Analysis of a ridge waveguide using overlapping T-blocks

1. Analysis of a ridge waveguide using overlapping T-blocks
Yong H. Cho and Hyo J. Eom
Department of Electrical Engineering
Korea Advanced Institute of Science and Technology
373-1, Kusong Dong, Yusung Gu, Taejon, Korea
Phone +82-42-869-3436 Fax +82-42-869-8036
E-mail : hjeom@ee.kaist.ac.kr
Abstract A T-block (TB) approach is proposed to analyze the dispersion relation of a
ridge waveguide. The eld representations of a TB are obtained with the Green's function
and mode-matching technique. Rigorous yet simple dispersion equations for symmetric and
asymmetric ridge waveguides are presented using a superposition of overlapping T-blocks.
The rapid convergence characteristics of the dispersion equation are illustrated in terms
of the cuto wavenumbers. A closed-form dispersion relation, based on a dominant-mode
approximation, is shown to be accurate for most practical applications such as coupler,
lter, and polarizer designs.
Index Terms Ridge waveguide, dispersion, Green's function, mode-matching technique,
superposition.
1 Introduction
The propagation and coupling characteristics of a ridge waveguide have been investi-
gated with various methods 1-7] because of its broad band, low cuto frequency, and
low impedance characteristics compatible with a coaxial line. When the geometry of a
waveguide is complicated such as a ridge waveguide, the formulation and dispersion analy-
sis becomes naturally involved. It is therefore desirable to develop an analysis scheme with
fast CPU time, increased accuracy, simple applicability, and wide versatility. To that end,
we propose a new approach using the T-block and superposition. In 8], a simple and new
equivalent network for T-junction, which is similar to T-block, is presented to obtain the
closed-form expression for open and slit-coupled E-plane T-junctions. In this paper, the
approach of the T-block and superposition is employed to divide a total region into several
overlapping T-blocks. It is possible to represent the eld within a T-block in simple and
numerically e cient series based on the Green's function and mode-matching method. The
Green's function approach allows us to reduce the number of unknown modal coe cients
and improve the convergence rate. Since the Green's function for the T-block is available,
the involved residue calculus as in 7] is unnecessary, thereby increasing computational ef-
ciency. The advantage of the T-block approach lies in substantially reducing the amount
of computational e ort. In this paper, we will analyze the propagation characteristics of a
1

2. double ridge waveguide by utilizing the T-block approach and superposition principle. It is
important to note that our T-block approach is applicable to other waveguide structures,
whose geometry can be broken into several overlapping T-blocks such as the nline, shielded
microstrip line, and multiconductor transmission line.
2 Analysis of T-Block
T
2a
b
dRegion (I)
Region (II)
x
y
z
Figure 1: Geometry of a T-block.
Assume that a TE-wave propagates along the z-direction inside a T-block (TB) in
Fig. 1. The phase factor ei( z !t) is omitted throughout. In region (I) ( d < y < 0), we
represent the Hz component as
HI
z(x;y) =
1X
m=0
qm cosam(x+ a)cos m(y + d)
h
u(x + a) u(x a)
i
(1)
where am = m =(2a), m =
q
k2
0
2 a2m, k0 = 2 = 0, and u( ) is a unit step function.
In order to represent the Hz eld in region (II) (0 < y < b), we divide region (II) into two
subregions such as Fig. 2(a) and 2(b). Based on the superposition, the eld in region (II)
is given by
HII
z (x;y) =
1X
m=0
sm
h
Hm(x;y) + RH
m(x;y)
i
(2)
where Hm(x;y) and RH
m(x;y) are the eld components within subregions of Fig. 2(a) and
2(b), respectively. Similar to the HI
z(x;y) component, we represent Hm(x;y) as
Hm(x;y) = cos m(y b)
m sin( mb) cosam(x+ a)
h
u(x+ a) u(x a)
i
(3)
Multiplying the Ex eld continuity at y = 0 between regions (I) and (II) by cosal(x + a)
and integrating over a < x < a, we obtain
sm = qm m sin( md) (4)
2

