New iterative equations for an E-plane T-junction in a parallel-plate waveguide using Green's function

1. 1
New iterative equations for E-plane
T-junction in parallel-plate waveguide using
Green's function
Yong H. Cho
Advanced Antenna Research Team
Advanced Radio Technology Department
Electronics and Telecommunications Research Institute
161 Gajeong-dong, Yuseong-gu, Daejeon, 305-350, Republic of Korea
Phone +82-42-860-6738 Fax +82-42-860-5199
E-mail: iloveah@mail.kaist.ac.kr
November 15, 2002 DRAFT

2. 2
Abstract
Re ection and transmission behaviors of an E-plane T-junction in a parallel-plate waveguide are
theoretically investigated. The Green's function and iterative procedure are used to obtain the iterative
equations for the Hz eld modal coe cients. Numerical computations illustrate the characteristics of
re ection and transmission powers in terms of frequency. A dominant-mode solution is presented and
compared with the higher-mode solutions.
Keywords
Parallel-plate waveguide, T-junction, scattering, Green's function, iteration
I. Introduction
A waveguide T-junction is a fundamental waveguide junction structure and has been
extensively studied 1-5]. The solution of the waveguide T-junction associated with the
generalized scattering matrix technique is utilized for the analysis of waveguide disconti-
nuity problems such as couplers, multiplexers, or power dividers 1-3]. In 4], a simple and
new equivalent network is presented to obtain the closed-form expression for open and
slit-coupled E-plane T-junctions. The analytic series solution for the E-plane T-junction
is also obtained using the Fourier transform and residue calculus 5]. In the present work,
we utilize the Green's function and iterative procedure in order to obtain the simple yet
rigorous representations for re ection and transmission powers of the E-plane T-junction
without recourse to the involved residue calculus 5]. The iterative method 6] similar to
the present work, is used to obtain an initial assumption on the electric eld distribution
in the plane of waveguide discontinuity.
II. Field Analysis
Consider the TM wave impingingon an E-plane T-junction in a parallel-platewaveguide
shown in Fig. 1. The time-factor e i!t is omitted throughout. The shaded area ( a <
x < a and 0 < y < b) in Fig. 1 is an overlapping region between region (I) and (II). In
region (I) ( a < x < a and y < b) and (II) (0 < y < b), the incident and re ected Hz
elds are
HI
0(x;y) = I0
s cosas(x + a) ei sy + ei s(2b y)] (1)
HII
0 (x;y) = J 0
s cos(bsy)ei s(x+a) (2)
November 15, 2002 DRAFT

3. 3
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 Hz elds from Port 1 and Port
2, respectively. The geometry in Fig. 1 is composed of the superposition of those in Fig.
2(a) and 2(b). In the iteration scheme, the HI
n 1(x;y) eld generates the HII
n (x;y) eld
and the HII
n (x;y) eld does the HI
n(x;y) eld in successive manners. Then, the total Hz
elds in region (I) and (II) are
HI
z(x;y) =
1X
n=0
HI
n(x;y) (3)
HII
z (x;y) =
1X
n=0
HII
n (x;y) (4)
Utilizing the Green's function relation in 7] gives
HII
n (x;y) = 1
0
r A z component
= r
Z
J(r0)GII(r;r0) dr0
z component
=
Z
HI
n 1(r0) @
@n
h
GII(r;r0)
i
dr0 (5)
HI
n(x;y) =
Z
HII
n (r0) @
@n
h
GI(r;r0)
i
dr0 (6)
where A is a magnetic vector potential, J(r0) = Hz(r0) ^az ^n, n is the outward normal
direction to r0 in Fig. 2(a) and 2(b),
GI(r;r0) = 1
a
1X
m=0
cosam(x + a)cosam(x0 + a)
m
e i my< sin m(y> b)
me i mb (7)
GII(r;r0) = 2
b
1X
m=0
cos(bmy)cos(bmy0)
m
ei mjx x0
j
2i m
(8)
0 = 2, m = 1 (m = 1;2; ), y> the greater of y or y0, and y< the lesser of y or
y0. The two dimensional Green's functions in (7) and (8) are derived from the standard
procedure in 8]. Note that the scattered elds HI
n(x;y) and HII
n (x;y) are thought of
as the elds produced to eliminate the surface currents J(r0) generated by the Hz eld
discontinuities. The Hz eld 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 eliminate the surface currents or the Hz eld discontinuities
at ( a < x < a, y = 0) and (x = a, 0 < y < b), we utilize the iteration scheme. Then,
HI
n(x;y) =
1X
m=0
cosam(x + a)
h
sgn(y)ei mjyj + ei m(2b y)
i
In
m (9)
November 15, 2002 DRAFT

