admittance of a cavity-backed annular slot antenna-3uq

portance of the edge singularities in the fields of apertures larger and smaller than resonant dimensions. The edge fields are so large for small apertures that radiation due to the terms having a singular behavior at the aperture edges, forms the major contribution to the fields scattered by such an aperture. For apertures of larger than resonant dimensions, however, it appears that the edge singularities have a n almost negligible effect on the total scattered field.

I t is also formally possible to evaluate the ponent of the aperture field. This may be expected to be small i n the case considered here, since the incident field contains no s conlponent and the edges of the aperture are either parallel or perpendicular to the incident electric field. The component of the aperture field may be represented by a series of functions satisfying the edge condition, and the two sides of ( l i j can be es- panded in pouyer series to obtain algebraic equations i n the unknown coefficients of the aperture field. Since the power series on each side of ( I f ) is differentiated, h o w ever, convergence is much slower and the solution cannot be obtained b>- simpl>. truncatillg the system of equations. The results gi\len here for scattering cross

section are not affected by neglecting the component of aperture field, since it cannot radiate for the range of aperture spacings considered.

IS. CONCI.ITSIOSS

The problem of diffraction b>r an aperture in the range of wavelengths corresponding to the resonance region requires a rigorous ,fornlulation as a boundary value problem. Physical approxilnations can be made i n both the long and the short wavelength limits, but these approximate solutions break down i n the resonance region i n most cases.

A rigorous fornlulation has been presented here for the problem of diffraction by ;I periodicall>- apertured conducting screen. The form of the solution is very com- plicated, but is suitable for numerical computation. From numerical solution, the aperture resonance has been demonstrated? and the relative importance of the singularities of the field which occur a t the edges of the apertures can be seen. The edge singularities have a pronounced effect on the scattered field for apertures smaller than resonant dimensions, but only a small effect for larger apertures.

Admittance of a Cavity-Backed Annular Slot Antenna*

J. GA4LEJSj', MEMBER, IRE, AKD T. W. THOMPSON,$ MEMBER, IRE

Summary-An annular slot antenna which is backed by a cylindrical or a coaxial cavity is excited by a current sheet in the slot plane which exhibits no azimuthal (6) variation. The integral equation which relates the radial electric field in the slot plane to the linear source current density is solved by variational techniques. The numerical calculations emphasize narrow slots and shallow cavities. The slot antennas may resonate with cavity depth z ~ < h / 4 . A resonant antenna exhibits nearly the same bandwidth as the slot which is backed by h/4 deep cavity. Dielectric cavity loading decreases the size of a resonant cavity, but it also decreases the antenna bandwidth.

I. INTKODL-CTIOK

ADIXTIOK characteristics of annular slot antennas have been considered b>+ several authors [1]-[3]. However, there is little information

available on the effects of a back-up cavity. The radiation conductance of an annular slot antenna which is backed by a hemispherical cavity has been recently

1962. This work was sponsored by the 1-S.AG Systems Command, Received February 2.3, 1962; rex-ised manuscript recei\.ed June

Rome -Air Development Center, Y . t Sylvania Electric Products, II'altham, Mass. 2 L3eparrment of Electrical Engineering. Cornell I-niversity.

I t h a m , 1'.

calculated [4]. This anal>-sis emphasizes the finite con- ductivit!. of the cavit>- and does not consider the slot susceptance.

The slot susceptance determines the bandwidth limitations of the antenna. Sizeable susceptances and narrow bandwidths may be expected for cavit>--backed slots of restricted size. Ob\-ious size restrictions arise in air- borne applications, but economical considerations m a > - also limit the size of ground based and possibly hardened antenna structures for use at H F or standard broadcast band.

This paper \ d l consider the complex admittance of a cavity-backed annular slot antenna. The conductance and the susceptance of such antennas which are excited b,. a voltage impressed across the slot can be calculated using the variational techniques oi Schwinger and Levine [SI. The calculations ma>- follow the general procedure of Levine and Papas [ 2 ] \vho have considered the principal mode adnlittance of a coaxial line that radiates in a semi-infinite space. The admittance of the slot as seen b?- the source is the ratio of the source current I to the voltage across the slot l,-. The magnetic

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fields in the cavity H+- and in the outside space H++ can be expressed as integrals of the electric field in the

density J,. The integrals pJH+& will differ bl- the source current I=psJ,d#. Now I and J , are expressed as integrals of the electric field E, in the slot. The integral equation for E , may be solved by the variational techniques used by Levine and Papas [2].

