Wavelets trong Electromagnetics và mô hình thiết bị P8 ppsx

CHAPTER NINE Wavelets in Packaging, Interconnects, and EMC In this chapter we will study multiconductor, multilayered transmission lines (MMTL) employing quasi-static, quasi-dynamic, and full-wave analyses. We extract from MMTL the distributed (parasitic) parameters in matrix form of the capacitance [C], inductance [L], resistance [R] and conductance [G], or the [Z]-parameters, [Y ]-parameters, or more generally the scattering matrix [S]. MMTL systems are commonly found in high-speed, high-density digital electronics at the levels of in- dividual chip carriers, printed circuit boards (PCBs), and more recently, multichip modules (MCMs). Previous methods for extraction of the distributed circuit param- eters include the quasi-TEM solutions [1–5], and more rigorous techniques [6–9]. They also included full-wave analysis algorithms [10–15]. We begin with the quasi-static formulation (QSF) [1], which provides the para- sitic capacitance [C], inductance [L], resistance [R], and conductance [G]. Due to the limitation of its assumptions, the QSF results for L, C, R,andG are independent of frequency values. This characteristic is accurate only under special circumstances. The comparison of the QSF solution with the full-wave finite element method (FEM) data indicates that the capacitance [C] values from the QSF are accurate to at least 50 GHz [16], while the [L] and [R] may have large errors. For most practical applica- tions, conductance [G]is negligibly small. Therefore, in the quasi-static formulations of Sections 9.1 and 9.2, we will focus mainly on capacitance extraction. In Section 9.3 we will introduce an intermediate formulation between that of the quasi-static and full-wave, referred to as the quasi-dynamic formulation (QDF). The QDF provides us with frequency-dependent parameters of the skin effect resistance and total (internal plus external) inductance. The comparison of the QDF with the FEM [17] and laboratory tests [18] reveals that the [L] and [R] matrices from the QDF are accurate from 1 MHz to at least 10 GHz. Following this we will present the full-wave analysis in Sections 9.4 and 9.5, from which we extract the scattering parameters [S]. The emphasis of this chapter will be 401 Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright ¶ 2003 John Wiley & Sons, Inc. ISBN: 0-471-41901-X 402 WAVELETS IN PACKAGING, INTERCONNECTS, AND EMC given to packaging and interconnects of high-speed digital circuits and systems and the implementation of numerical algorithms using wavelets. 9.1 QUASI-STATIC SPATIAL FORMULATION In this section the wavelet expansion method in conjunction with the boundary ele- ment method (BEM) is applied to the evaluation of the capacitance and inductance matrices of multiconductor transmission lines in multilayered dielectric media. The integral equations obtained by using a Green function above a grounded plane are solved by Galerkin’s method, with the unknown total charge expanded in terms of orthogonal wavelets in L 2 ([0, 1]). The unknown functions defined in finite intervals are expanded in terms of wavelets in L 2 ([0, 1]), as discussed in Chapter 4. Adopting the geometric representation of the BEM converts the 2D problem into a 1D prob- lem and provides a versatile and accurate treatment of curved conductor surfaces and dielectric interfaces. A sparse matrix equation is developed from the set of integral equations. This equation is extremely valuable for solving a large system of equa- tions. We will compare the numerical QSF results with previously published data and demonstrate good agreement between the two sets of results. Recently Nekhla reported in [19] that by modifying our wavelet-BEM ap- proach [20], “The proposed algorithm has a major impact on the speed and accuracy of physical interconnect parameter extraction with speedup reaching 10 3 for even moderately sized problems.” 