ab - connecting repositoriesdr. jan ter maten (nxp semiconductors, the netherlands) dr. bastiaan...
TRANSCRIPT
TKK Radio Science and Engineering Publications
Espoo, September 2008 REPORT R4
SCEE 2008 BOOK OF ABSTRACTS
The 7th International Conference on
Scientific Computing in Electrical Engineering (SCEE 2008),
September 28 – October 3, 2008,
Helsinki University of Technology,Espoo, Finland
Editors: Janne Roos and Luis R.J. Costa
AB TEKNILLINEN KORKEAKOULU
TEKNISKA HÖGSKOLAN
HELSINKI UNIVERSITY OF TECHNOLOGY
TECHNISCHE UNIVERSITÄT HELSINKI
UNIVERSITE DE TECHNOLOGIE D’HELSINKI
TKK Radio Science and Engineering Publications
Espoo, September 2008 REPORT R4
SCEE 2008 BOOK OF ABSTRACTS
The 7th International Conference on
Scientific Computing in Electrical Engineering (SCEE 2008),
September 28 – October 3, 2008,
Helsinki University of Technology,Espoo, Finland
Editors: Janne Roos and Luis R.J. Costa
This report contains abstracts of presentations given at the SCEE 2008 conference.
Helsinki University of Technology
Faculty of Electronics, Communications and Automation
Department of Radio Science and Engineering
Distribution:
Helsinki University of Technology
Department of Radio Science and Engineering
P.O. Box 3000
FI-02015 TKK
Tel. +358 9 451 2261
Fax +358 9 451 2267
E-mail: [email protected]
c© 2008 Janne Roos, Luis R.J. Costa, SCEE, and TKK
ISBN 978-951-22-9557-9 (printed)
ISBN 978-951-22-9563-0 (electronic)
http://lib.tkk.fi/Books/2008/isbn9789512295630.pdf
urn:nbn:fi:tkk-012356
ISSN 1797-4364 (printed)
ISSN 1797-8467 (electronic)
Picaset Oy
Helsinki 2008
Preface
Preface
SCEE is an international conference series dedicated to Scientific Computing in Electrical Engineering.It started as a national German meeting held in Darmstadt (1997) and Berlin (1998), both under theauspices of the Deutscher Mathematiker Verein. In 2000, the first truly international SCEE confer-ence was organized in Warnemunde by the University of Rostock, Germany. In 2002, the 4th SCEEconference was jointly organized by the Eindhoven University of Technology and Philips ResearchLaboratories Eindhoven, The Netherlands. In 2004, the 5th SCEE conference took place in CapoD’Orlando, Italy, organized by Universita di Catania and Consorzio Catania Ricerche. Most recently,in 2006, the 6th SCEE conference took place in Sinaia, Romania, organized by Politehnica Universityof Bucharest.
The 7th International Conference on Scientific Computing in Electrical Engineering (SCEE 2008)in Espoo, Finland, is organized by the Helsinki University of Technology (TKK); Faculty of Electron-ics, Communications and Automation (ECA); Department of Radio Science and Engineering (RAD);Circuit Theory Group. (SCEE 2008 web site: http://www.ct.tkk.fi/scee2008/).
The aim of the SCEE 2008 conference is to bring together scientists from academia and industrywith the goal of intensive discussions on modeling and numerical simulation of electronic circuits andof electromagnetic fields. The conference is mainly directed towards mathematicians and electricalengineers. The SCEE 2008 conference has the following four main topics:
1. Computational Electromagnetics (CE),2. Circuit Simulation (CS),3. Coupled Problems (CP),4. Mathematical and Computational Methods (CM).
The selection of abstracts in this book was carried out by the Program Committee; each abstractwas reviewed by two or three reviewers. The authors of all accepted abstracts were invited to submitan extended full paper, which will be reviewed as well. The accepted full papers will be published ina separate post-conference book.
On behalf of the Local Organizing Committee, we would like to thank the invited speakers forgraciously agreeing to partcipate in the conference, the authors of contributed abstracts for their well-written scientific texts, the members of the Program Committee for their efforts and time, and those ofthe Scientific Advisory Committee for their support. Also, we are grateful to the financial and materialsupport received from our sponsors.
Finally, we would like to warmly welcome all participants to SCEE 2008.
Espoo, September 12, 2008
Janne Roos, SCEE 2008 chairman
Luis Costa, SCEE 2008 secretary
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Committees
Committees
Local Organizing Committee
Dr. Janne Roos (chairman, TKK/RAD)Mr. Luis Costa (secretary, TKK/RAD)Prof. Martti Valtonen (TKK/RAD)Mr. Mikko Honkala (TKK/RAD)Mr. Jan Fagerstrom (TKK)Mr. Vesa Linja-aho (TKK)Ms. Jenni Tulensalo (TKK)Ms. Tuija Karviala (Congreszon)Ms. Katri Luomanpaa (Congreszon)
Program Committee
Prof. Gabriela Ciuprina (Politehnica University of Bucharest, Romania)Dr. Georg Denk (Qimonda, Germany)Prof. Michael Gunther (University of Wuppertal, Germany)Dr. Jan ter Maten (NXP Semiconductors, The Netherlands)Dr. Bastiaan Michielsen (ONERA, France)Prof. Ursula van Rienen (University of Rostock, Germany)Prof. Vittorio Romano (University of Catania, Italy)Dr. Janne Roos (Helsinki University of Technology, Finland)Prof. Wil Schilders (TU Eindhoven & NXP Semiconductors, The Netherlands)Prof. Jan Sykulski (University of Southampton, United Kingdom)Prof. Thomas Weiland (TU Darmstadt & CST, Germany)
Scientific Advisory Committee
Dr. Andreas Blaszczyk (ABB Corporate Research, Switzerland)Prof. Roland Freund (University of California, Davis, USA)Dr. Francois Henrotte (IEM RWTH, Germany)Prof. Ralf Hiptmair (ETH Zurich, Switzerland)Prof. Daniel Ioan (Politehnica University of Bucharest, Romania)Prof. Lauri Kettunen (Tampere University of Technology, Finland)Prof. Wolfgang Mathis (Leibniz Universitat Hannover, Germany)Dr. Irina Munteanu (Computer Simulation Technology, Germany)Prof. Caren Tischendorf (Universitat zu Koln, Germany)
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Sponsors
Sponsors
Industrial sponsors
Nokia http://research.nokia.com/
STMicroelectronics http://www.st.com/
ABB http://www.abb.com/
Computer Simulation Technology (CST) http://www.cst.com/
Applied Wave Research (AWR) http://www.awrcorp.com/
MunEDA http://www.muneda.com/
Other sponsors
Academy of Finland http://www.aka.fi/en-gb/A/
Helsinki University of Technology http://www.tkk.fi/en/
City of Espoo http://english.espoo.fi/
CoMSON Research Training Network http://www.comson.org/
Elektroniikkainsinoorien seura http://www.eis.fi/
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Conference program
Conference program
Sun 28.09. Mon 29.09. Tue 30.09. Wed 01.10. Thu 02.10. Fri 03.10.
09:00–09:45Invited CE 1I
09:00–09:45Invited CP 1Ia
09:00–09:45Invited CE 2I
09:00–09:45Invited CS 2I
09:00–09:45Invited CM 2I
09:45–10:15Oral CE 1A
09:45–10:15Oral CP 1A
09:45–10:15Oral CE 2A
09:45–10:15Oral CS 2A
09:45–10:15Oral CM 2A
10:15–10:45 Coffee break10:45–11:15Oral CE 1B
10:30–11:15Invited CP 1Ib
10:45–11:15Oral CE 2B
10:45–11:15Oral CS 2B
10:45–11:15Oral CM 2B
11:15–11:45Oral CE 1C
11:15–11:45Oral CP 1B
11:15–11:45Oral CE 2C
11:15–11:45Oral CS 2C
11:15–11:45Oral CM 2C
11:45–12:15Oral CE 1D
11:45–12:15Oral CP 1C
11:45–12:15Oral CE 2D
11:45–12:15Oral CS 2D
11:45–12:15Oral CM 2D
12:15–12:45Short poster
presentation 1
12:15–12:45Short poster
presentation 2
12:15–12:45Oral CE 2E
12:15–12:45Short poster
presentation 3
12:15–12:45Closing
12:45–14:00 Lunch14:00–14:45
Invited CS 1I14:00–14:45
Invited CM 1I14:00–14:45
Invited CS 3I14:45–15:15Oral CS 1A
14:45–15:15Oral CM 1A
14:45–15:15Oral CS 3A
15:15–15:30 Coffee break Coffee break15:30–16:00Oral CS 1B
15:30–16:00Oral CM 1B
14:00–22:0015:30–16:00Oral CS 3B
16:00–16:30Oral CS 1C
16:00–16:30Oral CM 1C
Excursion16:00–16:30Oral CS 3C
16:00–18:00Registration
16:30–17:00Oral CS 1D
16:30–17:00Oral CM 1D
&16:30–17:00Oral CS 3D
17:00–18:00Poster P1
17:00–18:00Poster P2
Dinner17:00–18:00Poster P3
18:00–19:00Openingsession
Invited CS 0I
Walk toRadisson SASHotel Espoo
19:00–21:00Welcomecocktail
18:45–21:00
Espoo cityreception
Invited talks
CS 0I Prof. Peter Benner (TU Chemnitz, Germany)Advances in balancing-related model reduction for circuit simulation
CE 1I Dr. Sergey Yuferev (Nokia, Finland)Challenges and approaches in EMC/EMI modeling of wireless devices
CS 1I Prof. Qi-Jun Zhang (Carleton University, Canada)ANN/DNN-based behavioral modeling of RF/microwave components/devices and circuit blocks
CP 1Ia Dr. Wim Schoenmaker (MAGWEL, Belgium)Evaluation of the EM coupling between microelectronic device structures using computational electrodynamics
CP 1Ib Prof. Ansgar Jungel (TU Wien, Austria)Thermal effects in coupled circuit-device simulations
CM 1I Dr. David Levadoux (ONERA, France)New trends in the preconditioning of integral equations of electromagnetism
CE 2I Prof. Jan Hesthaven (Brown University, USA)High-order discontinuous Galerkin methods for computational electromagnetics and uncertainty quantification
CS 2I Dr. Emira Dautbegovic (Qimonda, Germany)Wavelets in circuit simulation
CS 3I Prof. Daniel Ioan (Politehnica University of Bucharest, Romania)Parametric reduced-order models for passive integrated components coupled with their EM environment
CM 2I Dr. Galina Benderskaya (CST, Germany)Numerical time integration in computational electromagnetics
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Contents
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Committees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Sponsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Conference program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Invited talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Session: CS 0
CS 0I Advances in balancing-related model reduction for circuit simulationPeter Benner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Session: CE 1
CE 1I Challenges and Approaches in EMC/EMI Modeling of Wireless Consumer DevicesSergey Yuferev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
CE 1A Computation of Eigenmodes in Dispersive MaterialsBastian Bandlow, Rolf Schuhmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
CE 1B Region-oriented BEM formulation for numerical computations of electric and magneticfieldsAndreas Blaszczyk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
CE 1C Tearing and Interconnecting Methods for Non-Elliptic OperatorsMarkus Windisch, Olaf Steinbach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
CE 1D Parametric Models of Transmission Lines Based on First Order SensitivitiesAlexandra Stefanescu, Daniel Ioan, Gabriela Ciuprina . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Session: CS 1
CS 1I ANN/DNN-based Behavioral Modeling of RF/Microwave Components/Devices andCircuit BlocksQ.J. Zhang, Shan Wan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
CS 1A Model Order Reduction for Nonlinear IC models with PODA. Verhoeven, M. Striebel, E.J.W. ter Maten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
CS 1B Model Order Reduction for Systems with Coupled ParametersLihong Feng, Peter Benner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
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CS 1C Model order and terminal reduction approaches via matrix decomposition and lowrank approximationAndre Schneider, Peter Benner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
CS 1D Model order reduction for large resistance networksJoost Rommes, Peter Lenaers, Wil H.A. Schilders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Session: P 1
P 1.1 Integral equation method for magnetic modelling of ciliaMihai Rebican, Daniel Ioan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
P 1.2 Electromagnetic modeling of 3D bodies in layered earths using integral equationsR. Grimberg, Adriana Savin, Rozina Steigmann, Alina Bruma, S.S. Udpa, Lalita Udpa 33
P 1.3 Electromagnetic field diffraction on subsurface electrical barrier. Application tonondestructive evaluationAdriana Savin, Rozina Steigmann, Alina Bruma, Raimond Grimberg . . . . . . . . . . . . . . . 35
P 1.4 Relativistic High Order Particle Treatment for Electromagnetic Particle-In-CellSimulationsMartin Quandt, Claus-Dieter Munz, Rudolf Schneider . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
P 1.5 A Simulation-Based Genetic Algorithm for Optimal Antenna Pattern DesignYi-Ting Kuo, Yiming Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
P 1.6 Probabilistic characterization of resonant EM interactions with thin-wires: varianceand kurtosis analysisO.O. Sy, J.A.H.M. Vaessen, M.C. van Beurden, B.L. Michielsen, A.G. Tijhuis . . . . . . 41
P 1.7 Domain Partitioning Based Parametric Models for Passive On-chip ComponentsGabriela Ciuprina, Daniel Ioan, Diana Mihalache, Ehrenfried Seebacher . . . . . . . . . . . . 43
P 1.8 Parametric Reduced Compact Models for Passive ComponentsDiana Mihalache, Daniel Ioan, Gabriela Ciuprina, Alexandra Stefanescu . . . . . . . . . . . . 45
P 1.9 An Effective Technique for Modelling Nonlinear Lumped Elements Spanning MultipleCells in FDTDLuis R.J. Costa, Keijo Nikoskinen, Martti Valtonen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
P 1.10 Mixed-Derating of Three-Phase Distribution Transformer under Unbalanced SupplyVoltage and Non-Linear Load Conditions Using Finite Element MethodJawad Faiz, M. Ghofrani, B.M. Ebrahimi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
P 1.11 Design Optimization of Radial Flux Permanent MagnetWind Generator for HighestAnnual Energy Input and Lower Magnet VolumesJawad Faiz, M. Rajabi-Sebdani, B.M. Ebrahimi, M.A. Khan . . . . . . . . . . . . . . . . . . . . . . 53
P 1.12 Finite-element discretization for analyzing eddy currents in the form-wound statorwinding of a cage induction motorM. Jahirul Islam, Antero Arkkio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Session: CP 1
CP 1Ia Evaluation of the electromagnetic coupling between microelectronic device structuresusing computational electrodynamicsWim Schoenmaker, Peter Meuris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
CP 1A Simulations of a electron-phonon hydrodynamical model based on the maximumentropy principleV. Romano, C. Scordia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
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CP 1Ib Thermal Effects in Coupled Circuit-Device SimulationsMarkus Brunk, Ansgar Jungel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
CP 1B Automatic Thermal Network Extraction and Multiscale Electro-Thermal SimulationMassimiliano Culpo, Carlo de Falco, Georg Denk, Steffen Voigtmann . . . . . . . . . . . . . . . 71
CP 1C Canonical Stochastic Electromagnetic FieldsB.L. Michielsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Session: CM 1
CM 1I New trends in the preconditioning of integral equations of electromagnetismDavid P. Levadoux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
CM 1A A New Approach to Modeling Multi-Port Scattering ParametersSanda Lefteriu, Athanasios C. Antoulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
CM 1B Numerical and Mathematical Properties of Electromagnetic Surface IntegralEquationsPasi Yla-Oijala, Matti Taskinen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
CM 1C Nonlinear models for silicon semiconductorsSalvatore La Rosa, Giovanni Mascali, Vittorio Romano . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
CM 1D Hyperbolic PDAEs for semiconductor devices coupled with circuitsGiuseppe Alı, Giovanni Mascali, Roland Pulch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Session: P 2
P 2.1 Consistent Initialization for Coupled Circuit-Device SimulationSascha Baumanns, Monica Selva, Caren Tischendorf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
P 2.2 Heating of semiconductor devices in electric circuitsMarkus Brunk, Ansgar Jungel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
P 2.3 Surrogate Modeling of Low Noise Amplifiers Based on Transistor Level SimulationsLuciano De Tommasi, Dirk Gorissen, Jeroen Croon, Tom Dhaene . . . . . . . . . . . . . . . . . . 93
P 2.4 On Local Handling of Inner Equations in Compact ModelsUwe Feldmann, Masataka Miyake, Takahiro Kajiwara, Mitiko Miura-Mattausch . . . . . 95
P 2.5 Transient Analysis of Nonlinear Circuits Based on WavesCarlos Christoffersen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
P 2.6 Time-Domain Preconditioner for Harmonic Balance JacobiansMikko Honkala, Ville Karanko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
P 2.7 Evaluation of Oscillator Phase Transfer FunctionsM.M. Gourary, S.G. Rusakov, S.L. Ulyanov, M.M. Zharov, B.J. Mulvaney, K.K.Gullapalli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
P 2.8 Quasiperiodic steady-state analysis of electronic circuits by a spline basisHans Georg Brachtendorf, Angelika Bunse-Gerstner, Barbara Lang, Siegmar Lampe . . 103
P 2.9 Accurate Simulation of the Devil’s Staircase of an Injection-Locked Frequency DividerTao Xu, Marissa Condon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
P 2.10 Nonlinear distortion in differential circuits with single-ended and balanced driveTimo Rahkonen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
P 2.11 Speed-up techniques for time-domain system simulationsTimo Rahkonen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
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P 2.12 Lookup-Table Based Settling Error Modeling in SIMULINKMarko Neitola, Timo Rahkonen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
P 2.13 BRAM - Backward Reduced Adjoint MethodZ. Ilievski, W.H.A. Schilders, E.J.W. ter Maten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
P 2.14 Applications of Eigenvalue Inclusion Theorems in Model Order ReductionE. Fatih Yetkin, Hasan Dag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
P 2.15 Hierarchical Model-Order Reduction FlowMikko Honkala, Pekka Miettinen, Janne Roos, Carsten Neff . . . . . . . . . . . . . . . . . . . . . . . 117
P 2.16 Partitioning-based RL-in–RL-out MOR MethodPekka Miettinen, Mikko Honkala, Janne Roos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
P 2.17 Nonlinear model order reduction based on trajectory piecewise linear approach:comparing different linear coresKasra Mohaghegh, Michael Striebel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
P 2.18 On Model Order Reduction for Perturbed Nonlinear Neural Networks with FeedbackMarissa Condon, Georgi G. Grahovski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Session: CE 2
CE 2I High-order discontinuous Galerkin methods for computational electromagnetics anduncertainty quantificationJ.S. Hesthaven, T. Warburton, C. Chauviere, L. Wilcox . . . . . . . . . . . . . . . . . . . . . . . . . . 127
CE 2A Using Nudg++ to Solve Poisson’s Equation on Unstructured GridsChristian R. Bahls, Gisela Poplau, Ursula van Rienen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
CE 2B From the Boundary Element DDM to Trefftz Finite Element Methods on ArbitraryPolyhedral MeshesDylan Copeland, Ulrich Langer, David Pusch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
CE 2C Reduced Basis Method for Electromagnetic Field ComputationsJan Pomplun, Frank Schmidt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
CE 2D Surface Integrated Field Equations Method for solving 3D Electromagnetic problemsZhifeng Sheng, Patrick Dewilde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
CE 2E A Novel Staggered Finite Volume Time Domain MethodThomas Lau, Erion Gjonaj, Thomas Weiland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Session: CS 2
CS 2I Wavelets in circuit simulationEmira Dautbegovic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
CS 2A Hybrid Analysis of Nonlinear Time-Varying Circuits Providing DAEs with at MostIndex 1Satoru Iwata, Mizuyo Takamatsu, Caren Tischendorf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
CS 2B Convergence Issues of Ring Oscillator Simulation by Harmonic Balance TechniqueM.M. Gourary, S.G. Rusakov, S.L. Ulyanov, M.M. Zharov, B.J. Mulvaney . . . . . . . . . 145
CS 2C Stochastic Models for Analysing Parameter SensitivitiesRoland Pulch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
CS 2D Simultaneous Step-size and Path Control for Efficient Transient Noise AnalysisWerner Romisch, Thorsten Sickenberger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
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Contents
Session: CS 3
CS 3I Parametric Reduced Order Models for Passive Integrated Components Coupled withtheir EM EnvironmentDaniel Ioan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
CS 3A Stability of the Super Node Algorithm for EMC Modelling of ICsM.V. Ugryumova, W.H.A. Schilders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
CS 3B GABOR: Global-Approximation-Based Order ReductionJanne Roos, Mikko Honkala, Pekka Miettinen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
CS 3C Passivity-preserving balanced truncation model reduction of circuit equationsTatjana Stykel, Timo Reis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
CS 3D A new approach to passivity preserving model reductionRoxana Ionutiu, Joost Rommes, Athanasios C. Antoulas . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Session: P 3
P 3.1 Analysis of a PDE Thermal Element Model for Electrothermal Circuit SimulationGiuseppe Alı, Andreas Bartel, Massimiliano Culpo, Carlo de Falco . . . . . . . . . . . . . . . . . 165
P 3.2 Efficient simulation of large-scale dynamical systems using tensor decompositionsF. van Belzen, S. Weiland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
P 3.3 Magnetic Force Calculation applied to Magnetic Force MicroscopyThomas Preisner, Michael Greiff, Josefine Freitag, Wolfgang Mathis . . . . . . . . . . . . . . . . 169
P 3.4 EM Scattering Calculations using PotentialsMagnus Herberthson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
P 3.5 Simulation of large interconnect structures using a ILU-type preconditionerD. Harutyunyan, W. Schoenmaker, W.H.A. Schilders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
P 3.6 Fast Linear Solver for Vector Fitting MethodM.M. Gourary, S.G. Rusakov, S.L. Ulyanov, M.M. Zharov, B.J.M. Mulvaney, B. Gu 175
P 3.7 Evaluation of Domain Decomposition Approach for Compact Simulation of On-ChipCoupled ProblemsMichal Dobrzynski, Jagoda Plata, Sebastian Gim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
P 3.8 A Novel Graphical Based Tool for Extraction of Magnetic Reluctances betweenOn-Chip Current LoopsSebastian Gim, Alexander Vasenev, Diana Mihalache, Alexandra Stefanescu, SebastianKula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
P 3.9 Multiobjective Optimization Applied to Design of PIFA AntennasStefan Jakobsson, Bjorn Andersson, Fredrik Edelvik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
P 3.10 Space Mapping with Difference Method for OptimizationN. Serap Sengor, Murat Simsek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
P 3.11 Modification of Space Mapping with Difference Method for Inverse ProblemsMurat Simsek, N. Serap Sengor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
P 3.12 Exploiting Model Hierarchy in Semiconductor Design using Manifold-MappingD.J.P. Lahaye, C. Drago . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
P 3.13 An hp-Adaptive Strategy for the Solution of the Exact Kernel Curved WirePocklington EquationD.J.P. Lahaye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
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Contents
P 3.14 Sensitivity Analysis of Capacitance for a TFT-LCD PixelTa-Ching Yeh, Hsuan-Ming Huang, Yiming Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Session: CM 2
CM 2I Numerical Time Integration in Computational ElectromagneticsGalina Benderskaya, Oliver Sterz, Wolfgang Ackermann, Thomas Weiland . . . . . . . . . . 195
CM 2A Co-Simulation of Circuits Refined by 3-D Conductor ModelsSebastian Schops, Andreas Bartel, Herbert De Gersem, Michael Gunther . . . . . . . . . . . . 197
CM 2B Robust FETI solvers for multiscale elliptic PDEsClemens Pechstein, Robert Scheichl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
CM 2C Functional Type A Posteriori Error Estimates for Maxwell’s EquationsAntti Hannukainen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
CM 2D Large-Scale Atomistic Circuit-Device Coupled Simulation of Intrinsic Fluctuations inNanoscale Digital and High-Frequency Integrated CircuitsChih-Hong Hwang, Ta-Ching Yeh, Tien-Yeh Li, Yiming Li . . . . . . . . . . . . . . . . . . . . . . . . 203
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
xii
Session
CS 0
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 0I
Advances in balancing-related model reduction for circuit
simulation
Peter Benner
TU Chemnitz, Fakultat fur Mathematik, 09107 Chemnitz, Germany [email protected]
Summary. We discuss methods related to balancedtruncation for model reduction of linear systems.These methods are known to have good global ap-proximation properties and to preserve importantsystem properties. A computable error bound allowsto choose the order of the reduced-order model adap-tively. We will emphasize those aspects that maketheir application to models arising in circuit simula-tion a non-straightforward task. In recent years, theseissues have been addressed by several authors. Wewill survey these developments and demonstrate thatthese techniques are now suitable for the large-scale,sparse systems encountered in circuit simulation.
1 Introduction
Model order reduction (MOR) is an ubiquitoustool in the design and analysis of integrated cir-cuits (ICs) and circuit simulation in general. Inmany situations, only the use of MOR techniquesallows the numerical simulation of the usuallyvery large systems of ordinary differential anddifferential-algebraic equations used to describe(parts of) complex circuit layouts. MOR has beenparticularly successful in reducing the complexityof large linear subcircuits modeling parasitic ef-fects of interconnect etc., and it is becoming anincreasingly useful tool also in other areas of cir-cuit design.
We consider linear descriptor systems
Ex(t) = Ax(t) +Bu(t), y(t) = Cx(t), (1)
with A,E ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n andcorresponding transfer function G(s) = C(sE −A)−1B, resulting from describing the input-to-output map u → y in frequency domain.1 Onedifficulty for balancing related model reductionmethods arises from E being singular as it is usu-ally the case in circuit simulation — a part ofthe talk and the full paper will be devoted to theadvances made for resolving this issue.
The model reduction problem now consists offinding a reduced-order system,
1 Note that frequently in the area of circuit simula-tion, different notation is used: there E,A, and Cbecome C,G, and LT , respectively. The notationused here is standard in systems theory.
E ˙x(t) = Ax(t) + Bu(t), y(t) = Cx(t), (2)
of order r, r ≪ n, with the same number of inputsm, the same number of outputs p, and associatedtransfer function G(s) = C(sE − A)−1B, so thatfor the same input function u ∈ L2(0,∞; Rm), wehave y(t) ≈ y(t).
2 Review of MOR in IC design
First attempts of employing MOR in circuit sim-ulation used the so called asymptotic waveformevaluation (AWE) technique [6]. This is based onthe following observation: for s0 6∈ Λ(A,E), whereΛ(A,E) denotes the set of generalized eigenvaluesof the matrix pencil A− λE, we can re-write thetransfer function of (1) and expand it into a power(Laurent) series as follows:
G(s) = C(I + (s− s0)(s0E −A)−1E
)−1
×(s0E −A)−1B
= M0 +M1(s− s0) +M2(s− s0)2 + . . .
For s0 = 0, the coefficient matrices in the powerseries are given by Mj := C(A−1E)jA−1B andare called moments.
As reduced-order model we can now use therth Pade approximant G to G, defined by theproperty G(s) = G(s) +O((s− s0)
2r), i.e., Mj =
Mj for j = 0, . . . , 2r − 1. Thus, Pade approxi-mation is based on moment matching. AWE nowcomputes the moments explicitly and obtains thecoefficients of the enumerator and denominatorpolynomials of G by solving a linear system ofequations. It was soon observed that this pro-cedure is numerically unstable as computing themoments can be interpreted as kind of a poweriteration so that the computed moments becomenumerically linearly dependent quite fast.
A breakthrough for Pade approximation meth-ods then was the observation that the momentsneed not be computed explicitly as moment match-ing is equivalent to projecting the state-spaceonto V = K(A, B, r), along W = K(AH , CH , r).Here, A = (s0E − A)−1E, B = (s0E − A)−1B,and K(M, b, ℓ) denotes the Krylov subspace
3
spanb,Mb,M2b, . . . ,M ℓ−1b.Bi-orthogonal bases of these two (block-)Krylovsubspaces can be computed via the unsymmet-ric (block, band) Lanczos method [2–4]. In ad-dition, a realization of the reduced-order modelis computed as a by-product of the unsymmetricLanczos process. The computation of a reduced-order model based on the moment matching ideaimplemented as unsymmetric Lanczos process iscalled the (matrix) Pade-via-Lanczos ((M)PVL)method [2,3]. PVL is popular in circuit simulationand micro electronics as it can be applied with-out changes to descriptor systems with E singularwhich quite commonly arise in these applications.
Pade-type methods are also based on the mo-ment matching property, but the approximationsneed not match the maximum possible numberof moments. One such method is PRIMA [5]which employs the numerically stable Arnoldiprocess to compute an orthogonal bases of Vas above and performs an orthogonal projectiononto V . PRIMA has been a success story in MORfor circuit simulation as besides having moment-matching properties, it preserves stability andpassivity of RLC circuit models.
Despite all the success with Pade(-type) ap-proximation techniques based on the moment-matching properties of Krylov subspace methods,some major difficulties of this approach persist:
1. So far there exists in general no computableerror estimate or bound for ‖y − y‖2.
2. The reduced-order model provides a good ap-proximation quality only locally.
3. The preservation of physical properties likestability or passivity can only be shown invery special cases; usually some post process-ing which (partially) destroys the momentmatching properties, is required.
There are many recent advances with respect toitems 1.–3. discussed in the recent literature, dueto space limitations we can not discuss all thesenew developments here.
3 Balancing-related MOR
All the above problems of moment-matching meth-ods are avoided when using balanced truncation(BT) or its relatives. Computable error boundsexist and come for free as by-product of the com-putational procedures for obtaining the reduced-order model. Stability of the linear system is pre-served for all variants, other properties like pas-sivity (which is important for passive devices) canbe preserved, depending on which variant of BT isused. The methods have good global approxima-tion properties and thus, the reduced-order model
can serve as surrogate for a large frequency range.In the simplest case of balanced truncation, themain computational task is the solution of thedual Lyapunov equations
AP + PAT +BBT = 0,ATQ+QA+ CTC = 0.
(3)
The reduced-order model is then obtained by fur-ther manipulations from the solutions P,Q of (3).Other variants of BT replace the Lyapunov equa-tions by algebraic Riccati equations which seemsto be even more challenging from the computa-tional point of view. It has been common beliefuntil recently that BT-related methods are notapplicable in circuit simulation due to the O(n3)complexity of usual matrix equation solvers. Butadvances in numerical linear algebra nowadays al-low to compute solutions to those Lyapunov andRiccati equations arising in BT-related methodsfor linear system in circuit simulation at a compu-tational cost that scales with the cost for solvinglinear systems of equations with coefficient ma-trix A+s0E. Thus these methods can now be ap-plied to systems of order O(106). Moreover, mostof the difficulties resulting from a singular E ma-trix have now also been overcome. Many of thesedevelopments can already be found in [1] and ref-erences therein. In the talk and the full paper, wewill survey the most important advances usefulfor MOR of linear systems in circuit simulationand will also highlight some recent improvements.
References
1. P. Benner, V. Mehrmann, and D.C. Sorensen, ed-itors. Dimension Reduction of Large-Scale Sys-tems, volume 45 of Lecture Notes in Computa-tional Science and Engineering. Springer-Verlag,Berlin/Heidelberg, Germany, 2005.
2. P. Feldmann and R.W. Freund. Efficient linear cir-cuit analysis by Pade approximation via the Lanc-zos process. IEEE Trans. Comput.-Aided Des. In-tegr. Circuits Syst., 14:639–649, 1995.
3. R. Freund. Model reduction methods based onKrylov subspaces. Acta Numerica, 12:267–319,2003.
4. K. Gallivan, E. Grimme, and P. Van Dooren.Asymptotic waveform evaluation via a Lanczosmethod. Appl. Math. Lett., 7(5):75–80, 1994.
5. A. Odabasioglu, M. Celik, and L.T. Pileggi.PRIMA: passive reduced-order interconnectmacromodeling algorithm. In Tech. Dig. 1997IEEE/ACM Intl. Conf. CAD, pages 58–65. IEEEComputer Society Press, 1997.
6. L.T. Pillage and R.A. Rohrer. Asymptotic wave-form evaluation for timing analysis. IEEE Trans.Comput.-Aided Design, 9:325–366, 1990.
4
Session
CE 1
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CE 1I
Challenges and Approaches in EMC/EMI Modeling of
Wireless Consumer Devices
Sergey Yuferev
Nokia Corp., Visiokatu 3, 33720 Tampere, Finland, [email protected]
Summary. The paper contains overview of key tasksin numerical modeling of EMC (Electromagnetic Com-patibility) problems of mobile phones. Special focusis given to following questions: overcoming computa-tional difficulties caused by multi-scale feature, repre-sentation of different types of EMI sources in numer-ical models using measured data and methods of val-idation of numerical results. Examples of numericalmodeling of real problems are included in full versionof the paper.
1 Introduction
With decreasing design circles in modern elec-tronic industry, simulations play more and moreimportant role as an alternative to traditionalway of making prototypes and measurements.High frequency wireless consumer devices likemobile phones are becoming increasingly com-plex since they are actually combinations of otherproducts such as cameras, sensors, radio, comput-ers etc. At the same time, they are getting smallerso complying with EMC regulations is now chal-lenging task. At least some of potential EMCproblems can be predicted well before physicalprototypes are built by application of numericalanalysis at early design stage. Significant effortshave been recently made by IEEE EMC Societyto develop standards and recommended practiceof the use of computational packages for simula-tion of real EMC problems [1].
In spite of so obvious benefits, CAD simula-tion packages have not become yet everyday toolsof EMC designers unlike antenna, thermal or sig-nal integrity designers of wireless devices. Evennow, EMC problems of wireless devices are fre-quently considered as ”black magic area” whererigorous numerical analysis is impossible due tohigh complexity of the problems. In the presentpaper some key challenges in numerical analy-sis of EMC problems of wireless devices are dis-cussed.
2 Tasks for EMC Simulations
EMC regulations are imposed as maximum ac-ceptable values (limits) of the electromagnetic
field at certain distances from the product. Lim-iting values of the EM field parameters are alsospecified inside the phone around modules con-taining strong EM radiators (emission interoper-ability limits) and modules sensitive to excessivelevels of the EM field (immunity interoperabilitylimits). Compliance with interoperability limitsensures functioning of the device without noisesand interferences, and, finally, effects on the prod-uct quality.
Obviously those limits are expressed in termsof absolute numbers and do not depend on con-ditions of neither measurements nor numericalmodeling. Thus EMC simulation technology shouldalso provide numerical solution in the form of ab-solute numbers to be compared with EMC stan-dards and interoperability limits.
3 Multi-Scale Feature
Characteristic size of the device is about 5-10 cmwhereas characteristic size of the PCB details isless than 1 mm. Difference between the charac-teristic dimensions is two orders of magnitudethat leads to very serious computational prob-lem, namely: to provide necessary resolution insimulation of the PCB details, the average size ofthe cell in computational mesh used by the soft-ware for three-dimensional electromagnetic simu-lations should be hundreds or even tens microns.Thus discretization of the computational spaceincluding the device by cells of this size leadsto such a mesh that no one modern computerwill be able to simulate such a device ”as is” inreasonable time. This problem got the name of”multi-scale” feature is well known in computa-tional electromagnetics [2].
Computational problems related with ”multi-scale” features may be overcomed using so-calledmulti-stage modeling: the problem is divided intosub-problems, each of them is analyzed by spe-cial software and the results are combined [3]. Inthe case of wireless devices like mobile phones,basic idea of this approach is separate numericalsolution of Maxwell’s equations in domain occu-pied by the printed circuit board and in rest of
7
the device. Realization of the idea can be done byiterations; each of them consists of the followingtwo steps:
a. The electromagnetic behavior of the printedcircuit board is analyzed separately from thedevice (the device is ignored) and near fieldradiation of the board is calculated; then theelectromagnetic field distribution around theboard is transferred to the next iteration tobe used as input data (equivalent source orboundary condition).
b. Three-dimensional problem of the electromag-netic field distribution is solved everywhereinside the device excepting domain occupiedby the printed circuit board: the PCB domainis eliminated from the numerical procedureand replaced by the field distribution at theinterface (obtained at the previous iteration).
Main advantage of the described procedureis that cells for modeling the PCB and the de-vice are not combined in the same mesh. Instead,computer grids for the PCB and device are usedseparately at different stages so the multi-scaleproblem is resolved and three-dimensional distri-bution of the electromagnetic field inside and out-side the device can be simulated in wide range offrequencies. However, practical realization of theapproach requires high level of compatibility be-tween codes applied at steps a and b (files pro-duced by one code should be used as input datain another one). Another issue requiring furtherinvestigation is convergence of the iterative pro-cedure. Intuitively clear that convergence will de-pend on characteristic scales of the problem: sizeof device, size of PCB and distance between them.Full version of the paper contains analysis of con-ditions of applicability and numerical example.
4 Representation of EMC/EMISources
Main EMI sources in the wireless devices are an-tennas, RF modules located on the board and cur-rents flowing in the PCB traces. Electromagneticbehaviour of most of them cannot be describedwithout SPICE-like models. Rigorous consider-ation requires combination of 3D EM (device),2.5D EM (PCB) and circuit simulations using realcharacteristics of RF ICs. Alternative approach isto approximate real sources by combinations oflumped ports and calibrate their parameters us-ing measured data, for example - near field distri-bution over IC. Advantages, limitations and ex-amples of both approaches are considered in fullversion of the paper.
5 Validations of Numerical Results
Mixing circuit (lumped) level, field level and evenmolecular level models may naturally lead to sig-nificant errors so the problem of validation ofnumerical results becomes central problem whennew simulation approaches are developed. In thearea of board level EMC problems neither ana-lytical solutions nor estimations based on previ-ous experience may be applied for validations someasurements are the only approach to obtainreference data. Full version of the paper is fo-cused on the near/far fields and scattering param-eters measurements, and methodologies to repre-sent measured data in the form suitable for com-parison with numerical results and EMC limits.
References
1. A.L. Drozd. Progress on the development of stan-dards and recommended practices for CEM com-puter modeling and code validation. In Proceedingsof 2003 IEEE International Symposium on Elec-tromagnetic Compatibility, pp.313-316, 8-22 Aug.2003.
2. A. Yilmaz, M.L. Choi, J. Jin, E. Michielsson andA.C. Cangellaris. Multi-scale hybrid electromag-netic modeling and transient simulation of multi-layered printed circuit boards. In Proceedings ofProgress in Electromagnetic Research Symposium(PIERS), 28-31 Mar. 2004.
3. B. Archambeault, O.M. Ramahi and C. Brench,EMI/EMC Computational Modeling Handbook.Kluwer Academic Publishers, 1998.
8
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CE 1A
Computation of Eigenmodes in Dispersive Materials
Bastian Bandlow and Rolf Schuhmann
Theoretische Elektrotechnik, Universitat Paderborn, Warburger Str. 100, D-33098 Paderborn, [email protected], [email protected]
Summary. In dispersive materials, the material prop-erties depend on the operating frequency. Eigenvalueproblems including dispersive materials occur whencomputing the dispersion relations of photonic crys-tal and metamaterial unit cells in the THz spectrum.One possible procedure for solving is a fixed pointiteration, but we show, that a reformulation as poly-nomial eigenvalue problem is also possible.
1 Introduction
In Fig. 1 there is exemplarily shown a metamate-rial unit cell consisting of two rectangular struc-tures of silver and a dielectric spacing in between.The fundamental eigenmode of such an unit cellis the focus of our interest. Especially in theTHz frequency range, noble metals show a dis-persion of their dielectric constant which followsthe Drude model [3]. Thus, solving for eigenmodesof electromagnetic structures can be a quite chal-lenging task when frequency dispersive materialsare involved. A first approach is to evaluate thefrequency dependent material properties at a spe-cific frequency and to perform a fixed point itera-tion over several frequencies which may convergeto the desired eigenmode of interest. But sincethe dependency of the material parameters on thefrequency is explicitly known, the problem can bereformulated leading to a polynomial eigenvalueproblem (PEP) which can be solved in differentways.
2 Formulation
¿From Maxwell’s equations we derive the eigen-mode formulation for the electric field strengthE
rot1
µrot E = ω2ǫE, (1)
whereµ = µ0µr ǫ = ǫ0ǫ(ω). (2)
We refer the following results to a relative per-meability of µr = 1. The dependency on the fre-quency of the relative permittivity ǫ(ω) is givenfor example by a general 2nd order model
ǫ(ω) = ǫ∞ − β0 + iωβ1
α0 + iωα1 − ω2, (3)
with the real-valued parameters ǫ∞, α0, α1, β0
and β1. Note that all underlined symbols are com-plex quantities. Then we insert equation (3) intoequation (1) and formulate a PEP in ω
(ω4A4 + ω3A3 + ω2A2 + ωA1 +A0)E = 0 (4)
A4 = −ǫ∞, A3 = i(α1ǫ∞ + β1),
A2 = ǫ∞α0 + β0 + Acc,
A1 = −iα1Acc, A0 = −α0Acc,
Acc = rot1
µrot.
Using the Finite Integration Technique (FIT) [4]this representation of the PEP can be trans-formed into a matrix-vector formulation in a straightforward manner
(ω4MA4+ω3MA3+ω
2MA2+ωMA1+MA0)e = 0,
(5)where e is a large algebraic vector of the electricgrid voltages.
Fig. 1. Sample unit cell structure with two layers ofDrude-dispersive silver and a spacing in between.
3 Strategies for Solving
3.1 Linearization of the PEP
A PEP can be linearized be the cost of multiply-ing the dimension of matrices by the order l ofthe polynomial, generating a generalized eigen-value problem Ax = λBx of high dimension.
9
A =
0 I · · · 0...
.... . .
......
...... I
−MA0 −MA1 · · · −MAl−1
, (6)
B =
I. . .
IMAl−1
This approach leads to (l − 1) times additionaleigenvalues which are not all solutions of the origi-nal PEP. Moreover, the system (6) is often knownto be ill-conditioned [1].
3.2 Jacobi-Davidson for PEP
A promising approach was recently published in[2]. The PEP is kept in the form (5), the di-mension is not blown up and a Jacobi-Davidsonmethod is established. So far, we implemented thelinearized harmonic Rayleigh-Ritz variant of themethod in [2] which is intended to find one ormore interior eigenvalues of the spectrum near agiven estimation. Note, that the linearized har-monic Rayleigh-Ritz method fails to produce anexact eigenvalue, since only the linear part of aTaylor expansion is incorporated. Nevertheless,the obtained results are quite promising.
4 Numerical Example
As numerical example we choose a rectangularwaveguide resonator. The resonator is filled ho-mogeneously with a Drude-dispersive material,leading to a PEP of third order. The advantageof this rather simple setup is that the computa-tion of the fundamental mode can be done forcomparison purposes
1. analytically (including the lattice dispersionequation [4]),
2. using a fixed point iteration around ω,3. using the blown-up linearized approach,4. using the JD approach.
4.1 Results
For the analytical solution we obtain the referenceresult
ωanalytical = 2π(24.49 × 109 + 66.32 × 103i) s−1.(7)
Note, that there is a factor 106 between real andimaginary part of the result.
For the first setup with full matrix-formulationwe perform the fixed point iteration until the
residual is smaller than 10−12. We obtain a de-viation to the analytical solution of
|ωmeshed,fixedpoint − ωanalytical||ωanalytical|
= 3.38 × 10−8.
(8)The blown-up linearized approach of equation (6)is solved by using Matlab’s eigs and yields
|ωmeshed,linearized − ωanalytical||ωanalytical|
= 1.07 × 10−4.
(9)This rather poor accuracy is due to the bad con-dition of system (6). It can easily be improvedby scaling and shifting the matrices in (6) whichleads to
|ωmeshed,linearized,scaled − ωanalytical||ωanalytical|
= 5.31×10−8.
(10)The JD method based on linearized harmonicRayleigh-Ritz method produces an approxima-tion only which leads to a deviation of
|ωmeshed,JD − ωanalytical||ωanalytical|
= 7.57 × 10−4 (11)
which is in the same order of magnitude as theblown up linearized approach without any addi-tional preconditioning operations.
5 Summary
We have shown how to get a PEP from an elec-tromagnetic structure containing some materialfollowing general 2nd order dispersive models. Asa first result, This PEP can be solved using aJacobi-Davidson method from [2]. Further detailswill be given in the full paper.
References
1. Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, andH. van der Vorst, editors. Templates for the solu-tion of Algebraic Eigenvalue Problems: A PracticalGuide. SIAM, Philadelphia, 2000.
2. Michiel E. Hochstenbach and Gerard L. G. Slei-jpen. Harmonic and refined rayleigh-ritz for thepolynomial eigenvalue problem. Numerical LinearAlgebra with Applications, 15(1):35–54, 2008.
3. P. B. Johnson and R. W. Christy. Opticalconstants of the noble metals. Phys. Rev. B,6(12):4370–4379, Dec 1972.
4. T. Weiland. Time domain electromagneticfield computation with finite difference meth-ods. International Journal of Numerical Mod-elling, 9(4):295–319, 1996.
10
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CE 1B
Region-oriented BEM formulation for numerical computations
of electric and magnetic fields
Andreas Blaszczyk
ABB Corporate Research, 5405 Baden-Daettwil, Switzerland [email protected]
Summary. The paper presents a concept of theregion-oriented 3D formulation for the boundary ele-ment method (BEM) applied to computation of elec-tric and magnetic fields. Basic equations are included.An example shows the advantages of the new formu-lation over the classical approach used for dielectriccomputation of high voltage power equipment.
1 Introduction
The calculation of electric fields based on theboundary element method became during last 10years a defacto standard in design of high voltagedevices [1, 2]. The fundamental advantage of theBEM approach is related to the fact that only sur-face mesh on boundaries of solid parts is needed.The modeling of so called “airbox” as well as thegeneration of solid mesh is not required, whichsignificantly simplifies the discretization of com-plex models.
The classical formulation of the indirect BEMfor electrostatic problems introduces a single layerof virtual charges (free and bound) located on sur-faces of solid bodies [3]. The field and potential inarbitrary points can be computed as a superposi-tion of all the charges. This approach provides agood accuracy for dielectric problems with typicalvalues of the relative permittivity (in the rangebetween 1 and 10). For values of material prop-erties far beyond the typical range significant nu-merical errors may occur (see example in section4).
In this paper the region-oriented formulationconcept has been introduced to improve the nu-merical performance of the BEM approach. Thistype of formulation has been successfully appliedin the charge simulation method in the 1990s[4, 5].
The concept of the region-oriented BEM canbe also used for the computation of magneticfields [6]. However in this abstract only the for-mulation and examples for electric fields are pre-sented.
2 Concept
The basic concept is shown in Fig. 1. The modelspace has been divided into regions in which thefield and potential is calculated based on a uniqueset of charges located on the region boundaries.Consequently, on boundaries between two regionsa double layer of charges is defined while in triplepoints three different charge values are specified.In contrast to the region-oriented charge simula-tion the regions can consist of many domains withdifferent material properties.
Fig. 1. Basic concept of the region-oriented formu-lation: (a) geometrical configuration (b), (c) and (d)charges assigned to regions 1, 2 and 3 respectively.
3 Formulation
In order to find the values of charges the followingtypes of equations are formulated:
• fixed potential on electrodes boundaries in re-gion k:
∑
j∈kpij · σjk + Φrk = Φi (1)
• potential continuity on interfaces between re-gions – this equation is formulated for eachregion l that meets with region k at point i:∑
j∈kpij · σjk + Φrk =
∑
j∈lpij · σjl + Φrl (2)
11
• electric displacement continuity at the bound-ary point i (one equation for all regions thatmeet at point i)
∑
k∈iεknik
∑
j∈k∇pij · σjk = 0 (3)
• flux (charge) compensation for closed regions:
∑
i∈knik∑
j∈k∇pij · σjk = 0 (4)
where εk are electric permittivities and nik arenormal vectors. The unknowns in (1)–(4) arecharge densities σik at collocation points i (meshcorner nodes) and reference potentials Φrk (bothassigned to region k). The potential pij and fieldcoefficients ∇pij between the collocation point iand the integration points j are calculated in thesame way as in the classical BEM approach [1,3].
4 Example
Figure 2 shows an example of surge arrester. Itconsists of 3 sections of zinc oxide cylinders sup-ported by porcelain tubes. In spite of a simplegeometry this example makes some difficulties forthe classical BEM approach because of high valueof electric permittivity of the zinc oxide.
Fig. 2. Example of surge arrester. This example isused by IEC as a benchmark model for arrester cal-culations. For the benchmark configuration the wholearrangement is surrounded by a grounded cylinderwith 8 m diameter.
The goal of designers is calculation of thepotential distribution along the arrester axis forthe purely capacitive case. In order to keep theproperties of zinc oxide in linear range a uni-form potential gradient is required for normaloperation. The result in Fig. 3 shows significantdifferences between the classical and the region-oriented BEM approaches. The better accuracy ofthe region-oriented result could be achieved dueto explicit formulation of potential continuity (2)
as well as the charge compensation for the highpermittivity regions (4). In this way the resultdoes not depend on precision of the numerical in-tegration.
Fig. 3. Potential distribution along the axis of thesurge arrester from Fig. 2.
5 Conclusion
The region-oriented BEM formulation is suitablefor the calculation of electric fields in arrange-ments with extreme differences in material prop-erties. The new approach provides a good founda-tion for handling of complex geometries as well asnonlinear and quasi-static problems. The detailson numerical performance of this formulation willbe presented in the extended version of this pa-per.
References
1. N. De Kock, M. Mendik, Z. Andjelic and A.Blaszczyk. Application of 3D boundary elementmethod in the design of EHV GIS components.IEEE Magazine on Electrical Insulation., Vol.14,No. 3, pp. 17–22, May/June 1998
2. A. Blaszczyk, H. Ketterer and A. Pedersen. Com-putational electromagnetism in transformer andswitchgear design: Current trends. SCEE 2000.Lecture Notes on Computational Science and En-gineering, vol. 18, Springer Verlag, ISBN 3-540-42173-3, pp.55–62
3. Z. Andjelic, B. Krstajic, S. Milojkovic, A.Blaszczyk, H. Steinbigler and M. Wohlmuth. Inte-gral methods for the calculation of electric fields.Scientific Series of the International Bureau Re-search Center Juelich, 1992 (ISBN 3-89336-084-0).
4. A. Blaszczyk and H. Steinbigler. Region-crientedcharge simulation. IEEE Trans. on Magnetics, vol30, no. 5, September 1994, pp. 2924–2927.
5. A. Blaszczyk. Computation of quasi-static elec-tric fields with region-oriented charge simulation.IEEE Trans. on Magnetics, vol. 32. no. 3, May1996, pp. 828–831.
6. A. Blaszczyk. Formulations based on region-oriented charge simulation for electromagnetics.IEEE Trans. on Magnetics, vol 36, no. 4, July2000, pp. 701–704.
12
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CE 1C
Tearing and Interconnecting Methods for Non-Elliptic
Operators
Markus Windisch and Olaf Steinbach
Institut fur Numerische Mathematik [email protected], [email protected]
Summary. Finite and boundary element tearing andinterconnecting methods are well established domaindecomposition solvers for elliptic operators. Contrary,for non-elliptic operators much less is known. Herewe discuss some approaches for both the Helmholtzequation and the Maxwell system, to construct effi-cient related parallel solvers.
1 Introduction
The use of domain decomposition methods canhave different reasons, for example jumping co-efficients, the use of different methods on differ-ent subdomains, or just because of computationalarguments, such as preconditioning. As a modelproblem we first consider the Helmholtz equation
−∆u(x) − κ2u(x) = f(x) for x ∈ Ω,γ0u(x) = g(x) for x ∈ ΓD,
γ1u(x) =∂u
∂n(x) = h(x) for x ∈ ΓN ,
where ∂Ω = ΓD ∪ ΓN .
2 Domain decomposition
We decompose the domain Ω in several non-overlapping bounded and simply connected sub-domains Ωi. If present, Ω0 may denote the exte-rior of some bounded domain, in addition.
The interface and the skeleton of the domaindecomposition are defined by
ΓI :=⋃
i,j∈IΓ ij \ ΓD,
ΓS :=⋃i∈I
Γi = ΓI ∪ ∂Ω.
3 Steklov-Poincare operator
The Dirichlet to Neumann map including the Ste-klov-Poincare operator plays a key role in ouranalysis. Analytically it is just defined by themapping of some Dirichlet data γ0u of a func-tion which fulfills the Helmholtz equation to its
Neumann data γ1u. This operator is well de-fined and invertible as long the wave number κis not a Dirichlet (resp. Neumann) eigenvalue ofthe Laplace operator (this is always the case, ifthe domain Ωi is small enough). There are differ-ent possibilities to discretize this operator, herewe want to mention the approaches based on fi-nite and boundary element methods. In the finiteelement approach one assemblies the local stiff-ness matrices and eliminates all interior degreesof freedom to obtain
S := AΓΓ −AΓIA−1II AIΓ .
In the boundary element framework we can real-ize the Steklov-Poincare operator by the so-calledsymmetric representation
S := Dκ + (1
2+K ′
κ)V−1κ (
1
2+Kκ)
where Vκ is the single layer, Kκ is the doublelayer, and Dκ is the hypersingular potential op-erator.
4 Deduction of coupled systems
In this section we deduce different equivalent for-mulations of our problem. First we localize thepartial differential equation.
Find u ∈ H1(Ω) such that
−∆ui − κ2iui=0 in Ωi,γ1ui=−γ1uj on Γij ,γ0u= g on ΓD,γ1u=h on ΓN ,
where ui = u|Ωi .By using the local Steklov-Poincare operator,
it is possible to reduce the problem to functionsdefined on the boundary:
Find γ0u ∈ H1/2(ΓS) so that
Siγ0u|Γi + Sjγ0u|Γj = 0 on Γijγ0u = g on ΓD,
Siγ0u = h on ΓN ∪ Γi.(1)
This system is the starting point of the algebraicapproach in the next section.
13
For a second approach we introduce bound-ary functions for the local subdomains and theNeumann data λ ∈ H−1/2(Γ ) on the skeleton.
Find γ0ui ∈ H1/2(Γi), λ ∈ H−1/2(ΓS) suchthat
µij · Siγ0ui = λ on Γij ,γ0u = g on ΓD,λ = h on ΓN ,
γ0ui = γ0uj on Γij
(2)
with µij := sign(i− j).
5 Algebraic approach
After discretizing (1) we end up with the algebraicsystem
Shu =
p∑
i=1
ATi Sh,iAiu =
p∑
i=1
ATi f i.
Here, one entry on an interface between two do-mains has the form
Sh[m,n]un = 〈(Si + Sj)un, φm〉.
By splitting the degrees of freedom and enforcingthe continuity again by an additional equation(this is done for the whole column), this leads to
〈Siui,n, φm〉 + 〈Sjuj,n, φm〉 = rhsi + rhsj ,ui,n − uj,n = 0.
By splitting the test function and reinforcing thecontinuity of the related Neumann data we endup with
〈Siui,n, φi,m〉 + λ = rhsi,〈Sjuj,n, φj,m〉 − λ = rhsj ,
ui,n − uj,n = 0.
By repeating these operations for all columns onefinally obtains
Sh,1 BT1. . .
...Sh,p B
Tp
B1 . . . Bp 0
u1
...up
λ
= rhs.
This is the same system which is usually de-duced in the elliptic case, but we didn’t use anyminimization arguments as it is done in the ellip-tic case.
6 Continuous approach
A more analytical approach is based on system(2), the variational formulation of it is
∑ij
∫Γij
(γ0ui − γ0uj)µ(x)dsx
+∫
ΓN∩Γi
(Siγ0ui − h(x)) · vidsx
+∑
ij,i>j
(∫Γij
(Siγ0ui − λ(x)) · vidsx
+∫Γij
(Sjγ0uj + λ(x)) · vjdsx)
= 0.
By introducing related bilinear forms we getthe mixed formulation:
Find u ∈ ΠiH1/2(Γi) and λ ∈ H−1/2(ΓS)
such that
a(u, v) + b(v, λ) = rhsb(u, µ) = rhs
where b(u, µ) fulfills the BBL-condition. A Galerkindiscretization will now lead to a similar system asabove, but the connectivity matrices Bi have tobe replaced by related mass matrices.
7 Alternative interface conditions
It is not always possible or practical to decom-pose the domain Ω in domains Ωi such that eachdomain is small enough to avoid eigenvalues ofthe Laplace operator. Another way to avoid thiscritical eigenvalues are alternative interface con-ditions. One possible choice are Robin-type inter-face conditions. In the finite element case this ap-proach was introduced in [1], in the case of bound-ary elements one has to find a way to treat theinverse of the single layer potential which also ap-pears when solving Robin-type problems. At leastin the case of the domain Ω0 it is no problemto use the boundary element method (which isthe favourable method for unbounded domains)also for Dirichlet or Neumann interfaces whenusing regularized boundary integral formulations,see [2].
Another possibility is to use interface condi-tions which are based on the representation for-mula, see [3]
References
1. O. Cessenat and B. Despres. Application of anultra weak variational formulation of elliptic PDEsto the two-dimensional Helmholtz problem. SIAMJ. Numer. Anal., 35(1):255–299 (electronic), 1998.
2. S. Engleder and O. Steinbach. Modified boundaryintegral formulations for the Helmholtz equation.J. Math. Anal. Appl., 331(1):396–407, 2007.
3. R. Hiptmair. Boundary element methods for eddycurrent computation. In Boundary element analy-sis, volume 29 of Lect. Notes Appl. Comput. Mech.,pages 213–248. Springer, Berlin, 2007.
14
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CE 1D
Parametric Models of Transmission Lines Based on First
Order Sensitivities
Alexandra Stefanescu, Daniel Ioan, and Gabriela Ciuprina
Politehnica University of Bucharest, Electrical Engineering Dept., Numerical Methods Lab., Spl.Independentei 313, 060042, Romania [email protected]
Summary. One of the major goals in designing trans-mission lines is to handle variability. This paper pro-poses a method to parameterize the p.u.l. parametersof TL models w.r.t. the geometric parameters, subjectto large or small variations. Examples of such varia-tions are either due to design or technology (processor lithography), respectively. The accuracy of the sim-plest first order models is studied and validated ex-perimentally for both affine and rational variabilitymodels.
1 Introduction
While short interconnects have simple circuit mod-els (e.g. a lumped capacitance), interconnects longerthan the wave length are critical for a proper func-tionality of the designed system. In this case theextracted model has to consider also the effect ofthe distributed parameters. Fortunately, in mostcases, the long interconnects have the same cross-sectional geometry along their extension. If not,they may be decomposed in straight parts con-nected together by junction components. The for-mer are represented as transmission lines, whilethe latter are modeled as common passive 3Dcomponents.This paper focuses on the variability of the nu-merical extracted models for interconnects withrespect to the geometric parameters. Such an ap-proach represents one of the issues of the re-search carried out within the European projectFP6/IST/Chameleon [1].
2 Parametric Models Based onFirst Order Sensitivities
The parameter variability is computed using thefirst order Taylor series expansion. This needsthe computation of the derivatives of the devicecharacteristics with respect to the design param-eters [2]. Let y be the device characteristic whichdepends on the design parameters (p1, p2, . . . , pn).The quantity y may be, for instance the real orthe imaginary part of the device admittance ata given frequency. The parameter variability is
thus completely described by the real functiony : S → ℜ, defined over the design space S, asubset of ℜn. Denoting by p0 = (p01, p02, . . . , p0n)the nominal values of the design parameters andby y0 = y(p0), the device characteristic for thenominal set of parameter, if y is smooth enoughthen its truncated Taylor series expansion is thebest polynomial approximation in the vicinity ofthe expansion point p0. The first order truncationof the Taylor series is the affine function:
y(p1, p2, . . . , pn) = y0 +n∑
k=1
Spk(pk − p0k), (1)
where Spk = ∂y∂pk
(p0) are the first order absolutesensitivities, defined as partial derivatives of thedevice characteristics, with respect to the designparameters, computed for the nominal values ofthe parameters. This definition is valid not onlyfor real valued characteristics, but also when y isa complex number, a vector or a matrix.
3 Results
The test consists of a microstrip (MS) transmis-sion line having one Aluminum conductor em-bedded in a SiO2 layer. The line has a rect-angular cross-section, parameterized by the pa-rameters p2 and p3 (Fig. 1). Its position is pa-rameterized by p1. The return path is on thegrounded surface placed at y = 0. The nomi-nal values used are: xmax = 20µm, h2 = 10µm,h3 = 5µm, h0 = 1µm, p1 = 1µm, p2 = 0.67µm,p3 = 3µm,σAl = 3.3MS/m, ǫr−SiO2 = 3.9F/m.For the nominal case, by using dFIT + dELOB,at low frequencies, the following values are ob-tained:
R = 18.11kΩ/m L = 322nH/mC = 213pF/m
(2)
At high frequencies, the per unit length resis-tance and inductance are frequency dependent,and they can be computed for instance with themethod described in [3].
15
Fig. 1. Stripline parameterized conductor
A. Affine or Rational Models for One ParameterCase
The first order sensitivities are essential for theanalysis of the parameter variability in the timedomain. In order to evaluate the error introducedby the first order TS expansion, let us consideronly one parameter (n = 1). The variabilitymodel based on (1) defines an affine or additivemodel (A) since each new parameter consideredadds a contributing term to the sum.The affine models for the p.u.l. parameters con-sidered are first order truncation of the TaylorSeries:
X = X0(1 + SXp δp) (3)
where X is either R, L or C. The rational modelsare first order Taylor series approximations of theinverse quantity, 1/X :
X =X0
(1 + S1Xp δp)
(4)
where it can be easily shown that the reverse rel-
ative sensitivity is S1Xp = −SX0
p . The sensitivitiesare computed using the CHAMY software [1], bythe differential method (DM) applied to the statespace equations [2].
6 6.5 7 7.5 8
x 10−7
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8x 10
4
Parameter p2 [m]
p.u.
l. re
sist
ance
[Ohm
/m]
frequency f=1GHz
Variation of p.u.l. resistanceFirst TS approximation − directFirst TS approximation − inverse
Fig. 2. Reconstruction of the per unit length R at1GHz, from TS first order expansion
Figure 2 illustrates the effect of variation ofparameter p2 between 0.57µm and 0.77µm. As
expected, the reverse approximation of R is bet-ter than the direct Taylor series approximation.Using this graph, the validity range for an im-posed accuracy can be identified.
B. Models for Multi Parametric Case
For the example considered above, in order to ex-tend its validity domain a rational model (R) canbe used:
X =X0(1 + SXp1δp1)
(1 + S1Xp2 δp2)(1 + S
1Xp3 δp3)
(5)
where nominal values of per length parametersare given by (2). A multi - parametric model canbe obtained by multiplication of affine models forseveral parameters (model M):
X = X0
n∏
k=1
(1 + SXpkδpk) (6)
while the additive (affine) models A are
X = X0(1 +
n∑
k=1
SXpkδpk) (7)
First order multi - variable model is obtainedby addition of variations along several directions.The main difference between A and M models arethe ”cross terms”, which are neglected in the Amodel. In the full paper we will address this issue,being a completion to [4].The paper analyzes variability models for TLstructures considering the dependency of p.u.l.parameters w.r.t. geometric parameters, at a givenfrequency. A symbolic approach will also be con-sidered in a future research. As an alternative tothe affine dependence, a rational approximationof first order is proposed.
References
1. CHAMELEON-RF site: www.chameleon-rf.org2. G. Ciuprina, D. Ioan, D. Niculae, J. F. Villena,
L. M. Silveira Parametric models based on sensi-tivity analysis for passive components IntelligentComputer Techniques in Applied ElectromagneticsStudies in Computational Intelligence vol. 119, pp.17-22, Springer, 2008.
3. D. Ioan, G. Ciuprina, S. Kula, Reduced OrderModels for HF Interconnect over Lossy Semicon-ductor Substrate Proceedings of the 11th IEEEWorkshop on Signal Propagation on Interconnects,May 2007, pp.233-236
4. A. Stefanescu, G. Ciuprina, D. Ioan, Models forVariability of Transmission Line Structures Pro-ceedings of the 12th IEEE Workshop on SignalPropagation on Interconnects, May 2008
16
Session
CS 1
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 1I
ANN/DNN-based Behavioral Modeling of RF/Microwave
Components/Devices and Circuit Blocks
Q.J. Zhang and Shan Wan
Dept. of Electronics, Carleton University, 1125 Colonel By Dr., Ottawa, Canada K1S [email protected], [email protected]
Summary. This paper provides a tutorial overviewof ANN/DNN for RF and microwave modeling anddesign. We will describe neural network structuressuitable for representing high-speed/high-frequencybehaviors in components and circuits, ANN trainingexploiting RF/microwave device and circuits data,formulation of ANN/DNN for passive and active de-vice modeling, and behavioral modeling of nonlinearcircuit blocks, and use of ANN/DNN models for highlevel RF/microwave simulation and design optimiza-tion.
1 Introduction
Artificial neural networks (ANNs) have gainedrecognition as an emerging vehicle in enhancingthe effectiveness of computer-aided modeling anddesign of RF and microwave circuits and sys-tems [1–6]. ANNs can be trained to learn electro-magnetic/circuit behaviors from component data.Trained ANNs can be used in high-level circuitand system simulation and optimization provid-ing accuracy and/or speed advantages in design.The learning capabilities of ANN can be used forenhancing the existing CAD models of passiveand active components [7], and thereby extend-ing our ability of describing component behaviorsto be even closer towards reality. ANNs can beused to improve speed, accuracy and flexibility ofmicrowave modeling and CAD. This is made pos-sible because of their established network struc-tures, universal approximation property, and theability to integrate with circuit knowledge. ANNshave been applied to modeling and design of mi-crostrip and CPW circuits, multilayer intercon-nects, embedded passive components, printed an-tennas, semiconductor devices, filters, amplifiersand so on. Further advances in embedding mi-crowave information into neural networks lead tonew knowledge-based methods for robust elec-trical modeling [8]. There are increased initia-tives for integration of neural network capabili-ties into circuit design and test processes. For ex-ample, recent reports include embedding neuralnetworks in circuit optimization, statistical de-sign, global modeling, computational electromag-
netics, measurement standards, and nonlinear cir-cuits and system level design. Automatic modelgeneration algorithms allow systematic and com-puterized model creation, complimenting exist-ing human based approaches in developing RFand microwave models [9]. The development ofan advanced ANN structure, called dynamic neu-ral network (DNN) [10, 11], is one of the majorrecent directions where dynamic behavior of non-linear circuits is modeled by ANN-learning of cir-cuit input-output data.
2 Tutorial Overview ofMethodology and Application
This paper will describe the fundamentals of us-ing ANN/DNN for RF and microwave model-ing and design, and highlight its recent appli-cations. The major topics will be ANN struc-tures, ANN training, formulation of ANN forRF/microwave modeling, and use of ANN mod-els for RF/microwave design. Several ANN struc-tures will be described, including multiplayer per-ceptron (MLP), radial basis function (RBF) net-works, knowledge-based neural networks (KBNN),recurrent neural networks (RNN) and dynamicneural networks (DNN). The knowledge basedneural networks combine conventional empirical/equivalent models with ANN to achieve accuratemodel with less training data. Several methodsfor knowledge based networks will be highlightedsuch as different method, prior-knowledge-inputmethod, space mapped neural network method,and KBNN method. By establishing relationshipsbetween buffered history of input-output signals,or set of time-derivatives of input-output sig-nals, the RNN and DNN can represent the dy-namic relationship between input and output sig-nals in circuit blocks. Formulations of RNN andDNN for behavioral modeling of nonlinear de-vices and circuit blocks will be presented. Wealso introduce automated model generation al-gorithms (AMG) which perform ANN trainingincluding selection of training samples and neu-ral network size adjustment. An adaptive sam-
19
pling algorithms are used during the training pro-cess to decide how many training data is needed,and how the data should be distributed/sampledin the training space. The neural network size(such as the number of hidden neurons) is de-termined during the training process accordingto a set of under-learning and over-learning cri-teria. The AMG algorithm aims to achieve re-quired model accuracy with minimum amount oftraining data. Application examples such as fastparametric modeling of EM structures, modelingof semiconductor devices, behavioral modelingof high-speed driver/receiver buffers, behavioralmodeling of power amplifiers will be presented.Examples of applying the ANN/DNN models incircuit simulation will be presented.
References
1. Q.J. Zhang and K.C. Gupta. Neural Networksfor RF and Microwave Design. Artech House,Boston, Massachusetts, 2000.
2. Q.J. Zhang, K.C. Gupta and V.K. Devabhaktuni.Artificial neural networks for RF and microwavedesign: from theory to practice. IEEE Trans. Mi-crowave Theory Tech., 51:1339-1350, 2003.
3. J.E. Rayas-Sanchez. EM-based optimization ofmicrowave circuits using artificial neural net-works: the state-of-the-art. IEEE Trans. Mi-crowave Theory Tech., 52:420-435, 2004.
4. X. Ding, V.K. Devabhaktuni, B. Chattaraj,M.C.E. Yagoub, M. Doe, J.J. Xu and Q.J. Zhang.Neural network approaches to electromagneticbased modeling of passive components and theirapplications to high-frequency and high-speednonlinear circuit optimization. IEEE Trans. Mi-crowave Theory Tech., 52:436-449, 2004.
5. Y. Fang, M.C.E. Yagoub, F. Wang and Q.J.Zhang. A new macromodeling approach for non-linear microwave circuits based on recurrent neu-ral networks. IEEE Trans. Microwave TheoryTech., 48:2335-2344, 2000.
6. H. Zaabab, Q. Zhang and M. Nakhla. A neu-ral network modeling approach to circuit opti-mization and statistical design. IEEE Trans. Mi-crowave Theory Tech., 43:1349-1358, 1995.
7. L. Zhang, J.J. Xu, M. Yagoub, R. Ding and Q.J.Zhang. Efficient analytical formulation and sen-sitivity analysis of neuro-space mapping for non-linear microwave device modeling. IEEE Trans.Microwave Theory Tech., 53:3752-3767, 2005.
8. F. Wang and Q.J. Zhang. Knowledge-based neu-ral models for microwave design. IEEE Trans.Microwave Theory Tech., 45:2333-2343, 1997.
9. V.K. Devabhaktuni, B. Chattaraj, M.C.E.Yagoub and Q.J. Zhang. Advanced microwavemodeling framework exploiting automatic modelgeneration, knowledge neural networks and spacemapping. IEEE Trans. Microwave Theory Tech.,51:1822-1833, 2003.
10. J.J. Xu, M.C.E. Yagoub, R. Ding and Q.J. Zhang.Neural-based dynamic modeling of nonlinear mi-crowave circuits. IEEE Trans. Microwave TheoryTech., 50:2769-2780, 2002.
11. Y. Cao, R.T. Ding and Q.J. Zhang. State-spacedynamic neural network technique for high-speedIC applications: modeling and stability analysis.IEEE Trans. Microwave Theory Tech., 54: 2398-2409, 2006.
20
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 1A
Model Order Reduction for Nonlinear IC models with POD
A. Verhoeven1, M. Striebel2,3, and E.J.W. ter Maten3,4
1 VORtech Computing, Delft, The Netherlands [email protected] Technical University Chemnitz, Chemnitz, Germany [email protected] NXP Semiconductors, Eindhoven, The Netherlands [email protected] Technische Universiteit Eindhoven, The Netherlands
Summary. Reduced order modelling of large nonlin-ear system is of large importance for industry. Despitethe linear case where great progress is being recorded,methodologies for nonlinear problems is only begin-ing to develop. Approaches based on linearisation arepromising. We present a technique based on singularvalue decomposition that keeps the nonlinear charac-teristic and yet avoids the need to evaluate the non-linear functions describing the complete system.
1 Introduction
The developement of fail-safe circuit designs reliesheavily on computer simulations. The demand forshort time-to-market circles requires both accu-rate and fast algoritms.
Model Order Reduction (MOR) aims at accel-erating the process of simulation by streamliningthe models to be treated in such a way that thebasic characteristics of interest are conserved.
Due to the ever increasing refinement in mod-elling physical effects in circuit design, networkequations exhibit a more and more nonlinearcharacteristic.
As the packaging density and therefore thecomplexity of network models is increasing, too,reduced order modelling of large nonlinear sys-tems is of large practical importance for industry.
For linear RLC circuits there exist alreadyquite a lot of MOR techniques [1]. However, cor-responding techniques for nonlinear problems isonly beginning to develop.
2 Pojection Based Nonlinear MOR
In general an electrical network is described bythe nonlinear differential-algebraic equation ofthe form
d
dt[q(x(t))] + (x(t)) +Bu(t) = 0, (1)
where x(t) ∈ Rn represents the unknown vectorof circuit variables at time t; q, : Rn → Rn de-scribe the contribution of reactive and nonreac-tive elements, respectively. B ∈ Rn×n distributesthe input u(t) ∈ Rm.
Projection based MOR techniques use Petrov-Galerkin projection to obtain a order reducedmodel for (1) of the form
d
dt[WT q(V z)] +WT (V z) +WTBu = 0, (2)
where W,V ∈ Rn×r. The original state can be
obtained by x = V z. With r ≪ n it is clear thatthe order of (2) is much smaller than the size ofthe original system (1).
But, using any implicit integration scheme,e.g., a Backward Differentiation Formula (BDF),to solve (2) numerically on some discrete timescale t0, t1, · · · necessitates the Jacobians
WT · ∂∂xq(x(ti)) · V and WT · ∂
∂x(x(ti)) · V,
(3)evaluated at x(ti) = V z(ti) for i = 1, 2, · · · .Clearly, still the complete functions q and haveto be evaluated. Furthermore, matrix-vector prod-ucts have to be evaluated and the Jacobians mostlikely become dense. Actually, except from thelower dimension, no reduction is achieved.
2.1 Adapted POD
To overcome these problems, we suggest to adaptthe POD approach such that only a few compo-nents of q and have to be determined i.e., eval-uated.
We assume that we have a benchmark solu-tion x(t) of (1) to some input u(t) and a matrixX ∈ R
n×N of N snapshots. We determine theeigenvalue decomposition of the correlation ma-trix
W =1
NXXT =
1
NUΣΣTUT . (4)
Here the matrix U ∈ Rn×n is orthonormal. Theproduct ΣΣT ∈ Rn×n is a positive real diagonalmatrix, and therefore we can write ΣΣT = Γ 2
for a real positive diagonal matrix Γ ∈ Rn×n.In contrast to POD we introduce the orthogonalmatrix L = UΓ ∈ Rn×n, such that W = 1
NLLT .
We perform a projection of (1):
d
dt
[LT q(Ly)
]+ LT (Ly) + LTBu = 0, (5)
21
with x = Lu. We approximate L ≈ LPTr Pr andLT ≈ LTPTg Pg where Pr ∈ 0, 1r×n and Pg ∈0, 1g×n select the r and g columns of L andLT with largest norm, respectively. That means,L and LT are replaced by matrices where all butthe r and g, respectively, most dominant columnsare set to zero.
After multiplication of (5) with Pr, i.e. trun-cation according to the r most dominant singularvalues of X and proper scaling we arrive at thereduced system of order r ≪ n
d
dt[Wrg q(Urz)] +Wrg (Urz) +Bru = 0, (6)
with reduced state z(t) ∈ Rr, Ur ∈ R
n×r con-taining the r most dominant columns of U likein classical POD, Wrg = UTr P
Tg ∈ Rr×g and
Br = UTr B ∈ Rr×m. The state vector of the fullsystems can be recoverd by x(t) = Urz(t).
We stress that the principal reduction takesplace in the nonlinear functions q(·) = Pgq(·) ∈Rg and = Pg(·) ∈ Rg, meaning that instead ofthe complete functions, just g ≪ n componentshave to be evaluated.
2.2 Trajectory PieceWise Linear approach
The basic idea of the Trajectory PieceWise Linearapproch (TPWL, [2, 3]) is to replace the nonlin-ear problem (1) with a convex linear combinationof linear systems which are reduced by means ofsome linear MOR techniques:
s∑
i=1
wi(z) · [V Tr CiVr z+V Tr GiVrz+V Tr Biu(t)] = 0,
where the weights wi(z) ≥ 0 with∑wi(z) = 1
determine, which of the local linear subsystemsthat arise from linearisation of (1) around a typ-ical trajectory are contributing to the dynamicsclose to the current state.
3 Numerical Results
We consider the academic diode chain modelshown in Fig. 1 with 300 nodes. The currenttraversing a diode with potential Va and Vb at theinput- and output-node, respectively is describedby the nonlinear equation
q(Va, Vb) =
Is(e
Va−VbVT − 1) if Va − Vb > 0.5
0 otherwise
The voltage source is described by
u(t) =
20 if t ≤ 10ns170 − 15 · 109 · t if 10ns < t ≤ 11ns5 if t > 11ns
Fig. 1. Testcircuit: Diode chain
0 10 20 30 40 50 60 7010
−16
10−14
10−12
10−10
10−8
10−6
10−4
10−2
100
time [ns]
erro
r
TPWL: order 50POD: order 30,dim f 35
Fig. 2. Relative errors for TPWL and POD
Table 1. Comparison of adapted POD and TPWL
Method r g Extr. time Sim. time
Original 302 0 79sTPWL 50 202s 2.4sadapt. POD 30 35 107 s 10.2s
Figure 2 shows the relative error and Table 1reflects the time needed to extract and resimulatea reduced model. This clearly shows that PODapproach still is a promising approach for MORof nonlinear problems.
Acknowledgement. This work is funded by the Marie-Curie projects O-MOORE-NICE! and COMSON.
References
1. A. Antoulas. Approximation of Large-Scale Dy-namical Systems, Advances in Design and Control6, SIAM, 2005
2. M. J. Rewienski. A Trajectory Piecewise-LinearApproach to Model Order Reduction of NonlinearDynamical Systems. Ph.D. thesis, MassachusettsInstitute of Technology, USA, 2003.
3. T. Voß, R. Pulch, J. ter Maten, A. El Guennouni.Trajectory piecewise linear approach for nonlin-ear differential-algebraic equations in circuit sim-ulation. Scientific Computing in Electrical Engi-neering. Mathematics in Industry,, G. Ciuprina,D. Ioan (editors), Vol. 11, Springer, pp. 167-174,2007.
