guaranteed passive parameterized admittance-based macromodeling
TRANSCRIPT
IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 33, NO. 3, AUGUST 2010 623
Guaranteed Passive ParameterizedAdmittance-Based Macromodeling
Francesco Ferranti, Luc Knockaert, Senior Member, IEEE, and Tom Dhaene, Senior Member, IEEE
Abstract—We propose a novel parametric macromodeling tech-nique for admittance and impedance input–output representationsparameterized by design variables such as geometrical layout orsubstrate features. It is able to build accurate multivariate macro-models that are stable and passive in the entire design space. Anefficient combination of rational identification and interpolationschemes based on a class of positive interpolation operators, en-sures overall stability and passivity of the parametric macromodel.Numerical examples validate the proposed approach on practicalapplication cases.
Index Terms—Interpolation, parametric macromodeling, pas-sivity, rational approximation.
I. INTRODUCTION
E FFICIENT design space exploration, design optimiza-tion and sensitivity analysis of microwaves structures
call for the development of robust parametric macromodelingtechniques. Parametric macromodels can take multiple designvariables into account, such as geometrical layout or substratefeatures.
Recently, a multivariate extension of the orthonormal vectorfitting (OVF) technique was presented in [1] and [2]. ThisMOVF method is able to compute accurate parametric macro-models based on parameterized frequency responses whichexhibit a highly dynamic behavior. Unfortunately, the algorithmdoes not guarantee stability and passivity of the parametricmacromodel. In [3] the stability problem is addressed bycomputing a parametric macromodel with barycentric interpo-lation of univariate stable macromodels. It is shown that theoverall stability of the parametric macromodel is guaranteed.An enforcement scheme for the passivity of the parametricmacromodel is proposed by perturbation of the barycentricweights. This technique has some limitations: 1) the conver-gence of the passivity enforcement procedure is not guaranteed,2) the passivity violations must be reasonably small, 3) a densesweep in the design space is needed to detect possible passivityviolations, with a computational cost that increases exponen-tially with the number of design variables, 4) the data samples
Manuscript received April 11, 2009; revised July 14, 2009. First publishedOctober 13, 2009; current version published August 04, 2010. This work wassupported by the Research Foundation Flanders (FWO). This paper was recom-mended for publication by Associate Editor M. Nakhla upon evaluation of thereviewers comments.
The authors are with the Department of Information Technology (INTEC),Ghent University-IBBT, 9000 Ghent, Belgium (e-mail: [email protected]; [email protected]; [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TADVP.2009.2029242
cannot be scattered in the design space, but must be locatedon a fully filled, not necessarily equidistant, rectangular grid.A method that overcomes the restriction on the data samplesordering and uses the flexibility of least-squares fitting, whilepreserving stability was proposed in [4]. More recently, a noveltechnique that combines the advantages of [1] and [4] waspresented in [5]. The hybrid technique is able to calculate morecompact macromodels without compromising the accuracy ofthe results. It is less sensitive to the sample density and theoverall stability of the poles is preserved.
This paper presents a novel technique to build accurate mul-tivariate rational macromodels that are stable and passive in theentire design space, for admittance and impedance rep-resentations. It combines rational identification and interpola-tion schemes based on a class of positive interpolation operators[6], [7], to guarantee overall stability and passivity of the para-metric macromodel. The technique starts by computing mul-tiple univariate frequency domain macromodels using the (or-thonormal) vector fitting ((O)VF) technique [8], [9] for differentcombinations of design variables, as in [3]. In the paper we referto these initial univariate macromodels as root macromodels. Asimple pole-flipping scheme is used to enforce stability [8] foreach root macromodel, while passivity is checked and enforcedby means of standard techniques (see e.g., [10]–[12]). Next,a multivariate macromodel is obtained by combining all rootmacromodels using an interpolation scheme that preserves sta-bility and passivity properties over the complete design space.The proposed technique is validated by some numerical appli-cation examples.
II. PARAMETRIC MACROMODELING
This section explains how the proposed technique builds amultivariate representation which models accurately alarge set of data samples and guar-antees overall stability and passivity in the design space. Thesedata samples depend on a complex frequency , and sev-eral design variables . The design variables de-scribe e.g., the metallizations in an EM-circuit (such as lengths,widths, etc.) or the substrate parameters (like thickness, dielec-tric constant, losses, etc.). Two data grids are used in the mod-eling process: an estimation grid and a validation grid. The firstone is utilized to build the root macromodels which, combinedwith an interpolation scheme, provide the parametric macro-model. The second grid, more dense than the previous one, isutilized to assess the interpolation capability of the parametricmacromodel, its capability of describing the system under studyin points of the design space previously not used for the con-struction of the root macromodels.