9.1.1 What Is Quasi-static? In digital and microwave circuits and systems, the electromagnetic (EM) modeling was based on the quasi-static method. The distributed circuit parameters obtained are inductance L(H/m), capacitance C(F/m), resistance R(/m), and conductance G(S/m), all expressed per unit length. These parameters are frequency-independent under the quasi-static assumption. The quasi-static method assumes: (1) The wavelength of interest is much greater than the dimensions of the cir- cuit/subsystems under consideration. Typically f < 3 ∼ 5 GHz. (2) The longitudinal fields and transverse currents are negligible, which leads to k 2 = k 2 x + k 2 y + k 2 z ≈ k 2 z ,wherek z is the wavenumber in the direction of propagation. (3) Ohmic loss is low so that small perturbation is applicable. (4) The linear dimension of the transmission line cross section is much greater than δ (skin depth). As a result current flows only on the conductor surface. Equivalently the microstrip thickness t and width w satisfy w  t  δ,and thus internal inductance L int can be neglected, and L = L ext . These assumptions no longer hold for high-speed electronic packaging applications. For instance, for typical multichip module (MCM) structures, the cross section of QUASI-STATIC SPATIAL FORMULATION 403 the transmission lines is w × t = 8 × 6 µm. For such a structure the dc resistance ≈ 400 /m at 1 GHz with copper of conductivity σ = 5.8 ×10 7 S/m and skin depth δ = 1/ √ π f µσ ≈ 2 µm. The signal frequency bandwidth ranges from 10 MHz to 10 GHz, and the corresponding skin depths are from δ = 20 to δ = 0.7 µm. Thus the surface resistance formula R s = 1 σδ =  π f µ σ is not applicable, since we do not have w  t  δ. In addition the small perturbation approach does not apply due to relatively high ohmic losses. Nonetheless, the quasi- static approximation is still widely used, in particular, for capacitance computations. The wave phenomena are governed by the Helmholtz equation ( 2 + k 2 )φ( x , y, z) = 0, (9.1.1) where φ(x, y, z) is the potential, k = ω √ µ =  k 2 x + k 2 y + k 2 z is the wavenumber. Let φ(x, y, z) = V (x, y)e ±jk z z , (9.1.2) where V (x, y) is the potential profile in the transverse plane. Substituting (9.1.2) into (9.1.1), we obtain  ∂ 2 ∂x 2 + ∂ 2 ∂y 2  + (k 2 − k 2 z )  V (x, y) = 0. (9.1.3) Under quasi-static assumption (2), one has k ≈ k z . Hence (9.1.3) becomes  ∂ 2 ∂x 2 + ∂ 2 ∂y 2  V (x, y) = 0. (9.1.4) Equtation (9.1.4) is a 2D Laplace equation, which is much simpler then the Helm- holtz equation (9.1.1). The static nature of (9.1.4) gives the name of this approach as quasi-static. The prefix “quasi-” is necessary because the wave does propagate along the ∓ˆz direction. The quasi-static (quasi-TEM) method is very popular because of its simplicity in mathematics. 