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A Closed-Form Solution of Vertical Dipole Antennas above a Dielectric
Halfspace
Article in IEEE Transactions on Antennas and Propagation · January 1994
DOI: 10.1109/8.273319 · Source: IEEE Xplore
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION,VOL. 41, NO. 12. DECEMBER 1993
[8] S. L. Ray, “Grid decoupling in finite element solutions of Maxwell’s
equations,” IEEE Trans. Antennas and Propagation, vol. AP-40,
[9]
[lo]
[I 11
[I21
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the condensed node TLM mesh,” IEEE Trans. Magnetics, vol. 27,
no. 5, pp. 3982-3985, Sept. 1991.
R. W. Noack, “Time domain solutions of Maxwell’s equations
using a finite-volume formulation,” Ph.D. dissertation, Univ. Texas
at Arlington, May 1991.
R. Holland, “Finite-difference solution of Maxwell’s equations in
generalized nonorthogonal coordinates,” IEEE Trans. Nucl. Sci.,
vol. NS-30, no. 6, pp. 4589-4591, Dec. 1983.
N. K. Madsen and R. W. Ziolkowski, “A modified finite-volume
technique for Maxwell’s equations,” E/ectromagnet., vol. 10,
no. 1-2, pp. 147-161, Jan.-June, 1990.
T. D. Deveze, L. Beaulieu, and W. Tabbara, “An absorbing boundary condition for the fourth order FDTD scheme,” presented at
IEEE Antennas Propagat. Soc. Int. Symp. (Chicago, IL),
pp. 342-345, July 18-25, 1992.
A Closed-Form Solution of Vertical Dipole Antennas
above a Dielectric Half-space
R. M. Shubair and Y. L. Chow
Abstract-This paper derives a convenient closed-form expression for
the input impedance of a vertical antenna above a dielectric half-space.
The expression is obtained from the induced EMF method using a
complex-image spatial Green’s function. It is found that the effect of the
dielectric half-space can be modeled by a short image array of three to
five complex image antennas. Numerical results verify the accuracy and
convenience of the method for a vertical antenna at any height above a
dielectric half-space with and without loss.
combining it with the induced EMF method [7]-[9] to derive an
expression for the input impedance of the vertical antenna
above the dielectric half-space. Based on the Green’s function
properties mentioned above, the effect of the dielectric halfspace on the vertical antenna can be modeled through the
complex image technique by the introduction of an image array
consisting of only a few quasidynamic and complex images of
the whole original vertical antenna. The image array elements
are all located in homogeneous free-space and, except for complex amplitude changes, have the same current distribution as
the original antenna.
In addition, as one complex-image array is found for a given
antenna height above the dielectric half-space, then for a different antenna height, say upwards, the complex-image array
remains unchanged except that it is bodily translated downwards
by the same distance. That is the complex images behave just
like a classical image in front of a mirror. This means that as the
induced EMF method is applied once, then at other heights the
convergence is even faster since the complex images need not to
be recalculated.
The outline of this paper is as follows. Section 11-A describes
the induced EMF method for a vertical antenna above a dielectric half-spaces. Section 11-Bbriefly describes the complex image
technique for the derivation of the vector potential Green’s
function. Section 11-C shows that with the use of the complex
image Green’s function, an expression for the input impedance
is obtained as the superposition of the antenna’s self-impedance,
and the mutual impedances due to the quasidynamic and complex image antennas. Numerical results are obtained in Section
I11 showing the image coefficients of the image array, as well as
the input impedance as a function of the antenna height above
the dielectric half-space, with and without loss. Finally, conclusions are given in Section IV.
I. INTRODUCTION
11. THEORY
The electromagnetic modeling of the radiation by vertical
antennas above a dielectric half-space is an important problem
[l]. In a previous paper by Shubair and Chow [2], the input
impedance has been obtained from the moment method using
closed-form spatial Green’s function for the vector and scalar
potentials [3], [4]. These Green’s functions are given in closedform by the superposition of contributions due to only a few
(3 to 5) quasidynamic and complex images.