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624 IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 33, NO. 3, AUGUST 2010
A. Root Macromodels
Starting from a set of data samples afrequency dependent rational model is built for all grid pointsin the design space by means of (O)VF. A pole-flipping schemeis used to enforce stability [8] and passivity enforcement canbe accomplished using one of the robust standard techniques[10]–[12]. The result of this initial procedure is a set of ra-tional univariate macromodels, stable and passive, that we callroot macromodels being the starting points to build a parametricmacromodel.
B. 2-D Macromodeling
First, we discuss the representation of a bivariate macromodeland afterwards the generalization to more dimensions. Oncethe root macromodels are built, the next step is to find a bi-variate representation which models the set of datasamples and preserves stability and pas-sivity over the entire design space. The bivariate macromodelwe adopt can be written as
(1)
where each interpolation kernel is a scalar function satis-fying the following constraints:
(2)
(3)
The model in (1) is a linear combination of stable and passiveunivariate models by means of positive interpolation kernels [6],[7]. The positiveness of the interpolation kernels is fundamentalto preserve passivity in the design space, while stability is auto-matically preserved as (1) is a weighted sum of stable rationalmacromodels. The proof of the passivity preserving property ofthe proposed technique in the entire design space is given inSection II-D.
C. N-D Macromodeling
The bivariate formulation can easily be generalized to themultivariate case by using multivariate interpolation methods.Multivariate interpolation can be realized in different forms: bymeans of tensor product [13], [14] and algorithms for scattereddata as well-known Shepard’s method [6], [7], [15].
1) Tensor Product Multivariate Interpolation: The tensorproduct multivariate interpolation suffers from the curse of di-mensionality. The data samples have to be located on a fullyfilled, but not necessarily equidistant, rectangular grid. In manycases, this corresponds to the most practical way how multi-variate data samples are organized and computed by a numericalsimulation tool. The multivariate model can be written as
(4)
where each respects both constraints(2) and (3). A suitable choice is to select each set as inpiecewise linear interpolation
(5a)
(5b)
otherwise (5c)
that yields to an interpolation scheme in (4) called piecewisemultilinear interpolation. It can be also seen as a recursive im-plementation of simple piecewise linear interpolation [16], [17].
2) Shepard’s Multivariate Interpolation: Shepard’s methodis a standard algorithm for interpolation at nodes having noexploitable pattern, referred to as scattered or irregularly dis-tributed data. The corresponding multivariate model is writtenin a barycentric form as
(6)
(7)
where . The case is of particular impor-tance, since the interpolation kernels are then infinitely differen-tiable. The interpolation kernels of Shepard’s formula also re-spect both constraints (2) and (3) [7]. Unfortunately Shepard’sscheme presents the occurrence of flat spots at the grid pointswhen since its gradient vanishes, and it is not differ-entiable if giving a generally unsatisfactory internodalbehavior [6], [18]. Shepard’s method in one dimension can bealso extended to more dimensions by using the tensor productformulation, leading to a different Shepard’s multivariate inter-polation scheme not related to scattered data.
In this paper we use the piecewise multilinear interpolationmethod based on a fully filled data grid in the design space, that,as mentioned before, in many cases represents the structure ofmultivariate data samples computed by a numerical simulationtool. It is a local method, because each interpolated value doesnot depend on all the data and it avoids unsatisfactory internodaloscillations as present in Shepard’s method. The scheme is easyto implement and provides accurate results. It is clear that moredata samples in the estimation grid are needed in the case of highdynamics induced by the design parameters on the frequencybehavior of the system than in the case of low dynamics, leadingto an increased computational cost to obtain the multivariatemodel . We note that the kernel functionswe propose only depend on the data grid points and their compu-tation does not require the solution of a linear system to imposean interpolation constraint. The proposed technique is generaland any interpolation scheme that leads to a parametric macro-model composed of a weighted sum of root macromodels withnonnegative weights can be utilized.
FERRANTI et al.: GUARANTEED PASSIVE PARAMETERIZED ADMITTANCE-BASED MACROMODELING 625
Fig. 1. Cross section of the microstrip.
D. Passivity Preserving Interpolation
When performing transient analysis, stability and passivitymust be guaranteed. It is known that, while a passive systemis also stable, the reverse is not necessarily true [19], which iscrucial when the macromodel is to be utilized in a general-pur-pose analysis-oriented nonlinear simulator. Passivity refers tothe property of systems that cannot generate more energy thanthey absorb through their electrical ports. When the system isterminated on any arbitrary passive loads, none of them willcause the system to become unstable [20], [21]. A linear net-work described by admittance matrix is passive if [22],[23]
1) for all , where “ ” is the complex conju-gate operator;
2) is analytic in ;3) is a positive-real matrix, i.e.,
: and anyarbitrary vector .
