Efficient techniques for accurate modeling and simulation of substrate coupling in mixed-signal IC''s.
Industry trends aimed at integrating higher levels of circuit functionality have triggered a proliferation of mixed analog-digital systems. Magnified noise coupling through the common chip substrate has made the design and verification of such systems an increasingly difficult task. In this paper we present a fast eigendecomposition technique that accelerates operator application in BEM methods and avoids the dense-matrix storage while taking all of the substrate boundary effects into account explicitly. This technique can be used for accurate and efficient modeling of substrate coupling effects in mixed-signal integrated circuits.
Introduction
Industry trends aimed at integrating higher levels of circuit functionality resulting from an
emphasis on compactness in consumer electronic products and a widespread growth and interest
in wireless communications, have triggered a proliferation of mixed analog-digital systems. Single
chip mixed-signal designs combining digital and analog blocks built over a common substrate,
provide reduced levels of power dissipation, smaller package count, as well as smaller package
interconnect parasitics. The design of such systems however, is becoming an increasingly difficult
task owing to the various coupling problems that result from the combined requirements for high-speed
digital and high-precision analog components. Noise coupling through the common chip
substrate, caused by the nonideal isolation provided by the common substrate has been identified
as a significant contributor to the coupling problem in mixed-signal designs [3, 2, 11, 1]. Fast
switching logic components inject current into the substrate causing voltage fluctuation which
can affect the operation of sensitive analog circuitry through the body-effect, since the transistor
threshold is a strong function of substrate bias.
Perhaps the most common way to deal with these problems is to resort to costly trial and
error techniques. Clearly such a methodology, which requires the ability to fabricate multiple
version of a design and relies heavily on the expertise and experience of the designer, is not
adequate in the face of rising fabrication costs and increasing demands for shorter design cycle
times [1]. Several approaches have been presented in the past to attempt to quantify the effects
of noise coupling through the substrate in order to account for and deal with it without the
need for expensive redesigns and multiple fabrication runs. Of these methods, those based on
an appropriate formulation of the substrate electromagnetic interactions which rely on detailed
numerical analysis are in general more reliable and accurate. Examples of 3D techniques include
Finite Element (FEM) and Finite Diference (FD) numerical methods for computing all the
currents and voltages in the substrate [2, 3, 4, 5, 6]. These techniques perform a full domain
discretization on the large but bounded substrate and can easily handle irregular substrates
(wells, doping profiles, etc). Unfortunately such methods are impractical for anything but simple
problems, since the number of unknowns resulting from the discretization is too large because
of volume-meshing of the entire substrate. Device simulators such as MEDICI and PISCES can
also be used for this task. However they are in general too slow, since they are meant to simulate
the drift-diffusion phenomenon in semiconductors, while we are interested only in the simple
resistive effects. Thus such methods are not efficient nor versatile enough for implementation in
standard CAD systems.
Boundary-Element BEM (methods) have been applied with some success to the problem of
modeling substrate coupling. BEM methods are very appealing for the solution of this type of
problems because by requiring only the discretization of the relevant boundary features they
dramatically reduce the size of the matrix to be solved. In [7] a Green's function for a two-layer
substrate without backplane is used. In effect this approach amounts to analyzing a chip with infinite
dimensions in the lateral directions. In [8] a distinct approach is taken in that point to point
impedances are precomputed which are then used to find the admittance matrix that describes
the contacts configuration. An interpolation process is used to fit the preprocessed impedance
