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- Spheroidal Wave Functions in Electromagnetic Theory
Le-Wei Li, Xiao-Kang Kang, Mook-Seng Leong
Copyright 2002 John Wiley & Sons, Inc.
ISBNs: 0-471-03170-4 (Hardback); 0-471-22157-0 (Electronic)
6
Spheroidal Antennas
6.1 INTRODUCTION
In this chapter, a solution of EM radiation from a prolate spheroidal an-
tenna, excited by a voltage across an infinitesimally narrow gap somewhere
around the antenna center, is obtained. Three specific cases are considered:
an uncoated antenna, a dielectric-coated antenna, and an antenna enclosed
in a confocal radome. The method used is that of separating the scalar wave
equation in prolate spheroidal coordinates and then representing the solution
in terms of prolate spheroidal wave functions. A simplified version of the so-
lution, after taking account of the fact that the antenna is symmetrical in the
&direction, can then be used to obtain the electric and magnetic fields. The
Mathematics code written allows a user to model each of the three types of
antenna radiation problem discussed in this chapter.
The type of antenna used in this chapter is a prolate spheroid excited by
a slot cut through the spheroid. Axial symmetry prevails regardless of the
location of the slot. In aircraft applications, the antenna used is generally
mounted on the nose of the aircraft, and the type of antenna used is often a
slot antenna. Therefore, it is possible to model this antenna configuration as a
slot antenna mounted on a spheroid. The effects of a protecting coating layer
or radome on such a configuration can also be investigated and considered in
the optimized operation.
145
- 146 SPHEROIDAL ANTENNAS
6.2 PROLATE SPHEROIDAL ANTENNA
6.2.1 Antenna Geometry
A perfectly conducting prolate spheroidal antenna excited by a specified field
over an aperture on its surface and immersed in a homogeneous, isotropic
medium is a convenient introductory problem. It is also assumed that the
surrounding medium is nonconducting and nonmagnetic. To simplify the
situation, it is assumedthat symmetry about the axial direction prevails.
The geometry of the prolate antenna is shown in Fig. 6.1. The semimajor
and semiminor axes of the spheroid are designated a and b, respectively, and
the interfocal distance d, as indicated in previous chapters. In general, the
surface of the spheroid is < = 51, and the excitation gap can be located
anywhere, say at 7 = ~0.
6.2.2 Maxwell’s Equations for the Spheroidal Antenna
For the symmetrical situation (i.e., a/&$ = 0), Maxwell’s equations in free
space, relating E and H, take the form
(6.la)
1 a(P%J (6.lb)
hrE77 = -GT’
(6.1~)
(6.ld)
1 a(Pw (6.le)
hrH,, = -jClowpdSy
achvH,) w-QH< -icowhth,E4,
- > (6.lf)
at drl =
where p = dJ(1 - q2)(S2 - 1)/2, ht and h, have been defined in Chapter 2.
Also, ~0 and ~0 are the permittivity and permeability of free space, respec-
t ively.
From Eqs. (6.la) to (6.lf) it is observed that the problem may be split into
two parts. If the applied field on the aperture has only an Eq component, the
excited magnetic field has only the H$ component and E+ = 0. On the other
hand, if the applied field has only an E4 component, the excited electric field
has only an E+ component and H& = 0.
- PROLATE SPHEROlDAL ANTENNA 147
iv=+1
I
:
b I
:
I
i T -- 1
Fig. 6.1 Prolate spheroid model of an antenna.
