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A Fast Surface Method to Model Skin Effect inTransmission Lines with Conductors of Arbitrary
Shape or Rough ProfileUtkarsh R. Patel and Piero Triverio
Edward S. Rogers Sr. Department of Electrical and Computer EngineeringUniversity of Toronto, Toronto, Canada
Emails: [email protected], [email protected]
Abstract—Accurate models for on-chip interconnects are re-quired to study signal integrity issues in electronic systems. Thispaper presents a fast surface method to calculate the impedanceof transmission lines with conductors of arbitrary shape, in-cluding those with irregular profile because of roughness. Thepresented technique is over 140 times faster than finite elements,which makes it an ideal tool to perform sensitivity analyses. Themethod is used to study the impact of conductors’ geometricalvariations and conductors’ surface roughness on the impedanceparameters.
I. INTRODUCTION
Accurate transmission line models of interconnects arerequired to study and mitigate signal integrity issues such ascrosstalk and distortion in high-speed electronic systems [1].To create transmission line models, one requires an accuratecalculation of the per-unit length (p.u.l.) impedance and ad-mittance parameters of the interconnects, often over a widefrequency range from DC up to several GHz. Over such awide frequency range, skin effect leads to large changes inthe impedance, which are not trivial to predict.
In the literature, two class of techniques to calculateimpedance parameters have been explored: volumetric andsurface methods. Volumetric techniques based on the finiteelement method (FEM) [2], conductor partitioning [3], orvolumetric integral equations [4] require a fine discretiza-tion of the entire cross-section of the conductor in order tocapture skin effect. Such techniques lead to a large numberof unknowns and long computation times. Therefore, volu-metric methods become impractical for certain problems, forexample when multiple impedance evaluations are required,as in the case of sensitivity [5] or stochastic analysis [6].Surface methods such as [7]–[10] achieve the same resultby taking only the fields on the conductors’ boundary asunknown. An explicit discretization of the conductors’ volumeis thus avoided. These techniques are more efficient thanvolumetric techniques, as they only introduce unknowns alongthe boundary of the conductors. However, these techniquesalso require the calculation of several kernel matrices whichare dependent on the volumetric mesh. Recently, a surfacemethod that altogether avoids volumetric discretization wasproposed in [11]. This technique can efficiently calculatethe impedance parameters of transmission lines made up of
conductors of irregular shapes, such as an on-chip interconnectwith trapezoidal lines, valley-shaped microstrip lines [12],and conformal interconnects [13]. This approach generalizesthe surface admittance operator concept [14], which waspreviously restricted to conductors with canonical shapes, suchas those with circular [14], [15], tubular [16], rectangular [14],or triangular [17] cross section.
In this paper, we make three contributions. First, we showhow the method of [11] can be used to robustly computethe eigenfunctions of the Helmholtz operator for arbitraryconductors. Such functions provide an insight on which cur-rent distributions are most relevant for capturing skin andproximity effect in a conductor. The eigenvalue decompositionalso demonstrates how the presented method generalizes thesurface admittance formulation of [14]. Second, we showhow the proposed method can be used to quickly assess theimpact of geometrical variations on the line impedance. Suchanalyses are becoming increasingly relevant in signal integrityengineering, due to the growing impact of manufacturingvariations on transmission lines performance. Unfortunately,when performed with existing volumetric methods like FEM,sensitivity analyses are notoriously time consuming, as theyrequire many impedance evaluations for different parametervalues. Finally, we demonstrate that the ability of predictingskin effect in conductors of arbitrary cross section can beeffectively used to assess the impact of surface roughness onattenuation [18] over a broad frequency range.
II. IMPEDANCE CALCULATION BASED ON THE SURFACEADMITTANCE OPERATOR
First, we briefly review the method proposed in [14] andthen generalized in [11] for impedance computation. Weconsider a multiconductor transmission line composed of Pconductors of arbitrary shape with conductivity σ, permittivityε, and permeability µ. The conductors are assumed to belongitudinally invariant and are surrounded by a stratifiedmedium. Our goal is to calculate the p.u.l. resistance R(ω) andinductance L(ω) that appear in the Telegrapher’s equation [19]
∂V
∂z= − [R(ω) + jωL(ω)] I , (1)
1
γ(p)
n′t′
O
n t
~r ′
~r~d
σ, µ, ε
µ0, ε0
γ(p)
J(p)s (~r)
µ0, ε0
µ0, ε0
Fig. 1. Left panel: Sample geometry of conductor with arbitrary cross-section.In this diagram, we assume that the surrounding medium is free space. Rightpanel: equivalent configuration obtained after replacing the conductor by thesurrounding (lossless) medium and equivalent current density J(p)
s (~r) on thecontour γ(p).
