IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 53, NO. 4, NOVEMBER 2011
959
Multipoint Full-Wave Model Order Reduction for
Delayed PEEC Models With Large Delays
Francesco Ferranti, Member, IEEE, Michel S. Nakhla, Fellow, IEEE, Giulio Antonini, Senior Member, IEEE,
Tom Dhaene, Senior Member, IEEE, Luc Knockaert, Senior Member, IEEE, and Albert E. Ruehli, Life Fellow, IEEE
Abstract—The increase of operating frequencies requires 3-D
electromagnetic (EM) methods, such as the partial element equivalent circuit (PEEC) method, for the analysis and design of highspeed circuits. Very large systems of equations are often produced
by 3-D EM methods and model order reduction (MOR) techniques are used to reduce such a high complexity. When signal
waveform rise times decrease and the corresponding frequency
content increases, or the geometric dimensions become electrically
large, time delays must be included in the modeling. A PEEC formulation, which include delay elements called τ PEEC method,
becomes necessary and leads to systems of neutral delayed differential equations (NDDE). The reduction of large NDDE is still
a very challenging research topic, especially for electrically large
structures, where delays among coupled elements cannot be neglected or easily approximated by rational basis functions. We
propose a novel model order technique for τ PEEC models that
is able to accurately reduce electrically large systems with large
delays. It is based on an adaptive multipoint expansion and MOR
of equivalent first-order systems. The neutral delayed differential
formulation is preserved in the reduced model. Pertinent numerical examples based on τ PEEC models validate the proposed MOR
approach.
Index Terms—Delayed partial element equivalent circuit
(PEEC) method, model order reduction (MOR), neutral delayed
differential equations (NDDE).
I. INTRODUCTION
HE increasing demand for performance of ICs pushes operation to higher signal bandwidths and accurate modeling
of previously neglected second-order effects, such as crosstalk,
T
Manuscript received December 1, 2010; revised March 30, 2011; accepted
April 30, 2011. Date of publication June 27, 2011; date of current version
November 18, 2011. This work was supported by the Research Foundation
Flanders (FWO) and by the Italian Ministry of University (MIUR) under a Program for the Development of Research of National Interest (PRIN) under Grant
2006095890.
F. Ferranti, T. Dhaene, and L. Knockaert are with the Department of Information Technology, Internet Based Communication Networks and Services
(IBCN), Ghent University-IBBT, Gaston Crommenlaan 8 Bus 201, B-9050
Gent, Belgium (e-mail: francesco.ferranti@ugent.be; tom.dhaene@ugent.be;
luc.knockaert@ugent.be).
M. S. Nakhla is with the Department of Electronics, Carleton University,
Ottawa, ON K1S 5B6, Canada (e-mail: msn@doe.carleton.ca).
G. Antonini is with the UAq EMC Laboratory, Dipartimento di Ingegneria Elettrica e dell’Informazione, Università degli Studi dell’Aquila, 67100
L’Aquila, Italy (e-mail: giulio.antonini@univaq.it).
A. E. Ruehli is with the IBM T. J. Watson Research Center, Yorktown Heights,
NY 10598 USA, and also with the Missouri University of Science and Technology, Rolla, MO 65409 USA (e-mail: albert.ruehli@gmail.com).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TEMC.2011.2154335
reflection, delay, and coupling, becomes increasingly important
during circuit and system simulations [1], [2]. Electromagnetic
(EM) 3-D methods [3]–[5] have become increasingly indispensable analysis and design tools for a variety of complex
high-speed systems. Large systems of equations are usually
generated by the use of these methods, and model order reduction (MOR) techniques are crucial to reduce the complexity
of EM models and the computational cost of the simulations,
while retaining the important physical features of the original
system [6]–[9]. The development of reduced-order models of
EM systems has become a topic of intense research over the last
years, with applications to vias, high-speed packages, interconnects, and on-chip passive components [1], [10]–[12]. Among
all EM methods, the partial element equivalent circuit (PEEC)
method [4] has been found particularly useful for modeling
printed circuit boards, interconnects, and power systems. The
PEEC method uses a circuit interpretation of the electric field
integral equation (EFIE) [13], thus, being especially suitable to
problems involving both EM fields and circuits [2], [4], [14].