- 148 SPHEROIDAL ANTENNAS
Here we consider only the former case; that is, the applied field has only an
I$, component and is considered in the subsequent analysis. Then, following
Schelkunoff [35], we set
A
H4=p’ (6 .2)
where A is an auxiliary scalar wave function to be defined later. Therefore,
from Eqs. (6.la) and (6.lb), we obtain
4 1 dA
Et = - (6.3a)
&icowd2 Jcrz - l)(
- PROLATE SPHEROIDAL ANTENNA 149
The orthogonality condition for m = 1 is provided as follows:
(6.8)
In general, in the exterior region (< > = -@&)I~=& and ~1 has been replaced by ~1 = ~~1~0,to
allow for any dielectric medium. ~1 is the relative dielectric constant of
the medium. To obtain a,, the orthogonality condition of Eq. (6.8) can be
applied to Eq. (6.11). This is done by first multiplying both sidesof Eq. (6.11)
bY c2 - v2 and then integrating r) from -1 to +l. The outcome of this
manipulation is:
jqwd2
an = - 1’ E~({l,~)/ff-J$Vn(V) dV- (6.12)
4Nl,nu~(tl) -1
Equation (6.12) can also be written in the form
(6.13)
where
(6.14)
and 770 a fixed angular coordinate where the excitation gap is located. The
is
V is a generalized voltage. In the case of a delta gap, E: is zero
- 150 SPHEROIDAL ANTENNAS
everywhere on the spheroid surface except at some infinitesimally thin ring at
q = 70. Then, in this limiting case,
II
r7o+&lo
Vn =
-
-
-[i
V?
??o--nr)o
q5 - Wh, d77
bo-+O
the voltage across the slot. (6.15)
6.2.5 Far-Field Expressions
In the far field (5 ---) oo), by substituting Eq. (6.9) into Eq. (6.2), the following
approximate expressions of the magnetic and electric fields can be obtained:
2e-jklr O”
H4 x x &(+)ntZ~Vn(?j), (6.16)
IclrdJr--;r;i n=l2
9
where ICI = ,/Gko = w,/m, and
(6.17)
where q = cos8.
6.2.6 Numerical Computations and Mathematics Code
Using Eqs. (6.13) and (6.15), the coefficients an can be obtained. The sub
script n can range from 1 to 00, but it is possible to obtain truncated coeffi-
cients up to only 30. In the Mathematics code, the number of coefficients used
in the actual computation of the magnetic field can be even smaller. These
values can then be substituted into Eq. (6.16) to compute the magnetic field.
The entire code is created as a Mathematics package called SpheroidAn-
tenna.nb. In this package there are three main modules, one for each of the
three problems: the uncoated antenna, the dielectric-coated antenna, and the
antenna enclosed in a radome. The second and third modules are discussed
in the following sections, respectively.
The first main module, for the uncoated spheroidal antenna, is a Math-
ematica module called Uncoated[ul-,erl-,1-,vO-,Vapplied-1, which com-
putes the radiation pattern. There are five arguments and they are specified
as follows:
l ul: the radial coordinate representing the boundary between the spher-
oidal antenna and the outer region
l erl: the dielectric constant of the outer region
l 2: the semi-interfocal distance
- PROLATE SPHEROIDAL ANTENNA 151
l ~0: the angular coordinate on which the excitation gap lies
l Vapplied: the applied voltage
The arguments ~1 and I together define the actual size and shape of the
antenna.
To gain brief insight into the workings of the module, a summary is provided
below:
l Based on the input arguments, the parameter cl is calculated.
l Using the value of cl, the expansion coefficients dp=(cl) and the eigen-
values X,, (cl) are computed using the support modules Getdmn and
EigenCommon, respectively.
l The spheroidal radial functions of the first, second, and fourth kinds and
their derivatives are evaluated next. This is achieved through the sup
port modules PSpheroidRl, PSpheroidRZ, PSpheroidRZforsmallc,
and PSpheroidR2forlargec.
l Next, the spheroidal angular function of the first kind is computed via
the PSpheroidS module.
l With the radial and angular functions computed and tabulated, it is
then possible to compute the expansion coefficients using a submodule
called [a- n]. In this submodule, the support modules V[m-,n,,c-,v-1,
NormFactor[m-,n-,c-1, and UPrime[m-,n-,c-,u,] are called.to com-
pute the angular function, the normalization factor N,,(c), and the
derivative of the radial function of the fourth kind, respectively.
l Prior to computing the &, pattern, the expansion coefficients are first
checked to remove those coefficients with magnitudes that are lessthan
lo? The reason for this is that these coefficients are too insignificant
to affect the computation of the H+ pattern and are removed to reduce
computation time.
l A table for the &, pattern is then computed by calling the support
module Hphi[c-,coeff-,theta].