The derivation of the complex image Green’s functions for
microstrip substrates was presented by Chow et a1 and co-workers
[5] as well as recently by Aksun and Mittra [6]. There, it was
shown that the closed-form complex image Green’s function has
the advantage of reducing the computation time of evaluating
the Sommerfeld integral and, thus, the fill-in time of the
impedance matrix generated for the moment method. For a
uertical antenna, the convergence is further enhanced by the fact
that the scalar potential is directly proportional to the vector
potential [2], and hence the former can be grouped with the
latter. This property does not occur for the horizontal antenna.
This paper provides another application of the complex image
Green’s function, with still faster convergence. This is done by
Manuscript received December 12, 1992; revised June 1, 1993.
The authors are with the Department of Electrical and Computer
Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1,
Canada.
IEEE Log Number 9214100.
A. The Induced EMF Method for a Vertical Dipole Antenna aboue
a Dielectric Half-space
Fig. 1 shows a center-fed vertical dipole antenna of length (21)
located at a height ( h ) above a dielectric half-space with parameters ( p 0 , ereo).Note that the height h is measured from lower
end of the vertical antenna. In the following development, the
time variation e’“ is assumed and suppressed.
The electric field due to a surface current J flowing along the
antenna is obtained from Pocklington’s integral [71:
where k , is the free-space wavenumber ( k , = w&).
r and
r‘ are tbe position vectors of the field and source points, respectively. CAis the dyadic Green’s function for the vector potential
due to an infinitesimal unit-strength vertical dipole. Gq is t&e
Green’s function for the scalar potential, and is related to GA
through the Lorentz gauge:
Sommerfeld’s choice for the vector potential due to an &finitesimal dipole oriented along the vertical direction z gives G$??G,“,
and according to (2), Gq is obtained such that JGJdz’ =
-dG,’I/dz. This means that the scalar potential is directly
proportional to the vector potential, except for a constant of
0018-926X/93$03.00 0 1993 IEEE
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 41, NO. 12, DECEMBER 1993
1738
Sommerfeld-type integral [lo]:
kEz
where k , is the spectral radial wavenumber, the radial distance
d(.x
+
=
-x02
( y - y’)’, Hd2) is the Hankel function of
order zero second kind, the constant K = ( 1 - e r ) / ( 1 E,)
corresponds to the amplitude of the quasidynamic image, and
the spectral function f is given by
p
+
(7)
Eo
The z-directed wavenumbers k,, and k,, are defined such that
hl
k:,
Fig. 1. A vertical dipole antenna of length (21) at a height ( h ) above a
dielectric half-space.
integration. This nice property exists only for vertical antennas.
In this case, the z-component of the electric field, which is the
parallel component along the vertical antenna, is obtainable
from ( 1 ) in terms of G T only:
The electric field is then used to derive the following induced
EMF expression for the input impedance [8]:
+ ki = ki,
k:,
+ k:
=
E,ki.
(8)
The Sommerfeld integral in (6) is well known to be slowly
convergent which makes its numerical evaluation time-consuming. However, one can use the complex image technique [2]-[6]to
evaluate the Sommerfeld integral analytically. In so doing, the
spectral function f ( k , , ) is approximated by of a short series of
exponential functions of the complex spectral variable k,,,, i.e.,
f(k,,)
=
a l e b l k z+
~ ~a2ebzkro +
...
-
(9)
As pointed out in [2], [5] a value of N = 3 5 is usually
sufficient to ensure a highly accurate approximation of f (i.e.,
error within 1%).The complex coefficients a; and 6, are obtained from Prony’s method [13], which involves the solution of
an Nth order difference equation and an N X N system of
linear equations. By substituting (9) into (6), one can use Sommerfeld identity [12] to perform the resulting integral analytically in terms of spherical wave components:
(4)
Kz’)
=Z
,
sin k , [ l - Iz’ - (1
+ hill,
h I z’ I 21
+ h,
(5)
where I, is a constant current. The above sinusoidal current
approximation has been validated for a dipole antenna in freespace especially when the antenna radius is small [7]. Moreover,
we have found that the same sinusoidal current assumption
frequently has an error in the input impedance that is approximately 1% for different antenna lengths. The sinusoidal current
assumption deteriorates a little when the vertical antenna is
close to the half-space interface (i.e., for h < 0.15h0). Despite
this slight inaccuracy, the uan’ational formulation of the induced
EMF still gives very accurate radiation impedance of the antenna (error 2%).
