Fourier-transform analysis of a ridge waveguide and a rectangular coaxial line

1. Fourier-transform analysis of ridge waveguide and
rectangular coaxial line
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 novel technique based on the Fourier transform is applied to analyze
a ridge waveguide and rectangular coaxial line. Based on the image theorem, the
ridge waveguide and rectangular coaxial line are transformed into an in nite num-
ber of multiple groove guide. The dispersion relations for the ridge waveguide and
rectangular coaxial line are obtained in rigorous, yet simple series forms. Numerical
computations show that our series solutions converge fast and agree with others.
1 Introduction
Ridge waveguides have been utilized in microwave communication due to their wide
band and low impedance characteristics Pyle, 1966]. A rectangular coaxial line has
been also studied in Gruner, 1983] to assess its guidance characteristics in view of a
circular coaxial line. Although the guidance characteristics for a ridge waveguide and
a rectangular coaxial line are well understood, it is of theoretical interest to obtain
their dispersion relations in analytic and rigorous form. The purpose of the present
1

2. paper is to show that a new, simple, numerically e cient, yet rigorous analysis is
possible for a ridge waveguide and a rectangular coaxial line. The present paper
utilizes the Fourier transform approach that was used in Lee et al., 1995]. In order
to apply the Fourier transform to a ridge waveguide or a rectangular coaxial line,
the image theorem must be invoked beforehand. A use of the image theorem allows
us to transform the ridge waveguide and rectangular coaxial line into an equivalent
in nite number of multiple groove guide. The present paper shows how to combine
the Fourier transform with the image theorem to obtain the dispersion relation. This
means that the present paper o ers a new technique based on the Fourier transform
and image theorem. This paper o ers a new technique by converting a shielded trans-
mission line into an equivalent periodic transmission line structure such as the in nite
number of multiple groove guide. Our approach is di erent from other existing ap-
proaches based on Floquet's theorem or a Fourier series expansion. While Floquet's
theorem is often used for the eld expansion of a periodic structure, our approach
based on the Fourier transform and image theorem is a new concept dealing with
a periodic function without recourse to Floquet's theorem. We divide the periodic
transmission line structures into open and closed regions. We then use the Fourier
transform to represent the scattered eld in the open region. The residue calculus is
performed to evaluate the integral representation for the scattered eld in the open
region. Our approach allows us to obtain the simultaneous equations for the modal
coe cients corresponding the closed region, thereby reducing the number of unknown
modal coe cients and improving the convergence of series. This implies that our ap-
proach is numerically more e cient than other approach based on the mode-matching
technique. In this paper, we claim that the presented approach is applicable to other
passive components that are shielded by rectangular conductors. For example, our
approach is applicable to the shielded suspended substrate microstrip line, shielded
microstrip line, shielded strip line, and n line. In the next sections, we will derive the
2

3. dispersion relations for the ridge waveguide and rectangular coaxial line by applying
the Fourier transform to the multiple groove guide.
2 Analysis of Ridge Waveguide
Consider a double-ridge waveguide in Fig. 1(a). The side walls at x = a=2 and a=2 T
in a double-ridge waveguide are the electric mirrors, thus rendering their mirror-
images in nitely along the x-direction. This means that a double-ridge waveguide in
Fig. 1(a) is equivalent to an in nite number of groove guides in Fig. 1(b), where
the tangential E- eld at x = a=2 nT (n = 0;1;2; ) vanishes. Although the
analysis of a nite number of groove guides is available in Eom and Cho, 1999],
we repeat a similar analysis for an in nite number of groove guides for the sake of
easiness. A TE-wave is assumed to propagate along the z-direction in Fig. 1(b) such
as H(x;y;z) = H(x;y)ei z with the e i!t time convention. In region (I) ( d < y < 0),
(II) (0 < y < b), and (III) (b < y < b + d), we represent the Hz components as
HI
z(x;y) =
1X
n= 1
1X
m=0; even
qn
m cosam(x nT)cos m(y + d)
u(x nT) u(x nT a)] (1)
HII
z (x;y) = 1
2
Z 1
1
~H+
z ei y + ~Hz e i y]e i xd (2)
HIII
z (x;y) =
1X
n= 1
1X
m=0; even
sn
m cosam(x nT)cos m(y b d)
u(x nT) u(x nT a)] (3)
where am = m =a, m =
q
k2
a2
m
2
, =
pk2 2 2
, k = 2 = 0 and u( )
is a unit step function. The enforcement of the boundary conditions on the eld
continuities is required to determine the modal coe cients qn
m and sn
m. Enforcing the
Ex continuity at y = 0 gives
EII
x (x;0) =
8
<
:
EI
x(x;0) for nT < x < nT + a
0 otherwise
(4)
3

