1792 I E E E TRANSACTIONS O N MICROWAVE THEORY A N D TECHNIOUFS, VOL. 38, NO. 12, DFCEMBFR 1990
Full-Wave Analysis of Lossy Quasi-Planar
Transmission Line Incorporating
the Metal Modes
CHING-KUANG C. TZUANG, MEMBER, IEEE, CHU-DONG CHEN, STUDENT MEMBER, IEEE,
AND SONG-TSUEN PENG, FELLOW, IEEE
Abstract -We present a novel and accurate full-wave mode-matching approach to analyze the dispersion characteristics of millimeter-wave and microwave transmission lines with finite conductivity, metallization thickness, and holding grooves. The approach is quite general but only the results for a unilateral finline are presented. The accuracy of the solution depends primarily on the correct and complete description of eigenfunction expansions in each of the uniform (stratified) o r nonuni- form layer regions. The latter consists of metallized strips of finite conductivity, which in turn produce the so-called metal modes (eigen- modes). The metal mode exists in the metallized region with high conductivity for the most part and decays sharply in the air region. Without incorporating the metal modes, the convergence studies will fail and the accuracy of the field theory solution deteriorates.
Since the accuracy of the present approach is established, the compos- ite effects of the finite conductivity and metallization thickness can be studied rigorously. A numerical limiting case analysis shows that the mode conversion between the dominant finline mode and the dielectric- slab-loaded waveguide mode may happen through the reduction of the metallization thickness. The theoretical results for the dispersion pa- rameters of the dominant mode propagation constant and the character- istic impedance are reported. The effects of the conductor losses using various metallizing materials are also presented.
I. INTRODUCTION
H E ANALYSIS of conductor losses on integrated
T
millimeter-wave and microwave transmission lines plays an important role in the accurate computer-aided design (CAD) modeling required in many demanding applications. Three mechanisms constitute the attenua- tion in the transmission lines, namely, conductor loss, dielectric loss, and radiation loss [l]. The effects of dielec- tric losses on transmission lines have been analyzed thor- oughly [2], [3]. This paper will focus on the propagation characteristics of an electrically shielded transmission line for use in millimeter-wave and microwave component design. Therefore, we will restrict our attention to con- ductor loss. Recently, a few methods have been devel-Manuscript received March 29, 1990; revised August 20, 1990. This work was supported in part by the Taiwan National Science Council under Grant NSC79-0404-E009-29 and Contract D78026.
C.-K. C. Tzuang and C.-D. Chen are with the Institute of Communi- cation Engineering and the Center for Telecommunication Research, National Chiao Tung University, No. 75, P o Ai Street, Hsinchu, Taiwan, Republic of China.
S.-T. Peng is with the Electrical Engineering Department, New York Institute of Technology, Old Westbury, NY 11568.
I E E E Log Number 9040054.
oped for this, among them a combined surface integral equation method [4], a phenomenological loss equivalence method [5], and a modified mode-matching method [6].
In formulating the combined surface integral equation [4], prior knowledge of quasi-TEM analysis of the field penetration into the microstrip line is required for later formulation. The phenomenological loss equivalence method is developed for a transmission line which sup- ports a quasi-TEM mode with conductor thickness of the order of the skin depth. For an integrated finline, how- ever, the above methods need modifications because the dominant mode is not quasi-TEM. In certain applications when a low-impedance microstrip line needs to operate at very high frequency, the quasi-TEM assumption fails to model the microstrip line faithfully.
