This paper presents a design optimization approach for electromagnetic systems using parametric macromodels. The parametric macromodels are generated using an efficient sequential sampling of the design space of interest which ensures optimal sample selection for a required level of accuracy. The proposed method is validated on a microwave notch filter example for which the parametric macromodel is used in a minimax optimization algorithm so that the design parameters are optimized for some specific electrical design performances

Chemmangat Manakkal Cheriya, Krishnan

Dhaene, Tom

Ferranti, Francesco

Knockaert, Luc

English

Krishnan Chemmangat

Francesco Ferranti

Tom Dhaene

Luc Knockaert

Crossref

Optimization of high-speed electromagnetic systems with accurate parametric macromodels generated using sequential sampling of the design space

Ghent University Academic Bibliography

Optimization of High-Speed ElectromagneticSystems with Accurate Parametric MacromodelsGenerated using Sequential Sampling of the DesignSpace.Krishnan Chemmangat∗ Francesco Ferranti∗ Tom Dhaene∗ Luc Knockaert∗Abstract — This paper presents a design optimiza-tion approach for electromagnetic systems usingparametric macromodels. The parametric macro-models are generated using an efficient sequentialsampling of the design space of interest which en-sures optimal sample selection for a required levelof accuracy. The proposed method is validated on amicrowave notch filter example for which the para-metric macromodel is used in a minimax optimiza-tion algorithm so that the design parameters areoptimized for some specific electrical design perfor-mances.1 INTRODUCTIONDesign optimization of high-speed microwave sys-tems focuses on calculating the optimal values ofthe design variables for which the system responsessatisfy the design specifications. Optimal values ofthe design variables are often determined using op-timization algorithms (optimizers) which drive theelectromagnetic (EM) simulator to obtain the re-sponses and their sensitivities in consecutive op-timization iteration. Unfortunately, multiple con-secutive EM simulations are often computationallyexpensive.In recent years, design optimization of high-speedmicrowave systems using parametric macromodelsattained considerable interest [1, 2]. These para-metric macromodels accurately describe the param-eterized frequency behavior of EM systems for theentire design space of interest, thereby acting as areplacement model for the original and expensiveEM simulators [3, 4, 5, 6, 7]. However, as pointedout in [2], one of the key issues in these modelingapproaches, rarely addressed in the literature, is theoptimal selection of data samples in order to limitthe total number of expensive EM simulations.In this work, we focus on using efficient sequentialsampling schemes to generate parametric macro-models which are used in design optimization of∗Department of Information Technology, Ghent Uni-versity - IBBT, Gaston Crommenlaan 8 Bus 201, B-9050,Gent, Belgium., e-mail: krishnan.cmc@intec.ugent.be,francesco.ferranti@intec.ugent.be, tom.dhaene@intec.ugent.be, luc.knockaert@intec.ugent.be, tel.: +32 9 33148 91, fax: +32 9 331 48 99.microwave systems. Parametric macromodels areused in conjunction with standard optimization al-gorithms [8], resulting in an optimal design suchthat all the design specifications are satisfied.2 PROPOSED METHODThe flowchart of Fig. 1 describes the main idea ofthe proposed optimization method based on para-metric macromodels generated by means of sequen-tial sampling. The method begins by identifyingthe design parameters ~g = (g(n))Nn=1, which aretuned to optimize certain cost functions or per-formance indexes. Once the ranges of the designparameters are defined, some initial design sam-ple points ~gk, k = 1, 2, ...K and the correspond-ing frequency response H(si, ~gk), i = 1, 2, ...Ns aregenerated. Here si represents the complex Laplacevariable. The Vector Fitting (VF) method [9, 10]is used to build frequency-dependent rational mod-els called root macromodels R(s,~gk) at each initialsample points.Once the root macromodels are built at the initialdesign sample points, the next step is to parameter-ize them with respect to the design parameters ~g.In [4], a parametric macromodel is built by interpo-lating a set of root macromodels at an input-outputlevel, while in [5, 6], both poles and residues areparameterized by interpolating the internal state-space matrices, resulting in a higher modeling ca-pability with respect to [4]. In [7], a novel en-hanced interpolation of root macromodels at an in-put/output level is described, which is based on theuse of some coefficients: one coefficient as a mul-tiplicative factor at the input/output level of thesystem and the other coefficient as a compressionor expansion term for the Laplace variable s. Itresults in high modeling capability and robustnessand it is used in this paper.After generating an initial parametric macro-model, the accuracy of this parametric macromodelis checked using some predefined error criteria atcarefully chosen locations of the design space. Ifthe accuracy is found to be inadequate, additionalFigure 1: Flowchart of the proposed optimization.samples are generated in the design space and theparametric macromodel is updated with the gen-eration of root macromodels on the new samplesfollowed by a parameterization using [7]. This pro-cess continues until the required level of accuracyis achieved.The efficient and accurate parametric macro-model generated is used in a minimax optimizationalgorithm which finds the optimal values of the de-sign parameters such that certain performance in-dexes are minimized.3 SEQUENTIAL SAMPLING ALGO-RITHMOnce the ranges of the design parameters ~g are de-fined, a sequential sampling algorithm is used tobuild an accurate parametric macromodel choosingthe data samples at optimal locations in the designspace. Here, the sequential sampling method is ex-plained for a two design parameter case, though itis applicable to any number of design parameters.The design space of the two parameters (g(1), g(2))is defined by the four corners of a rectangular re-gion as shown in Fig. 2.a. Using the corner pointsof the design space, a parametric macromodel isbuilt. Please note that the sequential sampling al-gorithm described here is general so that severalparametric macromodeling schemes based on thelocal N-box regions of the design space can be ap-plied [4, 5, 6, 7]. Here, the method of [7] is selectedbecause of its modeling capability and robustness.Figure 2: Design space during sequential sampling.Next step is to check the accuracy of the gen-erated parametric macromodel at the center of aparticular subspace shown by the gray circle in Fig.2.b. If the accuracy of the parametric macromodelis found to be not satisfactory, four new subspacesare generated, new sample points are added andthe parametric macromodel is updated. This re-finement step continues (Fig. 2.c) till the algorithmterminates by satisfying the accuracy at each localregion of the design space as shown in Fig. 2.d.4 DESIGN OPTIMIZATIONOnce the parametric macromodel is built, it can beused in the optimization process of EM systems.Considering microwave filters, a typical optimiza-tion process begins by defining passband and stop-band specifications in terms of the frequency re-sponses, which are reformulated in the form of acost function Fi(~g), to be minimized at optimiza-tion frequency samples si, i = 1, 2, ..Ns:Fi(~g) = RiL −R(si, ~g) or R(si, ~g)−RiU, (1)In (1), RiL and RiU represents lower and upperfrequency response thresholds, respectively, at fre-quency samples si, i = 1, 2, ..., Ns, spread over thefrequency range of interest. A negative value for thecost function indicates that the corresponding spec-ification is satisfied, while a positive value denotesthat the specification is violated. The minimiza-tion (1) can be performed by several state-of-the-art optimization algorithms. In this paper, we use aminimax optimization algorithm [8] which uses thecost function (1) with respect to design parameter~g giving the optimum design parameters ~g∗ as~g∗ = argmin~g{maxi[Fi(~g)]}. (2)5 NUMERICAL RESULTSA folded stub microwave notch filter on a substratewith relative permitivity r = 9.6 and a thickness of0.635 mm is modeled in this example. The layoutof this filter is shown in Fig. 3. The spacing Sand the length L of the stub are chosen as designvariables in addition to frequency whose ranges areS ∈ [0.5, 1.0] mm, L ∈ [5.0, 10.0] mm and frequency∈ [2, 4] GHz. The design specifications of this notchfilter are given in terms of the scattering parameter|S21| ≤ −20 dB for freq ∈ [2.925, 3.075] GHz (3)From the design specifications (3), considering aresonance frequency of 3 GHz, a cost function (1)is formulated in terms of S21 and ~g = (S,L).Figure 3: Layout of the folded stub notch filter.The scattering matrix S(s, S, L) has been com-puted using the ADS Momentum1 software. Thesequential sampling algorithm has been imple-mented in Matlab R2010a2 and used to drive theADS Momentum simulations to generate S(s, S, L)at selected samples. The number of frequency sam-ples were chosen to be equal to 41. The Mean Ab-solute Error (MAE) given byE(~g) =Pin∑i=1Pout∑j=1Ns∑k=1|Ri,j(sk, ~g)−Hi,j(sk, ~g)|PinPoutNs(4)with number of input ports Pin, output ports Poutand frequency samples Ns is used as an error mea-sure to assess the accuracy at each region of the de-sign space during the sequential sampling process.The final design space generated with the sequen-tial sampling is shown in Fig. 4.As shown in Fig. 4, the algorithm selected 65sample points in the design space of interest shownby the black dots. The validation samples are gen-erated at each and every step of the sequentialsampling and are used in further refinement, asshown in Fig. 2.b. However, for the terminal sub-spaces, since no further refinements are performed,the validation samples are not used in the model-ing but stored in the data base for later use. Fig. 4shows 46 validation sample points for the terminalsubspaces, shown by gray asterisks. The averageADS Momentum simulation time for each designspace point (S,L) has been found to be equal toTSimAvg = 127.70 seconds on a Windows 7 platformon Intel(R) Core(TM)2 Duo P8700 2.53 GHz ma-chine with 2 GB RAM. The total simulation time1Momentum