New Single-Source Surface Integral Equation for ...

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New Single-Source Surface Integral Equation for Magneto-Quasi-Static Characterization of Transmission Lines Situated in Multilayered Media Shucheng Zheng, Student Member, IEEE, Anton Menshov, Student Member, IEEE, and Vladimir I. Okhmatovski, Senior Member, IEEE

Abstract— We recently proposed a novel single-source integral equation (SSIE) for accurate broadband resistance and inductance extraction and current flow modeling in 2-D conductors. The new surface integral equation is advantageous compared with the traditional volume electric field integral equation (V-EFIE) used for the inductance extraction, since the unknown function is defined on the surface of conductors as opposed to the volumetric unknown current density in V-EFIE. The new SSIE is also more suitable for the solution of inductance extraction problems than the traditional surface integral equation formulations, as it features only a single unknown surface function as opposed to having the unknown equivalent electric and magnetic surface current densities. The new equation also features only the electric field Green’s functions unlike the previously known SSIE formulations. The latter property makes the new SSIE equation particularly suitable to the inclusion of the multilayered substrate effect into the inductance extraction model. This paper describes the generalization of the new SSIE formulation to the case of transmission line models embedded into the multilayered lossy substrates. This paper also shows how the matrix sparsity in the method of moments discretization of the novel integral equation can be exploited to accelerate its numerical solution and reduce associated memory use. This sparsity arises due to the skin-effect-based attenuation of the fields in conductors’ cross sections leading to vanishing levels of the matrix elements corresponding to the distant interactions. Typical examples of inductance extraction in complex interconnects situated in lossy substrate are considered to validate the proposed techniques against traditional approaches. Index Terms— Inductance extraction, multiconductor transmission lines (MTLs), multilayered media, single-source integral equations (SSIEs).

Manuscript received July 2, 2016; revised October 11, 2016 and October 26, 2016; accepted October 27, 2016. This work was supported by a Collaborative Research and Development Grant from the Natural Sciences and Engineering Research Council (NSERC) and by the Manitoba HVDC Research Center of Manitoba Hydro International. An earlier version of this paper was presented at the IEEE MTT-S International Microwave Symposium, San Francisco, CA, USA, May 22–27, 2016. S. Zheng and V. I. Okhmatovski are with the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB R3T 5V6, Canada (e-mail: [email protected]; [email protected]). A. Menshov is with the Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78712 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2016.2623625

I. I NTRODUCTION

T

RANSIENT analysis of multiconductor transmission lines (MTLs) plays an important role in the design of high-speed digital interconnects [1]–[3], analysis of microwave and millimeter wave circuits [4], simulation of power systems [5]–[7], and various other areas. Under the assumption of quasi-TEM wave propagation along an MTL situated in lossy dielectric substrate, the Maxwell equations are simplified to the system of Telegrapher’s equations governing the wave propagation along the line, and the decoupled problems of electro- and magneto-quasi-static governing cross-sectional components of the electric and magnetic fields, respectively [4], [8], [9]. Solution of the electro- and magneto-quasi-static problems yields frequency-dependent per-unit-length (p.u.l.) capacitance (C), conductance (G), inductance (L), and resistance (R) matrices, which upon substitution into the Telegrapher’s equations enable transient analysis of signal propagation along the MTL conductors. This paper presents a new surface integral equation formulation for magneto-quasi-static analysis of the MTL embedded into lossy dielectric substrates. Such analysis is traditionally done via solution of the volume electric field integral equation (V-EFIE) under magneto-quasi-static approximation [3], [10]. In our previous works [11]–[13], we proposed novel surface-volume-surface EFIEs (SVS-EFIE) for the broadband network characterization and current flow modeling in 2-D conductors and 3-D interconnects of arbitrary cross section. The novel equations are derived from the classical V-EFIE [14] through the representation of the electric field inside a conductor as a superposition of the cylindrical waves emanating from the conductor’s surface. Thus, derived singlesource integral equations (SSIEs) feature only the derivativefree electric field kernel and confine the unknowns in the method of moments (MoM) discretization to the contour of the conductor. The latter property greatly reduces the computational complexity of the numerical solution compared with the solution of the traditional V-EFIE. Alternative single-source surface integral equations [15], [16] feature similar number of unknowns as the proposed SVS-EFIE, though they introduce large number of

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integral operator products, that makes their numerical solution difficult to compute. The novel SVS-EFIE formulation features a single product of integral operators mapping the field from the conductor’s contour to its volume and back from its volume to the contour. All three integral operators entering into the proposed SVS-EFIE contain derivative-free electric field Green’s functions only. The absence of derivatives greatly simplifies the introduction of the multilayered media substrate into the transmission line model. Green’s function of such substrates must be computed numerically and as such exhibits notable numerical error. The traditional surface integral equations featuring derivatives acting on Green’s function greatly amplify this numerical error and make inclusion of the multilayered substrate into the surface integral formulation particularly challenging. In this paper, we generalize previously derived SVS-EFIE formulation to the case of the MTL embedded in layered lossy substrates by taking advantage of its derivative-free nature. To compute the layered media Green’s function, we utilize the finite-difference (FD) approach described in [9]. The samples of Green’s function are stored in a database with subsequent interpolation used for the evaluation of Green’s function values needed for computation of the MoM matrix elements. The new model is shown to accurately account for the inclusion of the lossy substrate into the MTL model and modification of the current flow in its conductors. The comparisons are made against traditional V-EFIE solution independently obtained by different authors in [17] with the layered substrate Green’s function computed via direct evaluation of the Sommerfeld integrals [18]. The method for the evaluation of the layered media Green’s function is chosen due to its robustness over fitting techniques, such as the discrete complex image method [19], [20] and the rational function fitting method [21] and numerical efficiency over direct integration methods. In the SVS-EFIE, both contour and volume meshes are involved into the MoM discretization of the pertinent integral operators. As the number of surface and volume elements dictated by the skin depth increases with frequency, the solution of the SVS-EFIE at high frequencies becomes computationally challenging. In this paper, we demonstrate how the solution efficiency of the SVS-EFIE can be improved when the skin effect becomes sufficiently developed. This efficiency improvement stems from the fact that the kernels of the integral operators constituting the SVS-EFIE include Green’s function of the conductor’s medium exponentially attenuating with an increase in distance. This leads to sparsification of the matrices arising from the MoM discretization of the SVS-EFIE based on the prescribed tolerance. This, in its turn, makes the network parameter extraction via SVS-EFIE at high frequencies more efficient. The numerical results showing the impact of utilizing the sparsity pattern on the computational time and accuracy of the solution are demonstrated. It is shown that a significant computational advantage can be achieved without loss of accuracy from the utilization of the pertinent matrix sparsity. This paper is an extended version of the conference paper published in the proceedings of IMS 2016 [22] and is focused

