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[10.1109/LMWC.2003.819378]
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IEEE
472 IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 13, NO. 11, NOVEMBER 2003
Package Macromodeling via
Time-Domain Vector Fitting
S. Grivet-Talocia, Member, IEEE
Abstract—This paper addresses the construction of lumped
macromodels for package structures. A technique named Time-
Domain Vector Fitting (TD-VF) is introduced for the identification
of the dominant poles of the structure. This method uses as raw
data transient excitations and responses at the ports of the struc-
ture. These responses are easily obtained from transient full-wave
electromagnetic solvers based, e.g., on Finite Differences. The
rational approximation can be easily synthesized into a SPICE-
compatible subcircuit providing a broadband approximation to
the input-output behavior of the package.
Index Terms—Circuit extraction, macromodeling, time-domain
vector fitting, vector fitting.
I. INTRODUCTION
T
HE design of modern high-speed electronic systems
requires careful modeling for the assessment of Signal
Integrity (SI) and Electromagnetic Compatibility (EMC) issues.
Indeed, the ever increasing clock speed of digital systems re-
quires consideration of parasitic effects that were not significant
at lower operation frequencies. As an example, it is well known
that electrical interconnects at chip, multichip, package, and
board level are one of the most critical parts for SI and EMC
due to mutual couplings, transmission-line effects, frequency-
dependent conductor losses, etc. Accurate modeling of such
structures can be achieved by full-wave electromagnetic
solvers, which are however too complex for global system-level
simulations.
Macromodeling techniques allow us to tackle the system-
level modeling problem by splitting the entire system into
subparts interacting with each other only through well-defined
ports. Each part is characterized separately using specific tools,
with the aim of generating an equivalent circuit providing an
approximation of its input/output port behavior. This equivalent
circuit is used for subsequent system-level simulation within a
standard circuit analysis environment. At this stage, inclusion
of nonlinear/dynamic terminations like drivers and receivers
can be done easily.
Several macromodeling techniques are available in the liter-
ature (see, e.g., [2], [3], [5], [7]). Each of these techniques is
tailored to the specific form in which the structure under in-
vestigation is characterized. We focus here on linear macro-
Manuscript received March 17, 2003; revised July 10, 2003. This work
was supported in part by the Italian Ministry of University (MIUR) under a
Program for the Development of Research of National Interest (PRIN Grant
#2 002093437), and in part by CERCOM, Center for Multimedia Radio
Communications of the Electronics Department, Politecnico di Torino. The
review of this letter was arranged by Associate Editor Dr. Shigeo Kawasaki.
The author is with the Dipartimento di Elettronica, Politecnico di Torino,
10129, Torino, Italy (e-mail: grivet@polito.it).
Digital Object Identifier 10.1109/LMWC.2003.819378
modeling from transient port responses, and we present a new
macromodeling technique that we denote Time-Domain Vector
Fitting (TD-VF). The algorithm is an extension of the well-
known standard Vector-Fitting (VF) technique [5], whose ac-
curacy and efficiency is widely recognized. Standard VF oper-
ates entirely in frequency domain, providing rational approxi-
mations of transfer functions starting from frequency-domain
samples. Conversely, the proposed TD-VF technique produces
rational approximations directly from transient input/output re-
sponses. A specific application where this approach is conve-
nient is the equivalent circuit extraction for three-dimensional
interconnect structures (like, e.g., electronic packages or con-
nectors), that are characterized using a full-wave electromag-
netic solver based, e.g., on the Finite-Difference Time-Domain
(FDTD) method [8]. Due to the high computational cost, it is de-
sirable to terminate the full-wave analysis before all transients
have extinguished, thus obtaining truncated time responses. The
presented algorithm is ideally suited for this type of native data,
whereas a frequency-domain approach would not be feasible.
Section II presents the formulation of the TD-VF algorithm. Nu-
merical examples are presented in Section III.
II. T
IME-DOMAIN VECTOR FITTING
We consider a package structure with an arbitrary number
of ports. Usually, a transient characterization of such a multi-
port structure is obtained by exciting one port at the time and
computing/measuring the responses at all ports. As a result, the
raw dataset for the package characterization is a matrix of re-
sponse waveforms
at port , due to excitation at
port
. We remark that this type of dataset is the natural out-
come of time-domain full-wave electromagnetic solvers. The
objective is the derivation of a rational approximation to the ma-
trix transfer function
(1)
Note that the
poles are common to all matrix entries, whereas
the direct coupling term and the residues are
matrices.
The raw transient responses satisfy the relation
(2)
where
is the inverse Laplace operator and denotes con-
volution.
The TD-VF algorithm is a time-domain formulation of the
well-known standard Vector Fitting algorithm [5]. It is based on
acombinationof digital filteringandlinear least squares approx-
imations. A first step allows the identification of the dominant
1531-1309/03$17.00 © 2003 IEEE
GRIVET-TALOCIA: PACKAGE MACROMODELING VIA TIME-DOMAIN VECTOR FITTING 473
poles of the structure. Once these poles are known, the matrices
of residues are computed. Finally, a state-space realization is
derived, leading to a direct equivalent circuit synthesis. These
three steps are detailed below.
A. Poles Identification
We introduce, as for standard VF [5], a scalar weight function
(3)
with known (initial) poles
and unknown residues .
Let us assume now that the following approximation holds:
(4)
Since the right-hand-side has the same poles
as the weight
function, a cancellation between the zeros
of and the
poles
of must occur. This condition provides indeed
a way to estimate these poles by solving (4) for the unknown
residues
, computing the zeros using standard tech-
niques [5], and by enforcing
. This procedure, named
pole relocation, avoids use of ill-conditioned nonlinear least
squares algorithms for the direct approximation of (1). Also,
the poles relocation can be iterated using the estimated poles
as starting poles for the new iteration. Convergence is usually
reached in few iterations, as discussed in Section III.