Nonlinear circuit devices, such as drivers and receivers, are usually connected to PEEC equivalent circuits using a time-domain
circuit simulator (e.g. SPICE [15]). However, inclusion of the
PEEC model directly into a circuit simulator may be computationally intractable for complex structures because the number
of circuit elements can be in the tens of thousands. The quasistatic PEEC formulation [4], which approximates the full-wave
PEEC approach [16], yields an equivalent RLC circuit by neglecting the time delays between the elements in the full-wave
PEEC formulation. Systems of ordinary differential equations
(ODE) with constant coefficients in the time domain and complex algebraic systems of equations with frequency-independent
matrices in the frequency domain are obtained. Standard MOR
techniques for ODE systems can be used to reduce the size of a
quasi-static PEEC model [9], [17], [18]. The quasi-static PEEC
formulation is reasonable for problems with electrically small
geometries. When signal waveform rise times decrease and the
corresponding frequency content increases, or the geometric dimensions become electrically large, time delays must be taken
into account and included in the modeling.
A PEEC formulation, which includes delay elements called
τ PEEC method [19], becomes necessary and leads to systems
of neutral delayed differential equations (NDDE) [16] with constant coefficients and constant delay times in the time domain
and to complex algebraic systems of equations with frequencydependent matrices in the frequency domain. Simply using
quasi-static PEEC models can result in significant errors and
artifacts in the modeling [20]. While several successful MOR
0018-9375/$26.00 © 2011 IEEE
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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 53, NO. 4, NOVEMBER 2011
methods for large ODE systems have been proposed over the
years, the reduction of large NDDE systems is still a very challenging research topic since standard MOR techniques for ODE
systems cannot be directly applied to NDDE systems. Especially, the reduction of electrically large structures, where delays
among coupled elements cannot be neglected or easily approximated by rational basis functions needs to be investigated and
addressed.
Some techniques for the reduction of NDDE systems have
been proposed over the years [21]–[26]. In [23], an equivalent
first-order system is computed by means of a Taylor expansion,
and then MOR Krylov subspace methods [8], [9] are applied.
The NDDE formulation is not preserved in the reduction process. In [26], an equivalent first-order system is computed by
means of a single-point Taylor expansion [23], and a corresponding orthogonal projection matrix is computed by means
of a block Arnoldi algorithm [9]. Then, an orthogonal projection
matrix for the original NDDE system is extracted and a reduced
NDDE system is obtained.
This paper presents a novel MOR method for τ PEEC models that is able to accurately reduce electrically large structures,
where delays among coupled elements cannot be neglected or
easily approximated by rational basis functions. It is based on
an adaptive multipoint expansion and MOR of equivalent firstorder systems [26]. An adaptive algorithm to select the expansion points is presented. The NDDE formulation is preserved in
the reduced model. It should be noted that the proposed MOR
technique is not bound to the τ PEEC method since it can be
applied to NDDE systems obtained by other methods.
This paper is organized as follows. Section II describes
the modified nodal analysis (MNA) equations of the τ PEEC
method. Section III describes the proposed MOR method for
NDDE systems. Finally, some pertinent numerical examples
based on the τ PEEC method validate the proposed technique in
Section IV.
II. DELAYED PEEC FORMULATION
The PEEC method [4] stems from the integral equation form
of Maxwell’s equations.
The main difference of the PEEC method with other integralequation-based techniques such as the method of moments [3]
resides in the fact that it provides a circuit interpretation of the
EFIE [13] in terms of partial elements, namely, resistances, partial inductances, and coefficients of potential. Thus, the resulting
equivalent circuit can be studied by means of SPICE-like circuit
solvers [15] in both time and frequency domain.
Over the years, the PEEC method has been extended to
nonorthogonal geometries [19] and modified to incorporate
ideal and lossy dielectrics [27], [28].
Following the standard approach [4], volumes and surfaces
are discretized into elementary regions, hexahedra and patches,
respectively [19]. The current and charge densities are expanded
into a series of basis functions, which are usually assumed as
pulse basis functions. The choice of pulse basis functions implies to assume constant current and charge densities over the
elementary volume (inductive) and surface (capacitive) cells,
respectively.
Conductors are modeled by their ohmic resistance, and dielectrics by the excess capacitance [27]. Hence, magnetic and
electric field coupling are modeled by partial inductances and
coefficients of potential, respectively.
The magnetic field coupling between two inductive volume
cells α and β is modeled by the partial inductance
1
μ 1
duα duβ
(1)
Lpα β =
4π aα aβ u α u β Rα β
where Rα β is the distance between any two points in volumes
uα and uβ , with aα and aβ their cross sections. The electric
field coupling between two capacitive surface cells γ and δ is
modeled by the coefficient of potential
1
1
1
dSγ dSδ
(2)
Pγ δ =
4πε Sγ Sδ S γ S δ Rγ δ
where Rγ δ is the distance between any two points on surfaces
γ and δ, while Sγ and Sδ denote the area of their respective
surfaces.