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on the inductance extraction in layered media as opposed to our overview conference paper published in conference proceedings of EMTS 2016 [23]. The extension consists of description of the method used for the computation of the multilayered media Green’s functions, description of MoM discretization which accounts for the matrix sparsity arising from the skin-effect attenuation of the fields, and various numerical results, including comparison of the proposed method against independent integral equation reference solution [17] and finite-element solution computed by COMSOL commercial software. This paper is organized as follows. In Section II, we present the expanded description of the proposed SVS-EFIE. Section III describes the FD methodology for the computation of the multilayered media Green’s function. Section IV introduces a technique for the sparsification of the matrix elements based on the information about field attenuation in the conductors due to skin effect. In Section V, numerical validation of Green’s function computations is presented as well as the validation of the proposed MoM solution of the SVS-EFIE compared with the traditional MoM solution of the V-EFIE as well as the finite-element solution computed by COMSOL software. Reduction in the CPU time and memory due to the utilization of the matrix sparsity is also demonstrated in Section V. II. SVS-EFIE F ORMULATION IN M ULTILAYERED M EDIA The problem of magneto-quasi-static determining the crosssectional components of the magnetic field Hx and H y and longitudinal component of the electric field E z inside the MTL conductors directed along z coordinate is traditionally cast into the form of V-EFIE [14] jz (ρ) + iω G (ρ, ρ ) jz (ρ )ds = Vp.u.l., ρ ∈ S (1) σ S where jz is the unknown current density inside the conductors related to the electric field by Ohm’s law jz = σ E z . Here, σ is the bulk conductivity of the conductors occupying crosssectional area S. In (1), G (ρ, ρ ) is Green’s function of the stratified medium described by the Helmholtz equation ∇ 2 G (ρ, ρ ) + k2 (y)G (ρ, ρ ) = −μ0 δ(ρ − ρ )

(2)

where k (y) = ω((y)ε0 μ0 )1/2 , ω = 2π f , f being frequency of the time-harmonic fields, and (y) = ε − i σ /(ωε0 ), y ≤ y ≤ y+1 is the complex relative permittivity of the layered medium, where = 1, . . . , L, L is the total number of layers. Green’s function G (ρ, ρ ) in (2) has the physical meaning of the magnetic vector potential A(ρ) = zˆ A z (ρ) produced at the location ρ by the z-directed filament of current (i.e., 2-D point source) situated at the location ρ . To satisfy the boundary condition of continuity for the tangential electric and magnetic fields produced by the filament of current, Green’s function must be a subject to the following boundary conditions at the material interfaces with elevations y : ∂ G ∂ G G |y+ = G |y− , = (3) ∂y y + ∂y y −

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and G = 0 at the perfectly electrically conducting (PEC) planes, if present. The theory of SSIEs demonstrates that the electric field inside the conductor can be represented as a superposition of the waves emanating from its surface and weighted with a single unknown fictitious current density Jz instead of traditional equivalence principle representation featuring both equivalent electric and magnetic currents as follows: E z (ρ) = −

∂ G σ (ρ, ρ ) i ωμ0 aG σ (ρ, ρ ) − b Jz (ρ )dρ ∂n ∂S (4)

III. L AYERED M EDIA G REEN ’ S F UNCTION C OMPUTATION A. Analytic Form in Spectral Domain The sought expression for the spatial domain Green’s function G (ρ, ρ ) = G (|x − x |, y, y ) can be written in the form of the inverse Fourier transform of Green’s function over coordinate x +∞ 1 Gˆ (λ, y, y )eiλ|x−x | dλ (10) G (|x − x |, y, y ) = 2π −∞ where λ is the spectral variable. The spectrum of Green’s function Gˆ (λ, y, y ) satisfies the following 1-D ordinary differential equation (ODE): d 2 Gˆ + [k2 (y) − λ2 ]Gˆ = −μ0 δ(y − y ) d y2

(2)

where ρ ∈ S, G σ (ρ, ρ ) = −(i /4)H0 (kσ |ρ − ρ |) is Green’s function of the homogeneous medium of the conductor with wavenumber kσ = (ωμ0 σ/2)1/2 (1 − i ), σ is the conductor (2) bulk conductivity, and H0 is the second kind Hankel function of the zeroth order. In (4), a and b are arbitrary constants, which are selected in such a way that spurious resonances are eliminated. Since the internal resonances in the volumes of the wires occur at significantly higher frequencies than the frequencies of interest, we can chose constants a = 1 and b = 0 from the consideration of the simplicity of the formulation and convenience of the numerical solution. In the previously known SSIEs, the superposition representation of the field (4) is constrained by the traditional surface equivalence principle [15] to determine the fictitious current Jz . In our recent works [11]–[13], we showed that the superposition of the waves (4) can be constrained using the volume equivalence principle instead. For that purpose, field representation (4) should be substituted into the V-EFIE (1) and the latter restricted to the observation points approaching the boundary of the conductor from inside its volume. This yields the following new SSIE with respect to the unknown auxiliary surface current density Jz :

G σ (ρ, ρ )Jz (ρ )dρ + σ ω μ0 ∂ S × G (ρ, ρ )G σ (ρ , ρ )ds Jz (ρ )dρ

−i ωμ0

∂S

2

⎡

e Gˆ (λ, y, y ) = μ0 ⎣ (5)

The derived integral equation (5) is identical to the SVS-EFIE in [11] except that Green’s function of the integral operator relating volumetric current density of the magnetic vector potential is that of the lossy layered media. It can be written in a concise operator form Tσ∂ S,∂ S ◦ Jz + σ T∂ S,S ◦ TσS,∂ S ◦ Jz = Vp.u.l..