The possible #+dependence of the excited fields which is caused by a nonuniform distribution of the source current density J , is ignored. Therefore, the results of the present calculations are strictly applicable only to an idealized source current density Js=1 / (2 rp ) , but involve approximations to other forms of excitation. The electric field in the slot plane E, is computed in Section I1 from a formal solution of the integral equation as a series in powers of pn, where is the radius and where the integers n 2 1). However, the analysis becomes rather complex with increasing values of n and the numerical computations are restricted to n = 1 and 0. The slot admittance is discussed in Section 111. for cylindrical back-up cavities and in Section IV for for coaxial back-up cavities. When the cavity walls and the conducting screen are perfect electric conductors, only the upper half space reflects a conductance G into the slot plane. The upper half space and the cutoff cavity modes reflect a capacitive susceptance Beep into the slot plane, while the propagating cavity modes reflect an inductive susceptance Bind. The cavity depths where the antenna becomes self-resonant (B,,, -Bind) and the corresponding GIB,,, or G / [ B i n d [ ratios are also calculated in Sections I11 and 1 1 7 . Self-resonant antennas exhibit about the same bandwidth as slots backed by X/4 deep cavities unless resonance occurs for cavity depths <X/30. The antenna bandwidth is significantly decreased by decreasing the cavity depth below its value at antenna resonance. Dielectric cavity loading is shown to decrease the G/B,,, ratio. However, dielectric loading permits a decrease of the radius in a cylindrical cavity and decrease of the depth in a coaxial cavity. For small slots of radius the cylindrical back-up cavity exhibits a larger G / [ B ratio which is still smaller than GIB,,, of a top loaded vertical electrical dipole of comparable volume.

11. INTEGRAL EQUATION

The geometry of the slotted coaxial and cylindrical cavities is shown in Fig. 1. The subsequent development will be carried out for both cavity types. For an exp

time variation of the symmetrical fields the magnetic field in the cavity H,-(p, is given by [ 6 ]

Ao H+-(p, z ) (eiklz B0-j~1~) A.R,(p)

7l=l

[exp (\An2 K12 Z) B n exp dXn2 k1' z)] (1)

Fig. 1-Geometry of cylindrical and coaxial cavities.

where k l = u d G l . For a cylindrical cavity [6]

-40

R,,(P) X J l ( X n P ) (3)

where Jm(x) is the Bessel function of the first kind of order m,

Jo(X,b)

and

N , - l ( b " 2 ) J 1 ' ( X , , b ) . ( 5 )

For a coaxial cavity [Z ]

A Y n [ J l ( h n p ) I'o(Xna) J o ( L a ) Y ~ ( L P ) ] (6)

where Ym(.t-) is the Bessel function of second kind of order m,

Y o ( A n a ) J o ( X n b ) Jo(Xna.) J'O(Xn6) (7)

and

The radial electric field component E, is computed from as

The condition E, 20) is met with

exp (2jKlzo) (10)

B, exp k I 2 201.

The amplitudes A, are related to the excitation field of the cavity E,(p, 0) Multiplying 0), as computed from (9), with R n ( p ) p d p and integrating from

a to 6(a 0, for a cylindrical cavity) gives

- j m € l & ( p ) R n ( p ) p & A,, l:

e24AIa-k12 4 / x n 2 K12


196.2 Galejs and Thompson Cacity-Backed Annular Slot Antenna 673

For a coaxial cavity. multiplying E,(p, 0) with p d p and integrating from p a to b gives

For 0, H,-(p, of (1) becomes

where Bo, -4,> and B, are given by ( l o ) , (12) and respectivel!.. --lo and R,, are given b!- (2) and (3) for cylindrical cavities and by (13) and ( 6 ) for coaxial cavities.

The fields excited by an aperture in a n infinite conducting plane may be determined from the distribution of the tangential electric fields in the aperture. annular aperture is assumed to be excited b,, a radial electric field which exhibits no variations. The resultant magnetic field which has only a component is given for a n exp (jut) time variation of the fields by 121, [41, [51 as

HQ'(p,

where

and k = u x / , i i , where p 1 and p r are the inner and outer radii of the slot respectively. The last integral can be expressed in terms of Bessel functions [X] . After evaluating the 6 integral this gives

H , + ( p , j L ! e o J p ~ 2 & ( p ~ ) p ' d p CO

0

esp k ? ] J 1 ( [ p ) J 1 ( [ p ' ) l d ( (17)

where arg t/c2- kz 0, for k , k , respectively. The tangential fields H,-(p, 0) and H,+(p, 0) differ by

linear source current density J,(p) across the slot. Eqs. ( 1 4 ) and (17) result in the integral equation

where the first term of the left-hand side is equal to zero for cylindrical cavities according to ( 2 ) , and where

The radial electric field distribution &(p ) and also the source current I are determined from a solution of (18). A variational principle for determining & ( p ) may be con- structed by multiplying (18) with p & ( p ) and then integrating the resultant equation from p = p l to p = p z . Llssunling t h a t

1 " & ( P ) C amprn, (20)

P m=O

V J p : * 6 ( p ) d p -1 Qnpn p z

n=l p 1

The following notation will be introduced to sinlplif\r (21).


