This document presents a rigorous solution for analyzing the radiation of a coaxially-fed monopole antenna into a parallel-plate waveguide. Fourier transform representations are used to model the scattered fields. Boundary conditions are enforced to obtain equations for the modal coefficients. A rapidly convergent series solution for the reflection coefficient is obtained and shown to agree well with other results. Input impedance and reflected power characteristics are calculated and presented.

1. Monopole Antenna Radiation into a Parallel-Plate
Waveguide
Hyo J. Eom, Yong H. Cho, and Min S. Kwon
Department of Electrical Engineering
Korea Advanced Institute of Science of Technology
373-1, Kusong-Dong, Yusong-Gu, Taejon, Korea
Phone: +82-42-869-3436, Fax: +82-42-869-8036
e-mail: hjeom@ee.kaist.ac.kr
Abstact A rigorous solution of coaxially-fed monopole antenna radiation into a
parallel- plate waveguide is obtained. The Fourier transform/series representations
are used to represent scattered elds and the boundary conditions are enforced to ob-
tain the simultaneous equations for discrete modal coe cients. Fast convergent series
of the re ection coe cient is obtained and compared with other existing results.
1 Introduction
Coaxially-fed monopole antenna radiation into a parallel-plate waveguide is of practi-
cal interest in microwave engineering and its characteristics are relatively well under-
stood 1-2]. An equivalent circuit representations for coaxially-fed monopole antenna
was obtained and compared with experimental data 1]. More recently, a modal
expansion analysis for coaxially-fed monopole antenna has been presented and its
impedance characteristics were numerically investigated 2]. In this paper, we intend
to present a rigorous, yet numerically-e cient solution for coaxially-fed monopole
antenna radiation into a parallel-plate waveguide. We use Fourier series/transform
representations for the scattered elds and apply the boundary conditions to obtain
a rigorous solution. Note that the Fourier series/transform approach was previously
1

2. used in 3] to obtain an exact solution for radiation from a dielectric- lled edge slot
antenna.
2 Field Analysis
Consider the problem of a coaxially-fed monopole antenna radiating into a parallel-
plate waveguide as shown in Fig.1. Assume that an incident TEM-wave with e i!t
time convention impinges upon a junction. In region (III), the total eld consists of
the incident, re ected, and scattered components as:
Hi = e ik3z=( r); (1)
Hr = eik3z=( r); (2)
EIII
z (r;z) = 2
i! 3
Z 1
0
R( r)cos( z)d : (3)
In regions (I) and (II), the scattered elds are
EI
z(r;z) = i
! 1
1X
m=0
pm 1m cos(h1mz)J0( 1mr); (4)
EII
z (r;z) = i
! 2
1X
m=0
qm 2m cos(h2mz)H(1)
0 ( 2mr); (5)
where hpm = m
hp
; pm =
q
k2
p h2
pm; =
q
k2
3
2
; kp = !p p; =
q
= 3; R( r) = J0( r) ~E+
( ) N0( r) ~E ( ); H(1)
0 ( ) = J0( ) + iN0( ), J0( ) is the
0th
order Bessel function, and N0( ) is the 0th
order Neumann function. The eld
representations (1) through (5) are somewhat similar to (1) through (7) in 3]. To de-
termine the modal coe cients pm and qm, we enforce the boundary conditions of the
Ez and H eld continuities. Applying the Fourier cosine transform (
R 1
0 ( )cos( z)dz)
to the Ez continuity at r = a1, Ei
z + Er
z + EIII
z = EI
zjr=a1 yields
R( a1) =
1X
m=0
pm
3
1
1mJ0( 1ma1)F1
m( ): (6)
Similarly from Ez continuity at r = a2,
R( a2) =
1X
m=0
qm
3
2
2mH(1)
0 ( 2ma2)F2
m( ); (7)
2

3. where
Fp
m( ) = ( 1)m sin( hp)
2
h2
pm
: (8)
Multiplying the H continuity at r = a1, for 0 < z < h1, by cos(n =h1)z and
integrating yield
I1 + pn
h1
2
J1( 1na1) n = 2 F1
n(k3)
a1
; (n = 0;1; ) (9)
where
I1 = 2 Z 1
0
1R0( a1)F1
n( )d ; (10)
0 = 2, n = 1 (n = 1;2; ), and R0( ) = dR( )=d( ). Using the residue calculus, it is
possible to transform (10) into a rapidly-convergent series. First identifying the simple
poles in the integrand of (10) , applying the residue theorem 4], and performing a
lengthy algebraic manipulation yields
I1 =
1X
m=0
( 1)m+n pm
3
1
1mJ0( 1ma1)I1 + qm
3
2
2mH(1)
0 ( 2ma2)I12]: (11)
Similarly from H continuity at r = a2 for 0 < z < h2,
I2 + qn
h2
2 H(1)
1 ( 2na2) n = 2 F2
n(k3)
a2
; (n = 0;1; ) (12)
where
I2 = 2 Z 1
0
1R0( a2)F2
n( )d
=
1X
m=0
( 1)m+n qm
3
2
2mH(1)
0 ( 2ma2)I2 pm
3
1
1mJ0( 1ma1)I21]; (13)
Ip = hp
2
p mn m
pm ( pm)
i( 1)pk3
2ap ln a2
a1
1 ei2k3hp
(k2
3 h2
pm)(k2
3 h2
pn)
i
ap
1X
v=1
v
1 (J2
0 ( va2)
J2
0 ( va1)
)( 1)p
]
1 ei2 vhp
( 2
v h2
pm)( 2
v h2
pn); (14)
Ipq = Xpq
ik3
2ap ln a2
a1
eik3jh2 h1j eik3(h2+h1)
(k2
3 h2
pn)(k2
3 h2
qm)
i
ap
1X
v=1
v
J0( va1)
J0( va2)
J0( va2)
J0( va1)
]
ei vjh2 h1j ei v(h2+h1)
( 2
v h2
pn)( 2
v h2
qm)
; (15)
X12 =
8
>>>><
>>>>:
2( 1)m
ap 2
2m ( 2m)
h2m sin(h1
h2
m )
h2
2m h2
1n
(h2 > h1)
2( 1)n
ap 2
1n ( 1n)
h1n sin(h2
h1
n )
h2
1n h2
2m
(h2 < h1);
(16)
3

