Iterative Green's function analysis of an H-plane T-junction in a parallel-plate waveguide

1. 1
Iterative Green's function analysis of an
H-plane T-junction in a parallel-plate
waveguide
Yong H. Cho
Division of Information Communication & Radio Engineering
Mokwon University
800 Doan-dong, Seo-gu, Daejeon, 302-729, Republic of Korea
Phone: +82-42-829-7675 Fax: +82-42-825-5449
Email: yhcho@mokwon.ac.kr
February 26, 2004 DRAFT

2. 2
Abstract
Scattering solutions of an H-plane T-junction in a parallel-plate waveguide are investigated. The
iterative procedure and Green's function relation are used to obtain the iterative equations for the Ez
eld modal coe cients, thus resulting in matrix solutions. The scattering characteristics of re ection and
transmission powers are presented and compared with other existing results.
Keywords
Parallel-plate waveguide, T-junction, scattering, Green's function, iteration
I. Introduction
A waveguide T-junction is a fundamental waveguide junction structure for coupler, l-
ter, multiplexer, and power divider and has been extensively studied 1-5]. In 4], the
T-junction is used for a key-building block of multi-aperture coupler structures. The rig-
orous series solution of the H-plane T-junction is obtained using the Fourier transform
and residue calculus 3]. In this paper, we use the same technique in 5] to obtain scat-
tering solutions of the H-plane T-junction. Utilizing the iterative procedure and Green's
function relation, we obtain the simple yet rigorous solutions for re ection and transmis-
sion powers of the H-plane T-junction. The algebraic series expressions of the iterative
Green's function approach are di erent from those of the Fourier-transform method 3]
and mode-matching technique 4].
II. Field Analysis
The H-plane T-junction in a parallel-plate waveguide is shown in Fig. 1. The incident
TE-waves from Port 1 and 2 impinge on the waveguide junction. The time-factor e;i!t
is omitted throughout. In region (I) (;a < x < a and y < b) in Fig. 2(a) and (II)
(0 < y < b) in Fig. 2(b), the incident and re ected Ez elds are
EI
0(x y) = I0
s sinas(x + a) ei sy ;ei s(2b;y)] (1)
EII
0 (x y) = J;0
s sin(bsy)ei s(x+a) (2)
where as = s =(2a), s =
q
k2
0 ;a2
s, bs = s =b, s =
q
k2
0 ;b2
s, and k0 = !p
0 0. Note
that I0
s and J;0
s are the modal coe cients of the incident Ez elds from Port 1 and 2,
respectively. The geometry in Fig. 1 is composed of the superposition of those in Fig.
February 26, 2004 DRAFT

3. 3
2(a) and 2(b). In the iteration scheme, the EI
n;1(x y) eld generates the EII
n (x y) eld
and the EII
n (x y) eld generates the EI
n(x y) eld in successive manner. Then, the total
Ez elds in region (I) and (II) are
EI
z(x y) = EI
0(x y) +
1X
n=1
EI
n(x y) (3)
EII
z (x y) = EII
0 (x y) +
1X
n=1
EII
n (x y) (4)
Utilizing the Green's function relation in 6] gives
EII
n (x y) = i!A z component
= ;i! 0
Z
J(r0)GII(r r0) dr0
z component
= ;
Z
@
@n0
h
EI
n;1(r0)
i
GII(r r0) dr0 (5)
EI
n(x y) = ;
Z
@
@n0
h
EII
n (r0)
i
GI(r r0) dr0 (6)
J(r0) =
8
><
>:
Hy(r0)^ay ^n for GII(r r0)
Hx(r0)^ax ^n for GI(r r0)
(7)
where A is a magnetic vector potential, n is the outward normal direction to r0 in Fig.
2(a) and 2(b),
GI(r r0) = 1
a
1X
m=1
sinam(x + a)sinam(x0 + a)
e;i my< sin m(y> ;b)
; me;i mb (8)
GII(r r0) = 2
b
1X
m=1
sin(bmy)sin(bmy0)
ei mjx;x0
j
;2i m
(9)
y> the greater of y or y0 and y< the lesser of y or y0. The scattered elds EI
n(x y)
and EII
n (x y) are thought of as the elds produced to eliminate the surface currents J(r0)
generated by the Hx and Hy elds discontinuities. The Hx and Hy elds discontinuities
appear in the intervals (;a < x < a, y = 0) and (x = a, 0 < y < b) in Fig. 1,
respectively, due to the presence of the PEC wall in Fig. 2(a) and 2(b). In order to
February 26, 2004 DRAFT