22
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 1B
Model Order Reduction for Systems with Coupled Parameters
Lihong Feng and Peter Benner
Mathematics in Industry and Technology, Faculty of Mathematics, Chemnitz University of Technology,D-09107 Chemnitz, Germany [email protected],[email protected]
Summary. We consider parametric model order re-duction of systems with parameters which are non-linear functions of the frequency parameter s. Suchsystems result from, for example, the discretization ofelectromagnetic systems with surface losses [3]. Sincethe parameters are functions of the frequency s, theyare highly coupled with each other. We see them as in-dividual parameters when we implement model orderreduction. By analyzing existing parametric model or-der reduction methods for computing the projectionmatrix V , we show the applicability of each methodand propose an optimized method for the parametricsystem considered in this paper.
1 Problem Description
The transfer function of the parametric systemsconsidered here takes the form
H(s) = sBT(s2In − 1/√sD + A)−1B, (1)
where A,D and B are n×n and n×m matrices,respectively, and In is the identity of suitable size.To apply parametric model order reduction to (1),we first expand H(s) into a power series. Usinga series expansion about an expansion point s0,and defining σ1 := 1
s2√s− 1
s20√s0
, σ2 := 1s2 − 1
s20,
G := I − 1s20
√s0D + 1
s20A, we get:
H(s) = sBT(s2I − 1/√sD +A)−1B
= 1sB
T(I − 1s2
√sD + 1
s2A)−1B
= 1sB
T(G− σ1D + σ2A)−1B= 1
sBT(I − σ1G
−1D + σ2G−1A)−1G−1B
= 1sB
T[I − (σ1G−1D − σ2G
−1A)]−1G−1B
= 1sB
T∞∑i=0
(σ1G−1D − σ2G
−1A)iG−1B.
(2)We may use the three different parametric modelreduction methods below to compute a projectionmatrix V (which is independent of s) from (2) andget the reduced-order transfer function
H(s) = sBT(s2Ir − 1/√sD + A)−1B,
where A = V TAV , B = V TB, etc., and V isan n × r projection matrix with V TV = Ir. Tosimplify notation, in the following we use BM :=G−1B, M1 := G−1D, and M2 := −G−1A.
Notice that (1) is also discussed in [3], whereonly conventional non-parametric model order re-duction method is considered. The generated pro-jection matrix V is dependent on a particularvalue of s: s0, which may cause large error forother values of s which are far away from s0. Allmethods presented in this paper are parametricmodel order reduction methods. The basic dif-ference of parametric model reduction from non-parametric model reduction is that the computedprojection matrix V is independent of any specialvalue of s, which produces a reduced model withevenly distributed small error.
2 Different Methods of ComputingProjection matrix V
2.1 Directly Computing V
A simple and direct way for obtaining V is tocompute the coefficient matrices in the series ex-pansion
H(s) = 1sB
T[BM + (M1BMσ1 +M2BMσ2)+ (M2
1BMσ21 + (M1M2 +M2M1)BMσ1σ2
+ M22BMσ
22) + (M3
1BMσ31 + . . .) + . . .],
(3)by direct matrix multiplication and orthogonal-ize these coefficients to get the matrix V [1]. Af-ter the coefficients BM , M1BM ,M2BM , M2
1BM ,(M1M2 + M2M1)BM , M2
2BM , M31BM , . . . are
computed, the projection matrix V can be ob-tained by
rangeV = orthogonalizeBM ,M1BM ,. . . , (M1M2 +M2M1)BM , . . . (4)
Unfortunately, the coefficients quickly become lin-early dependent due to numerical instability. Inthe end, the matrix V is often so inaccurate thatit does not possess the expected theoretical prop-erties.
2.2 Recursively Computing V
The series expansion (3) can also be written intothe following formulation:
23
H(s) = 1s [BM + (σ1M1 + σ2M2)BM
+ . . .+ (σ1M1 + σ2M2)iBM + . . .]
(5)
Using (5), we define
R0 = BM ,R1 = [M1,M2]R0,
...Rj = [M1,M2]Rj−1,....
(6)
We see that R0, R1, . . . , Rj , . . . include all thecoefficient matrices in the series expansion (5).Therefore, we can use R0, R1, . . . , Rj , . . . to gen-erate the projection matrix V :
rangeV = colspanR0, R1, . . . , Rm. (7)
Here, V can be computed by employing the re-cursive relations between Rj , j = 0, 1, . . . ,mcombined with the modified Gram-Schmidt pro-cess [2].
3 Optimized Method of ComputingProjection Matrix V
Note that the coefficientsM1M2BM andM2M1BMare two individual terms in (6), which are com-puted and orthogonalized sequentially within themodified Gram-Schmidt process. Observing thatthey are actually both coefficients of σ1σ2, theycan be combined together as one term duringthe computation as in (4). Based on this, we de-velop an algorithm which can compute V in (4)by a modified Gram-Schmidt process. By thisalgorithm, the matrix V is numerically stablewhich guarantees the accuracy of the reduced-order model. Furthermore, the size of the reduced-order model is smaller than that of the reduced-order model derived by (7). The improved algo-rithm is well suited for the parametric systemfrom computational electromagnetics consideredin this paper. Its performance will be illustratedusing an industrial test example.
Acknowledgement. This research is supported by theAlexander von Humboldt-Foundation and by the re-search network SyreNe — System Reduction for NanoscaleIC Design within the program Mathematics for Inno-
vations in Industry and Services (Mathematik fur In-novationen in Industrie und Dienstleistungen) fundedby the German Federal Ministry of Education andScience (BMBF).
References
1. L. Daniel, O.C. Siong, L.S. Chay, K.H. Lee, andJ. White. A multiparameter moment-matching
model-reduction approach for generating geomet-rically parameterized interconnect performancemodels. IEEE Trans. Comput.-Aided Des. Integr.Circuits Syst., 22(5):678–693, 2004.
2. L. Feng and P. Benner. A robust algorithm forparametric model order reduction. In Proc. Appl.Math. Mech., 2008 (to appear).
3. T. Wittig, R. Schuhmann, and T. Weiland. Modelorder reduction for large systems in computationalelectromagnetics. Linear Algebra Appl., 415(2-3):499–530, 2006.
24
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 1C
Model order and terminal reduction approaches via matrix
decomposition and low rank approximation
Andre Schneider and Peter Benner
Technische Universitat Chemnitz, Fakultat fur Mathematik, 09107 Chemnitz, [email protected], [email protected]
Summary. We discuss methods for model order re-duction (MOR) of linear systems with many inputsand outputs, arising in the modeling of linear (sub)circuits with a huge number of nodes and a large num-ber of terminals, like power grids. Our work is basedon the approaches SVDMOR and ESVDMOR pro-posed in recent publications. In particular, we discussefficient numerical algorithms for their implementa-tion. Only using efficient tools from numerical linearalgebra techniques, these methods become applicablefor truly large-scale problems.
1 Introduction
One issue in MOR for VLSI design is the reduc-tion of parasitic linear sub circuits. These circuitsform substructures in the design of ICs and con-tain linear elements with little or no influence onthe result of the simulation. In some applications,the structure of these parasitic linear sub circuitshas recently changed in the following sense. Sofar, the number of elements in these sub circuitswas significantly larger than the number of con-nections to the whole circuit, the so called pinsor terminals. This assumption is no longer validin all cases. Circuits with a lot of elements needan extra power supply network, so called powergrids. In clock distribution networks, the clocksignal is distributed from a common point to allthe elements that need it for synchronization. Forsimulating these circuits new methods are needed.We want to introduce an efficient algorithm han-dling this problem.
2 SVDMOR and ESVDMOR
Recent studies have shown that we can make useof a large degree of correlation between the plural-ity of input and output terminals. We use the sin-gular value decomposition (SVD) based methodSVDMOR [2, 8] as well as an extended versionof SVDMOR, the so called ESVDMOR [4–6, 8],which is the foundation for our work and will beexplained in the following.
2.1 Extended-SVDMOR
We assume that the linear system to be reducedhas the following transfer function in frequencydomain:
H(s) = L(sC +G)−1B, (1)
with C,G ∈ Rn×n, B ∈ Rn×min , and L ∈Rmout×n. We assume that there is a differencebetween the number of inputs min and the num-ber of outputs, here mout. Consider the i-th blockmoment of (1)
mi =
mi1,1 mi
1,2 . . . mi1,min
mi2,1 mi
2,2 . . . mi2,min
......
. . ....
mimout,1 m
imout,2 . . . m
imout,min
, (2)
where mi is amout×min matrix. The ESVDMORapproach uses the information of these momentsto create a decomposition of (1) and consequentlya new internal transfer function Hr(s),
H(s) ≈ H(s) = VOroLr(G+ sC)−1Br︸ ︷︷ ︸
:=Hr(s)
V TIri. (3)
The matrices Lr and Br are approximations of Land B computed with the help of the SVD, whichalso generates VOro
and V TIri. Note that ri and ro
are the numbers of the reduced virtual input andoutput terminals. This terminal reduced transferfunction is now reduced to Hr(s) by a well knownestablished MOR method, e. g., balanced trunca-tion or Krylov subspace methods. At the end weget a very compact terminal and reduced ordermodel
H(s) ≈ VOroHr(s)V
TIri. (4)
Note that SVDMOR can be considered as a spe-cial of ESVDMOR, using only one moment, e.g.,m0 and one SVD.
Drawbacks and solutions
For very large sub circuits the (E)SVDMOR ap-proaches are not suitable because of using the
25
SVD. Therefore we combine the (E)SVDMOR ap-proach with cheaper matrix decomposition meth-ods, like the truncated SVD (TSVD), which justcomputes the singular values and the correspond-ing singular vectors we need. Also other ideas tocheaply compute a truncated SVD-like decompo-sition like [1, 3, 7] can be used here.
To perform the TSVD we do not computethe moments in (2) explicitly. This would be nu-merically unstable and too expensive. The TSVDcan be computed, e.g., with the implicit restartedArnoldi method as implemented in the Matlab
function svds or with the Jacobi-Davidson SVD[3]. Thus, we only need to provide a function ap-plying mi to a vector. We will give more detailsof this approach in the full paper.
3 Numerical Results
The spectrum of the singular values of the mo-ment used for computing the SVD is essential forthis kind of MOR, so we firstly concentrate onthis issue. Figure 1 shows the decrease of the sin-gular values of a circuit with 3916 nodes and 1905terminals, provided by NEC Labs in St. Augustin,Germany. We can see, that there are about 130significant singular values. That means, after thereduction we have 130 virtual input and outputpins instead of 1905 terminals originally. The rel-
0 50 100 150 200 250 300 350 400 450 50010
−2
10−1
100
101
102
Index i of the singular value σi
Val
ue o
f σi
circuit 3
Fig. 1. Range of the largest 500 singular values ofcircuit3.
ative approximation error for a circuit with 141nodes and 70 terminals is shown in Figure 2. Wereduced to just one virtuel terminal and we cansee that the error is sufficiently small up to theGigahertz range which is enough for the applica-tion behind this problem.
Acknowledgement. The work reported in this paperwas supported by the German Federal Ministry of Ed-ucation and Research (BMBF), grant no. 03BEPAE1.
10−2
100
102
104
106
108
1010
10−7
10−6
10−5
10−4
10−3
10−2
RC549
Frequency ω
Rel
ativ
e er
ror
σm
ax(H
(jω)
− H
HA
T(jω))
/σm
ax(H
(jω))
m0
MI / O
(ESVD)
ms0
, s0=108
Fig. 2. Relative error ǫrel of RC549 with the help ofSVDMOR (m0, ms0
) and ESVDMOR (MI/O).
Responsibility for the contents of this publicationrests with the authors.
References
1. M. W. Berry, S. A. Pulatova, and G. W. Stew-art. Algorithm 844: Computing sparse reduced-rank approximations to sparse matrices. ACMTrans. Math. Softw., 31(2):252–269, 2005.
2. P. Feldmann and F. Liu. Sparse and efficient re-duced order modeling of linear subcircuits withlarge number of terminals. In ICCAD ’04: Proceed-ings of the 2004 IEEE/ACM International con-ference on Computer-aided design, pages 88–92,Washington, DC, USA, 2004. IEEE Computer So-ciety.
3. M. E. Hochstenbach. A Jacobi–Davidson typeSVD method. SIAM J. Sci. Comput., 23(2):606–628, 2001.
4. P. Liu, S. X. D. Tan, H. Li, Z. Qi, J. Kong, B. Mc-Gaughy, and L. He. An efficient method for termi-nal reduction of interconnect circuits consideringdelay variations. In ICCAD ’05: Proceedings ofthe 2005 IEEE/ACM International Conference onComputer-Aided Design, pages 821–826, Washing-ton, DC, USA, 2005. IEEE Computer Society.
5. P. Liu, S. X. D. Tan, B. Yan, and B. McGaughy.An efficient terminal and model order reductionalgorithm. Integr. VLSI J., 41(2):210–218, 2008.
6. P. Liu, S.X.-D. Tan, B. Yan, and B. Mcgaughy.An extended SVD-based terminal and model or-der reduction algorithm. In Proceedings of the2006 IEEE International, Behavioral Modelingand Simulation Workshop, pages 44–49, 2006.
7. M. Stoll. A Krylov-Schur approach to the trun-cated SVD. Submitted to SIAM J. Sci. Comput.,2008.
8. S. Tan and L. He. Advanced Model Order Reduc-tion Techniques in VLSI Design. Cambridge Uni-versity Press, New York, NY, USA, 2007.
26
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 1D
Model order reduction for large resistance networks
Joost Rommes1, Peter Lenaers2, and Wil H.A. Schilders1
1 NXP Semiconductors, HTC 37, 5656 AE Eindhoven, The Netherlands [email protected],[email protected]. Supported by O-MOORENICE! (MCA FP6 MTKI-CT-2006-042477).
2 Mathematics for Industry, Eindhoven University of Technology, P.O.Box 513, 5600 MB, Eindhoven, TheNetherlands [email protected]
Summary. Electro Static Discharge analysis is timeconsuming due to the increasing size of resistance net-works. We propose an algorithm for the reduction oflarge resistance networks to much smaller equivalentnetworks. Experiments show reduction and speed-upsup to a factor 10.
1 Introduction
Electro Static Discharge (ESD) analysis [4] re-quires knowledge on how fast electrical charge onthe pins of a package can be discharged. In manycases, the discharge is done through the powernetwork and the substrate, which both are resis-tive. Diodes are used to protect transistors on achip against peak charges. The discharge paths,that consist of very large resistance networks con-nected through diodes, must be of low resistanceto allow for sufficient discharge.
In order to analyze the discharge behavior, de-signers are interested in the path resistance be-tween all IC-pins. To this end, the substrate andresistance network are modeled by resistors. Theresulting resistive network may contain up to mil-lions of resistors, hundreds of thousands of inter-nal nodes, and thousands of external nodes (nodeswith connections to diodes). Simulation of suchlarge networks within reasonable time is not pos-sible, and including such networks in full systemsimulations may be even unfeasible. Hence, thereis need for much smaller networks that accuratelyor even exactly describe the resistive behavior ofthe original network, but allow for fast analysis.
In this paper we describe a new approach forthe reduction of large resistance networks. Weshow how insights from graph theory, numeri-cal linear algebra, and matrix reordering algo-rithms can be used to construct an equivalentnetwork with the same number of external nodes,but much less internal nodes and resistors. Thisequivalent reduced network exactly describes thebehavior of the original network. The approach isillustrated by numerical results.
2 Properties of resistance networks
A resistance network consists of internal nodes,external nodes (or terminals), and resistors. Fig-ure 1 shows a simple resistance network with ex-ternal nodes Z (the input node), A, B, and C (theoutput nodes), and internal nodesX and Y (thereare five resistors). Of interest are the path resis-tances from Z to A, B, and C. This small exam-
1
23
45
6
7
8
9
10
11
12
13
14
1516
17
18
19
20
21
22
23
24
25
26
27
28
29
3031
3233
34
35
36
37
38
39
Fig. 1. Simple resistance network (left) with externalnodes Z, A, B, and C, and realistic network (right,squares are external nodes). Of interest are the pathresistances between external nodes.
ple is purely for illustrational purposes; in real-lifeapplications the number of nodes and resistors ismuch larger. In the following it will be assumedthat the network has n > 0 internal nodes, m > 0external nodes, and r > 0 resistors.
2.1 Mathematical formulation
Using Ohm’s Law for resistors and Kirchhoff’sCurrent Law [2], the electrical behavior of a re-sistance network can be described by
i = Y · v, (1)
where i,v ∈ RN and Y ∈ RN×N (with N = n +m) contain the unknown inflowing currents, nodevoltages, and conductances, respectively.
27
We distinguish between internal and external nodes:
[ieii
]=
[Yee YeiY Tei Yii
] [vevi
],
where ie,ve ∈ Rm and ii,vi ∈ Rn correspond toexternal and internal nodes, respectively, and Yis partitioned accordingly. Note that ii = 0 sincecurrents can only be injected in external nodes.
All diagonal elements of Y are strictly positiveand all off-diagonal elements are negative or zero.The conductance matrix Y = (yij) is symmetricand (after grounding) positive-definite (xTY x >0 for all x ∈ Rn). In most of the applications, theconductance matrix Y is very sparse, typicallyhaving O(1) nonzeros per row.
The impedance matrix Z can be obtained byinverting Y : Z = Y −1. For large networks this isnot possible due to memory and CPU limitations,and it is neither necessary since usually only spe-cific elements are needed: the path resistances, forinstance, can be found on the diagonal of Z.
2.2 Problem formulation
The problem is: given a large resistance networkdescribed by (1), find an equivalent network with(a) the same external nodes, (b) exactly the samepath resistances between external nodes, (c) n≪n internal nodes, and (d) r ≪ r resistors.
Simply eliminating all internal nodes will leadto an equivalent network that satisfies conditions(a)–(c), but violates (d): for large numbers m ofexternal nodes, the number of resistors r = m2/2in the dense reduced network is in general muchlarger than the number of resistors in the sparseoriginal network (r = O(n)), leading to increasedmemory and CPU requirements.
3 Improved approach
Knowing that eliminating all internal nodes isnot an option, we use concepts from matrix re-ordering algorithms such as AMD [1] and BBBD[5], usually used as preprocessing step for LU-factorization, to determine which nodes to elimi-nate. The fill-in reducing properties of these meth-ods also guarantee sparsity of the reduced net-work. For related work, see [6].
The first step is to bring the conductancematrix Y into Balanced Border Block Diagonal(BBBD) form [1, 3, 5], see Figure 2. In this form,the matrix consists of two parts: the main bodyA11 and the border blocks A12, A21 = AT12 andA22. The blocks Aij can be partitioned into sub-blocks Bkl. The second step is to eliminate theinternal nodes in block A11. Since the ordering is
Fig. 2. Matrix in BBBD-form (left) with subblocks.
chosen to minimize fill-in, the resulting reducedmatrix is sparse. Finally, the reduced conductancematrix can be realized as an reduced resistancenetwork that is equivalent to the original network.
4 Numerical results and conclusions
Initial results for realistic resistance networks areshown in Table 1. The number of nodes is reducedby a factor > 10 and the number of resistors bya factor > 3. As a result, the computing time forcalculating path resistances is 10 times smaller.
Table 1. Results of reduction algorithm
Network I Network IIOriginal Reduced Original Reduced
#nodes 5558 516 99112 6012#resistors 8997 1505 161183 62685CPU time 10 s 1 s 67 hrs 7 hrs
By using insights from matrix reordering algo-rithms, the proposed algorithm can reduce largeresistance networks to small equivalent networks,that allow for faster simulation and can efficientlybe used in Electro Static Discharge analysis. Fur-ther improvements are under development.
References
1. P. R. Amestoy, T. A. Davis, and I S. Duff. An Ap-proximate Minimum Degree Ordering Algorithm.SIAM J. Matrix Anal. Appl., 17(4):886–905, 1996.
2. L. O. Chua and P. Lin. Computer aided analysisof electric circuits: algorithms and computationaltechniques. Prentice Hall, first edition, 1975.
3. I. S. Duff, A. M. Erisman, and J. K. Reid. Directmethods for sparse matrices. Oxford: ClarendonPress, 1986.
4. J. M. Kolyer and D. Watson. ESD: From A To Z.Springer, 1996.
5. A. I. Zecevic and D. D. Siljak. Balanced decom-positions of sparse systems for multilevel prallelprocessing. IEEE Trans. Circ. Syst.–I:Fund. The-ory and Appl., 41(3):220–233, 1994.
6. Q. Zhou, K. Sun, K. Mohanram, and D. C.Sorensen. Large power grid analysis using domaindecomposition. In Proc. Design Automation andTest in Europe, pages 27–32, 2006.
28
Session
P 1
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 1.1
Integral equation method for magnetic modelling of cilia
Mihai Rebican and Daniel Ioan
University Politehnica of Bucharest, Splaiul Independentei 313, Bucharest, Romania [email protected]
Summary. The paper presents an original and ac-curate method to model the magnetic behavior of thecilia. The method is based on Maxwell equation, for-mulated as integral equations for the magnetization.
1 Introduction
Micro-Fluidics is the science and technology ofmanipulating and analysing fluid flow in struc-tures of sub-millimetre dimensions. The availabi-lity of micro-fluidics technology is essential for thedevelopment of advanced products in a variety ofapplication areas, the most important of which isthe biomedical field.
Many industrial research groups are study-ing ways of micro-fluidic manipulation. A methodof fluid manipulation technology in micro-fluidicssystems, inspired by nature is based on cilia. Thisapproach enables effective local fluid manipula-tion, and the possibility to generate complex flowpatterns.
The movement of the artificial cilia can be ac-tively controlled, preferably using a magnetic fieldor an electrical field.
In this paper, an original, accurate and veryfast method to model the magnetic behavior ofthe cilia is proposed. The method is based on theintegral equations of the magnetic field, in termof the magnetization.
2 Integral equations method
Starting from Maxwell equations, the integralequation for the magnetization is:
M +χ
4π
∫
Ω2
R∇M
R3dv = χH0 + Mp. (1)
Let us consider the cilium divided in n ele-ments (1D finite elements), each being uniformlymagnetised along the cilium direction (Fig. 1).Using this assumption, the integral equation (1)becomes a linear system of n equations [1]:
Mj +χ g∆z
4π
n∑
i=1
(Mi −Mi−1) cos(∆βi/2)
aij√
(∆z/2)2 + a2ij
=
= χH0 +Mp, j = 1, n. (2)
Figure 2 shows the magnetic flux density alonga rigth cilium, discretized in 10 finite elements.The cilium (µr = 100) is situated into an uniformmagnetic field (B0 = 10mT), oriented along theciliaum. The cilium has a paralelipipedic geome-try (100x20x2µm).
Fig. 1. Discretization of the cilium for the magneticfield computation
Fig. 2. Magnetic field along a right cilium situatedin a uniform magnetic field
References
1. D. Ioan, I.F. Hantila, M. Rebican, and C. Con-stantin. FLUXSET Sensor Analysis Based on Non-linear Magnetic Wire Model of the Core. In Elec-tromagnetic Nondestructive Evaluation (II), pages160–169. IOS Press, Amsterdam, 1998.
31
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 1.2
Electromagnetic modeling of 3D bodies in layered earths using
integral equations
R. Grimberg1, Adriana Savin1, Rozina Steigmann1, Alina Bruma1, S.S. Udpa2, and Lalita Udpa2
1 National Institute of Research and Development for Technical Physics, Iasi, [email protected], [email protected], [email protected], [email protected]
2 Michigan State University, East Lansing, MI, USA [email protected], [email protected]
Consider a 3D body in an n-layered host. Thetotal electric and magnetic fields as functionsof position r for an ejωt time dependence obeyMaxwell’s equation
−∇× E = zH + Mi
∇× H = J + Ji(1)
where J = yE is the total current density, y =σ + jωε is the admittivity, z = jωµ at any pointand Ji and Mi are impressed electric and mag-netic source current, σ , ǫ and µ are conductivity,permittivity and permeability.
In our model, the 3D body is replaced by anequivalent scattering current distribution. The to-tal fields
(E, H
)in any layer are hence decom-
posed into an incident set(Ei, Hi
)due to Ji and
Mi and scattered set(ES , HS
)contributed by the
body.The scattered fields can be expressed as
∫V
↔
GE
l (r|r′) JS (r′) dr′
HS (r) =∫V
↔
GH
l (r|r′) JS (r′) dr′(2)
The quantities↔
GE
l (r|r′) and∫V
↔
GH
l (r|r′) are 3x3
dyadic Green’s functions relating a vector fieldat r′ in layer l to a current element at in layerj, including l=j. A matrix solution of (2) can befound using the method of collocation with pulsesubsection basis functions [1].
The body is approximated by N rectangu-lar prismatic cells, each of dimensions MXn∆n ×MY n∆n ×MZn∆n , where MXn,MY n and MZn
are positive integers and ∆n is the size of a cu-bic sub cell in cell n. Over each cell, the bodyconductivity and total electric field are presumedconstant.
The total electric field at the center of cell mdue to all N cells is approximated by
E (rm) = Ei (rm)+
N∑
n=1
(σn − σj)↔
ΓE
l (rm|rn) E (rn)
(3)
in which the electric Green’s dyadic function fora rectangular prism of current is
↔
ΓE
l (rm|rn) =
∫
Vn
↔
GE
l (rm|r′) dr′ (4)
Calculation of↔
ΓE
l (rm|rn) from↔
GE
l (r|r′) forthe matrix elements requires same care. When thein homogeneity does not cut across layer inter-faces, l=j. Each cell in this case is coupled to ev-ery other cell by a Green’s function compared of aprimary on free space component and a reflectedcomponent
↔
ΓE
l (m,n) = P↔
ΓE
l (m,n) + S↔
ΓE
l (m,n) (5)
Each matrix element is given by a summationover the cubic subcells. If the body cuts acrosslayer interfaces cells in different layers are coupledonly by secondary electric Green’s function.
An efficient evaluation of the dyadic Green’sfunctions is necessary to avoid prohibitive com-puter time. The primary solutions are analyticexpressions and present no problems, but the sec-ondary Green’s function requires Hankel transfor-mation of complicated kernel functions. Followingthe procedure described above, the forward prob-lem was solved for the situations:
• the incident field is created by planar rectan-gular coil with 1m outer side, 0.004X0.006mm2
transversal section of conductor, 0.006 stepand 10turns, current density 108A/m
2and fre-
quency 10kHz.• scattering electric field was determined by us-
ing one array 5X5 rectangular coil sensors• 3D body conductive sphere with 1m diameter,
106 S/m electric discretized in 53 cells, thecenter of sphere is at 10m under the surfaceof soil
Acknowledgement. This paper is partially supportedby the Romanian Ministry of Education, Researchand Youth under Excellence Research Program CEEXunder contracts: 9 I 03 / 06.10.2005 nEDA, Grantno.706/CNCSIS.
33
Fig. 1. The image of a conductive sphere
References
1. R.F. Herrington. Field computation by momentmethods, McMillan, NY, 1968
2. R.Grimberg , L.Udpa, SS Udpa. Electromagnetictransducer for the determination of soil condition,accepted for publication in International Journalfor Applied Electromagnetics and Mechanics
34
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 1.3
Electromagnetic field diffraction on subsurface electrical
barrier. Application to nondestructive evaluation
Adriana Savin, Rozina Steigmann, Alina Bruma, and Raimond Grimberg
National Institute of Research and Development for Technical Physics, Iasi, Romania [email protected],[email protected], [email protected], [email protected]
Consider a half-space conductor which contains adeep dielectric barrier whose edge lies a depth hbelow the surface of the conductor. The boundarycondition for scattered magnetic field on the sur-face of the conductor will match by introducingan image of the barrier (Figure 1).
Fig. 1. Problem geometry
The incident magnetic field in each conductinghalf-space is
Ψ (i) ∼−e−jkz z < 0−ejkz z > 0
(1)
The diffracted field, which is determined bya generalization of Fermat’s principles, has theform [1]
Ψ (S) (0, z) ∼−e−jkz , z < −h−ejkz , z > h
∂Ψ (S)(0,z)∂x = 0, |z| < h
(2)
The diffraction functions for the lower and up-per edges, K1 and K2 respectively are defined asfollows
K1 (r1, θ1, ϕ1) =
= − ejkr1
2
[w(√
2jkr1 cos 12 (θ1 + ϕ1)
)+
+ w(√
2jkr1 cos 12 (θ1 − ϕ1)
)]
K2 (r2, θ2, ϕ2) =
= − ejkr2
2
[w(−√
2jkr2 cos 12 (θ2 + ϕ2)
)+
+w(√
2jkr2 cos 12 (θ2 − ϕ2)
)]
(3)
where w is the Fadeeva’s function [2].The single scattered field is
ψ(S)S = ψ
(S)1S + ψ
(S)2S (4)
whereψ
(S)1S = ejkhK1 (r1, θ1, 0)
ψ(S)2S = ejkhK2 (r2, θ2, π)
(5)
The multiple scattering may be treated sys-tematically in the following manner. There arefour distinct ”types” of multiply scattered field,tabulated in Table 1.
Table 1. Types of multiple scattered field
type Initialscatteringat edgenumber
Scattering at each edge Finalscatteringat edgenumber
Lower edge 1 Upper edge 2
A 1 n+1 n 1
B 1 n n 2
C 2 n n+1 2
D 2 n n 1
The sum over all scattered fields can be writ-ten as follows
ψ(S)n = −ejkhK1 (2h, 0, 0) [1 −K1 (2h, 0, 0)] ·
· [K1 (r1, θ1, 0) −K2 (r2, θ2, π)] ··
∞∑n=1
K2n−21 (2h, 0, 0)
(6)The multiply scattered field has the same spa-
tial form as the simply scattered field. The totalscattered field is simply the sum of the simply andmultiply scattered fields The sum over all scat-tered fields can be written as follows
ψ(S) = ψ(S)S + ψ(S)
n = (1 +M)ψ(S)S (7)
where
M = −K1(2h,0,0)1+K1(2h,0,0)
=e2jkhw(
√4jkh)
1−e2jkhw(√
4jkh)w (z) = e−z
2
erfc (−jz)(8)
35
If the electrical conductivity of the conductinghalf-space is and the relative magnetic permeabil-ity is r=1 (amagnetic conductor), the standardpenetration depth is defined as
δ =
√2
ωµσ(9)
where ω is the angular frequency of the incidentfield and µ = µrµ0.
The contour plot of real and imaginary com-ponent of the total magnetic field, for which mul-tiple scattering has been taken into account, onthe dielectrically barrier placed at 4mm below thesurface of an aluminium alloy 4032T6 half spaceat 10kHz, the frequency of the incident electro-magnetic field are presented in Figure 2 and re-spective Figure 3.
Fig. 2. Real component of the total magnetic fieldon the subsurface dielectric barrier
Fig. 3. Imaginary component of the total magneticfield on the subsurface dielectric barrier
Acknowledgement. This paper is partially supportedby the Romanian Ministry of Education, Researchand Youth under Excellence Research Program CEEXunder contracts: 9 I 03 / 06.10.2005 nEDA and 6110/2005SINERMAT; Grant no.586/CNCSIS and PNII -71-016/2007 MODIS.
References
1. J.von Blodel. Electromagnetic field, Sec.Ed., IEEEPress Series on Electromagnetic Wave Theory, NY,2007
2. M. Abramowitz, I. Stegun. Handbook of Math-ematical Functions: with Formulas, Graphs, andMathematical Tables , National Bureau of Stan-dards,1964
3. R. Grimberg, A.Savin, E.Radu, O.Mihalache.Nondestructive Evaluation of the Severity of Dis-continuities in Flat Conductive Materials Usingthe Eddy Current Transducer with OrthogonalCoils. IEEE Trans on Mag. 36, 1, (2000), 299-307
36
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 1.4
Relativistic High Order Particle Treatment for
Electromagnetic Particle-In-Cell Simulations
Martin Quandt1, Claus-Dieter Munz1, and Rudolf Schneider2
1 Institut fur Aerodynamik und Gasdynamik, Universitat Stuttgart [email protected],[email protected]
2 Forschungszentrum Karlsruhe, Institut fur Hochleistungsimpuls und [email protected]
Summary. A recently developed high order fieldsolver for the complete Maxwell equations providesall informations needed by a new relativistic particlepush method based on a truncated Taylor series ex-pansion up to the desired order of convergence. Thecapability of this approach is compared with otherclassic schemes for different numerical experiments.
1 Abstract
The deeper physical understanding of systems likemicrowave devices and pulsed plasma thrustersrequires the numerical modeling and simulation ofhighly rarefied plasma flows. Mathematically, thedescription of such flows demand a kinetic formu-lation which is established by the time-dependentBoltzmann equation. An attractive numerical ap-proach to tackle this complex non-linear prob-lem consists in a combination of the Particle-in-Cell (PIC) and Direct Simulation Monte Carlo(DSMC) methods extended by a PIC-based Fok-ker-Planck model which are coded in the hy-brid PIC/DSMC simulation program named Pi-cLas [1]. In this code the DSMC module accom-modates the charged-neutral and neutral-neutralparticle interactions as well as the plasma chem-istry. The new developed Fokker-Planck solver[5] takes into account the long range intra- andinter-species charged particle Coulomb collisions.Finally, the Maxwell-Vlasov module allows theself-consistent simulation of charged particles inelectro-magnetic fields. Note, the proposed highorder particle (HIOP) treatment is an essentialpart in the latter building block of the PicLascode.In essence, the operating mode of the Maxwell-Vlasov solver is as follows: at each time step theelectromagnetic fields are obtained by the numer-ical solution of the full set of the nonstationaryMaxwell equations, where different kind of meth-ods like finite volume or dicontinuous Galerkinschemes of free selectable order of convergenceare applied [3, 4]. Note, that the Maxwell part ofthis solver comprises additionally a purely hyper-
bolic divergence correction mechanism [2] to en-sure the constrain of charge conservation duringthe simulation. Subsequently, these fields are in-terpolated to the actual locations of the chargedplasma particles which are then pushed accord-ing the Lorentz force and redistributed in phasespace according to the usual laws of dynamics.Afterwards, the particles have to be located withrespect to the computational grid in order toassign the contribution of each charge to thechanged charge and current density to the nodesof the mesh. These densities are the sources forthe Maxwell equations for the subsequent itera-tion cycle which finally guarantee a self-consistentcomputation of the interaction of the electromag-netic fields with the charged plasma particles.To sustain the high order of the Maxwell-Vlasovmodule, we introduce a new particle treatmentbased on a Taylor series expansion of the phasespace variables in time up to the desired orderof the field solver, where high order time deriva-tives of the electro-magnetic fields occur. To re-place these derivatives by known spatial deriva-tives of the fields given by the Maxwell solver, weapply a Cauchy-Kovalevskaya procedure to theLorentz force equation. To demonstrate the capa-bility and reliability of the HIOP procedure dif-ferent numerical simulations are performed. Theresults are compared with those obtained fromthe classic second order Boris Leap-Frog scheme.The achieved effectiv order of convergence is alsotested for non-relativistic as well as relativisticcases and listed in charts.
Acknowledgement. We gratefully acknowledge the Lan-desstiftung Baden-Wurtemberg for funding the presentproject for two years up to September 2006. Further-more, the authors would like to thank the DeutscheForschungsgemeinschaft for financial support of thiswork.
37
References
1. D. Petkow, M. Fertig, T. Stindl, M. Auweter-Kurtz, M. Quandt, C.-D. Munz, S. Roller, D.D’Andrea and R. Schneider, Development of a3-Dimensional, Time Accurate Particle Methodfor Rarefied Plasma Flows, AIAA-2006-3601, Pro-ceedings of the 9th AIAA/ASME Joint Thermo-physics and Heat Transfer Conference, San Fran-cisco, USA, 2006
2. C.-D. Munz, P. Omnes, R. Schneider, E. Son-nendrucker and U. Voß, Divergence correctiontechniques for Maxwell solvers based on a hyper-bolic model, J. Comput. Phys.,2000
3. J.S. Hesthaven and T. Warburton, Nodal high-order methods on unstructured grids J. Comput.Phys.,181:186–221,2002
4. T. Schwartzkopff, F. Lorcher, C.-D. Munz, andR.Schneider, Arbitrary High Order Finite-VolumeMethods for Electromagnetic Wave PropagationComputer Physics Communications,2006
5. D. D’Andrea, C.-D. Munz and R. Schneider, Mod-eling of Electron-Electron Collisions for Particle-In-Cell Simulations, FZKA 7218 Research Re-port, Forschungszentrum Karlsruhe – in derHelmholtz-Gemeinschaft,2006
38
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 1.5
A Simulation-Based Genetic Algorithm for Optimal Antenna
Pattern Design
Yi-Ting Kuo and Yiming Li1
National Chiao Tung University, 1001 Ta-Hsueh Rd., Hsinchu 300, Taiwan [email protected]
Summary. In this paper, a simulation-based opti-mization method for the design of antenna pattern inmobile broadcasting, multi-bandwidth operation andthe 802.11a WLAN is presented. A simulation-basedgenetic algorithm (GA) is advanced for the antennadesign automation with requested specifications. Thecorresponding cost function in optimization is eval-uated by external numerical electromagnetic (EM)solver, and the communication of the GA and EMsolver is provided on our unified optimization frame-work (UOF). Our preliminary numerical results con-firm the robustness and efficiency of the proposedsimulation-based optimization method.
1 IntroductionA trend of portable device integration has risenlast decade due to the massive growth of wire-less communications, and products are expectedto have multiple wireless service. However, howto adjust the geometry to get the best design is astate-of-art and needs more experience throughtrial-and-error process. Recently, the optimiza-tion scheme, genetic algorithm (GA), is consid-ered to be a efficiency approach to optimize theantenna to satisfy requested specifications andmake design procedure fully automatic. However,the notch and slit-loaded entries provide extraresonant frequencies and introduce new band intooriginal antenna [1, 2]. Hence, to make GA moreflexible in planer antenna design optimization, weprovide a new approach to optimize the planerantenna.
In this work, we implement an optimizationmethod to enhance the receiving and transmittingabilities of the given antenna with GA and nu-merical simulation. Based on our experience andtechnique of using GA to optical proximity cor-rection (OPC) [3] and transistor model param-eter extraction [4], we for the first time advancethe idea of pattern correction in OPC for antennadesign optimization. The proposed method parti-tions the edges of the antenna into some sections(small segments) and then those sections can beshifted in the normal direction of the edges to im-prove the return loss of the antenna. This novelapproach can extend the geometry without lim-ited boundaries and can construct geometry with-out specific form. The involved simulator solvesMaxwell’s equations by finite element method.
Maxwell’s equations govern the performance inelectric and magnetic fields and the simulated re-sults are applied to evaluate the new generatedantenna.
Partition the edges of the antenna into segments
Construct the geometry of the new antenna
Simulate the new antenna to evaluate the performance
Satisfy the specifications?
Postprocess
No
Yes
Move the segments by GA
Initial antenna
Fig. 1. A flowchart for the proposed optimizationapproach in antenna design.
2 The Intelligent Methodology
As a designer drew a blueprint of an antenna forthe specific purpose, a fine tune process shouldbe performed repeatedly for further improvement.We already know the geometry of the antennaaffects its performance. However, the major dif-ficulty is how the antenna shape affects its per-formance. For an antenna designer, when one finetunes the shape, it is just like making a wild guess.To automate the search for the optimized antennashape in an efficient way, GA is a good candi-date of optimization methods in the simulation-based procedure. There are several componentsin the GA, such as problem definition, encod-ing method, fitness evaluation, selection method,crossover procedure, and mutation scheme haveto be performed for an evolutionary process. Fig-ure 1 shows the flowchart of the proposed op-timization approach. During the procedure, theoriginal antenna shape is divided into small seg-ments. The movements of those segments are thenoptimized with respect to the calculated resultsusing GA. Then the GA is applied to correct theantenna shape and outputs a new geometry ofantenna. To evaluate the cost function in GA,the external EM solver is used to simulate thenew antenna. This procedure will stop till the allspecifications are satisfied by the optimized an-tenna. Figure 2 shows the geometry and the par-titioned segements for a specified antenna shape.We have implemented this optimization proce-
39
dure in the developed unified optimization frame-work (UOF) [5]. We notice that UOF has alreadyprovided an efficient way in optical proximity cor-rection problem using a simulation-based opti-mization scheme [3].
1
2
3
4
5
6
7
8
9
10
11 12 13 14 15
16 17 18 19 20
P min P max P
1
2
3
4
5
6
7
8
9
10
11 12 13 14 15
16 17 18 19 20
P min P max P
Fig. 2. The original geometry of examined antennaand the partitioned segments. The black rectangularframe near the top line is the contact for input exci-tation.
3 Results and Discussion
X
Y
Fig. 3. Obtained antenna patterns in the GA proce-dure with 5, 10, 20, and 30 evolutionary generations.
Figure 2 shows the original shape of the exam-ined antenna. The studied structure is a Z-shapedantenna and has radiation elements and groundplane on the same plane. As shown in Fig. 2, thetop line is the ground plane, the bottom rectan-gle represents the radiation element, and the seg-ments 1-20 are the partitions for evolution. Thedual operating frequencies are set to 2.6 and 4.9GHz, where the return loss of the original an-tenna are about -14 and -10 dB, respectively. Thefrequency 2.6 GHz is referred as the ”S-band”used for mobile broadcasting and the frequency4.9 GHz is used by the 802.11a WLAN protocol.Our target is to improve the return loss for bothoperating frequencies 2.6 and 4.9 GHz. Apply GAto search the better antenna patterns automati-cally, the antenna patterns in some generationsare shown in Fig. 3. Figure 4 shows the corre-sponding return losses for the antenna patternsin each generation. The number on the subplotsin Fig. 3 represents the number of generations inGA procedure. Figure 4 shows that the returnloss of the second band has been improved as thenumber of generations increases. When the an-tenna pattern generated by GA satisfies the de-
sign specifications, it will be the optimized pat-tern. The return loss on the frequencies of 2.6 and4.9 GHz is down to -20 dB for the optimized an-tenna which has a significant improvement.
Frequency (GHz)
2 3 4 5 6
Ret
urn
Loss
(dB
)
-40
-30
-20
-10
0
5th generation10th generation20th generation30th generation
Fig. 4. The return losses corresponding to genera-tions of antenna patterns in GA procedure.
4 ConclusionIn this paper, simulation-based optimization methodfor the design of antenna has been presented. GAworks well for the antenna design automation onour UOF [5] and communicated with an externalnumerical EM solver. This method minimizes thereturn loss under the frequencies 2.6 and 4.9 GHz.Improved return loss is observed, and the effectof this method for the optimized configuration isexamined and discussed. Furthermore, UOF alsohas capability to connect with other external soft-ware through the script file, which can generatethe antenna geometry. We believe that this op-timization method can benefit the RF antennadesign applied for mobile broadcasting and the802.11a WLAN.
Acknowledgement. This work was supported in partby Taiwan National Science Council (NSC) underContract NSC-96-2221-E-009-210.
References
1. F. Castellana, F. Bilotti, and L. Vegni. Automateddual band patch antenna design by a geneticalgorithm based numerical code. IEEE Anten-nas and Propagation Society International Sym-posium, 4:696–699, 2001.
2. F. J. Villegas, T. Cwik, Y. Rahmat-Samii, andM. Manteghi. A parallel electromagnetic genetic-algorithm optimization (EGO) application forpatch antenna design, IEEE Transaction on An-tennas and Propagation, 52:2424–2435, 2004.
3. S.-M. Yu and Y. Li. A computational intelligentoptical proximity correction for process distortioncompensation of layout mask in subwavelengthera, 10th International Workshop on Computa-tional Electronics, 179–180, 2004.
4. Y. Li and Y.-Y Cho. An automatic parameterextraction technique for advanced CMOS devicemodeling using genetic algorithm, Microelectron.Eng., 84:206–272, 2007.
5. Y. Li, S.-M. Yu and Y.-L. Li. Application of a uni-fied optimization framework to electronic designautomation, Mathematics and Computers in Sim-ulation, To appear.
40
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 1.6
Probabilistic characterization of resonant EM interactions
with thin-wires: variance and kurtosis analysis
O.O. Sy1, J.A.H.M. Vaessen1, M.C. van Beurden1, B.L. Michielsen2, and A.G. Tijhuis1
1 Eindhoven University of Technology, Den Dolech 2, 5600 MB, Eindhoven, The Netherlands [email protected],[email protected], [email protected], [email protected]
2 ONERA, 2, av E. Belin, 31055 Toulouse Cedex, France [email protected]
Summary. A probabilistic characterization of ran-dom electromagnetic interactions affected by reso-nances is presented. It hinges on the analysis of thevariance and the kurtosis. It is illustrated by the caseof a randomly undulating thin wire over a plane.
1 Introduction
Interactions between electronic devices and elec-tromagnetic sources in their environment are ofprime importance in EMC models for design ormaintenance studies. The range of validity ofthese models depends on their ability to accu-rately represent an ensemble of configurations.For non-resonant systems, the study of a few con-figurations provides a good picture of the overallinteraction. However, for resonant phenomena, astochastic approach yields a more suitable quanti-tative and qualitative model. Stochastic methodsare frequently used in fields as diverse as rough-surface scattering problems [1] and Mode-stirred-Chamber theory [2]. In EMC, random modelshave been applied to undulating thin-wire setupsmodeled by transmission-line theory, or by inte-gral equations [3]. In all these cases the aim is toquantify the uncertainty of the response parame-ters or observables, by their average and variance.Although these statistics provide bounds for theobservable, they do not inform on the presence ofextreme values beyond these bounds.This paper proposes to characterize stochasti-cally, the voltage induced at the port of randomly-undulating thin-wire setups affected by resonances.In addition to the average and the variance, thefourth-order moment, or kurtosis, is computed byquadrature. The kurtosis measures the likelihoodof presence of extreme events. It is often used infinancial-risk analysis to foretell bankruptcy [4].
2 Deterministic configuration
The scattering configuration consists of a scatter-ing device and the incident field. The scattereris a perfectly electrically conducting (PEC) wireSα containing a port region, and, located over aPEC ground plane, as shown in figure 1.
Fig. 1. undulating thin-wire over a PEC plane
The vector α contains all the parameters control-ling the geometry of Sα. The parameters of theincident field Ei
β , such as its direction of propa-
gation or its amplitude, form the vector β. There-fore the vector γ = α⊕β contains all informationneeded to define the configuration. The couplingitself is observed through the voltage Ve(γ) in-duced at the port. It follows by first solving afrequency-domain electric-field integral equation(EFIE) depending on Sα, then performing a du-ality product between the solution and Ei
β [3].
3 Random parameterization
An ensemble Ωγ of configurations is considered.Computing Ve(γ) for each element γ of Ωγ canbe very costly numerically. Instead, the variationsof γ in Ωγ are viewed as random according to aknown distribution pγ . The voltage Ve(γ) thusbecomes a random variable, with statistical mo-ments, such as its mean E[Ve] and its standarddeviation σ [Ve], defined as
E[Ve] =
∫
ΩγVe(γ
′)pγ (γ′)dγ ′, (1)
σ [Ve] =√
E [|Ve|2] − |E [Ve] |2, (2)
where σ [Ve] measures the spread of Ve aroundE [Ve]. Extreme values of Ve, at least 4σ away fromE [Ve], are accounted for by the kurtosis κ[|Ve|]which is a dimensionless moment defined as
41
κ [|Ve|] = E
[( |Ve| − E [|Ve|]σ [|Ve|]
)4]≥ 0. (3)
Gaussian random variables, which have 97% oftheir values within 2σ of their average, have akurtosis of 3. Thus, the higher the value of κ [|Ve|]above 3, the more peak values of Ve are present.Since all the statistical moments are integrals overa known integrand depending on Ve(γ), and thesame support Ωγ . They can be computed byquadrature rules, which are cautiously chosen, toefficiently handle the dimension of Ωγ [3].
4 Results
With reference to figure 1, a roughly undulatingthin wire is studied with a geometry defined as
xα(y) = α1 sin(3πy), (4)
zα(y) = 5 + α2 sin(9πy) in cm (5)
α1 and α2 being independent and uniformly dis-tributed in Ωα1 = Ωα2 = [−5; 5] cm. The inci-dent field is a vertically-polarized plane-wave withan amplitude of 1 V.m−1, and propagating in thedirection θi = 45, φi = 0. Figure 2 depictsσ [Ve] and κ[|Ve|].
Fig. 2. σ [Ve] (dotted line) and κ [|Ve|] vs frequency
The graph of σ [Ve] shows several peaks which in-dicate an increased uncertainty of Ve, that canbe caused by a generalized spread Ve, or, by afew very large samples of Ve. The mitigation ofthese two cases is possible thanks to κ[|Ve|], whichis generally consistent with σ [Ve]. However, be-tween 230 MHz and 260MHz, although Ve is con-centrated around E [Ve] (σ [Ve] ≤ 2V ), extremesamples are present (κ[|Ve|] ≥ 15). The oppositesituation is observed between 330 MHz and 350MHz. To confirm these observations, 1000 sam-ples have been computed for the frequencies spec-ified in Table 1 . These samples are then normal-ized as follows
Vn =Ve − E[Ve]
σ[Ve], (6)
Table 1. Results at given frequencies
σ [Ve] κ[|Ve|]
f1 = 120 MHz 0.126 V 0.001f2 = 223 MHz 0.613 V 31f3 = 346 MHz 4.169 V 4.3
thus allows for comparisons between different fre-quencies, as in figure 3. The concentric circles re-sult from the normalization of circles centered atE[Ve] and with radii equal to multiples of σ[Ve].Figure 3 confirms the predictions based on theanalysis of σ [Ve] and κ [|Ve|].
Fig. 3. Normalized Samples Vn for f1, f2 and f3
More results will be shown at the conference forsetups with multiple undulating thin wires, underdeterministic and random incident fields.
5 Conclusion
The method presented in this abstract shows howthe efficient computation of the kurtosis can yieldvaluable statistical information. The kurtosis, in-forms qualitatively on the ability of the vari-ance to accurately represent stochastic electro-magnetic interactions hindered by resonances.
Acknowledgement. This work is funded by the DutchMinistry of Economic Affairs, in the Innovation Re-search Program (IOP) number EMVT 04302.
References
1. G. S. Brown. Simplifications in the stochasticfourier transform approach to random surface scat-tering. IEEE TAP, AP-33(1):48–55, 1985.
2. D.A. Hill. Plane wave integral representation forfield in reverberation chambers. IEEE Trans. EMC40(3):209–217, 1998.
3. Sy O.O. et. al. Probabilistic study of the couplingbetween deterministic electromagnetic fields and astochastic thin-wire over a pec plane. In ICEAA(pp. 1-4). Torino, Italy, 2007.
4. B. Mandelbrot R.L. Hudson. The (Mis)behaviourof Markets. Profile Business, 2004.
42
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 1.7
Domain Partitioning Based Parametric Models for Passive
On-chip Components
Gabriela Ciuprina1, Daniel Ioan1, Diana Mihalache1, and Ehrenfried Seebacher2
1 Politehnica University of Bucharest, Electrical Engineering Dept., Numerical Methods Lab., Spl.Independentei 313, 060042, Bucharest, Romania [email protected]
2 Austriamicrosystems, A 8141 Schloss Premstaetten, [email protected]
Summary. Models for passive integrated componentsthat take into consideration the variability of theirdesign can be obtained by partitioning the computa-tional domain into sub-domains, by using the elec-tromagnetic circuit element formulation. The sub-models can be interconnected afterwards to obtain aglobal parametric model that can be simulated or re-duced. The sub-models can be treated independentlyeither from the point of view of the variability, or fromthe point of view of electromagnetic field formulation.
1 Introduction
The design of the next-generation of integratedcircuits is challenged by an increased number ofdifficulties since electromagnetic field effects athigh frequencies are too relevant to be neglected.In this respect, one of the issues of the Europeanresearch project CHAMELEON-RF was to de-velop methodologies and tools able to simulateRF blocks up to 60 GHz by taking the electro-magnetic (EM) coupling and variability into ac-count (www.chameleon-rf.org). In this frame-work, the concept of magnetic terminals (”hooks”or ”connectors”) was used for the first time, to de-scribe the interaction of on-chip components withtheir environment [2]. These magnetic hooks arespecial boundary conditions that allow the exten-sion of the electric circuit element (ECE) to theelectromagnetic circuit element (EMCE). Suchan EMCE allows the connection to an externalmagnetic circuit, and thus inductive coupling ef-fects between the device and its environment canbe considered. Details on the implementation ofthese concepts when using the Finite IntegrationTechnique (FIT) are given in [1].
This paper shows how domain partitioning(DP) can be further exploited: on the one handby taking into account the variability, and on theother hand by using simplified models for sub-domains in which the full-wave (FW) electromag-netic field formulation is not strictly required.
2 Parametric Models Obtained byDomain Partitioning
The EM field effects at high frequencies are quan-tified by Maxwell equations in FW regime. Byapplying FIT to discretize these equations inthe case of the EMCE formulation, a paramet-ric semi-state space model can be obtained
C(p)dx
dt+ G(p)x = Bu, (1)
y = Lx, (2)
where u = [ue,um,y]T is the state space vector,consisting of electric voltages ue defined on theelectric grid, magnetic voltages um defined on themagnetic grid and output quantities y. Only twomatrices are affected by the parameters p.
The simplest way to analyze the parametervariability is to compute first order sensitivities,which are derivatives of the device characteristicwith respect to the design parameters. The com-putation of sensitivities of matrices is straight-forward in FIT, since the assembling of sensi-tivities is similar to the assembling of matrices,the only difference being that only the affectedcells add contributions to the sensitivities ma-trices ∂C/∂pk, ∂G/∂pk. Such parametric mod-
EMCE−FW,p
model
less no ofhooks
Simplified
EMCE−FW,p
EMCE−FWEMCE−FW
all possiblenode hooks
ECE−FW,p
211 2
1 2
Fig. 1. Domain partitioning of complex models: pa-rameters affect only some submodels while simplifiedformulations can be used for other submodels.
43
els are useful for a variety of available paramet-ric model order reduction procedures, and that iswhy, a standard for parametric representation ofsystems has been defined [3].
By means of electric and/or magnetic termi-nals the EMCE parametric models can be cou-pled, or the analysis of a more complex devicecan be divided into sub-parts that can be mod-eled independently. The latter idea is illustratedin Fig. 1 and numerical results are given in thenext section.
3 Numerical results and conclusions
The effectiveness of the proposed methodologywas tested at first on a simple test, consistingof two U-shape conductors, placed above a sili-con substrate. This domain was decomposed intwo parts: a bottom part (the substrate), the restbeing the top part. Table 1 gives the complex-ity of the models obtained by DP and couplingof sub-models. The reference taken is the resultobtained by using FW models and node-hooks onthe cut. It can be noticed that for the substratethe FW formulation can be substituted with asimpler one, a magnetostatic formulation (MS),case in which the size of the bottom model de-creases about 6 times. Figure 2 shows the parame-ter (conductor width) impact at 60 GHz, obtainedfor the parametric model obtained.
Table 1. Ucoupled test - complexity of models ob-tained by coupling.
Model Top Bottom Model Rel.size size size error
FW-top (5466;129) (5351;127) (10817;2) 0FW-bot
FW-top (5466;129) (1152;72)+ (7569;2) 5e-4EQS+MS bot +(950;55)=
=(2102;127)
FW top (5394;57) (950;55) (6344;2) 3e-3MS bot
The real test aimed consists of two coupledinductors, as shown in Fig. 3. Each spiral coilhave five metal turns, one terminal grounded andthe other excited by means of corresponding pads.The test structure is surrounded by a ground ring.Its domain was decomposed in three parts, thetop part (air) and the bottom part (Si) while themiddle part, which contain the coils is modeledwith FW. Results are shown in Fig. 4. Also thecoils width is considered variable and the influ-ence on the result is analysed.
2.5 3 3.5
x 10−6
2.465
2.47
2.475
2.48
2.485
2.49
2.495
2.5
2.505
2.51x 10
−3
Parameter g [m]
Imag
par
t of o
utpu
t 1 2
Ref (FIT) − 60 GHzTS for system matrices
Fig. 2. Parameter impact at 60 GHz.
Fig. 3. CHRF benchmark no 202 - two coupled in-ductors.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 1010
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
Frequency [Hz]
Imag
inar
y pa
rt−
−−
S12
CHRF202−measurements.s2psimulations−DD3.s2pabsolute error
Fig. 4. Comparison between measurements fromAMS and simulations for CHRF 202.
The full paper will describe in detail the mod-elling methodology and will prove its usefulness.
References
1. G. Ciuprina, D. Ioan, and D Mihalache. Magnetichooks in the finite integration technique: a waytowards domain decomposition. In Proceedings ofCEFC. Greece, 2008.
2. D. Ioan, W. Schilders, G. Ciuprina, N. Meijs, andW. Schoenmaker. Models for integrated compo-nents coupled with their em environment. COM-PEL Journal, 27(4):820–829, 2008.
3. L.M. Silveira et al. Report on parametricmodel order reduction techniques. Technical re-port, FP6/Chameleon-RF deliverable D1.3a, 2007.Available at http://www.chameleon-rf.org/.
44
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 1.8
Parametric Reduced Compact Models for Passive Components
Diana Mihalache, Daniel Ioan, Gabriela Ciuprina, and Alexandra Stefanescu
Politehnica University of Bucharest, Electrical Engineering Dept., Numerical Methods Lab., Spl.Independentei 313, 060042, Romania [email protected]
Summary. This paper describes a technique for ex-traction of electromagnetic models for passive inte-grate components, considering the variability of theirdesign (physical and geometrical) parameters.
1 Introduction
Actual fabrication of physical devices is prone tothe variation of certain circuit parameters due todeliberate adjustment of the process or randomdeviations inherent to this manufacturing. Thisvariability leads to a dependence of the circuit ex-tracted elements on several parameters which canhave either electrical or geometrical background.After the extraction stage a parametric system isobtained, often represented as a state-space sys-tem in its descriptor form. Finally, a compact cir-cuit has to be obtained.
Such an approach was the goal work of theframe of the FP6 / IST / Chameleon Europeanproject [1], which referred to developing method-ologies and prototype tools for a comprehensiveand highly accurate analysis of complete next-generation nanoscale functional IC blocks thatwill operate at RF frequencies of up to 60 GHz.
The problem of parametric model extractionis formulated in mathematical terms. In orderto allow the consistent coupling between deviceswith EM field effects and electric circuits, spe-cial boundary conditions called EMCE (Electro-magnetic Circuit Element) are used [2]. For nom-inal cases the reduction procedure based on Vec-tor Fitting (VFIT) [3] and circuit synthesis basedon Differential Equation Method (DEM) [4] werevery succsseful. The paper investigates if they canbe applied to parametric extraction as well.
2 Numerical Example
The study case consist of two rectangular, spiralcoils with five metal turns, each coil having onegrounded terminal and the other one excited. Theparametric model was obtained with the coarsestgrid and it included one parameter: the designparameter d - the distance between the coils. Thevalues of the varying parameter are 14µm and
56µm. Figure 1 shows the layout of the couplingof two spiral inductors. The benchmark is sym-metrically w.r.t. their terminals S11 = S22 andS12 = S21.
Fig. 1. CHRF 202 benchmarks layout (d=56µm)
In order to estimate the size of the reducedcompact model, VFIT procedure followed by DEMprocedure was used for the experimental dataavailable for both cases. Reduced models of or-der 8 have been obtained for both cases, with arelative error between measurements and reducedorder of 1%.
Since the order is the same, both circuits haveidentical topologies but different values for theparameters.
Also, numerical tests showed that the varia-tion of the output characteristic w.r.t. the pa-rameters is almost linear, especially at low fre-quencies. That is why, this paper will investi-gate if a parametric circuit can be obtained start-ing form the two reduced circuits obtained fromVFIT+DEM.
Since VFIT procedure starts from impedanceor admittance characteristics, the first thing to dois see how far are S parameters computed fromaveraged Z or averaged Y, to average value of S.Results are given in fig. 2 and the relative errorscomputed as:
err = rms||X −Xref || F/max||Xref || F ||
are:err(Sm, SmZ) = 0.124%
err(Sm, SmY ) = 0.120%.
45
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 1010
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Frequency [Hz]
Rea
l par
t−−
−S
12CHRF201− measuredCHRF202− measuredS−average from measurementsS−average from YS−average from Z
Fig. 2. Measurements vs. Average characteristic - Sparameter
This preliminary result shows that paramet-ric compact models can be obtained by combin-ing compact models from two simulations, whenthe variation of the interest quantity w.r.t. theparameter is linear.
Acknowledgement. The authors whish to acknowl-edge that the measurements have been carried out inthe frame of FP6/IST/Chameleon European projectwith support of Austria Micro Systems www.austriamicrosystems.com.
References
1. Chameleon RF Project site: www.chameleon-rf.org.
2. Daniel Ioan, Wil Schilders, Gebriela Ciuprina,Nick van der Meijs, Wim Schoenmaker. Modelsfor integrated components coupled with their EMenvironment. COMPEL, vol. 27, 820–829, 2008
3. Gustavsen and A. Semlyen. Rational Approxima-tion of Frequency Domain Responses by VectorFitting. IEEE Trans. Power Delivery, 14(3), 1052–1061, 1999.
4. T. Palenius and Janne Roos. Comparison ofreduced-order interconnect macromodels for time-domain simulation. IEEE Trans. Microwave The-ory and Techniques, 52(9), 2240–2250, 2004.
46
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 1.9
An Effective Technique for Modelling Nonlinear Lumped
Elements Spanning Multiple Cells in FDTD
Luis R.J. Costa, Keijo Nikoskinen, and Martti Valtonen
Dept. of Radio Science and Engineering. Faculty of Electronics, Communications and Automation, HelsinkiUniversity of Technology, Finland luis.costa, keijo.nikoskinen, [email protected]
Summary. A technique for modelling lumped ele-ments spanning multiple cells in FDTD is presented.The technique is applied to produce a stable LE-FDTD diode model that works well far beyond nor-mal operational voltage ranges. Simulation results arein good agreement with those obtained with the cir-cuit simulator Aplac and those in the literature [1].
1 Introduction
Simulating complex electronic systems often re-quires lumped elements to be embedded in elec-tromagnetic field simulators. This was accom-plished in [1] in a 3D finite-difference time domain(FDTD) simulation for the passive elements, theresistive voltage source, and the diode, each span-ning one cell, and the transistor spanning twocells. The lumped element–FDTD (LE-FDTD)method, as this technique is called, requires aniteration routine to solve the transcendental equa-tions resulting in the diode and transistor mod-els due their exponential current-voltage relation.The one-celled resistive voltage source LE-FDTDmodel was extended in [2] to span multiple cellsusing a linear equation solver to update the elec-tric field in the region occupied by the source.The authors presented a model for the resistivevoltage source that does away with the need fora linear equation solver [3]. The method, suit-ably modified, turns out to be an effective tech-nique for modelling nonlinear lumped elements,like the diode, spanning multiple cells withoutneeding to solve a system of equations, makingthe implementation simpler. The strongly nonlin-ear diode, however, requires iteration to obtain astable model over a very wide voltage range.
2 Modelling Technique
The LE-FDTD method splits the current densityterm in Ampere’s circuital law into the conduc-tion current density Jc = σE of the lossy mediumand the current density Jl due to the lumped el-ement. The FDTD method evaluates the E fieldat time n and the H field at n+ 1/2 [4]:
∮
C
Hn+ 12·dl=
∫
S
(ε∂En+ 1
2
∂t+σEn+ 1
2+Jn+ 1
2
l
)·ds, (1)
where En+ 12 is computed semi-implicitly as the
average (En+1 + En)/2. In the following, a z-directed lumped element current is assumed, with-out loss of generality, and (1) is discretised in thestandard manner, with some short-hand notationintroduced at the same time, to derive the updateequation for Ez . So, the left-hand side of (1), thecurrent due to the circulating magnetic field, is
Iz |i,j,k = (Hy |n+ 12
i+ 12 ,j,k+
12
−Hy|n+ 12
i− 12 ,j,k+
12
)∆y
− (Hx|n+ 12
i,j+ 12 ,k+
12
−Hx|n+ 12
i,j− 12 ,k+
12
)∆x, (2)
so that the update equation for the Ez field is
Ez|n+1i,j,k+ 1
2
=
(1 − σ∆t
2ε
1 + σ∆t
2ε
)Ez|ni,j,k+ 1
2+
(∆t
ε
1 + σ∆t
2ε
)
×Iz |i,j,k∆x∆y
− Il∆x∆y
(3)
The position indices of ε and σ are omitted forcompactness and must be added by the reader.
The voltage of the lumped element, connectedacross multiple cells, as shown in Fig. 1a, is
Vn+ 1
2
l = −b∑
m=a
Ez|n+1i,j,m + Ez|ni,j,m)
2∆z
= −b∑
m=a
2
1 + σ∆t
2ε
Ez |ni,j,k+ 12
+
∆t
εIz
∆x∆y
1 + σ∆t
2ε
∆z
2
+ Il
b∑
m=a
∆z
2∆x∆y
∆t
ε(1 + σ∆t
2ε
) . (4)
Above, Ez |n+1 is substituted using (3). Denoting
Rg =
b∑
m=a
∆z
2∆x∆y
∆t
ε(1 + σ∆t
2ε
) , (5)
(which is the grid resistance seen by the lumpedelement) dividing both sides of (4) by Rg, anddenoting the first summation on the right-handside of (4) divided by Rg as IEM, results in
47
Vn+ 1
2
l
Rg= IEM + Il, (6)
which is Kirchhoff’s current law for the circuit inFig. 1b. IEM is the current due to the FDTD grid.
f(Vl)Vl
Il
i, j, b
i, j, a
Ez IEM Rg f(Vl)Vl
Il
(a) (b)
Fig. 1. (a) Lumped element connected across threecells and (b) the circuit equivalent of Eq. (6).
For a linear element, Vl is found directly fromthe circuit in Fig. 1b and put into (6) to obtain itscurrent, which is then plugged into (3) to get theLE-FDTD model. This was done in [3] for theresistive voltage source. For nonlinear elements,however, Vl must be found with Newton-Raphsoniteration, for example, since otherwise the simu-lation can be rendered unstable – the stability isvery sensitive to the parameters used.Diode: A simple diode model is given by
Id = Is
(e
qkT Vd − 1
), (7)
where Is is the saturation current, q is the elec-tron charge, k is Boltzmann’s constant, and T isthe temperatue in Kelvin. The diode voltage Vd
is found from Fig. 1b using Newton-Raphson it-eration, typically in two to four iterations, afterwhich (6) gives the diode current Id (= −Il) thatis then plugged into (3). Alternatively, (7) is usedto compute Id.
To test this model, the microstrip circuit inFig. 2a was simulated. The microstrip had width6∆x, length 30∆y, and substrate height 3∆z,its characteristic impedance is 50 Ω. The FDTDspace was 64×50×12, terminated in a first-orderMur boundary condition, with ∆x = 0.4064 mm,∆y = 0.4233 mm and ∆z = 0.265 mm. The sourceresistance Rs = 50 Ω and frequency of the sinu-soid excitation was 2GHz. The simulation wasrun for 2286 time steps using a 10V and 2.5 kVpeak voltage to test the stability of the diodemodel. At 2.6 kV the simulation became unsta-ble, but the instability formed in the Newton-Raphson iteration, which was implemented with-out any time-step control. A more sophisticatedtime-step control will further extent the workingrange of this diode model. The simulation resultsare compared with those obtained with the cir-cuit simulator Aplac and are seen to be in goodagreement. At 200MHz, the 10V peak voltageyields the result obtained by [1], whose model isreported to be stable using a peak voltage of 15V.
+
−Vs
Rs
εr = 2.2
Microstrip line
Vd
(a)
0 0.5 1 1.5
−10
−8
−6
−4
−2
0
Time, ns
Vol
tage
Vd ,
V
LE−FDTDAplac
(b)
0 0.5 1 1.5
−2500
−2000
−1500
−1000
−500
0
Time, ns
Vol
tage
Vd ,
V
LE−FDTDAplac
(c)
Fig. 2. (a) Microstrip test circuit used to verify thediode model. (b) Diode voltage when source peakvoltage is 10V and (c) 2.5 kV. The simulation resultsare compared with those obtained with Aplac.
3 Conclusion
A technique for modelling lumped elements span-ning multiple cells in FDTD is presented. Thetechnique yields a novel diode model that is sta-ble even far beyond useful operational voltages.The model produces simulation results that arein agreement with those produced by the circuitsimulator Aplac.
References
1. M. Picket-May, A. Taflove, and J. Baron. FD-TDmodeling of digital signal propagation in 3-D cir-cuits with passive and active loads. IEEE Trans.Microwave Theory and Tech., MTT-42(8):1514–1523, August 1994.
2. J. Xu, A.P. Zhao, and A.V. Raisanen. A stable al-gorithm for modeling lumped circuit source acrossmultiple FDTD cells. IEEE Microwave and GuidedWave Letters, 7(9):308–310, September 1997.
3. L. J. Costa, K. Nikoskinen, and M. Valtonen. Mod-els for the LE-FDTD resistive voltage source span-ning multiple cells. In Proceedings of the 2007 Eu-ropean Conference on Circuit Theory and Design,ECCTD 2007, pages 671–674, August 26-30 2007.
4. A. Taflove and S.C. Hagness. Computational Elec-trodynamics: the Finite-Difference Time DomainMethod. 2nd edition, Artech House, Boston, 2000.
48
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 1.10
Mixed-Derating of Three-Phase Distribution Transformer
under Unbalanced Supply Voltage and Non-Linear LoadConditions Using Finite Element Method
Jawad Faiz, M. Ghofrani, and B.M. Ebrahimi
Department of Electrical and Computer Engineering University of Tehran, North Kargar Avenue, Tehran,Iran [email protected]
Summary. In this paper, a three-phase, three-legged20/0.4 KV, 50 KVA distribution transformer is ana-lyzed under unbalanced supply voltage in the pres-ence of non-linear load. The transformer is modeledusing time stepping finite element method (TSFEM),considering all geometrical and physical characteris-tics. Performance of the modeled transformer underunbalanced supply voltage and non-sinusoidal loadconditions has been studied individually. Then theeffect of unbalanced supply voltage in the presence ofnon-sinusoidal load current on the transformer perfor-mance is analyzed. It is shown that non-linear loadcauses the winding losses to increase considerablywhile the core loss approximately remains constant.According to IEEE standards and other papers, dis-tribution transformers are needed to be derated undernon-linear load conditions to prevent the insulationfatigue and the decrease of transformers life time. It isalso shown that unbalanced supply voltage increasesthe core and copper losses simultaneously due to anincrease in the amplitude of harmonic components. Sothe distribution transformers should be mix-deratedin order to account for both winding losses and corelosses increments.
1 Introduction
COMMON sources of harmonics in industrialelectrical systems are rectifiers, dc motor drives,uninterruptible power supplies, and arc furnaces.With the ever-increasing use of these non-linearloads, transformers are exposed to the higher loadlosses, early fatigue of insulation, premature fail-ure and reduction of the useful life. Therefore,distribution transformers should be derated ac-cording to IEEE C57-110 or other methods un-der non-sinusoidal load currents [1-2]. Anotherundesirable phenomenon that occurs in electricalpower systems is the voltage unbalancy. Electricaltrains, non-uniform distribution of loads on threephases of the power system and non-equal spacingof the transmission lines in non-transposed linesare the most common reasons of unbalanced volt-age occurrence[3]. Unbalanced voltage leads to anincrease in core loss as well as winding loss. So thestudying of distribution transformer performance
under the simultaneous presence of unbalancedsupply voltage and non-sinusoidal load currentscould be of great interest. Since the transformeroperates under various harmonic frequencies, thefrequency dependent parameters of transformer,winding skin effect and magnetic core nonlinear-ity must be taken into account. Using Finite El-ement Method (Opera-2D) satisfies all these cri-teria with an acceptable accuracy.
2 Modeling of transformer usingtime stepping finite element method
Calculating the magnetic field distribution withinthe transformer is essential for transformer anal-ysis. FEM is able to calculate the magnetic fielddistribution within the transformer from geomet-rical dimensions and magnetic parameters of thetransformer. Other quantities of the transformersuch as the induced voltage waveform, magneticflux density and forces exerted on the windingscan be determined knowing the magnetic fielddistribution [4]. Two-dimensional finite elementmethod solves the following Poisson equation inorder to evaluate and analyze the magnetic fluxdistribution:
∂
∂x(ℜ∂A
∂x) +
∂
∂y(ℜ∂A
∂y) = −ni
Se(1)
where A is the magnetic potential vector, R is thecore sheets reluctivity, n is the number of windingturns and Se is the cross-section of the conduc-tors. The principal equation of the electric circuitsis presented as follows:
Vs = Rsi+ Lsdi
dt(2)
where Vs is the input voltage of the transformer,Rs and Ls are the resistance and inductance ofthe supply. The matrix form of (1) and (2) is
[K][A] + [M ]∂
∂t[A] − [C][i] = 0 (3)
[D]T ∂
∂t[A] + [L]
∂
∂t[i] + [R][i] = [u] (4)
where [A] and [i] are the magnetic potential andcurrent vectors, [K], [M ], [C], [D], [L] and [R]
49
are the vector of coefficients and [U ] is the in-put vector. Now a non-linear equation must beintroduced that links the FE magnetic equationsto the electric circuit equations. This equation is
[P ] [A E i]T + [Q]∂
∂t[A E i]T = [S] (5)
where [P ] and [Q] are the coefficients vector and[S] is the transformer input vector. Solution of(5) gives magnetic potential vector [A] and trans-former current [i].
Having obtained parameter A from (5), distri-bution of flux density can be calculated in the coreas follows:
B = ∇×A (6)
3 Effects of nonlinear loads ondistribution transformerperformance
The non-linear modeled load is a six-pulsed powerelectronic converter. The current of this converteris of 6k±1 harmonic orders (k is an integer). Non-sinusoidal load currents increase the winding eddycurrent losses and RI2 losses of the transformer,while the core loss approximately remains con-stant. Table I summarizes the transformer losscalculations for linear load and three differentcases of non-linear loads using FEM.
Table 1. Performance of distribution transformer fornon-linear load conditions
Load THDI(%) PCu(w) PEc(w)
Linear - 1199 46.76
Non-linear (case1) 18.22% 1238.85 90.8
Non-linear (case2) 23.01% 1262.5 115.97
Non-linear (case3) 28.32% 1295.2 168.8
4 Effects of unbalanced voltage ondistribution transformerperformance
For analyzing performance of the transformer un-der the unbalanced voltage condition, three differ-ent cases of unbalanced supply voltage are consid-ered. FEM results show that core loss and wind-ing copper loss increase considerably while thewinding eddy current losses have a slight accre-tion.
Table 2. Table II: Performance of distribution trans-former for unbalanced (U) voltage conditions
Supply Phase(a-b-c) PCore(w) PCu(w) PEc(w)
Balanced 0%-0%-0% 210 1199 46.76
U(case1) 0%-5%-7.5% 228.9 1272.1 50
U(case2) 0%-7.5%-10% 256.18 1477.12 53
U(case3) 0%-10%-12.5% 270.2 1624 53.2
5 Effects of simultaneous presenceof unbalanced voltage and non-linearload conditions on distributiontransformer performance
Distribution transformer performance under indi-vidually applied unbalanced voltage or non-linearload condition has been studied using FEM. It isshown that the distribution transformer needs tobe derated considering the winding eddy currentlosses increase under the non-sinusoidal load con-dition. Unbalanced supply voltage in the presenceof the non-linear load adds to the amplitude ofharmonic components and increases the core lossas well as the winding losses. In order to con-sider the effect of both core and winding eddycurrent losses on the transformer performance,the transformer should be mix-derated. Trans-former mixed-derating requires the precise calcu-lation of the losses under the simultaneous pres-ence of unbalanced voltage and non-sinusoidalload current. The given transformer is modeledunder unbalanced supply voltage using FEM anda six-pulsed power electronic converter is con-nected to the external circuit as the transformerload. Table III and Table IV summarize the re-sults of transformer simulation under the abovementioned conditions.
Table 3. Table III: Performance of distribution trans-former for simultaneous presence of unbalanced volt-age (UV) and non-linear load (NL) conditions (threedifferent cases of unbalanced voltage)
UV & NL THDI() Phase(a-b-c) PCore(w) PCu(w) PEc(w)
Case1 28.32% 0%-5%-7.5% 230 1310 170
Case2 28.32% 0%-7.5%-10% 260.2 1480 172
Case3 28.32% 0%-10%-12.5% 276 1730 180
Table 4. Table IV: Performance of distribution trans-former for simultaneous presence of unbalanced volt-age (UV) and non-linear load (NL) conditions (threedifferent cases of non-linear load)
UV & NL THDI() Phase(a-b-c) PCore(w) PCu(w) PEc(w)
Case1 18.22% 0%-5%-7.5% 229 1245 100
Case2 23.01% 0%-5%-7.5% 229.8 1270 120
Case3 28.32% 0%-5%-7.5% 230 1310 170
6 Spectrum analysis of distributiontransformer
Fig. 1 presents the frequency spectrum of thetransformer secondary voltage for two differentcases: 1) balanced supply voltage and linear load.2) Unbalanced supply voltage and non-linear load.
50
Fig. 1. Fig. 1. Normalized plots of the line volt-age spectra of a transformer under balanced voltage-linear load and unbalanced voltage-non-linear loadwith 5th, 7th, 11th, 13th, 17th, 19th harmonics
References
1. ANSI/IEEE C57.110/D7, “Recommended prac-tice for establishing transformer capability whensupplying non-sinusoidal load currents,” IEEE,NY, Feb. 1998.
2. M.A.S. Masoum and E.F. Fuchs,“Derating ofanisotropic transformers under nonsinusoidal op-erating conditions,” Elect. Power energy Syst.,vol.25, pp. 1–12, 2003.
3. A.V. Jouanne, B. Banerjee, “Assessment of Volt-age Unbalance”, IEEE Trans. Power Delivery, Vol.16, No. 4, October 2001, pp. 782–790.
