IEEE MTT-V034-I08 (1986-08)

72 Pages·1986·7.646 MB·English

by  FANTOMASPING

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-34,NO.8,AUGUST 1986 845
Field Solution, Polarization, and Eigenmodes
of Shielded Microstrip Transmission Line
ESSAM E. HASSAN
Abstract —Application of the reciprocity theorem leads to a variational radiation from an axially slotted waveguide. Numerical
expression for the propagation constant of the fields inside shielded results using this method are presented for several cases,
microstrip-like transmission lines. The resntting eqnation involves both the
which compare well with the published data [9], [10], [17].
propagation constant and the tangential fields at the air-dielectric inter-
face. Using the Rayleigh-Ritz optimization technique, both the propa- The method, in addition, points out the close relationship
gation constant and the fields are completely determined. between the higher order modes and their counterparts of
De calculated results of the propagation constant compare well with the LSE and LSM modes of the dielectric-loaded wave-
other available data. Moreover, the field solution obtained is presented in guide. Finally, the technique can easily be extended to
the form of a polarization ratio relating the axiaf to the transverse electric
handle the coplanar stripline [14], [15] and the various
field. Resultscover both low and high frequencies, and the technique
proves vahd at both frequency ranges. The method may be extended to forms of microstrip transmission lines.
other configurations of planar striplines by proper adjustment of the
integration limits. II. FORMULATION OFTHE PROBLEM
I. INTRODUCTION A. Field Components
T HE problem of solving for’ the dispersion characteris- Fig. 1 shows a microstrip-like transmission line with two
tics of shielded rnicrostrip line has been tackled by lossless dielectric layers. Region I has a relative dielectric
many authors, either using the quasi-TEM theory [1]–[4] constant c, and Region II is free space. The metal strip is
or through more rigorous analytical techniques [5]–[8]. of width 2t and assumed to be perfectly conducting with
Results for higher order modes have been reported by negligible thickness. The metallic shield enclosure is as-
several authors [8]–[11]. Most of the techniques used in- sumed of infinite conductivity and of width 2L and height
volve some large set of equations to be numerically solved. h, and the dielectric layer of Region I is of height d.
In all data so far published in the literature, nothing or Although the technique presented here is quite general and
very little has been said about the form of the field inside could be easily applied to any number of inner conductors
the structure. This field is neither apurely TE nor apurely once the proper integral limits are considered, the analysis
TM mode, rather it is a hybrid mode and cannot be will be restricted to the case of a single conductor symmet-
directly obtained from a single scalar potential. A thor- rically placed inside the enclosure as show-n in Fig. 1. In
ough understanding of this field configuration is an im- this inhomogeneous structure, the fields cannot be as-
portant factor in both the efficiency and the proper oper- sumed to be purely TE or TM modes, rather a superposi-
ation of many devices employing the microstrip line. This tion of both TE and TM is unavoidable. Let the TE modes
paper presents a powerful method to obtain both the field be derived from a scalar potential ~: and the TM modes
distribution and the propagation modes supported by this be derived from a scalar potential T? (i= 1 or 2 dependi-
structure. The method is based on the application of the ng on the region). It is necessary to point out here that
reciprocity theorem in the two domains comprising the because of the symmetry of the structure, two orthogonal
shielded line. This leads to a variational expression for the sets of modes exist. The first has a symmetric E= and an
propagation constant in the form of an integral equation antisymmetric H, (this mode is known as E, even– H,
linking the fields at the air-dielectric interface: With a odd), and the other has an antisymmetric E, and a sym-
proper field expansion and the Rayleigh-Ritz optimization metric HZ (known as E, odd– Hz even). The TEM-like
technique, the relative amplitudes of the field components mode is the lowest order E, even– Hz odd mode [17]1.The
are determined, and the integral equation lends itself to a analysis will be carried out for the E, even– H, odd modes,
simple transcendental equation similar in nature to those and the necessary modification for the solution of the
of the LSE and LSM modes in the dielectric-loaded wave- other mode will be given whenever appropriate. Now let
guide. A similar technique has been employed earlier by the scalar potentials for the E= even– Hz odd modes be
Rumsey [12] and by Barrington [13] in their treatment of given by
yf = ~ A: sirtha~’jxsiny~yZX
Man’uscnpt receivedMay 30,1985;revisedFebruary 24, 1986. This ~=1
work was supported by the University of Petroleum and Minerals,
for Region I
Dhahran, Saudi Arabia.
