Fourier-transform analysis of a unilateral fin line and its derivatives

1. Fourier-transform analysis of a unilateral n line
and its derivatives
Yong H. Cho and Hyo J. Eom
Department of Electrical Engineering
Korea Advanced Institute of Science and Technology
373-1, Guseong-dong, Yuseong-gu, Daejeon, Republic of Korea
Phone +82-42-869-3436 Fax +82-42-869-8036
E-mail : hjeom@ee.kaist.ac.kr
Abstract A unilateral n line and its derivatives are analyzed with the Fourier trans-
form and mode-matching. The image theorem is utilized to transform a unilateral
n line into an in nite number of suspended substrate microstrip lines. New rigorous
dispersion relations are obtained in fast-convergent series which are e cient for nu-
merical computation. Calculations are performed to illustrate the rate of convergence
and accuracy of our series solution.
1 Introduction
The wave propagation characteristics for the planar waveguide structures have been
extensively studied and well understood. Various numerical and analytical techniques
1-7] have been utilized to obtain their dispersion relations in simple algebraic for-
mulas which give accurate values for practical purposes. Although their dispersion
relations for the planar waveguides are available in approximate simple formulas, it is
also of fundamental interest to obtain the dispersion relations in rigorous and analytic
1

2. forms. A standard mode-matching technique 4] has been used to rigorously derive
the exact dispersion relations for a certain class of planar waveguide structures. In
5,6], the spectral domain approach is utilized to calculate the dispersion relations of
a shielded microstrip line and bilateral n line. In the spectral domain approach, the
dispersion equations are obtained by the relationship of the nite Fourier-transformed
tangential E eld and transformed strip current density. Then, the Galerkin's method
and Parseval's theorem are applied to get the algebraic dispersion equations of a given
geometry. The motivation of the present paper is to show that another new approach
7], based on the image theorem, Fourier transform, and mode matching, yields a rig-
orous, analytic, yet numerically e cient dispersion relation. In our method, we take
the Fourier transform over an in nite range, thus converting the original problem to
the equivalent one. Then, we obtain the algebraic dispersion equations of the eld
continuities, irrespective of the current density. The residue calculus is utilized to
obtain fast-convergent series solutions. The Fourier transform approach 8] allows us
to obtain the simultaneous equations for the modal coe cients with respect to only
the closed regions, thereby reducing the number of unknown modal coe cients and
improving the convergence rate of series. This implies that our approach is numeri-
cally more e cient than other approaches based on the mode matching method. In
this paper, we will derive the rigorous dispersion relations for a certain class of planar
waveguides whose most representative form is a unilateral n line. We rst trans-
form a unilateral n line into equivalent multiple suspended substrate microstrip lines
using the image theorem, we then utilize a technique of the Fourier transform and
mode matching to obtain its dispersion relation. Our theoretical approach, based on
the Fourier transform, is novel in that the algebraic series expression of our solution
is di erent from any other existing ones. Although the expressions of our new rigor-
ous dispersion relation appear to be more involved than other existing solutions, our
rigorous, analytic solution is rapidly-convergent series, so that it is e cient for numer-
2

3. ical computation. We also present the dispersion relations for a shielded suspended
substrate microstrip line, shielded microstrip line, shielded strip line, slot line, and
coplanar waveguide, which are all derivatives of the unilateral n line. In the next
sections, we start with the eld representations for a unilateral n line and derive its
dispersion relation.
2 Field Representations
Consider a unilateral n line as shown in Fig. 1(a). Note that regions (I) through (IV)
are lled with the dielectric materials of permittivities 1 through 4, respectively. Due
to the electric walls placed at x = (a T)=2, it is possible to apply the image theorem
to transform the problem of Fig. 1(a) into an equivalent problem of Fig. 1(b). Fig.
1(b) illustrates an in nite number of suspended substrate microstrip lines. A hybrid
mode, consisting of the TM and TE waves transverse to the z-direction, is assumed to
propagate in regions (I) through (IV) in Fig. 1(b). With an e i!t time convention, we
represent the propagating longitudinal eld components as Ez(x;y;z) = Ez(x;y)ei z
and Hz(x;y;z) = Hz(x;y)ei z. Then in each region, we represent the guided waves
as
EI
z(x;y) = 1
2
Z 1
1
~E+
I ei 1y + ~EI e i 1y]e i xd (1)
HI
z(x;y) = 1
2
Z 1
1
~H+
I ei 1y + ~HI e i 1y]e i xd (2)
EII
z (x;y) =
1X
n= 1
1X
k=1
pn
k sin k(y + d) + rn
k sin( ky)]sinak(x nT)
u(x nT) u(x nT a)] (3)
HII
z (x;y) =
1X
n= 1
1X
m=0
qn
m cos m(y + d) + sn
m cos( my)]cosam(x nT)
u(x nT) u(x nT a)] (4)
EIII
z (x;y) = 1
2
Z 1
1
~E+
IIIei 3(y+d)
+ ~EIIIe i 3(y+d)
]e i xd (5)
HIII
z (x;y) = 1
2
Z 1
1
~H+
IIIei 3(y+d)
+ ~HIIIe i 3(y+d)
]e i xd (6)
3