4. Vector Field Software Documentation, 2005.
51
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 1.11
Design Optimization of Radial Flux Permanent MagnetWind
Generator for Highest Annual Energy Input and LowerMagnet Volumes
Jawad Faiz1, M. Rajabi-Sebdani2, B.M. Ebrahimi2, and M.A. Khan3
1 University of Tehran and Iran Academy of Sciences, Tehran, Iran [email protected] Department of Electrical and Computer Engineering, University of Tehran, Tehran, Iran3 Department of Electrical Engineering, University of Cape Town, Cape Town, South Africa
Summary. This paper presents a multi-objectiveoptimization method to maximize annual energy in-put (AEI) and minimize permanent magnet (PM) vol-ume in use. For this purpose, the analytical model ofthe machine is utilized. Effects of generator specifica-tions on the annual energy input and PM volume arethen investigated. Permanent magnet synchronousgenerator (PMSG) parameters and dimensions arethen optimized using genetic algorithm incorporatedwith an appropriate objective function. The resultsshow an enhancement in PMSG performance. Finally2D time stepping finite element method (2D TSFE)is used to verify the analytical results. Comparison ofthe results validates the optimization method.
1 Analytical design
The power captured by a wind turbine from theincident wind can be expressed as:
Pshaft =1
2ρairπR
2U3Cp, (1)
where ρair is the density of air at the height of theturbine hub and Cp is the dimensionless powerco-efficient or aerodynamic efficiency of the tur-binewhich is a function of the tip speed ratio (λ)at which the turbine is operating. This ratio isdefined as the ratio of blade tip speed to windspeed, and can be expressed as:
λ =ωrR
U(2)
Equations (1) and (2) can be used to determine anexpression for the torque applied to the turbineshaft, which can be expressed as:
Tshaft =Pshaft
ωr=
1
2ρairπR
3U2CT (3)
where CT = Cp/λ is the torque coefficient of theturbine.
The power captured by a wind turbine andturbine shaft speed for a PMSG is shown in Fig. 1.Generator output power can be expressed as fol-lows:
Pgen = ηgenPshaft (4)That power shaft is maximized when [1]:
dPgen
dωr= 0 (5)
Fig. 1. Variation of shaft power with turbine speed
or dPgen
dωr=
dPgen
dλ· dλ
dωr= 0 (6)
It is necessary to note thatdλ
dωr6= 0 → dPgen
dλ6= 0 (7)
So, the peak power shaft can be found as follows:
dPgen
dωr= 0 ↔ dPgen
dλ= 0 → dCp(β, λ)
dλ= 0 (8)
With defined Cp, the maximization Pgen andPshaft will be defined. The apparent power of aradial-flux machine at the air-gap can be writtenin terms of the main dimensions of the machineas follows:
Sg =1
2π2 ·Kw1 ·D2 · ℓ · nsSMLpk · SELpk (9)
where Kw1 is the fundamental winding factor, nsis the rotational speed in rps, SMLpk and SELpk
are the peak values of the specific magnetic andelectric loadings of the machine. The output elec-trical power of a radial-flux generator can be re-lated to its apparent airgap power by the follow-ing expression:
Pgen = 3VaIa cosφ = 3Ef
εIa cosφ (10)
where:Ef =
√2πfNphkω1φp (11)
where φp is the flux per pole due to the funda-mental space harmonic component of the excita-tion flux density distribution, f is the frequencyand Nph is the number of turns per phase. Theproduct D2ℓ, which is proportional to the rotorvolume of a radial-flux machine, determines itsoutput torque capability. This product can be de-termined by combining (10 10) and (11 11) asfollows:
D2ℓ =εPgen
0.5π2Kw1nsSMLpkSELok cosφ(12)
53
2 Optimization algorithm
Some of the typical dimensions of PMSG are se-lected as design variables where values are de-termined through an optimization procedure. Inthis paper, design variables are axial length, re-manent flux density of the PMs, specific electricloading, efficiency, power factor, terminal voltageand wind speed. To have a more realistic design,some constraints are applied to design variableslisted in Table 2. The pole pairs, rated frequency,number of blades, the main fixed specifications inthe design procedure are (250 rpm), 50 Hz and 3respectively. The operating frequency range forsmall PM wind generators is reported as typi-cally: 30-80Hz . Thus, the nominal frequency ofthe generator at rated wind speed is chosen to bewithin the range 45-65Hz.
Table 1. Design constraints
Parameter Minimum Value Maximum ValueTerminal Voltage 180 V 240 V
Axial Length 0.04 m 0.08 mOutput power 1625 W 3800 W
Frequency 45 Hz 65 HzNumber of pole pairs 4 24
SEL 10000 A/m 40000 A/mEfficiency 80% 100%
Remnant of flux density 0.8 T 1.4 TWind speed 7 m/s 13 m/s
To obtain an optimal design considering, ter-minal voltage, power factor and PM volume, thethree objective functions are defined as follows:
Jz(x1, . . . , xn) =AEI(x1, . . . , xn)
m
V PM(x1, . . . , xn)n(13)
where x1, x2, . . . , xn are design variables. As seenin (11) the importance of the terminal voltage,power factor and PM volume are adjusted bypower coefficients. Maximization for Jz fulfils si-multaneously three objectives of the optimiza-tion. Such as objective function provides a higherdegree of freedom is selecting appropriate designvariables. The results of optimization are summa-rized in Table 2.
Table 2. Specification of intial and optimal designedgenerator
Parameters Original PMSG ProposedOpt. Design
Output power (W) 1625 1785No. of turns per phase 370 816
Shaft torque (N.m) 112.4922 74.0794Remnant of flux dens. (T) 1.17 1.1512
Axial length (m) 0.0665 0.0581
Current dens. (A/mm2) 2.037 0.9336Pole pairs 4 12
Rotor speed (rpm) 157 250No. of turbine blade 12 3
frequency (Hz) 10.47 50Wind speed (m/s) 10 8.62
Ef (V) 72.89 247.0602Annual energy capture (Wh) 1.4756e+007 1.9911e+07
Efficiency 87.86 92.03Volume of PM (m3) 18.272e-005 14.824e-005
3 Time stepping finite elementanalysis
The first step in prediction and analysis of thePMSG is the precise modeling of the motor. Themotor is modeled taking into account the non-linearity of the ferromagnetic materials and thephysical conditions of the motor components. Thelocal total current density (J) is:
J = J0 − jωA (14)where J0 = σE0 is the current density due to thesupply voltage and A is the magnetic potentialvector. The total current in a conductor havingcross-section Ωb is:∫
Ωb
J · ds =
∫
Ωb
(J0 − jωσA) dξ (15)
where ω is the angular frequency, σ is the electri-cal conductivity of conductor and ξ is the cross-section of the conductor. Since E0 is constant overthe entire conductor cross section then:
I = J0
∫
Ωb
dξ − jω
∫
Ωb
(σA)dξ (16)
where I is the total current of the conductor. Forevery circuit:
νs = Rexti+ Lextdi
dt+ Es (17)
where νs is the impressed voltage, i is the phasecurrent, Rext is the stator phase resistance andLext is the end winding inductance and:
Es = ℓm
N∑
i=1
±Ei (18)
where lm is the length of the FE region, Zext is theexternal circuit impedance and N is the numberof turns. By the following definition:
νs =Vs
ℓm, Ze =
Zext
ℓm(19)
The following can be obtained:
νs =
N∑
i01
±Ei + Zei (20)
The matrix form of (20) is as follow:[Vs] = [D]
T[E] + [Zext] [Ib] (21)
By combining the above equations, the followinggeneral system is obtained:
AEs
jωIb
jω
=
[S] + jωσxy [T ] −jωσz [C] [0]
−jωσz [C] −jωσz [Ωb] −jω [D][0] −jω [D] −jω [Zext]
−1
00
−νs
(22)where [Ib] is the circuit vector (number of circuitsmultiplied by 1), [Es] is the bar voltage vector(number of bars multiplied by 1), [D] is the con-nection matrix of the bar (number of circuits mul-tiplied by the number of bars), [Ωb] is the diagonalmatrix of the cross-section of the bar (number ofthe bars multiplied by 1), [Zext] is the diagonalexternal impedance matrix and νs is the circuitvoltage vector (number of circuits multiplied by1). Fig. 2 shows the optimized excitation voltagecompared with the voltage of the original gener-ator.
54
Fig. 2. Variation of excitation voltage with time, (a)original PMSG and (b) propose optimal design
References
1. J. Clerk Maxwell. A Treatise on Electricity andMagnetism, 3rd ed., vol. 2. Oxford: Clarendon,1892, pp. 68-73.
55
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 1.12
Finite-element discretization for analyzing eddy currents in
the form-wound stator winding of a cage induction motor
M. Jahirul Islam and Antero Arkkio
Department of Electrical Engineering, Helsinki University of Technology, Finland [email protected],[email protected]
Summary. For the computation of the eddy-currentloss in the form-wound multi-conductor stator wind-ing accurately, finite element discretization (FED) isan important issue. The stator bars of the studied mo-tor must be discretized detailed enough to computethe eddy-current effects accurately. Two main supplyconditions of a cage induction motor, sinusoidal andPWM, are considered.
1 Introduction
The finite element analysis (FEA) is one of themost popular and reliable methods to considereddy-currents in the form-wound multi-conductorstator windings. To model the eddy currents inthe stator winding with FEA, all the stator barsconnected in series and parallel should be con-sidered. As a result, finite element discretization(FED) of every bar is an essential requirement[1].In an induction motor, high-frequency harmon-ics are generated from the internal and externalsources. When the supply of the motor is takenfrom grid, the source is almost sinusoidal. Theharmonics are generated from the internal sourcesdue to the complex geometry of electrical motor.The eddy-current loss can be reduced by liftingup the stator coils from the air-gap since thoseharmonics mainly stay close to the air-gap [2].Due to the requirement of the speed control ofelectrical motor, the application of frequency con-verter supply has increased in recent decades. Thesupply from the freq uency converter is stronglynon-sinusoidal and contains a set of additionalharmonics (external source) [3]. The presence ofthese high-frequency components means that theskin depth becomes shorter [4]. To consider theeddy-current effects of those high-frequency com-ponents accurately, the FED should be detailedenough. A more detailed discretization means anincrease in the number of equations to be solved.As a result, the computational complexity andtime is increased.The main objective of this work is to study the ac-curacy of the FED for the eddy-current loss com-putation in the form-wound stator winding of a
cage induction motor. The following two condi-tions are considered for the study:1. The motor is supplied from a sinusoidal voltagesource2. The motor is supplied from a pulse-width-modulated (PWM) voltage source.
2 Methods
To model the eddy current losses, time steppingfinite element analysis has been used in whichthe backward Euler method is used for time dis-cretization. The circuit equations of the cage ro-tor, stator phase and stator bars are strongly cou-pled with the electromagnetic field equations andthe total system of equation is solved altogether[1-3], [5]. The stator resistive loss in the slot em-bedded stator bars after considering the eddy-current loss is
Psb =Qsleσ
Qsb∑
j=1
∫
Ω
ηsiJ
2dΩ (1)
where the current density in the stator bar i is
J = −σ ∂A∂t + σus
iez
le, Qs is the number of symme-
try sectors, le is the length of the slot embeddedstator bars, σ is the conductivity, Ω is the crosssectional area of a stator bar, ηs
i is the param-eter for correspondence between the points theequation is applied and the stator bar i, A is thevector potential, us
i is the potential difference ofthe stator bar i and ez is the unit vector in the zdirection.
The eddy-current loss arises due to the unevencurrent distribution in the bars [6]. This causesthe effective bar resistance to exceed its DC value.This effective resistance is defined as the AC re-sistance [1]. The eddy factor (keddy ) is definedas
keddy =Psb
i2RDC=i2RAC
i2RDC=RAC
RDC(2)
3 Results
A 3-phase, 50 Hz, 690 V, delta-connected, 1250kW cage induction machine with double layer
57
winding is studied [1-3]. The number of rotor slotsis adjusted to reduce the symmetry sector.A bar is that part of a strand that is embedded ina slot. The bars are connected in series in a partic-ular strand. Each effective turn consists of threecopper strands connected in parallel. There aresix effective turns in a slot. Therefore, there are18 bars in a slot. The height of a bar is 3.2 mm andwidth is equal to 10.6 mm [1-3]. To study the ac-curacy of the FED, the number of finite elementsis increased from 4 to 48 successively. Based onthe number of FED layers, the discretization inthe direction of bar height can be divided intothree cases. Fig. 1 presents the finite-element dis-cretization for all the three cases. For case A,there is only one layer over the height. In a simi-lar way, by making two and three layer FED overthe bar height leads to case B and C. For eachcases, the number of finite elements is increasedby making more subdivisions in the bar width.Second-order isoparametric triangular finite ele-ments have been used to simulate the electricalmach ine. Fig. 2 presents the eddy-current effects(RAC
RDC−1) as a function of the number of elements
in a stator bar. The results have been computedwhen the motor is supplied from a sinusoidal ora PWM voltage source.
Case A
Case B
Case C
4 8
8 16 32
12 24 48
36
Fig. 1. Finite-element discretization of a bar.
4 Discussion
For the sinusoidal supply, the eddy-current effects(keddy − 1) remains almost constant for the casesA, B and C (Fig. 2). Fine discretization seemsjust to increase the computational complexityand time. For this supply, the main componentfor the eddy-current loss in the stator winding isleakage flux that passes from one stator tooth toanother.The frequency of this component is equalto the fundamental.When the motor is supplied from a PWM voltagesource, a single layer FED (case A) is not accurate
0.0
0.3
0.6
0.9
1.2
1.5
0 10 20 30 40 50
Number of finite elements in a bar
RA
C/R
DC-1
A-Sin B-Sin C-Sin
A-PWM B-PWM C-PWM
Fig. 2. Eddy-current effects ( RACRDC
− 1) are presentedas a function of the number of elements in a bar forsinusoidal and PWM supply.
enough (Fig. 2). The PWM supply injects higherharmonics to the supply. The high-frequency fluxpasses circumferentially from one stator tooth toanother. The two and three layer FEDs result inalmost identical losses.For the sinusoidal supply, case A, where the num-ber of finite elements is 4, and for the PWM sup-ply, case B, where the number of elements is 8,compromises regarding the computational com-plexity and accuracy.
References
1. M. J. Islam, J. Pippuri, J. Perho and A. Arkkio.Time-harmonic finite-element analysis of eddy cur-rents in the form-wound stator winding of a cageinduction motor, IET Electric Power Application,vol. 1, issue 5, September 2007, pp. 839-846.
2. M. J. Islam and A. Arkkio. Time-stepping finite el-ement analysis of eddy currents in the form-woundstator winding of a cage induction motor sup-plied from a sinusoidal voltage source, IET ElectricPower Application, In press.
3. M. J. Islam and A. Arkkio. Effects of pulse-width-modulated supply voltage on eddy currents in theform-wound stator winding of a cage inductionmotor, Under review.
4. A. D. Podoltsev, I. N. Kucheryavaya and A.B. Levedev. Analysis of effective resistance andeddy-current losses in multiturn winding of high-frequency magnetic components, IEEE Transac-tions on Magnetics, vol. 39, issue 1, January 2003,pp. 539-848.
5. I. Yatchev, A. Arkkio and A. Niemenmaa. Eddy-current losses in the stator winding of cage in-duction motors, Laboratory of Electromechanics,Helsinki University of Technology, Report 47, Fin-land 1995, p. 34.
6. A. Konrad. Integrodifferential finite element for-mulation of two-dimensional steady-state skin ef-fect problems, IEEE Transactions on Magnetics,vol. 18, Issue 1, January 1982, pp. 284-292.
58
Session
CP 1
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CP 1Ia
Evaluation of the electromagnetic coupling between
microelectronic device structures using computationalelectrodynamics
Wim Schoenmaker and Peter Meuris
MAGWEL NV Martelarenplein 13, B-3000 Leuven, Belgium [email protected], [email protected]
Summary. Electromagnetic coupling between devicesin an microelectronic layout can become a serious de-sign concern. In this paper, the problem of electro-magnetic coupling is addressed from field computa-tional point of view. Approximation schemes are jus-tified by evaluating dimensionless parameters in theset up of the field equations and scale considerationsof devices.
1 Introduction
With the use of increasing frequency ranges, elec-tromagnetic coupling becomes more pronounceddesign concern because induced electric fields areproportional to rate of change of the magnetic in-duction. However, not only the pace of time vari-ations are determining for including electromag-netic coupling but also the problem scale and theintensity of the currents that are responsible forthe induced fields must be considered. An overallpicture of scaling arguments is presented in sec-tion 2 which is valuable for computational consid-erations. Once we note from scale considerationsthat electromagnetic coupling terms represent anon-negligible contribution to the full system ofequations, we move on the the solution of theseequations. In section 3, we review and update ofthe approach that was proposed some years agoby the author and co-workers [1–3]. We will referto this approach as ’computational electrodynam-ics’. At several occasions we were inquired if thismethod is equivalent to the method based on Ned-elec’s edge elements [4]. To the best of our knowl-edge, we believe it is not. The main differencebeing that we do not refer to test functions at all.Our method is more related to finite-integrationtechniques (FIT). Scale considerations are notonly an issue for deciding if some terms in thefull system of Maxwell equation and constitutivelaws can be neglected. When discussing the cou-pling of devices, it is also important to realizethat different devices can have an intrinsic or geo-metrical scales that differ orders of magnitude. Insuch scenarios the coupled problem is most easilysplit in computational domains. Computational
electrodynamics gives rather straightforwardly aseries of prescriptions for matching the interfaceconditions of the various domains.
Electromagnetic coupling of microelectronicdevices is an RF issue and is most convenientlymeasured using s-parameters. In section 4, wepresent our method to compute these matrixelements. In fact, s-parameter extraction is astraightforwardly achieved as a post-processingof the results of a computational electrodynam-ics problem with the appropriate setting of theboundary conditions.
In section 5 we will present a few of coupledproblems, that we have addressed recently.
2 Scaling rules the Maxwellequations
The use of scale arguments is definitely not new tothe field of computing in electromagnetic model-ing. Well-known approximations are the so-calledEQS (electro-quasi static) and MQS (magneto-quasi static) approximations. Approximations canbe put in a different perspective by consideringthe scaling step that is necessary when convert-ing the full set of equations to dimensionless equa-tions before computing can start. For our presentargument it suffices to consider insulators andmetals only. Semiconductors can be easily added.
Jc = σE (1)
D = ε0εrE (2)
H =1
µ0µrB (3)
B = ∇× A (4)
E = −∇V − ∂A
∂t(5)
Consider the Maxwell equation is the potentialformulation. The Poisson equation is used to solvethe scalar field in insulators and semiconduct-ing regions and the current-continuity equation isused in metals to find the scalar potential there.The electric system is :
61
∇. [ε (∇V + iωA)] + ρ = 0
∇. [(σ + iωε) (∇V + iωA)] = 0 (6)
The Maxwell-Ampere equation is :
∇×(
1
µ∇× A
)− (σ + iωε) (−∇V − iωA) = 0
(7)This system must be completed with a gauge con-dition
∇ ·A + iωξεµV = 0 (8)
We introduced a parameter ξ that allows us toslide over different gauge conditions. Now let Lbe the ’natural’ length scale of the problem thatis considered. For example L = 1µm. Further-more, let T , be the natural time scale. For exam-ple T = 10−9 sec. It is possible to reformulate theequations ( 6, 7) in dimensionless variables V andA and the set of equations is controlled by twodimensionless variables, K and ν.
∇. [εr (∇V + iωA)] + ρ = 0
∇. [(σ + iωεr) (∇V + iωA)] = 0 (9)
and
∇×(
1
µr∇× A
)− Kω2 (εr − i ν)A
−i ωK (εr − iν)∇V = 0 (10)
The constants K = ε0µ0L2/T 2 and ν = σT/ε0.
Note that for σ = 104 S/m, we obtainKν = 10−5.This number enters into the Maxwell-Ampereequation and suggests that in this scenario themagnetic sector is negligible. However, one caneasily find cases with metallic conductances of 107
S/m, that magnetic effects are important.
3 Discretization
In our earlier work, we presented a discretizationmethod that decided for each variable where onthe grid it belongs. It was concluded that the ge-ometrical and physical meaning of variables playsa key role. For instance, a scalar variable, e.g. thePoisson potential, V , is a number assigned to eachspace location and for computational purpose, itdiscretized value should be assigned to the nodesof the grid. On the other hand the vector poten-tial A is a vector variable of the same character as∇V and should therefore be assigned to the linksof the computational grid. The conversion of con-tinuous variables to discrete variables on the com-putation grid has also consequences for the partic-ular discretization route that is followed when im-plementing discrete versions of the Maxwell equa-tions. Gauss’ law is discretized by considering the
elementary volumes around the nodes of the gridand next perform an integration of Gauss’ lawover the volume cell. The flux assigned to eachsegment of the enclosing surface is assumed tobe constant which allows for expressing this (con-stant) flux in terms of the node variables and linkvariables. This scheme has been the key to thesuccess of the simulation of the semiconductor de-vices. The Scharfetter-Gummel formulation of thediscretized currents can be set up following aboveapproach. Since links variables are essentially dif-ferent from node variables, we expect that the dis-cretization of the Maxwell-Ampere equation hasto be done taking this geometrical aspect into ac-count. Whereas it was quite ’natural’ to regardsnode variables as a representative of some vol-ume element, in the same way we consider a linkvariable representing some area element. Thus toeach link is associated an area element and in or-der to discretize the Maxwell-Ampere equation ona grid we now apply Stokes’ law to arrive at thediscretized equations.
Once that we have found the way to dis-cretize the Maxwell equations, we proceed withexpanding then into a small signal analysis. Thismeans that each variable is written as a time-independent part and an harmonic part.
X = X0 +X1eiωt (11)
If we apply boundary conditions of a similar formand collect terms independent of ω and terms pro-portional to eiωt and omit terms proportional toX2
1 then we obtain a system of equations for thephasors X1.
Of particular interest is the treatment of thespurious modes in the fields. These modes can beeliminated by selecting a ’gauge tree’ in the mesh,adding a ghost field to the equation system or ap-ply a projection method while iterating towardsthe solution. However, we can also apply a gaugecondition and construct discrete operators thatresemble a continuous operator as close as pos-sible including having a semi-definite spectrum.Using a two-fold application of Stokes’ law, the
term ∇×(
1µr
∇× A)
appears in the discretized
formulation as a collection of closed-loop circu-lations. By subtracting a discretized version of∇ (∇ · A) we arrive at a operator that resembles−∇2A. However, since A is a vector field, thelatter can only have meaning in terms of the fore-going expressions. The discretization of the firstterm in (7) can be illustrated as shown in Fig. 1.
4 S-parameters
In order to determine the S matrix, a ratherstraightforward procedure is followed. For that
62
Fig. 1. Discretized version of the regularized curl-curloperator acting on a vector field.
purpose a collection of ports is needed and eachport consists of two contacts. A contact is de-fined as a collection of nodes that are electricallyidentified. A rather evident appearance of a con-tact is a surface segment on the boundary of thesimulation domain. A slightly less trivial contactconsists of two or more of these surfaces on theboundary of the simulation domain. The nodesthat are found on these surface are all havingthe same electrical potential. Therefore, althoughthere may be many nodes assigned to a singlecontact, all these nodes together generate onlyone potential variable to the system of unknowns.Of course, when evaluating the current enteringor leaving the contact, each node in the contactcontributes to the total contact current. Assign-ing prescribed values for the contact potential canbe seen as applying Dirichlet’s boundary condi-tions to these contact. This is a familiar techniquein technology CAD. Outside the contact regions,Neumann boundary conditions are applied. Un-fortunately, since we are now dealing with the fullsystem of Maxwell equations, providing boundaryconditions for the scalar potential will not suffice.We also need to provide boundary conditions forthe vector potential. Last but not least, since thethe set of variable V and A are not independent,setting a boundary condition for one variable hasan impact on the others. Moreover, the choice ofthe gauge condition also participates in the ap-pearance of the variables and their relations. Aconvenient set of boundary conditions is given bythe following set of rules :
• contact surface: V = V |ic. To each contact areais assigned a prescribed potential value.
• outside the contact area on the simulation do-main : Dn = 0. There is no electric field com-ponent in the direction perpendicular to thesurface of the simulation domain.
• For the complete surface of the simulation do-main, we set Bn = 0. There is no magnetic
induction perpendicular to the surface of thesimulation domain.
We must next translate these boundary condi-tion to restriction on A. Let us start with thelast one. Since there is no normal B, we may as-sume that the vector potential has only a normalcomponent on the surface of the simulation do-main. That means that the links at the surface ofthe simulation domain do not generate a degreeof freedom. It should be noted that more generaloptions exist. Nevertheless, above set of boundaryconditions provide the minimal extension of theTCAD boundary conditions if vector potentialsare present.
In order to evaluate the scattering matrix, sayof an N-port system, we iterate over all ports andput a voltage difference over one port and putan impedance load over all other ports. Thus thepotential variables of the contacts belonging toall but one port, become degrees of freedom thatneed to be evaluated. The following variable arerequired to understand the scattering matrices,where Z0 is a real impedance that is usual takento be 50 Ohms.
ai =Vi + Z0Ii
2√Z0
(12)
bi =Vi − Z0Ii
2√Z0
(13)
The variables ai represent the voltage waves inci-dent on the ports labeled with index i and thevariables bi represent the reflected voltages atports i. The scattering parameters sij describethe relationship between the incident and re-flected waves.
bi =
N∑
j=1
sijaj (14)
The scattering matrix element sij can be foundby putting a voltage signal on port i and place animpedance of Z0 over all other ports. Then aj iszero by construction, since for those ports we havethat Vj = −Z0Ij Note that Ij is defined positive ifthe current is ingoing. In this configuration sij =bi/aj . In a simulation setup, we may put the inputsignal directly over the contacts that correspondto the input port. This would imply that the inputload is equal to zero. The s-parameter evaluationset up is illustrated in Fig. 2.
5 Applications
Using the solver based on computational electro-dynamics, we are able to compute the s-parameters
63
Fig. 2. Set up of the s-parameter evaluation: 1 portis excited and all others are floating.
by setting up a field simulation of the full struc-ture. This allows us in detail to study the phys-ical coupling mechanisms. As an illustration, weconsider two inductors which are positioned ona substrate layer separated with a distance of 14micron. This structure was processed and char-acterized and the s-parameters were obtained. Itis quite convenient to study a compact modelparameters to obtain a quick picture of the be-haviour of the structure. For this device a conve-nient variable is the ’gain’, which corresponds tothe ratio of the injected injected power and thedelivered power over an output impedance [5].
G =PinPout
(15)
The structure is shown in Fig. 3 and a three-dimensional view using editEM (MAGWEL) isshown in Fig. 4.
Fig. 3. View on the coupled spiral inductor using theVirtuosa design environment.
When computing the s-parameters, we put thesignal source on one spiral (port 1) and place50 Ohm impedance over the contacts of the sec-ond spiral (port 2).The s11-parameter is shown inFig. 5 and the s12-parameter is shown in Fig. 6.Finally, the gain plot is shown in Fig. 7. This
results shown here have been obtained withoutany calibration of the material parameters. Thesilicon is treated ’as-is’. This means that the sub-strate and the eddy current suppressing n-wellsare dealt with as doped silicon.
Fig. 4. View on the coupled spiral inductor using theMAGWEL structure editor editEM.
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0 2e+09 4e+09 6e+09 8e+09 1e+10 1.2e+10 1.4e+10 1.6e+10
f [Hz]
S11-abs-arg
S11 meas absS11 mgw absS11 meas argS11 mgw arg
Fig. 5. Comparison of the experiment and simulationresults for s11 .
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
0 2e+09 4e+09 6e+09 8e+09 1e+10 1.2e+10 1.4e+10 1.6e+10
f [Hz]
S12-abs-arg
S12 meas absS12 simul absS12 meas argS12 simul arg
Fig. 6. Comparison of the experiment and simulationresults for s12 .
6 conclusions
In this paper we presented a version of compu-tational electrodynamics which is based on thescalar and vector potential formulation. Whereasthe finite-integration technique deals directly withthe field intensity quantities E and B, our for-mulation deals with the more fundamental gaugefields. It should be emphasized that the fieldquantities are derived variables and one that thepotentials have been computed, all other variablesare obtained by ’post-processing’. Our approach
64
-100
-90
-80
-70
-60
-50
-40
-30
-20
1e+07 1e+08 1e+09 1e+10 1e+11
mag
nitu
de
frequency
Gain in dB
Gain measGain mgw
Fig. 7. Comparison of the experimental and simula-tion results for the gain.
is a discrete implementation of the geometricalinterpretation of electrodynamics [6]. Accordingto this interpretation, the field intensities corre-spond to the curvature and the potentials areconnections in the geometrical sense. The practi-cal capabilities of our method are comparable toother field solvers that focus directly on the fieldsE and B, with one exception: if the potential areneeded in the evaluation of the constitutive re-lations then our method has a clear advantage.This happens if the true semiconductor modelingis needed and one can not mimic the semicon-ductor with moderately conductive material. An-other area of application is the unified solving ofquantum problems and magnetic induction prob-lems. Our approach will be an excellent startingpoints. We have shown with a realistic applicationthat the method is capable of producing fairlygood results. The deviations at higher frequencyare an indication that adaptive meshing methodsare mandatory.
Acknowledgement. This work was performed in theEU funded projects CODESTAR and CHAMELEON-RF. The authors the numerous stimulating discus-sions with the partners in this project.
References
1. P. Meuris, W. Schoenmaker and W. MagnusStrategy in electromagnetic interconnect model-ing. IEEE Trans. on CAD of Integr. Syst., 20:739-752, 2001
2. W. Schoenmaker, Peter Meuris ”ElectromagneticInterconnects and Passives Modeling: Software Im-plementation Issues Software Implementation Is-sues. IEEE Trans. on CAD of Integr. Syst.,21:534-543, 2002
3. W. Schoenmaker, W. Magnus and P. Meuris GhostFields in Classical Gauge Theories Phys. Rev.Lett., 21:181602-1 - 181602-4, 2002
4. J.F. Lee, D.K. Sun and Z.J. Cendes. Tangen-tial vector finite elements for electromagnetic field
computation. IEEE Transactions on Magnetics,27:4032-4035, 1991
5. A. M. Niknejad and R. G. Meyer Analysis, Designand Optimization of Spiral Inductors and Trans-formers for Si RF ICs. IEEE Journ. of Solid-StateCircuits, 33:1470–1481, 1998.
6. T. Frankels. The Geometry of Physics CambridgeUniversity Press, Cambridge, UK, 1997
65
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CP 1A
Simulations of a electron-phonon hydrodynamical model based
on the maximum entropy principle
V. Romano1 and C. Scordia2
1 Department of Mathematics and Computer Science, University of Catania, Italy [email protected],2 University of Wuppertal [email protected]
Summary. Recently a hydrodynamical and energy-transport models have been formulated based on themaximum entropy principle for the coupled phonon-electron system in silicon in order to cope with theeffects of heating of the crystal lattice. Here the sim-ulations of some standard devices are presented inorder to asses the validity of the model.
1 Introduction
At macroscopic level, several heuristic models forthe thermal effects in the crystal lattice have beenproposed. They are represented by the crystal lat-tice energy balance equation and differ for theproposed form of thermal conductivity and en-ergy production, e.g. [1–5]. A critical review canbe also found in [6].
Recently, a consistent hydrodynamical modelfor charge carriers has been developed startingfrom the moment system associated with thetransport equations, obtaining the closure rela-tions with the maximum entropy principle (here-after MEP) [7–11]. The same approach has beenadopted in [12] for the electron-phonon system,obtaining also an energy-transport and a drift-diffusion model under appropriate scalings.
The electrons are described with the 8-momentsystem as in [7–9]. The phonons are consideredas two populations: acoustic and non polar opti-cal. The non-polar optical phonons are describedwith a Bose-Einstein distribution while the acous-tic ones are described by the MEP distributionfunction in the 9-moment approximation alreadyintroduced in [13].
The simulations of some benchmark devicesare presented and compared with those obtainedwith the standard models existing in the litera-ture. The numerical discretization has been per-formed along the guide lines reported in [14, 15].
2 The model
The kinetic description of the electron-phononsystem is based on the Bolzmann equations for
electrons distribution f and phonons distribu-tions, both acoustic and non-polar optical, g(ac), g(no),coupled to the Poisson electron for the electric po-tential
∂f
∂t+ vi(k)
∂f
∂xi− eEi
~
∂f
∂ki= C[f, g(ac), g(no)],(1)
∂g(ac)
∂t+∂ω(ac)
∂qi∂g(ac)
∂xi= S(ac)[g(ac), g(no), f ],(2)
∂g(no)
∂t= S(no)[g(ac), g(no), f ], (3)
Ei = − ∂φ
∂xi, ǫ∆φ = −e(ND −NA − n). (4)
k and q are the wave vectors of electrons andphonons respectively. E is the electric field ande the elementary charge. The non polar opticalphonons are assumed to be described by the Bose-Einstein distribution.
The direct solution of the system (1)-(4) isa daunting computational task and requires longCPU times, not yet suitable for CAD purposes.
By considering the 8-moment model for elec-tron and the 9-moment model for phonons of theprevious sections, the following complete set ofmacroscopic balance equations is obtained
∂n
∂t+∂(nV i
)
∂xi= 0, (5)
∂(nP i
)
∂t+∂(nU ij
)
∂xi+ neEi = nCip, (6)
∂ (nW )
∂t+∂(nSj
)
∂xi+ neVkE
k = nCW , (7)
∂(nSi
)
∂t+∂(nF ij
)
∂xi+ neEjG
ij = nCiW , (8)
∂u
∂t+Qk = Pu, (9)
∂pi∂t
+1
3
∂u
∂xi+∂N〈jk〉∂xk
= Pi, (10)
∂N〈ij〉∂t
+∂M〈ij〉k∂xk
= P〈ij〉. (11)
n, P i, W and Si are the electron density, averagecrystal momentum and average energy-flux whileu, pi and N〈ij〉 are the phonon energy, averagemomentum and deviatoric part of the average fluxof momentum.
67
In (9)-(11) each term is given by the sum ofthe contribution of the acoustic and the non-polaroptical phonons.
Since the number of unknowns exceeds thenumber of equations, closure relations must be in-troduced. The maximum entropy principle (here-after MEP) gives a systematic way for obtainingconstitutive relations on the basis of informationtheory. It has been used in [12] to close the pre-vious electron-phonon system as in [7–9].
All the coefficients depend on the electron en-ergy W and crystal temperature TL. Their ana-lytical expressions are explicitly given and relatedto the scattering parameters (see [12] for more de-tails).
In particular the energy phonon productionterms can be written as
Pu = −nCW , (12)
that is the opposite of the electron production.In Fig. 1 we plot the energy relaxation time
defined as τW = −W− 32kBTL
nC(e)W
versus the electron
energy, for different values of the crystal temper-ature from 300o K to 330o K.
Under an appropriate scaling the followingcrystal energy balance equation is derived
ρcV∂TL∂t
− div [k(TL)∇TL] = H, (13)
where ρ is the silicon density, k(TL) the thermalconductivity,
H = −nCW + c2div(τR nc
(p)11 V + τRnc
(p)12 S
)(14)
with nc(p)ij related to the phonon production terms.
Equation 13 generalizes the models proposedin [1–5].
0 0.1 0.2 0.3 0.4 0.5 0.60
0.2
0.4
0.6
0.8
1
300 K306 K312 K318 K324 K330 K
Fig. 1. energy relaxation time versus the electronenergy, for different values of the crystal temperaturefrom 300o K to 330o K
Acknowledgement. The authors acknowledge the fi-nancial support by the EU Marie Curie RTN project
COMSON grant n. MRTN-CT-2005-019417. V.R.acknowledges also the contribution by M.I.U.R. (PRIN2007 Equazioni cinetiche e idrodinamiche di sistemicollisionali complessi), the P.R.A. University of Cata-nia (ex 60 %).
References
1. S. P. Gaur abd D. H. Navon, Two-dimensionalcarrier flow in a transistor structure under non-isothermal conditions, IEEE trans. Electron. De-vices, ED-23 50–57, 1976.
2. D. K. Sharma and K. V. Ramanthan, Modelingthermal effetcs on MOS I-V characteristics, IEEEElectron. Device Lett., EDL-4, 362–364, 1983.
3. M. S. Adler, Accurate calculations of the forwarddrop and power dissipation in thyristors, IEEETrans. Electron. Devices, ED-25, 16–22, 1979.
4. A. Chryssafis and W. Love, A computer-aidedanalysis of one dimensional thermal transient inn-p-n power transistors, Solid-State-Electron., 22,249-256, 1978.
5. G. Wachuka, Rigorous thermodynamic treatmentof heat generation and conduction in semiconduc-tor device modeling, IEEE Trans. on Computer-Aided Design, 9 1141–1149 (1990).
6. S. Selberherr, Analysis and simulation of semicon-ductor devices, Wien - New York, Springer-Verlag,1984.
7. A.M. Anile, V. Romano, Non parabolic band trans-port in semiconductors: closure of the momentequations, Continuum Mech. Thermodyn., 1999,11: 307-325
8. V. Romano, Non parabolic band transport in semi-conductors: closure of the production terms inthe moment equations, Continuum Mech. Thermo-dyn., 2000, 12: 31-51
9. V. Romano, Non parabolic band hydrodynamicalmodel of silicon semiconductors and simulation ofelectron devices, Mathematical Methods in the Ap-plied Sciences 24 439 (2001).
10. E. T. Jaynes, Information theory and statisticalmechanics, Physical Review, Vol. 106, Nr. 4, 620(1957).
11. N. Wu, The maximum entropy method, Springer-Verlag Berlin Heidelberg, 1997
12. V. Romano and M. Zwierz, Electron-phonon hy-drodynamical model for semiconductors, preprint(2008).
13. W. Dreyer, H. Struchtrup, Heat pulse experimentrevisited, Continuum Mech. Thermodyn., 1993, 5:3-50
14. V. Romano, 2D simulation of a silicon MES-FET with a non-parabolic hydrodynamical modelbased on the maximum entropy principle, Journalof Computational Physics 176 70 (2002).
15. V. Romano, 2D numerical simulation of the MEPenergy-transport model with a finite differencescheme, J. Comp. Physics 221 439-468 (2007).
68
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CP 1Ib
Thermal Effects in Coupled Circuit-Device Simulations
Markus Brunk1 and Ansgar Jungel2
1 Norwegian University of Science and Technology, Department of Mathematical Sciences, 7491 Trondheim,Norway; [email protected],
2 Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstr. 8-10,1040 Wien, Austria; [email protected]
Summary. A coupled semiconductor-circuit modelincluding thermal effects is proposed. The semicon-ductor device is modeled by the energy-transportequations, and the electric circuit is described bythe network equations from modified nodal analysis.The resulting system of nonlinear partial differential-algebraic equations is solved by mixed finite elementsin space and the BDF2 method in time. Numericalresults for a rectifying circuit and a high-pass filterincluding a laser diode are presented.
1 Introduction
In industrial application, complex semiconductordevice models are usually substituted by circuitsof basic network elements resulting in simplercompact models. Such a strategy was advanta-geous up to now since integrated circuit simula-tions were possible without computationally ex-pensive device simulations. Parasitic effects andhigh frequencies in the circuits, however, requireone to take into account a very large numberof basic elements and to adjust carefully a largenumber of parameters in order to achieve the re-quired accuracy. Moreover, device heating cannotbe easily modeled by this approach. Therefore,it is preferable to model those semiconductor de-vices which are critical for the parasitic effects bysemiconductor transport equations.
We include the device model into the net-work equations directly and employ the energy-transport equations to model the electron tem-perature. A coupled model including the latticetemperature is work in preparation. As the devicemodel is described by partial differential equa-tions, and the network equations are given bydifferential-algebraic equations, this results in acoupled system of nonlinear partial differential-algebraic equations.
The (scaled) energy-transport equations con-sist of conservation laws for the electron densityn and electron energy density w,
nt − div Jn = −R(n, p),
wt − div Jw = −Jn · ∇V +W (n, T )− 3
2TR(n, p),
with constitutive relations for the particle cur-rent density Jn and the energy current densityJw, derived from the semiconductor Boltzmannequation:
Jn = µn
(∇n− n
T∇V
),
Jw =3
2µn(∇(nT ) − n∇V
),
where R(n, p) denotes the generation-recombina-tion term, p is the hole density, which is computedfrom the drift-diffusion equations, W (n, T ) is therelaxation-time term, and µn the (scaled) elec-tron mobility [3]. The equations are coupled tothe Poisson equation for the electric potential V :
λ2∆V = n− p− C(x),
where λ is the scaled Debye length and C(x) thedoping concentration. The equations are supple-mented by initial conditions for n and w and ap-propriate boundary conditions (see [1, 2] for de-tails).
2 Numerical approximation
The energy-transport model is discretized in onespace dimension by mixed-hybrid finite elementsof Marini-Pietra type. These elements instead ofstandard RaviartThomas elements are employedsince this guarantees the M-matrix property ofthe global stiffness matrix, even in the presenceof zeroth-order terms.
The network equations as well as the parabolicequations are discretized in time by the 2-stagebackward difference formula (BDF2) since thisscheme allows one to maintain the M-matrix prop-erty. Then the semidiscretized equations are ap-proximated by the mixed finite-element method,and static condensation is applied to reduce thenumber of variables. The final nonlinear discretesystem is solved by Newton’s method.
For a large applied bias, we observe stronggradients of the temperature at the boundary.Employing Robin boundary conditions for the
69
electron density and energy density instead ofstandard Dirichlet conditions help to avoid theboundary layer [1].
3 Numerical results
First, we consider a rectifying circuit containingfour pn diodes (see Figure 1). Each of the diodeshave the length 0.1µm and a maximal doping of1022 m−3. The voltage source is given by v(t) =U0 sin(2πω) with U0 = 5 V and ω = 10 GHz.
Vin
Vout
Fig. 1. Rectifier circuit.
The current through one of the diodes is pre-sented in Figure 2. The figure clearly shows therectifying behavior of the circuit. The largest cur-rent is obtained from the drift-diffusion (DD)model since we have assumed a constant electronmobility such that the drift is unbounded with re-spect to the modulus of the electric field. The sta-tionary energy-transport (ET) model is not ableto catch the capacitive effect at the junction.
0 20 40 60 80 100−6
−4
−2
0
2
4
6
8
10
12x 10
−3
time [ps]
curr
ent [
mA
]
ET transient
ET stationary
DD transient
Fig. 2. Current through a 0.1µm diode in a 10 GHzcircuit.
Second, we consider a laser and photo diodeas part of a small circuit (see Figure 3). The laserdiode is biased with a digital input signal withbias 3V and frequency 5GHz. The photo diodereceives the transmitted signal and is coupled toa high-pass filter only passing frequencies largerthan the cutoff frequency. The filter consists of
Uin
UoutRL
R1
R2
+
−
Fig. 3. Photo diode with a high-pass filter.
the photo diode, a capacitor, and three resistors.The photo diode is reversely biased with 0.2V.
The output of the high-pass filter is shownin Figure 4 for stationary and transient devicesimulations. The results for the high-pass outputvoltage differ significantly as the frequency is toolarge to be resolved by the stationary model.
0.2 0.4 0.6 0.80
0.5
1
1.5
2
time [ns]
Uou
t [V]
ET transientET stationaryDD transient
Fig. 4. Output of the high-pass filter for a 5 GHzdigital input signal with bias 3V.
Acknowledgement. The authors acknowledge partialsupport from the German Federal Ministry of Ed-ucation and Research (BMBF), grant 03JUNAVN,and from the FWF Wissenschaftskolleg “Differentialequations”.
References
1. M. Brunk and A. Jungel. Numerical couplingof electric circuit equations and energy-transportmodels for semiconductors. SIAM J. Sci. Comput.30 (2008), 873-894.
2. M. Brunk and A. Jungel. Simulation of ther-mal effects in optoelectronic devices using coupledenergy-transport and circuit models. To appear inMath. Models Meth. Appl. Sci., 2008.
3. A. Jungel. Quasi-hydrodynamic SemiconductorEquations. Birkhauser, Basel, 2001.
70
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CP 1B
Automatic Thermal Network Extraction and Multiscale
Electro-Thermal Simulation
Massimiliano Culpo1, Carlo de Falco2, Georg Denk3, and Steffen Voigtmann3
1 Bergische Universitat Wuppertal [email protected],2 Dublin City University [email protected] Qimonda AG Munich [email protected], [email protected]
Summary. A strategy to perform coupled electro-thermal simulations is presented. The electrical be-haviour of each circuit element is modeled by stan-dard compact models with an added temperaturenode. Mutual heating is accounted for by a 2-D or3-D diffusion reaction PDE which is coupled to theelectrical network by enforcing instantaneous energyconservation. A suitable spatial discretization schemeis adopted which allows efficient meshing of large do-mains with details at a much smaller scale.
1 Introduction
In this communication, we present a tool to au-tomatically extract a thermal network model fora CHIP level electro-thermal simulation. Startingfrom layout geometry and CHIP material data, itproduces an n-port thermal device model, possi-bly non-linear, that can be coupled to the electri-cal circuit network via an extra temperature nodein the electrical device compact models. Com-pared to other similar tools [2, 3], our tool doesnot work by fitting the parameters of a givennetwork of 0D thermal resistors and capacitorsbut rather it produces a full 2D or 3D discretiza-tion of the heat equation on the whole CHIP andtransforms it into a form which is amenable forcoupling to a standard circuit simulator. If thematerial properties are assumed to be indepen-dent from the temperature, the resulting modelis linear and the number of state variables maybe reduced by means of standard Model OrderReduction techniques.
2 Electro-Thermal Network Model
Below we briefly outline how the model for thethermal network is computed and how it is cou-pled to the rest of the system. We will refer tothe case of transient simulation, but the modelwe produce can be used in DC, AC and HarmonicBalance simulations as well. The global system ofequation describing our circuit is constructed fol-lowing the well known MNA approach:
Nel∑
i=1
(Jik(Reie, R
θi θ, ri
)+AEikri
)= 0 (1a)
fil(Reie, R
θi θ, ri
)+ AIil ri = 0 (1b)
where in (1a) k = 1 , . . . , Nnodes while in (1b)i = 1 , . . . , Nel ; l = 1 , . . . , Nintvar (i). Here thenumber of elements and nodes in the circuit aredenoted with Nel and Nnodes, respectively, whilethe number of internal variables describing the i-th element of the network will be Nintvar (i), e (t)represents the set of unknown voltages, θ (t) theset of unknown temperatures, ri (t) the i-th ele-ment internal variables. Furthermore Rei and Rθiare the electrical and thermal incidence matrix as-sociated with element i, respectively, and AEik andAIil are the coefficients of the differential term.The solution of (1) requires, at each time step,the resolution of a system of non-linear algebraicequations which is usually solved via Newton it-erations. At each step of the Newton method thefull jacobian matrix and the residual of the com-plete system are built assembling together the el-ement matrices Jik,Re
i e, Jik,Rθ
i θ, fil,Rei e
, fil,Rθ
i θ,
AEik and AIil and the element residuals Jik, fil.For electrical devices the latter represent essen-tially the currents and charges at the device con-tacts and the former the conductance and induc-tance and capacitance matrices. For the n−portthermal element, the local residual and matricesrepresent the net joule power in each device andits derivative with respect to each device temper-ature. The following sections describe how thesequantities are computed.
2.1 Thermal network model
For sake of simplicity in this section we assumethe distributed thermal network to be the elementnumbered as 1 in the circuit netlist. Let Ω be thewhole CHIP domain, Ωq the q-th active region onΩ, pq the Joule power per unit Area on Ωq and θkthe temperature at the k-th external pin. Denotewith J1k the power flux entering the device fromnode k and with T (x, t) the temperature at a
71
point x ∈ Ω at the time point t. If Ωq is connectedto pin k:
J1k =
∫
Ωq
pq dΩq = pq |Ωq|
while, at the external boundary:
J1k =
∫
∂Ω
T − θkR
dγ
where R is the CHIP to ambient thermal re-sistance. The above equations must be comple-mented by constitutive relations of the form:
f1q = θk −1
|Ωq|
∫
Ωq
T (x, t) dΩq = 0
for each active region Ωq, assuming θk to be thetemperature node connected to Ωq in the netlist.The temperature T is the solution of the followingPDE boundary value problem:
cv T + div (−κ (T )∇T ) − r = 0 in Ω
−κ (T )∂T
∂n=T − θext
Ron ∂Ω
(2)
where r = pq in Ωq and zero otherwise, θext is thetemperature of the pin attached to the externalboundary of Ω.
2.2 Space discretization
The method of Patches of Finite Elements (PFE)[1] allows us to discretize the heat equation (2)on a set of unstructured triangulations Thq eachcovering the subdomain Ωq that are allowed tocommunicate with each other via a coarser trian-gulation TH covering the whole CHIP domain Ω.By allowing Thq to completely overlap with THthis method allows to simulate characteristics ofequation (2) that appear at completely differentscales (CHIP scale/device scale) without impos-ing too many requirements on meshing softwareas the coarse mesh step H and the finer mesh steph can be completely unrelated.
Briefly the PFE method can be described asfollows. Let Vhq be the standard piecewise linearconforming finite element function space on thetriangulation Thq and VH the space of piecewiselinear conforming finite element functions definedon TH . The PFE discretization reads:find THh ∈ VHh such that
a(THh, vHh) = (f, vHh) ∀vHh ∈ VHh
with
VHh ≡(VH +
∑
q
Vhq
)
where a(·, ·) is the bilinear form representing theweak form of the differential operator in (2), (·, ·)is a scalar product in VHh and the known functionf accounts for the source term r as well as forthe boundary fluxes and the terms containing Tevaluated at previous times. Notice that the sumabove is not a direct sum, therefore forming abasis of VHh requires some extra care.
3 Workflow for Electro-thermalSimulation
Below we summarize the expected workflow us-ing the tool we presented for a coupled electro-thermal simulation:
• In designing the IC the thermally active re-gions are defined by adding an extra masklayer to the layout
• A 2D or 3D mesh of the IC layout is formedautomatically
• A passive thermal element is attached to thetemperature nodes of the circuit elements,ambient temperature is set via additional tem-perature sources
• The full system can be simulated by a stan-dard circuit simulator, producing as extra out-put the average temperature in each device aswell as the full multidimensional temperaturefield in the IC
4 Acknowledgments
The first author was supported by the EuropeanCommission in the framework of the CoMSONRTN project. The second author was supportedby the Mathematics Applications Consortium forScience and Industry in Ireland (MACSI) underthe Science Foundation Ireland (SFI) mathemat-ics initiative.
References
1. Roland Glowinski, Jiwen He, Jacques Rappaz, andJoel Wagner. Approximation of multi-scale ellipticproblems using patches of finite elements. C. R.Math. Acad. Sci. Paris, 337(10):679–684, 2003.
2. P.M. Igic, P.A. Mawby, M.S. Towers, andS. Batcup. Dynamic electro-thermal physicallybased compact models of the power devices for de-vice and circuit simulations. Semiconductor Ther-mal Measurement and Management, 2001. Sev-enteenth Annual IEEE Symposium, pages 35–42,2001.
3. V. Szekely, A. Poppe, A. Pahi, A. Csendes, G. Ha-jas, and M. Rencz. Electro-thermal and logi-thermal simulation of vlsi designs. VLSI Systems,IEEE Transactions on, 5(3):258–269, Sep 1997.
72
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CP 1C
Canonical Stochastic Electromagnetic Fields
B.L. Michielsen
ONERA, 2 avenue Edouard Belin, Toulouse, France [email protected]
Summary. The interaction between electromagneticfields and material systems is often subject to un-controlled fluctuations. As an alternative to tryingto model this by choosing a number of “representa-tive” cases, we propose methods to directly computestatistics from a stochastic field model. An impor-tant statistic is the standard deviation which dependson the covariance operator of the stochastic field. Inthis paper, we show the existence of stochastic fieldssuch that the covariance operator is natural in a sensemade precise below.
1 Introduction
In this paper, we are concerned with the prob-lem of uncertainty in the modelling of the in-teraction between electromagnetic fields and sys-tems. ¿From the physical point of view this in-teraction is represented for by “observables” onan incident electromagnetic field. If the incidentfield is not exactly controlled, deterministic mod-els are not the most appropriate. In particular, ifwe limit our simulations to some interaction con-figurations, which we think “representative,” wecan easily miss the effects of resonant coupling forparticular incidences. A more comprehensive wayto handle uncertainty is to model the interactionproblem as a problem to statistically characteriseobservables on stochastic fields.
In order to allow for an immediate interpreta-tion of our considerations, we briefly present theparticular case of the Thevenin model for a linearmulti-port system
V = V + Z ∗ I
ou V, I,V ∈ C(R,Rn) are Rn-valued functionsof time representing the port-voltages, the port-currents and the open-port voltages, or Theveninsources, respectively. The Thevenin sources aredefined through distributions on the fields (inte-gral representations). For example, V = 〈JP , E〉 =∫P E, where P is a curve defining a port. Now,
we are interested in the time covariance of suchobservable functions of time. Supposing that theaverage field is zero, we get in the general case(Xis the time reversal of X),
E(V (t1)V (t2)) = E(〈JP,t1 , E〉〈JP ;t2 , E〉)
Introducing CE , the covariance operator of thestochastic field E, we get
E(V (t1)V (t2)) = 〈JP,t1 , CE(JP,t2)〉
This expression motivates the search for covari-ance operators of stochastic fields. However, inthis paper we consider the inverse problem of find-ing a field such that the covariance operator is aknown and “natural” operator. This is a devel-opment analoguous to a time-harmonic analysispresented before (see [1]).
2 Basic field theory
The heart of our reasoning consists of a pair ofintegral relations between causal and anti-causalsolutions of the Maxwell equations
dH − ε0∂t ∗ E = ∗JdE + µ0∂t ∗H = 0
We can derive integral relations between two dif-ferent electromagnetic states, i.e., solutions ofMaxwell’s partial differential equations, by meansof Stokes’ theorem on a cylindrical domain D ×T ⊂ R3 × R.
∫
D×t1(µ0H
a ∧ ∗Hb + ε0Ea ∧ ∗Eb)
= −∫
D×T(Ea ∧ ∗Jb + Eb ∧ ∗Ja) (1)
where we used the vanishing of the integral over∂D × T for large enough D. Observe that whenone of the two states is in fact an anti-causal state,say Jc, Ec, Hc (we use the over-line to indicatetime-reversal), the domain of integration can bechosen such that also the contribution of the D×∂T integrals vanish (see Fig. 1). This leads to
0 =
∫
D×T(Ea ∧ ∗Jc + Ec ∧ ∗Ja) (2)
If we choose the time interval T = (t0, t1) suchthat supp(Ja) does not intersect D×T , we get athird integral relation
73
t
∂D
TJb
Ja
C+supp(Ja)
C−
supp(Jb)
D
Fig. 1. Illustration of a causal and an anti-causallight cone bundle
∫
D×t0(µ0H
a ∧ ∗Hc + ε0Ea ∧ ∗Ec)
= −∫
D×TEa ∧ ∗Jc (3)
Making the substitutions Jc = Jb (and thus Jc =
Jb) and Ec = E(Jb) = Eac(Jb) (the anti-causalsolution for a given Jb) we get for (3)
∫
D×t0(µ0H
a ∧ ∗Hac(Jb) + ε0Ea ∧ ∗Eac(Jb))
= −∫
D×TEa ∧ ∗Jb (4)
whereas adding (2), with the same substitutions,to (1) gives
∫
D×t1(µ0H
a ∧ ∗Hb + ε0Ea ∧ ∗Eb)
= −∫
D×TC(Jb) ∧ ∗Ja (5)
where C(J) = E(J) − Eac(J) is the electric fieldpropagator anti-symmetrised with respect to timereversal, i.e.,
C(J) = −C(J)
This operator C can be shown to define a naturalenergy emission correlation operator.
3 Observables on canonicalstochastic time-domain fields
We shall now consider the general problem of twoobservables, A and B, defined by electric cur-rent distributions, Ja and Jb respectively. We
consider the values of these observables on astochastic electromagnetic field. The covarianceof the stochastic values of these observables isthen given by
E(AB) = 〈Ja, CE(Jb)〉Comparing this with (5), leads to the idea to de-fine a stochastic field such that its covariance op-erator be proportional to the emission correlationoperator.
In order to achieve that, we define a directsum Hilbert space H , with scalar product
〈ψa, ψb〉H =
∫
R3
(ε0Ea ∧ ∗Eb + µ0H
a ∧ ∗Hb)
where E and H are the two components of ψ ∈H . On this space a probability measure exists ofwhich the covariance operator is proportional tothe metric. That means that a stochastic distri-bution exists ψ = (e0, h0) such that
∀f, g ∈ H E(〈ψ, f〉〈ψ, g〉) = σ2〈f, g〉HNow, using (4) we derive (explaining the detailsexceeds the limits of this summary),
∫
D×TCE0(J
a) ∧ ∗Jb
= σ2
∫
D×TC(Ja) ∧ ∗Jb (6)
where E0 is the electric field corresponding to thestochastic distribution on D × t0. Because thetwo distributions Ja and Jb can be chosen arbi-trarily, we obtain
CE0 = σ2C
So, σ2C is the electric field covariance operator of(E0, H0) the Maxwellian field with initial values(e0, h0) (the stochastic distribution defined by thespecific probability measure on H ).
4 Conclusion
In this proposal, we have outlined the electro-magnetic theory for the definition of a canonicalstochastic electromagnetic field with a known co-variance operator. In the final paper, we shall givethe mathematical details and show the computa-tion of the auto-covariance of induced Theveninsources in linear interconnect systems placed insuch a canonical stochastic field.
References
1. Bas Michielsen and Cecile Fiachetti. Covarianceoperators, green functions and canonical stochasticfields. Radio Science, 40(5):1–12, 2005.
74
Session
CM 1
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CM 1I
New trends in the preconditioning of integral equations of
electromagnetism
David P. Levadoux12
1 ONERA, Chemin de la Huniere, 91 761 Palaiseau, France [email protected] University Paris XI, Laboratory of Mathematics, 91405 Orsay, France
The pertinence of integral equation methods forsolving scattering Maxwell problems in harmonicregime requires no further proof. Using them incombination with a rapid multipole algorithmand an iterative solver makes them efficient andprecise methods that can solve problems involvinghundreds of thousands of unknowns. But the ef-ficiency of iterative methods depends on the con-ditioning of the linear systems, so it is absolutelycrucial to have either high-performance precon-ditioners or intrinsically well-conditioned integralformulations. It is in this field, a strategic onebecause necessary for solving very large calcu-lation configurations, that the electromagnetismand radar department of ONERA has been con-ducting fundamental research for the past fewyears [2].
The main research theme, begun in 1998, isbased on a generalized combined source integralequation (GCSIE) [4]. The master idea is to finda way to incorporate information on the solutionthat can be extracted beforehand in the integralformalism using the means offered by asymptotictheories (physical and/or mathematical). Thusthe GCSIE equation depends on the choice of anoperator whose purpose is to approximate the ad-mittance of the diffracting body as best as pos-sible. In the limit case where this approximationis exact, the integral operator to be inverted be-comes the identity. We thus try to construct ap-proximations of the admittance such that the op-erator underlying the equation appears as a math-ematically controllable perturbation (e.g. com-pact) of the identity. When the GCSIE is dis-cretized, we arrive at a linear system that is byessence well conditioned. From a theoretical view-point, this formalism is now well understood [1],as one of its foremost features is that it has noresonance frequency. As for experimental results,we point out for example that the Channel cav-ity (Fig. 1, right) modeling an aircraft air intakewas processed at 7 GHz (300,000 unknowns) withthe GCSIE in half as much computation time asis needed in production with a classical equation(Fig. 2).
A second research theme concerns the con-struction of non-standard preconditioners. Con-trary to the preconditioners commonly used inindustry, which are often algebraic and conse-quently leave out the mathematical propertiesof the equation from which the linear systemderives, we construct analytical preconditionersfrom the parametrix (pseudo-inverse) of the equa-tion. The result is new preconditioners, each ded-icated to a given equation. The preconditioner ofthe CFIE, for example, gives encouraging results.Furthermore, the theoretical consequences of thiswork have established an original proof of thenumerical convergence of the CFIE equation [3]which was still recently an open problem.
Fig. 1. Translucent view of the spherical cavity andChannel used for the tests.
References
1. F. Alouges, S. Borel, and D. P. Levadoux. A stablewell-conditioned integral equation for electromag-netism scattering. J. Comp. Appl. Math, 204:440–451, July 2007.
2. D. P. Levadoux. Stable integral equations forthe iterative solution of electromagnetic scatteringproblems. C. R. Physique, 7(5):518–532, 2006.
3. D. P. Levadoux. Some preconditioners for theCFIE equation of electromagnetism. Math. Meth.Appl. Sci., 2008, to appear.
4. D. P. Levadoux and B. L. Michielsen. An effi-cient high-frequency boundary integral equation.Antennas and Propagation Society InternationalSymposium, 2:1170–1173, Orlando, FL, USA, July11–16 1999.
77
0 20 40 60 80angle (degree)
0
1000
2000
3000
4000
5000
CP
U ti
me
(s)
CFIECFIE + prec
GCSIE
0 10 20 30 40angle (degree)
0
1000
2000
3000
4000
5000
CP
U ti
me
(s)
CFIECFIE + prec
GCSIE
Fig. 2. Solution times as a function of the angle ofpresentation of the target. Top: spherical cavity at 2.8GHz with 264,186 unknowns. Bottom: Channel at 7GHz with 309,711 unknowns.
78
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CM 1A
A New Approach to Modeling Multi-Port Scattering
Parameters
Sanda Lefteriu and Athanasios C. Antoulas
Rice University, 6100 Main Street, Houston, TX, 77005, USA [email protected], [email protected]
Summary. This paper addresses the issue of inter-polating frequency response data by means of a lin-ear dynamical system (macromodel). The proposedalgorithm adaptively chooses the measurements to beused in building the system by minimizing the errorin the singular values. Our method is suited for de-vices with a large number of ports due to the fact thatdata matrices are collapsed into vectors by taking ap-propriate linear combinations of the rows or columns.
1 Introduction and Motivation
The problem of building a system which approxi-mates given frequency response measurements ofa device is known as the rational interpolationproblem and has been studied thoroughly (see [1]for a survey). This is related to the constructionof state-space models of linear dynamical systemswhich are consistent with the data. Previous workincludes several optimization methods [2], as wellas the well-known vector fitting method [4].
Current applications in electromagnetic de-vices require a model of reasonable dimensionswhich approximates the response accurately inthe desired frequency range. In this paper wepresent a novel approach for generating stableand passive macromodels of devices with a largenumber of ports, for which only a set of frequency-domain measurements (S-parameters) are avail-able. Vector fitting and optimization-based meth-ods are prohibetively expensive for this kind ofdevices. The proposed algorithm is based on asystem-theoretic tool developped in [5]. It con-structs the model by adaptively choosing mea-surements from the set of available ones to min-imize the difference between the singular valuesof the data matrix and those of the transferfunction of the intermediate model. The perfor-mance is compared to the popular vector fitting(VF) method in terms of the computational timeneeded to generate a model and the accuracy ofthe fit. Our technique proves to be both fasterand more accurate.
2 Describing the Algorithm
The proposed algorithm constructs a macromodelof order k by building the state-space matrices
E, A ∈ Rk×k, B ∈ R
k×Np , C ∈ RNp×k, D =
0Np×Np such that the tranfer function H(s) =
C (sE− A)−1
B + D of the system, when evalu-ated at the Ns available frequency samples, ap-proximates the S-parameter matrix measured atthe same frequency ωi: H(jωi) ≈ Si ∈ C
Np×Np
Our approach is based on the theory of tan-gential interpolation. The key and novel toolsare the Loewner matrix together with the shiftedLoewner matrix associated with the data. For de-tails and proofs, we refer to [5].
We generate random vectors ri ∈ RNp×1, set
li = rTi ∈ R1×Np (where (·)T stands for transpo-
sition) and, for each sample jωi, we collapse theNp × Np S-parameter matrix to a vector (wi =
Siri and vi = liSi, where (·) stands for complexconjugation). This idea makes the method suit-able for devices with a large number of ports.
The right interpolation data is chosen as
Λ = diag [jω1, · · · , jωNs ] ∈ CNs×Ns , (1)
R = [r1, · · · , rNs ] ∈ RNp×Ns , (2)
W = [w1, · · · , wNs ] ∈ CNp×Ns , (3)
while the left interpolation data is constructed as
M = diag [−jω1, · · · , − jωNs ] ∈ CNs×Ns ,(4)
L = [l1, · · · , lNs ]T ∈ R
Ns×Np , (5)
V = [v1, · · · , vNs ]∗ ∈ C
Ns×Np . (6)
Afterwards, the Loewner and shifted Loewnermatrices are built as follows for m, n = 1, . . . , Ns
Lmn =vmrn − lmwn
µm − λn, (7)
σLmn =µmvmrn − λnlmwn
µm − λn(8)
In practice, the number of measurements Nsis much larger than the order of the minimal re-alization of the underlying system, so the pen-cil (σL,L) is singular. In the case of noise-freemeasurements, the underlying system is obtainedby projecting out the singular part, to obtain theregular part of the pencil. In our method, this sin-gularity is overcome by using only selected mea-surements chosen adaptively.
79
The algorithm starts by selecting Np measure-ments from the Ns available and constructing asystem of orderNp by building the matrices Λ,M ,R, L, W, V, L and σL as presented in (1)-(8) andsetting E as −L, A as −σL, B as V, C as W andD as 0. We assess the error between the currentinterpolant and the data by computing, at eachfrequency sample, the difference between the Npsingular values of the S-parameter data matricesand the Np singular values of the transfer func-tion of the intermediary model evaluated at thatparticular frequency. For each singular value, thenext measurement is chosen where the maximumerror occurs. At the next step, a system of order2 × Np is constructed and the error in the sin-gular values is computed again. The next set ofNp measurements will be picked according to thesame criterion and the procedure continues untilthe desired accuracy or the desired order of thereduced model has been reached.
3 Numerical Example
The data set was provided by CST Ltd and itcomes from a 14-port device. For each one of the1001 frequency samples in the range 6MHz to6GHz, a matrix of dimension 14×14 with complexentries is given.
There were two error measures employed: theH∞-norm of the error system defined as
H∞ error = maxi=1...Ns
maxk=1...Np
σ1 (Si − H(jωi))
and the H2-norm of the error system normalizedwith respect to the number of samples and thenumber of ports of the device as used in [3, 4]
H2 error =1
2π
√∆ω
NsN2p
‖Si − H(jωi)‖2F
where ∆ω = ωi+1 − ωi, for i = 1, . . . , Ns − 1(assuming equally spaced frequency samples).
Tables 1 and 2 compare the proposed algo-rithm with vector fitting. They show that ourmethod is faster and yields better approximantsfor the same order of the desired system.
Table 1. Results obtained with our method
Order Time (s) H∞ error H2 error
14· 3=42 2.04 5.7×10−2 7.81 ×10−5
14· 4=56 3.35 1.17 ×10−2 1.34 ×10−5
14· 5=70 5.39 1.04 ×10−2 8.1 ×10−6
14· 6=84 8.01 1.02 ×10−2 7.81 ×10−6
Table 2. Results obtained with VF
Order Time (s) H∞ error H2 error
14· 3=42 7.23 1.97 ×10−1 1.01 ×10−3
14· 4=56 8.92 7.96 ×10−2 3.23 ×10−4
14· 5=70 11.31 3.65 ×10−2 1.25 ×10−4
14· 6=84 13.51 2.04 ×10−2 4.89 ×10−5
Figures 1(a) and 1(b) present the response ofthe order k = 56 interpolating system obtainedwith the proposed algorithm and vector fitting,respectively. They support the errors presentedin the tables.
10−2
10−1
100
101
102
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
Frequency (rad/sec)
Mag
nitu
de(d
B)
Interpolating System obtained with our method
datamodel
(a) Order 56 system obtained with our method
10−2
10−1
100
101
102
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
Frequency (rad/sec)
Mag
nitu
de(d
B)
Interpolating system obtained with VF
datamodel
(b) Order 56 system obtained with vector fitting
Fig. 1. Comparison of our method and vector fittingagainst the data; plots have a logarithmic frequencyaxis scaled by 109.
References
1. A. C. Antoulas. Approximation of Large-Scale Dy-namical Systems. SIAM, Philadelphia, 2005.
2. C. P. Coelho, J. R. Phillips, and L. M. Silveira.Optimization based passive constrained fitting. InProc. IEEE/ACM International Conference onComputer-Aided Design(ICCAD’02), pages 775–780, 2002.
3. Ioan D. and Ciuprina G. Reduced order modelsof on-chip passive components and interconnects,workbench and test structures. In Worst H. v. d.and Schilders W., editors, Reduced Order Model-ing. Springer, 2008.
4. B. Gustavsen and A. Semlyen. Rational approxi-mation of frequency domain responses by vectorfitting. IEEE Transactions on Power Delivery,14:1052–1061, 1999.
5. A. J. Mayo and A. C. Antoulas. A framework forthe solution of the generalized realization problem.Linear Algebra and Its Applications, 405:634–662,2007.
80
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CM 1B
Numerical and Mathematical Properties of Electromagnetic
Surface Integral Equations
Pasi Yla-Oijala and Matti Taskinen
Department of Radio Science and Engineering, Helsinki University of Technology, [email protected], [email protected]
Summary. Properties of electromagnetic surface in-tegral operators, equations and formulations are in-vestigated. It is shown that the two fundamental nu-merical properties, accuracy of the solution and con-vergence of an iterative solver, seems to be almostopposite properties. Reasons and remedies for thisproblem are discussed.
1 Introduction
The surface integral equation method is an effi-cient numerical technique for solving time-harmonicelectromagnetic field problems. By using surfaceintegral representations the original boundary valueproblem for Maxwell’s equations can be trans-formed onto another equivalent problem of solv-ing equivalent sources from surface integral equa-tions. The method essentially reduces the dimen-sionality of the problem by one and can straight-forwardly deal with unbounded domains.
Because discretization of an integral equationleads to a dense system matrix, the traditionalsurface integral equation methods are limited torather small scale problems. Recently so calledfast methods [1] have been developed to signifi-cantly enlarge the size of the problems that canbe solved with integral equation methods. Manyof the fast integral equation solvers, like the mul-tilevel fast multipole algorithm (MLFMA), arebased on the iterative solution of the matrix equa-tion. In practical problems the bottleneck in thesimulations can be the poor convergence of theiterative solution.
Recently, new surface integral equation formu-lations giving rise to better conditioned matrixequations and more efficient iterative solutionshave been developed [2], [3]. However, many ofthese new formulations lead to lower solution ac-curacy. The fundamental problem with the elec-tromagnetic surface integral equations seems tobe that the solution accuracy and the convergenceof iterative methods are almost opposite proper-ties [4]. In this paper, reasons and remedies forthe this problem are discussed and the propertiesof the surface integral operators, equations andformulations are studied.
2 Surface Integral Equations
Consider time-harmonic (time-factor e−iωt) elec-tromagnetic scattering by a bounded and homo-geneous object D in a homogeneous medium. Letεo/i, µo/i denote the constant electromagnetic pa-rameters outside (o) and inside (i) the object.
For a given primary field Ep,Hp the task isto find the secondary fields Es
o/i,Hso/i so that
they satisfy Maxwell’s equations and the radia-tion conditions at the infinity and the total fieldssatisfy boundary conditions on the surface S.
Using the surface integral representations
E =−1
iωε
(∇∇ · +k2
)S(J) −∇× S(M) (1)
H =−1
iωµ
(∇∇ · +k2
)S(M) + ∇× S(J) (2)
the above-mentioned boundary value problem canbe reformulated as surface integral equations. HereJ = n ×H ,M = −n × E and S is the single-layer operator
S(F )(r) =
∫
S
G(r, r′)F (r′) dS(r′) (3)
with the homogeneous space Green’s function
G(r, r′) =eik|r−r
′|
4π|r − r′| (4)
and the wavenumber k = ω√εµ.
From (1) and (2) two different types of surfaceintegral equations can be derived:
1. Tangential T-equations are obtained by op-erating with −n× n× to (1) - (2) on S.
2. Rotated tangential N-equations are obtainedby operating with n× to (1) - (2) on S.
By properly combining these equations, derivedseparately for both the primary and secondaryfields, and inside and outside the object, and byapplying the boundary conditions on the surface,several alternative surface integral formulationscan be obtained.
81
3 Properties of Integral Operators
Consider a linear surface integral operator A
A(F )(r) =
∫
S
K(r, r′)F (r′) dS(r′). (5)
Here K is the kernel of A satisfying
|K(r, r′)| ≤ C
|r − r′|2−c (6)
with C > 0. Depending on the value of the coef-ficient c, K and A have different properties:
1. If c > 0 K is weakly singular and operator A
is a smoothing operator.2. If c ≤ 0 K is hyper-singular and operator A
is a non-smoothing operator.
For a mathematical point of view the nicest prop-erties has a Fredholm’s integral equation of thesecond kind, i.e., a compact operator plus theidentity operator. Because the derivatives in (1)and (2) increase the singularity of the kernel,in electromagnetics the surface integral operatorsare compact only in very rare cases.
4 Discretization and Results
In practical implementations the used numeri-cal methods (discretization) plays also an im-portant role in determining the properties of theequations. For example, with Galerkin’s methodand divergence conforming functions the hyper-singular integral operator ∇∇ ·S can be regular-ized and reduced onto a operator with a weaklysingular kernel.
Next the properties of the electromagneticsurface integral equations are studied by consid-ering planewave scattering from a metallic (per-fectly conducting, PEC) cube. The problem is for-mulated using a combined field integral equation(CFIE) defined as a combination of T electric fieldintegral equation (T-EFIE) and N magnetic fieldintegral equation (N-MFIE)
CFIE :=α
ηoT-EFIE + (1 − α)N-MFIE. (7)
Here 0 ≤ α ≤ 1 is a coupling coefficient andηo =
√µo/εo. Equation (7) is discretized with
Galerkin’s method using low order divergenceconforming vector functions defined on planar tri-angles. Figures 1 and 2 show the solution (backscat-tered radar cross section) and the number of GM-RES iterations with different discretization den-sities as functions of α.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 111
11.5
12
12.5
13
13.5
α
Back
scat
tere
d RC
S
m = 10m = 14m = 16
Fig. 1. Backscattered RCS for a λ×λ×λ PEC cube.m is the number of subdivisions on the edges.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
20
40
60
80
100
120
140
160
180
200
α
Num
ber o
f ite
ratio
ns
m = 10m = 14m = 16
Fig. 2. Number of GMRES iterations for a λ×λ×λPEC cube with stopping criterion 10−6.
The results show that the solution accuracyand convergence of the iterative solver behave al-most opposite as functions of α and mesh den-sity. CFIE (7) with α = 0 is an integral equa-tion of the second kind containing an operatorwith a hyper-singular kernel and the identity op-erator, whereas with α = 1, the equation is ofthe first kind containing the single-layer opera-tor, i.e., a smoothing operator, only. Hence, theexistence of a smoothing/non-smoothing opera-tor improves/decreases the solution accuracy, butdecreases/improves the iteration convergence.
References
1. W.C. Chew, J.-M. Jin, E. Michielssen and J. Song,Fast and Efficient Algorithms in ComputationalElectromagnetics. Artech House, Boston, 2001.
2. P. Yla-Oijala, M. Taskinen and S. Jarvenpaa,Surface integral equation formulations for solvingelectromagnetic scattering problems with iterativemethods, Radio Science, 40:RS6002, 2005.
3. M. Taskinen and P. Yla-Oijala, Current and chargeintegral equation formulation, IEEE Trans. An-tennas Propag., 54:58–67, 2006.
4. P. Yla-Oijala, M. Taskinen and S. Jarvenpaa,Analysis of surface integral equations in electro-magnetic scattering and radiation problems, Eng.Analysis Boundary Elements, 32:196–209, 2008.
82
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CM 1C
Nonlinear models for silicon semiconductors
Salvatore La Rosa1, Giovanni Mascali2, and Vittorio Romano1
1 Dipartimento di Matematica e Informatica, Universita di Catania, viale A. Doria 6, 95125 Catania, [email protected], [email protected]
2 Dipartimento di Matematica, Universita della Calabria and INFN-Gruppo c. Cosenza, 87036 Cosenza, [email protected]
Summary. Exact closures are obtained of the 8-moment and the 9-moment models for silicon semi-conductors based on the maximum entropy principle.The validity of these models is assessed by numericalsimulations of some test problems.
1 Introduction and mathematicalmodel
Modern electronic devices require increasingly ac-curate models of charge transport in semiconduc-tors in order to describe high-field phenomenasuch as hot electron propagation, impact ioniza-tion and heat generation.
Moreover, in many applications in optoelec-tronics, it is necessary to describe the transient in-teraction of electromagnetic radiation with carri-ers in complex semiconductor materials. In thosecases the characteristic times are of the order ofthe electron momentum or the energy flux relax-ation times. These are some of the main reasonsof the necessity of developing models which incor-porate a number of moments of the distributionfunction higher than those in the drift-diffusionand the energy transport models.
These extended models, generally called hy-drodynamical models, are usually derived fromthe infinite hierarchy of the moment equationsof the Boltzmann transport equation by suitabletruncation procedures. One of the most success-ful among these procedures is that based on theMaximum Entropy Principle (MEP), see [1] for acomplete review both for Si and GaAs semicon-ductors.
The models comprise the balance equations ofelectron density, energy density, average velocity,energy flux and possibly also of higher scalar andvector moments which do not have an immedi-ate physical interpretation. These equations arecoupled to the Poisson equation for the electricpotential. Apart from the Poisson equation, thesystem is hyperbolic in the physically relevant re-gion of the field variables.
The MEP approach leads to a constrained op-timization problem which is handled by resort-
ing to the Lagrangian multipliers method. Re-cently [2], this problem has been solved numeri-cally without resorting to asymptotic procedures.In this way the model is expressed in terms of theLagrangian multipliers and the constitutive rela-tions are given by integral expressions that do notallow an efficient numerical tabulation but requirethe use of suitable quadrature formulas in the halfspace.
