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1ISQED 2005 Tutorial IIFrom Field Solvers toFrom Field Solvers toParameterized Reduced Order ModelsParameterized Reduced Order ModelsLuca Daniel, [email protected] to:Jacob White, M.I.T.Joel Phillips, Cadence Berkeley Labs, Zhenhai Zhu, IBM T.J.Watsonwww.rle.mit.edu2ISQED 2005 Tutorial IIElectronic Systems on a Integrated Circuit (IC) Electronic Systems on a Integrated Circuit (IC) or on a Multior on a Multi--Chip Module (MCM)Chip Module (MCM)Courtesy of Harris semiconductorRF InductorsMEMresonatorsPicture by Z. ZhuOn-Chip Interconnectand SubstrateSource: Rabaey, Chandrakasan, NicolicIC PackageSource: CoventorModern electronic systems consist of several circuit components for instance digital circuits, analog RF or mixed signal circuits, RF inductors, Micro-Electro-Mechanical resonators.These components are assembled over a semiconductive substrate or over a package (Multi-Chip-Module) and live inside a very complicated network of wires.3ISQED 2005 Tutorial IIElectronic Systems on a Integrated Circuit (IC) Electronic Systems on a Integrated Circuit (IC) or on a Multior on a Multi--Chip Module (MCM)Chip Module (MCM)Courtesy of Harris semiconductorRF InductorsMEMresonatorsOn-Chip Interconnectand SubstratedtdEHdtdHEεµ=×∇−=×∇dtdEHdtdHEεµ=×∇−=×∇dtdEHdtdHEεµ=×∇−=×∇4224220()welecauuuEISFppdyxxtρ∂∂∂−=+−−∂∂∂∫3()((16))12puKupptµ∂∇⋅+∇=∂RF InductorsMEMresonatorsOn-Chip Interconnectand SubstrateIC PackageThe designers of these Systems on Chip or Systems on Package are well aware that the performance of their systems depend critically on what they call “second order effects” (e.g. capacitive coupling, inductive coupling, electromagnetic fullwave coupling, skin effect, proximity effect, substrate noise, package resonances.)These second order effects can be described accurately only starting from the underling partial differential equations (Maxwell, or Navier-stokes).4ISQED 2005 Tutorial IISource: CoventordWFrom Field Solvers From Field Solvers to Parameterized Model Order Reduction (PMOR).to Parameterized Model Order Reduction (PMOR).dtdEHdtdHEεµ=×∇−=×∇1M equations)()(),(tuBtxdtdxdWE+=10 equations)(ˆ)(ˆˆ),(ˆtuBtxdtxddWE+=PMORPMORField Solvers discretize Field Solvers discretize geometry and produce large geometry and produce large systemssystemsPMOR produces a dynamical model: PMOR produces a dynamical model: –– automaticallyautomatically–– match port impedance match port impedance –– small (10small (10--15 ODEs)15 ODEs)In the previous talk we have seen how the field solver based parasitic extraction tools can efficiently assemble a very accurate model describing the input out behavior of the system components.The model typically consist of a set of ordinary differential equations whose coefficients could in general depend on layout parameters such as wire width W and wire separation d.The task of the Parameterized Model Order Reduction is to produce a dynamical system model automatically, with same input out behavior but much smaller number of ODE (e.g. 10-15), and that can still be instantiated quickly for different values of the layout parameters W and d.5ISQED 2005 Tutorial IIOutlineOutlinen Introductionn Parameterized Model Order Reduction Classificationn From Field Solvers to Parameterized Modelsn Case 1: Model Reduction with Geometrical Parametersn Case 2: Model Reduction with Frequency Parametern ConclusionsHere is an outline for the remaining part of this talk.We will first try to classify the Parameterized Model Order Reduction (PMOR) problem.Then we will see in a simple example how one can assemble a large dynamical linear system model from the output of a field solverFinally we will present techniques for reducing the size of the model.We will have to distinguish two important cases: the case where the system parameters are geometrical (e.g. wire width and separation)and the case where the parameter is frequency.6ISQED 2005 Tutorial IIParameterized model order reduction.Parameterized model order reduction.Problem classification [Rutenbar DAC02]Problem classification [Rutenbar DAC02])()(),...,,(21tuBtxdtdxsssEp+=linearitymatrix size# parametersThe level of difficulty of a parameterized model order reduction problem can be classified according to Rutenbar using 3 main axis:the number of parameters the number of equations (or size of the system)and how linear those equations are7ISQED 2005 Tutorial IIParameterized Model Order ReductionParameterized Model Order ReductionProblem Classification [D. APS04]Problem Classification [D. APS04]matrix size# parameterslinearityNon-Linear SystemsLinear Time Invariantlinearlyparameterizednon-linearly parameterized( )=0,...,,,,,21 pdtdxsssuxE)()(),,(1tuBtxdtdxssEp+=K)()(...2211tuBtxdtdxEsEsEspp+=+++)()(),...,,(21tuBtxdtdxsssEp+=A linear system is a system for which-if for instance I apply double the input I double the output-if I sum two inputs the output is the some of their separate outputs.However let me introduce a further distinction WITHIN the LINEAR systems.The coefficients of the equations of a linear system could -either depend linearly on the parameters-or could depend in a nonlinear way on the parameters8ISQED 2005 Tutorial IIParameterized Model Order Reduction.Parameterized Model Order Reduction.ApplicationsApplicationsinterconnectRF inductorsmatrix size# parameterslinearitylinearlyparameterizednon-linearly parameterizedLinear Time InvariantNon-Linear SystemsLOLNAADCMEMSPackagesHere is where some of the typically electronic components can be situated according to such classification:-the systems generated by field solvers applied on interconnects are