The author is with the Department of Electrical Engineering, Univer-
sity of Petroleum and Minerals, Dhahran 31261, Saudi Arabia. ~: = ~ A:cosha:l)x COS yny~x
IEEE Log Number 8608832. ~=1
0018 -9480/86/0800-0845 $01.00 01986 IEEE
846 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-34, NO. 8, AUGUST 1986
Y
I
I
I
/
II I
i h
A
‘r
Y *..--––. m
Fig. 2. Cross section of slotted waveguide.
q
Fig. 1. Microstrip configuration.
II
and 7
h-d
‘2 = so
—M 2 ~
1
I d
c1 = ‘o ‘r
-?-
for Region II ‘
+----2’---4
*J= ~ B;cosha:2)(h – X) COSYnyiix (1) Fig. 3. Microstrip transmission line with M metal strips,
~=1
B. Variational Expression
where the factor expj( at – #z ) is understood and
Rumsey, in his treatment of the leaky wave slotted
yn= (2n –l)rI/2L, n=l,2,3, ... waveguide [12], has shown that a stationary expression for
the propagation constant ~ of the cross section shown in
Fig. 2 is
I#I=j_w;2[Ey(rHz -’H,)+ E,(’HY -’w,)] dy= O (3)
A!, A;, B;, and B; are unknown coefficients
where EY and EZ are the components of the assumed
electric field in the slot and ‘H and ‘H are the internal and
c~ and p. are the permittivity and the magnetic permeabil- the external magnetic fields which fit the assumed EY and
ityy of free space, and ~, is the relative permittivity of E=. LJsing the same approach, a similar expression could be
medium I. With u = 21T~ the operating frequency (rad/s), applied to the configuration of Fig. 1. Further, the integra-
/3 is the desired propagation constant. For the EZ odd-HZ tion could be taken along as many slots as there are in the
even modes the term sin y.y replaces cosy.y, and vice interface between the two regions. Thus, for the configura-
versa. Also, y~= nn/L, and the summation should start tion of Fig. 3, (3) becomes
from Oto co.
The hybrid field components are derived from the above .:lmd’=eH’JE+4’%%-’)1
‘Y=o
scalar potential in a way similar to that presented in [8].
Applying the appropriate boundary conditions of the
(4)
tangential electric and magnetic fields along the plane
x = d and using the orthogonality of sinusoidal functions, where the subscript w refers to the rnth slot of a total of
the coefficients -4~, A;, B:, and B: may all be obtained in M slots. For the configuration of Fig. 1, using the field
terms of the integrals I,(n) and ~z(n) defined as [8]
components as obtained from the scalar potentials (1)
where the set of constants A!, A~, B$, and B; are all in
lY(n)=J~EY sin y.ydy terms of lY(n) and 1=(n) and substituting in (3), one gets,
t d=~
after some manipulation, an expression of the form
Iz(n)=/~Ez COSYnydy. (2)
t ~ad
By substituting this set of coefficients in (l), all field
components in either region are readily obtained in terms
of the values of EY and E, along the plane x = d.
HASSANS:HIELDEDMICROSTIUTPRANSMISSIOLNINE
847
where Substituting back in (6) and using the result in (5), the
latter will take the form
sn=– —&[(kw)GAn)+(l P-K;)G2(n)]
,l~l[(~;- ~2)G~(n)+(~~-P2 )GZ(n)]l~(n)
vn=– &[ G,(n) +G,(n)] - B’
1
$ly~(G~(n)+,G’( n))lY(n)lZ(n) 2
un=— 2J [( Kf-y:)G,(n)+(Ki-Y~ )G2(~)] {[
upOL
LcNo (
~/–i?)Gl(n)
with
/ ?7=1
Gl(n ) = cothc#Jd/a;l) + (K&yj’)G2(n)] l;(n)} =0. (9)
G2(n) = coth(x~’~(h – d)/a\2).