By way of example, Fig. 1 shows an electrically shielded symmetrical lossless microstrip line integrated on a 100- Fm-thick GaAs substrate ( E ~ = 13) that is analyzed by the spectral-domain approach (SDA) using a highly effective set of basis functions [7]. When operating at 150 GHz, the results indicate that the longitudinal and transverse sur- face current densities,
J,
and J,, are comparable in magnitude for impedances lower than 17.0 R. For the 32.4 R microstrip line, the transverse current component is about 1/15 the longitudinal component. Thus, depend- ing on the structure and operating frequency, the quasi- TEM assumption may not apply for millimeter-wave cir- cuit designs. Another full-wave approach for analyzing a transmission line with conductor losses is the perturba- tional method [8], [9]. This approach assumes that the quasi-planar transmission lines have infinitely thin metal- lization with conductivity of infinite value. After the loss- less full-wave solution is obtained, a perturbational ex- pression is invoked for computing the conductor loss, e.g. [8, eq. (ll)]. The assumption is apparently valid for struc- tures with small losses.It is difficult, however, to track the tangled effects of the finite conductivity, finite metallization thickness, and broad operating frequencies covered in the millimeter- wave and microwave regimes without skillfully managing the above-mentioned assumptions or simplifications. This paper presents a reliable and accurate method for solving the above problems by the full-wave mode-matching 0018-9480/90/1200-1792$01.00 01990 IEEE
TZUANG et al.: FULL-WAVE ANALYSIS OF LOSSY QUASI-PLANAR TRANSMISSION LINE 1 II I II W=500 p m 2,=17.0 n W=200 pm 2,=32.4 fl W=lOO pm 2.=47.4 n
__---
__---
W=500 p m 2,=17.0 n W=200 pm 2,=32.4 fl W=lOO pm 2.=47.4 n 1- b-
-1 1 -0.5 0.0 0.5x/w
Fig. 1. Longitudinal ( J : ) and transverse ( J , ) surface current densities of a lossless microstrip line integrated on a GaAs substrate at 150 GHz. h = 10 mm, h l = 100 p m , and h, = 1 mm.
method [6], [lo], [ll]. The solutions are correct in the sense that the numerically truncated solutions should satisfy the so-called relative and absolute convergence criteria [lo]-[ 131 for various lossless waveguide structures. The convergence study is further complicated by the inclusion of the metal modes in the mode-matching for- mulation [14]. In this paper a series of convergence stud- ies is presented for a unilateral finline to illustrate the convergence properties of the present formulation taking into account the conductor losses. Such an accurate field theory approach enables us to investigate important prop- agation characteristics of transmission lines involving fi- nite metallization thicknesses and conductivities.
11. FORMULATION: A MODE-MATCHING METHOD
INCORPORATING THE METAL MODES
Fig. 2 illustrates a particular example to be analyzed rigorously. The quasi-planar transmission line is sur- rounded by a perfectly conducting enclosure. The wave- guide cross section is subdivided into six regions with their respective relative dielectric constants. The subscript of each relative dielectric constant is the name of that region. Thus, region 31 is the region with relative dielec- tric constant E ~ ~ . Throughout this paper, the e j w f P y z fac- tor is assumed. Therefore, for a lossy transmission line, we will expect a complex propagation constant y ( y = a
+
j P ) to exist. For a metallized region of finite conductivity, E ~ , = E, - ju/coEo, where cr is the conductivity of the metallization. If regions 1 and 4 are air-filled and region 2 is the supporting dielectric substrate, Fig. 2 may become two different quasi-planar transmission lines. When re- gions 31 and 33 are metallized and region 32 is air-filled, the structure is a unilateral finline. The reverse of this is a suspended microstrip line. This paper will focus on the analysis of the unilateral finline to study the important1793
Y
t
2 - b - L -
gl gz
Fig. 2. A unilateral finline with finite conductivity, metallization thick- ness, and holding grooves. h = 3.556 mm, c = 1.6002 mm, d = 1.9558 mm, g l = g , = O , h , = 3 . 4 9 2 5 mm, h 2 = 0 . 1 2 7 mm, t + h,=3.4925 mm, E , = t4 = 1, E , = 2.22, c 3 , = = 1 - j u / w e , , and U = 3.333 X 10' mhos/rn.
physical characteristics of the lossy finline without loss of generality.
The mode-matching formulation based on the TM-to-x and TE-to-x eigenfunction expansions for all regions is derived. They are summarized as follows.
Region 1: (3) NZ
T
!
= sin[
x+
g , ) ] n = l (4)1194 I F F E TRANSACTIONS ON M I C R O W A V E THEORY A N D TEC H N I O U E S . vol. 38. NO 12. DFCFMBER 1990 Region 3: Region 4: (11) N1 cos
[
P k ( h l + h ,+
t - Y ) ] ( a 4 n = n . s r / b ) . n = l cos[
Because the sidewalls of the waveguide housing are assumed to be perfect electric conductors, the biorthogo- nality relationship holds for eigenfunctions in region 3, although it contains a lossy conductor layer [151. The biorthogonality relationship in this region reads as sin
[
P?hn(hl+ h2+
t - Y11
+
F,"}
(6) (6) sin[
~ t n t ] N 3 = N31 + N 3 2 + N33 (x)@,:,( X )dx
= 6: (12)dx
= 8; (13)where N31, N32, and N33 can be either the number of metal modes or the number of air modes. When analyzing of metal modes associated with the metallized fins, whereas N32 is the number of air modes corresponding to
the gap:
Fig. 2 as a unilateral finline, N31 and N3, are the numbers
where 6; is the Kronecker delta function.
