This paper proposes a parameterized model order reduction technique for efficient global sensitivity analysis of coupled coils over a design space. It uses parameterized models of the electromagnetic matrices and Krylov matrices from the original and adjoint systems, derived using interpolation. Numerical results confirm the efficiency and accuracy of the proposed method for sensitivity analysis across the entire design space of interest.

1. Global Adjoint Sensitivity Analysis of Coupled
Coils using Parameterized Model Order Reduction
Luca De Camillis and Giulio Antonini
Dipartimento di Ingegneria Industriale
e dell'Informazione e di Economia
Francesco Ferranti
Department of Fundamental Electricity
and Instrumentation
Albert E. Ruehli
EMC Lab.
Missouri Univ. of Sci. & Technol.,
Rolla, MO, USAVia G. Gronchi, 18,67100, L'Aquila, ITALY
Email: giulio.antonini@univaq.it
Vrije Universiteit Brussel
B-1050 Brussels, Belgium
Email: francesco.ferranti@vub.ac.be
Email: albert.ruehli@gmail.com
Abstract-This paper presents a parameterized model order
reduction technique for eft'cient global sensitivity analysis of
coupled coils. It Is based on the use of parameterized models
for the electromagnetic matrices and the Krylov matrices of
the original and corresponding adjoint systems, and congruence
transformations. Numerical results conf rm the eft'ciency and
accuracy of the proposed method for global sensitivity analysis
over the complete design space of interest.
Index Terms-Partial element equivalent circuit, sensitiv-
ity analysis, parameterized model order reduction, frequency-
domain circuit simulation.
I. INTRODUCTION
Full wave numerical techniques are often used to model
electromagnetic (EM) systems and are often employed in their
optimization process to achieve particular performances or
mitigating electromagnetic compatibility (EMC) and signal
integrity (SI) problems. Effcient and accurate sensitivity infor-
mation with respect to design parameters is important, since
it provides designers with valuable information in terms of
identifying critical parameters in the design. Furthermore, sen-
sitivity analysis can be used with gradient-based optimization
techniques to speed-up the design process, to perform statis-
tical and yield analysis, and to evaluate the impact of process
variability on the design. Traditional EM-based optimization
techniques estimate the responses sensitivities required by the
optimizer through a f nite-difference approach, which invokes
the EM simulator repeatedly for perturbed values of the design
variables [1]. The perturbative approach is computationally
expensive and often inaccurate, thus impractical, when the size
of the model and number of design parameters for optimization
becomes large.
Among EM methods, the partial element equivalent circuit
(PEEC) [2]-[4] has become increasingly popular because of its
capability to transform the EM system under examination into
a RLC equivalent circuit. PEEC uses a circuit interpretation
of the electric f eld integral equation (EFIE) [5], thus allowing
the handling of complex problems involving EM f elds and
circuits using a unique circuit environment [6], [7].
Although the PEEC method is very popular for its accuracy
and versatility, its use to solve EM equations can result in very
large system of equations, which are often computationally
978-1-5090-1442-2/16/$31.00 ©2016 IEEE 90
expensive to solve. This problem becomes even worse when
one considers a typical design process, which requires repeated
simulations of the same structure for different frequency and
design parameters values.
Model-order reduction (MOR) techniques [8]-[11] and their
parameterized versions (PMOR) [12]-[14] have been proposed
to minimize the computational complexity of PEEC simu-
lations, while retaining accuracy. MOR techniques perform
model reduction only with respect to frequency, while PMOR
techniques that can reduce large systems of equations with
respect to frequency and other design parameters of the circuit,
such as geometrical layout characteristics.
More recently, the PEEC method has also been used to
perform parameterized sensitivity analyses of multiport sys-
tems that depend on multiple design parameters [15]. An
interpolation process provides parameterized models of these
matrices as functions of design parameters. The proposed
interpolation scheme is able to compute derivatives of EM
matrices, which are needed to perform the system sensitivity
analysis [13]. The limitation of such an approach relies on the
high computational cost when the size of the EM matrices of
the PEEC circuits becomes large.
In this paper, we propose a new PMOR technique to
effciently perform global sensitivity analysis of coupled coils
over the complete design space of interest. It is based on the
use of parameterized models for the electromagnetic PEEC
matrices and the Krylov matrices of the original and cor-
responding adjoint systems, and congruence transformations.
Of course, the proposed methodology can also be applied to
different and more complex problems.
II. SENSITIVITY FORMULATION
Consider the following system of equations in the Laplace
domain and in an admittance representation:
{ sCX(s) = -GX(s) +BVp(s) (1)
Ip(s) = LTX(s)
where C E lR.n• xn., G E lR.n• xn., B E lR.n.xnp e L E
lR.n• xnp , ns the number of states and np the number of ports.
In [12], [13], such a formulation is obtained applying the
modifed nodal formulation (MNA) to the circuit equations