3. 2a
b
dRegion (I)
Region (II)
x
y
z
(a) Subregion for Hm(x;y)
T
b
Region (II)
x
y
z
2a
(b) Subregion for RH
m(x;y)
Figure 2: Subregions of a T-block.
The component of Hm( a;y) results in the Hz discontinuity at x = a. In order to enforce
the Hz continuity at x = a, we next consider a contribution from RH
m(x;y). Utilizing the
Green's function relation 8] gives
RH
m(x;y) = 1
0
r A z component
= r
Z
GH(r;r0)J(r0) dr0
z component
=
Z
@
@n
h
GH(r;r0)
i
Hm(r0) dr0 (5)
where A is a magnetic vector potential, J(r0) = Hm(r0) ^az ^n, n is the outward normal
direction to r0 in Fig. 2(a),
GH(r;r0) = 2
b
1X
v=0
cos( vy)cos( vy0)
v
g(x;x0; v) (6)
g(x;x0; ) = sin (x< + T=2)sin (T=2 x>)
sin( T) (7)
x> the greater of x or x0, x< the lesser of x or x0, 0 = 2; m = 1 (m = 1;2; ),
v = v =b, v =
q
k2
0
2 2v, GH(r;r0) is shown in 9], and g(x;x0; v) isa one dimensional
Green's functionwithan electric wallat x = T=2. Note that r denotes an observation point
(x;y) and r0 is a source point (x0;y0) = ( a;0 < y0 < b). The scattered component, RH
m(x;y)
is thought of as the eld produced to eliminate the surface current, J(r0) = Hm(r0) ^az ^n.
This means that RH
m( a;y) produces the Hz( a;y) discontinuity at x = a, which is an
inverse sign to Hm( a;y0). Integrating (5) with respect to y0 from 0 to b, we obtain
RH
m(x;y) =
Z b
0
@GH(r;r0)
@x
cos m(y0 b)
m sin( mb) x0
= a
dy0
( 1)m
Z b
0
@GH(r;r0)
@x
cos m(y0 b)
m sin( mb) x0
=a
dy0
= 1
b
1X
v=0
cos( vy)
v( 2v a2m)
h
fH(x; a; v) ( 1)mfH(x;a; v)
i
(8)
3

4. where
fH(x;x0; ) =
sgn(x x0)
h
ei jx x0
j ( 1)mei (T jx x0
j)
i
1 ( 1)mei T (9)
and sgn(x) = 2u(x) 1. The total longitudinal magnetic eld is therefore given as
TH(x;y) = HI
z(x;y) + HII
z (x;y) (10)
Note that the Ex(x;0) eld, which is produced by Rm(x;0), is zero. Hence, the Ex and Hz
eld continuities are satis ed except for the Hz eld discontinuity at y = 0 and a < x < a.
By enforcing the continuity of TH(x;0) for a < x < a, it is possible to determine the
dispersion relation for a TB. If a waveguide structure can be divided into a number of TB,
it is possible to use (10) directly in the derivation of dispersion relation. The advantage of
our TB approach lies in its computational simplicity since (10) can be repeatedly used for
each TB. In the next section, we will show how to obtain the dispersion relation for a ridge
waveguide from (10).
Similarly from the analysis of the TE mode, the Ez components for the TM mode are
represented as
EI
z(x;y) =
1X
m=1
pm sinam(x + a)sin m(y + d)
h
u(x + a) u(x a)
i
(11)
EII
z (x;y) =
1X
m=1
pm sin( md)
h
Em(x;y) + RE
m(x;y)
i
(12)
where
Em(x;y) = sin m(b y)
sin( mb) sinam(x+ a)
h
u(x+ a) u(x a)
i
(13)
RE
m(x;y) =
Z
GE(r;r0) @
@n0
h
Em(r0)
i
dr0
= ami
b
1X
v=1
v sin( vy)
v( 2v a2m)
h
fE(x; a; v) ( 1)mfE(x;a; v)
i
(14)
GE(r;r0) = 2
b
1X
v=1
sin( vy)sin( vy0)g(x;x0; v) (15)
fE(x;x0; ) = ei jx x0
j + ( 1)mei (T jx x0
j)
1 ( 1)mei T (16)
Then, we obtain the total longitudinal electric eld as
TE(x;y) = EI
z(x;y) + EII
z (x;y) (17)
Contrary to the TE mode, the total eld, TE(x;y) is continuous at y = 0, while the Hx
eld is discontinuous. The dispersion relation can be obtained by enforcing the Hx eld
continuity at y = 0 for a < x < a. Although RH
m and RE
m are represented in series
forms, their convergence rate is very fast, thus e cient for numerical computations. Note
that the convergence characteristics of the HII
z and EII
z are independent of T in Fig. 1.
These characteristics are di erent from the standard mode-matching technique, where its
convergence rate is mainly determined by T. When T ! 1, our solutions converge to those
of a rectangular groove guide with an electric conductor cover at y = b in 10].
4

