Abstract-A simple method for formofating the dyadic Green&apos;s functions in the spectral domain is presented for generalized printed traaaroiasion fines wbieh contain severaf dielectric layers and conductors appearing at several dielectric interfaa+s. The method is based on the transverse equivalent tranamiwion line for a spectraf wave and on a simple coordinate transformation. Tfds formulation process is so simple that often it is accompfiibed afmost by inspection of the physicaf cross-aectionaf sbwture of the transmission fine. Tbe method is applied to a new versatile transmission Iinq a mfcrastrfp-slot~me, and some numerical results are presented

SENIOR MEMBER, IEEE Tatsuo Itoh

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MIT-28, NO. 7, JULY 1980 733
Spectral Domain Immitance Approach for
Dispersion Characteristics of Generalized
Printed Transmission Lines
TATSUO ITOH, SENIOR MEMBER, IEEE
Abstract-A simple method for formofating the dyadic Green’s func-
tions in the spectral domain is presented for generalized printed traaaroia-
sion fines wbieh contain severaf dielectric layers and conductors appearing
at several dielectric interfaa+s. The method is based on the transverse
equivalent tranamiwion line for a spectraf wave and on a simple coordinate
transformation. Tfds formulation process is so simple that often it is
accompfiibed afmost by inspection of the physicaf cross-aectionaf sbwture
of the transmission fine. Tbe method is applied to a new versatile transmis-
sion Iinq a mfcrastrfp-slot ~me, and some numerical results are presented.
I. INTRODUCTION
A FEW YEARS AGO, a method called the spectral-domain technique was developed for efficient
numerical analyses for various planar transmission lines
and successfully applied to a number of structures [1], [2].
One difficulty in applying this technique is that a lengthy
derivation process is required in the formulation stage,
especially for the more complicated structures such as the
one recently proposed by Aikawa [3], [4] in which more
than one conductor are located at different dielectric
interfaces. This paper presents a simple method for deriv-
ing the dyadic Green’s functions (immittance functions)
which is based on the transverse equivalent circuit con-
cept as applied in the spectral domain in conjunction with
a simple coordinate transformation rule. This technique is
quite versatile and the formulation of the Green’s function
may be done almost by inspection in many structures. It is
noted that symmetry in the structure is not required and
that the analysis can be extended to finite circuit ele-
ments, such as the disk resonator.
In what follows, we first illustrate the formulation pro-
cess for the microstrip line and subsequently extend it to a
more general microstrip-slot structure. Numerical results
for the microstrip-slot structure are also presented.
II. ILLUSTRATION OF THE FORMULATION PROCESS
To illustrate the formulation process, we will use a
simple shielded microstrip line shown in Fig. 1. In conven-
tional space-domain analysis [5], this structure may be
analyzed by first formulating the following coupled homo-
geneous integral equations and then solving for the un-
Manuscript received October 26, 1979; revised February 2, 1980. This
work was supported in part by U.S. Army Research Officx under Grant
DAA29-78-G-0145.
The author is with the Department of Electrical Engineering, Univer-
sity of Texas, Austin, TX 78712.
Fig. 1. Cross section of a rnicrostfip line.
known propagation constant fl:
J[ (2== x - X’, d).lz(x’) + Zzx(x – X’,d).lx(x’) ] dx’ =0
(1)
J-[ (2X= X --X’,d)~z(X’) +Zxx(x –X’, d)~x(X’) ] dx =0
(2)
where JX and J= are unknown current components on the
strip and the Green’s functions (impedance functions) 2=,,
etc., are functions of unknown ~ as well. The integration
is over the strip, and (1) and (2) are valid on the strip. The
left-hand sides of these equations give E= and EX compo-
nents on the strip and, hence, are required to be zero to
satisfy the boundary condition at the perfectly conducting
strip. These equations may be solved provided that Z==,
etc., are given. However, for the inhomogeneous struc-
tures, these quantities are not available in closed forms.