9.1.2 Formulation Figure 9.1 shows the transmission line system under consideration. An arbitrary number of conductors N c is embedded in a dielectric slab consisting of an arbitrary number of individual layers N d . A perfectly conducting ground plane extends from x =−∞to x =∞. The system is uniform in the y direction. The conductors are 404 WAVELETS IN PACKAGING, INTERCONNECTS, AND EMC GND ε d ε d d ε d ε 0 xO z m+1 m m-1 m-2 m+1 m m-1 m+1 m m-1 FIGURE 9.1 Geometry of multiconductor multilayer transmission lines (MMTL). perfectly lossless and can possess either a finite cross section or be infinitesimally thin. The integral equation formulation for this system is derived in [1]. For ease of reference, we briefly repeat the basic formulation here. The integral equations solved for the unknown total charge distribution σ T ( ␳ ) can be obtained as follows: 1 2π 0 J  j=1  l j σ T ( ␳  ) ln | ␳ − ␳  | | ␳ − ␳  | dl  = V c ( ␳ ) = const. (9.1.5) on the conductor surfaces, and  + ( ␳ ) +  − ( ␳ ) 2 0   + ( ␳ ) −  − ( ␳ )  σ T ( ␳ ) + 1 2π 0 J  j=1  l j − σ T ( ␳  ) ·  ␳ − ␳  | ␳ − ␳  | 2 − ␳ − ␳  | ␳ − ␳  | 2  ·ˆn( ␳ ) dl  = 0 (9.1.6) on the dielectric-to-dielectric interface. Here ρ = √ x 2 + z 2 , l j is the contour of the jth interface above the ground plane, ␳  is the image point of ␳  about the ground plane, and J is the total number of the interfaces (including conductor-to- QUASI-STATIC SPATIAL FORMULATION 405 dielectric interfaces and dielectric-to-dielectric interfaces);  − denotes the Cauchy principal value of the integral, and ˆn( ␳ ) is the unit normal vector at ␳ . The side of the curve l j is referred to as the “positive” side if ˆn( ␳ ) points away from the curve, while the other side is called its “negative” side;  + ( ␳ ) and  − ( ␳ ) denote the permittivity on the positive and negative sides, respectively, of the interface that ␳ approaches. In order to obtain the capacitance matrix [C], the integral equations (9.1.5) and (9.1.6) must first be solved for the total charge distribution σ T ( ␳ ), with V c assigned as a unity voltage on each particular conductor surface l j as zero voltage on the other conductors. After obtaining the total charge distribution σ T ( ␳ ), the free charge distribution σ F ( ␳ ) on the conductors can be evaluated by σ F ( ␳ ) = ( ␳ )  0 σ T ( ␳ ) for the conductors of finite cross section, and σ F ( ␳ ) =  + ( ␳ ) +  − ( ␳ ) 2 0 σ T ( ␳ ) +  + ( ␳ ) −  − ( ␳ ) 2π 0 J  j=1  l j − σ T ( ␳  ) ·  ␳ − ␳  | ␳ − ␳  | 2 − ␳ − ␳  | ␳ − ␳  | 2  ·ˆn( ␳ ) dl  (9.1.7) for infinitesimally thin strips. The total free charge Q i (per unit length in the z di- rection) on conductor l i corresponding to this potential distribution yields the ele- ment C ij (i, j = 1, 2, ,N c ) of the capacitance matrix. The external inductance matrix [L] is related to the vacuum capacitance matrix [C v ] by the simple formula [L]= 0 µ 0 [C v ] −1 . The vacuum capacitance matrix [C v ] itself is the capacitance matrix of the same conductor system where all dielectrics have been replaced by a vacuum. The previous integral equations, (9.1.5) and (9.1.6), need to be solved numerically for the unknown charge distribution σ T ( ␳ ). This distribution on each interface is expanded in terms of basis functions σ T ( ␳ )  M  m=1 g m−1 ( ␳ )σ Tm , (9.1.8) where g m−1 ( ␳ )(m = 1, 2, ,M) are the basis functions, σ Tm (m = 1, 2, ,M) are the unknown coefficients to be determined, and M is the total number of the bases. We use Galerkin’s method for the testing procedure. Using (9.1.8), a set of linear algebraic equations in matrix form can be derived from integral equations (9.1.5) and (9.1.6) [1] as [ A nm ][ σ Tm ] = [ B n ] , (9.1.9) where the elements of the matrices are 406 WAVELETS IN PACKAGING, INTERCONNECTS, AND EMC A nm = J  j1=1  l j1 g n−1 ( ␳ ) ·  1 2π 0 J  j2=1  l j2 g m−1 ( ␳  ) · ln  | ␳ − ␳  | | ␳ − ␳  |  dl   dl (9.1.10) B