Due to the finite value of the speed of light, partial inductances
and coefficients of potentials relate causes and effects delayed
in time
diβ (t − τα β )
dt
(t) = Pγ δ qδ (t − τγ δ )
vL ,α (t) = Lpα β
(3)
vP ,γ
(4)
where τα β = Rα β /c0 and τγ δ = Rγ δ /c0 are the center-tocenter distances between the corresponding basis-function domains, and c0 denotes the free-space speed of light. Hence,
partial inductance and coefficient of potential matrices act as
a delay operator for time derivatives of currents and charges,
respectively,
di(t − τ L )
p (τ L ) di(t)
=L
dt
dt
(τ C ) q(t)
v(t) = Pq(t − τ C ) = P
vL (t) = Lp
(5a)
(5b)
where τ L and τ C denote the center-to-center delay matrices for
the magnetic and electric field couplings, respectively.
Generalized Kirchoff’s laws, for conductors, can be rewritten
as follows:
dq(t)
− AT i(t) + ie (t) = 0
dt
p (τ L ) di(t) − Ri(t) = 0
− Av(t) − L
dt
(6a)
(6b)
where A is the connectivity matrix, v(t) denotes the node potentials to infinity, and i(t) and ie (t) represent the currents flowing
in volume cells and the external currents, respectively.
Equation (6b) has to be modified when dielectrics are considered since the resistance R is substituted by the excess capacitance, which relates the polarization charge and the corresponding voltage drop as vd (t) = C−1
d qd (t) [27]. Hence, for
FERRANTI et al.: MULTIPOINT FULL-WAVE MODEL ORDER REDUCTION FOR DELAYED PEEC MODELS WITH LARGE DELAYS
into account, read as follows:
⎡
0n n ,n i
0n n ,n d
In n ,n n
⎢
p (τ L ) 0n ,n
⎢ 0n i ,n n L
i
d
⎢
⎢
0
0
C
⎣ n d ,n n
n d ,n i
d
0n p ,n n
0n p ,n i
⎤
0n n ,n p
⎡
q(t)
0n p ,n p
is (t)
x(t)
C
⎡
dielectric elementary cells, (6) become
dq(t)
− AT i(t) + ie (t) = 0
dt
p (τ L ) di(t) − vd (t) = 0
− Av(t) − L
dt
dvd (t)
.
i(t) = Cd
dt
T
ie (t) = K is (t).
+
(7a)
0n n +n i +n d ,n p
−In p ,n p
0n n ,n d
KT
Φ
0n i ,n p
0n d ,n d
0n d ,n p
0n p ,n d
0n p ,n p
⎤ ⎡
⎤
q(t)
⎥ ⎢
⎥ ⎢ i(t) ⎥
⎥
⎥·⎢
⎥ ⎣ v (t) ⎥
⎦
d
⎦
is (t)
x(t)
G
· [ vp (t) ]
(9)
u(t)
B
(7b)
(7c)
A selection matrix K is introduced to define the port voltages
by selecting node potentials. The same matrix is used to obtain
the external currents ie (t) by the currents is (t), which are of
opposite sign with respect to the port currents ip (t)
vp (t) = Kv(t)
−AT
0n n ,n n
⎢
(τ C )
R
⎢ AP
= −⎢
⎢ 0
−ΦT
⎣
n d ,n n
(τ C ) 0n ,n
−KP
p
i
Fig. 1. Illustration of τ PEEC circuit electrical quantities for a conductor
elementary cell.
⎤
⎥ ⎢
i(t) ⎥
0n i ,n p ⎥
⎥
⎥ d ⎢
⎢
⎥
⎥
0n d ,n p ⎦ dt ⎣ vd (t) ⎦
0n p ,n d
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(8a)
(8b)
An example of τ PEEC circuit for a conductor elementary cell
is illustrated, in the Laplace domain, in Fig. 1, where the currentcontrolled voltage sources sLp,ij Ij and the charge-controlled
current sources sQi model the magnetic and electric field couplings, respectively.
A. Descriptor Representation of τ PEEC Circuits
We assume that the system under analysis consists of conductors and dielectrics. Let the current and charge densities be
defined in volumes and surface of conductors and dielectrics,
respectively. The Galerkin approach is applied to convert the
continuous EM problem described by the EFIE to a discrete
problem in terms of electrical circuit quantities. Let us denote
with nn the number of nodes and ni the number of branches
where currents flow. Among the latter, we denote with nc and nd
the number of branches of conductors and dielectrics, respectively. Furthermore, let us assume to be interested in generating
an admittance representation having np output currents ip (t) under voltage excitation vp (t). Since dielectrics require the excess
capacitance to model the polarization charge [27], additional
nd unknowns are needed in addition to currents. Hence, if the
MNA approach [29] is used, the global number of unknowns is
nu = ni + nd + nn + np . In a matrix form, (6)–(8), taking (5)
where In p ,n p is the identity matrix of dimensions equal to the
number of ports. Matrix Φ is
0n c ,n d
Φ=
.