In the spectral domain formulation, we use Fourier
transforms of (1) and (2) and deal with algebraic equa-
tions
~zz(a, d)~=(a,d) + ZzX(a, d)~X(a>d) = ~.(a>d) (3)
~Xz(a,d)~z(a,d) + zXX(a,d)~X(a,d) = ~x(a,d) (4)
instead of the convolution-type coupled integral equations
(1) and (2). In (3) and (4), quantities with - are Fourier
transforms of corresponding quantities without -. The
Fourier transform is defined as
~(a) = ~@ ~(x)~tidx. (5)
—m
Notice that the right-hand sides of (3) and (4) are no
longer zero because they are the Fourier transforms of E=
and EX on the substrate surface which are obviously
nonzero except on the strip. Hence, algebraic equations
(3) and (4L contai~ four unknowns ~z, ~., ~z, and &
However, E= and EX will be eliminated later in the solu-
tion process based on the Galerkin’s procedure.
0018-9480/80/0700-0733$00.75 @1980 IEEE
734 IEEE TRANSACTIONS ON MICROWAVE THEORY ANO TECHNIQUES, VOL. MTT-28, NO. 7, JULY 1980
x
k
YTMi=:=J=, i=l,2 (11)
v Yi
v
YTEi=–~=J, i=l,2
jup
(12)
8
‘here y=ti is the propagation constant in
z the y direction in the ith region. All the boundary condi-
tions for the TE and TM waves are incorporated in the
equivalent circuits. For instance, the ground planes at
u y = O and h are represented by short-circuits at respective
Fig. 2. Coordinate transformation.
places. The electric fields fic and ~U are continuous at
y = d and are related to the currents via
y.h
FIR
#o(a, d) = ~e(cr, d)~o(a,d)
Yl, ‘TM1 )’1 I ‘TEI
&(a,d) = ~fi(a, d)j.(a,d).
~,d 2’ and ~h are the input impedances looking
72’ =TM2
~v(a, d) fu(a,d)
X’ZTE2
equivalent circuits at y = d and are given by
~ ~=fJ ~
Fig. 3. Equivalent transmission lines for the rnicrostrip line.
The closed forms of Green’s impedance functions ~zz, ~k(a, d)= ~h: yh
etc., can be derived by first writing the Fourier transforms 1 2
of field components in each region in terms of superposi- where Y: and Y; are input admittances looking down and
tion of TM-to-y and TE-to-y expressions by way of up at y = d in the TM equivalent circuit and Y; and Y~
Maxwell’s equations.
iy((X,y)=~ ‘coshyl y, O<y<d
= Becoshy2(h ‘y), d<y <h
ii-y(a,y) =A ~sinhyl y, O<y<d
= l?~sinhy2(h –y), d<y <h
-r -Ty,–a +@ –ek2 y2– a +~ –kz
Next, we match tangential (x and z) components
interface and apply appropriate boundary conditions at ante matrix-elements in (3) and (4) are found to be
(13)
(14)
into the
(15)
(16)
are those in the TE circuit:
Y;= Y~~l cothyl(h – d) Y:= Y~M2 cot’ y2d (17)
(6) Y~ = Y~~l cothyl(h – d) Y;= YTH2 coth yzd. (18)
The final step consists of the mapping from the (u, v) to
(x, z) a coordinate system for the spectral wave corre-
(7) spending to each /3 given by a and j3. Because of the
(8)
coordinate transform (10), EX and E= are linear combina-
tions of Eu and En. Similarly, JX and J, are superpositions
at the of JU and J,,. When these relations are used, the imped-
the strip [1], [2]. By eliminating A‘, B‘, A‘, and Bh from
these conditions, we obtain expressions for Green’s im-
~=z(a,d) = N:.&(a,d) + A!@(rx,d) (19)
pedance functions ~zz, etc. ~zX(a,h) = ~X=(a,d) = N#Vz[ – ie(a,d) + ~h(a,d)]
In the new formulation process we will make use of
equivalent transmission lines in the y direction. To this
(20)
end, we recognize that from ~XX(a, d) = N:~e(a,d) + A@h(a,d) (21)
EY(x,y)e ‘~ez = & ~_w EY(a,y)e ‘~(a+flz) da (9)
m
all the field components are a superposition of inhomoge-
neous (in y) waves propagating in the direction of O from
the z axis where d = Cos- 1(@/&), .$= ~a~. Foraeach (9,
waves may be~de$omposed into TM-to-y (~Y, J.?O,HU), and
TE-to-y (fiY, EU, Ho) where the coordinates v and u are as
shown in Fig. 2 and related with (.x, z) via
u=zsinO—xcos O
v=zc0s13+xsin8. (lo)
We recognize that ~. current creates only the TM fields
and ~U the TE fields. Hence, we can draw equivalent
circuits for the TM and TE fields as in Fig. 3. The
characteristic admittances in each region are
where
~.= ~a- =Sine N,= ~a~ =COS6.(22)
Notice that Z“ and ~h are functions of az + ~ 2 and the
ratio of a to ~ enters only through NX and N=.
