HIGH speed interconnects both on-board and on-chip that

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1 Eigenmode-based Capaitane Calulations with Appliations in Passivation Layer Design Thomas Demeester and Daniël De Zutter, Fellow, IEEE Abstrat The design of high-speed metalli interonnets suh as mirostrips requires the orret haraterization of both the ondutors and the surrounding dieletri environment, in order to aurately predit their propagation harateristis. A fast boundary integral equation approah is obtained by modeling all materials as equivalent surfae harge densities in free spae. The apaitive behavior of a finite dieletri environment an then be determined by means of a transformation matrix, relating these harge densities to the boundary value of the eletri potential. In this paper a new alulation method is presented for the important ase that the dieletri environment is omposed of homogeneous retangles. The method, based on a surfae harge expansion in terms of the Robin eigenfuntions of the onsidered retangles, is not only more effiient than traditional methods, but also more aurate, as shown in some numerial experiments. As an appliation, the design and behavior of a mirostrip passivation layer is treated in some detail. Index Terms Interonnets, apaitane, Robin eigenfuntions, passivation layer, mirostrip design. I. INTRODUCTION HIGH speed interonnets both on-board and on-hip that display substantial wave effets, are often long enough to be modeled as two-dimensional transmission line strutures, suh as mirostrips or striplines, both on the printed iruit board (PCB) level, as on-hip. For suh models, the so-alled RLGC iruit matries, i.e., R, L, G, C, respetively, the resistane, indutane, ondutane, and apaitane matries, in ombination with the telegrapher s equations desribe the behavior of the fundamental propagation modes along the lines. It is shown in [] that for typial high-speed interonnet appliations, the so-alled quasi transverse magneti (quasi- TM) onditions remain valid even for the highest operating frequenies. This is the ase, as long as the orresponding longitudinal wavelength remains enough longer than the transverse dimension over whih the fields are relevant. If this were not the ase, the struture would no longer be apt for an effiient data transmission. Full-wave solutions to Maxwell s equations as a basis for the RLGC matrix extration have been proposed as well [2]. These however lead to different models (whih all oinide in the quasi-tm frequeny range), and they loose a major advantage of quasi-tm models: the quasi-tm onditions lead to a deoupling of the total eletromagneti field problem into (i) a quasi-stati omplex apaitane problem (to determine the omplex apaitane C + G/jω, with ω the angular frequeny) and (ii) a quasi-tm omplex indutane problem (to alulate L + R/jω). The authors are with the Department of Information Tehnology, Ghent, Sint-Pietersnieuwstraat 4, B-9000 Gent, Belgium. Tel: , Fax: thomas.demeester@inte.ugent.be, daniel.dezutter@inte.ugent.be. The numerial solution of a apaitane problem has been under investigation for a long time now and many different solution methods have been developed. Some of these, as well as a further overview of the existing literature, are found in [3] []. Even today, the quasi-stati approah to determine the apaitive behavior of many pratial interonnets remains valid. The urrent paper proposes a new numerial solution method that is similar to the reent tehnique desribed in [], in the sense that it allows for an effiient boundary integral equation based solution. The onsidered onfigurations are not restrited to infinitely wide layered media but require that the ross-setion an be divided into an ensemble of homogeneous retangles, whih is often the ase. As remarked by one of the reviewers, a solution tehnique using a standard surfae integral equation for the ondutor-dieletri eletrostati problem, see e.g. [2] setion 3.2., is a valid alternative for the tehnique presented in this paper and an be applied to arbitrary shapes. An example of the use of this tehnique an be found in [3]. In our tehnique the use of the more singular derivative of the Green s funtion is avoided. In setion II, some elements of the method of [] are briefly summarized, and from these the new method is developed, with a mathematial desription in Setion III. In the numerial examples of Setion IV, firstly some properties of the new field expansion funtions are