In [2], we assessed the validity of the 8-momentmodel and we showed that the full nonlinear MEPmodel (hereafter NLMEP) is more accurate thanthe semi-linear MEP one (hereafter SLMEP).However, we found that for devices with a chan-nel length smaller than 0.3 microns in some re-gions of the devices there is a loss of integrabilityof the fully nonlinear MEP distribution function,see Figure 1. This probably happens because themoments cross the boundary of the region of re-alizability as already known in classical gas dy-namics. In this paper we tackle the problem todetermine how the region of realizability changesby adding a further scalar moment. The modelis given by the following system of balance equa-tions
∂n
∂t+∂(nV i)
∂xi= 0, (1)
∂(nV i)
∂t+∂(nU ij)
∂xj+ neEj H
ij = nCV i , (2)
∂(nW )
∂t+∂(nSi)
∂xi+ neViE
i = nCW , (3)
∂(nSi)
∂t+∂(nF ij)
∂xj+ neEjG
ij = nCSi . (4)
∂(nW2)
∂t+∂(nSi2)
∂xi+ 2neEi S
i = nCW2 , (5)
where e is the absolute value of the electroncharge and E the electric field. The macroscopicquantities, which are involved in the balance equa-tions, are related to the electron distribution func-tion f(x,k, t) by the definitions
n =
∫
R3
fdk, electron density,
V i =1
n
∫
R3
fvidk, average electron velocity,
83
W =1
n
∫
R3
E(k)fdk, average electron energy,
W2 =1
n
∫
R3
E2(k)fdk, average electron
energy square,
Si =1
n
∫
R3
fviE(k)dk, energy flux,
Si2 =1
n
∫
R3
fviE2(k)dk, flux of the electron
energy square,
U ij =1
n
∫
R3
fvivjdk, velocity flux,
Hij =1
n
∫
R3
1
~f∂
∂kj(vi)dk,
F ij =1
n
∫
R3
fvivjE(k)dk, flux of the energy
flux,
Gij =1
n
∫
R3
1
~f∂
∂kj(Evi)dk,
CV i =1
n
∫
R3
C[f ]vidk, velocity production,
CW =1
n
∫
R3
C[f ]E(k)dk, energy production,
CSi =1
n
∫
R3
C[f ]viE(k)dk, energy flux
production,
CW2 =1
n
∫
R3
C[f ]E2(k)dk, electron energy
square production.
The numerical simulations will give informationabout the integrability region. In particular if itis larger than that found in the 8-moment caseor vice versa. ¿From a theoretical point of viewthe question is rather involved. Indeed the addi-tional Lagrangian multipliers, associated to newmoments corresponding to weight functions rep-resented by powers of energy with an exponentgreater than one, are zero at equilibrium. There-fore the equilibrium states are located at theboundary of the realizability region. This impliesthat small perturbations can have both positiveand negative sign causing a loose of integrabil-ity, limiting the validity of the nonlinear models.Preliminary numerical experiments confirm sucha conjecture (see Figure 2).
Acknowledgement. The authors acknowledge the fi-nancial support by M.I.U.R. (PRIN 2007 Equazionicinetiche e idrodinamiche di sistemi collisionali ecomplessi), P.R.A. University of Catania and Univer-sity of Calabria (ex 60 %), CNR (grant n. 00.00128.ST74),the EU Marie Curie RTN project COMSON grant n.MRTN-CT-2005-019417.
0 0.1 0.2 0.3 0.4 0.50
5
10
15
20
25
30
35
40
micron
inte
grab
ility
con
ditio
n
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−5
0
5
10
15
20
25
30
35
40
45
micron
inte
grab
ility
con
ditio
n
Fig. 1. Devices with channel length 0.3µm and0.2µm respectively in the 8-moment case: integrabil-ity is guaranteed if the plotted function is greater thanzero (see [2])
0 0.1 0.2 0.3 0.4 0.5−40
−20
0
20
40
60
80
100
120
140
160
microm
inte
grab
ility
con
ditio
n
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−50
0
50
100
150
200
microm
inte
grab
ility
con
ditio
n
Fig. 2. Integrability in the devices with chan-nel length 0.3µm and 0.2µm respectively in the 9-moment case
References
1. A.M. Anile, G. Mascali and V. Romano, Re-cent developments in Hydrodynamical Modelingof semiconductors. In A. M. Anile, editor, Math-ematical Problems in Semiconductor Physics, Lec-ture Notes in Mathematics 1823. Springer, Berlin,2003.
2. S. La Rosa, G. Mascali, V. Romano. Exact Max-imum Entropy Closure of the HydrodynamicalModel for Si Semiconductors: the 8-Moment Casepreprint (2008).
84
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CM 1D
Hyperbolic PDAEs for semiconductor devices coupled with
circuits
Giuseppe Alı1, Giovanni Mascali1, and Roland Pulch2
1 Dipartimento di Matematica, Universita della Calabria and INFN-Gruppo c. Cosenza, 87036 Cosenza, [email protected], [email protected]
2 Department of Mathematics and Sciences, Chair of Applied Mathematics and Numerical Analysis,Bergische Universitat Wuppertal, Germany [email protected]
Summary. In this paper, we address the problem ofcoupling a system of network equations correspond-ing to an electric circuit and the partial differentialequations modelling a separate device.
1 Model of Electric Circuit
We consider an electric circuit, which, by adopt-ing the formalism of Modified Nodal Analysis [1]can be described by a system of differential alge-braic equations (DAEs) in the form
0 = ACddtqC(A⊤
Cu(t), t) + ARr(A⊤Ru(t), t)
+ ALL(t) + AV V (t) + AIı(t) + Aλλ(t),
0 = ddtφL(L(t), t) − A⊤
Lu(t),
0 = A⊤V u(t) − v(t).
(1)The unknown functions are the state variables y :=(u, L, V )⊤, where u comprises the node volt-ages, L the currents through inductances, andV the currents through voltage sources. The cur-rents λ represent coupling variables, since theyleave external devices and enter the electric net-work (or vice versa).
2 Refined Model for Devices
Complex elements like semiconductors in an elec-tric circuit may require sophisticated modelling,see [2] for a review. We apply a 1-dimensional hy-drodynamical model for semiconductor devices.For simplicity, we restrict to the case of purelynegative doped regions in the semiconductor. Com-pactly the system of equations can be written inthe form
∂U
∂t+∂F (U)
∂x= B(U , E), (2)
where
U =
nnVnWnS
, F (U) =
nVn( 1
m∗G1 − 2αG2)nSnG2
,
B(U , E)
=
0d11(W )nV + d12(W )nS + q
(2αG3 − 1
m∗
)nE
−nW−W 0
τ(W ) − qnV E
d21(W )nV + d22(W )nS − qnG3E
.
The respective variables and constants of the hy-drodynamical model are given in Table 1.
The system (2) is coupled with the Poissonequation for the electric potential
−ǫ∂2ψ
∂x2= q(N+(x) − n), E = −∂ψ
∂x. (3)
Table 1. Variables related to PDE system.
n number densityV average velocityW average energyS energy flux
Gijl fluxes
dij velocity and energy flux production coefficientsW 0 equilibrium energyτ energy relaxation timeψ electric potentialE electric fieldǫ dielectric constantq unit chargeN+ donor concentrationα non-parabolicity factorm∗ effective electron mass
The following initial values are considered
n(x, 0) = N+(x), W (x, 0) = W 0,
V (x, 0) = S(x, 0) = 0. (4)
The corresponding boundary conditions read
85
n(0, t) = N+(0), n(L, t) = N+(L),
W (0, t) = W (L, t) = W 0,
∂V
∂x(0, t) =
∂V
∂x(L, t) =
∂S
∂x(0, t) =
∂S
∂x(L, t) = 0
with device length L. The boundary conditionsfor the Poisson equation are determined by cou-pling the device to the system of DAEs, whichmodels the connected electric circuit.
3 Coupling Conditions
We consider a device with just two connections tothe surrounding circuit. Figure 1 illustrates thetopological relations. The incidence matrix Aλ
specifies the position of the device within the elec-tric network. In case of a single device, it holds
Aλλ = Aλ
(λlλr
)≡ ADI, (5)
with I := λl = −λr, and AD := Aλ
(1−1
). In
the case of just one device coupled to the elec-tric circuit, the matrix AD reduces to a columnvector.
The difference of the two boundary potentials,is given by
ψ(0, t) − ψ(L, t) =
(1−1
)⊤A⊤λu(t) = A⊤
Du(t).
(6)We consider a diode of type n+/n/n+ in the
following and apply the corresponding hyperbolicmodel (2) in one space dimension. Then we canderive the expression
I(t) =ǫA
LA⊤D
d
dtu(t) + J(t), (7)
where J(t) := − qA
L
∫ L
0
n(x, t)V (x, t) dx.
ul
λ l λ r
ur
Fig. 1. Coupling of circuit and device by node volt-ages ul, ur and branch currents λl, λr (l: left, r: right).
4 Numerical results
The coupled system of PDAE’s is numericallysolved in the case of a n+ − n − n+ diode cou-pled with a inductor, a capacitor and a resistor.
The results for the device voltage and current arerepresented in the following Figures
0 50 100 150 200 250 300 350 400 4500.92
0.94
0.96
0.98
1
1.02
1.04
1.06
1.08
1.1
t
pote
ntia
l
Fig. 2. Voltage (V) vs time (ps)
0 50 100 150 200 250 300 350 400 4500
1000
2000
3000
4000
5000
6000
7000
8000
t
curr
ent
Fig. 3. Current (A) vs time (ps)
Acknowledgement. The authors acknowledge the fi-nancial support by P.R.A. University of Calabria (ex60 %) and by the EU Marie Curie RTN project COM-SON grant n. MRTN-CT-2005-019417.
References
1. M. Gunther, U. Feldmann, E.J.W. ter Maten,Modeling and discretization of circuit problems.In W.H.A. Schilders, E.J.W. ter Maten, editors,Handbook of Numerical Analysis. Special VolumeNumerical Analysis of Electromagnetism. ElsevierNorth Holland, Amsterdam, 2005.
2. A.M. Anile, G. Mascali and V. Romano, Re-cent developments in Hydrodynamical Modelingof semiconductors. In A. M. Anile, editor, Math-ematical Problems in Semiconductor Physics, Lec-ture Notes in Mathematics 1823. Springer, Berlin,2003.
86
Session
P 2
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.1
Consistent Initialization for Coupled Circuit-Device
Simulation
Sascha Baumanns, Monica Selva, and Caren Tischendorf
Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 [email protected], [email protected], [email protected]
Summary. A coupled circuit-device simulation leadsto equation systems involving non-dynamic equationsresulting from Kirchhoff’s laws. Consequently, initialvalues can not be chosen arbitrarily – they have to beconsistent. Here, we present an opportunity to calcu-late consistent initial values for the numerical integra-tion based on a method of lines approach discretizingthe circuit-device system in space first.
1 Model Equations for a CoupledCircuit-Device Simulation
Nowadays circuits are present in all parts of ourlive. ¿From the practical point of view, the in-terest in circuit modeling is to replace as muchlaboratory testing as possible by numerical simu-lation in order to minimise costs.
1.1 The Circuit Equations
We focus here onto the modifified nodal analy-sis (MNA) for modeling the network equationssince it is used in most commercial circuit simula-tors. Furthermore, we restrict ourselves to circuitscontaining resistors, capacitors, inductors, inde-pendent voltage sources and independent currentsources as well as semiconductors modeled by thedevice equations described in Section 1.2. Thisleads to a differential-algebraic equation (DAE)which is an equation of the form (see e.g. [2])
ACd
dtqC(A⊤
Ce) +ARg(A⊤Re) +ALjL
+AV jV +ASjS +AIis(t) = 0 (1)
d
dtΦ(jL) −A⊤
Le = 0 (2)
A⊤V e− vs(t) = 0 (3)
Here, the unknowns are the node potentials e,the currents jL through inductors, the currentsjV through voltage sources and the currents jSthrough semiconductors.
1.2 The Device Equations
Companion models for semiconductor devices con-taind hundreds of parameters today and it is a
costly task to tune all parameters. On the otherhand the companion models reflect the physicalbehavior only up to a certain level. The idea wewant to follow here is to avoid companion mod-els and model the semiconductors directly by asystem of partial-differential equations (PDE). Astandard approach here is to use the drift diffu-sion model
−∇(εS∇ψ) = q(p− n+ C) onΩS(4)
−∇(εO∇ψ) = 0 on ΩO (5)
-∂tn+1
q∇Jn(n,p,ψ) = R(n,p) on ΩS (6)
∂tp+1
q∇Jp(n,p,ψ) = −R(n,p) on ΩS (7)
with
Jn(n,p,ψ) = qµn(UT∇n− n∇ψ)
Jp(n,p,ψ) = −qµn(UT∇p− p∇ψ)
Here, the unknowns are the electrostatic potentialψ, the electron concentration n and the hole con-centration p. The field ΩO describes the oxid areaof the semiconductor. Correspondingly, the areaof semiconducting material is denoted by ΩS .
The drift diffusion equations can be coupledwith the MNA equations and we receive a so-called partial differential algebraic equation sys-tem (PDAE). The coupling conditions are givenby the applied node potentials and the currentthrough the semiconductor:
-jSk=
∫
Γk
(Jn + Jp − εS∂t∇ψ)νdx, Γk ⊆ ΓDS(8)
-jSk=
∫
Γk
(−εS∂t∇ψ)νdx, Γk ⊆ ΓDO (9)
whereas ΓDO consists of all contact boundaries ofthe oxyd and ΓDS involves all contact boundariesof the semiconductor area (cf. [1]).
1.3 Space Discretized Circuit-DeviceSystem
After space discretisation of the drift diffusionequations (4)-(7) and the semiconductor current
89
equations (8)-(9) using a finite element method,the coupled system becomes a DAE of the form
0 = MAN,ANA⊤S e+ qS
+HANXAN,DN(ψDN +H⊤DNA
⊤S e)
+HANXAN,FNX−1FN,FN [qCφ −XFN,DNψDN ]
+ qHANXAN,FNX−1FN,FN
· [ZFN,FNS(PFNS −NFNS)
+ ZFN,DNS(pDNS − nDNS )] (10)
0 = jS +HAN [Jnφ + Jpφ] +d
dtqS (11)
0 = XFN,FNΨFN − qCφ
+ qZFN,FNS(NFNS − PFNS)
+ XFN,DN(ψDN +H⊤DNA
⊤S e)
+ qZFN,DNS(nDNS − pDNS) (12)
0 =d
dtYFNS ,FNSNFNS +
1
qJnθ +Rθ (13)
0 =d
dtYFNS ,FNSPFNS − 1
qJnθ +Rθ (14)
with Ψ , N and P are vectors representing thecoefficents in the linear combinations of the FEMbasis functions for ψ, n and p, respectively. Fur-themore, we used the indexes AN , FN , DN ,ANS , FNS , DNS to indicate that the corre-sponding matrices vectors have to be evaluatedat all nodes, all free nodes, all Dirichlet boundarynodes, the nodes in the semiconducting area, thefree nodes in the semiconducting area, and theDirichlet boundary nodes in the semiconductingarea, respectively.
For the coupled circuit-device simulation wehave to solve the DAE system resulting from (1)-(3) and (10)-(14) (cf. [4] for one-dimensional semi-conductor models).
2 Consistent Initial Values
Some components of the solution are determinedby constraints. For initial value problems, theseconstraints restrict the choice of initial values,since there is not a solution through every giveninitial value.
One of the difficult parts in solving DAEs nu-merically is to determine a consistent set of ini-tial conditions in order to start the integration.A set of initial conditions are called consistent ifthere exists a solution of the DAE passig throughthe conditions. The problem of computing initialvalues can formulate as follow: Given some userdefined initial values for the DAE not necessarilyconsistent. Modify them in such a way that thereexists a solution passing through them.
To calculate consistent initial values for theDAE system (1)-(3) and (10)-(14) we exploit its
special structure and obtain the following resultwhich is an extension of the results in [3] to the2D and 3D case.
Theorem 1. Let all capacitances, inductances andresistances in the network are passive. Then, theDAE system (1)-(3) and (10)-(14) has index 1if and only if the circuit does not involve cutsetsof current sources and inductances and loops ofvoltages sources, capacitances and semiconductorbranches.For a consistent initialization of this DAE sys-tem, initial values for the current through induc-tors jL, the capacitive branch voltages A⊤
Ce, thesemiconductor charges qS, the concentration ofelectrons N and the concentration of holes P canbe freely chosen. Every solution of the system
Q⊤CARg(A
⊤Re) +Q⊤
CALjL +Q⊤CAV jV
+Q⊤CASjS +Q⊤
CAI is(t) = 0,
(3), (10) and (12) with jL = jL0, A⊤Ce = A⊤
Ce0,qS = qS0, N = N0 and P = P0 provides a con-sistent initial value for the coupled system (1)-(3)and (10)-(14).
For index-2-DAEs or higher, so-called hiddenconstraints appear. Hence consistent values haveto be chosen in such a way that not only theexplicit equations of the DAE but additionallythese hidden constraints have to be fulfilled aswell. Thus a proper characterization of the hid-den constraints becomes necessary. The study ofthe index 2 case is subjected to ongoing research.
References
1. M. Bodestedt and C. Tischendorf. PDAE modelsof integrated circuits and index analysis. Math.Comput. Model. Dyn. Syst., 13(1):1–17, 2007.
2. D. Estevez Schwarz and C. Tischendorf. Struc-tural analysis of electric circuits and consequencesfor MNA. Int. J. Circ. Theor. Appl., 28:131–162,2000.
3. M. Selva. An index analysis from coupled circuitand device simulation. Scientific Computing inElectrical Engineering, Proceedings of the SCEE-2004 Conference held in Capo d’ Orlando, Italy,Springer Verlag, Berlin, pages 121–128, 2006.
4. M. Selva and C. Tischendorf. Numerical analysisof DAEs from coupled circuit and semiconductorsimulation. Appl. Num. Math., 53(2-4):471–488,2005.
90
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.2
Heating of semiconductor devices in electric circuits
Markus Brunk1 and Ansgar Jungel2
1 Norwegian University of Science and Technology, Department of Mathematical Sciences, N-7491Trondheim, Norway [email protected]
2 Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstr. 8-10,1040 Wien, Austria [email protected]
Summary. A coupled model of thermal network andsemiconductor model equations is proposed. The de-vice is modeled by the energy-transport equations andan heat flow equation for the temperature of the crys-tal lattice. An energy conserving source term for theheat-flow equation is presented. The thermal networkis described by a system of simplified heat flow equa-tions. Device and network are thermally coupled viathe lattice temperature and hot charge carrier effects.Numerical results for a pn diode and a frequency mul-tiplier are presented.
1 Introduction
In application, thermal modeling of electric cir-cuits has been taken into account by accompany-ing thermal networks consisting of one-dimensio-nal and lumped heat flow equations. Parasitic ef-fects and high frequencies in the circuits, however,require a more detailed consideration of thermaleffects in semiconductor devices.
We apply the energy-transport equations as(electric) semiconductor model to allow for localthermal effects in terms of the charge carrier tem-perature Tn. The (scaled) energy-transport equa-tions consist of conservation laws for the electrondensity n and electron energy density w,
nt − div Jn = −R(n, p),
wt−div Jw = −Jn ·∇V +Wn(n, Tn)−3
2TnR(n, p),
with constitutive relations for the particle currentdensity Jn and the energy current density Jwn :
Jn = µn
(∇n− n
Tn∇V
),
Jwn =3
2µn(∇(nTn) − n∇V
),
where R(n, p) denotes the generation-recombina-tion term, p is the hole density, which is computedfrom analogue equations, Wn(n, Tn) is the relaxa-tion-time term, and µn the (scaled) electron mo-bility. The equations are coupled to the Poissonequation for the electric potential V :
λ2∆V = n− p− C(x),
where λ is the scaled Debye length and C(x) thedoping concentration. The lattice temperature TLis modeled by the (scaled) heat flow equation
ρLcL∂tTL − div(κL∇TL) = H,
with material density ρL, heat capacity cL, heatconduction κL and an appropriate heat sourceH .
2 Heat source and thermal coupling
We present a heat source term that ensures energy-conservation for the coupled system of energy-transport and heat flow equation:
H = −Wn −Wp +R(Eg − TLE′g +
3
2(Tn + Tp))
+α(Jn · ∇[Ec − TLE′c] + Jp · ∇[Ev − TLE
′v]),
with scaling constant α, the band edges Ec andEv, and the energy gap Eg. Tp denotes the holetemperature, Wp the corresponding relaxation-time term for the holes and Jp denotes the holecurrent density. In order to take into account thethermal radiation to environment, for simulationsthe term SL(TL − Tenv) with the environmentaltemperature Tenv and the transmission functionSL = SL(x) has to be added to the source term.
Coupling of the device and thermal networkequations is taken into account in terms of bound-ary conditions for TL and the semiconductor heatflux, that enters the surrounding thermal net-work. A quasi stationary approach leads to the(scaled) semiconductor heat flux
JSth = β(−κL∇TL − [EcJn +EvJp]− Jwn + Jwp),
where Jwp denotes the energy flux density of theholes. β denotes a scaling constant, see [3]. Ther-mal interaction in terms of the lattice tempera-ture as well as hot carrier effects is considered.
The device model is described by partial dif-ferential equations. The thermal network equa-tions result in partial differential (algebraic) equa-tions - depending on the topology of the network.For thermal-electric simulation of circuits theseequations are coupled with the differential alge-braic equations resulting from MNA, thus thata coupled system of nonlinear partial differential-algebraic equations has to be solved, see [1, 3].
91
3 Numerical results
As numerical examples we consider a 100 nm pndiode with a maximum doping of 5 · 1023 m−3.For simplicity we model the positive (minority)charge carriers by the simpler drift-diffusion model.Moreover, assuming it to be a non-degenerate ho-mostructure device and neglecting the direct de-pendency of the energy gap on TL a simplifiedheat source term is used. For the contribution ofthe drift-diffusion model to the heat source weuse the widespread term −JpV . For radiation weassume a value of SL = 4 · 1015 W/(m3K).
Fig. 1. Lattice temperature in a 0.1µm pn diodebiased with 1.5 V.
Firstly, we bias the device with a bias of 1.5 Vstarting in thermal equilibrium. In Figure 1 thelattice temperature is depicted and we see that inconverges to the stationary state where the latticeheats up by 25 K.
0 0.5 1 1.5 2 2.50
1
2
3
4
5
6
7
8x 10
−4
applied voltage [V]
curr
ent [
A]
ET with var. TL
ET, TL = 300 K
DD, TL = 300K
Fig. 2. Current-voltage curve of a 0.1µm pn diodefor different models.
In Figure 2 we see the stationary computedIV-characteristic of the considered device. Wecompare the drift-diffusion model with the energy-transport model with and without lattice heat-ing. Obviously, the thermal effects influence theelectric behavior of the considered device. Finally
Vin
VoutR1 R2
C2L2C1
L1
Fig. 3. Frequency multiplier.
we simulate the frequency multiplier depicted inFigure 3 containing the considered device, wherethe parameters are chosen such that the reso-nance frequency is given at 3.2 GHz. The cir-cuit is biased with a sinusoidal signal of v(t) =3 · sin(3.2 ·2π ·109Hzt)V. The lattice temperaturein the device is depicted in Figure 4 and a strongdevice heating can be observed.
Fig. 4. Lattice temperature in a 0.1µm pn diode inthe frequency multiplier.
Acknowledgement. The authors acknowledge partialsupport from the German Federal Ministry of Ed-ucation and Research (BMBF), grant 03JUNAVN,and from the FWF Wissenschaftskolleg “Differentialequations”.
References
1. M. Brunk and A. Jungel. Numerical couplingof electric circuit equations and energy-transportmodels for semiconductors. SIAM J. Sci. Comput.30 (2008), 873-894.
2. M. Brunk and A. Jungel. Simulation of ther-mal effects in optoelectronic devices using coupledenergy-transport and circuit models. To appear inMath. Models Meth. Appl. Sci., 2008.
3. M. Brunk. Numerical coupling of thermal-electricnetwork models and energy-transport equa-tions including optoelectronic semiconductor de-vices. PhD-thesis. Johannes Gutenberg-university,Mainz, 2008.
92
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.3
Surrogate Modeling of Low Noise Amplifiers Based on
Transistor Level Simulations
Luciano De Tommasi1, Dirk Gorissen2, Jeroen Croon3, and Tom Dhaene2
1 University of Antwerp, Department of Mathematics and Computer Science, Middelheimlaan 1, B-2020Antwerp, Belgium and NXP Semiconductors, High Tech Campus 37, PostBox WY4-01, NL-5656AEEindhoven, Netherlands [email protected], [email protected]
2 Ghent University - IBBT, Depart. of Information Technology (INTEC), Gaston Crommenlaan 8 Bus 201,B-9050 Ghent, Belgium [email protected], [email protected]
3 NXP-TSMC Research Center, High Tech Campus 37, PostBox WY4-01, NL-5656AE Eindhoven,Netherlands [email protected]
Summary. The paper deals with the identificationof surrogate models of RF low noise amplifiers. Thebehavior of such circuit blocks can be evaluated viaaccurate transistor-level simulations. Since a transis-tor level simulation is quite a time-consuming process,the possibility of extensive trade-off investigations isoften limited during the design process.Surrogate modeling involves the construction of aglobal model based onto a certain number of transis-tor level simulations. It provides a good-enough ap-proximate description of the circuit block over thedesign space, but being much faster to evaluate thanan additional transistor level simulation.
1 Introduction
Accurate surrogate models for single RF andmicrowave devices have been already developed(e.g. using ANNs [1]). In this work, we do notmodel a single device like a MOSFET, but a com-plete (though simple) RF circuit block: a LowNoise Amplifier (LNA). Future work will includethe analysis of other circuit blocks (e.g. mixers,VCOs, ...).An LNA (see fig. 1) is the typical first stageof a receiver, having the main function of pro-viding the gain needed to suppress the noise ofsubsequent stages, such as a mixer. In additionit has to give negligible distortion to the sig-nal while adding as little noise as possible [2].The behavior of an LNA can be fully describedby means of the admittance and noise functions,which are evaluated via accurate transistor-levelsimulations. Such functions can be easily used todetermine the performance figures (gain, inputimpedance, noise figure and power consumption)used by designers [3]. Each simulation typicallyrequires one-two minutes, which is a too long timeto allow a designer to explore how performancefigures of LNA scale with key circuit-design pa-rameters, such as the dimensions of transistors,passive components, signal properties and bias
conditions. Therefore, the transistor level modelcan be usefully replaced with an accurate surro-gate model (based on transistor level simulations)which is much cheaper to evaluate.Our previous investigations [4], [5] exploited afirst order analytical model, which enabled to runa lot of different experiments with several differ-ent model types, being much cheaper to evalu-ate than a transistor level model. Such experi-ments have shown that rational functions, ANNsand Kriging models are the most promising modeltypes for the admittance and noise functions of anLNA.The goal of this work is the validation of the re-sults obtained with the simple analytical model,when a more accurate transistor-level model isused. In particular, we aim to study the accuracyof surrogate models as a function of the numberof design variables (inputs) and number of datasamples in the design space.
Fig. 1. A narrowband Low Noise Amplifier.
2 Experimental Setup
The surrogate modeling approach developed inthis paper, is based on the SUrrogate MOdeling(SUMO) Matlab Toolbox [4]. It is linked with the
93
Cadence SPECTRE simulator, which runs thetransistor level simulations, so providing the sam-ples of the LNA decribing functions.Models types considered in this work are Artifi-cial Neural Networks (ANNs) and Rational Func-tions (RFs). The ANNs implementation includedin the SUMO Toolbox is built on the MatlabNeural Network Toolbox. The network is trainedwith Bayesian regularization and its topology isdetermined by a Genetic Algorithm (GA). RFshave two different implementations to optimizethe complexity, one based on a custom stochastichill climber and another one based on a GA.The modeling loop starts with an initial Latin hy-percube design. Then, the adaptive sampling loopstarts. In order to improve the accuracy, new sam-ples are selected using one of the adaptive sam-pling algorithm available in the SUMO toolbox.Experience has showed that gradient sample se-lector [6] usually gives the best results. It per-forms a trade off between a density based sampleselection and error-based sample selection: boththe undersampled and non-linear regions are au-tomatically identified and sampled more densely.In order to judge the quality of the models, k-foldcrossvalidation is applied. The termination crite-ria can be chosen as follows: the modeling stopseither when a predefined maximum number sam-ples is reached or when the root relative squareerror is below a predefined threshold.
3 Results
The authors performed a preliminary analysis ofthe achievable accuracy as function of the numberof samples and the number of design parameters,based on a first order analytical model of the LNA(fig. 2) [7].
Fig. 2. Root relative square error as function of thenumber of samples and number of design parameters.Surrogate models were built from an approximate,analytical description of LNA.
Results shown in fig. 2 will be validated re-placing the approximate analytical model with
the one provided by transistor level simulator.Figure 3 shows a preliminary result based on tran-sistor level simulations: the model of the outputnoise figure, built with ANNs and 113 samples. Avery good agreement between the model and thesamples (crosses) is achieved.
Fig. 3. Surrogate model of output noise figure asfunction of device width W and inductance Ls builtusing SUMO Toolbox and SPECTRE simulations.
Acknowledgement. This work was supported in partby the European Commission through the Marie CurieActions of its Sixth Program under the contract num-ber MTKI-CT-2006-042477.
References
1. Q. J. Zhang and K. C. Gupta. Neural Networksfor RF and Microwave Design. Artech House Inc.,2000.
2. Thomas H. Lee. The Design of CMOS Radio-Frequency Integrated Circuits. Cambridge Univer-sity Press, 2004.
3. J.A. Croon, D.M.W. Leenaerts, D.B.M. Klaassen.Accurate Modeling of RF Circuit Blocks: Weakly-Nonlinear Narrowband LNAs. Proceedings ofthe IEEE Custom Integrated Circuits Conference(CICC), 2007.
4. D. Gorissen, L. De Tommasi, J. Croon and T.Dhaene. Automatic Model Type Selection withHeterogeneous Evolution: An application to RFcircuit block modeling. Proceedings of the IEEECongress on Evolutionary Computation (WCCI),2008.
5. D. Gorissen, L. De Tommasi, W. Hendrickx, J.Croon and T. Dhaene. RF circuit block modelingvia Kriging surrogates. Proceedings of the 17th In-ternational Conference on Microwaves, Radar andWireless Communications (MIKON), 2008.
6. Karel Crombecq. A gradient based approach toadaptive metamodeling. Technical Report, Univer-sity of Antwerp, 2007.
7. D. Gorissen, L. De Tommasi, K. Crombecq, T.Dhaene. Sequential Modeling of a Low Noise Am-plifier with Neural Networks and Active Learning.Technical Report, University of Antwerp, 2008.
94
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.4
On Local Handling of Inner Equations in Compact Models
Uwe Feldmann, Masataka Miyake, Takahiro Kajiwara, and Mitiko Miura-Mattausch
USDL-AdSM, Hiroshima-University,1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8530 [email protected], miyake-m053223,ancient-future,[email protected]
Summary. The burden of solving inner equations incompact semiconductor models is often shifted to thehost circuit simulator. Schur complement techniquesmay help to reduce the size of both the model stampand memory needs, and – at least for large circuits –overall CPU time. Some practical aspects of applyingthese concepts in compact modeling are discussed.
1 Introduction
State-of-the art models of semiconductor devicesin circuit simulation exhibit an increasing num-ber of implicit model internal equations, eitherby introducing intrinsic circuit nodes, or by us-ing auxiliary (nonlinear and/or differential) equa-tions for more accurate description of device be-havior. These equations are often exported to thehost simulator. This is easy to implement, in gen-eral, but also may give rise to a huge model stampand suffers from the overlinear complexity of thesparse solver in memory and CPU time, unlesssophisticated ordering strategies or hierarchicalsolver concepts are employed. Furthermore, ro-bustness and efficiency (parallel processing!) areto a large extent simulator dependent.
Our objective is to implement compact mod-els with local solver concepts, such that the deviceinternal equations are as far as possible hiddenfrom the host simulator. The benefits are obvious:Small model stamps, reduced simulator depen-dence, and possibly higher efficiency due to higherdegree of locality and parallelism. Of course, thereare also risks: Higher expense per model evalua-tion, convergence problems, and much higher ef-forts for model development. The first risk shouldbe overcompensated by better overall efficiency,at least for large circuits. Due to the second riskwe pursue only hierarchical versions of a singlelevel Newton method, with a special focus tomake sure that convergence is (almost) the sameas if the equations were exported to the host simu-lator. And finally, the third risk can be reduced bythe concept to perform the local handling in somekind of intermediate layer between the model it-self and the host simulator. The second and thethird item distinguish this approach from previ-ous attempts to apply local solver concepts formodel evaluation [2].
The techniques employed here are not new atall, cf. [1,3]. However, to our knowledge they havenot yet been used directly in compact model de-velopment. This contribution makes a first stepinto this direction. First, Schur complement tech-niques are shortly reviewed. Then some special as-pects of their usage in SPICE like simulators willbe discussed. A simple MOSFET model serves asan example. Finally, the current state and futurework will be described.
2 Review of Schur ComplementTechniques
We formulate the problem for the DC and (aftertime discretization) transient analysis: Solve thenonlinear coupled system
f i(xi,xm) = 0 (1)
fm(xi,xm) = 0 (2)
where index i denotes the model internal equa-tions and variables, and m denotes the outerequations from Modified Nodal Analysis and net-work variables. (For the ease of notation a circuitwith one single device is considered.) Applicationof a single level Newton method yields the linearsystem for the Newton corrections xi, xm:
(∂f i
∂xi
∂f i
∂xm∂fm
∂xi
∂fm
∂xm
)(xixm
)= −
(f ifm
)(3)
From the upper equation we get
xi = −(∂f i∂xi
)−1(f i +
∂f i∂xm
xm
)(4)
and plug into the lower equation yields:[∂fm∂xm
− ∂fm∂xi
(∂f i∂xi
)−1∂f i∂xm
]xm (5)
= −[fm − ∂fm
∂xi
(∂f i∂xi
)−1
f i
]
Equation (5) is the Schur equation which the hostsimulator has to solve. Once the latter has com-puted a solution xm, the Newton correction forthe inner system can be calculated from (4).
95
3 Aspects of Implementation inCompact Modeling
Before applying Schur’s techniques in compactmodeling, we have to take some specialities ofmany SPICE like simulators into account:
• The linear system is established directly forxnew = x + x rather than for x.
• Model evaluation is usually based on branchvoltages u rather than node voltages:f = f(u(x))
• Often a branch oriented limiting ulimiter→ u is
used, with f(u) ≈ f(u) + ∂f∂u
∣∣u=u
(u− u)
To this end a set of branch voltages
ui = ui(xi,xm) um = um(xi,xm) (6)
is introduced, such that
f i = f i(ui) fm = fm(um) (7)
Noting that the branch voltages are a linear com-bination of the node voltages with constant coeffi-cients, we get after some algebraic manipulations:
∂f i∂xi
xnewi = −
(f i(ui) −
∂f i∂ui
ui
)
︸ ︷︷ ︸feqi
− ∂f i∂xm
xnewm (8)
∂fm∂xm
− ∂fm∂xi
(∂f i∂xi
)−1∂f i∂xm︸ ︷︷ ︸
P
xnew
m (9)
= −(fm(um) − ∂fm
∂umum
)
︸ ︷︷ ︸feqm
+∂fm∂xi
(∂f i∂xi
)−1
f eqi
P is in case of convergence of the inner equa-tions just ∂xi
∂xm. For the implementation we resolve
(8) for xnewi , and introduce another intermedi-
ate quantity xeqi
def= −
(∂f i
∂xi
)−1
f eqi to get finally:
xnewi = xeq
i + Pxnewm (10)
(∂fm∂xm
+∂fm∂xi
P
)xnewm = −f eq
m − ∂fm∂xi
xeqi (11)
xeqi and P are stored over the Newton iterations
such that xnewi can be updated using (10). The
next steps are as usual: Calculation of ui, um,branch limiting, and model evaluation. Finally,the model stamp is computed from (11).
4 Example: A Simple MOS Model
When we consider the stamp for the simple MOS-FET model of Fig. 1 then
xi = (vdi vsi )T xm = (vd vg vs vb )T
fi =
(1/RD(vdi − vd) + Ids + qdi
1/RS(vsi − vs) − Ids + qsi
)(12)
fm =
1/RD(vd − vdi) + qfdqg − qfd − qfs
1/RS(vs − vsi) + qfsqb
(13)
where Ids = Ids(vdi − vsi, vg − vsi, vb − vsi),
Ids
b
RS RDqsi
qb
qdi
si di
CfdCfs
g
s d
gq
Fig. 1. MOSFET model with two internal nodes andouter fringing capacitances
qk = qk(vdi−vsi, vg−vsi, vb−vsi) k = di, g, si, b,qfd = Cfd · (vd − vg), and qfs = Cfs · (vs − vg). Thecalculus described above can now be applied aftertime discretization without any difficulties.
5 Current State and Future Work
The implementation of the high voltage MOSmodel HiSIM-LDMOS [4] with local handling ofthe inner equations is almost completed. If all par-asitic effects (selfheating, nonquasistatic behav-ior, gate and bulk resistance network) are acti-vated then the model has up to 10 internal equa-tions; however, not all of them are supported yet.From a comparison with the same model export-ing all equations to the host simulator, we expectto get information on how to setup the architec-ture of future compact models.
References
1. M. Gunther, U. Feldmann and E.J.W. ter Maten.Modelling and discretization of circuit problems.In W. Schilders and J. ter Maten, editors, Hand-book numerical analysis vol. XIII. Elsevier, Ams-terdam, 2005.
2. M. Braack, U. Feldmann and U. Wever. About lo-cal iteration for calculating transistor characteris-tics in circuit simulation. Proc. ITG Diskussion-sitzung ANALOG’94, Bremen, 165–170, 1994.
3. N.B.G. Rabbat, A.L. Sangiovanni-Vincentelli andH.Y. Hsieh. A multilevel Newton algorithm withmacromodeling and latency for the analysis oflarge-scale nonlinear circuits in the time domain.IEEE Trans. Circ. Syst. CAS, 26:733–741, 1979.
4. M. Yokomichi et al.. Laterally diffused metal oxidesemiconductor model for device and circuit opti-mization. Jpn. J. Appl. Phys., 47:2560-2563, 2008.
96
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.5
Transient Analysis of Nonlinear Circuits Based on Waves
Carlos Christoffersen
Department of Electrical Engineering, Lakehead University, 955 Oliver Road, Thunder Bay, Ontario P7B5E1, Canada [email protected],
Summary. A new approach for transient analysisof nonlinear circuits is presented. The circuit equa-tions are formulated as functions of waves in fictitioustransmission lines. The waves are calculated followinga procedure that resembles the actual signal propaga-tion in a circuit and is fully parallelizable. A stronglynonlinear circuit is used as a case study.
1 Introduction
Circuit-level simulation of complex systems is achallenging task in terms of memory and CPUtime. It is thus of great interest to find more effi-cient methods for circuit-level simulation. At thecore of nonlinear circuit analysis is the solution ofa system of nonlinear algebraic equations. Solvingthis system of equations using Newton methodrequires the decomposition of a large (normallysparse) matrix for each iteration. This matrix isparticularly large in the case of harmonic balance(HB) or techniques based on multiple time dimen-sions [1].
This paper presents a transient analysis ap-proach that requires only one matrix decomposi-tion for a given time step size. This approach wasinspired in the multiple reflections technique [2]and wave digital simulation of circuits [3,4]. Wavedigital filters [5] were developed to replace analogfilters with a digital structure. Several methodshave been proposed for transient simulation usingwaves but they have limitations handling nonlin-earities or do not scale well with the size of thecircuit. Felderhoff [3] proposed a convergent re-laxation method that can treat several nonlineardevices and is easily parallelizable. The methodproposed in this paper is also parallelizable andcan handle arbitrary static and dynamic nonlin-earities.
2 Formulation
Assume that the total number of ports of all de-vices in a given circuit is equal to n. For eachport, we associate a reference impedance Zj. Thevoltage and current at Port j can be expressed as
vj = v+j + v−j (1)
ij =v+j − v−jZj
, (2)
where v+j and v−j are the incident and reflected
waves at Port j as seen from the devices as shownin Fig. 1. The circuit topology defines the rela-
Devices
Topology
[S]
Zn
Z2
Z1v1
− +
v2
vn
−
−
...
v2
vn
+
+
v1
Fig. 1. Circuit partition
tionship between the vector of incident and re-flected waves, v+ and v−, respectively. Let Q andB be the full cut-set and loop-set matrices for agiven tree in the circuit. The vectors of all portcurrents (i) and port voltages (v) satisfy
Qi = 0 (3)
Bv = 0 . (4)
Combining (1) and (2) with (3) and (4) the fol-lowing equation is obtained:
[QG−B
]v+ =
[QGB
]v− ,
where G is a diagonal matrix that has the recip-rocal of the reference impedaces (Zj) in its diag-onal. Matrices Q, G and B are sparse and thusv+ can efficiently be obtained for large circuits.
The reference impedance at sources and lineardevices can be chosen such that there are no re-flections from the device back to the network [4].Nonlinear devices will cause reflections. In thispaper it is proposed to calculate these reflectionsusing Newton’s method. For example, supposethe current in a nonlinear device is given by
ij = f(vj) ,
97
with f() a nonlinear function. An error functionis defined as follows:
F (v+j ) = Zjf(v+
j + v−j ) − v+j + v−j = 0 .
This can easily be generalized for multi-port non-linear devices, both static and dynamic. More-over, the same idea can be applied to a completesubcircuit.
Using this formulation, only one large matrixdecomposition is necessary for the complete sim-ulation. This decomposition could eventually beeliminated if an automatic way to decompose thetopology in adaptors is developed. The compu-tation performed at each iteration is completelyparallelizable: propagation of waves through thetopology is essentially a matrix-vector productand the Newton method for each device is inde-pendent of the rest of the circuit. This iterativeprocess resembles the actual propagation of sig-nals in a physical circuit.
3 Case Study
The circuit shown in Fig. 2 was simulated to testthe approach proposed in this paper. The cir-cuit parameters are: C = 4 µF, RS = 50 Ω,RL = 5 kΩ. The source is sinusoidal with a peakof 3 V and a frequency of 500 Hz. The diode pa-rameters are IS = 1 fA, N = 1, Cj = 100 nF,Mj = 0.5, Vj = 1 V and FC = 0.5. The refer-ence impedances were chosen to avoid reflectionswhere possible and a value of 50 Ω was used forthe diode ports. A transient simulation with a du-
C
CC
RL
Rs
Vs
Fig. 2. Nonlinear circuit
ration of 8 ms and a time step equal to 50 µs wasperformed. The simulation results are comparedwith a Spice simulation in Fig. 3.
The convergence of this method as describedhere is linear, but it can be made superlinear us-ing Steffensen-like updates. An average of 30 iter-ations per sample point were necessary to achievea tolerance of 10−8. The number of iterations wasat least 11 and at most 42 for each sample point.
Several issues must still be addressed. Themost important is a rigorous convergence anal-ysis. Another issue is that the choice of reference
0
1
2
3
4
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008
Out
put V
olta
ge (
V)
Time (s)
Wave AnalysisSpice
Fig. 3. Comparison of voltages at load resistor
impedance in each nonlinear port affects the con-vergence rate, but currently there is no methodto predict the optimum value.
The Spice simulation is significantly faster inthis example, but the proposed method was notoptimized for speed. Many improvements are stillpossible to reduce the computational cost of eachiteration. If a comparable performance can beachieved for small circuits, then due to its parallelnature the proposed method may become moreefficient for very large circuits or when appliedto analyses where several samples/harmonics arecalculated at once such as in HB or techniquesbased on multiple time dimensions [1].
Acknowledgement. This work was funded by the Nat-ural Sciences and Engineering Research Council ofCanada (NSERC).
References
1. J. Roychowdhury. Analyzing circuits with widelyseparated time scales using numerical PDE meth-ods. IEEE Trans. on Circuits and Systems—I:Fundamental Theory and Applications, Vol. 48,No. 5, May 2001, pp. 578–594.
2. A. R. Kerr. A technique for determining the localoscillator waveforms in a microwave mixer. IEEETrans. on Microwave Theory and Techniques, Oc-tober 1974, pp. 828–831.
3. T. Felderhoff. Jacobi’s method for massive paral-lel wave digital filter algorithm. Proc. of the IEEEConference on Acoustics, Speech and Signal Pro-cessing, vol. 3, May 1996, pp. 1621–1624.
4. A. Fiedler and H. Grotstollen. Simulation ofpower electronic circuits with principles used inwave digital filters. IEEE Trans. on Indutry Ap-plications, Vol. 33, No 1, Jan./Feb. 1997, pp. 49–57.
5. A. Fettweis. Wave Digital Filters: Theory andPractice. IEEE Proceedings, Vol 74, Feb. 1986,pp. 270–327.
98
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.6
Time-Domain Preconditioner for Harmonic Balance Jacobians
Mikko Honkala1 and Ville Karanko2
1 Helsinki University of Technology, Faculty of Electronics, Communications and Automation, Department ofRadio Science and Engineering, P.O.Box 3000, FI-02015 TKK, Finland [email protected]
2 AWR-APLAC Corporation, P.O. Box 284, FI-02600 Espoo, Finland [email protected]
Summary. Preconditioning is crucial for the conver-gence of GMRES iteration in the harmonic balanceanalysis of RF and microwave circuits. This paperpresents a variant of the time-domain preconditionerfor harmonic balance Jacobians. The efficiency of thepreconditioner is demonstrated with realistic simula-tion examples.
1 Introduction
The harmonic balance (HB) [1] method is a well-established analysis method for RF and microwavecircuits. It is a frequency-domain technique forperiodic and quasi-periodic steady-state analysis.
In HB analysis, the circuit equations are rep-resented in terms of Fourier coefficients. This re-sults in a system of nonlinear algebraic equations,which are usually solved using the inexact New-ton method, i.e., the Newton–Raphson methodwith an iterative linear solver. The typical choicefor the linear solver is GMRES.
The efficiency of a linear iterative solver de-pends heavily on the preconditioner used. Natu-rally, most preconditioners for HB equations arein the frequency domain, but for highly nonlinearcircuits, especially for frequency dividers, time-domain preconditioners [2–4], which take nonlin-ear behavior better into account, become attrac-tive.
This paper proposes a variant of the time-domain preconditioner.
2 Harmonic Balance
The nodal harmonic balance equations are
F(u) = I(u) + jΩQ(u) + U, (1)
where u is the nodal voltage vector, I is the non-linear nodal current vector, Ω is the frequency do-main differentiation matrix, and U is the vectorof excitation currents. The nonlinear equation (1)is typically solved using iterative methods like in-exact Newton with a preconditioned linear solverGMRES.
The inexact-Newton iteration requires a Jaco-bian matrix
J = jΩC + G, (2)
where C and G are the nodal capacitance andconductance matrices, respectively.
In time domain, the Jacobian (no oversam-pling) is
J = Γ−1JΓ = DC + G, (3)
where
G := Γ−1GΓ =
g1 0 0 . . . 00 g2 0 . . . 00 0 g3 . . . 0
. . .
0 0 0 . . . gn
, (4)
and, similarly,
C := Γ−1CΓ, (5)
where gk, ck are block matrices and the differenceoperator D = Γ−1jΩΓ is a matrix having generalform
D =
0 α1I α2I . . . α−1Iα−1I 0 α1I . . . α−2Iα−2I α−1I 0 . . . α−3I
. . .
α1I α2I . . . α−1I 0
. (6)
The matrix I is the (m ×m) identity matrix (mis the number of nodes in the circuit or in oneblock), and coefficients αk are the weights of time-domain difference operator.
Since, for strongly nonlinear circuits the resis-tive nonlinearities are dominant, it is tempting toapproximate the equations by considering themin this form and further approximating the differ-ence operator D by some typical finite difference.
3 Time-domain preconditioners
Time-domain preconditioners for single-tone HBJacobians have the form
99
K = Γ(DC + G)Γ−1. (7)
As stated above, the main problem when con-structing a time-domain preconditioner is how toefficiently approximate the difference matrix. Oneof the simplest and the most frequently used, butnot most efficient, way is to use backward-Euler(BE) differentiation. Then, the difference matrixis
D =
I∆t1
0 0 . . . −I∆t1−I
∆t2I∆t2
0 . . . 0
0 −I∆t3
I∆t3
. . . 0
. . .. . .
0 0 . . . −I∆tn
I∆tn
. (8)
Since there is a nonzero block in the upper-rightcorner, the matrix is not cheaply invertible and,therefore, this block is ignored. Then, the equa-tion (DC + G)x = b can be solved efficiently ina block-wise manner.
The disadvantage of ignoring the upper-rightblock can be diminished, e.g., by iteration asshown in Ref. [4]. It, however, increases the costand there is no guarantee that the iteration con-verges.
The variant of the time-domain preconditionerproposed in this paper compensates the abscenceof the upper-right block by applying forwar-Eulerintegration at the first time point. Then, D is
D =
−I∆t1
I∆t1
0 . . . 0−I∆t2
I∆t2
0 . . . 0
0 −I∆t3
I∆t3
. . . 0
. . .. . .
0 0 . . . −I∆tn
I∆tn
. (9)
4 Simulation results
The preconditioners were implemented in the in-house development version of the APLAC circuitsimulator. The tests were run on several relevantindustrial circuits with an oversampling factor of2. The circuit details are presented in Table 1.
Table 1. Number of nodes, harmonics used, resis-tors, capacitors, inductances, BJTs, MOSFETs, anddiodes of example circuits.
Circ. nodes harmonics R C L BJT Mos. D.
1 272 16 72 36 - - 288 -2 9 16 3 1 - 1 - -3 31 64 2 - - 1 12 -4 80 8 14 1 3 8 - -
The simulation results with different precon-ditioners are presented in Table 2. FD, TD1, and
TD2 stands for the standard frequency-domainblock Jacobi preconditioner, the BE precondi-tioner, and the time-domain preconditioner pro-posed, respectively. A dash indicates that the HBsimulation, i.e., inexact Newton iteration, did notconverge.
Table 2. Number of GMRES iterations NGMRES,number of HB iterations NHB and simulation times twith different preconditioners.
c. Prec. NGMRES NHB t/s
1 FD 4721 59 195TD1 637 20 46TD2 - - -
2 FD 2451 23 0.51TD1 297 23 0.14TD2 196 23 0.10
3 FD 1376 49 19.1TD1 177 29 7.3TD2 - - -
4 FD 1462 26 4.6TD1 1528 26 6.5TD2 695 20 3.1
5 Conclusion
A variant of a time-domain preconditioner thattries to diminish the lack of periodicy in the BEpreconditioner was proposed. In some cases, itworked fine, but unfotunately, in some cases not.
References
1. Kenneth S. Kundert, Jacob K. White, and Al-berto Sangiovanni-Vincentelli. Steady-State Meth-ods for Simulating Analog and Microwave Circuits.Kluwer Academic Publishers, Boston, 1990.
2. D. Long, R. Melville, K. Ashby, and B. Horton.Full-chip harmonic balance. In Proceedings of theIEEE 1996 Custom Integrated Circuits Confer-ence, pages 379–382, May 1997.
3. O. Nastov. Spectral methods for Circuit Analysis.PhD thesis, MIT, EECS, 1999.
4. Fabrice Veerse. Efficient iterative time precondi-tioners for harmonic balance RF circuit simula-tion. In Proceedings of the 2003 IEEE/ACM Inter-national Conference on Computer-Aided Design,pages 251–254, Washington, DC, USA, 2003. IEEEComputer Society.
100
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.7
Evaluation of Oscillator Phase Transfer Functions
M.M. Gourary1, S.G. Rusakov1, S.L. Ulyanov1, M.M. Zharov1, B.J. Mulvaney2, and K.K. Gullapalli2
1 IPPM, Russian Academy of Sciences, Moscow, Russia [email protected] Freescale Semiconductor Inc., Austin, Texas, USA [email protected]
Summary. A general expression for the phase trans-fer functions of an oscillator for frequencies close toharmonics of oscillator fundamental is derived. Thenumerical testing and comparison with some knownresults are performed.
1 Introduction
In this paper we obtain the phase transfer func-tions of an oscillator through the analysis of asymp-totic behavior of Linear Periodically Time-Varying(LPTV) solution of oscillator circuit. The analysisis performed in frames of Harmonic Balance (HB)technique. Below we present basic expressions ofLPTV HB analysis.
The LPTV analysis is performed after steady-state solution of an oscillator is obtained. The so-lution vector X involves nodal harmonics of os-cillator fundamental ω0 representing coefficientsof Fourier series of periodic waveforms x(t). Vec-tor X consists of components Xkl, where k, l, areharmonic and nodal indexes respectively.
The quasi-periodic small signal model of oscil-lator is similar to the cyclostationary HB systemfor forced circuits [1]
J(∆ω) ·∆X = B (1)
Here a component Bkl of rhs vector B repre-sents harmonic signal with frequency kω0 + ∆ωapplied to the lth circuit node. The vector ∆Xdefines the small signal solution, and J(∆ω) is aconversion matrix for the given frequency offset∆ω
J(∆ω) = G+ j(Ω +∆ω · E)C (2)
Here G, C are block Toeplitz matrices of har-monics of nodal conductances and capacitances,Ω is a block-diagonal matrix of harmonic fre-quencies Ω = diag[. . . ,−kω0, . . . , 0, . . . , kω0, . . .].The matrix (2) at zero offset coincides with theHB Jacobian matrix at the steady-state solutionJ(0) = J0 = G+ jΩ ·C. So (2) can be written as
J(∆ω) = J0 + j∆ω · C (3)
The Jacobian matrix J0 is singular, and so thereexists the eigenvector U corresponding to zero
eigenvalue and the eigenvector V of the trans-posed Jacobian matrix such that
J0 · U = 0, JT0 · V = V T · J0 = 0 (4)
The eigenvector U is a frequency domain repre-sentation of time derivatives of the large signaloscillator solution dx/dt. The eigenvector V is afrequency domain representation of the pertur-bation projection vector (PPV) introduced in [2].The traditional normalization of V is defined asfollows
V TCU = 1 (5)
2 Evaluation of Transfer Function
To obtain the transfer function it is needed toconsider only one component in rhs vector. Sowe assume that the vector B contains only onenonzero component corresponding to harmonic kand node l
B = e(kl) (6)
where unit vector e(kl) selects kl component ofrhs.
To analyze the solution of (1, 6) at small off-set we transform (1, 6) into the equivalent linearsystem with nonsingular matrix by the methodproposed in [3]. First we form a new equation byleft multiplying (1) by PPV. Taking into account(3, 4, 6) we obtain
j∆ωV TC∆X = Vkl or V TC∆X =Vklj∆ω
(7)
Then we replace the equation (1) correspondingwith the only nonzero component of the excita-tion (6) by the obtained equation (7). To simplifynotations we assume the equation to be replacedis the last one in (1). Thus we obtain the trans-formed linear system
J∆X =Vklj∆ω
e(kl), (8)
where J(∆ω) =
[J
V TC
]. Here J is matrix J
without the last row.
101
Note that in general case the matrix J(∆ω) isnot singular with ∆ω = 0. So at small offset onecan neglect the matrix dependence on the offset
J(∆ω) = J0 +O(∆ω) ≈ J0 =
[J0
V TC
](9)
So the solution of (8) can be represented as
∆X(∆ω) =Vklj∆ω
J−10 · e(kl) (10)
From the definition of U (4) and J0 (9) it can be
concluded that the matrix-vector product J0 · Ucontains only one nonzero component correspond-ing with replaced row
J0 · U = V TCU · e(kl) = e(kl) (11)
The last equality in (11) follows from (5). Thus
J−10 · e(kl) = U , and (10) can be written as
∆X(∆ω) =Vklj∆ω
U (12)
The component∆Xmn(∆ω) of the vector (12) de-fines the magnitude of the harmonic mω0+∆ω atnth node. So the time domain waveform at noden is defined as the sum of all harmonics at thenode
∆xn(t) =∑
m
∆Xmn(∆ω) exp j(mω0 +∆ω)t
= exp j∆ωt ·∑
m
∆Xmn(∆ω) exp jmω0t (13)
After substituting (12) into (13) we obtain
∆xn(t) =
„
Vkl
j∆ωexp j∆ωt
«
·X
m
Umn exp jmω0t (14)
It has been pointed out that Unm are harmon-ics of large signal time derivatives. So the sum in(14) represents Fourier series of dxn/dt. Thus
∆xn(t) =dxndt
· Vklj∆ω
exp j∆ωt (15)
This expression gives the voltage waveform at nthnode resulted from the unit excitation (6). To ob-tain the phase transfer factor we can consider thewaveform due to the phase modulation of oscil-lator solution x(t). Supposing the magnitude ofthe phase modulation Φ to be small we can applyfirst order Taylor expansion
xn (t+Φ
ω0exp j∆ωt
)= xn(t)
+dxndt
· Φω0
exp j∆ωt
or ∆xn(t) =dxndt
· Φω0
exp j∆ωt (16)
From comparison of (15) with (16) the phasemagnitude Φ can be determined. Its value is equalto the transfer factor Hφ
kl from the sideband fre-quency of kth harmonic of the excitation at lthnode to the baseband frequency of the phase
Hφkl(∆ω) = Φ =
ω0
j∆ωVkl (17)
The evaluation of the transfer function Hφ(∆ω)can be easily implemented in a circuit simulator.PPV harmonics V can be obtained by the algo-rithm [4].
3 Validation
At low input frequency (ω ≪ ω0) the transferfactor is defined by (17) with k = 0. It corre-sponds with the traditional VCO representationas an ideal integrator in PLL macromodels [5]φ = KVCOνinp/jω, where KVCO is the DC sen-sitivity of oscillation frequency. The sensitivityanalysis of the steady-state solution confirms thatKVCO = ∂ω0/∂νinp = ω0V0,inp.
Phase noise analysis can be done using (17).When unmodulated white noise source is attachedto lth node the phase PSD is evaluated by
Sφ = SX
k
˛
˛
˛Hφ
kl(∆ω)˛
˛
˛
2
=
Sω2o
X
k
|Vkl|2
!
/∆ω2 (18)
where S is the input PSD. This expression corre-sponds with time domain expressions in [4]. Nu-merical experiments performed by SPICE simu-lations also confirm the correctness of (17).
References
1. V. Rizzoli, F. Mastri, and D. Masotti. Generalnoise analysis of nonlinear microwave circuits bythe piecewise harmonic balance technique. IEEETrans. Microwave Theory Tech., 42:807–819, May1994.
2. A. Demir, A. Mehrotra, and J. Roychowdhury.Phase Noise in Oscillators: A Unifying Theory andNumerical Methods for Characterization. IEEETrans. on Circuits and Systems - I, 47:655–674,May 2000.
3. M.M. Gourary, S.G. Rusakov, S.L. Ulyanov, M.M.Zharov, B.J. Mulvaney, and K.K. Gullapalli. NewNumerical Technique for Cyclostationary NoiseAnalysis of Oscillators. Proc. of the 37th EuropeanMicrowave Conference, Munich, 2007, pages 1173–1176.
4. A. Demir, D. Long, and J. Roychowdhury.Computing Phase Noise Eigenfunctions Directlyfrom Steady-State Jacobian Matrices. Int. Conf.Computer-Aided Design, 283–288, Nov. 2000.
5. Ken Kundert. Predicting the Phase Noise and Jit-ter of PLL-Based Frequency Synthesizers.www.designers-guide.org/Analysis/PLLnoise+jitter.pdf
102
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.8
Quasiperiodic steady-state analysis of electronic circuits by a
spline basis
Hans Georg Brachtendorf1, Angelika Bunse-Gerstner2, Barbara Lang2, and Siegmar Lampe2
1 University of Applied Science of Upper Austria [email protected],2 University of Bremen, Germany [email protected]
Summary. Multitone Harmonic Balance (HB) is wid-ely used for the simulation of the quasiperiodic steady-state of RF circuits. HB is based on a Fourier expan-sion of the waveforms. Unfortunately, trigonometricpolynomials often exhibit poor convergence proper-ties when the signals are not quasi-sinusoidal, whichleads to a prohibitive run-time even for small circuits.Moreover, the approximation of sharp transients leadsto the well-known Gibbs phenomenon, which cannotbe reduced by an increase of the number of Fouriercoefficients.
In this paper we present alternative approachesbased on cubic or exponential splines for a (quasi-) pe-riodic steady state analysis. Furthermore, it is shownbelow that the amount of coding afford is negligibleif an implementation of HB exists.
1 Summary
Electronic circuits lead to a system of generallynonlinear differential-algebraic equations (DAEs)of first order of dimension N
f(v(t), t) = i(v(t)) +d
dtq(v(t)) + b(t) = 0 (1)
v : R → RN is the vector of the unknown nodevoltages and branch currents. q : RN → RN isthe vector of charges and magnetic fluxes, i :RN → RN the vector of sums of currents enter-ing each node and branch voltages. Furthermoreb(t) : R → RN is the vector of input sources.
Circuit designers are often interested in thesteady-state of the circuit, driven by a multi-tonesignal. For clarity, we restrict below to the 2-tonecase
b(t) =
∞∑
k1=−∞
∞∑
k2=−∞Bs(k1, k2)·e(jk1ω1t)e(jk2ω2t) (2)
In [1–3] it has been shown that a reformu-lation of the underlying ordinary DAE (1) intoan appropriate partial DAE system eases the nu-merical treatment of the multitone problem. Thismethod has been widely accepted by different re-search groups, i.e. [4–7].
Theorem:Consider the system of ordinary differential-algebraic equations (1) with quasi-periodicstimulus (2) and the partial DAE system
f(v(t1, t2)) = i(v(t1, t2)) +∂
∂t1q(v(t1, t2))
+∂
∂t2q(v(t1, t2)) + b(t1, t2) = 0 (3)
where the quasiperiodic stimulus b(t1, t2) isdefined by
b(t1, t2) :=
∞∑
k1=−∞
∞∑
k2=−∞B(k1, k2) · e(jk1ω1t1) e(jk2ω2t2)
with Fourier coefficients B(k1, k2).Then
v(t) =
∞∑
k1=−∞
∞∑
k2=−∞V (k1, k2) · e(jk1ω1+jk2ω2)t
is a steady-state solution of the ordinary differ-ential-algebraic equation (1), if and only if
v(t1, t2) =
∞∑
k1=−∞
∞∑
k2=−∞V (k1, k2)·e(jk1ω1t1) e(jk2ω2t2)
(4)is a steady-state solution of the partial differ-ential-algebraic equation as well. The two so-lutions are related by v(t) = v(t, t) for allt ∈ R and the relation V (k1, k2) = V (k1, k2)holds.
The theorem states, that a solution of the un-derlying ordinary DAE can be obtained along acharacteristic of the partial DAE.
Consider the two-tone signal v(t) depicted infig. 1. Due to Nyquist’s theorem, the time-stepfor a numerical solution of the underlying par-tial DAE should be at least twice the highest fre-quency of the waveform. However, the simulationinterval on the other hand should be reciprocal tothe lowest frequency of interest. Hence, the run-time depends highly on the ratios of the frequen-cies involved. In practical applications, the enve-lope is in the kHz or MHz range and the carrierfrequency is often several GHz or higher. Simpletransient simulation is therefore prohibitive.
Alternatively, consider the corresponding mul-titone signal v(t1, t2) shown in fig. 2 with peri-odic boundary conditions along the t1, t2 axes.The orginal signal v(t) is regained along the spe-cial characteristic t1 = t2 = t, i.e. v(t) = v(t, t).
103
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10−6
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t=s
x(t)V
Fig. 1. Quasiperiodic signal example.
00.2
0.40.6
0.81
x 10−6
0
1
2
3
4
x 10−8
−1
−0.5
0
0.5
1
t1=st2=sx(t1;t2)V
Fig. 2. Multitone signal example.
Please note the different scalings of the axes. Thestiffness of the underlying ordinary DAE is castedinto these different time-scales. This allows fora discretization with different grid-spacings andtherefore a more appropriate discretization. Thepartial DAE formulation circumvents thereforethe limitations of Nyquist’s theorem.
Another way to consider the method is in thefrequency domain as shown in fig. 3. The wave-form v(t) has a broad but very sparse spectrumwhich is clustered along integer multiples of thecarrier f0. On the other hand, a reordering leadsto the artifical two-dimensional spectrum illus-trated in fig. 4, which is now a compact repre-sentation of the signal.
The spectrum of the waveform v(t1, t2) isclustered and can therefore be compactly repre-sented using a 2-dimensional Fourier transform.
The Fourier expansion used by the HarmonicBalance method (HB) of the unknown waveforms(4) is not appropriate for waveforms with sharpedges, because the approximation leads to thewell-known Gibb’s phenomenom. An approxima-tion of the waveforms by a cubic or exponen-tial spline basis is often better suited due to thefinite support of the basis functions. We showthat the coding of the spline approximation intoan HB simulator is simple. Moreover, designersare mainly interested in Fourier coefficients and
jY (f)j
f2f1 3f1 f0 f1 f0 f0 + f1 3f02f0 f1 2f0 + f12f0f1Fig. 3. Signal spectrum of v(t).
fxfy
Fig. 4. Artifical spectrum of v(t1, t2).
not in the coefficients of the spline basis. We de-rive further a simple map from the spline to theFourier basis.
References
1. H. G. Brachtendorf. Simulation des eingeschwun-genen Verhaltens elektronischer Schaltungen.Shaker, Aachen, 1994.
2. H. G. Brachtendorf. On the relation of certainclasses of ordinary differential algebraic equa-tions with partial differential algebraic equa-tions. Technical Report 1131G0-791114-19TM,Bell-Laboratories, 1997.
3. H. G. Brachtendorf, G. Welsch, R. Laur, andA. Bunse-Gerstner. Numerical steady state anal-ysis of electronic circuits driven by multi-tone sig-nals. Electronic Engineering, 79(2):103–112, April1996.
4. R. Pulch and M. Gunther. A method of character-istics for solving multirate partial differential equa-tions in radio frequency application. Preprint 07,IWRMM, Universitat Karlsruhe, 2000.
5. J. Roychowdhury. Efficient methods for simulatinghighly nonlinear multi-rate circuits. In Proc. IEEEDesign Automation Conf., pages 269–274, 1997.
6. J. Roychowdhury. Analyzing circuits with widelyseparated time scales using numerical pde meth-ods. IEEE Transactions on Circuits and SystemsI - Fundamental Theory and Applications, 48:578–594, May 2001.
7. F. Constantinescu, M. Nitescu, F. Enache, 2D timedomain analysis of nonlinear circuits using pseudo-envelope initialization., Proceedings of the 2-nd In-ternational Conference on Circuits and Systemsfor Communication, Moscow, 2004.
104
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.9
Accurate Simulation of the Devil’s Staircase of an
Injection-Locked Frequency Divider
Tao Xu and Marissa Condon
Research Institute for Networks and Communications Engineering (RINCE), School of ElectronicEngineering, Dublin City University, Dublin 9, Ireland [email protected], [email protected]
Summary. The Devil’s Staircase of an Injection-Locked Frequency Divider (ILFD) is simulated in anovel and efficient manner in this contribution. In par-ticular, the method of Multi-Phase Conditions-BasedEnvelope Following (MPCENV) is utilised. The lock-ing range of the ILFD is then determined from theDevil’s Staircase. The proposed method is applied toan LC oscillator based ILFD and the results are val-idated by comparison with experimental results.
1 Introduction
In general, Injection-Locked Frequency Dividers(ILFD) are used in the negative feedback of a fre-quency synthesizer as a prescaler to divide thefrequency by a fixed number. In comparison withthe traditional static [1] frequency dividers, theILFDs consume less power but at the expense of anarrow locking range. Hence, the accurate deter-mination of the locking range is important in thedesign of ILFDs. The Devil’s Staircase [2] was in-troduced as an experimental method to measurethe locking range. However, simulation would bea cheaper and preferable means of determiningthe locking range.
Since the ILFD is an oscillator with an in-jected signal, it can take thousands of cycles tolock the oscillator. Hence, traditional transientanalysis takes far too long to be of use. However,the envelope-following methods provide an effec-tive alternative. Here, we use the Multi-PhaseConditions-Based Envelope Following [3] (MP-CENV) method to determine the locking rangein an efficient manner.
2 Determination of the LockingRange
2.1 MPCENV Approach
The basis for envelope following methods is asfollows:
The circuit solution is assumed to be com-posed of fast oscillations whose amplitude and
frequency varies much more slowly than the oscil-lations themselves. Let the period of the fast os-cillation be T . In the case of oscillators, this willvary slowly. Let Tenv be the envelope timestepover which the response of the system can be ex-trapolated.
x(t+Tenv+T)
t=t t0 t1
x0
Xstart
Fig. 1. Backward-Euler-based envelope-followingmethod
As can seen in Fig. 1, let x0 be the state att0 = t+ Tenv. Now x(t+ Tenv + T ) is determinedfrom x0 using an accurate simulation of one pe-riod T . Using the Implicit Euler Method for sta-bility purposes, the envelope following process isdescribed by
x(t+ Tenv + T ) − x0
T=x0 −Xstart
Tenv. (1)
Since Tenv and T are also unknowns two fur-ther equations are required. They are shown asbellow:
x0l = c (2)
x(t+ Tenv + T )l = c (3)
where l denotes the lth state variable and c is aconstant. In our implementation, c is obtained bya short traditional transient simulation.
2.2 Devil’s Staircase
The Devil’ Staircase [2] is a plot of finj/f0 againstfinj. Figure 2 shows the experimental results for
105
1 1.5 2 2.5 3
x 106
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
finj
f inj/f
o
Staircase with Amplitude of Vinj
= 1 v
Fig. 2. The staircase obtained from experiment.
the ILFD. For simulation purposes, the injectedfrequency, finj is varied and f0 is determined fromthe MPCENV as:
f0 =1
T. (4)
3 Case Study and NumericalResults
The LC oscillator-based ILFD is selected as anexample. The schematic is shown as bellow.
R
M1 M2
C
L
M3 M4
M6
CINJVINJ
M5
IBIAS RS
VDD
Fig. 3. Circuit schematic
The governing equations are:
CdVCdt
= IL − (A+ daVinj)VC +A+ daVinj
V 2DD
V 3C ,
(5)
LdILdt
= −ILR− VC , (6)
where A and da are the coefficients obtained fromthe negative resistance [4].
The result obtained from the proposed methodis shown in Fig. 4. Both the simulation and ex-perimental results concur that the ILFD is lockedwhen finj/f0 is an even number [4] i.e., 2 and 4as shown in Fig. 2 and 4. Table 1 shows the lock-ing ranges captured from the staircases. The error
between the simulation and experimental resultsis less than 5%.
1 1.5 2 2.5 3
x 106
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
finj
f inj/f
o
Staircase with Amplitude Vinj
= 1 V
Fig. 4. The staircase obtained from simulation.
finj/f0=2 finj/f0=4
Vinj S (MHz) E (MHz) S (MHz) E (MHz)
1V 0.1 0.099 0.18 0.165
1.5V 0.14 0.146 0.235 0.241Table 1. Locking range captured from staircases,where S and E represent Simulation and Exper-iment
4 Conclusions
A technique based on the Multi-Phase Conditions-Based Envelope Following (MPCENV) algorithmhas been proposed for the determination of thelocking range of ILFDs. The simulation techniqueis advantageous for design and analytical work asit is far less costly than experimental determi-nation. Results for an LC-oscillator based ILFDconfirm its efficacy.
References
1. U. Singh, and M. Green, “Dynamics of high-freuqncy CMOS dividers”, In Proc. IEEE ISCAS,vol.5, pp.421–424, May 2002.
2. D. O’Neill, D. Bourke and M. P. Kennedy,“The devil’s staircase as a method of comparinginjection-locked frequency divider topologies,” InProc. ECCTD, vol.3, pp.317–320, Sept. 2005.
3. T. Mei and J. Roychowdhury, “A time-domainoscillator envelope tracking algorithm employ-ing dual phase conditions”, IEEE Trans. onComputer-Aided Design of Integrated Circuits andSystems, Vol. 27, No. 1, January 2008.
4. T. Xu, Z. Ye, and M.P. Kennedy, “MathematicalAnalysis of Injection-Locked Frequency Dividers ”,In Proc. NOLTA, pp.639-642, Sept. 2006.
106
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.10
Nonlinear distortion in differential circuits with single-ended
and balanced drive
Timo Rahkonen
University of Oulu, Finland [email protected]
Summary. Nowadays most of the high-resolutionanalog components have a fully balanced structure,i.e. they have differential input and output signals.This improves the tolerance against common modeinterference and doubles the dynamic range, but alsobrings some new problems. The circuit now needsimpedance match and stability analysis of both thedifferential and common mode operation, and alsomode conversions from common mode to differentialand vice versa may be of importance.
In circuit simulations the use of differential excita-tion is not a problem, but measuring instruments lagbehind (at least if you do not buy new equipment): of-ten the devices are characterised by driving the circuitfrom one port at a time with single-ended signals, andthe differential or common mode behaviour is solvedmathematically. Hence it is important to know whenyou can rely on single-ended measurements, and whenthese will give a different result compared to real bal-anced drive.
Circuit theory includes a lot of tricks used tostudy balanced circuits. For RF circuits, the formal-ism was presented by Bockelman and Eisenstadt [1].They assumed that the balanced circuit(let us assumeit is a fully differential amplifier, though the techniquecan handle any N-port with a mixture of single-endedand balanced signals) is measured by single-endedmeasurements as a linear 4-port, and presented thelinear matrix transforms needed to calculate differ-ential and common mode drives. This technique iscalled mixed-mode presentation, and it converts a 4-port s-parameter matrix to four 2-port matrices, onedescribing the truly differential operation, one com-mon mode behaviour, and two remaining ones themode conversions between these two. The latter areimportant for example when studying the rejection ofeven order distortion or other common mode interfer-ence.
Bockelman presented his formulation to s param-eters, but the idea can naturally be extended toany linear parameters (see [2]), and normal designmethods can be used - for example, one can pickthe common-mode only presentation, and calculatethe stability circles for the common mode match-ing impedances. The above needs only some matrixalgebra, but due to simulator support it is usuallyavoided by simulating the device so that the de-vice to be simulated is embedded between two idealbaluns (balanced-to-unbalanced transformers) and itis driven successively by common mode and differen-tial signals. In more expensive simulations, however,
it would be helpful to have a sufficient signal pro-cessing functions that would enable the designer todrive the circuit with a single test signal containingboth modes, and based on this simulation, track theirmode changes inside the circuit.
The previous analysis assumes superposition to bevalid and obviously does not operate in non-linear cir-cuits. As the single-ended driving technique is largelyused, however, it is valuable to know what exactlyhappens to the nonlinear distortion, when we drivethe circuit with single-ended and balanced signals.For this analysis, we need a tool capable for plottingdifferent distortion mechanisms separately.
Heiskanen et al. [3] have written a standard ACVolterra analysis software so that it stores all the dif-ferent contributions of distortion in vector form - dis-tortion from each nonlinear device and mixing mech-anism can be plotted separately, and their phase re-lation is immediately visualised. The tool is writtenso that it calculates the nonlinear spectrum by a con-volution of lower-order spectra, and this is performedso that mixing from different harmonic bands is cal-culated and stored separately. This makes it possibleto track how much third-order distortion is actuallygenerated by down or up-conversion from DC or 2ndharmonic bands. This feature is employed in the fol-lowing analysis.
The purpose of this paper is the following: a fullybalanced amplifier is driven both with a balanced anda single-ended test signal, and the differences in thegenerated distortion are recognised and explained. Toillustrate it, the term-wise Volterra analysis softwaredescribed in [3] is used.
The mechanism to be illustrated is the following.It is well known that a common-mode signal easilygenerates even-order distortion. This can be seen e.g.in 1, that models a bipolar differential pair, where I0 isthe bias current and Ro is the ouput impedance of thecurrent source. In balanced mode v1 = −v2 and thebias current remains constant. However, with single-ended drive the common mode variation also modu-lates the bias current, causing additional even-orderdistortion, that is normally not seen in the tanh() re-sponse. The generated 2nd-order distortion will thenbe coupled back to the input via the DC feedbackof the amplifier, and mix to the fundamental band,generating 3rd-order distortion. Hence, with single-ended drive a portion of the compression/expansionbehaviour is caused by even-order distortion, thatnormalli is very small in a device that is driven withbalanced input signal.
107
i(t) = (I0 +v1 + v2Ro
) · tanh(v1 − v2Vt
) (1)
The contribution of this paper is to illustrate theabove phenomenon using the Volterra analysis soft-ware [3]. It can plot all the 2nd and 3rd-order dis-tortion components in a vector form, with magnitudeand phase shown. This helps in understanding whichphenomena are compressing, which expansive, and italso capable of illustrating the numerous cancellingeffects inside a balanced circuit.
References
1. D. Bockelman, W. Eisenstadt. Combined differen-tial and common mode scattering parameters: the-ory and simulation. IEEE Trans. on MicrowaveTheo. and Tech.., , vol. 43, no 7, July 1995, pp.1530-1539.
2. T. Rahkonen, J. Kortekangas. Mixed-mode pa-rameter analysis of fully differential circuits. IEEEInt’l Symposium on Circ. and Syst., Vancouver,Canada, May 23-26 2004, vol. 1, pp. 269-272.
3. A. Heiskanen, T. Rahkonen. 5th-order electro-thermal multi-tone Volterra simulator with com-ponent level output. IEEE Int’l Symposium onCirc. and Syst., Bangkok, Thailand, May 25-282003, vol. 4, pp. 612-615.
108
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.11
Speed-up techniques for time-domain system simulations
Timo Rahkonen
University of Oulu, Finland [email protected]
Summary. Often, hardware description languageslike Verilog-A or VHDL-AMS are used to build be-havioral analog models and hence speed up the sim-ulations of systems that contain a lot of DSP, longdata sequences, fandew few analog parts. However,these models still rely on standard numerical tran-sient analysis, that is slow in applications, that forsome reason require much smaller time step than thesample duration.