where
The above equations indicate that 12 sets of unknown coefficients exist. These coefficients can be eliminated by matching all the necessary tangential boundary conditions at each interface and applying the biorthogonality rela- tionship governed by (12) and (13). Finally a nonstandard eigenvalue equation is derived, i.e.,
(a!31n)2 ='3lP,' - ( P ! 3 n ) ' + Y 2
(
a 3 2 n i)L
' 3 2 P i - ( P i n ) ' ' Y 2( a ~ n ) ' = E , , P ; - ( P i n ) ' + y 2 [A(Y)l[XI = [OI ( 14)
P i =
W ' ~ ~ E ~i ,= e or h . (9) where the column vector is [x]=[C,h 0,' E,h F,'IT, whichcontains the remainder sets of coefficients. The matrix
[AI has the size 2(N2
+
N,
+
1) by 2(N2+
N3+
1). The can be obtained in the same way as in a dielectric-slab- roots of the equation det([A(y)]) = 0 give rise to theloaded waveguide problem and will not be repeated here solutions for the complex propagation constants. The
~ 5 1 . nontrivial solution directly leads to the solution for
[XI.
TZUANG et ul.: FULL-WAVE ANALYSIS O F LOSSY QUASI-PLANAR T K A N S M I S S I O N I I N L 1795
TM
modeTE
modeO("sln/Bo
&n/&
a & n / B o aXhgln/Bo a&/B0ahdB0
~0 2.73~1 d -p.73xld 1 .59xitiz+p.a4xi8 2.73xld -j2.73x1 d
AIR
MODE &=I 2.73x103-j2.73x1d 1.05~10' +jl.16x1E4 2.73x103-j2.73x1d 2.73x103-j2.73xld 1.05~10' +Jl.28x1$2.73~1 O3-j2.73x1 d 273x103-j2.73x1 O3 2.10xld +j5.81 x l b 2.73~1 03-j2.73xld 2.73x103-j2.73x1d 2.10~10'
+
J2.57xl# 2.73x1O3-j2.73x1dNSi=l 1 . 1 7 ~ 1 0 ~ - ~ . ~ x ~ ~ " 2 . 7 3 x l O 3 + ~ . 7 3 x l d 1.1 7x1
8
-jl .92xlti4 2 . 3 4 ~ 1 8 -j3.19x1-d42.73x103+j2.73x12
2.34x10°-j4.1 5 ~ 1 6 ~ N91=2 3.51 xlOo -j3.19x1t1' 273x1 03+j2.73x1d 3.51 x1 0°-8.40x164 4 . 6 8 ~ 1 8 -j6.38x1a4 2.73~1 03+j2.73x1 d 4.68x1O0-j6.88x1c4
%1=3 5.85~10' -j5.32x~d'2.73x103+ j2.73~13 5.85~1 @ - p . 8 4 x l r 1 ~ 7 . 0 2 ~ 1 0 ~ - ~ . ~ ~ x ~ ~ 4 4 2 . 7 3 x 1 0 3 + j . 7 3 x 1 0 3 7 . 0 2 ~ 1 0 ~ - j 9 . 8 9 ~ 1 5 ~Given the solution for
[XI,
the characteristic impedance can be obtained in the same way as in [121.111. METAL MODES A N D A I R MODES
Section I1 presents a classical mode-matching method which is distinguished from the conventional one by the inclusion of the metal modes. Regions 1, 2, and 4 are expanded in a way similar to that in [12]. The eigenfunc- tions in these regions are classified as air modes. For a lossy unilateral finline, regions 31, 32, and 33 should have both air modes and metal modes to provide the mode completeness required in the mode-matching method. The first few TM-to-x air modes in the air region 32 and TM-to-x metal modes in the lossy conductor region 31 were reported in [14, figs. 2 and 31 and will not be repeated here. T o further clarify the concept of using both air modes and metal modes, Table I lists the values of the first three eigenvalues at 40 GHz for air modes of region 32 and metal modes of region 31 for the structural parameters shown in Fig. 2. When the conductivity is high, the eigenfunction corresponding to the metal mode confines itself in the metal region and decays abruptly in the air region. The eigenfunction corresponding to the air mode is much more familiar to us. In contrast to the metal mode, the air mode resides mostly in the air (di- electric) region bounded by good metals.