illustrated, seondly a omparison between the method of [] and the new tehnique is made, and finally, a passivation layer design example is disussed, as the new method appears to be well-suited for thin layers. II. CAPACITANCE CALCULATIONS For ondutors in a lossy dieletri environment that exists of finite homogeneous subregions, a fast solution for the apaitane problem, based on a boundary integral equation, was presented in []. Some aspets of this method are briefly reapitulated here, as they are needed in the sequel. First, all materials are replaed by equivalent surfae harge densities ρ in free spae, suh that the quasi-stati 2-D Green s funtion G 0 of free spae an be used to relate the eletri potential φ to these harges, aording to φ(r) = G 0 (r r )ρ(r )d(r ) () i in whih the integral runs over the boundary i of eah homogeneous subregion i and with G 0 (r r ) = 4πɛ 0 ln( r r ). (2)

2 2 Now, ρ and φ are disretized as ρ(r) m R m b m (r) and φ(r) m Φ m b m (r) (3) for r on the onsidered boundary, and {b m } a set of appropriate basis funtions. A Galerkin weighing of (), evaluated on the material boundaries, yields a first set of relationships between the unknowns R m and Φ m, in the form [ ] [ ] [ ] V G G d R =, (4) Φ d G d G dd R d in whih the expansion oeffiients for the harge, respetively, eletri potential are taken together for the ondutors in R and V, and for the dieletris in R d and Φ d. Furthermore, the onstant voltage V on the ondutors boundaries is used as exitation, and the equivalent harges on the boundary of dieletri subregion i an be related to the boundary value of its potential, through ρ(r) = (ɛ i ɛ 0 ) φ(r) n, r i (5) = (ɛ i ɛ 0 ) D(r,r )φ(r )d(r ), i (6) withd the Dirihlet-to-Neumann operator (DtN), mappingφ i onto its normal derivative along i, and with ɛ i the permittivity of material i. In its disretized form, and written for all dieletris together, (6) beomes R d = DΦ d (7) whih allows, upon substitution into (4), to determine all unknown oeffiients. The total harge on eah ondutor, leads to the entries of the apaitane (and ondutane) matrix if eah ondutor is set in turn to V while the others are kept on 0V. The method as explained above works very well, but has one major drawbak due to the disretization of the DtN operator into the DtN matrix D. The alulation of D in the quasi-stati frequeny range is based on an expansion of φ over the dieletri subregions (disretized in retangles or triangles) in terms of sine basis funtions on eah side, while enforing the exat solution for φ whih, in the quasi-tm limit, satisfies Laplae s equation 2 φ(x,y) = 0. Its normal derivative is then determined as the superposition of the normal derivatives of eah basis funtion. This is desribed in [4] and [5], for retangular, respetively, triangular geometries, but in this paper we fous on the important ase of retangular dieletri subregions. A Galerkin weighing proedure yields an expansion of φ with a minimal quadrati error along the boundary. The small omponents in φ, whih vary rapidly along the boundary, get a more important weight in φ/ n. On the one hand, enough higher harmonis are required for the representation of any boundary value of φ, but on the other hand, a small error on these omponents leads to a strongly inreased error in the normal derivative, as demonstrated further on. It espeially deteriorates the auray of ρ near the orners, beause of an often imperfet estimation of φ at the orner points, used to eliminate the Gibb s effet [4]. In Setion III, this problem is solved by means of an expansion with a minimal quadrati error for ρ instead of φ over the onsidered dieletri regions. This has as a diret onsequene that the apaitane alulations, by integrating ρ, are muh more aurate. Indiretly, the error on φ, in turn no longer minimal in a quadrati sense, remains small as we integrate ρ to obtain φ, and beause φ is generally muh smoother than ρ. Of ourse, the boundary value of φ on a dieletri blok annot be determined unambiguously from φ/ n. Therefore, we will first extrat a onstant φ 0 from φ, by averaging over the boundary. For the remaining omponent(φ φ 0 ), its relation to φ/ n is bijetive, and we will onstrut an operator B whih maps φ/ n on boundary i onto (φ φ 0 ), suh that with (5), φ(r) = φ 0 + B(r,r ) ρ(r )d(r ), r i. (8) i ɛ i ɛ 0 The disretized form of (8), taken together for all dieletris, beomes Φ d = Φ 0 +BR d (9) in whihφ 0 is a olumn vetor, for eah dieletriiontaining a different unknown onstant