n = J  j1=1  l j1 g n−1 ( ␳ )V c ( ␳ ) dl, (9.1.11) for those g n−1 ( ␳ ) defined on the conductor-to-dielectric interfaces, and A nm = J  j1=1  l j1 g n−1 ( ␳ ) ·   + ( ␳ ) +  − ( ␳ ) 2 0   + ( ␳ ) −  − ( ␳ )  g m−1 ( ␳ ) + 1 2π 0 J  j2=1  l j2 − g m−1 ( ␳  ) ·  ␳ − ␳  | ␳ − ␳  | 2 − ␳ − ␳  | ␳ − ␳  | 2  ·ˆn( ␳ ) dl   dl (9.1.12) B n = 0, (9.1.13) for those g n−1 ( ␳ ) defined on the dielectric-to-dielectric interfaces. After A nm (n = 1, 2, ,M;m = 1, 2, ,M) and B n (n = 1, 2, ,M) have been calculated, (9.1.9) produces M simultaneous equations in M unknowns, σ Tm (m = 1, 2, ,M). These simultaneous equations can then be solved for σ Tm (m = 1, 2, ,M) in terms of the potential V c ( ␳ ) on the conductors. 9.1.3 Orthogonal Wavelets in L 2 ([0, 1]) Orthogonal periodic wavelets in L 2 ([0, 1]) were studied in great detail in Chapter 4. We will review the relevant material briefly here. Given a multiresolution analysis with scalet ϕ(x) and wavelet ψ(x) in L 2 (R), the wavelets in L 2 ([0, 1]) are ϕ per m,n (x) =  k∈Z ϕ m,n (x + k), (9.1.14) ψ per m,n (x) =  k∈Z ψ m,n (x + k), (9.1.15) and V per m = clos L 2 ([0,1]) {ϕ per m,n (x) : n ∈ Z}, W per m = clos L 2 ([0,1]) {ψ per m,n (x) : n ∈ Z}. It can be shown that V per m are all identical one-dimensional spaces containing only the constant functions for m ≤ 0, and W per m ={∅}for m ≤−1. Thus we only need to study V per m and W per m for m ≥ 0. Moreover it can easily be verified that V per m+1 = V per m ⊕ W per m QUASI-STATIC SPATIAL FORMULATION 407 and clos L 2   m∈N V per m  = L 2 ([0, 1]), where N is the set of nonnegative integers. Hence there is a ladder of multiresolution spaces V per 0 ⊂ V per 1 ⊂ V per 2 ⊂··· with successive orthogonal complement W per 0 , W per 1 , W per 2 , , and orthonormal bases {ϕ per m,n (x)} n=0, ,2 m −1 in V per m , {ψ per m,n (x)} n=0, ,2 m −1 in W per m for m ∈ N .In particular, that {ϕ per 0,0 }  {ψ per m,n : m ∈ N , n = 0, ,2 m − 1} constitute an orthonormal basis in L 2 ([0, 1]). For simplicity, we relabel this basis as follows: g 0 (x) = ϕ per 0,0 (x) = 1 g 1 (x) = ψ per 0,0 (x) g 2 (x) = ψ per 1,0 (x) g 3 (x) = ψ per 1,1 (x) = g 2  x − 1 2  . . . g 2 m (x) = ψ per m,0 (x) . . . g 2 m +n (x) = ψ per m,n (x) = g 2 m (x − n2 −m ), 0 ≤ n ≤ 2 m − 1 . . . These Daubechies periodic scalets were illustrated in Fig. 4.7. For any f (x) ∈ L 2 ([0, 1]), the approximation at the resolution 2 m can be defined as the projection in V per m , f (x)  P m f (x) = 2 m −1  k=0 f k g k (x) where P m is the orthogonal projection operator onto V per m and f k is the inner product of f (x) and g k (x). 408 WAVELETS IN PACKAGING, INTERCONNECTS, AND EMC 9.1.4 Boundary Element Method and Wavelet Expansion Geometrical Representation Before considering the details of this problem, we will assume that most curves {l j } are closed for the purpose of expressing the unknown charge distribution. Roughly speaking, there are four types of contours: (1) the contour of the conductor with finite cross section, (2) the contour along the infinitesimally thin metal strip, (3) the contour along the dielectric-to-dielectric interface from −∞ to +∞, and (4) the contour along the dielectric-to-dielectric interface from −∞ to +∞, with some spaces of discontinuity wherever there is a conductor along the interface. We will examine the four types of contours one by one. In the first place, all the contours except type (4) are geometrically continuous. Moreover the contour of type (1) is closed geometrically. The contour of type (2) can be considered to be