Similar results are valid for a linear network described byimpedance matrix .
Concerning the root macromodels, conditions 1) and 2) arealways satisfied since all complex poles/residues are alwaysconsidered along with their conjugates and strict stability isimposed by pole-flipping. Condition 1) is preserved in (1) andthe proposed multivariate extensions, as they are weightedsums with real nonnegative weights of systems respectingthis first condition. Condition 2) is preserved in (1) and theproposed multivariate extensions, as they are weighted sums ofstrictly stable rational macromodels. Condition 3) is enforced,if needed, on the root macromodels by using a standard pas-sivity enforcement technique. To prove that our parameterizedmacromodeling technique preserves overall passivity, we referto the following theorem [24]:
Theorem 1: Any nonnegative linear combination of positivereal matrix is a positive real matrix.
Since (1) and the proposed multivariate extensions areweighted sums with real nonnegative weights of passivemacromodels (root macromodels), condition 3) is satisfiedby construction. We have proven that all the three passivityconditions for admittance and impedance representations arepreserved in our parametric macromodeling algorithm.
III. NUMERICAL EXAMPLES
This section presents two numerical examples related to in-terconnection systems that validate the proposed approach onapplication cases. During the construction of the root macro-models a weighting function equal to
(8)
TABLE IPARAMETERS OF THE MICROSTRIP STRUCTURE
Fig. 2. Magnitude of the parametric macromodel of � ����� ��(� � ��� �m).
is used in the VF fitting process for each entry of the admittanceor impedance matrix. where is the numberof system ports. This approach gives increased weight to smallfunction values [25], thus tending to provide a fitting with a highrelative accuracy rather than a high absolute accuracy.
The weighted rms-error for the parametric macromodels isdefined as
(9)
The worst case rms-error over the validation grid is chosen toassess the accuracy and the quality of parametric macromodels
(10)
(11)
and it is used in the numerical examples. The number of polesfor each root macromodel is selected adaptively in VF by abottom-up approach, in such a way that the correspondingweighted rms-error is smaller than .
A. One Stripline With Variable Width and Height Substrate
In this example, a microstrip transmission line (lengthcm) has been modeled. The cross section is shown in Fig. 1. A
trivariate macromodel is built as a function of the width of thestrip and the height of the substrate in addition to frequency.Their corresponding ranges are shown in Table I.
The admittance matrix has been computed basedon the quasi-TEM model discussed in [26] over a validationgrid of 250 70 40 samples . We have builtroot macromodels for 24 values of the width and 14 values ofthe height substrate by means of VF. The passivity of each
626 IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 33, NO. 3, AUGUST 2010
Fig. 3. Magnitude of the parametric macromodel of � ����� �� (� ���� �m).
Fig. 4. Magnitude of the parametric macromodel of � ����� �� (� ���� �m, � � � �m).
model has been verified by checking the eigenvalues of theHamiltonian matrix [27] and enforced if needed. A trivariatemacromodel is obtained by piecewise multilinear interpolationof the root macromodels. The passivity of the parametricmacromodel has been checked by the Hamiltonian test on adense sweep over the design space and the theoretical claim ofoverall passivity has been confirmed. Figs. 2 and 3 show themagnitude of the parametric macromodel of for
m and m, respectively. The worst caserms-error defined in (11) is equal to and it occursfor . Figs. 4–7 compare
, and their macromodels for thewidth and height substrate values corresponding to . Asclearly seen, a very good agreement is obtained between theoriginal data and the proposed passivity preserving macromod-eling technique. The parametric macromodel captures veryaccurately the behavior of the system, preserving stability andpassivity properties over the entire design space.
Fig. 5. Phase of the parametric macromodel of� ����� �� (� � ����m,� � � �m).
Fig. 6. Magnitude of the parametric macromodel of � ����� �� (� ���� �m, � � � �m).
Fig. 7. Phase of the parametric macromodel of� ����� �� (� � ����m,� � � �m).
FERRANTI et al.: GUARANTEED PASSIVE PARAMETERIZED ADMITTANCE-BASED MACROMODELING 627
Fig. 8. Cross section of the two coupled microstrips.
TABLE IIPARAMETERS OF THE TWO COUPLED MICROSTRIPS STRUCTURE
Fig. 9. Magnitude of the parametric macromodel of� ��� ��.
Fig. 10. Magnitude of the parametric macromodel of � ��� �� (� ���� �m).
