SSN 208, T. Itoh and R. Mittra, Analysis of Modes in a Finite-Width Parallel

SSN 208, T. Itoh and R. Mittra, Analysis of Modes in a Finite-Width Parallel
Sensor and Simulation Noted
Note 208
January 1975
Analysis of Modes in a Finite-Width
Parallel-Plate Waveguide
T. Itoh and R. Mittra
Electromagnetic Laboratory
Department of Electrical Engineering
University of Illinois at Urbana-Champaign
Urbana, Illinois 61801
!3.1::-.::::
~~fl
~{j~;~~
[;~:~:.::
pL/p’
5A”/7;
Abstract
An efficient method has been developed for analyzing modal characteristics of a finite–width parallel-plate waveguide. The method is based on
an extension of Galerlcinlsprocedure applied in the Fourier transform
domain. Numerical values of propagation constants and field distributions
have been obtained for various structural and modal parameters.
-. --
!
‘rhisstudy was performed under subcontract to
The Dikewood Corporation
1009 Bradbury Drive, S.E.
University Research Park
Albuquerque, New Mexico 87106
..——.
-.
,- —.. . ..—
d.~-n
.—
-——-—.
. .
.-
---
Section
1
11
J2u!2
INTRODUCTION
5
. FORMULATION OF THE PROBLEM
7
111
METHOD OF SOLUTION
13
IV
NUMERICAL PROCEDURE
16
NUMERICAL RESULTS
19
CONCLUSIONS
22
REFERENCES
23
v
VI
2
.—
—r
.—
———
.
.
.
. .
——
—--
----
.
,.
.
.
.
—___
-,-————
.——
—
. .
.
.
.
.-
ILLUSTRATIONS
22.!ss
1
(a) Cross Section of a Finite-Width Parallel-Plate Waveguide
(b),.(c) Equivalent Structure
6
2
Classification by Symmetry
18
3
Field Distribution in the Cross Section of the Waveguide
21
,
3
.—
.
—.-..
..___
- -——..— —~
,.. ,
—.
.._
.-: —-:
.—
—--J
-4A------
. ..-.
—..
—.
. .. —-. —
- .
Table
1
Classification by Symmetricity
Computer Solutions of Propagation Characteristics
2
4
——.
-
. . ..
.
-,...-.—.——-
—.
—.--.—
.-—.
—____
SECTION I
INTRODUCTION
Parallel-plate transmission lines are often employed as a guiding
structure for electromagnetic pulse (EMP) simulators. This type of transmission
line, as shown in figure 1, has in addition to the dominant TEMmode an infinite
number of higher-order modes.
Because of the open nature of the structure, the
propagation constants of these higher-order modes are usually complex,
representing the propagation as well as radiation loss of the modes.
When
an object to be tested is placed in the present structure, it is illuminated
by electromagneticwaves consisting of a combination of TEM and a number of
higher-order modes. The scattered field is also a superposition of TEM and
higher-order modes.
Hence, the nature of the higher-order modes in the
parallel-plate transmission line is worth investigating and the development of a computer program to obtain the propagation constant and the modal
field distribution is important for EMP studies.
The problem of parallel-plate transmission lines has been studied by
several worlcers(refs. 1 and 2)
using various approaches. In this paper,
a new method is presented for attaclcingthis problem. The method is an
extension of Galerkinls procedure in the spectral domain. The original
version of this method, which has been applied to many microstrip-type
transmission line structures (refs. 3 and 4), is extended here to apply to
the structure with complex transverse propagation constants. In the
following sections, the formulation of the problem, numerical procedures,
and some results are presented;
,
5
—
-.. .—.<——..-— . . . ..~:.
. .=
‘
‘“
‘A---
.=
A-
. . . . . _——
-..
-. .—.
.—-.
. . .. . . . . . . .
. ..-—
. ..
.-
-
.
t
2d
+
(a)
@
t
I
Y’
(D
ELECTRIC
~CONDUCTOF?7
x
(b)
@
t
[email protected]
———
/—.—
.—
‘1-.
x
MAGNETIC
CONDUCTOR
——— — T–
(d
Figure 1.
(a) Cross Section of a Finite-Width Parallel-Plate Waveguide
(b), (c) Equivalent Structure.
6
. —
..———.——
-—
-.’,:
--
-..
.