(6)

The operators entering in (6) are defined for (5) as follows: Tσ∂ S,∂ S ◦ Jz = −i ωμ0 G σ (ρ, ρ )Jz (ρ )dρ , ρ ∈ ∂ S (7) ∂ S S,∂ S ◦ Jz = −i ωμ0 G σ (ρ, ρ )Jz (ρ )dρ , ρ ∈ S (8) Tσ ∂ S T∂ S,S ◦ jz = i ω G (ρ, ρ ) jz (ρ )ds , ρ ∈ ∂ S. (9) S

where = 1, . . . , L, L is the total number of layers, and Gˆ = 0 at the PEC planes. The solution of 1-D ODE (11) which satisfies the boundary conditions (12) can be found analytically for any lossy substrate with arbitrary number of layers [14]. In the simple case of the two-layer medium consisting of the upper half-space occupied by free space and lower half-space occupied by lossy material with permittivity s and conductivity σs and dielectric interface between them situated at y2 = 0 m, as shown in Fig. 5, the analytical solution [17] for Green’s function spectrum Gˆ is as follows: √ 2 2 − λ2 −k02 y λ −ks y e e Gˆ (λ, y, y ) = μ0 , y < 0 (13) λ2 − ks2 + λ2 − k02 − λ2 −k02 |y−y |

+

−e

− λ2 −k02 (y +y)

2 λ2 − k02

S

= Vp.u.l., ρ ∈ ∂ S.

(11)

and the spectral domain counterpart of the boundary conditions (3) d Gˆ d Gˆ ˆ ˆ G y+ = G |y− , = (12) d y y+ d y y−

and

3

e λ2

− λ2 −k02 (y +y)

− ks2

⎤

⎦, 2 2 + λ − k0

y≥0 (14)

where the source is situated above the material interface, i.e., y ≥ 0, k0 = (ω2 μ0 ε0 )1/2 is the wavenumber of the free space, and ks = (ω2 μ0 εs − i ωμ0 σs )1/2 is the wavenumber of the lower half-space lossy material. The inverse Fourier transform with the spectrum given by (13) and (14) is complicated, can be slowly convergent, and does not allow for analytic evaluation in a closed-form. Various techniques have been developed in the past for the evaluation of such integrals, including direct integration approach, fitting of the spectrum with rational functions amenable to analytic evaluation of the resultant integrals [19]–[21], transformation techniques [24], fast Fourier transform approach [25], and others.



and at the bottom. In this situation, a grid point is introduced on the PEC plane and boundary condition Gˆ = 0 is enforced at it. When the layered substrate is terminated with an unbounded layer stretching to infinity over y an extra care is required to impose the termination condition on the FD grid. Consider unbounded two-layer media shown in Fig. 1 (note that PEC planes at the top and the bottom are situated at infinity). In order to terminate the FD grid at the upper and lower halfspaces, we introduce buffer layers above and below the region of interest spanning interval y ∈ [0, Y ]. Within the upper buffer layers, the following coordinate transformation [26] is introduced: y(t) =

Fig. 1. Depiction of the two-conductor MTL situated in unbounded two-layer medium consisting of air half-space and lossy half-space and its partitioning into layers. The layered medium is terminated by PEC planes situated at y = ±∞ to enable termination of the FD grid while imitating unbounded medium. In the region of interest y ∈ [y2 , y5 ], where y2 = 0 and y5 = Y , the FD discretization of spectral domain 1-D ODE (11) governing spectrum of Green’s function Gˆ is performed. Infinite buffer layers y ∈ (−∞, y2 ) and y ∈ (y5 , +∞) are mapped to finite regions of parametric variable t ∈ (0, D) via coordinate transformation (16) and (20). The FD discretization of the 1-D ODE (17) governing the spectrum of Green’s function in the buffer regions is performed over parametric variable t.

B. Numerical Evaluation in Spatial Domain In this paper, we utilize the semianalytical approach [9], which performs numerical solution of the 1-D ODE (11) and represents the spectral domain Green’s function Gˆ in the poleresidual form allowing for analytic evaluation of the inverse Fourier transform. FD discretization of (11) on nonuniform grid is introduced, as shown in Fig. 1, for the unbounded two-layer medium. Such discretization yields the following relationship between the samples of Green’s function spectrum Gˆ n = Gˆ (λ, yn , ym ) at the nth grid point yn , its sample above Gˆ n+1 = Gˆ (λ, yn+1 , ym ) at the (n + 1)th grid point yn+1 = yn + h + , and its sample below Gˆ n−1 = Gˆ (λ, yn−1 , ym ) at the (n − 1)th grid point yn−1 = yn − h − :

+ − 1 ˆ 1 1 2 h +h ˆ G n+1 − G n + + − + λ h+ h h 2 − − + i ωμ0 (σ h + σ h + ) + 2 ω2 μ0 (ε− h − + ε+ h + ) − 2 1 −μ0 , n = m (15) + − Gˆ n−1 = h 0, n = m where y = ym is the source point situated at grid location ym , and + , σ + , and − , σ − are permittivity and conductivity on the grid intervals y ∈ [yn , yn+1 ] and y ∈ [yn−1 , yn ], respectively. C. Unbounded Media Handling Truncation of the FD discretization domain is trivial when the layered media is terminated by the PEC planes at the top

tL + Y, t ∈ (0, D) D−t

(16)

which maps the finite interval from 0 to D of parameter t to the infinite interval of physical coordinate y ∈ (Y, +∞). The parameter L in (16) controls the rate of increase in coordinate y throughout the interval t ∈ (0, D) [26] and is used to control the FD grid step in the buffer layer y ∈ (Y, +∞) near Y to ensure smooth transitioning of Green’s function spectral solution from the domain of interest y ∈ [0, Y ] to the buffer region y ∈ (Y, +∞) [see layers y ∈ (y1 , y2 ) and layer y ∈ (y5 , y6 )] in Fig. 1. The change of variable in ODE (11) results in the following ODE over parametric variable t:

2 t 4 d 2 Gˆ (λ, y(t), y ) D 1− L D dt 2

3 ˆ d G (λ, y(t), y ) 2D t − 2 1− L D dt 2 2 − (λ + i ωμ0 σ − ω μ0 ε)Gˆ (λ, y(t), y ) = 0 (17) where it is assumed that the source point elevation is situated in the region of interest y ∈ [0, Y ], hence, yielding zero righthand side in the ODE (17). A uniform FD discretization of the ODE (17) on the interval of parametric variable t ∈ (0, D), tl = (l + 1)h t , l = 0, . . . , P − 1, P being the total number of samples between 0 and D, h t = D/(P + 1), produces the following relationship between the samples of Green’s function spectrum: 2 t 4 D 1− − 4 L D 2 2 2 2 2 + i 2ωμ0 σ h t − 2h t ω μ0 ε + 2h t λ Gˆ l 2 D 1− + 2 L 2 D + 2 1− L

t 3 ˆ 1− G l+1 D

2D t 4 t 3 ˆ + ht 2 1 − G l−1 = 0 D L D (18) t D

4

− ht

2D L2

where Gˆ l is Green’s function Gˆ (λ, y(tl ), ym ) at the lth grid point tl , its sample above is Gˆ l+1 = Gˆ (λ, y(tl+1 ), ym ) at the (l + 1)th grid point y(tl+1 ) = y(tl + h t ), and its sample below is Gˆ l−1 = Gˆ (λ, y(tl−1 ), ym ) at the (l − 1)th grid point y(tl−1 ) = y(tl − h t ).


5

At the grid point y = Y situated at the junction of the physical domain y ∈ [0, Y ] and the infinite buffer layer y ∈ (Y, +∞), the FD relationship between Green’s function samples is the following: 1 1 D − − − i ωμ0 σ h − + ω2 μ0 εh − − h − λ2 Gˆ l − ht L h +

1 ˆ 1 D ˆ G l−1 + G l+1 = 0. (19) − h ht L

In the lower buffer layer mapping the finite interval from 0 to D of parameter t to the infinite interval of physical coordinate y ∈ (−∞, 0), the following coordinate transformation is used: y(t) = −

tL , t ∈ (0, D). D−t

(20)

Modification of the FD equations in the lower infinite half-space buffer layer similar to (18) and (19) is obtained analogously with how they were derived for the case of the upper buffer layer. D. Pole-Residual Approximation in Spectral Domain and Spatial Solution Enforcement of the boundary conditions at the PEC planes terminating the buffer layers, interfaces between the layers, and FD equations at the grid points within the layers produces the system of linear algebraic equations (SLAE) with respect to the sought samples of Green’s function Gˆ (A + λ2 B) · Gˆ = b.

(21)

The SLAE (21) can be rewritten in the following form, which is amenable to the creation of the pole-residual representation of the spectral domain Green’s function: (C + λ2 E) · Gˆ = d

(22)

where C = B−1 · A, d = B−1 · b, and E is the identity matrix. Eigenvalue decomposition of matrix C = T · S · R, where R = T−1 , S = diag(S j ), j = 0, . . . , N − 1 is the diagonal matrix of the eigenvalues, yields the sought pole-residual form −1 N Tn j R j m dm Gˆ λ, yn , ym = S j + λ2

(23)

j =0

where Tn j is the element in the nth row and the j th column of matrix T storing eigenvectors, R j m is the element in the j th row and the mth column of matrix R, and N is the number of samples in the FD discretization of Green’s function spectrum. Substitution of the representation (23) into the inverse Fourier transform (10) −1 N Tn j R j m dm G |x − x |, yn , ym = j =0

×

1 2π

∞ −∞

1 eiλ|x−x | dλ S j + λ2

(24)

Fig. 2. Volume and contour meshes utilized in the MoM discretization of the SVS-EFIE (5). The plot shows contributing and noncontributing surface and volume elements if βδ( f, σ ) sparsification criteria are used in (32), (33), (36), and (38).

followed by the analytic evaluation of the Fourier transform integrals yields the sought approximation of Green’s function: √ −1 N e− S j |x−x | G |x − x |, yn , ym = Tn j R j m dm . (25) 2 Sj j =0 It is important to note here that the number of terms N in the series representation of Green’s function (25) is strongly dependent on the lateral distance |x − x | between the source and observation points. Specifically, prior to the evaluation of Green’s function, all the eigenvalues S j are sorted in the increasing order of the real part ( S j ) similar to the procedure in [27]. Subsequently, for any given lateral distance |x − x |, the sum √ in (25) is truncated at such index ν that produces exp[−( Sν |x − x |)] < τ , τ being the desired error in the computation of Green’s function. The remaining terms ν < j ≤ N − 1 do not produce contributions into Green’s function value beyond the requested error τ . IV. S PARSITY-AWARE MoM D ISCRETIZATION OF SVS-EFIE To formalize the structure of the matrices arising from the MoM discretization of (5) and show the computational benefits gained by utilizing the sparsity pattern, we consider a pair of 2-D conductors A and B of conductivity σ shown in Fig. 2. The cross sections of the conductors are discretized with a volumetric mesh consisting of N = N A + N B triangular elements. The contours of the conductors ∂ S A and ∂ S B forming the cross sections are discretized with M A and M B linear elements, respectively, as shown in Fig. 2. The total number of contour elements M and volume elements N grows with frequency in order to capture the behavior of the volumetric current density according to the skin effect. In most cases, N >> M, as the number of elements in the volumetric mesh√grows proportionally to the frequency f as f dependence for the number of contour opposed to the elements also representing the number of unknowns in the set of linear algebraic equations. The optimal ratio relation √ between N ∼ M has been studied in [11]. The MoM discretization of (5) has been extensively discussed in [12] for the case of a coaxial transmission line.