-
B. Complex Image Green’s Function for the Vector Potential
As seen from (31, the calculation of the electric field component E, requires knowledge of the vector potential Green’s
function Gjz. For an infinitesimal vertical dipole located at
(x’,y’, z’) above a dielectric half-space of permittivity E , € , , the
vector potential GT at (x,y , z ) takes the form of a
Ri
=
[ p 2 + ( z + z’ - j b j ) ’ ]
1/2
.
(11)
It should be noted that the spectral function f in (7) is independent of the vertical coordinates z and z’, and so does the
subsequent complex image coefficients a, and 6; of (9). This
means that for a dielectric half-space with a specific er, Prony’s
method needs only to be employed once to obtain the coefficients a, and b,. For different source and observation points, the
same set of coefficients can therefore be used to calculate the
vector potential GT, e.g., at different antenna heights and for
couplings between antennas in an antenna array.
It is easy to demonstrate that in the limit of infinite dielectric
constant, the quasidynamic image amplitude becomes K = - 1,
and the spectral function f in (7) vanishes so that the complex
image expression for G‘; in (10) reduces to the classical image
solution of an infinitesimal vertical dipole above a perfectly
conducting half-space, i.e.,
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. VOL. 41, NO. 12, DECEMBER 1993
C. The Expression for the Input Impedance
original
antenna
The advantage of the complex image Green's function is
evident when (10) for Gj' is substituted in (3). This Green's
function allows the resulting expression for E, to be integrated
analytically and simply into the three-term form of Schelkunoff
[7]. The simplified E, is then substituted in (4) so that the
resulting expression can again be integrated analytically to obtain the input impedance of the vertical antenna. The input
impedance is then found in closed-form as the superposition of
only few terms corresponding to the antenna's self impedance
and the mutual impedances due to the image antennas, i.e.,
1739
0
realquasi-dynamic
image ~
1'
Id]
dq = I+h
dl = I+h-jb,
'
d2= l+h-jb,
IV
Z,"
=
z,
-
c a,Z,,
E, +
I=
(13)
1
where the self impedance Z , results from the source term of G<*
in (lo), the mutual impedance 2, results from the quasidynamic
image term of GjZ in (IO), and the mutual impedance Z , results
from the i-th term of the complex image contribution of Gf;"
in
(10). These impedances are given by:
e -Jk
e -1k
11 R ,
U
. sin k,[l
-
e -1k
RI
Zs.,,,
= - j v " ~ 2 ~ + h j 2 c o s~ ( + ~
Iz - (1
oK:
-
+ h)ll dz,
1
complex
/I
Id..
Fig. 2. The equivalent model for a vertical antenna above a dielectric
half-space consists of the original antenna and its quasidynamic and
complex images all located in free-space.
(14)
where similar to the one antenna in Fig. 1:
TABLE I
C O M P L E X I M A G E COEFFICIENTS O F T H E SPECTRAL FUNCTlONf
FOR A
DIELECTRIC
H A L F - S P A C E WITH E , = 16 - j16.
(NUMBER
OF COMPLEX
IMAGES
N = 3)
1/2
[
R 2 = a 2 + ( z - z2)*]
1
Amplitudes U ,
(For Unit Source)
Locations b,
below the Quasi-Dynamic
Image (Meters)
1.4062 + j1.2901
-0.5213 - j1.0248
0.0996 - j0.1423
-0.2912 + j0.3841
0.1931 + j0.1519
0.0649 + j0.0288
and a is the antenna radius. For the images in Fig. 2, we have
for Z , ,
l+h
for Z ,
h,
1
2
3
jb, - ( I
+h)
for Z , ,
jb,
21+h
-
h,
+h)
for Z , ,
for Z,.