5. 3 Analysis of Ridge Waveguide Using Two TB
It is possible to apply the T-block approach to the dispersion analysis for a double ridge
waveguide. We rst divide a double ridge waveguide in Fig. 3 into two overlapping
T
2a
b
d
x
y z
(1)
(1)
b(2)
T(2)
T
(2)
/
(x, y)
/
T(1) /
(x, y)
/
Figure 3: Geometry of a double ridge waveguide.
T
2a
b
d
x
y z
(1)
(1)
b(2)
T
(2)
T(1) /
(x, y)
/
/
/ 2a
d
T(2) /
(x, y)
/
x
yz /
/
+
Figure 4: Superposition of two T-blocks.
T-blocks as shown in Fig. 4. The Hz eld in a ridge waveguide is represented as
Hz(x;y) = T(1)
H (x;y) + T(2)
H ( x; y d) (18)
Note that the (x;y) coordinate in Fig. 3 is the global coordinate system for a double ridge
waveguide and the (x0;y0) in Fig. 4 is the local coordinate system for subregions of T-
blocks. The enforcement of the boundary conditions on the Hz eld continuities is required
to determine the relations of modal coe cients, q(1)
m and q(2)
m . In matching the Hz continuity,
it is expedient to introduce a general integration form as
I(x;y;c) =
Z x+c
x c
h
HI
z(x0;y) T(1)+T(2)
HII
z (x0;y) T(1)+T(2)
i
coscl(x0 x+ c) dx0
5

6. =
1X
m=0
h
q(1)
m I(1)
H (x;y;c) + q(2)
m I(2)
H ( x; y d;c)
i
(19)
where cl = l =(2c) and
IH(x;y;c) =
Z x+c
x c
n
cosam(x0 + a)cos m(y + d) u(x0 + a) u(x0 a)] u(y + d) u(y)]
+ m sin( md) Hm(x0;y) + RH
m(x0;y)] u(y) u(y b)]
o
coscl(x0 x+ c) dx0 (20)
Since the integrand of IH(x;y;c) is composed of elementary functions, the evaluation of
(20) is trivial. In order to satisfy the Hz continuity at ( a < x < a;y = 0) and ( a <
x < a;y = d), we put (x;y) = (0;0) and (0; d) into (19), respectively, and obtain the
dispersion relation for a double ridge waveguide as
1X
m=0
h
q(1)
m I(1)
H (0;0;a) + q(2)
m I(2)
H (0; d;a)
i
= 0 (21)
1X
m=0
h
q(1)
m I(1)
H (0; d;a) + q(2)
m I(2)
H (0;0;a)
i
= 0 (22)
When a double ridge waveguide is symmetric with respect to the y direction, q(1)
m = q(2)
m .
Note that sign denotes even and odd modes with respect to the y direction, respectively.
Then, the dispersion equations, (21) and (22) reduce to a simpli ed one as
1X
m=0
q(1)
m
h
I(1)
H (0;0;a) I(2)
H (0; d;a)
i
= 0 (23)
In a dominant mode approximation (m = 0; l = 0), (23) further simpli es to
I(1)
H (0;0;a) I(2)
H (0; d;a) m=0; l=0
= 2a
h
cos(kcd) 1
i
+ kc sin(kcd)
"
2a
kc tan(kcb)
2i
b
1X
v=0
1 ei v2a + ei vT ei v(T 2a)
v 3v(1 ei vT)
#
= 0 (24)
where kc =
q
k2
0
2 and v =
p
k2c (v =b)2. Similarly, utilizing the approach of T-block
and superposition, we obtain the TM dispersion relation of a double ridge waveguide as
1X
m=1
h
p(1)
m I(1)
E (0;0;a) p(2)
m I(2)
E (0; d;a)
i
= 0 (25)
1X
m=1
h
p(1)
m I(1)
E (0; d;a) p(2)
m I(2)
E (0;0;a)
i
= 0 (26)
where
I(x;y;c) =
Z x+c
x c
h
@
@yEI
z(x0;y) T(1)+T(2)
@
@yEII
z (x0;y) T(1)+T(2)
i
sincl(x0 x+ c) dx0
6