Hence, only the key ideas important for the efficiency improvement based on the MoM matrix sparsity are left in this paper. The domain discretization of the integral operators (7)–(9) is performed using the piecewise basis functions, and the testing is done via point-matching [12], [28]. The radius vectors on the mth line element of the surface ∂ Sα are defined as follows: α,2 α,1 ρ αm (l) = v α,1 (26) m + l v m − v m , l ∈ [0, 1] where α = A, B is the index identifying the conductor contour ∂ Sα forming the cross section of the conductor, and α,2 v α,1 m and v m are the end and start points of the mth element on the surface ∂ Sα , m = 1, . . . , Mα . The radius vector on the nth triangular element of the cross section Sα in barycentric coordinates is given by α,2 α,3 ρ αn (ξ, η) = v α,1 n ξ + v n η + v n (1 − ξ − η) , ξ, η ∈ [0, 1] (27)

where α = A, B is the index identifying the conductor cross α,2 α,3 section Sα , and v α,1 n , v n , and v n are the three vertices of the nth triangle in the discretization of the cross section Sα , n = 1, . . . , Nα . The MoM discretization of the surface-to-surface integral operator Tσ∂ S,∂ S (7) and the surface-to-volume integral operator TσS,∂ S (8) results in the following matrix forms as shown in [12]: Z∂σ S A ,∂ S A 0 ∂ S,∂ S = (28) Zσ 0 Z∂σ S B ,∂ S B S A ,∂ S A Z 0 σ ZσS,∂ S = (29) 0 ZσS B ,∂ S B and the matrix elements are defined as

1 1 ∂ Sα ,∂ Sα α α α Z σ,mm l = −i ωμ L dl G ρ , ρ (30) 0 m σ m m 2 0

1 1 1 Sα ,∂ Sα α α α , dl G σ ρ n Z σ,n m = −i ωμ0 σ L m , ρ m (l ) 3 3 0 (31) where α = A, B is the index identifying the conductor, L αm is the length of the mth linear element in the discretization of the contour ∂ Sα , m, m , m = 1, . . . , Mα , and n = 1, . . . , Nα . The evaluation of the line integral in (31) can be performed after subtraction of the singular ln-part of the integral kernel via Gauss–Legendre quadrature rule [29], while the singular part can be integrated over the line element analytically [30]. The kernel of the integral operators (7) and (8) is Green’s function of the conductor medium G σ that exponentially attenuates with distance at a given analysis frequency. Thus, if the frequency is relatively low and the skin effect is not developed, then the blocks Z∂σ S A ,∂ S A , Z∂σ S B ,∂ S B , ZσS A ,∂ S A , and ZσS B ,∂ S B in (28) and (29), correspondingly, are dense. The higher is the frequency, the more elements in the MoM discretization of (7) and (8) can be neglected, and the usage of the sparse matrix format [29] is preferred. We will define the discretized versions of the integral operators (7), (8), where the matrix elements corresponding to the interaction between the elements that


are distant from each other more than β skin-depths are preemptively zeroed out as

⎧ α 1 1 ⎨0, α > βδ ρm − ρ m ∂ Sα ,∂ Sα 2 2 Z σ,mm (βδ) = ⎩ ∂ Sα ,∂ Sα Z σ,mm , otherwise (32)

⎧ 1 1 1 ⎨0, α ρ α Sα ,∂ Sα n 3 , 3 − ρ m 2 > βδ Z σ,n m (βδ) = ⎩ Sα ,∂ Sα Z σ,n m , otherwise (33) where δ( f, σ ) = [ (kσ ( f, σ ))]−1 is the skin depth at frequency f for the conductivity σ . The line elements and triangular elements that are at the distance larger than βδ( f, σ ) from the selected linear element m = 1 in the surface mesh of ∂ S A are shown in Fig. 2 with empty circles and thin lines. Thus, the corresponding elements in the matrices (32) and (33) will be zeroed out. The MoM discretization of the volume-to-surface integral operator T∂ S,S (9) results in the following matrix form: ∂ S ,S ∂ S A ,S B A A Z Z (34) Z∂ S,S = Z∂ S B ,S A Z∂ S B ,S B and the matrix elements are defined as 1 1−η ∂ Sα ,Sα α Z ,mn = i ω2 An G

0 0 1 , ρ αn (ξ , η ) dξ dη × ρ αm 2

(35)

where α, α = A, B is the index identifying the conductor, and Aαn is the area of the nth triangular element in the discretization of the conductor cross section Sα , m = 1, . . . , Mα , and n = 1, . . . , Nα . The matrix Z∂ S,S (34) can be sparsified despite the fact that the integral operator (9) features nonattenuating Green’s function of the multilayered substrate G in its kernel. The SVS-EFIE in the operator form (6) reveals that the resultant operator is formed by the product of integral operators (8), (9) represented in the MoM discretized form by the matrices (29) and (34). Since the discretized surface-to-volume operator can be sparsified according to the β skin-depth criteria (33): the columns in Z∂ S,S that will be multiplied by zeroed rows in ZσS,∂ S can be zeroed out as well. These columns in Z∂0 S,S will correspond to the triangles in the volume mesh of the conductor that are located at the distance larger than βδ( f, σ ) from all contour elements (in Fig. 2, those triangles are located inside the dashed circle in the cross section ∂ S B ) ∂ Sα ,Sα Z ,mn (βδ) ⎧ ⎨0, = ⎩ ∂ Sα ,Sα Z ,mn ,

α 1 1 1 α ρ m 2 − ρ n 3 , 3 > βδ ∀m

(36)

otherwise

where α, α = A, B is the index identifying the conductor, m = 1, . . . , Mα , and n = 1, . . . , Nα . In (28), (29), and (34), we obtained the matrix forms of the MoM discretized integral operators Tσ∂ S,∂ S , TσS,∂ S ,