This paper studies several situations, that arecomputationally expensive, and shows some ways tospeed up the long transient simulations. Especially inthe following applications, we need to simulate longsequences of data to catch some statistical propertiesof the signal, while very high oversampling is neededto model the non-ideal effects to be studied:
• Settling errors of the switched-capacitor (SC) in-tegrators in A/D converters
• Small timing errors and slew rate errors (i.e.,glitches) in the output of transmitter D/A con-verters
• Behaviour of modulated, linearly assisted switch-mode power supplies
The basic analysis techniques suited for the aboveapplications have reported before, for example in [1], [2] and [3]. In this paper, the two of these techniqueswill be slightly expanded, verified more thoroughly,and applied for a more complex system.
Vset
VPA
RFin RFout
+VDD
Fig. 1. Structure of an envelope tracking (ET) RFpower amplifier.
Envelope tracking (see Fig. 1) is a popular tech-nique to enhance the power efficiency of an RF trans-mitter. Here the supply voltage of the RF power am-plifier is modulated by combination of a switch-mode
DC/DC downconverter, and an assisting linear am-plifier. The input signal VSET is sampled, but the lin-ear settling within the sample period, high slew-ratesof the signals controlling the switches, as well as theasychronous control of the DC/DC converter force usto use a time step much smaller than the samplingperiod. This results in long simulation times. In [1]the simulation speed was enhanced by modeling thelinear settling within one sample period by a linearstate model of the switcher/amplifir combination
sX = A ·X +B · U
Y = C ·X +D · U. (1)
This traditional state model has the following gen-eral solution
x(t) = φ(t) · x(0) +
Z t
0
φ(t− τ )Bu(τ )dτ (2)
where x(0) is the initial state vector and
φ(t) = expm(At)
= I +At+A2t2
2!+ · · · (3)
Assuming that the input signals U are stepwiseconstant (output of a D/A converter, for example),(2) can be simplified into form
x(t) = φ(t) · x(0) +A−1 · (φ(t) − I) ·B · U. (4)
This is readily solvable at any time point with nor-mal matrix operations, without iteration or interme-diate time points. Hence, this approach can be usedto analyse the linear settling of filters, switchers andother piecewise linear circuits without oversampling,but at the sample rate of the piece-wise constant inputsignal. One practical problem related to this approachis the formulation of the state matrix presentation.
Sometimes the response of the switched systemsis not linear, and this is typically the case in SC am-plifiers and integrators. Now the linear solution ( 4)can not be used. Several nonlinear behavioural modelshave been proposed, but they typically are simplified,for example excluding the effects of output relatednonlinearities. However, the nonlinearity is determin-istic, and can be pre-characterised for a given set ofinput signals. Such an approach is used in [3] to makea realistic but fast model of the first integrator in asigma-delta ADC.
109
∆t
x0
∆t = x0/SR
x0
x0 x0
Fig. 2. Skew and slew rate errors in the output of aD/A converter
Yet another very expensive phenomenon to modelis the small timing skew in a segmented D/A structure(illustrated in Fig. 2): here a portion of the outputgets delayed by some picoseconds, resulting spuriousspectral responses. Even more difficult to model is afinite slew rate in the output, because one needs avery high oversampling ratio to catch the shape ofthe slew-rate phenomenon. In [2] standard Fourieranalysis was used to show that correct spectral re-sponse is actually achieved by using a lumped errormodel, where the area of the error is simply integratedand summed up to the correct value. Below are thelumped models of a timing skew with amplitude x0
and duration of ∆t, and slew-rate error with ampli-tude x0 and duration of ∆t.
eskew = x0 ·∆t
eSR =x0 ·∆t
2(5)
In this paper, the above analysis ideas will bestudied a bit further by building up a simulation offollowing test cases:
• Comparison of lumped skew and slew rate errormodel and a precise linear settling result.
• System simulation of transmitter consisting of twotime-interleaved D/A converters and a linear re-construction filter.
• Finally, a technique for compressing a standardMNA matrix to a state matrix presentation issought.
The first case is an extension of [2] where a sim-ple numerical overampling was used as a referencefor checking the validity of the lumped error models.Here, the lumped model will be compared against thesolution given by the linear, time-varying state mod-els. The test setup consists of a D/A converter withtime skew errors, and a reconstruction filter, and theresults of a lumped model and a linear solution cal-culated piecewise over all small time-skewed periodsare compared.
The second case (see Fig. 3) is a previously pre-sented D/A conversion system [4], that achieves lin-ear phase image filtering by extending the hold dura-tion of the samples: when the length of the hold time
equals the sample period, the notch of the sample-and-hold sinf/f frequency response is at the sam-ple frequency. By extending the hold time (unfortu-nately, this requires the use of two parallel and time-interleaved D/A converters that can overlap their out-puts) the notch can be brought down to filter the firstimage band. This operation has been verified in prac-tice, but its simulation is quite lengthy, too, and hereit serves as a test bench for really comparing the statemodel and lumped model presentation.
Last, ways to easily create the state model fromnetlist or a standard MNA matrix presentation arestudied.
As a conclusion, this paper experiments methodsto speed-up long time-domain system simulations byemploying linear state modeling and lumped model-ing of time skew errors. The experimented methodswill be verified by Matlab simulations.
DAC1
+
+
DAC2
yo(t)
ye(t)
go(t)
ge(t)
DIV2clk
data
(a)
Fig. 3. Structure of a time-interleaved D/A converterpair.
References
1. T. Rahkonen, O-P Jokitalo. Design of a linearlyassisted switcher for a supply modulated RF trans-mitter. Springer journal on Analog integrated circ.and signal proc., , vol. 54, no 2, February 2008, pp.113-119.
2. T. Rahkonen, H. Repo. Efficient behavioral mod-elling of small timing errors in A/D and D/A con-verters. Kluwer Academic journal on Analog Inte-grated Circuits and Signal Processing, vol. 46, no.1, January 2006, pp. 29-36.
3. M. Neitola, T. Rahkonen. Look-up based settlingerror modeling in Simulink. Submitted to SCEE2008 conference.
4. T. Rahkonen, J. Aikkila. Linear phase reconstruc-tion filtering using hold time longer than one sam-ple period. IEEE Trans. Circ. Syst. I, vol. 51, no.1, January 2004, pp.178-181.
110
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.12
Lookup-Table Based Settling Error Modeling in SIMULINK
Marko Neitola and Timo Rahkonen
University of Oulu, Finland [email protected], [email protected]
Summary. Behavioral modeling of circuit nonlin-earities in a discrete-time simulator is essential espe-cially in case of large mixed-signal systems. A delta-sigma (DS) converter is a good example of such sys-tem, because it is inherently a nonlinear and it needslong time-domain simulations to ensure the stability.
There are many publications, e.g [1–3], on howto model not only the settling error but most domi-nant non-idealities in a single SIMULINK behavioralmodel. This work concentrated only on the settlingerror part and provided a very simple, amplifier topol-ogy independent methodology. The behavioral modelis based on characterization simulations using a com-prehensive set of initial input and output transients.The resulting settling error table will be integratedinto a SIMULINK-model using a lookup-table (LUT)block. The error model can be either one- or two-dimensional according to the capacitor network.
In the characterization of an amplifier, the settlingerror data can be originated from any simulator, andthe amplifier detail level can be just as comprehensiveas needed. In this work, we will show an examplewhich used transient simulation data from a nonlinearstate-space MATLAB-model of a two-pole amplifier.This amplifier is used in a switched-capacitor (SC)integrator.
Once the amplifier model is constructed, a de-signer may re-characterize and simulate the integra-tor quickly within a parameter sweep. Moreover, thisapproach avoids constructing complex output relatedanalytical equations which are typically modeled asgroup of conditional function blocks in SIMULINK.
The behavioral SIMULINK-simulations of DS A/D-converter were made by sweeping the parametersthat affect the slew-rate and phase margin of a SC-integrator. This type of behavioral simulation enablesthe designer to establish performance boundaries thatmakes less conservative parameter dimensioning pos-sible.
The lookup-table approach is not tied to any spe-cific SC-circuit model. In this work, we use a popu-lar SC-integrator model of a parasitic insensitive SC-integrator as a case study. The initial voltage at theinput of the amplifier is recognized by studying pas-sive charge transfer in the capacitive network aroundthe amplifier. All capacitors are initially charged tosome voltage. When the capacitors are reconnected,the total charge is passively re-distributed, resultingin initial voltages vi0 and vo0 at the input and outputof the operational amplifier. It is quite straightfor-ward to construct charge-sharing equations for bothphases of an SC-integrator.
Two switching capacitance networks of course re-sults on two settling transients. Therefore, the settlingerror modeling is based on the error table acquiredfrom two separate characterizations. This table is ei-ther one- or two-dimensional depending on the ca-pacitor network topology. In our case study, the sam-pling phase φ1 requires a two-dimensional error modelas the charge transfer phase φ2 only requires a one-dimensional settling error model. This means that inphase φ1, all integrator input signals and one out-put signals contribute to the initial input and outputvoltages.
The amplifier settling error is an error table ob-tained from settling simulations. These simulationscan be performed by a simulation platform of userschoice. This can be Spice, Verilog-A, VHDL-AMS,MATLAB, etc. The amplifier connected to a capaci-tance network is first exited with full range of inputsignals. Then, the simulated settling errors for bothintegrator phases and for each initial condition pairsare tabulated. Here we used MATLAB to create arealistic nonlinear state-space model of the amplifier.The settling error-table is obtained by simulating thestate-space model defined by a group of differentialequations and some nonlinear behavior.
In a single characterization shown in Fig. 1, wehave a set of amplifier output voltage settling simula-tions for a hundred different initial input and outputvoltages. The settling errors are then compared toideal results and the settling error is stored. In ourcase study, he settling error ǫ1 from phase φ1, is de-pendent on vi0 and vo0. For φ2, the error ǫ2 is onlydependent on vo0.
The error tables are now used in SIMULINKs one-and two-dimensional lookup tables called ”LookupTable” and ”Lookup Table (2-D)” [4]. In the sam-pling phase (phase φ1) the LUT is two-dimensionaland produces settlig error ǫ1(n). In the charge trans-fer phase, the LUT is one-dimensional and producessettling error ǫ2(n).
The objective for the behavioral model was to ob-serve delta-sigma A/D-converters performance as afunction of slew-rate and closed-loop phase margin ofan integrator. The model of a second order DS A/D-converter is presented in Fig. 2. It is a one-bit DSconverter with oversampling ratio of 32. Sampling fre-quency is 10MHz and the integrator open-loop band-width is 65MHz. The LUT-block first calculates theinitial voltages according to input and output sig-nals using the charge sharing equations. After this,the lookup-tables are used to determine the settlingerrors ǫ1(n) and ǫ2(n). In our case study, the errorǫ2(n) is summed to the output of the integrator and
111
Fig. 1. a) Settling curves and b) the settling errors.
the sampling phase error ǫ1(n) is a two-dimensionalerror-model summed to the integrator input. The lat-ter was found dominant settling error model through-out the parameter sweep.
Fig. 2. SIMULINK-model of a second order DS A/Dconverter.
A full sweep contained 1024 characterizations andsimulations. For each iteration, SNDR and SFDR per-formances were calculated. The object was to findperformance boundaries as a function of slew-rate andphase margin. The result shown in Fig. 3 shows the 6dB deterioration boundary of SFDR and 3 dB bound-ary of SNDR.
Initially, the DS converter had one-bit quantizer,and the boundary is shown in black graph in Fig. 3 aand b. The gray graph is the boundary for a three-bitDS-converter. The three-bit converter had the sameconverter topology with seven quantizer levels andcoefficients optimized for three-bit quantizer. Concen-trating only on the performance boundaries, the dif-ference between these two converters can be seen intwo areas:
1. With small slew-rates the three-bit convertersSNDR and SFDR performance deteriorates lesseasily. This is explained by the fact that thethree-bit feedback signal has significantly smaller
step-size, making the amplifier less susceptible toslew-rate.
2. In case of low phase margin and large slew-rate,the SNDR boundary of one-bit system is at lowerphase margins simply due to some 25 dB lowermaximum SNDR performance.
Fig. 3. Performance boundaries: a) 6 dB SFDR andb) 3 dB SNDR performance deterioration.
References
1. W. M Koe and J. Zhang. Understanding the Effectof Circuit Non-idealities on Sigma-Delta Modula-tor. Proceedings of the 2002 IEEE InternationalWorkshop on Behavioral Modeling and Simulation,94–101, 2002.
2. A. Fornasari, P. Malcovati, F. Maloberti. ImprovedModeling of Sigma-delta Modulator Non-idealitiesin Simulink. IEEE International Symposium onCircuits and Systems, Vol.6:5982–5985, 2005.
3. H. Zare-Hoseini, I. Kale and O. Shoaei. Modelingof Switched-Capacitor Delta-Sigma Modulators inSimulink. IEEE tr. Instrumentation and measure-ment, Vol.54:1646–1654, 2005.
4. The Mathworks inc. Simulink Release Notes, ver-sion R14 and higher.
112
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.13
BRAM - Backward Reduced Adjoint Method
Z. Ilievski1, W.H.A. Schilders1,2, and E.J.W. ter Maten1,2
1 CASA - Technische Universiteit Eindhoven, 5600 MB Eindhoven [email protected] NXP Semiconductors, Eindhoven
Summary. This paper describes the current avail-able methods for the transient sensitivity analysis ofelectronic circuits and demonstrates how, by theiradaptation through the inclusion of model order re-duction techniques, the computational time neededto obtain satisfactory sensitivity values can be vastlyreduced. The discussion is limited to linear problems.
1 Circuit Analysis
1.1 Circuit Equations
Equation (1) shows the Differential-Algebraic Equa-tion that describes the behaviour of a standardelectrical circuit that, on solving, will give the val-ues of all the circuit states,
d
dt[q(x(t,p),p)] + j(x(t,p),p) = s(t,p) (1)
Here x(t,p) ∈ RN is the state vector whichrepresents the node voltages and the currentsthrough the voltage defined elements. The vectorfunction j ∈ RN describes the current behaviourand q ∈ RN the charge and flux behaviour, allsource values can be found in s(t,p) ∈ RN
The vector p ∈ RP represents the parametersof a component, such as the width and length ofa resistor, on which a state x(t,p) could dependupon.
Equation (1) can be rewritten as system (2),in which C = ∂q/∂x and G = ∂j/∂x,
Cx(t,p) + Gx(t,p) = s(t,p) (2)
For the linear case that we are considering thecoefficient matrices C & G are constant in time,they themselves are also dependent on the pa-rameters p ∈ RP . For electrical circuits, C is asingular matrix.
1.2 Circuit Observation Functions
In addition to observing each individual circuitstate the observation of the total performance ofa circuit or a subcircuit can be done. For example,in an amplifier you may want to observe its total
gain; or perhaps take a closer look at the energyused by of one of its resistors.
Equation (3) shows the form of the energy ob-servation function. Relevant node or state valuesare already available for selection from the anal-ysis in section 1.1,
GF(x(p),p) =
∫ T
0
F(x(t,p),p) · dt (3)
2 Sensitivity Types
2.1 State Sensitivity
Each state in (1) may have a dependence on oneof the parameters in p, this means it is sensitiveto any parameter changes. Equation (4) shows anexpression for these state sensitivities,
x(t,p) ≡ ∂x(t,p)
∂p∈ RN×P (4)
2.2 Observation Funtion Sensitivity
The function sensitivities are found in a similarway, they are the differentials of the observationfunction with respect to the parameters p, (5)shows this sensitivity and also reveals the pres-ence of the state sensitivitiy x,
dGF
dp=
∫ T
0
(∂F
∂x· x +
∂F
∂p
)· dt (5)
3 Sensitivity Analysis
Generally it is the calculation of the sensitivityof the observation function that is most useful, ifpossible with the avoidence of the calculation ofthe state sensitivities. In the following, in addi-tion to N and P , F is the number of observationfunctions.
113
3.1 Direct Forward Method
In this approach, after an inital forward state so-lution by a Newton-Raphson procedure involv-ing the coefficient matrix Y = 1
∆tC + G , eachstate sensitivity value is explicitly calculated byrecursion [1, 2]. To complete the calculation ofthe observation function sensitivities these statesensitivities are substituted into the integrand in(5), the total computational cost [3, SCEE, 2006]of one time integrand step is O(min(F, P )N2 +FPN) operations.
3.2 Adjoint Transient Sensitivity Method
In [3] we give a full description of an approachbased on (backward) adjoint integration [4], wewere able to re-express the observation functionintegrand in terms of a function λ⋆(t) ∈ RF×N .
After an initial transient analysis, and suit-ably chosen boundary conditions a second equa-tion (6), the Backward Adjoint Equation, yieldsunique values of λ(t),
C⋆ dλ
dt− G⋆λ = −(
∂F
∂x)⋆. (6)
The integrand of (5) has been reduced to animpressive O(PN +FP ). When solving (6), eachtime integration requires W + O(FN2) opera-tions.
The remaining burden is the W = O(Nα) (1 ≤ α ≤ 3 ) work needed for the LU-decompositionswhen integrating backward in time.
3.3 Backward Reduced Adjoint Method
We have noticed that (6) has the same form as theoriginal circuit system (2), which suggests thatboth the forward and backward steps of section3.2 could share important basic characteristics.We have carried out a more thorough analysissince our initial findings in [3], and these will bedescribed in this presentation.
Proper Orthogonal Decomposition
The Proper Orthogonal Decomposition (POD) isa method that is popular for forming a descrip-tion and obtaining the basis U of a system bycollecting its output behaviour X ∈ Rn,m,
X = UΣVT (7)
1
NXXT =
1
NUΣVTVΣUT =
1
NUΣ2UT (8)
Backward Reduction Step
The important step here is that we are able toreuse the effort in the forward analysis to con-struct projection matrices that can gives a re-duced W = O(Nα
k ) (mentioned in section 3.2)where k is the number of dominant eigenvaluesrequired for a satisfactory approximation.
This immediately gives rise to an attractiveBRAM method, we will give a more vigourousexplanation of the BRAM method in our paper.
Other Considerations
We will pay extra attention to the following points.
• We have noticed that if we replace a compo-nent with an ever increasing equivalent seriesof components and then interpolate the statesan eigenvalue stretch phenomenon is observed.
• The use of the projection matrix P, itself de-pendant (and sensitive) on circuit parameters,reveals an extra term in the state sensitiv-ity expression when projecting back in to theoriginal space, as shown in (9). Here x is theapproximated state.
∂x
∂p(t,p) ≈ ∂P(p)
∂p· x(t,p) + P(p) · ∂x
∂p(t,p)(9)
Acknowledgement. The authors acknowledge the sup-port by TU Eindhoven, NXP and by the EU MarieCurie RTN project COMSON grant n. MRTN-CT-2005-019417.
References
1. L. Daldoss, P. Gubian, M. Quarantelli: Multi-parameter time-domain sensitivity computation,IEEE Trans. on Circuits and Systems - I: Fund.Theory and Applics, Vol. 48-11, pp. 1296–1307,2001.
2. Y. Favennec, M. Girault, D. Petit: The ad-joint method coupled with the modal identifica-tion method for nonlinear model reduction, InverseProbl. in Science and Engng., Vol. 14, No. 3, 153–170, 2006.
3. Z. Ilievski, H. Xu, A. Verhoeven, E.J.W terMaten, W.H.A. Schilders, R.M.M. Mattheij, Ad-joint transient sensitivity analysis in circuit simu-lation SCEE2006, Scientific Computing in Electri-cal Engineering, Springer ISBN 978-3-540-71979-3,p183 -189,
4. Y. Cao, S. Li, L. Petzold, R. Serban: Adjointsensitivity for differential-algebraic equations: theadjoint DAE system and its numerical solution,SIAM J. Sci. Comput., Vol. 24-3, pp. 1076–1089,2002.
114
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.14
Applications of Eigenvalue Inclusion Theorems in Model
Order Reduction
E. Fatih Yetkin1 and Hasan Dag2
1 Istanbul Technical University, Informatics Institute Istanbul, Turkey [email protected],2 Kadir Has University, Istanbul, Turkey [email protected]
Summary. We suggest a simple and efficient methodbased on the Gerschgorin eigenvalue inclusion theo-rem to compute a Reduced Order Model (ROM) tolower greatly the order of a state space system. Thismethod is especially efficient in multi time scaled sys-tems but it also works in other cases under some as-sumptions.
1 Introduction
The computational cost of the simulation of to-day’s technological equipment can be very high.Hence the model order reduction methods havevery wide application areas especially in sub-micron electronic device and microelectronic me-chanical system (MEMS) modeling and simula-tion [1]. In general, a single input single output(SISO) system can be defined with the state equa-tions in the standard form as below.
x = Ax + bu
y = cTx + du (1)
where A ∈ Rnxn, b, c ∈ Rn and d is a scalar.Here, the dimension of the state space n is verylarge and the model order reduction techniquesare employed to build an mth order system wherem ≪ n. The reduced system can be given as be-low,
x = Ax + bu
y = cTx + du (2)
where A ∈ Rmxm, b, c ∈ Rm and d is a scalar [2].Using the dominant eigenmodes is one of the bestways to build a reduced state space but it isnot computationally feasible due to huge cost tocompute all eigenmodes of the system at hand.Therefore, computing only the dominant poles ofa transfer matrix can also be an effective wayfor computing the reduced system matrices [3].In this study we suggest a new method based onthe eigenvalue inclusion theorems such as that ofGerschgorin’s. Gerschgorin discs are very usefuland there are computationally efficient methodsto determine the area of possible eigenvalue loca-tions of a given matrix.
First a possible eigenvalue location and thenumber of eigenvalues are determined using one ofthese theorems. Then, the random complex num-bers laid in Gerschgorin discs can be used insteadof the original eigenvalues of A. These randomeigenvalue estimates can be used to build a pro-jection matrix to reduce the order of the systemat hand with a reasonable frequency domain er-ror. The approximate eigenvectors of the matrixA form the transformation matrix.
Gerschgorin discs are also used in conjunctionwith Singular Perturbation Approximation for re-ducing a system order [4]. This method requiresthat the system matrix A lend itself to partition-ing so that it can be partitioned into two subsys-tems called fast and slow subsystems. They canbe separated by their proximity to the origin ofeigenvalues. To identify and separate these sub-systems Gerschgorin theorem can be applied.
2 Method and Algorithm
Singular perturbation analysis based methods givevery rapid and accurate results for small andmulti-time scaled systems. But if system sizegets larger, computational cost of the matrix in-verses becomes infeasible. Therefore, more effi-cient strategies have to be developed. A givensystem’s characteristic is determined by its eigen-value. If we are interested in a reduced ordermodel of a stytem at hand it is natural to select agroup of dominant eigenvalues and construct thereduced model using those eigenvalues. On theother hand, computing a set of dominant eigenval-ues of a matrix itself is a computationaly expen-sive work. Thus, we choose to find approximationto the dominant eingevalues. In order to have agood approximation to the eigenvalues of interestwe use the Gerchgorin disks since all eigenvalueslie within the union of those disks. After select-ing a group of approximate eigenvalues randomlywithin the union of the disks the transform ma-trices have to be computed to round all systemtriple into a reduced order.
If an approximate eigenvalue set is estimatedwith the Gerschgorin or Brauer theorem then the
115
approximate eigenvectors associated with theseapproximate eigenvalues can also be estimated.With this estimated set of eigenvectors we canbuild a transformation matrix to convert wholesystem to a reduced order system.
The last part of the work is estimating theeigenvectors corresponding to approximate eigen-values. These eigenvectors have to be orthonor-malized after estimation. The orthonormalizedeigenvectors form a transformation matrix thatis used to reduce the system order.
Ar = V TAV
br = V Tb
cTr = cTV (3)
3 Numerical Examples
A fifth order random system is selected to makea comparison between suggested method and sin-gular perturbation approximation. Using the Ger-schgorin theorem, one can says that there are3 separate eigenvalues very close to the origin.Then (A,b, c) triple have to be reorganized dueto eigenvalues and the associated rows. Then non-intersecting sets of Gerschgorin circles has to beemployed to divide matrices in blocks. In this casewe have three non-intersecting sets but we onlyconsider to smallest set. Therefore the other twosets can be written as one set. We can separatesystem triple (A,B,C) into blocks to apply singu-lar perturbation approximation.
The original and the reduced systems fre-quency responses are given in Fig. 1 and 2.
10−3
10−2
10−1
100
101
102
103
−225
−180
−135
−90
−45
Phase
(deg)
Bode Diagram
Frequency (rad/sec)
−60
−50
−40
−30
−20
−10
0
10
20
Magnitu
de (
dB
)
original
reduced
Fig. 1. Frequency plots of the original and reducedsystems with singular perturbation approximation.
4 Conclusions and Future Work
In this study, dominant poles of the linear statespace system are estimated using the Gerschgorin
10−3
10−2
10−1
100
101
102
103
−225
−180
−135
−90
−45
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
−60
−40
−20
0
20
Magnitude (
dB
)
originalreduced
Fig. 2. Bode diagrams of reduced and full systemwith suggested new method.
theorem. This estimation is then used for pro-ducing the associated approximate eigenvectors.Then orthonormalized set of these approximateeigenvectors are used as a projection matrix toreduce the system order. The method gives veryaccurate results especially for multi-time systems-meaning that when the eigenvalues of matrix Agrouped in separate Gerschgorin discs. In thesetype of systems singular perturbation approxi-mations also can be very successful. However, ifwhole eigenvalues of system lied in same locationof the complex plane and Gerschgorin discs arenested, singular perturbation approximation doesnot work. The suggested method works well forboth cases, but for the second case one has to runthe algorithm several times for a satisfactory re-sult. Therefore, more sophisticated eigenvalue in-clusion theorems have to be employed for the sec-ond case. Future work is focused on using Brauertheorem and Gleyse method in algorithm to pro-duce more accurate results.
References
1. Meijs, N.P. , Smedes, T., Accurate InterconnectModeling: Towards Multi-million Transistor ChipsAs Microwave Circuits. Int. Conf. On CAD, Proc.of ICCAD’96 p. 244-251, 1996.
2. Tan, S. X. D, He, L., Advanced Model Order Re-duction Techniques in VLSI Design, CambridgeUniversity Press, Cambridge, 2007.
3. Rommes, J.: Methods for Eigenvalue Problemswith Applications in Model Order Reduction,Ph.D. Thesis, Univ. of Utrecht, 2007.
4. Ansary, A., De Abreu-Garca, J.A.: A Simple Al-gorithm for Time Scale Separation, Proceedingsof the 15th International Conference on Indus-trial Electronics, Control, and Instrumentation(IECON’89), Vol. 2, p. 445, Philadelphia, PA,November 6-10, 1989.
116
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.15
Hierarchical Model-Order Reduction Flow
Mikko Honkala1, Pekka Miettinen1, Janne Roos1, and Carsten Neff2
1 Helsinki University of Technology, Department of Radio Science and Engineering, P.O.Box 3000, FI-02015TKK, Finland mikko,pekka,[email protected]
2 NEC Laboratories Europe, IT Research Division, NEC Europe Ltd., Rathausallee 10, D-53757 St.Augustin, Germany [email protected]
Summary. In this abstract, a Hierarchical Model-Order Reduction (HMOR) flow is proposed, wherethe linear parts of a circuit are hierarchically dividedinto independently reducable subcircuits. The impactof the hierarchical structure and circuit partitioningto three MOR methods are discussed, and some sim-ulation results are presented.
1 Introduction
In this abstract the Hierarchical Model-Order Re-duction (HMOR) of very large linear blocks ofnonlinear circuits is considered. The HMOR hier-archically divides the circuit into subcircuits thatcan be independently reduced and then put to-gether. The benefits of the HMOR are:
1. Very large circuits can be simulated in limitedcomputer resources.
2. Circuit partitioning makes possible to applyparallel processing in a natural way.
3. The computational cost can be minimized byanalyzing the repetitive structures only once.
4. The most suitable MOR method can be cho-sen for each subcircuit independently.
In the litterature, HMOR is considered previ-ously, e.g., in Ref. [1].
2 Hierarchical MOR flow
Here, a hierarchical netlist-in–netlist-out MORflow is presented. The flow utilizes hMETIS [2]partitioning algorithms for circuit partitioningand uses the MOR methods in a hierarchical man-ner. The MOR methods considered are PRIMA[3], (modified) Liao–Dai [4], and TICER [5].
The HMOR flow proposed is presented brieflyin the following.
1. Netlist parsing: extract the graph presenta-tions of all the RLC circuits from the ‘messy’(hierarchical) nonlinear SPICE netlist.
2. Circuit partitioning:a) separate disconnected parts.
b) divide each RLC graph (using hMETIS)into appropriate subcircuit graphs andmap the graphs back into circuit netlists.
3. Matrix construction: build, for each subcir-cuit, G, C, and B, the conductance, capac-itance, and selector matrix of the MNA for-mulation, respectively.
4. Model-order reduction: use an appropriate MORmethod for each subcircuit: PRIMA for RLCblocks, and Liao–Dai or TICER for RC blocks.
5. Macromodel realization: synthesize each re-duced subcircuit using resistors, capacitors,and, if needed, Voltage-Controlled CurrentSources (VCCSes).
6. Netlist reconstruction: include each macro-model in the proper position in the finalnetlist to achieve the original hierarchical struc-ture.
3 Hierarchy and circuit partitioning
Consider a netlist consisting of linear and nonlin-ear components defined in hierarchical subcircuits(see Fig. 1). The netlist has, in addition to themain-level circuit (treated as a subcircuit equalto other subcircuits in the following), three dif-ferent types of subcircuits. Each subcircuit mayhave nonlinear components (MOSFETs, diodes,etc.) and/or linear RLC components.
The subcircuit may have several disconnectedRLC parts, and the same subcircuit can be placedseveral times in different levels of hierarchy.
S u b c i r c u i t 1
S u b c i r c u i t 2
S u b c i r c u i t 5
D i s c o n n e c t e dp a r t 1
M a i n c i r c u i t
D i s c o n n e c t e dp a r t 2
S u b c i r c u i t 3
S u b c i r c u i t 4 S u b c i r c u i t 4
Fig. 1. The hierarchical structure of a circuit.
117
If the netlist is processed hierarchically, eachsubcircuit needs to be analysed only once, com-pared to a typical approach, where the wholenetlist is first flattened with all the hierarchicalstructures written out for each reference. For ex-ample, Subcircuit 4 in Fig. 1 needs to be analyzedonly once.
The subcircuit may contain disconnected RLCparts (e.g., see Subcircuit 5 in Fig. 1). If they arenot separated into different subcircuits, numericalproblems may arise. Even if no problems wouldoccur, the disconnection is, in any case, a naturallocation for further partitioning.
Some MOR methods are based on partition-ing, like Liao–Dai that employs low-order macro-models. For methods for which the partitioningis not an essential part of the algorithm (e.g.,PRIMA) it is not clear whether further partition-ing improves the performance. However, the par-titioning is useful for large subcircuits in order tobe able to reduce them at all with limited com-puter resources.
The goal of the circuit partitioning regardingMOR is to obtain such subcircuits that have alarge number of internal nodes compared to ex-ternal port nodes. Partitionings fulfilling this cri-terion are best suitable for MOR, since the be-haviour of internal nodes are not interest and theymay be reduced.
4 MOR methods
The three methods in the HMOR flow are chosenfor their different approaches to MOR.
The well-known RLC MOR method PRIMA[3] employs congruence transformations to projecta large system of equations onto a smaller sub-space, so that passivity is preserved during re-duction.
The Liao–Dai method [4] divides the circuitinto smaller subcircuits, computes the first twomoments of y-parameters of each subcircuit, andrealizes the y-parameters by matching momentsof RC macromodels.
TICER [5] is a nodal-elimination-based RCMOR method.
5 Simulation example
The HMOR flow has been implemented using Cand MATLAB.
The Liao–Dai algorithm is tested with a largeRC netlist. The results with several partitions arepresented in Table 1, where Np, n, R, C, Etr, Ttr,and ttr are the average partition size, number of
nodes, number of resistances, number of capaci-tances, error of transient analysis, CPU time oftransient analysis, and normalized CPU time oftransient analysis, respectively.
Table 1. Liao–Dai results for RC circuit with severalpartitions.
Np n R C Etr/% Ttr/s ttrorig. 1537 10432 197 - 4.04 1.00
106 556 8515 573 0.009 5.77 1.4353 237 3261 255 0.009 1.35 0.3326 81 543 95 0.008 0.37 0.0913 34 137 54 0.008 0.27 0.077 27 109 47 0.008 0.26 0.062 17 62 35 0.008 0.24 0.061 15 44 29 0.010 0.24 0.06
As can be seen from Table 1, the partition-ing plays the key role in controlling the size andaccuracy of the reduced circuit.
6 Conclusions
The hierarchical MOR flow was presented. It wasshown how to reduce very large circuits using hi-erarchical circuit division. Without division thecircuit may be so large that it is impossible tomanage the computational cost.
In the planned full paper, the MOR methodswill be presented in detail and their suitability onHMOR will be discussed. More simulation resultswill be presented.
References
1. Y.-M. Lee, Y. Cao, T.-H. Chen, J. M. Wang, andC. C.-P. Chen. HiPRIME: Hierarchical and Passiv-ity Preserved Interconnect Macromodeling Enginefor RLCK Power Delivery. IEEE Transactions onComputer-Aided design of Integrated Circuits andSystems, 24:797–806, 2005.
2. G. Karypis and V. Kumar. hMETIS, A HypergraphPartitioning Package Version 1.5.3.
3. A. Odabasioglu, M. Celik, and L. T. Pi-leggi. PRIMA: passive reduced-order interconnectmacromodeling algorithm. IEEE Transactions onComputer-Aided design of Integrated Circuits andSystems, 17:645–654, 1998.
4. H. Liao and W. W.-M. Dai. Partitioning andreduction of RC interconnect networks basedon scattering parameter macromodels. In Digestof Technical Papers of IEEE/ACM InternationalConference on Computer Aided Design, pages 704–709, 1995.
5. B. N. Sheehan. Realizable Reduction of RCNetworks. in IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems,26:1393–1407, 2007.
118
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.16
Partitioning-based RL-in–RL-out MOR Method
Pekka Miettinen, Mikko Honkala, and Janne Roos
Helsinki University of Technology, Faculty of Electronics, Communications and Automation, Department ofRadio Science and Engineering, P.O.Box 3000, FI-02015 TKK, Finland pekka,mikko,[email protected]
Summary. This abstract proposes a netlist-in–netlist-out-type Model-Order Reduction (MOR) method suit-able for the reduction of very large RL circuit blocks.The method relies on partitioning the circuit into sub-circuits that can be approximated efficiently with low-order macromodels. The efficiency of the method isdemonstrated with realistic simulations.
1 Introduction
Although the study of linear MOR with inter-connect circuits has been centered mainly on RCand RLC circuits, some pure RL MOR algorithmshave also been presented, e.g., in Refs. [1] and [2].The demand for RL MOR arises in certain situa-tions, such as when modeling a conductor’s skineffect with lumped elements or when operatingat high frequencies, where the capacitances playa minor part.
Furthermore, one important motivation forthe RL MOR presented in this abstract is thepossibility to use it (linked with circuit partition-ing) on a single RL block appearing inside a muchlarger RLC circuit. In this case, the RL MORmethod is used as one of the many specializedmethods in a complete MOR tool.
The basic idea behind the proposed RL MORmethod is to partition the circuit into smaller sub-circuits, which may then be approximated withrelatively simple low-order macromodels, and fi-nally, combined back together. The concept oflow-order macromodels via partitioning was firstpresented in Ref. [3] for RC circuits. This wasfurther studied and refined in Ref. [4] with sup-port for RLC circuits by using also PRIMA [5]for the partitioned subcircuits. However, simula-tions with the original PRIMA algorithm resultedoften with numerical stability issues.
In this abstract, a stable RL MOR methodfor the special case of RL circuits is proposed, in-spired by the original RC MOR method in Ref.[3]. The proposed RL-in–RL-out MOR methodmay be conceptually divided into three steps:circuit partitioning (Sect. 2), calculation of y-parameter moments (Sect. 3), and macromodelsynthesis (Sect. 4).
2 Circuit partitioning
Since the RL MOR method is based on approx-imating interconnects between port nodes withlow-order macromodels, it is necessary to performa partition on the large RL(C) circuit prior tomacromodel synthesis. The quality of the subcir-cuits is essential; if the subcircuits are too com-plex, the low-order macromodel used later is notaccurate enough to model the partition, while ifthe subcircuit is too simple, the magnitude of thereduction is small and the MOR is of little use.
METIS is an algorithm package for parti-tioning large irregular graphs, partitioning largemeshes, and computing fill-reducing orderings ofsparse matrices. The use of METIS and hMETISalgorithms especially in circuit partitioning wasstudied in Ref. [4], where it was noted that theyboth produced excellent partitionings of equalsize. In this abstract, we consider the hMETISalgorithm as a partitioning method in the com-plete MOR flow [4].
3 Calculation of y-parametermoments
Once the RL(C) circuit is partitioned into RLsubcircuits, the y-parameters are needed to ob-tain the corresponding macromodel. The Laplace-domain circuit equations for an RL circuit can beexpressed as
(G + 1
sΓ)x(s) = Bu(s)
i(s) = LTx(s),(1)
where x denotes the nodal voltages and port cur-rents, u denotes the port voltages, and i denotesthe port currents. Here, B = L is a selector ma-trix consisting of ones, minus ones, and zeroes,
G =
[G11 Mu
−MTu 0
]Γ =
[Γ11 00 0
]. (2)
Matrices G11 and Γ11 contain the conductanceand inductance element stamps, and Mu con-sist of the stamps for the port-voltage sources.Solving now for the port currents and defining
119
A ≡ −G−1Γ and R ≡ −G−1B results in thefollowing y-parameter matrix:
Y(s) = LT(I− 1
sA)−1R, (3)
Finally, the term (I − 1sA)−1 can be expanded
into a Neumann series to obtain
Y(s) = M0 + M11
s+ M2
1
s2+ · · · . (4)
The block moments of Y can be calculated us-ing the relation Mi = LTAiR. Note that thedimension (N -by-N) of the block moments Mi
is the same as the number (N) of ports in the(sub)circuit.
4 Macromodel synthesis
For an N -port, the admittance between the ithport and ground is given by the sum of the ith row(or column) of its Y-matrix, Y(s). The admit-tance connecting port-i and port-j is −yij . Thus,the circuit synthesis problem amounts to synthe-sizing admittances between pairs of ports and be-tween a port and ground with lumped R and Lelements. Once M0 and M1 have been calculated,each element of Y(s) can be approximated as
yij ≈ mij0 +mij
1 s. (5)
Using a direct synthetization, the first two mo-ments are realized with parallel R and L elements.For off-diagonal elements yij(i 6= j),
Rij = − 1
mij0
and Lij = − 1
mij1
. (6)
For diagonal elements yii,
Rii =1
mi00
and Lii =1
mi01
. (7)
5 Simulation results
The RL MOR algorithm was verified and sim-ulated with several interconnect RL circuits, ofwhich two representative cases are shown in Ta-ble 1. Here, n, R, L, cp, Etr and Ttr mean thenumber of nodes, resistances, inductances, ele-ment reduction ratio, relative transient analysiserror, and transient analysis CPU time, respec-tively. From the table it can be seen that the al-gorithm achieves excellent reduction of CPU timewith only a minimal error. Figure 1 shows thecomparison of transient simulations with the orig-inal and reduced circuit c2. In the planned fullpaper, the RL MOR method is presented in moredetail, and more types of different RL circuits aresimulated, along with a comparison to other MORmethods.
Table 1. Simulation results
circuit n R L cp/% Etr/% Ttr/s
orig. c1 2000 2000 2000 - - 3.25reduc. c1 63 34 41 97.6 0.198 0.10
orig. c2 10000 10000 10000 - - 123reduc.c2 154 99 102 99.0 0.354 0.16
Fig. 1. Transient simulation of original and reducedcircuit c2 (the curves are nearly indistinguishable).
6 Conclusions
In this abstract, a new RL MOR method capableof efficient reduction of very large RL circuits wasproposed. Using partitioning to generate smallersubcircuits, a method of calculating the block mo-ments of the y-parameters was described and amacromodel realization for the first two momentsderived. Finally, simulation results showing excel-lent reduction were presented.
References
1. S. Mei and Y.I. Ismail. Modeling Skin Effect withReduced Decoupled R-L Circuits. Proceedings ofthe 2003 International Symposium on Circuits andSystems, 4:588–591, 2003.
2. Q. Yu and E.S. Kuh. Accurate Reduced RL Modelfor Frequency Dependant Transmission Lines. The9th International Conference on Electronics, Cir-cuits and Systems, 2:761–764, 2002.
3. H. Liao and W.W.-M. Dai. Partitioning andReduction of RC Interconnect Networks Basedon Scattering Parameter Macromodels. Digestof Technical Papers of IEEE/ACM InternationalConference on Computer Aided Design, 704–709,1995.
4. P. Miettinen. Hierarchical Model-Order ReductionTool for RLC Circuits. Master’s thesis, HelsinkiUniversity of Technology, 2007.
5. A. Odabasioglu, M. Celik, and L.T. PileggiPRIMA: Passive Reduced-Order InterconnectMacromodeling Algorithm. IEEE Transactions onComputer-Aided design of Integrated Circuits andSystems, 17:645–654, 1998.
120
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.17
Nonlinear model order reduction based on trajectory
piecewise linear approach: comparing different linear cores
Kasra Mohaghegh1,3 and Michael Striebel2,3
1 Bergische Universitat Wuppertal [email protected] TU Chemnitz [email protected] NXP Semiconductors, Eindhoven
Summary. Refined models for MOS-devices and in-creasing complexity of circuit designs create the needfor Model Order Reduction (MOR) techniques thatare capable of treating nonlinear problems. In time-domain simulation the Trajectory PieceWise Linear(TPWL) approach is promising as it is designed touse MOR methodologies for linear problems as thecore of the reduction process. We compare differentlinear approaches with respect to their performancewhen used as kernels for TPWL.
1 Introduction
The tendency to analyze and design systems ofever increasing complexity is becoming more andmore a dominating factor in progress of chip de-sign. Along with this tendency, the complexityof the mathematical models increases both instructure and dimension. Complex models aremore difficult to analyze, and due to this it isalso more difficult to develop control algorithms.Therefore Model Order Reduction is of utmostimportance. For linear systems, quite a num-ber of approaches are well-established and haveproved to be very useful. However, accurate mod-els for MOS-devices introduce highly nonlinearequations. And, as the packing density in circuitdesign is growing, very large nonlinear systemsarise. Hence, there is a growing request for re-duced order modeling of nonlinear problems. Intransient analysis the Trajectory PieceWise Lin-ear approach [5] is a promising technique as itmakes use of linear MOR methods. A brief intro-duction to TPWL is given below.
Analysing the TPWL approach, we are inter-ested in how different linear MOR techniques per-form when used as a linear kernel, how robust thereduced model are and how they behave whencombined to more complex systems.
2 MOR for linear problems
A continuous time-invariant (lumped) multi-inputmulti-output linear dynamical system is of theform:
C dx(t)
dt = −Gx(t) +Bu(t)y(t) = Lx(t) +Du(t), x(0) = x0,
(1)
where x(t) ∈ Rn is the inner state; u(t) ∈ Rm
is the input, y(t) ∈ Rp is the output. The di-mension n of the state vector is called the orderof the system. C, G, B, L and D are the statespace matrices. The dimension n of the systemis of the order of elements contained in the cir-cuit. As VLSI systems exhibit a large density ofelements, n≫ max(m, p) is usually very large.
Basically, MOR techniques aim to derive asystem:
C dx(t)
dt = −Gx(t) + Bu(t), x(t) ∈ Rq
y(t) = Lx(t) + Du(t), x(0) = x0(2)
of order q with q ≪ n that can then replace theoriginal high-order system (1) in a sense, that theinput-output behaviour, described by the transferfunction in the frequency domain, of both systemsis similar.
The reduction can be carried out by meansof different techniques. Aiming at a combinationwith TPWL we concentrate on projection basedapproaches, namely SPRIM [2], PRIMA [3], andPMTBR [4]. The first two rely on Krylov sub-space methods, which provide numerically robustalgorithms for generating approximations to thetransfer function of the full system (1) that matcha certain number of its moments. Owing to theirrobustness and low computational cost, Krylovsubspace algorithms proved suitable for reductionof large-scale systems, and gained considerablepopularity, especially in electrical engineering.
PRIMA combines the moment matching ap-proach with projection to arrive at a reduced sys-tem of type (2). Its main features is that it pro-duces provably passive order reduced models.
SPRIM which can preserve reciprocity or theblock structure of the circuit matrices, inherentto RLC circuits is also studied. SPRIM gener-ates provably passive and reciprocal macromod-els of multiport RLC circuits. The SPRIM modelsmatch twice as many moments as the correspond-ing PRIMA models obtained with same amount
121
of computational work, also SPRIM is less restric-tive to C & G in system (1), see [1].
PMTBR is a projection MOR technique thatexploits the direct relation between the multi-point rational projection framework and the Trun-cated Balanced Realization (TBR). This approachcan take advantage of some a priori knowledge ofthe system properties, and it is based in a statis-tical interpretation of the system Gramians.
Example 1. [6] We consider the RLC ladder net-work shown in Fig. 1. The MNA modeling leedsto odd n. We take n = 401 and arrange R1 = 1
2 ,R2 = 1
5 and all inductors and capacitors have unitvalue.
L(n−1)/2 L1
C1C2C(n+1)/2 R1
R2y
u
Fig. 1. RLC Circuit of order n.
The original transfer function is reduced bythree linear techniques (PRIMA, SPRIM andPMTBR) from order 401 to order 37. Figure 2presents the most dominant error parts for thesemethods.
10−2
10−1
100
101
102
103
−350
−300
−250
−200
−150
−100
−50
0
50
Frequency (rad/sec)
Err
or(
dB
)
PRIMAPMTBRSPRIM
Fig. 2. Corresponding error plot.
3 MOR for nonlinear systems –TPWL
An electrical circuit that contains nonlinear ele-ments, like MOS-transistors, can be described bya nonlinear differential-algebraic equation (DAE)of the form
d
dt[q(x(t))] + (x(t)) +Bu(t) = 0 (3)
with q, : Rn → Rn, u : R → Rm, B ∈ Rn×m.Again, one is interested in replacing (3) by a sys-tem of much smaller dimension. At present, how-ever, only linear MOR techniques are well-enoughdeveloped.
TPWL [5, 7], the trajectory piecewise linearapproach, allows MOR techniques developed forthe linear problem (1) to be applied to nonlinearsystem (3). This is done by linearizing (3) sev-eral times along a trajectory, reducing the linearlocal ersatz systems and recomposing these mod-els again. The latter step is done by means of aweighted sum, i.e. a linear combination of all thesubmodels available. To represent the nonlinearcharacteristics by a series of linear models, typicalinput signals u(t) are applied, giving typical tra-jectories which are then used in the described pro-cedure. Hence, TPWL leads to an order-reducedmodel that is trained to replace the full nonlinearsystems for a certain class of input signals.
We compare the behaviour of TPWL, whenPRIMA, SPRIM and PMTBR are used as reduc-tion kernel. Furthermore, we study the sensitivityof the reduced models w.r.t. different inputs. Weapply the TPWL to a chain of nonlinear invert-ers. Observations made lead to considerations onimprovements of the weighting, i.e. the process ofrecombining the local reduced linear models.
Acknowledgement. The work presented is supportedby the Marie Curie RTN and ToK projects COMSONand O-MOORE-NICE!, respectively.
References
1. R.W.Freund. Structure-Preserving Model OrderReduction of RCL Circuit Equations. In: ModelOrder Reduction: Theory, Research Aspects andApplications, H.A. van der Vorst, W.H.A.Schilders& J.Rommes (eds.): Springer-Verlag 2008, to ap-pear.
2. R.W.Freund. SPRIM: structure preservingreduced-order interconnect macromodeling. Proc.ICCAD, pp. 80-87, 2004.
3. A.Odabasioglu, M.Celik, L.Paganini. PRIMA:Passive reduced-order interconnect macromodel-ing algorithm. IEEE TCAD of Integ. Circuits andSystems, vol. 17(8), pp. 645-654, 1998.
4. J.Phillips, L.M.Silveira. Poor’s man TBR: a simplemodel reduction scheme. Proc. DATE, vol. 2, pp.938-943, 2004.
5. M.J.Rewienski. A Trajectory Piecewise-LinearApproach to Model Order Reduction of Nonlin-ear Dynamical Systems. Ph.D. thesis, MIT, USA,2003.
6. D.C.Sorensen. Passivity Preserving Model Reduc-tion via Interpolation of Spectral Zeros: Systemand Control Letters, vol. 54(4), pp. 347-360, 2005.
7. T.Voß, R.Pulch, J.ter Maten, A.El Guennouni.Trajectory piecewise linear approach for nonlineardifferential-algebraic equations in circuit simula-tion. In: G.Ciuprina, D.Ioan (eds.):Proc. SCEE.Mathematics in Industry, vol. 11, Springer, pp.167-174, 2007.
122
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 2.18
On Model Order Reduction for Perturbed Nonlinear Neural
Networks with Feedback
Marissa Condon1 and Georgi G. Grahovski1,2
1 School of Electronic Engineering,Dublin City University, Glasnevin, Dublin 9, [email protected], [email protected]
2 Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences,72 Tzarigradsko chaussee, 1784 Sofia, Bulgaria [email protected]
Summary. The paper addresses the dynamical prop-erties of large-scale perturbed nonlinear systems ofthe Hopfield type with feedback. In particular, it fo-cuses on the hyperstability of the equilibria of the sys-tem. It proceeds to examine the effect of the empir-ical balanced truncation model reduction techniqueon the hyperstability properties. The results are il-lustrated using a numerical test case.
1. Introduction/Motivation
Neural networks have attracted the attention ofthe scientific community for several decades [2,7]and of these, one of the most important nonlinearneural networks is the Hopfield model [5,6] whichwas introduced by J. J. Hopfield in the 1980s. Ithas been extensively studied (see, e.g., [8] and thereferences therein) and has found many impor-tant applications in diverse areas such as patternrecognition, associative memory and combinato-rial optimisation.
The study of the stability of the equilibriumpoints is an important area which has been the fo-cus of study over the past number of years. One ofthe reasons for its importance is that if an equi-librium of a Hopfield neural network is globallyasymptotically stable, then the domain of attrac-tion of this point is the entire state space [8].
A special subclass of the Hopfield neural net-works are Hopfield networks models with feed-back. In this case, an additional functional de-pendence is imposed between the inputs and thecorresponding outputs. This is related to the pas-sivity property of the considered system [9].
In most of the applications involving neuralnetworks, the model equations form a large-scalesystem (see e.g. [8] and the references therein).For example, there are approximately 1012 neu-rons in the human brain [2]. As a rule, this leadsto costly and inefficient computations. Therefore,model reduction is of paramount importance. Thereduced model must mirror the properties of theoriginal system if it is to be of practical utility.
In the present article, the effect of model re-duction on the hyper/stability properties of the
nonlinear Hopfield model with feedback is is stud-ied.
2. Perturbed Hopfield model with feedback
Consider the following system of non-linear ODE’s(known as Hopfield models [5, 6]) in the form:
xi = −bixi +
N∑
j=1
AijGj(xj) + Ui(t), (1)
where i = 1, . . . , n, bi’s are constants, Aij form aconstant matrix and the external inputs Ui(t) arefunctions of the time variable t. The functions Gjare, in general, nonlinear with respect to the statevariables xj , j = 1, . . . N (here N is the numberof the neurons in the network). The second termin (1) gives the interconnection between the neu-rons.
The corresponding perturbed version of theHopfield model (1) takes the form
xi = −bixi +
N∑
j=1
AijGj(xj) + Ui(t), (2)
where bi = bi + ∆bi, Aij = Aij + ∆Aij (note,that ∆Aij does not need to be a symmetric) and
Gj(uj) = Gj(uj) + ∆Gj(uj), 1 ≤ i ≤ n. Here∆bi, ∆Aij and ∆Gj(uj) are considered as (small)perturbations of the system (1).
Model order reduction (Empirical Balance trun-cation) is applied to the model equations (1), andthe paper studies the qualitative behaviour of thesolutions of the reduced perturbed model. Spe-cial attention is paid to the Popov hyper-stabilityproperties.
The feedback in the Hopfield model (1) is in-troduced by defining a function F relating the in-puts Ui(t) with the corresponding outputs yi(t):Ui = F (yi) (see section 3 below).
123
3. Model Reduction
The Hopfield model corresponds to a nonlinearsystem of the generic form:
x(t) = f(t,x(t)) + B(t)u(t) (3)
y(t) = h(t,x(t))
where f : IRn → IRn and h : IRn → IRq are non-linear functions, the vector u(t) ∈ IRn is the inputto the system (3), while the vector y(t) ∈ IRq isregarded as an output. For the model equations(1), we identify
f(t,x(t)) = −bixi +
n∑
j=1
AijGj(xj)
and the feedback is given by a nonlinear func-tion F (y) relating the output y(t) of the system(3) and its input u(t): u(t) = −F (y). The block-scheme of such a model is depicted on fig. 1.
Fig. 1. A block-scheme of a nonlinear systems (3)with feedback
The reduced model (via empirical balancedtruncation) that corresponds to (3) has the form:
z(t) = ΠT f(t, T−1Π∗z(t)) +ΠTB(t)u(t)
y(t) = h(t, T−1Π∗z(t)) (4)
where T is the transformation matrix which castsinto a diagonal form both the empirical controlla-bility and observability gramians, associated withthe nonlinear system (3), and Π is a Galerkinprojection [3].
4. Hyperstability of Neural Netwotks withFeedback
The hyperstability property of dynamical systemsis a generalisation of Lyapunov stability. It givesthe most general conditions to be imposed on thesystem in (3) in order to ensure that the solutionsare bounded.
For a completely controllable and completelyobservable linear time-varying system, the seriesof conditions that guarantee hyperstability are:the Kalman-Yakubovich equation and the Popovintegral inequality [9].
For general nonlinear systems in the formof (3) (as opposed to the linear time-varyingsystem), this paper will provide nonlinear ana-logues of the Kalman-Yakubovich equation andthe Popov inequality through the correspondingnonlinear controllability and nonlinear observ-ability gramians [3].
5. Model Reduction and Hyperstability ofNeural Networks
Since most of the nonlinear neural networks withfeedback that are of interest are large-scale sys-tems, it is important to determine whether modelorder reduction for such systems will affect its hy-perstability. In this paper, it is shown that empir-ical balanced truncation preserves hyperstability.
Finally, these results are illustrated by a numeri-cal example.
Acknowledgement. The authors wish to acknowledgethe financial support of Science Foundation Irelandfor this research.
References
1. A. C. Antoulas, Approximation of Large-Scale Dy-namical Systems (Advances in Design and ControlSeries), SIAM, Philadelphia (2003).
2. Borisyuk A., Friedman A., Ermentrout B. andTermanD., Tutorials in Mathematical BiosciencesI: Mathematical Neuroscience, Lecture Notes inMathematics 1860, Springer-Verlag, Berlin, 2005.
3. Condon M. and Ivanov, R. Nonlinear systems-algebraic gramians and model reduction. J. Nonl.Sci. 14 (2004), 405–414.
4. M. Condon, G. G. Grahovski, Stability and ModelOrder Reductions of Nonlinear Systems with Ap-plications to Neural Networks I: Hopfiel TypeModels, Preprint 2008.
5. J. J. Hopfield Neural Networks and PhysicalSystems with Emergent Collective ComputationalAbilities, Proc. Nat. Acad. Sci. USA 79 (1982),2554–2558;Neurons with graded response have collective com-putationa properties like those of two-state neu-rons, Proc. Nat. Acad. Sci. USA 81 (1984), 3088-3092.
6. J. J. Hopfield and D. W. Tank, Computing withneural circuits: A model, Science, vol. 233, pp. 625-633, 1986.
7. Leah Edelstein-Keshet, Mathematical Models inBiology, Classics in Applied Mathematics 46,SIAM, Philadelphia, 2005.
8. A. N. Michel, D. Liu, Qualitative Analysis andSynthesis of Recurrent Neural Networks, MarcelDekker Inc., New York, 2004.
9. V. M. Popov, Hyperstability of Control Sys-tems, Die Grundlehren der Mathematischen Wis-senschaften 245, Springer-Verlag, Berlin, 1973.
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Session
CE 2
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CE 2I
High-order discontinuous Galerkin methods for computational
electromagnetics and uncertainty quantification
J.S. Hesthaven1, T. Warburton2, C. Chauviere3, and L. Wilcox4
1 Division of Applied Mathematics, Brown University, Providence, RI 02912, [email protected]
2 Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005, [email protected]
3 Laboratoire de Mathematiques, Universite Blaise Pascal, 63177 Aubie re, [email protected]
4 Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin, Austin, TX78712, USA [email protected]
Summary. We discuss time-domain discontinuousGalerkin methods for the high-order accurate and ge-ometrically flexible modeling of electromagnetic prob-lems, touching also on extensions to frequency do-main formulations. Using such techniques, we furtherexplore efficient probabilistic ways of dealing with un-certainty and uncertainty quantification in electro-magnetics applications. Whereas the discussion oftendraws on scattering applications, the techniques areapplicable to general problems in CEM.
1 Introduction
As successful at the classic finite-difference time-domain method has been for solving EM prob-lems, its limitations are emerging as a bottleneckin emerging applications. This is particularly thecase for problems characterized by being electri-cally large scale, for problems where interactionswith complex geometries and/or materials are ofimportance or for problems with a high dynamicrange in which case high accuracy is required.
To address these shortcomings, several alter-natives have emerged during the last decade, mostnotably finite-element time-domain methods andhybrid finite-elememt, finite-difference methods.While these offer geometric flexibility and evenhigh-order accuracy, they also result in implicitcomputational methods for time-dependent meth-ods. Yet another alternative is finite-volume meth-ods which has the geometric flexibility and ex-plicit nature but suffers from difficulties whenhigh-order accuracy is requested.
2 DG Methods for EM Problems
By combining the benefits of finite-element andfinite-volume methods, one recovers what has be-come known as discontinuous Galerkin methods
(DG) [6] which has gained increasing interest dur-ing the last decade as a suitable general model-ing tool for EM problems, primarily in the time-domain [3,4] but also for frequency domain prob-lems [5].
We consider the time-domain Maxwell’s equa-tions as a conservation law
Q∂q
∂t+ ∇ · F(q) = 0, (1)
where q is the state vector given by q = (E,H)T
and the components of the tensor F are
Fi(q) =
(−ei × Hei × E
), (2)
where ei denotes the Cartesian unit vectors andwe have the material matrix Q. We assume thatthe computational domain, Ω, is tessellated bytriangles in two spatial dimensions and tetrahe-drons in three spatial dimensions. Given an ele-ment D of the tessellation, the represent the localsolution qN restricted to D is given as
qN (x, t) =
N∑
i=1
qi(t)Li(x), (3)
where qi reflects nodal values, defined on the el-ement. The function Li(x) signifies an pth orderLagrange polynomial. The discrete solution, qN ,of Maxwell’s equations is required to satisfy
∫
D
(Q∂qN∂t
+ ∇ ·F(qN ) − SN
)Li(x)dx(4)
=
∮
∂D
n · [F(qN ) − F∗]Li(x)dx.
In (4), F∗ denotes a numerical flux, whose ex-pression can be found in [4], and n is an outwardpointing unit vector defined at the boundary ∂Dof the element D. Note that this is an entirely
127
local formulation, and relaxing the continuity ofthe elements decouples the elements, resulting ina block-diagonal global mass matrix which can betrivially inverted in preprocessing. The local na-ture of the scheme allows for the use of an explicitRunge-Kutta method for the time discretization.
As an application example, computed usingthis approach, we show in Fig. 1 the scatteredfield on a missile-like structure with a 1GHz inci-dent field.
Fig. 1. 1Gz scattered field on missile computed usinga DG time-domain method.
3 Accounting for Uncertainty inEM using DG
While computational methods have become in-creasingly accurate, their reliance on exact data(e.g. the geometry of the objects, the materialparameters, the sources terms, etc.) is becominga bottleneck in the modeling of complex prob-lems. A standard way to deal with this lackof knowledge or uncertainty, is to assume thatsome of the parameters are random and computemacroscopic quantities (e.g. means and variances)through Monte-Carlo sampling with the advan-tage of its simplicity. However, the rate of con-vergence of Monte-Carlo is quite slow since it isproportional to
√M, where M is the number of
samples. Therefore, designing more efficient nu-merical methods for the solution of stochastic par-tial differential equations with random inputs orrandom coefficients is meeting growing interest.
As an illustration of the versatility of the DGapproach, we shall discuss how ideas of probabilis-tic modeling through Wiener Chaos expansionscan be combined to enable the accurate and effi-cient modeling of large scale EM problems withuncertainty [1, 2]. In Fig. 2 we illustrate this by
showing the bistatic radar cross section (RCS)for a ka = π PEC sphere with a uniformly dis-tributed 10% uncertain radius, clearly showingthe sensitivity of the radar signature. These re-sults are obtained at a fraction of the cost asso-ciated with a brute-force Monte Carlo approach.
-20
-10
0
10
20
30
RC
S(i
nd
b)
45 90 135 180scattering angle θ
RCS averageRCS+sqrt(varience)RCS-sqrt(variance)
Fig. 2. Radar cross section and sensitivity for ka = πsphere of uncertain radius.
Acknowledgement. Partial support for the this workby the National Science Foundation, the Airforce Of-fice of Scientific Research, and the US Airforce Test& Evaluation Program is acknowledged.
References
1. C. Chauviere, J.S. Hesthaven, and L. Lurati.Computational Modeling of Uncertainty in Time-Domain Electromagnetics. SIAM J. Sci. Comp.28:751–775, 2006.
2. C. Chauviere, J.S. Hesthaven, and L. Wilcox. Effi-cient Computation of RCS from Scatterers of Un-certain Shapes. IEEE Trans. Antennas Propagat.55:1437–1448, 2007.
3. J. S. Hesthaven. High-Order Accurate Methods inTime-Domain Computational Electromagnetics. AReview. Advances in Imaging and Electron Physics127:59–123, 2003.
4. J. S. Hesthaven and T. Warburton. High-OrderNodal Methods on Unstructured Grids. I. Time-Domain Solution of Maxwell’s Equations. J. Com-put. Phys., 181:186–221, 2002.
5. J.S. Hesthaven and T. Warburton. High Or-der Nodal Discontinuous Galerkin Methodsfor theMaxwell Eigenvalue Problem. Royal Soc. LondonSer A 362:493-524, 2004.
6. J.S. Hesthaven and T. Warburton, Nodal Discon-tinuous Galerkin Methods: Algorithms, Analysis,and Applications. Springer Texts in Applied Math-ematics 54, Springer Verlag, New York, 2008.
128
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CE 2A
Using Nudg++ to Solve Poisson’s Equation on Unstructured
Grids
Christian R. Bahls, Gisela Poplau, and Ursula van Rienen
Universitat Rostock, A.-Einstein-Str. 2, D-18051 Rostock [email protected],[email protected], [email protected]
Summary. In this paper we explore the viability ofusing nodal discontinuous Galerkin methods as imple-mented in the software library Nudg++ [4] to com-pute the space charge field of an electron or positronbunch. We benchmark this method by comparing re-sults from this scheme with solutions obtained fromother sources.
1 Introduction
Being known from the early 70’s [6] discontinu-ous Galerkin finite element methods (DG-FEM)have only recently become popular as a methodfor the numerical solution of partial differentialequations arising in computational fluid dynam-ics or computational electromagnetism.
Though being primarily developed with hy-perbolic partial differential equations in mind,with some small modifications DG-FEM can alsobe used to solve elliptic PDEs.
The objective of this paper is to apply DG-FEM to the calculation of space charge fieldsof particle bunches. The efficient computation ofthese fields is one of the primary focuses of beamdynamics simulations as needed for the design offuture accelerator facilities like the XFEL [1] orthe ILC [2].
2 Using DG-FEM for the Solutionof Elliptic PDEs
Our aim is the estimation of the solution u(x) ofPoisson’s equation in the domain Ω ⊆ R3:
−∆u(x) = f(x), ∀x ∈ Ω. (1)
We use Dirichlet boundary conditions on ∂ΩDand Neumann boundary conditions on ∂ΩN :
u(x) = gD(x), ∀x ∈ ∂ΩD, (2)
∂u(x)
∂n(x)= gN(x), ∀x ∈ ∂ΩN . (3)
In the context of space charge calculations u(x)denotes the potential and f(x) the source termf(x) = ρ(x)/ε0 with the charge density ρ(x) andthe vacuum permittivity ε0.
Applying DG-FEM to Poisson’s Equation
When looking at the operator resulting from thediscretisation of Poisson’s equation (1) using DG-FEM with a central flux we find that it has a non-trivial kernel. This problem can be mitigated byusing an altered flux that penalizes discontinuitiesat element boundaries as described in [3].
To numerically approximate the solution ofPoisson’s equation (1) we use the special class ofnodal discontinuous Galerkin methods providedby the C++ library Nudg++ [4].
3 Numerical Investigations
We assess the viability of this approach to com-pute the space charge field of charged particles bycomputing the electric field of different test con-figurations. Further we compare these solutionswith results derived from analytic considerationsand from computations with other methods [5].
Acknowledgement. This work is supported by theDFG under contract number RI 814/18-1 and partlyby DESY, Hamburg.
References
1. DESY, Hamburg, Germany. Das europaischeRontgenlaserprojekt XFEL, 2005.
2. GDE (Global Design Effort), www.
linearcollider.org. What is the ILC?3. J.S. Hesthaven and T. Warburton. Nodal Discon-
tinuous Galerkin Methods: Algorithms, Analysis,and Applications, volume 54 of Texts in AppliedMathematics. Springer Verlag, New York, 2008.
4. N. Nunn and T. Warburton. Nudg++: a NodalUnstructured Discontinuous Galerkin framework.www.nudg.org.
5. G. Poplau and U. van Rienen. A self-adaptivemultigrid technique for 3D space charge calcula-tions. IEEE Transactions on Magnetics, 44(4):toappear, 2008.
6. W.H. Reed and T.R. Hill. Triangular mesh meth-ods for the neutron transport equation. Tech. re-port LA-UR-73-479, Los Alamos Scientific Labo-ratory, 1973.
129
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CE 2B
From the Boundary Element DDM to Trefftz Finite Element
Methods on Arbitrary Polyhedral Meshes
Dylan Copeland1, Ulrich Langer1,2, and David Pusch2
1 Institute of Computational Mathematics, Johannes Kepler University Linz, Altenbergerstr. 69, A-4040Linz, Austria [email protected], [email protected]
2 RICAM, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040 Linz, Austria [email protected]
Summary. We derive and analyze new boundary el-ement (BE) based finite element discretizations ofpotential-type, Helmholtz and Maxwell equations onarbitrary polygonal and polyhedral meshes. The start-ing point of this discretization technique is the sym-metric BE Domain Decomposition Method (DDM),where the subdomains are the finite elements. Thiscan be interpreted as a local Trefftz method thatuses PDE-harmonic basis functions. This discretiza-tion technique leads to large-scale sparse linear sys-tems of algebraic equations which can efficiently besolved by Algebraic Multigrid preconditioned conju-gate gradient methods in the case of the potentialequation and by Krylov subspace iterative methodsin general.
1 Introduction
We introduce new finite element methods basedon the symmetric boundary element domain de-composition method presented in [3], which canbe applied with general polygonal or polyhedralmeshes. Each element of the mesh may be anypolygon or polyhedron, as we treat the elementsas subdomains. There are many important prac-tical applications where one wants to discretizePDEs on such kinds of meshes without further de-composition of the polyhedra. Boundary integraloperators are utilized to obtain a method whichsolves for traces of the solution on the elementsurfaces, from which the solution may be obtainedvia a representation formula. Simple, low-orderboundary element spaces are used to approximatetraces on the element surfaces, yielding a finiteelement method with PDE-harmonic basis func-tions.
Since boundary integral operators are usedonly locally, piecewise constant coefficients areadmissible, and the coupling of boundary elementfunctions is local. Consequently, sparse linear sys-tems are obtained, which can be solved by Kryloviterative methods. In the case of the potentialequation, the resulting system is symmetric andpositive definite, and algebraic multigrid is a veryeffective preconditioner in the conjugate gradientsolver.
2 The Potential Equation
Let Ω ⊂ Rd be a bounded domain with a polyg-onal (d = 2) or polyhedral (d = 3) Lipschitzboundary Γ = ∂Ω. As a model problem, we con-sider the potential equation
−div(a(x)∇u(x)) = f(x) for x ∈ Ω (1)
with the Dirichlet boundary condition u = g onΓ . We assume that the coefficient a is piecewiseconstant, f ∈ L2(Ω), and g ∈ H1/2(Γ ). Fur-ther, we suppose that there is a non–overlappingdecomposition of our domain Ω into eh shape-regular polygonal elements Ωi such that Ω =∪ehi=1Ωi, Ωi ∩ Ωj = ∅ for i 6= j, Γi = ∂Ωi,
Γ ij = Γ i ∩ Γ j and a(x) = ai > 0 for x ∈ Ωi,i = 1, . . . , eh. Under the assumptions made above,there obviously exists a unique weak solutionu ∈ H1(Ω) of the BVP (1).
Using the local Dirichlet-to-Neumann map
ai∂u/∂νi = aiSiu|Γi−Nif on Γi, (2)
we observe that the variational formulation of (1)is equivalent to the associated variational formu-lation on the skeleton ΓS = ∪eh
i=1Γi (see, e.g., [4]):find u ∈ H1/2(ΓS) with u = g on Γ such that
eh∑
i=1
∫
Γi
ai(Siui) vidsx =
eh∑
i=1
∫
Γi
(Nif)vidsx (3)
for all v ∈ H1/20 (ΓS), where ui and vi denote
the traces of u and v on Γi, respectively. TheSteklov–Poincare operator Si and the Newton po-tential operator Ni have different representations(see again [4]). Here we are using the symmetricrepresentation
Si = Di + (0.5I +K ′i)V
−1i (0.5I +Ki) (4)
of the local Steklov–Poincare operator Si via thelocal single layer potential integral operator Vi,the local double layer potential operator Ki, itsadjoint K ′
i, and the local hypersingular boundaryintegral operatorDi, see, e.g., [5] for the definition
131
and properties of these boundary integral opera-tors. The operator Ni = V −1
i Ni,0 is defined bythe Newton potential operator
(Ni,0f)(x) =
∫
Ωi
U∗(x − y)f(y)dy, x ∈ Γi, (5)
where U∗(x) = 1/(4π|x|) denotes the fundamen-tal solution of the Laplace operator −∆ for d = 3.
For simplicity we use continuous piecewise lin-ear boundary element functions for approximat-ing the potential u on the skeleton ΓS and piece-wise constant boundary element functions for ap-proximating the normal derivatives ti = ∂u/∂νion the boundary Γi of the polygonal element Ωi.This yields the element stiffness matrices Si,h =
aiDi,h+ai(0.5 I⊤i,h+K⊤
i,h
)(Vi,h
)−1(0.5 Ii,h+Ki,h
)
and the element vectors fi,h = I⊤i,h(Vi,h
)−1fNi,h,
where the matrices Vi,h, Ki,h, Di,h and Ii,h arisefrom the BE Galerkin approximation to the localboundary integral operators Vi, Ki, Di, and tothe identity operator Ii living on Γi, respectively.Ii,h is nothing but the mass matrix. The vectorfNi,h is defined by the Newton potential identity
(fNi,h, ti,h) =
∫
Γi
∫
Ωi
U∗(x−y)f(y)dy th,i(x)dsx (6)
for all vectors ti,h corresponding to the piecewiseconstant functions th,i on Γi. Now, we obtain theBE-based FE system
Shuh = fh (7)
by assembling the stiffness matrix Sh and the loadvector fh from the element stiffness matrices Si,hand the element load vectors fi,h, respectively,and by incorporating the Dirichlet boundary con-dition as usual.
The solution of (7) provides an approximationto the Dirichlet trace of the solution to (1) on theboundary ∂Ωi of all elements Ωi, i = 1, . . . , eh.Applying the Dirichlet-to-Neumann map locally(i.e. element-wise), we may obtain an approxi-mate solution uhto u in each element Ωi via therepresentation formula (see, e.g., [4] or [5]).
Following [3] we immediately obtain the dis-cretization error estimate O(h3/2) in the mesh-dependent norm ‖v‖2
h :=∑eh
i=1 ‖v|Γi‖2H1/2(Γi)
for
a sufficiently (piecewise) smooth solution u, whereuh is the continuous piecewise linear function onthe skeleton ΓS,h corresponding to the Dirichletnodal values and to the nodal values from the so-lution vector uh of (7). This yields the usual O(h)estimate of the discretization error u− uh in theH1(Ω)-norm.
We refer the reader to [2] for more detaileddescription and for the results of our numericalexperiments.
3 Generalization to the Helmholtzand Maxwell Eqautions
Consider the interior Dirichlet boundary valueproblem for the Helmholtz equation
−∆u(x) − κ2u(x) = f for x ∈ Ω, (8)
with u = g on Γ . We assume that the wavenum-ber κ > 0 is constant, or piecewise constant andnot an interior eigenvalue, and g ∈ H1/2(Γ ).
The BE-based FE method for the Helmholtzequation is formally identical to the method pre-sented in the previous section for the potentialequation (with a = 1). One only needs to usedifferent operators Di, Ki, and Vi based on thefundamental solution U∗(x) = eiκ|x|/(4π|x|) forthe Helmholtz operator, see, e.g., [5].
We also study a similar method for the time-harmonic Maxwell equation
curl curl u− κ2u = 0 in Ω, (9)
with the Dirichlet boundary condition γtu :=u×n = g on Γ , where n is the outward unit nor-mal vector. The BE-based FE method for (9) in-volves quite technical trace spaces and boundaryintegral operators. For details and results, see [1].
Acknowledgement. The authors gratefully acknowl-edge the financial support of the FWF research projectP19255.
References
1. D. Copeland. Boundary-element-based finite ele-ment methods for Helmholtz and Maxwell equa-tions on general polyhedral meshes. Technical Re-port 2008-11, RICAM, Linz, 2008.
2. D. Copeland, U. Langer, and D. Pusch. From theboundary element ddm to local trefftz finite el-ement methods on arbitrary polyhedral meshes.Technical Report 2008-10, RICAM, Linz, 2008.
3. G.C. Hsiao and W.L. Wendland. Domain decom-position in boundary element methods. In Pro-ceedings of the Fourth International Symposium onDomain Decomposition Methods for Partial Dif-ferential Equations (ed. by R. Glowinski and Y.A.Kuznetsov and G. Meurant and J. Periaux andO. B. Widlund), Moscow, May 21-25, 1990, pages41–49, Philadelphia, 1991. SIAM.
4. U. Langer and O. Steinbach. Coupled finite andboundary element domain decomposition meth-ods. In ”Boundary Element Analysis: Mathemat-ical Aspects and Application”, ed. by M. Schanzand O. Steinbach, Lecture Notes in Applied andComputational Mechanic, Volume 29, pages 29–59, Berlin, 2007. Springer.
5. O. Steinbach. Numerical Approximation Methodsfor Elliptic Boundary Value Problems. Finite andBoundary Elements. Springer, New York, 2008.
132
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CE 2C
Reduced Basis Method for Electromagnetic Field
Computations
Jan Pomplun and Frank Schmidt
Zuse Institute Berlin, Takustrasse 7, D-14195 Berlin, Germany [email protected], [email protected]
Summary. We explain the reduced basis (RB) methodapplied to electromagnetic field computations withthe finite element method. Rigorous numerical sim-ulations for practical applications often become verytime consuming. The RB method allows to split upthe solution process off an e.g. geometrically param-eterized problem into an expensive offline and a fastonline part. For an actual simulation only the fast on-line part is evoked. We apply the RB method to thesimulation of a parameterized phase shift mask.
1 Introduction
The finite element method (FEM) has been suc-cessfully applied to a large number of nano-opticalproblem classes [2, 5]. The computation time ofa single FEM simulation however can becomevery long especially for 3D problems. Design, op-timization and inverse problems usually involve alarge number of such simulations having an un-derlying layout with a few varying geometricalparameters. The RB method can be applied tothis setup [4]. In the offline step the underlyingmodel is solved rigorously several times for differ-ent values of the geometrical parameters. Thesesolutions built the reduced basis. The full param-eterized problem is projected onto the RB whichresults in a significant reduction of the problemsize. In the online step the reduced problem issolved.
2 Reduced Basis Method
For electromagnetic field computations Maxwell’sequations have to be solved:
∇× µ−1∇× E − ω2ǫpE = 0. (1)
Since we consider geometrically parameterizedproblems the material distribution described bythe permittivity ǫp is parameter dependent whichis denoted by the subscript p. Discretizing (1)with the finite element method leads to a linearsystem of equations [3] with parameter dependentcoefficients and solution:
Apup = f. (2)
In practice the number of unknowns (dimensionof up) can be up to several millions. Now sup-pose that up stays on a low dimensional submanifold of the complete solution space if wevary the parameters p. Then it is reasonable tosolve (2) only on this subspace, i.e. the full sys-tem is projected onto a reduced basis U . Usu-ally the reduced basis is built by a number of socalled snapshot solutions, which are rigorous so-lutions of (2) for different parameter values pi:U = [up1 , up2 , . . . , upN ] . The reduced basis iscomputed in the offline step. The projected sys-tem reads:
[UHApU
]λp = UHf (3)
which is a system of dimension N (∝ 100) muchsmaller then the original problem. After solving(3) in the online step the reduced basis solutionis obtained by up = Uλp.
2.1 Affine Parameter Dependence
The expensive steps performing a reduced basiscomputation for an actual parameter set p is ac-cording to (3) the assembling of Ap and the pro-jection step onto the RB U . These steps can alsobe performed offline if the matrix Ap has an affineparameter dependence defined by:
Ap =M∑
i=1
gi(p)A(i) (4)
⇒[UHApU
]=
M∑
i=1
gi(p)[UHA(i)U
].
