Computation of Capacitance Coefficients of Rectangular Thick Patches on Multilayered Substrates

Gonzalo Plaza, Francisco Mesa, and Manuel Horno

Microwave Group, Department of Electronics and Electromagnetism, Avd. Reina Mercedes sln 41 01 2 Seville, Spain Received Janualy 5, 1996; revised May 13, 1996.

ABSTRACT

This article presents an alternative method to compute the capacitance coefficients of a system of rectangular thick patches on a multilayered substrate. Following the M-strips, previously developed to analyze infinite length transmission lines, the conductors are modeled by a set of zero-thickness plates at the same potential. The capacitance coefficients of the model structure are obtained after applying the Galerkin’s method in the spatial domain. A suitable approach for computing the static spatial Green’s function is also presented. 0 1996 John Wiley & Sons, Inc.

INTRODUCTION

In MMIC technology, lumped and distributed ele- ments have to be integrated together to build functional devices. As is well known, the simplest way to simulate a capacitor in this technology is by means of a rectangular microstrip patch. Thus, for CAD purposes, the development of efficient and accurate computer codes to analyze lumped capacitors is of great interest. Moreover, these codes could be used to compute the parasitic capacitances in high speed ICs (where the para- sitic capacitances of the interconnecting lines govern the device switching speed and delays) and could also be used in those practical situations, at low frequencies, in which planar microstrip cir- cuits may be accurately represented by lumped elements [l].

Zero-thickness patches have been previously analyzed by means of different techniques; for example, by solving the integral equation for the

free charge on the patches using the subareas model [2, 31, or by the method of moments using entire domain functions as basis functions and delta functions (point-matching) as test functions [4]. In ref. 5, zero-thickness rectangular patches on anisotropic substrates were treated using vari- ational techniques in the spectral domain. Finite- element methods have also been employed to compute the capacitance of zero-thickness con- ductors on a dielectric sheet in ref. 6, and itera- tive techniques applied to the computation of the capacitance of MMIC patches are presented in ref. 7. More recently, in ref. 8, several geometries of zero-thickness patches on multilayered sub- strates have been studied using the so-called MWM (modified Wolff model). However, in many practical situations, an accurate determination of the capacitances of rectangular patches requires taking into account the thickness of the conduc- tors, especially when coupling capacitances be- tween patches or lines have to be computed. In

International Journal of Microwave and Millimeter-Wave Computer-Aided Engineering, Vol. 6, No. 6, 445-453 (1996) 0 1996 John Wiley & Sons, Inc. CCC 1050-1S27/96/060445-09

445

446 Plaza et al.

spite of this, there are few works in the literature dealing with the computation of the capacitances when the thickness of the metallizations is consid- ered. For example, in ref. 9 capacitance coeffi- cients for VLSI metallization interconnecting thick lines are obtained by solving the integral equations using a piecewise linear finite-element technique.

In the present article, we propose a model to analyze rectangular thick microstrip patches on a multilayered anisotropic dielectric substrate. The thick patches are considered in this model as a set of zero-thickness plates at the same potential, reducing the analysis of the original three-dimen- sional (3D) structure to the analysis of a zero- thickness multipatch structure. The capacitance coefficients are then computed by solving the integral equation for the free charge in the model structure via the Galerkin method in the spatial domain. The corresponding spatial Green’s func- tion is computed following a suitable approach based on the spectral Green’s function for the multilayered substrate.

Note that because our model reduces the 3D surface charge distribution to several two-dimen- sional (2D) surface charge distributions, a spec- tral domain approach similar to that reported in ref. 5 could be applied. However, the computa- tion of the reaction spectral integrals appearing in ref. 5 requires performing double integrals from --oo to involving four Bessel functions. Due to the strong oscillatory behavior and the slow convergence of the integrands, the numeri- cal computation of these reaction integrals is very time-consuming, even though suitable asymptotic treatments were used. In our experience, the nu- merical computation of the reaction integrals is considerably enhanced by applying a spatial do- main scheme. In addition, the use of entire do- main functions for both basis and testing func- tions, after optimizing the spatial integrals, signif- icantly reduces the order of the linear system of equations to be solved (when compared with other techniques using piecewise basis functions) and makes the whole procedure fast and accurate.

ANALYSIS

Our goal will be the computation of the static capacitance coefficients of a set of thick rectangu- lar conducting patches printed on a multilayered anisotropic substrate as shown in Figure 1. The layers as well as the ground plane are of infinite

Figure 1. Rectangular thick patches on an anisotropic multilayered substrate.

extent and the inhomogeneous medium can be either isotropic or uniaxially anisotropic (provided that the optical axis is normal to the dielectric interface).