4. We apply the Fourier transform (
R 1
1(:)ei xdx) to (4) to obtain
~H+
z ~Hz =
1X
n= 1
1X
m=0; even
1
i qn
m m sin( md)Gn
m( ) (5)
where
Gn
m( ) = i 1 ( 1)mei a]
2
a2
m
ei nT (6)
Similarly applying the Fourier transform to the Ex continuity at y = b,
~H+
z ei b ~Hz e i b =
1X
n= 1
1X
m=0; even
1
i sn
m m sin( md)Gn
m( ) (7)
We multiply the Hz continuity at y = 0 by cosal(x pT) and integrate over pT <
x < pT + a to get
1X
m=0; even
n
qm m sin( md)I1 + a
2
cos( md) ml m] + sm m sin( md)I2 ]
o
= 0 (8)
where ml is the Kronecker delta, 0 = 2; m = 1 (m = 1;2; ),
I1 =
1X
n= 1
( 1)n
2
Z 1
1
cot( b)Gn
m( )Gp
l ( )d (9)
I2 =
1X
n= 1
( 1)n
2
Z 1
1
csc( b)Gn
m( )Gp
l ( )d (10)
When the number of a multiple groove guide Eom and Cho, 1999] approaches in nity,
the modal coe cients qn
m and sn
m in (1) and (3) become independent of the groove
location n. This means that for an in nite number of multiple groove guide, qn
m
becomes ( 1)nqm. Each sign in (9) and (10) denotes the even and odd number
of a half-wave variation from x = a=2 T through a=2. In order to obtain rapidly-
convergent series representations for (9) and (10), we identify the simple poles ( 2
a2
m = 0 and sin( b) = 0) in integrands of (9) and (10) Eom and Cho, 1999]. We
perform a contour integration based on the residue calculus to get
I1 = limN!1
NX
n= N
( 1)n
n
a
2
m ml np
m tan( mb)
i
b
1X
v=0
vf1( v)
v( 2
v a2
m)( 2
v a2
l )
o
(11)
I2 = limN!1
NX
n= N
( 1)n
n
a
2
m ml np
m sin( mb)
i
b
1X
v=0
( 1)v vf1( v)
v( 2
v a2
m)( 2
v a2
l )
o
(12)
f1( ) = ( 1)m+l + 1]ei vjn pjT ( 1)mei vj(n p)T+aj ( 1)lei vj(n p)T aj (13)
4

5. where v =
q
k2
(v =b)2 2
. Letting p = 0 and assuming that the medium
wavenumber k has a small positive imaginary part in order to guarantee the conver-
gence of a geometric series in (11) and (12), we obtain
I1 = a
2
m ml
m tan( mb)
i
b
1X
v=0
vf2 ( v)
v( 2
v a2
m)( 2
v a2
l ) (14)
I2 = a
2
m ml
m sin( mb)
i
b
1X
v=0
( 1)v vf2 ( v)
v( 2
v a2
m)( 2
v a2
l )
(15)
f2 ( ) = limN!1
NX
n= N
( 1)nf1( ) = 2 1 ei a] ei T ei (T a)
]
1 ei T (16)
Similarly from the Hz continuity at y = b between region (II) and (III), we get
1X
m=0; even
n
qm m sin( md)I2 ] + sm m sin( md)I1 + a
2 cos( md) ml m]
o
= 0(17)
The dispersion relation of the TE-mode is obtained from (8) and (17) as
j 1 + 2j j 1 2j = 0 (18)
where the elements of 1 and 2 are
1;ml = m sin( md)I1 + a
2 cos( md) m ml (19)
2;ml = m sin( md)I2 (20)
Note that j 1 + 2j = 0 (or j 1 2j = 0) represents the TE-mode dispersion
relation with an even (or odd) number of a half-wave variation in Hz along the
y-direction. This implies that j 1 + 2j = 0 gives the dispersion relation for a single-
ridge waveguide. When the TM-wave propagates along the z-direction in Fig. 1(b),
the Ez components are
EI
z(x;y) =
1X
n= 1
1X
m=2; even
pn
m sinam(x nT)sin m(y + d)
u(x nT) u(x nT a)] (21)
EII
z (x;y) = 1
2
Z 1
1
~E+
z ei y + ~Ez e i y]e i xd (22)
EIII
z (x;y) =
1X
n= 1
1X
m=2; even
rn
m sinam(x nT)sin m(y b d)
u(x nT) u(x nT a)] (23)
5

