Ferranti, Francesco

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Ghent University Academic Bibliography

Parametric Macromodeling with GuaranteedPassivity for S-parametersFrancesco FerrantiSupervisor(s): Luc Knockaert, Tom DhaeneI. INTRODUCTIONRobust parametric macromodeling is becom-ing increasingly important for efficient designspace exploration, design optimization and sen-sitivity analysis of microwave structures. Para-metric macromodels can take multiple designvariables into account, such as geometrical lay-out or substrate features, in addition to time orfrequency [1].This paper presents a novel technique tobuild accurate multivariate rational macromod-els for scattering representations that are stableand passive over the entire design space. Thetechnique is validated by a numerical example.II. PARAMETRIC MACROMODELINGThe goal of the proposed algorithm is tobuild a multivariate representation R(s,~g)which accurately models a large set ofKtot data samples {(s,~g)k,H(s,~g)k}Ktotk=1and guarantees stability and passivity over theentire design space. These data samples dependon the complex frequency s = jω, and sev-eral design variables ~g = (g(n))Nn=1, such asthe layout features of a circuit (e.g. lengths,widths,...) or the substrate parameters (e.g.thickness, dielectric constant, losses,...).A. Root MacromodelsStarting from a set of multivariate data sam-ples, a frequency dependent rational model in a—————————————————–F. Ferranti is with the Department of InformationTechnology (INTEC), Ghent University (UGent),Gent, Belgium. E-mail: francesco.ferranti@ugent.be.pole-residue form is built for all grid points inthe design space by means of the Vector Fitting(VF) technique [2]. A pole-flipping scheme isused to enforce strict stability [2] and passivityenforcement can be accomplished using one ofthe robust standard techniques [3]. The resultof this initial step is a set of rational univariatemacromodels, stable and passive, that we callroot macromodels being the starting points tobuild the global parametric macromodel.B. 2-D MacromodelingIn this section we discuss the represen-tation of a bivariate macromodel. Once theroot macromodels are available, the bivariatemacromodel R(s, g) can be written as:R(s, g) =K1∑k=1R(s, gk)`k(g) (1)where the interpolation kernels `k(g) are scalarfunctions satisfying the following constraints:`k(g) ≥ 0, `k(gi) = δk,i,K1∑k=1`k(g) = 1(2)A suitable choice is to select `k(g) as in piece-wise linear interpolation. The model in (1) isa linear combination of stable and passive uni-variate models by means of a class of positiveinterpolation kernels [4]. Stability is automati-cally preserved in (1), as it is a weighted sum ofstable rational macromodels. The proof of thepassivity preserving property of the proposedtechnique over the entire design space can befound in [5]. The generalization to a multi-variate model can be realized by a multivari-ate interpolation scheme called piecewise mul-tilinear interpolation. The proposed parametricmacromodeling technique is general and anyinterpolation scheme that leads to a parametricmacromodel composed of a weighted sum ofroot macromodels with weights satisfying (2)can be used.III. NUMERICAL EXAMPLEA. Double folded stub microstrip bandstop fil-terThe double folded stub microstrip bandstopfilter under study is shown in Fig. 1. The para-metric macromodel of the scattering matrix isbuilt as function of the varying length of eachfolded segment L ∈ [2.08 - 2.28] mm andvarying spacing between a folded stub and themain line S ∈ [0.091 - 0.171] mm over thefrequency range [5 - 20] GHz. All data is sim-ulated by ADS-Momentum1 over a referencegrid of 300× 60× 60 samples (freq, L, S).P1 P2SSLLFigure 1. Geometry of the double folded stub mi-crostrip bandstop filter.441 root macromodels are built for 21 values ofthe length (L) and 21 values of the spacing (S)by means of VF. The passivity of each modelis verified and enforced if needed. A trivariatemacromodel is obtained by piecewise multilin-ear interpolation of all 441 root macromodels.Fig. 2 shows the magnitude of the parametricmacromodels of S11(s, L, S) and S21(s, L, S)for the spacing value S = 0.091 mm. The—————————————————–1Momentum EEsof EDA, Agilent Technologies,Santa Rosa, CA.maximum absolute error over the reference gridis bounded by −64.4 dB.2.12.25 1015 2000.20.40.60.81Frequency [GHz]Length [mm]|S 11,|S 21|Figure 2. Magnitude of the trivariate models of S11(light grey surface) and S21 (dark grey surface)for S = 0.091 mm.IV. CONCLUSIONSWe have presented a new macromodelingtechnique for parameterized scattering repre-sentations. An efficient and reliable combina-tion of rational identification and interpolationschemes based on a class of positive interpo-lation operators guarantees the overall stabilityand passivity of the parametric macromodel.REFERENCES[1] D. Deschrijver, T. Dhaene, and D. De Zut-ter, “Robust parametric macromodeling usingmultivariate orthonormal vector fitting,” IEEETrans. Microw. Theory Tech., vol. 56, no. 7, pp.1661-1667, Jul. 2008.[2] B. Gustavsen and A. Semlyen, “Rational ap-proximation of frequency domain responses byvector fitting,” IEEE Trans. Power Del., vol.14, no. 3, pp. 1052-1061, Jul. 1999.[3] T. Dhaene, D. Deschrijver, N. Stevens, “Effi-cient algorithm for passivity enforcement of S-parameter based macromodels,” IEEE Trans.Microw. Theory Tech., vol. 57, no. 2, pp. 415-420, Feb. 2009.[4] G. Allasia, “Simultaneous interpolation and ap-proximation by a class of multivariate positiveoperators,” Numer. Algor., vol. 34, no. 2, pp.147-158, Dec. 2003.[5] F. Ferranti, L. Knockaert, T. Dhaene, “Parame-terized S-parameter based macromodeling withguaranteed passivity,” accepted for publicationon IEEE Microw. Wireless Compon. Lett.

Parametric macromodeling with guaranteed passivity for S-parameters

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Parametric macromodeling with guaranteed passivity for S-parameters

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