The analysis of the structure is based on the extension of the M-strips model (presented by the authors in refs. 10 and 11 to analyze general 2D multilayered and multiconductor transmission lines with finite thickness conductors) to a planar system of rectangular conductors (3D problem). Following the model, each rectangular thick patch is modeled as a set of Mk parallel rectangular zero-thickness plates at the same potential (see Fig. 2). As shown in Figure 2, the top plate of the set is at the top of the patch and the plates are evenly spaced.

Integral Equation Formulation

The integral equation for the model structure can be written in terms of the following double convo-

Figure 2. model.

Original rectangular thick patch and its

Capacitance of Rectangular Thick Patches 447

lution product: 9:; n , m ( X , Y )

where v’ is the potential of the ith zero-thick- ness plate modeling the Zth thick patch; G[ij(x, y ; x’, y’ ) is the static spatial Green’s func- tion for field points, (x, y , z) , on the ith zero- thickness plate modeling the Zth thick patch and source points, (x’, y ’, z’), on the j th zero-thick- ness plate modeling the kth thick patch; SF and pF(x, y ) represent the surface and free-charge density respectively on this latter plate; Mk is the number of zero-thickness plates used to model the kth thick patch in the original structure; and Np is the total number of thick patches.

As mentioned previously, all zero-thickness plates modeling the same thick patch must be at the same potential:

vk = V k , i = l . . . M k , k = 1 ,... Np ( 2 )

where Tn(.) are the Chebyshev polynomials of the first kind; (xf, y:) are the coordinates of the cen- ter of the Ith thick patch, and wL,wj are the patch dimensions. These basis functions have been previously employed in ref. 5 and although they do not present the right singularity at the comers, they do provide the proper R1/* singularity at the edges of the zero-thickness plates [121. So, a small number of basis functions would be required to achieve accurate results for many practical structures where the contribution of the edge singularities is more significant than that of the corner singularities. Note that the above set of basis functions verify the following normalization condition:

The resulting eq. (1) with boundary conditions (2) is solved applying the Galerkin method in the spatial domain.

where “so,n is the Kronecker delta. After applying the corresponding inner prod-

ucts required by the Galerkin method, the follow- ing spatial integrals are obtained:

rl;i,nm k ; j 7 p q = jLF $,,,(x, y ) [ // G[;’;(x,y;x’,y’)

X 9 : ; n , m ( X ’ , y ’ ) h ’ d y ’ (6)

Galerkin Method

procedure, the charge density on the ith zero- To solve the above equation using the Galerkin

thickness plate modeling the Zth thick patch is expanded into basis functions:

Sf

F:x F [ y

n = l m = l

/ I and the linear system to determine the coeffi- cients aXP,q can be written in matrix form as [ r] A = B, where the elements of vector B are the potentials of the zero-thickness plates ”;“so, S o : q , [TI is a matrix whose elements are the spatial

P ! ( x , Y ) = C C a i ; n , m ( P i ; x , n ( X ) ( P t ; y , r n ( Y ) ( 3 )

where ~ p / ; ~ , ~ ( x ) and 9;; ,,,(y) denote the basis functions, F i f x is the highest Order Of the integrals [eq. (6)] and the elements of vector A functions 9; x , n ( x ) (Pi’, y , m ( Y ) ) , and ‘I; n, m are the are the unknown coefficients a,k; n m . Finally, vet- unknown coefficients to be determined. As can be seen in eq. (3), the basis functions have been chosen as the product Of functions, each Of

them varying only with respect to One SPecificallY, we have chosen the following Set of basis functions:

tor A can be formally written as A = [r]-iB. The capacitance matrix [C] can be easily com-

puted by setting the potential of the set of zero- thickness plates modeling one thick patch to 1 (say kth) and the remaining ones to 0. Consider- ing now the normalization condition [eq. (511, the

448 Plaza et al.

kth column of matrix [C] is obtained as:

MI

C,, = C U ~ ; ~ , ~ , 1 = 1 ,... Np (7) i = l

Spatial Green’s Function Approach One of the most important points in the practi- cal application of the integral equation method turns out to be the computation of the spatial Green’s function G(x , y , z ; x‘, y ’ , 2‘1. For the multi-layered structure shown in Figure 1, G(x, y , z ; x’, y ’ , z ’ ) can be expressed as a series by means of the image method [13]. However, the computation of this series requires a considerable analytical effort, specially for substrates with more than two or three layers.