If a further simplification is made and we consider only
Equation (5) has three unknowns, the propagation con-
the n = 1 term of (9), the resulting expression will satisfy
stant ~ and the slot electric fields EY and E=. As men-
the characteristic equations for the LSE and the LSM of
tioned previously, (5) is a stationary expression for ~, and
the corresponding partially dielectric-filled waveguide,
reasonable assumed values for EY and E, when substituted
namely [16],
in it will yield a very close approximation to the true value
of ~. The electric field components at x = d are to be
c#) cot afl)d = – af2Jcot af2J(h – d) for LSE
expanded in a set of orthogonal functions with unknown
coefficients (aP’s and b~’s) in the form and -
(2p-l) m(y_t) cot af)d ._ cotaf)(h–d)
EYIX=~= ~ aPsin (, @ 42) for LSM. (10)
~=~ 2(L–t)
(2q-l)7r This establishes analytically the closeness of the solution of
EzIX=~= E b~cos 2(L_t) (Y–-t) (6)
the propagation mode for the LSE and LSM of the clielec-
~=1
tric loaded waveguide and the corresponding shielded mi-
with al normalized to unity. crostrip line, a fact that has been observed numerically [9].
Substituting (6) in (5), one gets the stationary expression The analysis for the E odd-H even modes follows the
of ~ in terms of the coefficients up and b~.Now, since the same steps and will yield an expression similar to (9).
assumed field expansion is a complete set, the Raylei@– However, the sine and cosine “terms of the assumed slot
Ritz optimization procedure could be applied to the result- field, (6), should be interchanged. Further, in (1) the
ing equation by successively differentiating +( n) with re- summation over n starts at n = O, and the n = O term
spect to aP’s and b~’s and setting the result to zero, i.e., should be halved.
d+(n) III. SOLUTION OF THE PROBLEM
p=2,3,4, ... (7a)
daP ‘0’ As explained earlier, the solution is based on combining
(5) along with (7). This results in the following set of
d+(n) =0
equations:
q=l,2,3, .... (7b)
db~ ‘
~ 2SJi(n) ~ a,f,(n)+f~fi(n) ~ b,g,(n)
This optimization technique must, in principle, obtain
!1=1 ~=’ n ~=1
the true values of the field as the number of terms in the
N
expansion (6) approaches infinity. The successive differ-
=- ~ 2S~fi(n)f1(n), i=2,3,..., P
entiations as given by (7a) and (7b) when truncated at P
~=1
and Q, respectively, will result in a set of (P+ Q – 1)
equations in the unknown coefficients a‘, a~,. .”, ap and and
bl, bz,”.“,b~ along with ~. This seti along with (5), maybe
solved as shown later to determine both /3 and the un-
known coefficients. Once these coefficients are known, all
field components are readily obtained. Detailed discussion N
of the solution is presented in Section III of this paper. =- ~ ~gj(n)fl(n), j=l,2,3,0.., Q (11)
It is interesting at this stage to examine the very simple n=l
case of P = Q =1. In such case, with al = 1 one gets where
]-sinYnY}
=~L{sin[(2p-l)~J(Lj’~)
fp(~) I@
848 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-34, NO. 8, AUGUST 1986
and w?f.18 _____
T Wf. 9 _
1.5
ref. 8 ___
+1*1
&J~) = JL(COS [(2@ )fi;:--_’j ]“cosYnY) ~1. mI-=5 0.5 llus Wrk _
@/K. &3 +7
These are (~ – 1+ Q) simultaneous linear equations in 3-0
the unknowns az, aq,. . ..aP, bl, bz,. . ..bQ. Notice that the 2.9
propagation constant ~ is still unknown and is part of the / /f”
2.8
coefficients of the set (11). To complete the solution, (5) is
(,’
added to the set (11). The solution for ~ is sought first, 2“7
/
and once it is known the field coefficients are readily z.b
obtained. The technique adopted is straightforward. We
assume a value for ~ which is chosen close to the true 2“5 ~ 2“’”
value. (This is chosen in the vicinity of the value of the *A
propagation constant for the corresponding mode of a
similar dielectric loaded waveguide, for the non-TEM 1 2 34510 20 50 100 f(m)
mode, and is chosen to be ~-KO for the TEM Fig,,4. variation of ~/K. with frequency compared with other meth-
mode.) The coefficients up and b~ are then obtained from ods. Units are in millimeters.
