ESPRIT: A Compact Reluctance Based Interconnect
Model Considering Lossy Substrate Eddy Current
Rong Jiang Charlie Chung-Ping Chen
Electrical and Computer Engineering Graduate Institute of Electronics Engineering
College of Engineering & Department of Electrical Engineering
University of Wisconsin, Madison, WI 53706 National Taiwan University, Taipei 106, Taiwan
Abstract— With the advancement of radio frequency mixed- capture the inductance loss due to the formation of eddy
signal ICs, lossy silicon substrate has signiﬁcant impact on the currents in the conductive substrate. Although several works
already complicated interconnect modeling issue. To account for have been proposed to resolve this issue by constructing three
the substrate loss, the traditional electromagnetic methods are dimensional linear substrate models –, most of these
often computationally prohibitive for large scale VLSI geome- approaches employ a numerical ﬁnite difference based method
tries. In this paper, we extend the traditional PEEC model to
by spatially discretizing a large volume of silicon bulk under
consider the substrate eddy current loss based on the complex
the conductor system and hence will lead to equivalent circuits
image theory and the skin and proximity effects by discretization
of conductors. To deal with even larger scale of interconnects, prohibitive in size.
we present a reluctance based model, ESPRIT, to enhance the In this paper, we propose an accurate and efﬁcient method
extended PEEC model to use reluctance by equipping it with to extend the PEEC model to consider the substrate eddy
an advanced windowing algorithm to further reduce the model current loss based on the complex image theory , which
size and runtime. Detail comparisons with state-of-the-art tools has been recently used in RFIC regime to accurately capture
such as FastHenry and Momentum demonstrate that ESPRIT is line impedances of microstrips   and spiral inductors
within 1% accuracy while providing over 100X speedup.  on lossy silicon substrates. The complex image theory
generates the complex images of interconnects based on the
I. I NTRODUCTION
conﬁguration of substrate structure instead of discretizing the
Due to the proliferation of mixed analog-digital system and substrate and hence can result in very compact models for
radio frequency integrated circuit (RFIC), the development interconnects.
of efﬁcient interconnect models for such a system is made To deal with millions of interconnects and their images, we
difﬁcult because of the lossy nature of the silicon substrate. enhance the extended PEEC model to use reluctance element
In particular, the creation of eddy currents in the conductive with an extended window searching reluctance extraction
silicon substrate can lead to signiﬁcant interconnect inductance algorithm. Finally, since this new model, ESPRIT, includes
loss. An interconnect system analysis without considering the mutual resistances and reluctances, in order to be applicable
lossy substrate effect will result in an over-designed network to general circuit simulators, SPICE compatible models for
and waste chip resources. mutual resistance and reluctance are also provided. Detail
With the increasing clock frequency and integration density, comparisons with state-of-the-art tools such as FastHenry and
intentional and unintentional inductance effects gradually rise. Momentum demonstrate that ESPRIT is within 1% accuracy
One major problem of inductance analysis is the unknown while providing over 100X speedup.
current return path. Fortunately, the PEEC (Partial Equivalent
Element Circuit) method has been widely adopted to deal with
this issue . However, since PEEC model assumes current II. C OMPLEX I MAGE T HEORY
return paths at inﬁnity, extremely dense partial inductance
matrices are usually generated which dramatically increases For frequencies up to a few Giga Hertz, the wavelength of
both model size and simulation runtime. the magnetic ﬁelds far exceeds a typical die’s dimension. Thus
For this reason, various inductance sparsiﬁcation techniques we can make magneto-quasi-static approximations.
