , is com­

pared with the fIrst-order terminal admittance Y tl

in figures 4,5 , and 6 for cases 1a and lb . There fl.re no apprecifl.ble differences between Y IO and Y ll

except for the difference b etween R e Y IO fl.nd R e Y tl

near the r esonance of case 1a in fi.g:ure 4. Thi difference is decreased for a lossy dielectric in figure 5. There are no such resonance phenomena for case lb . The terminal admi ttance for case 1b in figure 6 is twice the free-space terminal admittance of the wide-angle biconical antenna of T ai [1949].

The inpu t admi ttance Yj is related by Lhe Lrans­mission line equaLion (1) to the terminal admiLLance Y e• The calculated values of B t= Im Y t , G;= R e Y i

= Grfor tan 0= 0, Gi= R e Y i= Gr+ GLfor tan 0[ = 0.01 are plotted in figures 7 to 9. There are addiLional transmission line resonances in fi.gures 7 to 9. For increased terminal capacities of the biconical trans­mission line (8[ large) the im.aginary part of the denominator in (1 ) becomes zero at smaller valu es of k2a. The fU'st resonance of figure 8 (81 = 66 °) occurs therefore at a smaller k2a value than in figure 7 (01 = 39 .2°) . The differences b etween the input admittance plots of case 1a in figures 7 and 8 and case 1 b in figure 9 are due to the differences between the terminal admittances in figures 4 or 5 and figure 6.

The characteristics of small biconical antennas (k2a< O.2) are examined more closely in figures 10 to 17.

The radiation conductance Gr is plotted in figures 10 and 11. For antennas of k2a< O.05 , GT is propor­tional to (lc2a) 4. Over this range GT is essentially independent of the internal antenna structure and of the dielectric loading. Larger differences between the various antenna types occur for larger values of 01 when the transmission line resonance occurs at smaller values of lC2a, as may b e seen from a com­parison of figures 7 and .

The antenna susceptance B = Im Y t has been plotted in figures 12 to 14 for fl / f O= 1 and 2.8. The

169

u Z « S u

~

-5 x 10 - 3

91 = 39 . 2"

90 = 10- 3 RAD

' / '0=2.8

- · - ·G. FOR TAN 5 ·0

- - - - G.' FOR TAN o=ODI --- b;'

-4 5 x 10

w U z « ~ u :::> o Z o u

- 5 x 10- 4

_I x 10- 2 '----'----'-__ -cL,----_-'--_---,c"--__ '--_ _=' - I x 10-3 0.5 1. 0 1. 5 2. 0 2.5 3.0

FIGU RE 7.

-5 x 10 - 3

N O RMA LI ZE D RADIUS - k20

J nput admittance of a half-buried cap-loaded biconical antenna (case 1a) 01= 39. 2° .

9 1 = 66"

90 = 10- 3 RAD

'/' 0 = 2.8

_ . _ .G; FOR TAN 51 = 0

____ G; FOR TAN 0tO. OI ___ 8;

.£ E

0-w U

0 Z ~ u ::::> Cl Z 0 u

-5 x 10 - 4

-2 ~-'---;::-1-=-----,-L;,-----:-'-:---~-___;:_'_;_-__:_'-1 x 10-3 -I x 10 0 0 .5 1.0 1.5 2. 0 2. 5 3. 0

NORMALIZED RADIUS - k20

FIGURE 8. - I nput admittance of a half-buried cap-loaded biconical antenna (case 1 a) 01 = 66° .