values computed at various contact distances to a power series of the distance between contacts.
In order to avoid edge effects the authors also assume infinite lateral chip dimensions. This technique
can be very efficient because numerical acceleration techniques can be applied to speedup
the computation of the impedance matrix coefficients. Accuracy can be compromised due to
the assumption of infinite lateral dimensions if some of the contacts are placed near the physical
walls of the substrate. In [10] a substrate Green's function is derived that takes into account the
actual properties of the domain and the problem. Repeated evaluation of this Green's function
is performed in order to obtain an accurate substrate model. However, direct computation of
the substrate Green's function is avoided and instead the computation is performed by means
of a 2D DCT (Discrete Cosine Transform), which in turn can be efficiently computed with an
FFT. Despite these improvements, the computational effort involved in computing a model using
this approach is still considerable and the method has difficulty handling problems with a large
number of contacts.
The matrices produced by BEM methods are dense which has limited the usage of such methods
to small to medium size problems. However, substrate coupling is in general a global problem
and its effect cannot be accurately predicted by analyzing small subsets of the layout. Therefore
speeding up the computations encountered in BEM formulations is of paramount importance
if accurate models for large substrate coupling problems are to be obtained. In this paper we
present a novel eigendecomposition method, used in a Krylov subspace solver, that eliminates
dense-matrix storage and speeds up operator-vector application significantly. This method is used
to speedup the computations necessary for computing substrate models in a BEM formulation
and allows for the extraction of substrate models in problems containing several hundred surface
unknowns.
In Section 2 we present some background into the problem of modeling substrate coupling and
specifically on BEM methods. Then, in section 3 we present our algorithm based on a functional
eigendecomposition of the substrate current to voltage operator and show how to use it to speed
up substrate extraction. Then we show that the resulting model is in a form that makes it easy
to incorporate into standard circuit simulators such as spice or spectre to perform coupled
circuit-substrate simulation. In section 4 we include examples that illustrate the efficiency and
accuracy of the techniques described. Finally, in section 5, we present some conclusions from our
work.
Background
2.1 Problem Formulation
For typical mixed-signal circuits operating at frequencies below a few gigahertz, the substrate
behaves resistively [11, 8]. Assuming this electrostatic approximation, the substrate is therefore
usually modeled as a stratified medium composed of several homogeneous layers characterized
by their conductivity, as shown in Figure 2.1-a).
On the top of this stack of layers a number of ports or contacts are defined, which correspond
to the areas where the designed circuit interacts with the substrate. Examples of these contacts
include possible noise sources or receptors, such as contacts between substrate or wells to supply
lines, drain/source/channel areas of transistors, etc. Figure 2.1-b) exemplifies the typical model
assumed for the substrate and examples of contact areas or terminals. The contacts on the
substrate top are usually assumed to be planar (bidimensional). The bottom of the substrate is
either attached through some large contact to some fixed voltage (usually ground) or left floating.
Substrates with backplane connectors are known to provide better isolation against coupling, but
are also more expensive [3].
In the electrostatic case the governing equations reduce to the well known Laplace equation
inside the substrate volume, where OE is the electrostatic potential and oe is the substrate con-
ductivity, which is assumed constant in each layer. Application of Green's theorem, assuming
a modified Green's function G which accounts for the problem's boundary conditions, gives the
potential at some observation point r due to a unit current injected at some source point r 0
Z
ae(r
(a) (b)