have been introduced to alleviate this problem –. In par-
Under this assumption, for a z-direction current, only the
ticular, the reluctance-based method   has been proposed
z-component of the magnetic potential A is nonzero, thus
by Hao Ji et al. Since reluctance has higher degree of locality
the substrate diffusion equation can be reduced to a two
similar to capacitance, only a small number of neighbors need
dimensional EM problem:
to be considered, and hence reluctance matrix for circuit sim-
ulation is very sparse compared to partial inductance matrix. ∂
▽2 Az (x, y) − µσ Az (x, y) = 0 (1)
Moreover, the traditional PEEC approach does not take ∂t
the substrate loss effect into consideration and hence cannot By constructing the Green’s function, the solution can be
expressed as: Since for every metal layer of the on-chip conductor system,
only the ﬁrst term dMi in Eq. 8 is different, a common complex
Az (x, y) = G(x, y|x′ , y ′ )Jz (x′ , y ′ )dx′ dy ′ (2) image plane is shared by all metal layers. Based on the method
of image, the common complex image plane can be substituted
where Jz (x, y) is the current distribution of a line current by image conductors which are at a distance 2hi f below the
located above the substrate. physical conductors in metal layer i.
Without loss of generality, we assume that a unit line current Besides the lossy substrate effect, as the frequency goes
is located at (x′ = 0, y ′ ), with the consideration of the ﬁnite high, the current in a physical conductor is no longer evenly
thickness of the substrate and the presence of a ground plane, distributed, which leads to signiﬁcant changes in resistance
the Green’s function G(x, y|x′ = 0, y ′ ) can be expressed as and inductance values. In order to obtain wide band accuracy,
: those effects, namely skin effect and proximity effect, also
G(x, y|x′ = 0, y ′ ) = need to be modeled. For capturing both skin and proximity
effects, conductors have to be discretized into ﬁlaments so as
µ0 ∞ e−k|y−y |
p − k kd e−k(y+y +d) to account for the non-uniform distribution of current within
− e cos(kx)dk (3)
2π 0 k p+k k conductors .
where The extended PEEC model, which is shown in Fig. 1, is
obtained by the application of complex image theory and the
γ = jωµ0 σsi (4) discretization of both the physical and image conductors into
q(k) = k2 + γ 2 (5) ﬁlaments. Physical Conductors
p(k) = q(k)coth[q(k)hsi ] (6) . . . .
. . . . . . . .
µ0 is the permeability of free space, σsi is the bulk conductiv- . . . .
ity, hsi is the thickness of the substrate, while coth[x] is the
hyperbolic cotangent function. Effective
. . . .
The kernel of the integral in Eq. 3 has two terms. The ﬁrst . . . . . . . .
. . . .
term can be attributed to the physical line current located at
(x′ = 0, y ′ ), while the second term is due to an image line Image Conductors
current located at y = −(y ′ + d). This approximation holds
when the coefﬁcient of the second term, p−k ekd , is approxi-
Fig. 1. Extended PEEC model
mated by constant one. By applying the Taylor expansion of The complex inductance matrix for the conductor system
p−k kd with n ﬁlaments is given by:
p+k e at k = 0 and neglecting high order terms, we obtain
that this requirement can be satisﬁed when L(hef f ) = Lf reespace − Limage (9)
(1 + j)hsi
d = (1 − j) · δsi · tanh (7) Lf reespace is the inductance matrix without considering the
δsi lossy substrate, i.e. in free space. Limage is the mutual
where δsi = 1/ πf µ0 σsi is the skin depth of the bulk silicon inductance matrix between physical and image ﬁlaments.
and tanh[x] is the hyperbolic tangent function. The calculation of Limage depends on the effective complex
Thus the eddy current effect in the lossy substrate and distance hef f , thus L(hef f ) will be frequency and process
the ground plane can be approximated by an image current parameters dependent.
located at the complex distance d below the substrate surface. Since L(hef f ) = L(ω)+R(ω)/jω, the complex inductance
Alternatively, an image ground plane can be placed at d/2 matrix can be interpreted as follows:
below the surface to represent the currents both in the substrate
L(ω) = Real[L(hef f )] (10)
and the ground plane.