The matrices[UHA(i)U
]can be assembled and
projected offline. For finite elements on a rectan-gular grid such an affine parameter dependence isgiven [6]. For unstructured triangular grids how-ever the matrix entries of Ap are rational expres-sions in the parameters with different roots inthe denominator leading to a non-affine param-eter dependence. This non-affine dependence isdue to the term gp = 1
detJpJTp Jp assembling the
local finite element matrices. Jp is the Jacobianof the transformation of a reference triangle ontoa parameter dependent physical triangle of thecomputational domain [3].
133
2.2 Empirical Interpolation
Empirical interpolation [1] is a method to decom-pose a matrix with non-affine parameter depen-dence into a form according to (4):
Ap ≈L∑
j=1
αj(p)Aj . (5)
The basic idea is to assemble a few snapshot ma-trices Aj for certain parameters pj and to deter-mine the coefficients αj(p) by a fit of the non-affine function gp with functions gpj :∣∣∣∣gp − αj(p)gpj
∣∣∣∣ != min .
Increasing the number of interpolation matrices,i.e. L should allow a more accurate approximationof Ap (5) and therewith a more accurate reducedbasis solution.
3 Example
As an example we compute transmission of lightthrough a phase shift mask [6]. The parameter-ized geometry is shown in Fig. 1. To quantify the
Fig. 1. Parameterized phase shift mask for reducedbasis computations.
quality of the reduced basis solution for a certainparameter set p = d1, d2, d3 we compare it tothe exact finite element solution for this param-eter set. Figure 2 shows the convergence of thereduced basis solution for increasing dimension ofthe reduced basis and different number of param-eter sets L for empirical interpolation. The erroris computed in the L2 norm. For comparison thereduced basis solution without empirical interpo-lation is shown. We observe that the reduced ba-sis solution converges exponentially towards theexact finite element solution. For fixed L the con-vergence is as fast as without empirical interpola-tion but stops at a specific level. However taking amore accurate empirical interpolation dimensionL this level can be lowered.
0 20 40 60 80 100dimension of reduced basis
10-5
10-4
10-3
10-2
10-1
100
L2
erro
r
L=30
L=50
L=70
no empirical interpolation
Fig. 2. Convergence of reduced basis solution for in-creasing dimension of reduced basis and different di-mension of empirical interpolation ansatz L.
4 Conclusions
We have shown that the reduced basis method isvery well suited for electromagnetic field compu-tations with a parameterized geometry. Exponen-tial convergence was observed for transmission oflight through a phase shift mask. For finite ele-ments on an unstructured grid a non-affine pa-rameter dependence of the system matrix is ob-tained. However with empirical interpolation theassembling of the reduced basis system can beperformed offline.
Acknowledgement. We acknowledge fruitful and stim-ulating discussions with Zhenhai Zhu from CadenceDesign Systems, Inc.
References
1. M. Barrault, N.C. Nguyen, Y. Maday, and A.T.Patera. An ”empirial interpolation” method: Ap-plication to efficient reduced-basis discretizationof partial differential equations. C. R. Acad. Sci.Paris, I(339):667–672, 2004.
2. S. Burger, R. Kohle, L. Zschiedrich, W. Gao,F. Schmidt, R. Marz, and C. Nolscher. Bench-mark of FEM, Waveguide and FDTD Algorithmsfor Rigorous Mask Simulation. In J. T. Weed andP. M. Martin, editors, Photomask Technology, vol-ume 5992, pages 378–389. Proc. SPIE, 2005.
3. Peter Monk. Finite Element Methods forMaxwell’s Equations. Oxford University Press,2003.
4. A.T. Patera and G. Rozza. Reduced Basis Approx-imation and A Posteriori Error Estimation forParametrized Partial Differential Equations. MITPappalardo Graduate Monographs in MechanicalEngineering, 1. edition, 2006.
5. J. Pomplun, S. Burger, F. Schmidt, F. Scholze,C. Laubis, and U. Dersch. Metrology of EUVmasks by EUV scatterometry and finite elementanalysis. In Photomask and NGL Mask Technol-ogy XV, volume 7028, page 24. Proc. SPIE, 2008.
6. Z. Zhu and F. Schmidt. An efficient and robustmask model for lithography simulation. volume6925, page 126. Proc. SPIE, 2008.
134
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CE 2D
Surface Integrated Field Equations Method for solving 3D
Electromagnetic problems
Zhifeng Sheng1 and Patrick Dewilde2
1 Circuits and Systems, EEMCS, TUDelft, The Netherlands [email protected],2 Circuits and Systems, EEMCS, TUDelft, The Netherlands [email protected]
Summary. This paper describes how the Surface In-tegrated Field Equations method (SIFE) can be im-plemented to solve 3D Electromagnetic (EM) prob-lems on substrates in which high contrast materialsoccur. It gives an account of the promising resultsthat are obtained with it when compared to tradi-tional approaches. Advantages of the method are thehighly improved flexibility and accuracy for a givendiscretization level, at the cost of higher computa-tional complexity.
Key words: Integrated Field Equations, timedomain, high contrasts, computational method
1 Method Outline
In our previous work, we have used the SurfaceIntegrated Equations method for solving 2D elec-tromagnetic problems [2], in which domains arepresent that exhibit highly contrasting materialproperties (electric and/or magnetic) with eachother. In this paper, we develop the method fur-ther to solve 3D electromagnetic problems.
In strongly heterogeneous media such as mod-ern chips, the constitutive parameters can jumpby large amounts upon crossing the material in-terfaces. On a global scale, the EM field compo-nents are not differentiable and Maxwell’s equa-tions in differential form cannot be used, one hasto resort to the original integral form of the EMfield relations as the basis for the computationalmethod.
Let D be the domain of interest with bound-ary ∂D, S be any (sufficiently smooth and small)surface (S ∈ D) with boundary ∂S in D, T be atime interval with boundary points ∂T . For any Sand T , Maxwell’s equations in surface integratedform are (using Einstein’s summation conventionfor repeated indices, which is consistently usedthroughout this paper):Z
t∈T
dt
I
x∈∂S
Hk(x, t)dlk =
Z
x∈S
Dq(x, t)dsq
˛
˛
˛
˛
t∈∂T
,
Z
t∈T
dt
I
x∈∂S
Ek(x, t)dlk = −
Z
x∈S
Bq(x, t)dsq
˛
˛
˛
˛
t∈∂T
,
where dlk = τkdl, dsq = νqds, νq is the unit vec-tor normal to the bounded surface area S. This
vector is oriented toward the side of advance of aright-hand screw as it is turned around ∂S in thedirection of the unit vector τk.
The constitutive relations are:
Dk(x, t) =
Z ∞
t′=0
ηk,r(x, t′)Er(x, t− t′)dt′ + P ext
k (x, t),
Bj(x, t) =
Z ∞
t′=0
ζj,p(x, t′)Hp(x, t− t′)dt′ +Mext
j (x, t).
where P extk is the (field-independent) external
electric polarization(C/m2); Mextj is the (field-
independent) external magnetization(T); Assum-ing instantaneously reacting isotropic media, theconstitutive relations take the form:
ηk,r(x, t) = ǫ(x)δ(t) + σ(x)δk,rH(t)
ζk,r(x, t) = µ(x)δ(t) + κ(x)δk,rH(t)
where κ is the linear magnetic hysteresis (H/m.s),δ(t) the Dirac delta distribution in time t, H(t)the Heaviside unit step funciton and δk,r the sym-metrical unit tensor of rank 2 (Kronecker tensor).
To satisfy the partial continuity conditions onmaterial interfaces, we construct a so called ’con-sistent linear discretization schema’ that meetsthe continuity requirements across interfaces ex-actly, using a 3D tetrahedra mesh combined witha consistent linear interpolation [1] of electric andmagnetic field strengths.
Furthermore, we use in this paper simple bound-ary conditions to truncate the computational do-main (our goal being to show the validity of themethod, we are developing a package with generalboundary conditions).
2 Discretization schema
The electric and magnetic field strengths are in-terpolated with linear nodal expansion functionsin homogeneous subdomains, where the EM fieldis totally continuous. At material interfaces, how-ever, the field strengths are interpolated withedge based linear expansion functions. We referto [2] for details on this novel kind of approxi-mation that allows for partial discontinuities at
135
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−3
−2
−1
0
1
2
3
Number of elements ( base 10 logarithmic axis )
Relat
ive m
ean s
quare
s erro
r ( ba
se 10
loga
rithmi
c axis
)
RMSE in subdomain0 : Hybrid LSFIMRMSE in subdomain1 : Hybrid LSFIMRMSE in subdomain0 : Nodal LSFIMRMSE in subdomain1 : Nodal LSFIMRMSE in subdomain0 : Galerkin(w=0.3)RMSE in subdomain1 : Galerkin(w=0.3)
Fig. 1. Relative mean squares error (RMSE) in Hcomputed with SIFE method and Galerkin’s method.
−10.6−10.5−10.4−10.3−10.2−10.1−10−9.9−9.8−9.7−9.6−2.5
−2
−1.5
−1
−0.5
0
time step size ( base 10 logarithmic axis )
Spac
e−tim
e RMS
E ( b
ase 1
0 log
arithm
ic ax
is )
RMSE in E : computed with GalerkinRMSE in H : computed with GalerkinRMSE in E : computed with LSFIMRMSE in H : computed with LSFIM
Fig. 2. Mean square error Vs time step size compar-ison of the methods tested
0.20.40.60.811.21.41.61.82x 10
−10
0
1000
2000
3000
4000
5000
6000
7000
Time step size (s)
total
numb
er of
Iterat
ions n
eede
d
Total number of iterations needed by GalerkinTotal number of iterations needed by LSFIM
Fig. 3. Total number of iterations Vs time step size
interfaces, in particular the discontinuity of nor-mal components. We use, in addition, a regulartime stepping schema with time step choosen inaccordance with the space discretization, and thetrapezium rule to approximate the time integrals.
After properly assembling the local matriceswe obtain an overdetermined system of equationsthat we solve numerically using the Bicg-stabmethod. We have studied the source of overde-termination, which can be reduced to numericaltransformations of local quantities and can hencebe controlled adequately (this issue is beyond thescope of the present paper - here we suffice withcomputing a least squares solution directly usinga pre-conditioner). The pre-conditioner that weuse incorporates incomplete Cholesky factoriza-tion on the system matrix, which may introducea lot of fill-ins in the incomplete Cholesky factor.Reordering of the system matrix significantly re-duces the fill-ins. For time domain problems, thesolution of the previous time instant is used forthe initial guess of the current time instant.
3 Numerical experiments
We present a number of numerical experimentsfor the 3D case that show the superior perfor-mance of the method. In particular:
• We verify the accuracy and convergence ofspacial discretization schema through 3D mag-netostatic problems with extreme high con-trast in permeability, for which an analyticsolution is known. As indicated in Fig. 1, theSIFE method based on hybrid linear finite ele-ment delivers accurate solutions while the con-ventional Galerkin method does not.
• We verify the accuracy and stability of themethod in the time domain by comparingour method with the conventional Galerkinmethod on computing 3D EM fields in in-homogeneous media in the time domain forwhich analytic solution are available. As in-dicated in Fig. 2 and Fig. 3, the time do-main convergence rate of the SIFE methodis higher than Galerkin’s method without in-creasing compuational cost.
• We are working on a software package thatshall demonstrate the practical usage of ourmethod in VLSI circuit simluation by simlu-ating EM wave propagation through the sub-strate and the interconnect in the time do-main.
4 Conclusions
The SIFE method holds considerable promise tosolve three dimensional time domain electromag-netic problems, in which high contrasts betweendifferent types of materials exist and irregularstructures are present. It handles irregular struc-tures and partial discontinuities in a systemati-cally correct and elegant fashion. Its accuracy andstability has now been demonstrated and veri-fied with numerical experiments. With these basicproperties being established, we are now workingon improvements of the computational propertiesof the method in terms of numerical complexityand versatility.
References
1. Gerrit Mur. The finite-element modeling of three-dimensional electromagnetic fields using edge andnodal elements. IEEE tranactions on antennasand propagation, 41(7), July 1993.
2. Zhifeng Sheng, Rob Remis, and Patrick Dewilde.A least-squares implementation of the field inte-grated method to solve time domain electromag-netic problems. Computational Electromagneticsin Time-Domain, 2007. CEM-TD 2007. Workshopon, pages 1–4, 15-17 Oct. 2007.
136
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CE 2E
A Novel Staggered Finite Volume Time Domain Method
Thomas Lau, Erion Gjonaj, and Thomas Weiland
Institut fur Theorie Elektromagnetischer Felder,Technische Universitat Darmstadt, Darmstadt, Germany [email protected], [email protected], [email protected]
Summary. This work presents a novel finite volumetime domain method on Cartesian grids. In compar-ison to the finite difference time-domain scheme ofYee the method offers a better accuracy while stillmaintaining a compact stencil. Additionally, mimeticdiscrete vectoranalytical operators for the method areconstructed.
1 Introduction
In the majority of finite volume time-domain(FVTD) methods the electric degrees of freedom(DOF) and the magnetic DOF are co-locatedon the same spatial position [2]. Unfortunately,compared to the classical finite difference time-domain (FDTD) scheme of Yee [4] this co-locationreduces the accuracy of the FVTD method. Hence,one approach to increase the accuracy of theFVTD method, motivated by the Yee scheme, isto stagger the DOFs. However, the staggering canbe performed in three different ways: First, stag-gering the electric and magnetic fields, only. Sec-ond, staggering the vectorial components of theDOFs, only. Third, combining both approachesand staggering the electric and magnetic DOFsand each of their vectorial components. The lastapproach corresponds to the staggering applied inthe Yee scheme.
In this work the authors restrict their investi-gation to the second strategy, and only considerthe specific staggering, shown in figure 1.
2 Construction of the FVTDmethod
Starting point for constructing the FVTD method,neglecting electric currents for the sake of brevity,is the volume integral formulation of Maxwell’sequations [1] over a domain Ω:
∀V ⊆ Ω,d
dt
∫
V
µHdV = −∫
∂V
dA × E, (1)
∀V ⊆ Ω,d
dt
∫
V
ǫEdV =
∫
∂V
dA× H (2)
x
y
z
Ex
Hx
Ey
Hy
Ez
Hz
Fig. 1. Staggering of the field DOFs. The DOFs aredefined, according to the vectorial component theyrepresent, on three different dual volumes (shaded)with respect to the primary grid cell (gray).
x
y
z
Ex Hz
HyEx
a) b)
Fig. 2. The time evolution of the integral of Ex overits control volumes (shaded cell) solely depends on thefluxes trough the y and z faces, respectively. Figurea) shows the flux through the y face (red) which isuniquely defined by the values of Hz on its controlvolumes (solid cells). Figure b) shows the flux throughthe z face (red) of the same cell which is uniquelydefined by the values of Hy on its control volumes(solid cells).
with the electric field, E, and the magnetic field,H. Different materials in Ω are characterized bytheir permeability, µ, and permitivity, ǫ, respec-tively. The domain Ω is discretized by restrictingthe arbitrary volumes V in equation (1) and (2) toa finite set of specific control volumes Vx, Vyand Vz for each vectorial component respec-tively. For a primary Cartesian grid, xi, yj, zkwith i = 1 . . . I, j = 1 . . . J , k = 1 . . .K, the vol-
137
umes are defined by
[Vx]ijk = [xi, xi+1] [yj , yj+1] [zk, zk+1] , (3)
[Vy]ijk = [xi, xi+1] [yj , yj+1] [zk, zk+1] , (4)
[Vz ]ijk = [xi, xi+1] [yj , yj+1] [zk, zk+1] , (5)
xi =(xi + xi−1)
2, . . . . (6)
The field DOF are defined by the volume av-erages of the electric and magnetic field compo-nents eα and hα. From equation (2) the updateequation for Ex is calculated as
d
dt
∫
Vx
Ex dV =
∫
∂Vx
dAyHz −∫
∂Vx
dAzHy. (7)
Assuming that the field components are constantinside their control volumes, respectively, it ispossible to calculate the fluxes directly. This isillustrated in figure 2 for the component Ex. Inthe same manner the equations for the othercomponents of the electric and magnetic fieldsare derived. Assigning constant, effective mate-rial parameters ǫ and µ to each control volume asemi discrete approximation of the equations (1)and (2) is obtained.
3 Properties of the FVTD method
It is possible to define, in analogy to the FITmethod [3], discrete vector analytical operatorsdiv, grad, rot which mimics the algebraic proper-ties of their continuous counterparts.
The dispersion relation of the FVTD methodon a Cartesian grid with grid spacing ∆ and timestep ∆t is
sin(∆tω
2) = σ2(KYee − δK),
KYee =
(sin2(
βx2
) + sin2(βy2
) + sin2(βz2
)
),
δK =1
2
∑
γ 6=δsin2(
βγ2
) sin2(βδ2
),
σ = c∆t/∆(Courant number),
with γ, δ ∈ x, y, z. A more detailed analysis re-veals that the dispersion properties of the FVTDare in average superior to that of the Yee scheme.This is illustrated in figure 3 which shows therelative error in the phase velocity for five gridpoints per wavelength at the stability limit ofboth methods, respectively. The maximal stabletime step of the FVTD method is roughly 1.7times higher than that of the Yee scheme on ahomogeneous mesh. Both schemes offer a magictime step at their stability limit. For the Yeescheme waves along the space diagonals and forthe FVTD method waves along the axes are prop-agated exactly.
-0.04-0.02
00.02
0.04x
-0.04-0.02
00.02
0.04y
-0.04
-0.02
0
0.02
0.04
z
Yee
FVTD
Fig. 3. Relative error in the phase velocity of the Yeescheme and the FVTD method for five grid points perwavelength at their stability limit, respectively.
References
1. E. J. Rothwell and M. J. Cloud. Electromagnetics.CRC Press, Boca Raton, 2001.
2. V. Shankar, W. F. Hall, and A. H. Mohammadian.A time-domain differential solver for electromag-netic scattering problems. Proc. of IEEE, 77:709–721, 1989.
3. T. Weiland. A discretisation method for the so-lution of maxwell’s equations for six-componentfields. Elektrotechnik und bertragungstechnik (AE),31:116–120, 1977.
4. K. S. Yee. Numerical solution of initial valueboundary problems involving maxwell’s equa-tions in isotropic media. IEEE Trans. AntennasPropagt., 14:302–307, 1966.
138
Session
CS 2
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 2I
Wavelets in circuit simulation
Emira Dautbegovic
Qimonda, Am Campeon 1-12, 85579 Neubiberg, Germany [email protected]
Summary. Wavelet theory is a relatively recent areaof scientific research, with a very successful applica-tion in a broad range of problems such as image andsignal processing, mathematical modeling, electro-magnetic scattering and many more. The key waveletproperty contributing to this success is the capabilityof a simultaneous time and frequency representationof a signal. The potential exploitation of this prop-erty for a next-generation, wavelet-based simulationof analog circuits is discussed in this paper.
1 Circuit simulation
Analog circuit simulation is the standard indus-try approach to verify an integrated circuit (IC)operation at the transistor level before commit-ting a designed circuit to the expensive manu-facturing process. A starting point for an ana-log circuit simulator (e. g. SPICE and its deriva-tives) is usually a text file (netlist) that describesthe circuit elements (resistors, capacitors, tran-sistors, etc.) and their connections. This input isthen parsed and translated to a data format re-flecting the underlying mathematical model of thesystem. This is done by applying the basic phys-ical laws (such as energy and charge conserva-tion laws) onto network topology and taking intoaccount characteristic equations for the networkelements. The most used “translation” approachis the modified nodal analysis (MNA). This au-tomatic modeling approach aims to preserve thetopological structure of the network instad of de-scribing a system with a minimal set of unknowns.The resulting mathematical model is in form ofan initial-value problem of differential-algebraicequations (DAEs). For example, network equa-tions in charge/flux oriented MNA formulationhave a following semi-implicit nonlinear form:
Adq(x)
dt+ f(x) = s(t). (1)
The incidence matrixA has entries 0, 1,−1 andis in general singular. x is the vector of node po-tentials and specific branch voltages and q is thevector of charges and fluxes. f comprises staticcontributions, while s contains the contributionsof independent sources. t is the time.
The underlying DAE system (1) is then solvedusing Newton’s method, implicit integration meth-
ods and sparse matrix techniques. In general, (1)is a stiff system (i. e. contains characteristic con-stants of several orders of magnitude) and suffersfrom poor smoothness properties of modern tran-sistor model equations. Furthermore, if more gen-eral models for network elements are utilized, orrefined models are used to include second orderand parasitic effects, an ill-conditioned problemmay arise and very special care must be taken toavoid divergence during the numerical solution ofsuch a system.
Modern industrial circuit simulators are fac-ing two serious challenges today: the simulationof very large circuits with millions of transistors(e. g. memory chips) on the one hand, and sim-ulation of RF circuits comprising multitone os-cillatory circuits with widely-separated tones andhigh amount of digital content, on the other hand.
The sheer size of the simulation of large cir-cuits containing millions of transistors yields sim-ulations that are impractical (they can last weeks,even longer than a month) or infeasible due to ex-treme memory and computational requirements.
On the other hand, when simulating RF cir-cuits or circuits that contain RF portions stan-dard analog circuit simulators are both inefficient,due to the presence of multitone signals withwidely separated tones, and inadequate since lin-earization around DC operating point done by astandard circuit simulator causes loss of informa-tion on non-linear effects. Therefore, a special-ized RF simulator based on either the frequency-domain Harmonic Balance or the time-domainShooting algorithm [1] is employed for the sim-ulation of RF circuits. Both methods make ana-priori assumption that the signals present in acircuit are (quasi) periodic and have a narrow-band spectrum (i. e. contain only a small num-ber of frequencies). With the first assumption, anRF simulator gains efficiency of simulations butloses generality (i. e. ability to simulate non-RFcircuits). Furthermore, the second assumption isbeing considerably violated lately, since modernRF designs feature an ever-increasing number offrequencies present in their spectrum (due to asharp increase in digital content), as well as in-clude intrinsically broadband circuit parts (e. g.high-ratio dividers).
141
Currently, the RF front-end is usually de-signed by employing a frequency domain RF sim-ulator, while the rest of the circuit is designed inthe time domain employing the standard circuitsimulation techniques. Due to this separation dur-ing the design process, these RF/non-RF circuitsare usually not realized on the same die in orderto keep spurious couplings between them as smallas possible, since they cannot be easily character-ized in a common simulation environment. How-ever, with the trend towards ever-decreasing chipsize, integration of RF and non-RF part on a samedie is eminent and new simulation tools that cansupport these designs are urgently needed.
2 Wavelets in circuits simulation
Wavelet-based algorithms are already in produc-tive use in a broad range of scientific computa-tion applications, such as image and signal pro-cessing (signal and image compression, nonlinearfiltering, etc.), numerical analysis (solving oper-ator equations, boundary value problems, largematrix equations, etc.), the compression of theFBI fingerprints data, to name just few. Factorscontributing to this success are the intrinsic prop-erty of the localized support of a wavelet basis,the numerically efficient Multi-Resolution Analy-sis (MRA) framework and the addaptive approx-imation approach, which leads to a uniform errordistribution over approximation interval.
The name wavelet or ondelette was coined inthe early 1980’s by French researchers [2] meaningthe small wave. Smallness refers to the fact thata wavelet is a locally supported function (i. e. oflimited duration) and wave refers to the conditionthat it has to be an oscillatory function (i. e. hasan average zero value).
In circuit simulation wavelets may be usedas basis functions to represent electrical signals.The wavelet basis is formed via translations anddilatations from a single wavelet function Ψ(x),called mother wavelet, and noted as:
Ψs,τ (x) =1√sΨ
(x− τ
s
), (2)
where (s, τ) ∈ IR+ × IR. All wavelets from aspecific basis are shifted (parameter τ) and di-lated/compressed (by factor s) versions of thismother wavelet. The translation parameter τ isresponsible for the localization in time of a cor-responding wavelet. The scaling parameter s, orthe scale, is related to the frequency in that it isthe frequency inverse. Therefore, the high scalecorresponds to low frequencies or a global view ofthe signal and low scales correspond to high fre-quencies or a detailed view of the signal. Finally,
the factor 1/√s is used for energy normalization
across different scales. From the definition it isclear that a wavelet basis intrinsically supportsa simultaneous time-frequency representation ofsignal.
The ongoing investigation into use of wavelets[3–7] has shown that their intrinsic propertiesmake wavelets the natural candidate for a success-ful successor of time-marching (transient anal-ysis) and/or frequency domain (Harmonic Bal-ance analyses) paradigms used in circuit simula-tion today. However, although valuable as a proof-of-concept, current wavelet approaches to circuitsimulation are essentially academic in scope, withalgorithms tested only on small and simple sam-ple circuits.
In conclusion, a substantial effort is neces-sary to drive further advances in wavelet-basedsimulation techniques to the point when waveletanalysis is mature enough to be driven into pro-ductive use within industrial simulators. Howeverthe expected benefit that a wavelet-based simula-tion engine could bring both in quantitative terms(reduction in computation time and memory re-quirements) as well as qualitative terms (analyz-ing signals with resolutions adapted to a problemat hand) is well worth allocating effort in a bidto develop the next-generation circuit simulators,capable of answering industrial challenges of to-morrow.
References
1. K. Kundert. Simulation methods for RF in-tegrated circuits. Proc. IEEE/ACM Int. Conf.Computer-Aided Design, pages 752–765, 1997.
2. A. Grossmann and J. Morlet. Decomposition ofhardy functions into square integrable wavelets ofconstant shape. SIAM Journal of MathematicalAnalysis, 15:723–736, 1984.
3. D. Zhou and W. Cai. A fast wavelet collocationmethod for high-speed circuit simulation. IEEETrans. Circuit and Systems, 46:920–930, 1999.
4. N. Soveiko, E. Gad, and M. Nakhla. A wavelet-based approach for steady-state analysis of non-linear circuits with widely separated time scales.IEEE Microwave and Wireless Components Let-ters, 17:451–453, 2007.
5. C.E. Christoffersen and M.B. Steer. State-variable-based transient circuit simulation usingwavelets. IEEE Microwave and Wireless Compo-nents Letters, 11:161–163, 2001.
6. E. Dautbegovic, M. Condon, and C. Brennan.An efficient nonlinear circuit simulation technique.IEEE Trans. Microwave Theory and Techniques,53:548–555, 2005.
7. A Bartel, S. Knorr, and R. Pulch. Wavelet-basedadaptive grids for multirate partial differential-algebraic equations. submitted to Appl. Numer.Math., 2007.
142
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 2A
Hybrid Analysis of Nonlinear Time-Varying Circuits
Providing DAEs with at Most Index 1
Satoru Iwata1, Mizuyo Takamatsu2, and Caren Tischendorf3
1 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, [email protected]
2 Department of Mathematical Informatics, Graduate School of Information Science and Technology,University of Tokyo, Tokyo 113-8656, Japan [email protected]
3 Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Koln, [email protected]
Summary. Commercial packages for a transient cir-cuit simulation are often based on the modified nodalanalysis which allows an automatic setup of modelequations and needs a nearly minimal number of vari-ables. However, it may lead to DAEs with higher in-dex. Here, we present a hybrid analysis for nonlineartime varying circuits leading to DAEs with at mostindex 1. This hybrid analysis is based on the networktopology only. Consequently, an automatic setup ofthe hybrid equations from netlists is possible.
1 Introduction
When modelling electrical circuits for a transientsimulation, one has to regard Kirchhoff’s laws forthe network and the constitutive laws for the dif-ferent types of network elements. They are orig-inally based on the branch voltages u und thebranch currents i existing in the network. Theyform the basis for all modelling approaches as forinstance the popular modified nodal analysis.
Concerning the huge number of variables in-volved (all branch voltages and branch currents)one is interested in a reduced system reflecting thecomplete circuit behaviour that can be generatedautomatically. Whereas the modified nodal anal-ysis focusses on a description depending mainlyon nodal potentials, our hybrid analysis approachhere bases on certain branch voltages and branchcurrents obtained from a construction of normaltrees, i.e. from trees containing all independentvoltage sources, no independent current source,a maximal number of capacitive branches and aminimal number of inductive branches.
Such normal trees have already been used in[1] for state approaches for linear RLC networks.The results have been extended in [4] for linearcircuits containing ideal transformers, nullors, in-dependent and controlled sources, resistors, in-ductors, and capacitors, and, under a topologicalrestriction, gyrators. Furthermore, it formed thebasis for developing a hybrid approach for RLC
networks providing a DAE system with minimalindex [3].
Here, we follow the hybrid approach in [3]but use projection techniques (cf. [2]) in orderto prove the index results for general nonlineartime-varying circuit systems.
2 Hybrid Analysis
Here, we consider nonlinear time-varying circuitscomposed of resistors, inductors, capacitors, volt-age and current sources. For simplicity, we restrictto independent sources only. The case of depen-dent sources is under current research but seemsno to form a limit of the approach when certaintopological criteria are satisfied.
Let Γ = (W,E) be the network graph withvertex set W and edge set E. An edge in Γ cor-responds to a branch that contains one elementin the circuit. We denote the set of edges corre-sponding to independent voltage sources and in-dependent current sources by Ev and Ej , respec-tively. We split Evj := E \ (Ev ∪Ej) into Ey andEz, i.e., Ey ∪ Ez = Evj and Ey ∩ Ez = ∅. A par-tition (Ey , Ez) is called an admissible partition, ifEy includes all the capacitances, and Ez includesall the inductances.
We call a spanning tree T of Γ a referencetree if T contains all edges in Ev, no edges inEj , and as many edges in Ey as possible. Notethat a reference tree T may contain some edgesin Ez . The cotree of T is denoted by T = E \ T .A reference tree is called normal if it contains asmany edges corresponding to capacitors and asfew edges corresponding to inductors as possible.
We denote the vector of currents through allbranches of the circuit by i, and the vector of volt-ages across all branches by u. Let V , J , C, and Ldenote the sets of independent voltage sources, in-dependent current sources, capacitors, and induc-tors, respectively. Moreover, G and R denote the
143
sets of conductors and resistors in Ey and Ez , re-spectively. Given an admissible partition (Ey , Ez)and a normal reference tree T , we split i and uinto
i = (iV , iτC , i
τG, i
λC , i
λG, i
τR, i
τL, i
λR, i
λL, iJ)
and
u = (uV ,uτC ,u
τG,u
λC ,u
λG,u
τR,u
τL,u
λR,u
λL,uJ)
where the subscripts correspond to elements andthe superscripts τ and λ designate the tree T andthe cotree T .
The physical characteristics of elements deter-mine constitutive equations. Conductors includedin Ey are modelled by constitutive equations ofthe form
iτG = gτ (uτG,uλG) and iλG = gλ(uτG,u
λG). (1)
Resistors included in Ez are modelled by
uτR = rτ (iτR, iλR) and uλR = rλ(iτR, i
λR). (2)
All capacitors are given by
iτC =d
dtqτ (uτC ,u
λC) and iλC =
d
dtqλ(uτC ,u
λC).
(3)All inductors are given by
uτL =d
dtφτ (iτL, i
λL) and uλL =
d
dtφλ(iτL, i
λL).
(4)Independent voltage and current sources simplyread as
uV = vs(t) and iJ = js(t). (5)
Kirchhoff’s current law may be written as
Qi = 0
where Q denotes the fundamental cutset matrix
iV iτC i
τG i
λC i
λG i
τR i
τL i
λR i
λL iJ
I 0 0 AV C AV G 0 0 AV R AV L AV J
0 I 0 ACC ACG 0 0 ACR ACL ACJ
0 0 I 0 AGG 0 0 AGR AGL AGJ
0 0 0 0 0 I 0 ARR ARL ARJ
0 0 0 0 0 0 I 0 ALL ALJ
,
Kirchhoff’s voltage law provides
Bu = 0
with B being the fundamental loop matrix
uV uτC u
τG u
λC u
λG u
τR u
τL u
λR u
λL uJ
-A⊤V C -A⊤
CC 0 I 0 0 0 0 0 0
-A⊤V G -A⊤
CG -A⊤GG 0 I 0 0 0 0 0
-A⊤V R -A⊤
CR -A⊤GR 0 0 -A⊤
RR 0 I 0 0
-A⊤V L -A⊤
CL -A⊤GL 0 0 -A⊤
RL -A⊤LL 0 I 0
-A⊤V J -A⊤
CJ -A⊤GJ 0 0 -A⊤
RJ -A⊤LJ 0 0 I
.
3 Hybrid Equations
Performing the hybrid analysis presented in [3],we obtain the hybrid equation system
−A⊤CRu
τC −A⊤
GRuτG −A⊤
RRrτ + rλ = A⊤
V Rvs(t)
−A⊤CLu
τC −A⊤
GLuτG −A⊤
RLrτ
−A⊤LLφ
τ + φλ = A⊤V Lvs(t)
ACGgλ +ACRi
λR +ACLi
λL
+ qτ +ACC qλ = −ACJjs(t)
gτ +AGGgλ +AGRi
λR +AGLi
λL = −AGJjs(t)
with
qτ = qτ (uτC , A⊤V Cvs(t) +A⊤
CCuτC),
qλ = qλ(uτC , A⊤V Cvs(t) +A⊤
CCuτC),
gτ = gτ (uτG, A⊤V Gvs(t) +A⊤
CGuτC +A⊤
GGuτG),
gλ = gλ(uτG, A⊤V Gvs(t) +A⊤
CGuτC +A⊤
GGuτG),
rτ = rτ (−ARRiλR −ARLiλL −ARJjs(t), i
λR),
rλ = rλ(−ARRiλR −ARLiλL −ARJjs(t), i
λR),
φτ = φτ (−ALLiλL −ALJjs(t), iλL),
φλ = φλ(−ALLiλL −ALJjs(t), iλL).
This hybrid equation system depends on the branchvoltages uτC of capacitances of the tree, the branchvoltages uτG of conductances of the tree, the branchcurrents iλR of the resistances of the cotree andthe branch currents iλL of the inductances of thecotree only.
Theorem 1. The hybrid equation system has atmost index 1.
The proof bases on the tractability index con-cept using a projector based analysis.
Higher index variables as known from themodified nodal analysis do not appear in the hy-brid system. If one is interested in e.g. a currentthrough a loop of voltages and capacitances, onecan calculate it easily after the numerical integra-tion of the hybrid system. This way, the usuallylarger error in the higher index variables does notinfluence the integration process.
References
1. P. R. Bryant. The order of complexity of electricalnetworks. Pro. IEE (GB), Part C, 106:174–188,1959.
2. D. Estevez Schwarz and C. Tischendorf. Struc-tural analysis of electric circuits and consequencesfor MNA. Int. J. Circ. Theor. Appl., 28:131–162,2000.
3. S. Iwata and M. Takamatsu. Index minimizationof differential-algebraic equations in hybrid analy-sis for circuit simulation. Math. Programming, toappear.
4. G. Reißig. Extension of the normal tree method.Int. J. Circ. Theor. Appl., 27:241–265, 1999.
144
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 2B
Convergence Issues of Ring Oscillator Simulation by
Harmonic Balance Technique
M.M. Gourary1, S.G. Rusakov1, S.L. Ulyanov1, M.M. Zharov1, and B.J. Mulvaney2
1 IPPM, Russian Academy of Sciences, Moscow, Russia [email protected] Freescale Semiconductor Inc., Austin, Texas, USA [email protected]
Summary. Poor convergence properties of harmonicbalance algorithm for ring oscillators are explained bythe existence of multiple solutions of the algebraic sys-tem corresponding with each physical oscillation. Theapproach to improve the convergence is proposed.
1 Introduction
Difficulties in Harmonic Balance (HB) simula-tion of ring oscillators (RO) with large numberof stages were reported in [1]. We performed aninvestigation of the problem on the base of HBsimulations of CMOS RO with the continuationalgorithm with frequency adjusting [2,3]. The in-vestigation results are presented in this paper.
The continuation algorithm with frequencyadjusting is the homotopy algorithm specially de-signed to prevent the convergence of Newton iter-ations to DC point. It is based on the incorporat-ing of an artificial voltage source (probe) into thecircuit. The probe magnitude corresponding tothe oscillation voltage is obtained by varying themagnitude from zero (DC solution) until probecurrent is null. The probe frequency is varying insuch a way that the phase of probe current is zero.The initial value of the frequency ω0 is defined byAC oscillation conditions at unstable DC point
ℑm(Z(ω0)) = 0 ℜe(Z(ω0)) < 0 (1)
where Z(ω)is the input impedance.The convergence difficulties in the simulation
of CMOS RO are illustrated with the followingexamples. The continuation curves (probe currentvs probe voltage) have simple form for 3- and 5-stage oscillators (see Fig. 1).
But for 7-stage RO the simulation exposes acomplex continuation process (Fig. 2) and failsafter some thousand steps. A similar effect is ob-served in RO with larger number of stages.
Fig. 1. Continuation curve. 5-stage RO
(a)
(b)
Fig. 2. Continuation curve. (a) 7-stage RO (b) frag-ment
The analysis of (1) for 7-stage RO shows thatthere exists an additional frequency value satisfy-ing conditions (1). The simulation with this ini-tial frequency is successfully performed providingthe simple continuation curve. However, simula-tion results essentially differ from the transientsimulation; the frequency (2.33GHz) is almostfour times higher than in transient simulation(570.4MHz) and the sinusoidal output waveformevidently differs from the pulsed one obtained bytransient simulation. Thus the simulation deter-mines another RO solution. Note that no addi-tional solutions for 3 and 5 stage oscillators exist.
One can suppose the existence of two solu-tions to be the source of poor convergence dueto the appearance of additional attractive pointsfor continuation trajectories. However this cannotexplain the successful convergence to the secondsolution in spite of the existence of the first so-lution. More detailed analysis of the problem isconsidered below.
2 Explanation
The existence of additional (unstable) solutionsin RO is well-known [4]. It can be understood byconsidering n-stage RO containing ideal inverterswith equal delay time τ . In this case the keepingof self-oscillations requires the output signal ofthe chain to be delayed from the input signal byinteger number of periods: nτ = T/2+mT . Thusthe frequency of mth solution is unambiguouslydefined by the frequency of the first solution
f (n)m = (2m− 1)f
(n)1 m = 1, 2, 3, . . . (2)
In real RO the number of possible periodic os-cillations cannot be infinite because at high fre-quency the inverter amplifying property decrease,
145
and generation conditions (1) cannot be met. Thepoor HB convergence cannot be explained by ad-ditional solutions only. The explanation can beobtained if one will take into account the effect ofmultiple numerical representations of the periodicsolution. The effect can be described as follows.
Each T-periodic function can be also consid-ered as the function with period kT and frequencyf/k, where k is an integer. HB method for os-cillators reduces the steady-state problem to thesolving of nonlinear algebraic system with respectto the vector containing unknown oscillation fre-quency and harmonics of circuit variables:
X = [f,A1, A2, . . . , Ai, . . .] (3)
Here Ai is the vector of ith harmonics of circuitvariables. But if the vector (3) defines the periodicstate of the oscillator then this state can be alsodefined by vectors
X′
=[f2 , 0, A1, 0, A2, . . . , 0, Ai, . . .
]
X” =[f3 , 0, 0, A1, 0, 0A2, . . . , 0, 0, Ai, . . .
] (4)
We shall refer vectors (4) as the images of thebasic solution vector (3).
Thus for each steady-state behavior of the os-cillator there exists infinite set of images repre-senting parasitic solutions of HB system. It isnatural to assume that the effect of the image de-pends on the deviation of its frequency from thefrequency of the solution to be found. In the caseof the single steady state solution its own imageshave no effect on the convergence because theirfrequencies are essentially less than the basic fre-quency. But if there exists a number of solutionsthen their images can complicate the continua-tion process when the frequency of the sought-forsolution is close to the frequency of any image ofother solution.
Presented considerations explain the pointedabove behavior of the continuation process for 7-stage RO. The second solution is easily simulatedbecause its frequency 2.33GHz is much higherthan the frequencies of images of the first solu-tion. But the continuation process for the firstsolution with frequency 570MHz is deteriorateddue to the 4th image of the second solution withfrequency 2.33 GHz/4 = 582.5 MHz.
Note that in ideal RO the frequency of anyadditional solution is a multiple of the basic fre-quency (2). Taking into account that real RO ap-proaches the ideal one on the increase of the num-ber of stages we can explain poor HB convergenceof RO with large number of stages.
The presented explanation was also confirmedby simulations of another types of RO. Particu-larly the method proposed in [1] for the simula-tion of RO with identical stages was analyzed.
The results of numerical experiments are in goodagreement with the proposed theory.
3 Convergence Improvement
From the presented analysis we conclude that toimprove the convergence we need to decrease theeffect of foreign images on the desired solution.
One can pointed out that under truncated setof harmonics an image is not an exact solutionof HB system, and we expect that this effect willdecay with the reducing of the number of consid-ering harmonics. The most decay (or full elimina-tion) of the effect can be awaited when the num-ber of harmonics is less than image index, becausefirst l − 1 harmonics of lth image are zeroes.
This assumption was supported by the simu-lations with small number of harmonics. The ex-periments showed that HB simulation with onlyone harmonic provides the evaluated oscillationfrequency to be close to its true value. No conver-gence difficulties were observed in experiments,and continuation trajectories have simple formsimilar with Fig. 1. Obtained results allowed topropose the following modification of continua-tion method for RO.
1. The continuation algorithm is performed forthe state vector containing only one harmonic.
2. The obtained truncated solution is consideredas the initial guess of Newton iterations.
3. Newton iterations for HB autonomous systemare performed with required number of har-monics.
The implementation of the algorithm into HBsimulator allowed to evaluate RO with consideredCMOS inverter up to 15 stages. Besides compu-tational efforts were essentially reduced.
References
1. X. Duan and K. Mayaram. An Efficient and Ro-bust Method for Ring-Oscillator Simulation Us-ing the Harmonic-Balance Method. IEEE Trans.Computer-Aided Design, 24(8):1225–1233, 2005.
2. M. Gourary, S. Ulyanov, M. Zharov, S. Rusakov.K. K. Gullapalli, and B. J. Mulvaney. Simulationof high-Q oscillators. IEEE/ACM InternationalConference on Computer-Aided Design, pp. 162–169, Nov. 1998.
3. M. Gourary, S. Ulyanov, M. Zharov, S. Rusakov,K. K. Gullapalli, and B. J. Mulvaney. A robust andefficient oscillator analysis technique using har-monic balance. Computer Methods in Applied Me-chanics and Engineering, 181:451.-466, Jan. 2000.
4. P. Selting, Q. Zheng. Numerical stability analysisof oscillating integrated circuits. ELSEVIER J. ofComputational and Applied Mathematics, 82:367–378, 1997.
146
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 2C
Stochastic Models for Analysing Parameter Sensitivities
Roland Pulch
Lehrstuhl fur Angewandte Mathematik und Numerische Mathematik, Bergische Universitat Wuppertal,Gaußstr. 20, D-42119 Wuppertal [email protected]
Summary. Mathematical modelling of electric cir-cuits yields systems of differential algebraic equations.Often some technical parameters exhibit uncertain-ties. We arrange a stochastic model by consideringthese parameters as random variables. The resultingstochastic processes provide global information on theparameter sensitivity. We apply the generalised poly-nomial chaos to construct efficient numerical methodsfor solving the model. According numerical simula-tions are presented.
1 Stochastic Modelling
We consider an initial value problem of a systemof differential algebraic equations (DAEs)
A(p)x(t) = f(t,x(t),p), x(t0) = x0 (1)
with unknown solution x : [t0, t1] → Rk and amatrix A(p) ∈ Rk×k. In the DAE (1), the para-meters p ∈ Rl are included and thus the solutionbecomes parameter dependent: x(t) = x(t;p).Partial derivatives with respect to p yield localinformation on the sensitivity of the solution.
Assuming some uncertainties, we replace theparameters by random variables p = p(ω) ona probability space (Ω,A, P ). Thereby, a rela-tively large range of the variables can be chosento achieve global information on the dependence.Consequently, the solution of the system (1) be-comes a stochastic process X : [t0, t1] ×Ω → Rk.The according stochastic system reads
A(p(ω))X(t, ω) = f(t,X(t, ω),p(ω)),
X(t0, ω) = x0.(2)
Crucial data like expected values and variances ofthe process have to be determined efficiently bynumerical methods.
This stochastic approach can be used for anarbitrary DAE system. We focus on DAEs result-ing from mathematical models of electric circuits.In particular, oscillators are considered, whereanalysing the stability of periodic solutions rep-resents an important task.
2 Application of Polynomial Chaos
The data of the stochastic model from Sect. 1can be computed by Monte-Carlo simulation orits more sophisticated variants. However, a hugenumber of simulations based on (2) often has tobe performed to achieve sufficiently accurate re-sults. Alternatively, the approach of the gener-alised polynomial chaos yields efficient numericalmethods for stationary problems, see [2].
We apply the generalised polynomial chaos tothe transient problem (2). Assuming finite secondmoments with respect to a specific Sobolev norm,the stochastic process exhibits the representation
X(t, ω) =∞∑
i=0
vi(t) Φi(p(ω)) (3)
with coefficient functions vi : [t0, t1] → Rk andorthogonal basis polynomials Φi : Rl → R. Theorthogonality is based on the inner product fol-lowing from the expected value denoted by 〈·〉,i.e., 〈ΦiΦj〉 = 〈Φ2
i 〉δij .To construct a numerical technique, the se-
ries (3) is truncated and inserted in the stochas-tic system (2), which causes a residual. Using theGalerkin approach, we obtain a larger system ofDAEs
m∑
i=0
〈Φl(p)Φi(p)A(p)〉vi(t)
=
⟨Φl(p) f
(t,
m∑
i=0
vi(t)Φi(p),p
)⟩ (4)
for l = 0, 1, . . . ,m, where the unknowns are thetime-dependent functions vi. In this system, theexpected values are evaluated component-wise.We solve the system (4) with according initialvalues by standard integrators. Boundary valueproblems can be investigated in case of oscilla-tors, which imply periodic solutions.
Given a numerical solution of (4), the ex-pected value and the variance of the process areapproximated directly via
〈X(t, ·)〉 .= v0(t),
Var(X(t, ·)) .=
m∑
i=1
vi(t)2〈Φi(p)2〉, (5)
147
where the operations are done component-wiseagain. Thus we have to solve the extended sys-tem (4) only once.
For integrating the system (4), evaluations ofthe expected values in the right-hand side are re-quired at each used time point. Applying multidi-mensional quadrature formulas demands a largecomputational effort. In this case, the approachof stochastic collocation becomes superior to theabove strategy, see [3]. Nevertheless, the nonlinearfunction f may exhibit multiplicative structures,which allow for an efficient method based on thepolynomial chaos.
The approach of the polynomial chaos can beapplied in many applications, see [1]. Random os-cillators modelled by ordinary differential equa-tions are considered in [4]. The use of the poly-nomial chaos seems to be a relatively open taskin case of DAEs. For example, the index of theextended system (4) with respect to the index ofthe original system (1) has to be analysed.
3 Numerical Simulation
As test example, we apply the circuit of a Schmitttrigger given in Fig. 1. This circuit transforms asinusoidal input signal to a digital output signal.Mathematical modelling yields a nonlinear sys-tem (1) of dimension k = 5. We substitute the ca-pacitance included in the matrix A by a uniformlydistributed random variable in [10−9F, 10−7F].Since a relatively large range is specified, a localsensitivity analysis is not sufficient here.
0
u
uop
in
uout
Fig. 1. Schmitt trigger circuit.
We use the stochastic model from Sect. 1 andthe approach of the polynomial chaos from Sect. 2with the expansion (3) based on Legendre polyno-mials up to degree m = 3. Trapezoidal rule solvesan initial value problem of the system (4). Wedetermine data applying (5). Figure 2 illustratesthe resulting expected value of the output volt-age and three samples for different capacitances.The corresponding standard deviation is shownin Fig. 3.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
time [ms]
expe
cted
val
ue o
f out
put [
V]
Fig. 2. Expected value (solid line) and three samples(dashed lines) of output voltage.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.005
0.01
0.015
0.02
0.025
time [ms]
stan
dard
dev
iatio
n of
out
put [
V]
Fig. 3. Standard deviation of output voltage.
We recognise that the variation of the capac-itance does not influence the upper values of thedigital output signal but the lower values. Thestandard deviation or variance, respectively, indi-cate directly the subintervals, where the solutionis sensitive with respect to the parameter, and thecorresponding magnitudes.
References
1. F. Augustin, A. Gilg, M. Paffrath, P. Rentrop andU. Wever. Polynomial chaos for the approximationof uncertainties: chances and limits. To appear in:Euro. Jnl. of Applied Mathematics.
2. R.G. Ghanem and P.D. Spanos. Stochastic FiniteElements. (Rev. Ed.) Dover Publications Inc., Mi-neola, New York, 2003.
3. A. Loeven, J. Witteveen and H. Bijl. Probabilisticcollocation: An efficient non-intrusive approach forarbitrarily distributed parametric uncertainties. InM.B. Bragg, editor, Proceedings of 45th AerospaceSciences Meeting and Exhibit, 1–14, 2007.
4. D. Lucor, C.H. Su and G.E. Karniadakis. General-ized polynomial chaos and random oscillators. Int.J. Numer. Meth. Engng., 60:571–596, 2004.
148
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 2D
Simultaneous Step-size and Path Control for Efficient
Transient Noise Analysis
Werner Romisch1 and Thorsten Sickenberger2
1 Institute of Mathematics, Humboldt-Universitat zu Berlin, Unter den Linden, D-10099 Berlin, [email protected]
2 Department of Mathematics, Universitat zu Koln, Weyertal 86-90, D-50931 Cologne, [email protected]
Summary. Transient noise analysis requires an effi-cient numerical time integration by mean-square con-vergent numerical methods. We develop a strategyfor controlling the step-size and the number of solu-tion paths simultaneously in one approximation. Thestrategy is based on the sampling error of the mean-square approximation of the local discretisation errorand yields an approximate solution which consists ofa tree of solution paths that is extended, reduced orkept fixed adaptively. An algorithm to generate thesetrees is described.
1 Transient noise analysis in circuitsimulation
Transient noise analysis means time domain sim-ulation of noisy electronic circuits. The currenttechnological progress in microelectronics is drivenby the desire to decrease feature sizes, increasefrequencies and the need for low supply volt-ages. Integrated circuits become much more com-plex, parasitic effects become predominant andthe signal-to-noise ratio decreases. To address thesignal-to-noise ratio the modelling and the simu-lation can be improved by taking the inner elec-trical noise into account.
We deal with the thermal noise of resistorsas well as the shot noise of semiconductors. Weconsider mathematical models where the noise istaken into account by means of sources of Gaus-sian white noise that are added to the deter-ministic network equations. This leads to sys-tems of stochastic differential algebraic equations(SDAEs) of the form
Ad
dtq(x(t)) + f(x(t), t) +
m∑
r=1
gr(x(t), t)ξr(t) = 0 ,
where gr(x, t) denotes the vector of noise in-tensities for the r-th noise source, and ξ is anm-dimensional vector of independent Gaussianwhite noise sources.
A crucial property of the arising SDAEs is thelarge number of small noise sources that are in-cluded. Fortunately, even in current chip designs,
the noise level is still smaller in magnitude com-pared to the wanted signal, which can be used forthe construction of efficient numerical schemes.
The focus here is on numerical methods tosimulate solution paths, i.e., strong approxima-tions of the solution of the arising large system ofindex-1 SDAEs, since only such paths can revealthe phase noise. The calculation of hundreds oreven a thousand solution paths are necessary forgetting sufficient numerical confidence about thephase or the opening of eye-diagrams.
For this task we present adaptive linear multi-step Maruyama schemes, together with a newstep-size and path selection control.
2 Adaptive numerical methods
We consider stochastic analogues of the variablecoefficient two-step backward differentiation for-mula (BDF2) and the trapezoidal rule, whereonly the increments of the driving Wiener pro-cess are used to discretize the diffusion part. Boththe BDF2-Maruyama method and the stochastictrapezoidal rule of Maruyama type have only anasymptotic order of strong convergence of 1/2.
However, the noise intensities contain smallparameters and the error behaviour is much bet-ter. In fact, the errors are dominated by the de-terministic terms as long as the step-size is largeenough. Also the smallness of the noise allows spe-cial estimates of the local error terms, which canbe used to control the step-size. We aim at an ef-ficient estimate of the mean-square of local errorsby means of a number of simultaneously com-puted solution paths. This leads to an adaptivestep-size sequence that is identical for all paths.
Depending on the available information wewill monitor different quantities to satisfy accu-racy requirements, such that the designer of a cir-cuit can choose whether he would like to controlthe accuracy of the charges and fluxes or of thenodal potentials and branch current throughoutthe numerical integration.
149
3 A solution path tree algorithm
In the analysis so far, the number of simultane-ously computed sample paths has not been spec-ified. It influences the sampling error in the ap-proximation of the L2-norm in the error estimate.Our aim in tuning the number of paths is to bal-ance the local error and the sampling error.
The best number of paths Mℓ depends on thetime-point tℓ and is realized by approximate so-lutions generated on a tree of paths that is ex-tended, reduced or kept fixed adaptively. Figure1 gives an impression, how a solution path treelooks like. At each time-step the optimal expan-
t tt 21
x
ttt0
solution path tree
3 4
3 31 1
(x ,1 12 2
1 11 1(x ,
1 1(x ,0 0
(x ,
π )
π )
π )
π )
Fig. 1. A solution path tree: Variable time-points tℓ,solution states xi
ℓ and path weights πiℓ.
sion or reduction problem is formulated by meansof combinatorial optimization models. The pathselection is modelled as a mass transportationproblem in terms of the L2-Wasserstein metric.This solution path tree algorithm has been im-plemented in practice. The results presented inthe next section show its performance.
4 Numerical results
We consider a model of an inverter circuit with aMOSFET transistor, under the influence of ther-mal noise (see Fig. 2). The thermal noise of theresistor and of the MOSFET is modelled by addi-tional white noise current sources that are shuntin parallel to the original, noise-free elements. Tohighlight the effect of the noise, we scaled the dif-fusion coefficient by a factor of 1000.
We have applied the solution path tree al-gorithm to this example. The upper graph in
Uop
Uin
R
C
12
3
Fig. 2. Thermal noise sources in a MOSFET invertercircuit
Fig. 3 shows the computed solution path treetogether with the applied step-sizes. The lowergraph shows the simulation error (solid line), itserror bound (dashed line) and the used numberof paths (marked by×), vs. time. The results in-
e1
5
4
3
2
1
0time t2.5*10-82*10-81.5*10-81*10-80.5*10-80
step-size
8*10-9
6*10-9
4*10-9
2*10-9
0
step-sizesolution path tree
error
10-16
10-17
time t2.5*10-82*10-81.5*10-81*10-80.5*10-80
M
100
10
used number of paths Mcalculated error bound
estimated sampling error
Fig. 3. Simulation results for the noisy inverter cir-cuit: Solution path tree and step-sizes (top), samplingerror, its error bound and the number of paths (bot-tom).
dicate that there exists a region where we have touse very much paths. This is exactly the area inwhich the MOSFET is active and the input signalis inverted. Outside this region the algorithm pro-poses approximately 70 simultaneously computedsolution paths.
Acknowledgement. We wish to thank Dr. Renate Win-kler (Uni Wuppertal) and Dr. Georg Denk (Qimonda)for the excellent cooperation. Support of BMBF throughproject 03RONAVN and of the EU throuph the projectICESTARS is gratefully acknowledged.
References
1. G. Denk, W. Romisch, Th. Sickenberger, andR. Winkler. Efficient Transient Noise Analysis inCircuit Simulation. In W. Jager, H.-J. Krebs, edi-tors, Mathematics - Key Technology for the Future.Springer, Berlin, 2008.
2. Th. Sickenberger. Efficient transient noise analy-sis in circuit simulation. Phd thesis, Humboldt-Universitat zu Berlin, Berlin, 2008.
3. Th. Sickenberger and E. Weinmuller and R. Win-kler. Local error estimates for moderately smoothproblems: Part II - SDEs and SDAEs with smallnoise. Preprint 07-07, Inst. fur Mathematik, HUBerlin, 2007. Submitted for publication.
4. Th. Sickenberger and R. Winkler. Adaptive Meth-ods for Transient Noise Analysis. In G. Ciup-rina and D. Ioan, editors, Scientific Computingin Electrical Engineering SCEE 2006, p. 403–410.Springer, Berlin, 2007.
150
Session
CS 3
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 3I
Parametric Reduced Order Models for Passive Integrated
Components Coupled with their EM Environment
Daniel Ioan
Politehnica University of Bucharest, Spl. Independentei 313, 060042, Romania [email protected]
Summary. The paper focuses on the implementa-tion and use of a new type of boundary conditions -Electric and Magnetic Hooks - in the Finite Integra-tion Technique applied for the simulation of passivedevices in full-wave electromagnetic field regime. Thistype of boundary conditions offers a new and effectivepossibility to carry out domain decomposition for themodelling of very large scale problems, such as com-plex on-chip passive devices.
1 Introduction
While next-generation integrated circuits designswill always be challenged by an increased numberof trouble spots, surely the most important onein this case is related to the nature of the electro-magnetic coupling among the down-scaled indi-vidual devices to be integrated on one chip. Newcoupling mechanisms, including EM field cou-pling, are becoming too strong to be neglected [1].
To address these problems, a European re-search project entitled: Comprehensive High Ac-curacy Modelling of Electromagnetic - Chameleon-RF (www.chameleon-rf.org) was started withinFP6. The general objective of the Chameleon-RF project is that of developing methodology andprototype tools that take a layout description oftypical RF functional blocks that will operate atRF frequencies up to 60GHz and transform theminto sufficiently accurate, reliable electrical simu-lation models taking EM coupling and variabilityinto account.
2 Boundary conditions and hooksdefinition
One of the main theoretical problems encoun-tered in the modelling of RF components is thedifficulty to define a unique terminal voltage, in-dependent of the integration path. The solutionfound in our approach is to adopt appropriateboundary conditions (EMCE) for the field prob-lem associated to the analysed components, in or-der to allow the consistency with the electricalcircuit which contains this component. The cor-rect definition of the component terminals (usedfor intentional interconnections) and connectors(”hooks” which represent parasitic couplings) wasan important challenge of our research [2].
3 Domain Partitioning
The geometric complexity of nowadays designscan be handled only by Domain Decomposition(DD). Interface Relaxation (IR) is a frequentlyencountered iterative form of DD solvers [3]. Itassumes a splitting of the domain into a set ofnon-overlapping sub-domains and considers theassociated PDE problems defined on each one ofthem. The novelty of this paper is that it anal-yses the convergence properties of new interfaceapproximations, based on electric and magnetichooks, which were proposed the first time in [4]to describe the parasitic EM interaction betweencomponents in RF-ICs.
The hooks technique has practical importancewhen their number is reduced to 1 . . . 10. Withsuch values, the sub-domains having different shapescan be modeled independently in parallel. After-wards the reduced size models (represented asmatrices - frequency dependent circuit functions,state equations or reduced order Spice circuits)are interconnected, aiming to obtain a model forthe global system. Unlike DD, which is basicallyan iterative process, in the proposed approach ofDomain Partitioning (DP), due to the reducednumbers of hooks, the interface iterations can beremoved and the resulted technique is a ”direct”,not an iterative one, as DD is.
4 Numerical approach
This concept of EMCE (Electro-Magnetic CircuitElement) was implemented on a prototype soft-ware tool developed in the frame of the FP6/Cha-meleon-RF project. The numerical method usedfor the discretization of the EM field equationis the Finite Integration Technique (FIT), whoseone of the first promoters was T. Weiland [5]. FITuses two orthogonal staggered grids, the electricgrid, and the magnetic grid on which electricaland magnetic quantities are defined, respectively(Fig.1). An electric terminal is defined on the elec-tric grid, so it is placed exactly on the domainboundary, whereas a magnetic terminal is definedon the magnetic grid, so it is placed a bit underthe geometrical boundary.
153
Fig. 1. Electric terminal on the electric grid (left);Magnetic terminal on magnetic grid (right).
FIT is applied for the EMCE formulation,in order to obtain the following discrete time-domain model
Cdx
dt+ Gx = Bu, y = Lx, (1)
where x is the state space vector, u is the vectorof input quantities and y is the vector of out-put quantities. The input/output quantities aresolely related to the terminals. Each terminal in-troduces one input and one output quantity. Forinstance, if an electric terminal is voltage excited,its voltage is a component of the input vector andthe current flowing through it (entering in the do-main) is an output quantity. Similarly, if a mag-netic terminal is excited in magnetic voltage, thisone will be an input quantity and the magneticflux entering in the domain through this terminalwill be an output quantity. Thus, the number ofinputs is always equal to the number of outputs.
5 Validation and numerical results
The method described above was validated on aseries of test structures: CHRF benchmarks [6]. Inthe case of a coupled spiral coils the reference re-sult was obtained by simulating the coupled coilsas a whole device. Two separate EMCE modelswere extracted, each corresponding to a singlecoil, but with hooks on the cutting interface.
Figure 2 shows the results obtained from thesimulation of the coupled coils without DD andwith DD decomposition in three cases: with allpossible electric and magnetic hooks (case in whicheach electric and magnetic node on the commonsurface represents a distinct terminal - the sur-face is in this case transparent for the EM field),only with electric hooks, and only with magnetichooks. The full simulation perfectly overlaps thesimulation with DD and with node-by-node in-terconnection between the models thus validat-ing the theory of hooks and its implementationin the framework of the FIT numerical meth-ods. The use of magnetic hooks is very importantto catch the output behaviour on the whole fre-quency range, whereas the electric hook becomemore important at higher frequencies, due to the
Fig. 2. Imaginary part of the admittance obtainedfrom full simulation, and simulation using DD.
capacitive effects that cannot be neglected anymore.
6 Conclusions
The paper illustrated how a new type of boundaryconditions (hooks) can be implemented in FITand used in EM coupling simulations to obtain in-dependent models of sub-problems. The parasiticand intentional coupling between either time do-main or frequency domain models was modeled.
If only node-hooks are used, the interface con-ditions are imposed node by node, and the cou-pled numerical model is equivalent to the nu-merical model obtained with FIT applied for thewhole domain, providing that the grid used is theunion of grids used for each sub-problem. Alter-native solutions include iterative procedures asin DD or a convenient reduction of the numberof hooks and thus number of terminals of sub-models. In the latter case, the magnetic terminalsare surface-hooks, each hook including more thanone node of the discretization grid.
References
1. T. Yao, M.Q. Gordon, K.K.W.Tang, K.H.K. Yau,M-T Yang, P.Schvan, S.P. Voinigescu AlgorithmicDesign of CMOS LNAs and Pas for 60 GHz RadioIEEE Journal on Solid State Circuits vol.42, no.5,pp.1044-1048, 2007.
2. D. Ioan, G. Ciuprina, L.M. Silveira Effective Do-main Partitioning with Electric and MagneticHooks Proceedings of the CEFC 2008.
3. Domain decomposition web-page. www.ddm.org.4. D. Ioan, W. Schilders, G. Ciuprina, N. Meijs, W.
Schoenmaker Models for integrated componentscoupled with their EM environment COMPELJournal vol. 27, no.4, pp.820-828, 2008.
5. T. Weiland A discretization method for the solu-tion of Maxwell’s equations for 6 component fieldsAEU, Electronics and Communication vol. 31, pp.116-120, 1977.
6. Chameleon-RF website, www.chameleon-rf.org.
154
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 3A
Stability of the Super Node Algorithm for EMC Modelling of
ICs
M.V. Ugryumova and W.H.A. Schilders
Eindhoven University of Technology, CASA, Den Dolech 2, 5612 AZ Eindhoven [email protected]
Summary. The super node algorithm performs modelorder reduction based on physical principals. Althoughthe algorithm provides us with compact models, itsstability has not thoroughly been studied yet. Theloss of stability is a serious problem because simu-lations of the reduced network may encounter arti-ficial behavior which render the simulations useless.We prove that the algorithm delivers unstable modelsand we will present a way to overcome this problem.
1 Introduction
FASTERIX is a layout simulation tool for EMC(electromagnetic compatibility) modelling. It trans-forms PCB properties into an equivalent RLC cir-cuit model which is then reduced into a compactone with approximately the same behavior. As areduction technique it uses the so called ’supernode algorithm’.
The approach of compact equivalent circuitsmakes FASTERIX an excellent and fast programfor the frequency analysis of the electromagneticbehavior for PCB’s. The reader is referred to [1],[3], for more information about the methods inFASTERIX. Although the super node algorithmprovides us with compact models, the problem ofcarrying out simulations in the time domain stillexists. One can think of using another reductiontechniques instead of the super node algorithm,which can preserve stability and passivity. Theadvantage of the super node algorithm is that it isinspired by physical insight into the models, andproduces reduced RLC circuits depending on themaximum predefined frequency. This fact makesthe algorithm an attractive technique for EMCmodelling.
We briefly review the algorithm, and give asketch of the proof that it delivers unstable re-duced models. Corresponding example is consid-ered.
2 Derivation of the model used inFASTERIX
FASTERIX translates electromagnetic propertiesof a PCB into a circuit model which is describedby the system of Kirchhoff’s equations
(R + sL)I − PV = 0 (1)
PT I + sCV = J (2)
where R ∈ ℜε×ε is the resistance matrix,L ∈ ℜε×ε is the inductance matrix, P ∈ ℜε×ηis an incidence matrix, C ∈ ℜη×η is the capac-itance matrix, I ∈ Cǫ is the vector of currentsflowing in the branches, V ∈ Cη is a vector ofvoltages at the nodes. Vector J ∈ Cǫ collects theterminal currents flowing into the interconnectionsystem. Value s is a complex number with nega-tive imaginary part: s = −jω. Matrices R, L, Care symmetric and positive definite.
In order to obtain an equivalent network model,the set of all nodes in the circuit described by(1)-(2) is subdivided into two subsets N and N ′
where N denotes the subset of nodes which areretained in the reduced circuit, and N ′ is for allother nodes. Due to this, vectors V , J and matri-ces P, C are partitioned into blocks, see [1], [4](chapter 8).
If we consider the voltage at the nodes (fromthe subset N) as an input v(p), and currents flow-ing into the system through the nodes as an out-put i(p), we come to the following system:
(R −PN ′
PTN ′ 0
)
︸ ︷︷ ︸G
+s
(L 00 CN ′N ′
)
︸ ︷︷ ︸C
x =
=
(PN
−sCN ′N
)v(p), (3)
i(p)(s) =(PTN sCT
N ′N
)x+ sCNNv
(p), (4)
where x =(I , VN ′
)T. It should be noted that
in (3) the matrix G is positive real, and matrixC is positive semi-definite. It is enough to provethat the unreduced system is stable.
The currents i(p) obtained each time undera unity voltage v(p) at a particular input ter-minal, the others being grounded, constitute thecolumns of the desired admittance matrix Y. Thefollowing holds
155
Yv = i. (5)
Matrix Y describes the reduced circuit at aparticular frequency ω.
3 Super Node Algorithm
The super node algorithm constructs an approxi-mation of Y, which describes the reduced circuitat any frequency 0 < ω < Ω as
Y ∼ (s)−2YR + (s)−1YL + YG + (s)YC . (6)
Each pair of nodes in the reduced circuit con-tains a branch, Fig. 1. Resistance R, inductanceL, capacitance C, and resistance of conductanceG can be found from the admittance matricesYR, YL, YG and YC in (6). Thus the reducedcircuit described by (6) can be used in a circuitanalysis program.
L R
C
G
i j
Fig. 1. High frequency branch between two nodes
Stability
For carrying out reliable simulations in the timedomain, first it should be checked that the systemis stable [2].
An illustration of possible stability violation isgiven below. Figure 2 shows an example of simu-lations in the time domain for lowpass filter modelgenerated by FASTERIX. This model is supposedto be valid up to 10 GHz. The reason why the
Fig. 2. Simulation in the time domain
transient analysis fails is explained by the non-stability of the reduced model.
To show that the algorithm does not preservestability, we suggest to follow all reduction stepsand check stability at the each step. At the firststep, the system (3)-(4) is subdivided into twosystems by introducing the following expansionsof I and VN ′ in powers of (ik0h)
VN ′ = V0 + V1(ik0h), (7)
I = I0 + I1(ik0h), (8)
where k0 is the free space wave number. Pairs(I0, V0) and (I1, V1) are obtained from two sets ofequations which are found from (3)-(4) by gath-ering appropriate terms with orders (ik0h)
0 and(ik0h)
1. It can be checked that the two new sys-tems are again stable.
At the second step, the state space vectors ofthe systems from the previous step are expandedthrough frequency independent quantities, see [1],[4] (chapter 8), for details. In fact, (6) is a sum ofoutputs of the two systems under the expansions.Thus, it is important to investigate whether thestate space vectors of the systems have unstablepoles. The following representation of the statespace vector of the 1-st system has been obtained
x1 =(A1(sM1 + M2)
−1 + A2(sM3 + I)−1)A3×
×(G1 + sC1)−1Bi1V
(p)N . (9)
Taking into account the properties of the ma-trices: C1 and M3 are positive semi-definite, Iand G1 are positive real and M1 is indefinite, ingeneral, we conclude that (9) does not correspondto analytic function for all s with Re(s) > 0. Sim-ilar result can be obtained for the state space vec-tor of the second system. Thus, this fact provesnon-stability of the reduced model described by(6). We will present changes in the algorithm, inorder to guarantee its stability.
References
1. R.Du Cloux, G.P.J.F.M Maas, and WachtersA.J.H. Quasi-static boundary element method forelectromagnetic simulations of pcbs. Philips J.Res., 48:117–144, 1994.
2. E.A. Guillemin. Synthesis of Passive Networks.Wiley, New York, 1957.
3. R.F. Milsom. Rf simulation of passive ics usingfasterix. Report rp3506, Philips Electronics, 1996.
4. W.H.A. Schilders and E.J.W ter Maten. Specialvolume : numerical methods in electromagnetics.Elsevier, Amsterdam, 2005.
156
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 3B
GABOR: Global-Approximation-Based Order Reduction
Janne Roos, Mikko Honkala, and Pekka Miettinen
Helsinki University of Technology, Faculty of Electronics, Communications and Automation, Department ofRadio Science and Engineering, P.O.Box 3000, FI-02015 TKK, Finland janne,mikko,[email protected]
Summary. This abstract proposes a new approachfor the Model-Order Reduction (MOR) of RLC-circuitblocks. Instead of Taylor-series-like local fitting usingimplicit moment matching, a global approximation ofthe matrix-valued s-domain transfer function is gen-erated. Then, the Krylov-like subspace spanned bythe ‘moments’ of this approximation is used to set upthe projection matrices needed for MOR. The pro-posed MOR approach preserves passivity, reciprocity,and the properties of the global approximation.
1 Introduction
Typical Krylov subspace Model-Order Reduction(MOR) methods [1–3] are based on implicit mo-ment matching. This approach results in a Taylor-series-like approximation, which is exact at theexpansion point (e.g., at the origin of the complexplane), but which looses accuracy when movingfar away (e.g., towards high frequencies). There-fore, one avenue to reduce (or, at least, to spreadmore equally) the approximation error could beto base the MOR on a global approximation.
This abstract proposes a new MOR approachfor RLC-circuit blocks: Global-Approximation-Based Order Reduction (GABOR). GABOR pre-serves passivity and reciprocity, and matches the‘moments’ of the global approximation.
2 Preliminaries
PRIMA [1] operates with Y-parameters; SPRIM[2] and ENOR [3], in turn, are formulated usingZ-parameters. Due to lack of space, but withoutloss of generality, let us limit the discussion totreatment of Z-parameters, only.
Let us consider an RC-only circuit with nnodes and N port nodes, excited by N currentsources for obtaining the Z-parameter matrix.Now, a PRIMA-like formulation results in
Z(s) = LT(G + sC)−1B
= LT(I + sG−1C)−1G−1B
, LT(I − sA)−1R
= LTR + LTARs+ LTA2Rs2 + . . .
, M0 + M1s+ M2s2 + . . .
(1)
where G and C are n-by-n conductance and ca-pacitance matrices, respectively, L = B is an n-by-N selector matrix, and Mi = LTAiR are themoments. In particular, Z(0) = M0 = LTG−1B.
For an RL-only circuit, with inductances treatedas in Ref. [3], we get the dual version of (1):
Z(s) = LT(G + 1sΓ)−1B
= LT(I + 1sG
−1Γ)−1G−1B
, LT(I − 1sA)−1R
= LTR + LTAR1s + LTA
2R 1s2 + . . .
, M0 + M−11s + M−2
1s2 + . . .
(2)
Now, we have Z(∞) = M0 = LTG−1B.
3 Derivation of GABOR
Applying nodal formulation (that is, not MNA)to an RLC circuit, we obtain [3]
Z(s) = LT(sC + G + 1
sΓ)−1
B
= LT(sG−1C + I + 1
sG−1Γ
)−1G−1B
, LT(sD + I + 1
sE)−1
R(3)
When truncated to a finite number of mo-ments, (1) and (2) are, still, accurate near s = 0and s = ∞, respectively. The idea behind GA-BOR is to (try to) combine the best properties of(1) and (2) for an RLC circuit. In particular, theterm (. . . )−1 in (3) is approximated by “a combi-nation of the moment series in (1) and (2)”:
Z(s) = LT(sD + I + 1
sE)−1
R =
· · · + M−21s2 + M−1
1s + M0 + M1s+ M2s
2 + . . .(4)
Let
Mi = LTNi, i = . . . ,−2,−1, 0, 1, 2, . . . (5)
Combining (4) and (5) and some algebra gives(sD + I + 1
sE)
(· · · + N−1
1s + N0 + N1s+ . . .
)= R
(6)
Equating the negative/zero/positive powers of sresults in the following matrix equation (trun-cated to ±3 terms for convenience):
157
2
6
6
6
6
6
6
6
6
4
I DE I D
E I DE I D
E I DE I D
E I
3
7
7
7
7
7
7
7
7
5
2
6
6
6
6
6
6
6
6
4
N3
N2
N1
N0
N−1
N−2
N−3
3
7
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
6
4
000R000
3
7
7
7
7
7
7
7
7
5
(7)
Equation (7) can be solved semi-analyticallywith two sets of recursions (shown for ±3 terms):
↓ A3 =−D ↑ N3 =A3N2
↓ A2 =−(I + EA3)−1D ↑ N2 =A2N1
↓ A1 =−(I + EA2)−1D ↑ N1 =A1N0
→ →N0 =(. . . )−1R↑ A−1=−(I + DA−2)
−1E ↓ N−1=A−1N0
↑ A−2=−(I + DA−3)−1E ↓ N−2=A−2N−1
↑ A−3=−E ↓ N−3=A−3N−2
(8)
That is, starting from A−3 and A3, one recur-sively obtains A−1 and A1 that are used to solve
N0 = (DA−1 + I + EA1)−1R (9)
Then, starting from N0, one obtains all the Ni
terms, which, together with (5), give the negativeand positive ‘moments’ up to the desired order.
Note that with (3)–(5), (8), and (9), a globalapproximation is created, since M0 = LTN0 (andall the other ‘moments’, Mi) depend on the orderof reduction, (q−, q+).
The next step is to find the projection matri-ces needed for MOR. To start, let us define thefollowing Krylov-like subspace:
Kr(Ai,N0, q−, q+) ≡ colspanA−q . . .A−1N0, . . . ,A−2A−1N0,A−1N0,
N0,A1N0,A2A1N0, . . . ,Aq . . .A1N0(10)
In GABOR, a PRIMA-like (but ‘bidirectional’)block-Arnoldi method is used to obtain
X = [X−q, . . . ,X−1,X0,X1, . . . ,Xq] (11)
such that the orthonormalized blocks Xi spanKr(Ai,N0, q−, q+). Finally, the desired reduced-order matrices of GABOR are obtained by setting
G = XTGX, C = XTCX, Γ = XTΓX,
B = XTB, L = XTL(12)
It can be shown [3] that any reduced-ordermodel represented by (12) preserves the passiv-ity and reciprocity of the original RLC circuit.The planned full paper will contain a (rather longbut already derived) proof showing that GABORalong with (12) matches the Z-parameter block‘moments’ M−q, . . . ,M−1,M0,M1, . . . ,Mq of theglobal approximation (4).
Here, it is fair to mention that GABOR hassome problematic features, too. First, in (8), all
the Ai terms (or the related LU factorizations)must be precomputed into the memory space.Second, the conductance matrix G in (3) be-comes singular for real-life RLC interconnect cir-cuits. Therefore, an ENOR-like frequency scal-ing [3] must be applied. The planned full paperwill discuss the frequency scaling in more detail.
4 Simulation example
A dispersive transmission line was modeled with50 LRCG sections, each having L = 1 nH, R =1 mΩ, C = 1 pF, and G = 1 mS. This two-portRLC circuit was reduced using a MATLAB/C im-plementation of GABOR with (q−, q+) = (10, 10)and σ = −(s − s0)/s0 [3], where s0 = 5 GHz (tobe explained in the full paper). Figure 1 shows|Z21(f)| in the frequency range ]0, 5] GHz. Theoriginal and reduced circuit resulted in 101-by-101 and 20-by-20 circuit matrices, respectively.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 515
20
25
30
35
40
45
f/GHz
|Z21
|
Fig. 1. Orig. (dashed) and reduced (solid) |Z21(f)|.
5 Conclusions
This abstract proposed a Global-Approximation-Based Order Reduction (GABOR), which pre-serves the passivity and reciprocity of the orig-inal RLC circuit, and matches the ‘moments’ ofthe underlaying global approximation. The cor-rect operation of GABOR was verified with onesimulation example. The planned full paper willdiscuss the advantages, limitations, and moment-matching property of GABOR in more detail.
References
1. A. Odabasioglu, M. Celik, and L.T. Pileggi.PRIMA: passive reduced-order interconnectmacromodeling algorithm. IEEE Trans. CAD ofInt. Circ. and Syst., 17:645–654, 1998.
2. R.W. Freund. SPRIM: Structure-PreservingReduced-Order Interconnect Macromodeling.Proc. ICCAD 2004, 80–87, 2004.
3. B.N. Sheehan. ENOR: model order reduction ofRLC circuits using nodal equations for efficientfactorization. Proc. DAC 1999, 17–21, 1999.