=
770
-[y+
4.rr
In(2.rr)
-
Ci(2.rr) +jSi(Z.rr)]
means that only the real part of the image locations changes
while the imaginary part remains fixed.
111. NUMERICAL
RESULTS
Following [9], the integral in (14) is easily evaluated in terms of
the simple sine and cosine integrals [lo], which are computationally rapidly-convergent. The simplest of these is the integral for
the self-impedance Z, for a half-wavelength dipole (21 = A,,/2):
Z,
~
for Z,
(16)
(jb, - (21
~
(01,
(17)
where 77" is the free-space intrinsic impedance, y is the Euler's
constant ( y = 0.57721, and Si and Ci are the sine and cosine
integrals [IO].
The solution in (13) means that the effect of the dielectric
half-space is modeled by the introduction of a short image array
consisting of the quasidynamic and complex images of the vertical antenna, all in homogeneous free-space. Fig. 2 illustrates
that the array element spacings are determined by the complex
image locations b,'s. The input impedance of the vertical antenna is then determined as the superposition of the self-impedance (ZJ, the mutual impedance (2,) due to the quasidynamic
image antenna, and the mutual impedances ( 2 , ' s ) due to the
complex image antennas. As the height h changes, the image
array bodily moves by the same displacement in the opposite
direction, just like a classical images. As observed in (16), this
Image Array of a Vertical Antenna aboue a Lossy Dielectric
Half-Space: For a lossy dielectric half-space with E, = 16 - j16,
a sample solution for the complex image coefficients a, and 6,
from Prony's method is given in Table I. The coefficients a,
define the current amplitudes of the images. Moreover, the
coefficients b, define the locations of the corresponding "N"
complex images (antennas) below the quasidynamic image, as
shown in Fig. 2.
It should be pointed out that the image coefficients a, and b,
listed in Table I are not unique. A different set of coefficients is
obtained for a different number of images, or a different approximation path [14]. Despite this, since the boundary conditions at
the half-space interface are satisfied, they all give the correct
image field according to the uniqueness theorem.
The Input Impedance: The input impedance of a half-wavelength vertical antenna is calculated from (13). Figs. 3 and 4
show the input impedance versus the antenna height above a
dielectric half-space with E , = 4 and E , = 16 - j16, respectively.
In these figures, the impedance values from (13) using the
complex image/induced EMF approach of this paper are compared to the values obtained directly from the moment method
[2]. It is found that the for heights above h = OSA,,, the input
IEEE TRANSACTIONS ON ANTENNASAND PROPAGATION, VOL. 41, NO. 12, DECEMBER 1993
I740
,
110
I
moment method
~
- _ _ _ induced EMF method
N
1
60
Gz
50
a
01
40
E
30
4
20
a
.rl
;
10
Y
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
antenna height, h/Ao
Fig. 3. Input impedance versus height above a dielectric half-space of a
half-wavelength vertical dipole antenna (er = 4).
h
110
v)
E
100
Y
2
90
E
80
7
70
+
and the mutual impedances due to the quasidynamic and complex image antennas. The derived expression for the input
impedance is convenient and highly accurate and can be used as
an alternative to the numerical solution from the moment
method.
It should be pointed out that for a horizontal, say an
x-directed antenna, the scalar potential G4 in the dielectric
half-space problem is linked through the Lorentz gauge (2) to
two vector potential components G T and
[lo]. This means
that the integrand of (1) to obtain the electric field component
parallel to the horizontal antenna involves two potential components rather than one. Hence, a similar image array model with
the induced EMF method cannot be developed for the horizontal dipole antenna above a dielectric half-space.