III. E XTENDED PEEC M ODEL
R(ω) = −ωImag[L(hef f )] + RDC (11)
For interconnects within metal layer i, which has a distance where L(ω) and R(ω) are the frequency dependent partial
dMi above the substrate, according to the complex image inductance and resistance matrix respectively. RDC is a diag-
theory, the lossy silicon substrate effect can be approximated onal matrix including DC resistances of the physical ﬁlaments.
by placing a complex image plane below metal layer i at an It can be seen that R(ω) contains off diagonal terms which
effective complex distance, hi f . If we denote the thickness
ef represent mutual resistances. We will address the mutual
of oxide and silicon bulk as hox and hsi respectively, by using resistance modeling problem in the following section.
Eq. 7, the effective complex distance of metal layer i is given
by: IV. SPICE C OMPATIBLE R ELUCTANCE -BASED M ODEL
1−j (1 + j)hsi In the previous section, we present how to obtain par-
hi f = dMi + hox +
ef · δsi · tanh (8) tial inductance matrix L(ω) and resistance matrix R(ω) by
using complex image theory. However, L(ω) and R(ω) are • Effective Search Window (ESW): Extend the physical
extremely dense due to the globe effect of partial inductance aggressor along its length by a window extension factor
coupling. Therefore, a more practical modeling approach is (WEF) and obtain the effective window width (EWW).
necessary to obtain circuit model of manageable size. Then, the ESW is deﬁned by sweeping in the direction
perpendicular to the length of the aggressor to inﬁnity
A. Physical Meaning of Partial Reluctance
with the EWW.
Reluctance based methods have been extensively used re- • Conductor Shielding Level (CSL): The CSL of the aggres-
cently because reluctance has better locality than inductance. sor is deﬁned as 0, which is the highest level. Conductors
The partial reluctance matrix K is deﬁned as the inverse of outside ESW are of CSL ∞, the lowest level. A conductor
the partial inductance matrix L. i is directly shielded by conductor j if conductor j can be
K = L−1 (12) reached by some points along the length of conductor i
within ESW without encountering any other conductors.
Since LI = Φ and by applying the Stoke’s theorem: A conductor is of CSL k+1, if the minimum CSL of
conductors directly shielding it is k.
Φ= Bds = ▽ × Ads = Adl (13) • Conductor Group of CSL k: Conductor group of CSL k
contains two parts. The physical part includes the physical
the partial inductance matrix for a system including n con- aggressor and its victim conductors of CSL no larger than
ductors will be: k. The image part includes images of physical conductors
in the physical part. The union of these two parts gives
L11 L12 · · ·
L21 L22 · · · . =
. (14) conductor group of CSL k.
Ln1 Ln2 Lnn In An dln We illustrate our extended window selection algorithm
through a small example shown in Figure 2. If the current
where Ai is the vector potential in conductor i. Hence the Effective Window Width
partial reluctance matrix can be obtained as follows: 1
A dl I
K11 K12 · · · 1 1 1
. = .
K21 K22 · · · . . . (15)
Kn1 Kn2 Knn An dln In
The globe coupling effect of partial inductance is caused
by the artiﬁcial assumption that the current return path is 8 9
at inﬁnite. During partial inductance extraction, we apply a
unit current source on the aggressor conductor at inﬁnity and Fig. 2. Extended Window Selection Algorithm
force the currents in victim conductors to be zero by applying aggressor is conductor 1, its CSL is 0. Conductor 3 and 4
zero current sources at inﬁnity. Since in this scenario the only are of CSL 1; Conductor 5, 6, 7 and 8 are of CSL 2 while
magnetic ﬁeld is generated by the current in the aggressor and conductor 9 is of CSL 3. Conductor 2 is outside the ESW
no other magnetic ﬁelds cancel its effect, it can propagate far and hence its CSL is ∞. Conductor group of CSL 1 includes
away and give rise to a dense partial inductance matrix. physical conductors 1, 3, 4 and image conductors 1′ , 3′ and
However, when calculating the self and mutual reluctances 4′ .