Figure

1: Cross-section of substrate showing: a) a 3D model as an homogeneous multilayered
system, b) the layered model, and some points of contact between the circuit and the substrate.
where ae is the source current density. For this case, all the source and observation points are
at the substrate contacts which are on the top of the substrate and are assumed bidimensional.
The above volume integral thus reduces to a surface integral
Z
ae s
ae s is now the surface current density.
The usage of the medium's Green's function greatly simplifies the problem by implicitly taking
into account the boundary conditions, making it unnecessary to discretize the boundaries. The
substrate Green's function has been previously computed in analytical form and shown to be [11]
fmn Cmn cos( m-x
a
a
where a and b are the substrate lateral dimensions and the fmn can be computed with the help of
recursion formulas. In order to keep the description simple, we will refrain from reproducing the
exact formulas as they are somewhat cumbersome. For the exact expressions and their derivation
see [10] or [11].
2.2 Computing a Resistive Model
Once the Green's function is known, Eqn. (3) can be used to compute the potential at any
point with any current distribution on the substrate contacts. Given a set of S contacts, we
seek a model that relates the currents on those contacts, I c to their voltage distribution V c . In
practice, for reasons of accuracy it is necessary to discretize each of the contacts into a series of
panels. A set of equations relating the currents and potentials on all panels in the system is then
formulated
Each entry in this impedance matrix is of the form
Z
Z
are the surface areas of panels i and j respectively.
Using (4) directly to compute the elements of Z, leads to a doubly infinite series which exhibits
slow convergence and is therefore computationally expensive. In [10] it was shown that this
computation can be performed in a very efficient way by truncating the series and noticing that
the resulting equation can be rewritten such that each of the matrix elements can be obtained
from careful combinations of appropriate terms of a two-dimensional DCT. Since a DCT can be
efficiently computed with an FFT, this techniques leads to a significant speedup in computation
time.
The overall algorithm proceeds as follows: one of the contacts, j, is placed at one Volt, while
all the others contacts are set to zero Volt. This implies that all panels that are contained in
contact j will be set to a voltage of one Volt and the other to ground. Then Eqn. (5) can be
solved to obtain the corresponding current distribution. By appropriately adding the current
corresponding to panels on the same contact one obtains the equivalent contact currents. This
procedure is equivalent to computing one column of the admittance matrix Y c defined as
I
By repeating this procedure for every contact j in the problem the substrate admittance model,
Y c can be readily constructed. It should be noted that the model given by Eqn. (7) is a simple
resistive network where the contacts are the network nodes and entry (i;
the conductance between nodes i and j. Thus, inclusion of such a model in a standard circuit
simulator such as spice or spectre is a trivial task.
The method outlined above can quickly become computationally very costly if the number of
panels is large. Also the cost of storing the dense matrices resulting from the discretization is
in itself a problem. For anything but very small problems one of the dominant factors in the
computational cost will be that of solving the system in Eqn (5) S times. If Gaussian elimination
(i.e. LU-factorization) is used then this cost will be O(n 3 ) which is overwhelming for typical values
of n. Methods for speeding up the solution of this problem are however well known and have in
fact been applied to substrate extraction [8]. Iterative algorithms and namely Krylov-subspace
algorithms can be used to speedup the computation of (5). An example of such methods is the
Generalized Minimum Residual algorithm, GMRES [12]. GMRES solves the linear system by
minimizing the norm of the residual r
p at each iteration k, of the iterative process.
The major cost of these algorithms is the computation of a matrix-vector product which is
required at each iteration. Thus, if the number of iterations does not grow too rapidly and is
kept small, the total cost of obtaining the substrate admittance model is O(S KG
is the average number of GMRES iterations per solution. In [8] this cost is further decreased
to O(S K P n) by acceleration of the matrix-vector product using the hierarchical multipole
algorithm [13]. However this is only possible due to the simplifications assumed in computing
Z, specifically the translation invariance of the "implied" Green's function in this case, and is
thus not a general procedure. Therefore general methods have to be devised to accelerate the
computation of Eqn. (5) in order to be able to handle problems involving several hundreds of
substrate contacts with high accuracy.
3 Sparsification via Eigendecomposition
The extraction algorithm that we propose in this paper for computing a substrate model is
based on the above method for computing an admittance representation of the substrate. The
iterative algorithm GMRES is used to solve Eqn. (5) but direct computation of the matrix-vector
product, ZI k
p is avoided. This operation corresponds in essence to computing a set of
average panel potentials given a substrate injected current distribution. This can readily be done
by means of an eigenfunction decomposition of the linear operator that relates injected currents