potential φ 0,i. Substitution of (9) into (4) leads to a system of equations to be solved with respet to the unknown R, R d and the mean dieletri boundary potentials φ 0,i. This system has to be ompleted with as many extra equations as there are unknowns φ 0,i, expressing the requirement that the total equivalent harge on eah dieletri has to remain zero. III. ROBIN EIGENFUNCTION EXPANSION OF THE CHARGE DENSITIES In this setion, we will explain how the relation φ(r) φ 0 = B(r,r ) φ(r ) n d(r ), r (0) is disretized along the boundary of a retangular area S { x 0 x x 0, y 0 y y 0 }, for a quantity φ satisfying Laplae s equation inside S. We will need the following expansion φ(r) N n = χ n ψ n (r), r () n= with N high enough (see Setion IV), and in whih the basis funtions ψ n satisfy 2 ψ n (r) = 0, r S (2) ψ n (r) = λ n ψ n (r), r (3) n ψ n (r)ψ m (r)d = δ nm (4) with δ nm = if n = m, and else δ nm = 0. Beause of the soalled Robin boundary ondition (3), the orthonormal set of eigenfuntions we wish to onstrut, are Robin eigenfuntions of S. Separation of the variables (SoV) in this ase (omitting the index n), and already taking into aount (3) on the sides

3 3 x = 0 and y = 0, yields the following four independent sets of solutions { } { } osγx oshγy ψ(x,y) = C (5) sinγx sinh γy with a normalization oeffiient C yet to be determined. The four equations are obtained by ombining eah of the two terms between the first brakets with eah of the terms between the seond brakets. The remaining onditions, 0 = [ ψ x λψ ] x=x 0 and 0 = [ ψ y λψ lead with (5), and by elimination of λ, to { } { } tanγx0 tanhγy0 = otγx 0 othγy 0 ] y=y 0 (6) (7) 0 0 ψ 0 (λ = 0) ψ (λ = 0.768π) ψ 2 (λ = 0.486π) ψ 0 (λ =.737π) ψ (λ =.763π) ψ 2 (λ = π) ψ 20 (λ = π) ψ 2 (λ = π) ψ 22 (λ = π) displaying four different equations, orresponding to the different ombinations for ψ in (5). Four sets of solutions for γ an now be alulated, whih are all real and positive (as the orresponding negative solutions lead to the same eigenfuntions and are thus omitted), with the orresponding eigenfuntion given by (5) and the eigenvalues by { } tanγx0 λ = γ. (8) otγx 0 We see that, if we replaeγ in (7) byjγ, the same eigenvalue equations are found, provided x 0 and y 0 are exhanged. The same observation holds for (5). As a result, the algorithm to solve (7) an be used again, with x 0 and y 0 exhanged. The real solutions one finds now, are in fat the remaining stritly imaginary solutions of (7). The numerial proedure to solve (7) is straightforward. For the examples treated in this paper, the seant method is used, whih onverges rapidly and to an arbitrary preision, due to the regularity in the zeros, and the asymptoti behavior for large values of γ. Combining (2) and (3) with Gauss law, shows that two eigenfuntionsψ i andψ j are always orthogonal with respet to integration overif they have different eigenvaluesλ i λ j. In some ases, different funtions are found with the same eigenvalues, e.g., for x 0 = y 0, but in these ases those funtions are orthogonal as well. A more rigorous mathematial analysis would be outside the sope of this paper, however. Finally, for eah eigenfuntion ψ i, the normalization oeffiient C i an be determined by equating ψi 2, after integration over, to one. As an illustration, the boundary value of a few Robin eigenfuntions is shown in Fig., for a retangle with dimensions 2, with onseutive sides, respetively,, 2, 3, and 4. The eigenvalues are given as well. For the higher values of γ n, (8) shows that λ γ, and from (7) it then follows that γx 0 (2k + )π/4, k (as is the ase for ψ 22 ), or, for the seond set of eigenvalues,γy 0 (2k+)π/4 (as for ψ 20 and ψ 2 ). Beyond the first 20 eigenvalues, these approximations form a very good starting guess in the root finding proedure, allowing to reah a suffiient auray within very few iterations. Notie the presene of the onstant eigenfuntion ψ 0, orresponding to the zero eigenvalue. ψ 0 is Fig. : Boundary value of some Robin eigenfuntions, for a retangle with dimensions 2. not present in (), as the mean of φ/ n has to be zero, but in a general expansion it is of ourse required. Consider again the expansion (), in whih φ/ n is disretized as φ(r) n = m E m b m (r), r (9) For the basis funtions b m along, pulses are used in the examples in Setion IV, but other hoies are equally possible. The Galerkin weighing proedure leads from () and (9) to the disretized form X = P T E (20) in wih X and E ontain the oeffiients χ n (see ()), respetively, E m, and [ ] P = b m (r)ψ n (r)d. (2) mn If we expand φ on S in terms of the new eigenfuntions, we find that φ(r) = N α n ψ n (r), r S (22) n=0 α 0 ψ 0 = C φ(r)d(r) def = φ 0 (23) with C the perimeter of S. Comparing the normal derivative of (22) to (), shows with (23) that φ(r) φ 0 = N n= χ n λ n ψ n (r), r (24)