158
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 3C
Passivity-preserving balanced truncation model reduction of
circuit equations
Tatjana Stykel and Timo Reis
Institut fur Mathematik, MA 4-5, TU Berlin, Straße des 17. Juni 136, D-10623 Berlin, [email protected], [email protected]
Summary. We consider passivity-preserving modelreduction of circuit equations using the bounded realbalanced truncation method applied to the Moebius-transformed system. This method is based on balanc-ing the solutions of the projected Lur’e and Lyapunovmatrix equations. We also discuss their numerical so-lution exploiting the underlying structure of circuitequations.
1 Introduction
A modified nodal analysis (MNA) for linear RLCcircuits yields a linear system of differential-algebraicequations (DAEs)
Ex(t) = Ax(t) +Bu(t),y(t) = Cx(t),
(1)
where
E=
AC C A
TC 0 0
0 L 0
0 0 0
, B=−
AI 00 00 I
=CT,
A=
−AR R −1
ATR −AL −AVATL 0 0
ATV 0 0
.
(2)Here AC , AL , AR , AV and AI are the incidencematrices describing the circuit topology, and R ,L and C are the symmetric, positive definite re-sistance, inductance and capacitance matrices, re-spectively. Linear RLC circuits are often used tomodel interconnects, transmission lines and pinpackages in VLSI networks. The number n ofstate variables is called the order of the MNAsystem (1). It is related to the number of circuitelements and usually very large. This makes theanalysis and numerical simulation of circuit equa-tions unacceptably time consuming. Therefore,model order reduction is of great importance.
A general idea of model reduction is to ap-proximate the large-scale system (1) by a redu-ced-order model
E ˙x(t) = A x(t) + B u(t),
y(t) = C x(t),(3)
where E, A ∈ Rℓ,ℓ, B ∈ Rℓ,m, C ∈ Rm,ℓ andℓ≪ n. It is required that the approximate system(3) captures the input-output behavior of (1) toa required accuracy and preserves passivity. Thisproperty is important in circuit simulation. Gene-rally speaking, passivity means that system doesnot produce energy. System (1) is passive if andonly if its transfer functionG(s) = C(sE−A)−1Bis positive real, i.e., G is analytic in the openright half-plane C+ and G(s) +GT (s) is positivesemidefinite for all s ∈ C+, see [1].
2 Passivity-preserving balancedtruncation
Balanced truncation model reduction is based onthe transformation of the dynamical system intoa balanced form whose controllability and ob-servability Gramians are both equal to a diago-nal matrix. Then a reduced-order model is deter-mined by the truncation of the states correspond-ing to small diagonal elements of the balancedGramians. Depending on system properties, dif-ferent types of balancing may be introduced.
A passive reduced-order system can be com-puted using the bounded real balanced truncationmethod applied to a Moebius-transformed system
G(s) =(I −G(s)
)(I +G(s)
)−1
= −2C(sE −A+BC)−1B + I.
Note that G(s) is positive real if and only if G(s)is bounded real, i.e., G is analytic in C+ and thematrix I −G(s)GT (s) is positive semidefinite forall s ∈ C+, see [1]. In the bounded real balancedtruncation method for G, the bounded real Grami-ans are defined as minimal solutions X and Y ofthe projected Lur’e equations
EXAT+ AXET+ 2PlBBTPTl = −2KcK
Tc ,
EXCT − PlBMT0 = −KcJ
Tc ,
JcJTc = I −M0M
T0 , X = PrXP
Tr ,
and
159
ETY A+ ATY E + 2PTr CTCPr = −2KT
o Ko,
−ETY B + PTr CTM0 = −2KT
o Jo, (4)
JTo Jo = I −MT0 M0, Y = PTl Y Pl,
where A = A − BC, Pr and Pl are the spectralprojectors onto the right and left deflating sub-spaces of the pencil λE − A corresponding to thefinite eigenvalues and M0 = lims→∞ G(s). Fur-ther, we define the improper controllability andobservability Gramians Ximp and Yimp of G asunique symmetric, positive semidefinite solutionsof the projected discrete-time Lyapunov equations
AXimpAT− EXimpE
T = 2QlBBTQTl ,
Ximp = QrXimpQTr ,
and
ATYimpA − ETYimpE = 2QTr CTCQr,
Yimp = QTl YimpQl,(5)
where Ql = I − Pl and Qr = I − Pr , see [4]. Thereduced-order model (3) can then be computedby projecting
E = WTET, A = WTAT, B = WTB, C = CT,
where the projection matrices W,T ∈ Rn,ℓ deter-mine the left and right subspaces correspondingto the dominant bounded real characteristic va-lues and non-zero improper Hankel singular va-lues of G defined as the square roots of the posi-tive eigenvalues of XETY E and XimpA
TYimpA,respectively. One can show that the reduced-ordersystem is passive. Furthermore, if
‖I +G‖H∞(πℓf +1 + . . .+ πnf) < 1,
where πj are the truncated characteristic valuesof G, then we have the following error bound
‖G−G‖H∞ ≤ ‖G+ I‖2H∞
(πℓf +1 + . . .+ πnf).
Since the MNA matrices in (2) satisfy
ET = SES, AT = SAS, BT = S1CS,
where S = diag(I,−I,−I) and S1 = diag(I,−I)are partitioned in accordance with A and C, re-spectively, we find that
Pl = SPTr S, X = SY ST , Ximp = SYimpST .
Thus, for circuit equations, it is enough to solveonly one Lur’e equation and also one Lyapunovequation. The solutions of the dual equations aregiven for free.
If Do = I −MT0 M0 is nonsingular, the pro-
jected Lur’e equation (4) can be rewritten as theprojected algebraic Riccati equation
ETY Ao +ATo Y E + 2ETY BD−1o BTY E
+ 2PTr CTD−1
c CPr = 0, Y = PTl Y Pl,
where Ao = A − BC − 2PlBD−1o MT
0 CPr andDc = I − M0M
T0 . This equation can be solved
using Newton’s method [2]. In each step of thismethod, the projected continuous-time Lyapunovequation of the form
ETZF+FTZE+PTrQTQPr=0, Z=PTl ZPl
has to be solved for Z. For this propose, wecan use the generalized alternating direction im-plicit method [5]. The projected discrete-timeLyapunov equation (5) can be solved using thegeneralized Smith method [5].
Note that the solution of the large projectedRiccati equation has often a low numerical rank.Such a solution can be well approximated bya matrix of low rank. Moreover, this low-rank ap-proximation can be constructed in factored formY ≈ LLT with L ∈ R
n,k that significantly reducesthe memory requirements when k ≪ n.
A major difficulty in the numerical solutionof the projected Lyapunov and Riccati equationswith large matrix coefficients is that the spectralprojectors Pl and Pr are required. Fortunately,the matrix coefficients (2) in the MNA circuitequations have some special block structure. Wecan exploit this structure to construct the re-quired projectors in explicit form using a matrixchain approach from [3]. Furthermore, we givean explicit formula for the matrix M0 and presentnecessary and sufficient conditions for invertibili-ty of Do = I − MT
0 M0 in terms of the circuittopology.
Acknowledgement. This work is supported by theDFG Research Center Matheon in Berlin and theBMBF, Grant 03STPAE3.
References
1. B. Anderson and S. Vongpanitlerd. Network Ana-lysis and Synthesis. Prentice Hall, EnglewoodCliffs, NJ, 1973.
2. P. Benner, R. Byers, E. Quintana-Ortı, andG. Quintana-Ortı. Solving algebraic Riccati equa-tions on parallel computers using Newton’smethod with exact line search. Parallel Comput.,26:1345–1368, 2000.
3. R. Marz. Canonical projectors for linear diffe-rential algebraic equations. Comput. Math. Appl.,31:121–135, 1996.
4. T. Stykel. Gramian-based model reduction for des-criptor systems. Math. Control Signals Systems,16:297–319, 2004.
5. T. Stykel. Low-rank iterative methods for projec-ted generalized Lyapunov equations. Electr. Trans.Numer. Anal., to appear.
160
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CS 3D
A new approach to passivity preserving model reduction
Roxana Ionutiu1, Joost Rommes2, and Athanasios C. Antoulas1
1 William Marsh Rice University, Department of Electrical and Computer Engineering, MS-380, PO Box1892, Houston, TX, 77251-1892, USA [email protected], [email protected]
2 NXP Semiconductors, Corporate I&T/DTF, High Tech Campus 37, 5656 AE Eindhoven, The [email protected]
Summary. A new model reduction method for cir-cuit simulation is presented, which preserves passivityby interpolating dominant spectral zeros. These arecomputed as poles of an associated Hamiltonian sys-tem, using an iterative solver: the subspace acceler-ated dominant pole algorithm (SADPA). Based on adominance criterion, SADPA finds relevant spectralzeros and the associated invariant subspaces, whichare used to construct the passivity preserving projec-tion. Our method reduces all passive circuits, includ-ing those not suitable for reduction with PRIMA.
1 Introduction
Model reduction for circuit simulation has beenextensively studied (see [3] for a selection of re-cent results). In transmission lines, interconnec-tions are modeled as systems with millions of in-ternal variables, that are impossible to simulate infull dimension. A system of smaller dimension isneeded to approximate the behavior of the origi-nal and replace it during simulation. RLC circuitsdescribing the interconnect are passive systems,with positive real transfer functions [1], thus thereduced models should also passive.
The framework involves approximation of anoriginal dynamical system described by a set ofdifferential algebraic equations in the form (1),
Ex(t)=Ax(t)+Bu(t), y(t)=Cx(t)+Du(t), (1)
where the entries of x(t) are the system’s inter-nal variables, u(t) is the system input and y(t) isthe system output, with dimensions x(t) ∈ Rn,u(t) ∈ Rm, y(t) ∈ Rp. Correspondingly, E ∈Rn×n, A ∈ Rn×n, (A,E) is a regular pencil,B ∈ Rn×m, C ∈ Rp×n, D ∈ Rp×m. The dimen-sion of Σ is n, usually very large. In circuit sim-ulation, internal variables are chosen as the nodevoltages and loop currents: x(t) = [v(t), i(t)]T ,and the system original Σ(E,A,B,C,D) is sta-ble and passive. We aim for a reduced order modelΣ(E, A, B, C,D), which similarly to (1), satisfies:
E ˙x(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t),
where x ∈ Rk, E ∈ Rk×k, A ∈ Rk×k, B ∈ Rk×m,
C ∈ Rp×k, D ∈ R
p×m. Σ is obtained by project-ing the internal variables of the original systemx onto a subspace ColSpan V ⊂ R
n×k, along
Null W∗ ⊂ Rk×n. The goal is to construct V
and W, such that Σ is stable and passive. Addi-tionally, V and W should be computed efficiently.The reduced matrices are obtained as follows:
E = W∗EV, A = W∗AV, B = W∗B, C = CV.(2)
2 The method
We present the dominant spectral zero interpola-tion method (dominant SZM), a passivity preserv-ing model reduction method that extends beyondthe scope of existing methods such as PRIMA [5].Imposing no constraints on the system matrices,dominant SZM can reduce passive systems withan underlying possibly indefinite E matrix, trans-mission line models with controlled sources, orcircuits containing susceptance elements. The in-gredient for passivity preservation are the spectralzeros of Σ(E,A,B,C,D), defined as follows:
Definition 1. For system Σ with transfer func-tion: H(s) := C(sE − A)−1B + D, the spectralzeros are all s∈C such that H(s) + H∗(−s) = 0,where H∗(−s)=B∗(−sE∗−A∗)−1C∗+D∗.
According to [2, 8], the spectral zero methoduses a rational Krylov approach to construct theV and W which interpolate spectral zeros of Σ.Spectral zeros are solved efficiently from an asso-ciated Hamiltonian eigenvalue problem. In [2, 8],the selection of spectral zeros was still an openproblem. We propose a solution as follows: weextend the concept of dominance from poles [7]to spectral zeros, and adapt the iterative solverSADPA for the computation of dominant spec-tral zeros. The corresponding invariant subspacesare obtained as a by-product of SADPA, and areused to construct the passivity preserving project-ing matrices V and W. The procedure for passivereduction with dominant SZM follows [4]:
1) Given Σ(E,A,B,C,D), construct the associ-ated Hamiltonian system Σs described by (3),whose poles are the spectral zeros of Σ.
As =
0
@
A 0 B
0 −A∗ −C∗
C B∗ D+D∗
1
A ,Es =
0
@
E 0 0
0 E∗ 0
0 0 0
1
A ,Bs =
0
@
B
−C∗
0
1
A Ds
Cs =−Ds
`
C B∗ 0´
, Ds =(D+D∗)−1 (3)
161
2) Solve the Hamiltonian eigenvalue problem(Λ,R,L) = eig(As,Es), i.e. AsR = EsRΛ,L∗As=ΛL∗Es. Eigenvalues Λ=diag(s1, . . . , sn,−s∗1, . . . ,−s∗n) are the spectral zeros of Σ, andR=[r1, . . . , r2n], L=[l1, . . . , l2n].
3) Compute residues Rj associated with the sta-ble1 spectral zeros sj , j = 1 . . . n as follows:Rj=γjβj , γj=Chrj(l
∗jEhrj)
−1, βj= l∗jBh.4) Sort spectral zeros descendingly according to
dominance criterion|Rj |
|Re(sj)| [7, Chapter 3],
and reorder right eigenvectors R accordingly.5) Retain the right eigenspace R = [r1, . . . , rk] ∈
C2n×k, corresponding to the stable k most
dominant spectral zeros.5 Construct passivity projection matrices V and
W from the rows of R: V = R[1:n,1:k], W =
R[n+1:2n,1:k], and reduce Σ according to (2).
As explained in [2,4,8], by projecting with (2), Σinterpolates the k most dominant spectral zeros ofΣ, guaranteeing passivity and stability. For large-scale applications, a full solution to the eigen-value problem in step 2), followed by the dom-inant sort 3)-4) is computationally unfeasible. In-stead, the iterative solver SADPA [7, Chapter 3]is applied with appropriate adaptations for spec-tral zero computation [4], and the k most dom-inant spectral zeros and associated 2n × k righteigenspace R are obtained automatically.
3 Preliminary results
We reduce an RLC transmission line model withvoltage controlled current sources [4]. The sys-tem has an associated singular and indefinite ma-trix E, thus Σ has poles at infinity. PRBT [6]does not apply to E singular, and a deflation ofinfinite poles is required to make E invertible.For large systems this can be computationallyunfeasible, or must exploit a specific structureof E and A. PRIMA [5] on the other hand re-quires a definite E and an A matrix with specialblock symmetries. To apply PRIMA, stamps inE and A corresponding to the loop current vari-ables should be negated, so that E becomes defi-nite. This requires knowledge of how the simula-tor generates states internally and makes reduc-tion non-automatic and cumbersome in practice.Dominant SZM is more general, avoiding boththe deflation step required by PRBT and the signchange in PRIMA.
Fig. 1 shows the reduced models from domi-nant SZM, modal approximation [7] and PRIMA.Already for k = 2, dominant SZM gives a passivereduced model which is indistinguishable from theoriginal, while the other methods are less accu-rate. Table 1 summarizes approximation errors
1 s ∈ C is stable if Re(s) < 0.
and CPU times. SADPA computed two dominantspectral zeros, enabling efficient reduction withdominant SZM, when reduction with PRBT iscomputationally unfeasible. Passivity is preservedautomatically even when E is indefinite.
0 2 4 6 8 10 12 14−60
−50
−40
−30
−20
−10
0
10
Frequency (rad/s)
Mag
nitu
de (
db)
Frequency responseSZM with SADPA, Modal Approxiation, PRIMA
n = 1501, k = 2
OriginalReduced(SZM)Reduced(MA)Reduced(PRIMA)
Fig. 1. Original and reduced models with dominantSZM, modal approximation and PRIMA.
Table 1. Transmission line reduction n=1501 k=2
SADPA based Error Time(s) Iterations
Dominant SZM 1.89 · 10−4 0.8 14
MA 1.24 0.98 35
Other methods Error Time(s) Constraints
PRIMA 9.98 · 10−1 0.01E definite
otherwise unstable
PRBT - -E singular
deflation unfeasible
References
1. A. C. Antoulas. Approximation of Large-Scale Dy-namical Systems. SIAM, Philadelphia, PA, 2005.
2. A. C. Antoulas. A new result on passivity preserv-ing model reduction. Systems and Control Letters,54:361–374, 2005.
3. G. Ciuprina and D. Ioan, editors. Scientifc Com-puting in Electrical Engineering SCEE 2006, vol-ume 11 of Mathematics in Industry. Springer,Berlin, Germany, 2007.
4. R. Ionutiu, J. Rommes, and A.C. Antoulas. Pas-sivity preserving model reduction using dominantspectral zero interpolation. Submitted for publica-tion in IEEE Trans. Comput.-Aided Design Integr.Circuits Syst., 2007.
5. A. Odabasioglu, M. Celik, and L.T. Pillegi. Prima.passive reduced-order interconnect macromod-elling algorithm. 17:645–654, 1998.
6. J.R. Phillips, L. Daniel, and L.M. Silveira.Guaranteed passive balancing transformations formodel order reduction. 22(8):1027–1041, 2003.
7. Joost Rommes. Methods for eigenvalue problemswith applications in model order reduction. PhDthesis, Utrecht University, The Netherlands, 2007.
8. D.C. Sorensen. Passivity preserving model reduc-tion via interpolation of spectral zeros. Systemsand Control Letters, 54:347–360, 2005.
162
Session
P 3
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 3.1
Analysis of a PDE Thermal Element Model for
Electrothermal Circuit Simulation
Giuseppe Alı1, Andreas Bartel2, Massimiliano Culpo2, and Carlo de Falco3
1 Universita della Calabria [email protected] Bergische Universitat Wuppertal [email protected], [email protected],3 Dublin City University [email protected]
Summary. In this work we address the well-posed-ness of the steady-state and transient problems stem-ming from the coupling of a network of lumped elec-tric elements and a PDE model of heat diffusion inthe Chip substrate.
1 Introduction
Due to downscaling, power densities become moreimportant [5] and therefore thermal models to re-solve the geometric layout, which fit seamless intothe circuit design are necessary. A method forautomatically deriving a thermal network modelfrom the layout of an IC and substrate materialproperties was introduced in [1] and the numeri-cal validation is in progress [2].
The novelty of this method compared to otherexisting aproaches [3, 4] is that it does not workby fitting the parameters of a given networktopology, but rather it consists of a parabolicPDE which can be connected to a network oflumped (electrical) device models to perform cou-pled system-level electrothermal simulation witha standard spice-like circuit simulator. The cou-pling is performed by controlling the average tem-perature of some substrate regions via a set of(controlled) voltage sources and the total powerdissipated in some other regions via a set of (con-trolled) current sources. As an initial step towardsthe analysis of the coupled electro-thermal sys-tem, in this work we consider the equations ob-tained when the thermal element is controlled bya set of independent sources with finite internalresistance or conductance.
2 Statement of the problem
Let the domain Ω ⊂ Rd, with d = 1, 2, 3, modelthe IC substrate and Ωk be the active region ofthe k-th circuit element. We assume that Ω beLipschitz and that the family Ωk, k = 1, . . . , nsatisfies the requirements:
1.Ωk 6= ∅, Ωk ⊂ Ω ∀k = 1, . . . , n
2. Ωk ∩ Ωj = ∅ ∀j, k ∈ 1, . . . , n, k 6= j.
We denote by u(x, t) the temperature at an in-stant t at each point x in Ω, we let qk(t)/ |Ωk|be the average temperature in the region Ωk attime t and pk(t) the instantaneous Joule powerper unit length, area or volume (in case d = 1, 2or 3, respectively) dissipated by element k. Theheat diffusion in the substrate is governed by theheat equation
∂u
∂t−∆u+ αu = f(p), in Ω × (0, T ) (1)
denoting by f(p) the function
f(x,p) =
n∑
k=1
pk1Ωk(x),
where the vector function p is defined as
p(t) = (p1(t), . . . , pn(t)) ,
and the symbol 1Ωkrepresents the indicator func-
tion of the set Ωk. Equation (1) is supplementedwith initial-boundary conditions
u(x, 0) = u0(x), in Ω, (2)
u+ β∂u
∂n= g(t), on ∂Ω × (0, T ), (3)
where g is the ambient temperature. Further-more, the k-th average device temperature qk isconnected to u by the relation
∫
Ωk
u(x, t) dΩ = qk(t). (4)
Finally, to close the system, we need to state con-stitutive relations for the vector of average tem-peratures
q(t) = (q1(t), . . . , qn(t)) ,
and for the vector of instantaneous powers p interms of the electrical variables in the circuit. Asanticipated in the introduction, in the presentwork we make the simplifying assumption thatthe thermal network be controlled via indepen-dent sources. Under such an assumption the con-stitutive relations can be cast into the form
165
akpk + bkqk = ck(t), k = 1, . . . , n (5)
where the ck are given functions and ak and bkdenote constant coefficients. Notice that ak = 0for a given k indicates that the k-th region is at-tached to a voltage source fixing the value of itsaverage temperature, while bk = 0 indicates thatthe Joule power dissipated in the k-th region hasbeen assigned by attaching it to a current source.Summarizing, the problem we intend to investi-gate reads:
Given initial datum u0(x) and ambient temper-ature g(t), find u(x, t), p(t) and q(t) such thatequations (1)–(5) are satisfied, where α, β, ak, bkare known quantities and α, β ≥ 0.
The above problem is related to the following ker-nel problem:Given q and g, find u(x) and p such that:
−∆u+ αu = f(p), in Ω,
u+ β∂u
∂n= g, on ∂Ω,
∫
Ωk
u dΩ = qk k = 1, . . . , n.
(6)
We sketch below our approach to prove the well-posedness of the kernel problem (6), such proofwill be used, in the complete version of this work,as the basis for the analysis of the time-dependentproblem (1)–(5).
Sketch of the solution
Since the differential operator in (6) is linear, wecan represent the general solution, for any p, as
u = u∗ +
n∑
k=1
pkuk, (7)
with u∗(x) solution of the problem
−∆u∗ + αu∗ = 0, in Ω,
u∗ + β∂u∗∂n
= g, on ∂Ω,(8)
and uk(x), k = 1, . . . , n, solution of the problem
−∆uk + αuk = 1Ωk, in Ω,
u+ β∂uk∂n
= 0, on ∂Ω,. (9)
Both problems (8) and (9) are uniquely solvable.Then we can write the conditions for qk in (6)3as a linear algebraic system with variables pk:
n∑
k=1
∫
Ωj
uk dΩ
pk = qj −
∫
Ωj
u∗ dΩ, (10)
with j = 1, . . . , n. This system is uniquely solvedif and only if
the matrix
(∫
Ωj
uk dΩ
)is nonsingular. (11)
By multiplying equation (9)1 by uj and inte-grating on Ω, we find (β > 0)
∫
Ω
(∇uk · ∇uj + αukuj) dΩ
+
∫
∂Ω
1
βukuj dΩ =
∫
Ωk
uj dΩ, (12)
and we can write(∫
Ωk
uj dΩ
)=
(〈uj , uk〉
), (13)
where 〈·, ·〉 is the scalar product defined by theleft-hand side of (12). By the Cauchy-Schwartzinequality,
det
(〈uj, uk〉
)≥ 0, (14)
and the equality holds if and only if the func-tions are linearly dependent. This occurrence isexcluded by condition 2 on the subdomains Ωk.
3 Acknowledgments
The third author was supported by the EuropeanCommission in the framework of the CoMSONRTN project. The fourth author was supportedby the Mathematics Applications Consortium forScience and Industry in Ireland (MACSI) underthe Science Foundation Ireland (SFI) mathemat-ics initiative.
References
1. Massimiliano Culpo and Carlo de Falco. dynami-cal iteration schemes for coupled simulation. 2008.
2. Massimiliano Culpo, Carlo de Falco, Georg Denk,and Steffen Voigtmann. Automatic thermal net-work extraction and multiscale electro-thermalsimulation. Work in progress 2008.
3. T. Grasser and S. Selberherr. Fully coupled elec-trothermal mixed-mode device simulation of sigehbt circuits. Electron Devices, IEEE Transactionson, 48(7):1421–1427, Jul 2001.
4. P.M. Igic, P.A. Mawby, M.S. Towers, andS. Batcup. Dynamic electro-thermal physicallybased compact models of the power devices for de-vice and circuit simulations. Semiconductor Ther-mal Measurement and Management, 2001. Sev-enteenth Annual IEEE Symposium, pages 35–42,2001.
5. International roadmap commitee. Internationaltecnology roadmap for semiconductors 2007. Tech-nical report, ITRS, 2007.
166
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 3.2
Efficient simulation of large-scale dynamical systems using
tensor decompositions
F. van Belzen and S. Weiland
Eindhoven University of Technology, Department of Electrical Engineering, P.O. Box 513, 5600 MBEindhoven, The Netherlands [email protected], [email protected]
Summary. Tensors are the natural mathematicalobjects to describe physical quantities that evolveover multiple independent variables. This abstractconsiders the computation of empirical projectionspaces by decomposing the tensor that can be as-sociated with the measurement data. We show howthese projection spaces can be used to derive reducedorder models. The procedure is applied to a two-dimensional heat diffusion problem.
1 Introduction
Most model reduction techniques such as bal-anced truncation, Krylov methods, and ProperOrthogonal Decompositions (POD) [5] are pro-jection based methods. In this abstract, we ex-amine the POD method to reduce the complexityof systems with both space and time as indepen-dent variables. The POD method leads to sim-plified models by applying a Galerkin projectionon both the signals and the equation residuals ofa dynamical model. The POD method is partic-ularly popular in computational fluid dynamicsapplications where it simplifies spatial-temporalsystems by inferring projection spaces from col-lections of spatially distributed data. These pro-jection spaces are computed via singular value de-compositions of matrices that have the total meshsize of the spatial geometry as its dimension. Forspatial geometries that are two, three or largerdimensional, these computations become partic-ularly cumbersome in combination with fine grid-ded mesh-sizes. Moreover, this computation gen-erally neglects structures such as symmetries orCartesian structures that may be present in thespatial geometry.
In this abstract we propose a method to com-pute data-dependent projection spaces that leavesCartesian structures in multidimensional arraysof measured data intact. For this, we propose anextension of the concept of SVD to tensors.
2 Model reduction by projections
Consider an arbitrary linear N -dimensional sys-tem described by the Partial Differential Equa-
tion (PDE)
R
(∂
∂x1, . . . ,
∂
∂xN
)w = 0 (1)
in which w : X ⊂ RN → R is the (unknown)solution in N independent variables. Here, R ∈R·×1[ξ1, . . . , ξN ] is a real matrix valued polyno-mial in N indeterminates. Weak solutions of (1)are generally defined with respect to a Hilbertspace H of functions on a subset X′ of X accord-ing to
〈R(
∂
∂x1, . . . ,
∂
∂xN
)w,ϕ〉 = 0 ∀ϕ ∈ H. (2)
Given a basis ϕn of H and a finite dimensionalsubspace Hr = spanϕ1, . . . , ϕr of H, a reducedorder model is defined by the collection of solu-tions wr ∈ Hr that satisfy (2) with H replaced byHr. This is usually called a Galerkin projection ofthe original model [4].
3 Tensor decompositions
Suppose that the domain X of (1) has an Eu-clidean structure X = X1 × . . . × XN . In thissection we propose a construction of the projec-tion space Hr that is inferred from a measuredor simulated solution w of (1). More specifically,suppose that, for n = 1, . . . , N , the domain Xn
is gridded into a finite set of Ln elements and letXn := RLn . Suppose that w is a Finite Element(FE) solution of (1). Then w defines a multidi-mensional array W ∈ RL1×...×LN with wℓ1...ℓNbeing the (ℓ1, . . . , ℓN )th entry.
At a more abstract level W defines a tensor.An order-N tensor T is a multilinear functionalT : X1 × · · · × XN → R operating on N vectorspaces X1, . . . , XN . Elements of T , tℓ1···ℓN , canbe defined with respect to bases for X1, . . . , XN
according to tℓ1···ℓN = T (eℓ11 , · · · , eℓNN ), whereeℓnn , ℓn = 1, . . . , Ln is a basis for Xn, n =1, . . . , N .
A FE solutionw, or its associated multidimen-sional array W , therefore defines the tensor
167
W =
L1∑
ℓ1=1
. . .
LN∑
ℓN=1
wℓ1···ℓN eℓ11 ⊗ · · · ⊗ eℓNN (3)
where e1 ⊗ · · · ⊗ eN is shorthand for the rank-1tensor E(x1, . . . , xN ) := ΠN
n=1e⊤n xn.
The tensor (3) associated with the FE solutiondefines suitable projection spaces as follows. Wepropose the construction of orthonormal basesmℓn
n , ℓn = 1, . . . , Ln of Xn such that a coor-dinate change of W with respect to these basesachieves that the truncated tensor
WT :=
R1∑
ℓ1=1
. . .
RN∑
ℓN=1
wℓ1,...,ℓNmℓ11 ⊗ . . .⊗mℓN
N (4)
with Rn ≤ Ln, n = 1, . . . , N , will minimize theerror ‖W−WT ‖, in a suitable tensor norm, [2,3].For order-2 tensors (matrices) this is achievedby the SVD. For higher-order tensors, it is notstraightforward how to construct proper sets oforthonormal bases with this property. Differentmethods exist, including the Higher-Order Sin-gular Value Decomposition [1] and the TensorSVD [3].
Note that projection spaces are not only de-fined by the orthonormal bases. The truncationlevels, Rn ≤ Ln should also be chosen appropri-ately. This choice involves making a trade-off be-tween complexity and accuracy.
4 2D heat diffusion example
Consider the following model of a heat transferprocess on a rectangular plate of size Lx × Ly:
0=−ρcp∂w∂t +κx
∂2w∂x2 +κy
∂2w∂y2 . (5)
Here, w(x, y, t) denotes temperature on position(x, y) and time t and the rectangular spatial ge-ometry defines the Cartesian product X × Y :=[0, Lx]× [0, Ly]. Let Hr = XR1 ×YR2 with XR1 ⊆X = L2(X) and YR2 ⊆ Y = L2(Y) be finite di-mensional subspaces spanned by R1 and R2 or-thonormal bases functions ϕℓ1 and ψℓ2, re-spectively. Solutions of the reduced model arethen given by wr(x, y, t) =
∑ij aij(t)ϕi(x)ψj(y)
with aij(t) = [A(t)]ij a solution of the matrix dif-ferential equation
0 = −ρcpA+ κxFA+ κyAP. (6)
Here, F and P are defined as:
F=
2
6
6
4
〈ϕ1,ϕ1〉 ... 〈ϕ1,ϕR1〉...
...〈ϕR1 ,ϕ1〉 ... 〈ϕR1 ,ϕR1〉
3
7
7
5
P=
2
6
6
4
〈ψ1,ψ1〉 ... 〈ψ1,ψR2〉...
...〈ψR2 ,ψ1〉 ... 〈ψR2 ,ψR2〉
3
7
7
5
A FE solution of (5) is computed with gridsizes (L1, L2, L3) = (50, 100, 500), see Fig. 1. Or-thonormal bases were computed using the TensorSVD [3]. Basis functions are displayed in Fig. 2.
020
4060
80100
0
20
40
600
2000
4000
6000
8000
10000
12000
Fig. 1. Time slice of the FE solution of (5).
0 20 40 600.05
0.1
0.15
0.2
0.25
0 20 40 60−0.1
0
0.1
0.2
0.3
0.4
0 20 40 60−0.4
−0.2
0
0.2
0.4
0 20 40 60−0.5
0
0.5
0 50 1000
0.05
0.1
0.15
0.2
0.25
0 50 100−0.1
0
0.1
0.2
0.3
0 50 100−0.4
−0.2
0
0.2
0.4
0 50 100−0.6
−0.4
−0.2
0
0.2
0.4
Fig. 2. First basis functions for X (left) and Y (right),computed using tensorial SVD.
5 Conclusion
In this abstract we considered model reduction formultidimensional systems using the POD method.For the computation of empirical projection spaceswe proposed a method using tensor decomposi-tions. The techniques proposed were applied to atwo-dimensional heat diffusion problem. In the fu-ture, we plan to test the method on more complexexamples and aim to compare different tensorialdecompositions to assess accuracy, computationaleffort and reliability.
Acknowledgement. This work is supported by theDutch Technologiestichting STW under project num-ber EMR.7851.
References
1. L. de Lathauwer et al. A Multilinear Singu-lar Value Decomposition. SIAM J. Matrix Anal.Appl., Vol. 21 (4), 2000.
2. T.G. Kolda. Orthogonal tensor decompositions.SIAM J. Matrix Anal. Appl., Vol. 23 (1), 2001.
3. F. van Belzen, S. Weiland and J. de Graaf. Singu-lar value decompositions and low rank approxima-tions of multi-linear functionals. Proc. 46th IEEEConf. on Decision and Control, 2007.
4. S. Volkwein and S. Weiland. An Algorithm forGalerkin Projections in both Time and Spatial co-ordinates. Proc. 17th MTNS, 2006.
5. M. Kirby. Geometric Data Analysis. John Wiley,2001
168
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 3.3
Magnetic Force Calculation applied to Magnetic Force
Microscopy
Thomas Preisner, Michael Greiff, Josefine Freitag, and Wolfgang Mathis
Leibniz Universitaet Hannover, Appelstrasse 9A, 30167 Hannover, Germany [email protected],[email protected], [email protected], [email protected]
Summary. In IC failure analysis the detection ofcurrents is often used to indicate the presence of adefective device. One method used for this analysis isthe Magnetic Force Microscopy (MFM). The manu-factured tips used for MFM measurements frequentlyshow fabrication errors. Hence, in this work a the-oretical model of the MFM was developed to verifyand improve the results of laboratory MFM measure-ments. Furthermore, the influence of the thickness ofthe magnetic tip coating as well as different kindsof asymmetrical geometries are numerically investi-gated.
1 Introduction
Due to technical advances in the developmentof integrated circuits a reduction of the dimen-sions of electronic devices and structures is feasi-ble. Consequently, the IC failure analysis, whichmakes use of occurring currents as a possible ev-idence for defective devices, becomes more com-plex. The magnetic field caused by these currentscan be detected by using special methods such asthe Magnetic Force Microscopy (MFM). In [1] itis demonstrated that the MFM technique is appli-cable for the detection of currents down to 1µA.Furthermore, a comparison between laboratorymeasurements and a theoretical consideration inwhich the magnetic coated tip is approximated byseveral magnetic moments is shown. The result-ing values of both approaches in [1] are on a pi-cometer scale but the amplitudes differ from eachother. As solely reason for these differences onlythe orthogonally assumed tip magnetization inthe theoretical consideration is denoted; but somefurther effects must be considered. In an ideal casethe tip is atomically sharp and features the ge-ometry of a symmetric cone. In fact, a couple ofatoms are located at the tip apex and even dou-ble tips might emerge depending on the mannerof preparation. In addition a further reason forthe differences is the technical unattainable geo-metric symmetry of the tip. Hence, in this studyan improved model with more degrees of freedomwas developed by using the finite element method(FEM). The main objectives are the verification
of the laboratory MFM measurement results andthe investigation of the influences and effects dueto variations of the tip geometry. For that pur-pose different force calculation methods like theVirtual Work Principle and the Maxwell StressTensor are implemented and compared with eachother. It is a fact, both methods are often used toobtain the total force of an object under investi-gation, but the force distribution strongly differfrom each other [4,5]. For this reason the VirtualWork Principle is implemented in our study insuch a manner as it show in [4, 5] to obtain localforces with physical meaning.
2 Micromagnetic Model
In order to develop an applicable theoretical modelof the MFM it is necessary to describe the causesfor the magnetic fields. Therefore two differentsources must be considered, the current densityJ and the material magnetization M. The funda-mental expression for such problems is the wellknown curl-curl equation
∇× 1
µ(∇× A) = ∇× µ0
µM + J, (1)
whereas A is the magnetic vector potential, µ isthe material permeability and µ0 is the perme-ability of free space. For the purpose of ensur-ing the uniqueness of the magnetic vector poten-tial and making the solution of the coupled sys-tem numerically stable [2] the Coulomb gauge isadded. Applying the method of weighted residu-als and using Gauss law and a vector identity,the weak formulation can be obtained. As anexample for a three dimensional model used inthis work we consider a current carrying micro-conductor which is scanned by a cantilever hold-ing the magnetic coated tip underneath. This ar-rangement is pictured in Fig. 1. In this approachwe assume a conically shaped tip with an angle of30. This tip consists of a cobalt-chromium com-pound, which is orthogonally magnetized with re-spect to the sample surface. Analogous to [3] weset the value for the magnetization in negative
169
5 µm
2.5 µm
4 µm
8 µma
x
y
z
I
Fig. 1. Configuration of the 3D-model.
y-direction to M = 749kAm . Furthermore, we as-sume a hard ferromagnetic material, so that thepermanent permeability of the conical magneticcoating is approximately µr ≈ 1. The thicknessof this magnetic coating amounts to 60nm. Themicro-conductor features a width of 5µm, a thick-ness of 2.5µm and carries a current of 1mA.
3 Results
The occurring forces during the scanning processbetween tip and the micro-conductor are calcu-lated in a span of 55µm. Therefore different for-mulations are applied and compared. The firstone is the Virtual Work Principle, the secondone the surface integration over Maxwells StressTensor. Furthermore, the Virtual Work Principleis implemented in such a manner as it is showin [4, 5] in order to obtain the force distributionon the magnetic coating of the tip. Therefore wesolve (2) at the elements e corresponding to anode k in a direction i
Fik = −∑
ek
[ ∫
Ωek
(B− Br)T
µ0J−1 δJ
δsiB |J| dΩek
−∫
Ωek
(B − Br)T
(B− Br)
2µ0
δ |J|δsi
dΩek
](2)
where B is the magnetic flux density, Br is theremanence flux density and J is the Jacobian ma-trix. Finally we withdraw the intrinsic forces [4,5].Concerning the total magnetic force, all threemethods are in excellent agreement with eachother. All the values are located in a pN -range.The active force with respect to the tip axis anda simulated MFM image is shown in Fig. 2.
In [1] the magnetically coated tip is approxi-mated by using several magnetic moments and iscompared to laboratory measurement values. Thepresented numerical results are perfectly concor-dant with these considerations. In Fig. 3 the forcedistribution of the magnetic coating at the point3.5µm to the right of the current carrying micro-conductor is shown.
−30 −20 −10 0 10 20 30−8
−6
−4
−2
0
2
4
6
8
X [ µm ]
Y [
pN
]
Total Force
VW 1VW 2MST
Fig. 2. Total force acting on the cantilever (left) anda simulated MFM image (right).
−3
−2
−1
0
1
2
3x 10−6
−3−2
−10
12
3
x 10−6
0
1
2
3
4
5
6
7
8
9
x 10−6
X [ m ]
Force Distribution
Z [ m ]
Y [
m ]
Fig. 3. Force distribution of the magnetic coated tip.
In this abstract only forces and simulations of atotally symmetric tip are shown. In the full pa-per the influences of tip asymmetry and differentthicknesses of the magnetically coated film will bepresented.
References
1. A. Pu, A. Rahman, D.J. Thomson and G.E.Bridges. Magnetic force microscopy measumentsof current on integrated circuits. IEEE CCECE2002, 1:439–444, 1997.
2. K. Preis, I. Bardi, O. Biro, C. Magele, W. Ren-hart, K.R. Richter and G. Vrisk. Numerical analy-sis of 3D magnetostatic fields. IEEE Trans. Magn.,27:3798–3803, 1991.
3. A. Carl, J. Lohau, S. Kirsch and E.F. Wassermann.Magnetization reversal and coercivity of magneticforce microscopy tips. J. Appl. Phys., 89:6098–6104, 2001.
4. L.H. de Medeiros, G. Reyne, G. Meunier and J.P.Yonnet. Distribution of electromagnetic force inpermanent magnets. IEEE Trans. Magn., 34:3012–3015, 1998.
5. L.H. de Medeiros, G. Reyne and G. Meunier.About the distribution of forces in permanentmagnets. IEEE Trans. Magn., 36:3345–3348, 1999.
170
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 3.4
EM Scattering Calculations using Potentials
Magnus Herberthson1,2
1 Department of Sensor Informatics, Swedish Defence Research Agency, Box 1165, SE - 581 11 Linkoping,Sweden [email protected]
2 Department of Mathematics, Linkopings Universitet, 581 83 Linkoping, Sweden [email protected]
Summary. EM scattering from PEC surfaces aremostly calculated through the induced surface currentJ. In this paper, we consider PEC surfaces homeomor-phic to the sphere, apply Hodge decomposition theo-rem to a slightly rewritten surface current, and showhow this enables us to replace the unknown currentwith two scalar functions which serve as potentials forthe current.
Implications of this decomposition are pointedout, and numerical results are demonstrated.
1 Introduction
There are numerous ways to address the prob-lem of calculating the radar cross section of PECsurfaces [1, 2]. One method is to solve the elec-tric field integral equation, (EFIE), where a stan-dard reference is [3]. In frequency domain, takingthe incoming field Ei to be a plane wave we haveEi = E0e
−ik·r. By choosing an adapted ON basis,the EFIE becomes [1, 3, 4],
∀r ∈ S : E0e−ikzx = (1)
ikcµ0(I +1
k2∇∇·)
∫
S
g(r, r′)J(r′)dS′
where J is the surface current on S, g is the
Greens function g(r, r′) = eik|r−r′|
4π|r−r′| , and where =
means tangential equality (on S). k, k, µ0, c, and∇ have their usual meanings.
Since the electric field is a covariant vectorfield (i.e. a one-form) rather than a contravariantvector field the equality in (1) is, for each r ∈ S,evaluated in T ∗
p S, the cotangent space at p. It istherefore natural to use the theory of forms, andin particular Hodge decomposition theorem whenaddressing (1). For simplicity, we will assume thatthe surface S is closed and homeomorphic to asphere.
2 Reformulation of EFIE
One first notes that the LHS of (1) is not an ex-act one-form. However, by multiplying both sideswith eikz, the LHS is just the tangential part ofx, i.e., the one-form dx, which is exact, i.e., a
gradient of a scalar function. For this reason, weintroduce the following functions
h(r, r′) = g(r, r′)eik(z−z′),K(r′) = eikz
′
J(r′) (2)
After multiplication with eikz, (1) becomes
e0∇x−∇[∫
S
hz ·KdS′ +i
k
∫
S
K·∇hdS′]
= (3)
ik
∫
S
hKdS′ − z
[ik
∫
S
hz ·KdS′ −∫
S
K·∇ hdS′]
This formulation is still in vector calculus nota-tion, and theoretical advantage is that the LHSof (3) is now an exact one-form. Note, however,that the fictive current K is a ’down sampled’version of J, i.e., the oscillatory part e−ikz is fac-tored out. In practice, this may therefore allowfor a much sparser sampling of the ’current’ K,and therefore reduced numerics. Also, note thatcomplicated objects may require dense samplingover areas of resonance.
To proceed we now use the Hodge decom-position [5] applied to K. The Hodge decom-position theorem asserts that, when S is com-pact, any one-form K can be written uniquely asK = dΦ+ β + δψ where Φ is a scalar, β is a har-monic one-form and ψ is a two-form. Moreover,d is the exterior derivative, the coderivative δ isδ = − ∗ d∗, where ∗ is the Hodge star. However,β ∈ Harm1(S) is zero since there are no nontriv-ial harmonic one-forms on S. Thus, with Ψ = ∗ψ,so that Ψ is a scalar function, we have that
K = dΦ+ δ ∗ Ψ ∼= ∇SΦ+ n×∇SΨ
on S, which means that K is expressed throughthe two scalar potentials Φ and Ψ . Here, n is aunit normal to S and ∇S stands for the intrinsicgradient operator on S. Equation (3) now reads
∇[e0x−
∫
S
(<d′z′,K > +i
k∆′SΦ)h dS′
]= (4)
i
∫
S
(kK− z [k <d′z′,K > +i∆′SΦ])h dS′
where ∆S is the intrinsic laplace operator on S.Depending on approach, equation (4) can be
addressed in several ways. In the next section, wewill consider a few of these.
171
3 Calculational benefits
The factorization K(r′) = eikz′
J(r′) may leadto sparser sampling. Since the external appliedfield is −E0x e
−ikz we can expect that the in-duced current J largely contains the oscillatorypart e−ikz. Therefore, in non-resonant areas, the’current’ K resembles an envelope, which can besampled much sparser than J. For high frequen-cies, this can reduce the numerical problem sub-stantially.
Another way of using (4) is to note that theleft hand side is an exact one-form on S. Namely,by a discretization of K = dΦ+ δ∗Ψ which givesΦ and Ψ n degrees of freedom each, we must inprinciple produce and solve a 2n × 2n system ofequations
(X XX X
)([Φ]n×1
[Ψ ]n×1
)=
(v1v2
)(5)
where [Φ]n×1, [Ψ ]n×1 are n× 1 vectors represent-ing the fields Ψ and Φ, where v1, v2 also are n× 1vectors and where X stands for various matricesof order n × n. Instead of solving (5), it is obvi-ously much cheaper to solve a system which takesthe form
(X XX X
)([Φ]n×1
[Ψ ]n×1
)=
(v10
)(6)
since one can first solve the lower block row of(6) and then substitute in the upper block row.In principle, one 2n×2n inversion is replaced withtwo n× n inversions. One can go from (5) to (6)in several ways. Since the LHS of (4) is exact, onecan take the exterior derivative and coderivativeof (5) to get (6). One can also get (6) by testing(4) against suitable test functions directly.
4 Numerical results
Although this approach has been reported ear-lier, the full implementation using the scalar po-tentials Φ and Ψ has not been demonstrated sofar. Considering the space available, we will onlyexemplify the above approach in the most famil-iar case, i.e., the sphere. Illuminating a sphereof radius 1 m with a plane wave with λ =1 m,propagating along the z axis and with the electricfield parallel to the x axis, the resulting potentialsand currents are illustrated in Fig. 1-3. In Fig. 1,the scalar potential Φ is shown as an intensitymap over the surface. |Φ| is given by the bright-ness, and the phase information is encoded in thecolour. The gradient ∇Φ is shown with real (blue)and imaginary (red) parts. In Fig. 2, Ψ and theco-gradient n × ∇Ψ are displayed. In Fig. 3, the
x
z
Fig. 1. The potential Φ and its current.
x
z
Fig. 2. The potential Ψ and its (co-)current.
x
z
Fig. 3. The current K.
total current K, from which the physical currentJ(r) = e−ikzK(r) is obtained, is drawn.
A more detailed analysis will follow, but pre-liminary, the outcome of the calculations are ingood agreement with expected results.
References
1. Harrington, R. F., Field Computation by MomentMethods, IEEE Press, 1993.
2. Chew, W. C. et al, editors, Fast and efficient algo-rithms in computational electromagnetics, ArtechHouse, 2001.
3. Rao, S. M., Wilton, D.R., Glisson, A.W., ”Elec-tromagnetic Scattering by Surfaces of ArbitraryShape”, IEEE Trans. Antennas Propag., vol. AP-30, pp. 409-418, May 1982
4. Kristensson, G., ”Spridningsteori med anten-ntillampningar”’, Studentlitteratur, Lund, 1999
5. Gockeler, M., Schucker, T., ”Differential geometry,gauge theories, and gravity”, Cambridge Univer-sity Press, 1987
172
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 3.5
Simulation of large interconnect structures using a ILU-type
preconditioner
D. Harutyunyan1,3, W. Schoenmaker2, and W.H.A. Schilders1,3
1 Technische Universiteit Eindhoven, Den Dolech 2, 5612 AZ Eindhoven, The Netherlandsd.harutyunyan,[email protected],
2 Magwel NV, Martelarenplein 13, B-3000 Leuven, Belgium [email protected] NXP Semiconductors, High Tech Campus 37, 5656 AE Eindhoven, The Netherlandsdavit.harutyunyan,[email protected]
Summary. For a fast simulation of on-chip inter-connect structures we consider preconditioned iter-ative solution methods of large complex valued lin-ear systems. The matrices are ill-conditioned and ef-ficient preconditioners are indispensable to solve thelinear systems accurately. We analyze the dual thresh-old ILUT preconditioner with BICGSTAB iterativesolver and show that this preconditioner provides avery accurate solution for real life complicated prob-lems in an efficient way.
1 Introduction
With the increasing complexity of on-chip inter-connect structures more robust and fast simu-lation methods are necessary to understand thebehavior of electromagnetic fields in such com-plex structures. Field simulation approaches pro-vide more insight and preserve important physi-cal characteristics of electromagnetic fields at adiscrete level. The behavior of the electromag-netic fields is governed by the Maxwell equations.For many applications the potential formulationof the Maxwell equations is used which has severaladvantages. In particular, for on-chip intercon-nect structures the potential formulation allowsseparate modeling of fields on dielectric, semicon-ductor and metallic regions.
After discretization of time harmonic Maxwellequations we obtain a complex valued linear sys-tem of equations
Ax = b. (1)
Rapid changes in material properties can affectthe numerical properties of the matrix A and mayresult into slow convergence for iterative solvers.To overcome these problems good preconditionersare required to solve the linear system of equa-tions accurately.
For ill-conditioned matrices the standard ILUfactorization requires a small drop tolerance to
achieve good accuracy. However, with a smalldrop tolerance, the fill-in of the L and U matri-ces is not predictable and for large difficult prob-lems it becomes almost impossible to constructthe ILU factorization due to memory limitations.
To overcome the memory problems we use theILUT(p, τ) preconditioner [2], where p is a param-eter which controls the number of fill-ins in the Land U matrices, and τ is the drop tolerance. Thispreconditioner nowadays is widely used and hasproved to be a very efficient and robust methodfor many complicated problems.
2 Potential formulation of theMaxwell equations
The potential formulation of the Maxwell equa-tions in the frequency domain can be writtenas [1]:
∇× 1
µ∇×A− k(−jωA−∇V ) = Jdiff , (2a)
∇ · (ε(∇V + jωA)) = −ρ, in insulators (2b)
and semiconductors,
∇ · (k(∇V + jωA)) = 0 in metals, (2c)
where k = (σ + jωε). This formulation allows usto deal with different materials separately.
For the unique solution of (2) a gauge conditionis necessary [3]. The following gauge condition isconsidered:
1
µ0∇ (∇ · A) + jωεξ∇V = 0 (3)
where 0 ≤ ξ ≤ 1. Equation (3) resembles to theCoulomb gauge for ξ = 0 and to the Lorentzgauge for ξ = 1.
We note that the vector potential A is a 1-formand the potential V is a 0-form. For the discretiza-tion of the vector potentials we apply the ’finite-surface method’ [4], whose origin is found in the
173
Stokes’ theorem. This discretization method pre-serves important physical characteristics of theelectromagnetic fields at the discrete level and weobtain physically relevant solutions.
3 Numerical experiments
For a solution of large ill-conditioned linear sys-tems (1) we apply the dual threshold incompleteILUT preconditioner. The computational time re-quired for the ILUT factorization can be sig-nificantly reduced by proper reordering of thematrix elements. We compare two common re-ordering methods: the symmetric reverse Cuthill-McKee reordering (SYMRCM) and the approxi-mate minimum degree (AMD) reordering.
We present the numerical experiments on the in-terconnect structure given in Fig. 1. The structuredimension in micrometers is 4.4 × 5.5 × 4.24 andit consist of 3 metallic layers. The contacts be-tween semiconductor and the first metallic layeris Tungsten and all the other metals and vias arecopper.
Fig. 1. Interconnect structure.
The frequency of interest is 500MHz and we gen-erate a mesh which results into a system with141513 unknowns. The convergence diagram ofthe relative residual of the BICGSTAB method isgiven in Fig. 2. In Fig. 3 eigenvalue distributionof the 25 smallest eigenvalues using the Jacobi-Davidson method are shown. It is clear that thepreconditioner shifts the smallest eigenvalues farfrom the origin.
Acknowledgement. The first author was financed bythe European Commission through the Marie CurieActions of its Sixth Program under the project O-MOORE-NICE, contract number MTKI-CT-2006 -042477.
0 100 200 300 400 500 600 700
10−12
10−8
10−2
Iterations
Rel
ativ
e re
sidu
al
p=100, τ=10−7
p=100, τ=10−8
p=150, τ=10−7
p=150, τ=10−8
p=200, τ=10−6
0 50 100 150 200 250
10−12
10−8
10−4
10−1
Iterations
Rel
ativ
e re
sidu
al
p=100, τ=10−7
p=100, τ=10−8
p=150, τ=10−7
p=150, τ=10−8
p=200, τ=10−6
Fig. 2. Convergence diagram of relative residual.Top: SYMRCM reordering, Bottom: AMD reorder-ing.
−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6−0.02
0
0.02
0.04
0.06
0.08
0.1
Ax=λ xAx=λ LUx
Fig. 3. The smallest 25 eigenvalues distribution ofthe original matrix and the preconditioned matrix.
References
1. P. Meuris, W. Schoenmaker, and W. Magnus,Strategy for electromagnetic interconnect model-ing, IEEE TCAD for IC 20 (2001), 753–762.
2. Yousef Saad, ILUT: a dual threshold incompleteLU factorization, Numer. Linear Algebra Appl. 1(1994), no. 4, 387–402.
3. W. Schoenmaker and P. Meuris, Geometryand computational electrodynamics-II selection ofgauge condition, Personal communication.
4. W. Schoenmaker, P. Meuris, W. Schilders, K.-J.van der Kolk, and N. van der Meijs, Maxwell equa-tions on unstructured grids using finite-integrationmethods, Proceedings of the 12th SISPAD 2007,Vienna.
174
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 3.6
Fast Linear Solver for Vector Fitting Method
M.M. Gourary1, S.G. Rusakov1, S.L. Ulyanov1, M.M. Zharov1, B.J.M. Mulvaney2, and B. Gu2
1 IPPM, Russian Academy of Sciences, Moscow, Russia [email protected] Freescale Semiconductor Inc., Austin, Texas, USA [email protected]
Summary. A new algorithm to solve the least squaresproblem in Vector Fitting (VF) method is proposed.The algorithm is based on QR-decomposition that ex-ploits the specific structure of the VF matrix. Numer-ical experiments confirmed the essential reduction ofcomputational efforts. Besides a new solver providesbetter accuracy and requires much less memory thanthe standard Matlab sparse solver.
1 Introduction
The Vector Fitting (VF) method [1] approximatesexperimental data of vector transfer functions(TF) by rational functions defined by partial frac-tions
fn(s) =∑
m
rmnspm
+ dn + shn (1)
VF is the iteration method. It iterates valuesofpoles by solving the least squares (LS) problemwith respect to the residues rmn, dn, hn, rm(∑
m
rmsk − pm
+1
)·fn(sk)=
∑
m
rmnsk − pm
+dn+skhn
(2)Here m = 1...Np is pole index, n = 1...NTF
is the index of TF component of the vector, skare frequencies of measured or simulated data(fn(sk)), k = 1...Nf .
In the paper we consider the most frequentcase when for each frequency sk samples fn(sk)are obtained for all TF (n). In this case the sizeof (2) is Ns× (Np +NTF(Np +2)) where Ns is thetotal number of samples.
After solving (2) new values of poles are ob-tained as zeroes of σ-function
σ(s) =∑
m
rmsk − pm
+ 1 (3)
But the application of VF to large size prob-lems meet difficulties due to essential growth ofcomputational efforts in solving (2) under the us-age of the standard Matlab sparse solver. In thispaper we present special-purpose algorithm to re-duce the efforts. Reducing is achieved by takinginto account specific structure of the linear sys-tem (2).
2 Algorithm Description
The system (2) can be represented as follows
Ax = b, where x =[xF xS
]T, (4)
xF is the vector of values rmn, dn, hn and xS isthe vector of residues of σ(rm).
The system (4) is the overdetermined linearsystem, and its solution corresponds to the leastsquares (LS) problem ‖Ax−b‖2 → min. The Mat-lab solver obtains the solution by QR decomposi-tion A = Q ·R with subsequent solving the uppertriangle system Rx = QT · b.
The effectiveness of the solver can be essen-tially improved due to the structure of A in (4)
A =[AF AS
](5)
where Ns ×Nf (Np +2) matrix AF has the block-diagonal form with NTF identical blocks
AF = diag(a, a, . . . , a) (6)
The size of a is Nf × (Np + 2). Due to such formthe QR decomposition AF = QF ·RF can be eas-ily performed by QR decomposition of each blocka = q · r taking into account that
RF =diag(r, r, . . . , r) (7)
QF =diag(q, q, . . . , q). (8)
For convenience of the description we assumethat the matrix AS in (5) and rhs vector b alsohave block form (each block contains Nf rows)
AS =[A
(1)S A
(2)S . . . A
(NTF)S
]T
b =[b(1) b(2) . . . b(NTF)
]T , (9)
The following algorithm solves the problem (4).Algorithm with simple orthogonalization
1. QR decomposition of block a
a = q · r2. Orthogonalize matrix with respect to QF
H = QTF · AS(H(k) = qT · A(k)S )
The result of orthogonalization is the matrixBS
BS = AS −QF ·H(B(k)S = A
(k)S − q ·H(k))
175
3. QR decomposition of BS
BS = QS · RS
4. Solve upper triangle linear system
RS · xS = bS where bS = QTS · b
To perform VF iteration (3) it is sufficient to de-termine only subvector xS, so after step 4 the al-gorithm can be stopped.
The experiments showed poor numerical prop-erties of the algorithm due to insufficient accuracyof orthogonalization in step 2. In accurate arith-metic the orthogonalization must provide zeromatrix QTF · QS = 0. But actually the values ofentries of the matrix noticeably differ from zero,especially at higher orders. The reorthogonaliza-tion allows to essentially decrease values of matrixentries. In experiments with Np = 20 the maxi-mum entry of the matrix product was reducedfrom 0.35 to 6 · 10−17.
Thus, the algorithm was modified to includethere orthogonalization.Algorithm with reorthogonalization
First 3 steps are the same as in the Algorithmwith simple orthogonalization.
4. Reorthogonalization.
H1 = QTF ·QS, BS1 = QS −QF ·H1
5. QR decomposition of BS1
BS1 = QS1 ·RS1
6. Compute new upper triangle matrix
RS2 = RS1 ·H1
7. Solve upper triangle linear system
RS2 · xS = QTS1 · b
3 Testing Results
Experiments were performed for two data sets:1. The s-parameters of transmission line obtainedby EM simulation. LS systems in the tests corre-spondwith initial poles. The systems are singularat these points.2. The artificially generated TF with known polesand residues. LS systems in the test correspondwithoriginal poles, so the accurate solution mustprovide zero residual norm. The systems are non-singular.
The results are presented in Table 1, 2 forNf = 200 and different Np, NTF. The prefix atthe value of NTF indicates the data set: trans-mission line (T) or generated (G). The results of
the Matlab sparse solver and the new solver arepresented in the form Matlab/new = ratio.
Table 1 presents the CPU time to perform onesolving. One can see that new solver allows toreduce CPU time by the factor 50–100 for suffi-ciently large sizes of LS system. For very smallsizes the Matlab solver provides less efforts thanthe new one but it occurs only in cases with CPUtime less than 0.1sec.
Table 2 presents residual norms (LS values)for both solvers. For the generated example normsare given without factor 10−12 due to space short-age. Note that LS algorithm minimizes the resid-ual norm, so less value of the norm corresponds tothe more accurate solution of the problem. Thusfrom Table 1 one can conclude that in all casesnew solver provides better accuracy. The advan-tages of new solver increase with increasing thesize of the systems.
Table 1. CPU Time, sec
NTF Np = 10 Np = 20 Np = 40
T5 .01/.04 = 0.25 .37/.09 = 4.1 1/0.22 = 4.4
T10 0.6/0.09 = 6.8 1.6/0.17 = 9.2 4.8/0.4 = 12
T25 3.4/0.54 = 6.2 20/0.5 = 40 69/1.5 = 46
G6 0.1/0.05 = 2 0.3/0.1 = 3 0.7/0.33 = 2.1
G20 1.9/0.17 = 11.5 5.3/0.47 = 11 21/1 = 21
G30 5.2/0.3 = 17 20/0.95 = 21 96/1.4 = 68
Table 2. Residual norm (‖Ax− b‖)
NTF Np = 10 Np = 20 Np = 40
T5 2/1.5 = 1.4 3.5/3.2 = 1.1 6.9/6.8 = 1.0
T10 3.3/2.3 = 1.45 5.6/5.0 = 1.1 11/0.43 = 25
T25 6.5/5.5 = 1.2 1.3/1.2 = 1.1 26/1.2 = 21
G6 2.4/0.9 = 2.6 11/3.4 = 3.2 15/5.1 = 2.9
G20 6.8/2.1 = 3.2 60/8.0 = 7.5 88/11 = 8.0
G30 17/4.3 = 3.9 111/9 = 12.6 197/21 = 9.3
The comparisons for higher sizes of LS prob-lem were not performed due to insufficient mem-ory in the run of the Matlab sparse solver. A newsolver could be run for considerably higher sizes.Thus a new solver essentially reduces memory re-quirements.
References
1. B. Gustavsen and A. Semlyen. Rational approxi-mation of frequency domain responses by vectorfitting. IEEE Trans. Power Delivery, vol. 14, no.3, pp. 1052–1061, 1999.
176
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 3.7
Evaluation of Domain Decomposition Approach for Compact
Simulation of On-Chip Coupled Problems
Michal Dobrzynski, Jagoda Plata, and Sebastian Gim
Numerical Methods Laboratory, Faculty of Electrical Engineering, Politehnica University of BucharestSpl. Independentei 313, 060042 Bucharest, Romania ( d michal, plata, seb) @lmn.pub.ro
Summary. Continued device scaling into the nanome-ter region has given rise to new effects that previ-ously had negligible impact but now present greaterchallenges to successful design of mixed-signal silicon.This paper evaluates Domain Decomposition strate-gies for compact simulation of on-chip coupled prob-lems from a computational perspective using the re-cently completed CHAMELEON-RF software proto-type on several standard benchmark structures.
1 Introduction
Incessant miniaturization of the transistor ac-cording to Moore’s Law has lead to generationalimprovements in microprocessor technology [1].However, continued scaling of devices into thenanometer region has given rise to new effectsthat had previously negligible impact but nowpresent challenges to continued scaling. This hasresulted in an increased complexity in engineer-ing resources essential for a successful design. TheITRS roadmap suggests extreme scaling of CMOStechnology until the 10 nm region and operatingfrequencies of up to 60 GHz in future generationdevices [2]. At such close dimensions, fabricationprocess variations, substrate noise and EM cou-pling between circuit components make mixed-signal RF silicon designs extremely challenging.
Because of this, the CHAMELEON-RF projectwas conceived as a part of an initiative to addressthese issues [3]. The project is a research plat-form for the development of prototype tools andmethodologies for comprehensive high accuracymodeling of on-chip electromagnetic effects usingthe Domain Decomposition (DD) approach andthe concept of electromagnetic interconnectorsor ’hooks’. They are essentially boundary condi-tions between ElectroMagnetic Circuit Elements(EMCEs), in order to manage the unprecedentedcomplexity faced when designing next generationhighly integrated mixed-signal architectures andSystem on Chip (SoC) applications.
The CHAMELEON-RF nano-EDA researchplatform incorporates a novel dual Finite IntegralTechnique (dFIT) EM field simulator with sys-tematic All Level Model Order Reduction to keep
manageable the IC’s complexity [4]. The methodallows tractable multi-scale parameterized modelextraction of coupled structures with the possibil-ity of sensitivity analysis [5]. This approach hasadvantages over alternative approaches in full 3Dfield simulators - such as FEM, BEM etc. - which,although enable greater accuracy, are intractablefor most practical real world designs.
In this paper, we evaluate the DD strategyfor compact simulation of on-chip coupled prob-lems from a computational perspective. By de-composing an intractable simulation domain intodistinct domains connected by ’hooks’, computa-tional savings can be obtained by simulating nonessential domains, such as the substrate or air lay-ers, in just the MagnetoStatic (MS) regime, com-pared with a Full Wave (FW) simulation. Thisapproach results in state space matrices with areduced number of Degrees Of Freedom (DoFs).
The remaining sections of this paper are or-ganized as follows. Section 2 presents the keypoints of the DD approach to manage the designcomplexity with a user specified accuracy. Vali-dation results from a benchmark test case usingthe recently completed CHAMELEON-RF soft-ware prototype are presented in Section 3. Thecomputational savings of the DD strategy is thendemonstrated. The paper is finally concluded inSection 4.
2 Domain Decomposition Approach
Fig. 1. Air domain
177
Fig. 2. SiO2 domain
Fig. 3. Substrate domain
3 Validation Results
Fig. 4. Parameter S11
Fig. 5. Parameter S21
Fig. 6. Parameter S12
Fig. 7. Parameter S22
4 Conclusions
Acknowledgement. The authors would like to grate-fully acknowledge support and insightful discussionwith Prof. Daniel Ioan, Asc. Prof. Gabriela Ciup-rina and the LMN Chameleon project team. The au-thors would also like to acknowledge the financialsupport offered by the European Commission underthe Marie Curie Fellowship programme (FP6/EST3,FP6/ToK4nEDA).
References
1. E. Mollick, Establishing Moore’s Law, IEEE An-nals of the History of Computing, vol. 28, pp. 62-75, 2006.
2. ITRS Roadmap, in www.itrs.net.3. J. Niehof, H. Janssen, and W. Schilders, Compre-
hensive High-Accuracy Modeling of ELectromag-netic Effects in Complete Nanoscale RF blocks:CHAMELEON RF, presented at IEEE Workshopon Signal Propagation on Interconnects, 2006.
4. D. Ioan, G. Ciuprina, M. Radulescu, and E. See-bacher, Compact modeling and fast simulation ofon-chip interconnect lines, IEEE Transactions onMagnetics, vol. 42, pp. 547-550, 2006.
5. D. Ioan, G. Ciuprina, and M. Radulescu, Theoremsof parameter variations applied for the extractionof compact models of on-chip passive structures,presented at International Symposium on Signals,Circuits and Systems, 2005.
178
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 3.8
A Novel Graphical Based Tool for Extraction of Magnetic
Reluctances between On-Chip Current Loops
Sebastian Gim, Alexander Vasenev, Diana Mihalache, Alexandra Stefanescu, and Sebastian Kula
Politehnica University of Bucharest, Electrical Engineering Dept., Numerical Methods Lab., Spl.Independentei 313, 060042, Romania [email protected]
Summary. Continued device scaling into the nanome-ter region has given rise to new effects that previouslyhad negligible impact but now present greater chal-lenges and unprecedented complexity to designingsuccessful mixed-signal silicon. This paper presents anovel graphical tool for semi automatic extraction ofmagnetic reluctances between on-chip current loops.The novel graphical tool seamlessly integrates withinthe workflow of the CHAMELEON-RF software pro-totype developed.
1 Introduction
One of the major challenges in the nano electronicdesign industry (EDA) is the management of de-sign complexity. As integrated circuits are beingscaled into the nanometer scale, more and morefunctionality can be integrated on-die . However,this greater integration has also leaded to a num-ber of design challenges, amongst them, the mu-tual coupling between interconnects, sub-circuitsand functional blocks. All these need to be effec-tively managed by the EDA software in an effi-cient but intuitive manner to ensure a successfuldesign. A coherent framework and comprehensiveworkflow model is needed.There are several reasons to find an optimal pro-cedure to support communication between thevarious data formats within a framework’s workflow model and automating the tasks. The twomost pressing reasons are reduction of human ef-fort and elimination of user errors. Furthermorefor the effective use of simulation software it isnecessary to find a proper way for correct repre-sentation and transformation of initial data fromschematic parameters and layout to simulation.In this paper, we present a novel graphical tooland a proposed workflow model for the extractionof magnetic reluctances between on-chip currentloops that integrates seamlessly with the Chamyelectromagnetic simulator. Based on the princi-ples identified previously, code for a novel graphi-cal tool called Mag-Tris (Magnetic Tetris) was de-veloped in LayoutEditorTM and MatlabTM . Thisnovel graphical tool that is specially designed forChamy software [1] is based on Matlab and can
effectively work with Matlab standard files (.mat)and ASCII files as an import format.A few rules should be observed to solve this task.First, the solution should comply with require-ments of Chamy. For example, it must supportany Manhattan geometry structure and otherprinciples, defined by the software. Secondly, itmust be based on Graphical User Interface (GUI).A designer as the user can easily make modifica-tion, see and examine changes. Finally, implemen-tation should be done by using non-commercialgeneral public license.The remaining sections of this paper are orga-nized as follows. Section 2 describes the key fea-tures of Mag-Tris and presents the proposed work-flow model and seamless integration with Chamysoftware. This approach is then applied to ex-tracting the magnetic reluctances between fun-damental current loops in a 24 GHz LNA. Theresults from this extraction process are then pre-sented in Section 3.
2 Workflow for Multi-scaleCompact Modeling
Fig. 1. Dataflow of Mag-Tris
179
3 Calculation of MagneticReluctances
The fluxes and magnetic voltages of the magneticterminals are related by means of the linear rela-tions:
ϕ = Pvm (1)
where P is is the nodal magnetic permeance ma-trix of magnetic terminals. The matrix P is a sym-metric, diagonal dominant and positive definedmatrix with positive diagonal entries and negativeoff-diagonal terms. By expressing voltage betweentwo terminals as potentials difference, the branchreluctances in the complete polygonal topologyare obtained
ϕkj =∑nj,k=1 pjkvk =
∑nj,k=1Gmjk(vj − vk)
Gmkj = −pkj > 0, Gmk0 =∑ni=1 pki > 0
Rmkj = 1/Gmkj , Rmk0 = 1/Gmk0(2)
The test case consists on a 24GHz Car RadarLNA (Fig. 2). According to the modeling proce-dure described on short above, first a system ofindependent fundamental loops has to be identi-fied before computation of magnetic reluctances.After that, in order to model the parasitic inducedvoltages, a dual magnetic circuit needs to be cou-pled to the original electrical circuit by meansof controlled sources. The demo Chameleon soft-
Fig. 2. Identified fundamental current loops in a 24GHz LNA
ware prototype [1] was developed to extract theself and mutual reluctances between the hooks ofManhattan shapes (union of rectangles). The val-ues are summarized in the tables below.
Acknowledgement. The authors would like to grate-fully acknowledge support and insightful discussionwith Prof. Daniel Ioan, Asc.Prof. Gabriela Ciuprina.The authors would also like to acknowledge the fi-nancial support offered by the European Commission
Fundamental Loop Reluctance [1/H]
11 2.40630e+010
22 3.14921e+011
33 4.30807e+010
44 2.29307e+011
12 2.53936e+011
13 3.83304e+010
14 2.00067e+011
23 3.97307e+011
24 1.97317e+012
34 2.76450e+011Table 1. Extracted self and mutual reluctances
under the Marie Curie Fellowship (FP6) programmeand also the Romanian Cercetare de Excelenta inCEEX/eEDA program. The views stated herein re-flect only the author’s views and the European Comis-sion is not liable for any use that may be made of theinformation contained.
References
1. CHAMELEON - RF website: www.chameleon-rf.org
2. E. Mollick, Establishing Moore’s Law IEEE An-nals of the History of Computing vol. 28, pp. 62-77, 2006.
3. J. Nieholf, H. Janssen and W. Schilders, Compre-hensive High - Accuracy Modeling of Electromag-netic Effects in Complete Nanoscale RF blocks:CHAMELEON RF presented at Signal Propaga-tion on Interconnects, IEEE Workshop, 2006
180
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 3.9
Multiobjective Optimization Applied to Design of PIFA
Antennas
Stefan Jakobsson, Bjorn Andersson, and Fredrik Edelvik
Fraunhofer-Chalmers Research Centre, Chalmers Science Park, SE-412 88 Goteborg, [email protected], [email protected],[email protected]
Summary. In this paper we apply multiobjectiveoptimization to antenna design. The optimization al-gorithm is a novel respond surface method based onapproximation with radial basis functions. This algo-rithm is combined with CAD and mesh generationsoftware, electromagnetic solvers and a decision mak-ing platform for postprocessing. To demonstrate theprocedure we optimize the geometric design and lo-cation of PIFA antennas on a ground plane.
1 Introduction
In many engineering applications there are often,at least partly, conflicting requirements. For ex-ample, for cars it is desirable to have both low fuelconsumption and high effect. In antenna designthe requirements can be based on S-parameters,functions of the directivity of the antenna, band-width, input impedance and/or other character-istics of the antenna.
In a current project, which is a collaborationbetween the Fraunhofer-Chalmers Centre and theAntenna Research Centre at Ericsson AB, we aredeveloping new efficient optimization algorithmswith the purpose of studying communication per-formance possibilities and limitations for multi-ple antennas within a limited area, such as ahandheld terminal. The antenna elements are so-called printed inverted “F” antenna (PIFA) de-signs that have low profile, good radiation charac-teristics and wide bandwidths. This makes theman attractive choice for antenna designs of vari-ous wireless systems. In some recent works geneticand evolutionary algorithms have been applied tosingle objective optimization of PIFAs [3, 4].
In multiobjective optimization some require-ments are formulated as objective functions andthe ultimate goal is to find all Pareto optimal so-lutions : a solution is Pareto optimal if there is noother solution which is better in all objectives. Asin other types of optimization, some requirementsmay also be formulated as constraints.
We have developed a multiobjective optimiza-tion algorithm based on radial basis functions tofind an approximation of the Pareto front (the set
of Pareto solutions). In this paper the algorithmis demonstrated for optimization of the design ofa PIFA antenna on a ground plane. The objectivefunctions that should be minimized are the maxi-mum return loss in a frequency band, the antennaheight and the enclosed antenna area. The designparameters describe the geometry and location ofthe PIFA on the ground plane. The electromag-netic simulations are performed with the MoMsolver from the software package efield R© [1].
In post processing of results from multiob-jective optimization it is interesting to analyzethe trade-off between the different objectives. Themain advantage of multiobjective optimizationis that the decision making process takes placewhen all possibilities and limitations are known.A graphical tool to assist the decision maker inthis process will be presented at the conference.
2 A multiobjective optimizationalgorithm
The algorithm, called qualSolve, is a respond sur-face method based on interpolation/approximationwith radial basis functions and is described indetail in [2]. In each iteration of the optimiza-tion an interpolation/approximation (also calledsurrogate model) of every objective function ismade based on all previous evaluations of thegoal functions. By using for example evolution-ary algorithms, an approximation of the Paretofront for the surrogate models is made. In the sec-ond step, a new evaluation point is chosen suchthat it maximizes a quality measure. This qual-ity measure is a function both of the quality ofthe approximation (measured by the distant to analready evaluated point in the parameter space)and the interest to further explore an area of theparameter space. The latter is measured in termsof the distance to the Pareto front for the sur-rogate models in the objective space. The resultsof the algorithm are both a finite set of approxi-mate Pareto points and coefficients for RBF ex-pansions of the objective functions that can be
181
used for further post-processing. This enables thealgorithm to use less evaluations of the objectivefunctions compared to genetic algorithms. Sincethe evaluations of the objective functions involvetime-consuming simulations this fact can greatlyimprove efficiency.
3 Results
To demonstrate the performance we perform anoptimization of a PIFA antenna element locatedon a 100 × 45 × 2 mm ground plane, see Fig.1. For
Fig. 1. The PIFA design used for optimization.
this case we have two objective functions whichare the enclosed antenna area (convex hull) andthe maximum return loss in the frequency in-terval 700 - 800 MHz. Both objective functionsshould be minimized. The optimization is car-ried out with respect to four design parameters:the size of the patch (length and width), the dis-tance to the patch and the position of the feed,see Fig.1. All parameters are subjected to simplebox constraints. Simulations are performed usingthe efield R© MoM solver for five frequencies in theabove given interval. The Pareto front is shownin Fig.2, where it can be clearly seen that thesetwo objectives are conflicting.
−2 −1.5 −1 −0.5 0 0.50
200
400
600
800
1000
1200
S11 [dB]
Con
vex
hull
area
[mm
2 ]
Fig. 2. The Pareto front for the RBF approximationsof the maximum return loss in the frequency band andthe enclosed antenna area.
This section will be extended with results forcases with more design parameters and also addi-tional objectives such as the antenna height. Tobe able to change also other parameters of the an-tenna is clearly needed to e.g. obtain lower returnloss.
4 Conclusions
A key property of the proposed algorithm is thatthe result is both a set of approximately Pareto-optimal solutions and also approximations of allobjective function as expansions in radial ba-sis function which can be used for further post-processing. This enables the algorithm to use lessevaluations of the objective functions comparedto e.g. genetic algorithms. Another key aspectis that all numerical simulations contain errors(noise) and to replace interpolations with approx-imations is a feature of the algorithm to make itmore robust. Finally, by avoiding a weight-basedtrial-and-error strategy, where the objectives areweighted to form a single objective function, thedecision of the optimal solution is postponed untilall possibilities and limitations are known.
Future work includes to study multiple anten-nas on the same ground plane. The performancein the frequency band 2500-2700 MHz will alsobe optimized. Furthermore, the efficiency of theoptimization algorithm in finding new evaluationpoints needs to be improved for cases with manydesign parameters.
Acknowledgement. Financial support for this workhas been provided by the Swedish Foundation forStrategic Research (SSF) through the GothenburgMathematical Modeling Centre (GMMC).
References
1. efieldR©. http://www.efieldsolutions.com.2. S. Jakobsson, M. Patriksson, J. Rudholm, and
A. Wojciechowski. A method for simulation basedoptimization using radial basis functions. Submit-ted for publication.
3. D.Y. Su, D.M. Fu, and Yu D. Genetic algorithmsand Method of Moments for the design of pi-fas. Progress in Electromagnetics Research Letters,1:9–18, 2008.
4. L. Zhan and Y. Rahmat-Samii. Optimizationof PIFA-IFA combination in handset antenna de-signs. IEEE Trans. on Antennas and Propagation,53:1770–1778, 2005.
182
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 3.10
Space Mapping with Difference Method for Optimization
N. Serap Sengor and Murat Simsek
Istanbul Technical University, Electrical and Electronics Engineering Faculty, Maslak, Istanbul, [email protected], [email protected]
Summary. The surrogate optimization techniqueshave to combine the high fidelity of fine models withspeed of coarse models. In Space Mapping (SM) tech-niques, this is achieved by a mapping from the finemodel input space to the coarse model input space.Space Mapping with Difference (SM-D) method is avariant of SM and in this work, how SM-D can beused in optimization will be presented.