where V =[V1 V2 . . . VP
]Tand I =[
I1 I2 . . . IP]T
contain, respectively, the scalarpotential Vp and current Ip in each conductor. Theimpedance parameters will be calculated based on thesurface admittance operator [14] obtained through the contourintegral method [11].
A. Surface Admittance Operator Obtained Using the ContourIntegral Method
We discuss the surface admittance formulation by consider-ing p-th conductor of an arbitrary shape enclosed by contourγ(p), as shown in Fig 1. In order to calculate the p.u.l.impedance parameters, we expand the electric field on γ(p)
using pulse basis functions as follows
E(p)z (~r) =
Np∑n=1
e(p)n Π(p)n (~r) , (2)
where Π(p)n (~r) is the n-th pulse basis function that is equal
to one on the n-th segment of the p-th conductor, and is zerootherwise. Likewise, we also expand the tangential magneticfield on the contour of the p-th conductor using pulse basisfunctions
H(p)t (~r) =
Np∑n=1
h(p)n Π(p)n (~r) . (3)
To simplify the notation, we collect the expansion coefficients
in (2) and (3) into vectors E(p) =[e(p)1 e
(p)2 . . . e
(p)Np
]Tand H(p) =
[h(p)1 h
(p)2 . . . h
(p)Np
]T.
Under the quasi-TM assumption, the electric field E(p)z (~r)
and tangential magnetic field H(p)t (~r) on γ(p) are related by
the contour integral equation (CIM) [20]
E(p)z (~r) =
j
2
˛γ(p)
[∂G(~r, ~r ′)
∂n′E(p)z (~r ′)
−jωµG(~r, ~r ′)H(p)t (~r ′)
]dr′ , (4)
where G(~r, ~r ′) is the Green’s function of the two-dimensionalHelmholtz equation and is given by
G(~r, ~r ′) = C0J0(kd)− jY0(kd) (5)
where k =√ωµ(ωε− jσ) is the wavenumber inside the
conductor, and J0(.) and Y0(.) are the zeroth order Bessel andNeumann functions [21], respectively. The distance between ~rand ~r ′ is denoted as
~d = ~r ′ − ~r , (6)
and d =∣∣∣~d ∣∣∣. Theoretically, CIM in (4) is valid for any value
of C0. However, for numerical robustness, we use C0 = 106
for low frequency and C0 = 1 for high frequency, as discussedin [11].
Next, we substitute (2) and (3) into (4), following whichwe apply the method of moments with point-matching [22] toobtain
P−1UE(p) = H(p) . (7)
The equation above relates the discretized electric and mag-netic fields through matrices P and U that can be calculatedfrom (4), as shown in [11].
Next, we apply the equivalence theorem [23] to replacethe conductor by the surrounding medium, and the equivalentcurrent density
J (p)s (~r) =
Np∑n=0
j(p)n Π(p)n (~r) (8)
on the contour γ(p) in order to ensure that the field outsidethe contour γ(p) remains unchanged [14]. According to theequivalence principle [23], we set
J (p)s (~r) = H
(p)t (~r)− H(p)
t (~r) , (9)
where H(p)t (~r) is the tangential magnetic field on γ(p) after
the conductor is replaced by the surrounding medium, i.e. forthe configuration shown in the right panel of Fig. 1. Magneticfield H(p)
t (~r) is also discretized using pulse basis functions asfollows
H(p)t (~r) =
Np∑n=0
h(p)n Π(p)n (~r) . (10)
The expansion coefficients of equivalent current J(p)s (~r)
and tangential magnetic field H(p)t (~r) are collected into
vectors J(p) =[j(p)1 j
(p)2 . . . j
(p)Np
]Tand H(p) =[
h(p)1 h
(p)2 . . . h
(p)Np
]T. The vector counterpart of (9) is
given byJ(p) = H(p) − H(p) . (11)
We again relate the electric field E(p)z (~r) and magnetic field
H(p)t (~r) using the CIM (4), except we replace k in (5) by k0
which is the wavenumber of the surrounding layer. Followingthe method of moments procedure, we obtain
P−1UE(p) = H(p) , (12)
which is analogous to (7) for the equivalent configuration inthe right panel of Fig. 1. Next, we substitute (7) and (12)
2
into (11) to relate the equivalent current to the electric fieldon the contour γ(p)
J(p) = Y(p)s E(p) , (13)
whereY(p)s = P−1U− P−1U (14)
is the so-called surface admittance operator introduced in [14].It is important to emphasize that the above derivation isapplicable to any conductor shape, unlike prior works inthe literature which are only applicable to round [14], [16],rectangular [14], and triangular conductors [17].
