Quasi-TEM analysis of reciprocal/nonreciprocal planar lines with polygonal cross-section conductors

Quasi-TEM Analysis of Reciprocal / Nonreciprocal Planar Lines with Polygonal Cross-Section Conductors

Gonzalo Plaza, Francisco Mesa, and Manuel Horno

Microwave Group, Department of Electronics and Electromagnetism, University of Seville, Avda. Reina Mercedes s / n, 41 01 2 Seville, Spain Received October 22, 1993; revised May 11, 1994.

ABSTRACT

Multilayered multiconductor transmission lines with polygonal cross-section-perfect conductors are analyzed under a quasi-TEM approach. According to a model previously proposed by the authors, each polygonal cross-section conductor is modeled as a set of zero-thickness strips. The substrates can be lossy, and present possible gyromagnetic (longitudinal magnetization) characteristics as well as dielectric anisotropic properties. The method of moments of Galerkin is applied in the spectral domain to analyze the model structure. Special attention has been paid to the fast and accurate computation of spectral integrals. Thus, the possible symmetries of the Galerkin matrix are emphasized and recurrence formulas to compute the tail integrals have been developed. 0 1994 John Wiley & Sons, Inc.

1. INTRODUCTION

Microstrip line is commonly used as the basic structure in the design of different types of reciprocal (filters, directional couplers, etc.), and nonreciprocal (isolators, phase shifters, etc.) microwave devices. Moreover, as a consequence of the increasing development of the integrated cir- cuit technology, the interconnects between VLSI devices in high-speed digital computers have to be considered as strip-like lines [l].

Most works published on multilayered multiconductor microstrip lines are based on the zero strip thickness approach. Generally, this approach provides accurate results in the analysis of MIC

Supported by the DGICYT, Spain (Project No. TIC91- 1018).

lines, but it is not sufficiently valid to analyze MMIC circuits, since the ratio of strip thickness to strip width may be significant [2] . In addition, the strips tend to have trapezoidal cross-sections as a result of the technological process [3, 41. These effects should be taken into account to predict the performance of the circuits accurately. On the other hand, the anisotropic nature of the substrates should be also incorporated in the analysis to account for the possible crystalline anisotropy as well as to study the potential applications of nonreciprocal gyrotropic media (e.g., ferrites biased by an external DC magnetic field).

Commonly, the typical dimensions and operation frequencies used in MIC and MMIC technology make the quasi-TEM approach sufficient for an accurate characterization of the propagation parameters of the lines. A validity criterion for the quasi-TEM model, assuming perfect conduc-

International Journal of Microwave and Millimeter-Wave Computer-Aided Engineering, Vol. 4, No. 4, 363-373 (1994) 0 1994 John Wiley & Sons, Inc. CCC 10.50-1827/94/040363-11

363

364 Plaza, Mesa, and Homo

tors, is obtained in refs. 5 and 6. This criterion is obtained from dimensional analysis of the Maxwell equations and it emphasizes that the restriction of the quasi-TEM model comes from the possible dispersion due to the substrate inho- mogeneity. In this article, planar lines with arbitrarily shaped perfect conductors will be then analyzed under the quasi-TEM approach. This approach can be also extended to deal with actual conductors provided that the magnetic energy storage inside the conductors is much less than the magnetic energy storage outside (usually strong skin regime).

Microstrip structures with finite metallization thickness have been analyzed previously under quasi-TEM and full-wave approaches. In the quasi-TEM frame, refs. 7, 8 analyzed single microstrip lines on a dielectric layer. More general structures with polygonal conductors in multilayered media were studied in refs. 9-11. More recently, an efficient method to analyze arbitrarily shaped conductors in multilayered media was presented [12]. Most of that article dealt only with isotropic substrates. There are also several re- ports dealing with microstrip lines with nonzero- thickness conductors under the full-wave approach. In this way, microstrip lines with rectangular cross-sections were analyzed in refs. 13-15. Similar structures but with semicircular edges for the strip were considered in ref. 16. Refs. 2 and 17 addressed the trapezoidal cross-section conductors case, and recently, more general structures with arbitrary cross-section conductors embedded in multilayered media were studied in refs. 1 and 18.