(11) for the assumed value of ~, and the result is sub-
stituted in (5), which must be zero for the true value of /?. This wor~
The procedure is iterated until the zero is captured. The B/K. 1 Ref. 9 _-
value of/3 is then obtained, and the coefficients a~ and b~,
I 2t=2 .54
hence the field solutions, are readily available. In the “K
following ,section, some results are presented and dis-
cussed.
m
IV. RESULTSAND DISCUSSION
A. TEikf Mode
Figs. 4 and 5 represent the normalized propagation
constant ~/KO as a function of frequency, compared with
results published in [8], [9], and [18]. The results of this
paper are presented for frequencies up to 50 MHz and, if
desired, could be easily extended to higher frequencies.
—
The results are shown to be almost ident~cal with ~hose of 10 20 30 40 50 f(GHz)
the nonuniform discretization of integral equation [9] and Fig. 5. Variation of /3/K0 with frequency. (r= 8.875, 2L = 12.7 mm,
very close to the results of [10] in all frequency spectrums h=12.7 mm, d = 1.27 mm.
tested up to 50 MHz. This verifies the accuracy of the
technique. Such good agreement with [9], [10] seems to TABLE I
CONVERGENCE OF THE ROOT /l/KO AGAINST THE NUMBER OF
have its importance at the high frequency region where
EXPANSION TERMS
discrepancy is noticed between our work and other meth-
ods presented here [8], [18] or presented in [9]. The accu- P Q B/K.
racy of the empirical formulas [19], [20] seems to be
inaccurate by a considerable margin at high frequencies 3 3 3.1535
[9]. 6 6 2.9405
Convergence towards the true value of ~ could be 10 10 2.8593
obtained in a relatively small number of expansion terms.
20 20 2.7968
When the expansion of (6) is truncated at p = q = 10, a
3-4 percent discrepancy of the true value of ~ is noticed. 22 22 2.7906
Such convergence with a low number of terms and subse- 23 23 2.7885
quently low computational time is indeed expected to be ~=15 GHz, 2t=l.905 mm, 2L =12.7 mm, c=8.875, d=l 27 mm,
due to the stationary nature of the expression. If higher h =12.7 mm.
accuracy is desired, a higher number of expansion terms
could be used. In the results presented here, an initial a result for the specific case indicated. The number of
iteration at p = q = 8 is executed starting at a value of ~ expansion terms used is comparable with that of the
={-*KO. When the root is captured, it is used as integral equation method of [9]. Examination of the results
a starting iteration for a second stage at p = q = 20 to 22. shows that the value of ~/KO seems to tend to a constant
In all cases examined, such anumber of terms would result value as the frequency increases. Investigating several re-
in a stable value of ~. Table I presents an example of such sults shows that this l@iting value approaches A. For
HASSAN: SHIELDED MICROSTRIP TRANSMSSS1ON LINE 849
IEZ l/lEY] + field. It is found that inclusion of higher number of terms
q does not alter the value of R appreciably. Fig. 6 shows a
1 :;;;.,,5: ;
T plot of the polarization lEzl/ IEYI versus frequency for two
b.635 11.43
Ref. 17 _ different values of c, as compared with the only data
L
1.27 available to the author. This comparison is the reason to
IO-12 .7-A T consider IEzl/ IEYI and not the axial-to-total field ratio
(i.e., lE.1/{m as considered in Fig. 7). Results
0.1 xx show that the axial component of the electric field E= is
x 0
0 negligibly small at low frequencies where the field ap-
x 0
proaches true TEM. In fact, the quasi-TEM theory as
x 0
shown in [17] is highly accurate at this region. At higher
/
frequencies, however, the value of the axial field compo-
nent E, increases and the field deviates from the TENI-like
mode. Fig. 6 shows this general trend for two different
5 10 15 20 25 GHZ
values of dielectric constant ~, compared with data pub-
Fig. 6. Polarization versus frequency compared with data published in
[17]. Units are in millimeters. lished in [17]. The agreement is quite remarkable. It is
shown that the polarization ratio decreases as c, decreases,
which is expected since as C, approaches unity one should
2t.2.54 m — expect a true TEM mode inside the structure. In Fig. 7, the
=1.905mm .--. –
polarization ratio R versus frequency is presented for
=1.27 mm
several different strip widths. The same general trend of
R
r
very low R at low frequency is observed. Also, it seems
0.6
2t=2.54 mm that the polarization ratio tends towards a limit at wave-
I
0.5 length comparable to the structure dimensions. Further
0.4 ,~/ / 2t=l .27 mm work is under way to investigate analytically such limit. It
0.3 2t=l.905 // is clear that the ratio R is insensitive to the strip width, but
# is primarily a function of frequency and dielectric constant
0.2 ,//
as evident from Figs. 6 and 7.