7. Table 1: Cuto wavenumbers (rad=m) of the rst eight TE-modes for a single ridge waveg-
uide.
Mode number 1 2 3 4 5 6 7 8
m = 0 91.34 333.1 379.9 524.7 664.7 690.5 744.8 828.0
m = 0; 2 92.40 333.2 380.7 525.8 665.3 692.1 745.4 828.9
m = 0; 2; 4 92.60 333.2 380.8 526.0 665.3 692.1 745.5 829.2
m = 0; 2; 4; 6 92.66 333.2 380.9 526.1 665.3 692.1 745.5 829.2
5] 92.6 333.2 381.1 526.3 665.3 691.6 745.3 829.5
Parameters : a = 1:7 mm; T(1) = T(2) = 19 mm; d = 0:3 mm; b(1) = b(2) = 9:35 mm
Table 2: Cuto wavenumbers (rad=m) of the rst eight TM-modes for a single ridge waveg-
uide.
Mode number 1 2 3 4 5 6 7 8
m = 2 471.2 471.4 741.1 741.7 748.3 748.8 940.5 942.4
m = 2; 4 471.1 471.4 741.0 741.7 748.2 748.8 940.3 942.3
m = 2; 4; 6 471.1 471.4 741.0 741.6 748.2 748.7 940.2 942.3
m = 2; 4; 6; 8 471.1 471.4 741.0 741.6 748.2 748.7 940.2 942.3
5] 471.1 471.4 741.0 741.6 748.1 748.7 940.0 942.2
Parameters : a = 1:7 mm; T(1) = T(2) = 19 mm; d = 0:3 mm; b(1) = b(2) = 9:35 mm
=
1X
m=0
h
p(1)
m I(1)
E (x;y;c) p(2)
m I(2)
E ( x; y d;c)
i
(27)
IE(x;y;c) =
Z x+c
x c
n
sinam(x0 + a)sin m(y + d) u(x0 + a) u(x0 a)] u(y + d) u(y)]
sin( md) Em(x0;y) + RE
m(x0;y)] u(y) u(y b)]
o
sincl(x0 x + c) dx0 (28)
In a dominant mode approximation (m = 2; l = 2) for a symmetric single ridge waveguide,
(25) and (26) reduce to
I(1)
E (0;0;a) I(2)
E (0; d;a) m=2; l=2
= a 2
h
cos( 2d) 1
i
+ sin( 2d)
"
a 2
tan( 2b)
2a2
2i
b
1X
v=1
2
v
h
1 ei v2a + ei vT ei v(T 2a)
i
v( 2v a2
2)2(1 ei vT)
#
= 0 (29)
where 2 =
q
k2
0
2 a2
2. Tables 1 and 2 show the convergence characteristics of the TE
and TM modes in a single ridge waveguide, respectively. The agreement of our solution
7

8. with 5] is excellent, and even a dominant-mode solution (m = 0) is accurate within 1%
error. This means that the dominant-mode dispersion relations, (24) and (29), are useful for
most practical applications. Fig. 5 represents the cuto wavenumber characteristics of the
0.5 0.6 0.7 0.8 0.9 1
80
100
120
140
160
180
d = 0.3 mm
b /
(1)
(T - d)
(1)
Cutoffwavenumber[rad/]mm
d = 1.2 mm
m = 0
m = 0,2
m = 0,2,4
m = 0,2,4,6
Figure 5: Cuto wavenumber of the TE10 mode in a double ridge waveguide versus the
position of the ridge. ( a = 1:7 mm; T(1) = T(2) = 19 mm; and b(1) + b(2) + d = T(1))
TE10 mode in a single ridge waveguide versus the position of a ridge. As the width of the
ridge, d, increases, the cuto wavenumber of the TE10 mode decreases. This means that the
bandwidth of a single-mode operation increases. The dominant-mode solution agrees well
with the higher-mode solutions, thus con rming the fast convergence. As b(1)=(T(1) d)
approaches one, the solution converges to that of a single TB with b(1) = T(1) d.
4 Analysis of Ridge Waveguide Using Four LB
For the analysis of an asymmetric double ridge waveguide of Fig. 6, it is convenient to
use a L-block (LB) as shown in Fig. 7. When the mode number, m, is even, the TE and
TM mode characteristics of a TB represent those of a LB. An asymmetric double ridge
waveguide in Fig. 6 is divided into four overlapping LB as in Fig. 8. The Hz eld in an
asymmetric double ridge waveguide is represented as the superposition of four LB. Then,
Hz(x;y) = L(1)
H (x;y) + L(2)
H (x T + a(1) +a(2);y)
+ L(3)
H (a(3) a(1) x;b y) + L(4)
H (T a(1) a(4) x;b y) (30)
where LH(x;y) = TH(x+a;y) m = even
. Matching points, (x;y) for the Hz eld continuities
are (0;0), (T a(1) a(2);0), (a(3) a(1);b), and (T a(1) a(4);b) in the global coordinates
8

