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Fig. 1. (a) Coaxially fed monopoles in a parallel-plate waveguide (b) single
monopole in a parallel-plate waveguide.
Coupling Between Coaxially Fed Monopoles in a
Parallel-Plate Waveguide
Mi Y. Park, Hyo J. Eom, and Yong B. Park
Abstract—Electromagnetic coupling between coaxially fed monopoles in
a parallel-plate waveguide is investigated. The boundary-value problem of
monopoles in a parallel-plate waveguide is solved using the Fourier transform, mode matching, and superposition. Residue calculus is used to represent the solution in a fast convergent series. Our numerical results agree
favorably with Computer Simulation Technology Microwave Studio (CST
MWS) simulation results.
Index Terms—Coaxially fed monopole, Fourier transform, mode
matching technique.
I. INTRODUCTION
Coaxially fed monopoles have been widely used as feeding structures in microwave devices and antennas. The problems of a coaxially fed monopole in free space or in a parallel-plate waveguide have
been studied using various analytic techniques [1]–[6]. Radiation from
a single coaxially fed monopole in a parallel-plate waveguide has been
considered in [7] using the Fourier transform and mode matching. The
motivation of the present communication is to develop a formulation
for coupling between coaxially fed monopoles in a parallel-plate waveguide. The boundary-value problem of coaxially fed monopoles in a
parallel-plate waveguide will be solved by utilizing the Fourier transform, mode matching, and superposition. Although the derivation procedure is rather complicated, our approach gives an analytic, fast convergent series solution. When the outer radius a2 of a monopole is small
Manuscript received October 20, 2004; revised March 23, 2005. This research was supported by the ElectroMagnetic Environment Research Center
(EMERC), Chungnam National University, one of IT Research Centers.
M. Y. Park and H. J. Eom are with the Department of Electrical Engineering
and Computer Science, Korea Advanced Institute of Science and Technology,
Daejeon 305-701, Korea (e-mail: hjeom@ee.kaist.ac.kr).
Y. B. Park is with the Convergence Laboratory, Korea Telecom, Seoul 137792, Korea.
Digital Object Identifier 10.1109/TAP.2005.854546
compared to the wavelength (a2 = 1), the azimuthal field variation within the outer radius (j < a2 ) is negligible (@=@j 0). Our
analysis assumes no azimuthal field variation within the outer radius,
thereby using a TM0m mode series. Our theory is valid only when the
outer radius of a monopole is small compared to the wavelength. The
notations used in this paper somewhat follow those in [7].
II. FIELD REPRESENTATIONS
Consider coaxially fed monopoles in a perfectly conducting parallelplate waveguide, as shown in Fig. 1(a). For illustration, two monopoles
are shown. For the convenience of analysis, the region is divided into
Ij (j < a1 , 0 < z < h1j ), II, and IIIj (a1 < j < a2 , 0 <
z < 1), where region II is the intersection of regions IIj (j >
a2 , 0 < z < h2 ) of all monopoles and j = 1; 2; . . . ; N . The time
convention e0i!t is suppressed throughout. Assume that a transverse
electromagnetic (TEM) mode excites each monopole with a different
amplitude sj . The total field in region IIIj consists of the incident,
reflected, and scattered components as in [7]
0ikz
e
i
H (j ; z) = 0
sj
(1)
j
ikz
e
r
H (j ; z) = 0
sj
(2)
j
1
2
III
Ez
(j ; z) =
Rj (j ) cos(z)d
(3)
i!
0
where (j ; j ; z) is cylindrical coordinates centered at the j th
p
k2 0 2 ,
monopole, k = ! is the wavenumber, =
+
=
= is the intrinsic impedance, Rj (j ) = J0 (j )Ej () 0
0
+
0
N0 (j )Ej (), Ej () and Ej () are the unknown coefficients, and
the notations, J0 (1) and N0 (1), are the Bessel functions of the first and
second kinds, respectively. The scattered field in region Ij is
I
Ez (j ; z) =
i
!
1
m=0
pjm 1jm cos(h1jm z)J0 (1jm j )
(4)
2 . No
where h2j = h2 , hpjm = m=hpj , and pjm = k2 0 hpjm
azimuthal j variation is assumed in the scattered field expressions, (3)
and (4).