In this article, the M-strips model [19], together with the quasi-TEM approach, are applied to the spectral domain analysis (SDA) of multiconductor lines in multilayered media with polygonal cross-section perfect conductors. We can so complement the advantages of the SDA (namely, the easy way of dealing with an arbitrary number of layers, conductors, and with dielectric/magnetic anisotropy) with the possibility of studying nonplanar conductors (note that this latter point is usually an important drawback of the SDA). According to the M-strips model, in the quasi- TEM frame, each polygonal conductor is modeled as a set of compact zero-thickness strips at the same potential. The model structure is then analyzed using Galerkin’s method in the spectral domain. Owing to the possible great number of conductors involved in the model structure, a careful and efficient numerical treatment has been incorporated in the analysis.

2. ANALYSIS

Figure l a shows the cross section of the structure under study. In accordance with the M-strips model presented elsewhere 1191, each polygonal conductor has been modeled by a set of infinitely thin strips at the same potential shown in Figure lb. This model structure is viewed as a new multilayered, multiconductor line where fictitious layers arise from the manner in which the conductors have been modeled. These new layers will be considered as different ones in the following analysis.

Under the quasi-TEM approach, assuming perfect conductors, the transmission line propagation characteristics (namely, the modal propagation constants and characteristic impedances) can be obtained from the structure capacitance matrix per unit length (p.u.1.) [C], and inductance matrix p.u.1. [L]. The inductance matrix, [L] can be found as the inverse of an equivalent capacitance matrix, [C,,] [20]. This latter capacitance matrix refers to a new structure whose layers are now characterized by equivalent dielectric tensor permittivity related to the magnetic tensor permeability of the original structure [20].

Any of the [C] and [C,,] capacitance matrices is computed by solving the integral equation for

Figure 1. (a) Cross-section of the subject structure with N,, conductors. (b) Cross-section of the model structure. Each conductor is modeled by a set of infinitesimally thin strips.

Quasi- TEMAnalysis of Potygonal Conductors 365

the charge density p.u.1. on the strips in the model structure:

where and p j ( x ’ ) designate the potential and charge density at the ith and j th metallized interfaces, respectively; G,,, the corresponding el- ement of the Green’s matrix of the multilayered structure; and N,, the total number of metallized interfaces in the model structure.

To apply Galerkin’s method to the problem at hand, we must expand the charge density p.u.1. on the strips into basis functions. In this way, the charge density on the pth strip, which models the kth thick conductor, can be written as:

where N’; is the greatest order of the basis functions used for the charge density on the strip, cp” and wp”, the abscissa of the center of the strip and its width, and T’(.) is the nth Chebyshev polynomial of first kind. The coefficients 2/(wpk~) are introduced for the integrals of the basis functions verify:

where So, are the Kronecker delta. Galerkin’s matrix elements can be expressed as integrals in the spectral domain via Parseval’s theorem (apart from constants) as:

k,wl x J , ( 9) d k x ( C f - C , k ) dk, . (4)

where the subscript i = i ( p , k ) { j = j(q, Z)} is the interface in the model structure where the p{q)th strip modeling the k{Z}th thick conductor lies. The elements of the spectral Green’s matrix in (4) can be obtained using the algorithm shown elsewhere [20, 211.

The capacitance matrix is computed by impos- ing potential unity on those strips which model the kth polygonal conductor, and potential zero on the rest. Thus, the elements of the k-th column are numerically equal to the total charge on each set of thin strips. Due to the normalization condition imposed on the basis functions, capacitance coefficients of the kth column are obtained as:

where M, is the total number of strips used to model the Zth conductor in the original structure, and N,, the total number of conductors in the original structure.

3. NUMERICAL PROCEDURE

It is important to note that the efficiency of applying Galerkin’s method in the spectral domain basically lies in the fast and accurate computation of the spectral integrals. Although this fact is relevant for the analysis of typical lines (with no more than three conductors), it becomes crucial in the present case, since the model structure may involve a great number of conductors. Note that the number of integrals are quadratically dependent on the number of conductors. A first way of reducing the numerical computations is to exploit all the symmetries of the elements of the Galerkin matrix to determine the minimum number of integrals to be computed. This minimum number depends on the magnetic and dielectric properties of the layers, as well as the geometrical symmetry of the conductors. The following notation will be used: 1;;: for the spectral integrals of (4) and RfPp“n for the part of these integrals extended from zero to infinity.