(10)
In d ,n d
(τ C ) is applied to the first equation in
If the delay operator P
(9), the system can be recast as follows:
⎡
⎤ ⎡
⎤
(τ C ) 0n ,n
P
0n n ,n d 0n n ,n p
q(t)
n
i
⎢
⎥ ⎢
p (τ L ) 0n ,n
⎢ 0n i ,n n L
i(t) ⎥
0n i ,n p ⎥
⎥
i
d
⎢
⎥ d ⎢
⎢
⎥
⎢
⎥
0n ,n
Cd
0n ,n ⎦ dt ⎣ vd (t) ⎦
⎣ 0n ,n
d
n
d
0n p ,n n
i
0n p ,n i
d
0n p ,n d
p
0n p ,n p
C
⎡
0n n ,n n
⎢ ⎢ AP (τ C )
= −⎢
⎢
⎣ 0n d ,n n
(τ C )
−KP
⎡
q(t)
⎤
(τ C ) AT
−P
0n n ,n d
R
Φ
−Φ
T
0n p ,n i
0n d ,n d
is (t)
x(t)
(τ C ) KT ⎤
P
⎥
0n i ,n p ⎥
⎥·
⎥
0n d ,n p ⎦
0n p ,n d
0n p ,n p
G
⎢ i(t) ⎥ 0n +n +n ,n ⎢
⎥
n
i
p
d
·⎢
· [ vp (t) ] .
⎥+
⎣ vd (t) ⎦
−In p ,n p
u(t)
is (t)
B
(11)
x(t)
In a more compact form, (11) can be rewritten as follows:
C (τ )
dx(t)
= −G (τ ) x(t) + Bu(t)
dt
ip (t) = LT x(t)
(12a)
(12b)
where x(t) = [q(t) i(t) vd (t) is (t)]T ∈ n u ×1 and τ ∈
n τ ×1 contains all delays τ L , τ C . Since this is an np -port formulation, whereby the only sources are the voltage sources at
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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 53, NO. 4, NOVEMBER 2011
the np -port nodes, B = L, where B ∈ n u ×n p . Each delayed
entry of matrices C (τ ) and G (τ ) act as a delay operator for the
corresponding entry of vector x(t). Hence, (12) can be rewritten
in the Laplace domain as follows:
sC(s)X(s) = −G(s)X(s) + BVp (s)
Ip (s) = BT X(s)
C(s) = C0 +
nτ
(13)
(14)
Ck e−sτ k
(15)
Gk e−sτ k .
(16)
k =1
G(s) = G0 +
nτ
k =1
The solution of (12) can be carried out using direct or iterative
solvers. Among the direct methods, the LU decomposition has
been widely used [30]. If the transient analysis is carried out
for nt time steps, the complexity of the direct solution scales as
O(nt n3u ), where nu is the total number of unknowns. Hence,
for a large number of unknowns nu , equations (12) cannot be
solved directly and iterative solvers are needed. Assuming that
an average number niter of iterations per time step is required for
the iterative solver to converge to a specified residual, and n2u is
the computational cost for the matrix–vector products involved,
the iterative solution for all time steps nt scales as O(niter nt n2u )
[31]. It is evident that the MOR becomes fundamental to provide
a significant acceleration in time-domain analyses of large EM
delayed systems.
III. MOR ALGORITHM
In [23], an equivalent first-order system is computed by means
of a Taylor expansion, and then it is reduced by means of MOR
Krylov subspace methods. The NDDE formulation is not preserved. In [26], an equivalent first-order system is computed
by means of a single-point Taylor expansion [23], and a corresponding orthogonal projection matrix is computed by means
of a block Arnoldi algorithm [9]. Then, an orthogonal projection matrix for the original NDDE system is extracted and a
reduced NDDE system is obtained. The NDDE formulation is
preserved in the reduction process. The proposed novel MOR
algorithm is based on an adaptive multipoint expansion and
MOR of equivalent first-order systems. As in [26], the NDDE
formulation is preserved. The multipoint expansion feature allows to reduce electrically large structures with large delays
(2πfreqm ax τm ax > 10) [25] that cannot be neglected or easily
approximated by a single-point expansion and rational functions. The equivalent first-order system obtained after the singlepoint Taylor expansion of exponential terms has an order equal
to qnu , where q is the order of the Taylor expansion and nu
the order of the original NDDE system [26]. Since exponential
terms with large delays need many terms in the Taylor expansion
to be accurately approximated, the reduction of equivalent firstorder systems becomes computationally expensive and sometimes unfeasible. The multipoint expansion [32], [33] addresses
this issue and is able to accurately reduce NDDE systems with
large delays since a small expansion Taylor order can be used
for each expansion point and the accuracy of the reduced model
is increased by adding new expansion points. An adaptive algorithm is proposed to choose the expansion points, assuming that
the order of the Taylor expansion is fixed for each expansion
point.