,
IV. CONVERGENCE STUDIES FOR A PARTICULAR
LOSSY FINLINE
As pointed out by various authors interested in analyz-
ing the propagation characteristics of lossless waveguide structures using the mode-matching method, the relative convergence criterion should be satisfied to obtain good field matchings at discontinuities o r interfaces [lo]-[13]. Failing to do this will result in inaccurate field solutions. In the case of a lossy quasi-planar transmission line, the convergence study is further complicated by the existence of the metal modes described in Section 111. Since most millimeter-wave and microwave integrated transmission lines are gold-plated to reduce the conductor losses that inevitably exist in these structures, we restrict our atten- tion to the convergence study for a transmission line with a good conductor coating. Under this condition and using
0.3 0.35 0.4 0.45 0.6
x/b
Fig. 3 . Relative convergence studies of the tangential electric field E , at the interfaces y = ( h i
+
h z+
t ) - and y = ( h ,+
h 2+
t ) + for vari- ous numbers of metal modes N,,, (N,,, = N31+
N33) at 40 GHz. NI =N2 = N4 = 160, N, = N32 = 16, t = 5 p m , and W / b = 0.1. (Other
structural and material parameters are listed in Fig. 2.)
the structural and material parameters listed in Fig. 2, Figs. 3 to 5 summarize the results for the convergence
study.
The number of expansion terms used in the formula- tion in regions 1 to 4 are N , , N 2 , N,,, N32, N,,, and N 4 , respectively. In the particular case study for a symmetrical lossy finline, the sum of N,, and
N,,
is renamedN,,
which represents the number of metal modes. N32 is renamed No, i.e., the number of air modes. When analyz-
ing the complex modes
of
a lossless bilateral finline, it was found that the relative convergence criterion should be simultaneously satisfied at the interfaces near the dual slots to obtain the best field matchings in the interfaces[ 121. Since the symmetrical unilateral finline under inves- tigation assumes 5-Fm-thick gold-plated strips and the
1796 IEEE TRANSACTIONS O N MICROWAVE THEORY A N D T K H N I Q U E S , VOL. 38, NO. 12, DECEMBER 1990
200
4
0 40 80 120 160 200 240 280 Relative and absolute convergence studies of the normalized propagation constant for the dominant mode versus the value of N,, under different controlling parameter:
N,
Fig. 4. N I = N, = N4 = 40 N, = N, = N4 = 80 N, = N, = N4 = 120 N, = N, = N4 = 160 N, = N 2 = N4 = 200 N, = 4 N, = 8 N,, = 12 No = 16 N, = 20.W / b = 0.1, t = 5 p m , and frequency = 40 GHz. (Other structural and material parameters are listed in Fig. 2.)
1.2
I I I I I I I I
Fig. 5. Absolute convergence studies of the complex propagation con- stant for the dominant mode versus the normalized fin gap W / b ; t = 5 p m and frequency = 40 GHz. T h e N,,, values are determined by the aspect ratio discussed in Figs. 3 and 4. (Other structural and material parameters are listed in Fig. 2.)
skin depth at 40 GHz is about 0.4 pm, it is plausible to expect that the value of Nu should be closely related to the proper aspect ratio as reported in [ll] and [121. By having N, = N2 = N4 = 160, and W / b = 0.1, we set Nu = 16. Varying the value of N, from 132 to 142, 144, 146, and 156, respectively, Fig. 3 plots the tangential electric field E, at the y = ( h ,
+
h ,+
t ) - and y = ( h ,+
h ,+
t)'interfaces. The tangential fields just underneath the 5-pm-thick metal strip do not match well with those just above the metallized strip for N, = 132 and 142. When N, = 146 and 156, the interface field matchings seem to
be good, but the nearly singular property imposed on the corner
(x
= 0.45b) of the good rectangular strip begins to degrade as the value ofN,
is increased. Notice that the field matching properties can change substantially by sub- tracting or adding just two terms toN,
= 144. This is by no means a coincidence. It follows the rule of aspect ratio, i.e., N , , = N , , + N3,=160X(1-O.1)=144.To give a broader idea on the effects of the relative convergence criterion for the particular case study, Fig. 4 plots the normalized propagation constant against the number of metal modes
N,,
using N 2 ( N I = N2 = N4) as the controlling parameter. The data points under the circle(0)
signs are for those obeying the relative conver- gence criterion confirmed earlier in Fig. 3. For each value of N,, the normalized propagation constant may change drastically as N, changes. The contour of the circle signs, however, represents a smooth convergence property against N, and converges quickly, as shown in Fig. 4, for the normalized propagation constant.Both the relative and absolute convergence studies depicted in Figs. 3 and 4 can be illustrated together as shown in Fig. 5, which plots the normalized propagation constant and conductor loss on both the left and right axes against the normalized gap width (W/b), respec- tively, using a different number of expansion terms N 2 (NI = N, = N4) and
4,.