For cylindrical cavities

0

P, 0

with N , given by (5). For coaxial cavities

cy, :I7,Yo(h,a) (33)

Pq RJo(h,a) (34)

with N , given by (S). Applying (25) to (29) to i t follows that

X I m

After defining

(35) may be rearranged into

rv BOO 2 bnHon 4- bnbmHnm. (37)

CO m m

2rjwelao2 ,,=l n=l m=l

The expression ( I / V ) , which may be computed from or (37), can be shown to be stationary with respect

to the first variation in or in bn [9]. Setting

i t follows that

m m 1 3 dV lL=l bnHon bnbtnHnm]-

db ,

m= 1

Substituting (39) in (37)

I t is seen that (39) represents an infinite set of non- linear equations for determining bm. Approximate values of the coefficients b, may be obtained by assuming b, 0 for m and g, 0 for Q. High JI and Q values will result in more accurate expressions for b,. Because of the algebraic effort associated with the evaluation of b, and H,,, coefficients, X will be restricted to 1 although higher Q values wil l be considered in the examples.

The simplest solution is obtained by assuming that all the coefficients b, 0. In this particular case (35) and

result in

I’ Yo

with Hoo determined from (25). If b1ZO and (35) result in

bl PI) Hol log ( P B / P ~ )

(42) a11 log Hol(pe P I )

the admittance Y is obtained from as

or from (37) as

Hoo 2blHo1 b1’Ell P Y1

b l ( m l 2 111. ADMITTANCE WITH A CTLINDRICXL CAVITY

The admittance with a cylindrical cavity will be calculated only for the zero order approximation of the field distribution across the slot (bn=O). Eq. (41) may be rewritten as

The admittance reflected to the slot plane by the outside space is

y+

[log

and the admittance reflected by the cavity is

The symbols N, and f~(x,> are defined by (22), (5) and respectively.

Y+ of is easily related to the principal mode admittance seen by a coaxial line that radiates in a half space. Comparison of with of Levine and Papas shows that

P l J

where Y(0 ) ; Y 0 ] C denotes the normalized admittance of the coaxial line [2 ] and where the asterisk denotes the


1 2

2 3 1 (49)

where

log by gives Substituting (5) and (23) in (49) and approximating

where < O . A slot of will be considered as an example. For the slot admittance is Y+= (4.8+j37.5) millimhos. The susceptance B Im( I,'++ E') is calculated for Q= 100 and is plotted in Fig. 2. E'or ( b i X ) <0.384 all the cavity modes are below cutoff and the cavity reflects a capacitive susceptance. By increasing the cavity radius to (bihj 0.384 one of the cavit!; modes wi l l propagate. This mode reflects inductive susceptance which is indicated by dashes in Fig. 2. The inductive susceptance is suf- ficient to tune out the capacitive parts BCaI, of I"and E'+ if the propagating mode is near its cutoff. I t is seen that i Bindl at resonance (B =0) is only slightly- less than U for unless resonance occurs for

Hence shallow self-resonant cavities exhibit somewhat lower G:'B,,, ratios than ca\lities the depth of which approaches

The admittance ratio of G= ReY to Be,,= -Bind BI res at resonance is plotted in Fig. 3 and the cor-

responding cavity depth is shown in Fig. 4. The effect of dielectric cavity loading > E O ) is indicated b!, the dotted and dashed curves. For zo;'X=const, the slots of larger radii require a slight increase in cavity radius b? but the\- provide a significant increase of the G.': BI res

ratio. G/i B' ratios of more than 5 per cent are achieved with and b Dielectric cavitl- loading decreases the cutoff radius of the first propagating cavity mode which is indicated bs- vertical dashed lines in Figs. 3 and 4. The cavity radius b is decreased, but the G.1 U1 ratio is also decreased by dielectric loading of the cavity.

Fig. 2-Slot susceptanre \vith c>.lintirical cavity.

675


6'76

= b

Fig. 5-Admittance ratio G/B of annular slots backed bl- cylindrical cavities. propagating modes in the cavity.

For cavities of radius b <0.384X all the cavity modes are below cutoff and they reflect a capacity into the slot plane. Such cavities will be examined in conjunction with slots which have a circumference 2xp2 comparable to a wavelength X and with slots of p,<<X.