9. x
y
z
L(2) /
(x, y)
/
L(1) /
(x, y)
/
L(3) /
(x, y)
/
L(4) /
(x, y)
/
T
b
Figure 6: Geometry of an asymmetric double ridge waveguide.
T
2a
b
d Region (I)
Region (II)
x
y
z
Figure 7: Geometry of a L-block.
of Fig. 6. Using the same procedure as in Sect. 2, the dispersion relations of an asymmetric
double ridge waveguide are obtained. For instance, the dispersion equation corresponding
to the matching point (0;0) is obtained as
I(0;0;a(1)) =
Z a(1)
a(1)
h
HI
z(x0;0) L(1)+L(2)+L(3)+L(4)
HII
z (x0;0) L(1)+L(2)+L(3)+L(4)
i
cosa(1)
l (x0 + a(1)) dx0
=
1X
m=0
h
q(1)
m I(1)
LH(0;0;a(1)) + q(2)
m I(2)
LH(a(1) +a(2) T;0;a(1))
+ q(3)
m I(3)
LH(a(3) a(1);b;a(1)) + q(4)
m I(4)
LH(T a(1) a(4);b;a(1))
i
= 0 (31)
where ILH(x;y;c) = IH(x+a;y;c) m = even
. It is straightforward to obtain the remaining
three dispersion equations corresponding to (x;y) = (T a(1) a(2);0), (a(3) a(1);b), and
(T a(1) a(4);b). In order to verify the validity of the approach based on LB, in Table 3 we
show the cuto wavenumber of an asymmetric ridge waveguide of Fig. 3. For comparison
purpose, we also show the result with the TB approach for the same geometry in Table
4. Since the width of LB, 2a, is twice that of TB in our computation, the convergence
characteristics of TB is better. This is because more higher modes are needed to obtain
9

10. d
(1)
2a
(1)
L(1) /
(x, y)
/
x
y
z
/
/
L(2) /
(x, y)
/
x
y
z
/
/
d
(2)
2a
(2)
+
+
+ +
L(3) /
(x, y)
/ L(4) /
(x, y)
/
x
y
z
/
/
x
y
z
/
/
d
(4)d
(3)
2a
(3) 2a
(4)
Figure 8: Superposition of four L-blocks.
convergence as the width 2a in region (I) becomes wider. It is interesting to note that
the cuto wavenumbers of the second and third modes in Table 4 approximately agree
with those of a rectangular waveguide, 78.54 and 104.7, respectively. In Fig. 9(a), the
Hz eld distribution of the second mode in Table 4 is illustrated. The Hz eld is almost
concentrated within the left cavity of Fig. 3, whose eld distribution is very similar to that
of a rectangular waveguide. Since the propagation constant for an asymmetric double ridge
waveguide is quite di erent from the rectangular waveguide similar to the right cavity, the
wave within the right cavity becomes evanescent. The Hz eld plot of a symmetric ridge
waveguide is shown in Fig. 9(b). The Hz eld is distributed in two cavities as it should be.
Table 3: Cuto wavenumbers (rad=m) of the three TE-modes for an asymmetric double
ridge waveguide using two LB.
Number of modes used 1 4 5
1 37.25 125.4 164.8
3 40.79 137.8 165.2
5 41.06 138.1 165.3
7 41.14 138.2 165.3
Parameters : a(1) = 10 mm; d(1) = 10 mm; a(2) = 7:5 mm; d(2) = 15 mm; b = 5 mm; T =
45 mm
10

11. Table 4: Cuto wavenumbers (rad=m) of the rst ve TE-modes for an asymmetric double
ridge waveguide using two TB.
Number of modes used 1 2 3 4 5
1 40.96 80.36 107.0 137.7 165.3
2 41.15 80.46 107.1 138.2 165.3
3 41.19 80.49 107.2 138.3 165.4
6] 41.31 80.56 107.2 138.5 165.5
Parameters : a = 5 mm; d = 10 mm; T(1) = 30 mm; b(1) = 20 mm; T(2) = 40 mm; b(2) =
15 mm
5 Conclusion
A novel T-block approach is proposed for analyzing symmetric and asymmetric ridge waveg-
uides. Simple closed-form dispersion relations for ridge waveguides are expressed in rapidly-
convergent series. Computed results indicate that our method, based on a superposition of
overlapping T-blocks, is accurate and numerically-e cient. A dominant-mode approxima-
tion for a ridge waveguide is shown to be valid and useful for most practical cases. It is
possible to extend our theory to the analysis of other complex waveguide structures that
can be divided into a superposition of overlapping T-blocks. For example, the shielded
microstrip line, nline, nonradiative dielectric guide, etc. are some typical waveguides that
our T-block approach can be applied to obtain their dispersion relations.
11

12. xy
zH
(a) Asymmetric double ridge waveguide
xy
zH
(b) Symmetric double ridge waveguide
Figure 9: Hz eld distributions of TE modes using two TB.
12

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