..-,
——
-x——
-—
———.—
--
—--
---—
SECTION II
FORMUiLA1’ION
OF THE PROBLEM
The cross section of the parallel-plate transmission line is shotm in
figure 1. Assume that the plates are infinitely thin and that both the plates
and the medium are lossless. It is well knoom that the dominant mode in this
structure is TEM with the propagation constant identical to the free-space
wave number. All of the higher-order modes may be classified into the two
sets, TM and TE, with respect to the z-direction. Assuming exp(jwt - j~z)
variation, where the propagation constant 6 is complex, in general, all of
the field components of the TM and TE modes may generally be expressed in
terms of scalar potentials as
~2
Ez=j’-6
2
‘+(:%Y)
B
(1)
TE modes
Hz=j
k2 -62
f3”
$(x,,,)
‘PO w
E .
‘Tax
Y
Ex=— ‘po~
B ay
Hx=~
.
(2)
H .$
.1
Y
where k = 2n1A is the free-space wave number, and co and P. are the permittivity
and permeability of free space, respectively. The common factor exp(jut - j~z)
has been and will be omitted throughout this paper.
7
..
—.—...
.. .... .-.. .
.. .. .._--:..
-4—
–———-
---
-.
——.–
—-
—.,————
.—
--—-.
—..
—
-----
—.
.
In addition to the TE and TM classification, the symmetry of the structure
allows us to further subclassify the modal spectrum. For instance, the
symmetricity of the direction of the z-component of induced current on the
plates allows us to subclassify the spectrum into the four cases listed in
table 1.
The detailed formulation will now be given for Cases 1 and 3 of
TM modes only (odd TM modes), and only the resultant equations will be
summarized for the rest of the cases. Because of the nature of the present
method of analysis, the distinction between Cases 1 and 3 (and between
Cases 2 and 4) is not necessary in the formulation process.
Table 1
CLASSIFICATION BY SYMMETRICITY
x>
Case
-1-
0, y > ()
x<
(),
y >
1
+
-1-
2
-1-
+
3
i-
4
+
0
X<o,y<o
X>o,yco
+.
-+
+-
+
Jz flows in the positive z direction
Jz flows in the negative z direction
Such distinction is undertaken only at the stage of preparation for numerical
computation.
In the odd TM mode cases, it is only necessary to consider the equivalent
structure shown in figure l(b) where the y = O plane is an electric conductor.
Since the structure is infinite in the x-direction, the electromagnetic
boundary value problem is formulated in the spectral or I?ouriertransform
domain as opposed to the conventional space domain formulation (refs. 3 and 4).
8
——
-—
.
,-.-7--
—r
,.
—
-.
.—..
—
,.:.~.
—~.
---.,.
.
—-
.—.
——
-
.
.
.
.
——
.
To this end, let us define the Fourier transform ~(a,y) of the scalar potential
f$(x,y)
via
ii(%Y) = l’ @i(x,Y)
-w
ejctx
~x
(3)
where i = 1 and 2 designates the regions O < y < d and y > d, respectively.
The transforms of field components may be defined from equations (1) and (3)
as
—
(4)
Since @i satisfies the wave equation, @i is a solution of
u
d2
—_
dy2
where
72
ii(a,y) = o
(5)
(6)
y2=a2+62-k2
Because of the boundary conditions E= = Ex ==O at y = O and the radiation
condition at y + +CO , the solution of equation (5) is
~l(a,Y) ‘A(a)
i2(%Y)
= B(a)
O<y<d
sinhyy
(7a)
y>d
exp[-y(y - d)]
(7b)
where A and B are unlcnoorns.Note that Re y > 0 and Im y > 0 are to be
satisfied so that equation (7b) represents a valid form for y + +~
9
-...
.—
.
..—
-.. ,
..r
_ ..-
‘. —’. —---
:...:!-.--—— —
—.. ~...
:.. .”-.
.”...—
...-
The next step is to apply the interface conditions at y = d in the
transform domain.
Since
Ezl(x,d-) = Ez2(x,d+)
all x
Efi(x,d-) = Ez2(x,d+)
all x
Hxl(X,d-)
Hx2(x,d-)
=
1X1 >W
the interface conditions expressed in the transform domain are
(8)
~a(a,d-)
= fix2(a,d+-)
(9)
fixl(~,d-)
- fix2(a,d+-)
= ~z(a)
where jz is the transform of the z-directed, unknovm,
plate at y = d.
induced current on the
Substitution of equatioils(4) and (7) into equation (8) gives
the relation between A and B.
A and B.
(lo)
Equation (9) is automatically satisfied for
If these quantities are substituted in equation (10), A or B is
expressed in terms of another unknown jz(a).
.