Taking into account the characteristics of the substrates, and assuming that the nature of the layered medium is given by the most complex layer, the following relations hold:


0 Computation of matrix [ C ] :

--Lossless iso/anisotropic dielectric layered medium:

In this case, only N X (N + 1)/2 integrals must be computed from 0 to rn (N is the order of the Galerkin matrix, super- script * denotes complex conjugate).

--Lossy iso/anisotropic dielectric layered medium:

where N x ( N + 1)/2 integrals must be performed from --oo to rn.

0 Computation of matrix [C,,]

--Lossless nonmagnetic and isotropic magnetic layered medium: This case is equivalent to the lossless iso/anisotropic dielectric medium case [20], and the above relations apply.

-Lossless gyromagnetic layered medium:

N x ( N + 1)/2 integrals, from --oo to 00,

are to be computed.

--Lossy gyromagnetic layered medium: Due to the nonreciprocity of the media, and since Gij(k,) are complex functions, all the spectral integrals must be computed.

The numerical computation of the spectral inte-

grals is carried out using the following scheme:

where the functions G;(k,) represent the asymptotic behavior of the Green's functions, deter- mined to be:

(12)

In (11) and (12), the superscripts + and - refer to the upper and lower layers, respectively, adja- cent to the ith metallized interface of the model structure. If the capacitance matrix p.u.l., [C], is to be computed, the diagonal elements of the permittivity tensor in (12) will include an imagi- nary part to account for dielectric losses. When computing matrix [C,,] for magnetic media, the elements of the tensor permittivity in these ex- pressions are obtained from the Polder's tensor, as shown elsewhere [20].

The constant u in the integrals (9) is suitably chosen to reach a good tradeoff between accuracy and CPU time. As a general criterion, we assume u - 0.5/{Re(Sh)},in, where {Re(Sh)},, is the minimum value of the product between the height and the module of parameter S-see expression

Quasi- TEMAnalysis of Polygonal Conductors 367

(ll)-for each layer in the model line. We have not used any asymptotic behavior for the Green's functions 6 i , j ( k x ) corresponding to different metallization levels, since its convergence is quicker than the convergence of GJk, ) . Thus, the two first integrals appearing in (9) are readily computed since the first term is a definite integral that can be computed using a simple Gauss-Legendre quadrature and the second one (extended to infinite) goes quickly to zero. The main numerical problem and the most consuming CPU part is then related to the proper computation of the tail integrals. These integrals [i.e., the third term in (911 can be generically expressed as:

where a and b stand for the semiwidth of the strips, and s for the difference between the ab- scissas of its centers. This latter integral presents two main problems: (1) very slow convergence, and (2) numerical problems related to the computation of high order Bessel functions. The first problem can be treated using a similar scheme as that proposed in ref. 22 employing analytical pre- processing as well as asymptotic techniques. The second problem can be overcome by developing recurrence formulas to calculate the integral tails for high orders (greater than 1) of the Bessel functions. Thus, the following recurrence formulas have been obtained, after performing an inte- gration by parts and employing the recurrence formulas for the Bessel functions together with the relations for their derivatives:

+jsA;-l - eb A"'] (rn # 1) 2(rn - 1)

1

-.(I + 2(n 5 + 1) )A;+' ] ( n f -1) (14)

with

( = n - m + 3

Based on (141, we have developed a simple algorithm to obtain recurrently any spectral integral tail A;, from the value of four initial integrals; namely, A:, A ; , A:, and A ; . These initial integrals are efficiently computed following the scheme proposed in the Appendix of ref. 22. Once these initial integrals are performed, the CPU time required to compute the remaining tails becomes negligible. In consequence, the total CPU time used to analyze the structure does not change considerably when the total number of basis functions is increased. It should be noted that a similar recurrence scheme could be used for a different set of basis functions.

4. NUMERICAL RESULTS

Based on the numerical procedure illustrated in the previous section, a Fortran computer code has been developed to analyze a transmission line like that shown in Figure la. The numerical data shown in this section have been computed in an Apollo 9000-730 HP workstation, with four significant digits. Therefore, the CPU times refer to this machine under these conditions.