It is easily shown that ( 19)–(21) are identical to those
previously derived by means of boundary value problems
imposed on the field expression [1], [2].
HI. EXTENSION TO THE MICROSTRIP-SLOT
STRUCTURE
The method presented in the previous section may be
extended to more complicated structures such as the mi-
crostrip-slot line structure in Fig. 4. This structure is
ITOH: DISPERSION CHARACTERISTICS OF TRANSMISSION LINES 735
T 0=
Fig. 4. Cross section of a rnicrostrip-slot line.
“’ZTM’IZ:LP’ZTE’
Fig. 5. Equivalent transmission lines for the rnicrostnp-slot line.
believed to be useful in many microwave integrated circuit
designs because there is an additional degree of freedom
in design due to the existence of the slot [3], [4], [6]. The
characteristic impedance and the propagation constant
may be altered from those in the microstrip line by
changing the slot width in the new structure.
By comparing the new structure with the rnicrostrip line
in Fig. 1, we may draw equivalent circuits in Fig. 5. From
Fig. 5, we get
Eo(cr,d+ t) = Z~l~O(a,d+ t) + i~&(a,d) (23a)
~a(a,d+ t) = ~l,~u(a,d+ t) + ~f2~U(a,d) (23b)
io(a,d) = Zfl.fo(a,d+t) + Z~2Jo(a,d)(24a)
l?U(a,d) = ~;,~U(a,d+ t) + ~22~U(a,d) (24b)
where ~~1 is the driving point input impedance at y = d+ t
and ~12 is the transfer impedance which expresses the
contribution of the source at y = d to the field at y = d+ t.
Other quantities may be similarly defined. Specifically
(25)
Y:= YT~lcothyl(h– d– t) (26)
YTM2 + Y; coth y2t
y;~ = ‘TM2 Y;+ YTM2 coth Y2t (27)
where
Y;= YT~3 cothy~d. (28)
It is readily seen that Y; and Y~~ are input impedances
looking down at y = d and d+ t, respectively, while Y: is
the one looking upward at y = d+ t. On the other hand
YTM2/sinhy2t
(29)
‘;2 = Y;; Y;u Y:+ YT~2 cothyzt “
Here
YTM2 + Y; coth y2t
y;. = ‘TMZ Y;+ YTM2 cothyzt
is the input admittance looking upward at y = d. We
recognize that Zf2 is the transfer impedance from Port 2
to Port 1 in the TM equivalent circuit. All other imped-
ance coefficients in (23) and (24) may be similarly derived.
Impedance-matrix elements may be derived by the co-
ordinate transform identical to the one used in the micro-
strip case. Some of the results are
(31)
The subscripts, say ZX, indicate the direction of the field
(Ez) caused by that of the contributing current (JX). The
superscripts, say 12, signify the relation between the inter-
face where the field is observed (1) and the one where the
current is present (2).
IV. SOME FEATURES OF THE METHOD
The method presented here is useful in solving many
printed line problems. We will summarize the procedure
for the formulation. 1) When the structure is given, we
first draw TM and TE equivalent circuits. Each layer of
dielectric medium is represented by different transmission
lines and whenever conductors are present at particular
interfaces, we place current sources at the junctions be-
tween transmission lines. At the ground planes, these
transmission lines are shorted. 2) We derive driving point
and transfer impedances from the equivalent circuits. 3)
They are subsequently combined according to the sub-
and superscript conventions described in the previous
section, and we obtain the necessary impedance matrix
elements.
The method has certain attractive features:
1) When the structures are modified, such changes are
easily accommodated. For instance, when our structure
has sidewalls, at say x= + L, to completely enclose the
printed lines, all the procedures remain unchanged pro-
vided the discrete Fourier transform is used
~(a)=/L @(x)#xdx, nw
a=%”
(33)
–L
On the other hand, when the top wall is removed, we only
replace the shorted transmission line for the top-most
layer with a semi-infinitely long one extending to y+ + co.
2) The formulation is independent of the number of
strips and their relative location at each interface. Infor-
mation on these parameters is used in the Galerkin’s
procedure to solve equations such as (3) and (4).