The plot of the GIB ratio of the larger slots in Fig. 5 is about the same as the G! I B res ratio in Fig. 3 for slots which have back-up cavities of radius b>0.384h. An increase of the cavity radius b from b = p 2 to b may make the antenna self-resonant and obviate the need for a tuning inductance, but i t does not provide an increase of GIB or of the antenna bandwidth.

The admittance ratio GIB is decreased for the small slots and it is less than 10" for b=p2<0.0ZX, as seen from Fig. 6. For small cavities kl<<Xq= ( q - $ ) x i b . G/B will become approximately proportional to if

<<pJn, while B is only slightly dependent on if >p?/n. An upper bound to the possible increase of GIB of a small slot with an increasing cavit>- size can be obtained from the admittance of the same slot in an infinite conducting screen G,+iB, which may be readily computed from the impedance of a comple- mentary loop antenna [ lo]. I t follows that

G G Gs a ( k P d 3

B B .max 2B.3 l2 log (2%) P1

which has been also indicated in Fig. 6. The G/B ratio of the small slots of Fig. 6 is compared in Fig. 7 with the GIB ratio of a circular top loaded dipole, which is given by 11 as

B 3E,

where E , is the effective relative dielectric constant of the antenna, h is the antenna height and a is its effective radius. The effective radius a becomes equal to the actual radius of the antenna disk p z if a<<k. The plot

I

IC! OA06

Iz$A

Fig. 6-Admittance ratio G / B of small annular slots backed by shallow cylindrical cavities.

0.006

Fig. 7-Admittance ratio G I B of a top loaded dipole and of an annular slot backed by a cylindrical cavity.

of GIB of (52) for er= 1 in Fig. 7 indicates that the top loaded dipole exhibits higher values of G/B than the slot with a cylindrical cavity of comparable size.

ADMITTANCE WITH A COAXIAL CAVITP

The admittance with a coaxial cavity will be calculated for the zero order (b ,= and the first order ( b l # O ) field distribution across the slot. For the zero order field distribution the admittance Yo is given by (45). The admittance reflected to the slot plane by the outside space Y+ is the same as in (46) and (49). The admittance reflected bs; the cavity is computed from


Fig. 8-Slot susceptance n.ith a coaxial cavity.

where < O . The susceptance B I m ( Y++ I-), as well as the inductive susceptance reflected by the propagating cavit\- mode Bind are plotted in Fig. 6. Xgain, the cavity resonance is defined by the condition B = O or Bind= -Bcap. I t is seen that Bin,+] a t resonance is near]). the same as B for except when resonance occurs for This implies that a rela- tively shallow self-resonant cavity ( B 0) will exhibit nearly the same G/Bcap ratio as a deep cavitli which will require additional inductive tuning. When the cavity depth is less than the resonant depth the slot susceptance becomes inductive, the slot requires additional capacitive tuning, and the Bind! ratio is decreased relative to its value at the self-resonance. -4 self-resonant antenna will exhibit near117 the same bandwidth as a slot with a deep cavit>-. The bandwidth of the antenna is significantly decreased by decreasing the cavity depth helow its value at self-resonance.

The admittance ratio of G= Re I r to Bcnp -Bind B res a t resonance is plotted in Fig. 9 and the cor-

responding cavitJ- depth is shown in Fig. 10. G!! BI ratios of more than 5 per cent are achieved with b and (b:a) 2. G/ B of nearly 10 per cent is reached with b=X/4 and 1.5. The correspond-

IO OF b h

Fig. 9->1dmittanre ratio G / I B a t Coaxial cavity.

r

p I

n

u - 0

Fig. 10-Resonance depth of a coaxial cavity.

2

m i

z u

k

500 3

u

Fig. ll--\.ariation of the slot susceptance with slot width.

ing cavitJ- depth where the smaller depth figures are associated with the larger ( b / a ) ratios. For a given G/l I? ratio the slots are of approximately the same size for coaxial and cylindrical cavities. However, the radius of the resonant coaxial cavit?. is almost one half of the radius of the c>-lindrical cavity. The effect of a dielectric cavity loading (e l is indicated bq; the dotted and dashed curves. Dielectric loading decreases the resonant cavit)? depth, but i t also decreases the G;] B,,,j ratio. The effects oi different slot widths is indicated in Fig. 11.


For very small slots of (pp/X)<<l the slot susceptance is approximated by the first term of (53). Numerical calculations show that this approximates B within a few per cent if <0.02. Sow B is inductive and inde- pendent of the dielectric loading. The resulting GIB is given by

G 20 log (b;’a)

6 (54)

which is seen to be much less than GIB in Fig. 2 or G/B l m a x of (51) of the cylindrical cavity.