0018-926X/$20.00 © 2005 IEEE
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 9, SEPTEMBER 2005
Consider a coaxially fed monopole in a parallel-plate waveguide,
as shown in Fig. 1(b). The scattered field in region IIj (j > a2 ,
0 < z < h2 ) is
1
EzII (j ; z ) = i
q cos(h2jm z )H0(1) (2jm j )
! m=0 jm 2jm
(5)
where H0 is the zeroth-order Hankel function of the first kind. To
solve the problem of a linear array of coaxially fed monopoles in a
shorted parallel-plate waveguide [8], the coaxially fed monopole array
has been replaced by equivalent currents and the superposition has been
applied to the currents. The superposition principle is applicable to our
problem if every coaxial aperture and monopole is approximately replaced by a magnetic frill current and an induced electric current, respectively. All those currents are approximately j -independent since
the outer radius of a coaxial line is assumed to be small compared to
the wavelength. By the superposition principle, the scattered field in
region II of Fig. 1(a) is the sum of each single coaxial line contribution
of Fig. 1(b) as
(1)
EzII =
N
`=1
Fig. 2.
Graf’s addition theorem.
Substituting (12) into (9) and applying the Fourier cosine transform
gives (13), shown at the bottom of the page. Multiplying the tangential
magnetic field continuity at (j = a1 and 0 < z < h1j ) by cos(h1jn z )
for n = 0; 1; . . . and integrating with respect to z gives [7]
I1j (n) + pjn h1j n J1 (1jn a1 ) = 0
s:
(14)
a1 j
Similarly, the tangential magnetic field continuity j = a2 yields
I (n) + q h2 H (1) ( a )
(6)
2j
jn
m=0
where
p
Fmj
( ) =
1
pjm 1jm J0 (1jm a1 )Fmj
( )
0
( 1)
m
2
(7)
sin(hpj ) :
0 hpjm
(8)
2
Similarly, the tangential electric field continuity at j = a2 is given by
II
EzIII (a2 ; z ) = Ez (j = a2 ; z );
0;
for 0 < z < h2
otherwise
(9)
where
EzII = EzII (j ; z ) +
N
`=1
`6=j
EzII (` ; z ):
+
1
n=01
H0(1) (2`m ` ) H0(1) (2`m dj` )J0 (2`m a2 ):
m=0
1
2jn 2
q`n h2 n H0(1) (2`n dj` )J1 (2`n a2 )
2
nj (k )
sj
0 2Fa
2
=
(15)
2
where 0 = 2, n = 1 (n = 1; 2; . . .)
11 0
Ipj (n) = 0 2
(16)
R (a )F p ( )d
0 j p nj
and Rj0 (ap ) = @Rj (j )=@ (j )j =a . Applying residue calculus
to (16) gives fast convergent series (17) and (18) shown at the bottom
of the following page, where I jp (m; n) and I jpq (m; n) are given by
[7, (14) and (15)] with the parameters hp , hpm , and pm replaced by
hpj , hpjm , and pjm , respectively. Equations (14) and (15) constitute a
set of simultaneous equations for the modal coefficients pjm and qjm .
The sum of reflected and scattered TEM fields at z = 1 in region IIIj
is given by
ikz
Hr (j ; z ) + HIII (j ; z ) = 0 e (sj + L0j 0 M0j )
j
(19)
where, see (20) and (21) at the bottom of the following page. When
= 1, our final results (14) and (15) are reduced to [7, eqs. (9) and
(12)].
)
IV. NUMERICAL RESULTS
(11)
where dj` is the distance between the j th and `th monopoles. In view of
(11), the scattered field in region II, EzII , is shown to have an azimuthal
j dependence. Since the outer radius of a monopole is small compared
to the wavelength (a2 ), the expression (11) at j = a2 becomes
1
n
N
Hn(1) (2`m dj` )Jn (2`m j )ein( 0
Rj (a2 ) = 0
`=1
`6=j
(10)
Graf’s addition theorem [9] in Fig. 2 gives
H0(1) (2`m ` ) =
2
N
The enforcement of boundary conditions is necessary to obtain a set
of simultaneous equations for the modal coefficients pjm and qjm . Ap1
plying the Fourier cosine transform 0 (1) cos( 0 z )dz to the tangential electric field continuity at j = a1 yields [7]
Rj (a1 ) = 0
1
2
EzII (` ; z ):
III. ENFORCEMENT OF BOUNDARY CONDITIONS
1
2Fnj (k )
(12)
qjm H0(1) (2jm a2 ) +
N
`=1
`6=j
To check the validity of our formulation, our results for nine
monopoles in a parallel-plate waveguide are compared to the Computer Simulation Technology Microwave Studio (CST MWS) [10]
simulation results. The problem geometry is shown in Fig. 3. When
a TEM wave excites the first monopole, our computed S -parameter
magnitudes are shown in Fig. 4. Minima jS11 j and maxima jS`1 j
2
q`m H0(1) (2`m dj` )J0 (2`m a2 ) 2jm Fmj
( ):
(13)
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 9, SEPTEMBER 2005
3111
Fig. 3. Nine coaxially fed monopoles in a parallel-plate waveguide.