4. 4
HII
n (x;y) =
1X
m=0
cos(bmy)
h
sgn(x + a)ei mjx+ajJ n
m sgn(x a)ei mjx ajJ+n
m
i
(10)
where n = 1;2; , sgn( ) = 2u( ) 1, u( ) is the unit step function,
In
m = 1
2a m
Z a
a
cosam(x0 + a)HII
n (x0;0) dx0
= i
2a m
1X
v=0
v
h
( 1)mei2 va 1
i
2
v a2
m
h
J n
v + ( 1)mJ+n
v
i
(11)
J n
m = 1
b m
Z b
0
cos(bmy0)HI
n 1( a;y0) dy0
= i
b m
1X
v=0
( 1)v v(ei2 vb 1)
2
v b2
m
In 1
v (12)
I0
m = I0
s ms, J 0
m = J 0
s ms, J+0
m = 0, and ms is the Kronecker delta. In a dominant-mode
approximation (m = 0 and s = 0) for I0
0 = 1 and J 0
0 = 0, we get
I0 = (1 ei2k0a)(ei2k0b 1)
4abk2
0 + (ei2k0a 1)(ei2k0b 1) (13)
J0 = i2k0a(ei2k0b 1)
4abk2
0 + (ei2k0a 1)(ei2k0b 1)
(14)
where Im = P1
n=1 In
m and Jm = P1
n=1 J n
m . Similarly, we obtain the solution for I0
0 = 0
and J 0
0 = 1 as
J0 = (ei2k0a 1)(1 ei2k0b)
8abk2
0 + 2(ei2k0a 1)(ei2k0b 1) (15)
I0 = ik0b(ei2k0a 1)
4abk2
0 + (ei2k0a 1)(ei2k0b 1) (16)
When I0
s = 1 and J 0
s = 1, the time-averaged incident powers from Port 1 and Port 2 are
Pi1 = (a s s)=(2! 0) and Pi2 = (b s 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
2
Re
"
Z a
a
Ex(x;y)Hz(x;y) dx
#
= a
2
M1X
m=0
m m
! 0
(ei2 mb 1)Im + ei2 sbI0
s ms
2
(17)
P2 = 1
2Re
"
Z b
0
Ey(x;y)Hz(x;y) dy
#
November 15, 2002 DRAFT

5. 5
= b
4
M2X
m=0
m m
! 0
ei maJ+
m e i maJm
2
(18)
P3 = b
4
M2X
m=0
m m
! 0
ei ma
h
Jm + J 0
s ms
i
e i maJ+
m
2
(19)
where M1 = 2ak0= ], M2 = bk0= ], and x] denotes the maximuminteger less than x. Fig.
3 shows the convergence characteristics of our iterative solution (19) versus the number of
iteration n. Note that a dominant-mode solution (m = 0) is obtained using (15), (16), and
(19). When n > 20, the transmission power to Port 3 is almost unchanged with respect to
n. Fig. 4 illustrates the scattering characteristics for the E-plane T-junction, con rming
that the proposed solution agrees well with 5] when m 1. Our computational experience
indicates that in a low frequency limit, a dominant-mode solution (m = 0) with (15) and
(16) is almost identical to a more accurate solution including three higher-modes (m = 3).
III. Conclusions
Scattering analysis of an E-plane T-junction in a parallel-plate waveguide is analytically
shown using the Green's function and iterative procedure. Simple yet rigorous scattering
relations for the E-plane T-junction in parallel-plate waveguide are presented and com-
pared with other results. The behaviors of the re ection and transmission powers are
studied in terms of frequency. A dominant-mode solution is shown to be accurate and
useful for the analysis of the E-plane T-junction in a low frequency limit.
November 15, 2002 DRAFT

6. 6
References
1] F. Arndt, 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] J. M. Rebollar, J. Esteban, and J. E. Page, Fullwave analysis of three and four-port rectangular waveguide
junctions," IEEE Trans. Microwave Theory Tech., vol. 42, no. 2, pp. 256-263, Feb. 1994.
3] T. Sieverding, 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.
4] P. Lampariello and A. A. Oliner, New equivalent networks with simple closed-form expressions for open and
slit-coupled E-plane tee junctions," IEEE Trans. Microwave Theory Tech., vol. 41, no. 5, pp. 839-847, May
1993.
5] K. H. Park, H. J. Eom, and Y. Yamaguchi, An analytic series solution for E-plane T-junction in parallel-plate
waveguide," IEEE Trans. Microwave Theory Tech., vol. 42, no. 2, pp. 356-358, Feb. 1994.
6] M. F. Iskander and M. A. K. Hamid, Iterative solutions of waveguide discontinuity problems," IEEE Trans.
Microwave Theory Tech., vol. 25, no. 9, pp. 763-768, Sept. 1977.
7] Y. H. Cho and H. J. Eom, Analysis of a ridge waveguide using overlapping T-blocks," IEEE Trans. Microwave
Theory Tech., vol. 50, no. 10, pp. 2368-2373, Oct. 2002.
8] R. E. Collin, Field Theory of Guided Waves, 2nd ed., New York: IEEE Press, pp. 78-86, 1991.
November 15, 2002 DRAFT

7. 7
2a
b y
x
z
Region (I)
Region (II)
PEC
Port 1
Port 3Port 2
Fig. 1. Geometry of T-junction.
November 15, 2002 DRAFT

8. 8
y
xz
2a
b
Region (I)
(a) Subregion for region (I)
y
xz
2a
b Region (II)
(b) Subregion for region (II)
Fig. 2. Subregions of T-junction.
November 15, 2002 DRAFT

9. 9
1 10 20 30 40 50
0.4
0.44
0.48
0.52
0.56
0.6
Number of iteration
Normalizedpower,P3
/Pi2
m = 0
m = 1
m = 2
m = 3
Fig. 3. Behavior of normalized transmission power P3=Pi2 versus number of iteration n with f = 10 GHz,
a = 3:5 mm, and b = 7 mm.
November 15, 2002 DRAFT

10. 10
5 10 15 20
0
0.2
0.4
0.6
0.8
1
Frequency [GHz]
Normalizedpower
P2
/Pi2
P1
/Pi2
P3
/Pi2
m = 0
m = 1
m = 2
m = 3
[5]
Fig. 4. Behavior of normalized re ection and transmission powers versus frequency with n = 30, a =
3:5 mm, and b = 7 mm.
November 15, 2002 DRAFT