First, and in order to validate our M-strips model, Table I shows the convergence of the capacitance values when the number of strips is increased for a rectangular ( t / w = 0.5) microstrip in vacuum. This structure has been analyzed previously in ref. 12 by a reliable space-domain analysis. Our computed data (obtained using five basis function for each thin strip) con- verge monotonically, although we observe a slight discrepance (- 0.3%) between our data computed using 30 strips and the most accurate da- tum in Table I11 of ref. 12. Nevertheless, an acceptable result (error - 1.5%) is obtained when


TABLE I. Capacitance for a Rectangular Perfect Conductor in Vacuum over a Ground Plane Obtained Using Different Number of Strips

2a

l a

Number of

Strips

2

5

10

15

20

25

30 ~

Data of [12]

Capacitance (F/m)

4.3462 e-11

4.4468 e-11

4.5026 e-11

4.5136 e-11

4.5189 e-11

4.5218 e-11

4.5237 e-11

4.5377 e-11

the rectangular conductor is modeled by five thin strips. Clearly, a great number of strips in our model would be only necessary in those cases with high t / w ratios. A second comparison is reported in Table 11, which shows our computed data together with the static values presented in ref. 18 for the modal normalized propagation constants, P / k , (where k , is the vacuum wave number), and the modal impedances for two coupled microstrips on a dielectric layer. Five basis functions are used to expand the charge distribution on each thin strip. A six-strip model (for each thick strip) provides a satisfactory convergence for the propagation characteristics of both modes. Our results agree acceptably with those reported in Figure 8 of ref. 18 (these latter data have been extracted from a figure, and may be affected by certain error).

Although in most cases strip-like conductors can be efficiently characterized employing a few thin strips model (namely, no more than three thin-strips), this model could not be sufficient in those cases where edge coupling effects are significant. This effect is analyzed in Table 111, which

shows the capacitance values of the same structure as in Table II for two different values of the ratio s /w. As can be seen in Table 111, the percentage variation between the results obtained with two and five strips amounts to 15% when coupling effects are specially significant (i.e., odd mode with s / w = 0.3). On the contrary, this dis- crepancy remains lesser than 1% for the even modes in all cases. Thus, modeling coupled strips could require the use of more than two thin strips for each thick conductor.

A comparison with a polygonal-shaped conductor is provided in Table IV. We have computed the values of characteristic impedance and propagation constant of a hexagonal cross-section wire on an isotropic layer previously analyzed in ref. 18 (static results for this line have been interpolated from Fig. 9 of this reference). We have modeled the hexagonal wire by seven thin strips. The Val- ues in Table IV have been computed using a different number of basis functions and they show a very good convergence. The number of integrals to compute increases quadratically with respect to the number of basis functions employed. This behavior should also be expected regarding CPU time. However a linear increase is achieved owing to the use of recurrence formulas.

In the following, some novel results will be presented for lines with polygonal cross-section conductors. First, Figure 2 shows the transmission line characteristics of a single microstrip on a gyromagnetic layer. The values of the effective magnetic permeability, p,, , and the characteristic impedance, Z , , of the line have been plotted versus the frequency. Three different shapes of conductor cross-section have been analyzed: rectangular (cp = 909, trapezoidal (cp = 135"), and inverted trapezoidal (cp = 45"). The strip thickness is assumed to be 20% of the rectangular strip width. We have also plotted the experimental values of peff reported in ref. 23 for the same line, although the thickness of the strip corresponding to these experimental values is not spec- ified. A good agreement is observed with the experimental values for the effective permeability throughout the frequency range. The effective permeability (defined as L/L,; Lo represents the vacuum inductance of the line) does not differ appreciably for the various cases shown in Fig- ure 2, since the changes in the values of L and Lo due either to the thickness or to the edge shapes are generally similar. On the contrary, the characteristic impedance, 2, = m, does de-

Quasi-TEMAnalysis of Polygonal Conductors 369

Even mode

TABLE 11. Normalized Propagation Constants and Modal Impedances for Two Coupled Microstrips on an Isotropic Layer, as Studied in Figure 8 of Ref. 18

Odd mode I Number I C.P.U

W S W LI 1 -1- h _I- *.- A 1

Zo(Q) 32.15

31.83

31.70

31.64

31.60

of Strips Time (sec.)