7

T∂ S,S , respectively, that form the SVS-EFIE (5). Thus, the SVS-EFIE (5) is reduced to the set of M linear algebraic equations (SLAE), as it has been described in [11]–[13] ∂ S,∂ S (37) + σ Z∂ S,S · ZσS,∂ S · I = V. Zσ The MoM discretized integral operators in the sparse matrix form (using β skin-depth sparsification criteria) (32), (33), (36) result in the similar SLAE that can be solved numerically using sparse matrix techniques [29] ∂ S,∂ S Zσ (βδ) + σ Z (βδ)∂ S,S · ZσS,∂ S (βδ) · I = V. (38) It is important to point out that similar sparsification patterns apply to the SVS-EFIE in 3-D [31] with a substantially larger potential for computational time improvements. V. N UMERICAL R ESULTS A. Computation of Green’s Function Databases Filling the MoM matrix Z∂ S,S (34) requires special care as the integrals must be computed using Green’s function of the multilayered substrate. To perform such computations of the layered media Green’s function G (ρ, ρ ) = G (|x −x |, y, y ) efficiently for the arbitrary positions of the source point vector ρ and observation point ρ, we create far-field- and near-fielddatabases of Green’s function samples prior to computation of the MoM matrix elements Z∂ S,S . The far-field-database of Green’s function samples is precomputed separately for each frequency of analysis. They are intended for computation of all the Z∂ S,S matrix elements, except the ones corresponding to the nearly interacting ones (|(ρ − ρ ) · x| ˆ < and |(ρ − ρ ) · yˆ | < ). is a frequency-dependent and conductivitydependent parameter defining the characteristic size of the mesh elements discretizing the cross section of the MTL. At the low frequencies where the skin depth substantially exceeds geometric features of the MTL cross section, the characteristic mesh size is determined from the consideration of accurate representation of the geometry. At all the other frequencies, the mesh size is determined by the skin depth as = δ( f, σ )/2. For a given MTL, the samples G (κi , y j , y j ) are stored in the far-field-database, such that κi = i h κ , y j = ymin + j h, and y j = ymin + j h , where i = 0, . . . , I + 1, and j, j = 0, . . . , J + 1, with I = |x max − x min|/(/3) − 1, J = |ymax − ymin |/(/3) − 1, h κ = |x max − x min |/(I + 1), and h, h = |ymax −ymin |/(J +1), such that κ0 = 0 and κ I +1 = |x max −x min|, where x min and x max correspond to the minimum and maximum bounds of the MTL over x coordinate, while y0 and y J +1 correspond to the minimum and maximum bounds ymin and ymax of the MTL over the y coordinate (in the two-conductor MTL example shown in Fig. 6, κ0 = 0 μm, κ I +1 = 60 μm, y0 = 50 μm, and y J +1 = 56 μm). We would like to point out here that more economical sampling strategies that account for the fact that Green’s function varies much more rapidly near the source point can be introduced as well. In such schemes, the step of the sampling may increase nonuniformly along the x coordinate. The near-field-database of the layered media Green’s function is also precomputed at each frequency and is used to facilitate rapid and accurate evaluation of the Z∂ S,S

Fig. 3. Spectral domain Green’s function Gˆ (λ, y, y ) of two-layer medium computed analytically at frequency f = 5 GHz using (13) and (14) and using FD approximation (23) versus transversal coordinate y for point source location y = 52.37 μm and spectral variable λ equal to 103 104 , 105 , and 106 m−1 .

matrix elements corresponding to the mesh elements situated ˆ < within near-field range of each other (|(ρ − ρ ) · x| and |(ρ − ρ ) · yˆ | < ). In this database, the samples of G (κi , y j, j , y j ) are stored according to κi = i /30, y j = ymin + j h , and y j, j = y j − 2 + j h, where i = 0, . . . , 30, j = 0, . . . , J + 1, and j = 0, . . . , F + 1, with J = |ymax − ymin |/(/3)−1, h = |ymax − ymin|/(J +1), and h = |ymax − ymin |/(K + 1) with K = |ymax − ymin |/(/30) − 1, F is chosen such that y0, j = y j −2 and y F +1, j = y j +2. The near-field-database enables accurate sampling of the singular behavior of Green’s function near the source. B. Green’s Function Validation To demonstrate the accuracy of the pole-residual approximation for the spectrum of the layered medium Green’s Gˆ , we consider unbounded two-layer medium consisting of a layer of air situated above lossy substrate with σs = 1 · 104 S/m, εs = 12ε0 , and material interface located at y = 50 μm (Fig. 1). The FD grid is introduced over the interval of interest y ∈ [0, Y ], Y being 65 μm with the uniform step sizes of 0.55 μm on the interval y ∈ [0 m, 50 μm], 0.158 μm on the interval y ∈ [50 μm, 56 μm], and 0.099 μm on the interval y ∈ [56 μm, 65 μm]. The upper and lower buffer regions y ∈ (Y, ∞) and y ∈ (−∞, 0) are mapped to the parametric variable domain t ∈ (0, D) via transformations (16) and (20), respectively, and uniformly discretized with L = 200 samples resulting in grid step h t = 0.32 μm over the parametric variable t. The analytic solution for the spectral domain Green’s function in this example is known analytically and is given by formulas (13) and (14) for the case of the filament of current situated on and above the material interface, i.e., y ≥ 50 μm. The FD solution (23) and the analytic solutions (13) and (14) are plotted in the same Fig. 3 versus transversal coordinate y for the spectral variable λ equal to 103 , 104 , 105 , and 106 m−1 at frequency f = 5 GHz. The point source is situated above the dielectric interface at the location y = 52.37 μm. The layered media Green’s function G (|x − x |, y, y ) in spatial domain produced by the closed-form expression (25)



Fig. 4. Space domain Green’s function G (|x − x |, y, y ) of two-layer medium computed at frequency f = 5 GHz to controlled precision via direct integration of the inverse Fourier transform (also known as Sommerfeld integral) (10) using analytic spectra (13) and (14) and using FD approximation (25) versus transversal coordinate y for point source location y = 52.37 μm and lateral distances |x − x | equal to 0.01, 0.1405, 1.124, and 10.12 μm.

Fig. 6. Normalized current density |twjz /I | in the two-conductor MTL and the lossy substrate (with conductor width w = 20 μm, depth t = 6 μm, and frequency f = 5 GHz). The current distribution computed using SVS-EFIE is shown with dots. Dashed-dotted contour lines and solid contour lines show the current distribution computed via traditional MoM solution of V-EFIE (1) and via FEM COMSOL multiphysics package simulation.