4. EIV
z (x;y) = 1
2
Z 1
1
~E+
IV ei 4(y+d+b3)
+ ~EIV e i 4(y+d+b3)
]e i xd (7)
HIV
z (x;y) = 1
2
Z 1
1
~H+
IV ei 4(y+d+b3)
+ ~HIV e i 4(y+d+b3)
]e i xd (8)
where am = m =a, m =
q
k2
2 a2
m
2
, q =
q
k2
q
2 2
, kq = !p q (q =
1;2;3;4), and u( ) is a unit step function.
3 Boundary Condition Enforcement
The tangential E and H elds must be continuous at the boundaries y = b1, 0,
d; ( d b3), and ( d b3 b4). The boundary conditions to be enforced at each
boundary are given as
EI
x;z(x;b1) = 0 (9)
EI
x;z(x;0) =
8
<
:
EII
x;z(x;0) for nT < x < nT + a
0 otherwise
(10)
HI
x;z(x;0) = HII
x;z(x;0) (11)
EIV
x;z(x; d b3 b4) = 0 (12)
EIII
x;z (x; d b3) = EIV
x;z(x; d b3) (13)
HIII
x;z (x; d b3) = HIV
x;z(x; d b3) (14)
EIII
x;z (x; d) =
8
<
:
EII
x;z(x; d) for nT < x < nT + a
0 otherwise
(15)
HII
x;z(x; d) = HIII
x;z (x; d) (16)
From (9),
~EI = ~E+
I ei2 1b1
(17)
~HI = ~H+
I ei2 1b1
(18)
Applying the Fourier transform (
R 1
1( )ei xdx) to (10) yields
~E+
I + ~EI =
1X
n= 1
1X
k=1
pn
k sin( kd)Fn
k ( ) (19)
4

5. ~H+
I ~HI =
1X
n= 1
1X
k=1
1
i 1
pn
kA1
k sin( kd)Gn
k( )
1X
n= 1
1X
m=0
1
i 1
qn
mB1
m sin( md)Gn
m( ) (20)
where Aq
k =
k2
q k2
2
k2
2
2
ak
! ; Bq
m =
k2
q
2
k2
2
2 m; Fn
k ( ) = ak ( 1)kei a 1]
2
a2
k
ei nT, and
Gn
m( ) = i 1 ( 1)mei a]
2
a2
m
ei nT: By substituting ~EI and ~HI from (17) through (20)
into (11), multiplying (11) by cosal(x pT), and integrating from pT < x < pT +a,
we obtain
1X
k=1
pkA1
k sin( kd)J+
1
1X
m=0
n
qmB1
m sin( md)J+
1 + qm cos( md) + sm]a
2 m ml
o
= 0 (21)
1X
k=1
n
pk sin( kd)K+
1 + pk cos( kd) + rk]a
2
C1
k ks
o
+
1X
m=0
qm cos( md) + sm]a
2
D1
m ms = 0 (22)
where ml is the Kronecker delta, 0 = 2; m = 1 (m = 1;2; ); Cq
k =
k2
q
2
k2
2
2
2
q
k,
and Dq
m =
k2
q k2
2
k2
2
2
am
! q
. The explicit representations of J+
1 and K+
1 are given in
Appendix. From (12) through (14),
~EIV = ~E+
IV e i2 4b4
(23)
~HIV = ~H+
IV e i2 4b4
(24)
~E+
IIIe i 3b3
+ ~EIIIei 3b3
= ~E+
IV + ~EIV (25)
~H+
IIIe i 3b3
+ ~HIIIei 3b3
= ~H+
IV + ~HIV (26)
~H+
IIIe i 3b3 ~HIIIei 3b3
= A! 3
~E+
IV + ~EIV ] + B 4
3
~H+
IV ~HIV ] (27)
~E+
IIIe i 3b3 ~EIIIei 3b3
= A! 3 3
~H+
IV + ~HIV ] + B 4 4
3 3
~E+
IV ~EIV ] (28)
5