The procedure we propose next will avoid the above drawback, since it provides an accurate approach of the spatial Green’s function as a finite series with no more than five or six terms regardless of the number of layers of the sub- strate. Let us consider this approach for the generic structure shown in Figure 1. Taking Fourier transforms in both x and y variables and solving the Poisson equation for field and source points in the same isotropic medium of permit- tivity E , E ” above the patch interface, the resul- ting spectral Green’s function can be written as follows:

(8)

with y = d w , and k,, k, being the spec- tral variables. The function ( 1 / 2 ) y ~ , e - ~ ~ * - ~ ’ ~ is the spectral representation of the Green’s func- tion in an isotropic homogeneous medium, and G b ( y ) for the spectral Green’s function with source and field points in the same interface where the patches are placed. Owing to the cylin- drical symmetry of the dielectric properties of the substrate, &( y , z , z ‘1 depends on k, and k, vari- ables only via y . Therefore, we can readily obtain the spectral Green’s function, Gb( y ) , by making use of the algorithms developed in refs. 14 and 15

for analyzing 2D multilayered structures and re- placing the 2D spectral variable in 14 and 15 with the 3D spectral variable y = d w .

By expressing the spectral Green’s function as an infinite series of exponentials divided by the spectral variable, as shown in ref. 16 using the spectral images representation, we can obtain a good representation of eq. (8) by approximating d b ( y ) with a finite series of exponentials of the same type. The coefficients and exponents of the exponentials can be obtained via either the Prony’s method [171 or the matrix pencil method “1. So, we can write:

where ai and pi are, in general, complex num- bers with Re( pi) > 0, NI is the number of terms chosen to approach the infinite series, and Eb = ,/=, with cZz and E,, = E,, being the compo- nents of the permittivity tensor of the layer just below the patches. In this work, we have used Prony’s method interpolating with no more than two or three exponentials (i.e., four or six match- ing points, respectively). In most cases, this num- ber or exponentials suffices to get accurate re- sults. As a general rule, we use the interval (0, - l /h) as the interpolation interval, where h is the height of the top layer of the substrate.

Combining now eqs. (8) and (9), we can write the following approximate representation of the spectral Green’s function of our problem:

1 + -e-Y 12-2’1

2Y%

where E, = ( E , + E b ) / 2 .

Considering the following relation between the spectral and spatial terms of the expansion of the Green’s function:

1 (11)

where r = d ( x - x’)’ + ( y - y’)’ + d 2 , the

Capacitance of Rectangular Thick Patches 449

corresponding spatial representation of the Green’s function can be readily found to be:

with:

2 2 Y g = d ( x - x‘) + (y - y r ) + ( z - Z ’ l 2

rl = j / ( x - X r I 2 + ( y - y r y + (2 + zr)2

di = J ( x -xr )2 + ( y - y y + ( z + z r + (13)

As a general rule, no more than four terms are required to represent G,(-y) accurately (four sig- nificant digits, at least in most cases), and there- fore the final expression (12) consists of usually less than six terms.

Using the above approach to obtain the spatial Green’s function, the integrals in eq. (6) can be accurately evaluated, as will be shown later. In the computation of these integrals, it is important to remark that we can get a significant speedup by treating properly the singularities of the inte- grand. These singularities appear when we com- pute the potential at a plate due to itself (self- reaction). They appear when the source and field points coincide [ro = 0 in eq. (1311. A simple but efficient treatment of these singularities was found to be the use of Chebyshev quadratures of difler- ent pan’ty when computing the inner and convolu- tion products appearing in eq. (6). In this way, the edges singularities are directly considered and we avoid the coincidence of equal source and field quadrature points. Specifically, the computation of the convolution product in a self-reaction inte- gral requires using quadratures of about 200 points in both x and y directions to get sufficient accuracy ( - 0.2%). This number of points de- creases drastically when the singularity is not present (cross-reaction, i.e., integrals involving two different zero-thickness plates), especially if we are dealing with two zero-thickness plates that belong to different thick patches (in this case five quadrature points suffices). In any event, the inte- grand of the inner products does not present singularities and they can be computed accurately using 5 to 10 quadrature points.

NUMERICAL EXAMPLES

As a first example to check the validity of the above procedure, we computed the capacitance of a zero-thickness squared patch printed on a di- electric sheet. To study the suitability of our spatial Green’s function approach [eq. (12)], in Table I we compared our result with those ob- tained in ref. 19 via the Galerkin method in the spectral domain using the exact spectral Green’s function and the same basis functions. As in ref. 19, we employ three basis functions in both x and y directions to expand the charge density on the plate (which means a total of nine basis func- tions). It can be seen in Table I that the use of just four terms to approximate the function Gb(y)--see eq. (9)-provides a complete agree- ment with the data of ref. 19.