A. Single-Point Expansion Reduction
Performing the MOR on the NDDE system (13) and (14)
requires the computation of an orthogonal basis for the first nn pr block moments of X(s), i.e., its first nn pr scaled derivatives
w.r.t. s, where nr represents the order of the reduced system.
Denoting the orthogonal basis as Q ∈ n u ×n r , the reducedorder system is given by
sCr (s)χ(s) = −Gr (s)χ(s) + Br Vp (s)
Ip (s) =
BTr
(17)
χ(s)
Cr (s) = Cr,0 +
(18)
nτ
Cr,k e−sτ k
(19)
Gr,k e−sτ k
(20)
k =1
Gr (s) = Gr,0 +
nτ
k =1
where the following congruence transformations are used:
Cr,i = QT Ci Q,
i = 0, . . . , nτ
(21a)
Gr,i = QT Gi Q,
i = 0, . . . , nτ
(21b)
T
Br = Q B
(21c)
Lr = QT L
(21d)
and χ(t) is a vector containing the state variables in the reduced
domain.
The reduced-order system (17) and (18) can be efficiently
incorporated into SPICE-like simulators for efficient and accurate transient analysis. The construction of an orthogonal basis
Q represents a key issue in obtaining a compact and accurate
reduced-order model over a wide frequency range. The Arnoldi
algorithm [9] is adopted in this paper to carry out the computation of Q since it is numerically reliable and robust. The
numerical robustness of the Arnoldi algorithm stems from its
capability to obtain the orthogonal basis for the transfer function
moments without computing these moments explicitly. Unfortunately, adapting the Arnoldi algorithm for NDDE systems is not
a straightforward task, and therefore, the original NDDE system has to be transformed in a suitable form to apply standard
Arnoldi-based reduction. To address this issue, an equivalent
first-order system is computed by expanding the exponential
factors e−sτ k in a Taylor series form and using a companion
form [26]. Using the Arnoldi algorithm on the first-order equivalent system, a corresponding orthogonal projection matrix is
computed. Then, an orthogonal projection matrix Q for the
original NDDE system is extracted and a reduced NDDE system is obtained [26]. This single-point expansion-based MOR
algorithm for NDDE systems was proposed assuming s = 0 as
an expansion point. If another expansion point s = s1 , s1 = 0,
is selected, setting s = s1 + σ, where s1 is a frequency shift
and σ is the new Laplace variable, the NDDE system (13), (14)
FERRANTI et al.: MULTIPOINT FULL-WAVE MODEL ORDER REDUCTION FOR DELAYED PEEC MODELS WITH LARGE DELAYS
963
becomes
σ C(σ)X(σ)
= −G(σ)X(σ)
+ BVp (σ)
T
Ip (σ) = B X(σ)
0 +
C(σ)
=C
nτ
(22)
(23)
k e−σ τ k
C
(24)
k e−σ τ k
G
(25)
k =1
0 +
G(σ)
=G
nτ
k =1
Fig. 2. Pseudocode for computing the orthogonal basis Q in the multipoint
expansion case.
where
0 = C0
C
(26a)
0 = G0 + s1 C0
G
(26b)
k = Ck e−s 1 τ k ,
C
k = 1, . . . , nτ
k = (Gk + s1 Ck )e−s 1 τ k ,
G
k = 1, . . . , nτ
(26c)
(26d)
and the algorithm described in [26] can be applied. Although
both real and imaginary shiftings have been investigated in literature [33], we focus on the minimization of the frequencyresponse error between reduced and full-order NDDE systems,
and therefore, imaginary expansion points are considered. This
single-point MOR approach [26] is able to preserve the NDDE
formulation, but it may be not able to reduce NDDE systems
with large delays since the reduction of equivalent first-order
systems becomes computationally expensive and sometimes
unfeasible.