Standard VF solves (4) via linear least squares using the avail-
able raw data
at given frequency points . Instead,
we reformulate this condition in time domain by applying (4) to
the vector of input signals
and applying inverse Laplace
transform. The resulting condition reads componentwise:
(5)
where the transient waveforms
(6)
(7)
are convolutions resulting from inverse Laplace transform of
each partial fraction in the expansions (3)–(4). These wave-
forms are easily obtained by applying a suitable discretization
of the convolution integrals. We use here a linear interpolation
between the raw samples, leading to
(8)
corresponding to a first-order IIR filter with weights
(9)
A similar expression holds for the output responses
.
Condition (5) is enforced in least squares sense using raw and
filtered input/output sequences. The residues
of the weight
function are common in all the
independent equations in (5),
which result coupled and not independent. These residues are
the only quantities that are needed for further processing.
Equation (5) is presented assuming that all available output
responses at all ports are used for the poles estimation. How-
ever, sufficiently accurate poles estimates can often be obtained
by processing a significant subset of (dominant) port responses,
e.g., only diagonal entries with
. This is often the case for
packages exploiting weak coupling between nonadjacent pins.
In such case, the overall system results much smaller. An ex-
ample will be provided in Section III . We remark that the raw
and filtered waveforms can be sub-sampled in order to reduce
the size of the size of the system (5). In fact, the subsampling
rate can be related to the effective bandwidth of the excitation
waveform (usually a Gaussian) via the Nyquist rule. As a result,
for typical package structures, the resulting equivalent samples
count per signal is about 50–100. Note also that the sparse struc-
ture of the system matrix in (5) can be exploited to optimize the
performance of the algorithm.
B. Residues
Once the poles
are known, the residues and the di-
rect coupling matrix
in (1) are computed componentwise:
(10)
with
defined as in (6) with replaced by the esti-
mated poles
.
C. State-Space Realization
The final step in the algorithm is the identification of a
state-space realization
(11)
from the poles and residues representation in (1) Once the state
matrices are found, they can be used to synthesize SPICE-like
equivalent circuits through well known techniques. The partic-
ular form of the state matrices depends on some arbitrary choice,
since several realizations are possible. We can use one of the
simplest possibilities, namely a realization in Jordan canonical
form. This amounts to setting
. Therefore
(12)
where
and are the columns of and the rows of , re-
spectively. This expression indicates that residues
have a theoretical unitary rank. However, since the actual esti-
mates come from a least squares fit, the numerical rank will be
generally larger than one. Therefore, some matrix factorization
can be applied to
to find suitable minimal realizations that
retain the accuracy of the performed fit. In particular, we adopt
here the procedure in [1], which is based on the singular value
decomposition. The direct coupling matrix is
.
A final remark on passivity. It is well-known that only strictly
passive macromodels can be effectively used, since stable but
474 IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 13, NO. 11, NOVEMBER 2003
Fig. 1. Package structure used for illustration of the TD-VF algorithm.
Bondwires (not shown in the picture) were also included in the model.
nonpassive circuits may lead to instabilities depending on the
termination networks. The proposed TD-VF algorithm is not
passive by construction, as it provides an approximation to the
input-output responses of the structure. Therefore, the passivity
of the generated macromodel will depend both on the passivity
of the raw transient data and on the accuracy of the rational ap-
proximation algorithm. The former condition is usually satisfied
if a stable field solver is used for the generation of the raw data.
We will show through the examples in Section III that the level
of accuracy of TD-VF is excellent. Therefore, if some passivity
test is applied to the macromodel and if some passivity violation
occurs, this violation will be small. In this case, some standard
passivity compensation can be applied without difficulties. See
[4] and [6] for details.
III. N
UMERICAL EXAMPLES
In order to test the TD-VF algorithm, a commercial 14-pin
surface mount package has been meshed and analyzed through a
full-wave electromagnetic solver based on the Finite-Difference
Time-Domain method [8]. The structure, depicted in Fig. 1,
has 28 ports, since each pin leads to one port on the board side
(odd-numbered) and one port on the die side (even-numbered).
As a result, a complete set of 28
28 responses (transient
scattering waves with reference load
) have been
obtained, using a unitary Gaussian pulse as excitation, with
a 20 dB bandwidth of 30 GHz. Only the selected responses
, i.e., reflected and transmitted wave on
each excited pin, plus near-end crosstalk, were used for the
determination of the dominant poles, with dynamic order
ranging from 11 up to 30. Fig. 2 depicts the maximum approx-
imation error among all responses. Although no theoretical
relation between maximum approximation error and order
of the macromodel can be argued, the figure shows that the
rational approximation is convergent with an increasing order.
There is a clear tradeoff between macromodel complexity
and accuracy. The macromodel order should therefore be
determined based on the accuracy requirements for the specific
application it is being derived for. As an example, Fig. 3 shows
a comparison between some transient scattering responses of
the original FDTD simulation and the corresponding responses
of the macromodel of order
. No difference can be seen
on this scale between corresponding waveforms. This example
illustrates that the proposed TD-VF technique leads to very ac-
curate macromodels of structures with a possibly large number
Fig. 2. Macromodeling of a commercial 14-pin package. Maximum approx-
imation error among all 28
2
28 port responses for various approximation
orders
N
.
Fig. 3. Macromodeling of a commercial14-pin package. Comparison between
original and macromodel responses for selected transient scattering waveforms.
of ports. The core of the algorithm is very straightforward and
simple, yet robust and reliable. Further application examples
are being developed and will be published elsewhere.
R
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