By applying the above procedure to each conductor in thesystem, we obtain the following global surface admittancerelationship
J = YsE , (15)
where J and E collect equivalent currents and electric fields onthe contour of each conductor. The discretized surface operatorYs is a block diagonal matrix with each block being Y
(p)s .
B. Electric Field Integral Equation
Next, we relate the electric field and equivalent currents tothe p.u.l. impedance parameters by the electric field integralequation [14]
E(p)z (~r) = jωµ0
P∑q=1
˛γ(q)
J (q)s (~r ′)G0(~r, ~r ′)dr′− ∂Vp
∂z, (16)
where G0(~r, ~r ′) is the 2D Green’s function of the surroundingmedium.
Following the procedure outlined [11], we substitute (2), (8)and (1) into (16), and then apply the method of moments toobtain
E = jωµ0G0J + Q [R(ω) + jωL(ω)] I , (17)
where Q is a block diagonal matrix with P blocks, each blockof size Np × 1 and entries set to 1. The current I and currentdensity J are also related by
I = QTWJ , (18)
where W is a diagonal matrix with entries equal to thewidth of segment of each pulse basis function. As shown in[11], by substituting the surface operator (15) into (17) andmanipulating the resulting equation we obtain the partial p.u.l.resistance
R(ω) = Re
[QTW (1− jωµ0YsG0)
−1YsQ
]−1,
(19)and the partial p.u.l. inductance
L(ω) = ω−1Im[
QTW (1− jωµ0YsG0)−1
YsQ]−1
.
(20)
Fig. 2. The electric field corresponding to the three most dominant Helmholtzoperator eigenfunctions for a round conductor.
III. EIGENFUNCTION EXPANSION OF THE SURFACEADMITTANCE OPERATOR
The method in [11] is based on the surface admittanceoperator theory that was originally developed in [14] forcanonical conductor geometries, whose Helmholtz operatoreigenfunctions are known analytically. In this section, we showhow to calculate the eigenfunctions for an arbitrary geometryin a robust way, starting from the surface admittance operatorfrom [11].
A. Eigenvalue Decomposition
We apply the eigenvalue decomposition on the surfaceadmittance operator Y(p)
s , so that
Y(p)s = ADA−1 , (21)
where A =[A1 A2 . . . ANp
]contains the eigenvectors
and the diagonal matrix D contains the corresponding eigen-values λi.
Substituting (21) into (13) and left multiplying the resultingequation by A−1, we obtain
A−1J(p) = DA−1E(p) . (22)
We now define the following transformations
J(p) = AJ(p) , (23)
E(p) = AE(p) , (24)
so that we can rewrite (22) as
J(p) = DE(p) , (25)
which is of the same form as the surface admittance opera-tor (13). Transformations (23) and (24) suggest that the electricfield on the boundary γ(p) and inside the p-th conductor canbe represented as a superposition of the eigenfunctions of theHelmholtz operator. For canonical shapes, these eigenfunctionscan be obtained analytically [14]. For arbitrary shapes, theircomputation is not a trivial task, and is prone to numericalerrors. The proposed procedure provides a way to obtain themin a robust way, even for conductors with sharp angles and thincross section.
B. Eigenfunctions of a Round Conductor
We apply the eigenvalue decomposition on the surfaceadmittance matrix Ys calculated at f = 2 MHz for a singleround copper conductor of radius a = 1 mm and conductivityσ = 5.8 · 107 S/m. The eigenfunctions corresponding to the
3
Fig. 3. The electric field corresponding to the four most dominant Helmholtzoperator eigenfunctions for a square conductor.
the top three dominant eigenvectors are shown in Fig. 2. Theseeigenfunctions resemble those obtained analytically in [14].