B(a) =
where
Go(a,13)=
i++a) 3Z(U)
-f3
(IICoy[l
+ coth
(11)
yd]
(12)
Now, the final boundary condition Ez(x;d) = O for lx] < w is imposed in
the transform domain.
Since Ez(x,d) is unknown but nonzero for lx] > w, it
can be written as
1.0
.——
.. ..-
——
—---
(13)
Hence, the transform is
~z(a,d) = j
1C2- $2 “
~
u (a)
(14)
where
co
-v
C(a) =/
u(x) ejaxdx + ~ u(x) eJax dx
-OJ
V7
Eliminating B(a) from equations (8), (11) and (14), obtain
Go(a,f3)3Z(U) = ti(ci)
(15)
It should be mentioned that equation (15) is the transform of the integral
equation of the convolution form encountered in many conventional space domain
analyses. It may also be worthwhile to mention that equation (15) contains
two unkno~,msj~z and fi;however, it is possible that in the solution one of
the unknowns, U, may be eliminated and that equation (15) is solved for
~z only.
Before concluding this section, let us summarize the resultant equations
for other symmetries and polarizations.
TM, even in y (Cases 2 and 4 in table 1)
(16a)
Ge = ticOy[l+-6 tanh yd]
(16bj
11
.
—
io(a,
io(a)
f3) ~z(a) = V(a)
YB
=
(17a)
(17b)
j(k2 - f32)[l+ cothyd]
TE,
even in y
[email protected],B)
Ze(a)
3=(CX) = i(a)
y$
=
(18a)
(18b)
j(k2 - 62)[1+
tanhyd]
For TE cases,
-t7
~(a)
=/
v(x)
ejaxdx+~m
-co
v(y)
. ejaxdx
(19a)
w
(19b)
12
—.— _________
.—..
—- —..
---.
-—— ..-.—--—
.. --—-——. .— .-
-
SECTION 111
METHOD OF SOLUTION
In this section, a method of solving algebraic equation (15) is
discussed. The method, which is applicable to solving equations (16a), (17a),
and (18a) as well, is based on Galerkints procedure applied in the Fourier
transform domain.
The first step expands the unknown ~z(u) in terms of known basis
functions ~n(u), n = 1, . . .,N.
(20)
where c 1s are unknown coefficients to be determined. The choice of ~n(a)is
n
is such that they are the Fourier transforms of appropriate functions with
finite support, viz., Jn(x)ts, the inverse transforms of ~n(a)ls, are zero
for lx! > v7.
Substituting equation (20) into equation (15) and taking an inner
product of the resultant equation with one of ~m(a)’s, m = 1,2, . . .,N,
one obtains
~ K (6)cn =0,
n=l m
m=l,2,
(21)
. . ..N
where
K
(6) = ~w ~m(a) ~o([email protected])
-w
(22)
~n(a) da
The right-hand side of equation (21) is zero using Parsevalls relation
6
~= J (-x)“E (x,d) dx >0
$= Sin(a)ti(a)da =
z
-m
j (lcz- f32)‘-m m
because Jm and Ez are nonzero only over complementary regions of x.
13
.—
.-
..
—
..
.—.-
-—
When Jn(x)’s, whose transforms are to be used as basis functions ~n(a)fs,
are chosen, they must satisfy certain symmetry requirements in addition to
being zero for lx! > v7. For instance, in Case 1 in table 1, Jn(x)ls must be
symmetric with respect to the y axis, while in Case 3 they are required to be
antisymmetric. Furthermore, it is desirable to use Jn(x)ls which well
represent the edge condition at x = tw where the actual z-directed current
shows square integrable singularity. Using these basis functions, equation (21)
is solved for unknown propagation constant 6 which is usually a complex number.
After 6 is obtained, the field distribution may be calculated as follows.
Except for the normalization factor, the ratio of all Cn’s is determined, which
gives the current distribution
J(x)
z
The field distribution of Ez maybe
=~c
n=l
(23)
J(x)
nn
obtained from equations (7), (11), (15)
and (20)-
Ez(x,y) =
j
k’-~’
~
N
~
n=l
c ~rnsinhyy
n _m sinh yd
.
Go(a) ~n(a) e-Jax da
(24a)
rj<y<d
,
~2 _B2N
j’fi
[
~ cnJmexp[-Y(Y
-m
n=l
-jcixda
- d)] ~O(a) in(a) e
(24b)
y>d
All the higher-order TE and TM modes in the present structure may be
designated by a set of numbers (p,q) for each symmetric subgroup. The number
p is associated with the field variation in the x direction and number q for
the variation in the y direction. In the present method, p can be predetermined
by the appropriate choice of basis functions ~n for the current distribution
14
——
——
-
.