4 4 and if we define Φ m def = Φ m φ 0, and weigh (24) and (3) with eah of the basis funtions b m, we find the diret disretization of (0), Φ = BE, with B = T PΛ P T (25) in whih the olumn vetor Φ ontains the oeffiients Φ m, the diagonal matrix Λ ontains the eigenvalues λ n, (n =,...,N) on its diagonal, and [ ] T b m (r)b m (r)d(r). (26) m m = For the alulation of the DtN matrix D as in [4], analogous matries as P and P T are required, but instead of the diagonal matrix Λ in (25) a muh slower matrix multipliation with a large non-sparse matrix is required, to transform the sine expansion into the orresponding normal derivative. Here a numerial root finding proedure is required to determine the diagonal elements of Λ. Globally, the determination of B remains more effiient, apart from the fat that the results are by far more aurate (see Setion IV-B). It is worth mentioning that these Robin eigenfuntions represent a very natural basis to alulate the matrix B from. The reason is, that its eigenvetors are a disretized version of the eigenfuntions ψ n, with eigenvalues approximately /λ n. This diretly follows from the boundary ondition (3) in omparison with the eigenvalue equation for B. As an illustration, the first and eleventh eigenvetors of B are plotted in Fig. with small ross symbols, and ompared with ψ and ψ, respetively, where the retangle was disretized with 38 intervals along its boundary. In order to ompensate for the different normalization, they were resaled with a salar fator of / ( k d kv k ) (with d k the interval width, and V k the entries of the onsidered eigenvetor with unit eulidean norm), and the orrespondene is indeed good. IV. NUMERICAL EXAMPLES A. Investigation of the Expansion Properties As a first, theoretial, example, the numerial error on φ/ n is ompared as obtained from the DtN matrix, with respet to the new expansion in Robin eigenfuntions. Consider again the retangle S from Setion II, this time disretized with 38 intervals along its boundary, as shown in the inset of Figure 2. Take an arbitrary funtion φ(x,y) = osh( x/x 0 )sin(y/x 0 ) (with x 0 = and y 0 = 0.5) that satisfies Laplae s equation. In Fig. 2(a), the ontinuous value of φ/ n is shown as a referene, as well as two pulsebased approximations (displaying the oeffiients in the middle of the intervals). These were obtained as follows. From a Galerking weighing of the exat boundary value of φ, the vetor Φ was obtained, and the oeffiients displayed with x -symbols, are found as DΦ from the DtN matrix D. The oeffiients shown in dots are found from an expansion in the Robin eigenfuntions, with the same number of basis funtions as used to alulate the DtN matrix. From E, the best possible disretization (in a least squares sense) of φ/ n, we obtain the oeffiients of the Robin funtions as P T E. Again projeting these on the basis of pulses yields normal derivative φ/ n abs. error on disret. of φ/ n 2 exat via DtN matrix via Robin expansion (a) (b) 3 E D E B Fig. 2: (a) Boundary valueφ (ontinuous and disretized) for a retangle with dimensions 2, (b) normal derivative φ/ n (ont., disret., and approximations via the DtN matrix and via the Robin expansion, both with the same amount of expansion funtions), and (b) absolute error on both approximations. T PP T E, shown with dots in Fig. 2. The absolute value of the differene between the ideal oeffiients E and both approximations is shown in Fig. 2(b). The overall auray using the Robin eigenfuntion expansion is, as expeted, muh better than with the DtN matrix. The reason is primarily due to the inaurate treatment near the orners, and to the inreased weight of higher order ontributions in the DtN approah. In this example,304 expansion funtions were used (to both alulate D and P), or 8 times the number of disretization intervals. Let us in general find out how the error behaves in terms of the number N of basis funtions (for Dirihletvs. Robin expansion), and in relation to the number M of disretization intervals. In Fig. 3, the error is shown for varying N (normalized by M), for two different disretizations, i.e., M = 38 (as for Fig. 2) and M = 98. The shown error is a relative root mean square (RMS) error, defined as error = ( [ φ/ n]approx. φ/ n ) 2 d ( ) 2d (27) φ/ n in whih pulses are used to disretize the ontinuous quantities. We observe a minimum in the error for the DtN approah, when about as many expansion funtions are used as the number of disretization segments. If more expansion funtions are used, the error grows, due to the reinforement of inauraies in the alulation of the normal derivatives. In pratie, the boundary value of φ might vary muh faster over the sides 3 4 4