,//
0.1 r ,Y
I B. Higher Order Modes
10
20 30 40 50 f(GHz)
Higher order modes were also investigated with respect
Fig. 7. Variation ofpolarization ratio R with frequency. Parameters are
to both their propagation constant and polarization ratio.
those of Fig. 5.
Figs. 9 and 10 give the value of ~/KO compared with data
published in the literature. Results are almost typical of
example, the result shown in Fig. 5 for c, =8.875 tends to those of [9]. However, this technique has the advantage of
about 2.96 while & is 2.979. Other results available to the being highly convergent. Table 11 shows a sample of the
author (not shown here) showed that for c,=5 and 12 the results available where a total of only five terms was
limiting’ values are around 2.18 and 3.45; respectively, sufficient to obtain a value for ~/KO with an error less
which arevery close to fi =2.23 and @=3.46. It is also than 0.4 percent. This was noticed in all results examined
found that in such case, the higher the value of c,, the for the first as well as the higher-order modes. The similar-
closer the ratio (limiting value of [~/KO]/&) is to unity. ity of the ~ – ~ characteristics of these curves and the LSE
This result indicates that the field in the very-high- or LSM is evident and is discussed elsewhere [8], [9]. The
frequency case would possibly concentrate in the dielectric study of the polarization factor (Fig. 11) reveals the same
region acquiring its mode of propagation. general characteristics observed in the TEM-like modes.
Turning our attention now to the field polarization, The value of R is low at low frequencies but, compared to
Figs. 6 and 7 present some examples of the results ob- the TEM mode, it increases appreciably as the frequency
tained. These results are obtained by evaluating the coeffi- increases. They seem to tend to a limit. However, extra
cients aP and b~ of the expansion (6) as indicated in the work is needed to investigate such limiting value. Further,
theory. Numerical evaluation shows that alarge number of the value of R decreases—as expected—with the lower
terms of the expansion (6) are needed to obtain stabilized value of c.. The convergence of the co(;fficients of expan-
values for the first few terms. Fig. 8 shows the convergence sion (6) is found to be much faster here than in the TEM
of the first term of each of the aP’s and b~’s. It is clear that case. An example of this convergence is given in Tablle III.
they are slowly approaching a limit as the number of terms Such difference in convergence may be attributed to the
increases. In all calculations of the polarization ratio R of difference in behavior between the TEM and the higher
Fig. 7 (defined as lE21/~m) or polarization P of order modes. While the expansion (6) is most suited for the
Fig. 6, up to 22 terms of each expansion are considered to higher order modes, it might be more appropriate (for fast
approach as accurately as possible the true value of the convergence) to assume an expansion for the TEM which
850 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT:34, NO. 8, AUGUST 1986
a2 %
—7— Coefficient a2 _
al al Coefficient bl ___
-i
6
5
4
~--
//
3
0’
2 /
,0’
1
/“
0
I I ,/ 1 a
36912151821 24 Number of terms (P, Q)
Fig. 8. Convergence of az and bl. Parameters are c,= 8.875, j_=15 GHz, 2?=1.905, 2L =12.7, h =12.7, d=1.27. All
dimensions are in millimeters.
B/K.
2.5 Ref. 9
This technique 000
2.0
LiJT
o
11.43
0.635
++ /
1.5 ~=. 1 0
1.27 x’ .