.
.._-
.=
.—z.
,..
,-
.
on the plate. However, there is no built-in process to choose q in the
analysis procedure. Ratherjq is controlled in the numerical process of
finding 13by the judicious choice of a starting point in the root-seeking
algorithm.
15
.. —.
.—
. ...:
‘.-
”---
.=-’’.
.....
.”_
. ..
. ..--=’
.’. ___
-----
.
—..—.
—.
.
_-.-.,_,’._
----
____
.. -,._
.
SECTION IV
NUMERICAL PROCEDURE
A numerical algorithm has been developed for TM modes of Case 1 with
the mode index p = O.
This choice of p corresponds to the higher-order
modes with the least field variation in the x-direction. The basis functions
have been selected so that the qualitative nature of the actual current is
well represented. Specifically the following functions have been employed:
Jl(x) =
{
1
1X1 <w
o
otherwise
(25)
(26)
(.0
otherwise
The basis functions for equation (21) are the Fourier transforms of
equations (25) arid(26)swhich are
ii(a) =
~2(a) =
2 sin aw
(27)
a
4 sin aw _— 2
2
a
(1 -
(28)
Cos (w)
Although any number of functions similar to these given by the above equations
could be used as a set of basis functions, N in “equation(21) was set to two.
Only two basis functions (27) and (28) were employed for economic reasons,
that is, for minimizing the computer time.
It was found, however, that quite
satisfactory answers were obtained for many microstrip problems by using only
on~.or two basis functions of the type similar to equations (27) and (28)
(ref. 4).
16
.—.—
,.—..- ,,
—.——,
-—--—
—- .—-—— -.. .—
—..—-
Equations (27) and (28) are substituted into equation (22) to numerically
compute K~ls,
and complex roots 6 of
det Km(Ol
(29)
= O
are found by a complex root-seeking algorithm. This algorithm finds the
closest zero from a given starting point. At this stag~only the mode
number p is given and another mode number q is left undecided. It is possible
to correlate the value of q and the appropriately chosen starting paint of the
algorithm.
The starting values of the route-seeking routine have been chosen in the
following way.
The present structure can be viewed as a fictitious closed
vaveguide with sidewalls with complex surface impedance. For well-guided
modes, these sidewalls may be very close to the magnetic walls since,for
such modes,the radiation loss may be quite small. Hence, the propagation
constant 6 of waveguides with magnetic sidewalls may be chosen as a starting
s
point of the algorithm. (See figure 2.)For the present mode spectrum of
Case 1 and TM, 6s may be given from figure 2a
,=4C2
s
s
(30)
‘f=-[:12-1%2~
p=0,1,2~o**
q =1,
●
2, . . .
Note that we are concerned with p = O modes in the present numerical computations using equations (27) and (28) for basis functions.
It is hoped in the numerical algorithm that the zero of equation (29)
closest to the value of $s be obtained for a given q; such a zero is called
of the TM
mode of Case 1.
the propagation constant ~
Pq
Pq
17
—-
..
—-...
.-.
.. ...’
.,
.-,
.– -——
—-
1
-, :.. ‘- .’-. :
. . . ..
..
J----
..-
I
I
I
-’i-i
.--.-—
—
I
1
1
I
1
I
I
I
x
.. .
-.-—
.-
- .
x
(b)
(cd
rl
I
I
I
t
----
-i
(d
Figure 2.
Classification by Symmetry for the Determination
of the Starting Value of B.
18
..
——
-—-y
-—
.---— —.-.———...–—
———
-.
SECTION V
NUMERICAL RESULTS,
Some typical computer solutions are summarized in table 2 with d, w and
the wavelength A being input parameters. Also,the p = O and q values are
specified.
In many FJ4Pproblems, the so-called transverse propagation constant a
Pq
is more preferable than the propagation constant 6Pq (ref. 1).
The
definition of u
is
Pq
,~=fi
Pq
(31)
Pq
here corresponds to pn given in
The transverse propagation constant a
Pq
ref. 1.
as 6 carries tT70
Notice, however, that unlike.in ref; 1,-cias T17ell
indices p and q because the structure in the present case has non-negligible
plate width.
The magnitudes of the Ez field in the waveguide cross section, which
are computed using equation (24), are plotted in figure 3.
It is clear that
the field decays away from the waveguide in both the x- and y-directions.
Although IEZI must be zero on the plate at y = d, the numerical results did
not predict that it would be zero, but would approach zero.