6. where A = k2
3 k2
4
k2
4
2
and B = k2
3
2
k2
4
2
. By taking the Fourier transform (
R 1
1( )ei xdx)
of (15), we obtain
~E+
III + ~EIII =
1X
n= 1
1X
k=1
rn
k sin( kd)Fn
k ( ) (29)
~H+
III ~HIII =
1X
n= 1
1X
k=1
1
i 3
rn
kA3
k sin( kd)Gn
k( )
+
1X
n= 1
1X
m=0
1
i 3
sn
mB3
m sin( md)Gn
m( ) (30)
It is possible to solve a set of simultaneous equations (23) through (30) for ~EIII and
~HIII. The results are
~E+
III ~EIII =
1X
n= 1
1X
k=1
rn
k sin( kd)
n
i M( )
( ) Fn
k ( ) + A3
kA! 3
N( )
3 ( )Gn
k( )
o
1X
n= 1
1X
m=0
sn
m sin( md)B3
mA! 3
N( )
3 ( )
Gn
m( ) (31)
~H+
III + ~HIII =
1X
n= 1
1X
k=1
rn
k sin( kd)
n
A3
k
L( )
3 ( )Gn
k( ) + iA!
N( )
( ) Fn
k ( )
o
1X
n= 1
1X
m=0
sn
m sin( md)B3
m
L( )
3 ( )Gn
m( ) (32)
Substituting ~EIII and ~HIII into (16), we obtain
1X
k=1
rk sin( kd) A3
kL+
A ak
! N+
]
1X
m=0
n
qm + sm cos( md)]a
2 m ml + smB3
m sin( md)L+
o
= 0 (33)
1X
k=1
n
pk + rk cos( kd)]a
2
C3
k ks + rk sin( kd) M+
A3
kA as
! 3
N+
]
o
+
1X
m=0
n
qm + sm cos( md)]a
2D3
m ms + sm sin( md)B3
mA as
! 3
N+
o
= 0 (34)
where the explicit representations of L+
; M+
, and N+
are given in Appendix. Since
the number (N) of multiple suspended substrate microstrip lines shown in Fig. 1(b)
approaches in nity (N ! 1), the modal coe cients pn
k; rn
k; qn
m, and sn
m become inde-
pendent of the microstrip location n (thus, pn
k ! ( 1)nkpk; , and sn
m ! ( 1)nmsm).
Note that (21), (22), (33), and (34) is the dispersion relation for a unilateral n line,
6

7. constituting a system of simultaneous equations for the unknown modal coe cients,
pk; rk; qm; and sm. In order to check the validity of our dispersion relation, we
compare our computational results with 9]. The comparison is shown in Fig. 2,
thus con rming an excellent agreement between them. Our computational experi-
ence indicates that two modes (k = 1;2) are enough to achieve the convergence in
the propagation constant . This means that our dispersion relation is accurate yet
numerically e cient for practical purpose, although the expressions for the dispersion
relation are rather involved.
4 Special Cases
Under special conditions, the dispersion relation, (21), (22), (33), and (34) for a
unilateral n line, may be simpli ed to those of other planar structure waveguides.
If the indices m and k in (21), (22), (33), and (34) are chosen to be even, then they
become the dispersion relation for a shielded suspended substrate microstrip line (see
an inset in Fig. 3 with 3 6= 4). Note that each sign in Appendix should be used to
take into account the even and odd modes propagating along the x-direction. If we
choose 3 = 4, then (21), (22), (33), and (34) simplify to the dispersion relation for
a shielded microstrip line (see an inset in Fig. 3 with 3 = 4) with A = 0; L = J3 ,
and M = K3 as
1X
k=2; even
pkA1
k sin( kd)J1
1X
m=0; even
n
qmB1
m sin( md)J1 + qm cos( md) + sm]a
2 m ml
o
= 0 (35)
1X
k=2; even
n
pk sin( kd)K1 + pk cos( kd) + rk]a
2
C1
k ks
o
+
1X
m=0; even
qm cos( md) + sm]a
2D1
m ms = 0 (36)
1X
k=2; even
rkA3
k sin( kd)J3
7