The value of N, is determinedby the aspect ratio discussed in Fig. 3. When the finline has a narrow gap width, and consequently a smaller W / b
ratio, our formulation requires that N2 be over 80 for solutions with better convergence, as shown in Fig. 5. The fact that the conductor loss converges slower than that of the propagation constant is clear in these plots. Each conductor loss plot is distinguishable from others, whereas the plot for the propagation constant overlaps for N, greater than 120. Higher loss for the finline with smaller gap is understandable since the dominant mode electro- magnetic field line is concentrated near the gap of the finline. This in turn will cause higher conductor loss associated with the metal fins.
-.
V. THE COMBINED EFFECTS OF FINITE
CONDUCTIVITY AND THICKNESS
ON A LOSSY FINLINE
The power of the full-wave approach presented in this paper will be investigated by a case study, which investi- gates the same unilateral finline with the structural pa- rameters given in Fig. 2. The test conditions for our formulation are shown in Fig. 6. Given an operating frequency of 40 GHz, the solid lines in Fig. 6 are the corresponding plots of the normalized propagation con- stant
( p / p o )
and conductor loss with respect to the metallization thickness t . The conductor loss increases gradually as t is reduced from 10 to 0.01 pm. Here caution should be exercised. The absolute lower limit to the macroscopic domain imposed on the Maxwe! equa- tions with a continuous dielectric constant is 100 A (lo-' pm) 1161. When t reaches a value of p m , it is regarded as a numerical limiting case analysis of the present formulation for an infinitely thin conductor. If theTZUANC et al. : FUI.I.-WAVT ANALYSIS OF I.OSSY QUASI-PLANAR TRANSMISSION L I N E 1797
Fig. 6. Complex propagation constant of the dominant mode versus the metallization thickness t . v = 3 . 3 3 3 ~ 10' mhos/m, W / b = 0.1, and frequency = 40 GHz. (Other structural and material parameters are listed in Fig. 2.)
physical environment were to support the Maxwell equa- tions in the limiting case analysis, we would not only test the validity of the present formulation but also establish the validity of using the assumption of an infinitely thin
perfect conductor for analyzing lossless planar or quasi-
planar structures [81, [91.
The skin depth obtained for the test case is 0.436 p m .
As the value of t is reduced from 1 p m (2.36) to 0.01 p m (0.0236), the excited currents start to distribute them- selves throughout the entire cross section of the thin rectangular metal strip. The cross-sectional area also be- comes smaller as t is reduced. Thus the ohmic loss increases. The slope for the conductor loss curve is nearly -1. This is a manifestation of the fact that the electro- magnetic fields are uniformly distributed inside the rect- angular strip and consequently the conductor loss is in- versely proportional to the thickness t .
When t is reduced from l o p 2 to p m (1
A),
the conductor loss rises and declines. This is the region where a mode conversion takes place gradually. When the thick- ness t is reduced further froml o p 4
to p m , the conductor loss becomes smaller. This region corresponds to the familiar first LSE mode region; i.e., the finline essentially becomes a dielectric-slab-loaded waveguide. The field distribution is no longer that of a dominant finline mode, which carries most of the electromagnetic energy in the vicinity of the gap between the metal fins. Therefore the loss is smaller.Next we test our formulation under the extreme condi- tion that simulates the situation as an infinitely thin per-
fect conductor. This is done by increasing the conductivity
by more than five orders of magnitude and reducing the thickness by five orders of magnitude from 1 pm. The results indicated by the triangle, circle, and cross signs show that the normalized propagation constant is very close to the solid line in the dominant mode region, and the conductor loss is decreased by increasing the conduc- tivity from 3.333 X 10" mhos/m to 6.666 x lo'* mhos/m.