4. mn is the Kronecker delta, ( ) = J0( a1)N0( a2) J0( a2)N0( a1), 1 = J0( 1ma2)
N1( 1ma1) J1( 1ma1)N0( 1ma2), 2 = J1( 2ma2)N0( 2ma1) J0( 2ma1)N1( 2ma2),
pm =
q
k2
3 h2
pm, v is given by ( v) = 0, and v =
q
k2
3
2
v. The re ected plus
scattered TEM eld at z = 1 is
Hr + HIII = eik3z
r (1 + L0 M0); (17)
where
L0 = i
k3 ln a2
a1
1X
m=0
pm
3
1
1mJ0( 1ma1)F1
m(k3); (18)
M0 = i
k3 ln a2
a1
1X
m=0
qm
3
2
2mH(1)
0 ( 2ma2)F2
m(k3): (19)
Fig. 2 (a) illustrates the comparison of antenna input admittance Yin =
2
ln(a2=a1)
1 + (1 + L0 M0)ei2k3h2
1 (1 + L0 M0)ei2k3h2
between ours and 2], indicating a favorable
agreement. We used 10 terms (30 terms) in the series of (18) and (19) to achieve
numerical convergence to within 1% error in conductance (susceptance) computation,
when h2 < 1:4 0. Fig. 2 (b) shows the normalized re ected power (j1 + L0 M0j2
)
versus the antenna length (h2 h1) for three di erent frequencies. No re ection takes
place when (h2 h1) = 4.1, 5.8, 9.3 mm at 15, 10, 5 GHz, where the corresponding
=4 antenna lengths are 3.5, 5.3, 10.6 mm, respectively. The number of modes used
in computation is 5, indicating fast numerical convergence.
3 Conclusion
A simple series solution for monopole antenna radiating into a parallel plate is ob-
tained. Our solution is compared with other existing one and is shown to be accurate
and e cient for numerical evaluation.
References
1] A.G. Williamson, Radial-line/coaxial-line junctions : analysis and equivalent
circuits," Int. J. Electronics, vol. 58, no. 1, pp. 91-104, 1985.
4

5. 2] Z. Shen and R.H. MacPhie, Modal expansion analysis of monopole antennas
driven from a coaxial line," Radio Science, vol. 31, no. 5, pp. 1037-1046, Sept.-
Oct. 1996.
3] J.K. Park and H.J. Eom, Fourier transform analysis of dielectric- lled edge-
slot antenna," Radio Science, vol. 32, no. 6, pp. 2149-2154, Nov.-Dec. 1997,
Correction to Fourier transform analysis of dielectric- lled edge-slot antenna,"
Radio Science, vol. 33, no. 3, p. 631, May-June 1998.
4] G.B. Arfken and H.J. Weber, Mathematical methods for physicists, Academic
Press, 1995, pp. 414-439.
5

6. PEC
0 a
Region (II)
a
PEC
Region (II)
Region (III)
1 2
zincidence
h2
h1
PEC
rr
µ, ε2
µ, ε3
µ, ε1
Region(I)
Figure 1: Geometry of the monopole antenna.
6

7. 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
5
10
15
20
25
30
35
40
inputadmittance[m]
Ω
0
10
20
30
40
antenna length, (h - h ) /12 λ0
0.1 0.3 0.5 0.7
susceptance
conductance
Shen’s resultso x
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
normalizedreflectedpower
0
0.2
0.4
0.6
0.8
antenna length, (h - h ) [12
2 4 6 8
5 GHz
10 GHz
15 GHz
mm ]
10
1
(b)
Figure 2: (a) Input admittance of a monopole antenna fed by a coaxial line (a1 =
0:0509 0; a2 = 1:187a1; h1 = 0; 1 = 2 = 3 = 0). (b) Normalized re ected power
of a parallel-plate waveguide fed by a coaxial line (a1 = 0:635mm; a2 = 2:055mm;
h2 = 10:2mm; 1 = 3 = 1:99 0; 2 = 0).
7