6.2.7 Results and Discussion
In this section we present and discuss results obtained for prolate spheroidal
antennas with different parameters. Here the semi-interfocal distance is indi-
cated as 1 = d/2. To verify the Mathematics code written for the uncoated
case, results were obtained and compared to those obtained by Weeks [135].
Figure 6.2 shows the radiation patterns for spherical antennas of radii
0.81X0 and 2.1x0 with different excitation gaps. The dotted points stand
for results from Weeks [135]. The results obtained in this chapter are in good
- 152 SPHEROIDAL ANTENNAS
agreement with existing results. Therefore, the accuracy of the code has been
verified. The results also suggest that the number of lobes increases with ra-
dius. Furthermore, for the asymmetrically excited spheres, it can be seen that
increasing radius sharpens the main lobe located near the excitation gap.
Figure 6.3 shows the radiations patterns from prolate spheroidal antennas
excited at 8 = 30’ (Q = 0.866) and 0 = 90° (7 = 0), where semi-interfocal dis-
tances are Z/X0 = 5/27r and 47r, respectively. Again, the results are compared
with those of Weeks to show the agreement between results. One similar ob-
servation as in Fig. 6.2 is that the number of lobes increases with increasing
dimensions of the spheroid, indicated by the semi-interfocal distances.
Figure 6.4 shows the variation in radiation patterns when the shape of the
prolate spheroidal antennas is changed. From the first set of patterns it can
be observed that the effect of varying the radial coordinate of the spheroid
from 1.005 to 1.10 is insignificant. On the other hand, from the second set
of patterns, when larger variations were made, the number of lobes in the
patterns is seen to increase with the radial coordinate of the antenna. This
is expected because as
- DIELECTRIC-COATED PROLATE SPHEROIDAL ANTENNA 153
,+ R=0.81
__zff_ R=2.41
0.8
0 R=0.81 (Weeks)
A R=2.41 (Weeks)
0.6
s-
z
0.4
0.2
0.0
0 90 180
@ to)
(a) q~ = 0, and R = r/X0
0.8
0.6
0.4
0 90 180
@ to)
(b) q) = -0.940, and R = r/X0
Fig. 6.2 Radiation patterns of I&,1 versus 7, due to spherical antennas with different
conducting sphere radii excited by slots at (a) ~0 = 0 and (b) ~0 = -0.940.
- 154 SPHEROIDAL ANTENNAS
1.0
0.8
0.6
I 0 ~0=0.866 (Weeks)
A TJO=O (Weeks)
0 90 180
@ to)
(a) t1 = 1.077, and Z/X0 = 5/2n
1.0
0.8
0.6
0.4
0.2
0.0 I I
I
0 90 180
@ co)
04 51 = 1.077, and Z/AC) = 4n
Fig. 6.3 Radiation patterns of prolate spheroidal antennas excited by slots at 70 =
0.866 and 0 with different semi-interfocal distances are 1/X0 = (a) 5/27~ and (b) 47~
- DIELECTRIC-COATED PROLATE SPHEROIDAL ANTENNA 155
1.0
0.8
0.6
0.4
0.2
0.0
0 90 180
6, to)
(a) t1 = 1.005, 1.077, and 1.1; and Z/X0 = 0.25
1.0
0.8
0.6
0.4
0.2
0.0
0 90 180
0 to)
u-4 45 = 1.1, 2, and 5; and ~/XC) = 0.25
Fig. 6.4 Radiation patterns of prolate spheroidal antennas excited by slots at q = (
with different radial coordinates of (a)
- 156 SPHEROIDAL ANTENNAS
1.0
0.8
0.6
0.4
0.2
0.0
0 so 180
0 co)
(a)
- DIELECTRIC-COATED PROLATE SPHEROIDAL ANTENNA 157
1 .o
0.8
0.6
0.4
0.2
0.0
0 90 180
e (0)
(4 51 = 1.077, and 1/X0 = 0.25
0.8
0.6
0.0
0 90 180
e (0)
(b) 51 = 2, and I/& = 0.25
Fig. 6.6 Radiation patterns of prolate spheroidal antennas with different excitation
slots locations and radial coordinates (a)
- 158 SPHEROlDAL ANTENNAS
surroundings, the result of which may be corrosion of the antenna. To get
around this problem, the antenna is often coated with a protective layer.