It is observed that for the dominant mode the electricfield is uniform along the boundary. In absence of proximityeffects, only this mode is excited, and leads to an exactmodeling of skin effect inside the conductor. In the presenceof other conductors, proximity effect exists, and higher ordereigenfunctions are excited.
C. Eigenfunctions of a Square ConductorNext, we carry out the same analysis at 1 MHz for a square
copper conductor with side length 1 mm. The eigenfunctionsassociated with the four most-dominant eigenvalues are shownin Fig. 3. In [14], it is shown that the electric field distributionat the edges of a square conductor can be expanded into aseries of sinusoids with different harmonic frequency. Theeigenfunctions in Fig. 3 clearly exhibit the same behaviour,which confirms how the proposed method generalizes [14].
D. Eigenfunctions of a Trapezoidal ConductorNext, we consider the case of a trapezoidal copper conductor
with top edge length of 0.5 mm and bottom edge length of1 mm. The eigenfunctions of the Helmholtz operator for atrapezoidal conductor are not known analytically. However, wenumerically extracted these eigenfunctions using the presentedmethod. The electric field distribution corresponding to thefour dominant eigenfunctions are shown in Fig. 4. The firsteigenfunction corresponds to a uniform electric field on theboundary, while higher-order eigenfunctions describe fieldvariations along the edges, and are important to capture theeffect of corners, as well as proximity effect.
E. Eigenfunctions of a Valley-shaped ConductorFinally, Fig. 5 shows the electric field corresponding to four
relevant eigenfunctions for a valley-shaped conductor whosedimensions are given in Sec. IV-A. It can be seen from Fig. 5that none of the eigenfunctions lead to a uniform electric fielddistribution on the edges. Among the eigenfunctions in thefigure, two describe a current flowing in the two upper arms,while the other two describe the current in the V-shaped region.The functions in the left panels have a relatively smoothvariation over space, while those in the right panels have anull at the midpoint of the upper arms and of the V-shapedregion, respectively.
IV. NUMERICAL RESULTS
In this section, we consider two examples to demonstratethe accuracy, computational speed, and the practical usefulnessof a technique that can handle arbitrary cross sections.
Fig. 4. The electric field corresponding to the four most dominant Helmholtzoperator eigenfunctions for a trapezoidal conductor.
Fig. 5. The electric field corresponding to the four most dominant Helmholtzoperator eigenfunctions for a valley-shaped conductor.
A. Valley-shaped Interconnect
We consider the valley-shaped interconnect shown in Fig. 6,where the two lower conductors form the reference line. Theconductivity of each conductor in the system is σ = 5.8 ·107 S/m.
We analysed the sensitivity of the p.u.l. resistance andinductance to three parameters: width of the ground plane w,spacing between the signal conductor and reference groundplane h, and edge extension of the signal conductor l, thatchanges the angle of the V-shaped section. The nominal valuesof the three parameters were set to h = 3 µm, w = 0 µm,and l = 0 µm. For the nominal values, we also calculated theimpedance at 61 frequency points with FEM [24] to evaluatethe accuracy and computational speed of the proposed method.
First, we varied length l between −1 µm and 1 µm withstep size of 0.1 µm, and calculated the impedance at 61frequency points for each step. The superimposed resistanceand inductance curves for different values of l are shownin Fig. 7. Varying l has a significant impact on inductance,and a very small influence on resistance. The results obtainedwith FEM match very well those obtained with the presentedtechnique. The presented technique took just 11.10 minutesto run this parametric sweep. Next, we swept w from 0 µmto 1.5 µm with step size of 0.1 µm. Figure 8 depicts theresistance and inductance curves for this case, and showsthe significant influence of w on inductance. The presentedtechnique took 7.90 minutes to run this parametric sweep.Finally, parameter h was swept from 2 µm to 4 µm with stepsize of 0.1 µm, and inductance results are shown in Fig. 9. Asexpected, the inductance varies significantly when the distancebetween the signal and reference lines is increased.
This study shows that a small variation in geometricalparameters can lead to a large variation in line impedance.Simulating this transmission line with FEM took 73.8 s perfrequency point. In contrast, the presented technique requiredonly 0.52 s per frequency, for a speed up of 140X. Therefore,
4
131311
7.57.511
10
hw
l
Fig. 6. Valley microstrip line considered in Sec. IV-A. All dimensions arein micrometers. We consider three parameter variations: l, w and h whichare as labelled in this figure. The nominal values for these parameters arel = 0 µm, w = 0 µm, and h = 3 µm.