4. 4
eliminate the Hx and Hy elds discontinuities, we utilize the iteration scheme. Then,
EI
n(x y) =
1X
m=1
sinam(x + a)
h
ei mjyj ;ei m(2b;y)
i
In
m (10)
EII
n (x y) =
1X
m=1
sin(bmy)
h
ei mjx+ajJ;n
m ;ei mjx;ajJ+n
m
i
(11)
where n = 1 2 ,
In
m = i
2 ma
Z a
;a
sinam(x0 + a) @
@y0EII
n (x0 0) dx0
= iam
2 ma
1X
v=1
bv
h
(;1)mei2 va ;1
i
2
v ;a2
mh
J;n
v + (;1)mJ+n
v
i
(12)
J n
m = i
mb
Z b
0
sin(bmy0) @
@x0EI
n;1( a y0) dy0
= ibm
mb
1X
v=1
( 1)vav(ei2 vb ;1)
2
v ;b2
m
In;1
v (13)
I0
m = I0
s ms, J;0
m = J;0
s ms, J+0
m = 0, and ms is the Kronecker delta. Applying matrix
operations to (12) and (13), we obtain
In
m] = Amv] J;n
v ] + (;1)mAmv] J+n
v ] (14)
J n
m ] = ( 1)vBmv] In;1
v ] (15)
where
Amv =
iambv
h
(;1)mei2 va ;1
i
2 ma( 2
v ;a2
m)
(16)
Bmv = ibmav(ei2 vb ;1)
mb( 2
v ;b2
m)
(17)
and Amv] and In
m] denote m v matrix and mth order column vector, respectively. Solving
the simultaneous equations for In
m] and J n
m ] gives
Im] =
n
I ; Cml]
o;1
n
Cml] I0
l ] + Aml] J;0
l ]
o
(18)
February 26, 2004 DRAFT

5. 5
Jm] = ( 1)vBml]
n
I ; Cml]
o;1
n
I0
l ] + Aml] J;0
l ]
o
(19)
where Im = P1
n=1 In
m, Jm = P1
n=1 J n
m , I is an identity matrix, and
Cml = 1 + (;1)m+l]
1X
v=1
AmvBvl (20)
(18) and (19) are obtained from the matrix series based on the Cayley-Hamilton theorem
such as
1X
n=0
Cml
n
=
n
I ; Cml]
o;1
(21)
In this manner, the in nite iteration reduces to the matrix inversion. In a dominant-mode
approximation (m = 1 and s = 1) for I0
1 = 1 and J;0
1 = 0, we get
I1 = (a1b1)2(1 + ei2 1a)(ei2 1b ;1)
K (22)
J1 = ia1b1 1a(k2
0 ;a2
1 ;b2
1)(ei2 1b ;1)
K (23)
where
K = 1a 1b(k2
0 ;a2
1 ;b2
1)2
; (a1b1)2(1 + ei2 1a)(ei2 1b ;1) (24)
Similarly, we obtain the solution for I0
1 = 0 and J;0
1 = 1 as
J1 = (a1b1)2(1 + ei2 1a)(ei2 1b ;1)
2K (25)
I1 = ia1b1 1b(a2
1 + b2
1 ;k2
0)(1 + ei2 1a)
2K (26)
The time-averaged incident powers from Port 1 and 2 are Pi1 = (a sI0
s)=(2! 0) and Pi2 =
(b sJ;0
s )=(4! 0), respectively. Let the time-averaged power Pi (i = 1 2 3) denote the
transmission power to Port i. Then, we obtain
P1 = 1
2Re
"
;
Z a
;a
Ez(x y)Hx(x y) dx
#
= a
2
M1X
m=1
m
! 0
(ei2 mb ;1)Im + ei2 sbI0
s ms
2
February 26, 2004 DRAFT

6. 6
(27)
P2 = 1
2
Re
"
Z b
0
Ez(x y)Hy(x y) dy
#
= b
4
M2X
m=1
m
! 0
ei maJ+
m ;e;i maJ;
m
2
(28)
P3 = b
4
M2X
m=1
m
! 0
ei ma
h
J;
m + J;0
s ms
i
;e;i maJ+
m
2
(29)
where M1 = 2ak0= ], M2 = bk0= ], and x] denotes the maximum integer less than x.
M1 and M2 indicate the number of propagation modes within Port 1 and 2, respectively.
Fig. 3 shows the scattering characteristics for the H-plane T-junction of Port 2 incidence,
con rming that the proposed solution agrees well with 2, 3] when m 2. Note that
M1 = M2 due to 2a = b. Our computational experience indicates that a dominant-
mode solution (m = 1) with (27), (28), and (29) is similar to a more accurate solution
(m = 4). The modal index m in the legend of Fig. 3 denotes the number of modes used
for numerical computations. The characteristics for the H-plane T-junction of Port 1
incidence are shown in Fig. 4. Fig. 4 also indicates that our solutions are fast convergent
and numerically e cient.
III. Conclusions
Scattering analysis of an H-plane T-junction in a parallel-plate waveguide is analytically
shown using the iterative procedure and Green's function relation. Simple yet rigorous
scattering relations for the H-plane T-junction in a parallel-plate waveguide are presented
and compared with other results.
February 26, 2004 DRAFT