The geometry of the coated prolate spheroidal antenna is similar to that in
Fig. 6.1, but with a confocal dielectric layer. The thickness of the dielectric
layer along the semiminor axis is indicated as t. A detailed illustration of the
antenna and coating geometry is provided in Fig. 6.7. The first region (region
I) contains a coating with a dielectric constant of ~~1. It is bounded by 5 = ci
and { = &. The second region (region II), defined by c > c2, has a dielectric
constant of ~2, which is actually unity in most cases, to represent free space.
There are two regions with different characteristics. Therefore, it is expected
that the auxiliary wave function in each region will be different, too.
6.3.2 Obtaining the Auxiliary Wave Functions
6.3.2.1 Region I ((1 < c < 62) We shall first look at the region imme-
diately adjacent to theantenna surface (i.e., the region from & to
- DIELECTRIC-COATED PROLATE SPHEROIDAL ANTENNA 159
jq=+l
I
I
I
:
:
I
I -- 1
i v- --
v-
Fig. 6.7 Prolate spheroid model of a coated antenna.
- 160 SPHEROIDAL ANTENNAS
The coefficients in Eq. (6.21) can then be calculated from
jqwd2
J d ’ Ea rt - q2
Pn = - -Vn (Cl 3 rl) drl7 (6.22)
4Nl,n(Cl) -1 ’ IL - V2
which in the case of an infinitesimally thin slot is given by
jcowd
V(Cl, rlo)K (6.23)
pn z zNl,n(Cl)
where Vs is the voltage applied acrossthe slot. Next, we substitute Eq. (6.20)
into Eq. (6.18a), to obtain
4
(6.24)
Comparing Eq. (6.24) with Eq. (6.20), it can be seen that
1 t
pn = M~~~(cl~t) + NnTn(clYC)* (6.25)
Equation (6.25) can also be rewritten as
- Pn - M~u~(cl~tl)
N1
n- (6.26)
Ty&l,&) l
Substituting Eq. (6.26) into Eq. (6.21), Eq. (6.20) becomes
Al = 2 K(Cl7rl)
n=1,2
(6.27)
6.3.2.2 Region /I (e >, 62) On the other hand, for the region from 52 to 00,
the appropriate representation for A2 is given by
A2 = 2 MzUn(C2,
- DIELECTRIC-COATED PROLATE SPHEROIDAL ANTENNA 161
6.3.3 Imposing the Boundary Conditions
The boundary conditions, which require continuity of the tangential fields,
can be obtained by comparing Eq. (6.18a) with Eq. (6.29a) and Eq. (6.18b)
with Eq. (6.2913)at < = 52. Aft er simplification, the boundary conditions can
be stated as follows:
Al = A2)5=62 7 (6.30a)
(6.30b)
It is observed that the left- and right-hand sidesof Eqs. (6.30a) and (6.30b)
cannot be matched term by term due to the different Vn’s on both sides.
However, Vn can be expressed as follows:
(6.31)
where the coefficients d.)n(cl) are nonzero when T is even (odd) and 72is odd
(even). P!+,(q) is the usual associated Legendre function. Since the same
P:+,(q) exists on both sides of the equations now, it is possible to make use
of the orthogonality of Legendre polynomials or to compare both sides of the
equations using the coefficients of P:+,.(v). Therefore, Eqs. (6.27) and (6.28)
can now be written in the respective forms
US(c1 9 ~l)Tn(% 0 + pnTn(c1,t)
Un(Cl,
- 162 SPHEROIDAL ANTENNAS
from those of even order. This implies that the problem can be broken via
decomposition into two parts: solving for the even-ordered coefficients
solving for the odd-*ordered coefficients.