It is hoped that
these values approach zero as the number of basis functions are increased.
It is also seen that the number of peaks in the y direction for O < y < d
coincides with the given value of q.
19
.-
.
.
. . . ...
--2- A---- . ..- . .
<,.
. . . -.-.
.
..—
.
...-..’
‘-
$’.....
. . .. —”-.
-, +.-.
.
.
Table 2
COMPUTER SOLUTIVLvSOF PROPAGATION CHARACTERISTICS
(0,1) MODE
d
2$~d
T
2apqd
No.
Iter.
2
24.32 - j2.432
21.76 - jO.114
-.1972 -ij12.58
10
1
10.88 - jl.088
10.82 - jO.0652
-.1104 +j6.382
5
2
24.32 - j2.432
21.72 - jO.0448
-.0728 -1j12.644
12
3.33
3
37.17 - j3.717
20.82 - jO.0335
-.0222 +j31.428
45
2.5
4
49.84- j4.984
55.92 - j6.68
-27.28 + j13.696
15
2
5
62.5 - j6.25
54.1 - jO.163
-.276 +j31.96
24
5
10
125.5 - j12.55
107.2 -.j14.72
-22.28 +j70.80
7
5
5
62.5 - j6.25
59.6 - j1415
-28.46 l-j30.36
25
45.8 - jll.78
-25.52 -1-j21.14
29
2.5
5
3.33
41.4 - j4.14
5
2.5
30.8 - j3.08
5
2
24.4 - j2.44
.34.4 - j3.94
16
5
8
-15.92 +j8.52
21.8 - jO.045
-.078 +j12.50
12
5
1.67
19.98 - jl.998
16.74 - jO.060
-.080 +j12.58
12
5
1.11 .12.46 - jl.246
12.36 - jO.021
-.040 +j6.50
5
5
1
10.88 - jl.088
10.78 - jO.044
-.074 -1j6.46
5
2
1
10.88 - jl.088
10.29 - jO.044
-.063 + j7.22
8
1
2
24.32 - j2.432
19.28 - jO.068
-.081 +j16.12
15
0.67
3
37.2 - j3.72
50.88 - j12.66
-36.3 + j17.70
27
-7.6 X 10-7
12
(0,2) MODE
10
1
0.078 - jO.8~,,
0.2 - jO.02*
x 10--’
*The
~~tu~l
z~~d
~,7~~
0.0 _
+ j9.56
However, 2B~d was shifted to 0.2 - jO.02
joooa
for numerical convenience.
20
-——---
.
-–,——~.,—--—--—---
----- .—.
AT X=o
—
---x=o
‘i
().2
(M
0.6
I
I
I
0.2
0.4
0.6
0.8
)fId
10
.
(0,1)MODE
(0,2)MODE
[.2
5~
=4
—
ld
—3 “
2–
I–
‘0
Figure 3.
1.4
I
0.8
X/b’i!
1.0
1.2
1.4
[.6
i
1.6
Field distribution in the Cross Section of the Waveguide
21
—
..—..... -.—.
-;-.
—.———-
.-.’..-’-:..
...,
-.
-.-.
.
—.--—
,,
— <—.
—-.
. .. —. —-—
_._—.
_.
—.:
SECTION VI
CONCLUSIONS
A simple and efficient numerical method has been developed for analyzing
a finite-width parallel-plate waveguide.
Sample computations based on this
method predicted the propagation constants of the modes in such a structure
and the field plots so obtained have shown the expected physical nature of
these modes.
22
——-——
—-
--— ..--=
------ —
—-— ...-.
.=-— .,,
.. .:-=-.
——-——--———-—
—_
—__
pREFERENCES
1.
Marin, L., “Modes on a Finite-Width, Parallel-Plate Simulator. I. Narrow
Plates,” Sensor and Simulation Notes No. 201, Dikewood Corporation,
September 1974.
2.
Liu, T. K., “Impedance and Field Distributions of Curved Parallel-Plate
Transmission-Line Simulators,” Sensor and Simulation Notes No. 170,
Dikewood Corporation, February 1973.
3.
Itoh, T. and Mittra, R., “Dispersion Characteristics of the Slot Lines,”
Electronics Letters, ~, 13, pp. 364-365, July 1971.
4.
Itoh, T. and Mittra, R., “Spectral Domain Approach for Calculating the
Dispersion Characteristics of Microstrip Lines,” IEEE Trans. Microwave
Theory Tech., MTT-21, 7, pp. 496-499, July 1973.
23
—-
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