5 relat. error on approx. of φ/ n via Robin expansion via DtN matrix M = 38 M = 98 ɛ oat ɛ sub tanδ sub σ σ w s W t sig h t ref Fig. 4: Coated differential pair (not shown on sale), with t sig = 0.5oz (oz = 34.8µm), t ref = oz, h = 5mils (mil = 25.4µm), s = 5mils, σ = 58MS/m (opper), ɛ sub = 4.3ɛ 0 and tanδ sub = Variable parameters are w, and ɛ oat N/M Fig. 3: Relative RMS error on both approximations of φ/ n (via DtN matrix, and via Robin expansion), for the onfiguration from Fig. 2. N is the number of expansion funtions, M the number of disretization intervals. than in this theoretial example. In that ase, a number of sine funtions equal to the number of intervals along that side ould be insuffiient to expand the boundary value, and the error would onsequently inrease. If too many expansion funtions are hosen, the error beomes muh larger than the monotonially dereasing error of the Robin expansion (see the results in IV-B). The behavior for a finer disretization is qualitatively the same, but with an overall lower error. The numerial errors shown in Fig. 2 and Fig. 3 are an inomplete way to ompare the auray of both methods. The error on φ/ n for the DtN expansion is neessarily larger than for the Robin expansion, as the former one involves a hange of basis, subsequently the transformation from φ to φ/ n and then again a hange of basis, whereas the latter expansion is only a asade of two basis transformations. One ould put forward the question how large the error on φ would be in the Robin ase, inluding the transformation φ/ n φ, but then again, it is not possible to diretly ompare this error to the ones shown in Fig. 3. Therefore, in the next paragraphs a diret omparison between both methods is performed on the apaitane level. B. Comparison DtN vs. Robin Method Consider the differential pair struture shown in Fig. 4, used in this setion without the dieletri oating ( = 0), and with w = 6.56mils. The apaitane problem is solved by both the DtN method and the new Robin method. To investigate the influene of the disretization and the number of expansion funtions used inside the substrate, its width W is varied from narrow (W = 2w+s = 8.mils = 460µm, only underneath the differential pair) up to very wide (W = 00 mils = 2540 µm). As a referene, the same struture was simulated with the Capad software developed by F. Olyslager, whih uses a boundary integral equation approah, based on the Method of Moments with the Green s funtion of an infinitely wide multilayered medium [6]. That program only deals with perfet ondutors (inluding an infinitely wide perfet onduting ground plane), but for the quasi-stati apaitane problem, the finite ondutivity of the ondutors is irrelevant. Nonetheless, this only yields the asymptoti solution for a very wide substrate. In order to learn as muh as possible about the numerial behavior of both methods, simulations were done for three different disretizations. In the onfigurations designated as oarse, medium and fine, the number of segments along the width of eah ondutor are 6, 2 and 24, respetively (and 50 for the referene simulation with Capad). For eah simulation, the disretization is realulated automatially by our software and hene, the segment width slightly inreases for wider substrates. In order to keep the omparison between the different simulations transparant, the disretization per side is hosen to be fairly uniform (whereas normally it would be muh denser near orners in order to inrease the auray for a lower number of unknowns). For both methods, the parameter α is defined as the number of expansion funtions (Dirihlet or Robin) over the substrate, divided by the number of disretization segments along its boundary. The different values of this parameter that are used in the simulations are α =, 2, 4, 8 and 6. Note that for a finer disretization, a proportionally larger number of basis funtions is used for the same α. In Fig. 5, the self-apaitane