6. Utilizing the same method as was used in the previous TE-mode analysis, we obtain
the double-ridge waveguide dispersion relation for the TM-mode as
j 3 + 4j j 3 4j = 0 (24)
where
3;ml = sin( md)I3 + a
2 m cos( md) ml (25)
4;ml = sin( md)I4 (26)
I3 = a
2
m ml
tan( mb) amal
2i
b
1X
v=1
(v
b )2
f2 ( v)
v( 2
v a2
m)( 2
v a2
l ) (27)
I4 = a
2
m ml
sin( mb) amal
2i
b
1X
v=1
( 1)v(v
b )2
f2 ( v)
v( 2
v a2
m)( 2
v a2
l ) (28)
Similar to the TE-mode, j 3 + 4j = 0 (or j 3 4j = 0) represents the dispersion
relation associated with the even (or odd) number of a half-wave variation in Ez along
the y-direction. In addition, the analysis of an L-shaped waveguide in Fig. 1(c) is
possible by extending the results of double-ridge waveguide analysis. The TE- and
TM-mode dispersion relations of an L-shaped waveguide are given by j 1 + 2j = 0
with I+
1;2 and j 3 + 4j = 0 with I+
3;4, respectively. Fig. 2 shows the normalized cuto
wavelength versus various dimensions, indicating that our series solution (18) agrees
well with Pyle, 1966]. In our computations, two modes (m = 0; 2) are enough to
achieve the convergence of our series solution, when b=(b+2d) > 0:25. Although our
rigorous solution, (18) and (24), is represented in double-series, its convergence rate
is fast, thus e cient for numerical computation.
3 Analysis of Rectangular Coaxial Line
Consider a rectangular coaxial line in Fig. 3(a). Using the same procedure based on
the image theorem as in Sec. 2, it is possible to transform a rectangular coaxial line
in Fig. 3(a) into an in nite number of rectangular strips in Fig. 3(b). Note that
the tangential E- eld at x = a=2 nT (n = 0;1;2; ) vanishes in Fig. 3(b). We
6

7. express the elds in regions (I) and (III) as continuous modes in terms of the Fourier
transform. We represent the Hz eld of the TE-mode in region (II) as a superposition
of discrete modes as
HII
z (x;y) =
1X
n= 1
1X
m=0; even
qn
m cos m(y + d) + sn
m cos( my)]cosam(x nT)
u(x nT) u(x nT a)] (29)
By matching the boundary conditions on eld continuities at y = 0 and d, we obtain
the dispersion relation for the TE-mode as
1 2
2 3
= 0 (30)
where the elements of 1, 2, and 3 are
1;ml = m sin( md)J1 + a
2 cos( md) m ml (31)
2;ml = a
2 m ml (32)
3;ml = m sin( md)J3 + a
2 cos( md) m ml (33)
Jp = I1 + I2
= a
2
m ml
m tan( mbp)
2i
bp
1X
v=0
vf2 ( v)
v( 2
v a2
m)( 2
v a2
l )
(34)
and v =
q
k2
(v =bp)2 2
. We represent the Ez eld of the TM-mode as
EII
z (x;y) =
1X
n= 1
1X
m=2; even
pn
m sin m(y + d) + rn
m sin( my)]sinam(x nT)
u(x nT) u(x nT a)] (35)
By enforcing the boundary conditions at y = 0 and d, we obtain the dispersion
relation for the TM-mode
4 5
5 6
= 0 (36)
where
4;ml = sin( md)K1 + a
2 m cos( md) ml (37)
7