We consider now the convergence of the model. Because, to the authors’ knowledge, there are no analytical results for thick patches for which to compare, we employ the same structure as in the above study but now assume a thick patch. Thus, Figure 3 shows convergence of the model with respect to the number of zero-thickness plates used to model the thick patch. The results of this figure were obtained using four terms to approxi- mate the function G b ( 7 ) and 5 X 5 basis func- tions to expand the charge density on each of the M zero-thickness plates (for M = 1 the zero-

TABLE I. Normalized Capacitance of a Square Thick Patch on a Dielectric Sheet Using a Different Number of Terms [ N = NI + 1 see Eq. (911 to Approach the Spectral Green’s Function G,(y

E , = 9.6, w, = wy = 1 mm, h = 1 mm, and t = 0 mm.

450 Plaza et al.

3 x 3

F x F 4 x 4

5 x 5

1

1 2 3 4

17.075 18.803 18.981 19.050

17.085 18.808 18.987 19.061

17.087 18.810 18.989 19.063

.01

Ref.[9]

Ref. [20]

Ref. [21]

1 2 3 4 5 6 7 0 9

Number of plates M

Figure 3. Variation percentage 100 X [C(10) - C(M)]/C(lO) for the patch shown in Table I versus the number of zero-thickness plates, M , used to model a patch with t = 0.1 mm.

18.025

18.219

19.6

thickness plate is located at the bottom of the original thick patch). In Figure 3, C ( M ) is the patch capacitance computed using M zero-thick- ness plates. As can be seen in Figure 3, the percentage difference between M = 1 (zero- thickness case) and M = 2 is about 2.5%, whereas the difference between M = 2 and M = 10 is less than 1%. This fact points out that the use of only two plates to model the patch provides an accept- able value of the capacitance. The computed data also show a good convergence and suggest that no more than three zero-thickness plates are neces- sary to get errors below 0.25% (with respect to our best result corresponding to M = 10). The CPU times (evaluated on an HP Apollo 9000/730 workstation) range from 16 seconds for M = 1 to 35 seconds for M = 10. These CPU time values show that the most time-consuming part of the procedure is related to the computation of the self-reaction terms, and so the addition of more zero-thickness plates does not significantly in- crease the total CPU time. Moreover, the short times required in the above computations suggest that the present method could be efficiently used as a CAD tool.

As a second example we analyzed a conducting bar embedded in a dielectric medium over a ground plane. This example could also be used to compute the parasitic capacitance of an intercon- necting finite line in a VLSI circuit. As shown in Table 11, the capacitance of this line is computed

TABLE XI. Parasitic Capacitance of a Finite Intercon- necting Line Embedded in a Dielectric Medium (E, = 3.9) Filling All the Space

M I

M: number of plates to model the thick line; F X F : number of basis functions for the free charge density on each zero-thickness plate. Dimensions: w = 5 pm, I = 100 wm, t = 1 pm, and h = 2 pm.

for several basis functions and a different number of plates to model the original thick line. The comparison of our results with those obtained by other investigators shows an acceptable agree- ment. As in the previous example, our results corresponding to only one zero-thickness plate located at the bottom of the original thick bar differs by a 10% from our best result, whereas the use of two plates differs by only 1.5%. Thus, the use of two plates, even with considerably thick conductors ( w / t = 5 in this example), still pro- vides accurate enough results.

Table I11 shows the results for two coupled rectangular finite-length lines. These results are computed for different numbers of basis func-

Capacitance of Rectangular Thick Patches 451

TABLE 111. Capacitance Coefficients of a Pair of Couple Interconnecting Lines Embedded in a Dielec- tric Medium (e, = 3.9) Filling All the Space

F x F

M

I 1 l 2 1 3 1 4

Cn + (fF.) in this work

33.689

Cll + ClP (fF.)

M : number of plates to model each thick line; F x F : number of basis functions for the free charge density on each zero-thickness plate. Dimensions: w = 5 pm, I = 100 pm, f = 0.5 pm, s = 5 pm, and h = 0.8 pm.

tions and plates to model the thick lines. The comparison of our results with some previously reported data show an acceptable agreement (- 2.4%), although our numerical results are consistently smaller than those computed by other investigators. The use of only two plates and 4 X 4 basis functions again suffices to obtain an accurate enough result (percentage difference - 1% with respect to our best value), whereas the use of only one zero-thickness plate (i.e., neglecting the thickness) provides a percentage difference of about 5%.