0. 015 ~.'-- 10- 5

I \ I '\ I I I \ ., I \

I '\ I " I .,

I I I '\/ I 10-6 I ~ / 0 0 . 01 I I

I ::; I \ I

0 I il I _ . - . G; FOR TAN 61 = 0 0 "

~

I 0 I .~ I --- G; FOR TAN 61 = 0. 01 w ::E I U

::E I \ / z -- 8; ::

0 I ~.,. .

u 10-7 ::>

w 0. 005 I 0 u Z z

I 0 <{ U U I z ::>

0 I 0 z ~ -- '';'0= I 0 ! u 0

10-8 - -- '';'0 = 2.8 f ~ I

91 = 39 . 2° 9 1 = 66°

'';'0 . 2.8

10-9

0.02 0 .03 0 . 05 0 . 1 0 . 2 0. 3

- 0 . 005 NORMA LI ZED RADIUS k 2 0

0 0.5 1. 0 1. 5 2. 5

NORMALIZED RADIUS - k20

FIG UHE 9. Inp ld admittance of a half-buried wide-an(fle , FIGURE J 1. Radiation conductance of cap-loaded and wide-biconical antenna (case ib) 01= 39.2°. angle biconical antennas (cases ia, lb, Sa, and 3b).

10-5 ,------,---,-------.:-::~"-:::::-;-;-,----, 30 'Ir lb

10-'1 ~3b

I I.

I~ I~ II

- - - '';'0 = 2. 8

1 0 - 9 '-------'----'------'------'---~ 0.02 0 . 03 0 . 05 O. I 0 . 2 0 . 3

NORMALIZED RADIUS - k20

FIGUHE 10. Radiation conductance of cap-loaded and wide­angle biconical antennas (wses la, lb , Sa, and Sb).

o <0 I

Z ::E <{

~ ~ <.) ~ 10- 3

u u Z Z ~ ~ u ::> o Z o u

91 = 39 . 2°

90 = 10- 3 RAD

G(TAN 01 FOR, ';'0 = 2.8

8 FOR '/'0 = 2.8

8 FOR '';'0 = I

10-4 L-_--" ___ L-___ --'-____ L-_-'

0.02 0. 03 0. 05 0 . 1 0. 2 0. 3

NORMALIZED RADIUS k20

FIGURE 12. Loss conductance and susceptance (~f rap-loaded biconical antennas (cases 1 a and SalO l = :39.2°.

170

0 I ::E

vO 0 I

Z ::E <{

S m

l? w

~ V Z z :': [ v ::> 0 v Z ~ 0 V ~ ~

9 G L/ TANb l FOR 'I / '0·2.8

--- B FOR '/ '0= 2. 8

_ ._. B FOR,/,O= I

10- 4 '-_......l. ___ L-___ ....L. ___ ---l __ -1

0.02 0 . 03 0 . 05 0 . 1 0 . 2 0 . 3

NORMA LI ZED RAD IUS k2 0

FIGURE ] :1. Loss conductance and susceptnnce of wide-ana1e biconical antennas (cases Ib and .'lib) 0, = 39.2°.

C I

'f vO 0 Z I <{ ::E S l? w v ~ Z <{ Z >- <{ v ~ ::> 0

~ z 0 c; v ~ ~

9

el = 39.2·

eo ~ 10-3 RAD

GL/TAN 51 FOR '1 / '0 = 2.8

BFOR'/'0=2 . B

B FOR , / ' 0 = I

10-4 L_---l ___ L-___ ......l.. ____ L-_~

0 . 02 0. 03 0 . 05 O. I 0 . 2 0 . 3

NORMALIZED RAD IU S k2 0

F IGUHE 1 4. L oss conductance and susceptance of cap-loaded biconical antennas (cases 1 a an(1 .'Ia) 0, = 66° .

171

10- 3 ,-------,---,-------,T7T-r---r---

el = 39.2 ·

eo = 10- 3 RAD

--- ' /'0= I

- - -- '/'0 = 2 . 8

10-6 L_......l. ___ L-___ ....L ____ L-__

0 . 02 0 . 03 0 . 05 0 .1 0 . 2 0 . 3

NORMA LI ZED RAD IUS - k20

FIGlJlU; 1.5. Radiation powel fart ors of cap-loaded biconital antennas (cases 1 a (md 3a) 0, = ::\9 .2° .

el • 66·

eo = 10-3 RAD

I 'II

'II, '/ I 30

/7<10 II

,/1 ,/1

II II

/1 II

II /

--- '/' 0 ' 2.8

10-6 L_......l. ___ L-___ ......l.. ____ L-__

0. 02 0 . 03 0 . 05 O. I 0. 2 0 . 3

NORMA LI ZED RAD IUS - k20

FIG URE 16. Radi ation power fact ol's of cap-loaded biconical anlennas (cases 1 a and 3a) 0, = 66°.