For the first order field distribution (b1#0) the admittance Y1 has been calculated from (44). A rather lengthy calculation results for el eo and b in

B B i n d

A Y = Y1-

x? (55)

2 ? r

where

2 ( k p p ) (56) Pl 1

The correction to the admittance AY may be expected to be small for narrow slots since it depends on the square of Considering 20, b/a= 2 and b / X = 5 , ( 5 5 ) and (56) result in A II

(7.3 mho which is negligible with respect to Yol 5 mho.

V. DISCVSSIOX

A . Conclusions

The admittance of Y=G+jB of annular slots was calculated for cylindrical and coaxial back-up cavities. The antenna self-resonance -Bind or B 0 ) for a given ratio of G/B,,, can be achieved with a smaller cavity volume in the coaxial geometq-. The self- resonant antennas exhibit about the same G:Bca,, ratio as slots backed by deep cavities, unless the cavity resonates for a cavity depth <X,!30. further decrease of the cavity depth deteriorates GjB,,, ratio. Dielectric cavity loading decreases the G,:B,,, ratio, although it decreases the cavity volume which is re- quired for achieving self-resonance.

The radius of the cylindrical cavity may be decreased to a point where all the cavity modes are below cutoff and reflect a capacity in the slot plane. The resulting

G/Beap ratio may be comparable to the G/B,,, ratio of a self-resonant coaxial cavity antenna of the same volume.

In the antenna of a small size the susceptance B is capacitive with a cylindrical cavity and inductive with a coaxial cavity. The cylindrical cavity exhibits the larger G;B ratio, which is still smaller than GIB,,, of a top loaded vertical electric dipole antenna of comparable volume.

B. Limitations of the analysis

The above analysis tends to illustrate the bandwidth limitations of annular slot antennas which are backed by shallow cavities. However, the assumed excitation by a uniform current sheet in the slot plane is an ideali- zation and the result of the calculations can not be used for estimating the admittance for excitation which exhibit a sizeable azimuthal field variation along the slot.

The variational formulation of the slot admittance formally specifies the field distributions across the slot with an arbitrary accuracy. However, the solution is rather simple only for the zero order field distribution (b,) 0). There are already computational difficulties for the first order field distribution (bJ#O) , and the present method does not appear suited for determining highly accurate field distributions across the slot which would involve a large number of b,’s for very narrow slots.

VI. AACKNOWLEDGMENT

The authors appreciate the aid of P. Kimball, E. Larsen and Mrs. J. Van Horn in the computer pro- gramming and numerical computations.

REFERENCES A. A. Pistolkors, “Theory of the circular diffraction antenna,‘’ PROC. IRE, vol. 36, pp. 54-60; January, 1948. H. Le\;ine and C. H. Papas, “Theory of circular diffraction antenna, J. A$@. Plzys., vol. 22, pp. 29-43; January, 1951. (Eq. (6.12) should be multiplied with a factor 2[(T-1)/(7+1)]*).

bIarcuvitz, “\\.aveguide Handbook,” McGraw-Hill Book Co., IIK., h-ew York, Y., Sec. 4.16; 1951. J. R. \Vait, ‘‘-4 lowfrequency annular-slot antenna,” J. Res.

multiplied with 2). NBS. vol. 60, pp. 59-64; January, 1958. (Eq. (12) should be

H. Levine and J. Schwinger, “On the theory of electromagnetic

screen,” Comm. Pu.re. App l . X&., vol. 3, pp. 355-391; Decem- wave diffraction by an aperture in an infinite plane conducting

ber, 1950. S. Marcuvitz, “iYaveguide Handbook,” iVcGraw-Hill Book Co., Inc., Sew I7ork, N. Y . , Sec. 2.3 and 1951. Levine and Papas, up. tit., see ( A S ) and (A.12). -4.. Sommerfeld, “Partial Differential Equations,” Academic Press Inc., Sew York, Y., 1949. Eqs. (31.14) and (21.3a). P. 31. Morse and H. Feshbach, “Methods of Theoretical Phys- ics,“ hIcGraw-Hill Book Co., Inc., New York, IC. Y., pp. 1108- 1109; 1953. J. D. Kraus, “-Antennas,” McGraw-Hill Book Co., Inc., Sew York, Y., pp. i69-371; 1950. H. A. \\:heeler, Fundamental limitations of small antennas,“ PROC. IRE, vol. 35, pp. 1479-1484; December, 1947.


admittance of a cavity-backed annular slot antenna-3uq

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