As a final example, in Table IV we present new results for a thick patch printed on an anisotropic multilayered substrate (to our knowledge, there is no previous data regarding this structure). The capacitance of the patch is computed using a different number of zero-thickness plates to model

TABLE lV. Capacitance of a Square Thick Patch over a Multilayered Anisotropic Substrate

F x F

1 x 1

1 x 1

1 x 1

2 x 2

3 x 3

4 x 4

- M

2

3

4

4

4

4

-

-

Capaci tance(fF)

47.9934

48.0218

48.0297

48.6558

48.6 876

48.6964

-1 I

11.8

15.8

20.3

26.4

32.9

39.6

M : number of zero-thickness plates used to model the patch; F X F : number of basis functions for the charge density on each zero-thickness plate. e r , , = 10, er, = 2.5, E , ; ~ ; ~ ~ = E , , ~ ; ~ ~ = 9.4 and E , , ~ , il = 11.6. Dimensions: w, =

wy = 200 pm, t = 5 pm, h , = h, = h, = 100 prn.

the patch until convergence is achieved (three significant digits). Then the number of basis func- tions is increased until convergence is obtained once again. The CPU time on an HP Apollo 9000/730 workstation is also shown. This time is on the order of 1 minute, which can be consid- ered a short time in the framework of the analysis of 3D structures with thick metallizations.

CONCLUSIONS

In this article we have presented an alternative spatial-domain analysis of rectangular thick patches on a multilayered iso-anisotropic sub- strate using the M-strip model. Thus, this model previously developed by the authors to analyze transmission lines with arbitrary cross-section conductors (2D problem) has been now extended to treat problems of finite-length metallizations (3D problem). The computation of the capaci- tance coefficients has been carried out by solving

452 Plaza et al.

the corresponding integral equation for the free- charge density via the Galerkin procedure using entire-domain basis functions. We have also pre- sented an accurate approach for the static spatial Green’s function starting from the static spectral Green’s function of the multilayered structure. The main advantages of the above method are:

1. The M-strips model reduces directly the analysis of the original structure with finite thickness patches to a simpler analysis of zero-thickness patches. Moreover, the use of more than one patch to simulate the thick conductor does not imply a significant increase in the computational effort.

2. The cases studied show the possibility of computing accurate enough results in short CPU times using a few zero-thickness plates to model the rectangular patches as well as a reduced number of basis functions for the free charge density. These facts are essen- tial for the model to be suitable for CAD purposes.

3. The approach presented revealed that the static spatial Green’s function expressed in terms of infinite series can be substituted with a finite series consisting of a small number of terms.

ACKNOWLEDGMENTS

This work was supported by CICYT, Spain (Proj. No. TIC95-0447).

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14. R. Crampagne, M. Ahmadpanah, and J. L. Guiraud, “A Simple Method for Determining the Green’s Function for a Large Class of MIC Lines Having Multilayered Dielectric Structures,” ZEEE Trans. Microwaue Theory Tech., Vol. 26, No. 2, Feb. 1978, pp. 82-87.

15. M. Horno, F. Mesa, F. Medina, and R. Marques, “Quasi-TEM Analysis of Multilayered, Multicon- ductor, Coplanar Structures with Dielectric and Magnetic Anisotropy Including Substrate Losses,” IEEE Trans. Microwave Theory Tech., Vol. 38, No, 8, Aug. 1990, pp. 1059-1068.

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Capacitance of Rectangular Thick Patches 453

20. T. Sakurai and K. Tamuru, “Simple Formulas for 21. A. E. Ruehli and P. A. Brennan, “Efficient Capaci- Two and Three Dimensional Capacitances,” IEEE tance Calculations for Three-Dimensional Multi- Trans, Electron. Deu., Vol. D-30, Feb. 1983, conductor Systems,” IEEE Trans. Microwave The- pp. 183-185. ory Tech., Vol. 21, Pt. 2, Feb. 1973.

BIOGRAPHIES

Gonzalo Plaza was born in CBdiz, Spain, in November 1960. He obtained the li- cenciado degree in June 1986 and the doctoral degree in March 1995, both in physics from the University of Seville, Spain. He is currently an assistant profes- sor in the Department of Applied Physics at the University of Seville (Spain). His present interests lie in the area of elec-

tromagnetic propagation in planar lines with general bian- isotropic materials.

Francisco Mesa was born in CBdiz, Spain, in April 1965. He received the licenciado degree in June 1988 and the doctoral degree in December 1991, both in physics, from the University of Seville, Spain. He is currently associate 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’7.5) was born in Torre del Campo, JaBn, 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- tronic and Electromagnetism at the Uni- versity 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.