closed, since the charge distribution has the same behavior (singularity) at its two edge points. Similarly the contour of type (3) can also be viewed as closed since no charge exists at infinity, and thus the charge distribution gives the same value of zero at the two ends (−∞ and +∞) of the contour. In the case of of type (4), the contour intersects the conductor at two points if the conductor is lying along the contour and creates a discontinuity space for that contour. We must employ intervallic wavelets, instead of periodic wavelets. Since the periodized wavelets are defined in L 2 ([0, 1]), one must map each of the contours {l j } onto the interval [0, 1]. For an arbitrary contour l j , we take two steps: (1) Use the conventional boundary element method to discretize the contour into a series of boundary elements, and then map each of the boundary elements onto 1D standard elements through the shape functions or interpolation func- tions [3, 22]. (2) Map the standard elements into corresponding portions of interval [0, 1].A linear map is sufficient for this step. This procedure can be precisely formulated in mathematical language as well. In step (1), the global coordinates ␳ are expressed in terms of the local coordinate ξ of a standard element [3]: ␳ = M e  i=1 N i (ξ) ␳ i =  1 (ξ), (9.1.16) where M e is the number of the interpolation nodes in the local standard element, N i (ξ) is the shape function referred to node i of the local standard element, and ␳ i are the global coordinates of node i of the actual element. The shape functions { N i (ξ) } are given in standard finite element or boundary element books and literature (e.g., [3, 22]). Upon inspecting (9.1.16), we can conclude that (9.1.16) maps the standard ele- ment in local coordinates onto the actual element, which may have a quite arbitrary or distorted shape, in global coordinates. The node ␳ i in the actual element corre- QUASI-STATIC SPATIAL FORMULATION 409 sponds to the node i in the standard element (by definition, N i (ξ) is assumed to have a unity value at node i and zero at all other nodes of the element). In step (2), the standard elements corresponding to the actual elements from con- tour l j are mapped into the subintervals [ζ 0 ,ζ 1 ], [ζ 1 ,ζ 2 ], ,[ζ K j −1 ,ζ K j ] of in- terval [0, 1],whereK j is the number of the elements from contour l j and 0 = ζ 0 <ζ 1 <ζ 2 < ··· <ζ K j = 1 (e.g., one can simply assume that ζ k = k/K j , k = 1, ,K j − 1). The map between the local coordinate ζ in interval [0, 1] and the local coordinate ξ in the kth standard element of contour l j can be written as ζ = ζ k−1 + (ζ k − ζ k−1 ) · ξ, (9.1.17) or ξ = ζ − ζ k−1 ζ k − ζ k−1 , (9.1.18) where k = 1, 2, ,K j . Combining (9.1.16) and (9.1.18), we obtain a map between the global coordinates ␳ and the local coordinate ζ in interval [0, 1]: ␳ =  1  ζ − ζ k−1 ζ k − ζ k−1  =  2 (ζ ). (9.1.19) The maps (9.1.16) through (9.1.19) establish the conversions among the local coor- dinate ξ, the local coordinate ζ and the global coordinates ␳ . Source Representation Now we may define the basis functions {g m−1 ( ␳ )}.For simplicity and generality, the basis functions will not be directly defined over all the contours in terms of a set of global coordinates, but rather over interval [0, 1] since each of the contours can be related to interval [0, 1] through the map described by (9.1.19). By using the conversion between the global coordinates ␳ and the local coordinate ζ for each individual contour, we can easily obtain the basis functions of the