8. 1X
m=0; even
n
qm + sm cos( md)]a
2 m ml + smB3
m sin( md)J3
o
= 0 (37)
1X
k=2; even
n
pk + rk cos( kd)]a
2C3
k ks + rk sin( kd)K3
o
+
1X
m=0; even
qm + sm cos( md)]a
2D3
m ms = 0 (38)
Fig. 3 represents the convergence characteristics of the normalized guided wavelength
for a shielded microstrip line, indicating that our solution is fast-convergent and
accurate. If we choose 1 = 2 = 3 = 4, then (21), (22), (33), and (34) simplify to the
dispersion relationfor a shielded strip line with Aq
k = Dq
m = A = 0. Note that (21) and
(33) yield the dispersion relation for the TE mode, whereas (22) and (34) for the TM
mode. The dispersion relations for the TE and TM modes agree with (15) and (21) in
7], respectively, thus we do not show their explicit representations. If the number of
slots (N) in region (II) in Fig. 1(b) is chosen to be nite (N = 1;2), then ml ! np ml,
ks ! np ks, f2 ( ) ! f1( ), and (21), (22), (33) and (34) are transformed into the
dispersion relations for a slot line (N = 1) and coplanar waveguide (N = 2) with the
upper shielding at y = b1 and conductor backing at y = ( d b3 b4). Our approach
is based on the open con guration such as a rectangular groove guide 8, 11]. In
11], the multiple rectangular groove guide is analyzed with the Fourier transform
and mode matching. The analysis of a nite number of slots is similar to that in 11].
Their explicit representations for the dispersion relations are
N 1X
n=0
1X
k=1
pn
kA1
k sin( kd)J1
N 1X
n=0
1X
m=0
n
qn
mB1
m sin( md)J1 + qn
m cos( md) + sn
m]a
2 m np ml
o
= 0 (39)
N 1X
n=0
1X
k=1
n
pn
k sin( kd)K1 + pn
k cos( kd) + rn
k]a
2C1
k np ks
o
+
N 1X
n=0
1X
m=0
qn
m cos( md) + sn
m]a
2D1
m np ms = 0 (40)
N 1X
n=0
1X
k=1
rn
k sin( kd) A3
kL A ak
! N]
8

9. N 1X
n=0
1X
m=0
n
qn
m + sn
m cos( md)]a
2 m np ml + sn
mB3
m sin( md)L
o
= 0 (41)
N 1X
n=0
1X
k=1
n
pn
k + rn
k cos( kd)]a
2
C3
k np ks + rn
k sin( kd) M A3
kA as
! 3
N]
o
+
N 1X
n=0
1X
m=0
n
qn
m + sn
m cos( md)]a
2D3
m np ms + sn
m sin( md)B3
mA as
! 3
N
o
= 0 (42)
In order to check the validity of (39) through (42), we compute the dispersion relation
for a slot line (see an inset in Fig. 4 for a slot line) using k = 1 and m = 0. The
comparison between our computation and 12] is illustrated in Fig. 4, con rming a
favorable agreement.
5 Conclusion
Fast-convergent series solutions for a wide class of planar structure waveguides are
presented. Although our series solutions are rather involved, they are rigorous yet e -
cient for numerical computation. The dispersion relations for a unilateral n line and
its derivatives (shielded suspended substrate microstrip line, shielded microstrip line,
shielded strip line, slot line, and coplanar waveguide) are presented. The comparisons
to other existing solutions indicate that our rigorous series solutions are accurate and
numerically e cient. It is also possible to extend our proposed approach to analyzing
the characteristics of multi-conductor transmission lines.
9