to panel potentials. As we shall see, this computation can be performed very efficiently by means
of 2D DCT's.
3.1 Computation of the Operator Eigenpairs
The first step in our method is to discretize the substrate top surface in small rectangular
panels, where small means that the current that flows across them can be considered uniformly
distributed in each one. Considering the top of the substrate as a 2D surface, this discretization
leads to an M \Theta N set of panels or cells (usually since the substrate is square, but for
the sake of generalization in our derivation we will allow different discretizations in each axis).
With this approximation the current distribution in the M by N panels can be represented by
the following equation (in the remainder we will use the notation q for current instead of the
more usual i due to the obvious parallel with the capacitive problem in the electrostatic case.)
where q mn is the total current at panel (m; n), -(x; y) is a square-bump function, with 0 value
throughout the plane except in a rectangle defined by (\Gammaa=2M; \Gammab=2N ); (\Gammaa=2M; b=2N); (a=2M; \Gammab=2N )
and (a=2M; b=2N ), where it takes a constant amplitude of MN=ab so that it has unit volume.
In essence it serves as an averaging function.
Let us now assume that the current distribution function q(x; y) can be represented (decom-
posed) by a sum of functions of the form
are the functions and a ij are the coefficients of the decomposition.
are the eigenfunctions of the linear operator L which takes us from currents to
potentials, then by definition, the potential can be written as
are the eigenvalues of L. Therefore, if the eigenpairs (eigenfunctions and eigenvalues)
of the operator L implied by Poisson's equation are known, and an eigendecomposition of the
injected substrate currents can be obtained such as (9), then the potentials are trivially obtained
from (10).
For the substrate problem, the eigenfunctions of the impedance operator L can be derived
from Poisson's equation and the knowledge of the boundary conditions. In fact, since the first
derivative of the potential across the lateral boundaries must be 0, cosines are good candidates
to represent the current distribution. The particular form of the functions would then be
a
cos
To see if these functions are indeed eigenfunctions of L, we substitute the current distribution in
this equation to find if the potential can be written as in (10). Since all panels are at the top of
the substrate, the current distribution on the contacts is represented along the z axis as a delta
function at z = 0. Poisson's equation can then be written as
dy 2
amn' mn (x; y)
oe
where the potential is a function of x; y and z. The 3D potential is
-mn (z) evaluated at are the eigenvalues of L. Substituting (13) in (12), computing the
derivatives, multiplying both sides by cos
a
cos
and integrating in x from 0 to a and
in y from 0 to b, we finally get a second order differential equation for the eigenvalues -mn (z)
mn -mn
oe
where
The solution of Eqn. (14) can be readily obtained to be, for m;n 6= 0,
oe L fl mn (fi L cosh(fl mn d)
where L is the number of layers in the substrate profile with resistivities oe
its thickness. For
oe L fi L
The values of \Gamma L and fi L can be computed in a recursive manner as in [10]. It should come as no
surprise that the derivations are fairly similar given the definition of Green's function. However,
in [10] they were used to evaluate the Green's function between two panels, while here we use
them directly to expand global functions defined over the entire substrate.
At this point we have shown that the candidate functions introduced in Eqn. (11) are indeed the
impedance operator's eigenfunctions for the substrate problem. Furthermore we have computed
the eigenvalues of the operator to be given by (16) and (17).
3.2 Eigendecomposition Representation for Panel Potentials
Given this information we can now go back to Eqn. (9) and obtain the eigenfunction decomposition
for the current distribution. To that end, we need to determine a ij , which we can easily
do by multiplying both sides of (9) by ' kl (x; y) followed by integration in x between 0 and a and
in y between 0 and b. This yields
Z aZ bq(x; y) cos
a
cos
dxdy (18)
with
Since q(x; y) is constant by rectangles as given by (8), replacing we obtain, after some algebra,
cos
with
ab-j sin
ab-i sin
sin
Now that the potential distribution on the top of the substrate, given by Eqn. (10) is known,
the average potential in each panel can easily be computed. In fact, the average potential in
any panel can be obtained by taking the inner product between Eqn.(10) and the square-bump
function supported over the given panel. The result of this operation is easily shown to be
\Phi pq =X
cos
with
-j sin
-i sin
sin
Numerical evaluation of the average panel potential, amounts to truncating Eqn. (21) in order
to compute a finite number of products . The size of the summation is controlled by
the number of coefficients a ij available, and therefore by the number of cosine modes used in the
eigendecomposition.
3.3 Efficient Computation of the Panel Potentials
Equation (21) allows us to compute the average panel potentials given any arbitrary current
distribution on the top of the substrate. In order to accomplish this task, one has to compute the
individual a ij coefficients using (19), multiply then appropriately by the operator's eigenvalues
and compute the double summation indicated in (21). Inspection of Eqn. (19) reveals that, for
the coefficients a ij are the result of a 2D type-2 DCT on the
set q mn . Such an operation can be efficiently performed by means of an FFT. Furthermore, after