108
109
1010
1011
103.2
103.4
103.6
Frequency [Hz]
Res
ista
nce
p.u
.l. [Ω
/m]
106
108
1010
1012
0.23
0.24
0.25
0.26
0.27
0.28
Frequency [Hz]
Induct
ance
p.u
.l. [µ
H/m
]
l = −1µ m
l = 1µ m
Fig. 7. P.u.l. resistance and inductance of the valley-shaped interconnectconsidered in Sec. IV-A for l = [−1µm, 1µm], w = 0 µm, and h = 3 µm.The FEM results are marked by (×).
performing such a sensitivity analysis with FEM or anothervolumetric method is very time consuming, as it would haverequired 71 hours with FEM to perform this study comparedto less than 30 minutes required with the proposed technique.
B. Two Rectangular Conductors with Triangular RoughnessProfile
Finally, we consider two rectangular conductors, each of di-mension 127 µm×15 µm, spaced 330 µm apart. This exampleis a simplified version of the test case presented in [18] to studyroughness. We use a simple triangular roughness profile tostudy the impact of surface roughness on the attenuation con-stant of the transmission line. A sample roughness profile of
108
109
1010
1011
103.2
103.4
103.6
Frequency [Hz]
Res
ista
nce
p.u
.l. [Ω
/m]
106
107
108
109
1010
1011
0.22
0.23
0.24
0.25
0.26
0.27
0.28
Frequency [Hz]In
du
ctan
ce p
.u.l
. [µ
H/m
]
w = 0µm
w = 1.5µm
Fig. 8. P.u.l. resistance and inductance of the valley-shaped interconnectconsidered in Sec. IV-A for w = [0, 1.5µm], l = 0 µm, and h = 3 µm.The FEM results are marked by (×).
106
108
1010
0.22
0.24
0.26
0.28
0.3
Frequency [Hz]
Induct
ance
p.u
.l. [µ
H/m
]
h = 4µm
h = 2µm
Fig. 9. P.u.l. inductance of the valley-shaped interconnect considered inSec. IV-A for h = [2 µm, 4µm], w = 0 µm, and l = 0 µm. The FEMresults are marked by (×).
Ta
Fig. 10. Rectangular conductor with triangular roughness profile consideredin Sec. IV-B
one of the conductors is shown in Fig. 10. We let T = 9.07 µmand we vary the roughness amplitude a from 0 µm to 3.4 µm,as roughness can introduce peaks with amplitude of several
5
5 10 15 20 25 300.2
0.4
0.6
0.8
1
Frequency [GHz]
Att
enuat
ion C
onst
ant
[Np/m
]
a = 3.4µm
a = 0µm
Fig. 11. Attenuation constant of line considered in Sec. IV-B for differentamplitude of roughness a.
micrometers [18]. We calculated the impedance parametersfor different values of a using the presented technique. Theline capacitance was calculated with FEM [24]. Figure 11shows that the amplitude of the triangular grooves significantlyinfluences the attenuation constant of the transmission line,especially at high frequency. Each simulation took an averageof 2.16 s per frequency point, which is very reasonable giventhe irregular shape of the conductor.
V. CONCLUSION
We presented a fast surface method to model skin effect inconductors of arbitrary cross section, and accurately estimatethe per-unit-length impedance over a broad frequency range.The method relies on the surface admittance operator toefficiently model how current redistributes inside a conductoras skin effect develops. Previously, such method was restrictedto canonical conductor shapes (circular, rectangular, triangu-lar), for which the Helmholtz equation eigenfunctions areknown analytically. Using a contour integral formulation, wedemonstrated that the eigenfunctions can be reliably computednumerically for any cross section, and provide an interestinginsight on the most relevant current distributions inside aconductor in presence of skin and proximity effect. The com-putational efficiency of the proposed method was demonstratedthrough a sensitivity analysis, which otherwise would havebeen very time consuming if performed with existing methodssuch as finite elements. Finally, the presented method wasapplied to a conductor with irregular profile, to estimate howattenuation increases with surface roughness. This latter testdemonstrated the usefulness of the novel formulation, whichcan handle conductors of arbitrary profile.
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