10. References
1] P. Pramanick and P. Bhartia, Accurate analysis equations and synthesis tech-
nique for unilateral nlines," IEEE Trans. Microwave Theory Tech., 1985, MTT-
33, (1), pp. 24-30
2] Y. H. Shu, X. X. Qi, and Y. Y. Wang, Analysis equations for shielded sus-
pended microstrip line and broadside-coupled stripline," IEEE MTT-S Int. Mi-
crowave Symp. Digest, 1987, pp. 693-696
3] R. Garg and K. C. Gupta, Expressions for wavelength and impedance of a slot-
line," IEEE Trans. Microwave Theory Tech., 1976, MTT-24, (8), p. 532
4] R. Vahldieck and J. Bornemann, A modi ed mode-matching technique and its
application to a class of quasi-planar transmission lines," IEEE Trans. Microwave
Theory Tech., 1985, MTT-33, (10), pp. 916-926
5] T. Itoh and R. Mittra, A technique for computing dispersion characteristics of
shielded microstrip lines," IEEE Trans. Microwave Theory Tech., 1974, MTT-22,
(10), pp. 896-898
6] L.-P. Schmidt and T. Itoh, Spectral domain analysis of dominant and higher
order modes in n-lines," IEEE Trans. Microwave Theory Tech., 1980, MTT-28,
(9), pp. 981-985
7] Y. H. Cho and H. J. Eom, Fourier-transform analysis of ridge waveguide and
rectangular coaxial line," Radio Science, 2001, 36, (4), pp. 533-538
8] B. T. Lee, J. W. Lee, H. J. Eom and S. Y. Shin, Fourier-transform analysis for
rectangular groove guide," IEEE Trans. Microwave Theory Tech., 1995, MTT-43,
(9), pp. 2162-2165
10

11. 9] E. Yamashita and K. Atsuki, Analysis of microstrip-like transmission lines by
nonuniform discretization of integral equations," IEEE Trans. Microwave Theory
Tech., 1976, MTT-24, (4), pp. 195-200
10] R. Mittra and T. Itoh, A new technique for the analysis of the dispersion
characteristics of microstrip lines," IEEE Trans. Microwave Theory Tech., 1971,
MTT-19, (1), pp. 47-56
11] H. J. Eom and Y. H. Cho, Analysis of multiple groove guide," Electron. Lett.,
1999, 35, (20), pp. 1749-1751
12] E. A. Mariani, C. P. Heinzman, J. P. Agrios, and S. B. Cohn, Slot line character-
istics," IEEE Trans. Microwave Theory Tech., 1969, MTT-17, (12), pp. 1091-1096
11

12. Appendix : Evaluation of Integrals
Jq and Kq are represented in integrals as
Jq =
1X
n= 1
( 1)n
2
Z 1
1
( 1)nmGn
m( )Gp
l ( )
q tan( qbq)
d (43)
Kq =
1X
n= 1
( 1)n
2
Z 1
1
( 1)nk q
tan( qbq)Fn
k ( )Fp
s ( ) d (44)
Applying the residue calculus to Jq and Kq yields
Jq = a
2
m ml
m tan( mbq)
i
bq
1X
v=0
vf2 ( v)
v( 2
v a2
m)( 2
v a2
l ) (45)
Kq = a
2
k ks
tan( kbq) akas
i
bq
1X
v=1
(v
bq
)2
f2 ( v)
v( 2
v a2
k)( 2
v a2
s) (46)
f2 ( ) = limN!1
N=2
X
n= N=2
( 1)nf1( )
= ( 1)m+l + 1](1 ( 1)mei T) ( 1)m + ( 1)l](ei a ( 1)mei (T a)
)
1 ( 1)mei T
(47)
f1( ) = ( 1)m+l + 1]ei vjn pjT ( 1)mei vj(n p)T+aj ( 1)lei vj(n p)T aj (48)
where v =
q
k2
q (v =bq)2 2
and m =
q
k2
q a2
m
2
. Note that the number
of slots (N) is one or two in case of a slot line or coplanar waveguide. Similarly,
L ; M , and N are
L =
1X
n= 1
( 1)n
2
Z 1
1
( 1)nm L( )
3 ( ) Gn
m( )Gp
l ( ) d
= a
2
L(am)
m (am) m ml + i
1X
v=1
2
v L( v)f2 ( v)
3
0( v)( 2
v a2
m)( 2
v a2
l )
(49)
M =
1X
n= 1
( 1)n
2
Z 1
1
( 1)nk 3 M( )
( ) Fn
k ( )Fp
s ( ) d
= a
2
k M(ak)
(ak) ks + akasi
1X
v=1
3 M( v)f2 ( v)
0( v)( 2
v a2
k)( 2
v a2
s)
(50)
N =
1X
n= 1
( 1)n
2
Z 1
1
( 1)nk N( )
( ) Gn
k( )Gp
s( ) d
= a
2
N(ak)
(ak) ks + i
1X
v=1
2
v N( v)f2 ( v)
0( v)( 2
v a2
k)( 2
v a2
s)
(51)
12