B. Multipoint Expansion Reduction
The equivalent first-order system obtained after the singlepoint Taylor expansion of exponential terms has an order equal
to qnu , where q is the order of the Taylor expansion and nu
the order of the original NDDE system [26]. The reduction of
equivalent first-order systems becomes computationally expensive and sometimes not feasible when large delays are involved
since exponential terms with large delays need many terms in
the Taylor expansion to be accurately approximated. The multipoint expansion [32] addresses this issue and is able to accurately reduce NDDE systems with large delays since a small
expansion Taylor order can be used for each expansion point
and the accuracy of the reduced model is increased by adding
new expansion points. Assuming that the order of the Taylor
expansion is fixed for each expansion point, an adaptive algorithm is used to choose the expansion points. As in [26], the
NDDE formulation is preserved in the reduced model. At each
expansion point, the MOR algorithm described in [26] is applied
and the corresponding projection matrix Qi , i = 1, . . . , np oints
is computed, where np oints denotes the number of expansion
points. The final projection matrix Q is based on the orthogonalization of the stack column collection of all single expansion
point projection matrices. The computation of Q in the case of
multipoint expansion is explained in the flowchart in Fig. 2. The
MOR algorithm described in [26] is called one point DM OR
in Fig. 2, where nr represents the reduced order for each expansion point. Once Q is computed, it is applied to the original
NDDE system (13), (14) and a reduced NDDE system (17), (18)
is obtained.
C. Algorithm for the Selection of Expansion Points
Assuming that the order of the Taylor expansion q is fixed
for each expansion point, an algorithm is needed to choose the
location of the expansion points. An adaptive and iterative algorithm is proposed to determine the expansion points. It is
based on an iterative comparison between reduced and original model. It starts from two expansion points sm in = jωm in
and sm ax = jωm ax located at the minimum and maximum frequency of interest, respectively, and the corresponding reduced
model is compared with the original model in the midpoint between sm in and s, , therefore sm id,1 = (sm in + sm ax )/2. If an
error threshold is satisfied, then the algorithm stops, otherwise
this midpoint is considered as a new expansion point and a new
reduced model is computed, which contains information from
all the expansion points sm in , sm id,1 , and sm ax , and it is compared with the original model in two new points, namely, the
midpoints of the intervals [sm in , sm id,1 ] and [sm id,1 , sm ax ]. If no
comparison point exceeds the error threshold, then the algorithm
stops, otherwise the new expansion points for the next iteration
are chosen as the comparison points that exceed the error threshold. If some comparison points satisfy the error threshold, the
corresponding intervals defined by expansion points are considered to be accurately described and are not checked for accuracy
in the next iterations. The comparison points at each iteration
are the midpoints of all intervals defined by expansion points,
except those ones considered accurate in the previous iterations.
Concerning the error criterion, let us define the weighted rms
error as follows:
Err
=
(n p ) 2 K s
i=1
k =1
|wY i (sk )(Yr,i (sk ) − Yi (sk ))|2
(27)
(np )2 Ks
with
wY i (s) = |(Yi (s))−1 |.
(28)
In the comparison step at each iteration, the error is computed
on each comparison point separately; therefore, Ks = 1 and the
accuracy threshold is chosen equal to 0.05.
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Fig. 3.
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 53, NO. 4, NOVEMBER 2011
Cross section of the multiconductor system.
TABLE I
MOR PARAMETERS
Fig. 4.
Magnitude of Y 1 1 .
Fig. 5.
Phase of Y 1 1 .
Fig. 6.
Magnitude of Y 1 6 .
IV. NUMERICAL RESULTS
Two numerical examples based on τ PEEC systems validate
the proposed MOR approach for NDDE systems. The error defined in (27) is used to assess the accuracy of reduced models
over a sampled bandwidth of interest. The proposed MOR algorithm was implemented in MATLAB R2009A [34] and all
experiments were carried out on Windows platform equipped
with Intel Core2 Extreme CPU Q9300 2.53 GHz and 8GB RAM.
A. Multiconductor System
A multiconductor system composed of six conductors with a
length = 7 cm, a width W = 1 mm, a thickness t = 0.25 mm,
and a horizontal Sx = 3 mm and a vertical Sy = 2 mm spacing
has been modeled in this example. Fig. 3 shows its cross section.
The six ports of the system are defined between a conductor and
the corresponding one above. The system is analyzed on the
frequency range [0.0001, 20] GHz.
The order of the original τ PEEC model is equal to nu = 3366,
the number of delays is equal to 664, and the largest delay gives
2πfreqm ax τm ax = 29.5. Table I shows some parameters of the
proposed MOR method using the adaptive algorithm for the
selection of the expansion points.
Figs. 4–7 compare the magnitude and phase of the original
and reduced NDDE model of Y11 (s) and Y16 (s). As clearly
seen, the presented MOR method can reduce a large NDDE
system with large delay terms, while preserving the behavior
of the original system. The proposed adaptive algorithm for the
selection of the expansion points is able to converge, producing
a compact and accurate reduced-order model.