~12.7+ ~
./0” /
1.0
0.5
10 15 20 25 Frequency :GHZ)
Fig. 9. The ~ –/3 characteristics of rnicrostrip transmission line with strip width 0.635 mm for E even– H odd modes
compared with other published data (dimensions are in millimeters).
Ref. 9 _
This technique ~ o ~
“KO 2.5 ml
2.0
1.5
1.0
10 15 20 25 Frequency (GHz)
Fig. 10. The ~ –/3 characteristics of microstrip transmission line with strip width 1.27 mm for E even– H odd modes
compared with other published data (dimensions are in millimeters).
HASSAN: SHIELDED MICROSTIUP TRANSMISSION LINE 851
TABLE II
CONVERGENCE OFTHEPROPAGATION CONSTANT.
1
f = 15 GHz f = 25 GHz
m. of w M 3, 3 6, 6 9, 9 3, 3 6, 6 9, 9 12, 12
Root $/Ko 0.780 0. 784 0.785 1.754 1.7592 1.7580 1.7576
Parameters are, those of Fig. 9.
RI Z==12
.—
./.-
/’
./.
0.8 .
./ ‘r=8.875
/
./.
0.6
/
./.
/
0.4 . /.
/. _ --. — S==2
.-”----
0.2
4
15 20 25 30 f(G51z)
Fig. 11. Polarization Rofthefirst mode fordifferent vahsesof c,. Parameters arethoseof Fig.9.
TABLE III order modes were also investigated, and the similarity
EXAMPLE OFCONVERGENCE OFTHECOEFFICIENTS OF
between such modes and those of the LSE and LSM was
EXPANSION (8)
shown both analytically and numerically.
Nsr&r Ofterms M =2 ‘3 % b2 b3 ACKNOWLEDG~NT
The research of this paper was carried out in the Electric-
(3, 3) -0.627 0.124 -2.5 1.20 0.2
al Engineering Department, University of Petroleum and
(6, 6) -0.66 0.088 -2.57 1.27 0.21 Minerals, Dhahran, Saudi Arabia.
(8, 8) -0.684 0.076 -2.61 1.30 0.22
(10,10) -0.694 0.067 -2.61 1.317 0.224 REFERENCES
(12,12) -0.700 0.062 -2.61 1.320 0.227 [1] H. A. Wheeler, “Transmission-line properties of parallel strips
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This paper presents a variational technique to solve for
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[12] V. Rumsey, “Traveling wave slot antennas,” .J. App/. I’hys., vol. Essm’ E. Hassan was born in Alexandria, Egypt
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[13] R, F. Barrington, “Propagation along a slotted cylinder,” .J. Appl. engineering (with honors) from Alexandria Uni-
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[14] C. P. Wen, “Coplanar waveguide: A surface strip transmission line in electrical engineering from the University of
suitable for non-reciprocal gyromagnetic device applications,” IEEE Manitoba, Winnipeg, Canada, in 1977 and 1978,
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1969. From 1977 to 1979, he was a Senior Research
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Microwave Theoty Tech., vol. MTT-17, pp. 768-778, Oct. 1969. digitaJ switching division, where he was responsi-
[16] R. F. Barrington, Time-Harmonic Electromagnetic Fiela5-. &ew ble for design and testing of digitaf networks. In
York: McGraw-Hill, 1961. 1979.,.he ioined the Department of Electrical Enxineerinz at the Univer-
[17] L. Young and H. Sobol, Eds., Advances in Microwaves, vol. 8. sity of Petroleum and ‘Mnerak, Dhahran, Saudi Arabia, and since then
New York: Academic Press, 1974. he has been active in the fields of communication, microwave communi-
[18] W. J. Getsinger, “Microstrip dispersion model; IEEE Trans. Mi- cation, and electromagnetic field theory.
crowaue Theory Tech., vol. MTT-21, pp. 34–39, Jan. 1973. Dr. Hassan is an author or coauthor of severaf technical papers in
[19] M. V. Schneider, “Microsttip dispersion,” Proc. IEEE, vol. 60, pp. electromagnetic and in communication aswell asone book in communi-
144-146, Jan. 1972. cation theory.