Fig. 5. Geometry of the two-conductor copper transmission line situated above the two-layer unbounded lossy substrate.

is shown in Fig. 4 versus transversal coordinate y ∈ [50 μm, 56 μm] for the same above-described two-layer medium configuration, frequency, source location, and FD discretization at progressively increasing lateral distances from the source |x −x | equal to 0.01, 0.1405, 1.124, and 10.12 μm. C. Method of Moments Solution As an example of MTL featuring lossy substrate, we consider two-conductor transmission line with identical conductors having rectangular cross section and made of copper (σ = 5.8 · 107 S/m). The conductors are situated in the air layer on top of the material interface between the air half-space and the lossy half-space substrate with σs = 1 · 104 S/m and εs = 12ε0 , as shown in Fig. 5. The MoM discretization (37) and (38) of the SVS-EFIE (5) was performed with the frequency adaptive mesh featuring two samples per skin depth, i.e., = δ( f, σ )/2. This results in the MoM solution utilizing the meshes consisting of N1 = 612, N2 = 3002, and N3 = 5918 triangular elements, and M1 = 104, M2 = 224, and M3 = 320 linear elements at the frequencies f 1 = 1 GHz, f 2 = 5 GHz, and f 3 = 10 GHz, respectively. The required values of G (ρ, ρ ) for arbitrary source and observation points ρ and ρ are computed using 3-D linear interpolation [32] over |x − x |, y, and y coordinates from Green’s function samples precomputed and stored in the databases, thus, allowing to fill out the Z∂ S,S matrix (34). Subsequent computation of the SVS-EFIE MoM matrices

Z∂ S,∂ S in (28) and ZS,∂ S in (29) and their substitution into SLAE (37) produces the vector of sought coefficients I in the expansion over the pulse basis functions of the fictitious surface current Jz [11]. The latter upon substitution into the superposition integral (4) with a = 1 and b = 0 yields the electric field E z and volumetric current density jz = E z σ inside conductors of the MTL. The computed volumetric current density jz in the two-conductor MTL at f = 5 GHz in the presence of the unbounded two-layer medium is shown in Fig. 6. The values of Green’s function in the near-fielddatabase and far-field-database were precomputed using formula (25) with the number of terms in its sum representation truncated according to the lateral distance |x − x | between the source and observation points. For comparison in the same Fig. 6, the current distribution obtained using MoM solution of the traditional V-EFIE and the COMSOL commercial software for the two-conductor MTL in the presence of the lossy unbounded substrate is shown. Once the volumetric current density jz in the conductors of the MTL is made available through MoM solution of the SVS-EFIE, one can analyze the distribution of induced current jzs in the multilayered substrate through the relationship jzs (ρ) = (ks2 /μ0 )A(ρ) and integral representation of the magnetic vector potential in the substrate in terms of the volumetric current density in the MTL conductors and the substrate Green’s function G A z (ρ) = G (ρ, ρ ) jz (ρ ) ds (39) S


9

S ,∂ S

α α Fig. 8. Magnitude of the matrix elements Z σ,n m (31) for the 2-D transmission line example at frequencies f = 1 GHz, f = 5 GHz, f = 10 GHz, m = 21, and n = 1 . . . N .

Fig. 7. p.u.l. self-resistance R11 and self-inductance L 11 in two-conductor MTL shown in Fig. 5 as functions of frequency for various levels of the substrate conductivity σs . Parameters extracted using the proposed SVS-EFIE solution are compared against independent reference solution [17] and FEM simulation in the COMSOL multiphysics package.

for any observation point ρ. The distribution of the induced current in the substrate computed using (39) with jz obtained from the fictitious current Jz and Green’s function representation (25) is plotted in Fig. 6. For comparison, the plot of the induced current computed in the substrate using (39) with jz obtained via traditional MoM solution of V-EFIE (1) is shown together with the current distribution obtained via COMSOL multiphysics simulation at frequency f = 5 GHz. D. Frequency-Dependent Resistance and Inductance The MTL network parameters of interest in the magnetoquasi-static analysis are the p.u.l. inductance and resistance matrices [10]. For the two-conductor transmission line situated on top of the lossy half-space (Fig. 5), the self-resistance R11 and the self-inductance L 11 are shown in Fig. 7, for varying levels of conductivity σs of the substrate. The extracted p.u.l. R11 and L 11 are compared against the COMSOL commercial software and the independent V-EFIE MoM solution obtained by different authors of the same problem provided in [17]. Close agreement of R11 and L 11 is observed in the wide range of frequencies and substrate conductivity values. E. Skin-Effect Driven Meshing and MoM Matrix Sparsity The magnitudes of the matrix elements in ZσS,∂ S (31) corresponding to the line-to-triangle interactions in the two-conductor MTL of Fig. 5 are plotted for different frequencies in Fig. 8 as a function of distance |ρ αn (1/3, 1/3)− ρ αm (1/2)| from the centroid of source line element m to the centroid observation triangle n . For this experiment, the source line element in the discretization of the conductor surface ∂ S is fixed. It is seen that for the frequencies

Fig. 9. Relative error in self-resistance and inductance extracted with MoM solution (38) of the SVS-EFIE (5) with respect to the MoM solution (37) of the SVS-EFIE for the 2-D transmission lines depicted in Fig. 5 for different frequencies and sparsification criteria β, where β is the number of skin depths within which the elements are deemed to have sufficiently strong interaction.

f 2 = 5 GHz and f 3 = 10 GHz at which the skin effect is well developed, the magnitude of the matrix elements drops dramatically with distance. Thus, the usage of the SVS-EFIE in the sparse matrix form (38) is preferred. Next, we study the effect of the sparsification criteria β on the accuracy of the SVS-EFIE solution. The dependence of the relative error |R11 − R˜ 11 |/|R11 | in the extracted selfresistance R˜ 11 and relative error |L 11 − L˜ 11 |/|L 11 | in the self-inductance L˜ 11 using sparse matrix form of the SVS-EFIE (38) with respect to the network parameters R11 and L 11 obtained using dense matrix form of the SVS-EFIE (37) is shown as a function of β for different frequencies in Fig. 9. The plot demonstrates that the choice of β ≥ 4 for the MoM discretized SVS-EFIE in the sparse matrix form (38) provides relative error below 2% in the extracted self-resistance and inductance in comparison with the dense matrices solution (37). The results were obtained using a C++ implementation of the MoM solutions (37), (38) on a single core of Intel Core i7 processor running at 2.7 GHz. The unparalleled version of Intel Math Kernel Library was used for matrix–matrix products