10-6 1-----'--------'---------'---------'------' 0. 02 0. 03 0 . 05 0 . 1 0.2 0.3

NORMA LI ZED RAD IU S - k2Q

FIGURE 17. Radiation power jactors cap-loaded and wide-angle biconical antennas (cases la, lb, Sa, and Sb) 81= 39. 2° .

same figures contain also plots GL/tan 0 for El / EO = 2.8 which is equal to the susceptance B j=WOi associated with fields inside the dielectric. Band GL are pro­portional to k 2a for k2a< <1. In the configuration of case 3 there is less fringing of the fields , and Gd tan 0 differs l ess from B than in case 1. However, Gd tan 01 may exceed IBI for values of k 2a near the resonance of the transmission lin e. The resonance is approached for k2a> 0.1.

The plots of figures 10 to 14 provide the data for computing th e radiation efficiency

(38)

the relative antenna bandwidth

(39)

or the radiation power fac tor

p = (BW) . (eff) = fl· (40)

Calculated values of the radiation power factor p are shown in figures 15 to 17. The power factors P are proportional to (k2a) 3. The power factors pare largest for Ed Eo = l; Pa for 80< < (JI (configuration a) exceeds Pb for (JO= (JI (configuration b) because of Ga< Gb for Ga "" Gb • The power factor PI of case 1 exceeds P3 of case 3 because of larger total antenna volume for a given value of k 2a, as pointed out in section 3. However, the antenna of case 1 exhibits th e smaller power factor if the volume of the antenna of case 3 is made equal in volume to the antenna of case 1. This is achieved by increasing (1c2a) of case 3 by a factor of 21/ 3 "" 1.26. The above argument will be illustrated with an example for (J I =39.2° and 1c2a = 0.05. The calculations of eff, BW, and p will be made in the cases of la, 1b, 3a, 3b and for antennas of case 3 with a doubled volume (cases 4a and 4b). The results listed in table 1 are also in accord with the discussion of section 3.

T A BLE 1. Power jactoTs, bandwidth, and efficiency figures above-the-ground and half-buried antennas

Configurations p DW efT

Case l a 5.25X IO-' 5.8XIO-' 9. 1 X IO-3 3a 4. 22X IO-5 6.5XIO-' 6.5 XJO-3 4a 8. 44XIO-5 6.5XIO-' 13 XIO-3

Case Ib 4. 69XlO-5 6.3XIO-3 7. 5 XlO-3 3b 2.6 X IO-5 7. 8XIO-3 3. 33XIO-3 4b 5.2 XIO-' 7.8XIO-' 6. 66X IO-3

Appreciation is expressed to T. W. Thompson for checking the derivations and to Miss E . Marley for computational assistance.

5. References

Galejs, J. (Sept.-Oct. 1962), Dielectric loading of electri c dipole antennas, J . R es. NBS 66D (R adio Prop.) , No . 5, 557- 562.

Galejs, J. , a nd T . W. Thompson (Nov. 1962), Admittance of a cavity-backed annular slot antenna, IRE Trans. Ant. Prop. AP - 10, 671- 678.

Schelkunoff, S. A. (1943), Electromagnetic Waves (D . Van Nostrand Co., New York, N.Y.).

T ai, C. T . (Nov. 1949), Application of a variational principle of bi conical antennas, J. App. Phys. 20, No. 11, 1076- 1084..

Wheeler, H . A. (Dec . 1947), Funda mental li mitations of small antenn as, Proc. IRE 35, No. 12, 1479-1484.

(Paper 68D2- 329)

172