B. Three-Port Microstrip Power-Divider Circuit
A three-port microstrip power-divider circuit [35] has been
modeled in this example. The structure is shown in Fig. 8. The
dimensions of the circuit are [20, 20, 0.5] mm in the [x, y, z]
directions and the width of the microstrips is equal to 0.8 mm.
The relative dielectric constant is equal to r = 2.2. The system
is analyzed on the frequency range [0.0001, 20] GHz.
The order of the original τ PEEC model is equal to nu = 4149,
the number of delays is equal to 692, and the largest delay gives
2πfreqm ax τm ax = 12. Some parameters of the proposed MOR
method are shown in Table II.
FERRANTI et al.: MULTIPOINT FULL-WAVE MODEL ORDER REDUCTION FOR DELAYED PEEC MODELS WITH LARGE DELAYS
Fig. 7.
Phase of Y 1 6 .
Fig. 8.
Structure of the three-port microstrip power-divider circuit.
Fig. 9.
Fig. 10.
Phase of Y 1 1 .
Fig. 11.
Magnitude of Y 1 3 .
Fig. 12.
Phase of Y 1 3 .
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Magnitude of Y 1 1 .
TABLE II
MOR PARAMETERS
V. CONCLUSION
Figs. 9–12 compare the magnitude and phase of the original
and reduced NDDE model of Y11 (s) and Y13 (s). As in the
previous example, the proposed MOR method is able to accurately reduce an NDDE system with large delay terms using an
adaptive algorithm for the selection of the expansion points.
We have presented a new MOR technique for large NDDE
systems with large delays, which is applicable to τ PEEC models. It is able to accurately reduce electrically large structures,
where delays among coupled elements cannot be neglected or
easily approximated by rational basis functions. It is based on
an adaptive multipoint expansion and MOR of equivalent firstorder systems. An adaptive algorithm to select the expansion
points is discussed. The NDDE formulation is preserved in the
966
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 53, NO. 4, NOVEMBER 2011
reduced model. Numerical examples based on τ PEEC models
have validated the proposed MOR approach, showing that it is
able to accurately reduce large NDDE systems with large delays.
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Francesco Ferranti (M’10) received the B.S. degree (summa cum laude) in electronic engineering
from the Università degli Studi di Palermo, Palermo,
Italy, in 2005, and the M.S. degree (summa cum
laude and honors) in electronic engineering from
the Università degli Studi dell’Aquila, L’Aquila,
Italy, in 2007, and the Ph.D. degree in electrical
engineering from the University of Ghent, Ghent,
Belgium, in 2011.
He is currently a Post-Doctoral Research Fellow with the Department of Information Technology
(INTEC), Ghent University, Ghent, Belgium. His research interests include
parametric macromodeling, parameterized model order reduction, electromagnetic compatibility numerical modeling, and system identification.
Michel S. Nakhla (S’73–M’75–SM’88–F’98) received the Ph.D. degree in electrical engineering from
the University of Waterloo, Waterloo, ON, Canada,
in 1975.
From 1976–1988, he was a Senior Manager of the
computer-aided engineering group at Bell-Northern
Research, Ottawa, ON. In 1988, he joined Carleton
University, Ottawa, Canada, as a Professor and the
holder of the Computer-Aided Engineering Senior Industrial Chair established by Bell-Northern Research
and the Natural Sciences and Engineering Research
Council of Canada, where he currently a Chancellor’s Professor of Electrical Engineering and the founder of the high-speed CAD research group. His research
interests include modeling and simulation of high-speed circuits and interconnects, nonlinear circuits, parallel processing, multidisciplinary optimization,
and neural networks.
Dr. Nakhla is on various international committees, including the standing
committee of the IEEE International Signal Propagation on Interconnects Workshop, the technical program committee of the IEEE International Microwave
Symposium, the technical program committee of the IEEE Conference on
FERRANTI et al.: MULTIPOINT FULL-WAVE MODEL ORDER REDUCTION FOR DELAYED PEEC MODELS WITH LARGE DELAYS
Electrical Performance of Electronic Packaging, and the CAD committee of the
IEEE Microwave Theory and Techniques Society (MTT-1). He is an Associate
Editor of the IEEE TRANSACTIONS ON ADVANCED PACKAGING. He was as Associate Editor of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS. He has
also been a member of many Canadian and international government-sponsored
research grants selection panels He is a technical consultant for several industrial organizations and is the principal investigator for several major sponsored
research projects.
Giulio Antonini (M’94–SM’05) received the
Laurea degree (summa cum laude) in electrical engineering from the Università degli Studi dell’Aquila,
L’Aquila, Italy, in 1994, and the Ph.D. degree in electrical engineering from the University of Rome “La
Sapienza,” Rome, Italy, in 1998.