Increasing the conductivity by another five orders of mag- nitude, to 6.666 x 10'' mhos/m, the resulting dashed line represents the lossless case.
It is obvious, based on the analyses presented above, that care should be exercised in extracting design parame- ters such as the propagation constant from a full-wave field-theory analysis of a quasi-planar transmission line assuming infinitely thin perfect conductor strips. It is im- plied in Fig. 6 that the thickness t should be about three skin depths or more to have the smallest conductor loss and to have the propagation constant closer to the theo- retical prediction when assuming infinitely thin perfect
conductors.
The electric field patterns corresponding to the points
P , Q, and R in Fig. 6 in the dominant finline mode,
transition, and the first LSE mode regions are plotted in, respectively, parts (a), (b), and (c) of Fig. 7. It is easy to identify that parts (a) and (c) are for the dominant finline mode and the familiar first LSE mode of a dielectric- slab-loaded waveguide, respectively. On the other hand, part (b) gives no clear indication of which mode the field pattern represents. Therefore point Q belongs to the transition region.
VI. OTHER THEORETICAL RESULTS
A comparison between the results obtained by this
paper and the perturbational method [8], [9] is illustrated in Fig. 8. The results based on the perturbational method include the conductor losses of the waveguide housing and fins. In the limiting case where W / b = 1, i.e., a dielectric-slab-loaded WR-28 waveguide, the conductor loss given by the dashed line [81 seems to be more accurate. As W / b is reduced, the conductor loss in the broken line is close to the measured value [9]. Since our results in the solid line consider only the conductor losses of the metal fins, an accuracy comparison will be difficult. When the W / b ratio is less than 0.3, our results are approximately twice those obtained by [8]. As W / b ap- proaches unity, the conductor loss in the solid line re- duces to zero. This validates our solutions for conductor losses since the waveguide housing is assumed to be a perfect conductor in our analysis.
Fig. 9 plots the real parts of the characteristic impedance under the power-voltage definition [ 11, [12] and the normalized propagation constant against the nor- malized gap width. The data points with the circle signs are obtained by assuming lossless conductors. The imagi- nary parts of these curves are at least four orders of magnitude smaller and are not reported here. It is impor- tant to emphasize again that in practice a circuit designer should choose a good, thick metal coating for finline and other quasi-planar transmission lines if the actual physical design parameters are not to deviate from those obtained by assuming lossless conductors. Finally, Fig. 10 shows the effects of different conductivities on the conductor losses of the finline. As expected, lower conductivity will result in higher ac resistance for the metallization and conse- quently higher conductor losses for the finline.
1798 l C r E TRANSAC‘TION\ O N M I C R O W A V E T H E O R Y A N D T E C H N I Q U E S , VOL. 38, NO. 12, D E C E M B E R 1990
Fig. 7. Electric field patterns of the different modes at ( a ) point P , (b) point Q , (c) point R of Fig. 6 . v = 3.333X lo7 mhos/m, W / h = 0.1, and frequency = 40 GHz. (Other 3tructural and material parameter5 are listed in Fig. 2.)
11 I I I I I l l I I
-
-
-
--
-
‘ope-.-.-
- - - - 0.0 0.2 0.4 0.8 1.0W/b
Fig. 8. Conductor loss 01, of the dominant mode versus the normalized fin gap W / h; U = 3.333 X lo7 mhos/m and frequency = 40 GHz. (Other structural and material parameters are listed in Fig. 2.)
VII. CONCLUSION
Full-wave theoretical analyses of a gold-plated unilat- eral finline have been presented. The mode-matching full-wave approach incorporating the metal modes has proved to be very accurate and reliable for analyzing lossy millimeter-wave quasi-planar transmission lines. Without including the metal modes or satisfying the relative con- vergence criterion for the particular case study on a unilateral finline with good, thick (one order of magni- tude greater than the skin depth) metal coating, the tangential electric field matchings become poor and result in inaccurate electromagnetic solutions.
A limiting case study for the particular finline, under
the condition that the metallization thickness t of Fig. 2
approaches a value far below the limit of the Maxwell equations 1161, shows that a mode conversion between the dominant finline mode and the first LSE dielectric-slab- loaded waveguide mode is numerically possible. The same
600 500 1.1 1.0 0.9 0 Q \ 0.8Q 2oo,
/;/’,
;?