13. where
( ) = 3 tan( 3b3) + B 4 tan( 4b4)] 3 tan( 4b4) + B 4
3
4 tan( 3b3)]
+ Ak3
2
tan( 3b3)tan( 4b4) (52)
L( ) = 3 B 4 tan( 3b3)tan( 4b4)] 3 tan( 4b4) + B 4
3
4 tan( 3b3)]
Ak3
2
tan2
( 3b3)tan( 4b4) (53)
M( ) = 3 tan( 3b3) + B 4 tan( 4b4)] B 4
3
4 3 tan( 3b3)tan( 4b4)]
+ Ak3
2
tan( 4b4) (54)
N( ) = 3 sec2
( 3b3)tan( 4b4) (55)
0( ) = d ( )=d , m =
q
k2
3 a2
m
2
, and v is obtained by ( v) = 0.
13

14. A Figure Captions
Figure 1 : Geometry of a unilateral n line.
(a) Unilateral n line (b) In nite number of suspended substrate microstrip lines
Figure 2 : Normalized propagation constant versus frequency for a unilateral n line
with 1 = 2 = 4 = 0; 3 = 9:35 0; a = 2mm; d = 0mm; b1 = b4 = 4:5mm, and
b3 = 1mm.
Figure 3 : Normalized guided wavelength versus frequency for a shielded microstrip
line with 1 = 2 = 0; 3 = 4; a = 11:43mm; d = 0mm; T = 12:7mm; b1 =
11:43mm; b3 = 1:27mm, and b4 = 0mm.
Figure 4 : Normalized guided wavelength versus frequency for a slot line with 1 =
2 = 4 = 0; a = 0:61mm; b1 = b4 = 4b3; d = 0mm; (a) 3 = 20:3 0; b3 = 3:5mm,
(b) 3 = 14:5 0; b3 = 1:8mm.
14

15. d
Region (I)b1
b3
T
y
b4
Region (III) a
Region (IV)
Region (II)
x
z
ε1
ε3
ε4
ε2
PEC
(a) Unilateral n line
a
b1
T b3
. . . . .. . . . . d
b4
Region (II)
Region (III)
Region (IV)
Region (I) y
x
z
PEC
PEC
(b) In nite number of suspended substrate microstrip lines
Figure 1: Geometry of a unilateral n line.
15

16. 1 2 5 10 20
0
0.5
1
1.5
2
2.5
1 2 5 10 20
0
0.5
1
1.5
2
β/βNormalizedphaseconstant,
Yamashita’s result in [9]
Frequency [GHz]
o o
0
2.5
T = mm10
mm20
mm40
mm60
Figure 2: Normalized propagation constant versus frequency for a unilateral n line
with 1 = 2 = 4 = 0; 3 = 9:35 0; a = 2mm; d = 0mm; b1 = b4 = 4:5mm, and
b3 = 1mm.
16

17. 5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
2 5 10 15
0
0.2
0.4
0.6
λ/λNormalizedguidedwavelength,
Mittra’s result in [10]
Frequency [GHz]
o o
0
ε = 2.65 ε
g
3 0
ε = 8.875 ε3 0
two modes
three modessingle mode
Τ
a/2 a/2
d
b4
b3
b1
Figure 3: Normalized guided wavelength versus frequency for a shielded microstrip
line with 1 = 2 = 0; 3 = 4; a = 11:43mm; d = 0mm; T = 12:7mm; b1 =
11:43mm; b3 = 1:27mm, and b4 = 0mm.
17

18. 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4
0.3
0.34
0.38
0.42
0.46
2 3
0.3
0.34
0.38
0.42
λ/λNormalizedguidedwavelength,
experiment [12]
Frequency [GHz]
o o
0g
(a)
(b)
a
d
b4
b3
b1
theory [12]x x
0.46
4
Figure 4: Normalized guided wavelength versus frequency for a slot line with 1 =
2 = 4 = 0; a = 0:61mm; b1 = b4 = 4b3; d = 0mm; (a) 3 = 20:3 0; b3 = 3:5mm,
(b) 3 = 14:5 0; b3 = 1:8mm.
18