multiplication by the eigenvalues, computation of the average potentials from (21), assuming
truncation of the summation, again amounts, up to a scaling factor, to the computation of an
inverse 2D type-2 DCT on the set C
If at some point it becomes necessary to increase the accuracy of the potential computation,
the standard way to accomplish this is to further refine the substrate discretization. However this
directly affects the number of panels in the system and would increase total computation time
and storage. In our method, higher accuracy is obtained, without refining the discretization, by
increasing the size of the eigendecomposition, i.e., by employing more cosine modes. Apparently
such an alternative would preclude the usage of the efficient FFT algorithm for Eqn. (19) since
there would now be more a ij coefficients than panels (i.e. q mn coefficients). However, by using
the symmetry properties of the DCT, it can be shown that all a ij can be related
to the first MN cosines modes, a This process is termed
unfolding. Thus by simple computation of the DCT implied in (19), it is possible to obtain an
arbitrary number of cosine mode coefficients without incurring in any substantial extra cost. By
a similar argument refolding of these coefficients can be performed to obtain the average panels
potentials from (21). Specifically,
cos
is the 2D DCT of q mn as seen in (19) and
sin 2
sin 2
-o
ab- 4
sin 2
sin 2
and similarly for -
-, where an appropriate number of terms is
used.
3.4 Memory and Cost Comparison
The eigendecomposition algorithm just described is extremely efficient both in terms of memory
usage as in terms of computational cost when compared to the Green's function based algorithm
for similar accuracies. Without loss of generality and in order to simplify the description we will
now assume that the discretization of the substrate is such that the number of panels in each
direction is the same, that is
The memory requirements for the eigendecomposition method are O(k M 2 ) space to store the
eigenvalues of the system, the DCT coefficients and the vectors necessary for the computation
of the GMRES algorithm, and O(S 2 ) for the final admittance model (assuming S contacts are
being used). These requirements should be compared to the Green's function based methods.
The storage requirements for those methods are O(P is the size
of the 2D grid used in the DCT and resulting from the panel discretization, n is the number of
panels and S the number of contacts (usually n AE P and n AE S). In [10] it is indicated that
Z does not need to be explicitly computed, thus reducing the memory requirements. However
this is done at the expense of increasing the computation cost because the elements
to be repeatedly assembled from the result of the 2D DCT sequence. The interesting case to
consider is when the density of contacts is large, as is common in typical designs where area is a
major concern. In that case n can be a large percentage of P 2 , which is the maximum number of
contacts defined on a P \Theta P grid. Typically M and P are of comparable magnitude for reasons
of accuracy, even though as we saw, in the eigendecomposition method one can increase accuracy
increasing M . But even assuming that M and P are comparable, if n - 20%P 2 which
is in fact a small density of contacts, the storage requirements of the Green's function methods
are then O(P 2 For large enough P , the last term will dominate
the storage cost and the Green's function method will require substantially more space that the
eigendecomposition method whose cost is always O(P 2 ), albeit with a large constant.
The computational cost of both methods can also be compared. For the Green's function
method this cost is O(2P 2 log(P
where the first term corresponds to the 2D type-1 DCT and the second term corresponds to
stenciling the Z matrix and to the multiple system solutions if GMRES is used. It is easily seen
that the second term always dominates the total cost. In the eigendecomposition method, the
total cost is O(2M 2 log(M) S KE ). Experience shows that KE and KG , the average number of
GMRES iterations per contact in each method, are comparable. Thus the difference between
the cost of the two methods rely on the comparison of M 2 log(M) to n As we saw
previously, even for sparse designs in the number of contacts (ff - 1%) n 2 AE M 2 , which implies
that the eigendecomposition method is almost always much more efficient.
4 Experimental Results
In this section we present examples that show the accuracy and efficiency of the substrate
coupling extraction algorithm presented in this paper.
We will use as an example a layout from a simple mixed-signal circuit. Figure 4-a) shows the
layout for the example problem where the substrate contacts are marked and numbered. As can
be seen from the figure this example has 52 contacts, so it would fit in the category of small to
medium size problem.
Two experiments, using different substrate profiles were conducted on this example layout in
order to test the versatility, accuracy and efficiency of the extraction algorithm. The profiles
used were taken from [10] and are described in Figure 4-b). The high-resistivity substrate is used
in various BiCMOS processes, while the low-resistivity substrate is used in CMOS due to their
latch-up suppressing properties. For each of the substrate profiles, extraction was performed and
a resistive model was obtained in the form of a matrix relating the resistance from every contact
to every other contact.291113
28 29
43 44
p-type (0.1 Ohm.cm)
p-type (15 Ohm.cm)
p-type (1 Ohm.cm)
backplane backplane
(a) (b)