individual contour in the set of global coordinates. For the unknown charge distribution along contour l j , expansion (9.1.8) can now accurately be written as the projection in V per m h (about ζ ): σ T ( ␳ )  P m h σ T ( ␳ ) = M j  m=1 g m−1   −1 2 ( ␳ )  σ Tm , (9.1.20) where  −1 2 denotes the inverse map of  2 , g m−1 (ζ ) represents the orthogonal wavelets in L 2 ([0, 1]),andM j = 2 m h is the number of the wavelet bases used for expressing the unknown charge distribution on contour l j . Because  −1 2 maps contour l j into interval [0, 1], the basis functions {g m−1 [ −1 2 ( ␳ )]} are well defined. It has been shown [24] that if σ T is smooth with a finite number of discontinuities, the error between σ T (ζ ) and P m h σ T (ζ ) is bounded: ||σ T (ζ ) − P m h σ T (ζ ) || ≤ C2 −m h s , (9.1.21) 410 WAVELETS IN PACKAGING, INTERCONNECTS, AND EMC where C and s are some positive constants, respectively, relating to ||σ T (ζ ) ||and the smoothness of σ T (ζ ). The function σ T (ζ ) with higher-order (piecewise) continuity has larger s value and thus faster error decay. Moreover the approximation error of expansion (9.1.20) can be estimated as ||σ T ( ␳ ) − P m h σ T ( ␳ ) || ≤ C d ||σ T (ζ ) − P m h σ T (ζ ) || ≤ CC d 2 −m h s , where C d is the tight upper bound of the Jacobian of the transformation  2 (ζ ).That is, the approximation error of (9.1.20) has exponential decay with respect to the resolution level m h . Matrix Equation Based on the preceding source expansion, a set of linear al- gebraic equations is obtained from integral equations (9.1.5) and (9.1.6) by using Galerkin’s method. This set is matrix form described by (9.1.9) if the elements of the matrices are computed by replacing {g m−1 ( ␳ )} with {g m−1 [ −1 2 ( ␳ )]} in equations (9.1.10) through (9.1.13), namely A nm = J  j1=1  l j1 g n−1   −1 2 ( ␳ )  ·  1 2π 0 J  j2=1  l j2 g m−1   −1 2 ( ␳  )  · ln  | ␳ − ␳  | | ␳ − ␳  |  dl   dl, (9.1.22) B n = J  j1=1  l j1 g n−1   −1 2 ( ␳ )  V c ( ␳ ) dl, (9.1.23) for those g n−1 [ −1 2 ( ␳ )] defined on the conductor-to-dielectric interfaces, and A nm = J  j1=1  l j1 g n−1   −1 2 ( ␳ )  ·   + ( ␳ ) +  − ( ␳ ) 2 0   + ( ␳ ) −  − ( ␳ )  g m−1   −1 2 ( ␳ )  + 1 2π 0 J  j2=1  l j2 −g m−1   −1 2 ( ␳  )  ·  ␳ − ␳  | ␳ − ␳  | 2 − ␳ − ␳  | ␳ − ␳  | 2  ·ˆn( ␳ ) dl   dl, B n = 0, (9.1.24) for those g n−1 [ −1 2 ( ␳ )] defined on the dielectric-to-dielectric interfaces. [...]... in the problem, resulting in much smaller impedance matrix (2) Only standard wavelets are employed to expand the free surface charges on closed contours of the conductor surfaces No periodic or intervallic wavelets are necessary, and so a much simpler treatment is possible (3) Replacing the Daubechies wavelets with Coifman wavelets allows singlepoint quadrature and leads to fast matrix filling 416 −3.580... presented in this subsection When using wavelets on the real line to solve problems with finite intervals, improper selection of the wavelets can result in nonphysical solutions In contrast, any type of wavelets on the real line can be used for the construction of the wavelets in L 2 ([0, 1]), although there may be some discrepancy in their smoothness, as seen in Chapter 4 However, since the derivatives... the integral equations under consideration, a set of basis functions with C 0 continuity is sufficient to yield a convergent solution In the following computations the Daubechies wavelets are employed to construct the orthogonal wavelets in L 2 ([0, 1]) Example 1 Thin microstrip line of width W above a dielectric