for conductor j, we need to set a unit magnetic ﬂux for the Our frequency dependent reluctance-based interconnect
j th conductor, and zero ﬂux for all others. In order to satisfy model, ESPRIT, is based on the combination of the extended
this conﬁguration, we need to apply an unit vector potential PEEC and the above window selection algorithm. For each
on the aggressor and at the same time pose negative vector conductor, we search its conductor group of CSL k and
potentials on victims to cancel the magnetic ﬁeld generated by calculate the small L(ω) and R(ω) for this conductor group
the aggressor. Therefore, the currents ﬂowing in aggressor and after proper discretization according to conductor skin depth.
victims are basically of opposite direction and the magnetic Then the small L(ω) for this conductor group is inverted
ﬁeld of the aggressor is mostly cancelled by victim magnetic to obtain the small K(ω) matrix. The ﬁnal circuit model is
ﬁelds and cannot propagate faraway. This explains why partial assembled by using those small K(ω) and R(ω) matrices.
reluctance has better locality than partial inductance. Since ESPRIT includes mutual resistances and reluctances,
in order to avoid signiﬁcant modiﬁcations on general simula-
B. Extended Window Selection Algorithm tion tools, we need to consider their SPICE compatible models,
In stead of directly calculating partial inductance matrix and which can be obtained from their branch equations respec-
inverting it to obtain partial reluctance matrix, most existing tively. The branch equation of self and mutual resistances is
reluctance extraction tools are based on window selection given by
algorithms, such as . Here we propose an extended window n n
selection algorithm to consider both physical conductors and Vi = Rij Ij = Rii Ii + Rij Ij (16)
their images. j=1 j=1,j=i
where Rii is self resistance and Rij is the mutual resistance The waveforms of the enhanced PEEC at different frequen-
between Rii and Rjj . Eq. 16 can be rewritten as cies are shown in Fig. 4.(a). Also the responses in Fig. 4.(b)
n demonstrate that ESPRIT has much smaller model size while
Rij maintaining less than 3% error compared to the enhanced
Vi = Rii Ii + (Rjj Ij ) (17)
Rjj PEEC model.
j=1,j=i 1.4 1.4
If we view Rii Ii as the voltage drop across the self 1.2 1.2
resistance Rii , Vi is then equal to the sum of the voltage drop 1.0 1.0
on a self resistance Rii and serially connected voltage control
voltage sources (VCVS). These VCVSs are controlled by 0.6 0.6
voltages on other self resistances which originally have mutual 0.4 Free Space
Lossy at 4gHz
resistances with Rii . Therefore, Eq. 17 can be used to construct 0.2 Lossy at 10gHz
Lossy at 20gHz
0.2 Enhanced PEEC
SPICE compatible model for mutual resistances, which is 0.0
0.05 0.10 0.15 0.20
0.05 0.10 0.15 0.20
R n Rij
shown in Figure 3.(a), where VV CV S = j=1,j=i Rjj Vjj .
ni ni ni ni
Fig. 4. The Enhanced PEEC vs. ESPRIT
Rii 1 / K ii
Rii K ii VI. C ONCLUSION
A new reluctance-based interconnect model ESPRIT con-
nj nj nj nj sidering the loss substrate effect is presented in this paper.
(a) (b) It’s obtained by combining an enhanced PEEC model with
an extended window-based reluctance extraction algorithm.
Fig. 3. SPICE Compatible Model for (a) mutual resistance (b) reluctance
Extensive simulation results demonstrate that ESPRIT has
SPICE compatible model for reluctance can be derived by extremely high accuracy and signiﬁcantly small model size.
similar method. It includes a self inductance 1/Kii and serial
K n Kij
VCVSs VV CV S = − j=1,j=i Kjj Vjj shown in Figure 3.(b). R EFERENCES
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