8. 5;ml = a
2 m ml (38)
6;ml = sin( md)K3 + a
2 m cos( md) ml (39)
Kp = I3 + I4
= a
2
m ml
tan( mbp)
amal
2i
bp
1X
v=1
(v
bp
)2
f2 ( v)
v( 2
v a2
m)( 2
v a2
l )
(40)
Table 1 shows the normalized cuto wavelength for the rst six TE-modes, indicat-
ing that our solution converges fast and agrees well with Navarro and Such, 1992].
The evaluation of eld variation within a coaxial line is trivial once the propagation
constant is determined. Some plots of eld variation are available in Navarro and
Such, 1992].
4 Conclusion
The analysis of a ridge waveguide and rectangular coaxial line is shown, using the
Fourier transform and image theorem. Simple yet rigorous dispersion relations for the
ridge waveguide and rectangular coaxial line are presented and compared with other
solutions. Our closed-form series solutions converge fast and are accurate enough
to use in most practical applications. It is possible to extend our theory to other
shielded waveguide structures including the shielded suspended substrate microstrip
line, shielded microstrip line, shielded strip line, and n line.
8

9. References
1] Eom, H.J. and Y.H. Cho, Analysis of multiple groove guide," Electrons Lett.,
vol. 35, no. 20, pp. 1749-1751, Sept. 1999
2] Gruner, L., Higher order modes in square coaxial lines," IEEE Trans. Microwave
Theory Tech., vol. 31, no. 9, pp. 770-772, Sept. 1983
3] Lee, B.T., J.W. Lee, H.J. Eom, and S.Y. Shin, Fourier-transform analysis for
rectangular groove guide," IEEE Trans. Microwave Theory Tech., vol. 43, no. 9,
pp. 2162-2165, Sept. 1995.
4] Navarro, E.A. and V. Such, Study of TE and TM modes in waveguides of
arbitrary cross-section using an FD-TD formulation," IEE Proc., pt. H, vol. 139,
no. 6, pp. 491-494, Dec. 1992
5] Pyle, J.R., The cuto wavelength of the TE10 mode in ridged rectangular waveg-
uide of any aspect ratio," IEEE Trans. Microwave Theory Tech., vol. 14, no. 4, pp.
175-183, April 1966
9

10. 5 Figure Captions
Figure 1: Geometry of a ridge waveguide.
Figure 2: Normalized cuto wavelength of the TE10 mode with an aspect ratio (b +
2d)=T = 0:9.
Figure 3: Geometry of a rectangular coaxial line.
10

11. 6 Table Caption
Table 1: Normalized cuto wavelengths c=T of the TE-mode for a square coaxial
line with a=T = 0:5.
11

12. T
a/2
a/2
d
d
b
y
x
z
Region (III) (III)
Region (II)
Region (I)(I)
(a) Double-ridge waveguide
. . . . .. . . . . y
x
z
a
T
b
d
d
Region (II)
Region (I)
Region (III)
(I)(I)
(III)(III)
(b) In nite number of groove guides
d
b/2
a/2
T/2
(c) L-shaped waveguide
Figure 1: Geometry of a ridge waveguide.
12

13. 0 0.2 0.4 0.6 0.8 1
2
2.5
3
3.5
4
4.5
5
5.5
0.2 0.4 0.6 0.8
2
3
4
5
Normalizedcutoffwavelength,
10
(T - a) / T
λc/T
x x
0.6
0.5
0.35
0.25
0.15
0.1
b / (b + 2d) =
theory
Pyle[ , 1966]
Figure 2: Normalized cuto wavelength of the TE10 mode with an aspect ratio (b +
2d)=T = 0:9.
13

14. d
b
Region (I)
Region (II)
Region (III)
(II)
b a/2 a/2
T1
3
y
x
z
(a) Rectangular coaxial line
. . . . .
a
b1
y
x
z
T b3
. . . . . d
Region (III)
Region (II)
Region (I)
(II)
(b) In nite number of rectangular strips
Figure 3: Geometry of a rectangular coaxial line.
14

15. Table 1: Normalized cuto wavelengths c=T of the TE-mode for a square coaxial
line with a=T = 0:5.
Mode number 1 2 3 4 5 6
m = 0 2.840 1.543 1.329 0.976 0.691 0.640
m = 0; 2 2.805 1.540 1.302 0.975 0.675 0.640
m = 0; 2; 4 2.799 1.539 1.298 0.974 0.674 0.640
Navarro and Such;1992] 2.807 1.538 1.290 0.970 0.672 0.637
15