1 Introduction
In Space Mapping (SM) techniques a mappingfrom the fine model input space to the coarsemodel input space [1] is constructed to obtain asurrogate model which would lessen the computa-tional burden during design procedure. In SpaceMapping with Difference (SM-D) method thismapping is revised by enlarging the dimension ofthe domain of the SM function, P (.), with the finemodel output. With this adjustment, the need toevaluate the fine model was reduced and the sim-ulation results obtained for different applicationsrevealed that the extrapolation capability of themodels obtained with SM-D were improved [2].Enlarging the dimension of the SM function P (.)with the already existing knowledge simplifies theprocess of constructing the function P (.). In pre-vious works on SM-D [2], the SM function is con-structed using feedforward Artificial Neural Net-work (ANN) structures and the method is usedonly to obtain surrogate models. In ! this work, inorder to reveal the efficiency of the SM-D methodand to achieve a simpler structure to be used inoptimization processes during desing procedurean affine function is constructed instead of ANN.In the next section, the SM-D method will be in-troduced and in the last section, the method willbe used to solve wedge problem [1] and the resultswill be compared with classical SM, Agressive SM(ASM) and Linear Inverse SM (LISM) [1, 3, 4].
2 Space Mapping with DifferenceMethod
In most of SM techniques as ASM, and SM amapping, P (.), from the fine model input space
to the coarse model input space is constructed asfollowing:
xc = P (xf ) (1)
such thatRc(P (xf )) ≈ Rf (xf ) (2)
where, xf , xc, Rf (.) and Rc(.) are fine and coarsemodel design parameters and fine and coarsemodel responses, respectively [1]. As it can befollowed from the block diagram given in Fig. 1in SM-D method Pd(.) maps fine model outputRf (xf ) and the fine model design parameter xfto the difference between fine and coarse modeldesign parameters xd [2]. Thus a mapping fromxf to xc is formed as following:
xc = Pd(Rf (xf ), xf ) + xf (3)
Since the fine model response Rf (xf ) is alreadyused, using it does not give rise to an extra com-putational burden. The steps to find the optimum
Fig. 1. The block diagram of SM-D method
design parameters of fine model xf giving riseto optimum fine model response Yf using coarsemodel responses Rc(.) in SM-D method are thefollowing:
• step 1: choose Y ∗c = Yf
• step 2: find x∗c from x∗c = minxc‖Y ∗c −Rc(xc)‖
• step 3: set x(i)f = x∗c
• step 4: find Y(i)f = R(x
(i)f )
• step 5: if ‖Y (i)f − Y ∗
c ‖ ≤ ε then xf = x(i)f else
go to step 6
• step 6: find x(i)c from
x(i)c = minxc‖Y (i)
f −Rc(xc)‖
183
• step 7: form P(i)d = QD−1 where
Q.= [xc − xf ], D
.= [1 xf ]
T
• step 8: i = i+ 1
• step 9: set x(i)f = (P
(i)d )−1(x∗c) from
x(i)f = minxf
‖x∗c − Pd(xf )‖ and go to step 4
3 Simulation Results
In order to demonstrate the efficiency of the SM-D method in optimization a simple problem de-fined in [1] and given in Fig. 2 is solved withclassical SM, LISM, ASM and SM-D and com-pared with the results obtained with SM-D. Inall these techniques linear SM mapping is con-structed. The results are sumarized in Table.
Fig. 2. Wedge problem [1]: (a) Fine model (b) Coarsemodel.
Fig. 3. The difference between the fine model andcoarse model optimum values.
Since the difference between the fine model andcoarse model optimum value is large as shown inFig. 3, the convergence of SM and LISM meth-ods took longer than ASM and SM-D methods.Even though the results of ASM and SM-D meth-ods are very much alike, ASM method diverge forsmall ε values.
4 Conclusions
In this work, SM-D method is utilized for op-timization. The simulation results obtained for
Table 1. Comparison of SM-D with SM, LISM andASM
Methods xf = 8 |xf− xopt| Yf = 28
SM 7.9997 3.32e-004 27.9990Iteration 123
LISM 7.9997 3.31e-004 27.9990Iteration 123
ASM 8.0002 1.50e-004 28.0005Iteration 6
SM-D 7.9997 2.73e-004 27.9992Iteration 3
Fig. 4. Convergence of ASM and SM-D.
wedge problem shows that the method is effi-cient compared to classical SM, LISM, ASM.Since Pd(.) is an affine function, the computa-tional complexity of the method is less than theoriginal SM-D proposed in [2]. The efficiency ofthe method will be further investigated with dif-ferent applications.
References
1. J.W. Bandler, Q.S. Cheng, S.A. Dakroury,S.A.Mohamed, M.H. Bakir, K.Madsen and J. Son-dergaard. Space Mapping : The State of the Art.IEEE Trans. on Microwave Theory and Tech.,52:337–361, 2004.
2. M. Simsek and N.S. Sengor. A New ModellingMethod Based on Difference between Fine andCoarse Models Using Space Mapping. AbstractBook of 2nd Int. Workshop SMSMEO-06, 64:65,2006.
3. J.W. Bandler, R.M. Biernacki, S.H. Chen, R.M.Hemmers and K. Madsen. Electromagnetic Op-timization Exploiting Aggressive Space Mapping.IEEE Trans. on Microwave Theory and Tech.,43:2847–2882, 1995.
4. J.E. Rayas-Sanchez, F. Lara-Rojo and E.Martinez-Guerrero. A Linear Inverse Space-Mapping (LSIM) Algorithm to Design Linearand Nonlinear RF and Microwave Circuits.IEEE Trans. on Microwave Theory and Tech.,53:960–368, 2005.
184
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 3.11
Modification of Space Mapping with Difference Method for
Inverse Problems
Murat Simsek and N. Serap Sengor
Istanbul Technical University, Electrical and Electronics Engineering Faculty, Maslak, Istanbul, [email protected], [email protected]
Summary. The surrogate methods have been usedto ease the computational burden in different disci-plines. In this work, a previously proposed surrogatemethod, namely, Space Mapping with Difference ismodified to solve the inverse problems. The efficiencyof the method is demonstrated solving the shape re-construction of a conducting cylinder.
1 Introduction
The distinctive feature of surrogate methods aretheir capability of combining the computationalefficiency of a coarse model with the accuracy ofthe fine model and in Space Mapping (SM) tech-nique this is provided through a mapping fromthe fine model input space to the coarse modelinput space [1]. In Space Mapping with Differ-ence (SM-D) method this mapping has been ad-justed by enlarging the dimension of the domainof the mapping with the coarse model output.With this adjustment, the need to evaluate thefine model was reduced and the simulation re-sults obtained for different applications revealedthat the extrapolation capability of the modelsobtained with SM-D were improved [2,3]. In thiswork, SM-D method is modified for inverse prob-lems and this new method is named Space Map-ping with Inverse Difference (SM-ID). In SM-ID,instead of coarse model an inverse coarse modelwhich is generated as a multilayer perceptron isused. The mappi! ng between coarse model andfine model parameter spaces are constructed sim-ilar to Linear Inverse Mapping (LISM) algorithmgiven in [4] but in the proposed method, parame-ter extraction step is no longer necessary to buildan appropriate space mapping function P (.). Inthe next section, the proposed method will be in-troduced and in the third section, the method willbe used to solve shape reconstruction of a con-ducting cylinder [5].
2 Space Mapping with InverseDifference
In design problems, the main concern is to deter-mine the design parameters xdesign which mini-
mize an objective function defined over responsesR(xdesign) of design parameters xdesign. In in-verse problems, main concern is to determine thesome parameters x of the problem given responsesR(x). As both problems are synthesis problemsthe solution is not unique and furthermore forthe inverse problems it can be ill-possed.
It has been shown that, SM-D method solvesthe design problems with efficiency. It is based onforming a mapping between fine and coarse modeldesign parameters xf and xc, respectively. Whileforming this mapping, a function Pd(.) is con-structed which maps coarse model output Rc(xf )and the fine model design parameter xf to thedifference between fine and coarse model designparameters xd [2, 3]. Thus the mapping formedmaps xf to xc via Pd(.) as following:
xc = Pd(Rc(xf ), xf ) + xf (1)
Fig. 1. The block diagram of SM-ID method
As it can be followed from Fig. 1, SM-Dmethod is modified in two ways; first inversecoarse model is used instead of coarse model. Inmost applications, there will not be possibility ofobtaining an inverse coarse model, in such casesusing a well-known feedforward neural networkstructure as multilayer perceptron would be suit-able. The second modification is in constructingthe SM function P (.), as the relation in SM-IDwill be the inverse of the relation set up in SM-D. SM function iPd(.) will resemble that of LISMalgorithm [4] but since inverse coarse model is
185
used there is no need for parameter extractionstep [1, 4] The SM function iPd(.) maps Yf , xcto xd as following:
xf =i Pd(Yf , xc) + xc (2)
In this work, the SM function iPd(.) is a linearmapping. As iPd(.) is linear and since there is noneed for parameter extraction the computationalburden is decreased compared to other SM meth-ods.
3 Simulation Results forReconstruction of a ConductingCylinder
The reconstruction of a conducting cylinder prob-lem is considered and the results obtained arecompared with results obtained by Artificial Neu-ral Networks (ANN).The inverse coarse model inSM-ID is trained with different number of data,and in the figures the test set results are exposed.The ANN’s trained have 20 inputs, nine outputs,where the inputs are the real amd imaginary com-ponents of scattered electric field obtained at 10different positions and the outputs are the Fourierseries coefficients of the geometrical shape of theconducting cylinder. It can be followed from
Table 1. Comparison of SM-ID results with ANN
Max Error Mean Error
IANN-50 0.22067 0.06715
SM-ID-50 0.03762 0.00868iteration : 5
IANN-100 0.11889 0.05266
SM-ID-100 0.00389 0.00081iteration : 4
IANN-200 0.10386 0.04285
the table and figures that SM-ID results outper-forms the ANN results and as the number of dataused increases, the iterations needed to constructiPd(.) decreases
Acknowledgement. The authors thank Necmi SerkanTezel for introducing the inverse scattering problem.
References
1. J.W. Bandler, Q.S. Cheng, S.A. Dakroury,S.A.Mohamed, M.H. Bakir, K.Madsen and J. Son-dergaard. Space Mapping : The State of the Art.
Fig. 2. On the left ANN result and on the right SM-ID result is given for 50 data.
Fig. 3. On the left ANN result and on the right SM-ID result is given for 100 data
Fig. 4. Only the ANN result is given for 200 data asSM-ID fits almost perfectly for 100 data
IEEE Trans. on Microwave Theory and Tech.,52:337–361, 2004.
2. M. Simsek and N.S. Sengor. A New ModellingMethod Based on Difference between Fine andCoarse Models Using Space Mapping. AbstractBook of 2nd Int. Workshop SMSMEO-06, 64:65,2006.
3. M. Simsek. Novel Methods for Surrogate Op-timization and Modeling Using Space MappingTechniques. Ph. D. Thesis Report 2, IstanbulTechnical University, 2007.
4. J.E. Rayas-Sanchez, F. Lara-Rojo and E.Martinez-Guerrero. A Linear Inverse Space-Mapping (LSIM) Algorithm to Design Linearand Nonlinear RF and Microwave Circuits.IEEE Trans. on Microwave Theory and Tech.,53:960–368, 2005.
5. N.S. Tezel and C. Simsek. Neural Network Ap-proach to Shape Reconstruction of a ConductingCylinder. Istanbul University, Journal of Electricaland Electronics Eng., 7:299–304, 2004.
186
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 3.12
Exploiting Model Hierarchy in Semiconductor Design using
Manifold-Mapping
D.J.P. Lahaye1 and C. Drago2
1 DIAM - Delft Institute of Applied Mathematics, Technical University of Delft , Mekelweg 4, 2628 CD Delft,The Netherlands [email protected],
2 Dipartimento di Matematica e Informatica, Universita di Catania, Citta Universitaria, viale Andrea Doria6, 95125 Catania, Italy [email protected]
Summary. In this work we exploit a model hierar-chy for the optimal control of semiconductors usingthe manifold-mapping technique. Herein the drift dif-fusion and energy transport models act as coarse andfine model, respectively. The advantage of this ap-proach is that it allows for the efficient optimizationof the energy transport model without having to im-plement its adjoint.
1 Introduction
The interest in optimal control for semiconductordesign has attracted considerable recent attentionin both the engineering and applied mathematicscommunity. A major objective in the design isto improve the current flow over some contacts,for fixed applied voltages, by a slight change ofthe device doping profile. In most applicationsthe design problem is addressed empirically, onthe knowledge and experience of electrical engi-neer. Although this problem can be clearly tack-led by an optimization approach, only recentlyefforts were made to solve the design problemvia optimization techniques. Approaches rangefrom the standard black box optimization meth-ods, which in general have an high computationalcost, since they require many solvers of the for-ward model, to the adjoint method, recently pro-posed in the field of optimal semiconductor de-sign, which provides good results by drasticallyreducing the amount of computational efforts.The adjoint model has been proposed and an-alyzed for the drift-diffusion model. Meanwhilethe same approach was extended to the energy-transport model. Comparisons between the newoptimal designs with the ones obtained based onthe drift diffusion model were presented in [3].Moreover in [3] the idea to exploit this clas-sical model hierarchy to speed up the conver-gence of the optimization algorithms by usingthe space mapping approach was introduced forthe first time in the field of optimal control insemiconductor design. In this work we replace
the agressive space-mapping technique by the re-cently proposed and equally efficient manifold-mapping variant [2] as it has better theoreticalproperties.
2 Semiconductors Models: DriftDiffusion and Energy Transport
Presently, there is a whole hierarchy of semi-conductor models available ranging from micro-scopic models, like the Boltzmann-Poisson or theWigner-Poisson model, to macroscopic ones, likethe drift diffusion, the energy transport and thehydrodynamic model. The drift diffusion model isthe simplest and most popular macroscopic one,widely used in commercial simulation packagesas it allows for efficient numerical study of chargetransport in many case of practical relevance.The unipolar Drift Diffusion model (DD) consistsof continuity equations for electron density n cou-pled to Poisson’s equation for the electrostatic po-tential V
divJn = 0, Jn = (∇n− n∇V )
λ2V = n− C
where C is the doping profile, λ2 the Debye lengthand Jn denotes the electron current density.On the other hand, in today semiconductor tech-nology, the miniaturization of devices is more andmore progressing. As a consequence the simu-lation of sub-micron semiconductor devices re-quires advanced transport models. Because of thepresence of very high and rapidly varying elec-tric fields, phenomena occur which cannot be de-scribed by means of drift-diffusion model, thatdoes not incorporate energy as a dynamical vari-able. The energy-transport model takes into ac-count also the thermal effects related to the elec-tron flow through the semiconductor crystal.The dimensionless Stratton’s energy transport(ET) model consists of continuity equations forelectron density n and the temperature T , cou-pled to Poisson’s equation reads
187
divJn = 0, Jn = −(∇n− n
T∇V
)
divS = Jn · ∇V +W (n, T )
S = −3
2(∇(nT ) − n∇V )
λ2∆V = n− C
where Jn is the carrier flux density, S the en-ergy flux density, W the energy relaxation term
W (n, T ) = − 32n(T−1)τ0 .
3 The Design Problem
Let C be a given reference doping profile and letΓO be a portion of the Ohmic contacts ΓD, atwhich we can measure the total density current J ,obtained by solving the DD or ET system. At thecontact ΓO we prescribe a gained current I∗ andallow deviations, in some suitable norm, of thedoping profile from C in order to gain this currentflow. In other words we intend to minimize costfunctionals of the form [3]
1
2
∣∣∣∣∫
Γ0
Jndν − I∗∣∣∣∣2
+γ
2
∫
Ω
|∇(C − C)|2dx
where γ > 0 allows to balance the effective cost. Centers as a source term in the DD and ET modelsand plays the role of our design varible. The DDand ET models are interpreted as a constraint, tothe minimization problem, determining the den-sity current Jn, by the state variables (n, V ) or(n, T, V ) respectively.
4 The Manifold-Mapping Technique
The manifold-mapping technique [2] exploits coa-rse model information and defines a surrogate op-timization problem whose solution do coincidewith x∗f . The key ingredient is the manifold-mapping function between the coarse and finemodel image spaces c(X) ⊂ Rm and f(X) ⊂ Rm.This function S : c(X) 7→ f(X) maps the pointc(x∗
f ) to f(x∗f ) and the tangent space of c(X) at
x∗f to the tangent space of f(X) at x∗
f . It allowsthe surrogate model S(c(x)) and the manifold-mapping solution to be defined as follows: findx∗mm ∈ X such that
x∗mm = argmin
z∈X‖S(c(z)) − y‖ . (1)
The manifold-mapping function S(x) is approx-imated by a sequence Sk(x)k≥1 yielding a se-quence of iterands xk,mmk≥1 converging to x∗
mm.The individual iterands are defined by coarsemodel optimization: find x∗
k,mm ∈ X such that
xk,mm = argminz∈X
‖Sk(c(z)) − y‖ . (2)
At each iteration k, the construction of Sk re-quires the singular value decomposition of thematrices Ck and Fk of size m × min(k, n)whose columns span the coarse and fine modeltangent space in the current iterand, respectively.Denoting these singular value decompositions by
Ck = Uk,cΣk,c VTk,c Fk = Uk,f Σk,f V
Tk,f ,
we introduce the updated objective yk as
yk = c(xk) −[Ck F †
k + (I − Uk,c UTk,c)]·
·(f(xk) − y) ,
where superscript † denotes the pseudo-inverse.With this notation, the problem (2) can shown beto be asymptotically equivalent to: find x∗
k,mm ∈X such that
xk,mm = argminz∈X
‖c(z) − yk‖ (3)
By construction x∗mm = x∗f .
5 Numerical Results
We test the performance of the MM algorithmfor a one-dimensional n+−n−n+ ballistic diode,which is a simple model for the channel of a MOStransistor. We tried to achieve an amplificationof the current of 50% for different value of theapplied voltage ranging between U = 0.026 V andU = 0.52 V. The current voltage characteristics(IVC) before and after optimization are shown inFig. 1.
5 10 15 200
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Scaled voltage
Cur
rent
before optimizationobjectiveafter optimization
Fig. 1. Resulting IVCs
References
1. Bandler et al. Space Mapping: The State ofthe Art IEEE Trans. on Microwave Theory andTechniques,52(1):337–361, 2004
2. Echeverrıa, D. and Hemker, P. W. Space Mappingand Defect Correction Comp. Methods in Appl.Math., 5(2):107–136, 2005.
3. Drago C. D. and Pinnau, R. Optimal Dopant Pro-filing based on Energy Transport SemiconductorModels, Math. Mod. Meth. Appl. Sc. 18(2):195–214, 2008.
188
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 3.13
An hp-Adaptive Strategy for the Solution of the Exact Kernel
Curved Wire Pocklington Equation
D.J.P. Lahaye
DIAM - Delft Institute of Applied MathematicsDepartment of Electrical Engineering, Mathematics and Computer ScienceTU Delft, Mekelweg 4, 2628 CD Delft, The [email protected]
Summary. In this paper we introduce an adaptivemethod for the numerical solution of the Pockling-ton integro-differential equation with exact kernel forthe current induced in a smoothly curved thin wireantenna. The hp-adaptive technique is based on therepresentation of the discrete solution, which is ex-panded in a piecewise p-hierarchical basis. The keyelement in the strategy is an element-by-element cri-terion that controls the h- or p-refinement. Numericalresults demonstrate both the simplicity and efficiencyof the approach.
1 Introduction
In this study we treat electric field scatteringfrom thin curved wire antennas. The current thatan incident electrical field induces in the an-tenna is computed by solving the Pocklingtonintegro-differential equation [4]. In engineeringliterature the reduced kernel approximation istypically used. However, if very fine meshes areused for the discretization, the ill-posedness of theresulting problem causes spurious oscillations inthe numerical solution, which prevents the com-putation of highly accurate solutions [3, 5]. Wetherefore treat the computationally more chal-lenging exact kernel model [1].
We consider a smoothly curving, non self-intersect-ing thin wire antenna of arbitrary shapewith total length L, and circular cross-sectionwith radius a, where a ≪ L. We denote the arclength variables along the axis and the bound-ary of the cross-section by s and θ, respectively.Let the symbols R, s and d
ds represent the dis-tance between two points on the lateral surface ofthe wire, the unit tangential vector and derivativealong the wire, respectively. The wire antenna isassumed to be illuminated by a time-harmonicelectrical field plane wave with angular frequencyω, wavenumber k = ω
c and wavelength λ = 2πk .
The wavelength is assumed to be large comparedwith wire radius, i.e. a/λ ≪ 1. The tangentialcomponent of the incident field will be denotedby Eincs (s, θ). We furthermore introduce the inte-gration kernel
G(s, θ, s′, θ′) = G(R(s, θ, s′, θ′)) =exp (− k R)
4 π R,
where 2 = −1. The Pocklington’s integral equa-tion for the current induced in the antenna thenreads:
k2
∫
s′
∫
θ′s · s′ I(s
′, θ′)
2 πG(‖r − z‖) ds′ dθ′
+ (s · ∇r)
∫
s′
∫
θ′
∂∂s′ I(s
′, θ′)
2 πG(‖r − z‖) ds′ dθ′
= − ω ǫ s · Ein(r)
The finite element (FE) technique proposedin this paper achieves high accuracy at moder-ate computational cost by automatic adaption ofboth the mesh width (h-adaptation) and the poly-nomial degree (p-adaptation) of the approxima-tion to the local smoothness of the solution. Thediscrete solution is expanded in a piecewise hier-archical basis [6], consisting of the standard lin-ear FE shape functions enriched with higher orderbubble functions . Apart from the treatment of asmooth arbitrarily curved wire, a new aspect inthis work is the element-by-element criterion forh- or p- refinement. As shown below, this crite-rion is based on the behaviour of the coefficientsof the discrete solution in the hierarchical basisrepresentation.
2 The hp-Adaptive Strategy
The hp-adaptive strategy in this paper relies onan p-hierarchical basis and makes use of the factthat for smooth functions the expansion coeffi-cients –on a single element in a piecewise polyno-mial approximation– are supposed to decrease asa geometric sequence.
This observations lead to the following hp-adaptive strategy. On a (non-uniform) mesh wedetermine in each interval a p-hierarchical repre-sentation of the solution with a sufficient numberof terms. On each interval the convergence of theapproximation is studied. If the rate of decrease
189
1 24 48 72 96
2
4
6
8
10
12
Element number
Mes
h re
finem
ents
and
pol
. deg
ree
mesh refinementspolynomial degree
(a) Final mesh
0 0.25 0.5 0.75 1
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Position on wire [m]
Rea
l and
imag
par
t cur
rent
[mA
]
real partimag part
(b) Computed current
Fig. 1. Results for a receiving straight wire antenna.Left: the number of mesh refinement steps and thepolynomial degree yielding an L1-error smaller thanTOL = 5e-8Am. Right: the real and imaginary partof the computed current.
of the coefficients is fast enough, then the seriesis truncated such that an a-priori given tolerancecriteria is satisfied. If the rate of decrease is notfast enough, then the interval is split into twosmaller intervals of equal size and a new approxi-mation is computed. The process will stop after afinite number of iterations, except in those areaswhere derivatives are unbounded. At such singu-lar locations the process can be stopped by intro-ducing a limit for the minimal allowed mesh-size.Details of the strategy are explained in [2].
3 Numerical Results
3.1 Current in Receiving Antenna
In the first experiment, we consider the compu-tation of the current in a straight wire antennaof length L = 1 m and radius a = 0.001 m, in-duced by a plane wave exitation of frequencyf = 500 MHz and propagation vector perpen-dicular to the axis of the antenna. We use anL1-error bound introduced as the stopping cri-terium in the mesh refinement process. As initialdiscretization we choose a uniform mesh with 16second degree elements having in total 33 (i.e., 17+ 16) degrees of freedom. Imposing a toleranceTOL = 5e-8 Am, this results in eight adaptive re-finement steps and a mesh of 96 elements with422 degrees of freedom. In Figure 1(a) for the fi-nal mesh we plot the number of local mesh refine-ments and polymomial order. In Figure 1(b) weshow the computed current. The mesh shows tohave been h-refined near the end-point singulari-ties and to have been p-refined where the currentis a smooth function.
3.2 Current in Emitting Antenna
This time we consider the computation of the cur-rent in an emitting straight antenna. Length andradius are as in the previous example. Now the
1 11 22 33 45
2
4
6
8
10
12
Element number
Mes
h re
finem
ents
and
pol
. deg
ree
mesh refinementspolynomial degree
(a) Final mesh
0 0.5 1−1.5
−1
−0.5
0
0.5
1
1.5
Position on wire [m]
Rea
l and
imag
par
t cur
rent
[A]
real partimag part
(b) Computed current
Fig. 2. Results for an emitting straight wire antenna.Left: the number of mesh refinement steps and thepolynomial degree yielding an L1-error smaller thanTOL = 3e-3Am. Right: the real and imaginary partof the computed current.
antenna has a gap of 0.01 m in its middle where acurrent source of 1 A is impressed. Starting froma piecewise second degree solution on a uniformmesh with eight elements on either side of thegap and imposing a tolerance TOL = 1e-3 Am,seven adaptive refinement steps result in a meshof 45 elements with 154 degrees of freedom. Thefinal hp-mesh and computed current are shownin Figure 2. Both the end-point singularities andthe singularities at the boundaries of the gap areclearly resolved by low order small elements.
Numerical results for a smoothly curved an-tenna can be found in [2].
Acknowledgement. This research was supported bythe Dutch Ministry of Economic Affairs through theproject IOP-EMVT 04302.
References
1. P. J. Davies, D. B. Duncan, and S. A. Funken.Accurate and Efficient Algorithms for FrequencyDomain Scattering from a Thin Wire. J. Compu-tational Phyiscs, 168:155–183, 2001.
2. P. W. Hemker and P. Lahaye, D. J. An hp-Adaptive Strategy for the Solution of the Ex-act Kernel Curved Wire Pocklington Equation.CMAM, 8(1):39–59, 2008.
3. B. P. Rynne. The Well-Posedness of the IntegralEquations for Thin Wire Antennas. IMA J. Ap-plied Math., 49:35–44, 1992.
4. A. G. Tijhuis, P. Zhongqiu, and A. R. Bretones.Transient Exitation of a Straight-Wire Segment:a New Look to an Old Problem. IEEE Trans.Antennas Propagat., 40(10):1132–1146, 1992.
5. M. C. van Beurden and A. G. Tijhuis. Analy-sis and Regularization of the Thin-Wire IntegralEquation With Reduced Kernel. IEEE Trans. An-tennas Propagat., 55(1):120–129, 2007.
6. H. Yserentant. Hierarchical basis. In R.E.O’Malley, editor, Proceedings of the Second In-ternational Conference on Industrial and AppliedMathematics (ICIAM 91), pages 256–276, 1992.
190
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 P 3.14
Sensitivity Analysis of Capacitance for a TFT-LCD Pixel
Ta-Ching Yeh, Hsuan-Ming Huang, and Yiming Li1
National Chiao Tung University, 1001 Ta-Hsueh Rd., Hsinchu 300, Taiwan [email protected]
Summary. In this paper, a computational statisticsmethod for sensitivity analysis of capacitance is im-plemented for a thin-film transistor liquid-crystal dis-play (TFT-LCD) Pixel. Based on 3D TCAD simula-tion and design of experiment (DOE), we constructcapacitances’ 2nd response surface model (RSM). Bythis model, we analyze the sensitivity of capacitanceswith design parameters under Gaussian distribution.This approach is cost effective and has its engineeringaccuracy, compared with 3D field simulation.
1 Introductions
Thin film transistors (TFTs) find wide usage inactive matrix liquid crystal displays (LCDs) [1-2]. The basic principle of operation of the LCDpanel is to control the transparency of each pixelportion by bus lines to charge the pixel elec-trode. The structure capacitance of each TFT-LCD pixel plays very important role in designinghigh quality display panel. However, the capaci-tance of a pixel is very hard to be analyzed ana-lytically because of the complex geometry struc-ture. In this work, we implement a systematicalway to analysis the effect of designing parame-ters on the structure capacitances of TFT-LCDpixel, as shown in Fig. 1. A computational statis-tics technique [3] is developed which consists ofthe 3D TCAD field simulator, the design of ex-periment (DOE) and the second order responsesurface model (RSM). By considering the design-ing parameters as changing factors, listed in Tab.1, according to the DOE, we construct a RSMfor capacitances of the explored TFT-LCD pixel.The RSM highly explains the behavior of capaci-tances for the investigated TFT-LCD pixel. WithGaussian distribution, the developed model al-lows us to analysis the sensitivity of capacitancesin a TFT-LCD pixel with respect to the afore-mentioned factors efficiently.
2 Computational Technique
The proposed computational statistics method-ology that accounts for the characteristic sensi-tivity is depicted in Fig. 2. A Plackett-Burmandesign is adopted to screen and determine major
interactions. Based on the results of the screen-ing experiment, parameters in need of furtherstudy are identified. With central composite de-sign (CCD) DOE technique, a 3D TCAD simula-tion is performed to calculate each capacitance forRSM construction. The model adequacy checkingis necessary to check the model assumption andaccuracy verification which is to examine the val-ues that we are interested in the accuracy of themodel in-between our high and low level settings.A 2nd order model is then established betweencapacitances (i.e., responses) and design parame-ters (i.e., factors). The sensitivity of capacitancesthen can be explored in a computationally cost-effective way.
ITO
Data line
Gate line
TFT
Shield metal
Common
x
y (a)
(b)
(d)
(f)
(c)
(e)
(a) (b)
Fig. 1. (a) 3D schematic plot of TFT-LCD pixel and(b) a top-view of the structure.
3 Results and Discussion
We construct 2nd order RSM using 149-runs withCCD for the ten capacitances in the investigatedTFT-LCD pixel. Without loss of generality, wepartially list a model for Cac which is the capac-itance between part(a) and part(c) in Fig. 1(b)
ln Cac = −0.498 + 0.321A+ 0.021B + 0.027C
+0.029D+ 0.011E + 0.337G
+0.473J − 0.045GJ + · · · (1)
where A to J are designing parameters, as listedin Tab. 1. The model predicted and the TCAD
191
Variable Screening
Central composite design
Model construction
Sensitivity analysis
Verify Model and Accuracy
TCAD simulation
Yes
No
Fig. 2. The computational procedure.
Table 1. The upper and lower bounds of the TFT-LCD designing parameters.
Variable Parameters Variation Range(um)
Size VariationA Gate line [0,10] in yB Shield metal(left) [0,3] in xC Shield metal(left) [0,3] in yD Shield metal(right) [0,3] in xE Shield metal(right) [0,3] in yF ITO electrode [0,5] in xG ITO electrode [0,5] in y
Position ShiftH Shield metal(left) [0,5] in xI ITO electrode [-2,2] in xJ ITO electrode [-2,2] in y
Table 2. The R-square of the capacitance models.
Response R-Square Response R-Square
Cab 0.9832 Cad 0.9701Cae 0.9986 Cac 0.9936Cfe 0.9935 Cbf 0.9624Cce 0.9563 Cdf 0.9472Caf 0.9938 Cbd 0.9778
simulated actual result, shown in Fig. 3, validatesthe accuracy of the constructed model (symbolslocate in the line very well). The R-square for thismodel is 0.9936, and the others are also listed inTab. 2. Figure 4 is the relationship between Cacand parameter J, given in Tab. 1, where the otherparameters are set to nominal values, and un-der Gaussian distribution (with more than 1000trails and σ is determined by process variation),the inset is the estimated standard deviation ofCac due to the parameter J variations of the ex-amined pixel TFT-LCD. A 0.1051 fF increase ofσCac when the ITO shift varies from -1.5 um to1.5 um. The increase of σCac is mainly due torelatively large variation of circuit performancewhen the ITO shift increase in the y direction.
Fig. 3. The model predicted vs. actual result for Cac.
Fig. 4. Plot of Cac versus J. The inset shows thestandard deviation of Cac with respect to J.
4 Conclusions
We have established RSMs for capacitance sensi-tivity analysis of a TFT-LCD pixel based upona computational statistics technique. The modelcould also be used for design optimization.
Acknowledgement. This work was supported by Tai-wan National Science Council under Contracts NSC-96-2221-E-009-210 and NSC-96-2752-E-009-003-PAE.
References
1. I.-W. Wu. Polycrystalline silicon thin film transis-tors for liquid crystal displays. Solid State Phe-nom., 37-38:553–564, 1994.
2. Y. Li. A Two-Dimensional Thin Film TransistorSimulation Using Adaptive Computing Technique.Appl. Math. Comput., 184:73–85, 2007.
3. Y. Li, Y.-L. Li and S.-M. Yu. Design Op-timization of a Current Mirror AmplifierIntegrated Circuit Using a ComputationalStatistics Technique. Math. Comput. Simul.,doi:10.1016/j.matcom.2007.11.002, 2008.
192
Session
CM 2
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CM 2I
Numerical Time Integration in Computational
Electromagnetics
Galina Benderskaya1, Oliver Sterz1, Wolfgang Ackermann2, and Thomas Weiland2
1 CST GmbH, Bad Nauheimer Str. 19, D-64289 Darmstadt, [email protected], [email protected]
2 Technische Universitat Darmstadt, Institut fur Theorie Elektromagnetischer Felder, Schlossgartenstr. 8,D-64289 Darmstadt, [email protected], [email protected]
Summary. In time domain, mathematical models ofvarious electromagnetic devices operating at low fre-quencies generally represent a system of first orderordinary differential equations or index 1 differential-algebraic equations. Numerical schemes employed tofollow the time evolution of the transient processesfall into explicit and implicit time integration algo-rithms. In the paper, the types of the implicit timeintegration are considered and illustrated with thecorresponding numerical experiments.
1 Introduction
Electromagnetic low frequency time domain mod-eling can be performed using the regimes of qua-sistatic approximations or the full set of Maxwell’sequations. Naturally, the usage of the full infor-mation provided by the Maxwell’s equations leadsto more accurate numerical simulation resultswhich are, however, accompanied by higher com-putational costs. Under certain conditions, thequasistatic approximations can be applied whichallows to obtain simulation results of compara-ble accuracy but considerably reduces the com-putational costs. In Section 2, a classification ofthe electromagnetic simulation regimes is given.The material highlighted in this paper is focusedon the quasistatic electromagnetic formulationswhich are described with a set of first order or-dinary differential equations (ODEs) or index 1differential-algebraic equations (DAEs). The in-dispensibe spatial discretization is performed e.g.,with the Finite Integration Technique (FIT) orthe Finite Element Method (FEM). Pros and consof different time integration methods used to re-solve the time dependencies appearing in theseformulations are discussed in Section 3. The fol-lowing Sections 4 and 5 provide a deep insightinto implicit time integration methods and dis-cuss the possible problems associated with theirimplementation. Finally, in the full paper the dis-cussed numerical schemes are illustrated with thetest examples in Section 6.
2 Quasistatic Approximation orExact Formulation
Maxwell’s equations govern the complete behav-ior of electromagnetic fields and represent a sys-tem of coupled differential equations. As a result,the full spectrum ranging from statics to high fre-quency can be modeled entirely. This approach al-lows to obtain precise simulation results but maybe equipped with high computational costs. Un-der certain conditions, however, it is possible topartially omit the time dependencies in the elec-tromagnetic field quantities without losing toomuch accuracy within the numerical simulations.
Excluding the time derivatives of the elec-tric or magnetic flux density from the systemof Maxwell’s equations leads to the magneto- orelectroquasistatic time domain approximations,respectively. The necessary condition for any qua-sistatic approximation to be valid is that the timerequired for an electromagnetic wave to propa-gate at the velocity c = 1/
√µǫ over the largest
length of the system consisting of a material withpermittivity ǫ and permeability µ is much shorterthan the characteristic time constant of the globalfield variation. In other words, this condition isfulfilled if the time rates of the changes are slowenough so that time delays due to the propagationof the electromagnetic waves can be neglected.The quasistatic approximation regime is furthersubdivided into electro- and magnetoquasistaticones depending on the dominant effect [1].
Resolving the space dependencies appearingin the quasistatic formulations using the FIT ap-proach results in the following discrete formula-tions
−SMεGdφ(t)
dt− SMσ(Gφ(t))Gφ(t) = 0 (1)
for the electroquasistatic simulation regime and
Mσd
dta(t) + CMµ−1(a(t))Ca(t) =
j s(t) (2)
195
for the magnetoquasistatic counterpart. In (1),
S represents the discrete dual divergence opera-tor, whereas the material parameters are gath-ered in the discrete permittivity matrix Mε andthe discrete conductivity matrix Mσ. The matrixG describes a discrete gradient operator and thescalar unknows are represented by the vector ofdiscrete potentials φ. Equation (2) is formulated
in terms of the discrete curl operators C and C,the discrete inverse permeability Mµ−1 , the im-
posed electrical grid currents
j s(t) and the vec-tor a(t) of magnetic vector potentials integratedalong the primal grid edges.
3 Implicit versus Explicit TimeIntegration
Since in electroquasistatic formulations the per-mittivity ε never vanishes, the matrix SMεG in(1) is always regular. As a consequence, the elec-troquasistatic formulation (1) represents a non-linear ODE system and all available explicit timeintegration schemes can be applied. If (1), how-ever, becomes stiff implicit time integration meth-ods are more appropriate.
In magnetoquasistatic formulations, the con-ductivity σ obviously equals zero exactly for thoseparts of the model which are filled with non-conductive materials. In this case, the diagonalmatrix Mσ contains zero entries and is singular.Then, (2) represents a nonlinear DAE system ofindex 1 [1]. Such problems are extremely stiff andexplicit integration methods will stuck. When en-countering stiffness implicit schemes offer a muchbetter choice.
There is another important criterion countingin favor of the implicit time integration schemeswhen they are applied to treat the time dependen-cies in parabolic partial differential equations: thestability limit ∆t ≤ O(∆x2) put on the size of thetime step in the explicit time integration methodsimplies that an enormous amount of time steps isnecessary to follow the evolution of the time pro-cess if one reduces the spatial resolution ∆x toimprove the overall accuracy of the numerical so-lution [2].
Typically, the electro- and magnetoquasistaticformulations (1) and (2) are written in the follow-ing general form
Md
dtx(t) + K(x(t))x(t) = r(t) (3)
where the corresponding unknowns are composedinto a vector x. The matrices M and K are squarematrices of real-valued entries and t is a real-valued variable denoting the time.
4 Types of Implicit TimeIntegration
Numerical time integration methods fall into twocategories: those which use only one starting valueat each integration step are called one-step meth-ods and those which are based on several val-ues of the solution calculated at the previoustime instants are called multistep methods. Inthis paper we concentrate only on the one-stepmethods. Among the most popular one-step im-plicit time integration schemes are the so calledθ-method, the Runge-Kutta methods [3] and theRosenbrock-type methods.
5 Adaptive Stepsize Control
Embedded implicit Runge-Kutta and Rosenbrock-type methods offer an effective way of estimatingthe local error of the method. Such integrationschemes deliver for each time step a solution x(p)
of a given order p and an embedded solution x(p)
of a lower order p. The difference between two vec-tors is used to build an error vector whose normis then employed to measure the local error fora given time step. The main and the embeddedmethods share the same coefficient matrix of theButcher table. Thus, the computation of the em-bedded solution does not require the solution ofa nonlinear system, but just the superposition ofthe already calculated stage values using merelydifferent weight factors.
6 Numerical Simulations
In the full paper, numerous numerical examplesdemonstrating the application of the implicit adap-tive methods to the quasistatic formulations de-scribed by (3) calculated using the CST EMSTUDIOTM software package will be presented.
References
1. G. Benderskaya. Numerical Methods for TransientFiled-Circuit Coupled Simulations Based on theFinite Integration Technique and a Mixed CircuitFormulation. PhD thesis, Darmstadt University ofTechnology, Darmstadt, 2007.
2. K.W. Morton, and D.F. Mayers. Numerical Solu-tions of Partial Differential Equations. CambridgeUniversity Press, Cambridge, 2005.
3. A. Nicolet and F. Delince. Implicti Runge-KuttaMethods for Transient Magnetic Field Computa-tion. IEEE Trans. Magn., 32:3:1405–1408, 1996.
196
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CM 2A
Co-Simulation of Circuits Refined by 3-D Conductor Models
Sebastian Schops1, Andreas Bartel1, Herbert De Gersem2, and Michael Gunther1
1 Bergische Universitt Wuppertal schoeps,bartel,[email protected] Katholieke Universiteit Leuven [email protected]
Summary. In this paper a coupling approach for thesolution of circuits refined by 3-D conductor modelsis proposed. The circuit is given in terms of the mod-ified nodal analysis and the field is discretized by thefinite integration technique. The coupled system isintroduced and analyzed; first numerical results areobtained by a co-simulation approach.
1 Introduction
Magnetoquasistatic (MQS) field problems natu-rally occur in circuit analysis. They are commonlysolved by either a circuit or a field approach [3].Any of them may become physically inaccurate orcomputationally inefficient. Consequentially, co-simulation, i.e., the simultaneous use of a cir-cuit and field method in different model parts,is beneficial [2]. Furthermore it provides a nat-ural support for diversifying integration methodsand time-stepping rates towards distinct subprob-lems.
2 Electric Circuit
Large circuits are simulated by the modified nodalanalysis (MNA) technique (notation from [7])
ACd
dtq +ARr(A
TRe, t) +ALıL
+AVıV +AIı(t) +Aλıλ = 0 , (1a)
d
dtΦ−ATLe = 0 , (1b)
ATVe− v(t) = 0 , (1c)
q − qC(ATCe, t) = 0 , (1d)
Φ− ΦL(ıL, t) = 0 , (1e)
with incidence matrices A, node potentials e,currents through voltage and flux controlled el-ements ıV and ıL, voltage dependent charges andfluxes q and Φ, voltage dependent resistors r, volt-age and current dependent charge and flux func-tions qC and ΦL, independent current and voltagefunctions ı and v, controlled voltage sources ıλ.
The numerical properties of (1) are well known,e.g. the index of this differential-algebraic equa-tion (DAE) has been shown to equal 1 assumingseveral topological conditions [6].
Fig. 1. Circuit symbol of a network element with 2terminals embedded in a field model
3 Magnetoquasistatic Field
The finite integration technique (FIT) translatesthe MQS subset (without displacement) of Max-well’s equations into
Ce = − d
dt
b , C
h = ,
S
d = q , S
b = 0 ,
(2)
which is accomplished by the material relations
b = Mµ
h ,
d = Mεe ,
= Mσe , (3)
with discrete curl operators C and C, divergenceoperators S and S, electric and magnetic fieldstrength e and
h, source current density, dis-crete magnetic and electric flux
,
b and
d.The material matrices Mµ, Mε and Mσ repre-sent the permeabilities, permittivities and con-ductivities, with Mσ singular in general due tonon-conducting regions [4].
A single equation of curl-curl form (4) is ob-tained from (2)+(3) by algebraic manipulations
CMνCa +Mσ
d
dta =
, (4)
where a denotes the Magnetic Vector Potential(MVP). This is a special index-1 DAE, becauseit has non-unique solutions. But for the followinganalysis a gauge can be assumed.
4 Coupling
In [5] field models for solid and stranded conduc-tors have been developed for the use in circuits.We use the symbol of Fig. 1 for a (n-terminal) de-vice of several solid and stranded conductors. Inthe case where the field model contains multiplecurrent-driven solid and voltage-driven strandedconductors, the field model (4) is excited by
197
= MσQsolusol +Qstrıstr , (5)
where usol and ıstr are unknowns that are de-scribed by the two additional coupling equations
Gsolusol −QTsolMσd
dta = ısol , (6)
Rstrıstr +QTstrd
dta = ustr , (7)
where Qsol, Qstr are coupling matrices, usol, ustr
denote branch voltages, and ısol, ıstr currentsthrough both conductor types; Gsol and Rstr de-note the solid conductor conductance and thestranded conductor resistance, respectively.
Assuming a regularization we can show that(4)–(7) is still index-1 and that the algebraic partof the MVP vanishes, if the discretization grid issufficiently fine such that no smearing of the twoconductor volumes occurs.
Assigning the voltages to differences of nodepotentials and replacing the controlled currentsource by the current through the conductors in(1a) yields the coupling conditions
usol = ATsole , ustr = ATstre , ıλ =
(ısolıstr
). (8)
Finally the combined system consists of (1), (4)–(8). A simple numerical analysis of this system ispresent, both for the case of a monolythically so-lution as for co-simulation. It turns out that themonolithic system is still index-1 and the conver-gence of the co-simulation is guaranteed accord-ing to the results of [1].
5 Numerical Experiments
Our software is implemented on top of the COMSONDP1. First numerical simulations are obtained forthe simple but effective RLC circuit in Fig. 2(a),where the inductance L is represented by a fieldmodel consisting of one stranded and one solidconductor.The field problem has been constructedusing the CST EMstudio software package2.
The co-simulation converges without any ad-ditional iteration of the individual time frames.The computational results are given in the plotFig. 2(b). They are comparable to the solutionsobtained from the other approaches, the mono-lithic and the plain MNA case. Examination ofthe time-scales in the co-simulation showed thatthe adaptive method for the integration of the cir-cuit part used the same step size as the integratorof the field part. This corresponds to the obser-vation that the dynamics in the field equal those
1 www.comson.org2 www.cst.com
v(t)
C
R
(a) Circuit
0 0.02 0.04 0.06 0.08 0.10
50
100
150
200
250
300
350
400
450
500
e1
e2
e4
(b) Node Voltages
Fig. 2. Refined RLC circuit with an inductance rep-resented by a field model and node potentials com-puted by co-simulation
of the circuit and hence no multi-rate potentialis recognizable in this particular example. This isa reasonable result since the circuit has no partswith additional dynamics.
6 Conclusions
The monolithic field-circuit coupled system isindex-1 and the convergence of the proposed co-simulation approach is guaranteed, as documentedby the simulation of a RLC circuit where the in-ductance is represented by a field model. The co-simulation approach allows for the use of differ-ent, existing software packages and enables theapplication of multi-rate time-stepping schemes.
Acknowledgement. This work was partially supportedby the European Commission in the framework of theCOMSON RTN project.
References
1. A. Bartel. PDAE Models in Chip Design - Ther-mal and Semiconductor Problems. PhD thesis, TUMunchen, 2004. VDI Verlag.
2. G. Bedrosian. A new method for coupling finiteelement field solutions with external circuits andkinematics. IEEE Trans. Magn., 29(2): 1664 –1668, 1993.
3. G. Benderskaya, H. De Gersem, T. Weiland. Inte-gration over discontinuities in field-circuit coupledsimulations with switching elements. IEEE Trans.Magn., 42(4): 1031 – 1034, 2006.
4. M. Clemens, T. Weiland. Discrete electromag-netism with the finite integration technique. PIER,32: 65 – 87, 2001.
5. H. De Gersem, K. Hameyer, T. Weiland. Field-circuit coupled models in electromagnetic simula-tion. J. Comp. Appl. Math., 168: 125 – 133, 2004.
6. D. Estevez Schwarz, C. Tischendorf. Structuralanalysis of electric circuits and consequences forMNA. Int. J. Circ. Theor. Appl., 28: 131 – 162,2000.
7. M. Gunther, P. Rentrop. Numerical simulation ofelectrical circuits. GAMM-Mitteilungen, 1/2: 51 –77, 2000.
198
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CM 2B
Robust FETI solvers for multiscale elliptic PDEs
Clemens Pechstein1 and Robert Scheichl2
1 SFB F013, Johannes Kepler University Linz, Austria [email protected],2 Department of Mathematical Sciences, University of Bath, United Kingdom [email protected]
Summary. FETI methods are efficient parallel do-main decomposition solvers for large-scale finite ele-ment equations. In this work we investigate the ro-bustness of FETI methods in case of heterogeneouscoefficients. Our main application are magnetic fieldcomputations where both large jumps and large vari-ation in the reluctivity coefficient may arise. We givetheoretical bounds and numerical results.
1 Introduction
Finite element tearing and interconnecting (FETI)methods due to Farhat and Roux [1, 7] are par-allel solvers for large-scale finite element systemsarising from partial differential equations (PDEs).We briefly describe the method. As a model prob-lem we consider the finite element discretizationof the Poisson-type problem
−∇ · (α∇u) = f (1)
in the bounded domain Ω ⊂ Rd, d = 2 or 3, sub-ject to suitable boundary conditions. The domainΩ is partitioned into N non-overlapping subdo-mains Ωi, i = 1, . . . , N , such that an effectivedirect solve of local subdomain problems is possi-ble. The FETI method is a special preconditionedconjugate gradient (CG) method, built from localDirichlet and Neumann solves, as well as a coarseproblem which corresponds to a sparse linear sys-tem of dimension O(N). The spectral conditionnumber κ of the preconditioned system can beestimated by
κ ≤ C∗(α)(1 + log(H/h)
)2, (2)
where the constant C∗(α) is independent of thesubdomain diameter H , the mesh parameter h,and the number N of subdomains. If α is (glob-ally) constant, then C∗(α) ∼ 1. The total compu-tational complexity in a parallel scheme is
O((D(N) + D(Nloc)) log(ε−1)
√κ), (3)
where Nloc is the maximal number of unknownsper subdomain, D(·) is the cost of the directsolver, and ε is the desired relative error.
However, in many applications the originalsystem matrix is ill-conditioned due to hetero-geneous coefficient distributions. For instance, in
Ωiηi
Ω ηi, i
interior
10
100
1000
10 100 1000
cond
ition
num
ber
H/eta
case 1case 2linear
Fig. 1. Left: Subdomain boundary layer. Right: Es-timated condition numbers for varying η, fixed h.
magnetic field computations one may have (i)large jumps in the reluctivity coefficient due todifferent materials, and (ii) smooth but large vari-ation in the same coefficient due to nonlineareffects. We are now interested in the questionwhether/how the condition number κ of the pre-conditioned system is affected by this. If the het-erogeneities are resolved by the subdomain par-tition (i. e., α constant on each Ωi), then, usinga special diagonal scaling, Klawonn and Widlund[2] proved that C∗(α) ∼ 1. However, in general,using classical proof techniques, we only get
C∗(α) ≤ C maxi=1,...,N
maxx,y∈Ωi
α(x)
α(y), (4)
with C independent of α, i. e., the bound is pro-portional to the maximum local variation of α onthe subdomains. As noticed by several authors[3, 6] this asymptotic bound is in general far toopessimistic, and robustness is observed for manyspecial kinds of coefficient distributions. The aimof the present contribution is to give more theo-retical insight on the coefficient-dependency.
2 Variation in subdomain interiors
In this section we give a sharper estimate than(4). On each subdomain Ωi with diameter Hi, wechoose a width ηi ∈ [hi, Hi] and define the bound-ary layerΩi,ηi by the agglomeration of those finiteelements which have distance at most ηi from theboundary, cf. Fig. 1, left. Our new bound reads
C∗(α) ≤ CN
maxk=1
(Hk
ηk
)2 Nmaxi=1
maxx,y∈Ωi,ηi
α(x)
α(y), (5)
i. e., it involves only the variation of α in theboundary layer Ωi,ηi . For ηi ≃ Hi we reproduce
199
the known estimate (4). Again our estimate is ro-bust with respect to large jumps across the subdo-main interfaces. However, if α exhibits large (evenarbitrary) variation in the interior Ωi \ Ωi,ηi ofthe subdomains, but varies little in the boundarylayers, our new bound (5) is in general far bet-ter/sharper than (4). Moreover, if in addition thecoefficient is larger in the interior Ωi \Ωi,ηi thanin the boundary layer on each subdomain, thenthe quadratic factor (Hi/ηi)
2 reduces to a linearfactor Hi/ηi. The detailed analysis will be givenin an upcoming paper [5].
Fig. 1, right, shows some numerical resultsfor a two-dimensional example. We set α = 105
(Case 1) and α = 10−5 (Case 2) in the subdo-main interiors, and α = 1 on the rest. We choosethe distance between the “material” jump and thesubdomain interfaces to be η, and set H/h = 512.We see that our asymptotic bound is sharp forCase 1, but still slightly pessimistic for Case 2.
3 Boundary layer variation
In the case of nonlinear magnetostatics in twodimensions (transverse magnetic mode), we haveto solve
−∇ · [ν(|∇u|)∇u] = f , (6)
subject to suitable boundary conditions, whereu is the z-component of the magnetostatic vec-tor potential. The reluctivity ν depends nonlin-early on the magnetic flux density |B| = |∇u|. Ifwe apply Newton’s method to (6), the linearizedsystem in each Newton step is of similar formas (1). However, the variation of the coefficientdepends essentially on the variation of |B|. In-deed, in many applications |B| may vary stronglyalong subdomain boundaries. Numerical experi-ments reveal that in this case, our new bound (5)is still too pessimistic: The constant C∗(α) de-pends only mildly on the variation of α near sub-domain interfaces. This effect has already beenreported in [3, 4].
Large values of |B| appear mostly at singular-ities of u, e. g., near material corners. Contrary tothe usual suggestion to choose subdomain parti-tions that resolve material interfaces in order toobtain robustness, our new bound (5) suggeststhat it might be more advantageous to put eachpeak of |B| and thus each material corner intothe center of a subdomain. Fig. 2 shows two suchexamples. In both cases, the coefficient variationis approximately 7×103 and our FETI solver per-forms extremely well (Case 1: condition number8.5, Case 2: condition number 13.7, compared to8.3 in case of a constant coefficient). In Case 1,the boundary layer variation is small, whereas itis large in Case 2.
100
1000
10000
100000
1e+06
1e+07
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
nu(|
B|)
x
coefficient along subdomain boundary 5
100
1000
10000
100000
1e+06
1e+07
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
nu(|
B|)
x
coefficient along subdomain boundary 5
Fig. 2. Upper: Case 1. Lower: Case 2. Left: |B|-field,subdomain partition. Right: Boundary variation of ν.
We are currently trying to extend our analysisto also cover this case and to explain why in manysituations even a variation in the boundary layerdoes not affect the robustness of our FETI solver.
Acknowledgement. We would like to thank UlrichLanger for his encouragement. The first author ac-knowledges the financial support by the Austrian Sci-ence Funds (FWF) under grant SFB F013.
References
1. C. Farhat and F.-X. Roux. A method of finite el-ement tearing and interconnecting and its parallelsolution algorithm. Int. J. Numer. Meth. Engrg.,32:1205–1227, 1991.
2. A. Klawonn and O. B. Widlund. FETIand Neumann-Neumann iterative substructuringmethods: connections and new results. Comm.Pure Appl. Math., 54(1):57–90, 2001.
3. U. Langer and C. Pechstein. Coupled finiteand boundary element tearing and interconnectingsolvers for nonlinear potential problems. ZAMMZ. Angew. Math. Mech., 86(12), 2006.
4. U. Langer and C. Pechstein. Coupled FETI/BETIsolvers for nonlinear potential problems in(un)bounded domains. In G. Ciuprina andD. Ioan, editors, Scientific Computing in ElectricalEngineering, volume 11 of Mathematics in Indus-try: The European Consortium for Mathematics inIndustry, Berlin, Heidelberg, 2007. Springer.
5. C. Pechstein and R. Scheichl. Analysis of FETImethods for multiscale elliptic PDEs. (in prepara-tion).
6. D. Rixen and C. Farhat. A simple and efficientextension of a class of substructure based pre-conditioners to heterogeneous structural mechan-ics problems. Internat. J. Numer. Methods Engrg.,44:489–516, 1999.
7. A. Toselli and O. B. Widlund. Domain Decoposi-tion Methods – Algorithms and Theory, volume 34of Springer Series in Computational Mathematics.Springer, Berlin, Heidelberg, 2005.
200
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CM 2C
Functional Type A Posteriori Error Estimates for Maxwell’s
Equations
Antti Hannukainen
Institute of Mathematics, Helsinki University of Technology, P. O. Box 1100, FIN–02015 [email protected]
Summary. In this work, we study functional type aposterior error estimation for eddy-current and time-harmonic approximations of Maxwell’ equations. Thepresented estimates can be applied to any admissi-ble approximation, hence they are not tied to anyparticular numerical scheme. The behavior of theseestimates is numerically tested.
1 Introduction
Majority of the existing research on a posteri-ori error estimates for Maxwell’s equations hasfocused on residual based error estimation tech-niques (see [1, 5, 8]). Such techniques require theapproximate solution E to satisfy a certain Galerkinorthogonality property, which in practice meansthat residual based estimates are applicable onlyin the context of the finite element methods. Inaddition, they contain Clement interpolation con-stants, which are mesh dependent and difficult toestimate sharply (see [2]). Hence, current residualbased a posterior error estimates cannot deliversharp upper bound for the error and serve onlyas error indicators.
In this note, we will focus on functional typea posteriori error estimation approach (see [3,4, 6]), which gives practically computable upperbounds. First functional type error estimates forthe eddy-current problem were presented in [7].Error bound for the time-harmonic problem waspresented by the author in the ENUMATH con-ference. In this talk, we derive an upper bound inboth cases and present numerical examples, whichverify the effectivity of our approach.
2 Model problem
The model problem is : find E ∈ H0 (Ω, curl) suchthat
(∇× E,∇× v) + β (E,v) =
(J,v) ∀v ∈ H0 (Ω, curl) , (1)
where Ω ⊂ R3 satisfies sufficient regularity as-sumptions and J ∈ L2(Ω). The spaceH0 (Ω, curl)
denotes the standard Sobolev space with zeroboundary condition n × E = 0.
We study the above problem in two differentcase, β ∈ R, β > 0 and β ∈ C,ℑβ ≥ 0. Forpositive parameter values (1) is the weak form ofthe eddy-current problem in a domain with posi-tive conductivity and PEC boundary conditions.For complex-valued parameters (1) is the time-harmonic approximation of Maxwell’s equationsin a domain with PEC boundary conditions.
3 Error bounds
In the following, we consider some approximationE ∈ H0 (Ω, curl) for E and derive an upper boundfor the error e = E− E in a suitable norm.
An upper bound for the error also yields anestimate for the relative error.
Lemma 1. Let ‖ · ‖ be any norm and M be anupper-bound to ‖e‖2, such that M ≤ ‖E‖. Thenthere applies
‖e‖‖E‖ ≤ M
‖E‖ −M.
3.1 Eddy-current case
In the eddy-current case, the bilinear form is el-liptic and induces a natural norm
||| · ||| :=√
(∇× ·,∇× ·) + β (·, ·),which provides a direct connection between theerror and the weak problem. In this norm, weobtain the following error bound.
Theorem 1. Let E ∈ H0 (Ω, curl) be the solutionto (1) with β > 0. For any E,y∗ ∈ H0 (Ω, curl)there applies
|||e|||2 ≤ ‖β−1/2(J − βE −∇× y∗
)‖2
+ ‖∇× E− y∗‖2. (2)
This estimate is sharp in the sense that choosingthe parameter y∗ as y∗ = ∇ × E yields equal-ity. One appropriate choice of y∗ is discussed inSection 4.
201
3.2 Time-harmonic case
In time-harmonic case, the sesquilinear form isnot elliptic and does not induce any natural norm.We will follow [5] and measure the error in theL2(Ω) - norm.
Theorem 2. Let E ∈ H0 (Ω, curl) be the solu-tion to (1) with β ∈ C,ℑβ ≥ 0. For any E,y∗ ∈H0 (Ω, curl) there applies
‖e‖ ≤ C(‖J − βE−∇× y∗‖
+ ‖y∗ −∇× E‖).
in which the constant C > 0 depends only on theparameter β and the domain.
4 Numerical example
The upper-bound for the eddy-current problem istested in the case Ω = (−1, 1)/[0, 1], β = 1 andf = [1, 1, 1]T . As the domain contains a re-entrantcorner at the origin (0, 0, 0), a boundary singular-ity can be expected near this point. Our aim is tostudy, how well does the upper bound detect thesingularity in adaptive refinenement process.
The problem (1) was solved using the finite el-ement method with lowest order Nedelec (With-ney) elements on tetrahedral mesh. The adaptiveprocedure is based on using the elementwise con-tributions of the upper bound as local error mea-sures.
The parameter y∗, required in Theorem 1, waschosen as a minimizer of (2) from the same finiteelement space Xh as the approximate solution.The minimizer is obtained by solving the problem: find y∗ ∈ Xh such that
(β−1∇× y∗,∇× v
)+ (y∗,v) = (∇× u,v)
+(β−1/2 (f − βu) ,∇× v
)∀v ∈ Xh,
which is also an eddy current problem, solvedwith the same finite element solver as the orig-inal problem.
The results are plotted in Fig. 1. As one canclearly observe, the adaptive refinement based onthe presented upper bound improves the conver-gence rate compared to uniformly refined mesh.This means that the adaptive refinement processcan detect the singularity and improve the qual-ity of the approximation. Since the true error isunknown, we have used a reference solution froma very dense mesh to compute estimate for theerror.
103
104
105
10−1
100
Fig. 1. Behavior of the error in adaptive refinementas a function of degrees of freedom. Diamonds denotethe estimated error, stars the reference error and cir-cles error in a solution from uniformly refined mesh.The dotted line is the optimal convergence rate O(h).
Acknowledgement. The author has been supportedby the project no. 211512 from the Academy of Fin-land.
References
1. R. Beck, R. Hiptmair, R. Hoppe, andB. Wohlmuth. Residual based a-posteriorierror estimators for eddy current computation.Math. Model. Numer. Anal, 34(1):159–182, 2000.
2. C. Carstensen and S. A. Funken. Constants inClement-interpolation error and residual based aposteriori error estimates in finite element meth-ods. East-West J. Numer. Math., 8:153–175, 2000.
3. A. Hannukainen and S. Korotov. Computationaltechnologies for reliable control of global and lo-cal errors for linear elliptic type boundary valueproblems. JNAIAM, 2(3–4):157–176, 2007.
4. S. Korotov. Two-sided a posteriori error estimatesfor linear elliptic problems with mixed boundaryconditions. Appl. Math., 52(3):235–249, 2007.
5. P. Monk. A posteriori error indicators forMaxwell’s equations. Journal of Computationaland Applied Mathematics, 100:173–190, 1998.
6. P. Neittaanmaki and S. Repin. Reliable Methodsfor Computer Simulation: Error Control and APosteriori Error Estimates. Elsevier, Amsterdam,2004.
7. S. Repin. Functional a posteriori estimates for themaxwell’s problem. J. Math. Sci., 1:1821–1827,2007.
8. J. Schoberl. Commuting quasi-interpolation op-erators for mixed finite elements. Math. Comp.(submitted), 2006.
202
SCEE 2008, Espoo, Finland, Sept. 28 – Oct. 3, 2008 CM 2D
Large-Scale Atomistic Circuit-Device Coupled Simulation of
Intrinsic Fluctuations in Nanoscale Digital andHigh-Frequency Integrated Circuits
Chih-Hong Hwang, Ta-Ching Yeh, Tien-Yeh Li, and Yiming Li1
National Chiao Tung University, 1001 Ta-Hsueh Rd., Hsinchu 300, Taiwan [email protected]
Summary. Investigation of device variability is cru-cial for the design of state-of-art nanoscale digitaland high-frequency circuits and systems. This studyprovides a large-scale statistically sound ”atomistic”circuit-device coupled simulation approach to explor-ing the random-dopant-induced characteristic fluctu-ations in nanoscale CMOS integrated circuits con-currently capturing the discrete-dopant-number- anddiscrete-dopant-position-induced fluctuations.
1 Introduction
As dimension of semiconductor scales down, char-acteristic fluctuation is more pronounced and be-come crucial for circuit design. Diverse approacheshave recently been reported to study fluctuation-related issues in semiconductor devices and cir-cuit [1, 2]. However, due to the randomness ofdopant position in device, the fluctuation of de-vice gate capacitance is nonlinear and hard tobe modeled in current compact models [2]. Inthis study, a large-scale statistically sound circuit-device coupled simulation approach is thus pro-posed to analyze the discrete-dopant-induced char-acteristic fluctuations in nanoscale MOSFET cir-cuit. Based on the statistically (totally randomly)generated large-scale doping profiles, the devicesimulation is performed. In estimation of the cir-cuit fluctuations, to capture the nonlinearity ofgate capacitance fluctuation, a circuit-device cou-pled simulation [5] with discrete dopant distribu-tion is conducted, which concurrently considersthe discrete-dopant-number- and discrete-dopant-position-induced fluctuations.
2 Simulation Technique
Figure 1 illustrates the developed simulation flowof the proposed approach. To consider the effectof random fluctuation of the number and loca-tion of discrete channel dopants in channel re-gion, 758 dopants are randomly generated in a 80nm3 cube, in which the equivalent doping concen-tration is 1.48×1018 cm−3 (the nominal channeldoping concentration), as shown in Fig. 1(a). The80 nm3 is then partitioned into sub-cubes of 16nm3. The number of dopants may vary from zeroto 14, and the average number is six, as shown
Dopants in (16nm3) Cube
0 5 10 15
His
tog
ram
(n
um
be
r)
0
5
10
15
20
25-3σ +3σ0 14
Mean = 6
Dopants in (16nm3) Cube
0 5 10 15
His
tog
ram
(n
um
be
r)
0
5
10
15
20
25-3σ +3σ0 14
Mean = 6
14 dopants in 16 nm3 cube
16nm
16nm
80nm
80
nm
1.48 1018 cm-3
758 dopants in 80 nm3 cube80nm
16nm
16nm
16n
m16
nm
zero dopants in16 nm3 cube
oxide
Channel Length
Gate
drainsource
(e)
(f)
Solving a set of 3D drift-diffusion equation with
quantum correction
Device characterization
Analog Circuit Logic Circuit
(a)
(c)
(b)
(d)
VDD
VIN
Planar MOSFET
VOUT
CR2
R1
Device equations coupled with circuit nodal equations
for mixed-mode device/ circuit simulation
Device and circuit characterization
VOUTVIN
Fig. 1. (a) Discrete dopants randomly distributedin the 80 nm3 cube with the average concentration1.48×1018 cm−3. The dopants in sub-cubes of 16 nm3
may vary from zero to 14 (the average number is six),[(b), (c), and (d)]. The sub-cubes are equivalentlymapped into channel region for discrete dopant de-vice and circuit-device coupled simulations as shownin (e) and (f).
in Figs. 1(b), 1(c) and 1(d). These sub-cubesare then equivalently mapped into the channel re-gion of the device for the 3D device simulation, asshown in Fig. 1(e).The device simulation is per-formed by solving a set of 3D density-gradientequation coupling with drift-diffusion equations[3], which is conducted using a parallel comput-ing system [4]. To investigate the random discretedopant induced circuit fluctuations, and due to
203
the lack of reasonable compact model for describ-ing device gate capacitance [1, 2], the aforemen-tioned device equations directly coupled with cir-cuit equations and then solved simultaneously [5].A common-source amplifier and an inverter) areused as the test circuits for high-frequency charac-teristic and timing variation estimation, as shownin Fig. 1(f). We noted that the proposed simula-tion technique is statistical sound and computa-tionally cost-effective for random dopant fluctua-tion characterization.
3 Results and Discussion
VG (V)0.0 0.5 1.0
Gate
Cap
acitan
ce
(fF
)
0.004
0.005
0.006
0.007
(b)VG (V)
0.0 0.5 1.0
I D (
A)
10-10
10-9
10-8
10-7
10-6
10-5
Ioff fluctuation
Vth fluctuation
Ion fluctuation
(a)
Lateral shift
change shape
Fig. 2. The (a) ID-VG curves (b)and gate capaci-tance fluctuations of the studied 16-nm-gate planarMOSFET
Figure 2(a) show the ID-VG characteristicsof the discrete-dopant-fluctuated 16 nm planarMOSFETs, where the solid line shows the nom-inal case (1.48×1018 cm−3 continuously dopingconcentration), and the dashed lines are random-dopant-fluctuated devices. The fluctuation of DCcharacteristics are observed. The capacitance-voltage (C-V) characteristics of the discrete-dopant-fluctuated 16 nm planar MOSFETs are investi-gated in Fig. 2(b). The lateral shift and changeof shape for the C-V characteristics are observed.The variation of shape of C-V curves is resultedfrom the position of discrete dopants in channeland therefore is hard to be described in presentcompact model [3]. Therefore, to investigate thecircuit level characteristic fluctuations, the circuit-device coupled simulation is used for obtainingcircuit characteristics. The transition character-istics of the inverter circuit are investigated inFigs. 3(a). The fluctuations of rise time and falltime for digital circuits are observed, where therise(fall) time is defined as the time required forthe output voltage to go from 10%(90%) of thelogic ”0”(”1”) level to 90%(10%) of the logic”1”(”0”). For the high-frequency charateristic sim-ulation, the high-frequency response are investi-gated in Fig. 3(b). The fluctuation of high fre-quency circuit gain, 3dB bandwidth, and unity-gain bandwidth are estimated. Fig. 3(c) summa-rized the fluctuations of DC, timing, and high-frequency characteristics. The accuracy of timingand stability of frequency characteristics are cru-cial for the state-of-art circuits and systems usingnanoscale transistor.
Time (s)30x10-12 50x10-12
VO
UT (
V)
0.0
0.6
1.2
(a)
Frequency (Hz)
100x106 1x109 10x109 100x109
Hig
h F
requ
ency C
ircu
it G
ain
(dB
)
0
2
4
6
8
10
12
fluctuation of circuit gain fluctuation
of unity-gain bandwidth
fluctuation of 3dB bandwidth
(b)
Time (s)0 100x10-12 200x10-12
VO
UT (
V)
0.0
0.6
1.2
VIN
(V
)
0.0
0.6
1.2
Time (s)100x10-12 120x10-12
VO
UT (
V)
0.0
0.6
1.2Fall time fluctuation Rise time fluctuation
(c)
10.4%2.81x1011Unity-gain
bandwidth (Hz)
14.1%6.84x10103dB bandwidth
(Hz)
5.7%8.138Gain (dB)
9.0%6.04x1012Fall time
3.1%5.97x1012Rise time
54.4%1.74x10-8Ioff (A)
7.9%8.46x10-6Ion (A)
41.7%0.14Vth (V)
VariationNominal
10.4%2.81x1011Unity-gain
bandwidth (Hz)
14.1%6.84x10103dB bandwidth
(Hz)
5.7%8.138Gain (dB)
9.0%6.04x1012Fall time
3.1%5.97x1012Rise time
54.4%1.74x10-8Ioff (A)
7.9%8.46x10-6Ion (A)
41.7%0.14Vth (V)
VariationNominal
Fig. 3. (a)The input and output signal for the stud-ied discrete-dopant-fluctuated 16-nm-gate inverter.(b) the high-frequency response of the common-source circuits. (c) summarized DC, timing, and high-frequency characteristic fluctuations
4 Conclusion
In this paper, a large-scale 3D ”atomistic” circuit-device coupled simulation approach has been pro-posed to investigate the random-dopant-inducedcharacteristic fluctuations in nanoscale CMOSdigital and high-frequency integrated circuits. Theapproach successfully estimated the timing andhigh-frequency characteristics fluctuations. Thefluctuation suppression technique may be furtherverified and developed from the viewpoint of cir-cuit operation. We believe that the proposed ap-proach will benefit the development of nanoscalecircuits and systems.
Acknowledgement. This work was supported by Tai-wan National Science Council under Contract NSC-96-2221-E-009-210, NSC-96-2752-E-009-003-PAE andby the Taiwan Semiconductor Manufacturing Com-pany under a 2007-2008 grant.
References
1. Y. Li, and C.-H Hwang. Discrete-dopant-inducedcharacteristic fluctuations in 16 nm multiple-gate silicon-on-insulator devices. J. Appl. Phy.,8:084509, 2007.
2. A. Brown and A. Asenov. Capacitance fluctua-tions in bulk MOSFETs due to random discretedopants. J. Comp. Elect., DOI:10.1007/s10825-008-0181-y, 2008.
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204
Author Index
Author Index
Ackermann, Wolfgang, 195Alı, Giuseppe, 85, 165Andersson, Bjorn, 181Antoulas, Athanasios C., 79,
161Arkkio, Antero, 57
Bahls, Christian R., 129Bandlow, Bastian, 9Bartel, Andreas, 165, 197Baumanns, Sascha, 89van Belzen, F., 167Benderskaya, Galina, 195Benner, Peter, 3, 23, 25van Beurden, M.C., 41Blaszczyk, Andreas, 11Brachtendorf, Hans Georg, 103Bruma, Alina, 33, 35Brunk, Markus, 69, 91Bunse-Gerstner, Angelika, 103
Chauviere, C., 127Christoffersen, Carlos, 97Ciuprina, Gabriela, 15, 43, 45Condon, Marissa, 105, 123Copeland, Dylan, 131Costa, Luis R.J., 47Croon, Jeroen, 93Culpo, Massimiliano, 71, 165
Dag, Hasan, 115Dautbegovic, Emira, 141De Gersem, Herbert, 197De Tommasi, Luciano, 93Denk, Georg, 71Dewilde, Patrick, 135Dhaene, Tom, 93Dobrzynski, Michal, 177Drago, C., 187
Ebrahimi, B.M., 49, 53Edelvik, Fredrik, 181
Faiz, Jawad, 49, 53
de Falco, Carlo, 71, 165Feldmann, Uwe, 95Feng, Lihong, 23Freitag, Josefine, 169
Ghofrani, M., 49Gim, Sebastian, 177, 179Gjonaj, Erion, 137Gorissen, Dirk, 93Gourary, M.M., 101, 145, 175Grahovski, Georgi G., 123Greiff, Michael, 169Grimberg, Raimond, 33, 35Gu, B., 175Gullapalli, K.K., 101Gunther, Michael, 197
Hannukainen, Antti, 201Harutyunyan, D., 173Herberthson, Magnus, 171Hesthaven, J.S., 127Honkala, Mikko, 99, 117, 119,
157Huang, Hsuan-Ming, 191Hwang, Chih-Hong, 203
Ilievski, Z., 113Ioan, Daniel, 15, 31, 43, 45, 153Ionutiu, Roxana, 161Islam, M. Jahirul, 57Iwata, Satoru, 143
Jakobsson, Stefan, 181Jungel, Ansgar, 69, 91
Kajiwara, Takahiro, 95Karanko, Ville, 99Khan, M.A., 53Kula, Sebastian, 179Kuo, Yi-Ting, 39
La Rosa, Salvatore, 83Lahaye, D.J.P., 187, 189Lampe, Siegmar, 103
Lang, Barbara, 103Langer, Ulrich, 131Lau, Thomas, 137Lefteriu, Sanda, 79Lenaers, Peter, 27Levadoux, David P., 77Li, Tien-Yeh, 203Li, Yiming, 39, 191, 203
Mascali, Giovanni, 83, 85ter Maten, E.J.W., 21, 113Mathis, Wolfgang, 169Meuris, Peter, 61Michielsen, B.L., 41, 73Miettinen, Pekka, 117, 119, 157Mihalache, Diana, 43, 45, 179Miura-Mattausch, Mitiko, 95Miyake, Masataka, 95Mohaghegh, Kasra, 121Mulvaney, B.J., 101, 145, 175Munz, Claus-Dieter, 37
Neff, Carsten, 117Neitola, Marko, 111Nikoskinen, Keijo, 47
Pechstein, Clemens, 199Plata, Jagoda, 177Poplau, Gisela, 129Pomplun, Jan, 133Preisner, Thomas, 169Pulch, Roland, 85, 147Pusch, David, 131
Quandt, Martin, 37
Rahkonen, Timo, 107, 109, 111Rajabi-Sebdani, M., 53Rebican, Mihai, 31Reis, Timo, 159van Rienen, Ursula, 129Romano, Vittorio, 67, 83Romisch, Werner, 149Rommes, Joost, 27, 161
205
Author Index
Roos, Janne, 117, 119, 157Rusakov, S.G., 101, 145, 175
Savin, Adriana, 33, 35Scheichl, Robert, 199Schilders, W.H.A., 27, 113, 155,
173Schmidt, Frank, 133Schneider, Andre, 25Schneider, Rudolf, 37Schoenmaker, Wim, 61, 173Schops, Sebastian, 197Schuhmann, Rolf, 9Scordia, C., 67Seebacher, Ehrenfried, 43Selva, Monica, 89Sengor, N. Serap, 183, 185Sheng, Zhifeng, 135Sickenberger, Thorsten, 149Simsek, Murat, 183, 185
Stefanescu, Alexandra, 15, 45,179
Steigmann, Rozina, 33, 35Steinbach, Olaf, 13Sterz, Oliver, 195Striebel, Michael, 21, 121Stykel, Tatjana, 159Sy, O.O., 41
Takamatsu, Mizuyo, 143Taskinen, Matti, 81Tijhuis, A.G., 41Tischendorf, Caren, 89, 143
Udpa, Lalita, 33Udpa, S.S., 33Ugryumova, M.V., 155Ulyanov, S.L., 101, 145, 175
Vaessen, J.A.H.M., 41Valtonen, Martti, 47
Vasenev, Alexander, 179Verhoeven, A., 21Voigtmann, Steffen, 71
Wan, Shan, 19Warburton, T., 127Weiland, S., 167Weiland, Thomas, 137, 195Wilcox, L., 127Windisch, Markus, 13
Xu, Tao, 105
Yeh, Ta-Ching, 191, 203Yetkin, E. Fatih, 115Yla-Oijala, Pasi, 81Yuferev, Sergey, 7
Zhang, Q.J., 19Zharov, M.M., 101, 145, 175
206
1. L. Jylha: Modeling of electrical properties of composites, March 2008.
2. I. Torniainen: Multifrequency studies of gigahertz-peaked spectrum sources andcandidates, May 2008.
3. I. Hanninen: Analysis of electromagnetic scattering from anisotropic impedanceboundaries, June 2008.
4. J. Roos, L.R.J. Costa (eds.): SCEE 2008 book of abstracts, September 2008.
ISBN 978-951-22-9557-9 (printed)
ISBN 978-951-22-9563-0 (electronic)
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