EEsof EDA, Agilent Technologies, SantaRosa, CA.2The Mathworks, Inc., Natick, MA, USAFigure 4: Design space generated using the sequen-tial sampling algorithm.to perform the ADS Momentum simulations and tobuild the parametric macromodel using the sequen-tial sampling is found to be equal to Ttot = 13966.21seconds of which TADS = 13920.68 seconds is spendon generation of data samples using ADS Momen-tum simulations. The target accuracy was set to−55 dB and the worst case accuracy was found tobe −55.21 dB for the parametric macromodel.Parameter ValueNumber of function evaluations 320Initial design point (0.750 mm,(S0, L0) 7.500 mm)Optimal design point (0.995 mm,(S∗, L∗) 6.089 mm)ParametricOptimization macromodel 23.5 stime ADSMomentum 35075.7 sTable 1: Optimization results.The cost function (1) calculated using the para-metric macromodel generated, have been suppliedto the minimax optimization algorithm (2), result-ing in the optimum design parameter values S∗ andL∗. Table 1 lists important parameters for the op-timization. As seen in Table 1, there is a consid-erable gain in terms of the computation time. Wenote that the generation of the parametric macro-model requires some initial ADS Momentum sim-ulations and therefore an initial computational ef-fort. However, once the parametric macromodelis generated and validated, it acts as an accurateand efficient surrogate of the EM solver and canbe used for multiple design optimization scenarios,for instance, changing filter specifications as well asother design activities such as design space explo-ration and sensitivity analysis. Therefore, multipleuses of the parametric macromodel makes the ini-tial computation effort negligible.Figure 5: Magnitude of S21 with optimization.Fig. 5 shows the magnitude of S21 as a functionof frequency before and after optimization alongwith the design specifications for the filter. As seenin Fig. 5 , all the specifications are met for the op-timal design parameter values (S∗, L∗). The actualADS Momentum simulation is also compared withthe parametric macromodel at the optimal solutionand a good agreement between the two responsescan be observed.6 ConclusionsWe have presented a new design optimization ap-proach for EM systems using parametric macro-models, generated based on efficient sequentialsampling of the design space. A minimax optimiza-tion was performed using the parametric macro-model on a microwave notch filter example forwhich the design parameters are optimized for someelectrical performances. The presented numericalresults validate the proposed method.AcknowledgmentsThis work was supported by the Research Founda-tion Flanders (FWO).References[1] F. Ferranti, T. Dhaene, L. Knockaert, G. An-tonini, and A. C. Scogna, “Scalable compactmodels for fast design optimization of complexelectromagnetic systems,” International Jour-nal of RF and Microwave Computer-Aided En-gineering, vol. 22, no. 1.[2] K. Chemmangat, F. Ferranti, T. Dhaene,and L. Knockaert, “Gradient-based opti-mization using parametric sensitivity macro-models,” International Journal of Numer-ical Modelling: Electronic Networks, De-vices and Fields, 2012. [Online]. Available:http://dx.doi.org/10.1002/jnm.839[3] D. Deschrijver, T. Dhaene, and D. De Zut-ter, “Robust parametric macromodeling us-ing multivariate orthonormal vector fitting,”IEEE Transactions on Microwave Theory andTechniques, vol. 56, no. 7, pp. 1661 –1667, July2008.[4] F. Ferranti, L. Knockaert, and T. Dhaene,“Parameterized S-parameter based macro-modeling with guaranteed passivity,” IEEEMicrowave and Wireless Component Letters,vol. 19, no. 10, pp. 608–610, Oct. 2009.[5] P. Triverio, M. Nakhla, and S. Grivet-Talocia,“Passive parametric macromodeling from sam-pled frequency data,” IEEE InternationalConference on Signal Propagation and Inter-connects, pp. 117 –120, May 2010.[6] F. Ferranti, L. Knockaert, T. Dhaene, andG. Antonini, “Passivity-preserving parametricmacromodeling for highly dynamic tabulateddata based on Lur’e equations,” IEEE Trans-actions on Microwave Theory and Techniques,vol. 58, no. 12, pp. 3688 –3696, Dec. 2010.[7] F. Ferranti, L. Knockaert, and T. Dhaene,“Passivity-preserving parametric macromodel-ing by means of scaled and shifted state-spacesystems,” IEEE Transactions on MicrowaveTheory and Techniques, vol. 59, no. 10, pp.2394 –2403, Oct. 2011.[8] D. Z. Du and P. M. Pardalos, Minimax andApplications. Dordrecht, The Netherlands:Kluwer Academic Publishers, 1995.[9] B. Gustavsen and A. Semlyen, “Rational ap-proximation of frequency domain responses byvector fitting,” IEEE Transactions on PowerDelivery, vol. 14, no. 3, pp. 1052–1061, July1999.[10] D. Deschrijver, M. Mrozowski, T. Dhaene, andD. De Zutter, “Macromodeling of multiportsystems using a fast implementation of thevector fitting method,” IEEE Microwave andWireless Components Letters, vol. 18, no. 6,pp. 1587–1592, June 2008.

https://biblio.ugent.be/publication/3175306/file/3175347.pdf

Optimization of high-speed electromagnetic systems with accurate parametric macromodels generated using sequential sampling of the design space

Abstract

Similar works

Full text

Available Versions

Crossref

Ghent University Academic Bibliography