The approach described in this paper can be very useful for
finding the mutual impedance between different vertical antenna elements (e.g., in an antenna array) above a dielectric
half-space. As the effect of the ground on each vertical antenna
is modeled by a short image array, the coupling between the
antenna elements can be conveniently calculated in closed-form
using the induced EMF approach.
Finally, although the theory in this paper is developed for a
single-layer ground, it can be extended to derive the input
impedance of a vertical antenna located above or buried in a
multilayere6 ground. This is done by modifying the spectral
function f of (7) to account for the multilayered ground [12],
[14]. The above discussion points to the fact that a buried
vertical antenna can also be analyzed by the same complex
image/induced EMF approached, provided that the antenna is
located entirely within one layer of ground, i.e., does not penetrate into another layer [2].
~
N
60
moment method
EMF method
- - - - induced
c.= 16-jl6
1
R
0
50
a
w
a
E
.d
3
4
40
30
20
10
U
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
antenna height, h/Ao
Fig. 4. Input impedance versus height above the dielectric half-space of
a half-wavelength vertical antenna ( E , = 16 - j16).
impedance values from the two methods are in excellent match.
For heights below h = 0.15h0, the impedance values from the
two methods are within 2% difference. As pointed out in Section
IIA, the difference in the induced EMF method is due to the
slight inaccuracy of the sinusoidal current assumption for small
antenna heights.
IV. CONCLUSIONS
In this paper, through a complex image/induced EMF approach, we have shown that the problem of a vertical dipole
antenna above a dielectric half-space can be replaced by the
equivalent problem of a short array of vertical antennas radiating in homogeneous free-space. The array consists of the vertical
antenna and its quasidynamic and complex image antennas. A
closed-form expression for the input impedance of the original
vertical antenna above the dielectric half-space is obtained from
the induced EMF method of the image array. The expressions
represents the superposition of the antenna’s self-impedance,
REFERENCES
J. R. Wait, Electromagnetic Waues in Stratifed Media. New York:
Pergamon, 1970.
R. M. Shubair and Y. L. Chow, “A simple and accurate complex
image interpretation of vertical antennas in contiguous dielectric
half-spaces,” to appear in IEEE Trans. Antennas Propagation.,
June 1993, vol. AP-41, pp. 806-812.
D. G. Fang, J. J. Yang, and G. L. Delisle, “Discrete image theory
for horizontal electric dipoles in a multilayered medium,” IEE
Proc., vol. 135, pp. 297-303, May 1988.
R. M. Shubair and Y. L. Chow, “A simple and accurate approach
to model the coupling of vertical and horizontal dipoles in layered
media,” 1992 IEEE Antennas and Propagation Symposium Digest,
Chicago, IL, July 18-25, 1992.
Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “Closedform Green’s functions for the thick microstrip substrate,” IEEE
Trans. Microwaue Theory Technol., vol. MTT-39, pp. 588-592, Mar.
1991.
M. I. Aksun and R. Mittra, “Derivation of closed-form Green’s
functions for a general microstrip geometry,” IEEE Trans. Microwaue Theory Technol., vol. MTT-40, pp. 2055-2062, NOV.1992.
S. A. Schelkunoff and H. T. Friis, Antennas: Theory and Practice.
New York Wiley, 1952, pp. 369-370.
J. D. Kraus, Antennas. New York McGraw-Hill, 1988, pp.
413-422.
C. A. Balanis, Antenna Theory: Analysis and Design. New York:
Harper & Row Publishers, 1982, pp. 290-295.
A. Banos, Dipole Radiation in the Presence of a Conducting Half
Space. New York Pergamon, p. 35, 1969.
R. W. Hamming, Numerical Methods for Scientists and Engineers.
New York: Dover, 1973, pp. 620-622.
A. Sommerfeld, Partial Differential Equations in Physics. New
York Academic, 1949.
H. Abramowitz and A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York:
Dover, 1972, pp. 378-379.
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. VOL. 41. NO. 12. DECEMBER 1993
[I41 R. M. Shubair. "Efficient analysis of vertical and horizontal electric dipoles in multilayered dielectric media," Ph.D. dissertation,
Univ. Waterloo, Feb. 1993.