I L j 0 . 7 lossless case 100 0.6 0.0 0.2 0.4 0 6 0.8 1.0 W/bFig. 9. Characteristic impedance Re(Z,,) and the normalized phase constant P / P , , of the dominant mode versus the normalized fin gap
W / h. t = 5 p m , and U = 3.333 x 10’ mhos/m. (Other structural and material parameters are listed in Fig. 2.)
limiting case study exposes two interesting aspects of the present full-wave formulation.
First, when the good conductor coating is of the order of a skin depth or less, the conductor loss is essentially ohmic with current flowing evenly inside the metal strip. Second, the commonly accepted assumption of an in-
finitely thin perfect conductor for use in many field theory
analyses of millimeter-wave quasi-planar transmission lines is validated and proved to be a numerical limiting case of the present formulation.
It is also confirmed that the dispersion parameters of a
lossy finline are in good agreement with those obtained by assuming lossless metallizations if the lossy finline is formed by a good, thick metal coating. In practice, a circuit designer should keep this in mind to reduce con- ductor loss and manage to use the dispersion parameters generated by a field theory package assuming infinitely thin perfect conductors.
TZUANG et al.: FUL1:WAVE ANALYSIS OF LOSSY QUASI-PLANAR TRANSMISSION LINE 1799
h l
o o ;
Fig. 10. Conductor loss of the dominant mode versus the normalized fin gap W / h under different controlling parameters. uNIC, = 0.1OX lo7 mhos/m, up, = 1.03 X 10’ mhos/m, U,,= 3.82 X lo7 mhos/m, a,, =
4.10x lo7 mhos/m, uA, = 6 . 1 7 ~ lo7 mhos/m. Frequency = 40 GHz, and t = 5 p m . (Other structural and material parameters are listed in Fig. 2.)
ACKNOWLEDGMENT
The authors would like to thank J.-T. Kuo, who verified data shown in Fig. 1 by the SDA.
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Ching-Kuang C. Tzuang (S’84-M’86) was born in Taiwan on May 10, 1955. H e received the B.S. degree in electronic engineering from the National Chiao Tung University, Hsinchu, Tai- wan, in 1977 and the M.S. degree from the University of California at Los Angeles in 1980. From February 1981 to June 1984, he was with T R W , Redondo Beach, CA, working on analog and digital monolithic microwave inte- grated circuits. He received the Ph.D. degree in electrical engineering in 1986 from the Univer- sity of Texas at Austin, where he worked on high-speed transient analyses of monolithic microwave integrated circuits. Since September 1986, he has been with the Institute of Communication Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C. His research activities involve the design and development of millimeter-wave and microwave active and passive circuits and the field theory analysis and design of various quasi-planar integrated circuits.
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Chu-Dong Chen (S’89) was born in Taiwan on March 14, 1956. H e received the B.S. degree in electronic engineering from Chung Yuan Uni- versity in 1978 and the M.S. degree in electronic engineering from the National Chiao Tung Uni- versity, Hsinchu, Taiwan, in 1980.
From 1981 to 1988, he was with the Chung Shan Institute of Science and Technology, Lung Tan, Taiwan, as an assistant researcher. He had been involved in the design and development of millimeter-wave and microwave 111-V com- pound semiconductor devices. Since 1988, he has been pursuing the Ph.D. degree at the National Chiao Tung University. His current re- search interests include the rigorous field analysis of quasi-planar trans- mission lines and the design of millimeter-wave components.
Song-Tsuen Peng (M’74-SM’82-F’88) was born in Taiwan, Republic of China, on February 19, 1937. He received the B.S. degree in electrical engineering from the National Cheng-Kung University in 1959 and the M.S. degree in electronics from the National Chiao-Tung University in 1961, both in Taiwan, and the Ph.D. degree in electrophysics from the Polytechnic Institute of Brooklyn, Brooklyn, NY, in 1968.
From 1968 t o 1983, he held various research positions with the Polytechnic Institute of Brooklyn. Since 1983, he has been a Professor of Electrical Engineering and Director of the Electromagnetics Laboratory at the New York Institute of Technology, Old Westbury, NY. H e has been active in the fields of wave propagation, radiation, diffraction, and nonlinear electromagnetics and has published numerous papers on electromagnetics, optics, and acoustics.