2 0.3

3 0.6

4 1 .o 5 1.6

6 2.3

P e I4 2.635

2.628

2.625

2.623

2.621

38.0f .5 2.60f.02 2.28f .02

39.60 2.328

39.38 2.314

39.30 2.308

39.23 2.306

39.20 2.305

31.7 f .5 Data from [18]

Strips

2

C J ~ O Co/~a CJEO COl~a 25.074 27.279 21.503 72.884

Our data have been obtained using a different number of strips to model the conductors. h = 0.635 mm, w = 1.0 mm, s = 1.0 mm, t = 0.3 mm, E , = 9.8.

TABLE 111. Modal Capacitances of the Same Structure as in Table I1 for Different Values of the Ratio s / w

II I U

(%) I 0.4 1 0.6 11 0.3 15.

The percentage variation (%) between the results for two and five strips are shown in the last row.

pend on the shape of the strip since the changes in L and C are opposite. The thick strip was modeled using two, three, and four strips in the rectangular, trapezoidal, and inverted trapezoidal cases, respectively. Each curve of Figure 2 was obtained from 50 values in the frequency range,

and using three basis functions for the charge density on each strip. The CPU time required for all these computations was 4.9 seconds for the rectangular case, 10.6 seconds for the trapezoidal case, and 18.5 seconds for the inverted trapezoidal case.

As a second example, we analyzed an MIS slow-wave structure with a single perfect strip in the range of validity of the quasi-TEM approach. Figure 3 shows the computed values for the slow- wave factor and the attenuation constant for three different cross sections of the microstrip: rectangular, trapezoidal, and inverted trapezoidal. The values of the characteristic impedance of these lines are presented in Figure 4. The field in these structures is highly concentrated under the conductor width adjoining the substrate. Therefore, only the inverted trapezoidal case differs appreciably, since it has a different width conductor adjoining the substrate [l]. The curves in Figures 3 and 4 have been plotted using four strips to model the thick conductor in every case, and three basis functions for the charge distribution


1.2054

1.2054

TABLE IV. Quasi-TEM Normalized Propagation Constant and Characteristic Impedance for a Wire as Studied in Figure 9 of Ref. 18

124.75 0.4

124.70 0.5

2a -

Basis Functions

in each strip (sec.)

2

3

1.2054

1.2054

h = 3 mm, H = 4.5 mm, a = 0.825 mm, E , = 4.

on each strip in the model line. Each structure was analyzed in 100 points, consuming about 79 seconds CPU time (0.79 seconds/point).

To demonstrate the potential applications of the implemented method, Figure 5 shows the dispersion curves for the two fundamental modes of a line composed by two identical hexagonal cross-section perfect conductors, embedded in a ferrite-dielectric composite medium. The ferrite medium is assumed to be fully saturated and biased by the longitudinal application of an external DC magnetic field. The line has been modeled using seven strips for each hexagonal conductor and five basis function for the charge distribution. All the computations to obtain the characteristic parameters of the line at a specific operation frequency requires about 4 seconds.

5. CONCLUDING REMARKS

Multilayer and multiconductor lines with perfect and arbitrary polygonal cross-section conductors have been analyzed under the quasi-TEM approach, by extending the model previously proposed in ref. 19. This model could be also extended to deal with actual conductors, provided there is a strong skin effect regime. Following this

I I I I I I

2 0 . 7 %

0.6

0.5

9 10 1 1 12 5 6 7 8 Freq( GHz)

80

70

60

50

e I N"

Figure 2. Effective magnetic permeability peff, and characteristic impedance Z, versus frequency, for a microstrip with rectangular, trapezoidal, and inverted trapezoidal cross-sections, on a partially magnetized garnet substrate: h = 0.25 in., w / h = 0.25, E , = 15.5, t / w = 0.2, 4 n M s = 1780 G, 4 n M , = 780 G. ( 0 ) Experimental data from ref. 23.

