An efficient integral method for capacitance extraction of multiconductor microstrip lines

A new integral method is developed to analyse multiconductor microstrip lines in quasi-static. The capacitance matrix of two and eight transmission lines has been successfully calculated by using our approach. The developed method is based on the generalised equivalent circuits technique and using transverse admittance operators. The obtained integral equations are solved using Galerkin’s method. This approach evaluates directly the accurate determination of capacitance matrix characterised by variational form. To improve the computation efficiency and the accuracy, the unknown charges are expanded in terms of compactly-supported wavelets. Thus highly sparse matrixes are generated and a significant reduction of computational time and memory storage are obtained. Numerical results are in good agreement with those in previous publications.


INTRODUCTION
With the growing speed and complexity of modern digital circuits and microwave devices, crosstalk and signal distortion become the dominant factors limiting the performance of microelectronic integrated circuits.A key step to optimize electrical performance of the integrated circuits is the extraction of the self and coupling capacitances of multilayer and multiconductor interconnects.To analysis this type of structures different methods has been reported in [1].These includes conformal transformation method [2,3], the finite difference method [4], and spectral domain approach [5].
Capacitance extraction is an important problem that has been extensively studied.Over the last decade some efforts have been made to accelerate the procedure [6] [7] [8] [9].Ongoing research at the present time is focused on devising methods which can be applied to more general geometries than the ones considered in the past and at the same time, improving the computational efficiency and accuracy of these methods.In the recent years, within new contributions, the measured equation of invariance on surface MEI method has been used in the capacitance matrix calculation of IC interconnects [10].This technique improved the computational efficiency through shrinking the boundary.The spectral domain approach combined with Galerkin method, variational form or iterative techniques is probably the most simple and widely used tool.This approach retains the simplicity of conventional moment methods and optimizes them by recasting all matrix elements into rapidly converting series [11].
In this paper, an original integral method is developed to calculate the capacitance matrix of multi conductor transmission lines.It is based on generalized equivalent circuit technique and using transverse admittance operators.The set of integral equations under consideration is solved by Galerkin method.Then it is demonstrated that matrix capacitance has variational form.
In first time, it is shown that this technique is simple and very efficient with an appropriate choice of basis functions such rooftop functions.In second time, a multiresolution approach is employed.Compactly-supported wavelets are used directly as trial functions to model the charge distribution on the strips.Numerical results will be presented for two dimensional structures and compared with other available data.It is shown that the application of the Discrete Wavelet Transform (DWT) highly sparse admittance matrixes and permits a significant reduction of computational time and of memory storage.

FORMALISM
The studied structures are composed of a number of conductors placed on one interface.We restrict ourselves to quasi-static domain although the present method can readily be extended to few dispersive domains.
Figure 1 shows the cross section of shielded coupled micro strip lines.There are N strips with arbitrary width of W i , air gaps of S i between them, located at x i (i = 1….N).The dimensions of the metallic shield are A and B = l 1 + l 2 , the thickness of the substrate is 1 1 with dielectric constant ε 1 , The dielectric substrate is assumed to be isotropic and homogeneous.Losses in the dielectric and in the conductors as well as the metal thickness are neglected.The structure is considered infinite in z direction.An efficient integral method for capacitance extraction of multiconductor microstrip lines Cross-sectional view of multiconductor transmission lines.

PRINCIPLES
The first point of this method consists in the determination of the equivalent diagrammt of a metallized interface on the transverse plane.The aim is to express the unknown charge density of the problem considering the potential as source excitation.
In the transverse resonance method (assuming propagation in y-direction), one considers transmission plane waveguide xoy connected by adjustable sources describing the electromagnetic state of the adjustings surfaces [12].In our work, the equation of resonance expressing the boundary conditions will be formulated by transverse admittance operators that binder the charge density ρ and the potential V [13] given by Eqn 1. (1) The adjustable virtual sources are the trial functions describing the electromagnetic state of the problem while considering the boundary conditions.These functions are going to permit the decomposition of ρ (or V).One will have the equivalent circuit given in Figure 2 verifying: jωρ = 0 on the isolant domain And the dual of jωρ, V' = 0 on the metal.
(2) Figure 3 Equivalent diagram with potential virtual source.
Otherwise, one can notice that this problem can be modelled considering the equivalent diagram where the virtual source translated the potential defines on the dielectric as shown in figure 3 2

.2. DETERMINATION OF THE ADMITTANCE OPERATOR
Let E y represents a longitudinal electric field (transverse plane xoz).according to the boundary conditions on the metal domain given by Figure 2, the charge distribution ρ is defined by the following relation:

E y E y − =
One can multiply Eqn 3 by jω.So we consider: In fact, we replace ρ by jωρ to verify the dimension equation given in Eqn 1.The current on the plane (y = 0) is generated by TM modes (ƒ k TM ) according to y-direction of the empty guide of the structure.So: (5) Where E → T is the transverse electric field [14].Taking in account Eqns 3, 5 and divE → = 0, we find for each mode designed by k, the following relation: • i designes the half space 1 and 2 indicated in Figure 2 • V k is the component potential decomposed on the TMy mode: • Y ik defined as the mode admittance given by Eqn 8: In Eqn 8: β denotes the propagation constant along the z-axis.
P ik is the propagation constant along the y-axis From Eqns 6, 7 and considering the contribution of all the modes ƒ k TM (k = 1..NB), we obtain: (10) Where Y ik is the short-circuit admittances (i = 1,2 : top and bottom of the shielding box) on the plane y = 0:

EXPRESSION OF THE CAPACITANCE MATRIX
Taking in account the equivalent circuit in the interface y = 0 for one strip given in Figure 2, the structure given in Figure 1 can be modelled with equivalent diagram shown in Figure 4: where : Y ˆ1 and Y ˆ2 are the short-circuit admittances (top and bottom of the shielding box) V i : is the known potential on the i-th strip V' i : dual potential on the insulator jωρ i : is the current on the interface defined by the charge distribution ρ i on the i-th strip.
Taking in account Kirchhoff laws we obtain: Assuming the charge distribution in terms of basis functions g n (i) (n = 1 ..NE) (14) x n (i) (n = 1..NE) are the unknown coefficients to be determined, and NE is the total number of the bases.Then Eqn 13 becomes:

1
Galerkin's technique is applied by multiplying the equation by the basis function.A set of linear algebric equations is obtained from integral equation.The matrix form is described by Eqn 16: (16) where: Thus we obtain a matrix form with the unkown vector[X] : [M] is a matrix with size (ns,ns) with ns = N*NE To obtain the expression of the capacitance matrix, we consider the free charge Q i in the i-th interface : D M : metallic domain (i-th strip) The elements of the capacitance matrix are given by: ( Therefore, we obtain a variational form of the capacitance matrix: . With an arbitrary excitation V 0 = 1 volt on the metal domains, we note: (25) In this paper, a static model is considered then β → 0 given in Eqn 8.

CHOICE OF SOURCES AND TRIAL FUNCTIONS
The modelling of a planar circuit consists of determining its behaviour by calculation and programming.Efficiency and accuracy of the computation can be improved by the choice of appropriate basis functions.
In our work different sets of basis functions have been tested, namely, wavelet bases, rooftop, U, and sinusoidal functions.In this paper, we choose compactly-supported wavelet and rooftop functions which verify the criterion established by [15], to avoid the appearance of spurious solutions, and which permit accurate results The rooftop expansion functions approach is mostly developed for complex structures.Therefore, this approach leads to a large and dense matrix [16].To increase the efficiency of such methods, the use of wavelet bases in multi-resolution approaches (MRA) is often proposed [17].The use of the wavelet in L2 has two advantages: No truncation of the wavelet is needed at the boundary, and the wavelet offers a complete basis in the region of interest.
A wavelet system consists of a mother scaling function φ(x) and a mother wavelet function ψ(x).These functions generate a family of basis functions .
Where n is the translation factor, and j is the resolution level [18].An approximate of the charge distribution at a resolution k can be written as the sum of two mutually orthogonal functions namely smooth (ρ s ,macroscale) and detail (ρ d ,microscale) components.We have: complex matrices arising in numerical solutions to integral equations [19].Then many matrix elements are negligible.A thresholding procedure is applied to set to zero all elements in the matrix whose magnitudes fall below the threshold.This threshold is adjusted to get an error of 1 percent or less in the results.A compression rate is defined as follows [20], [21]: (29 where NbZero is the number of elements set to zero in the compression matrix and ns is the size of the linear system given in Eqn 21.

SYSTEM WITH TWO STRIP CONDUCTORS
In order to assess its effectiveness, the proposed approach is applied for a two conductors structure N = 2.The normalised parameters' structure are the same as [22]: Since the developed technique uses located basis functions, the matrix given in Eqn 20 is sparse.However, a highly sparse matrix is generated when using wavelet functions.Figure 4 shows a colormap plot obtained by taking the elements of the matrix given in Eqn 20.This clearly demonstrates that the nonnegligible elements are located on or near the diagonal, which correspond to self-interactions or near-functions interactions.However, the use of wavelet functions allows a considerable reduction of nonnegligible elements.
Furthermore, as shown by table 1, for an error of 1% or less in the results, the compression rate is 40% with rooftop functions and 88% with wavelets trial functions.Thus, the use of wavelets expansion decreases the computation time by a factor of three and the required memory by a factor of six.
In addition, we verify that the charge distribution computed with our approach and a compression rate of 88% satisfies the boundary conditions (figure 5).

MULTICONDUCTORS STRUCTURE
In order to further prove the good performances of the presented method, we consider the application of the proposed approach for extracting the coupling capacitances of an eightconductors structure.The parameters of the structure are: N = 8 conductors, W 1 = … = W 8 = 1, l 1 = 16, S 1 = … = S 8 = 1, A = 175, l 2 = 100 and ε r =12.9.
The coefficients of the capacitance matrix (normalized to ε 0 ) computed with our approach are compared with those published by Oprea [23] reported by Amirhosseini in [22] and Table 2 shows that our results are in perfect agreement with those given by the references.The computing error is within 2%.

CONCLUSION
A new accurate quasi static integral method for extraction capacitance matrix of multiconductor transmission lines is presented.The method is based on the generalised equivalent circuits technique and using transverse admittance operators.This approach possesses the properties of rapid convergence.It has been validated by comparing our results with those published by other authors.
Taking full advantages of multiresolution analysis, it has been demonstrates that highly sparse matrix is obtained.Thus the calculation time and the memory storage are significantly reduced.

Figure 2
Figure 2 Interface domain and the equivalent diagram with current virtual source.

Figure 4
Figure 4 Equivalent circuit of the studied structure.
Eqns 27, 28 denotes the inner product of L2(R) and j(k) is the reference smoothing resolution.Wavelets with N vanishing moments have the potential to sparsify dense

268Figure 5
Figure 5 Normalized charge density on the x-direction (null on the insulator domain).
the permittivity of air).Two different types of basis functions have been used to model the charge distribution; Daubechies wavelet and rooftop functions.For the first kind, we exploit the Discrete Wavelet transform DWT to compute the inner products.A convergence study is necessary.Table1shows that, the results converge from NB = 1000 modes TMy and NE = 15 rooftop functions.Using Daubechies wavelet (db1), NB = 1900 modes and NE = 16 trial functions are needed to obtain the convergence.

Table 2
Normalized capacitance for configuration of 8 conductors