Figure

2: Example problem showing a significant number of substrate contacts in a mixed-signal
design: a) symbolic layout; b) Substrate cross-sections used profiles for low and high resistivity
substrates.

Table

1 shows a selected set of relevant resistances computed using both the method based on
direct application of the substrate Green's function and our eigendecomposition based method
for the case of the low-resistivity profile. As seen from the table the accuracy of both methods is
comparable. Similar accuracies were noted when the high-resistivity profile was employed. We
would like to point out that in fact it is possible to obtain for both methods the exact same
values, up to machine precision, by using the same uniform discretization.
As can be seen from Figure 4-a) the contacts for this problem are of varying dimensions, which
is typical of mixed-signal designs. Accuracy constraints will limit the discretization employed for
producing the set of panels that describe the problem. In this example, for both substrate profiles,
usage of a uniform discretization for the Green's function method would produce a problem with
too many panels and the computation time and memory requirements would be overwhelming.
Therefore an efficient non-uniform discretization algorithm was devised and employed in this
problem. The results in Table 2 show that the discretization using this algorithm produces a
Name Contact 1 Contact 2 Green's func Eigendecomp.
non-unif. disc. unif. disc.
R400 9 21 5.55422e+06 7.46023e+06

Table

1: Selected set of extracted resistances for the example layout using the low-resistivity
substrate. The node numbers refer to the respective contact numbers and the node called
refers, in this example, to the grounded backplane contact.
relatively small number of panels.
This is a very important observation because it implies that an efficient discretization algorithm
has to be developed if the Green's function method is to be applied for extraction. On the other
hand, the eigendecomposition method, as described in section 3, requires a uniform discretization
which is easier to implement and, as we shall see does not imply any loss of efficiency. Although
perhaps not the most advantageous from our point of view, this comparison seems fair and closer
to reality.