1741
closed form solution can be obtained, which may be used as a
first guess in an iterative numerical solution.
The function f ( N ) has the following useful asymptotic properties:
First.
A Procedure for Determining the Largest
Computable Order of Bessel Functions
of the Second Kind and
Hankel Functions
2
f ( N ) E -N
e
(4)
for N >> 1.
Second, inspection of the logarithm of f ( N ) , viz.
C. F. du Toit
Abstract-Routines for the computation of Bessel functions of the
second kind Y , ( z ) , and Hankel functions H,!')(z) and H j 2 ' ( z ) , may
exhibit numerical overflow problems for large orders n and small
complex or real arguments z . In this communication, an upper limit for
the largest calculable order of these functions for a given argument, is
derived.
reveals that the first term on the right dominates the other
terms when N is not much bigger or smaller than one and
In(M) ,> 1. Hence, for N = 1
I. 1NTRODUCTION
Numerical solutions of physical problems formulated in cylindrical coordinates [ l , Ch. 51 usually require routines for evaluating Bessel functions or Hankel functions 121-[SI. For small
values of IzI and large order n x- IzI, lYx(z)l, IH;')(z)l, and
IHA2'(z)l, are very large, and in an attempt to compute their
values, numerical overflow problems may be cncountered. A
simple way of preventing these problems is describcd in this
communication. The analysis is done in tcrms of Y , , ( i ) only, but
it also applies to the Hankel functions, sincc the latter are
dominated for large orders by the jY,,(z)-terms in their definitions.
11. A ~ A L Y S I S
Let M be the largest magnitude of Y,,(z), which may still be
processed safely in a particular numerical application, without
incurring numerical overflow problems. In other words, for a
given complex or real argument z , all useable values of n would
satisfy
lY,(z)l 5 h4.
(1)
Because I Y,(z)l increases monotonically with increasing n for
J z J In\, an upper bound N of n may in principle be obtained
from (1) by assuming equality and solving for n = N. Working
towards an approximate solution, it is noticed first that the
asymptotic expansion of Y,(i) for large n 19, Eq. 9.3.11 is
applicable here, due to the fact that Izl
INI. In other words,
According to these properties, it is therefore possible to solve (3)
for N in terms of 121, when N >> 1, or when N = 1. These
solutions may now be used to construct a crude, but more
general solution:
where g ( l z / ) is a function which is approximately equal to one
when (z1 is near zero (1 z+ J z J= O), otherwise it has to dampen
the effect of the first term on the right-hand side. This will
ensure that h = f - ' . Numerical experiments suggested the following empirical form for g(lzl):
where a and b are constants. As long as a and b are finite, the
condition limlzl+,) g(lzl) = 1 is satisfied. Solving for a and b to
satisfy the arbitrary conditions
h ( f ( 1 0 0 ) ) = 100,
(9)
h ( f ( 4 0 0 ) ) = 400,
(10)
yields the values a = 3.077 and b = - 1.420 for M = 10'").
Successively better approximations for N may be obtained
iteratively via the following formula:
Solving for IzI yields
Due to the transcendental nature of (31, it cannot be solved
algebraically for N in terms of 121. However, an approximate
for k = 1,2,3;.., starting with N , = h(lz1).
Values of N,, N , , and N3 are tabulated for various 121, with
M = IO"" in Table I. The exact value of N is also shown for
comparison. It is clear that only one or two steps in the iteration
are needed.
Manuscript received April 5, 1993; revised August 13. 1993.
The author is with University of Stellcnbosch, Department of Electrical and Electronic Engineering, Stellenbosch 7600, South Africa.
IEEE Log Number 9214101.
A quick and simple way of determining the highest order of
Y,(z), H ; ' ) ( z ) , and HA2)(z)which has a magnitude no larger
111. ANA P P R O X I M A T L SOI.UTION
FOR N
Iv. CONCLUSIONS
0018-926X/93$03.00 6 1993 IEEE
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