7. 7
References
1] Arndt, F., I. Ahrens, U. Papziner, U. Wiechmann, and R. Wilkeit, Optimized E-plane T-junction series
power dividers," IEEE Trans. Microwave Theory Tech., vol. 35, no. 11, pp. 1052-1059, Nov. 1987.
2] Liang, X. P., K. A. Zaki, and A. E. Atia, A rigorous three plane mode-matching technique for characterizing
waveguide T-junction and its application in multiplexer design," IEEE Trans. Microwave Theory Tech., vol.
39, no. 12, pp. 2138-2147, Dec. 1991.
3] Park, K. H. and H. J. Eom, An analytic series solution for H-plane waveguide T-junction," IEEE Microwave
Guided Wave Lett., vol. 3, no. 4, pp. 104-106, April 1993.
4] Sieverding, T., U. Papziner, and F. Arndt, Mode-matching CAD of rectangular or circular multiaperture
narrow-wall couplers," IEEE Trans. Microwave Theory Tech., vol. 45, no. 7, pp. 1034-1040, July 1997.
5] Cho, Y. H., New iterative equations for an E-plane T-junction in a parallel-plate waveguide using Green's
function," Microwave Optical Tech. Lett., vol. 37, no. 6, pp. 447-449, June 2003.
6] Cho, Y. H. and H. J. Eom, Analysis of a ridge waveguide using overlapping T-blocks," IEEE Trans. Mi-
crowave Theory Tech., vol. 50, no. 10, pp. 2368-2373, Oct. 2002.
Figure Captions
Fig. 1 Geometry of a T-junction in a parallel-plate waveguide.
Fig. 2 Subregions of a T-junction in a parallel-plate waveguide for region (I) and (II).
Fig. 3 Behavior of normalized re ection and transmission powers versus frequency with
I0
1 = 0, J;0
1 = 1, a = 3:5 mm, and b = 7 mm.
Fig. 4 Behavior of normalized re ection and transmission powers versus frequency with
I0
1 = 1, J;0
1 = 0, a = 3:5 mm, and b = 7 mm.
February 26, 2004 DRAFT

8. 8
;;;;;;;;;;;;;;
b
Port 1
e0
;;;
;;;
;;;
;;;
;;;
;;;
PEC
x
y
z
a2
Port 2 Port 3
Fig. 1. Geometry of a T-junction in a parallel-plate waveguide.
;;;
;;
;;b
Region
(I) ;;
;;x
y
z
a2
n^n^
(a)
;;;;;;b
a2
Region (II)
;;;;;;x
y
z
n^
(b)
Fig. 2. Subregions of a T-junction in a parallel-plate waveguide for region (I) and (II).
February 26, 2004 DRAFT

9. 9
25 30 35 40
0
0.2
0.4
0.6
0.8
1
Frequency [GHz]
Normalizedpower
P1
/Pi2
P2
/Pi2
P
3
/P
i2
m = 1
m = 2
m = 3
m = 4
[2]
[3]
Fig. 3. Behavior of normalized re ection and transmission powers versus frequency with I0
1 = 0, J;0
1 = 1,
a = 3:5 mm, and b = 7 mm.
25 30 35 40
0
0.2
0.4
0.6
0.8
1
Frequency [GHz]
Normalizedpower
P
1
/P
i1
P2
/Pi1
m = 1
m = 2
m = 3
m = 4
Fig. 4. Behavior of normalized re ection and transmission powers versus frequency with I0
1 = 1, J;0
1 = 0,
a = 3:5 mm, and b = 7 mm.
February 26, 2004 DRAFT

10. 10
Yong H. Cho was born in Daegu, Korea, in 1972. He received the B.S. degree in
electronic engineering from the Kyungpook National University, Daegu, Korea, in 1998,
the M.S. and Ph. D. degrees in electrical engineering from the Korea Advanced Institute of
Science and Technology (KAIST), Daejeon, Korea, in 2000 and 2002, respectively. From
2002 to 2003, he was a senior research sta with Electronics and Telecommunications
Research Institute (ETRI). In 2003, he joined the division of information communication
and radio engineering, Mokwon University, Daejeon, Korea, where he is currently a full-
time lecturer. His research interests include wave scattering and dispersion characteristics
of waveguides.
February 26, 2004 DRAFT