Since 1998, he has been with the UAq Electromagnetic Compatibility (EMC) Laboratory, Department
of Electrical Engineering, University of L’Aquila,
where he is currently an Associate Professor. He is
the author or coauthor of more than 170 technical papers and 2 book chapters, and holds one European patent. His research interests
include EMC analysis, numerical modeling, and signal integrity for high-speed
digital systems.
Dr. Antonini has given keynote lectures and chaired several special sessions
at international conferences. He has been the recipient of the IEEE Transactions
on Electromagnetic Compatibility Best Paper Award in 1997, the Computer
Simulation Technology University Publication Award in 2004, the IBM Shared
University Research Award in 2004, 2005, and 2006, The Institution of Engineering and Technology—Science, Measurement & Technology Best Paper
Award in 2008, and a Technical Achievement Award from the IEEE EMC Society “for innovative contributions to computational electromagnetic on the Partial
Element Equivalent Circuit technique for EMC applications” in 2006.
Tom Dhaene (SM’05) was born in Deinze, Belgium,
on June 25, 1966. He received the Ph.D. degree in
electrotechnical engineering from the Ghent University, Ghent, Belgium, in 1993.
From 1989 to 1993, he was a Research Assistant
in the Department of Information Technology, Ghent
University, where he was engaged in research on different aspects of full-wave electromagnetic (EM) circuit modeling, transient simulation, and time-domain
characterization of high-frequency and high-speed
interconnections. In 1993, he joined the EDA company Alphabit (now part of Agilent). He was one of the key developers of the
planar EM simulator ADS Momentum. Since September 2000, he has been a
Professor in the Department of Mathematics and Computer Science, University
of Antwerp, Antwerp, Belgium, and since October 2007, a Full Professor in
the Department of Information Technology (INTEC), Ghent University. He is
the author or coauthor of more than 150 peer-reviewed papers and abstracts in
international conference proceedings, journals, and books, and is the holder of
3 US patents.
967
Luc Knockaert (SM’00) received the M.Sc. degree
in physical engineering, the M.Sc. degree in telecommunications engineering, and the Ph.D. degree in
electrical engineering from Ghent University, Ghent,
Belgium, in 1974, 1977, and 1987, respectively.
From 1979 to 1984 and from 1988 to 1995, he
was at the Universities of the Democratic Republic
of the Congo and Burundi, where he was involved in
North–South cooperation and development projects.
He is currently with the Interdisciplinary Institute for
BroadBand Technology, Ghent University, where he
is also a Professor in the Department of Information Technology. He is the
author or coauthor of more than 100 international journal and conference publications. His current research interests include the application of linear algebra
and adaptive methods in signal estimation, model order reduction, and computational electromagnetics.
Dr. Knockaert is a member of the Mathematical Association of America and
the Society for Industrial and Applied Mathematics.
Albert E. Ruehli (LF’03) received the Ph.D. degree in electrical engineering from the University of
Vermont, Burlington, VT, Canada, in 1972, and an
honorary doctorate from the Lulea University, Lulea,
Sweden, in 2007.
He has been a Manager of a very large scale integration (VLSI) design and CAD group, and a member
of various projects with IBM, including mathematical
analysis, semiconductor circuits, and devices modeling. Since 1972, he has been at IBM T. J. Watson
Research Center, Yorktown Heights, NY, where he
was a Research Staff Member in the Electromagnetic Analysis Group, and is
currently an Emeritus of IBM Research. He is also an Adjunct Professor in the
electromagnetic compatibility (EMC) area at the Missouri University of Science
and Technology, Rolla. He is the author or coauthor of more than 190 technical
papers, and is the editor of two books Circuit Analysis, Simulation and Design
(New York: North Holland, 1986 and 1987).
Dr. Ruehli has served in numerous capacities for the IEEE. In 1984 and
1985, he was the Technical and General Chairman, respectively, of the ICCD
International Conference. He has been a member of the IEEE Administrative
Committee for the Circuit and System (CAS) Society. He has been an Associate Editor for the IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN. He has
given talks at universities including keynote addresses and tutorials at conferences, and has organized many sessions. He received IBM Research Division or
IBM Outstanding Contribution Awards in 1975, 1978, 1982, 1995, and 2000. In
1982, he received the Guillemin-Cauer Prize Award for his study on waveform
relaxation, and in 1999, he received a Golden Jubilee Medal, both from the
IEEE CAS Society. In 2001, he received a Certificate of Achievement from the
IEEE EMC Society for Inductance Concepts and the Partial Element Equivalent
Circuit method. He received the 2005 Richard R Stoddart Award, and in 2007,
he received the Honorary Life Member Award from the IEEE EMC Society for
outstanding technical performance. He is a member of the Society for Industrial
and Applied Mathematics.