entries C are displayed. Using a dashed line, the referene value for an infinitely wide substrate is indiated on eah figure. The left olumn displays the results for the DtN method, with a finer disretization from top to bottom. In eah ase, the apaitane results inrease for a higher α. This onfirms our assumption that one annot add too many higher order Dirihlet funtions in the expansion of φ, as small errors on φ lead to unaeptable errors on φ/ n. Clearly, the results are better for a finer disretization, and the algorithm is normally used for a fine disretization and α around 3. For the Robin method, displayed on the right-hand side, the behavior is different. On the one hand, a fator α = leads to unreliable results, indiating that the same number of expansion funtions as the number of boundary segments is not suffiient to alulate φ from φ/ n. As soon as α >, the results are very onsistent, and not only for

6 6 Capaitane C (pf/m) DtN Robin 0 Re(Z diff ) as a funtion of f and oarse medium fine α = α = α = W (mils) W α = α = α = α = W (mils) α = α = Fig. 5: Capaitane C for the struture of Fig. 4 (with = 0, w = 6.56mils), with the DtN method (left), and the Robin method (right), for 3 different disretizations ( oarse, medium and fine ), as a funtion of the substrate width W and the parameter α, the ratio of the number of expansion funtions to disretization segments. The dashed line is the referene value for an infinitely wide substrate. Fig. 6: Simulated geometry, onsisting of 3 retangular bloks (shown on sale, but displaying only half of the simulated substrate width). The dimensions are as in Fig. 4, with w = 6.56mils, and = 0.5oz. the fine disretization (although in the oarse ase, there is a.3% underestimation of the result, due to the inaurate disretization). Two major advantages of the Robin method have ome forward. On the one hand, the results onverge to a fixed value, as soon as suffiient Robin funtions are used (whih is not the ase for the DtN method). On the other hand, the fator α is about the same as for the DtN method, but it allows for a oarser disretization. This means that the absolute number of Robin funtions required is lower than the required number of Dirihlet funtions, and the total number of unknowns is smaller than for a similar auray with the DtN method. C. Modeling of a Passivation Layer As an appliation example, we will study the effet of a thin dieletri passivation layer on top of the mirostrip struture Re(Zdiff) (Ω) f = 0MHz f = 00MHz f = GHz f = 0GHz f = 00GHz f = 50GHz Coating thikness (oz) Fig. 7: Re(Z diff ) (Ω) for the struture of Fig. 4, with w = 6.56mils and ɛ oat = 3.3ɛ 0, for variable oating thikness, and shown at several frequenies. The dash-dot line at 50GHz is used as a referene for further simulations. used in the previous example. Suh a oating onsists of an inert dieletri material, and serves the purpose of proteting the surfae against humidity and orrosion [6]. This however leads to an inreased total amount of dieletri material near the traes, and hene a lower harateristi impedane. This effet is often ompensated by reduing the trae width, whih in turn leads to an inreased resistane and hene higher propagation losses. The new tehnique developed in this paper to deal with dieletris, is well-suited for the simulation of a thin passivation layer, due to its auray, as seen in Setion IV-B. We will explore in detail how the harateristi impedane of a differential mirostrip pair is affeted by the thikness and dieletri onstant of the passivation layer, in ombination with the trae width. The purpose is to demonstrate our tehnique by means of simulation results that are useful for the highspeed digital designer. The harateristi impedane of a symmetrial onfiguration is in reality more often used in the design of transmission line strutures than separate values of the apaitane, ondutane, indutane, and resistane, beause it diretly determines the modal refletion oeffiients at both ends of the line, depending on the soure and load impedane. For a symmetrial struture with two signal lines and a referene ondutor, the differential impedane is defined as twie the odd mode impedane, or hene [2], Z diff = 2 jω(l s L m )+(R s R m ) jω(c s C m )+(G s G m ) (28) with the subsript s denoting the diagonal elements of the 2 2 iruit matries, m the non-diagonal elements, and C, G, L, and R, respetively, the apaitane, ondutane, indutane and resistane elements. Note that C m and G m are always negative while L m and R m are positive. The apaitane and ondutane matries are found with the quasi-stati tehnique desribed in Setion II and III. The