(` = 2; 3; . . . ; 9) occur when the lengths of monopoles are odd
multiples of quarter wavelengths (h2 0 h1j = n=4, n = 1; 3; . . .).
The comparison between our results and CST MWS simulation results
shows good agreement up to 8 GHz (a2 = = 0:129). Our results
start to deviate from the CST MWS simulation results beyond 8
GHz because our assumption (a2 = 1) is no longer valid. The
convergence of the S -parameter magnitudes is checked in terms of the
mode numbers m in regions Ij and IIj . Table I shows the convergence
behavior of the S -parameter magnitudes for nine monopoles. Note
that m = 10 is sufficient to achieve convergence to within 2% error in
the S -parameter magnitudes. Ten modes are used in our computation.
In the CST MWS simulation, the finite-sized parallel plates (750
mm 2 750 mm) are assumed and the automatic mesh generation is
used with a 030 dB accuracy solver setting. Our additional computation shows that the admittances p
of two monopole antennas in a
parallel-plate waveguide with h12 = 2 [6] compare favorably with
those of two monopole antennas on a ground plane radiating into a
half-space [11, Figs. 4–6].
I1j (n) =
1
m=0
+
I2j (n) =
0
+
L0j
=
M0j
=
0 m n pjm jm J ( jm a )I j
( 1)
1
1
m=0
m=0
i
k ln
1
a
a
a
a
1
1(
jm a2 ) +
(1)
0 ( 2
+
1
0 m n qjm H
+
1
m=0
1
m=0
0
1
1
jm a2 ) +
(1)
0 ( 2
-parameter magnitudes versus frequency.
= 2 1 mm,
=
= 15 mm,
= 90 mm,
= 15 mm (
=
1 2 . . . 9). The solid lines represent our numerical results and the dash lines
represent the CST MWS simulation results.
2 301
,
V. CONCLUSION
The problem of coupling between coaxially fed monopoles in a
parallel-plate waveguide has been solved using the Fourier transform,
mode matching, and superposition. A simple, numerically efficient
solution has been obtained when the outer radius of a coaxial line
m; n)
0 m n pjm jm J ( jm a )I j
( 1)
i
0
+
( 1)
1
1
0 m n qjm H
( 1)
m=0
k ln
+
Fig. 4.
N
`=1
`6=j
21 (
q`m H0(1) (2`m dj` )J0 (2`m a2 )
1 jm I j
12 (
1 jm I j
2(
2
m; n)
m; n)
N
`=1
`6=j
q`m H0(1) (2`m dj` )J0 (2`m a2 )
2
m; n)
1
pjm 1jm J0 (1jm a1 )Fmj
(k )
qjm H0(1) (2jm a2 ) +
(17)
N
`=1
`6==j
(18)
(20)
q`m H0(1) (2`m dj` )J0 (2`m a2 )
1 jm Fmj (k):
2
2
(21)
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 9, SEPTEMBER 2005
TABLE I
CONVERGENCE BEHAVIOR OF THE -PARAMETER MAGNITUDES VERSUS THE
= 2 1 mm,
= 2 301
, = 15 mm,
MODE NUMBERS .
= 90 mm,
= 15 mm ( = 1 2 . . . 9),
= 6 7 GHz
[2]
[3]
[4]
[5]
[6]
[7]
is small compared to the wavelength. Our solution agrees well with
the CST MWS simulation results. Our analytic formulation is useful
for estimating coupling between coaxially fed monopoles in a parallel-plate waveguide.
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