Table

2 summarizes the relevant parameters obtained for the extraction applied to both the
low and high-resistivity profile examples, such as the number of panels obtained, the memory
used, the CPU time necessary, etc. For the examples shown, in order to compute the solution
using the Green's function based method, GMRES was used to solve Eqn. (5) since usage of LU
factorization would take too long. Also, in order to maintain similar accuracy between methods
the minimal discretization used for each method was different. In particular a larger DCT was
necessary for the Green's function based method. However, the cost of computing the DCT is
not very relevant relative to the total cost, as seen in Table 2.
As can be seen from Table 2 the memory requirements (shown for the high-resistivity case) for
the eigendecomposition algorithm are much smaller. A factor of almost 6 was obtained in terms
of memory savings. For the low-resistivity profile similar results are to be expected.
In terms of computational cost a factor of over 6 speedup was obtained for the low-resistivity
substrate profile, and a speedup of almost 15 was recorded for the high-resistivity substrate, which
leads to a harder numerical problem. We point out that for the Green's function method, the Z
matrix relating the direct couplings between panels was explicitly computed and stored. While
Low-resistivity profile High-resistivity profile
Value Green's func Eigendecomp. Green's func Eigendecomp.
non-unif. disc. uniform disc. non-unif. disc. uniform disc.
Number of contacts 52 52 52 52
Number of panels 2547 17764 2547 17764
Average # panels/contact 50 341 50 341
Size of DCT 512 \Theta 512 256 \Theta 256 512 \Theta 512 256 \Theta 256
Memory usage - 144.6MB 25.5MB
Number of GMRES iterations 1238 1868 8030 2930
Average per contact 23 35 154 56
Computation Times (seconds on an Ultra Sparc 1)
discretization 0.06 0.54 0.06 0.54
Green's function DCT 12.90 N/A 10.33 N/A
Total setup time 14241.5 12.5 14241.1 9.92
Solve cost (GMRES) 16656.8 4111.4 107278 6450.05
Total extr. time 30965.5 4994.9 123630 8405.64

Table

2: Summary of the relevant parameters obtained for the extraction of the example problem
for both substrate profiles.
this is inefficient in terms of memory, the savings in computational time more than compensate
for it, so a compromise was taken here. As can be seen in the tables, the cost of computing
this matrix, which dominates the "Total setup cost" values shown, is non-trivial, specially in the
low-resistivity substrate. In either case a significant speedup was obtained using the eigendecomposition
method, which coupled with its efficient memory usage, makes it a very appealing
method for medium to large high-density substrate extraction problems.
Conclusions
In this paper we reviewed some of the commonly used techniques for extracting and generating
accurate models for substrate coupling, that are amenable to efficient circuit level simulation.
A new eigendecomposition-based technique to perform this extraction was presented. Examples
that show the relevance, accuracy and efficiency of this substrate coupling extraction algorithm
were presented. A speedup of over 15 was obtained when comparing the new extraction method
with direct usage of the problem's Green's function for some substrate profiles. This result coupled
with significant reductions in memory usage make the method presented here very interesting
for this problem.


--R

How to deal with substrate noise in analog cmos circuits.
Chip substrate resistance modeling technique for integrated circuit design.
Experimental results and modeling techniques for substrate noise in mixed-signal integrated circuits
A methodology for rapid estimation of substrate-coupled switching noise
Addressing substrate coupling in mixed-mode ic's: Simulation and power distribution systems

Extraction of circuit models for substrate cross-talk
Verification techniques for substrate coupling and their application to mixed-signal ic design
Modeling and analysis of substrate coupling in integrated circuits.
Modeling and analysis of substrate coupling in integrated circuits.
Modeling and Analysis of Substrate Coupling in Integrated Circuits.
GMRES: A generalized minimal residual algorithm for solving non-symmetric linear systems
Fast capacitance extraction of general three-dimensional structures
--TR
GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
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--CTR
Joo Paulo Costa , Mike Chou , L. Miguel Silveira, Efficient techniques for accurate extraction and modeling of substrate coupling in mixed-signal IC's, Proceedings of the conference on Design, automation and test in Europe, p.84-es, January 1999, Munich, Germany
Discrete Cosine Transform;mixed-signal;eigenfunction;substrate coupling;Fast Fourier Transform;eigenvalue;eigenpair