Quasi-TEMAnalysis of Polygonal Conductors 371

n

2 a Y z;1

W

c)

Lc

B 5i 0

Freq(GHz)

n 1 !3 0.1 b

a a 0.01 w

0 1 0 - ~ -= a 10-4 2

a2

10-5 3 1 o-6

Figure 3. Slow wave factor P / k , and attenuation constant for an MIS structure with rectangular, trapezoidal, and inverted trapezoidal cross-sections where h, = 200 pm, ha = 0.5 pm, t = 5 pm, w = 80 pm, (T = 0.05 (0 - mrnl-', el,, = 12.9, and E ~ , ~ = 4.

E? W

2 "" N v

22

20

18

16

14

12

10

' . r n I- (0=43-

c V

15

10

5

0

N v

E CII

1 o - ~ 0.01 0.1 1 10 Freq(GHz)

Figure 4. Characteristic impedance for the same structures as those presented in Figure 4.

model, each polygonal cross-section conductor is modeled applying a set of infinitesimally thin strips at the same potential. The multilayered medium is considered nonrestricting on those dielectric, magnetic, and gyrotropic properties compatible with the quasi-TEM approach. Thus, the possibil- ities of the quasi-TEM approach to compute the transmission line characteristics of lines are ex- tensively exploited. Owing to the planar characteristics of the model structure, analysis has been

accomplished by applying Galerkin's method in the spectral domain.

To enhance the efficiency of the numerical procedure analysis, special attention has been paid to those critical points in the computational aspects. Starting from the symmetrical properties of Green and Galerkin matrices, the minimum number of spectral integrals necessary to be computed has been deduced. In addition, recurrence formulas have been developed to compute the

372 Plaza, Mesa, and Horno

mode 1 ,mode 2

h

l o E 1 E 0.1 ;3i = 0.01 - 1 0 - ~ 1 0-4 2

2 10-’4

6 Q ) 10-

1 2 3 4 5 6 7 8 9 1 0 Freq( GHz)

Figure 5. Dispersion curves and attenuation constant for the transmission modes of a line with two identical hexagonal conductors in a multilayered lossy fully saturated gyromagnetic medium where: h , = 200 pm, h , = 200 pm, d = 20 pm, a = 58 pm, 4 r M , = 870 G, H, = 2000 Oe, AH, = 50 Oe, E,, = 13, er2 = 2.32.

integral tails concerning high order basis functions (greater than 1). This latter procedure makes the CPU time required by the implemented numerical algorithms less sensitive to an increase in the number of basis functions.

Original results have been presented, and several lines have also been analyzed to compare with published data. An acceptable agreement was achieved in all cases.

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BIOGRAPHIES

Gonzalo Plaza was born in Cidiz, Spain, in November 1960. He obtained the licenciado degree in June 1986 from the University of Seville, Spain. He is currently an assistant professor in the De- partment of Applied Physics at the Uni- versity of Seville (Spain). His present in- terests lie in the area of electromagnetic propagation in planar lines with general

anisotropic materials.

Francisco Mesa was born in Cidiz, Spain, in April 1965. He received the licenciado degree in June 1989 and the doctoral degree in December 1991, both in physics, from the University of Seville, Spain. He is currently assistant professor in the De- partment of Applied Physics at the Uni- versity of Seville (Spain). His research interest focuses on electromagnetic prop-

agation/radiation in planar lines with general anisotropic materials.

Manuel Homo (M’75) was born in Torre del Campo, JaCn, Spain. He received the degree of licenciado in physics in June 1969, and the doctoral degree in physics in January 1972, both from the University of Seville, Spain. Since October 1969 he has been with the Department of Elec- tricity and Electronics at the University of Seville, where he became an assistant

professor in 1970, associate professor in 1975, and professor in 1986. He is a member of the Electromagnetism Academy of MIT (Cambridge, MA). His main fields of interest include boundary value problems in electromagnetic theory, wave propagation through anisotropic media, and microwave integrated circuits. He is presently engaged in the analysis of planar transmission lines embedded in anisotropic materials, multiconductor transmission lines, and planar slow-wave structures.

Quasi-TEM analysis of reciprocal/nonreciprocal planar lines with polygonal cross-section conductors

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