2. provided by the PEEC method. If we include the dependence
on a design parameter A, the system (1) becomes
{ sC(A) X(s, A) = -G(A) X(s, A) + B Yp(s) (2)
Ip(s, A) = LTX(s, A)
The PEEC matrices B, L do not depend on the design
parameters. We note that for ease of notation we consider one
design parameter in what follows, but the proposed method
can be applied to multiple design parameters. Starting from
(1), one of the most effcient method to determine sensitivity
information is the adjoint technique [16]. The system
{sCa(A)Xa(s, A) = -Ga(A)Xa(s, A) + BaYp(s) (3)
where Ca = CT, G a = GT, Ba = L is defned as
adjoint system. Given the original and adjoint systems, and
being A a design parameter, the sensitivity of the admittance
representation with respect to Acan be written in the Laplace
domain as
y = -X~ . (sa+G) .x (4)
where X and Xa are the solution of the original and adjoint
systems:
(sC+G)X=B
(sCa +G a) Xa = Ba
(5)
The symbol ~ denotes the derivative with respect to A. The
solution of equation (4) yields the admittance sensitivity with
respect to A. The solution of this equation can be computa-
tionally expensive, therefore a model order reduction technique
will be shown in the next Section to reduce the complexity of
the calculation.
III. MODEL ORDER REDUCTION
The purpose of MOR techniques is to reduce large systems
of equations, while preserving the behavior of the original
model. The reduced system must be an approximation of the
original system, in a sense that its input-output behavior is
comparable to the original one, within a certain accuracy. In
[17], an effcient method to reduce the set of equations (1-4)
was presented. The method is based on the calculation and
use of two projection matrices transformations Y,Ya
X=YXr
Xa = YaXr,a
(6)
The Y matrix is the projection matrix of the original system
(1) that can be obtained using a projection-based MOR algo-
rithm [8], [9]. The Y a matrix is instead the projection matrix
obtained applying the same algorithms to the adjoint system
(3) [17]. In this paper, the Laguerre-SVD technique [8] has
been chosen to compute the projection matrices. Using these
projection matrices, the matrices Xr and Xr,a are smaller in
size with respect to the original ones. From (5) Xr and Xr,a
can be rewritten as
Xr = (sCr +Gr)-l . Br
Xr,a = (sCr,a +Gr,a)-l . Br,a
(7)
(8)
91
where
Cr,a = Y~ . Ca . Y ac.r = yT·C·Y
G r = yT·G·Y
Br = yT·B
Gr,a = YJ . Ga . Y a (9)
Br,a = YJ ·Ba
The sensitivity of the admittance representation based on the
reduced matrices can be written as
(10)
where
~ T ~
Cr=Ya ·C·y
Gr =Y~ ·G·y
(11)
The sensitivity information can be found by solving equation
(10) upon the knowledge of derivatives of the PEEC matrices
C, Gwith respect to the design parameters. We note that the
MOR technique in [17] is limited to a local sensitivity analysis
around a design space nominal point. In the next Section, we
will discuss how to build parameterized models that allow
computing the derivatives of the PEEC matrices C, G and
how to extend this MOR algorithm to take into account design
parameters and perform a global sensitivity analysis over the
complete design space of interest.
IV. PARAMETERIZED MODELS AND MODEL ORDER
REDUCTION
In this Section, we extend the MOR algorithm [17] towards
a PMOR algorithm. The frst step is to generate a set of
PEEC matrices {C(Ak), G (Ak), B, L}:;,!']t from a set of initial
design space samples Ak [12], [13], [15]. The sampling grid
in the design space can be chosen to be uniform or adaptive.
Then, parameterized models {C(A), G(A))} are built using
interpolation schemes that allow computing derivatives [13],
[15]. Here, the multivariate cubic spline interpolation method
[18] is used, which is well-known for its stable and smooth
characteristics. This interpolation scheme is continuous in the
f rst and second order derivatives and therefore can be used
to compute the f rst order derivatives of the PEEC matrices
needed to solve equation (10). This interpolation scheme can
be used in the general case of an M-dimensional (M-D) design
space.
The second step involves the computation of the projection
matrices Y (Ak),Y a(Ak) for each point of the design space
sampling grid. The projection matrices are computed using
the Laguerre-SVD technique [8]. This technique f rst generates
the Krylov matrices Kq(Ak),Kq,a(Ak) based on the PEEC
matrices and then the projection matrices Y(Ak),Ya(Ak)
using a Singular Value Decomposition (SVD) decomposition
of Kq(Ak) and Kq,a(Ak), respectively [8]. Once the set of
Krylov matrices is gathered for each point of the design
space sampling grid (Ak,k = 1, ..., K tot ), the parameterized
models Kq(A),Kq,a(A) based on spline interpolation are built.
In theory, it is also possible to model directly the projection
matrices and create the models Y(A),Ya(A). However, this
choice might lead to inaccurate results due to the fact that
the projection matrices Y(Ak),Ya(Ak) are obtained from