TABLE I CPU T IME AND M EMORY U SED BY C++ I MPLEMENTATION OF THE MoM D ISCRETIZED SVS-EFIE W ITH AND W ITHOUT S PARSITY AND MoM D ISCRETIZED V-EFIE

tangential components of the magnetic vector potential to zero at the boundary). The entire geometry is discretized by free triangular meshes. The discretization of cross sections of two conductors is performed according to the skin effect that is two triangular elements per skin depth are used to capture the variation of the current density. We use external current density excitation for the conductors to add an externally generated current density jez , which appears in the righthand side of differential equations. The value of the external current density could be selected as jez /σ = Vp.u.l., where Vp.u.l. is the per unit length voltage drop on the right-hand side of the integral (5). The choice of impressed current excitation in COMSOL results in Helmholtz equation with respect to magnetic vector potential A, which would lead to the same integral equation with respect to the electric field inside the conductors as V-EFIE and subsequently the SVS-EFIE. Hence, the numerical results from COMSOL can be used to verify the proposed SVS-EFIE solution. From Figs. 6 and 7, we observe that the results obtained using FEM with COMSOL multiphysics have good agreement with the results obtained using the proposed SVS-EFIE. VI. C ONCLUSION

and direct solution of the SLAE. The network parameters were first extracted via the MoM solution of the SVS-EFIE using sparse matrices (38) with sparsification parameter β = 4, and then the execution time compared with the MoM solution of the SVS-EFIE in dense matrix form (37). For comparison, in Table I, we provide the matrix fill time, matrix-vector product time, and matrix storage for both version of MoM discretized SVS-EFIE solutions (37), (38) as well as for the MoM discretized solution of V-EFIE (1). The matrix fill time for the MoM solution of the SVS-EFIE (5) using sparse matrices is seen to be significantly reduced at frequencies 5 and 10 GHz for which the skin effect is well developed.

This paper proposes a new SSIE for solution of timeharmonic current flow in 2-D MTLs embedded in multilayered substrates. Detailed description of Green’s function formulation used in the solution is provided. The closed-form Green’s function of the multilayered medium in the spatial domain is obtained via FD solution of 1-D spectral domain ODE governing the spectrum of Green’s function. The new surface integral equation formulation enables efficient resistance and inductance extraction in complex 2-D MTLs with consideration of the current flow in complex lossy substrate. It is also shown how to utilize the sparsity pattern in the matrices arising from the MoM discretization of the novel SVS-EFIE allowing to greatly reduce both computational time and memory requirements.

F. COMSOL Simulation Setup Figs. 6 and 7 validate the proposed solution using finite-element method (FEM) implemented in the COMSOL multiphysics package. In COMSOL, a 2-D model was created by selecting ac/dc module and magnetic and electric fields interface. In the proposed method, the domain of air and the substrate extends to infinity. Exact replication of the model is not possible in COMSOL, however, due to finite region of field computations in FEM. In our simulation, we enclosed the entire model geometry by a circle shield with a radius of 30 mm. The interface between air domain and the substrate domain divides the circle into two half-circle regions each imitating unbounded half-space of the air and the substrate. Two rectangular conductors arranged in parallel are located in the center of the circle, and just above the interface. The resultant geometry mimics that shown in Fig. 5. With this configuration, the air and substrate domains are extended far enough, so that the influence of the terminating shield becomes negligible. The boundary condition for the shield is magnetic insulation (e.g., nˆ × A = 0, which sets the

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Shucheng Zheng (S’16) was born in Fujian, China, in 1987. He received the B.Sc. degree with Distinction in electrical and computer engineering from the University of Manitoba, Winnipeg, MB, Canada, in 2016, where he is currently pursuing the M.Sc. degree in electrical and computer engineering. His current research interests include computational electromagnetics, multilayered media Green’s functions, modeling of high-speed interconnects, and transient analysis of power systems.

Anton Menshov (S’12) was born in Kovrov, Russia, in 1988. He received the B.S. degree in automatics and management from the Moscow Institute of Electronic Technology, Moscow, Russia, in 2010, and the M.Sc. degree in electrical and computer engineering from the University of Manitoba, Winnipeg, MB, Canada, in 2014. He is currently pursuing the Ph.D. degree in electrical and computer engineering at The University of Texas at Austin, Austin, TX, USA. His current research interests include fast algorithms for computational electromagnetics and inverse problems.

Vladimir I. Okhmatovski (M’99–SM’09) was born in Moscow, Russia, in 1974. He received the M.S. degree (Hons.) in radiophysics and the Ph.D. degree in antennas and microwave circuits from the Moscow Power Engineering Institute, Moscow, in 1996 and 1997, respectively. In 1997, he joined the Radio Engineering Department, Moscow Power Engineering Institute, as an Assistant Professor. He was a Post-Doctoral Research Associate with the National Technical University of Athens, Athens, Greece, from 1998 to 1999, and the University of Illinois at Urbana–Champaign, Champaign, IL, USA, from 1999 to 2003. From 2003 to 2004, he was with the Department of Custom Integrated Circuits, Cadence Design Systems, San Jose, CA, USA, as a Senior Member of Technical Staff, and from 2004 to 2008 as an Independent Consultant. In 2004, he joined the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, Canada, where is currently an Associate Professor. His current research interests include the fast algorithms of electromagnetics, high-performance computing, modeling of interconnect, and inverse problems. Dr. Okhmatovski is a Registered Professional Engineer in the Province of Manitoba, Canada. He was a recipient of the 1995 Scholarship of the Government of Russian Federation and the 1996 Scholarship of the President of the Russian Federation. He was a recipient of the 1996 Best Young Scientist Report of the VI International Conference on Mathematical Methods in Electromagnetic Theory. He was also a co-recipient of the Best Paper Award of the 3rd Electronic Packaging Technology Conference in 2001 and the Outstanding ACES Journal Paper Award in 2007.