In principle, the problem of the coated spheroidal antenna is solved once
the Mz coefficients are obtained. These coefficients can then be substituted
into
(6.34)
to calculate the magnetic field.
6.3.4 Numerical Computations
To solve for unknown coefficients Mi and M#, it is possible to cast the equa-
tions in matrix form. To do so, it is convenient to define some intermediate
terms as follows:
In U#A 9 tl)Ta(cl 7 t2)
A(r,n) = dr* (~1) K&I&) - ? (6.35a)
q(cl,cl) >
In &h2Tn(clk2J
B(r, n) = dr9 (~1) (6.3513)
q!Jcl,tl) ’
In
c(r, n, = d,) (c2)u-(c2Y C2)9 (6.35~)
In u;(cly &)q!&(c19 cd
D(r, n) = dr9 (Cl) us - 9 (6.35d)
T;(c1,
In pnTA(c13 t2)
E(r, n) = dry (cl) (6.35e)
T;(cl,tl) ’
F(r, n) = 3yc2)U~(c2,
- DIELECTRIC-COATED PROLATE SPHEROIDAL ANTENNA 163
x03
x23
x05
x25
..
..
... , n odd
x43 x45
...
1
x12 x16
x32 x36
, n even
x52 x56 . l .
. . .
with X being either A, C, D, or F;
cy 2n cy 47-h
. . .
T
> , n odd
T
> , n even
3n 5n . . .
cy cy
with Y being either B or E; and
-
22 24 26 l * )‘,n even
with 2 being either M1 or M 2. The matrix equation (6.37) can then be solved
twice, for the even and odd cases.
In this section, the truncation number is taken to be 30. This means that
there are a total of 60 coefficients to be computed (30 each for MA and M..).
However, since the odd and even casesare decoupled from each other due to
the nature of the coefficients diy”, each case will have 30 coefficients to be
solved for. For the odd case, there have to be 30 equations in order to have
30 unique solutions. Since there are two equations for every value of r, r will
have to range from 0 to 28 in steps of 2. For the even case, T will range from
1 to 29 in steps of 2. To test for convergence of the results, computations
were performed for truncation numbers of 30 and 34. Upon comparing the
results, it is observed that the two sets of results match at least to three
significant figures, and the smallest value in the solution can be considered
to be negligible compared to the largest value in the same solution. For the
largest value in the solution set, the results match up to 13 significant figures.
Therefore, a truncation number of 30 is sufficient to obtain highly accurate
results.
6.3.5 Mathematics Code
The module for the dielectric-coated spheroidal antenna is called Coated[ul-,
u2-, erl-, er2-, l,, vO,, Vapplied-1. There are seven arguments and they
are specified as follows:
- 164 SPHEROIDAL ANTENNAS
l ul: the radial coordinate representing the boundary between the spher-
oidal antenna and region I
l u2: the radial coordinate representing the boundary between the regions
I and II
l erl: the relative dielectric constant of region I
l er2: the relative dielectric constant of region II
l 1: the semi-interfocal distance
l ~0: the angular coordinate on which the excitation gap lies
l Vupplied: the applied voltage
This module developed for computation can be summarized as follows:
Based on the input arguments, the parameters cl and c2 are calculated.
Using the values of cl and ~2, the expansion coefficients drmn(c) and
the eigenvalues A,, (c) are computed for each c using the supporting
modules Getdmn and EigenCommon, respectively.
The spheroidal radial functions of the first, second, third, and fourth
kinds and their derivatives are evaluated subsequently. This is achieved
through the supporting modules PSpheroidRl, PSpheroidR2, PSph-
eroidR2forsmallc, and PSpheroidR2forlargec. These are computed
for all combinations of c and
nguon tai.lieu . vn