3. Fig. 1. Coupled coils.
independent SVD operations for k = 1, ..., K tot and since
the SVD is not unique in general. V(>'k) and Va(>'k) might
result nonsmooth functions of >..
Finally. the system sensitivities for any point in the design
space are computed by evaluating the parameterized models
of the PEEC and Krylov matrices at that specif c point of
interest in the design space. performing an SVD step on
the interpolated Krylov matrices. performing the congruence
transformations (9). and solving equation (10) with the reduced
matrices.
V. NUMERICAL RESULTS
In this Section, the performances of the proposed algorithm
are demonstrated via its application to the sensitivity analysis
of the coupled coils depicted in Fig. 1. The sensitivity infor-
mation is computed with respect to two design parameters,
namely d being the distance between the two spirals and 9
being the gap of the single spiral. The sensitivity is calculated
considering 1x = 1y = 100 mm. t = s = 1 mm. A set of
PEEC matrices is computed over a grid of 6 x 6 samples for
values of d E [20 - 60] mm and 9 E [2 - 8] mm. Then. the
proposed algorithm is used to generate the parameterized mod-
els of the PEEC matrices and Krylov matrices of the original
and corresponding adjoint systems. Finally. the parameterized
models can be used along with congruence transformations to
carry out an eff cient global sensitivity analysis.
A parameterized frequency-domain sensitivity analysis is
performed for values of d E [24 - 56] mm and 9 E
[2.6 - 7.4] mm. In order to validate the algorithm, these values
have been chosen differently from the design space samples
used to create the parameterized models. The proposed tech-
nique has been compared with the technique in [15] (called
Full Parameterized in what follows) and with a perturbative
approach using a 1% step. Numerical simulations have been
performed on a Linux platform on an AMD FX(tm)-6100 Six-
Core Processor 3.3 GHz with 16 GB RAM.
Figures 2 and 4 show the sensitivities of the admittances
Y12. Y 22 with respect to the two design parameters. Figures 3
and 5 show the relative error of the sensitivities computed by
the proposed PMOR technique and the perturbation approach
92
considering as a reference the sensitivities computed by the
Full Parameterized approach. These error plots show a high
accuracy of the proposed technique, while the perturbative
approach leads to poorer results in terms of accuracy. Table
I clearly shows the computational advantage of the proposed
PMOR technique. For the Full Parameterized and proposed
PMOR models, the model evaluation CPU time indicates the
average time needed to evaluate the corresponding parameter-
ized models in a point of the validation grid in order to obtain
a set of PEEC matrices. Moreover, for the proposed PMOR
model. the SVD operation is also part of the model evaluation.
For the Perturbative Approach. the model evaluation CPU
time refers to the average time needed to compute a set of
PEEC matrices by a PEEC solver at and around a point in
the validation grid. which are then used for a f nite difference
calculation. Once the parameterized models are evaluated, or
PEEC matrices have been computed (perturbative approach).
they can be used to carry out sensitivity analysis in frequency-
and time-domain. For each of the three methods, the average
time needed to perform the sensitivity analysis in a point of
the validation grid is denoted as simulation CPU time.
The CPU time is considerably improved with respect to
the two other approaches for a sensitivity analysis around a
nominal design point and therefore also for a global sensitivity
analysis. It is important to note that both the proposed PMOR
and Full Parameterized methods require a one-time effort to
build the corresponding parameterized models that then allow
global sensitivity simulations. The CPU time for the PMOR
is equal to 4821 s and for the Full Parameterized method is
equal to 4572 s.
Fig. 2. Magnitude of the sensitivity of Y12 with respect to d.
VI. CONCLUSIONS
In this paper. a novel PMOR method to perform global
sensitivity analysis of EM systems has been presented. The
key features of this technique are parameterized models for
the electromagnetic matrices and the Krylov matrices of the
original and corresponding adjoint systems. and congruence
transformations. The proposed method has been adopted for

4. Fig. 3. Relative error of the sensitivity of Y12 with respect to d.
Fig. 4. Magnitude of the sensitivity of Y22 with respect to g.
Fig. 5. Relative error of the sensitivity of Y22 with respect to g.
93
the sensitivity analysis of two coupled coils. The numerical
results have confnned the high modeling capability and the
improved eff ciency of the proposed approach with respect to
existing sensitivity analysis methods.
ACKNOWLEDGMENT
This work has been funded by the Research Foundation
Flanders (FWO-Vlaanderen). the Strategic Research Program
of the VUB (SRP-19) and the Belgian Federal Government
(IUAP VII). Francesco Ferranti is a Post-Doctoral Research
Fellow of FWO-Vlaanderen.
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TABLE I
SIMULATION TIME COMPARISON. THE CPU TIME INFORMATION RELATED
TO FREQUENCY-DOMAIN ANALYSIS REFERS TO AN AVERAGE VALUE OF
THE CPU TIME NEEDED TO PERFORM THE SENSITIVITY ANALYSIS IN A
POINT OF THE VALIDATION GRID.
Proposed [15] Perturbative
model eval 78 68 3818
simulation 578 3068 2148
total 64 s 312 s 595 s

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