7 00 7 oating permittivity ɛoat 4ɛ 0 3ɛ 0 2ɛ 0 Re(Z diff ) (Ω) as a funtion of ɛ oat and Coating thikness (oz) Fig. 8: Re(Z diff ) (Ω) for the struture of Fig. 4, with w = 6.56mils and at f = 50GHz, for a variable oating permittivity ɛ oat and thikness. The dash-dot line (ɛ oat = 3.3ɛ 0 ) orresponds to the 50GHz urve of Fig trae width w (mils) Re(Z diff ) (Ω) as a funtion of w and Coating thikness (oz) Fig. 9: Re(Z diff ) (Ω) for the struture of Fig. 4, with ɛ oat = 3.3ɛ 0 and at f = 50GHz, for a variable trae width w and thikness. The dash-dot line orresponds to the dash-dot lines in Fig. 7 and resistane and indutane matries are determined aurately with the tehnique desribed in [], up to high skin effet frequenies. Some useful information onerning differential pair interonnets from the viewpoint of the designer an be found in [7] and a similar geometry is used here, as shown in Fig. 4. The values of material and geometry parameters are given in the aption, apart from the spaing s between the traes, the oating thikness and the oating permittivity ɛ oat, whih are varied in the following simulations. Note that the metal thikness is expressed in oune ( oz ), as is ommon pratie amongst designers ( oz = 34.8 µm). The other dimensions are expressed in mils (mil = 25.4µm), following [7]. Furthermore, the total width of the simulated struture isw = 40mils, suh that the end effet of the finite substrate has a negligible influene on the differential behavior of the line. The oating thikness is assumed onstant along the surfae (as is indiated for the right ondutor in Fig. 4). The Robin eigenfuntion tehnique desribed in Setions II and III is applied by modeling the oating as a sequene of 9 onneted retangular dieletri bloks. Fig. 6 shows the 3 retangular simulation bloks that have been used: 9 passivation bloks, 2 ondutor bloks, a dieletri substrate and a ground plane blok. The geometry that we will use as the baseline throughout the different simulations, is the struture as shown in Fig. 4, with w = 6.56mils, ɛ oat = 3.3ɛ 0, and a varying oating thikness. As seen from Fig. 7, the dimensions are hosen suh, that its high-frequeny differential impedane Z diff beomes 00Ω in the ase without oating ( = 0). The overall behavior is as expeted, with a lower impedane as the oating thikness inreases. In a following simulation, in order to find out how Z diff depends on the oating permittivity ɛ oat, the frequeny is fixed at 50GHz, and ɛ oat is swept from.5ɛ 0 tot 4.3ɛ 0, with the result shown in Fig. 8. The dash-dot line shows the result for ɛ oat = 3.3ɛ 0, whih orresponds to the dash-dot urve at 50GHz in Fig. 7. The results displayed in Fig. 8 an be easily interpreted. For an inreasing oating thikness, the impedane dereases faster for a higher oating permittivity. The question is, how the geometry should be modified to ompensate for the effet of the oating. In the following experiment, we will redue the trae width w, keeping the spaing between the traes onstant. The resulting Z diff is shown in Fig. 9 as a funtion of w and. As a referene for the reader, the dash-dot line again orresponds to the result for the baseline geometry, already displayed in Fig. 7 and 8. The bold line shows the required ompensation of w as a funtion of the oating thikness, in order to maintain a 00Ω differential impedane. V. CONCLUSION This paper presents a new tehnique for fast apaitane alulations, based on the Robin eigenmode expansion of the eletri potential in homogeneous retangular dieletri subregions. A areful omparison with the Dirihlet-to-Neumann approah shows that the new tehnique performs better both in terms of auray and effiieny. As a result, hallenging onfigurations with very thin dieletri layers an be dealt with. This is demonstrated with the analysis of a differential mirostrip interonnet, oated with a dieletri passivation layer. ACKNOWLEDGMENT The authors would like to thank dr. Bart Baekelandt for his useful advie on some pratial aspets of high-speed interonnet design. REFERENCES [] T. Demeester and D. De Zutter, Quasi-TM transmission line parameters of oupled lossy lines based on the Dirihlet to Neumann boundary operator, IEEE Trans. Mirow. Theory Teh., vol. 56, no. 7, pp , Jul