3) For some structures such as fin lines [8], it is more
advantageous to use admittance matrix which provides
the current on the fins due to the slot field. The formula-
tion in this case almost parallels the present one. Instead
of the current sources, we need to use voltage sources in
the equivalent circuits.
4) It is easily shown that the method is applicable to
finite structures such as microstrip resonators and anten-
nas. Instead of (5), we need to use double Fourier trans-
forms in x and z directions so that only they dependence
remains to allow the use of equivalent circuit concept.
736 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TSCHNIQUBS, VOL. MIT-28, NO. 7, JOLY 1980
5) Certain physical information is readily extracted.
For instance, it is clear that denominators of typical
impedance matrix elements give the transverse resonance
equation when equated to zero. This implies that for
certain spectral waves determined by a and & surface
wave poles may be encountered. How strongly the surface
wave is excited, or if it is excited at all, is determined by
the structure.
V. NUMERICAL EXAMPLE
Although the intention of this paper is to show the
formulation process, the additional steps required to ob-
tain numerical results are discussed for the sake of com-
pleteness. We computed dispersion characteristics of the
microstrip-slot line with sidewalls at x = * L by the pre-
sent formulation followed by a Galerkin’s procedure re-
peatedly used in the spectral-domain method.
In the previous section, the problem is formulated by
using the impedance matrix with elements Z~~, (i,j = 1,2
and p, q = x, z) and we presumed that the current compo-
nents on the conductors are unknown. It is more advanta-
geous in numerical calculation if we choose the current
components on the strip ~z(cr, d + t) and ~X(a, d + t) and
the aperture fields in the slot ~z(a, d) and ~X(a, d) for
unknowns in the Galerkin’s procedure. This is because the
aperture field in the slot can be more accurately ap-
proximated than the current on the conductor at y = d [4],
[7]. To this end, we rearrange the impedance matrix
equation to the one in which the above four unknown
quantities are on the left-hand side. This modification can
be readily accomplished. In the Galerkin’s method, these
unknowns are expressed in terms of known basis func-
tions. Finally, we obtain homogeneous linear simulta-
neous equations as the right-hand side becomes identi-
cally zero by the inner product process [1], [2]. By equat-
ing the determinant to zero, we find the eigenvalue ~.
There are two types of modes in the structure. One of
them is a perturbed microstrip mode and another is a
perturbed slot mode. For the perturbed microstrip quasi-
TEM mode, we have computed dispersion relations by
choosing only one basis function each for four unknowns.
They are chosen such that appropriate edge conditions are
satisfied at the edges of strip and slot. For instance, we
can choose as the basis functions the Fourier transforms
of
Jz(x, d+t)=
A
JX(x, d+t)=x_
Ez(x, d) = V= EX(x, d) =
&“
It is readily seen that Fourier transforms of these func-
tions are analytically given in terms of Bessel functions.
Fig. 6 shows some numerical examples of dispersion char-
acteristics. The present results for a small slot width are
compared with those of a shielded microstrip line [1]. It is
clear that as the frequency increases, the presence of
nonzero slot width becomes more significant. It is also
050r
Present ReSUII~
---- Itoh and M$tfro
( No slot]
—. I
0351 I I 1
0 5 10 15
FREQUENCY (G Hz)
Fig. 6. Dispersion characteristics of rnicrostnp-slot lines L = 6.35 mm,
d=ll.43 mm, t=l.27 mm, h=24.13 mm, w= O.635 mm, <,=8.875.
seen that, as the slot width increases, the guide wavelength
becomes larger because the effect for the free space below
the slot is more pronounced. This suggests that the guide
wavelength is adjustable by two means, one by changing
the strip width and another by varying the slot width.
VI. CONCLUSIONS
We presented a simple method for formulating the
eigenvalue problems for dispersion characteristics of gen-
eral printed transmission lines. The method is intended to
save considerable analytical labor for these types of prob-
lems. In addition, the method provides certain unique
features. The method is applied to the problem of micro-
strip-slot line believed useful in microwave- and millimet-
er-wave integrated circuits. Numerical results are also
presented.
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
REFERENCES
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T. Itoh, “Analysis of microstrip resonators; IEEE Trans. Micra-
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M. Aikawa, “Microstrip line directional coupler with tight coupling
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Spectral domain immitance approach for dispersion characteristics of generalized printed transmission lines

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Spectral domain immitance approach for dispersion characteristics of generalized printed transmission lines

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