B. Two Coupled Microstrips With Variable Spacing
A three-conductor transmission line (length cm) withfrequency-dependent per-unit-length parameters has been mod-eled. It consists of two coplanar microstrips over a ground plane.The cross sections is shown in Fig. 8. The conductors have width
m and thickness m. The dielectric is 300m thick and characterized by a dispersive and lossy permit-
tivity which has been modeled by the wideband Debye model
Fig. 11. Phase of the parametric macromodel of� ��� �� (� � ��� �m).
Fig. 12. Magnitude of the parametric macromodel of � ��� �� (� ���� �m).
[28]. A bivariate macromodel is built as a function of the spacingbetween the microstrips in addition to frequency. The ranges
of frequency and spacing are shown in Table II.The frequency-dependent per-unit-length parameters have
been evaluated using a commercial tool [29] over a valida-tion grid of 250 80 samples, for frequency and spacingrespectively. Then, the admittance matrix has beencomputed using transmission line theory (TLT) [30]. Wehave built root macromodels for 30 values of the spacing bymeans of VF. The passivity of each model has been verifiedby checking the eigenvalues of the Hamiltonian matrix andenforced if needed. A bivariate macromodel is obtained bypiecewise multilinear interpolation of the root macromodels.The passivity test on a dense sweep over has confirmedthe theoretical claim of overall passivity. Fig. 9 shows themagnitude of the parametric macromodel of . Theworst case rms-error defined in (11) is equal to , andit occurs for . Figs. 10–13 compare
and their macromodels for the spacingvalue corresponding to . As in the previous example, theparametric macromodel describes very accurately the behavior
628 IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 33, NO. 3, AUGUST 2010
Fig. 13. Phase of the parametric macromodel of� ��� �� (� � ��� �m).
of the system, guaranteeing stability and passivity propertiesover the entire design space.
IV. CONCLUSION
We have presented a new method for the generation of pa-rameterized macromodels of admittance and impedance repre-sentations. The overall stability and passivity of the parametricmacromodel is guaranteed by an efficient and reliable combina-tion of rational identification and interpolation schemes basedon a class of positive interpolation operators. Numerical exam-ples have validated the proposed approach on practical applica-tion cases, showing that it is able to build very accurate para-metric macromodels, while guaranteeing stability and passivityover the complete design space.
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Francesco Ferranti received the B.S. degree(summa cum laude) in electronic engineering fromthe Università degli Studi di Palermo, Palermo,Italy, in 2005, and the M.S. degree (summa cumlaude and honors) in electronic engineering from theUniversità degli Studi dell’Aquila, L’Aquila, Italy,in 2007. He is currently working toward the Ph.D.degree in the Department of Information Technology(INTEC), Ghent University, Ghent, Belgium.
His research interests include robust parametricmacromodeling, rational least-squares approxima-
tion, system identification, and broadband macromodeling techniques.
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Luc Knockaert (SM’00) received the M.Sc. degreein physical engineering, the M.Sc. degree in telecom-munications engineering, and the Ph.D. degree inelectrical engineering from Ghent University, Ghent,Belgium, in 1974, 1977, and 1987, respectively.
From 1979 to 1984 and from 1988 to 1995 he wasworking in North-South cooperation and develop-ment projects at the Universities of the DemocraticRepublic of the Congo and Burundi. He is presentlyaffiliated with the Interdisciplinary Institute forBroadBand Technologies and a Professor at the
Department of Information Technology, Ghent University. His current interestsare the application of linear algebra and adaptive methods in signal estimation,model order reduction and computational electromagnetics. As author or coau-thor he has contributed to more than 100 international journal and conferencepublications.
Dr. Knockaert is a member of MAA and SIAM.
Tom Dhaene (SM’06) was born in Deinze, Belgium,on June 25, 1966. He received the Ph.D. degree inelectrotechnical engineering from the University ofGhent, Ghent, Belgium, in 1993.
From 1989 to 1993, he was Research Assistant atthe University of Ghent, in the Department of Infor-mation Technology, where his research focused ondifferent aspects of full-wave electromagnetic circuitmodeling, transient simulation, and time-domaincharacterization of high-frequency and high-speedinterconnections. In 1993, he joined the EDA com-
pany Alphabit (now part of Agilent). He was one of the key developers of theplanar EM simulator ADS Momentum. Since September 2000, he has beena Professor in the Department of Mathematics and Computer Science at theUniversity of Antwerp, Antwerp, Belgium. Since October 2007, he is a FullProfessor in the Department of Information Technology (INTEC) at GhentUniversity, Ghent, Belgium. As author or coauthor, he has contributed tomore than 150 peer-reviewed papers and abstracts in international conferenceproceedings, journals, and books. He is the holder of three U.S. patents.