8 8 [2] F. Olyslager, Eletromagneti Waveguides and Transmission Lines. Oxford, U.K.: Oxford University Press In., 999. [3] E. Denlinger, A frequeny dependent solution for mirostrip transmission lines, IEEE Trans. Mirow. Theory Teh., vol. 9, no., pp , Jan. 97. [4] C. Wei, R. Barrington, J. Mautz, and T. Sarkar, Multiondutor transmission lines in multilayered dieletri media, IEEE Trans. Mirow. Theory Teh., vol. 32, no. 4, pp , Apr [5] W. Delbare and D. De Zutter, Spae-domain green s funtion approah to the apaitane alulation of multiondutor lines in multilayered dieletris with improved surfae harge modeling, IEEE Trans. Mirow. Theory Teh., vol. 37, no. 0, pp , Ot [6] F. Olyslager, N. Fahé, and D. De Zutter, New fast and aurate line parameter alulation of general multiondutor transmission lines in multilayered media, IEEE Trans. Mirow. Theory Teh., vol. 39, no. 6, pp , jun 99. [7] K. S. Oh, D. Kuznetsov, and J. Shutt-Aine, Capaitane omputations in a multilayered dieletri medium using losed-form spatial Green s funtions, IEEE Trans. Mirow. Theory Teh., vol. 42, no. 8, pp , Aug [8] J. Bernal, F. Medina, and M. Horno, Quik quasi-tem analysis of multiondutor transmission lines with retangular ross setion, Mirowave Theory and Tehniques, IEEE Transations on, vol. 45, no. 9, pp , Sep [9] W. Shu and S. Xu, Capaitane extration of multiondutor transmission lines in multilayered dieletri media using numerial green s funtion, in Antennas and Propagation Soiety International Symposium, IEEE, vol., Jun. 2003, pp vol.. [0] L. Shujing and Z. Hanqing, An effiient algorithm for the parameter extration of multiondutor transmission lines in multilayer dieletri media, in Antennas and Propagation Soiety International Symposium, 2005 IEEE, vol. 3A, Jul. 2005, pp vol. 3A. [] S. Musa and M. Sadiku, Appliation of the finite element method in alulating the apaitane and indutane of multiondutor transmission lines, in Southeaston, IEEE, Apr. 2008, pp [2] J. Van Bladel, Eletromagneti Fields, 2nd ed. The IEEE Press Series on Eletromagneti Wave Theory, Wiley - Intersiene, [3] P. Kok, F. Olyslager, J. Van Hese, L. Van Hauwermeiren, M. Botte, and D. De Zutter, Quasi-TEM modelling of two and three dimensional highspeed interonnetion strutures, in Pro. International Eletronis Pakaging Conferene (IEPS 90), Sep. 990, pp [4] T. Demeester and D. De Zutter, Constrution and appliations of the Dirihlet to Neumann operator in transmission line modeling, J. Ele. Eng. Comp. Si., vol. 7, no. 3, pp , Nov [5], Constrution of the Dirihlet to Neumann boundary operator for triangles and appliations in the analysis of polygonal ondutors, IEEE Trans. Mirow. Theory Teh., vol. 58, no., pp. 6 27, Jan [6] H. Johnson, Passivation and solder mask, Eletronis Design, Strategy, News (EDN), Jun [Online]. Available: [7], Differential trae impedane, High-Speed Digital Design - online newsletter, vol. 5, no. 2, Jan [Online]. Available: 2.htm Thomas Demeester was born in 982 in Ghent, Belgium. He reeived his M. S. Degree in eletrial engineering from Ghent University in 2005, after a one-year period at ETH Zürih for his master thesis in the field of time-domain eletromagnetis. In 2009 he obtained the Ph.D. degree in eletrial engineering, as a Researh Fellow of the Fund for Sientifi Researh, Flanders. His researh ranges from eletromagneti field alulations in the presene of highly lossy media and the development of transmission line models for interonnets, to mahine learning tehniques for optimization and searh methods. Daniël De Zutter was born in 953. He reeived his M. S. Degree in eletrial engineering from Ghent University in 976. From 976 to 984 he was a researh and teahing assistant at the same university. In 98 he obtained a Ph.D. degree and in 984 he ompleted a thesis leading to a degree equivalent to the Frenh Aggrégation or the German Habilitation. From 984 to 996 he was with the National Fund for Sientifi Researh of Belgium. He is now a full professor of eletromagnetis. Most of his earlier sientifi work dealt with the eletrodynamis of moving media. His researh now fousses on all aspets of iruit and eletromagneti modelling of high-speed and high-frequeny interonnetions, pakaging, on-hip interonnet and on numerial solutions of Maxwell s equations. As author or o-author he has ontributed to more than 200 international journal papers. In 2000 he was eleted to the grade of Fellow of the IEEE. He served as dean of the Faulty of Engineering of Ghent University and as assoiate editor for the MTT-Transations. At present he is the head of the Department of Information Tehnology of Ghent University.