substrate of thickness H and r = 6 We have studied this example of an infinitesimally thin... surprise that the wavelet expansions converge more quickly; that is, fewer coefficients are required by wavelets to represent a given function than by other expansions, since this is a well-known result from wavelet theory and has been extensively studied in Chapter 2 One of the most attractive features of wavelets is that they give completely local information on the functions analyzed It can be shown that... function does not have uniform smoothness, for instance, if a smooth function possesses discontinuities, there is an optimal way to approximate the function using low resolution wavelets everywhere and adding high resolution wavelets near the singularities [24] Example 2 Multiconductor Transmission Lines above a Thick Substrate Shown in Fig 9.2 is a 10-conductor transmission line system This problem... Gaussian quadrature [25] 412 WAVELETS IN PACKAGING, INTERCONNECTS, AND EMC 9.1.5 Numerical Examples Based on the technique presented in the preceding subsections, a program has been designed to compute the capacitance and external inductance matrices of multiconductor transmission lines in multilayered dielectrics Two numerical examples are presented in this subsection When using wavelets on the real line... utilizes the arc length ζ , varying from 0 to in circumference; it is in [0, 1] after normalization In principle, one needs to utilize periodic wavelets in L2 ([0, 1]) when the domain of the problem is over a finite interval Nevertheless, we find that the standard wavelets are sufficient to represent the contours In fact, we now deploy the wavelet bases one by one on the contour that has been mapped by the... Considering the case that a unit source is in layer m (see Fig 9.1) The 3D Green’s function satisfies Poisson’s equation 2 G 3D (x, y, z | x0 , y0 , z 0 ) = 1 δ(x − x0 ) δ(y − y0 ) δ(z − z 0 ) (9.2.2) 418 WAVELETS IN PACKAGING, INTERCONNECTS, AND EMC Spatial domain and spectral domain Green’s functions are related by the 2D Fourier transform pair as G 3D (x, y, z | x0 , y0 , z 0 ) = ∞ 1 (2π)2 ∞ −∞ −∞ dα... For n < m, z < z 0 , ˜ G(z | z 0 ) = A− m,n γ z [e + ˜ n,n−1 e2γ dn−1 −γ z ], 2 mγ − − − Ai,i−1 = Ai,i Si,i−1 , A− = A− m,n m,m m − Si,i−1 , n+1 where A− = Mm [e−γ z 0 + ˜ m,m+1 e−2γ dm +γ z 0 ] m,m 420 WAVELETS IN PACKAGING, INTERCONNECTS, AND EMC In the previous formulas ˜ i,i+1 = + Si,i+1 = + ˜ i+1,i+2 e2γ (di −di+1 ) , 1 + i,i+1 ˜ i+1,i+2 e2γ (di −di+1 ) i,i+1 1− i+1,i Ti,i+1 , ˜ i+1,i+2 e2γ (di −di+1... 2 + z2 , we can write the approximated Green’s function for 2D and 3D cases as G 3D (r| r0 ) = 1 4π m G 2D (␳ | ␳0 ) = − For the 2D case, 4 f j3D,± (r |r0 ), j=1 1 2π m 4 j=1 f j2D,± (␳ | ␳0 ), , 422 WAVELETS IN PACKAGING, INTERCONNECTS, AND EMC +,∞ f j2D,+ (␳ | ␳0 ) = K m,n, j ln + Nm,n, j + (x − x0 )2 + Z + 2 j +,i Cm,n, j ln +,i (x − x0 )2 + (Z + + am,n, j )2 , j (9.2.12) i=1 −∞ f j2D,− (␳ | ␳0 . emphasis of this chapter will be 401 Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright ¶ 2003 John Wiley & Sons, Inc. ISBN: 0-471-41901-X 402 WAVELETS IN PACKAGING, INTERCONNECTS,. discontinuity space for that contour. We must employ intervallic wavelets, instead of periodic wavelets. Since the periodized wavelets are defined in L 2 ([0, 1]), one must map each of the contours. subsection. When using wavelets on the real line to solve problems with finite intervals, improper selection of the wavelets can result in nonphysical so- lutions. In contrast, any type of wavelets on the